Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD

Abstract

We link the spontaneous breakdown of chiral symmetry in Euclidean QCD to the collision of spectral shock waves in the vicinity of zero eigenvalue of Dirac operator. The mechanism, originating from complex Burger's-like equation for viscid, pressureless, one-dimensional flow of eigenvalues, is similar to recently observed weak-strong coupling phase transition in large N c Yang-Mills theory. The spectral viscosity is proportional to the inverse of the size of the random matrix that replaces the Dirac operator in the universal (ergodic) regime. We obtain the exact scaling function and critical exponents of the chiral phase transition for the averaged characteristic polynomial for N c ≥ 3 QCD. We reinterpret our results in terms of known properties of chiral random matrix models and lattice data.

Unveiling the significance of eigenvectors in diffusing non-Hermitian matrices by identifying the underlying Burgers dynamics

“…In this way, not only Airy but also the Pearcey functions are captured in the same formalism. A similar Burgers equation was also obtained for the Wishart ensemble and chiral GUE yielding a universal scaling associated with the Bessoid function [11]. The equivalent phenomenon appears also at the level of CUE diffusion, providing new insight for order-disorder transition of Wilson loops in Yang-Mills theory [12].…”

Universal Spectral Shocks in Random Matrix Theory --- Lessons for QCD

“…of the Banks Casher formula, in the limit when the volume of the Euclidean space-time tends to infinity, a dramatic accumulation of small eigenvalues has to take place in the vicinity of zero. We argue, that this sudden increase in the density is achieved by the formation of a spectral shock wave at zero eigenvalue [13]. In order to demonstrate this phenomenon, we have to tune the random matrix model in such a way that it includes chiral properties, i.e.…”

“…where the entries of K, an M ×N (M ≥ N ) matrix, evolve in time t according to a Brownian walk [13,14]. The fundamental object of our studies is, again, the characteristic polynomial…”

Diffusion method in random matrix theory

“…It serves as a fast and powerful framework useful when dealing with powers and ratios of characteristic polynomials averaged over external source gaussian measures. It began as a byproduct of QCD considerations made several years ago [4] and was thereafter successfully applied to hermitian, Wishart and chiral models [5,6,7]. The method uses a Dyson-like picture of dynamical matrices to achieve this goal.…”

“…It serves as a fast and powerful framework useful when dealing with powers and ratios of characteristic polynomials averaged over external source gaussian measures. It began as a byproduct of QCD considerations made several years ago [4] and was thereafter successfully applied to hermitian, Wishart and chiral models [5,6,7]. The method uses a Dyson-like picture of dynamical matrices to achieve this goal.Studying characteristic polynomials in the RMT community is now a prolific topic with many branches, its root however can be traced back to a remarkable relation connecting them to orthogonal polynomials det(z − X) X = π N (z).Such objects were considered in [8] and in many areas of application such as zeroes of Riemann ζ function [9], eigenvalue statistics in quantum chaotic systems [10] and matrix models of QCD [11].…”