Threshold factorization of the Drell-Yan quark-gluon channel and two-loop soft function at next-to-leading power
Alessandro Broggio, Sebastian Jaskiewicz, Leonardo Vernazza
Abstract
A bstract We present a factorization theorem of the partonic Drell-Yan off-diagonal processes $$ g\overline{q}(qg)\to {\gamma}^{\ast }+X $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mfenced> <mml:mi>qg</mml:mi> </mml:mfenced> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>γ</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>X</mml:mi> </mml:math> in the kinematic threshold regime $$ z={Q}^2/\hat{s}\to 1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:mover> <mml:mi>s</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> </mml:math> at general subleading powers in the (1 − z ) expansion. Focusing on the first order of the expansion (next-to-leading power accuracy with respect to the leading power $$ q\overline{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> channel), we validate the bare factorization formula up to $$ \mathcal{O}\left({\alpha}_s^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mfenced> </mml:math> . This is achieved by carrying out an explicit calculation of the generalized soft function in d -dimensions using the reduction to master integrals and the differential equations method. The collinear function is a universal object which we compute from an operator matching equation at one-loop level. Next, we integrate the soft and collinear functions over the convolution variables and remove the remaining initial state collinear singularities through PDF renormalization. The resulting expression agrees with the known cross section in the literature.