Effective-Range Approach to the Low-Energyp-Wave Pion-Nucleon Interaction

The theory of $p$-wave pion-nucleon scattering is reexamined using the formalism recently proposed by one of the authors (F.E.L.). On the basis of the cut-off Yukawa theory without nuclear recoil it is found, for not too high values of the coupling constant, that: (a) For each $p$-wave phase shift a certain function of the cotangent should be approximately linear at low energies and should extrapolate to the Born approximation at zero total energy. The value of the renormalized unrationalized coupling constant determined in this way from experiment is $f^2=0.08\mathrm{}$. A special feature of the predicted energy dependence of the phase shifts is that ${\delta }_{33}$ is positive and the other $p$ phase shifts are negative. (b) The so-called "crossing theorem" requires a relation between the four $p$ phase shifts, so that in addition to the coupling constant only two further constants are needed to completely specify the low-energy behavior. (c) The direction of the energy variation in the (3,3) state is such that a resonance will occur for a sufficiently large cut-off ${\omega }_{max}$. Rough estimates indicate that ${\omega }_{max}\approx 6$ will produce a resonance at the energy required by experiment. It is argued that the results (a) and (b) are very probably also consequences of a relativistic theory but that (c) may not be.