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

Authors: N. V. Samsonenko; Adamou Ousmane Manga; Almoustapha Aboubacar; Aboubacar Moussa
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

The differential cross section of electron inelastic scattering by nuclei followed byγ 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.


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## Body

## 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 θW [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]
(2)∑MiMf|Mfi|2=12Ji+1×∑MiMf|∑Mn〈JfMf|Hγ(Ωγ,σ)|JnMn〉hhhhhhhhhhh×〈JnMn|Hee′(q→,se)|JiMi〉∑Mn〈JfMf|Hγ(Ωγ,σ)|JnMn〉|2,
where Mni and Mfn are the matrix elements of |JiMi〉→|JnMn〉 and |JnMn〉→|JfMf〉 transitions. Ji, Jn, and Jf are, respectively, the spins of the initial, intermediate (excited), and the final states of the nucleus. Let us consider
(3)Hγ(Ωγ,σ)=-2πeωΩ∑J1≥1(-i)J1[J1](σT^J1σm-T^J1σe),
the interaction Hamiltonian for the emission of gamma rays with polarization σ in the direction Ωγ. Here [J1]=(2J1+1)1/2. The magnetic and electric multipole operators T^J1σm and T^J1σe 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 wherez-axis is oriented along the transferred momentum q→ we perform a rotation of the system of coordinates X′Y′Z′ by angles φγ and θγ. So we obtain the relation [18]:
(4)T^J1σm(e)(k)=∑M1T^J1M1m(e)DM1σJ1(φγ,θγ,φγ).
By using (3) and (4) and applying the Wigner-Eckart theorem the matrix element of the Hamiltonian Hγ can take the following form:
(5)〈JfMf|Hγ(Ωγ,σ)|JnMn〉=-2πeωΩ×∑J1≥1M1(-i)J1[J1]DM1σJ1(φγ,θγ,φγ)hhhhhhh×(Jf-MfJ1M1JnMn)〈Jf||σT^J1σm-T^J1σe||Jn〉.
So the matrix element for the nuclear transition from the initial state |JiMi〉 to the intermediate state |JnMn〉 is given by
(6)Mni=〈JnMn|H^ee′|JiMi〉=CγlμγJμγ+CWlμZJμZ,
where
(7)lμZ=u-(p′)γμ(aV+aAγ5)u(p),(8)Jμ(q)=〈JnMn|∫dx→exp⁡(-iq→x→)J^μh(x→)|JiMi〉
are the leptonic and hadronic currents. Cγ=4πα/qμ2 and CW=-GF/2 are related, respectively, to the electromagnetic and weak interactions. qμ=(p′-p)μ is the transferred 4-momentum, α is the fine structure constant, and GF is the Fermi coupling constant for weak interaction. The constants aV and aA take specific values depending on the process. In the Weinberg-Salam model, for electron or muon scattering processes we have according to [19] aV=-1+4sin2θW and aA=-1.The density of hadronic currentJ^μh is composed by vector and axial-vector currents with isoscalar (τ=0) and isovector (τ=1) components [18]:
(9)J^μh=βV(τ)(J^μ)τMτ+βA(τ)(J^μ5)τMτ,
where βV(τ) and βA(τ) are the linking constants whose values depend on the process considered. For the electromagnetic interactions Mτ=0, βV(τ)=1, and βA(τ)=0(τ=0,1) and the hadronic current is given by
(10)Jμγ=(Jμ)00+(Jμ)10.
In the case of processes with weak neutral currents, Mτ=0(τ=0,1) so the hadronic current is given by
(11)JμZ=βV(0)(Jμ)00+βA(0)(Jμ5)00+βV(1)(Jμ)10+βA(1)(Jμ5)10.
In the Weinberg-Salam model, for scattering processes according to [19] we have
(12)βV(0)=-2sin2θWβA(0)=0,βV(1)=1-2sin2θWβA(1)=1,
where θW is the Weinberg angle.The square of the matrix element (7) for electron electroweak scattering by nuclei takes the following form:
(13)|Mni|2=Cγ2{(lμγ*lνγ)(Jμγ*Jνγ)+2ρRe(lμγ*lνZJμγ*JνZ)},
where ρ=CW/Cγ.In (13) we neglect the term proportional to ρ2. In the ultrarelativistic longitudinally polarized electrons case we obtain the relation
(14)lμγ*lνZ=(aV-aAse)lμγ*lνγ,
where the tensor lμγ*lνγ is given by
(15)lμγ*lνγ=δμ2EE′{δμν(pp′)-pμpν′-pνpμ′-seεμναβpαpβ′}.
Here E and p (E′ and p′) are the energy and the 4-momentum of the initial (final) electron; se=±1 is the helicity of the initial electron. The factor δμ takes the following values:
(16)δμ={-1forμ=1,2,3+1forμ=4.
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:
(17)dσdΩe′dΩγ=Γγn→fΓTotaln→f×∑ττ′(τnττi-Mτn0Mτi)hhhhhhhh×(τnτ′τi-Mτn0Mτi)2ασee′W0t(qγ)fn×{∑L≥30ddfL(n→f+γ)PL2(cos⁡θγ)cos⁡2φγR~L21+1R00(q)nihhhhhi×∑L≥2evenfL(n→f+γ)(PL(cos⁡θγ)RL0hhhhhhhhhhhhhhhhhhh+PL1(cos⁡θγ)cos⁡φγRL1hhhhhhhhhhhhhhhhhhh+PL2(cos⁡θγ)cos⁡2φγRL2)hhhhhi+sγR00(q)ni∑L≥1oddfL(n→f+γ)(PL(cos⁡θγ)R~L0hhhhhhhhhhhhhhhhhhhhhhhhh+PL1(cos⁡θγ)cos⁡φγR~L1)hhhhhi+sγR00(q)ni∑L≥3oddfL(n→f+γ)PL2(cos⁡θγ)cos⁡2φγR~L2},
where Γγn→f and Γtotaln→f indicate, respectively, the relative and total disintegration widths of the excited nucleus state and PLm(cos⁡θγ) is Legendre polynomial. The angles θγ and φγ determine the photon direction, sγ is photon helicity, and σee′ is the differential cross section of the electron scattering by unpolarized nuclei given by
(18)σee′=4πσMR00(q)nifrec-1,
where σM and frec are, respectively, the Mott cross section and the nuclear correction factor given by
(19)σM=(Zαcos⁡(θ/2)2Esin2(θ/2))2,frec=1+2EMsin2θ2,
where M is the mass of the nucleus and Z is the atomic number.The functionsRLm and R~Lm are given by the following formulae:
(20)RL0=X1(vlWLl+vtWLt)+X2vt′WLa,RL1=X2vacWLac+X3vtl′WLal+X4vtlWLtl,RL2=X1vttWLtt,R~L0=X1vt′WLt′+X2(vtWLa′+vl′WLl′),R~L1=X2vac′WLac′+X3vtlWLal′+X4vtl′WLtl′,R~L2=X2vttWLtt′,X1=1+2ρβV(τ′)(aV-aAse),X2=2ρβA(τ′)(aV-aAse),X3=2ρβA(τ)(aV-aAse),X4=(1+ρ(βV(τ)+βV(τ′))(aV-aAse)).
Here vi(i=l,t,…,ac′) are the leptonic functions and WLi 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≤2Jn. The functions fL(n→f+γ) are defined as follows:
(21)fL(n→f+γ)={WLt(qγ)W0t(qγ)forLevenWLt′(qγ)W0t(qγ)forLodd,
where
(22)WLt(qγ)=-(-)Jf+Jn×∑J1′J1(2L+1)(2J1+1)(2J1′+1)hhhhhhhh×(J1′J1L1-10){J1′J1LJnJnJf}×{PJ1′+J1+(FEJ1′FEJ1+FMJ1′FMJ1)hhhhhh+PJ1′+J1-(FEJ1′FMJ1-FMJ1′FEJ1)}.
Hence qγ=Eγ is the gamma transition energy.Quantum numbersJ1 and J1′ are given by
(23)|Jn-Jf|≤J1≤Jn+Jf,|Jn-Jf|≤J1′≤Jn+Jf,WLt′(qγ)=WLt(qγ)forLodd,PJ1′+J1+=12(-1)(1/2)(J1′-J1)(1+(-1)J1′+J1),PJ1′+J1-=12(-1)(1/2)(J1′-J1+1)(1-(-1)J1′+J1).


## 3. Study of the TransitionsJi=0→Jn=1→Jf=0

As an example of transitions let us consider the following process:(24)So the quantum numbersL, J, and J′ are defined as
(25)0≤L≤2Jn⟹0≤L≤2;|Ji-Jn|≤J≤Ji+Jn⟹J=1;|Ji-Jn|≤J′≤Ji+Jn⟹J′=1.
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
(26)dσdΩe′dΩγ=Σ0{R00+f2(n→f+γ)[P2(cos⁡θγ)R20+P21(cos⁡θγ)cos⁡φγR21hhhhhhhhhhhhhhhhhh+P22(cos⁡θγ)cos⁡2φγR22]hhhhhhh+sγf1(n→f+γ)[P1(cos⁡θγ)R~10hhhhhhhhhhhhhhhhhhh+P11(cos⁡θγ)cos⁡φγR~11]},
where
(27)Σ0=13FM12(qγ)σee′,f1(n→f+γ)=-32,f2(n→f+γ)=12,R00(q)=133FM1×[vtFM1(1+2ρβV(1)(aV-aAse))hhhhhh+2vt′FE15ρβA(1)(aV-aAse)],R20(q)=12R00(q),R21(q)=133vacFM1ρβA(1)(aV-aAse)(FC15-q0qFL15),R22(q)=-166vttFM12(1+2ρβV(1)(aV-aAse)),R~10(q)=-132FM1×[vt′FM1(1+2ρβV(1)(aV-aAse))hhhhh+2vtFE15ρβA(1)(aV-aAse)],R~11(q)=-13vac′FM1ρβA(1)(aV-aAse)(FC15-q0qFL15).
The matrix elements FM1, FC15, FL15, and FE15, calculated in the shell model with a harmonic oscillator potential, are given by [21]:
(28)FM1=ψ6πqMe-y(F1-μ(2-y));FC15=-ψ32πqMe-y(32FA-(1-y)(q0Fp-2MFT));FL15=-2ψ3πe-y(1-y)(FA-q22MFp);FE15=-ψ3πe-yFA(2-y),
where ψ=-0.003 [19], μ=F1+2MnF2, 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:
(29)dσdΩe′dΩγ=133FM1Σ0(Σ+seΔ+sγΔ′+sesγΔ′′),
where
(30)Σ=(1+12P2(cos⁡θγ))×(vtFM1(1+2ρβV(1)aV)-2v~tFE15ρβA(1)aA)-12P21(cos⁡θγ)cos⁡φγ×v~acρβA(1)aA(FC15-q0qFL15)-14P22(cos⁡θγ)cos⁡2φγvttFM1(1+2ρβV(1)aV),Δ=ρ{12(2+P2(cos⁡θγ))(v~tFE15βA(1)aV-vtFM1βV(1)aA)hhhhh+12P21(cos⁡θγ)cos⁡φγhhhhh×v~acβA(1)aV(FC15-q0qFL15)hhhhh+12P22(cos⁡θγ)cos⁡2φγvttFM1βV(1)aA},Δ′=ρ{(FC15-q0qFL15)32P1(cos⁡θγ)(vtFE15βA(1)aV-v~tFM1βV(1)aA)hhhhh+3P11(cos⁡θγ)cos⁡φγhhhhh×vac′βA(1)aV(FC15-q0qFL15)},Δ′′=34P1(cos⁡θγ)(v~tFE15(1+2ρβV(1)aV)-2vtFM1ρβA(1)aA)-3P11(cos⁡θγ)cos⁡φγvac′ρβA(1)aA(FC15-q0qFL15).
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:
(31)AR,L=dσ(se=+1)-dσ(se=-1)dσ(se=+1)+dσ(se=-1),(32)BR,L=dσ(sγ=+1)-dσ(sγ=-1)dσ(sγ=+1)+dσ(sγ=-1).
Assuming that the photon is emitted in the direction of the incident electron we have
(33)cos⁡θγ=E′cos⁡θ-Eq,φγ=0,sinθγ=E′sinθq,
and after averaging on the photon polarization, the asymmetry coefficient for the electron defined by (31) is given by
(34)AR,L=ΔΣ.
Averaging the differential cross section (29) on the electron polarization, the photon polarization degree defined by (32) takes the form
(35)BR,L=Δ′Σ.
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 sin2θW=0.23125 from [8] and formulas (12) relative to the coupling constants of standard model.(a) Asymmetry coefficient for the electron. (b) Photon polarization degree.
(a)
(b)Figure1(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.Figures2(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 aA and βA(1). Electron asymmetry coefficient is very sensitive to change of these parameters, but it could be sensitive to the change of sin2θW which depend on these parameters in the standard model.Asymmetry coefficient for the electron.
(a)
(b)
(c)
(d)
(e)
(f)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.



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