Constraint on the Double-Pion-Production Amplitude in Pion-Nucleon Collisions Implied by a Partially Conserved Axial-Vector Current

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Abstract
A constraint between the "nonpole" part of the pion-production amplitude and the pion-nucleon elastic scattering amplitude has been derived using the hypothesis of partially conserved axial-vector current (PCAC). An attempt has been made to evaluate this nonpole part of the pion-production amplitude, at the unphysical values of the invariants, under the assumption that the important processes are the one-pion-exchange process and the processes involving at least one nucleon pole. This allows us to evaluate the off-mass-shell pion-pion amplitude. We have shown that in the isobaric-spin-1 and -2 states our evaluated off-shell pion-pion amplitudes are not inconsistent with those evaluated on the basis of the algebra of currents and PCAC. In the isobaric-spin-0 state our results, with our approximations, are at variance with those of the algebra of currents and PCAC. We have shown that if the nonpole pion-production amplitude were assumed to be dominated by the one-pion-exchange process alone our results would agree remarkably well with those obtained using algebra of currents and PCAC.