Muon-electron backward scattering: a prime example for endpoint singularities in SCET

TL;DR

This paper studies muon-electron backward scattering as a model to understand endpoint divergences and resummation of large logarithms in Soft-Collinear Effective Theory, revealing complex patterns and the role of collinear anomalies.

Contribution

It demonstrates how endpoint refactorization and consistency relations reproduce known Bessel function results within SCET, highlighting the complexity of resummation in this process.

Findings

Endpoint divergences lead to iterative convolution integrals in SCET.

Rapidity logarithms generate an infinite tower of collinear-anomaly exponents.

Refactorization conditions reproduce the Bessel function in the effective theory.

Abstract

We argue that energetic muon-electron scattering in the backward direction can be viewed as a template case to study the resummation of large logarithms related to endpoint divergences appearing in the effective-theory formulation of hard-exclusive processes. While it is known since the mid sixties that the leading double logarithms from QED corrections resum to a modified Bessel function on the amplitude level, the modern formulation in Soft-Collinear Effective Theory (SCET) shows a surprisingly complicated and iterative pattern of endpoint-divergent convolution integrals. In contrast to the bottom-quark induced $h \to \gamma\gamma$ decay, for which a renormalized factorization theorem has been proposed recently, we find that rapidity logarithms generate an infinite tower of collinear-anomaly exponents. This can be understood as a generic consequence of the underlying $2\to 2$ kinematics. Using endpoint refactorization conditions for the collinear matrix elements, we show how the Bessel function is reproduced in the effective theory from consistency relations between quantities in a "bare" factorization theorem.