The Infrared Limit of the QCD Dirac Spectrum and Applications oc chiral Random Matrix Theory to QCD

TL;DR

This paper explores the infrared behavior of the QCD Dirac spectrum, linking it to chiral Random Matrix Theory, and examines its applications to QCD at nonzero temperature and chemical potential, highlighting symmetry patterns and phase transitions.

Contribution

It demonstrates that the low-energy QCD Dirac spectrum is governed by chiral Random Matrix Theory and extends this framework to finite temperature and chemical potential scenarios.

Findings

Dirac spectrum fluctuations match chRMT predictions below the Thouless energy.

ChRMT models capture the phase diagram of QCD at nonzero temperature and chemical potential.

No localization transition occurs with light quarks in the studied regimes.

Abstract

In the first part of these lectures we discuss the infrared limit of the spectrum of the QCD Dirac operator. We discuss the global symmetries of the QCD partition function and show that the Dirac spectrum near zero virtuality is determined by the pattern of spontaneous chiral symmetry breaking of a QCD-like partition function with additional bosonic valence quarks and their super-symmetric partners. We show the existence of an energy scale below which the fluctuations of the QCD Dirac spectrum are given by a chiral Random Matrix Theory (chRMT) with the global symmetries of the QCD partition function. Physically, for valence quark masses below this scale the partition function is dominated by the zero momentum modes. In the theory of disordered systems, this energy scale is known as the Thouless energy. In the second part of these lectures we discuss chRMT as a schematic model for the QCD partition function at nonzero temperature and chemical potential. We discuss novel features resulting from the non-Hermiticity of the Dirac operator. The analysis by Stephanov of the failure of the quenched approximation, the phase diagram of the chRMT partition function, as well as the properties of Yang-Lee zeros are discussed. We argue that a localization transition does not occur in the presence of light quarks. Several results will be derived in full detail. We mention the flavor symmetries of the QCD Dirac operator for two colors, the calculation of the valence quark mass dependence of the chiral condensate and the reduction of the chRMT partition function to the finite volume partition function.