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

Technical Summary: Heavy-quark production in deep-inelastic scattering – Mellin moments of structure functions

Problem and Motivation
Heavy-quark production in deep-inelastic scattering (DIS) is a critical process for testing Quantum Chromodynamics (QCD) and constraining parton distribution functions (PDFs), particularly the gluon PDF at small Bjorken-$x$. While next-to-leading order (NLO) QCD corrections to heavy-flavour structure functions are known numerically, analytic results with full dependence on the heavy-quark mass $m$ have been limited. Specifically, analytic NLO results for the gluon coefficient functions $C_{k,g}$ with full mass dependence were previously unknown. Furthermore, the precision of experimental data from HERA and anticipated future data from the Electron-Ion Collider (EIC) necessitates theoretical predictions beyond NLO, requiring Next-to-Next-to-Leading Order (NNLO) calculations. To facilitate the extension to NNLO, the authors aim to compute Mellin moments of the heavy-quark DIS structure functions $F_2$ and $F_L$ at NLO while retaining the full heavy-quark mass dependence.

Methodology
The calculation employs the optical theorem and the operator product expansion (OPE) to relate the hadronic tensor to the forward scattering amplitude. The authors compute fixed Mellin moments $N$ of the structure functions by projecting the forward scattering amplitude onto specific tensor structures using harmonic projection operators.

Key methodological steps include:

1. Formalism: The hadronic tensor is related to the imaginary part of the forward scattering amplitude $T_{\mu\nu}$. Using the OPE, the amplitude is expanded in terms of local twist-two operators. The Mellin moments are extracted by applying harmonic projection operators to the partonic invariants, effectively isolating the coefficient functions $C_{N,i,k}$.
2. Renormalisation and Mass Factorisation: The calculation is performed in dimensional regularisation ($D=4-2\epsilon$) within the $\overline{\text{MS}}$ scheme. The authors handle the mixing of singlet operators and the decoupling of heavy quarks from the running coupling using appropriate renormalisation constants and decoupling relations.
3. Integral Reduction: Two distinct approaches are utilized to compute the necessary Feynman integrals:
   • Harmonic Projection with Momentum Expansion: This method expands the forward amplitude in powers of the external parton momentum $p$. While effective for lower moments, it encounters bottlenecks for large $N$ due to the factorial growth of generated terms and the complexity of tensor reductions.
   • Alternative Approach (No Momentum Expansion): To overcome the bottlenecks of the first method for higher moments, the authors generate amplitudes without expanding in $p$. They utilize packages such as QGRAF, tapir, exp, and KIRA to reduce the amplitudes to master integrals. This approach exploits differential equations in the variable $\omega = 1/z$ (where $z$ is the parton momentum fraction) to calculate high moments, mapping the results to the integral families used in the first approach.
4. Master Integrals: The calculation reduces to a set of master integrals involving massive propagators. These are solved using the method of differential equations, rationalising square roots via variable transformations, and expressing results in terms of harmonic polylogarithms (HPLs).

Key Contributions and Results

• Analytic NLO Results: The paper presents the first analytic results for the NLO gluon coefficient functions $C_{N,2,g}$ and $C_{N,L,g}$ with full heavy-quark mass dependence for Mellin moments $N=2$ to $N=22$.
• Complete Set of Moments: Full analytic expressions for the Mellin moments of the transverse ($F_2$) and longitudinal ($F_L$) structure functions are provided for $N=2, \dots, 22$. These results are available in an ancillary file.
• Validation:
  • The results reproduce known expressions in the asymptotic limit $Q^2 \gg m^2$.
  • The one-loop results match exact expressions derived from Mellin moments of known NLO coefficient functions.
  • A numerical comparison with the semi-analytical parametrisations of Ref. [3] (RSvN) shows agreement at the permille level for the gluon channel, validating the existing parametrisations.
• Technical Framework: The authors demonstrate a robust computational setup capable of handling massive integrals and high Mellin moments, which serves as a direct pathway for extending these calculations to NNLO.

Significance and Claims
The authors state that this work provides a necessary foundation for extending heavy-quark DIS calculations to NNLO accuracy. By computing the Mellin moments with full mass dependence at NLO, they verify the accuracy of existing parametrisations and establish a formalism that naturally separates heavy-quark DIS coefficient functions from heavy-quark operator matrix elements. The paper claims that the developed framework is directly applicable to the three-loop level, although it acknowledges that NNLO calculations will face significant technical challenges, particularly regarding the Integration-by-Parts (IBP) reduction of three-loop integral families with internal masses and the potential emergence of elliptic structures (specifically in the gluon channel). The work does not claim to have solved the full NNLO problem but rather provides the essential first step and the computational tools required to pursue it.