Evaluation of the axial-vector commutator sum rule for pion-pion scattering

We consider the sum rule proposed by one of us (S. L. A.), 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 6%, and we give detailed estimates of the experimental and extrapolation errors in the closure discrepancies.