The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x,Q^2)$ and $g_1(x,Q^2)$

TL;DR

This work delivers the first complete three-loop (NNLO) single-mass heavy-flavor corrections to the DIS structure functions $F_2(x,Q^2)$ (unpolarized) and $g_1(x,Q^2)$ (polarized) in the asymptotic regime $Q^2 \gg m_Q^2$, enabling precise QCD analyses and consistent extractions of $\alpha_s(M_Z^2)$, $m_c$, and PDFs. The authors compute the heavy-flavor Wilson coefficients using massive operator matrix elements (OMEs) and massless Wilson coefficients, detailing the contributions $H_g^{\rm S}$, $H_q^{\rm PS}$, $L_q^{\rm NS}$, $L_g^{\rm S}$, and $L_g^{\rm PS}$ for $F_2$, and analogous polarized quantities $\Delta H_g^{\rm S}$, $\Delta H_q^{\rm PS}$, $\Delta L_q^{\rm NS}$, $\Delta L_g^{\rm S}$, and $\Delta L_g^{\rm PS}$ for $g_1$, up to $O(a_s^3)$. The results reveal sizable heavy-flavor effects, especially at small $x$ for $F_2$ and across a broad $x$ range for $g_1$, and they provide a fast, public Fortran library, WILS3HQ, to evaluate these coefficients in data analyses. This work thus supports high-precision global fits of DIS data and improved determinations of fundamental QCD parameters, including the charm mass and strong coupling, at current and future facilities such as the EIC.

Abstract

We present quantitative results on the single-mass heavy-flavor contributions in the region of large virtualities $Q^2$ up to three-loop order to the unpolarized structure function $F_2(x,Q^2)$ and the polarized structure function $g_1(x,Q^2)$ for the first time. These results are relevant for precision QCD analyses of the World deep-inelastic data and the data taken at future colliders, such as the Electron--Ion Collider, since the scaling violations due to massless and massive Wilson coefficients are significantly different. In order to measure the strong coupling constant $α_s(M_Z^2)$ and the twist-2 parton distribution functions consistently at highest precision, the next-to-next-to-leading order corrections have to be taken into account. Furthermore, the complete three-loop corrections will allow to reduce the present theory error of the charm mass $m_c$ as measured form deep-inelastic data. We provide a fast and precise public numerical code for the unpolarized and polarized massive Wilson coefficients in the asymptotic region.

Figures (8)

• Figure 1: The structure function $F_2(x,Q^2)$ at NNLO.
• Figure 2: The different heavy-flavor contributions due to charm quarks 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 $O(a_s^3)$.
• Figure 3: The heavy-flavor contributions due to charm and bottom quarks to the structure function $F_2(x,Q^2)$ at NNLO.
• Figure 4: The ratio of the heavy-flavor contributions due to charm and bottom quarks to the structure function $F_2(x,Q^2)$ at NNLO.
• Figure 5: The structure function $g_1(x,Q^2)$ at NNLO.