Research Article Influence of Neutral Currents on Electron and Gamma Polarizations in the Process e+n e +N+γ

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1 Advances in High Energy Physics Volume 014, Article ID 17389, 8 pages Research Article Influence of Neutral Currents on Electron and Gamma Polarizations in the Process e+n e +N+γ N. V. Samsonenko, 1 Adamou Ousmane Manga, Almoustapha Aboubacar, and Aboubacar Moussa 3 1 Department of Theoretical Physics, Faculty of Sciences, Russian Friendship University, 3 Ordjonikidze, Moscow 11793, Russia Department of Physics, Faculty of Sciences and Techniques, Abdou Moumouni University, 1066 Niamey, Niger 3 Department of Mathematics and Computer Science, Faculty of Sciences and Techniques, Abdou Moumouni University, 1066 Niamey, Niger Correspondence should be addressed to Adamou Ousmane Manga; manga adamou@yahoo.com Received 5 January 014; Accepted 9 March 014; Published 5 June 014 Academic Editor: Duncan. Carlsmith Copyright 014 N. V. Samsonenko et al. This is an open access article distributed under the Creative Commons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP 3. The differential cross section of electron inelastic scattering by nuclei followed by γ radiation is calculated using the multipole decomposition of the hadronic currents and by taking into account the longitudinal polarization of the initial electron and the circular polarization of theγ radiation. We performed the analysis of the angular and energy dependence of the degree of electron and photon polarization which can yield information on values of weak neutral currents parameters. 1. Introduction For many years lepton scattering by nucleon and nuclei has been playing a key role in the determination of the nuclear electromagnetic form factors and in testing the standard model of electroweak interactions [1 4]. During the last decade there has been a considerable effort to measure parityviolating asymmetries in electron scattering [5 7]. These asymmetries can improve the precision of the determination of the mixing angle θ W [8 11]. Electron scattering has also been used for the determination of the vector strange-quark matrix element [11 16]. In this work we study the influence of weak neutral currents parameters on the spin asymmetry coefficient of electron and photon polarization in the electron inelastic scattering by nuclei followed by gamma radiation. This process is illustrated by the following equation: e +(A,Z) e +(A,Z) (1) (A, Z) + γ R. Differential Cross Section In the first order perturbation theory the square of matrix element of the process (1) can be expressed as [17] M 1 fi = M i M J f i +1 M i M f J f M f H γ (Ω γ,σ) J nm n M n J n M n H ee ( q, s e) J im i, () where M ni and M fn are the matrix elements of J i M i J n M n and J n M n J f M f transitions. J i, J n,andj f are, respectively, the spins of the initial, intermediate (excited), and the final states of the nucleus. et us consider H γ (Ω γ,σ)= πe ωω ( i) J 1 [J 1 ](σ T m J 1 σ T e J 1 σ ), (3) J 1 1

2 Advances in High Energy Physics the interaction Hamiltonian for the emission of gamma rays with polarization σ in the direction Ω γ.here[j 1 ] = (J 1 +1) 1/. The magnetic and electric multipole operators T m J 1 σ and T e J 1 σ are defined in a system of coordinates X Y Z where z-axis is oriented along the momentum k of the photon (Figure 3). In order to define all the multipole operators in a system of coordinates where z-axis is oriented along the transferred momentum q we perform a rotation of the system of coordinates X Y Z by angles φ γ and θ γ.soweobtaintherelation [18]: T m(e) J 1 σ (k) = T m(e) J 1 M 1 D J 1 M 1 σ (φ γ,θ γ,φ γ ). M 1 By using (3)and(4)andapplyingtheWigner-Eckarttheorem the matrix element of the Hamiltonian H γ can take the following form: J f M f H γ (Ω γ,σ) J nm n = πe ωω J ( i) 1 [J 1 ]D J 1 M 1 σ (φ γ,θ γ,φ γ ) J 1 1 M 1 ( J f M f J 1 M 1 (4) J n ) J M f n σ T m J 1 σ T e J 1 σ J n. (5) So the matrix element for the nuclear transition from the initial state J i M i to the intermediate state J n M n is given by M ni = J n M n H ee J im i =C γ l γ μ Jγ μ +C Wl Z μ JZ μ, (6) where l Z μ = u(p )γ μ (a V +a A γ 5 )u(p), (7) J μ (q) = J n M n d x exp ( i qx) Ĵh μ ( x) J im i (8) are the leptonic and hadronic currents. C γ = 4πα/q μ and C W = G F / are related, respectively, to the electromagnetic and weak interactions. q μ =(p p) μ is the transferred 4-momentum, α is the fine structure constant, and G F is the Fermi coupling constant for weak interaction. The constants a V and a A take specific values depending on the process. In the Weinberg-Salam model, for electron or muon scattering processes we have according to [19] a V = 1+4sin θ W and a A = 1. The density of hadronic current Ĵ h μ is composed by vector and axial-vector currents with isoscalar (τ =0)andisovector (τ =1)components[18]: Ĵ h μ =β(τ) V ( J μ ) τmτ +β (τ) A ( J 5 μ ) τm τ, (9) where β (τ) V and β(τ) A are the linking constants whose values depend on the process considered. For the electromagnetic interactions M τ =0, β (τ) V =1,andβ(τ) A =0(τ=0,1)and the hadronic current is given by J γ μ =(J μ) 00 +(J μ ) 10. (10) In the case of processes with weak neutral currents, M τ = 0 (τ = 0, 1) so the hadronic current is given by J Z μ =β(0) V (J μ) 00 +β (0) A (J5 μ ) 00 (11) +β (1) V (J μ) 10 +β (1) A (J5 μ ). 10 In the Weinberg-Salam model, for scattering processes according to [19]we have β (0) V = sin θ W β (0) A =0, (1) β (1) V =1 sin θ W β (1) A =1, where θ W is the Weinberg angle. The square of the matrix element (7) for electron electroweak scattering by nuclei takes the following form: M ni =C γ {(lγ μ lγ ] )(Jγ μ Jγ ] )+ρre (lγ μ lz ] Jγ μ JZ ] )}, (13) where ρ=c W /C γ. In (13) we neglect the term proportional to ρ.inthe ultrarelativistic longitudinally polarized electrons case we obtain the relation l γ μ lz ] where the tensor l γ μ lγ ] is given by =(a V a A s e )l γ μ lγ ], (14) l γ μ lγ ] = δ μ EE {δ μ] (pp ) p μ p ] p ]p μ s eε μ]αβ p α p β }. (15) Here E and p (E and p ) are the energy and the 4-momentum of the initial (final) electron; s e = ±1 is the helicity of the initial electron. The factor δ μ takes the following values: 1 for μ = 1,, 3 δ μ ={ +1 for μ=4. (16) The differential cross section, calculated by multipole decomposition of matrix elements [19, 0] and by taking into

3 Advances in High Energy Physics 3 account the circular polarization of the emitted photon, is given by the following formula: The functions R m and R m aregivenbythefollowingformulae: dσ dω e dω γ = Γn f γ Γ n f Total ττ ( τ n τ τ i M τn 0 M τi ) { 1 1+ { R0 0(q) ni { ( τ n τ τ i M τn 0 M τi )ασ ee W t 0 (q γ) fn f even (P (cos θ γ )R 0 +P 1 (cos θ γ) cos φ γ R 1 R 0 =X 1 (V l W l + V tw t )+X V t W a, R 1 =X V ac W ac +X 3V tl W al R =X 1V tt W tt, +X 4V tl W tl, R 0 =X 1V t W t +X (V t W a + V l Wl ), R 1 =X V ac W ac +X 3 V tl W al R =X V tt W tt, X 1 =1+ρβ (τ ) V (a V a A s e ), X =ρβ (τ ) A (a V a A s e ), X 3 =ρβ (τ) A (a V a A s e ), +X 4 V tl W tl, X 4 =(1+ρ(β (τ) V +β(τ ) V )(a V a A s e )). (0) s γ + R0 0(q) f ni 1 odd +P (cos θ γ) cos φ γ R ) (P (cos θ γ ) R 0 +P 1 (cos θ γ) cos φ γ R 1 ) Here V i (i = l,t,...,ac ) are the leptonic functions and W i are the hadronic functions. The expressions of the hadronic and the leptonic functions by taking into account the longitudinal polarization of electrons are given in the Appendix. The quantum number is defined by the relation 0 J n.thefunctionsf are defined as follows: + s γ R 0 0 (q) ni 3 odd f P (cos θ γ) cos φ γ R }, } } (17) where Γ n f γ and Γ n f total indicate, respectively, the relative and total disintegration widths of the excited nucleus state and P m (cos θ γ) is egendre polynomial. The angles θ γ and φ γ determine the photon direction, s γ is photon helicity, and σ ee is the differential cross section of the electron scattering by unpolarized nuclei given by where f = W t (q γ)= ( ) J f+j n W t (q γ) { W0 t (q γ) W t (q γ) { { W0 t (q γ) for even for odd, (1) σ ee =4πσ M R 0 0 (q) ni f 1 rec, (18) where σ M and f rec are, respectively, the Mott cross section and the nuclear correction factor given by Zα cos (θ/) σ M =( E sin (θ/) ), f rec =1+ E θ M sin, (19) where M isthemassofthenucleusandz is the atomic number. ( + 1) (J 1 +1)(J1 +1) J 1 J 1 ( J 1 J ){ J 1 J 1 J n J n J f } {P + J 1 +J 1 (F EJ 1 F EJ 1 +F MJ 1 F MJ1 ) +P J 1 +J 1 (F EJ 1 F MJ 1 F MJ 1 F EJ1 )}. Hence q γ =E γ is the gamma transition energy. ()

4 4 Advances in High Energy Physics Quantum numbers J 1 and J 1 are given by J n J f J 1 J n +J f, W t (q γ)=w t (q γ) J n J f J 1 J n +J f, for odd, P + J 1 +J 1 = 1 ( 1)(1/)(J 1 J 1) (1 + ( 1) J 1 +J 1 ), P J 1 +J 1 = 1 ( 1)(1/)(J 1 J 1+1) (1 ( 1) J 1 +J 1 ). (3) 3. Study of the Transitions J i =0 J n =1 J f =0 As an example of transitions let us consider the following process: e + 1 C(0 + 0) e + 1 C (1 + 1) 1 C(0 + 0) + γ R (4) So the quantum numbers, J,andJ are defined as where Σ 0 = 1 3 F M1 (q γ)σ ee, R 0 0 (q) = F M1 f = 1, f 1 = 3, [V t F M1 (1 + ρβ (1) V (a V a A s e )) + V t F 5 E1 ρβ(1) A (a V a A s e )], R 0 (q) = 1 R0 0 (q), R 1 (q) = V ac F M1 ρβ (1) A (a V a A s e )(F 5 C1 q 0 q F5 1 ), R (q) = V ttf M1 (1 + ρβ(1) V (a V a A s e )), R 0 1 (q) = 1 3 F M1 [V t F M1 (1 + ρβ (1) V (a V a A s e )) +V t F 5 E1 ρβ(1) A (a V a A s e )], R 1 1 (q) = 1 3 V ac F M1ρβ (1) A (a V a A s e )(F 5 C1 q 0 q F5 1 ). (7) 0 J n 0 ; J i J n J J i +J n J = 1; J i J n J J i +J n J =1. (5) The differential cross section of the process (4)whichtakes into account the longitudinal polarization of the electron and the circular polarization of the photon is given by the formula dσ dω e dω γ =Σ 0 {R 0 0 +f(n f+γ) [P (cos θ γ )R 0 +P1 (cos θ γ) cos φ γ R 1 +P (cos θ γ) cos φ γ R ] +s γ f 1 [P 1 (cos θ γ ) R 0 1 +P 1 1 (cos θ γ) cos φ γ R 1 1 ]}, (6) The matrix elements F M1, F 5 C1, F5 1,andF5 E1,calculatedinthe shell model with a harmonic oscillator potential, are given by [1]: F M1 = ψ q 6 π M e y (F 1 μ( y)); F 5 C1 = ψ q 3 π M e y ( 3 F A (1 y)(q 0 F p MF T )) ; F 5 1 = ψ 3 π e y (1 y) (F A q M F p); F 5 E1 = ψ 3 π e y F A ( y), (8) where ψ = 3 [19], μ=f 1 +M n F, q 0 = ΔEis the energy of the transition, and y = (bq/),whereb is the parameter of the harmonic oscillator. Writing explicitly the electron and photon polarization, the differential cross section (6) takes the following form: dσ dω e dω γ = F M1Σ 0 (Σ + s e Δ+s γ Δ +s e s γ Δ ), (9)

5 Advances in High Energy Physics Photon polarization degree E = 100 MeV E=500MeV (a) E = 100 MeV E=500MeV (b) Figure 1: (a) Asymmetry coefficient for the electron. (b) Photon polarization degree. where Σ=(1+ 1 P (cos θ γ )) (V t F M1 (1 + ρβ (1) V a V) Ṽ t F 5 E1 ρβ(1) A a A) 1 P1 (cos θ γ) cos φ γ Ṽ ac ρβ (1) A a A (F 5 C1 q 0 q F5 1 ) 1 4 P (cos θ γ) cos φ γ V tt F M1 (1 + ρβ (1) V a V), Δ=ρ{(+P (cos θ γ )) (Ṽ t F 5 E1 β(1) A a V V t F M1 β (1) V a A) + 1 P1 (cos θ γ) cos φ γ Ṽ ac β (1) A a V (F 5 C1 q 0 q F5 1 ) + 1 P (cos θ γ) cos φ γ V tt F M1 β (1) V a A}, Δ =ρ{ 3 P 1 (cos θ γ )(V t F 5 E1 β(1) A a V Ṽ t F M1 β (1) V a A) + 3P 1 1 (cos θ γ) cos φ γ V ac β (1) A a V (F 5 C1 q 0 q F5 1 )}, Δ = 3 4 P 1 (cos θ γ )(Ṽ t F 5 E1 (1 + ρβ(1) V a V) V t F M1 ρβ (1) A a A) 3P 1 1 (cos θ γ) cos φ γ V ac ρβ (1) A a A (F 5 C1 q 0 q F5 1 ). (30) et us consider now two coefficients of asymmetry which can be experimentally determined, that is, the electron polarization ratio and the photon circular polarization degree: A R, = dσ (s e =+1) dσ(s e = 1) dσ (s e =+1)+dσ(s e = 1), (31) B R, = dσ (s γ =+1) dσ(s γ = 1) dσ (s γ =+1)+dσ(s γ = 1). (3) Assuming that the photon is emitted in the direction of the incident electron we have cos θ γ = E cos θ E q, φ γ =0, sin θ γ = E sin θ, q (33) and after averaging on the photon polarization, the asymmetry coefficient for the electron defined by (31)isgivenby A R, = Δ Σ. (34) Averaging the differential cross section (9) on the electron polarization, the photon polarization degree defined by (3) takes the form B R, = Δ Σ. (35) Figures 1(a) and 1(b) show the dependence on the diffusion angle θ of the asymmetry coefficient for electron and photon

6 6 Advances in High Energy Physics sin (θ W ) = (a) sin (θ W ) = (c) sin (θ W ) = a A = 1.0 β (1) A = 1.0 a A = 1.05 β (1) A = 1.05 a A = 1.07 β (1) A = 1.07 (e) sin (θ W ) = 0. E=500MeV (b) sin (θ W ) = 0.3 E=500MeV (d) sin (θ W ) = E=500MeV a A = 1.0 β (1) A = 1.0 a A = 1.05 β (1) A = 1.05 a A = 1.07 β (1) A = 1.07 (f) Figure : Asymmetry coefficient for the electron. polarization degrees for three values of the energy for incident electron. Calculation is done by using sin θ W = from [8] and formulas(1) relative to the coupling constants of standard model. Figure 1(a) shows for E = 300 MeV and E = 500 MeV variation of asymmetry coefficient. This variation is greater for electron scattering around value of angle θ=45.however, the photon degree of polarization undergoes greater variation for large values of scattering angle. Figures (a) (f) represent the angular dependence of asymmetry coefficient for electron with energy values E = 300 and E = 500 MeV and for different values of parameters

7 Advances in High Energy Physics 7 The hadronic functions are given by the following formulae: e z q W l (q) = A () 0;0 P+ J +J F CJ F CJ, p k W t (q) = A () 1; 1 {P+ J +J (F EJ F EJ +F MJ F MJ ) P J +j (F EJ F MJ F MJ F EJ )}, θ γ W tt (q) = A () 1;1 {P+ J +J (F EJ F EJ F MJ F MJ ) θ +P J +j (F EJ F MJ +F MJ F EJ )}, φ γ e y p p W tl = A () 0;1 F CJ (P+ J +J F EJ +P J +J F MJ), ex ( p p) q p W a (q) = A () 1; 1 {P+ J +J (F5 EJ F MJ +F 5 MJ F EJ) Figure 3: System of coordinates. a A and β (1) A. is very sensitive to change of these parameters, but it could be sensitive to the change of sin θ W which depend on these parameters in the standard model. 4. Conclusion The differential cross section of the inelastic electron scattering by nuclei is calculated and the expressions of the asymmetries coefficients are obtained; their angular-energy dependence is analyzed and some results are carried out. The experimental study of the electron scattering processes accompanied by gamma radiation can play an important role in research on the parameters of the weak neutral currents. Appendix By taking into account the longitudinal polarization of electronsweobtainfortheleptonicfunctionsthefollowing expressions: V l =( q μ q ), V t = 1 q μ q + θ tan, V tt = 1 q μ q, V tl = 1 q μ V l = q μ μ q (q q + θ tan ) 1/ q, V t = s 1 e (q q + θ tan ) q μ, V tl = 1 s e q tan θ, μ 1/ tan θ =s eṽ t, V ac =s e tan θ =s eṽ ac, V ac =( q 1/ μ q + θ tan ). (A.1) W ac W tt W al = W l = +P J +j (F5 EJ F EJ +F 5 MJ F MJ)}, A () 0;1 F CJ (P+ J +J F5 MJ +P J +J F5 EJ ), A () 0;1 (F5 CJ q 0 q F5 J )(P+ J +J F MJ P J +J F EJ), = A () 0;0 P J +J (F5 CJ q 0 q F5 J )F CJ, (q) = A () 1;1 {P+ J +J (F5 EJ F MJ F 5 MJ F EJ) W t (q) = Wt (q) ; W al P J +j (F5 EJ F EJ +F 5 MJ F MJ)}, Wtl W a (q) = Wa (q) ; (q) = W al (q) ; Wac (q) = Wtl (q) ; (q) = W ac (q). (A.) The coefficients A () m m and P± J +J aregivenbythefollowing formulae: A () m m = ( 1)J i+j n [J ] [J][]( (+M)! 1/ ( M)! ) ( J J m m M ){ J J J n J n J i }, P + J +J = 1 ( 1)(1/)(J J) (1 + ( 1) J +J ), P J +J = 1 ( 1)(1/)(J J+1) (1 ( 1) J +J ). (A.3)

8 8 Advances in High Energy Physics The functions F CJ, F J, F MJ,andF EJ (F 5 CJ,F5 J,F5 MJ, and F5 EJ ) are the matrix elements of the multipole vector (axial-vector) Coulomb, longitudinal, transverse magnetic, and transverse electric operators defined as follows: coul F CJ = J f M i ; F EJ = J f T el i ; F 5 CJ = J f F 5 EJ = J f i M coul5 i ; el5 i T J J i ; Conflict of Interests F J = J f J J i ; F MJ = J f i T m i ; F 5 J = J f i 5 J J i ; F 5 MJ = J f T m5 i. (A.4) The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] J. V. Allaby, U. Amaldi, G. Barbiellini et al., A precise determination of the electroweak mixing angle from semileptonic neutrino scattering, Zeitschrift für Physik C, vol. 36, no. 4, pp , [] P. G. Reutens, F. S. Merritt, M. J. Oreglia et al., A measurement of the neutral current electroweak parameters using the fermilab narrow band neutrino beam, Zeitschrift für Physik C, vol. 45, no. 4, pp , [3] C.E.Jones,E.J.Beise,J.E.Belzetal., 3 He( e, e ) quasielastic asymmetry, Physical Review C,vol.47, no.1,pp ,1993. [4] J. D. Walecka, Electron Scattering for Nuclear and Nucleon Structure, Cambridge University Press, Cambridge, UK, 001. [5] M.B.Barbaro,J.A.Caballero,T.W.Donnelly,andC.Maieron, Inelastic electron-nucleus scattering and scaling at high inelasticity, Physical Review C,vol.69,no.3,ArticleID03550,004. [6] S. Baunack, Asymmetries in polarized electron scattering and the strangeness content of the nucleon, Modern Physics etters A, vol. 4, no , pp , 009. [7] K. S. Kim and Y. Ko, Parity violation in electron quasielastic scattering, Few-Body Systems, vol.54,no.1-4,pp , 013. [8] K.S.Kumar,SonnyMantry,W.J.Marciano,andP.A.Souder, ow energy measurements of the weak mixing angle, Annual Review of Nuclear and Particle Science, vol. 63, pp , 013. [9] R. González-Jiménez,J.A.Caballero,andT.W.Donnelly, Parity violation in elastic electron-nucleon scattering: strangeness content in the nucleon, Physics Reports,vol.54,no. 1,pp.1 35, 013. [10] J. Erler and M. J. Ramsey-Musolf, Weak mixing angle at low energies, Physical Review D, vol. 7,ArticleID073003, 005. [11] J. iu, R. D. McKeown, and M. J. Ramsey-Musolf, Global analysis of nucleon strange form factors at low Q, Physical Review C,vol.76,no.,ArticleID050,007. [1] K. A. Aniol, D. S. Armstrong, T. Averett et al., Parity-violating electroweak asymmetry in ep scattering, Physical Review C,vol. 69, no. 6, Article ID , 004. [13] F. E. Maas, P. Achenbach, K. Aulenbacher et al., Measurement of strange-quark contributions to the nucleon's form factors at Q = (GeV/c), Physical Review etters, vol.93,no., Article ID 000, 004. [14] D. S. Armstrong, J. Arvieux, R. Asaturyan et al., Strange-quark contributions to parity-violating asymmetries in the forward G0 electron-proton scattering experiment, Physical Review etters,vol.95,articleid09001,005. [15] F. E. Maas, K. Aulenbacher, S. Baunack et al., Evidence for strange-quark contributions to the nucleon's form factors at Q = 8 (Ge V/c), Physical Review etters, vol.94,no.15, Article ID 15001, 005. [16] A. Acha, K. A. Aniol, D. S. Armstrong et al., Precision measurements of the nucleon strange form factors at Q 0.1 GeV, Physical Review etters, vol. 98, Article ID 03301, 007. [17] T. W. Donnelly, A. S. Raskin, and J. Dubach, Prospects for using the (e, e γ) reaction to explore nuclear structure, Nuclear Physics A,vol.474,no.,pp ,1987. [18] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, NJ, USA, [19] T. W. Donnelly and R. D. Peccei, Neutral current effects in nuclei, Physics Reports,vol.50,no.1,pp.1 85,1979. [0] T. W. Donnelly and A. S. Raskin, Considerations of polarization in inclusive electron scattering from nuclei, Annals of Physics,vol.169,no.,pp ,1986. [1] N. V. Samsonenko, C.. Kathat, M. A. Ousmane, and A.. Samgin, Polarisation effects of second-class currents in the direct and inverse β decay of nuclei, JournalofPhysicsG,vol. 15,no.9,pp ,1989.

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