Low-Energy QCD in the Delta Regime

TL;DR

This paper analyzes low-energy QCD in a finite volume across various temperatures, solving the effective quantum mechanics for different chiral symmetry groups to explore the mass gap, condensate, and symmetry restoration.

Contribution

It provides analytical and numerical solutions for the quantum mechanics of low-energy QCD in the delta regime, interpolating between different regimes and validating results against known limits.

Findings

Computed the mass gap and chiral condensate in finite volume.

Demonstrated the transition from rotor to harmonic approximation.

Validated condensate results against epsilon regime predictions.

Abstract

We investigate properties of low-energy QCD in a finite spatial volume, but with arbitrary temperature. In the limit of small temperature and small cube size compared to the pion Compton wavelength, Leutwyler has shown that the effective theory describing low-energy QCD reduces to that of quantum mechanics on the coset manifold, which is the so-called delta regime of chiral perturbation theory. We solve this quantum mechanics analytically for the case of a $U(1)_L \times U(1)_R$ subgroup of chiral symmetry, and numerically for the case of $SU(2)_L \times SU(2)_R$. We utilize the quantum mechanical spectrum to compute the mass gap and chiral condensate, and investigate symmetry restoration in a finite spatial volume as a function of temperature. Because we obtain the spectrum for non-zero values of the quark mass, we are able to interpolate between the rigid rotor limit, which emerges at vanishing quark mass, and the harmonic approximation, which is referred to as the p-regime. We find that the applicability of perturbation theory about the rotor limit largely requires lighter-than-physical quarks. As a stringent check of our results, we raise the temperature to that of the inverse cube size. When this condition is met, the quantum mechanics reduces to a matrix model. The condensate we obtain in this limit agrees with that determined analytically in the epsilon regime.