Automation of Beam and Jet functions at NNLO

Abstract

We present a novel framework to streamline the calculation of jet and beam functions to next-to-next-to-leading order (NNLO) in perturbation theory. By exploiting the infrared behaviour of the collinear splitting functions, we factorise the singularities with suitable phase-space parametrisations and perform the observable-dependent integrations numerically. We have implemented our approach in the publicly available code pySecDec and present first results for sample jet and beam functions.

Automated Calculation of Beam Functions at NNLO

“…We start with transverse-momentum resummation, which is a SCET-2 observable and hence corresponds to the case n = 0. According to the collinear-anomaly relation (14), we need to combine the bare beam-function matching kernels with the corresponding soft function that we obtain from SoftSERVE. Our results for the collinear-anomaly exponent d 2 are shown in Table 1, and the non-logarithmic terms of the Mellin-space matching kernels are displayed in the upper panels of Figure 1.…”

“…Their explicit form up to two loop-order can be found e.g. in [14] for Z B q and in [15] for Z f k←j . From the pole terms of the bare matching kernels we then extract the non-cusp anomalous dimension γ B m , and from the finite terms we obtain the nonlogarithmic coefficients I (m) i←j (N), which we sample for different values of the Mellin parameter N.…”

“…An automated approach for the calculation of beam functions at next-to-next-to-leading order (NNLO) in perturbation theory has been initiated only recently [14,15], and in this article we report on the status of these developments. An automated framework for the calculation of soft functions at NNLO is already available through the public package SoftSERVE [16][17][18].…”

“…The beam functions are, moreover, defined as proton matrix elements of collinear field operators, and they need to be matched onto the standard parton distribution functions to extract the relevant perturbative information. By employing suitable phase-space parametrisations, sector-decomposition techniques and non-linear transformations, we recently set up a similar automated framework for the calculation of the beam-function matching kernels [14]. As a first application of our approach, we computed the quark beam function for jet-veto resummation [15], and in this work we present our results for transverse-momentum resummation and the event-shape variable N-jettiness.…”

The NNLO quark beam function for jet-veto resummation

“…Our calculation is based on a novel framework that aims at automating the computation of NNLO jet and beam functions for a broad class of observables [53,54]. We indeed checked our numerical predictions against the analytic results for transverse-momentum resummation from [30], before implementing the phase-space constraints that are imposed by the jet veto.…”