Charged Current Quasi Elastic Scattering of Muon Anti-Neutrino Off
Total Page:16
File Type:pdf, Size:1020Kb
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 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 12 into account the nuclear effects in anti-neutrino - 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 energy spectra in beta (BNL) [15], Neutrino Oscillation MAgnetic Detector decay process, the neutrinos have been a major field of (NOMAD) [16], Main INjector ExpeRiment for ν -A research. Neutrinos exist in three flavors (electron, muon (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 standard model anti-neutrinos off various nuclear targets. GGM stud- of particle physics 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 weak interaction, i.e. charged current: via as target, NOMAD executed the studies on carbon, exchange of W +=W − boson and neutral current: 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 pion 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 lepton 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 −jk j); ~ 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 +jk~0j); 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 .