Current Algebra, Unsubtracted Dispersion Relations, and Radiative Decays of Strange Mesons

Analyzing the matrix elements related to the decays $K^{+}\to l^{+}+\nu +\gamma$ and $K^{+}\to {\pi }^0+l^{+}+\nu +\gamma$ under the assumptions of (i) equal-time current commutation relations and (ii) unsubtracted dispersion relations, we obtain a Fubini-type sum rule between weak $K$-decay form factors, meson-meson couplings, and photon-meson couplings. Because of lack of knowledge about strange axial-vector meson ($Q$) couplings, we use the hypothesis of partially conserved axial-vector current (PCAC) to eliminate such couplings and thereby obtain a sum rule between pseudoscalar meson-decay constants ($F_K,F_{\pi }$), $K_{l3\mathrm{}}$-decay form factors ($f_{\pm }$), and $\mathrm{VVP}$ and $\mathrm{VP}\gamma$ couplings. The sum rule obtained is in excellent agreement with the experimental data.