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

Abstract

Analyzing the matrix elements related to the decays ${K}^{+}\ensuremath{\rightarrow}{l}^{+}+\ensuremath{\nu}+\ensuremath{\gamma}$ and ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}+{l}^{+}+\ensuremath{\nu}+\ensuremath{\gamma}$ 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 (${F}_{K},{F}_{\ensuremath{\pi}}$), ${K}_{l3}$-decay form factors (${f}_{\ifmmode\pm\else\textpm\fi{}}$), and $\mathrm{VVP}$ and $\mathrm{VP}\ensuremath{\gamma}$ couplings. The sum rule obtained is in excellent agreement with the experimental data.