#  $\bar{\nu}_{\mu}$ induced quasielastic production of hyperons leading to
pions

A. Fatima<EMAIL_ADDRESS>Department of Physics, Aligarh Muslim
University, Aligarh-202002, India  M. Sajjad Athar  Department of Physics,
Aligarh Muslim University, Aligarh-202002, India  S. K. Singh  Department of
Physics, Aligarh Muslim University, Aligarh-202002, India

###### Abstract

The quasielastic production cross sections and polarizations of the hyperons
induced by ${\bar{\nu}}_{\mu}$ on the free nucleon as well as from 40Ar in the
sub-GeV energy region has been reviewed Fatima:2018wsy ; Fatima:2018tzs ;
Akbar:2016awk ; Fatima:2018gjy ; Akbar:2017qsf . The effects of the second
class currents in the axial vector sector with and without T-invariance as
well as the effect of SU(3) symmetry breaking are also studied. We find that
the cross sections and the various polarization components can effectively be
used to determine the axial vector transition form factors in the strangeness
sector and to test the validity of various symmetries of the weak hadronic
currents like G-invariance, T-invariance and SU(3) symmetry.
These hyperons decay dominantly into pions giving an additional contribution
to the weak pion production induced by the antineutrinos. In the case of
nuclear targets like 40Ar, this contribution is shown to be significant when
compared with the the pion production by the $\Delta$ excitations in the
energy range of $E_{\bar{\nu}_{\mu}}\leq 0.7$ GeV Fatima:2018wsy . This study
could be useful for the DUNE experiment where argon will be used as the target
material and the electronic imaging of particles is possible and the particle
tracks can be identified.

##  I Introduction

The goal of the present day experimenters using accelerator neutrino and
antineutrino beams at the ongoing experiments like T2K, NOvA, etc., or the
planned experiments like DUNE at the Fermilab fermi , T2HK in J-PARC JPARC ,
is to determine the neutrino oscillation parameters with greater precision,
and to search for the CP violation in the leptonic sector.

For this, the simultaneous knowledge of neutrino and antineutrino cross
sections in the same energy region for a given nuclear target is required.
Almost all the current generation neutrino experiments are using moderate to
heavy nuclear targets, where the (anti)neutrino-nucleus cross section gets
modified due to the nuclear medium effects. In the few GeV energy region of
neutrinos and antineutrinos, the contribution to the total scattering cross
section comes from the quasielastic, inelastic and deep inelastic scattering
of (anti)neutrinos with the nuclear targets. In this energy region, the
inelastic channel is dominated by the single pion production, followed by the
kaon production, associated particle production, multi-pion production, etc.
In the case of antineutrinos, apart from the aforementioned processes, the
quasielastic hyperon ($\Lambda$, $\Sigma^{0}$ and $\Sigma^{-}$) production
also contributes to the total scattering cross section. The produced hyperons
then decay into a nucleon and a pion, thus, giving additional contribution to
the single pion production. This mode of pion production is important in the
low energy region, even though, the dominant contribution to the single pion
production comes from the $\Delta$ excitations. We have estimated that in the
low energy region of $E_{{\bar{\nu}}_{\mu}}$ $\leq$ 0.7 GeV, the pions arising
from the hyperon decays are comparable to the pions arising from the $\Delta$
excitations Fatima:2018wsy . This is because the pion production arising from
the $\Delta$ excitations is considerably suppressed due to the nuclear medium
effects (NME) and the final state interaction (FSI) effects as the $\Delta$
properties like its mass and decay width are modified in the nuclear medium.
Furthermore, these deltas decay instantly to pions, which travel through the
nucleus and undergo final state interactions with the residual nucleus
resulting in pion absorption or charge exchange reactions. On the other hand,
the nuclear medium effects on the hyperon production cross sections are quite
small. Moreover, the final state interactions of the produced hyperons, due to
the strong interactions, in the presence of the nucleons in the nuclear medium
leading to elastic and charge exchange reactions like $\Sigma
N\rightarrow\Lambda N$ and $\Lambda N\rightarrow\Sigma N$ affects only the
relative yield of various hyperons ($\Lambda$ and $\Sigma$). The total number
of pions produced through the different $Y\rightarrow N\pi$ decay modes is,
therefore, least affected by the $YN\rightarrow YN$ final state interaction
effects. In addition, these pions are expected to be less affected by the
final state interaction with the residual nucleus. This is because the hyperon
decay widths are highly suppressed in the nuclear medium making them live
longer enabling them to travel through most of the nuclear medium before they
decay. Therefore, the cumulative effects of the lower threshold energy for the
hyperon production compared to the delta production, and the near absence of
the FSI for the pions coming from the hyperon decay compensate for the Cabibbo
suppression as compared to the pions coming from the $\Delta$ excitations.
This makes the study of hyperon production processes important in the context
of oscillation experiments with antineutrinos in the sub-GeV energy region.

The study of single hyperon production is important in its own right as it
provides information about the nucleon-hyperon transition form factors at high
$Q^{2}$, which is presently known only at low $Q^{2}$ from the study of the
semileptonic decays of neutron and hyperons. The symmetries of the weak
hadronic current like the T-invariance, G-invariance and SU(3) symmetry can
also be tested, and some information on the physics beyond the standard model
may also be obtained. In the past, various calculations on the charged current
induced (anti)neutrino and electron scattering from the free nucleon and
nuclear targets, have been done by many authors in which both the production
cross sections as well as the polarization observables of muons and baryons
were studied Akbar:2016awk ; Fatima:2018gjy ; Akbar:2017qsf ; Bilenky:2013fra
; Alam:2014bya ; Bilenky:2013iua ; Graczyk:2017rti ; Kuzmin:2003ji ;
Wu:2013kla ; Mintz:2001jc ; Mintz:2002cj ; Mintz:2004eu . With the advancement
in the detection technology and the observation of the $\tau$ neutrino
($\nu_{\tau}$) experimentally, some calculations for the polarization
observables of the $\tau$ leptons produced in the
$\nu_{\tau}(\bar{\nu}_{\tau})$ induced reactions have also been performed in
the weak processes Kuzmin:2003ji ; Fatima:2020pvv ; Kuzmin:2004ke ;
Graczyk:2004vg ; Hagiwara:2004gs ; Graczyk:2004uy ; Valverde:2006yi .
Recently, Thorpe et al. Thorpe:2020tym  have also studied the effect of second
class currents, axial dipole mass and SU(3) symmetry violation on the total
and differential scattering cross sections in the antineutrino induced single
hyperon production. We have also studied the dependence of different vector
and axial vector current form factors including the second class current form
factors on the total and differential cross section and have tested the degree
of time reversal and G-parity violations in the antineutrino as well as
electron induced single hyperon production Fatima:2018wsy ; Fatima:2018tzs ;
Akbar:2016awk ; Fatima:2018gjy ; Akbar:2017qsf ; Fatima:2020pvv . In this
work, we have extended our model Fatima:2018tzs ; Fatima:2018gjy  for the
production cross section and polarization observables of the hyperons produced
in the antineutrino and electron induced reactions by taking into account the
effect of SU(3) symmetry breaking, following the formalism of Faessler et al.
Faessler:2008ix  in the determination of the vector and axial vector form
factors in the strangeness sector.

We have studied the polarization components of the hyperons produced in the
antineutrino induced reactions and the effect of the various weak form factors
on these polarization observables. The longitudinal ($P_{L}$) and
perpendicular ($P_{P}$) components of polarization lie in the plane while the
transverse ($P_{T}$) component of polarization lies perpendicular to the
reaction plane and is forbidden by T-invariance. The experimental measurement
of the non-zero value of the transverse component of the polarized hyperon
will directly show the violation of time reversal in the weak sector. Recently
developed liquid argon time-projection chamber (LArTPC) technology which is
being used in the MicroBooNE and ICARUS experiments at the Fermilab and
planned to be used in the proposed DUNE (The Deep Underground Neutrino
Experiment) experiment DUNE ; Acciarri:2016ooe , with an expected fiducial
mass of about 40kT, holds very high promises as LArTPC provide excellent
tracking and calorimetry, real time full imaging of events where the detectors
would be capable of identifying and measuring precisely the neutrino events
over a wide range of energies. It would be possible to study the hyperon
($\Lambda,\Sigma$) production and its polarization by observing the asymmetry
in the angular distribution of the pions when the hyperons decay into $N\pi$,
as LArTPC gives the 3-dimensional track of the interaction. Therefore, it is
feasible to study the physics of T-violation in the leptonic sector by
measuring the polarization components of the hyperons produced in the
antineutrino reactions Fatima:2018wsy ; Fatima:2018tzs ; Fatima:2020pvv .

In this work, we have focused our attention especially on the $\Lambda$
production in the antineutrino–nucleon/nucleus reactions, which is the most
feasible channel to study experimentally. In section-II, we describe, in
brief, the formalism for calculating the cross sections for the single hyperon
production and the delta production from the free nucleon target. The
expression for the polarization components viz., $P_{L}$, $P_{P}$ and $P_{T}$
of the hyperon produced in the quasielastic reactions, are given in section-
II.2. Section III deals with the effect of the nuclear medium on the $\Delta$
and the hyperon productions, and the final state interactions of the hyperons
in the nuclear medium. The final state interactions of the produced pions are
discussed in section-IV. In section-V, we present our results and finally in
section-VI, we summarize and conclude the findings.

##  II Formalism

###  II.1 Hyperon production off the free nucleon

####  II.1.1 Matrix element and form factors

The general expression of the differential scattering cross section
corresponding to the process

$\displaystyle\bar{\nu}_{\mu}(k)+N(p)$ $\displaystyle\rightarrow$
$\displaystyle\mu^{+}(k^{\prime})+Y(p^{\prime});\qquad\quad N=n,p;\qquad\quad
Y=\Lambda,\Sigma$ (1)

in the rest frame of the initial nucleon, is written as:

$\displaystyle d\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\frac{1}{4E_{\bar{\nu}_{\mu}}M}\delta^{4}(k+p-k^{\prime}-p^{\prime})\frac{d^{3}k^{\prime}}{2E_{k^{\prime}}}\frac{d^{3}p^{\prime}}{2E_{p^{\prime}}}\overline{\sum}\sum|{\cal{M}}|^{2},$
(2)

where the transition matrix element squared is expressed as:

$\overline{\sum}\sum|{\cal{M}}|^{2}=\frac{G_{F}^{2}\sin^{2}\theta_{c}}{2}\cal{L}_{\mu\nu}\cal{J}^{\mu\nu},$
(3)

with $G_{F}$ being the Fermi coupling constant and $\theta_{C}$ being the
Cabibbo angle. The leptonic ($\cal{L}_{\mu\nu}$) and the hadronic
($\cal{J}^{\mu\nu}$) tensors are given by

$\displaystyle\cal{L}^{\mu\nu}$ $\displaystyle=$
$\displaystyle~{}\mathrm{Tr}\left[\gamma^{\mu}(1+\gamma_{5})\Lambda(k)\gamma^{\nu}(1+\gamma_{5})\Lambda(k^{\prime})\right],$
(4) $\displaystyle\cal{J}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{Tr}\left[\Lambda({p^{\prime}})J_{\mu}\Lambda({p})\tilde{J}_{\nu}\right],\qquad\quad\tilde{J}_{\nu}=\gamma^{0}J^{\dagger}_{\nu}\gamma^{0}.$
(5)

The leptonic current is given by

$l^{\mu}=\bar{u}(k^{\prime})\gamma^{\mu}(1+\gamma_{5})u(k),$ (6)

and the hadronic current is defined as

$\displaystyle{{J}}_{\mu}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\left(\left[\gamma_{\mu}f_{1}^{NY}(Q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M+M^{\prime}}f_{2}^{NY}(Q^{2})+\frac{2~{}q_{\mu}}{M+M^{\prime}}f_{3}^{NY}(Q^{2})\right]\right.$
(7) $\displaystyle-$
$\displaystyle\left.\left[\gamma_{\mu}\gamma_{5}g_{1}^{NY}(Q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M+M^{\prime}}\gamma_{5}g_{2}^{NY}(Q^{2})+\frac{2~{}q_{\mu}}{M+M^{\prime}}\gamma_{5}g_{3}^{NY}(Q^{2})\right]\right)u(p),~{}~{}~{}$

with $M$ and $M^{\prime}$ being the masses of the initial nucleon and the
final hyperon. $q_{\mu}(=k_{\mu}-k_{\mu}^{\prime}=p_{\mu}^{\prime}-p_{\mu})$
is the four momentum transfer with $Q^{2}=-q^{2},~{}Q^{2}\geq 0$.
$f_{1}^{NY}(Q^{2})$, $f_{2}^{NY}(Q^{2})$ and $f_{3}^{NY}(Q^{2})$ are the
vector, weak magnetic and induced scalar form factors and $g_{1}^{NY}(Q^{2})$,
$g_{2}^{NY}(Q^{2})$ and $g_{3}^{NY}(Q^{2})$ are the axial vector, induced
tensor (or weak electric) and induced pseudoscalar form factors, respectively.

Using the above definitions, the $Q^{2}$ distribution is written as

$\frac{d\sigma}{dQ^{2}}=\frac{G_{F}^{2}~{}sin^{2}\theta_{c}}{16\pi
M^{2}E_{\bar{\nu}_{\mu}}^{2}}N(Q^{2}),$ (8)

where the expression of $N(Q^{2})$ is given in the Appendix-A of Ref.
Fatima:2018tzs .

For the determination of the vector and axial vector $N-\Lambda$ transition
form factors, we take the following considerations into account:

  1. a)

The assumption of the SU(3) symmetry Cabibbo:2003cu  and the conserved vector
current (CVC) hypothesis leads to $f_{3}^{N\Lambda}(Q^{2})=0$ and the two
vector form factors viz. $f^{N\Lambda}_{1}(Q^{2})$ and
$f^{N\Lambda}_{2}(Q^{2})$ are determined in terms of the electromagnetic form
factors of the nucleon, i.e. $f_{1}^{N}(Q^{2})$ and
$f_{2}^{N}(Q^{2}),~{}N=(p,n)$ as

$\displaystyle f_{1,2}^{p\Lambda}(Q^{2})$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{2}}~{}f_{1,2}^{p}(Q^{2}).$ (9)

The electromagnetic form factors are expressed in terms of the Sachs’ electric
and magnetic form factors $G_{E}^{p,n}(Q^{2})$ and $G_{M}^{p,n}(Q^{2})$ of the
nucleons, for which various parameterizations are available in the literature
and in our numerical calculations, we have used the parameterization given by
Bradford et al. Bradford:2006yz .

  2. b)

Assuming the SU(3) symmetry, the axial vector form factors
$g_{1,2}^{p\Lambda}(Q^{2})$ are rewritten in terms of $g_{1,2}^{pn}(Q^{2})$
and
$x_{1,2}(Q^{2})=\frac{F^{A}_{1,2}(Q^{2})}{F^{A}_{1,2}(Q^{2})+D^{A}_{1,2}(Q^{2})}$
as

$\displaystyle g_{1,2}^{p\Lambda}(Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{6}}(1+2x_{1,2})g_{1,2}^{np}(Q^{2}).$ (10)

We further assume that $F^{A}_{1,2}(Q^{2})$ and $D^{A}_{1,2}(Q^{2})$ have the
same $Q^{2}$ dependence, such that $x_{1,2}(Q^{2})$ become constant given by
$x_{1,2}(Q^{2})=x_{1,2}=\frac{F^{A}_{1,2}(0)}{F^{A}_{1,2}(0)+D_{1,2}^{A}(0)}$.

For the axial vector form factor $g_{1}^{pn}(Q^{2})$, a dipole
parameterization has been used:

$\displaystyle
g_{1}^{pn}(Q^{2})=g_{A}(0)\left(1+\frac{Q^{2}}{M_{A}^{2}}\right)^{-2},$ (11)

where $M_{A}$ is the axial dipole mass and $g_{A}(0)$ is the axial charge. For
the numerical calculations, we have used the world average value of
$M_{A}=1.026$ GeV. $g_{A}(0)$ and $x_{1}$ are taken to be 1.2723 and 0.364,
respectively, as determined from the experimental data on the $\beta-$decay of
neutron and the semileptonic decay of hyperons.

$g_{2}^{pn}(Q^{2})$ is taken to be of the dipole form, in analogy with
$g_{1}^{np}(Q^{2})$, i.e.,

$\displaystyle
g_{2}^{pn}(Q^{2})=g_{2}^{pn}(0)\left(1+\frac{Q^{2}}{M_{2}^{2}}\right)^{-2}.$
(12)

In the numerical calculations we have taken real as well as imaginary values
of $g_{2}^{pn}(0)$, with $|g_{2}(0)|$ varying in the range $0-2$
Fatima:2018tzs .

  3. c)

The pseudoscalar form factor $g_{3}^{N\Lambda}(Q^{2})$ is proportional to the
lepton mass and the contribution is small in the case of muon antineutrino
scattering. However, in the numerical calculations, we have taken the
following expression given by Nambu Nambu:1960xd  using the generalized
Goldberger-Treiman relation Goldberger:1958vp ,

$\displaystyle
g_{3}^{N\Lambda}(Q^{2})=\frac{(M+M^{\prime})^{2}}{2~{}(m_{K}^{2}+Q^{2})}g_{1}^{N\Lambda}(Q^{2}),$
(13)

where $m_{K}$ is the mass of the kaon.

For the discussion of $N-\Sigma$ transition form factors, please see Refs.
Fatima:2018tzs ; Fatima:2018gjy .

####  II.1.2 SU(3) symmetry breaking effects

We have included the effect of SU(3) symmetry breaking on the vector and axial
vector form factors following Ref. Faessler:2008ix , where the main features
of the model may be summarized as:

  * i)

At the leading order, there is no symmetry breaking effect for the vector form
factor $f_{1}^{N\Lambda}(Q^{2})$ because of the Ademollo-Gatto theorem
Ademollo:1964sr .

  * ii)

The weak magnetic ($f_{2}^{N\Lambda}(Q^{2})$) and axial vector
($g_{1}^{N\Lambda}(Q^{2})$) form factors transform as

${\cal
F}^{N\Lambda}=-\sqrt{\frac{3}{2}}\left(F+\frac{D}{3}+\frac{1}{9}\left(H_{1}-2H_{2}-3H_{3}-6H_{4}\right)\right),$
(14)

where $F$ and $D$ are the SU(3) symmetric couplings while $H_{i};i=1-4$ are
the SU(3) symmetry breaking couplings. The values of these parameters are
given in Ref. Faessler:2008ix , and are here quoted for
$f_{2}^{N\Lambda}(Q^{2})$:

$\displaystyle D$ $\displaystyle=$ $\displaystyle 1.237,\qquad\quad
F~{}=~{}0.563,\qquad\quad H_{1}~{}=~{}-0.246$ $\displaystyle H_{2}$
$\displaystyle=$ $\displaystyle 0.096,\qquad\quad
H_{3}~{}=~{}0.021,\qquad\quad H_{4}~{}=~{}0.030$

and for $g_{1}^{N\Lambda}(Q^{2})$:

$\displaystyle D$ $\displaystyle=$ $\displaystyle 0.7505,\qquad\quad
F~{}=~{}0.5075,\qquad\quad H_{1}~{}=~{}-0.050$ $\displaystyle H_{2}$
$\displaystyle=$ $\displaystyle 0.011,\qquad\quad
H_{3}~{}=~{}-0.006,\qquad\quad H_{4}~{}=~{}0.037.$

For the $Q^{2}$ dependence of these SU(3) breaking couplings, we assume the
dipole parameterization of the form

$H_{i}^{V,A}=\frac{H_{i}}{\left(1+\frac{Q^{2}}{M_{V,A}^{2}}\right)^{2}}$ (15)

where $V,A$, respectively, represents the vector and axial vector form factor,
$M_{V}=0.84$ GeV and $M_{A}=1.026$ GeV represent the corresponding vector and
axial dipole masses.

  * iii)

In the case of the second class currents, we have not considered any SU(3)
symmetry breaking effects.

  * iv)

Since the pseudoscalar form factor is parameterized in terms of the axial
vector form factor, therefore, it receives the SU(3) symmetry breaking effect
from $g_{1}^{N\Lambda}(Q^{2})$.

###  II.2 Polarization of the hyperon

Using the covariant density matrix formalism, the polarization 4-vector
($\xi^{\tau}$) of the hyperon produced in the reaction given in Eq. (1) is
written as Bilenky :

$\displaystyle\xi^{\tau}$ $\displaystyle=$
$\displaystyle\left(g^{\tau\sigma}-\frac{p^{\prime\tau}p^{\prime\sigma}}{{M^{\prime}}^{2}}\right)\frac{{\cal
L}^{\alpha\beta}\mathrm{Tr}\left[\gamma_{\sigma}\gamma_{5}\Lambda(p^{\prime})J_{\alpha}\Lambda(p)\tilde{J}_{\beta}\right]}{{\cal
L}^{\alpha\beta}\mathrm{Tr}\left[\Lambda(p^{\prime})J_{\alpha}\Lambda(p)\tilde{J}_{\beta}\right]}.$
(16)

Figure 1: Polarization observables of the hyperon. $\hat{e}_{L}$,
$\hat{e}_{P}$ and $\hat{e}_{T}$ represent the orthogonal unit vectors
corresponding to the longitudinal, perpendicular and transverse directions
with respect to the momentum of the final baryon.

One may write the polarization vector $\bm{\xi}$ in terms of the three
orthogonal vectors $\hat{e}_{i}~{}(i=L,P,T)$, i.e.

$\bm{\xi}=\xi_{L}\hat{e}_{L}+\xi_{P}\hat{e}_{P}+\xi_{T}\hat{e}_{T},$ (17)

where $\hat{e}_{L}$, $\hat{e}_{P}$ and $\hat{e}_{T}$ are chosen to be the set
of orthogonal unit vectors corresponding to the longitudinal, perpendicular
and transverse directions with respect to the momentum of the hyperon,
depicted in Fig. 1, and are written as

$\hat{e}_{L}=\frac{\vec{p}^{\,\prime}}{|\vec{p}^{\,\prime}|},~{}~{}~{}~{}~{}\hat{e}_{P}=\hat{e}_{L}\times\hat{e}_{T},~{}~{}~{}~{}\hat{e}_{T}=\frac{\vec{p}^{\,\prime}\times\vec{k}}{|\vec{p}^{\,\prime}\times\vec{k}|}.$
(18)

The longitudinal, perpendicular and transverse components of the polarization
vector $\bm{\xi}_{L,P,T}(Q^{2})$ using Eqs. (17) and (18) may be written as:

$\xi_{L,P,T}(Q^{2})=\bm{\xi}\cdot\hat{e}_{L,P,T}~{}.$ (19)

In the rest frame of the initial nucleon, the polarization vector $\bm{\xi}$
is expressed as

$\bm{\xi}=A(Q^{2})~{}\vec{k}+B(Q^{2})~{}\vec{p}^{\,\prime}+C(Q^{2})~{}M(\vec{k}\times\vec{p}^{\,\prime})$
(20)

and is explicitly calculated using Eq. (16). The expressions for the
coefficients $A(Q^{2})$, $B(Q^{2})$ and $C(Q^{2})$ are given in the Appendix-A
of Ref. Fatima:2018tzs .

The longitudinal ($P_{L}(Q^{2})$), perpendicular ($P_{P}(Q^{2})$) and
transverse ($P_{T}(Q^{2})$) components of the polarization vector in the rest
frame of the final hyperon are obtained by performing a Lorentz boost and are
written as Fatima:2018tzs :

$P_{L}(Q^{2})=\frac{M^{\prime}}{E^{\prime}}\xi_{L}(Q^{2}),~{}~{}~{}~{}~{}~{}~{}P_{P}(Q^{2})=\xi_{P}(Q^{2}),~{}~{}~{}~{}~{}~{}~{}P_{T}(Q^{2})=\xi_{T}(Q^{2}).$
(21)

The expressions for $P_{L}(Q^{2})$, $P_{P}(Q^{2})$ and $P_{T}(Q^{2})$ are then
obtained using Eqs. (18), (19) and (20) in Eq. (21) and are expressed as

$\displaystyle P_{L}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{M^{\prime}}{E^{\prime}}\frac{A(Q^{2})\vec{k}\cdot\hat{p}^{\prime}+B(Q^{2})|\vec{p}^{\,\prime}|}{N(Q^{2})},$
(22) $\displaystyle P_{P}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{A(Q^{2})[(\vec{k}.\hat{p}^{\prime})^{2}-|\vec{k}|^{2}]}{N(Q^{2})~{}|\hat{p}^{\prime}\times\vec{k}|},$
(23) $\displaystyle P_{T}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{C(Q^{2})M|\vec{p}^{\,\prime}|[(\vec{k}.\hat{p}^{\prime})^{2}-|\vec{k}|^{2}]}{N(Q^{2})~{}|\hat{p}^{\prime}\times\vec{k}|}.$
(24)

To study the dependence of the polarization observables
${P}_{L}(E_{\bar{\nu}_{\mu}})$, ${P}_{P}(E_{\bar{\nu}_{\mu}})$ and
${P}_{T}(E_{\bar{\nu}_{\mu}})$ on $E_{\bar{\nu}_{\mu}}$, we have integrated
$P_{L}(Q^{2}),~{}P_{P}(Q^{2})$ and $P_{T}(Q^{2})$ over $Q^{2}$ and obtained
the expressions for ${P}_{L,P,T}(E_{\bar{\nu}_{\mu}})$ as:

$\displaystyle{P}_{L,P,T}(E_{\bar{\nu}_{\mu}})$ $\displaystyle=$
$\displaystyle\frac{\int_{Q^{2}_{min}}^{Q^{2}_{max}}P_{L,P,T}(Q^{2})\frac{d\sigma}{dQ^{2}}dQ^{2}}{\int_{Q^{2}_{min}}^{Q^{2}_{max}}\frac{d\sigma}{dQ^{2}}dQ^{2}}.$
(25)

If the T-invariance is assumed then all the vector and the axial vector form
factors are real and the expression for $C(Q^{2})$ vanishes which implies that
the transverse component of polarization, $P_{T}(Q^{2})$ perpendicular to the
production plane, vanishes.

###  II.3 $\Delta$ production off the free nucleon

In the intermediate energy region of about $0.5-1$ GeV, the antineutrino
induced reactions on a nucleon is dominated by the $\Delta$ excitation, given
by the reactions:

$\displaystyle\bar{\nu}_{\mu}(k)+n(p)$ $\displaystyle\rightarrow$
$\displaystyle\mu^{+}(k^{\prime})+\Delta^{-}(p^{\prime}),$ (26)
$\displaystyle\bar{\nu}_{\mu}(k)+p(p)$ $\displaystyle\rightarrow$
$\displaystyle\mu^{+}(k^{\prime})+\Delta^{0}(p^{\prime}),$ (27)

and the matrix element for the process given in Eq. (26) is written as
Athar:2007wd :

$T=\sqrt{3}\frac{G_{F}}{\sqrt{2}}\cos{\theta_{c}}~{}l_{\mu}~{}J^{\mu},$ (28)

where the leptonic current $l_{\mu}$ is defined in Eq. (6) and the hadronic
current $J^{\mu}$ is given by

$\displaystyle J^{\mu}$ $\displaystyle=$
$\displaystyle\overline{\psi}_{\alpha}(p^{\prime})O^{\alpha\mu}u(p).$ (29)

In the above expression, ${\psi_{\alpha}}(p^{\prime})$ is the Rarita Schwinger
spinor for $\Delta$ of momentum $p^{\prime}$ and $u(p)$ is the Dirac spinor
for the nucleon of momentum $p$. $O^{\alpha\mu}$ is the $N-\Delta$ transition
operator which is the sum of the vector ($O^{\alpha\mu}_{V}$) and the axial
vector ($O^{\alpha\mu}_{A}$) pieces, and the operators $O^{\alpha\mu}_{V}$ and
$O^{\alpha\mu}_{A}$ are given by:

$\displaystyle O^{\alpha\mu}_{V}$ $\displaystyle=$
$\displaystyle\left(\frac{C^{V}_{3}(q^{2})}{M}(g^{\alpha\mu}{\not
q}-q^{\alpha}{\gamma^{\mu}})+\frac{C^{V}_{4}(q^{2})}{M^{2}}(g^{\alpha\mu}q\cdot{p^{\prime}}-q^{\alpha}{p^{\prime\mu}})\right.$
(30) $\displaystyle+$
$\displaystyle\left.\frac{C^{V}_{5}(q^{2})}{M^{2}}(g^{\alpha\mu}q\cdot
p-q^{\alpha}{p^{\mu}})+\frac{C^{V}_{6}(q^{2})}{M^{2}}q^{\alpha}q^{\mu}\right)\gamma_{5}$
$\displaystyle O^{\alpha\mu}_{A}$ $\displaystyle=$
$\displaystyle\left(\frac{C^{A}_{3}(q^{2})}{M}(g^{\alpha\mu}{\not
q}-q^{\alpha}{\gamma^{\mu}})+\frac{C^{A}_{4}(q^{2})}{M^{2}}(g^{\alpha\mu}q\cdot{p^{\prime}}-q^{\alpha}{p^{\prime\mu}})+C^{A}_{5}(q^{2})g^{\alpha\mu}\right.$
(31) $\displaystyle+$
$\displaystyle\left.\frac{C^{A}_{6}(q^{2})}{M^{2}}q^{\alpha}q^{\mu}\right).$

A similar expression for $J^{\mu}$ is used for the $\Delta^{0}$ excitation
without a factor of $\sqrt{3}$ in Eq. (28). $C^{V}_{i}~{}(i=3-6)$ are the
vector and $C^{A}_{i}(i=3-6)$ are the axial vector transition form factors,
which are discussed in Ref. paschos2 .

The differential scattering cross section for the reactions given in Eqs. (26)
and (27) is given by Akbar:2017qsf ; AlvarezRuso:1998hi ; ruso :

$\frac{d^{2}\sigma}{dE_{k^{\prime}}d\Omega_{k^{\prime}}}=\frac{1}{64\pi^{3}}\frac{1}{MM_{\Delta}}\frac{|{\vec{k}^{\prime}|}}{E_{k}}\frac{\frac{\Gamma(W)}{2}}{(W-M_{\Delta})^{2}+\frac{\Gamma^{2}(W)}{4}}{|{T}|^{2}},$
(32)

where $M_{\Delta}$ is the mass of $\Delta$ resonance, $\Gamma$ is the Delta
decay width, $W$ is the center of mass energy i.e. $W=\sqrt{(p+q)^{2}}$ and

$\displaystyle{|{T}|^{2}}$ $\displaystyle=$
$\displaystyle\frac{{G_{F}}^{2}\cos^{2}{\theta_{c}}}{2}L_{\mu\nu}J^{\mu\nu}$
$\displaystyle J^{\mu\nu}$ $\displaystyle=$
$\displaystyle\overline{\Sigma}\Sigma
J^{\mu\dagger}J^{\nu}=\frac{1}{2}Tr\left[\frac{(\not
p+M)}{2M}{\tilde{\mathcal{O}}}^{\alpha\mu}{\it
P}_{\alpha\beta}{\mathcal{O}}^{\beta\nu}\right].$ (33)

In the above expression $L_{\mu\nu}$ is given by Eq. (4),
${\tilde{\mathcal{O}}}^{\alpha\mu}=\gamma^{0}{{\mathcal{O}}^{\alpha\mu}}^{\dagger}\gamma^{0}$,
${\mathcal{O}}^{\alpha\mu}=O^{\alpha\mu}_{V}+O^{\alpha\mu}_{A}$ and
$P^{\mu\nu}$ is the spin $\frac{3}{2}$ projection operator defined as

$P^{\mu\nu}=\sum_{spins}\psi^{\mu}{\overline{\psi}^{\nu}}=-\frac{\not{p^{\prime}}+M_{\Delta}}{2M_{\Delta}}\left(g^{\mu\nu}-\frac{2}{3}\frac{p^{\prime\mu}p^{\prime\nu}}{M_{\Delta}^{2}}+\frac{1}{3}\frac{p^{\prime\mu}\gamma^{\nu}-p^{\prime\nu}\gamma_{\mu}}{M_{\Delta}}-\frac{1}{3}\gamma^{\mu}\gamma^{\nu}\right).~{}~{}$
(34)

In Eq. (32), the Delta decay width $\Gamma$ is taken to be an energy dependent
P-wave decay width given by a4 :

$\displaystyle\Gamma(W)=\frac{1}{6\pi}\left(\frac{f_{\pi
N\Delta}}{m_{\pi}}\right)^{2}\frac{M}{W}|{{\vec{q}}_{cm}|^{3}}\Theta(W-M-
m_{\pi}),$ (35)

where $f_{\pi N\Delta}$ is the $\pi N\Delta$ coupling constant taken as 2.12
for numerical calculations and $|{\vec{q}}_{cm}|$ is defined as

$|{\vec{q}}_{cm}|=\frac{\sqrt{(W^{2}-m_{\pi}^{2}-M^{2})^{2}-4m_{\pi}^{2}M^{2}}}{2W}.$

The step function $\Theta$ in Eq. (35) denotes the fact that the width is zero
for the invariant masses below the $N\pi$ threshold, and ${|\vec{q}_{cm}|}$ is
the pion momentum in the rest frame of the $\Delta$ resonance.

##  III Nuclear medium effects

###  III.1 Hyperons produced inside the nucleus

When the antineutrino induced hyperon production takes place on the nucleons
which are bound in the nucleus, Fermi motion and Pauli blocking effects of
initial nucleons are considered. In the present work, the Fermi motion effects
are calculated in a local Fermi gas model (LFGM), and the cross section is
evaluated as a function of local Fermi momentum $p_{F}(r)$ and integrated over
the whole nucleus. The incoming antineutrino interacts with the nucleon moving
inside the nucleus of density $\rho_{N}(r)$ such that the differential
scattering cross section inside the nucleus is expressed as

$\frac{d\sigma}{dQ^{2}}=2{\int
d^{3}r\int\frac{d^{3}p}{{(2\pi)}^{3}}n_{N}(p,r)\left[\frac{d\sigma}{dQ^{2}}\right]_{\bar{\nu}N}},$
(36)

where $n_{N}(p,r)$ is the occupation number of the nucleon. $n_{N}(p,r)=1$ for
$p\leq p_{F_{N}}(r)$ and is equal to zero for $p>p_{F_{N}}(r)$, where
$p_{F_{N}}(r)$ is the Fermi momentum of the nucleon and is given as:

${p_{F}}_{p}(r)=\left(3\pi^{2}\rho_{p}(r)\right)^{\frac{1}{3}};\qquad\qquad{p_{F}}_{n}(r)=\left(3\pi^{2}\rho_{n}(r)\right)^{\frac{1}{3}},$

with $\rho_{p}(r)$ and $\rho_{n}(r)$ are, respectively, the proton and the
neutron densities inside the nucleus and are, in turn, expressed in terms of
the nuclear density $\rho(r)$ as

$\displaystyle\rho_{p}(r)$ $\displaystyle\rightarrow$
$\displaystyle\frac{Z}{A}\rho(r);\qquad\qquad\rho_{n}(r)\rightarrow\frac{A-Z}{A}\rho(r).$

In the above expression, $\rho(r)$ is determined in the electron scattering
experiments vries .

The produced hyperons are further affected by the FSI within the nucleus
through the hyperon-nucleon elastic processes like $\Lambda
N\rightarrow\Lambda N$, $\Sigma N\rightarrow\Sigma N$, etc. and the charge
exchange scattering processes like $\Lambda+n\rightarrow\Sigma^{-}+p$,
$\Lambda+n\rightarrow\Sigma^{0}+n$, $\Sigma^{-}+p\rightarrow\Lambda+n$,
$\Sigma^{-}+p\rightarrow\Sigma^{0}+n$, etc. Because of such types of
interaction in the nucleus, the probability of $\Lambda$ or $\Sigma$
production changes and has been taken into account by using the prescription
given in Ref. Singh:2006xp .

###  III.2 Delta produced inside the nucleus

When an antineutrino interacts with a nucleon (Eq.26) inside a nuclear target,
nuclear medium effects like Fermi motion, Pauli blocking, etc., come into
play. The produced $\Delta$s have no such constraints in the production
channel but their decay is inhibited by the Pauli blocking of the final
nucleons. Also, there are other disappearance channels open for $\Delta$s
through particle hole excitations and this leads to the modification in the
mass and width of the propagator defined in Eq.(35).

To take into account the nuclear medium effects, we have evaluated the cross
section using the local density approximation, following the same procedure as
mentioned in section-III.1, and the differential scattering cross section for
the reactions given in Eqs. (26) and (27) is defined as :

$\displaystyle\left.\frac{d^{2}\sigma}{dE_{k^{\prime}}d\Omega_{k^{\prime}}}\right|_{\bar{\nu}A}$
$\displaystyle=$ $\displaystyle\int
d^{3}r\frac{1}{64\pi^{3}}\frac{1}{MM_{\Delta}}\frac{|{\vec{k}^{\prime}|}}{E_{k}}\frac{\left(\frac{\tilde{\Gamma}(W)}{2}-Im\Sigma_{\Delta}\right)}{\left(W-M_{\Delta}-Re\Sigma_{\Delta}\right)^{2}+\left(\frac{\tilde{\Gamma}(W)}{2}-Im\Sigma_{\Delta}\right)^{2}}$
(37) $\displaystyle\times$
$\displaystyle\left(\rho_{n}(r)~{}+~{}\frac{1}{3}\rho_{p}(r)\right){|{T}|^{2}}.$

The modifications in the mass and decay width of $\Delta$ due to the nuclear
medium effects, are given by

$\frac{\Gamma}{2}\rightarrow\frac{\tilde{\Gamma}}{2}-Im\Sigma_{\Delta},\qquad\quad
M_{\Delta}\rightarrow\tilde{M}_{\Delta}=M_{\Delta}+Re\Sigma_{\Delta}.$ (38)

The expressions of $Re\Sigma_{\Delta}$ and $Im\Sigma_{\Delta}$ are given in
Ref. a4 .

##  IV Final state interaction effect

###  IV.1 Hyperon production

The hyperons decay to a pion and a nucleon viz., $\Lambda\rightarrow
p\pi^{-},n\pi^{0}$, $\Sigma^{0}\rightarrow p\pi^{-},n\pi^{0}$ and
$\Sigma^{-}\rightarrow n\pi^{-}$. However, when the hyperons are produced in a
nuclear medium, some of them disappear through the hyperon-nucleon interaction
processes like $YN\rightarrow NN$, though it is suppressed due to the nuclear
effects Holstein ; Oset:1989ey . Moreover, the pionic decay modes of hyperons
are Pauli blocked as the momentum of the nucleons available in these decays is
considerably below the Fermi level of energy for most nuclei leading to a long
lifetime for the hyperons in the nuclear medium Holstein ; Oset:1989ey .
Therefore, the hyperons which survive the $YN\rightarrow NN$ transition in the
medium live long enough to travel the nuclear medium and decay outside the
nucleus. In view of this, we have assumed no final state interaction of the
produced pions with the nucleons inside the nuclear medium. In a realistic
situation, all the hyperons produced in these reactions will not survive in
the nucleus, and the pions coming from the decay of hyperons will undergo FSI
Singh:2006xp . A quantitative analysis of the hyperon disappearance through
the $YN\rightarrow NN$ interaction and the pions having FSI effect, will
require a dynamic nuclear model to estimate the nonmesonic and mesonic decays
of the hyperons in a nucleus which is beyond the scope of the present work.
Our results in the following section, therefore, represent an upper limit on
the production of pions arising due to the production of hyperons.

###  IV.2 Delta production

The pions produced in the $\Delta$ excitation processes inside the nucleus may
re-scatter or may produce more pions or may get absorbed while coming out from
the final nucleus. We have taken the results of Vicente Vacas et al. a5  for
the final state interaction of pions which is calculated in an eikonal
approximation using probabilities per unit length as the basic input. The
details are given Ref. a5 .

##  V Results and discussions

Figure 2: Results for $\pi^{-}$ (left panel) and $\pi^{o}$ (right panel)
production in 40Ar with and without NME+FSI. The results are presented for the
pion production from the $\Delta$ and $\Lambda$ with and without NME+FSI.
Notice that in the case of $\pi^{-}$ production, the results of $\Delta$
without NME+FSI are suppressed by a factor of 6 and the results for $\Delta$
with NME+FSI are suppressed by a factor of 3 to bring them on the same scale
while in the case of $\pi^{0}$ production, the results of $\Delta$ without
NME+FSI are suppressed by a factor of 3, and there is no scale factor when
NME+FSI effects are included.

###  V.1 Total scattering cross sections

In Fig. 2, the results for the total scattering cross sections
$\sigma(E_{\bar{\nu}_{\mu}})$ vs $E_{{\bar{\nu}}_{\mu}}$ are presented for
${\bar{\nu}}_{\mu}$ scattering off 40Ar nuclear target, for the $\pi^{-}$(Left
panel) and $\pi^{o}$(Right panel) productions, respectively. These results are
shown for the cross sections obtained without and with nuclear medium and
final state interaction (NME+FSI) effects for the pion production arising due
to the $\Delta$ excitation and the $\Lambda$ production. Note the different
scales used in Fig. 2 for displaying the results for $\pi^{-}$ and $\pi^{0}$
production cross sections as explained in the respective figure captions. We
have earlier discussed in some detail, the nuclear medium and final state
interaction effects on the production of $\Delta$ and hyperon
($\Lambda+\Sigma$) in nuclei as well as the effect of nuclear absorption and
charge exchange scattering of $\pi^{0}$ and $\pi^{-}$ in the nuclear medium
Fatima:2018wsy ; SajjadAthar:2009rd . We show, in Fig. 2, the numerical
results for 40Ar nucleus relevant in the low energy region of the DUNE
experiment. We find that in the case of pions produced through $\Delta$
excitations, NME+FSI leads to a reduction of around $55-60\%$ in the $\pi^{-}$
production from the case when there is no NME+FSI, for the antineutrino
energies 0.6 $\leq E_{\bar{\nu}_{\mu}}\leq$ 1 GeV. Similarly, in the case of
$\pi^{0}$ production from the $\Delta$ excitation, there is a reduction of
about $50-60$% when NME+FSI effects are included, in the energy range 0.6 GeV
$\leq E_{\bar{\nu}_{\mu}}\leq 1$ GeV. This reduction decreases with the
increase in antineutrino energies SajjadAthar:2009rd . In the case of
$\pi^{-}$ production from $\Lambda$ decays, due to the inclusion of the FSI
effect, there is a small enhancement at lower energies, which increases to
$\sim 35\%$ at $E_{\bar{\nu}_{\mu}}=1$ GeV. However, in the presence of NME
and FSI effects, the $\pi^{-}$ production from $\Delta$ decays dominates for
$E_{\bar{\nu}_{\mu}}>0.7$ GeV, below which $\pi^{-}$ from $\Lambda$ decays is
quite comparable to the $\pi^{-}$ from $\Delta$ decays. Similarly, in the case
of $\pi^{0}$ from the $\Lambda$ decays, the NME+FSI increases the production
cross section by about 40% at $E_{\bar{\nu}_{\mu}}=1$ GeV, and this
enhancement increases with the increase in antineutrino energy. Moreover, it
may be noticed from the figure that the $\pi^{0}$ production from $\Lambda$
decays is comparable to $\pi^{0}$ from $\Delta$ excitations in the region from
threshold up to $E_{\bar{\nu}_{\mu}}\leq 0.75$ GeV, while at
$E_{\bar{\nu}_{\mu}}<0.75$ GeV, $\pi^{0}$ from the $\Delta$ decays dominates.
It should be noted that the enhancement due to nuclear medium of $\pi^{-}$ and
$\pi^{0}$ yield from the hyperon ($\Lambda+\Sigma$) decays is mainly due to
$YN\rightarrow YN$ final state interactions in which the $\Sigma$ hyperons get
converted to $\Lambda$ hyperons.

Figure 3: Results for the ratio $R_{\pi^{0}}$ (solid line) and $R_{\pi^{-}}$
(dashed line) in 40Ar with NME+FSI.

Figure 4: $\sigma~{}vs.~{}E_{\bar{\nu}_{\mu}}$ for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free case (left panel)
and for the proton bound inside an Argon nucleus (right panel), at the
different values of $g_{2}^{R}(0)$ viz. $g_{2}^{R}(0)=$ 0 (solid line), $-1$
(dashed line), $-2$ (dashed-dotted line), $+1$ (double-dashed-dotted line) and
$+2$ (dotted line).

Figure 5: $\sigma~{}vs.~{}E_{\bar{\nu}_{\mu}}$ for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free case (left panel)
and for the proton bound inside an Argon nucleus (right panel), at the
different values of $g_{2}^{R}(0)$ and the axial dipole mass, $M_{A}$, viz.
$g_{2}^{R}(0)=0$ and $M_{A}=1.026$ GeV (solid line), $g_{2}^{R}(0)=+1$ and
$M_{A}=1.026$ GeV (solid line with circle), $g_{2}^{R}(0)=+2$ and
$M_{A}=1.026$ GeV (solid line with diamond), $g_{2}^{R}(0)=0$ and $M_{A}=1.1$
GeV (dotted line), and $g_{2}^{R}(0)=0$ and $M_{A}=1.2$ GeV (double-dotted-
dashed line).

Figure 6: $\sigma~{}vs.~{}E_{\bar{\nu}_{\mu}}$ for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free case (left panel)
and for the proton bound inside an Argon nucleus (right panel), at the
different values of $g_{2}^{I}(0)$ and the axial dipole mass, $M_{A}$, viz.
$g_{2}^{I}(0)=0$ and $M_{A}=1.026$ GeV (solid line), $g_{2}^{I}(0)=1$ and
$M_{A}=1.026$ GeV (solid line with circle), $g_{2}^{I}(0)=2$ and $M_{A}=1.026$
GeV (solid line with diamond), $g_{2}^{I}(0)=0$ and $M_{A}=1.1$ GeV (dotted
line), and $g_{2}^{I}(0)=0$ and $M_{A}=1.2$ GeV (double-dotted-dashed line).

To quantify the hyperon production cross sections and to compare the pions
arising due to hyperons with the pions arising due to delta, in Fig. 3, we
have obtained the results for the ratio of delta to hyperon production cross
sections, with NME+FSI, for $\pi^{-}$ and $\pi^{o}$ by defining

$\displaystyle R_{\pi^{-}}$ $\displaystyle=$
$\displaystyle\frac{\sigma(\Delta\rightarrow N\pi^{-})}{\sigma(Y\rightarrow
N\pi^{-})}$ $\displaystyle R_{\pi^{o}}$ $\displaystyle=$
$\displaystyle\frac{\sigma(\Delta\rightarrow N\pi^{o})}{\sigma(Y\rightarrow
N\pi^{o})}$

It may be observed from these curves that in the case of $\pi^{-}$ production,
at low energies, say around $E_{\bar{\nu}_{\mu}}\sim 0.5$ GeV, the ratio
$R_{\pi^{-}}$ is approximately unity which implies that the number of
$\pi^{-}$ coming from the hyperons is almost comparable to the number of
$\pi^{-}$ coming from $\Delta$. However, with the increase in
$E_{\bar{\nu}_{\mu}}$, $\pi^{-}$ coming from $\Delta$ starts dominating and
becomes $\sim$ 4 times the number of $\pi^{-}$ coming from the hyperons at
$E_{\bar{\nu}_{\mu}}=1$ GeV. In the case of $\pi^{0}$ production, as observed
from Fig. 3, the ratio $R_{\pi^{0}}$ is less than one almost up to
$E_{\bar{\nu}_{\mu}}\leq 0.8$ GeV, which means that the $\pi^{0}$ arising from
the hyperons has a larger contribution than the $\pi^{0}$ coming from
$\Delta$, in the low energy region. However, with the increase in
$E_{\bar{\nu}_{\mu}}$, the $\pi^{0}$ contribution from $\Delta$ starts
dominating.

In Fig. 4, we present the results for
$\sigma(E_{\bar{\nu}_{\mu}})~{}vs.~{}E_{\bar{\nu}_{\mu}}$ for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free nucleon (left
panel) and for the argon nucleus (right panel), at the different values of
$g_{2}^{R}(0)$ viz. $g_{2}^{R}(0)=$ 0 , $\pm 1$, $\pm 2$. In the case of the
free nucleon target, we find that a non-zero value of $g_{2}^{R}(0)$ results
in an increase in the cross section, which has been shown to be slightly
larger for $g_{2}^{R}(0)<0$ than for $g_{2}^{R}(0)>0$ and this enhancement
increases with the increase in the value of $g_{2}^{R}(0)$ and also with the
increase in the energy of antineutrinos with a fixed $|g_{2}^{R}(0)|$. For
example, with $g_{2}^{R}(0)=+1$ this enhancement from $g_{2}^{R}(0)=0$ is
about $1-2\%$, while for $g_{2}^{R}(0)=-1$ the enhancement is about $3\%$, and
with $g_{2}^{R}(0)=+2$ this enhancement from $g_{2}^{R}(0)=0$ is about $4\%$,
while for $g_{2}^{R}(0)=-2$ the enhancement is about $8\%$ in the region of
$E_{\nu_{\mu}}\sim 3$ GeV. While in the case of argon nucleus, we find that
the difference in the numerical values of the cross sections calculated for
the positive and negative values of $g_{2}^{R}(0)$, becomes negligible.

Recently, in the literature there has been a lot of discussion on a larger
value of the axial dipole mass $M_{A}$, compared to its world average value,
in the quasielastic scattering of (anti)neutrinos on the nuclear targets in
order to explain the MiniBooNE, K2K and other results from the accelerator
experiments. A higher value of $M_{A}$, almost $10-20\%$ deviation from the
world average value was suggested in the various analyses for explaining the
results from the MiniBooNE experiment. For a detailed discussion, please see
Refs.Alvarez-Ruso:2017oui ; Katori:2016yel . Therefore, to study whether a
non-zero value of $g_{2}^{R}(0)$ has some effect on the total cross section
for ${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ scattering on the free
nucleon (left panel) and the nuclear target (right panel), which can simulate
the effect of using a higher value of $M_{A}$, we have taken a few non-zero
values of $|g_{2}^{R}(0)|$ viz. 1 and 2 and varied $M_{A}$ in the range 1.026
GeV to 1.2 GeV. It may be observed from Fig. 5, that $g_{2}^{R}(0)=2$ and
$M_{A}=1.026$ GeV, simulate the cross section for $g_{2}^{R}(0)=0$ and
$M_{A}=1.2$ GeV in the case of a free nucleon target. In the case of Argon
nucleus, one may notice that the results obtained with $g_{2}^{R}(0)=1$ and
$M_{A}=1.026$ GeV simulate the results obtained with $g_{2}^{R}(0)=0$ and
$M_{A}=1.1$ GeV, while the results obtained with $g_{2}^{R}(0)=2$ and
$M_{A}=1.026$ GeV are larger than the results obtained with $g_{2}^{R}(0)=0$
and $M_{A}=1.2$ GeV. However, the results with $g_{2}^{R}(0)=1.5$ and
$M_{A}=1.026$ GeV may simulate the results obtained with $g_{2}^{R}(0)=0$ and
$M_{A}=1.2$ GeV. It suggests that a nonzero value of $g_{2}^{R}(0)$, can lead
to a smaller value of $M_{A}$.

We have also considered T-violation and taken a non-zero imaginary value of
$g_{2}(0)$, and obtained the results for $\sigma~{}vs.~{}E_{\bar{\nu}_{\mu}}$,
which are shown in Fig. 6, for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free nucleon (left
panel) and for the argon nucleus (right panel), with $M_{A}=1.026$ GeV, 1.1
GeV and 1.2 GeV with $g_{2}^{I}(0)=0,1$ and 2. It may be observed that the
cross section not only increases with the increase in the value of $M_{A}$,
but it also increases with the increase in the value of $g_{2}^{I}(0)$. The
results of the cross section with $M_{A}=1.026$ GeV and $g_{2}^{I}(0)=2$,
simulate the results with $M_{A}=1.2$ GeV with $g_{2}^{I}(0)=0$, i.e., when
the second class current is switched off. It may be observed from the figure
that this enhancement in the cross section with $M_{A}=1.026$ GeV and
$g_{2}^{I}(0)=2$, is more in the case of a nuclear target.

Figure 7: $\sigma~{}vs.~{}E_{\bar{\nu}_{\mu}}$ for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free case (left panel)
and for the proton bound inside an Argon nucleus (right panel), with SU(3)
symmetry breaking effect and in the presence of second class current form
factor assuming T invariance.

In Fig. 7, we have presented the results for
$\sigma~{}vs.~{}E_{\bar{\nu}_{\mu}}$ for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free nucleon (left
panel) and for the argon nucleus (right panel), with SU(3) symmetry breaking
effects and in the presence of second class current form factor assuming T
invariance. We have already shown (Fig. 4) that the presence of the second
class current form factor $g_{2}(Q^{2})$ with or without T invariance
increases the value of the cross section both for the free nucleon as well as
for the argon nucleus. It may be observed from Fig. 7 that the effect of SU(3)
symmetry breaking in the absence of the second class curent is to decrease the
total cross section for the free nucleon as well as for the argon nucleus.
Therefore, the two effects tend to cancel one another. In the case of the free
nucleon, we observe that the results obtained with SU(3) symmetry breaking
effect and $g_{2}^{R}(0)=-1$ is similar to the results obtained with SU(3)
symmetry in the absence of the second class currents. In the case of argon
nucleus, the results obtained with SU(3) symmetry breaking with
$g_{2}^{R}(0)=+1(-1)$ are similar to the results obtained assuming SU(3)
symmetry in the absence of second class currents.

Figure 8: $\sigma~{}vs.~{}E_{\bar{\nu}_{\mu}}$ for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free case (left panel)
and for the proton bound inside an Argon nucleus (right panel), with SU(3)
symmetry breaking effect and in the presence of second class current form
factor assuming T violation.

In Fig. 8, we have presented the results for
$\sigma~{}vs.~{}E_{\bar{\nu}_{\mu}}$ for the process
${\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda}$ for the free nucleon (left
panel) and for the argon nucleus (right panel), with SU(3) symmetry breaking
effects and in the presence of second class current form factor assuming T
violation. In this figure, we observe similar results as in Fig. 7. In the
case of both free nucleon as well as the argon nucleus, the results obtained
with SU(3) symmetry breaking and $g_{2}^{I}(0)=1$ is similar to the results
obtained with SU(3) symmetry in the absence of second class currents.

###  V.2 Polarization observables

Figure 9: $P_{L}(E_{\bar{\nu}_{\mu}})$ (left panel) and
$P_{P}(E_{\bar{\nu}_{\mu}})$ (right panel) for the process
$\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda$ with $g_{2}^{R}(0)=$ 0 (solid
line), 0.5 (dashed line), 1 (dash-dotted line), -0.5 (double-dotted-dashed
line) and -1 (double-dashed-dotted line).

In Fig. 9, we have presented the results for the longitudinal
($P_{L}(E_{\bar{\nu}_{\mu}})$) and perpendicular
($P_{P}(E_{\bar{\nu}_{\mu}})$) components of the polarization for the
polarized $\Lambda$ produced in the reaction
$\bar{\nu}_{\mu}+p\longrightarrow\mu^{+}+\Lambda$ and studied the effect of
the nonvanishing real values of the weak electric form factor viz.,
$g_{2}^{R}(0)=0,~{}\pm 0.5$ and $\pm 1$ on the polarization observables. It
may be observed from the figure that both $P_{L}(E_{\bar{\nu}_{\mu}})$ and
$P_{P}(E_{\bar{\nu}_{\mu}})$ are quite sensitive to the variation of
$g_{2}^{R}(0)$ for both positive as well as negative values. Qualitatively,
for both the positive and negative values of $g_{2}^{R}(0)$, the longitudinal
component of polarization shows similar trend while the perpendicular
component of polarization shows different trends for positive and negative
values of $g_{2}^{R}(0)$.

Figure 10: $P_{L}(E_{\bar{\nu}_{\mu}})$ (left panel),
$P_{P}(E_{\bar{\nu}_{\mu}})$ (middle panel) and $P_{T}(E_{\bar{\nu}_{\mu}})$
(right panel) for the process $\bar{\nu}_{\mu}+p\rightarrow\mu^{+}+\Lambda$ at
different values of $g_{2}^{I}(0)$ viz., $g_{2}^{I}(0)=$ 0 (solid line), 0.5
(dashed line) and 1 (dashed-dotted line).

In Fig. 10, we have presented the results for the longitudinal
($P_{L}(E_{\bar{\nu}_{\mu}})$), perpendicular ($P_{P}(E_{\bar{\nu}_{\mu}})$)
and transverse ($P_{T}(E_{\bar{\nu}_{\mu}})$) components of the polarized
$\Lambda$ produced in the reaction
$\bar{\nu}_{\mu}+p\longrightarrow\mu^{+}+\Lambda$ and studied the effect of
the non-zero imaginary values of the weak electric form factor viz.,
$g_{2}^{I}(0)=0,0.5$ and 1 on the polarization observables. The results for
$P_{L}(E_{\bar{\nu}_{\mu}})$ and $P_{P}(E_{\bar{\nu}_{\mu}})$ remain unchanged
when the negative values of $g_{2}^{I}(0)$ are taken as they depend on
$g_{2}^{R}(Q^{2})$ and $|g_{2}^{R,I}(Q^{2})|^{2}$ (see Appendix-A of Ref.
Fatima:2018tzs ). While in the case of $g_{2}^{I}<0$,
$P_{T}(E_{\bar{\nu}_{\mu}})$ just changes sign but the magnitude remains the
same, and therefore, the results have not been depicted in the figures. From
the figure, it may be noted that the longitudinal as well as the perpendicular
components of polarization are insensitive to the variation in $g_{2}^{I}(0)$,
but the transverse component of polarization, which appears only in the
absence of T invariance, is quite sensitive to $g_{2}^{I}(0)$. Even at
$g_{2}^{I}=0.5$, it could be around $5-7\%$ in the energy range of 1 to 3 GeV
and increases with increase in the $g_{2}^{I}(0)$.

##  VI Summary and conclusions

In this review, we have studied the antineutrino induced quasielastic
production of hyperons and their polarization components from the free nucleon
as well as from the argon nucleus. These results may be useful in determining
the axial vector transition form factors in the strangeness sector specially
the form factor corresponding to the second class currents with and without
T-invariance. We have also studied the effect of SU(3) symmetry breaking on
the cross section for hyperon production.

We summarize our results in the following:

  * i)

In the low energy region of antineutrinos, i.e., $E_{\bar{\nu}_{\mu}}<0.7$
GeV, the pions produced by the decay of hyeprons ($\Lambda+\Sigma$) is
comparable to the production of pions from the $\Delta$ decays. This is
because the weak production of $\Delta$ in this energy region is inhibited by
the threshold effect and is reduced by the nuclear medium effects. Moreover,
the pions produced by the $\Delta$ decays undergo absorption and other final
state interaction effects in the nuclear medium leading to a substantial
reduction in the pion yield. These effects compensate for the Cabibbo
suppression of the hyperon production as well as the reduction due to the
nuclear medium and final state interactions in the case of the pions produced
by the hyperon decays.

  * ii)

The total cross sections for the single hyperon production both from the free
nucleon and from the argon nucleus are quite sensitive to the variation in the
G violating axial vector form factor associated with the second class
currents.

  * iii)

The non-zero value of the second class current form factor with and without T
invariance may lead to a smaller value of the axial dipole mass in the case of
$\nu_{\mu}(\bar{\nu}_{\mu})$ induced $\Delta S=0$ and $\Delta S=1$
quasielastic reactions from free as well as bound nucleons.

  * iv)

The presence of the second class current form factor with or without T
invariance increases the total cross section with increase in
$E_{\bar{\nu}_{\mu}}$, while the inclusion of the effect of the SU(3) symmetry
breaking, in the case of $\Lambda$ production, decreases the cross section
with the increase in $E_{\bar{\nu}_{\mu}}$, tending towards a cancellation due
to these effects.

  * v)

In the case of T invariance, the polarization components
$P_{L}(E_{\bar{\nu}_{\mu}})$ and $P_{P}(E_{\bar{\nu}_{\mu}})$ are found to be
sensitive to the variation in the value of $g_{2}^{R}(0)$, even for
$g_{2}^{R}(0)=\pm 0.5$, and this sensitivity increases with the increase in
$g_{2}^{R}(0)$.

  * vi)

In the case of T violation, the polarization components
$P_{L}(E_{\bar{\nu}_{\mu}})$ and $P_{P}(E_{\bar{\nu}_{\mu}})$ do not show much
sensitivity to the variation in $g_{2}^{I}(0)$, while the transverse component
of polarization is found to be quite sensitive to $g_{2}^{I}(0)$ variation.
This can provide an opportunity to study T violation by measuring the
transverse polarization of $\Lambda$ in imaging detectors like the one planned
to be used in DUNE.

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