Macroscopic Universality : Why QCD in Matter is Subtle?

4 Pages·0.15 MB·English

by  Romuald A. Janik

Macroscopic Universality :
Why QCD in Matter is Subtle ?
Romuald A. Janik1, Maciej A. Nowak1,2 , G´abor 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, E¨otv¨os 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 con-
nected. 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 bro-
ken 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 sig-
nal 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 op-
erator 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.
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