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

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.

💡 Research Summary

The paper presents a comprehensive calculation of Mellin moments of the heavy‑quark structure functions (F_2) and (F_L) in deep‑inelastic scattering (DIS) at next‑to‑leading order (NLO) in QCD, keeping the full dependence on the heavy‑quark mass (m). Using the optical theorem to relate the hadronic tensor to the discontinuity of the forward scattering amplitude, the authors employ the operator product expansion (OPE) to express the amplitude as a sum over local twist‑2 operators. The Mellin moments are then obtained by projecting the OPE coefficients onto the moments of the structure functions, leading to a compact relation (Eq. 18) that connects the even‑(N) moments of (F_{2,L}) to the product of operator matrix elements (A_N^k) and coefficient functions (C_{N,i}^k).

Two independent computational strategies are pursued. The first expands the forward amplitude in the external momentum and projects onto harmonic polynomials, reducing the problem to a set of master integrals via integration‑by‑parts (IBP) identities. This method encounters bottlenecks for large (N) due to the growth of polynomial degree. To overcome this, a second approach performs the IBP reduction without any momentum expansion, using the Laporta algorithm to obtain master integrals directly. Both methods yield identical results for fixed moments (N=2,\dots,22), providing a stringent cross‑check.

Dimensional regularisation ((D=4-2\varepsilon)) and the modified minimal subtraction ((\overline{\text{MS}})) scheme are used throughout. The bare operators are renormalised through a mixing matrix (Z_{ij}) that incorporates the one‑ and two‑loop anomalous dimensions (\gamma^{(0)}{ij},\gamma^{(1)}{ij}) and the QCD (\beta)‑function coefficients. The renormalisation constants are expanded in both the strong coupling (a_s) and (\varepsilon), leading to explicit expressions (Eqs. 27a‑d). After renormalisation of both the operators and the coupling, the finite, renormalised forward amplitude (Eq. 28) is matched to the diagrammatic calculation, yielding the desired coefficient functions.

The resulting Mellin moments are verified against two benchmarks. In the asymptotic limit (Q^2\gg m^2) they reproduce known analytic expressions from the literature, confirming the correct handling of the large‑scale limit. Moreover, the moments agree numerically with the Mellin‑space parametrisation of the NLO coefficient functions provided in Ref.