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

TL;DR

This paper demonstrates that leading-power factorization in SCET remains stable under certain photon propagator modifications in strong electromagnetic backgrounds, with background effects only appearing beyond leading power.

Contribution

It shows that the set of leading regions and the LP factorization structure are preserved under photon propagator numerator modifications in SCET.

Findings

Leading regions are unchanged under propagator numerator modifications.

LP soft kernel reduces to vacuum if background enters only through certain contractions.

Background sensitivity appears only beyond leading power for specific polarization modifications.

Abstract

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.