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

TL;DR

The paper shows that leading-power collinear factorization is stable under photon propagator numerator modifications that preserve the vacuum pole structure, provided no new analytic singularities or scales are introduced. The LP soft kernel depends on background data only through the longitudinal contraction $n^\mu \bar n^\nu \Delta_{\mu\nu}(k)$, enabling a practical LP projection criterion: if this contraction vanishes (or is power-suppressed for soft momenta), the LP soft function equals its vacuum form. In an occupancy-number background model with the physical polarization sum $g_{T\mu\nu}(k;n)$, transversality enforces the LP criterion, pushing genuine background sensitivity to beyond LP where NLP effects, including transverse contractions and subleading SCET interactions, can contribute. This clarifies how background electromagnetic environments influence high-energy factorization, predicting background effects at subleading orders while preserving LP structure in the leading regime.

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 $Δ_{μν}(k)$. 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 $n^μ\bar n^νΔ_{μν}(k)$ 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 $g_{Tμν}(k;n)$, transversality implies $n^μ\bar n^νΔ_{μν}=0$, so genuine background sensitivity starts only beyond LP.