Radiative corrections to the ${\eta }^{(\mathrm{\prime })}$ Dalitz decays

We provide the complete set of radiative corrections to the Dalitz decays ${\eta }^{(\mathrm{\prime })}\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}\gamma$ beyond the soft-photon approximation, i.e. over the whole range of the Dalitz plot and with no restrictions on the energy of a radiative photon. The corrections inevitably depend on the ${\eta }^{(\mathrm{\prime })}\to {\gamma }^{\ast }{\gamma }^{(\ast )}$ transition form factors. For the singly virtual transition form factor appearing e.g. in the bremsstrahlung correction, recent dispersive calculations are used. For the one-photon-irreducible contribution at the one-loop level (for the doubly virtual form factor), we use a vector-meson-dominance-inspired model while taking into account the $\eta$-${\eta }^{\mathrm{\prime }}$ mixing.