Collinear Subtractions in Hadroproduction of Heavy Quarks

TL;DR

This work develops a detailed, convolution-based derivation of collinear subtraction terms needed to define a massive general-mass variable-flavour-number scheme for one-particle inclusive heavy-quark hadroproduction. By factorizing partonic cross sections into universal partonic PDFs and FFs, and carefully matching the massive calculation to the massless MSbar theory in the $m\to0$ limit, the authors show how the finite mass-effects can be absorbed into perturbative fragmentation functions, preserving the correct $m$-dependent structure. They provide explicit NLO subtraction terms for all relevant partonic channels, verify consistency with previous results (up to minor scheme-related differences), and highlight the universality and process independence of the partonic FFs. The framework clarifies the gauge- and scheme-dependent aspects of heavy-quark mass factorization and lays out a path for extending GM-VFNS to additional processes and higher orders, with practical implications for precise heavy-flavour phenomenology. All critical equations are expressed with proper mass-factorization and scale dependencies, enabling straightforward implementation in hadroproduction calculations.

Abstract

We present a detailed discussion of the collinear subtraction terms needed to establish a massive variable-flavour-number scheme for the one-particle inclusive production of heavy quarks in hadronic collisions. The subtraction terms are computed by convoluting appropriate partonic cross sections with perturbative parton distribution and fragmentation functions relying on the method of mass factorization. We find (with one minor exception) complete agreement with the subtraction terms obtained in a previous publication by comparing the zero-mass limit of a fixed-order calculation with the genuine massles results in the MSbar scheme. This presentation will be useful for extending the massive variable-flavour-number scheme to other processes.

Figures (17)

• Figure 1: Feynman diagrams for the LO gluon-gluon fusion process $g + g\rightarrow Q + \overline{Q}$.
• Figure 2: The LO quark-antiquark annihilation process $q + \overline{q} \rightarrow Q + \overline{Q}$.
• Figure 3: Sketch of kinematics of mass factorization for (a) upper incoming line (b) lower incoming line and (c) outgoing line.
• Figure 4: Feynman diagrams representing (a) $f_{g\to g}^{(1)}(x_1) \otimes {\operatorname{d}} \hat{\sigma}^{(0)}(g g \to Q \overline{Q})$ and (b) $f_{g\to g}^{(1)}(x_2) \otimes {\operatorname{d}} \hat{\sigma}^{(0)}(g g \to Q \overline{Q})$. The fermion loops on the external gluon lines are heavy-quark loops.
• Figure 5: Feynman diagrams representing ${\operatorname{d}} \hat{\sigma}^{(0)}(g g \to Q \overline{Q}) \otimes d_{Q\to Q}^{(1)}(z)$.