Table Of Content

Macroscopic Universality : Why QCD in Matter is Subtle ? Romuald A. Janik1, Maciej A. Nowak1,2 , Gábor Papp3 and Ismail Zahed4 1 Department of Physics, Jagellonian University, 30-059 Krakow, Poland. 2 GSI, Plankstr. 1, D-64291 Darmstadt, Germany & Institut fu¨r Kernphysik, TH Darmstadt, D-64289 Darmstadt, Germany 3GSI, Plankstr. 1, D-64291 Darmstadt, Germany & Institute for Theoretical Physics, Eötvös University, Budapest, Hungary 4Department of Physics, SUNY, Stony Brook, New York 11794, USA. (February 1, 2008) chiral symmetry would be restored. This expectation 7 Weuseachiralrandommatrixmodelwith2N flavorsto 9 f is not borne out by direct numerical calculations which mock up the QCD Dirac spectrum at finite chemical poten- 9 show that chiral symmetry is restored for any nonzero tial. We show that the 1/N approximation breaks down in 1 chemicalpotentialinthechirallimitandforlargematrix thequenchedstatewith spontaneouslybrokenchiralsymme- sizes[12],inqualitativeagreementwiththequenchedlat- n try. The breakdown condition is set by the divergence of a a tice simulations. The numericalresults have been shown two-pointfunction,thatisshowntofollowthegeneralloreof J to follow from a modified replica trick [12,15], in agree- macroscopic universality. Inthisstate, thefermionic fluctua- 0 ment with [3]. tions are not suppressed in thelarge N limit. 1 In this letter, we reanalyze the spectral density with 2Nf conventional quarks. In section 2, we introduce 2 the model and discuss the 1/N expansion around the v quenched state with spontaneously broken chiral sym- 9 1. Inthe presenceofachemicalpotential,the lattice metry. In section 3, we evaluate a pertinent two-point 2 3 QCD Dirac operator is non-hermitean. As a result, the correlationinthesamestate,andshowthatitdivergesin 6 fermionicdeterminant(probabilitymeasure)oftheQCD the eigenvalue plane in the small eigenvalue region. The 0 Dirac operator is a complex number. Since Monte-Carlo support for the ensuing spectral distribution is in agree- 6 simulationsdemandapositivedefinitemeasure,straight- ment with a recent result [12] using different arguments. 9 forwardalgorithmshavefailed [1]. UnconventionalQCD We explain why. Our result for the two-pointcorrelator, / h algorithms have been devised, leading to a chiral phase extends the macroscopic universality argument in ran- p transitionatfinitechemicalpotentialandstrongcoupling dom matrix models [16] to the chiral case. In section 4 - (g =∞)[1–3]. Inthe quenchedapproximation,however, we discuss the difference between the unquenched and p e the restoration of chiral symmetry was found to set in quenchedspectraldensities. Wearguethatthequenched h at surprisingly small chemical potentials of the order of statewithspontaneoussymmetrybreakingdoesnotsup- : half the pion mass, thereby vanishing in the chiral limit press the fermionic fluctuations. Our conclusions are v i [4]. The originofthe discrepancywastracedbackto the summarized in section 5. X phase of the Dirac operator [5], as demonstrated by in 2. Amodelformofthe QCDHamiltonianinthe chi- r [3] using lattice simulations. a ralbasiswithfinitechemicalpotentialµandzerocurrent Whilemanyofthebasicissuesrelatedtothequenched quark mass is calculations are known on the lattice [2–5], we feel the need for a better understanding using simple models. QQCD ≡M= 0 µ + 0 −iA (1) ThespontaneousbreakingofchiralsymmetryinQCDre- 5 (cid:18)−µ 0(cid:19) (cid:18)iA† 0 (cid:19) flects the spectral distribution of quark eigenvalues near zero-virtuality[6]. Forsufficientlyrandomgaugeconfigu- whereonlyconstantquarkmodeshavebeenretained[13]. rations,theDiracoperatorcanbeapproximatedbyachi- In the context of four dimensional field-theories Q5 is ralmatrixwithrandomentries[7,8],inqualitativeagree- the Hamiltonian of a Dirac particle in five dimensions. ment with a number of lattice simulations for both the The translationofthis operatorto chiralrandommatrix bulkeigenvaluedistributions[9],andcorrelations[10,11]. modelsisachievedby assumingthatthegluonicconfigu- Chiral random matrix models with finite chemical po- rations A are sufficiently randomso that A→R, where tentialhave been discussedrecentlyin [12–14]using var- R is an N ×N complex random matrix with Gaussian ious methods. For zero chemical potential, the ground weight. With this in mind, (1) is the sum of a determin- state breaks spontaneously chiral symmetry. The low- istic D and random (chiral) piece R. The deterministic lying spectrum is that of constituent quarks with no piece is non-hermitian for µ 6= 0, with complex ±i|µ| Fermisurface. Naivelyonewouldexpectthatforachem- eigenvalues. icalpotentialof the orderofthe constituent quarkmass, Now, consider the partition function associated with (1) for 2N flavors f 1 Z(2N ,µ)=<det2Nf(z−M)> (2) 3. The breakdown can be probed either by looking f at the nonplanar corrections to (6), or by testing the with V2Nf = −logZ/2Nf playing the role of a complex the stability content of the quenched state against local potential. For z = im, (2) mocks up the partition func- correlations. One pertinent correlation function is given tion of 2Nf quarks of equal mass m. The averaging in by the two-point function (2) is carried using a gaussian weight N2C(z,z¯)=<Tr(z−M)−1Tr(z¯−M†)−1 > (8) 1 c <...>≡ ...e−NΣTrRR†dR (3) Z Z in the quenched state. The lower script in (8) is for connected. Thephysicalinterpretationof(8)willbeclarified with Σ setting the scale of chiral symmetry breaking in the nextsection. For largevalues of z,(8) may be an- (throughout Σ is set to 1 for simplicity, unless indicated alyzed using again a 1/N expansion [16–18]. Following otherwise). The gaussian weight (3) is an idealization of the arguments in [16], the result is [21], the gluonaction, anddefines the quenchedaveragingsto bediscussedbelow. Forsufficientlysmallquarkeigenval- 1 2 2 2 4 2 ues (here constantmodes), the gluon interactionmay be N C(z,z¯)=− ∂z∂z¯Log{[(H−µ ) −|z−G| ]/H } (9) 4 thoughtasrandom[7]. Thegaussianweightfollowsfrom the principle of maximum entropy. where H = |z −G|2/|G|2 and G is a solution to Pas- First, let us consider the spectral distribution associ- tur’s equation (6). For µ = 0 and z = z¯ outside the ated to (2) in the quenched approximation. With V2Nf cut, (9) reduces to N2C(z,z) = 1/z2(z2−4)2, which is playing the role of a complex potential, the spectral dis- different from the hermitean (nonchiral) result of [16], tribution follows from Gauss law [15] but in agreement with the one of [22]. The result (9) showsthatinourcasethe two-pointcorrelationfunction 1 ν(z,z¯)=Nlfim→0ρ(z,z¯)=Nlfim→0−πN∂z¯∂zV2Nf (4) itsenadlsinogatmheenmabalcerotsocotphiecounneiv-perosianltitfyuanrcgtiuomn,entthedriescbuysseexd- originally in [16] to the chiral case. Explicitly, πν(z,z¯)=∂G/∂z¯, with the resolvent From (9) we observe that the two-point correlation 1 function diverges in the eigenvalue-plane in the domain G(z,z¯)= Tr(z−M)−1 (5) N prescribed by the zero of the logarithm, (cid:10) (cid:11) We have retained (z,z¯) in both (4) and (5) to allow for |z−G|2(1−|G|2)−µ2|G|2 =0 (10) the possibility that holomorphic symmetry may be broken in the thermodynamical limit by the quenched aver- For µ = 0, this condition is fulfilled for G = z, which is aging (3). The factor of 1/N in (4) implies that ν(z,z¯) a trivial (unphysical) solution to Pastur’s equation, and is normalized to one on its pertinent support. for |G|2 = 1 a nontrivial (physical) solution to Pastur’s For largez and to leading orderin 1/N,G obeys Pas- equation along the cut. The latter when supplemented tur’s equation (planar approximation) [13,19], withtheconditionlim G∼1/zintheoutside,yields z→∞ Wigner’s distribution for the eigenvalue density. G((z−G)2+µ2)−z+G=0 (6) Figure 1 shows the envelope in dashed line for which the condition (10)is met in the w =iz plane, for several The solution to (6) is a holomorphic function of z, ex- values of the chemical potential µ. A structural change ceptforlinediscontinuities,withend-pointsgivenbythe occurs at µ2 = 1. The solid lines are the mean-field zeroes of the discriminant solution following from (7). They lie in the eigenvalue- 4µ2z4+z2(8µ4−20µ2−1)+4(µ2+1)3 =0 (7) domain where the 1/N fluctuations are dominant, a signal that chiral symmetry is restored. Since (8) mixes z For µ = 0, there is a cut along the real axis be- andz¯,holomorphicsymmetryisspontaneouslybrokenin tween −2 and +2. The discontinuity along the cut is the quenched state. One can recast (10) using (6) into Wigner’s semicircle, with a quenched and stable state (iw=x+iy) that breaks spontaneously chiral symmetry. With in- creasing µ2 → 1/8, the two roots approach each other, (µ2−x2)2y2 = 4µ4(1−µ2)−(1+4µ2−8µ4)x2−4µ2x4 , (11) followedby the emergenceoftwo new rootsfromrealin- (cid:2) (cid:3) finity. For µ2 =1/8 they coalesce pairwise, and move to Theresult(11)isinagreementwiththeonein[12],where the complex plane for µ2 > 1/8. This behavior suggests thequenchedspectraldensitywasevaluatedintheinside that the quenched state with ν(+i0,−i0) 6= 0, supports (small eigenvalues) using a pair of conjugate quarks in a chiral condensate up to µ2 =0.125, after which a first the quenched approximation(essentially |z−M| in (2)). order transition is observed. This conclusion, however, Conventional quenched QCD in the 1/N approximation does not hold since the 1/N approximationbreaks down worksas well outside. The two approachesagree on (11) for small eigenvalues in the present model, as we now which sets their domain of validity. To be able to de- show. scribethespectraldistributioninsideusingthequenched 2 z−M version of (2) requires a different method than the 1/N (z−M)2 =|z−M|2×( ) (12) approximation we have used. z¯−M† Intermsof(2)and(12),theunquenchedspectraldensity 2 µ2=0.125 2 µ2=0.8 forfiniteN andNf isnowgivenbyρ(z,z¯). From(2)the contribution of the modulus is 1 1 0 0 NfN ρ (z,z¯)=ν(z,z¯)− C(z,z¯) mod 2π -1 -1 2N -2-1 0 1 -2-1 0 1 + Nf <Trδ(z−M)ln det|z−M|>c +... (13) 2 2 µ2=1. µ2=1.2 while the contribution of the phase is 1 1 N N f 0 0 ρph(z,z¯)= C(z,z¯)+... (14) 2π -1 -1 The density ρ(z,z¯) is the sum of (13) and (14) modulo -2 -2 anextra contribution fromthe crossedterms, essentially -1 0 1 -1 0 1 FIG.1. Theenvelope(dashedlines) isfrom eq.(10) inthe the last term in (13) to the order quoted. We note that plane w=iz while the cuts (solid lines) are from eq. (7), for the connected two-point function (8) is essentially the different µ. Shaded islands represent “other vacuum”. fluctuation of the phase of the fermion determinant in thequenchedstate. Itappearswithoppositesignsinthe Figure2 showsthe analyticalbehavior(solidlines)for modulusandthe phase,andcancelsinthesum. Thedif- the imaginary part of the resolvent (5-6) and the corre- ferenceρ(z,z¯)−ν(z,z¯)accountsforthe effectsofthe sea lation function (9) for two values of w = iz and fixed µ2 = 2, that is w = i4 − y (left) and w = i0.02− y fermions (unquenching). This difference involves nonlo- cal correlation functions of z and z¯, of which (8) is a (right) with y≥0. The dashed curves are a comparison genericexample. Thefermioniceffectsaredownby1/N, to a numerical estimate using an ensemble of 200 chi- provided that the correlation functions are stable. This ral gaussian plus deterministic matrices with dimension is not the case in the quenched state with spontaneous 100×100. Note that in the right figures, two different symmetry breaking. solutions to (6) have to be used while crossing the re- Further insights to the above results can be achieved, gionofnon-analyticity(shadedregion). Inthe“bounded if were to note that (8) may be rewritten as region” between the islands of Figure 1 the asymptotic condition limz→∞G ∼ 1/z is no longer required. We N2C(z,z¯)=hq†q Q†Qic (15) remark that ImG=0 for z = 0, and chiral symmetry is restored. with 1 1 0 num num qq† = and QQ† = (16) exact 2 exact z−M z¯−M† -0.05 G 1 G playing the role of “propagators”. For µ = 0, hq†qi and m m I-0.1 0 I hQ†Qi are nonzero for z ∼i0, and the vacuum contribu- x=4, µ2=2 -1 x=0.02, µ2=2 tions cancel out. Typically, ρ(z,z¯)−ν(z,z¯)∼O(Nf/N) -0.15 for z ∼ i0. For µ = 0 and large N, the fermionic fluc- 0 1 2 3 4 0 1 2 3 4 10-2 104 tuations are suppressed in the quenched state. The lat- Eq. (12) Eq. (12) N=100 N=100 NN==25000 102 ter breaks spontaneously chiralsymmetry, andpreserves -2N C(z,z)10-3 1 -2N C(z,z) hoFloomrosrmpahlilcµsy,m(1m5)ettrhyro(qug↔h Q(1)0.) diverges when closing 10-2 on the dashed curve of Figure 1 from the outside. This 10-4 x=4, µ2=2 10-4 x=0.02, µ2=2 is a signal that (15) is receiving increasingly large con- 0 1 y2 3 4 0 1 y2 3 4 tributionsfromthe“mixedcondensates”hq†Qiandtheir FIG.2. Imaginarypartoftheresolvent(upperfigures)and conjugates(other “vacuum”),sothe fluctuations arenot the correlation function (lower figures) for w = ix−y with suppressed any more. As a result ρ(z,z¯) − ν(z,z¯) ∼ x = 4, 0.02 and y ≥ 0 at µ2 = 2. The solid curves are O(N N0)forsmallz. Inthequenchedstatewithsponta- f analytical while thedashed curves are numerical (see text). neous symmetry breaking, the fermionic fluctuations are not suppressed in the large N limit. 4. A dequenching of the spectral density (4) can be simply achieved by noting that since (z − M) is non- 5. Using a chiral random matrix model with a fi- hermitean, we can split it into a phase and a modulus nite chemical potential µ, we have shown that the small through and quenched eigenvalue distribution of the Dirac operator is fluctuation driven, with a size in the complex 3 plane conditioned by the divergence of the connected [22] J.Ambjørn,C.F.KristjansenandY.M.Makeenko,Mod. two-point correlation function (8). In this domain the Phys. Lett. A7(1992) 3187. 1/N approximation breaks down and the effects from the fermionic fluctuations do not decouple in the ther- modynamicallimit. The fluctuation drivenphase breaks spontaneously holomorphicsymmetry and preserveschi- ral symmetry, as originally shown in [12]. Acknowledgments We thank Jurek Jurkiewicz and Jac Verbaarschot for discussions. M.A.N. thanks the INT at the Univer- sity of Washington for its hospitality during the com- pletion of this work. This work was supported in part by the US DOE grant DE-FG-88ER40388, by the Pol- ish Government Project (KBN) grants 2P03B19609 and 2P03B08308andbytheHungarianResearchFoundation OTKA. [1] K.Kanaya, Nucl. Phys.(Proc. Suppl.) B47 (1996). [2] E.Dagotto, A.Moreo andU.Wolff,Phys.Rev.Lett.57 (1986) 1292; F. Karsch and K.H. Mutter, Nucl. Phys. B313 (1989) 541; I.M. Barbour, C.T.H. Davies and Z. Sabeur,Phys. Lett. B215 (1988) 567. [3] A.Gocksch, Phys. Rev.Lett. 61 (1988) 2054. [4] I. Barbour et. al. , Nucl. Phys. B275[FS17] (1986) 296; C.T.H. Davies and E.G. Klepfish, Phys. Lett. B256 (1991) 68; J.B. Kogut, M.P. Lombardo and D.K. Sin- clair, Phys.Rev. D51(1995) 1282. [5] P.E. Gibbs, Phys. Lett. B182 (1986) 369. [6] T. Banks and A. Casher, Nucl.Phys. B169 (1986) 103. [7] E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. A560 (1993) 306; J.J.M. Verbaarschot and I. Zahed, Phys.Rev.Lett. 70 (1993) 3852. [8] J.J.M. Verbaarschot, Phys.Rev.Lett. 72 (1994) 2531. [9] M.A. Nowak, G. Papp and I. Zahed, eprint hep- ph/9604235; J. Jurkiewicz, M.A. Nowak and I. Zahed, eprint hep-th/9603308. [10] J.J.M. Verbaarschot, Phys.Lett. B368 (1996) 137. [11] M.A. Halasz and J.J.M. Verbaarschot, Phys. Rev. Lett. 74 (1995) 3920. [12] M. Stephanov,Phys.Rev.Lett. 76 (1996) 4472. [13] R. A. Janik, M.A. Nowak and I. Zahed, eprint hep- ph/9604253, and references therein. [14] M.A. Halasz, A.D. Jackson and J.J.M. Verbaarschot, in preparation; T. Guhr and T. Wettig, eprint hep- th/9605110. [15] H.-J. Sommers, A. Crisanti, H. Sempolinsky and Y. Stein,Phys. Rev.Lett. 60 (1988) 1895. [16] J. Ambjørn, J. Jurkiewicz and Y.M. Makeenko, Phys. Lett. B251 (1990) 517; E. Brézin and A. Zee, eprint cond-mat/9507032. [17] J.J.M. Verbaarschot, H.A. Weidenmu¨ller and M. Zirn- bauer, Ann.Phys. 153 (1984) 367. [18] E. Brézin and A. Zee, Phys. Rev. E49 (1994) 2588; E. Brézin, S. Hikamiand A.Zee, eprint hep-th/9412230. [19] L.A. Pastur, Theor. Mat. Phys. (USSR)10 (1972) 67. [20] G. ’t Hooft, Nucl. Phys. B75, 464 (1974). [21] R. A. Janik, M.A. Nowak, G. Papp and I. Zahed, to be published. 4