# Hyperon production in quasielastic $\bar{\nu}_{\tau}-$nucleon scattering

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

###### Abstract

The theoretical results for the total cross sections and polarization
components of the $\tau^{+}$ lepton produced in the charged current induced
$|\Delta S|=1$ quasielastic $\bar{\nu}_{\tau}-N$ scattering leading to
hyperons ($\Lambda,\Sigma$) have been presented assuming T invariance. The
theoretical uncertainties arising due to the use of different vector, axial
vector and pseudoscalar form factors as well as the effect of SU(3) symmetry
breaking have been studied. We have also presented, for the first time, a
comparison of the total cross sections for the production of $e,\mu,\tau$
leptons to facilitate the implications of lepton flavor universality (LFU) in
the $|\Delta S|=1$ quasielastic reactions induced by the antineutrinos of all
flavors i.e., $\nu_{l};~{}l=e,\mu,\tau$.

###### pacs:

12.15.-y, 13.15.+g, 13.88.+e

##  I Introduction

The ${\tau}$ neutrinos ($\nu_{\tau}$) were experimentally observed for the
first time by the DONUT collaboration Kodama:2000mp ; Kodama:2007aa  by
observing the $\tau^{-}$ leptons produced through the charged current
scattering of $\nu_{\tau}$ on the nucleons in the reaction,
$\nu_{\tau}+N\longrightarrow\tau^{-}+X$, where $N=n$ or $p$, and $X$
represents hadron(s) in the final state. Since then, a few $\tau^{-}$
production events have been observed by the OPERA collaboration at CERN
Agafonova:2014bcr ; Agafonova:2015jxn ; Agafonova:2018auq  using accelerator
neutrinos and by the SuperKamiokande Li:2017dbe ; Abe:2012jj  as well as the
IceCUBE Aartsen:2019tjl  collaborations using the atmospheric neutrinos where
the $\nu_{\tau}$s are assumed to be produced through the
$\nu_{\mu}\longrightarrow\nu_{\tau}$ oscillation. Since these experiments have
observed very few events of the $\tau$ lepton production, new experiments for
producing larger number of $\tau$ leptons have been proposed by the SHiP
Yoon:2020yrx ; Anelli:2015pba ; Alekhin:2015byh , DsTau Aoki:2019jry , DUNE
Machado:2020yxl ; Strait:2016mof ; Abi:2018rgm  and FASER$\nu$ FASER:2020gpr
collaborations in which the number of $\tau$ lepton events are expected to
reach a few hundreds during the running time of 3–5 years of the experiment.
The results from these experiments would provide the desired data with
reasonable statistics on the total and differential scattering cross sections
as well as on the polarization components of the $\tau$ lepton, which would
enable a reliable study of the various aspects and properties of $\nu_{\tau}$
and $\tau$ leptons.

The $\tau$ lepton production in $\nu_{\tau}-N$ scattering has a threshold of
3.5 GeV in the charged current induced quasielastic (CCQE) reactions i.e.,
$\nu_{\tau}(\bar{\nu}_{\tau})+N\longrightarrow\tau^{-}(\tau^{+})+N^{\prime}$
($N,N^{\prime}=n$ or $p$). As the energy increases, the production of the
$\tau$ leptons is accompanied with the inelastic and deep inelastic production
of hadrons. In the energy region of $\nu_{\tau}~{}(\bar{\nu}_{\tau})$
experiments, where the energy of the produced $\tau$ lepton is not too large
compared to its rest mass $m_{\tau}=1.776$ GeV, the $\tau$ leptons would not
be completely longitudinally polarized, and would also have transverse
component of the polarization Fatima:2020pvv ; Kuzmin:2003ji ; Kuzmin:2004ke ;
Hagiwara:2003di ; Hagiwara:2004gs ; Hagiwara:2004xe ; Sobczyk:2019urm ;
Valverde:2006yi . Moreover, any presence of the polarization component
perpendicular to the reaction plane would provide information about the T
noninvariance in $\nu_{\tau}-N$ interactions Fatima:2018gjy ; Fatima:2018tzs ;
Fatima:2018wsy ; Fatima:2021ctt . The polarization state of the $\tau$ lepton
affects the total and differential cross sections and is, therefore, an
important observable in the study of $\nu_{\tau}-N$ interactions. Thus, it is
highly desirable that a comprehensive study of the $\tau$ polarization along
with the cross sections be made in the quasielastic, inelastic and deep
inelastic reactions induced by $\nu_{\tau}$ and $\bar{\nu}_{\tau}$ on nucleons
in order to understand the $\nu_{\tau}-N$ interactions.

In the case of Standard Model (SM) of particle physics, all the leptons
interact with each other through the purely leptonic processes and with the
quarks through the various semileptonic processes, having the same strength
for each leptonic flavor. This is called the lepton flavor universality (LFU)
and is an essential feature of the SM.

The validity of LFU has been experimentally studied in the purely leptonic
decays of $\mu$, $\tau$ leptons as well as in the various semileptonic decays
of mesons and baryons. The LFU seems to work quite well in the case of purely
leptonic decays of $\mu$ and $\tau$ leptons and $W$ boson ATLAS:2021icw . In
the case of semileptonic decays of mesons and baryons like $K$, $D$,
$D_{s}^{\star}$, $\Lambda$ and $\Lambda_{c}$, etc., involving quark
transitions between medium heavy and light quarks i.e., $s\longrightarrow
ul\bar{\nu}_{l}$, $s\longrightarrow dl\bar{l}$, $c\longrightarrow
sl\bar{\nu}_{l}$, $c\longrightarrow dl\bar{\nu}_{l}$, etc., the available
experimental results are in agreement with the prediction of SM within
statistical uncertainties and no evidence of LFU violation (LFUV) has been
reported Zhang:2019kky ; Yang:2018qdx ; Fleischer:2019wlx ; Golz:2021yqz ;
BESIII:2021ynj ; BESIII:2018ccy ; Mezzadri:2018axu . However, the recent
indications of LFUV in semileptonic decays in the heavy quark sector involving
transitions like $b\longrightarrow sl\bar{l}$ and $b\longrightarrow
cl\bar{\nu}_{l}$ at BaBar, LHCb, BESIII and Belle collaborations
Bifani:2018zmi ; Albrecht:2021tul ; LHCb:2021trn ; deSimone:2020kwi ;
Belle:2019rba ; Celani:2021hni  have generated great interest in studying the
origins of LFUV. Further hints for the LFUV Crivellin:2021sff ; Bryman:2021teu
have been inferred from the recent measurements of the anomalous magnetic
moment ($g-2$) of the muons, at the Fermilab Muong-2:2021ojo , as well as the
anomalies reported in the recent measurements of the Cabibbo angle
Coutinho:2019aiy  and production of lepton pairs through the process of
$q\bar{q}\longrightarrow l\bar{l}$ ($l=e$ and $\mu$) in $pp$ collisions at
CERN CMS:2021ctt . In the case of $b\longrightarrow sl\bar{l}$ decays, LHCb
LHCb:2021trn  collaboration has reported the ratio of the branching fractions
in $l=\mu$ mode to $l=e$ mode to be 2.6 $\sigma$ lower than the SM prediction.
While in the case of $b\longrightarrow cl\bar{\nu}_{l}$ decays, all the three
collaborations Bifani:2018zmi  have reported the ratio of the branching
fraction in $l=\tau$ mode to $l=\mu$ mode, which challenge LFU at the level of
four standard deviation. This has led to extensive theoretical work in
constructing the models of new physics (NP) going beyond the standard model
(BSM) to explain these experimental results on LFUV in the $b$ quark sector
Allwicher:2021ndi ; Crivellin:2021bkd ; London:2021lfn . In view of these
developments in $b$ quark sector, it is natural to apply these theoretical
models Allwicher:2021ndi ; Crivellin:2021bkd ; London:2021lfn  in the medium
heavy quark sector involving $c$ and $s$ quarks decays like $c\longrightarrow
sl\bar{\nu}_{l}$, $c\longrightarrow dl\bar{\nu}_{l}$, $s\longrightarrow
ul\bar{\nu}_{l}$, $s\longrightarrow dl\bar{l}$ and critically study the LFUV
effects and search for them experimentally Zhang:2020dla ; Becirevic:2020rzi ;
Leng:2020fei ; Zhang:2019tcs . While these studies are being pursued with some
interest, the need for further efforts in this direction has been emphasized
recently BESIII:2021ynj .

Notwithstanding the above efforts in the study of LFUV in the semileptonic
decays, there have been almost no study of LFUV effects in the (anti)neutrino-
nucleus scattering. The only exception is comparative study of the
quasielastic $\nu_{e}(\bar{\nu}_{e})$-nucleon/nucleus and
$\nu_{\mu}(\bar{\nu}_{\mu})$-nucleon/nucleus scattering and analyzing the
differences in the cross sections of these processes arising due to the lepton
mass and the radiative corrections and the contributions due to the
pseudoscalar and second class current form factors Akbar:2015yda ; Day:2012gb
. A comparison with the experimental results would give information about the
presence or absence of any LFUV effects, however, the present results on the
cross sections in the neutrino energy region of a few GeV are not precise
enough to conclude about the presence of LFUV effects. Such studies of LFUV
effects in the (anti)neutrino scattering with
$\nu_{l}(\bar{\nu}_{l});(l=e,\mu,\tau)$ in the strange, charm, bottom and top
quark sectors have not been done. With the aim of exploring the presence of
such effects in the strangeness sector, we have studied in some detail the
quasielastic scattering process of $\bar{\nu}_{l}+N\longrightarrow
l^{+}+\Lambda(\Sigma);(l=e,\mu,\tau)$ corresponding to $u\longrightarrow s$
transition.

In this work, we report on the results of the theoretical calculations within
the ambit of the SM with implicit lepton flavor universality for the total
cross section, differential cross section and the $\tau$ lepton polarization
in the reaction $\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda(\Sigma)$,
using various parameterizations for the vector and axial vector form factors
using SU(3) symmetry. The uncertainties in the numerical values of these
observables due to the use of different parameterizations of the form factors
are discussed. We have also studied the effect of SU(3) symmetry violation
using the various weak form factors which have been used to study the SU(3)
symmetry violating effects in the semileptonic hyperon decays of $\Lambda$ and
$\Sigma$ Schlumpf ; Faessler:2008ix  and have discussed the uncertainties
associated with SU(3) violations. Any deviation of the experimental results on
the cross sections and $\tau$ polarization to be obtained in future will be a
signal of LFUV. The results presented here would facilitate the study of LFUV
effects in $\bar{\nu}_{l}~{}(l=e,\mu,\tau)$ induced processes in the
strangeness sector corresponding to $u\longrightarrow s$ transition and
compliment such LFUV studies in the semileptonic decays of strange particles
He:2019xxp .

Figure 1: Feynman diagram for the process
$\bar{\nu}_{\tau}(k)+N(p)\rightarrow\tau^{+}(k^{\prime})+Y(p^{\prime})$, where
$N=p,n$; $Y=\Lambda,\Sigma^{0},\Sigma^{-}$ and the quantities in the bracket
represent four momenta of the corresponding particles.

In section II, we present the formalism for calculating the differential and
total scattering cross sections and using them the polarization observables of
the $\tau$ lepton produced in the quasielastic $|\Delta S|=1$ scattering
processes have been discussed in section-III. In section IV.1, we present and
discuss the numerical results obtained for the differential cross section and
polarization observables of the $\tau$ lepton and study the dependence of the
different parameteric forms of the vector and axial vector form factors as
well as the effect of SU(3) symmetry breaking on these observables. Similar
effects are studied in the case of total cross sections and average
polarizations in section IV.2. Further, we have also studied the lepton flavor
universality in the case of $e-\mu$ and $e-\mu-\tau$ sectors in the total
cross sections and the numerical results are presented for the ratios
$R_{1}=\frac{\sigma(\bar{\nu}_{\mu}+p\longrightarrow\mu^{+}+\Lambda)}{\sigma(\bar{\nu}_{e}+p\longrightarrow
e^{+}+\Lambda)}$ and
$R_{2}=\frac{2\sigma(\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda)}{\sigma(\bar{\nu}_{\mu}+p\longrightarrow\mu^{+}+\Lambda)+\sigma(\bar{\nu}_{e}+p\longrightarrow
e^{+}+\Lambda)}$, in Section IV.3. Section V summarizes the results and
conclude our findings.

##  II Formalism

###  II.1 Matrix element and weak form factors

####  II.1.1 Matrix element

The transition matrix element for the quasielastic hyperon production
processes depicted in Fig. 1, given by

$\displaystyle\bar{\nu}_{\tau}(k)+N(p)$ $\displaystyle\longrightarrow$
$\displaystyle\tau^{+}(k^{\prime})+Y(p^{\prime}),\qquad N=p,n;\qquad
Y=\Lambda,\Sigma^{0},\Sigma^{-},$ (1)

is written as

$\displaystyle{\cal{M}}=\frac{G_{F}}{\sqrt{2}}\sin\theta_{c}~{}l^{\mu}{{J}}_{\mu},$
(2)

where the quantities in the brackets of Eq. (1) represent the four momenta of
the respective particles, $G_{F}$ is the Fermi coupling constant and
$\theta_{c}~{}(=13.1^{\circ})$ is the Cabibbo mixing angle. The leptonic
current $l^{\mu}$ is given by

$l^{\mu}=\bar{u}(k^{\prime})\gamma^{\mu}(1+\gamma_{5})u(k).$ (3)

The hadronic current ${J}_{\mu}$ is expressed as:

${{J}}_{\mu}=\bar{u}(p^{\prime}){\Gamma_{\mu}}u(p)$ (4)

with

${\Gamma_{\mu}}=V_{\mu}-A_{\mu}.$ (5)

The vector ($V_{\mu}$) and the axial vector ($A_{\mu}$) currents are given by
Fatima:2018tzs ; Fatima:2018gjy :

$\displaystyle\langle Y(p^{\prime})|V_{\mu}|N(p)\rangle$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\left[\gamma_{\mu}f_{1}^{NY}(Q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M+M_{Y}}f_{2}^{NY}(Q^{2})+\frac{2~{}q_{\mu}}{M+M_{Y}}f_{3}^{NY}(Q^{2})\right]u(p),$
(6) $\displaystyle\langle Y(p^{\prime})|A_{\mu}|N(p)\rangle$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\left[\gamma_{\mu}\gamma_{5}g_{1}^{NY}(Q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M+M_{Y}}\gamma_{5}g_{2}^{NY}(Q^{2})+\frac{2~{}q_{\mu}}{M+M_{Y}}g_{3}^{NY}(Q^{2})\gamma_{5}\right]u(p),$
(7)

where $M$ and $M_{Y}$ are the masses of the initial nucleon and the final
hyperon. $q(=k-k^{\prime}=p^{\prime}-p)$ 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 $N-Y$ transition 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.

####  II.1.2 Weak transition form factors

The weak vector and axial vector form factors are determined using the
following assumptions, which are consistent with the constraints due to the
symmetry properties of the weak hadronic currents Marshak ; Pais:1971er ;
LlewellynSmith:1971uhs :

  * (a)

T invariance implies that all the vector ($f_{i}^{NY}(Q^{2})$; $i=1-3$) and
axial vector ($g_{i}^{NY}(Q^{2})$; $i=1-3$) form factors are real.

  * (b)

The hypothesis that the charged weak vector currents and its conjugate along
with the isovector part of the electromagnetic current form an isotriplet
implies that the weak vector form factors $f_{1}^{NY}(Q^{2})$ and
$f_{2}^{NY}(Q^{2})$ are related to the isovector electromagnetic form factors
of the nucleon. The hypothesis ensures conservation of vector current (CVC) in
the weak sector.

  * (c)

The hypothesis of CVC of the weak vector currents implies that
$f_{3}(Q^{2})=0$.

  * (d)

The principle of G-invariance implies the second class current form factors to
be zero, i.e., $f_{3}^{NY}(Q^{2})=0$ and $g_{2}^{NY}(Q^{2})=0$.

  * (e)

The hypothesis of partially conserved axial vector current (PCAC) relates the
pseudoscalar form factor ($g_{3}^{NY}(Q^{2})$) to the axial vector form factor
($g_{1}^{NY}(Q^{2})$), through the Goldberger-Treiman (GT) relation.

  * (f)

The assumption of SU(3) symmetry of the weak hadronic currents implies that
the vector and axial vector currents transform as an octet under the SU(3)
group of transformations.

The determination of all the weak form factors is based on the symmetry
properties discussed above, and the details are given in Ref. Fatima:2018tzs ;
Fatima:2018gjy . The explicit expressions of the vector and axial vector form
factors for the different $N-Y$ transitions, assuming the SU(3) symmetry are
given in Section II.1.3, while the effects of SU(3) symmetry breaking on these
form factors are discussed in Section II.1.4.

####  II.1.3 Form factors with SU(3) symmetry

The weak vector and the axial vector currents corresponding to the $\Delta
S=1$ currents whose matrix elements are defined between the initial
($|N\rangle$) and final ($|Y\rangle$) states in Eq. (1) are assumed to belong
to the octet representation of the SU(3). Since $|N\rangle$ and $|Y\rangle$
also belong to the octet representation under SU(3), each of these form
factors are described in terms of the functions $D(Q^{2})$ and $F(Q^{2})$
corresponding to the symmetric (S) and antisymmetric (A) couplings and the
SU(3) Clebsch-Gordan coefficients. Explicitly, the form factors can be
expressed as (for details, see Ref. Fatima:2018tzs ):

$\displaystyle f_{i}(Q^{2})$ $\displaystyle=$ $\displaystyle
aF_{i}^{V}(Q^{2})+bD_{i}^{V}(Q^{2})$ (8) $\displaystyle g_{i}(Q^{2})$
$\displaystyle=$ $\displaystyle aF_{i}^{A}(Q^{2})+bD_{i}^{A}(Q^{2})\qquad
i=1,2,3$ (9)

The Clebsch-Gordan coefficients $a$ and $b$ are calculated for each $N-Y$
transitions and are given in Table 1.

Transitions |  $a$ |  $b$
---|---|---
$p\rightarrow\Lambda$ |  $-\sqrt{\frac{3}{2}}$ |  $-\frac{1}{\sqrt{6}}$
$p\rightarrow\Sigma^{0}$ |  $-\frac{1}{\sqrt{2}}$ |  $\frac{1}{\sqrt{2}}$
$n\rightarrow\Sigma^{-}$ |  $-1$ |  1
Table 1: Values of the coefficients $a$ and $b$ given in Eqs. (8)$-$(9).

From Table 1, we see that the SU(3) symmetry predicts a relation between the
vector and axial vector form factors for the transitions
$p\longrightarrow\Sigma^{0}$ and $n\longrightarrow\Sigma^{-}$, which implies
that

$\left[\frac{d\sigma}{dQ^{2}}\right]_{p\longrightarrow\Sigma^{0}}=\frac{1}{2}\left[\frac{d\sigma}{dQ^{2}}\right]_{n\longrightarrow\Sigma^{-}},$
(10)

and

$\left[P_{L,P}\right]_{p\longrightarrow\Sigma^{0}}=\left[P_{L,P}\right]_{n\longrightarrow\Sigma^{-}}.$
(11)

  * (i)

Vector form factors:
In the case of vector form factors, the functions $D_{i}^{V}(Q^{2})$ and
$F_{i}^{V}(Q^{2})$ are determined in terms of the nucleon electromagnetic form
factors $f_{i}^{n,p}(Q^{2});~{}(i=1,2)$, following the same method as
discussed above (Eq. (8)) in the case of electromagnetic interactions, i.e.

$\displaystyle D_{i}^{V}(Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{3}{2}f_{i}^{n}(Q^{2})\qquad\qquad~{}~{}~{}~{}~{}~{}\qquad
i=1,2$ (12) $\displaystyle F_{i}^{V}(Q^{2})$ $\displaystyle=$ $\displaystyle
f_{i}^{p}(Q^{2})+\frac{1}{2}f_{i}^{n}(Q^{2})\qquad~{}~{}\qquad i=1,2$ (13)

Using the expressions of $D_{i}^{V}(Q^{2})$ and $F_{i}^{V}(Q^{2})$ obtained
above and the values of the coefficients $a$ and $b$ from Table 1, the vector
form factors $f_{1,2}^{NY}(Q^{2})$ are expressed in terms of the nucleon
electromagnetic form factors as:

$\displaystyle f_{1,2}^{p\Lambda}(Q^{2})$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{2}}~{}f_{1,2}^{p}(Q^{2}),$ (14) $\displaystyle
f_{1,2}^{n\Sigma^{-}}(Q^{2})$ $\displaystyle=$
$\displaystyle-\left[f_{1,2}^{p}(Q^{2})+2f_{1,2}^{n}(Q^{2})\right],$ (15)
$\displaystyle f_{1,2}^{p\Sigma^{0}}(Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left[f_{1,2}^{p}(Q^{2})+2f_{1,2}^{n}(Q^{2})\right].$
(16)

The electromagnetic nucleon form factors, in turn, are expressed in the terms
of the Sachs’ electric ($G_{E}(Q^{2})$) and magnetic ($G_{M}(Q^{2})$) form
factors of the nucleons, for which various parameterizations are available in
the literature Bradford:2006yz ; Bosted:1994tm ; Budd:2004bp ; Alberico:2008sz
; Kelly:2004hm ; Galster:1971kv ; Platchkov:1989ch ; Punjabi:2015bba . For the
numerical calculations, we have used the parameterization given by Bradford et
al. (BBBA05) Bradford:2006yz  unless stated otherwise.

  * (ii)

Axial vector form factors:
We express $g_{1}^{NY}(Q^{2})$ in terms of $g_{1}(Q^{2})$ and $x_{1}(Q^{2})$,
which are defined as

$\displaystyle g_{1}(Q^{2})$ $\displaystyle=$ $\displaystyle
F_{1}^{A}(Q^{2})+D_{1}^{A}(Q^{2}),$ (17) $\displaystyle x_{1}(Q^{2})$
$\displaystyle=$
$\displaystyle\frac{F_{1}^{A}(Q^{2})}{F_{1}^{A}(Q^{2})+D_{1}^{A}(Q^{2})};$
(18)

where $F_{1}^{A}(Q^{2})$ and $D_{1}^{A}(Q^{2})$ are the antisymmetric and
symmetric couplings of the two octets, determined from the semileptonic decays
of hyperons at very low $Q^{2}$ $\approx 0$. It may be pointed out that there
is no information available in the literature, for the $Q^{2}$ dependence of
these parameters. Therefore, phenomenologically same $Q^{2}$ dependence for
$F_{1}^{A}$ and $D_{1}^{A}$, that is the dipole form is assumed, such that the
parameter $x_{1}(Q^{2})$ becomes a constant, i.e., $x_{1}(Q^{2})\approx
x_{1}(0)=0.364$.

The explicit expressions of $g_{1}^{NY}(Q^{2})$ for $p\longrightarrow\Lambda$,
$p\longrightarrow\Sigma^{0}$ and $n\longrightarrow\Sigma^{-}$ are given as

$\displaystyle g_{1}^{p\Lambda}(Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{6}}(1+2x_{1})g_{1}(Q^{2}),$ (19) $\displaystyle
g_{1}^{n\Sigma^{-}}(Q^{2})$ $\displaystyle=$
$\displaystyle(1-2x_{1})g_{1}(Q^{2}),$ (20) $\displaystyle
g_{1}^{p\Sigma^{0}}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(1-2x_{1})g_{1}(Q^{2}),$ (21)

where

$g_{1}(Q^{2})=\frac{g_{A}(0)}{\left(1+\frac{Q^{2}}{M_{A}^{2}}\right)^{2}},$
(22)

with $g_{A}(0)=1.267$ Cabibbo:2003cu  and $M_{A}=1.026$ GeV Bernard:2001rs .

  * (iii)

Pseudoscalar form factor:
The contribution of $g_{3}^{NY}(Q^{2})$ in $\nu_{\tau}~{}(\bar{\nu}_{\tau})-N$
scattering is significant due to the high value of $m_{\tau}$. In literature,
there exists two parameterizations, given by Marshak et al. Marshak  and by
Nambu Nambu:1960xd , for the pseudoscalar form factor in the $\Delta S=1$
channel. In order to study the effect of the pseudoscalar form factor on the
cross section and polarization observables, we have used both the
parameterizations. The expression of the pseudoscalar form factor
parameterized by Nambu Nambu:1960xd  is given as:

$g_{3}^{NY}(Q^{2})=\frac{(M+M_{Y})^{2}}{2(m_{K}^{2}+Q^{2})}g_{1}^{NY}(Q^{2}),$
(23)

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

In the parameterization of Marshak et al. Marshak , the expression for the
pseudoscalar form factor is given as:

$g_{3}^{NY}(Q^{2})=\frac{(M+M_{Y})^{2}}{2Q^{2}}\frac{g_{1}^{NY}(Q^{2})(m_{K}^{2}+Q^{2})-m_{K}^{2}g_{1}^{NY}(0)}{m_{K}^{2}+Q^{2}}.$
(24)

####  II.1.4 Form factors with SU(3) symmetry breaking effects

In the literature, SU(3) symmetry breaking effects have been studied by
various groups Wang:2019alu ; Shanahan:2015dka ; Yang:2015era ;
Becirevic:2004bb ; Lacour:2007wm ; Pham:2012db ; Pham:2014lla ; Ledwig:2014rfa
; Chang:2014iba  especially in the case of semileptonic decays of hyperons. In
this work, we have studied the effect of SU(3) symmetry breaking parameterized
in the two models by Faessler et al. Faessler:2008ix  and Schlumpf Schlumpf :

  * (A)

Faessler et al. Faessler:2008ix :
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 $f_{1}^{NY}(Q^{2})$ because of the Ademollo-Gatto theorem
Ademollo:1964sr .

    * ii)

In the presence of SU(3) symmetry breaking, the value of $f_{2}^{NY}(Q^{2})$
is modified from its SU(3) symmetric value $f_{2}^{NY}(Q^{2})$ to ${\cal
F}_{2}^{NY}(Q^{2})$, as

$\displaystyle f_{2}^{p\Lambda}(Q^{2})$ $\displaystyle\longrightarrow$
$\displaystyle{\cal
F}_{2}^{p\Lambda}(Q^{2})=f_{2}^{p\Lambda}(Q^{2})-{\frac{1}{3\sqrt{6}}}\left[H_{1}^{V}(Q^{2})-2H_{2}^{V}(Q^{2})-3H_{3}^{V}(Q^{2})-6H_{4}^{V}(Q^{2})\right],$
(25) $\displaystyle f_{2}^{n\Sigma^{-}}(Q^{2})$ $\displaystyle\longrightarrow$
$\displaystyle{\cal
F}_{2}^{n\Sigma^{-}}(Q^{2})=f_{2}^{n\Sigma^{-}}(Q^{2})-\frac{1}{3}\left[H_{1}^{V}(Q^{2})+H_{3}^{V}(Q^{2})\right],$
(26) $\displaystyle f_{2}^{p\Sigma^{0}}(Q^{2})$ $\displaystyle\longrightarrow$
$\displaystyle{\cal
F}_{2}^{p\Sigma^{0}}(Q^{2})=f_{2}^{p\Sigma^{0}}(Q^{2})-\frac{1}{3\sqrt{2}}\left[H_{1}^{V}(Q^{2})+H_{3}^{V}(Q^{2})\right],$
(27)

where $f_{2}^{NY}(Q^{2})$ for the different $N-Y$ transitions are given in
Eqs.(14)–(16) and $H_{i}^{V}(Q^{2});i=1-4$ are the SU(3) symmetry breaking
terms. Since the symmetry breaking effects, in this model, are studied for the
semileptonic decays of hyperons at very low $Q^{2}$, i.e., $Q^{2}\simeq 0$,
therefore, no information about the $Q^{2}$ dependence of $H_{i}^{V}(Q^{2})$
is available in the literature. For simplicity, a dipole parameterization is
assumed

$H_{i}^{V}(Q^{2})=\frac{H_{i}^{V}(0)}{\left(1+\frac{Q^{2}}{M_{V}^{2}}\right)^{2}},\qquad\quad
i=1-4$ (28)

where $M_{V}=0.84$ GeV is the vector dipole mass, and the values of the
couplings $H_{i}^{V}(0)$ are given in Ref. Faessler:2008ix , and are here
quoted as:

$\displaystyle H_{1}^{V}(0)~{}=~{}-0.246,\qquad\quad
H_{2}^{V}(0)~{}=~{}0.096,\qquad\quad H_{3}^{V}(0)~{}=~{}0.021,\qquad\quad
H_{4}^{V}(0)~{}=~{}0.030$

Similarly, the axial vector form factor $g_{1}^{NY}(Q^{2})$, in the presence
of SU(3) symmetry breaking, is modified to ${\cal G}_{1}^{NY}(Q^{2})$ as

$\displaystyle g_{1}^{p\Lambda}(Q^{2})$ $\displaystyle\longrightarrow$
$\displaystyle{\cal
G}_{1}^{p\Lambda}(Q^{2})=g_{1}^{p\Lambda}(Q^{2})-{\frac{1}{3\sqrt{6}}}\left[H_{1}^{A}(Q^{2})-2H_{2}^{A}(Q^{2})-3H_{3}^{A}(Q^{2})-6H_{4}^{A}(Q^{2})\right],$
(29) $\displaystyle g_{1}^{n\Sigma^{-}}(Q^{2})$ $\displaystyle\longrightarrow$
$\displaystyle{\cal
G}_{1}^{n\Sigma^{-}}(Q^{2})=g_{1}^{n\Sigma^{-}}(Q^{2})-\frac{1}{3}\left[H_{1}^{A}(Q^{2})+H_{3}^{A}(Q^{2})\right],$
(30) $\displaystyle g_{1}^{p\Sigma^{0}}(Q^{2})$ $\displaystyle\longrightarrow$
$\displaystyle{\cal
G}_{1}^{p\Sigma^{0}}(Q^{2})=g_{1}^{p\Sigma^{0}}(Q^{2})+\frac{1}{3\sqrt{2}}\left[H_{1}^{A}(Q^{2})+H_{3}^{A}(Q^{2})\right],$
(31)

with $g_{1}^{NY}(Q^{2})$ defined in Eqs.(19)–(21) and a dipole
parameterization is assumed for $H_{i}^{A}(Q^{2})$ as

$H_{i}^{A}(Q^{2})=\frac{H_{i}^{A}(0)}{\left(1+\frac{Q^{2}}{M_{A}^{2}}\right)^{2}};\qquad\quad
i=1-4$ (32)

where the couplings $H_{i}^{A}(0)$ are Faessler:2008ix

$\displaystyle H_{1}^{A}(0)~{}=~{}-0.050,\qquad\quad
H_{2}^{A}(0)~{}=~{}0.011,\qquad\quad H_{3}^{A}(0)~{}=~{}-0.006,\qquad\quad
H_{4}^{A}(0)~{}=~{}0.037.$

    * iii)

Since the pseudoscalar form factor is parameterized in terms of the axial
vector form factor (Eqs. (23) and (24)), therefore, it receives SU(3) symmetry
breaking effect via., $g_{1}^{NY}(Q^{2})$.

  * (B)

Schlumpf Schlumpf :
Schlumpf Schlumpf  has studied SU(3) symmetry breaking in the hadronic current
containing vector ($f_{1}$) and axial vector ($g_{1}$) form factors using
relativistic quark model, and this symmetry breaking in the model originates
from the mass difference between $m_{s}$ and $m_{u/d}$ quarks. The modified
$f_{1}$ and $g_{1}$ form factors are given by

$\displaystyle f_{1}(Q^{2})$ $\displaystyle\rightarrow$ $\displaystyle
f_{1}^{\prime}(Q^{2})=\alpha f_{1}(Q^{2})$ $\displaystyle g_{1}(Q^{2})$
$\displaystyle\rightarrow$ $\displaystyle g_{1}^{\prime}(Q^{2})=\beta
g_{1}(Q^{2}),\text{where}$ (33)

$\alpha$=0.976, 0.975 and 0.975; $\beta$=1.072, 1.051 and 1.056, respectively
for $p\longrightarrow\Lambda$, $p\longrightarrow\Sigma^{0}$ and
$n\longrightarrow\Sigma^{-}$ transitions. Since the induced pseudoscalar form
factor $g_{3}$ is related to the axial vector form factor $g_{1}$, therefore,
this modification is also applicable to the terms containing $g_{3}$ through
Eqs. (23) and (24).

##  III Cross section and polarization observables of the final lepton

###  III.1 Cross section

The general expression of the differential cross section for the processes
given in Eq. (1), in the laboratory frame, is given by

$\displaystyle d\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\frac{1}{4ME_{\bar{\nu}_{\tau}}}\delta^{4}(k+p-k^{\prime}-p^{\prime})\frac{d^{3}k^{\prime}}{2E_{k^{\prime}}}\frac{d^{3}p^{\prime}}{2E_{p^{\prime}}}\overline{\sum}\sum|{\cal{M}}|^{2}.$
(34)

Using Eqs. (2)–(4), the transition matrix element squared is obtained as:

$\overline{\sum}\sum|{\cal{M}}|^{2}=\frac{G_{F}^{2}\sin^{2}\theta_{c}}{2}\cal{J}^{\mu\nu}\cal{L}_{\mu\nu},$
(35)

where the hadronic ($\cal{J}_{\mu\nu}$) and the leptonic ($\cal{L}_{\mu\nu}$)
tensors are obtained using Eqs. (3) and (4) as

$\displaystyle\cal{J}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\overline{\sum}\sum
J_{\mu}J_{\nu}^{\dagger},\qquad\quad{\cal{L}}^{\mu\nu}~{}=~{}\overline{\sum}\sum
l_{\mu}l_{\nu}^{\dagger}.$ (36)

Following the above definitions, the differential scattering cross section
$d\sigma/dQ^{2}$ for the processes given in Eq. (1) is written as

$\frac{d\sigma}{dQ^{2}}=\frac{G_{F}^{2}\sin^{2}\theta_{c}}{8\pi{M}^{2}{E^{2}_{\bar{\nu}_{\tau}}}}N(Q^{2}),$
(37)

where $N(Q^{2})=\cal{J}^{\mu\nu}\cal{L}_{\mu\nu}$ is obtained from the
expression given in Appendix-A of Ref. Fatima:2018tzs  with the substitution
of $M^{\prime}=M_{Y}$ and $m_{\mu}=m_{\tau}$.

###  III.2 Polarization of the final lepton

Using the covariant density matrix formalism, the polarization 4-vector
($\zeta^{\tau}$) of the $\tau$ lepton produced in the final state in reactions
given in Eq. (1) is written as Bilenky

$\zeta^{\tau}=\frac{\mathrm{Tr}[\gamma^{\tau}\gamma_{5}~{}\rho_{f}(k^{\prime})]}{\mathrm{Tr}[\rho_{f}(k^{\prime})]},$
(38)

and the spin density matrix for the final lepton $\rho_{f}(k^{\prime})$ is
given by

$\rho_{f}(k^{\prime})={\cal J}^{\alpha\beta}\text{
Tr}[\Lambda(k^{\prime})\gamma_{\alpha}(1\pm\gamma_{5})\Lambda(k)\tilde{\gamma}_{\beta}(1\pm\tilde{\gamma}_{5})\Lambda(k^{\prime})],$
(39)

with $\tilde{\gamma}_{\alpha}=\gamma^{0}\gamma^{\dagger}_{\alpha}\gamma^{0}$
and $\tilde{\gamma}_{5}=\gamma^{0}\gamma^{\dagger}_{5}\gamma^{0}$.

Figure 2: (a) Momentum and polarization directions of the final lepton
produced in the reaction
$\bar{\nu}_{\tau}(k)+N(p)\longrightarrow\tau^{+}(k^{\prime})+Y(p^{\prime})$.
(b) $\hat{e}_{L}^{l}$, $\hat{e}_{P}^{l}$ and $\hat{e}_{T}^{l}$ represent the
orthogonal unit vectors corresponding to the longitudinal, perpendicular and
transverse directions with respect to the momentum of the final lepton.

Using the following relations:

$\Lambda(k^{\prime})\gamma^{\tau}\gamma_{5}\Lambda(k^{\prime})=2m_{\tau}\left(g^{\tau\sigma}-\frac{k^{\prime\tau}k^{\prime\sigma}}{m_{\tau}^{2}}\right)\Lambda(k^{\prime})\gamma_{\sigma}\gamma_{5}$
(40)

and

$\Lambda(k^{\prime})\Lambda(k^{\prime})=2m_{\tau}\Lambda(k^{\prime}),$ (41)

$\zeta^{\tau}$ defined in Eq. (38) may also be rewritten as

$\zeta^{\tau}=\left(g^{\tau\sigma}-\frac{k^{\prime\tau}k^{\prime\sigma}}{m_{\tau}^{2}}\right)\frac{{\cal
J}^{\alpha\beta}\mathrm{Tr}\left[\gamma_{\sigma}\gamma_{5}\Lambda(k^{\prime})\gamma_{\alpha}(1+\gamma_{5})\Lambda(k)\tilde{\gamma}_{\beta}(1+\tilde{\gamma}_{5})\right]}{{\cal
J}^{\alpha\beta}\mathrm{Tr}\left[\Lambda(k^{\prime})\gamma_{\alpha}(1+\gamma_{5})\Lambda(k)\tilde{\gamma}_{\beta}(1+\tilde{\gamma}_{5})\right]},$
(42)

where $m_{\tau}$ is the mass of the $\tau$ lepton. In Eq. (42), the
denominator is directly related to the differential cross section given in Eq.
(37).

With ${\cal J}^{\alpha\beta}$ and ${\cal L}_{\alpha\beta}$ given in Eq. (36),
an expression for $\zeta^{\tau}$ is obtained. In the laboratory frame where
the initial nucleon is at rest, the polarization vector $\vec{\zeta}$,
assuming T invariance, is calculated to be a function of 3-momenta of incoming
antineutrino $({\vec{k}})$ and outgoing lepton $({\vec{k}}\,^{\prime})$, and
is given as

$\vec{\zeta}=\left[{A^{l}(Q^{2})\vec{k}}+B^{l}(Q^{2}){\vec{k}}\,^{\prime}\right],$
(43)

where the expressions of $A^{l}(Q^{2})$ and $B^{l}(Q^{2})$ are obtained from
the expression given in Appendix-B of Ref. Fatima:2018tzs  with the
substitution $M^{\prime}=M_{Y}$ and $m_{\mu}=m_{\tau}$.

One may expand the polarization vector $\vec{\zeta}$ along the orthogonal
directions, ${\hat{e}}_{L}^{l}$, ${\hat{e}}_{P}^{l}$ and ${\hat{e}_{T}^{l}}$
in the reaction plane corresponding to the longitudinal, perpendicular and
transverse directions of the final lepton ($l$), as depicted in Fig. 2 and
defined as

$\hat{e}_{L}^{l}=\frac{\vec{k}^{\,\prime}}{|\vec{k}^{\,\prime}|},\qquad\hat{e}_{P}^{l}=\hat{e}_{L}^{l}\times\hat{e}_{T}^{l},\qquad{\rm
where}~{}~{}~{}~{}~{}\hat{e}_{T}^{l}=\frac{\vec{k}\times\vec{k}^{\,\prime}}{|\vec{k}\times\vec{k}^{\,\prime}|}.$
(44)

We then write $\vec{\zeta}$ as:

$\vec{\zeta}=\zeta_{L}\hat{e}_{L}^{l}+\zeta_{P}\hat{e}_{P}^{l}+\zeta_{T}\hat{e}_{T}^{l},$
(45)

such that the longitudinal and perpendicular components of the $\vec{\zeta}$
in the laboratory frame are given by

$\zeta_{L}(Q^{2})=\vec{\zeta}\cdot\hat{e}_{L}^{l},\qquad\zeta_{P}(Q^{2})=\vec{\zeta}\cdot\hat{e}_{P}^{l}.$
(46)

From Eq. (46), the longitudinal $P_{L}(Q^{2})$ and perpendicular
$P_{P}(Q^{2})$ components of the polarization vector defined in the rest frame
of the outgoing lepton are given by

$P_{L}(Q^{2})=\frac{m_{\tau}}{E_{k^{\prime}}}\zeta_{L}(Q^{2}),\qquad
P_{P}(Q^{2})=\zeta_{P}(Q^{2}),$ (47)

where $\frac{m_{\tau}}{E_{k^{\prime}}}$ is the Lorentz boost factor along
${\vec{k}}\,^{\prime}$. Using Eqs. (43), (44) and (46) in Eq. (47), the
longitudinal $P_{L}(Q^{2})$ and perpendicular $P_{P}(Q^{2})$ components are
calculated to be

$\displaystyle P_{L}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{m_{\tau}}{E_{k^{\prime}}}\frac{A^{l}(Q^{2})\vec{k}.\vec{k}^{\,\prime}+B^{l}(Q^{2})|\vec{k}^{\,\prime}|^{2}}{N(Q^{2})~{}|\vec{k}^{\,\prime}|},$
(48) $\displaystyle P_{P}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{A^{l}(Q^{2})[|\vec{k}|^{2}|\vec{k}^{\,\prime}|^{2}-(\vec{k}.\vec{k}^{\,\prime})^{2}]}{N(Q^{2})~{}|\vec{k}^{\,\prime}|~{}|\vec{k}\times\vec{k}^{\,\prime}|},$
(49)

where $N(Q^{2})=\cal{J}^{\mu\nu}\cal{L}_{\mu\nu}$ is obtained from the
expression given in Appendix-A of Ref. Fatima:2018tzs .

##  IV Results and discussion

In this section, we present and discuss the results of the differential
(Section IV.1) and total (Section IV.2) scattering cross sections as well as
the polarization observables of the final $\tau$ lepton produced in the
$|\Delta S|=1$ quasielastic scattering of $\bar{\nu}_{\tau}$ from nucleons. We
also present a comparison of the total cross section for the production of
$e$, $\mu$ and $\tau$ leptons in the quasielastic scattering of
$\bar{\nu}_{e}$, $\bar{\nu}_{\mu}$ and $\bar{\nu}_{\tau}$ to demonstrate the
implications of LFU in these processes (Section IV.3).

###  IV.1 Differential scattering cross section and polarization observables

Figure 3: $\frac{d\sigma}{dQ^{2}}$ (left panel), $P_{L}(Q^{2})$ (middle panel)
and $P_{P}(Q^{2})$ (right panel) versus $Q^{2}$ for the process
$\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda$ at $E_{\bar{\nu}_{\tau}}$
= 4 GeV (upper panel), 5 GeV (middle panel) and 10 GeV (lower panel). The
calculations have been performed using the SU(3) symmetry with the axial
dipole mass $M_{A}=1.026$ GeV and for the different parameterizations of the
nucleon vector form factors viz., BBBA05 Bradford:2006yz  (solid line), BBA03
Budd:2004bp  (dashed line), Alberico et al. Alberico:2008sz  (dashed-dotted
line), Bosted Bosted:1994tm  (dotted line), Galster et al. Galster:1971kv
(double-dotted-dashed line) and Kelly Kelly:2004hm  (double-dashed-dotted
line).

Figure 4: $\frac{d\sigma}{dQ^{2}}$ (left panel), $P_{L}(Q^{2})$ (middle panel)
and $P_{P}(Q^{2})$ (right panel) versus $Q^{2}$ for the process
$\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda$ at $E_{\bar{\nu}_{\tau}}$
= 4 GeV (upper panel), 5 GeV (middle panel) and 10 GeV (lower panel). The
calculations have been performed using the SU(3) symmetry with the electric
and magnetic Sachs form factors parameterized by Bradford et al.
Bradford:2006yz  and for the axial form factor, the different values of
$M_{A}$ have been used viz. $M_{A}=$ 0.9 GeV (solid line), 1.026 GeV (dashed
line), 1.1 GeV (dashed-dotted line), 1.2 GeV (dotted line) and 1.3 GeV
(double-dotted-dashed line).

Figure 5: $\frac{d\sigma}{dQ^{2}}$ (left panel), $P_{L}(Q^{2})$ (middle panel)
and $P_{P}(Q^{2})$ (right panel) versus $Q^{2}$ for the process
$\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda$ at $E_{\bar{\nu}_{\tau}}$
= 4 GeV (upper panel), 5 GeV (middle panel) and 10 GeV (lower panel). The
calculations have been performed using the SU(3) symmetry with the electric
and magnetic Sachs form factors parameterized by Bradford et al.
Bradford:2006yz , $M_{A}=1.026$ GeV, and for the different parameterizations
of the pseudoscalar scalar form factor, viz. the parameterizations given by
Nambu Nambu:1960xd  (solid line) and by Marshak et al. Marshak  (dashed line).

Figure 6: $\frac{d\sigma}{dQ^{2}}$ (left panel), $P_{L}(Q^{2})$ (middle panel)
and $P_{P}(Q^{2})$ (right panel) versus $Q^{2}$ for the process
$\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda$ at $E_{\bar{\nu}_{\tau}}$
= 4 GeV (upper panel), 5 GeV (middle panel) and 10 GeV (lower panel). The
calculations have been performed with $M_{A}=1.026$ GeV and BBBA05
Bradford:2006yz  for the vector form factors with SU(3) symmetry (solid line),
SU(3) symmetry breaking effects parameterized by Faessler et al.
Faessler:2008ix  (dashed line) and the symmetry breaking effects parameterized
by Schlumpf Schlumpf  (dashed-dotted line).

Figure 7: $\frac{d\sigma}{dQ^{2}}$ (left panel), $P_{L}(Q^{2})$ (middle panel)
and $P_{P}(Q^{2})$ (right panel) versus $Q^{2}$ for the process
$\bar{\nu}_{\tau}+n\longrightarrow\tau^{+}+\Sigma^{-}$ at
$E_{\bar{\nu}_{\tau}}$ = 5 GeV (upper panel) and 10 GeV (lower panel). The
calculations have been performed using the SU(3) symmetry with the axial
dipole mass $M_{A}=1.026$ GeV and for the different parameterizations of the
nucleon vector form factors viz., BBBA05 Bradford:2006yz  (solid line), BBA03
Budd:2004bp  (dashed line), Alberico et al. Alberico:2008sz  (dashed-dotted
line), Bosted Bosted:1994tm  (dotted line), Galster et al. Galster:1971kv
(double-dotted-dashed line) and Kelly Kelly:2004hm  (double-dashed-dotted
line).

Figure 8: $\frac{d\sigma}{dQ^{2}}$ (left panel), $P_{L}(Q^{2})$ (middle panel)
and $P_{P}(Q^{2})$ (right panel) versus $Q^{2}$ for the process
$\bar{\nu}_{\tau}+n\longrightarrow\tau^{+}+\Sigma^{-}$ at
$E_{\bar{\nu}_{\tau}}$ = 5 GeV (upper panel) and 10 GeV (lower panel). The
calculations have been performed using the SU(3) symmetry with the electric
and magnetic Sachs form factors parameterized by Bradford et al.
Bradford:2006yz  and for the axial form factor, the different values of
$M_{A}$ have been used viz. $M_{A}=$ 0.9 GeV (solid line), 1.026 GeV (dashed
line), 1.1 GeV (dashed-dotted line), 1.2 GeV (dotted line) and 1.3 GeV
(double-dotted-dashed line).

Figure 9: $\frac{d\sigma}{dQ^{2}}$ (left panel), $P_{L}(Q^{2})$ (middle panel)
and $P_{P}(Q^{2})$ (right panel) versus $Q^{2}$ for the process
$\bar{\nu}_{\tau}+n\longrightarrow\tau^{+}+\Sigma^{-}$ at
$E_{\bar{\nu}_{\tau}}$ = 5 GeV (upper panel) and 10 GeV (lower panel). The
calculations have been performed with $M_{A}=1.026$ GeV assuming SU(3)
symmetry (solid line), as well as with SU(3) symmetry breaking effects
parameterized by Faessler et al. Faessler:2008ix  (dashed line) and by
Schlumpf Schlumpf  (dashed-dotted line).

We have used Eqs. (37), (48) and (49), respectively, to numerically evaluate
the differential scattering cross section $d\sigma/dQ^{2}$, the longitudinal
($P_{L}(Q^{2})$) and the perpendicular ($P_{P}(Q^{2})$) components of
polarization of $\tau$ lepton. The Dirac and Pauli form factors
$f_{1,2}^{N}(Q^{2});~{}(N=p,n)$ are expressed in terms of the electric and
magnetic Sachs’ form factors, for which the various parameterizations
Bradford:2006yz ; Budd:2004bp ; Bosted:1994tm ; Alberico:2008sz ; Kelly:2004hm
; Galster:1971kv  available in the literature, have been used. For
$g_{1}(Q^{2})$ a dipole parameterization has been used (Eq. (22)), with the
world average value of the axial dipole mass $M_{A}=$ 1.026 GeV. For the
pseudoscalar form factor $g_{3}(Q^{2})$, the parameterizations given by Nambu
Nambu:1960xd  and Marshak et al. Marshak  have been used. The numerical
results of the differential scattering cross section and polarization
observables obtained assuming SU(3) symmetry and the results with SU(3)
symmetry breaking effects, using the prescriptions of: (i) Faessler et al.
Faessler:2008ix , and (ii) Schlumpf Schlumpf , are presented separately for
the $\Lambda$ and $\Sigma^{-}$ productions from the nucleons. The results for
the $\Sigma^{0}$ production can be expressed in terms of the $\Sigma^{-}$
production in the SU(3) symmetric limit (Eqs. (10 and 11)) and are not
presented separately.

####  IV.1.1 $\Lambda$ production

In Fig. 3, we present the results for the $Q^{2}$ distribution i.e.
$\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$ and $P_{P}(Q^{2})$ vs $Q^{2}$ for
$\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda$ process at the three
different values of energy viz. $E_{\bar{\nu}_{\tau}}=$ 4 GeV, 5 GeV and 10
GeV, assuming SU(3) symmetry with $M_{A}=1.026$ GeV and using the different
parameterizations of the nucleon vector form factors viz., BBBA05
Bradford:2006yz , BBA03 Budd:2004bp , Alberico Alberico:2008sz , Bosted
Bosted:1994tm , Galster Galster:1971kv  and Kelly Kelly:2004hm . We see that
at low ${\bar{\nu}}_{\tau}$ energies, for example at $E_{\bar{\nu}_{\tau}}=4$
GeV, there is considerable dependence of the different parameterizations of
the vector form factors on $\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$ and
$P_{P}(Q^{2})$ distributions. However, with the increase in
${\bar{\nu}}_{\tau}$ energy, this difference decreases and becomes almost
negligible at higher energies, like at $E_{\bar{\nu}_{\tau}}=10$ GeV,
especially for $\frac{d\sigma}{dQ^{2}}$ and $P_{L}(Q^{2})$ distributions.

To study the effect of the variation in $M_{A}$ (in the range 0.9–1.3 GeV) on
the differential cross section and polarization observables, we present, in
Fig. 4, the results for $\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$ and
$P_{P}(Q^{2})$ distributions at $E_{\bar{\nu}_{\tau}}$ = 4 GeV, 5 GeV and 10
GeV. We find that at low ${\bar{\nu}}_{\tau}$ energies, there is a significant
dependence of these distributions on the choice of $M_{A}$. With the increase
in ${\bar{\nu}}_{\tau}$ energy, this dependence on the variation in $M_{A}$
decreases, especially for $\frac{d\sigma}{dQ^{2}}$ and to some extent for
$P_{L}(Q^{2})$ but not for $P_{P}(Q^{2})$ distribution. Moreover, it is
important to point out that in the case of
$\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda$ reaction, with the
increase in $M_{A}$, $\frac{d\sigma}{dQ^{2}}$ decreases (0.9 GeV to 1.1 GeV),
but with the further increase in $M_{A}$ (1.1 GeV to 1.3 GeV),
$\frac{d\sigma}{dQ^{2}}$ increases, which is not generally the case in
${\nu}_{l}+n\longrightarrow l^{-}+p;(l=e,\mu,\tau)$ scattering. Moreover, in
the case of $\bar{\nu}_{l}+p\longrightarrow l^{+}+n$, we have observed that
with the increase in $M_{A}$, $\frac{d\sigma}{dQ^{2}}$ decreases (from 0.9 GeV
to 1.1 GeV) and with further increase in $M_{A}=1.2$ GeV,
$\frac{d\sigma}{dQ^{2}}$ increases Fatima:2020pvv . In the present work, for
$\Lambda$ production we observe similar trend as in the case of
$\bar{\nu}_{\tau}$ induced CCQE reaction Fatima:2020pvv . In the
$\bar{\nu}_{\tau}$ induced reactions because of the production of massive
$\tau$ lepton in the final state, the pseudoscalar form factor becomes
significant. The variation in $\frac{d\sigma}{dQ^{2}}$ observed in Fig. 4
arises due to interference of the pseudoscalar form factor with axial vector
and vector form factors.

Figure 10: $\sigma$ (left panel), $\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$
(middle panel), and $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ (right panel) vs
$E_{\bar{\nu}_{\tau}}$ for $\bar{\nu}_{\tau}+p\rightarrow\tau^{+}+\Lambda$
process. The calculations have been performed using the SU(3) symmetry with
electric and magnetic Sachs form factors parameterized by Bradford et al.
Bradford:2006yz  and for the axial form factor, the different values of
$M_{A}$ have been used viz. $M_{A}=$ 0.9 GeV (solid line), 1.026 GeV (dashed
line), 1.1 GeV (dashed-dotted line), 1.2 GeV (dotted line) and 1.3 GeV
(double-dotted-dashed line).

To see the dependence of $\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$ and
$P_{P}(Q^{2})$ on the pseudoscalar form factor $g_{3}^{NY}(Q^{2})$, we have
used the two parameterizations of $g_{3}^{NY}(Q^{2})$ given in Eqs. (23) by
Nambu Nambu:1960xd , and (24) by Marshak et al. Marshak , and show the
numerical results in Fig. 5. It may be observed that at low
${\bar{\nu}}_{\tau}$ energies, there is a large dependence on the choice of
$g_{3}^{NY}(Q^{2})$. While with the increase in $E_{\bar{\nu}_{\tau}}$, this
dependence on the choice of $g_{3}^{NY}(Q^{2})$ becomes almost negligible for
the $\frac{d\sigma}{dQ^{2}}$ distribution, whereas for $P_{L}(Q^{2})$ and
$P_{P}(Q^{2})$ distributions, they are found to be quite significant, even for
the higher values of $E_{\bar{\nu}_{\tau}}$, say $E_{\bar{\nu}_{\tau}}=10$
GeV.

In Fig. 6, we present the results for $\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$
and $P_{P}(Q^{2})$ vs $Q^{2}$ at $E_{\bar{\nu}_{\tau}}$ = 4 GeV, 5 GeV and 10
GeV, when the SU(3) symmetry breaking effects are taken into account following
the prescription of Faessler et al. Faessler:2008ix  and Schlumpf Schlumpf .
We observe that at low ${\bar{\nu}}_{\tau}$ energies, for example at
$E_{\bar{\nu}_{\tau}}=4$ GeV, there is some effect of SU(3) symmetry breaking
on the $\frac{d\sigma}{dQ^{2}}$ distribution if the parameterization by
Schlumpf Schlumpf  is used, while there is almost no effect if one uses the
parameterization of Faessler et al. Faessler:2008ix . However, in the case of
$P_{L}(Q^{2})$ and $P_{P}(Q^{2})$, there is some effect of SU(3) breaking
using either the prescription of Faessler et al. Faessler:2008ix  or Schlumpf
Schlumpf . Moreover, with the increase in ${\bar{\nu}}_{\tau}$ energy, the
difference in $\frac{d\sigma}{dQ^{2}}$ due to the use of the two
parameterizations of SU(3) symmetry breaking becomes almost negligible at
higher energies. In the case of polarization observables, the two
parameterizations of SU(3) symmetry breaking give different results for all
the values of $Q^{2}$.

####  IV.1.2 $\Sigma^{-}$ production

In Fig. 7, we present the results for $\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$
and $P_{P}(Q^{2})$ vs $Q^{2}$ for the process
$\bar{\nu}_{\tau}+n\longrightarrow\tau^{+}+\Sigma^{-}$ at the two different
value of energies viz. $E_{\bar{\nu}_{\tau}}$ = 5 GeV and 10 GeV, using the
different parameterizations of the nucleon vector form factors viz., BBBA05
Bradford:2006yz , BBA03 Budd:2004bp , Alberico Alberico:2008sz , Bosted
Bosted:1994tm , Galster Galster:1971kv  and Kelly Kelly:2004hm . We observe
that at low ${\bar{\nu}}_{\tau}$ energies, for example at
$E_{\bar{\nu}_{\tau}}=5$ GeV, there is considerable dependence of the
different parameterizations of the vector form factors on
$\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$ and $P_{P}(Q^{2})$ distributions.
However, unlike the case of $\Lambda$ production (Fig. 4), with the increase
in ${\bar{\nu}}_{\tau}$ energy, this difference further increases and becomes
quite significant at higher energies, especially for the distributions of the
polarization observables $P_{L}(Q^{2})$ and $P_{P}(Q^{2})$.

The effect of $M_{A}$ variation on the differential cross section and
polarization observables are presented in Fig. 8, at $E_{\bar{\nu}_{\tau}}$ =
5 GeV and 10 GeV by varying $M_{A}$ in the range 0.9–1.3 GeV. We find that at
low ${\bar{\nu}}_{\tau}$ energies, there is large dependence of these
distributions on the choice of $M_{A}$, which increases with increase in
antineutrino energy.

In Fig. 9, we show the effect of SU(3) symmetry breaking by taking into
account the parameterizations given by Faessler et al. Faessler:2008ix  and
Schlumpf Schlumpf  on $\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$ and
$P_{P}(Q^{2})$ vs $Q^{2}$ at $E_{\bar{\nu}_{\tau}}$ = 5 GeV and 10 GeV. From
the figure, it may be observed that in the case of $\frac{d\sigma}{dQ^{2}}$,
at low energies, say $E_{\bar{\nu}_{\tau}}=5$ GeV, the effect of SU(3)
symmetry breaking is quite small. While at higher energies, say
$E_{\bar{\nu}_{\tau}}=10$ GeV, the effect of SU(3) breaking is seen in the
model of Faessler et al. Faessler:2008ix  but not in the model of Schlumpf
Schlumpf . Whereas the polarization observables (both $P_{L}(Q^{2})$ and
$P_{P}(Q^{2})$) are quite sensitive to SU(3) symmetry breaking effect, and
there is considerable change in the results obtained using the two different
choices of parameterizing the SU(3) symmetry breaking effect.

Figure 11: $\sigma$ (left panel), $\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$
(middle panel), and $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ (right panel) vs
$E_{\bar{\nu}_{\tau}}$ for $\bar{\nu}_{\tau}+p\rightarrow\tau^{+}+\Lambda$
process. The calculations have been performed using the SU(3) symmetry with
electric and magnetic Sachs form factors parameterized by Bradford et al.
Bradford:2006yz  and for the axial form factor, $M_{A}=1.026$ GeV is used,
with the different parameterizations of the pseudoscalar form factor viz.,
using the parameterizations given by Nambu Nambu:1960xd  (solid line) and
Marshak et al. Marshak  (dashed line).

Figure 12: $\sigma$ (left panel), $\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$
(middle panel), and $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ (right panel) vs
$E_{\bar{\nu}_{\tau}}$ for $\bar{\nu}_{\tau}+p\rightarrow\tau^{+}+\Lambda$
process. The calculations have been performed using the SU(3) symmetry (solid
line), the SU(3) symmetry breaking effects parameterized by Faessler et al.
Faessler:2008ix  (dashed line) and by Schlumpf Schlumpf  (dashed-dotted line),
with electric and magnetic Sachs form factors parameterized by Bradford et al.
Bradford:2006yz  and for the axial form factor, $M_{A}=1.026$ GeV is used.

Figure 13: $\sigma$ (left panel), $\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$
(middle panel), and $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ (right panel) vs
$E_{\bar{\nu}_{\tau}}$ for $\bar{\nu}_{\tau}+n\rightarrow\tau^{+}+\Sigma^{-}$
process. The calculations have been performed using the SU(3) symmetry with
the electric and magnetic Sachs form factors parameterized by Bradford et al.
Bradford:2006yz  and for the axial form factor, the different values of
$M_{A}$ have been used viz. $M_{A}=$ 0.9 GeV (solid line), 1.026 GeV (dashed
line), 1.1 GeV (dashed-dotted line), 1.2 GeV (dotted line) and 1.3 GeV
(double-dotted-dashed line).

Figure 14: $\sigma$ (left panel), $\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$
(middle panel), and $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ (right panel) vs
$E_{\bar{\nu}_{\tau}}$ for $\bar{\nu}_{\tau}+n\rightarrow\tau^{+}+\Sigma^{-}$
process. The calculations have been performed using the SU(3) symmetry (solid
line), the SU(3) symmetry breaking effects parameterized by Faessler et al.
Faessler:2008ix  (dashed line) and by Schlumpf Schlumpf  (dashed-dotted line),
with electric and magnetic Sachs form factors parameterized by Bradford et al.
Bradford:2006yz  and for the axial form factor, $M_{A}=1.026$ GeV is used.

###  IV.2 Total scattering cross section and average polarizations

To study the dependence of the total scattering cross section
$\sigma(E_{\bar{\nu}_{\tau}})$ and the average polarizations
$\overline{P}_{L,P}(E_{\bar{\nu}_{\tau}})$ on $E_{\bar{\nu}_{\tau}}$, we have
integrated $d\sigma/dQ^{2}$ and $P_{L,P}(Q^{2})$ over $Q^{2}$, and obtained
the expressions for $\sigma(E_{\bar{\nu}_{\tau}})$ and
$\overline{P}_{L,P}(E_{\bar{\nu}_{\tau}})$ i.e.:

$\displaystyle\sigma(E_{\bar{\nu}_{\tau}})$ $\displaystyle=$
$\displaystyle\int_{Q^{2}_{min}}^{Q^{2}_{max}}\frac{d\sigma}{dQ^{2}}dQ^{2},$
(50) $\displaystyle\overline{P}_{L,P}(E_{\bar{\nu}_{\tau}})$ $\displaystyle=$
$\displaystyle\frac{\int_{Q^{2}_{min}}^{Q^{2}_{max}}P_{L,P}(Q^{2})\frac{d\sigma}{dQ^{2}}dQ^{2}}{\int_{Q^{2}_{min}}^{Q^{2}_{max}}\frac{d\sigma}{dQ^{2}}dQ^{2}}.$
(51)

In this section, we present the results for the total cross section (Eq. (50))
and average polarizations (Eq. 51) of the tau lepton produced in
$\bar{\nu}_{\tau}+N\longrightarrow\tau^{+}+Y$ reaction, separately for
$\Lambda$ and $\Sigma^{-}$ productions, in Sections IV.2.1 and IV.2.2,
respectively.

####  IV.2.1 $\Lambda$ production

In Fig. 10, the results are presented for $\sigma$ as well as for
$\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$ and
$\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ obtained using the different values
of $M_{A}$ viz. $M_{A}=$ 0.9, 1.026, 1.1, 1.2 and 1.3 GeV for the process
$\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda$. As expected, $\sigma$
increases with the increase in antineutrino energy as well as it increases in
magnitude with higher values of $M_{A}$. For example, at
$E_{\bar{\nu}_{\tau}}=10$ GeV, $\sigma$ increases by about 25%, when the value
of $M_{A}$ is increased from its world average value ($M_{A}=1.026$ GeV) by
30% and a decrease in the value of $M_{A}$ by 10% from the world average
value, decreases the cross section by $\sim 7\%$. At low antineutrino energies
$E_{\bar{\nu}_{\tau}}=4$ GeV, $\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$
increases with the increase in $M_{A}$, which is almost 40% when $M_{A}$ is
varied from 0.9 to 1.3 GeV. However, with the increase in
$E_{\bar{\nu}_{\tau}}$, this variation in
$\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$ decreases and becomes almost
negligible at $E_{\bar{\nu}_{\tau}}=10$ GeV. We observe some dependence of
$M_{A}$ on $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ in the entire range of
$E_{\bar{\nu}_{\tau}}$. Note that in the case of $\Lambda$ production,
$\sigma$, $\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$ and
$\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ are almost insensitive to the
different parameterizations of the Sachs’ form factors, and are not shown here
explicitly.

In Fig. 11, we show the dependence of $\sigma$ and
$\overline{P}_{L,P}(E_{\bar{\nu}_{\tau}})$ on the pseudoscalar form factor,
when the parameterizations by Marshak et al. Marshak  and Nambu Nambu:1960xd
are used in the numerical calculations. It may be observed from the figure
that $\sigma$ as well as the average polarizations show some dependence on the
pseudoscalar form factor.

In Fig. 12, we present the results for $\sigma$,
$\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$ and
$\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ with SU(3) symmetry as well as when
the SU(3) symmetry breaking effects are taken into account following the
prescriptions of Faessler et al. Faessler:2008ix  and Schlumpf Schlumpf . We
find that there is not much effect of SU(3) symmetry breaking on $\sigma$,
while there is some effect on $\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$ whereas
in the case of $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$, the effect is found
to be significant.

####  IV.2.2 $\Sigma^{-}$ production

The dependence of the total cross section and the average polarizations on
$M_{A}$ are shown in Fig. 13, where we present the results for $\sigma$ and
$\overline{P}_{L,P}(E_{\bar{\nu}_{\tau}})$ for the process
$\bar{\nu}_{\tau}+n\longrightarrow\tau^{+}+\Sigma^{-}$ by varying $M_{A}$ in
the range, $M_{A}=$ 0.9 to 1.3 GeV. The results are qualitatively similar to
the results obtained in Fig. 10 for the $\bar{\nu}_{\tau}$ induced $\Lambda$
production. $\sigma$ increases with increase in the value of $M_{A}$ and shows
considerable variation at higher antineutrino energies. The average
polarizations are quite sensitive to the variation in $M_{A}$, especially at
low $E_{\bar{\nu}_{\tau}}$.

In Fig. 14, we show the results for $\sigma$,
$\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$ and
$\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ with SU(3) symmetry as well as when
the SU(3) symmetry breaking effects are taken into account. We find the effect
of SU(3) symmetry breaking to be quite small on $\sigma$. In the case of
$\overline{P}_{L}(E_{\bar{\nu}_{\tau}})$, the effect of SU(3) symmetry
breaking is more when Faessler et al. Faessler:2008ix  prescription is used.
While in the case of $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$, the symmetry
breaking effects using the model of Faessler et al. Faessler:2008ix  increases
the magnitude of $\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ while the model of
Schlumpf Schlumpf  decreases the magnitude of
$\overline{P}_{P}(E_{\bar{\nu}_{\tau}})$ by about 25% in the entire range of
$E_{\bar{\nu}_{\tau}}$.

Figure 15: $R_{1}$ as a function of $E_{\bar{\nu}}$ (left panel) with
$M_{A}=1.026$ GeV (solid line), $M_{A}=1.1$ GeV (dashed line) and $M_{A}=1.2$
GeV (dashed-dotted line). $R_{2}$ as a function of $E_{\bar{\nu}}$ (middle
panel) with $M_{A}=1.026$ GeV (solid line), $M_{A}=1.1$ GeV (dashed line) and
$M_{A}=1.2$ GeV (dashed-dotted line). $R_{2}$ as a function of $E_{\bar{\nu}}$
(right panel) with $g_{3}^{NY}(Q^{2})=0$ (solid line), the parameterization of
$g_{3}^{NY}(Q^{2})$ by Marshak et al. Marshak  (dashed line) and by Nambu
Nambu:1960xd  (dashed-dotted line).

###  IV.3 Implications of lepton flavor universality in neutrino scattering

The lepton flavor universality in the $e-\mu$ sector was proposed long time
back in 1947 by Pontecorvo Pontecorvo:1947vp  and has been established
phenomenologically by studying the weak processes of $\mu$ decay, $\mu^{-}$
capture and $e^{-}$ capture from nucleons and nuclei Athar:2020kqn . After the
discovery of $\tau$ lepton and the analyses of its leptonic and semileptonic
decays, the principle of lepton flavor universality was extended to the
$e-\mu-\tau$ sector. As mentioned in the introduction in Section I, recently
there has been considerable work in studying the implications of lepton flavor
universality in weak interactions in the $e-\mu-\tau$ sector, which have
focused mainly on the decay processes Zhang:2019kky ; Yang:2018qdx ;
Fleischer:2019wlx ; Golz:2021yqz ; BESIII:2021ynj ; BESIII:2018ccy ;
Mezzadri:2018axu ; Bifani:2018zmi ; Albrecht:2021tul ; LHCb:2021trn ;
deSimone:2020kwi ; Belle:2019rba ; Celani:2021hni . While there has been very
little work on the study of lepton flavor universality in the scattering
processes induced by $\nu_{e}$, $\nu_{\mu}$ and ${\nu}_{\tau}$. Only two
groups have investigated the implications of lepton flavor universality in the
$e-\mu$ sector by studying the quasielastic scattering induced by
$\nu_{e}~{}(\bar{\nu}_{e})$ and $\nu_{\mu}~{}(\bar{\nu}_{\mu})$ from the free
nucleons Day:2012gb  and nuclei Akbar:2015yda . They have reported the results
comparing the cross sections of quasielastic scattering of
$\nu_{e}(\bar{\nu}_{e})$ and $\nu_{\mu}(\bar{\nu}_{\mu})$ with free nucleons
Day:2012gb  and with nuclear targets Akbar:2015yda  induced by the $\Delta
S=0$ weak charged current reactions. In Ref. Day:2012gb , Day and McFarland
have assumed the LFU and studied the differences in the electron and muon
production cross section arising due to the lepton mass effect and other
effects that depend upon the lepton mass like the radiative corrections, the
pseudoscalar form factor as well as the form factors associated with the
second class currents. In Ref. Akbar:2015yda , Akbar et al. have studied these
differences in the neutrino-nucleus cross sections including the nuclear
medium effects, which are important in the intermediate energy region where
most of the present neutrino experiments are being done.

In this section, we study the implications of LFU in the $\Delta S=1$ sector
of antineutrino scattering, and compare the total cross sections for $e$,
$\mu$ and $\tau$ productions in the quasielastic scattering of
$\bar{\nu}_{e}$, $\bar{\nu}_{\mu}$ and $\bar{\nu}_{\tau}$ from the nucleons
induced by the weak charged currents. Specifically, we study the ratios of the
total cross sections $R_{1}$ and $R_{2}$, defined as:

$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\frac{\sigma(\bar{\nu}_{\mu}+p\longrightarrow\mu^{+}+\Lambda)}{\sigma(\bar{\nu}_{e}+p\longrightarrow
e^{+}+\Lambda)},$ (52) $\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\frac{2\sigma(\bar{\nu}_{\tau}+p\longrightarrow\tau^{+}+\Lambda)}{\sigma(\bar{\nu}_{\mu}+p\longrightarrow\mu^{+}+\Lambda)+\sigma(\bar{\nu}_{e}+p\longrightarrow
e^{+}+\Lambda)},$ (53)

as a function of antineutrino energy $E_{\bar{\nu}}$ and investigate the
effect of axial dipole mass $M_{A}$ and the pseudoscalar form factor
$g_{3}^{NY}(Q^{2})$ assuming the SU(3) symmetry. The numerical results are
calculated by taking $M_{A}=1.026$ GeV and the parameterization of
$g_{3}^{NY}(Q^{2})$ by Nambu, unless stated otherwise, and presented in Fig.
15. We have also studied the effect of changing the vector form factors as
well as the effect of SU(3) symmetry breaking. These effects are found to be
quantitatively very small on $R_{1}(E_{\bar{\nu}})$ and
$R_{2}(E_{\bar{\nu}})$, and are not presented here.

We see from Fig. 15 that

  * (i)

the ratios $R_{1}(E_{\bar{\nu}})$ (Eq. (52)) and $R_{2}(E_{\bar{\nu}})$ (Eq.
(53)) have very little dependence on the choice of $M_{A}$, which is almost
negligible in the case of $R_{1}(E_{\bar{\nu}})$. It may be noticed that in
the kinematic region of $\tau$ production i.e., $E_{\bar{\nu}}\simeq 4$ GeV,
the ratio $R_{1}(E_{\bar{\nu}})$ is almost unity. While in the case of
$R_{2}(E_{\bar{\nu}})$, the ratio is highly suppressed due to the threshold
effects and becomes more than 0.5, only for $E_{\bar{\nu}}>10$ GeV. Thus, any
deviation of $R_{1}(E_{\bar{\nu}})$ and $R_{2}(E_{\bar{\nu}})$ from the values
shown in Fig. 15, would be a possible signal for the violation of LFU.

  * (ii)

the ratio $R_{2}(E_{\bar{\nu}})$ has some dependence on the choice of the
pseudoscalar form factor $g_{3}^{NY}(Q^{2})$. If we take $g_{3}^{NY}(Q^{2})=0$
or the parameterization of $g_{3}^{NY}(Q^{2})$ given by Marshak et al. Marshak
, the value of $R_{2}(E_{\bar{\nu}})$ increases as compared to the value
obtained by using the parameterization of $g_{3}^{NY}(Q^{2})$ given by Nambu
Nambu:1960xd . For example, at $E_{\bar{\nu}}=10$ GeV, when we compare the
results obtained with $g_{3}^{NY}(Q^{2})$ using the parameterization of
Marshak et al. Marshak , the value of $R_{2}(E_{\bar{\nu}})$ increases by 5%,
which becomes 18% at $E_{\bar{\nu}}=5$ GeV, from the results using
$g_{3}^{NY}(Q^{2})$ from the results obtained using $g_{3}^{NY}(Q^{2})$
parameterized by Marshak et al. Marshak . Therefore, an experimental
determination of $R_{2}(E_{\bar{\nu}})$ with a precision of 20% or higher
would be able to show any evidence of the violation of LFU in the $e-\mu-\tau$
sector.

##  V Summary and conclusions

In this work, we have presented the results for the $|\Delta S|=1$ hyperon
production in the quasielastic ${\bar{\nu}}_{\tau}$-nucleon scattering and
obtained the differential ($\frac{d\sigma}{dQ^{2}}$) and total ($\sigma$)
scattering cross sections as well as the longitudinal ($P_{L}$) and
perpendicular ($P_{P}$) components of the polarized $\tau^{+}$ lepton produced
in these reactions. We studied theoretical uncertainties arising due to the
use of different vector, axial vector and pseudoscalar form factors as well as
the effect of SU(3) symmetry breaking on these observables.

Our observations are the following:

  * (i)

In the case of $\Lambda$ production, the total cross section as well as the
average polarizations increases with increase in $M_{A}$ at low antineutrino
energies, say at $E_{\bar{\nu}_{\tau}}=5$ GeV. However, with the increase in
$E_{\bar{\nu}_{\tau}}$, $\sigma$ further increases with $M_{A}$ while
$\overline{P}_{L,P}(E_{\bar{\nu}_{\tau}})$ saturates with increase in $M_{A}$.
Moreover, in the case of $\Sigma^{-}$ production, the results for $\sigma$ as
well as $\overline{P}_{L,P}(E_{\bar{\nu}_{\tau}})$ are qualitatively similar
to the results obtained for $\Lambda$ production.

  * (ii)

There is not much effect of SU(3) symmetry breaking on $\sigma$ for both
$\Lambda$ and $\Sigma^{-}$ productions. However, we observe some dependence of
the symmetry breaking effect on $\overline{P}_{L,P}(E_{\bar{\nu}_{\tau}})$,
which are qualitatively different for $\Lambda$ and $\Sigma^{-}$ productions.

  * (iii)

In the case of $\Lambda$ production, at low ${\bar{\nu}}_{\tau}$ energies,
specifically near the threshold energy, the effect of the different
parameterizations of vector form factors on $\frac{d\sigma}{dQ^{2}}$,
$P_{L}(Q^{2})$ and $P_{P}(Q^{2})$ distributions is large, which decreases with
the increase in $E_{{\bar{\nu}}_{\tau}}$. However, in the case of $\Sigma^{-}$
production, $\frac{d\sigma}{dQ^{2}}$ shows appreciable dependence on the
different parameterizations of the vector form factors for all values of
$E_{{\bar{\nu}}_{\tau}}$, while $P_{L}(Q^{2})$ and $P_{P}(Q^{2})$
distributions are quite sensitive to the choice of the vector form factors.

  * (iv)

The effect of variation in the axial dipole mass $M_{A}$ on
$\frac{d\sigma}{dQ^{2}}$, $P_{L}(Q^{2})$ and $P_{P}(Q^{2})$, when $\Lambda$ is
produced in the final state, is significant at low $E_{{\bar{\nu}}_{\tau}}$,
which decreases with the increase in $E_{{\bar{\nu}}_{\tau}}$ for $Q^{2}$ and
$P_{L}(Q^{2})$ distributions but for $P_{P}(Q^{2})$ distribution still remains
significantly different even at high ${\bar{\nu}}_{\tau}$ energies. While
there is a large dependence of $M_{A}$ on $\frac{d\sigma}{dQ^{2}}$,
$P_{L}(Q^{2})$ and $P_{P}(Q^{2})$ at all values of $Q^{2}$ and
$E_{\bar{\nu}_{\tau}}$, when $\Sigma^{-}$ is produced in the final state.

  * (v)

In the case of $\Lambda$ production, the effect of SU(3) symmetry breaking on
$\frac{d\sigma}{dQ^{2}}$ is very small at all values of $E_{\bar{\nu}_{\tau}}$
and $Q^{2}$ while there is a sizeable effect of SU(3) symmetry breaking on
$P_{L}(Q^{2})$ and $P_{P}(Q^{2})$. However, in the case of $\Sigma^{-}$
production, the effect of SU(3) symmetry breaking on $\frac{d\sigma}{dQ^{2}}$
is small at low $E_{\bar{\nu}_{\tau}}$, and increases with the increase in
energy. Moreover, different results for the polarization observables are
obtained when the SU(3) symmetry breaking using the prescriptions by Schlumpf
Schlumpf  and Faessler et al. Faessler:2008ix  are taken into account.

  * (vi)

We have also tested the lepton flavor universality in the antineutrino induced
$\Lambda$ production by calculating the ratios (given in Eqs. (52) and (53))
of the total cross sections in the $e-\mu$ and $e-\mu-\tau$ sectors. We find
that there is no dependence of $M_{A}$ or the SU(3) symmetry breaking on
$R_{1}$. However, $R_{2}$ shows some dependence on the choice of $M_{A}$ as
well as to the different parameterizations of the pseudoscalar form factor.
Thus, the experimental observation of $R_{2}(E_{\bar{\nu}})$ with a precision
of 20% or higher would be able to show any evidence of the violation of LFU in
the $e-\mu-\tau$ sector.

## Acknowledgment

A. F. and M. S. A. are thankful to the Department of Science and Technology
(DST), Government of India for providing financial assistance under Grant No.
SR/MF/PS-01/2016-AMU.

## References

  * (1) K. Kodama et al. [DONUT Collaboration], Phys. Lett. B 504, 218 (2001).
  * (2) K. Kodama et al. [DONuT Collaboration], Phys. Rev. D 78, 052002 (2008).
  * (3) N. Agafonova et al. [OPERA Collaboration], Phys. Rev. D 89, 051102 (2014).
  * (4) N. Agafonova et al. [OPERA Collaboration], Phys. Rev. Lett. 115, 121802 (2015).
  * (5) N. Agafonova et al. [OPERA Collaboration], Phys. Rev. Lett. 120, 211801 (2018).
  * (6) Z. Li et al. [Super-Kamiokande Collaboration], Phys. Rev. D 98, 052006 (2018).
  * (7) K. Abe et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. 110, 181802 (2013).
  * (8) M. G. Aartsen et al. [IceCube Collaboration], Phys. Rev. D 99, 032007 (2019).
  * (9) C. Yoon [SHiP Collaboration], PoS NuFact2019, 103 (2020).
  * (10) M. Anelli et al. [SHiP Collaboration], [arXiv:1504.04956 [physics.ins-det]].
  * (11) S. Alekhin et al., Rept. Prog. Phys. 79, 124201 (2016).
  * (12) S. Aoki et al. [DsTau Collaboration], JHEP 2001, 033 (2020).
  * (13) P. Machado, H. Schulz and J. Turner, [arXiv:2007.00015 [hep-ph]].
  * (14) J. Strait et al. [DUNE Collaboration], [arXiv:1601.05823 [physics.ins-det]].
  * (15) B. Abi et al. [DUNE Collaboration], [arXiv:1807.10340 [physics.ins-det]].
  * (16) H. Abreu et al. [FASER Collaboration], [arXiv:2001.03073 [physics.ins-det]].
  * (17) A. Fatima, M. Sajjad Athar and S. K. Singh, Phys. Rev. D 102, 113009 (2020).
  * (18) K. S. Kuzmin, V. V. Lyubushkin and V. A. Naumov, Mod. Phys. Lett. A 19, 2815 (2004), [Phys. Part. Nucl. 35, S133 (2004)].
  * (19) K. S. Kuzmin, V. V. Lyubushkin and V. A. Naumov, Mod. Phys. Lett. A 19, 2919 (2004).
  * (20) K. Hagiwara, K. Mawatari and H. Yokoya, Nucl. Phys. B 668, 364-384 (2003).
  * (21) K. Hagiwara, K. Mawatari and H. Yokoya, Nucl. Phys. Proc. Suppl. 139, 140 (2005).
  * (22) K. Hagiwara, K. Mawatari and H. Yokoya, Phys. Lett. B 591, 113-118 (2004).
  * (23) J. E. Sobczyk, N. Rocco and J. Nieves, Phys. Rev. C 100, 035501 (2019).
  * (24) M. Valverde, J. E. Amaro, J. Nieves and C. Maieron, Phys. Lett. B 642, 218-226 (2006).
  * (25) A. Fatima, M. Sajjad Athar and S. K. Singh, Eur. Phys. J. A 54, 95 (2018).
  * (26) A. Fatima, M. Sajjad Athar and S. K. Singh, Phys. Rev. D 98, 033005 (2018).
  * (27) A. Fatima, M. Sajjad Athar and S. K. Singh, Front. in Phys. 7, 13 (2019).
  * (28) A. Fatima, M. Sajjad Athar and S. K. Singh, doi:10.1140/epjs/s11734-021-00272-0 [arXiv:2106.14590 [hep-ph]].
  * (29) G. Aad et al. [ATLAS Collaboration], Nature Phys. 17, 813 (2021).
  * (30) S. F. Zhang [BESIII Collaboration], [arXiv:1906.08912 [hep-ex]].
  * (31) Y. H. Yang [BESIII Collaboration], [arXiv:1812.00320 [hep-ex]].
  * (32) R. Fleischer, R. Jaarsma and G. Koole, Eur. Phys. J. C 80, 153 (2020).
  * (33) M. Golz, PoS CHARM2020, 051 (2021).
  * (34) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 127, 121802 (2021).
  * (35) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 122, 011804 (2019).
  * (36) G. Mezzadri [BESIII Collaboration], PoS BEAUTY2018, 025 (2018).
  * (37) S. Bifani, S. Descotes-Genon, A. Romero Vidal and M. H. Schune, J. Phys. G 46, 023001 (2019).
  * (38) J. Albrecht, D. van Dyk and C. Langenbruch, Prog. Part. Nucl. Phys. 120, 103885 (2021).
  * (39) R. Aaij et al. [LHCb Collaboration], [arXiv:2103.11769 [hep-ex]].
  * (40) P. de Simone [LHCb Collaboration], EPJ Web Conf. 234, 01004 (2020).
  * (41) G. Caria et al. [Belle Collaboration], Phys. Rev. Lett. 124, 161803 (2020).
  * (42) S. Celani [LHCb Collaboration], [arXiv:2111.11105 [hep-ex]].
  * (43) A. Crivellin and M. Hoferichter, Science 374, 1051 (2021).
  * (44) D. Bryman, V. Cirigliano, A. Crivellin and G. Inguglia, [arXiv:2111.05338 [hep-ph]].
  * (45) B. Abi et al. [Muon g-2 Collaboration], Phys. Rev. Lett. 126, 141801 (2021).
  * (46) A. M. Coutinho, A. Crivellin and C. A. Manzari, Phys. Rev. Lett. 125, 071802 (2020).
  * (47) A. M. Sirunyan et al. [CMS Collaboration], JHEP 07, 208 (2021).
  * (48) L. Allwicher, G. Isidori and N. Selimovic, [arXiv:2109.03833 [hep-ph]].
  * (49) A. Crivellin, M. Hoferichter, M. Kirk, C. A. Manzari and L. Schnell, JHEP 10, 221 (2021).
  * (50) D. London and J. Matias, doi:10.1146/annurev-nucl-102020-090209 [arXiv:2110.13270 [hep-ph]].
  * (51) L. Zhang, X. W. Kang, X. H. Guo, L. Y. Dai, T. Luo and C. Wang, JHEP 02, 179 (2021).
  * (52) D. Bečirević, F. Jaffredo, A. Peñuelas and O. Sumensari, JHEP 05, 175 (2021).
  * (53) X. Leng, X. L. Mu, Z. T. Zou and Y. Li, Chin. Phys. C 45, 063107 (2021).
  * (54) S. Zhang [BESIII Collaboration], SciPost Phys. Proc. 1, 016 (2019).
  * (55) F. Akbar, M. Rafi Alam, M. Sajjad Athar, S. Chauhan, S. K. Singh and F. Zaidi, Int. J. Mod. Phys. E 24, 1550079 (2015).
  * (56) M. Day and K. S. McFarland, Phys. Rev. D 86, 053003 (2012).
  * (57) F. Schlumpf, Phys. Rev. D 51, 2262 (1995).
  * (58) A. Faessler, T. Gutsche, B. R. Holstein, M. A. Ivanov, J. G. Korner and V. E. Lyubovitskij, Phys. Rev. D 78, 094005 (2008).
  * (59) X. G. He, J. Tandean and G. Valencia, JHEP 07, 022 (2019).
  * (60) R. E. Marshak, Riazuddin, C. P. Ryan, Theory of Weak Interactions in Particle Physics (Wiley-Interscience, 1969).
  * (61) A. Pais, Annals Phys. 63, 361 (1971).
  * (62) C. H. Llewellyn Smith, Phys. Rept. 3, 261 (1972).
  * (63) R. Bradford, A. Bodek, H. Budd and J. Arrington, Nucl. Phys. (Proc. Suppl.) 159 127 (2006).
  * (64) P. E. Bosted, Phys. Rev. C 51, 409 (1995).
  * (65) H. Budd, A. Bodek and J. Arrington, Nucl. Phys. (Proc. Suppl.) 139, 90 (2005).
  * (66) W. M. Alberico et al., Phys. Rev. C 79, 065204 (2009).
  * (67) J. J. Kelly, Phys. Rev. C 70, 068202 (2004).
  * (68) S. Galster et al., Nucl. Phys. B 32, 221 (1971).
  * (69) S. Platchkov et al., Nucl. Phys. A 510, 740 (1990).
  * (70) V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash and C. E. Carlson, Eur. Phys. J. A 51, 79 (2015).
  * (71) N. Cabibbo, E. C. Swallow, R. Winston, Annu. Rev. Nucl. Part. Sci. 53, 39 (2003).
  * (72) V. Bernard, L. Elouadrhiri, U. G. Meissner, J. Phys. G 28, R1 (2002).
  * (73) Y. Nambu, Phys. Rev. Lett. 4, 380 (1960).
  * (74) R. M. Wang, M. Z. Yang, H. B. Li and X. D. Cheng, Phys. Rev. D 100, no.7, 076008 (2019).
  * (75) P. E. Shanahan, A. N. Cooke, R. Horsley, Y. Nakamura, P. E. L. Rakow, G. Schierholz, A. W. Thomas, R. D. Young and J. M. Zanotti, Phys. Rev. D 92, 074029 (2015).
  * (76) G. S. Yang and H. C. Kim, Phys. Rev. C 92, 035206 (2015).
  * (77) D. Becirevic, D. Guadagnoli, G. Isidori, V. Lubicz, G. Martinelli, F. Mescia, M. Papinutto, S. Simula, C. Tarantino and G. Villadoro, Eur. Phys. J. A 24S1, 69 (2005).
  * (78) A. Lacour, B. Kubis and U. G. Meissner, JHEP 10, 083 (2007).
  * (79) T. N. Pham, Phys. Rev. D 87, 016002 (2013).
  * (80) T. N. Pham, Nucl. Part. Phys. Proc. 258, 102 (2015).
  * (81) T. Ledwig, J. Martin Camalich, L. S. Geng and M. J. Vicente Vacas, Phys. Rev. D 90, 054502 (2014).
  * (82) H. M. Chang, M. González-Alonso and J. Martin Camalich, Phys. Rev. Lett. 114, 161802 (2015).
  * (83) M. Ademollo and R. Gatto, Phys. Rev. Lett. 13, 264 (1964).
  * (84) S. M. Bilenky, Basics of Introduction to Feynman Diagrams and Electroweak Interactions Physics (Editions Frontieres, Gif-sur-Yvette Cedex, 1994).
  * (85) B. Pontecorvo, Phys. Rev. 72, 246 (1947).
  * (86) M. Sajjad Athar and S. K. Singh, The Physics of Neutrino Interactions (Cambridge University Press, Cambridge, England, 2020).