11institutetext: 1Department of Physics, Aligarh Muslim University,
Aligarh-202002, India
2Departamento de Física Atómica, Molecular y Nuclear, and Instituto de Física
Teórica y Computacional Carlos I, Universidad de Granada, Granada 18071, Spain

# Deep inelastic (anti)neutrino-nucleus scattering

V. Ansari1 M. Sajjad Athar1 H. Haider1 email<EMAIL_ADDRESS>I. Ruiz
Simo2 S. K. Singh1 F. Zaidi1

###### Abstract

The present status of the field theoretical model studies of the deep
inelastic scattering induced by (anti)neutrino on the nuclear targets in a
wide range of Bjorken variable $x$ and four momentum transfer square $Q^{2}$,
has been reviewed Haider:2011qs ; Haider:2012nf ; Haider:2016zrk ;
Zaidi:2019mfd ; Zaidi:2019asc ; Ansari:2020xne . The effect of the
nonperturbative corrections such as target mass correction and higher twist
effects, perturbative evolution of the parton densities, nuclear medium
modifications in the nucleon structure functions, nuclear isoscalarity
corrections on the weak nuclear structure functions have been discussed. These
structure functions have been used to obtain the differential scattering cross
sections. The various nuclear medium effects like the Fermi motion, binding
energy, nucleon correlations, mesonic contributions, shadowing and
antishadowing corrections relevant in the different regions of $x$ and $Q^{2}$
have been discussed. The numerical results for the structure functions and the
cross sections are compared with some of the available experimental data
including the recent results from MINERvA. The predictions are made in argon
nuclear target which is planned to be used as a target material in DUNE at the
Fermilab.

##  1 Introduction

In recent years the need for a better understanding of neutrino interaction
cross section with nucleon and nuclear targets in the few GeV region of
(anti)neutrino energies has been emphasized in order to reduce the systematic
uncertainty in the analysis of neutrino oscillation parameters of the
Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS matrix). The lack in the
understanding of (anti)neutrino-nucleon/nucleus cross section adds to
25-30$\%$ uncertainty to the systematics and a considerable reduction in this
uncertainty is required for a precise measurement of PMNS matrix or in the
study of CP violation in the leptonic sector or in the determination of the
neutrino mass hierarchy (normal or inverted). In the few GeV region of
(anti)neutrino energies the contribution to the total (anti)neutrino cross
section comes from the quasielastic, inelastic as well as the deep inelastic
scattering processes on the nuclear targets. In this review, we shall focus on
the understanding of nuclear medium effects in the weak interaction induced
deep inelastic scattering(DIS) processes in a theoretical model using field
theory. For the details on the current status readers are referred to
Ref.SajjadAthar:2020nvy .

There is intrinsic interest to understand DIS processes induced by the charged
leptons and (anti)neutrinos on nucleons and nuclear targets as they are
important tools to study the quark parton structure of the free nucleons and
the nucleons when they are bound in a nucleus. The observation of EMC
(European Muon Collaboration) effect in the DIS of muon from iron target
showed that the quark parton distributions of free nucleons are considerably
modified when they are bound in a nucleus which was later confirmed in several
other experiments by using electron, muon, (anti)neutrino and other particle
beams on the various nuclear targets. The DIS experiments done with lepton
beams make measurements of the DIS cross sections which are expressed in terms
of the nucleon structure functions. These structure functions are determined
by making Rosenbluth-like separations of the measured cross section. The
electromagnetic DIS cross section induced by the charged leptons are generally
expressed in terms of the two structure functions
$F_{iN}^{\text{EM}}(x,Q^{2})$ $(i=1,2)$, which are functions of the Bjorken
scaling variable $x=\frac{Q^{2}}{2M_{N}\nu}$ (0 $\leq\;x\;\leq\;$1); and
$Q^{2}$ (where $Q^{2}\geq$ 0, is the four momentum transfer square and $\nu$
is the energy transferred to the target i.e. $\nu=E_{\nu}-E_{l}$,
$E_{\nu}(E_{l})$ is the energy of the incoming(outgoing) lepton, and $M_{N}$
is the nucleon mass). For the weak DIS process induced by (anti)neutrinos the
cross sections are given in terms of three structure functions
$F_{iN}^{\text{Weak}}(x,Q^{2})$ $(i=1,2,3)$, in the limit of the vanishing
lepton mass. However, in the case of DIS induced by the weak charged current
of $\nu_{\mu}$ and $\nu_{\tau}$, where the lepton mass in the final state
could be non-negligible as compared to $Q^{2}$ in some kinematic regions, then
two additional structure functions $F_{4N}^{Weak}(x,Q^{2})$ and
$F_{5N}^{Weak}(x,Q^{2})$ also contribute to the cross sections. In the exact
limit of Bjorken scaling i.e. $\nu\rightarrow\infty$,
$Q^{2}\rightarrow\infty$, such that $x=\frac{Q^{2}}{2M_{N}\nu}$, remains
fixed, the structure functions scale and become functions of only one variable
$x$. These structure functions are not all independent when calculated in the
quark-parton model and satisfy certain relations given by Callan-Gross
Callan:1969uq  and Albright-Jarlskog Albright:1974ts . Consequently, the
electromagnetic cross sections are given in terms of only one structure
function chosen to be $F_{2N}^{EM}(x,Q^{2})$ while the weak cross sections are
given in terms of two structure functions taken to be $F_{2N}^{Weak}(x,Q^{2})$
and $F_{3N}^{Weak}(x,Q^{2})$.

The study of the nucleon structure functions gives important information about
the structure of the nucleon and provides opportunity to test the predictions
of the perturbative Quantum Chromodynamics(QCD). Depending upon the kinematic
region of the centre of mass energy ($W$) and the four momentum transfer
square($Q^{2}$), it can also provide some important information about the non-
perturbative QCD. In case of the electromagnetic(EM) DIS reactions induced by
electrons and muons there is considerable experimental data on the EM nucleon
structure functions viz. $F_{1N}^{EM}(x,Q^{2})$ and $F_{2N}^{EM}(x,Q^{2})$ and
the nuclear structure functions $F_{1A}^{EM}(x,Q^{2})$ and
$F_{2A}^{EM}(x,Q^{2})$ enabling us to study the nuclear medium effects on the
electromagnetic nucleon structure functions by making a comparative study.
This is not so in the case of weak nucleon structure function
$F_{1N}^{Weak}(x,Q^{2})$, $F_{2N}^{Weak}(x,Q^{2})$ and
$F_{3N}^{Weak}(x,Q^{2})$, where there is almost no experimental data on free
nucleons. The weak nucleon structure functions have to be extracted from the
(anti)neutrino DIS experiment on heavy nuclear targets like freon, freon-
propane, etc. MINERvA at Fermilab Mousseau:2016snl  has performed experiment
with $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ beams in the wide range of $x$ and
$Q^{2}$ using several nuclear targets like carbon, iron, lead, etc. and the
aim is to understand nuclear medium effects by doing EMC kind of measurements.
Theoretically, a better understanding of nuclear medium effects on the weak
structure functions is also required.

Generally there are two approaches in order to understand the medium effects
in the weak nuclear structure functions, one is phenomenological and the other
is theoretical. In the phenomenological analysis, the general approach is that
nuclear parton distribution functions(PDFs ) are obtained using the charged
lepton-nucleus scattering data and analyzing the ratio of the structure
functions e.g. $\frac{F_{2A}^{EM}}{F_{2A^{\prime}}^{EM}}$,
$\frac{F_{2A}^{EM}}{F_{2D}^{EM}}$, where $A,A^{\prime}$ represent any two
nuclei and $D$ stands for the deuteron, nuclear correction factor is
determined. The same correction factor is then used for the weak structure
functions $F_{1A}^{Weak}(x,Q^{2})$ and $F_{2A}^{Weak}(x,Q^{2})$. For
$F_{3A}^{\text{Weak}}(x,Q^{2})$, the information is inferred from
(anti)neutrino scattering data Berge:1989hr ; Oltman:1992pq ; Tzanov:2005kr .
The other phenomenological approach is to directly extract the nuclear PDFs by
analyzing the experimental data i.e. without using nucleon PDFs or nuclear
correction factor. This approach has been recently used by nCTEQ
Kovarik:2015cma  group in getting $F_{2A}^{EM}(x,Q^{2})$,
$F_{2A}^{Weak}(x,Q^{2})$ and $F_{3A}^{Weak}(x,Q^{2})$ nuclear structure
functions by analyzing together the charged lepton-$A$ DIS data and Drell-Yan
$p$-$A$ data sets, and separately analyzing $\nu(\bar{\nu})-A$ DIS data sets.
Their observation is that the nuclear medium effects on $F_{2A}^{EM}(x,Q^{2})$
in electromagnetic interaction are different from $F_{2A}^{Weak}(x,Q^{2})$ in
the weak interaction specially at low $x$.

On the other hand, theoretically, there have been very few calculations to
study nuclear medium effects in the weak structure functions. One is by us
(Aligarh-Valencia collaboration Haider:2011qs ; Haider:2012nf ; Haider:2016zrk
; Zaidi:2019mfd ; Zaidi:2019asc ; Ansari:2020xne ; SajjadAthar:2009cr ), and
the other is by Kulagin and Petti Kulagin:2007ju ; Kulagin:2004ie . The
present review is based on our theoretical works performed in the last several
years Haider:2011qs ; Haider:2012nf ; Haider:2016zrk ; Zaidi:2019mfd ;
Zaidi:2019asc ; Ansari:2020xne ; SajjadAthar:2009cr , on the electromagnetic
and weak interaction induced DIS processes on the free nucleon and nuclear
target in the wide region of $x$ and $Q^{2}$. Recently we have extended our
study for $\nu_{\tau}/\bar{\nu}_{\tau}$ scattering on the free nucleon and
obtained the structure functions as well as the differential and total
scattering cross sections by including various perturbative and non-
perturbative effects Ansari:2020xne . Work is in progress to understand
nuclear medium effects in $\nu_{\tau}(\bar{\nu}_{\tau})$-nucleus interactions
and will be reported elsewhere.

In the present paper, we review the work on the nuclear medium effects in the
structure functions using a microscopic approach based on field theoretical
formalism Marco:1995vb . A relativistic nucleon spectral function has been
used to describe the energy and momentum distribution of the nucleons in
nuclei FernandezdeCordoba:1991wf . This is obtained by using the Lehmann’s
representation for the relativistic nucleon propagator and nuclear many body
theory is used to calculate it for an interacting Fermi sea in nuclear matter.
A local density approximation is then applied to translate these results to
finite nuclei. Furthermore, we include the contributions of meson clouds and
include the pion and rho contributions in a many body field theoretical
approach. For the shadowing and anti-shadowing corrections which have been
found to be effective in the low region of $x$($x\leq 0.2$), we follow the
works of Kulagin and Petti Kulagin:2007ju ; Kulagin:2004ie . In the present
work the numerical results for various structure functions and the cross
sections have been presented and compared with experiments. Predictions have
been made for ${}^{40}Ar$, relevant for the upcoming DUNE
experimentAbi:2018alz  at the Fermilab.

The plan of the paper is the following. In section-2, we describe, in brief
the formalism for calculating $\nu_{l}({\bar{\nu}}_{l})$-nucleon scattering
cross section and in section-3 we discuss the nuclear medium effects in the
evaluation of structure functions and differential scattering cross section
for $\nu_{l}({\bar{\nu}}_{l})$-nucleus scattering. In section-4, we present
and discuss the results and section-5, summarizes our study.

##  2 Formalism

###  2.1 (Anti)neutrino-nucleon cross section and the structure functions

The basic reaction for the (anti)neutrino induced charged current deep
inelastic scattering process on a free nucleon target is given by

$\displaystyle\nu_{l}(k)/\bar{\nu}_{l}(k)+N(p)\rightarrow
l^{-}(k^{\prime})/l^{+}(k^{\prime})+X(p^{\prime});\;\;\;l=e/\mu/\tau,\;\;N=n,p,$
(1)

where $k$ and $k^{\prime}$ are the four momenta of incoming and outgoing
lepton, $p$ and $p^{\prime}$ are the four momenta of the target nucleon and
the jet of hadrons produced in the final state, respectively. This process is
mediated by the $W$-boson ($W^{\pm}$) and the invariant matrix element
corresponding to the reaction given in Eq.1, is written as

$-i{\cal
M}=\frac{iG_{F}}{\sqrt{2}}\;l_{\mu}\;\left(\frac{M_{W}^{2}}{q^{2}-M_{W}^{2}}\right)\;\langle
X|J^{\mu}|N\rangle\;,$ (2)

where $G_{F}$ is the Fermi coupling constant, $M_{W}$ is the mass of $W$
boson, and $q^{2}=(k-k^{\prime})^{2}$ is the four momentum transfer square.
$l_{\mu}$ is the leptonic current and $\langle X|J^{\mu}|N\rangle$ is the
hadronic current for the neutrino induced reaction.

The general expression of the double differential scattering cross section
(DCX) corresponding to the reaction given in Eq. 1 (depicted in Fig. 1) in the
laboratory frame is expressed as:

$\frac{d^{2}\sigma}{dxdy}=\frac{yM_{N}}{\pi}~{}\frac{E_{\nu}}{E_{l}}~{}\frac{|{\bf
k^{\prime}}|}{|{\bf
k}|}\;\frac{G_{F}^{2}}{2}~{}\left(\frac{M_{W}^{2}}{Q^{2}+M_{W}^{2}}\right)^{2}~{}L_{\mu\nu}~{}W^{\mu\nu}_{N},$
(3)

Figure 1:  $\nu_{l}({\bar{\nu}}_{l})-N$ inclusive scattering, where the
summation sign represents the sum over all the hadronic states such that the
cross section($d\sigma$) for the deep inelastic scattering $\propto
L_{\mu\nu}W_{N}^{\mu\nu}$.

where $x$ and $y$ are the scaling variables which lie in the following ranges:

$\frac{m_{l}^{2}}{2M_{N}(E_{\nu}-m_{l})}\leq x\leq 1;~{}~{}~{}a-b\leq y\leq
a+b,$ (4)

with

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1-m_{l}^{2}\Big{(}\frac{1}{2M_{N}E_{\nu}x}+\frac{1}{2E_{\nu}^{2}}\Big{)}}{2\Big{(}1+\frac{M_{N}x}{2E_{\nu}}\Big{)}},\;\;b=\frac{\sqrt{\left(1-\frac{m_{l}^{2}}{2M_{N}E_{\nu}x}\right)^{2}-\frac{m_{l}^{2}}{E_{\nu}^{2}}}}{2\Big{(}1+\frac{M_{N}x}{2E_{\nu}}\Big{)}},$
(5)

where $m_{l}$ is the charged lepton mass. The leptonic tensor $L_{\mu\nu}$ is
given by

$\displaystyle L_{\mu\nu}$ $\displaystyle=$ $\displaystyle
8(k_{\mu}k^{\prime}_{\nu}+k_{\nu}k^{\prime}_{\mu}-k.k^{\prime}g_{\mu\nu}\pm
i\epsilon_{\mu\nu\rho\sigma}k^{\rho}k^{\prime\sigma})\,,$ (6)

with +ve sign for antineutrino and -ve sign for neutrino in the antisymmetric
term.

The hadronic tensor $W_{N}^{\mu\nu}$ is written in terms of the weak nucleon
structure functions $W_{iN}(\nu,Q^{2})~{}(i=1-6)$ as

$\displaystyle W_{N}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\frac{q^{\mu}q^{\nu}}{q^{2}}-g^{\mu\nu}\right)\;W_{1N}(\nu,Q^{2})+\frac{W_{2N}(\nu,Q^{2})}{M_{N}^{2}}\left(p^{\mu}-\frac{p.q}{q^{2}}\;q^{\mu}\right)$
(7) $\displaystyle\times$
$\displaystyle\left(p^{\nu}-\frac{p.q}{q^{2}}\;q^{\nu}\right)-\frac{i}{2M_{N}^{2}}\epsilon^{\mu\nu\rho\sigma}p_{\rho}q_{\sigma}W_{3N}(\nu,Q^{2})+\frac{W_{4N}(\nu,Q^{2})}{M_{N}^{2}}q^{\mu}q^{\nu}$
$\displaystyle+\frac{W_{5N}(\nu,Q^{2})}{M_{N}^{2}}(p^{\mu}q^{\nu}+q^{\mu}p^{\nu})+\frac{i}{M_{N}^{2}}(p^{\mu}q^{\nu}-q^{\mu}p^{\nu})W_{6N}(\nu,Q^{2})\,.$

The contribution of the term with $W_{6N}(\nu,Q^{2})$ vanishes when contracted
with the leptonic tensor. In the limit of high $Q^{2}$ and $\nu$, the
structure functions $W_{iN}(\nu,Q^{2});~{}(i=1-5)$ are generally expressed in
terms of the dimensionless nucleon structure functions $F_{iN}(x),\;\;i=1-5$.
However, as we move towards the region of low and moderate $Q^{2}$, the
dimensionless nucleon structure functions show $x$ as well as $Q^{2}$
dependence. Hence, $F_{iN}(x,Q^{2}),\;\;i=1-5$ are defined as

$\displaystyle F_{1N}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
W_{1N}(\nu,Q^{2}),\;\;F_{2N}(x,Q^{2})=\frac{Q^{2}}{2xM_{N}^{2}}W_{2N}(\nu,Q^{2}),\;\;F_{3N}(x,Q^{2})=\frac{Q^{2}}{xM_{N}^{2}}W_{3N}(\nu,Q^{2}),$
$\displaystyle F_{4N}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{2M_{N}^{2}}W_{4N}(\nu,Q^{2}),\;\;F_{5N}(x,Q^{2})=\frac{Q^{2}}{2xM_{N}^{2}}W_{5N}(\nu,Q^{2}).\;\;\;$
(8)

The expression for the differential scattering cross section for the
$\nu_{l}/{\bar{\nu}}_{l}-N$ scattering given in Eq. 3 is written by using Eqs.
6 and 2.1 as:

$\displaystyle\frac{d^{2}\sigma}{dxdy}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}M_{N}E_{\nu}}{\pi(1+\frac{Q^{2}}{M_{W}^{2}})^{2}}\Big{\\{}\Big{[}y^{2}x+\frac{m_{l}^{2}y}{2E_{\nu}M_{N}}\Big{]}F_{1N}(x,Q^{2})+\Big{[}\Big{(}1-\frac{m_{l}^{2}}{4E_{\nu}^{2}}\Big{)}-\Big{(}1+\frac{M_{N}x}{2E_{\nu}}\Big{)}y\Big{]}F_{2N}(x,Q^{2})$
(9) $\displaystyle\pm$
$\displaystyle\Big{[}xy\Big{(}1-\frac{y}{2}\Big{)}-\frac{m_{l}^{2}y}{4E_{\nu}M_{N}}\Big{]}F_{3N}(x,Q^{2})+\frac{m_{l}^{2}(m_{l}^{2}+Q^{2})}{4E_{\nu}^{2}M_{N}^{2}x}F_{4N}(x,Q^{2})-\frac{m_{l}^{2}}{E_{\nu}M_{N}}F_{5N}(x,Q^{2})\Big{\\}}.\;\;\;~{}~{}~{}~{}~{}$

In general, the dimensionless nucleon structure functions are derived in the
quark-parton model assuming Bjorken scaling in which they scale and are
functions of only one variable $x$. In this limit, these structure functions
obey Callan-Gross Callan:1969uq  and Albright-Jarlskog Albright:1974ts
relations

$\displaystyle F_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{F_{2}(x)}{2x}\;;\;\;F_{5}(x)=\frac{F_{2}(x)}{2x},$

and at the leading order, the structure functions are written in terms of the
parton distribution functions $q_{i}(x)$ and $\bar{q}_{i}(x)$ as:

$\displaystyle F_{2}(x)$ $\displaystyle=$
$\displaystyle\sum_{i}x[q_{i}(x)+\bar{q}_{i}(x)]\;;\;xF_{3}(x)=\sum_{i}x[q_{i}(x)-\bar{q}_{i}(x)]\;;\;\;F_{4}(x)=0.$
(10)

For example, in the case of $\nu(\bar{\nu})$-proton scattering above the charm
production threshold, $F_{2,3}(x)$ are given by:

$\displaystyle F_{2p}^{\nu}(x)$ $\displaystyle=$ $\displaystyle
2x[d(x)+s(x)+\bar{u}(x)+\bar{c}(x)]\;;\;\;F_{2p}^{\bar{\nu}}(x)=2x[u(x)+c(x)+\bar{d}(x)+\bar{s}(x)];$
$\displaystyle xF_{3p}^{\nu}(x)$ $\displaystyle=$ $\displaystyle
2x[d(x)+s(x)-\bar{u}(x)-\bar{c}(x)]\;;\;\;xF_{3p}^{\bar{\nu}}(x)=2x[u(x)+c(x)-\bar{d}(x)-\bar{s}(x)]\;\;\;\;\;\;\;~{}~{}$
(11)

and for the $\nu(\bar{\nu})$-neutron scattering $F_{2,3}(x)$ are given by

$\displaystyle F_{2n}^{\nu}(x)$ $\displaystyle=$ $\displaystyle
2x[u(x)+s(x)+\bar{d}(x)+\bar{c}(x)]\;;\;\;F_{2n}^{\bar{\nu}}(x)=2x[d(x)+c(x)+\bar{u}(x)+\bar{s}(x)];$
$\displaystyle xF_{3n}^{\nu}(x)$ $\displaystyle=$ $\displaystyle
2x[u(x)+s(x)-\bar{d}(x)-\bar{c}(x)]\;;\;\;xF_{3n}^{\bar{\nu}}(x)=2x[d(x)+c(x)-\bar{u}(x)-\bar{s}(x)].\;\;\;\;\;\;\;$
(12)

For an isoscalar nucleon target, we use

$F_{iN}=\frac{F_{ip}+F_{in}}{2}.~{}~{}~{}(i=1-5)$ (13)

In the present formalism we have performed numerical calculations in the
three($u$, $d$ and $s$) as well as four($u$, $d$, $s$ and $c$) flavor schemes
by taking $u$, $d$ and $s$ to be massless and the charm quark to be massless
as well as massive. In the region of low and moderate $Q^{2}$, the
perturbative and nonperturbative QCD corrections such as $Q^{2}$ evolution of
parton distribution functions from leading order to higher order terms (next-
to-leading order (NLO), next-next-to-leading order (NNLO), …), the effects of
target mass correction due to the massive quarks production (e.g. charm,
bottom, top) and higher twist (twist-4, twist-6, …) because of the multiparton
correlations, become important. These nonperturbative effects are specifically
important in the kinematical region of high $x$ and low $Q^{2}$. The $Q^{2}$
evolution of structure functions is determined by the DGLAP evolution equation
Dokshitzer:1977sg ; Gribov:1972ri ; Altarelli:1977zs ; Lipatov:1974qm . The
parton distribution functions for the nucleon have been determined by various
groups and we have taken the parameterization of MMHT Harland-Lang:2014zoa  in
our numerical calculations up to NNLO following Ref. Vermaseren:2005qc ;
Moch:2004xu ; Moch:2008fj . The nonperturbative higher twist effect is
incorporated by using the renormalon approach Dasgupta:1996hh  and the target
mass correction is included following the works of Schienbein et al.
Schienbein:2007gr . The incorporation of the contribution from gluon emission
induces the $Q^{2}$ dependence of the nucleon structure functions. The details
of the discussion are given in Ref. Zaidi:2019asc .

###  2.2 (Anti)neutrino-nucleus cross section and structure functions

The differential scattering cross section for the charged current inclusive
$\nu_{l}/\bar{\nu}_{l}$-nucleus deep inelastic scattering process (depicted in
Fig. 2):

$\nu_{l}/\bar{\nu}_{l}(k)+A(p_{A})\rightarrow
l^{-}/l^{+}(k^{\prime})+X(p^{\prime}_{A})$ (14)

Figure 2:  Feynman diagrams for the $\nu_{l}/\bar{\nu}_{l};~{}(l=e,\mu,\tau)$
induced DIS process off nuclear target ($A$).

is expressed in terms of the leptonic tensor $L_{\mu\nu}$ and the nuclear
hadronic tensor $W^{\mu\nu}_{A}$ as:

$\displaystyle{d^{2}\sigma_{A}\over dxdy}$ $\displaystyle=$
$\displaystyle\left({G_{F}^{2}yM_{N}E_{\nu}\over 2\pi
E_{l}}\right){\left(M_{W}^{2}\over M_{W}^{2}+Q^{2}\right)^{2}}\;{|{\bf
k^{\prime}}|\over|{\bf k}|}\;L_{\mu\nu}\;W^{\mu\nu}_{A},$ (15)

where the physical quantities have their usual meanings. The expression of
$L_{\mu\nu}$ is given by Eq.6. The nuclear hadronic tensor $W^{\mu\nu}_{A}$ is
written in terms of the weak nuclear structure functions
$W_{iA}(\nu_{A},Q^{2})$ ($i=1-6$) as:

$\displaystyle W_{A}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\frac{q^{\mu}q^{\nu}}{q^{2}}-g^{\mu\nu}\right)\;W_{1A}(\nu_{A},Q^{2})+\frac{W_{2A}(\nu_{A},Q^{2})}{M_{A}^{2}}\left(p^{\mu}_{A}-\frac{p_{A}.q}{q^{2}}\;q^{\mu}\right)$
(16) $\displaystyle\times$
$\displaystyle\left(p^{\nu}_{A}-\frac{p_{A}.q}{q^{2}}\;q^{\nu}\right)\pm\frac{i}{2M_{A}^{2}}\epsilon^{\mu\nu\rho\sigma}p_{A\rho}q_{\sigma}W_{3A}(\nu_{A},Q^{2})+\frac{W_{4A}(\nu_{A},Q^{2})}{M_{A}^{2}}q^{\mu}q^{\nu}$
$\displaystyle+\frac{W_{5A}(\nu_{A},Q^{2})}{M_{A}^{2}}(p^{\mu}_{A}q^{\nu}+q^{\mu}p^{\nu}_{A})+\frac{i}{M_{A}^{2}}(p^{\mu}_{A}q^{\nu}-q^{\mu}p^{\nu}_{A})W_{6A}(\nu_{A},Q^{2})\,,$

where $M_{A}$ is the mass of the nuclear target. The nuclear structure
functions $W_{iA}(\nu_{A},Q^{2})~{}(i=1-5)$ are written in terms of the
dimensionless nuclear structure functions $F_{iA}(x_{A},Q^{2})~{}(i=1-5)$ as:

$\displaystyle F_{1A}(x_{A},Q^{2})$ $\displaystyle=$ $\displaystyle
W_{1A}(\nu_{A},Q^{2})\;;\;\;F_{2A}(x_{A},Q^{2})=\frac{Q^{2}}{2xM_{A}^{2}}W_{2A}(\nu_{A},Q^{2})\;;\;\;F_{3A}(x_{A},Q^{2})=\frac{Q^{2}}{xM_{A}^{2}}W_{3A}(\nu_{A},Q^{2});$
$\displaystyle F_{4A}(x_{A},Q^{2})$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{2M_{A}^{2}}W_{4A}(\nu_{A},Q^{2})\;;\;\;F_{5A}(x_{A},Q^{2})=\frac{Q^{2}}{2xM_{A}^{2}}W_{5A}(\nu_{A},Q^{2}),$
(17)

where $\nu_{A}$(=$\frac{p_{{}_{A}}\cdot q}{M_{{}_{A}}}(=q^{0})$ ) is the
energy transferred to the target in the Lab frame and $x_{A}(=x/A)$ is the
Bjorken scaling variable given by:

$\displaystyle x_{A}$ $\displaystyle=$ $\displaystyle\frac{Q^{2}}{2p_{A}\cdot
q}=\frac{Q^{2}}{2p_{A}^{0}q^{0}}=\frac{Q^{2}}{2A~{}M_{N}q^{0}}=\frac{x}{A}.$
(18)

The expression for the differential cross section for the
$\nu_{l}/{\bar{\nu}}_{l}-A$ scattering can be obtained using Eqs. 6, 16 and
2.2 in Eq. 15 as

$\displaystyle\frac{d^{2}\sigma_{A}}{dxdy}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}M_{N}E_{\nu}}{\pi(1+\frac{Q^{2}}{M_{W}^{2}})^{2}}\Big{\\{}\Big{[}y^{2}x+\frac{m_{l}^{2}y}{2E_{\nu}M_{N}}\Big{]}F_{1A}(x,Q^{2})+\Big{[}\Big{(}1-\frac{m_{l}^{2}}{4E_{\nu}^{2}}\Big{)}-\Big{(}1+\frac{M_{N}x}{2E_{\nu}}\Big{)}y\Big{]}F_{2A}(x,Q^{2})$
(19) $\displaystyle\pm$
$\displaystyle\Big{[}xy\Big{(}1-\frac{y}{2}\Big{)}-\frac{m_{l}^{2}y}{4E_{\nu}M_{N}}\Big{]}F_{3A}(x,Q^{2})+\frac{m_{l}^{2}(m_{l}^{2}+Q^{2})}{4E_{\nu}^{2}M_{N}^{2}x}F_{4A}(x,Q^{2})-\frac{m_{l}^{2}}{E_{\nu}M_{N}}F_{5A}(x,Q^{2})\Big{\\}}.~{}~{}~{}\;\;\;$

For $\nu_{e}/\bar{\nu}_{e}$ and $\nu_{\mu}/\bar{\nu}_{\mu}$ charged current
interactions, in the limit $m_{l}\to 0$, only the first three terms of Eq. 19,
i.e. the terms with $F_{1A}(x,Q^{2})$, $F_{2A}(x,Q^{2})$ and $F_{3A}(x,Q^{2})$
structure functions would contribute. We consider the scattering process in
the laboratory frame, where target nucleus is at rest i.e.
$p_{A}=(p_{A}^{0},~{}{\bf p_{A}}=0)$ and the momentum of the nucleon in the
nucleus (${\bf p_{N}}$) is non-zero and the motion of such nucleons
corresponds to the Fermi motion.

If the momentum transfer is along the $Z$-axis then
$q^{\mu}=(q^{0},0,0,q^{z})$ and the Bjorken variable $x_{N}$ is written as:

$x_{N}=\frac{Q^{2}}{2p_{N}\cdot
q}=\frac{Q^{2}}{2(p_{N}^{0}q^{0}-p_{N}^{z}q^{z})}.$ (20)

The nuclear medium effects such as Fermi motion, binding, nucleon correlations
incorporated through nucleon spectral function, meson cloud contribution and
shadowing effect are discussed in the following subsections 2.2.1, 2.2.2 and
2.2.3.

####  2.2.1 Effect of Fermi motion, binding and nucleon correlation

We start by writing the scattering cross section ($d\sigma_{A}$) for small
elemental volume ($dV$) inside the nucleus in terms of the probability of
neutrino interaction with a bound nucleon per unit time ($\Gamma$).
Probability times the differential of area ($dS$) defines the cross section
Marco:1995vb , i.e.

$d\sigma_{A}=\Gamma dtdS=\Gamma\frac{E_{\nu}}{\mid{\bf
k}\mid}d^{3}r,~{}~{}~{}~{}~{}~{}~{}~{}~{}\Big{[}\because~{}~{}dtdS=\frac{dV}{v}=\frac{E_{\nu}}{\mid{\bf
k}\mid}d^{3}r\Big{]}$ (21)

where $v$ is the velocity of the incoming neutrino. $\Gamma$ is related to the
imaginary part of the $\nu_{l}$ self energy ($\Sigma(k)$) as Marco:1995vb :

$-\frac{\Gamma}{2}=\frac{m_{\nu}}{E_{\nu}({\bf k})}\;Im\Sigma(k).$ (22)

It may be pointed out that the neutrino self energy $\Sigma(k)$ has two parts,
the real part of “neutrino self energy” modifies the lepton mass and imaginary
part i.e. $Im\Sigma(k)$ gives information about the total number of neutrinos
interactions that yield the final state leptons and hadrons.

From Eq.21 and Eq.22, we get

$d\sigma_{A}=-2\frac{m_{\nu}}{\mid{\bf k}\mid}Im\Sigma(k)d^{3}r.$ (23)

Figure 3: Diagrammatic representation of (a) the neutrino ($\nu_{l}$) self-
energy and (b) the intermediate vector boson ($W$) self-energy.

$\Sigma(k)$ is evaluated corresponding to the diagram shown in Fig.3 (left
panel) using the Feynman rules and on applying the Cutkowsky rules the
imaginary part of neutrino self energy is obtained as Haider:2016zrk :

$Im\Sigma(k)=\frac{G_{F}}{\sqrt{2}}{4\over
m_{\nu}}\int\frac{d^{3}k^{\prime}}{(2\pi)^{4}}{\pi\over E({\bf
k^{\prime}})}\theta(q^{0})\left(\frac{M_{W}}{Q^{2}+M_{W}^{2}}\right)^{2}\;Im[L_{\mu\nu}^{WI}\Pi^{\mu\nu}(q)],$
(24)

where $\Pi^{\mu\nu}(q)$ is the $W$-boson self-energy (as shown in Fig. 3(b)),
which is generally written in terms of the nucleon propagator ($G_{l}$) and
meson propagator ($D_{j}$) corresponding to Fig. 3(b), as:

$\displaystyle\Pi^{\mu\nu}(q)$ $\displaystyle=$
$\displaystyle\left(\frac{G_{F}M_{W}^{2}}{\sqrt{2}}\right)\times\int\frac{d^{4}p}{(2\pi)^{4}}G(p)\sum_{X}\;\sum_{s_{p},s_{l}}\prod_{i=1}^{N}\int\frac{d^{4}p^{\prime}_{i}}{(2\pi)^{4}}\;\prod_{{}_{l}}G_{l}(p^{\prime}_{l})\prod_{{}_{j}}\;D_{j}(p^{\prime}_{j})\;$
(25)
$\displaystyle<X|J^{\mu}|N><X|J^{\nu}|N>^{*}(2\pi)^{4}~{}\delta^{4}\Big{(}k+p-k^{\prime}-\sum^{N}_{i=1}p^{\prime}_{i}\Big{)},\;\;\;$

where $s_{p}$ and $s_{l}$ are the spin of the initial state nucleon and the
final state fermions, the indices $l$ and $j$ are respectively, for the
fermions and bosons in the final hadronic state, $<X|J^{\mu}|N>$ represents
the hadronic current and
$\delta^{4}(k+p-k^{\prime}-\sum^{N}_{i=1}p^{\prime}_{i})$ ensures the
conservation of four momentum. $G(p)$ gives the information about the
propagation of the nucleon from the initial state to the final state or vice
versa.

To obtain the relativistic nucleon propagator $G(p^{0},{\bf p})$ in a nuclear
medium we start with the relativistic free nucleon Dirac propagator
$G^{0}(p^{0},{\bf p})$ which is written in terms of the Dirac spinors $u({\bf
p})$ and $v({\bf p})$. This includes the contribution from positive and
negative energy components of the nucleon and the negative energy contribution
is suppressed while the positive energy contribution survives Marco:1995vb ;
FernandezdeCordoba:1991wf . Considering only the positive energy part of the
nucleon propagator $G^{0}(p^{0},{{\bf p}})$, we write

$G^{0}(p^{0},{{\bf p}})={\not p+M_{N}\over p^{2}-M_{N}^{2}+i\epsilon}+2\
i\pi\theta(p^{0})\delta(p^{2}-M_{N}^{2})n({\bf p})(\not p+M_{N}).$ (26)

In the nuclear medium considered as an interacting Fermi sea, $G(p^{0},{\bf
p})$ is written in terms of the nucleon self energy
$\Sigma^{N}(p^{0},\bf{p})$, which contains all the information on single
nucleon. Then in nuclear medium the interaction is taken into account through
Dyson series expansion, which is in principle an infinite series in
perturbation theory. This perturbative expansion is summed in a ladder
approximation as Marco:1995vb :

$\displaystyle G(p)$ $\displaystyle=$ $\displaystyle\frac{M_{N}}{E({\bf
p})}\frac{\sum_{r}u_{r}({\bf p})\bar{u}_{r}({\bf p})}{p^{0}-E({\bf
p})-\sum_{r}\bar{u}_{r}({\bf p})\Sigma^{N}(p^{0},{\bf p})u_{r}({\bf
p})\frac{M_{N}}{{E({\bf p})}}}.\;\;\;$ (27)

One may notice from the expression for the nucleon propagator $G(p)$ given in
Eq.27 that it contains nucleon self energy $\Sigma^{N}(p_{0},{\bf p})$. The
nucleon self-energy is written using the techniques of the standard Many-Body
Theory FernandezdeCordoba:1991wf . The inputs required for the NN interaction
are incorporated by relating them to the experimental elastic NN cross
section. Furthermore, RPA-correlation effect is taken into account using the
spin-isospin effective interaction as the dominating part of the particle-hole
(ph) interaction. Using the modified expression for the nucleon self energy,
the imaginary part of it is obtained Haider:2015vea . These considerations
lead to a dressed nucleon propagator in the nuclear matter which is given by
Marco:1995vb ; FernandezdeCordoba:1991wf :

$\displaystyle G(p)=$ $\displaystyle\frac{M_{N}}{E({\bf p})}\sum_{r}u_{r}({\bf
p})\bar{u}_{r}({\bf
p})\left[\int^{\mu}_{-\infty}d\omega\frac{S_{h}(\omega,{\bf{p}})}{p^{0}-\omega-i\eta}+\int^{\infty}_{\mu}d\omega\frac{S_{p}(\omega,{\bf{p}})}{p^{0}-\omega+i\eta}\right],\;\;~{}~{}~{}$
(28)

where the expressions for the hole $S_{h}(\omega,{\bf{p}})$(for
$p_{0}\leq\mu$) and the particle $S_{p}(\omega,{\bf{p}})$(for $p_{0}\geq\mu$)
spectral functions and the nucleon self energy $\Sigma^{N}(p^{0},{\bf{p}})$
are taken from Ref. FernandezdeCordoba:1991wf . For an inclusive process, only
the hole spectral function contributes. In the above expression, $\mu$ is the
chemical potential given by
$\mu=\frac{p_{F}^{2}}{2M}~{}+~{}Re\Sigma^{N}\left[\frac{p_{F}^{2}}{2M},p_{F}\right]$
and $\omega=p^{0}-M_{N}$ is the removal energy. $\eta$ is the infinitesimal
quantity i.e. $\eta\to 0$. $p_{F}$ is the Fermi momentum of the nucleon in the
nucleus. In the local Fermi gas model the Fermi momentum is a function $r$,
the point at which the interaction in the nucleus takes place and is given by
$p_{{}_{F}}(r)=\left[\frac{3}{2}\pi^{2}\rho(r)\right]^{1/3}$, where $\rho(r)$
is the nucleon charge density inside the nucleus, the parameters of which are
determined from electron scattering experiments DeVries:1987 . The spectral
function for an isoscalar nuclear target is normalized to the number of
nucleons $(A)$ in the nucleus i.e.

$\displaystyle 4\int
d^{3}r\;\int\frac{d^{3}p}{(2\pi)^{3}}\;\int_{-\infty}^{\mu}\;S_{h}(\omega,{\bf
p},\rho(r))\;d\omega=A.\;$ (29)

Some of the properties of the spectral function may be found in the appendix
of Ref. Haider:2015vea .

Then using Eqs. 23 and 24 the expression for the differential cross section is
written as

$\frac{d\sigma_{A}}{dxdy}=-\frac{G_{F}^{2}\;M_{N}\;y}{2\pi}\;\frac{E_{l}}{E_{\nu}}\;\frac{|\bf{k^{\prime}}|}{|\bf{k}|}\left(\frac{M_{W}^{2}}{Q^{2}+M_{W}^{2}}\right)^{2}L_{\mu\nu}\int
Im\Pi^{\mu\nu}(q)d^{3}r.$ (30)

Comparing Eq.30, with Eqs.15, 25 and 28, the nuclear hadronic tensor (for
isospin symmetric nucleus) can be expressed in terms of the nucleon hadronic
tensor and the hole spectral function and is given as Haider:2015vea

$W^{\mu\nu}_{A}=4\int\,d^{3}r\,\int\frac{d^{3}p}{(2\pi)^{3}}\,\frac{M_{N}}{E({\bf
p})}\,\int^{\mu}_{-\infty}dp^{0}S_{h}(p^{0},{\bf
p},\rho(r))W^{\mu\nu}_{N}(p,q),\,$ (31)

a factor of 4 is because of the spin-isospin degrees of freedom of the
nucleon.

For a non-isoscalar nuclear target like 56Fe and ${}^{208}Pb$, the spectral
functions for the proton ($Z$) and neutron ($N=A-Z$) numbers in a nuclear
target which are the function of local Fermi momenta
$p_{{}_{{F}_{p,n}}}(r)=\left[3\pi^{2}\rho_{p(n)}({r})\right]^{1/3}$, are
normalized separately such that

$\displaystyle 2\int
d^{3}r\;\int\frac{d^{3}p}{(2\pi)^{3}}\;\int_{-\infty}^{\mu_{p}}\;S_{h}^{p}(\omega,{\bf
p},\rho_{p}(r))\;d\omega$ $\displaystyle=$ $\displaystyle Z\;,$ (32)
$\displaystyle 2\int
d^{3}r\;\int\frac{d^{3}p}{(2\pi)^{3}}\;\int_{-\infty}^{\mu_{n}}\;S_{h}^{n}(\omega,{\bf
p},\rho_{n}(r))\;d\omega$ $\displaystyle=$ $\displaystyle N\;,$ (33)

where the factor of 2 is due to the two possible projections of nucleon spin,
$\mu_{p}(\mu_{n})$ is the chemical potential for the proton(neutron), and
$S_{h}^{p}(\omega,{\bf p},\rho_{p}(r))$ and $S_{h}^{n}(\omega,{\bf
p},\rho_{n}(r))$ are the hole spectral functions for the proton and neutron,
respectively. The proton and neutron densities $\rho_{p}(r)$ and $\rho_{n}(r)$
are related to the nuclear density $\rho(r)$ as Haider:2016zrk ;
Haider:2015vea :

$\displaystyle\rho_{p}(r)$ $\displaystyle=$
$\displaystyle\frac{Z}{A}\;\rho(r)~{};\hskip
14.22636pt\rho_{n}(r)=\frac{(A-Z)}{A}\;\rho(r).$

Hence for a nonisoscalar nuclear target, the nuclear hadronic tensor is
written as

$W^{\mu\nu}_{A}=2\sum_{\tau=p,n}\int\,d^{3}r\,\int\frac{d^{3}p}{(2\pi)^{3}}\,\frac{M_{N}}{E_{N}({\bf
p})}\,\int^{\mu_{\tau}}_{-\infty}dp^{0}S_{h}^{\tau}(p^{0},{\bf
p},\rho_{\tau}(r))\;W^{\mu\nu}_{\tau}(p,q).\,$ (34)

To evaluate the weak dimensionless nuclear structure functions by using Eq.31,
the appropriate components of nucleon ($W^{\mu\nu}_{N}$ in Eq.7) and nuclear
($W^{\mu\nu}_{A}$ in Eq.16) hadronic tensors along the $x$, $y$ and $z$ axis
are chosen. For example, the expression of $F_{1A,N}(x_{A},Q^{2})$ is obtained
by taking the $xx$ components, $F_{2A,N}(x_{A},Q^{2})$ by taking the $zz$
components, $F_{3A,N}(x_{A},Q^{2})$ by taking the $xy$ components, etc.
Zaidi:2019asc , and for an isoscalar nuclear target, the expressions for the
three nuclear structure functions viz. $F_{iA,N}(x_{A},Q^{2})\;\;i=1-3$ i.e.
for the massless leptons as are obtained as:

$\displaystyle F_{iA,N}(x_{A},Q^{2})$ $\displaystyle=$ $\displaystyle
4\int\,d^{3}r\,\int\frac{d^{3}p}{(2\pi)^{3}}\,\frac{M_{N}}{E_{N}({\bf
p})}\,\int^{\mu}_{-\infty}dp^{0}~{}S_{h}(p^{0},{\bf p},\rho(r))~{}\times
f_{iN}(x,Q^{2}),~{}~{}~{}\;\;\;$ (35)

where

$\displaystyle f_{1N}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
AM_{N}\left[\frac{F_{1N}(x_{N},Q^{2})}{M_{N}}+\left(\frac{p^{x}}{M_{N}}\right)^{2}\frac{F_{2N}(x_{N},Q^{2})}{\nu_{N}}\right],$
(36) $\displaystyle f_{2N}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\left[\frac{Q^{2}}{(q^{z})^{2}}\left(\frac{|{\bf
p}|^{2}~{}-~{}(p^{z})^{2}}{2M_{N}^{2}}\right)+\frac{(p^{0}~{}-~{}p^{z}~{}\gamma)^{2}}{M_{N}^{2}}\left(\frac{p^{z}~{}Q^{2}}{(p^{0}~{}-~{}p^{z}~{}\gamma)q^{0}q^{z}}~{}+~{}1\right)^{2}\right]~{}$
(37)
$\displaystyle~{}~{}~{}~{}\times\left(\frac{M_{N}}{p^{0}~{}-~{}p^{z}~{}\gamma}\right)\times
F_{2N}(x_{N},Q^{2}),$ $\displaystyle f_{3N}(x,Q^{2})$ $\displaystyle=$
$\displaystyle A\frac{q^{0}}{q^{z}}\;\times\left({p^{0}q^{z}-p^{z}q^{0}\over
p\cdot q}\right)F_{3N}(x_{N},Q^{2}).$ (38)

Similar expression is obtained for a non-isoscalar nuclear target
Zaidi:2019asc .

The nonperturbative effects of the target mass corrections and the higher
twist Dasgupta:1996hh  have been incorporated in the free nucleon structure
functions and then we have convoluted these nucleon structure functions with
the spectral function in order to evaluate the nuclear structure functions
(Eq.35). Using the nuclear structure functions, we have obtained the
differential scattering cross sections for the
$\nu_{l}(\bar{\nu}_{l})-A,\;\;l=e,\mu$ DIS process (Eq.19).

The calculations are performed in the four flavor MS-bar scheme. We have
considered two cases: (i) In the case of massless leptons
($e^{\pm},~{}\mu^{\pm}$), all the four quarks, i.e., $u,~{}d$, $s$ and $c$ are
treated to be massless while (ii) For massive lepton ($\tau^{\pm}$), light
quarks $u,~{}d$ and $s$ are treated as massless but charm quark $c$ as a
massive object for which we define

$F_{iA}(x,Q^{2})=F_{iA}^{n_{f}=4}(x,Q^{2})=\underbrace{F_{iA}^{n_{f}=3}(x,Q^{2})}_{\textrm{for
massless($u,d,s$) quarks}}+\underbrace{F_{iA}^{n_{f}=1}(x,Q^{2})}_{\textrm{for
massive charm quark}}.$ (39)

Figure 4: Neutrino self energy diagram accounting for neutrino-meson DIS (a)
the bound nucleon propagator is substituted with a meson($\pi$ or $\rho$)
propagator with momentum $p$ represented here by dashed line (b) by including
particle-hole $(1p–1h)$, delta-hole $(1\Delta–1h)$, $1p1h-1\Delta 1h$, etc.
interactions.

The nucleons which are bound inside the nucleus may interact with each other
via meson exchange such as $\pi,~{}\rho,$ etc. This mesonic effect has been
incorporated and is discussed in the next sub-subsection 2.2.2.

####  2.2.2 Effect of mesonic cloud contribution

Associated with each bound nucleon there are virtual mesons (pion, rho meson,
etc.) and because of the strong attractive nature of the nucleon-nucleon
interaction, the probability of a $W$-boson interaction with the mesonic cloud
becomes high. We have discussed the $\pi$ and $\rho$ meson contributions in
our earlier works Haider:2016zrk ; Zaidi:2019mfd ; Zaidi:2019asc ;
Haider:2015vea ; Haider:2016tev . The contribution from heavier mesons like
$\omega$ meson are expected to be very small due to their significantly higher
masses. The mesonic contribution is significant in the intermediate region of
$x~{}(0.2<x<0.6)$. For the medium nuclei like ${}^{4}He$, ${}^{12}C$, etc.,
mesonic contribution is small Haider:2011kvi  while it becomes pronounced in
heavier nuclear targets Haider:2012nf  such as ${}^{56}Fe$, ${}^{208}Pb$, etc.
To incorporate the mesonic effect we draw a diagram similar to Fig.3, but here
a nucleon propagator is replaced by a meson propagator. The meson propagator
corresponds to the virtual mesons arising due to the nuclear medium effects
and can arise through the particle-hole $(1p–1h)$, delta-hole $(1\Delta–1h)$,
$2p-2h$, etc. interactions as shown in Fig.4 FernandezdeCordoba:1991wf .

The mesonic structure functions
$F_{{iA,a}}(x_{a},Q^{2}),~{}~{}(i=1,2;a=\pi,\rho)$ are obtained as
Zaidi:2019asc :

$\displaystyle F_{{iA,a}}(x_{a},Q^{2})$ $\displaystyle=$
$\displaystyle-6\kappa\int\,d^{3}r\int\frac{d^{4}p}{(2\pi)^{4}}\theta(p^{0})~{}\delta
ImD_{a}(p)\;2m_{a}~{}\;f_{ia}(x_{a}),$ (40)

where

$\displaystyle f_{1a}(x_{a})$ $\displaystyle=$ $\displaystyle
AM_{N}\left[\frac{F_{1a}(x_{a})}{m_{a}}~{}+~{}\frac{{|{\bf
p}|^{2}~{}-~{}(p^{z})^{2}}}{2(p^{0}~{}q^{0}~{}-~{}p^{z}~{}q^{z})}\frac{F_{2a}(x_{a})}{m_{a}}\right],~{}~{}~{}~{}~{}$
(41) $\displaystyle f_{2a}(x_{a})$ $\displaystyle=$
$\displaystyle\left[\frac{Q^{2}}{(q^{z})^{2}}\left(\frac{|{\bf
p}|^{2}~{}-~{}(p^{z})^{2}}{2m_{a}^{2}}\right)+\frac{(p^{0}~{}-~{}p^{z}~{}\gamma)^{2}}{m_{a}^{2}}\times\left(\frac{p^{z}~{}Q^{2}}{(p^{0}~{}-~{}p^{z}~{}\gamma)q^{0}q^{z}}~{}+~{}1\right)^{2}\right]$
(42) $\displaystyle\times$
$\displaystyle\left(\frac{m_{a}}{p^{0}~{}-~{}p^{z}~{}\gamma}\right)~{}F_{{2a}}(x_{a}).~{}~{}~{}~{}~{}~{}$

In Eqs. 40 and 41, $\kappa=1(2)$ for pion(rho meson), $\nu=\frac{q_{0}(\gamma
p^{z}-p^{0})}{m_{a}}$, $x_{a}=-\frac{Q^{2}}{2p\cdot q}$, $m_{a}$ is the mass
of the meson($\pi$ or $\rho$). $D_{a}(p)$ is the meson($\pi$ or $\rho$)
propagator in the nuclear medium and is written as

$D_{a}(p)=[p_{0}^{2}-{\bf{p}}\,^{2}-m_{a}^{2}-\Pi_{a}(p_{0},{\bf p})]^{-1}\,,$
(43)

with

$\Pi_{a}(p_{0},{\bf
p})=\frac{f^{2}}{m_{\pi}^{2}}\;\frac{C_{\rho}\;F^{2}_{a}(p){\bf{p}}\,^{2}\Pi^{*}}{1-{f^{2}\over
m_{\pi}^{2}}V^{\prime}_{j}\Pi^{*}}\,,$ (44)

where $C_{\rho}=1(3.94)$ for pion(rho meson).
$F_{a}(p)={(\Lambda_{a}^{2}-m_{a}^{2})\over(\Lambda_{a}^{2}-p^{2})}$ is the
$\pi NN$ or $\rho NN$ form factor, $\Lambda_{a}$=1 $GeV$ SajjadAthar:2009cr ;
Haider:2015vea  and $f=1.01$. $V_{j}^{\prime}$ is the longitudinal(transverse)
part of the spin-isospin interaction for pion(rho meson), and $\Pi^{*}$ is the
irreducible meson self energy that contains the contribution of particle-hole
and delta-hole excitations. For the pions, we have used the PDFs
parameterization given by Gluck et al.Gluck:1991ey  and for the $\rho$ mesons
same PDFs as for the pions have been used as there is no available explicit
parameterization for the $\rho-$meson PDFs in the literature.

####  2.2.3 Shadowing and Antishadowing effects

In this work the shadowing and antishadowing effects are taken into
consideration following the works of Kulagin and Petti Kulagin:2007ju ;
Kulagin:2004ie  who have used the original Glauber-Gribov multiple scattering
theory. For example, with these effects, the nuclear structure function is
given by

$F_{iA,shd}(x,Q^{2})=\delta R(x,Q^{2})\times F_{iN}^{WI}(x,Q^{2})\;\;\;i=1-3,$
(45)

where the expression for $\delta R(x,Q^{2})$ used in the present numerical
calculations is given in Ref. Kulagin:2004ie .

Now, using the present formalism, we have presented the results for the weak
structure functions and scattering cross sections for both the free nucleon
and nuclear targets in the next section.

Figure 5:  Results for the free nucleon weak structure functions
$F_{iN}(x,Q^{2})$;$(i=1-5)$(Top to Bottom) at the different values of $Q^{2}$=
2 GeV2(Left) and 5 GeV2 (Right) are shown. These results are obtained at NLO
by using MMHT nucleon PDFs parameterization Harland-Lang:2014zoa . The results
are shown without the TMC effect (double dashed-dotted line), with the TMC
effect in the 3-flavor(nf3) scheme (dashed-dotted line) as well as four
flavor(nf4) scheme(dotted line), with TMC and HT effects in the 3-flavor(nf3)
scheme (dashed line) as well as four flavor(nf4) scheme(solid line). Figure 6:
$\frac{\sigma}{E}$ vs $E$ for $\nu/{\bar{\nu}}_{\tau}$ interactions on a free
nucleon target, with a center of mass energy ($W$) cut of 1.4 GeV(dashed line)
and 2 GeV(solid line), for tau type neutrinos(Top panel) and
antineutrinos(Bottom panel) with the TMC Kretzer:2003iu  and higher twist
Dasgupta:1996hh  effects. These results are compared with the results of
various theoretical models available in the literature Kretzer:2002fr ;
Jeong:2010nt ; Hagiwara:2003di ; Conrad:2010mh ; Paschos:2001np ; Gazizov:2016
; Li:2017dbe ; Anelli:2015pba  as well as with the Monte Carlo generators
GENIE GENIE  and NuWroNuWro .

##  3 Results and Discussion

First, we present the results for the free nucleon weak structure functions
$F_{iN}(x,Q^{2})$;$(i=1-5)$, and using them the results for the total
scattering cross section $\sigma(E_{\nu})$ are obtained and presented for
$\frac{\sigma(E_{\nu})}{E_{\nu}}$ vs $E_{\nu}$ in the limit $m_{l}\neq 0$.

Then we have presented the results for the weak nuclear structure functions
$F_{iA}(x,Q^{2})$;$(i=1-3)$ in the limit $m_{l}=0$, when the calculations are
performed using the spectral function (SF) only. Furthermore, the contribution
from the meson clouds as well as the shadowing effect are taken into account
and this corresponds to the full model (Total) results. The expression of
total nuclear structure functions in the full theoretical model is given by

$\displaystyle
F_{iA}(x,Q^{2})=F_{iA,N}(x,Q^{2})+F_{iA,\pi}(x,Q^{2})+F_{iA,\rho}(x,Q^{2})+F_{iA,shd}(x,Q^{2}),\;\;\;\;\;$
(46)

where $i=1,2$. $F_{iA,N}(x,Q^{2})$ are the weak nuclear structure functions
which have contributions from the spectral function only,
$F_{iA,\pi/\rho}(x,Q^{2})$ take into account mesonic contributions.
$F_{iA,shd}(x,Q^{2})$ has contribution from the shadowing effect.

Figure 7: Results of EM(Left panel) and Weak(Right panel) nuclear structure
functions in ${}^{56}Fe$(isoscalar) obtained using spectral function(dashed
double-dotted line), including mesonic contribution(double dashed-dotted
line), full model(solid line). These results are presented for different
$Q^{2}$ and are compared with the available experimental data of JLab(solid
circle) Mamyan:2012th , CDHSW(semi solid circle) Berge:1989hr , CCFR(triangle
up) Oltman:1992pq , NuTeV(diamond) Tzanov:2005kr .

In this model, the full expression for the parity violating weak nuclear
structure function $F_{3A}(x,Q^{2})$ is given by,

$\displaystyle F_{3A}(x,Q^{2})=F_{3A,N}(x,Q^{2})+F_{3A,shd}(x,Q^{2}).$ (47)

Notice that this structure function has no mesonic contribution. The
contribution to the nucleon structure function mainly comes from the valence
quarks distributions. For $F_{3A,shd}(x,Q^{2})$ similar definition has been
used as given in Eq.(45) following the works of Kulagin et al. Kulagin:2004ie
.

Figure 8: Results for $F_{2A}^{\nu+\bar{\nu}}(x,Q^{2})$ (top panel) and
$xF_{3A}^{\nu+\bar{\nu}}(x,Q^{2})$ (bottom panel) vs $Q^{2}$ are shown at
$x=0.275$ and 0.45 in 56Fe . The results are obtained with the spectral
function only (dashed line) and with the full model (solid line) at NNLO and
are compared with the results of the available experimental data Berge:1989hr
; Oltman:1992pq ; Tzanov:2005kr . Iron is treated as an isoscalar nuclear
target.

###  3.1 Nucleon Structure Function and Cross Section

In Fig. 5, the results for the free nucleon weak structure functions
$2xF_{1N}(x,Q^{2})$, $F_{2N}(x,Q^{2})$, $xF_{3N}(x,Q^{2})$, $F_{4N}(x,Q^{2})$
and $2xF_{5N}(x,Q^{2})$ (from the top to bottom) are shown at the two
different values of $Q^{2}$ viz. $Q^{2}=2$ GeV2 (left panel) and $Q^{2}=5$
GeV2 (right panel). The nucleon structure functions are presented at NLO
without the TMC effect (double dash-dotted line), with the TMC effect in
3-flavor(dash-dotted line:nf3) and 4-flavor(dotted line:nf4) schemes, with TMC
and HT effects in 3-flavor(dashed line: nf3) and 4-flavor(solid line:nf4)
schemes. From the figure, it may be noticed that the TMC effect is dominant in
the region of high $x$ and low $Q^{2}$ and it becomes small at low $x$ and
high $Q^{2}$. Quantitatively, the TMC effect is found to be different in
$F_{2N}(x,Q^{2})$ from $F_{1N}(x,Q^{2})$ while the TMC effect in
$F_{5N}(x,Q^{2})$ is similar to the effect in $F_{2N}(x,Q^{2})$. However, in
the case of $F_{4N}(x,Q^{2})$ the whole contribution, arises in the leading
order due to the TMC effect at mid and high $x$. $F_{4N}(x,Q^{2})$ contributes
to the cross section due to the non-vanishing lepton mass when it is large,
and contributes only in the region of $x\leq 0.2$. We find that at NLO,
$F_{4N}(x,Q^{2})$ becomes almost negligible in the region of $x>0.2$ when TMC
effect is not incorporated but with the inclusion of TMC effect a nonzero
though small contribution in the region of high $x$ and low $Q^{2}$ has been
found. The difference in the results of free nucleon structure functions
$F_{iN}(x,Q^{2});~{}~{}(i=1-5)$ evaluated at NLO with and without the TMC
effect at $x=0.3$ is $5\%(3\%)$ in $2xF_{1N}(x,Q^{2})$, $2\%(<1\%)$ in
$F_{2N}(x,Q^{2})$, $7\%(\sim 3\%)$ in $xF_{3N}(x,Q^{2})$ and $4\%(\sim 2\%)$
in $2xF_{5N}(x,Q^{2})$ for $Q^{2}=2(5)$ GeV2.

For the first three structure functions ($F_{iN}(x,Q^{2});~{}~{}i=1-3$), the
HT effect has also been studied. This is found to be comparatively small in
$F_{1N}(x,Q^{2})$ and $F_{2N}(x,Q^{2})$ than in $F_{3N}(x,Q^{2})$. Due to
higher twist(HT) corrections, we have observed a decrease in the value of
$F_{3N}(x,Q^{2})$, which becomes small with the increase in $Q^{2}$. To show
the effect of massive charm on the nucleon structure functions, we have
compared the results obtained with the TMC and HT effects for the three flavor
of massless quarks ($n_{f}=3$) with the results when additional contribution
from the massive charm quark ($n_{f}=4$) have also been considered. It is
found that massive charm effect is almost negligible in the kinematic region
of low $Q^{2}$ and high $x$ while it increases with the increase in $Q^{2}$
and is appreciable at low $x$.

Figure 9: $\frac{d\sigma_{A}/dx}{d\sigma_{CH}/dx}~{}(A=$ 56Fe, 208Pb) vs $x$
for incoming neutrino beam of energies $E=7$ GeV and $25$ GeV. The numerical
results are obtained with the full model (solid line: $E=25$ GeV, solid line
with star: $E=25$ GeV and double dash-dotted line: $E=7$ GeV) at NNLO and are
compared with the simulated results Mousseau:2016snl . The solid squares are
the experimental points of MINERvA Mousseau:2016snl . The results are shown
for the isoscalar nuclear targets.

In Fig. 6, we have shown our results for the total scattering cross section
$\sigma/E_{\nu}$ vs $E_{\nu}$, for the DIS of $\nu_{\tau}$ and
${\bar{\nu}}_{\tau}$ from the nucleons and compared them with the results of
the different models available in the literature like that of Pashchos et al.
Paschos:2001np  (dashed line with diamond), Kretzer et al. Kretzer:2002fr
(solid line with right triangle without a cut on $W$; dotted line with a cut
of $W>1.4$ GeV), Jeong et al.Jeong:2010nt  (dash-dotted line), Gazizov et
al.Gazizov:2016  (solid line with down triangle), Hagiwara et
al.Hagiwara:2003di  (solid line with cross symbol), Anelli et
al.Anelli:2015pba  (double dash-dotted line) and Li et al. Li:2017dbe  (solid
line with circles) as well as with the predictions made by the Monte Carlo
generator GENIE GENIE  and NuWro NuWro . The results are presented for the two
cases of cut on the center of mass energy(W), taken to be W=1.4 GeV(shown by
dashed line) and 2 GeV(shown by the solid line). The results are presented by
incorporating the target mass correction and higher twist effects at NLO in
the four flavor scheme. Our results with a cut of $W=1.4~{}GeV$ (shown by
dashed line) is in good agreement with the result of Kretzer et al.
Kretzer:2002fr  (shown by the dotted line) while there are significant
differences from the result of Jeong et al. Jeong:2010nt  (shown by the dash-
dotted line). Notice that the results of the total scattering cross section
with the same CoM energy cut reported by Kretzer and Reno Kretzer:2002fr  and
Jeong and Reno Jeong:2010nt  are also different with each other. The
difference is mainly due to the choice of lower cuts on $Q^{2}$ in the
evaluation of PDFs. It is important to point out that the results given by the
different models Kretzer:2002fr ; Jeong:2010nt ; Hagiwara:2003di ;
Conrad:2010mh ; Paschos:2001np ; Gazizov:2016 ; Li:2017dbe ; Anelli:2015pba
have significant differences due to their choice of different kinematic
conditions. Furthermore, we have observed that the dependence of the cross
section on the effect of CoM energy cut is more pronounced in the case of
$\bar{\nu}_{\tau}-N$ DIS than in $\nu_{\tau}-N$ DIS process. Moreover, one may
also notice that the total scattering cross section is quite sensitive to the
kinematic cut on the CoM energy. It implies that a suitable choice of the CoM
energy cut as well as the four momentum transfer square ($Q^{2}$) cut to
define the deep inelastic region and use them to calculate the nucleon
structure functions, differential and total scattering cross sections is quite
important. These kinematic considerations should be kept in mind while
comparing the predictions of the cross sections in various theoretical models.

Figure 10: Predictions for the differential scattering cross section vs y, at
the different values of x for 6.25 GeV $\nu$-Ar treated as an isoscalar
target. The results are obtained with a $Q^{2}\geq 1.0~{}GeV^{2}$ cut by the
Aligarh-Valencia model using CTEQ 6.6 nucleon PDFs Nadolsky:2008zw  at NLO in
the MS-bar scheme (solid line). The nCTEQnu nuclear PDFs private  based
prediction is the blue dash-dotted line.

###  3.2 Nuclear Structure Function and Cross Section

In Fig. 7, we present the results for the weak nuclear structure function
$F_{2A}^{Weak}(x,Q^{2})$ and compare them with the electromagnetic structure
function $F_{2A}^{EM}(x,Q^{2})$ as a function of $x$ for the two different
values of $Q^{2}$ viz. 2 and 20 $GeV^{2}$. In this figure, we show the curves
for $F_{2A}^{EM,Weak}(x,Q^{2})$ obtained using the spectral function only,
also including the mesonic contribution, and finally using the full model
which also includes shadowing. The use of nuclear spectral function leads to a
reduction of $\sim 8\%$ at $x=0.1$; $\sim 18\%$ at $x=0.4$; $\sim 3\%$ at
$x=0.7$ in $F_{2A}^{EM}(x,Q^{2})$ as well as in $F_{2A}^{Weak}(x,Q^{2})$
nuclear structure functions as compared to the free nucleon case. The
inclusion of mesonic contributions from pion and rho mesons leads to an
enhancement in these structure functions at low and medium values of $x$. For
example, the enhancement is $\sim 30\%$ at $x$=0.1; $\sim 15\%$ at $x$=0.4;
$\sim 0.3\%$ at $x$=0.7. The inclusion of shadowing effects further reduces
these structure functions and are effective in low region of
$x~{}(~{}<~{}0.1)$. For example, the reduction is $\sim 10\%$ at $x$=0.05 and
$\sim 5\%$ at $x$=0.1.

In Fig. 8, we present the results for the (anti)neutrino induced processes in
$F_{2A}^{\nu+\bar{\nu}}(x,Q^{2})$ and $xF_{3A}^{\nu+\bar{\nu}}(x,Q^{2})$ vs
$Q^{2}$ in 56Fe by treating it as an isoscalar nuclear target at the different
values of $x$ using the full model at NNLO and compared them with the
available experimental data from CCFR Oltman:1992pq , CDHSW Berge:1989hr  and
NuTeV Tzanov:2005kr  experiments. We find a good agreement between the
theoretical results for $F_{2A}^{\nu+\bar{\nu}}(x,Q^{2})$ and reasonable
agreement for $F_{3A}^{\nu+\bar{\nu}}(x,Q^{2})$ with the experimental data. It
can be seen that the description of the nuclear medium effect in our model is
able to reproduce the experimental results.

Furthermore, in Fig.9, we have presented the results for the ratio of the
differential scattering cross sections, for the different nuclear targets viz.
A= 56Fe and 208Pb to the hydrocarbon(CH) target i.e.
$\frac{d\sigma_{A}/dx}{d\sigma_{CH}/dx}$ (A= 56Fe, 208Pb) vs $x$ at the two
different values of neutrino energies viz. $E_{\nu}=7$ GeV and $E_{\nu}=25$
GeV and compared them with the corresponding experimental data of MINERvA
Mousseau:2016snl . It may be noticed that MINERvA’s experimental data have
large error bars and the wide band around the simulation is due to the
systematic errors which shows an uncertainty up to $\sim 20\%$
Mousseau:2016snl .

Fig. 10 shows predictions for the differential scattering cross section for
$\nu_{\mu}$ induced reaction on 40Ar nuclear target at $E_{\nu_{\mu}}$=6.25
GeV. The numerical calculations are done by using the CTEQ6.6 PDFs
parameterization Nadolsky:2008zw . Our theoretical results obtained using the
full model are compared with the results of nCTEQnu nPDFs private . As $y$
decreases, the nPDF approach predicts lower cross sections than the
theoretical approach. For high-$x$ ($\gtrapprox$ 0.5), the nPDF approach and
the theoretical approach both predict quite similar cross sections while for
the low-$x$ ($\lessapprox$ 0.3) region the nPDF approach predicts a lower
cross section than our theoretical model.

###  3.3 Nonisoscalarity Effect

For a nonisoscalar nuclear target like ${}^{56}Fe$ and ${}^{208}Pb$, where
$(A-Z)>Z$, we obtain the normalized spectral function to the proton
($S_{h}^{p}$) and neutron ($S_{h}^{n}$) numbers (Eqs.34, 32 and 33). For
details please see the discussion in Ref.Zaidi:2019mfd .

Figure 11: $\frac{F_{iA}(x,Q^{2})}{F_{iA^{\prime}}(x,Q^{2})};~{}(i=1-3$;
$A=^{56}$Fe, 208Pb; $A^{\prime}=^{12}$C) vs $x$ are shown at $Q^{2}=5$ GeV2.
The results are obtained using the full model at NNLO by treating 56Fe, 208Pb
to be isoscalar (solid line) as well as nonisoscalar (dashed line) nuclear
targets.

In Fig.11, we have presented the results for the ratios of nuclear structure
functions
$\frac{F_{iA}(x,Q^{2})}{F_{iA^{\prime}}(x,Q^{2})};~{}(i=1,2,3;~{}A=^{56}Fe,~{}^{208}Pb~{}\textrm{and}~{}A^{\prime}=^{12}C)$
vs $x$ at $Q^{2}=5$ GeV2. The present numerical results are obtained by using
the full model at NNLO and treating iron and lead to be isoscalar as well as
nonisoscalar nuclear targets. We find that the ratios
$\frac{F_{iFe}(x,Q^{2})}{F_{iC}(x,Q^{2})}$ and
$\frac{F_{iPb}(x,Q^{2})}{F_{iC}(x,Q^{2})}$ deviate from unity in the entire
range of $x$ which implies that nuclear medium effects are $A$ dependent, and
the medium effects become more pronounced with the increase in the nuclear
mass number. This enhancement in the ratio is more at high $x$.

##  4 Summary

  1. 1.

The inclusion of perturbative and nonperturbative effects is quite important
in the evaluation of nucleon structure functions. These effects are both $x$
and $Q^{2}$ dependent.

  2. 2.

The difference in the results of all the free nucleon structure functions
$F_{iN}(x,Q^{2});~{}~{}(i=1-5)$ evaluated at NLO with and without the TMC
effect is non-negligible. In the case of $F_{4N}(x,Q^{2})$ this difference is
quite large when TMC effect is considered, specially at mid and high $x$. The
higher twist(HT) effect has not much effect on $F_{1N}(x,Q^{2})$ and
$F_{2N}(x,Q^{2})$, but affects $F_{3N}(x,Q^{2})$. We find that the difference
due to HT effect is somewhat larger for $F_{3N}(x,Q^{2})$ at low $x$ and low
$Q^{2}$. With the increase in $x$ the effect of HT increases.

  3. 3.

The total cross section $\sigma$ for $\nu_{\tau}$ and ${\bar{\nu}}_{\tau}$
interactions on the free nucleon target shows significant dependence on the
model used to calculate the nucleon structure functions which implies that
there are uncertainties even at the level of free nucleon level and more work
is needed.

  4. 4.

NME in $F_{2A}^{EM}(x,Q^{2})$ and $F_{2A}^{Weak}(x,Q^{2})$ in iron nucleus are
almost the same for $x$ i.e $x~{}>~{}0.3$, but are different in small $x$
region($x<0.3$). However, this difference is found to be very small for
isoscalar nuclei, and is similar to the case of free nucleon.

  5. 5.

The nuclear structure functions obtained with spectral function only are
suppressed as compared to the free nucleon case in the entire region of $x$.
Whereas, the inclusion of mesonic contributions results in an enhancement in
the nuclear structure functions in the low and intermediate region of $x$.
Mesonic contributions are observed to be more pronounced with the increase in
mass number and they decrease with the increase in $x$ and $Q^{2}$. The
results for the nuclear structure functions $F_{2A}^{Weak}(x,Q^{2})$ and
$F_{3A}^{Weak}(x,Q^{2})$ with the full theoretical model show good agreement
with the experimental data of CCFR Oltman:1992pq , CDHSW Berge:1989hr  and
NuTeV Tzanov:2005kr  especially at high $x$ and high $Q^{2}$.

  6. 6.

The present theoretical results for the ratio
$\frac{d\sigma_{A}^{Weak}/dx}{d\sigma_{CH}^{Weak}/dx}~{}(A=^{56}Fe,~{}^{208}Pb)$
when compared with the different phenomenological models and the recent
MINERvA’s experimental data on $\nu_{l}-A$ scattering, imply that a better
understanding of nuclear medium effects is required for the
$\nu_{l}(\bar{\nu}_{l})-$nucleus deep inelastic scattering process.

  7. 7.

We find the non-isoscalarity effect to be non-negligible which increases with
the non-isoscalarity $\delta(=\frac{N-Z}{A})$ and $x$.

  8. 8.

Predictions are also made for $\nu_{\mu}(\bar{\nu}_{\mu})$ interactions on
${}^{40}Ar$ that may be useful in analyzing the experimental results of DUNE
Abi:2018alz .

## Acknowledgment

F. Zaidi is thankful to the Council of Scientific & Industrial Research
(CSIR), India, for providing the research associate fellowship with award
letter no. 09/112(0622)2K19 EMR-I. I. Ruiz Simo acknowledges support from
Spanish Ministry of Science and ERDF under contract FIS2017-85053-C2-1P, and
from Junta de Andalucia through grant No FQM-225.

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