Influence of Neutral Currents on Electron and Gamma Polarizations in the Process e + N → e

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 the γ 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.


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][2][3][4].During the last decade there has been a considerable effort to measure parityviolating asymmetries in electron scattering [5][6][7].These asymmetries can improve the precision of the determination of the mixing angle   [8][9][10][11].Electron scattering has also been used for the determination of the vector strange-quark matrix element [11][12][13][14][15][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: (A, Z) +  RL e − + (A, Z) → e − + (A, Z) *   (1)

Differential Cross Section
In the first order perturbation theory the square of matrix element of the process (1) can be expressed as [17] ∑ Advances in High Energy Physics the interaction Hamiltonian for the emission of gamma rays with polarization  in the direction Ω  .Here [ 1 ] = (2 1 + 1) 1/2 .The magnetic and electric multipole operators T  1  and T  1  are defined in a system of coordinates       where -axis is oriented along the momentum ⃗  of the photon (Figure 3).
In order to define all the multipole operators in a system of coordinates where -axis is oriented along the transferred momentum ⃗  we perform a rotation of the system of coordinates       by angles   and   .So we obtain the relation [18]: By using ( 3) and ( 4) and applying the Wigner-Eckart theorem the matrix element of the Hamiltonian   can take the following form: So the matrix element for the nuclear transition from the initial state |    ⟩ to the intermediate state |    ⟩ is given by where are the leptonic and hadronic currents.  = 4/ 2  and   = −  / √ 2 are related, respectively, to the electromagnetic and weak interactions.  = (  − )  is the transferred 4-momentum,  is the fine structure constant, and   is the Fermi coupling constant for weak interaction.The constants   and   take specific values depending on the process.In the Weinberg-Salam model, for electron or muon scattering processes we have according to [19] The density of hadronic current Ĵℎ  is composed by vector and axial-vector currents with isoscalar ( = 0) and isovector ( = 1) components [18]: where  ()  and  ()  are the linking constants whose values depend on the process considered.For the electromagnetic interactions   = 0,  ()  = 1, and  ()  = 0 ( = 0, 1) and the hadronic current is given by In the case of processes with weak neutral currents,   = 0 ( = 0, 1) so the hadronic current is given by In the Weinberg-Salam model, for scattering processes according to [19] we have where   is the Weinberg angle.
The square of the matrix element (7) for electron electroweak scattering by nuclei takes the following form: where  =   /  .
In (13) we neglect the term proportional to  2 .In the ultrarelativistic longitudinally polarized electrons case we obtain the relation where the tensor ℓ  *  ℓ  ] is given by Here  and  (  and   ) are the energy and the 4-momentum of the initial (final) electron;   = ±1 is the helicity of the initial electron.The factor   takes the following values: The differential cross section, calculated by multipole decomposition of matrix elements [19,20] and by taking into account the circular polarization of the emitted photon, is given by the following formula: where Γ  →   and Γ

𝑛 → 𝑓
total indicate, respectively, the relative and total disintegration widths of the excited nucleus state and    (cos   ) is Legendre polynomial.The angles   and   determine the photon direction,   is photon helicity, and    is the differential cross section of the electron scattering by unpolarized nuclei given by where   and  rec are, respectively, the Mott cross section and the nuclear correction factor given by where  is the mass of the nucleus and  is the atomic number.
The functions    and R  are given by the following formulae: Here V  ( = ℓ, , . . .,   ) are the leptonic functions and    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 ≤  ≤ 2  .The functions  ( → +)  are defined as follows: for  even where Hence   =   is the gamma transition energy.

Study of the Transitions
As an example of transitions let us consider the following process: 12 C(0 + 0) +  RL e − + 12 C(0 + 0) → e − + 12 C * (1 + 1) So the quantum numbers , , and   are defined as The differential cross section of the process (24) which takes into account the longitudinal polarization of the electron and the circular polarization of the photon is given by the formula where The matrix elements  1 ,  5 1 ,  5 1 , and  5 1 , calculated in the shell model with a harmonic oscillator potential, are given by [21]: where  = −0.003[19],  =  1 + 2   2 ,  0 = −Δ is the energy of the transition, and  = (/2) 2 , where  is the parameter of the harmonic oscillator.Writing explicitly the electron and photon polarization, the differential cross section (26) takes the following form: where Let us consider now two coefficients of asymmetry which can be experimentally determined, that is, the electron polarization ratio and the photon circular polarization degree: Assuming that the photon is emitted in the direction of the incident electron we have and after averaging on the photon polarization, the asymmetry coefficient for the electron defined by ( 31) is given by Averaging the differential cross section (29) on the electron polarization, the photon polarization degree defined by (32) takes the form polarization degrees for three values of the energy for incident electron.Calculation is done by using sin 2   = 0.23125 from [8] and formulas (12) relative to the coupling constants of standard model.
Figure 1(a) shows for  = 300 MeV and  = 500 MeV variation of asymmetry coefficient.This variation is greater for electron scattering around value of angle  = 45 ∘ .However, the photon degree of polarization undergoes greater variation for large values of scattering angle.  and  (1)   .Electron asymmetry coefficient is very sensitive to change of these parameters, but it could be sensitive to the change of sin 2   which depend on these parameters in the standard model.

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.

Figures 1 (Figure 2 :
Figures 1(a) and 1(b) show the dependence on the diffusion angle  of the asymmetry coefficient for electron and photon