Off-the-Mass-Shell Correction in Pion-Pion Scattering

A discussion is given of the off-the-mass-shell correction to the PCAC (partially conserved axial-vector current) relation for pion-pion scattering. In particular, special attention is drawn to the fact that the usual $s$ and $u$ cuts of the pion-pion amplitude give rise to cuts in the external pion mass extended far below the cut due to the three-pion intermediate state. An estimate is made of the off-the-mass-shell correction due to these induced cuts. It is shown that this correction is not likely to be significant as long as the $S$-wave pion-pion interaction is relatively weak, implying that the use of the PCAC relation for pion-pion scattering without the off-the-mass-shell correction is not expected to be any worse than that for any other PCAC relation.