Matching effective chiral Lagrangians with dimensional and lattice regularization

TL;DR

This paper establishes relations between effective chiral Lagrangians regularized via dimensional and lattice methods, enabling translation of physical results like the mass gap in QCD-like theories between these frameworks.

Contribution

It provides a systematic matching of 4-derivative couplings in effective theories across dimensional and lattice regularizations, facilitating the extraction of low energy constants from lattice QCD data.

Findings

Derived relations between couplings in different regularizations.

Validated the relations with a direct mass gap computation.

Applied results to estimate low energy constants in QCD.

Abstract

We compute the free energy in the presence of a chemical potential coupled to a conserved charge in effective O($n$) scalar field theory (without explicit symmetry breaking terms) to NNL order for asymmetric volumes in general $d$--dimensions, using dimensional (DR) and lattice regularizations. This yields relations between the 4-derivative couplings appearing in the effective actions for the two regularizations, which in turn allows us to translate results, e.g. the mass gap in a finite periodic box in $d=3+1$ dimensions, from one regularization to the other. Consistency is found with a new direct computation of the mass gap using DR. For the case $n=4, d=4$ the model is the low-energy effective theory of QCD with $N_{\rm f}=2$ massless quarks. The results can thus be used to obtain estimates of low energy constants in the effective chiral Lagrangian from measurements of the low energy observables, including the low lying spectrum of $N_{\rm f}=2$ QCD in the $\delta$--regime using lattice simulations, as proposed by Peter Hasenfratz, or from the susceptibility corresponding to the chemical potential used.