[a]Arie Bodek

# Axial and Vector Structure Functions for Lepton-Nucleon Scattering, NuFact
2021 Update

Un-ki Yang  Yang Xu

###### Abstract

We report on an update (2021) of a phenomenological model for inelastic
neutrino- and electron-nucleon scattering cross sections using effective
leading order parton distribution functions with a new scaling variable
$\xi_{w}$. Non-perturbative effects are well described using the $\xi_{w}$
scaling variable in combination with multiplicative $K$ factors at low
$Q^{2}$. The model describes all inelastic charged-lepton-nucleon scattering
data (HERA/NMC/BCDMS/SLAC/JLab) ranging from very high $Q^{2}$ to very low
$Q^{2}$ and down to the $Q^{2}=0$ photo-production region. The model has been
developed to be used in analyses of neutrino oscillation experiments in the
few$\;{\mathrm{GeV}}$ region. The 2021 update accounts for the difference
between axial and vector structure functions which brings it into much better
agreement with neutrino-nucleon total cross section measurements. The model
has been developed primarily for hadronic final state masses $W$ above 1.8
GeV. However with additional parameters the model also describes the $average$
neutrino cross sections in the resonance region down to $W$=1.4 GeV.

##  1 Introduction

In the few $\;{\mathrm{GeV}}$ region there are contributions from several
kinds of lepton-nucleon interaction processes as defined by the final state
invariant mass $W$ and square of the momentum transfer $Q^{2}$. These include
quasi-elastic reactions ($W<1.07~{}\;{\mathrm{GeV/c^{2}}}$), the
$\Delta(1232)$ region ($1.1<W<1.4~{}\;{\mathrm{GeV/c^{2}}}$ ), higher mass
resonances ($1.4<W<2.0~{}\;{\mathrm{GeV/c^{2}}}$), and the inelastic continuum
region ($W>2.0~{}\;{\mathrm{GeV/c^{2}}}$). At low momentum transfer the
inelastic continuum is sometimes referred to as "shallow inelastic", and at
high momentum transfer it is referred to as "deep inelastic". It is quite
challenging to disentangle each of those contributions separately, and in
particular the contribution of resonance production and the inelastic
scattering continuum. At low $Q^{2}$ there are large non-perturbative
contributions to the inelastic cross section. These include kinematic target
mass corrections, dynamic higher twist effects, higher order Quantum
Chromodynamic (QCD) terms, and nuclear effects in nuclear targets.

In the Bodek-Yang model we focus on the inelastic part of the cross sections
above the region of the $\Delta$(1232) resonance (i.e. the higher mass
resonances, and the inelastic continuum). The model is duality based model of
neutrino interactions using effective leading order parton distribution
functions (PDFs). Earlier versions of the model[1, 2] have been incorporated
into several Monte Carlo generators of neutrino interactions including NEUT,
NEUGEN, NUANCE and GENIE. The current version of GENIE is using the NUINT04[2]
version of the model. These early versions assume that the axial structure
functions are the same as the vector structure functions. The model is based
on parameters extracted from electron scattering data. The leading order GRV98
PDFs are used with modifications that include a new scaling variable
($\xi_{w}$) to account for deviations from Bjorken scaling at lower values of
$Q^{2}$ and low $Q^{2}$ K factors that extend the validity of the model down
the $Q^{2}$=0 photo-production limit. Figure 1 shows a comparison of electron
scattering structure functions to predictions of the GRV98 PDFs, with and
without our modifications.

In this conference report we present a short a summary of the results of a
2021 update (details in ref.[3] in which we further refine the model and
$also$ account for the difference between axial and vector structure functions
at low values of $Q^{2}$. We refer to the version of the model which assumes
that vector and axial structure functions are the same as "Type I". The "Type
I" version should be used to model electron and muon scattering. We refer to
the updated version of the model that accounts for the difference in vector
and axial structure functions as "Type II". "Type II" model should be used to
model neutrino scattering.

Figure 1 shows the Bodek-Yang model with effective LO PDF model compared to
charged-lepton ${\cal F}_{2}$ experimental data (SLAC, BCDMS, NMC). Left:
${\cal F}_{2}$ proton. Right ${\cal F}_{2}$ deuteron (per nucleon). The solid
lines are the model, and the dashed lines are predictions of the GRV98 PDFs
without our low $Q^{2}$ modifications

Figure 1: The Bodek-Yang model with effective LO PDF model compared to
charged-lepton ${\cal F}_{2}$ experimental data (SLAC, BCDMS, NMC). Left:
${\cal F}_{2}$ proton. Right ${\cal F}_{2}$ deuteron (per nucleon). The solid
lines are the model, and the dashed lines are predictions of the GRV98 PDFs
without our low $Q^{2}$ modifications.

Figure 2 shows the model predictions (per nucleon) for neutrino total cross
sections on an isoscalar target compared to measurements. $\sigma_{\nu}$/E
data as a function of energy are shown on the left and $\sigma_{\bar{\nu}}$/E
are shown on the right. The green points are MINOS $\sigma_{\nu}$/E. data, the
blue points are NOMAD, and the yellow crosses are BNL82. The MINERvA and T2K
data are shown in purple and brown, respectively. The Gargamelle and ArgoNeut
measurements of $\sigma_{\bar{\nu}}$/E per nucleon are identified on the
figure. There is good agreement of the the Type II (A$>$V) model predictions
with neutrino and antineutrino total cross section measurements.

Figure 2: Model predictions (per nucleon) for neutrino total cross sections on
an isoscalar target compared to measurements. $\sigma_{\nu}$/E data as a
function of energy is shown on the left and $\sigma_{\bar{\nu}}$/E data are
shown on the right. The green points are MINOS data, the blue points are NOMAD
and the yellow crosses are BNL82. The MINERvA and T2K data are shown in purple
and brown, respectively. The Gargamelle and ArgoNeut measurements of
$\sigma_{\bar{\nu}}$/E per nucleon are identified on the figure. There is good
agreement of the Type II (A$>$V) model predfictions with neutrino and
antineutrino total cross section measurements.

Figure 3 shows the ratio of charged-current neutrino and antineutrino
differential cross sections $d^{2}\sigma/dxdy$ on lead from CHORUS (blue
points) to the Type II (A$>$V) default model predictions. The ratios are shown
for energies of 15 and 25 $GeV$. On the left side of each panel we show the
comparison for neutrino cross sections and on the right side we show the
comparisons for antineutrinos. The black line is the ratio of the predictions
of the Type I (A=V) model for which the axial structure functions are set
equal to the vector structure functions, to the predictions of the Type II
(A$>$V) default model. The differential cross section data favor the Type II
(A$>$V) model.

Figure 3:  The ratio of charged-current neutrino and antineutrino differential
cross sections $d^{2}\sigma/dxdy$ on lead from CHORUS (blue points) to the
Type II (A$>$V) default model predictions. The ratios are shown for energies
of 15 and 25 $GeV$. On the left side of each panel we show the comparison for
neutrino cross sections and on the right side we show the comparisons for
antineutrinos. The black line is the ratio of the predictions of the Type I
(A=V) model for which the axial structure functions are set equal to the
vector structure functions, to the predictions of the Type II (A$>$V) default
model. The differential cross section data favor the Type II (A$>$V) model.

In conclusions, the predictions of the 2021 update of the Bodek-Yang model are
in excellent agreement with both neutrino and charged lepton cross section
measurements on nuclear targets

## References

  * [1] A. Bodek and U.K. Yang (NUINT01),Nucl. Phys. Proc. Suppl.112, 70 (2002), arXiv:hep-ex/0203009; A. Bodek and U. K. Yang (NUINT02), arXiv:hep-ex/0308007.
  * [2] A. Bodek , Ikyu Park and U. K. Yang, (NUINT04) Nucl. Phys. Proc. Suppl.139, 113 (2005), arXiv:hep-ph/0411202.
  * [3] Arie Bodek, Un Ki Yang, Yang Xu, "Inelastic Axial and Vector Structure Functions for Lepton-Nucleon Scattering, 2021 Update",. arXiv:2108.09240v2 [hep-ph] (2021).