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

Abstract

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.

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 parity-violating 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: (1)

2. Differential Cross Section

In the first order perturbation theory the square of matrix element of the process (1) can be expressed as [17] _{𝑀𝑖𝑀𝑓}𝑀_𝑓𝑖²=12𝐽_𝑖+1×_{𝑀𝑖𝑀𝑓𝑀𝑛}𝐽_𝑓𝑀_𝑓𝐻_𝛾Ω_𝛾,𝜎𝐽_𝑛𝑀_𝑛×𝐽_𝑛𝑀_𝑛𝐻_𝑒𝑒′^→𝑞,𝑠_𝑒𝐽_𝑖𝑀_𝑖², (2) where 𝑀_𝑛𝑖 and 𝑀_𝑓𝑛 are the matrix elements of |𝐽_𝑖𝑀_𝑖⟩→|𝐽_𝑛𝑀_𝑛⟩ and |𝐽_𝑛𝑀_𝑛⟩→|𝐽_𝑓𝑀_𝑓⟩ transitions. 𝐽_𝑖, 𝐽_𝑛, and 𝐽_𝑓 are, respectively, the spins of the initial, intermediate (excited), and the final states of the nucleus. Let us consider 𝐻_𝛾Ω_𝛾,𝜎=−2𝜋𝑒√𝜔Ω_𝐽1≥1(−𝑖)^𝐽1𝐽₁𝜎𝑇^𝑚_𝐽1𝜎−𝑇^𝑒_𝐽1𝜎, (3) the interaction Hamiltonian for the emission of gamma rays with polarization 𝜎 in the direction Ω_𝛾. Here [𝐽₁]=(2𝐽₁+1)^1/2. The magnetic and electric multipole operators 𝑇^𝑚_𝐽1𝜎 and 𝑇^𝑒_𝐽1𝜎 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. So we obtain the relation [18]: 𝑇_𝐽^𝑚(𝑒)_1𝜎(𝑘)=_𝑀1𝑇_𝐽^𝑚(𝑒)_1𝑀1𝐷^𝐽1_𝑀1𝜎𝜑_𝛾,𝜃_𝛾,𝜑_𝛾. (4) By using (3) and (4) and applying the Wigner-Eckart theorem the matrix element of the Hamiltonian 𝐻_𝛾 can take the following form: 𝐽_𝑓𝑀_𝑓𝐻_𝛾Ω_𝛾,𝜎𝐽_𝑛𝑀_𝑛=−2𝜋𝑒√𝜔Ω×_{𝐽1𝑀≥11}(−𝑖)^𝐽1𝐽₁𝐷^𝐽1_𝑀1𝜎𝜑_𝛾,𝜃_𝛾,𝜑_𝛾×𝐽_𝑓−𝑀_𝑓𝐽₁𝑀₁𝐽_𝑛𝑀_𝑛𝐽_𝑓𝜎𝑇^𝑚_𝐽1𝜎−𝑇^𝑒_𝐽1𝜎𝐽_𝑛. (5) So the matrix element for the nuclear transition from the initial state |𝐽_𝑖𝑀_𝑖⟩ to the intermediate state |𝐽_𝑛𝑀_𝑛⟩ is given by 𝑀_𝑛𝑖=𝐽_𝑛𝑀_𝑛𝐻_𝑒𝑒′𝐽_𝑖𝑀_𝑖=𝐶_𝛾ℓ^𝛾_𝜇𝐽^𝛾_𝜇+𝐶_𝑊ℓ^𝑍_𝜇𝒥^𝑍_𝜇, (6) where ℓ^𝑍_𝜇=𝑢𝑝^′𝛾_𝜇𝑎_𝑉+𝑎_𝐴𝛾₅𝑢𝑝,𝒥_𝜇𝑞=𝐽_𝑛𝑀_𝑛𝑑𝑥^→exp−𝑖^→𝑞𝑥^→𝒥𝑥^ℎ_𝜇^→𝐽_𝑖𝑀_𝑖 (7)(8) are the leptonic and hadronic currents. 𝐶_𝛾=4𝜋𝛼/𝑞²_𝜇 and 𝐶_𝑊=−𝐺_𝐹/√2 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+4sin²𝜃_𝑊 and 𝑎_𝐴=−1.

The density of hadronic current 𝒥^ℎ_𝜇 is composed by vector and axial-vector currents with isoscalar (𝜏=0) and isovector (𝜏=1) components [18]: 𝒥^ℎ_𝜇=𝛽_𝑉^(𝜏)𝐽̂_{𝜇𝜏𝑀𝜏}+𝛽_𝐴^(𝜏)𝐽̂⁵_{𝜇𝜏𝑀𝜏}, (9) where 𝛽_𝑉^(𝜏) and 𝛽_𝐴^(𝜏) are the linking constants whose values depend on the process considered. For the electromagnetic interactions 𝑀_𝜏=0, 𝛽_𝑉^(𝜏)=1, and 𝛽_𝐴^(𝜏)=0(𝜏=0,1) and the hadronic current is given by 𝐽^𝛾_𝜇=𝐽_𝜇00+𝐽_𝜇10. (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_𝑉⁽⁰⁾𝐽_𝜇00+𝛽_𝐴⁽⁰⁾𝐽⁵_𝜇00+𝛽_𝑉⁽¹⁾𝐽_𝜇10+𝛽_𝐴⁽¹⁾𝐽⁵_𝜇10. (11) In the Weinberg-Salam model, for scattering processes according to [19] we have 𝛽_𝑉⁽⁰⁾=−2sin²𝜃_𝑊𝛽_𝐴⁽⁰⁾𝛽=0,_𝑉(1)=1−2sin²𝜃_𝑊𝛽_𝐴⁽¹⁾=1, (12) 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_𝑛𝑖²=𝐶²_𝛾ℓ_𝜇^𝛾∗ℓ^𝛾_𝜈𝐽_𝜇^𝛾∗𝐽^𝛾_𝜈+2𝜌Reℓ_𝜇^𝛾∗ℓ^𝑍_𝜈𝐽_𝜇^𝛾∗𝐽^𝑍_𝜈, (13) where 𝜌=𝐶_𝑊/𝐶_𝛾.

In (13) we neglect the term proportional to 𝜌². In the ultrarelativistic longitudinally polarized electrons case we obtain the relation ℓ_𝜇^𝛾∗ℓ^𝑍_𝜈=𝑎_𝑉−𝑎_𝐴𝑠_𝑒ℓ_𝜇^𝛾∗ℓ^𝛾_𝜈, (14) where the tensor ℓ_𝜇^𝛾∗ℓ^𝛾_𝜈 is given by ℓ_𝜇^𝛾∗ℓ^𝛾_𝜈=𝛿_𝜇2𝐸𝐸^′𝛿_𝜇𝜈𝑝𝑝^′−𝑝_𝜇𝑝^′_𝜈−𝑝_𝜈𝑝^′_𝜇−𝑠_𝑒𝜀_{𝜇𝜈𝛼𝛽}𝑝_𝛼𝑝^′_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: 𝛿_𝜇=−1for𝜇=1,2,3+1for𝜇=4. (16) 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: 𝑑𝜎𝑑Ω_𝑒′𝑑Ω_𝛾=Γ_𝛾^𝑛→𝑓Γ^𝑛→𝑓_Total×_𝜏𝜏′𝜏_𝑛𝜏𝜏_𝑖−𝑀_𝜏𝑛0𝑀_𝜏𝑖×𝜏_𝑛𝜏^′𝜏_𝑖−𝑀_𝜏𝑛0𝑀_𝜏𝑖2𝛼𝜎_𝑒𝑒′𝑊^𝑡₀𝑞_𝛾𝑓𝑛×11+𝑅⁰₀𝑞_𝑛𝑖×_𝐿≥2even𝑓_𝐿^{(𝑛→𝑓+𝛾)}𝑃_𝐿cos𝜃_𝛾𝑅⁰_𝐿+𝑃¹_𝐿cos𝜃_𝛾cos𝜑_𝛾𝑅¹_𝐿+𝑃²_𝐿cos𝜃_𝛾cos2𝜑_𝛾𝑅²_𝐿+𝑠_𝛾𝑅⁰₀𝑞_{𝑛𝑖𝐿≥1odd}𝑓_𝐿^{(𝑛→𝑓+𝛾)}𝑃_𝐿cos𝜃_𝛾𝑅⁰_𝐿+𝑃¹_𝐿cos𝜃_𝛾cos𝜑_𝛾𝑅¹_𝐿+𝑠_𝛾𝑅⁰₀𝑞_{𝑛𝑖𝐿≥3odd}𝑓_𝐿^{(𝑛→𝑓+𝛾)}𝑃²_𝐿cos𝜃_𝛾cos2𝜑_𝛾𝑅²_𝐿, (17) where Γ_𝛾^𝑛→𝑓 and Γ^𝑛→𝑓_total indicate, respectively, the relative and total disintegration widths of the excited nucleus state and 𝑃^𝑚_𝐿(cos𝜃_𝛾) is Legendre 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 𝜎_𝑒𝑒′=4𝜋𝜎_𝑀𝑅⁰₀𝑞_𝑛𝑖𝑓⁻¹_rec, (18) where 𝜎_𝑀 and 𝑓_rec are, respectively, the Mott cross section and the nuclear correction factor given by 𝜎_𝑀=𝑍𝛼cos(𝜃/2)2𝐸sin²(𝜃/2)²,𝑓_rec=1+2𝐸𝑀sin²𝜃2, (19) where 𝑀 is the mass of the nucleus and 𝑍 is the atomic number.

The functions 𝑅^𝑚_𝐿 and 𝑅^𝑚_𝐿 are given by the following formulae: 𝑅⁰_𝐿=𝑋₁𝑣_ℓ𝑊^ℓ_𝐿+𝑣_𝑡𝑊^𝑡_𝐿+𝑋₂𝑣_𝑡′𝑊^𝑎_𝐿,𝑅¹_𝐿=𝑋₂𝑣_𝑎𝑐𝑊_𝐿^𝑎𝑐+𝑋₃𝑣_𝑡ℓ′𝑊_𝐿^𝑎ℓ+𝑋₄𝑣_𝑡ℓ𝑊_𝐿^𝑡ℓ,𝑅²_𝐿=𝑋₁𝑣_𝑡𝑡𝑊_𝐿^𝑡𝑡,𝑅⁰_𝐿=𝑋₁𝑣_𝑡′𝑊^𝑡′_𝐿+𝑋₂𝑣_𝑡𝑊^𝑎′_𝐿+𝑣_ℓ′𝑊^ℓ′_𝐿,𝑅¹_𝐿=𝑋₂𝑣_𝑎𝑐′𝑊^𝑎𝑐′_𝐿+𝑋₃𝑣_𝑡ℓ𝑊^𝑎ℓ′_𝐿+𝑋₄𝑣_𝑡ℓ′𝑊^𝑡ℓ′_𝐿,𝑅²_𝐿=𝑋₂𝑣_𝑡𝑡𝑊^𝑡𝑡′_𝐿,𝑋₁=1+2𝜌𝛽^(𝜏′)_𝑉𝑎_𝑉−𝑎_𝐴𝑠_𝑒,𝑋₂=2𝜌𝛽^(𝜏′)_𝐴𝑎_𝑉−𝑎_𝐴𝑠_𝑒,𝑋₃=2𝜌𝛽_𝐴^(𝜏)𝑎_𝑉−𝑎_𝐴𝑠_𝑒,𝑋₄=1+𝜌𝛽_𝑉^(𝜏)+𝛽^(𝜏′)_𝑉𝑎_𝑉−𝑎_𝐴𝑠_𝑒. (20) Here 𝑣_𝑖(𝑖=ℓ,𝑡,…,𝑎𝑐^′) are the leptonic functions and 𝑊^𝑖_𝐿 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 𝐿 is defined by the relation 0≤𝐿≤2𝐽_𝑛. The functions 𝑓_𝐿^{(𝑛→𝑓+𝛾)} are defined as follows: 𝑓_𝐿^{(𝑛→𝑓+𝛾)}=𝑊^𝑡_𝐿𝑞_𝛾𝑊^𝑡₀𝑞_𝛾for𝐿even𝑊^𝑡′_𝐿𝑞_𝛾𝑊^𝑡₀𝑞_𝛾for𝐿odd, (21) where 𝑊^𝑡_𝐿𝑞_𝛾=−(−)^{𝐽𝑓+𝐽𝑛}×_{𝐽′1𝐽1}(2𝐿+1)2𝐽₁+12𝐽^′₁×𝐽+1_′1𝐽₁𝐿𝐽1−10^′1𝐽₁𝐿𝐽_𝑛𝐽_𝑛𝐽_𝑓×𝑃⁺_{𝐽′1+𝐽1}𝐹_𝐸𝐽′1𝐹_𝐸𝐽1+𝐹_𝑀𝐽′1𝐹_𝑀𝐽1+𝑃⁻_{𝐽′1+𝐽1}𝐹_𝐸𝐽′1𝐹_𝑀𝐽1−𝐹_𝑀𝐽′1𝐹_𝐸𝐽1. (22) Hence 𝑞_𝛾=𝐸_𝛾 is the gamma transition energy.

Quantum numbers 𝐽₁ and 𝐽^′₁ are given by 𝐽_𝑛−𝐽_𝑓≤𝐽₁≤𝐽_𝑛+𝐽_𝑓,𝐽_𝑛−𝐽_𝑓≤𝐽^′₁≤𝐽_𝑛+𝐽_𝑓,𝑊^𝑡′_𝐿𝑞_𝛾=𝑊^𝑡_𝐿𝑞_𝛾for𝐿odd,𝑃⁺_{𝐽′1+𝐽1}=12(−1)^{(1/2)(𝐽′1−𝐽1)}1+(−1)^{𝐽′1+𝐽1},𝑃⁻_{𝐽′1+𝐽1}=12(−1)^{(1/2)(𝐽′1−𝐽1+1)}1−(−1)^{𝐽′1+𝐽1}. (23)

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: (24)

So the quantum numbers 𝐿, 𝐽, and 𝐽^′ are defined as 0≤𝐿≤2𝐽_𝑛𝐽⟹0≤𝐿≤2;_𝑖−𝐽_𝑛≤𝐽≤𝐽_𝑖+𝐽_𝑛𝐽⟹𝐽=1;_𝑖−𝐽_𝑛≤𝐽^′≤𝐽_𝑖+𝐽_𝑛⟹𝐽^′=1. (25) 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 𝑑𝜎𝑑Ω_𝑒′𝑑Ω_𝛾=Σ₀𝑅⁰₀+𝑓₂^{(𝑛→𝑓+𝛾)}𝑃₂cos𝜃_𝛾𝑅⁰₂+𝑃¹₂cos𝜃_𝛾cos𝜑_𝛾𝑅¹₂+𝑃²₂cos𝜃_𝛾cos2𝜑_𝛾𝑅²₂+𝑠_𝛾𝑓₁^{(𝑛→𝑓+𝛾)}𝑃₁cos𝜃_𝛾𝑅⁰₁+𝑃¹₁cos𝜃_𝛾cos𝜑_𝛾𝑅¹₁, (26) where Σ₀=1√3𝐹²_𝑀1𝑞_𝛾𝜎_𝑒𝑒′,𝑓₁^{(𝑛→𝑓+𝛾)}√=−3√2,𝑓₂^{(𝑛→𝑓+𝛾)}=1√2,𝑅⁰₀𝑞=13√3𝐹_𝑀1×𝑣_𝑡𝐹_𝑀11+2𝜌𝛽_𝑉⁽¹⁾𝑎_𝑉−𝑎_𝐴𝑠_𝑒+2𝑣_𝑡′𝐹⁵_𝐸1𝜌𝛽_𝐴⁽¹⁾𝑎_𝑉−𝑎_𝐴𝑠_𝑒,𝑅⁰₂𝑞=1√2𝑅⁰₀𝑞,𝑅¹₂𝑞=13√3𝑣_𝑎𝑐𝐹_𝑀1𝜌𝛽_𝐴⁽¹⁾𝑎_𝑉−𝑎_𝐴𝑠_𝑒𝐹⁵_𝐶1−𝑞₀𝑞𝐹⁵_𝐿1,𝑅²₂𝑞1=−6√6𝑣_𝑡𝑡𝐹²_𝑀11+2𝜌𝛽_𝑉⁽¹⁾𝑎_𝑉−𝑎_𝐴𝑠_𝑒,𝑅⁰₁𝑞1=−3√2𝐹_𝑀1×𝑣_𝑡′𝐹_𝑀11+2𝜌𝛽_𝑉⁽¹⁾𝑎_𝑉−𝑎_𝐴𝑠_𝑒+2𝑣_𝑡𝐹⁵_𝐸1𝜌𝛽_𝐴⁽¹⁾𝑎_𝑉−𝑎_𝐴𝑠_𝑒,𝑅¹₁𝑞1=−3𝑣_𝑎𝑐′𝐹_𝑀1𝜌𝛽_𝐴⁽¹⁾𝑎_𝑉−𝑎_𝐴𝑠_𝑒𝐹⁵_𝐶1−𝑞₀𝑞𝐹⁵_𝐿1. (27) The matrix elements 𝐹_𝑀1, 𝐹⁵_𝐶1, 𝐹⁵_𝐿1, and 𝐹⁵_𝐸1, calculated in the shell model with a harmonic oscillator potential, are given by [21]: 𝐹_𝑀1=𝜓6√𝜋𝑞𝑀𝑒^−𝑦𝐹₁;𝐹−𝜇2−𝑦_5𝐶1𝜓=−3√𝑞2𝜋𝑀𝑒^−𝑦32𝐹_𝐴−1−𝑦𝑞₀𝐹_𝑝−2𝑀𝐹_𝑇;𝐹⁵_𝐿1√=−2𝜓3√𝜋𝑒^−𝑦1−𝑦𝐹_𝐴−𝑞²𝐹2𝑀^𝑝;𝐹⁵_𝐸1𝜓=−3√𝜋𝑒^−𝑦𝐹_𝐴2−𝑦, (28) where 𝜓=−0.003 [19], 𝜇=𝐹₁+2𝑀_𝑛𝐹₂, 𝑞₀=−Δ𝐸 is the energy of the transition, and 𝑦=(𝑏𝑞/2)², where 𝑏 is the parameter of the harmonic oscillator.

Writing explicitly the electron and photon polarization, the differential cross section (26) takes the following form: 𝑑𝜎𝑑Ω_𝑒′𝑑Ω_𝛾=13√3𝐹_𝑀1Σ₀Σ+𝑠_𝑒Δ+𝑠_𝛾Δ^′+𝑠_𝑒𝑠_𝛾Δ^′′, (29) where 1Σ=1+2𝑃₂cos𝜃_𝛾×𝑣_𝑡𝐹_𝑀11+2𝜌𝛽_𝑉⁽¹⁾𝑎_𝑉−2𝑣_𝑡̃𝐹⁵_𝐸1𝜌𝛽_𝐴⁽¹⁾𝑎_𝐴−1√2𝑃¹₂cos𝜃_𝛾cos𝜑_𝛾×𝑣̃_𝑎𝑐𝜌𝛽_𝐴⁽¹⁾𝑎_𝐴𝐹⁵_𝐶1−𝑞₀𝑞𝐹⁵_𝐿1−14𝑃²₂cos𝜃_𝛾cos2𝜑_𝛾𝑣_𝑡𝑡𝐹_𝑀11+2𝜌𝛽_𝑉⁽¹⁾𝑎_𝑉,Δ=𝜌2+𝑃₂cos𝜃_𝛾𝑣_𝑡̃𝐹⁵_𝐸1𝛽_𝐴⁽¹⁾𝑎_𝑉−𝑣_𝑡𝐹_𝑀1𝛽_𝑉⁽¹⁾𝑎_𝐴+1√2𝑃¹₂cos𝜃_𝛾cos𝜑_𝛾×𝑣̃_𝑎𝑐𝛽_𝐴⁽¹⁾𝑎_𝑉𝐹⁵_𝐶1−𝑞₀𝑞𝐹⁵_𝐿1+12𝑃²₂cos𝜃_𝛾cos2𝜑_𝛾𝑣_𝑡𝑡𝐹_𝑀1𝛽_𝑉⁽¹⁾𝑎_𝐴,Δ^′3=𝜌2𝑃₁cos𝜃_𝛾𝑣_𝑡𝐹⁵_𝐸1𝛽_𝐴⁽¹⁾𝑎_𝑉−𝑣_𝑡̃𝐹_𝑀1𝛽_𝑉⁽¹⁾𝑎_𝐴+√3𝑃¹₁cos𝜃_𝛾cos𝜑_𝛾×𝑣_𝑎𝑐′𝛽_𝐴⁽¹⁾𝑎_𝑉𝐹⁵_𝐶1−𝑞₀𝑞𝐹⁵_𝐿1,Δ^′′=34𝑃₁cos𝜃_𝛾𝑣_𝑡̃𝐹⁵_𝐸11+2𝜌𝛽_𝑉⁽¹⁾𝑎_𝑉−2𝑣_𝑡𝐹_𝑀1𝜌𝛽_𝐴⁽¹⁾𝑎_𝐴−√3𝑃¹₁cos𝜃_𝛾cos𝜑_𝛾𝑣_𝑎𝑐′𝜌𝛽_𝐴⁽¹⁾𝑎_𝐴𝐹⁵_𝐶1−𝑞₀𝑞𝐹⁵_𝐿1. (30) 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: 𝐴_𝑅,𝐿=𝑑𝜎𝑠_𝑒=+1−𝑑𝜎𝑠_𝑒=−1𝑑𝜎𝑠_𝑒=+1+𝑑𝜎𝑠_𝑒,=−1𝐵_𝑅,𝐿=𝑑𝜎𝑠_𝛾=+1−𝑑𝜎𝑠_𝛾=−1𝑑𝜎𝑠_𝛾=+1+𝑑𝜎𝑠_𝛾.=−1 (31)(32) Assuming that the photon is emitted in the direction of the incident electron we have cos𝜃_𝛾=𝐸^′cos𝜃−𝐸𝑞,𝜑_𝛾=0,sin𝜃_𝛾=𝐸^′sin𝜃𝑞, (33) and after averaging on the photon polarization, the asymmetry coefficient for the electron defined by (31) is given by 𝐴_𝑅,𝐿=ΔΣ. (34) Averaging the differential cross section (29) on the electron polarization, the photon polarization degree defined by (32) takes the form 𝐵_𝑅,𝐿=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 polarization degrees for three values of the energy for incident electron. Calculation is done by using sin²𝜃_𝑊=0.23125 from [8] and formulas (12) relative to the coupling constants of standard model.

(a)

(b)

Figure 1 

(a) Asymmetry coefficient for the electron. (b) Photon polarization degree.

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 2(a)–2(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 𝑎_𝐴 and 𝛽_𝐴⁽¹⁾. Electron asymmetry coefficient is very sensitive to change of these parameters, but it could be sensitive to the change of sin²𝜃_𝑊 which depend on these parameters in the standard model.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Asymmetry coefficient for the electron.

Figure 3 

System of coordinates.

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 electrons we obtain for the leptonic functions the following expressions: 𝑣_ℓ=𝑞²_𝜇𝑞²²,𝑣_𝑡=12𝑞²_𝜇𝑞²+tan²𝜃2,𝑣_𝑡𝑡1=−2𝑞²_𝜇𝑞²,𝑣_𝑡ℓ=1√2𝑞²_𝜇𝑞²𝑞²_𝜇𝑞²+tan²𝜃2^1/2,𝑣_𝑡ℓ′=1√2𝑠_𝑒𝑞²_𝜇𝑞²𝜃tan2,𝑣_ℓ′=𝑞²_𝜇𝑞²,𝑣_𝑡′=−𝑠_𝑒1√2𝑞²_𝜇𝑞²+tan²𝜃2^1/2𝜃tan2=𝑠_𝑒𝑣_𝑡̃,𝑣_𝑎𝑐=𝑠_𝑒𝜃tan2=𝑠_𝑒𝑣̃_𝑎𝑐,𝑣_𝑎𝑐′=𝑞²_𝜇𝑞²+tan²𝜃2^1/2. (A.1) The hadronic functions are given by the following formulae: 𝑊^ℓ_𝐿𝑞=_𝐽′𝐽𝐴^(𝐿)_0;0𝑃⁺_𝐽′+𝐽𝐹_𝐶𝐽′𝐹_𝐶𝐽,𝑊^𝑡_𝐿𝑞=−_𝐽′𝐽𝐴^(𝐿)_1;−1𝑃⁺_𝐽′+𝐽𝐹_𝐸𝐽′𝐹_𝐸𝐽+𝐹_𝑀𝐽′𝐹_𝑀𝐽−𝑃⁻_𝐽′+𝑗𝐹_𝐸𝐽′𝐹_𝑀𝐽−𝐹_𝑀𝐽′𝐹_𝐸𝐽,𝑊_𝐿^𝑡𝑡𝑞=−_𝐽′𝐽𝐴^(𝐿)_1;1𝑃⁺_𝐽′+𝐽𝐹_𝐸𝐽′𝐹_𝐸𝐽−𝐹_𝑀𝐽′𝐹_𝑀𝐽+𝑃⁻_𝐽′+𝑗𝐹_𝐸𝐽′𝐹_𝑀𝐽+𝐹_𝑀𝐽′𝐹_𝐸𝐽,𝑊_𝐿^𝑡ℓ=2√2_𝐽′𝐽𝐴^(𝐿)_0;1𝐹_𝐶𝐽′𝑃⁺_𝐽′+𝐽𝐹_𝐸𝐽+𝑃⁻_𝐽′+𝐽𝐹_𝑀𝐽,𝑊^𝑎_𝐿𝑞=−_𝐽′𝐽𝐴^(𝐿)_1;−1𝑃⁺_𝐽′+𝐽𝐹⁵_𝐸𝐽′𝐹_𝑀𝐽+𝐹⁵_𝑀𝐽′𝐹_𝐸𝐽+𝑃⁻_𝐽′+𝑗𝐹⁵_𝐸𝐽′𝐹_𝐸𝐽+𝐹⁵_𝑀𝐽′𝐹_𝑀𝐽,𝑊_𝐿^𝑎ℓ=2√2_𝐽′𝐽𝐴^(𝐿)_0;1𝐹_𝐶𝐽′𝑃⁺_𝐽′+𝐽𝐹⁵_𝑀𝐽+𝑃⁻_𝐽′+𝐽𝐹⁵_𝐸𝐽,𝑊_𝐿^𝑎𝑐=_𝐽′𝐽𝐴^(𝐿)_0;1𝐹⁵_𝐶𝐽′−𝑞₀𝑞𝐹⁵_𝐿𝐽′𝑃⁺_𝐽′+𝐽𝐹_𝑀𝐽−𝑃⁻_𝐽′+𝐽𝐹_𝐸𝐽,𝑊^ℓ′_𝐿=_𝐽′𝐽𝐴^(𝐿)_0;0𝑃⁻_𝐽′+𝐽𝐹⁵_𝐶𝐽′−𝑞₀𝑞𝐹⁵_𝐿𝐽′𝐹_𝐶𝐽,𝑊^𝑡𝑡′_𝐿𝑞=_𝐽′𝐽𝐴^(𝐿)_1;1𝑃⁺_𝐽′+𝐽𝐹⁵_𝐸𝐽′𝐹_𝑀𝐽−𝐹⁵_𝑀𝐽′𝐹_𝐸𝐽−𝑃⁻_𝐽′+𝑗𝐹⁵_𝐸𝐽′𝐹_𝐸𝐽+𝐹⁵_𝑀𝐽′𝐹_𝑀𝐽,𝑊^𝑡′_𝐿𝑞=𝑊^𝑡_𝐿𝑞;𝑊^𝑡ℓ′_𝐿𝑞=−𝑊_𝐿^𝑡ℓ𝑞;𝑊^𝑎′_𝐿𝑞=𝑊^𝑎_𝐿𝑞;𝑊^𝑎ℓ′_𝐿𝑞=−𝑊_𝐿^𝑎ℓ𝑞;𝑊^𝑎𝑐′_𝐿𝑞=−𝑊_𝐿^𝑎𝑐𝑞. (A.2) The coefficients 𝐴_𝑚^(𝐿)_′𝑚 and 𝑃^±_𝐽′+𝐽 are given by the following formulae: 𝐴_𝑚^(𝐿)_′𝑚=(−1)^{𝐽𝑖+𝐽𝑛}𝐽^′𝐽𝐿]()![][𝐿+𝑀(𝐿−𝑀)!^1/2×𝐽^′𝑚𝐽𝐿_′𝐽𝑚𝑀𝐽^′𝐽𝐿_𝑛𝐽_𝑛𝐽_𝑖,𝑃⁺_𝐽′+𝐽=12(−1)^{(1/2)(𝐽′−𝐽)}1+(−1)^𝐽′+𝐽,𝑃⁻_𝐽′+𝐽=12(−1)^{(1/2)(𝐽′−𝐽+1)}1−(−1)^𝐽′+𝐽. (A.3) The functions 𝐹_𝐶𝐽, 𝐹_𝐿𝐽, 𝐹_𝑀𝐽, and 𝐹_𝐸𝐽(𝐹⁵_𝐶𝐽, 𝐹⁵_𝐿𝐽, 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: 𝐹_𝐶𝐽=𝐽_𝑓𝑀^coul_𝐽𝐽_𝑖;𝐹_𝐿𝐽=𝐽_𝑓𝐿_𝐽𝐽_𝑖;𝐹_𝐸𝐽=𝐽_𝑓𝑇^el_𝐽𝐽_𝑖;𝐹_𝑀𝐽=𝐽_𝑓𝑖𝑇^𝑚_𝐽𝐽_𝑖;𝐹⁵_𝐶𝐽=𝐽_𝑓𝑖𝑀^coul5_𝐽𝐽_𝑖;𝐹⁵_𝐿𝐽=𝐽_𝑓𝑖𝐿⁵_𝐽𝐽_𝑖;𝐹⁵_𝐸𝐽=𝐽_𝑓𝑖𝑇^el5_𝐽𝐽_𝑖;𝐹⁵_𝑀𝐽=𝐽_𝑓𝑇_𝐽^𝑚5𝐽_𝑖. (A.4)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.