Calculating soft radiation at one loop

Abstract

We present an efficient way to calculate the effect of soft QCD radiation at one loop, which is needed for predictions at next-to-next-to-leading logarithmic accuracy. We use rapidity coordinates and isolate the divergences in the integrand. By performing manipulations with cumulative variables, we avoid complications from plus distributions. We address rapidity divergences, divergences with an azimuthal dependence, complicated jet boundaries and multi-differential measurements. The process and measurements can be easily adjusted, as we demonstrate by reproducing many existing soft functions. The results for a general LHC process with multiple Wilson lines are obtained by treating Wilson lines that are not back-to-back using a boost. We also obtain, for the first time, the N -jettiness soft function for generic jet angularities, and the collinear-soft function for the measurement of two angularities.

Generic dijet soft functions at two-loop order: correlated emissions

“…We in fact already presented the SCET-1 NLO master formula in [42], and an earlier derivation along similar lines -although less general -was given in [45]. A similar NLO formula, valid also for non-back-to-back configurations, was presented in [50]. For SCET-2 soft functions with n = 0, it is evident from (3.9) that the y k -integration produces a 1/α pole.…”

The transverse momentum distribution of hadrons within jets

“…We usê S i (b, µ, ν) to denote the global soft function in Fourier transform space. Without any phase space constraints on the soft radiation, we obtain the following expression for the global soft function in momentum space [29,36]…”

The energy distribution of subjets and the jet shape

“…Performing the calculation using ref. [79], and noting that the jet region corresponds to rapidity y > − ln(R/2) with respect to the jet axis, we obtain [77,78]. In the original frame these corresponds to emissions outside the jet described by Hij.…”