Revisiting ν μ ( ν ¯ μ ) and ν e ( ν ¯ e ) Induced Quasielastic Scattering from Nuclei in Sub-GeV Energy Region
F. Akbar, M. Rafi Alam, M. Sajjad Athar, S. Chauhan, S. K. Singh, F. Zaidi
aa r X i v : . [ nu c l - t h ] F e b Revisiting ν µ ( ¯ ν µ ) and ν e ( ¯ ν e ) Induced Quasielastic Scatteringfrom Nuclei in Sub-GeV Energy Region F. Akbar , M. Rafi Alam, M. Sajjad Athar, S. Chauhan, S. K. Singh and F. Zaidi
Department of Physics, Aligarh Muslim University, Aligarh - 202 002, IndiaE-mail: [email protected] (Received : August 13, 2018)We present the results of charged current quasielastic(CCQE) scattering cross sections from free aswell as bound nucleons like in C , O , Ar and Pb nuclear targets in E ν ( ¯ ν ) ≤ ff ect. Thedi ff erences those may arise in the electron and muon production cross sections due to the di ff erentlepton mass, uncertainties in the axial dipole mass M A and pseudoscalar form factor, and due to theinclusion of second class currents have been highlighted for neutrino / antineutrino induced processes. KEYWORDS:
Lepton mass, Nuclear medium effects, Second class currents, Local densityapproximation.
1. Introduction
Precise calculations for quasielastic reactions in nuclei induced by ν µ (¯ ν µ ) and ν e (¯ ν e ) are requiredto study the CP violation and mass hierarchy in present experiments done in the sub-GeV energyregion. Therefore, in this energy region, it is important to understand the di ff erences that may arisein the electron vs muon production cross sections due to the lepton mass, the axial dipole mass M A ,pseudoscalar form factor and the inclusion of second class currents. Here we present the results of astudy performed using a local Fermi gas model(LFG) with RPA e ff ect to take into account nuclearmedium e ff ects, and obtained the ratio σ ν e /σ ¯ ν e , σ ν µ /σ ¯ ν µ , σ ν e /σ ν µ and σ ¯ ν e /σ ¯ ν µ in nuclei like C , O and Ar . The uncertainties due to pseudoscalar form factor and its deviation, if any, from the PCACand pion pole dominance value as well as the second class current form factors within the limits ofpresent allowed constraints have been discussed. The details of the calculations are given in Ref. [1].
2. Formalism
For neutrino / antineutrino induced CCQE process ( ν l / ¯ ν l ( k ) + n / p ( p ) → l − / l + ( k ′ ) + p / n ( p ′ )),the general expression of di ff erential cross section is d σ d Ω l dE l = | ~ k ′ | π E ν E n E p ¯ ΣΣ |M| δ [ q + E n − E p ] (1)where |M| is the matrix element square which is written as |M| = G F θ c L µν J µν (2)with leptonic tensor L µν = Σ l µ l † ν and hadronic tensor J µν = ¯ ΣΣ j µ j ν † . he leptonic current is given by l µ = ¯ u ( k ′ ) γ µ (1 ± γ ) u ( k ) , (3)where ( + ve) − ve sign is for (antineutrino)neutrino. J µ is the hadronic current given by [2] J µ = ¯ u ( p ′ ) " F V ( Q ) γ µ + F V ( Q ) i σ µν q ν M + F A ( Q ) γ µ γ + F P ( Q ) q µ M γ + F V ( Q ) q µ M + F A ( Q ) ( p + p ′ ) µ M γ u ( p ) , (4)where F V , ( Q ) are the isovector vector form factors and F A ( Q ), F P ( Q ) are the axial and pseu-doscalar form factors, respectively. F V ( Q ) and F A ( Q ) are respectively associated with the vectorpart and the axial vector part of the second class current.The hadronic current contains isovector vector form factors F V , ( Q ) of the nucleons, which aregiven as F V , ( Q ) = F p , ( Q ) − F n , ( Q ) (5)where F p ( n )1 ( Q ) and F p ( n )2 ( Q ) are the Dirac and Pauli form factors of proton(neutron) which inturn are expressed in terms of the experimentally determined Sach’s electric G p , nE ( Q ) and magnetic G p , nM ( Q ) form factors as F p , n ( Q ) = + Q M ! − " G p , nE ( Q ) + Q M G p , nM ( Q ) (6) F p , n ( Q ) = + Q M ! − h G p , nM ( Q ) − G p , nE ( Q ) i (7) G p , nE ( Q ) and G p , nM ( Q ) are taken from BBBA05 parameterization [3].The isovector axial form factor is obtained from the quasielastic neutrino and antineutrino scat-tering as well as from pion electroproduction data and is parameterized as F A ( Q ) = F A (0) + Q M A − ; F A (0) = − . . (8)The pseudoscalar form factor is determined by using PCAC which gives a relation between F P ( Q )and pion-nucleon form factor g π NN ( Q ) [2], and is given by F P ( Q ) = M F A (0) Q F A ( Q ) F A (0) − m π ( m π + Q ) g π NN ( Q ) g π NN (0) ! , (9)where m π is the pion mass and g π NN (0) is the pion-nucleon strong coupling constant. F P ( Q ) is dom-inated by the pion pole and is given in terms of axial vector form factor F A ( Q ) using the Goldberger-Treiman(GT) relation F P ( Q ) = M F A ( Q ) m π + Q (10) F V ( Q ) which is associated with the vector part of the second class current is taken as [2] F V ( Q ) = . F V ( Q ) (11) σ / N ( - c m ) ν e LFG ν µ LFG ν e RPA ν µ RPA ν e LFG ν µ LFG ν e RPA ν µ RPA σ / N ( - c m ) ν (GeV)00.20.40.60.81 σ / N ( - c m ) ν (GeV)00.10.20.30.4 ν - C ν - C ν - O ν - O ν - Ar ν - Ar Fig. 1.
Total scattering cross section per interacting nucleon for neutrino / antineutrino induced CCQE processfor C , O and Ar nuclear target. The cross sections are evaluated using local Fermi gas model(LFG) andLFG with RPA e ff ect(RPA). The form factor associated with the parity violating term of the second class current F A ( Q ) is takenas F A ( Q ) = . F A ( Q ) . (12)When the reaction, ν l / ¯ ν l + n / p → l − / l + + p / n takes place inside the nucleus, then due to nuclearmedium e ff ects scattering cross section gets modified and in the local Fermi gas model with RPAe ff ects are obtained as σ ( E ν ) = − G F cos θ c Z r max r min r dr Z k ′ max k ′ min k ′ dk ′ Z Q max Q min dQ E ν E l L µν J µν RPA
ImU N [ E ν − E l − Q r − V c ( r ) , ~ q ] , (13) where ImU N is the imaginary part of the Lindhard function, Q r is the Q − value of the reaction, V c is the Coulomb potential and J µν RPA is the modified hadronic tensor when RPA correlations are takeninto account.To observe the sensitivity of di ff erence in lepton production cross section due to di ff erent valuesof axial dipole mass, we define ∆ ( E ν ) = σ ν µ ( M modi f iedA ) − σ ν e ( M modi f iedA ) σ ν e ( M modi f iedA ) ; ∆ ( E ν ) = σ ν µ ( M A = W A ) − σ ν e ( M A = W A ) σ ν e ( M A = W A ) , ∆ M A = ∆ ( E ν ) − ∆ ( E ν ) . (14)where M A = W A = M modi f iedA = ff ect of second classcurrent on the ν e /ν µ and ¯ ν e / ¯ ν µ cross sections we study the di ff erences of the following ratios ∆ ( E ν ) = σ ν µ ( F i , − σ ν e ( F i , σ ν e ( F i ,
0) ; ∆ ( E ν ) = σ ν µ ( F i = − σ ν e ( F i = σ ν e ( F i =
0) (15) ∆ F i = ∆ ( E ν ) − ∆ ( E ν ) . (16)where i = V or A . ν (GeV)012345 R a ti o Free C O Ar0.2 0.4 0.6 0.8 1E ν (GeV)03691215 0.2 0.4 0.6 0.8 1E ν (GeV)03691215 0.2 0.4 0.6 0.8 1E ν (GeV)0246810 σ(ν e ) / σ ( ν e ) σ(ν µ ) / σ ( ν µ ) σ(ν e ) / σ ( ν µ ) σ(ν e ) / σ ( ν µ ) Fig. 2.
Ratio of scattering cross sections for free nucleon case and for bound nucleons using LFG with RPAe ff ect in C , O and Ar .
3. Results and discussions
In Fig. 1, we have presented the results of total scattering cross section for neutrino / antineutrinoinduced CCQE process in C , O and Ar using local Fermi gas model(LFG) with and withoutRPA e ff ect. As compared to the free nucleon cross section, Pauli blocking and Fermi motion, reducethe total scattering cross section significantly, particularly at low energies. Inclusion of RPA e ff ectfurther reduces the cross section considerably and the reduction is more in heavier nuclear targets.The suppression due to nuclear medium e ff ects is larger in the case of antineutrinos as compared tothe neutrino induced processes.In Fig. 2, we have shown the ratio of total scattering cross sections for electron and muon typeneutrinos / antineutrinos i.e. σ ν e /σ ¯ ν e , σ ν µ /σ ¯ ν µ , σ ν e /σ ν µ and σ ¯ ν e /σ ¯ ν µ for CCQE scattering process infree nucleon target and for C , O and Ar nuclear targets using local Fermi gas model(LFG) withRPA e ff ect. Q-value of the reaction for ν µ − Ar reaction is much smaller than in C and O nuclei.Furthermore, Coulomb energy correction is large for Ar . Thus the results in Ar is di ff erent innature than in C and O nuclei at low energies. E ν (GeV) -0.006-0.004-0.00200.0020.0040.006 ∆ M A ν (M A =0.9 GeV) ν (M A =1.2 GeV) ν (M A =0.9 GeV) ν (M A =1.2 GeV) E ν (GeV) -0.006-0.004-0.00200.0020.0040.0060.008 0.2 0.4 0.6 0.8 1 E ν (GeV) -0.006-0.004-0.00200.0020.0040.006 Free
LFG LFG+RPA
Fig. 3. E ff ect of axial dipole mass on the cross section(from left to right): on free nucleon; LFG, with andwithout RPA e ff ect on Ar target. 4 e have also studied the sensitivity of the di ff erence in electron and muon production crosssections due to the uncertainty in the choice of axial dipole mass M A . For this we define ∆ M A in Eq.14 and the results for free nucleon and Ar nuclei are shown in Fig. 3. The percentage di ff erencein electron and muon production cross sections due to uncertainty in axial dipole mass is more inthe case of nuclear targets as compared to free nucleon target but always remains less than 1%. Thedi ff erence increases with the increase in mass number.The fractional di ff erence in the cross sections due to the presence of pseudoscalar form factor ismore in the case of ¯ ν µ induced CCQE process than ν µ induced process for the free nucleon case aswell as in nuclear targets. This di ff erence vanishes with the increase in energy (not shown here).We have also studied the individual sensitivity due to F V and F A in the electron and muon pro-duction cross sections for free nucleon as well as for C , Ar and Pb nuclear targets and resultsare presented in Fig. 4. We find that the sensitivity is non-negligible at low energies and becomesalmost negligible beyond E ν = . GeV . For example, for neutrino induced reactions, ∆ F V is 3% forfree nucleon, ∼
4% for C & Ar and ∼
2% for Pb at E ν = . GeV . For antineutrino inducedreactions at E ν = . GeV , ∆ F V is ∼
9% for free nucleon, ∼
12% for C , Ar and Pb targets.The sensitivity in the di ff erence between the electron and muon production cross sections due to F A ν (GeV)-0.12-0.09-0.06-0.030 ∆ F V ν -N ν -N ν - C ν - C ν - Ar ν - Ar ν - Pb ν - Pb ν (GeV)00.0010.0020.0030.0040.0050.006 ∆ F A ν -N ν -N ν - C ν - C ν - Ar ν - Ar ν - Pb ν - Pb Fig. 4.
The di ff erence of fractional changes ∆ F V and ∆ F A for free nucleon case, and for bound nucleons usingLFG with RPA e ff ect in C , Ar and Pb nuclear targets. is very small as compared to sensitivity due to F V for free nucleon as well as for nuclear targets. It isalso non-zero at low energies and becomes almost negligible beyond E ν = . GeV . References [1] F. Akbar, M. R. Alam, M. Sajjad Athar, S. Chauhan, S. K. Singh and F. Zaidi, Int. Jour. Mod. Phys. E 24,11 (2015); M. Sajjad Athar, S. Chauhan and S. K. Singh, Eur. Phys. J. A 43, 209 (2010); M. Sajjad Atharand S. K. Singh, Phys. Rev. C , 028501 (2000).[2] M. Day and K. S. McFarland, Phys. Rev. D , 053003 (2012).[3] R. Bradford, A. Bodek, H. S. Budd and J. Arrington, Nucl. Phys. Proc. Suppl.159