Renormalization of U(1) Gauge Boson Kinetic Mixing

TL;DR

The paper tackles the renormalization of kinetic mixing between two U(1) gauge bosons, using a QED extension with a dark photon and Stückelberg mass. It analyzes two renormalization schemes—one with kinetic mixing and one in a basis where the mixing is rotated away—within a general $R_\xi$ gauge at one loop and demonstrates that gauge-dependent off-diagonal contributions cancel, yielding equivalent descriptions for physical observables. Explicit one-loop renormalization constants, pole masses, resummed Green's functions, and LSZ-corrected amplitudes are computed in both bases, and a detailed matching shows consistent RG flows across schemes. The results establish the renormalizability of kinetically mixed U(1) systems at one loop and provide a robust framework for comparing dark-photon phenomenology across different renormalization bases, with plans to extend to higher orders.

Abstract

Quantum field theories containing fields with the same quantum numbers allow for mixed kinetic terms in the Lagrangian, leading to off-diagonal elements in the tree-level two-point function. After removing the mixing by a field rotation, the off-diagonal UV divergences cannot be subtracted by a counterterm, still one can show that the theory is renormalizable. We study kinetic mixing of $U(1)$ gauge bosons in an extension of QED with a massive "dark" photon at one-loop order. In general covariant $R_ξ$-gauge, the gauge-fixing function naively obstructs the removal of tree-level mixing but we show that these off-diagonal gauge-dependent contributions cancel. We compare two renormalization schemes: one with and one without kinetic mixing, and relate them via a scale-dependent field transformation, showing that the schemes are equivalent.