Sum Rules forP-Wave Pion-Nucleon Scattering Lengths from Current Algebras

Using current commutation relations and the hypothesis of a partially conserved axial-vector current, approximate sum rules are derived relating pion-nucleon $P$-wave scattering lengths to the $S$-wave scattering lengths, the axial-vector renormalization constant $g_A$, the pion-nucleon coupling constant, the nucleon isovector magnetic moment, and the contribution of the low-energy $N^{*}$ resonances to the matrix element (between nucleon states) of the product of two axial-vector currents. The sum rules are compared with the available experimental data.