# Evaluation of the Axial Vector Commutator Sum Rule for Pion-Pion Scattering

## TL;DR

This paper evaluates a sum rule relating pion decay constants to pion-pion cross sections using recent scattering data, confirming the sum rule's accuracy within six percent and discussing related extrapolation issues.

## Contribution

It provides a precise evaluation of the axial vector commutator sum rule for pion-pion scattering using current data, and discusses the related pion-nucleon sum rule and off-shell extrapolation.

## Key findings

- Sum rule satisfied within 6% accuracy
- Recent data enables precise sum rule evaluation
- Analysis of off-shell extrapolation effects

## Abstract

We consider the sum rule proposed by one of us (SLA), obtained by taking the expectation value of an axial vector commutator in a state with one pion. The sum rule relates the pion decay constant to integrals of pion-pion cross sections, with one pion off the mass shell. We remark that recent data on pion-pion scattering allow a precise evaluation of the sum rule. We also discuss the related Adler--Weisberger sum rule (obtained by taking the expectation value of the same commutator in a state with one nucleon), especially in connection with the problem of extrapolation of the pion momentum off its mass shell. We find, with current data, that both the pion-pion and pion-nucleon sum rules are satisfied to better than six percent, and we give detailed estimates of the experimental and extrapolation errors in the closure discrepancies.