Finite Volume Dependence of the Quark-Antiquark Vacuum Expectation Value

TL;DR

This paper derives a new formula for how vacuum expectation values depend on finite volume, involving pion matrix elements, and evaluates higher-order corrections to the quark condensate, comparing with existing Luscher formula results.

Contribution

It introduces a novel finite volume dependence formula for vacuum expectation values that simplifies calculations by avoiding kinematic integrals and relates to pion sigma terms.

Findings

Derived a general finite volume dependence formula for vacuum expectation values.

Calculated full order p^6 correction to the quark condensate <ar q q>.

Compared the correction with Luscher formula, finding no definitive accuracy conclusion.

Abstract

A general formula is derived for the finite volume dependence of vacuum expectation values analogous to Luscher's formula for the masses. The result involves no integrals over kinematic quantities and depends only on the matrix element between pions at zero momentum transfer thus presenting a new way to calculate the latter, i.e. pion sigma terms. The full order $p^6$ correction to the vacuum condensate $<\bar q q>$ is evaluated and compared with the result from the Luscher formula. Due to the size of the $p^6$ result no conclusion about the accuracy of the Luscher formula can be drawn.