Relation Between the Pion-Nucleon and theρ-Meson Coupling Constants from Pion Scattering Lengths

The scattering length of a pion on an isospin-bearing target can be computed either by applying the U(2){}U(2) current algebra or by assuming ρ dominance in the t channel. Comparing the two approaches, we establish $({{{f}_{{ρ}}}²}{4{π}})=({{{G}_{{π}NN}}²}{4{π}}){({{g}V}{{g}A})}²({{{m}_{{ρ}}}²}{2{{m}N}²}),$ a relation previously obtained on the basis of current algebra applied to ${ρ}{→}2{π}$.