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T) and (v\,U2,vz) a r e related via a unitary mixing matrix. For simplicity consider a two-neutrino scenario in which we ignore (say) the electron neutrino. Then the mass and flavour eigenstates are related via a simpler (unitary) matrix :
103
104 cosO sinO —sinO cosO
Z) •
(I
»
If one produces a beam of one flavour (say v^) then after a distance L the probability that it has become a vT is : P{yT,L)
= sin2 (26) x sin2 (nL/Losc),
(2)
where the characteristic oscillation length is given by : Losc = (47r^)/Am 2 ,
(3)
and Am2 = m\—m\.
(4)
This equation is in natural units (h = c = 1 ) . If we measure L in km, E in GeV and Am 2 in eV 2 , then : Losc » ( 2 . 5 ^ ) / A r a 2 .
(5)
By conservation of particles, the probability that the v^ remains a v^ is : P(I/„,L)=1.0-P(I/T,I).
(6)
3. The N O M A D and CHORUS Experiments at CERN These experiments were situated in the West Area Neutrino Facility (WANF) at CERN. Neutrinos were produced by allowing a beam of 450 GeV/c protons to strike a Berylium target. The resulting spectrum of neutrinos at NOMAD is shown in Fig. 1. The spectrum at CHORUS is slightly harder as the crosssectional area of that experiment is smaller than NOMAD and neutrinos near the beam axis have slightly higher energies. The methods used by NOMAD and CHORUS are completely complementary, and they are described below. Both seek to observe the appearance of vT via the reaction : vT + N->T~ 3.1. The NOMAD
+ X.
(7)
Experiment
A full description of the NOMAD detector can be found in Altegoer et al.1. NOMAD does not have the spatial resolution to see the track of a T~ which typically travels less that 1 mm. Instead it relies on Kinematic Cuts, Topological Constraints and Likelihood Methods. And it uses the novel (at
105
Neutrino Flux
80
100
120
Neutrino Energy (GeV)
Figure 1.
The energy spectra of the various neutrino flavour states at NOMAD.
least to HEP) techniques of Blind Analyses and Frequentist Statistics. A full description and final results can be found in Astier et al.2. In summary NOMAD sees no evidence for oscillations. The number and shape of the observed events are consistent with background. The number of candidates is 52 events, to be compared to a total predicted background of 51.1 ± 5.4 events. The signal and background are compatible in each decay channel. In the most sensitive region (i.e. that with the smallest background) NOMAD observes 1 event, as compared to a calculated background of 1.62 (+1.89-0.38).
3.2. The CHORUS
Experiment
CHORUS had a target of photographic emulsion with a total mass of 770 kg. The spatial resolution is very good and would allow the tracks produced by
106
a T~ to be seen. Potential r vertices are located via a Fiber target tracker, with an angle resolution of about 2 mrad and a position resolution of 150 mm. A calorimeter and spectrometer are situated downstream of the target and tracker. More details can be found in Eskut et al.3 CHORUS has completed what they term their "Phase I" analysis, and no candidates were observed with negligible backgrounds, see Eskut et al.4 A complete rescanning of the emulsion (Phase II) is underway. This will allow a signicant improvement of their current published limits. 3.3. The CERN
Results
The final NOMAD limit and the Phase I CHORUS limits are summarised in the plot below in Fig. 2.
T
1—i
i ill"
^Tio> u
1 0 '2
_ NOMA©
10
1
T
V —» V
H x 90% C.L. CDIIS .1
10
10
j
i
i i i i i il
10
-J
• • i t •i
10
-1
sin2 28 Figure 2.
The regions of parameter space excluded by the two CERN experiments.
As can be seen from the exclusion plot, at high values of Am 2 the NOMAD
107
experiment produced a limit about an order of magnitude better than previous results. CHORUS (Phase I) is not quite as good but in Phase II it is expected to better the NOMAD limit. It is also clear that the CERN experiments are only sensitive for values of Am 2 > 1 eV 2 . For smaller mass differences significant oscillations would not have had time to develop.
4. Experiments with Stopped Pion Beams Two experiments have searched for oscillations using neutrinos produced by stopping a beam of n mesons in a massive target. The results are controversial. The LSND experiment 5 at Los Alamos in the U.S.A. claims to have seen a signal. The KARMEN experiment 6 at the Rutherford Appleton Laboratory in the U.K. does not see any sign of oscillations A careful comparison (see Fig. 3) shows that the two groups are not quite in conflict. There is a very small region of parameter space where a signal from LSND cannot be excluded by KARMEN.
Figure 3.
Results from the KARMEN and LSND experiments.
However the jury is still out. A new experiment MiniBoone, which will use the 8 GeV booster at Fermilab in the U.S.A., should resolve this issue soon. It has a custom-designed beam stopper, as does KARMEN, whereas the beam stopper at Los Alamos was not optimised for neutrino oscillation searches.
108
5. Long Baseline Experiments The low values of Am 2 suggested by positive results for oscillations using neutrinos produced in the Sun and in the atmosphere have led to a number of proposals to send accelerator-produced neutrinos to distant detectors. Two of these, from Fermilab to the Soudan mine and from CERN to Gran Sasso in Italy, have yet to collect data. Coincidently both have baselines of about 750 km. One experiment in Japan is already producing results, and will be discussed here. 5.1.
K2K
The K2K experiment 7 sends neutrinos produced in a 12 GeV proton accelerator at the KEK laboratory to the giant SuperKamiokande (SuperK) detector in western Japan. The baseline is about 250 km. Unfortunately K2K has ceased operations due to an accident at SuperK in which many photomultipliers were destroyed. The prior results were tantalising. The mean energy of the neutrinos (mostly u^) produced at KEK means that, should they oscillate into i/T, then those vT would be below threshold for producing charged tauons via a charged current reaction. K2K is thus essentially a disappearance experiment in which the "signal" would be too few neutrinos arriving at SuperK. The preferred values of the oscillation parameters to explain the SuperK atmospheric neutrino data are Am 2 = 0.3 x 1 0 - 3 eV2 with maximal mixing. If those values are assumed then the K2K experiment expects to observe a deficit of events at a neutrino energy of about 600 MeV. Preliminary data, see Fig. 4, show just such a deficit. As noted K2K has been halted by the accident at SuperK. Reconstruction of that detector is proceeding at pace, and K2K hopes to resume operations by the end of 2002. 6. Reactor Experiments Nuclear reactors produce copious numbers of neutrinos, in particular the anti-iv In the absence of oscillations the flux should fall off like an inverse square law. A departure from such behaviour would indicate oscillations into another type of neutrino which, at these low energies, would not interact. The two most powerful reactors being used are at CHOOZ in France and at Palo Verde in the U.S.A. Figure 5 shows the CHOOZ detector. Neither experiment sees a departure from the fluxes predicted in a no-oscillation scenario.
109
m
15
Note: Am2=3xMF aV2 corresp onds to 600 MeV Ev
1
10
4 Figure 4. The spectrum of i/ M Charged Current neutrino interactions observed in the SuperKamiokande detector using a ^ b e a m from KEK. See the text for a discussion.
Results from CHOOZ 8 are shown below in Fig. 6. The Palo Verde experiment has very similar limits. 7. Conclusions Positive results (discussed by other speakers at this workshop)from experiments studying neutrinos produced in the Sun and in the Earth's atmosphere do disappear before arriving at a detector. By far the most likely scenario is that they have oscillated into another neutrino species which cannot be detected. These experiments indicate that the differences in the neutrino masses (or more precisely the quantities Am 2 ) are small. Short baseline experiments at high energy accelerators have all yielded negative results, which is now thought to be because they cannot explore the region of small Am 2 . Planned long baseline experiments at CERN and Fermilab should (just) be able to explore that region.
110
CZZZZZTZ. .'*>'
1
.. j 2
~ n x r z r r r i . . ._..; „u „;.:;:•••. :i 3
4
5
6fn
Figure 5. The CHOOZ detector in France
The one positive hint of an oscillation signal has come from a low-energy accelerator produced beam in a medium baseline experiment from KEK to SuperK. The resumption of that experiment is eagerly awaited. Finally it should be noted that n o experiment has seen the appearance of a new species of neutrino in a beam that was originally of a different species. When, or if, observed that will be the decisive evidence that oscillations do occur.
Acknowledgements I would like to thank Dr. Jaap Panman, the spokesman of the CHORUS experiment for providing me with the powerpoint version of a recent CHORUS presentation.
111
— analysis C YA 90% CL Kamiokande (multi-GeV) H 90% CL Kamiokande (sub+multi-GeV) I . . . .
0
i
0.1
i . . , . i .. • .
0.2
Figure 6.
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sin2(28)
Oscillation limits from CHOOZ
References 1. 2. 3. 4. 5. 6. 7.
J. Altegoer et al., NIM A 4 0 4 , 96 (1998). P. Astier et al., Nucl. Phys. B 6 1 1 , 3 (2001). E . E s k u t et a.l, NIM A 4 0 1 , 7 (1997). E . E s k u t et al., Phys. Lett. B 4 9 7 , 8 (2001). C . A t h a n a s s o p o l u s et al., Phys. Rev. C 5 4 , 2658 (1996). B . A r m b r u s t e r et al., Phys. Rev. C 5 7 , 3414 (1998). Y. O y a m a , K E K P r e p r i n t 2001-7, copy of a talk a t t h e "Cairo Int. Conference", J a n u a r y 2001. 8. M. Apollonio et al., Phys. Lett. B 4 6 6 , 415 (1999).
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4. Hadron Structure and Properties
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LATTICE QCD A N D H A D R O N S T R U C T U R E
A. W. T H O M A S Special Research Centre for the Subatomic Structure of Matter and Department of Physics and Mathematical Physics, Adelaide, SA 5005, Australia E-mail: [email protected]
One of the great challenges of lattice QCD is to produce unambiguous predictions for the properties of physical hadrons. We review recent progress with respect to a major barrier to achieving this goal, namely the fact that computation time currently limits us to large quark mass. Using insights from the study of the lattice data itself and the general constraints of chiral symmetry we demonstrate that it is possible to extrapolate reliably from the mass region where calculations can be performed to the chiral limit.
1. Introduction At the present time we have a wonderful conjunction of opportunities. Modern accelerator facilities such as Jlab, Mainz, DESY and CERN are providing data of unprecedented precision over a tremendous kinematic range at the same time as numerical simulations of lattice QCD are delivering results of impressive accuracy. It is therefore timely to ask how to use these advances to develop a new and deeper understanding of hadron structure and dynamics 1 . Let us begin with lattice QCD. Involving as it does a finite grid of spacetime points, lattice QCD requires numerous extrapolations before one can compare with any measured hadron property. The continuum limit, a —• 0 (with a the lattice spacing), is typically under good control 2 . With improved quark and gluon actions the 0(a) errors can be eliminated so that the finite-a errors are quite small, even at a modest lattice spacing 3 - say 0.1 fm. In contrast, the infinite volume limit is much more difficult to implement as the volume, and hence the calculation time, scales like TV4. Furthermore, this limit is inextricably linked to the third critical extrapolation, namely the continuation to small quark masses (the "chiral extrapolation"). The reason is, of course, that chiral symmetry is spontaneously broken in QCD, with the pion being a massless Goldstone boson in the chiral limit. As the lattice volume must contain the pion cloud of whatever hadron is under study one expects that the box size, L, should be at least 4m" 1 . At the physical pion mass this is a box
115
116
5.6 fm on a side, or a 56 4 lattice, with a — 0.1 fm. This is roughly twice as big as the lattices currently in use. Since the time for calculations with dynamical fermions (i.e. including quark-antiquark creation and annihilation in the vacuum) scale4 as m~36, current calculations have been limited to light quark masses 6-10 times larger than the physical ones. With the next generation of supercomputers planned to be around 10 Teraflops, it should be possible to get as low as 4 times the physical quark mass, but to actually reach the physical mass on an acceptable volume will need at least 500 Teraflops. This is 10-20 years away. Since a major motivation for lattice QCD must be to unambiguously compare the calculations of hadron properties with experiment, this is somewhat disappointing. The only remedy for the next decade at least is to find a way to extrapolate masses, form-factors, and so on, calculated at a range of masses considerably larger than the physical ones, to the chiral limit. In an effort to avoid theoretical bias this has usually been done through low-order polynomial fits as a function of quark mass. Unfortunately, this is incorrect and can yield quite misleading results because of the Goldstone nature of the pion. Once chiral symmetry is spontaneously broken, as we have known for decades that it must be in QCD and as it has been confirmed in lattice calculations, all hadron properties receive contributions involving Goldstone boson loops. These loops inevitably lead to results that depend on either logarithms or odd powers of the pion mass. The Gell-Mann-Oakes-Renner relation, however, implies that mn is proportional to the square root of mq, so logarithms and odd powers of m , are non-analytic 5 in the quark mass, with a branch point at mq = 0. One simply cannot make a power series expansion about a branch point. On totally general grounds, one is therefore compelled to incorporate the non-analyticity into any extrapolation procedure. The classical approach to this problem is chiral perturbation theory, an effective field theory built upon the symmetries of QCD 6 . There is considerable evidence that the scale naturally associated with chiral symmetry breaking in QCD, A X SB, is of order 4irfn, or about 1 GeV. Chiral perturbation theory then leads to an expansion in powers of m ff /A x sB and P / A X S B , with p a typical momentum scale for the process under consideration. At 0(p4), the corresponding effective Lagrangian has only a small number of unknown coefficients which can be determined from experiment. On the other hand, at 0(p6) there are more than 100 unknown parameters, far too many to determine phenomenologically. Another complication, not often discussed, is that there is yet another mass scale entering the study of nucleon (and other baryon) structure 7,8 . This scale is the inverse of the size of the nucleon, A ~ R"1. Since A is naturally more like
117 a few hundred MeV, rather t h a n a 1 GeV, the n a t u r a l expansion parameter, mn/A, is of order unity for mn ~ 2 - 3mP h y s - the lowest mass scale at which lattice d a t a exists. This is much larger than mK/AxsB ~ 0.3 — 0.4, which might have given one some hope for convergence. As it is, the large values of m,r / A at which lattice d a t a exist make any chance of a reliable expansion in chiral perturbation theory (xPT) fairly minimal 9 . Even though one has reason to doubt the practical utility of xPT, the lattice d a t a itself does give us some valuable hints as to how the dilemma might be resolved. T h e key is to realize t h a t , even though the masses may be large, one is actually studying the properties of Q C D , not a model. In particular, one can use the behaviour of hadron properties as a function of mass to obtain valuable new insights into hadron structure. T h e first thing t h a t stands out, once one views the d a t a as a whole, is just how smoothly every hadron property behaves in the region of large quark mass. In fact, baryon masses behave like a + bmq, magnetic m o m e n t s like (c + t / m q ) - 1 , charge radii squared like (e + / m q ) _ 1 and so on. T h u s , if one defined a light "constituent quark mass" as M = Mo + cmq (with c ~ 1), one would find baryon masses proportional to M (times the number of u and d quarks), magnetic moments proportional to M _ 1 and so on - just as in the constituent quark picture. There is simply no evidence at all for the rapid, nonlinearity associated with the branch cuts created by Goldstone boson loops. Indeed, there is not even any evidence for a statistically significant difference between properties calculated in quenched versus full Q C D ! How can this be? T h e n a t u r a l answer is readily found in the additional scale, A ~ Br1, mentioned earlier. In Q C D (and quenched Q C D ) , Goldstone bosons are emitted and absorbed by large, composite objects built of quarks and gluons. Whenever a composite object emits or absorbs a probe with finite m o m e n t u m one m u s t have a form-factor which will suppress such processes for m o m e n t a greater t h a n A ~ Br1. Indeed, for m* > A we expect Goldstone boson loops to be suppressed as powers of A / m r , not mn/A (or mn/Axss). Of course, this does not mean t h a t one cannot in principle carry through the program of xPT. However, it does mean t h a t there may be considerable correlations between higher order coefficients and t h a t it may be much more efficient to adopt an approach which exploits the physical insight we j u s t explained. Over the past three years or so, our group in Adelaide has worked with a number of colleagues around the world to do just this. T h a t is, we have developed an efficient technique, using very few free parameters, to extrapolate every h a d r o n property which can be calculated on the lattice from the large mass region to the physical quark mass - while preserving the most i m p o r t a n t non-analytic behaviour of each of those observables. This task is not trivial,
118
in that various observables need different phenomenological treatments. On the other hand, there is a unifying theme. That is, pion loops are rapidly suppressed for pion masses larger than A (mff > 0.4 — 0.5 GeV). In this region the constituent quark model seems to represent the lattice data extremely well. However, for mw below 400-500 MeV the Goldstone loops lead to rapid, nonanalytic variation with mq and it is crucial to preserve the correct leading non-analytic (LNA) and sometimes the next-to-leading non-analytic (NLNA) behaviour of xPT. In order to guide the construction of an effective, phenomenological extrapolation formula for each hadron property, we have found it extremely valuable to study the behaviour in a particular chiral quark model - the cloudy bag model (CBM) 8 ' 10,11 . Built in the early 80's it combined a simple model for quark confinement (the MIT bag) with a perturbative treatment of the pion cloud necessary to ensure chiral symmetry. The consistency of the perturbative treatment was, not surprisingly in view of our earlier discussion, a consequence of the suppression of high momenta by the finite size of the pion source (in this case the bag). Certainly the MIT bag, with its sharp, static surface, has its quantitative defects. Yet the model can be solved in closed form and all hadron properties studied carefully over the full range of masses needed in lattice QCD. Provided one works to the appropriate order the model preserves the exact LNA and NLNA behaviour of QCD in the low mass region while naturally suppressing the Goldstone boson loops for m^ > A ~ R~1. Finally, it actually yields quite a good description of lattice data in the large mass region for all observables in terms of just a couple of parameters. With this lengthy explanation of the physics which underlies the superficially different extrapolation procedures, we now summarise the results for some phenomenologically significant baryon properties. 2. Electromagnetic Properties of Hadrons While there is only limited (and indeed quite old) lattice data for hadron charge radii, recent experimental progress in the determination of hyperon charge radii has led us to examine the extrapolation procedure for extracting charge radii from the lattice simulations. Figure 1 shows the extrapolation of the lattice data for the charge radius of the proton, including the In mff (LNA) term in a generalised Pade approximant 12 ' 13 : {r}
*~
l + c2ml
•
{i}
Here ci and c2 are parameters determined by fitting the lattice data in the large mass region (m 2 > 0.4 GeV 2 ), while \i, the scale at which the effects of
119 pion loops are suppressed, is not yet determined by the data but is simply set to 0.5 GeV. The coefficient \ is model independent and determined by chiral perturbation theory. Clearly the agreement with experiment is much better if, as shown, the logarithm required by chiral symmetry is correctly included rather than simply making a linear extrapolation in the quark mass (or ml). Full details of the results for all the octet baryons may be found in Ref. [12]. The situation for baryon magnetic moments is also very interesting. The LNA contribution in this case arises from the diagram where the photon couples to the pion loop. As this involves two pion propagators the expansion of the proton and neutron moments is: MP(")
= A (J ( n ) T am, + O K )
(2)
Here //Q is the value in the chiral limit and the linear term in mn is proportional to m | , a branch point at m g = 0. The coefficient of the LNA term is a = 4.4//jvGeV _1 . At the physical pion mass this LNA contribution is 0.6pjv, which is almost a third of the neutron magnetic moment. 1.2 1.0 0.8
*A
v C o
"o u
0.6
-
0.4
•--+-
—•
- - 1 - * - . -
0.2 0.0 }r
A J_
-0.2 0.0
0.2
0.4 m
2
0.6 (GeV 2 )
0.8
1.0
Figure 1. Fits to lattice results for the squared electric charge radius of the proton - from Ref. [12]. Fits to the contributions from individual quark flavours are also shown (the u-quark results are indicated by open triangles and the d-quark results by open squares). Physical values predicted by the fits are indicated at the physical pion mass. The experimental value is denoted by an asterisk.
Just as for the charge radii, the chiral behaviour of fj.p^ is vital for a correct extrapolation of lattice data. As shown in Fig. 2, one can obtain a very
120
satisfactory fit to some rather old data, which happens to be the best available, using the simple Pade approximant 14 : pin)
^
=
li-jfom^+^m?
(3)
Existing lattice data can only determine two parameters and Eq. (3) has just two free parameters while guaranteeing the correct LNA behaviour as m^ —¥ 0 and the correct behaviour of HQET at large ml. The extrapolated values of fj,p and n" at the physical pion mass, 2.85 ± 0.22/JJV and —1.90 ± 0.15pjv are currently the best estimates from non-perturbative QCD 14 . (Similar results, including NLNA terms in chiral perturbation theory, have been reported recently by Hemmert and Weise15.) For the application of similar ideas to other members of the nucleon octet we refer to Ref. [16], while for the strangeness magnetic moment of the nucleon we refer to Ref. [17]. The last example is another case where tremendous improvements in the experimental capabilities, specifically the accurate measurement of parity violation in ep scattering 18 , is giving us vital information on hadron structure. Hs-
3 +->
^ome
c
-s o -1-5
CI)
aM
3.5 3.0 2 5 2.0 1.5 1.0 0.5 ().()
cO
-0 5 -1 0 C o -1.5 CD U -2 0
S
3 55
-2.5
Figure 2. Extrapolation of lattice QCD magnetic moments for the proton (upper) and neutron (lower) to the chiral limit. The curves illustrate a two parameter fit to the simulation data, using a Pade approximant, in which the one-loop corrected chiral coefficient of mn is taken from xPT. T h e experimentally measured moments are indicated by asterisks. The figure is taken from Ref. [14].
121 In concluding this section, we note t h a t the observation t h a t chiral corrections are totally suppressed for m^ above about 0.5 GeV and t h a t the lattice d a t a looks very like a constituent quark picture there suggests a novel approach to modelling hadron structure. It seems t h a t one might avoid m a n y of the complications of the chiral quark models, as well as m a n y of the obvious failures of constituent quark models by building a new constituent quark model with u and d masses in the region of the strange quark - where SU(3) s y m m e t r y should be exact. Comparison with d a t a could then be m a d e after the same sort of chiral extrapolation procedure t h a t has been applied to the lattice d a t a . Initial results obtained by Cloet et al. for the octet baryon magnetic m o m e n t s using this approach are very promising indeed 1 9 . We note also the extension to A-baryons, including the NLNA behaviour, reported for the first time in these proceedings 2 0 .
3. M o m e n t s of Structure Functions T h e m o m e n t s of the parton distributions measured in lepton-nucleon deep inelastic scattering are related, through the operator product expansion, to the forward nucleon m a t r i x elements of certain local twist-2 operators which can be accessed in lattice simulations 1 . T h e more recent d a t a , used in the present analysis, are taken from the Q C D S F 2 1 and M I T 2 2 groups and shown in Fig. 3 for the n = 1, 2 and 3 moments of the u — d difference at N L O in the MS scheme. To compare the lattice results with the experimentally measured m o m e n t s , one must extrapolate in quark mass from about 50 MeV to the physical value. Naively this is done by assuming t h a t the moments depend linearly on the quark mass. However, as shown in Fig. 3 (long dashed lines), a linear extrapolation of the world lattice d a t a for the u — d m o m e n t s typically overestimates the experimental values by 50%. This suggests t h a t important physics is still being omitted from the lattice calculations and their extrapolations. Here, as for all other hadron properties, a linear extrapolation in m ~ m 2 must fail as it omits crucial nonanalytic structure associated with chiral symmetry breaking. T h e leading nonanalytic (LNA) t e r m for the u and d distributions in the physical nucleon arises from the single pion loop dressing of the bare nucleon and has been shown 2 3 , 2 4 to behave as m 2 log mn. Experience with t h e chiral behaviour of masses and magnetic m o m e n t s shows t h a t the LNA t e r m s alone are not sufficient to describe lattice d a t a for m^ > 200 MeV. T h u s , in order to fit the lattice d a t a at larger m^, while preserving the correct chiral behaviour of m o m e n t s as mn —>• 0, a low order, analytic expansion in m 2 is also included in the extrapolation and the m o m e n t s of u — d are fitted
122 1
0.4
1
1
QCDSF Gockeler e t al.. 1996 QCDSF Gockeler et al., 1997 QCDSF Best et al., 1997 MIT-DD60Q (Quenched) MIT-SESAM (Full)
Experiment
Meson Cloud Model
0.3
0.1 •+-
h
-l
h
-i
h
0.12-
0.04 H
1-
•H
h
0.06-
-jbn-iL0.02-
o.o
0.2
0.4
0.6
0.8
[GeV2]
1.0
Figure 3. Moments of the u — d quark distribution from various lattice simulations. The straight (long-dashed) lines are linear fits to this data, while the curves have the correct LNA behaviour in the chiral limit — see the text for details. The small squares are the results of the meson cloud model and the dashed curve through them best fits using Eq. (4). The stars represent the phenomenological values taken from NLO fits in the MS scheme. The figure is taken from Ref. [25].
with the form 25 : (xn)u-d
= an + K ml + an c L N A ml In
ml
ml + (i
• 7T+ 7' p. The Particle Data Group 1 provides the range 3.7-7.5 /z;v for the A + + magnetic moment with the most recent experimental result 2 of 4.52 ± 0.50 ± 0.45 p.^. In principle, the A + magnetic moment can be obtained from the reaction y p —^ 7r° 7' p, as demonstrated at the Mainz microtron 3 . An experimental value for the A + magnetic moment appears imminent. Recent extrapolations of octet baryon magnetic moments 4,5,6 have utilized an analytic continuation of the leading nonanalytic (LNA) structure of Chiral Perturbation Theory (xPT), as the extrapolation function. The unique feature of this extrapolation function is that it contains the correct chiral behaviour
125
126
as mq —y 0 while also possessing the Dirac moment mass dependence in the heavy quark mass regime. The extrapolation function utilized here has these same features, however we move beyond the previous approach by incorporating not only the LNA but also the next to leading nonanalytic (NLNA) structure of x P T in the extrapolation function. Incorporating the NLNA terms contributes little to the octet baryon magnetic moments, however it proves vital for decuplet baryons. The NLNA terms contain information regarding the branch point at m^ = M A — MN, associated with the A —> Nn decay channel and play a significant role in decuplet-baryon magnetic moments. 2. Leading and Next-to-Leading Nonanalytic Behaviour We begin with the chiral expansion for decuplet baryon magnetic moments 7 . The LNA and NLNA behaviour is given by • rn.Tr.
127
Hence the LNA behaviour of decuplet magnetic moments is given by XTT mn + XK mK + X'„ ^(-^N,mn,n7r)
+ X'K
F{-6N,mK,nK),
(4)
where Xvr
=
MNH2
MNC2
XK
* CM 2
108 2
MNn
*(fi Nir. The physics behind the cusp is intuitively revealed by the relation between the derivative of the magnetic moment with respect to ml and the derivative with respect to the momentum transfer q2, provided by the pion propagator l/(q2 + ml) in the heavy baryon limit. Derivatives with respect to q2 are proportional to the magnetic charge radius in the limit q2 -> 0, (r2M) = -^GM(q2)\q,=0.
(18)
If we consider for example A++ ->• pn+ with | j , rrij) = |3/2, 3/2), the lowestlying state conserving parity and angular momentum will have a relative Pwave orbital angular momentum with \l,mj) — |1,1). Thus the positivelycharged pion makes a positive contribution to the magnetic moment. As the opening of the p n+ decay channel is approached from the heavy quark-mass regime, the range of the pion cloud increases in accord with the Heisenberg uncertainty principle, AE At ~ h. Just above threshold the pion cloud extends
132 6
M
i
—i
1
1
1
1
0.3
0.4
0.5
r
-
0.0
0.1
0.2
mj1 (GeV2)
0.6
0.7
Figure 3. The extrapolation function fit for A++ and A+ magnetic moments. The magnetic moments given by the CQM either side of the SU(3)-flavour limit are indicated by dots (•) and the theoretical prediction for each baryon is indicated at the physical pion mass by a star (*). The only available experimental d a t a is for the A + + and is indicated by an asterisk (*). The proton extrapolation 4 (dashed line) is included to illustrate the effect of the open decay channel, A —¥ N TT, in the A"*" extrapolation. The presence of this decay channel gives rise to a A+ moment smaller than the proton moment.
towards infinity as AE -» 0 and the magnetic moment charge radius diverges. Similarly, (d/drn^)GM —*• —oo. Below threshold, GM becomes complex and the magnetic moment of the A is identified with the real part. The imaginary part describes the physics associated with photon-pion coupling in which the pion is subsequently observed as a decay product. It is the NLNA terms of the chiral expansion for decuplet baryons that contain the information regarding the decuplet to octet transitions. These transitions are energetically favourable making them of paramount importance in determining the physical properties of A baryons. The NLNA terms serve to enhance the magnitude of the magnetic moment above the opening of the decay channel. However, as the decay channel opens and an imaginary part develops, the magnitude of the real part of the magnetic moment is suppressed. The strength of the LNA terms, which enhance the magnetic moment magnitude as the chiral limit is approach, overwhelms the NLNA contributions such that the magnitude of the moments continues to rise towards the chiral limit. The inclusion of the NLNA structure into octet baryon magnetic moment
133
-1.0 -1.5
^r -2.o c
6 o-2.5 -3.0 -3.5 0.0
0.1
0.2
0.3 0.4 0.5 m 2 (GeV2)
0.6
0.7
Figure 4. The extrapolation function fit for the A - magnetic moment. The magnetic moments given by the CQM either side of the SU(3)-flavour limit are indicated by dots (•) and the theoretical prediction is indicated at the physical pion mass by a star (*). There is currently no experimental value for the A - magnetic moment.
extrapolations is less important for two reasons. The curvature associated with the NLNA terms is negligible for the N and E baryons and small for the A and S baryons. More importantly one can infer the effects of the higher order terms of xPT, usually dropped in truncating the chiral expansion, through the consideration of phenomenological models. If one incorporates form factors at the meson-baryon vertices, reflecting the finite size of the meson source, one finds that transitions from ground state octet baryons to excited state baryons are suppressed relative to that of xPT to finite order, where point-like couplings are taken. In xPT it is argued that the suppression of excited state transitions comes about through higher order terms in the chiral expansion. As such, the inclusion of NLNA terms alone will result in an overestimate of the transition contributions, unless one works very near the chiral limit where higher order terms are indeed small. For this reason octet to decuplet or higher excited state transitions have been omitted in previous studies 4 ' 5 ' 6 . In the simplest CQM with mu = md, the A + and proton moments are degenerate. However, spin-dependent interactions between constituent quarks will enhance the A + relative to the proton at large quark masses, and this is supported by lattice QCD simulation results 9 . As a result, early lattice QCD predictions based on linear extrapolations 9 report the A + moment to be
134 greater t h a n the proton moment. However with the extrapolations presented here which preserve the LNA behaviour of x P T , the opposite conclusion is reached. We predict the A + and proton magnetic m o m e n t s of 2.58 //JV a n d 2.77 /ijy respectively. The proton magnetic moment extrapolation 4 is included in Fig. 3 as an illustration of the importance of incorporating the correct nonanalytic behaviour predicted by %PT in any extrapolation to the physical world. An experimentally measured value for the A + magnetic m o m e n t would offer i m p o r t a n t insights into the role of spin-dependent forces and chiral nonanalytic behaviour in the quark structure of baryon resonances.
5.
Conclusion
An extrapolation function for the decuplet baryon magnetic m o m e n t s has been presented. This function preserves the leading and next-to-leading nonanalytic behaviour of chiral perturbation theory while incorporating the Dirac-moment dependence for moderately heavy quarks. Interesting nonanalytic behaviour of the magnetic m o m e n t s associated with the opening of the n N decay channel has been highlighted. It will be interesting to apply these techniques to existing and forthcoming lattice Q C D results, and research in this direction is currently in progress. An experimental value exists only for the A + + magnetic m o m e n t where P A + + = 4.52 ± 0.50 ± 0.45 (IN- This value is in good agreement with the prediction of 5.39 HN given by our AccessQM as described above. Arrival of experimental values for the A + and A - magnetic m o m e n t s are eagerly anticipated and should be forthcoming in the next few years. More importantly, these techniques may be applied to the decuplet hyperon resonances where the role of the kaon cloud becomes i m p o r t a n t . We look forward to new J H F results in this area in the future.
Acknowledgement This work was supported by the Australian Research Council.
References 1. Particle Data Group, Eur. Phys. J., C15, (2000). 2. A. Bosshard et al., Phys. Rev. D44, 1962 (1991). 3. M. Kotulla [TAPS and A2 Collaborations], Prepared for Hirschegg '01: Structure of Hadrons: 29th International Workshop on Gross Properties of Nuclei and Nuclear Excitations, Hirschegg, Austria, 14-20 Jan 2001. 4. I. C. Cloet, D. B. Leinweber and A. W. Thomas, Phys. Rev. C65, 062201 (2002). 5. D. B. Leinweber, D. H. Lu and A. W. Thomas, Phys. Rev. D60, 034014 (1999).
135 6. E. J. Hackett-Jones, D. B. Leinweber and A. W. Thomas, Phys. Lett. B489, 143 (2000). 7. M. K. Banerjee and J. Milana, Phys. Lett. D54, 5804 (1996). 8. H. Leutwyler, Phys. Lett. B378, 313 (1996). 9. D.B. Leinweber, R.M. Woloshyn, T. Draper, Phys. Rev. D46, 3067 (1992).
LATTICE QCD, G A U G E FIXING A N D THE T R A N S I T I O N TO THE P E R T U R B A T I V E REGIME
A. G. W I L L I A M S A N D M. S T A N F O R D Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, SA 5005, Australia E-mail: Anthony. [email protected]; [email protected]
The standard definition of perturbative QCD uses the Faddeev-Popov gauge-fixing procedure, which leads to ghosts and the local BRST invariance of the gauge-fixed perturbative QCD action. In the nonperturbative regime, there appears to be a choice of using nonlocal Gribov-copy free gauges (e.g. Laplacian gauge) or of attempting to maintain local BRST invariance at the expense of admitting Gribov copies and somehow summing or averaging over them. It should be recognized that the standard implementation of lattice QCD corresponds to the former choice even when only physical (i.e. colour singlet) observables are being calculated. These issues are introduced and briefly explained.
1. Introduction Perturbative quantum chromodynamics (QCD) is formulated using the Faddeev-Popov gauge-fixing procedure, which introduces ghost fields and leads to the local BRST invariance of the gauge-fixed perturbative QCD action. These perturbative gauge fixing schemes include, e.g. the standard choices of covariant, Coulomb and axial gauge fixing. These are entirely adequate for the purpose of studying perturbative QCD, however, they fail in the nonperturbative regime due to the presence of Gribov copies, i.e. gauge-equivalent gaugefield configurations survive in the gauge functional integration after gauge fixing. One could define nonperturbative QCD by imposing a non-local Gribovcopy free gauge fixing (such as Laplacian gauge) or, alternatively, one could attempt to maintain local BRST invariance at the cost of admitting Gribov copies. We will see that by definition the standard ensemble-averaging technique of lattice QCD corresponds to the former definition. We begin by reviewing the standard arguments for constructing QCD perturbation theory, which use the Faddeev-Popov gauge fixing procedure to construct the perturbative QCD gauge-fixed Lagrangian density. The naive La-
136
137
grangian density of QCD is £QCD
= —F^F^
+ -£qj(iP-
msMh
(1)
where the index / corresponds to the quark flavours. The naive Lagrangian is neither gauge-fixed nor renormalized, however it is invariant under local SU(3)C gauge transformations g(x). For arbitrary small uia{x) we have
g(x) = expi-igs
(j)
ua(x)\
£ SU(Z),
(2)
where the \a/2 = ta are the generators of the gauge group SU(3) and the index a runs over the eight generator labels a = 1, 2,..., 8. Consider some gauge-invariant Green's function (for the time being we shall concern ourselves only with gluons)
wnmm = sv;™£*.
...
where 0[A] is some gauge-independent quantity depending on the gauge field, A^(x). We see that the gauge-independence of 0[A] and S[A] gives rise to an infinite quantity in both the numerator and denominator, which must be eliminated by gauge-fixing. The Minkowski-space Green's functions are defined as the Wick-rotated versions of the Euclidean ones. The gauge orbit for some configuration A^ is defined to be the set of all gauge-equivalent configurations. Each point A9 on the gauge orbit is obtained by acting upon A^ with the gauge transformation g. By definition the action, S[A], is gauge invariant and so all configurations on the gauge orbit have the same action, e.g. see the illustration in Fig. 1.
.gauge orbit Figure 1. Illustration of the gauge orbit containing Ap and indicating the effect of acting on A^ with the gauge transformation p. The action S[A] is constant around the orbit.
138
The integral over the gauge fields can be written as the integral over a full set of gauge-inequivalent (i.e. gauge-fixed) configurations, fVAs{, and an integral over the gauge group fVg. In other words, fVAst is an integral over the set of all possible gauge orbits and fDg is an integral around the gauge orbits. Thus we can write
I VA= fvA6t
fvg.
(4)
To make integrals such as those in the numerator and denominator of Eq. (3) finite and also to study gauge-dependent quantities in a meaningful way, we need to eliminate this integral around the gauge orbit, fT>g. 2. Gribov Copies and the Faddeev-Popov Determinant Any gauge-fixing procedure defines a surface in gauge-field configuration space. Figure 2 is a depiction of these surfaces represented as dashed lines intersecting the gauge orbits within this configuration space. Of course, in general, the gauge orbits are hypersurfaces and so are the gauge-fixing surfaces. Any gaugefixing surface must, by definition, only intersect the gauge orbits at distinct isolated points in configuration space. For this reason, it is sufficient to use lines for the simple illustration of the concepts here. An ideal (or complete) gaugefixing condition, F[A] — 0, defines a surface that intersects each gauge orbit once and only once and by convention contains the trivial configuration A,, = 0. A non-ideal gauge-fixing condition, F'[A] = 0, defines a surface or surfaces which intersect the gauge orbit more than once. These multiple intersections of the non-ideal gauge fixing surface(s) with the gauge orbit are referred to as Gribov copies 1 ' 2,3,4 ' 5 ' 6 . Lorentz gauge (dflA^(x) = 0) for example, has many Gribov copies per gauge orbit. By definition an ideal gauge fixing is free from Gribov copies. We refer to the ideal gauge-fixing surface ^[^4] = 0 as the Fundamental Modular Region (FMR) for that gauge choice. Typically the gauge fixing condition depends on a space-time coordinate, (e.g. Lorentz gauge, axial gauge, etc.), and so we write the gauge fixing condition more generally as F([Aj;a:) = 0. Let us denote one arbitrary gauge configuration per gauge orbit as A° and let this correspond to the "origin" of gauge configurations on that gauge orbit, i.e. to g — 0 on that orbit. Then each gauge orbit can be labelled by A° and the set of all such A°„ is equivalent to one particular, complete specification of the gauge. Under a gauge transformation, g, we move from the origin of the gauge orbit to the configuration, A3^, where by definition A° - ^ A9^ — gA°^g^ - (i/gs)(d^g)g^. Let us denote for each gauge orbit the gauge transformation, g = #[J4°], as the transformation which takes us
139
F'[A]=0 Figure 2.
Ideal, F[A], and non-ideal, F'[A], gauge-fixing.
from the origin of that orbit, A° , to the configuration, Aff-, which lies on the ideal gauge-fixed surface specified by F([A];x) = 0. In other words, we have F{[A];x)\Ai = 0 for A« = A**- G FMR. The inverse Faddeev-Popov determinant is defined as the integral over the gauge group of the gauge-fixing condition, i.e.
-,l[A^f} =
jvg8[F[A}]jv98{g-g) det
fSF([A];x" V Sg(y)
-1
(5)
Let us define the matrix Mf [A] as SFa([A];x) (6) Sgb(y) Then the Faddeev-Popov determinant for an arbitrary configuration A^ can be defined as Ap[A] = |det Mf[^4]|. (The reason for the name is now clear). Note that we have consistency, since A ^ 1 ^ 6 ^ ] = A ^ 1 ^ ] = fVg S(g — g)AFl\A\. We have 1 = fVg AF[A] £[F[y4]] by definition and hence MF([A];x,y)ab
I VAef-
= f VA^- f Vg AF[A] S[F[A]] = f VA AF[A] S[F[A]].
(7)
Since for an ideal gauge-fixing there is one and only one g per gauge orbit, such that -F([A|; x)\§ = 0, then |detMf[^4]| is non-zero on the FMR. It follows that if there is at least one smooth path between any two configurations in the FMR and since the determinant cannot be zero on the FMR, then it cannot change sign on the FMR. The Gribov horizon is defined by those configurations with det M F [ A ] = 0 which lie closest to the FMR. By definition the determinant can change sign on or outside this horizon. Clearly, the FMR is contained within the Gribov horizon and for an ideal gauge fixing, since the sign of the determinant cannot change, we can replace IdetM^I with det M^, [i.e. the overall sign of the functional integral is normalized away in Eq. (3)].
140
These results are generalizations of results from ordinary calculus, where d e t
(|^)
_ = jdx1---dxnS^(f(x)),
(8)
and if there is one and only one x which is a solution of f(x) — 0 then the matrix Mjj = dfi/dxj is invertible (i.e. non-singular) on the hypersurface f(x) = 0 and hence det M ^ 0. 3. Generalized Faddeev-Popov Technique Let us now assume that we have a family of ideal gauge fixings F([i4];i) = f([A];x) — c(x) for any Lorentz scalar c(x) and for /([A]; x) being some Lorentz scalar function, (e.g. d^A^x) or n^A^x) or similar or any nonlocal generalizations of these). Therefore, using the fact that we remain in the FMR and can drop the modulus of the determinant, we have fvA«f
= JVA det MF[A] S[f[A] - c].
(9)
Since c(x) is an arbitrary function, we can define a new "gauge" as the Gaussian weighted average over c(x), i.e., fvA**-
oc fvcexpl-^-
fd4xc(x)2\
oc / x > y l d e t M F [ ^ ] e x p | - ^ f ex jvAVXVx
expl.-ijd4xd4y
fVA detMF[A] S[f[A] - c] d4xf{[A];x)2\ x x(x)MF([A];x,y)X{y)\
xexp{~Jd4xf([A];x)2},
(10)
where we have introduced the anti-commuting ghost fields \ and \. Note that this kind of ideal gauge fixing does not choose just one gauge configuration on the gauge orbit, but rather is some Gaussian weighted average over gauge fields on the gauge orbit. We then obtain
where \Fa,u,F%
- 1 (f([A];x))2
+ J d4xdAy x(x)MF([A]; x, y)X(y).
+ Y,9f(ip-
"»/)«/ (12)
141
4. S t a n d a r d G a u g e Fixing We can now recover standard gauge fixing schemes as special cases of this generalized form. First consider standard covariant gauge, which we obtain by taking f([A];x) = dliA>1(x) and by neglecting the fact that this leads to Gribov copies. We need to evaluate Mf[A] in the vicinity of the gauge-fixing surface for this choice: b MJIWx MF([A],x,y) vY -
8 F a
^
x
Sgb{y)
)
-- *[^A«>(x)-c{x)\ Sgb{y)
m ) -_0 „ ^SA°»{*) • (13) ( t f ) ]
Under an infinitesimal gauge transformation g we have A$(x)Sm-^9
(A%(x)
= Al(x) + 9sfab^b(x)Al(x)
- d^a{x)
+
0(J){\A)
and hence in the neighbourhood of the gauge fixing surface (i.e. for small fluctuations along the gauge orbit around A& f •), we have
MF{[A];x,yyb=d*JAail{x)
(15)
6u»(y)} u-0
= d; ( [ - c W +g,fabeAcHx)]
x S^(x
- y)) .
We then recover the standard covariant gauge-fixed form of the QCD action
+(dl>Xa)(d"6ab-gfabcA2)Xb.
(16)
However, this gauge fixing has not removed the Gribov copies and so the formal manipulations which lead to this action are not valid. This Lorentz covariant set of naive gauges corresponds to a Gaussian weighted average over generalized Lorentz gauges, where the gauge parameter £ is the width of the Gaussian distribution over the configurations on the gauge orbit. Setting £ — 0 we see that the width vanishes and we obtain Landau gauge (equivalent to Lorentz gauge, dl'All(x) = 0). Choosing £ = 1 is referred to as "Feynman gauge" and so on. We can similarly recover the standard QCD action for axial gauge, where n^A^(x) = 0. Proceeding as for the generalized covariant gauge, we first identify /([.A]; a;) = ntiA>i(x) and obtain the gauge-fixed action
SdQ,q,A] = 0 we select nltA,t(x) = 0 exactly and recover the usual axial-gauge fixed Q C D action. Axial gauge does not involve ghost fields, since in this case MF{[A^);x,yYb=n,8-£^
_ = „ „ {[-d^Sab]S^(x
- yj)
,
(18)
which is independent of A^ since n^Aff(x) = 0. In other words, the gauge field does not appear in Mp[A] on the gauge-fixed surface. Unfortunately axial gauge suffers from singularities which lead to significant difficulties when trying to define perturbation theory beyond one loop. A related feature is t h a t axial gauge is not a complete gauge fixing prescription. While there are complete versions of axial gauge on the lattice, these always involve a nonlocal element, or reintroduce Gribov copies at the boundary so as not to destroy the Polyakov loop. 5. D i s c u s s i o n a n d C o n c l u s i o n s There is no known Gribov-copy-free gauge fixing which is a local function of A^x). In other words, such a gauge fixing cannot be expressed as a function of A^(x) and a finite number of its derivatives, i.e. F([yt];;c) ^ F{d,j,,A^{x)) for all x. Hence, the gauge-fixed action, 5 j [• • •], in Eq. (12) becomes non-local and gives rise to a nonlocal q u a n t u m field theory. Since this action serves as the basis for the proof of the renormalizability of Q C D , the proof of asymptotic freedom, local BRS symmetry, and the Schwinger-Dyson equations (to n a m e but a few) the nonlocality of the action leaves us without a reliable basis from which to prove these features of Q C D in the nonperturbative context. It is well-established t h a t Q C D is asymptotically free, i.e. it has weakcoupling at large m o m e n t a . In the weak coupling limit the functional integral is dominated by small action configurations. As a consequence, momentum-space Green's functions at large m o m e n t a will receive their dominant contributions in the p a t h integral from configurations near the trivial gauge orbit, i.e. the orbit containing A^ = 0, since this orbit minimizes the action. If we use s t a n d a r d gauge fixing, which neglects the fact t h a t Gribov copies are present, then at large m o m e n t a J VA will be dominated by configurations lying on the gauge-fixed surfaces in the neighbourhood of each of the Gribov copies on the trivial orbit. Since for small field fluctuations the Gribov copies cannot be aware of each other, we merely overcount the contribution by a factor equal to the number of copies on the trivial orbit. This overcounting is normalized away by the ratio in Eq. (3) and becomes irrelevant. T h u s it is possible t o understand why Gribov copies can be neglected at large m o m e n t a and why it is sufficient to use standard gauge fixing schemes as the basis for calculations
143
in perturbative QCD. Since renormalizability is an ultraviolet issue, there is no question about the renormalizability of QCD. Lattice QCD has provided numerical confirmation of asymptotic freedom, so let us now turn our attention to the matter of Gribov copies in lattice QCD. Since the observable 0[A] and the action are both gauge-invariant it does not matter whether we sample from the FMR of an ideal gauge-fixing or elsewhere on the gauge orbit. The trick is simply to sample at most once from each orbit. Since there is an infinite number of gauge orbits (even on the lattice), no finite ensemble will ever sample the same orbit twice. This makes Gribov copies and gauge-fixing irrelevant in the calculation of colour-singlet quantities on the lattice. The calculation of gauge-dependent Green's functions on the lattice does require that the gauge be fixed. The standard choice is naive lattice Landau gauge, which selects essentially at random between the Landau gauge Gribov copies for the gauge orbits represented in the ensemble. This means that, while the gauge fixing is well-defined in that there are no Gribov copies, the Landau gauge-fixed configurations are not from a single connected FMR. For this reason lattice studies of gluon and quark propagators are now being extended to Laplacian gauge for comparison. Laplacian gauge is interesting because it is Gribov-copy-free (except on a set of configurations of measure zero) and it reduces to Landau gauge at large momenta. Lattice calculations of the Laplacian gauge and Landau gauge quark and gluon propagators converge at large momenta and hence are consistent with this expectation. In conclusion, it should be noted that throughout this discussion there has been the implicit assumption that nonperturbative QCD should be defined in such a way that each gauge orbit is represented only once in the functional integral, i.e. that it should be defined to have no Gribov copies. This is the definition of nonperturbative QCD implicitly assumed in lattice QCD studies. We have seen that this assumption destroys locality and the BRS invariance of the theory. An equally valid point of view is that locality and BRS symmetry are central to the definition of QCD and must not be sacrificed in the nonperturbative regime, (see, e.g. Ref. [3,4,5,6]). This viewpoint implies that Gribov copies are necessarily present, that gauge orbits are multiply represented, and that the definition of nonperturbative QCD must be considered with some care. Since these nonperturbative definitions of QCD appear to be different, establishing which is the one appropriate for the description of the physical world is of considerable importance.
144
References 1. L. Giusti, M. L. Paciello, C. Parrinello, S. Petrarca and B. Taglienti, Int. J. Mod. Phys. A16, 3487 (2001) [arXiv:hep-lat/0104012] and references therein. 2. P. van Baal, arXiv:hep-th/9711070. 3. H. Neuberger, Phys. Lett. B183, 337 (1987). 4. M. Testa, Phys. Lett. B429, 349 (1998) [arXiv:hep-lat/9803025]. 5. M. Testa, arXiv:hep-lat/9912029. 6. R. Alkofer and L. von Smekal, Phys. Rep. 353, 281 (2001) [arXiv:hep-ph/0007355] and references therein.
Q U A R K MODEL A N D CHIRAL S Y M M E T R Y A S P E C T S OF EXCITED B A R Y O N S
A. H O S A K A Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0047 Japan E-mail: [email protected]
We investigate properties of excited baryons from two different points of view. In one of them, the argument is based on a quark model with an emphasis on flavour independent nature of the baryon spectra. This easily indicates positions of missing resonances. In the other, we discuss how baryon resonances are classified by the chiral symmetry group. It is shown that there are two baryon representations. We present nir production experiments to distinguish the two baryon representations.
1. Introduction It is important that we are able to control meson-baryon dynamics quantitatively as much as possible. This will provide not only a clue to understand fundamentals of hadron physics but also a basis for the description of more complicated system such as hadronic and/or quark matter in extreme conditions. The latter is one of the important subjects of the JHF project 1 . Quark models provide now standard methods to describe hadrons, as we know empirically that they work well over a wide range of phenomena 2 . Flavour symmetry is implemented by regarding quarks as fundamental representations of SU(6) spin-flavour symmetry. By choosing constituent quark masses appropriately, baryon magnetic moments are well reproduced. Introducing another scale parameter for quark distributions (oscillator parameter for a quark confining force), numbers of excited states are predicted which can be compared with data, although fine tuning is necessary to get reasonable results. We also know that chiral symmetry dictates much of hadron dynamics 3 . Chiral symmetry with its spontaneous breaking is indeed one of important aspects of low energy QCD. Interactions of pions and kaons, which are the Nambu-Goldstone bosons, are described well by the current algebra and chiral perturbation theory 4 . To the present date, it is not obvious how chiral symmetry and the quark
145
146
model aspects can merge into a single framework. Nevertheless, here we attempt to demonstrate examples based on the above two aspects for excited baryons. In Sec. 2, we study masses and transition amplitudes of excited baryons in a quark model. We point out that baryon spectra resemble very much rotational bands of a deformed harmonic oscillator model. As an illustration, we compute not only masses but also pion transition amplitudes between the same rotational band. In Sec. 3, we propose another point of view where positive and negative parity baryons are considered as members of a multiplet of chiral symmetry. Two distinguished assignments of chiral symmetry of baryons are discussed from a group theoretical point of view. Experiments which can observe the two assignments are proposed. The final section is devoted to a brief summary of this report. 2. Quark Model Description and Deformed Excited States 2.1.
Masses
Let us first look at experimental data as shown in Fig. 1. We take 49 states out of 50 states of three and four stars, and several states with one and two stars 5 . In showing experimental data we follow the prescriptions: (1) Masses are measured from the ground states in each spin-flavour multiplet, in order to subtract the spin-flavour dependence which is well established by the GellMann-Okubo mass formula. (2) Masses of 2 8 M S , 4&MS states for positive parity, and of 48JWS states for negative parity are reduced by 200 MeV. Then the resulting mass spectra show a very simple systematics, which remarkably resembles rotational bands. It is also important that the regularity is common to all channels independent of flavour. Hence, we consider a quark model with a deformed harmonic oscillator potential 6 : HDOQ -
7^ + 2 m^lx2i + "&? + w ^. 2 )
22 »=l
L
(1)
J
Here we ignore interactions due to gluons and mesons. The only dynamics here is the shape change which is described by the deformed oscillator potential, u)x ^ uiy 7^ <JJZ . The Hamiltonian looks very simple, but it can indeed reproduce the structure of the mass spectra in Fig. 1. After removing the centre of mass motion, we find an intrinsic energy Eint(Nx,Ny,Nz) = (Nx + l)cux + [Ny + l)w„ + {Nz + l)w z , where Nx,Ny, Nz are the sum of principal quantum numbers for the internal degrees of freedom for the p and A coordinates. Imposing the volume conservation condition LOxwyu)z = w 3 , the minimum energy occurs when the system is deformed. When
147
Nx = Ny = 0,NZ = N, one obtains a prolate deformation. Our discussions in what follows are based on this prolate deformation, since it is energetically most favourable. Physical states are obtained by rotating the deformed states, whose energies are computed by the standard cranking method 6 JtT)N 1
Total Cross Section ~ 60 - 40 <j
| 20 H
0 540
560 580 Pen, ( M e V / 500 MeV, these rapid-varying chiral effects must be incorporated phenomenologically. Within the quenched approximation dynamical sea quarks are absent from the simulation. As a consequence the structure of meson-loop contributions is modified. In the physical theory of QCD, meson-loop diagrams can be described by two topologically differing types. A typical meson-loop diagram may be decomposed into those where the loop meson contains a sea quark, such as Fig. 1(a), and those where the loop meson is comprised of pure valence quarks, see Fig. 1(b). These diagrams involving the sea quark, type (a), are obviously absent in the quenched approximation and consequently only
157
(a)
Figure 1.
(b)
Quark flow diagrams of pion loop contributions appearing in QCD.
(a)
(b)
Figure 2. Quark flow diagrams of chiral vf loop contributions appearing in QQCD: (a) axial hairpin, (b) double hairpin.
a subset of the contributions to the physical theory are included. This type of argument, together with SU(6) symmetry, is precisely that described for the evaluation of non-analytic contributions to baryon magnetic moments by Leinweber6. This quark flow approach is analogous to the original approach to chiral perturbation theory for mesons performed by Sharpe 7 ' 8 . In addition to the usual pion loop contributions, quenched QCD (QQCD) contains loop diagrams involving the flavour singlet rf which also give rise to important non-analytic structure. Within full QCD such loops do not play a role in the chiral expansion because the rf remains massive in the chiral limit. On the other hand, in the quenched approximation the rf is also a Goldstone boson 8 and the rf propagator has the same kinematic structure as that of the pion. As a consequence there are two new chiral loop contributions unique to the quenched theory. The first of these corresponds to an axial hairpin diagram such as that indicated in Fig. 2(a). This diagram gives a contribution to the chiral expansion of baryon masses which is non-analytic at order m^. The second of these new rf loop diagrams arises from the double hairpin vertex as pictured in Fig. 2(b). This contribution is particularly interesting because it involves two Goldstone boson propagators and is therefore the source of more singular non-analytic behaviour - linear in m^. In studying the extrapolation of quenched lattice results it is essential to treat these contributions on an equal footing to the pion-loop diagrams discussed earlier.
158
3. Chiral Extrapolations In general, the coefficients of the leading (LNA) and next-to leading nonanalytic (NLNA) terms in a chiral expansion of baryon masses are very large. For instance, the LNA term for the nucleon mass is 8rrvN ' = —5.6 m% (with m , and 5mN in GeV). With mw = 0.5GeV, quite a low mass for current simulations, this yields Sm^N ' — 0.7 GeV — a huge contribution. Furthermore, in this region hadron masses in both full and quenched lattice QCD are found to be essentially linear in rn\ or equivalently quark mass, whereas 8rrvN ' is highly non-linear. The challenge is therefore to ensure the appropriate LNA and NLNA behaviour, with the correct coefficients, as mn —• 0, while making a sufficiently rapid transition to the linear behaviour of actual lattice data, where ra, becomes large. A reliable method for achieving all this was proposed by Leinweber et al.9 They fit the full (unquenched) lattice data with the form: MB =aB+pBml
+J2B(mn,A),
(1)
where Y,B is the total contribution from those pion loops which give rise to the LNA and NLNA terms in the self-energy of the baryon. The extension to the case of quenched QCD is achieved by replacing the self-energies, S s , of the physical theory by the corresponding contributions of the quenched theory 2 ,
tB. The linear term of Eq. (1), which dominates for mn S> A, models the quark mass dependence of the pion-cloud source — the baryon without its pion dressing. This term also serves to account for loop diagrams involving heavier mesons, which have much slower variation with quark mass. The diagrams for the various meson-loop contributions are evaluated using a phenomenological regulator. This regulator has the effect of suppressing the contributions as soon as the pion mass becomes large. At light quark masses the self-energies, T,B, provide the same non-analytic behaviour as xPT, independent of the choice of regulator. Therefore the functional form, Eq. (1), naturally encapsulates both the light quark limit of %PT and the heavy quark behaviour observed on the lattice. We consider the leading order diagrams containing only the lightest Goldstone degrees of freedom. These are responsible for the most rapid non-linear variation as the quark mass is pushed down toward the chiral limit. In the physical theory we consider only those diagrams containing pions, as depicted in Fig. 3. In QQCD there exist modified pion loop contributions and the additional structures arising from the rf behaving as a Goldstone boson. The diagrams contributing to the nucleon self-energy in the quenched approximation are shown in Fig. 4, the A can be described by analogous diagrams.
159
+ EJV=
L
:
'
+ F i g u r e 3 . Illustrative view of the meson-loop self-energies, S s , infullQCD. These diagrams give rise to the LNA and NLNA contributions in the chiral expansion. Single (double) lines denote propagation ofaJV (A).
+ -'N
+ —
-
*
—
—
0. These results are of particular relevance for heavy-ion collision experiments.
'speaker at the workshop 'permanent address
164
8PP,
165
1. Introduction The QCD phase diagram has come under increasing experimental and theoretical scrutiny over the last few years. On the experimental side, very recent studies of compact astronomical objects have suggested that their cores contain "quark matter", i.e. QCD in a new, unconfined phase where the basic units of matter are quarks, rather than nuclei or nucleons 1 . More terrestrially, heavy ion collision experiments, such as those performed at RHIC and CERN, are also believed to be probing unconfined QCD 2 . On the theoretical side, the study of QCD under these extreme densities and temperatures has proceeded along several fronts. One of the most promising areas of research is the use of lattice techniques to study either QCD itself, or model theories which mimic the strong interaction 3 . Clearly the most satisfying approach would be the former, i.e. a direct lattice study of QCD at various coordinates (T, p) in its phase space (fi is the chemical potential for the quark number). However, until recently, this has proved intractable at a practical level for very fundamental reasons. This is because the Monte Carlo integration technique, which is at the heart of the (Euclidean) lattice approach, breaks down when \i ^ 0. This work summarises one new approach which overcomes this problem and has made progress for /i ^ 0 and T =£ 0. In the next section a summary is given of the lattice technique and the problem incurred when p ^ 0. Section 3 describes the method used to overcome these difficulties, and Sec. 4 outlines the simulation details. The next two sections apply the method to variations in m and fi, and section Sec. 7 describes calculations of the pressure and energy density as functions of fi. A full account of this work is published elsewhere4.
2. Lattice Technique On the lattice, the quark fields, 4>(x), are defined on the sites, x, and the gluonic fields, U^(x), on the links x —• x + fi. Observables are then calculated via a Monte Carlo integration approach :
} which are selected with probability proportional to the Boltzmann weight P{{U,il>,xl>}) oc e-s({u,i>,i>}) w ith S a suitably defined (Euclidean) gaugeinvariant action. The fermionic part of this action is
166
SF = YJ^)iP+rn)^{x).
(2)
= M For /i = 0 it can be shown that this action produces a (real-valued) positive Boltzmann weight. Calculations at non-zero temperature, T ^ 0, can be performed by using a lattice with a finite temporal extent of Nta — 1/T, where Nt is the number of lattice sites in the time dimension. In practice, T is varied by changing the gauge coupling, go, and hence (through dimensional transmutation) the lattice spacing, a, rather than by changing Nt (which can only be changed in discrete steps!). The chemical potential is introduced into the system via an additional term in the quark matrix M, proportional to the Dirac gamma matrix, 70,
M^M + mo-
(3)
For fi ^ 0, this leads to a complex-valued Boltzmann "weight" which can therefore no longer be used as a probability distribution, and, hence, the Monte Carlo integration procedure is no longer applicable. This is known as the Sign Problem and has plagued more than a decade of lattice calculations of QCD at fj, 7^ 0. 3. Reweighting This section outlines the Ferrenberg-Swendsen reweighting approach 5 which is used to overcome the sign problem detailed in the previous section. Observables at one set of parameter values {fi,m,pi) (where /? = Q/gl, and m is the quark mass) can be calculated using an ensemble generated at another set of parameters (/3o,mo,fio) as follows,
iM
_ ((ftp + Qifi + ft2p2) expjRin + R2H2 ~ Ag g )) (/?0i(x)(r^Fllvip(x)
,
(1)
where S\y is the standard Wilson action and Csw is the clover coefficient which can be tuned to remove 0(a) artifacts. Nonperturbative (NP) 0(a) improvement 3 tunes Csw to all powers in g2 and displays excellent scaling, as shown by Edwards et al.4 who studied the scaling properties of the nucleon and vector meson masses for various lattice spacings (see also Sec. 4 below). In particular, the linear behaviour of the NP-improved clover actions, when plotted against a2, demonstrates that 0(a) errors are removed. It was also found in Ref. [4] that a linear extrapolation of the mean-field improved data fails, indicating that 0(a) errors are still present. A drawback to the clover action, however, is the associated problem of exceptional configurations, where the quark propagator encounters singular behaviour as the quark mass becomes small. In practice, this prevents the use of coarse lattices (/3 < 5.7 ~ a > 0.18 fm) 5 ' 6 . Furthermore, the plaquette version of F^, which is commonly used in Eq. (1), has large 0(a2) errors, which can lead to errors of the order of 10% in the topological charge even on very smooth configurations7. The idea of using fat links in fermion actions was first explored by the MIT group 8 and more recently has been studied by DeGrand et al.6'9, who showed that the exceptional configuration problem can be overcome by using a fat-link (FL) clover action. Moreover, the renormalization of the coefficients of action improvement terms is small. A drawback to conventional fat-link techniques, however, is that in smearing the links gluon interactions are removed at the scale of the cutoff. While this has some tremendous benefits, the short-distance quark interactions are lost. As a result decay constants, which are sensitive to the wave function at the origin, are suppressed. A solution to these problems is to work with two sets of links in the fermion action. In the relevant dimension-four operators, one works with the untouched
176
links generated via Monte Carlo methods, while the smeared fat links are introduced only in the higher dimension irrelevant operators. The effect this has on decay constants is under investigation and will be reported elsewhere. In this paper we present the first results of simulations of the spectrum of light mesons and baryons at light quark masses using this variation of the clover action. In particular, we will start with the standard clover action and replace the links in the irrelevant operators with APE smeared 10 , or fat links. We shall refer to this action as the Fat-Link Irrelevant Clover (FLIC) action 11 . 2. Gauge Action The simulations are performed using a tree-level (9(a 2 )-Symanzik-improved 12 gauge action on a 163 x 32 lattice at 0 = 4.60, providing a lattice spacing a = 0.125(2) fm determined from the string tension with yfa = 440 MeV. A total of 50 configurations are used in this analysis, and the error analysis is performed by a third-order, single-eliminationjackknife, with the \ 2 P e r degree of freedom (Nj^p) obtained via covariance matrix fits. Further details of this simulations may be found in Ref. [11]. 3. Fat-Link Irrelevant Fermion Action Fat links 6 ' 9 are created by averaging or smearing links on the lattice with their nearest neighbours in a gauge covariant manner (APE smearing). The smearing procedure 10 replaces a link, U^x), with a sum of the link and a times its staples 4
+Ul(x - ua)Ufi(x - va)Uu{x - ua + fia) ,
(2)
followed by projection back to SU(3). We select the unitary matrix J/J L which maximizes Heti(U^U',!),
(3)
by iterating over the three diagonal SU(2) subgroups of SU(3). We repeat this procedure of smearing followed immediately by projection n times. We create our fat links by setting a = 0.7 and comparing n — 4 and 12 smearing sweeps. The mean-field improved FLIC action now becomes S & = Sw - ^
J
^x)a^F^(x)
,
(4)
where F^ is constructed using fat links, and where the mean-field improved Fat-Link Irrelevant Wilson action is
177 Table 1. The value of the mean link for different numbers of smearing sweeps, n. n
KL)4
"o
0 4 12
0.88894473 0.99658530 0.99927343
0.62445197 0.98641100 0.99709689
(5) X
In x,p — V
-*
FL U0
ip(x+n)-
4>{X +fl)+
ip(x - p.) «o
wo
„FLF L u;0
nX •/*)
(6)
with K = l/(2m + 8r). We take the standard value r = 1. The 7-matrices are hermitian and a^ = [7^, 7„]/(2i). As reported in Table 1, the mean-field improvement parameter for the fat links is very close to 1. Hence, the mean-field improved coefficient for Csw is expected to be adequate a . In addition, actions with many irrelevant operators (e.g. the D234 action) can now be handled with confidence as treelevel knowledge of the improvement coefficients should be sufficient. Another advantage is that one can now use highly improved definitions of F^ (involving terms up to UQ2), which give impressive near-integer results for the topological charge 13 . In particular, we employ an 0(a4) improved definition13 of F^ in which the standard clover-sum of four l x l Wilson loops lying in the fi, v plane is combined with 2 x 2 and 3 x 3 Wilson loop clovers. Work by DeForcrand et a/.14 suggests that 7 cooling sweeps are required to approach topological charge within 1% of integer value. This is approximately 15 16 APE smearing sweeps at a = 0.7. However, achieving integer topological charge is not necessary for the purposes of studying hadron masses, as has been well established. To reach integer topological charge, even with improved definitions of the topological charge operator, requires significant smoothing and associated loss of short-distance information. Instead, we regard this as an upper limit on the number of smearing sweeps. "Our experience with topological charge operators suggests that it is advantageous to include «o factors, even as they approach 1.
178
Using unimproved gauge fields and an unimproved topological charge operator, Bonnet et al.7 found that the topological charge settles down after about 10 sweeps of APE smearing at a — 0.7. Consequently, we create fat links with APE smearing parameters n = 12 and a = 0.7. This corresponds to ~ 2.5 times the smearing used in Refs. [6,9]. Further investigation reveals that improved gauge fields with a small lattice spacing (a = 0.125 fm) are smooth after only 4 sweeps. Hence, we perform calculations with 4 sweeps of smearing at a = 0.7 and consider n = 12 as a second reference. Table 1 lists the values of UQ L for n = 0, 4 and 12 smearing sweeps. We also compare our results with the standard Mean-Field Improved Clover (MFIC) action. We mean-field improve as defined in Eqs. (4) and (6) but with thin links throughout. The standard Wilson-loop definition of F^ is used. A fixed boundary condition is used for the fermions by setting Ut{x, nt) = 0 and UfL(x, nt) = 0
Vx ,
(7)
in the hopping terms of the fermion action. The fermion source is centred at the space-time location (x,y,z,t) — (1,1,1,3), which allows for two steps backward in time without loss of signal. Gauge-invariant gaussian smearing 16 in the spatial dimensions is applied at the source to increase the overlap of the interpolating operators with the ground states. 4. R e s u l t s Hadron masses are extracted from the Euclidean time dependence of the calculated two-point correlation functions. The effective masses are given by M(t + 1/2) = log[G( & *
IO.
• 0
_ji
i
0.0
0.2
0.4 m/
0.6 (GeV8)
-
~
FLIC4 Wilson i
0.8
1.0
Figure 1. Masses of the nucleon, A and p meson versus m\ for the FLIC4, FLIC12 and Wilson actions.
reassuring that all actions give the correct mass ordering in the spectrum. The value of the squared pion mass at m^/mp = 0.7 is plotted on the abscissa for the three actions as a reference point. This point is chosen in order to allow comparison of different results by interpolating them to a common value of m^/mp = 0.7, rather than extrapolating them to smaller quark masses, which is subject to larger systematic and statistical uncertainties. The scaling behaviour of the different actions is illustrated in Fig. 2. The present results for the Wilson action agree with those of Ref. [4]. The first feature to observe in Fig. 2 is that actions with fat-link irrelevant operators perform extremely well. For both the vector meson and the nucleon, the FLIC actions perform significantly better than the mean-field improved clover action. It is also clear that the FLIC4 action performs systematically better than the FLIC12. This suggests that 12 smearing sweeps removes too much shortdistance information from the gauge-field configurations. On the other hand, 4 sweeps of smearing combined with our 0(a4) improved F^„ provides excellent results, without the fine tuning of Csw m the NP improvement program. Notice that for the p meson, a linear extrapolation of previous mean-field improved clover results in Fig. 2 passes through our mean-field improved clover result at a2a ~ 0.08 which lies systematically low relative to the FLIC actions. However, a linear extrapolation does not pass through the continuum limit result, thus confirming the presence of significant 0(a) errors in the mean-field improved clover fermion action. While there are no NP-improved clover plus improved gluon simulation results at a2
a
NP Clover
•» FLIC4, 200conf |» i Opr jTilgon I ,
0.00
0.05
,
* ,
Np
,
Clover+imp glue I , , , , I
0.10 a2 a
0.15
Figure 2. Nucleon and vector meson masses for the Wilson, NP-improved and FLIC actions obtained by interpolating our results of Fig. 1 to mTjmp = 0.7. Results from the present simulations are indicated by the solid points. The fat links are constructed with n = 4 (solid squares) and n = 12 (stars) smearing sweeps at a = 0.7.
obtained with a NP-improved clover fermion action. Having determined FLIC4 is the preferred action, we have increased the number of configurations to 200 for this action. As expected, the error bars are halved and the central values for the FLIC4 points move to the upper end of the error bars on the 50 configuration result, further supporting the promise of excellent scaling. Finally, in order to search for exceptional configurations by pushing the bare quark mass down, we would like our preferred action to be efficient when inverting the fermion matrix. In Fig. 3 we compare the convergence rates of the different actions by plotting the number of stabilized biconjugate gradient 17 iterations required to invert the fermion matrix as a function of mn/mp. For any particular value of mv/mp, the FLIC actions converge faster than both the Wilson and mean-field improved clover fermion actions. Also, the slopes of the FLIC lines are smaller in magnitude than those for Wilson and meanfield improved clover actions, which provides great promise for performing cost effective simulations at quark masses closer to the physical values. Problems with exceptional configurations have prevented such simulations in the past. The ease with which one can invert the fermion matrix using FLIC fermions leads us to attempt simulations of three lighter quark masses corresponding
181 450
1
1
-
400
1
- - - FLIC12 ---FLIC4 ---Wilson - - - MFIC
N
\
350
\
-
x
M300
\
-
.2 250 xx xx
£200
a 1.2
1
$ 1 i
0.2
0.4 TO '
i
0.6 (GeV2)
' 0.8
1.0
Figure 4. Masses of the nucleon, A, E and H versus m\ for the FLIC4 fermion action.
improvement on mean field-improved gluon configurations. Simulations are possible and the results are competitive with nonperturbative-improved clover results on plaquette-action gluon configurations. We have found that minimal smearing holds the promise of better scaling behaviour. Our results suggest that too much smearing removes relevant information from the gauge fields, leading to poorer performance. Fermion matrix inversion for FLIC actions is more efficient and results show no sign of exceptional configuration problems down to mn/mp = 0.45. However we encounter divergences in the pion correlator at m^/rrip = 0.36 on 1% of the configurations analysed on this particular lattice. This work paves the way for promising future studies. It will be of great interest to consider different lattice spacings to further test the scaling of the fat-link actions. Current work is under way to further explore the exceptional configuration problem where a precision field-strength tensor and additional smearing hold promise. A study of the spectrum of excited hadrons using the fat-link clover actions is currently in progress 19 . Acknowledgements This work was supported by the Australian Research Council. We would also like to thank the National Computing Facility for Lattice Gauge Theories for the use of the Orion Supercomputer. W.M. and F.X.L. were partially supported by the U.S. Department of Energy contract DE-AC05-84ER40150, under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility (Jefferson Lab).
183
References 1. K.G. Wilson, in New Phenomena in Subnuclear Physics, Part A, A. Zichichi (ed.), Plenum Press, New York, p. 69, 1975. 2. B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259, 572 (1985). 3. M. Luscher et al., Nucl. Phys. B478, 365 (1996) [hep-lat/9605038]; M. Luscher et ai, Nucl. Phys. B491, 323 (1997) 323 [hep-lat/9609035]. 4. R.G. Edwards, U.M. Heller and T.R. Klassen, Phys. Rev. Lett. 80, 3448 (1998). [hep-lat/9711052]; see also R.D. Kenway, Nucl. Phys. Proc. Suppl. 73, 16 (1999) [hep-lat/9810054] for a review. 5. W. Bardeen et al., Phys. Rev. D57, 1633 (1998) [hep-lat/9705008]; W. Bardeen et al., Phys. Rev. D57, 3890 (1998). 6. T. DeGrand et al. (MILC Collaboration), [hep-lat/9807002]. 7. F.D. Bonnet et al., Phys. Rev. D62, 094509 (2000) [hep-lat/0001018]. 8. M.C. Chu et al., Phys. Rev. D49, 6039 (1994) [hep-lat/9312071]. 9. T. DeGrand (MILC collaboration), Phys. Rev. D60, 094501 (1999) [heplat/9903006]. 10. M. Falcioni et al., Nucl. Phys. B251, 624 (1985); M. Albanese et al., Phys. Lett. B192, 163 (1987). 11. J.M. Zanotti et al. Phys. Rev. D60, 074507 (2002) [hep-lat/0110216]; Nucl.Phys.Proc.Suppl. 109, 101 (2002) [hep-lat/0201004]. 12. K. Symanzik, Nucl. Phys. B226, 187 (1983). 13. S. Bilson-Thompson et al., Nucl.Phys.Proc.Suppl 109 116 (2002) [heplat/0112034]; hep-lat/0203008. 14. P. de Forcrand et al., Nucl. Phys. B499, 409 (1997) [hep-lat/9701012]; P. de Forcrand et ai, [hep-lat/9802017]. 15. F.D. Bonnet et al., [hep-lat/0106023]. 16. S. Gusken, Nucl. Phys. Proc. Suppl. 17, 361 (1990). 17. A. Frommer et al., Int. J. Mod. Phys. C 5 , 1073 (1994) [hep-lat/9404013]. 18. M.D. Morte et al., hep-lat/0111048. 19. W. Melnitchouk et al., Nucl. Phys. Proc. Suppl 109, 116 (2002), hep-lat/0201005; hep-lat/0202022.
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5. Nuclear and Nucleon Structure Functions
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SMALL-X N U C L E A R EFFECTS IN PARTON D I S T R I B U T I O N S
V. GUZEY Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, Adelaide, 5005, Australia E-mail: [email protected]
We summarize the status of nuclear parton distribution functions in nuclei at low x and show t h a t they are not constrained well enough by the available data. Measurements of Drell-Yan dimuon pair production on nuclei at the JHF is singled out as a good candidate to significantly increase information about antishadowing and shadowing of valence and antiquark parton distributions in nuclei. A brief description of the leading twist approach to nuclear shadowing is given and outstanding theoretical problems are reviewed.
1. Parton Distributions in Nuclei It is now very well established that the distributions of partons (quarks and gluons) in nuclei differ from those in free nucleons. Experimental information on nuclear parton distribution functions (nPDFs) is obtained from the following fixed nuclear target experiments: inclusive deep inelastic scattering (DIS) of leptons, Drell-Yan dimuon production by hadron beams, and lepto- and hadroproduction of vector mesons. Each of the afore mentioned processes is sensitive to a particular type or combination of nPDFs. Inclusive DIS, l + A—tl' + X, measures the nuclear structure function F£, which, to leading order, takes the form F2A(x,Q*)=Yje](f?(x,Q2)
+ fi(x,Q*)).
(1)
i
Here f* is the quark nPDF; f£ is that of the antiquark. Inclusive DIS constrains best the combination f£ + f£. It is important to note that the fixedtarget kinematics correlates the values of x and Q 2 entering Eq. (1). For instance, for the NMC experiment 1 , the condition Q2 > 1 GeV 2 means that x > 5 x 1 0 - 3 . Hence the data set eligible for the perturbative QCD analysis becomes reduced, and the reliable extraction of nPDF, especially at low x, becomes more difficult. Once F£ is measured, the gluon distribution in nuclei, gA(x,Q2), can be
187
188
determined indirectly and approximately through scaling violations of FA :
^^?f
2 l s W
>.
(2)
However, the accuracy and extent in x and Q2 variables of the available data do not offer enough statistics to pin down gA{x, Q2) reliably enough. Alternatively, gA{x,Q2) can be studied directly via the "proton-gluon fusion" mechanism : 7 *-|V
">•//* + A
(3)
which dominates lepto- and hadroproduction of vector mesons (J/^f, W, T) at high energies. Another important source of information on nPDFs is Drell-Yan high-mass dimuon production, p + A —> fi+^~ + X. To leading order, the differential cross section of this process reads
£k - C £ E - J ( ' . < ' ' > * >+tt-itfc))'
(4)
i
where y/s is the centre-of-mass energy; x\ and x-i are the Bjorken variables of the projectile and target, respectively. In the kinematics, where X\ ^> x%, the first term dominates, and Eq. (4) gives a measure of the distribution of antiquarks in nuclei. How well do we know nPDFs after almost 20 years of experiments? The dispersion of predictions of modern models for nPDFs indicates that nPDFs are not constrained sufficiently. Figures 1 and 2 demonstrate predictions for the ratios uA/(AuN) and gA/(AgN) at Q = 1.5 GeV as a function of x for 208 the heavy nucleus of Pb. The curves are results of Frankfurt et al.2 (solid curve labelled FGMS), Eskola et al.3 (dotted curve labelled ESK98), Li and Wang 4 (dotted curve labelled HIJING), and Kumano et al.5 (dash-dotted curve labelled HKM). While the solid curve is a theoretical prediction made within the framework of a leading twist approach (see Sec. 3), the other three curves are fits to the available data with several simplifying assumptions. There is no region in x, where the four results would be consistent with each other. 2. Drell-Yan Dimuon Production on Nuclei at JHF The Japan Hadron Facility (JHF) offers a unique possibility to perform very precise measurements of Drell-Yan high-mass dimuon production on nuclei, and thus to determine the nuclear sea and valence quark distributions 6 . Let us briefly discuss relevant kinematics of the reaction (see Eq. (4)). To leading order, Drell-Yan dimuon production can be envisioned as annihilation
189
£•••••
-
I
**f 0.9
0.8
0.7
/
. . /
J ^ 0.6
"
/
y
FGMS ESK98 HIJING HKM
Q-1.5 GeV 208
0.5
Pb
'/ i
Figure 1.
The ratio of the antiquark n P D F , uA/(AuN),
i
within four different models.
of a quark (antiquark) from the beam by an antiquark (quark) from the target into a virtual photon with subsequent production of a pair of oppositely charged muons. The Bjorken variable x\ refers to the beam (proton), while xi refers to the nuclear target. These variables are related by x\X2 = Q2/s, where Q2 is the virtuality of the photon. Q2 is equal to the invariant mass squared of the detected dimuon pair, and, since one wants to stay away from the n+(i~ resonances, Q2 is large, Q2 > 16 GeV 2 . Drell-Yan dimuon production on nuclei was studied by the Fermilab E772 experiment with 800 GeV/c protons 7 . In this case, 0.05 < x 2 < 0.3 and the second term in Eq. (4) can be safely neglected. Thus, the experiment gave a direct measure of the antiquark nPDFs in several nuclei with the conclusion that antiquarks are not modified in nuclei for 0.1 < x < 0.3. This finding imposes powerful constraints on models of nuclear shadowing and antishadowing for antiquarks in nuclei: the scenario of Kumano et al. is ruled out. Also, the E772 result rules out models of the EMC effect with significant pion excess. JHF will have two advantages over the E772 experiment. First, the beam energy is lower, Ep = 50 GeV, which results in an increase by factor 16 of the
190
FGMS ESK98 Q=1.5GeV
HIJING HKM
208pb
Figure 2.
The ratio of the gluon n P D F , gA/(Ag
), within four different models.
differential cross section in Eq. (4) because of the factor 1/s. Second, a much higher flux will give another factor 16 gain. The values of xi will be higher, #2 > 015 — 0.2, which means that both terms in Eq. (4) should be retained. This will allow one to study both antiquark and valence quark nPDFs. One should note that the depletion of nPDFs at low x (nuclear shadowing), is followed by some enhancement (antishadowing) at 0.05 < x < 0.2 (see Fig. 2). Thus, in the available domain of X2, JHF will be able to determine the pattern of antishadowing with much better accuracy than any previous experiment. Among other exciting possibilities one should also mention measurements of hotly debated parton energy loss, using Drell-Yan dimuon production 6 . Besides very high statistics, the values of x-i values are higher at JHF than at Fermilab (Fermilab experiment 8 E866), which would enable a much cleaner analysis of the data.
191
3. Leading Twist Nuclear Shadowing and Antishadowing of Singlet P D F s In Figs. 1 and 2 we presented our predictions for the singlet (antiquark and gluon) nPDFs. Inspecting Fig. 2 one notes that the ratio gA/(AgN) < 1 for 4 10~ < x < 0.03, which means that the gluon distribution is depleted, or shadowed, in nuclei. For 0.03 < x < 0.2, gA/(AgN) > 1, which represents an enhancement, which is termed antishadowing. Both effects are characteristic to small x and are believed to arise from the coherent (simultaneous) interaction of the projectile with several nucleons in the target. In what follows we shall give a brief account of the derivation of our leading twist result. For more details, we refer the reader to the original publication 2 . Our leading twist approach to nuclear shadowing in DIS on nuclei is based on the space-time picture of hadron-deuteron scattering developed by Gribov 9 . It was observed that at high energies the nuclear shadowing correction to the total cross section, which arises from the simultaneous interaction of the projectile with both nucleons, is dominated by the excitation of diffractive intermediate states. Ultimately this enables one to relate nuclear shadowing in hadron-deuteron scattering to diffraction in hadron-nucleon scattering. These ideas can be generalized to lepton DIS on any nucleus 10 . Moreover, using the QCD factorization theorems for inclusive and hard diffractive processes, nuclear shadowing can be formulated for each nuclear parton distribution, fj/A, separately. Introducing the shadowing correction Sfj/A as Sfj/A = fj/A - Afj/N, we obtain Sfj/A =
A i
\
^lQxRe
/•OO
(1-itf
P dxrfflrl(P,Q'>,xr,0)
/»00
/
pA{b,z^) pA{b,*,) u emerges naturally since the probability of finding the Fock state \TIK+) is much larger than that of | A + + T T _ ) . For the strange content of the nucleon sea, the important baryon-meson components are AA' and EA'. The non-perturbative contributions to the strange and anti-strange distributions in the proton can be written as 1 1 dy
fBK{y)sBC-)
Jx
y
(4) y
*"(*) = fdJ-fKB{y)sK{X-),
(5)
Jx y y where /sjf(y) = / K B ( 1 — y) (y and 1 — y being the longitudinal momentum fractions of the baryon and meson) are the fluctuation functions, and s (B = A, E) and sK are the s and s distributions in the A (E) baryons and K meson, respectively. The fluctuation functions are calculated 11 from an effective meson-baryon-nucleon interaction Lagrangian using time-ordered perturbation theory in the infinite momentum frame, _ 9NBK P (»)-l6^-y0
*MCM,.A
fBK
dk
l G%K(y,kl) {ymN-mB)2 , ( 1 _ y) {m% - m%KY y
+ kl '
(6)
where gNBK is the effective coupling constant; mBK — (m2B + k]_)/y+ (m2K + &j_)/(l—y) is the invariant mass squared of the BK Fock state; and GBK{V^ k\) is a phenomenological vertex form factor for which we adopt an exponential form GBK{y, fc±) = exp
N-mBK(y,kl) 2A?
(7)
with Ac being a cut-off parameter. From Eqs. (4) and (5) we know that the non-perturbative contributions to s and s distributions in the nucleon are different, and the difference s — s depends on both the fluctuation functions (/BK ano> IKB) and the parton distributions in the baryon and meson (sB and sK). Due to the A (S) baryons being heavier than the K meson, fBx(y) is harder than fxB(y), which suggests giv y gN -m £ n e j a r g e x r e g i o n . On the other hand, the s distribution of the K meson (sK(x)) is generally believed to be harder than the s distribution of
198
the baryon (sB(x)) as the baryon contains one more valence quark than the meson, which implies sN < sN in the large x region. The final prediction of the s-s asymmetry will depend on these two competing effects. We employed the following two prescriptions for the strange and antistrange distributions in the A (£) baryons and K meson : (1) Use SU(3) symmetry for the parton distribution functions of the baryons, i.e. sA = s s = \uN where the MRST98 17 parameterization for uN is adopted, and GRS98 18 parameterization for s , which is obtained by connecting sK to the valence quark distribution in the pionic meson based on the constituent quark model; (2) Calculate the strange distributions in the bag model in order to take into account the SU(3) symmetry breaking effect in the PDFs of the baryons, sA ^ s s ^ \uN, which could be important 19 . The numerical results are given in Fig. 1. It can be seen that the two prescrip0.0015
--
0.001 0.0005 0
-0.0005 0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 1. x(s — s) versus x. The solid and dotted curves are the results using prescription (1) and (2) for sB and SK, respectively.
tions give very different predictions for the s-s asymmetry. The calculations also depend on the cut-off parameter Ac introduced in Eq. (7). The baryon and meson production data can impose some constrains on this parameter and a value of about 1.1 GeV is preferred 20 . We studied the dependence on this parameter by allowing Ac to change from 0.8 GeV to 1.5 GeV. The probabilities of finding the AA' (£A') Fock states in the proton wavefunction change from 0.18% (0.09%) to 5.4% (3.5%). Consequently the independence and the magnitude of the s-s asymmetry change dramatically (see Fig. 2). The Ac-dependence of (Ss) is plotted in Fig. 3. (6s) depends on Ac much less dramatically than the difference x(s — s) because of the cancellation
199 SU3
0
0.1
0.2
0.3
Bag
0.4
0.5
0.1
0.6
0.2
0.3
0.4
0.5
0.6
Figure 2. x(s — s) versus x. The solid, dashed and dotted curves are the results with A c = 1.5, 1.08 and 0.8 GeV, respectively.
0.8
0.9
1.1
1.3
1.5
Figure 3. (Ss) versus A c . T h e dotted and solid curves are the results using prescription (1) and (2) for s and s , respectively.
between the contributions in the small x and large x regions (see Fig. 2). (Ss) changes from being positive to negative with the increase of the cut-off Ac and has a maximum at about Ac = 1.1 GeV. So the theoretical estimates for (Ss) lie in the range from -0.0005 to 0.0001, which is about an order of magnitude smaller than that is needed to explain the NuTeV anomaly. 3. C h a r g e S y m m e t r y B r e a k i n g It has been generally believed that charge symmetry is highly respected in the nucleon system. However some theoretical calculations 12 have suggested that the charge symmetry breaking (CSB) in the valence quark distributions may be as large as 2% - 10%, which is rather large compared to the low-energy
200
results. Any unexpected large CSB will greatly affect our understanding of non-perturbative dynamics and hadronic structure, and also the extraction of sin 2 6w from neutrino scattering. CSB in both the valence and sea quark distributions comes from nonperturbative dynamics. Thereby the meson cloud model (MCM), describing the non-perturbative structure of the nucleon, can provide a natural explanation of CSB in the valence and sea quark distributions of the nucleon 13 . The fluctuations we consider include TV —>• Nir and N —»• An. The baryons (mesons) in the respective virtual Fock states of the proton and neutron may carry different charges. If we neglect the mass differences between these baryons (mesons), the fluctuation functions for the proton and neutron will be the same, and the contributions to the parton distributions of the nucleon from these fluctuations will be charge symmetric. The electromagnetic interaction induces mass differences3, among these baryons (mesons) 21 , trip — f^n = —1-3 MeV, mAo - m A + = 1.3 MeV,
mn± — m„o = 4.6 MeV, mA- - mAo = 3.9 MeV.
(8)
Due to these mass difference the probabilities of corresponding fluctuations for the proton and neutron may be different, thereby the contributions to the PDFs of the proton and neutron from these fluctuations may be different. The numerical result for CSB in the valence quark sector is given in Fig. 4. We find that xSdv and xSuv have similar shapes and both are negative, which °T •S § -0.0002a* u xSdv f
-0.0004-
>
-0.0006-
x5uv
u -0.0008 -• 0
0.1
0.2
0.3
0.4
0.5
X
Figure 4.
Charge symmetry breaking in the valence sector.
is quite different from the quark model prediction 12 for x6dv being positive for a
Unlike the other mass splittings, the splitting between charged and uncharged pion masses is not a violation of charge symmetry - it is a violation of isospin symmetry.
201
most values of x. Our numerical results are about 10% of the quark model estimates 12 . We did not find any significant large-x enhancement of the ratio Rmin = 8dv/d%, which is predicted in the quark model calculations 12 . The smallness of any CSB effect as x -^ 1 is natural in the MCM, as all the fluctuation functions go to zero as y —>• 1, and hence there is no non-perturbative contribution to the parton distributions at large x. We estimate (Su)—{Sd) is about —0.0003, which is much smaller than required to explain the NuTeV anomaly. Considering both s-s symmetry breaking and charge symmetry breaking we have (Su) — (Sd) — (5s) ca 0.0002 0.0004, which is an of magnitude order smaller than needed to explain the NuTeV discrepancy. 4. Summary The NuTeV measurement of sin2 6W being 0.2277 ±0.0013(stat.)±0.0009{sy si.) is about 3 standard deviations above the standard model prediction. Possible symmetry breaking in the parton distribution functions of the nucleon, strangeantistrange symmetry breaking and charge symmetry breaking may affect the interpretation of NuTeV result. We reported theoretical calculations for these symmetry breakings using a meson cloud model, which could naturally explain flavour symmetry breaking in the nucleon sea (d > u). It was found that the corrections from the two symmetry breakings would not be significant enough to alter the NuTeV result. Moreover there is no established experimental evidences for or against quark-antiqu ark symmetry and charge symmetry in the parton distribution functions of the nucleon. More studies are needed to explain the NuTeV anomaly. Acknowledgments This work was partially supported by the Science and Technology Postdoctoral Fellowship of the Foundation for Research Science and Technology, and the Marsden Fund of the Royal Society of New Zealand. References 1. G. P. Zeller et al, NuTeV Collaboration, Phys. Rev. Lett. 88, 091802 (2002). 2. 3. 4. 5. 6.
E. A. Paschos and L. Wolfenstein, Phys. Rev. D7, 91 (1973). G. A. Miller and A. W. Thomas, hep-ex/0204007. S. Davidson et al., hep-ph/0112302. G. P. Zeller et al., NuTeV Collaboration, hep-ex/0203004. E. Ma and D. P. Roy, Phys. Rev. D65, 075021 (2002); S. Barshay and G. KreyerhofT, Phys. Lett. B535, 201 (2002); C. Giunti and M. Laveder, hepph/0202152.
202 7. 8. 9. 10. 11. 12.
13. 14. 15. 16. 17. 18. 19. 20. 21.
A. O. Bazarko et al., CCFR Collaboration, Z. Phys. C65, 189 (1995). M. Goncharov et al., NuTeV Collaboration, Phys. Rev. D64, 112006 (2001). V. Barone, C. Pascaud and F. Zomer, Eur. Phys. J. C12, 243 (2000). A. I. Signal and A. W. Thomas, Phys. Lett. B191, 205 (1987); S. J. Brodsky and B. Q. Ma, Phys. Lett. B381, 317 (1996). F.-G. Cao and A. I. Signal, Phys. Rev. D60, 074021 (1999). For a recent review see J. T. Londergan and A. W. Thomas, in Progress in Particle and Nuclear Physics, Volume 41, P. 49, ed. A. Faessler (Elsevier Science, Amsterdam, 1998); E. Sather, Phys. Lett. B274, 433 (1992); E. Rodionov, A. W. Thomas and J. T. Londergan, Mod. Phys. Lett. A9, 1799 (1994). F.-G. Cao and A. I. Signal, Phys. Rev. C62, 015203 (2000). E. A. Hawker et al., E866/NuSea Collaboration, Phys. Rev. Lett. 80, 3715 (1998); For a recent review see J.-C. Peng and G. T. Garvey, hep-ph/9912370. A. W. Thomas, Phys. Lett. B126, 97 (1983). J. D. Sullivan, Phys. Rev. D5, 1732 (1972). A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thome, Eur. Phys. J. C5, 463 (1998). M. Gliick, E. Reya and M. Stratmann, Eur. Phys. J. C2, 159 (1998). C. Boros and A. W. Thomas, Phys. Rev. D60, 074017 (1999). H. Holtmann, A. Szczurek and J. Speth, Nucl. Phys. A569, 631 (1996). D. E. Groom et al., Particle Data Group, Eur. Phys. J. C15, 1 (2000).
N U C L E O N S AS RELATIVISTIC T H R E E - Q U A R K STATES
M. OETTEL Max-Planck~Institut fur Metallforschung Heisenbergstr. 3, 70569 Stuttgart, Germany E-mail: [email protected]
A covariant nucleon model is formulated which uses dressed quarks compatible with recent lattice d a t a and Dyson-Schwinger results. Two—quark correlations are modelled as a series of quark loops in the scalar and axial vector channel. Faddeev equations for nucleon and delta are solved and nucleon form factors are calculated in a fully covariant and gauge-invariant scheme. The insufficience of a pure valence quark description becomes apparent in results for the electric form factor of the neutron and the ratio HGE/GM for the proton.
1. Introduction to the Model Virtually all intricacies and complications of QCD are present in the analysis of the simplest baryonic bound state, the nucleon. Spectroscopy suggests that it is mainly composed of three confined constituent quarks of mass ~ 0.33 GeV. Probing its structure in deep inelastic scattering (DIS) reveals a completely different picture: Depending on the energy of the incoming photon, the proton appears to be a complicated mixture of a three-quark valence core supplemented with higher qq and gluonic components. To be precise, such a Fock-state picture is rigorously valid only in the light-cone gauge, and the quark states should correspond to the nearly massless current quarks from the QCD Lagrangian. From both points of view, the nucleon is a complicated relativistic bound state which calls for a covariant description since already the simplest processes which are described by the various nucleon form factors involve considerable momentum transfers. It has become common wisdom that massive constituent quarks are generated by the spontaneous breaking of the approximate chiral symmetry of QCD. Assuming that the gluon propagator and the quark-gluon vertex possess enough integrated strength, Dyson-Schwinger (DS) equation studies 1 ' 2 have predicted a running quark mass which is about the constituent quark mass in the infrared and drops to the current mass in the ultraviolet. This has been confirmed on the lattice 3 ' 4 . Although all these results have been obtained
203
204
0
02
0.4
0.6
1
0.8
1.2
1.4
1.6
1.8
rfm
Figure 1. Lattice results for the static three-quark potential. The d a t a points for the potential (little dashes with error bars) are compared to static, string-like potentials of Y and A type between the three quarks. (Adapted from Ref. [8].)
in Landau gauge and not in the light-cone gauge, we may assume that a good part (though by no means all) of the complicated Fock state structure of the proton is buried in the structure of the constituent quark. Covariant bound states wave functions may be obtained by use of the covariant Bethe-Salpeter (BS) equation. The properties of pseudoscalar mesons are described successfully in combined DS/BS-studies 2 which emphasize their dual role as both q — q bound states and Goldstone bosons. Along these lines the nucleon's bound state amplitude can be obtained by solving a relativistic Faddeev equation if irreducible three-quark correlations are neglected. This is a strong assumption which has been tested on the lattice only for static quarks, see Fig. 1. The Faddeev equation needs as input the full solution for the q — q scattering kernel (whose determination is itself almost an unsurmountable task). Truncating the interaction between the two quarks to lowest order (one effective gluon which generates the constituent quark mass) one finds diquark poles with scalar (0 + ) diquarks (RS 0.7 — 0.8 GeV) and axial vector (1 + ) diquarks (m 0.9 — 1.0 GeV) having the lowest masses 2,5 . Although the poles may not survive in higher orders 6 the dominance of scalar and axial vector correlations in the q — q matrix has also be seen on the lattice (in Landau gauge), see Fig. 2. Despite these signals for pronounced q — q correlations, no evidence for compact (i.e. pointlike) diquarks has been seen in DIS or other processes.
205 40
i
••
p(o>)
I 1
35 30
.
25
'
1
-i
1
1
scalar (303) diquark axial vector (613) diquark
! "mass" peak K = 0.147
20 15
.
10
•
5
K
(a) 0
0.5
1
/ 1.5
continuum ••-.,. 2.5
0)[GeV] 3.5
Figure 2. Lattice results for the spectral function in the colour antitriplet (3) q — q channel. There is an almost perfect mass peak in the scalar diquark channel while for the axial vector diquark the mass peak and the continuum are of equal importance. The spectral functions of other diquarks is of no significance. (Adapted from Ref. [7].)
Putting all these motivations together, we formulate a covariant model which incorporates 9 • quark propagators with a running mass function in accordance with DS and lattice results. A parameterization in terms of entire functions is used which has been fitted to a number of low-energy meson observables 10 . Although the general shape agrees with recent lattice data a (see Fig. 3) the latter suggest a somewhat broader mass function. The relevance of this for the electromagnetic form factors will be discussed. • separable q — q correlations truncated to the 0 + and 1 + channels, see Fig. 3. Their parameterization involves diquark vertex functions x> which are described by their dominant Dirac structure, and a scalar function F, which assigns a width wo+[i+] to the diquarks. The diquark propagator is obtained as a sum of quark polarization loops. An external probe (e.g. a photon) will resolve the quark substructure of the diquarks by coupling to the quarks in the loop. • strict covariance in solving the resulting Faddeev equations. Thanks to the separability assumption the Faddeev equations reduce to an a
N o t e t h a t the fit has been obtained some years earlier than lattice d a t a became available.
206
A —
" Z 2€>=CC
Lattice, m ^ = 91 MeV u/d quark parametrizatkm m K . . = 5MeV
a=/0+,l+}
= A,, */"'"
/=Po+(75C)F(»o+)
Figure 3. Left panel: Parametrized quark mass function compared to lattice d a t a from Ref. [3]. Right panel: The separability assumption for the q — q t matrix.
effective diquark-quark Bethe-Salpeter equation where the attractive interaction is effected through the quark exchange between the quark and diquark, see Fig. 4. With the masses of the nucleon and delta as input parameters, we can vary only the width of the scalar diquark which in turn determines the diquark masses through the zeros of the inverse diquark propagators. The rich structure of the BS wave functions and the numerical procedure to solve the BS equation can be found in Refs. [11,12] respectively. Exemplary width parameters for which a solution can be found together with the resulting diquark masses are given in Table 1. We see that for a scalar diquark width of ~ 0.3 fm the diquark masses are in agreement with the position of the lattice peaks. Smaller widths give approximately equal masses for the scalar and axial vector diquark while larger widths tend to increase the mass difference. We remark that the width of the nucleon BS wave function (which can be interpreted as a quark-diquark separation) is of about 0.4 fm. On the one hand, this indicates no strong diquark clustering (for which there is no evidence anyway). On the other hand the diquark width should not exceed the quark-diquark width for
a,i={0, f } Figure 4. The Bethe-Salpeter equation for the effective baryon-quark-diquark vertex function »o+[i+l.
207
our diquark-quark picture to make sense. Table 1. Parameter sets which give a solution to the BS equation with the physical mass of nucleon and delta. Set I II III
w0+ wl+ -fm0.34 0.25 0.28 0.17 0.25 0.14
m0+ 0.75 0.80 0.86
TOj+ Mff -GeV0.92 0.89 0.94 0.87
M&
1.32
2. Electromagnetic Form Factors The so-called gauging method 13 gives the correct, gauge invariant prescription how to treat processes where an external photon couples to a bound state whose wave function in turn is described by the solution of an integral equation. It basically consists of coupling the photon to all momentum dependent terms (propagators, vertices) in the kernel of the integral equation. Consequently we have impulse approximation diagrams where the external photon couples to the spectator quark and the quarks within the diquark. Inspecting the kernel of the BS equation, we find furthermore diagrams where the photon couples to the exchange quark and directly to the quark-diquark vertices (seagull graphs). The most important diagram, though, is the impulse approximation quark diagram, and its central element is the quark-photon vertex. It consists of a longitudinal and a transverse part,
r£ = r£ L + r £ T )
(l)
where the longitudinal part T^ L (the Ball-Chiu vertex), which is fixed by the Ward-Takahashi identity, is entirely determined by the form of the quark propagator13. The remaining transverse part might receive dynamical contributions of which the p — w meson poles in the q — q vector channel are presumably the most important ones. The most thorough study of these contributions has been made in Ref. [14] where the quark-photon vertex is analysed in the DS/BS framework. It was shown/that the transverse vertex contributes about one half to the pion charge radius and that it is essential for reproducing the b
T o be precise, the Ball-Chiu vertex contains a fixed transverse part since it should be free of kinematical singularities.
208
Figure 5. Electric form factor of the proton (left panel) and neutron (right panel). T h e small dotted curve in the right panel refers to a calculation from Ref. [15], assuming an electromagnetically compact (pointlike) diquark.
experimental results for the pion form factor. Following the analysis of Ref. [14] we write for the transverse vertex r
I * r f = ^ Q ^ ^ ^
W = *-P).
(2)
Here the BS wave function of the vector meson ^ is modelled by its two dominant structures. It is properly normalized and reproduces the experimental decay constant fp. The exponential describes an off-shell damping of the vector meson propagator, and we fit the constant a to the pion form factor, for details see Ref. [15]. The resulting electric form factors for proton and neutron are plotted in Fig. 5. We find for all sets that the transverse part of the quark-photon vertex contributes about 25% to the proton electric charge radius. Not surprisingly the form factor becomes softer with increasing diquark size which needs to be above 0.3 fm to bring the theoretical curve close to the data. Turning to the neutron, we see that all sets give a positive GE (a relativistic effect due to lower components in the BS wave function) but miss the experimental low-Q 2 behaviour completely. For comparison we have given results from a calculation with pointlike diquarks which are roughly compatible with the data. It appears that the latter assumption mimics non-valence effects that may arise if e.g. a pion-cloud is added to the nucleon. A pion-cloud would not only influence neutron's GE but also induce a mass shift for the nucleon between 16 200 and 300 MeV and somewhat less for the delta. We therefore solved the BS equations for higher core masses of the nucleon and delta and found quantitatively little difference to the old form factor results. Especially
209
•
Jones etal.PRL 84(2000) 1398
0.1 0I
0
Figure 6.
,
1 0.5
Proton's ratio HGE/GM
,
1 1
.
1 1.5
,
1 2
in comparison with the experimental d a t a .
neutron's GE remains quenched. Thus, a covariant calculation of pionic effects would clearly be desirable. For realistic diquark sizes, proton's ratio /J,GE/GM is underestimated in our calculations. This discrepancy can be traced back to the structure of the Ball-Chiu vertex. After some reshuffling of its tensor structure 9 one can isolate a transverse term which is proportional to the difference in the quark mass function for the outgoing and the incoming quark, i.e. ~ [M(k2) — M(p2)]/[k2 — p 2 ] . This term (which is absent in c o n s t a n t - m a s s constituent models) produces quite large negative contributions to GE for intermediate Q2 and therefore quenches the ratio. To investigate this in more detail, we replaced the fit of the running quark mass in Fig. 3 by a fit to the chiral extrapolation of latest lattice d a t a 4 , see the left panel of Fig. 7. Replacing the quark mass in the Ball-Chiu vertex by (a) the new fit and (b) by a constant we find results for (IGE/G/M which are depicted in the right panel of Fig. 7. Surprisingly enough the constant constituent mass does the best j o b but also the new fit to the lattice d a t a causes some noticeable change compared to the results with the original meson fit. Since the running quark mass is now well established and should be an element of any serious nucleon model we therefore conclude t h a t the observed ratio is most likely a consequence of a subtle interplay of various contributions such as a realistic quark propagator, vector mesons in the q u a r k - p h o t o n vertex and possibly also pions from the cloud. Further work which analyses these contributions in much more detail t h a n could be done here is clearly required. To summarize, we have investigated a covariant nucleon model involving three constituent quarks whose propagators capture the essential features of lattice simulations and DS calculations. Three-quark irreducible interactions
210
p[GeV]
o*[GeV*]
Figure 7. Left panel: lattice d a t a and the two fits to the quark mass function. Right panel: the ratio I*GE/GM for calculations with different quark mass functions in the Ball-Chiu vertex. The parameter for the scalar diquark width is here w0+ = 0.30 fm which results in the diquark masses rn0+ = 0.77 GeV and mj+ = 0.91 GeV.
were discarded and the problem was reduced to an effective q u a r k - d i q u a r k problem. T h e form factor results, especially for neutron's GE, point to the necessity of incorporating meson cloud contributions. T h e observed ratio [IGE/GM eludes a simple interpretation and is possibly a consequence of a number of mechanisms.
Acknowledgments T h e a u t h o r wants to thank the Special Research Centre for the Subatomic Structure of M a t t e r (CSSM) in Adelaide where most of this study was conducted. He is also grateful to the Alexander-von-Humboldt foundation which supported this research by a Feodor-Lynen grant. Special thanks go t o Reinhard Alkofer with whom the work was done and to Tony T h o m a s for m a n y insightful remarks.
References 1. R. Alkofer and L. von Smekal, Phys. Rep. 353, 281 (2001). 2. C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45, SI (2000). 3. J. Skullerud, D. B. Leinweber and A. G. Williams, Phys. Rev. D64, 074508 (2001). 4. P. O. Bowman, U. M. Heller and A. G. Williams, hep-lat/0203001. 5. P. Maris, nucl-th/0204020. 6. A. Bender, W. Detmold, C. D. Roberts and A. W. Thomas, nucl-th/0202082. 7. C. Alexandrou, P. De Forcrand and A. Tsapalis, Phys. Rev. D65, 054503 (2002). 8. I. Wetzorke and F. Karsch, hep-lat/0008008. 9. M. Oettel and R. Alkofer, hep-ph/0204178. 10. C. J. Burden, C. D. Roberts and M. J. Thomson, Phys. Lett. B371, 163 (1996).
211 11. M. Oettel, G. Hellstern, R. Alkofer and H. Reinhardt, Phys. Rev. C58, 2459 (1998). 12. M. Oettel, L. von Smekal and R. Alkofer, Comp. Phys. Comm. 144, 63 (2002). 13. A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C60, 044003 (1999). 14. P. Maris and P. C. Tandy, Phys. Rev. C 6 1 , 045202 (2000). 15. M. Oettel, R. Alkofer and L. von Smekal, Eur. Phys. J. A 8 , 553 (2000). 16. M. Oettel and A. W. Thomas, nucl-th/0203073.
(POLARIZED) H A D R O P R O D U C T I O N OF O P E N C H A R M AT THE JHF IN NLO QCD
I. B O J A K CSSM, University of Adelaide Adelaide, SA 5005, Australia E-mail: [email protected]
We present the complete next-to-leading order QCD corrections to (polarized) hadroproduction of heavy flavors and investigate how they can be studied experimentally in (polarized) pp collisions at the JHF in order to constrain the (polarized) gluon density. It is demonstrated that the dependence on the unphysical renormalization and factorization scales is strongly reduced beyond the leading order. We also briefly discuss how the high luminosity of the JHF can be used to control the remaining theoretical uncertainties.
1. Introduction Although we have gained precise information concerning the total quark spin contribution to the nucleon spin in the last decade, the spin-dependent gluon density Ag remains elusive, see Fig. 1. Hence current and future experiments focus strongly on the issue of constraining Ag. The JHF could play a prominent role in these efforts using production of open charm. The gluon participates are dominantly in heavy flavor pair creation in longitudinally polarized pp collisions. Figure 1 also shows that even the unpolarized gluon distribution has considerable uncertainties at large x, and hence could be pinned down by a corresponding unpolarized measurement. In leading order (LO), heavy flavor pair production in hadron-hadron collisions proceeds through two parton-parton subprocesses, gg^QQ
and
qq^QQ
.
(1)
Gluon-gluon fusion is by far the most dominant mechanism for charm and bottom production in the unpolarized case in all experimentally relevant regions of the phase space 1,2 ' 3,4 . This should hold true in the polarized case unless Ag is very small. However, it is necessary to include next-to-leading order (NLO) QCD corrections for a reliable description. The LO results depend strongly on the arbitrary factorization and renormalization scales. Furthermore, in the unpolarized case the NLO corrections are known to be large 1,2 . Finally, new
212
213
1.2 1 0.8 0.6
xAG(x)
NLO
5 GeV2 x G ( x ) (GRV)
0.4 0.2 0 10
1 0 —x-
^T
1 0 "*
1
x Figure 1. Uncertainty in Ag (shaded la error bands) and in g at large a; (hatched). This is a copy of Fig. 5 of t h e analysis of Bliimlein and Bottcher 7 (BB).
processes with a single light quark in the initial state contribute for the first time in NLO. Unpolarized NLO results 1 ' 2,3,4 and polarized LO expressions 5 ' 6 have been available before, but the complete NLO results are presented here for the first time.
2. Technical F r a m e w o r k The 0{oP$) NLO QCD corrections to heavy flavor production consist of the one-loop virtual and the real "2 —• 3" corrections, the latter include the new production mechanism
