The factorization of amplitudes into hard, soft and collinear parts is known
to be violated in situations where incoming particles are collinear to outgoing
ones. This result was first derived by studying limits where non-collinear
particles become collinear. We show that through an effective field theory
framework with Glauber operators, these factorization-violating effects can be
reproduced from an amplitude that is factorized before the splitting occurs. We
confirm results at one-loop, through single Glauber exchange, and at two-loops,
through double Glauber exchange. To approach the calculation, we begin by
reviewing the importance of Glauber scaling for factorization. We show that for
any situation where initial state and final state particles are not collinear,
the Glauber contribution is entirely contained in the soft contribution. The
contributions coming from Glauber operators are necessarily non-analytic
functions of external momentum, with the non-analyticity arising from the
rapidity regulator. The non-analyticity is critical so that Glauber operators
can both preserve factorization when it holds and produce
factorization-violating effects when they are present.Comment: 55 Pages, 5 figure
