Random matrix theory and spectral sum rules for the Dirac operator in QCD1
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Random matrix theory and spectral sum rules for the Dirac operator in QCD1
arXiv:hep-th/9212088v1 14 Dec 1992SUNY-NTG-92/45Random matrix theory and spectral sum rules for the Dirac operator in QCD1E.V. Shuryak and J.J.M.
VerbaarschotDepartment of PhysicsSUNY, Stony Brook, New York 11794AbstractWe construct a random matrix model that, in the large N limit, reduces to the lowenergy limit of the QCD partition function put forward by Leutwyler and Smilga. Thisequivalence holds for an arbitrary number of flavors and any value of the QCD vacuumangle.In this model, moments of the inverse squares of the eigenvalues of the Diracoperator obey sum rules, which we conjecture to be universal.
In other words, the validityof the sum rules depends only on the symmetries of the theory but not on its details. Toillustrate this point we show that the sum rules hold for an interacting liquid of instantons.The physical interpretation is that the way the thermodynamic limit of the spectral densitynear zero is approached is universal.
However, its value, i.e. the chiral condensate, is not.SUNY-NTG-92/45December 19921Dedicated to Hans Weidenm¨uller’s 60th birthday.
1. IntroductionSince our main understanding of nonperturbative phenomena in QCD comes fromnumerical simulations analytical results in this direction are most welcome [1, 2].Inparticular the issue of quark confinement has not been resolved.
What is better known[3] is the mechanism of chiral symmetry breaking. In the chiral limit the QCD partitionfunction is invariant under SU(Nf)×SU(Nf)×U(1), whereas the axial U(1) symmetry isbroken explicitly by quantum fluctuations.
For more than one flavor (Nf > 1) the chiralsymmetry is spontaneously broken down to SU(Nf) × U(1) by the formation of quarkcondensates. According to Goldstone’s theorem this leads to N2f −1 massless excitations.At low energy, these are the relevant degrees of freedom of the QCD partition function, andit is possible to write down an effective Lagrangian for an arbitrary value of the vacuumangle θ in terms of these degrees of freedom only [4, 5, 2].
The partition function canbe simplified further by suppressing the space-time dependence of the Goldstone modesaltogether.Recently, Leutwyler and Smilga [2] evaluated the QCD partition function in this limit.This enabled them to write down sum rules for the spectrum of the Dirac operator. Whatenters in these sum rules is what we call the microscopic spectral density as opposed to thecontinuum spectral density that enters in the calculation of the condensate [6].
The latterone is related to the thermodynamic limit of the spectral density, whereas the former oneprovides information on the way the thermodynamic limit is approached. The questionwe would like to address is in how far the sum rules are specific for the QCD partitionfunction, and in what sense they are universal.
Since the partition function of [2] onlyinvolves constant fields one would suspect that the detailed structure of the QCD vacuumis not important, and that the sum rules only depend on the symmetries of the theory.This adagio is taken to the extreme by constructing a model with the correct chiralstructure, but that apart from this does not contain any other information. As is wellknown from random matrix theory, the minimization of information leads to gaussianrandom matrix ensembles [7].
The appropriate ensemble will be constructed in section 2,and with the help of mathematical techniques developed in the framework of Andersonlocalization and compound nucleus scattering [8, 9] we are able to derive the partitionfunction that was obtained by Leutwyler and Smilga (see section 3). Sum rules for one2
and two flavors are derived in section 4.The structure of the random matrix modelwas inspired by the semiclassical approximation to the QCD partition function whereinstantons are the main degrees of freedom [10, 11, 12]. Therefore, it is natural to checkthe validity of these sum rules for an instanton liquid model [13] of the QCD vacuum, atask which is carried out in section 5.
Concluding remarks are made in section 6. Finally,in appendix A sum rules for a finite number of degrees of freedom are derived.2.
Random matrix modelThe Euclidean QCD partition function for Nf flavors and nonzero vacuum angle θ canbe written asZQCD =Xνeiθν
Here and below, the mass matrix is taken to be diago-nal. According the the Atiyah-Singer theorem, the Dirac equation in a gauge field withtopological quantum number ν has exactly ν zero eigenvalues.
However, configurationswith zero total topological charge may actually be composed of spatially well-separatedcomponents with opposite topological charge. Such configurations will give rise to al-most zero modes and thus will play an important role in the chiral dynamics of the QCDpartition function.
Our starting point is that the chiral properties of the QCD partitionfunction are determined by zero modes and almost zero modes only. Although it is notnecessary for the sake of the argument, it is instructive to think of the field configurationsas a superposition of N+ instantons and N−anti-instantons.
Each isolated instanton oranti-instanton has exactly one fermionic zero mode with a definite chirality [14]. In totalwe have ν = N+ −N−exact zero modes.
At finite separation of instantons and anti-instantons the remaining modes are no longer exact zero modes of the Dirac equation andgive rise to nonzero overlap matrix elements T of the Dirac operator [15].A model describing the zero mode part of the QCD partition function with N zero3
modes or almost zero modes in the Euclidean volume V is defined byZ(θ) =XNµ(N)XN+ NN+!eiθ(N+−N−)ZDTP(T)NfYfdet mfiTiT †mf!. (2.2)Zero modes of each chirality are treated as independently distributed identical particles,hence the binomial factor.
The distribution function µ(N) of the total number of zeromodes N = N++N−is peaked at some average value N with a width that is much smallerthan N. As long as a large number of different values of N contribute to the partitionfunction with roughly equal probability2, our results do not depend on the detailed shapeof µ(N). The average over all gauge field configurations in eq.
(2.1) is replaced by anaverage over gaussian distributed overlap matrix elements with distribution function givenbyP(T) = exp(−N2λ2TT †). (2.3)The integration measure DT is the Haar measure.
The structure of the overlap matrix,with off-diagonal blocks T and T † and diagonal blocks equal to the quark masses mf timesthe identity, is dictated by the chirality of the zero modes. The matrix T is a N+ × N−matrix, and for zero quark masses, the total overlap matrix has |N+ −N−| exact zeroeigenvalues.
The density N/V of the total number of zero modes is kept fixed. As inthe instanton liquid approximation to the QCD partition function [16], it is consideredto be an external parameter.
The thermodynamic limit can thus be taken by lettingN →∞, where here and below we will omit the bar. A similar model with N+ = N−wasconsidered in [17, 18].The order parameter in the study of chiral symmetry breaking is the quark condensate< ¯qfqf > defined by< ¯qfqf >= limmf →0 limN→∞−1Nddmflog Z(θ),(2.4)where the order of the limits should be taken as indicated in the formula.
In general,< ¯qfqf > depends on θ. Its value at θ = 0 in the limit where for all flavors mf →0 isapproached from above is denoted by −Σ.
By writing the determinant as the product2If µ(N) ∼δ(N −N) the parity of N+ −N−is the same as the parity of N = N+ + N−, and thepartition function has the additional symmetry Z(θ = 0) = Z(θ = π).4
Q′(λ2n + m2f) one obtains the Banks-Casher formula [6] for ΣΣ = π < ρC(0) >Z(θ=0),(2.5)where the average < · · · >Z(θ) is with respect to the partition function (2.2) (or (2.1) inthe case of QCD). The continuum spectral density is defined byρC(λ) =limall mf ↓0 limN→∞1N ρ(λ),(2.6)and the spectral density ρ(λ) isρ(λ) =Xδ(λ −λn),(2.7)where the eigenvalues ±λn are the nonzero eigenvalues of the overlap matrix in the chirallimit.We will also consider a different limit of the derivative with respect of m of the partitionfunction, namely,limN→∞Nmf fixed1Npdpdmpflog Z(θ).
(2.8)Let us consider the case p = 1 in more detail. Again writing the determinant as a productover eigenvalues, one findslimN→∞Nmf fixed*Xn>0Nmfλ2nN2 + m2fN2+Z(θ)=limN→∞Nmf fixedZ ∞0dx 1N ρ( xN )Z(θ)Nmfx2 + m2fN2.
(2.9)What enters in this expression is what we call the microscopic spectral density defined byρS(x) =limN→∞Nmf fixed1N ρ( xN ),(2.10)as opposed to the continuum spectral density defined by in eq. (2.6).
In this limit, thespectral density function near zero is enlarged proportional to the size (given by N) ofthe system. Note that ρS(x) depends on Nmf.5
3. Calculation of the partition functionIn order to evaluate the partition function (2.2) the determinant is written as anintegral over Grassmann variablesYfdet mfiTiT †mf!=Z YfDψfDφf expXf ψf∗φf∗!
mfiTiT †mf! ψfφf!,(3.1)where the measure of the Grassmann integration is as usualDψf =Yidψfi dψf∗i ,(3.2)and the conjugation ∗is the conjugation of the second kind (i.e., ψ∗∗= −ψ) [9].
Theintegral over T is gaussian and can be performed easily. In the partition function thisresults in the factorexp 2λ2N ψf∗i ψgi φg∗j φfj ,(3.3)which represents a 4-fermion interaction.The quartic term can be written as a sum of two squaresψf∗i ψgi φg∗j φfj = 14(ψf∗i ψgi + φf∗i φgi )(ψg∗j ψfj + φg∗j φfj ) −14(ψf∗i ψgi −φf∗i φgi )(ψg∗j ψfj −φg∗j φfj ).
(3.4)Each of the two squares can be linearized with the help of a Hubbard-Stratonovich trans-formation [9]. This allows us to perform the Grassmann integrations at the expense ofthe introduction of the new real valued integration variables σfg and ¯σfg, respectively.Apart from an irrelevant overall constant, the partition function reduces toZ =XNµ(N)XN+ NN+!eiθ(N+−N−)ZDσD¯σdetN+(σ + i¯σ + m)detN−(σ −i¯σ + m)×exp −N2λ2Tr(σ + i¯σ)(σ −i¯σ).
(3.5)As always, the measure of the integral over the matrices σ and ¯σ is the Haar measure.The diagonal mass matrix is denoted by m.The complex matrix σ + i¯σ can be decomposed in ’polar coordinates’ as [19]σ + i¯σ = UΛV −1,(3.6)6
where U and V are unitary matrices and Λ is a diagonal real positive definite matrix. Sincethe r.h.s has Nf more degrees of freedom than the l.h.s., one has to impose constraintson the new integration variables.
This can be achieved [19] by restricting U to the cosetU(Nf)/U(1)Nf, where U(1)Nf is the diagonal subgroup of U(Nf). In terms of the newvariables the partition function readsZ=XNµ(N)XN+ NN+!
ZJ(Λ)DΛDUDV×detN+(UΛV −1 + meiθ/Nf )detN−(V ΛU−1 + me−iθ/Nf ) exp(−N2λ2TrΛ2), (3.7)where the integral over U is over U(Nf)/U(1)Nf and the integral over V is over U(Nf).A phase factor exp(−iθNf) has been absorbed in the integration over V .For Nf flavors we have Nf condensates which break down the symmetry of the actionto U(Nf)/U(1)Nf leaving us with N2f −1 Goldstone modes for N+ ̸= N−. When the totaltopological charge is zero the phase of the determinant also cancels which provides uswith an additional zero mode.The term proportional to m plays the role of a small symmetry breaking term.
Theintegrals over the nonzero modes will be performed by a saddle point integration atm = 0, whereas the integrals over the soft modes will be accounted for exactly at a fixedvalue of mN. There are two types of nonzero modes, the phase exp iα of the determinantdet V U−1 for N+ ̸= N−, and the eigenvalues Λ.
The partition function at m = 0 factorizesaccordingly,Z(m = 0) =XNµ(N)Zdα(exp(iα) + exp(−iα))NZJ(Λ)dΛdetNΛ exp(−N2λ2TrΛ2). (3.8)The leading order contribution in 1/N of the integral over Λ can be obtained by a saddlepoint approximation.
The saddle point equations for the Λ integrals readΛi = ±λ. (3.9)The negative solution is not inside the integration manifold and can be omitted.The integral over α, which ranges from 0 until 2π , can be executed either before orafter the summation over N+.
In the first case we find zero for odd values of N while for7
even values of N only the term N+ = N−= N/2 contributes in which case the α integralbecomes soft. However, in the second case the interference between contributions of alldifferent topologies results in the integrand (2 cos α)N which allows us to perform theintegral by a saddle point approximation in order to obtain the leading order contributionin 1/N.
The saddle points are located at α = 0 and α = π. It is at this point that thedistribution function µ(N) plays a role: the contribution of the latter saddle point, (−1)N,can be ignored after the summation over N, whereas the contribution at α = 0 yieldsa overall factor µ(N) that does not contribute to the m dependent part of the partitionfunction to be discussed below.At the saddle points in Λ and α, the U−dependence can be absorbed into V .
The theU−integration yields a finite irrelevant constant. Since the phase exp iα has already beenextracted the remaining integral over V is over SU(Nf) instead of U(Nf).
We treat mas a small parameter and expand the determinants up to first order in m. At the saddlepoint α = 0 the result for m dependent part of the partition function isZ(m, θ)Z(m = 0, θ) =Zdet V =1 DV exp( N2λTr(mV −1 exp(−iθ/Nf) + mV exp(iθ/Nf)),(3.10)which coincides with the result derived by Smilga and Leutwyler [2] for the QCD partitionfunction using chiral perturbation theory.The value of Σ at θ = 0 can be obtained from eq. (2.4)Σ(θ = 0) = limm→0 limN→∞12λ1Nf< Tr(V + V −1) >Z(θ=0) .
(3.11)For Nf = 1 the integral over V is absent, and the sign of the quark condensate is inde-pendent of the sign of m. For more than one flavor the order of the limits allows us toperform the V integral by a saddle point approximation. In the case of equal positivemasses the saddle point for θ = 0 is at V = 1 which allows us to identify the parameterλ asλ =1Σ(θ = 0),(3.12)which completes the calculation of the partition function.As observed in [2], for more than one flavor the value of the condensate depends onthe sign of the quark mass.
For example, in the case of two flavors with equal negativemasses and θ = 0, the saddle point is at V = −1.8
It is also possible to introduce a complex mass in eq. (2.2) with the mass in the lowerblock equal to the complex conjugate of the mass in the upper block of the overlap matrix.In this case the final result for the partition function depends only on the combinationmeiθ and its complex conjugate which makes it possible [2] to derive relations betweenthe m and the θ derivatives of the partition function.4.
Sum rules for one and two flavorsSum rules for moments of the inverse squares of the eigenvalues of the Dirac operatorcan be derived for an arbitrary number of flavors and for any value of the total topologicalcharge [2]. In this section we only present a derivation for the simplest two cases of oneflavor and of two flavors with equal masses.
Proofs of the general results can be found in[2].For Nf = 1 there is no integration and the result for the ratio of the massive andmassless partition function is particularly simpleZNf =1 = exp(NmΣ cos θ),(4.1)which was first obtained in [2].In the case of two flavors the integral over SU(2) can be performed for an arbitrarymass matrix. Here, we restrict ourselves to the simpler case of equal quark masses.
Inthis case the ratio of the massive and the massless partition function reduces toZNf =2 =Z 2π0dφπ sin2 φ2 exp(2NmΣ cos φ2 cos θ2),(4.2)where the factor sin2 φ2 results from the invariant measure of SU(2).The integral iselementary and results inZNf=2 = I1(2mNΣ cos θ2)mNΣ cos θ2. (4.3)The partition function can also be evaluated by writing the fermion determinant as aproduct over eigenvalues.
For Nf = 1, we obtain sum ruleexp(ΣmN cos θ) =< mν Yn′(1 + m2λ2n) >Z(m=0),(4.4)9
where the average is with respect to the partition function Z with m = 0, which alsoincludes the weight factorQ ′λ2n involving the product of the nonzero eigenvalues of theDirac operator. The exclusion of zero eigenvalues in the product is denoted by a prime.For two flavors a similar sum rule can be derivedYx1n≥0(1 + 4m2N2 cos2 θ2x21n) =< m2ν Yn′(1 + m2λ2n)2 >Z(m=0) .
(4.5)Here, the x1n is the n’th zero of the Bessel function J1. By expanding both the r.h.s.and the l.h.s.
of eqs. (4.4) and (4.5) in powers of m we obtain sum rules for the inversemoments of the eigenvalues λn.
Additional sum rules for Nf = 2 can be derived fromthe expansion in powers of both mu and md of the general expression for the partitionfunction.The sum rules (4.4) and (4.5) are for a fixed value of θ. However, we will investigatethe validity of the Leutwyler-Smilga sum rules in an interacting instanton vacuum where,for technical reasons, the total topological is taken to be zero.
The corresponding partitionfunction Z0 is given by the Fourier transform of Z(θ),Z0 =Z 2π0dθ2πZ(θ). (4.6)which can be derived from the the decompositionZ =XνeiνθZν.
(4.7)For one and two flavors, the integrals over θ are elementary. The resultsZ0=I0(NmΣ)forNf = 1,(4.8)Z0=I20(NmΣ) −I21(NmΣ)forNf = 2,(4.9)were first derived by Leutwyler and Smilga [2].
Sum rules are obtained by equating (4.8)and (4.9) to the r.h.s. of (4.4) and (4.5) restricted to zero topological charge, respectively.For Nf = 1 one findsI0(NmΣ) =
(4.11)10
The expansion of both sides of this equation in powers of mN yields sum rules for momentsof the inverse eigenvalues of the Dirac operator. The first two sum rules are
(4.13)As was shown by Leutwyler and Smilga, both sum rules hold for arbitrary Nf and alsobe worked out for arbitrary topological charge. The only modification is to replace Nfby Nf + |ν|.
In the case of two or more flavors additional sum rules can be derived byvarying the quarks masses independently [2]. We only quote the result<1N4λ4n>ν=0=Σ416Nf(N2f −1).
(4.14)Again, the result for arbitrary topological charge is obtained by replacing Nf by Nf + |ν|.Formally, this sum rule diverges for Nf = 1. That this is indeed the case can be understoodfrom the behavior of the spectral density at small λ.
For N+ = N−we expect ρ(λ) ∼λNf+2, where the factor λNf originates from the fermion determinant and the factor λ2from the Jacobian of the unitary transformation that diagonalizes the overlap matrix.The above sum rules diverge for Nf = 0. Consequently, we expect that in this case, andtherefore in the quenched approximation, that the thermodynamic limit of the spectraldensity near zero is approached in a completely different way.The sum rules (4.12-14) can be expressed in the microscopic spectral density by writingthe sum over the eigenvalues as the integralX1Npλpn =Z dxxp ρS(x)|mN=0.
(4.15)The second sum rule expresses an integral over the correlation function < ρS(x)ρS(x′) >ν=0in terms of integrals over the average microscopic spectral density < ρS(x) >ν=0.Finally, let us write down a sum rule for nonzero values of mN. In the case of oneflavor one obtains from the infinite product expansion for the Bessel function I0,XnmNΣ2x20n + m2N2Σ2 =Z ∞0dx < ρS(x)|mN fixed >ν=0mNx2 + m2N2,(4.16)11
where x0n is the n’th zero of the Bessel function J0. One is tempted to invert this equalityin order to obtain an analytical answer for the microscopic spectral density.
Since ρSdepends on mN there is no unique solution, but a trivial solution can be written downreadily,ρS(x) =Xnδ(x −x0nΣ ). (4.17)Because there is no reason to believe that the dispersion (4.13) is zero, we do not expectthat this is the exact microscopic level density.In appendix A it is shown that for one flavor sum rules can also be derived at finiteN.
These results provide an explicit proof that the thermodynamic limit has been takencorrectly.5. Sum rules in the instanton liquidThe partition function (2.1) can be approximated semiclassically by a liquid of in-stantons.
Instead of averaging over all gauge field configurations, we average over thecollective coordinates of the instantons only, whereas 1-loop quantum fluctuations aboutthe instantons are included in the measure. The action in (2.1) is the instanton action,which also includes the interaction between instantons.
We use the so called streamline[20, 21] interaction supplemented by a core in order to stabilize the instanton liquid. Thefermion determinant is calculated in the space spanned by the fermionic zero modes withoverlap matrix elements that can be derived from the streamline configuration [22].
Moredetails on the above instanton liquid model can be found in [13].The numerical simulations were carried out for a liquid of 64 instantons in a Euclideanspace time volume of (2.378)3 × 4.756 in units of Λ−4.Averages were obtained from125 statistically independent configurations. Our main results are presented in Table 1.Calculations were done for one, two and three massless flavors (see heading).
The firstrow shows the smallest eigenvalue and its average level motion (between brackets). Thecondensate Σ (row 2) is obtained from an extrapolation of the spectral density to λ = 0.The results for the sum rule (4.12) (row 3) are compared to the analytical result (row 4).We find complete agreement inside the error bars.
The remaining rows involve numerical12
and theoretical values for the quantities S1 and S2 defined byS1 =< (P 1λ2n)2 >ν=0 −< P 1λ4n >ν=0< P 1λ2n >2ν=0,S2 =< (P 1λ2n)2 >ν=0< P 1λ2n >2ν=0. (5.1)Also in this case we find excellent agreement with the theoretical valuesSth1 =NfNf + 1,Sth2 =N2fN2f −1,(5.2)where the derivation for S2 does not hold for Nf = 1.
We consider this a strong argumentin favor of the universality of the sum rules.6. Discussion and conclusionsThe main conclusion of this work is that the low energy chiral limit of the QCDpartition function as derived by Leutwyler and Smilga coincides with a large N limit of arandom matrix theory.
Therefore all conclusions of [2] also pertain to the chiral randommatrix theory proposed in section 2. The random matrix theory resembles the partitionfunction of a gas of instantons.
The main difference is that instead of inheriting its disorderfrom the distribution of the collective coordinates, the overlap matrix elements are takento be independently distributed according to a gaussian random matrix ensemble.From random matrix theory it is know that [23] microscopic level correlations such asfor example the nearest neighbor level spacing distribution or the variance of the numberof levels as a function of the average number of levels in a given interval are universal.They only depend on the symmetries of the Hamiltonian but not on the details of thematrix elements. The spectral sum rules put forward by Smilga and Leutwyler involve themicroscopic level density.
Since they can be obtained from a chiral random matrix theorywe expect that the sum rules are also universal, and that they do not depend on the detailsof the low-energy structure of the theory but only on its symmetries. In particular, theydepend on the number of flavors and, for example, diverge in the quenched approximation.This conclusion has been confirmed by simulations of an instanton liquid with overlapmatrix elements that differ strongly from those of a gaussian random matrix ensemble.Inside the statistical uncertainty of the calculations all sum rules were reproduced.
The13
physical interpretation is that the way the thermodynamic limit of the spectral densitynear zero is approached is universal. However, its value, i.e.
the chiral condensate, dependson the details of the theory.The advantage of a random matrix model is that it is amenable to powerful mathemat-ical techniques [24, 9] that makes it possible to obtain explicit analytical results for e.g.the microscopic spectral density, which, by virtue of the universality arguments discussedabove, also hold for the QCD partition function. Work in this direction is under way.AcknowledgementsThe reported work was partially supported by the US DOE grant DE-FG-88ER40388.We acknowledge the NERSC at Lawrence Livermore where most of the computationspresented in this paper were performed.
We would like to thank A. Smilga, M.A. Nowakand I. Zahed for useful discussions.Appendix AIn this appendix we derive the sum rules (4.12) and (4.13) for one flavor withoutrelying on a saddle point approximation.
For N+ = N−= N/2 and Nf = 1 the partitionfunction simplifies toZ =Zdσd¯σ(σ + i¯σ −m)N2 (σ −i¯σ −m)N2 exp(−N2λ2(σ + i¯σ)(σ −i¯σ)),(A.1)where an irrelevant overall constant has been suppressed. At finite N the pre-exponentialfactors can be expanded as a binomial series which provides us with an expansion inpowers of m. The coefficients are elementary integrals, and for the m−dependent part ofthe partition function we findZ(m)Z(0) = 1 + m2N2Σ24+ m4N4Σ464(1 −2N ) + · · · .
(A.2)It should be noted that no approximations have been made. On the other hand, thefermion determinant can be written as a product over the eigenvalues which leads to the14
expansionZ(m)Z(0) = 1 + m2*Xn1λ2n+ν=0+ m412* Xn̸=n′1λ2nλ2n′+ν=0+ · · · ,(A.3)where the average is with respect to the massless partition function. By equating thecoefficients of the powers of m2 we obtain sum rules for the inverse powers of the eigen-values that are valid for any value of N. The sum rules (4.12) and (4.13) for Nf = 1are reproduced by keeping only the leading order terms in 1/N.
We observe that thefirst sum rule is valid for any value of N. The second sum rule is modified by the factor(1−2/N). For N = 2 we find zero which is correct because in this case there are no termsthat contribute to the sum n′ ̸= n in (A.3).15
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Nf = 1Nf = 2Nf = 3λ10.0255(98)0.0531(164)0.1078(335)Σ1.90(5)1.33(5)0.90(10)P 1/λ2n0.89(3)0.21(1)0.066(2)Σ2/4Nf0.90(3)0.22(1)0.068(7)S10.53(3)0.70(5)0.83(7)Sth11/22/33/4S22.0(4)1.20(12)1.24(12)Sth2∞4/39/8Table 1. Numerical results for the spectral properties of the Dirac operator in the gaugefield of a liquid of instantons.
From the comparison of the 3rd and 4th, the 5th and 6thand the 7th and 8th rows we conclude that the Leutwyler-Smilga sum rules are observedby this model. For the definition and further discussion of the observable, we refer tosection 5.18
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