Radiative corrections to the Dalitz plot of $K_{l3}^{\pm }$ decays

We calculate the model-independent radiative corrections to the Dalitz plot of $K_{l3}^{\pm }$ decays to the order of $(\alpha /\pi )(q/M_1)$, where $q$ is the momentum transfer and $M_1$ is the mass of the kaon. The final results are presented, first, with the triple integration over the variables of the bremsstrahlung photon ready to be performed numerically and, second, in an analytical form. These two forms are useful to cross-check on one another and with other calculations. This paper is organized to make it accessible and reliable in the analysis of the Dalitz plot of precision experiments and is not compromised to fixing the form factors at predetermined values. It is assumed that the real photons are kinematically discriminated. Otherwise, our results have a general model-independent applicability.

Figure 1

Kinematical region for $K_{l3}^{\pm }$ decays as a function of the energies of the charged lepton and the emitted pion, $E$ and $E_2$, respectively, for (a) $K_{e3}^{\pm }$ and (b) $K_{\mu 3}^{\pm }$ decays. In this work, areas TBR and FBR are loosely referred to as the three- and four-body regions of the Dalitz plot, respectively.

Figure 2

Feynman diagrams, to first order in $\alpha$, which yield virtual radiative corrections in $K_{l3}^{+}$ decay. The wavy, broken, and continuous lines represent virtual photons, pseudoscalar mesons, and fermions, respectively. The blobs represent the effects of the strong interactions, and, at the weak vertex, they also represent the effects of details of the weak interactions.

Figure 3

Feynman diagrams which yield bremsstrahlung radiative corrections in $K_{l3}^{+}$ decay. The wavy lines represent real photons, the broken lines represent pseudoscalar mesons, and the continuous lines represent fermions.

Figure 4

Plot of $k_2$ and $k_4$ as functions of ${\theta }_k$ and ${\phi }_k$ for $K_{e3}^{+}$ decay, at $E=111.1\mathrm{MeV}$ and $E_2=222.2\mathrm{MeV}$. The uppermost surface represents $k_2$ so $k_4$ lies right below it.

Figure 5

Plot of $k_2$ and $k_4$ as functions of ${\phi }_k$ for ${\theta }_k=\pi /3$, at the same values of $(E,E_2)$ as in Fig. 4. The continuous and broken lines represent $k_2$ and $k_4$, respectively. The shaded area displays the accessible values of $k$ which lead to allowed values of $\cos {\theta }_2$.