Microscopic spectrum of the QCD Dirac operator with finite quark masses

Abstract

We compute the microscopic spectrum of the QCD Dirac operator in the presence of dynamical fermions in the framework of random-matrix theory for the chiral Gaussian unitary ensemble. We obtain results for the microscopic spectral correlators, the microscopic spectral density, and the distribution of the smallest eigenvalue for an arbitrary number of flavors, arbitrary quark masses, and arbitrary topological charge.

Power law deformation of Wishart–Laguerre ensembles of random matrices

“…In WL the gap probability E(s) and the first eigenvalue distribution p(s) are explicitly known and universal in the microscopic large-N limit for all ν at β = 2, for odd values of ν and 0 at β = 1, and for ν = 0 at β = 4. This has been shown by various authors independently [16,38,39,37]. In some cases only finite-N results are know in terms of a hypergeometric function of a matrix valued argument [40,41], from which limits are difficult to extract.…”

Smallest Dirac eigenvalue distribution from random matrix theory

“…The alternative technique of determining E(s) as a τ -function of a system of integrable partial differential equations [13] does not simplify the situation either. Consequently, the explicit calculation for the massive case has been done so far solely for the chiral Gaussian unitary ensemble [7]. The result contains an explicit factor of exp(−ζ 2 /4) which at first sight is due to the Gaussian potential.…”

“…If not, large-N random matrix theory, which in principle is foreign to the pertinent field theory language, would seem to be a new ingredient required to describe the observed spectral correlators. It has recently been shown that a description entirely in terms of finite-volume partition functions is indeed also possible [10].In order to confirm by numerical simulations that the low-lying spectra of QCD Dirac operators can be described alternatively by large-N random matrix theories, it is in practice most convenient to measure the distribution of the smallest eigenvalue [2] and compare that to the random matrix prediction [11,7]. Since the smallest eigenvalue distribution largely consists of the first peak of the microscopic spectral density (see Fig.…”

“…In order to confirm by numerical simulations that the low-lying spectra of QCD Dirac operators can be described alternatively by large-N random matrix theories, it is in practice most convenient to measure the distribution of the smallest eigenvalue [2] and compare that to the random matrix prediction [11,7]. Since the smallest eigenvalue distribution largely consists of the first peak of the microscopic spectral density (see Fig.…”

“…The suitably rescaled (microscopic) spectral correlation functions thus seem to provide exact finite-size scaling functions for QCD in a finite volume. Very recently, the microscopic spectral correlators have been calculated from random matrix theories that include the effect of fermion determinants with masses m ≃ O(1/V 4 ) [5][6][7] (see also [8]). When λ and m are measured in units of the mean level spacing at zero virtuality, all the random matrix predictions turn out to be universal, i.e., insensitive to the details of the random matrix potential [4][5][6].…”