On the Stability of Leading-Power Factorization under Photon Propagator Numerator Modifications

We study collinear factorization in strong electromagnetic backgrounds within SCET for a class of modifications where the photon propagator keeps the vacuum pole structure and $i\varepsilon$ prescription, while the background enters only through a numerator tensor $\Delta_{\mu\nu}(k)$. We show that the set of Landau pinch surfaces and leading momentum regions is unchanged, so the leading-power (LP) factorized form is preserved. Moreover, the LP cusp kernel depends on the background solely through the longitudinal contraction $n^\mu\bar n^\nu\Delta_{\mu\nu}(k)$ in the soft region; if it vanishes (or is power suppressed), the LP soft kernel reduces to the vacuum. As an application, for an occupancy-number modification with the physical polarization-sum tensor $g_{T\mu\nu}(k;n)$, transversality implies $n^\mu\bar n^\nu\Delta_{\mu\nu}=0$, so genuine background sensitivity starts only beyond LP.