The complete three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients

TL;DR

The paper provides a comprehensive treatment of three-loop heavy-flavor corrections in deep-inelastic scattering in the asymptotic regime $Q^2 \gg m_Q^2$, for both unpolarized and polarized cases, including single- and two-mass corrections. It details the calculation framework for massive operator matrix elements $A_{ij}$, the generation and reduction of Feynman diagrams, and advanced techniques (Mellin-space recurrences, differential equations in a generating parameter, and $x$-space representations) to obtain analytic and numerical results. Key contributions include analytic and numerical three-loop single- and two-mass corrections, the construction of asymptotic massive Wilson coefficients, and the provision of fast $x$-space representations and public Fortran codes for massless coefficients, splitting functions, and target-mass corrections. These results enable NNLO analyses of $F_2$ and $g_1$ at large $Q^2$, improve precision in determining $\alpha_s(M_Z^2)$ and heavy-quark masses, and supply the computational tools needed for current and future DIS studies, including EIC and LHeC programs.

Abstract

We report on the three-loop unpolarized and polarized massive operator matrix elements, with single- and two-mass corrections, and the associated deep-inelastic massive Wilson coefficients in the region $Q^2 \gg m_Q^2$, the calculation of which has been completed recently. We also provide fast and precise numerical representations of the massless Wilson coefficients, splitting functions to tree-loop order, and target-mass corrections in $x$-space well suited for QCD-fitting codes.

Figures (3)

• Figure 1: Left panel: the constant part of the massive OME $A_{Qg}^{(3)}$, $a_{Qg}^{(3)}(x)$, as a function of $x$, rescaled by the factor $x(1-x)$ in the region of smaller values of $x$. Full line (red): $a_{Qg}^{(3)}(x)$; dashed line (blue): leading small-$x$ term $\propto \ln(x)/x$Catani:1990eg; light blue region: estimates of Kawamura:2012cr; gray region: estimates of Alekhin:2017kpj; from Ref. Ablinger:2024xtt. Right panel: the charm contributions to the structure function $F_2(x,Q^2)$ by the Wilson coefficients $H_g^{\rm S}, H_q^{\rm PS}, L_q^{\rm NS}, L_g^{\rm S}$ and $L_g^{\rm PS}$ at $Q^2 = 100~\rm GeV^2$ at different orders in the strong coupling constant up to $O(a_s), O(a_s^2)$ and $O(a_s^3)$; from Ref. Ablinger:2025awb. For the notations of the massive Wilson coefficients see Ref. Bierenbaum:2009mv.
• Figure 2: The ratio of the three-loop two-mass contributions to $A_{gq,Q}$ to the complete gq $O(T_F^2)$ corrections. Dotted line: $Q^2 = 30~\rm GeV^2$; Full line: $Q^2 = 50~\rm GeV^2$; Dashed line: $Q^2 = 100~\rm GeV^2$; Dashed-dotted: $Q^2 = 1000~\rm GeV^2$.
• Figure 3: Left panel: The structure function $F_2(x,Q^2)$ at NNLO containing the massless and the asymptotic single-mass contributions due to charm and bottom quarks as a function of $x$ and $Q^2$, using the parton distribution functions of Ref. Alekhin:2017kpj; from Ref. Ablinger:2025awb. Right panel: The relative contribution of the heavy-flavor contributions due to $c$ and $b$ quarks to the structure function $F_2^{\rm NS}$; dashed lines: $100~\rm GeV^2$; dashed-dotted lines: $1000~\rm GeV^2$; dotted lines: $10000~\rm GeV^2$. massless: N$^3$LO contributions and single- and two-mass contributions: $O(a_s^3)$; from Ref. Blumlein:2021lmf.