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 the Dirac operator. The mechanism, originating from complex Burgerʼs-like equation for viscid, pressureless, one-dimensional flows of eigenvalues, is similar to the recently observed weak-strong coupling phase transition in large Nc Yang–Mills theory. The spectral viscosity is proportional to the inverse 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 Nc⩾3 QCD. We reinterpret our results in terms of known properties of chiral random matrix models and lattice data.