X-ray Edge Exponent in Fermi Liquids

We investigate how interactions modify the Fermi-edge singularity—traditionally studied under the assumption of non-interacting electrons. Using linear-response theory, perturbation theory, and standard spectral-function techniques, we analyze corrections to this singularity when conduction electrons are treated as interacting within Fermi-liquid theory. We compare the interacting case with the well-established non-interacting scenario from prior literature. To gain deeper insight into the Fermi-liquid description, we incorporate renormalization-group methods, treating Fermi-liquid theory as the low-energy effective framework for interacting fermions. Our analysis clarifies the role of electron interactions in shaping the Fermi-edge singularity and provides corrections to the scaling dimension of the impurity operator in terms of Landau Fermi-liquid parameters.