One-loop radiative correction to the Toshev relation for neutrino oscillations in matter

The one-loop electroweak radiative corrections to coherent forward neutrino scattering in a medium slightly violates the tree-level universality of neutral-current contributions of three neutrino flavors to the matter potential that is relevant to neutrino oscillations in matter. We examine this small but nontrivial quantum effect by deriving the differential equations of those effective neutrino oscillation quantities with respect to the electron number density of matter. The tree-level Toshev relation $\sin 2\tilde{\theta}_{23} \sin\tilde{\delta} = \sin 2\theta_{23} \sin\delta$, which links the fundamental flavor-mixing and CP-violating parameters