  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.
