Determine the hard-collinear B2(z) correction for the considered parton showers

Determine the function B2(z) that enters the effective coupling in the Sudakov exponent for the PanGlobal family of colour-dipole parton showers, specifying its z-dependent form for these showers to complement the known analytic-resummation results and thereby fix the shower-specific hard-collinear O(αs^2) correction beyond its integral constraint.

Background

In the paper’s formulation, the effective coupling used in the Sudakov exponent includes hard-collinear corrections via B2(z), which affect the hard-collinear region and tend to zero in the soft limit. While analytic resummation typically represents this contribution as a constant multiplying δ(1−z) and specific schemes have computed the corresponding constants, the shower-specific z-dependent form is not established for the PanGlobal shower variants examined here.

The authors achieve NNLL accuracy for event shapes by constraining only the integral B2^int through relations derived from NLO 1→2 calculations and drift effects, and they adopt a simple ansatz for B2(z) for practical implementation. However, they note that determining the meaningful z-dependence requires incorporating triple-collinear splitting functions in the shower, which are not yet included. Establishing B2(z) for these showers would solidify the hard-collinear component of the NNLL-accurate shower framework.