Two-loop spacelike splitting amplitude for N=4 Super-Yang-Mills theory

The study of collinear behavior for gauge theories in the spacelike region is of great phenomenological and theoretical importance. We analytically calculate the two-loop spacelike splitting amplitude for the full color <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi mathvariant="script">N</a:mi><a:mo>=</a:mo><a:mn>4</a:mn></a:mrow></a:math> super-Yang-Mills theory. The result is derived by two complementary methods starting from the known amplitude: one is based on a discontinuity analysis, while the other one is based on calculating Feynman integrals analytically in the collinear limit. Our result explicitly shows terms that violate naive factorization. However we show that factorization is restored at the level of color-summed unpolarized squared amplitudes at next-to-next-to-next-to leading order. We conjecture that the two-loop tripole terms in the generalized splitting amplitudes in quantum chromodynamics are identical to what we obtain in <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="script">N</d:mi><d:mo>=</d:mo><d:mn>4</d:mn></d:math> super Yang-Mills theory.