CONTINUOUS ADVANCES IN QCD 2004
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Proceedings of the Conference on
CONTINUOUS ADVANCES IN QCD 2004 William I. Fine Theoretical Physics Institute 13 - 16 May 2004 Minneapolis, USA
Editor
T. Gherghetta University of Minnesota, USA
NEWJERSEY * LONDON
-
K0World Scientific SINGAPORE * BElJlNG
*
SHANGHAI
-
HONG KONG
TAIPEI
-
CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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CONTINUOUS ADVANCES IN QCD 2004 Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-256-072-6
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FOREWORD The William I. Fine Theoretical Physics lnstitute hosted a workshop on the “Continuous Advances in QCD 2004”, from May 13-16, 2004. This biennial workshop was the sixth meeting of the series held at the University of Minnesota, Minneapolis, and marked the loth anniversary since the first such workshop was organised by the Institute in 1994. The workshop gathered together over sixty leading experts in the field to discuss the latest results and exchange ideas in Quantum Chromodynamics and non-Abelian gauge theories in general. In fact this year there were a record number of participants. The talks were organized into plenary sessions in the morning, which included reviews of the newest and most interesting research topics, while for the first time parallel sessions were scheduled in the afternoon devoted to shorter, original presentations. This lead to a broad scope of topics discussed. Amongst the hot topics this year included the discussion of the experimental evidence for pentaquarks, strong interactions under extreme conditions related to the ongoing experiments at RHIC, and new string theory inspired approaches to gauge theories. There were also very interesting presentations associated with perturbative and nonperturbative dynamics, heavy hadron decays, topological field configurations, as well as supersymmetry and novel theoretical methods. This Proceedings volume represents the current state of the QCD research frontier. It is hoped that the articles presented here convey some of the excitement of these topics as well as provide a useful review and reference for the reader in the years to come. Finally, on behalf of the organizers, I would like to take this opportunity to express my gratitude to Sally Menefee and Catharine Grahm for their dedicated efforts on the organization of numerous aspects of the workshop. T . Gherghetta
V
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CONTENTS 1. Perturbative and Nonperturbative QCD
1
Advances in Generalized Parton Distribution Study A . V. Radyushkin
3
QCD Dirac Spectra and the Toda Lattice K. Splittorff und J. J. M. Verbaarschot
15
Conformal Symmetry as a Template for QCD S.J. Brodsky
30
A Variational Look a t QCD J. Guilherme Milhano
44
Baryons, I N c . R.F. Lebed
54
Quark Correlations and Single-Spin Asymmetries M . Burkardt
64
2. Heavy Quark Physics
75
Soft-Collinear Factorization and the Calculation of the B M. Neubert
-+
X,y Rate 77
Status of Perturbative Description of Semileptonic Quark Decays A . Czarnecki
91
Heavy Quark Expansion in Beauty: Recent Successes and Problems N. Uraltsev
100
Polarization, Right-Handed Currents, and CP Violation in B A.L. Kugan
115
vi i
VV
viii
Lifetimes of Heavy Hadrons A.A. Petrov
129
Inclusive B-Decay Spectra and IR Renormalons E. Gardi
139
Summing Logs of the Velocity in NRQCD and Top Threshold Physics 150 A.H.-Hoang
B
+ 7r,K , r] Decay Formfactors from Light-Cone Sum Rules
160
P. Ball and R. Zwicky Understanding D$(2317), D,~(2460) F. De Fazio Search for Dark Matter in B M. Pospelov
+ S Transitions with Missing Energy
170
180
3. Exotic Hadrons
189
Exotica R.L. Jaffe
191
Quark Structure of Chiral Solitons D. Diakonov
215
Do Chiral Soliton Models Predict Pentaquarks? I.R. Klebanov and P. Ouyang
227
Baryon Exotics in the l/Nc Expansion E. Jenkins
239
Large N, QCD and Models of Exotic Baryons T.D. Cohen
251
4. QCD Matter at High Temperature and Density
259
Baryon and Anti-Baryon Production at RHIC J.I. Kapusta
261
ix
QCD at the Boiling Point: What Does Hadron Production at RHIC Tell Us? R. J. Fries
273
Nuclear Physics a t Small z D.E. Kharzeeu
283
Saturation Physics Meets RHIC Data Y. V. Kovchegou
293
Confinement and Chiral Symmetry A. ~ ~ ' c s y
303
Gapless Superconductivity in Dense QCD I . A . Shovkouy
313
Inhomogeneous Color Superconductivity G. Nardulli
323
Spontaneous Rotational Symmetry Breaking and Other Surprises in c-Model at Finite Density V.A. Miransky
335
Analytical Approach to Yang-Mills Thermodynamics R. Hofmann
346
Exotic Superfluids: Breached Pairing, Mixed Phases and Stability E. Gubankoua
357
5. Topological Field Configurations
367
Quantum Weights of Monopoles and Calorons with Non-trivial Holonomy D. Diakonou
369
Noncommutative Solitons and Instantons F. A. Schaposnik
381
X
Breather Solutions in Field Theories V. Khemani
393
Quantum Nonabelian Monopoles K. Konishi
403
Dilute Monopole Gas, Magnetic Screening and K-Tensions in Hot Gluodynamics C.P. Korthals Altes
416
Supersizing Worldvolume Supersymmetry: BPS Domain Walls and Junctions in SQCD A. Ritz
428
New Topological Structures in QCD F. Bruckmann, D. Ndgra'di and P. van Baal
440
6. Supersymmetry and Theoretical Methods
453
Viscosity of Strongly Coupled Gauge Theories P. Kovtun, D. T. Son and A . 0. Starinets
455
AdS/CFT Duality for States with Large Quantum Numbers A.A. Tseytlin
464
Planar Equivalence: From Type 0 Strings to QCD A. Armoni, M. Shifman and G. Veneziano
478
Gauge Theory Amplitudes, Scalar Graphs and Twistor Space V. V. Khoze
492
Weak Supersymmetry and its Quantum-Mechanical Realization A. V. Smilga
504
Nonperturbative Solution of Yukawa Theory and Gauge Theories J. R. Hiller
515
Fermionic Theories in Two-Dimensional Noncommutative Space E.F. Moreno
527
xi
Nonabelian Superconformal Vacua in Theoreis
= 2 Supersymmetric
537
R. Auzzi Predictions for QCD from Supersymmetry F. Sannino
547
Self-Duality, Helicity and Background Field Loopology G. V. Dunne
557
The Optical Approach to Casimir Effects A. Scardicchio
569
At The Crossroad: Perpetual Questions of the Fundamental Physics Here and Now A. Loseu
579
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SECTION 1. PERTURBATIVE AND NONPERTURBATIVE QCD
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ADVANCES IN GENERALIZED PARTON DISTRIBUTION STUDY
A. V. RADYUSHKIN172>* Physics Department, Old Dominion University, Norfolk, VA 23529, USA Theory Group, Jefferson Lab, Newport News, VA 23606, USA E-mail:
[email protected] The basic properties of generalized parton distributions (GPDs) and some recent applications of GPDs are discussed.
1. Introduction
The concept of Generalized Parton distribution^^^^^^ is a modern tool to provide a more detailed description of hadronic structure. The need for GPDs is dictated by the present-day situation in hadron physics, namely: i) The fundamental particles from which the hadrons are built are known: quarks and gluons. ii) Quark-gluon interactions are described by QCD whose Lagrangian is also known. iii) The knowledge of these first principles is not sufficient a t the moment, and we still need hints from experiment to understand how QCD works, and we must translate information obtained on the hadron level into the language of quark and gluonic fields. One can consider projections of combinations of quark and gluonic fields onto hadronic states IP) : ( 0 I qa(zl)qp(z2) I P ) , etc., and interpret them as hadronic wave functions. In principle, solving the bound-state equation H I P ) = E I P ) one should get complete information about hadronic structure. In practice, the equation involving infinite number of Fock components has never been solved. Moreover, the wave functions are not directly accessible experimentally. The way out is to use phenomenological functions. Well known examples are form factors, usual parton densities, and distribution amplitudes. The new functions, Generalized Parton D i s t r i b u t i o n s l ~ ~(for > ~ recent reviews, see4i5), are hybrids of these “old” *Also at Laboratory of Theoretical Physics, JINR, Dubna, Russia
3
4
functions which, in their turn, are the limiting cases of the ‘hew” ones. 2. Form factors, usual and nonforward parton densities
The nucleon electromagnetic form factors measurable through elastic eN scattering (Fig. 1, left) are defined through the matrix element
where r = p - p‘, t = r 2 . The current is given by the sum of its flavor eaF1,2,(t). components J,”(z) = e,&(z)rp$,(z), hence, F1,2(t)=
c,
P
P’
Figure 1. Left: Elastic e N scattering in one-photon approximation. Right: Lowest order pQCD factorization for DIS.
The parton densities are defined through forward matrix elements of quark/gluon fields separated by lightlike distances. In the unpolarized case,
and fa(a) (x)is the probability to find a (a)-quark with momentum xp in a nucleon with momentum p . One can access fa(a) (x)through deep inelastic scattering (DIS) y*N -+ X . Its cross section is given by imaginary part of the forward virtual Compton scattering amplitude. For large Q 2 E - q 2 , the perturbative QCD (pQCD) factorization works, and the leading order handbag diagram (Fig. 1, right) measures parton densities at the point x = X B ~E Q 2 / 2 ( p q ) . Note, that form factor deals with a point vertex instead of a light-like separation for the parton densities, and that p # p‘. Let us now “hybridize” form factors with parton densities by writing form factor components FI,(t) as integrals over the momentum fraction x
(3)
5
k
P = ( p + p ' ) /2 Figure 2.
t
J
Form factor and WACS amplitude in terms of nonforward parton densities.
(see Fig. 2, left). The nonforward parton densities (NPDs) Fa(a)(z,t), coincide in the forward t = 0 limit with the usual densities: t = 0) = fa(ii) (x). A nontrivial question is the interplay between z and t dependence. The simplest factorized ansatz F,(z, t ) = fa(x)F1(t) satisfies both the forward constraint and the local constraint (3). However, using the Gausk~1/ziX2]suggests7i6 sian light-cone wave functions !Q(zi,k i l ) exp[- C i Fa(%, t ) = f a ( z ) e Z t / 2 z XTaking Z. fa(z)from existing parametrizations and X2 generating the standard value (k:) M (300MeV)2 for quarks gives a reasonable description6 of FP(t) for -t 1 - 10GeV2. For small z, the usual parton densities have a Regge behavior f(z) zPa(O).For t # 0, this suggests F ( z , t ) z-a(t)or, for a linear Regge trajectory .F,(x, t ) = fa(z) z-~'~ With . the Regge slope a' 1GeV2, this model (Fig. 3, dotted lines) allows to obtain correct charge radii for the proton and neutron'. At large t , the form factor behavior is determined by This correthe z 1 behavior of f,(z),giving t-(n+')if fa(z) (1 - z)~. lation is different from the Drell-Yan-West relation, which gives t - ( n + 1 ) / 2 . One can conform with DYW without changing small-z behavior by taking = f,(~)z-"'('-")~. To apply this model to Fz(t), modified ansatz Fa(z,t) one needs unknown magnetic parton densities K , ( X ) . To produce a faster large-t fall-off of Fz(t) compared to Fl(t),one can take functions R , ( x ) having extra powers of (1 - z). With ~ ~ ( z (1 ) - z)qafa(z)one gets F2,(t)/Fla(t) l/t"'. The choice' qZl= 1.52, q d = 0.31 allows to fit the JLab polarization transfer datag on the ratio FZ(t)/Fl(t) for the proton, and also provides rather good fits for all four nucleon electromagnetic form factors, see solid line curves on Fig. 3.
-
-
N
-
-
-
-
-
3. Wide-angle Compton scattering
NPDs also appear in the wide-angle real Compton scattering (WACS). The handbag term (Fig. 2, right) is now given by the 1/z moment of F ( z ,t )
6 "0
.
L B
[3
1.2 1
0.8
0.6 0.4
0.08 0.07 0.06 0.05 0.04
0.03 0.02 0.01 0
10
-'
1
10
-t (GeV')
Figure 3.
Nucleon form factors in Regge-type models for nonforward parton densities.
and the amplitude of the Compton scattering off an elementary fermion. The cross section then can be expressed in terms of F a ( x ,t) and the KleinNishina (KN) cross section for the Compton scattering off an electron:
The approach6?l0based on handbag dominance gives (with the Gaussian NPDs fixed from the F l ( t ) form factor fitting) the results close both to old Cornell data" and the new preliminary d ata l2 ~ l3of JLab E-99-114 experiment. The predictions based on pQCD two-gluon hard exchange mechanism depend on the proton wave function and the value of a,. For the standard choice as = 0.3, the pQCD curves (see Ref.14 for the latest calculation) are well below the data even if one uses extremely asymmetric distribution amplitudes (DAs). Increasing a, to 0.5 gives a better agreement, but then pQCD predictions for Fl (t) form factor overshoot the data. To remove the overall normalization uncertainty, one can consider the ratio [ ~ ~ d u / d t ] / [ t ~ F 1sensitive ( t ) ] ~ only to the shape of the proton DA. The pQCD results for this ratio presented in Ref.14 are a n order of magnitude below the data for all DAs considered: unlike the GPD approach, pQCD cannot simultaneously describe form factor and WACS cross section data.
7
I Polarizationtransfer coefficient K~~ I
I Cross section scaling parameter 9.5
rn
E99-114(preliminsr/) Cornell
6.5
two duon exchange
5.5 60
Figure 4.
m
80
w
Ldeg
100
110
I 0
Comparison of preliminary JLab data with theoretical predictions.
Furthermore, hard pQCD and soft handbag mechanism give drastically different predictions1°~14for the polarization transfer coefficient KLL. The preliminary results (Fig. 4, left) of E-99-114 experiment13 strongly favor handbag mechanism that predicts a value close to the asymmetry for the Compton scattering on a single free quark. Another ratio-type prediction of pQCD is based on the dimensional quark counting rules, which give for WACS d a / d t s-"f(Oc,) with n = 6 for all center-of-mass angles OCM. The handbag mechanism corresponds to a power n depending on O C M , in agreement with the preliminary E-99-114 data13 (see Fig. 4, right). N
4.
Distribution amplitudes and pion form factors
Distribution amplitudes describe the hadron structure in situations when pQCD factorization is applicable for exclusive processes. They are defined through matrix elements (01.. . lp) of light cone operators. For the pion,
1 1
( 0 I d,i(-Z/2)rsy"$,zl(z/2)
I r + ( p ) ) = i#fir
-1
e-i"(pz)/2p,(a)
da 7 (5)
with 21 = (1+ a ) / 2 , 2 2 = (1 - a ) / 2 being the fractions of the pion momentum carried by the quarks. The simplest case is y*y -+ ro transition. Its large-Q2 behavior is light-cone dominated: there is no competing Feynmantype soft mechanism. The handbag contribution for y*y-+ ro (Fig. 5 , left) is proportional to the 1/(1- a 2 )moment of p x ( a )which allows for an experimental discrimination between the two popular models: asymptotic cpv(a)= i ( 1 - a 2 )and Chernyak-Zhitnitsky DA cpZz(a)= y a 2 ( 1 - a 2 ) . Comparison with data favors DA close to pF;s(cr). An important point is
8
Figure 5. Lowest-order pQCD factorization for y'y the pion EM form factor.
-+ K O
transition amplitude and for
-
that pQCD works here from rather small values Q2 2 GeV2, just like in DIS, which is also a purely light-cone dominated process. Another classic application of pQCD to exclusive processes is the pion electromagnetic form factor. With the asymptotic pion DA, the hard pQCD contribution (Fig. 5, right) to Q2Fn(Q2)is 2a,/n x 0.7GeV2, less than 1/3 of experimental value which is close to VMD expectation l / ( l / Q 2 l/mz). The suppression factor 2a,/n reflects the usual a,/n per loop penalty for higher-order corrections. The competing soft mechanism is zero order in a, and dominates over the pQCD hard term at accessible Q2. Just like in the case of F f ( t ) , the soft contribution for Fn(Q2) can be modeled by nonforward parton densities and easily fits the data (see Ref.15).
+
5.
Hard electroproduction processes and generalized parton distributions
A more recent attempt to use pQCD to extract information about hadronic structure is the study of deep exclusive photon2i3 or meson3J6 electroproduction. When both Q2 and s ( p q)2 are large while t ( p - P ' ) ~is small, one can use pQCD factorization of the amplitudes into a convolution of a perturbatively calculable short-distance part and nonperturbative parton functions describing the hadron structure. The hard subprocesses in these two cases have different structure (Fig. 6). For deeply virtual Compton scattering (DVCS), hard amplitude has structure similar to that of the y*yno form factor: the pQCD hard term is of zero order in a,, and there is no competing soft contribution. Thus, we can expect that pQCD works from Q2 2GeV2. On the other hand, the deeply virtual meson production process is similar to the pion EM form factor: the hard term has 0(as/7r) 0.1 suppression factor. As a result, the dominance of the hard pQCD term may be postponed to Q2 5 - 10GeV2. Just like in case of pion and nucleon EM form factors, the competing soft mechanism
+
-
=
-
-
9
Figure 6.
Hard subprocesses for deeply virtual photon and meson production.
can mimic the power-law Q2-behavior of the hard term. Hence, a mere observation of a “correct” power behavior of the cross section is not a proof that pQCD is already working. One should look a t several characteristics of the reaction to make conclusions about the reaction mechanism. To visualize DVCS’s specifics, take the y*N center-of-mass frame, with the initial hadron and the virtual photon moving in opposite directions along the z-axis. Since t is small, the hadron and the real photon in the final state also move close to the z-axis. This means that the virtual photon momentum q = q/ - z ~ j (where p X B ~= Q2/2(pq) is the same Bjorken variable as in DIS) has the component - z ~ j pcanceled by the momentum transfer T . In other words, T has the longitudinal component T + = z ~ j p + , and DVCS has skewed kinematics: the final hadron’s “plus” momentum (1 - [)p+ is smaller than that of the initial hadron (for DVCS, [ = X B ~ ) . The plus-momenta Xp+ and ( X - [)p+ of the initial and final quarks in DVCS are also not equal. Furthermore, the invariant momentum transfer t in DVCS is nonzero. Thus, the nonforward parton distributions (NFPDs) F c ( X ; t ) describing the hadronic structure in DVCS depend on X , the fraction of p+ carried by the initial quark, on [, the skewness parameter characterizing the difference between initial and final hadron momenta, and on t , the invariant momentum transfer. In the forward T = 0 limit, we have a reduction formula Ff=o(X,t = 0) = f a ( X ) relating NFPDs with the usual parton densities. The nontriviality of this relation is that &(X;t ) appear in the amplitude of the exclusive DVCS process, while the usual parton densities are extracted from the cross section of the inclusive DIS reaction. In the limit of zero skewness, NFPDs correspond to nonforward parton densities F f z 0 ( X ,t ) = P ( X ,t). The local limit results in a formula similar to Eq.(3) : X integral of F f ( X ,t ) - FF(X,t ) gives Fl,(t). The NFPD convention uses the variables most close to those of the usual parton densities. To treat initial and final hadron momenta symmetrically, Ji proposed2 the variables in which the plus-momenta of the hadrons are (1+ [)P+ and (1 - [ ) P + , and those of the active partons are (z [)P+ and (x - [)P+, with P = ( p p‘)/2 (Fig. 7). Since [p+ = T+ = 2[P+,
+
+
10
Figure 7. Comparison of N F P D s and OFPDs.
y f ~ t L Q ( O ) P a ( Y I )7I p , s )(16)
which has again a very physical interpretation: the average kl is obtained by summing over the I impulse caused by the color-Coulomb field (since we solved the constraint equations only to lowest order) of the spectators. One immediate consequence of this result is that the total Sivers effect (for the gluon Sivers effect see Refs. 13914) summed over all quarks and gluons with equal weight is zero
c
(kf) = 0
c=q,g
(by symmetry). One can show l5 that this result holds beyond lowest order in perturbation theory. It should emphasized that Eq. (17) is not a trivial consequence of momentum conservation since kl in the Sivers effect is not the momentum of the partons before the collision (which also enters the Nother momentum). Instead the I momentum in the Sivers effect is the sum of the momentum the partons have before being ejected plus the momentum they acquire due to the FSI. Since the momenta before the partons are ejected add up to zero, Eq. (17) is thus a statement about the net momentum due to the FSI: the net (summed over all partons) I momentum due to the FSI is zero, which is a nontrivial result since what one adds up here is not the Imomenta of all fragments in the target but
70 only the I momenta in the current fragmentation region. Eq. (17) is therefore a nontrivial and useful constraint on parameterizations of Sivers distributions Eq. (16) is also very useful for practical evaluation of the Sivers effect from light-cone wave functions. The original expression (8) involved the gauge field a t x- = 00, which is very sensitive to the regularization procedure, we have succeeded t o express the asymmetry in terms of degrees of freedom at finite x-, i.e. I color density-density correlations in the I plane. Eq. (16) can be directly applied t o light-cone wavefunctions, without further regularization. If we want to proceed further, we need a model for the light-cone wave function. Here we do not want to consider a specific model, but rather the whole class of valence quark models, which may be useful for intermediate and larger values of x. In a valence quark model, since the color part of the wave function factorizes, one can replace the color density-density correlations by neutral density-density correlations 14315y16.
and therefore
4.2. Connection with GPDs
From studies of generalized parton distributions (GPDs) it is known that the distribution of partons in the Iplane q(x,b l ) is significantly deformed for a transversely polarized target 17. The mean displacement of flavor q (Iflavor dipole moment) is given by
d;
's
IEP
- dxE,(x,O,O) = 2. (20) '-2M 2M
5Jdxpblq(x,bl)b
-
The I E ~= O(1- 2 ) are the anomalous magnetic moment contribution from each quark flavor to the anomalous magnetic moment of the nucleon (with charge factors taken out), i.e. Fz(0)= 3nU 2 - ?1 I E ~ Z1 K , .... This yields Id:[ = O ( 0 . 2 f m ) , where u and d quarks have opposite signs. This is a sizeable effect as is illustrated in Fig. ( 2 ) . The physical origin of this distaortion is that due t o the kinematics of DIS it is the j + = j o + j " density of the quarks which couples to the
71
Figure 2. Distribution of the j + density for u and d quarks in the Iplane ( x ~ =j 0.3 is fixed) for a nucleon that is polarized in the x direction in the model from Ref. 17. For other values of x the distortion looks similar.
electron: the electron in DIS couples more strongly to quarks which move towards the electron rather than away from it because if the quarks move towards (collision course) the electron the electric and magnetic forces add up, while if they move away the electric and magnetic forces act in opposite directions. For relativistic particles electric and magnetic forces are of the same magnitude. As a consequence, if the i axis is in the direction of the momentum of the virtual photon then the virtual photon couples only to the j + component of the quark current. Even though the j o component of the current density is the same on the +6 and -6 sides of the nucleon, the j z component has opposite signs on the +$ and -6 sides if the quarks have orbital angular momentum. Therefore the reason for the distortion is a combination of the fact that the electron ‘sees’ oncoming quarks better and the presence of orbital angular momentum. While Eq. (20) is a rigorous result regarding the average distortion of quarks with flavor q relative to the center of momentum, it still does not tell us exactly what the density density correlations are. However, qualitatively we expect that the sign and magnitude of the distortion is correlated with the sign of the density-density correlation. Using Eq. (19) we therefore expect for the resulting Sivers effect
for a proton polarized in +5 direction and we expect them to be roughly of the same magnitude.
72
Figure 3. The transverse distortion of the parton cloud for a proton that is polarized into the plane, in combination with attractive FSI, gives rise to a Sivers effect for u ( d ) quarks with a Imomentum that is on the average up (down).
The interpretation of these results is as follows: the FSI is attractive and thus it “translates” position space distortions (before the quark is knocked out) in the +fj-direction into momentum asymmetries that favor the -6 direction and vice versa (Fig. 3) 18. At least in a semi-classical description, this appears to be a very general observation, which is why we expect that the signs obtained above are not affected by higher order effects. 5 . Summary
Wilson line gauge links in gauge invariant Sivers distribution are a formal tool to include the final state interaction in semi-inclusive DIS experiments. The average transverse momentum due to these Wilson lines is obtained as the correlation between the quark density and the impulse from the spectators on the active quark is it escapes along its (almost) light-like trajectory. In light-cone gauge A+ = 0 only the gauge link at infinity contributes and careful regularization of the zero-modes is necessary. However, we succeeded in expressing the net asymmetry in terms of color density-density correlation in the Iplane. For a transversely polarized target the quark distribution in impact parameter space is transversely distorted due to the presence of quark orbital angular momentum: the j + current density is enhanced on the side where the quark orbital motion is head-on with the virtual photon. As the struck quark tries t o escape the target, one expects on average an attractive force from the spectators on the active quark, i.e. the FSI convert a left-right asymmetry for the quark distribution in impact parameter space into a right-left asymmetry for the I momentum of the active quark (Sivers effect). The sign of the distortion in impact parameter space, and hence the sign of the Sivers effect, for each quark flavor is determined by the sign of the anomalous magnetic moment contribution (of course with the electric
73 charge factored out) from t h a t quark flavor t o t h e anomalous magnetic moment of t h e nucleon.
References 1. N.C.R. Makins (HERMES collaboration), talk at eRHIC workshop, BNL, Jan. 2004; see also http://www-hermes.desy.de 2. D.W. Sivers, Phys. Rev. D 43,261 (1991). 3. J.C. Collins, Acta Phys. Polon. B 34,3103 (2003). 4. S.J. Brodksy, D.S. Hwang, and I. Schmidt, Phys. Lett. B 530,99 (2002). 5. M. Burkardt and D.S. Hwang, Phys. Rev. D 69, 074032 (2004); F. Yuan, Phys. Lett. B575,45 (2003); A. Bachetta, A. Schafer, and J.-J. Yang, Phys. Lett. B 578,109 (2004). 6. X. Ji and F. Yuan, Phys. Lett. B 543,66 (2002); A. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B 656,165 (2003). 7. J.C. Collins, Phys. Lett. B 536, 43 (2002); D. Boer, P.J. Mulders, and F. Pijlman, Nucl. Phys. B 667,201 (2003); see also R.D. Tangerman and P.J. Mulders, Phys. Rev. D 51,3357 (1995). 8. J.W. Qiu and G. Sterman, Phys. Rev. Lett. 67,2264 (1991). 9. A. Schafer et al., Phys. Rev. D 47,1 (1993). 10. D. Boer, P.J. Mulders, and F. Pijlman, Nucl. Phys. B 667,201 (2003). 11. M. Burkardt, Adv. Nucl. Phys. 23, l(1996). 12. M. Burkardt, Phys. Rev. D 69,057501 (2004). 13. P.J. Mulders and J. Rodrigues, Phys. Rev. D 63,094021 (2001). 14. D. Boer and W. Vogelsang, hep-ph/0312320. 15. M. Burkardt, Phys. Rev. D 69,091501 (2004). 16. M. Anselmino, M. Boglione, and F. Murgia, Phys. Rev. D 60,054027 (1999); M. Anselmino, U. D’Alesio, and F. Murgia, Phys. Rev. D 67,074010 (2003). 17. M. Burkardt, Int. J. Mod. Phys. A 18, 173 (2003). 18. M. Burkardt, Nucl. Phys. A 733,185 (2004).
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SECTION 2. HEAVY QUARK PHYSICS
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SOFT-COLLINEAR FACTORIZATION AND THE CALCULATION OF THE B -+ X,-y RATE
MATTHIAS NEUBERT* Institute for High-Energy Phenomenology Newman Laboratory for Elementary-Particle Physics, Cornell University Ithaca, IVY 14853, U . S . A .
Using results on soft-collinear factorization for inclusive B-meson decay distributions, a systematic study of the partial B --t X,y decay rate with a cut E, 2 Eo on photon energy is performed. For values Eo 5 1.9 GeV the rate can be calculated without reference to shape functions. The result depends on three large scales: mb, and A = m b - 2Eo. The sensitivity to the scale A M 1.1GeV (for Eo M 1.8 GeV) introduces significant uncertainties, which have been ignored in the past. Our new prediction for the B + X,y branching ratio with E, 2 1.8 GeV is where the errors refer to perturbative Br(B -+Xsy) = ( 3 . 4 4 f 0 . 5 3 f 0 . 3 5 ) x and parameter uncertainties, respectively. The implications of larger theory uncertainties for New Physics searches are explored with the example of the type-I1 two-Higgs-doublet model.
a,
1. Introduction
Given the prominent role of B + X,y decay in searching for physics beyond the Standard Model, it is of great importance to have a precise prediction for its inclusive rate and CP asymmetry in the Standard Model. The total inclusive B + X,y decay rate can be calculated using a conventional operator-product expansion (OPE) based on an expansion in logarithms and inverse powers of the b-quark mass. However, in practice experiments can only measure the high-energy part of the photon spectrum, E-, 2 Eo, where typically EO = 2GeV or slightly below (measured in the B-meson rest frame).li2 With E-, restricted to be close to the kinematic endpoint at M B / ~the , hadronic final state X , is constrained to have large energy EX M B but only moderate invariant mass Mx (MBAQCD)lI2. In this kinematic region, an infinite number of leading-twist terms in the OPE need
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*Work supported by the National Science Foundation under grant PHY-0355005
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to be resummed into a non-perturbative shape function, which describes the momentum distribution of the b-quark inside the B m e s ~ n . ~ ? ~ Conventional wisdom based on phenomenological studies of shapefunction effects says these effects are important near the endpoint of the photon spectrum, but they can be ignored as soon as the cutoff EO is lowered below about 1.9 GeV. In other words, there should be an instantaneous transition from the “shape-function region” of large non-perturbative corrections to the “OPE region”, in which hadronic corrections to the rate are suppressed by at least two powers of AqcDlmb. Below, we argue that this notion is based on a misconception. While it is correct that once the cutoff EO is chosen below 1.9GeV the decay rate can be calculated using a local short-distance expansion, we show that this expansion involves three “large” scales. In addition to the hard scale T n b , an intermediate scale corresponding to the typical invariant mass of the hadronic final state X,, and a low scale A = mb - 2Eo related to the width of the energy window over which the measurement is performed, become of crucial importance. The precision of the theoretical calculations is ultimately determined by the value of the lowest short-distance scale A, which in practice is of order 1GeV or only slightly larger. The theoretical accuracy that can be reached is therefore not as good as in the case of a conventional heavy-quark expansion applied to the B system. More likely, it is similar to (if not worse than) the accuracy reached, say, in the description of the inclusive hadronic decay rate of the T lepton. While we are aware that this conclusion may come as a surprise to many practitioners in the field of flavor physics, we believe that it is an unavoidable consequence of our analysis. Not surprisingly, then, we find that the error estimates for the B + X,y branching ratio that can be found in the literature are, without exception, too optimistic. Since there are unknown a : ( A ) corrections a t the low scale A 1GeV, we estimate the present perturbative uncertainty in the B -+X,y branching ratio with Eo in the range between 1.6 and 1.8GeV to be of order 10-15%. In addition, there are uncertainties due to other sources, such as the b- and c-quark masses. The combined theoretical uncertainty is of order 15-20%, about twice as large as what has been claimed in the past. While this is a rather pessimistic conclusion, we stress that the uncertainty is limited by unknown, higher-order perturbative terms, not by non-perturbative effects, which we find to be under good control. Therefore, there is room for a reduction of the error by means of well-controlled perturbative calculations.
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79 2. QCD factorization theorem
Using recent results on the factorization of inclusive B-meson decay distribution^,^?^ it is possible to derive a QCD factorization formula for the integrated B + X,y decay rate with a cut E7 2 EO on photon energy. In the region of large Eo, the leading contribution to the rate can be factorized in the form7
where AE = M B - 2Eo is twice the width of the window in photon energy over which the measurement of the decay rate is performed. The variable P+ = E x - IFXI is the “plus component” of the 4-momentum of the hadronic final state X,, which is related to the photon energy by P+ = M B - 2E7. The endpoint region of the photon spectrum is defined by the requirement that P+ 5 A E AQCD. Technically, the kinematic power corrections correspond to subleading jet functions arising in the matching of QCD onto higher-dimensional SCET operators, as well as subleading shape functions arising in the matching of SCET onto HQET operators. The corresponding terms are known in fixed-order perturbation theory, without scale separation and RG r e ~ u r n m a t i o n . ’ ~To > ~perform ~ a complete RG analysis of even the first-order terms in A/mb is beyond the scope of our discussion. Since for typical values of Eo the power corrections only account for about 15% of the B + X,y decay rate, an approximate treatment suffices at the present level of precision. Details of how these corrections are implemented can be found in Ref. 7. 4. Ratios of decay rates
.
The contributions from the three different short-distance scales entering the central result (4) and the associated theoretical uncertainties can be disentangled by taking ratios of decay rates. Some ratios probe truly shortdistance physics (i.e., physics above the scale p h mb) and so remain unaffected by the new theoretical results presented above. For some other ratios, the short-distance physics associated with the hard scale cancels to a large extent, so that one probes physics a t the intermediate and low scales, irrespective of the short-distance structure of the theory.
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Ratios insensitive to low-scale physics: Physics beyond the Standard Model may affect the theoretical results for the B -+ X,y branching ratio and CP asymmetry only via the Wilson coefficients of the various operators in the effective weak Hamiltonian. As a result, the ratio of the B + X,y decay rate in a New-Physics model relative to that in the Standard Model
85 remains largely unaffected by the resummation effects studied in the present work. F'rom (4),we obtain rB--tX.TlNP rB--tX,-ylSM
- IWPh)I2,P IH'i'(Ph)l$M
+
power corrections.
(7)
The power corrections would introduce some mild dependence on the intermediate and low scales pi and PO, as well as on the cutoff Eo. Another important example is the direct CP asymmetry in B 3 X,y decays, for which we obtain
. .
where p7 is obtained from H , by CP conjugation, which in the Standard Model amounts to replacing the CKM matrix elements by their complex conjugates. It follows that predictions for the CP asymmetry in the Standard Model and various New Physics scenarios15 remain largely unaffected by our considerations.
Ratios sensitive to low-scale physics: The multi-scale effects studied in this work result from the fact that in practice the B + X,y decay rate is measured with a restrictive cut on the photon energy. These complications would be absent if it were possible to measure the fully inclusive rate. It is convenient to define a function F(E0) as the ratio of the B + X,y decay rate with a cut EO divided by the total rate,
Because of a logarithmic soft-photon divergence for very low energy, it is con~entional'~ to define the "total" inclusive rate as the rate with a very low cutoff E, = rnb/20. The denominator in the expression for F(E0) can be evaluated using the standard OPE, which corresponds to setting all three matching scales equal to ph. The numerator is given by our expression in (4),supplemented by power corrections. Another important example of a ratio that is largely insensitive to the hard matching contributions is the average photon energy (E,), which has been proposed as a good way to measure the b-quark mass.16 The impact of shape-function effects on the theoretical prediction for this quantity has been investigated and was found to be significant.14J7 Here we study the average photon energy in the MSOPE region, where a modelindependent prediction can be obtained. It is structurally different from
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the one found using the conventional OPE in the sense that contributions associated with different scales are disentangled from each other. We stress that the hard scale ph mb affects the average photon energy only via second-order power corrections. This shows that it is not appropriate to compute the quantity ( E y )using a simple heavy-quark expansion a t the scale mb, which is however done in the conventional approach.16 This observation is important, because information about moments of the B + X,y photon spectrum is sometimes used in global fits to determine the CKM matrix element IVcbl. Keeping only the leading power corrections, which is a very good approximation, we find that ( E 7 ) only depends on physics a t the intermediate and low scales pi and po. For EO = 1.8GeV, we obtain ( E 7 ) NN [2.222 0.2540,(=) 0.009as(A)] GeV M 2.30 GeV.
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5 . Numerical results
We are now ready to present the phenomenological implications of our findings. A complete list of the relevant input parameters and their uncertainties is given in Ref. 7, where we also explain our strategy for estimating the perturbative uncertainty as well as the uncertainty due to parameter variations. We begin by presenting predictions for the CP-averaged B -+ X,y branching fraction with a cutoff Ey 2 EO applied on the photon energy measured in the B-meson rest frame. Lowering Eo below 2GeV is challenging experimentally. The first measurement with EO = 1.8GeV has recently been reported by the Belle Collaboration.' It yieldsa E,>1.8 GeV
Eo=1.8 GeV
= (3.38 f 0.30 f 0.29) . l o v 4 , = (2.292 f 0.026 f 0.034) GeV
For Eo = 1.8GeV we have A M 1.1GeV, which is sufficiently large to apply the formalism developed in the present work. (For comparison, the value EO= 2.0 GeV adopted in the CLEO analysis' implies A x 0.7 GeV, which we believe is too low for a short-distance treatment.) We find E0=1.8GeV
= (3.44 f0.53 [pert.] f0.35 [pars.]) x
, (11)
where the first error refers to the perturbative uncertainty and the second one to parameter variations. The largest parameter uncertainties are due &To obtain the first result we had to undo a theoretical correction accounting for the effects of the cut E, > 1.8GeV, which had been applied to the experimental data.
87 to the b- and c-quark masses. Our result is in excellent agreement with the experimental value shown in (10). Comparing the two results, and naively assuming Gaussian errors, we conclude that Br(B
+ X s ~ ) e x-pBr(B + X,Y)SM< 1.4. lop4
(95% CL) .
(12)
Mainly as a result of the enlarged theoretical uncertainty, this bound is much weaker than the one derived in Ref. 18, where this difference was found to be less than 0.5. Hence, we obtain a much weaker constraint on New Physics parameters. For instance, for the case of the type-I1 twoHiggs-doublet model, we may use the analysis of Ref. 19 to deduce mH+
> (slightly below) 200GeV
(95% CL) ,
(13)
which is significantly weaker than the constraints mH+ > 500 GeV (at 95% CL) and mH+ > 350GeV (at 99% CL) found in Ref. 18. The function F(E0) provides us with an alternative way to discuss the effects of imposing the cutoff on the photon energy. In contrast to the branching ratio, it is independent of several input parameters (e.g., mb(%b), Iv,',&bl,T B , XI,^), and it shows a very weak sensitivity to variations of the remaining parameters. We obtain F(1.8GeV) = (92f,; [pert.] f 1[pars.])%.
(14)
This is the first time that this fraction has been computed in a model independent way. The result may be compared with the values (95.8?:':)% and (95 f 1)%obtained from two studies of shape-function model^,^^^^^ in which perturbative uncertainties have been ignored. We obtain a significantly smaller central value with a much larger uncertainty. The last quantity we wish to explore is the average photon energy. As mentioned above, this observable is very sensitive to the interplay of physics at the intermediate and low scales. The study of uncertainties due to parameter variations exhibits that the prime sensitivity is to the b-quark mass, which is expected, since (E-,) = mb/2 . . . to leading order. The nextimportant contribution to the error comes from the HQET parameter X1. To a very good approximation, we have
+
where the error accounts for the perturbative uncertainty. The quantities 6mb and 6x1 parameterize possible deviations of the relevant input parameters from their central values mb = 4.65 GeV and A 1 = -0.25 GeV2. Our
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prediction is in excellent agreement with the Belle result in (10). This finding provides support to the value of the b-quark mass in the shape-function scheme extracted in Ref. 5. We stress, however, that the large perturbative uncertainties in the formula for (E T ) impose significant limitations on the precision with which mb can be extracted from a measurement of the average photon energy. Our estimate above implies a perturbative uncertainty of bmb[pert.] = ':::MeV. This is in addition to twice the experimental error in the measurement of ( E y ) ,which at present yields bmb[exp.] = 86 MeV.
6. Conclusions and outlook
We have performed the first systematic analysis of the inclusive decay B -+ X,y in the presence of a photon-energy cut ET 2 Eo, where EOis such that A = mb - 2Eo can be considered large compared to AQCD, while still A > A,,,) functions can be applied. However, the second condition (A $' problem for the transitions into the charm P-wave states is discussed.
Heavy quark physics, in particular electroweak decays of beauty particles, is now a well developed field of QCD. The most nontrivial dynamic predictions are made for sufficiently inclusive heavy flavor decays admitting the local operator product expansion (OPE). These predictions are phenomenologically important - they allow to reliably extract the underlying CKM mixing angles lV&,l and lvubl with record accuracy from the data, or the fundamental parameters like mb and m,. At the same time heavy quark theory yields informative dynamic results for a number of exclusive transitions as well. Recent years have finally witnessed a more united approach to inclusive and exclusive decays which previously have been largely isolated. In this talk I closely follow the nomenclature of the review' where the principal elements of the heavy quark theory can be found. For a number of years there has been a wide spread opinion that the predictions of the dynamic QCD-based theory were not in agreement with the data, a sentiment probably still felt today by many. The situation, in fact, has changed over the past few years. A better, more robust approach to the analysis has been put forward', made more systematic3 and applied in p r a ~ t i c e ~ !The ~ . perturbative corrections for all inclusive semileptonic characteristics have finally been ~ a l c u l a t e d Experiments ~~~. have accumulated data sets of qualitatively better statistics and precision. Critically reviewing the status of the theory when confronted with the
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data, we find that the formerly alleged problems are replaced by impressive agreement. Theory often seems to work even better than can realistically be expected, when pushed to the hard extremes. Old problems are left in the past. 1. Inclusive semileptonic decays: theory vs. data
The central theoretical result’ for the inclusive decay rates of heavy quarks is that they are not affected by nonperturbative physics at the level of AQcD/rnQ (even though hadron masses, and, hence the phase space itself, are), and the corrections are given by the local heavy quark expectation values - p: and p& to order l/m$, etc. Today’s theory has advanced far beyond that and allows, for instance, to aim at an 1% accuracy in lvcbl extracted from &@). A similar approach to Ivubl is more involved since theory has to conform with the necessity for experiment to implement significant cuts which discriminate against the b + cCu decays. Yet the corresponding studies are underway and a 5% accuracy seems realistic. There are many aspects theory must address to target this level of precision. One facet is perturbative corrections, a subject of controversial statements for many years. The reason goes back to rather subtle aspects of the OPE. It may be partially elucidated by Figs. 1 which shows the relative weight of gluons with different momenta Q affecting the total decay rate and the average hadronic recoil mass squared ( M i ) ,respectively. The contributions in the conventional ‘pole’-type perturbative approach have long tails extending to very small gluon momenta below 500MeV, especially for ( M i ) ; the QCD coupling a,(&) grows uncontrollably there. These tails would be disastrous for precision calculations manifest, for instance, through a numerical havoc once higher-order corrections are incorporated. Yet applying literally the Wilsonian prescription for the OPE with an explicit separation of scales in all strong interaction effects, including the perturbative contributions, effectively cuts out the infrared pieces! Not only do the higher-order terms emerge suppressed, even the leading-order corrections become small and stable. This approach, applied to heavy quarks long agog implies that the precisely defined running heavy quark parameters mb(p),K ( p ) , p ; ( p ) , ... appear in the expansion, rather than ill-defined parameters like pole masses, K, --A1 employed by HQET. Then it makes full sense to extract these genuine QCD objects with high precision.
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Figure 1. The role of the gluons with different momenta in rSland in
(M:),for b - t c t v .
The most notable of all the alleged problems for the OPE in the semileptonic decays was, apparently, the dependence of the fi0.4 m nal state invariant hadron mass on 0.2 the lower cut EfUtin the lepton Lepton Energy Cut (GeV) 0 energy: theory seemed to fall far 0.8 1 1.2 1.4 1.6 off lo of the experimental data, see Figure 2. Ref.lo predictions for (M;) Fig. 2. The robust approach, On (red triangles), with the authors' theory the contrary appears to describe error bars. Black squares are preliminary it well1', as illustrated by Figs. 3. (2002) BaBar data points. The second moment of the same distribution also seems to perfectly fit theoretical expectation^^.^ obtained using the heavy quark parameters extracted by BaBar from their data5. The second moment in the 'inapt' calculations by Bauer et aL1', on the contrary showed unphysical growth with the increase of E&, in clear contradiction with expectations and data.
,
I
Figure 3.
Hadron mass moments dependence on the lepton energy cut.
The comprehensive data analysis is now in the hands of professionals (experimentalists) armed with the whole set of the elaborated theoretical expressions. They are able to perform extensive fits of all the available data from different experiments, and arrive at rather accurate values of the heavy quark parameters, still observing a good consistency of data with
103 theory. A number of such analyses are underway12. Another possible discrepancy between data and theory used to be an inconsistency between the values of the heavy quark parameters extracted from the semileptonic decays and from the photon energy moments13 in B + X , fy. It has been pointed out, however,l' that with relatively high experimental cuts on E, the actual 'hardness' Q significantly degrades compared to mb, thus introducing the new energy scale with Q N 1.2 GeV at E&, = 2 GeV. Then the terms exponential in Q left out by the conventional OPE, while immaterial under normal circumstances, become too important. This is illustrated by Figs. 3 showing the related 'biases' in the extracted values of mb and p:. Accounting for these effects appeared to turn discrepancies into a surprisingly good agreement between all the measurements".
Figure 4. 'Exponential' in Q biases in mb and p : due to the lower cut on photon energy inB-tX,+y.
The problem of deteriorating hardness with high cuts and of the related exponential biases raised in'' was well taken by many experimental groups. BELLE have done a very good job14 in pushing the cut on E, down to 1.8 GeV, which softens the uncertainties in the biases:
(E,) = 2.292 f 0.026,tat f 0.034,,,
GeV
((E,-(E,))2) = 0.0305 5 0.0074,tat f 0.0063,,, GeV2.
(1)
The theoretical expectations based on the central BaBar values of the parameters with mb = 4.612GeV, p: = 0.40GeV2, for the moments with E&, = 1.8 GeV are
(E,) N 2.316GeV, ( ( E , - ( E , ) ) 2 )cz 0.0324GeV2 , (2) again in a good agreement. Although the heavy quark distribution functions governing the shape of the decay distribution in the b c and b 4 u or b + s transitions are different, the Wilsonian OPE ensures that the nonperturbative part of the moments in all these decays is given by the same heavy quark expectation values. This fact appears very important in practical studies aimed
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at extracting IVubl from the inclusive B Xulv rates, since the accuracy in constraining the heavy quark parameters achieved in the b + c l u measurements is significantly higher than direct constraints from the radiative decays. According to experimental analyses, incorporating the former information brings the currently achievable accuracy for extracting 1VubI close to the 5% goal. As a brief summary, the data show good agreement with the properly applied heavy quark theory. In particular, it appears that 0 Many underlying heavy quark parameters have been accurately determined directly from experiment. 0 Extracting lVcbl from r,l(B) has high accuracy and rests on solid grounds. 0 We have precision checks of the OPE-based theory at the level where nonperturbative effects play the dominant role. In my opinion, the most nontrivial and critical test for theory is the consistency found between the hadronic mass and the lepton energy moments, in particular ( M i )vs. (El). This is a sensitive check of the nonperturbative sum rule for M ~ - m b , at the precision level of higher power corrections. It is interesting to note in this respect that a particular combination of the quark masses, mb-Q.74mc has been determined in the BaBar analysis with only a 17MeV error bar! This illustrates how lVCblcan be obtained with high precision: the semileptonic decay rate rs1(B)is driven by nearly the same combination15.
1.1. Comments on the literature a) Semileptonic decays The developed, thorough theoretical approach to the inclusive distributions has not escaped harsh criticism from Ligeti et al. which amounted to the strong recommendations not to use it for data analysis, with the only legitimate approach assumed to be that of Ref.l'. It was claimed, in particular that the observed correct E&,-dependence of ( M ; ) is lost once the complete cut-dependence of the perturbative corrections is included, being offset by the growth in the latter. We showed these claims were not true: the perturbative corrections remain small for the whole interval of Ecutup to 1.4GeV, and actually are practically flat, Fig. 1 of Ref.6. Moreover, the figure shows that these perturbative corrections with the full Ecutdependence in the traditional pole scheme decrease for larger Ecut,in agreement with intuition. The problems in the calculations of Ref.1° have not been traced in detail; its general approach has a number of vulnerable elements, and the
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calculations themselves were not really presented. There are reasons to believe they actually contained plain algebraic mistakes. There is a deeper theoretical reason to doubt the validity of the approach adopted in Ref.” based on the so-called “1s”scheme which is pushed for the analysis of B decays somewhat beyond reasonable limits. In respect to the OPE implementation, it differs little, if any from the usual pole scheme. Only a t the final stage are the observables, like the total width which depends on the powers of mb, re-expressed in terms of the so-called ‘1,”’b mass. The latter is basically in perturbation theory. Since no ‘ T ( 1 S )c-quark mass’ exists, for b+c decays the scheme intrinsically relies on the pole mass relations, in particular to exclude m, from consideration. Use of the l / m , expansion certainly represents a weak point whenever precision predictions are required. In fact, there is a more serious concern about the legitimacy of the perturbative calculations in the ‘1s’scheme, whose working tool is the socalled ‘Upsilon expansion’16. Surprisingly, it is often not appreciated that this is not the conventional perturbative expansion based on the algebraic rules for the usual power series in an expansion parameter like as(mb). This framework rather involves more or less arbitrary manipulations with the conventional perturbative series, referred to as a ‘modified’ perturbative expansion. The rationale behind such manipulations is transparent: the Coulomb binding energy of two massive objects starts with a: terms, hence mis differs from the usual pole mass only to the second order in a , :
The last IR divergent terms 0; a: ln a, simply signify that the Coulomb bound state of a heavy quark Q has an intrinsically different, lower momentum scale a,mQ; in a sense, this scale is zero in the conventional perturbative expansion which assumes a series expansion around a, +0. Hence the relation is not infrared-finite, in the conventional terminology. On the other hand, since the leading, O(a:) term in Eq. (3) comes without PO,the ‘1s’ b quark mass in all available perturbative applications to B decays has to be equated with the pole mass m;’le, whether or not a few BLM corrections are included. To get around this obvious fact, the ‘Texpansion’ postulated considering a number of terms appearing to the lc-th order in perturbation theory, ckat(mb),to be actually of a lower order, c,ay(mb) with n < k . Since the power of the strong coupling is explicit, this is done by introducing an ad hoc factor 6 = 1 , making use of the property that unity remains unity raised
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to arbitrary power. This ad hoc reshuffling constitutes the heart of the ‘T expansion’ and of using the ‘1s’b quark mass mis in B decays.a The scale a,mQ naturally appears in bound-state problems for heavy quarks since the perturbative expansion parameter for nonrelativistic particles is not necessarily a,, but rather runs in powers of a , / ~ where , v is their velocity. The analogue of the ‘bound-state’ mass then naturally appears there, since powers of velocity make up for the missing powers of a,. Yet nothing of this sort is present in B mesons or in their decays, the E parameters introduced by the ‘Yexpansion’ is unity and can be placed ad hoc at any arbitrary place. One clearly should not make up for the numerically larger than a,(mb) value of a,(&) at the smaller momentum scale Q = a,mb by equating at will terms of explicitly different orders in as in the usual perturbative expansion. The ‘Yexpansion’ would be meaningless already in the simplest toy analogue of the B decays, muon @-decay. It is then difficult to count on this approach to be sensible for more involved B decays where real OPE has to be used for high precision. More recently, when this contribution was in writing, the new paper by Bauer et al. appearedlg. Claiming now to describe the cut-dependence of (M:), this paper came up with new statements aimed at discrediting the Wilsonian approach and its implementation. The authors assert that the approach we follow suffers from a large ‘scale-dependence’ when varying the Wilsonian separation scale p. In addition, the authors state they cannot reproduce the hadronic moments calculated in ref^.^>^ used by experimental groups for the data analysis. Once again I have to refute the criticism the p-dependence turns out weak, actually far below the expected level, as illustrated, for instance by Figs. 5 for ( E l ) and ( M i ) . The change in the moments from varying p corresponds to the variation in, say mb of only 4 MeV and 1MeV, respectively! (Ref.3 allowed an uncertainty of 20 MeV due to uncalculated higher-order perturbative corrections.) It looks probable that the authors of Ref.lg simply were not able to perform correctly the calculations in the Wilsonian ‘kinetic’ scheme, at least for the hadronic mass moments. In fact, the suppressed dependence of the observables on the separation scale p is a routine check applied to the calculations. The two facts together are then rather suggestive. Varying p represents a useful - if limited - probe of the potential impact *The original paper16 presented some arguments calling upon the so-called “large n f expansion” supposed to justify reshuffling the orders. I believe that the reasoning was wrong ab initio missing the basics of the renormalon c a l c ~ l u s ’ ~ ~ ~ ~ .
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-7Y 1.387
1.3861 1.386
\
1.384
0.6
0.8
1.2
1.4
Figure 5. Dependence of ( E l ) (left) and of ( M i ) (right) on the separation scale p . The green vertical bars show the change in the moments when mb is varied by f l MeV.
of the omitted higher-order corrections. Clearly, not varying p but fixing its value once and for all, one does not see any scale-dependence (the pole scheme simply amounts to setting p=O). In this respect hints a t an absent p-dependence in the pole-type schemes like ‘IS’smell suspiciously. And, certainly, the absence of an explicit separation scale is not an advantage. The analogous sensitivity to the actually used scale is of course present in the approach of ref^.^^,^^, and the related uncertainties can be easily revealed. The ‘1s’scheme ad hoc postulates using mtS, a half of the “(1s)mass. However, on the same grounds m;b, half of the mass of the ground-state bottomonium, qb(lS) can be used. Even accepting the arbitrary counting rules of the ‘T-expansion’ , all the theoretical expressions used in the analyses, are identical for mp and mis - the masses differ only to order a: (without DO). At the same time, the two b quark masses do differ numerically by at least 20 to 30MeV! This is significantly larger than the criticized p-dependence of the ‘kinetic’ Wilsonian scheme, and it, in any case, should be included as the minimal theory uncertainty of every calculation based on the ‘T’-mass of the b quark (it has not been, of course). The theory error estimates of ref^.^^!^^, upon inspection, look unrealistic, significantly underestimating many potential corrections. The numerical outcome of the fit for lV&l looks close to the value obtained by experimental groups in our approach, within the error estimates we believe are right. This impression would be superficial - the two calculations share many common starting assumptions; therefore, they must yield - if performed correctly - much closer results. One can state they do differ a t a level which is significant theoretically. In this respect I would urge experiments to refrain from averaging the results obtained in the two approaches. It is never a good idea to combine correct results with those based on a potentially flawed calculations. In my opinion, those relying on the ‘Upsilon expansion’ can be considered as such. For instance, the authors of Ref.”, according to their Eq. (26) and
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Table I increase the value of lVcal due to electroweak corrections (the same appears t o apply t o the recent Ref.lg). The fact is the electroweak factor qaEDincreases the width and, therefore suppresses the extracted value of lVcbl by an estimated 0.7%. It is curious to note that, assuming this is just a mistake rather than yet another ad hoc postulate of the ‘Upsilon expansion’, correcting for it would make the lVcbl value of Ref.lg nearly identical to the result obtained by BaBar. Whether such a correspondence is inevitable, or is a matter of coincidence, is not obvious at the moment. The existing PDG reviews on the subject have been so far based exclusively on the questionable papers ignoring more thorough existed analyses, and may therefore represent a not too trustworthy source of information.
b) b + s+-y and b + u+Lv There is a subtlety in accounting for the perturbative effects in the heavy-to-light decays which we do not see in b c &. The radiated gluons can be emitted with sufficiently large energy yet at a very small angle, so that their transverse momentum is only of order Phadr or even lower. This is a nonperturbative regime, and it may generate a new sort of the nonperturbative corrections. These are physically distinct from the Fermi motion encoded in the distribution function of the heavy quark inside the B meson. A dedicated discussion can be found in the recent paper20. Such contributions may indicate that the so-called ‘soft-collinear effective theories’ (SCET), in all their variety, may not truly represent an effective theory of actual QCD, not having the identical nonperturbative content. It has been shown in Ref.20 that this physics, nevertheless do not affect the moments of the decay distributions, in particular the photon energy moments (at a low enough cut). The relation of the moments to the local heavy quark expectation values remains unaltered: the perturbative corrections have the usual structure and include only truly short-distance physics. In this respect, we do not agree with the recent claims found in the literature that the usual OPE relations for the moments in the light-like distributions do not hold where perturbative effects are included. Our analysis does not support large uncertainties in the b -+ s y moments reported by Neubert at the Workshop, see also Refs.21. (It is curious to note the increase in ( E 7 )when lowering Ecutobtained by the author). I would disagree already with the starting point of that approach. On the contrary, applying the Wilsonian approach we find 22 quite accurate, stable (and physical as well) predictions whenever the cut on the photon energy is sufficiently low to cover the major part of the distribution function domain.
+
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2.
A ‘BPS’ expansion
The heavy quark parameters as they emerge from the fit of the data are close to the theoretically expected values, mb(1 GeV) N 4.60 GeV, p:(1 GeV) 21 0.45 GeV2, p& (1 GeV) N0.2 GeV3. The precise value, in particular of p:, is of considerable theoretical interest. It is essentially limited from below by the known chromomagnetic expectation value 23 : &(p) > &(p), & ( l GeV) 11 0.35::;; GeV2, (4) and experiment seem to suggest that this bound is not too far from saturation. This is a peculiar regime where the heavy quark sum rules’, the exact relations for the transition amplitudes between sufficiently heavy flavor hadrons, become highly constraining. One consequence of the heavy quark sum rules is the lower bound24 on the slope of the 1W function e2> They also provide upper bounds which turn out quite restrictive once is close to &, say
i.
e2-
i 5 0.3
if pE(l GeV)-&(l GeV) 6 0.1 GeV2. (5) This illustrates the power of the comprehensive heavy quark expansion in QCD: the moments of the inclusive semileptonic decay distributions can tell us, for instance, about the formfactor for B+ D or B - i D*decays. Another application is the B + D e u amplitude near zero recoil. Expanding in & - & an accurate estimate was obtained25
f+(O) = 1.04 f 0.01 f 0.01 MB MD In fact, p: 1~& is a remarkable physical point for B and D mesons, since the equality implies a functional relation ZbbjiblB) = 0. Some of the Heavy Flavor symmetry relations (but not those based on the spin symmetry) are then preserved to all orders in l/mQ. This realization led to a ‘BPS’ e x p a n ~ i o nwhere ~~>~ the ~ usual heavy quark expansion was combined with an expansion around the ‘BPS’ limit &bjib(B)=O. There are a number of miracles in the ‘BPS’ regime. They include e2= 2 and p i s = -p$; a complete discussion can be found in Ref.25. Some intriguing ones are2’: No power corrections to the relation M p = mQ + and, therefore to mb-m, = M B - M D . For the B + D amplitude the heavy quark limit relation between the two formfactors 2m
+
110
does not receive power corrections. 0 For the zero-recoil B -+ D amplitude all b l l m k terms vanish. 0 For the zero-recoil formfactor f+ controlling decays with massless leptons
holds t o all orders in l/mQ. 0 At arbitrary velocity, power corrections in B + D vanish,
so that the B-+ D decay rate directly yields the Isgur-Wise function [(w). Since the ‘BPS’ limit cannot be exact in actual QCD, we need to understand the accuracy of its predictions. The dimensionless parameter ,B describing the deviation from the ‘BPS’ limit is not tiny, similar in size to the generic l/m, expansion parameter, and relations violated to order ,B may in practice be more of a qualitative nature. However, the expansion parameters like p: -& 0: ,B2 can be good enough. One can actually count together powers of l/m, and ,B to judge the real quality of a particular heavy quark relation. In fact, the classification in powers of ,B to a l l o r d e r s in l / m is ~ possible.25 Relations (7) and (9) for the B -+ D amplitudes at arbitrary velocity can get first order corrections in ,B, and may be not very accurate. Yet the slope e2 of the IW function differs from only a t order ,B2. Some other important ‘BPS’ relations hold up to order ,B2: M B - M D = mb-m, and M D = m c + x Zero recoil matrix element (DlEyoblB) is unity up to O(,B2) The experimentally measured B -+ D formfactor f+ near zero recoil receives only second-order corrections in ,B to all orders in l/mQ:
2
The latter is an analogue of the Ademollo-Gatto theorem for the ‘BPS’ expansion, and is least obvious. The ‘BPS’ expansion turns out more robust than the conventional l / m Q one which does not protect the decay against the first-order corrections. As a practical application, Ref.25 derived an accurate estimate for the formfactor f+(O) in the B + D transitions, Eq. ( 6 ) , incorporating terms through l/mz,b. The largest correction, +3% comes from the short-distance
111
perturbative renormalization; power corrections are estimated to be only about 1%.
3. The
‘5 > %’problem
So far mostly the success story of the heavy quark expansion for semileptonic B decays has been discussed. I feel obliged to recall the so-called ‘ $ > $, puzzle related to the question of saturation of the heavy quark sum rules. It has not attracted due attention so far, although it had been raised independently by two teams28*1t29 including the Orsay heavy quark group, and it has been around for quite some time. A useful review was recently presented by A. Le Yaouanc30; here I briefly give a complementary view. There are two basic classes of the sum rules in the Small Velocity, or Shifrnan-Voloshin (SV) heavy quark limit. First are the spin-singlet sum rules which relate e2, p:, p g , ... to the excitation energies E and transition amplitudes squared 1rI2for the P-wave states. Both and $ P-wave states, i.e. those where the spin j of the light cloud is 3 or $, contribute to these sum rules. The second class are ‘spin’ sum rules, they express similar relations for e2- h-2C7 p:-p’$, etc. These sum rules include only states. The spin sum rules strongly suggest that the $ states dominate over states, having larger transition amplitudes ~ 3 / 2 In . fact, this automatically happens in all quark models respecting Lorentz covariance and the heavy quark limit of QCD; an example are the Bakamjian-Thomas-type quark models developed at 0rsayB1,or the covariant models on the light front32. The lowest $ P-wave excitations of D mesons, D1 and 05 are narrow and well identified in the data. They seem to contribute to the sum rules too little, with 1 ~ ~ / ~ 1 ~according ~ 0 . 1 5to Ref.33. Wide states denoted by D,*and D; are possibly produced more copiously; they can, in principle, saturate the singlet sum rules. However, the spin sum rules require them to be subdominant to the states. The most natural solution for all the SV sum rules would be if the lowest $ states with ~ 3 / 2N 450 MeV have 1 ~ ~ M/ 0.3, while for the states 1 ~ ~ M/ 0.07 ~ 1 to0.12 ~ with ~ 3 / 2% 300 to500 MeV. Strictly speaking, higher P-wave excitations can make up for the wrong share between the contributions of the lowest states. This possibility is disfavored, however. In most known cases the lowest states in a given channel tend to saturate the sum rules with a reasonable accuracy. It should be appreciated that the above sum rules are exact for heavy quarks. Likewise, the discussed consequences rely on the assumptions most
x,
3
i,
3
3
3
112
robust among those we usually employ in dealing with QCD. Therefore, the problem we examine is not how in practice 7 3 1 2 might turn out less than r1/2. Rather it is why, in spite of the actual hierarchy between 7312 and r112the existing extractions seem to indicate the opposite relation. In fact, the recent pilot lattice indicated the right scale for both ~ 3 / 2and r112and, taken at face value, showed a reasonable saturation of the spin sum rule by the lowest P-wave clan. Similar predictions had been obtained in the relativistic quark model from Orsay31 and in the light-cone quark models32. Experimentally 7312 and r112can be extracted from either nonleptonic decays B + D** T assuming factorization and the absence of the final state interactions, or directly from their yield in B -+ D**lv decays. The former way suffers from possible too significant corrections to factorization, in particular for the case of excited charm states. Such decays also depend on the amplitude at the maximal recoil, kinematically most distant from the small recoil we need the amplitude at; we know that the slopes of the formfactors are quite significant even with really heavy quarks. The safer approach is the direct yield in the semileptonic decays. The data interpretation is obscured, however by the significant corrections to the heavy quark limit for charm mesons. For instance, the classification itself over the light cloud angular momentum j relies on the heavy quark limit. However, one probably needs a good physical reason to have the hierarchy between the finite-m, heirs of the $ and states inverted, rather than only reasonably modified compared to the heavy quark limit. Yet, as has been shown, these corrections are generally significant and may noticeably affect the extracted 1r3p12. It has also been routinely assumed that the slope of the formfactors are similar for the and for the D**, something which is not expected to hold in QCD. The existing models likewise predict a large slope for the $ mesons and a moderate one for the states. This clearly enhances the actual extracted value of 1 ~ ~ / ~ ( ~ . The experimental situation in respect to the wide $ charm states still remains uncertain. It cannot be excluded that their actual yield is smaller, and a t the same time it can be essentially enhanced compared to the largem, limit. To summarize, we do not have a definite answer to how this apparent contradiction of theory with data is resolved. Considering all the evidence, the scenario seems most probable where all the above factors contribute coherently, suppressing the yield of the states more than expected and
+
3
113
i
enhancing the production of the states. First of all, this refers to the size of the power corrections in charm. Secondly, the effect of significant formfactor slopes for the $ states. Finally, it seems possible that the actual branching fraction of the P-wave states in the semileptonic decays would be eventually below 1%level. In my opinion it is important to clarify this problem.
i
Conclusions. The dynamic QCD-based theory of inclusive heavy flavor decays has finally undergone and passed critical experimental checks in the semileptonic B decays at the nonperturbative level. Experiment finds consistent values of the heavy quark parameters extracted from quite different measurements once theory is applied properly. The heavy quark parameters emerge close to the theoretically expected values. The perturbative corrections to the higher-dimension nonperturbative heavy quark operators in the OPE have become the main limitation on theory accuracy; this is likely to change in the foreseeable future. Inclusive decays can also provide important information for a number of individual heavy flavor transitions. The B + D lv decays may actually be accurately treated. The successes in the dynamic theory of B decays put a new range of problems in the focus; in particular, the issue of the saturation of the SV sum rules requires close scrutiny from both theory and experiment. Acknowledgments
I am grateful to D. Benson, I. Bigi, P. Gambino, M. Shifman, A. Vainshtein, 0. Buchmueller and P. Roudeau, for close collaboration and discussions. This work was supported in part by the NSF under grant number PHY0087419. References 1. N.Uraltsev, in Boris I o f e Festschrift “At the Frontier of Particle Physics -- Handbook of QCD”, Ed. M. Shifman (World Scientific, Singapore, 2001), Vol. 3, p. 1577; hep-ph/0010328. 2. N.Uraltsev, Proc. of the 31st International Conference on High Energy Physics, Amsterdam, The Netherlands, 25-31 July 2002 (North-Holland Elsevier, The Netherlands, 2003), S. Bentvelsen, P. de Jong, J. Koch and E. Laenen Eds., p. 554; hep-ph/0210044. 3. P. Gambino and M. Uraltsev, Europ.Phys. Journ. C 34 (2004) 181. 4. M. Battaglia e t al., Phys. Lett. B 556 (2003) 41. 5. B. Aubert et al., BaBar Collaboration, Phys.Rev. Lett. 93 (2004) 011803.
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6. N. Uraltsev, hep-ph/0403166; to appear in IJMPA. 7. M. Trott, hep-ph/0402120. 8. I. Bigi, N. Uraltsev and A. Vainshtein, Phys. Lett. B293 (1992) 430; B. Blok and M. Shifman, Nucl. Phys. B399 (1993) 441 and 459. 9. N.G. Uraltsev, Int. J . Mod. Phys. A l l (1996) 515; Nucl. Phys. B491 (1997) 303; I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, Phys. Rev. D56 (1997) 4017. 10. C.W. Bauer et al., Phys. Rev. D67 (2003) 054012. 11. 1.1.Bigi and N. Uraltsev, Phys. Lett. B579 (2004) 340. 12. 0. Buchmueller, private communication. 13. Z. Ligeti, M. E. Luke, A. V. Manohar and M. B. Wise, Phys. Rev. D60 (1999) 034019. 14. P. Koppenburg et al., Belle collaboration, Phys. Rev.Lett. 93 (1999) 061803. 15. D. Benson, I. Bigi, Th. Mannel and N. Uraltsev, Nucl. Phys. B665 (2003) 367. 16. A.H. Hoang, Z. Ligeti and A.V. Manohar, Phys. Rev. Lett. 82 (1999) 277. 17. A.I. Vainshtein and V.I. Zakharov, Phys. Rev.Lett. 73 (1994) 1207. 18. Yu.L. Dokshitzer and N.G. Uraltsev, Phys. Lett. B380 (1996) 141. 19. C.W. Bauer et al., hep-ph/0408002. 20. N. Uraltsev, hep-ph/0407359. 21. M. Neubert, hep-ph/0408179; hep-ph/0408208. 22. D. Benson, I. Bigi and N. Uraltsev, UND-HEP-04-BIGO3, LNF04/17(P), in preparation. 23. I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, Int. Joum. Mod. Phys. A9 (1994) 2467; M. Voloshin, Surv. High En. Phys. 8 (1995) 27. 24. N. Uraltsev, Phys. Lett. B501 (2001) 86; J . Phys. G27 (2001) 1081. 25. N. Uraltsev, Phys. Lett. B585 (2004) 253. 26. N. Uraltsev, Phys. Lett. B545 (2002) 337. 27. N. Uraltsev, hep-ph/0309081, eConf C030603, JEU05 (2003). Talk at Int. Conference “Flavor Physics & CP Violation 2003” , June 3-6 2003, Paris. 28. I. Bigi, M. Shifman and N.G. Uraltsev, Ann. Rev.Nucl. Part. Sci. 47 (1997) 591. 29. A. Le Yaouanc et al., Phys. Lett. B480 (2000) 119. 30. A. Le Yaouanc, hep-ph/0407310. 31. A. Le Yaouanc, L. Oliver, 0.Pene and J.C. Raynal, Phys. Lett. B365 (1996) 319; V. Morenas et al., Phys. Rev. D56 (1997) 5668. 32. H.Y. Cheng, C.K. Chua and C.W. Hwang, Phys. Rev. D69 (2004) 074025. 33. A.K. Leibovich, Z. Ligeti, I.W. Stewart and M.B. Wise, Phys. Rev.Lett. 78 (1997) 3995; Phys. Rev. D57 (1998) 308. 34. D. Becirevic et al., hep-lat/0406031.
POLARIZATION, RIGHT-HANDED CURRENTS, AND CP VIOLATION I N B VV ---f
ALEXANDER L. KAGAN Department of Physics University of Cincinnati Cincinnati, Ohio 45221, U.S.A. A detailed analysis of B ---t VV polarization in QCD factorization reveals that the low longitudinal polarization fractions f ~ ( 4 K ' = ) 50% can be accounted for in the SM via a QCD penguin annihilation graph. The ratio of transverse rates rl/rll % provides a sensitive test for new right-handed currents in the four-quark operators. CP violation measurements in B -+ VV decays can discriminate between new contributions to the dipole and four quark operators
1. Introduction
Polarization measurements in B -+ VV decays should be sensitive to the V - A structure of the Standard Model. This issue has recently been studied in the QCD factorization framework in and the results are summarized in this contribution. We have found that certain non-factorizable annihilation graphs can easily account for the small B -+ 4K* longitudinal polarization fractions observed a t the B factories. A direct test for right-handed currents in the four-quark operators emerges from the ratio of the transverse perpendicular and parallel transversity rates. In the event that non-Standard Model CP-violation is confirmed, e.g., in the B -+ q5K3 time-dependent CP asymmetry, an important question will be whether it arises via New Physics contributions to the four-quark operators, the b -+ sg dipole operators, or both. We will see that this question can be addressed by comparing CP asymmetries in the different transversity final states in pure penguin B + VV decays, e.g., B -+$K* and Bh -+ K*'p+. The underlying reason is large suppression of the transverse dipole operator matrix elements. It is well known that it is difficult to obtain new O(1) CP violation effects a t the loop-level from the dimension-six four-quark operators. Thus, this information could help discriminate between scenarios in which New Physics effects are induced via loops versus a t tree-level.
115
116
Extensions of the Standard Model often include new b -+ S R righthanded currents. These are conventionally associated with opposite chirality effective operators Qi which are related to the Standard Model operators Qi by parity transformations, 0
QCD Penguin operators Q3,5 Q4,6
0
= ( S ~ ) V - A( ( ? ? ) v F A = (3ibj)V-A (4jQi)VrA
= (Sb)v+A (qq)VfA
Q4,6
-+
(%bj)V+A (qjqi)V&A
Chromo/Electromagnetic Dipole Operators
+~5)biFpu + y5)tabGEu
= &mb%a’l’(l &ag = ambSaPu(1 Q7y
0
Q3,5
+
Q7y
-+
-i
Qgg
= &mbSial’lv(l = &mbSafi’(l
-
ys)U‘p”
- y5)tabGEu
Electroweak Penguin Operators
Q7,9
&S,lO
= Z ( S ~ ) V - Aeq ( & ) V f A = ; ( S i b j ) V - A eq ( q j % ) V * A
+
+
Q7,9 Q8,lO
= ; ( S ~ ) V + A eq ( q Q ) v r A = ;(SZbj)V+A eq ( q j q i ) V r A
Examples of New Physics which could give rise to right-handed currents include supersymmetric loops which contribute to the QCD penguin or chromomagnetic dipole operators. Figure 1 illustrates the well known squarkgluino loops in the squark mass-insertion approximation. For example, the down-squark mass-insertion bm? - (bm!*- ) would contribute to Qsg bRSL
SRbL
whereas bmiLiL(SmiRLR) would contribute t o Q3,..6 (Q3,..,6). Righthanded currents could also arise via tree-level contributions to the QCD or electroweak penguin operators, e.g. , due t o flavor-changing 2’ couplings 2 , R-parity violating couplings 3, or color-octet exchange 4 . (Qgg),
Figure 1. Down squark-gluino loop contributions t o the Standard Model and opposite chirality dipole operators in the squark mass insertion approximation.
We recall that there are three helicity amplitudes Ah ( h = 0, -, +) in VV decays: A’, in which both vectors are longitudinaly polarized; A-, in which both vectors have negative helicity; and A+, in which both
B
-+
117
vectors have positive helicity. In the transversity basis are given by,
A l J = (A- F A + ) / & ,
5,
the amplitudes
A0 = A 0
(1)
In B decays, Al,ll = (A+ ~ A - ) / f i . The polarization fractions are defined as fi = ri/rtota1, i = 0, I,11, where rtotal is the total decay rate. Under parity, the effective operators transform as Qi c) Qi. The New Physics amplitudes, for final states f with parity Pf, therefore satisfy
(fIQiP)= -(-)Pf(fIQilB)
* ANp(B
-+
f) K CifTp(pb)- (-)pfCFp(pb),
(2)
where CY' and 6Yp are the new Wilson coefficient contributions to the i'th pair of Standard Model and opposite chirality operators 6 . In B -+ VV decays the Itransversity and 0, 11 transversity final states are P-odd and P-even, respectively, yielding AiN P ( B + VV)o,ll K CifTP(pb)-C'FP(pb), ANp(B -+ V V ) I K CyP(pb)+CyP(pb).
(3) The modes of particular interest in the search for new physics in rare B decays are those which are pure-penguin or penguin-dominated in the Standard Model. This is because they necessarily have 0 0
null decay rate CP-asymmetries, A c p ( f ) 1%,or null deviations of the time-dependent CP-asymmetry coefficient Sf,, from (sin 2P)J/,pK, in decays to CP-eigenstates, (->cpsfcpl 1% or l(sin2P)J/*Ks null B + VV triple-product CP-asymmetries, A$ll(f) 1%in the Standard Model. N
+
0
N
The CP-violating triple-products
2. Polarization in
B
.--)
(related t o
4'.
x
Z2)
are given by
VV decays
A discussion of polarization in B -+ VV decays has been presented in in the framework of QCD factorization. Here we summarize some of the results. Sensitivity to the V - A structure of the Standard Model is due to
118
the power suppression associated with the ‘helicity-flip’ of a collinear quark. For example, in the Standard Model the factorizable graphs for B 4 4K* are due to transition operators with chirality structures (Sb)V-A(Ss)V?A, see Figure 2 . In the helicity amplitude A- a collinear s or 3 quark with positive helicity ends up in the negatively polarized 4, whereas in A+ a second quark ‘helicity-flip’ is required in the form factor transition. Collinear quark helicity flips require transverse momentum, kl , implying a suppression of O(AQcD/mb)per flip. In the case of new right-handed currents, e.g., ( S ~ ) V + A ( S S ) V ~ the A , helicity amplitude hierarchy would be inverted, with A+ and A- requiring one and two helicity-flips, respectively.
Figure 2. Quark helicities (short arrows) for the B -+ q5K* matrix element of the operator ( B b ) v - A ( S s ) v - A in naive factorization. Upward lines form the q5 meson.
In naive factorization the B -+ $K* helicity amplitudes, supplemented by the large energy form factor relations 8, satisfy 2
A’
0: f$mB(11
(r
and
K’ 1
-f@mq5mB2cf*,
A-
c,” are the B
t
A+ 0: -fdm+mB2cLK’rlK’ .
V form factors in the large energy limit
8.
(5) Both
scale as mb3’2 in the heavy quark limit, implying A-/A’ = O ( m + / m g ) . r l parametrizes form factor helicity suppression. It is given by
where Al,2 and V are the axial-vector and vector current form factors, respectively. The large energy relations imply that r l vanishes at leading power, reflecting the fact that helicity suppression is O ( l / m b ) . Thus, A+/A- = O(f!QcD/mb). Light-cone QCD sum rules ’, and lattice form
119
factor determinations scaled to low q2 using the sum rule approach l o , give rf' M 1 - 3%; QCD sum rules give r f * M 5 % ll; and the BSW model gives rf' M 10%12. The polarization fractions in the transversity basis (1) therefore satisfy
fi/flJ = 1 + 0 (l/mb>
1 - f L = 0 (1/mi) I
(7)
7
in naive factorization, where the subscript L refers to longitudinal polarization, fi = J?i/rtotal, and f~ f i fll = 1. The measured longitudinal fractions for B -+ pp are close to 1 This is not the case for B -+ +K*O for which full angular analyses yield
+ + 16117.
f~ = 0 . 5 2 5 . 0 7 f .02, fr. = .52 f .05 rt .02, For B*
-+$K**,
f i= . 3 0 f fi = .22 rt
. 0 7 f .03
.05 & .02
'* 19.
(8) (9)
Belle measures l8
fL = .49 f
.13 f .05,
fi = .12::
f .03.
(10)
and BaBar measures f~ = .46 f 0.12 f 0.03 17. Naively averaging the Belle and BaBar $K*O measurements (without taking correlations into account) yields f i / f l l = 0.92 f .31. Finally, the longitudinal fraction for B* -+ K*'p* has also been measured at BaBar 2o and Belle21, yielding f~ = 0.79 f 0.08 f 0.04 f 0.02, and 0.5 f o.19tbqi7, respectively, which averages t o f~ = 0.74 f .08. We must go beyond naive factorization in order to determine if the small values of ~ L ( $ K *could ) simply be due to the dominance of QCD penguin operators in A S = 1 decays, rather than New Physics. In particular, it is necessary to determine if the power counting in (7) is preserved by non-factorizable graphs, i.e., penguin contractions, vertex corrections, spectator interactions, annihilation graphs, and graphs involving higher Fock-state gluons. This question can be addressed in QCD factorization '. In QCD factorization exclusive two-body decay amplitudes are given in terms of convolutions of hard scattering kernels with meson light-cone distribution amplitudes At leading power this leads to factorization of short and long-distance physics. This separation breaks down a t subleading powers with the appearance of logarithmic infrared divergences, 1 e.g., dxlx InrnBlAh, where x is the light-cone quark momentum fraction in a final state meson, and Ah AQ~D is a physical infrared cutoff. Nevertheless, the power-counting for all amplitudes can be obtained. The extent to which it holds numerically can be determined by assigning large 13314115.
so
N
N
120 uncertainties t o the logarithmic divergences. Fortunately, certain polarization observables are less sensitive t o this uncertainty, particularly after experimental constraints, e.g., total rate or total transverse rate, are imposed.
b
Figure 3. Quark helicities in B -+ q5K* matrix elements: the hard spectator interaction for the operator ( S b ) v - A ( S S ) V F A (left), and annihilation graphs for the operator ( ( t b ) S - p ( B d ) S + p with gluon emitted from the final state quarks (right).
Examples of logarithmically divergent hard spectator interaction and QCD penguin annihilation graphs are shown in Figure 3, with the quark helicities indicated. The power counting for the helicity amplitudes of the annihilation graph, including logarithmic divergences, is
The logarithmic divergences are associated with the limit in which both the s and 3 quarks originating from the gluon are soft. The annihilation topology implies an overall factor of l / r n b . Each remaining factor of l/mb is associated with a quark helicity flip. In fact, adding up all of the helicity amplitude contributions in QCD factorization formally preserves the naive factorization power counting in (7) Recently, the first relation in (7) has been confirmed in the soft collinear effective theory 23. However, as we will see below, it need not hold numerically because of QCD penguin annihilation. 22y1.
2.1. Numerical results f o r polarization
'.
The numerical inputs are given in The logaritmic divergences are modeled as in l4?l5. For example, in the annihilation amplitudes the quantities
121
X A are introduced as
This parametrization reflects the physical ~ ( R Q c Dcutoff, ) and allows for large strong phases E [0,2n] from soft rescattering. The quantities X A (and the corresponding hard spectator interaction quantities X H ) are varied independently for unrelated convolution integrals. The predicted longitudinal polarization fractions f~(p-po) and f ~ ( p - p + ) are close to unity, in agreement with observation and with naive power counting (7). The theoretical uncertainties are small, particularly after imposing the branching ratio constraints, due to the absence of (for p-po) or CKM suppression of (for p-p+) the QCD penguin amplitudes. Averaging the Belle and BaBar B -+ 4K*O measurements yields = 0.52 h 0.04 and 1O6BreXP= 9.7 f 0.9, or lo6 B r y P = 5.1 f 0.6 and lo6 B r y P = 4.7f0.6. BrL and BrT = B r l +Brll are the CP-averaged longitudinal and total transverse branching ratios, respectively. In the absence l61l7
18,19124
fFp
+6.79+.88
of annihilation, the predicted branching ratios are lo6 BrL = 5.15- 4.66-.81 and lo6 BrT = .61?:2:?:;,, where the second (first) set of error bars is due to variations of XH (all other inputs). However, the ( S P ) ( S - P ) QCD penguin annihilation graph in Figure 3 can play an important role in both 21' and 21- due to the appearance of a logarithmic divergence squared ( X z ) , the large Wilson coefficient c6, and a l/Nc rather than 1/NZ dependence. Although formally O(l/m2), see (ll),these contributions can be O(1) numerically. This is illustrated in Figure 4, where BrL and BrT are plotted versus the quantities p; and p i , respectively, for B ---f q5K*O. p i and p i enter the parametrizations (12) of the logarithmic divergences appearing in the longitudinal and negative helicity ( S P ) ( S - P ) annihilation amplitudes, respectively. As p 2 - increase from 0 to 1, the corresponding annihilation amplitudes increase by more than an order of magnitude. The theoretical uncertainties on the rates are very large. Furthermore, the largest input parameter uncertainties in BrL and BrT are a priori unrelated. Thus, it is clear from Figure 4 that the QCD penguin.annihilation amplitudes can account for the q5K*O measurements. Similarly, the measurements of ~ L ( $ K * *M) 50% can be accounted for. Do the QCD penguin annihilation amplitudes also imply large transverse polarizations in B + pK* decays? The answer depends on the pattern of s U ( 3 ) flavor ~ symmetry violation in these amplitudes. For light mesons containing a single strange quark, e.g., K * , non-asymptotic effects
+
+
122
10
10
5
5
0
0
0.2
0.6
0.4
0.8
1
0
0
0.2
0.6
0.4
0.8
1
PA
POA
Figure 4. BrL(dK*O) vs. p i (left), BrT(dK*O) vs. p a (right). Black lines: default inputs. Blue bands: input parameter variation uncertainties added in quadrature, keeping default annihilation and hard spectator interaction parameters. Yellow bands: additional uncertainties, added in quadrature, from variation of parameters entering logarithmically divergent annihilation and hard spectator interaction power corrections. Thick line: BrFax under simultaneous variation of all inputs.
shift the weighting of the meson distribution amplitudes towards larger strange quark momenta. As a result, the suppression of ss popping relative to light quark popping in annihilation amplitudes can be 0(1),which is consistent with the order of magnitude hierarchy between the B -+ Doro and B -+ D$K- rates 25. (See 26 for a discussion of other sources of S U ( 3 ) violation). In the present case, this implies that the longitudinal polarizations should satisfy f ~ ( p * K * ' ),< f ~ ( q 5 K *in) the Standard Model '. Consequently, f ~ ( p * K * ' )M 1 would suggest that U-spin violating New Physics entering mainly in the b + sss channel is responsible for the small f ~ ( q 5 K * ) . One possibility would be right-handed vector currents; they could interfere constructively (destructively) in A 1 (&) transversity amplitudes, see (3). Alternatively, a parity-symmetric scenario (CTp M a t the weak scale) would only significantly affect A*. A more exotic possibility would be tensor currents; they would contribute to the longitudinal and transverse amplitudes a t sub-leading and leading power, respectively, opposite to the vector currents. The current experimental average for f ~ ( K * ~ p is * )2 , . 5 ~ larger than f ~ ( q 5 K *but ) , it lies significantly below the naive factorization prediction. We should mention that our treatment of the charm (and up) quark loops in the penguin amplitudes follows the usual perturbative approach used in QCD factorization The authors of 23 have argued that the
eTp
13914,15.
123
region of phase space in which the charm quark pair has invariant mass q2 4mz, and is thus moving non-relativistically, should be separated out into a long-distance 'charming penguin' amplitude 27. NRQCD arguments are invoked to claim that such contributions are O ( v ) ,where u M .4-.5, so that they could effectively be of leading power. Furthermore, it is claimed that the transverse components may also be of leading power, thus potentially accounting for f ~ ( c $ K * )Also . see 28. However, a physical mechanism by which a collinear quark helicity-flip could arise in this case without power suppression remains t o be clarified. Arguments against a special treatment of this region of phase space l 3 ? l 4are based on parton-hadron duality. More recently, arguments supporting the power suppression of long-distance charming penguin effects have been presented in 29. N
3. A test for right-handed currents Does the naive factorization relation fi/fll= 14- O(hQcD/mb)(7) survive in QCD factorization? This ratio is very sensitive t o the quantity TI defined = .05 f .05 in (6). As TI increases, f i / f l l decreases. The range spanning existing model determinations is taken in '. In Figure 5 (left) the resulting predictions for fi/fll and BrT are studied simultaneaously for in the Standard Model. Note that the theoretical uncertainty B -+ for f i / f l l is much smaller than for f ~ Evidently, . the above relation still holds, particularly a t larger values of BrT where QCD penguin annihilation dominates both B r l and Brll. A ratio for f ~ / f l l in excess of the Standard Model range, e.g., f i / f l l > 1.5 if TI > 0, would signal the presence of new right-handed currents. This is due to the inverted hierarchy between A- and A+ for righthanded currents, and is reflected in the sign difference with which the Wilson coefficients (?i enter and All. For illustration, new contributions to the QCD penguin operators are considered in Figure 5 (right). At the New Physics matching scale M , these can be parametrized as
TI"'
(-)
(-1
(-)
(-)
CS = -3C5 = -3C3 =
(-)
. For simplicity, we take M
MW and consider two cases: K = -.007 or new left-handed currents (lower bands), and R = -.007 or new right-handed currents (upper bands), corresponding to c,",;;(mb) or (?Gc)(mb)M .18Cfg(mb),and c,",c)(mb)or (?,",$(mb) M .25 C,"t;',(mb). Clearly, moderately sized right-handed currents could increase f i / f l l well beyond the Standard Model range if TI 2 0. However, new left-handed currents would have little effect. The experimental average for f l / f j r in B -+ c$K*' lies close to unity, and thus gives no C4 =
K
M
124
indication for sizable right-handed currents.
1.5
1
3
4
5
6
l o 6 BrT
7
I
I
8
3
I
3.5
4
4.5
5
5.5
6
10 BrT
Figure 5. f l / f l l vs. BTT in the SM (left), and with new RH or LH currents (right). Black lines, blue bands, and yellow bands are as in Figure 4. Thick lines: (f~/fll)"'"" in the Standard Model for indicated ranges of rf' under simultaneous variation of all inputs. Plot for r y * > 0 corresponds to BrFaX in Figure 4.
4. Distinguishing four-quark and dipole operator effects
The O(a,) penguin contractions of the chromomagnetic dipole operator Qsg are illustrated in Figure 6. a4 and a6 are the QCD factorization coefficients of the transition operators ($)"-A 8 (.&)V-A and (Qb)s-p 8 (Dq)S+p1 respectively, where q is summed over u , d , s l49l5. Only the contribution on the left ( a q ) to the longitudinal helicity amplitude A' is non-vanishing In particular, the chromo- and electromagnetic dipole operators QS9 and Qyy do not contribute to the transverse penguin amplitudes a t O(a,) due to angular momentum conservation: the dipole tensor current couples to a transverse gluon, but a 'helicity-flip' for q or in Figure 6 would require a longitudinal gluon coupling. Formally] this result follows from Wandura-Wilczek type relations among the vector meson distribution amplitudes, and the large energy relations between the tensor-current and vector-current form factors. Transverse amplitudes in which a vector meson contains a collinear higher Fock state gluon also vanish at O(a,), as can be seen from the vanishing of the corresponding partonic dipole operator graphs in the same momentum configurations. Furthermore, the transverse O(a:) contributions involving spectator interactions are highly suppressed.
'.
125
Figure 6. Quark helicities for the O(cys)penguin contractions of &Q. The upward lines form the q4 meson in B -+ q4K* decays.
This has important implications for New Physics searches. For example, in pure penguin decays to CP-conjugate final states f , e.g., B .+ 4(K*' -+ KSn-O),if the transversity basis time-dependent C P asymmetry parameters (S,)l and ( S j ) , ,are consistent with ( s i n 2 P ) ~ , ~ K ,and , ( S ~ )isOnot, then this would signal new CP violating contributions to the chromomagnetic dipole operators. However, deviations in (Sf)lor ( S f )11 would signal new CP violating four-quark operator contributions. If the triple-products A$ and A,11 (4) do not vanish and vanish, respectively, in the pure-penguin decays B #K*'lf, K*Op*, then this would also signal new CP violating contributions to the chromomagnetic dipole operators. This assumes that a significant strong phase difference is measured between All and A l l for which there is some experimental indication However, non-vanishing A,,II or non-vanishing transverse direct CP asym--f
19118.
metries, e.g.,Acp(4K*'~*)ll,l,A C P ( K * ' ~ * ) I ( Iwould , ~ , signal the intervention of four-quark operators. The above would help to discriminate between different explanations for an anomalous S+K, or S q / ~ which ,, fall broadly into two categories: radiatively generated dipole operators, or tree-level induced four-quark operators. Finally, a large value for f i / f l l would be a signal for right-handed four-quark operators. 5. Conclusion
Polarization measurements in B decays to light vector meson pairs offer a unique opportunity to probe the chirality structure of rare hadronic B decays. A Standard Model analysis which includes all non-factorizable graphs in QCD factorization shows that the longitudinal polarization formally satisfies 1 - f~ = O ( l / m 2 ) ,as in naive factorization. However, the contributions of a particular QCD penguin annihilation graph which is formally O ( l / m 2 )can be O(1) numerically in longitudinal and negative helicity A S = 1 B decays. Consequently, the observation of f~(qW*'?-) M 50%
126 can be accounted for, albeit with large theoretical errors. The expected pattern of s U ( 3 ) ~ violation in the QCD penguin annihilation graphs, i.e., large suppression of sS relative t o uii or dd popping, implies that the longitudinal polarizations should satisfy f ~ ( p * K * ' ),< f ~ ( 4 K *in)the Standard Model. Consequently, f ~ ( p * K * ' )F=: 1 would suggest that U-spin violating New Physics entering mainly in the b SSS channel is responsible for the small values of f ~ ( 4 K * )There . is currently no experimental indication for such effects a t the B factories. The ratio of transverse rates in the transversity basis satisfies f ~ / f r = 1 O(l/m), in agreement with naive power counting. A ratio in excess of the predicted Standard Model range would signal the presence of new righthanded currents in dimension-6 four-quark operators. The maximum ratio attainable in the Standard Model is sensitive to the B -i V form factor combination r L , see (6), which controls helicity suppression in form factor transitions. All existing model determinations give a positive sign for r l , which would imply f i ( $ K * ) / f l l ( + K *7.This conspiracy of two in the ratio of the Ab and B-meson lifetimes. Since all three heavy mesons belong to the same S U ( 3 ) triplet, their lifetimes are the same at order l / m t . The computation of the ratios of heavy meson lifetimes is equivalent to the computation of U-spin or isospin-violating corrections. Both l/mz-suppressed spectator effects and our corrections computed in the previous sections arise from the spectator interactions and thus provide a source of U-spin or isospin-symmetry
-
-
-
136
400
200 0
0.75 0.8 0.85 0.9 0.950.75 0.8 0.85 0.9 0.95
0.75 0.8 0.85 0.0 0.95
T(Ab)/T(Bd)
Histograms sh_owing the random distributions around the central values of the and p; parameters contributing to T ( R b ) / T ( B d ) . Three histograms are f ~ m b~, B,, 6, shown for the scales p = m b / 2 (a), p = m b (b), and p = 2 m b ( c ) . Figure 2.
-1.04 1.06 1.08
1.1 1.12 1.14
1.04 1.08 1.08 1.1 1.12
1.04 1.08 1.08 1.1 1.12
T(Bu)/T(Bd)
Figure 3.
Same as Fig. 2 for T(B,)/T(B~).
breaking. We shall, however, assume that the matrix elements of both l / m i and l / m t operators respect isospin. The ratio of lifetimes of B, and Bd mesons involves a breaking of U-spin symmetry, so the matrix elements of dimension-6 operators could differ by about 30%. We shall introduce different B- and €-parameters to describe B, and Bd lifetimes. In order to obtain numerical estimate of the effect of l/mb corrections to spectator effects, we adopt the statistical approach for presenting our results and generate 20000-point probability distributions of the ratios of lifetimes obtained by randomly varying the parameters describing matrix elements within a &30% interval around their “factorization” values7, for three different scales p. The decay constants . f ~ and~b-quark pole mass mb are taken to vary within la interval indicated above. The results are presented in Figs. 2, 3, and 4. These figures represent our main result. We also performed studies of convergence of l/mb expansion by computing a set of l/mt-corrections to spectators effects and estimating their size in factorization7. The expansion appears to be well-convergent for the bflavored hadrons. Due to the relative smallness of m, (and thus vicinity
137
400
200 0 0.940.980.98
11.021.04 0.940.980.98
1 1.021.04 0.940.980.98
1 1.021.04
T(B*)/T(Bd)
Figure 4.
Same as Fig. 2 for T(B~)/T(B~).
of the sector of QCD populated by the light quark resonances'') it is not clear that the application of these findings to charmed hadrons will produce quantitative, rather than qualitative results. 4. Conclusions
We computed subleading l/mb and l / m t corrections to the spectator effects driving the difference in the lifetimes of heavy mesons and baryons. Thanks to the same 16r2 phase-space enhancement as l/mz-suppressed spectator effects, these corrections constitute the most important set of l/m:-suppressed corrections. The main result of this talk are Figs. 2, 3, and 4, which represent the effects of subleading spectator effects on the ratios of lifetimes of heavy mesons and baryons. We see that subleading corrections to spectator effects affect the ratio of heavy meson lifetimes only modestly, at the level of a fraction of a percent. On the other hand, the effect on the Ab-Bd lifetime ratio is quite substantial, at the level of -3%. This can be explained by the partial cancellation of WS and P I effects in hb baryon and constructive interference of l/mb corrections to the spectator effects. There is no theoretically-consistent way to translate the histograms of Figs. 2, 3, and 4 into numerical predictions for the lifetime ratios. As a useful estimate it is possible to fit the histograms to gaussian distributions and extract theoretical predictions for the mean values and deviations of the ratios of lifetimes. Predictions obtained this way should be treated with care, as it is not expected that the theoretical predictions are distributed according to the gaussian distribution. This being said, we proceed by fitting the distributions to gaussians and, correcting for the small scale uncertainty, extract the ratios T(B~)/T(B~) = 1.09 & 0.03, 7(Bs)/7(Bd) = 1.00 f 0.01, and T(Ab)/T(Bd) = 0.87 f 0.05. This brings the experimental
138
and theoretical ratios of baryon and meson lifetimes into agreement. I would like to thank F. Gabbiani and A. I. Onishchenko for collaboration on this project, and N. Uraltsev and M. Voloshin for helpful discussions. It is my pleasure to thank the organizers for the invitation to this wonderfully organized workshop. References I. I. Bigi et al., arXiv:hep-ph/9401298; M. B. Voloshin, arXiv:hep-ph/0004257. M. Neubert and C. T. Sachrajda, Nucl. Phys. B 483,339 (1997). J. L. Rosner, Phys. Lett. B 379,267 (1996). S. Eidelman et al., Phys. Lett. B592, 1 (2004); E. Barberio, presented at the Workshop on the CKM Unitary Triangle, http://ckm-workshop.web.cern.ch; M. Battaglia et al., arXiv:hep-ph/0304132. 5. J. Rademacker [On behalf of the CDF Collaboration], arXiv:hep-ex/0406021. See also: LEP B Lifetime Working Group, http://lepbosc.web.cern.ch/LEPBOSC/lifetimes/lepblife.html. 6. M. Ciuchini, E. Franco, V. Lubicz, and F. Mescia, Nucl. Phys. B 625, 211 (2002); E. Franco, V. Lubicz, F. Mescia, and C. Tarantino, ibid. 633, 212 (2002); M. Beneke, G. Buchalla, C. Greub, A. Lenz, and U. Nierste, Nucl. Phys. B 639,389 (2002). 7. F. Gabbiani, A. I. Onishchenko and A. A. Petrov, arXiv:hep-ph/0407004; F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, Phys. Rev. D 68,114006 (2003). 8. M. A. Shifman and M. B. Voloshin, Sov. J. Nucl. Phys. 41,120 (1985) [Yad. Fiz. 41, 187 (1985)]; I. Bigi, M. Shifman, N. Uraltsev, and A. Vainshtein, Phys. Rev. Lett. 71,496 (1993); B. Guberina, S. Nussinov, R. D. Peccei, and R. Ruckl, Phys. Lett. B 89,111 (1979); N. Bilic, B. Guberina, and J. Trampetic, Nucl. Phys. B 248, 261 (1984); B. Guberina, R. Ruckl, and J. Trampetic, Z. Phys. C 33, 297 (1986); B. Guberina, B. Melic, and H. Stefancic, Phys. Lett. B 484,43 (2000). 9. M. Beneke, G. Buchalla, and I. Dunietz, Phys. Rev. D 54, 4419 (1996). M. Beneke, G. Buchalla, A. Lenz, and U. Nierste, Phys. Lett. B 576, 173 (2003). 10. J. Chay, A. F. Falk, M. E. Luke, and A. A. Petrov, Phys. Rev. D 61,034020 (2000); see also4 and M. Di Pierro and C. T. Sachrajda [UKQCD Collaboration], Nucl. Phys. B 534, 373 (1998); M. Di Pierro, C. T. Sachrajda, and C. Michael [UKQCD collaboration], Phys. Lett. B 468, 143 (1999); P. Colangel0 and F. De Fazio, ibid. 387,371 (1996); M. S. Baek, J. Lee, C. Liu, and H. S. Song, Phys. Rev. D 57,4091 (1998); J. G. Korner, A. I. Onishchenko, A. A. Petrov, and A. A. Pivovarov, Phys. Rev. Lett. 91,192002 (2003). 11. A. F. Falk et. al, Phys. Rev. D 69, 114021 (2004); A. F. Falk et. al, Phys. Rev. D 65,054034 (2002); E. Golowich and A. A. Petrov, Phys. Lett. B 427, 172 (1998); A. A. Petrov, Phys. Rev. D 56,1685 (1997).
1. 2. 3. 4.
INCLUSIVE B-DECAY SPECTRA AND IR RENORMALONS
E. GARDI Cavendish Laboratory, University of Cambridge Madingley Road, Cambridge, CB3 OHE, UK I illustrate the role of infrared renormalons in computing inclusive B-decay spectra. I explain the relation between the leading ambiguity in the definition of Sudakov form factor exp(NAlA4) and that of the pole mass, and show how these ambi-
-
guities cancel out between the perturbative and non-perturbative components of the b-quark distribution in the meson.
1. Introduction
-
B-decay physics is gradually turning into a field of precision phenomenology. Inclusive decay measurements provide some of the most robust tests of the standard model. Classical examples are the rate of B X,y decays [l] and constraints on the unitarity triangle through the measurement of Vub from charmless semileptonic decays [2]. The advantage of inclusive measurements over exclusive ones is that the corresponding theoretical predictions are, to large extent, free of hadronic uncertainties. QCD corrections to total decay rates are dominated by short distance scales, of order of the heavy-quark mass m, and are therefore primarily perturbative. Confinement effects appear as power corrections in A / m . Moreover, the Operator Product Expansion (OPE) allows one to estimate these power corrections by relating them to specific matrix elements of local operators between B-meson states, which are defined in the infinite-mass limit in the framework of the heavy-quark effective theory (HQET) [3]. These matrix elements can either be computed on the lattice or extracted from experimental data. In reality, however, experiments cannot perform completely inclusive measurements. Precise measurements are restricted t o certain kinematic regions where the background is sufficiently low. The experimentally accessible region in B X,y is where the photon energy E7 in the B rest frame is close t o its maximal possible value, M / 2 , (A4is the B meson mass),
-
139
140
-
or, equivalently, x = 2E,/M is near 1,which is the endpoint. Similarly, the accessible region in the CKM-suppressed B X,lV decay is where the lepton energy fraction is near maximal, or where the invariant mass of the hadronic system is small. Out of this region this decay mode is completely overshadowed by the decay into charm. As a consequence, precision phenomenology must rely on detailed theoretical understanding of the spectrum [4].Of particular importance is the spectrum near the endpoint. It turn out, however, that the endpoint region is theoretically much harder to access as both the perturbative expansion and the OPE break down there. In the large-z region gluon emission is restricted to be soft or collinear to the light-quark jet. While the associated singularities cancel with virtual corrections (decay spectra being infrared and collinear safe) large Sudakov logarithms of (1- x) appear in the expansion, which must therefore be resummed. Moreover, the OPE breaks down since the hierarchy between operators scaling with different powers of the mass is lost when (1 - z ) M become as small as the QCD scale. Physically this reflects the fact that the spectrum in the endpoint region is driven by the dynamics of the light degrees-of-freedom in the meson. The lightcone-momentum distributiona of the b-quark in the B-meson has a particularly important role in the endpoint region [5-131. It has been shown that up to subleading corrections O(A/m) the physical spectrum can be obtained as the convolution between a perturbatively-calculable coefficient function and the QDF, where the latter essentially determines the shape for x 1. The key point is that the QDF is a property of the B meson, not of the particular decay mode considered, so it can be measured in one decay and used in another. Moreover, a systematic analysis of the QDF in the HQET highlights the significance of a few specific parameters which constitute the first few moments of this function: most importantly = M - m, the difference between the meson mass and the quark pole mass, and then XI corresponding to the kinetic energy of the b quark in the meson. Nevertheless, the dependence of theoretical predictions for the spectra on the QDF is still a major source of uncertainty. Apart from identifying its first few moments, very little is known about this function, so the phenomenology of decay spectra in the immediate vicinity of the endpoint
-
*We shall define this function in full QCD, and call it Quark Distribution Function (QDF). This should be distinguished from the common practice to define it directly in the HQET, where the name “Shape Function” is often used.
-
141
(z 1) remains, to large extent, model dependent. On the other hand, successful precision phenomenology can well be expected for more moderate (yet large) z values, corresponding to the region where the distribution peaks. Here the main obstacle has been in combining [10,14] perturbative Sudakov effects with the HQET-based non-perturbative treatment discussed above. It has recently been shown [15] that the resolution of this problem is firmly connected with infrared renormalons (for general review of renormalons see [16]). Since the formulation of the HQET as well as the perturbative calculation of decay spectra rely on the concept of an on-shell heavy quark, both ingredients suffer from renormalon ambiguities. These ambiguities cancel out, of course, in the physical spectra. It is therefore useful to traceb the precise cancellation of ambiguities: the use of the HQET brings about dependence on the quark pole mass, which has a linear renormalon ambiguity [17-201. This ambiguity cancels against the leading renormalon ambiguity in the Sudakov exponent [15]. In order to achieve power-like separation between perturbative and non-perturbative contributions to decay spectra, one must therefore compute the Sudakov exponent as an asymptotic expansion, thus replacing the standard Sudakov resummation with fixed logarithmic accuracy by Dressed Gluon Exponentiation (DGE) [15,21-261. In what follows we illustrate the role of renormalons in the QCD description of decay spectra. We begin by briefly reviewing the HQET analysis for the QDF where we identify dependence on the quark pole mass. We recall that the pole mass suffers from an infrared renormalon ambiguity and show how this affects the QDF [15].We then consider inclusive B-meson decays within perturbation theory, review the relevant results on large-z factorization and Sudakov resummation [13], and then show that renormalon ambiguities appear in the Sudakov exponent [15], which, we emphasize, is a general phenomenon rather than a peculiarity of B decays. Finally, we combine the perturbative and non-perturbative ingredients recovering an unambiguous answer for the QDF in the meson and consequently for decay spectra. We conclude by shortly discussing the implications for precision phenomenology in inclusive decays.
bThis can be understood in analogy with factorization scale dependence, the main difference being that here the interest is in power terms.
142
2. Heavy-quark effective theory and the QDF
We define the QDF f ( z ;p ) as the Fourier transform of the forward hadronic matrix element of two heavy-quarks fields on the lightcone (y2 = 0):
where a path-ordered exponential between the fields is understood, p~ is the B-meson momentum ( p i = M 2 ) , z is the fraction of the momentum component carried by the b-quark field and p is the renormalization scale of the operator. Decay spectra can be computed as a convolution between a perturbatively calculable coefficient function and f ( z ; p ) . Let us first analyze f ( z ;p ) non-perturbatively - we denote it fNp(z)- suppressing any perturbative corrections. These will be recovered later on. Since the b-quark mass is large, the heavy quark is not far from its mass shell. This observation is the basis of the HQET. The momentum of the heavy quark is p = mu Ic where u is the hadron four velocity, v = p~ J M , and Ic is a residual momentum, llcl ~ > detailed ~ discussion of the rationale of LCSRS and of the more technical aspects of the method can be found e.g. in Ref.8. The formfactors in question can be defined as ( q = p~ - p )
The starting point for the calculation of e.g. f; is the correlation function
i
/
d4YeiqY(~(p)lT[~,Y~b](y)[mbbzysq](0))0) =
+ ...,
(4)
where the dots stand for other Lorentz structures. For a certain configuration of virtualities, namely mg - p i 2 O ( A Q c D m b ) and mi - q2 L O ( h Q c D m b ) , the integral is dominated by light-like distances and can be expanded around the light-cone:
As in Eq. (l),n labels the twist of operators and p~ denotes the factorisation scale. The restriction on q 2 , mz - q2 2 O ( L i Q c D m b ) , implies that is not accessible at all momentum-transfers; to be specific, we restrict ourselves to 0 5 q2 5 14GeV2. As 11+ is independent of p ~ the , above formula implies that the scale-dependence of T(n)must be canceled by that of the DAs # ( n ) . In Eq. (5) it is assumed that 11+ can be described by collinear factorisation, i.e. that the only relevant degrees of freedom are the longitudinal momentum fractions u carried by the partons in the T , and that transverse momenta can be integrated over. Hard infrared (collinear) divergences occurring in T(")should be absorbable into the DAs. Collinear factorisation is trivial at tree-level, where the b quark mass acts effectively as regulator, but
f?
163 can, in principle, be violated by radiative corrections, by so-called "soft" divergent terms, which yield divergences upon integration over u.Actually, however, it turns out that for all formfactors calculated in Ref.' the T are nonsingular a t the endpoints u = 0,1, so there are n o soft divergences, independent of the end-point behavior of the distribution amplitudes. In Ref.' Eq. ( 5 ) has been demonstrated to be valid t o O(a,) accuracy for twist-2 and twist-3 contributions for all correlation functions II+,o,Tfrom which to determine the formfactors f+,O,T. As for the distribution amplitudes (DAs), they have been discussed intensively in the literatureg. For pseudoscalar mesons, there is only one DA of leading-twist, i.e. twist-2, which is defined by the following light-cone matrix element ( x 2 = 0):
-XI
where C = 2u - 1 and we have suppressed the Wilson-line [ x , to ensure gauge-invariance. The higher-twist DAs are of type
needed
where u is a number between 0 and 1 and r a combination of Dirac matrices. The sum rule calculations performed in Refs.ll2y3include all contributions from DAs up t o twist-4. The DAs are parametrized by their partial wave expansion in conformal spin, which t o NLO provides a controlled and economic expansion in terms of only a few hadronic parametersg. The LCSR for f; is derived in the following way: the correlation function II+,calculated for unphysical p i , can be written as dispersion relation over its physical cut. Singling out the contribution of the B meson, one has
H+ = f ; ( q 2 )
m;fB 2
mg
-Pi
+ higher poles and cuts,
(7)
where f B is the leptonic decay constant of the B meson, fBm$ = mb(Bl6iysdlO). In the framework of LCSRs one does not use (7) as it stands, but performs a Borel transformation, l / ( t - p i ) + 2l / ( t - p $ ) = 1/M2 exp(-t/M2), with the Borel parameter M 2 ; this transformation enhances the ground-state B meson contribution to the dispersion representation of II+ and suppresses contributions of higher twist to the light-cone expansion of II+. The next step is to invoke quark-hadron duality to approximate the contributions of hadrons other than the ground-state B me-
164
son by the imaginary part of the light-cone expansion of II+, so that
Subtracting the 2nd term on the right-hand side from both sides, one obtains
Eq. (9) is the LCSR for f;. SO is the so-called continuum threshold, which separates the ground-state from the continuum contribution. At tree-level, the continuum-subtraction in (9) introduces a lower limit of integration in u,the momentum fraction of the quark in the 7r: u 2 (m: - q 2 ) / ( s o - q 2 ) , in (5), which behaves as 1 - h q c D / m b for large m b and thus corresponds t o the dynamical configuration of the Feynman-mechanism, as it cuts off low momenta of the u quark created a t the weak vertex. At O ( a s ) ,there are also contributions with no cut in the integration over u,which correspond to hard-gluon exchange contributions. The task now is to find sets of parameters M 2 (the Bore1 parameter) and SO (the continuum threshold) such that the resulting formfactor does not depend too much on the precise values of these parameters. 2. Results For a detailed discussion of the procedure used to determine the hadronic and sum rule specific input parameters we refer to Ref.l. One main feature is that f ~ the , decay constant of the B meson entering Eq. (9) is calculated from a sum rule itselflo, which reduces the dependence of the resulting formfactors on the input parameters, in particular mb, which is the one-loop pole mass and taken to be (4.80 f 0.05) GeV. This procedure does not, however, reduce the formfactors’ dependence on the parameters describing the twist-2 DAs, which turns out to be rather crucial. Despite much effort spent on both their calculation from first principles and their extraction from experimental data, these so-called Gegenbauer moments, a1 (only for K ) , a2 and a4 (for all P ) are not known very precisely. Figure 1 shows the dependence of f+(O) on a2 and a4; the dots represent different determinations of these parameters and illustrate the resulting spread in values of the formfactor. The situation is even more disadvantageous for
165
Figure 1. Dependence of fT(0) on a2 and a 4 , for central values of input parameters. The lines are lines of constant fT(0). The dot labeled BZ denotes our preferred values of a2,4, BMS the values from the nonlocal condensate modelll and BF from sum rule caIculationsQ.
Figure 2. (a) Dependence of f f ( 0 ) on the Gegenbauer moment a l . (b) f 2 ( q 2 ) as function of q2 for different values of a l : solid line: a? = 0.17, short dashes: uf = 0, long dashes: a? = -0.18.
the K , whose formfactors depend on the SU(3) breaking parameter u f , whose size and even sign are under discussion12: at present, values as different as -0.18 and +0.17 (at p = 1GeV) are being quoted. Figure 2 shows the dependencies of (a) f$(O) and (b) f F ( q 2 ) on this parameter; evidently it is very important to determine its value more precisely. Summarizing the detailed analysis of the uncertainties induced by both external input and LCSR parameters, the final results for the formfactors at zero momentum transfer obtained in Ref.l are:
fT(0) = 0.258 f0.031, f+K(O) = 0.331 f 0.041 0.256a1, fT(0) = 0.275 f0.036,
+
f;(O)
= 0.253 f 0.028,
f$(O)
= 0.358 f 0.037
f;(O)
= 0.285 f 0.029.
+ 0.31&, ,
daI is defined as u f ( 1 GeV) - 0.17, i.e. the deviation of uf from the central value used in Ref.l. For f T > q the total theoretical uncertainty ranges between 10% to 13%, for f K it is 12%, plus the uncertainty in u l , which hopefully will be clarified through an independent calculation in the not
166
Figure 3. f+ (solid lines), fo (short dashes) and f~ (long dashes) as functions of q2 for n, K and q. The renormalisation scale of f T is chosen t o be m b .
too far future. The intrinsic, irreducible uncertainty of the sum rule calculation is related to the dependence of the result on the sum rule specific parameters M 2 and SO and estimated to be 7%. Turning t o the q2-dependence of formfactors, it has to be recalled that LCSRS are only valid if the energy Ep of the final state meson, measured in the rest frame of the decaying B , is large, i.e. if q2 = m i - 2 r n ~ E pis not too large; specifically, we choose Ep > 1.3 GeV, i.e. q2 5 14 GeV2. The resulting formfactors are plotted in Fig. 3, using central values for the input parameters. In order to allow a simple implementation of these results in actual applications, and also in order to provide predictions for the full physical regime 0 5 q2 5 (mB - m p ) 2 M 25GeV2, it is necessary to find parametrizations of f ( q 2 ) that N
0
reproduce the data below 14 GeV2 with good accuracy; provide a n extrapolation to q2 > 14GeV2 that is consistent with the expected analytical properties of the formfactors and reproduces the lowest-lying resonance (pole) with J p = 1- for f+ and fT .a
aFor fo, the lowest pole with quantum numbers O+ lies above the two-particle threshold
167 Table 1. Fit parameters for f ( q 2 ) . ml is the vector m e o n mass in the corresponding channel: m:9q = mgr = 5.32 GeV and mf = mB; = 5.41 GeV. The scale of fT is p = 4.8 GeV.
f;
I Tl 1 0.744 0 1.387 0.162 0 0.161 0.122 0 0.111
rz -0.486 0.258 -1.134 0.173 0.330 0.198 0.155 0.273 0.175
(m~)'
(my)' (m?)'
(rn7)'
m?, 40.73 33.81 32.22
37.46 (mf)2 (rn?)' 31.03 (m:)2
As shown in Ref.', the following parametrizations are appropriate: 0
for f-;,T:
0
where my is the mass of B*(l-), my = 5.32 GeV; the fit parameters are r-1, 1-2 and mfit; for f+,;.: Kv
0
where ml is the mass of the 1- meson in the corresponding channel, i.e. 5.32 GeV for 77 and 5.41 GeV for K; the fit parameters are r1 and 7-2; for fo:
the fit parameters are 7-2 and mfit. The central results for the fit parameters are collected in Tab. 1. The quality of all fits is very good and the maximum deviation between LCSR and fitted result is 2% or better. The impact of the extrapolation of the fit formulas to q2 > 14GeV2 is of phenomenological relevance mainly for B --+ r e v , relevant for the determination of lVubl from experiment. We have starting at (mB
+ mp)'
and hence is not expected to feature prominently.
168
estimated the effect of the extrapolation on the decay rate by implementing different parametrisations for f;, which all fit the LCSR result very well for q2 < 14 GeV2, but differ for larger q 2 , the main distinguishing feature being the positions of the poles. We find that for reasonable parametrisations, that is such that do not exhibit too strong a singularity a t q2 = mp, the total rates differ by not more than 5%, the difference becoming smaller if an cut-off on the maximum invariant mass of the lepton pair is implemented, which implies that the extrapolation is well under control.
3. Summary & Conclusions LCSRs provide accurate results for weak decay formfactors of the B meson into light mesons, in particular 7r, K and q. The results depend on sum rule specific input parameters which generate an irreducible “systematic” uncertainty of the approach estimated to be 7%. Additional uncertainties are induced by imprecisely known hadronic input parameters, in particular the Gegenbauer moments a1,2,4 describing the leading-twist light-meson distribution amplitudes. An improved determination of these parameters would be very welcome. The present total uncertainty of the formfactors a t zero momentum transfer varies between 10 and 13%, but becomes smaller a t larger q 2 . LCSR calculations require the energy of the final state meson to be large in the rest-frame of the decaying B and hence are valid only for not too large momentum transfer q2; the maximum eligible q2 is chosen to be 14 GeV2. The q2-dependence of the formfactors can be cast into simple parametrizations in terms of two or three parameters, which also capture the main features of the analytical structure and are expected to be valid in the full kinematical regime 0 5 q2 5 ( m-~ m ~ )The ~ total . uncertainty introduced by the extrapolation of the formfactors to q2 larger than the sum rule cut-off 14 GeV2 is estimated to be 5% for the semileptonic rate N
N
r ( B t 7rev). Ref.l also contains a detailed breakdown of the dependence of the formfactors on the Gegenbauer moments, which not only allows one to recalculate the formfactors once these parameters are determined more precisely, but also makes it possible t o consistently assess their impact on nonleptonic decay amplitudes ( e g . B + m )treated in QCD factorisation. The LCSR approach is complementary to standard lattice calculations, in the sense that it works best for large energies of the final state meson (i.e. small q 2 ) , whereas lattice calculations work best for small energies - a situation that may change in the future with the implementation of moving
169
NRQCD13. Previously, the LCSR results for f-;,o at small and moderate q2 were found to nicely match14 t h e lattice results obtained for large q2. T h e situation will have to be reassessed in view of our new results and i t will be very interesting to see if a n d how i t will develop with further progress in both lattice a n d LCSR calculations. References 1. 2. 3. 4.
5.
6. 7.
8. 9. 10.
11. 12.
13. 14.
P. Ball and R. Zwicky, arXiv:hep-ph/0406232. P. Ball, JHEP 9809 (1998) 005 [hep-ph/9802394]. P. Ball and R. Zwicky, JHEP 0110 (2001) 019 [arXiv:hep-ph/0110115]. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385; ibd. 147 (1979) 448. V.L. Chernyak and A.R. Zhitnitsky, J E T P Lett. 25 (1977) 510 [Pisma Zh. Eksp. Teor. Fiz. 25 (1977) 5441; Sov. J. Nucl. Phys. 31 (1980) 544 [Yad. Fiz. 31 (1980) 10531; A.V. Efremov and A.V. Radyushkin, Phys. Lett. B 94 (1980) 245; Theor. Math. Phys. 42 (1980) 97 [Teor. Mat. Fiz. 42 (1980) 1471; G.P. Lepage and S.J. Brodsky, Phys. Lett. B 87 (1979) 359; Phys. Rev. D 22 (1980) 2157; V.L. Chernyak, A.R. Zhitnitsky and V.G. Serbo, J E T P Lett. 26 (1977) 594 [Pisma Zh. Eksp. Teor. Fiz. 26 (1977) 7601; Sov. J. Nucl. Phys. 31 (1980) 552 [Yad. Fiz. 31 (1980) 10691. V.L. Chernyak and I.R. Zhitnitsky, Nucl. Phys. B 345 (1990) 137. P. Ball and V.M. Braun, Phys. Rev. D 55 (1997) 5561 [arXiv:hepph/9701238]; A. Khodjamirian et al., Phys. Lett. B 402 (1997) 167 [arXiv:hep-ph/9702318]; E. Bagan, P. Ball and V.M. Braun, Phys. Lett. B 417 (1998) 154 [hep-ph/9709243]; A. Khodjamirian, R. Ruck1 and C.W. Winhart, Phys. Rev. D 58 (1998) 054013 [arXiv:hep-ph/9802412]; P. Ball and V. M. Braun, Phys. Rev. D 58 (1998) 094016 [arXiv:hep-ph/9805422]; A. Khodjamirian et al., Phys. Rev. D 62 (2000) 114002 [arXiv:hepph/0001297]; P. Ball and E. Kou, JHEP 0304 (2003) 029 [arXiv:hepph/0301135]; P. Ball, arXiv:hep-ph/0308249. P. Colangelo and A. Khodjamirian, hep-ph/0010175; A. Khodjamirian, hepph/0108205. V. M. Braun and I. E. Filyanov, Z. Phys. C 44, 157 (1989) [Sov. J. Nucl. Phys. 501; P. Ball, JHEP 9901 (1999) 010 [hep-ph/9812375]. For instance, T. M. Aliev and V. L. Eletsky, Sov. J. Nucl. Phys. 38 (1983) 936 [Yad. Fiz. 38 (1983) 15371; E. Bagan et al., Phys. Lett. B 278 (1992) 457. A. P. Bakulev et al., arXiv:hep-ph/0405062. P. Ball and M. Boglione, Phys. Rev. D 68 (2003) 094006 [arXiv:hepph/0307337]. C.H. Davies, talk at UK BaBar meeting, Durham April 2004. D. Becirevic, arXiv:hep-ph/0211340.
UNDERSTANDING 0; (2317), O,j(2460)
F. DE FAZIO Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Via Orabona, 4 I-70186 Bari, ITALY E-mail: fuluia. defazioQba.infn.it
I briefly review the experimental observations concerning the charmed mesons D:J(2317), 0,~(2460) and survey on some of the interpretations proposed in order to understand their nature. I present an analysis of their decay modes in the hypothesis that they can be identified with the scalar and axial vector sf = states of cS spectrum (D:o, Oil). The method is based on heavy quark symmetries and Vector Meson Dominance ansatz. Comparison with present data supports the interpretation.
4'
1. Introduction
In April 2003, the BaBar Collaboration reported the observation of a narrow peak in the D$r0 invariant mass distribution, corresponding to a state of mass 2317 MeV, denoted as DtJ(2317) The state is produced from charm continuum and the observed width is consistent with the resolution of the detector, I? 5 10 MeV. A possible quantum number assignment to D,*,(2317) is J p = Of, as suggested by the angular distribution of the meson decay with respect to its direction in the e+ - e- center of mass frame. This assignment can identify the meson with the scalar D:o state in the spectrum of the cS system. Considering the masses of the other observed states belonging to the same system, D,1(2536) and D,~ (2 5 7 3 ),the mass of the scalar DZ0 meson was expected in the range 2.45 - 2.5 GeV, therefore 150 MeV higher than the observed 2.317 GeV. A D;, meson with such a large mass would be above the threshold M D K = 2.359 GeV to strongly decay by S-wave K m n emission to D K , with a consequent broad width. For a mass below the D K threshold the meson has to decay by different modes, namely the isospinviolating D,7ro mode observed by BaBar, or radiatively. The J p = O+ assignment excludes the final state D,y, due to angular momentum and
'.
-
170
171
parity conservation; indeed such a final state has not been observed. On the other hand, for a scalar cs meson the decay DQ0 4 D,*y is allowed. However, no evidence is reported yet of the D,yy final state resulting from the decay chain D:o t DGy + D,yy. Later on, in May 2003, CLEO Collaboration confirmed the BaBar observation of D,*,(2317) with the same features outlined above; furthermore, CLEO Collaboration observed a second narrow peak, corresponding to a state with mass 2460 MeV decaying t o DGxo '. Again, the width is compatible with the detector resolution. Evidence of this second state was present in the first analysis by BaBar Collaboration, which gave subsequent confirmation of the CLEO observation3. BELLE Collaboration has confirmed both states4, observing their production both from charm continuum, both in B decays; more recently also FOCUS Collaboration5 has detected of a narrow peak at 2323 f 2 MeV, slightly above the values obtained by the other three experiments for Di,(2317). The observation of the decay D,~ (2 4 6 0 ) -+ D,*xo suggests that D,~(2460)has J p = I f . This assignment is supported also by the observation of the mode D,~ (2 4 6 0 )-+ D,y, forbidden to a O+ state, and by the angular analysis performed by BELLE6. Such an analysis was carried out for D,~(2460)produced in B decays and favours the identification of D,~ (2 460)with an axial-vector particle. Production of D0,5(2460)in B decays was observed also by BaBar'. However, as in the case of D,*,(2317), the measured mass is below theoretical expectations for the 1+ cg state DLl and the narrow width contrasts with the expected broadness of the latter. These peculiar features of D,*,(2317) and D,~ (2 4 6 0 )have prompted a number of analyses, aimed either at refining previous results in order to , at explaining support the cg interpretation of D,*,(2317) and D,~ ( 2 4 6 0 )or their nature in a different context. The various interpretations are reviewed in Ref. 8. Among the non standard scenarios, it has been often considered the possibility of a sizeable four-quark component in D,*,(2317) and D,J( 2460). Four-quark states could be baryonum-like or molecular-like, if they result from bound states of quarks or of hadrons, respectively, and examples of the ) insecond kind of states are the often discussed fo(980) and ~ ( 9 8 0 when terpreted as K r molecules. In the molecular interpretation, the D,*,(2317) could be viewed as a D K moleculeg, an interpretation supported by the fact that the mass 2.317 GeV is close to the DK threshold, or as a D,x atomlo. Analogously, D , ~( 246 0 )would be a D * K molecule. Mixing between ordinary cS state and a composite state has also been considered". No definite
172
answer comes from lattice QCD, since, according t o Ref. 12, lattice predictions are inconsistent with the simple qtj interpretation for DiJ(2317), while in Ref. 13 no exotic scenario is invoked to interpret this state. QCD sum rules are compatible with the cB interpretati~n'~. To understand the structure of a particle one needs to analyse its decay modes under definite assumptions and compare the result with the experimental measurements. In the following we present an analysis based on such a strategy t o discuss whether the identification of D,',(2317) and D , ~ ( 2 4 6 0 )with the two states (D,*,,DL,) is supported by data. To this end, we compute the decay modes of a scalar and an axial-vector particle with masses of 2317 MeV and 2460 MeV respectively, and check whether they can be predicted in agreement with the experimental findings presently available. In particular, the isospin violating decays t o D$*).rroshould proceed a t a rate larger than the radiative modes, though not exceeding the experimental upper bounds on the total widths. 2. Hadronic Modes In order to analyze the isospin violating transitions DHo -+ D,ro and DLl t D,'ro, one can use a formalism that accounts for the heavy quark spin-flavour symmetries in hadrons containing a single heavy quark, and the chiral symmetry in the interaction with the octet of light pseudoscalar states. In the heavy quark limit, the heavy quark spin ZQ and the light degrees of freedom total angular momentum Z l are separately conserved. This allows t o classify hadrons with a single heavy quark Q in terms of se by collecting them in doublets the members of which only differ for the relative orientation of ZQ and Ze. The doublets with J p = (0-, 1-) and J p = (0+, 1+) (corresponding to = 1- and = ;+, respectively) can be described by the effective fields
SF
SF
where v is the four-velocity of the meson and a is a light quark flavour index. In particular in the charm sector the components of the field Ha are P,'*' = D(*)O,D(*)+and D?' (for a = 1 , 2 , 3 ) ;analogously, the components of Sa are P,*, = D:', D;', D,*, and Pi, = Die, D';t, DLl. In terms of these fields it is possible to build up an effective Lagrange
173
density describing the low energy interactions of heavy mesons with the pseudo Goldstone 7r, K and q b o ~ ~ n ~ ~ ~ * ~ ~ ~
-
+
L = i Tr{HbZfPDpbaHa} cTr{6’PCapct} f Tr{Sb (i
V’Dpba
-
8
dba A)sa}
+ + [i h T ~ ( S b 7 p ~ 5 A i ~ +Z ah.c.1 )
9 Tr{HbypY5Atapa} i 9’ Tr{SbYpY5Atasa}
sa
(3)
In (3) Ba and are defined as p a = yoHAyo and 3, = yoSfLyo;all the heavy field operators contain a factor and have dimension 312. The parameter A represents the mass splitting between positive and negative parity states. The T , K and q pseudo Goldstone bosons are included in the effective lagrangian (3) through the field t = e y that represents a unitary matrix describing the pseudoscalar octet, with
The strong interactions between the heavy Ha and Sa mesons with the light pseudoscalar mesons are thus governed, in the heavy quark limit, by three dimensionless couplings: g, h and 9’. In particular, h describes the coupling between a member of the Ha doublet and one of the Sa doublet to a light pseudoscalar meson, and is the one relevant for our discussion. Isospin violation enters in the low energy Lagrangian of 7r, K and q mesons through the mass term
with m, the light quark mass matrix:
mu 0 0 m q = ( 0 m0d m, 0 )
~
174
In addition to the light meson mass terms, L,,, contains an interaction term between no ( I = 1) and r] ( I = 0) mesons: LmiIing = $ m d ~ m ~ n O r ] which vanishes in the limit mu = m d . Let us focus on the mode D,*o -+ D,no. As in the case of D,* ---f D,n' studied in Ref. 19, the isospin mixing term can drive such a transition". The amplitude A(DZ0 -+ D,no) is simply written in terms of A(D,*,4 D,r]) obtained from (3), A(v + no)from (6) and the r] propagator that puts the strange quark mass in the game. The resulting expression for the decay amplitude involves the coupling h and the suppression factor (md - mu)/(m,- ""$"Y) accounting for isospin violation, so that the width I'(Dz0 + D,no) reads:
As for h, the result of QCD sum rule analyses of various heavy-light quark current correlators is Ih( = 0.6 f 0.2 17. Using the central value, together with the factor (md -mu)/(ms- m d $ m = ) fi 1 43.7 2o and f = ffl= 132 MeV we obtain21:
I'(Dzo 3 D,no) = 7 f 1 K e V .
(8)
The analogous calculation for Oil -+ D,*noprovides the result':
rp;, + D,*T") = 7 f 1 K e V .
(9)
3. Radiative Modes Let us now turn to the calculation of radiative decay rates. We describe the procedure considering the mode DQ0 + D,*y, the amplitude of which has the form:
A(DBo -D,*y) = e d [ ( ~ * . r ] * ) ( p . I c ) - ( r ] * . p ) ( ~ * . k ) ] , (10) where p is the Dz0 momentum, E the D,* polarization vector, and Ic and r] the photon momentum and polarization. The corresponding decay rate is:
I'(D,*, -+ D z y ) = ald121L13 .
(11)
The parameter d gets contributions from the photon couplings to the light quark part e,%y,s and to the heavy quark part e,Cy,c of the electromagnetic aElectromagnetic contributions to Dfo -+ D s x o are expected to be suppressed with respect t o the strong interaction mechanism considered here.
175
current, e, and e, being strange and charm quark charges in units of e. Its general structure is:
where A, (a = c, s) have dimension of a mass. Such a structure is already known from the constituent quark model. In the c a e of M1 heavy meson transitions, an analogous structure predicts a relative suppression of the radiative rate of the charged D* mesons with respect t o the neutral onez2123~z4y25, suppression that has been experimentally confirmedz6. From (11,12) one could expect a small width for the transition D& --+ DZr,t o be compared t o the hadronic width D,*o-+ D,r0 which is suppressed as well. In order to determine the amplitude of DE0 -+ DQ*y we follow a method based again on the use of. heavy quark symmetries, together with the vector meson dominance (VMD) a n ~ a t z We ~ ~first ~ ~consider ~ . the coupling of the photon t o the heavy quark part of the e.m. current. The matrix element (DH(v‘, ~)l~?~~clD (v,~v’~meson ( v ) ) four-velocities) can be computed in the heavy quark limit, matching the QCD Ey,c current onto the corresponding HQET e x p r e ~ s i o n ~ ~ :
JfLHQE= T&,[v,
i + -( i +a, - ca,) + B , , , ( ~+ ~ %,,) + . . . ] h ,
(13) 2mQ 2mQ where h, is the effective field of the heavy quark. For transitions involving DHo and DQ*, and for v = v’ (v . v’ = l), the matrix element of vanishes. The consequence is that dh)is proportional to the inverse heavy quark mass mQ and presents a suppression factor since in the physical case v . v’ = (m2 m & ) / 2 m p o m ~ D=: 1.004. Therefore, we neglect d h )in
JFQET
D:O
+
(12).
To evaluate the coupling of the photon to the light quark part of the electromagnetic current we invoke the VMD ansatz and consider the contribution of the intermediate 4(1020): (Dg*(v’,f)JSrPslD;o(v)) =
(14)
with k 2 = 0 and (OJSy,sl+(k, €1)) = M4 f461,. The experimental value of f4 is f+ = 234 MeV. The matrix element (D;(v’,~ ) $ ( kE, ~ ) I D Q * ~describes (V)) the strong interaction of a light vector meson ( 4 ) with two heavy mesons (0: and DQo). It can also be obtained through a low energy lagrangian in which the heavy fields Ha and S, are coupled, this time, t o the octet
176 of light vector mesonsb. The Lagrange density is set up using the hidden gauge symmetry method16, with the light vector mesons collected in a 3 x 3 matrix ipanalogous to M in (4). The lagrangian' reads as28:
L'
=if
+
i ~ r { S , ~ b a ~ ~ ~ ~ ,h.c., (p)~,}
(15)
+
with V x u ( p ) = &,pu - dypx [px,py] and px = i%bx, gv being fixed to gv = 5.8 by the KSRF rule2'. The coupling fi in (15) is constrained to fi = -0.1GeV-1 by the analysis of the D -+ K' semileptonic transitions induced by the axial weak current28t1s. The resulting expression for is:
&
The numerical result for the radiative width2,:
l?(Dt04 Dty) = 0.85 f 0.05 KeV
(17)
shows that the hadronic D& -+ D,*7ro transition is more probable than the radiative mode D,*o + Dfy, a t odds with the case of the D,* meson, where the radiative mode dominates the decay rate. In particular, if we assume that the two modes essentially saturate the DZO width, we have l?(DfO)= 8 5 1 KeV. As for the two radiative modes allowed for D,~(2460), one findss:
r ( D ; , -+ D,y) = 3.3 f 0.6 KeV
I'(D;,
-+
D t y ) = 1.5 KeV (18)
which in turn give a total width r(DA,) = 12 f 1 KeV. 4. Comparison with other approaches
The results of the previous two sections show that, within the described approach, the observed hierarchy of hadronic versus radiative modes is realized, supporting the identification of D,*,(2317) and D,.~(2460)with ( D & ,D t l ) . Other analyses have followed the same strategy of computing decay rates of the two narrow states in order t o understand their structure. In Table 1 we compare our results with the outcome of other approaches based on the c s picture as well. Analyses in which the states are assumed to have an exotic structure provide larger values for the widths (O(10') KeV)30. bThe standard W 8 - wo mixing is assumed, resulting in a pure 3s structure for 4. 'The role of other possible structures in the effective lagrangian contributing to radiative decays is discussed in Ref. 25.
177 Table 1. Estimated width (KeV) of DS'o and DL1, using the cB picture. The results in column [35]are obtained using experimental inputs from Belle (Focus). Decay mode
[31]
D:, -+D,T'
21.5
D,*o4 D:?
1.74
[32]
[21,81
10
751
21
1.9
[33]
l6 0.851k0.05 0.2
D ~ , - + D , ' T ~ 21.5
2110
7fl
D:,
+ D,y
5.08
6.2
3.3f0.6
Dil
-+
D:y
4.66
5.5
1.5
32
1341 129 f 43 (109616) < 1.4 187 f 73 (7.4312.3) 5 5
In particular, we observe that conclusions analogous to those presented above have been reached in Ref. 31, which is based on the observation that heavy-light systems should appear as parity-doubled, i.e. in pairs differing for parity and transforming according to a linear representation of chiral symmetry. In particular, the doublet composed by the states having J p = (Of, 1+) can be considered as the chiral partner of that with J p = (0-, 1-) '. Since our calculation is based on a different method, the sep = 1- and s c ='f doublets being treated as uncorrelated multiplets, we find the agreement noticeable. 5 . Conclusions and perspectives
We presented the calculation of hadronic and radiative decay rates of D,*,(2317) and D,~(2460)in a framework based on heavy quark symmetries and on the Vector Meson Dominance ansatz. This analysis shows that the observed narrow widths and the enhancement of the DL*)7rodecay modes are compatible with the identification of Dz,(2317) and D,~(2460) with the states belonging to the J E = (O+, l+); doublet of the cs spectrum. Nevertheless, unanswered questions remain, such as the near equality of the masses of DzJ(2317) and D,~(2460)with their non-strange partners. The missing evidence of the radiative mode D:,(2317) -+ D:-y is another puzzling aspect deserving further experiment a1 investigations. The quantum number assignment has a rather straightforward consequence concerning the doublet of scalar and axial vector mesons in the b3 spectrum. Since the mass splitting between B and D states is similar to the corresponding mass splitting between B, and D, states, such mesons dThis idea was first suggested in Ref. 35 in order to obtain a consistent implementation of chiral symmetry and reconsidered also in Ref. 36.
178
should be below t h e B K and B*K thresholds, thus producing narrow peaks in B,xo and B,*nomass distributions.
Acknowledgments I warmly t h a n k the Organizers for inviting m e to present this talk and for their very kind hospitality. I also thank P. Colangelo and R. Ferrandes for collaboration o n t h e topics discussed above. Partial support from t h e EC Contract No. HPRN-CT-2002-00311 (EURIDICE) is acknowledged. References B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 90,242001 (2003). D. Besson et al. [CLEO Collaboration], Phys. Rev. D68, 032002 (2003). B. Aubert et al. [BABAR Collaboration], Phys. Rev. D69,031101 (2004). K. Abe et al., Phys. Rev. Lett. 92,012002 (2004). E. Vaandering, arXiv:hepex/0406044. P. Krokovny et al. [Belle Collaboration], Phys. Rev. Lett. 91,262002 (2003). P. Krokovny, arXiv:hep-ex/0310053. 7. G. Calderini [BaBar Collaboration], arXiv:hep-ex/0405081. 8. P. Colangelo, F. De Fazio and R. Ferrandes, arXiv:hepph/0407137. 9. T. Barnes, F. E. Close and H. J. Lipkin, Phys. Rev. D68, 054006 (2003); H. J. Lipkin, Phys. Lett. B580,50 (2004); P.Bicudo, arXiv:hep-ph/0401106. 10. A. P. Szczepaniak, Phys. Lett. B567,23 (2003). 11. T. E. Browder, S. Pakvasa and A. A. Petrov, Phys. Lett. B578,365 (2004); S. Nussinov, arXiv:hepph/0306187. 12. G. S. Bali, Phys. Rev. D68,07150 (2003). 13. A. Dougall, R. D. Kenway, C. M. Maynard and C. McNeile [UKQCD Collaboration], Phys. Lett. B569,41 (2003). 14. P. Colangelo, F. De Fazio and N. Paver, Phys. Rev. D58, 116005 (1998); Y. B. Dai, C. S. Huang, C. Liu and S. L. Zhu, Phys. Rev. D68, 114011 (2003). 15. M. B. Wise, Phys. Rev. D45,R2188 (1992); G.Burdman and J. F. Donoghue, Phys. Lett. B280, 287 (1992); P. Cho, Phys. Lett. B285, 145 (1992); H.Y.Cheng, C.-Y. Cheung, G.-L. Lin, Y. C. Lin and H.-L. Yu, Phys. Rev. D46, 1148 (1992). 16. R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Lett. B292,371 (1992). 17. P. Colangelo, F. De Fazio, G. Nardulli, N. Di Bartolomeo and R. Gatto, Phys. Rev. D52, 6422 (1995); P.Colangela and F. De Fazio, EZLT. Phys. J. C4, 503 (1998). 18. For a review see: R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281,145 (1997). 19. P. L. Cho and M. B. Wise, Phys. Rev. D49,6228 (1994). 20. J. Gasser and H. Leutwyler, Nucl. Phys. B250,465 (1985).
1. 2. 3. 4. 5. 6.
179 21. P. Colangelo and F. De Fazio, Phys. Lett. B570, 180 (2003). 22. E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Phys. Rev. D21, 203 (1980). 23. J. F. Amundson et al., Phys. Lett. B296,415 (1992); P. L. Cho and H. Georgi, Phys. Lett. B296,408 (1992) [Erratum-ibid. B300, 410 (1993)]. 24. P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett. B334, 175 (1994). 25. P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett. B316, 555 (1993). 26. K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D66, 010001 (2002). 27. A. F. Falk, B. Grinstein and M. E. Luke, Nucl. Phys. B357, 185 (1991). 28. R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Lett. B299, 139 (1993). 29. K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16,255 (1966); Riazuddin and Fayazuddin, Phys. Rev. 147, 1071 (1966). 30. H. Y. Cheng and W. S. Hou, Phys. Lett. B566, 193 (2003); S. Ishida, M. Ishida, T. Komada, T. Maeda, M. Oda, K. Yamada and I. Yamauchi, arXiv:hep-ph/0310061. 31. W. A. Bardeen, E. J. Eichten and C. T. Hill, Phys. Rev. D68,054024 2003). 32. S . Godfrey, Phys. Lett. B568, 254 (2003). 33. Fayyazuddin and Riazuddin, arXiv:hep-ph/0309283. 34. Y. I. Azimov and K. Goeke, arXiv:hep-ph/0403082. 35. M. A. Nowak, M. Rho and I. Zahed, Phys. Rev. D48, 4370 1993); W. A. Bardeen and C. T. Hill, Phys. Rev. D49, 409 (1994). 36. M. A. Nowak, M. Rho and I. Zahed, arXiv:hep-ph/0307102.
SEARCH FOR DARK MATTER IN B + S TRANSITIONS WITH MISSING ENERGY
M. POSPELOV Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada E-mail: pospelov@uvic. ca Dedicated underground experiments searching for dark matter have little sensitivity to GeV and sub-GeV masses of dark matter particles. We show that the decay of B mesons to K ( K * ) and missing energy in the final state can be an efficient probe of dark matter models in this mass range. We analyze the minimal scalar dark matter model to show that the width of the decay mode with two dark matter scalars B + K S S may exceed the decay width in the Standard Model channel, B ---t KuD, by up to two orders of magnitude. Existing data from B physics experiments almost entirely exclude dark matter scalars with masses less than 1 GeV. Expected data from B factories will probe the range of dark matter masses up to 2 GeV.
1. Introduction
Although the existence of dark matter is firmly established through its gravitational interaction, the identity of dark matter remains a big mystery. Of special interest for particle physics are models of weakly interacting massive particles (WIMPS), which have a number of attractive features: well-understood mechanisms of ensuring the correct abundance through the annihilation a t the freeze-out, milli-weak t o weak strength of couplings t o the “visible” sector of the Standard Model (SM), and as a consequence, distinct possibilities for WIMP detection. The main parameter governing the abundance today is WIMP annihilation cross section directly related to the dark matter abundance. In order to keep WIMP abundance equal or smaller than the observed dark matter energy density, the annihilation cross section has t o satisfy the lower bound, oannvre1 >, 1 pb, (see e.g. I ) . In all WIMP models studied to date the annihilation cross section is suppressed in the limit of very large or very small mass of a WIMP particle S . This confines the mass of a stable WIMP within a certain mass range, mmin 5 ms 5 mmax,which we refer to as the Lee-Weinberg window ’.
180
181
This window is model-dependent and typcally extends from a few GeV to a few TeV. If the neutralino is the lightest stable supersymmetric particle, mmin_N 5 GeV but in other models of dark matter mmin can be lowered 495
Recently, WIMPs with masses in the GeV and sub-GeV range have been proposed as a solution to certain problems in astrophysics and cosmology. For example, sub-GeV WIMPs can produce a high yield of positrons in the products of WIMP annihilation near the centres of galaxies 6 , which may account for 511 KeV photons observed recently in the emission from the Galactic bulge GeV-scale WIMPs are also preferred in models of self-interacting dark matter that can rectify the problem with over-dense galactic centers predicted in numerical simulations with non-interacting cold dark matter. Dedicated underground experiments have little sensitivity to dark matter in the GeV and sub-GeV range. Direct detection of the nuclear recoil from the scattering of such relatively light particles is very difficult because of the rather low energy transfer to nuclei, AE v2mi/mNuc15 0.1 KeV, which significantly weakens experimental bounds on scattering rates below r n of ~ few GeV, especially for heavy nuclei. Indirect detection via energetic neutrinos from the annihilation in the centre of the Sun/Earth is simply not possible in this mass range because of the absence of directionality. Therefore, the direct production of dark matter particles in particle physics experiments stands out as the most reliable way of detecting WIMPs in the GeV and sub-GeV mass range. The purpose of this work is to prove that B decays can be an effective probe of dark matter near the lower edge of the Lee-Weinberg window. K decays can also be used for this purpose, but B decays have far greater reach, up to m s 2.6 GeV. In particular, we show that pair production of WIMPs in the decays B -+ K ( K * ) S Scan compete with the Standard Model mode B -+ K(K*)vF. In what follows, we analyze in detail the “missing energy” processes in the model of the singlet scalar WIMPS and use the existing data from B physics experiments to put important limits on the allowed mass range of scalar WIMPs. The main advantage of the singlet scalar model of dark matter is its simplicity,
’.
-
-
4,9110
xs 4
= -S
4
1
-t- -(m: 2
x + Xv‘&,)S2 + XvEWS’h + -S2h2, 2
182
where H is the SM Higgs field doublet, WEW = 246 GeV is the Higgs vacuum expectation value (vev) and h is the field corresponding t o the physical Higgs, H = (0, (WEW h ) / a ) . It is important to recognize that the physical mass of the scalar S receives contributions from two terms, m i = m; and can be small, even if each term is on the order ~ O ( W ’ &Although ~). admittedly fine-tuned, the possibility of low ms is not a priori excluded and deserves further studies as it also leads t o Higgs decays saturated by the invisible channel, h 4 SS and suppression of all observable modes of Higgs decay a t hadronic colliders ‘. The minimal scalar model is not a unique possibility for light dark matter, which can be introduced more naturally in other models. If for example, the dark matter scalar S couples to the Hd Higgs doublet in the two-Higgs modification of (l), XS2HlHd, the fine-tuning can be relaxed if the ratio of the two electroweak vevs, t a n p = (Hu)/(Hd) is a large parameter. A well-motivated case of tanp 50 corresponds t o ( H d ) 5 GeV, and only a modest degree of cancellation between mg and X(Hd)’ would be required to bring m S in the GeV range. More model-building possibilities open up if new particles, other than electroweak gauge bosons or Higgses, mediate the interaction between WIMPs and SM particles. If the mass scale of these new particles is smaller than the electroweak scale 5, sub-GeV WIMPs are possible without fine-tuning.
+
+
N
-
Figure 1. Feynman diagrams which contribute t o B meson decays with missing energy.
2. Pair-production of WIMPs in B decays The Higgs mass mh is heavy compared to ms of interest, which means that in all processes such as annihilation, pair production, and elastic scattering of S particles, X and mh will enter in the same combination, X2mh4. In what follows, we calculate the pair-production of S particles in B decays in terms of two parameters, A2/mi and ms, and relate them using the dark matter abundance calculation, thus obtaining the definitive prediction for
183 the signal as a function of ms alone. At the quark level the decays of the B meson with missing energy correspond to the processes shown in Figure 1. The SM neutrino decay channel is shown in Figure l a and l b . The b -+ s Higgs penguin transition, Figure l c , produces a pair of scalar WIMPS S in the final state, which likewise leave a missing energy signal. In this section, we show that this additional amplitude generates b --f sSS decays that can successfully compete with the SM neutrino channel. A loop-generated b - s-Higgs vertex at low momentum transfer can be easily calculated by differentiating the two-point b s amplitude over D E W . We find that t o leading order the b + sh transition is given by an effective interaction --f
Using this vertex, Eq. (1) and safely assuming mh >> mb, we integrate out the massive Higgs boson to obtain the effective Lagrangian for the b --+ s transition with missing energy in the final state: Lb,,&
1 2
= -CDMmbsLbRS2
-
C,gLy,bLfi~,v 4-(h.c.).
(3)
Leading order Wilson coefficients for the transitions with dark matter scalars or neutrinos in the final state are given by
where xt = m:/M&. We would like to remark at this point that the numerical value of CDM is a factor of few larger than C,,
-
if AmL2 O(g&MG2). This happens despite the fact that the effective bsh vertex is suppressed relative to bsZ vertex by a small Yukawa coupling mb/vEw. The 1 / v in~ (2) ~ is compensated by a large coupling of h to S2, proportional to AVEW, and mb is absorbed into the definition of the dimension 6 operator mbBLbRS2. N
184
We concentrate on exclusive decay modes with missing energy, as these are experimentally more promising than inclusive decays and give sensitivity to a large range of ms. A limit on the branching ratio has recently been at 90% c.1. 'I, reported by BABARcollaboration, BrB+--rK+vu< 7.0 x which improves on a previous CLEO limit 12, but is still far from the SM prediction Br(B + KvP) N (3 - 5) x (See, e.g. 1 3 ) . We use the result for Lb,,@ along with the hadronic form factors determined via light-cone sum rule analysis in l4 to produce the amplitude of B 4 K S S decay,
) ~ the form factor for B -+ K transition is where q2 = d = ( p - ~ p ~ and approximated as fo N 0.3exp{0.63dMB2 - O.095i2ME4 O.591d3Mi6}. The differential decay width to a K meson and a pair of WIMPS is given by
+
where I ( ; ) reflects the available phase space,
+
I ( ; , m s ) = [i2 - 2 2 ~ ~M2; ) +
+ [I- 4mS/d] +.
( M B - hfK)2]
From Eq. (7) and the prediction for the SM neutrino channel, we obtain the total branching ratio for the B+ to K+ decay with missing energy in the final state,
+
Brg+-+K++F= B ~ B + - + K +B~B++K+ss ~~
4
+ 2.8 x 1 0 - 4 K 2 ~ ( m s ) .
Eq. (8) uses the parametrization of X2mi4, K 2 = A 2 ( 100mGeV h
(8)
), 4
(9)
and the available phase space as a function of the unknown m s , F ( m s ) = / ' m ~ ( d ) 2 1 ( d , m s > dd Smin
[
iz!(i)21(d, 0) dd
1
-l
Notice that F ( 0 ) = 1 and F ( m s ) = 0 for m s > $(mB - mK) by construction. Similar calculations can be used for the decay B K*SS,
185
with an analogous form factor. For light scalars, ms few 100 MeV, and K O(1) the decay rates with emission of dark matter particles are 50 times larger than the decay with neutrinos in the final state! This is partly due to a larger amplitude, Eq. (5), and partly due to phase space integral that is a factor of a few larger for scalars than for neutrinos if m S is small. N
N
N
3. Abundance calculation and Comparison with
Experiment The scalar coupling constant X and the scalar mass ms are constrained by the observed abundance of dark matter. For low ms, as shown in 4 , the acceptable value of K is K O(1). Here we refine the abundance calculation for the range 0 < m S < 2.4 GeV in order to obtain a more accurate quantitative prediction for K . The main parameter that governs the energy density of WIMP particles today, which we take to be equal to the observed value of RoMh2 0.13 15, is the average of their annihilation cross section a t the time of freeze-out. This cross section multiplied by the relative velocity of the annihilating WIMPS is fixed by S ~ D Mand can be conveniently expressed as N
N
Here rGx denotes the partial rate for the decay, 4 X , for a virtual Higgs, h, with the mass of mh = 233s N 2ms. Notice that Eq. (11) contains the same combination X2rnh4as (8). The zero-temperature width rGx was extensively studied two decades ago in conjunction with searches for light Higgs 16,17,18 For ms larger than mT the annihilation to hadrons dominates the cross section, which is therefore prone t o considerable uncertainties. At a given value of ms, we can predict rhx within a certain range that reflects these uncertainties. With the use of (ll),this prediction translates into the upper (A) and the lower (B) bounds on K(ms),which we insert into Eq. (8) and plot the resulting Brg++K++e in Figure 2 . In the interval 150 MeV 5 ms mo) in curve A. Both curves include the T threshold. There are no tractable ways of calculating the cross section in the intermediate region 650 MeV 5 ms 5 1 GeV. However, there are no particular reasons to believe that the annihilation into hadrons will be significantly enhanced or suppressed relative to the levels in adjacent domains. In this region, we interpolate between high- and low-energy sections of curves A and B. Thus, the parameter space consistent with the required cosmological abundance of S scalars calculated with generous assumptions about strong interaction uncertainties is given by the area between the two curves, A and B. Figure 2 presents the predicted range of BrB++K++F as a function of ms and is the main result of our paper. The SM “background” from B --t KuF decay is subdominant everywhere except for the highest kinematically allowed domain of ms. To compare with experimental results ”~”, we must convert the limit on BrB+-+K+VD t o a more appropriate bound on Brg+,K++F according to the following procedure. We multiply the experimental limit of 7.0 x by a ratio of two phase space integrals, F ( m s ,imin)/F(ms,iexp), where seXpis determined by the minimum Kaon momentum considered in the experimental search, namely 1.5 GeV. This produces an exclusion curve, nearly parallel to the ms axis at low ms, and almost vertical near the experimental kinematic cutoff. The current BABARresults (curve I) exclude ms < 430 MeV, as well as the region 510 MeV< ms < 1.1 GeV, and probe the allowed parameter space for dark matter up t o ms 1.5 GeV. Generalized model with N component dark matter scalar gives N2-fold increase in the branching ratio 4 , and thus greater sensitivity to ms. The B factories will soon have larger data samples and can extend the
-
N
187
w
+ 1: 1 0 - ~
M
a
6i
10-~ 0
0.5
1
1.5
2 MdGeV)
Figure 2. Predicted branching ratios for the decay B + K+ missing energy, with current limits from Babar (I) CLEO (111) and expected results from BABAR(11). Parameter space above curves I and I11 is excluded. The horizontal line shows the SM B -+ KvD signal. Parameter space to the left of the vertical dashed line is also excluded by K i + T+$.
search to lower Kaon momenta. The level of sensitivity expected from an integrated luminosity L of 250 fb-I and momentum cutoff of 1 GeV is shown by curve 11, which assumes that the sensitivity scales as L-l/’, as suggested by the analysis in In reality, the experimental limit will extend to Kaon momenta below 1 GeV where the sensitivity will gradually degrade due t o increasing backgrounds; however, we expect the implication of curve I1 t o remain valid, namely that the B factories will probe scalar dark matter up to 2 GeV. If ms 5 150 MeV, the decay K+ -+ T+SS becomes possible. The width for this decay can be easily calculated in a similar fashion to b 4 s transition. The concordance of the observed number of events with the SM prediction 2o rules out scalars in our model with ms < 150 MeV. This exclusion limit is shown by a vertical line in Figure 3. It is below mminof 350 MeV. 4. Conclusions
To conclude, we have demonstrated that the b 4 s transitions with missing energy in the final state can be an efficient probe of dark matter when pair production of WIMPS in B meson decays is kinematically allowed. In particular, we have shown that the minimal scalar model of dark matter
188
with the interaction mediated by the Higgs particle predicts observable rates for B f -+K+ and missing energy. A large portion of the parameter space 1 GeV is already excluded by current BABARlimits. New with m S experimental data should probe a wider range of masses, up t o ms 2 GeV. The limits obtained in this paper have important implications for Higgs searches, as the existence of relatively light scalar WIMPs leads t o the Higgs decays saturated by invisible channel. Given the astrophysical motivations for GeV and sub-GeV WIMPs combined with insensitivity of dedicated dark matter searches in this mass range, it is important t o extend the analysis of b -+ s transition with missing energy onto other models of light dark matter. N
Acknowledgments We thank Misha Voloshin for valuable discussion. This research is supported in part by NSERC of Canada and PPARC UK.
References 1. E.W. Kolb and M.S. Turner, The Early Universe, Addison-Wesley (1990). 2. B. W. Lee and S. Weinberg, Phys. Rev. Lett. 39, 165 (1977). 3. G. Belanger, F. Boudjema, A. Cottrant, A. Pukhov and S. Rosier-Lees, JHEP 0403, 012 (2004). 4. C. P. Burgess, M. Pospelov and T. ter Veldhuis, Nucl. Phys. B 619, 709 (2001). 5 . P. Fayet, Phys. Rev. D 70,023514 (2004). 6. C. Boehm, D. Hooper, J. Silk and M. Case, Phys. Rev. Lett. 92, 101301 (2004). 7. P. Jean et al., Astron. Astrophys. 407, L55 (2003). 8. D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84, 3760 (2000). 9. V. Silveira and A. Zee, Phys. Lett. B 161, 136 (1985). 10. J. McDonald, Phys. Rev. D 50, 3637 (1994). 11. B. Aubert et al. [BABAR Collaboration], arXiv:hep-ex/0304020. 12. T. E. Browder et al. [CLEO Collaboration], Phys. Rev. Lett. 86, 2950 (2001). 13. G. Buchalla, G. Hiller and G. Isidori, Phys. Rev. D 63, 014015 (2001). 14. A. Ali, P. Ball, L. T. Handoko and G. Hiller, Phys. Rev. D 61,074024 (2000). 15. D. N. Spergel et al., Astrophys. J . Suppl. 148, 175 (2003). 16. M. B. Voloshin, Sou. J . Nucl. Phys. 44, 478 (1986). 17. S. Raby and G. B. West, Phys. Rev. D 38, 3488 (1988). 18. T. N. Truong and R. S. Willey, Phys. Rev. D 40, 3635 (1989). 19. K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D 66, 010001 (2002). 20. S. Adler et al. [E787 Collaboration], Phys. Rev. Lett. 88, 041803 (2002).
SECTION 3.
EXOTIC HADRONS
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EXOTICA
R. L. JAFFE Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology Cambridge, M A 02139, USA E-mail: jaffeamit. edu The sudden appearance of narrow, prominent exotic baryons has re-invigorated light hadron spectroscopy. At present the experimental situation is confused; so is theory. The recent discoveries are striking. So too is the complete absence of exotic mesons, and, except for the recent discoveries, of exotic baryons as well. Whether or not the new states are confirmed, the way we look at complicated states of confined quarks and gluons has changed. Perhaps the most lasting result, and the one emphasized in these notes, is a new appreciation for the role of diquark correlations in QCD.
1. Introduction The absence of exotics is one of the most obvious features of QCDa. In the early years experimenters searched hard for baryons that cannot be made of three quarks or mesons that cannot be made of qq. Exotic mesons seemed entirely absent. Controversial signals for exotic baryons known as 2's came, and usually went, never rising to a level of certainty sufficient for the Particle Data Group's tables2. In the 1990's the subject of exotic baryons did not receive much attention except from a small band of theorists motivated by the predictions of chiral soliton model^^^^^^. Then, in January of 2003 evidence was reported of a very narrow baryon with strangeness one and charge one, of mass M 1540 MeV, now dubbed the O+, with minimum quark content u ~ d d S ~The > ~ first . experiment was followed by evidence for other exotics: a strangeness minus two, charge minus two particle now officially named the @-- by the PDG, with minimum quark content ddsscg at 1860 MeV , and an as-yet nameless charm exotic (uuddc)" at 3099 MeV. Theorists, myself included, descended upon these aThis paper is a condensation of material that can be found in Ref.
191
'.
192
reports and tried to extract dynamical insight into QCD. Other experimental groups began searches for the O+ and its friends. As time has passed the situation has become more, rather than less, confusing": several experiments have now reported negative results in searches for the O+13; no one has confirmed either the @--(1860) or the uuddc(3099); and theorists have yet to find a compelling (to me at least) explanation for the low mass or narrow width of the O f . The existence of the O+ is a question for experimenters. Theorists simply do not know enough about QCD t o predict without doubt whether a light, narrow exotic baryon exists. Whether or not the Of survives, it is clear that exotics are very rare in QCD. Perhaps they are entirely absent. This remarkable feature of QCD is often forgotten when exotic candidates are discussed. The existence of a handful of exotics has to be understood in a framework that also explains their overall rarity. Along the same line, the aufbau principle of QCD differs dramatically from that of atoms and nuclei: to make more atoms add electrons, to make more nuclei, add neutrons and protons. However in QCD the spectrum - with the possible exception of a few states like the O+ - seems to stop at qqq and qij. Thinking about this problem in light of early work on multiquark correbegan t o re-examine the role of lations in QCD 14, Frank Wilczek and 115b diquark correlations in QCD. Diquarks are not new; they have been chamWe l~~. pioned by a small group of QCD theorists for several d e c a d e ~ ~ ~ already knew14 that diquark correlations can naturally explain the general absence of exotics and predict a supernumerary nonet of scalar mesons which seems to exist. We quickly learned that they can rather naturally accommodate exotics like the Of. They also seem to be important in dense quark matter20, to influence quark distribution21 and fragmentation functions, and to explain the systematics of non-leptonic weak decays of light quark baryons and mesons22. Whether or not the O+ survives, diquarks are here to stay. In the first part of this talk, after looking briefly at the history of exotics, I assume that the O+ exists, and see how well it fits with other features of light quark spectroscopy. I will take a look at the O+ from several perspectives: general scattering theory, large N,, chiral soliton models, quark models, and lattice calculations. In general these exercises raise more questions than they answer. In brief A light, narrow exotic is inconvenient but not impossible for QCD spectroscopy. bClosely related ideas have been explored by NussinovlG and by Karliner and LipkinL7.
193 The later half of the talk is devoted to diquarks. I define them carefully and then describe how diquark correlations in hadrons can explain qualitatively most of the puzzles of exotic hadron spectroscopy: first, why exotics are so rare in QCD; next, why there is an extra nonet of scalar mesons; third, why an exotic baryon antidecuplet containing the O+ would be the only prominent baryon exotic; and finally, why non-strange systems of 6, 9, 12, . . .quarks form nuclei not single hadrons. “Qualitatively” is an important modifier, however: like all quark model ideas, this one lacks a quantitative foundation: the need for a systematic and predictive phenomenological framework for QCD spectroscopy has never been greater. Spectroscopy was at the cutting edge of high energy physics in the ‘60’s and ‘70’s. A great deal of effort and sophisticated analysis was brought to bear on the study of the hadron spectrum, and the conclusions remain important. In the decade that followed the first conjectures about quarks experimental groups studied meson-baryon and meson-meson scattering, and extracted the masses and widths of meson and baryon resonances. Resonances were discovered in nearly all non-exotic meson and baryon channels, but no prominent exotics were found. The zeroth order summary prior to January 2003 was simple: no exotic mesons or baryons. In fact the only striking anomaly in low energy scattering was the existence of a supernumerary (ie not expected in the quark model) nonet of scalar, (J” = Of) mesons with masses below 1 GeV: the f0(600),n(800), f0(980), and ao(980) that is now widely considered to contain important ijGqq components23 When the Of was first reported, several groups re-examined the old K N scattering data and interpreted the absence of any structure near 1540 MeV as an upper limit on the width of the O+’6924>25>26. The limits range from 0.8 MeV through “a few” MeV. It is important to remember that these are not sightings of a narrow O+, rather they are reports of negative results expressed as an upper limit on the width of the O+. Space and time do not permit me to present and review all the reports of exotics since January of 2003. Summaries of the experimental sightings of the O+ can be found in many recent reviews27. Table 1 is a summary of the properties of the reported states. The baryons in Table 1 can be classified in the i-6or 8 and representations of SU(3)f Although they could be in higher representations, the is the simplest that can accomodate both the O+ and the @--. I have not attempted to summarize the searches that fail to see the Of or the other new exotics. These require a careful discussion, which can be found, for example in Ref. 13.
s
194
Isospin
Decays
K+n, KSP
I
a--
I
I
I
~0/~0141
a- /=-I41
1860 1860 1855
I
< 18 < 18 < 18
From analysis of K N scattering; 1’ dence for J
I
I I
?
I
?
I ? E*O( 1 5 3 0 ) ~ -
from direct detection of 8 ; 131
Weak evi-
= 112, no information on parity; 141 QifI = 3/2, 9 i f I = l/2;[’]I
=
112favored if decay to 8’0(1530) is correct.
There are several puzzling aspects of the data: the variation of the O+ mass, the claims of HERMES and ZEUS to have measured a non-vanishing width, and the apparent inconsistency of several positive experiments with the limits obtained in other experiments, for example. The interested reader should consult the talks by Nakano” and Dzierba13 and other presentations at QNP2004. There are many predominantly experimental issues that I have not covered here: proposals to measure the parity of the O+, limits on the production of other exotics like a @++, and reports of other bumps with the quantum numbers of the O+ at higher mass, to name only a few. 2. Theoretical perspectives 2.1. Insight
from scattering theory
The small width of the O+ poses a challenge for any theoretical interpretation. The O+ is unique among hadrons in that its valence quark configuration, uudds, already contains all the quarks needed for it to decay into K N . Non-exotic hadrons like the ~ ( 7 7 0 or ) A(1520) in their valence quark configuration can only couple to their decay channels (TT for the former, KN for the latter) by creating quark pairs. The suppression of quark pair creation, known as the Okubo, Zweig, Iizuki (OZI) Rulez8, is often invoked as an explanation for the relative narrowness of hadronic resonances. Some other, as yet unknown mechanism would have to be responsible for the narrowness of the O+. General principles of scattering theory allow one to get at least a qualitative answer to the question: ‘ L Hunusual ~ ~ is the width of the 0+?’’29.
195
There are two ways t o make a resonance in low energy scattering, either (a) the resonance is generated by the forces between the scattering particles, or (b) it exists in another channel, which is closed (or confined), and couples to the scattering channel by some interaction. Potential scattering resonances (case (a)) are generated by the interplay between attraction due to interparticle forces and repulsion, usually due t o the angular momentum barrier. These resonances subside into the continuum as the interaction is turned off. An example of case (b) is a bound state in a closed or confined channel that couples to a scattering channel by an interaction. These states decouple, i e their widths vanish, as the interaction is turned off. In the case of a qqq baryon coupling to the meson(@q)-baryons(qqq)continuum, quark pair creation is the interaction. The O+ is unusual: Because its valence quark content, uudds, is the same as the valence quark content of K N , the possibility that it arises from the K N potential cannot be excluded a priori and has to be analyzed. The lesson of this exercise is qualitative (and here I quote from Ref. ”): If the O+ appears in a low partial wave (l = 0 or l), schemes which hope to produce it from K N forces seem doomed from the start; schemes which introduce confined channels (quark models are an example, where reconfiguration of the quark substructure of the Of could be required for it to decay) are challenged to find a natural physical mechanism that suppresses the Of decay more effectively than the OZI rule suppresses the decay of the A(1520).
2.2. Large Nc a n d chiml soliton models
The number of colors is the only conceivable parameter in light quark QCD, so it is natural t o consider exotic baryon dynamics as an expansion in l,”,. It is known from the work of ‘t Hooft, Witten, and many others, that as N, + 00 QCD reduces t o a theory of zero width @q mesons with AQCLI~’and heavy baryons with masses N,AQCO in which masses quarks move in a mean (Bartree) field31. This is far from a complete description even at the heuristic level: Almost nothing is known about the spectrum of 4q mesons from large N, except for the pseudoscalar Goldstone bosons required by chiral symmetry. The dynamics of baryons is accessible only through a loose association with the chiral soliton model (CSM). It is important to remember that the CSM has not be derived from large N, QCD. Its appeal is based on its proper implementation of chiral symmetry and anomalies, and on the resemblance of collectively quantized solitons to N
N
196
the lightest positive parity baryons. Dashen, Jenkins, and M a n ~ h a made r ~ ~ the application of large N, ideas to baryons more precise and extended their work t o exotic baryons33. Their ideas were presented a t this conference by E. Jenkins. They cannot determine whether the @+ is light enough to be narrow and prominent, or even whether it is the lightest qqqqtj state. However, if the existence of the @+ is fed into their machinery, one can predict its properties and the masses and properties of other states in the qqqqtj spectrum. One result of their analysis that is particularly troubling is also a general feature of all quark model/QCD treatments of the exotic baryons: the most natural candidate for the @+ (all quarks in the lowest Hartree level) has negative parity, corresponding to the K N s-wave, where it is very hard to explain its narrow width. The positive parity @+, which Jenkins and Manohar study in detail, require one quark to be in an excited orbital. This state has internal structure which might sequester it from the K N channel enough to help explain its long lifetime. So all QCD approaches consistent with large N , should share a fundamental difficulty a t the outset: If the quark configuration is “natural”, it’s hard to explain why the O+ should be narrow. To have any hope t o obtain a narrow @+, strong quark forces must make a state of mixed spatial symmetry the lightest. Understanding these strong quark correlations and their consequences then becomes a central issue. Much of the discussion of exotic baryons over the past decade4i6 and, in particular, a remarkable prediction of the mass and width of the @+ by Diakonov, Petrov, and Polyakov 5 , has been carried out in the context of chiral soliton models. This is not surprising since CSM’s are teeming with exotics. Collective quantization of a classical soliton solution t o a chiral field theory of pseudoscalar bosons in SU(2)f or SU(3)f, consistent with anomaly constraints, yields towers of baryons only the lightest of which are not e x o t i ~ ~ The ~ i ~simplest ~. case is SU(2)f, where the spectrum of baryons begins with a rotational band of positive parity states with I = J = 1/2,3/2,5/2,. . .. Other excitations, radial for example, are heavier, separated from the ground state band by order ( N : ) . The lack of evidence for a I = 5/2 baryon resonance led most workers to dismiss the heavier states as artifacts of large N,. The generalization of the CSM to three flavors with broken SU(3)f has always been controversial. Guadagnini’s original approach (the “rigid rotor” (RR) approach) was to quantize in the SU(3)f limit and introduce SU(3)f violation p e r t ~ r b a t i v e l y ~ Alternatively, ~. Callan and Klebanov
197
quantized the S U ( 2 ) f soliton and constructed strange baryons as kaon bound states (the "bound state (BS) a p p r ~ a c h ) ~Although ~. different in principle, the two approaches give roughly the same spectrum for the octet and decuplet. When generalized to three flavors the rotational band of the 3+
1 '
I+
3+
103 , 10' , 275 ' 5 , . . .. Diakonov, Petrov, and Polyakov5 estimated its mass and width of the --1/2+ first exotic multiplet in this tower, 10 . They argued that it should be light and narrow37. Their work stimulated the experimenters who found the first evidence for the Of7. Soon afterwards Weigel showed that it is inconsistent in the RR approach to ignore the mixing between the and radial excitations excited by order (Nf) above the ground state6. After the first reports of the Of, other groups found the mass splitting between the ground state and the 10 to be order (Nf)40739.Cohen pointed out that the width of the Of does not vanish as N, -+ ca in contrast to non-exotic states like the A39, making it hard to understand the very small value of I'(O)/r(A) obtained in Ref. 5 . Furthermore, the O+ does not exist in the BS approach unless the mass of the kaon is of the order of 1 GeV6y4O. In fact, in the BS approach the force between the collectively quantized two-flavor soliton and the kaon is repulsive for physical kaon masses. Chiral soliton models describe at best a piece of QCD: Their picture of the nucleon and A (or 84' and 10;' in S U ( 3 ) f )is internally consistent and predictive. Some progress has been made in the description of baryon resonances41. However the incorporation of strangeness is not satisfactory and still c o n t r o ~ e r s i a land ~ ~ , the CSM gives no insight at all into the meson spectrum. As for exotics, the candidate for the O+ is controversial and there is no insight into the striking absence of exotic mesons and baryons in general. The prediction of a narrow width for the O+ is very controversial.
RR approach becomes B;',
2.3. Quark models
The quark model in its many variations has been by far the most successful tool for the classification and interpretation of light hadrons. It predicts the principal features and many of the subtleties of the spectrum of both mesons and baryons, and it matches naturally onto the partonic description of deep inelastic phenomena. The limitations of the quark model are, however, as obvious as its successes. It has never been formulated in a way that is fully consistent with confinement and relativity. Of course quarks can move relativistically, governed by the Dirac equation, in first quantized models like the MIT bag, but
198
there is no fully relativistic, second quantized version of the quark model. Furthermore, quark models are not the first term in a systematic expansion. No one knows how to improve on them. Nevertheless all hadrons can be classified as relatively simple configurations of a few confined quarks, and there is no reason to be expect the Of to be an exception. So looking for a natural quark description of the Of is a high priority, and if there were none, it would be most surprising.
2.3.1. Generic features of a n uncorrelated quark model Although quark models differ in their details, the qualitative aspects of their spectra are determined by features that they share in common. They need not be correct but they form the context in which other proposals have to be considered. Certainly they do a good job for mesons, baryons, and even tetraquark (ie. tjtjqq) spectroscopy. Here is a summary of the basic i n g r e d i e n t ~ that ~~t~ can ~ be applied to qqqqq states. (1) The spectrum can be decomposed into sectors in which the numbers of quarks and antiquarks, ng and nq, are good quantum numbers - the OZI rule28. (2) Hadrons are made by filling quark and antiquark orbitals in a hypothetical mean field. The total angular momentum of the (relativistic) quark in the lowest orbital is 1/2. Its parity is even (relative to the proton). Typically the first excited orbitals have negative parity and total angular momentum 1/2 and 3/2 because orbital excitations are invariably less costly than radial. (3) The ground state multiplet is constructed by putting all the quarks and antiquarks in the lowest orbital - the “single mode configuration”. This is a natural assumption, but one which must fail for qqqqq if the O+ has positive parity (see below). (4) As the number of quarks and antiquarks grows, the number of p c q n q eigenstates proliferates wildly. Although the qqqq and qqqqq states are stationary states in a potential or bag, they do not in general correspond to stable hadrons or even resonances. In a fairly precise way the @?jqq state can be considered a piece of the mesonmeson continuum that has been artificially confined by an inappropriate confining potential or boundary condition or potential43. Unless the multiquark state is unusually light or sequestered (by the spin, color and/or flavor structure of the wavefunction) from the scattering channel, it is just an artifact of a silly way of enu-
199
merating the states in the continuum.
2.3.2. Pentaquarks an the uncorrelated quark model The parity of the ground state of nq quarks and nq antiquarks in the lowest orbital is (-l)ng. When a quark or antiquark is excited, the parity flips. Light meson and baryon multiplets do (roughly) alternate in parity - one of the remarkably simple and successful predictions of the quark model. Accordingly the pentaquark ground state has negative parity. This makes the existence of the @+ embarrassing for this model for several reasons: first, there is no evidence for a negative parity, non-strange @-analogue state, uudd(ti, 2) which should be lighter than the @+. Second, the ground state multiplet of qqqqq contains 1260 states. Although most will be heavy enough to disappear into the continuum, it is hard to imagine that only the antidecuplet should be seen. Finally, a negative parity @+ would have to appear in the s-wave of K N scattering, where its narrow width would be very hard t o explain. All in all, the uncorrelated quark model gives little reason to expect a light, narrow, exotic baryon with the quantum numbers of the @+. Later I will describe the somewhat more positive perspective of correlated quark models.
2.4. Early lattice results
The first lattice studies of the @+ appeared soon after the first experimental r e p o r t ~ ~ ~Both v ~ ~groups . agreed that there is evidence for the K N threshold and for a state in the uuddg J" = 1/2- channel. They also reported evidence for a state in the 1/2+ channel, but a t a higher mass. Subsequent work by the Kentucky does not find a state in either parity channel. These results are troubling: lattice calculations have, in the past, got the quantum numbers of the ground state correct in each sector of QCD. In this case the calculations support negative parity (Ref. 46 excepted), which, as we have seen, is hard to reconcile with the narrow width of the @+. Lattice studies of qqqqq are only beginning. They are still far from the chiral limit and confined to small lattices. Advocates of a positive parity @+ can hope that better sources and better approximations to the chiral limit will reverse the order of states.
200
3. Diquarks An uncorrelated quark model leads to a negative parity ground state multiplet, which contains 1/2- and 3/2- candidates for the However the very narrow width of the Of seems to be an insuperable difficulty. So a quark description of the O+ must look to some correlation to invert the naive ordering of parity supermultiplets. This is where diquarks enterl5iI6. QCD phenomena are dominated by two well known quark correlations: confinement and chiral symmetry breaking. Confinement hardly need be mentioned: color forces only allow quarks and antiquarks correlated into color singlets. Chiral symmetry breaking can be viewed as the consequence of a very strong quark-antiquark correlation in the color, spin, and flavor singlet channel: [qq]lclfo.The attractive forces in this channel are so strong that [qq]lClf0condenses in the vacuum, breaking SU(Nf)r,x SU(Nf)R chiral symmetry. The “next most attractive channel” in QCD seems to be the color antitriplet, flavor antisymmetric (which is the 3f for three light flavors), spin singlet with even parity: [qq]zc3fo+. This channel is favored by one gluon exchange and by instanton interactions. It will play the central role in the exotic drama to follow. The classification of diquarks is not entirely trivial. Operators that will create a diquark of any (integer) spin and parity can be constructed from two quark fields and insertions of the covariant derivative. We are interested in potentially low energy configurations, so we omit the derivatives. There are eight distinct diquark multiplets (in colorxflavor xspin) that can be created from the vacuum by operators bilinear in the quark field’. However, the interesting candidates can be pared down quickly: Color 6, diquarks have much larger color electrostatic field energy. All models agree that this is not a favored configuration. Odd parity diquarks require quarks to be excited relative to one another. This leaves only two diquarks consistent with fermi statistics,
e+.
(hl3)c c 4 S f ( 4Of@)) ) { 4 d %(A) 6f(S>l+(S))
(1) where A or S denotes the exchange symmetry of the preceding representation. Both of these configurations are important in spectroscopy. In what follows I will refer to them sometimes as the “scalar” and “vector” diquarks, or more suggestively, as the “good” and “bad” diquarks. Remember, though, that there are many “worse” diquarks that we are ignoring entirely.
201 Models universally suggest that the scalar diquark is lighter than the vector. For example, one gluon exchange evaluated in a quark model gives rise t o a color and spin dependent interaction that favors the scalar diquark. The matrix elements of this interaction in the “good” and “bad” diquark states are -8MoO and +8/3Moo respectively, where Moo is model dependent. To set the scale, the A-nucleon mass difference is 16MO0, so the energy difference between good and bad diquarks is $ ( M A- M N ) 200 MeV. Not a huge effect, but large enough to make a significant difference in spectroscopy. After all, the nucleon is stable and the A is 300 MeV heavier and has a width of 120 MeV!
-
N
3.1. Characterizing diquarks
The good scalar and bad vector diquarks are our principal subjects. Since the good diquarks are antisymmetric in flavor they lie in the 3 representation of S U ( 3 ) f . We will denote them by [ q l ,421 : { [u, d] [d, s] [s, a ] } when flavor is important and by when it is not. Under flavor S U ( 3 ) transformations they behave exactly like antiquarks, [u, d] c-) 3, [d,s] t--f ‘L1, [s, u] H 2. The bad diquarks are symmetric in flavor, forming the 6 representation of SU(3)f. The notation {q1,q2} : {{u,u}{u,d} (64( 4 s ) {s,s} { s , u } } will do. Although diquarks are colored states, their properties can be studied in a formally correct, color gauge invariant way on the lattice. To define the non-strange diquarks, introduce an infinitely heavy quark, Q, ie a Polyakov line. Then study the qqQ correlator with the qq quarks either antisymmetric ([u, d ] Q ) or symmetric ( { u ,d } Q ) in flavor. The results, M [ u ,d] and M { u , d } - labelled unambiguously - are meaningful in comparison, for example, with the mass of the lightest qQ meson, M ( u ) = M ( d ) . M { u , d } - M [ u ,d] is the good-bad diquark mass difference for massless quarks. It is a measure of the strength of the diquark correlation. The diquark-quark mass difference, M [ u , d ]- M ( u ) , is another. The same analysis can be applied to diquarks made from one light and one strange quark giving M [ u ,s] and M { u , s}. These mass differences are fundamental characteristics of QCD, which should be measured carefully on the lattice. In practice we can estimate these masses by replacing the infinitely heavy quark by the physical charm or bottom, or even the strange quark. The analysis is complicated by the fact that the spin interactions between the light quarks and the s, c or b quark are not negligible. Of course the scalar diquark has no spin interaction with the spectator heavy quark (Q),
202 but the vector diquark does. We parameterize it by K(Q, {ql, q z } ) , for Q = s, c, b. The light antiquark and heavy quark in a 4Q meson has a similar interaction, K(Q, q ) . In order to obtain estimates of diquark mass differences, it is necessary to take linear combinations of baryon and meson masses that eliminate these spin interactions'. Among the non-strange quarks, we obtain 1 M { u , d)lQ - M [ u ,dllQ = 3 (2M(C;)
+ M ( C Q ) )- M(AQ)
1 = M(AQ)- 4 ( M ( D Q )+3M(DG)) 1 K(Q,{ u , d ) ) = j ( M ( C ; ) - M ( ~ Q ) )
M[u,dllQ - M(u)IQ
(2)
To obtain useful information from the BQ and RQ (0 = ( Q s s ) ~ = ' / ~ ) states, it is necessary to assume that both the bad diquark mass and the spin interaction are linear functions of the strange quark mass, M { s , S } ~ Q M { u ,~ ) I Q = 2M{u, S ) ! Q and K(Q,{s, S}) + K(Q,(21, d ) ) = 2K(Q, {u, s)), amounting to first order perturbation theory in m,. With this we can deduce,
+
hf{'LL, S } / Q
- hf[u, S ] ~ Q=
2
j ( h f ( E 6 ) + h f ( C Q ) + hf(RQ))
M [ u , s ] l Q - M(s)lQ = M ( E Q )
- h f ( E Q )-
hf(z:(Q)
1
+ M ( Z b ) - Z ( M ( C Q )+ M(RQ))
1 - -4( M ( D ~ Q )+ 3 M ( D , * Q ) ) 1 K(Q,('4 s)) = - (2M(EG) - M ( ~ Q-)M ( ~ Q .) ) 6
(3)
When we substitute numbers into eqs. (2)-(3), quite a consistent picture of diquark mass differences and diquark-spectator interactions emerges: First,
M[u,d]l,- M ( u ) l s = 321 MeV M { u , d ) l , - M[u , d] J= , 205 MeV M[u,dlIc- M(u )l c = 312 MeV M{u,d)Ic - M[u,d]Ic= 212 MeV M[u,dllb - M(u)lb = 310 MeV
(4) shows that the properties of hypothetical non-strange diquarks are the pretty much the same when extracted from the charm and bottom, and
203 even strange, baryon sectors. Second,
M { u , s}lc - M [ u ,s]Ic = 152 MeV M [ u , s ] lc- M ( s ) l , = 498 MeV
(5)
shows that the diquark correlation decreases when one of the light quarks is strange. This is certainly to be expected, since it originates in spin dependent forces. As the correlation decreases, the mass difference between the scalar and vector diquarks decreases (-210+-150 MeV) and the mass difference between the scalar diquark and the antiquark increases (-310+-500 MeV). Finally,
K ( s ,{ u , d } ) = 64 MeV K ( c ,{u,d } ) = 21 MeV K ( c ,{u,s}) = 24 MeV shows that the non-strange vector diquark interaction with the spectator charm quark is significantly weaker than with a spectator strange quark, as expected from heavy quark theory. The only mildly surprising result is that the { u , s} and { u , d } vector diquarks have roughly the same interaction with the charm spectator. It will be very interesting to compare these results with further measurements in the b-quark sector and, of course, with the results of lattice calculations. 4. Diquarks and Exotics
4.1. A n overview
I want to look at exotics assuming little more than that two quarks prefer to form the good, scalar diquark when possible. States dominated by that configuration should be systematically lighter, more stable, and therefore more prominent, than states formed from other types of diquarks. This qualitative rule leads to qualitative predictions - all of which seem to be supported by the present state of experiment. This is clearly an idealization - a starting place for describing exotic spectroscopy. To learn the real dominance will require more models and more information from experiment. The qualitative ideas explored here are not powerful enough to fix the overall mass scale of any given sector in QCD. So we cannot predict the existence of (nearly) stable exotic pentaquarks. As was the case of the large N,-dynamics of Jenkins and Manohar, once a particle like the O+ is found, it sets the scale, and leads to many interesting predictions.
204
dominance are simple, and striking. The predictions that follow fro of exotic spectroscopy and provide They capture all the important fe the conceptual basis of a unified description of this sector of QCD. ould be no (light, prominent) exotic mesons: The good flavor 3, just like the antiquark. Tetraquarks, 4444, poexotics in 27,10, and 10 representations of flavor S U ( 3 ) . contains only non-exotic representations, 1 and 8, just like q @ q: q3 @ q3 = ( t j q ) l @ compared with Other diquark-antidiquark mesons are heavie in the meson-meson continuum. As described in Section 1I.C probably they are not just “broad”, but in fact absent43. [b] The only prominent tetraquark mesons should be an SU(3) nonet with Jn = Of. This prediction - a simple corollary of the one just above - dates back to the late 1 9 7 0 ’ ~Since ~ ~ . the good diquarks, bosons, the spinparity of the lightest nonet is J” = O+. Over the years evidence has accumulated that the nine O+-mesons with masses below 1 GeV (the f0(600), ~ ( 8 0 0 )f0(980), , and ~ ( 9 8 0 )have ) important tetraquark componentsz3. Space does not permit me to present the evidence here. The interested reader can find more in Ref. [c] If there are any exotic pentaquark baryons, they lie in a positive parity of SU(3)f. This is also a simple consequence of combining good diquarks. To make pentaquarks it is necessary to combine two diquarks and an antiquark. The result is 3€338 3 = 1 @ 8 @ 8 @ 10. The only exotic is the 10. Other exotic flavor multiplets, like the 27 and 35, which occur in the uncorrelated quark picture and/or the chiral soliton models, should be heavier and most likely lost in the meson-baryon continuum. [d] Nuclei will be made of nucleons. To a good approximation, nuclei are made of nucleons - a fact which QCD should explain. If diquark correlations dominate, systems of 3A quarks should prefer to form individual nucleons, not a single hadron. The argument is based on statistics: Good diquarks are spinless color anti-triplet bosons. Only one, [u,d], is non-strange. A six-quark system made of three of these, antisymmetrized in color to make a color singlet, would have to have fully antisymmetric space-wavefunction to satisfy Bose statistics. The simplest would be a triple-scalar product, $1 . $2 x $3, which should be much more energetic than two separate, color-singlet nucleons in an s-wave (eg. the deuteron). The argument generalizes to heavy nuclei. Of course it does not explain nuclear binding or the rich phenomena of nuclear physics.
’.
205 4.2. Pentaquarks from diquarks I: The general idea
Again, we assume that the scalar diquark dominates the spectrum. The rest follows from rather simple considerations of the symmetry of the function in color, flavor, and space (the spi function is trivial)15J6. The good diquark is a spinless boson so the wavefunction must be symmetric under interc The two diquarks must couple to a color 3,, wavefunction is antisymmetric in color. Two choices remain: It can be (a) antisymmetric in flavor and symmetric in space; or (b) symmetric in flavor and antisymmetric in space. Symmetric in flavor means 6 and antisymmetric means 3: p C3 = p 8 A = 3. Symmetry in space means even parity and a tower of states presumably beginning with e = 0. Antisymmetry in space means odd parity and a tower beginning with l= 1. So the candidates for light pentaquarks in the diquark scheme fall into two categories: (a) A negative parity nonet with J" = 1/2-. Overall the four quarks carry no spin and form a flavor triplet. [The three states are [[u, d], [d, s]], [[d,s ] , [s,u],and [[s, u], [u, 611.3 They combine with the antiquark to form nine states with J" = 1/2+; and (b) A positive parity 18-plet (an octet and antidecuplet) with J" = 1/2+ & 3/2+. With e = 1 the four quarks couple to the antiquark to make J" = 1/2-, or 3/2-. The six four quark states, [u, d2,{[u, d], [u, s ] } , etc, combine with the antiquark to make eighteen states that include both a flavor octet and antidecuplet. The quark content of the eighteen states is summarized in Fig. 1, where ideal mixing has been assumed. In Fig. 1 the SU(3)f weight diagrams of the unmixed octet and antidecuplet are shown on the left. The results after ideal mixing are shown on the right. The exotics in the antidecuplet do not mix with the octet. Isospin symmetry precludes mixing between the A and the Cos or between the Eoy- and the a0>-. The other states, the N's and the C's, mix to diagonalize the number of 5s pairs. One set has hidden strangeness, the other does not. Which multiplet, the odd parity nonet or the even parity 18-plet, is lighter depends on the quark model dynamics. This is exactly the same question we encountered in the large N, classification of Jenkins and Manohar. Color-spin interactions modeled after one gluon exchange favor the s-wave, ze the odd parity nonet, but there may be Pauli blocking in this state as in the H . This effect would elevate the mass of the negative parity nonet. Flavor-spin interactions, modeled after pseudoscalar meson exchange48,
q
s,
q
206
I
*+
1
(‘1)
Octe\ and Anticlccupirt
(b) Ideally hlivcd Quark Contcnt
Figure 1. Even parity pentaquark 18-plet: diquark pairs in the Gf combine with an The S U ( 3 ) weight diagram for the 8fand antiquark in the & to make a 8fand i&. is shown at left, where the unmixed states are named (the decuplet in black, the octet in grey). The ideally mixed states, some with their valence quark content, are shown a t right. The exotics (O+, @--, and @+)and certain octet states (A, So, Z-) do not mix if isospin is a good symmetry.
mf
apparently favor the pwave (in contrast to the qq, qqq, and qQq sectors where the ground state is always the s-wave), making the even-parity 18plet the lightest. Whichever way, the diquark picture leads to clear predictions for the light pentaquarks: The only potentially light, prominent exotic multiplet is the antidecuplet, which contains candidates for the O+, the @--, and an as yet unreported @+. The exotics are accompanied by an non-exotic octet, which mixes with the antidecuplet to give several non-exotic (or “cryptoexotic”) analogue states, for example a [u,dl[u,d&and [u,dl[u,d]dpair, which should be lighter than the O+. There are no other light, prominent exotics, like the 27 that figures prominently in the chiral soliton model. The O+ should have positive parity. The exotics should come in spin-orbit pairs with J“ = and %.
207 More predictions include SU(3)f mass splittings and the existence of charm and bottom analogue states discussed in detail in Ref. I will have little further to say about the negative parity nonet. These states couple strongly to the meson-nucleon s-wave. The non-strange members of the multiplet contain an Ss pair and should therefore couple to N q and A K , not to N n . Unless these states were below fall apart decay threshold they would be lost in the meson-nucleon continuum. The absence of candidates in the PDG tables should not be surprising. A word about complications that I have ignored in this presentation: First are the states constructed from the other diquarks: Residual interactions will certainly mix them into the Qq states, but a t zeroth order the goodxbad and badxbad states are -200 and -400 MeV heavier than the goodxgood states. If the lightest states in each family are the s-waves - as QCD based interactions prefer -then these states are all well above threshold to fall apart into meson and baryon, and disappear into the continuum. Among them are many exotics, but only one candidate for a negative parity antidecuplet. Goodxbad states lie in the 3@6@3 of SU(3)f which includes the exotic 27. Badxbad states lie in the 6 @ 6 @ 5 and include a negative parity antidecuplet (as well as the 35). So the first candidate for a negative parity 0’ lies in the “bad-bad” sector and furthermore is created by the same operator that creates K N in an s-wave1. So the diquark picture is quite firm that a negative parity O+ is much heavier and strongly coupled to the K N s-wave continuum. Second is mixing between qqqqq states and ordinary qqq baryons. Mixing is possible when the qqqqq states are not exotic, especially if there are qqq states with the same quantum numbers nearby. Mixing will alter both the spectrum and the decay widths that would otherwise be determined by S U ( 3 ) flavor symmetry.
’.
4.3. Pentaquarks from diquarks 11: A mom detailed look at the positive parity octet and antidecuplet
If the O+ and its brethren are confirmed, and if they have positive parity, then the diquark based pentaquark picture seems like a strong candidate for a quark description of the structure. The diquark model predictions for masses, mixings, and flavor selection rules can be found in Refs. and are reviewed in Ref. l. The spectrum is summarized in Table 2. The SU(3) violating effects of 15747,49
208
the s-quark mass have been included to lowest order in perturbation theory, which has been perfectly adequate for all qqq baryons and Qq mesons in the past. For reasons discussed in Ref. it is reasonable, to lowest order, to assume that the symmetry breaking Hamiltonian acts independently in the four quark 6 and the antiquark 3. This leaves a mass formula that depends mass, p the matrix on only three parameters, Mo, the unperturbed 8 and element of m,ss, and a , the mass difference M [ u , s ]- M[u,d]. In Ref. l5 (labelled Mass I in the table) p was taken from the @-Ropermass difference. a was taken from a full quark model analysis of the C - A mass difference, which goes beyond the diquark hypothesis, and led to a prediction of 1750 MeV for the Q,--. At the time the prediction of a relatively light Q, was rather daring. Now that the Q,-- has reported a t 1860 MeV, it seemed appropriate to re-examine these predictions and assignments. In retrospect taking a from the A-C system may not have been particularly appropriate. The value extracted in Ref. l5 assumed the full color-spin Hamiltonian of the quark model. Instead in Ref. 49 Wilczek and I propose to identify C(l660), a 1/2+ resonance given three stars by the PDG12, with the C states in the 18-plet. This choice is motivated by a global fit to baryon resonances5’ and is labelled “Mass II” in Table 2.
There is an important, qualitative difference between the diquark picture with its 18-plet and other models of the @+ with an antidecuplet alone, which will help sort out the correct physical picture of the exotics. In an antidecuplet-only picture the @+, NE, CE, and Q, must be spaced a t equal mass intervals. In their original paper5 Diakonov, Petrov, and Polyakov identify the Nm with the N(1710), which puts the Q, at 2070 MeV, much higher than the quark content (uudds + ddssti) would suggest15. If the
209 @ lies at 1860, the N m and Cm must lie near 1650 MeV and 1750 MeV respectively. In their revised discussion of the spectrum, Diakonov and Petrov, fit the @(1860)and predict 1/2+ N and C resonances in the intervals 1650-1690 and 1760-1810 respectivelys1. In contrast with the 1 - o n l y , the 18-plet picture suggested by diquark arguments allows the O+ and to be interior to the multiplet, with the C,(uussS) and N(uuddG) at the top and bottom respectively. As noted in the Table, there are possible candidates for all the 18-plet states.' Time will tell which of these qualitatively different spectra are closest to Nature - provided, of course, the exotics survive the next round of experiments. 4.4. Pentaquark from diquarks 111: Charm and bottom
analogues Charm and bottom analogues of the O+ can be obtained by substituting the heavy or 6 quark for the 3 in the O+, @: =
I[u,dl"L1,dlc)
@- = I[% dlb, 46)
(7)
=
The existence of the 0: O+(1540) fixes the mass scale for exotics and leads to rather robust predictions of the masses of the O: and O t . The simplest, though not necessarily the least accurate approach, is to find an Q system. The obvious analogy among qqQ baryons and apply it to the choice is the AQ, which has the quark content The heavy anti-quark inglet, color 3,spin singlet pair in the OQ sits in the background of an sits in a background identical of diquarks. The heavy quark in the in isospin, color, and spin. The only difference is that the spin of the 0 in the OQ can interact with the orbital angular momentum (! = 1) in the OQ and this interaction is not would expect the relations M ( is M ( @ ) - M ( O $ ) = M ( A , ) - M ( A ) to be nearly exact. QCD spin-orbit interactions are not strong, so these rules should not be badly violated. The differences among the predictions of various QCD based quark models reflect the different ways that the residual interactions are treated. From the relation quoted Wilczek and I estimated, Ad(@:) = 2710 MeV and M ( O z ) = 6050 MeV. CAlthough the width of the Roper presents a problem5', suggesting that non-exotic qqqqg states may mix significantly with qqq states.
210
If these estimates are correct, the 0: and 0: will be stable against strong decay. The lightest strong decay channel for the 0: is N D with a threshold at 2805 MeV, and for the @ ,: it is N B with a threshold at 6220 sec. MeV. They would have to decay weakly with lifetimes of order How did this happen? The 0: is light, but it is not stable. The reason lies not in the linear scaling of the masses of the heavy pentaquarks with the heavy quark mass, but rather in the non-linear scaling of the pseudoscalar meson masses, which determine the strong decay thresholds. Consider the four analogue states, [u,d][u,d]Q, with Q = u , s , c , ~and , identify the 00, with the Roper as I advocated earlier. Then
+ N7r 0: + N K 0: + N D 0: + N B 0;
has decay Q-value
Q
M
350 MeV
has decay Q-value
Q
M
100 MeV
has decay Q-value
Q
M
-100 MeV
has decay Q-value
Q x -150 MeV
(8)
The 0;, ie the Roper, is unstable because the pion is anomalously light, a consequence of approximate chiral symmetry. The effect is still significant enough for the kaon to make the @$unstable. The D and B-meson masses are not significantly lowered by chiral symmetry, the thresholds are proportionately higher, and the @: and @: are stable. The details are model dependent. Other model estimates are generally higher than the simple scaling law described here53, some predict stable c and b-exotics, others predict light and narrow, but not stable states. The exotic charm baryon reported by H1 is not bound. With a mass of 3099 MeV, it is much too heavy to be the 0: as I have described it. The width is reported to be less than 12 MeV. It has been observed through its strong decay into D*-p, into which it has a Q value of 150 MeV. If it were the @!, and if it has J" = 1/2+, it would have a significant decay into D-p (which would not have been seen at H l ) , with a partial width that can be related to the width of the 0: by scaling pwave phase space. The result is l?(3099)/r(0$) M 15, barely consistent with the H1 limit if the width of the 0: is 1 MeV. An interesting possibility - if the 3099 state should be confirmed - is that it is an L = 2 Regge excitation of the 0: with J" = 312-. This object can decay into D*-p in the s-wave, but D-p in the d-wave, accounting perhaps for its surprisingly narrow width. Should this assignment prove correct, there must be many other excited charm exotic baryons awaiting discovery. Clearly, if the initial reports are confirmed, there is a fascinating spec-
21 1
troscopy of heavy exotic baryons awaiting us. But it is a big “if”!
5 . Conclusions
There are two distinct, but related issues at the core of this discussion: first, a question: are there light, prominent exotic baryons, and if so, what is the best dynamical framework in which to study them? and second, a proposal: diquark correlations are important in QCD spectroscopy, especially in multiquark systems, where they account naturally for the principal features. I believe the case for diquarks is already quite compelling. There are many projects ahead: re-evaluating the 444 spectrum5’; systematically exploring the role of diquarks in deep inelastic distribution and fragmentation functions, and in scaling violation; seeing if diquarks can help in other areas of hadron phenomenology like form-factors, low p~ particle production, and polarization phenomena; developing a more sophisticated treatment of quark correlations, recognizing that diquarks are far from pointlike inside hadrons; establishing diquark parameters and looking for diquark structure in hadrons using lattice QCD; and - the holy grail of this subject -seeking a more fundamental and quantitative phenomenological paradigm for light quark dynamics at the confinement scale. Diquark advocates have considered many of these issues in the pastlg. No doubt many other important contributions, like the diquark analysis of the AI = 1 / 2 - r ~ l e have ~ ~ , already been accomplished. We can hope eventually to have as sophisticated an understanding of diquark correlations as we have of iiq correlations, as expressed in chiral dynamics. The situation with the O+ is less clear. Of course it will eventually be clarified by experiment - a virtue of working on QCD as opposed to string theory! However, theorists’ attempts to understand the O+ have raised more questions than they have answered. To wit: (a) A negative parity ( K N s-wave) O+ is intolerable to theorists, but that is what lattice studies find, if they find anything at all. (b) No one has come up with a simple, qualitative explanation for the exceptionally narrow width of the O+. (c) The original prediction of a narrow, light O+ in the chiral soliton model does not appear to be robust. (d) Quark models can accomodate the O+, but only by reversing the naive, and heretofore universal, parity of the qnqQllq- ground state. It is necessary to excite the quarks in order to capture the correlation energy of the good diquarks. This does not sound like a way to make an exceptionally light and stable pentaquark. (e) When
212
models are adjusted to accomodate the O+, they predict the existence of other states that should have been observed by now: The diquark picture wants both a of+and a O$+; the CSM and large N, want a relatively light 27, which includes an I = 1 triplet: O*O, O*+, O*++. None of these problems seems insuperable. Indeed, there are papers appearing every day that propose an interesting solution to one or another. Taken together, however, they are an impressive set. They leave us in limbo: Either the O+ will go away, or it will force us to rewrite several chapters of the book on QCD. 6. Acknowledgements
Many of these ideas were developed in collaboration with Frank Wilczek. This work is supported in part by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818.
References 1. R. L. Jaffe, hep-ph/0409065. 2. M. Roos et al, “Review Of Particle Properties. Particle Data Group,” Phys. Lett. B 111,1 (1982). 3. G. P. Yost et al, “Review Of Particle Properties: Particle Data Group,” Phys. Lett. B 204, 1 (1988). 4. A. V. Manohar, Nucl. Phys. B 248, 19 (1984); M.Chemtob; L. C. Biedenharn and Y . Dothan, Print-84-1039 (DUKE) in E. Gotsman, G. Tauber, From SU(3) To Gravity p. 15; M. zalowicz, in Skyrmions and Anomalies, M. Jezabek and M. Praszalowicz, eds., World Scientific (1987), p. 112;H. Walliser, Nucl. Phys. A 548 649 (1992). 5. D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359, 305 (1997) [arXiv:hep-ph/9703373]. 6. H. Weigel, Eur. Phys. J. A 2, 391 (1998) [arXiv:hep-ph/9804260]. 7. T. Nakano et a1 [LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003) [arXiv:hep-ex/0301020]. 8. For references, see for example, Ref. 9. C. Alt e t a1 “A49 Collaboration], Phys. Rev. Lett. 92, 042003 (2004) [arXiv:hep-ex/0310014]; but see also H. G. Fischer and S. Wenig, arXiv:hepex/0401014 and K. T. Knopfle, M. Zavertyaev and T. Zivko [HERA-B Collaboration], J. Phys. G 30,S1363 (2004) [arXiv:hep-e~/0403020]. 10. A. Aktas e t a1 [Hl Collaboration], arXiv:hep-ex/0403017, but see also K. Lipka [Hl Collaboration], arXiv:hep-ex/0405051. 11. For a recent overview of the experimental situation, see G. Trilling in Ref. l 2 12. S. Eidelman et al [Particle Data Group Collaboration], “Review of particle physics,” Phys. Lett. B 592, 1 (2004).
’.
213 13. For a recent summary of negative results see A. Dzierba, talk presented at at the International Conference on Quarks and Nuclear physics, QNP2004, http://k9.physics.indiana.edu/.ueric/QNP/QNP /QNP-2004-talks/plenary-Friday /dzierba5q-qnp2004.pdf. 14. R. L. Jaffe and K. Johnson, Phys. Lett. B 60,201 (1976). R. L. Jaffe, Phys. Rev. D14 267, 281 (1977). 15. R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003) [arXiv:hepph/0307341]. 16. S. Nussinov, arXiv:hep-ph/0307357. 17. M. Karliner and H. J. Lipkin, arXiv:hep-ph/0307243. 18. M. Ida and R. Kobayashi, Prog. Theor. Phys. 36 (1966) 846; D.B. Lichtenberg and L.J. Tassie, Phys. Rev. 155 (1967) 1601. 19. For a review an further references, M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson and D. B. Lichtenberg, Rev. Mod. Phys. 65, 1199 (1993), or M. Anselmino, E. Predazzi, eds. International Workshop on Diquarks and Other Models of Compositeness: Diquarks III, Turin, Italy, 28-30 Oct 1996 (World Scientific, 1998). 20. M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B 422,247 (1998) [arXiv:hep-ph/9711395]; for a review with further references, see K. Rajagopal, “Color superconductivity,” Prepared f o r Cargese S u m m e r School o n Q C D Perspectives o n Hot and Dense Matter, Cargese, France, 6-18 Aug 2001. 21. See, for example, F. E. Close and A. W. Thomas, Phys. Lett. B 212, 227 (1988). 22. M. Neubert and B. Stech, Phys. Lett. B 231,477 (1989), Phys. Rev. D 44, 775 (1991). 23. C. Amsler and N. A. Tornqvist, Phys. Rept. 389, 61 (2004); F. E. Close and N. A. Tornqvist, J. Phys. G 28, R249 (2002) [arXiv:hep-ph/0204205]; For a recent reconsideration, see L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, arXiv:hep-ph/0407017. 24. R. N. Cahn and G. H. Trilling, Phys. Rev. D 69,011501 (2004) [arXiv:hepph/0311245]. 25. R. A. Arndt, I. I. Strakovsky and R. L. Workman, Phys. Rev. C 68,042201 (2003) [Erratum-ibid. C 69,019901 (2004)] [arXiv:nucl-th/0308012]. 26. A. Sibirtsev, J. Haidenbauer, S. Krewald and U. G. Meissner, arXiv:hepph/0405099. 27. See, for example, T. Nakano, talk presented at the International Conference on Quarks and Nuclear physics, QNP2004, http: //www.qnp2004.org/5q~talks/T_Nakano.ppt. 28. S. Okubo, Phys. Lett. 5, 165 (1963); G.Zweig, CERN Report No. 8419 T H 412, 1964 (unpublished); reprinted in Devel- opments in the Quark Theory of Hadrons, edited by D. B. Lichtenberg and S. P. Rosen (Hadronic Press, Massachusetts, 1980); J. Iizuka, Prog. Theor. Phys. Suppl. 37,21 (1966). 29. This section is based on R. L. Jaffe and A. Jain, arXiv:hep-ph/0408046, where more details can be found . 30. G. ’t Hooft, Nucl. Phys. B 72,461 (1974).
214 31. E. Witten, Nucl. Phys. B 160,57 (1979). 32. R. F. Dashen, E. Jenkins and A. V. Manohar, Phys. Rev. D 49,4713 (1994) [Erratum-ibid. D 51,2489 (1995)l [arXiv:hep-ph/9310379]. 33. E. Jenkins and A. V. Manohar, Phys. Rev. Lett. 93, 022001 (2004) [arXiv:hep-ph/0401190]; JHEP 0406,039 (2004) [arXiv:hep-ph/0402024]. 34. G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B 228,552 (1983). 35. E. Guadagnini, Nucl. Phys. B 236,35 (1984). 36. C. G. Callan and I. R. Klebanov, Nucl. Phys. B 262,365 (1985). 37. Ref. quotes a width of order 15 MeV, however the calculation on which that claim is based contains an error, which corrected yields an estimate of -30 MeV 38. 38. R. L. Jaffe, Eur. Phys. J. C 35, 221 (2004) [arXiv:hep-ph/0401187]; D. Diakonov, V. Petrov and M. Polyakov, arXiv:hep-ph/0404212; R. L. Jaffe, arXiv:hep-ph/0405268. 39. T. D. Cohen, Phys. Lett. B 581, 175 (2004) [arXiv:hep-ph/0309111], Phys. Rev. D 70, 014011 (2004), arXiv:math-ph/0407031. For further discussion, see A. Cherman, T. D. Cohen and A. Nellore, arXiv:hep-ph/0408209. 40. N. Itzhaki, I. R. Klebanov, P. Ouyang and L. Rastelli, Nucl. Phys. B 684,264 (2004) [arXiv:hep-ph/0309305]; I. R. Klebanov and P. Ouyang, arXiv:hepph/0408251. 41. M. P. Mattis and M. Karliner, Phys. Rev. D 31, 2833 (1985), Phys. Rev. Lett. 56,428 (1986), Phys. Rev. D 34,1991 (1986). 42. R. L. Jaffe, K. Johnson and Z. Ryzak, Annals Phys. 168,344 (1986). 43. R. L. Jaffe and F. E. Low, Phys. Rev. D 19,2105 (1979). For a pedagogical introduction, see R. L. Jaffe, “HOWto analyse low energy scattering”, in H. Guth, K. Huang, and R. L. Jaffe, eds. Asymptotic Realms of Physics, Essays in Honor of Rancis E. Low, (MIT Press, Cambridge, 1983). 44. S. Sasaki, arXiv:hep-lat/0310014. 45. F. Csikor, Z. Fodor, S. D. Katz and T. G. Kovacs, JHEP 0311,070 (2003) [arXiv:hep-lat/0309090]. 46. N. Mathur e t al, arXiv:hep-ph/0406196. 47. R. Jaffe and F. Wilczek, arXiv:hep-ph/0401034. 48. F. Stancu and D. 0. Riska, Phys. Lett. B 575, 242 (2003) [arXiv:hepph/0307010]. 49. R. Jaffe and F. Wilczek, Phys. Rev. D 69, 114017 (2004) [arXiv:hepph/0312369]. 50. A. Selem and F. Wilcxek, to be published. 51. D. Diakonov and V. Petrov, Phys. Rev. D 69, 094011 (2004) [arXiv:hepp h/ 03 102 121. 52. T. D. Cohen, arXiv:hep-ph/0402056. 53. For references and a review of heavy exotic baryons and their excitations, see K. Maltman, arXiv:hep-ph/0408145.
QUARK STRUCTURE OF CHIRAL SOLITONS
DMITRI DIAKONOV Thomas Jefferson National Accelerator Facility, Newport News, V A 23606, USA NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark St. Petersburg Nuclear Physics Institute, Gatchina, 188 900, St. Petersburg, Russia There is a prejudice that the chiral soliton model of baryons is something orthogonal to the good old constituent quark models. In fact, it is the opposite: the spontaneous chiral symmetry breaking in strong interactions explains the appearance of massive constituent quarks of small size thus justifying the constituent quark models, in the first place. Chiral symmetry ensures that constituent quarks interact very strongly with the pseudoscalar fields. The “chiral soliton” is another word for the chiral field binding constituent quarks. We show how the old S U ( 6 ) quark wave functions follow from the %oliton”, however, with computable relativistic corrections and additional quark-antiquark pairs. We also find the 5-quark wave function of the exotic baryon O+.
1. The necessity of quantum field theory
It has been known since the work of Landau and Peierls (1931) that the quantum-mechanical wave function description, be it non-relativistic or relativistic, fails a t the distances of the order of the Compton wave length of the particle. Measuring the electron position with an accuracy better than cm produces a new electron-positron pair, by the uncertainty principle. One observes it in the Lamb shift and other radiative corrections. Fortunately, the atom size is cm, therefore there is a gap of three orders of magnitude where we can successfully apply the Dirac or even the Schrodinger equation. In baryons, we do not have this luxury. Measuring the quark position with an accuracy higher than the p i o n Compton wave length of 1.4fm produces a pion, i.e. a new quark-antiquark (QQ) pair, whereas the baryon size is 0.8 fm. Therefore, there seems t o be no room for the quantum-mechanical wave function description of baryons a t all. To describe baryons, one needs a quantum field theory from the start, with a varying number of QQ pairs, because of the spontaneous chiral symmetry breaking which makes pions light.
215
216
Ignoring quantum field theory where it cannot be ignored, causes multiple problems. Let me mention just two paradoxes of the standard constituent quark models, out of many. The first is the value of the so-called nucleon sigma term It is experimentally measured in low-energy nN scattering, and its definition is the scalar quark density in the nucleon, multiplied by the current (or bare) quark masses,
’.
mu
+m d
+
< Nlau &IN >= 67 f 6 MeV. 2 The standard values of the current quark masses are mu N 4MeV, md N 7MeV (and m, 2: 150 MeV). In the non-relativistic limit, the scalar density is the same as the vector density; therefore, in this limit the matrix element above is just the number of u, d quarks in the nucleon, equal to 3. If u,d quarks are relativistic, the matrix element is strictly less than three. Hence, in the naive constituent quark model o=
cquarks
MeV * 3 = 17.5 MeV, 5 MeV -I2
that is four times less than experimentally! Three quarters of the (T term is actually residing not in the three constituent quarks but in the additional quark-antiquark pairs in the nucleon. The second paradox which is probably less known, arises when one attempts t o extract quark distributions as function of Bjorken z from a constituent quark model, be it any variant of the bag model or any variant of the potential models with any kind of correlations between quarks. If the three quarks are loosely bound, their distribution function is just 6 (z each quark carrying 1/3 of the nucleon momentum in the infinite momentum frame. As quarks become more bound, this &function is smeared around 1/3. However, higher quark velocities imply that the “lower” component of the Dirac bispinor wave function increases (it is zero in the extreme non-relativistic case), a t the expense of the decrease of the “upper” component. It means that if quarks are moving inside a nucleon, there are less than three quarks in the nucleon. Since the number of quarks minus the number of antiquarks is the conserved baryon number, it automatically means that the number of antiquarks is negative ’. It is a n inevitable mathematical consequence of the Dirac equation. The paradox is cured by adding the Dirac sea t o valence quarks; only then the antiquark distribution becomes positive-definite, and satisfies the general sum rules Thus, a field-theoretic description of baryons is a must if one does not wish to violate general theorems, and also for practical reasons.
i),
’.
217
I present below a relativistic field-theoretic model of baryons where the above paradoxes are resolved, together with the well-known “spin crisis” paradox. Actually, one has to be surprised not by why the constituent quark approach is a failure but rather why does it work a t all in a variety of cases. The model will answer this question, too. 2. The chiral quark - soliton model The most important happening in QCD from the point of view of the light hadron structure is the Spontaneous Chiral Symmetry Breaking (SCSB): as its result, almost massless u , d, s quarks get the dynamical momentumdependent masses MU,d,+(p),and the pseudoscalar mesons 7r,K , q become light (pseudo) Goldstone bosons. At the same time, pseudoscalar mesons are themselves bound QQ states. How to present this queer situation mathematically? There is actually not much freedom here: the interaction of pseudoscalar mesons with constituent quarks is dictated by chiral symmetry. It can be written in the following compact form 3:
L,ff = 4 [i@-
M exp(i y5 7rAXA/F,)] q,
7rA
= 7 r , K , 77.
(1)
Since Eq.(l) is an effective low-energy theory, one expects formfactors in the constituent quark - pion interaction; in particular, M ( p ) is momentumdependent and provides an UV cutoff. In fact, Eq.( 1)is written in the limit of zero momenta. A possible wave-function renormalization factor Z ( p ) can be also admitted but it can be absorbed into the definition of the quark field. Notice, that there is no kinetic-energy term for pseudoscalar fields in Eq.(l). It is in accordance with the fact that pions are not “elementary” but a composite field, made of constituent quarks. The kinetic energy term (and all higher derivatives) for pions appears from integrating out quarks, or, in other words, from quark loops, see Fig. 1.
Figure 1. The effective chiral lagrangian is the quark loop in the external chiral field, or the determinant of the Dirac operator (1). Its real part is the kinetic-energy term for pions, the Skyrme term and, generally, an infinite series in derivatives of the chiral field. Its imaginary part is the Wess-Zumino-Witten term (with the correct coefficient), plus also an infinite series in derivatives *.
218
An interesting question is, how does the effective lagrangian (1) “know” about the confinement of color? One writes Eq.(l) from the general chiral symmetry considerations, and only the formfactors e.9. the dynamical mass M ( p ) are subject to dynamical details. The difference between a confining and a non-confining theory is hidden in the subtleties of the analytical behavior of M ( p ) and possible other formfactor functions in the Minkowski domain of momenta. Specifically, the instanton model of the spontaneous chiral symmetry breaking leads to such M ( p ) that there is no real solution of the mass-shell equation p 2 = M 2 ( - p 2 ) , meaning that quarks cannot be observable, only their bound states! However, this is not the only confinement requirement. Unfortunately, the instanton model’s M ( p ) has a cut at p 2 = 0 corresponding to massless gluons left in the model. In the true confining theory there should be no such cuts. In the bound states problems, however, quarks’ momenta are space-like. Therefore, one can use any reasonable falling function M ( p ) reproducing the phenomenological value of F, constant and of the chiral condensate ‘. As a matter of fact, instantons do it phenomenologically very satisfactory. Constituent u,d, s quarks necessarily have to interact with the 7r, K , 77 fields according to Eq.(l), and the dimensionless coupling constant is actually very large: gTqq(0)= M(O) 21 4, where the constituent quark mass F, M ( 0 ) N_ 350MeV and F, N_ 93MeV are used. The chiral interactions of constituent quarks in baryons, following from Eq.(l), are schematically shown in Fig. 2. Antiquarks are necessarily present in the nucleon as pions propagate through quark loops. The nonlinear effects in the pion field are essential since the coupling is strong. I would like to stress that this picture is a model-independent consequence of the spontaneous chiral symmetry breaking. One cannot say that quarks get a constituent mass but throw away their strong interaction with the pion field. In principle, one has to add perturbative gluon exchange on top of Fig. 2. However, a , is never really strong, such that gluon exchange can be disregarded in the first approximation. The large value of the pion-quark coupling suggests that Fig. 2 may well represent the most essential forces
Figure 2. Quarks in the nucleon (solid lines), interacting via pion fields (dash lines).
219
inside baryons. No ‘‘confiningstrings” are expected in the real world where it is energetically favorable to break an expanded string by creating light pions. Although the low-momenta effective theory (1) is a great simplification as compared to the microscopic QCD, as it uses the right degrees of freedom appropriate a t low energies, it is still a strong-coupling relativistic quantum field theory. Summing up all interactions inside the nucleon of the kind shown in Fig. 2 is a difficult task. Maybe some day it will be solved numerically, e.g. by methods presented by John Hiller in these Proceedings ‘. In the meanwhile, it can be solved exactly in the limit of large number of colors N,. With N, colors, the number of constituent quarks in a baryon is N,, and all quark loop contributions in Fig. 2 are also proportional t o N,. Therefore a t large N,, quarks inside the nucleon create a large, nearly classical pion field: quantum fluctuations about the mean field are suppressed as l / N c . The same field binds the quarks; therefore it is called the self-consistent field. [A similar idea is exploited in the shell model for nuclei and in the Thomas-Fermi approximation to atoms.] The problem of summing up all diagrams of the type shown in Fig. 2 is reduced to finding a classical self-consistent pion field. As long as l / N c corrections to the mean field results are under control, one can use the large-N, logic and put N , to its real-world value 3 a t the end of the calculations. The model of baryons based on these approximations has been named the Chiral Quark Soliton Model (CQSM) ‘. The “soliton” is another word
-
- - _ - - - - - ---_, a_ _ _ - - _ - - _ _ _ -
-- - --
-
i* BI
E-sea
r
i
m
--
E = -lM
Dirac sea
Figure 3. If the trial pion field is large enough (shown schematically by the solid curve), there is a discrete bound-state level for three ‘valence’ quarks, Eval.One has also to fill in the negative-energy Dirac sea of quarks (in the absence of the trial pion field it corresponds to the vacuum). The continuous spectrum of the negative-energy levels is shifted in the trial pion field, its aggregate energy, as compared to the free case, being Es,,,. The nucleon mass is the sum of the ‘valence’ and ‘sea’ energies, multiplied by three The self-consistent pion field binding quarks colors, M N = 3 (E,,l[n(z)] Esea[n(z)]). is the one minimizing the nucleon mass.
+
220 for the self-consistent pion field in the nucleon. However, the model operates with explicit quark degrees of freedom, which enables one to compute any type of observables, e.g. relativistic quark (and antiquark!) distributions inside nucleons 2 , and the quark light-cone wave functions ‘. In contrast to the naive quark models, the CQSM is relativistic-invariant. Being such, it necessarily incorporates quark-antiquark admixtures to the nucleon. Quark-antiquark pairs appear in the nucleon on top of the three valence quarks either as particle-hole excitations of the Dirac sea (read: mesons) or as collective excitations of the mean chiral field. There are two instructive limiting cases in the CQSM: 1. Weak T ( Z ) field. In this case the Dirac sea is weakly distorted as compared to the no-field and thus carries small energy, E,,, 21 0. Few antiquarks. The valence-quark level is shallow and hence the three valence quarks are non-relativistic. In this limit the CQSM becomes very similar to the constituent quark model remaining, however, relativistic-invariant and well defined. 2. Large T ( X ) field. In this case the bound-state level with valence quarks is so deep that it joins the Dirac sea. The whole nucleon mass is given by E,,, which in its turn can be expanded in the derivatives of the mean field, the first terms being close to the Skyrme lagrangian. Therefore, in the limit of large and broad pion field, the model formally reduces to the Skyrme model. The truth is in between these two limiting cases. The self-consistent pion field in the nucleon turns out to be strong enough to produce a deep relativistic bound state for valence quarks and a sufficient number of antiquarks, so that the departure from the non-relativistic constituent quark model is considerable. At the same time the self-consistent pion field is spatially not broad enough to justify the use of the Skyrme model which is just a crude approximation to the reality, although shares with reality some qualitative features. The CQSM demystifies the main paradox of the Skyrme model: how can one make a fermion out of a boson-field soliton. Since the %oliton” is nothing but the self-consistent pion field that binds quarks, the baryon and fermion number of the whole construction is equal to the number of quarks one puts on the valence level created by that field: it is three in the real world with three colors.
3. Baryon excitations There are excitations related to the fluctuations of the chiral field about its mean value in the baryons. In the context of the Skyrme model many
221 resonances were found and identified with the existing ones in Ref. 7,8 and quite recently in Ref. '. As I said before, the Skyrme model is too crude, and one expects only qualitative agreement with the Particle Data. The same work has to be repeated in the CQSM but it has not been done so far. There are also low-lying collective excitations related to slow rotation of the self-consistent chiral field as a whole both in ordinary and flavor spaces. The result of the quantization of such rotations was first given by Witten lo. The following SU(3) multiplets arise as rotational states of a chiral soliton: , (10, , , (27, , (27,;') ... They
$+) (m, a+)
(8,i')
:+)
are ordered by increasing mass, see Fig. 4. The first two (the octet and the decuplet) are indeed the lowest baryons states in nature. They are also the only two that can be composed of three quarks. However, the fact that one can manage to obtain the correct quantum numbers of the octet and the decuplet combining only three quarks, does not mean that they are made of three quarks only. The difficulties of such an interpretation have been mentioned in the beginning.
__
4
_ _ _ _ .z
..__ .__ &-. _ ._ 4_ ..._. _ _ ._ _ 4_ _
f
-=
,
,
I:
z+ Q
_-=i - , , - =o
i
-
=+
n(8,1/2)
(10:3/2)
(iO,l/2,
Figure 4. The lowest baryon multiplets which can be interpreted as rotational states in ordinary and 3-flavor spaces, shown in the Y - T3 axes.
Therefore, one should not be a priori confused by the fact that higherlying multiplets cannot be made of three quarks: even the lowest ones are not. A more important question is where to stop in this list of multiplets. Apparently for sufficiently high rotational states the rotations become too fast: the centrifugal forces will rip the baryon apart. Also the radiation of pions and kaons by a fast-rotating body is so strong that the widths of the corresponding resonances blow up Which precisely rotational excitation is the last to be observed in nature, is a quantitative question: one needs to compute their widths in order to make a judgement. If the width turns out to be in the hundreds of MeV, one can say that this is where the rotational sequence ceases to exist. An estimate of the width of the lightest member of the antidecuplet,
222
shown at the top of the right diagram in Fig. 4, the Of, gave a surprisingly small result: I?@ < 15MeV 12. This result obtained in the CQSM, immediately gave credibility to the existence of the antidecuplet. It should be stressed that there is no way to obtain a small width in the oversimplified Skyrme model. In pentaquarks forming the antidecuplet shown on the right of Fig. 4, the additional QQ pair is added in the form of the excitation of the (nearly massless) chiral field. Energy penalty would be zero, had not the chiral field been restricted to the baryon volume. Important, the antidecupletoctet splitting is not twice the constituent mass 2M but less. In the case of a large-size baryon it costs a vanishing energy to excite the antidecuplet 13. 4. Quark wave functions
The wave function of baryons in the CQSM has been derived recently by Petrov and Polyakov in the infinite momentum frame. Here I translate it to the baryon rest frame. We shall see how easily one can get the nonrelativistic SU(6) wave functions for ordinary octet and decuplet baryons “from the soliton”. Next, I derive the new result for the antidecuplet 5quark wave functions. Let a , at(p) and b, b+(p)be the annihilation-creation operators of quarks and antiquarks (respectively) satisfying the usual anticommutator algebra. The vacuum (00> is such that a,b(Oo >= 0. According to the CQSM, a baryon is N, “valence” quarks on a discrete level created by the selfconsistent pion field, plus the negative-energy Dirac sea of quarks, distorted as compared to the free case by the same self-consistent pion field, see Fig. 2. At large N,, the Dirac sea is given by the coherent exponent coherent exponent = exp (/(dp)(dp’) at (p)W(p, p’)bt (p’)) 100>, (2) where (dp) = ~ i ~ p / ( 2 and 7 ~ )W(p1,pz) ~ is the finite-time quark Green function at equal times in the static external field of the chiral “soliton”, to be specified below. The valence quark part of the wave function is given by a product of N , quark creation operators that fill in the discrete level: NC
valence =
IT /@PI
F(P) J(P),
(3)
color= 1
~ ( p=/(dp’) ) [ . * ( P > f i e v ( P ) ( 2 ~ ) 3 ~ ( p - p ’ ) - ~ (P’)V*(P‘)fiev(-P)] pi ,(4)
223 where fiev(p) is the Fourier transform of the wave function of the level. The second term in Eq.(4) is the contribution of the distorted Dirac sea to the one-quark wave function; I shall neglect it for simplicity in what follows. With the same accuracy, the discrete level’s wave function can be approximated by the upper component (as if it was non-relativistic):
where h(r) is the L = 0 solution of the bound-state Dirac equation with energy E E [-M, M ] for the given profile function of the soliton P ( T )4:
+ ( E + Mcos P )j , j ’ + -2 = ( M cos P - E ) h + M s i n P j . r Ph
h’ = -&‘sin 2.
In the non-relativistic limit the L= 1 function j ( r ) is neglected in Eq.(5). In Eq.(5) i = 1 , 2 are spin and j = 1 , 2 are isospin indices; e i j is the antisymmetric tensor. The QQ pair wave function W(p1, p2) determines the structure of the Dirac continuum; it is also a matrix in both spin and isospin indices. I denote by ( i , j ) those of the quark and by (z’,j’) those of the antiquark. We shall need the Fourier transforms of all odd (IT) and all even (C) powers of the self-consistent pion field:
J
I!, (9)= dr e--i(q.r)(n. TI;,sin ~ ( r, ) 3
c:, (9) = /dr
e--i(q.r)d;,
(cos P ( r ) - 1).
+
Correspondingly, W = W(”) W(’) can be divided into two pieces,
W ; ; p ) ( p , p ’ ) = ul:!“’”(p,p’)~(C);,(p
+ p’),
where iW,’) =
Wi’
+ E’)
2(€ l
MM‘ d €E’(M + € ) ( M I+ € 1 )
[(P.P’) - ( M
+
+ € ) ( M I+ €91bit + iEpqrTpP;(OT):’,
with E = dM2(p) p2, the primed variables being related to the antiquark. In the coordinate space the pair wave function is given by a convo-
224 lution of the self-consistent chiral field and the Fourier transforms of
~ ( ~ 1 ’ ) :
These functions can be computed numerically once the profile function of the self-consistent chiral field is known. Eqs.(9,10) give the amplitudes of various spin, isospin and orbital QQ states inside a baryon. The partial waves depend on the QQ coordinates (r,r‘) with respect to the baryon center of mass. To get quark wave functions inside a particular baryon, one has to rotate all the isospin indices j’s, both in the discrete level and in the QQ pairs, by an SU(3) matrix R f , f = 1,2,3, j = 1 , 2 , and to project it to the
4’)
(m, 4’).
, (10,;’) or “Project” means specific baryon from the (8, integrating over the SU(3) rotation matrices R with a Haar measure normalized to unity. In full glory, the quark wave function inside a particular baryon B with spin projection k is given by Q.~B =
s
~ R D ,*(R)E--aNc B
fj/(dpn)
Rf;Fi”jqpn)a; ,, ,i,(pn)
n= 1
.exp ( / ( d p ) ( d p ’ ) a t u , i ( p ) f~w j jj‘~ i , ( p , p ’ ) R ~ ~ ’ ~ t ~ f lao> ’ z ’ ~ p. ’ (11) )) Here Q stands for color, f for flavor and i for spin indices. Let me give a few examples of the baryons’ (conjugate) rotational wave functions DB* (R): neutron, spin projection IC : 3 A++ , spin projection - : 2
+
A’, spin projection
+ -21 :
o+,spin projection k :
D;* = d ‘ i i E k l ~ I 1 ~ , 3 , DA++ TT *
- mRf,2Rf,2Ri2,
D p o* = O R ; 2(2Rf,2Ri
D? * = d36R;R;R;.
(12)
(13)
+Ri’Rf, ’),( 14) (15)
If the coherent exponent with Q o pairs is ignored, one gets from the general Eq.(ll) the 3-quark Fock component of the octet and decuplet baryons. It depends on the quark “coordinates”: the position in space (r), the color ( a ) ,the flavor ( f ) and the spin (i), and also on the baryon spin
225
k. For example, the neutron 3-quark wave function turns out to be ( in >k ) f 1 f 2 f 3 ,i1izi3 (rl, r2, r3) = ~ f l f zeiliz bzf3 :6 h ( ~ l ) h ( ~ 2 ) h ( ~ 3 ) +permutations of 1,2,3,
(16)
antisymmetrized in color. It is better known in the form InT> = 2 d f (rl)df ( T 2 ) u l (T3)-dT permutations of T I , ~ 2 , ~
+
(T1)uf ( T 2 ) d i
(T3)-uT ( r l ) d i (r2)dT ( T 3 )
3 ,
(17)
which is the well-known non-relativistic SU(6) wave function of the nucleon! Petrov and Polyakov have obtained the corresponding SU(6) function in the infinite-momentum frame. Performing the group integration with the decuplet rotational functions (13,14) one also gets the well-known S U ( 6 ) wave functions in the nonrelativistic limit. Relativistic corrections to those wave functions are easily computable from Eq.(4), as are the 5-quark Fock components of the usual octet and decuplet baryons. To find those, one needs to expand the coherent exponent in Eq.(ll) to the linear order in the additional QQ pair, and perform the S U ( 3 ) projecting. The result will be given in a subsequent publication. Here I shall go straight to the 0+. Projecting the three quarks from the discreet level on the 0 rotational function (15) gives an identical zero, in accordance with the fact that the 0 cannot be made of 3 quarks. The non-zero projection is achieved when one expands the coherent exponent to the linear order. One gets then the 5-quark component of the 0 wave function: f fz f3 f4,ii iz i3 i4 (rl . . . r 5 ) = E f fz ef3 f4 63 + >f5,i5 1
lQk
I
€ i i2~
f5
. h ( ~ l ) h ( ~ 2 ) hWi:2(rq, ( ~ 3 ) r5) + permutations of 1,2,3.
(18)
The color structure of the antidecuplet wave function is ~ ~ ~ ~ Indices ~ ~ ~ 1 to 4 refer to quarks and index 5 refers to the antiquark, in this case S thanks to 6j5. The quark flavor indices are f1-4 = 1 , 2 = u, d. Naturally, we have obtained 0+ = uudd3. We see that the first two valence u , d quarks from the discrete level form a spin- and isospin-singlet diquark (although not correlated in space), like in the nucleon, see Eq.(16). However, the other pair of quarks do not form a similar spin-zero diquark. For example, in the “C” part of the wave function the 0 spin k is determined by the spin of the third quark from the discrete level. Since in the CQSM the functions h(r1,2,3)and W(r4, r5) are known, Eq.(18) gives the complete color, flavor, spin and space 5-quark
b
226
wave function of the O+ in its rest frame. The 5-quark wave functions of other members of the antidecuplet can be obtained in a similar manner. For the computation of the 0 width, this wave function is, however, inadequate as a matter of principle. As explained in Refs. the only consistent way to compute the width is using the infinite momentum frame where there is no pair creation or annihilation, and the Fock decomposition is well defined. In that frame, the decay of the Of goes into the fiwe-quark component of the nucleon only. It is first of all suppressed to the extent the 5-quark component of the nucleon is less than its 3-quark component. An additional suppression comes from the spin-flavor overlap. A preliminary crude estimate shows that the O+ width can be extremely small. 14915,
I thank the organizers of the Continuous Advances for hospitality, and V. Petrov and M. Polyakov for numerous discussions. This work has been supported in part by the DOE under contract DE-AC05-84ER40150. References 1. M.M. Pavan, R.A. Arndt, 1.1. Strakovsky and R.L. Workman, TN Newslett. 16, 110 (2002), hep-ph/0111066. 2. D. Diakonov, V. Petrov, P. Pobylitsa, M. Polyakov and C. Weiss, Nucl. Phys. B 4 8 0 , 341 (1996), hep-ph/9606314; Phys. Rev. D 5 6 , 4069 (1997), hep-ph/9703420. 3. D. Diakonov and V. Petrov, Nucl. Phys. B272, 457 (1986). 4. D. Diakonov and V. Petrov, JETP Lett. 43, 75 (1986) [PismaZh. Elcsp. Teor. Fiz.43, 57 (1986)l; D. Diakonov, V. Petrov and P.V. Pobylitsa, Nucl. Phys. B306, 809 (1988); D.Diakonov and V. Petrov, in Handbook of QCD, M. Shifman, ed., World Scientific, Singapore (2001), vol. 1, p. 359, hep-ph/0009006. 5. J . Hiller, these Proceedings, hep-ph/0408131. 6. V. Petrov and M. Polyakov, hep-ph/0307077. 7. A. Hayashi, G. Eckart, G. Holzwarth and H. Walliser, Phys. Lett. B147, 5 (1984). 8. M. Karliner and M.P. Mattis, Phys. Rev. D31, 2833 (1985); ibid. 34, 1991 (1986). 9. N. Itzhaki, I.R. Klebanov, P. Ouyang and L. Rastelli, Nucl. Phys. B684, 264 (2004), hep-ph/0309305. 10. E. Witten, Nucl. Phys. B160, 433 (1983). 11. D. Diakonov and V. Petrov, hep-ph/0312144. 12. D. Diakonov, V. Petrov and M. Polyakov, 2. Phys. A359, 305 (1997), hep-ph/9703373; hep-ph/0404212. 13. D. Diakonov and V. Petrov, Phys. Rev. D69, 056002 (2004), hep-ph/0309203. 14. D. Diakonov and V. Petrov, Phys. Rev. D69, 094011 (2004), hep-ph/0310212. 15. D. Diakonov, hep-ph/0406043.
DO CHIRAL SOLITON MODELS PREDICT PENTAQUARKS ?
IGOR R. KLEBANOV AND PETER OUYANG Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544, USA We reconsider the relationship between the bound state and the S U ( 3 ) rigid rotator approaches to strangeness in chiral soliton models. For non-exotic S = -1 baryons the bound state approach matches for small m K onto the rigid rotator approach, and the bound state mode turns into the rotator zero-mode. However, for small m K , there are no S = -tl kaon bound states or resonances in the spectrum. This shows that for large N and small m K the exotic state is an artifact of the rigid rotator approach. An S = +1 near-threshold state with the quantum numbers of the O+ pentaquark comes into existence only when sufficiently strong S U ( 3 ) breaking is introduced into the chiral lagrangian. Therefore, pentaquarks are not generic predictions of the chiral soliton models.
1. Introduction These lecture notes are largely based on our paper with N. Itzhaki and L. Rastelli'. Recently there has been a flurry of research activity on exotic pentaquark baryons, prompted by reports2v3y4of the observation of the S = tl baryon O+ (1540). The original photoproduction experiment2 was largely motivated by theoretical work5 in which chiral soliton models were used to predict a rather narrow I = 0, J P = $+ exotic S = +1 baryon whose minimal quark content is uudds. The method used in t o predict the baryon spectrum is the S U ( 3 ) collective coordinate quantization of chiral soliton^^?^. This approach predicts the well-known 8 and 10 S U ( 3 ) multiplets of baryons, followed by an exotic 10multiplets~g~'O whose S = +1 member is the Of. The fact that the exotic 10multiplet is found simply by exciting the soliton to the next rotational energy level after the well-known decuplet has led to a widespread belief that the pentaquarks are a robust prediction of chiral soliton models, independent of assumptions about the dynamics. In this talk we argue that this belief is not well-founded. In-
227
228 stead, we will conclude that in soliton models the existence or non-existence of the pentaquarks very much depends on the details of the dynamics, i.e. the structure of the chiral lagrangian. Thus, the soliton models do not produce any miracles that are not obvious from general priciples of QCD. Neither in QCD nor in chiral soliton models is there anything that a priori guarantees the existence of narrow pentaquarks. Indeed, we will show that with the standard set of the Skyrme model parameters, a resonance with the quantum numbers of the O+ does not form. This fact should be kept in mind as some of the more recent searches" have failed to confirm the existence of the O+ or other pentaquarks. Our theoretical discussion follows the basic premise6 that the semiclassical quantization of chiral solitons corresponds to the 1/N expansion for baryons in QCD generalized to a large number of colors N . It is therefore important to generalize the discussion of exotic collective coordinate states carried out for N = 3 in to large N . The allowed multiplets must contain states of hypercharge N / 3 , i.e. of strangeness S = 0. In the notation where S U ( 3 ) multiplets are labeled by ( p , q ) , the lowest multiplets one finds8~14~15~16~17~18 are (1,n) with J = ?jand ( 3 , n - 1) with J = $. These are the large N analogues of the octet and the decuplet. Exactly the same multiplets appear when we construct baryon states out of N quarks. The splittings among them are of order 1/N, as is usual for soliton rotation excitations. The large N analogue of the exotic antidecuplet is the representation (0,n 2) with J = and one finds that its splitting from the lowest multiplets is O ( N o )in the rotator approximation. The fact that the mass splitting is of order one, comparable to the energy of mesonic fluctuations, raises question^^^^^^^^^ about the validity of the rigid rotator approach to these states. Instead, a better treatment of these states is provided by the bound state approach20 where one departs from the rigid rotator ansatz and adopts more general kaon fluctuation profiles; in this approach one describes the O+ as a kaon-skyrmion resonance or bound state of S = +1, rather than by a rotator state (a similar suggestion was made independently by Cohen17.) In the non-exotic sector, as we take the limit m K 4 0 the bound state description of low-lying baryons smoothly approaches the rigid rotator description, and the bound state wavefunction approaches a zeromode. However, in contrast with the situation for S = -1, for S = +1 there is no fluctuation mode that in the m~ + 0 limit approaches the
+
i,
229
rigid rotator mode. Thus, for large N and small S U ( 3 ) breaking, the rigid rotator state with S = +1 is an artifact of the rigid rotator approximation (we believe this to be a general statement that does not depend on the details of the chiral lagrangian.) Next we ask what happens as we increase the SU(3) breaking by varying parameters in the effective lagrangian (such as the kaon mass and the weight of the Wess-Zumino term) and find that a substantial departure from the SU(3)-symmetric limit is necessary to stabilize the kaon-skyrmion system. We reach a conclusion that, at least for large N, the exotic S = +1 state exists only due to the S U ( 3 ) breaking and disappears when the breaking is too weak. 2. The rigid rotator vs. the bound state approach
Our discussion of chiral solitons is mainly carried out in the context of the Skyrme model, but our conclusions will not be tied to a specific model. The Skyrme approach to baryons begins with the Lagrangian12
+ T r ( M ( U + Ut - 2 ) ) ,
(2.1)
where U ( x p ) is a matrix in SU(3) and M is proportional to the matrix of quark masses. There is an additional term in the action, called the Wess-Zumino term, whose normalization is proportional to N. Skyrme showed that there are topologically stabilized static solutions in which the radial profile function F ( r ) of hedgehog form, Uo = ei.r..FF(T), satisfies the boundary conditions F ( 0 ) = r,F(cm) = 0. The non-strange low-lying excitations of this soliton are given by rigid rotations of the pion field A ( t ) E S U ( 2 ) :
U ( Z ,t ) = A(t)UoA-'(t).
(2.2)
For this ansatz the Wess-Zumino term does not contribute. By expanding the Lagrangian about Uo and canonically quantizing the rotations, one finds that the Hamiltonian is
where J is the spin and the c-numbers Mcl and R are functionals of the
230
soliton profile. For vanishing pion mass, one finds numerically that 107 R2Mcl 21 3 6 . 5 5 , e3fn For N = 2n 1, the low-lying quantum numbers are independent of the integer n. The lowest states, with I = J = and I = J = %,are identified and e with the nucleon and A particles respectively. Since fn 1 / a , the soliton mass is N, while the rotational splittings are 1/N. Adkins, Nappi and Witten13 found that they could fit the N and A masses with the parameter values e = 5.45, fT = 129 MeV. In comparison, the physical value of f n = 186 MeV. A generalization of this rigid rotator treatment that produces SU(3) multiplets of baryons is obtained by making the collective coordinate A ( t ) an element of SU(3). Then the WZ term makes a crucial constraint on allowed m u l t i p l e t ~ ~The ~ ~ large-N ~ ~ ~ ~ .treatment of this 3-flavor Skyrme model is more subtle than in the 2-flavor case. When N = 2n 1 is large, even the lowest lying (1,n) S U ( 3 ) multiplet contains (n 1)(n 3) states with strangeness ranging from S = 0 to S = -n - 1 16. When the strange quark mass is turned on, it introduces a splitting of order N between the lowest and highest strangeness baryons in the same multiplet. Thus, SU(3) is badly broken in the large N limit, no matter how small m, is 16. It is helpful to think in terms of SU(2)x U(1) flavor quantum numbers, which do have a smooth large N limit.. In other words, we focus on low strangeness members of these multiplets, whose I , J quantum numbers have a smooth large N limit, and identify them with observable baryons. N strange quarks, Since the multiplets contain baryons with up to the wave functions of baryon with fixed strangeness deviate only an amount 1 / N into the strange directions of the collective coordinate space. Thus, to describe them, one may expand the SU(3) rigid rotator treatment around the SU(2) collective coordinate. The small deviations from SU(2) may be assembled into a complex SU(2) doublet K ( t ) . This method of 1/N expansion was implemented in 19, and reviewed in 16. From the point of view of the Skyrme model the ability to expand in small fluctuations is due to the Wess-Zumino term which acts as a large magnetic field of order N. The method works for arbitrary kaon mass, and has the correct limit as m K -+ 0. To order O(No) the Lagrangian has the formlg
+
- a, -
N
+
N
+ +
N
-
N L = 4akW + i - ( ~ +-kk 2
+~ - r) KtK.
(2.5)
231 The Hamiltonian may be diagonalized:
where
N2 M,2 = 1 6 . ~ (2.7) The strangeness operator is S = btb - at,. All the non-exotic multiplets contain at excitations only. In the SU(3) limit, w- -+ 0, but w+ t $ N o . Thus, the “exoticness” quantum number mentioned in l8 is simply E = btb here, and the splitting between multiplets of different “exoticness” is &, in agreement with results found from the rigid rotator14?15f16p17>18. The O ( N o )splittings predicted by the rigid rotator are, however, not exact: this approach does not take into account deformations of the soliton as it rotates in the strange direction^^^^'^^^^. Another approach to strange baryons, which allows for these deformations, and which proves to be quite successful in describing the light hyperons, is the so-called bound state method“. The basic strategy is to expand the action to second order in kaon fluctuations about the classical hedgehog soliton. Then one can obtain a linear differential equation for the kaon field, incorporating the effect of the kaon mass, which one can solve exactly. The eigenenergies of the kaon field are then the O ( N o ) differences between the masses of the strange baryons and the classical Skyrmion mass. It is convenient to write U in the form U = f l U ~ a where , U, = exp[2iXj7rj/fT] and UK = exp[2iXaKa/f,] with j running from 1 to 3 and a running from 4 to 7. The A, are the standard SU(3) Gell-Mann matrices. We will collect the K“ into a complex isodoublet K : Wf
=
N (J1+ 8@
( r n K / M 0 ) 2 f 1)
,
-
K=’(
K 4 - iK5 K6 - iK7
)=(::).
Expanding the Wess-Zumino term to second order in K , we obtain
ZN
L~~ = -w
f:
( K ~ D , K- ( D , K ) ~ K )
where
and B, is the baryon number current. Now we decompose the kaon field into a set of partial waves. Because the background soliton field is invariant
232
+
under combined spatial and isospin rotations T = I L, a good set of quantum numbers is T ,L and T,, and so we write the kaon eigenmodes as K = k(r, t)YTLT,. Substituting this expression into L S k y r m e LWZ we obtain an effective Lagrangian for the radial kaon field k ( r ,t ) of the form21
+
d
d
-h(r)-kt-k dr dr
- ktk(m;
+I&(?-))
The formula for the effective potential V,ff(r)appears in equation of motion for k is
-f(r)k
The resulting
21y1.
+ 2iA(r)k + Ok = 0 ,
(2.12)
1
C? E --arh(r)r2ar - m& - ~ ( r ) . r2
The eigenvalue equations are
+ 2X(T)W, + O)kn = 0 ( f ( r ) G z - 2 ~ ( r )+~ o$, , =o (f(r)w:
( S = -1)
,
( S = +I),
(2.13)
with w,,G, positive. Crucially, the sign in front of A, which is the contribution of the WZ term, depends on whether the relevant eigenmodes have positive or negative strangeness. It is possible t o examine these equations analytically for m~ = 0. Then one finds that the S = -1 equation has an exact solution with w = 0 and k ( r ) sin(F(r)/2), which corresponds to the rigid rotator zero-mode21. As m K is turned on, this solution turns into an actual bound state 20,21. On the other hand, the S = +1 equation does not have a solution with G= and k(r) sin(F(r)/2). This is why the exotic rigid rotator state is not reproduced by the more precise bound state approach to strangeness. In section 3 we further check that, for small m ~there , is no resonance corresponding to the rotator state of energy in the S U ( 3 ) limit. The lightest S = -1 bound state is in the channel L=l, T = $, and its mass is Mcl 0.218 efr = 1019 MeV. This state gives rise to the A(1115), C(1190), and C(1385) states, where the additional splitting arises from S U ( 2 ) rotator corrections20y21.There is also a L = 0, T = bound state corresponding to the negative parity hyperon A(1405). The natural appearance of the A(1405) is a major success of the bound state a p p r ~ a c h ~
-
&
N
2
+
233 6 4
1 ~
0
0.2
0.4
0.6
'
0.8
1
W
s,
Figure 1. Phase shift as a function of energy in the L = 2, T = S = -1 channel. The energy w is measured in units of e f n (with the kann mass subtracted, so that w = 0 at threshold), and the phase shift 6 is measured in radians. Here e = 5.45 and fT = 129 MeV.
The same method can be applied also to states above threshold. Such states will appear as resonances in kaon-nucleon scattering, which we may identify by the standard procedure of solving the appropriate kaon wave equation and studying the phase shifts of the corresponding solutions as a funcbion of the kaon energy. In the L = 2, T = channel there is a resonance a t Mcl 0.7484 ef,=1392 MeV (see Figure 1). Upon the SU(2) collective coordinate quantization, it gives rise t o three states22with ( I ,J ) given by (0, :), (1, (1, with masses 1462 MeV, 1613 MeV, and 1723 MeV respectively (see Table 2). We see that these correspond nicely to the known negative parity resonances h(1520) (which is Do3 in standard notation), C(1670) (which is 0 1 3 ) and C(1775) (which is 0 1 5 ) . As with the bound states, we find that the resonances are somewhat overbound (the overbinding of all states is presumably related to the necessity of adding an overall zero-point energy of kaon fluctuations), but that the mass splittings within this multiplet are accurate to within a few percent. In fact, we find that the ratio
+
4
g), g),
(2.14) while its empirical value is 1.70.
234 3. Baryons with S=+1?
For states with positive strangeness, the eigenvalue equation for the kaon field is the same except for a change of sign in the contribution of the WZ term. This sign change makes the WZ term repulsive for states with 3 quarks and introduces a splitting between ordinary and exotic baryons20. In fact, with standard values of the parameters (such as those in the previous section) the repulsion is strong enough to remove all bound states and resonances with S = 1, including the newly-observed O+. It is natural to ask how much we must modify the Skyrme model to accommodate the pentaquark. The simplest modification we can make is to introduce a coefficient a multiplying the WZ term. Qualitatively, we expect that reducing the WZ term will make the S = +1 baryons more bound, while the opposite should happen to the ordinary baryons. The most likely channel in which we might find an exotic has the quantum numbers L = 1, T = f , as in this case the effective potential is least repulsive near the origin. For fn= 129, 186, and 225 MeV, with e 3 f n fixed, we have studied the effect of lowering the WZ term by hand. Interestingly, in all three cases we have to set a 21 0.39 to have a bound state at threshold. If we raise a slightly, this bound state moves above the threshold, but does not survive far above threshold; it ceases to be a sharp state for a N 0.46. We have plotted phase shifts for various values of a in Figure 2.a Assuming that the parameters of the chiral lagrangian take values such that the Of exists, we can then consider the SU(2) collective coordinate quantization of the state, in a manner analogous to the treatment of the S = -1 bound states2'. Here we record our results, assuming that a = 0.39 and fn = 129 MeV, and refer the reader to our original paper' for details. and positive The lightest S = +1 state we find has I = 0, J = parity, i e . it is an S = +1 counterpart of the A. This is the candidate O+ state. Its first S U ( 2 ) rotator excitations have I = 1, J p = and I = 1, Jp = (a relation of these states to O+ also follows from general large N relations among baryon^^^^'^). The counterparts of these J p = $+, states in the rigid rotator quantization lie in the 27-plets of SU(3) We find that the I = 1, J p = $' state is 148 MeV heavier than the W,
i
:+
4'
:
26y27.
N
aWhen the state is above the threshold, we do not find a full 7r variation of the phase. Furthermore, the variation and slope of the phase shift decrease rapidly as the state moves higher, so it gets too broad to be identifiable. So, the state can only exist as a bound state or a near-threshold state.
235
:+
while the I = 1 , J p = state is 289 MeV heavier than the O+. We may further consider I = 2 rotator excitations which have J p = ;+,:+. Such states are allowed for N = 3 (in the quark language the N
charge +3 state, for example, is given by uuuus). The counterparts of these J p = states in the rigid rotator quantization lie in the 35plets of SU(3)26t27.We find
z+, 5’
M ( 2 , $) - M ( 0 ,
i)
M(2,g)- M(0,;)
N
N
494 MeV, 729 MeV
.
(3.15)
Although the specific mass splittings which we have computed depend on the choice of parameters in the chiral lagrangian, it turns out that we may form certain combinations of masses of the exotics which rely only the existence of the S U ( 2 ) collective coordinate:
+ M ( 1 ,!j)- 3M(O,3) = 2(MA - M N ) = 586 MeV , z M ( 2 , z) + M ( 2 , ; ) - g M ( 0 , i)= MA - M N ) = 1465 MeV , M ( 2 , S) - M ( 2 , E) = 4 ( M ( 1 ,3) - M ( l , z)) , (3.16) 2M(l,z)
3
where we used M a - M N = &. These “model-independent” relations have also been derived using a different method 2 5 . 6
a=0.4 1.5
/----
---______-
---
a=O.6
0.5
a
Figure 2. Phase shifts 6 as a function of energy in the S = +1, L = 1, T = channel, for various choices of the parameter a (strength of the WZ term). The energy w is measured in units of efir ( e = 5.45, fir = f~ = 129 MeV) and the phase shift 6 is measured in radians. w = 0 corresponds to the K - N threshold.
As another probe of the parameter space of our Skyrme model, we may vary the mass of the kaon and see how this affects the pentaquark. As
236 observed in Section 3, in the limit of infinitesimal kaon mass, there is no resonance in the S = +1, L = l , T = channel. We find that to obtain a bound state in this channel, we must raise m K to about 1100 MeV. Plots of the phase shift vs. energy for different values of m K may be found in l. 4. Discussion
The main implication of our analysis is that in chiral soliton models there is no “theorem” that exotic pentaquark baryons exist, nor is there a theorem that they do not exist. The situation really depends on the details of the dynamics inherited from the underlying QCD. The statements above apply to general chiral soliton models containing various lagrangian terms consistent with the symmetries of low-energy hadronic physics. In the bound state approach to the Skyrme model we saw that an S = +1 near-threshold state is absent when we use the standard parameters, but comes into existence only at the expense of a large reduction in the Wess-Zumino term. It is doubtful that such a reduction is consistent with QCD. However, one can and should explore other variants of chiral soliton models. For example, in 28 the exotic S = +1 resonances were studied in a model containing explicit K* fields. This model contains a coupling constant which is, roughly speaking, the analogue of the coefficient of the WZ term, a , in our approach. The findings of 28 are largely parallel to ours. For a wide range of values of this coupling, the repulsion is too strong, and no S = +1 resonances can form. When this coupling is very small, then there exists an S = +1 bound state. There is also a narrow intermediate range where this bound state turns into a near-threshold resonance. An important question is whether chosing parameters to lie in this narrow range is consistent with the empirical constrains on the effective lagrangian. If not, then one may have to conclude that chiral soliton models actually predict the absence of pentaquarks.
Acknowledgments We are grateful to N. Itzhaki and L. Rastelli for collaboration on a paper reviewed here, and to E. Witten for discussions. This material is based upon work supported by the National Science Foundation Grants No. PHY0243680 and PHY-0140311. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
237 References 1. N. Itzhaki, I. R. Klebanov, P. Ouyang and L. Rastelli, “Is Theta(1540)+ a kaon Skyrmion resonance?,” Nucl. Phys. B 684, 264 (2004) [arXiv:hep-
ph/0309305]. 2. T. Nakano et al. [LEPS Collaboration], “Evidence for a narrow S = +1 baryon resonance in photoproduction from the neutron,” Phys. Rev. Lett. 91,012002 (2003) [arXiv:hep-ex/0301020]. 3. S. Stepanyan et al. [CLAS Collaboration], “Observation of an exotic S = +1 baryon in exclusive photoproduction from the deuteron,” arXiv:hepex/03070 18. 4. J. Barth et al. [SAPHIR Collaboration], “Observation of the positivestrangeness pentaquark Theta+ in photoproduction with the SAPHIR detector at ELSA,” arXiv:hep-ex/0307083. 5 . D. Diakonov, V. Petrov and M. V. Polyakov, “Exotic anti-decuplet of baryons: Prediction from chiral solitons,” Z. Phys. A 359, 305 (1997) [arxiv:hep-ph/9703373]. 6. E. Witten, “Global Aspects Of Current Algebra,” Nucl. Phys. B 223, 422 (1983); “Current Algebra, Baryons, And Quark Confinement,” Nucl. Phys. B 223,433 (1983). 7. E. Guadagnini, “Baryons As Solitons And Mass Formulae,” Nucl. Phys. B 236,35 (1984). 8. A. V. Manohar, “Equivalence Of The Chiral Soliton And Quark Models In Large N,” Nucl. Phys. B 248, 19 (1984). 9. M. Chemtob, “Skyrme Model Of Baryon Octet And Decuplet,” Nucl. Phys. B 256,600 (1985). 10. M. Praszalowicz, “SU(3) Skyrmion,” TPJU-5-87 Talk presented at the Cracow Workshop on Skyrmions and Anomalies, Mogilany, Poland, Feb 20-24, 1987
11. K. T. Knopfle, M. Zavertyaev and T. Zivko [HERA-B Collaboration], “Search for Theta+ and Xi(3/2)- pentaquarks in HERA-B,” J. Phys. G 30, S1363 (2004) [arXiv:hep-ex/0403020]; R. Mizuk [Belle collab.], talk at PENTA04; V. Halyo [Babar collab.], talk at PENTA04. 12. T. H. Skyrme, “A Nonlinear Field Theory,” Proc. Roy. SOC.Lond. A 260, 127 (1961); “A Unified Field Theory Of Mesons And Baryons,” Nucl. Phys. 31,556 (1962). 13. G. S. Adkins, C. R. Nappi and E. Witten, “Static Properties Of Nucleons In The Skyrme Model,” Nucl. Phys. B 228,552 (1983). 14. V. Kaplunovsky, unpublished. 15. Z. Dulinski and M. Praszalowicz, “Large N(C) Limit Of The Skyrme Model,” Acta Phys. Polon. B 18, 1157 (1988). 16. I. R. Klebanov, ‘[Strangeness In The Skyrme Model,” PUPT-1158 Lectures given at N A T O A S 1 on Hadron and Hadronic Matter, Cargese, fiance, Aug 8-18, 1989.
17. T. D. Cohen, “Chiral soliton models, large N(c) consistency and the Theta+
238
18. 19. 20.
21.
22. 23. 24.
25. 26. 27. 28.
exotic baryon,” arXiv:hep-ph/0309111; T. D. Cohen and R. F. Lebed, “Partners of the Theta+ in large N(c) QCD,” arXiv:hep-ph/0309150. D. Diakonov and V. Petrov, “Exotic baryon multiplets at large number of colours,” arXiv:hep-ph/0309203. D. B. Kaplan and I. R. Klebanov, “The Role Of A Massive Strange Quark In The Large N Skyrme Model,” Nucl. Phys. B 335,45 (1990). C. G. Callan and I. R. Klebanov, “Bound State Approach To Strangeness In The Skyrme Model,” Nucl. Phys. B 262,365 (1985); C. G. Callan, K. Hornbostel and I. R. Klebanov, “Baryon Masses In The Bound State Approach To Strangeness In The Skyrme Model,” Phys. Lett. B 202, 269 (1988). N. N. Scoccola, “Hyperon Resonances In SU(3) Soliton Models,” Phys. Lett. B 236, 245 (1990). D. 0. Riska and N. N. Scoccola, “Anti-charm and anti-bottom hyperons,” Phys. Lett. B 299, 338 (1993). C. L. Schat, N. N. Scoccola and C. Gobbi, “Lambda (1405) in the bound state soliton model,’’ Nucl. Phys. A 585, 627 (1995) [arXiv:hep-ph/9408360]. E. Jenkins and A. V. Manohar, “l/N(c) expansion for exotic baryons,” JHEP 0406,039 (2004) [arXiv:hep-ph/0402024]. H. Walliser and V. B. Kopeliovich, “Exotic baryon states in topological soliton models,” arXiv:hep-ph/0304058. D. Borisyuk, M. Faber and A. Kobushkin, “New family of exotic Theta baryons,” arXiv:hep-ph/0307370. B. Y . Park, M. Rho and D. P. Min, “Bound state approach to pentaquark states,” arXiv:hep-ph/0405246.
BARYON EXOTICS IN THE 1/Nc EXPANSION
ELIZABETH JENKINS Department of Physics, 9500 Gilman Drive, University of California San Diego, La Jolla, CA 92093-0319, USA E-mail:
[email protected] The implications of the l / N c expansion for the spin-flavor properties of qqqqq pentaquark exotics are presented. The masses, axial couplings and decay widths of pentaquark baryons containing only light quarks are discussed. In addition, qqqqQ pentaquark baryons containing a single heavy antiquark are studied.
1. Introduction Normal QCD baryons are three-quark qqq states which are completely antisymmetric in the color indices of the quarks. In the past year and a half, experimental evidence has been reported for exotic baryon^^^^^^ - baryons whose flavor quantum numbers forbid their classification as qqq states. All presently observed exotic baryons can be classified as qqqqq E q44 pentaquarks. Experimental evidence has been reported for three different pentaquark and 0,-(3099). The Of baryons: 0+(1540), @(1860)(also designated +) is thought to be a u u d d s bound state with strangeness S = +1 (since it contains a strange antiquark), isospin I = 0 and spin J = :. The @ has strangeness S = -2, isospin I = and spin J = so the manifestly exotic @-- state is an ssddii bound state. The 0, is the anticharmed analogue of the Of:it is an isosinglet u u d d c bound state. The @ has been observed by a single experiment2. This observation has not been confirmed, and indeed a number of experiments have failed to observe the states even though they have greater statistical sensitivity. Thus, it does not appear that the evidence for the Q, will survive. The 0,-reported by the H1 experiment3 also is unconfirmed. Thus, the predominant evidence for pentaquarks is for the state Of. The 0+ is seen by eight different experiments'. The experimental situa-
4
3,
239
240
tion is confused, however, since a number of experiments (ALEPH, BaBar, CDF) do not see it. The experiments which do report evidence for the O+ see a narrow state with a width I? < 15 MeV. A lower bound on the width has not been set; however, studies of K N scattering data and other data indicates that the width must be of order 1 MeV to have escaped d e t e ~ t i o n ~The > ~ O+ . is seen primarily in photoproduction, but it has also been seen in deep inelastic scattering. The parity of the state is unmeasured. The O+ is observed to decay to nucleon and kaon, either n K + or pKg. The experimental data in support of the O+ is summarized in Table 1. Table 1. Experiments reporting evidence of the Q+. The mass and width measurements, observed decay channel, production mechanism and statistical significance of the reported signal axe given. Expt . LEPS DIANA CLAS SAPHIR SVD HERMES COSY-TOF ZEUS
Mass (MeV)
Width (MeV)
1540 f 10 1539 f 2 1542 f 5 1540 f 4 f 2 1526 f3 f3 1528 f 2.6 f 2.1 1530 f 5 1521.5 f 1.5:;:;
< 25 1.5 GeV (Fig. 3)11. This is at odds with ex0.2 as it has pectations from pQCD, that the ratio should be roughly e- collisions at LEP. It turns been observed at similar energies in e+ out that this is a consistent feature of baryon enhancement at intermediate PT (from roughly 1.5 to 5 GeV/c) and is also confirmed for jj, A and Ll2. The anomalous baryon enhancement can also be seen in the nuclear suppression factor RAA. While pions and kaons show values as low as 0.2 at intermediate PT, protons and baryons have values between 0.8 and 1. One immediately asks the question whether energy loss, which is believed to be responsible for the suppression, can be partonic in nature, if baryons and mesons are affected differently. The standard theory is that a fast parton is created in a hard QCD process, propagates through the medium and loses energy while interacting with it, and finally fragments into hadrons. If this picture applies the energy loss should be equal for all kinds of hadrons.
+
-
4. Hadronization by Quark Recombination
As a solution, several groups suggested, that hadronization at intermediate PT in Au+Au collisions is not governed by fragmentation of hard partons, as it is in e+ + e- and p f j collisions, but by recombination of quarks from the surface of the firebal113~14~15~16. The basic idea is that if partons
+
278
1.8 1.6 1.4 1.2 +k 1.0 \ Q 0.8 0.6 0.4 0.2 0.0
'
'
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 PT (GeV)
Figure 3. The ratio of p and 7 ~ f in central (filled circles) and very peripheral (open circles) Au+Au collisions. The ratio in central collisions, where we expect a quark gluon plasma, follows predictions from a purely thermal model (dash-dotted line). In peripheral collisions, where thermal and chemical equilibrium is not reached, the data is closer to the pQCD prediction (solid line). Recombination can explain why protons 4 GeV/c. Data from the PHENIX are so effectively produced thermally at high PT collaboration. N
are available in phase space, in particular in a situation close to kinetic equilibrium, they might simply coalesce or recombine into hadrons. This idea was first advocated outside of heavy ion physics to describe hadron production, in particular the so called leading particle effect17. In a heavy ion collision, the partonic degrees of freedom, when approaching T, from above, are essentially constituent quarks, since the average virtualities are below any perturbative scale. So gluons and sea quarks are taken care of by dressing the constituent quarks and the data indeed seems to indicate that hadrons are only recombining using the lowest Fock state, i.e. valence quarks. How can recombination help to explain the anomalous baryon enhancement seen at RHIC? Let us first see how the hadron yield is usually calculated in the case of a meson. The Wigner function W of the valence quarks in the parton phase is convoluted (using relative momentum and relative distance) with the Wigner function of the meson. The spectrum is then
279
obtained by integrating over the hadronization hypersurface C
E dN d3P
d a u ( a )" ( 2 ~ ) ~
It is easy to prove that recombination is more effective than fragmentation on a thermal parton distribution, while a power-law shaped parton spectrum favors fragmentation at large enough PT. It is also easy to see that recombination from a thermal distribution treats mesons and baryons on equal footing, leading to a thermal distribution of hadrons in the final state, while it is known that fragmentation disfavors baryons. When doing the numerics one finds that a thermal parton distribution, recombining at a temperature of T = 175 MeV and with a radial flow of 0 . 5 5 ~together with fragmentation from hard processes gives an excellent description of all hadron production data above a PT of 1.5 GeV/c. Recombination from the fireball is dominant up to 4 GeV/c for mesons and up to about 6 GeV/c for baryons, while fragmentation from hard processes takes over at higher PT. The calculations show good agreement with the measured hadron spectra, hadron ratios and RAAratios. If already dominant at intermediate PT, recombination should be even more important at low PT where the bulk of hadrons is produced. This is probably true, but it is obscured by several things. Firstly, it can be shown that at intermediate PT and for thermal parton distributions the hadron Wigner functions (or equivalently the hadron wave function) can be integrated trivially regardless of their shape. This means that the nonperturbative dynamics of the bound state is not important in this case. This is very different at low PT. Even worse, the kinematics must be different to accommodate energy conservation, involving at least 2 -+ 2 and 3 -+ 2 processes instead of 2 -+ 1 and 3 -+ 1 as in the case of the simple minded coalescence. This would also cure the problem of entropy non-conservation. Finally, interactions also occur in the hadronic phase after the phase transition. They are very important at low PT and have important consequences on the observed hadron spectra.
5. A New Quark Counting Rule for Elliptic Flow In the last section we have seen that we have a theory involving partonic degrees of freedom, thermalization and collectivity, and which can nicely describe the data. Nevertheless it would be good to have more direct evidence
280 for subhadronic degrees of freedom showing thermalization or collectivity. This was provided by a stunning result coming from measurements of the elliptic flow coefficient u2 as a function of PT. Measurements carried out at intermediate PT were a t first disappointing, since different hadrons led to very different results, making this another RHIC puzzle. However, it soon became clear that there is again a systematic pattern, distinguishing between mesons and baryons. It was proposed that this can again be understood in terms of parton recombinationls. Starting from the expansion in Eq. (2) one finds that the elliptic flow of a hadron is connected to the elliptic flow of the partons before hadronization by a simple scaling law13918
where n is the number of valence quarks. One can rephrase this t o make it look like one of the classical quark counting rules
As Fig. 4 shows, the experimental data follows this quark counting rule with impressive p r e c i s i ~ n Deviations ~ ~ ~ ~ ~ . for the pion are understood since most of them, even at intermediate PT, are coming from resonance decays15. Besides being a strong direct indication of quark degrees of freedom, the scaling law permits, for the first time, direct access to a n observable in the QGP phase, namely the elliptic flow of quarks.
6. Summary
I have discussed evidence from measurements of hadrons, that matter created in central Au+Au collisions a t RHIC exhibits partonic degrees of freedom which show signs of thermalization and collectivity. This is an indication for a deconfinement phase transition a t temperatures around 170-180 MeV. Further signatures including jet quenching, the absence of back-to-back jet correlations, saturation of the hydrodynamic limit, d+Au control experiments and others confirm this picture. However, they could only be touched shortly or had to be completely omitted in these notes. An overview of recent developments can be found in the proceedings of the last Quark Matter conference21.
281
0.1 2
0.1
A
T+ (PHENIX) p(PHENIX) Ko,(STAR)
I
0.08
5
0.06
0.04
1
0.02
0.0- 0.5
1.0 1.5 2.0 PT/n [GeV]
2.5
3.0
Figure 4. The scaled elliptic flow coefficient v z / n as a function of scaled transverse momentum P T / n for different hadron species where n is the number of valence quarks. The v2 data have been measured by the PHENIX and STAR collaborations at RHIC. The data points for all hadron species lie on top of each other after scaling. This is interpreted as the universal curve for the elliptic flow of u , d and s quarks (and their antiquarks) before hadronization. Pions are pushed to lower PT because they come mainly from decays of p mesons.
Acknowledgments I want to t h a n k t h e organizers for a n inspiring workshop. T h e author is supported by DOE grant DE-FG02-87ER40328. References 1. Official web site http://www. bnl.gov/rhic. 2. F. Karsch, Nucl. Phys. A698, 199 (2002); Lect. Notes Phys. 583,209 (2002); C.R. Allton et al., Phys. Rev. D66, 074507 (2002). 3. W. Broniowski and W. Florkowski, Phys. Rev. Lett. 87, 272302 (2001); W.Broniowski, A. Baran and W. Florkowski, Acta Phys. Polon. B33, 4235 (2002). 4. P. Braun-Munzinger, D. Magestro, K. Redlich and J. Stachei, Phys. Lett. B518, 41 (2001). 5. N. Xu and M. Kaneta, Nucl. Phys. A698, 306 (2002). 6. P. Huovinen, P. F. Kolb, U. W. Heinz, P. V. Ruuskanen and S. A. Voloshin, Phys. Lett. B503, 58 (2001); P.F. Kolb, U. W. Heinz, P. Huovinen, K. J. Eskola and K. Tuominen, Nucl. Phys. A696, 197 (2001); D. Teaney, J. Lauret and E. V. Shuryak, preprint nucl-th/0110037; T. Hirano, J. Phys. G30,(2004). 7. S. Voloshin and Y. Zhang, Z. Phys. C70,665 (1996).
282 8. J. D. Bjorken, FERMILAB-PUB-82-059-THY (1982); M. H. Thoma and M. Gyulassy, Nucl. Phys. B 351,491 (1991); X. N. Wang and M. Gyulassy, Phys. Rev. Lett. 68,1480 (1992); M. Gyulassy and X. Wang, Nucl. Phys. B 420,583 (1994); R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. B 483,291 (1997); B.G. Zakharov, J E T P Lett. 65,615 (1997) M. Gyulassy, P. Levai and I. Vitev, Phys. Rev. Lett. 85,5535 (2000); U.A. Wiedemann, Nucl. Phys. B 588,303 (2000); R. Baier, Y. L. Dokshitzer, A. H. Mueller and D. Schiff, JHEP 0109,033 (2001). 9. K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 88, 022301 (2002); C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 90,082302 (2003). 10. X. N. . Wang, Phys. Lett. B579,299 (2004). 11. S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 172301 (2003). 12. H. Long [STAR Collaboration], J. Phys. G30,S193 (2004). 13. R. J. Fries, B. Muller, C. Nonaka and S. A. Bass, Phys. Rev. Lett. 90,202303 (2003); Phys. Rev. C68,044902 (2003). 14. R. J. Fries, J . Phys. G30,S853 (2004). 15. V. Greco, C. M. KO and P. Levai, Phys. Rev. Lett. 90,202302 (2003); Phys. Rev. C68,034904 (2003). 16. R. C. Hwa and C. B. Yang, Phys. Rev. C67,034902 (2003); preprint hepph/0312271; preprint nucl-th/0401001. 17. J. D. Bjorken and G. R. Farrar, Phys. Rev. D9, 1449 (1974). K. P. Das and R. C. Hwa, Phys. Lett. B68,459 (1977); Erratum-ibid. B73,504 (1978); E.Braaten, Y. Jia and T. Mehen, Phys. Rev. Lett. 89,122002 (2002). 18. S. A. Voloshin, Nucl. Phys. A715, 379 (2003); Z. W. Lin and C. M. KO, Phys. Rev. (265,034904 (2002); D.Molnar and S. A. Voloshin, Phys. Rev. Lett. 91,092301 (2003). 19. S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 182301 (2003). 20. J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92,052302 (2004). 21. Proceedings of Quark Matter 2004, Oakland, J. Phys. G30.
NUCLEAR PHYSICS AT SMALL X*
D. E. KHARZEEV Physics Department Brookhaven National Laboratory, Upton, NY 11777, USA E-mail:
[email protected] Some of the recent work on the theory of nuclear processes at high energies and small Bjorken 2 is reviewed, with an emphasis on the physics of nucleus-nucleus and hadron-nucleus collisions. In particular, we discuss the influence of parton saturation in the Color Glass Condensate on the nuclear dependence of hard processes, approach to thermalization and the recent insights on the properties of the Quark-Gluon Plasma.
1. Introduction
Recent years have seen an increase of interest in high energy nuclear physics. To large extent, this is due to an intense stream of new data from Relativistic Heavy Ion Collider and elsewhere; moreover, for a theorist nuclear physics a t small Bjorken z is appealing because it is placed at the intersection of three different, and equally interesting, directions in contemporary theoretical research: i) high parton density QCD; ii) non-equilibrium field theory; and iii) phase transitions in strongly interacting matter. Indeed, understanding the evolution of a heavy ion collision requires a working theory of initial conditions, of the subsequent evolution of the produced partonic system, and of the phase transition(s) to the deconfined phase. This talk is an attempt to capture some of the recent changes and developments in the theoretical picture of these phenomena.
*Work supported by the US Department of Energy under under Contract No. DE-ACOZ98CH10886.
283
284
2. Initial conditions and global observables
2.1. The r81e of coherence
Not so long ago, before the advent of RHIC, it was widely believed that at collider energies the total multiplicities will become dominated by hard incoherent processes. The very first data from RHIC (see and references
1" 0
"
100
"
200
"
300 (N
"
400
0
100
200
300 400 (Nwtit)
Figure 1. Centrality dependence of the charged particle multiplicity near mid-rapidity Au collisions at fi = 20 and 200 GeV; fromlO. in Au
+
therein in this volume) provided a lot of food for new thought: the measured charged hadron multiplicities in Au - Au collisions appeared much smaller than expected on the basis of incoherent superposition of hard processes. Given that any inelastic rescatterings in the final state can only increase the multiplicity", we have an experimental proof of a high degree of coherence in multi-particle production in nuclear collisions at RHIC energies. 2.2. Semi-classical QCD and hadron multiplicities
Combining the idea of coherence with the parton model, we have to consider the initial parton wave functions of the colliding nuclei as coherent superpositions of the wave functions of the constituent nucleons. Since at small Bjorken x all of the partons in the nucleus a t a fixed transverse coordinate participate in a hard scattering process, this treatment naturally leads to the notion of parton density in the transverse plane Q: - a new dimensionful scale of the problem. Once this scale becomes comparable to the resolution scale determined by the kinematics of the hard scattering, the amplitude of the process is severely affected by the coherence. The aFor statistical systems, this is due to the second law of thermodynamics
285 limit on the parton density is reached when the occupation numbers of the gluon field modes with transverse momenta p~ < Q, reach the value nk l/cys(Qs), characteristic for classical gauge fields - this is the phenomenon known as ”parton saturation”2, leading to a coherent state of gluons - Color Glass Condensate (for reviews, see Since the integrated multiplicities are dominated by momenta p~ 5 Q, and parton density in the transverse plane scales as Q: Npart 113 (where N
314,5t6,7).
N
NpaTtis the number of nucleons which participate in the process), Color Glass Condensate leads to a simple predictions for the centrality dependence of hadron multiplicity in heavy ion collisions:
Combined with the dependence of the gluon structure function on Bjorken x known from HERA, which implies Q:(x) l / d ,one can generalize this formula to predict the energy, centrality, rapidity, and atomic number dependencies of hadron multiplicities8. Additional information on the dynamics of the collision can be inferred from the numerical lattice simulationsg. So far this approach has been quite successful in predicting the multiplicities measured a t RHIC; a recent important example is given at Fig.1 which shows the evolution of centrality dependence with energy in the entire RHIC range between fi = 20 and 200 GeV. One can see that the shape of the centrality dependence changes very little over a large energy range, in which the perturbative minijet cross section grows by over an order of magnitude. The prediction of the saturation model is seen to agree with the data reasonably well; this indicates the possibility that parton saturation sets in in heavy ion collisions already at moderate energies. We do not expect the method to apply below fi = 20 GeV however, since at lower energies the coherence length becomes shorter than the nuclear radius. N
2.3. High p~ hadron suppression at forward rapidities, and quantum evolution i n the Color Glass Condensate
Parton saturation at transverse momenta kT 5 &, at sufficiently small x appears to have non-trivial consequences also for the nuclear dependence of the semi-hard processes. At very small 5, when a , In 1/x 1, a semiclassical description has to be modified due to the quantum evolution. Small x evolution introduces anomalous dimension y 21 1/2 in the gluon densities, so that the dependence on the momentum scale Q is modified, to Q 2 -+ Q2’. Since in the vicinity of the saturation boundary the only dimensionful scale N
286
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2
3
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Figure 2. Nuclear modification factor in dAu collisions as a function of transverse momentum for different rapidities; the data from BRAHMS Collaboration”, theoretical calculations from16.
characterizing the system is the saturation momentum Q:, the cross section of semi-hard scattering should scale as a function of Q:/Q2 - it was found is consistent with HERA data that this ”geometrical scaling” on deep-inelastic scattering. Combining these two observations with the A dependence of the saturation momentum Q: All3 we come to the c o n c l ~ s i o nthat ~ ~ ~at~sufficiently ~ small z and moderate kT the nuclear dependence of hard processes in A A collisions should change from S A Q ~ (where SA N,”$ is the overlap area) to SAQ;? Npart. In p A (or dA) collisions the nuclear dependence is then SAQ?? A5I6,so there has to be a suppression as well. This suppression has also been found17 in the numerical solution of the Balitsky-Kovchegov equation, as well as in18; for recent work, see also The experimental test of these ideas has been performed shortly af11112913314
-
-
--
191’20721.
-
287 terwards - it has been established (for a review, see 43 in this volume) that at mid-rapidity y = 0 there is no high kT suppression in dAu data; this means that the suppression observed in AuAu collisions has to come from the final-state effects, which will be discussed below. The data thus rule out the possibility1' that 2 is small enough for quantum evolution to develop already at mid-rapidity at RHIC. Nevertheless, the presented arguments should apply at sufficiently small z. This is why the data on high kT hadron production at forward rapidities giving access to much smaller values of IC were eagerly awaited. The results from the BRAHMS experiment22 demonstrated a strong suppression of high kT hadrons; moreover, the centrality dependence appeared consistent with the predicted &A N~::{~scaling, in a dramatic contrast to the increasing Cronin enhancement observed at mid-rapidity. Complementary results have been reported in23127124.STAR Colaboration has also reported2' on an observation of a predicted26 nuclear-dependent weakening of the back-to-back correlations for hadrons separated by several units of rapidity. It will be interesting to check if the suppression extends to charm hadrons at forward rapidities28; at mid-rapidity, the first results have been reported inz9. Alternative explanations based on "conventional" shadowing and multiple scattering (for a review, see30) are also being explored.
-
3. Approach to thermalization, and the r6le of classical fields
There is by now an ample evidence of the importance of final state interactions in heavy ion collisions. Among the bulk observables, the azimuthal anisotropy of hadron production is a most spectacular evidence of this - indeed, if all of the elementary nucleon-nucleon collisions were independent, the produced hadrons would not be correlated with the nucleus-nucleus reaction plane. The observed azimuthal anisotropy (or the "elliptic flow", for a review) indicates the existence in the parlance of the field; see of a correlation between the geometry of the nucleus-nucleus collision and the momenta of the emitted hadrons. An economical way of describing the evolution of a large number of particles in space and momentum is provided by relativistic hydrodynamics, which transforms the gradients of the initial parton density into the momentum flow of the produced hadrons. Hydrodynamical description is valid when the mean free path of partons is much smaller than the size of the system, i.e. when the system is sufficiently thermalized. The free expansion (or "inflation") of the produced system 31932
288
with time reduces the density gradients, so the magnitude of the elliptic flow crucially depends on the thermalization time when a hydrodynamical calculation is initiated. It appears that to describe the elliptic flow of the
I'reliminxq
7, 1 -0.021.., I ' ,, I 0 0.2 0.4 0.6 0.8 "
cross section [%I
,
s
*
Preliminary I
'
I
a
,
1 1.2 1.4 1.6 1.8 2
pt [GeVkI
Figure 3. Elliptic flow near mid-rapidity in Au + Au collisions as a function of centrality (left) and transverse momentum (right); from Ref.31.
observed magnitude 31, one has to assume that the thermalization time is Such a short very short, about Ttherm N_ 0.5 fm (for a review, see thermalization time presents a problem both for the traditional perturbative and non-perturbative treatments. Indeed, in perturbative QCD the rescattering amplitudes are suppressed by powers of the coupling a,, so the thermalization time appears long, on the order of 10 fm, which makes the description of the elliptic flow problematic 35. In non-perturbative approaches, the interactions can be assumed strong, but the typical time scale of an interaction is l / h ~ c 1~fm, so it is difficult to expect that several interactions needed for thermalization will occur during qherm 21 0.5 fm. The coherent classical fields present in the Color Glass Condensate (CGC) scenario may eventually provide a solution to this puzzle, since in this case the multi-gluon scattering amplitudes A ( n + rn) from n l / a , to m l / a , gluons are not suppressed. An approach to thermalization in this scenario was explored in Ref. 36; recently, an attention was brought also to the role of instabilities in the equilibration process 3 7 . The use of CGC initial conditions of Ref. * in a hydrodynamical approach 38 has led to a successful description of the RHIC data. Nevertheless, much work will have to be done to understand the thermalization process. 33134).
N
-
-
-
4. Hydrodynamical evolution: more fluid than water
Since hydrodynamical description relies on the direct use of the equation of state, the data can be used to extract an information on the properties
289
of the medium. It appears that the quark-gluon plasma equation of state as measured on the lattice (for a review, see 39) is successful in describing the data. However the data can tell even more about the properties of the medium, if one considers the influence of viscous corrections on various observables 40. Viscosity of the medium appears to affect the observables in a very significant way; in fact, one can deduce an upper limit 40,34 on the ratio of shear viscosity r] to the entropy density s, r]/s 5 0.1 - much smaller than the same ratio for the water! Such a small value of viscosity, which reflects the dissipation of energy in a hydrodynamical evolution, contradicts the picture of weakly coupled quark-gluon plasma, and is more indicative of a strongly coupled quark-gluon liquid. A calculation of shear viscosity in the strong coupling regime of QCD is still beyond the reach; however it has been made in N = 4 supersymmetric Yang-Mills theory 41 - the result is a small ratio of r]/s = 1/4n, comparable to the one inferred from RHIC data. A small value of viscosity in the strongly coupled quark-gluon plasma a posteriori justifies the use of the approach to hadron multiplicities assuming the proportionality of the number of measured hadrons to the number of the initially produced partons. This assumption would be unnatural if the evolution of the plasma were accompanied by parton multiplication, but is justified if the viscosity is small and evolution of the system is close to isenthropic. 5. High p~ hadron suppression, jet quenching, and heavy quarks The suppression of high p~ hadrons in Au - Au collisions is certainly one of the most spectacular new results at RHIC (for a comprehensive review of the data, see 43 in this volume). Such an effect has not been seen at lower energiesb; moreover, the results from the dAu run at RHIC indicate that at pseudo-rapidity r] = 0 the observed suppression is entireIy due to the final state effects, very likely a jet quenching in the quark-gluon plasma (for an overview, see 4 4 ~ 4 5 ) . Alternative scenarios, e.g. the absorption in a dense hadron gas, seem unlikely in view of the high energy density 6 20 GeV/fm3 (see e.g. 8, achieved in the collisions. Nevertheless, additional experimental checks have to be performed; an important additional test of the jet quenching scenario involves the measurement of the suppression
-
bA moderate amount of suppression in the SPS results however cannot be excluded due to uncertainties in the reference pp data 42
290 for heavy hadrons containing c or b quarks. If the suppression of high pt particles is indeed due to the induced radiation of gluons by fast partons, heavy quarks should lose significantly less energy than the light ones due to the "dead cone" effect 46. This prediction seems to be in accord with the first RHIC data, which within the error bars indicate no quenching effect on the spectra of open charm, as inferred from the decay electrons 47; however more precise data are desirable. Several other calculations of the energy loss of heavy partons have been performed (see Refs.48~49~50 and papers in this volume); while they differ in the formalisms used, they all find a reduced energy loss for the heavy quarks. On the other B , ...) have a typical large size determined hand, since heavy mesons (D, by the presence of the light quark in their wave functions, in the hadronic absorption mechanism one would expect that heavy mesons interact with about the same probability as the light ones. 51352
Figure 4. Nuclear modification factor for charged hadrons and neutral pions (left and right panels, respectively) in Au Au collisions a t b , = 200 GeV; from Ref.43
+
6. Summary The first years of the experiments at RHIC have changed in a dramatic way the theoretical picture of dense and hot parton systems. The evidence for the existence of new states of QCD matter is mounting53, and a consistent description of the observed phenomena has started to emerge. Nevertheless, hot and dense QCD is still in its infancy - and we have every reason to expect new surprises ! I am grateful to the Organizers for a most stimulating and inspiring meeting.
291
References 1. 2. 3. 4. 5. 6. 7. 8.
9.
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(2001). P. Arnold, J. Lenaghan and G. D. Moore, JHEP 0308,002 (2003). T. Hirano and Y. Nara, arXiv:nucl-th/0403029. F. Karsch and E. Laermann, arXiv:hep-lat/0305025. D. Teaney, Phys. Rev. C 68,034913 (2003). G. Policastro, D. T. Son and A. 0. Starinets, Phys. Rev. Lett. 87, 081601 (2001). D. d’Enterria, arXiv:nucl-ex/0403055. D. d’Enterria, arXiv:nucl-ex/0406012. M. Gyulassy, arXiv:nucl-th/0403032. P. Jacobs and X. N. Wang, arXiv:hep-ph/0405125. Y. L. Dokshitzer and D. E. Kharzeev, Phys. Lett. B 519, 199 (2001). K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 88,192303 (2002). M. Djordjevic and M. Gyulassy, Phys. Lett. B 560,37 (2003). N. Armesto, C. A. Salgado and U. A. Wiedemann, Phys. Rev. D 69,114003 (2004). R. Thomas, B. Kampfer and G. Soff, arXiv:hep-ph/0405189. N. Armesto, C. A. Salgado and U. A. Wiedemann, arXiv:hep-ph/0405184. A. Dainese [ALICE Collaboration], arXiv:nucl-ex/0405008. M. Gyulassy and L. McLerran, arXiv:nucl-th/0405013.
SATURATION PHYSICS MEETS RHIC DATA
YURI V. KOVCHEGOV Department of Physics, University of Washington Seattle, W A 98195-1 560, USA E-mail:
[email protected]. edu We review the derivation of the expression for the cross section of single inclusive gluon production in proton-nucleus collisions ( P A )and in deep inelastic scattering (DIS) at very high energy, when the parton saturation/Color Glass Condensate effects become important in the target’s wave function, both at the classical and quantum levels. We study the resulting particle spectra concentrating on the nuclear modification factor R P A for gluon production. We show that a t moderately high energy/rapidity the nuclear modification factor R P A exhibits Cronin enhancement at transverse momenta p~ of the order of the saturation scale Q s . As the energy/rapidity increases, RPA decreases. At sufficiently high energy/rapidity R P A becomes less than 1 for all values of p~ indicating the onset of suppression of gluon production due to small-s evolution effects. Our prediction of suppression is confirmed by the recently reported BRAHMS collaboration data on particle production at forward rapidity in dAu collisions a t RHIC.
1. Introduction
The results presented here are based on the work done in collaboration with Dmitri Kharzeev and Kirill Tuchin in Refs. 1, 2, 3. Recently there has been a surge of interest in particle production in proton-nucleus (PA)and deuteron-nucleus (dA) collisions at high energies. The interest was inspired by the new data produced by the d A program at RHIC, which should enable one to separate the contributions of the initial state effects, such as parton saturation, from the final state effects, such as jet quenching and energy loss in the quark-gluon plasma (QGP), to the suppression of high transverse momentum particles observed in Au - Au collisions at RHIC. However, particle production in p ( d )A collisions is very interesting by itself. The transverse and longitudinal momentum spectra of the produced particles are related to the small-a: wave function of the nucleus. One can, therefore, gain important insights in the dynamics of small-x partons in the
293
294
nuclear wave function by studying particle production in p ( d ) A collisions. The hadronic and nuclear wave functions at small-a: are characterized by extremely high parton density. The density of gluons in it can become very large, corresponding to very strong gluonic fields, A, l/g, which is the strongest possible gluon field in QCD. At the same time the strong coupling constant in these systems is small, because of the presence of the saturation scale Q,, which is a large momentum scale arising due to a high density of color charges (partons) in the transverse plane. Therefore, the partonic systems in the hadronic and nuclear small-a: wave functions have very high density with weak interactions at the partonic level. This means that high parton density leads to complicated nonlinear effects, which can nevertheless be studied analytically due to the weakness of individual partonic interactions characterized by the small parameter a,(&,) > 1 limit of A l l 3 . For a cylindrical nucleus the Eq. (1) and using the fact that Q:o impact parameter b integration would just give a factor of SA. As was shown in Ref. 6 in the approximation when the logarithmic dependence of the powers of the exponents in Eq. (1) on the transverse size is neglected, x21n(l/xTA) x g2, the x~ and y~ integrations in Eq. (1) can be done exactly. (Note that Eq. (12) was obtained in the same approximation.) Using the result of the integrations along with Eq. (12) in Eq. (11) we then obtain' N
+ Ei ( g)]} , (13) Q:o
where Ei(z) is the exponential integral. The ratio R P A ( k ~is ) plotted in Fig. 1for A = 0.2 QSo.It clearly exhibits an enhancement at high-kT typical of Cronin effect (see also Ref. 8). Analyzing Eq. (13) and remembering
RPA t
2-
1.5.
1. 0.5. -.
~ the o quasi-classical Figure 1. The ratio RPA for gluons plotted as a function of k ~ / Q in McLerran-Venugopalan model as found in Ref. 6. The cutoff is A = 0.2Qs.
--
that Q20 one can easily see that the height of the Cronin maximum, scaling like In Q s 0 , and its position, scaling like Qs0, are both increasing functions of A, or, equivalently, of the centrality of the collision.' N
298 3. Gluon Production Including Small-z Evolution
3.1. Gluon Production Cross Section
As the energy of the collisions increases quantum evolution corrections become important. The evolution equation in rapidity Y for the forward dipole-nucleus scattering amplitude N ( r ,b, Y) closes only in the large-N, limit of QCD and readsg
with the initial condition given by Eq. (4)with the saturation scale Q:o replaced by Q;suark2 = ( C F / N ~02,. ) Equation (14) resums leading powers of aSY and all multiple rescatterings on the target.g Incorporating the effects of evolution (14) in the expression for the single inclusive gluon production cross section is rather intricate and tedious. It has been done for the case of DIS in Ref. 2 and we refer the interested reader to Ref. 2 for more details. The result, however, is very straightforward and can be easily generalized to the case of p A to give'
-duPA -
d21c d y
-
1
cF / d 2 B d 2 b d 2 z V ~ n G ( z l b - - D , Y -y)ePik'" as7r ( 2 ~ b2 ) ~
v: NG (2,b, 9 )
7
(15)
where Y is the total rapidity interval between the proton and the nucleus. In Eq. (15) n ~ ( z , Y - y ) is evolved by the linear part of Eq. (14) (the BFKL equation") with Eq. ( 6 ) as the initial condition and NG = 2N - N 2 with N taken from Eq. (14). The effect of quantum evolution (14) is to leave Eq. (7) essentially unchanged evolving only the amplitudes NG and in it. The IcT-factorization expression (10) is still valid after inclusion of evolution (14) (see also Ref. 11).
b-a,
3.2. High-pT S u p p r e s s i o n in p A Collisions We will analyze RPA given by Eq. (15) in three separate regions of the transverse momentum of the produced gluon. We will first note that, as
299
was shown in Ref. 12, apart from the saturation scale Qs(y) there exists another scale in the problem - the geometric scaling scale, which was It signifies the boundary approximately found to be kgeom x Q;(y)/Q;,,. of the region where geometric scaling is valid: for k ~ s thek gluon ~ distribution is the function of a single variable k ~ / Q $ ( y ) .We will, therefore, consider three cases: (i) double logarithmic region k~ > kgeom,(ii) extended geometric scaling region Q,(y) < k~ s k g e o mand (iii) the saturation region kT
s
Qs(Y).
3.2.1. Double Logarithmic Region In the double logarithmic region k~ >> Qs(y) and the logarithm of the transverse momentum becomes important as well. At very high k~ only the linear part of the evolution equation (14) contributes. The resulting evolution is equivalent to the double logarithmic approximation (DLA) to the BFKL evolution, the solution of which is known exactly. In the end we write for the nuclear modification factor1
A1/3A2, for a large enough nucleus the power of the expoSince QZo nent in Eq. (16) becomes negative and large, giving R P A ( k ~ , y < ) 1 and, thus, leading to suppression of the gluon production in the double logarithmic region. Note that here saturation effects come in only through the logarithmic cutoff Qso, which is still sufficient to cause suppression. N
3.2.2. Extended Geometric Scaling Region In the extended geometric scaling region Qs(y) < kT 5 kgeom12 the effects of gluon anomalous dimension become important. One can show that in this region we obtain the following asymptotic expression for the nuclear modification factor calculated for simplicity at mid-rapidity ('y = Y/2)l
In Eq. (17) we assume that the collision energy is high enough for the gluon distribution function in the proton to also be in the extended geometric scaling region. From Eq. (17) we conclude that in the extended geometric scaling region at asymptotically high energies, RPA saturates to
~
~
300 a parametrically small lower bound, RPA A-l16 0. When p2 < m2, the SU(2) x U(1)y x SO(3) symmetry is exact. Of course in this case a confinement dynamics for three SU(2) vector bosons takes place and it is not under our control. However, taking p2 m2 and choosing m to be much larger than the confinement scale Asu(2),we get controllable dynamics at large momenta k of order m. It includes three massless vector bosons A; and two doublets, ( K + , K o )and ( K - , K o ) . The spectrum of the doublets is qualitatively the same as that in model (1): the chemical potential leads to splitting the masses (energy gaps) of these doublets and, in tree approximation, their masses are m - p and m p, respectively (see Sec. 2). In order to make the tree approximation to be reliable, one should take X to be small but much larger than the value of the running coupling g4(m) related to the scale m [smallness of g2(m) is guaranteed by the condition m >> A s u ( ~ ) assumed above]. The condition g4(m) A S a r m a , BCS state wins, i.e., system prefers the BCS ground state over the Sarma state. At fixed Fermi momenta p ~ , pthe ~ ,thermodynamic potential as a function of the gap has two minima - normal (N) and the BCS (absolute min) states, and one maximum - the Sarma state. Hence the Sarma state is metastable. We consider the pressure versus the Fermi momentum mismatch for states which are solutions of the gap equation, where normalized pressure is defined through the condensation free energy as Ps = - ( ( a s ) (Ro))/l( ( 0 ~ ~ (Ro))~ s ) where ( 0 0 ) is the free energy of the normal state at Sp = 0 and the BCS condensation energy is (OBCS)- (no)= - N ( O ) A ; / 2 , A0 is the BCS gap and N ( 0 ) is the density of states at the Fermi surface. In the leading order A Sp p~ and f(p) = 0 for p 5 PA. The gap parameter, defined
363
Gap
20 E
30 E
n
n
nb
nb
Figure 4. Gap, dispersions and occupation numbers for the cut-off (left) and the twobody potential (right) interactions.
as A
s
=
g s d 3 p / ( 2 n ) 3 f ( p ) ( $ ~ p $ ~ - psatisfies ), the gap equation A =
1/29 d 3 q / ( 2 n ) 3 A f ( q ) / d G where momentum integration is performed outside the breach region. The occupation numbers n A , TZBshow the evidance that it is a breached paired state, Figure 4. This state is an absolute minimum of the thermodynamic potential, hence we obtained a stable BP state. 11. Spherically symmetric static two-body potential. With attractive potential V ( x - d), interaction is HI = J d 3 p / ( 2 n ) 3 d 3 q / ( 2 n ) 3 V ( p- q ) $ l p $ L . - p $ B - q $ A q and the gap parameter acquires a momentum dependence, Ap = d 3 q / ( 2 n ) 3 V ( p- Q ) ( + B ~ $ A - ~ ) .
s
The gap equation is written Ap = 1 / 2 J d ’ q / ( 2 ~ ) ~ w (-p q ) A , / J G ,
+
and quasiparticle dispersion relations are Epf = E; f JE$ A;. We take a gaussian potential for numerical simulations. Due to the BCS instability,
364
A p picks at the effective Fermi surface given by the pole of the gap equation at A = 0, = 0. Therefore A p supports the BCS-like pairing around PO, and allows free dispersion relations, and hence free Fermi surfaces, outside the breached region, Figure 4. It is, however, difficult to varify that this state is an absolute minimum of the thermodynamic potential since instead of a number, A, we have a function Ap in the variational ansatz. We performed minimization of the thermodynamic potential numerically using different potentials. Generally, there is a central strip of fully gapped BCS phase about P A = p~ with normal unpaired phase outside. Depending on the parameters of interaction, these phases may be separated by a region of gapless BP superfluid phase, Figure 5. Conditions to have BP phase are as follows. At P A = p g there is standard BCS, which is a stable fully gapped solution. By adjusting the chemical potentials so as to increase the Fermi surface p g , we stress the system and low the pressurerelative to the normal phase. Eventually, either before or after a transition to a BP state, the pressure becomes negative and there is a first order phase transition t o the normal phase. At the point just before transition: if Ape is sufficiently large, the state is fully gapped (BCS) and no BP state will occur; if Ape is small, then it will not appreciably affect the dispersions and one finds a gapless Fermi surface coexisting with the superfluid phase. As long as A p falls off sufficiently quickly, one can choose large ratio mB/mA >> 1 so that the transition will occur with Ape small enough to support the BP phase. States shown at Figure 5 have mg/mA = 10. For a wider in q-space interaction, larger mass ratio is needed.
&to
I
Pb
I
I
pb N
I
/
20
Figure 5. Phase diagram of possible homogeneous phases in coordinates of the Fermi momenta (PA,p ~ for) the cut-off (left) and two-body potential (right) interactions.
365 4. Conclusion
We considered a Fermi system with weak attractive interaction between species A and B, where the BCS state forms at equal number densities, = n B . What is the ground state of this system when n A
# nB?
There is a range of parameters, where a breached pair phase exist. Breached pair state is a homogeneous phase where superfluid and normal components coexist. This state is stable and can be found provided there is a momentum structure of interaction and large enough mass ratio of two species. nA
Acknowledgments This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement DF-FC0294ER40818.
References 1. For a review, see Nature 416, 205 (2002). 2. For a review, see K Rajagopal and F. Wilczek, hep-ph/0011333. 3. W. V. Liu and F. Wilczek, Phys. Rev. Lett. 90, 047002 (2003), condmat/0208052. 4. G. Sarma, Phys. Chem. Solid 24, 1029 (1963); A. A. Abrikosov, “Foundations of the theory of metalls”. 5 . A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, “Methods of quantum field theory in statistical physics”, Dover Publications, Inc., New York. 6. W. V. Liu, F. Wilczek, and P. Zoller (2004), cond-mat/0404478. 7. M. Alford, C. Kouvaris, and K. Rajagopal (2003), hep-ph/0311286. 8. E. Gubankova, W. V. Liu, and F. Wilczek, Phys. Rev. Lett. 91,032001 (2003), hep-ph/0304016. 9. I. Shovkovy and M Huang, Phys. Lett. B564,205 (2003), hep-ph/0302142. 10. P. F. Bedaque, H. Caldas, and G Rupak, Phys. Rev. Lett. 91, 247002 (2003), cond-mat/0306694. 11. M. M. Forbes, E. Gubankova, W. V. Liu, and F. Wilczek, hep-ph/0405059.
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SECTION 5. TOPOLOGICAL FIELD CONFIGURATIONS
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QUANTUM WEIGHTS OF MONOPOLES AND CALORONS WITH NON-TRIVIAL HOLONOMY *
DMITRI DIAKONOV Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA NORDITA, Blegdamsuej 17, DK-2100 Copenhagen, Denmark St. Petersburg Nuclear Physics Institute, Gatchina, 188 300, St. Petersburg, Russia
Functional determinant is computed exactly for quantum oscillations about periodic instantons with non-trivial values of the Polyakov line at spatial infinity (or holonomy). Such instantons can be viewed as composed of the BogomolnyiPrasad-Sommerfeld (BPS) monopoles or dyons. We find the weight or the probability with which dyons occur in the pure Yang-Mills partition function. It turns out that dyons experience quantum interactions having the familiar “linear plus Coulomb” form but with the “string tension” depending on the holonomy. We present an argument that at temperatures below the critical one computed from A,,, , trivial holonomy becomes unstable, with instantons “ionizing” into separate dyons. It may serve as a microscopic mechanism of the confinement-deconfinement phase transition.
1. Introduction
Several years ago a new self-dual solution of the Yang-Mills equations at non-zero temperatures has been found independently by Kraan and van Baal [2] and Lee and Lu [3]. I shall call them for short the KvBLL calorons. In the case of the simplest SU(2) gauge group to which I restrict myself in this paper, the KvBLL caloron is characterized by 8 parameters or collective coordinates, as it should be according to the general classification of the self-dual solutions with a unity topological charge. The most interesting feature of the KvBLL calorons is that they can be viewed as composed of two “constituent” dyons; one is the standard BPS monopole [4,5] and the other is the so-called Kaluza-Klein monopole [6]. I denote them as M , L dyons; explicitly their fields can be found e.g. in the Appendix of Ref. [7]. *Based on the work in collaboration with Nikolay Gromov, Victor Petrov and Sergey Slizovskiy [ 11.
369
370
The collective coordinates or the moduli spa,ceof the KvBLL caloron can be chosen in various ways, however the physically most appealing choice is the six coordinates of the two dyons’ centers in space 2 1 , 2 , and two compact “time” variables. When the spatial separation of two constituent dyons 7-12 = 1 2 1 - 7 2 1 is larger than the compactification circumference 1/T the caloron action density becomes static and is reduced to the sum of the static action densities of the two dyons. At r l 2 T 5 1 the two dyons merge, and the action density becomes a time-dependent 4d lump, see Fig. 1.
Figure 1. The action density of the KvBLL caloron as function of z , t at fixed x = y = 0, with the asymptotic value of A4 at spatial infinity v = O.QnT,V = 1.lnT. It is periodic 2 in the t direction. At large dyon separation the density becomes static (left, ~ 1 = 1.5/T). As the separation decreases the action density becomes more like a 4d lump 2 0.6/T). In both plots the dyons are centered at 21 = -vr12/2nT, z2 = (right, ~ 1 = vr12/2nT, % i , z = y i , z = 0. The axes are in units of temperature T.
We use the gauge freedom to choose the gauge where the A4 component of the Yang-Mills field is static and diagonal at spatial infinity, A4 -+ ~ T ~ vvE, [0,27rT].The Polyakov line or the holonomy at spatial infinity is then -1T r L = - T1r P e x p 2 2
(
ii1’TdtA4)
4 C O SV 5
E [-1,1].
(1)
At the end points (v = 0, 2x7’) the holonomy belongs to the group center, iTrL = fl,and is said to be ‘trivial’. In this case the KvBLL caloron is reduced to the standard periodic instanton, also called the HarringtonShepard caloron [8]. It has been known since the work of Gross, Pisarski and Yaffe [9] that gauge configurations with non-trivial holonomy, iT r L # f l , are strongly suppressed in the Yang-Mills partition function. Indeed, the 1-loop effective action obtained from integrating out fast varying fields where one keeps all
371
powers of
A4
but expands in (covariant) derivatives of
A4
has the form [lo]
~~(2nT-v)' mod 27rT
where the perturbative potential energy term P(A4) has been known for a long time [9,11],see Fig. 2. The zeros of the potential energy correspond to 4Tr L = fl,ie. to the trivial holonomy. If a dyon has v # 27rTn a t spatial infinity the potential energy is positive-definite and proportional to the 3d volume. Therefore, dyons and KvBLL calorons with non-trivial holonomy seem to be strictly forbidden: quantum fluctuations about them have an unacceptably large action. PT '
Figure 2. Potential energy as function of v/T. Two minima correspond t o $TrL = fl,the maximum corresponds to TrL = 0. The range of the holonomy where dyons experience repulsion is shown in dashing.
Meanwhile, precisely these objects determine the physics of the supersymmetric YM theory where in addition to gluons there are gluinos, i.e. Majorana (or Weyl) fermions in the adjoint representation. Because of supersymmetry, the boson and fermion determinants about dyons cancel exactly, so that the perturbative potential energy (2) is identically zero for all temperatures, actually in loops. Therefore, in the supersymmetric theory dyons are openly allowed. [To be more precise, the cancellation occurs when periodic conditions for gluinos are imposed, so it is the compactification in one (time) direction that is implied, rather than physical temperature which requires antiperiodic fermions.] Moreover, it turns out [12] that dyons generate a non-perturbative potential having a minimum a t v = 7rT, i e . where the perturbative potential would have the
372 maximum. This value of A4 corresponds to the holonomy Tr L = 0 at spatial infinity, which is the “most non-trivial” ; as a matter of fact < Tr L >= 0 is one of the confinement’s requirements. In the supersymmetric YM theory there is a non-zero gluino condensate whose correct value is reproduced by saturating it by the L , M dyons’ zero fermion modes [12]. On the contrary, the saturation of the (square of) gluino condensate by instanton zero modes gives the wrong result, namely that of the correct value [13]. Recently it has been observed [7] that the square of the gluino condensate must be computed not in the instanton background but in the background of exact solutions “made of” L L , M M and LM dyons. The first two are the double-monopole solutions and the last one is the KvBLL caloron. As the temperature goes to zero, the L L and M M solutions have locally vanishing fields, whereas the KvBLL LM solution reduces to the instanton field up to a locally vanishing difference. Therefore, naively one would conclude that the dyon calculation of the gluino condensate, which is a local quantity, should be equivalent to the instanton one, but it is not. The fields vanishing as the inverse size of the system have a finite effect on such a local quantity as the gluino condensate! This is quite unusual. The crucial difference between the (wrong) instanton and the (correct) dyon calculations is in the value of the Polyakov loop, which remains finite. In the N = 1 SUSY theory, as one increases the compactification circumference 1/T, the average < A4 >= 7rT at infinity decreases, however the theory always remains in the Higgs phase, in a sense. Instantons do not satisfy this boundary conditions whereas dyons and calorons with the non-trivial holonomy do satisfy them. In the supersymmetric YM theory configurations having Tr L = 0 at infinity are not only allowed but dynamically preferred as compared to those with L = fl.In a non-supersymmetric theory it looks as if it is the opposite. Nevertheless, it has been argued in Ref. [14] that the perturbative potential energy (2) which forbids individual dyons in the pure YM theory might be overruled by non-perturbative contributions of an ensemble of dyons. For fixed dyon density, their number is proportional to the 3d volume and hence the non-perturbative dyon-induced potential as function of the holonomy (or of A4 at spatial infinity) is also proportional to the volume. It may be that at temperatures below some critical one the nonperturbative potential wins over the perturbative one so that the system prefers < Tr L >= 0. This scenario could then serve as a microscopic mechanism of the confinement-deconfinement phase transition [14]. It should be
373 noted that the KvBLL calorons and dyons seem to be observed in lattice simulations below the phase transition temperature [15,16,17]. To study this possible scenario quantitatively, one first needs to find out the quantum weight of dyons or the probability with which they appear in the Yang-Mills partition function. At-the - second stage, one has to consider the statistical mechanics of the L , M , L , M dyons for fixed Ad at infinity and find the free energy of the system as function of v = Finally, one has t o study this free energy as function of v a t different temperatures, to see what value of v (equivalent to the holonomy according to the formula L = cos(v/2T)) is preferred by the theory. Unfortunately, the single-dyon measure is not well defined: it is too badly divergent in the infrared region owing to the weak (Coulomb-like) decrease of the fields. What makes sense and is finite, is the quantum determinant for small oscillations about the KvBLL caloron made of two dyons with zero combined electric and magnetic charges. Knowing the weight of the electric- and magnetic-neutral KvBLL caloron one can read off the individual L , M dyons’ weights and their interaction as one moves the two dyons apart. The problem of computing the effect of quantum fluctuations about a caloron with non-trivial holonomy is of the same kind as that for ordinary instantons (solved by ’t Hooft [18]) and for the standard HarringtonShepard caloron (solved by Gross, Pisarski and Yaffe [9]) being, however, technically much more difficult. I remind the results of the above two calculations in the next two sections. In Section 4 I report on the very recent result for the KvBLL caloron [l].Remarkably, the quantum weight of the KvBLL caloron can be computed exactly. It becomes possible because we are able t o construct the exact propagator of spin-0, isospin-1 field in the KvBLL background, which is some achievement by itself. It turns out from the exact calculation of the KvBLL weight that dyons experience a familiar “linear plus Coulomb” interaction at large separations. That is why the individual dyon weight is ill-defined: their interaction grows with the separation. The sign of the interaction depends critically on the value of the holonomy. If the holonomy is not too far from the trivial such that 0.787597 < iITrLI < 1, corresponding t o the positive second derivative P”(v) (see Fig. 2) the L and M dyons experience a linear attractive potential. Integration over the separation 7-12of dyons inside a caloron converges. We perform this integration in Section 6 assuming calorons are in the “atomic” phase, estimate the free energy of the neutral caloron gas and conclude that the trivial holonomy (v = 0,27~T)is unstable a t temperatures
~~l,l,+m.
iTr
374 below T, = 1.125Rm, despite the perturbative potential energy P(v). In the complementary range $ITrLI < 0.787597, the second derivative P”(v) is negative, and dyons experience a strong linear-rising repulsion. It means that for these values of v, integration over the dyon separations diverges: calorons with holonomy far from trivial “ionize” into separate dyons. 2. Ordinary instantons at
T =0
The usual Belavin-Polyakov-Schwartz-Tyupkin instanton [19]has the field
A, = AEta =
-ip2U[a,(z - z)+ - (z - Z),]U+ , .( - z)2[p2 (z - z)2]
+
of = (l,*i?).
The moduli or parameter space is described by the center z, (4), size p (l), and orientation U (3). The action density TrF$ is O(4) symmetric, see Fig. 3. As it is well known [18],the calculation of the l-loop quantum weight of a Euclidean pseudoparticle consists of three steps: i) calculation of the metric of the moduli space or, in other words, calculation of the Jacobian composed of zero modes, needed to write down the pseudoparticle measure in terms of its collective coordinates, ii) calculation of the functional determinant for non-zero modes of small fluctuations about a pseudoparticle, iii) calculation of the ghost determinant resulting from background gauge fixing in the previous step. In fact, for self-dual fields problem ii) is reduced to iii) since for such fields Det(W,,) = Det(-D2)4, where W,, is the quadratic form for spin-1, isospin-1 quantum fluctuations and D2 is the covariant Laplace operator for spin-0, isospin-1 ghost fields [20]. Symbolically, one can write
J
pseudoparticle weight = d(col1ective coordinates). Jacobian.Det-l(-D2), The functional determinant is normalized to the free one (with zero background fields) and UV regularized by the standard Pauli-Villars method. The l-loop quantum weight of the BPST instanton has been computed by ’t Hooft [18]. If p is the Pauli-Villars mass, i.e. the UV cutoff, and g 2 ( p ) is the gauge coupling given at this cutoff, the instanton weight is
CO
= exp
16 log2 [-9 3 + -9- - - 3
210g(2n)
3
375 The last factor (in the curly brackets) is due to the regularized smalloscillation determinant; all the rest is actually arising from the 8 zero 22
-*
modes. The combination p T e 9 ( P ) = A? is the scale parameter which is renormalization-invariant at one loop. Since the action density is O ( 4 ) symmetric, the quantum weight depends only on the dimensionless quantity p A where p is the instanton size, and even this dependence follows trivially from the known renormalization properties of the theory. Therefore, only the overall numerical constant Co is the non-trivial result of the calculation. The prefactor g(p)-8 is not renormalized at one loop. At two loops the instanton weight can be obtained without further calculations [21] if one demands that it should be invariant under the simultaneous change of the cutoff p and g2(p) given at this cutoff, such that the 2-loop scale parameter
11 , b l = -N 3
b2
34 = -N
3
2
,
remains fixed. The 2-loop instanton weight computed from this requirement is [22]
3. Quantum weight of the periodic instanton with trivial holonomy
The Harrington-Shepard caloron [8] is an immediate generalization of the ordinary Belavin-Polyakov-Schwartz-Tyupkin instanton [19],continued by periodicity in the time direction. The l-loop quantum weight of the periodic instanton with the trivial holonomy has been computed by Gross, Pisarski and Yaffe [9]. The weight is a function of the instanton size p, temperature T and, after the renormalization, of the scale parameter A. In fact, the dependence on A follows from the renormalization properties of the theory, therefore the caloron weight is a non-trivial function of one dimensionless variable, p T . At p T < I it reduces to the 't Hooft's result for the ordinary BPST instanton [18]. At p T > l the caloron weight is [9]
376
0
Figure 3. Action densities of the ordinary BPST instanton (left) and of the periodic Harrington-Shepard instanton with trivial holonomy (right) as function of z, t at fixed 2 = y = 0. The former is O(4) and the latter is O ( 3 ) symmetric. The size of the latter is p = 0.8 2T. ’
The last factor suppresses large calorons: it is the consequence of the Debye screening mass which is nothing but the second derivative of the potential energy P(A4) a t zero. The vacuum made of these calorons was built using the variational principle in Ref. [23]. It turns out that the average of the Polyakov line grows rapidly from 0 to 1 near T M [24], however strictly speaking, there is no mass gap and no confinement-deconfinement phase transition. 4. Q u a n t u m weight of the caloron w i t h non-trivial
holonomy We define the quantum weight of the KvBLL caloron in the same way as it is done in the case of ordinary instantons and the periodic instantons, see eq.(3). The problem of writing down the Jacobian related to the caloron zero modes has been actually solved already by Kraan and van Baal [2]. Therefore, to find the quantum weight of the KvBLL caloron, only the ghost determinant needs to be computed. The KvBLL caloron has only the O(2) symmetry corresponding to the rotation about the line connecting the dyon centers, and the determinant is a non-trivial function of three variables: the holonomy at spatial infinity encoded in the asymptotic value
377 of ~ ~ ~ l = ~v, the l temperature + m T, and the separation between the two dyons 7-12. Computing this function of three variables looks like a formidable problem, however it has been solved exactly in Ref. [l]. We have followed Zarembo [26] and first found the derivative of the determinant Det(-D2) with respect to the holonomy or, more precisely, to v. [The holonomy is $TrL = cos(v/2T)]. The derivative dDet(-D2)/av is expressed through the Green function of the ghost field in the caloron background [26]. If a self-dual field is written in terms of the AtiyahDrinfeld-Hitchin-Manin-Nahm construction, and in the KvBLL case it basically is [2,3], the Green function is generally known [27-291 and we build it explicitly for the KvBLL case. Therefore, we are able to find the derivative dDet(-D2)/dv. Next, we reconstruct the full determinant by integrating over v using the determinant for the trivial holonomy (4) as a boundary condition. This determinant at v = 0 is still a non-trivial function of the caloron size p related t o the dyon separation according to 7-12 = Iz1-221 = np2T, and the fact that we match it from the v # 0 side is a serious check. Actually we need only one overall constant factor from Ref. [9] in order to restore the full determinant a t v # 0, and we make a minor improvement of the Gross-Pisarski-Yaffe calculation as we have computed the needed constant analytically. Depending on the holonomy, the M , L dyon cores are of the size $ and 1 c, respectively, where v = 2nT - v. At large dyon separations, 7-12 >> 1/T, the l-loop KvBLL caloron weight can be written in a compact form in terms of the coordinates of the dyon centers [l]:
= 1.031419972084,
P ( v ) = -v2ij-2, 12.rr2T
P”(v)
=
d2 -P(v). dv2
This expression is valid a t 7-12T>> 1 but arbitrary holonomy, v, V E [0,2.rrT1, meaning that it is valid also for overlapping dyon cores.
378
5. Dyon interaction At large separations, dyons (curiously) have the familiar “linear plus Coulomb” interaction:
V(r12)= r12T22n
(-
4
7-12
37r
+
log [ v ( l - v ) ( 2 ~ 1 2 T ) ~1.946 ]
+.. . ,
v
V =27rT E [0,11.
This interaction is a purely quantum effect: classically dyons do not interact a t all since the KvBLL caloron is a classical solution whose action is independent of the dyon separation. When the holonomy is not too far from trivial, 0.788 < LI < 1, such that P”(v) > 0, dyons inside the KvBLL caloron attract each other, and calorons can be stable. At v -+ 0 this attraction is in fact the wellknown effect of the suppression of large-size calorons owing to the non-zero Debye mass, cf. eq.(4):
iITr
More generally, the coefficient in the linear term is the second derivative of the potential energy P”(v). Therefore, in the complementary range, +ITr LI < 0.788, where the second derivative changes sign, dyons experience a strong linearly rising repulsion, see Fig. 2. For these values of v, integration over the dyon separation diverges: calorons with holonomy far from trivial “ionize” into separate dyons. 6. Caloron free energy and instability of the trivial
holonomy Let us make a crude estimate of the free energy of the non-interacting N+ calorons and N- anti-calorons a t small v < 7rT 1 - - where the integral over dyon separation inside the KvBLL caloron converges:
(
1
03
fugacity C N = exp
drl2
[-VT3F(v,T ) ]
(dyon weight) exp
2
379 where 4x2
V
v =27rT ’
F(u,T ) = - 3- - - v ~ ( ~ - u-)2~ m i.e. there is a mass gap in the linear spectrum. < m i.e. the potential is non-linear in such a way
that it supports oscillations with fundamental frequency in the mass gap. 2wB > Jm2 AX)^ i.e. the discreteness of the system imposes an upper bound on the linear spectrum, which allows the harmonics of the non-linear oscillation to lie beyond the linear spectrum.
+
398 I refer to this stabilization scenario as the frequency-mismatch mechanism. 3. Approximate Breathers in Continuum Field Theories
We1 have fruitfully adopted the ideas of discrete breathers into continuum field theory and find that the frequency-mismatch mechanism persists in a variety of models. Consider the strong inter-oscillator coupling limit of the chain of anharmonic oscillator discussed in the previous section. This corresponds to the continuum #4 theory with the equation of motion
4- + A#(& #I/
- w2) = 0 .
(17)
The mass gap in the linear spectrum persists and we have a theory of elementary bosons with mass
rn=dSw.
(18)
However, the linear spectrum has no upper bound now. For small curvature configurations, we may treat #(O,t) as a decoupled oscillator in a double-well potential. Large amplitude oscillation of #(O, t ) has a fundamental frequency, W B , in the mass gap suggesting stability against decay by elementary boson radiation. However, the very non-linearity that stabilizes the oscillation has a destabilization mechanism built in - harmonics of the fundamental frequency must be present and these can couple to linear modes. So, we retain two out of the three crucial ingredients that drive breathers as we transition to the continuum and at best we can expect to find approximate breathers in the continuum. These will not be absolutely stable, but as I show later, they are unnaturally long-lived. This heuristic of frequency mismatch with the linear spectrum does not require solitons to exist in the theory. This allows us to explore a large class of theories with massive particles and appropriate non-linear scalar potentials, including the physically interesting case of gauge theories with the Higgs mechanism of spontaneous symmetry breaking in 3+1 dimensions. In the rest of the talk I demonstrate how these ideas do indeed work and argue that there could be approximate breathers in the electroweak theory. 3.1. 1+1 Dimensions First consider scalar field theories in one spatial dimension with the potential
x ($2 - w2) 2 + -.w2$($ 1 Va($)= 4 2
- w)2 .
(19)
399 This describes a self-interacting scalar with mass
m=J X ' v . We choose the dimensionless vev v to be 1. We also choose X = 1 and SO all dimensionful quantities are measured in units of 6. For Q = 0, the asymmetry vanishes and the potential reduces to the symmetric doublewell, in which case we already know of approximate breathers from soft kink-antikink collisions. In order to emphasize that the frequency-mismatch mechanism of stability does not rely on underlying solitons forming bound configurations, we consider the case of CY > X/4 for which we have an asymmetric single-well at q5 = u and there are no solitonic solutions. We choose CY = XI2 and the initial static configuration
+(x,0) = tanh2(x/2).
(21)
Then we numerically evolve the configuration according to the equation of motion. q5(0,t) starts with a large deviation from Y and so the nonlinear potential could drive a low frequency oscillation (wg < m) with the configuration relaxing to an approximate breather. Since the initial configuration is a n even €unctions of z, it remains even throughout the time evolution according to the equation of motion. So we consider only the positive half-line with vanishing spatial derivative boundary condition at the origin. (In higher dimensions this condition is required for regular configurations.) In Fig. 1 I display some characteristics of the evolution of the initial configuration. In the left panel I plot the value of the field a t the origin as a function of time, towards the end of the time of evolution considered. This oscillates about q5 = u (asymmetrically because the potential is asymmetric). The fundamental period of oscillation is approximately obtained by measuring the interval between peaks and the corresponding fundamental frequency turns out to be W B M 1.43. This is lower than the mass of the scalar particle, m M 1.58, as expected. In the right panel I plot the total energy, Et, on the half-line as a function of time. This is of course conserved through the evolution. I also plot the energy, El, localized between x = 0 and 2 = 10. This falls very slowly and after a time of 10,000, only about 4% of the total energy has dissipated out of the local region. We do not yet understand what sets the scale for this decay rate. But it is clear that naturalness arguments are violated in this example and the energy is localized for time periods much longer than all dimensionfull scales in
400
the problem. The existence of such approximate breathers seems to be a generic feature of the theory, and we have successfully constructed several such configurations with different frequencies and amplitudes.
1.05 1.1
1.5
-
1 -
-*
0
8
W
1
0.9 -
0.95
L
0.5 0.85
I 9970
9980
9990
-
2-------I,
0.8
10000
t
Figure 1. Time evolution of the initial configuration in Eq. 21 in a 1+1 dimensional scalar theory with the potential in Eq. 19. In the left panel is the oscillatory d(0, t ) . In the right panel is the total energy, Et, on the positive half-line and the local energy, El, between z = 0 and x = 10, as a function of time. All dimensionful quantities are in units of fi,the vev v is 1, and the asymmetry parameter a is 0.5X.
Next consider 0 < a < X/4, which corresponds to an asymmetric double-well potential with again no solitonic solutions. We find that the initial configuration in Eq. 21 again relaxes to an approximate breather with all the signatures of the frequency-mismatch mechanism. The same happens in the $4 theory ( a = 0). Thus, instead of colliding kink-antikink to find an approximate breather (as discussed in Sec. 2.2), we obtain them using large amplitude, low frequency oscillations of $(O, t ) as above. We extend the $4 theory by considering complex $ and gauging the global U(1) symmetry to obtain the Abelian Higgs model in one spatial dimension. It is described by the action
S[$,A,] = / d 2 z [ D p $ t D p $ -
- v")'
-
;FpuFpu l
l
,
(22)
where
Spontaneous symmetry breaking results in a Higgs scalar with mass mH = 6
v ,
(24)
401
and the gauge boson eats the Goldstone degree of freedom to acquire mass
So the linear spectrum has mass gaps, as required. We find that the breathers in the real q54 theory are stable against perturbations in the complex gauged theory, as long as the U(1) charge is large enough so that the fundamental breather frequencies are lower than the mass of the gauge boson. However, when we reduce the charge so that the fundamental breather frequency falls within the spectrum of small gauge field oscillations, the breathers dissipate their energies rapidly. Again this is completely consistent with our picture. 3 . 2 . Towards the Electroweak Theory
The existence of approximate breathers is not a low-dimensional, toy-model peculiarity. We have found them within a spherical ansatz in 3+1 dimensions in the 44 theory, with all the signatures of the frequency-mismatch mechanism. In fact Gleiserg has demonstrated their existence in scalar field theories in 3+1 dimensions, for both symmetric and asymmetric double-well potentials. Now that we understand the mechanism that allows long-lived, approximately spatially localized, temporally periodic configurations to exist in a classical field theory, we can ask if there is any possibility for such objects to exist in the Standard Model. The answer is yes. When we ignore the hypercharge gauge fields in the electroweak theory and consider the S U ( 2 ) Higgs theory, there is no unbroken subgroup, and all three W gauge bosons are massive. So, the theory fulfills the first requirement of having a mass gap in the linear spectrum. Also, the Higgs potential is the usual doublewell potential and so it seems likely that we could set up large-amplitude configurations with oscillatim frequencies in the mass gap. The theory appears to have all the ingredients required for the existence of approximate breathers. When we include the U(1) sector, the photons remain massless after symmetry breaking. However, it is conceivable that the breathers can be made electrically neutral so that they don’t dissipate by electromagnetic radiation. 4. Conclusions and Discussion
We are currently investigating whether approximate breathers exist in the electroweak theory. They could have many significant implications. Firstly,
402
since the lifetimes of the breathers is expected to be many orders of magnitude larger than all scales in the problem, they could shed light on the notion of naturalness in physics. Secondly, if these breathers exist just after the electroweak phase transition, they would set up out-of-equilibrium regions in space, with implications for electroweak baryogenesis. There are several other intriguing questions to be answered. What sets the lifetime of approximate breathers? Is there a quantum particle association with these approximately periodic configurations? We continue to explore these issues.
Acknowledgments This work has been done in collaboration with E. Farhi and N. Graham. I am supported in part by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818.
References 1. E. Farhi, N. Graham, and V. Khemani. In preparation. 2004. 2. R. Rajaraman. Solitons and Instantons. Elsevier Science B. V, The Netherlands, paperback, fourth impression edition, 2003. 3. James N. Hormuzdiar and Stephen D. H. Hsu. Pion breather states in QCD. Phys. Rev., C59:889-893, 1999, hep-phJ9805382. 4. David K. Campbell, Jonathan F. Schonfeld, and Charles A. Wingate. Resonance structure in kink - antikink interactions in phi**4 theory. Physica, 9D:1, 1983. 5. Roger F. Dashen, Brosl Hasslacher, and Andre Neveu. The particle spectrum in model field theories from semiclassical functional integral techniques. Phys. Rev., D11:3424, 1975. 6. H. Segur and M. D. Kruskal. Nonexistence of small amplitude breather solutions in phi**4 theory. Phys. Rev. Lett., 58:747-750, 1987. 7. James N. Hormuzdiar and Stephen D. H. Hsu. On spherically symmetric breathers in scalar theories. 1999, hep-th/9906058. 8. D. K. Campbell, S. Flach, and Y. S. Kivshar. Localizing energy through nonlinearity and discreteness. Physics Today, page 43, January, 2004. 9. Marcel0 Gleiser. Pseudostable bubbles. Phys. Rev., D49:2978-2981, 1994, hepph/9308279.
QUANTUM NONABELIAN MONOPOLES
K. KONISHI Dipartimento d i Fisica, ‘%. Fermi” Universitci d i Pisa, Via Buonamoti, 2, Ed. C 56127 Pisa, Italy E-mail:
[email protected] We discuss quantum mechanical and topological aspects of nonabelian monopoles. Related recent results on nonabelian vortices are also mentioned.
1. Prologue There are several reasons to be interested in quantum, nonabelian monopoles. First, if confinement of QCD is a sort of dual superconductor, it is likely to be one of nonabelian variety. Then the effective degrees of freedom involve nonabelian, and not, better understood abelian monopoles. Second, the phenomenon of confinement has t o do with fully quantum mechanical, and not semi-classical, behavior of the monopoles. Thirdly, the very concept of nonabelian monopoles is, as we shall see, intrinsically quantum mechanical, in contrast t o that of the ’t Hooft-Polyakov monopole carrying an abelian charge only. A semi-classical consideration only might easily lead us astray. Finally, some recent developments on nonabelian BPS vortices provide further hints on the subtle nature of nonabelian monopoles and related dual gauge transformations. These considerations are sufficient motivations to give a renewed look on the topological as well as dynamical aspects of these soliton states, in particular in relation t o N = 2 gauge theories.
2.
Confinement in S U ( N ) YM Theory
The test charges in S U ( N ) YM theory take values in ( Z L ! ) ,ZLE))where ZN is the center of S U ( N ) and ZkM’,Z?) refer t o the magnetic and electric center charges. (ZLM’,Z?’) classification of phases follows Namely, 192.
403
404
(1) If a field with z
=
( a ,b ) condenses, particles X = ( A ,B ) with
( z , X )3 U B- b A # 0 ( m o d N ) are confined. (e.g. (q5(o,1)) # 0 -+ Higgs phase.) ( 2 ) Quarks are confined if some magnetically charged particle x condenses, (X(1,O)) # 0. (3) In the softly broken N = 4 (to N = 1) theory (often referred to as N = 1”)all different types of massive vacua, related by SL(2,Z ) , appear; the chiral condensates in each vacua are known. (4) Confinement index is equal to the smallest possible r E Zf’for which Wilson loop displays no area law. For instance, for S U ( N ) YM, r = N in the vacuum with complete confinement; r = 1 in the totally Higgs vacuum, etc. (5) In softly broken N = 2 gauge theories, dynamics turns out to be particularly transparent. We are particularly interested in questions such as: What is x in QCD? How do they interact? Is chiral symmetry breaking related to confinement? A familiar idea is that the ground state of QCD is a dual superconductor Although there exist no elementary nor soliton monopoles in QCD, monopoles can be detected as topological singularities (lines in 4 0 ) of Abelian gauge fixing, SU(3) + U(1)2. Alternatively, one can assume that certain configurations close to the Wu-Yang monopole ( S U ( 2 ) )
’.
A;
N
(a,n x n)a+ . . . ,
r n(r)= r
*
rj
AUa = caz3 r3
dominate 4 . Although there is some evidence in lattice QCD for “Abelian dominance”, there remain several questions to be answered. Do abelian monopoles carry flavor? What is L , f f ? What about the gauge dependence of such abelian gauge-fixed action? Most significantly, does dynamical S U ( N ) -+ U ( l ) N - ’ breaking occur? That would imply a richer spectrum of mesons (TI # T2, etc) not seen in Nature and not expected in QCD. Both in Nature and presumably in QCD there is only one “meson” state, EL1I qi qi), i e . , 1 state vs states. Note that it is not sufficient to assume the symmetry breaking S U ( N ) -+ U ( l ) N - ’ x Weyl symmetry, with an extra discrete symmetry, to solve the problem: the multiplicity would be wrong. If nonabelian degrees of freedom are important, after all, how do they manifest themselves?
[g]
405 3. “Semiclassical” Nonabelian Monopoles
Let us review briefly the standard results about nonabelian monopoles 9-18. One is interested in a system with gauge symmetry breaking
where H is non abelian. Asymptotic behavior of scalar and gauge fields (for a finite action) are:
4 A:
-
U . &Ut
N
4
U . ( 4 ) .U-’ Fij
-
N
rk
EijkT(D.
II2(G/H)= IIl(H); Hi E Cartan S.A. of G.
H),
Topological quantization then leads to 2 a . P c Z,
cfr. 2gegrn = n
pi = weight vectors of
(1)
(= dual of H ) ,
namely, the nonabelian monopoles are characterized by the weight vectors of the dual group H . A general formula for the semiclassical monopole solutions (set ( $ 0 ) = h . H) is given in terms of various broken S U ( 2 ) subgroups, 1
s1 = m(Ea
+ E-a);
S2 =
i --(Ea
- E-a);
S3
1CY*
.H;
the nonabelian monopoles are basically an embedding of the ’t HooftPolyakov monopoles in such S U ( 2 ) subgroups:
-
A i ( r ) = Aq(r,h * a ) Sa; $(r) = Xa(r,h a ) S,
+ [ h - (h . a )a*] H, (2) *
where (a*= a / ( a .a ) )
r-l A:(r) = ~ , i j7 A ( r ) ;
ra X”(r) = T x ( r ) l
x(00)
= h . a.
The mass and U(1) flux can be easily calculated:
M=
I
dS*Tr4B,
U ( 1 ) flux (for instance, for S U ( N
+ 1) + S U ( N ) x U ( 1 ) )is
Example of dual groups (defined by a H a*)are:
406
'
I
-
\
,
4. Some Examples The simplest system with nonabelian monopoles involves the gauge symmetry breaking,
The monopole solutions are
-;vo
0
$(r) and A ( r ) are BPS 't Hooft's functions with $(m) = 1, $(O) 0, A(m) = -l/r, where S is an S U ( 2 ) subgroup
=
or an analogous one in the (2 - 3) raws and columns. So in this case there are two degenerate S U ( 3 ) solutions. The generalization to the case with symmetry breaking SU(N /u
is straightforward. rows/ columns: then
0
0
+ 1)
...
+
S U ( N ) x U(1) ZN
,
0 \
Consider a broken S U ( 2 ) , Si living in (1,N
.. .
...
0
0
.. ...
+(N+I)u(S.F)+(r),
+ 1)
407
+
gives a monopole solution of SU ( N 1) equations of motion. By considering various SU(2) subgroups living in (ilN 1) rows/columns, i = 1 , 2 , . . . N , one is led t o N degenerate solutions. 5. Homotopy Groups in Sytems G
+
H
----t
Let us consider now the relevant homotopy groups. The short exact sequence
0 -+ .rrz(G/H)f .rrl(H)
-+
.rrl(G)
-+
0.
tells us that regular (BPS) monopoles represent .rr2(G/H) C n l ( H ) c 7r1 (G). Alternatively (Coleman) one can say that regular monopoles correspond to the kernel of mapping .rrl(H) -+ .rrl(G).In general, BPS monopoles belong t o a Ic t h tensor irrep of Ic E .rrl(H). The relation between 't Hooft-Polyakov (regular) monopoles and Dirac (singular) monopoles is illustrated in Figure l , which schematically represents the exact sequence above.
-
'T HooftlPolyakov
/
Dirac Figure 1
6 . Monopoles are multiplets of H
A crucial fact for us is that monopoles are multiplet of H and not of the original gauge group H . This is most clearly seen in the case of USp(2N 2 ) - 4 USp(2N) x U(1) where we find 2N 1 degenerate monopoles (of USp(2N)= SO(2N 1) !), or in the system SO(2n 3) -+ SO(2n 1) x U(l), where the multiplicity of degenerate monopoles is 2N (a right number for the fundamental representation of SO(2N 1 ) = USp(2N).) We have recently re-examined the possible irreducible representation (of the dual group H ) to which monopoles belong, in various cases. The results are shown in Table 1 taken from 19.
+
+
+
+
+
+
408
7. Why Nonabelian Monopoles are Intrinsically Quantum Mechanical Nonabelian monopoles turn out to be essentially quantum mechanical. In fact, finding semiclassical degenerate monopoles, as reviewed above, is not sufficient for us t o conclude that they form a multiplet of H , as H can break itself dynamically at lower energies and break the degeneracy among the monopoles. We must ensure that this does not take place. Nonabelian monopoles are in this sense never really semi-classical, even if (4) >> AH : (e.g., Pure N = 2, SU(3) ). In this connection, there is a famous “no go theorem” which states that there are no “colored dyons”16. For instance, in the background of the monopole arising from the breaking s usu ( 2 3) ) ~ u ( 1no ) , global T 1 ,T 2 ,T 3 isomorphic t o SU(2) can be shown to exist. These results have somewhat obscured the whole issue of nonabelian monopoles for some time. Do they not exist? Are they actually inconsistent? The way out of this impasse is actually very simple: nonabelian monopoles are multiplets of the dual H group, and the results of l6 does not exclude existence of sets of monopoles transforming as members of a dual multiplet (even if a.t present the explicit form of such nonlocal transformations are not known; see however below). Nevertheless, the no-go theorem implies that the true gauge group of the system is not
Ggauge #H8
409
as sometimes suggested, but H or H or something else, according t o which degrees of freedom are effectively present. (See also 12). 8. Phases of Softly Broken N = 2 Gauge Theories
Fully quantum mechanical results about the phases of SU(n,), U S p ( 2 n C ) and SO(n,) theories with nf hypermultiplets (quarks), perturbed by the superpotential
W ( 4 ,Q , Q ) = p”rTrQr2+ miQiQi, are known
20,21.
Deg.Freed. monopoles monopoles NA monopoles rel. nonloc.
mi -+
0
(See Table). Eff. Gauge Group U (1 y - l U (1 p - 1 S U ( T )x U ( l ) n c - r
Phase Confinement Confinement Confinement Confinement
Global Symmetry
Wnf) U ( n f - I) x U(1) U ( n f - T-) x U ( T ) U ( n f / 2 )x U ( n f / 2 )
NA monopoles
SU(&) x U ( l ) n c - n ~ Free Magnetic
Deg.Freed. rel. nonloc. dual quarks
Eff. Gauge Group
Phase
Global Symmetry
USp(25,) x U(l)nc-nc
Confinement Free Magnetic
SO(2 n f )
U(nf)
Wn,)
From these results we learn, in particular, that the spectrum of the “dual quarks” in the infrared theory (charges, multiplicity, flavor) is identical to what is expected from the semiclassical abelian or nonabelian monopoles. We note in particular that the r- vacua (i.e. vacua with a low-energy effective SU(T) gauge group) exist only for T < %, namely as long as the sign-flip of the beta function occurs:
bpa’)-2r + nf > 0 , 0:
bo
0:
-2n,
+ n f < 0.
Indeed, analogous r vacua exist semiclassically for all values up to min(nf,nc),but quantum mechanically, only those with T 5 n f / 2 give rise to vacua with nonabelian gauge symmetry. Also, when the sign flip is not possible (e.g. N = 2 YM or on a generic point of the quantum moduli space) dynamical Abelianization is expected and does take place! These observations led us to conclude that the “dual quarks” belonging to the fundamental representation of the infrared SU(T) gauge group,
410
actually are the Goddard-Nuyts-Olive-Weinberg monopoles, which have become massless by quantum effects 22. Most importantly, we are led to the general criterion for nonabelian monopoles to survive quantum effects: the system must produce, upon symmetry breaking, a sufficient number of massless flavors to protect H from becoming too strongly-coupled. Natural embedding in N = 2 systems for various cases in Table 1 has been discussed in Ref.19 A very subtle hint about the nature of the nonabelian monopoles come from the recent discovery of nonabelian vortices. 9. Vortices
Vortices occur in a system where a gauge group H is broken to some discrete group
such that I I l ( H / C ) is not trivial. Gauge field behaves far from the vortex axis as
Quantization condition reads a! . p dual of H Some known cases are: 0
c Z where pi
are weight vectors of H ,
H = U(1): in this case vortices correspond to the wellknown Abrikosov-Nielsen-Olesen vortices, representing elements of IIl(U(1)) = Z. According to the parameters appearing in the system they yield Type I, Type I1 or BPS superconductors; The case H = S U ( N ) / Z N yields ZN vortices. These are non BPS and are difficult to analyse (model dependence), although there are some interesting work on the tension ratios, the sine formula ( T k 0: sin $), etcZ3
10. Nonabelian Vortices
Truely nonabelian vortices (ie., with a nonabelian flux) have recently been constructed In the simplest case, we consider the system 24125.
41 1
The high-energy theory has monopoles; the low-energy theory (monopoles heavy) has vortices. We are here mainly interested in the low-energy theory ( s u ( 2ZZ) x u ( 1 )3 0,). We embed the system in a N = 2 model with number of flavor, 4 5 n f 5 5, so as to maintain the “unbroken” subgroup SU(2) non asymptotically free. We shall take the bare mass m and the adjoint scalar mass p a2,so that 212 = E = &Ti N f 2 2N, there appear vortices with 2 ( N - 1) - parameter family of zero modes, parametrizing
+
they nicely match the space of (quantum) states of a particle in the fundamental representation of an S U ( N ) . (Actually, for N f > N there are other vortex zero modes (semilocal strings), not related to the unbroken, exact S U ( N ) ~ + symmetry. F Those are related to the flat directions.) )x U(1) Furthermore, vortex dynamics ( S U ( 2zz --f 8) n -+ n(z,t)
can be shown to be equivalent to:
Spsl) = p
/
1
d2x 5
+ fermions :
an O ( 3 ) = CP' sigma model It has two vacua; no spontaneous breaking of S U ( ~ ) C +occurs; F Also, there is a close connection between the 2D vortex sigma model dynamics and the 4D gauge theory dynamics: they are dual to each other 29. In N = 2 theory, due to the presence of two independent scales which we take very different ( p N f 2 2 N , is fundamental. If N j were less than 2 N , the subgroup S U ( N ) would become strongly coupled, and break itself dynamically. Nonabelian vortices do not exist quantum mechanically in such a system. We are then led to the following relation between the vortex zeromodes and fi transformation of monopoles. Consider a configuration consisting of a monopole (of G / H ) and an infinitely long vortex, which carries away the full monopole flux. At small distances r from the monopole center, T O(l/m), HE theory is a good approximation and the monopole flux looks isotropic; at a much larger distances of order of, T > one sees the vortex of LE theory. The energy of the configuration is unchanged if the whole system is rotated by the exact HC+F transformation. This is a nonlocal transformation. The end point monopole is apparently transformed by the HC part only, but, since in order to keep the energy of the whole system unchanged it is necessary to transform the whole system, it is not a simple gauge transformation H of the original theory. It is in this sense that the nonlocal, global HC+F transformations can be interpreted as the dual transformation H acting on the monopole, at the endpoint of the vortex. -
+
+
-
i,
414
12. To conclude: where do we stand ? 0 0
0 0
0
Nonabelian monopoles are intrinsically quantum mechanical; Massless flavors are important for (i) keeping H unbroken; and (ii) for providing enough global symmetry giving rise to exact vortex zeromodes: these can be interpreted as the dual gauge transformation acting on the monopoles at the ends of the vortex; One has a nice ”model” of monopole confinement by vortices. Light nonabelian monopoles appear as IR degrees of freedom (examples in N = 2 models). Are there light nonabelian monopoles in some other N = 1 theories? Do some vacua of N = 2 theories, especially those based on “almost superconformal vacua” 30 provide a good model of confinement in
QCD? Acknowledgment The results reported here are fruits of enjoyable collaboration I had with Roberto Auzzi, Stefan0 Bolognesi, Jarah Evslin, Alexei Yung and Hitoshi Murayama. I thank Arkady Vainshtein and the organizers of the Workshop for their kind hospitality and for providing us with an occasion for interesting discussions with many participants.
References 1. G. ’t Hooft, Nucl. Phys. B138 (1978) 1; ibid B153 (1979) 141, B190 (1981) 455. 2. G. ’t Hooft, Nucl. Phys. B190 (1981) 455; S. Mandelstam, Phys. Lett. 53B (1975) 476. 3. F. Cachazo, N. Seiberg and E. Witten, JHEP 0302 (2003) 042, hepth/0301006. 4. T.T. Wu and C.N. Yang, in “Properties of Matter Under Unusual Conditions”, Ed. H. Mark and S. Fernbach, Interscience, New York, 1969, Y.M Cho, Phys. Rev. D21 1080 (1980), L.D. Faddeev and A.J. Niemi, Phys. Rev. Lett. 82 (1999) 1624, hep-th 9807069. 5. L. Del Debbio, A. Di Giacomo and G. Paffuti, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 231; A. Di Giacomo, hep-lat/0206018. 6. R. Dijkgraaf and C.Vafa, hep-th/0208048; F. Cachazo, M. R. Douglas, N. Seiberg and E. Witten, JHEP 0212 (2002) 071, hep-th/0211170. 7. T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845. 8. G. ’t Hooft, Nucl. Phys. B79 (1974) 276; A.M. Polyakov, JETP Lett. 20 (1974) 194. 9. E. Lubkin, Ann. Phys. 23 (1963) 233. 10. C. Montonen and D. Olive, Phys. Lett. 72 B (1977) 117. 11. P. Goddard, J. Nuyts and D. Olive, Nucl. Phys. B125 (1977) 1.
41 5 12. F.A. Bais, Phys. Rev. D18 (1978) 1206; B.J. Schroers and F.A. Bais, Nucl. Phys. B512 (1998) 250, hep-th/9708004; Nucl. Phys. B535 (1998) 197, hep-th/9805163. 13. E. J. Weinberg, Nucl. Phys. B167 (1980) 500; Nucl. Phys. B203 (1982) 445. 14. S. Coleman, “The Magnetic Monopole Fifty Years Later,” Lectures given at Int. Sch. of Subnuclear Phys., Erice, Italy (1981). 15. Chan Hong-Mo and Tsou Sheung Tsun, Nucl. Phys. B189 (1981) 364. 16. A. Abouelsaood, Nucl. Phys. B226 (1983) 309; P. Nelson and A. Manohar, Phys. Rev. L e t t . 50 (1983) 943; A. Balachandran, G. Marmo, M. Mukunda, J. Nilsson, E. Sudarshan and F. Zaccaria, Phys. Rev. Lett. 50 (1983) 1553; P. Nelson and S. Coleman, Nucl. Phys. B227 (1984) 1. 17. K. Lee, E. J. Weinberg and P. Yi, Phys. Rev. D 54 (1996) 6351, hepth/9605229. . 18. C. J. Houghton, P. M. Sutcliffe, J.Math.Phys.38 (1997) 5576, hepth/9708006. 19. R. Auzzi, S. Bolognesi, Jarah Evslin, K. Konishi, H. Murayama, hepth/0405070. 20. A. Hanany and A. Oz, Nucl. Phys. B452 (1995) 283, hep-th/9505075, P. Argyres, M. Plesser and N. Seiberg, Nucl. Phys. B471 (1996) 159, hep-t h/9603042, 21. G. Carlino, K. Konishi and H. Murayarna, JHEP 0002 (2000) 004, hepth/0001036; Nucl. Phys. B590 (2000) 37, hep-th/0005076; G. Carlino, K. Konishi, Prem Kumar and H. Murayama, Nucl. Phys. B608 (2001) 51, hep- th/Ol04064. 22. S. Bolognesi and K. Konishi, Nucl. Phys. B645 (2002) 337, hepth/0207161. 23. M. Douglas and S. Shenker, Nucl. Phys. B447 (1995) 271-296, hepth/9503163, A. Hanany, M. Strassler and A. Zaffaroni, Nucl.Phys. B513 (1998) 87, hep-th/9707244. B. Lucini and M. Teper, Phys. Rev. D64 (2001) 105019, hep-lat/0107007, L. Del Debbio, H. Panagopoulos, P. Rossi and E. Vicari, Phys. Rev. D65 (2002) 021501, hep-th/0106185; JHEP 0201 (2002) 009, hep-th/0111090, C. P. Herzog and I. R. Klebanov, Phys. L e t t . B526 (2002) 388, hep-th/0111078, R. Auzzi and K. Konishi, New J. Phys. 4 (2002) 59 hep-th/0205172. 24. A, Hanany and D. Tong, JHEP 0307 (2003) 037, hep-th/0306150. 25. R. Auzzi, S. Bolognesi, Jarah Evslin, K. Konishi and A. Yung, Nucl. Phys. B to appear, hep-th/0307287. 26. V. Novikov, M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rep C 116 (1984) 103. 27. K. Hori and C. Vafa, hep-th/0002222. 28. R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, Nucl. Phys. B686 (2004) 119, hep- t h/03 12233. 29. A. Hanany and D. Tong, JHEP 0404 (2004) 066, hep-th/0403158; M. Shifman and A. Yung, hep-th/0403149. 30. R. Auzzi, R. Grena and K. Konishi, Nucl. Phys. B653 (2003) 204, hepth/0211282, R. Auzzi and R. Grena, hep-th/0402213.
DILUTE MONOPOLE GAS, MAGNETIC SCREENING AND K-TENSIONS IN HOT GLUODYNAMICS
C. P. KORTHALS ALTES
Centre Physique The‘orique au CNRS, Case 907, Luminy, F13288, Marseille, France
E-maihltes @cpt.univ-rnrs.fr A dilute monopole gas explains, in quarkless gluodynamics, the small ratio 6 between the square of magnetic screening mass mM and spatial Wilson loop tension. This ratio is 0.0895 for T = 0 to 0.0594 a t T = ca for any number of colours with order N-’ corrections and equals up to a numerical factor of 0(1)the diluteness. The monopoles have a size 1~ = m;’. The GNO classification tells us they are in a representation of the magnetic SU(N) group. Choosing the adjoint for the dilute gas predicts the k-tensions to scale as k ( N - k ) , within a percent for high T, and a few percent for low T for the seven ratios determined by lattice simulation. The transition is determined by the transition of the dilute Bose gas at Tc = 0 . 1 7 4 m ~ , and the transition is that of a non- or nearly-relativistic Bose gas.
1. Introduction
An ancient idea in QCD is that monopoles are responsible for flux tube formation through a dual superconductor mechanism. Another mechanism, that of the “Copenhagen vacuum” of the early eighties 2 , proposes macroscopic Z(N) Dirac strings or Z(N) vortices. In this talk I will discuss a specific model of the first type 4 , that works surprisingly well in its most direct applications. A straightforward way to see its workings is to start from the plasma phase. That sounds at first sight self-defeating as there is no confinement in this phase. But it will turn out that precisely the absence of confinement renders the detection of these monopoles, or perhaps more appropriately, magnetic quasi-particles, so straightforward. We know that electric quasi-particles, the gluons, are approximately free at very high temperature. This follows from the Stephan-Boltzmann law. We may guess that the same happens to the magnetic monopoles that were condensed in the cold phase. What kind of monopoles can we expect? In the absence of any spon-
41 6
417
taneous breaking the GNO classification applies. For a theory with only Z(N) neutral fields, like the adjoint, the magnetic group is S U ( N ) . SO we can have monopoles in the fundamental, adjoint or any representation we like. One of the consequences of the idea of a monopole condensate is the screening of the magnetic Coulomb force between two static magnetic sources. This is a straightforward generalization of the screening of the Coulomb force between two heavy quarks. There is now ample affirmation of screening l 3 21 from lattice simulations. It is reasonable to think of the monopole as having a size of about the screening length 111.1 = rn;. We will assume that its size is much smaller than the inter-monopole distance. That, together with the choice of multiplet, will fix the ratios of spatial string tensions. The reader who is only interested in how one computes these ratio’s should read sections 2,4 and 5. A more complete version appeared in hep-ph/0408301. 2. Electric flux loops
In this section we discuss the behaviour of electric flux loops in the plasma, as it will explain the basic features that are permeating the talk. 2.1. &ED
Consider a plasma of ions and electrons. We will take the ions to have the same but opposite charge of the electron. Suppose we want to compute the electric flux going through some large (with respect to the atomic size) closed loop L with area A(L). Normalize the flux @ = Js ds’ . l? by the electron charge e and define :
V ( L )= exp i27r@/e.
(1)
Of course, at T below the ionization temperature no flux would be detected by the loop, because there are only neutral atoms moving through the loop. Let us now raise the temperature above Tionisation. What will happen? Both electrons and ions are screened. For simplicity we will take the ions to have the opposite of one electron charge. We are going to make the following simplification. The charged particles are supposed to shine their flux through the loop if they are within distance ID from the minimal area of the loop. This defines a slab of thickness 210. Here we will keep the factor 2.
418
Then one electron (ion) on the down side of the loop will contribute +1/2(-1/2) to the flux, and with opposite sign if on the up side of the loop. That is: V(L)lonecharge = -1. This result is independent of the sign of the charge! The plasma is overall neutral, the loop is sensitive only to charge fluctuations. For 1 charges inside the slab the flux adds linearly and we find:
V(L)II = (-l)!
(2)
Assuming that all charges move independently, the average of the flux loop V ( L ) is determined by the probability P(1) that 1 electrons (ions) are present in the slab of thickness 21D around the area spanned by the loop. Taking for P(1) the Poisson distribution exp -1- dis the average number of electrons (ions) in the slab- we find for the thermal average of the loop due to both ions and electrons:
i(d)'
(v(L))T
=
C P ( ~ ) V ( L )=J ~C ~ ( l ) ( - ) ' I
= exp -41.
(3)
1
Now = A ( L ) ~ Z D ~ ( Tso) , the electric flux loop obeys an area law exp -p(T)A(L), with a tension
p(T) = 81~7%(T).
(4)
This area law distinguishes the behaviour of the loop in the plasma from that in the normal unionized state. The flux loop has a very useful alternative formulation as a Dirac flux along the border L. 2.2. Electric flux in gluodynarnics
Is there an S U ( N ) generalization of the QED case? In fact yes. The closed Dirac string in QED is replaced by a Z(N) vortex of strength k, the 't Hooft loop3. We introduce in the Lie algebra of traceless hermitean N x N matrices a basis for the Cartan subalgebra, the (N-1) x (N-1) dimensional subspace of diagonal matrices. This basis of N - 1 diagonal matrices Yk is chosen such that in exponentiated form exp ( - i27~Yk)it gives the N - 1 centergroup elements e x p i k 9 . A simple choice is 1 N
Yk = --diag(N
- k, . . , ...., N
- k, -k,. . ....., -k).
(5)
The entry N - k comes in k times, and -k comes in N - k times, so the trace is 0.
419
The flux operator becomes in this notation:
It does correspond to the vortex operator of 't Hooft with strength Ic. The next question is: does the gas of deconfined gluons induce an area law in this operator? The answer is yes, and the reasoning is as before. The charge of a gluon with respect to the charge Y k is found from the form Y k takes in the adjoint representation. This is easy: we have charge f l like in QED, but unlike in QED we have 2k(N - Ic) gluon species in the gluon multiplet with such a charge. All other gluons have vanishing charge so do not contribute to the flux. Since we take the gluons to be statistically independent, the charged ones all contribute a factor to ( v k ) , and this factor is the same by S U ( N ) symmetry: exp -21~n. Conclusion: the expectation value of the loop is
with the tension
So this is the k-scaling law for the electric flux loop. It does obey large N factorization:
This computation is corroborated by low order perturbation theory for the loop. The expectation value of the loop can be computed from a tunneling process between adjacent vacua of an effective potential with a Z ( N ) symmetry 1 9 , 2 0 . This potential has, because of the Z ( N ) symmetry, degenerate minima in the centergroup elements. The propagators feel the coloured background, so the double line representation does not apply. So the expectation of pure gluodynamics that only 1 / N 2 corrections appear is not valid. 3. Some basic facts in quarkless Yang-Mills
In this section we gather a few facts, partly empirical, from lattice simulations, and partly from simple physics arguments.
420
3.1. Magnetic screening
A feature that sets gluodynamics apart from QED is the appearance of magnetic screening. When we put two heavy monopoles far apart at a distance T we find a Yukawa type potential: C
V ( T )= - exp -mMr. T
(10)
A simple argument shows
that the magnetic mass is the mass of a scalar excitation of a Hamiltonian in a world with two large space dimensions and one periodic mod 1/T. The space symmetries in such a world are SO(2) x P x C x R. We have the rotation group in the two large dimensions, C is charge conjugation, P is parity in the large 2d space, and R is the parity in the periodic direction. As T + 0 the rotation group becomes S 0 ( 3 ) ,and R and P become related through a rotation, so at T = 0 the Of+ glueball mass results. 3.2. Spatial tension
The spatial tension is defined through a spatial loop L, on which an ordered product P exp i $ d$.AR lives. R labels the representation for the vector potential we have chosen. The spatial Wilson loop is then defined as the trace over colour: WR(L) = T r P e x p i
dZ.AR.
The path integral average gives an area law: (WdL)) = Cexp ( - d T ) A ( L ) )
(12)
for all temperatures. At zero temperature the spatial tension equals the string tension a from a time-space loop because of Euclidean rotation invariance. But as function of temperature they behave differently. The string tension suffers a correction at low temperature of -:T2 due to the excitations contributing the Luescher term. Above T, it vanishes. The spatial tension stays flat, see fig. (1). Like the magnetic mass it does not feel the Z ( N ) transition and starts to scale like (g2T)2well above the critical temperature. The dimensionless ratio of the spatial tension in the fundamental representation and the square of the magnetic mass is small, on the order of a few percent. In two extreme cases the ratio 6(T) is accurately known from simulations : at T = 00 in the 3d theory, and at aIn what follows we reserve the name string tension for the time-space loop.
42 1
T = 0 lo. In between there are simulations l 3 21 that are consistent with a slowly varying S(T). It is very important to have this verified with more precision.
TC
Tg
T
Magnetic mass mM and tension u as function of temperature, schematically. mo++and from ref. (13): Tc = 0.174 mo++.The temperature Tq 5 mo++ is where the de Broglie thermal wave length becomes equal to the magnetic screening length. For the calculation of the tension it is below Tq that quantum statistics applies, above classical statistics applies as in section 5. :
3.3. Dependence of the tension on the representation, and N-allity
There is widespread agreement on the dependence of the tension and the string tension on the representation R. It is only its N-allity k that counts. If we think of R being built up by f fundamental and f anti-fundamental representations, the N-allity is just the difference:
k = f -J.
(13)
To get a more precise idea, let us imagine the Wilson loop is formulated in a periodic box, with the 3 space dimensions being of macroscopic length L. The fourth dimension is of length l/T. The loop is now replaced by
422
two Polyakov lines of opposite orientation in the x-direction. They are a distance r apart- and carry the reprsentation R. The expectation value of the loop is then parametrized as:
( W R ( L )= ) Cexp -LVT(T).
(14)
If the distance T is very small, asymptotic freedom tells us:
where C2(R) is the quadratic Casimir operator. In three dimensions the Coulomb force is logarithmic. The Casimir operator can be related to our N x N matrices Y k if R corresponds to a Young tableau, with the first row having w1 more boxes than the second. The second row has w2 more boxes than the third, and so on. By definition, the numbers 'Wk are never negative. There is a one-to-one correspondence between Young diagrams and irreducible representations of SU(N ) . Define the highest weight HR of R as k
as we already alluded to at the end of section 2.2 , Then, if Y = Yk : 1 C2(R) = ~ ( T T H ; ~ T T Y H R ) .
ck
+
(17)
If the distance is long enough ( a somewhat ambiguous criterion!) the string regime sets in and we expect: v(T)
=ak(T)T.
(18)
The tension Ck only depends on the N-allity k. String formation between the two sources renders this dependence very plausible. The tension (TI of the string formed in between two fundamental sources can form a bound state in the string with tension u2. We can go on like this and the question is, what is the dependence on k? It should obey certain a priori criteria. For large number of colours we expect factorization ffk
= ka1.
From general arguments one also expects ffk
= kffl
(19) l4
in that same limit:
+ o(l/N2).
This is discussed in depth by Misha Shifman in this volume.
(20)
423
3.4. Flux representation of the Wilson loop
If we want to proceed with the Wilson loops, as we did with the 't Hooft loops, we need clearly a representation similar to eq.(6). Of course a natural guess is to take this electric flux formula, replace E' by 6, a = g 2 / 4 r by l / a to get:
This formula cannot be true 20. But it is true in a dynamical sense: consider the limit of a path integral average in the 3d theory with an adjoint Higgs, the electrostatic QCD Lagrangian. For a given representation R with highest weight H R with stability group S (i.e. S H R S - ~= HR)one finds: (wR)
=< J DRexp (ig
J~s.(z,
- -1f a b c f i n b A f i j n , ) Xa ~ r - - ~ ~ R> ~ .+ )
9
2
(22) with na = T T % R H R R ~the Higgs field's angular part that parametrizes the coset S U ( N ) / S ( H R ) .The second term in the exponent is the source term for the monopoles in the sense that it carries no long range effects in the original Higgs phase. The average of the r.h.s is the 3d Yang-Mills average. The integration is over all gauge transforms with an arbitrary number of hedge hog configurations. The r.h.s. is obtained l6 by taking the average of the magnetic charge operator in the 3d Higgs phase characterized by H R . Then the the VEV of the Higgs field is let to zero, which introduces the fluctuations over na, the angular components of the Higgs field. Finally one lets the mass of the Higgs become infinite, which suppresses the radial integrations. On the other hand l 5 the integrand of the r.h.s., without the brackets, was shown by Diakonov and Petrov to equal WR,by one dimensional quantum mechanics methods. It involves a limiting procedure which is procured in the path integral by starting out from the Higgs phase and moving to the symmetic phase as described above. With this specification we will use eq. (21) without quotation marks in what follows. 4. Predictions for the Wilson loops
Once we are given a dilute gas of screened monopoles, we only need to specify the representation of the magnetic group for our monopoles. Then the calculation of the tension is done with our flux representation for the
424
average of the loop, eq.(22). But in practice we can use the simple formula eq.(21). The reason is that we put the contribution of the monopole gas in by hand. That takes care of the singular gauge transforms in eq.(22). The remaining regular ones can be dropped because we compute something gauge invariant. So the average of a k-loop in the totally anti-symmetric representation is computed from (23) For the dilute gas in the adjoint representation the computation is not different from the one with gluons in section 2.2. The adjoint monopoles have a magnetic charge equal to f?, or 0. Only the former contribute and their individual contribution to the loop in eq. (23) is -1. The diluteness, and classical statistics (as the thermal wave length is T-l and the typical interparticle distance is ( g 2 T ) - l this is justified, see also fig. 1.) then give for every charged species the same contribution exp -2lMnM, nM being the common density of a given species. As there are 2k(N - k) charged species in the adjoint, the total results in the k-tension being: LTk
= 4k(N
- k)lMnM.
(24)
One can do a similar calculation for monopoles in the fundamental representation. The counting goes now as follows. Recall eq.(5), Y k = $diag(N - Ic, ....N - k,- k , .... - k) with trace zero. The Y k charge of the highest Ic components of the N components of the column spinor is The remaining N - k components have charge given by It then follows from the use of the Poisson distribution that the flux of a given component is contributing c o s ( n 9 ) or cos(r6). Taking into account the degeneracies the final result for the tension becomes:
9.
-6.
k N
ck = dMnM(N - k cos(np ( N - k)) - ( N - k) cos(n-)).
N
Both tensions give ratios
L T ~ / L Tthat ~
(25)
behave like k for large N and finite
k. This factorization is expected for a k-loop tension caused by screened particles , But for large N and 5 = N/2 the ratio for the fundamental multiplet is a factor 2 larger. bFactorization is not obvious if uncorrelated Z(N) vortices cause the area law: uk N (1 - c o s ( 2 k a / N ) ) . This is due to the macroscopic size of the vortex perimeter causing long range correlations between loops.
425
5. Comparison to lattice simulations
We have been discussing a model at very high temperature. Hence it is tested in 3d lattice simulations. The ratios found l 2 for the totally antisymmetric irreps are close - within a percent for the central value - as far as the adjoint multiplet of magnetic quasi-particles is concerned : SU(4) : 02/01
= 1.3548 f 0.0064
adjoint : 1.3333
fundamental : 1.8182
SU(6) : 02/01
= 1.6160 f 0.0086
adjoint : 1.6000
fundamental : 1.9686
adjoint : 1.8000
fundamental : 2.3635
uS/u1 = 1.808 f 0.025
The results are that precise, that you see a two standard deviation from the adjoint, except for the second ratio of SU(6). This deviation is natural, since the diluteness of the magnetic quasi-particles is small, on the order of a couple of percent, as we will explain at the end of this subsection. So we expect corrections on that order to our ratios. There is a less precise determination of the ratio 02/01 = 1.52 f 0.15 in SU(5) 17. But the central value is within 1 to 2% of the predicted value 3/2 from the adjoint. The fundamental gives a ratio 1.8231. The SU(8) ratios are known on a rather course lattice l7 and using a different algorithm: 02/01
= 1.692(29)
adjoint : 1.714
fundamental : 2.106
03/01
= 2.160(64)
adjoint : 2.143
fundamental : 2.958
04/01
= 2.26(12)
adjoint : 2.286
fundamental : 3.256
In conclusion: the seven measured ratios are consistent with the quasiparticles being independent, as in a dilute gas and in the adjoint representation. The number of quasi-particle species contributing to the k-tension is 2k(N - k). This number happens to coincide with the quadratic Casimir operator of the anti-symmetric representation. The fundamental monopoles are clearly disfavoured by the data. 6. Conclusions
There is remarkable agreement with an accuracy of about a percent between numerical simulation at high T and the dilute adjoint monopole gas. This diluteness appears as a small parameter for every SU(N) theory. Some parameters, like the mass of the magnetic quasi-particle, are still within a
426
large range, although its size is 1 ~ Is. it heavy, on the order of the lowest , the transition is that of a non-relativistic glueball mass (i.e. m ~ )then Bose gas. Is it light (with respect to its size) then the transition is that of a near-relativistic BE transition. Perhaps the best way to summarize our approach is to return to fig. 1. It is clear that the relation: CJ
-
(26)
ZMnM
implies that Z&CJ is the diluteness 6, and must be small for consistency. And lattice data have borne that out! Now , once we accept the idea, that on the high temperature side of the transition a dilute gas describes tensions so well, the constancy of that diluteness down till T = 0 suggests that all what changes is the thermal wavelength AB(T). At temperatures on the order of the glueball mass or higher it is clear that the interparticle distance O(l/g2T)is larger than the thermal wave length 1/T, because the coupling is so small. But at a temperature Tq the thermal wave length takes over with Tq mmonopo~e. It is very unlikely that Tq will be below T,,since that would imply the BE transition as a second one below T,. So the monopole mass should not be lower than T,, so the transition will then be a non- or near-relativistic BE transition. Below Tq the dilute gas gives a contribution to our ratios, but now determined by Bose statistics. For an analysis of the cold phase ratios see ref. 18. Of course for most observables in the low T phase the Bose statistics is all-important. Thus the string tension will become non-zero below T,, the character of the transition, i.e. a jump or continous behaviour in the occupation fraction of the p’ = 0 states will be crucial to know and and is calculable in this model. Realistic QCD involves quarks, and there is all reason t o believe they couple strongly to our monopoles. After all, the latter are bound states of magnetic gluons. Note that a real time picture of our quasi-particles is lacking. According to our Euclidean picture they become, at very high T , a 3d gas of particles with small size ZM and small interparticle distance. The real time picture is a challenge. Adi Armoni, Luigi del Debbio, Pierre Giovannangeli, Christian Hoelbling, Dima Kharzeev, Alex Kovner, Mikko Laine, Biagio Lucini, Harvey Meyer, Misha Shifman, Rob Pisarski, and Mike Teper provided me with
-
427 useful comments. I thank t h e organizers for their invitation, a n inspiring meeting, and for wonderful hospitality.
References 1. G. 't Hooft, in High Energy Physics, ed. A. Zichichi (Editrice Compositori Bologna, 1976); S. Mandelstam, Phys. Rep. 23C (1976), 245 2. H. B. Nielsen, P. Olesen, Nucl. Phys.B61, 45,(1973); Nucl. Phys.Bl60, 380 (1979). 3. G.'t Hooft, Nucl.Phys.Bl38, 1 (1978). 4. P. Giovannangeli, C.P. Korthals Altes, Nucl.Phys.B608:203-234,2001; hepph/0102022. 5. P. Goddard, J. Nuyts, D. Olive, Nucl.Phys B125,1 (1977). S.Coleman, Erice lectures 1981. 6. G. 't Hooft, Nucl. Phys.B 105, 538 (1976). 7. P. de Forcrand, C. P. Korthals Altes, 0. Philipsen, in preparation 8. C. P. Korthals Altes, 2003 Zakopane lectures,hep-ph/0406138. 9. M. Teper, Phys.Rev.D59, 014512, 1999; hep-lat/9804008. 10. B. Lucini, M. Teper, JHEP 0106,050 (2001), M. Teper, hep-th/9812187 . 11. B. Lucini, M. Teper, U. Wenger, JHEP 0401:061,2004, hep-Iat/0307017. 12. M. Teper, B. Lucini, Phys.Rev. D64 (2001) 105019. 13. C. Hoelbling, C. Rebbi, V.A. Rubakov, Phys. Rev.D63 :034506,2001; heplat/0003010. 14. A. Armoni, M. Shifman, Nucl.Phys.BB71, 67, 2003; hep-th/0304127, hepth/0307020. 15. D. Diakonov, V. Petrov, Phys. Lett.B242, 425 (1990); see also heplat/0008004. 16. C.P. Korthals Altes, A. Kovner, Phys.Rev.D62:096008, 2000; hepph/0004052 17. H. Meyer, private communication and hep-lat/0312034. 18. C. P. Korthals Altes, H. Meyer, to be published. 19. T. Bhattacharya, A. Gocksch, C. P. Korthals Altes and R. D. Pisarski, Phys.Rev.Lett.66, 998 (1991); Nuc1.Phys.B 383 (1992), 497. 20. C.P. Korthals Altes, A. Kovner and M. Stephanov, hep-ph/99, Phys.Lett. B469, 205 (1999), hep-ph/99095 16. 21. Ph. de Forcrand, M. D'Elia, M. Pepe, Phys.Rev.Lett.86:1438,2001;heplat /0007034. 22. For a related problem see: A. Kovner, B. Rosenstein, 1nt.J.Mod. Phys. A7(1992), 7419.
SUPERSIZING WORLDVOLUME SUPERSYMMETRY: BPS DOMAIN WALLS AND JUNCTIONS IN SQCD
A. RITZ Theory Division, Department of Physics, CERN, Geneva 23, CH-1211, Switzerland We study the worldvolume dynamics of BPS domain walls in N= 1 SQCD with gauge group SU(N) and Nf = N flavors, and exhibit an enhancement of supersymmetry for the reduced moduli space associated with broken flavor symmetries. The worldvolume superalgebra corresponds to an N= 2 Kahler sigma model in 2+1D deformed by a potential, given by the norm squared of a U(1) Killing vector, resulting from the flavor symmetries broken by unequal quark masses. This framework leads t o a description of 1/4-BPS two-wall junction configurations as l/P-BPS worldvolume kinks.
1. Introduction and Summary
Solitonic solutions of supersymmetric field theories generally exhibit a moduli space M which locally admits the decomposition
M
2~
Msusy x
G.
(1)
Here Msusy refers t o the sector associated with bosonic generators in the supersymmetry (SUSY) algebra which are broken by the soliton, by virtue of the introduction of central charges [l],and in flat space always includes a translational component Rd c M S U S Ywhere , d is the codimension. The realization of supersymmetry in this sector, associated with the unbroken generators, is then fixed by the kinematics of the bulk superalgebra. In contrast, G - the ‘reduced moduli space’ - is not directly associated with broken super-translation generators. This implies that the realization of worldvolume supersymmetry in this sector is less constrained and, as discussed below, can exhibit more supersymmetry (at least in the twoderivative sector) than the translation sector associated with Msusy. The origin of this supersymmetry enhancement is that not all the supercharges which are realized on the worldvolume of the soliton lift to supercharges in the full theory. The additional charges arise purely due to geometric
428
429
features, e.g. a Kahler structure, of the reduced moduli space. As a simple exmaple, we will consider BPS kinks in an N = 1 sigma model on S3, where the kink profile lies entirely within an S1 fibre of S 3 . The kinks then 21 S2 which is the base of this exhibit a nontrivial reduced moduli space fibration. The latter manifold is Kahler and thus the low energy dynamics exhibits enhanced N=2 supersymmetry, despite the soliton being 1/2-BPS. A primary motivation for studying this phenomenon is that it arises rather naturally in the context of BPS domain walls in N=1 SQCD with a sufficiently large number of flavors. With gauge group S U(N) accompanied by N f = N funadamental flavors, with masses small compared to the dynamical scale AN, the low energy description on the Higgs branch in terms of meson and baryon moduli corresponds to a massive Kahler sigma model on the manifold determined by the constraint [2]
c
detM - B B
= ACN
.
(2)
The theory possesses N quantum vacua, distinguished by the phase of the superpotential which is a multiple of 27r/N [3,4,5,6,7], and between which 1/2-BPS domain walls can interpolate [8,11,12,13,14,15]. In this theory, BPS k-walls (walls which interpolate between vacua differing in phase by 27rk/N) were shown [9] to exhibit a nontrivial classical reduced moduli space c k due to localized Goldstone modes associated with the flavor symmetries which are broken by the wall solution. The corresponding coset is a complex Grassmannian [9],
One can then formally deduce that the multiplicity of k-walls, vk [lo], is given by the worldvolume Witten index for this Grassmannian sigma model, which depends only on the topology of the space, and is given by the Euler characteristic (see also [16]),
The fact that the reduced moduli space is Kahler is significant here. Since the worldvolume theory lives in 2+1D, the dual constraints of (i) a Kahler target space, and (ii) Lorentz invariance, imply that the low energy dynamics must preserve N= 2 supersymmetry, namely four supercharges! Since only two of the bulk supercharges are preserved by a 1/2-BPS state, we see that this implies sueprsymmetry enhancement on M k . We will show
-
430 below that, for the simplest SU(2) example, this feature can be straightforwardly understood as an extension of the enhancement observed for kinks in the S3 sigma model. The relevant point is that the nontrivial part of the Higgs branch (2) is T * ( S 3 )and , BPS walls can be understood as kinks embedded in the S3 base of this manifold. In the next three sections we describe several aspects of this phenomenon (see [17] for further details). In section 2, supersymmetry enhancement on the reduced moduli space of BPS kinks in the perturbed S3sigma model is illustrated in detail. We then turn in Section 3 t o BPS walls in hl= 1 SQCD and describe how a similar structure arises for gauge group SU(2) with two flavors. In this case, detuning the two quark masses, as is required for the vacua to remain a t weak coupling, has the effect of inducing a potential on the moduli space given by the norm-squared of a U(l) Killing vector. This potential is nonetheless consistent with the enhanced supersymmetry. Finally, in Section 4 we consider the worldvolume interpretation of intersecting domain walls, as worldvolume BPS kinks, and verify that the tension deduced in this way is equal to the bulk central charge. Although we will focus here just on BPS kinks and walls, it seems likely that this mechanism for supersymmetry enhancement will arise in other contexts. For example, N= 1 perturbations of the vortices recently studied in N= 2 SQCD [18], should still preserve a Kahler reduced moduli space, and it would be intersting to verify whether the worldvolume theory realizes (0,2) or an enhanced (2,2) supersymmetry in this case. More generally, the “unwinding” of the target space fibration by the soliton described above as the underlying mechanism for the enhancement in the models considered here has some analogy with gauging the U(1) isometry of the fibre. This again leads to a Kahler sigma model a t low energies, albeit now in the same dimension. This unwinding is also known to occur under T-duality, and it would be interesting t o explore this connection in more detail. 2. Supersymmetry enhancement and worldvolume moduli We can exhibit the enhancement phenomenon for the low energy dynamics on M rather transparently in a simple lfl-dimensional model. Consider an N= 1 sigma model with target space S 3 , with coordinates qP = (13,E , 4 } , ds2 = T [do2
+ sin2 I3 (dJ2 + sin2 Jdq52)] .
(5)
We also turn on a (real) superpotential,
W ( $ )= mcosl9
,
(6)
43 1
which depends on only one of the angular coordinates parametrizing the S 3 . The theory then has two vacua a t 0 = 0 , ~ . Classical BPS kinks exist which interpolate between the two vacua, having mass Msol = 2 = 2rn and satisfying the Bogomol’nyi equation, = g a b a b w ( $ ) . Solutions have the sine-Gordon form
a,@
osoi ( z ) = 2 arctan
[exp
m (- r
(2
- .a))],
tsoi =
to,
= $0
, (7)
exhibiting three bosonic moduli { ZO,6 ,$o}. These bosonic moduli are Goldstone modes for the symmetries broken by the wall: zo is associated with the breaking of translation invariance; and $0 arise from the SO(3) global symmetry of the target space which is preserved in the vacua but broken to SO(2) by the kink solution. &, and $0 thus coordinatize the coset S 0 ( 3 ) / S 0 ( 2 ) 2~ S 2 , as may be verified by computing the induced metric for the bosonic zero modes [19,20],
d.sL = 2m d z i
+ hijdxidxi
= 2m d z i
2r2 +[dci + sin2 Jd&] m
,
(8)
where hij is the metric of the reduced moduli space %. The bosonic moduli space is thus
M
=R
x
xi = IR x S 2 ,
(9)
with the natural metric on each factor. are Let us now consider the fermionic sector. The S3 coordinates partnered under N=1 SUSY by a set of two-component Majorana spinors, $:, a = 1,2. For each bosonic zero mode 22, one finds a corresponding (one-component) fermionic partner vi in the lower component of $,:
$tol= r l i z (;) +- non-zero modes . Only one of these modes is guarunteed to exist by virtue of the fact that the solution is classically 1/2-BPS and thus breaks one of the two supercharges. The broken supercharge is realized as &I = 2 2 f in terms of this ‘goldstino’ mode. Here $ is the superpartner of zo. The novel feature of this system is that the reduced moduli space % is a Kahler manifold and, since the bosonic and fermionic zero-modes are paired, exhibits N= 2 supersymmetry! One of these supercharges is Q 2 , the unbroken charge present in the bulk theory, while the second which we will call &2 exists only due to the complex structure J associated with %.
432
We can represent the supercharges in the rest frame as
and, noting that {qi,$} = hij, one can verify that they satisfy the algebra of N = 2 SQM,
{Q',
eJ>=
WSQMGI~
,
(12)
where WSQM = ( M - Z ) is the worldline Hamiltonian. Introducing the complex coordinate CO w = eido tan -
(13)
2
on
c, and its fermionic partner (14)
we can rewrite the algebra in the form
{ Q , Q*> = WSQM ,
(Q)2
= ( Q * l 2=
o,
(15)
where
In this specific example, one can show that on quantization there are no supersymmetric vacua, and thus no quantum BPS kinks, since ( Q 2 ) 2 is bounded from below by the scalar curvature R of which is clearly positive [20]. Nonetheless, this argument for SUSY enhancement clearly generalizes straightforwardly to other sigma models with targets which are (real) fibrations over Kahler manifolds. One particular generalization will be relevant here, where we embed this model in a Kahler N = (2,2) sigma model with target space T * ( S 3 ) which , arises in context of N= 1 SQCD with gauge group SU(2). 3. Domain Wall Moduli in n/=1 SQCD
Worldvolume supersymmetry enhancement has an interesting application in the context of BPS domain walls in SQCD. Restricting ourselves to the simplest example, consider N= 1 SQCD with gauge group SU(2) and two
433 fundamental flavors. This matter content is sufficient to fully Higgs the gauge group in any vacuum in which the matter fields have a nonzero vacuum expectation value. One can then write down a low energy description in terms of meson moduli fields. Limiting attention to the 'hen-baryonid' branch of the moduli space, the low energy superpotential is given by [2]
W = Tr(rizM)+ X (detM - A:),
(17)
in terms of the meson matrix M , the dynamical scale A2, and a Lagrange multiplier X which enforces a reduced (non-baryonic) form of the general quantum moduli space constraint [2]. The mass matrix 7j7. = (ml,mz} should be hierarchical, m2 >> ml, to ensure that the ensuing vacua (M:) = f(mz/m1)1/2A; lie at weak coupling. However, for part of the analysis below, we will temporarily ignore this constraint in order to focus on the most symmetric regime. These two vacua allow for the presence of BPS domain walls interpolating between them. To proceed, it is useful to introduce a dimensionless meson field Z = MAZ2, and a convenient basis is then provided by the following decomposition,
z= u,,-,, (zon+ ~ Z ~ O ~ ) U , ~ - , ~ ,U, = exp (i a c 3 ) ,
(18)
where the (axial) rotation angle is the relative phase of the two quark masses; m k = lmkleiak for k = 1,2. In this basis,
W
+ iArn&] + X
= eiYh; [%ZO
+
where y = (a1 a 2 ) / 2 and and Am = lm2l - Irnll. Cz=,2," = 1, describing as the deformed conifold.
(1,
22
-
)
1
,
(19)
+
the (real) mass parameters are m = Iml I Im2 I The moduli space constraint takes the form, a smooth complex submanifold of C4, known This manifold is symplectically equivalent to
T*(S3). We now observe that this system contains the simple model of the previous section as a subsector. Setting Am = 0, the two vacua, 2 0 = f l , now lie at the poles of the S3 which forms the real section of the surface C;=,Z," = 1. As explained in [17], it is sufficient here to use the classical metric on the base S3,i.e. the metric induced by the classical Kahler potential, and thus in spherical polar coordinates (0, E , +}, ds&,,
= A; (do2
+ sin20 (dE2 + sin2 @id2)),
(20)
434
while
W = eiYpA;TrZ
-
2eaYpA; cos 0 ,
(21)
where p = d m = f i / 2 , and we have set Am = 0. This is equivelant, up to normalization, to the superpotential of the S3 model analyzed in Sec. 2. We conclude that the bosonic moduli space for BPS walls in this theory is the same as that obtained for kinks within the S3 model, namely MN=2= R x CP’. Matching the scales, we obtain
d s h = T I d.zi
+ h i j d x i d x j = T Id z i + R,-
( d J i 3- sin2 J0d&)
,
(22)
with T I = 4 4 ; the wall tension and R,- = A ; / p the scale of the reduced moduli space. The underlying Kahler structure of the bulk theory does however leave an imprint on the realization of supersymmetry in the reduced moduli space. The first point to note, following the comments at the end of Sec. 2, is that the present system has twice as many fermions as the S3 model considered earlier. The second set of fermions arise from the cotangent directions of T * ( S 3 ) .We can choose a basis where the complex fermions lying in the chiral multiplet Z decompose into two (real) sets, one $1, the N = 1 partner of the S3 coordinates of the base, and the other $2, the N= 1partner of the cotangent directions. One then finds that a second set of fermionic zero modes arise from $21. The fermionic mode decomposition takes the form
where {&}, for A = 1 , 2 , are two sets of fermionic operators satisfying
1 {&,rljB} = hZj6AB , (24) Tl where hij is the reduced moduli space metric. Thus we now find in full a one-to-two matching between the number of bosonic versus fermionic zero modes. It is important that since the worldvolume is now 2+1-dimensional, this matching condition is a requirement of Lorentz invariance - a constraint that was not present in our earlier discussion of 1+1D kinks. The complex structure on CP1 again leads to an enhancement of supersymmetry. Indeed, it is clear that essentially the same construction as before, now augmented with two-component spinors q’, will lead to the dynamics admitting N= 2 supersymmetry in 2+1D, or four supercharges, only two of which can be identified with the unbroken generators of the bulk superalgebra.
{772,&}
= - ~ A ,B
435
If we drop the translational zero modes, and restrict Qa to the reduced moduli space, with xi = {&,,$o}, then we discover that there is a second unbroken spinor supercharge, existing by virtue of the complex structure J associated with .6?= S2. We can then form a complex spinor charge
Qi
and these charges satisfy the algebra of N= 4 SQM or more importantly, when lifted back to 2+1D, the N=2 superalgebra. The worldvolume theory is then an N = 2 @PI sigma model and the Witten index for this theory is equal to two, implying that there are two supersymmetric vacua and thus two BPS domain walls. The doubling of fermions in this system relative to the worldline theory of the S3 kink is crucial in generating these supersymmetric vacua. In the preceding discussion, we abstracted slightly in ignoring the deformation imposed by considering a hierarchical mass matrix for the quarks. The effect of this deformation at linear order in
corresponds to "twisting" the worldvolume supercharges by a U( 1 ) Killing vector G = Gi& for rotations in $ 0 ,
1 Gi = -.lrAmdi40 . (27) 2 An important feature of this particular deformation on the worldvolume is that it preserves the enhanced N = 2 SUSY [21]. In particular, the worldvolume supercharges persist but pick up corrections due to this twist [17],and we can write them in the form
+
Q L = hut@[@~$JL TEma$R],
Q R = bur@
[W$JR - I T E ~ ~ ~ $ L ] ,
(28)
with the remaining supercharges given by Q L and QR. The Fubini-Study metric is
In the Lagrangian, this twist amounts to the introduction of a potential given by the norm-squared of the Killing vector, AV = hijGiGj/2, although strictly speaking this is now a second order effect and thus could be accompanied by further corrections.
436
4. 1/4-BPS Wall Intersections
As an application of the worldvolume theory described above, we can explore the structure of wall intersections - namely 1/4-BPS junctions of two walls which are possible by virtue of the nonzero multiplicity in each charge sector. These configurations thus constitute a novel class of junctions, distinct from those generically present in theories where the vacua spontaneously break a ZN symmetry. Junction sources are supported by two types of central charge, 2 w and 2 s , transforming respectively in the (0,l) and (1/2,1/2) representations of the Lorentz group. The first of these is supported by BPS walls, while the second is associated with string-like sources. The Bogomol'nyi bound in 2+1D, with one dimension compactified on a circle of circumference L , takes the form [22,23]
M > J2wlL+-% ,
(30)
and 1/4-BPS junction configurations are required t o saturate this bound.
4.1. Junction tension for N f = 1 We first consider the hierarchical regime for gauge group SU(2), and integrate out the second flavor. It is then useful to introduce another dimensionless field Y in the form Y = a ( R : m , 1 ) - 1 / 4 , such that after decoupling
W=
Jm,h: (Y' + Y-') ,
and
Kclass =
d=yY
.
(31)
Provided we take ml ~express * the gauge field, fermion zero-mode, its density, and the action density in terms of fz as defined in Eq. (12). Fourier transformation turns the matrix fz into a Green's function fz(z,2 ) = it(z)fz(z,z')ij(z'), where fz(z, z') satisfies
1
{ -2
-I- V ( Z ; 2)
with
V ( z ;2) = 47r222(z; 2)
f&,
z') = 42nk6(z-z'),
(19)
+ 27r C S ( z - pm)Sm, sm= ij(/Lm)Sm$(pm), m
R j ( z ; Z )3 " j - (27ri)-lij(*)Aj(z)ij+(z).
(20)
Note that the Sm play the role of "impurities" and that
$ ( z ) 3 exp(27ri( n, and @ k = 4 k ?&+3, k = 1 , 2 , 3 . Below we will consider “far-from-BPS” operators like t r ( @ p@$ ...) ... where J1 JZ >> 1. The type IIB string action in Ad& x S5 space has the following structure
+
+
+
-
1 I = --T/dT 2
27r
+
du ( d P Y p d P Y V ~ , , dPXmdpXnG,,
+++
+ ...)
, (1)
where YpY”qpV= -1, XmX”S,, = 1 , qpV = (+-), T = and dots stand for the fermionic terms that ensure that this model defines a 2-d conformal field theory. The closed string states can be classified by the values of the Cartan charges of the obvious symmetry group S0(2,4) x S 0 ( 6 ) ,i.e. (El5’1,s~; J1, Jz, J3), i.e. by the Ad& energy, two spins in Ads5 and 3 spins in S 5 . The mass shell condition gives a relation E = E ( Q ,T ) . Here T is the string tension and Q = (S1,S2, 51, J2, J3; Tzk) where stand for higher conserved charges (analogs of oscillation numbers in flat space). According to AdS/CFT duality quantum closed string states in Ads5 x S5 should be dual to quantum SYM states at the boundary R x S 3 or, via radial quantization, to local single-trace operators at the origin of R4.
466
The energy of a string state should then be equal to the dimension of the corresponding SYM operator, E ( Q ,T ) = A(Q, A), where on the SYM side the charges Q characterise the operator. By analogy with flat space and ignoring a‘ corrections (i.e. assuming R -+ co or a‘ + 0) the excited string states are expected to have energies E 1 A l l 4 which represents a non-trivial prediction for strong-coupling asymptotics of SYM dimensions. In general, the natural (inverse-tension) perturbative expansion on the string side will be given by En &, while on the SYM side the usual planar perturbation theory will give the eigenvalues of the anomalous dimension matrix as A = C n u n A n . The AdS/CFT duality implies that the two expansions are to be the strong-coupling and weak-coupling asymptotics of the same function. To check the relation E = A is then a non-trivial problem. On the symmetry grounds, this can be shown in the case of 1 / 2 BPS (chiral primary) operators dual to supergravity states (“massless” or ground state string modes) since their energies/dimensions are protected from corrections. For generic non-BPS states the situation looked hopeless before a remarkable suggestion ‘v4 that progress is checking duality can be made by concentrating on a subsector of states with large ( LLsemiclassical’l) values of quantum numbers, Q T f i (here Q stands for generic quantum number like spin in Ads5 or S5 or a n oscillation number) and considering the new limit limit
-a-
N
-
A
A == fixed , Q2
where on the string side
5 -& plays the role of the semiclassical pa=
rameter (like rotation frequency) which can then be taken to be large. The energy of such states happens to be E = Q + f ( Q , A). The duality implies that such semiclassical string states as well as near-by fluctuations should be dual to “longrrSYM operators with large canonical dimension, i.e. containing large number of fields or derivatives under the trace. In this case the duality map becomes more explicit. The simplest possibility is to start with a BPS state that carries a large quantum number and consider small fluctuations near it, i.e. a set of nearBPS states all characterised by a large parameter ’. The only non-trivial example of such BPS state is described by a point-like string moving along geodesic in S5 with large angular momentum Q = J = J1. Then E = J and the dual operator is t r a J , = & + id’. The small (nearly pointlike) closed strings representing near-by fluctuations are ultrarelativistic,
467
i.e. their kinetic energy is much larger than their mass. They are dual t o SYM operators of the form tr(@J . . . ) where dots stand for a small number of other fields and/or covariant derivatives (and one needs to sum over different orders of the factors to find an eigenstate of the anomalous dimension matrix). The energies of the small fluctuations happen to be E =J N, O( -$). One can argue in general and check explicitly that higher-order quantum string sigma model corrections to the leading square root term in the fluctuation energy are indeed suppressed in the limit (2), i.e. in the large J , fixed 5 = -$= A’ limit. The remarkable feature of this expression is that E is analytic in i, suggesting direct comparison with perturbative SYM expansion in A. Indeed, it can be checked directly that the first two and terms in the expansion of the square root agree precisely with the one and two lo (and also three loop terms in the anomalous dimensions of the corresponding operators. there is also a general argument l3 (for a 2-impurity case) suggesting how the full expression can appear on the perturbative SYM side. However, the general proof of the consistency of the BMN limit on the SYM side (i.e. that the usual perturbative expansion can be rewritten as an expansion in iand remains to be given; also, to explain why the string and SYM expressions match one should show that the string limit (first J -+ 00, then i= -+ 0) and the SYM limit (first X + 0, then J + m) produce the same expressions for the dimensions (cf. l4,l5>l6). If one moves away from the near-BPS limit and considers, e.g., a nonsupersymmetric closed string state with a large angular momentum Q = S in AdS5 4 , a direct quantitative check of the duality is no longer possible: here the classical energy is not analytic in X and quantum corrections are no longer suppressed by powers of $. However, it is still possible t o demonstrate the remarkable qualitative agreement between the string energy and SYM anomalous dimension as far as the dependence on the spin S is concerned. The energy of a folded closed string rotating a t the center of Ads5 which is dual t o the twist 2 operators on the SYM side (tr(@:D’@h), D = D1 +iDz and similar operators with spinors and gauge bosons that mix at higher loops 19,20) has the form (when expanded at large S): E = S+f(X)lnS+ .... On the string side f(X)x,l = c o f i + c l + a + ..., Jj; In 2 is the l-loop coefficient. where co = is the classical and c1 = On the gauge theory side one finds the same S-dependence of the anomalous dimension with the perturbative expansion of the In S coefficient being f(X),,, = a1X azX2 a 3 X 3 ..., where a1 = 18, a2 = 19, and ‘9’
+ Jm +
697
899
x2
11i12)
J
m
3)
-2
2
+
+
+
&
-&
468
a3 = -20. Like in the case of the SYM entropy 21, here one expects the existence of a smooth interpolating function f ( X ) that connects the two perturbative expansions. One could still wonder if examples of quantitative agreement between string energies and SYM dimensions observed for near-BPS (BMN) states can be found also for more general non-BPS string states. Indeed, it was noticed already in that a string state that carries large spin in Ads5 as well as large spin J = 0 in S5 has, in contrast to the above J = 0 case, an analytic expansion of its energy in i= -$, just as in the BMN case with N,, S. It was observed in 22 that in general semiclassical string states carrying several large spins (with at least one of them being in S 5 ) have regular expansion of their energy in square of effective tension or in powers of iand it was suggested, by analogy with the near-BPS case, that the corresponding coefficients can be matched with the coefficients in the perturbative expansion for the SYM dimensions. For a classical rotating closed string solution in S5 one has E = f i & ( w i ) , J; = f i w ; so that E = E ( J ; , X ) and the key property is that there is no fifactors in the expression of the energy (as it was in the case of a single spin in AdS5)
-
X
A2
E = J+ci-+c:!-+...= J J3
J [1+c1i+czi2+...
1,
(3)
xi=l
A where J = J i , X- = 7 and c, = c n ( $ ) are functions of ratios of the spins which are finite in the limit Ji >> 1 , X =fixed. The simplest example of such a solution is provided by a circular string rotating in two orthogonal planes in S3 part of S5 with the two angular momenta being equal J1 = J2 22: XI XI i x 2 = cos(na) eiwT, ~2 G ~3 i x 4 = sin(na) eiwT, with the global Ads5 time being t = ICT (Y5 iY0 = eit). The conformal gauge constraint implies K' = w2 n2 and thus E = dor E = J(1+ ...), where J = J1+ J2 = 2J1. For fixed J the energy thus has a a regular expansion in tension (in contrast t o what happens in
+
+
i n 2 i in4x2+
+
F
+
flat space where E = 7J ) . Similar expressions (3) are found also for more general multispin circular strings In particular, for a folded string string rotating in one plane of S5 and with its center of mass orbiting along big circle in another plane 24 the coefficients c , are transcendental functions (expressed in terms of elliptic integrals). More generally, the 3-spin solutions are described by a n integrable Neumann model and the coefficients c, in the energy are expressed in terms of genus two hyperelliptic functions. 22f23124925926~27.
25726
469
To be able t o compare the classical energy to the SYM dimension one should be sure that string a’ corrections are suppressed in the limit J + 00, 5 =fixed. Formally, this should be the case since a’ 2 - but, fi J f i ’ what is more important, the $ corrections are again analytic in i23, i.e. the expansion in large J and small is well-defined on the string side,
-
N
di d2 l + X ( c i + j + ...)+ X 2 ( c 2 + -J+
1
...)+... ,
(4)
with the classical energy (3) being the J -+00 limit of the exact expression.a Similar expressions are found for the energies of small fluctuations near a given classical solution: as in the BMN case, the fluctuation energies are ...) i 2 ( k 2 7 suppressed by extra factor of J , i.e. 6E = X ( K 1
+ + +
...) + ....
+ +
Assuming that the same limit is well-defined also on the SYM side, one should then be able to compare the coefficients in (4) to the coefficients in the anomalous dimensions of the corresponding SYM operators 22 t r ( @ F @ 2 @ 2 ) ... (and also do similar matching for near-by fluctuation modes). In practice, it is known at least in principle how t o compute the dimensions in a different limit: first expanding in X and then expanding in -$. One may expect that this expansion of anomalous dimensions takes the form equivalent to (4), i.e.
+
A = J + X ( -a1 + J
bl
a2 -+...) + A 2 ( - + J2 J3
b2 +...) +... , J4
(5)
and, moreover, the respective coefficients in (4) and (5) agree with each other. The subsequent work did verify the structure of (5) and moreover established the general agreement between the two leading coefficients cl, c2 in (4) and the “one-loop” and “two-loop” coefficients a l l a2 in ( 5 ) . To compute (5) one is first to solve a technical problem of how to diagonalize anomalous dimension matrix defined on a set of long scalar operators. The crucial step was made in 28 where it was observed that the one-loop 28329130931732j33*15*34335
aThe reason for this particular form of the energy (4) can be explained as follows we are computing string sigma model loop corrections to the mass of a stationary solitonic solution on a 2-d cylinder (no IR divergences). This theory is conformal (due to the crucial presence of fermionic fluctuations) and thus does not depend on UV cutoff. The As a result, the inrelevant fluctuations are massive and their masses scale as w :’2p‘
- 3.
verse mass expansion is well-defined and the quantum corrections should be proportional to positive powers of X.
470
planar dilatation operator in the scalar sector can be interpreted as a Hamiltonian of an integrable SO(6) spin chain and thus can be diagonalized even for large length L = J by the Bethe ansatz method. In the simplest case of (closed) “SU(2)” sector of operators t r ( @ p @ $ ) ... built out of two chiral scalars the latter can be interpreted as spin up and spin down states of periodic XXX1/2 spin chain with length L = J = J1 J2, Then the 1-loop dilatation operator becomes equivalent t o the Hamiltonian of the ferromagnetic Heisenberg model
+
+
Using this relation and considering the thermodynamic limit ( J + m) of the Bethe ansatz the proposal of 22 was confirmed at the leading order of expansion in 5 in 29,30. Namely, it was found that for eigen-operators with both J1 and J2 being comparable and large A - J = A? .,. and a remarkable agreement was found between a1 and the coefficient c1 (which both are non-trivial functions of $) in the energies of various 2-spin string solutions. As in the BMN case, it was possible also to match the energies of fluctuations near the circular J1 = J2 with the corresponding eigenvalues of ( 6 ) 29. Similar leading-order agreement between string energies and SYM dimensions was observed also in other sectors of states with large quantum numbers: (i) for specific solutions in the SU(3) sector of states with 3 spins in S5 dual to tr(@?@$@$) ... operators (ii) for a folded string state belonging t o the SL(2) 37 sector of states with one spin in Ads5 and one spin in S5 (with E = J + S -i-$c1(5) ... 6 * 2 2 ) dual to t r ( D S a J ) ... 30; (iii) in a “subsector” of SO(6) states containing pulsating (and rotating) solutions which again have regular energy expansion in the limit of large oscillation number, e.g., E = L c1$ ... 38.
+
22925926
+
32936;
+
+
29932
+
+
2. Effective actions for coherent states
The observed agreement between energies of particular semiclassical string states and dimensions of the corresponding “long” SYM operators leaves many questions, in particular: (i) How to understand this agreement beyond specific examples, i.e. in a universal way? Can one derive a relevant limit of string sigma model action directly from SYM dilatation operator? (ii) Which is a precise relation between profiles of string solutions and the structure of the dual SYM operators? (iii) How to characterise the set
47 1
of semiclassical string states and dual SYM operators for which the correspondence should work? (iv) Why agreement works, i.e. why the two limits (first J + 00, and then i-+ 0, or vice versa) taken on the string and SYM sides give equivalent results? Should it work to all orders in expansion a n alin i(and $)? The questions (i),(ii) were addressed in ternative approach based on matching the general solution (and integrable structure) of the string sigma model with that of the thermodynamic limit of the Bethe ansatz was developed in 34. The question (iii) was addressed in 43,44,45,42 , and the question (iv) - in One key idea was that instead of comparing particular solutions one should try t o match effective sigma model actions which appear on the string side (in the limit J -+ 00, i-+ 0) and the SYM side (in the limit 5 -+ 0, J -+m). Another related idea was that since “semiclassical” string states carrying large quantum numbers are represented in the quantum theory by coherent states, one should be comparing coherent string states t o coherent SYM (spin chain) states. The crucial point is that because of the ferromagnetic nature of the dilatation operator (6) in the thermodynamic limit J = J1 J2 -+ m with fixed number of impurities it is favorable to form large clusters of spins and thus a “low-energy” approximation and continuum limit should apply, leading t o an effective “non-relativistic” sigma model for a coherent-state expectation value of the spin operator. Taking the “large energy” limit directly in the string action gives a reduced “non-relativistic” sigma model that describes in a universal way the leading-order O ( i ) corrections to the energies of all string solutions in the two-spin sector. The resulting action agrees exactly 33 with the semiclassical coherent state action describing the S U ( 2 ) sector of the spin chain in the J -+ 00, i=fixed limit. This demonstrates how a string action can directly emerge from a gauge theory in the large-N limit and provides a direct map between the “coherent” SYM states and all two-spin classical string states, bypassing the need to apply the Bethe ansatz to find anomalous dimension for each particular state. Furthermore, the correspondence established at the level of the action implies also the matching of fluctuations around particular solutions. Let us briefly review the definition of coherent states. Starting with the SU(2) algebra [S3,S*] = fS*, [S+,S-] = 2S3 and considering the s = 1/2 representation where S = f 3 one can define spin coherent state as a linear superposition of spin up and spin down states: Iu) = R(u)JO), where R = eus+-u’s- , 10) = 1 f , and u is a complex number that can be parametrized as u = i 8 e i @ . An equivalent way to label the coherent state 33135*40941142;
15116917.
33935942
2
+
-4
4)
472
is a by a unit 3-vector 5 defining a point of S 2 . Then 15) = R(5)lO)where 10) corresponds t o a 3-vector (0, 0 , l ) along the 3-rd axis ( 5 = UtSU, U = (211,212)) and R(5) is an SO(3) rotation from a north pole to a generic point of S2. The key property of the coherent state is that ii determines the expectation value of the spin operator: (filslfi)= f f i . In general, one can rewrite the usual phase space path integral as a path integral over the overcomplete set of coherent states (for the harmonic oscillator this is simply the change of variables u = ’ ( q + i p ) ) : 2 = J [ d u ]eiSI”].The action J;i is S = J d t ( ( u l i ~ l u ) - ( u I H I u ) ) where , the first (WZ or ‘‘Berry phase”) term is the analog of the usual pq term in the phase-space action. Applying this t o the case of the Heisenberg spin chain Hamiltonian (6) one ends up with with the following action for the coherent state variables & ( t )at sites 1 = 1,..., J (see also, e.g., 39): S = C{=,I d t [ ( ? ( n l ) . % - & (a) fi~+l - i i l ) ’ ] . Here d C = e i j k n i d n j A d n k , i.e. (? is a monopole potential on s2.In the limit J + 00 with fixed = -$ we are interested in one can take a continuum limit by introducing the 2-d field 5(t,u ) = {5(t,?I)}. Then
S=J/dt12a
2 [B .
1&fi - -8X ( & f i ) 2
1
+ ... ,
(7)
5.
where dots stand for higher derivative terms suppressed by Since in the limit we are interested in J -+ 00 all quantum corrections are also suppressed by $, and thus the above action can be treated classically. The corresponding equation of motion n; = i i e ; j k n j n i are the Landau-Lifshitz equation for a classical ferromagnet. The action (7) should be describing the coherent states of the Heisenberg spin chain in the above thermodynamic limit. One may wonder how a similar “non-relativistic” action may appear on the string side where one starts with a usual sigma model (1)which is quadratic in time derivatives. However, to obtain an effective action that can be compared t o the spin chain one is first t o perform the following procedure (i) isolate a “fast” coordinate (Y whose momentum p , is large in our limit; (ii) gauge fix t 7- and p , J or ii u where ii is “T-dual” to a; (iii) expand the action in derivatives of “slow” or ‘transverse” coordinates (to be identified with G ) . To see how this procedure can be implemented explicitly let us consider the SU(2) sector of string states carrying two large spins in S5, with string motions restricted t o S3 part of S5. The relevant part of the A d s 5 x S5 metric is then ds2 = -dt2 d X i d X f , with X ; X f = 1, and we can set X 1 = X1 iX2 = uleiLI, X2 = X3 iX4 = u3eia, u;uf = 1. Here (Y will be a collective coordinate associated to the total (large) 33135*42:
N
-
N
+
+
+
473 spin in the two planes (which in general will be the sum of orbital and internal spin). ui will be “slow” coordinates determining the “transverse” string profile (ui are defined modulo U(1) gauge transformation). Then dXidX,t = (da C)’ DuiDuf, where C = -iuTdui, Dui = dui - iCui and the second term represent the metric of C P 1 (this parametrisation corresponds t o Hopf fibration S3 S 1 x S 2 ) . Introducing 5 = UtZU, U = ( 2 ~ 1 , ~ ’ we ) get d X i d X r = (Da)’ ;(dii)’, Da = da! C ( n ) . Writing the resulting sigma model action in phase space form and imposing the (non-conformal) gauge t = T, p , =const= J one gets the action (7) with the WZ term 6 . 8,s originating from the p,Da term in the phase-space Lagrangian. This conforms to its origin on the spin chain side as an analog of the ‘pq’ in the phase space action. Equivalent approach is based on first applying a 2-d duality (or “T-duality”) a + 6 and then choosing the “static” gauge t = 7, 5 = ( f i ) - ’ o , = Applying T-duality ~~~) Thus we get L = - ~ J - 9 9 P ~ ( - a , t ~ , t + d , 5 ~ , 5 + D , ~ T D +@%‘,aq& the “T-dual” background has no off-diagonal metric component but has a non-trivial NS-NS 2-form coupling in the (6,ui) sector. Eliminating the 2-d metric g p q we then get the Nambu-type action L = P C p d , 6 where h = ldet h,,l and h,, = -aptaqt 8,6aq6 D~,uTD,)ui. If we now fix the static gauge we finish, to the leading order in A, with
+
+
-+
+
(A)-’ 5.
+
+
6,
which is the same as the C P 1 Landau-Lifshitz action ( 7 ) when written in terms of 5. We thus uncover 42 the origin of the string-theory counterpart of the WZ term in the spin-chain coherent state effective action (7): it comes from the 2-d NS-NS WZ term upon the static gauge fixing in the “T-dual” 6 action. The agreement between the low-energy actions on the spin chain and the string side explains not only the matching between energies and coherent states for configurations with two large spins (and near-by fluctuations) but also the matching of integrable structures observed on specific examples in 31,32. To summarize: (i) (t15)play the role of longitudinal coordinates that are gauge-fixed (with 5 playing the role of string direction or spin chain direction on the SYM side); (ii) U = (ul,u2) or 6 = U t X J are “transverse” coordinates that determine the semiclassical string profile and also the structure of the coherent operator on the SYM side, t r IIo(ui@i). This leading-order agreement in SU(2) sector has several generalizations. First, we may include higher-order terms on the string side. System-
474
atically expanding in iand eliminating higher powers of time derivatives by field redefinitions (note that leading-order equation of motion is 1st order in time) we end up with 35
The same i2 term is obtained 35 in the coherent state action on the spin chain side by starting with the sum of the 1-loop dilatation operator (6) and the 2-loop one l1
This explains the matching of energies and dimensions t o the first two orders, as first observed on specific examples using Bethe ansatz in 15. Equivalent general conclusion about 2-loop matching was obtained in the integrability-based approach in 34. The order-by-order agreement seems to break down at h3 (%loop) order, which has a natural explanation 15*16in that the string limit (first J -+ 00, then X -+ 0) and the SYM limit (first X -+ 0, then J + 00) need not be the same.b One can also generalize the above leading-order agreement to the SU(3) sector of states with three large S5 spins J i , i = 1,2,3, finding the CP2 analog of the CP1 “LandauLifshitz” Lagrangian in (7),(8) 41 C = --iufd~ui- iiIDlui12 on both string and spin chain sides. Similar conclusion is found 41 in the SL(2) sector of ( S ,J ) states. Finally, one can also consider pulsating string states 42. One may try also to go further and use the duality t o string theory as a tool t o determine the structure of planar SYM theory to all orders in X by imposing the exact agreement with particular string solutions. For example, demanding the consistency with the BMN scaling limit determines (along with the superconformal algebra) the structure of the full %loop SYM dilatation operator in the SU(2) sector l1sl2. One can also use the BMN limit t o fix a part of the dilatation operator but to all orders in X 47. Generalizing (6),(10) and the 3- and 4-loop expressions in l1>l2one can organize the dilatation operator as an expansion in powers of &k,l = I - a‘k . & which reflects interactions between spin chain sites, D = C Q C Q Q + C QQQ+ ..., where the products Q...Q are “irreducible”, i.e. each 40t41
+
bSuggestions how t o “complete” the gauge-theory answer t o restore the agreement with disagreement string theory appeared in l 6 , I 7 . This may also resolve the order i3 corrections t o the BMN spectrum. between string a n d gauge theory predictions for
5
475
site index appears only once. The QQ-terms first appear at 3 loops, QQQterms - a t 4 loops, etc. l 1 > l 2 Concentrating . on the order Q part D(’) of D and demanding the BMN-type scaling limit (and agreement with the BMN square root spectrum) one finds, in the limit of large L , i.e. when D acts on “long” operators,
goes rapidly to zero at large k , so we effectively have a spin chain with short range interactions. The function fk(X) smoothly interpolates between the usual perturbative expansion at small X and 6 at strong X (which is the expected behaviour of anomalous dimensions of LLlong’l operators dual to states). Similar interpolating functions are expected to appear in anomalous dimensions of other SYM operators. Acknowledgments We are grateful to M. Kruczenski, A. Ryzhov and B. Stefanski for collaborations on the work described above. This work was supported by the DOE grant DE-FG02-91ER40690, the INTAS contract 03-51-6346 and the RS Wolfson award. fk
N
References 1. A. M. Polyakov, “Gauge fields and space-time,” Int. J. Mod. Phys. A 17S1, 119 (2002) [hep-th/0110196]. 2. D. Berenstein, J. M. Maldacena and H. Nastase, “Strings in flat space and pp waves from N =4 super Yang Mills,” JHEP 0204,013 (2002) [hep-th/0202021]. 3. R. R. Metsaev and A. A. Tseytlin, “Type IIB superstring action in AdS(5) x S(5) background,” Nucl. Phys. B 533, 109 (1998) [hep-th/9805028]. 4. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “A semi-classical limit of the gauge/string correspondence,” Nucl. Phys. B 636,99 (2002) [hep-th/0204051]. 5. R. R. Metsaev, “Type IIB Green-Schwarz superstring in plane wave RamondRamond background,” Nucl. Phys. B 625, 70 (2002) [hep-th/0112044]. R. R. Metsaev and A. A. Tseytlin, “Exactly solvable model of superstring in plane wave Ramond-Ramond background,” Phys. Rev. D 65, 126004 (2002) [hepth/0202109]. 6. S. Frolov and A. A. Tseytlin, “Semiclassical quantization of rotating superstring in Ads5 x S5 ,” JHEP 0206, 007 (2002) [hep-th/0204226]. 7. A. A. Tseytlin, “Semiclassical quantization of superstrings: Ads5 x S5 and beyond,” Int. J. Mod. Phys. A 18,981 (2003) [hepth/0209116].
476 8. A. Parnachev and A. V. Ryzhov, “Strings in the near plane wave background and AdS/CFT,” JHEP 0210,066 (2002) [hep-th/0208010]. 9. C. G. Callan, H. K. Lee, T. McLoughlin, J. H. Schwarz, I. Swanson and X. Wu, “Quantizing string theory in AdS(5) x S**5: Beyond the pp-wave,” Nucl. Phys. B 673, 3 (2003) [hep-th/0307032]. 10. D. J. Gross, A. Mikhailov and R. Roiban, “Operators with large R charge in N = 4 Yang-Mills theory,’’ Annals Phys. 301, 31 (2002) [hep-th/0205066]. 11. N. Beisert, C. Kristjansen and M. Staudacher, “The dilatation operator of N = 4 super Yang-Mills theory,” Nucl. Phys. B 664,131 (2003) [hep-th/0303060]. 12. N. Beisert, “The su(213) dynamic spin chain,” hep-th/0310252. 13. A. Santambrogio and D. Zanon, “Exact anomalous dimensions of N = 4 Yang-Mills operators with large R charge,” Phys. Lett. B 545, 425 (2002) [hep-th/0206079]. 14. I. R. Klebanov, M. Spradlin and A. Volovich, “New effects in gauge theory from ppwave superstrings,’’ Phys. Lett. B 548, 111 (2002) [hep-th/0206221]. 15. D. Serban and M. Staudacher, “Planar N = 4 gauge theory and the Inozemtsev long range spin chain,” JHEP 0406, 001 (2004) [hepth/0401057]. 16. N. Beisert, V. Dippel and M. Staudacher, “A novel long range spin chain and planar N = 4 super Yang-Mills,” arXiv:hep-th/0405001. 17. G. Arutyunov, S. Frolov and M. Staudacher, “Bethe ansatz for quantum strings,” arXiv:hepth/0406256. 18. D. J. Gross and F. Wilczek, “Asymptotically Free Gauge Theories. I,” Phys. Rev. D 8, 3633 (1973). 19. A. V. Kotikov, L. N. Lipatov and V. N . Velizhanin, “Anomalous dimensions of Wilson operators in N = 4 SYM theory,” Phys. Lett. B 557, 114 (2003) [hepph/0301021]. 20. A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko and V. N. Velizhanin, “Three-loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model,’’ arXiv:hep-th/0404092. 21. S. S. Gubser, I. R. Klebanov and A. A. Tseytlin, “Coupling constant dependence in the thermodynamics of N = 4 supersymmetric Yang-Mills theory,” Nucl. Phys. B 534, 202 (1998) [hep-th/9805156]. 22. S. Frolov and A. A. Tseytlin, “Multi-spin string solutions in Ads5 x S5 ,” Nucl. Phys. B 668, 77 (2003) [hepth/0304255]. 23. S. Frolov and A. A. Tseytlin, “Quantizing three-spin string solution in Ads5 x S5 ,” JHEP 0307,016 (2003) [hep-th/0306130]. 24. S. Frolov and A. A. Tseytlin, “Rotating string solutions: AdS/CFT duality in non-supersymmetric sectors,” Phys. Lett. B 570, 96 (2003) [hep-th/0306143]. 25. G. Arutyunov, S. Frolov, J. Russo and A. A. Tseytlin, “Spinning strings in Ads5 x S5 and integrable systems,” Nucl. Phys. B 671, 3 (2003) [hepth/0307191]. 26. G. Arutyunov, J. Russo and A. A. Tseytlin, ‘Spinning strings in Ads5 x S5 : New integrable system relations,” Phys. Rev. D 69, 086009 (2004) [hepth/0311004]. 27. A. A. Tseytlin, “Spinning strings and AdS/CFT duality,” hep-th/0311139. 28. J . A. Minahan and K. Zarembo, “The Bethe-ansatz for N = 4 super Yang-
477 Mills,” JHEP 0303,0 13 (2003) [hep-t h/02 122081. 29. N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, “Stringing spins and spinning strings,” JHEP 0309,010 (2003) [hep-th/0306139]. 30. N. Beisert, S. Frolov, M. Staudacher and A. A. Tseytlin, “Precision spectroscopy of AdS/CFT,” JHEP 0310,037 (2003) [hep-th/0308117]. 31. G. Arutyunov and M. Staudacher, “Matching higher conserved charges for strings and spins,” JHEP 0403,004 (2004) [hep-th/0310182]. “Two-loop commuting charges and the string / gauge duality,” hep-th/0403077. 32. J. Engquist, J. A. Minahan and K. Zarembo, “Yang-Mills duals for semiclassical strings on Ads5 x S5 ,” JHEP 0311,063 (2003) [hep-th/0310188]. 33. M. Kruczenski, ‘‘Spin chains and string theory,” hepth/0311203. 34. V. A. Kazakov, A. Marshakov, J. A. Minahan and K. Zarembo, “Classical/quantum integrability in AdS/CFT,” hep-th/0402207. 35. M. Kruczenski, A. V. Ryzhov and A. A. Tseytlin, “Large spin limit of Ads5 x S5 string theory and low energy expansion of ferromagnetic spin chains,” h e p th/0403 120. 36. C. Kristjansen, “Three-spin strings on AdS(5) x S**5 from N = 4 SYM,” Phys. Lett. B 586, 106 (2004) [hep-th/0402033]. L. Freyhult, “Bethe ansatz and fluctuations in SU(3) Yang-Mills operators,’’ hep-th/0405167. C. Kristjansen and T. Mansson, “The Circular, Elliptic Three Spin String from the SU(3) Spin Chain,” hep-th/0406176. 37. N. Beisert and M. Staudacher, “The N=4 SYM integrable super spin chain”, Nucl. Phys. B 670, 439 (2003) [hep-th/0307042]. 38. J. A. Minahan, “Circular semiclassical string solutions on Ads5 x S5 ,” Nucl. Phys. B 648, 203 (2003) [hepth/0209047]. 39. E. H. Fradkin, “Field Theories Of Condensed Matter Systems,” Redwood City, USA: Addison-Wesley (1991) 350 p. (Frontiers in physics, 82). 40. R. Hernandez and E. Lopez, “The SU(3) spin chain sigma model and string theory,” JHEP 0404,052 (2004) [hep-th/0403139]. 41. B. J . Stefanski, Jr. and A. A. Tseytlin, “Large spin limits of AdS/CFT and generalized Landau-Lifshitz equations,” JHEP 0405, 042 (2004) [ h e p th/0404133]. 42. M. Kruczenski and A. A. Tseytlin, “Semiclassical relativistic strings in S5 and long coherent operators in N = 4 SYM theory,” arXiv:hepth/0406189. 43. D. Mateos, T. Mateos and P. K. Townsend, “Supersymmetry of tensionless rotating strings in Ads5 x S5 , and nearly-BPS operators,” hepth/0309114. “More on supersymmetric tensionless rotating strings in Ads5 x S5 ,” hepth/0401058. 44. A. Mikhailov, “Speeding strings,” JHEP 0312,058 (2003) [hep-th/0311019]. 45. A. Mikhailov, “Slow evolution of nearly-degenerate extremal surfaces,” h e p th/0402067. “Supersymmetric null-surfaces,” hepth/0404173. 46. J . A. Minahan, “Higher loops beyond the SU(2) sector,” hep-th/0405243. 47. A. V. Ryzhov and A. A. Tseytlin, LLTowards the exact dilatation operator of N = 4 super Yang-Mills theory,” arXiv:hep-th/0404215.
PLANAR EQUIVALENCE: FROM TYPE 0 STRINGS TO QCD
A. ARMONI Department of Physics, Theory Division C E R N , CH-1211 Geneva 23, Switzerland M. SHIFMAN * William I. Fane Theoretical Physics Institute, University of Minnesota, Minneapolis, M N 55455, USA G. VENEZIANO Department of Physics, Theory Division C E R N , CH-1211 Geneva 23, Switzerland
This talk is about the planar equivalence between N = 1 gluodynamics (superYang-Mills theory) and a non-supersymmetric “orientifold field theory.” We outline an “orientifold” large N expansion, analyze its possible phenomenological consequences in one-flavor massless QCD, and make a first attempt a t extending the correspondence to three massless flavors. An analytic calculation of the quark condensate in one-flavor QCD starting from the gluino condensate in N = 1 gluodynamics is thoroughly discussed.
1. Genesis of the idea
Kachru and Silverstein studied various orbifolds of R6 within the framework of the AdS/CFT correspondence. Starting from N = 4,they obtained distinct four-dimensional (daughter) gauge field theories with matter, with varying degree of supersymmetry, N = 2,1,0, all with vanishing ,8 functions. Shortly after, Bershadsky and Johansen abandoned the string theory set-up altogether. They proved that non-supersymmetric large-N orbifold *conference speaker
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field theories admit a zero beta function in the framework of field theory per se. The first attempt to apply the idea of orbifoldization to non-conformal field theories was carried out by Schmaltz who suggested a version of Seiberg duality between a pair of non-supersymmetric large-N orbifold field theories. After a few years of a relative oblivion, the interest in the issue of planar equivalence was revived by Strassler 4 . In the inspiring paper entitled “On methods for extracting exact non-perturbative results in nonsupersymmetric gauge theories” he shifted the emphasis away from the search of the conformal daughters, towards engineering QCD-like daughters. Unfortunately, it turned out that Strassler conjecture could not be valid. The orbifold daughter theories “remember” that they have fewer vacua than the parent one, which results in a mismatch in low-energy theorems. In string theory language, the killing factor is the presence of tachyons in the twisted sector. This is clearly seen in light of the calculation presented in Ref. ‘. 596
2. Orientifold field theory and
N = 1 gluodynamics
Having concluded that, regretfully, the planar equivalence of the orbifold daughters does not extend to the non-perturbative level, we move on to another class of theories, called orientifolds, which lately gave rise to great expectations. We will argue that in the N + 03 limit there is a sector in the orientifold theory exactly identical to N = 1 SYM theory, and, therefore, exact results on the IR behavior of this theory can be obtained. This sector is referred to as the common sector. The parent theory is N = 1 SUSY gluodynamics with gauge group SU(N). The daughter theory has the same gauge group and the same gauge coupling. The gluino field A; is replaced by two Weyl spinors 77[ijl and [rijl, with two antisymmetrized indices. We can combine the Weyl or !PIij]. Note that the number spinors into one Dirac spinor, either 9[ij] of fermion degrees of freedom in is N2 - N , as in the parent theory in the large-N limit. We call this daughter theory orientifold A . There is another version of the orientifold daughter - orientifold S. Instead of the antisymmetrization of the two-index spinors, we can perform symmetrization, so that A; -+ ( r ] { i j ) , [iij)). The number of degrees of freedom in P{ij) is N 2 N. The field contents of the orientifold theories
+
480
is shown in Table 1. We will mostly focus on the antisymmetric daughter since it is of more physical interest; see Sect. 4. Table 1. The field content of the orientifold theories. Here, 77 and E are two Weyl fermions, while A, stands for the gauge bosons. In the left (right) parts of the table the fermions are in the two-index symmetric (antisymmetric) representation of the gauge group SU(N).
Adj
0
0
The hadronic (color-singlet) sectors of the parent and daughter theories are different. I n the parent theory composite fermions with mass scaling as N o exist, and, moreover, they are degenerate with their bosonic SUSY counterparts. In the daughter theory any interpolating color-singlet current with the fermion quantum numbers (if it exists at all) contains a number of constituents growing with N . Hence, at N = co the spectrum contains only bosons. 2.1. Perturbative equivalence
Let us start from perturbative considerations. The Feynman rules of the planar theory are shown in Fig. 1. The difference between the orientifold theory and N = 1 gluodynamics is that the arrows on the fermionic lines point in the same direction, since the fermion is in the antisymmetric representation, in contrast to the supersymmetric theory where the gaugino is in the adjoint representation and the arrows point in opposite directions. This difference between the two theories does not affect planar graphs, provided that each gaugino line is replaced by the sum of q[,,] and ]6 ( N ) ( ( 9 2 )
1-6(N)
-
QQ) =
-6(N - 2) A$
wN)
7
(4.10)
where p is some fixed normalization point; the correction factor
K ( p , N = 00) = 1 ,
(4.11)
simultaneously with 6 ( N = a)= 0. At finite N the correction factor K ( p , N ) - 1 = 0(1/N) and K depends on p in the same way as [ g 2 ( p ) ]6 ( N ).
489
-
1-6(N)
The combination (g2) QQ on the left-hand side is singled out because of its RG invariance. Equation (4.10) is our master formula. The factor N - 2 on the right-hand side of Eq. (4.10), a descendant of N l--b(N) in Eq. (4.1), makes (g2) QQ vanish at N = 2. This requirement is obvious, given that at N = 2 the antisymmetric fermion loses color. In fact, we replaced N in Eq. (4.1) by N - 2 by hand, assembling all other 1/N corrections in K, in the hope that all other 1/N corrections collected in K are not so large. There is no obvious reason for them to be large. Moreover, we can try to further minimize them by a judicious choice of p . It is intuitively clear that 1/N corrections in K will be minimal, provided that p presents a scale “appropriate to the process”, which, in the case at hand, is the formation of the quark condensate. Thus, p must be chosen as low as possible, but still in the interval where the notion of g 2 ( p ) makes sense. Our educated guess is
(
)
[g2(p)]6(3)M 4.9.
(4.12)
As a result, we arrive at the conclusion that in one-flavor QCD (4.13) Empiric determinations of the quark condensate with which we will confront our theoretical prediction are usually quoted for the normalization point 2 GeV. To convert the RGI combination on the left-hand side of Eq. (4.13) to the quark condensate at 2 GeV, we must divide by [g2(2 GeV)]1-6(3) x 1.4. Moreover, as has already been mentioned, we expect non-planar corrections in K to be in the ballpark f l / N . If so, three values for K ,
K
(4.14)
= {2/3, 1, 4/31 7
give a representative set. Assembling all these factors together we end up with the following prediction for one-flavor QCD:
( $[ijlQ[ijl)
2 GeV
= - (0.6 to 1.1)
Ak.
(4.15)
Next, our task is to compare it with empiric determinations, which, unfortunately, are not very precise. The problem is that one-flavor QCD is different both from actual QCD, with three massless quarks, and from quenched QCD, in which lattice measurements have recently been carried out 1 4 . In quenched QCD there are no quark loops in the running of a,; thus, it runs steeper than in one-flavor QCD. On the other hand, in threeflavor QCD the running of a, is milder than in one-flavor QCD.
490
To estimate the input value of X m (the ’t Hooft coupling) we resort to the following procedure. First, starting from a,(M,) = 0.31 (which is (3) close to the world average) we determine Am Then, with this A used as an input, we evolve the coupling constant back to 2 GeV according to the one-flavor formula. In this way we obtain X(2 GeV) = 0.115.
(4.16)
A check exhibiting the scatter of the value of X(2 GeV) is provided by lattice measurements. Using the results of Ref. l5 referring to pure YangMills theory one can extract a,(2 GeV) = 0.189. Then, as previously, we find Am, ( 0 ) and evolve back to 2 GeV according to the one-flavor formula. The result is X(2 GeV) = 0.097.
(4.17)
The estimate (4.17) is smaller than (4.16) by approximately one standard deviation CJ. This is natural, since the lattice determinations of a , lie on the low side, within one o of the world average. In passing from Eq. (4.13) to Eq. (4.15) we used the average value X m ( 2 GeV) = 0.1. One can summarize the lattice (quenched) determinations of the quark condensate, and the chiral theory determinations extrapolated to one flavor, available in the literature, as follows: 2 GeV, “empiric”
= - (0.4 to 0.9) A&.
(4.18)
We put empiric in quotation marks, given all the uncertainties discussed above. Even keeping in mind all the uncertainties involved in our numerical estimates, both from the side of supersymmetry/planar equivalence 1/N corrections, and from the “empiric” side, a comparison of Eqs. (4.15) and (4.18) reveals an encouraging overlap. References 1. S. Kachru and E. Silverstein, Phys. Rev. Lett. 80, 4855 (1998) [hepth/9802183]. 2. M. Bershadsky and A. Johansen, Nucl. Phys. B 536, 141 (1998) [hepth/9803249]. 3. M. Schmaltz, Phys. Rev. D 59, 105018 (1999) [hep-th/9805218]. 4. M. J. Strassler, On methods for extracting exact non-perturbative results in non-supersymmetric gauge theories, hep-th/0104032, unpublished. 5. A. Gorsky and M. Shifman, Phys. Rev. D 67, 022003 (2003) [hepth/0208073].
491
6. D. Tong, JHEP 0303,022 (2003) Ihep-th/0212235]. 7. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 229, 381 (1983); Phys. Lett. B 166,329 (1986). 8. 0. 't Hooft, Nucl. Phys. B 72, 461 (1974). 9. E. Witten, Nucl. Phys. B 160,57 (1979). 10. E. Corrigan and P. Ramond, Phys. Lett. B 87, 73 (1979). 11. G. Veneziano, Nucl. Phys. B 117,519 (1976). 12. E. Witten, Nucl. Phys. B 156,269 (1979). 13. G. Veneziano, Nucl. Phys. B 159,213 (1979). 14. L. Giusti, C. Hoelbling and C. Rebbi, Phys. Rev. D 64,114508 (2001), (E) D65, 079903 (2002) [hep-lat/0108007]. 15. M. Liischer, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B 413, 481 (1994) [hep-lat/9309005]; M. Guagnelli, R. Sommer and H. Wittig [ALPHA collaboration], Nucl. Phys. B 535,389 (1998) [hep-lat/9806005].
GAUGE THEORY AMPLITUDES, SCALAR GRAPHS AND TWISTOR SPACE
VALENTIN V. KHOZE Department of Physics a n d IPPP University of Durham. Durham, DH1 3LE United Kingdom E-mail:
[email protected] We discuss a remarkable new approach initiated by Cachazo, Svrcek and Witten for calculating gauge theory amplitudes. The formalism amounts to an effective scalar perturbation theory which in many cases offers a much simpler alternative t o the usual Feynman diagrams for deriving n-point amplitudes in gauge theory. At tree level the formalism works in a generic gauge theory, with or without supersymmetry, and for a finite number of colours. There is also a growing evidence that the formalism works for loop amplitudes.
1. Introduction
In a recent paper1 Cachazo, Svrcek and Witten (CSW) proposed a new approach for calculating scattering amplitudes of n gluons. In this approach tree amplitudes in gauge theory are found by summing tree-level scalar diagrams. The CSW formalism' is constructed in terms of scalar propagators, l/q2, and tree-level maximal helicity violating (MHV) amplitudes, which are interpreted as new scalar vertices. This novel diagrammatic approach follows from an earlier construction2 of Witten which related perturbative amplitudes of conformal N = 4 supersymmetric gauge theory to Dinstanton contributions in a topological string theory in twistor space. The key observation2y1is that tree-level and also loop diagrams in SYM posess a tractable geometric structure when they are transformed from Minkowski to twistor space. The results2i1 have been tested and further developed in gauge theory3>4,5,6,897,9and in string theoryl0,11,12,13,14,15,16 The CSW diagrammatic approach' was extended to gauge theories with fermions3, and it was also shown that supersymmetry is not required for
492
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the construction to work. At tree level the scalar graph formalism works in supersymmetric and non-supersymmetric theories, including QCD. Using this approach all tree-level antianalytic MHV (or googly) aplitudes were calcualted4 in complete agreement with known results. Recursive relations for constructing generic tree-level non-MHV amplitudes in the CSW formalism were derived5 and further progress was made in Refs.6f8where generic n-point amplitudes with three negative helicities were calculated at treelevel. In fact, all tree-level amplitudes in 0 N _< 4 gauge theories can be obtained directly from the scalar graph approach of CSW. By starting with amplitudes containing fermions', the reference spinor for the negative helicity gluons can be chosen to be that of the negative helicity fermion. As a consequence, the amplitudes are free of unphysical singularities for generic phase space points and no further helicity-spinor algebra is required to convert the results into an immediately usable form. The gluons only amplitudes can then be simply obtained as sums of fermionic amplitudes using the supersymmetric Ward identity. These amplitudes are therefore also immediately free of unphysical poles. Expressions for n-point amplitudes with three negative helicities carried by fermions and/or gluons were derived' in this way. The next logical step is to extend the formalism to the computation of loop graphs. When gauge theory amplitudes at 1-loop level are Fourier transfomed to twistor space, their analytic structure again acquires geometric meaning7. In Ref.g the CSW diagramatic approach' is used at loop level leading to a remarkable agreement with the known results17 for 1-loop MHV amplitudes in N = 4 SYM. It will be very interesting to extend the resultsg and to see how the CSW formalism will work in general settings, i.e. at 1-loop and beyond, for N 5 4 supersymmetry, and for non-MHV amplitudes.
21 and are called maximal helicity violating (MHV) amplitudes. In the spinor helicity formalism22~20~21 an on-shell momentum of a massless particle, p,p, = 0, is represented as
++
pa& = p,afa = A a i a ,
(1)
where A, and i& are two commuting spinors of positive and negative chirality. Spinor inner products are defined by (X,X’) = EabXaAtb and - [X,A’] = ~,~;\ai’b.A scalar product of - -two null vectors, pa& = A a i & and qa& = A b i 6 , becomes p,q, = $(A, A’)[A, A’]. An MHV amplitude A, = A1+2rnwith 1 gluons and 2 m fermions in N 5 1 theories exists only for m = 0,1,2. This is because it must have precisely n - 2 particles with positive and 2 with negative helicities, and our fermions always come in pairs with helicities ff.Hence, there are three types of MHV tree amplitudes in N 5 1 theories:
An(g,,gs)
7
A n ( g t , K , A b )7
Suppressing the overall factor,
A n ( A t , A Q , K , A : ).
(2)
( 2 ~6(4)(Cy=1 ) ~ A i a i i & ) , the MHV
aIn the N = 1 theory this is also correct to all orders in the loop expansion and nonperturbatively.
495
purely gluonic amplitude reads20i21:
The MHV amplitude with two external fermions and
where the first expression corresponds to T < s and the second to s < T (and t is arbitrary). The MHV amplitude with four fermions and n - 4 gluons on external lines is
This corresponds to t < s < r < q , and there are other similar expressions, obtained by permutations of fermions, with the overall sign determined by the ordering. Expressions (4), (5) can be derived from supersymmetric Ward i d e n t i t i e ~ l ~ t ~and ~ > we ' ~ ,will have more to say about this in section 3. The MHV amplitude can be obtained, as always, by exchanging helicities H - and (i j ) ++[i j ] .
+
2. Gluonic NMHV amplitudes and the CSW method
The formalism of CSW was developed' for calculating purely gluonic amplitudes at tree level. In this approach all non-MHV n-gluon amplitudes (including MHV) are expressed as sums of tree diagrams in an effective scalar perturbation theory. The vertices in this theory are the MHV amplitudes (3), continued off-shell as described below, and connected by scalar propagators l / q 2 . It was s h ~ w nthat ~ ? the ~ same idea continues to work in theories with fermions and gluons. Scattering amplitudes are determined from scalar diagrams with three types of MHV vertices, (3),(4) and (5), which are connected to each other with scalar propagators l / q 2 . When one leg of an MHV vertex is connected by a propagator to a leg of another MHV vertex, both legs become internal to the diagram and have to be continued off-shell. Off-shell continuation is defined as follows': we pick an arbitrary spinor and define A, for any internal line carrying a momentum qaa by A, = qaa[R,f. External lines in a diagram remain onshell, and for them X is defined in the usual way. For the off-shell lines, the same tRef is used in all diagrams contributing to a given amplitude.
tief
496
&f
For practical applications CSWl have chosen to be equal to x a of one of the external legs of negative helicity, e.g. the second one, JRef = @. This corresponds to identifying the reference spinor with one of the kinematic variables of the theory. The explicit dependence on the reference spinor disappears and the resulting expressions for all scalar diagrams in the CSW approach are the functions only of the kinematic variables X i a and This means that the expressions for all individual diagrams automatically appear to be Lorentz-invariant (in the sense that they do not depend on an external spinor J&,) and also gauge-invariant (since the reference spinor corresponds to the axial gauge fixing n,A, = 0, where nab = J R e f a J R e f b ) . There is a price to pay for this invariance of the individual diagrams. Off-shell continuation described above leads to unphysical singularities which occur for the whole of phase space and which have to be cancelled between the individual diagrams. The result for the total amplitude is, of course, free of these unphysical singularities, but their cancellation and the retention of the finite part requires some work, see' and section 3.1 of Ref.3. However, these unphysical singularities are specific to the three-gluon MHV vertices and, importantly, they do not occur in any of the MHV vertices involving a fermion field3t8. Using supersymmetric Ward identitieslg one can obtain purely gluonic amplitudes from a linear combination of amplitudes with one fermionantifermion pair':
4fB
N
VAVB
7
A?+
N
1:
i
gi+
1 7
N
(9) with expressions for the remaining hi with A = 2 , 3 , 4 written in the same manner as the expression for A, in (9). The first MHV amplitude (3) is derived from (7) by using the dictionary (9) and by selecting the (qr)4(q8)4 term in (7). The second amplitude (4) follows from the (qt)4(qT)3(qs)1 term in (7); and the third amplitude (5) is an (qr)3(q8)1(qt)3(qq)1 term. There is a large number of such component amplitudes for an extended susy Yang-Mills, and what is remarkable, not all of these amplitudes are MHV. The analytic amplitudes of the N = 4 SYM obtained from ( 7 ) , (9) are8: An(g-,g-) , An(g-,A,,AA+) > An(A,,A,,AA+,AB+) 7 A3+, A4+) , A,(g-, A'+, A2+, A3+, A4+) , A n ( A i ,AA+,A1+,A2+, An(A 1-k 7 A2+ 7 AS+, Ad+,A1+,A2+, A4-k) An(?AB,A A+ > A B + A l + , A 2 f , A 3 + , A 4 + ) , An(g-,$AB,$AB)
7
An(g-r$AB,AA+iAB+)
An(&, 4 A B , $ B C , A C + )1 An@,
4, $, 4)
1
,
An($,
41 $7
7
An(Ai,Aii4AB)
I
A n ( ~ , l $ B C I A A + , ~ B + l ~ C7+ )
A+,A+)
7
An($,
$1
A+,A+,A+,A+)
7
(10)
498 where it is understood that $AB = ~ E A B C D $ In ~ ~Eqs. . (10) we do not distinguish between the different particle orderings in the amplitudes. The labels refer to supersymmetry multiplets, A , B = 1,. . . , 4. Analytic amplitudes in (10) include the familiar MHV amplitudes, (3), (4), ( 5 ) , as well as more complicated classes of amplitudes with external gluinos hA,RBZAl etc, and with external scalar fields $ A B . The second, third and fourth lines in (10) are not even MHV amplitudes, they have less than two negative helicities, and nevertheless, these amplitudes are non-vanishing in N = 4 SYM. The conclusion we drawg is that in the scalar graph formalism in N 5 4 SYM, the amplitudes are characterised not by a number of negative helicities, but rather by the total number of q’s associated to each amplitude via the rules (9). All the analytic amplitudes listed in (10) can be calculated directly from (7), (9). There is a simple algorithm for doing thiss. (1) For each amplitude in (10) substitute the fields by their 7expressions (9). There are precisely eight q’s for each analytic amplitude. (2) Keeping track of the overall sign, rearrange the anticommuting 7’s into a product of four pairs: (sign) x qfq; qZ$q$q: q:qt. (3) The amplitude is obtained by replacing each pair q$$ by the spinor product (i j ) and dividing by the usual denominator,
The vertices of the scalar graph method are the analytic vertices (10) which are all of degree-8 in q and are not necessarily MHV. These are component vertices of a single analytic supervertexb (7). The analytic amplitudes of degree-8 are the elementary blocks of the scalar graph approach. The next-to-minimal case are the amplitudes of degree-12 in q, and they are obtained by connecting two analytic vertices” with a scalar propagator l / q 2 . Each analytic vertex contributes 8 q’s and a propagator removes 4. Scalar diagrams with three degree-8 vertices give the degree-12 amplitude, etc. In general, all n-point amplitudes are characterised by a degree 8,12,16,. . . , (4n - 8) which are obtained from scalar diagrams with bThe list of component vertices (10) is obtained by writing down all partitions of 8 into groups of 4, 3, 2 and 1. For example, A,(g-,qAB,AA+,AB+)follows from 8 = 4 + 2 + 1 1.
+
499
1,2,3, . . . analytic vertices.c In Ref.8 we have derived a simple expression for the first iteration of the degree-8 vertex. This iterative process can be continued straightforwardly to higher orders. 4. Calculating Simple Antianalytic Amplitudes
To show the simplicity of the scalar graph method and to test its results, in this section we will calculate simple antianalytic amplitudes of r]-degree-12. We work in N = 1, N = 2 and N = 4 SYM theories, and study
using the scalar graph method with analytic vertices. The labels N = 1 , 2 , 4 on the three amplitudes above corresponds to the minimal number of supersymmetries for the given amplitude. In this section the N-supersymmetry labels A , B are shown as ( A ) and ( B ) . In all cases we will reproduce known results for these antianalytic amplitudes, which implies that at tree level the scalar graph method appears to work correctly not only in N = 0 , l theories, but also in full N = 2 and N = 4 SYM. In particular, the N = 4 result (34) for the amplitude (12c) will verify the fact that the building blocks of the scalar graph method are indeed the analytic vertices (lo), which can have less than 2 negative helicities, i.e. are not MHV. = as in the section We will be using the of-shell prescription tRef 3. Since in our amplitudes, the reference spinor always corresponds to a gluino A-, rather than a gluon g-, there will be no singularities in our formulae at any stage of the calculation. 4.1. Antianalytic N = 1 amplitude
There are three diagrams contributing to the first amplitude, Eq. (128). The first one is a gluon exchange between two 2-fermion MHV-vertices. This diagram has a schematic form,
=In practice, one needs to know only the first half of these amplitudes, since degree(4n- 8) amplitudes are anti-analytic (also known as googly) and they are simply given by degree-8* amplitudes, similarly degree-(4n - 12) are given by degree-12*, etc.
500
Here gf and g I I are off-shell (internal) gluons which are Wick-contracted -
+
via a scalar propagator, and I = (3,4), which means, X I = (p3 p 4 ) * i 2 . The second and the third diagrams involve a fermion exchange between a 2-fermion and a 4-fermion MHV vertices. They are given, respectively by
with I = (2,4), and
with I = (3,5). Both expressions, (14) and (15), are written in the form which is in agreement with the ordering prescription3 for internal fermions, ket+ ket-. All three contributions are straightforward to evaluate using the relevant expressions for the component analytic vertices. These expressions follow from the algorithm (11). 1. The first contribution, Eq. (13), is
-(1 2)2
1
((2 3 ) 31~+ (2 4 ~ 42 1 1 ~ ) 11~ (3 4 ~ 413 [2 412(1 2)2
-
(4 3)[2 412 (16)
-
1341((2 3 ) 31~+ (2 4 ) 4 ~1 ~ 15 ) 11~* 2. The second diagram, Eq. (14), gives
-(2 3 ) 2 1 ((2 3)[2 31 + (2 4 ) P 41)(3 4) (5 1)[5 11 -[2 512(2 3)2 [2 11[5 11((2 3 ) 31~+ (2 4 ~ 42 1 ~ 4) 3 3. The third contribution, Eq. (15), is
-(2. 1)
(5 ~2 5i2 [2 11
(17)
*
(3 1)2[2 11 (3 (18) [2 11 (1 2 ) 21~ (3 4 ~ 1)5 [2 11(3 4)(5 1) . Now, we need to add up the three contributions. We first combine the expressions in (16) and (17) into 1
using momentum conservation identitites, and the fact that (2 3)[2 31 + (2 4)[2 41 = -(3 4)[3 41 (5 1)[5 11. Then, adding the remaining contribution (18) we obtain the final result for the amplitude,
+
501
which is precisely the right answer for the antianalytic 5-point 'mostly minus' diagram. This can be easily verified by taking a complex conjugation (parity transform) of the corresponding analytic expression. 4.2. Antianalytic N = 2 amplitude
There are three contributions to the amplitude (12b) The first contribution is a scalar exchange between two analytic vertices,
4p4)are off-shell (internal) scalars which are Wick- =-
Here $-I(12) and 4 ( qI
-
contracted. The external index I = (1,2), which implies XI = (p1+pz).i2 = pl . i z . The second contribution to (12b) is a fermion exchange,
with external index I = (3,4), that is, XI = ( p 3 +p4) . x 2 . The final third contribution is again a fermion exchange,
+
with I = (2,3), and XI = (pz p3) . i 2 = p3 . XZ. As before, all three contributions are straightforward to evaluate using the rules (11). 1. The first contribution, Eq. (21), is
1
1 (1 2)[1 21
-(3 5)(3 1 ) P 11 - (3 5)(3 1) . (4 5)(5 1)[1 21 (4 5)(5 1) [2 11 2. The second contribution (22) gives
( l 2,
*
(3 4)'[2 41' 1 4 1 2) (4 3 ) 31~ (3 4 ~ 413 (5 1 ) 11~ 3. The third contribution (23), is
(1 2) P 4i2 . (5 1) [1 21[2 3113 41
(24)
(25)
Now, we add up the three contributions in Eqs. (24), (25), (26) and using the momentum conservation identities obtain
which is the correct result for the antianalytic amplitude.
502 4.3. Antianalytic N = 4 amplitude
The amplitude Af=4(Ac, 1, AG, 2 , A& 3 , A& 4 , g z ) receives contributions only from diagrams with a scalar exchange. There are three such diagrams. The first one is
d?;) and 4y) are off-shell (internal) scalars which are Wickcontracted and XI = (PI + p 2 + p 5 ) . x 2 = (PI + p 5 ) . i2. Here
The second contribution to (12c) is
with external index I = (1,2), that is, XI = (pl + p2) . The third diagram gives,
x2 = p l . x2.
+
with I = (2,3), and XI = (p2 p 3 ) 1 2 = p3 . x 2 . 1. The first contribution, Eq. (28), is
2. The second contribution (29) gives 1
2,
.
(12;p 21
.
3. The third contribution ( 1 4)2 ( 1 4)2 (33) (4 5)(5 1) (4 5)(5 1)[2 31 ’ We add up the three contributions (31), (32), (33) and using the momentum conservation identities obtain (2 3,
’
1 (2 3)[2 31
which is again the correct answer for this amplitude, as it can be easily seen from taking a complex conjugation of the corresponding analytic expression.
Acknowledgements. I am grateful to George Georgiou and Nigel Glover for an enjoyable collaboration and for greatly contributing to my
503 understanding of these topics. I thank t h e organizers for a n excellent conference and Zvi Bern, Lance Dixon, Misha Shifman, Andrei Smilga, Gabriele Travaglini and Arkady Vainshtein for useful discussions and comments. This work is supported by a PPARC Senior Fellowship.
References 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11.
12. 13. 14. 15. 16. 17.
18. 19.
20. 21. 22.
23. 24.
F. Cachazo, P. Svrcek and E. Witten, hep-th/0403047. E. Witten, hep-th/0312171. G. Georgiou and V. V. Khoze, JHEP 0405 (2004) 070, hep-th/0404072. C. J. Zhu, JHEP 0404 (2004) 032 hep-th/0403115; J. B. Wu and C. J. Zhu, hep-th/0406085. I. Bena, Z. Bern and D. A. Kosower, hep-th/0406133. D. A. Kosower, hep-th/0406175. F. Cachazo, P. Svrcek and E. Witten, hep-th/0406177. G. Georgiou, E. W. N. Glover and V. V. Khoze, hep-th/0407027. A. Brandhuber, B. Spence and G. Travaglini, hep-th/0407214. N. Berkovits, hep-th/0402045; N. Berkovits and L. Motl, JHEP 0404 (2004) 056 hep-t h/0403 187. R. Roiban, M. Spradlin and A. Volovich, JHEP 0404 (2004) 012 hepth/0402016; hep-th/0403190. A. Neitzke and C. Vafa, hep-th/0402128; N. Nekrasov, H. Ooguri and C. Vafa, hepth/0403167. E. Witten, hep-th/0403199. S. Gukov, L. Motl and A. Neitzke, hep-th/0404085. W. Siegel, hep-th/0404255. N. Berkovits and E. Witten, hep-th/0406051. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 425 (1994) 217 hep-ph/9403226; Nucl. Phys. B435:59 (1995) hep-ph/9409265; Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. Proc. Suppl. 51C:243 (1996) hep-ph/9606378. V. P. Nair, Phys. Lett. B 214 (1988) 215. M. T. Grisaru, H. N. Pendleton and P. van Nieuwenhuizen, Phys. Rev. D 15 (1977) 996; M.T. Grisaru and H. N. Pendleton, Nucl. Phys. B 124 (1977) 81. S. J. Parke and T. R. Taylor, Phys. Rev. Lett. 56 (1986) 2459. F. A. Berends and W. T. Giele, Nucl. Phys. B 306 (1988) 759. F. A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T. T. Wu, Phys. Lett. B 103 (1981) 124; P. De Causmaecker, R. Gastmans, W. Troost and T. T. Wu, Nucl. Phys. B 206 (1982) 53; R. Kleiss and W. J. Stirling, Nucl. Phys. B 262 (1985) 235; J. F. Gunion and Z. Kunszt, Phys. Lett. B 161 (1985) 333. M. L. Mangano and S. J. Parke, Phys. Rept. 200 (1991) 301. L. J. Dixon, hep-ph/9601359.
WEAK SUPERSYMMETRY AND ITS QUANTUM-MECHANICAL REALIZATION
A. V. SMILGA SUBATECH, Universite‘ de Nantes 4 rue Alfred Kastler, BP 20722, Nantes 44307,France. * E-mail: smilgaOsubatech.in2p3.fr
We explore “weak” supersymmetric systems whose algebra involves, besides Poincare generators, extra bosonic generators not commuting with supercharges. This allows one to have inequal number of bosonic and fermionic 1-particle states in the spectrum. Coleman-Mandula and Haag-Lopuszanski-Sohnius theorems forbid the presence of such extra bosonic charges in interacting theory for d 2 3. However, these theorems do not apply in one or two dimensions. For d = 1, we construct a nontrivial interacting system characterized by weak supersymmetric algebra. It is related to “n-fold” supersymmetric systems and to quasi-exactly solvable systems studied earlier.
1. Introduction
The basic defining feature of any standard supersymmetric system is double degeneracy of all excited states. This follows from the minimal supersymmetry algebra Q2
= Q 2 = 0,
{Q,Q}+ = 2H ,
(1)
where Q is a complex conserved [this is a corollary of Eq.(l)] supercharge. If the superalgebra describing symmetries of the system includes (1) as a subalgebra, double degeneracy of all excited levels (if supersymmetry is spontaneously broken, also the ground state is doubly degenerate) follows. An interesting question is whether some other “weak” supersymmetric algebras involving Poincare generators and conserved supercharges, but not including (1) as subalgebra, are possible. At the algebraic level, the answer is trivially positive. It is easy also to construct Lagrangians enjoying weak * On leave of absence from ITEP, Moscow, Russia.
504
505 supersymmetry. Indeed, the Lagrangian 1 c = -(aw$)2+i$ffwaw$ (2) 2 (4 is a real scalar and $ is a Weyl spinor) is invariant with respect to supersymmetry transformations a
The corresponding supercharges are
Now, Q , and Q" are conserved, but their anticommutators involve besides Pa" also extra terms. In particular, { Q a , &a} # 0. The resulting superalgebra does not include the subalgebra (1) and the number of bosonic and fermionic 1-particle states might be different. And it is: the Lagrangian (2) describes a free real boson (one state IB) for each 3-momentum 3 and a free Weyl fermion (two states IF*)). It is interesting (and important !) that, in the sector with given 6, the state pairing is restored for two-particle excitations and higher. Thus, at the two-particle level, there are two boson states IBB) and IF+F-) and two fermion states IBF+) and IBF-). Actually, any Lagrangian involving some number of free bosonic and some number of free fermionic fields is supersymmetric. There are a lot of such supersymmetries: each bosonic field can be mixed with each fermionic field independently of the others. However, this is only true for free theory. As soon as the interaction is switched on, supersymmetry (strong or weak) is broken. Indeed, nonvanishing (Q,, &a} implies the presence of an extra conserved bosonic charge in the representation (1,O) of the Lorentz group. It is none other than the self-dual part of the fermion spin operator S,p. Spin is not conserved, however, in interacting theories. Actually, in use the standard Weyl notation where the dotted and undotted indices are raised and lowered with eao(&o) and eap(e&b) = - & ( e a o ) ; $6 = (&)t and $& = - ( Q a ) t ;
4' = $&$&; up = (1,a)and ot = (1,-m). of the consequencies of this is the presence of the so called quasigoldstino branch in the spectrum of collective excitations in quark-gluon plasma l. 11' = &$" and
506 any theory involving mass gap and a nontrivial S-matrix, the presense of extra nonscalar conserving charges is ruled out by the Coleman-Mandula theorem 2 . Interacting supersymmetric theories can only involve, besides the Poincare generators P,, M,, and supercharges Q i , central charges commuting with everything and some extra global symmetry generators, which can have nontrivial commutators with Q and between themselves, but they cannot appear in this case in the anticommutators of supercharges 3 . This is true if the dimension of space-time is 3 or more. In two dimensions, where scattering can be only forward or backward, the theorems do not apply. In particular, one can have an infinite number of conserved bosonic charges (like, e.g., it is the case in the Sine-Gordon model). Seemingly, nothing prevents one to have an interacting 2d theory enjoying a version of weak supersymmetry. We tried to construct one, but failed. Probably, one should try harder. What we were able to construct is a weakly supersymmetric quantum mechanical system. It has two complex conserved supercharges with nontrivial anticommutators involving besides H four other bosonic generators which are not central charges - their commutators with supercharges and between themselves do not vanish. The boson-fermion degeneracy is there starting from the second excited level. But not for the first excited level and not for vacuum. In Sect. 2, we describe the simplest such system - the weak supersymmetric oscillator. In Sect. 3, we present a nontrivial weak supersymmetric Hamiltonian. We find that previously studied quantum systems with socalled “2-fold supersymmetry” are in fact weak supersymmetric systems in disguise. We briefly discuss their relationship to quasi-exactly solvable models Sect. 4 is reserved for discussion, conclusions, and acknowledgements. 2t3
‘.
2. Weak supersymmetric oscillator
Consider the Lagrangian
a = 1,2. It can be obtained out of the massive version of the free field theory Lagrangian (2) by dimensional reduction. It is invariant with respect ~~
CFormassless theories, Poincare group can be extended to conformal and super-Poincare - to superconformal.
507
to supersymmetry transformations
The corresponding supercharges are
where p = j: is the bosonic canonical momentum. The canonical Hamiltonian is 1 2
H = -p2
l , , m - + -m x + y($a$a +$a$a) 2
.
(8)
The quantum Hamiltonian can be written by replacing p by -idIda: and by a/d$,. Supercharges commute with the Hamiltonian. On the other hand,
where
+ $'p$a) = y($a$a +$a$a)
zap = m($a$p
y
m - -
9
(10)
The operators Zap and Y commute with H and with each other, but the cammutator s
and also
508
are nontrivial. The algebra can be presented in a little bit more convenient form if introducing S, = Q , - Q,, Sa = Q" Qff. Then {S,
S p } = 4H6:
+ - 2Y6: + 22:
,
[Sa,2071 = m ( E m a s 7 + GYrSL3) , [Sa,~ p , ] = m (6Z.S; + 6,"Sp) ,
[So,YI = -m&, [Sa,Y]= mS"
,
(13)
to which the commutator (12) should be added. The subalgebra (12) is none other than sZ(2) , which can be readily seen by identification 211
2ima+,
Z22
= 2ima-,
212
-mn3 .
= 221
All other commutators and the anticommutator {S,, So} vanish. It is not difficult to find the spectrum of H . The eigenstates are ;
@: = +,In)
,
(14)
where In) are the bosonic oscillator eigenstates. Their energies are
E F = ( - - ; + n ) m , E: =
(: ) -+n
m , E;4 =
(i +
n ) m .(15)
The spectrum is drawn in Fig. 1. We see that there is one vacuum state (its energy can be brought to zero by adding the constant m/2 to the Hamiltonian, but for the weak supersymmetric systems with algebra (13), Eva,= 0 is an option rather than requirement). There are three first excited states: a bosonic and two fermionic. Starting from the second excited state, there are 2 bosonic and 2 fermionic states at each level. The eigenvalues of the operator Y is -m for the leftmost tower, 0 for two central and +m for the rightmost one. In other words, the operator Y / m (or rather Y / m 1 ) plays the role of the fermionic charge. The operators Zap annihilate the states @$while the states @: form doublet representations of the s l ( 2 ) algebra (12). To acquire further insights, it is instructive to write the action of the operators S,, Sa on the states (14):
+
S"@? = 0 ,
S,@? = 2Jmn+:-l,
so@; = -2&€,p@y-l, S,@T = 0
Sv; =2 , Sa@; = 2
J m 6 p n f l , J
m
~
~
~
.@
g( 1 6+)
~
509
Figure 1. Spectrum of weak supersymmetric oscillator.
We see that S, annihilates the states from the rightmost column and brings the states from the ieft columns to the right. The action of is opposite. Now, one can divide all eigenstates in two sets: (2) the states CP? and and (ii) the states CP; and CP?. The states from the subset (i) form the Hilbert space of the = 1 supersymmetric oscillator, with S1 playing the role of the supercharge. The same applies to the subset (iz) with the supercharge 5'2. For ordinary N = 2 supersymmetric quantum mechanics, the Hilbert space can also be divided into two N = 1 subspaces, but the specifics of a weak supersymmetric system is that two sets of states are shifted with respect to each other, i.e. the Harniltonian for the right subset differs from the Hamiltonian for the left subset by a constant. 3. A class of interactive weak supersymmetric systems
Consider the Lagrangian j.2
-
L = - + i$"$, 2
.
v2
V'
2
2
B' 2
- - - - ($2 + 42) - - $ 2 4 2
,
(17)
510
where V(x) is an arbitrary function and C is an arbitrary constant. One can observe that the corresponding action is invariant with respect to the supersymmetry transformations
s x = S$
S$, SG"
+
$€
,
+ S,(V + B $ J -~ 2B ) + ($E)$"] = iPX - E,(V + B q 2 )- 2B [ P ( $ $+ ) $a(S$)]
,
= -i~aX
.
(18)
When V(x) = mx and C = m, Eq.(17) is reduced to the oscillator Lagrangian (5) considered above. The first four terms in Eq.(17) represent a rather natural generalization of Eq.(5), like in Witten's supersymmetric quantum mechanics 6 . In our case, we are obliged to add also a 4-fermion term in the Lagrangian and extra nonlinear terms in the transformation law. The canonical classical supercharges and Hamiltonian are
Qa = p$, Q" = pq"
+ iV$, + iB$'&
, + iV$" + iBq2$," ,
{au,
(19)
The Poisson brackets {Q,, H } ~ . Band . H}P.B.vanish. A certain care is required when quantizing this theory. We want to fix the ordering ambiguities in Q and H such that classical supersymmetry were not spoiled at the quantum level. An experience acquired by fiddling with supersymmetric a-models and gauge theories teaches us that proper quantum supercharges should be obtained by Weyl ordering procedure from the classical expressions (19). In other words, (19) should be Weyl symbols of the quantum supercharges. This does not necessarily apply to the Hamiltonian. Indeed, if choosing the quantum Hamiltonian such that (20) represents the Weyl symbol of &qu, the Weyl symbol of the commutator [ Q a ,&] would be given by the Moyal bracket of Q$ and Hc', 877
[Qa,&] W =
{Q,,H}M.B.=
51 1
TOcompensate that, we should add to Hc' the term B2/2. The quantum supercharges and Hamiltonian thus obtained are Qa
= Inl,
+ iV& + iB
4"= p$" + iV$"
+ iB
+ &)
,
- $a)
(22)
and
v2 V' B' B2 C 2 (q2 q2) 2 2 ($'q' - 2$4 1) 2 2 (23) where we added for convenience the constant C/2 in the Hamiltonian. Direct calculation of the commutators (or, which is simpler, of the Moyal brackets of the classical expressions) leads to a remarlable conclusion: the algebra (13, 12) derived for the oscillator is valid also in the interactive case, with m C, H -+ H - C/2 and Z,p,Y having the same form (10) as before. As earlier, the quantum states can be divided into three classes: (i) the states I-) 0: 1 - $'/2, (ii) the states la) o( qff (they are present in two copies as the Hamiltonian (23) is not sensitive to the index a of the fermion state) and (iii) the states I+) oc 1+$2/2. These states are characterized by a definite value of the "fermion charge" Y : Y* = f m and Ya = 0. In each such sector, we have an ordinary Schrodinger equation with the potentials
a
=
+y
+
+
1 u" ---(W"WL) 2
u,
1
= z(w:
+ +
+
=
+
1 -cw:-w;>+c, 2
+ w;) + c ,
where W*=V&B.
(25)
It is clear now that we are dealing with two superimposed ordinary Witten's SQM systems. The states I-) and 11) are described by such system with superpotential W- and the states 12) and I+) - by the system with superpotential W+, with the constant C added to the Hamiltonian. Excited states are mostly 4-fold degenerate as for usual N = 2 SQM. The ground state is not necessarily degenerate. If exp{ - J W- (z)ds} is normalizable, this (being multiplied by 1 - +'/2) determines the wave function of the unique vacuum state. With the chosen normalization of the Hamiltonian [the term C/2 in Eq.(23) !] it has zero energy. Further,
512 if exp{ - 1W+(z)dx} is normalizable, there is also a unique zero-energy ground state for Witten’s Hamiltonian with superpotential W+. Thus, we obtain a state in the sector 12) with energy C. Due to 12) +) 11) and 11) +) I-) degeneracies, we have altogether three states with energy C at the first excited level, and the picture is the same as for the oscillator (see Fig.1). We have obtained free of charge a wide class of quasi-exactly solvable potentials U - ( x ) for which the energy of the ground state (Eo = 0) and of the first excited state El = C are exactly known. They depend on an arbitrary function V ( x )and an arbitrary constant C with a certain restriction: both exp{W - ( x ) d z } and exp{- W+(x)dz} should be normalizable. Probably, the simplest nontrivial choice is V ( z )= mx ax3 with C = m (m,a > 0). Note that the potential U-(x)is not polynomial in this case. The potentials U* in Eqs.(24, 25) were discussed before (see Eq.(4.18) in Ref.g and Eq.(50) in Ref.” ) in association with the so called 2-fold supersymmetry construction developped in 4 . N-fold supersymmetry is a supersymmetry where supercharges are not linear in momentum, but present polynomials of power N . In our case, one can define the quadratic in p operator
s
s
& =
+
s1s2
el+)
a
(26)
with the action = Q ( a )= 0 and &I-) = I+). acts in the opposite direction: GI+) = I-), Q1a) = GI-) = 0. The operators &, commute with H (as S1,2 and S1>’do). If disregarding the states la) annihilated by both Q and and considering only the sectors I+) and I-), one can deduce
a
{Q,Q} = 16H(H- C) .
(27)
The quadratic polynomial of H appearing on the right side is characteristic of 2-fold supersymmetry. The full algebra of the weak supersymmetry (12, 13) displays itself only if the “central” sector la) is brought into consideration. 4. Discussion
Our original motivation was the quest for nontrivial supersymmetric systems with mismatch between bosonic and fermionic degrees of freedom. Let us note here that, while it is difficult to find such field theory systems, their presence in quantum mechanics was known for a long time. Most popular SQM systems (Witten’s quantum mechanics, standard supersymmetric n models, etc) have an equal number of bosonic and fermionic phase
513
space coordinates. But the SQM system describing planar motion in transverse magnetic field involves two pairs of bosonic variables and only one pair of fermionic variables. A class of nonstandard “symplectic” N = 2 supersymmetric B models involving 3r bosonic variables and 2r fermionic variables (r is an integer) was constructed in Ref.”. The Diaconescu-Entin N = 4 symplectic B model l2 generalized in l 3 involves 5r bosonic and 4r fermionic variables. An industrial method to construct SQM models where the number of bosonic variables is less than the number of fermionic ones was suggested in 1 4 . In SQM, an equal number bosonic and fermionic degrees of freedom is not required by supersymmetry. What is required is the equal number of bosonic and fermionic quantum states. But in field theory, any bosonic or fermionic dynamical field correspond to an asymptotic state (a particle), and bosons and fermions should normally be matched. We notice that this matching can be absent if relaxing the requirement that the anticommutator of supercharges involves only the Hamiltonian, momentum, and central charges. A lot of free weak supersymmetric models can be written, but, for d > 3, interactive weak supersymmetric theories do not exist. This follows from the Haag-Lopuszanski-Sohnius theorem. This theorem does not apply to 2 dimensions, however, and the existence of interactive weak supersymmetric theories cannot be ruled out. Our quest for such theories was not successful, but it certainly pays to try harder. The main positive result of this paper is the system (17) which enjoys a weak supersymmetry algebra (12), (13). It describes quantum systems which were studied before, but from a different perspective. It would be interesting to construct and study other, more complicated weak supersymmetric models, especially the models involving several bosonic degrees of freedom. This would allow one to construct new examples of multidimensional quasi-exactly solvable models. I am indebted to N. Dorey, M. Henneaux, N. Nekrasov, M. Plyushchay, M. Shifman, and A. Vainshtein for illuminating discussions and correspondence.
References 1. V.V. Lebedev and A.V. Smilga, Ann. Phys. 202, 229 (1990). 2. S.R. Coleman and J. Mandula, Phys. Rev. 159,1251 (1967). 3. R.Haag, J.T. Lopuszanski, and M. Sohnius, Nucl. Phys. B88,257 (1975).
4. A.A. Andrianov, M.V. Ioffe, and V.P. Spiridonov, Phys. Lett. A174, 273 (1993). 5. A. Turbiner, Commun. Math. Phys. 118,467 (1988); A. Ushveridze, Sov. J .
514
Part. Nucl. 20, 504 (1989); for a review see M. Shifman, ITEP lectures on particle physics and field theory, World Scientific, 1999, p. 775. 6. E. Witten, Nucl. Phys. B188, 513 (1981). 7. A.V. Smilga, Nucl. Phys. B292, 363 (1987). 8. I.E. Moyal, Proc. Cambr. Phil. SOC.45,99 (1949). 9. H. Ayoama, M. Sato, and T. Tanaka, Nucl. Phys. B619, 105 (2001) [arXive: quant-ph/0106037]. 10. A.A. Andrianov and A.V. Sokolov, Nucl. Phys. B660, 25 (2003) [arXive: hep-th/0301062]. 11. A.V. Smilga, Nucl. Phys. B291, 241 (1987); E.A. Ivanov and A.V. Smilga, Phys. Lett. B257, 79 (1991). 12. D.-E. Diaconescu and R. Entin, Phys. Rev. D56, 8045 (1997) [arXive: hepth/9706059]. 13. A.V. Smilga, Nucl. Phys. B652, 93 (2003) [arXive:hep-th/0209187]. 14. E.A. Ivanov, S.O. Krivonos, and A.I. Pashnev, Class. Quant. Grav. 8 , 19 (1991); A. Losev and M. Shifman, Mod. Phys. Lett. A16, 2529 (2001) [arXive: hep-th/0108151].
NONPERTURBATIVE SOLUTION OF YUKAWA THEORY AND GAUGE THEORIES
JOHN R. HILLER Department of Physics University of Minnesota-Duluth Duluth, M N 55812 USA E-mail:
[email protected] Recent progress in the nonperturbative solution of (3+1)-dimensional Yukawa theory and quantum electrodynamics (QED) and (l+l)-dimensional super Yang-Mills (SYM) theory will be summarized. The work on Yukawa theory has been extended to include two-boson contributions to the dressed fermion state and has inspired similar work on QED, where Feynman gauge has been found surprisingly convenient. In both cases, the theories are regulated in the ultraviolet by the inclusion of Pauli-Villars particles. For SYM theory, new high-resolution calculations of spectra have been used to obtain thermodynamic functions and improved results for a stress-energy correlator.
1. Introduction Numerical techniques can be successfully applied to the nonperturbative solution of field theories quantized on the light c ~ n e Unlike . ~ ~lattice ~ ~ gauge ~ theory: wave functions are computed directly in a Hamiltonian formulation. The properties of an eigenstate can then be computed relatively easily. There have been a number of successes in two-dimensional t h e ~ r i e s ,but ~ in three or four dimensions the added difficulty of regulating and renormalizing the theory has until recently limited the success of the approach. Here we discuss recent progress with two different yet related approaches to regularization. One is the use of Pauli-Villars (PV) r e g u l a r i ~ a t i o n ~ ~ and the other supersymmetry.1° The particular applications to be discussed are to Yukawa theory and &ED in 3fl dimensions with P V fields and to super Yang-Mills (SYM) theory in 1+1 dimensions. In the latter case, extension to 2+1 dimensions has already been done;" however, the most recent developments have used two dimensions as a proving ground. There we consider in particular a stress-energy c ~ r r e l a t o r land ~ ~ analysis ~ ~ > ~ ~of finite-temperature effects.15
515
516
The light-cone coordinates1 that we use are defined by xf = xo f x3, 21 = (x1,x2),with the expression for xf divided by & in the case of supersymmetric theories. Light-cone three-vectors are denoted by a n underline: p = (p+ ,pi). The keyelements of the PV approach are the introduction of negative metric P V fields to the Lagrangian, with couplings only to null combinations of P V and physical fields; the use of transverse polar coordinates in the Hamiltonian eigenvalue problem; and the introduction of special discretization of this eigenvalue problem rather than the traditional momentum grid with equal spacings used in discrete light-cone q u a n t i ~ a t i o n . The ~ ? ~ choice of null combinations for the interactions eliminates instantaneous fermion terms from the Hamiltonian and, in the case of QED, permits the use of Feynman gauge without inversion of a covariant derivative. The transverse polar coordinates allow use of eigenstates of J, and explicit factorization from the wave function of the dependence on the polar angle; this reduces the effective space dimension and the size of the numerical calculation. The special discretization allows the capture of rapidly varying integrands in the product of the Hamiltonian and the wave function, which occur for large PV masses. For supersymmetric theories, the technique used is supersymmetric discrete light-cone quantization (SDLCQ) ,l6>lowhich is applicable to theories with enough supersymmetry to be finite. .This method uses the traditional DLCQ grid in a way that maintains the supersymmetry exactly within the numerical approximation. The symmetry is retained by discretizing the supercharge Q- and computing the discrete Hamiltonian P- from the superalgebra anticommutator {Q- , Q - } = Z A P - . To limit the size of the numerical calculation, we work in the large-l\r, approximation; however, this is not a fundamental limitation of the method. 2. Yukawa theory
The Yukawa action with a PV scalar and a PV fermion is
517
From this we obtain the light-cone Hamiltonian (2)
+
c
/d&{
[v--*2s(p,g)+ V z s ( p + q-, g ) ] bJ,s(p)a:(g)bi,-s(E+g)
i,j,k,s
+ [W_P,g)+
+ g 4 ] b;,,(E)ab(g)bi,s(p + g) + h.c.}
7
where antifermion terms have been dropped. No instantaneous fermion terms appear because they are individually independent of the fermion mass and cancel between instantaneous physical and P V fermions. The vertex functions are
with
-32,
- L ( 2 s , i ) . The nonzero (anti)commutators are fi
We construct a dressed fermion state, neglecting pair contributions; it takes the form
The wave functions f...s(xn,fin) satisfy the coupled system of equations that results from the Hamiltonian eigenvalue problem P+P-@+ = M2@+. Each wave function has a total L, eigenvalue of 0 (1)for s = +1/2 (-1/2). The coupled equations are P+
mSzi
+ C(-~)~’+jp+ J d q { f i j j - ( g ) [ V + ( L :-g,g> + v-*(~fl~)I (6) i’ , j
+ fitj+(Q)[Ui,(P - g,g) + K(P,Q)l)= M 2 G ,
518
and
We consider truncations of the system.
A truncation to one boson leads to an analytically solvable problem dThe one-boson wave functions are
Substitution into Eq. (6) yields
519
with
The presence of the P V regulators allows 10 and J to satisfy the identity & J ( M 2 ) = M210(M2). With M held fixed, the equations for zi can be viewed as a n eigenvalue problem for g2. The solution is
_ z1 -F mo = - ( M F mo)(M F m1) (14) (m1- mo)(Pol1 f M I o ) zo M F 7% * An analysis of this solution is given in Ref. 7. In a truncation to two bosons, we obtain the following reduced equations for the one-boson-one-fermion wave functions:8 92
= f i j + ( y , q ~ ) ,@fij-(g) = fij-(y,ql)ei', I is an where mfij+(g) analytically computable self-energy, and J(")is a kernel determined by nboson intermediate states. These reduced integral equations are converted to a matrix equation via quadrature in y' and q y . The matrix is diagonalized to obtain g2 as an eigenvalue and the discrete wave functions from the eigenvector. A useful set of quadrature schemes is based on Gauss-Legendre quadrature and particular variable transformations. The transformation for y' is motivated by the need for an accurate approximation to the integral J . This integral appears implicitly in the product of the Hamiltonian and the eigenfunction and is largely determined by contributions near the endpoints whenever the P V masses are large. The transformation for the transverse integral is chosen to reduce the range from infinite to finite, so that no momentum cutoff is needed.
520 From the wave functions we can extract a structure function fBs(y),
n=l
n=l
defined as the probability density for finding a boson with momentum fraction y while the constituent fermion has spin s. Typical results are plotted in Fig. 1 .
0.10
0.08 0.06 >\
W
l4
cc
0.04 0.02 0.00 0.0
0.2
0.4
0.6
0.8
1 .o
Y Figure 1. Bosonic structure functions in Yukawa theory, with a two-boson truncation (fs+:solid; fs-: long dash) and a one-boson truncation ( f ~ + :short dash; fs-: dotted). The constituent masses had the values mo = -1.7~0, ml = p1 = 1 5 ~ 0 .The resolutions used in the Gauss-Legendre method are K = 20 and N = 30.
521 3. Feynman-gauge QED We apply these same techniques to QED.g The Feynman-gauge Lagrangian is
which is independent of A and can therefore be solved without inverting a covariant derivative. We then obtain the Hamiltonian without antifermion terms as being
+ bf,s(P)b3.-s(Q)~~,-zs(p_,g)]
- p ) + h.c.} ,
where e p = (-1, 1 , 1 , 1 ) . The vertex functions U and V are given in Ref. 9. The dressed electron state, without pair contributions and truncated to one photon, is
I+) = C z i b i + ( P ) I O ) + i
c1
dkf? - %is (k)bf,(kb!”(P - IG)IO),
S,p,iJ
(21)
522 with one-photon-one-electron wave functions
Substitution into P+P-I$) = M21$) yields
This is the same form as in the one-boson Yukawa problem, with g2 -+ 2e2 and I1 + -2I1, and a n analytic solution is again obtained. From this solution we can compute various quantities, including the anomalous magnetic m ~ m e n t . ~ 4. A correlator in n / = ( 2 , 2 ) SYM theory Reduction of N=1 SYM theory from four to two dimensions provides the action we need. In light-cone gauge ( A - = 0) it is
+-(a-A+)2 1 + gA+ J+ 2
+ h g B T ~ ~ P r [ OR] x r ,+ g XI, 2 XJ]’
1
.
Here the trace is over color indices, the X I are the scalar fields and the remnants of the transverse components of the four-dimensional gauge field A,, the two-component spinors OR and BL are remnants of the right-moving and left-moving projections of the four-component spinor in the fourdimensional theory. We also define J+ = i [ X ~ , d - X r ] 2 @ 3 ~ , PI (TI, P2 ( ~ 3 and , €2 -ia2. The stress-energy correlation function for N=(S,S)SYM theory can be calculated on the string-theory side:12 (T++(x)T++(O))= (N2’2/g)x-5. We find numerically that this is almost true in N=(2,2)SYM theory.14 To compute the correlator,13 we fix the total momentum P+, compute the Fourier transform, and express the transform in a spectral decomposed
+
=
523 form
1 F(P+,x+) = -(T++(P+, x+)T++(-P+, 0)) 2L
The position-space form is recovered by Fourier transforming with respect to the discrete momentum P+ = K7r/L, where K is the integer resolution and L the length scale of DLCQ.2 This yields
We then continue to Euclidean space by taking r = d E to be real. The is independent of L. Its form can be matrix element (L/7r)(OIT++(K)li) substituted directly to give an explicit expression for the two-point function. The correlator behaves like l / r 4 at small r :
For arbitrary r , it can be obtained numerically by either computing the entire spectrum (for “small” matrices) or using Lanczos iterations (for large).13 In Fig. 2, we plot the log derivative of the scaled correlation function14 -
f
2
47r25-4 (3) N,”(2% + nf)
(T++(X)T++(O))
’
At small r , the results for f match the expected (1 - 1 / K ) behavior. At large r the behavior is different between odd and even K , but as K increases, the differing behavior is pushed to larger r . For even K , there is exactly one massless state that contributes to the correlator, while there is no massless state for odd K . The lowest massive state dominates for odd K a t large r ; however, this state becomes massless as K -+ co. In the intermediate-r region, the correlator behaves like r-4,75,or almost T - ~ . The size of this intermediate region increases as K gets larger. 5.
N=(l,l)SYM theory
at finite temperature
In this case, the Lagrangian is L: = Tr (-$F,,Fp” + i$’y,D”iP), with F,, = a,A, - &A, ig[A,,A,] and D, = 8, ig[A,]. The supercharge in light-cone gauge is &- = 23/49 dx- (i[+, &+] 2 $ 4 ~ a:)’$.
+
s
+
+
524
log,&)
Figure 2. The log derivative of the scaled correlation function f defined in Eq. (30) of the text. The resolution K ranges from 3 to 12. For even K , f becomes constant at large r and the derivative goes to zero.
From the discrete form we can compute the spectrum, which at large-Nc represents a collection of noninteracting modes. With a sum over these modes, we can construct the free energy at finite temperature from the partition function17J5 e - p o l T . The one-dimensional bosonic free energy is
and the fermionic free energy is
The contributions from the K - 1 massless states in each sector are
The total free energy, with the logs expanded as sums and the PO integral already performed, is
The sum over 1 is well approximated by the first few terms. We can represent the sum over n as an integral over a density of states: E n -+
525
s p( M ) d M and approximate p by a continuous function. The integral over
-
M can then be computed by standard numerical techniques. We obtain p by a fit to the computed spectrum of the theory and find p ( M ) exp(M/TH), with TH 0 . 8 4 5 d m , the Hagedorn temperature.18 From the free energy we can compute various other thermodynamic functions up to this temperature.15 N
6 . Future work
Given the success obtained to date, these techniques are well worth continued exploration. In Yukawa theory, we plan to consider the two-fermion sector, in order to study true bound states. For QED the next step will be inclusion of two-photon states in the calculation of the anomalous moment. For SYM theories, we are now able to reach much higher resolutions, by computing on clusters. This will permit continued reexamination of theories where previous calculations were hampered by low resolution, particularly in more dimensions. Earlier work on inclusion of fundamental matterlg can be extended to three dimensions and modified to include finite-N, effects, such as baryons with a finite number of partons and the mixing of mesons and glueballs. For all of this work, the ultimate goal is, of course, the development of techniques sufficient to solve quantum chromodynmics. Acknowledgments
The work reported here was done in collaboration with S.J. Brodsky, G. McCartor, V.A. Franke, S.A. Paston, and E.V. Prokhvatilov, and S. Pinsky, N. Salwen, M. Harada, and Y. Proestos, and was supported in part by the US Department of Energy and the Minnesota Supercomputing Institute. References P.A.M. Dirac, Rev. Mod. Phys. 21,392 (1949). H.-C. Pauli and S.J. Brodsky, Phys. Rev. D 32, 1993 (1985); 2001 (1985). S.J. Brodsky, H.-C. Pauli, and S.S. Pinsky, Phys. Rep. 301,299 (1997). I. Montvay and G. Miinster, Quantum Fields on a Lattice (Cambridge U. Press, New York, 1994); J. Smit, Introduction t o Quantum Fields on a Lattice (Cambridge U. Press, New York, 2002). 5. W. Pauli and F. Villars, Rev. Mod. Phys. 21,4334 (1949). 6. S.J. Brodsky, J.R. Hiller, and G. McCartor, Phys. Rev. D 5 8 , 025005 (1998) [arXiv:hep-th/9802120]; 60, 054506 (1999); 64, 114023 (2001): Ann. Phys. 296,406 (2002). 7. S.J. Brodsky, J.R. Hiller, and G. McCartor, Ann. Phys. 305,266 (2003).
1. 2. 3. 4.
526
8. S.J. Brodsky, J.R. Hiller, and G. McCartor, in preparation. 9. S.J. Brodsky, V.A. Franke, J.R. Hiller, G. McCartor, S.A. Paston, and E.V. Prokhvatilov, arXiv:hep-ph/0406325. 10. 0. Lunin and S. Pinsky, in New Directions in Quantum Chromodynamics, edited by C.-R. Ji and D.-P. Min, AIP Conf. Proc. No. 494 (AIP, Melville, NY, 1999), p. 140, [arXiv:hep-th/9910222]. 11. F. Antonuccio, 0. Lunin, and S. Pinsky, Phys. Rev. D 59, 085001 (1999); P. Haney, J.R. Hiller, 0. Lunin, S. Pinsky, and U. Trittmann, Phys. Rev. D 62,075002 (2000);J.R. Hiller, S. Pinsky, and U. Trittmann, Phys. Rev. D 64, 105027 (2001). 12. F. Antonuccio, A. Hashimoto, 0. Lunin, and S. Pinsky, JHEP 9907, 029 (1999). 13. J.R. Hiller, 0. Lunin, S. Pinsky, and U. Trittmann, Phys. Lett. B 482,409 (2000); J.R. Hiller, S. Pinsky, and U. Trittmann, Phys. Rev. D 63, 105017 (2001). 14. M. Harada, J.R. Hiller, S. Pinsky, and N. Salwen, to appear in Phys. Rev. D, arXiv:hep-ph/0404123. 15. J.R. Hiller, Y. Proestos, S. Pinsky, and N. Salwen, arXiv:hep-th/0407076. 16. Y. Matsumura, N. Sakai, and T. Sakai, Phys. Rev. D 52,2446 (1995). 17. S. Elser and A.C. Kalloniatis, Phys. Lett. B 375,285 (1996). 18. R. Hagedorn, Nuovo Cimento Suppl. 3, 147 (1965); Nuovo Cimento 56A, 1027 (1968). 19. J.R. Hiller, S.S. Pinsky, and U. Trittmann, Nucl. Phys. B 661, 99 (2003); Phys. Rev. D 67,115005 (2003).
FERMIONIC THEORIES IN TWO-DIMENSIONAL NONCOMMUTATIVE SPACE
E. F. MORENO* Departamento de Fisica Universidad Nacional de La Plata C.C. “ 6 7 - 1900 La Plata - Argentina E-mail:
[email protected] We analyze the connection between Wess-Zumino-Witten and free fermion models in two-dimensional noncommutative space. Starting from the computation of the determinant of the Dirac operator in a gauge field background, we derive the corresponding bosonization recipe. Concerning the properties of the noncommutative Wess-Zumino-Witten model, we construct an orbit-preserving transformation that maps the standard commutative WZW action into the noncommutative one.
1. Introduction
Noncommutative field theories have recently attracted much attention in connection with the low-energy dynamics of D-branes in the presence of a background B field Concerning two-dimensional noncommutative field theories, both bosonic and fermionic models have been recently investigated 4-9. In this paper we pursue the analysis of two-dimensional noncommutative models by carefully studying the fermion effective action. ll2i3.
2. Fermionic and gauge fields in d = 2 noncommutative
space We work in two-dimensional Euclidean space and define the tween functions $(x) and x ( x ) in the form
*Associated with CONICET.
527
* product
be-
528
where 6,, = OED, with 6 a real constant. Then, the Moyal bracket is defined as {+(Z)I
x ( x ) ) = +(XI
* X(”)
- x(x)
* +(x)
which implies a noncommutative relation for space-time coordinates
{x,, 5,)
= ie,,
(2) zp,
(3)
In the case of gauge theories, noncommutativity leads to the definition of the curvature F,, in the form
F,, = a,A, - &A, - ie{A,, A,}
(4)
Gauge transformations are defined in the form
* g-’(x) + ig-’(x)a,y(z)
A:(%)= g(x) * A,(x) where g(x) is represented by a
g ( x ) = exp,(iX(z))
(5)
* exponential, 1 = 1+ iX(x) - -X(x) * X(x) + . . . 2
(6)
with X = X a t a taking values in the Lie algebra of U ( N ) . The covariant derivative D, [A]implementing gauge transformations takes the form
D,X = a X ,
- ie{A,, A}
(7)
Given a fermion field $(x), three alternative infinitesimal gauge transformations can be considered lo
* $(x) z$(x) * 4x1
S,$ S,$
= i+)
(8)
=
(9)
S,$
= i{+),
-’
$(.))
(10)
In this respect, we should refer to fermions in the fundamental f (eq.(8)), ‘anti-fundamental’ f (eq.(9)) and ‘adjoint’ ad (eq.(lO)) representations. The associated covariant derivative are defined accordingly,
Dj$+”)
= a D $ ( 4 - ieA,(x)
D $ 4 1 $ ( 4 = a,$(.) D 3 A I $ ( x ) = D,$(x)
* $(x)
+ ie$(x)A,(x)
(11) (12)
(13)
Using each one of these three covariant derivatives we can construct three different gauge invariant Dirac actions for fermions
529
The corresponding gauge effective action is given by
3. Perturbative effective action
Let us start by computing the quadratic part of the effective action defined by eq.(15). The interaction term SI of the action S[$,$,A] (eq.(14)) takes the form, in momentum representation,
sr = 2ie
l,
q ( P ) y p $ ( q ) ~ , ( - p- 4 ) f ( q A P )
(16)
with f(p A q) =
{
fund ei P A q -e-i pAq antif 22 sin(p A q ) adj
,
P A q = 8’l”Ppq”
(17)
The quadratic part r(’) of the effective action is given by the vacuum polarization tensor, thus giving
Clearly, for the fundamental and anti-fundamental representation, If ( q A p)I2 = 1 and we obtain the standard (commutative) result. For the adjoint representation, using the identity sin(a)2 = 1/2 cos(2a)/2 we can extract form eq. (18) the so-called planar contribution (corresponding to the factor 1/2) and the non-planar contribution (corresponding to the factor - cos(2a)/2:
The planar contribution to the diagram is the standard (commutative) one and can be computed using for example dimensional regularization (in this case the infrared an ultraviolet divergences cancel each other). One has
It is worthwhile to mention that this result is twice the effective action in the fundamental and anti-fundamental representations (that is, taking the Dirac operator as defined either by (11) or by (12)), as it can be easily
530 seen by noticing that in the later the diagram has a vertex contribution of eipAq e-apAq = 1 (there is no non-planar contribution) while in the former we have a contribution -(2isin(p A q ) 2 = 4(1/2 - 1/2cos(2 p A 4)). Thus, for the planar part of the diagram we have (2) Adj = 2 r(2)F u n d (21) ‘planar Concerning the non-planar contribution to I?(’) (see for a detailed computation) , it can be shown using Schwinger parameters that it vanishes. That is, up to quadratic order in the fields, the effective action is given by the planar part:
Therefore, assuming that the higher point contributions to the effective action are the minimal necessary to recover gauge invariance (we will prove this statement in the next section), the effective action in the adjoint representation is twice the effective action in the fundamental representation. Hence, a relation like (21) should hold for the complete effective action I? rAdj = 2 rFund (23) This computation can be easily generalized to U ( N ) . The planar and non-planar contributions are the same as in the U(1) case except for the group theoretical prefactors. Taking into account that the non-planar contribution vanishes we finally have for U ( N )in the adjoint representation
This result is 2N times the quadratic effective action in the fundamental representation. 4. The fermion determinant in the U(1) case
Here we shall briefly describe the exact calculation of the effective action for noncommutative U(1) fermions in the fundamental and adjoint representations by integrating the chiral anomaly. In two dimensions a gauge field A, can always be written in the form 415
1 .
4 = - -e ( V w 4 ,771) * m 4 , 771
(25)
with
+
U [ 4 ,rll = exP*(Y54
277)
7
(26)
531 so one can relate the fermion determinant in a gauge field background A, with that corresponding to A, = 0 by making a decoupling change of variables in the fermion fields. The appropriate change of fermionic variables is
llt = U[+>rll * x
4 = x * u-l[+,rll
7
(27)
in the fundamental representation or
llt = ey5{4?'}* x = x + r5{+, XI
1 + Ti+, {+, X I ) +
* *
in the adjoint. One gets
where J f [t+,A] is the Fujikawa Jacobian associated with a transformation Ut such that (0 5 t 5 1)
uo = 1
7
Ul = U[+,rll
Let us briefly describe the calculation presented in 4,5 based in the the evaluation of the chiral anomaly. Considering an infinitesimal local chiral transformation of the fermion fields the chiral anomaly A = A?", 13,j,"" = A"[A] ,
(30)
where j: is the chiral current
j: =
{
llta * $p (r5 Y,)Ba *
fund (31)
(r5 r?pa
adj
can be calculated from the F'ujikawa Jacobian J [ EA,] , associated with infinitesimal chiral transformation, log J [ E A,] , = -2tr'
sd
'ZA[[A]E(Z)
(32)
Using the heat-kernel regularization, the anomaly can be written as
A[A]= lim Tr M2+Co
(33)
The covariant derivative in the regulator has to be chosen among those defined by eqs.(ll)-(13) according to the representation one has chosen for
532 the fermions. Concerning the fundamental representation, it can be shown but that the anomaly gives a result similar to the commutative case with the noncommutative version of FPu given in (4). e A f [ A ]= -&PUFP,, 4n (34) 111475
Analogously, one obtains for the anti-fundamental representation e A f [ A ]= --&P"FPU (35) 4n For the adjoint representation, however we have different result. In fact, expanding equation (33) in a planar wave basis, keeping the proper powers of M , taking the trace and integrating over d 2 k we arrive to the following expression for the anomaly
Notice in this expression that if we take the 8 -+ 0 limit before taking the M -+ 00 limit Aadjvanishes and the Jacobian is 1, so we recover the standard (commutative) result which corresponds to a trivial determinant for the trivial U(1) covariant derivative. However the limits 8 -+ 0 and M + 00 do not commute so that if one takes the M 00 limit at fixed 8 one has the result -+
"s
Aadj= _2n _ d2x&PuFLu* (I
(37)
which is twice the result of the fundamental representation. The integration in t can be done in a very similar way to the commutative case (see and for details). The result can be put in a more suggestive way in the light cone gauge where
Indeed, in this gauge one can see that the logarithm of the Jacobian becomes log
(w) det
iC
= log J [ $ ,A] =
+--&ijktrC 12n
d3yg-'
C
--trc 8n
/
d 2 x (dPg-')
* (aPg)
* ( a i g ) * 9-l * ( a j g ) * g - l *
(&g)
(39)
with c = 1 in the fundamental and anti-fundamental representations and c = 2 in the adjoint representation. Here we have written d 3 y = d 2 x d t so
533
that the integral in the second line runs over a 3-dimensional manifold B which in compactified Euclidean space can be identified with a ball with boundary S 2 . This is precisely the noncommutative extension of the WessZumino-Witten action. This result can be straightforwardly generalized to the group U ( N ) ,with the only difference that a trace over the color indices has to be performed and c = 2N in the adjoint representation4. 5. The bosonization rules Once one has gotten an exact result for the fermion determinant, one can derive the path-integral bosonization recipe. That is, a mapping from the two-dimensional noncommutative fermionic model onto an equivalent noncommutative bosonic model. This implies a precise relation between the fermionic and bosonic Lagrangian, currents, etc. The basic procedure to obtain this bosonization recipe parallels that already established in the ordinary commutative case. For the noncommutative case, it was developed in detail in (see also 5 ) . Summarizing we obtain, analogously to the commutative case, the following bosonization rules
We have then an equivalence between a two-dimensional noncommutative free fermion model and the non-commutative Wess-Zumino-Witten model. Now, on one hand we know that, being quadratic, the action for noncommutative free fermions coincides with that for ordinary (commutative) ones. On the other hand we know that ordinary free fermions are equivalent to a bosonic theory with ordinary WZW action. The situation can be represented in the following figure
Jd2x
4i$$
I
-
WZW[g]
534
So, we were able to pass from the noncommutative WZW theory with action WZW[ij]to the commutative one, WZW[g]by going counterclockwise through the fermionic equivalent models. One should then be able to find a mapping ij -+ g , analogous to the one introduced in for noncommutative gauge theories. In the present case, the mapping should connect exactly WZW[ij]with WZW[g]filling the cycle in the figure. Next section is devoted to the construction of such a mapping. 6. Mapping of the Wess-Zumino-Witten actions
The Seiberg-Witten map is a mapping between gauge theories defined in different noncommutative spaces, so is natural to try adapt this construction to noncommutative WZW fields. Consider a WZW action defined in a non-commutative space with deformation parameter 0. The action is invariant under chiral holomorphic and anti-holomorphic transformations ij
-+
iI(5)ij R(z)
(41)
so in analogy with the Seiberg-Witten mapping we will look for a trans-
formation that maps respectively holomorphic and anti-holomorphic orbits into orbits. Of course the analogy breaks down at some point as this holomorphic and anti-holomorphic “orbits” are not equivalence classes of physical configurations, but just symmetries of the action. However we will see that such a requirement is equivalent, in some sense, to the “gauge orbits preserving transformation condition” of Seiberg-Witten. Thus, we will find a transformation that maps a group-valued field g’ defined in noncommutative space with deformation parameter 0’ to a groupvalued field i j , with deformation parameter 0. We demand this transformation to satisfy the condition
iI’(z)*’ 4’
*’ R‘(z) + iI(z)* ij * R(z)
where the primed quantities are defined in a O‘-noncommutative space and the non primed quantities defined in a 0-noncommutative space. In particular this mapping will preserve the solutions of the equations of motion:
4’ = a’(z)*’ pyz)
ij = a(5)* p ( z )
(43) After some trial and error it can be shown that the following transformations -+
535 preserve condition (42) for functions R(z) and deed we have, for example
= (i* O ( z ) ) * a,
n(z)independent of 8. In-
(4 * R(z))-l * &(i* R(z))
(45)
and a similar equation for the anti-holomorphic transformation. The next step is to see how does the WZW action transforms under this mapping. Let us define w = 4-1
* dij
(46)
where 6 is any variation that does not acts on 8, and the holomorphic current j ,
j , = 4-1
* a,ij
(47)
It can be shown that
dw _ de - - a , ~* j ,
-j,
* aZw
4, = -a& -
* j , - j , * a& = -az(jZ) (49) d8 Similarly we can find the variations for a = dij * 4-l and the antiholomorphic current j,. Now, instead of studying how does the &map acts on the WZW action, it is easy to see how does the mapping acts on the variation of the WZW action with respect to the fields. In fact, we have k
dW[4 = 7T
/
d2x tr ( & j , w )
(50)
where j , and w are the quantities defines in eqs.(46) and (47) and there is no *-product between them in eq.(50) in virtue of the quadratic nature of the expression. Thus, a simple computation shows that
and we have a remarkable result: the transformation (44),integrated between 0 and 6 maps the standard commutative WZW action into the noncommutative WZW action. That is, we have found a transformation mapping orbits into orbits such that it keeps the form of the action unchanged. This should be contrasted with the 4 dimensional noncommutative YangMills case for which a mapping respecting gauge orbits can be found (the
536
Seiberg-Witten mapping) but the resulting commutative action is not the standard Yang-Mills one. However, one can see that the mapping (44) is in fact a kind of Seiberg-Witten change of variables. Indeed if we consider the WZW action as the effective action of a theory of Dirac fermions coupled t o gauge fields, as we did in previous sections, instead of an independent model we can relate the group valued field g to gauge potentials. As we showed in eq.(38), this relation acquires a very simple form in the light-cone gauge A+ = 0 where
A-
*
= ij(z) a-ij-’(z)
.
(52)
But notice that in this gauge, A- coincides with jzrso we have
6 ~ = - 68 ( a z ~*-A-
+ A- * a Z ~ - )
(53)
which is precisely the Seiberg-Witten mapping in the gauge A+ = 0.
Acknowledgments
I am very grateful to the organizers of the conference “Continuous Advances in QCD-2004” for their invitation to participate in the conference. References 1. A.Connes, M.R. Douglas and A.S. Schwarz, JHEP 02 (1998) 003. 2. M.R. Douglas and C. Hull, JHEP 02 (1998) 008. 3. N. Seiberg and E. Witten, JHEP 09 (1999) 032. 4. E. Moreno and F.A. Schaposnik, JHEP 0003 (2000) 032.
5. 6. 7. 8. 9.
E. Moreno and F.A. Schaposnik, Nucl. Phys. B596 (2001) 439. C.S. Chu, Nucl.Phys. B580 (2000) 352. L. Dabrowski, T. Krajewski and G. Landi, hep-th/00003099. K. F’uruta and T. Inami, hep-th/0004024. C. Nhiez, K. Olsen and R. Schiappa, JHEP 0007 (2000) 030. 10. J.M. Garcia-Bondia and C.P. Martin, Phys.Lett. B479 (2000) 321. Phys. Rev. D21 (1980) 2848. 11. F. Ardalan and N. Sadooghi, hep-th/0002143
NONABELIAN SUPERCONFORMAL VACUA IN n/ = 2 SUPERSYMMETRIC THEORIES
ROBERTO AUZZI Scuola Normale Superiore, Pisa Istituto Nazionale d i Fisica Nucleare - Sezione d i Pisa Piazza dei Cavalieri 8, Pisa, Italy E-mail:
[email protected] The dynamics of some confining vacua which appear as deformed superconformal theories with a nonabelian gauge symmetry is studied by taking concrete examples in N = 2, S U ( 3 ) and USp(4) gauge theories with n f = 4 and equal quark masses. The low-energy degrees of freedom consist of some nonabelian monopoles and dyons with relatively non-local charges. The mechanism of cancellation of the beta function is studied, in analogy to an abelian superconformal vacuum studied by Argyres and Douglas. Study of our SCFT theories as a limit of colliding vacua suggests that confinement in the present theories occurs in an essentially different manner from those vacua with dynamical Abelianization, and involves strongly interacting nonabelian magnetic monopoles.
1. Introduction
The most interesting vacua found in the moduli space of N = 2 theories are the superconformal ones. The low-energy degrees of freedom are relatively nonlocal dyons. Electric particles couple to the vector potential A , magnetic particles couple to the magnetic potential AD and there is no simple local relation between A and A D . The first example of these vacua has been discovered by Argyres and Douglas in N = 2 S U ( 3 ) Super Yang-Mills (see 14); other examples have been discovered in S U ( 2 ) theories with flavor for particular values of quarks bare masses (see "). A classification of these vacua has been developed in 12. In the Argyres-Douglas vacuum the degrees of freedom are relatively non-local abelian dyons. We will study two examples in which relatively non-local nonabelian dyons are involved. Perturbing the theory with a small adjoint mass p, confinement and chiral symmetry breaking ensue, as can be demonstrated indirectly through various considerations, such as the study of large p effective action '. Confinement is described in this way in
537
538 many vacua of n/ = 2 theories with N j quarks hypermultiplets. The analysis of these nonabelian vacua has been developed in 5: these results are reviewed here.
and in
2. A nonabelian vacuum in S U ( 3 ) Nf = 4 theory
We study the r = 2 vacuum of S U ( 3 ) N = 2 gauge theory with 4 quarks hypermultiplets. The Seiberg-Witten curve 1,2 of this theory for equal bare quark masses is (by setting 2 h = 1) 3
y2 =
U(Z- 4i12 - (x + m14 = (x3 - uz - v ) -~ + m14. (Z
(1)
i=l
The T = 2 vacuum correspond to the point, diag 4 = (-m, -m, 2m),i.e.,
u = 3m2; v = 2m3,
(2)
corresponding to an unbroken S U ( 2 ) symmetry. This vacuum splits to six separate vacua when the quark masses are taken to be slightly unequal and generic. The conformally invariant vacua occur at the points where more than one singularity loci in the u - u plane, corresponding to relatively nonlocal massless states, coalesce l 1 > l 2 . The nature of the low-energy degrees of freedom at the SCFT vacua can be determined by studying the monodromy matrices around each of the singularity curves. In the case of the r = 2 vacua of S U ( 3 ) theory with n j = 4, it is necessary to study the behavior of the theory near the point Eq.(2). Set
u=3m2+u,
v=2m3+u,
The discriminant of the curve factorizes
l2
x+m-+x.
(3)
as
A = A, A+ A _ ,
(4)
where the squark singularity” corresponds to the factor
A,
= (mu - u ) ~ ;
(5)
where the fourth zero represents the flavor multiplicity n f = 4;
A+ = 4 m u
+ 36m2u + 108m3u+ 108m4u + u2 + 24mu2+ 36 m2u2+
4~~-4u-36mu-108m~u-108m~u-18uu-27u~,
(6)
aAt large m hence at large U and V it represents massless quarks and squarks; as is well known, at small m it becomes monopole singularity, due to the fact that the corresponding singularity goes under certain cuts produced by other singularities 15.
539
A- = - 4 m u
+ 36m2u - 108 m3u + 108m4u + u2 - 24mu2 + 36m2u2 + + 108m2w - 108m3w + 18uv - 27v2, (7)
4u3 + 4 w - 36mw
represent the loci where some other dyons become massless. The m = 0 case will be studied in some detail in the remaining part of this section. In order t o define uniquely the monodromies around the three curves near the SCFT point we consider the intersections of these curves with the S3 sphere 1612
+ ( G I 2 = 1.
(8)
It is convenient to make first a stereographic projection from S3 --t R3 after which the intersection curves take the form of the three linked rings (see figure 1).
Figure 1. Zero loci of the discriminant of the curve of N = 2, S U ( 3 ) , nf = 4 theory at m = 0.
We consider now various closed paths in the space (u,w), starting from a fixed reference point (for instance lying above the page), encircling various parts of the rings and coming back to the original point. As the branch points move, the integration contour cycles ai’s and pi’s over the SeibergWitten curve get entangled in a non-trivial way. The relation between the monodromy matrix and the corresponding massless charge (with magnetic charge m and electric charge q ) is the following (I6):
540
Indeed, various monodromy transformations are related by the conjugations, for example: = Mg1A5M6, A2 = kfT1Mikf2,
. ..
(10)
as can be easily verified by looking at Fig. 1. By knowing any three of them, for instance MI,M2, M6 above, these relations yield uniquely all the other monodromy matrices. The self-consistency of these relations is a non-trivial check for the numerical calculation of the first three matrices MI,Mz, M6. Using formula (9), it’s possible to calculate the the magnetic and electric charges {ml, m2;91,92): Table 1. Matrix
Charge
Note that only the members of the same doublet are relatively local, i.e., have a vanishing relative Dirac unit
c( 2
NO =
- qAi g B i
gAi 9Bi
).
(11)
i= 1
It’s interesting at this point to compute the low-energy abelian effective coupling matrix r i j (see for the details of the calculation). For m = 0 we have the following Riemann surface: y 2 = (x3 - ux - w)2 - x4 = (x3
+ x2 - ux - w)(x3 - x2 - ux - w).
(12)
At u = 0, w = 0 the branch points are at: 21
= x2 = 2 3 = 2 4 = 0,
25
= -1,
x6 = 1.
(13)
A similar degeneration of genus two Riemann surfaces was studied in the period matrix in a particular base r i j splits as:
17;
541 7-22 is the modulus of the “large” torus (given by the modulus of the genus one surface with two colliding branch points in 0 and other two branch points in 51; in our case 5-22 = 0 because this U(1) factor is infrared free) and 5-11 is the modulus of the “small” torus (given by the modulus of the genus one surface formed by the four colliding branch points; it is well defined only if in our limit the angles formed by the four colliding points are kept constant). It is convenient to introduce the variables ~ , (see p 14, 4 ) , given by:
2 v = E ;
u=Ep.
(15)
In these coordinates 5-11near the conformal point is an holomorphic function which depends only on p and can be written in terms of Jacobi Theta functions (see for the details). The function 5-11(p) has the following singularities:
p=
00
+ v = 0.
(17)
Thus around the points +2, -2,2i, -2i, 00 the function 7-11(p) has nontrivial monodromies, which correspond to the Ui(1) c S U ( 2 ) charges of the previous section.
Figure 2.
The p plane.
A crucial observation is that in the (u, v) space there are infinite number of copies of p plane, corresponding to different phases of E . Namely, by varying the phase of E the linked rings in Fig. 1 are cut in different sections (different copies of p plane). It is therefore natural to identify the set of singularities in each section with the monodromy matrices of Table 1. Physics near the conformal vacuum depends only on p; different sections of the moduli space at different phase of E give rise to the same dynamical
542
picture in different basis. In a given section at constant E the degrees of freedom are the following: a SU(2) x U(1) gauge theory, with 4 magnetic monopole doublets (61, @I) = ( f l , 0), a non-abelian electric doublet with charges (rh1,Ql) = ( 0 , f l ) and a non-abelian dyon doublet with charges (fi1,gl) = ( f 2 , hl) . The four magnetic doublets have U(1) abelian magnetic charge equal to f i z = 1; the other two doublets have abelian magnetic charge equal to f i 2 = 2. Changing the phase of E the p plane cuts the knot in Figure 1 at different points and the charges of our degrees of freedom can be different, but each od these description is equivalent to the one we have given modulo an S p ( 4 , Z)transformation. The SU(2) factor defines an interacting conformal theory. The p function cancellation is reproduced by the following guess, which is a generalization of an idea of Argyres and Douglas 14. The four magnetic monopole doublets cancel the contribution of the dual SU(2) gauge bosons and of their supersymmetric partners as in a local N = 2, SU(2) gauge theory with n f = 4. The non-trivial part of the cancellation occurs between the non-abelian electric and dyonic doublets. By considering the U(1) subgroup of the SU(2), their contribution to the first term of the beta function cancel if 14:
a
With the dyon charges works if
(fi1,Ql) =
(0, f l ) ,( f 2 , f l ) at our disposal, this
(19)
This calculation reproduces the value of ~ 1 1 ( p = 0) (see 4 ) . p = 0 seems to define the right superconformal limit; note that this value of p is the only one which does not break the anomaly free 2 4 symmetry (this discrete symmetry is contained in the anomalous U ( ~ ) R symmetry, which should be indeed somehow restored in all N = 2 scale invariant theories). 3. A nonabelian vacuum in U S p ( 4 ) N f = 4 theory
In a similar analysis is developed for a SU(2) superconformal vacuum in the U S p ( 4 ) theory with Nf = 4 with m = 0. With m # 0 the situation is very similar to the S U ( 3 ) vacuum analised in the previous section: the structure of the monodromies is the same as in Figure 1. In the m -+ 0 limit something new happens.
543
Figure 3. m + 0 limit for the monodromies structure near the superconformal vacuum of USp(4), Nf = 4 theory.
The Seiberg-Witten curve of the U S p ( 4 ) theory quarks, is (setting A = 1) y2
= x(x2 -
(16),
ux - vy - 4x3.
with nf massless (20)
where
u = $I + &,
v = -$&g*
(21)
$1 and $2 are the two components of the adjoint scalar field that breaks the gauge symmetry:
d i w d = ($1,
$2, -41, - 4 2 ) .
(22)
The behavior of the curve is highly singular at the two points U = f2, V = 0, where four of the five branchpoints coalesce. In these vacua we have $1 = 0, and so the gauge symmetry is broken to U S p ( 2 ) x U(1) 2: S U ( 2 ) x U(1). Giving equal masses to all of the quarks (ma = m ) the
544
U = -2 singularity splits into three singularities: two of them have an unbroken U(1)2, while the third has four colliding branchpoints. Considering the intersections of the zero-discriminant part of the moduli space with a little 3-sphere centered in the superconformal vacuum, the two rings shown in the third drawing of figure 3 are obtained. Monodromies around different elements of this knot configuration have been computed in 5 ; the results is that these elements are not in the form (9) and so charges cannot be computed directly. The approach used is to compute charges in the m 4 0 limit shown in figure 3: giving a mass to the quark hypermultiplets, one of the two rings splits in three disconnected pieces; monodromies for m # 0 are in the form (9) and so it is possible to compute the charges involved (see 5). The degrees of freedom are the following: a SU(2) x U(1) gauge theory, with 4 magnetic monopole doublets (rii1,Gl) = ( f l , 0), a non-abelian electric doublet with charges (riil,Q1) = (0, f l ) and two non-abelian dyon doublets with charges (rii1,Qll) = ( f 2 , f l ) . The four magnetic doublets have abelian magnetic charge equal to 1; the electric doublet and one of the dyonic doublet have abelian magnetic charge equal to 2; the other dyonic doublet has abelian magnetic charge equal to 0. The P-function cancellation condition (18) gives the following value of 7:
This result does not agree with the direct calculation done using the Seiberg-Witten curve (see 5 ) . With this approach the following value of 7 is found: 1 (24) 711 = -2' Neglecting the contribution of the electric doublet in (18) the condition becomes 1 7 - _ _ c (25) 2' and it would be satisfied by this vacuum; however, there is no a priori reason to neglect this particle in the calculation. 4. Superconformal Vacuum as Limit of Colliding Vacua
The superconformal limit may be approached breaking the flavor symmetry explicitly by unequal bare quark masses. For example, in the T = 2
545 vacuum of N = 2 S U ( 3 ) Nf = 4 theory with generic and small quark hypermultiplets masses, there are six nearby singularities. In the equal mass limit, these singularities coalesce and become the conformal vacuum. Each of the theories before the equal mass limit is taken is a local U(1)2 gauge theory, with precisely two massless hypermultiplets, each of which carrying only one of the U(1) charges. A partial support comes from the observation that the massless states at one singularity and those at another singularity are relatively non-local to each other (4). Nonetheless, there are reasons to believe that the mechanism of confinement in the superconformal theory cannot be understood this way. Adding an adjoint mass term p T r a 2 into each of these vacua the low energy degrees of freedom (monopoles, dyons) condense. However, in the superconformal limit all the condensates become zero (see the discussion of the SU(3) case in for more details). This is analogous to the phenomenon discussed in l8 at the Argyres-Douglas point of JV= 2, S U ( 2 ) gauge theory with nf = 1.
5 . Conclusions
The charges of the magnetic and dyonic degrees of freedom have been determined in two different superconformal vacua in SU(3) and USp(4) gauge theories; the direct calculation of some monodromies and the monodromomies algebra have been used and the result is self-consistent (this is a non-trivial check). These charges have been interpreted as non abelian degrees of freedom (in our vacua the gauge symmetry is broken to S U ( 2 ) x U(1)). The effective gauge coupling found with the Seiberg-Witten curve and the one found from the condition of the @-function cancellation at one loop have been compared. In the SU(3) case a perfect agreement is found, as in the abelian vacua studied by Argyres and Douglas. In the USp(4) case we have not found this agreement; the p function cancellation condition reproduces the value found with the Seiberg-Witten only if we neglect one of the dyons in the calculation. If an adjoint mass term in the superpotential AW = p T r Q 2 is added, confinement and chiral symmetry breaking occur. This can be seen in the limit p >> AN=^ ’. It’s difficult to study confinement and chiral symmetry breaking in the limit p
