Complete collection of one-loop triple-collinear splitting operators for dimensionally-regulated QCD
M. Czakon, Sebastian Sapeta
Abstract
A bstract We provide results for the one-loop triple-collinear color/spin-space splitting operators for the five possible processes, q → $$ {qq}^{\prime }{\overline{q}}^{\prime } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>qq</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:msup> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>′</mml:mo> </mml:msup> </mml:math> , q → $$ qq\overline{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>qq</mml:mi> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , q → qgg , g → $$ gq\overline{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>gq</mml:mi> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> and g → ggg . The expressions are exact in dimensionally-regulated massless QCD up to a single integral, which we expand to second order in the dimensional-regularisation parameter. We also evaluate the related splitting functions. Our results are both sufficient and indispensable for the construction of subtraction and integrated-subtraction terms for triple-collinear singularities of one-loop double-real-emission cross-section contributions as part of a next-to-next-to-next-to leading order subtraction scheme.