Quark Models, Current Algebras, and Radiative Corrections to Beta Decay

We study radiative corrections to semileptonic decays. Special commutation relations between the weak and electromagnetic currents reduce the question of finiteness of $i\to f+e^{-}+\overline{\nu }$ to that of the finiteness of $\frac{〈f\left|J_{\mu }^{}{}_{}{}^W\right|i〉}{\surd Z_2^{}{}_{}{}^e}$. In a quark model with interactions properly chosen, standard renormalization theory is then used to prove that these quantities, hence the order-$\alpha$ radiative corrections to any $\beta$ decay, are finite. We conclude that commutators are not enough and that some small amount of dynamics is needed to remove divergent radiative corrections.