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Heavy-quark production in deep-inelastic scattering -- Mellin moments of structure functions

Marco Klann, Sven-Olaf Moch, Kay Schönwald

TL;DR

The paper develops and implements a framework to compute NLO QCD corrections to heavy-quark DIS structure functions while retaining full heavy-quark mass dependence. By exploiting the optical theorem and operator product expansion, the authors derive analytic fixed Mellin moments for $F_2$ and $F_L$ in the range $N=2$ to $22$, using harmonic tensors and projection techniques, and validating against known asymptotic limits and RSvN parametrisations. The work establishes a robust method for mass-dependent coefficient functions and operator-matrix elements, with cross-checks including gauge invariance and mass-factorisation consistency, and provides a clear path toward NNLO accuracy. These results enhance precision predictions for heavy-flavour DIS and underpin future global PDF analyses and collider phenomenology, particularly in regions sensitive to small-$x$ gluons and heavy-quark thresholds. They also outline computational strategies to manage the complexity of higher-loop calculations, including fixed-moment reductions and differential-equation approaches for master integrals, with potential extensions to charged-current and polarized observables.

Abstract

We compute Mellin moments of the heavy-quark structure functions in deep-inelastic scattering at next-to-leading order in quantum chromodynamics, retaining their full dependence on the heavy-quark mass. Using the optical theorem and the operator product expansion, we derive analytic results for fixed Mellin moments $N = 2$ to $22$ of the structure functions $F_2$ and $F_L$. Our results reproduce the known expressions in the relevant asymptotic limits, in particular for virtualities of the exchanged photon $Q^2$ much larger than the heavy-quark mass squared $m^2$, and are in agreement with existing parametrisations of the next-to-leading-order coefficient functions. The computational set-up developed in this work also provides a direct pathway toward extending these calculations to next-to-next-to-leading order.

Heavy-quark production in deep-inelastic scattering -- Mellin moments of structure functions

TL;DR

The paper develops and implements a framework to compute NLO QCD corrections to heavy-quark DIS structure functions while retaining full heavy-quark mass dependence. By exploiting the optical theorem and operator product expansion, the authors derive analytic fixed Mellin moments for and in the range to , using harmonic tensors and projection techniques, and validating against known asymptotic limits and RSvN parametrisations. The work establishes a robust method for mass-dependent coefficient functions and operator-matrix elements, with cross-checks including gauge invariance and mass-factorisation consistency, and provides a clear path toward NNLO accuracy. These results enhance precision predictions for heavy-flavour DIS and underpin future global PDF analyses and collider phenomenology, particularly in regions sensitive to small- gluons and heavy-quark thresholds. They also outline computational strategies to manage the complexity of higher-loop calculations, including fixed-moment reductions and differential-equation approaches for master integrals, with potential extensions to charged-current and polarized observables.

Abstract

We compute Mellin moments of the heavy-quark structure functions in deep-inelastic scattering at next-to-leading order in quantum chromodynamics, retaining their full dependence on the heavy-quark mass. Using the optical theorem and the operator product expansion, we derive analytic results for fixed Mellin moments to of the structure functions and . Our results reproduce the known expressions in the relevant asymptotic limits, in particular for virtualities of the exchanged photon much larger than the heavy-quark mass squared , and are in agreement with existing parametrisations of the next-to-leading-order coefficient functions. The computational set-up developed in this work also provides a direct pathway toward extending these calculations to next-to-next-to-leading order.
Paper Structure (20 sections, 1 theorem, 137 equations, 2 figures, 1 table)

This paper contains 20 sections, 1 theorem, 137 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Set-up
  3. Hadronic Tensor
  4. Forward Scattering Amplitude
  5. Renormalisation
  6. Mellin moments for fixed values of $N$
  7. Harmonic Tensors
  8. Harmonic Projection
  9. Mass Factorisation
  10. Details of the Calculation
  11. Feynman Graphs
  12. Integral Families
  13. Master Integrals
  14. Mellin moments at large values of $N$
  15. Going beyond fixed moments
  16. ...and 5 more sections

Key Result

Lemma 1

Let $\mathcal{F}$ be a the disjoint union of the finite sets $\mathcal{U}$ and $\mathcal{D}$, and $\lambda$ be a traceless matrix satisfying $\tr_\mathcal{F}(\lambda) = 0$. Then,

Figures (2)

  • Figure 1: Collection of Feynman graphs showing examples for all flavour structures that appear in neutral current forward scattering amplitudes. Light quarks $\text{l}$ are depicted in black, heavy quarks $\text{h}$ in green, and vector bosons by $\text{v}$.
  • Figure 2: Independent two-point subtopologies and momentum routing at one and two loops, $\mathcal{R}_1$ and $\mathcal{R}_2$, see Eqs. (\ref{['eq:R1']}) and (\ref{['eq:R2']}).

Theorems & Definitions (2)

  • Lemma
  • proof