Goldstone Bosons in a Finite Volume: The Partition Function to Three Loops

TL;DR

This paper calculates the partition function of Goldstone bosons in a finite volume at finite temperature up to three loops, demonstrating renormalizability and consistency across different measures, with implications for various physical models.

Contribution

It provides a three-loop calculation of the partition function for Goldstone bosons in finite volume, confirming renormalizability and measure independence in the non-linear sigma model.

Findings

Partition function computed to 3 loops in finite volume.

Renormalizability confirmed with singularities absorbed by existing couplings.

Results consistent across different invariant measures.

Abstract

A system of Goldstone bosons - stemming from a symmetry breaking $O(N) \to O(N-1)$ - in a finite volume at finite temperature is considered. In the framework of dimensional regularization, the partition function is calculated to 3 loops for 3 and 4 dimensions, where Polyakov's measure for the functional integration is applied. Although the underlying theory is the non-linear $\sigma $ model, the 3 loop result turns out to be renormalizable in the sense that all the singularities can be absorbed by the couplings occuring so far. In finite volume, this property is highly non trivial and confirms the method for the measure. We also show that the result coincides with the one obtained using the Faddeev- Popov measure. This is also true for the maximal generalization of Polyakov's measure: none of the additional invariant terms that can be added contributes to the dimensionally regularized system. Our phenomenological Lagrangian describes e.g. 2 flavor chiral QCD as well as the classical Heisenberg model, but there are also points of contact with the Higgs model, superconductors etc. Moreover the finite size corrections to the susceptibility might improve the inerpretation of Monte Carlo results on the lattice.