Cabibbo Theory, Pole Dominance, and Sum Rules for Radiative Decays of Pseudoscalar Mesons

By extending the Cabibbo theory to the radiative leptonic decay ${\pi }^{+}\to {\pi }^0+l^{+}+{\nu }_l+\gamma$, we analyze the vector and axial-vector form factors which describe this decay. Assuming unsubtracted dispersion relations for these form factors at fixed momentum transfer as well as pole dominance, we obtain a number of sum rules relating the masses and couplings of the $\omega$, $\rho$, $A_1$, and $A_2$ mesons. In addition, we obtain, using current algebra and the hypothesis of a partially conserved axial-vector current, a soft-pion theorem relating the vector form factors of the decay ${\pi }^{+}\to {\pi }^0+l^{+}+{\nu }_l+\gamma$ to the axial-vector form factor associated with ${\pi }^{+}\to l^{+}+{\nu }_l+\gamma$. For those sum rules that can be tested, excellent agreement is obtained with the experimental data. A similar analysis is made of the decay $K^{+}\to {\pi }^0+l^{+}+{\nu }_l+\gamma$, and a corresponding set of sum rules is obtained relating the masses and couplings of the $\omega$, $K^{*}$, $K_V$, and $K_A$ mesons.