The O(α_s^3) Heavy Flavor Contributions to the Charged Current Structure Function xF_3(x,Q^2) at Large Momentum Transfer

TL;DR

This work computes the heavy-flavor non-singlet Wilson coefficients for the charged-current structure function xF3^{W^+ - W^-} in the asymptotic region Q^2 ≫ m^2 up to O(α_s^3), including both heavy-quark pair production and s → c flavor excitation. The authors derive Mellin-space expressions using nested harmonic sums and provide corresponding x-space results with harmonic polylogarithms, constructing the coefficients from massive operator matrix elements and massless Wilson coefficients. Numerical studies with m_c = 1.59 GeV and Alekhin PDFs show charm effects at the few-percent level relative to massless results, with mild scale dependence; the effects are small but potentially relevant for future high-precision DIS facilities. In the GLS sum rule, heavy-flavor corrections correspond to N_F → N_F+1 in the asymptotic limit, with CKM factors introducing only tiny deviations from the canonical result, and the d^{abc}d^{abc} contributions cancel in the final OME. These results enhance precision tests of QCD in charged-current DIS and underpin future determinations of α_s at next-generation neutrino facilities.

Abstract

We calculate the massive Wilson coefficients for the heavy flavor contributions to the non-singlet charged current deep-inelastic scattering structure function $xF_3^{W^+}(x,Q^2)+xF_3^{W^-}(x,Q^2)$ in the asymptotic region $Q^2 \gg m^2$ to 3-loop order in Quantum Chromodynamics (QCD) at general values of the Mellin variable $N$ and the momentum fraction $x$. Besides the heavy quark pair production also the single heavy flavor excitation $s \rightarrow c$ contributes. Numerical results are presented for the charm quark contributions and consequences on the Gross-Llewellyn Smith sum rule are discussed.

Figures (6)

• Figure 1: The corrections up to 3-loop order to the combination of the structure functions $xF_3^{W^+}(x,Q^2) + xF_3^{W^-}(x,Q^2)$ off a proton target, including both the massless and the charm contributions in the asymptotic approximation in the on-shell scheme for $m_c = 1.59~\rm GeV$Alekhin:2014sya as a function of $x$ and $Q^2$. The parton distribution functions of Ref. Alekhin:2013nda have been used with $\alpha_s(M_Z^2) = 0.1132$. These settings are the same in the subsequent Figures.
• Figure 2: Ratio of the structure functions $xF_3^{W^+}(x,Q^2) + xF_3^{W^-}(x,Q^2)$ off a proton target up to 3-loop order for the charm contribution and the massless terms for three flavors.
• Figure 3: Ratio of the structure functions $xF_3^{W^+}(x,Q^2) + xF_3^{W^-}(x,Q^2)$ off a proton target up to 3-loop order for the charm contribution and the massless terms for three flavors from tree-level to $O(a_s^3)$ at $Q^2 = 100~\rm GeV^2$.
• Figure 4: The ratio of the structure functions $xF_3^{W^+}(x,\mu^2) + xF_3^{W^-}(x,\mu^2)$ to $xF_3^{W^+}(x,Q^2) + xF_3^{W^-}(x,Q^2)$, varying $\mu^2 \in [Q^2/4, 4 Q^2]$ at $O(a_s^3)$ and $Q^2 = 100~\rm GeV^2$ as a function of $x$.
• Figure 5: The corrections up to 3-loop order to the combination of the structure functions $xF_3^{W^+}(x,Q^2) + xF_3^{W^-}(x,Q^2)$ off a nucleon in an isoscalar target, including both the massless and charm contributions in the asymptotic approximation.