^{12}C

Supported by Department of Science and Technology, New Delhi, India

^{12}C

^{3}and published under licence by 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

In this work, we study charged current quasi-elastic scattering (QES) of _{A}, and compare the results with data from the GGM, SKAT, BNL, NOMAD, MINER

Article funded by SCOAP^{3}

From their first postulation by Wolfgang Pauli in 1930, to explain the continuous energy spectra in the beta decay process, neutrinos have been a major field of research. Neutrinos exist in three flavors (electron, muon and tau neutrinos) along with their anti-particles, which are called anti-neutrinos. Searches for more neutrino flavors, called sterile neutrinos, are 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 ^{[1–10]}. Being neutral particles, (anti)neutrinos undergo only weak interactions (charged current, via exchange of

Precise knowledge of (anti)neutrino CCQES is crucial to high energy physics experiments studying neutrino oscillations and hence extracting the neutrino mass hierarchy, mixing angles etc. ^{[1–10]}. 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 ^{[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 a freon target. SKAT bombarded a wide energy band neutrino/anti-neutrino beam onto a heavy freon (CF_{3}Br) target. BNL used hydrogen (H_{2}) as a target, NOMAD used carbon, MINER_{A}, is presented in Ref. [

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 the electric and magnetic Sachʼs form factors of the nucleons. To incorporate the nuclear effects, in the case of ^{12}C, we use the Fermi gas model along with the Pauli suppression condition ^{[19,20,22–24]}. We calculate _{A} and compare the results with experimental data, with the goal of finding the most appropriate _{A} value. This work does not include the contribution from the 2N2h (two nucleons two holes) effect in QES, where the interaction involves two nucleons producing two holes in the nucleus. Studies of the 2N2h effect in QES are presented in Refs. [

The Llewellyn Smith model describes (anti)neutrino scattering using a plane wave impulse approximation and calculates the QES cross sections. The anti-neutrino–nucleon charged current quasi-elastic differential cross section for a free nucleon at rest is given as ^{[20]}:
_{N} is the nucleon mass, _{l} is the mass of the outgoing lepton.

The functions ^{[11,20,23]}:
_{A} is the axial form factor, _{P} is the pseudoscalar form factor and

The axial form factor _{A} is defined in the dipole form as ^{[30]}:
_{A} is the axial mass.

The pseudoscalar form factor _{P} is defined in terms of the axial form factor _{A} as ^{[31]}:

The vector form factors ^{[30,32]}:
^{[21]}, Budd et al. ^{[33]}, Bradford et al. ^{[34]}, Bosted ^{[35]} and Alberico et al. ^{[36]} provide parameterizations of these form factors by fitting the electron scattering data. For the present calculations, we are using Galster et al.ʼs parameterizations of these form factors.

The electric and magnetic Sachʼs form factors of the nucleons are defined as ^{[21]}:
^{[37]} parameterization as:
^{[30]}:

The total cross section of anti-neutrino–nucleon (free) quasi-elastic scattering is obtained by integrating the differential cross section, defined by Eq. (^{[23]}:
_{l} and _{Q} is defined as:

For studying anti-neutrino–nucleus quasi-elastic scattering, the nucleus can be treated as a Fermi gas ^{[19,20,22–24]}, where the nucleons move independently within the nuclear volume in an average binding potential generated by all nucleons. The 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 _{F} is equal to zero.

The differential cross section per proton for anti-neutrino–nucleus quasi-elastic scattering is defined as ^{[23,24]}:
_{p} is the momentum of the proton,

^{[23]}:
_{p} is the proton mass and ^{[23]}:
_{p} is the proton energy, defined as:

The Fermi distribution function ^{[24]}. The Fermi momentum _{F} for the carbon nucleus is 0.221 GeV ^{[38]}.

The Pauli suppression factor ^{[23,24]}:
_{n} is the final state neutron mass and _{B} is the binding energy. For the carbon nucleus, ^{[24]}.

The total cross section of anti-neutrino–nucleus quasi-elastic scattering is obtained by integrating the differential cross section, as defined by Eq. (

We calculated the charged current

(color online) Differential cross section _{A} and for anti-neutrino energy

Figure

(color online) Differential cross section

We compared the present calculations of ^{[18]}, measuring the muon anti-neutrino CCQES cross section off a mineral oil (carbon) target. The calculations are performed for different values of axial mass

(color online) Flux-integrated differential cross section ^{[18]}. The average anti-neutrino energy

Figure ^{[12]}, which studied quasi-elastic reactions of neutrinos and antineutrinos on a propane plus freon target. The calculations with axial mass

(color online) Differential cross section _{A} and for average anti-neutrino energy ^{[12]}.

Figure ^{[14]} studying the cross sections of neutrino and anti-neutrino quasi elastic interactions using a wide energy band (3-30 GeV) neutrino/anti-neutrino beam on a heavy freon (CF_{3}Br) target. The calculations with axial mass

(color online) Differential cross section _{A} and for average anti-neutrino energy ^{[14]}.

Figure ^{[17]} measuring muon anti-neutrino quasi-elastic scattering on a hydrocarbon target at

(color online) Differential cross section _{A} and for average anti-neutrino energy ^{[17]}.

We calculated the total cross section for charged current ^{[15]} and NOMAD ^{[16]} experiments. The calculation with axial mass

(color online) Total cross section _{A} compared with BNL ^{[15]} and NOMAD ^{[16]} data.

Figure

(color online) Total cross section

Figure ^{[12]}, GGM(1979) ^{[13]}, SKAT ^{[14]}, NOMAD ^{[16]} and MiniBooNE ^{[18]} experiments. The calculations with axial mass _{A} as a function of anti-neutrino energy, presented in Ref. [

(color online) Total cross section _{A} compared with GGM(1977) ^{[12]}, GGM(1979) ^{[13]}, SKAT ^{[14]}, NOMAD ^{[16]} and MiniBooNE ^{[18]} data.

We have presented a study on charged current anti-neutrino–nucleon and anti-neutrino–nucleus (carbon) quasi-elastic scattering using the Llewellyn Smith (LS) model. For the electric and magnetic Sachʼs form factors of nucleons, we used Galster et al.ʼs parameterizations. The Fermi gas model, along with the Pauli suppression condition, has been used to incorporate the nuclear effects in anti-neutrino–nucleus QES. We calculated _{A} and compared the obtained results with data from the GGM, SKAT, BNL, NOMAD, MINER