Entropy, confinement, and chiral symmetry breaking

Abstract

This paper studies the way in which confinement leads to chiral symmetry breaking (CSB) through a gap equation. We argue that entropic effects cut off infrared singularities in the standard confining effective propagator $1/p^4$, which should be replaced by $1/(p^2+m^2)^2$ for a finite mass $m\sim K_F/M(0)$ [$M(0)$ is the zero-momentum value of the running quark mass]. Extension of an old calculation of the author yields a specific estimate for $m$. This cutoff propagator shows semi-quantitatively two critical properties of confinement: 1) a negative contribution to the confining potential coming from entropic forces; 2) an infrared cutoff required by gauge invariance and CSB itself. Entropic effects lead to a proliferation of pion branches and a $\bar{q}q$ condensate, and contribute a negative term $\sim -K_F/M(0)$ to the effective pion Hamiltonian allowing for a massless pion in the presence of positive kinetic energy and string energy. The resulting gap equation leads to a well-behaved running constituent quark mass $M(p^2)$ with $M^2(0)\approx K_F/\pi$. We include one-gluon terms to get the correct renormalization-group ultraviolet behavior, with the improvement that the prefactor (related to $<\bar{q}q>$) can be calculated from the confining solution. We discuss an integrability condition that guarantees the absence of IR singularities at $m=0$ in Minkowski space through use of a principal-part propagator.