Target Mass Corrections to Electro-Weak Structure Functions and Perturbative Neutrino Cross Sections

TL;DR

The paper develops a complete framework to incorporate target mass corrections, NLO perturbative QCD, and heavy-quark mass effects into weak structure functions and neutrino cross sections within the twist-2 (leading) sector. It formulates the corrections via the operator-product expansion (OPE), derives explicit master formulas for structure functions in terms of Nachtmann xi scaling, and provides a detailed comparison with the collinear parton model, including a Monte Carlo implementation. Key contributions include the explicit tensor mixing in the OPE-based TMC, handling of charm production in charged-current processes, and a quantitative assessment of how these corrections affect observables like the Paschos-Wolfenstein relation. The results support more precise extraction of electroweak parameters from neutrino scattering and offer a robust foundation for neutrino cross-section modeling in oscillation and precision studies.

Abstract

We provide a complete and consistent framework to include subasymptotic perturbative as well as mass corrections to the leading twist (tau=2) evaluation of charged and neutral current weak structure functions and the perturbative neutrino cross sections. We revisit previous calculations in a modern language and fill in the gaps that we find missing for a complete and ready-to-use "NLO xi-scaling" formulary. In particular, as a new result we formulate the mixing of the partonic and hadronic structure function tensor basis in the operator approach to deep inelastic scattering. As an underlying framework we follow the operator product expansion a la Georgi & Politzer that allows the inclusion of target mass corrections at arbitrary order in QCD and we provide explicit analytical and numerical results at NLO. We compare this approach with a simpler collinear parton model approach to xi-scaling. Along with target mass corrections we include heavy quark mass effects as a calculable leading twist power suppressed correction. The complete corrections have been implemented into a Monte Carlo integration program to evaluate structure functions and/or integrated cross sections. As applications, we compare the operator approach with the collinear approximation numerically and we investigate the NLO and mass corrections to observables that are related to the extraction of the weak mixing angle from a Paschos-Wolfenstein-like relation in neutrino-iron scattering. We expect that the interpretation of neutrino scattering events in terms of oscillation physics and electroweak precision physics will benefit from our results.

Figures (5)

• Figure 1: ($a$) The charged current neutrino structure function $F_2$ at $Q^2=1\ {\rm GeV}^2$ evaluated with NLO $\xi$-scaling corrections (solid line) and in LO (dashed) and NLO (dot-dashed) under the neglect of target mass corrections. ($b$) Ratio of the LO $\xi$-scaling evaluation of $F_2$ as in ($a$) with $Q^2=1$ GeV$^2$ to the LO evaluation in the collinear approximation is shown by the solid line. The dot-dashed line shows the same ratio at NLO. The dashed line shows the ratio of the NLO target mass corrected $F_2$ to the NLO $F_2$ in the $M\rightarrow 0$ limit.
• Figure 2: Same as Fig. \ref{['fig:f21']} for $Q^2=4\ {\rm GeV}^2$.
• Figure 3: Same as Fig. \ref{['fig:f21']} for the structure function $xF_3$.
• Figure 4: Same as Fig. \ref{['fig:f31']} for $Q^2=4\ {\rm GeV}^2$.
• Figure 5: The neutrino (solid) and anti-neutrino (dashed) flux used in Eq. (\ref{['eq:sigint']}).