Critical scaling at zero virtuality in QCD

14 January 1999

Physics Letters B 446 Ž1999. 9–14

Critical scaling at zero virtuality in QCD Romuald A. Janik a

a,c

, Maciej A. Nowak

b,c

, Gabor ´ Papp

d,e

, Ismail Zahed

f

SerÕice de Physique Theorique, CEA-Saclay, F-91191, Gif-sur-YÕette, France ´ b GSI, Planckstr. 1, D-64291 Darmstadt, Germany c Department of Physics, Jagellonian UniÕersity, 30-059 Krakow, Poland d ITP, UniÕ. Heidelberg, Philosophenweg 19, D-69120 Heidelberg, Germany e Institute for Theoretical Physics, EotÕos ¨ ¨ UniÕersity, Budapest, Hungary f Department of Physics and Astronomy, SUNY, Stony Brook, NY 11794, USA Received 21 October 1998 Editor: J.-P. Blaizot

Abstract We show that at the critical point of chiral random matrix models, novel scaling laws for the inverse moments of the eigenvalues are expected. We evaluate explicitly the pertinent microscopic spectral density, and find it in agreement with numerical calculations. We suggest that similar sum rules are of relevance to QCD at the critical temperature, and even above if the transition is amenable to a Ginzburg-Landau description. q 1999 Elsevier Science B.V. All rights reserved. PACS: 11.30.Rd; 11.38.Aw; 64.60.Fr

1. A large number of physical phenomena can be modeled using random matrix models w1x. An important aspect of these models is their ability to capture the generic form of spectral correlations in the ergodic regime of quantum systems. This regime is reached by electrons traveling a long time in disordered metallic grains w2x or virtual quarks moving a long proper time in a small Euclidean volume w3x. In QCD, the ergodic regime is characterized by a huge accumulation of quarks eigenvalues near zero virtuality. This is best captured by the Banks-Casher w4x relation <² qq :< ' S s pr Ž0., where the nonvanishing of the chiral condensate in the vacuum signals a finite quark density r Ž l s 0. / 0 at zero virtuality. This behavior is at the origin of spectral sum rules w5x, which are reproduced by chiral random matrix

models w6x. These sum rules reflect on the distribution of quark eigenvalues and correlations w7x. If QCD is to undergo a second or higher order chiral transition, then at the critical point there is a dramatic reorganization of the light quark states near zero virtuality as the quark condensate vanishes. In Section 2, we suggest that such a reorganization is followed by new scaling laws, which are captured by a novel microscopic limit. In Section 3, we use a chiral random matrix model with a mean-field transition to illustrate our point. In Section 4, we explicitly construct the pertinent microscopic spectral distribution in the quenched case and compare it to numerical calculations. In Section 5 we argue that if QCD is to be characterized by mean-field universality then the present matrix model results are applicable. In

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 4 9 8 - 1

R.A. Janik et al.r Physics Letters B 446 (1999) 9–14

10

Section 6, we suggest that spectral correlations persist in the vicinity of the critical point from above with new sum rules. Our conclusions are in Section 7. 2. The nonvanishing of S in the QCD vacuum implies that the number of quark states in a volume V, N Ž E . s VHd lr Ž l. in the virtuality band E around 0 grows linearly with E, that is N Ž E . ; EV. As a result, the level spacing D s dErdN; 1rV for N ; 1, and the eigenvalues of the Dirac operator obey spectral sum rules w5x. During a second or higher order phase transition S vanishes in the chiral limit. Scaling arguments give S ; m1r d at the critical point, where m is the current quark mass w8x. It follows again from the Banks-Casher relation w4x, that for small virtualities l, r Ž l. ; < l < 1r d to leading order in the current quark mass w9x. Hence, N Ž E . ; VE 1q 1r d , and the level spacing is now D) ; 1rV d rŽ dq1. at N ; 1. For a mean-field exponent d s 3, and we have D) s Vy3 r4 , which is intermediate between Vy1 in the spontaneously broken phase and Vy1 r4 in free space. At the critical point there are still level correlations in the quark spectrum except for the free limit, corresponding formally to d s 1r3. We now conjecture that at the critical point, the rescaling of the quark eigenvalues through l ™ lrD) , yields new spectral sum rules much like the rescaling with D s 1rV in the vacuum w5x. The master formula for the diagonal sum rules is given by the Ždimensionless. microscopic density of states

n) Ž s . s lim Ž VD) . r Ž s D) . V™`

3. Consider the set of chiral random plus deterministic matrices Ms

ž

im t q A†

tqA im

/

Ž 2.

where A is an N = N complex matrix with Gaussian weight, m a ‘mass’ parameter, and t a ‘temperature’ parameter. Such matrices or variant thereof have been investigated by a number of authors in the recent past w10x. Their associated density of states is

r Ž l. s

1 2N

²Tr d Ž l y M . :

where the averaging is carried using the weight associated to the following partition function Z w m,t x s dAdA† det M N f eyN Tr A A

H

†

Ž 4.

For t s m s 0 the density of states is r Ž0. s 1rp , while zero for t G 1 and m s 0. At t s 1, r Ž l. s < l < 1r3, which indicates that the t-driven transition is mean-field with r Ž0. as an order parameter. In particular, at t s 1 the level spacing is D) s Ny3 r4 near zero. Standard bosonization of the partition function Ž4. yields Z w m,t x s dPdP † e N logdet w Ž mqP . Ž mqP

H

†

. qt 2 x yN Tr P P †

Ž 5. where P is an Nf = Nf complex matrix. We may shift P by the mass matrix P ™ Q s P q m and get

Ž 1.

and similarly for the off-diagonal sum rules in terms of the microscopic multi-level correlators. Since we lack an accurate effective action formulation of QCD at T s Tc Ža possibility based on mean-field universality is discussed below., the nature and character of these sum rules is not a priori known, but could easily be established using lattice simulations in QCD. Could these sum rules be shared by random matrix models? We will postpone the answer to this question till Section 5, and instead show in what follows that the present scaling laws hold at zero virtuality for chiral random matrix models with mean-field exponents.

Ž 3.

Z w m,t x s dQdQ† det Ž QQ † q t 2 .

H

N

† † 2 =eyN w Tr Q Q ym Tr Ž QqQ . qm x

Ž 6.

dropping the irrelevant normalization factor. We will now specialize to the critical temperature t s 1, and denote the rescaled mass by x s imrD) and eigenvalue by s s lrD) . This suggests the rescaling Q ™ Q˜ s N 1r4 Q, so that 1

˜ ˜ † .2

˜

˜† .

˜ ˜† ey 2 Tr Ž QQ e i x TrŽQqQ Z w x x s dQdQ

H

Ž 7.

reducing to 1

`

Zw x x s

y

H0 rdr e

2

r4

J0 Ž 2 xr .

Ž 8.

R.A. Janik et al.r Physics Letters B 446 (1999) 9–14

for one flavor. Expanding this integral in powers of x, `

Z w m x s Z w0x 1 q

2 k Ž mrD) .

Ý

2k

2

Ž k! . 'p

ks1

G

ž

kq1 2

be evaluated using supersymmetric methods w1,11x. For example, 1

/

n) ,0,0 Ž s . s y

Ž 9. gives rise to spectral sum rules for the moments of the reciprocals of the eigenvalues of M by matching the mass power in the spectral representation of the partition function, m2

¦ ž ;/

Z w m x rZ w 0 x s

1q

Ł

l k)0

l2k

Ž 10 .

0

where the averaging is done over the Gaussian randomness with the additional measure Łlk ) 0 l2k . Matching the terms of order m2 yields 1

¦Ý ; l k)0

l2k

2 s

0

Ž 11 .

'p D)2

and matching the terms of order m 4 gives

¦ž

1

Ý

2 l k)0 l k

2

1

/ ; ¦Ý ; y

0

4 l k)0 l k

0

1 s

D)4

Ž 12 .

The relations Ž11. – Ž12. are examples of microscopic sum rules at the critical point t s 1. One should note here that the preceding calculation has been performed for the gaussian matrix model. It turns out that only if we were to add terms of the form eyN g 0Tr Ž P P

† 2

. yN g 1ŽTr P P † . 2

11

Ž 13 .

248

p2

Ž k 2 Ž s . j0 Ž s . q k 0 Ž s . j2 Ž s .

yk 1 Ž s . j1 Ž s . .

Ž 14 .

where k nŽ s . and jnŽ s . are given by ` s 4 jn Ž s . s dz z 2 nq1 J0 2 z 3r4 eyz r2 2 0 s 4 k n Ž s . s dz z 2 nq2 K 1 2 z 3r4 e z r2 2 C

H

H

ž ž

/ /

Ž 15 . Ž 16 .

and where the integration contour C is the sum of two lines: C s wyŽ1 q i .`,0x j w0,Ž1 y i .`x. In this spirit, the first sum rule Ž11. reads `

H0

1 s

2

n) ,1,0 Ž s . ds s

2

'p

Ž 17 .

The second sum rule Ž12. involves the 2-level microscopic correlator for Nf s 1 and n s 0 which can be obtained using a similar reasoning. In Fig. 1 we compare the quenched Nf s 1 Žleft. and unquenched Nf s 1 Žright. results to the numerically generated microscopic spectral density at t s 1 using N s 100 size matrices. The agreement suggests that the present method of finding the scaling properties of microscopic spectral distributions can be used to accurately determine the value of the critical exponent d in lattice simulations. 5. In QCD the character of the finite temperature transition depends crucially on the number of flavors Nf and the fate of the UAŽ1. quantum breaking. For

to the measure in Ž5., they would influence the spectral sum rules. However for Nf s 1 this addition amounts just to a global shift x ™ xrŽ1 q 2 g 0 q 2 g 1 . which sets the normalization in the sum rules. Higher powers of PP † are subleading after rescaling and do not affect the sum rules in the large N limit. 4. The diagonal moments of the reciprocals of the rescaled eigenvalues are generated by the microscopic density Ž1., in the limit N ™ ` and l ™ 0 but s s lrD) fixed. The microscopic density Ž1. for Nf flavours in a fixed topological sector n: n) , N f , n , can

Fig. 1. left: n) ,0,0 Ž s . at t s1 and Nf s 0 for matrices of size Ns100 Ždots. and the theoretical prediction Ž14. Žsolid line.. right: n) ,1,0 Ž s . at t s1 and Nf s1 for matrices of size N s 20 Ždotted., N s 50 Ždashed. and N s100 Žsolid..

R.A. Janik et al.r Physics Letters B 446 (1999) 9–14

12

two light flavors, we may assume with Pisarski and Wilczek w12x that the transition is from an SUŽ2. spontaneously broken phase to Z2 = SUŽ2. = SUŽ2. ; O Ž4. w13x. Choosing the order parameter F s s q it P p Žvacuum analogous to a ferromagnet., implies for the Ginzburg-Landau potential V Ž F . s qm Tr Ž F † q F . q g 0 Ž T . Tr Ž F †F . 2

q g 1 Ž T . Ž Tr Ž F †F . . q . . .

Ž 18.

at zero vacuum angle u . The dots refer to marginal or irrelevant terms. For a second order transition, g 0 ŽT . ; T y Tc , which is negative below Tc and positive above. The O Ž4. critical exponents following from Ž18. hold within the pertinent GinzburgLandau window w14x and Ž18. implies a specific set of microscopic sum rules which are not amenable to the mean-field matrix model we have discussed. They may be readily established using Dirac spectra from Lattice QCD simulations. Outside the Ginzburg-Landau window the critical exponents are in general mean-field w14x and our matrix model arguments hold. Indeed, for T s Tc the potential Ž18. when reduced to the space of constant modes, is reminiscent of the one discussed above w9x. If we note that the measure on the manifold with restored symmetry is eyb V 3 V ' eyV V , we conclude that Ž18. is enough to accommodate for the level spacing D) s 1rV d rŽ dq1. with d s 3 Žmean-field.. After the rescaling x s imrD) we recover Ž7. with g 1 s 1r2 and the proper identification of the manifold. 6. For temperatures near Tc from above, we can use Ž18. to define new sum rules for the quark eigenvalues in the sector with zero winding number w5x. In particular Ž Nf s 2. 1 V

2

1

¦Ý ; 2 l k)0 l k

0

1 s

2p

H 2N 0

du 2p

f

2

iu

−ž

=

Nf

iu

Tr e F † q F ey N f

ž

//

< Ž 19 .

The rhs measures the variance in the scalar direction on an invariant OŽ4. manifold with eyV V as a

measure. As T ™ Tc from above, the scalar susceptibility averaged over ‘u-states’ Žrhs of Ž19.. diverges since s and p become degenerate. These modes are the analogue of the ones originally discussed by Hatsuda and Kunihiro w15x using an effective model of QCD. In the case where the fluctuations are not important Žmean-field., then Ž19. can be readily assessed by rescaling F ™ 'V F and noting that the quartic contribution in Ž18. becomes subleading in large V. Hence, 1 V

1

¦Ý ; 2 l k)0 l k

1

Ž 20 .

s 2 g0 Ž T .

0

Near Tc from above it is seen to diverge as 1rŽT y Tc . with the critical exponent g s 1 Žmean-field.. In Ž20. the eigenvalues are of order 1 due to the gap in the spectrum, hence 1rV normalizes a sum of V terms and the sum rules carries information on the entire spectrum, not just the microscopic part. They are not universal in the sense of matrix theory. The possibility of sum rules in the vicinity of the critical temperature from above reflects on the persistence of the level correlations in the Dirac spectrum in relation to the O Ž4. manifold despite the gap developing in the eigenvalue density. The latter is due to the fact that near zero virtuality the accumulation of eigenvalues is not commensurate with the volume V. At high temperature, the quark eigenvalues are typically of order T, and both p and s correlations are dissolved in the ‘plasma’ with trivial sum rules. In QCD we expect this to take place at a temperature T ; 3Tc w16x. These ideas can be tested in the context of a mean-field transition using again a chiral random matrix model with one flavor. We may use the bosonized form of the partition function Ž6., but now instead introduce the variable y s 'N im ; 'N l, and rescale Q by Q ™ Q˜ s Q'N . This leads to the following expression for the partition function ZŽ y. s t 2 N

` H0 rdre ž

p t2N s N

1

y 1y

P

t2 2

t y1

t2

/

r2

J0 Ž 2 yr . e y

1

ey t 2y1 y

2

2

Ž 21 .

Expanding the partition function in powers of y

R.A. Janik et al.r Physics Letters B 446 (1999) 9–14

13

leads again to modified sum rules, the simplest example being 1

¦Ý ; 2 l k)0 Nl k

1 s

0

Ž 22 .

2

t y1

which is seen to diverge at t s 1 as expected. The comparison to numerical simulation is shown in Fig. 2 using a random set of matrices M distributed with a Gaussian measure. For high temperatures or large matrix sizes the agreement is good. At the critical temperature, the finite size effects are important. Amusingly, we note the drop by two orders of magnitude at t ; 3 s 3t c . The Green’s function both for quenched Žthe mass playing the role of an external parameter. and one flavor chiral random matrix model in the rescaled variables may be readily obtained, GŽ y. s

1

'N Ýi

1 y y 'N l i

sy

1

y

'N

t2y1

Ž 23 .

Since for T ) Tc the eigenvalue spectrum develops a gap, there are no eigenvalues for small values of y, hence the resolvent is purely real. For finite sizes however, y may get out of the gap and our result breaks down as well as the scaling arguments. This is seen in Fig. 3, where for temperatures slightly above the critical one we are entering the nonzero eigenvalue density part of the spectra. This effect is shifted for higher values with increasing temperature and matrix size. 7. Using arguments based on universality we have suggested that the QCD Dirac spectrum may exhibit

Fig. 3. Scaled and quenched resolvent Ž23. Žsolid line. in comparison to a numerical simulation with N s 20 Žlong dashes., N s 50 Žshort dashes. and N s100 Ždotted line. random matrices.

universal spectral correlations at T s Tc that reflect on the nature of the chirally restored phase. We note that these correlations are different from the ones considered in w17x, which are valid for T - Tc but not in the vicinity of T s Tc where they vanish. To probe the correlations around the chiral phase transition, one has to consider the new scaling regime described in this paper. To emphasize its novelty, we will refer to it as ‘‘critical scaling’’. To illustrate our points, we have used a chiral random matrix model. Although the matrix model is based on a Gaussian weight, we provided qualitative arguments for why the results are insensitive to the choice of the weight at the critical point. In QCD this can be readily checked using our recent arguments w3x at finite temperature, since the closest singularity to zero in the virtuality plane is persistently ‘pionic’ for T F Tc w8x. The existence of a microscopic spectral density at T s Tc for QCD opens up the interesting possibility of measuring both the critical temperature Tc and the critical exponent d by simply monitoring the pertinently rescaled distribution n) Ž s . of eigenvalues in lattice QCD simulation. We have also suggested that spectral correlations persist near Tc from above. Clearly, most of our observations are subject to finite size effects whose analysis is beyond the scope of this work.

Acknowledgements

Fig. 2. The result Ž22. Žsolid line. checked against numerical simulations using N s 20 Žopen squares., N s 50 Žopen circles. and N s100 Žopen triangles. random matrices.

This work was supported in part by the US DOE grant DE-FG-88ER40388, by the Polish Government grant Project ŽKBN. grants 2P03B04412 and 2PB03B00814 and by the Hungarian grants FKFP0126r1997 and OTKA-T022931. RAJ is supported by the Foundation for Polish Science ŽFNP..

14

R.A. Janik et al.r Physics Letters B 446 (1999) 9–14

Note added After completing this work we noticed the paper by Brezin ´ and Hikami w18x, where the issue regarding a new universality class at the closure of the gap in the eigenvalue distribution is also discussed using different arguments. Their results are carried for non-chiral ensembles, and hence different from ours. However, the similarities between say our results Ž15., Ž16. and their integral forms suggest relationships between the chiral and non-chiral results, that are worth unravelling.

References w1x K. Efetov, Supersymmetry in Disorder and Chaos, Camb. UP, NY, 1997, and references therein. w2x D.J. Thouless, Phys. Rep. 13 Ž1974. 93. w3x R.A. Janik, M.A. Nowak, G. Papp, I. Zahed, Phys. Rev. Lett. 81 Ž1998. 264.

w4x w5x w6x w7x w8x w9x w10x

w11x w12x w13x w14x w15x w16x w17x

w18x

T. Banks, A. Casher, Nucl. Phys. B 169 Ž1980. 103. H. Leutwyler, A. Smilga, Phys. Rev. D 46 Ž1992. 5607. E. Shuryak, J. Verbaarschot, Nucl. Phys. A 560 Ž1993. 306. J. Verbaarschot, I. Zahed, Phys. Rev. Lett. 70 Ž1993. 3852. A. Kocic, J. Kogut, M. Lombardo, Nucl. Phys. B 398 Ž1993. 376. R.A. Janik, M.A. Nowak, I. Zahed, Phys. Lett. B 392 Ž1997. 155. E. Brezin, S. Hikami, A. Zee, Phys. Rev. E 51 Ž1995. 5442; A. Jackson, J. Verbaarschot, Phys. Rev. D 53 Ž1996. 7223; T. Wettig, A. Schafer, H. Weidenmuller, Phys. Lett. B 367 ¨ Ž1996. 28; M.A. Nowak, G. Papp, I. Zahed, Phys. Lett. B 389 Ž1996. 137. K. Efetov, Adv. Phys. 32 Ž1983. 53. R. Pisarski, F. Wilczek, Phys. Rev. D 29 Ž1984. 338. The Z2 factor is from the persistent UAŽ1. breaking effects. M. Rho, M.A. Nowak, I. Zahed, Chiral Nuclear Dynamics, WS 1996; and references therein. T. Hatsuda, T. Kunihiro, Phys. Rev. Lett. 55 Ž1985. 158. I. Zahed, Act. Phys. Pol. B 25 Ž1994. 99. A.D. Jackson, M.K. Sener, J.J.M. Verbaarschot, Nucl. Phys. B 479 Ž1996. 707; Nucl. Phys. B 506 Ž1997. 612; T. Guhr, T. Wettig, Nucl. Phys. B 506 Ž1997. 589. E. Brezin, S. Hikami, cond-matr9804023. ´