In this work, we perform a lattice QCD calculation of the branching ratios and the form factors of radiative leptonic decays $P\to \ell {\nu }_{\ell }\gamma$ ($P=\pi$, $K$) using $N_f=2+1$ domain wall fermion ensembles generated by the RBC and UKQCD Collaborations at the physical pion mass. We adopt the infinite-volume reconstruction (IVR) method, which extends lattice data to infinite volume and effectively controls the finite-volume effects. This study represents a first step toward a complete calculation of radiative corrections to leptonic decays using the IVR method, including both real photon emissions and virtual photon loops. For decays involving a final-state electron, collinear radiative corrections, enhanced by the large logarithmic factors such as $\ln (m_{\pi }^2/m_e^2)$ and $\ln (m_K^2/m_e^2)$, can reach the level of $O(10\%)$ and are essential at the current level of theoretical and experimental precision. After including these corrections, our result for $\pi \to e{\nu }_e\gamma$ agrees with the PIBETA measurement; for $K\to e{\nu }_e\gamma$, our results are consistent with the KLOE data and exhibit a $1.7\sigma$ tension with E36; and for $K\to \mu {\nu }_{\mu }\gamma$, where radiative corrections are negligible, our results confirm the previously observed discrepancies between lattice results and the ISTRA/OKA measurements at large photon energies, and with the E787 results at large muon-photon angles.

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• Cabibbo–Kobayashi–Maskawa matrix
• Lattice QCD
• Leptonic, semileptonic & radiative decays