Heavy-quark initiated charged-current deep-inelastic scattering coefficient functions through $\mathcal{O}(α_s^2)$

TL;DR

The paper addresses the NNLO ($oldsymbol{\mathcal{O}( olinebreak )}$) calculation of heavy-quark initiated charged-current DIS coefficient functions with full mass dependence. It employs a cut-based approach to isolate diagram contributions, derives differential equations for 111 master integrals, and transforms them to canonical $oldsymbol{psilon}$-form to obtain analytic results in terms of generalized polylogarithms. Calculations are performed in the decoupling scheme to match onto VFNS frameworks like ACOT and FONLL, with explicit Born, NLO, and NNLO contributions and careful treatment of soft singularities. The results, validated against massless limits and independent computations, provide essential inputs for precise PDF fits and intrinsic charm studies, and are complemented by accessible auxiliary files for implementation in phenomenological analyses.

Abstract

We compute the coefficient functions for heavy-quark initiated charged-current deep-inelastic scattering through $\mathcal{O}(α_s^2)$, retaining full mass dependence for a single heavy-quark flavor. The calculation employs a cut-based approach to isolate individual diagram contributions and to derive differential equations for the relevant master integrals, which are solved analytically in terms of generalized polylogarithms. The results are presented in the decoupling scheme for $n_L$ light active flavors, facilitating their direct implementation in variable-flavor number schemes such as ACOT and FONLL.

Figures (3)

• Figure 1: Typical Compton diagrams that appear in the calculations. Here, a solid line indicates a heavy quark propagator, a dashed line indicates a light quark propagator, a wavy line indicates the external boson, and a curly line represents a gluon propagator. Zigzag lines represent the cuts (cf. Tab. \ref{['tab:cutRules']}). The right diagram in the top row, and similar sea diagrams, appear in the coefficient functions proportional to $\omega^{\mu\nu}_{n,\mathrm{ud}}$, while other diagrams are typical for $\omega^{\mu\nu}_{n,\mathrm{cs}}$.
• Figure 2: Integration paths in the $(y_1,y_2)$ plane specific to the systems defined in the text. The $\mathrm{I}$- and $\mathrm{II}$-systems follow the same integration contour, differing only in their initial phase-space points.
• Figure 3: One of the diagrams in which each individual cut (cf. Tab. \ref{['tab:intermidStates']}) produces a large triple logarithm.