Weak quasielastic hyperon production leading to pions in the antineutrino-nucleus reactions

In this review, we have studied the quasielastic production cross sections and polarization components of $\Lambda$, $\Sigma^0$ and $\Sigma^-$ hyperons induced by the weak charged currents in the antineutrino reactions on the nucleon and the nuclear targets like $^{12}$C, $^{16}$O, $^{40}$Ar and $^{208}$Pb. It is shown that the energy and the $Q^2$ dependence of the cross sections and the various polarization components can be effectively used to determine the axial vector transition form factors in the strangeness sector and test the validity of various symmetry properties of the weak hadronic currents like G-invariance, T-invariance and SU(3) symmetry. In particular, the energy and the $Q^2$ dependence of the polarization components of the hyperons is found to be sensitive enough to determine the presence of the second class current with or without T-invariance. These hyperons decay dominantly into pions giving an additional contribution to the weak pion production induced by the antineutrinos. This contribution is shown to be quantitatively significant as compared to the pion production by the $\Delta$ excitation in the nuclear targets in the sub-GeV energy region relevant for the $\bar{\nu}_\mu$ cross section measurements in the oscillation experiments. We have also included a few new results, based on our earlier works, which are in the kinematic region of the present and future (anti)neutrino experiments being done with the accelerator (anti)neutrinos at T2K, MicroBooNE, MiniBooNE, NO$\nu$A, MINER$\nu$A and DUNE, as well as for the atmospheric (anti)neutrino experiments in this energy region.


I. INTRODUCTION
A simultaneous knowledge of the neutrino and antineutrino cross sections in the same energy region for the nuclear targets is highly desirable in order to understand the systematics relevant for the analyses of various neutrino oscillation experiments being done in search of CP violation in the leptonic sector and in the determination of neutrino mass hierarchy [1][2][3][4][5][6][7]. Experimentally there are many results available in the cross section measurements for the various weak processes induced by the neutrinos in nuclei in the sub-GeV and few-GeV energy region [8][9][10][11]. There are very few measurements reported for the processes induced by the antineutrinos in the same energy region specially around Eν µ ≈ 1 GeV [8,12]. Theoretically, however, there exists quite a few calculations for the antineutrino-nucleus cross sections and some of them have been incorporated in most of the neutrino event generators like GENIE [13], NEUT [14], NuWro [15] and GiBUU [16]. In this energy region of antineutrinos, Eν µ ≈ 0.5−1.2 GeV, the most important processes contributing to the nuclear cross sections are the quasielastic (QE) scattering and the inelastic scattering where the excitation of ∆ resonance is the dominant process contributing to the single pion production (CC1π). There is some contribution from the excitation of higher resonances and very little contribution from the deep inelastic scattering (DIS) [17][18][19][20][21].
It is well known that the cross sections for the various weak processes induced by the neutrinos and antineutrinos differ by the sign of the interference terms between the vector and the axial vector currents making the antineutrino cross sections smaller and fall faster with Q 2 as compared to the neutrino cross sections [22][23][24][25][26][27]. There is another difference between the neutrino and antineutrino induced processes on the nucleon and the nuclear targets which has not been adequately emphasized in the context of the discussion of the systematics in the neutrino oscillation experiments. This difference arises due to the phenomenological ∆S = ∆Q rule implicit in the standard model (SM) in the charged current sector which allows the quasielastic production of hyperons on nucleons induced by the antineutrinos, i.e.ν l + N → l + + Y ; N = n or p, Y = Λ, Σ 0 or Σ − , but not with the neutrinos i.e. ν l + N → l − + Y . The hyperon production process is Cabibbo suppressed and its cross section is generally small as compared to the quasielastic or ∆ production in the ∆S = 0 sector. However, in the lower energy region of the antineutrinos i.e. Eν µ << 1 GeV where the production of ∆ resonance is kinematically inhibited due to a higher threshold for the ∆ production as compared to Λ production, the hyperon production cross section may not be too small. These hyperons dominantly decay into π − and π o and give additional contribution to the pion production induced by the antineutrinos from the nucleon and the nuclear targets.
Since π − and π o are the largest misidentified background for theν µ disappearance andν e appearance channels in the present neutrino oscillation experiments with the antineutrino beams, the hyperon production becomes an important process to be considered in the accelerator experiments specially at T2K [8], MicroBooNE [9], MiniBooNE [10,12] and NOνA [28], where the antineutrino energies are in the sub-GeV energy region. Moreover, these experiments are being done using nuclear targets like 12 C, 16 O, 40 Ar, etc., where the pion production cross sections through the ∆ excitations are considerably suppressed due to the nuclear medium effect (NME) and the final state interaction (FSI) effect [17,18]. On the other hand, the pions arising from the hyperons are expected to be less affected by these effects due to the fact that the hyperon decay widths are highly suppressed in the nuclear medium making them live longer and travel through most of the nuclear medium before they decay [29,30]. Therefore, the two effects discussed above i.e. the lower threshold energy of the hyperon production and near absence of the FSI for the pions coming from the hyperon decay compensate for the tan 2 θ c suppression as compared to the pions coming from the ∆ production. This makes these processes important in the context of oscillation experiments with antineutrino beams in the sub-GeV energy region.
Notwithstanding the importance of the hyperon production in the context of present day oscillation experiments with the accelerator antineutrino beams at lower energies, the study of these processes is important in their own right as these processes give us an opportunity to understand the weak interactions at higher energies in the ∆S = 1 sector through the study of the nucleon-hyperon transition form factors at higher energy and Q 2 . The information about these form factors is obtained through the analysis of semileptonic hyperon decays which is limited to very low Q 2 [31][32][33]. It is for this reason that the work in the quasielastic production of hyperons induced by the antineutrino was started more than 50 years back and many theoretical papers have reported results for the cross section and the polarization of the hyperons in the literature [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. Experimentally, however, there are very few attempts made where the quasielastic production of Λ, Σ 0 , Σ − have been studied, like at CERN [54][55][56], BNL [57], FNAL [58,59] and Serpukhov [60]. With the availability of the high intensity antineutrino beams at JPARC [61] and FNAL [62] and the advances made in the detector technology, the feasibility of studying the quasielastic production of hyperons and their polarizations have been explored in many theoretical calculations [63][64][65][66][67][68][69][70][71][72][73][74][75][76]. Experimentally, while the MINERνA [77] collaboration has included the study of quasielastic production of hyperons in its future plans, some other collaborations are also considering the feasibility of making such measurements [78].
In this review, we have attempted to give an overview of the present and the earlier works done in the study of the quasielastic production of hyperons induced by the antineutrinos from the nucleon and the nuclear targets and its implications for the pion production relevant for the analysis of the oscillation experiments being done with the antineutrino beams in the sub-GeV energy region. Specifically, we describe the energy and the Q 2 dependence of the production cross section and polarizations of Λ, Σ 0 and Σ − hyperons in the reactions on the nucleon and the nuclear targets like 12 C, 16 O, 40 Ar and 208 Pb. We also take into account the nuclear medium effects on the production cross section of hyperons in a local density approximation [76,79]. The effect of the final state interaction of the hyperons on the production cross section and its Q 2 dependence arising due to the strong interactions in the presence of the nucleons in the nuclear medium leading to elastic and charge exchange reactions like ΣN → ΛN and ΛN → ΣN is also taken into account in a simple model [52]. The effect of the second class current with or without the presence of T-invariance [74,75] on the total and the differential cross sections, and the Q 2 dependence of the polarization components of the hyperons have also been presented.
These hyperons have considerably long lifetime (∼ 10 −10 sec), which then decay into pions via Y −→ N π, making the pions live long enough to pass through the nucleus and decay outside the nuclear medium. Hence, the pions produced by the hyperons are less affected by the strong interaction of nuclear field, and their FSI have not been taken into account. These pions are in addition to the pions produced from the ∆ excitation by the antineutrinos, The pions come from the ∆ 0 and ∆ − through the following channel, ∆ −→ N π. These pions are produced inside the nucleus which may re-scatter or may produce more pions or may get absorbed while coming out from the final nucleus and, therefore, we have taken into account the final state interaction of pions inside the nucleus.
In Section II, we describe in brief the formalism for calculating the cross sections and the polarization components of the Λ, Σ 0 and Σ − hyperons produced in the quasielastic reactions of the antineutrinos from the nucleons in the The quantities in the bracket represent four momenta of the corresponding particles. N stands for a n or p, Y may be a Λ or Σ 0 or Σ − and the ∆ may be a ∆ 0 or ∆ − depending upon the initial nucleon state.
presence of the second class currents. We also reproduce the essential formalism for the excitation of ∆ in this section and describe the process of pion production from the hyperon (Y) and ∆ decay. We describe in Section III the effect of the nuclear medium on the ∆ and the hyperon productions, and in Section IV final state interactions of the hyperons in the nuclear medium and the final state interactions on the production of pions as a result of the ∆ excitations. In Section V, we present our results and finally in Section VI conclude the findings.
with x 1,2 (Q 2 ) defined as 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 . c) For the axial vector form factor g pn 1 (Q 2 ), a dipole parameterization has been used: 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 β−decay of neutron and the semileptonic decay of hyperons.
d) The induced tensor form factor g pn 2 (Q 2 ) is taken to be of the dipole form, i.e., There is limited experimental information about g pn 2 (Q 2 ) which is obtained from the analysis of the weak processes made for the search of G-noninvariance assuming T-invariance which implies g pn 2 (0) to be real. A purely imaginary value of g pn 2 (0) implies T-violation [86]. In the numerical calculations we have taken real as well as imaginary values, with |g 2 (0)| varying in the range 0 − 3 [74].
e) The pseudoscalar form factor g N Y 3 (Q 2 ) is proportional to the lepton mass and the contribution is small in the case of antineutrino scattering with muon antineutrinos. However, in the numerical calculations, we have taken the following expression given by Nambu [87] using the generalized GT relation.
where m K is the mass of the kaon.

Cross section
The general expression of the differential cross section corresponding to the processes (6), (7) and (8), in the rest frame of the initial nucleon, is written as: where the transition matrix element squared is expressed as: The leptonic (L µν ) and the hadronic (J µν ) tensors are given by and J µ is defined in Eq. (11). Using the above definitions, the Q 2 distribution is written as where the expression of N (Q 2 ) is given in the Appendix.

Polarization of the hyperon
Using the covariant density matrix formalism, the polarization 4-vector (ξ τ ) of the hyperon produced in the reactions given in Eqs. (6), (7) and (8) is written as [88]: One may write the polarization vector ξ in terms of the three orthogonal vectorsê i (i = L, P, T ), i.e.
whereê L ,ê P andê 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. 2, and are written aŝ The longitudinal, perpendicular and transverse components of the polarization vector ξ L,P,T (Q 2 ) using Eqs. (39) and (40) may be written as: In the rest frame of the initial nucleon, the polarization vector ξ is expressed as and is explicitly calculated using Eq. (38). The expressions for the coefficients A(Q 2 ), B(Q 2 ) and C(Q 2 ) are given in the Appendix.
x z 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 [74]: The expressions for P L (Q 2 ), P P (Q 2 ) and P T (Q 2 ) are then obtained using Eqs. (40), (41) and (42) in Eq. (43) and are expressed as 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.
B. ∆ 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 ∆ excitation, presented in Fig. 1(b) and is given by: and the matrix element for the antineutrino charged current process on the free neutron is written as [26]: where the leptonic current l µ is defined in Eq. (10) and the hadronic current J µ is given by In the above expression, ψ α (p ′ ) is the Rarita Schwinger spinor for the ∆ of momentum p ′ and u(p) is the Dirac spinor for the nucleon of momentum p. O αµ is the N − ∆ transition operator which is the sum of the vector (O αµ V ) and the axial vector (O αµ A ) pieces, and the operators O αµ V and O αµ A are given by: and A similar expression for J µ is used for the ∆ 0 excitation from the proton target without a factor of √ 3 in Eq. (49). Here q(= p ′ − p = k − k ′ ) is the momentum transfer, Q 2 (= −q 2 ) is the momentum transfer square and M is the mass of the nucleon. C V i (i = 3 − 6) are the vector and C A i (i = 3 − 6) are the axial vector transition form factors which have been taken from Ref. [89] to be: C V 6 (q 2 ) and C A 6 (q 2 ) are determined using CVC and PCAC hypotheses to be C V 6 (q 2 ) = 0 and C A 6 (q 2 ) = 2. The differential scattering cross section for the reactions given in Eqs. (47) and (48) is given by [66,90,91]: where M ∆ is the mass of ∆ resonance, Γ is the delta decay width, W is the center of mass energy i.e. W = (p + q) 2 and In the above expression L µν is given by Eq.
projection operator defined as P µν = spins ψ µ ψ ν and is given by: In Eq. (55), the delta decay width Γ is taken to be an energy dependent P-wave decay width given by [92]: where f πN ∆ is the πN ∆ coupling constant taken as 2.12 for numerical calculations and | q cm | is defined as .
The step function Θ in Eq. (58) denotes the fact that the width is zero for the invariant masses below the N π threshold, | q cm | is the pion momentum in the rest frame of the resonance.
C. Pion production from the hyperons and ∆ The basic reactions for the charged current antineutrino induced one pion production off the nucleon N, arising from a hyperon in the final state are given by, where the quantities in the square brackets represent the branching ratios of the respective decay modes. The basic reactions for the charged current neutrino and antineutrino induced one pion production off the nucleon N (proton or neutron) through the production of ∆ are: where the quantities in the square brackets represent the respective Clebsch-Gordan coefficients for ∆ → N π channel.

A. Hyperons produced inside the nucleus
When the reactions shown in Eqs. (59), (60), (61) take place on 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 ρ N (r) such that the differential scattering cross section inside the nucleus is expressed in terms of the differential scattering cross section for an antineutrino scattering from a free nucleon (Eq. (37)) as where a factor of 2 is to account for the spin degrees of freedom. In a local density approximation it is assured that the nucleons in a nucleus (or nuclear matter) occupy one nucleon per unit cell in phase space so that the total number of nucleons N is given by All states up to a maximum momentum p F (p < p F ) are filled. The momentum states higher than p > p F are unoccupied such that the occupation number n( p) is defined as: In LFGM, the Fermi momentum is a function of r and is not a constant, protons and neutrons are supposed to have different Fermi sphere such that where, ρ p (r) and ρ n (r) are, respectively, the proton and the neutron densities inside the nucleus and are, in turn, expressed in terms of the nuclear density ρ(r) as In the above expression, ρ(r) is determined in the electron scattering experiments for the different nuclei [93]. The differential scattering cross section in this model using Eqs. (66) and (67) is given by where n N (p, r) is the occupation number of the nucleon. n N (p, r) = 1 for p ≤ p FN and is equal to zero for p > p FN , where p FN is the Fermi momentum of the nucleon. The produced hyperons are further affected by the FSI within the nucleus through the hyperon-nucleon elastic processes like ΛN → ΛN , ΣN → ΣN , etc. and the charge exchange scattering processes like Λ + n → Σ − + p, Because of such types of interaction in the nucleus, the probability of Λ or Σ production changes and has been taken into account by using the prescription given in Ref. [52]. In this prescription, an initial hyperon produced at a position r within the nucleus interacts with a nucleon to produce a new hyperon state within a short distance dl with a probability P = P Y dl, where P Y is the probability per unit length given by where f denotes a possible final hyperon-nucleon [Y f (Σ or Λ)+ N (n or p)] state with energy E in the hyperon-nucleon center of mass system, ρ n (r) (ρ p (r)) is the local density of the neutron (proton) in the nucleus, and σ is the total cross section for a charged current channel like Y (Σ or Λ) + N (n or p) → f [52]. Now a particular channel is selected, which gives rise to a hyperon Y f in the final state with the probability P . For the selected channel, the Pauli blocking effect is taken into account by first randomly selecting a nucleon in the local Fermi sea. Then a random scattering angle is generated in the hyperon-nucleon center of mass system assuming the cross sections to be isotropic. By using this information, hyperon and nucleon momenta are calculated and Lorentz boosted to the lab frame. If the nucleon in the final state has momentum above the Fermi momentum, we have a new hyperon type (Y f ) and/or a new direction and energy of the initial hyperon (Y i ). This process is continued until the hyperon gets out of the nucleus.

B. Delta produced inside the nucleus
When an antineutrino interacts with a nucleon (Eq.47) inside a nuclear target, nuclear medium effects come into play like Fermi motion, Pauli blocking, etc. The produced ∆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 ∆s through particle hole excitations and this leads to the modification in the mass and width of the propagator defined in Eq. (57).
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 A, and the differential scattering cross section for the reactions given in Eqs. (47) and (48)is defined as : In the nuclear medium the properties of ∆ like its mass and decay width Γ to be used in Eq. (58) are modified due to the nuclear medium effect. These are mainly due to the following processes: (i) In the nuclear medium, ∆s decay mainly through the ∆ → N π channel. The final nucleons have to be above the Fermi momentum p F of the nucleon in the nucleus thus inhibiting the decay as compared to the free decay of the ∆ described by Γ in Eq. (58). This leads to a modification in the decay width of delta which has been studied by many authors [92,[94][95][96]. We take the value given by Oset et al. [92] and write the modified delta decay widthΓ asΓ where F (p F , E ∆ , k ∆ ) is the Pauli correction factor given by [92]: In the nuclear medium there are additional decay channels open due to two and three body absorption processes like ∆N → N N and ∆N N → N N N through which ∆ disappears in the nuclear medium without producing a pion, while a two body ∆ absorption process like ∆N → πN N gives rise to some more pions. These nuclear medium effects on the ∆ propagation are included by describing the mass and the decay width in terms of the self energy of ∆ [92]. The real part of the ∆ self energy gives modification in the mass and the imaginary part of the ∆ self energy gives modification in the decay width of ∆ inside the nuclear medium. The expressions for the real and imaginary part of the ∆ self energy are taken from Oset et al. [92]: In the above equation C Q accounts for the ∆N → πN N process, C A2 for the two-body absorption process ∆N → N N and C A3 for the three-body absorption process ∆N N → N N N . The coefficients C Q , C A2 , C A3 and α, β and γ are taken from Ref. [92].
These considerations lead to the following modifications in the widthΓ and mass M ∆ of the ∆ resonance.

IV. FINAL STATE INTERACTION EFFECT
A. Pions produced inside the nucleus

Delta production
When the reactions, given in Eqs. (62)-(65) take place inside the nucleus, the pions may be produced in two ways, through the coherent channel and the incoherent channel. If the target nucleus stays in the ground state and does not loose its identity, giving all the transferred energy in the reaction to the outgoing pion, then the pion production process is called coherent pion production otherwise if the nucleus can be excited and/or broken up then it leads to the incoherent production of pions. The contribution of coherent pion production has been found to be less than 2 − 3% at the antineutrino energies of the present interest [18,19], and is not discussed here. We have not considered the contributions from the nonresonant background terms and higher resonances like P 11 (1440), S 11 (1535), etc.
The transition amplitude for an incoherent pion production process is given in Eq. 65 is written as [26]: where the symbols have the same meaning as in section-II B. Starting with the general expression for the differential scattering cross section in the lab frame and using the local density approximation, following the same procedure as mentioned in section-III A, we may write which gives where q 0 is the energy transferred to the target particle. Using Eq. (77) may also be written as where ρ N (r) is the nucleon density defined in terms of nuclear density ρ(r). In a nucleus, the contributions to π − and π o productions come from the neutron and proton targets. These are taken into account using the Clebsch-Gordan coefficients written in Eqs. (62)- (65). The total production cross section for π − and π o from a nucleus can be written by replacing ρ N as ρ N (r) = ρ n (r) + 1 9 ρ p (r) for π − production, ρ N (r) = 2 9 [ρ n (r) + ρ p (r)] for π o production.
The pions produced in these 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. [97] for the final state interaction of pions which is calculated in an eikonal approximation using probabilities per unit length as the basic input. In this approximation, a pion of given momentum and charge is moved along the z-direction with a random impact parameter b, with |b| < R, where R is the nuclear radius which is taken to be a point where nuclear density ρ(R) falls to 10 −3 ρ 0 , where ρ 0 is the central density. To start with, the pion is placed at a point (b, z in ), where z in = − R 2 − |b| 2 and then it is moved in small steps δl along the z-direction until it comes out of the nucleus or interact. If P (p π , r, λ) is the probability per unit length at the point r of a pion of momentum p π and charge λ, then P δl << 1. A random number x is generated such that x ∈ [0, 1] and if x > P δl, then it is assumed that pion has not interacted while traveling a distance δl, however, if x < P δl then the pion has interacted and depending upon the weight factor of each channel given by its cross section it is decided that whether the interaction was quasielastic, charge exchange reaction, pion production or pion absorption [97]. For example, for the quasielastic scattering where N is a nucleon, ρ N is its density and σ is the elementary cross section for the reaction π λ + N → π λ ′ + N ′ obtained from the phase shift analysis.
For a pion to be absorbed, P is expressed in terms of the imaginary part of the pion self energy Π i.e. P abs = − ImΠ abs (pπ) pπ , where the self energy Π is related to the pion optical potential V [97].

Hyperon production
The pions are produced as a result of hyperon decays as shown in Eqs. (59), (60) and (61). However, when the hyperons are produced in a nuclear medium, some of them disappear through the hyperon-nucleon interaction processes like Y N → N N , though it is suppressed due to nuclear effects [29,30]. The pionic 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 [29,30]. Therefore, the hyperons which survive the Y N → N N decay 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 [52]. A quantitative analysis of the hyperon disappearance through the Y N → N N interaction and the pions having FSI effect, will require a dynamic nuclear model to estimate the nonmesonic and mesonic decay 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.

V. RESULTS AND DISCUSSION
We present the numerical results for the total cross section(σ) and the differential scattering cross section( dσ dQ 2 ) for the charged currentν µ induced processes. These results are presented for the free nucleons as well as for some nuclear targets like 12 C, 16 O, 40 Ar and 208 Pb with and without the nuclear medium (NME) and final state interaction (FSI) effects. The results have also been presented for the longitudinal (P L (Q 2 )), perpendicular (P P (Q 2 )) and transverse (P T (Q 2 )) components of the polarization vector of the hyperon in the presence of the second class currents with and without T-invariance by taking the numerical values of g 2 (0) to be real and imaginary, respectively.
The following points describe the inputs used for the numerical calculations which have been done to obtain these results: 1. For the hyperon production cross section off the free nucleon target, we have integrated over Q 2 in Eq. (37) and obtained the results for the total scattering cross section. For the ∆ production cross section off the free nucleon target in the charged current neutrino and antineutrino induced reactions, we have used Eq. (55) and integrated over the final lepton kinematical variables.
2. In the presence of nuclear medium effects the expression of the cross sections given in Eqs. (69) and (70), respectively for the Y production and the ∆ production, have been used. In the case of hyperon production FSI arising due to the quasielastic and charge exchange hyperon nucleon scattering has been taken into account as described in section-III A.
3. For the pion production cross section from the hyperons, we have used the same expression (Eq. (69)) with the hyperon-nucleon interaction. For the pions arising from the ∆ decay with NME+FSI, we have used Eq.(78) with the pion FSI effect as described in section-IV A 1. Therefore, FSI effect in the case of pion production from the hyperons is different from the FSI effect for the pion production from the ∆ i.e. there is no pion absorption in the case of hyperons giving rise to pions, whereas there is pion absorption inside the nucleus when ∆s give rise to pions.
4. For the proton density, we have used ρ p (r) = Z A ρ(r) and for the neutron density ρ n (r) = A−Z A ρ(r), where ρ(r) is nuclear density taken as 3-parameter Fermi density for 12 C, 16  5. The results for the longitudinal (P L (Q 2 )), perpendicular (P P (Q 2 )) and transverse (P T (Q 2 )) components of the polarization of the Λ hyperon have been obtained using Eqs. (44), (45) and (46) respectively in the presence of second class currents with and without T-invariance. For the pseudoscalar form factor g N Y 3 (Q 2 ), Nambu's parameterization given in Eq. (32) has been used.
A. Hyperon and delta productions from free nucleons

Hyperon production
In Fig. 3, we have presented the results for the hyperon production cross sections from the free nucleons presented in Eqs. (6)-(8) as a function of antineutrino energies. These results are presented for the Λ and Σ − cross sections at the two values of M A viz. M A = 1.026 GeV and 1.2 GeV. We find that in this region there is very little dependence of M A on the cross section in the case of Σ − production, while in the case of Λ production, the cross section increases with energy and the increase is about 5% at Eν µ = 1 GeV. In the case of free nucleon, the cross sections forν µ +n → µ + +Σ − are related toν µ + p → µ + + Σ 0 by a simple relation and is given by σ(ν µ p → µ + Σ 0 ) = 1 2 σ(ν µ n → µ + Σ − ), while no Σ + is produced off the free nucleon target due to ∆S = ∆Q rule.
In Fig. 4, the results are presented for the differential cross section (dσ/dQ 2 ) as a function of Q 2 for the Λ and Σ − produced in the final state at the different antineutrino energies viz. Eν µ = 0.5 GeV, 0.75 GeV and 1 GeV at the two values of M A viz. M A = 1.026 GeV and 1.2 GeV. One may notice that the Q 2 -distribution is not much sensitive to the choice of M A .
Experimentally, one may get information about the polarization of hyperons through the structure of the angular distribution of the pions, which are produced by the hyperon decay via. Y → N π. The observation of the components of the polarization provide an alternative method to determine the axial dipole mass, M A , nature of the second class current (whether with or without TRI) and the pseudoscalar form factor independent of the total and the differential scattering cross sections. Moreover, the experimental observation of the transverse component of polarization can be used to study the physics of T-violation. In Fig. 5, we have made an attempt to explore the possibility of determining the pseudoscalar form factor g N Y 3 (Q 2 ) in |∆S| = 1 sector and studied the sensitivity of the Q 2 -dependence on the polarization components P L (Q 2 ), P P (Q 2 ) and P T (Q 2 ) using the expression of Nambu [87] in Eq. (32) for the process ν µ p → µ + Λ at Eν µ =0.4 and 0.7 GeV. We see that at Eν µ = 0.4 GeV, P L (Q 2 ), P P (Q 2 ) and P T (Q 2 ) are sensitive to g N Y 3 (Q 2 ), but with the increase in energy the difference in the results obtained with and without g N Y 3 (Q 2 ) are almost the same. It seems, therefore, possible in principle, to determine the pseudoscalar form factor in the Λ polarization measurements at lower antineutrino energies. The total cross section σ and the differential cross section dσ/dQ 2 are not found to be very sensitive to the values of g N Y 3 (Q 2 ) and are not shown here [76]. For the reactionν µ + p → µ + + Λ, we have also studied the dependence of the polarization components on the second class currents with T-invariance and showed the results for P L (Q 2 ) and P P (Q 2 ) as a function of Q 2 in Fig. 6. These results are presented for the polarization components using the second class current form factor in the presence of T invariance i.e. using the real values of g np 2 (0) = g R 2 (0) = 0, ±1 and ±3 and M 2 = M A in Eq. (31) at the different values of Eν µ = 0.4 and 0.7 GeV. We find that P L (Q 2 ) shows large variations as we change |g R 2 (0)| form 0 to 3 at high antineutrino energies, Eν µ (say 0.7 GeV) in comparison to the lower energies (say 0.4 GeV). For example, in the peak region of Q 2 , the difference is 80% at E νµ = 0.7 GeV and it is 20% at E νµ = 0.4 GeV as g R 2 (0) is changed from 0 to 3. In the case of P P (Q 2 ) also, the Q 2 dependence is quite strong and similar to P L (Q 2 ).
In Fig. 7, the results are presented for P L (Q 2 ), P P (Q 2 ) and P T (Q 2 ) as a function of Q 2 in the presence of the second class current without T-invariance, using the imaginary values of the induced tensor form factor i.e. g np 2 (0) = i g I 2 (0) =, where g I 2 (0) = 0, 1 and 3, at the different values of Eν µ = 0.4 and 0.7 GeV. We see that while P L (Q 2 ) is less sensitive to g I 2 (0) at Eν µ in the range 0.4 − 0.7 GeV. P P (Q 2 ) is almost insensitive to g I 2 (0) at the lower Eν µ , say at Eν µ = 0.4 GeV. However, at higher antineutrino energies, say at Eν µ = 0.7 GeV, P P (Q 2 ) is sensitive to g I 2 (0). Moreover, P T (Q 2 ) is sensitive to g I 2 (0) at all antineutrino energies. P T (Q 2 ) shows 8% and 25% variations at Q 2 = 0.08, and 0.25 GeV 2 at Eν µ = 0.4 and 0.7 GeV, respectively, when g I 2 (0) is varied from 0 to 3.

∆ production
The results for the ∆ production cross sections are presented in Fig. 8 for ν µ andν µ induced processes off the free nucleon target. For ν µ induced reaction, the leptonic current in Eq. (10) will read as l µ =ū(k ′ )γ µ (1 − γ 5 )u(k). In the case of ∆ production, the cross sections are inhibited by the threshold effect at lower energies, and there is almost no cross section until Eν µ = 0.4 GeV. In Fig. 9, the results for dσ/dQ 2 are presented for the ∆s produced in the final state at the different (anti)neutrino energies viz. E νµ,ν µ = 0.5, 0.75 and 1 GeV.

B. Hyperon and delta production from nuclei
In Fig. 10, we have presented the results for σ vs Eν µ , for the ∆ 0 produced off the proton bound in various nuclear targets like 12 C, 16 O, 40 Ar and 208 Pb with and without the NMEs. It may be noticed that the NMEs due to the modification of the ∆ properties in nuclei reduce the cross section. In the case of lighter nuclei like 12 C and 16 O, this reduction is about ∼ 35% at Eν µ =1 GeV. The reduction in the cross section increases with the increase in the nuclear mass number and decreases with the increase in energy. For example, it becomes ∼ 40% and 50% in 40 Ar and 208 Pb, respectively at Eν µ = 1 GeV. We find that the NMEs due to Pauli blocking and Fermi motion effects, in the case of hyperons in the final state, are negligibly small and therefore we have not discussed these effects and the results of the cross sections are almost the same as for the free hyperon case (Fig. 3). Moreover, when the hyperon-nucleon interaction i.e. the FSI effect in the hyperon production, is taken into account the overall change in the hyperon production cross section is very small. These results are used to obtain the ratio of total hyperon to ∆ production cross sections i.e. σY σ∆ which have been shown in the inset of these figures. It may be noticed that due to the threshold effect initially the hyperon production cross section dominates and with the increase in energy the ratio reduces. Due to the substantial reduction in the cross section for the ∆ production, the ratio increases when NME is taken into account in comparison to the free case. Moreover, this ratio is larger in heavy nuclei like 208 P b as NME increases with the nucleon number.

C. Pion production
In this section the results are presented for the π − and π o productions respectively in the nuclei like 12 C, 16 O, 40 Ar and 208 Pb. We give a preview of our main results for π − and π o productions before they are presented in detail in Figs. 11-14 and Figs. 15-18 for each case. These results are shown for the cross sections obtained without and with NME+FSI effect for the pion production arising due to the Λ production, total hyperon(Y) production and the ∆ production. In the case of hyperon production, NMEs in the production process as well as the FSI due to hyperonnucleon interactions have been taken into account. Moreover, we do not include the FSI of pions within the nuclear medium which are produced as a result of hyperon decays. This is because the decay width of pionic decay modes of hyperons is highly suppressed in the nuclear medium, making them live long enough to pass through the nucleus and decay outside the nuclear medium and thus less affected by the strong interaction of nuclear field. This is not the case  with the pion produced through strong decays of ∆, which are further suppressed by the strong absorption of pions in the nuclear medium. Therefore, in the low energy region the Cabibbo suppression in the case of pion production through hyperons get compensated by the threshold suppression as well as the strong pion absorption effects in the case of the pions produced through the delta excitation. However, FSI due to Σ − N and Λ − N interactions in various channels tend to increase the Λ production cross section and decrease the Σ − production cross section. The quantitative increase (decrease) in Λ(Σ) yield due to FSI increases with the increase in the nucleon number. The Σ − and Σ 0 productions are separately affected and the relation σ ν µ + p → µ + + Σ 0 = 1 2 σ (ν µ + n → µ + + Σ − ) which holds for the free case, is modified due to Y − N interaction in the nucleus. We must point out that although Σ + is not produced from a free nucleon but can be produced through the final state interactions like Λp → Σ + n and Σ 0 p → Σ + n, albeit the contributions would be small.
The results for ∆ (without NME+FSI) are suppressed by a factor of 8 208

Pb
The results for ∆ (with NME+FSI) are suppressed by a factor of 2 FIG. 14: Results for π − production in 208 Pb. Lines and points have the same meaning as in Fig.11. Notice that the results of ∆ without NME+FSI are suppressed by a factor of 8 and the results with NME+FSI are suppressed by a factor of 2 to bring them on the same scale.
Using the results of σ, we have obtained the results for the ratio of hyperon to delta production cross sections, with and without NME+FSI, for π − as well as π o productions for all the nuclear targets considered here by defining and This ratio directly tells us the enhancement of the ratio R A due to NME+FSI with the increase in the mass number of the nuclear targets as the pions getting produced through the ∆-resonant channel undergo a suppression due to NME+FSI effect, while the pions getting produced from the hyperons (all the interactions taken together i.e. Λ as well as Σ contributions) have comparatively small NME+FSI effect. In Fig. 11 and Fig. 12, we have presented the results for the total scattering cross section σ vs Eν µ , forν µ scattering off the nucleon in 12 C and 16 O nuclear targets giving rise to π − . The results are presented for the pion production from ∆, Λ and Y with and without NME and FSI. In the case of hyperon production for 12 C, the effect of FSI due to Y − N interaction leads to increase in the cross section of Λ production from the free case, which is about 23 − 24% for Eν µ = 0.6 − 1 GeV, while the change in the total hyperon production cross section results in a decrease in the cross section due to the FSI effect which is about 3 − 5% at these energies. We find that in the case of pions produced through ∆ excitations, NME+FSI lead to a reduction of around 50% in the π − production for the antineutrino The results for ∆ (without NME+FSI) are suppressed by a factor of 3 208 Pb FIG. 18: Results for π o production in 208 Pb. Lines and points have the same meaning as in Fig.15. Notice that the results of ∆ with and without NME+FSI are suppressed by a factor of 3. energies 0.6 < Eν µ < 1GeV. This results in the change in the ratio of R N (Eq. (80)) from 0.28 and 0.14 respectively at Eν µ =0.6 and 1GeV to R A (Eq. (81)) → 0.58 and 0.25 at these energies. In the case of 16 O nuclear target the observations are similar to what has been discussed above in the case of 12 C nuclear target. In Fig. 13, we have presented the results for σ vs Eν µ , forν µ scattering off 40 Ar nuclear target. In the case of Λ production, the effect of FSI leads to increase in the cross section which is about 34 − 38% for Eν µ = 0.6 − 1 GeV, however, the overall change in the π − production from the hyperons results in a net reduction in the cross section from the free case, which is about 6 − 8% at these energies. In the case of pions produced through ∆ excitations, NME+FSI leads to a reduction of around 55 − 60% in the π − production for the antineutrino energies 0.6 ≤ Eν µ ≤ 1 GeV, and the reduction is less at higher energies. This results in the change in the ratio of R N from 0.25 and 0.13 respectively at Eν µ = 0.6 and 1 GeV to R A , 0.6 and 0.26 at the corresponding energies.  19: Results for the νµ induced π + and π o production cross sections andνµ induced π − and π o production cross sections. For νµ scattering the contribution to the pions is coming from the ∆ only, while forνµ scattering the contribution to the pions is coming from the ∆ as well as the hyperons. The results are presented for 12 C, 16 O, 40 Ar and 208 Pb with NME+FSI. Notice that π + production cross section has been reduced by half to bring it on the same scale.
In the case of heavy nuclear target like 208 Pb, the change in the cross section due to NME+FSI is quite large and the results for σ vs Eν µ , forν µ scattering off the nucleon in 208 Pb nuclear target are shown in Fig.14. For example, the reduction in the cross section due to NME+FSI when a ∆ is produced as the resonant state, is about 75% at Eν µ = 0.6 GeV and 70% at Eν µ = 1 GeV from the cross sections calculated without the medium effect. The enhancement in the Λ production cross section is about 55 − 60% at these energies. While the overall change in the π − production from the hyperons results in a net reduction which is about 8 − 12%. This results in the change in the ratio of R N from 0.23 and 0.12 respectively at Eν µ = 0.6 and 1 GeV to R A → 0.86 and 0.35.
In Figs. 15, 16, 17 and 18, we have presented the results for the total scattering cross section σ vs Eν µ , forν µ scattering off nucleon in 12 C, 16 O, 40 Ar and 208 Pb nuclear targets giving rise to π o . These results are presented for the pion production from ∆, Λ and Y with and without NME+FSI. In the case of π o arising due to hyperon decay, the contribution comes from the Λ and Σ 0 decay, while there is no contribution from Σ − . Due to the FSI effect there is substantial increase in the Λ production cross section and reduction in the Σ 0 production cross section from the free case, which leads to an overall increase in the π o production. Therefore, unlike the π − production where there is overall reduction, in the case of π o production there is an increase in the cross section which is about 13 − 14% in 12 C and 16   specially in the case of heavier nuclear targets.
In Fig. 19, we have presented the results for the ν µ induced π + and π o productions andν µ induced π − and π o productions. For ν µ induced reactions, the leptonic current in Eq. (10) will read as l µ =ū(k ′ )γ µ (1 − γ 5 )u(k) and the expression of ρ N (r) in Eq. (79) will become ρ N (r) = ρ p (r) + 1 9 ρ n (r). These results are shown for 12 C, 16 O, 40 Ar and 208 Pb with NME+FSI. For ν µ scattering, the contribution to the pions is coming from ∆ only, while for ν µ scattering the contribution to the pions is coming from the ∆ as well the hyperons. Though in the case of the pions produced through the hyperon production, there is an overall suppression by a factor of sin 2 θ c but these are kinematically favored as the Λ production starts around Eν µ = 250 MeV, while Σ − and Σ 0 production start around Eν µ = 325 MeV, and there is overall no NME effect on the total hyperon production and no FSI effect on the outgoing pions, whereas the reduction is quite significant for the pions arising from the ∆s.
In Figs. 20, 21, 22 and 23, we have presented the results for the Q 2 distribution i.e. dσ dQ 2 vs Q 2 in 12 C, 16 O, 40 Ar and 208 Pb nuclear targets with NME+FSI at the different incident antineutrino energies ofν µ viz. Eν µ = 0.5 GeV, 0.75GeV and 1GeV. These results are presented for the π − contribution from the ∆s and the hyperons. It may be observed that at low Eν µ , π − has significant contribution from the hyperons, like at Eν µ = 0.5 GeV, in the peak region of Q 2 , hyperons contribute ∼40% in 12 C, 16 O and 40 Ar and 50% in 208 Pb of the total π − production, while with the increase in energy the contribution from the hyperons decreases, for example, at Eν µ = 1 GeV hyperons contribute 16% in 12 C, 16 O and 40 Ar and 24% in 208 Pb. The peak region of Q 2 for the hyperons shifts towards the lower Q 2 than the ∆s. In the case of π o (not shown here), the results are similar except that the contributions from the hyperons dominate at lower energies in all the nuclear targets in comparison to the ∆ contributions and the dominance increases with the increase in the nuclear mass number.

VI. SUMMARY AND CONCLUSIONS
In this review, we have presented the results for the quasielastic production cross sections (σ), Q 2 dependence of the cross sections (dσ/dQ 2 ) and the polarization components (P L (Q 2 ), P P (Q 2 ) and P T (Q 2 )) of the hyperons induced by the weak charged currents in the antineutrino reactions on the nucleon. The calculations have also been performed for σ and dσ/dQ 2 in nuclei like 12 C, 16 O, 40 Ar and 208 Pb, which are being used in the various oscillation experiments with the accelerator and the atmospheric antineutrino experiments in the sub-GeV energy region. These results may be helpful in determining the axial vector transition form factor in the strangeness sector specially for the pseudoscalar form factor and the form factor corresponding to the second class currents with and without T-invariance and test the validity of various symmetry properties of the weak hadronic currents. We have also studied the contribution of hyperons produced in these reactions towards theν induced pion production as they decay dominantly into pions.