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:>IF;,lUl.ll,.,CbUI..,C for
signal and backgrounds for the 8M Higgs though + ttVV for mH =120 GeV and integrated luminosity 30
50
3.2 Intermediate mass region (130 GeV< mH < 2mz) The most promising observation channel is the H -t Z Z* 4l. The main ba'C:K~[rOUna is the irreducible continuum which is known to NLO. The and tt backgrounds can be strongly reduced with isolation cuts in the calorimeter and by vertex constraints. For the region around 160 GeV the H has a very large, close to 100%, branching ratio. no mass reconstruction is possible because of the neutrinos and one has to on number counting and very precise knowledge of the The main backgrounds are the tt which can be with a mass and the WW continuum. Recently the gluon-gluon-t WW contribution has been and the single top production at the NLO leveL The VBF -t WW can also be added to obtain extra In VBF channels qqH -t qqT7 can be combined with the above two as contributes to about 3 a well. Each individual channel for a 130 GeV 1 ""ISHUIlCa,HC'''. All together give a 6 a significance at 10 fb- at 130 GeV.
Figure 5: The signal and the backgrounds for the H-+WW decay as a function of the transverse mass.
3.3
mass region (mH > 2mz)
The best discovery channel is the "golden H -t -t 41 chanwhere the background is very small due to the constraint of the two reconstructed masses to be both compatible with the Z-mass. For the very masses (mH >700GeV) the decays H -t WW -t ZZ lllllJ, H -t ZZ -t lljj, due to their will enhance the discovery sensitivity and compensate for the lower T"','V11""''' cross-sections. 6 shows the overall sensitivity covering the full mass range. lnt',PU,."j,,,,rl luminosity corresponding to one year of low data takline) almost the full range is covered, provided that the detector
51
is optimal and well understood, Figure 6 summarizes the discovery potential for ATLAS as was calculated already in 2003 based on LO cross-sections. The update of this plot is expected soon.
Figure 6: The signal significance for a SM Higgs boson in ATLAS as a function of the Higgs mass for two different integrated luminosities.
4
SUPERSYMMETRIC HIGGS SEARCHES
In most common supersymmetric scenarios, the Minimal tiupel:Symrnet;nc tension of the SM (MSSM), five bosons are odd), . The discovery limits are commonly in terms of uc;"uuU5 a where contours in mA and tan/3 are drawn. The LEP2 results have excluded a of the plane (the low tan/3 region). In addition to the SM channels described above, the most DH)mlSlIlIl: rf'"'NY""""" V,uo,.,U",.", in the tan/3 region, are the J-LJ-L channels and the charged decays. The - t 'T'T a rate compared to the J-LJ-L one, but it is more reconstructed. On the other hand, the J-LJ-L one produces a very clean O';
'"
t!l
3.5-
ATLAS
3
0
~
t1 ~
UJ
2.5 2" 1.5
0.5 2000 Invariant Mass (GeV)
2000 Invariant Mass (GeV)
Figure 3: Invariant mass reconstruction for an up type iso-singlet quark of mass 1 TeV using 300 fb- 1 integrated luminosity, Wb on the left and Zt on the right side.
New Leptons
New leptons, L, appear in various models [5,6,10]. The work in [11] concentrates on the lepton pairs produced from quark annihilation and from gluon fusion to a quark triangular loop. In both cases, the s channel contains the Z boson and a possible Z' as propagator. The first one can also propagate with a "y. The considered decay mode is L -> Zp, / Ze. The search was performed as a function of the new lepton and new heavy neutral gauge boson (Z') mass. The experimental reach is given in Fig. 4 for a Z' mass of 700 GeV. The lower (upper) curve is the reach for 10 (100) fb- 1 of integrated luminosity. A Z' of 2 Te V would increase the 5 cr reach from 800 Ge V to 1 Te V for 100 fb -1. Leptoquarks
Leptoquarks (LQ) are predicted by GUT and composite models. The study in [12] considered their pair production from gluon fusion and quark annihilation. The same work also covers the single production. The decay modes consist of electrons (type-I) or neutrinos (type-2) and a light jet. For both scalar and vector LQs, the mass scan was performed for different coupling coefficients.
59
'I8
7
200
eoo
6OC:
4(}O
lDC{1
1200
1400
M,
Figure 4: L reach as a function of its mass, low (high) luminosity corresponds to 10 (100) fb- 1 of integrated luminosity.
Fig. 5 summarizes the reach for 300 fb- 1 , showing that about 1.2 (1.5) TeV LQ can be discovered for scalar (vector) leptoquark models. til
105'~
~
_________
~
ATLAS
fLdt=3x10'W'
10 VlQType 1 _SLQTypel VLQ Type 2 SLQ Type 2
10
10
400
600
800
1000
1200
1400
1600
1800
Leptoquark mass, GeV
Figure 5: LQ reach as a function of its mass for 300 fb- 1 of integrated luminosity.
2.2
Searches for new gauge group structure
Embedding the 8M gauge group into a larger one brings additional gauge bosons, both neutral (Z/) and charged (WI). Additionally they appear in models with extra-dimensions (ED) as the Kaluza-Klein (KK) [13] excitations of their 8M counterparts.
60
Neutral gauge Hosons
A full GEANT MC simulation study was performed to investigate the Zl discovery potential of ATLAS using a generic parameterization called CDDT [14]. The CDDT parameterization classifies Zl searches into four distinct cases, depending on its coupling to the known fermions. In this study, a 1.5 and 4 TeV Zl produced by quark anti-quark annihilation was allowed to decay into e+epairs. The left side of Fig. 6 shows the ATLAS reach for 100 fb- 1 of integrated luminosity as a function of the ratio of the new gauge boson and its gauge coupling strength (Mz/g z ) and fermion coupling modification parameter (x) . A recent study investigated the discovery reach of the KK excitations of the Z boson, zn [15]. This model uses different parameterizations (A,B,C) to reproduce the known fermion masses and mixings. The right side of Fig. 6 shows the reconstruction of the zn invariant mass from e+e- pairs using a full GEANT based simulation together with the SM Drell-Yan background. The zn discovery reach for 100 fb- 1 integrated luminosity is up to 6 TeV, depending on the parameterization.
Figure 6: Left: generic Zl search with CDDT parameterization; Right: Zl invariant mass reconstruction for different fermion parameterizations. In both cases results with 100 fb- 1 are shown.
Charged gauge Hosons
Additional charged gauge bosons, W', appear in GUT, Little Higgs and ED models [6,8,18]. The quark anti-quark annihilation produces the W' that can be studied via its hadronic [16] or leptonic [17] decays. The important parameters are the W - W' mixing angle (cot B) and the mass of the W'. Fig. 7 shows the discovery reach for the WH search (Little Higgs model) , for 300 fb- 1 integrated luminosity in the cotB - mWH plane for the WH - t tb and W H - t ev modes.
Figure 7: WI discovery reach plane. The shaded area is from hadronic decay channel, dashed line is from electron decay.
Searches
new Electro-weak symmetry
mechanisms
variants of
Scalars
o
proposed for fermion and boson mass Cf""~"T·"i'·I"Yl The studies in the context F."'Lv"", and scalars can predicted by both Little
,"U(U
additional vector bosons.
'~l~"'r"" between the matter and force carrier solve the and fine tuning problems a candidate for the Dark Matter (DM) searches of
62
14()O
Figure 8: tl,±± search reach for single production on the left and for pair production on the right.
are expected to cascade decay down to the particle (LSP), the n jets + m leptons + channels are inv'es1;igl'1ted. The large number of free parameters can be reduced to 5 in case of [23] which proposes its LSP, the lightest neutralino, as the DM candidate. the reduced parameter set should also be consistent 9, left). A recent work has investigated the reach of with WMAP data pair production in the focus point scenario and ATLAS for and [24J. The result of the study is shown in background subtraction, for 10 fb- 1 of integrated significance.
""'''''T''.;"nrnn,,,t-,'l,,
Figure 9: Left: parameter space in mSUGRA. The medium gray region is consistent with WMAP data. Right: reconstructed 9 visible invariant mass for 1 fb- 1 of data.
63
2.4
Searches for new Dimensions
If the relative weakness of the gravitational force is attributed to the existence of extra dimensions (ED), the graviton becomes the object to search for. The graviton couples to all particles and can escape undetected. The most promising channels are gluon-gluon, quark-gluon fusion and quark anti-quark annihilation yielding one jet + missing E T . The experimental reach depends on the number of EDs and also the fundamental gravity scale. A study has shown that, for 100 fb- 1 of integrated luminosity, the reach would be about 9, 7 and 6 TeV, for 2, 3 and 4 additional dimensions [25]. Large (Te V- 1 ) EDs appearing in ADD models [26] predict KK excitations of gluons, g*, which would decay into heavy quark anti-quark pairs. A study evaluated the reach of ATLAS for the decay into bb and tf pairs [27]. Depending on the mass of the g*, it is possible to discover a g* with mass up to 3.3 TeV with an integrated luminosity of 300 fb- 1 • 3
Results and Conclusions
Although this note summarized only a selection of discovery possibilities, it has shown that ATLAS has a very rich discovery potential for physics beyond the SM. The differentiation between models and the possible boost to SM process cross sections from the particles proposed by the BSM physics were also not discussed. The preparation of the experimental apparatus for data taking is well underway, the new analyses with full simulation are also ongoing. These studies will immediately be applicable to first data from LHC. Acknowledgments
The author would like to thank A. Studenikin for his hospitality in Moscow and F. Ledroit and A. Parker for useful discussions. G.U.'s work is supported in part by U.S. Department of Energy Grant DE FG0291ER40679. References
[1] ATLAS Detector and Physics Performance Technical Design Report. CERN/LHCC/99-14/15. [2] LRiu, ATL-SLIDE-2007-05, proceedings of 15th IEEE Real Time Conference (2007). [3] E.J.Eichten, KD.Lane and M.E.Peskin Phys. Rev. Lett. 50, 811 (1983); L.Abbot, E.Farhi Phys.Lett., B 101,69 (1981). [4] A.Belyaev, C.Leroy, RMehdiyev, Eur.Phys.J. C 41, 1 (2005). [5] B.Holdom, JHEP 0608, 076 (2006); B.Holdom, JHEP 0703,063 (2007).
64
[6] F.Gursey, P.Ramond and P.Sikivie, Phys.Lett. B 60, 177 (1976); F.Gursey and M.Serdaroglu, Lett. Nuovo Cimento 21, 28 (1978). [7] RMehdiyev et al., Euro.Phys.J. C. 49, 613 (2007). [8] M.Schmaltz, Nucl.Phys. B 117, 40 (2003). [9] G.Azuelos et al., Euro.Phys.J. C. 39, Suppl.2, 13 (2005). [10] S.Dimopoulos Nucl.Phys. B 168, 69 (1981); E.Farhi, L.Susskind, Phys.Rev. D 20, 3404 (1979); J.Ellis et al., Nucl.Phys. B 182, 529 (1981). [11] C.Alexa, S.Dita, ATL-PHYS-2003-014 (2003). [12] A.Belyaevet al., J.H.E.P. 09, 005 (2005). [13] E.Witten, it Nucl.Phys. B186, 412 (1981). [14] F.Ledroit, B.'frocme, ATL-PHYS-PUB-2006-024, proceedings of TeV 4 LHC workshop, (2006). [15] F.Ledroit, G.Moreau, J.Morel, J.H.E.P. 09,071 (2007). [16] Gonzlez de la Hoz, S; March, L; Ros, E; ATL-PHYS-PUB-2006-003. [17] G.Azuelos et al., Eur.Phys.J. C 39,13 (2005). [18] L.Randall, RSundrum, Phys. Rev. Lett. 83, 3370 (1999). [19] LF.Ginzburg, M.Krawczyk, Phys.Rev. D72, 115013 (2005). [20] G.Azuelos, K.Benslama, J.Ferland, J.Phys. G 32, 73 (2006). [21] Y.Hosotani, Phys.Lett. B 126, 309 (1983) ; B.McInnes, J. Math. Phys. 31, 2094 (1990). [22] H.P.Nilles, Phys. Rev. 110, 1 (1984) and references therein. [23] A.H.Chamseddine, RArnowitt and P.Nath, Phys. Rev. Lett. 49 (1982) 970; H.P.Nilles, Phys.Rept. 110 (1984). [24] U.De Sanctis, T.Lari, S.Montesano, C.'froncon, arXiv:0704.2515, SNATLAS-2007-062, Eur.Phys.J. C52, 743 (2007). [25] L.Vacavant, LHinchliffe, J.Phys. G 27, 1839 (2001). [26] N.Arkani-Hamed, S.Dimopoulos, G.Dvali, it Phys.Lett. B 429, 96 (1998). [27] L.March, E.Ros, B.Salvachua, ATL-PHYS-PUB-2006-002 (2006).
THE STATUS OF THE INTERNATIONAL LINEAR COLLIDER B. Foster" Department of Physics, University of Oxford Denys Wilkinson Building, Keble Road, Oxford, OXl 3RH, UK Abstract. The status and prospects for the International Linear Collider are summarised.
The International Linear Collider (ILC) is accepted by the international community of particle physicists as the next major project in the field. The Global Design Effort (GDE), directed by Professor Barry Barish, has been charged by the International Committee for Future Accelerators with the task of preparing all necessary design and documentation to present a fully cos ted and robust design to the funding authorities by 2010, at which point the status of the Large Hadron Collider and other relevant information will be available to allow an informed decision on construction. The current status of the project is that the Baseline Design of January 2006 has been used as the basis for a greatly developed and refined reference design, summarised in the Reference Design Report [1], which in additional to descriptions of the major parts of the project, also gives a costing in ILC units ( = US$ 1 on Jan. 1st 2007) together with an estimate of the labour necessary to realised the project. The RDR estimates are 6.62 Billion ILCUs plus 14,100 person-years of effort. The Reference Design is based around a central campus containing a single interaction hall capable of hosting two detectors in a "push-pull" configuration, a damping ring complex for both electrons and positrons, the electron source and other central services. The damped electrons from the source are transported to the end of the superconducting modules and accelerated to an energy of 250 Ge V; part of the way along their path they are diverted through a wiggler which produces hard photons which are converted to electron-positron pairs. The positrons are selected and transported to the damping ring, damped and then transported to the other end of the machine to be accelerated in their turn and then collided with electrons at the interaction point. The RDR assumes that an accelerating gradient of 31.5 MV1m can be attained by each super conducting accelerating module. Many technical developments and specifications remain to be completed. The goal of 31.5 MV1m, although reached in some cavities, has not yet been achieved with the reproducibility and the yield necessary for the industrial production for the ILC. There are many remaining technical questions in areas such as the damping rings and a whole process of value engineering, optimisation and cost containment and reduction is required. This will be carried out ae-mail: [email protected]
65
66
in a third phase of the project, under the supervision of the GDE, known as the Engineering Design Report. This will be supervised by a Project management Team currently being set up and will begin in the autumn of 2007. At the same time as the technical and engineering developments for the ILC are progressing, it is important to develop political institutions and to explain the importance of the physics of the ILC both to politicians, other scientists and to the general public. The mechanism by which the site for the ILC will be bid for and chosen needs to be investigated and defined. The GDE welcomes the strong interest recently evinced by JINR Dubna in proposing Dubna as a possible site for the ILC and the interest expressed by the Russian Federation in exploring this possibility. It is also necessary to propose models and reach agreement on how a fully international project such as the ILC can be managed and governed to ensure accountability and transparency for stake-holders. The accomplishment of the technical aims of the EDR phase resulting in a proposal to construct the ILC in 2010 could lead, if prompt approval were granted, to ground breaking in 2012 and operation by 2019, allowing a substantial period of operation overlapping that of the LHC. The GDE is committed to maintaining such a timeline, defined as it is by the available effort and likely technical progress, while in parallel assisting the resolution of political questions and preparing an atmosphere conducive to approval of the ILC project. Reference
[1] G. Aarons et al., International Linear Collider Reference Design Report, available from http://www.linearcollider.org/cms/?pid=1000025, August, 2007.
REVIEW OF RESULTS OF THE ELECTRON-PROTON COLLIDER HERA
v. Chekelian (ShekeJyan) a Max Planck Institute for Physics, Foehringer Ring 6, 80805 Munich, Germany Abstract. A review of results of the electron-proton collider HERA is presented with emphasis on the structure of the proton and its interpretation in terms of
QeD.
1
Introduction
In summer 2007, after 15 years of successful operation, the first and only electron-proton collider HERA has finished data taking. The HERA collider project started in 1985 and produced the first ep collisions in 1992. It was designed to collide electrons with an energy of 27.5 GeV with protons with an energy of 920 GeV (820 GeV until 1997). This corresponds to a center of mass energy of 320 GeV. The maximum negative four-momentum-transfer squared from the lepton to the proton, Q2, accessible with this machine is as high as 100000 GeV2 . Two ep interaction regions were instrumented with the multi-purpose detectors of the HI and ZEUS collider experiments. In 2002, placing strong super-conducting focusing magnets close to the interaction points inside the HI and ZEUS detectors, the specific luminosity provided by the collider was significantly increased. At the same time spin rotators were installed in the HI and ZEUS detector areas, and since then a longitudinal polarisation of the lepton beams of 30-40% was routinely achieved. Over the 15 years of data taking, each collider experiment collected an integrated luminosity of :::::: 0.5jb- 1 , about equally shared between positively and negatively polarised electron and positron beams. The HERA physics program covers a broad spectrum of topics such as searches of new physics, hadron structure, diffractive processes, heavy flavour and jet production, vector meson production and many others. In this paper I will concentrate on results related to neutral current (NC) and charged current (CC) deep-inelastic scattering (DIS) and the QCD aspects of these measurements at HERA. 2
Deep-inelastic NC and CC ep scattering
The deep-inelastic NC scattering cross section can be written as (1) ae-mail: [email protected]
67
68 HERA II H1 Quark Radius Limit HERA 1+11 (417 pb")
"*
H1 e+pNC03..(l4(prel.)
t.
H1 e'p NC 2005 (pre •. )
g ...
o zeus e+p NC 2004 o
1.4
. ___ .~ ___• __ .. ___ .i
"0 -C
..... 8M e+p NC (CTEQ6M)
_
~
~ 1.2
ZEUS e'p NC 04-05 (prel.
~
0.8
~ ~ ~ ~ ~~;~~::t~oa~n~ncertainty _
0_6
'*
H1 e+p CC 03-04 (pre'.)
....
H1 e'p CC 2005 (prel.)
•
ZEUS e+p CC 2004
•
ZEUS e'p CC O4-OS (pre •. )
Rq =O.74'lO,'B m (95%CL)
f-~~~~~-~~~~~-~;-------I
1_4
1_' -I
.• _-- 8M e+p CC (CTEQ6M) -
..,~_
8M e-p NC (CTEC6M)
8M e'p CC (CTEQ6M)
L
0_'
y < 0.9 Pe=O
10-7b...L~J...i~~10L3-~~---'-'-LLLl1o'~~----'--.d
Q'(GeV')
Q'(GeV')
Figure 1: The Q2 dependence of the NC and CC cross sections du/dQ2 for e±p scattering (left). The NC cross section normalised to the Standard Model expectation (right).
where a*,-c is the cross section in a reduced form, a is the fine structure constant, x is the Bjorken scaling variable, and y characterises the inelasticity of the interaction. The helicity dependence is contained in Y± = 1 ± (1 _ y2). The generalised proton structure functions, F2 ,3, occurring in eq.(l) may be written as linear combinations of the hadronic structure functions F 2 , Fi and F2~3' containing information on the QeD parton dynamics and the el~c troweak couplings of the quarks to the neutral vector bosons. The function H is associated with pure photon exchange term, F~f correspond to photon-Z interference, and F.f,3 correspond to pure Z exchange terms. The longitudinal structure function FL may be decomposed in a similar way. The generalised proton structure functions depend on the charge of the lepton beam, on the lepton beam polarisation, defined as P = (NR - N L) / (NR + N L), where N R (Nd is the number ofright (left) handed leptons in the beam, and on the electroweak parameters Mz and sin 2 () (or Mw):
f,
+ k( -Ve =t= Pae)Fi z + k2(V; + a; ± 2Pvea e )Fl, k( -a e =t= Pve)xF;z + k2(2veae ± P(v; + a;»xFl.
F2± = F2
(2)
XF3± =
(3)
Here, k( Q2)
=
4 sin2
~ cos2 (1 Q2~~2 determines the relative amount of z
Z to I
exchange, Ve = -1/2 + 2 sin () and a e = -1/2 are the vector and axial-vector couplings of the electron to the Z boson, and () is the electroweak mixing angle. At leading order in QeD the hadronic structure functions are related to linear combinations of sums and differences of the quark and anti-quark momentum 2
69 HERA Charged Current
*
* Hl e+p94-00
Hl e'p
a ZEUSe-p98-99
[J
ZEUSe+p99-00
-
SM e-p (CTE06D)
-
SM e+p (CTEQ6D)
Figure 2: The NC (left) and CC (right) double differential cross sections d 2uldxdQ2 in a reduced form for e+p and e-p DIS scattering.
distributions xq(x, Q2) and xij(x, Q2) of the proton: z (F2 , Fi , F2Z) = x ~)e~, 2eqvq, v; (xF:;z,xF3Z)
+ a~)(q + ij),
= 2x ~)eqaq,vqaq)(q -
(4)
ij),
(5)
where Vq and a q are the vector and axial-vector couplings of the light quarks to the Z boson, and e q is the charge of the quark of flavour q. The deep-inelastic CC cross section can be expressed as d2a6c 27l'x dxdQ2 G}
(jac
[MrvMrv+Q2]2 =__acdx,Q ± 2 _ )-
1
2'
(±
±
Y+W2 =F LxW3 -
Y2W L±) ' (6)
where is the cross section in a reduced form, G F is the Fermi constant, and Mw is the mass of the W boson. W 2, XW3 and W L are the CC structure functions defined in a similar manner as for NC. In the quark parton model, where WL == 0, the structure functions W 2 and XW3 may be expressed as the sum and difference of the quark and anti-quark momentum distributions: + + W 2 = x(U + D), XW3 = x(D - U), W 2 = x(U + D), XW3 = x(U - D). The terms xU, xD, xU and xV are defined as the sums of up-type, of down-type and of their anti-quark-type distributions. The Standard Model (SM) predicts that in the absence of right-handed charged currents the e+p (e-p) CC cross section is directly proportional to the fraction of right-handed positrons (left-handed electrons) in the beam and can be expressed as (7)
70 1
N
~M
FL extraction from H1 data (for fixed W=276 GeV)
u.. X
2,
H1 +ZEUS Combined (pre)) 0.8
Q2=1500 GeV -
1.2
L.L'
1
•
Hl preliminary • Hl e·
H1 2000 PDF 0.8
0.6
[J
Hl e-
j
-
NLOa,fit(Hl) NLO fit (ZEUS)
-
NlO MRST 2001
-
NLO (Alekhin)
:
......
NNLO (Alekhi") 1
I
0.6
0.4 0.4 0.2
t'" 10
2
10
L~ ...
1
~ I
0.2
x
-I_" .""'" ._
.. ~;' ~t--;
•
.,
...............................................................................• 10
i.
Q'/GeV'
Figure 3: The combined HI and ZEUS measurements of the structure function xF;;z (left). Summary of FL measurements by HI at a fixed photon-proton center of mass energy W = 276 GeV, W ~ .;sy (right).
The HI and ZEUS measurements of the single differential NC and CC e±p cross sections dO' / dQ2 are summarised in Figure 1 (left). At low Q2 < 100 Ge V2 the cross section of the CC process mediated by the W boson, is smaller by 3 orders of magnitude compared to the NC process, due to the different propagator terms. At high Q2 ~ the cross section measurements are approaching each other demonstrating the unification of the weak and electromagnetic forces. From a comparison of the NC measurements at highest Q2 with the SM expectation (see Figure 1, right), a limit on the quark radius of 0.74.10- 18 m is obtained [1], proving a point-like behaviour of the quarks down to about 1/1000 of the proton radius. The double differential NC and CC cross sections measurements [2] are shown in Figure 2. HERA allows to enlarge the coverage of the NC measurements by more than two orders of magnitude both in Q2 and x. The CC data provide information about individual quark flavours as can be seen in Figure 2 (right), especially at large Q2.
Mi, Mar
3
Structure functions F 2 , XF3 and FL
The NC cross section is dominated by the F2 contribution, and the reduced cross section, shown in Figure 2 (left), is essentially the proton structure function F 2 . In the figure one can see Bjorken scaling behaviour at x ~ 0.13, positive scaling violation at higher x due to gluon radiation from the valence quarks and negative scaling violation for x < 0.13 due to sea quarks originated from gluons. At fixed Q2 one observes a steep rise of F2 towards low x. The region at low x is due to quarks which have undergone hard or multiple soft gluon radiation and which carry a low fraction of the proton momentum at the time of interaction. The rise of the proton structure function at low x is
71
~'';'~Oi3 10 • HI Data • ZEUS (pre!.) 39 pb- 1 .... MRST04 10
MRSTNNLO CTEQ6HQ HVQDIS + CTEQ5F4
x=O.032
t
i=O
1O~1~0~~~1O'2~~~IO"~
10'
10
~
Q2(GeV2)
Q2/Gey2
Figure 4: The measured F!j" (left) and F~b (right), shown as a function of Q2 for various x values.
one of the most surprising observations at HERA. It can be understood as an unexpected rapid increase of the gluon density towards low x. The structure function XF3 is obtained from the NC cross section difference between e-p and e+p data, i.e. XF3 = (y+/2Y_) [a-(x,Q2) -a+(x,Q2)]. The dominant contribution to XF3 arises from "(Z interference, which allows xF;z to be extracted according to xF;z ::= -xF3/kae neglecting the pure Z exchange contribution, which is suppressed by the small vector coupling Ve. This structure function is non-singlet and has little dependence on Q2. The measured xF;z at different Q2 values can thus be averaged taking into account the small Q2 dependence. The averaged xF;z determined for a Q2 value of 1500 Gey2 is shown in Figure 3 (left) [3]. In leading order QCD the interference structure function xF;z leads to the following sum rule:
r z~ 1 r 10 xF; -;- = 310 1
1
(2u v + dv)dx
5
= 3'
(8)
Higher order corrections to this are expected to be of order as /7r. In the range of acceptance, the integral of F;z is measured to be Jo~~625 F;z dx = 1.21 ± 0.09(stat) ± 0.08(syst), which is consistent with the results of the HI and ZEUS QCD fits [4] of 1.12 ± 0.02 and 1.06 ± 0.02, respectively, for the same x interval at Q2 = 1500 Ge y2 . Non-zero values of the longitudinal structure function FL appear in perturbative QCD due to gluon radiation. According to eq. 1, the FL contribution to the inclusive cross section is significant only at high y. A direct way to measure
72 Charged Current e~p Scattering e"p-JovX • H1 2005 (prel.) 0H198-99 '" ZEUS 04-05 (prel.) f:.ZEUS 98-99
e+p---JoVX • H199-04 ... ZEUS 06-07 (prel.) f:;ZEUS 99-00
60
..
r 0.8
f
0.6 0.4 0.2
CTEQ6D .... MRST2004
-0.2 -0.4
-0.6
-O.B
DoCo','-'-'--'-c~~-'---!--~~o!oD05~~' P,
0'
• A> • A
Hl 2000 PDF ZEUS-JETS PDF
10'
10
4
Figure 5: The dependence of the e+p and e-p CC cross-section on the lepton beam polarisation P (left). Measurements of the polarisation asymmetries A± in NC interactions (right).
FL is to explore the y dependence of the cross section at given x and Q2 by
changing the center of mass energy of the interaction. Such analysis at HERA is in progress now using dedicated data collected with lower proton beam energies of 460 and 575 GeV. Data at the nominal proton energy of 920 GeV have been used by the HI collaboration to determine FL which is responsible for the observed decrease of the Ne cross section at high y. A summary of these FL measurements by HI [5] is shown in Figure 3 (right). They are compared with QeD calculations and different phenomenological models, showing that already at the present level of precision the measurements can discriminate between different predictions.
4
Charm and bottom structure functions F~c,
F!/'
Heavy quark production is an important process contributing to DIS. It is expected to be well described by perturbative QeD at next-to-Ieading order (NLO), especially at values of Q2 greater than the square of the heavy quark masses. The charm and bottom contributions to the proton structure function Fie, F~b are shown in Figure 4 [6]. They are measured using exclusive D or D* meson production and using a technique based on the lifetime of the heavy quark hadrons. In the latter case all events containing tracks with vertex detector information are used. The charm contribution on average amounts to 20 - 25% of F2 . The bottom structure function F~b is measured at HERA for the first time. It is about 1/10 of the charm contribution and amounts to ~ 2.5% of F2 at Q2 = 650 GeV 2 • The data are well described by QeD calculations. The accurate measurement of these structure functions is important to test the reliability of the theoretical framework used for the QeD analysis of
73
inclusive data and of predictions for the forthcoming LHC data, because their contribution is expected to be much increased at scales relevant for the LHC. 5
5.1
Polarisation effects in NC and CC
Polarisation dependence of the CC cross section
Measurements of CC deep-inelastic scattering with polarised leptons on protons allows the HERA experiments to extend tests of the V-A structure of charged current interactions from low-Q2, performed in the late seventies by the CHARM collaboration, into the high-Q 2 regime. ± The total CC cross sections a~d' as a function of the polarisation, measured in the range Q2 > 400 GeV 2 and y < 0.9, are shown in Figure 5 (left) [7]. The measurements agree with the SM predictions and exhibit the expected linear dependence as a function of the polarisation. Linear fits provide a good description of the data, and their extrapolation to the point P = 1 (P = -1) yields a fully right (left) handed CC cross section for e-p (e+p) interactions which is consistent with the vanishing SM prediction. The corresponding upper limits on the total CC cross sections exclude the existence of charged currents involving right handed fermions mediated by a boson of mass below 180 208 Ge V at 95% confidence level, assuming SM couplings and a massless right handed Ve.
5.2
Polarisation asymmetry in NC
The charge dependent longitudinal polarisation asymmetries of the neutral current cross sections, defined as (9)
measure to a very good approximation the structure function ratio, proportional to combinations aeVq, and thus provide a direct measure of parity violation. In the Standard Model A+ is expected to be positive and about equal to -A-. At large x the asymmetries measure the diu ratio of the valence quark distributions according to A± ~ ±k(1 + dv /u v )/(4 + dv/u v ) . The combined HI and ZEUS data are shown in Figure 5 (right) [3]. The asymmetries are well described by the Standard Model predictions as obtained from the HI and ZEUS QCD fits [4]. The measured asymmetries A± are observed to be of opposite sign and the difference 6A = A+ - A- can be seen to be significantly larger than zero, thus demonstrating parity violation at very small distances, down to about 10- 18 m.
74
6: Parton distribution functions determined at HERA (left). Results on the weak neutral of the 'It quark to the Z boson as determined at HERA in comparison with similar results by the CDF experiment and the combined LEP experiments
Partonic structure of the
nJ'ntnn
The measurements of the full set of NC and CC douhle differential cross sections at HERA allow comprehensive QCD analyses to determine the and distributions inside the proton and the constant cross inclusive HERA measurements of the NC and CC HI and ZEUS performed NLO QCD fits [4), which lead to a decmnDositioln of the parton densities. In the fit their data are and gluon distributions obtained in the HI and fits are shown in 6 (left). The results agree within the error also agree with the parton densities from fits which bands. include not HERA but also fixed target DIS data as well as data from other processes sensitive to parton distributions, such as inclusive DI'()dllct:ion and the W-lepton asymmetry in collisions. The inclusive and CC cross sections are not distribution functions but also to the electroweak the NC cross section at depends on the weak vector (v q ) and axial-vector IlDI'HHno; of up- and down-type quarks to the Z boson via the structure functions. The longitudinal polarisation of the lepton beam additional to the couplings. This has been in a combined fit the PDFs and the electroweak parameters [8]. The fitted the u are shown in Figure 6 (right) in with similar suIts obtained the CDF experiment and the combined LEP The HERA determination has a better precision than that from the
75
tho UDeert.
HERA
~'~,~,-
-'--, • ZEUS (inclusive-jet NC DIS) .,. ZEUS (inclusive-jet yp) ... ZEUS (norm. dijet NC DIS) III HI (norm. inclusive-jet NC DIS) . HI (event shapes NC DIS)
-~
- ,
-~
Judu J/'ljJK~7r-, BO ----> J/'ljJK~, and Ab ----> J/'ljJA samples with equivalent luminosities significantly greater than that of the data analyzed. No indication of a mass peak is observed in the reconstructed J /'ljJSmass distributions. (4) The mass distributions of J /'ljJ, S-, and A are investigated by relaxing the mass requirements on these particles one at a time for events both in the Sb signal region and the sidebands. The numbers of these particles determined by fitting their respective mass distribution are fully consistent with the quoted numbers of signal events plus background contributions. (5)The robustness of the observed mass peak is tested by varying selection criteria within reasonable ranges. All studies confirm the existence of the peak at the same mass.
Interpreting the peak as Sb production, candidate masses are fitted with the hypothesis of a signal plus background model using an unbinned likelihood method. The signal and background shapes are assumed to be Gaussian and flat, respectively. The fit results in a Sb mass of 5.774 ± 0.011 GeV with a width of 0.037 ± 0.008 GeV and a yield of 15.2 ± 4.4 events. Unless specified, all uncertainties are statistical. Following the same procedure, a fit to the Me Sb events yields a mass of 5.839 ± 0.003 GeV, in good agreement with the 5.840 GeV input mass. The fitted width of the Me mass distribution is 0.035±O.002 GeV, consistent with the 0.037 GeV obtained from the data. Since the intrinsic decay width of the Sb baryon in the Me is negligible, the width of the mass distribution is thus dominated by the detector resolution. To assess the significance of the signal, the likelihood, Ls+b, of the signal plus background fit above is first determined. The fit is then repeated using only the background contribution, and a new likelihood Lb is found. The logarithmic likelhood ratio J2ln(L s+b/ Lb) indicates a statistical significance of 5.5u, corresponding to a probability of 3.3 x 10- 8 from background fluctuation for observing a signal that is equal to or more significant than what is seen in the data. Including systematic effects from the mass range, signal and background models, and the track momentum scale results in a minimum signicance of 5.3u and a Sb yield of 15.2 ± 4.4(stat.)!6:~(syst.).
Potential systematic biases on the measured Sb mass are studied for the event selection, signal and background models, and the track momentum scale (see more at [1]). So, the resulting measured Sb mass is: 5.774 ± O.Ol1(stat.) ± O.015(syst.) GeV.
A lot of thanks to my DO b-Physics group colleagues, the staffs at Fermilab and collaborating institutions.
89
Figure 1: Decay topology of the :=:;; --> J /1/1:=:- where J /1/1 -> J.t+ J.t- and :=:- --> A7r- --> (P7r-)7r~. The:=:- and A baryons have decay lengths of the order of cm; the :=:;; has an estImated decay length of the order of mm (IP is the primary Interaction Point).
~ lD13,1.310' ~ 400t t right-sign
2!
;-
o'
~300[
(a)
wrong-sign ,
:>
~ 350
D13, 1.310'
(b)
- Me: s~ signal ........ Data: wrong-sign
'H
I:
~ 200
W
1.28
1.3
1.32 1.34 1.36 M(An) [GeV)
Figure 2: (a) The effective mass distribution of the A7r pair before the :=:;; reconstruction. Filled circles are from the right-sign A7r- combinations showing a :=:- mass peak while the histogram is from the wrong-sign A7r+ combinations. (b) Distributions of the proton transverse momentum of the wrong-sign background events (dotted histogram) and Monte Carlo signal :=:;; events (solid histogram) after preselection. The signal distribution is scaled to the same number of background events.
90
-
(a)
>Q)
G It)
0121,1.3 fb· 1
(b)
wrong-sign DI2I, 1.3 fb· 1
S
0
...J Data ..... Fit
c:i
~ _6
J94 3
s:: Q) >
UJ
4 3
5i
Jj2
4
1
(d) 2
0
5.5
6.5 6 7 M(S~) (GeV]
Figure 3: (a) The invariant mass distribution of the :=:;; candidates after all selections. The dotted curve is an unbinned likelihood fit to the model of a constant background plus a Gaussian signal. (b - d) The (f.1,+ f.1,- ) (p-rr- )-rr- invariant mass distributions of the wrong-sign background, J/'I/J sideband, and:=:- sideband events.
References
[1] V.M. Abazov et ai. (DO Collaboration), Phys.Rev.Lett. 99, 1052001(2007). [2] J. Abdallah et al. (DELPHI Collaboration), Eur. Phys.J. C44, 299(2005); D.Buskulic et al. (ALEPH Collaboration), Phys.Lett.B 384, 449(1996). [3] V.M. Abazov et al. (DO Collaboration), Nucl. lnstrum. Methods A565, 463(2006). [4] T. Sjostrand et al., Comput. Phys. Commun. 135, 238 (2001). [5] D.J. Lange, Nucl. lnstrum. Methods A462, 152(2001). [6] R. Brun and F. Carminati, CERN Program Library Writeup W5013, 1993 (unpublished). [7] V.M. Abazov et al.(DO Collaboration), Phys.Rev.Lett. 98, 121801(2007). [8] W.-M. Yao et al., Journal of Physics G33, 1(2006).
SEARCH FOR NEW PHYSICS IN RARE B DECAYS AT LHCb V. Egorychev a on behalf of the LHCb collaboration Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia Abstract.We discuss the potential of the LHCb experiment to study rare B decays and their impact on various scenarios for New Physics. Some possible experimental strategies are presented.
1
Introduction
Rare decays in the beauty sector encompass a wide range of processes offering exceedingly valuable tool in the search for New Physics (NP) as well as in precision measurements of the Standard Model (SM) parameters, e.g. the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements. The focus of this paper is made on the processes with the final states containing photons or leptons in addition to daughter hadron(s). The examples considered include the electromagnetic or electroweak penguin decays B --+ K*" Bs --+ ¢,' B --+ K*f+Cand the dilepton decay Bs --+ p,+ p,-. Most of these rare decays correspond to diagrams with internal loops or boxes leading to effective flavor-changing neutral current (FCNC) transitions. Presence of new virtual particles (e.g. the supersymmetric ones) with masses of the order of 100 GeVjc 2 may manifest itself in altering the decay rate, C P asymmetry and other observable quantities. Exciting new perspectives for the B physics emerge owing to the large statistics to be collected by the LHCb experiment, which will enable to enter a new realm of high precision studies of rare B decays.
2
LHCb experiment
The LHCb experiment features a forward magnetic spectrometer with a polar angle coverage of 15-300 mrad and a pseudo-rapidity range of 1.8 < 'f} < 4.9 [1). In order to maximize the probability of a single interaction per bunch crossing, it was decided to limit the luminosity in the LHCb interaction region to '" 2 X 1032 cm- 2 8- 1 . This has the additional advantage of limiting the radiation damage due to the high particle flux at small angles. The bb cross-section at the nominal LHCb luminosity is large enough to produce'" 10 12 bb pairs per year (10 7 s). The detector consists of a silicon vertex locator, followed by a first Ring Imaging Cherenkov Counter (RICH), a silicon trigger tracker, a 4 Tm spectrometer dipole magnet, tracking chambers, a second RICH detector, a calorimeter system and a muon identifier. One of the main features is a versatile trigger with a 2 kHz output rate dominated by pp --+ bbX events. The reconstruction of rare a e-mail:
[email protected]
91
Figure 1: The integrated luminosity required to achieve a 3 K*y candidates (the blue filled histogram represents combinatorial background).
B decays at LHCb is a challenge due to their small rates and large backgrounds from various sources. The .B-mesons are separated from .the large background produced directly at the interaction point in a detached vertex analysis by exploiting the relatively long lifetime of B-meson and a large average transverse momentum (pr) of the B-meson decay products. Therefore, the signature for a B event is based upon selection of particles with high px coming from a displaced vertex. The most critical backgornnd is the combinatorial background from pp -> bbX events, containing secondary verteces and characterized by high charged and neutral multiplicities. 3 The search for Bs —> fi+fj,^ The decay Ba —> /A+ fi~ is highly suppressed in the SM, since it can only be produced through a box diagram or a Z penguin. The current SM prediction is Br{Ba —> (J,+IJ,~~) = (3.4±0.5) x 10^9 [2]. In some new physics scenarios, the branching fraction can be enhanced by a high power of tan/? (e.g. Br oc tan6 /?), where tan/? is the ratio of the Higgs vacuum expectation values. For large values of tan/?, the branching fraction could be enhanced by two orders of magnitude, which is currently within the reach of the CDF and DO experiments. The large background (unlike sign muons originating from B decays or B decays into hadrons which are misidentified as muons) expected in the search for this decay is kept under control thanks to an excellent tracking performance of LHCb (namely the invariant mass resolution for dimuons ~ 18 MeV/e2), a good particle identification and good vertex resolution. The LHCb sensitivity
93
as a function of integrated luminosity is shown in Fig 1. LHCb has the potential to claim a three standard deviation observation at the level of the 8M prediction with", 2jb- 1 whereas a five standard deviation observation would require about 10 jb- 1 [3]. 4
The search for NP in b ---; sry and b ---; s£+ £-
Phenomenologically the b ---; sry and b ---; s£+ £- decays are closely linked. 8M calculations for these rare decays are performed using an effective Hamiltonian that is written in terms of several short-distance operators [4]. The process b ---; sry is dominated by the photon penguin operator, with Wilson coefficient C7 , while b ---; s£+£- has contributions also from semileptonic vector and axialvector operators with Wilson coefficients C9 and C lO respectively. To further pin down the values of these coefficients, it is necessary to exploit interference effects between the contributions from different operators. This is possible in the exclusive decay B --7 K*£+£- decays by measuring the forward-backward asymmetry AFB(q2), the longitudinal polarisation fraction of the K*o FL and the second of the two polarisation amplitude asymmetries A~). 4·1
Electroweak penguin decay B~
--7
K* fJ.,+ fJ.,-
The decay B~ --7 K* fJ.,+ fJ.,- is loop-suppressed in the 8M, Br(B~ ---; K* fJ.,+ fJ.,-) = (1.22~g:~~ x 10- 6 ) [5]. NP contributions could drastically change the shape of the AFB(q2) curve. For example, the sign of AFB(q2) can be flipped, the zero-crossing point may be shifted, or AFB(q2) may not even cross zero [6J. The procedure is to measure the AF B asymmetry of the angular distribution of daughter fJ.,+ relative to the B direction in the fJ.,+ fJ.,- rest frame as a function of the fJ.,+ fJ.,- invariant mass. The expected number of events in one year of data taking (2 jb- 1 ) by LHCb is 7200 ± 2100 (the error is due to the branching ratio), with a background-to-signal ratio B / S < 0.5 [7]. LHCb expects to extract the C9 /C7 Wilson coefficients ratio from the value of the fJ.,+ fJ.,- invariant mass for which the AFB is equal to zero to a precision of 13% after 5 years of running (10 jb- 1 ). Taking into accout the expected background level, the resolution with 2jb- 1 of integrated luminosity is 0.016 in FL and 0.42 in A~) [8].
4.2
Radiative decays b --7 sry
The polarization of the photons emitted in the b --7 sry transition provides an important test of the 8M, which predicts most left-handed photons. In the LHCb experiment these radiative decays can be reconstructed in the modes Bd --7 K*ry, Bs --7 ¢ry or Ab --7 Ary. The reconstruction procedures for Bd,s --7 K*(¢h decays are similar. To suppress the background from Bd,s --7 K*(¢)7r° in which the 7r 0 is misidentified as a single photon, a cut on the angle between
94 Table 1: Annual yields and background-to-signal ratios for radiative Ab decays (upper limits calculated at 90 % C ... L)
channel Ab -; ky Ab -; A(1520)-"y Ab -; A(1670)-"y Ab -; A(1690)-"y
yield/2 jb- 1 750 4.2 x 10 3 2.5 x 103 4.2 x 103
B/S 42 10 18 18
< < <
7 respectively, leading to 0'(')') rv 4° under the assumption of perfect U-spin symmetrye. 4
Outlook
With an accumulated luminosity of 10 fb- 1 LHCb will be able to measure the angle 'Y to a precision 0' stat rv 5° from tree-mediated processes, 0' stat rv 2° from processes where NP could enter DO mixing, and O'stat rv 2° (under U-spin symmetry assumption) from processes involving penguin loops, thus providing a powerful probe for NP. The Bs mixing phase 1>s will be measured to a precision O'stat rv 0.01 providing a constraint on NP by comparing tree-mediated with pure penguin processes. References [1] LHCb Collaboration, LHCb Reoptimized Detector Design and Performance TDR, CERN LHCC 2003-30. [2] B. Spaan, talk at this conference. [3) V. Egorychev, talk at this conference. [4] G.F. Tartarelli, Eur.Phys.J.direct C4S1 (2002) 35. [5] Z. Ligeti et al., hep-ph/0604112. [6] S. Cohen, M. Merk, E. Rodrigues, CERN-LHCb-2007-041. [7] M. Gronau, D. London, Phys.Lett. B253 (1991) 483; M. Gronau, D. Wyler, Phys.Lett. B265 (1991) 172. [8] D. Atwood, I. Dunietz, A. Soni, Phys.Rev.Lett. 78 (1997) 3257. [9] A. Giri, Yu. Grossman, A. Soffer, J. Zupan, Phys.Rev. D78 054018 (2003). [10] M. Patel, CERN-LHCb-2007-043; V. Gibson, C. Lazzeroni, J. Libby, CERN-LHCb-2007-048; K. Akiba, M. Gandelman, CERN-LHCb-2007050; J. Libby, A. Powell, J. Rademacker, G. Wilkinson, CERN-LHCb2007-098. [11] R. Fleischer, Phys.Lett. B459 (1999) 306. [12] J. Nardulli, talk at the 4th Workshop on the CKM Unitarity Triangle, December 2006.
eSensitivity of the method degrades with the U-spin symmetry breaking [12]. With no constraints on OTrTr,K K and 20% breaking of d TrTr = dK K, O'(-y) "" 10°. The method is believed to fail for larger U -spin symmetry breaking.
COLLIDER SEARCHES FOR EXTRA SPATIAL DIMENSIONS AND BLACK HOLES Greg Landsberg a Brown University, Department of Physics, 182 Hope St., Providence, RI02912, USA Abstract. Searches for extra spatial dimensions remain among the most popular new directions in our quest for physics beyond the Standard Model. High-energy collider experiments of the current decade should be able to find an ultimate answer to the question of their existence in a variety of models. We review these models and recent results from the Tevatron on searches for large, TeV-1-size, and Randall-Sundrum extra spatial dimensions. The most dramatic consequence of low-scale (~ 1 TeV) quantum gravity is copious production of mini-black holes at the LHC. We discuss selected topics in the mini-black-hole phenomenology.
1
Models with Extra Spatial Dimensions
A new, string theory inspired paradigm [1] proposed by Arkani-Hamed, Dimopoulos, and Dvali (ADD) in 1998 suggested the solution to the hierarchy problem of the standard model (SM) by introducing several (n) spatial extra dimensions (ED) with the compactification radii as large as ~ 1 mm. These large extra dimensions are introduced to solve the hierarchy problem of the SM by lowering the Planck scale to a TeV energy range. (We further refer to this fundamental Planck scale in the (4+n)-dimensional space-time as MD') In this picture, gravity permeates the entire multidimensional space, while all the other fields are constrained to the 3D-space. Consequently, the apparent Planck scale M p1 = l/JGN only reflects the strength of gravity from the point of view of a 3D-observer and therefore can be much higher than the fundamental (4+n)-dimensional Planck Scale. The size of large extra dimensions (R) is fixed by their number, n, and the fundamental Planck scale MD. By applying Gauss's law, one finds [1,2]: M~l = 87rM£;+2 Rn. If one requires MD ~ 1 TeV and a single extra dimension, its size has to be of the order of the radius of the solar system; however, already for two ED their size is only ~ 1 mm; for three ED it is ~ 1 nm, i.e., similar to the size of an atom; for larger number of ED it further decreases to subatomic sizes and reaches ~ 1 fm for seven ED. Almost simultaneously with the ADD paradigm a very different low-energy utilization of the idea of compact extra dimensions has been introduced by Dienes, Dudas, and Gherghetta [3]. In their model, additional dimension(s) of the "natural" EWSB size of R ~ 1 TeV- 1 [4] are added to the SM to allow for low-energy unification of gauge forces. In conventional SM and its popular extensions, such as super symmetry, gauge couplings run logarithmically with energy, which is a direct consequence of the renormalization group evolution (RGE) equations. Given the values of the strong, EM, and weak couplings at low energies, all three couplings are expected to "unify" (i.e., reach the same ae-mail: [email protected]
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strength) at the energy'" 10 13 TeV, know as the Grand Unification Theory (GUT) scale. However, if one allows gauge bosons responsible for strong, EM, and weak interactions to propagate in extra dimension(s), the RGE equations would change. Namely, once the energy is sufficient to excite Kaluza-Klein (KK) modes of gauge bosons (i.e., '" 1I R '" 1 Te V), running of the couplings is proportional to a certain power of energy, rather than its logarithm. Thus, the unification of all three couplings can be achieved at much lower energies than the GUT scale, possibly as low as 10-100 TeV [3]. While this model does not incorporate gravity and thus does not explain its weakness relative to other forces, it nevertheless removes another hierarchy of a comparable size - the hierarchy between the EWSB and GUT scales. In 1999, Randall and Sundrum offered a rigorous solution [5] to the hierarchy problem by adding a single extra dimension (with the size that can range anywhere from", 1/MpI virtually to infinity) with a non-Euclidean, warped metric. They used the Anti-deSitter (AdS) metric (i.e. that of a space with a constant negative curvature) ds 2 = exp( -2kRI N, Dirt Other Data Data - Background Data
Bac1t[lrouna
2 2 -)8 _ Data +8 Background
[er]
200-300 284±25 26 258 115 20 99 24 375±19 9l±31 2.50
300-475 274±21 67 207 76 51 50 30 369±19 95±28 2.81
475-1250 358±35 229 129 62 20 17 30 380±19 22±40 0.50
bins, separated into various components. The low Ev region is dominated by NC 1To, radiative t:. decay, and dirt events, in contrast to the region above 475 MeV, which is dominated by intrinsic I/e events. All these events are measured and constrained by the MiniBooNE data with uncertainties quoted in Table l. The excess is attributed to either electrons or gammas, as the MinibooNE detector cannot distinguish an outgoing electron from a single gamma from a 1/ interaction. The excess events originate either from a background process not included in the prediction, or a new physics phenomenon. Both possibilities are being actively pursued in current analyses. Currently the MiniBooNE collaboration is analyzing off-axis neutrinos observed in the MiniBooNE detector originating from the NuMI/Minos beam. These events will provide an important complementary analysis of the 1/ spectrum since the energy and distance is similar to the Fermilab Booster 1/ beam. The NuMI events are dominated by intrinsics I/e in low Ev region and therefore subject to different systematics. In summary, while there is presently an unexplained excess of I/e-like events at low energy, the excellent agreement in the oscillation region permits MiniBooNE to rule out the LSND result interpreted as two neutrino 1//1 --) I/e oscillation described by the standard L/E dependence. References
[1] [2] [3] [4]
A.Aguilar et al.,Phys.Rev.D64 (2001) 112007. M.Sorel, J.M.Conrad, and M.Shaevitz,Phys.Rev.D70 (2004) 073004. Z. Djurcic, Nucl.Phys.Proc.Supp1.168 (2007) 309,arXiv:hep-ex/0701017. A.Aguilar et al.,Phys.Rev.Lett.98 (2007) 231801,arXiv:0704.1500.
MINOS RESULTS AND PROSPECTS J.P.Ochoa-Ricomf for the MINOS Collaboration Lauritsen Lab, California Institute of Technology, Pasadena, CA 91125 USA Abstract. We report on the updated measurement of muon neutrino disappearance observed in the MINOS detectors. These preliminary results are determined from an exposure of 2.5 x 10 20 protons on the NuMI target and incorporate several improvements to our analysis. From a maximum likelihood fit to the reconstructed vI' energy spectra we obtain the neutrino squared-mass difference l~m~21 = (2.38:+:g:ig) x 1O- 3 eV 2 and mixing angle sin2(2023) = 1.00-0.08 with errors quoted at the 68% confidence level. We also report on the outlook for future analyses such as the searches for electron neutrino appearance and sterile neutrinos, as well as muon anti-neutrino oscillations and transitions.
1
Introduction
Neutrinos are the most enigmatic particles in the Standard Model, as the basic questions concerning their masses and mixing remain unanswered. By studying the flavor composition of a beam of muon neutrinos as it travels through the earth the MINOS experiment is able to probe the nature of these elusive particles. The neutrino beam is produced at the NuMI [lJ facility, where 120 GeV protons extracted from the Fermilab Main Injector impinge on a graphite target. The particles produced in the collisions are then focused by two parabolic horns into a 675 m long, 2 m diameter, evacuated steel pipe. The decay of these particles produces a neutrino beam with a predicted charged current (CC) neutrino event yield of 92.9%vJ.t, 5.8%vJ.t, 1.2%ve and O.l%v e . By changing the separation between the target and the horns it is possible to modify the neutrino energy spectrum. Most of the MINOS data is taken in the "low energy" (LE) beam configuration, which optimizes neutrino production in the 1-3 GeV region where the largest oscillation effects are expected. The NuMI beam is sampled by the two MINOS detectors. The 0.98 kton Near Detector (ND) is located about 1 km downstream of the NuMI target at Fermilab and is used to study the beam composition and energy spectra. The 5.4 kton Far Detector (FD) is located in the Soudan Underground Laboratory at a distance of 735 km from the target and is used to look for oscillation effects. The detectors are toroidally magnetized iron-scintillator sampling calorimeters and are functionally identical. They are described in more detail elsewhere [2J. The primary goal of the experiment is to perform precision measurements of the muon neutrino disappearance phenomenon associated with the dominant vJ.t ---> Vr oscillation mode. Results on vJ.t disappearance based on an exposure of 1.27 x 10 20 protons on the NuMI target corresponding to the data collected during the first period of NuMI operations between May 2005 and February ae-mail: [email protected]
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2006 were reported in [3]. Here we present updated results obtained from a re-analysis of this data set along with an additional data set collected between June 2006 and March 2007, corresponding to a total exposure of 2.5 x 10 20 protons on target. The outlook for future measurements, including vi-' disappearance, is also discussed here. MINOS is sensitive to sub-dominant vi-' ---+ Ve oscillations as well as oscillations to sterile neutrinos. In addition, muon anti-neutrinos and neutrinos can be separated on an individual basis due to the magnetic field that pervades the two detectors. This allows for the possibility of searching for exotic vi-' ---+ l7i-' transitions as well as for directly measuring l7i-' disappearance. 2
2.1
vi-'
disappearance results
Analysis
This updated analysis follows the same strategy as in [3]. Charged Current (CC) vi-' events leave a nearly unmistakeable signature consisting of a negatively charged muon track with some hadronic activity at the vertex. Events with these characteristics and coincident with beam spills are selected in both detectors using a multivariate algorithm that relies on kinematic and topology variables to distinguish the CC vi-' signal events from the background of neutral current (NC) events. The unoscillated CC vi-' energy spectrum is then obtained from the ND data and its comparison with the oscillated FD spectrum allows for the extraction of the oscillation parameters. No anti-neutrinos are included in the analysis and the FD data were intentionally obscured until all selection, fitting and systematic error estimation procedures were finalized. This result incorporates several improvements with respect to the previously published analysis. The neutrino interaction simulation package features more accurate models of hadronization, intranuclear rescattering and deep inelastic scattering. A new track reconstruction algorithm is used which successfully finds and fits 4% more tracks from signal CC vi-' events. A 3% increase in acceptance is also achieved by an expansion of the FD fiducial volume along the beam direction. Finally, the multivariate algorithm used to separate the signal from the NC background now combines an increased number of observables and takes advantage of their correlations with event length. The predicted unoscillated FD neutrino energy spectrum is obtained by extrapolating the observed ND spectrum using a beam transfer matrix [4] that encapsulates the kinematic and geometry effects responsible for the small (up to ±30%) differences in shape between the two spectra. This extrapolation method is largely insensitive to mis-modeling of the neutrino flux and neutrino interaction cross-sections. In addition, the hadron production model and other elements of the simulation are tuned to ND data taken in different beam configurations in order to correct for higher order effects. The beam matrix
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Data Sample vI" (all E) vI" « 10 GeV) vI" « 5 GeV)
Observed 563 310 198
Expected (no osc.) 738 ±30 496 ± 20 350 ± 14
Observed/Expected 0.74(4.40-) 0.62(6.20-) 0.57(6.50-)
Table 1: Expected and observed CC vI' events in the FD for an exposure of 2.5 x 10 20 protons on target. The Monte Carlo expectation for the NC background in this sample is 5.6 ± 2.8 events.
FD prediction has been cross-checked by other extrapolation methods and the spread between different predictions found to be much smaller than the expected statistical error.
2.2
Oscillation results
The number of observed CC vI" events compared to the null oscillations expectation is shown in Table 1. In the low energy region of the spectrum the observed deficit of events is substantial. The oscillation parameters are extracted from a maximum likelihood fit to the observed FD spectrum. The fit is done under the framework of vI" ----> Vr oscillations and the unphysical region of sin 2(2B 23 ) > 1 is excluded. The impact of the systematic effects on the measurement is assessed by performing oscillation fits to simulated data sets with the corresponding systematics applied. The most significant sources of systematic error are found to be the uncertainty in the near to far normalization (4%), the absolute hadronic shower energy scale (10%) and the neutral current normalization (50%). These systematic uncertainties are incorporated in the oscillation fit as nuisance parameters. From the best fit values of the oscillation fit we obtain the neutrino squared-mass difference l~m~21 = (2.38:t.:g:ig) x 1O~3eV2 and mixing angle sin 2(2B 23 ) = 1.00~O.08 with errors quoted at the 68% confidence level. The best oscillation fit corresponds to X 2 = 41.2 for 34 degrees of freedom and is shown alongside the data in the left plot of Figure 1. The best fit point and the 68% and 90% confidence intervals in oscillation parameter space can be seen on the right plot of Figure 1. 3
Prospects
As the beam data continues to be collected we anticipate a significant increase to our vI" disappearance sensitivity, as shown on the left plot of Figure 2. Beyond these results, there is the possibility that MINOS could make the first measurement of a non-zero B13 if this mixing angle lies in the vicinity of the current experimental limit set by CHOOZ [8]. Even though MINOS does
116 Oscillation Results lor 2.50E20 POTs
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100
80 60 40 20 10
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Figure 1: (left) Reconstructed CC vJ.L energy spectrum in the FD for the null prediction (black), the best fit (red) and the data (points). (right) The new MINOS best fit point (star) along with the 68% and 90% contours. Overlaid are the 90% contours from the Super-Kamiokande zenith angle [5] and LIE analyses [6], as well as that from the K2K experiment [7].
not have the optimal granularity for separating between electromagnetic and hadronic showers several techniques have been devised that successfully select CC Ve events. The background consists primarily of NC and CC vI" events and is predicted using the ND data. With the existing data set our sensitivity to 813 is comparable or better to the current best limit as obtained by CHOOZ, and a first result is expected in 2008. With the full data set a discovery will be made or the current best limit reduced by about a factor of 2. By selecting for short and diffuse showers MINOS has the capability of identifying NC events with high efficiency (rv 90%) and purity (rv 60%). Given that NC events are unaffected by standard three-flavor neutrino oscillations any depletion of NC events at the FD detector would be an indication of oscillations to sterile neutrinos. The left plot of Figure 2 shows the MINOS sensitivity to the fraction of sterile mixing is defined as the fraction of disappearing vI-' 's that oscillate to sterile neutrinos. A result for this analysis is expected very soon. Its almost unprecedented ability for distinguishing between positive and negative neutrino induced muons makes MINOS an ideal ground for studying the physics of muon anti-neutrinos. For instance the FD data will be searched for exotic vI" ---+ TJI" transitions. Such transitions are predicted by some models beyond the Standard Model [9] and, it has been speculated, could explain the muon neutrino deficit observed in atmospheric neutrino experiments [10]. An anti-neutrino oscillation analysis is also in the works. Such a measurement would constitute a direct test of CPT conservation in the neutrino sector and could have a strong impact on CPT violating models introduced to, for example, expain the LSND signal [11]. In order to maximize the sensitivity to CC TJ I" disappearance we are currently studying the possibility of running with the horn current reversed for a small period of time. In such a configuration
117 MINOS Sensitivity as a function of Integrated POT
r
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10 GeV) in 1998-2002 are shown in Fig. 3. It was shown [12-14] that the size of a cone which contains 90% of signal strongly depends on neutralino mass. 90% C.L. flux limits are calculated as a function of neutralino mass using cones which collect 90% of the expected signal and are corrected for the 90% collection efficiency due to cone size. Also a correction is applied for each neutralino mass to translate from 10 GeV to 1 GeV threshold (thus modifying the results as presented earlier for 10 GeV threshold [15]). These limits are shown in Fig. 4. Also shown in Fig. 3, and Fig. 4 are limits obtained by Baksan [12], MACRO [13], Super-Kamiokande [14] and AMANDA (from the hard neutralino annihilation channels) [16]. 2.3
A search for fast magnetic monopoles
Fast magnetic monopoles with Dirac charge g = 68.5e are interesting objects to search for with deep underwater neutrino telescopes. The intensity of monopole Cherenkov radiation is ~ 8300 times higher than that of muons. Optical modules of the Baikal experiment can detect such an object from a distance up to hundred meters. The processing chain for fast monopoles starts with the selec-
124
tion of events with a high multiplicity of hit channels: Nhit > 30. In order to reduce the background from downward atmospheric muons we restrict ourself to monopoles coming from the lower hemisphere. For an upward going particle the times of hit channels increase with rising z-coordinates from bottom to top of the detector. To suppress downward moving particles, a cut on the value of the time-z-correlation, Ctz , is applied: "Nhit(t· - l)(z· - z)
Ctz =
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>0
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10. 10 -"'1~-'-!0."='8-'c-0~.6,....-~0•..,.4-'c-0~.2~-'-'-'='=-'-'~~~~
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Figure 5: C tz distributions for experimental events (triangles), simulated atmospheric muon events (solid), and simulated upward moving relativistic magnetic monopoles (dotted); mUltiplicity cut Nhit > 30.
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B = vIc Figure 6: Upper limits on the flux of fast monopoles obtained in this analysis (Baikal) and in other experiments.
In Fig. 5 we compare the Ctz-distribution for experimental data (triangles) and simulated atmospheric muon events (solid curve) with simulated upward moving monopole events (dotted curve). Within 1038 days of live time using in this analysis, we have selected 20943 events satisfying cut 0 (Nhit > 30 and C tz > 0). For further background suppression (see [17] for details of the analysis) we use additional cuts, which essentially reject muon events and at the same time only slightly reduce the effective area for relativistic monopoles* : (1) Nhit > 35 and Ctz > 0.4 -;- 0.6 (2) X2 determined from reconstruction has to be smaller than 3 (3) Reconstructed zenith angle () > 100° (4) Reconstructed track distance from NT200 center R rec > 10 -;- 25 m . • Different values of cuts correspond to different NT200 operation configurations.
125
No events from the experimental sample pass cuts (1)-(4). The acceptances f3 =1, 0.9 and 0.8 have been calculated for all NT200 operation configurations (various sets of operating channels). For the time periods included, AefJ varies between 3· lOB and 6· lO Bcm 2sr (for f3 = 1). From the non-observation of candidate events in NT200 and the earlier stage telescopes NT36 and NT96 [18], a combined upper limit on the flux of fast monopoles with 90% C.L. is obtained. Upper limit on a flux of magnetic monopoles with f3 1 is 4.6· 1O-17cm-2s-1scl. In Fig. 6 we compare our upper limit for an isotropic flux of fast monopoles obtained with the Baikal neutrino telescope to the limits from the underground experiments Ohya [19] and MACRO [20] and to the limit reported for the underice detector AMANDA B10 [21] and preliminary limit for AMANDA II [22]. AeJJ for monopoles with
2.4
A search for extraterrestrial high-energy neutrinos
The BAIKAL survey for high energy neutrinos searches for bright cascades produced at the neutrino interaction vertex in a large volume around the neutrino telescope [3]. We select events with high multiplicity of hit channels Nhib corresponding to bright cascades. To separate high-energy neutrino events from background events a cut to select events with upward moving light signals has been developed. We define for each event tmin = min(ti - tj), where ti, tj are the arrival times at channels i, j on each string, and the minimum over all strings is calculated. Positive and negative values of tmin correspond to upward and downward propagation of light, respectively. Within the 1038 days of the detector live time between April 1998 and February 2003, 3.45 x lOB events with Nhit ~ 4 have been recorded. For this analysis we used 22597 events with hit channel multiplicity Nhit >15 and tmin >-10 ns. We conclude that data are consistent with simulated background for both tmin and Nhit distributions. No statistically significant excess above the background from atmospheric muons has been observed. To maximize the sensitivity to a neutrino signal we introduce a cut in the (tmin, Nhit) phase space. Since no events have been observed which pass the final cuts upper limits on the diffuse flux of extraterrestrial neutrinos are calculated. For a 90% confidence level an upper limit on the number of signal events of n90% =2.5 is obtained assuming an uncertainty in signal detection of 24% and a background of zero events. A model of astrophysical neutrino sources, for which the total number of expected events, N m , is large than ngO%, is ruled out at 90% CL. Table 1 represents event rates and model rejection factors (MRF) ngo%/Nm for models of astrophysical neutrino sources obtained from our search, as well as model rejection factors obtained recently by the AMANDA collaboration [23-25].
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Figure 7: Left panel: all-flavor neutrino flux predictions in different models of neutrino sources compared to experimental upper limits to E-2 fluxes obtained by this analysis and other experiments (see text). Also shown is the sensitivity expected for 3 live years of the new telescope NT200+ [5,40]. Right panel: Baikal experimental limits compared to two model predictions. Dotted curves: predictions from model SS [26]' SeSi [32J and SS05 [27J. Full curves: obtained experimental upper limits to spectra of the same shape. Model SS is excluded (MRF=0.25), model SeSi is not (MRF=2.12).
For an E- 2 behaviour of the neutrino spectrum and a flavor ratio Ve : v M : V T = 1 : 1 : 1, the 90% C.L. upper limit on the neutrino flux of all flavors obtained with the Baikal neutrino telescope NT200 (1038 days) is: E 2 iJ? < 8.1 x 1O-7cm-2s-1sr-1GeV. (2) For the resonant process with the resonant neutrino energy Eo = 6.3 X 10 6 GeV the model-independent limit on ve is: iJ?ve < 3.3 x 1O-20cm-2s-1sr-1GeV-l. (3) Fig. 7 (left panel) shows our upper limit on the all flavor E- 2 diffuse flux (2) as well as the model independent limit on the resonant De flux (diamond) (3). Also shown are the limits obtained by AMANDA [23-25] and MACRO [33], theoretical bounds obtained by Berezinsky (model independent (B) [34] and for an E-2 shape of the neutrino spectrum (B(E-2)) [35], by Waxman and Bahcall (WB) [36], by Mannheim et al.(MPR) [31], predictions for neutrino fluxes from topological defects (TD) [32], prediction on diffuse flux from AGNs according to Nellen et al. (NMB) [37], as well as the atmospheric conventional neutrino fluxes [38] from horizontal and vertical directions ( (v) upper and lower curves, respectively) and atmospheric prompt neutrino fluxes (vpr ) obtained by Volkova et al. [39]. The right panel of Fig. 7 shows our upper limits (solid curves) on diffuse fluxes from AGNs shaped according to the model of Stecker and Salamon (SS, SS05) [26,27] and of Semikoz and Sigl (SeSi) [32], according to Table 1.
127 Table 1: Expected number of events N m and model rejection factors for model of astrophysical neutrino sources
Model 10 '0 X E'~ SS Quasar [26] SS05 Quasar [27] SP u [28] SP I [28] P PI [29] M PP+PI [30] MPR [31] SeSi [32]
LIe
BAIKAL + Llr ngo%/Nm 3.08 0.81 10.00 0.25 1.00 2.5 40.18 0.062 6.75 0.37 2.19 1.14 0.86 2.86 0.63 4.0 1.18 2.12
+
LlJ.L
AMANDA [23-25] ngo%/Nm 0.22 0.21 1.6 0.054 0.28 1.99 1.19 2.0 -
3 Towards a km3 detector in Lake Baikal The construction of NT200+ is a first step towards a km3-scale Baikal neutrino telescope. Such a detector could be made of building blocks similar to NT200+, but with NT200 replaced by a single string, still allowing separation of highenergy neutrino induced cascades from background. It will contain a total of 1300-1700 OMs, arranged at 90-100 strings with 12-16 OMs each, and a length of 300-350 m. Interstring distance will be R::100 m. The effective volume for detection of cascades with energy above 100 TeV is 0.5-0.8 km3 . The existing NT200+ allows to verify all key elements and design principles of the km3 (Gigaton-Volume) Baikal telescope. Next milestone of the ongoing km3-telescope research and development work (R&D) will be spring 2008: installation of a "new technology" prototype string as a part of NT200+. This string will consist of 12 optical modules and a FADC based measuring system. Three issues, discussed in the remainder of this paper, have been investigated in 2007, and will permit installation of this prototype string: (1) increase of underwater (uw) data transmission bandwidth, (2) in-situ study of FADC PMpulses, (3) preliminary selection of optimal PM. More details can be found in [6].
3.1 Modernization of data acquisition system The basic goal of the NT200+ DAQ modernization is a substantial increase of uw-data rate - to allow for transmission of significant FADC data rate, and also for a more complex trigger concept (e.g. lower thresholds and topological trigger). In a first step, in 2005 a high speed data/control tcp/ip connection between the shore station and the central uw-PCs (data center) had been established (full multiplexing over a single pair of wires, with a hot spare) [4,5,40]' based on DSL-modems (FlexDSL). In 2007, the communication on the remaining segment uw-PC - string controller was upgraded using the same approach.
128
The basic elements are new string-controllers (handling TDC/ ADC-readout) with an ethernet-interface, connected by a DSL-modem to the central uw-DSL unit (3 DSL modems, max. 2 Mbps each), connected by ethernet to the uwPCs. The significant increase in uw-data rate (string to uw-PC) provided the possibility to operate the new prototype FADC system. 3.2 Prototype on a FADe based system A prototype FADC readout system was installed during the Baikal expedition 2007. It should yield input for the design of the 2008 km3 prototype string (FADC), such as: optimal sampling time window, dynamic range, achievable pulse parameter precisions, algorithms for online data handling, estimation of true bandwidth needs. These data will also be useful to decide about the basic DAQ/Triggering approach for the km3-detector: at this stage, both a complex FADC based, as well as a classical TDCI ADC approach seem feasible. The FADC prototype is located at the top of the 2nd outer string. It includes two optical modules with up-looking PM R8055, a slow control module and a FADC sphere. The FADC sphere consists of two 250 MHz FADCs, with USB connection to an embedded PCI04 computer emETX-i701, and a counter board MPCI48. The standard string trigger (2-fold channel coincidence) is used as FADC trigger. Data are transfered via local ethernet and the DSL-link of the 2nd string. Data analysis from FADC prototype is in progress.
3. 3 PM selection for the km3 prototype string Selection of the optimal PM type for the km3 telescope is a key question of detector design. Assuming similar values for time resolution and linearity range, the basic criteria of PM selection is its effective sensitivity to Cherenkov light, determined as the fraction of registered photons per photon flux unit. It depends on photocathode area, quantum efficiency, and photoelectron collection efficiency. We compared effective sensitivities of Hamamatsu R8055 (13" photocathode diameter) and XP1807 (12") with QUASAR-370 (14.6") [41], which was successfully operated in NT200 over more than 15 years. In laboratory we used blue LEDs (470 nm), located at 150 em distance from the PM. Underwater measurements are done for 2 R8055 and 2 XP1807, installed permanently as two NT200-channels, which are illuminated by the external laser calibration source [40], located 160 - 180 m away. Preliminary results of these effective PM sensitivity measurements show relatively small deviations. Smaller size (R8055, XP1807) tends to be compensated by larger photocathode sensitivities. In addition, we emphasize the advantage of a spherical shape (as QUASAR-370); we are investigating the angular integrated sensitivity looses due to various deviations from that optimum. 4 Conclusion The Baikal neutrino telescope NT200 is taking data since April 1998. The upper limit obtained for a diffuse (ve + vJ1 + vT ) flux with E- 2 shape is
129
8.1 x 1O- 7 cm- 2 s- 1 sr- 1 GeV. The limits on fast magnetic monopoles and on additional muon flux induced by WIMPs annihilation at the center of the Earth belong to the most stringent limits existing to date. The limit on a 17e flux at the resonant energy 6.3x10 6 GeV is presently the most stringent. To extend the search for diffuse extraterrestrial neutrinos with higher sensitivity, NT200 was significantly upgraded to NT200+, a detector with about 5 Mton enclosed volume, which takes data since April 2005 [5,40]. The threeyear sensitivity of NT200+ to the all-flavor neutrino flux is approximately 2 x 1O- 7 cm- 2 s- 1 sr- 1 GeV for E >10 2 TeV (shown in Fig. 7). For a km3-scale detector in Lake Baikal, R&D-activities are in progress. The NT200+ detector is, beyond its better physics sensitivity, used as an ideal testbed for critical new components. Modernization of the NT200+ DAQ allowed to install a prototype FADC PM readout. Six large area hemispherical PMs have been integrated into NT200+ (2 Photonis XP1807/12" and 4 Hamamatsu R8055/13"), to facilitate an optimal PM choice. A prototype new technology string will be installed in spring 2008 and a km3-detector Technical Design Report is planned for fall 2008. E2if? =
Acknowledgments This work was supported by the Russian Ministry of Education and Science, the German Ministry of Education and Research and the Russian Fund of Basic Research (grants 05-02-17476, 05-02-16593, 07-02-10013 and 07-02-00791), and by the Grant of President of Russia NSh-4580.2006.2. and by NATO-Grant NIG-9811707(2005). References [1] 1. Belolaptikovet al. Astropart. Phys. 7, 263 (1997). [2] V. Aynutdinov et al. Nucl. Phys. (Proc. Suppl.) B143, 335 (2005). [3] V. Aynutdinov et al. Astropart. Phys. 25, 140 (2006). [4] V. Aynutdinov et al. (Proc. of V Int. Conf. on Non-Accelerator New Physics) June 7-10 (2005) Dubna Russia. [5] V. Aynutdinov et al.(NIM) A567, 433 (2006). [6] V. Aynutdinov et al. (Proc. 30th ICRC) (icrc1084), Merida, 2007; arXiv.org: astro-ph/0710.3063. [7] V. Aynutdinov et al. (Proc. 30th ICRC) (icrc1088), Merida, 2007; arXiv.org: astro-ph/0710.3064. [8] V. Balkanov et al. Astropart. Phys. 12, 75 (1999). [9] V. Aynutdinovet al. Int. J. Mod. Phys. B20, 6932 (2005). [10] V. Balkanov et al. Nucl.Phys. (Proc.Suppl.) B91, 438 (2001). [11] V. Agrawal, T. Gaisser, P. Lipari & T. Stanev Phys. Rev. D 53, 1314 (1996). [12] M. Boliev et al. Nucl. Phys. (Proc. Suppl.) 48, 83 (1996); O. Suvorova arXiv.org: hep-ph/9911415 (1999). [13] M. Ambrosio et al. Phys. Rev. D 60, 082002 (1999).
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[14] S. Desai et al. Phys. Rev. D 70, 083523 (2004); erratum ibid D, 70, 109901 (2004). [15] K. Antipin et al. (Proc. of the First Workshop on Exotic Physics with Neutrino Telescopes) Uppsala, Sweden, Sept. 20-22, 34 (2006). [16] J. Ahrens et al. arXiv.org: astro-ph/0509330 (2005). [17] K. Antipin et al. (Proc. of the First Workshop on Exotic Physics with Neutrino Telescopes), Uppsala, Sweden, Sept. 20-22,80 (2006). [18] r. Belolaptikov et. al. [Baikal collaboration] (26th ICRC) , Salt Lake City, V.2, 340 (1999). [19] S. Orito et. al. Phys. Rev. Lett. 66, 1951 (1991). [20] M. Ambrosio et. al. [MACRO collaboration] arXiv.org: hepex/02007020 (2002). [21] P. Niessen, C. Spiering [AMANDA collaboration] (27th ICRC), Hamburg, V.4, 1496 (2001). [22] H. Wissing et al. [Ice Cube Collaboration], (Proc. 30th ICRC), Merida, 2007. [23] M. Ackermann et al. Astropart. Phys. 22, 127 (2005); Astropart. Phys. 22, 339 (2005). [24] M. Ackermann et al. Astropart. Phys. 22, 339 (2005). [25] M. Ackermann et al.Phys. Rev. D 76, 042008 (2007). [26] F. Stecker and M. Salamon Space Sci. Rev. 75, 341 (1996). [27] F. Stecker Phys. Rev. D 72, 107301 (2005). [28] A. Szabo and R. Protheroe (Proc. High Energy Neutrino Astrophysics), ed. V.J. Stenger et al., Honolulu, Hawaii (1992). [29] R. Protheroe arXiv.org:astro-ph /9612213. [30] K. Mannheim Astropart. Phys. 3, 295 (1995). [31] K. Mannheim, R. Protheroe and J. Rachen Phys. Rev. D 63, 023003 (2001). [32] D. Semikoz and G. Sigl, arXiv.org:hep-ph/0309328. [33] M. Ambrosio et al. Nucl. Phys. (Proc. Suppl.), B110, 519 (2002). [34] V. Berezinskyet al. "Astrophysics of Cosmic Rays", (Elsevier Science, North-Holland) 1990. [35] V. Berezinsky arXiv.org: astro-ph/0505220 (2005). [36] E. Waxman and J. Bahcall Phys. Rev. D 59, 023002 (1999). [37] L. Nellen, K. Mannheim and P. Biermann Phys. Rev. D, 47, 5270 (1993). [38] L. Volkova Yad. Fiz. 31, 1510 (1980). [39] L. Volkova and G. Zatsepin Phys. Lett. B462, 211 (1999). [40] V. Aynutdinov et al. (Proc. 29th Int. Cosmic Ray Conf.) August 3-10 Pune India (2005); arXiv.org: astro-ph /0507715. [41] R. Bagduev et al., (NIM) A420 (1999) 138.
NEUTRINO TELESCOPES IN THE DEEP SEA Vincenzo Flaminio a on behalf of the ANTARES Collaboration Physics Department and INFN, University of Pisa, Largo Bruno Pontecorvo 3, 56117, Pisa, Italy
Abstmct. The present is a review of current experiments performed in the deep sea in a search for 1/8 of cosmic origin. After a short recollection of the historical background, we discuss experiments that are now under construction or in the data-taking phase.
1
Introduction
Our understanding of the highly energetic processes that take place in violent stellar processes, such as Supernovae explosions, Gamma-Ray Bursts, AGNs etc. has considerably improved over the last decades, thanks to the big technological progess in the field of X and l'-ray astronomy. Apart from the intrinsic limitations that to further advances in this field are placed by the absorption of X and l'-rays in the intergalactic medium, the information that electromagnetic radiation conveys is incomplete, in that such radiation is generated mainly by high-energy electrons and photons in the dense environments of stellar objects, while there is every reason to believe that, in most of these, hadronic processes play an important role. Information on such processes can only come from VB originating from the decay of shortlived hadrons produced in high-energy nuclear interactions [1-3]. So far, the only v 8 of extraterrestrial origin detected are the Solar VB [4] and a handful of VB produced in the Supernova 1987A [5]. Many groups have actively been pursuing the task of constructing large apparatus aimed at the detection of high-energy v 8 of cosmic origin. Because of the tiny cross section, and the consequent need of very large detector masses, these detectors have adopted the Cerenkov technique using as medium either large sea or lake volumes, or the Antarctic ice. The first suggestion to use sea water as a target-detector medium for high energy cosmic v 8 is due to M.A. Markov [6]. The detection principle is sketched in figure 1. Muons produced by up-going v 8 interacting in the Earth's crust underneath the instrumented volume are detected through the Cerenkov photons they emit in water. A large photomultiplier (PMT) array records position and time of arrival of the Cerenkov photons, thus allowing a precise reconstruction of the muon direction. The range (~ 1 km for a 200 GeV muon) and Cerenkov yield (about 3 x 104 photons/meter in the frequency sensitivity range of PMTs) of high energy muons in sea-water are both very large. In addition, the water transparency in this frequency range is excellent (Aab8 ~ 50 +- 60m is the typical ae-mail: [email protected]
131
Figure 1: Schematic view of an underwater neutrino detector. The v charged-current interaction occurs in the Earth's crust underneath the detector. Cereakov photons are emitted by the. nation, while crossing the instrumented region. Each detector registers the position and arrival time of the photons, thus allowing a reconstruction of the muon direction.
value In the deep_sea);_The angle 9 between the neutrino and muon directions is: 9 < 1.5°/y/Wv (TeV): hence at high energy the ji and v directions coincide. Besides the obvious requirement of a large detector volume, an additional one derives from the Heed of screening the PMTs from undesired backgrounds, such as skylight and Cerenkov light from atmospheric y?. This requires the detector to be installed at large depths*. However, even working at large depths the latter background source may complicate the data analysis. To further reduce the effect of this background, experiments are therefore optimised for the detection of upgoing onions, generated by neutrinos that have crossed the Earth underneath. The advantages that this choice provide are achieved at a price: the Earth is not transparent to very high energy neutrinos. Indeed, for energies of the order of 103 TeV the neutrino interaction length becomes comparable to the Earth diameter. A further, unavoidable background comes from neutrinos originated in the decay of shortlived particles produced by cosmic rays in the upper atmosphere. These "atmospheric neutrinos" have relatively low energies and their contribution can. be reduced by cuts on energy. In this talk I will summarise the experiments of this kind that have been or
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are being carried out using sea water as the medium. It is interesting to begin this review by briefly recalling the DUMAND experiment, that led the early pioneering studies and of whose experience all subsequent experiments took advantage.
2
Pioneering developments: DUM AND
The DUMAND project aimed at the installation of an underwater Cerenkov detector at a depth of about 4500 km, near the Hawaii islands. The experiment, funded in 1990, was terminated only six years later. For a comprehensive account of the historical developments and construction steps of DUM AND we refer to a paper published in 1992 by Arthur Roberts [7]. A large number of deployments were performed, some of which provided results on the atmospheric muon vertical intensity vs depth down to 5 km and on the muon angular distribution at 4 km. The DUMAND collaboration also set what at the time were the best limits on the flux of high-energy VB from AGNs. The shore station was located at the Natural Energy Laboratory of Hawaii (NELH) at Keahole Point. A cable to connect the detector to the shore station, comprising 12 single-mode optical fibers in a stainless-steel tube and surrounded by a copper sheath capable of transmitting 5 kW of electrical power, was designed and manufactured in the early 90's. In December 1993, the DUMAND collaboration successfully deployed the first major components of DUMAND, including the junction box, environmental monitoring equipment, and the shore cable, with one complete string equipped with 16" PMTs, attached to the junction box (JB). The data system could cope, with a negligible dead time, with the background rate from radioactivity in the water (primarily from natural 40 K and bioluminescence). The counting rate for a single PMT was of the order of 60 kHz, primarily due to trace 40 K in the huge volume of seawater viewed by each tube. Noise due to bioluminescence was episodic and expected to be unimportant after the array had been stationary on the ocean bottom for some time, since the light-emitting microscopic creatures are stimulated by motion. 40 K and bioluminescence contribute mainly 1 photoelectron hits distributed randomly in time over the entire array. Bioluminescence caused spikes in the singles rate which reached 100 kHz for periods on the order of seconds, but with a very low frequency of occurrence. The deployed string was used to record backgrounds and muon events. Unfortunately, an undetected flaw in one of the electrical penetrators (connectors) used for the electronics pressure vessels produced a small water leak. Seawater eventually shorted out the string controller electronics, disabling further observations after about 10 hours of operation. Recovery of the string was accomplished between 28-30 January 1994, about 44 days after it had been deployed. The developments took a long time. In retrospective it seems that this was
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mainly caused by the lack, at least in the initial development stages, of the necessary technological means, like a reliable fiber-optical technology, pressureresistant electro-optical connectors and Remote-Operated-Vehicles (ROVs) capable to operate at depths in excess of 4000 m, for underwater connections. These became gradually available in the process of detector design, and were eventually adopted in the final setup. In mid 1996, DOE determined that further support for DUMAND-II should be terminated. The same Russian groups that contributed to DUMAND in the early stages, started an analogous enterprise under lake Baikal. This relatively small detector has operated for many years and the construction of a larger detector is now underway. Because of time constraints we shall not discuss details of this, that is not properly an "undersea" experiment. 3
Experiments in the Mediterranean sea
The construction of different prototype-detectors has been pursued over the last decades at three different locations in the Mediterranean. Chronologically, the first of these has been initiated by the NESTOR collaboration in the Ionian Sea, off the coast of Pylos, Southwestern Greece, at a depth of approximately 4000 m. The second one, NEMO is a prototype meant to be the basic building block of a km 3 detector. The chosen location is off the southern coast of Sicily, at a depth of >:;;j 3500 m. The third one, ANTARES is a medium-sized experiment (the effective area is of the same order of magnitude as the one of AMANDA [8]) now being assembled in the Mediterranean, off the southern French coast, at a depth of about 2500 m. 4
The NESTOR experiment
This has involved a large international collaboration and results obtained during the initial tests have already been published [9,10]. The chosen site is located at a distance of about 30 km Southwest of the small harbour of Methoni, at a depth of 4000 -;- 5000 m where the water quality is excellent. The detector architecture is based on what the authors call" stars". A typical star, an example of which is shown in figure 2, consists of six 16 m long arms attached to a central casing. Two optical modules (15" photomultipliers, enclosed in spherical glass housings) are attached at the end of each of the arms, one facing upwards and the other facing downwards. The electronics for each star is housed in a one-meter diameter titanium sphere within the central casing. A full NESTOR tower would consist of 12 such floors stacked vertically with a spacing od 30 m between floors. As in other undersea detectors, data and
135
power transmission is provided by electro-optical cables linking the detector to the shore station. The architecture is conceived in such a way as to avoid
Figure 2: [Left] hotmrr:~nh of a full-size NESTOR star. The test described here used a smaller detector, [right] Zenith angular distribution of atmospheric unions measured in the NESTOR prototype star in March 2003. The black dots are Me MC predictions. The inset shows the same distribution on a linear scale.
underwater operations: all connections are performed in air using dry-mating connectors before deployment. Repair operations need the recovery of the entire tower and connection cables, a formidable task for a very large detector. So far only a single smaller (5 m long arms) star has been deployed and tested for a short period in March 2003, at a depth of 3800 HI. From the total data sample collected with a four-fold coincidence trigger, 45,800 events have been selected. The resulting zenith angular distribution is shown in figure 2, where it is compared with the result of a MonteCarlo (MC) simulation of atmospheric muons, based on the Okada parameterisation [11]. The agreement is good both in shape and absolute flux. 5
The NEMO experiment
This experiment is being carried out by a large Italian collaboration. The geometry adopted is that of a series of "towers". The structure of a single tower is sketched in figure 3. It consists of 16 arms ("floors") each 18 m in length and holding a pair of 10" PMTs at its ends. The various loors are held by tensioning ropes bound to a buoy at the top. The geometry is such as to facilitate transportation and deployment, since the different loors of a given tower can by folded one on top of another, achieving a compact structure that can afterward easily be unfolded for deployment. The propsed detector geometry consists of an array of 9 x 9 such towers, interspaced by 140 m, providing an effective area of over lfern2 for energies above 10 TeV. The site
136
chosen Is located South-East of Sicily (see the inset in igure 3) at less than 80 km from shore, at a depth of 3500 m. The sea properties of this site have been studied in detail over a period of several years, and they turn out to be Ideal [12], both in terms of water properties and of biolumlnescence, A small prototype ("minitower") has been successfully deployed and tested for a few months at a somewhat shallower (« 2000 m) depth, about 20 kin away from the Catania harbour, at the end of 2006 [13]. A junction box (JB) was deployed first and connected to a pre-existing 25 km long electro-optical cable linked to a shore station. A minitower, consisting of only four "arms", each 15 m long and holding two 10" Hamamatsu PMTs at each end, was then deployed and connected to the JB. The vertical distance between arms was 40 m. Several trigger schemes were tested at the same time and a large number of events, mostly due to atmospheric muons, were recorded. Figure 3 shows a typical reconstructed muon. At the saaie time a full-scale tower is being built by the collaboration. A 100 km long electro-optical cable has beea deployed, linking the shore station, located inside the harbour area of " Portopalo di Capo Passero" with the chosen site. The building to be used as the shore laboratory for a fern,3 size detector has been acquired and is currently being equipped. Deployment of a full-sized tower is foreseen for the end of 2008.
Figure 3: [Left] The, inset shows the Capopassero site where the NEMO fern3 detector should be Installed, The two drawings illustrate the structure of the NEMO tower. [.Right] One of the first atmospheric muon tracks reconstructed In the NEMO minitower.
137
6
The ANTARES experiment
Antares is a multidiseipiiiiaxy experiment, whose main aim, of detecting neutrinos of cosmic origin, is accompanied by parallel research interests in the fields of marine biology and geophysics [15,16]. It is being carried out by a large collaboration, including Research Institutions from France, Germany, Italy, The Netherlands, Spain and Romania. Being this the experiment that has made the most impressive progress over the last couple of years, I will go here a little bit more in detail. The detector, schematically shown in figure 4, consists of 12 lines, each holding 75 10" Hamamatsu PMTs arranged in triplets (storeys) and looking downward, at an angle of 45° to the vertical. The PMTs are housed in pressure-resistant glass spheres. The separation between storeys in each line is 14.5 m, starting 100 m from the sea-floor; the distance between pairs of strings in the horizontal plane is 60-70 m. Each PMT triplet is held in place by a
Figure 4: Sketch of the Antares detector.The inset shows an indiYictaai storey, with a titanium frame holding the three glass spheres, each housing a PMT
titanium frame, as shown in the inset of figure 4, attached to a vertical electrooptical cable used for data and clock signals as well as for power transmission. Digital data transmission uses optical fibres. At the center of the frame a titanium cylinder encloses the readout/control electronics, together with compasses/tiltmeters used for geometrical positioning. Some of the storeys also house LED beacons, each containing 38 LEDs, which provide very fast pulses used for timing calibrations. For readout purposes, each group (sector) of five storeys in any given line is treated separately.
138
A laser is located at the bottom of one of the lines, while a second one is located at the bottom of the instrumentation line (see below). These provide additional means for timing calibrations. Hydrophones, attached one per sector, are used, in conjunction with sonic transmitters located at fixed locations on the sea-floor and with the compasses-tilt meters installed in the LCMs, for precise position determinations. An additional line (Instrumentation Line), equipped with instruments used to monitor other important parameters, such as temperature, pressure, salinity, light attenuation length and sound speed, is an essential component of the detector. Each line is connected, via an electronics module located at its bottom, to a JB in turn connected via a 42 km long electro-optical cable (installed in November 2001) to the shore station. All data are here collected by a computer farm, where a fast processing of events satisfying predetermined trigger requirements is performed. Precise timing is provided by a 20 MHz high accuracy on-shore clock synchronised with the GPS, distributed via the electro-optical cable and the JB to each electronics module. The expected performance of the detector has been studied in detail using MC simulations. The effective area for neutrinos reaches a maximum of :::::: 30m 2 • For v 8 at small nadir angles there is a drastic decrease at very high energies, due to absorption by the Earth. The neutrino angular resolution is dominated by electronics at high energies, where it reaches a value of :::::: 0.2 -70.3 0 • At lower energies it is dominated by the kinematics of muon production by neutrinos. Following many tests carried out over several years, the installation of the detector in its final configuration started in December 2005 and has continued in 2006 and 2007. At present five of the lines are installed and data taking is going on smoothly C. The excellent performance of the detector, both in terms of its time and space resolution has been demonstrated using data obtained with the first lines installed [14]. A very large number of triggers has been collected using the present five-line detector. These are mainly due to atmospheric muons, together with a smaller number of v 8 • Figure 5 shows the ¢ and () (zenith) distributions for atmospheric muons, compared with the MC predictions. The anisotropy in the ¢ distribution reflects the non-uniform distribution of lines in the horizontal plane. The small discrepancy present between data and MC in the () distribution is due to a still inaccurate knowledge of the angular acceptance of the optical modules d. Figure 6 shows the (z-t) plot for a reconstructed muon moving upwards (due to a neutrino interaction). The top histogram shows the measured muon angular distribution, after the application of cuts designed to reduce the contribution of atmospheric muons. The neuC At the time of writing, five additional lines and an instrumentation line have been installed and connected dThe photomultipliers look downwards at an angle of 45° in such a way as to optimise the acceptance for muons moving upwards. Their acceptance for downgoing muons is therefore more limited
139 .,-o.OO7 c - - - - - - - - - - - - - - - - - , .,- O.02C--------------~ i~~~~~ I-·_m~~
i
N
0.005
&~
Antares Data "'"'' Monte Carlo
;; 0.016 ~ 0.014
1-
Antares Data .. ,"" Monte Carlo
0.01
0.00
0.00 ·150
-100
50
150 Azimuth angle [deg]
Figure 5: [Left] 4> and [Right]
(j
distributions for atmospheric muons obtained in the five-line detector.
trino sample is associated to muon events having cos () > 0; the corresponding rate is a few per day. 7
The future: conclusions
Following the pioneering DUMAND attempts and in parallel with analogous detectors now operational under the Antarctic ice, a number of undersea mediumto-large-scale experiments are under construction in the northern hemisphere. These are: NESTOR, NEMO, ANTARES. The latter, with five strings (375 PMTs) already installed e, is at present the largest running undersea experiment in the northern hemisphere. Recently the three collaborations have merged their efforts in an attempt to design and build a km 3 detector in the Mediterranean. A design study has been approved and financed by the EU [17] and work is in progress. References
M.D. Kistler and J.F. Beacom, Phys.Rev. D 74, 063007 (2006). F. W. Stecker, Phys.Rev. D 72, 107301 (2005). V. Cavasinni, D. Grasso and L. Maccione, Astrop. Phys. 26, 41(2006). For a comprehensive Review of the SSM and of the early solar neutrino experiments, see: J.N. Bahcall, "Neutrino Astrophysics", (Cambridge University Press) 1989. [5] K. Hirata et al., Phys.Rev. Lett. 58, 1490 (1987). R. M.Bionta et al., Phys.Rev. Lett. 58, 1494 (1987).
[1] [2] [3] [4]
eTen strings and 750 PMTs at the time of writing
the plot photons on the PMT and hlsl;ogl:am shows the measured Ue!llglled to further suppress atrnospn'3r1C
~Jt"j."lJI~{}V
DOUBLE BETA DECAY: PRESENT STATUS A.S. Barabash a
Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia Abstmct.The present status of double beta decay experiments is reviewed. The results of the most sensitive experiments, NEMO-3 and CDORICINO, are discussed. Proposals for future double beta decay experiments are considered.
1
Introduction
Interest in neutrino less double-beta decay has seen a significant renewal in recent years after evidence for neutrino oscillations was obtained from the results of atmospheric, solar, reactor and accelerator neutrino experiments. These results are impressive proof that neutrinos have a nonzero mass. The detection and study of Ov{3{3 decay may clarify the following problems of neutrino physics: (i) neutrino nature: whether the neutrino is a Dirac or a Majorana particle, (ii) absolute neutrino mass scale (a measurement or a limit on md, (iii) the type of neutrino mass hierarchy (normal, inverted, or quasidegenerate), (iv) CP violation in the lepton sector (measurement of the Majorana CP-violating phases). 2 2.1
Results of experimental investigations Two neutrino double beta decay
This decay was first recorded in 1950 in a geochemical experiment with 130Te [lJ; in 1967, it was also found for 82Se [2J. Only in 1987 2{3(2v) decay of 82Se was observed for the first time in a direct experiment [3]. Within the next few years, experiments employing counters were able to detect 2{3(2v) decay in many nuclei. In looMo [4-6], and 150Nd [7J 2{3(2v) decay to the 0+ excited state of the daughter nucleus was recorded. Also, the 2{3(2v) decay of 238U was detected in a radiochemical experiment [8], and in a geochemical experiment for the first time the ECEC process was detected in 130Ba [9J. Table 1 displays the present-day averaged and recommended values of T 1 / 2 (2v) from [10J (for looMo-100Ru(Ot) transition value is from [ll]). 2.2
Neutrinoless double beta decay
In contrast to two-neutrino decay, neutrinoless double-beta decay has not yet been observed b. ae-mail: [email protected] bThe possible exception is the result with 76Ge, published by a fraction of the HeidelbergMoscow Collaboration, Tl/2 ~ 1.2 . 1025 Y [12J and Tl/2 ~ 2.2· 1025 y [13J (see Table 2).
141
142 Table 1: Average and recommended Tl/2(2v) values [10). For looMo-lOORu(ot) transition value is from [11).
Isotope 48Ca 76Ge
82Se 96Zr 100Mo 100Mo-100Ru(Oi) 116Cd 128Te 130Te 150Nd 150Nd- 15 0Sm(Oi) 238U 130Ba; ECEC(2v)
1:6 .
4.3:: 10 19 (1.5 ± 0.1) . 10 21 (0.92 ± 0.07) . 1020 (2.0 ± 0.3) . 10 19 (7.1 ± 0.4) . 10 18 6.2~g:~ . 1020 (3.0 ± 0.2) . 10 19 (2.5 ± 0.3) . 1024 (0.9 ± 0.1) . 1021 (7.8 ± 0.7) . 10 18 1.4~g:~ . 1020 (2.0 ± 0.6) . 10 21 (2.2 ± 0.5) . 10 21
The present-day constraints on the existence of 2,B(Ov) decay are presented in Table 2 for the nuclei that are the most promising candidates. In calculating constraints on (mv), the nuclear matrix elements from QRPA calculations [1618] were used (3-d column). It is advisable to employ the calculations from these studies, because the calculations are the most thorough and take into account the most recent theoretical achievements. In column four, limits on (mv), which were obtained using the NMEs from a recent Shell Model (SM) calculations [19,20]. 3
Best present experiments
In this section the two large-scale running experiments NEMO-3 and CUORICINO are discussed. 3.1
NEMO-3 experiment !26,29j
Since June of 2002, the NEMO-3 tracking detector has operated at the Frejus Underground Laboratory (France) located at a depth of 4800 m.w.e. The detector has a cylindrical structure and consists of 20 identical sectors. A thin (about 30-60 mg/cm 2) source containing beta-decaying nuclei and having a The Moscow portion of the Collaboration does not agree with this conclusion [14) and there are others who are critical of this result [15). Thus at the present time this "positive" result is not accepted by the "2/3 decay community" and it has to be checked by new experiments.
143 Table 2: Best present results on 2/3(01/) decay (limits at 90% C.L.).
Isotope 76Ge
T 1/ 2 , Y
> 1.9.10 25
1.2. 1025 (7) 2.2 . 10 25 (7) > 1.6.10 25 > 3.10 24 > 5.8.10 23 > 4.5.10 23 > 2.1 . 1023 > 1.7.1023
~ ~
130Te lOoMo 136Xe 82Se 116Cd
(mvl' eV
(mvl' eV
[16-18] < 0.22 - 0.40 ~ 0.28 - 0.51(7) ~ 0.21 - 0.37(7) < 0.24 - 0.44 < 0.30 - 0.57 < 0.81 - 1.28 < 1.49 - 2.66 < 1.47 - 2.17 < 1.45 - 2.73
[19] < 0.76 ~ 0.96(7) ~ 0.71(7) < 0.83 < 0.75
< 2.2 < 3.4 < 2.1
Experiment HM [21] Part of HM [12] Part of HM [13] IGEX [22] CUORICINO [23] NEMO- 3 [25] DAM A [27] NEMO-3 [25] SOLOTVINO [28]
total area of 20 m 2 and a weight of up to 10 kg was placed in the detector. The energy of the electrons is measured by plastic scintillators (1940 individual counters), while the tracks are reconstructed on the basis of information obtained in the planes of Geiger cells (6180 cells) surrounding the source on both sides. In addition, a magnetic field of strength of about 25 G parallel to the detector axis is created by a solenoid surrounding the detector. At the present time, the investigations are being performed for seven isotopes; these are looMo (6.9 kg), 82Se (0.93 kg), 1l6Cd (0.4 kg), 150Nd (37 g), 96Zr (9.4 g), 130Te (0.45 kg), and 48Ca (7 g). The corresponding limits on Tl/2(0//) and (mv} for looMo and 82Se are presented in Table 2. T 1 / 2 (2//) for all seven isotopes have been measured (see [36]). The NEMO-3 experiment is on going and new improved results will be obtained in the near future. In particular, the sensitivity of the experiment to 2;3(0//) decay of lOoMo will be on the level of rv 2 . 10 24 y. This in turn means the sensitivity to (mv} will be on the level of rv 0.4 - 0.7 eV. 3.2
CUORICINO [24]
This program is the first stage of the larger CUORE experiment (see section 4). The experiment is running at the Gran Sasso Underground Laboratory. The detector consists of 62 individual low-temperature natTe02 crystals, their total weight being 40.7 kg. The energy resolution is 7.5-9.6 keY at an energy of 2.6 MeV. The experiment has been running since March of 2003. The corresponding limits on Tl/2(0//) and (mvl for 130Te are presented in Table 2. The sensitivity of the experiment to 2;3(0//) decay of 130Te will be on the
144
level of '" 5 . 1024 for 3 y of measurement. This in turn means the sensitivity to (my) is on the level of'" 0.2 - 0.6 eV. 4
Planned experiments
In this section, main parameters of five promising experiments which can be realized within the next five to ten years are presented. The estimation of the sensitivity in all experiments to the (my) is made using NMEs from [16-19]. Table 3: Five most developed and promising projects.
5
Experiment
Isotope
CUORE [30] GERDA [31]
130Te 76Ge
MAJORANA [32,33] EXO [34]
76Ge 130Xe
SuperNEMO [35,36]
82S e 150Nd
Mass of isotope, kg 200 40 1000 60 1000 200 1000 100-200
Sensitivity T 1L2, Y 2.1. 1026 2.10 26 6.10 27 2.10 26 6.10 27 6.4.10 25 2.10 27 (1 - 2) . 1026
Sensitivity (my), meV 35-90 70-230 10-40 70-230 10-40 120-220 20-40 45-110
Status accepted accepted R&D R&D R&D accepted R&D R&D
Conclusion
In conclusion, two-neutrino double-beta decay has so far been recorded for ten nuclei (48 Ca, 76Ge, 82Se , 96Z r , lOoMo, 116Cd, 128Te, 130Te, 150Nd, 238U). In addition, the 2f3(2v) decay of lOoMo and 150Nd to 0+ excited state of the daughter nucleus has been observed and the ECEC(2v) process in 130Ba was recorded. Neutrinoless double-beta decay has not yet been confirmed. There is a conservative limit on the effective value of the Majorana neutrino mass at the level of 0.75 eV. Within the next few years, the sensitivity to the neutrino mass in the CUORICINO and NEMO-3 experiments will be improved to become about 0.2 to 0.6 eV with measurements of 130Te and lOoMo. The next-generation experiments, where the mass of the isotopes being studied will be as grand as 100 to 1000 kg, will have started within three to five years. In all probability, they will make it possible to reach the sensitivity to the neutrino mass at a level of 0.1 to 0.01 eV.
145
References
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M.G. Inghram, J.H. Reynolds, Phys. Rev. 78,822 (1950). T. Kirsten, W. Gentner, O.A. Schaeffer, Z. Phys. 202, 273 (1967). S.R. Elliott, A.A. Hahn, M.K. Moe, Phys. Rev. Lett. 59, 2020 (1987). A.S. Barabash et al., Phys. Lett. B 345, 408 (1995). A.S. Barabash et al., Phys. At. Nucl. 62, 2039 (1999). L. De Braeckeleer et al., Phys. Rev. Lett. 86,3510 (2001). A.S. Barabash et al., JETP Lett. 79, 10 (2004). A.L. Turkevich, T.E. Economou and G.A. Cowan, Phys. Rev. Lett. 67, 3211 (1991). [9] A.P. Meshik et al., Phys. Rev. C 64, 035205 (2001). [10] A.S. Barabash et al., Czech.J. Phys. 56,437 (2006). [11] A.S. Barabash, AlP Conf. Proc. 942: 8 (2007). [12] H.V. Klapdor-Kleingrothaus et al., Phys. Lett. B 586, 198 (2004). [13] H.V. Klapdor-Kleingrothaus and LV. Krivosheina. Mod. Phys. Lett. A 21, 1547 (2006). [14] A.M. Bakalyarov et al., Phys. Part. Nucl. Lett. 2, 77 (2005); hepex/0309016. [15] A. Strumia and F. Vissani, Nucl. Phys. B 726, 294 (2005). [16] V. Rodin et al., Nucl. Phys. A 793, 213 (2007). [17] M. Kortelainen and J. Suhonen, Phys. Rev. C 75, 051303(R) (2007). [18] M. Kortelainen and J. Suhonen, Phys. Rev. C 76,024315 (2007). [19] E. Caurier et al., nucl-th/0709.2137. [20] E. Caurier et al., nucl-th/0709.0277. [21] H.V. Klapdor-Kleingrothaus et al., Eur. Phys. J. A 12, 147 (2001). [22] C.E. Aalseth et al., Phys. Rev. C 65, 09007 (2002). [23] A. Giuliani (CUORICINO Collaboration), report at TAUP'07 (Sendai, 11-13 September, 2007). [24] C. Arnaboldi et al., Phys. Rev. Lett. 95, 142501 (2005). [25] A.S. Barabash (NEMO Collaboration), hep-ex/0610025. [26] R. Arnold et al., Phys. Rev. Lett. 95, 182302 (2005). [27] R. Bernabei et al., Phys. Lett. B 546, 23 (2002). [28] F.A. Danevich et al., Phys. Rev. C 67, 035501 (2003). [29] R. Arnold et al., Nucl. Instr. Meth. A 536, 79 (2005). [30] C. Arnaboldi et al., Nucl. Instr. Meth. A 518, 775 (2004). [31] 1. Abt et al., hep-ex/0404039. [32] Majorana White Paper, nucl-ex/0311013. [33] C.E. Aalseth et al., Nucl. Phys. B (Pmc. Suppl.) 138, 217 (2005). [34] M. Danilovet al., Phys. Lett. B 480, 12 (2000). [35] A.S. Barabash, Czech. J. Phys. 52,575 (2002). [36] S. Soldner-Rembold (NEMO Collaboration), hep-ex/0710.4156.
BETA-BEAMS C. Volpe a
Institut de Physique Nucleaire Orsay, F-91406 Orsay cedex, FRANCE Abstract. Beta-beams is a new concept for the production of intense and pure neutrino beams. It is at the basis of a proposed neutrino facility, whose main goal is to explore the possible existence of CP violation in the lepton sector. Here we briefly review the original scenario and the low energy beta-beam. This option would offer a unique opportunity to perform neutrino interaction studies of interest for particle physics, astrophysics and nuclear physics. Other proposed scenarios for the search of CP violation are mentioned.
1
Introduction
The observations made by the Super-Kamiokande [1], the K2K [2], the SNO [3] and the KAMLAND [4] experiments have brought a breakthrough in the field of neutrino physics. The longstanding puzzles of the solar neutrino deficit [5] and of the atmospheric anomaly have been clarified: the expected fluxes are reduced due to the neutrino oscillation phenomenon, i.e. the change in flavour that neutrinos undergo while traveling [6]. The overall picture is now also confirmed by the recent mini-BOONE result [7]. Neutrino oscillations imply that neutrinos are massive particles and represent the first direct experimental evidence for physics beyond the Standard Model. Understanding the mechanism for generating the neutrino masses and their small values is clearly a fundamental question, that needs to be understood. On the other hand, the presently known (as well as unknown) neutrino properties have important implications for other domains of physics as well, among which astrophysics, e.g. for our comprehension of processes like the nucleosynthesis of heavy elements, and cosmology. An impressive progress has been achieved in our knowledge of neutrino properties. Most of the parameters of the Maki-Nakagawa-Sakata-Pontecorvo (MNSP) unitary matrix [8], relating the neutrino flavor to the mass basis, are nowadays determined, except the third neutrino mixing angle, usually called 813 . However, this matrix might be complex, meaning there might be (one or more) phases. A non-zero Dirac phase introduces a difference between neutrino and anti-neutrino oscillations and implies the breaking of the CP symmetry in the lepton sector. Knowing its value might require the availability of very intense neutrino beams in next-generation accelerator neutrino experiments, namely super-beams, neutrino factories or beta-beams. Besides representing a crucial discovery, the observation of a non-zero phase might help unraveling the asymmetry between matter and anti-matter in the Universe and have an impact in astrophysics, e.g. for core-collapse supernova physics [9]. ae-mail: [email protected]
146
147
Zucchelli has first proposed the idea of producing electron (anti)neutrino beams using the beta-decay of boosted radioactive ions: the "beta-beam" [lOJ. It has three main advantages: well-known fluxes, purity (in flavour) and collimation. This simple idea exploits major developments in the field of nuclear physics, where radioactive ion beam facilities under study such as the european EURISOL project are expected to reach ion intensities of 10 11 - 13 per second. A feasibility study of the original scenario is ongoing (2005-2009) within the EURISOL Design Study (DS) financed by the European Community. At present, various beta-beam scenarios can be found in the literature, depending on the ion acceleration. They are usually classified following the value of the Lorentz I boost factor, as low energy h = 6-15) [11-21,21-24]' original h ~ 60 - 100) [10,25-30], medium h of several hundreds) and high-energy h of the order of thousands) [31-35]. (For a review of all scenarios see [36].) An extensive investigation of the corresponding physics potential is being performed and new ideas keep being proposed. For example, a radioactive ion beam production method is discussed in [37] and will be investigate within the new "EuroNU" DS. Thanks to this method two new ions 8B and 8Li are being considered as candidate emitters, while the previous literature is mainly focussed on 6He and 18Ne. The corresponding sensitivity is currently under study (see e.g. [38]). 2
The original scenario
In the original scenario [10]' the ions are produced, collected, accelerated up to several tens GeV /nucleon - after injection in the Proton Synchrotron and Super Proton Synchrotron accelerators at CERN - and stored in a storage ring of 7.5 km (2.5 km) total length (straight sections). The neutrino beam produced by the decaying ions point to a large water Cerenkov detector [39] (about 20 times Super-Kamiokande), located at the (upgraded) FrEljus Underground Laboratory, in order to study CP violation, through a comparison of lie -+ 11/-1 and De -+ D/-I oscillations. This facility is based on reasonable extrapolation of existing technologies and exploits already existing accelerator infrastructure to reduce cost. Other technologies are being considered for the detector as well [40J. A first feasibility study is performed in [41]. The choice of the candidate emitter has to meet several criteria, including a high intensity achievable at production and a not too short/long half-life. The ion acceleration window is determined by a compromise between having the I factor as high as possible, to profit of larger cross sections and better focusing of the beam on one hand, and keeping it as low as possible to minimize the single pion background and better match the CP odd terms on the other hand. The signal corresponds to the muons produced by 11/-1 charged-current events in water, mainly via quasi-elastic interactions at these energies. Such events are
148 Table 1: Number of events expected after 10 years, for a beta-beam produced at CERN and sent to a 440 kton water Cerenkov detector located at an (upgraded) Frejus Underground Laboratory, at 130 km distance. The results correspond to ve (left) and Ve (right). The different 'Y values are chosen to make the ions circulate together in the ring [26] 18 Ne
6He
(r CC events (no oscillation) Oscillated (sin228l3 = 0.12, 15 Oscillated (15 = 90° ,8 13 = 3°) Beam background Detector backgrounds
= 0)
= 60) 19710 612 44
o 1
(r
= 100) 144784 5130 529 0 397
selected by requiring a single-ring event, with the same identification algorithms used by the Super-Kamiokande experiment, and by the detection of the electron from the muon decay. At such energies the energy resolution is very poor due to the Fermi motion and other nuclear effects. For these reasons, a CP violation search with 'Y = 60 - 100 is based on a counting experiment only. The beta-beam has no intrinsic backgrounds, contrary to conventional sources. However, inefficiencies in particle identification, such as single-pion production in neutral-current Ve (ve) interactions, electrons (positrons) misidentified as muons, as well as external sources, like atmospheric neutrino interactions, can produce backgrounds. The background coming from single pion production has a threshold at about 450 MeV, therefore giving no contribution for 'Y < 55. Standard algorithms for particle identification in water Cerenkov detectors are quite efficient in suppressing the fake signal coming from electrons (positrons) misidentified as muons. Concerning the atmospheric neutrino interactions, estimated to be of about 50/kton/yr, this important background is reduced to 1 event/440 kton/yr by requiring a time bunch length for the ions of 10 ns. The expected events from [26] are shown in Table 1, as an example. The discovery potential is analyzed in [10,25-30]. A detailed study of 'Y = 100 option is made for example in [29] based on the GLoBES software [42], including correlations and degeneracies and using atmospheric data in the analysis [33]. The fluxes are shown in Figure 1. Figure 2 shows the CP discovery reach as an example of the sensitivity that can be reached running the ions around 'Y = 100.
3
Low energy beta-beams
A low energy beta-beam facility producing neutrino beams in the 100 MeV energy range has been first proposed in [11]. Figure 3 shows the corresponding fluxes. The broad physics potential of such a facility, currently being analyzed, covers:
149 t..
x l 0 7 e - - - -_ _ _ _ _ _ _~
~ 8000
-
o
"'~ 7000
.€ ;>
2n 30 discovery of CP violation:
!:J.i (liep = 0, n):= 9
SPL v~
','" Q)
e:;;
2,Qx10·
s
20
40
60
80
100
Ey (MeV)
Figure 3: Anti-neutrino fluxes from the decay of 6He ions boosted at "! = 6 (dot-dashed line),,,! = 10 (dotted line) and,,! = 14 (dashed line). The full line presents the Michel spectrum for neutrinos from muon decay-at-rest.
I'"
03.4
3.5
4.0
Figure 4: eve Test : ~X2 obtained from the angular distribution of electron anti-neutrinos on proton scattering in a water Cerenkov detector in the cases when the statistical error only (solid), with 2% (dashed), 5% (dash-dotted) and 10% (dotted) systematic errors. The 1, Fp>. is a dual electromagnetic field tensor. It is obvious that the solution of equation (6) can be represented as a matrix exponent. It is not difficult to verify that the solution of equation (1) can take the form (4) only if the second invariant 12 = ~Fl1v H l1v of the tensor Fl1v is equal to zero, and the vector 111 is its eigenvector corresponding to the zero eigenvalue. So we come to the condition above (2), which was obtained only on the physical attends. Substitution of the expression (4) into equation (1) taking into account (6) leads to the relation
Nil = al1T = _
m(Jcp) q 2((cpq)2 - m 2cp2) 11
+ r~+cpl1 2(cpq)
3
m (Jcp) , (7) 2(cpq)((cpq)2 _ m 2cp2)
where cpl1 = 111 12 + HI1V qv 1m. Since 111, Nil = const we obtain the proper time = (N x) and the phase factor which determines energy shift of neutrino in matter F(x) = (Jx)/2. Hence the expression for the wave function takes the form T
\lI(x) =
~
2:: e-
i
(P,x)(l_
(1' 5 Stp )(1- (01'5S0)(q + m)'l/Jo,
(8)
(=±1
where
ql1(cpq)/m - cpl1m , ...j(cpq)2 _ cp2m2 12 - (NI1...j'-(cp-q-)-2---cp-2m-2/m.
11 _
Stp -
Pt = ql1 + r
(9) (10)
It is obvious that the system of solutions (8) is a complete system of solutions of equation (1), which is characterized by the kinetic momentum of the particle ql1 and the quantum number (0 = ±1, which can be interpreted as the neutrino spin projection on the direction sg at the moment T = (N x) = O. In the general case this system is not stationary. The received solutions are stationary only when = p • In this case the wave functions are the eigenfunctions of the spin projection operator on the direction p with the eigenvalues ( = ±1 and of the canonical momentum operator ia l1 with the eigenvalues The orthonormalized system of the stationary solutions of equation (1) can be written down in the following way:
sg st
st
Pt.
(11)
191
Here J is the transition Jacobian between the variables qll and J =
Pt=
2(1 + (fIlHJ1-l'ql'/2m -2 211 (ltp) (1 + (2J(tpq)2 - m 2tp2 ) J(tpq)2 - m tp2 ).
(12)
The structure of solution (11) directly leads us to the conclusion that when neutrinos move through a dense matter and an electromagnetic field which satisfy condition (2), they can behave as free particles, i.e. move with the constant group velocity opo Vgr
= --' oP, = ~ qO
(13)
and conserve the polarization. However in interactions with other particles the channels of reactions which are closed for a free neutrino can be opened (see, for example, [11,12]), as a result of difference of the dispersion law for the free neutrino p2 = m 2 from the one for neutrino in matter and electromagnetic field:
(14) where
FJ1- = A LJ.
Pt - fll/2, •
= SIgn
(1 +
~J1- = r/2+HJ1- l' Fl' /m,
211)
rfllHlll'ql'/2m J(cpq)2 - m 2cp2
'>
(15)
.
i
Here II = Fill' Fill' is the first invariant of the tensor Fill'. Let us discuss the physical meaning of the results obtained. For this purpose we shall consider vector and axial currents constructed with help of solution
(8):
Vil ~~
= ~(XhlllJ.f(X) = qll/l,
S/1 = -Sfp(SoStp)
All
= ~(Xh5,IlIJ.f(X) =
(0 ~SIl, q
+ [st +Sfp(SoStp)] cos 2B- ~elll'PAql'SOPStPA sin 2B, m
B = (Nx)J(cpq)2 - cp 2m 2/m.
(16)
(17)
~-----
Thus, solution (8) which is a linear combination of solutions (11) describes a spin-coherent state of neutrino, propagating with the velocity v = q/ qO . In these states neutrino spin rotation takes place. Therefore, neutrino state with rotating spin is a pure state. Existence of such solutions is the direct consequence of the neutrino state description in terms of kinetic momentum. It should be stressed that as the result of calculations we obtained the complete system of neutrino wave functions, which show spin rotation properties. Introduce a flight length L of a particle and an oscillations length Lose, using the relation B = 7r L / Lose. Since the scalar product (N x) = T can be interpreted as the proper time of a particle, then the oscillation length is defined as
192
In this formula we use gaussian units and restore the neutrino magnetic moment /10·
Hence if as a result of a certain process a neutrino arises with polarization (0, (the spin vector ( can be expressed in terms of the four-vector SJ-I components as ( = S - qSO /(qO + m)), after travelling the distance L the probability for the neutrino to have polarization -(0 is equal to 2
W s! = [(0 x (tp] sin 2 (7fL/Los c). (19) Consequently, if the condition (o(tp) = 0 is fulfilled, this probability can become unity, i.e. a resonance takes place. In this way we obtained the exact solutions of the Dirac-Pauli equation for neutrino in dense matter and electromagnetic field. It was demonstrated that if the neutrino production occurs in the presence of an external field and a dense matter, then its spin orientation is characterized by the vector Sip' Due to the time-energy uncertainty relation the considered states of neutrino can be generated only when the linear size of the area occupied by the electromagnetic field and the matter is comparable with the process formation length. This length is of the order of the oscillations length. Acknowledgments The authors are grateful to A.V. Borisov, O.F. Dorofeev and V.Ch. Zhukovsky for helpful discussions. This work was supported in part by the grant of President of Russian Federation for leading scientific schools (Grant SS 5332.2006.2) . References
[1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12]
B.W. Lee, R.E. Shrock, Phys. Rev. D 16, 1444 (1977). K. Fujikawa, R.E. Shrock Phys. Rev. Lett. 45,963 (1980). J. Schechter, J.W.F. Valle, Phys. Rev. D 24, 1883 (1981). M.B. Voloshin, M.I. Vysotsky, L.B. Okun, Zh. Eksp. Teor. Fiz. 91, 754 (1986); C.-S. Lim, W.J. Marciano, Phys. Rev. D 37, 1368 (1988); E.Kh. Akhmedov, Phys. Lett. B 213, 64 (1988). A.V. Borisov, A.I. Ternov, V.Ch. Zhukovsky, Izv. Vyssh. Uchebn. Zaved. Fiz. 31, No 3, 64 (1988); M. Dvornikov arXiv:0708.2328 [hep-ph] (2007). A.Yu. Smirnov, Phys. Lett. B 260, 161 (1991); E.Kh. Akhmedov, S.T. Petcov, A.Yu. Smirnov, Phys. Rev. D 48, 2167 (1993). L. Wolfenstein, Phys. Rev. D 17, 2369 (1978). S.P. Mikheyev, A.Yu. Smirnov, Yad. Fiz. 42, 1441 (1985). A.E. Lobanov, A.I. Studenikin, Phys. Lett. B 515, 94 (2001). A.E. Lobanov, O.S. Pavlova, Vestn. MGU. Fiz. Astron. 40, No 4, 3 (1999); A.E. Lobanov, J. Phys. A: Math. Gen. 39, 7517 (2006). A.E. Lobanov, Phys. Lett. B 619, 136 (2005). V.Ch. Zhukovsky, A.E. Lobanov, E.M. Murchikova, Phys. Rev. D 73, 065016 (2006).
PLASMA INDUCED NEUTRINO SPIN FLIP VIA THE NEUTRINO MAGNETIC MOMENT A.Kuznetsov a, N.Mikheev b Yaroslavl State P. G. Demidov University, Sovietskaya 14, 150000 Yaroslavl, Russia Abstract. The neutrino spin flip radiative conversion processes ilL -> IIR + 'Y. and + 'Y. -> IIR in medium are considered. It is shown in part that an analysis of the so-called spin light of neutrino without a complete taking account of both the neutrino and the photon dispersion in medium is physically inconsistent. ilL
1
Introduction
The most important event in neutrino physics of the last decades was the solving of the Solar neutrino problem. The Sun appeared in this case as a natural laboratory for investigations of neutrino properties. There exists a number of natural laboratories, the supernova explosions, where gigantic neutrino fluxes define in fact the process energetics. It means that microscopic neutrino characteristics, such as the neutrino magnetic moment, etc., would have a critical impact on macroscopic properties of these astrophysical events. One of the processes caused by the photon interaction with the neutrino magnetic moment, which could play an important role in astrophysics, is the radiative neutrino spin flip transition VL --t VRf' The process can be kinematically allowed in medium due to its influence on the photon dispersion, w = Iklln (here n =1= 1 is the refractive index), when the medium provides the condition n > 1. In this case the effective photon mass squared is negative, m; == q2 < O. This corresponds to the well-known effect of the neutrino Cherenkov radiation [1]. There exists also such a well-known subtle effect as the additional energy W acquired by a left-handed neutrino in plasma. This additional energy was considered in the series of papers by Studenikin et al. [2] as a new kinematical possibility to allow the radiative neutrino transition VL --t VRf' The effect was called the "spin light of neutrino". For some reason, the photon dispersion in medium providing in part the photon effective mass, was ignored in these papers. However, it is evident that a kinematical analysis based on the additional neutrino energy in plasma (caused by the weak forces) when the plasma influence on the photon dispersion (caused by electromagnetic forces) is ignored, cannot be considered as a physical approach. In this paper, we perform a consistent analysis of the radiative neutrino spin flip transition in medium, when its influence both on the photon and neutrino dispersion is taken into account. a e-mail:
[email protected] be-mail: [email protected]
193
194
2
Cherenkov process
VL ----) VR'Y
and its crossing
VL'Y ----) VR
Let us start from the Cherenkov process of the photon creation by neutrino, VL ----) VR'Y, which should be appended by the crossed process of the photon absorption VL'Y ----) VR. At this stage we neglect the additional left-handed neutrino energy W, which will be inserted below. For the VL ----) VR conversion width one obtains by the standard way:
r tot
VL-tl/R
where CrA) is the photon polarization vector, j"" is the Fourier transform of the neutrino magnetic moment current, pOI. = (E,p), p'OI. = (E',p') and qOl. = (w, k) are the four-momenta of the initial and final neutrinos and photon, respectively, A = t, C mean transversal and longitudinal photon polarizations, f'Y(w) = (e w / T _1)-1 is the Bose-Einstein photon distribution function, and Z~A) = (1 - 8II(A)/8w 2)-1 is the photon wave-function renormalization. The functions II(A) , defining the photon dispersion law:
(2) CrA) = II(A) C(A)OI.' The width r~~t--+VR can be rewritten to another form. Let us introduce the energy transferred from neutrino: E - E' = qo, which is expressed via the photon energy w(k) as qo = ±w(k). Then dqo dk (k == Ikl), one obtains: 2E+W-qo
E+W
J
2qOW) - ( 1- ~ where q2
{!(i) (qO,
dqo
J
dk [ T 1 +( f-y qo)]
(2E - qo) 2 q4
k) } ,
(6)
= q5 - k 2, and the photon spectral density functions are introduced: (7)
Our formula (6) having the most general form, can be used for neutrinophoton processes in any optically active medium. We only need to identify the photon spectral density functions {!(A).
4
Does the window for the "spin light of neutrino" exist?
To show manifestly that the case considered in the papers by Studenikin et al. [2], with taking the additional left-handed neutrino energy W in plasma and ignoring the photon dispersion, was really unphysical, let us consider the region of integration for the width r~~t--->VR in Eq. (6). In Fig. 1, the photon vacuum dispersion line qo = k is inside the allowed kinematical region (left plot), but the plasma influenced photon dispersion curve appears to be outside, if the neutrino energy is not large enough (right plot).
196
"-
"-
"-
"-
"-
"-
"-
"-
"-
"-
k
"-
o= k/
q
"-
"-
"-
"- "-
// ,
"-
,/
E
"-
"-
"-
"0 W
qo
k
"-
"-
,
"-
"-
"-
"-
"-
"-
"-
"-
"-
, /.
"-
"-
,/ ,/
':)
,// // '/
"- // 0
W
CJ)p
qo
Figure 1: The region of integration for the width r~7-+"R with the fixed initial neutrino energy E is inside the slanted rectangle shown by dashed line. The vacuum photon dispersion (if the medium influence is ignored) is shown by bold line in the left plot. The photon dispersion curve in plasma is shown by bold line in the right plot.
For the fixed plasma parameters, the threshold neutrino energy Emin exists for coming of the dispersion curve into the allowed kinematical region. Even for the interior of a neutron star this threshold neutrino energy is rather large: Emin '::::' w~/(2 W) '::::' 10 TeV, where Wp is the plasmon frequency. One could hope that the "spin light of neutrino" may be possible at ultra-high neutrino energies. However, in this case the local limit of the weak interaction is incomplete, and the non-local weak contribution into additional neutrino energy W must be taken into account. This contribution always has a negative sign, and its absolute value grows with the neutrino energy. One could only hope that the window arises in the neutrino energies for the process to be kinematically opened, Emin < E < Emax. For example, in the solar interior there is no window for the process with electron neutrinos at all. A more detailed analysis of this subject was performed in our papers [3,4J. Acknowledgements
A. K. expresses his deep gratitude to the organizers of the 13th Lomonosov Conference on Elementary Particle Physics for warm hospitality. The work was supported in part by the Russian Foundation for Basic Research under the Grant No. 07-02-00285-a. References
[1] [2J [3] [4J
W.Grimus, H.Neufeld, Phys. Lett. B 315, 129 (1993). A.Studenikin, e-print hep-ph/0611100, and the papers cited therein. A.V.Kuznetsov, N.V.Mikheev, Mod. Phys. Lett. A 21,1769 (2006). A.V.Kuznetsov, N.V.Mikheev, Int. J. Mod. Phys. A 22, 3211 (2007).
Astroparticle Physics and Cosmology
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INTERNATIONAL RUSSIAN-ITALIAN MISSION "RIM-PAMELA" 1 A.M.Galper , P.Picozza2 and O.Adriani3 , M.Ambriola4 , G.C.Barbarino 5 ,A.Basili6 , G.A. Bazilevskaj a lO , R.Beliotti 2, M.Boezio 5 , E.A.Bogomolov 9 , L.Bonechi 3 , M.Bongi 3 , L.Bongiorno 7 , V.Bonvicini 5 , A.Bruno\ F.Cafagna4 , D.Campana6 , 11 P.Carlson , M.Casolino 2, G.Castellini8 , M.P.De Pascale 2, G.De Rosa6 ,V.Di Felice2, 7 D.Fedele , P.Hofverberg 11 , L.A.Grishantseva 1 , S.V.Koldashovl, S.Y.Krutkov9 , A.N.Kvashnin 10 , J.Lundquist 5 , O.Maksumov lO , V.Malvezzi2, L.Marcelli 2 , I2 W.Menn , V.V.Mikhailovl, M.Minori 2 , E.Mocchiutti 5 , A.Morselli 2 , S.Orsi11, G.Osteria6 , P.Papini 3 , M.Pearce 11 , M.Ricci 7 , S.B.Ricciarini 3 , M.F.Runtso l , S.Russo6 , M.Simon I2 , R.Sparvoli 2 , P.Spillantini3 , Y.I.Stozhkov l , E.Taddei 3 , A.Vacchi 5 , E.Vannuccini 3 , G.Vasilyev 9 , S.A.Voronov 1 , Y.T.Yurkin l , G.Zampa5 , N.Zampa5 , V.G.Zverev l . 1 MEPhI, INFN f Roma2, 3 Florence, 4 Bari, 5 Trieste, 6 Napoli, 7 Prascati), 8 IFAC, 9 Ioffe Institute, 10 Lebedev Institute, 11 KTH, 12 University of Siegen.
Abstract. The successful launch of spacecraft "RESURS DK" 1 with precision magnetic spectrometer "PAMELA" onboard was executed at Baikonur cosmodrome 15 June 2006. The primary phase of realization of International Russian-Italian Project "RIM-PAMELA" with German and Swedish scientists' participation has begun since the launch of instrument "PAMELA" that has mainly been directed to investigate the fluxes of galactic cosmic rays. This report contains the main scientific Project's tasks and the conditions of science program's implementation after one year since exploration has commenced.
1
Introduction (scientific tasks)
a) Definitely one of the main problems of cosmology, physics of elementary particles and physics of cosmic rays, whose solutions are being worked on by many scientific groups today and who's every effort is being hampered by, is the nature of dark (latent) mass. Almost 25% of all the energy density of Universe is concentrated in the form of dark (latent) matter! The result of numerous theoretical researches comes to conclusion that dark matter consists of particles. However none of the known particles including neutrinos can not possibly be part of dark matter. They are absolutely new hypothetical particles. They can be either much lighter than electrons and they are called axions or much more heavy than nucleon and are called WIMPs (WIMP - weakly interactive massive particle). Practically all theoretical models assume that particles composing of dark matter were created early in the development of Universe. At present they appear like cooled (low energy) relic particles. A model of supersymmetry (SUSY) has had special development, there is a supersymmetric particle named neutralino and denoted by X possessing properties of WIMPs with mass 100 + 1000 bigger than a proton's mass [1]. In the case of neutralino X == X. Another model (Kaluza, Klein (KK)) is built on a multidimensional space-time and also supposes the existence of massive particles Bkk (kk- excitations) with mass :::: 5· 10 2 a proton's mass [2,3]. In the case of Bkk we have also Bkk == Bkk. 199
200
Interaction of both XX and BkkBkk can cause annihilation of WIMPs with creation of usual particles finally:
x + X -> bb, tt, TT, ZO ZO, Zo,
->,
+ ... , e± + ... ,pp + ... , dd + ...
This process is the basis for the search for X and Bkk at Project "RIMPAMELA". It is known that primary cosmic rays basically protons and nuclei are interacting with interstellar gas while going through cosmic space. This would result in a fragmentation of nuclei and creation of elementary particles including antiparticles p, e+ [4]. If to this source of antiprotons and positrons is added a source from process of annihilation of X and Bkk particles with creation of p and e+ then some peculiarities concerning process of WIMP annihilation will appear in spectra of antiparticles that are registering in near Earth cosmic. A peculiarity also could be seen in energy dependency ofp/p and e+ /e- ratios. Because masses of WIMPs are considerably bigger than mass of antiproton, some particular features should be seen at energies more than a few GeV. Precision measurements of antiprotons' and positrons' fluxes for searching of special peculiarities in spectra of antiparticles are main task of project "RIMPAMELA" - the clarification of nature of dark matter. b) The next task is searching for antihelium in cosmic rays. The probability of creation of antihelium nuclei in interaction of primary cosmic rays with interstellar gas is extremely small. Act of registration of such particles could allow us to understand better the nature of baryonic asymmetry of the Universe. It means existing antimatter's region in our Galaxy. The definition of the upper limit of c) Precise measurements of the energy spectra of known particles - protons, electrons, positrons and isotopes of light nuclei, are provided within the project RIM-PAMELA are essential for studying of the mechanisms and physical conditions of the generation in sources and acceleration during propagation of cosmic rays. Also important to study interaction of galactic cosmic rays in heliosphere and magnetosphere of the Earth. So these measurements are very interesting part of the scientific program of project "RIM-PAMELA". d) We would like to mention that we plan to study physics of acceleration processes during solar flares. e) Finally we plan to study flux of particles in nearest Earth cosmic space (albedo, captured particles). That is another important task of project "RIMPAMELA", because it is an applied aspect of the experiment.
201
2
Magnetic spectrometer PAMELA
The physical scheme of magnetic spectrometer " PAMELA" is presented in fig. 1. This set of detectors allow to measure following characteristics of particles such as charge, mass, momentum, energy, time and direction of arrival. Mode of operation of PAMELA is: when a particle fly inside the aperture of the device, it will be detected by the scintillation Time Of Flight system (TOF); the TOF system produces a signal, and gives the command to start registration of information for other detection systems of spectrometer - the coordinate system in magnet, calorimeter, shower detector, neutron detector and system of anticoincidence. This information is recording into the memory of spectrometer and then transferring to the recording device of spacecraft every three hours. The spectrometer has its own control system. This system can be reprogrammed by using commands from the Earth.
Fig. 1. The physical schematic of Magnetic spectrometer PAMELA: 1,3,7time of flight system; 2,4 - anticoincidence system; 5 - silicon strip tracker (6 double plates); 6 - magnet (5 sections); 8 - silicon strip imaging calorimeter; 9 - bottom scintillator 84; 10 - neutron detector; 11 - pressurized container.
The main physic-technical characteristics of spectrometer " PAMELA" are taken from the Monte-Carlo simulation and on-ground measurements in monochromatic beams of electrons and protons from the CERN accelerators. This data is shown in table 1.
3
Measurements. Analysis of scientific Information,
The satellite "Resurs DK" 1 was launched on the elliptic orbit with the next parameters: 380–800 km and inclination 70.4°. Magnetic spectrometer "PAMELA" was placed in hermetic container, the axis of the device is directed to the local zenith (see Fig 2). Scientific information is downloaded when the satellite is lying over the ground receiving station located in NTsOMZ (Otradnoe, Moscow). This process is performed 3-4 times per day.
Fig. 2. The spacecraft "RESURS DK" 1. The particular information from spectrometer PAMELA is extracted at the ground receiving station from the full data stream of information. Expressanalysis of data quality is performing at ground station PAMELA in NTsOMZ, If necessary it's possible to send special command to satellite for adjustments of PAMELA'S work. These commands are sending to satellite from Flight
203
Control Center (Korolev city). Raw data is transferring by the fast Internet link to MEPhI and then using the international scientific system GRID to the scientific information center of INFN in Bologna. After that, all scientific groups process the PAMELA data and perform scientific analysis (Fig.3.).
+ I
I
r-~--..,
Qualitive Separation of analysis of the PAMELA's I--____~ data being information received
-Ground - station - -of PAMELA - 1 T@Ils6!cQ. In_ rrna!iQ.n : .---'---.., Express-analysis of the whole
Preparation of Initial dati for spaaecmt's Work Program WIthIn PAMElA
I" I
The Aight Control Center
I: Express-processing I :_ __________, '--_---J
On-line
1,·,·_...,
' - -_ _ _...J
Expr.ss-diagnosHcs of detectors
ME PhI GRID
(CNAF)
*-----'
~
,---
Italy, Gennany, Sweden
I
Total processing of events
__.*___
...J
NEPhI, Ioffe institute, lebedev institute
Fig. 3. The physical scheme of receiving and processing of PAMELA data.
4
Preliminary results
Since the real beginning of experiment in July 2006 till August 2007 the spectrometer PAMELA has downlinked approximately 5 TB of raw data for analysis. Around 108 particles were chosen for the first phase of analysis. Mainly they are protons with energy range 108 -;- 2 . lOll eV, and also helium nuclei and nuclei of some other light elements, electrons, positrons and antiprotons. In contrast with previous measurements made, this magnetic spectrometer can register the fluxes of particles in the radiation belts of Earth. That allowed it to obtain the unique information about high-energy particles captured by magnetic field. The figure 4 shows 2 reconstructed events of antiproton and positron with high energy. In the diagrams (Fig. 5 and 6) are shown the distribution of particles' fluxes depending on rigidity (momentum) and dEjdx (losses of energy). This picture demonstrates the real capabilities of this scientific apparatus in flight. We consider that we will obtain new experimental information, which will be needed to help in the solving of the tasks declared in the introduction of this report.
204
Ge V antiproton (dark color of neutron detector means that there are registered neutrons), right - 92 Ge V nn.nt·.n'-A?"'')'"'' in the energy range 1-;-10 Ge V to the antlDroi~orls are created in thenteractions of nr.TYH.r" lU~'U-'''A''/H'. interstellar gas the next
205
7. The ratio
depending on kinetic energy
8. The flux depending on kinetic energy. It has detected several tens of thousands of leptons. That will soon make to estimate the ratio in the energy range 0.1+50 which is like the ratio with the problem of the nature of dark matter. has detected several millions of helium nuclei and has shown that ratio of antihelium to helium is rv • It is that this value will be in experiment . But the value conditions for models the there is no difference between energy Drcltoxls and helium nuclei within the range of measured
206
energies (fig. 8). But further investigation of these spectra could show the difference. Now we carefully study the ratio H e~j H e~ and BjC for different energies. These ratios are very important for understanding the process of interaction of high energy cosmic rays in galaxy with interstellar gas. In particular, it's very important for understanding of fluxes of secondary antiprotons and positrons. The fluxes He and protons have been measured in solar cosmic rays during the solar flare 13.12.2006. In particular, it has detected fluxes of He-4 nucleus with energy up to several GeV[6]. Finally by using large amounts of statistical material it is shown that there is a considerable predominance of electron fluxes over the amount of positron fluxes that appear in the energy range 50-;.-300 Me V in the radiation belts of the Earth. It is already known that positrons are dominant at the boundary of the radiation belt. 5
Conclusion
The ground station NTsOMZ is receiving approximately 15GB of scientific data information daily. This Russian-Italian Mission (RIM) will continue according to plan until the middle of 2009. However the main and new scientific results will be achieved earlier. Project RIM-PAMELA is supported by Russian Academy of Science and Russian Cosmic Agency, the Italian National Institute of Nuclear Physics, Italian Space Agency, German Space Agency, Swedish National Space Board and Swedish Research Council. 6
Acknowledgement
We would like to thank the Ministry of Education and Science of Russian Federation for financial support within grant RNP 2.2.2.2.8248. References
[1] [2] [3] [4] [5] [6]
L. Bergstrom et al., Phys.Rev. D 59 (1999) 43506. D. Hooper et al., Phys.Rev. D 71 (2005) 083503. T. Bringmann , JeAP 08 (2005) 006; astro-phj0506219 (2005). M. Simon et al., Astrophys. J. 499 (1998) 250. P. Picozza et al., Astroparticle Physics 27 (2007) 296. M. Casolino et al., in "Solar cosmic ray observations with PAMELA experiment" (Proc. 30th International Cosmic Ray Conference) ,Merida, Mexico (2007).
DARK MATTER SEARCHES WITH AMS-02 EXPERIMENT A.Malinin a, For AMS Collaboration [PST, University of Maryland, MD-20742, College Park, USA Abstmct. The Alpha Magnetic Spectrometer (AMS), to be installed on the International Space Station, will provide data on cosmic radiations in a large range of rigidity from 0.5 GV up to 2 TV. The main physics goals in the astroparticle domain are the anti-matter and the dark matter searches. Observations and cosmology indicate that the Universe may include a large amount of unknown Dark Matter. It should be composed of non baryonic Weakly Interacting Massive Particles (WIMP). A good WIMP candidate being the lightest SUSY particle in R-parity conserving models. AMS offers a unique opportunity to study simultaneously SUSY dark matter in three decay channels from the neutralino annihilation: e+, anti-proton and gamma. The supersymmetric theory frame is considered together with alternative scenarios (extra dimensions). The expected flux sensitivities in 3 year exposure for the e+ /e- ratio, anti-proton and gamma yields as a function of energy are presented and compared to other direct and indirect searches.
Introduction The first evidence for the existence of Dark Matter comes from the observation of rotation velocities across the spiral galaxies, derived from the variation in the red-shift. The rotation velocities rise rapidly from the galactic center, then remain almost constant to the outermost regions of a galaxy. The observations are consistent with the gravitation motion only if the matter in the Universe is mostly non luminous" dark matter". The recent WMAP results [1] confirm that about 83% of the matter in the Universe exists in the form of cold Dark Matter (DM). The mystery of the Dark Matter remains unsolved. Many candidates such as massive neutrino, Universal Extra Dimensions Kaluza-Klein states and Super Symmetry theory (SUSY) heavy neutralinos were proposed. If Dark Matter, or a fraction of it, is non-baryonic and consists of almost noninteracting particles like neutralinos, it can be detected in cosmic rays through its annihilation into positrons or anti-protons, resulting in deviations (in case of anti-protons) or structures (in case of positrons) to be seen in the otherwise predictable cosmic ray spectra [2]. Considering the hypothesis of a possible clumpy DM, the expected fluxes of such primary positrons, 'Y-s or anti-protons may be enhanced [3] since the annihilation rate is proportional to the square DM density contrary to the direct DM searches which will suffer from a decreased probability for the Earth to be contained into an eventual DM clump. ae-mail: [email protected]
207
208 1
AMS-02 Instrument
The Alpha Magnetic Spectrometer (AMS) is a particle physics experiment in space. Its initial space mission on board of the Space Shuttle Discovery (STS91) in June, 1998 confirmed the basic concept of the experiment [4]. During this short flight AMS measured of the GeV cosmic-ray fluxes over most of the Earth's surface [5-8], and provided the impetus to upgrade the instrument for the ISS 3 year mission (hereafter called AMS-02). These upgrades include among others a stronger, BL2 = 0.9T super conducting magnet to achieve the maximal detectable rigidity of 1 TV (the rigidity resolution better than 2% up to 20 GV) in the Silicon 'Thacker, as well as the addition of a 'Thansition Radiation Detector (TRD), a Ring Imaging Cherenkov (RICH) and an Electromagnetic Calorimeter (ECAL). The upgraded instrument will provide data on cosmic radiation in a large range of energy from a fraction of Ge V to 3 Te V with very high accuracy and free from the atmospheric corrections needed for balloon-born measurements. Its main physics goals in the astroparticle domain are the Antimatter and the Dark Matter searches as well as the cosmic ray composition and propagation study.
2
AMS-02 Sensitivity for DM search
The Monte Carlo study, based on the AMS-02 mathematical model, was performed to estimate the instrument sensitivity for the indirect DM search channels [9-13]. More than 109 events containing p+-, He, e+- and'Y at different energies have been fully simulated [14-17] passing through the detector and then reconstructed. The results of the study in the anti-proton, e+ and'Y cannels are presented in the figures 1-9. The background rejection factors up to 10- 6 necessary to extract the tiny anti-proton and e+ signals were achieved by combining the redundant information from the TRD, RICH and ECAL detectors. The selection criteria were tuned and the resulting efficiencies were used to simulate the measured spectra. Comparison with the existing data demonstrates that the AMS-02 will have an adequate sensitivity to address the enhancement in the positron fraction measurement reported by HEAT [9,10]' simultaneously constraining the DM signal parameter space by combining the anti-proton, e+ and 'Y channels [18]. The Galactic Center 'Y signal measurement by AMS-02 would provide 95% CL exclusion limits for several mSUGRA models in 3 years.
209 O.06
r------------_-.
~O.20
'5 § O.B
i
()
()
0::1 0.10
~ ·g.0.()5
Average Acc(1<E~O~l
'7'1
_'1
"1)
"
flf.SSOO 10 '
~
~Hr'
~!O'
.rJ"
t
"
~IU--l
_ "_ 10'
1 ~!
10
"
-'
K""-,~-KI.jn,,glmJ
~w'.o~~
lO '
,,' 10
Kinetic energy l GeV ]
Kinetic energy [GeV]
Figure 7: Combined example. The anti-proton flux as a function of kinetic energy, assuming 150 GeV SUSY neutralino mass (left plot) and 50 GeV Kaluza-Klein boson mass (right plot) for 3 years of AMS-02 data taking.
----Standatdsignnl
10-1
10
KalILl8·KJeinsignal
10
Energy [GeV ]
AMSOl projection (3 y.)
----StBndatdsignal
AMS02pcojeclion(3y.)
Susysigl1l!J
Energy [GeV 1
Figure 8: Combined example. The positron flux as a function of energy, assuming 150 GeV SUSY neutralino mass (left plot) and 50 GeV Kaluza-Klein boson mass (right plot) for 3 years of AMS-02 data taking.
----~~"
j(r!
A~QZ"' of. 0, the chiral symmetry is broken dynamically, and if < /::,. b > of. 0, the color symmetry is broken. The effective action for boson fields Seff can be expressed through the integral over quark fields, according to
In the mean field approximation, the fields a, if, /::,.b, /::,.*b can be replaced by their groundstate averages: < a >, < if >, < /::,. b > and < /::,. *b >, respectively. Let us choose the following ground state of our model: < /::,.1 > = < /::,. 2 > = < if > = 0, and denote < a >, < /::,.3 > of. 0, by letters a, /::,.. Evidently, this choice breaks the color symmetry down to the residual group SUc (2). Let us find the effective potential of the model with the global minimum point that will determine the quantities a and /::,.. By definition Seff = - Veff dD xFg, where
J
-
Veff
3
Sq = -~'
V
=
J
d D xF9.
(3)
Static Einstein universe
We will use the static D-dimensional Einstein universe as the simple example of curved spacetime. The line element is
(4) where a is the radius of the universe, related to the scalar curvature by the relation R = (D - l)(D - 2)a- 2 . The effective potential at finite temperature or thermodynamic potential may be obtained in the following form (for more details see [9]) ,,2 O(a, /::,.) = 3 ( 2G, N
00
2: dl {El + Tin (1 + e-t3CEI-lLl) + Tin (1 + e-t3CEI+lLl)}
-::,; (Nc - 2)
-::,; E N
1Ll.12) + e;-
00
+2TIn
dl
1=0 {
J(EI - J1.)2
+ 41/::,.1 2 + J(EI + J1.)2 + 41/::,.1 2 +
(1 + e- y'CE -lLl 2+41Ll.1 2) + 2T In (1 + e- t3 y'CE t3
1
where V is the volume of the universe V(a) El
=
V1( D-1)2 - 1+ - +a a 2 ' 2
2
1 +1L)2+41Ll.1
2 ) },
= 27l'D/2a D- 1 /r( ~), 1= 0, 1,2 ... ,
(5)
(6)
243
and d _
2[(D+l)/2 j qD
+l -
l!r(D _ 1)
I -
1) ,
(7)
where [xl is the integer part of x. 4
Phase transitions
In what follows, we shall fix the constant G 2 , similarly to what has been done in the flat case [5,10], by using the relation G 2 = ~Gl. For numerical estimates, let us choose the constant G 1 = 10 such that the chiral and/or color symmetries were completely broken. Moreover, let us now limit ourselves to the investigation of the case D = 4 only. In Fig. 1, the J1, - R-phase portrait of the system at zero temperature is depicted.
6 4
2 ~
o
2 ______ L-_____________ R 10
20
30
40
Figure 1: The phase portrait at T=O. Dashed (solid) lines denote first (second) order phase transitions. The bold point denotes a tricritical point.
For points in the symmetric phase 1, the global minimum of the thermodynamic potential is at (j = 0, t. = 0 (chiral and color symmetries are unbroken). In the region of phase 2, only chiral symmetry is broken and (j =1= 0, t. = O. The points in phase 3 correspond to the formation of the diquark condensate (color superconductivity) and the minimum takes place at (j = 0, t. =1= O. Moreover, the oscillation effect clearly visible in the phase curves of Fig. 1 should be mentioned. This may be explained by the discreteness of the fermion energy levels (6) in the compact space. This effect may be compared to the similar effect in the magnetic field H, where fermion levels are also discrete (the Landau levels). In Fig. 2, J1, - R- and T - J1,- phase portraits are depicted. It is clear from Fig.2 that with growing temperature both the chiral and color symmetries are restored. The similarity of plots in R - J1, and J1, - T axes leads one to the conclusion that the parameters of curvature R and temperature T play similar roles in restoring the symmetries of the system.
244
3.5
Jl
5
Jl
3
2.5 2
3
3
1.5 2 2
0.5 L---~~----'----R
4
6
8
10
12
14
16
'--------------L--T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Figure 2: The phase portraits at T=O.35 (left picture) and at R=3 (right picture).
We have considered the phase transitions in the static Einstein universe at finite temperature and chemical potentia!. In spite of the model character of the problem, we hope that the results of this paper may stimulate further investigations that are closer to realistic cosmological or astrophysical situations. Acknow ledgments
One of the authors (A.V.T.) is grateful to Prof. M. Mueller-Preussker for his attention and support of this work. This work was also supported by DAAD. References
[1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961); ibid. 124, 246 (1961); V. G. Yaks and A. I. Larkin, ZhETF 40, 282 (1961). [2] B.C. Barrois, Nuc!. Phys. B 129, 390 (1977). [3] D. Bailin and A. Love, Phys. Rept. 107, 325 (1984). [4] M. Alford, K. Rajagopal and F. Wilczek, Nuc!. Phys. B 537, 443 (1999); K. Langfeld and M. Rho, Nuc!. Phys. A 660, 475 (1999). [5] J. Berges and K. Rajagopal, Nuc!. Phys. B 538, 215 (1999). [6] T.M. Schwarz, S.P. Klevansky and G. Papp, Phys. Rev. C 60, 055205 (1999). [7] T. Inagaki, S.D. Odintsov and T. Muta, Prog. Theor. Phys. Supp!. 127, 93 (1997), hep-th!9711084 (see also further references in this review paper). [8] X. Huang, X. Hao, and P. Zhuang, hep-ph!0602186. [9] D.Ebert, A.V.Tyukov, and V.Ch. Zhukovsky, Phys. Rev. D76, 064029 (2007). [10] D. Ebert, V.V. Khudyakov, V.Ch. Zhukovsky, and K.G. Klimenko, Phys. Rev. D 65, 054024 (2002); D. Ebert, K.G. Klimenko, H. Toki, and V.Ch. Zhukovsky, Prog. Theor. Phys. 106,835 (2001).
CONSTRUCTION OF EXACT SOLUTIONS IN TWO-FIELDS MODELS Sergey Yu. Vernov a Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119991 Moscow, Russia Abstract.Dark energy model in the Friedmann Universe with a phantom scalar field, an usual scalar field and the polynomial potential has been considered. We demonstrate that the superpotential method is very effective to seek new solutions and to construct a two-parameter set of exact solutions to the Friedmann equations. We show that the standard formulation of superpotential method can be generalized.
1
Introduction
One of the most important recent results of the observational cosmology is the conclusion that the Universe expansion is speeding up rather than slowing down. The combined analysis of the type Ia supernovae, galaxy clusters measurements and WMAP (Wilkinson Microwave Anisotropy Probe) data gives strong evidence for the accelerated cosmic expansion [1-3]. To specify a component of a cosmic fluid one usually uses a phenomenological relation between the pressure p and the energy density {} corresponding to each component of fluid p = W{}, where W is called as the state parameter. The experimental data suggests that the present day Universe is dominated by a smoothly distributed slowly varying cosmic fluid with negative pressure, the so-called dark energy. Contemporary experiments [1-3] give strong support that the Universe is approximately spatially flat and the dark energy state parameter w DE is currently close to -1: w DE = -1 ± 0.2. The state parameter wDE == -1 corresponds to the cosmological constant. As it has been shown in [4] for a large region in parameter space an evolving state parameter wDE is favoured over w DE == -1. The standard way to obtain an evolving state parameter WDE is to include scalar fields into a cosmological model. Two-fields cosmological models, describing the crossing of the cosmological constant barrier wDE == -1, are known as quintom models and include one phantom scalar field and one usual scalar field. Nowadays the string field theory (8FT) has found cosmological applications related to the acceleration of the Universe. In phenomenological phantom models, describing the case WDE < -1, all standard energy conditions are violated and there are problems with stability (see [5] and references therein). Possible way to evade the instability problem for models with WDE < -1 is to yield a phantom model as an effective one, which arises from more fundamental theory with a normal sign of a kinetic term. In this paper we consider a 8FT inspired gravitational models with two scalar fields and a polynomial potentials. We propose new formulation of superpotential method, which is more suitable to construct models with a two-parameter set of exact solutions. ae-mail: [email protected]
245
246
2
String Field Theory Inspired Two-Fields Model
We consider a model of Einstein gravity interacting with a phantom scalar field ¢ and a standard scalar field ~ in the spatially flat Friedmann Universe. We assume that a phantom scalar field represents the open string tachyon, whereas the usual scalar field corresponds to the closed string tachyon [6-8]. Since the origin of the scalar fields is connected with the string field theory the action contains the typical string mass Ms and a dimensionless open string coupling constant go:
s=
JdxA (::1;R + :~ (~gI'V(al'¢av¢ 4
-
al'~av~) - V(¢,~))),
(1)
where Mp is the Planck mass. The Friedmann metric gl'v is a spatially flat:
(2) where a(t) is a scale factor. The coordinates (t, Xi) and fields ¢ and dimensionless. If the scalar fields ¢ and ~ depend only on time, then
H2 =
_1_ ( _
3m~
if =
¢ + 3H 4> =
~~
~ 12 2'1-'
~ c2
+ 2'" +
v) ,
2~~ (4)2 - e) ,
== V,;,
~ + 3He =
~
are
(3)
(4)
-
~~ == - V(
(5)
For short hereafter we use the dimensionless parameter mp: m~ = g;M~jM:. Dot denotes the time derivative. The Hubble parameter H == a(t)ja(t). Note that only three of four differential equations (3)-(5) are independent.
3
The Method of Superpotential
The gravitational models with one or a few scalar fields play an important role in cosmology and models with extra dimensions. One of the main problems in the investigation of such models is to construct exact solutions for the equations of motion. System (3)-(5) with a polynomial potential V(¢,~) is not integrable. The superpotential method has been proposed for construction of a potential, which corresponds to the particular solutions known in the explicit form [9]. The main idea ofthis method is to consider H(t) as a function (superpotential) of scalar fields: H(t) = W(¢(t),~(t)). (6) If we find such superpotential W (¢,~) that the relations
(7)
247
(8) are satisfied, then the corresponding cp, ~ and H are a solution of (3)-(5). The superpotential method separates system (3)-(5) into two parts: system (7), which is as a rule integrable for the given polynomial W (cp,~) and equation (8), which is not integrable if V(cp,~) is a polynomial, but has exact special solutions. The use of superpotential method does not include the solving of eq. (8). The potential V(cp,~) is constructed by means of the given W(cp,~). Relations (7) and (8) are sufficient, but maybe not necessary conditions to satisfy (3)-(5). To generalize them we assume that functions cp(t) and ~(t) are given by the following system of equations:
¢=
F(cp,~),
(9)
where F(cp,~) and G(cp,~) are some differentiable functions. We consider these functions as given ones and transform system (3)-(5) into equations in W(cp, ~):
W2 = _1_ ( _
3m~
W' F ¢
~ F2 2
+ W'G = E
~G2 +2 +
v) ,
_1_ (F2 _ G2 ) 2m2p '
(11)
F/pF + F~G + 3WF = V,;, G;PF + GeG + 3WG = -
(12)
vt
We differentiate equation (10) in
W ( W/p -
(10)
(13)
cp and use (12) to exclude VJ,:
2~~ F) = 6~~ (G¢ + FE) .
(14)
Similar manipulations give also
W(W~ + 2~~ G) = - 6~~ (G For any G(cp,~) and F(cp,~) such that G¢
2m;W';' = F,
=
¢
+ FE) .
-FE we find
2m;W~
= -G.
(15)
W(cp,~) using relations
(16)
These relations are equivalent to (7). In this case the equality G¢ = -FE is g~~ = g~~. So, we have shown that the relations (7) are necessary conditions if and only if the equality G¢ = -FE is satisfied. In other case one should solve nonlinear system (14)-(15) to find the corresponding superpotential W(cp,~). Note that in our formulation we do not use the explicit form of exact solutions to find potential. Note also that for one and the same one-parameter set of exact
248
solutions we can find different form of functions F and G and, therefore, the different form of potential V. We can conclude that the proposed formulation of the superpotential method is effective to seek potential V (¢, ~), which satisfies some conditions and corresponds to a two-parameter set of exact solutions. For example, if F and G are linear functions (A, Band C are constants): (17) then we obtain the fourth degree polynomial potential
V =
A2
2 - B2 ¢2+B(A+C)¢~+ B2 - C e + -3 -2 ( A¢2+2B¢~-Ce ) 2 . (18)
2
2
16mp
and two-parameter set of exact solutions for the Friedmann equations. For example, at C = 2B + A we obtain (Cl and C 2 are arbitrary constants) and
~=
(C + C;; + l
C2t)
e(A+B)t.
(19)
More nontrivial example with the sixth degree polynomial potential and a twoparameter set of kink-like (¢) and lump~like (~) solutions is presented in [8]. Acknowledgments
Author is grateful to LYa. Aref'eva for useful discussions. This research is supported in part by RFBR grant 05-01-00758 and by grant NSh-8122.2006.2. References
[1] A. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116, 1009-1038 (1998); astro-ph/9805201. [2] D.N. Spergel et al. [WMAP Collaboration]' Astrophys. J. Suppl. SeT. 170 377-408 (2007); astro-ph/0603449. [3] Tegmark et al. [SDSS Collaboration], Phys. Rev. D69 103501 (2004); astro-ph/0310723. [4] U. Alam, V. Sahni, T.D. Saina, A.A. Starobinsky, Mon. Not. Roy. Astron. Soc. 354, 275 (2004); astro-ph/0311364 [5] LYa. Aref'eva, LV. Volovich, 2006, hep-th/0612098. [6] LYa. Aref'eva, AlP Conf. Proc. 826301-311 (2006); astro-ph/0410443. [7] LYa. Aref'eva, A.S. Koshelev, S.Yu. Vernov, Phys. Rev. D72, 064017 (2005); astro-ph/0507067. [8] S.Yu. Vernov, 2006, astro-ph/0612487. [9] O. DeWolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Phys. Rev. D62, 046008 (2000); hep-th/9909134.
QUANTUM SYSTEMS BOUND BY GRAVITY Michael L. Fil'chenkov a , Sergey V. Kopylov b , Yuri P. Laptev C Institute of Gravitation and Cosmology, Peoples' Friendship University of Russia, Moscow Department of Physics, Moscow State Open University Department of Physics, Bauman Moscow State Technical University Abstract. Quantum systems contain charged particles around mini-holes called graviatoms. Electromagnetic and gravitational radiations for the graviatoms are calculated. Graviatoms with neutrino can form quantum macro-systems.
1
Introduction
As known, there exist bound quantum systems due to electromagnetic and strong interactions, e.g. atoms, molecules and atomic nuclei. If one component of the gravitationally bound system is assumed to be massive and the other is an elementary particle, then a quantum system can be formed, e.g. mini-holes in the early Universe [1-3]. Such systems are called graviatoms [4]. Another example of the quantum systems bound by gravity is macro-bodies capturing neutrinos having de Broglie's wave length of macroscopic value.
2
Theoretical solution to the graviatom problem
Schrodinger's equation for the graviatom [2]
~~
[r2 (dR pl )] r2 dr dr
2 2 _l(l r2+ 1) R P1+ 2m (E _ mc rQr g + mc r g ) 1i 2 4r2 2r
R 1 = 0 (1) P
describes a radial motion of a particle with the mass m in the mini-hole potential, where rg = 2GM/c 2 and M are the mini-hole gravitational radius and mass respectively. The energy spectrum is of hydrogen-like form
E 3
G2 M 2 m 3 21i 2 n 2
= ----,:--
(2)
Graviatom existence conditions
A graviatom can exist if the following conditions are fulfilled [4]: 1) the geometrical condition L > r 9 + R, where L is the characteristic size of the graviatom, R is that of a charged particle; 2) the stability condition: (a) Tgr < TH, where Tgr is the graviatom lifetime, TH is the mini-hole lifetime, (b) Tgr < T p , where Tp is the particle lifetime (for unstable particles); 3) the indestructibility condition (due to tidal forces and Hawking's effect) Ed < Eb, where Ed is the destructive energy, Eb is the binding energy.
249
250
L-,---~--_--
0.5
1.0
____
~
__
2.5
2.0
1.5
~
3.0
_ _ _ _ _ m·l0· 3.5
21
,g
4.0
Figure 1: The dependence of mini-hole masses on the charged particle masses satisfying the graviatom existence conditions. The light curves indicate the range of values related to the geometrical condition (the upper curve) and to Hawking's effect ionization one (the lower curve). The heavy curve is related to the particle stability condition (Tp = 1O- 22 s).
The charged particles able to be constituents of the graviatom are: the electron, muon, tau lepton, wino, pion and kaon. The conditions of existence the graviatoms reduce to the relation between the masses of the mini-hole and particle, with their product being approximately constant equal to the Planck mass squared. 4
Graviatom radiation
The intensity of the electric dipole radiation of a particle with mass m and charge e in the gravitational field of a mini-hole reads [4] 2
d
I fi
=
2fie w7f lif me3
'
(3)
where wif = (Ei - E f )/fi is the frequency of the transition i --> I and lif is the oscillator strength [5]. The electric quadrupole radiation intensity for the transition 3d --> Is is q
113
=
6fie 2 wg1 3 me
hd--+1s'
(4)
The gravitational radiation intensity for the graviatom performing the transition 3d --> Is reads 9 _ 6fiGMwg 1 113 3 hd--+1s' (5) e The mini-hole creates particles near its horizon due to Hawking's effect, its power [7].
251 Table 1: Parameters for graviatoms with the electron and wino [6].
electron 0.511
mc"'!', MeV
8
00
M, g L, cm
3.5.10 17 6.10 11 0.08 2. lOw
T,
hW12,
Id(2p
MeV 18), erg.
--t
8 ·1
wino 8 . lOt> 5. 10 -10 2.2. lOll 4.10 17 1.2. lOt> 4 .1O:.!",!.
The mini-holes being constituents of the graviatoms are formed due to Jeans' gravitational instability at the times about ~ = 10- 27 -;.-10- 21 8 from the initial c singularity. The mini-hole masses for the graviatoms involving electrons, muons and pions exceed the value of 4.38 . 1014g, which means that it is possible for such graviatoms to have existed up to now [7]. The quantity G~m = 0.608 -;.- 0.707 is a gravitational equivalent of the fine structure constant. The gravitational radiation intensities two orders exceed the electromagnetic ones. The graviatom dipole radiation energies and intensities have proved to be comparable with those for Hawking's effect of the mini-holes being constituents of the graviatoms. 5
Systems with neutrinos
De Broglie's wavelength for the neutrino with mass mv is
1) Graviatom The existence conditions: a~ = li.dB > 3rg, Tgr < energy: mvc2 rv 1 eV. Characteristic frequency:
TH.
Electron neutrino rest
Gravitational radiation: Igr =
Q9M9 m ll
c5 h lO
v
Mini-hole masses: 10 18 g < M < 10 23 g. For example, if M < 10 23 g, then hw~ < 0.2 eV,Igr size is about 10 1 -;.- 106 cm.
(8)
< 0.2 erg·8- 1 . System
252 2) Macroscopic system (comet nuclei, meteorites, small asteroids) Macro-bodies capture neutrinos onto both Bohr's hydrogen-like levels (outside the body) and Thomson's oscillatory ones (inside the body). Macro-body masses: 10 14 g < M < 10 19 g. Bohr's radius is about:
1 -;- 10 5 cm. The oscillation frequency w intensity
=
J!7fpG, the gravitational radiation (9)
where p is the macro-body density. Let consider the average density of a macro-body p equal to 4 g·cm- l . Then, we obtain the following parameters: fiw = 9.10- 19 eV, 1mb = 10- 104 erg· 8- 1 . It is of interest to note that the rotation curves of galaxies give an aI-most constant velocity v on their periphery, which for v 2 "" G M / R leads to the dependence of dark matter mass Mdm "" R, similar to the dependence of the mass of neutrinos on Bohr's radius L, since L = a~n2, and the total mass of all neutrinos on the nth level is equal to Mn = 2n 2 m v . Hence, we obtain Mn "" L. 6
Conclusion
The graviatom can contain only leptons and mesons. The observable stellar magnitude for graviatom electromagnetic radiation exceeds 23m. Stable graviatoms with baryon constituents are impossible. The internal structure of the baryons, consisting of quarks and gluons, should be taken into account. There occurs a so-called quantum accretion of baryons onto a mini-hole. The whole problem is solvable within the frame-work of quantum chromo dynamics and quantum electrohydrodynamics. Neutrinos can form quantum macro-systems. The description of gravitationally bound macro-systems with neutrinos may be helpful for solving the dark matter problem in the Universe. References
[1] [2] [3] [4]
A.B. Gaina, PhD Thesis, Moscow State University, Moscow, 1980. M.L. Fil'chenkov, Astron. Nachr. 311, 223 (1990). M.L. Fil'chenkov, 1zvestiya Vuzov, Fizika No.7, 75 (1998). Yu.P. Laptev, M.L. Fil'chenkov, Electromagnetic and Gravitational Radiation of Graviatoms/ / Astronomical and Astrophysical Transactions. 2006. v. 25, No.1, p. 33 - 42 [5] H.A. Bethe and E.E. Salpeter, "Quantum Mechanics of One- and TwoElectron Atoms", Springer-Verlag, Berlin, 1957. [6] M. Sher, hep-th/9504257. [7] V.P. Frolov, in "Einstein Col." 1975-1976, Nauka, Moscow, 1978, p. 82-151.
CP Violation and Rare Decays
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SOME PUZZLES OF RARE B-DECAYS A.B. Kaidalov a
Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia Abstract. It is emphasized that a study of rare B-decays provides an important information not only on CKM-matrix, but also on QCD dynamics. It is shown that some puzzles in B-decays can be explained by final state interaction (FSI). The model for FSI, based on Regge phenomenology of high-energy hadronic interactions is proposed. This models explains the pattern of phases in matrix elements of B ---> 'Tr'Tr and B ---> pp decays. These phases play an important role for CPviolation in B-decays. It is emphasized that the large distance FSI can explain the structure of polarizations of vector mesons in B-decays and very large branching ratio of of the B-decay to SeAc.
1
Introduction
This short review of some unusual properties of matrix elements in hadronic B-decays is based on papers with M.l. Vysotsky [1,2J. Detailed information on B-decays, obtained in experiments at B-factories [3J, provide a testing ground for theoretical models. Investigation of rare B-decays and CP violation in these decays provides not only an information on CKM matrix, but also on QCD dynamics both at small and large distances. One of the most interesting and still not solved problems in B-decays is the role of FSI. In this paper I shall demonstrate, that FSI play an important role ill hadronic B-decays and allow one to explain some puzzles observed in rare B-decays. In particular it will be shown that phases due to strong interactions are substantial in some hadronic B-decays. These phases are important for understanding the pattern of CP-violation in rare B-decays. The model for calculation of FSI will be formulated and compared to the data on B ----> 1r1r and B ----> pp decays. The model is based on regge-picture for high-energy binary amplitudes and allows also to explain a pattern of helicity non-conservation in some B-decays to vector mesons. Large distance interactions provide a simple explanation of anomalously large branching ratio of the B-decay to 3cAc. 2
B
----> 1r1r /
B
---->
pp puzzle
The probabilities of three B ----> 1r1r and three B ----> pp decays are measured now with a good accuracy. There is a large difference between ratios of the charged averaged Bd decay probabilities to the charged and neutral mesons.
ae-mail: [email protected]
255
256 It was demonstrated in refs. [1,2J that this difference is related to the difference of phases due to strong interactions for matrix elements of B ---; 1T1T and B ---; pp-decays. The matrix elements of these decays can be expressed in terms of amplitudes with isospin (I) zero and two with phases 6o and 62. To take into account differences in CKM phases for tree and penguin contributions we separate the amplitude with 1=0 into the corresponding parts A o and P. The contributions of P can be determined, using SU(3)-symmetry from decays Bu ---; KO* p+ and Bu ---; K01T+ [4] and turn out to be rather small compared to tree contributions. Note, however, that P determines magnitudes of direct CPV in hadronic decays. If we neglect the penguin contribution, then the difference of phases is expressed in terms of the branching ratios as follows
cos(oo -
on = -J3
B+_ - 2B oo
4 Jf!B+oJB+-
+ ~ I!.L B+o T+,
+ Boo - ~f!B+o
(2)
Using experimental information on branching ratios of B ---; 1T1T-decays [3J we obtain 160 - 5~1 = 48°. Penguin contributions to Bik do not interfere with tree ones because CKM angle ex = 1T - (3 - 'Y is almost equal to 1T /2. With account of P-term we get:
(3) This value agrees with the result of analysis of ref. [5J:
6o - 62 = 40° ± 7° ,
(4)
Thus the difference of phases of matrix elements with 1=0 and 1=2 is not small in sharp contrast with factorization approximation often used for estimates of heavy mesons decays. For B ---> pp-decays we obtain in analogous way: (5)
This phase difference is smaller than for pions and is consistent with zero. The fact that phases due to FSI are in general not small for heavy quark decays is confirmed by other D and B-decays. The data on D ---> 1T+1T-, D ---> 1T01TO and D± ---> 1T±1TO branching ratios lead to [6J:
(6) The last example is B ---; D1T decays. D1T pair produced in B-decays can have I = 1/2 or 3/2. From the measurement of the probabilities of B- ---> D01T-, BO ---> D-1T+ and BO ---; D01To decays in paper [7J the FS1 phase difference of these two amplitudes was determined: (7)
257
Thus experimental data indicate that the phases due to FSI are not small for heavy mesons decays.
3
Calculation of the FSI phases of B plitudes
--+
7r7r and B
--+
pp decay am-
Let me remind that for K --+ 7r7r decays there are no inelastic channels, MigdalWatson (MW) theorem is applicable and strong interaction phases of S-matrix elements of K --+ (27r) I decays are equal the phases of the corresponding 7r7r --+ 7r7r scattering amplitudes at E = m K. For B-mesons there are many opened inelastic channels and MW theorem is not directly applicable. Serious arguments that strong phases should disappear in the MQ --+ 00 limit were given by B.J. Bjorken [8J. He emphasized the fact that characteristic configurations of the produced in the decay light quarks have small size"" l/MQ and FSI interaction cross sections should decrease as 1/ M~. Similar arguments were applied in the analysis of heavy quark decays in the QeD perturbation theory [9J. These arguments can be applied to the total hadronic decay rates. For individual decay channels (like B --+ 7r7r), which are suppressed in the limit MQ --+ 00 the situation is more delicate. However even in these situations the arguments of Bjorken that due to large formation times the final particles are formed and can interact only at large distances from the point of the decay seem relevant. On the other hand a formal analysis of different classes of Feynman diagrams, including soft rescatterings [10,11]. show that the diagrams with pomeron exchange in the FSI-amplitudes do not decrease as MQ increases. Same conclusions follow from applications of generalizations of MW-theorem [12,13J. In the process of analysis of FSI in heavy mesons decays it is important to understand a structure of the intermediate multi particle states. It was shown in ref. [2] that the bulk of multiparticle states produced in heavy mesons decays has a small probability to transform into two-meson final state and only quasi two-particle intermediate states XY with masses M.k(Y) < MBA QCD « M~ can effectively transform into the final two-meson state. In refs. [1, 2J in calculation of FSI effects for B --+ 7r7r and B --+ pp decays we considered only two particle intermediate states with positive G-parity to which B-mesons have relatively large decay probabilities. Alongside with 7r7r and pp there is only one such state: 7ral. I shall use Feynman diagrams approach to calculate FSI phases from the triangle diagram with the low mass intermediate states X and Y . Integrating over loop momenta d 4 k one can transform the integral over ko and k z into the integral over the invariant masses of intermediate particles X and Y
J
dkodk z =
2~~
J
dsxdsy
(8)
258 and deform integration contours in such a way that only low mass intermediate states contributions are taken into account while the contribution of heavy states being small is neglected. In this way we get:
I M7r7r
=
M(O)I(" XY U7r XU7r Y
+ 2'TJ=O) XY->7r'lf
,
(9)
where Mflf are the decay matrix elements without FSI interactions and Tiyo....7r7r is the J = 0 partial wave amplitude of the process XY -+ 'lor (TJ = (SJ - 1)/(2i)) which originates from the integral over d2 kJ... For real T (9) coincides with the application of the unitarity condition for the calculation of the imaginary part of M while for the imaginary T the corrections to the real part of M are generated. This approach is analogous to the FSI calculations performed in paper [14]. However in [14] 2 -+ 2 scattering amplitudes were considered to be due to elementary particle exchanges in the t-channel. For vector particles exchanges s-channel partial wave amplitudes behave as sJ-1 cv sO and thus do not decrease with energy (decaying meson mass). However it is well known that the correct behavior is given by Regge theory: s"'i(0)-1. For p-exchange Cfp(O) r::::J 1/2 and the amplitude decrease with energy as 1/.fS. This effect is very spectacular for B -+ DD -+ 'WIT chain with D*(D2) exchange in t-channel: CfD*(O) r::::J -1 and reggeized D* meson exchange is damped as S-2 r::::J 10- 3 in comparison with elementary D* exchange (see for example [13]). For 71'-exchange, which gives a dominant contribution to pp -+ 71'71' transition (see below), in the small t region the pion is close to mass shell and its reggeization is not important. Note that the pomeron contribution does not decrease for MQ -+ 00, however it does not contribute to the difference of phases IJg -J~ I which we are interested in. So this phase difference is determined by the secondary exchanges p, 71' and it decreases at least as 1/MQ for large MQ in accord with Bjorken arguments. For phases J and J~ I separately the pomeron contribution does not cancel in general. If Bjorken arguments are valid for these quantities it can happen only under exact cancellation of different diffractively produced intermediate states and it does not happen in the model of refs. [1,2]. Let us calculate the imaginary parts of B -+ 71'71' decay amplitudes. In the amplitude pp -+ 71'71' of pp intermediate state in eq.(9) the exchange by pion trajectory in the t-channel dominates. It is determined by the well known constant gP->7r7r' This contribution is the dominant one for B -+ 71'71' decays due to a large probability of B -+ pp-transition. On the contrary 71'71' intermediate state plays a minor role in B -+ pp-decays. In description of 71'71' elastic scattering amplitudes in eq.(9) contributions of P, f and p regge-poles was taken from ref. [15]. Finally 71'a1 intermediate state should be accounted for. Large branching ratio of Bd -+ 71'±ai-decay ( Br(Bd -+ 71'±ai) = (40 ± 4) * 10- 6 ) is partially compensated by small P71'al coupling constant (it is 1/3 of p71'71' one). As a result the contributions of 71' a 1
o
259 intermediate state (which transforms into 7r7r by p-trajectory exchange in tchannel) to PSI phases equal approximately that part of 7r7r intermediate state contributions which is due to p-trajectory exchange. Assuming that the sign of the 7ral intermediate state contribution into phases is the same as that of elastic channel and taking into account that the loop corrections to B -. 7r7r decay amplitudes leads to the diminishing of the (real) tree amplitudes by ~ 30% we obtain: (10) The accuracy of this prediction is about 15°. For pp final state analogous difference is about three times smaller, og - o~ ~ 15°. Thus the proposed model for FSI allows us ti explain the B -. 7r7r / B -. pp puzzle. 4
Direct CPV in B -. 7r7r-decays and phases of the penguin contribution
The direct CP-violation parameter C+_ in B -. 7r7r-decays is proportional to the modulus of the penguin amplitude and is sensitive to the strong phases of Ao, A2 and penguin amplitudes. So far we have discussed phases of the amplitudes Ao, A 2. The penguin diagram contains a c-quark loop and has a nonzero phase even in the QCD perturbation theory. It was estimated in ref. [lJ and is about 10°. Note that in PQCD it has a positive sign. Let us estimate the phase of the penguin amplitude op considering charmed mesons intermediate states: B -. DD, 15* D, DD*, 15* D* -. 7r7r. In Regge model all these amplitudes are described at high energies by exchanges of D*(D 2)-trajectories. An intercept of these exchange-degenerate trajectories can be obtained using the method of [16J or from masses of D* (2007) - land D2 (2460) - 2+ resonances, assuming linearity of these Regge-trajectories. Both methods give aD- (0) = -0.8 -;- -1 and the slope a~" ~ 0.5GeV- 2 . The amplitude of D+ D- -. 7r+7r- reaction in the Regge model proposed in papers [17J can be written in the following form: 2
T DD-->1r7r (s, t) = -
g; e-
i1r
(t)r(l - aD" (t)) (S/ Sed)D" (t)
,
(11)
where r(x) is the gamma function. The t-dependence of Regge-residues is chosen in accord with the dual models and is tested for light (u,d,s) quarks. According to [17J Sed ~ 2.2GeV 2 . Note that the sign of the amplitude is fixed by the unitarity in the t-channel (close to the D*-resonance). The constant is determined by the width of the D* -. D7r decay: g6/(167r) = 6.6. Using (9) and the branching ratio Br(B -. DD) ~ 2 . 10- 4 we obtain the imaginary part of P and comparing it with the contribution of Pin B -. 7r+7r- decay probability we get Op ~ -3.5°.
95
260
The sign of op is negative - opposite to the positive sign which was obtained in perturbation theory. Since D V-decay channel constitutes only ~ 10% of all two-body charm-anticharm decays of Bd-meson, taking these channels into account we easily get (12) which may be very important for interpretation of the experimental data on direct CP asymmetry. It was shown in ref. [2], that assuming that phases satisfy to the conditions: 00 - 02 = 37°,02 > 0 and 0p > 0, it is possible to obtain the following inequality
C+_ > -0.18.
(13)
The experimental results obtained by Belle [18] and BABAR [19] are contradictory (14) c2~le = -0.55(0.09) , C2~BAR = -0.21(0.09), Belle number being far below (13). For non-perturbative phase of the penguin contribution (12) the value of theoretical prediction for C+- can be made substantially smaller and closer to the Belle result. 5
Polarizations of vector mesons in B --' VV -decays
Short distance contributions to vector meson production in B-decays lead to a dominance of the longitudinal polarization of vector meson. This is a general property valid in the large MQ- limit due to helicity conservation for vector currents and corrections should be '" M~ / M~. It is satisfied experimentally in B --' p+p_ decays, where the contribution of longitudinal polarization of p mesons is h = f L/f = 0.968 ± 0.023. On the other hand there are several B-decays to vector mesons, where longitudinal polarizations give only about 50% of decay rates. For example: for B+ --' K*op+ h = 0.48 ± 0.08, BO --' K*opo h = 0.57 ± 0.12, B+ --' ¢K*+ h = 0.50 ± 0.07, BO --' ¢K*o h = 0.491 ± 0.032 [3]. This is a real puzzle if only short distance dynamics for these decays is invoked. First let us note that in all decays, where f L ~ 50% penguin diagrams give dominant contribution. In this case a large contribution to the matrix elements of decays comes from DDs(D* V., DV;, .. ) intermediate states, which have large branching ratios. The amplitude of the binary reaction DVs --+ VV at high energies is dominated by the exchange of D* -regge trajectory and according to general rules for spin-structure of regge vertices (see for example [20]) final vector mesons are produced purely transversely polarized. Thus we expect a large fraction of transverse polarization of vector mesons in these decays. A value of h is sensitive to intercept of D*-trajectory [21]. If the
261
penguin contribution in the decays indicated above is dominant in the SU(3) limit we have:
and h in all these decays should be the same. These predictions agree with experimental data [3].
6
Puzzle of charm-anticharm baryons production
Large probability of B-decay to Ae3e has been observed recently: Br(B+ -; At3~ rv 10- 3 ) [3]. It is surprisingly large compared to the branching of Bdecay to Atf5 = (2.19 ± 0.8)10- 5 • From PQCD point of view both processes ~re described by similar diagrams with a substitution of ud (for p) by cs (for Be) and phase space arguments even favor p-production. On the other hand from the soft rescatterings point of view large DDs(.D* D s , D D; , .. ) intermediate states, considered in the previous section, can play an important role in B+ -; At3~-decays. For Atp final states corresponding two-meson intermediate states have smaller branchings and, what is even more important, have different kinematics. For DDs, .. intermediate states the momentum of these heavy states is not large (p ~ 1.8GeV) in B rest frame and all light quarks (u, d, d, s are slow in this frame. Final At3~ are also rather slow in the B-rest frame and thus all quarks have large projections to the wave functions of the final baryons. On the contrary for 7r D, pD, .. intermediate states in Atp-decays momenta of ii, d- quarks in light mesons are large and projections to the wave functions of final baryons have extra smallness. The resulting suppression can be estimated in regge-model of ref. [17] with nucleon trajectory exchange in the t-channel and is rv 10- 2 in accord with experimental observation,
7
Conclusions
FSI play an important role in two-body hadronic decays of heavy mesons. Theoretical estimates with account of the lowest intermediate states give a satisfactory agreement with experiment and provide an explanation of some puzzles observed in B decays.
Acknowledgments This work was supported in part by the grants: RFBR 06-02-17012, RFBR 06-02-72041-MNTI, INTAS 05-103-7515, Science Schools 843.2006.2 and by Russian Agency of Atomic Energy.
262
References
[1] [2] [3] [4] [5] [6] [7] [8] [9]
A.B. Kaidalov, M.I. Vysotsky, Yad. Fiz. 70,744 (2007). A.B. Kaidalov, M.l. Vysotsky, Phys. Lett. B652, 203 (2007). HFAG, http://www.slac.stanford.edu/xorg/hfag. M. Gronau, J.L. Rosner, Phys. Lett. B595, 339 (2004). C.-W. Chiang, Y.-F. Zhou, JHEP 0612,027 (2006). CLEO Collaboration, M. Selen et al., Phys. Rev. Lett. 71, 1973 (1993). CLEO Collaboration, S. Ahmed et al., Phys. Rev. D66, 031101 (2002). J.D. Bjorken, Nucl. Phys. (Proc. Suppl.) Bll, 325 (1989). M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, Nucl. Phys. B606, 245 (2001). [10] A. Kaidalov, Proceedings of 24 Rencontre de Moriond "New results in hadronic interactions", 391 (1989). [11] J.P. Donoghue, E. Golowich, A.A. Petrov and J.M. Soares, Phys. Rev. Lett. 77, 2178 (1996). [12] M. Suzuki, L. Wolfenstein, Phys. Rev. D60, 74019 (1999). [13] A. Deandrea et al., Int. J. Mod. Phys. A21, 4425 (2006). [14] H-Y. Cheng, C-K Chua and A. Soni, Phys. Rev. D71, 014030 (2005). [15] KG. Boreskov, A.A. Grigoryan, A.B. Kaidalov, I.I. Levintov, Yad. Fiz. 27,813 (1978). [16] A.B. Kaidalov, Zeit. fur Phys. C12, 63 (1982). [17] KG. Boreskov, A.B. Kaidalov, Sov.J.Nucl.Phys. 37, 109 (1983). [18] H.Ishino, Belle, talk at ICHEP06, Moscow (2006). [19] B.Aubert et ai, BABAR Collaboration, hep-ex/0703016 (2007). [20] A.B. Kaidalov, B.M. Karnakov, Yad. Fiz. 3,1119 (1966). [21] M. Ladissa, V. Laporta, G. Nardulli and P. Santorelli, Phys. Rev. D70, 114025 (2004).
MEASUREMENTS OF CP VIOLATION IN b DECAYS AND CKM PARAMETERS Jacques Chauveau a LPNHE,IN2P3/CNRS, Univ. Paris-6, case courrier 200, -4 place Jussieu, F-75252 Paris Cedex as, Prance Abstract. After a brief review of CP violation phenomenology in the standard model I depict recent measurements of the CKM angles. The emphasis is on the latest determinations of the angle (3 = 1 made using amplitude analyses of threebody final states. The results of the CKM fits to date are used to summarize the talk and put the subject into perspective.
1
CP Violation in the Standard Model
The B factories have established that the Standard Model (SM) accommodates CP violation via the Cabibbo Kobayashi Maskawa (CKM) formalism [1] where the couplings of the W boson to quarks include the elements of the fundamental unitary CKM matrix V. For three generations, V depends on four real parameters, one of which is an irreducible imaginary part that induces opposite sign weak phases for CP-conjugate transitions of quarks and antiquarks. The Wolfenstein parameterization [2], 1 - >-2/2 ->-
A>-3(1 - p - iT})
is the first term of an expansion in the small parameter A = sin Be (Be is the Cabibbo angle). By changing p and T} to 75 and 'fj defined in reference [3], an approximation good to 0(A6) is obtained which is widely used. The unitarity relation between the first and third columns of V is conveniently drawn on Figure 1 as a triangle, the Unitarity Triangle (UT), in the complex plane.
Figure 1: The CKM Unitarity Triangle UT. At 0(>-6), the apex coordinates are (p, 7i). Because of CP violation, the triangle is not squashed to the real axis. Its angles are constrained by CP violation experiments.
1\ .---------
A I
V.nv~
iVcdV.t
The apex of the UT has coordinates (75, Yj). Present measurements vindicate the SM with precisions of 0.5% on A, 2% on A, 20% on 75 and 7% on 'fj. These numbers are upper limits to possible New Physics corrections to the flavor sector of ae-mail: [email protected]
263
264 the SM. Such effects are actively searched for in an experimental programme seeking to overconstrain the position of the UT apex from all relevant measurements. In this talk I concentrate on the CP observables and the associated UT angles measurements. On August 21, 2007 the BABAR and Belle experiments had integrated luminosities of 469 and 710 fb -1. A B meson decay B ----> f ex(aJ hibits CP violation when (at least) two paths connect the initial and fib---IC nal states. One distinguishes three kinds of CP violating effects. Direct CP violation decay processes are such that the particle decay rate r(B ----> 1) differs from the antipar(bJ W ticle rate r(B ----> 1). Both neutral and charged B mesons can show b direct CP violation. CP violation in mixing stems from the misalignment of the neutral B CP eigenstates and propagation eigenstates (BH,L ex pBo ± qED, with masses mH,L). This effect which is domiFigure 2: Tree (a) and Penguin (b) b-quark nant for the neutral kaons is small decay diagrams. for the B mesons. Most important is the third category: CP violation in the interference between mixing and decay. The simplest case is a process where a neutral B decays to a CP eigenstate fop. The two paths, B O ----> fop and B O ----> EO ----> fop interfere with differing patterns for initial B O and EO resulting into a time dependent CP asymmetry. For golden modes, these asymmetries do not depend on strong phases and give clean experimental access to UT angles. Neglecting electroweak Penguin diagrams, a b-quark non leptonic decay amplitude (Table 1) is the sum of two terms which can be Tree-like or Penguin-like (Figure 2) and whose relevance is determined by the power of ..\ of their CKM factors. A decay channel where one term is dominant is a golden mode. At the B factories the B mesons are produced exclusively in BE pairs from the Y(4S) decays. The pairs of neutral B mesons are produced in an entangled quantum state. We select neutral pairs where one B mesod' decays to f and the other to a flavor specific mode. We measure 6.t the time difference between the two decays. The time dependent CP asymmetry Af is an oscillatory function of 6.t with a frequency given by the mass difference I::!.m = mH - mL: rBO--->f(l::!.t) - rB°--->f(6.t)
Af(l::!.t) bf
== r-
B0--->(
(6.) t
+ r B0--->f (6.t )
=
. Sf sm(6.m6.t) - C f cos(l::!.m6.t),
is the CP eigenstate, we drop the CP subscript for brevity.
265 'Sm>'t C 1-1>'11 2 . 9:At. . h S j -- 2 1+I>'tI 2 ' were j = 1+1>'tI 2 ' and the ratIo Aj = p At whIch compares mixing and decay amplitudes (Aj, Aj = A(BO, EO --> 1) have been introduced. For a golden mode in the 8M: Cj = 0 (no direct CP violation) and Sj = -7}j sin
sss). New Physics can contribute to the latter via virtual new particles in the loop.
Table 1: CKM structure of non leptonic b decay amplitudes. The amplitude for a b --> qlq-2q3 transition is written in terms of T and/or P amplitudes with the CKM factors shown explicitly. The power of >- governing the first and second terms are given. A golden channel leads to a pure measurement of a CKM phase or UT angle 'P. Only effective phases are accessible from the non golden channels. quark process Aces"" Veb Ve~Tees + Vub V':sPs Asss "" Veb Ve~Ps + Vub V':sP; Aced"" Veb Vcd Teed + "lltb '-"t'dPd Auud "" Vub V,:dTuud + Vtb Vt~Pd
2
1st term
2nd
>-~
>-4 >-4 >-3 >-3
>-2 >-3 >-3
example golden golden
Jj1/J K S,L ¢K£
D+D71"+ 71"- , pO pO
'P f3 f3 f3eff aeff
Recent Measurements of the angle /3
The most recent results are tabulated in reference [3]. Here I focus on the measurements of the b --> ccs and b --> sss channels, in particular on the golden modes. Over the last few years, much speculation was entertained by the observation that most Penguin dominated b --> sss final states were measured with sin2/3ejj lower than those from Tree dominated b --> ccs (Figure 3). A simple minded average of all the Penguin measurements fell lower than the Tree measurement with almost 3 standard deviation significance (Figure 4-b)). Figure 3 shows the latest measurements of A3'p(~t) with the golden channels B --> charmonium KO by BABAR [4] and J/1jJKo by Belle [5]: S S
= 0.714 ± 0.032 ± 0.018, C = 0.049 ± 0.022 ± 0.017 (BABAR), = 0.642 ± 0.031 ± 0.017, C = 0.018 ± 0.021 ± 0.014 (Belle),
where the first uncertainties are statistical and the second ones systematic. The average over all charmonium KO measurements is sin 2/3 = 0.678 ± 0.025 or, in the first quadrant of the (p,7}) plane: /31 = (21.3 ± 1.0)0 or /32 = (67.8 ± 1.0)0, /31 being favored by several measurements each with small statistical significance [6]. New this summer are the time dependent amplitude (Dalitz) analyses on Penguin golden channels K~h+ h- [7,8], where h refers to a 7r or a K meson.
888
Penguin channels.
direct measurements of the f3 (not a function final states as well as the non-resonant three"ll'OICl.l1lC, I focus on the After
are paraman isobar model. Each term is a ",,"ull.teA COmI)lCX (isobar) coefficient whose argument incor5 a)-c) show the two-body invariant and the components from >I",.rn11~1p·'.r"'" for the
enriched fit distributions for the time dependent Dalitz analysis of . The full fit distributions are superimposed over the data points three invariant mass spectra of the Dalitz plot. The shaded areas correspond to the background components, and the signal. Vetoes create holes the D and J /'Ij; The pO and fo peaks are in the 7r+7rThe corresponding dependent CP asymmetries for d) and are shown at the bottom of the
which makes the minded average sin over with the measurements from the Tree processes the result from the above pre4 to the older dataset 4
'AJ.tll,Jel>L,lUJlC
and measured B meson related There is no evidence for direct OP violation from the measured time
268 dependent CP asymmetries and no compelling hint for New Physics. 3
Recent Progress on the other UT Angles
Here, I have chosen to highlight the recent progress made on the GronauLondon (GL) analysis [9] of the B -+ pp channels. This charmless b -+ uud decay is not a golden mode (Table 1). The Penguin pollution introduces a phase shift on the determination of the UT angle a and one measure ael I instead of a. The GL method exploits the SU(2) symmetry to combine all charge states in B -+ pp and determine a-aell up to trigonometric ambiguities. It has been appreciated for some time that since the branching fraction for BO -+ pO pO is measurable with fair accuracy, the GL triangle can be constructed more precisely than in the founding case B -+ 7r7L Furthermore with four charged pions in the final state, the decay vertex can be accurately reconstructed and the time dependent CP asymmetry measured, an impossible feat for B -+ 7r 0 7r 0 . With high statistics it has been possible this year to measure Acp(t) for the longitudinally polarized pOpo pairs [10]. Including these results into the GL fit yields the confidence level profile for a - ael I shown on Figure 6. It is nomore fiat as was the case when no CP asymmetry measurement was included. Some discrimination between the mirror solutions already observed with previous spin-averaged measurements of Acp(t) can be seen. There is hope that an accurate determination of a will be obtained with the full data samples of the B factories.
Figure 6: Exclusion confidence level scan for Q QeJ J. The red (solid) curve corresponds to the recent measurement of the time dependent asymmetry for longitudinally polarized pO pO pairs in neutral B decays [10]. The one and two-sigma exclusion levels are shown as horizontal intermittent lines.
4
-!
~
1 ••••••. without C~Q and S~
0.8
with Cro and without S~
0 .•
0.4
0.2
·10
10
20
30
40 «- 30' level. We also search for CPV in DO -+ K+ K-, 7r+7r- decays. For CP-violation to occur, there must be at least two amplitudes with different strong and weak phases. In the standard model (SM), CPV asymmetries of the order of [10- 3 _ 10- 2 ]% are expected in these modes. [2J We determine the CP asymmetries to be afS/f = [0.00 ± 0.34(stat) ± 0.13(syst)J%, a cp = [-0.24 ± 0.52(stat) ± 0.22(syst)J% Finally, we compare the DO -+ K 7r7r 0 (WS) decay Dalitz plot with the DO -+ K 7r7r 0 (RS) decay Dalitz plot. The lifetime dependence of DO -+ K 7r7r0 (WS) decays is given by
ry(SI2' S13, t) = e- rt
{IAyI2 + IAyilAyl
[y" cos by - x" sin by] (rt) + ( Xff21yff2) IAyI2(rt)2}
where the first term describes DCS decays, the last term mixed decays and the remaining term the interference between the two. We find that x" = [2.39 ± 0.61(stat) ± 0.32(syst)J%, y" = [-0.14 ± 0.60(stat) ± OAO(syst)J%, Rmix = [2.9 ± 1.6(stat+syst)J x 10- 4 ,
in agreement with the world average.
3
Conclusions
The website http://www.slac.stanford.edu/xorg/hfag/ charm/index.html displays the latest results from BaBar combined with those from other experiments. We can state here that these various results are consistent with each other. Since the announcement of BaBar's result, various theoretical publications have emerged to interpret Do-If mixing. These use our results to constrain various models, such as certain SUSY models, Higgs models and excited vector boson models. One publication for instance, claims that "light non-degenerate squarks are unlikely to be observed at the LHC." [3J A more comprehensive recent review of a large list of models has also been published [4] and should be referred to for further details.
275
Acknowledgments Assistance of the BaBar collaboration, our PEP-II colleagues, and SLAC is deeply appreciated.
References [1] [2] [3] [4]
M. Staric et al. (Belle Collab.), Phys. Rev. Lett. 98,211803 (2007). F. Bucella, Phys. Rev. D51, 3478 (1995). M. Ciuchini, et al., Phys. Lett. B655, 162 (2007). E. Golowich, et al., Phys. Rev. D76, 095009 (2007).
SEARCH FOR DIRECT CP VIOLATION IN CHARGED KAON DECAYS FROM NA48/2 EXPERIMENT S.Balev a
Joint Institute for Nuclear Research, 141980 Dubna, Russia Abstmct.A high precision measurement of the asymmetry in Dalitz plot slopes Ag = (g+ - g-)/(g+ - g-) was performed by the experiment NA48/2. The obtained results, A~ = (-1.5 ± 2.1) . 10- 4 and A~ = (1.8 ± 1.8) . 10- 4 are based on record statistics - rv 3.1 .10 9 K± --t 7r±7r+7r- decays and rv 9.1 . 10 7 K± --t 7r±7r 0 7r 0 decays, correspondingly. The precision of the measurement is one order of magnitude better than the previous experiments and is limited by the statistical error.
1
Introduction
CP violation plays an important role in elementary particles physics: on one hand, its studies allow to make precise tests to the Standard Model (SM) and to search for new physics; on the other hand, this phenomenon is in the core of the baryogenesis according to modern cosmological models. Therefore, studies of each possible manifestation of CF-violation are tasks of fundamental importance. In kaons, besides the €' / € parameter in KO --+ 1m decays, promising complementary observables are the rates of GIM-suppressed rare kaon decays proceeding through neutral currents, and the asymmetry between K+ and Kdecays to three pions.
The K± --+ 37r matrix element can be parameterized by a polynomial expansion in two Lorentz-invariant variables u and v:
1M (u, v) 12 ex 1 + gu + hu 2 + kv 2 + ... ,
(1)
Ihl,lkl « Igl are the slope parameters, u = (83 - 80)/m;, v = (81 82)/m;, where m7r is the charged pion mass, 8i = (PK - Pi)2, 80 = L 8i/3 (i = 1,2,3), PK and Pi are kaon and i-th pion 4-momenta, respectively. The index i = 3 corresponds to the odd pion. The parameter of direct C P violation is usually defined as: Ag = (g+ - g-)/(g+ + g-), where g+ is the linear where
coefficient in (1) for K+ and g- - for K-. A deviation of Ag from zero is a clear indication for direct CP violation. The experimental precision before NA48/2 for both decay modes, K± --+ 7r±7r+ 7r- and K± --+ 7r±7r0 7r 0 , is at the level of 10- 3 [1], while SM predictions for Ag are below 10- 4 [2]. However, some theoretical calculations involving processes beyond the SM [3] predict substantial enhancements of the asymmetry, which could be observed in the present experiment.
276
277
FlIl~1
(,Alllinul!or Cll'allill~
l.OHilllnlolO
, KABES3
J_,_
r
Magnet §~ ~ ~ OCR! IOCR;: .... :;;:
M" I
I
X X:~======~'
--+--:~n~
:,.t==L
: fu, I argue that FSI does not increase the absolute value of this amplitude.
The essential progress in understanding the nature of the t11 = 1/2 rule in K -+ 2n decays was achieved in the paper [1], where the authors had found a considerable increase of contribution of the operators containing a product of the left-handed and right-handed quark currents generated by the diagrams called later the penguin ones. But for a quantitative agreement with the experimental data, a search for some additional enhancement of the < 2n; 1= otKO > amplitude produced by long-distance effects was utterly desirable. A necessity of additional enhancement of this amplitude due to long-distance strong interactions was also noted later in [2]. The attempts to take into account the long-distance effects were undertaken in [3] - [14]. In [3], the necessary increase of the amplitude < 2n; I = otKO > was associated with liN corrections calculated within the large-N approach (N being the number of colours). In [4], [5], the strengthening of the < 27r; I = OtK > amplitude arised due to a small mass of the intermediate scalar ()" meson. One more mechanism of enhancement of the < 2n; I = otKO > amplitude was ascribed to the final state interaction of the pions [6] - [14]. But as it will be shown, the unitarization of the K -+ 2n amplitude in presence of FSI leads to the opposite effect: a decrease of the < 2n; I = OtK O> amplitude. The result obtained in [1], [2] looks as
(1) where /'i, is a function of G F, F rr , Be and other numerical ingredients of a theory. The numerical values of /'i, obtained in [1] and [2] turned out to be insufficient for a reproduction of the observed magnitude of the < 27r; 1= OIKo > amplitude. To understand, could FSI occuring at long distances change the situation, I consider at first the elastic nn scattering itself. The elastic nn scattering. The general form of the amplitude of elastic nn scattering is T =< ndp~ )nl (p~)
tni (PI )nj (P2) >= A~,appr.
(5)
315 T=150. MeV T=100. MeV T=O.
0.24 0.22 0.2 0.18 <X: N
c.
/\
0.16
='
0. V
0.14 0.12 0.1
o
20
40
60
80
100
120
140
Figure 1: The energy dependence of the average transverse momentum squared for the uquark in a proton in nuclear matter at temperature T.
where < PZ >~ is the transverse momentum squared for the quark in a free hadron. More accurate calculational results for this quantity of the u quark in a proton < P;,t(x ::: 0) are presented in Fig.I. Let us estimate now the Pt distribution of hadron hI produced from a collision of two hadrons h inside the fireball. We shall explore the quark-gluon string model (QGSM) [16,17] or the dual parton model (DPM) [18] based on the liN expansion in the QeD [14,15]. To calculate the transverse momentum spectrum of hadron hI in the mid rapidity region, one needs to know the pt-dependence of the fragmentation function D~l. We assume the Gaussian dependence for D~l like as for ilvApt). However, the slope of this pcdependence 'Ye can differ from the slope 'Yq for constituent quark pt-distribution
>:
where 8' has been associated with the energy squared 8hh of colliding hadrons and r = 'Yelrq. Note, that JShh is not related directly to the initial energy of colliding heavy ions. We estimated the average value of transverse momentum squared for K+mesons produced in the nucleon-nucleon < P'i+,t >~1r and pion+nucleon < P'i+,t >~~ interactions in a fireball created in the central A - A collision as a function of JShh at T = 150 MeV for two cases when 'Ye » 'Yq and 'Ye = 3'Yq [19]. In Fig.2, the curves 1 and 2 correspond to < P'i+,t >~1r and
316 0.3
:>
0.28
~
0.26
1 2
3 4 5
Q)
LO
0.24
6
"
t-
0.22 0.2 0.18 0.16
+
""
Cl..
V
0.14
//~
.....................................•..........
0.12 0.1
o
20
40
60 Shh 1/2,
80 [GeV]
100
120
140
Figure 2: Average square for the transverse momentum of K+-meson produced from the interaction of two hadrons one of them is in the equilibrated fireball as a function of "fShh at T = 150 MeV.
< P~+ ,t
>~~, respectively, when IC > > Iq , whereas the curves 3 and 4 correspond to the same quantities with IC = 3 1q . The line 5 in Fig.2 corresponds to the average square for the transverse momentum of K+ produced in the free p + p collisions < p; >~r= 0.14 GeV/c 2 . As our calculations show, the temperature dependence for < P~+,t >~t is rather weak in the interval T = 100 - 150 MeV.
As is evident from Fig.2, the obtained results are sensitive to the mass value of a hadron which is locally equilibrated with the surrounding nuclear matter at ..jShh :::; 10(GeV). We found that the quark distribution in a hadron depends on the fireball temperature T. At any T the average transverse momentum squared of a quark grows and then saturates when ..jShh increases. Numerically this saturation property depends on T. It leads to a similar energy dependence for the average transverse momentum squared of hadron hI < pt,t >~t. The saturation property for < p~ 1, t >~t depends also on the temperature T and it is very sensitive to the dynamics of hadronization. As an example, we studied the energy dependence of the inverse slope of transverse mass spectrum of K-mesons produced in central heavy-ion collisions and got its energy dependence qualitatively similar observed to one experimentally. We guess that our assumption on the thermodynamical equilibrium of hadrons given by eq.(l) can be applied for heavy nuclei only and not for the early interaction stage.
317
AcknowledgIllents
The authors are grateful for very useful discussions with P.Braun-Munzinger, K.A.Bugaev, W.Cassing, A.V.Efremov, M.Gazdzicki, S.B.Gerasimov, M.I.Gorenstein, Yu.B.Ivanov, A.B.Kaidalov and O.V.Teryaev. This work was supported in part by RFBR Grant N 05-02-17695 and by the special program of the Ministry of Education and Science of the Russian Federation (grant RNP.2.1.1.5409). References
[1) L.Ahle et. at., E866 and E917 Collaboration, Phys. Let. B476, 1 (2000); ibid. B490, 53 (2000). [2) S.V.Afanasiev et at. (NA49 Collab.), Phys.Rev. C66, 054902(2002); C. Alt et al., J. Phys. G30, S119 (2004); M.Gazdzicki, et at., J. Phys. G30, S701 (2004). [3) C.Adleret at., STAR Collaboration, nucl-ex/0206008; O.Barannikova et at., Nucl. Phys. A715, 458 (2003); K.Filimonov et at., hep-ex/0306056; D.Ouerdane et at, BRAHMS Collaboration, Nucl. Phys. A715,478 (2003); J.H.Lee et at., J. Phys. G30, S85 (2004); S.S.Adler et at., PHENIX Collaboration, nucl-ex/030701O; nuclex/0307022. [4) E.V.Shuryak, Phys. Rep. 61, 71 (1980). [5) E.V.Shuryak and O.Zhirov, Phys. Lett. B89, 253 (1980); Yad. Fiz. 28, 485 (1978) [Sov. J. Nucl. Phys. 28,247 (1978). [6) L. van Hove, Phys. Lett. B118, 138 (1982). [7) M.Gorenstein, M.Gazdzicki and K.Bugaev, Phys. Lett. B567, 175 (2003). [8) B.Mohanty, et at., Phys. Rev. C68, 021901 (2003). [9) M.Gazdzicki et at., Braz. J. Phys. 34, 322 (2004). [10) J.Kuti and V.F.Weiskopf, Phys. Rev. D4, 3418 (1971). [11) G.I.Lykasov, A.N.Sissakian, A.S.Sorin, D.V.Toneev, in preparation. [12) A.Capella, V.J.Tran Than Van, Z.Phys.ClO, 249 (1981). [13) O.Benhar, S.Fantoni, G.I.Lykasov, N.V.Slavin, Phys. Rev. C55, 244 (1997). [14) G.'t Hooft, Nucl. Phys., B72, 461 (1974). [15) G.Veneziano, Phys. Lett., B52, 220 (1974). [16) A.B.Kaidalov and K.A.Ter-Martirosyan, Phys. Lett. B117, 247 (1982). [17) A.B.Kaidalov and O.I.Piskunova, Z. Phys. C30, 145 (1986). [18) A.Capella, U.Sukhatme, C.L.Tan, J. Tran Thanh Van, Phys.Rep. 236, 225 (1994). [19) G.I.Lykasov and M.N.Sergeenko, Z. Phys. C70, 455 (1996).
STRINGY PHENOMENA IN YANG-MILLS PLASMA V.1. Zakharov a
INFN, Sezione di Pisa, Largo PontecoTVo 3, 56127, Pis a, Italy ITEP, B.Cheremushkinskaya 25, Moscow, 117218, Russia Abstract. We review the grounds for and consequences from the hypothesis that at the point of the confinement-deconfinement phase transition both electric and magnetic strings are released into the Yang-Mills plasma. We comment also briefly on the averaged Polyakov line as an order parameter of the deconfinement phase transition.
1
Introduction
The goal of this talk is to substantiate a phenomenological stringy picture for the confinement-deconfinement phase transition. The stringy picture for the phase transition was advocated first long time ago [1] and the topic is, in its generality, too broad for such a talk. Thus, we will concentrate on a recent proposal [2,3] that there exists a magnetic component of the Yang-Mills plasma at temperatures close and above the critical temperature Te. While the main ideas are presented in the original papers [2], there appeared most recently results of dedicated lattice measurements [4,5] which support the picture proposed although much more remains to be done before one could really claim observation of the magnetic component of the plasma. Electric strings
2 2.1
Action vs entropy factoTs
Consider quark and anti-quark separated by distance x. To make the construct gauge invariant one has to connect the quarks by a string: (1)
The path-ordered exponent is our first image of what we would call electric string. If quarks develop in time, the string sweeps an area A. Let us consider the most primitive dynamics of a closed string. The string carries color charge and, therefore has a divergent self-energy. To regularize this divergence, introduce finite thickness of the string, TO , TO « Ixl. Then the corresponding action is of order bare Sstring
=
CIg 2(TO )Aj TO2
=
O"bare'
A
.
(2)
To evaluate the renormalized, or physical string tension O"ren one has to subtract from (2) the entropy factor (see, e.g., [6]): ae-mail: [email protected]
318
319
where Nstring is the number of various surfaces with the same area constant C2 is of pure geometrical origin. As a result b:
A,
(3) the
(4) Consider first ro = a, where a is the lattice spacing. In the limit of the large coupling, g2(a) » 1, the bare action factor prevails and the renormalized tension is positive. We have the strong-coupling confinement. This string is infinitely thin but theory is not realistic because of the strong-coupling limit. In the asymptotic-freedom case, g2(a) -+ 0 the tension (4) is negative and the string is unstable in the ultraviolet. In the ultraviolet, on the other hand, free gluons is the right approximation and strings with a negative tension is not a viable alternative. Next, we can still consider the asymptotic-freedom case but choose the thickness of the string ro rv AQ~D' Then g2(ro) can be large enough to make the renormalized tension (4) positive. Thus, we might have 'thick' strings which could be useful effective degrees of freedom in the infrared. At large temperatures g2 is limited by g2 (T), limT -+00 g2 (T) -+ 0 since the time extension of the lattice (Euclidean space-time) is liT. Thus, at an intermediate temperature the effective tension (4) vanishes and the electric strings percolate through the vacuum. 2.2
The Polyakov line
Continuing with the finite-temperature physics, another image for the electric strings is provided by the the Polyakov line which is a Wilson line winding once through the lattice in the periodic time direction:
r
llT
P == Trn
=
TrPexp}o
Ao(x,r)dr,
(5)
where the trace is taken in the fundamental representation. Imagine that we would like to use the Hamiltonian formalism and gauge Ao = O. Unlike the case of T = 0 it is not possible to fix Ao = 0 because of the periodicity in the time direction. In other words, the non-local variable (5) is gauge invariant and cannot be eliminated by gauge transformations. It is still possible to put Ao = 0 provided that the non-local degree of freedom (5) is added explicitly [1] into the partition function:
Z[n] = b Actually,
J
DAn(x,r)exp ( -
Jd3xdr((a~k)2+Ffl))
we oversimplify the estimate of the entropy greatly, see, in particular, [7].
(6)
320 where Ak(x, (3) = n- 1Adx, O)n + n-10kn(X). Note, however, that by introducing a new variable we admit extra ultraviolet divergences into the theory and make the model non-renormalizable, for further references see, e.g., [8]. 3
3.1
Magnetic strings
Topology of the magnetic strings
In Yang-Mills theories, one expects that the magnetic strings are no less fundamental than electric strings. Moreover, condensation of magnetic degrees of freedom is commonly believed to be responsible for the confinement. The scenario is realize in the Abelian case, [9]. The magnetic degrees of freedom are identified in this case through violations of the Bianchi identities:
(7) where j::,on is the monopole current. In the non-Abelian case, the gauge potential can be expressed in terms of the field strength tensor [10]:
(8) where G~l is the matrix inverse to the matrix of the field strength tensor. As far as (8) holds, the Bianchi identities are valid automatically. There might exist, however, such field configurations that the inversion (8) is not possible because the matrix G- 1 does not exist. Actually, it was noted from the very beginning [10] that the inversion (8) fails in 2d case, see also below. Alternatively, in 4d case there can exist 2d defects [11] along which the matrix G- 1 is singular. These 2d defects is our image for the magnetic strings. Magnetic strings are to be added as new degrees of freedom to the standard YM theories which assume Bianchi identities valid. 3.2
Surface operators, monopoles
The action associated with the 2d defects can be readily guessed on symmetry grounds. In fact, such surfaces were considered, for other reasons, in Refs. [12,13] (in the latter reference they were labeled as surface operators). Namely, consider a surface, with area element da 1"'-' and introduce the action: S s'ur face
= const
J J1.vG~v da
(no summation over
j-l,
v) .
(9)
The central point is that the action (9) respects non-Abelian invariance despite of the fact that it apparently carries a color index a. The reason is that one can
321
use gauge invariance to rotate one particular component of the field strength tensor to the Cartan subgroup:
(10) where for simplicity we considered the gauge group SU(2). In other words, non-Abelian fields living on a surface are in fact Abelian. The inversion (8), on the other hand, is specific for the non-Abelian case and fails in the Abelian case. Thus, the magnetic strings replace the magnetic monopoles (7) relevant to the Abelian case. It is worth emphasizing that to carry a finite magnetic flux the surfaces (9) are to be endowed with singular fields G~v. Note that gauge fixation (10) fails if G~v = o. Such defects are trajectories living on the surfaces and correspond to non-Abelian monopoles, for related discussion see [14].
3.3
Dual pictures of confinement
There is a deep relation between magnetic and electric strings. Namely, the expectation value of the Wilson line, < W > can be evaluated either in terms of electric strings open on the heavy quarks, or in terms of the linking number between electric and magnetic strings, [15,16]. It is useful to start with 3d and consider a tube of magnetic field which pierces a surface spanned on the Wilson line. Considering, for simplicity, the Abelian case one gets for the Wilson line:
exp(ifAJ.!dxJ.!)
=
exp(i ~ exp (-
Aminacon/)
,
(12)
where Amin is the minimal area spanned on the Wilson line. One can say that (12) means that the expectation value of the Wilson line is suppressed in the strongest possible way. To get such a suppression the magnetic flux carried by the magnetic strings is to be random [17]. This, in turn, implies that the magnetic strings percolate through the vacuum. For this to happen, the magnetic strings are to have a vanishing tension, (13) amagn = O. Eq. (13) means that the heavy-monopole potential is not confining, as it should be.
322 It is remarkable that we derived (13) starting with consideration of the Wilson line, not directly of the 't Hooft line. 4
Extra dimensions
The logic outlined in the preceding sections has a weak point since we mix up two different pictures for the confining string, that is, thin and thick strings. Indeed, the string which can be open on quarks is to be infinitely thin since quarks are point-like while the string which has tension in physical units, see Eq (4) has thickness of order AQ~D. This inconsistency is in fact difficult to remove and the way out which is becoming common nowadays introduces a novel notion of extra dimensions, or running string tension, for review see [18]. Roughly speaking, one is assuming the string to be infinitely thin but with tension depending on its size. For areas A :S AQ~D' (Jeff
rv
I/A
(14)
For larger areas, the string tension is frozen, (Jeff
~
(Jeanf
,
if A :::: AQ~D ,
(15)
where (Jeanf determines heavy-quark linear potential at large distances. Formulae (14), (15) are somewhat loose because area is not the only characteristic of a surface. It turns out that language of extra dimensions is much more adequate. In this framework, one postulates that there exists an extra dimension, z such that 'our' world corresponds to z = 0 while strings connecting quarks extend into z =I o. The action associated with the string is the same Nambu-Goto action which we actually discussed above but now the area is calculated with account of geometry which is a nontrivial function of z. In particular, assuming that the metric is (16) where R2 is a constant and Xi ,i = 1, .. ,4, are Euclidean 4d coordinates, one reproduces Coulomb-like heavy-quark potential. This is quite obvious from dimensional considerations and the metric (16) realizes the assumption (14). Concerning realization of (15) it is much more arbitrary since one introduces by hand a new parameter, AQCD . The following model
R2 2 ds = exp(cz 2 ) 2" (Jdt 2 +dx; +r1dz2), f(z) = 1-(nzTe)4, c ~ GeV 2, z (17) gives a reasonable description of broad variety of phenomena both at zero and finite temperatures [19].
323
As for the magnetic strings, they have another geometry and correspond to branes wrapped on extra compact dimensions, which are to be added to the five dimensions already introduced, for details see [18]. Magnetic monopoles, in this language, are Kaluza-Klein modes associated with the extra compact dimensions [11].
5
Stringy phenomena near the critical temperature
After all these preliminary remarks we are in position to make predictions specific for the string-based phenomenology of Yang-Mills theories.
5.1
Polyakov line as an order parameter
As argued first in Ref. [1], in pure Yang-Mills case (without quarks) the expectation value of the Polyakov line (5) serves as an order parameter:
(P) == 0 if T < Te .
(18)
In the explicit calculations [19] with the metric (15) the averaged Polyakov line contains at small temperatures exponentially small terms < P >~ exp( -constjT) and Eq. (18) does not hold. Although non-observance of (18) could well be a consequence of the approximations made, it might be useful to understand the reasons for this discrepancy. The proof of (18) exploits the center symmetry. Namely, the Polyakov line is changed by a phase factor under transformations belonging to the group center of the gauge group while the lattice Yang-Mills action can be formulated as symmetrical under the center transformations. However, the lattice action might not know about the center symmetry as well, (for recent discussion and references see [20]). There is no center-group symmetry in the stringy approach, based on (15) but probably there is nothing wrong about this. Thus, violations of (18) seemingly cannot be ruled out on general grounds. There are further interesting issues to discuss in this connection. In particular, the stringy formulation (15) leads to qualitative predictions which are in accord with the lattice data [22], like fast growing entropy in the system of heavy quarks towards T = Te. On pure theoretical side, dependence of continuum physics on details of the lattice regularization (whether we have the center symmetry or not) is most challenging. Because of space considerations, we cannot go into detailed discussion of these issues here, however.
5.2
Magnetic component of the Yang-Mills plasma
We have already mentioned that at the point of the phase transition, Te one expects [1] vanishing tension of the electric string:
(Jeleetr(T) 2: 0,
T 2: Te
(19)
324 On the other hand, tension of the confining string can be evaluated in terms of the magnetic strings, see subsection 3.3. Thus, Eq. (19) implies that magnetic strings acquire non-zero tension at T > Te:
(J"magn(T) 2: 0, T 2: Te .
(20)
Thus, in the deconfining phase the magnetic strings correspond to physical degrees of freedom and are to be present in the Yang-Mills plasma [2]. The question is, how to detect this effect. On the lattice, magnetic strings are identifiable directly, for details see [11]. And, indeed, the magnetic strings do not percolate at T > Te, for references see [15]. More quantitative predictions can be made in terms of the monopoles, which are, as explained above, particles living on the strings. The word 'particles' is to be perceived with some caution, however, since we are discussing now the lattice, or Euclidean formulation and the difference between virtual and real particles is not so obvious as in the Minkowski space. Nevertheless, one can argue [2] that the density of real (in the Minkowskian sense) particles is proportional to the density of the so called wrapped trajectories [21] which are trajectories stretching in the time direction from one boundary to the other:
Preal(T) '" Pwrapped(T) , T
> Te.
(21)
This relation implies, in turn, that the density pwrapped is to be in physical units, '" i\~CD and cannot depend on the lattice spacing. This is in fact a very strong constraint on the data. Which indeed turns to be true [5].
5.3
Ghost-like matter
Measurements on the magnetic strings, reveal [4] astonishingly enough, that both energy density and pressure associated with the magnetic strings are negative: tmagn(T) < 0, Pmagn(T) < 0, Te < T < 2Te . (22) There is a proposal [23] how to accommodate this observation within the stringy picture. The basic idea is that in 2d and 4d the conformal anomaly has opposite signs and this is responsible for the ghost-like sign in case of the 2d defects (22). Acknowledgments
I am indebted to O. Andreev, M.N. Chernodub, A. Di Giacomo, M. D'Elia, A.S. Gorsky for enlightening discussions. References
[1] A. M. Polyakov, Phys. Lett. B72, 477 (1978); "Confinement and liberat'ion", [arXiv:hep-th/0407209].
325 [2] M.N. Chernodub and V.1. Zakharov, Phys. Rev.Lett. 98, 082002 (2007); "Magnetic strings as part of Yang-Mills plasma ", [arXiv:hepphj0702245]. [3] Ch. P. Korthals Altes, "Quasi-particle model in hot QCD", [arXiv:hepphj0406138]; Jinfeng Liao and E. Shuryak, Phys. Rev. C75, 054907 (2007), [arXiv:hep-phj0611131]. [4] M.N. Chernodub et al., "Topological defects and equation of state of gluon plasma", [arXiv:0710.2547]. [5] A. D'Alessandro and M. D'Elia, "Magnetic monopoles in the high temperature phase of Yang-Mills theories", [arXiv:0711.1266]. [6] A.M. Polyakov, "Gauge Fields and Strings", Harvard Academic Publishers, (1987) . [7J A.B. Zamolodchikov, Phys. Lett. B117, 87 (1982). [8] J. C. Myers and M.C. Ogilvie, "New phases of finite temperature gauge theory from an extended action", [arXiv:0710.0674J; Ph. de Forcrand, A.Kurkela, A. Vuorinen, "Center-Symmetric Effective Theory for HighTemperature SU(2) Yang-Mills Theory" [arXiv:0801.1566]. [9] A.M. Polyakov, Phys. Lett. B59, 82 (1975); M. E. Peskin, Annals Phys. 113, 122 (1978). [lOJ M.B. Halpern, Phys. Rev. D16, 1798 (1977); ibid D19, 517 (1979). [l1J V.1. Zakharov, Braz. J. Phys. 37, 165 (2007), [arXiv:hep-phj0612342J. [12J M.N. Chernodub, F.V. Gubarev, M.1. Polikarpov, V. I. Zakharov, Nucl.Phys. B600, 163 (2001), [arXiv:hep-thjOOl0265]. [13J S. Gukov and E. Witten, "Gauge Theory, Ramification, And The Geometric Langlands Program", [arXiv:hep-thj0612073J. [14] G. 't Hooft, Nucl. Phys. B190 , 455 (1981). [15J J. Greensite, Prog. Part. Nucl. Phys. 51, 1 (2003), [arXiv:heplatj0301023]. [16J V.1. Zakharov, AlP Conf. Proc. 756, 182 (2005), [arXiv:hepphj0501011]. [17] A. Di Giacomo, H. G. Dosch, V.1. Shevchenko, Yu.A. Simonov, Phys. Rept. 372, 319 (2002), [arXiv:hep-phj0007223]. [18J O. Aharony et al., Phys. Rept. 323,1832000, [arXiv:hep-thj9905111J. [19J O. Andreev, V.1. Zakharov, Phys. Rev, D74, 025023 (2006),[arXiv:hepphj0604204]; Phys. Lett. B645, 437 (2007), [arXiv:hep-phj0607026]; JHEP, 0704:100 (2007), [arXiv:hep-phj0611304]. [20J G. Burgio, PoS(LAT2007), 292 (2007), [arXiv:0710.0476J. [21] V.G. Bornyakov, V.K. Mitrjushkin, M. Muller-Preussker , Phys. Lett. B284, 99 (1992). [22J P. Petreczky, Nucl. Phys. A 785, 10 (2007), [arXiv:hep-Iatj0609040]. [23J A. Gorsky, V. Zakharov, "Magnetic strings in Lattice QCD as Nonabelian Vortices", [arXiv:0707.1284J.
LATTICE RESULTS ON GLUON AND GHOST PROPAGATORS IN LANDAU GAUGE I.L. Bogolubsky Joint Institute for Nuclear Research, 141980 Dubna, Russia
V.G. Bornyakov a Institute for High Energy Physics, 142281 Protvino, Russia and Institute of Theoretical and Experimental Physics, Moscow, Russia
G. Burgio Universitiit Tilbingen, Institut filr Theoretische Physik, 72076 Tilbingen, Germany
E.-M. Ilgenfritz, M. Miiller-Preussker Humboldt-Universitiit zu Berlin, Institut filr Physik, 12489 Berlin, Germany
V.K. Mitrjushkin Joint Institute for Nuclear Research, 141980 Dubna, Russia and Institute of Theoretical and Experimental Physics, Moscow, Russia Abstract. We present clear evidence of strong effects of Gribov copies in Landau gauge gluon and ghost propagators computed on the lattice at small momenta by employing a new approach to Landau gauge fixing and a more effective numerical algorithm. It is further shown that the new approach substantially decreases notorious finite-volume effects.
1
Introduction
The gauge-variant Green functions, in particular for the covariant Landau gauge, are important for various reasons. Their infrared asymptotics is crucial for gluon and quark confinement according to scenarios invented by Gribov [1] and Zwanziger [2] and by Kugo and Ojima [3]. They have proposed that the Landau gauge ghost propagator is infrared diverging while the gluon propagator is infrared vanishing. The interest in these propagators was stimulated in part by the progress achieved in solving Dyson-Schwinger equations (DSE) for these propagators (for a recent review see [4]). Recently it has been argued that a unique and exact power-like infrared asymptotic behavior of all Green functions can be derived without truncating the hierarchy of DSE [5]. This solution agrees completely with the scenarios of confinement mentioned above. The lattice approach is another powerful tool to compute these propagators in an ab initio fashion but not free of lattice artefacts. So far, there is no consensus between DSE and lattice results. For the gluon propagator, the ultimate decrease towards vanishing momentum has not yet been established in lattice computations. Lattice results for the ghost propagator qualitatively agree with the predicted diverging behavior but show a substantially smaller infrared exponent [6]. The lattice approach has its own limitations. The effects of the finite volume might be strong at the lowest lattice momenta. Moreover, gauge fixing is ae-mail: [email protected]
326
327 not unique resulting in the so-called Gribov problem. Previously it has been concluded that the gluon propagator does not show effects of Gribov copies beyond statistical noise, while the ghost propagator has been found to deviate by up to 10% depending on the quality of gauge fixing [7,8]. Recently anew, extended approach to Landau gauge fixing has been proposed [9]. In this contribution we present results obtained within this new method and using a more effective numerical algorithm for lattice gauge fixing, the simulated annealing (SA) algorithm. Results for the gluon propagator have been already discussed in [10], while results for the ghost propagator are presented here for the first time. 2
Computational details
Our computations have been performed for one lattice spacing corresponding to rather strong bare coupling, at f3 == 4/ g6 = 2.20, on lattices from 84 up to 324. The corresponding lattice scale a is fixed adopting ..j(ia = 0.469 [ll] with the string tension put equal to a = (440 Me V)2. Thus, our largest lattice size 32 4 corresponds to a volume (6.7 fm)4. In order to fix the Landau gauge for each lattice gauge field {U} generated by means of a Me procedure, the gauge functional
(1) is iteratively maximized with respect to a gauge transformation g(x) which is usually taken as a periodic field. In SU(N) gluodynamics the lattice action and the path integral measure are invariant under extended gauge transformations which are periodic modulo Z(N),
g(x + Lv) = z"g(x),
z"
E Z(N)
(2)
in all four directions. Any such gauge transformation is equivalent to a combination of a periodic gauge transformation and a flip Ux " ----* z" Ux " for a 3D hyperplane with fixed X". With respect to the flip transformation all gauge copies of one given field configuration can be split into N 4 flip sectors. The traditional gauge fixing procedure considers one flip sector as a separate gauge orbit. The new approach suggested in [9] combines all N 4 sectors into one gauge orbit. Note, that this approach is not applicable in a gauge theory with fundamental matter fields because the action is not invariant under transformation (2), while in the deconfinement phase of SU(N) pure gluodynamics it should be modified: only flips in space directions are left in the gauge orbit. In practice, few Gribov copies are generated for each sector and the best one over all sectors is chosen by employing an optimized simulating annealing algorithm in combination with finalizing overrelaxation.
328 3
Results
Thus, we are looking for the gauge copy with the highest value of the gauge functional among gauge copies belonging to the enlarged gauge orbit as defined above. It is immediately clear that this procedure allows to find higher local maxima of the gauge functional (1) than the traditional ('old') gauge fixing procedures employing purely periodic gauge transformations and the standard overrelaxation algorithm. Obviously the two prescriptions to fix the Landau gauge, the traditional one and the extended one, are not equivalent. Indeed, for some modest lattice volumes and for the lowest momenta it has been shown in Ref. [9] that they give rise to different results for the gluon as well as the ghost propagators. Comparing results for different lattice sizes we found that the results seem to converge to each other in the large volume limit. It is important that results obtained with the new prescription converge towards the infinite volume limit much faster. In Fig. 1 the gluon propagator D(p2) is 9~~~~~~~~~~~~
••
......
8 7
N
>6 o --5 Q)
L = 1.7 fm 0 L = 2.5 fm L = 3.4 fm v L = 5.0 fm '" eL = 6.7 fm 0 0 does not differ much from the case /1 = O. 3
Phase transition
To obtain the curve of phase transition one needs to define pressure PI in the confined phase and PI I in the deconfined phase, taking into account that vacuum energy density contributes to the free energy, and hence to the pressure:
Having formulas for pressure (which contain parameter of may write for the phase transition curve T c (/1):
Lfund(X)
(7)) we
(12)
337
16
EfT4
14
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
Figure 4: The phase transition curve Tc(J.!) (in GeY) from (9) as function of quark chemical potential J.! (in GeY) for nf = 2 (upper curve) and nf = 3 (lower curve) and f1G2 = 0.0034 Gey4 from ( [8]).
Figure 3: Analytic and lattice ( [11]) curves for energy density of QGP with nf = 2 + 1 and nf = 3 from ( [9]).
0.40 0.35 0.30 0.25
nF2, Wilson: mps/mv=O.65. mpslmv=O.95 I SU(3) nF(2+1): staggered, mq•• >O nr=2: staggered, isentr. EoS
0.20 0.15
• ...
0.10 0.05 0.00 0.5
T ITO
1.0
Figure 5: Sound speed for J.!
=
1.5
2.0
2.5
3.0
0 and n f = 3 (blue dashed curve) compared to lattice data from ( [11]).
here D.cvac = Icvac - c~~~1 = (11 - ~nf )/32D.G 2 • In particular, for the expected value of D.G 2 /G 2 (stand) ;::::: 0.4 one obtains Tc = 0.27 GeV (nf = 0), 0.19 GeV (nf = 2), 0.17 GeV (nf = 3) in good agreement with lattice data.
338
4
Summary
The EoS of QGP is written, where the only np input is the Polyakov line. It should be stressed, that only the modulus of the Polyakov line enters in EoS due to gauge invariance. The phase transition curve Tc(J.l) and speed of sound c; (T) are obtained and agree well with lattice data. An important point of the work is that the only parameter used to receive the final physical quantities from the initial QeD Lagrangian is the Polyakov line taken from lattice data, and is in agreement with analytic estimate for T = Tc ( [5]). Acknowledgments
The financial support of RFFI grant 06-02-17012 is acknowledged. References
[1] M.Shifman, A.Vainshtein, V.Zakharov, Nucl. Phys. B147, 385, 448 (1979). [2] Yu.A.Simonov, JETP Lett. 54 (1991) 249. [3] M.D'Elia, A.Di Giacomo, E.Meggiolaro, Nucl. Phys. B 483, 371 (1997). [4] A.Di Giacomo, H.G.Dosch, V.I.Shevchenko, Yu.A.Siomonov, Phys. Rep. 372, 319 (2002). [5] Yu.A.Simonov, Phys. Lett. B 619, 293 (2005). [6] N.O.Agasyan, Yu.A.Simonov, Phys. Lett. B 693,82 (2006). [7] Yu.A.Simonov, Nonperturbative equation of state of QGP, arXiv:hepphj0702266v2, Ann. Phys (in press). [8] Yu.A.Simonov, M.A.Trusov, arXiv:hep-phj0703228v2, Phys. Lett.(2007) B, 650 (1), p.36-40. [9] E.V.Komarov, Yu.A.Simonov, arXiv:0707.0781v2, Ann. Phys (in press). [10] A.V.Nefediev, Yu.A.Simonov, arXiv:hep-phj0703306. [11] F.Karsch et al., hep-Iatj0312015, hep-Iatj0608003. [12] E.V.Komarov, Speed of sound in QGP (in preparation).
CHIRAL SYMMETRY BREAKING AND THE LORENTZ NATURE OF CONFINEMENT A.V.NefedieV'
Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya 25, 117218 Moscow, Russia Abstract.The Lorentz nature of confinement in a heavy-light quarkonium is investigated. It is demonstrated that an effective scalar interaction is generated selfconsistently as a result of chiral symmetry breaking, and this effective scalar interaction is responsible for the QeD string formation between the quark and the antiquark.
The question of the Lorentz nature of confinement is one of the most longstanding problems in QCD. Indeed, this question is very important for understanding the phenomenon of chiral symmetry breaking, for establishing the correct form of spin-dependent potentials in heavy quarkonia, for understanding the relation between chiral symmetry breaking and the QCD string formation between quarks, and so on. The latter question is discussed in this talk at the example of a heavy-light quarkonium. The approach of the QCD string [1] appears very convenient and successful in studies of various properties of hadrons, both conventional and exotic. This approach follows naturally from the Vacuum Background Correlators Method (VCM) [2]. The key idea of the approach is description of the gluonic degrees of freedom in hadrons in terms of an extended object - the QCD string - formed between colour sources. Nonexcited string is approximated by the straightline ansatz, so that only radial scratchings and rotations are allowed, whereas excitations of the string are described by adding constituent gluons to the system, so that a hybrid meson, for example, can be represented as the quarkantiquark pair attached to the gluon by two straight-line segments of the string. Consider the simplest case of the quark-antiquark conventional meson consisting of a quark and an antiquark connected by the straight-line Nambu-Goto string with the tension (T. The Lagrangian of such a system is L
= -m1 V1- £~ - m2V1- £~ - (Tr 11 d(3V1- [ii x ((3£1 + (1 - (3)£2)]2, (1)
r=
Xl - X2
n = rlr,
and one can proceed along the lines of Ref. [1] in order to arrive at the Hamiltonian. An important ingredient which distinguishes the given approach from the naive potential approach is the proper dynamics of the string which is encoded in the velocity-dependent interaction (the last term in Eq. (1)) and which translates into an extra inertia of the system with respect to the rotations. This ae-mail: [email protected]
339
340
extra inertia leads to the decrease of the (inverse) Regge trajectories slope and brings it to the experimentally observed value of 27f0" or 7f0", for a light-light and heavy-light systems, respectively [1,3]. This effect is unimportant for our present purposes and thus we neglect it, arriving a simple Salpeter equation (which is exact for the case of vanishing angular momentum, l = 0): H'Ij; = M'Ij;,
H
=
ViP' + mi + ViP' + m§ + O"r,
(2)
which is celebrated in the literature. The question we address in this talk is whether the confining interaction in Eq. (2) is of the scalar of vectorial nature. In order to make things as simple as possible, we consider the one-particle limit of Eq. (2) setting ml -* 00 and m2 == m. In this case, the resulting effectively single-particle system can be described by a Dirac-like equation and the question posed above translates into the form of the Dirac operator in this equation, that is whether the confining interaction is added to the energy or to the mass term in this operator. We choose the following strategy [4]. We start from the Euclidean Green's function of the given heavy-light quarkonium, SqQ(X, y) x exp
=
{-~
~c
J
D'Ij;D'lpt DAJJ- 'lj;t(x)SQ(x, YIA)'Ij;(y)
J
J
4 d xF:; -
d4x'lj;t( -if) - im - A)'Ij;} ,
(3)
and fix the modified Fock-Schwinger gauge [5],
(4) which leads to the static particle decoupling from the system. Then we perform integration over the gluonic field in Eq. (3) and arrive at the Dyson-Schwingertype equation derived in the Gaussian approximation for the QeD vacuum [2], that is only the bilocal correlator of gluonic fields is retained:
(-if)x - im)S(x, y) - i
J
d4zM(x, z)S(z, y) = O(4)(x - y),
(5)
with the mass operator -iM(x, z) = KJJ-v(x, z)')'JJ-S(x, z)')'v, Following Ref. [6] we approximate the interaction kernel as K 44 (X, y) == K(x, y) ~
O"(lxl + 1Y1 - Ix - YI),
K 4i (X, y) = 0, Kik(X, y)
= 0 (6)
and rewrite Eq. (5) in the form of a Schrodinger-type equation (in Minkowskii space):
«if; + i3m)lJI(x) + i3
J
d3 zM(x, Z)1JI(i) = EIJI(x),
(7)
341
with the mass operator
M(x, Z) =
-~K(X, £)(JA(x, Z),
A(x, Z) = 2i
J~~
S(w, x, Y)(J.
(8)
The question of the Lorentz nature of confinement can be now formulated as the question of the matrix structure of the quantity A [4]. Indeed, the phenomenon of spontaneous breaking of chiral symmetry (SBCS) means that a piece proportional to the matrix (J appears in A in a selfconsistent way. A convenient technique to deal with the phenomenon of chiral symmetry breaking is given by the chiral angle approach [7]. Let us parametrise the quark selfinteraction, described by the term V(x - Y) = alx - YI in the kernel (6),
"L,(j5) =
J
d4k
-iV(p - k)
7r
1'oko -1'k - m - "L,(k)
(2 )41'0
_-
- 1'0
-~
= [Ap - m] + ({j5)[Bp - p], (9)
by means of the so-called chiral angle '{Jp. Then the selfconsistency condition of such a parametrisation, Api Bp = tan'{Jp, takes the form (explicit expressions for the Ap and Bp follow from the term by term comparison of the l.h.s. and the r.h.s. of Eq. (9)):
psp-mcp =21
J
d k - [cksp-(pk)SkCp ~ :. ] ,splcp=sin/cos'{Jp, (10) (27r)3V(p-k) 3
which is known as the mass-gap equation [7]. Nontrivial solution to this equation behaves as '{Jp(p = 0) = 7r 12 and '{Jp(p ---4 (0) = O. This allows us to make definite conclusions concerning the Lorentz nature of confinement in Eq. (7) depending on the value taken by the quark relative momentum. Indeed,
and thus p---40 P ---4
00
scalar confinement vectorial confinement.
(12)
In the meantime, Eq. (7) admits a Foldy-Wouthuysen transformation, which can be performed in a closed form [8],
The resulting equation reads:
342
where Cp/Sp = cos/sin~(~-'Pp) and Ep = Ap sin 'Pp + Bpcos'Pp is the dressed-quark dispersive law which can be reasonably well approximated by the free-quark dispersion (this approximation fails for the chiral pion - see, for example, the discussion in Ref. [9]). For small values of the interquark momentum, when 'Pp :::::; 7r /2, Eq. (14) reduces to the one-particle limit of the Salpeter Eq. (2). Notice that, according to Eq. (12), this limit exactly corresponds to the purely scalar confinement in Eq. (7). We conclude therefore that the intrinsic Lorentz nature of the QCD string is scalar. As soon as chiral symmetry is broken spontaneously, the selfconsistently generated scalar part appear in the effective interquark interaction. If this interaction dominates, the system can be described by the Salpeter Eq. (2) or by its more sophisticated version which takes the proper string dynamics into account. Acknowledgments
This work was supported by the Federal Agency for Atomic Energy of Russian Federation and by grants NSh-843.2006.2, DFG-436 RUS 113/820/0-1(R), RFFI-05-02-04012-NNIOa, and PTDC /FIS /70843 /2006-Fisica. References
[1] A.Yu. Dubin, A.B. Kaidalov, and Yu.A. Simonov, Phys.Lett. B 323, 41 (1994); Phys.Lett. B 343, 310 (1995). [2] H.G. Dosch, Phys.Lett. B 190, 177 (1987); H.G. Dosch and Yu.A. Simonov, Phys.Lett. B 205, 339 (1988); Yu.A. Simonov, Nucl.Phys. B 307, 512 (1988). [3] V.L. Morgunov, A.V. Nefediev, and Yu.A. Simonov, Phys.Lett. B 459, 653 (1999). [4] Yu.A. Simonov, Yad.Fiz. 60, 2252 (1997); Phys.Rev. D 65, 094018 (2002). [5] I.I. Balitsky, Nucl.Phys. B 254, 166 (1985). [6] A.V.Nefediev and Yu.A.Simonov, Pisma Zh.Eksp. Teor.Fiz. 82, 633 (2005); Phys.Rev. D 76, 074014 (2007). [7] A.Amer, A.Le Yaouanc, L.Oliver, O.Pene, and J.-C. Raynal, Phys. Rev.Lett. 50, 87 (1983); A.Le Yaouanc, L. Oliver , O.Pene, and J.-C. Raynal, Phys.Lett. B 134, 249 (1984); Phys.Rev. D 29, 1233 (1984); A.Le Yaouanc, L.Oliver, S.Ono, O.Pene, and J.-C. Raynal, Phys.Rev. D 31, 137 (1985); P. Bicudo and J. E. Ribeiro, Phys.Rev. D 42, 1611, 1625, 1635 (1990); P. Bicudo, Phys.Rev.Lett. 72, 1600 (1994). [8] Yu.S. Kalashnikova, A.V. Nefediev, and J.E.F.T. Ribeiro, Phys.Rev. D 72,034020 (2005). [9] A.V. Nefediev and J.E.F.T. Ribeiro, Phys.Rev. D 70,094020 (2004).
STRUCTURE FUNCTION MOMENTS OF PROTON AND NEUTRON M.Osipenko a INFN, Sezione di Genova, 16146 Genova, Italy, Moscow State University, Skobeltsyn Institute of Nuclear Physics, 119992 Moscow, Russia Abstract. QCD-inspired phenomenological analysis of experimental moments of proton and deuteron structure functions F2 have been presented. The obtained results on the diu ratio at large-x, isospin dependence of higher twists and comparison with Lattice QCD calculations were discussed. We remind shortly these results: the obtained ratio is consistent with the asymptotic limit diu __ 0 at x -- 1, the total contribution of higher twists is found to be isospin independent and the non-singlet moments are in excellent agreement with the Lattice data. We present here some details of the analysis triggered by the public discussion.
Measurements of the nucleon structure function F2 provide the information about the longitudinal momentum distribution of partons. These distributions being governed by soft strong interactions cannot be described by perturbative QeD methods. Only Lattice QeD simulations allow to evaluate these quantities. Recent measurements of proton and deuteron structure function moments over wide Q2-interval [1,2] and the evaluation of neutron moments [3] allowed to improve the knowledge of these non-perturbative distributions. Detailed descriptions of these analyses are given in papers mentioned above, whereas in the present proceeding we develop further two arguments selected by the public discussion. Experimentally extracted moments of the proton and deuteron structure functions F2 were analyzed to separate leading twist (LT) and higher twist (HT) terms. This was performed by fitting the data Q2-dependence with the following expression:
where LTn is the LT part of the n-th moment evaluated at NLL accuracy, as is the running coupling constant, /1 2 is an arbitrary scale (taken to be 10 (GeV /C)2), a~ is the matrix element of corresponding QeD operator, I~ is its anomalous dimension, f30 = 11 - ~ N F with N F being number of active flavors, 7 is the order of the twist and k is the maximum HT order considered. The number of HT terms (k) in the expansion 1 is, of course, arbitrary because we don't know at which I/Q2 power the series converges. Moreover, anomalous dimensions of perturbative coefficients in front of HT terms are known in a very few cases [5,6]. Most of x-space analyses neglect this dependence assuming I~ = ae-mail: [email protected]
343
344
ofor T
> 2. In the presented analyses the anomalous dimensions were varied as free parameters and extracted from the best fit to the data. The results show a very strong sensitivity of the fit to the values of HT anomalous dimensions at low-Q2. Indeed, it can be seen in the comparison of two twist expansions shown in Fig. 1: one using HT anomalous dimensions as free parameters and another one assuming them to be zero. The lower limit of the fitted Q2_ interval was taken to be 1.2 (GeV jC)2 for the full fit. In the fit with fixed anomalous dimensions it was increased to 3.6 (GeVjc)2 by the condition of having the same X 2 per number of degrees of freedom. It is evident that only the variation of anomalous dimensions permits to describe the data until Q2 = 1.2 (GeVjc)2 by two HT terms. This observation emphasizes that the knowledge of perturbative anomalous dimensions of HT terms is crucial to single out individual HT operator matrix elements. 0.005 , . , . - - - , - - - - - - - - - - - - ,
0.005 r ; r - - . - - - - - - - - - - - - ,
0.004
0.004
0.003
0.003
0.002
0.002
LT\'........\ ....... .
J'.'" 0.001 HT '"
... . ~
O~----.~ ...~::~.:'~···~·------~ ,.....
-0.001 -0.002
/1"
!TW-6
o~--~~~~-------~ .................
TW-6 -0.001 -0.002
Figure 1: Fit of the structure function moment Ms with Eq. 1 using higher twist anomalous dimensions "I::; as free parameters (left) and assuming "I::; = 0 (right): dashed line - the leading twist contribution, dotted lines - twist-4 and twist-6 contributions, dot-dashed line - the total higher twist contribution, solid line - the total fit.
In the extraction of neutron moments we assumed the dominance of the Impulse Approximation (IA) in the LT part of deuteron moments and treated other nuclear effects as, model dependent, corrections to this approximation. This allowed for a simple extraction of LT neutron moments from the following algebraic relation:
(2) where MJ:, M;: and M;: are LT moments of the proton, neutron and deuteron, respectively. N;: is the moment of the nuclear momentum distribution fD
345
i.e. the structure function of the deuteron composed of point-like nucleons (see Ref. [3] for details). The dominance of IA in LT moments, however, implies that processes beyond IA contribute mainly to HT terms. These processes are the scattering off correlated nucleons (Final State Interaction (FSI)) and the scattering off a nuclear constituent different than the nucleon (Meson Exchange Current (MEC)). Only the lowest n = 2 LT moment, sensitive to the low-x dynamics, carries a small contribution from FSI and MECs estimated to be < 0.5 %, whereas it is found to be negligible for higher (n > 2) LT moments. In fact, the LT part of FSI shown in Fig.2 contaminates structure functions at very low x (x < 0.1) values because the nucleon spectator has a long time ~o ::::; 1/M x (here M is the nucleon mass) to interact with nuclear environment while awaiting return of the active quark [7]. However, for higher moments (n > 2) the mean x value is close to unity and therefore the time left for the interaction ~o ::::; 1/M < < 1/ m1f. Here we assume that nuclear interactions are carried mostly by pions. y
q
q
N
D
in
N
D
Figure 2: Example of FSI mechanism in the inclusive electron-deuteron scattering. The nuclear interaction between the nucleon spectator and nuclear medium is likely carried by a colorless pion.
Hence, the bulk of FSI was expected to give a contribution to the HT term of the moment expansion, where the current quark rescatters from the nuclear spectator. In order to test this assumption phenomenologically we used calculations of FSI in the quasi-elastic peak region based on the model from Ref. [4J. To this end, we computed moments of the modeled deuteron structure function F2 including and excluding FSI. The ratios between these two calculations for n = 2 and n = 8 are shown in Fig. 3. From the figure it follows that, indeed, FSI contribution disappears at large Q2 generating an additional HT term, while LT part is unaffected by this contribution. Moreover, the size of FSI contribution in moments does not exceed few % at lowest analyzed Q2, and it is much smaller than the total nucleon HT contribution estimated to be about ~ 25% at the same Q2. We speculate, therefore, that also the HT term has not more than 20% contamination from FSI, and Eq. 2 is also applied to this term within ~ 20% accuracy. This accuracy is comparable to the precision of the extracted total HT term [1,2].
346 1.05
1.04 1.03
1.02
......,
1.01 H
III
'" rr.~
\,
1
0.99 0.98 0.97 0.96 0.95 10
1 Q2
(GeV/c)
2
Figure 3: Ratio of the deuteron structure function moments calculated using the parameterization of Ff from Ref. [4J including and excluding FSI in the quasi-elastic channel: the solid line - n = 2, the dashed line - n = 8.
Summarizing, the presented analysis of the experimental moments of proton and deuteron structure functions F2 showed that: • knowledge of perturbative anomalous dimensions of higher twist terms is crucial to single out individual higher twist operator matrix elements; • for n > 2 FSI mechanism appears in the nuclear structure function moments as an additional higher twist term. Its partial estimates indicate that the relative contribution of FSI to the total higher twist term does not exceed 20%, comparable to the precision of the total higher twist extraction. References [1] [2] [3] [4] [5] [6] [7]
M. Osipenko et al., Phys.Rev. D 67, 092001 (2003). M. Osipenko et al., Phys.Rev. C 73, 045205 (2006). M. Osipenko et al., Nucl.Phys. A 766, 142 (2006). C. Ciofi degli Atti and S. Simula, Phys. Rev. C 53, 1689 (1996). E. Shuryak and A. Vainstein, Nucl. Phys. B 201, 141 (1982). H. Kawamura et al., Mod. Phys. Lett. A 12, 135 (1997). R.L. Jaffe, in M.B. Johnson and A. Pickleseimer, editors, Relativistic Dynamics and Quark-Nuclear Physics, pp. 1-82, Wiley New York, 1985.
HIGGS DECAY TO bb: DIFFERENT APPROACHES TO RESUMMATION OF QCD EFFECTS A.L. Kataev a Institute for Nuclear Research, 117312 Moscow, Russia V.T. Kim b St. Petersburg Nuclear Physics Institute, 188300 Gatchina, Russia Abstract.The comparison between parameterisations of the perturbation results for the decay width of the Standard Model Higgs boson to lib-quarks pairs, based on application of M S-scheme running quark mass and pole b-quark mass, are presented. In the case of the latter parameterisation taking into account of order O(a~) term is rather important. It is minimising deviations of the results obtained at the O(a;) level from the results, which follow from the running quark mass approach.
Decay widths and production cross-sections of scalar bosons are nowadays among the most extensively analysed theoretical quantities. In the case if the Standard Electroweak Model Higgs boson has the mass is in the region 115 GeV:S MH :S 2Mw , where the lower bound comes from the searches of Higgs boson at LEP2 e+e- -collider, it can be detectable in the LHC experiments through the mode H -; "("( and at Tevatron through the main decay mode H -; bb (see e.g. the review [1] ). Moreover, the decay H -; bb may be seen at TOTEM CMS LHC experiment, which is aimed for searches of Higgs boson through its diffraction production (see e.g. [2]). The detailed study of this mode is also useful for planning experiments at possible future linear e+e-colliders [3]. In the mentioned region of masses theoretical expressions for r(H -; bb) is dominating over expressions for other decay modes of the SM H-boson, and therefore is dominating in the denominators of various branching ratios, including Br(H -; "("(). In view of all these topics it is useful to estimate theoretical error bars of r Hbb. To consider this question we will compare parameterisations of QCD predictions for r Hbb' expressed through the running MS-scheme mass mb(MH ) and pole quark mass mb (at the order O(a;)-effects of perturbation theory the similar studies were made in Ref. [4]). Consider now the basic formula for r Hbb in the case of N f =5 number of active flavours [5]
ae-mail: [email protected] supported by RFBR Grants N 05-01-00992, 06-02-16659 be-mail: [email protected]
347
348
b
where rb ) = ¥J-GFMHml, mb=mb(MH), a s=a s (MH)=a s /7r, [3i, 'Yi are the coefficients of the QCD [3 and mass anomalous dimension functions ~r 1 was calculated in [6]. ~r2' ~r3, ~r4 were evaluated in [7], [8], [5]. The huge negative value of ~r 4 indicates, that the structure of perturbation series in the Minkowski region differs from the sign constant growth of perturbation QCD coefficients in the Euclidean region. The possibility of the manifestation of this effect at the a!-level was demonstrated previously in [9], [10]. Consider the renormalisation group (RG) equation
(4) The RG functions are known up to 4-loop level. The solution of Eq.( 4) is
_ (M) - ( ) (as(MH))"Yolf30 AD(as(MH)) mb H = mb mb as(mb) AD(as(mb))
(5)
and the coefficients ofthe polynomial AD(a s ) are expressed through the coefficients of RG-functions (see Ref. [11]). We will use the QCD coupling constant expanded in inverse powers of In(M~/ A~~5) 2) at the NLO, NNLO and N 3LO. At the a~-level the expression for r Hbb in terms of the quark pole mass and the MS-scheme coupling constant can be obtained by three steps. First, one should use the RG equation, which translates mb(MH ) to mb(mb). Second, one can use the relation
)3)
mb(mb)2 = ml(l- 2.67a s(mb) -18.57a s(mb)2 -175.79a s(m b
(6)
where the O(a;)-term was obtained in [12] and the order a~-term was calculated by semi-analytical methods in [13] (this result was confirmed soon in [14] by complete analytical calculation.) Finally, as(mb) should be transformed to as(MH)' The coefficients of the truncated in as(MH) series for r Hbb have the following numerical forms:
r Hbb = rbb) ( 1 + ~riOS) as + ~r~OS) a; + ~r~OS) a~ ) where ~riOS)
(7)
= (3 - 2L) wih L == In(MJdml) and
~r~OS)
= ( _
4.52 - 18.139L + 0.083L 2)
(8)
were previously taken into account in [4], while we will be interested in the effect of the next term. It reads
~r~OS) =
( -
316.906 - 133.421L - 1.153L 2 + 0.05L 3 )
.
(9)
349 The inclusion of the expressions for the two-loop diagrams with massive quark loop insertions, tabulated and taken into account in the RunDec Mathematica package of [15] leads to slight modification of the and a~-corrections to Eq.(7): c
a;-
( - 5.591 - 18.139L + 0.083L 2 )
(10)
( - 322.226 - 132.351L - 1.155L 2 + 0.05L 3 )
(11)
The constant term of .6.r~OS) is also affected by the contributions to the a~ coefficient of Eq.(6) of the diagrams with massive quark loop insertions, evaluated in [16]. However, even in the case of charm-quark loop, these extra terms are rather small. We will neglect these massive-dependent effects. Fig.1 demonstrate the of the ratio Rb(MH ) = r(Ho - t bb)/r~b) both in the case of running quark mass and pole quark mass parameterisations. The QeD parameters are fixed as: mb = 4.7 GeV and mb(mb) = 4.34 GeV [17], NLO: A 0 being the center-of-mass energy squared. It is worth noting here that R( s) vanishes identically for the energies below the two-pion threshold due to the kinematic restrictions, see also Ref. [7]. The mathematical implementation of the latter condition consists in the fact that II(q2) has the only cut q2 2': 4m; along the positive semiaxis of real q2 [1,7]. For practical purposes it is convenient to deal with the Adler function [1]
(2) where Q2 = _q2 2': 0 denotes a spacelike momentum. This function plays a crucial role for the congruous analysis of hadron dynamics in spacelike and timelike domains. In particular, the experimentally measurable R-ratio (1) and theoretically computable Adler function (2) can be expressed in terms of each other by making use of the relations (see Refs. [1,8]) S iE 1 de R(s) = -2. lim D( -C) -(' (3) 7fZ E--+O+ S+iE
l
where m7r :::::: 135 MeV is the mass of the 7f 0 meson. Although there are no direct measurements of the Adler function (2), it can be restored by employing the data on R-ratio (1). Specifically, in the integrand of the dispersion relation for D(Q2) (3) one usually approximates R(s) by its experimental measurements at low and intermediate energies, and by its theoretical expression at high energies. Computed in this way experimental prediction for the Adler function is presented in Fig. 1 by shaded band, see Refs. [6,9] for the details. As it has been noted above, the high-energy behavior of the Adler function (2) can be approximated by the power series in the running coupling oA Q2) in the framework of the perturbative approach. Specifically, at the £-loop level
(4) where a~l)(Q2) is the i-loop perturbative invariant charge and d 1 = 1/7f. As one may infer from Fig. 1, this approximation is reliable for the energies Q :2: 1.5 Ge V only. Besides, expansion (4) is incompatible with the dispersion relation (3) due to unphysical singularities of a 8 (Q2) in the IR domain. The latter also causes certain difficulties for processing the low-energy data.
353
3
Novel integral representation for Adler function
For practical purposes it proves to be convenient to express the Adler function (2) and R-ratio (1) in terms ofthe common spectral function. This objective can be achieved by employing relations (3), the parton model prediction Ro(s) = e(s - 4m;) [7], and the fact that the strong correction to the Adler function vanishes in the asymptotic ultraviolet limit Q2 --+ 00. Eventually one arrives at (see Refs. [6,9] for the details) (5) (6)
Here the spectral function PD (CJ) can be determined either as the discontinuity of a theoretical expression for the Adler function across the physical cut PD (CJ) = 1m D ( -CJ + iO+) /1r or as the numerical derivative of data on R-ratio PD (CJ) = -dR(CJ)/d InCJ. It is worth noting that Eq. (5) embodies the nonperturbative constraints on Adler function arising from the dispersion relation (3). Besides, Eq. (6) by construction properly accounts for the effects due to the analytic continuation of spacelike theoretical results into timelike domain. In order to compute the Adler function in the framework of the approach at hand, one first has to determine the spectral function PD(CJ). In what follows we restrict ourselves to the study of only perturbative contributions to the latter, namely p~fJrt(CJ) = ImD~?rt(-CJ+iO+)/1r, where D~?rt(Q2) is given by Eq. (4). Note that in the limit of massless pion (mrr = 0) the obtained expressions (5) and (6) become identical to those of the so-called Analytic perturbation theory (APT) [2], the perturbative spectral function Ppert(CJ) being assumed. For the illustration of the significance of the pion mass within the approach at hand, it is worth presenting Adler function (5) computed by making use of the perturbative spectral function Ppert(CJ) for both, massless and massive cases. The obtained results are presented in Fig. 1 by solid curves. In the case of the massless pion (which is identical to the APT [2]), one arrives at the result which is free of IR unphysical singularities, but fails to describe the Adler function for the energies Q ;S 1.0 GeV, see Fig. 1 A (see also paper [6] and references therein). At the same time, as one may infer from Fig. 1 B, for the case of the nonvanishing pion mass the representation (5) is capable of providing an output for the Adler function, which agrees with its experimental prediction in the entire energy range [6]. Moreover, the Adler function (5) is remarkably stable with respect to the higher loop corrections. Namely, the relative difference between the i-loop and (£+ I)-loop expressions for D(Q2) (5) is less than 4.9%, 1.5%, and 0.3% for £ = 1, £ = 2, and £ = 3, respectively, for 0 :::; Q2 < 00, see Refs. [6,9] for the details. It is worth noting also that
354 1.5
D(Q2) 1
1.0
0
0.5
Q,GeV 0.5
1.0
1.5
2.0
2.5
Q,GeV 1.0
1.5
2.0
2.5
Figure 1: The Adler function (5) (solid curves) calculated by making use of the perturbative spectral function ppe't(a) in the massless (A) and massive (B) cases. Numerical labels correspond to the loop level considered. The experimental prediction for D(Q2) is shown by the shaded band, whereas its perturbative approximation is denoted by the dashed curve.
the obtained results are supported by recent studies of meson spectrum in the framework of the Bethe-Salpeter formalism [4]. 4
Summary
The IR behavior of the Adler function is examined by employing representation (5), which retains the pion mass effects. The approach at hand possesses all the appealing features of the massless APT [2]: it supplies a self-consistent analysis of spacelike and timelike data; additional parameters are not introduced into the theory; the outcoming results possess no unphysical singularities and display enhanced higher loop stability. In addition, the developed approach provides a reliable description of the Adler function in the entire energy range. Acknowledgments
This work was supported by grants RFBR 05-01-00992 and NS-5362.2006.2. References
[1] S.L. Adler, Phys. Rev. DID, 3714 (1974). [2] D.V. Shirkov and I.L. Solovtsov, Phys. Rev. Lett. 79,1209 (1997); Theor. Math. Phys. 150, 132 (2007). [3] A.V. Nesterenko, Phys. Rev. D 62, 094028 (2000); 64, 116009 (2001). [4] M.Baldicchi, A.V.Nesterenko, G.M.Prosperi, D.V.Shirkov, and C.Simolo, Phys. Rev. Lett. (in press); arXiv:0705.0329 [hep-ph]. [5] A.C.Aguilar, A.V.Nesterenko, J.Papavassiliou, J. Phys. G31, 997 (2005). [6] A.V. Nesterenko and J. Papavassiliou, J. Phys. G 32, 1025 (2006); A.V. Nesterenko, arXiv:0710.5878 [hep-ph). [7) R.P. Feynman, "Photon-Hadron Interactions" (Benjamin, Mass.), 1972. [8) A.V.Radyushkin, Rep. JINR-2-82-159 (1982); JINR Rapid Comm. 78,96 (1996); N.V.Krasnikov and A.A.Pivovarov, Phys. Lett. B116, 168 (1982). [9] A.V. Nesterenko, in preparation.
QCD TEST OF z-SCALING FOR 1I"°-MESON PRODUCTION IN pp COLLISIONS AT HIGH ENERGIES M. Tokarev a and T. Dedovich Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia Abstract. Results of the next-to-Ieading order pQCD calculations of inclusive cross sections of nO mesons produced in pp and pp collisions over a wide range of collision energy (up to the LHC energy) and transverse momentum are compared with ISR and RHIC data. The dependence of the spectra in PT and z presentations for different parton distribution and fragmentation functions is studied. The sensitivity of obtained results to the choice of the renormalization (J.!R) , initial-state factorization (J.!F) and final-state factorization (fragmentation) (J.!H) scales is verified. It is shown that self-similar features of particle production dictated by the z-scaling give strong restriction on the asymptotic behavior of the inclusive spectra in high-PT region.
1
Introduction
Different approaches to description of particle production are used to search for regularities (Bjorken, Feynman and KNO scalings, quark-counting rules etc.) reflecting general principles in lepton, hadron and nucleus interaction at high energies. One of the most basic principles is the self-similarity of hadron production valid both in soft and hard constituent interactions. Other general principles are locality and fractality which can be applied to hard processes at small scales. These investigations have shown that the interactions of hadrons and nuclei can be described in terms of the interactions of their constituents. Fractality in hard processes is a specific feature connected with sub-structure of the constituents. This includes the self-similarity over a wide scale range. One of the scalings in high energy inclusive reactions is z-scaling observed in high-PT particle production (see [lJ and references therein). It is based on principles of locality, self-similarity and fractality of hadronic interactions. The scaling function 'I/J(z) and scaling variable z used for data presentation are expressed via experimental quantities such as the inclusive cross section Ed 3 (j/dp 3 and the multiplicity density dN/d'T/. Data z-presentation reveals properties of energy and angular independence with power law, 'I/J(z) rv z-f3, at high z. In the report we present results of analysis of new data on high-PT spectra of 11"0 mesons produced in PP collisions at the Relativistic Heavy Ion Collider (RHIC) in the framework of z-scaling. Properties of z presentation are used to predict spectra of 11"0 mesons produced in PP and PP collisions over a wide ae-mail:[email protected]
355
356
vIS and PT range. The obtained results are compared with NLO QeD calculations performed with different parton distribution and fragmentation functions (PDFs and FFs). The sensitivity of '¢(z) to renormalizaion, factorization and hadronization scales are studied as well. The results could give new constraints on PDFs and FFs, more deep understanding of phenomenological aspects of QeD and verification of flavor dependence of z-scaling [1,2]. 2
z-Scaling
The idea of z-scaling is based on the assumptions that gross feature of inclusive particle distribution of the process M1 + M2 -> m1 + X at high energies can be described in terms of the corresponding kinematic characteristics of the constituent subprocess written in the symbolic form (x1Mt) + (X2M2) -> m1 + (x1M1 + X2M2 + m2) satisfying the condition
(1) The equation is the expression of locality of hadron interaction at a constituent level. Here Xl and X2 are fractions of the incoming momenta P1 and P2 of the colliding objects with the masses M1 and M 2. They determine the minimum energy, which is necessary for production of the secondary particle with the mass m1 and the four-momentum p. The parameter m2 is introduced to satisfy the internal additive conservation laws (for baryon number, isospin, strangeness, and so on). The quantity n is introduced to connect kinematic (X1,2) and structural (8 1,2) characteristics of the interaction. It is chosen in the form
(2) where m is the mass of nucleon and 81 and 82 are factors relating to the fractal dimensions of the colliding objects. The fractions Xl and X2 are determined to maximize the value of n(X1,X2), simultaneously fulfilling the condition (1)
(3) The fractions X1,2 cover the full phase space accessible at any energy. According to the self-similarity principle the scaling function '¢ (z) is constructed as the function depending on the single dimensionless variable z expressed via dimensionless combinations of Lorentz invariants. It is written in the form 7rS -1 d3 (T (4) '¢ (z) = - (dN/dTJ )(T,n . J E dP3 Here, Ed3 (T/dp 3 is the invariant cross section, s is the center-of-mass collision energy squared, (Tin is the inelastic cross section, J is the corresponding Jacobian. The factor J is the known function of the kinematic variables, the
357
momenta and masses of the colliding and produced particles. The function 'I/J(z) is normalized as follows
1a';)O 'I/J(z)dz =
1.
(5)
The relation allows us to interpret the function 'I/J(z) as a probability density to produce a particle with the corresponding value of the variable z. According to the fractality principle the variable z is constructed as a fractal measure z = zon- 1 for the corresponding inclusive process. It reveals the property zen) -. 00 at n- 1 - . 00. The divergent part n- 1 describes the resolution at which the collision of the constituents can be singled out of this process. The n(X1,X2) represents relative number of all initial configurations containing the constituents which carry fractions Xl and X2 of the incoming momenta.
3
QeD test of z-scaling
Here we analyze the new data obtained by the STAR and PHENIX Collaborations [3,4] on high-PT spectra of nO mesons produced in PP collisions at VB = 200 GeV. The results are compared with the NLO QCD calculations in PT and z presentations. Figure lea) shows nO meson PT-spectra obtained at ISR (see [2] and references therein) and RHIC energies [3,4]. The strong dependence of cross sections on collision energy was experimentally observed. The scaling function 'I/J(z) for the same data are presented in Figure 1 (b). The shape of the scaling function for RHIC data (*,6) is found to be in good agreement with 'I/J(z) for the ISR data shown by the dashed line. The asymptotical behavior of 'I/J(z) is described by the power law, 'I/J(z) rv z-i3. The value of the slope parameter f3 is independent of kinematical variables. 10'~--------
10 • 10 1
10
~,
10
_2
"''''~'.,.
p-p
"~..
11'0
...';'....
10 -.
'M'
10'" . . 10 .... 10~
10 ., 10~ 1O~
s'IJ, GeV b.
200
" __ 90 G
"'.
--..,.~.
PHENIX
...
* 30-62 200 STAR \"'" ... ISR 10 -.. 10 _II 10 -.~~-"--~~.uL-~~.u.J 10
a)
10
I
b)
Figure 1: The PT (a) and z (b) presentations of experimental data on inclusive cross sections of rro mesons produced in pp collisions at the ISR and RHIC [3,4].
358
..
" ....-----------, ,,' , p+p",,,,O+X
"
10-'
..u 10-11
'> ~
::'"
17)1 1t 10· O~,~,,:-,O"';,':-',"';2~0"';2~'"';30~'':-'5"":"'0 p,.. GeV/e
b)
a) '0·...-----------, '0'
:u '>
10
1 10 "
10'"
~
p+p ... .,..O+X
p+p ....nO+X
~. ~::'.
",1. aA' -(3 >.2 ) , ->. 0 _0>.3 (3 >.2 0
C
C°
0>.3 (1- e- iO )
(5)
0 o >. 3 (-1 + ei6 ) 0 0 0
°
),
where A2 is the matrix (1) with 8=0. Then the transformation following form:
V ~ PRotPCP (1 - ~ [A2' All),
PRot
= e A2 ,
~6)
V takes the
PCP =
e
Al
,
(0 1)
(7)
CThe other possibility to separate the real and the imaginary parts would be the following: A2
0 =).,
(
1
-1 _0).,2 cos
(8)
0 (3).
0).,2
cos (8) )
-(3).,
0
and Ai
=
io).,3 s in8
0 1
0
0 0
0 0
362
where the commutator is of the order 0(>,4). Thus, in this approximation the transformation, corresponding to the quark mixing with CP violation, is composed of the purely rotational part PRot, which is related to the rotation matrix (2) via the formulae (3), (4) and the CP violating part PCP
PRo'
~ exp (~),
>.
-a >.3
a),3
-(3 >.2 0 (3 >.2 0
)
,
PCP
'in2~) COS 2~ 01 00 and the continuum limit is approached, therefore in the continuum limit of the theory Z2 electric strings occupy half of all lattice plaquettes and their density diverges as a- 4 . Nevertheless, the sum (8) remains well-defined at a -> 0 and in fact sums up to exp (-alI;e min!), which can only be explained by the exact cancellation L jt* of contributions from different surfaces with opposite signs of (_I)t*EV , i.e. due to the presence of Z2 topological monopoles. Indeed, if the term with jl* is omitted, the expression (8) can be considered as the partition function of Z2 lattice gauge theory with fluctuating, but always positive coupling. It can be shown that in the weak coupling limit electric strings in Z2 gauge theory also occupy half of all lattice plaquettes. The sum over such creased surfaces with positive weights can only lead to perimeter dependence of the Wilson loop, which is indeed the case for ZN lattice gauge theories in the weak coupling m[G]+
L
mp
limit. The terms Tr ( IT ) 91 and (-1) Ee min are also not likely to lEG contribute to the full string tension, since it can be shown that the expression (8) yields physical string tension even when these terms are omitted [16]. Thus it is reasonable to conjecture that in the weak coupling limit Z2 electric strings are confining due to the presence of topological monopoles with currents jl*. It could be interesting to study numerically the properties of such topological monopoles. To conclude, it was shown that unlike Z2 center vortices, which remain physical in the continuum limit [6,7], their duals - Z2 electric strings - can not be consistently described as random surfaces in the continuum theory. Instead, electric strings condense in a creased phase with infinite Hausdorf dimension, but nevertheless due to some cancelations between surfaces with positive and negative statistical weights the minimal surface I;e min dominates in the Wilson loop. In fact the formation of some creased structures is typical for subcritical strings [15]. For instance, sub critical Nambu-Goto strings exist only as branched polymers [15]. It was conjectured in [1] that such subcritical strings can be described as strings on AdS5 background, which hints at some possible relation with AdS/QCD.
385
Acknowledgments This work was partly supported by grants RFBR 05-02-16306, 07-02-00237-a, by the EU Integrated Infrastructure Initiative Hadron Physics (I3HP) under contract RII3-CT-2004-506078, by Federal Program of the Russian Ministry of Industry, Science and Technology No 40.052.1.1.1112 and by Russian Federal Agency for Nuclear Power. References [1] Alexander M. Polyakov. Confinement and liberation, 2004. [2] A. Karch, E. Katz, D. T. Son, and M. Stephanov. Linear confinement and AdS/QCD. Physical Review D, 2006. [3] G. S. Bali, K. Schilling, and C. Schlichter. Observing long colour flux tubes in SU(2) lattice gauge theory. Physical Review D, 51:5165, 1995. [4] P. Yu. Boyko, F. V. Gubarev, and S. M. Morozov. On the fine structure of QCD confining string, 2007. [5] Gerardus t' Hooft. On the phase transition towards permanent quark confinement. Nuclear Physics B, 138:1 - 25, 1978. [6] L. Del Debbio, M. Faber, J. Giedt, J. Greensite, and S. Olejnik. Detection of center vortices in the lattice Yang-Mills vacuum. Physical Review D, 58:094501, 1998. [7] F. V. Gubarev, A. V. Kovalenko, M. I. Polikarpov, S. N. Syritsyn, and V. I. Zakharov. Fine tuned vortices in lattice SU(2) gluodynamics. Physics Letters B, 574:136 - 140, 2003. [8] P. V. Buividovich and M. I. Polikarpov. Center vortices as rigid strings. Nuclear Physics B, 786:84 - 94, 2007. [9] A.Irback. A random surface representation of Wilson loops in Z(2) gauge theory. Physics Letters B, 211:129 - 131, 1988. [10] A. Ukawa, P. Windey, and A. H. Guth. Dual variables for lattice gauge theories and the phase structure of Z(N) systems. Physical Review D, 21:1013 - 1036, 1980. [11] P. Goddard, J. Nuyts, and D. Olive. Gauge theories and magnetic charge. Nuclear Physics B, 125:1 - 28, 1977. [12] P. Becher and H. Joos. The Dirac-Kahler equation and fermions on the lattice. Z. Phys. C, 15:343, 1982. [13] E. T. Tomboulis. 't Hooft loop in SU(2) lattice gauge theories. Physical Review D, 23:2371 - 2383, 1981. [14] V. G. Bornyakov, D. A. Komarov, and M. I. Polikarpov. P-vortices and drama of Gribov copies. Physics Letters B, 497:151, 2001. [15] J. Ambj0rn. Quantization of geometry. Lectures presented at the 1994 Les Houches Summer School, 1994. [16] J. Greensite, M. Faber, and S. Olejnik. Center projection with and without gauge fixing. JHEP, 9901:008, 1999.
UPPER BOUND ON THE LIGHTEST NEUTRALINO MASS IN THE MINIMAL NON-MINIMAL SUPERSYMMETRIC STANDARD MODEL S. Hesselbach, G. Moortgat-Pick IPPP, University of Durham, Durham, DH1 3LE, U.K. D. J. Miller, R. N evzorov a b Department of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, U.K. M. Trusov
Theory Department, ITEP, Moscow, 117218, Russia Abstract.We consider the neutralino sector in the Minimal Non-minimal Supersymmetric Standard Model (MNSSM). We argue that there exists a theoretical upper bound on the lightest neutralino mass in the MNSSM. An approximate solution for the mass of the lightest neutralino is obtained.
Super symmetric (SUSY) models provide an elegant explanation for the dark matter energy density observed in the Universe. To prevent rapid proton decay in SUSY models the invariance of the Lagrangian under R-parity transformations is usually imposed. As a consequence the lightest supersymmetric particle (LSP) is absolutely stable and can play the role of dark matter. In most super symmetric scenarios the LSP is the lightest neutralino, which provides the correct relic abundance of dark matter if it has a mass of 0(100 GeV). In this article we explore the neutralino sector in the framework of the simplest extension of the minimal SUSY model (MSSM) - the Minimal Nonminimal Supersymmetric Standard Model (MNSSM). The superpotential of the MNSSM can be written as follows [1-3]
(1) where WMSSM(/L = 0) is the superpotential of the MSSM without /L-term. The superpotential (1) does not contain any bilinear terms avoiding the /Lproblem. At the same time quadratically divergent tadpole contributions can be suppressed in the considered mdel so that ~ ::::; (TeV)2 [1,2]. At the electroweak (EW) scale the superfield 8 gets a non-zero vacuum expectation value ((8) = s/V2) and an effective /L-term (/Lett = AS/V2) is automatically generated. The neutralino sector of the MNSSM is formed by the superpartners of the neutral gauge and Higgs bosons. In the field basis (iJ, W3, iI~, iI~, S) the aO n leave of absence from the Theory Department, ITEP, Moscow, Russia. be-mail: [email protected]
386
387
neutralino mass matrix reads Mxo
= MI
0
-MzswcfJ
Mzsws fJ
0
0
M2
MzcwcfJ
-MzcwsfJ
0
-MzswcfJ
MzcwcfJ
0
-Meff
--sfJ
Mzsws fJ
-MzcwsfJ
-Meff
0
--cfJ
0
0
--sfJ
.xv
.xv --cfJ
0
V2
V2
.xv
~
(2)
V2
where MI and M2 are the U(l)y and SU(2) gaugino masses while Sw = sin ew , Cw = cosew , sfJ = sin/3, cfJ = cos/3 and Meff = .xs/V2. Here we introduce tan/3 = V2/VI and v = Jvr +v~ = 246 GeV, where VI and V2 are the vacuum expectation values of the Higgs doublets fields Hd and H u , respectively. From Eq.(2) one can easily see that the neutralino spectrum in the MNSSM may be parametrised in terms of
.x,
Me!! ,
tan/3 ,
(3)
In supergravity models with uniform gaugino masses at the Grand Unification scale the renormalisation group flow yields a relationship between MI and M2 at the EW scale, i.e. MI ~ 0.5M2 • The chargino masses in the MNSSM are also defined by the mass parameters M2 and Me!!' LEP searches for SUSY particles set a lower limit on the chargino masses of about 100 GeV restricting the allowed interval of IM21 and IMeffl above 90 -100GeV. In contrast with the MSSM the allowed range of the mass of the lightest neutralino in the MNSSM is limited. In Fig. 1 we plot the lightest neutralino mass Imx~ I in the MSSM and MNSSM as a function of M2 for different values of Me!!' From Fig. 1 it becomes clear that the absolute value of the mass of the lightest neutralino in the MSSM grows when IM21 and IMe!!1 increase while in the MNSSM the maximum value of Imx~1 reduces with increasing IM21 and IMeffl· In order to find the upper bound on Imx~ I it is convenient to consider the matrix MxoM1o whose eigenvalues are equal to the absolute values of the neutralino
masses squared. In the basis
(B, W -iI~sfJ + iI~cfJ' iI~cfJ + iI~sfJ' S) 3,
the
bottom-right 2 x 2 block of MxoM1o takes the form [4] (
IMe!!I~ + 01I Me!!
2
lI*l~elf!
),
(4)
where 0- 2 = M1 cos 2 2/3+11I12 sin 2 2/3, 1I = .xv/V2. Since the minimal eigenvalue of any hermitian matrix is less than its smallest diagonal element the lightest
388
neutralino in the MNSSM is limited from above by the bottom-right diagonal entry of matrix (4), i.e. Imx~ I ::; Ivl. At the same time since we can always choose the field basis in such a way that the 2 x 2 submatrix (4) becomes diagonal its minimal eigenvalue JL6 also restricts the allowed interval of Imx~ I, i.e.
Imx~ 12 ::; 11-6 = ~ [IJLe f f 12 + &2 + Ivl 2
(5)
.---------------~--------
12 + &2 + Iv12) 2- 4IvI2&2] .
(1l1-e f f
The value of JLa reduces with increasing IJLeff I. It reaches its maximum value, i.e. 11-6
min{ &2,
=
Ivl 2},
when l1-eff
O. Taking into account the restriction
-+
on the effective JL-term coming from LEP searches and the theoretical upper bound on the Yukawa coupling A which is caused by the requirement of the validity of perturbation theory up to the Grand Unification scale (A < 0.7) we find that Imx~ I does not exceed 80 - 85 Ge V at tree level [4,5].
"
\ 250 200
"..
70 r-----------'---,------------.-,
60
/
\ I
\
-----'~
I
\
150
100 50
-1000
-.-
Imx~1
-500
"..
-
,r 1/
./
.~
\
50
----
40 30 _ . _ . _ . _ . _ . -
-
20
10
,1
500
1000
-1000
-500
500
1000
Figure 1: Lightest neutralino mass versus M2 in the (a) MSSM and (b) MNSSM for tan.B = 3, >. = 0.7, Ml = 0.5Ml. Solid, dashed and dash-dotted lines correspond to J.LeJ J = 100 GeV, 200 GeV and 300 GeV, respectively.
Here it is worth to notice that at large values of JLef f the allowed interval of the lightest neutralino mass shrinks drastically. Indeed, for IJLef f 12 » M~ we have
1m 01 2< X,
-
2-2
IV 1 (J
(IJLe ff l2
+ &2 + IVI2)
•
(6)
Thus in the considered limit the lightest neutralino mass is significantly smaller than M z even for the appreciable values of A at tree level.
389 When the mass of the lightest neutralino is small one can also obtain an approximate solution for mx~' In general, the neutralino masses obey the characteristic equation det (Mxo - ",1) = 0, where", is an eigenvalue of the matrix (2). However, if '" - t 0 one can ignore all terms in this equation except the one which is linear with respect to '" and the ",-independent one which allows to solve the characteristic equation. This method can be used to calculate the mass of the lightest neutralino when M I , M2 and Ilef f » Mz because then the upper bound on Imx~ I goes to zero. We get in this limit (see [4,5]) \Ileff \v sin 213 Il;ff + v 2 2
Im Xl I 0
~
(7)
According to Eq.(7) the mass of the lightest neutralino is inversely proportional to Ilef f and decreases when tan 13 grows. At small values of A the lightest neutralino mass is proportional to A2 because the correct breakdown of electroweak symmetry breaking requires Ilef f to remain constant when A goes to zero. At this point the approximate solution (7) improves the theoretical restriction on the lightest neutralino mass derived above because for small values of A the upper bound (5)-(6) implies that \mxo\ ex A. Note, however, that the lightest neutralino is predominantly singlino if M I , M2 and Ileff » Mz which makes its direct observation at future colliders quite challenging.
Acknowledgment RN acknowledge support from the SHEFC grant HR03020 SUPA 36878.
References [1] C. Panagiotakopoulos, K. Tamvakis, Phys.Lett. B 469, 145 (1999). [2] C. Panagiotakopoulos, A. Pilaftsis, Phys.Rev. D 63, 055003 (2001). [3] A. Dedes, C. Hugonie, S. Moretti, K. Tamvakis, Phys.Rev. D 63, 055009 (2001). [4] S. Hesselbach, D. J. Miller, G. Moortgat-Pick, R. Nevzorov, M. Trusov, in preparation. [5] S. Hesselbach, D. J. Miller, G. Moortgat-Pick, R. Nevzorov, M. Trusov, arXiv:0710.2550 [hep-ph].
APPLICATION OF HIGHER DERIVATIVE REGULARIZATION TO CALCULATION OF QUANTUM CORRECTIONS IN N=l SUPERSYMMETRIC THEORIES K.Stepanyantz a
Faculty of Physics, Moscow State University, 119992 Moscow, Russia Abstract. We discuss the structure of quantum corrections in N = 1 supersymmetric theories, obtained with the higher covariant derivative regularization. In particular, we argue that all integrals, defining the Gell-Mann-Low function in supersymmetric theories, are integrals of total derivatives. As a consequence, there is an identity for Green functions, which does not follow from any known symmetry of the theory, in N = 1 super symmetric theories.
Investigation of quantum corrections in super symmetric theories is an interesting and sometimes nontrivial problem. For example, in N = 1 supersymmetric theories it is possible to suggest [1] the form of exact ,B-function
a2 ,B(a)
=-
[30
(1 - 'Y(a))]
OCR) 27r(1 - 02 a / 27r ) 2 -
.
(1)
To derive it by usual methods of the perturbation theory is a complicated problem. Here we will show that a derivation of this ,B-function (more exactly, the matter contribution to this function) can be made if a special new identity for the Green functions of the matter superfield takes place. We consider N = 1 super symmetric Yang-Mills theory, which is described by the action
Quantization of this model can be made by standard methods. In particular, we use the background field method, which allows preserving the background gauge invariance and considerably simplifies the calculations of quantum corrections. In order to regularize model (2) we add to its action the higher derivative term, which is invariant under the background gauge invariance, but breaks the BRST-invariance. Therefore, calculating quantum corrections it is necessary to use a special subtraction scheme, which restores the Slavnov-Taylor identities in each order [2]. In order to cancel the remaining one-loop divergences we should also insert to generating functional the Pauli-Villars determinants [3]. We will calculate a matter contribution to the Gell-Mann-Low function. If V denotes the background gauge field and
ae-mail: [email protected]
390
391
+41
J
4
d p d4 () ( ¢ + (-p, ()) ¢(p, ())¢-+ (-p, ()) ¢(p, -) (27f)4 ()) ZG(o:, J1-/p),
(3)
where 0: is a renormalized coupling constant and Z is the renormalization constant for the matter superfield, then the Gell-Mann-Low function {3(0:) and the anomalous dimension ')'(0:) are defined by
{3(d(O:,J1-/p))
= a~pd(O:'J1-/P);
')'(d(O:,J1-/p))
= - a~p In ZG(o:,J1-/p). (4)
Calculation of the matter contribution to the Gell-Mann-Low function can be made substituting solutions of Slavnov-Taylor identities to the SchwingerDyson equation for the two-point Green function of the gauge superfield [4]. The result (without subtraction diagrams) can be written as
(5)
where dots denote contributions of the gauge field and ghosts and (PV) denotes a contribution of the Pauli- Villars fields. The function G is defined by Eq. (3) and the function f is related with the three-point function
(6) where ¢o and ¢o are introduced in the generating functional by adding to the action the term
Actually the Green function (6) is very similar to the usual Green function, but one of the matter ends is not chiral. The first term in Eq. (5) is an integral of the total derivative and can be easily calculated using the identity
However, the explicit calculations [5,6] always show that the second term in Eq. (5) is also an integral of the total derivative and is always equal to O. This allows suggesting existence of the new identity
392
(9)
which is not a consequence of supersymmetric or gauge Slavnov-Taylor identities. (The derivative with respect to In A is needed in order to make all integrals well defined.) The new identity is nontrivial starting from the three-loop approximation (or the two-loop approximation for the Green function (6)). Its verification for the Abelian theory was made in three- and partially four-loop approximation [6]. A sketch of a possible proof exactly to all orders in Abelian case is made in Ref. [7]. However, it is necessary to verify if the new identity takes place in the non-Abelian case. For this purpose we consider [8] the three-loop diagram
and construct the corresponding function J, calculating the diagrams, which are obtained from it by cutting the matter line and attaching an external line of the background gauge field by all possible ways. After substituting the result to the left hand side of identity (9), we obtain
d4q d J(q2) (211")4 dInA q2 G(q2)
J8 {A x 8qJ.l
X
d [ dA (k
q2k2[2(k
2 2
=
a
11" O2
1 (02(R) - "202)
(k+q+l)J.I + q)2 (k + q + l)2 + k2n / A2n)
+ l)2 (1
(1
~ (k + l)2n / A2n ) ] } = O.
J
d4q d4k d4l (211")12
(1 + l2n / A2n)
X
X
(10)
Therefore, the new identity and the factorization of integrands to total derivatives also take place in the non-Abelian case. In order to check if the factorization of integrands to total derivatives is a general feature of supersymmetric theories, we calculate the two-loop ,6-function for the N = 1 supersymmetric Yang-Mills theory without matter. It well known that in this case
(11) Calculating two-loop diagrams, defining the Gell-Mann-Low function (so far without diagrams with insertions of counterterms), in the limit p -+ 0 we find (in the Euclidean space after the Weak rotation)
393 d - 8 2-
6
d
rr· rrQodlnA
x { ( (q
J
4 d kId (2rr)4k 2 dk2
J
4 d q (
2
(2rr)4 q (l+q
2n
2n) -1
/A)
+ k) 2 (1 + (q + k) 2n / A2n )) -1 [2 (n + 1) ( 1 + k2n / A2n ) -1
-2n( 1 + k 2n / A2n)
-2]}.
x
_
(12)
This integral can be calculated by Eq. (8). Then the two-loop Gell-Mann-Low function agrees with Eq. (1). After taking into account diagrams with counterterms insertions [9] with the higher derivative regularization we find that divergences are only in the one-loop approximations similar to the supersymmetric electrodynamics, while the Gell-Mann-Low function has corrections in all loops. This agrees with results of Ref. [10]. Therefore, factorization of integrands to total derivatives seems to be a general feature of all supersymmetric theories. However, the reason is so far unclear. Actually new identity for Green functions (9) is a consequence of this fact. Acknowledgment
This paper was supported by the Russian Foundation for Basic Research (Grant No. 05-01-00541). References
[1] V.Novikov, M.Shifman, A.Vainstein, V.Zakharov, Phys.Lett. 166B,329, (1985). [2] A.A.Slavnov, Phys.Lett. B 518, 195, (2001); Theor. Math. Phys. 130, 1, (2002); A.A.Slavnov, KV.Stepanyantz, Theor. Math. Phys. 135, 673, (2003); 139, 599, (2004). [3] L.D.Faddeev, A.A.Slavnov, "Gauge fields, introduction to quantum theory, second edition", Benjamin, Reading, 1990. [4] KV.Stepanyantz, Theor.Math.Phys. 142, 29, (2005); 150, 377, (2007). [5] A.A. Soloshenko, K V. Stepanyantz, hep-th/ 0304083. (Brief version of this paper is Theor.Math.Phys. 140, 1264, (2004).) [6] A.Pimenov, KStepanyantz, Theor. Math. Phys. 147, 687, (2006). [7] KStepanyantz, Theor.Math.Phys. 146, 321, (2006). [8] A.Pimenov, KStepanyantz, hep-th/0710.5040. [9] A.Pimenov, KStepanyantz, hep-th/0707.4006. [10] M.Shifman, A.Vainstein, Nucl.Phys. B277, 456, (1986).
NONPERTURBATIVE QUANTUM RELATIVISTIC EFFECTS IN THE CONFINEMENT MECHANISM FOR PARTICLES IN A DEEP POTENTIAL WELL K.A. Sveshnikov a , M.V. Ulybyshev b
Department of Physics, Moscow State University, 119991, Moscow, Russia Abstract. The properties of relativistic bound states of bosons and fermions confined in the deep potential well are considered within the framework of covariant hamiltonian formulation of the quasipotential approach. It is shown, that the main properties of such relativistic bound states like wavefunctions and the structure of energy spectrums turn out to be appreciably different from corresponding solutions of differential Schrodinger or Dirac equations for the static external potential of the same form.
1
Introduction
In this report the spectral problem for relativistic bound states of bosons and fermions in the one-dimensional potential well is considered within the framework of quasi potential approach [1] in the relativistic configuration representation (RCR) [2,3]. In this approach the kinetic term of the Hamiltonian contains operators of pure imaginary shift in the radial argument instead of differential ones [2,3], what allows for the study of some nonperturbative properties of relativistic bound states, that are invisible in the standard decription based on differential Schrodinger /Dirac equations in the static external potential of the same form. For a scalar particle in the RCR the quasipotential equation looks like the finite-difference analogue of the Schrodinger equation [2,4]:
~[¢(~ -
n
in) +
¢(~ + in) - 2¢(~)] + V(~)¢(~) = E¢(~)
,
(1)
where V(~) = V02 B(I~I
-
~o) .
In (1) the origin of the variable ~ is the group one, actually it is the eigenvalue of the (first) Lorentz group Kasimir operator and plays the role of the (dimensionless) particle relative coordinate in the center-of-mass system [2,3]. In our l+l-dimensional case the Lorentz group consists of only one Lorentz boost generator, whose spectrum coincides with the real axis, whence -00 :::; ~ :::; +00. In this case is also dimensionless and is nothing else, but the effective Planck constant of the system, which enters the r.h.s. of field and group algebra commutators and defines the magnitude of quantum effects with respect to the classical ones, that by definition are chosen to be 0(1). The regular dimensional c.m.s. coordinate x is connected with ~ via relation ~ = M x, where
n
a e-mail:
[email protected] be-mail: [email protected]
394
395 M is the total mass of the system. For x being the wavefunction argument, the imaginary shift in (1) should be hiM and so coincides with the (effective) Compton wavelength of the system. Here we'll use the first, more formal dimensionless treatment of ~ and h, thence all the other parameters should be considered as dimensionless too. The correct formulation of the problem (1) requires for a consistent definition of pure imaginary shifts in the argument of the wavefunction. For these purposes we'll consider in (1) only analytical in the strip 11m ~I < h functions [4]. In addition, we demand for the bound states wavefunctions to be square-integrable on the real axis. The physical sense of the analyticity condition is the convergence of the finite-difference analogue of the Schrodinger kinetic energy [4]: K[¢] = -
= -
;2 Jd~ [¢(~ -
;2 Jd~ [¢(~ +
ih) -
ih) -
¢(~)]*[¢(~ -
ih) -
¢(~)]
¢(~)]*[¢(~ + ih) - ¢(~)]
=
.
For a potential well this requirement removes also the ambiguity in solutions of finite-difference equations of such type. 2
The solution and main results
The general approach to solution of the spectral problem (1) is quite similar to the corresponding differential Schrodinger equation. It is solved firstly in spatial regions, where the potential V(~) is a constant, afterwards the obtained solutions are sewed together in order to provide analyticity of the wavefunction in the whole strip 11m ~I < h. The general form of solutions inside and outside the well is represented by the Dirichlet series with the following structure (here we show explicitly the even wavefunction inside the well): +00
¢in = 2
L
An cos(Wn~),
n=-oo
where the following parametrization of the energy is accepted: E
. 2 hw 2 4 . 2 hI), = -42 smh - = Vo - - 2 sm h
2
h
2 '
.
.
zWn = ZW+
27m
T.
(2)
The most effective method of treating the eq.(l) is transforming it to the integral form, what after some algebra combined with the analyticity condition mentioned above leads to the following infinite set of algebraic equations for
396 coefficients An [4]:
/1,8
=
27rS
/1,+
h'
S
= 0 ... 00.
(3)
The principal difficulty of algebraic systems like (3) is the existence of exponentially increasing with Inl factors in the coefficients. As a consequence, there doesn't exist any general method of solution for such systems. However, in the case under consideration it is possible to elaborate a specific nonperturbative method of "quasi-exact" solution [4], which allows to study the properties of spectrum and wavefunctions for a wide range of parameters of the problem in detail. The main condition for such" quasi-exact" solution to be valid reads
(4) what is quite consistent with the hadron structure at low energies. In particular, for ~o '::::!. n (the pion) we get oX '::::!. 10- 3 . Dropping the details of this solution given in [4], we present the final result: for the low-lying levels in a sufficiently deep well, for which the following conditions hold 2
Vo
4
»n2
7r
'
/1,
.
= h + za,
w«
a ,
na
-» 1 27r
(5)
the energy spectrum is defined from e2iwaeff
= -1,
aeff = 2~0 - 2~~0 ,
n na
~~o = - I n - . 7r
27r
(6)
The eq. (6) looks like the corresponding spectral equation for even solutions of ordinary differential Schrodinger equation for a particle in the infinitely deep well, but now the geometric width of the well 2~0 is replaced by the effective aeff, which is smaller than 2~0. As a result, the energy levels of relativistic bound states in a deep well lye higher, than their nonelativistic analogues. Although this effect can be rigorously proven for only such and parameters of the potential, which satisfy the following relation
n
1«
nVr)2 (T «
~
e2rren -
,
(7)
actually the interval, where the result (6) is valid, covers a rather wide range of values Vo. For example, for ~o '::::!. 2n the upper limit for (nVo)2 can be estimated as 10001000. In the fermionic case it is convenient to deal with the squared Dirac equation, since it provides the most straightforward way to achieve the energy spectrum.
397 Under the same conditions for low-lying levels in a deep well (5) we can show, that the spectral equation looks like the bosonic case, except additional phase factor S, which enters the r.h.s. of (6). Under our conditions it becomes approximately equal to ±1, namely
S = ±1
+ 0 [(In Va/Va)2]
.
So with increasing Va the spectral equation for fermions approximately coincides with the bosonic one for odd (even) levels, depending on the sign in S.
3
Conclusion
To conclude let us compare the bosonic and fermionic spectrums in the case of ordinary differential and finite-difference RCR-equations. In the case of the differential equations the energy levels in the fermionic case remarkably differ from the bosonic one by a constant shift due to different boundary conditions for infinitely deep potential well. Another picture we observe in the case of finite-difference RCR-equations for sufficiently, but not infinitely, deep well, which meets the conditions (7). Namely, with increasing Va the energy levels of fermions coincide with those of bosons, and an effective contraction of the well and the corresponding growth of the energy levels compared to the differential case takes place as well. The effect of coincidence of bosonic and fermionic spectrums for relativistic bound states can be qualitatively understood as follows. Boundary conditions here are replaced by condition of analyticity in the strip, therefore there is no discontinuity in the first derivative on the well boundary and the wavefunctions penetrate always into the forbidden region. This effect is quite similar both for bosons and fermions, so instead of different boundary conditions in the relativistic case for low-lying bound states of bosons and fermions in a deep well we have almost similar behavior of the wavefunctions with the same depth of penetration in the forbidden region, hence almost identical energy levels.
Acknowledgment This work has been supported in part by the RF President Grant NS-4476.2006.2.
References [1] A.A.Logunov, A.N.Tavkhelidze, Nuov. Cim. 29, 380 (1963). [2] V.G. Kadyshevsky, R. M. Mir-Kasimov, N. B. Skachkov, Part. and Nucl. 2, 635 (1972). [3] N.B.Skachkov, I.L.Solovtsov, Part. and Nucl. 9, 5 (1978). [4] K.A.Sveshnikov, P.K.Silaev, TMF 132, 408 (2002).
KHALFIN'S THEOREM AND NEUTRAL MESONS SUBSYSTEM a Krzysztof Urbanowski b University of Zielona Cora, Institute of Physics, ul. Prof. Z. Szafran a 4a, 65-516 Zielona Cora, Poland. Abstract.The consequences of Khalfin's Theorem are discussed.we find, eg., that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex can not be equal if CPT symmetry holds and CP symmetry is violated. Within a given model we examine numerically the Khalfin's Theorem and show in a graphic form how the Khalfin's Theorem works.
1
Introduction
One of the most interesting two state (or two particle) subsystems is the neutral mesons complex. The standard method used for the description of the properties of such complexes is the Lee-Oehme -Yang (LOY) approximation [2,3]. The source of this approximation applied by LOY to the description and analysis of the decay of neutral kaons is the well known Weisskopf-Wigner theory of the decay processes. The rigorous treatment of two particle complexes shows that there are some inconsistences in the LOY method. This problem is connected with the so-called Khalfin's Theorem [4-8].
2
Khalfin's Theorem and its implications
According to the general principles of quantum mechanics transitions of the system from a state I'lPI) E 11. at time t = to the state l'lh) E 11. at time t > 0, 11P1) ~ 11P2), are realized by the transition unitary unitary transition operator U(t) acting in 11.. The probability to find the system in the state l1Pj) at time t if it was earlier at instant t = in the initial state l1Pk) is determined by the transition amplitude Ajk(t),
°
°
(1)
where (j, k
= 1,2).
Khalfin's Theorem If [4-8]
= f 21 (t) ~f A21(t) A12(t) P=
const.
(2)
then there must be
R
= Ipi = 1.
aThis paper is a shortened version of [lJ be-mail: [email protected];[email protected]
398
(3)
399 As it was pointed out in [7] the only problem in the proof of this Theorem is to find conditions guaranteeing the continuity of hI (t) at t = O. This problem can be solved by taking into account properties of U(t). Namely quantum theory requires U(t) to have the form, U(t) = e- itH , (using units Ii = 1), where H is the total hermitian Hamiltonian of the system, (or, in the interaction picture
UI(t) = 11' e -i J~ H I (T) dT, where 11' denotes the usual time ordering operator and HI(T) is the operator H in the interaction picture). Using this observation one can easily verify that to assure the continuity of hI (t) at t = 0 it suffices that there exists such n ::::: 1 that ('l/J2I Hk l'l/JI) ('l/J2I Hn l'l/JI)
0,
(0
:s: k < n),
=I 0 and I('l/J2 IHnl'l/JI) I
0, then there must be (hll(t) - h22(t)) =/: 0 for t > O.
So, within the exact theory one can say that for real systems, the property (7) can not occur if CPT symmetry holds and CP is violated. This means that the relation (7) can only be considered as an approximation.
3
Model calculations
In this Section we discuss results of numerical calculations performed within the use of the symbolic and numeric package "Mathematica" for the model considered by Khalfin in [4,5], and by Nowakowski in [8] and then used in [11]. This model is formulated using the spectral language for the description of Ks,KL and KO, K, by introducing a hermitian Hamiltonian, H, with a continuous spectrum of decay products (for details see [1]). Assuming that CPT symmetry holds but CP symmetry is violated and using the experimentally obtained values of the parameters characterizing neutral kaon complex make it possible within this model to examine numerically the Khalfin's Theorem as well as other relations and conclusions obtained using this Theorem (for details see [8, lID. The results of numerical calculations of the modulus of the ratio ~~~m for some time interval are presented below in Fig. 1. Analyzing the results of these calculations one can find that for x E (0.01,10),
Ymax(X) - Ymin(X) ':::: 3.3 where, Ymax(X)
= Ir(t)lmax
and Ymin(X)
X
10- 16 ,
= Ir(t)lmin-
(10)
401
~~d~W~--~W~I~I~'i~-
··C.2:
..
4
10
Figure 1: Numerical examination of the Khalfin's Theorem. Here y(x)
= Ir(t)1 == 1~~~m I, x = If .t,
and x E (0.01,10).
Similarly, using "Mathematica" and starting from the amplitudes Ajk(t) and using the formulae for hll (t), h22(t) and the condition All (t) = A22(t) one can compute the difference (h ll (t) - h22 (t) for the model considered. Results of such calculations for some time interval are presented below in Fig. 2. An expansion of scale in the left panel of Fig. 2 shows that continuous fluctuations, similar to those in the right panel of Fig. 2, appear.
2.01' 10.
11
1.99"0-"+~--""--;;2-""'3--:----"5
x
1. 9S .10. 13 1.97 '10. 13
-1.5'10. 16
1.96 '10"l) 1.95-10. 13
Figure 2: The real part (left) and the imaginary part (right) of (hll(t) - h22(t))
*' .
There is y(x) = 3{(h ll (t) - h22(t) and y(x) = <J(hll(t) - h22(t) in Figs 2 t, x E (0.01,5.0) and 3{ (z) and <J (z) respectively. In these Figures x = denote the real and imaginary parts of z respectively and units on the y-axis are in [MeV]. 4
Final remarks
From (10) and Fig. 1 the conclusion follows that if one is able to measure the modulus of the ratio ~~~m only up to the accuracy 10- 15 then one sees this quantity as a constant function of time. The variations in time of I ~~~m I become detectable for the experimenter only if the accuracy of his measurements is of order 10- 16 or better. Within the standard approach the following parameters are used to describe the scale of CP- and possibly CPT - violation effects [3,12]' C ~f ~(cs
+ cl) ==
402 h - h A D- 1 (h 12 -h 21 ), and 0 clef = '12 ( Cs-Cl ) =~, where D clef = h 12+ h 21 +u/-L, and ~/-L = /-LS - /-LL· According to the standard interpretation following from the LOY approximation, C describes violations of CP-symmetry and 0 is considered as a CPT-violating parameter [12]. Such an interpretation of these parameters follows from the properties of LOY theory of time evolution in the subspace of neutral kaons [2,3,12]. From Conclusion 2 and from the results of the model calculations presented in Sec. 3 it follows that the parameter 0 should not be considered as the parameter measuring the scale of possible CPT violation effects: In the more accurate approach [13] and in the exact theory one obtains 0 =1= 0 for every system with violated CP symmetry and this property occurs quite independently of whether this system is CPT invariant or not. What is more, from the Conclusion 2 one finds that if CP symmetry is violated and CPT symmetry holds then there must be cl =1= Cs contrary to the standard predictions of the LOY theory. These conclusions are in full agreement with the results obtained in [14] within the quantum field theory analysis of binary systems such as the neutral meson complexes.
References [1] K. Urbanowski, arXiv: [hep-ph]0712.0328. [21 T. D. Lee, R. Oehme, C. N. Yang, Phys. Rev., 106 (1957) 340. [3] S. Eidelman et al, Review of Particle Physics, Phys. Lett. B 592, No 1-4, (2004). [4] L. A. Khalfin, Preprints of the CPT, The University of Texas at Austin: DOE-ER-40200-211, February 1990 and DOE-ER-40200-247, February 1991; (unpublished, cited in [6]), and references one can find therein. [5] L. A. Khalfin, Foundations of Physics, 27 (1997), 1549. [6] C. B. Chiu and E. C. G. Sudarshan, Phys. Rev. D42 (1990), 3712 [7] P. K. Kabir and A. Pilaftsis, Phys. Rev., A53, (1996), 66. [8] M. Nowakowski, Int. J. Mod. Phys. A14, (1999), 589. [9] K. Urbanowski, Phys. Lett., B 540, (2002), 89; hep-ph/0201272. [10] K. Urbanowski, Acta Phys. Polan. B 37 (2006) 1727. [111 J. Jankiewicz, Acta. Phys. Polan. B 36, (2005), 1901; Acta. Phys. Polan. B 38, (2007), 2471. [12] L. Maiani, in The Second Daif>ne Physics Handbook, vol. 1, Eds. L. Maiani, G. Pancheri and N. Paver, SIS - Pubblicazioni, INFN - LNF, Frascati, 1995; pp. 3 - 26. [13] J. Jankiewicz, K. Urbanowski, Eur. Phys. J. C 49, (2007), 721. [14] B. Machet, V. A. Novikov and M. 1. Vysotsky, Int. J. Mod. Phys. A 20, (2005), 5399; hep-ph/0407268. V. A. Novikov, hep-ph/0509126.
EFFECTIVE LAGRANGIANS AND FIELD THEORY ON A LATTICE Oleg V. Pavlovskya Institute for Theoretical Problems of Microphysics, Moscow State University, 119992 Moscow, Russia Abstract.esent developments in the Random Matrix and Random Lattice Theories give a possibility to find low-energy theorems for many physical models in the BornInfeld form [1]. In our approach that based on the Random Lattice regularization of QeD we try to used the similar ideas in the low-energy baryon physics for finding of the low-energy theory for the chiral fields in the strong-coupling regime.
1
Why we need in the Random Lattice QCD
The attempts to obtain a chiral effective lagrangian from lattice QCD had been performed many times a long ago. Using of the well-known Brezin&Gross trick [4] it could be possible to perform the link's matrix integration in strong coupling regime and obtain the various first order chiral effective theories [5]. In spite of first great success this approach had not been very popular and origin of this stems from the fact that the approaches from [5] does not take the possible to obtain any corrections to the first order results. The lattice regularization breaks a rotational symmetry of the initial theory from the continues rotation group to a discrete group of rotations on fixed angles. And the lattice regularization approach gives the correct results only such tensors which are invariant by respect to this discrete group. In particular using the ordinary Hyper-Cubical (HC) lattice one can obtains the only first order effective theory and for corrections this method generates non-rotational (non-lorentz) invariant terms. For generating of the high-order effective field theories more symmetrical lattice must be considered. Fortunately this conception is known for a long time and is called the Random Lattice Approach [6]. The ideas of the Random Lattice was proposed by Voronoi and Deloune and today this method is very widely used in the modern science and technology. For the quantum field theory method was modified by Christ, Friedberg and Lee [6]. In these articles have been shown that in order to obtain the restoration of the Lorentz (rotational) invariance, it is necessary to perform an average over an ensemble of random lattices. As result one get the averaging over all possible directions and it is intuitively clear that this procedure leads to the disappearance of the artifacts connected this violation of the group of the space rotation. But how to perform such random discretization? This procedure has the tree steps: 1) Draw N sites Xi at random in the volume V; 2) Associate with each Xi a so-called Voronoi cell Ci Ci = {xld(x, Xi) ~ d(x, Xj), Vj i- i} where d(x, y) is a distance between points X and y. It is means that Voronoi cell Ci ae-mail: [email protected]
403
404
consists of all points x that are closer to the center site Xi than any other site; 3) Constrict the dual Delaune lattice by linking the center sites of all Voronoi cells which share a common face. After this if one consider the the big ensemble of such Voronoi-Deloune random lattices based on various distributions of sites Xi, it possible to prove the origin rotational symmetry will restored [6]. In our work we use this procedure for obtaining of an effective chirallagrangian from lattice QCD. This methodological point of view this is a modification of the method proposed in [7] on the case of the Random Lattice approach.
2
From Lattice QeD to chiral lagrangians: step by step
Now let me briefly remand a general steps of the algorithm of the chiral lagrangian derivation from the lattice QCD that was proposed in [7]. The starting point of our analysis is a standard lattice action with Willson fermions
Z = j[DG][Di[J][D'¢] exp{ -Spl(G) - Sq(G, i[J, '¢) - SJ} where: 1) plaquette gauge field term is
BpI =
'7 2:
pl
Gx,J.t
[1 =
~c ReGx,J.tGx-J.t,vG~+v"Pt,v]
,
(1)
exp{ig flink dx~AJ.t(X')};
2) link fermions term is
Bq
=
I: Tr(AJ.t (x) GJ.t (x) + Gt(x)AJ.t(x)) x,J.t
AJ.t(x)g = i[Jb(X + /L)P:,¢a(x), AJ.t(x)g = i[Jb(x)P;:'¢a(x + /L)
P;-
and = ~(r ± IJ.t); 3) source term is
In order to realize the strong-coupling regime on the lattice let us consider the limit of the large coupling constant 9 (g -; 00). Main result is that in such limit integral over the gauge field can be performed. Let us consider the leading order contribution in this strong-coupling expansion. The integrals over the gauge degrees of freedom can be calculated into the large N limit by using of the standard procedure [5] and result of these calculations is following
Z = jlDi[J][D,¢]exP{-NI:Tr[F(),.(x,v))l- SJ} x,v
(2)
405 where Al/ = -M(x)P;; M(x
+ /I)P;;
and
~)]- Tr[log(l- ~~)]
F(A) = Tr[(l -
2
Now it would be very interesting to point out that the function F(A) has the typical form of the Born-Infeld action with first logarithmic correction. Our next step is the integration over the fermion degrees of freedom in (2). Using the source technics it was shown [7] that integral (2) can be re-written into the form of the integral over the unitary bosons matrix Mx
z=
J
(3)
DM eXPSeff(M).
As a matter of principle, we already perform the transformation from the color lattice degrees of freedom (G and 'IjJ) to the boson lattice degrees of freedom (M). Now our task is to realize the continuum limit of expression (3). The nest step of our analysis correspond with the studying of the stationary points of the lattice action Seff. Fortunately this is very well studied task [8]. This problem is connected with well-known investigations of the critical behavior of the chiral field on the lattice and with the problem of the phase transformation on the lattice (for references see the issue [9]). In [7], it was shown that for our task the stationary point is
Mo = uoi, uo(m q = 0, r = 1) = 1/4. Now one can expressed M(x) in terms of the pseudoscalar Goldstone bosons
1 + 15
M = uoexp(i7ri Ti'Y5/!7r) = uo[U(x)-2-
1 - 15 + U + (x)-2-]
and the effective action is given in the form of the Taylor expansion around this stationary point
(4) Let us consider the expansion of the chiral field U = exp(i7riTi/!7r) on the lattice around point x (by respect to the small step of the lattice a), U (x + /I) =
U(x) + a(ol/U(x)) +
a; (o~U(x)) + ....
And for components of the Taylor expansion (4) one obtain
Tr[(Al/(X) - AO)] Tr[(Al/(X) - Ao)2] Tr[(Al/(X) - AO)3]
- 2AoTr(a) 2 A'02Tr(",2) .... -2A~Tr(a3)
-
4,2Tr() AO a
(5)
406 Expressions (5) are very essential because these are a simplest illustration of all aspects of the violation of the rotational symmetry on the lattice. For this moment we specially say nothing about the structure of our lattice. The basic idea of the RL is the averaging over the big ensemble of various lattices with random distributions of sites and it possible to show that such averaging leads to the restoration of the rotational invariance. The basis of = = Sij / lij where Sij is a vectors in the CFL method is following volume of the corresponding 3-dimensional boundary surface of the Voronoi cells and lij = li'i - fj I is the length of link. Using the summation formulas from [6] one get that after the averaging only pairs are survive
1/
et
1/0
(6) a
pairings pairs
At other hand, the result (6) could be obtained by means of the following trick [10,11]. For beginning let us consider a lattice with fixed position (for simplicity it possible to use the trivial HC lattice) in a flat space. Now let us consider small deformations of the geometry of this space (Tij -; 9ij). Using of this idea one can rewrite the problem of the random lattice averaging in the terms of the random surface [10]. This is the standard quantum gravity task and using the methods of the Matrix Theory one can show that our result (6) is just the direct consequence of well-known Wick's Theorem about the pairings [12]. The expression (6) gives us possibility to calculate all terms in expansion (5). Let us consider just the first column in the expression (5). It is easy to show that either of these is proportional to some power of the leading order contribution tr [o/-,UoJ.L U+] tr[(oJ.L UoJ.LU+)2]
+ +
tr[(o/-,UoJ.LU+)n]
+
(7)
Substituting (7) into the (4) and collecting of all terms which depend on the power of the prototype lagrangian one obtain following expression for the effective chirallagrangian .ceff
rv
tr [IOg(l-
-tr [1- y'1- l/fJ2o/-,UoJ.LU+] -
~(1- y'1- 1/,820J.L UOJ.LU+))] + ...
(8)
where ... are all another terms (in particular the Skyrme term) and ,8 is a effective coupling constant that depend on the value of our stationary point Uo·
407 3
Conclusions
The aim of this paper is to derive the chiral effective lagrangian from QCD on the lattice at the strong coupling limit. We find that this theory looks like a Born-Infeld theory for the prototype chiral lagrangian. Such form of the effective lagrangian is expected. From the methodological point of view our consideration is very similar with the low-energy theorem in string theory that lead to the Born-Infeld action [1]. Moreover, in [13], it was shown that Chiral Born-Infeld Theory (without logarithmic corrections) has very interesting "bag" -like solution for chiral fields. It was additional motivation of our work. Acknowledgment
This work is partially supported by the Russian Federation President's Grant 195.2008.2. References
[1] A. A. Tseytlin, Nucl. Phys. B 501 (1997) 4l. [2] J. Ashman et al. [European Muon Collaboration], Phys. Lett. B 206, 364 (1988). [3] K. Kikkawa, T. Kotani, M. a. Sato and M. Kenmoku, Phys. Rev. D 19 (1979) lOll. [4] E. Brezin and D. J. Gross, Phys. Lett. B 97 (1980) 120. [5] H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B 190 (1981) 504. [6] N. H. Christ, R. Friedberg and T. D. Lee, Nucl. Phys. B 202, 89 (1982). [7] S. Myint and C. Rebbi, Nucl. Phys. B 421, 241 (1994) (arXiv:heplat/9401009]. [8] D. J. Gross and E. Witten, Phys. Rev. D 21, 446 (1980). [9] P. Rossi, M. Campostrini and E. Vicari, Phys. Rept. 302, 143 (1998) [arXiv:hep-lat/9609003]. (10] F. David, "Simplicial quantum gravity and random lattices," arXiv:hepth/9303127. (11] L. Bogacz, Z. Burda, J. Jurkiewicz, A. Krzywicki, C. Petersen and B. Petersson, Acta Phys. Polon. B 32, 4121 (2001) [arXiv:hep-lat/0110063]. (12] P. Di Francesco, arXiv:math-ph/9911002. (13] O. V. Pavlovsky, arXiv:hep-ph/0312349.
STRING-LIKE ELECTROSTATIC INTERACTION FROM QED WITH INFINITE MAGNETIC FIELD A.E. Shabad a P.N. Lebedev Physics Institute, Leninsky prospect 53, Moscow 119991, Russia
V.V. Usov b Center for Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel Abstract. In the limit of infinite external magnetic field B the static field of an electric charge is squeezed into a string parallel to B. Near the charge the potential grows like JX3J (In JX3J + canst) with the coordinate X3 along the string. The energy of the string breaking is finite and very close to the effective photon mass.
It is known that the attraction force between two colored charges, when calculated using the Wilson loop method on a lattice, is concentrated in a string with its width being of the order of the lattice spacing l. The string potential consists of three additive components: (i) the Coulomb term 1/ R that dominates at small distance R from the charge, (ii) a constant term, corresponding to the infinite mass renormalization (recall that l is the ultraviolet cutoff parameter in the lattice theory), turns to infinity as l/l, when l is taken close to zero, (iii) the term linearly growing with R corresponding to a constant string tension and providing the confinement according to the Wilson area criterion (see, e.g., [1]). We show that in quantum electrodynamics with external magnetic field B the Coulomb potential of a point-like static charge, when corrected by the vacuum polarization, acquires a similar string-like form in the infinite-magnetic-field limit, the electron Larmour radius LB = (vIeB)-l = (l/mv1J) ----) 0 playing, in a way, the same rOle as the spacing l does in the lattice theory. Here b stands for the magnetic field, b = B / B o, measured in the units of Bo = m 2 / e = 4.4 x 10 13 G, e and m are electron charge and mass, resp. Electric potential AD of a point charge q when calculated as a sum of chains of electron-positron loops in a magnetic field B, so strong that the Larmour radius is much smaller than the electron Compton length, LB « m- 1 (this implies B » B o), is represented as a sum Ao(x)
= A s .r . (x) + Al. r . (x).
(1)
The analytic expressions for these functions are given in [2,3]. The argument x here is the radius-vector with its origin in the charge, its components across and along the magnetic field being x = (Xl., X3). The function As.r. (x) has the exact scaling property A(x)
_________________A_s_.r_(X_)_=~, ae-mail: [email protected] be-mail: [email protected]
408
(2)
409 where the dimensionless function A contains the magnetic field through its argument x = xLi31 only. The function As.r.(x) is short-range: for distances from the charge, large in the Larmour scale, IX31 » L B, or Ix ~ I » LB, it reduces to the Yukawa law
As r (x) ..
~
_q_exp{ - (~)! Jii 41fLB Jii + ~
+ xD = .!L exp{ - (2ab/1f)! mlxl} 41f Ixl '
(3)
where a = e2/41f = 1/137. This equation reflects the Debye screening of the charge by the polarized vacuum. The effective photon mass (inverse Debye radius) in (3) M = (2a/1f)1/2 Li3 1 tends to infinity together with the magnetic field. The» photon mass M » in the dimensionless function A(x) is finite, M = (2a/1f)1/2, and corresponds to the topological photon mass in the twodimensional massless electrodynamics of Schwinger [4]. Anisotropic corrections to (3) were pointed in [5]. The second term in (1) is long-range: it slowly decreases at the distances of the order of the Compton length m- 1 and larger, following the anisotropic Coulomb law
(4) It represents the whole potential Ao(x) (1) there, since A s.r. (x) is already negligible at such distances. This potential decreases with the growth of perpendicular distance x~ from the charge faster than along the field. The equipotential surface is an ellipsoid (x~)2 + x~ = const, which contracts to a string in the limit b = 00. The family of electric lines of force, parameterized by the angle 2 ¢, is for each value of the magnetic field given as X3 = (x ~ )f3- tan ¢ (see Fig. 1). All of them gather together inside a string passing through the charge and directed along the external magnetic field. The short-range part of the potential has the following asymptotic expansion near the point X3 = X~ = 0, where the charge is located,
As.r.(x)
~ 4~
C!I -
2mCs.r .
+ o(x~) + O(X}J) .
(5)
For large magnetic fields b » 21f/a rv 10 3 we have 2mCs.r . ~ 0.9595 Li3 1J2a/1f. It is infinite for LB = 0 or b = 00, as stated at the beginning. The corresponding constant in the expansion of A(x) is finite: 28 = 0.9595 J2a/1f. The constant 0.9595 is calculated using the experimental value of a. For a = 0 it turns into unity. The growth of Cs.r. with the magnetic field following the square root law provides the narrowing of the potential and, in the end, the finiteness of the ground state energy of a hydrogen-like atom in infinite magnetic field (see [2,3]
410
(b)
1~~ 1
Figure
1:
Lines
2
345
of force of electric field coming from (a) No magnetic field, B=O, (b) B = 104 Bo
and also the discussion [6]). In the limit b potential becomes the Dirac 5-function:
=
a
6
point
charge.
00 the short-range part of the
q
(6)
As.r.(O,x)lb=oo = 2.178 27r5(x).
The behavior of the long-range part Al. r. (X3, X~ = 0) at the string X~ = 0 is shown in Fig.2. (See [3J for analytical equations). The limiting form of this function at LB = 0 is finite. It decreases in agreement with Eq. (4) at large distances, and has the following behavior near the charge, IX31 « m -1,
Al.r.(X3,0)lb=oo ~ Al.r.(O,O)lb=oo
+ ~:
(1 - ;f(a)) 2mlx31 [In(2m1X31) -
~ In 2 + l' -
1] ,
(7)
where l' = 0.577 is the Euler constant and the coefficient f depends on the fine structure constant, f (a = 1/137.036) = 4.533. It is nonanalytic in a : f(a)la-+o ~ -Ina. We see that the growth is not linear. It provides "confinement" within the Compton distances, where the approximation (7) is valid. The first constant term in (7) is defined by an integral depending on the fine structure constant. If calculated with its experimental value, a = 1/137.036, it makes Al. r. (0, O)lb=oo = 1.4152(qm/27r). The numerical coefficient here is rather close to v'2 = 1.4142. Bearing in mind that Al. r.(00,0) Ib=oo = 0 we note that Al. r.(0,0) Ib=oo is the increment ofthe long-range part of the potential along the string between the point where the charge is located and the infinitely remote point, i.e. is equal to the work needed for removing a unit test charge to infinity. It may be referred to as the energy of the string breaking. It is remarkable that this quantity is finite. If we set q = e, we find that the energy density
411 -0.8
-0.9
SN
-1
ts
~-1.1
---
" --,./. i-1.2 I
-1. 3
,;
-'1
-1. 4 I
o
0.2
0.4
0.6
0.8
IX31 [(2m)-']
Figure 2: Energy -eAl.r. (X3, 0) of the electron in the long-range part of the potential of the point charge q = eZ for b = 106 , 10 5 , 3 X 104 , 104 (dashed lines from bottom to top). The dashed thick line corresponds to the limit b = 00 and represents the string potential.
of breaking of a string associated with the electron is with surprising accuracy equal to the dimensionless photon mass: Al.r.(O,O)lb=oo 1m = 1.0007 M. The coincidence would be exact for the value of the fine-structure constant equal to 1/121. By postulating this coincidence as a a physical principle we may obtain an approach for calculating a, to which end an approximation, better than one-loop approximation referred to here, would be needed. Formation of a string in QED discussed in the present paper is not completely unexpected, since it was noted [1 J that the Abelian theory does have a topologically nontrivial two-dimensional sector (described by a nonlinear (J"model).
Acknowledgments Supported by Program No. LSS-4401.2006.2, RFBR Project No. 05-02-17217, and by the Israel Science Foundation ofIASH. One of the authors (A.E.S.) expresses his gratitude to Professor D.B.Melrose for hospitality at the University of Sydney and valuable discussions.
References [1] [2J [3J [4J [5J [6J
Kei-Ichi Kondo, Phys.Rev. D 5S, OS5013 (199S). A.E. Shabad and V.V. Usov, Phys. Rev. Lett. 98, lS0403 (2007). A.E. Shabad and V.V. Usov, arXiv: 0707.3475 [astro-ph]. J. Schwinger, Phys. Rev. 128, 2425 (1962). N. Sadooghi and A. Sodeiri Jalili, Phys.Rev. D 76, 065013 (2007). S.-Y. Wang, Phys. Rev. Lett. 99, 22S901 (2007); A.E. Shabad and V.V. Usov, Phys.Rev.Lett. 99, 22S902 (2007).
QFT SYSTEMS WITH 2D SPATIAL DEFECTS LV. Fialkovskya, V.N. MarkoV', Yu.M. Pismalf High Energy Physics Department, St-Petersburg State University, Russia Abstract.We present the general treatment of delta-potentials in the framework of quantum field theory. The path integral calculation technique is sketched by the example of scalar fields. Results of generalization of this approach to realistic fields are given: mean electromagnetic field for fermion system and Casimir energy for photo dynamics in presence of cylindrical shell are presented.
1
Introduction
In 1948 a prediction was made of a macroscopical attractive force between two neutral perfectly conducting parallel planes, the Casimir effect [1]. It appeared that imposing boundary conditions (BC) on quantum field changes its ground state, and consequently its (infinite) vacuum energy. The difference between unperturbed vacuum energy (i.e. as in empty space) and one of constrained vacuum (subject to boundary conditions) is finite and causes measurable force. Nowadays it is confirmed experimentally with 0,5% of total error [2]. Since the pioneering Casimir work, there was a lot of research done on the subject, for a review see for instance [3]. However, up to now the main approach to description of matter with sharp boundaries used in the literature was based on fixing the values of quantum fields (and/or their derivatives) on the surface with help of BC. From the physical point of view, BC cannot be fully accepted as they constrain all of the modes of the field while in nature high-frequency modes propagate freely through any material boundary. The most obvious generalization of BC in modeling sharp boundaries is to introduce into the theory interaction of quantum fields with static (classical) background. The simplest background is of delta-function profile supported on the defect, describing a thin film present in the system. Such delta-potentials in the framework of local, lorenz and gauge (if applicable) invariant, renormalizable QFT were first investigated by Simanzik in 1981 [4], our works [5,6] follow his approach. Since then, there were quite a number of Casimir calculations with delta-potentials done. However, until very lately the issue of renormalizability of such theories still invoked contradictions [7-9]. Still all of the existing papers are concerned with simplified scalar models treated via zeta-function technique or heat kernel expansion, which by definition are applicable in one-loop calculations only. Until [5,6] there were no attempts to construct a self-consistent QED model with a deltapotential interaction satisfying all QFT principles and allowing one to describe self-consistently all possible observable consequences of the presence of a defect. a e-mail:
[email protected] [email protected] c [email protected]
412
413
In this paper we will first sketch the basic ideas for calculations with deltapotentials by example of a scalar field. Then we will present our results for fermion fields interacting with an infinite plane, and for the electromagnetic field coupled to infinite cylindrical shell.
2
Scalar models
Any QFT model is completely determined by its action. Following Simanzik we construct the action of a model in two parts S = So + Sdef where So is the (standard) volume contribution as for quantum fields in empty space, and Sdef is the action of the defect with a delta-function potential. Both parts must satisfy all basic principles of QED - locality, gauge and Lorenz invariance, renormalizability. For scalar case it reads (1)
where K = -82 + m 2 - is standard kinetic operator for massive scalar fielcP) , and>' is the coupling constant of the interaction with the defect. Its surface S is defined by equation 4>(x) = 0, x = (XO,Xl,X2,X3). All physical phenomena can be easily described if the generating functional of the Green functions is known
Z[J] = N
J
Dtp exp {-S[tp]
+ Jtp}, N
= Det 1 / 2 (K)
(2)
for normalization of Z we choose the condition Z[O]I>.=o = 1 For explicit integration of functional integral we represent the contribution of the defect action with help of auxiliary fields 'l/J(x), defined in the surface of the defect xES. Then it is possible to perform integration coming to
here S - modified propagator of the model, D = K-l is free scalar propagator in empty space. Surface operator Q = 1 + 2>'(ODO) determines the Casimir energy of the system, Ecas = 1/ (2T) Tr Ln Q. Geometrical properties of the system are encoded in operator 0 which is a projector onto the surface of the defect J dyD(x, y)O(y, z) == D(x, z). This technique is generalized to other fields in our works [5,6,10,11]. dWe operate in Euclidian version of the theory, where
a2 == L::=o al
414
3
Quantum Electrodynamics
For QED fields, the principles of action construction lead unambiguously to the defect action as [5,6] SdeJ
=
J
d4xJ(<J>(x))
(~Q'!j!+aEJLVPO"aJL<J>(x)AVaPAO")
(4)
with Q - a linear combination of all 16 Dirac matrices with constant coefficients, and a - dimensionless constant. The standard QED part is obviously
(5) where A = ')'JLAJL, FlLv = avAIL - aIL A" , e and m - are unit charge and electron mass respectively. We can see that interaction of the electromagnetic (EM) field with the defect unavoidably brings parity violation into the theory. Sticking to parity even fermion models, one however should expect that all effects of the presence of the boundary will be suppressed at the distances much larger then inverse electron mass m. To prove this estimation, we consider [6,10J pure fermionic model with a= 0, Q = A+')'JLqJL and the defect on an infinite plane <J> = X3. We calculate (directly observable) mean electromagnetic field in the leading order in e. Extending the technique of the previous paragraph to spinor fields it is possible to derive the augmented free fermion propagator of the theory with defect. Then the tadpole graph gives non-trivial contribution to the current of fluctuating fermion fields, which produce non-vanishing mean EM field. The resulting electric and magnetic fields are constant at large distances from the defect plane as it should be in classical electrodynamics for uniformly charged plane with constant currents. This gives the normalization condition establishing correspondence between parameters of the model (A, qlL) and classical charge and current density distributions. At the distances of the order of picometers '" lO-lOcm and less the mean EM field possesses quantum corrections proportional to l/x§. These corrections are exponentially suppressed at larger scales with factor e- m \X3\ similar to the behavior of radiative corrections to the Coulomb law. For full details of the calculations see [10]. Thus, massive fermion fields cannot indeed contribute to the Casimir force which has macroscopical (experimentally verified) values at the scale of 10100nm. This means that to establish theoretically the Casimir effect in QED we unavoidably must consider parity-violating Chern-Simon term for EM field in (4). For the case of planar geometry it was considered in [5]. The nonuniversality of the Casimir force between two infinite planes and its sign change (depending on the value of the coupling constant a) was predicted.
415
xI
For the defect on infinite circular cylindrical shell = + x~ - R2 we considered [11] pure photodynamical model (4) with Q = 0, regularized within PauliVilars approach. Constructing vector analog of operator Q(x) from (3), we can derive Ecas as an integral over the corresponding phase space of In det Q(P). Application of the Abel-Plana summation formula and its generalizations [12] let us finally present the energy as 1 E = 47fR2 f (a)
M
+ RM3A3 + RAl
where M -+ 00 is the regularization parameter, f(a) is a particular finite function given explicitly in [11], and A l ,2 determine the counter-terms. Divergencies are removed by renormalization of corresponding parameters of classical part of the energy. In the limit a -+ 00 the finite part of the energy reproduces the r.esults for perfectly conducting defect [13]. Acknowledgments
This work is supported in part by RFRB grant 07-01-00692 (V.N. Markov and Yu.M. Pismak). References
[1] H. B. G. Casimir, Pmc. K. Ned. Akad. Wet. 51, 793 (1948). [2] G. L. Klimchitskaya, R. S. Decca, E. Fischbach, D. E. Krause, D. Lopez and V. M. Mostepanenko, Int. J. Mod. Phys. A20, 2205 (2005). [3] G. L. Klimchitskaya, V. M. Mostepanenko, Contemp.Phys. 47 (2006) 131-144, arXiv:quant-ph/0609145vl; [4] K. Symanzik, Nucl. Phys. B 190, 1 (1981). [5] V. N. Markov, Yu. M. Pis'mak, arXiv:hep-th/0505218; V. N. Markov, Yu. M. Pis'mak, J. Phys. A39 (2006) 6525-6532, arXiv:hep-th/0606058. [6] I. V. Fialkovsky, V. N. Markov, Yu. M. Pis'mak, Int. J. Mod. Phys. A21, No. 12, pp. 2601-2616 (2006), arXiv:hep-th/0311236. [7] N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, M. Scandurra and H. Weigel, Phys. Lett. B 572, 196 (2003), arXiv:hep-th/0207205. [8] K. A. Milton, J. Phys. A37 (2004) 6391-6406, arXiv:hep-th/0401090. [9] M. Bordag, D. V. Vassilevich, Phys. Rev. D70 (2004) 045003. [10] I. V. Fialkovsky, V. N. Markov, Yu. M. Pis'mak, J. Phys. A: Math. Gen. 39 (2006) 6357 - 6363. (11] I. V. Fialkovsky, V. N. Markov, Yu. M. Pis'mak, arXiv:0710A049. [12] A. A. Saharian, arXiv:0708.1187. I. V. Fialkovsky, arXiv:0710.5539. [13] L. L. DeRaad, Jr. and K. Milton, Ann. Phys. (N.Y.) 136, 229 (1981); K.A. Milton, A.V. Nesterenko, V.V. Nesterenko, Phys.Rev. D59 (1999) 105009, arXiv:hep-th/9711168v3.
BOUND STATE PROBLEMS AND RADIATIVE EFFECTS IN EXTENDED ELECTRODYNAMICS WITH LORENTZ VIOLATION LE.FrolovO, O.G.Kharlanov b , V.Ch.Zhukovskyc Faculty of Physics, Moscow State University, 119992 Moscow, Russia Abstract.Using the extended electrodynamics introducing the Lorentz violation of the minimal CPT-odd type, we discuss the electron bound states in a central potential and in a homogeneous magnetic field (taking the electron anomalous magnetic moment into account), including the corresponding eigenstate problems and the radiation angular distributions, in particular, for the synchrotron radiation.
1
Introd uction
In the present investigation, we will focus on the extended electrodynamics with the Lorentz violation of the minimal CPT-odd form [lJ:
,5
_ir°,l,2,3
with = and the electron charge qe = -e. b~ is the constant axial vector coupling (condensate) that encapsulates the Lorentz violation. Its timelike component bo has at present one of the weakest constraints compared to other similar couplings, e.g. lbol.$ 1O-2eV while Ibl .$ 1O-1ge V. Working within this theory, we will investigate the electron bound states both in the Coulomb potential and in a constant homogeneous magnetic field, discussing the one-particle integral.;; of motion, eigenstates, and spectrum in the external field, and the spontaneous mdiation (spectml-)angular di..;;tributions. 2
Hydrogen-like bound state
Let c = n = 1,0: = e2 /41[, ~ = {b o, O}, and consider the eigenstate problem for A~(x) = {cp(r), O}, i.e. in a central potential. Make the unitary tran..'lformation: (2) (3)
where HD is now an (approximately) P-even operator. The function 'lj;nljmj[bo=O, ·th quant urn numb 1 3 .-. 1 . Wl ers · J = 2' 2'···' mj = --J, J, = J. ± 21 and panty a e-mail: frolov_ieGmail.ru be-mail: okharlGmail.ru ce-mail: zhukovskOphys.msu.ru
416
417
Hn;
hO,
p = (_1)1, L"I the eigenfunction of both Hnlbn=O and and thus of this gives the solutions for Hn in the b5-approximation (O(b~) are omitted): e -ibn2\.
.1. 'f'nljm;
Ib,,=O , e-
-
E nlj = En1jlbn=o
2i bo A p'.1. u(/3,B)'
P ( /3 B) rr n"
= dW,ru = Frr (n,/3,B) dW,il
iJ>rr(/3,B)'
(2)
= dWr +dW,ru = Fu(n,/3,B) + Frr (n,/3,B) cos 2 B dWgl+dW,il
iJ>u(/3,B)+iJ>rr(/3,B)cos 2 B
Where dWqu, dWcl are the quantum and the classical angular distributions of SR respectively, B is the angle between the vector of the magnetic field and the direction of the radiation propagation. In equations (2) the following notations are used:
4+3q
4+q
iJ>u(/3, B) = 16(1 _ q)5/2' iJ>rr(/3, B) = 16(1 _ q)1/2 ' n
Fu,rr(n,/3,B) = LFu,rr(vjn,/3,B), v=l
8v 2 x[' 2
(x)
F ( /3 B) n n-v (/3) u V j n" = (2n + 1); (1 + p)2' Frr v; n, , B = 1- p
x
= v 1 + p'
P=
./
2vq
V1- 2n + l'
q
= /3 2 sin2 B,
4 2[2
v qp
()
X (in n-v + p)2
0~ q~
,
(3)
/3 2 < l.
Where [n,n-v(X), [~,n-v(x) are the Laguerre function and its derivative [1].
429
The remarkable thing is the following: at each fixed n the functions Pa ,7r (n, (J, 8) depend on one variable q only, where q = /3 2 sin 2 8:
Pa ,7r(n,{J,8)
= pJ':)(q).
(4)
The function P(n, /3, 8) dos not possess this property. At each fixed n these functions are strongly depend on both /3 and 8 and satisfy the inequality following from (2):
min{pJn)(q), pJn)(q)} ~ P(n,{J,8) ~ max{pJn) (q), pJn)(q)}.
(5)
The study of pJ~) (q) functions properties It is easy to see that in the nonrelativistic limit the (2, 3) from formulas (2, 3) imply
/3 « 1 (i. e.
q
< (J2
«
1)
(n)( 2n [ 37q ] (n) 2n [ 35 q ] Pa q):::::: 2n + 1 1 - 4(2n + 1) , P7r (q):::::: 2n + 1 1 - 4(2n + 1) , (6) P(n, /3, Therefore at
8) : : 2n2: 1 [1 -
q 4(2n + 1)
(35 + 1 + :OS2 8) ].
/3 « 1 we have p(n) (q) < P(n a
-
(.I
, p,
8) < p(n) (q) < ~ < l. 7r 2n + 1
(7)
The functions pJ~)(q), P(n,{J,8) decrease while q increases at fixed n. While n increases at fixed q these functions increase but stay lower than unity and at n -t 00 tend to unity. Replacing the summation on v by the corresponding integration in (3) and using well-known approximations [1] of Laguerre functions by McDonald's functions K 1/ 3(X), K 2/3(x) we could easily find the following expressions in the ultrarelativistic case:
2 3
Ho
m c = --. eli
It evidently follows from (5) that the pJ~) (q) are monotonically decreasing functions on J.L (therefore at fixed q monotonically increasing functions on n
430
and at fixed n monotonically decreasing functions on q) tending to zero at J.L -+ 00 follows from (8). At J.L < < 1 we have
p(n)( ) ~ 1 _ 320J.L p(n)() ~ 1- 256J.L • u q 2l7rv'3 ' ,.. q 157rv'3
(9)
It is obvious that inequality (7) changes to the opposite one:
1> pJn)(q) ~ P(n,(3,O) ~ pJn) (q).
(10)
It is obvious from (3) that the functions Fu,,..(n, (3, 0) are finite at any values of q (including q = 1). Hence, at q -+ 1 the following asympthotics always take place: pJn)(q) ~ Au(n)(l- q)5/2, pJn)(q) ~ A,..(n)(l- q)7/2. (11) Here Au,,..(n) are some numbers depending on n. This guaranties that at 1 q < < 1 the inequalities (10) hold and pJ~ (q), P(n, (3, 0) tend to zero at q -+ 1. The obtained results prove the validity of the following inequalities 0< min{pJn) (q), pJn)(q)} :S P(n,(3,O) :S max{pJn) (q), pJn)(q)} < 1. (12) In conclusion on the figures below we present the graphs of these functions for different n. 1.
1.
0.8
0.8
P( q- 24!SF3 q')]
= 1+
- interference of the charge radiation and radiation of intrinsic magnetic moment,
f )JL =
[1+'" 1+1-"'(~+ 2
9
2
9
385.J3 432'
J];:2
- magnetic moment radiation due to the Larmor precession
fTh
)J
= (1 + ,,' 7 + 1- ,{ 1 2
9
2
9
);2
'='
434 - magnetic moment radiation due to the Thomas precession,
=[_ 1+ ~~' 1 + 1- ~~' (_! _35-13 J];:2
jL-Th
2
fl
3
2
3
216 r;
':>
- interference of the Larmor and Thomas radiation,
fa
= 1+r;r;' [ 2
I'
a(,
r; 3
245-13 72
,2J_!!",2] + 1-r;r;' (49+ 175-13 J!!..e 9 2 6 r; 9
- radiation due to anomalous magnetic moment of an electron:
g-2 l1a = 11- 110 = -2-110 where Po formulae
=en / 2m oc is the Bohr magneton. Besides, everywhere in these
q=3110Hr=~nr2 =~~r 2 m oc
2 mocp
(2)
2H
is a quantum parameter well-known in the synchrotron radiation theory, and factors
=
r;,r;' ±1 correspond to the spin quantum numbers, magnetic field.
H*
is the Schwinger critical
3. Spin light in the classical theory of synchrotron radiation Here we will show that the purely classical theory can explain completely the origin of the spin light. The radiation of electrical charge, possessing also an intrinsic magnetic moment in the classical theory, is described by the Lienard- Wiechert potentials and Hertz tensor polarization potentials [17]
Aa
eva, QafJ =_~nafJ. c RP v P RP v P
=_!
Corresponding tensor of electromagnetic field is calculated by the formula
=L.~_A[,IlnV]_..!.. d Q[,IlU n c d'i c 2 d'i 2 2
HUV
n V]
U'
Using then the standard technique of classical theory of radiation with equations of an electron motion and spin precession in the homogeneous magnetic field in the linear approximation by 11 one can find the spectral-angular distribution of radiation power in the form
435
2 2 2 3 n { cos e 2 12 f.loOJ COS e '} dO. = 47r y4 f32 sin 2 eJ n + J n + 4~ eoc f3 sin e nJ n J n WSR eL
dwn
=
'
(3)
=
Here OJ eoH / mocy is the cyclotron frequency coincident at g 2 with the frequency of spin precession. This formula is a generalization of the Schott formula for spectral-angular distribution of synchrotron radiation with respect to radiation of intrinsic magnetic moment of an electron. It can be shown that this formula reproduces exactly all properties of spin light concerned with radiation at Larmor precession and described by the semi-classical theory (see also Ref. [8]). Here we will show this on the example of calculating of total radiation power in the most actual ultrarelativistic case when
f.loOJ _ f.lo _ 1 liOJ _ 1 H _ 1 ~ eoc - eop - 2 moc 2 - 2y H* - 3y2 . Towards that purpose one should sum up the expression in formula (3) over the spectrum and integrate it over the angles. As a result we find the same formulas for total synchrotron radiation power and its polarization components as in the semiclassical theory but without recoil effects, Thomas precession and anomalous magnetic moment of an electron
W = (1 +~~~ )W eL
SR '
eL ( 7 1 .;:) eL ( 1 1 ';:) Wa = 8+6~~ WSR ,W1/" = 8+6~~ WSR '
W;L =
(~ + ~ ~~ ) W
SR '
W;L
(4)
i
= ( + ~ ~~ ) WSR .
This result does not depend on sequence of foregoing operations. Thus, this radiation is non-polarized as one could expect from the origin of the spin light. The formula for spectral-angular distribution is eL
dW 27 2( 2)2{K213 2 +--2 x2 2 - - = - - 2 Y l+x K1I3 dxdy 167r 1+ X 2 f.lo y2 yx } + 6--~ ( )112 K1I3K2/3 WSR ' eOp 1+x2 Integration over the spectrum in this expression gives the angular distribution of synchrotron radiation power with additional term for the spin light
436
{3 [7 ---;;;- = 32 (1 + eL
dW
X2 y12
+
(1
2 5X + X2
Y'2
]
35
+ 16 ~~ (1 + X2 t 2 WSR ' X2}
If the expression (15) is integrated over the angles, one can find the spectral composition of this radiation eL
f
dW 9..[3 {co - - = - y K 5/3(X)d.x dy 81T y
f
2 co } +-~~ K I/3(X)d.x W SR '
3
y
The terms for the spin light in the last two formulas are equal to doubled components for linear polarization of radiation. Naturally, further integration in the last formulas over the angular parameter x or over the spectrum leads us to the formulae (4). This result can be shown by another method. According to the general theory of relativistic radiation of point like magnetic moment, the part of energy corresponding to the mixed synchrotron radiation emitted per unit proper time is determined by expression ([5],see also Ref. [18], formula (6.17»
ap 2 dpaJ = 2 eof.1o (d n w _~va dwp nJ'O'w __ 1 naPw w w p ). ( d-r 3 c4 d-r 2 p c2 d-r CT c2 P p Here nap is the dimensionless classical tensor of spin, pa is the four-dimensional momentum of radiation. Its zero component gives the power of mixed radiation
W eL
=.:.. dpo r d-r
Substitution of the corresponding solution of equation of motion, and averaging over period of charge motion and over the spin precession gives
W
eL
r JwSR'
z =(l+.!.;=n 3'='
where
n z = r~ . As to recoil effects and Thomas precession they can be completely
described by classical methods but with use of quantum laws of conservation.
4. Conclusion
Thus, we have shown that the classical and quantum theory of spin light are in agreement with each other at the first approximation by Plank's constant. A question is arising: is the correspondence principle fulfilled in higher-order approximation with respect to the Plank constant? According to the method described earlier the answer to this question is fairly evident: all depends on the possibility of neglecting of the quantum effects and other factors like the Thomas precession.
437
An extraordinary example is radiation of a neutron in a homogeneous magnetic field which arises exclusively due to the spin flip in the quantum theory. Relativistic quantum theory of neutron radiation was developed by the group of Russian scientists (I. M. Ternov, V. G. Bagrov and A. M. Hapaev) [21]. The classical theory of neutron radiation emitted at the spin precession, which was developed by V. A. Bordovitsyn with coauthors [17-20], turned out to be in full accordance with the quantum theory but differs by a constant coefficient equal to 4, which, as it turned out, is connected with specific properties of quantum transition with spin-flip ([8]). However such radiation in the classical theory does not exist in the common interpretation. Therefore the correspondence principle in this case is inapplicable. With regard to the synchrotron radiation of an electron the correspondence principle applied to radiation of the intrinsic magnetic moment works very well in the limit case p ~ 00 and on assumption that the value of anomalous magnetic moment is large enough to neglect the Thomas precession. Note that in the mixed synchrotron radiation the terms which are proportional to Plank constant and contain the anomalous magnetic moment are in full accordance with the classical theory. Apparently, this is connected with the fact that the anomalous magnetic moment does not undergo Thomas precession (see [22]) . It is easy to show that the developed here classical theory gives the same terms 2
for radiation without spin-flip and proportional to h as are derived by the semiclassical theory for spin radiation caused by Larmor precession. Thus, we have in detail considered here the spin light identification problem when the spin radiation proceeds against the background of powerful synchrotron radiation, recoil effects, and other relativistic phenomena. In its pure form the spin light contributes to the synchrotron radiation power as a small correction 2
proportional to h • At the present time the problem of spin light radiation of the relativistic magnetic moment is particularly urgent in connection with the construction of ultrahigh energy accelerators. The procedure for experimental observation of spin dependence of synchrotron radiation power was proposed in Budker Institute of Nuclear Physics (Novosibisk), and this experiment itself was described in [23-25]]. In this experiment synchrotron radiation power proportional to h was for the first time observed to be dependent on the spin orientation of a free electron moving in a macroscopic magnetic field. Now it is possible to carry out more detailed investigation of spin light. Acknowledgments
We thank Prof. Yu. L. Pivovarov. for interesting discussion on these problems and Prof. V.Ya. Epp for his help in improving of the paper. This work was supported by RF President Grant no. SS 5103.2006.2, and by RFBR grant no. 06-02-16 719.
438 References
[1] V. Bargmann, L. Michel, V. L. Telegdi, Phys. Rev. Lett. 2 (1959) 435. [2] A A Schupp, R. V. Pidd, H. R. Crane, Phys. Rev. 121 (1961) 1. [3] V. A Bordovitsyn, I. M. Ternov, V. G. Bagrov, SOY. Phys. Usp. 165 (1995) 1083 (in Russian). [4] V. A Bordovistyn, V. S. Gushchina, I. M. Ternov, Nucl. Instr. Meth. A 359 (1995) 34. [5] VA Bordovitsyn, Izv. Vuz. Fiz. 40, N22 (1997) 40 (in Russian). [6) G. N. Kulipanov, A E. Bondar, V. A Bordovitsyn et aI., Nucl. Instr. Meth. A405 (1998)191. [7] I. M. Ternov, Introduction to Spin Physics of Relativistic Particles, MSU Press (1997) 240 (in Russian). [8] Synchrotron Radiation Theory and its Development. Ed.V.ABordovitsyn, World Scientific, Singapore, 1999. See also: Radiation Theory of Relativistic Particles, Fizmatlit, Moscow, 2002 (in Russian). [9] VA Bordovitsyn, V.Ya. Epp, Nucl. Instr. Meth. A 220 (1998) 405. V. A Bordovitsyn, [10] A Lobanov, A Studenikin, Phys. Lett. B 564 (2003) 27. [11] A E. Lobanov, Phys. Lett. B 619 (2005) 136. [12] G. J. Bhabha, G.C. Corben, Proc. Roy. Soc. 178 (1941) 273. [13] A Bialas, Acta Phys. Polon, 22 (1962) 349. [14] M. Koisrud, E. Leer, Phys. Norv. 17 (1967) 181. [15] J .Cohn, H.Wiebe, J.Math. Phys. 17 (1976) 1496. [16] J. D. Jackson, Rev. Mod. Phys. 48 (1976) 417I. [17] V. A Bordovitsyn et aI., Izv. Vuz, Fiz.21, N25 (1978) 12; N210 (1980) 33. [18] V. A Bordovitsyn, G. K. Razina, N. N. Byzov, Izv. Vuz, Fiz. 23, N210 (1980) 33. [19] V. A Bordovitsyn, R. Torres, Izv. Vuz., Fiz. 29 N25 (1986) 38. [20] V. A Bordovitsyn, V.S.Guschina, Izv. Vuz., Fiz. 37, N21 (1994) 53. [21] I. M Ternov, V.G.Bagrov, A M. Khapaev, Zh. Exp. Teor. Fiz.48 919650 921 (in Russian), SOY. Phys, JETP 21 (1965) 613. [22] VA Bordovitsyn, V.V.Telushkin, Izv. Vuz., Fiz. 49, N2 (2006). [23] V.N.Korchuganov, G.N.Kulipanov, M.N.Mezentsev, et aI., Preprint INP 7783, INP, Novosibirsk (1977) . [24] AE.Bondar, E.L.Saldin, Nucl. Instr. Meth.195 (1982) 577. [25] S.ABelomestnykh, AE.Bondar, M.N.Yegorychev, et al. Nukl. Instr. Meth., 227 (1984) 173.
SIMULATION THE NUCLEAR INTERACTION Timur F. Kamalov a Physics Department, Moscow State Open University, 107966 Moscow, Russia Abstmct. Refined are the known descriptions of particle behavior with the help of Lagrange function in non-inertial reference systems depends of coordinates and their multiple derivatives. This entails existing of circumstances when at closer distances gravitational effects can prove considerably stronger than in case of this situation being calculated with the help of Lagrange function in inertial reference systems depends of coordinates and their first derivatives. For example, this may be the case if the gravitational potential is described as a power series in sir where s is a constant correspondence for the nuclei scale.
1
Simulation in real reference frame
1.1
Particles in real reference frame
Classical physics usually considers the motion of bodies in inertial reference systems. This is a simplified and approximate description of the real pattern of the motion, as it is practically impossible to get an ideal inertial reference system. Actually in any particular reference system there always exist minor influences. Let us consider the precise description of the dynamics of the motion of bodies taking into account complex non-inertial nature of reference systems. For this end, let us consider a body in a non-inertial reference system, denoting the position of the body as r and time as t. Then, expanding into Taylor series the function r = ret), we get
_ r - ro
at 2
1.
1 ..
3
1 . (n)
4
+ vt + - 2 + ,at + ,at + ... + ,a 3. 4. n.
t
n
+ ...
(1)
Let us compare this expansion with the well-known kinematical equation for inertial reference systems of Newtonian physics relating the distance to the acceleration a,
rNewton
at 2
= ro + vt + T·
(2)
Denoting the hidden variables accounting for additional terms in non-inertial reference systems with respect to inertial ones as qr, we get 1 .
3
qr = 3! at
1 ..
4
1 . (n)
+ 4! at + ... + n! a
n
t
+ ...
(3)
Then
r
=
rNewton
ae-mail: [email protected]
439
+q
(4)
440
For inertial reference systems the Lagrangian L is the function of only the coordinates and their first derivatives, L = L(t, r, r) For non-inertial reference systems, the Lagrangian depends on the coordinates and their higher deriva. .....
·(n)
tives as well as of the first one, i.e. L = L(t, r, r, r, r, ... , r ) Applying the principle of least action, we get [1]
J ......
JL.) ~
n dn 8L (5) -1) dt n (--:(;0 ) Jrdt = O. n=O 8 r Then, the Euler - Lagrange function for complex non-inertial reference systems takes on the form JS = J
·(n)
L(r, r, r, r, ... , r )dt
=
(6) Or
(7) Denoting
p = p(2)
aL p
Or'
=
aT
= a~, p(3) =
=aL
a.(4)' r
p(5)
= fLk-
, a·(20
cp
sn+l) (r-ro)n+l
7r+ VV experiment at CERN (20 min) 18.20 E.Shabalin (ITEP) Final state interaction in K->27r decay (15 min) 18.35 K.Urbanowski (Univ. of Zielona Gora) Khaljin's Theorem and neutral meson subsystem (20 min) 18.55 AAIi (DESY), ABorisov , M.Sidorova (MSU) Bilinear R-parity Violation in Rare Meson Decays (15 min) 19.10 O.Kosmachev (JINR) Nonstable leptons and (p-e-r)-universality (15 min) 26 August, SUN 9.00-19.00
Bus excursion to Sergiev Posad
466 27 August, MON MORNING SESSION (Conference Hall) 9.00 - 13.50 Chairman: ADelia Selva 9.00 D.Gorbunov (INR) Status o/UHECR (30 min) 9.30 H.Gemmeke (lnst. for Data Processing and Electronics, Research Center Karlsruhe) The Auger experiment (30 min) 10.00 H.Gemmeke (lnst. for Data Processing and Electronics, Research Center Karlsruhe) Radio detection o/ultra high energy cosmic rays (30 min) 10.30 V.Flaminio (Univ. of Pis a) Neutrino telescopes in the deep sea (30 min) 11.00 C.Volpe (lPN CNRS) Beta-beams (30 min) 11.30 - 11.55 Tea break Chairman: V.Flaminio 11.55 G.Landsberg (Brown Univ.) Search/or extra dimensions and black holes at colliders 12.25 _D.Polyakov (Center for Adv. Math. Sci. & American Univ. of Beirut) New discrete states in two-dimensional supergravity quantum systems bound by gravity (20 min) 12.45 M.Fil'chenkov, S.Kopylov, Yu.Laptev (Peoples' Friendship Univ. of Russia) Quantum systems bound by gravity (20 min) 13.05 R.Nevzorov, S.Hesselbach, D.l.Miller, G.Moortgat-Pick, M.Trusov (Univ. of Glasgow) Lightest neutralino in the MNSSM (20 min) 13.25 C.Heusch (Univ. of California, St.Cruz) High-energy e-e-, gamma-e-, gamma-gamma interactions (25 min) 13.50 - 15.00 Lunch 15.00 - 19.15 AFTERNOON SESSION (Conference Hall) Chairman: C.Heusch 15.00 A.Isaev (lINR) Algebraic approach to analytical evaluation 0/ Feynman diagrams 15.20 KStepanyantz (MSU) Application o/higher covariant derivative regularization to calculation 0/quantum corrections in N= 1 supersymmetric theories (20 min) 15.40 O.Kharianov, LFrolov, V.Zhukovsky (MSU) Bound state problems and radiative effects in extended electrodynamics with Lorentz violation (20 min) 16.00 ALobanov, AVenediktov (MSU) Triangle anomaly and radiatively-induced Lorentz and CPT violation in electrodynamics (15 min) 16.15 - 16.55 Tea break Chairman: ABorisov 16.55 S.Vernov (SlNP MSU) Construction o/exact solutions in two-fields models (20 min) 17.15 KSveshnikov, M.Ulybyshev (MSU) Nonperturbative quantum relativistic effects in the confinement mechanism/or particles in a deep potential well (15 min) 17.30 AMykhaylov, Yu.Mykhaylov (MSU) Linearized gravity in a stabilized brane world model in five-dimensional Brans-Dicke theory (15 min) 17.45I.Fialkovsky, V.Markov, Yu.Pis'mak (St. Petersburg State Univ.) Parity violating thin shells in the framework o/QED (15 min) 18.00 T.Kamalov (Moscow State Open Univ.) Simulation the nuclear interaction (15 min) 18.15 O.Olkhov (Semenov Inst. ofChem. Phys.) Unique geometrization o/material and electromagnetic wave fields (IS min) 18.30 Yu.Rybakov (Peoples' Friendship Univ. of Russia) Open and closed cosmic chiral strings in general relativity (15 min) 18.45 M.Georgieva (Offshore Tech. Development Pte Ltd, Singapore) The size 0/ a parton
467 19.00 S.Gladkov (Moscow State Regional Univ.) On nonlinear dispersion of electromagnetic spectrum (IS min) 28 August, TUE 9.00 - 14.00 MORNING SESSION (Conference Hall) Chairman: B.Spaan 9.00 A.Kaidalov (ITEP) Some puzzles in B-decays (25 min) 9.25 V.Zakharov (ITEP) Nonperturbative physics at short distances (25 min) 9.50 M.Polikarpov (ITEP) Low dimensional manyfolds in lattice QCD (25 min) 10.15 Yu.Simonov (lTEP) Dynamics ofQCD at nonzero T and density (20 min) 10.35 G.Lykasov, AN.Sissakian, AS.Sorin, V.D.Toneev (JINR) Thermal effects in heavy-ion collisions (20 min) 11.55 ABadalian (ITEP) Decay constants ofheavy-light mesons (15 min) 11.10 - 11.30 Tea break Chairman: AKaidalov 11.30 N.Mankoc (Univ. of Ljubljana) Properties offour families ofquarks and leptons within the approach unifying spins and charges (25 min) 11.55 ANesterenko (JINR) Adler function within the analytic approach to QCD (20 min) 12.15 I.Narodetskii (ITEP) P wave baryons within the Field Corre/ator Method in QCD (20') 12.35 ASidorov (JINR) Polarized parton densities and higher twist corrections in the light of the recent CLAS and COMPASS data (20 min) 12.55 ANefediev (ITEP) Chiral symmetry breaking and the Lorentz nature of confinement 13.10 M.Osipenko (SINP MSU & INFN) Experimental moments of the structure function F2 ofproton and neutron (15 min) 13.25 V.Braguta (IHEP) Double charm onium production at B-factories and charmonium distribution amplitudes (20 min) 13.45 V.Bomyakov (IHEP) Lattice results on gluon and ghost propagators in Landau gauge 14.00 -15.00
Lunch
15.00 - 19.20 AFTERNOON SESSION (Conference Hall) Chairman: G.Diambrini Palazzi 15.00 AKataev (INR), V.Kim (INP, Gatchina) Higgs-+bb decay and different QCD corrections (20 min ) 15.20 I.Bogolubsky, E.-M.Ilgenfritz, M.Muelier-Preussker, AStembeck (JINR) Gluon and Ghost propagators in SU(3) gluodynamics on large lattices (15 min) 15.35 M.Tokarev (JINR) QCD test ofz-scalingfor piO-mesonproduction (15 min) 15.50 ASafronov (SINP MSU) Analytic approach to constructing effective theory ofstrong interactions and its application to pion-nucleon scattering (IS min) 16.05 D.Ebert, ATyukov, V.Zhukovsky (MSU) Phase transitions in dense quark matter in a constant curvature gravitational field (15 min) 16.20 K.Zhukovskii (MSU) Quark mixing in the standard model and the space rotations (15 min) 16.35 O.Pavlovsky (MSU) Effective Lagrangians andfield theory on a lattice (15 min) 16.50-17.10
Tea break
Chairman: V.Zhukovsky
468 17.10 AShabad (Lebedev Physics Inst.) String-like electrostatic interaction in QED with infinite magnetic field (20 min) 17.30 V.Skvortsov (MIPT), N.Vogel (Univ. of Tech. Chemnitz) Nuclear reactions and accompanying physical phenomena in plasma oflaser-induced discharges (15 min) 17.45 E.Arbuzova (Intern. Univ. "Dubna"), G.Kravtsova (MSU), V.Rodionov (Russian State Geological Prospecting Univ.) Particles with low binding energy in the strong stationary magnetic field (15 min) 18.00 V.Bagrov (Tomsk State Univ.) New results of synchrotron radiation theory (20 min) 18.20 V.Telushkin, V.Bordovitsyn (Tomsk State Univ.) Coherent Spin Light (15 min) 18.35 V.Sharikhin (Moscow Power Engineering Inst.) Microdrops condensation ofsolar photons in strong magnetic field (15 min) 18.50 ARabinowitch (Moscow State Univ. ofInstrument Construction and Informatics) A new generalization ofDirac's equationfor nucleons (15 min) 19.05 V.Belov, E.Smimova (Moscow Inst. of Electronics and Math.) Semiclassical soliton type solution of the nonlocal Gross-Pitaevsky equation (15 min)
29 August, WED MORNING SESSION (Conference Hall) Round Table Discussion on "Dark Matter and Dark Energy: a Clue to Foundations of Nature" Chairman and convener: AStarobinsky 9.00 AGalper (MEPhI), P.Picozza (Univ. of Rome-II) Antimatter and dark matter research in space (30 min) 9.30 AStarobinsky (Landau Inst.) Dark energy: present observational status, scalar-tensor andf(R) models (30 min) 10.00 R.Bemabei (Univ. of Rome-II) Investigating the dark halo (30 min) 10.30 AMalinin (Univ. of Maryland) Dark matter searches with AMS-02 (30 min) 11.00 -11.30 Tea break 11.30 V.Dokuchaev (INR) Anisotropy of dark matter annihilation in the Galaxy (20 min) 11.50 V.Berezinsky (LNGS), Yu.Eroshenko (INR) Remnants of dark matter clumps in the Galaxy (20 min) 12.10 - 13.00 Discussion and conclusion 13.00 -14.30 Lunch
9.00-13.00
SEVENTH INTERNATIONAL MEETING ON PROBLEMS OF INTELLIGENTSIA: "Rights and Responsibility of the Intelligentsia" 14.30 -14.40 Opening (Conference Hall) Chairman: AStudenikin 14.40 V.Trukhin (Dean of the Faculty of Physics, Moscow State University) 15.00 S.Filatov (Found. for Social, Economical and Intellectual Programs) Rights and Responsibility of the Intelligentsia (30 min) 15.30 I.Bleimaier (Princeton) The Conscience of the Intelligentsia (30 min) 16.00 Discussion and conclusion Closing ofthe 13th Lomonosov Conference on Elementary Particle Physics and the 7th International Meeting on Problems of Intelligentsia SPECIAL SESSION (40 0)
List of participants of the 13th Lomonosov Conference on Elementary Particle Physics and the 7th International Meeting on Problems of Intelligentsia Andreotti Erica Arbuzova Elena Astafurov Vladimir Badalian Alia Bagrov Vladislav Balev Spasimir Barabash Alexander Barsuk Sergey Belokurov Vladimir Bernabei Rita Bleimaier John Bogolubsky Igor
Univ. of Insubria Int.Univ. ofDubna Group REI ITEP Univ.ofTomsk JINR ITEP LAL,Orsay MSU DAMA Princeton JINR
Borisov Anatoly Borisov Gennady Bornyakov Vitaly Braguta Victor Bunichev Viacheslav Chauveau Jacques Celnikier Ludwik Chekelian Vladimir Cherepashchuk Anatoly Cuhadar Donszelmann Tulay Curatolo Maria Della Selva Angelo Djurcic Zelimir Di Micco Biagio Di Ruzza Benedetto Diambrini Palazzi Giordano Dokuchaev Vladislav Egorychev Victor Eroshenko Yury Esposito Bellisario Fernandez Juan Pablo Fialkovsky Ignat Filatov Sergey
MSU Lancaster Univ. IHEP IHEP SINP Univ. Paris-VINII Observatoire de Meudon MPI SAl Univ. of British Columbia
Fil'chenkov Michael Flaminio Vincenzo Foster Brian Fujikawa Brian
INFN Frascati Univ. ofNeaples Columbia Univ. Univ .Rome-III Univ. of Trieste & INFN Trieste Univ.ofRome-I INR ITEP INR INFN Frascati CIEMAT St. Petersburg State Univ. Found. for Social, Economical and Intellectual Programs Peoples' Friendship Univ. of Russia Univ.ofPisa Univ.ofOxford LBNL
469
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected], [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] giordano.diarnbrini@ romal.infn.it [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
[email protected] [email protected] [email protected] [email protected]
470 Galper Arkady Gavrin Vladimir Gavryuseva Elena Gemmeke Hartmut
Gladkov Serguey Gorbunov Dmitry GrafKay Grats Yuri Greenberg Oscar Grigoriev Alexander Gutierrez Gaston Heusch Clemens Isaev Alexey Kadyshevsky Vladimir Kaidalov Alexei Kajino Toshitaka Kamalov Timur Kataev Andrei Kharlanov Oleg Kosmachev Oleg Kourkoumelis Christine Krasnikov Nikolay Kuznetsov Alexander Landsberg Greg Lobanov Andrei Lukash Vladimir Lykasov Gennady Madigozhin Dmitry Malinin Alexander Mankoc Borstnik Nonna Matveev Victor Mikhailin Vitaly Mikhailov Yuri Mikheyev Stanislav Mikheeva Elena Minakata Hisakazu Molokanova Natalia Murchikova Elena Narodetskii Ilya Nefediev Alexey Nesterenko Alexander Nevzorov Roman Nikishov Anatoly
MEPhl INR Inst. of Astrophys. and Space Research, Arcetri Inst. for Data Processing and Electronics, Research Center Karlsruhe Moscow State Regional Univ. INR Univ.ofErlangen-Nuremberg MSU Univ.ofMaryland MSU FNAL Univ. of California, St.Cruz JINR JINR ITEP Univ.ofTokyo Moscow State Open Univ. INR MSU JINR Univ. of Athenth INR Yaroslavl State Univ. Brown Univ. MSU LPI JINR JINR Univ.ofMaryland Univ.ofLjubljana INR MSU SINP INR LPI Tokyo Metropolitan Univ. JINR MSU ITEP ITEP JINR Univ.ofSouthampton LPI
amgalper@mephLru [email protected] [email protected] [email protected]
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] mikheyev@pcbail O.inr .ruhep.ru [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] nikishov@lpLru
471 Nones Claudia
Nozzoli Francesco Ochoa-Ricoux Juan Okun Lev Olkhov Oleg Osipenko Mikhail Panasyuk Mikhail Partridge Richard Pavlovsky Oleg Polikarpov Mikhail Polyakov Dimitri Potrebenikov Yury Pranko Alexander Purohit Milind Rabinowitch Alexander Ray Heather Rodionov Vasily Rybakov Yuri Safronov Arkady Sakashita Ken Savrin Vladimir Shabad Anatoly Shabalin Evgeny Sharikhin Valentin Shaibonov Bair Shirkov Dmitry Sidorov Alexander Sidorova Maria Siebel Martin Simonov Yuri Sissakian Alexey Skvortsov Vladimir Slavnov Andrey Smirnova Ekaterina Spaan Bernhard Spillantini Piero Starobinsky Alexei Starostin Alexander Stepanyantz Konstantin Studenikin Alexander Tavkhelidze Albert
Centre de Spectrometrie Nucleaire et de Spectrometrie de Masse Univ.ofRome-II California Inst. of Technology ITEP Semenov Inst. of Chern. Phys. SINP, INFN SINP& MSU Brown Univ. MSU ITEP Center for Adv. Math. Sci., American Univ. of Beirut JINR FNAL Univ. of South Carolina Moscow State Univ. ofInstrument Construction and Informatics LANL Moscow State Geological Prospecting Acad.
Peoples' Friendship Univ. of Russia SINP KEK SINP, Kurchatov Inst. Lebedev Phys. Inst. ITEP Moscow Power Engineering Inst. JINR JINR JINR MSU CERN ITEP JINR MIPT, Univ. of Tech. Chemnitz Steklov Math. Inst & MSU Moscow Inst. ofElectr. & Math. Univ.ofDortmund INFN-Florence LITP ITEP MSU MSU INR
[email protected], [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] shabad@lpLru [email protected] [email protected] bairsh@yandex. [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] smimova_ [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] a\[email protected]
472
Urbanowski Krzysztof
Tomsk State Univ. JINR Faculty of Physics, MSU MSU MSU CERN, Univ.ofCalifornia Univ. of Zielona Gora
Vannucci Francois Venediktov Artur Vemov Sergey Volpe Cristina Zakharov Valentin Zhukovsky Konstantin Zhukovsky Vladimir Zhuridov Dmitri
Univ. Paris 7 MSU SINP IPN CNRS ITEP MSU MSU MSU
Telushkin Valeriy Tokarev Mikhail Tmkhin Vladimir Tyukov Alexander Ulybyshev Maxim Unel Gokhan
[email protected] [email protected] [email protected] alex_ [email protected] [email protected] [email protected] [email protected]. zgora.pl [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
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