Lattice calculation of the Ds meson radiative form factors over the full kinematical range

We compute the structure-dependent axial and vector form factors for the radiative leptonic decays ${D}_{s}\ensuremath{\rightarrow}\ensuremath{\ell}{\ensuremath{\nu}}_{\ensuremath{\ell}}\ensuremath{\gamma}$, where $\ensuremath{\ell}$ is a charged lepton, as functions of the energy of the photon in the rest frame of the ${D}_{s}$ meson. We work in the electroquenched approximation, using gauge-field configurations with $2+1+1$ sea-quark flavors generated by the European Twisted Mass Collaboration and the results have been extrapolated to the continuum limit. For the vector form factor we observe a very significant partial cancellation between the contributions from the emission of the photon from the strange quark and that from the charm quark. The results for the form factors are used to test the reliability of various Anz\"atze based on single-pole dominance and its extensions, and we present a simple parametrization of the form factors which fits our data very well and which can be used in future phenomenological analyses. Using the form factors we compute the differential decay rate and the branching ratio for the process ${D}_{s}\ensuremath{\rightarrow}e{\ensuremath{\nu}}_{e}\ensuremath{\gamma}$ as a function of the lower cutoff on the photon energy. With a cutoff of 10 MeV for example, we find a branching ratio of $\mathrm{Br}({E}_{\ensuremath{\gamma}}>10\text{ }\text{ }\mathrm{MeV})=4.4(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ which, unlike some model calculations, is consistent with the upper bound from the BESIII experiment $\mathrm{Br}({E}_{\ensuremath{\gamma}}>10\text{ }\text{ }\mathrm{MeV})<1.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ at 90% confidence level. Even for photon energies as low as 10 MeV, the decay ${D}_{s}\ensuremath{\rightarrow}e{\ensuremath{\nu}}_{e}\ensuremath{\gamma}$ is dominated by the structure-dependent contribution to the amplitude (unlike the decays with $\ensuremath{\ell}=\ensuremath{\mu}$ or $\ensuremath{\tau}$), confirming its value in searches for hypothetical new physics as well as in determining the Cabibbo-Kobayashi-Maskawa parameters at $O({\ensuremath{\alpha}}_{\mathrm{em}})$, where ${\ensuremath{\alpha}}_{\mathrm{em}}$ is the fine-structure constant.