Annual modulation of event rate and electron recoil energy in inelastic scattering direct detection experiments

In 2020 the XENON1T experiment observed an excess of events with an electron recoil energy $E_R$ in the range of $2\,$--$\,3\,$keV. Such an excess can arise from a variety of sources such as solar axions or a neutrino magnetic moment, but also from inelastic scattering of dark matter off the xenon atoms. The recoil energy of the electron then depends on the mass difference of the dark particles. In this paper we show that the annual modulation of both the event rate and the electron recoil energy provide important additional information that allows to distinguish among different theoretical explanations of the signal. To this end, we first extend the formalism of annual modulation to electronic recoils, inelastic dark matter scattering and the electron recoil energy. We then study a concrete theoretical model with two Dirac fermions and a dark photon. We take into account all relevant cosmological and experimental constraints on this model and apply it to the XENON1T and and XENONnT experiments with realistic detection thresholds, efficiencies and energy resolutions, fitting the main physical parameters of the model, i.e. the mass splitting and the electron scattering cross section. The discriminatory power of the additional information from the annual modulation of both the signal rate and the electron recoil energy is then demonstrated for XENONnT with a simplified model based on these main physical parameters. This more sensitive procedure compared to time-only modulation analyses can also serve as a template for other theoretical models with different dark matter candidates, mediators and cosmology. For the $U(1)$ model with two Dirac fermions fitting the XENON1T excess and the experimental conditions of XENONnT, taking into account the annual variation of the signal rate and recoil energy allows for a faster and more precise determination of the free model parameters.