The fully-differential gluon beam function at NNLO
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Springer
Received: May 12, 2020 Accepted: June 30, 2020 Published: July 30, 2020
Jonathan R. Gaunta and Maximilian Stahlhofenb a
CERN Theory Division, 1211 Geneva 23, Switzerland b Albert-Ludwigs-Universit¨ at Freiburg, Physikalisches Institut, D-79104 Freiburg, Germany
E-mail: [email protected], [email protected] Abstract: The fully-differential beam function (dBF) is a universal ingredient in resummed predictions of hadron collider observables that probe the full kinematics of the incoming parton from each colliding proton — the virtuality and transverse momentum as well as the light-cone momentum fraction x. In this paper we compute the matching coefficients between the unpolarized gluon dBF and the usual parton distribution functions (PDFs) at the two-loop order. For observables probing both the virtuality and transverse momentum of incoming gluons, our results provide the part of the NNLO singular cross section related to collinear initial-state radiation, and are required for the resummation of large logarithms through N3 LL. Further to this, the dBF is closely linked to the beam function appearing in a generalized version of threshold factorization, via a simple integration. By performing this integration for the two-loop gluon matching coefficients, we also obtain the corresponding quantities for the generalized threshold beam function. Keywords: Jets, NLO Computations ArXiv ePrint: 2004.11915
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)234
JHEP07(2020)234
The fully-differential gluon beam function at NNLO
Contents 1
2 Definitions and properties of the dBF
3
3 Results for the dBF
7
4 Results for the generalized threshold beam function
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5 Conclusions
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A Tree-level and one-loop matching coefficients
10
B Perturbative ingredients B.1 Splitting functions B.2 Convolutions of one-loop functions
11 11 12
(2)
C Expressions for the ∆I˜gj
1
12
Introduction
Beam functions encode the effect of collinear initial state radiation (ISR) on the measurement of some observable T at a hadron collider [1, 2]. If T is a perturbative scale (T ΛQCD ), these functions can be expressed in terms of the convolution of the standard PDFs fj and perturbatively-calculable matching coefficients Iij . The indices i, j = {qi , q¯i , g} denote the different types of partons in QCD. In this paper we focus on the beam functions Bi (t, x, ~k⊥ ) [3],1 which can be thought of as a parton density completely differential in all of the kinematic variables of the ‘active’ parton that enters the hard process. It is a main ingredient in factorized cross sections for observables sensitive to ~k⊥ and t in the limit Q2 t ∼ ~k⊥2 Λ2QCD , where Q represents the scale of the hard interaction. The lightcone coordinates2 of the active parton are given in terms of the arguments of this beam function by (xP − , −t/[xP − ], ~k⊥ ), with P − the large component of the proton momentum. The Bi (t, x, ~k⊥ ) are often referred to as fully-differenti
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