Current Algebras, Regge Poles, and the Isovector Anomalous Magnetic Moment of the Nucleon

We discuss the sum rule for the isovector anomalous magnetic moment of the nucleon ${F_2}^V\mathrm{}\left(0\right)=1.85\mathrm{}$, which is obtained from the current commutation relation $\delta \left(x_0\right)\left[{A_0}^{+}\mathrm{}\right(x), {A_{\nu }}^{-}\mathrm{}(0\left)\right]=2\mathrm{}{V_{\nu }}^3\mathrm{}\left(x\right)\mathrm{}{\delta }^4\mathrm{}\left(x\right)\mathrm{}$ by use of the covariant method proposed by Fubini. We find that (1) the sum rule cannot be evaluated without explicit knowledge of one of the axial-vector-nucleon "scattering" amplitudes; (2) calculating the contributions from the $P_{33}\mathrm{}\left(1236\right)\mathrm{}$ and $D_{13}\mathrm{}\left(1525\right)\mathrm{}$ using a dispersion-pole model of the weak amplitude gives only ${F_2}^V\mathrm{}\left(0\right)=0.37\mathrm{}$, and (3) estimating the high-energy continuum contribution to the sum rule from Reggepole fits to $\pi p$ charge-exchange scattering increases the result to ${F_2}^V\mathrm{}\left(0\right)\mathrm{}\cong 1.0$. It seems that the sum rule is dominated by low- and high-energy continuum contributions, which must be more accurately known before the validity of the sum rule can be judged.