Charged current quasi elastic scattering of muon anti-neutrino off ^{12}C

In this work, we study charged current quasi elastic scattering of muon anti-neutrino off nucleon 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 nucleons. We use Fermi gas model along with Pauli suppression condition to take into account the nuclear effects in anti-neutrino - nucleus QES. We calculate muon anti-neutrino-p and muon anti-neutrino-^{12}C charged current quasi elastic scattering differential and total cross sections for different values of axial mass M_{A} and compare the results with data from GGM, SKAT, BNL, NOMAD, MINERvA and MiniBooNE experiments. The present theoretical approach gives an excellent description of differential cross section data. The calculations with axial mass M_{A} = 0.979 and 1.05 GeV are compatible with data from most of the experiments.


Introduction
From their first postulation by Wolfgang Pauli in 1930, to explain the continuous energy spectra in beta decay process, the neutrinos have been a major field of research. Neutrinos exist in three flavors (electron, muon and tau neutrinos) along with their anti-particles called anti-neutrinos. Search for more neutrino flavors called sterile neutrinos is still underway. The standard model of particle physics assumes (anti)neutrinos to be massless, however, several (anti)neutrino oscillation experiments have confirmed small but non zero (anti)neutrino masses [1][2][3][4][5][6][7][8][9][10]. Being neutral particles, (anti)neutrinos undergo only weak interaction, i.e. charged current: via exchange of W + /W − boson and neutral current: via exchange of Z boson, with matter through scattering processes such as quasi elastic scattering (QES), resonance pion production (RES) and deep inelastic scattering (DIS), for a review see Ref. [11]. In charged current (CC) quasi elastic scattering, an (anti)neutrino interacts with a (proton)neutron producing a corresponding lepton and the (proton)neutron changes to (neutron)proton. ν l + n → l − + p.
Precise knowledge of (anti)neutrino CCQES is crucial to high energy physics experiments studying neutrino oscillations and hence extracting neutrino mass hi-erarchy, mixing angles etc. [1][2][3][4][5][6][7][8][9][10]. Several experimental efforts such as studies at Gargamelle (GGM) [12,13], SKAT [14], Brookhaven National Laboratory (BNL) [15], Neutrino Oscillation MAgnetic Detector (NOMAD) [16], Main INjector ExpeRiment for ν -A (MINERνA) [17] and Mini Booster Neutrino Experiment (MiniBooNE) [18] etc. have been performed to describe the quasi elastic scattering of neutrinos and anti-neutrinos off various nuclear targets. GGM studied quasi elastic reactions of neutrinos and anti-neutrinos on propane along with freon target, SKAT bombarded a wide energy band neutrino/anti-neutrino beam onto heavy freon (CF 3 Br) target, BNL used hydrogen (H 2 ) as target, NOMAD executed the studies on carbon, MINERνA projected an anti-neutrino beam with average energy of 3.5 GeV onto a hydrocarbon target and MiniBooNE recorded the data on mineral oil target. A global analysis of neutrino and anti-neutrino QES differential and total cross sections along with the extraction of axial mass M A is presented in Ref. [19].
In this work, we study charged current anti-neutrino -nucleon and anti-neutrino -nucleus ( 12 C) QES. To describe CCQES, we use the Llewellyn Smith (LS) model [20] and parameterizations by Galster et al. [21] for electric and magnetic Sach's form factors of nucleons. For incorporating the nuclear effects, in case ofν µ scattering off 12 C, we use the Fermi gas model along with Pauli suppression condition [19,20,22]. We calculate ν µ − p andν µ − 12 C CCQES differential and total cross sections for different values of axial mass M A and compare the results with experimental data with the goal of finding the most appropriate M A value. This work does not include contribution from 2N2h (two nucleons two holes) effect in QES.
2 Formalism for quasi elasticν − N and ν − A scattering The anti-neutrino -nucleon charged current quasi elastic differential cross section for a free nucleon at rest is given by [20]: where M N is the nucleon mass, is the Fermi coupling constant, cosθ c (= 0.97425) is the Cabibbo angle and Eν is the anti-neutrino energy. In terms of the mandelstam variables s and u, the relation where Q 2 is the square of momentum transfer from anti-neutrino to the outgoing lepton and m l is the mass of the outgoing lepton.
The functions A(Q 2 ), B(Q 2 ) and C(Q 2 ) are defined as [20]: where . F A is the axial form factor, F P is the pseudoscalar form factor and F V 1 , F V 2 are the vector form factors.
The axial form factor F A is defined in the dipole form as [23]: where g A (= −1.267) is the axial vector constant and M A is the axial mass. The pseudoscalar form factor F P is defined in terms of axial form factor F A as [24]: where m π is the mass of pion. The vector form factors F V 1 and F V 2 are defined as [23,25]: where G p E is the electric Sach's form factor of proton, G n E is the electric Sach's form factor of neutron, G p M is the magnetic Sach's form factor of proton and G n M is the magnetic Sach's form factor of neutron. Several groups such as Galster et al. [21], Budd et al. [26], Bradford et al. [27], Bosted [28] and Alberico et al. [29] provide parameterizations of these form factors by fitting the electron scattering data. For present calculations, we are using Galster et al. parameterizations of these form factors.
The electric and magnetic Sach's form factors of nucleons are defined as [21]: We define the electric Sach's form factor of neutron using Krutov and Troitsky [30] parameterization as: where µ p (= 2.793) is the magnetic moment of proton, µ n (= −1.913) is the magnetic moment of neutron and G D (Q 2 ) is the dipole form factor defined as [23]: where M 2 v = 0.71 GeV 2 . The total cross section of anti-neutrino -nucleon (free) quasi elastic scattering is obtained by integrating the differential cross section defined by Eq. 3 over Q 2 as [31]: where Q 2 min and Q 2 max are defined as: Here, E l and k are the energy and momentum of the outgoing lepton and E Q is defined as: where s = M 2 N + 2M N Eν.

Nuclear modifications
For studying anti-neutrino -nucleus quasi elastic scattering, nucleus can be treated as a Fermi gas [19,20,22], where the nucleons move independently within the nuclear volume in an average binding potential generated by all nucleons. Pauli suppression condition is applied for the nuclear modifications which implies that the cross section for all the interactions leading to a final state nucleon with a momentum smaller than the Fermi momentum k F is equal to zero.
The differential cross section per proton for antineutrino -nucleus quasi elastic scattering is defined as: where the factor 2 accounts for the spin of the proton, V is the volume of the nucleus, k p is the momentum of the proton, dσ f ree dQ 2 is the differential cross section of the anti-neutrino quasi elastic scattering off free proton as defined by Eq. 3 and E ef f ν is the effective anti-neutrino energy in the presence of Fermi motion of nucleons.
E ef f ν is defined as: Here, M p is the proton mass and s ef f is defined as: where E p is the proton energy defined as: The Fermi distribution function f ( k p ) is defined as: where a = kT (= 0.020 GeV) is the diffuseness parameter [32]. The Fermi momentum k F for carbon nucleus is 0.221 GeV [33]. The Pauli suppression factor S(ν − ν min ) is defined as: where ν is the energy transfer in the interaction defined as: and ν min is defined as: Here, M n is the final state neutron mass and E B is the binding energy. For carbon nucleus, E B = 10 MeV [32]. The total cross section of anti-neutrino -nucleus quasi elastic scattering is obtained by integrating the differential cross section as defined by Eq. 20 over Q 2 , where Q 2 ranges from Q 2 min to Q 2 max defined by Eqs. 17 and 18 calculated with E ef f ν instead of Eν.

Results and discussions
We calculated the charged currentν − N andν − A quasi elastic differential scattering cross sections. Fig. 1 shows the present calculations ofν µ − p charged current quasi elastic differential scattering cross section dσ dQ 2 as a function of the square of momentum transfer Q 2 , for different values of axial mass (M A = 0.979, 1.05, 1.12 and 1.23 GeV) and for anti-neutrino energy Eν = 2 GeV. The value of differential cross section increases with increase in the value of axial mass. Fig. 2 shows the differential cross section dσ dQ 2 for ν µ − p andν µ − 12 C charged current QES as a function of the square of momentum transfer Q 2 , with axial mass M A = 1.05 GeV and anti-neutrino energy Eν = 2 GeV. The anti-neutrino -carbon cross section is lower than the anti-neutrino -proton cross section for smaller values of Q 2 due to nuclear effects. The cross sections gradually drop to zero with increase in the value of Q 2 .  We compared the present calculations ofν µ − 12 C charged current quasi elastic differential scattering cross section with experimental data from several collaborations. Fig. 3 shows the differential cross section dσ dQ 2 per proton for anti-neutrino -carbon CCQES as a function of the square of momentum transfer Q 2 , for different values of axial mass (M A = 0.979, 1.05, 1.12 and 1.23 GeV). The results obtained are compared with MINERνA data [17] measuring muon anti-neutrino quasi elastic scattering on a hydrocarbon target at < Eν > = 3.5 GeV. The calculation with axial mass M A = 0.979 GeV is compatible with data. Differential cross section dσ dQ 2 per proton forνµ− 12 C charged current QES as a function of the square of momentum transfer Q 2 , for different values of axial mass MA and for average anti-neutrino energy < Eν > = 3.5 GeV compared with MINERνA data [17].  Differential cross section dσ dQ 2 per proton forνµ− 12 C charged current QES as a function of the square of momentum transfer Q 2 , for different values of axial mass MA and for average anti-neutrino energy < Eν > = 2 GeV compared with GGM data [12].
Fig. 5 shows the differential cross section dσ dQ 2 per proton for anti-neutrino -carbon CCQES as a function of the square of momentum transfer Q 2 , for different values of axial mass (M A = 0.979, 1.05, 1.12 and 1.23 GeV) and for average anti-neutrino energy < Eν > = 3 GeV. The results obtained are compared with SKAT data [14] studying the cross sections of neutrino and antineutrino quasi elastic interactions using a wide energy band (3 -30 GeV) neutrino/anti-neutrino beam on heavy freon (CF 3 Br) target. The calculations with axial mass M A = 0.979 and 1.05 GeV are compatible with data. Fig. 6 shows flux-integrated differential cross section  Differential cross section dσ dQ 2 per proton forνµ− 12 C charged current QES as a function of the square of momentum transfer Q 2 , for different values of axial mass MA and for average anti-neutrino energy < Eν > = 3 GeV compared with SKAT data [14]. We performed the calculations of the total cross section for charged currentν−N andν−A quasi elastic scattering and compared the present results with data from several experiments. Fig. 7 shows the present calculations of the total cross section σ for anti-neutrino -proton CCQES as a function of the anti-neutrino energy Eν, for different values of axial mass (M A = 0.979, 1.05, 1.12 and 1.23 GeV). The value of total cross section increases with increase in the value of axial mass. We compared the obtained results with data from BNL [15] and NO-MAD [16] experiments. The calculation with axial mass M A = 1.05 GeV is compatible with data. Fig. 8 shows the total cross section σ forν µ − p and ν µ − 12 C charged current QES as a function of the antineutrino energy Eν, with axial mass M A = 1.05 GeV. The nuclear effects reduce anti-neutrino -carbon cross section compared to the anti-neutrino -proton cross section. Total cross section σ forνµ − p CCQES as a function of the anti-neutrino energy Eν , for different values of axial mass MA compared with BNL [15] and NOMAD [16] data. Total cross section σ per proton for νµ− 12 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. Fig. 9 shows the total cross section σ per proton for ν µ − 12 C charged current QES as a function of the antineutrino energy Eν, for different values of axial mass (M A = 0.979, 1.05, 1.12 and 1.23 GeV). The results obtained are compared with data from GGM(1977) [12], GGM(1979) [13], SKAT [14], NOMAD [16] and Mini-BooNE [18] experiments. The calculations with axial mass M A = 0.979 and 1.05 GeV are compatible with GGM(1977), GGM(1979) and SKAT data though the calculations overestimate the data at low anti-neutrino energies. The approach parameterizing axial mass M A as a function of anti-neutrino energy, presented in Ref. [34], can be used to get a better agreement with data at low anti-neutrino energies. The calculation with axial mass M A = 1.05 GeV is compatible with NOMAD data and the calculation with axial mass M A = 1.23 GeV is compatible with MiniBooNE data.

Conclusion
We presented a study on charged current antineutrino -nucleon and anti-neutrino -nucleus (carbon) quasi elastic scattering using Llewellyn Smith (LS) model. For electric and magnetic Sach's form factors of nucleons, we used Galster et al. parameterizations. Fermi gas model along with Pauli suppression condition has been used to incorporate the nuclear effects in antineutrino -nucleus QES. We calculatedν µ −p andν µ − 12 C charged current quasi elastic differential and total scattering cross sections for different values of axial mass M A and compared the obtained 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 M A = 0.979 and 1.05 GeV are compatible with data from most of the experiments.