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Electron Mass Singularities in Semileptonic Kaon Decays

We show that recent improvements in the $O(α)$ long-distance quantum electrodynamics (QED) corrections to the radiative inclusive $K_{e3}$ decay rate using the Sirlin representation are free from infrared divergences and collinear electron mass singularities in the limit $m_e\rightarrow0$, as predicted by the Kinoshita-Lee-Nauenberg theorem. We also verify that in massless QED with the simultaneous dimensional regularization of QED photon infrared divergences and electron mass singularities leads to the same result for the inclusive rate in the limit of four space-time dimensions. The equivalence of the two approaches results in part from an interesting interplay between a small chirality-breaking effect in the massless electron limit and the generalization of space-time algebra and phase space integrals to $d>4$ dimensions. Our finding supports the small theoretical uncertainty claimed for $K_{e3}$ radiative inclusive rates and reaffirms its utility in precision unitarity tests of the quark mixing matrix.