Off-shell form factor: Factorization is violated

We study the Sudakov form factor on the Coulomb branch of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">N</a:mi> <a:mo>=</a:mo> <a:mn>4</a:mn> </a:math> sYM which endows only external states with masses, and implies that the former is off-shell in the traditional sense. Our consideration is performed at three-loop order in the near mass-shell limit. We use a combination of tools to perform required calculations centered around the method of regions as the main go-to formalism for the asymptotic expansion of emerging parametric Feynman integrals. Explicit separation of quantum loops in terms of hard, collinear, and ultrasoft modes allows us to explore the factorization properties of this infrared-sensitive quantity. While the hard region is cleanly separated from the rest, the ultrasoft-collinear modes remain intertwined. We exhibit effects of factorization violation explicitly in the momentum space making use of the infrared power counting.