The renormalization-group running of neutrino mixing is undetectable

Abstract: Quantum corrections make the parameters of a theory depend on an arbitrary renormalization scale, $μ$. The renormalization-group (RG) running of neutrino masses and mixing is a standard tool to relate models defined at a high mass scale to measured values. A growing body of work holds that the running is also directly observable: a neutrino produced or detected at momentum transfer $Q$ would carry the mixing matrix evaluated at $μ= Q$. Proposed signatures include flavor change at zero distance from the source, shifts in the measured CP-violation phase, and an anomalous flavor composition of high-energy astrophysical neutrinos. We show that this premise is incorrect. The mixing is a property of the neutrino itself, fixed by its self-energy on the mass shell, not by the process that produces or detects it. Setting $μ= Q$ is legitimate, but then large logarithms in the self-energy undo the running; the premise incorrectly drops them. Hence, apart from small vertex and box corrections, every experiment measures the same mixing: the running is undetectable. We illustrate this in the Minimal Supersymmetric Standard Model, two seesaw models, and the Standard Model effective field theory. Independently, short-baseline data already limit the change of any one mixing angle between pion decay and detection at $Q$ of a few GeV to below $0.7^\circ$--$2.4^\circ$.