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

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.

Plain TeX file;minor changes; version to be published in Pys. Rev. D; corrected refs.12,13

References in corpus (2)

• Axial, induced pseudoscalar, and pion-nucleon form factors in manifestly Lorentz-invariant chiral perturbation theory
• An improved study of the kappa resonance and the non-exotic $s$ wave $\pi K$ scatterings up to $\sqrt{s}=2.1$GeV of LASS data

Cited by in corpus (2)

• The pion-pion scattering amplitude. III: Improving the analysis with forward dispersion relations and Roy equations
• Low energy scattering with a nontrivial pion