Existence and stability of multiple solutions to the gap equation

TL;DR

This paper demonstrates that the gap equation can have multiple solutions in certain regimes but only the Nambu solution remains stable for nonzero quark masses, simplifying hadron physics modeling.

Contribution

It applies the homotopy continuation method to find all solutions of the gap equation and introduces a stability criterion to identify the physically relevant solution.

Findings

Multiple solutions exist for the gap equation in certain parameter regimes.

Only the Nambu solution is stable for nonzero current-quark masses.

The existence of multiple solutions does not complicate hadron physics descriptions.

Abstract

We argue by way of examples that, as a nonlinear integral equation, the gap equation can and does possess many physically distinct solutions for the dressed-quark propagator. The examples are drawn from a class that is successful in describing a broad range of hadron physics observables. We apply the homotopy continuation method to each of our four exemplars and thereby find all solutions that exist within the interesting domains of light current-quark masses and interaction strengths; and simultaneously provide an explanation of the nature and number of the solutions, many of which may be associated with dynamical chiral symmetry breaking. Introducing a stability criterion based on the scalar and pseudoscalar susceptibilities we demonstrate, however, that for any nonzero current-quark mass only the regular Nambu solution of the gap equation is stable against perturbations. This guarantees that the existence of multiple solutions to the gap equation cannot complicate the description of phenomena in hadron physics.