The QCD Sign Problem for Small Chemical Potential

TL;DR

This paper investigates the behavior of the complex phase factor of the fermion determinant in QCD at small chemical potential, revealing a critical point at half the pion mass and the limitations of analytic continuation methods.

Contribution

It provides the first detailed calculation of the average phase factor in the microscopic domain of QCD at nonzero chemical potential, including its singularity structure and volume dependence.

Findings

Average phase factor is non-zero below half the pion mass.

The phase factor vanishes exponentially with volume above the critical chemical potential.

Analytic continuation from imaginary chemical potential is invalid at zero chemical potential.

Abstract

The expectation value of the complex phase factor of the fermion determinant is computed in the microscopic domain of QCD at nonzero chemical potential. We find that the average phase factor is non-vanishing below a critical value of the chemical potential equal to the half the pion mass and vanishes exponentially in the volume for larger values of the chemical potential. This holds for QCD with dynamical quarks as well as for quenched and phase quenched QCD. The average phase factor has an essential singularity for zero chemical potential and cannot be obtained by analytic continuation from imaginary chemical potential or by means of a Taylor expansion. The leading order correction in the $p$-expansion of the chiral Lagrangian is calculated as well.