12 * Charged current quasi elastic scattering of ν¯µ off C

Deepika Grover1 Kapil Saraswat1 Prashant Shukla2,3 Venktesh Singh1;1) 1 Department of Physics, Institute of Science, Banaras Hindu University, Varanasi 221005, India 2 Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 3 Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India

Abstract: In this work, we study charged current quasi elastic scattering ofν ¯µ off and nucleus using a formalism based on Llewellyn Smith (LS) model. Parameterizations by Galster et al. are used for electric and magnetic Sach’s form factors of . We use Fermi gas model along with Pauli suppression condition to take 12 into account the nuclear effects in anti- - nucleus QES. We calculateν ¯µ−p andν ¯µ− C charged current quasi elastic scattering differential and total cross sections for different values of axial mass MA and compare the results with data from GGM, SKAT, BNL, NOMAD, MINERνA and MiniBooNE experiments. The present theoretical approach

gives an excellent description of differential cross section data. The calculations with axial mass MA = 0.979 and 1.05 GeV are compatible with data from most of the experiments.

Key words: Cross section, Quasi elastic scattering, Axial mass, Form factors PACS: 13.15.+g, 25.70.Bc, 96.40.Tv

1 Introduction erarchy, mixing angles etc. [1–10]. Several experimen- tal efforts such as studies at Gargamelle (GGM) [12, From their first postulation by Wolfgang Pauli in 13], SKAT [14], Brookhaven National Laboratory 1930, to explain the continuous spectra in beta (BNL) [15], MAgnetic Detector decay process, the have been a major field of (NOMAD) [16], Main INjector ExpeRiment for ν -A research. Neutrinos exist in three flavors (, (MINERνA) [17] and Mini Booster Neutrino Experi- and tau neutrinos) along with their anti-particles called ment (MiniBooNE) [18] etc. have been performed to anti-neutrinos. Search for more neutrino flavors called describe the quasi elastic scattering of neutrinos and sterile neutrinos is still underway. The anti-neutrinos off various nuclear targets. GGM stud- of assumes (anti)neutrinos to be mass- ied quasi elastic reactions of neutrinos and anti-neutrinos less, however, several (anti)neutrino oscillation experi- on propane along with freon target, SKAT bombarded ments have confirmed small but non zero (anti)neutrino a wide energy band neutrino/anti-neutrino beam onto

masses [1–10]. Being neutral particles, (anti)neutrinos heavy freon (CF3Br) target, BNL used hydrogen (H2) undergo only , i.e. charged current: via as target, NOMAD executed the studies on carbon, exchange of W +/W − boson and : via MINERνA projected an anti-neutrino beam with aver- exchange of Z boson, with matter through scattering age energy of 3.5 GeV onto a hydrocarbon target and processes such as quasi elastic scattering (QES), reso- MiniBooNE recorded the data on mineral oil target. A nance production (RES) and deep inelastic scatter- global analysis of neutrino and anti-neutrino QES differ- arXiv:1807.08911v1 [hep-ph] 24 Jul 2018 ing (DIS), for a review see Ref. [11]. In charged current ential and total cross sections along with the extraction

(CC) quasi elastic scattering, an (anti)neutrino interacts of axial mass MA is presented in Ref. [19]. with a (proton)neutron producing a corresponding lep- In this work, we study charged current anti-neutrino ton and the (proton)neutron changes to (neutron)proton. - nucleon and anti-neutrino - nucleus (12C) QES. To describe CCQES, we use the Llewellyn Smith (LS) − νl +n → l +p. (1) model [20] and parameterizations by Galster et al. [21] + ν¯l +p → l +n. (2) for electric and magnetic Sach’s form factors of nucleons. For incorporating the nuclear effects, in case ofν ¯µ scat- Precise knowledge of (anti)neutrino CCQES is cru- tering off 12C, we use the Fermi gas model along with cial to high energy physics experiments studying neu- Pauli suppression condition [19, 20, 22]. We calculate trino oscillations and hence extracting neutrino mass hi-

∗ Supported by Department of Science and Technology, New Delhi, India 1) E-mail: [email protected] c 2018 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

1 12 ν¯µ − p andν ¯µ− C CCQES differential and total cross where gA (= −1.267) is the axial vector constant and MA sections for different values of axial mass MA and com- is the axial mass. pare the results with experimental data with the goal of The pseudoscalar form factor FP is defined in terms finding the most appropriate MA value. This work does of axial form factor FA as [24]: not include contribution from 2N2h (two nucleons two 2 M 2 holes) effect in QES. F (Q2) = N F (Q2), (8) P 2 2 A Q +mπ

2 Formalism for quasi elastic ν¯ − N and where mπ is the mass of pion. V V ν¯−A scattering The vector form factors F1 and F2 are defined as [23, 25]: The anti-neutrino - nucleon charged current quasi (" # elastic differential cross section for a free nucleon at rest V 2 1 p 2 n 2 F1 (Q ) = GE (Q )−GE (Q ) is given by [20]: (1+τ) " #) " dσfree M 2 G2 cos2θ + τ Gp (Q2)−Gn (Q2) , (9) = N F c A(Q2) M M dQ2 8πE2 ν¯ (" # # 1 B(Q2)(s−u) C(Q2)(s−u)2 F V (Q2) = Gp (Q2)−Gn (Q2) + + , (3) 2 (1+τ) M M M 2 M 4 N N " #) p 2 n 2 −5 −2 − GE (Q )−GE (Q ) , (10) where MN is the nucleon mass, GF (= 1.16×10 GeV ) is the Fermi coupling constant, cosθc (= 0.97425) is the p Cabibbo angle and Eν¯ is the anti-neutrino energy. In where GE is the electric Sach’s form factor of proton, n p terms of the mandelstam variables s and u, the relation GE is the electric Sach’s form factor of neutron, GM is 2 2 2 n s − u = 4MN Eν¯ − Q − ml , where Q is the square of the magnetic Sach’s form factor of proton and GM is the momentum transfer from anti-neutrino to the outgoing magnetic Sach’s form factor of neutron. Several groups and ml is the mass of the outgoing lepton. such as Galster et al. [21], Budd et al. [26], Bradford et The functions A(Q2),B(Q2) and C(Q2) are defined al. [27], Bosted [28] and Alberico et al. [29] provide pa- as [20]: rameterizations of these form factors by fitting the elec- tron scattering data. For present calculations, we are (" (m2 +Q2) using Galster et al. parameterizations of these form fac- A(Q2) = l (1+τ)F 2 −(1−τ)(F V )2 M 2 A 1 tors. N The electric and magnetic Sach’s form factors of nu- # V 2 V V cleons are defined as [21]: + τ(1−τ)(F2 ) +4τF1 F2 p 2 2 GE (Q ) = GD(Q ), (11) " m2 Gp (Q2) = µ G (Q2), (12) − l (F V +F V )2 M p D 4M 2 1 2 n 2 2 N GM (Q ) = µn GD(Q ). (13) #) We define the electric Sach’s form factor of neutron using + (F +2F )2 −4(1+τ)F 2 , (4) A P P Krutov and Troitsky [30] parameterization as: 2 0.942 τ 2 Q V V n 2 2 B(Q ) = F (F +F ), (5) GE (Q ) = −µn GD(Q ), (14) 2 A 1 2 (1+4.61 τ) MN " # 2 1 2 V 2 V 2 where µp (= 2.793) is the magnetic moment of proton, C(Q ) = FA +(F1 ) +τ(F2 ) , (6) 4 µn (= −1.913) is the magnetic moment of neutron and 2 GD(Q ) is the dipole form factor defined as [23]: Q2 1 where τ = 4M2 . FA is the axial form factor, FP is the 2 N GD(Q ) = , (15) pseudoscalar form factor and F V , F V are the vector form !2 1 2 Q2 1+ 2 factors. Mv The axial form factor FA is defined in the dipole form 2 2 as [23]: where Mv = 0.71 GeV . 2 gA The total cross section of anti-neutrino - nucleon FA(Q ) = , (7) Q2 2 (1+ 2 ) (free) quasi elastic scattering is obtained by integrating MA

2 2 the differential cross section defined by Eq. 3 over Q where Ep is the proton energy defined as: as [31]: q Q2 2 2 Z max dσfree(E ,Q2) Ep = kp +Mp . (23) σfree(E ) = dQ2 ν¯ , (16) ν¯ 2 Q2 dQ min ~ 2 2 The Fermi distribution function f(kp) is defined as: where Qmin and Qmax are defined as:

2 2 ~0 1 Qmin = −ml +2 Eν¯ (El −|k |), ~ f(kp) = k −k , (24) 2E2M −M m2 −E m2 −E 1+exp( p F ) = ν¯ N N l ν¯ l Q . (17) a 2Eν¯ +MN where a = kT (= 0.020 GeV) is the diffuseness parame- Q2 = −m2 +2 E (E +|k~0|), max l ν¯ l ter [32]. The Fermi momentum k for carbon nucleus is 2E2M −M m2 +E m2 +E F = ν¯ N N l ν¯ l Q . (18) 0.221 GeV [33]. 2E +M ν¯ N The Pauli suppression factor S(ν − νmin) is defined ~0 as: Here, El and k are the energy and momentum of the 1 outgoing lepton and EQ is defined as: S(ν −νmin) = , (25) 1+exp(− (ν−νmin) ) p 2 2 2 2 4 a EQ = Eν¯ (s−ml ) −2(s+ml )MN +MN , (19) where ν is the energy transfer in the interaction defined where s = M 2 +2M E . N N ν¯ as:

2 2 2 3 Nuclear modifications ν = (Q +Mn −Mp )/(2Mp). (26)

For studying anti-neutrino - nucleus quasi elastic and νmin is defined as: scattering, nucleus can be treated as a Fermi gas [19, p q 20, 22], where the nucleons move independently within ν = k2 +M 2 − k2 +M 2 +E . (27) the nuclear volume in an average binding potential gen- min F n p p B erated by all nucleons. Pauli suppression condition is applied for the nuclear modifications which implies that Here, Mn is the final state neutron mass and EB is the the cross section for all the interactions leading to a final binding energy. For carbon nucleus, EB = 10 MeV [32]. state nucleon with a momentum smaller than the Fermi The total cross section of anti-neutrino - nucleus momentum k is equal to zero. quasi elastic scattering is obtained by integrating the F 2 The differential cross section per proton for anti- differential cross section as defined by Eq. 20 over Q , 2 2 2 where Q ranges from Qmin to Qmax defined by Eqs. 17 neutrino - nucleus quasi elastic scattering is defined as: eff and 18 calculated with Eν¯ instead of Eν¯. dσnucleus(E ,Q2) 2V Z ∞ ν¯ = 2πk2dk d(cosθ) 2 3 p p dQ Z(2π) 0 4 Results and discussions ~ f(kp)S(ν −νmin) We calculated the charged currentν ¯ − N andν ¯ − A dσfree(Eeff (E ,k~ ),Q2) ν¯ ν¯ p , (20) quasi elastic differential scattering cross sections. Fig. 1 dQ2 shows the present calculations ofν ¯µ −p charged current dσ where the factor 2 accounts for the of the proton, quasi elastic differential scattering cross section dQ2 as a 2 V is the volume of the nucleus, kp is the momentum of function of the square of momentum transfer Q , for dif- dσfree the proton, dQ2 is the differential cross section of the ferent values of axial mass (MA = 0.979, 1.05, 1.12 and anti-neutrino quasi elastic scattering off free proton as 1.23 GeV) and for anti-neutrino energy Eν¯ = 2 GeV. The eff defined by Eq. 3 and Eν¯ is the effective anti-neutrino value of differential cross section increases with increase energy in the presence of Fermi motion of nucleons. in the value of axial mass. eff dσ Eν¯ is defined as: Fig. 2 shows the differential cross section dQ2 for 12 (seff −M 2) ν¯µ − p andν ¯µ− C charged current QES as a function eff p 2 Eν¯ = . (21) of the square of momentum transfer Q , with axial mass 2Mp MA = 1.05 GeV and anti-neutrino energy Eν¯ = 2 GeV. eff Here, Mp is the proton mass and s is defined as: The anti-neutrino - carbon cross section is lower than the ! anti-neutrino - proton cross section for smaller values of 2 seff = M 2 +2E E −k cosθ . (22) Q due to nuclear effects. The cross sections gradually p ν¯ p p drop to zero with increase in the value of Q2.

3 12 We compared the present calculations ofν ¯µ− C charged current quasi elastic differential scattering cross section with experimental data from several collabora- dσ 2.2 tions. Fig. 3 shows the differential cross section E = 2.0 GeV dQ2 ν per proton for anti-neutrino - carbon CCQES as a func- νµ - N CCQES 2 tion of the square of momentum transfer Q2, for differ- MA = 0.979 GeV

) 1.8 ent values of axial mass (MA = 0.979, 1.05, 1.12 and 2 MA = 1.05 GeV 1.6 1.23 GeV). The results obtained are compared with MA = 1.12 GeV /GeV MINERνA data [17] measuring muon anti-neutrino quasi 2 1.4 M = 1.23 GeV A elastic scattering on a hydrocarbon target at < Eν¯ > = cm 3.5 GeV. The calculation with axial mass M = 0.979 -38 1.2 A GeV is compatible with data.

(10 1 2

/dQ 0.8 free

σ 0.6 d 1.8 0.4 = 3.5 GeV 1.6 12 0.2 νµ - C CCQES MA = 0.979 GeV )

0 2 0 0.5 1 1.5 2 2.5 3 1.4 MA = 1.05 GeV

2 2 MA = 1.12 GeV

Q (GeV ) /GeV 2 1.2 M = 1.23 GeV dσ A Fig. 1. Differential cross section dQ2 forν ¯µ − p ν cm MINER A2013 : Hydrocarbon charged current QES as a function of the square of

-38 1 momentum transfer Q2, for different values of ax- (10

ial mass MA and for anti-neutrino energy Eν¯ = 2 2 0.8 GeV. /dQ 0.6 nucleus

σ 0.4 d

0.2 2.2 Eν = 2.0 GeV 0 2 MA = 1.05 GeV 0 0.5 1 1.5 2 2.5 3

νµ - N CCQES 2 2 1.8 Q (GeV )

) ν 12 2 µ - C CCQES dσ 1.6 Fig. 3. Differential cross section dQ2 per proton 12

/GeV forν ¯µ− C charged current QES as a function 2 1.4 of the square of momentum transfer Q2, for dif-

cm 1.2 ferent values of axial mass MA and for average -38 anti-neutrino energy < Eν¯ > = 3.5 GeV compared 1 (10 with MINERνA data [17]. 2 0.8 /dQ σ

d 0.6 dσ Fig. 4 shows the differential cross section dQ2 per pro- 0.4 ton for anti-neutrino - carbon CCQES as a function of 2 0.2 the square of momentum transfer Q , for different val- ues of axial mass (MA = 0.979, 1.05, 1.12 and 1.23 0 −2 −1 GeV) and for average anti-neutrino energy < E > = 2 10 10 1 ν¯ Q2 (GeV2) GeV. The results obtained are compared with data from Gargamelle (GGM) [12] studying quasi elastic reactions Fig. 2. Differential cross section dσ forν ¯ − p dQ2 µ of neutrinos and antineutrinos on propane plus freon tar- 12 andν ¯µ− C charged current QES as a function 2 get. The calculations with axial mass MA = 0.979 and of the square of momentum transfer Q , for axial 1.05 GeV are compatible with data. mass MA = 1.05 GeV and for anti-neutrino energy Eν¯ = 2 GeV.

4 1.8 1.8

= 2.0 GeV = 3.0 GeV 1.6 12 1.6 12 νµ - C CCQES νµ - C CCQES MA = 0.979 GeV MA = 0.979 GeV ) ) 2 2 1.4 MA = 1.05 GeV 1.4 MA = 1.05 GeV

MA = 1.12 GeV MA = 1.12 GeV /GeV /GeV

2 1.2 2 1.2 MA = 1.23 GeV MA = 1.23 GeV

cm GGM1977 : Propane + Freon cm SKAT1990 : CF3Br

-38 1 -38 1 (10 (10

2 0.8 2 0.8 /dQ /dQ 0.6 0.6 nucleus nucleus

σ 0.4 σ 0.4 d d

0.2 0.2

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Q2 (GeV2) Q2 (GeV2) dσ Fig. 5. Differential cross section dQ2 per proton 12 dσ forν ¯µ− C charged current QES as a function Fig. 4. Differential cross section 2 per proton dQ of the square of momentum transfer Q2, for dif- forν ¯ −12C charged current QES as a function µ ferent values of axial mass M and for average of the square of momentum transfer Q2, for dif- A anti-neutrino energy < E > = 3 GeV compared ferent values of axial mass M and for average ν¯ A with SKAT data [14]. anti-neutrino energy < Eν¯ > = 2 GeV compared with GGM data [12].

1.2 dσ = 0.665 GeV Fig. 5 shows the differential cross section dQ2 per pro- ν ν 12 ton for anti-neutrino - carbon CCQES as a function of µ - C CCQES 2 1 MA = 0.979 GeV the square of momentum transfer Q , for different val- ) 2 MA = 1.05 GeV ues of axial mass (MA = 0.979, 1.05, 1.12 and 1.23 MA = 1.12 GeV

GeV) and for average anti-neutrino energy < Eν¯ > = /GeV 2 0.8 MA = 1.23 GeV 3 GeV. The results obtained are compared with SKAT 12 data [14] studying the cross sections of neutrino and anti- cm MiniBooNE2013 : C -38 neutrino quasi elastic interactions using a wide energy 0.6 (10 band (3 - 30 GeV) neutrino/anti-neutrino beam on heavy 2

freon (CF3Br) target. The calculations with axial mass /dQ 0.4 MA = 0.979 and 1.05 GeV are compatible with data.

Fig. 6 shows flux-integrated differential cross section nucleus dσ σ 2 per proton for anti-neutrino - carbon CCQES as a d dQ 0.2 function of the square of momentum transfer Q2 corre- sponding to the MiniBooNE data [18], measuring the muon anti-neutrino CCQES cross section off mineral oil 0 (carbon) target. The calculations are performed for dif- 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 2 ferent values of axial mass (MA = 0.979, 1.05, 1.12 and Q (GeV ) 1.23 GeV). The average anti-neutrino energy < Eν¯ > = Fig. 6. Flux-integrated differential cross section dσ 12 665 MeV. The calculations with axial mass MA = 0.979 dQ2 per proton forν ¯µ− C charged current QES and 1.05 GeV are compatible with data. as a function of the square of momentum trans- fer Q2 corresponding to the MiniBooNE data [18]. The average anti-neutrino energy < Eν¯ > = 665 MeV.

5 We performed the calculations of the total cross sec- tion for charged currentν ¯−N andν ¯−A quasi elastic scat- 1.8 tering and compared the present results with data from M = 1.05 GeV several experiments. Fig. 7 shows the present calcula- 1.6 A tions of the total cross section σ for anti-neutrino - pro- νµ - N CCQES ton CCQES as a function of the anti-neutrino energy Eν¯, 1.4 12 νµ - C CCQES for different values of axial mass (MA = 0.979, 1.05, 1.12 and 1.23 GeV). The value of total cross section increases 1.2 ) with increase in the value of axial mass. We compared 2 1 the obtained results with data from BNL [15] and NO- cm

MAD [16] experiments. The calculation with axial mass -38 0.8

MA = 1.05 GeV is compatible with data. (10 Fig. 8 shows the total cross section σ forν ¯ −p and σ µ 0.6 12 ν¯µ− C charged current QES as a function of the anti- neutrino energy Eν¯, with axial mass MA = 1.05 GeV. 0.4 The nuclear effects reduce anti-neutrino - carbon cross section compared to the anti-neutrino - proton cross sec- 0.2 tion. 0 − 10 1 1 10 102

Eν (GeV)

12 Fig. 8. Total cross section σ forν ¯µ−p andν ¯µ− C charged current quasi elastic scattering as a func- 1.8 tion of the anti-neutrino energy Eν¯, for axial mass

νµ - N CCQES MA = 1.05 GeV. 1.6 MA = 0.979 GeV

MA = 1.05 GeV 1.4 MA = 1.12 GeV

MA = 1.23 GeV 1.2 BNL1980 : H2 )

2 NOMAD2009 : p 1.8 ν 12 1 µ - C CCQES cm M = 0.979 GeV 1.6 A -38 MA = 1.05 GeV 0.8 MA = 1.12 GeV (10 M = 1.23 GeV σ 1.4 A GGM1977 : Propane + Freon 0.6 GGM1979 : Propane + Freon 1.2 SKAT1990 : CF3Br ) 12 0.4 2 NOMAD2009 : C 12 1 MiniBooNE2013 : C cm

0.2 -38 0.8 (10

0 σ −1 2 10 1 10 10 0.6 Eν (GeV) 0.4

0.2 Fig. 7. Total cross section σ forν ¯µ − p CCQES as a function of the anti-neutrino energy E , for ν¯ 0 −1 2 different values of axial mass MA compared with 10 1 10 10

BNL [15] and NOMAD [16] data. Eν (GeV) Fig. 9. Total cross section σ per proton for 12 ν¯µ− C charged current QES as a function of the anti-neutrino energy Eν¯, for different values of axial mass MA compared with GGM(1977) [12], GGM(1979) [13], SKAT [14], NOMAD [16] and MiniBooNE [18] data.

6 Fig. 9 shows the total cross section σ per proton for 5 Conclusion 12 ν¯µ− C charged current QES as a function of the anti- neutrino energy Eν¯, for different values of axial mass We presented a study on charged current anti- (MA = 0.979, 1.05, 1.12 and 1.23 GeV). The results neutrino - nucleon and anti-neutrino - nucleus (car- obtained are compared with data from GGM(1977) [12], bon) quasi elastic scattering using Llewellyn Smith (LS) GGM(1979) [13], SKAT [14], NOMAD [16] and Mini- model. For electric and magnetic Sach’s form factors BooNE [18] experiments. The calculations with axial of nucleons, we used Galster et al. parameterizations.

mass MA = 0.979 and 1.05 GeV are compatible with Fermi gas model along with Pauli suppression condition GGM(1977), GGM(1979) and SKAT data though the has been used to incorporate the nuclear effects in anti- 12 calculations overestimate the data at low anti-neutrino neutrino - nucleus QES. We calculatedν ¯µ−p andν ¯µ− C

. The approach parameterizing axial mass MA as charged current quasi elastic differential and total scat- a function of anti-neutrino energy, presented in Ref. [34], tering cross sections for different values of axial mass MA can be used to get a better agreement with data at low and compared the obtained results with data from GGM, anti-neutrino energies. The calculation with axial mass SKAT, BNL, NOMAD, MINERνA and MiniBooNE ex-

MA = 1.05 GeV is compatible with NOMAD data and periments. The present theoretical approach gives an ex- the calculation with axial mass MA = 1.23 GeV is com- cellent description of differential cross section data. The patible with MiniBooNE data. calculations with axial mass MA = 0.979 and 1.05 GeV are compatible with data from most of the experiments.

References 17 L. Fields et al. [MINERvA Collaboration], Phys. Rev. Lett., 2013, 111: no. 2, 022501 1 M. H. Ahn et al. [K2K Collaboration], Phys. Rev. Lett., 2003, 18 A. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration], Phys. 90: 041801 Rev. D, 2013, 88: no. 3, 032001 2 E. Aliu et al. [K2K Collaboration], Phys. Rev. Lett., 2005, 94: 19 K. S. Kuzmin, V. V. Lyubushkin and V. A. Naumov, Eur. 081802 Phys. J. C, 2008, 54: 517 3 M. H. Ahn et al. [K2K Collaboration], Phys. Rev. D, 2006, 74: 20 C. H. Llewellyn Smith, Phys. Rept., 1972, 3: 261 072003 21 S. Galster, H. Klein, J. Moritz, K. H. Schmidt, D. Wegener and 4 Y. Ashie et al. [Super-Kamiokande Collaboration], Phys. Rev. J. Bleckwenn, Nucl. Phys. B, 1971, 32: 221 D, 2005, 71: 112005 22 R. A. Smith and E. J. Moniz, Nucl. Phys. B, 1972, 43: 605 5 Y. Takeuchi [Super-Kamiokande Collaboration], Nucl. Phys. Erratum: [Nucl. Phys. B, 1975, 101: 547] Proc. Supl., 2012, 79: 229 - 232 23 P. Stoler, Phys. Rept., 1993, 226: 103 6 P. Adamson et al. [NOvA Collaboration], Phys. Rev. Lett., 24 V. Bernard, L. Elouadrhiri and U. G. Meissner, J. Phys. G, 2016, 116: no. 15, 151806 2002, 28: R1 7 P. Adamson et al. [NOvA Collaboration], Phys. Rev. D, 2016, 25 H. S. Budd, A. Bodek and J. Arrington, hep-ex/0308005 93: no. 5, 051104 26 H. S. Budd, A. Bodek and J. Arrington, Nucl. Phys. Proc. 8 P. Adamson et al. [NOvA Collaboration], Phys. Rev. Lett., Suppl., 2005, 139: 90 2017, 118: no. 15, 151802 27 R. Bradford, A. Bodek, H. S. Budd and J. Arrington, Nucl. 9 P. Adamson et al. [NOvA Collaboration], Phys. Rev. Lett., Phys. Proc. Suppl., 2006, 159: 127 2017, 118: no. 23, 231801 28 P. E. Bosted, Phys. Rev. C, 1995, 51: 409 10 S. Ahmed et al. [ICAL Collaboration], Pramana, 2017, 88: no. 29 W. M. Alberico, S. M. Bilenky, C. Giunti and K. M. Graczyk, 5, 79 Phys. Rev. C, 2009, 79: 065204 11 J. A. Formaggio and G. P. Zeller, Rev. Mod. Phys., 2012, 84: 30 A. F. Krutov and V. E. Troitsky, Eur. Phys. J. A, 2003, 16: 1307 285 12 S. Bonetti, G. Carnesecchi, D. Cavalli, P. Negri, A. Pullia, 31 Amin A. Leghrouz, M. A. Abu-Samreh and A. M. Saleh, J. M. Rollier, F. Romano and R. Schira, Nuovo Cim. A, 1977, Al-Aqsa Unv., 2006, 10(S.E.) 38: 260 32 K. Saraswat, P. Shukla, V. Kumar and V. Singh, Indian J. 13 N. Armenise et al., Nucl. Phys. B, 1979, 152: 365 Phys., 2018, 92: no. 2, 249 14 J. Brunner et al. [SKAT Collaboration], Z. Phys. C, 1990, 45: 33 E. J. Moniz, I. Sick, R. R. Whitney, J. R. Ficenec, 551 R. D. Kephart and W. P. Trower, Phys. Rev. Lett., 1971, 26: 15 G. Fanourakis et al., Phys. Rev. D, 1980, 21: 562 445 16 V. Lyubushkin et al. [NOMAD Collaboration], Eur. Phys. J. 34 L. D. Kolupaeva, K. S. Kuzmin, O. N. Petrova and I. M. Shan- C, 2009, 63: 355 drov, Mod. Phys. Lett. A, 2016, 31: no. 12, 1650077

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