Two-Loop Effective Theory Analysis of $\pi$ $(K)\to e{\overline{\nu }}_e[\gamma ]$ Branching Ratios

We study the ratios $R_{e/\mu }^{(P)}\equiv \Gamma (P\to e{\overline{\nu }}_e[\gamma ])/\Gamma (P\to \mu {\overline{\nu }}_{\mu }[\gamma ])$ ($P=\pi$, $K$) in Chiral Perturbation Theory to order $e^2{p}^4$. We complement the two-loop effective theory results with a matching calculation of the counterterm, finding $R_{e/\mu }^{(\pi )}=(1.2352\pm 0.0001)\times {10}^{-4}$ and $R_{e/\mu }^{(K)}=(2.477\pm 0.001)\times {10}^{-5}$.

Figure 1

One- and two-loop 1PI topologies contributing to $R_{e/\mu }$ to order $e^2{p}^4$. Dashed lines represent pseudoscalar mesons, solid lines fermions and wavy lines photons. Shaded squares indicate vertices from the $O(p^4)$ effective Lagrangian.