The Lambert W Function, Laguerre Polynomials, and the Zeros of the QCD Partition Function

TL;DR

This paper explores the mathematical structure of phase transitions in a QCD model using the Lambert W function and Laguerre polynomials, providing explicit solutions and closed-form expressions for complex chemical potentials.

Contribution

It introduces explicit solutions for the transcendental equation using the Lambert W function and derives a closed-form partition function in terms of Laguerre polynomials for complex chemical potential.

Findings

Explicit Lambert W function solutions for phase transition points

Closed-form partition function expression with Laguerre polynomials

Insights into the zeros of the QCD partition function

Abstract

We study solutions of a transcendental equation for the complex chemical potential at which a random-matrix QCD model can undergo a phase transition at zero mass. An explicit solution is obtained in terms of the Lambert W function. We also provide a closed form expression for a QCD random matrix model partition function, as a sum of Laguerre polynomials, for complex chemical potential and non-zero mass.