Threshold factorization of the Drell-Yan process at next-to-leading power
Martin Beneke, Alessandro Broggio, Sebastian Jaskiewicz, Leonardo Vernazza
Abstract
A bstract We present a factorization theorem valid near the kinematic threshold $$ z={Q}^2/\hat{s}\to 1 $$ <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> of the partonic Drell-Yan process $$ q\overline{q}\to {\gamma}^{\ast }+X $$ <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: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> for general subleading powers in the (1 − z ) expansion. We then consider the specific case of next-to-leading power. We discuss the emergence of collinear functions, which are a key ingredient to factorization starting at next-to-leading power. We calculate the relevant collinear functions at $$ \mathcal{O}\left({\alpha}_s\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:mfenced> </mml:math> by employing an operator matching equation and we compare our results to the expansion-by- regions computation up to the next-to-next-to-leading order, finding agreement. Factorization holds only before the dimensional regulator is removed, due to a divergent convolution when the collinear and soft functions are first expanded around d = 4 before the convolution is performed. This demonstrates an issue for threshold resummation beyond the leading-logarithmic accuracy at next-to-leading power.