# Neutrino Cross Sections:
Interface of shallow- and deep-inelastic scattering for collider neutrinos

Yu Seon Jeong<EMAIL_ADDRESS>Chung-Ang University, High Energy Physics
Center, Dongjak-gu, Seoul 06974, Republic of Korea  Mary Hall Reno  mary-hall-
<EMAIL_ADDRESS>University of Iowa, Department of Physics and Astronomy, Iowa
City, Iowa 52242, USA

###### Abstract

Neutrino experiments in a Forward Physics Facility at the Large Hadron
Collider can measure neutrino and antineutrino cross sections for energies up
to a few TeV. For neutrino energies below 100 GeV, the inelastic cross section
evaluations have contributions from weak structure functions at low momentum
transfers and low hadronic final state invariant mass. To evaluate the size of
these contributions to the neutrino cross section, we use a parametrization of
the electron-proton structure function, adapted for neutrino scattering,
augmented with a correction to account for the partial conservation of the
axial vector current, and normalized to structure functions evaluated at next-
to-leading order in QCD, with target mass corrections and heavy quark
corrections. We compare our results with other approaches to account for this
kinematic region in neutrino cross section for energies between 10–1000 GeV on
isoscalar nucleon and iron targets.

##  I Introduction

Cross sections for neutrino interactions with nucleons and nuclei are
important ingredients to measurements of neutrino mixing angles in neutrino
oscillation experiments and for searches for physics beyond the standard model
(BSM) [1, 2, 3, 4]. The Deep Underground Neutrino Experiment (DUNE) will
intercept neutrinos and antineutrinos produced at Fermilab [5] with energies
up to more than 10 GeV to meet their physics objectives [6]. The cross
sections for neutrino energies of a few GeV have contributions from quasi-
elastic scattering (e.g., $\nu_{\mu}n\to\mu p$), resonant production (e.g.,
$\nu_{\mu}n\to\Delta^{0}\to\mu n\pi^{+}$) and inelastic scattering
$\nu_{\mu}N\to\mu X$, where $N=(p+n)/2$ represents an isoscalar nucleon target
and $X$ refers to all final state hadrons. The quasi-elastic and resonant
regions of the neutrino cross section are the focus of much of the theory
effort, however, inelastic scattering is also important for neutrino
interactions at DUNE [7]. More than 50% of DUNE events will be inelastic
scattering events with the hadronic final state invariant mass larger than the
mass of the $\Delta$ resonance, $m_{\Delta}(\approx 1.4~{}{\rm GeV})$ [8, 9].

Inelastic scattering can involve small momentum transfers $Q^{2}$. Quark-
hadron duality [10, 11] connects structure functions in the resonance region
with parton-model-based structure functions at higher $Q^{2}$ for inelastic
scattering [12, 13]. For $E_{\nu}=10$ GeV, the neutrino charged-current (CC)
cross section can have a contribution of $\sim 20\%$ from $Q^{2}<1$ GeV2 [14],
a $Q^{2}$ kinematic regime where a perturbative QCD parton distribution
function decomposition of the weak structure functions cannot necessarily be
performed.

The opportunity to measure neutrino and antineutrino cross sections and weak
structure functions will arise in experiments in the forward region at the
Large Hadron Collider (LHC). Recently, two neutrino experiments at the LHC,
FASER$\nu$ [15] and SND@LHC [16, 17] have been installed in existing caverns
along the beam-line direction, 480 m from the ATLAS interaction point, and are
taking data. Both experiments reported their first observations of collider
neutrino events with analysis of the data collected during 2022 [18, 19]. They
will operate continuously during Run 3 of the LHC. The next stage experiments
for the high luminosity phase of the LHC (HL-LHC) would run in the proposed
Forward Physics Facility (FPF). The FPF would house larger neutrino detectors
FASER$\nu$2, AdvSND and a liquid argon detector FLArE, and additional
detectors designed for BSM searches [14, 20]. In this configuration, a large
number of neutrinos from the ATLAS interaction point emitted in the very
forward region could be detected. With the increased luminosity and detector
size at the FPF, the expected number of neutrino events is much larger than in
FASER$\nu$ and SND@LHC. The estimated number of interactions are $\sim 10^{6}$
events for muon neutrinos and thousands for tau neutrinos in FPF experiments
[14].

Neutrinos produced at the LHC in the forward region are distributed in energy
up to a few TeV [14, 20, 21, 22, 23, 24, 25]. The TeV neutrino energy range
has not been reached directly in neutrino beams produced by accelerators. With
a large number of neutrino events, the FPF can provide a unique opportunity to
investigate neutrino interactions precisely at unexplored energy range for all
three flavors [14, 26, 27]. Although a large fraction of neutrinos is
distributed with energies above 100 GeV, also of interest are neutrinos with
energies below 100 GeV. There will be thousands of neutrinos in the 10’s of
GeV energy range [14, 28] where cross sections are sensitive to low $Q^{2}$
structure functions.

In this work, we evaluate the charged-current neutrino deep-inelastic
scattering (DIS) cross sections and investigate the size of the contribution
from the transition region from resonance meson/baryon production to DIS,
referred as the shallow inelastic scattering (SIS) region. We probe how the
neutrino DIS cross sections depend on the hadronic final state invariant mass
$W$ and on $Q^{2}$. We quantify the impact of ranges of these kinematic
variables relevant to the SIS region for muon neutrino and tau neutrino as
well as for antineutrino cross sections. In the next section, we outline
approaches to low $Q^{2}$ extrapolations of electromagnetic structure
functions and their translations to weak structure functions. In Section III,
we introduce our approximation to the contributions from the partially
conserved axial vector current. Section IV shows our results for cross
sections as a function of $W$ and $Q^{2}$, followed by a discussion in Section
V. We compare our structure function and cross section results with other
approaches [29, 30, 31, 32, 26]. For reference, the cross section formulas for
neutrino and antineutrino charge-current cross sections are collected in
Appendix A. Tables of cross section results are included in Appendix B.

##  II structure functions at Low $Q^{2}$

In evaluating the cross section of neutrino charged-current (CC) DIS
interaction, essential components are structure functions $F_{i,{\rm
CC}}(x,Q^{2})$ for $i=1-5$, as shown in detail in Appendix A. The structure
functions are described by the parton momentum fraction $x$ (or Bjorken
variable) and parton distribution functions (PDFs) $q(x,Q^{2})$ and
$\bar{q}(x,Q^{2})$ for quarks and antiquarks labeled by their flavors, written
schematically as

$\displaystyle F_{2,{\rm CC}}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\sum_{q,q^{\prime}}2x\bigl{(}q(x,Q^{2})+\bar{q}^{\prime}(x,Q^{2})\bigr{)}\,,$
(1) $\displaystyle F_{3,{\rm CC}}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\sum_{q,q^{\prime}}2\bigl{(}q(x,Q^{2})-\bar{q}^{\prime}(x,Q^{2})\bigr{)}\,,$
(2)

at leading order in perturbative QCD for $F_{2,{\rm CC}}$ and $F_{3,{\rm
CC}}$. The sum is over the relevant quark flavors for neutrino and
antineutrino scattering. For more details for the distinction between neutrino
and antineutrino structure functions, the reader is referred to Appendix A. By
comparison, the electromagnetic structure function $F_{2,{\rm EM}}(x,Q^{2})$
in the same approximation is

$F_{2,{\rm
EM}}=\sum_{q}e_{q}^{2}x\bigl{(}q(x,Q^{2})+\bar{q}(x,Q^{2})\bigr{)}\,.$ (3)

PDFs are extracted from analyses of lepton-hadron and hadron-hadron data for
ranges of the variables $x$ and $Q^{2}$ [33]. For the nCTEQ15 PDFs [34] used
in this work, the ranges of applicability of the PDFs are $5\times
10^{-6}<x<1$ and $1.3~{}{\rm GeV}<Q<10^{4}~{}{\rm GeV}$. Since perturbative
QCD is not applicable for $Q^{2}\lesssim 1~{}{\rm GeV^{2}}$, PDF-based
structure functions are not reliable in this low $Q^{2}$ region. The neutrino
cross sections are sensitive to such low $Q^{2}$ at low incident neutrino
energy, $E_{\nu}\lesssim{\cal O}(10)$ GeV, whereas the contributions from
$Q^{2}\lesssim 1~{}{\rm GeV^{2}}$ are negligible for neutrino energies above
several hundred GeV. This paper shows results for neutrino and antineutrino
cross sections from low $Q^{2}$ extrapolations of the CC structure functions
for incident energies between $10-1000$ GeV.

In a discussion of the structure functions, it is useful to relate $x$ and
$Q^{2}$ to the hadronic final state invariant mass $W$ defined by

$W^{2}=Q^{2}\Biggl{(}\frac{1}{x}-1\Biggr{)}+m_{N}^{2}\,,$ (4)

where $m_{N}$ is the mass of the nucleon. The DIS region is typically defined
in terms of the $W$ and $Q^{2}$ as $W\gtrsim$ 2 GeV and $Q^{2}\gtrsim 1-4$
GeV2. One can consider the SIS region as $m_{N}+m_{\pi}<W\lesssim 2$ GeV for
all $Q^{2}$ [8]. However, the boundary between the SIS and DIS regions is not
clearly defined. Inelastic scattering that includes $W>W_{\rm min}=1.4$ GeV
includes resonances above the $\Delta$ resonance. In the GENIE Monte Carlo
generator, the value of $W_{\rm min}$ is set to 1.7 GeV [35, 36].

In the context of the parton model for the lower range of the applicable
$Q^{2}$, target mass corrections (TMC) modify the simplified forms of eqns.
(1) and (2) [37, 38, 39, 40, 41]. Among the modifications is a replacement of
the Bjorken-$x$ variable in the PDFs with the Nachtmann variable $\xi$ defined
as

$\xi=\frac{2x}{1+\sqrt{1+4x^{2}m_{N}^{2}/Q^{2}}}\,.$ (5)

The Nachtmann variable corresponds to the light-front momentum fraction of the
struck parton in the massless parton limit, given the massive nucleon target.
When $m_{N}\to 0$, $\xi\to x$. Target mass corrections to the parton model
structure functions are largest for low $Q^{2}$ and for large $x$. The recent
work on the TMC for nuclear targets [41] shows that these corrections are
effectively independent of target mass number. We use here the TMC for
structure functions as outlined in refs. [39, 40].

In the non-perturbative regime, the structure functions can be
phenomenologically modeled by fitting to lepton-hadron scattering data. There
exist several models for such low $Q^{2}$ structure functions, e.g., refs.
[30, 32, 26, 42, 43]. A commonly used model is the Bodek-Yang model [30],
which provides effective PDFs at low $Q^{2}$ that can be used to construct
electromagnetic and weak structure functions according to, e.g., eqns. (1-3).
The Bodek-Yang approach uses GRV98 LO PDFs [44] evaluated with a modified
Nachtmann variable and frozen at $Q_{0}^{2}=0.8$ GeV2, then multiplied by
$Q^{2}$ dependent $K$-factors. The parameters in the Bodek-Yang model are
obtained by fitting inelastic charged lepton-nucleon scattering data [45, 46,
47]. The Bodek-Yang model is implemented in Monte Carlo event generators such
as GENIE [36], NEUT [48] and NuWro [49].

Figure 1:  The structure functions $F_{2}(x,Q^{2})$ for electron-nucleon
electromagnetic scattering for $Q^{2}$ = 1.69 GeV2 (upper) and $0.45$ GeV2
(lower). The CKMT [42], NLO and NLO+TMC structure functions in the upper panel
are compared with a parameterization of the electromagnetic scattering data by
Bosted and Christy (BC) [50]. The lower panel shows only the BC and CKMT
parameterizations.

Another approach and the one that we use here starts with the parameterization
of the structure function $F_{2,{\rm EM}}$ provided by Capella et al. (CKMT)
in ref. [42]. The CKMT parameterization [42, 43], obtained by fitting to
electromagnetic structure function data, has the form

$\displaystyle F_{2,{\rm EM}}^{\rm CKMT}(x,Q^{2})$ $\displaystyle=$
$\displaystyle F_{2}^{sea}(x,Q^{2})+F_{2}^{val}(x,Q^{2})$ (6) $\displaystyle=$
$\displaystyle Ax^{-\Delta(Q^{2})}(1-x)^{n(Q^{2})+4}$ $\displaystyle\times$
$\displaystyle\Biggl{(}\frac{Q^{2}}{Q^{2}+a}\Biggr{)}^{1+\Delta(Q^{2})}$
$\displaystyle+$ $\displaystyle
Bx^{1-\alpha_{R}}(1-x)^{n(Q^{2})}\Biggl{(}\frac{Q^{2}}{Q^{2}+b}\Biggr{)}^{\alpha_{R}}$
$\displaystyle\times$ $\displaystyle\Bigl{(}1+f(1-x)\Bigr{)}\ .$

The functions $n(Q^{2})$ and $\Delta(Q^{2})$ are

$\displaystyle n(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{3}{2}\Biggl{(}1+\frac{Q^{2}}{Q^{2}+c}\Biggr{)}\ ,$ (7)
$\displaystyle\Delta(Q^{2})$ $\displaystyle=$
$\displaystyle\Delta_{0}\Biggl{(}1+\frac{2Q^{2}}{Q^{2}+d}\Biggr{)}\ .$ (8)

The constants entering into the parameterization of the electromagnetic
structure function are in the first two rows of Table 1.

The functional form of eq. (6) accounts for terms based on Pomeron and Reggeon
contributions. At low $Q^{2}$, the distinction between sea and valence
contributions is not evident, but at large enough $Q^{2}$, a parton model
interpretation has the first term to be interpreted as the sea quark (and
antiquark) contribution and the second term as the valence contribution.

In electromagnetic scattering, in the limit of $Q^{2}\to 0$ with $W^{2}$
fixed, the real photon-proton cross section in the CKMT form is

$\displaystyle\sigma_{\gamma p}^{\rm tot}(W^{2})$ $\displaystyle=$
$\displaystyle\frac{4\pi^{2}\alpha_{\rm EM}}{Q^{2}}F_{2,{\rm EM}}^{\rm
CKMT}(x,Q^{2})\lvert_{Q^{2}\to 0}$ $\displaystyle\simeq$ $\displaystyle
4\pi^{2}\alpha_{\rm
EM}\Biggl{[}\frac{A}{a}\Biggl{(}\frac{W^{2}}{a}\Biggr{)}^{\Delta_{0}}$
$\displaystyle+$
$\displaystyle\frac{B(1+f)}{b}\Biggl{(}\frac{W^{2}}{b}\Biggr{)}^{\alpha_{R}-1}\Biggr{]}\,,$

where $W^{2}\simeq Q^{2}/x$ for $W^{2}\gg m_{N}^{2}$. This means that
$F_{2,EM}\sim Q^{2}$ for low $Q^{2}$.

The upper panel of Fig. 1 shows $F_{2,{\rm EM}}(x,Q^{2})$ for $Q^{2}=1.69\
{\rm GeV}^{2}$. The dot-dashed curve shows the parameterization of Jefferson
Laboratory electromagnetic structure function data by Bosted and Christy (BC)
[50]. The dashed line show the next-to-leading order (NLO) QCD evaluation of
the electromagnetic structure function using the nCTEQ15 proton PDF. The curve
labeled NLO+TMC also includes target mass [37, 38, 51]. The curve labeled CKMT
shows the result using the parameterization of eq. (6). In the lower panel of
Fig. 1, for $Q^{2}=0.45$ GeV2 where the PDF approach is not applicable, only
the BC and CKMT parameterizations of $F_{2,{\rm EM}}$ are shown.

To the left of vertical dashed lines in Fig. 1 are the regions of $x$ such
that $W$ is larger than the three values $W_{\rm min}=2,\ 1.7$ and 1.4 GeV
given $Q^{2}=1.69\ {\rm GeV}^{2}$ or $Q^{2}=0.45\ {\rm GeV}^{2}$. In the upper
panel, one can see that the NLO perturbative result overshoots the data for
$x\simeq 0.2-0.5$, while the NLO+TMC does somewhat better in the same range.
The CKMT parametrization agrees well with the BC parameterization for $W>2$
GeV, and roughly averages the resonance region for smaller values of $W$. The
lower panel shows that the CKMT structure function also roughly averages the
structure function in the resonance region at lower $Q^{2}$, here shown for
$Q^{2}=0.45$ GeV2.

For CC scattering in the Bodek-Yang approach, the effective PDFs are assembled
according to eqs. (1) and (2). For $F_{1,{\rm CC}}(x,Q^{2})$ we use the
relation

$R(x,Q^{2})=\frac{F_{2,{\rm CC}}(x,Q^{2})}{2xF_{1,{\rm
CC}}(x,Q^{2}}\Biggl{(}1+\frac{4m_{N}^{2}x^{2}}{Q^{2}}\Biggr{)}-1$ (10)

to write $F_{1,{\rm CC}}$ in terms of $F_{2,{\rm CC}}$ and $R$, where a
parameterization of $R$ is used.

For CC scattering in the CKMT approach, the same functional form as $F_{2,{\rm
EM}}$ is applied to $F_{2,{\rm CC}}$, with modifications to $A$, $B$ and $f$
to reflect the differences between eqs. (1) and (3) for the sea, overall
valence and relative importance of $u$ and $d$ valence contributions,
respectively [52, 53]. The functional form for $xF_{3,{\rm CC}}$ is the same
as for $F_{2,{\rm CC}}$, with the substitutions for $A$ and $B$ as indicated
in Table 1. For $F_{1,{\rm CC}}(x,Q^{2})$, we use the relation in eq. (10)
with the parameterization of Whitlow et al. [54] from electromagnetic
scattering, which applies for $Q^{2}>Q_{m}^{2}=0.3\ {\rm GeV}^{2}$. Below
$Q^{2}=Q_{m}^{2}$, we take $R(x,Q^{2})=R(x,Q_{m}^{2})\cdot Q^{2}/Q_{m}^{2}$.

For $\nu_{\tau}$ and $\bar{\nu}_{\tau}$ charged current scattering, in
principle two additional structure functions are required: $F_{4,{\rm CC}}$
and $F_{5,{\rm CC}}$. At leading order in QCD for massless partons,

$\displaystyle F_{4,{\rm CC}}(x,Q^{2})$ $\displaystyle=$ $\displaystyle 0\,,$
(11) $\displaystyle F_{5,{\rm CC}}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
F_{2,{\rm CC}}(x,Q^{2})/(2x)\,.$ (12)

These are the so-called Albright-Jarlskog relations [55]. We take into account
the NLO QCD, target mass and heavy quark mass corrections to $F_{4,CC}$ and
$F_{5,CC}$. For the low-$Q^{2}$ extrapolations, following the Albright-
Jarlskog relations, we set $F_{4,CC}$ =0 and extrapolate
$F_{5}(Q^{2}<Q_{0}^{2})$ with the functional form of the
$F_{2}(Q^{2}<Q_{0}^{2})$ extrapolation discussed below.

Figure 2:  Upper: Comparison of the NLO+TMC (dashed) and CKMT (solid)
evaluations of $F_{2,{\rm CC}}$ and $xF_{3,{\rm CC}}$. Lower: The ratios of
structure functions evaluated using NLO+TMC to CKMT for $Q^{2}=2\ {\rm
GeV}^{2}$.

In our evaluations of the neutrino and antineutrino cross section, we use
NLO+TMC evaluations of the structure functions that include heavy quark mass
corrections [56, 57, 39, 58] and nCTEQ15 PDFs for $Q^{2}\geq Q_{0}^{2}$. For
$Q^{2}<Q_{0}^{2}$, the CKMT neutrino structure functions are used. We will
take $Q_{0}^{2}=2,\ 4$ GeV2. Comparisons of NLO+TMC and CKMT evaluations of
$F_{2,{\rm CC}}$ and $xF_{3,{\rm CC}}$ at $Q^{2}=2$ GeV2 are shown in the
upper panel of Fig. 2. Numerically, the CKMT structure function $F_{2,{\rm
CC}}$ at $Q^{2}=2\ {\rm GeV}^{2}$ is within $+9\%$ to $-6\%$ of the NLO+TMC
corrected structure function evaluated using the nCTEQ15 PDFs for $x<0.5$. The
discrepancy is similar for $xF_{3,{\rm CC}}$, within $+11\%$ to $-8\%$.

For reference, in the cross section evaluations below, the $xF_{3,{\rm CC}}$
contribution to the $\nu_{\mu}N$ CC cross section is $\sim 10\%$ for
$E_{\nu}=10$ GeV, reducing to $\sim 0.4\%$ for $E_{\nu}=10^{2}$ GeV. For
$\bar{\nu}_{\mu}N$ CC scattering, the impact of $xF_{3,{\rm CC}}$ is larger
because of the minus sign in the differential cross section (see Appendix A).
Regarding the range of $x$ kinematically allowed for $Q^{2}>0.1$ GeV2, for
$E_{\nu}=10$ GeV, $x\gtrsim 6\times 10^{-3}$ in $\nu_{\mu}N$ CC scattering and
$x\gtrsim 0.2$ in $\nu_{\tau}N$ CC scattering [59]. For $E_{\nu}\lesssim 1$
TeV, relevant to the neutrinos that can be detected at the FPF, the neutrino
CC cross sections mostly come from $x\gtrsim 0.01$ [21]. For example, when
$E_{\nu}=1$ TeV, the contribution of $x>0.01$ is about 95% of the
$\bar{\nu}_{\mu}N$ cross section, and it is higher for the $\nu_{\mu}$,
$\nu_{\tau}$ and $\bar{\nu}_{\tau}$ cross sections.

To avoid discontinuities in the structure functions at $Q_{0}^{2}$, we
normalize the CKMT structure functions according to

$F_{i,{\rm CC}}(x,Q^{2})=F_{i,{\rm CC}}^{\rm CKMT}(x,Q^{2})\frac{F_{i,{\rm
CC}}^{\rm NLO+TMC}(x,Q_{0}^{2})}{F_{i,{\rm CC}}^{\rm CKMT}(x,Q_{0}^{2})}\,,$
(13)

and we label the CKMT structure functions normalized with NLO+TMC at
$Q_{0}^{2}$ by CKMT-NT. As noted above, we use the Albright-Jarlskog relation
to write $F_{5,{\rm CC}}^{\rm CKMT}$ in terms of $F_{2,{\rm CC}}^{\rm
CKMT}/(2x)$ to evaluate $F_{5,{\rm CC}}$ for $Q^{2}<Q_{0}^{2}$. The
normalization factors for $F_{2,{\rm CC}}$ and $xF_{3,{\rm CC}}$ are shown in
the lower panel of Fig. 2 for $Q_{0}^{2}=2$ GeV2. Our results are not very
sensitive to a choice of $Q_{0}^{2}$ between $2\ {\rm GeV}^{2}$ and $4\ {\rm
GeV}^{2}$ because the CKMT parameterization follows the $Q^{2}$ evolution of
$F_{2,{\rm CC}}$ reasonably well. As we show below, the results of $F_{2,{\rm
CC}}$ evaluated at NLO+TMC matched to the low-$Q$ extrapolation for
$Q_{0}^{2}=2$ GeV2 and $Q_{0}^{2}=4$ GeV2 according to eq. (13) agree within
$-6\%$ to $+2\%$ for $Q^{2}\gtrsim 1$ GeV2 (see Fig. 4). For $Q^{2}=0.1$ GeV2,
$F_{2,{\rm CC}}$ differs by up to $\sim 10\%$, however, because $F_{2,{\rm
CC}}^{\rm CKMT}\to 0$ as $Q^{2}\to 0$, this difference hardly impacts the
cross section evaluation.

Recently in ref. [26], low $Q^{2}$ CC structure functions (NNSF$\nu$) were
presented, modeled by a machine-learning parametrization of experimental data
for neutrino and antineutrino scattering on Ne, Fe, CaCO3 and Pb. Structure
functions for $F_{2,{\rm CC}},\ xF_{3,{\rm CC}}$ and $F_{L,{\rm CC}}$ are
provided in LHAPDF format, separately with low-$Q$ and high-$Q$ grids [60].
The low-$Q$ grid is intended for $Q^{2}\lesssim 500~{}{\rm GeV}^{2}$ and
$x\geq 10^{-3}$, while the high-$Q$ grid extends to $Q^{2}$ outside this
region. The low-$Q$ grid is thus suitable for the $x$ range for cross sections
for $E_{\nu}\lesssim 1$ TeV. Contributions from $Q^{2}>500~{}{\rm GeV}^{2}$
are less than 8% for both $\nu_{\mu}N$ and $\nu_{\tau}N$ cross sections at
$E_{\nu}=1$ TeV, and less than $\sim 1$% for their antineutrino cross
sections. Therefore, we use only the low-$Q$ grids for the NNSF structure
functions in our evaluations, and we label the results for deuterium structure
functions by NNSF$\nu$(D). For reference, the cross sections evaluated using
the low-$Q$ grid and the high-$Q$ grid agree within 1% at 10 TeV, and the
discrepancy becomes 2-3% at 1 TeV.

Figure 3:  The structure functions $F_{2,{\rm CC}}(x,Q^{2})$ (left) and $xF_{3,{\rm CC}}(x,Q^{2})$ (right) for neutrino-nucleon charged current scattering for $Q^{2}$ = 0.1, 1 and 2 $\rm GeV^{2}$. The parameterizations of the neutrino CC structure functions from CKMT-NT [42, 52], Bodek-Yang [30] and NNSF$\nu$(D) [26] are presented for comparison. The CKMT-NT structure functions are normalized at $Q_{0}^{2}$ = 2 GeV2 in the curves shown here. $\Delta_{0}$ | $\alpha_{R}$ | $a\ [{\rm GeV}^{2}]$ | $b\ [{\rm GeV}^{2}]$ | $c\ [{\rm GeV}^{2}]$ | $d\ [{\rm GeV}^{2}]$
---|---|---|---|---|---
0.07684 | 0.4150 | 0.2631 | 0.6452 | 3.5489 | 1.1170
| Process | $A$ | $B$ | $f$ |
| EM $F_{2}$ | 0.1502 | 1.2064 | 0.15 |
|  $\nu N$ $F_{2}$ | 0.5967 | 2.7145 | 0.5962 |
|  $\nu N$ $xF_{3}$ | 9.3955 $\times 10^{-3}$ | 2.4677 | 0.5962 |
|  $\bar{\nu}N$ $xF_{3}$ | 9.3955 $\times 10^{-3}$ |  $-$2.4677 | 0.5962 |
Table 1: Parameter values in Ref. [43] for CKMT parameterization of the
electromagnetic structure function $F_{2}$. The quantities $B$ and $f$ are
determined from the valence conditions at $Q^{2}=2\ {\rm GeV}^{2}$.

The two panels of Fig. 3 show a comparison of the two structure functions
$F_{2,{\rm CC}}$ and $xF_{3,{\rm CC}}$ for low $Q^{2}$ from the normalized
CKMT-NT for neutrino CC interactions, the Bodek-Yang model, and the
NNSF$\nu$(D) structure functions, each for several choices of $Q^{2}$: 0.1, 1
and 2 ${\rm GeV^{2}}$. One can see that the results from Bodek-Yang and CKMT-
NT are comparable except for $Q^{2}=0.1~{}{\rm GeV^{2}}$. The NNSF$\nu$(D)
structure functions are consistently larger than Bodek-Yang and CKMT-NT,
especially for $Q^{2}=0.1\ {\rm GeV}^{2}$. Both the Bodek-Yang and CKMT-NT
parameterizations have $F_{2,{\rm CC}}\to 0$ as $Q^{2}\to 0$ as in
electromagnetic interactions, while the NNSF$\nu$ approach does not a priori
require this. In fact, modifications of $F_{2,{\rm CC}}$ due to partial
conservation of the axial vector current (PCAC) change the $Q^{2}\to 0$ limit,
as we discuss below.

##  III Partially conserved axial vector current

There are recent investigations into modifications based on PCAC [61, 31, 32],
for which the weak structure function $F_{2,{\rm CC}}$ does not have the same
$Q^{2}\to 0$ limit as the corresponding electromagnetic structure function.
Indeed, for $F_{2,{\rm CC}}$ which includes both transverse and longitudinal
structure functions, the $Q^{2}\to 0$ limit of the longitudinal structure
function $F_{L}$ does not vanish. In ref. [61], Kulagin and Petti find that
for $F_{2,{\rm CC}}=(F_{T}+F_{L})/(1+4x^{2}m_{N}^{2}/Q^{2})$, PCAC corrections
to $F_{L}$ are

$\displaystyle F_{L}^{\rm PCAC}$ $\displaystyle=$
$\displaystyle\frac{f_{\pi}^{2}\sigma_{\pi}(W^{2})}{\pi}f_{\rm PCAC}(Q^{2})$
(14) $\displaystyle f_{\rm PCAC}(Q^{2})$ $\displaystyle=$
$\displaystyle\Biggl{(}1+\frac{Q^{2}}{M_{\rm PCAC}^{2}}\Biggr{)}^{-2}$
$\displaystyle\sigma_{\pi}$ $\displaystyle\simeq$ $\displaystyle
X(W^{2})^{\epsilon}+Y(W^{2})^{-\eta_{1}}\,.$

Here, $f_{\pi}=0.93m_{\pi}$ is the pion decay constant, $M_{\rm PCAC}=0.8$ GeV
[61]. The parameters in the $\pi p$ cross section $\sigma_{\pi}$ are
$\epsilon\simeq 0.09$, $\eta_{1}\simeq 0.36$, $X\simeq 12.1$ mb and
$Y_{1}\simeq 26.2$ mb, given $W^{2}$ in units of GeV. For $Q^{2}\to 0$ keeping
$W^{2}$ fixed, the longitudinal structure function written in terms of the
constants $a$ and $b$ in Table 1 are

$F_{L}(x,Q^{2})|_{Q^{2}=0}\simeq
A_{L}\Biggl{(}\frac{W^{2}}{a}\Biggr{)}^{\epsilon}+B_{L}\Biggl{(}\frac{W^{2}}{b}\Biggr{)}^{-\eta_{1}}\,,$
(15)

where $A_{L}\simeq 0.147$ and $B_{L}\simeq 0.423$.

Precisely how the full $x$ and $Q^{2}$ dependence of $F_{L}(x,Q^{2})$ proceeds
in weak interactions is still to be determined. To estimate how a non-zero
limit of $F_{2,{\rm CC}}(x,Q^{2})$ for $Q^{2}\to 0$ would affect the neutrino
and antineutrino cross section and to compare the NNSF$\nu$(D) structure
function, we parameterize $F_{2,{\rm CC}}^{\rm PCAC}(x,Q^{2})$ with

$\displaystyle F_{2,{\rm CC}}$ PCAC $\displaystyle=\Biggl{[}A^{\rm
PCAC}x^{-\Delta(Q^{2})}(1-x)^{n(Q^{2})+4}$ (16) $\displaystyle\times$
$\displaystyle\Biggl{(}\frac{Q^{2}}{Q^{2}+a}\Biggr{)}^{\Delta(Q^{2})}$
$\displaystyle+$ $\displaystyle B^{\rm
PCAC}x^{1-\alpha_{R}}(1-x)^{n(Q^{2})}\Biggl{(}\frac{Q^{2}}{Q^{2}+b}\Biggr{)}^{\alpha_{R}-1}$
$\displaystyle\times$ $\displaystyle\Bigl{(}1+f(1-x)\Bigr{)}\Biggr{]}f_{\rm
PCAC}(Q^{2})\ $

with $A^{\rm PCAC}=A_{L}\simeq 0.147$ and $B^{\rm PCAC}=B_{L}/(1+f)\simeq
0.265$. This form of $F_{2,{\rm CC}}^{\rm PCAC}$ reduces to eq. (15) in the
low $Q^{2}$ limit holding $W^{2}$ fixed, as long as we can approximate
$\epsilon\simeq\Delta_{0}$ and $\eta_{1}\simeq 1-\alpha_{R}$. This is not
exact, but it gives a good starting point for understanding how big the PCAC
corrections to the neutrino and antineutrino cross sections will be. We keep
$F_{1,{\rm CC}}$ and $F_{3,{\rm CC}}$ as above. With this approximation, the
PCAC correction to $F_{2,{\rm CC}}$ can be almost $\sim 10\%$ for some values
of $x$ at $Q^{2}=2$ GeV2, so we take the normalization scale $Q_{0}^{2}=4$
GeV2 where PCAC corrections are smaller, and we make the following
substitution for $Q^{2}<4$ GeV2 to include PCAC corrections. We set

$\displaystyle F_{2,{\rm CC}}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\Biggl{[}F_{2,{\rm CC}}^{\rm CKMT}(x,Q^{2})+F_{2,{\rm CC}}^{\rm
PCAC}(x,Q^{2})\Biggr{]}$ (17) $\displaystyle\times$
$\displaystyle\frac{F_{2,{\rm CC}}^{\rm NLO+TMC}(x,Q_{0}^{2})}{F_{2,{\rm
CC}}^{\rm CKMT}(x,Q_{0}^{2})}\,,$

and we designate the structure function normalized this way as CKMT+PCAC-NT.

Figure 4:  Upper: The structure functions $F_{2,{\rm CC}}(x,Q^{2})$ for
neutrino-nucleon charged current scattering for $Q^{2}$ = 0.1, 1 and 2 $\rm
GeV^{2}$ with the CKMT-NT form of the structure function (dashed) extrapolated
below $Q_{0}^{2}$ normalized according to eq. (13) and the CKMT+PCAC-NT
(solid) normalized according to eq. (17), both for $Q_{0}^{2}=4$ GeV2. For
$Q^{2}=2$ GeV2, the structure function for NNSF$\nu$(D) (dot-dashed) [26] and
for NLO+TMC (dotted) are presented for comparison. Lower: The structure
functions $F_{2,{\rm CC}}(x,Q^{2})$ with CKMT-NT for two normalization scales,
$Q_{0}^{2}$ = 2 and 4 GeV2, and the the ratio of the CKMT-NT results with
$Q_{0}^{2}$ = 4 GeV2 to $Q_{0}^{2}$ = 2 GeV2, for $Q^{2}$ = 0.1, 1 and 2 $\rm
GeV^{2}$.

Figure 4 shows a comparison of $F_{2,{\rm CC}}(x,Q^{2})$ for normalized CKMT
(eq. (13), CKMT-NT) and normalized CKMT+PCAC (eq. (17), CKMT+PCAC-NT), both
with $Q_{0}^{2}=4$ GeV2, for $Q^{2}$ = 0.1, 1 and 2 $\rm GeV^{2}$. Also shown
for reference are the NLO+TMC and NNSF$\nu$(D) structure functions for
$Q^{2}=2$ GeV2. Even with the PCAC correction, $F_{2,{\rm CC}}(x,Q^{2})$ in
our approach (CKMT+PCAC-NT) is much smaller than the NNSF$\nu$(D) structure
function for most of the range of $x$.

##  IV Neutrino cross sections

In this section, the impacts of $W$ and $Q^{2}$ kinematic regions on the
neutrino DIS CC cross sections are investigated using both the CKMT-NT and
CKMT+PCAC-NT extrapolations. All cross sections are for isoscalar nucleon
targets unless specified. We evaluate the $\nu_{\mu}$, $\bar{\nu}_{\mu}$,
$\nu_{\tau}$ and $\bar{\nu}_{\tau}$ CC cross sections with a default value of
hadronic final state invariant mass $W>W_{\rm min}=1.4$ GeV.

Results for $\sigma_{\rm CC}/E_{\nu}$ are shown in Fig. 5 for $\nu_{\mu}N$ and
$\nu_{\tau}N$ CC scattering (upper panel) and $\bar{\nu}_{\mu}N$ and
$\bar{\nu}_{\tau}N$ CC scattering (lower panel). Cross sections for CKMT-NT
evaluated with $Q_{0}^{2}=4\ {\rm GeV}^{2}$ agree with cross sections
evaluated with $Q_{0}^{2}=2$ GeV2 to within $1\%$ for $\nu_{\mu}N$ CC
scattering and to within $2\%$ for $\bar{\nu}_{\mu}N$ CC scattering, with
similar results for $\nu_{\tau}N$ and $\bar{\nu}_{\tau}N$ CC scattering with
$E_{\nu}\gtrsim 10$ GeV.

Figure 5:  The charged current cross sections per nucleon of muon neutrino and
tau neutrino (upper) and their antineutrino (lower) scattering with isoscalar
nucleons for $W_{\rm min}$ = 1.4 GeV for CKMT-NT and CKMT+PCAC-NT evaluations.
The predictions evaluated with the Bodek-Yang and NNSF$\nu$ with its error
band are presented for comparison.

Also shown in Fig. 5 are the cross sections evaluated using the Bodek-Yang
model using GRV98LO PDFs (dot-dashed) and with the NNSF$\nu$(D) structure
functions. There is very good agreement for $\sigma_{\rm CC}/E_{\nu}$ between
the Bodek-Yang and CKMT-NT evaluations. The CKMT+PCAC-NT cross sections are
larger than the CKMT-NT evaluations by $\sim 4(7)\%$ for $\nu N$ scattering
and $\sim 10(20)\%$ for $\bar{\nu}N$ scattering at $E_{\nu}=10$ GeV for muon
neutrinos (tau neutrinos). These cross sections are closer to each other at
higher energies, where larger $Q^{2}$ values contribute more to the cross
sections.

Given that the NNSF$\nu$(D) structure functions shown in Fig. 3 are higher
than the other curves, it is not surprising that the NNSF$\nu$(D) cross
sections are also larger than the other evaluations in Fig. 5. As noted in
ref. [26], the NNSF$\nu$(D) uncertainty bands for the CC cross section do not
overlap the Bodek-Yang cross section for $\nu_{\mu}N$ scattering for $W>2$
GeV. This is also the case for $W>1.4$ GeV. As shown in Fig. 5, the
NNSF$\nu$(D) uncertainty bands overlap neither our cross sections with the
CKMT+PCAC-NT nor the Bodek-Yang cross sections.

Figure 6:  The ratios of the charged current cross sections for muon neutrino
and antineutrino (upper) and tau neutrino and antineutrino (lower) scattering
with isoscalar nucleons as function of energy for the values of the minimum
hadronic final state invariant mass $W_{\rm min}$ = 1.4, 1.7 and 2 GeV to the
cross sections with $W_{\rm min}=m_{N}+m_{\pi}$ using CKMT+PCAC-NT with
$Q_{0}^{2}=4$ GeV2.

The remaining figures in this section come from evaluations with CKMT+PCAC-NT
with $Q_{0}^{2}=4$ GeV2. Fig. 6 shows the ratios of charged current cross
sections for $W_{\rm min}$ = 1.4 GeV, 1.7 GeV and 2.0 GeV with respect to the
cross section with $W_{\rm min}=m_{N}+m_{\pi}$, the minimum hadronic invariant
mass for inelastic scattering. The figure shows the results for muon neutrinos
and antineutrinos in the upper panel, and for tau neutrinos and antineutrinos
in the lower panel. The effect of $W_{\rm min}$ appears at energies below a
few hundred GeV, and it is more significant at lower energies. The cuts on
$W_{\rm min}$ reduce more the cross sections for tau neutrinos than the cross
sections for muon neutrinos for both neutrinos and antineutrinos. While the
cross sections with $W_{\rm min}=2.0~{}{\rm GeV}$, for example, are suppressed
by 3% compared to the results with $W_{\rm min}=m_{N}+m_{\pi}$ at
$E_{\nu}=100~{}{\rm GeV}$ for both muon neutrinos and tau neutrinos, the
suppression at 10 GeV is approximately 27% for muon neutrinos and 64% for tau
neutrinos. In addition, the impact of $W_{\rm min}$ is larger on the
antineutrino cross sections than the neutrino cross sections. The
corresponding suppression for $\bar{\nu}_{\mu}$ and $\bar{\nu}_{\tau}$ CC
cross sections is 6 – 7% at 100 GeV, and about 47% and 86% at 10 GeV,
respectively. Given the fact that for $\Delta(1232)$ resonance production,
$1.1~{}{\rm GeV}\lesssim W\lesssim 1.4~{}{\rm GeV}$, one can use $W>W_{\rm
min}=1.4$ GeV to exclude the $\Delta$ resonance region. For $W>W_{\rm
min}=1.4$ GeV, the cross sections are suppressed by $\sim 10$ (15)% and $\sim
25$ (40)% with respect to the results with $W_{\rm min}=m_{N}+m_{\pi}$ for
$\nu_{\mu}$ and $\nu_{\tau}$ (antineutrinos) at $E_{\nu}=10~{}{\rm GeV}$, and
such impacts are reduced to less than $\sim 2\%$ at $E_{\nu}=100~{}{\rm GeV}$.
The dependence of the cross sections on $W_{\rm min}$ is not very sensitive to
whether or not $F_{2,{\rm CC}}^{\rm PCAC}$ is included.

Figure 7:  The ratio of the CC cross sections with the cut of $Q_{\rm
min}^{2}$ = 1 and $1.3^{2}$ GeV2 to the ones with full $Q^{2}$ range for the
scattering of muon neutrino and antineutrino (upper), and tau neutrino and
antineutrino (lower) with nucleon for $W_{\rm min}=1.4~{}{\rm GeV}$ using
CKMT+PCAC-NT with $Q_{0}^{2}=4$ GeV2.

The $Q^{2}$ dependence of the neutrino and antineutrino CC cross sections are
shown in Fig. 7. In the figure, the ratios of the cross sections evaluated
with the minimum values of $Q^{2}_{\rm min}$ = 1 GeV2 and $1.69~{}{\rm
GeV}^{2}$ to those with full $Q^{2}$ range are presented for muon neutrinos
(upper) and tau neutrinos (lower) with $W_{\rm min}=1.4$ GeV as in Fig. 5.
Considering eq. (4), $W>W_{\rm min}$ enforces a minimum $Q^{2}$ with a value
greater than 0 as a function of parton momentum fraction $x$. Nonetheless, we
denote $Q_{\rm min}^{2}=0$ to indicate that there are no additional
restrictions on $Q^{2}$ beyond the requirement that $W>W_{\rm min}$. The value
of 1.3 GeV for $Q$ is selected from the minimum value of the nCTEQ15 PDFs used
in our evaluation.

From the upper panel of the figure for muon neutrinos (upper) and
antineutrinos (lower), one can see that the impact of $Q^{2}_{\rm min}$
appears over a wider energy range with more significant effects than the
impacts of $W_{\rm min}$. The figure indicates that for example, at
$E_{\nu}=10~{}{\rm GeV}$, 40% and 63% of the muon neutrino and antineutrino
cross sections come from the region of $Q^{2}\lesssim 1.69~{}{\rm GeV}^{2}$.
Such large effect is reduced with increasing energy, however, even for
$E_{\nu}$ = 100 GeV, the corresponding contributions to the $\nu_{\mu}N$ and
$\bar{\nu}_{\mu}N$ cross sections from $Q^{2}<Q^{2}_{\rm min}\simeq
1.69~{}{\rm GeV^{2}}$ are about 8% and 14%, respectively.

For tau neutrino scattering cross sections, the contributions from $Q^{2}\leq
1.69~{}{\rm GeV^{2}}$ are comparable to the muon neutrino case at $E_{\nu}$ =
100 GeV. On the other hand, the respective fractions of the cross sections at
10 GeV become 30% and 50%, which shows the impact of the cutoff on $Q^{2}$ is
lower than the impact on the muon neutrino and antineutrino cross sections
when $W_{\rm min}=1.4~{}{\rm GeV}$. This lower impact of $Q^{2}$ for tau
neutrinos and antineutrinos at low energies reflects kinematic constraints on
tau production given $W_{\rm min}$ [59], as also shown in Fig. 8.

Figure 8:  The ratio of the CC neutrino-isoscalar nucleon cross sections with
a minimum value of $Q^{2}$ to the cross section with all $Q^{2}$, as function
of $Q^{2}_{\rm min}$. The upper row shows the results for muon neutrinos
(left) and tau neutrinos (right) for the $W_{\rm min}=m_{N}+m_{\pi}$, 1.4 and
2 GeV, whereas the results for neutrino and antineutrino for $W_{\rm min}$ =
1.4 GeV are presented in the lower panels. Here, we use CKMT+PCAC-NT with
$Q_{0}^{2}=4$ GeV2.

Figure 8 presents the ratios as in Fig. 7 in more detail for selected
energies. The CC cross sections normalized to the cross sections with full
$Q^{2}$ range are shown, now as function of $Q^{2}_{\rm min}$. We show the
results for several values of $W_{\rm min}$ and for $E_{\nu}$ = 10, 50 and 100
GeV for muon neutrinos in the left panels and for tau neutrinos in the right
panels. From the upper panels, one can see that the impact of kinematic
restrictions appears more clearly at lower energies. First, the results with
the different values of $W_{\rm min}$ are distinguishable for
$E_{\nu}=10~{}{\rm GeV}$ while the corresponding results for
$E_{\nu}=100~{}{\rm GeV}$ almost overlap. The effect of the cutoff on $Q^{2}$
is more appreciable. For muon neutrinos, for example, setting $Q^{2}_{\rm
min}=1~{}{\rm GeV^{2}}$ suppresses the cross section by 23% at 10 GeV whereas
it reduces only 4% at 100 GeV when $W_{\rm min}$ = 1.4 GeV. For tau neutrinos,
the fractions with the same conditions are 12% and 3% at 10 GeV and 100 GeV,
respectively. In case of tau neutrinos, the influence of kinematic conditions
for tau production and $W_{\rm min}$ is included as shown in the low energy
results obviously in particular for 10 GeV. As shown in Fig. 6, the
restrictions for tau production and the cutoff on $W_{\rm min}$ already
significantly suppress the tau neutrino cross sections with full $Q^{2}$
range, and consequently the low $Q^{2}$ impact on the ratios diminishes as
$W_{\rm min}$ increases. In the lower panels of Fig. 8, the ratios of the CC
cross sections for antineutrinos are compared with the results for neutrinos
for $W_{\rm min}$ = 1.4 GeV. It is evident that the cutoff on the $Q^{2}_{\rm
min}$ has more significant impact on the antineutrino cross sections than on
the neutrino cross sections for both muon neutrinos and tau neutrinos. In
addition, we evaluated these ratios with different PDF sets, HERAPDF20 [62],
NNPDF4.0 [63], MSHT [64], and checked that the results are largely independent
of the PDF choice. For $\sigma_{cc}(Q^{2}_{\rm min})/\sigma_{cc}\gtrsim$ 0.1,
the ratios with the different PDFs differ by less than 10% from the one with
our default PDFs, nCTEQ15.

##  V Discussion

We have investigated contributions of the $W$ and $Q^{2}$ outside the DIS
region to the inelastic CC cross sections of neutrinos and antineutrinos
incident on isoscalar targets, using theoretically motivated and
phenomenological parameterizations of the low-$Q^{2}$ weak structure functions
as a starting point to developing a better understanding of this interesting
kinematic regime. As mentioned in section II, the DIS region is generally
defined as $W>2$ GeV and $Q^{2}>1~{}{\rm GeV^{2}}$, whereas in GENIE [36],
$W_{\rm min}=1.7$ GeV is used for the DIS contribution to the cross section.
Our interest in the low $W$ and low $Q^{2}$ region is in part motivated by
current and up-coming experiments to detect neutrinos produced in the forward
direction from LHC collisions with neutrino energies up to a few TeV.

We have focused here on the relative sizes of contributions of different
kinematic regions to the neutrino-nucleon CC cross sections. In our
evaluations using the CKMT extrapolation of $F_{2,{\rm EM}}$ [42, 43] adapted
to $F_{2,{\rm CC}}$ [52] and matched to NLO QCD with target mass corrections,
together with an approximation to the PCAC correction [61], the contributions
to the CC neutrino (antineutrino) cross sections from the range of
$m_{N}+m_{\pi}<W<2~{}{\rm GeV}$ are less than $\sim 3(7)\%$ at $E_{\nu}=100$
GeV for both muon neutrinos and tau neutrinos, and they are even less at
higher energies. The low-$Q^{2}$ region contributes a small fraction of
neutrino CC cross sections for $E_{\nu}>100$ GeV. For $W_{\rm min}=2~{}{\rm
GeV}$, 3(6)% of the $\nu_{\mu}(\bar{\nu}_{\mu})$ CC cross sections comes from
$Q^{2}<1~{}{\rm GeV^{2}}$ at $E_{\nu}=100~{}{\rm GeV}$. This is comparable to
results with $W_{\rm min}=1.4~{}{\rm GeV}$ as can be seen in Fig. 8. Slightly
lower CC cross section fractions come from $Q^{2}<1$ GeV2 for $\nu_{\tau}$ and
$\bar{\nu}_{\tau}$. Thus, our conclusion is that the impact of low $W$ and
$Q^{2}$ outside the range for DIS interaction on the neutrino and antineutrino
cross sections is at most a few percent level for a neutrino energy of 100
GeV, and negligible for neutrino energies above a few hundred GeV, the main
energy range that can be explored by the FPF.

Our CC cross sections differ from predictions using NNSF$\nu$(D) structure
functions [26]. Figure 9 shows $\sigma^{\rm CC}/A/E_{\nu}$ with theoretical
evaluations for $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ CC scattering on isoscalar
nucleons and data from neutrino experiments with a range of targets [65, 66,
67, 68, 69, 70, 71, 72, 73]. The dot-dashed curve shows the NNSF$\nu$(D) cross
section for $W_{\min}=1.4$ GeV and the dashed curve shows our CKMT+PCAC-NT
result, also with $W_{\min}=1.4$ GeV. For reference, we also show the CKMT-NT
and CKMT+PCAC-NT for $W_{\min}=m_{N}+m_{\pi}$, where the latter (solid curve)
represents our best approximation of the full CC cross section even at
$E_{\nu}=10$ GeV. For $E_{\nu}=200$ GeV where $W_{\rm min}$ is not important,
the NNSF$\nu$(D) cross section is a factor of $\sim$ 1.1 larger for
$\nu_{\mu}N$ CC scattering and a factor of $\sim 1.2$ larger for
$\bar{\nu}_{\mu}N$ CC scattering than the CKMT+PCAC-NT result. Cross sections
evaluated using the Bodek-Yang model agree well with the CKMT+PCAC-NT
evaluations, as shown in Fig. 5.

Figure 9:  The $\nu_{\mu}N$ and $\bar{\nu}_{\mu}N$ CC cross sections for
$W_{\rm min}=1.4$ GeV using CKMT+PCAC-NT (dashed) and NNSF$\nu$(D) structure
functions for deuterium (dot-dashed), and for $W_{\rm min}=m_{N}+m_{\pi}$ for
CKMT-NT (dotted) and CKMT+PCAC-NT normalized according to eq. (17) (solid).
Data are shown for neutrino experiments using a range of targets [65, 66, 67,
68, 69, 70, 71, 72, 73]. The light gray dashed lines are at 0.675 and 0.33.

Figure 10:  Upper: the structure functions $F_{2,{\rm CC}}(x,Q^{2})$ for
neutrino CC scattering with iron (Fe) for $Q^{2}$ = 0.1, 1 and 2 $\rm GeV^{2}$
with the CKMT+PCAC-NT (solid) and NNSF$\nu$(Fe) (dot-dashed). Lower: the
$\nu_{\mu}N$ and $\bar{\nu}_{\mu}N$ CC cross sections for $W_{\rm min}=1.4$
GeV using NNSF$\nu$(Fe) structure functions (dot-dashed) and CKMT+PCAC-NT
using nCTEQ15 iron PDFs, normalized according to eq. (17) (solid). Also shown
are the Bodek-Yang cross sections for comparison.  Figure 11:  A comparison of
the $\nu_{\mu}A$ and $\bar{\nu}_{\mu}A$ CC cross sections divided by incident
energy per $A$ for $W_{\rm min}=2$ GeV for $A=2$ (isoscalar nucleons, solid)
and $A=56$ (Fe, dashed) for CKMT+PCAT-NT and NNSF$\nu$ structure functions.
The results with NNSF$\nu$ are shown with uncertainty bands. The data are
shown as in Fig. 9.

As noted, the data shown in Fig. 9 come from neutrino experiments using a
range of targets. For example, the data for energies up to $\sim 350$ GeV from
CCFR [70] and NuTeV [65] come from neutrino and antineutrino scattering on
iron. The NNSF$\nu$(Fe) structure functions are more similar to the CKMT+PCAC-
NT structure functions evaluated with nCTEQ15 iron PDFs than the NNSF$\nu$(D)
structure functions are to the CKMT+PCAC-NT isoscalar nucleon structure
functions. The upper panel of Fig. 10 shows $F_{2,{\rm CC}}$ for iron, and the
lower panel shows the cross sections per nucleon, scaled by incident energy,
for $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ CC scattering given $W_{\rm min}=1.4$
GeV. The dashed curves show the same quantities for CKMT+PCAC-NT from
scattering with isoscalar nucleons. The agreement between the evaluations
using CKMT+PCAC-NT and NNSF$\nu$(Fe) for iron targets is better than for
isoscalar nucleon targets.

A similar agreement between CKMT+PCAC-NT and NNSF$\nu$(Fe) for iron targets is
seen in Fig. 11 where now $W_{\rm min}=2$ GeV. This is the minimum $W$ for the
NNSF$\nu$ structure function analysis. The NNSF$\nu$ error bands for isoscalar
and iron targets are also shown.

The disagreements between isoscalar nucleon results for CKMT+PCAC-NT and
NNSF$\nu$(D) CC cross sections persist for $W_{\rm min}=2$ GeV, even at high
energies where one expects that the PDFs rather than the low-$Q^{2}$ structure
function extrapolations account for most of the cross section. In our
approach, PDF uncertainties in the CC cross section cannot account for the
discrepancies. For example, for $E_{\nu}=100$ GeV using the default PDFs, PDF
uncertainties are $\sim 2\%$ for $\nu_{\mu}N$ and $\sim 4\%$ for
$\bar{\nu}_{\mu}N$ CC cross sections with $W_{\rm min}=2$ GeV. We also
compared with the results evaluated using the central set of other PDFs,
HERAPDF20 [62], NNPDF4.0 [63], MSHT [64], which agree with our results with
default PDFs, nCTEQ15 for the CKMT+PCAC-NT, to within $\pm$ 4% for both muon
neutrinos and antineutrinos for 10 GeV $\leq E_{\nu}\leq 1$ TeV.

The fits of NNSF$\nu$ to neutrino and antineutrino data are from scattering
with Ne, CaCO3, Fe and Pb targets. The good agreement between results for
scattering with Fe may suggest that the NNSF$\nu$ extrapolations to $A=2$ have
larger uncertainties than described in ref. [26]. One test of the NNSF
extrapolation to low $A$ could be made by performing an extraction of the
electromagnetic structure function (NNSF$\gamma$) for $A$ values comparable to
those used for NNSF$\nu$, then to compare its extrapolation to smaller $A$
with small $A$ electromagnetic scattering data. In any case, the CC cross
section comparisons presented here emphasize the value of new neutrino and
antineutrino scattering data in the 10 GeV – 1 TeV energy range.

In summary, using our approach to extrapolate PDF-based structure functions to
low-$Q^{2}$, for neutrino energies less than 100 GeV, the kinematic regions of
$W$ and $Q^{2}$ are more important than for higher incident neutrino energies,
the impact of which can be more than 50% for $E_{\nu}={\cal O}(10)$ GeV. As
indicated above, the low-$Q^{2}$ structure functions used in this work are
theoretically motivated, tied to the electromagnetic structure functions and
the partial conservation of the axial vector current, but more work studying
the low-$Q^{2}$ behavior of weak structure functions is merited and has begun
(e.g., in ref. [32]).

On the experimental side, neutrino experiments at the FPF will be able to
measure a large number of neutrinos up to a few TeV energies. The majority of
the neutrino events will be distributed in the energies above 100 GeV. An
estimate of the number of interacting $\nu_{\mu}+\bar{\nu}_{\mu}$ for the
FLArE detector at the FPF, with 3000 fb-1, is of order $1.5\times 10^{5}$ for
$E_{\nu}=100$ GeV$-1$ TeV [74]. However, simulations show that there can be a
considerable number of events at lower energies. For example, again for FLArE
at the FPF, of order $2\times 10^{4}$ $\nu_{\mu}+\bar{\nu}_{\mu}$ interactions
are anticipated for neutrino energies in the range of $50-100$ GeV [74]. An
estimate of 2,000 FLArE CC events in the resonance region, and an even larger
number of CC events in FASER$\nu$2, will be relevant for the SIS interactions
(see section 7.3 of ref. [14] and ref. [28]). From a combined analysis of the
neutrino fluxes and cross sections at the FPF [23], cross section measurements
will provide a unique opportunities to investigate neutrino interactions in
the kinematic region for SIS and DIS and at low $Q^{2}$, complementary to the
cross section analyses of DUNE in the future.

###### Acknowledgements.

This work is supported in part by US Department of Energy grants DE-SC-0010113
(MHR) and the National Research Foundation of Korea (NRF) grant funded by the
Korea government through Ministry of Science and ICT grant 2021R1A2C1009296
(YSJ). We thank F. Kling and A. Garcia for discussions, F. Kling and M.
Bustamante for providing the data files for the experiments in Fig. 9, and A.
Garcia for providing cross sections for Fig. 11. For facilitating portions of
this research, MHR wishes to acknowledge the Center for Theoretical
Underground Physics and Related Areas (CETUP*), The Institute for Underground
Science at Sanford Underground Research Facility (SURF), and the South Dakota
Science and Technology Authority for hospitality and financial support, as
well as for providing a stimulating environment.

##  Appendix A Deep inelastic scattering cross section

We summarize the formulas for neutrino
$\nu_{\ell}(k)+N(p)\to\ell(k^{\prime})+X$ and antineutrino
$\bar{\nu}_{\ell}(k)+N(p)\to\bar{\ell}(k^{\prime})+X$ charged-current (CC)
cross sections in this appendix. The differential cross sections for neutrino
and antineutrino CC scattering can be written

$\displaystyle\frac{d^{2}\sigma^{\nu(\bar{\nu})}}{dx\ dy}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}m_{N}E_{\nu}}{\pi(1+Q^{2}/M_{W}^{2})^{2}}\Biggl{(}(y^{2}x+\frac{m_{\ell}^{2}y}{2E_{\nu}m_{N}})F_{1,{\rm
CC}}$ (18) $\displaystyle+$
$\displaystyle\left[(1-\frac{m_{\ell}^{2}}{4E_{\nu}^{2}})-(1+\frac{m_{N}x}{2E_{\nu}})y\right]F_{2,{\rm
CC}}$ $\displaystyle\pm$
$\displaystyle\left[xy(1-\frac{y}{2})-\frac{m_{\ell}^{2}y}{4E_{\nu}m_{N}}\right]F_{3,{\rm
CC}}$ $\displaystyle+$
$\displaystyle\frac{m_{\ell}^{2}(m_{\ell}^{2}+Q^{2})}{4E_{\nu}^{2}m_{N}^{2}x}F_{4,{\rm
CC}}-\frac{m_{\ell}^{2}}{E_{\nu}m_{N}}F_{5,{\rm CC}}\Biggr{)}\,,$

for $x=Q^{2}/(2p\cdot q)$ and $y=p\cdot q/p\cdot k$, given $q=k-k^{\prime}$
and $Q^{2}=-q^{2}$. The sign of the $F_{3,{\rm CC}}$ term is $+(-)$ for
neutrinos (antineutrinos).

The CC structure function decomposition for $\nu p$ and $\bar{\nu}p$
scattering, at leading order in QCD for two generations of massless quarks, in
terms of proton parton distribution functions labeled by their flavors are

$\displaystyle F_{2,CC}$ $\displaystyle=$ $\displaystyle
2x(d+s+\bar{u}+\bar{c})\quad(\nu p)$ (19) $\displaystyle F_{3,CC}$
$\displaystyle=$ $\displaystyle 2(d+s-\bar{u}-\bar{c})\quad(\nu p)$ (20)
$\displaystyle F_{2,CC}$ $\displaystyle=$ $\displaystyle
2x(u+c+\bar{d}+\bar{s})\quad(\bar{\nu}p)$ (21) $\displaystyle F_{3,CC}$
$\displaystyle=$ $\displaystyle 2(u+c-\bar{d}-\bar{s})\quad(\bar{\nu}p)$ (22)

where the structure functions and parton distribution functions depend on $x$
and $Q^{2}$. For isoscalar nucleon targets, we equate the up quark PDF in the
neutron with the down quark PDF in the proton, and similarly for the up
antiquark, down quark and down antiquark PDFs in the neutron. Next-to-leading
order QCD corrections, quark mass corrections and target mass corrections
modify the expressions, as summarized in e.g, ref. [57, 39].

The limits of integration for eq. (18) are

$\displaystyle\frac{m_{\ell}^{2}}{2m_{N}(E_{\nu}-m_{\ell})}\ \leq\ x\ \leq\ 1\
,$ (23) $\displaystyle a\ -\ b\ \leq\ y\ \leq\ a\ +\ b\ ,$ (24)

where $a$ and $b$ are defined by

$\displaystyle a$ $\displaystyle=$
$\displaystyle\Biggl{[}1-m_{\ell}^{2}\Biggl{(}\frac{1}{2m_{N}E_{\nu}x}+\frac{1}{2E_{\nu}^{2}}\Biggr{)}\Biggr{]}/(2+m_{N}x/E_{\nu})\
,$ $\displaystyle b$ $\displaystyle=$
$\displaystyle\Biggl{[}\Biggl{(}1-\frac{m_{\ell}^{2}}{2m_{N}E_{\nu}x}\Biggr{)}^{2}-\frac{m_{\ell}^{2}}{E_{\nu}^{2}}\Biggr{]}^{1/2}/(2+m_{N}x/E_{\nu})\
.$

##  Appendix B Cross section tables

Table 2 – 4 show the charged current cross sections of neutrino scattering
with nucleons for muon neutrinos, tau neutrinos and their antineutrinos. We
present the results evaluated using CKMT+PCAC-NT and the nCTEQ15 PDFs for
$W_{\rm min}=m_{N}+n_{\pi}$, 1.4 and 2 GeV.

$E_{\nu}$ [GeV] |  $\sigma^{cc}(\nu N)/E_{\nu}~{}[10^{-38}{\rm cm^{2}/GeV}]$, $W_{\rm min}$ = $m_{N}+m_{\pi}$
---|---
$\nu_{\mu}$ | $\bar{\nu}_{\mu}$ | $\nu_{\tau}$ | $\bar{\nu}_{\tau}$
1.000E+01 | 6.818E-01 | 3.006E-01 | 1.335E-01 | 6.223E-02
1.259E+01 | 6.850E-01 | 3.062E-01 | 1.800E-01 | 8.408E-02
1.585E+01 | 6.847E-01 | 3.108E-01 | 2.282E-01 | 1.066E-01
1.995E+01 | 6.854E-01 | 3.152E-01 | 2.758E-01 | 1.286E-01
2.512E+01 | 6.849E-01 | 3.185E-01 | 3.215E-01 | 1.499E-01
3.162E+01 | 6.849E-01 | 3.218E-01 | 3.639E-01 | 1.694E-01
3.981E+01 | 6.833E-01 | 3.237E-01 | 4.028E-01 | 1.878E-01
5.012E+01 | 6.801E-01 | 3.253E-01 | 4.369E-01 | 2.047E-01
6.310E+01 | 6.787E-01 | 3.276E-01 | 4.682E-01 | 2.200E-01
7.943E+01 | 6.770E-01 | 3.298E-01 | 4.953E-01 | 2.343E-01
1.000E+02 | 6.732E-01 | 3.305E-01 | 5.176E-01 | 2.465E-01
1.259E+02 | 6.686E-01 | 3.309E-01 | 5.359E-01 | 2.576E-01
1.585E+02 | 6.662E-01 | 3.327E-01 | 5.524E-01 | 2.675E-01
1.995E+02 | 6.613E-01 | 3.331E-01 | 5.652E-01 | 2.763E-01
2.512E+02 | 6.584E-01 | 3.340E-01 | 5.759E-01 | 2.841E-01
3.162E+02 | 6.538E-01 | 3.345E-01 | 5.834E-01 | 2.906E-01
3.981E+02 | 6.477E-01 | 3.338E-01 | 5.881E-01 | 2.959E-01
5.012E+02 | 6.421E-01 | 3.347E-01 | 5.920E-01 | 3.017E-01
6.310E+02 | 6.370E-01 | 3.348E-01 | 5.938E-01 | 3.058E-01
7.943E+02 | 6.299E-01 | 3.342E-01 | 5.933E-01 | 3.090E-01
1.000E+03 | 6.200E-01 | 3.329E-01 | 5.892E-01 | 3.107E-01
1.259E+03 | 6.102E-01 | 3.321E-01 | 5.843E-01 | 3.127E-01
1.585E+03 | 5.993E-01 | 3.296E-01 | 5.771E-01 | 3.130E-01
1.995E+03 | 5.851E-01 | 3.260E-01 | 5.669E-01 | 3.119E-01
2.512E+03 | 5.725E-01 | 3.237E-01 | 5.568E-01 | 3.114E-01
3.162E+03 | 5.591E-01 | 3.216E-01 | 5.447E-01 | 3.101E-01
3.981E+03 | 5.400E-01 | 3.157E-01 | 5.282E-01 | 3.065E-01
5.012E+03 | 5.188E-01 | 3.090E-01 | 5.092E-01 | 3.014E-01
6.310E+03 | 4.965E-01 | 3.021E-01 | 4.887E-01 | 2.954E-01
7.943E+03 | 4.729E-01 | 2.939E-01 | 4.664E-01 | 2.885E-01
1.000E+04 | 4.480E-01 | 2.857E-01 | 4.424E-01 | 2.807E-01
Table 2:  The charged current cross sections of $\nu_{\mu}$, $\bar{\nu}_{\mu}$, $\nu_{\tau}$ and $\bar{\nu}_{\tau}$ scattering with isoscalar nucleon for $W_{\rm min}$ = $m_{N}+m_{\pi}$, evaluated using the nCTEQ15 PDFs and CKMT+PCAC-NT with $Q_{0}^{2}=4$ GeV2. $E_{\nu}$ [GeV] |  $\sigma^{cc}(\nu N)/E_{\nu}~{}[10^{-38}{\rm cm^{2}/GeV}]$, $W_{\rm min}$ = 1.4 GeV
---|---
$\nu_{\mu}$ | $\bar{\nu}_{\mu}$ | $\nu_{\tau}$ | $\bar{\nu}_{\tau}$
1.000E+01 | 6.240E-01 | 2.495E-01 | 1.033E-01 | 3.768E-02
1.259E+01 | 6.385E-01 | 2.641E-01 | 1.522E-01 | 5.968E-02
1.585E+01 | 6.490E-01 | 2.768E-01 | 2.037E-01 | 8.391E-02
1.995E+01 | 6.568E-01 | 2.875E-01 | 2.544E-01 | 1.082E-01
2.512E+01 | 6.628E-01 | 2.959E-01 | 3.036E-01 | 1.320E-01
3.162E+01 | 6.665E-01 | 3.038E-01 | 3.486E-01 | 1.545E-01
3.981E+01 | 6.686E-01 | 3.092E-01 | 3.905E-01 | 1.754E-01
5.012E+01 | 6.694E-01 | 3.140E-01 | 4.271E-01 | 1.943E-01
6.310E+01 | 6.691E-01 | 3.183E-01 | 4.596E-01 | 2.115E-01
7.943E+01 | 6.699E-01 | 3.223E-01 | 4.885E-01 | 2.273E-01
1.000E+02 | 6.681E-01 | 3.247E-01 | 5.124E-01 | 2.410E-01
1.259E+02 | 6.648E-01 | 3.265E-01 | 5.320E-01 | 2.530E-01
1.585E+02 | 6.624E-01 | 3.290E-01 | 5.489E-01 | 2.640E-01
1.995E+02 | 6.588E-01 | 3.302E-01 | 5.626E-01 | 2.734E-01
2.512E+02 | 6.561E-01 | 3.318E-01 | 5.737E-01 | 2.819E-01
3.162E+02 | 6.520E-01 | 3.328E-01 | 5.818E-01 | 2.887E-01
3.981E+02 | 6.464E-01 | 3.324E-01 | 5.868E-01 | 2.945E-01
5.012E+02 | 6.412E-01 | 3.333E-01 | 5.911E-01 | 3.003E-01
6.310E+02 | 6.360E-01 | 3.339E-01 | 5.929E-01 | 3.048E-01
7.943E+02 | 6.290E-01 | 3.335E-01 | 5.924E-01 | 3.083E-01
1.000E+03 | 6.194E-01 | 3.325E-01 | 5.887E-01 | 3.101E-01
1.259E+03 | 6.099E-01 | 3.317E-01 | 5.839E-01 | 3.124E-01
1.585E+03 | 5.990E-01 | 3.293E-01 | 5.768E-01 | 3.126E-01
1.995E+03 | 5.848E-01 | 3.256E-01 | 5.666E-01 | 3.116E-01
2.512E+03 | 5.724E-01 | 3.235E-01 | 5.567E-01 | 3.111E-01
3.162E+03 | 5.589E-01 | 3.214E-01 | 5.445E-01 | 3.100E-01
3.981E+03 | 5.398E-01 | 3.155E-01 | 5.281E-01 | 3.063E-01
5.012E+03 | 5.187E-01 | 3.092E-01 | 5.091E-01 | 3.013E-01
6.310E+03 | 4.964E-01 | 3.019E-01 | 4.886E-01 | 2.953E-01
7.943E+03 | 4.729E-01 | 2.939E-01 | 4.663E-01 | 2.884E-01
1.000E+04 | 4.480E-01 | 2.857E-01 | 4.424E-01 | 2.806E-01
Table 3: The charged current cross sections of $\nu_{\mu}$, $\bar{\nu}_{\mu}$, $\nu_{\tau}$ and $\bar{\nu}_{\tau}$ scattering with isoscalar nucleon for $W_{\rm min}$ = 1.4 GeV, evaluated using the nCTEQ15 PDFs and CKMT+PCAC-NT with $Q_{0}^{2}=4$ GeV2.  $E_{\nu}$ [GeV] |  $\sigma^{cc}(\nu N)/E_{\nu}~{}[10^{-38}{\rm cm^{2}/GeV}]$, $W_{\rm min}$ = 2.0 GeV
---|---
$\nu_{\mu}$ | $\bar{\nu}_{\mu}$ | $\nu_{\tau}$ | $\bar{\nu}_{\tau}$
1.000E+01 | 4.983E-01 | 1.586E-01 | 4.849E-02 | 8.933E-03
1.259E+01 | 5.370E-01 | 1.860E-01 | 9.893E-02 | 2.484E-02
1.585E+01 | 5.684E-01 | 2.104E-01 | 1.545E-01 | 4.699E-02
1.995E+01 | 5.930E-01 | 2.321E-01 | 2.104E-01 | 7.234E-02
2.512E+01 | 6.107E-01 | 2.493E-01 | 2.652E-01 | 9.869E-02
3.162E+01 | 6.259E-01 | 2.661E-01 | 3.161E-01 | 1.249E-01
3.981E+01 | 6.368E-01 | 2.787E-01 | 3.636E-01 | 1.500E-01
5.012E+01 | 6.436E-01 | 2.891E-01 | 4.048E-01 | 1.728E-01
6.310E+01 | 6.493E-01 | 2.985E-01 | 4.415E-01 | 1.941E-01
7.943E+01 | 6.530E-01 | 3.057E-01 | 4.730E-01 | 2.123E-01
1.000E+02 | 6.552E-01 | 3.119E-01 | 5.004E-01 | 2.286E-01
1.259E+02 | 6.545E-01 | 3.158E-01 | 5.223E-01 | 2.429E-01
1.585E+02 | 6.547E-01 | 3.207E-01 | 5.412E-01 | 2.560E-01
1.995E+02 | 6.524E-01 | 3.237E-01 | 5.565E-01 | 2.671E-01
2.512E+02 | 6.509E-01 | 3.266E-01 | 5.685E-01 | 2.768E-01
3.162E+02 | 6.477E-01 | 3.282E-01 | 5.777E-01 | 2.845E-01
3.981E+02 | 6.431E-01 | 3.292E-01 | 5.835E-01 | 2.913E-01
5.012E+02 | 6.383E-01 | 3.307E-01 | 5.882E-01 | 2.978E-01
6.310E+02 | 6.340E-01 | 3.318E-01 | 5.908E-01 | 3.028E-01
7.943E+02 | 6.273E-01 | 3.317E-01 | 5.908E-01 | 3.066E-01
1.000E+03 | 6.182E-01 | 3.309E-01 | 5.875E-01 | 3.089E-01
1.259E+03 | 6.091E-01 | 3.305E-01 | 5.831E-01 | 3.114E-01
1.585E+03 | 5.983E-01 | 3.283E-01 | 5.761E-01 | 3.118E-01
1.995E+03 | 5.843E-01 | 3.249E-01 | 5.660E-01 | 3.109E-01
2.512E+03 | 5.718E-01 | 3.230E-01 | 5.561E-01 | 3.106E-01
3.162E+03 | 5.585E-01 | 3.209E-01 | 5.441E-01 | 3.095E-01
3.981E+03 | 5.395E-01 | 3.151E-01 | 5.277E-01 | 3.059E-01
5.012E+03 | 5.186E-01 | 3.090E-01 | 5.089E-01 | 3.012E-01
6.310E+03 | 4.962E-01 | 3.017E-01 | 4.884E-01 | 2.950E-01
7.943E+03 | 4.728E-01 | 2.937E-01 | 4.662E-01 | 2.882E-01
1.000E+04 | 4.479E-01 | 2.856E-01 | 4.423E-01 | 2.805E-01
Table 4: The charged current cross sections of $\nu_{\mu}$, $\bar{\nu}_{\mu}$,
$\nu_{\tau}$ and $\bar{\nu}_{\tau}$ scattering with isoscalar nucleon for
$W_{\rm min}$ = 2.0 GeV, evaluated using the nCTEQ15 PDFs and CKMT+PCAC-NT
with $Q_{0}^{2}=4$ GeV2.

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