Heavy quark mass effects in charged-current deep-inelastic scattering at approximate NNLO in the Aivazis-Collins-Olness-Tung scheme

TL;DR

This work delivers a complete $NNLO$-level, heavy-quark-mass–aware description of deep-inelastic scattering within the ACOT framework by extending the aSACOT-χ scheme to charged-current processes and to the structure function $F_3$. The authors implement the approach in the APFEL++ code using precomputed interpolation tables, enabling fast, wide-range predictions for all nine DIS structure functions across neutral- and charged-current channels. They demonstrate the mass effects across kinematics relevant to HERA, the upcoming EIC, and neutrino-DIS experiments, finding pronounced mass sensitivities in neutrino data at low $Q^2$ and more modest effects in high-$Q^2$ charged-current measurements. The work provides a publicly available tool for precise, NNLO PDF extractions and clarifies where heavy-quark masses must be accounted to describe current and future DIS data accurately.

Abstract

The approximate SACOT-$χ$ scheme for heavy quark production in deep-inelastic scattering was initially formulated for the neutral current structure functions $F_2$ and $F_L$. We extend this approach to the charged current case (also including $F_3$), and thereby complete the definitions for the most relevant inclusive structure functions. Furthermore, we implement these structure functions in the open-source code $\texttt{APFEL++}$ which provides fast numerical evaluations over a wide kinematic range; this addition to the $\texttt{APFEL++}$ code is publicly available, with details provided in the appendix. This SACOT-$χ$ implementation enables, for the first time, detailed numerical insights on the mass dependence of the structure functions and cross sections in the $(x,Q^2)$-plane for both neutral and charged current processes. We consider kinematic regions relevant for the experimental measurements from fixed-target $ν$DIS experiments (NuTeV, CCFR and Chorus) and HERA, and also projections for the upcoming EIC. In particular, the $ν$DIS experiments reveal a surprisingly strong dependence on the mass effects, offering valuable insights that may help resolve long-standing challenges in accurately describing these datasets.

Figures (12)

• Figure 1: The ratio of neutral current structure functions $F_2,F_3$ and $F_L$ (from left to right) to the ZM NNLO-coefficient for $n=\{1,2,3\}$ (top to bottom). The ratio is defined in \ref{['eq:ratio_structure_fcn']}. The results are obtained with the CT18Hou:2019efy NNLO proton PDFs. We also indicate the mass thresholds for the bottom- and top-quark, as the ZM-coefficients are discontinuous at these values.
• Figure 2: The same as \ref{['fig:ratio_F23L_NC']} but for charged current with a $W^+$ exchange.
• Figure 3: The same as \ref{['fig:ratio_F23L_NC']} but for charged current with a $W^-$ exchange. The ratio for $F_3(W^-)$ exhibits a distinct feature at $x\sim0.005$. This is a numerical artifact, as the structure function turns negative below this $x$-value and the ratio is not well-defined in this point (see \ref{['subsec:comments_on_the_sign_of_F3']}).
• Figure 4: The reduced CC DIS cross-section measurements from HERA for an incoming $e^-$ (left, blue circles) and $e^+$ (right, green circles). The CMS energy is set to $s=318^2$ GeV${}^2$. From top to bottom we set the scaling variable $n=\{1,2,3\}$.
• Figure 5: We display pseudo-data for charged current interactions at the EIC AbdulKhalek:2021gbhKhalek:2021ulf. The left column assumes an incoming $e^-$ (location of pseudo-data as blue circles) and the right column an $e^+$ (green circles) with the CMS energy set to $s=140^2$ GeV${}^2$. The heat maps show \ref{['eq:ratio_sigma_sacot-chi_n_ZM_NNLO']} of aSACOT-χ against the ZM scheme (at all orders up to NNLO). From top to bottom we set the scaling variable to $n=\{1,2,3\}$.