Polarization observables in elastic lepton-deuteron scattering and their sensitivity to realistic NN potentials

Abstract

Polarization effects in the elastic lepton scattering on the deuteron process are studied in the one-photon-exchange Born approximation within the limit of zero mass of the lepton. Numerical estimations for the spin asymmetries with an unpolarized lepton beam and a tensor polarized target are given. The estimated results are analysed at various lepton beam energies and scattering angles. For the first time, the sensitivity of the results for tensor spin asymmetries to the choice of modern NN potential used for the deuteron wave function (DWF) is investigated. A considerable influence of the results on the realistic and modern DWFs at incident beam energies greater than 0.6 GeV and scattering angles greater than 30${}^{\circ }$ is obtained.

Keywords:

• Electromagnetic processes and properties
• elastic scattering
• electromagnetic form factors
• nucleon-nucleon interactions
• polarization phenomena in reactions

PACS numbers:

• 13.40.-f
• 13.85.Dz
• 13.40.Gp
• 13.70.Cs
• 24.70.+s

1. Introduction

Elastic lepton scattering on nuclei gives a very strong tool to investigate the internal hadron's structure and obtain valuable information about the structure of nucleons and nuclei. This reaction provides useful information about spin and electromagnetic structures of the neutron and proton. Thus, the investigation of spin observables in elastic scattering of leptons on nucleons and nuclei is one of the interesting topics in nuclear physics. This topic is still motivating and fascinating for researchers and deserves more investigations.

The study of light nuclei is one of the important points of interest in hadronic physics during the past decades [1–4]. For instance, the determination of the radii of light nuclei in elastic muon scattering on light nuclei is one of the main goals of the muon-proton scattering experiment (MUSE) [5–8]. Of particular interest is the experimental study on a few-nucleon scattering system, which is an attractive probe. This system plays a substantial role since the wave function of the few-nucleon system can be calculated without approximation by adapting a realistic nucleon-nucleon (NN) potential [9,10]. In particular, the lepton-deuteron elastic scattering process gives abundant information about the internal deuteron structure and hence on the NN potential.

The effects of radiative corrections on polarization observables in neutrino reactions off proton and deuteron were investigated in Refs. [11–17]. This study is closely related to the determination of the axial-vector coupling constant $g_A$, which receives radiative corrections of a constant term. It is well known that radiative corrections are particularly important for high resolution experiments. There are several relevant mechanisms such as the one-photon exchange mechanism and the two-photon contributions that can be considered in the analysis of radiative corrections. The interference between these mechanisms plays an important role.

The elastic lepton-deuteron scattering with polarization effects beyond the one-photon-exchange Born approximation (OPEA) was studied in Refs. [18–28]. Contributions from meson-exchange currents, retardation effects, two (or more) photon exchange, isobar configuration in the deuteron wave function (DWF), and relativistic (spin-orbit, Darwin-Foldy, and nuclear motion) corrections were examined. The influence of these various terms on the differential cross section and on polarization observables was presented.

Polarization effects in elastic lepton scattering on light nuclei have been the subject of many theoretical studies [17,29–34]. For instance, the elastic lepton-deuteron scattering with a polarized lepton and polarized deuteron in the OPEA was studied in Ref. [29]. The authors of Ref. [30] have reanalysed the existing, at that time, experimental data on the charge asymmetry of the lepton in both the elastic and the inelastic scattering of lepton on the nucleon and nuclei. The unpolarized and polarized cross sections were performed in Refs. [17,31,32]. Numerical estimations for spin correlation coefficients (SCCs) in the zero lepton mass limit were evaluated in Ref. [33] using the Reid-93 NN potential [35]. Numerical estimations for the SCCs in elastic lepton-deuteron scattering due to lepton beam and vector target polarizations in the OPEA were given in Ref. [34]. The sensitivity of the obtained results to the realistic NN potential employed for the DWF was investigated for the first time, and a considerable dependence was found at incident beam energies greater than 0.4 GeV and backward scattering angles.

We would like to emphasize that none of the works in Refs. [17,29–34] investigate the sensitivity of tensor target spin asymmetries (SAs) in elastic scattering of leptons on the deuteron to the choice of the realistic NN potential employed for the DWF. Indeed, the tensor SAs were chosen as a good tool for investigating the NN interaction at short distances [36–38]. This makes it possible to choose among different models of the NN interaction.

Therefore, we focus our attention in this work on tensor target SAs in elastic lepton-deuteron scattering in the OPEA neglecting the lepton mass. For the first time, we discuss the sensitivity of the obtained results for tensor SAs to the realistic NN potential employed for the DWF. For this study, we consider in the present work the realistic and high-precision Nijmegen-II [35], Argonne v18 [39], CD-Bonn [40], and Bonn-Q [41] NN potentials. These NN potentials are exceedingly employed for numerical estimations of electromagnetic reactions on the deuteron, give a precise characterization of the NN scattering data and phase shifts, and used to characterize the range of the NN interaction. The standard dipole parameterization for the nucleon form factors (NFFs) from Ref. [42] is considered in this work.

This paper is outlined as follows: in Section 2 we briefly describe the formalism for the elastic scattering of leptons on the deuteron in the OPEA with neglecting lepton mass. The numerical estimations of the tensor SAs are illustrated and discussed in Section 3. The last Section 4 is devoted to conclusions and outlook.

2. Formalism

Here, we briefly summarize the formalism for elastic scattering of leptons on the deuteron in the OPEA neglecting the lepton mass. This process can be written as (1) $\begin{array}{r}\ell (k_1)+d(p_1)\to \ell (k_2)+d(p_2),\ell =e,\mu ,\tau ,\end{array}$(1) where $k_2$ ($p_2$) and $k_1$ ($p_1$) are the four-momenta of the lepton (deuteron) in the final and initial states, respectively. In the present work, we perform our analysis in the laboratory frame. In this frame, the components of the four-momenta of the lepton and deuteron in the initial and final states are given by (2) $\begin{array}{r}\begin{array}{rl}& k_1=({\epsilon }_1,{\overrightarrow{k}}_1),k_2=({\epsilon }_2,{\overrightarrow{k}}_2),\\ & p_1=(M_d,0),p_2=(E_2,{\overrightarrow{p}}_2),\end{array}\end{array}$(2) where ${\epsilon }_1$ (${\epsilon }_2$) denotes the incident (scattered) lepton energy, $E_2$ is the energy of the final deuteron, and $M_d$ is the deuteron mass. Furthermore, the energy ${\epsilon }_2$ of the lepton in the final state with zero mass is given by (3) $\begin{array}{r}{\epsilon }_2={\epsilon }_1{(1+\frac{2{\epsilon }_1}{M_d}{\sin }^2\left(\frac{\theta }{2}\right))}^{-1},\end{array}$(3) where θ is the lepton scattering angle. The relation between $Q^2$ and ${\epsilon }_1$ is given by (4) $\begin{array}{r}{Q}^2=-q^2=4{\epsilon }_1^2{\sin }^2\left(\frac{\theta }{2}\right){(1+\frac{2{\epsilon }_1}{M_d}{\sin }^2\left(\frac{\theta }{2}\right))}^{-1}.\end{array}$(4) Thus, the incident lepton energy ${\epsilon }_1$ is given by (5) $\begin{array}{rl}{\epsilon }_1& =\frac{1}{4M_d}[q^2+{\csc }^2\left(\frac{\theta }{2}\right)\\ & \times \sqrt{q^4{\sin }^4\left(\frac{\theta }{2}\right)+4q^2{\sin }^2\left(\frac{\theta }{2}\right)M_d^2}].\end{array}$(5) In the OPEA, the scattering matrix elements of elastic scattering of leptons on the deuteron are given by (6) $\begin{array}{r}\mathcal{M}=\frac{e^2}{Q^2}{j}_{\mu }{J}_{\mu },j_{\mu }=\overline{u}(k_2){\gamma }_{\mu }u(k_1).\end{array}$(6) Following Refs. [17,31,32], the electromagnetic current for the spin-1 deuteron is given by (7) $\begin{array}{rl}{J}_{\mu }& =(p_1+p_2{)}_{\mu }[-G_1(Q^2)U_1\cdot U_2^{\ast }\\ & +\frac{G_3(Q^2)}{M_d^2}(U_1\cdot q{U}_2^{\ast }\cdot q-\frac{q^2}{2}{U}_1\cdot U_2^{\ast })]\\ & +G_2(Q^2)(U_{1\mu }{U}_2^{\ast }\cdot q-U_{2\mu }^{\ast }{U}_1\cdot q),\end{array}$(7) where $q=k_1-k_2=p_2-p_1$, $U_{1\mu }$ denotes the polarization four-vector for the deuteron in the initial state, and $U_{2\mu }$ is the polarization four-vector for the deuteron in the final state. The functions $G_1(Q^2)$, $G_2(Q^2)$, and $G_3(Q^2)$ denote the deuteron form factors (DFFs) which are connected to the standard deuteron charge monopole $G_C$, charge quadrupole $G_Q$, and magnetic dipole $G_M$ form factors by (8) $\begin{array}{r}\begin{array}{rl}{G}_C& =\frac{2}{3}\tau (G_2-G_3)+(1+\frac{2}{3}\tau )G_1,\\ G_Q& =G_1+G_2+2G_3,\\ G_M& =-G_2,\end{array}\end{array}$(8) with $\tau =Q^2/(4M_d^2)$.

To calculate the unpolarized cross sections and polarization observables, one needs information about the DFFs. In this work, we use the parameterization from Refs. [43,44] which reproduces well the experimental data. In this parameterization, the DFFs $G_C$, $G_Q$, and $G_M$ are given by (9) $\begin{array}{r}\begin{array}{rl}{G}_C& =G_E^S{C}_E,\\ G_Q& =G_E^S{C}_Q,\\ G_M& =2G_M^S{C}_S+G_E^S{C}_L,\end{array}\end{array}$(9) where $G_E^S=G_E^p+G_E^n$ ($G_M^S=G_M^p+G_M^n$) gives the charge (magnetic) isoscalar NFF and $G_{E,M}^{p,n}$ denote the electric and magnetic FFs of the proton and the neutron. The nonrelativistic formulas for the structure functions $C_E$, $C_Q$, $C_S$, and $C_L$, are calculated using the ${}^3{S}_1$- and ${}^3{D}_1$-state DWFs, $u(r)$ and $w(r)$, respectively (10) $\begin{array}{rl}{C}_E& ={\int }_0^{\mathrm{\infty }}\mathrm{d}r{j}_0\left(\frac{qr}{2}\right)[u^2(r)+w^2(r)],\end{array}$(10) (11) $\begin{array}{rl}{C}_Q& =\frac{3}{\sqrt{2}\tau }{\int }_0^{\mathrm{\infty }}\mathrm{d}r{j}_2\left(\frac{qr}{2}\right)[u(r)-\frac{w(r)}{\sqrt{8}}]w(r),\end{array}$(11) (12) $\begin{array}{rl}{C}_S& ={\int }_0^{\mathrm{\infty }}\mathrm{d}r\{[u^2(r)-\frac{1}{2}{w}^2(r)]j_0\left(\frac{qr}{2}\right)\\ & +\frac{1}{2}[\sqrt{2}u(r)w(r)+w^2(r)]j_2\left(\frac{qr}{2}\right)\},\end{array}$(12) (13) $\begin{array}{rl}{C}_L& =\frac{3}{2}{\int }_0^{\mathrm{\infty }}\mathrm{d}r{w}^2(r)[j_0\left(\frac{qr}{2}\right)+j_2\left(\frac{qr}{2}\right)],\end{array}$(13) with ${\int }_0^{\mathrm{\infty }}\mathrm{d}r[u^2(r)+w^2(r)]=1$ and $j_0(x)$ ($j_2(x)$) denotes the spherical Bessel function or order zero (two). For the calculation of the structure functions $C_E$, $C_Q$, $C_S$, and $C_L$, we adapt the realistic and high-precision Nijmegen-II [35], Argonne v18 [39], CD-Bonn [40], and Bonn-Q [41] NN potentials for the DWFs.

At $Q^2=0$, the standard DFFs are given by (14) $\begin{array}{r}\begin{array}{rl}{G}_C(Q^2=0)& =1,\\ G_Q(Q^2=0)& =M_d^2{Q}_d,\\ G_M(Q^2=0)& =(M_d/M_N){\mu }_d,\end{array}\end{array}$(14) where $M_N$ denotes the nucleon mass and $Q_d$ (${\mu }_d$) is the static deuteron charge quadrupole (magnetic dipole) moments.

In the OPEA for the characterization of lepton-deuteron elastic scattering, the lepton interacts with each nucleon in the deuteron through a virtual photon and the FFs of the active nucleon are considered to be the same as those for a free nucleon. There exist several models for the nucleon structure [42,45–57]. For the proton and neutron FFs $G_{E,M}^{p,n}$, we use the standard dipole fit (DFF) [42]. It is given by (15) $\begin{array}{r}{G}_D={(1+\frac{Q^2}{{\mathrm{\Lambda }}^2})}^{-2},\end{array}$(15) where ${\mathrm{\Lambda }}^2=0.71$ (GeV/c)${}^2$. The electric FFs of the proton (neutron) is given by $G_E^p=G_D$ ($G_E^n=0$), whereas the magnetic FFs of the proton (neutron) is given by the dipole parameterizations $G_M^p={\mu }_p{G}_E^p$ ($G_M^n={\mu }_n{G}_E^p$), with ${\mu }_p=2.7928$ and ${\mu }_n=-1.9130$. We would like to point out that the standard dipole parameterization is adopted in the present work for simplicity. The deviation from this approximation in the considered kinematical region may not be very large. Nevertheless, it is well known that the effect of the deviation of the neutron electric form factor $G_E^n(Q^2)$ from this approximation at large momentum transfer is more realistic [53].

The unpolarized differential cross section of elastic scattering of leptons on the deuteron in OPEA without the lepton mass is given in the laboratory frame by (16) $\begin{array}{r}\frac{\mathrm{d}{\sigma }_0}{\mathrm{d}\mathrm{\Omega }}={\left(\frac{\mathrm{d}\sigma }{\mathrm{d}\mathrm{\Omega }}\right)}_{Mott}\mathcal{S},\end{array}$(16) where (17) $\begin{array}{r}\mathcal{S}=A(Q^2)+B(Q^2){\tan }^2\left(\frac{\theta }{2}\right).\end{array}$(17) The Mott cross section is given by (18) $\begin{array}{r}{\left(\frac{\mathrm{d}\sigma }{\mathrm{d}\mathrm{\Omega }}\right)}_{Mott}=\frac{{\alpha }^2{\cos }^2\left(\frac{\theta }{2}\right)}{4{\epsilon }_1^2{\sin }^4\left(\frac{\theta }{2}\right)}{(1+2\frac{{\epsilon }_1}{M_d}{\sin }^2\left(\frac{\theta }{2}\right))}^{-1},\end{array}$(18) where $\alpha \simeq 1/137$. The structure functions $A(Q^2)$ and $B(Q^2)$ are given by (19) $\begin{array}{r}\begin{array}{rl}A(Q^2)& =G_C^2(Q^2)+\frac{8}{9}{\tau }^2{G}_Q^2(Q^2)+\frac{2}{3}\tau G_M^2(Q^2),\\ B(Q^2)& =\frac{4}{3}\tau (1+\tau )G_M^2(Q^2).\end{array}\end{array}$(19) By neglecting the lepton mass, $A(Q^2)$ and $B(Q^2)$ can be determined by measuring the unpolarized differential cross section at various values of θ and the same value of $Q^2$. Thus, one can calculate the magnetic form factor $G_M(Q^2)$ and the set $G_C^2(Q^2)+8{\tau }^2{G}_Q^2(Q^2)/9$ of the charge and quadrupole FFs. To separate the charge $G_C$ and quadrupole $G_Q$ FFs, the measurement of another observable is required. This observable must be a polarization observable. The determination of polarization observables seeks us to choose a specific system of coordinate. As in Ref. [31,32], we use in this work the laboratory frame in which the z axis is directed along ${\overrightarrow{k}}_1$. In this frame, the x-axis is chosen in order to form a left-handed coordinate system and the y axis is directed along the vector ${\overrightarrow{k}}_1\times {\overrightarrow{k}}_2$. Thus, the xz plane represents the reaction plane.

The present work focuses on the SAs which are caused by tensor polarized deuteron target. The differential cross section can be written as [31,32] (20) $\begin{array}{r}\frac{\mathrm{d}\sigma }{\mathrm{d}\mathrm{\Omega }}=\frac{\mathrm{d}{\sigma }_0}{\mathrm{d}\mathrm{\Omega }}[1+A_{xx}^{(0)}(Q_{xx}-Q_{yy})+A_{xz}^{(0)}{Q}_{xz}+A_{zz}^{(0)}{Q}_{zz}],\end{array}$(20) where $A_{ij}^{(0)}$ are the tensor SAs. As in Refs. [31,32], we considered that the tensor $Q_{ij}$ is symmetrical and traceless, i.e. $Q_{xx}+Q_{yy}+Q_{zz}=0$. Neglecting the lepton mass, the $A_{ij}^{(0)}$ asymmetries as functions of the DFFs are given by (21) $\begin{array}{rl}{A}_{xx}^{(0)}& =\frac{\tau }{2\mathcal{S}}\{(1+\frac{\tau M_d^2}{{\epsilon }_1^2})G_M^2\\ & +\frac{4G_Q}{1+\tau }[\tau (1+\frac{M_d}{{\epsilon }_1})(1-\tau \frac{M_d}{{\epsilon }_1})G_M\\ & +(1-\frac{\tau M_d^2}{{\epsilon }_1^2}-\frac{2\tau M_d}{{\epsilon }_1})(G_C+\frac{\tau }{3}{G}_Q)]\},\end{array}$(21) (22) $\begin{array}{rl}{A}_{xz}^{(0)}& =\frac{-1}{\mathcal{S}}\frac{{\epsilon }_2}{M_d}\frac{\tau \sin \theta }{1+\tau }\{(4+\frac{4M_d}{{\epsilon }_1})(G_C{G}_Q+\frac{\tau }{3}{G}_Q^2)\\ & +(1+\tau )(1+\frac{M_d}{{\epsilon }_1}){\tan }^2\left(\frac{\theta }{2}\right)G_M^2\\ & +2(1-\tau \frac{M_d}{{\epsilon }_1})\\ & \times [-1-\tau +2{\sin }^2\left(\frac{\theta }{2}\right)(1+\frac{{\epsilon }_1}{M_d}\\ & +\frac{{\epsilon }_1^2}{M_d^2}-\tau \frac{{\epsilon }_1}{M_d})](1+{\tan }^2\left(\frac{\theta }{2}\right))G_M{G}_Q\},\end{array}$(22) (23) $\begin{array}{rl}{A}_{zz}^{(0)}& =\frac{-\tau }{2\mathcal{S}}\{[\frac{6\tau }{1+\tau }\frac{{\epsilon }_1+{\epsilon }_2}{{\epsilon }_1}(1+\frac{M_d}{{\epsilon }_1})G_Q-G_M]G_M\\ & +{\tan }^2\left(\frac{\theta }{2}\right)[1-2\tau -6\tau \frac{M_d}{{\epsilon }_1}(1+\frac{M_d}{2{\epsilon }_1})]\\ & \times [G_M^2+\frac{4}{1+\tau }{\cot }^2\left(\frac{\theta }{2}\right)G_Q(G_C+\frac{\tau }{3}{G}_Q)]\}.\end{array}$(23) Thus, we consider in the present work three tensor SAs in lepton-deuteron elastic scattering. These are the $A_{xx}^{(0)}$, $A_{xz}^{(0)}$, and $A_{zz}^{(0)}$ asymmetries which are due to an unpolarized incident lepton and a tensor polarized target. The explicit expressions for these tensor target asymmetries are given in terms of the DFFs in Equations (21(21) $\begin{array}{rl}{A}_{xx}^{(0)}& =\frac{\tau }{2\mathcal{S}}\{(1+\frac{\tau M_d^2}{{\epsilon }_1^2})G_M^2\\ & +\frac{4G_Q}{1+\tau }[\tau (1+\frac{M_d}{{\epsilon }_1})(1-\tau \frac{M_d}{{\epsilon }_1})G_M\\ & +(1-\frac{\tau M_d^2}{{\epsilon }_1^2}-\frac{2\tau M_d}{{\epsilon }_1})(G_C+\frac{\tau }{3}{G}_Q)]\},\end{array}$(21) ), (22(22) $\begin{array}{rl}{A}_{xz}^{(0)}& =\frac{-1}{\mathcal{S}}\frac{{\epsilon }_2}{M_d}\frac{\tau \sin \theta }{1+\tau }\{(4+\frac{4M_d}{{\epsilon }_1})(G_C{G}_Q+\frac{\tau }{3}{G}_Q^2)\\ & +(1+\tau )(1+\frac{M_d}{{\epsilon }_1}){\tan }^2\left(\frac{\theta }{2}\right)G_M^2\\ & +2(1-\tau \frac{M_d}{{\epsilon }_1})\\ & \times [-1-\tau +2{\sin }^2\left(\frac{\theta }{2}\right)(1+\frac{{\epsilon }_1}{M_d}\\ & +\frac{{\epsilon }_1^2}{M_d^2}-\tau \frac{{\epsilon }_1}{M_d})](1+{\tan }^2\left(\frac{\theta }{2}\right))G_M{G}_Q\},\end{array}$(22) ), and (23(23) $\begin{array}{rl}{A}_{zz}^{(0)}& =\frac{-\tau }{2\mathcal{S}}\{[\frac{6\tau }{1+\tau }\frac{{\epsilon }_1+{\epsilon }_2}{{\epsilon }_1}(1+\frac{M_d}{{\epsilon }_1})G_Q-G_M]G_M\\ & +{\tan }^2\left(\frac{\theta }{2}\right)[1-2\tau -6\tau \frac{M_d}{{\epsilon }_1}(1+\frac{M_d}{2{\epsilon }_1})]\\ & \times [G_M^2+\frac{4}{1+\tau }{\cot }^2\left(\frac{\theta }{2}\right)G_Q(G_C+\frac{\tau }{3}{G}_Q)]\}.\end{array}$(23) ).

3. Results and discussion

Now, we give numerical estimations for the tensor SAs $A_{xx}^{(0)}$, $A_{xz}^{(0)}$, and $A_{zz}^{(0)}$ in the elastic scattering of leptons on the deuteron ignoring the lepton mass. We present results for these asymmetries as functions of ${ϵ}_1$ and θ using the DFF for NFFs [42]. To explore the sensitivity of the obtained results for these SAs to the DWF, the realistic Nijmegen-II [35], Argonne v18 [39], CD-Bonn [40], and Bonn-Q [41] NN potentials are considered. We present the results for the tensor SAs $A_{xx}^{(0)}$ (see Equation (21(21) $\begin{array}{rl}{A}_{xx}^{(0)}& =\frac{\tau }{2\mathcal{S}}\{(1+\frac{\tau M_d^2}{{\epsilon }_1^2})G_M^2\\ & +\frac{4G_Q}{1+\tau }[\tau (1+\frac{M_d}{{\epsilon }_1})(1-\tau \frac{M_d}{{\epsilon }_1})G_M\\ & +(1-\frac{\tau M_d^2}{{\epsilon }_1^2}-\frac{2\tau M_d}{{\epsilon }_1})(G_C+\frac{\tau }{3}{G}_Q)]\},\end{array}$(21) ) for its definition), $A_{xz}^{(0)}$ (see Equation (22(22) $\begin{array}{rl}{A}_{xz}^{(0)}& =\frac{-1}{\mathcal{S}}\frac{{\epsilon }_2}{M_d}\frac{\tau \sin \theta }{1+\tau }\{(4+\frac{4M_d}{{\epsilon }_1})(G_C{G}_Q+\frac{\tau }{3}{G}_Q^2)\\ & +(1+\tau )(1+\frac{M_d}{{\epsilon }_1}){\tan }^2\left(\frac{\theta }{2}\right)G_M^2\\ & +2(1-\tau \frac{M_d}{{\epsilon }_1})\\ & \times [-1-\tau +2{\sin }^2\left(\frac{\theta }{2}\right)(1+\frac{{\epsilon }_1}{M_d}\\ & +\frac{{\epsilon }_1^2}{M_d^2}-\tau \frac{{\epsilon }_1}{M_d})](1+{\tan }^2\left(\frac{\theta }{2}\right))G_M{G}_Q\},\end{array}$(22) ) for its definition), and $A_{zz}^{(0)}$ (see Equation (23(23) $\begin{array}{rl}{A}_{zz}^{(0)}& =\frac{-\tau }{2\mathcal{S}}\{[\frac{6\tau }{1+\tau }\frac{{\epsilon }_1+{\epsilon }_2}{{\epsilon }_1}(1+\frac{M_d}{{\epsilon }_1})G_Q-G_M]G_M\\ & +{\tan }^2\left(\frac{\theta }{2}\right)[1-2\tau -6\tau \frac{M_d}{{\epsilon }_1}(1+\frac{M_d}{2{\epsilon }_1})]\\ & \times [G_M^2+\frac{4}{1+\tau }{\cot }^2\left(\frac{\theta }{2}\right)G_Q(G_C+\frac{\tau }{3}{G}_Q)]\}.\end{array}$(23) ) for its definition) in the laboratory frame where the z axis is in the direction of the momentum of incident lepton. As an application, we show the case of muon-deuteron elastic scattering.

Figure 1 illustrates the results for $A_{xx}^{(0)}$ as a function of the muon beam energy ${ϵ}_1$ for muon scattering angles $\theta ={10}^{\circ }$, 70${}^{\circ }$, 110${}^{\circ }$, and 150${}^{\circ }$. The solid, dotted, dashed, and dash-double-dotted curves display the results for $A_{xx}^{(0)}$ using the Argonne v18, CD-Bonn, Nijmegen-II, and Bonn-Q NN potentials, respectively. We see that the results for $A_{xx}^{(0)}$ vanish at zero muon scattering angle and small values of muon beam energy. When the muon scattering angle and the incident muon energy increase, $A_{xx}^{(0)}$ becomes sizable. It begins with zero at ${ϵ}_1=0$ GeV and increases with increasing ${ϵ}_1$ until it reaches a maximum value at ${ϵ}_1\simeq 0.6$ GeV. Then, it decreases with increasing ${ϵ}_1$. The maximum value of $A_{xx}^{(0)}$ is shifted towards lower muon beam energy with increasing the muon scattering angle. At extremely forward muon scattering angles, the maximum value is not seen and $A_{xx}^{(0)}$ increases with increasing ${ϵ}_1$.

Figure 1. (Color online) Sensitivity of the tensor SA $A_{xx}^{(0)}$ in elastic muon-deuteron scattering as a function of the muon beam energy ${ϵ}_1$ at various fixed values of the muon scattering angle θ to realistic DWFs using the standard dipole fit for NFFs. The solid, dotted, dashed, and dash-double-dotted curves show the results for $A_{xx}^{(0)}$ using the realistic and high-precision Argonne v18, CD-Bonn, Nijmegen-II, and Bonn-Q NN potentials, respectively.

From Figure 1 it appears also that the sensitivity of the obtained results for $A_{xx}^{(0)}$ to the NN potential used for the DWF is obvious at large muon scattering angles, in particular at muon beam energies higher than 0.6 GeV. At extreme forward muon scattering angles, it is obvious that the results obtained for $A_{xx}^{(0)}$ using various realistic NN potentials are indistinguishable (see the top left panel in Figure 1). Similarly, at muon beam energies smaller than 0.6 GeV, one can see that the results for $A_{xx}^{(0)}$ obtained using various NN potentials are very close to each others. By increasing the muon scattering angle and beam energy, it is obvious that the estimations of $A_{xx}^{(0)}$ using various NN potentials differ from one another.

In Figure 2 we illustrate the results for $A_{xx}^{(0)}$ in a three-dimensional plot as a function of ${ϵ}_1$ and θ using the DFF NFFs. The left and right parts in Figure 2 show the results for $A_{xx}^{(0)}$(${ϵ}_1$, θ) using the Argonne v18 and Bonn-Q NN potentials for the DWF, respectively. We see that the $A_{xx}^{(0)}$ asymmetry vanishes at $\theta =0^{\circ }$ and at small muon beam energies. When the muon scattering angle and muon beam energy increase, $A_{xx}^{(0)}$ exhibits a peak near ${ϵ}_1=0.6$ GeV. The difference between the left and right parts in Figure 2, which highlights the sensitivity of the results to the DWF, can be seen by comparing the solid (Argonne v18) and the dash-double-dotted (Bonn-Q) curves in Figure 1. It is also clear that the sensitivity of $A_{xx}^{(0)}$ to the DWF is very clear at muon scattering angles greater than 30${}^{\circ }$ and at ${ϵ}_1>0.6$ GeV.

The results for $A_{xz}^{(0)}$ asymmetry as a function of ${ϵ}_1$ at the same values of θ as in the case of $A_{xx}^{(0)}$ asymmetry are shown in Figure 3. We see that the $A_{xz}^{(0)}$ asymmetry also vanishes at $\theta =0^{\circ }$ and small values of ${ϵ}_1$. By increasing ${ϵ}_1$ and θ, the asymmetry $A_{xz}^{(0)}$ becomes sizable and its absolute values are large compared with the ones for $A_{xx}^{(0)}$ with opposite behaviour. It vanishes at ${ϵ}_1=0$ GeV and decreases with increasing ${ϵ}_1$ until it reaches a minimum value at ${ϵ}_1\simeq 0.6$ GeV. Then, it increases with increasing ${ϵ}_1$ until it reaches a maximum value at ${ϵ}_1\simeq 1$ GeV and backward scattering angles and then decreases again. Figure 3 shows also that the sensitivity of the obtained results for $A_{xz}^{(0)}$ to the NN potential used for the deuteron wave function is more sizable at backward muon scattering angles and large values of ${ϵ}_1$. As in the case of $A_{xx}^{(0)}$ asymmetry, one can see at extreme forward muon scattering angles that the results obtained for $A_{xz}^{(0)}$ using various realistic NN potentials are very close to each others (see the top left panel in Figure 3). When ${ϵ}_1<0.4$ GeV, we see that the values for $A_{xz}^{(0)}$ obtained using various NN potentials are also very close to each others. By increasing ${ϵ}_1$ and θ, differences between the estimations of $A_{xz}^{(0)}$ using various NN potentials are obtained.

Figure 2. (Color online) A three-dimensional plot for the tensor SA $A_{xx}^{(0)}$ in elastic muon-deuteron scattering as a function of ${ϵ}_1$ and θ using the DFF nucleon form factors. The left and right parts show the results for $A_{xx}^{(0)}({ϵ}_1,\theta )$ using the Argonne v18 and Bonn-Q NN potentials for the DWFs, respectively.

Figure 4 displays the results for $A_{xz}^{(0)}$(${ϵ}_1$, θ) in a three-dimensional plot as a function of ${ϵ}_1$ and θ using the Argonne v18 (left part) and Bonn-Q (right part) NN potentials for the DWF. It is obvious that the $A_{xz}^{(0)}$ asymmetry exhibits a minimum value near ${ϵ}_1=0.6$ GeV at muon scattering angles greater than ${30}^{\circ }$.

Figure 3. (Color online) Same as in Figure 1 but for the tensor SA $A_{xz}^{(0)}$. Results at $\theta ={10}^{\circ }$ are multiplied by the factor in the parentheses.

The results for $A_{zz}^{(0)}$ as a function of ${ϵ}_1$ at the same values of the muon scattering angle as in the case of $A_{xx}^{(0)}$ and $A_{xz}^{(0)}$ asymmetries are displayed in Figure 5. The $A_{zz}^{(0)}$ asymmetry exhibits a different behaviour compared with $A_{xx}^{(0)}$ and $A_{xz}^{(0)}$ asymmetries. It vanishes at zero muon scattering angle and small values of muon beam energy. At forward scattering angles, $A_{zz}^{(0)}$ begins with zero at ${ϵ}_1=0$ GeV and decreases with increasing ${ϵ}_1$ until it reaches a minimum value at ${ϵ}_1\simeq 1$ GeV. Then, it increases with increasing ${ϵ}_1$. At extreme forward scattering angles, the minimum value is not seen (see the top left panel in Figure 5). At backward scattering angles, we see that $A_{zz}^{(0)}$ starts with zero and increases with increasing ${ϵ}_1$ until it reaches a maximum value at ${ϵ}_1\simeq 0.4$ GeV and then rapidly decreases until it reaches a minimum value at ${ϵ}_1\simeq 1$ GeV and increases again.

Figure 4. (Color online) Same as in Figure 2 but for the tensor SA $A_{xz}^{(0)}({ϵ}_1,\theta )$.

Figure 5. (Color online) Same as in Figure 1 but for the tensor SA $A_{zz}^{(0)}$.

With respect to the sensitivity of $A_{zz}^{(0)}$ asymmetry to the DWF, we see from Figure 5 that the results obtained for $A_{zz}^{(0)}$ using various realistic NN potentials are indistinguishable at extreme forward angles and at ${ϵ}_1<$0.4 GeV. At higher muon beam energy and $\theta >$30${}^{\circ }$, we obtain differences between the estimations of $A_{zz}^{(0)}$ using various NN potentials. A three-dimensional plot for the results of $A_{zz}^{(0)}$(${ϵ}_1$, θ) is displayed in Figure 6 using the Argonne v18 (left part) and Bonn-Q (right part) NN potentials for the DWF. It is obvious that the $A_{zz}^{(0)}$ asymmetry exhibits a minimum value at forward scattering angles and large beam energy, whereas a maximum value near ${ϵ}_1=0.4$ GeV is obtained.

Figure 6. (Color online) Same as in Figure 2 but for the tensor SA $A_{zz}^{(0)}$(${ϵ}_1$,θ).

The origin of the differences obtained using various realistic DWFs maybe due to the tensor force between two nucleons. The authors of Refs. [58,59] were expressed the measure of the tensor force strength in terms of the D-state probability $P_D$ obtained for the deuteron. The $P_D$ values for the realistic NN potentials used in this work are 5.64% for Nijmegen-II, 5.76% for Argonne v18, 4.85% for CD-Bonn, and 4.38% for Bonn-Q. The dependence of the $\gamma d\to {\pi }^{0}d$ and $ed\to e^{\prime }{d}^{\prime }$ observables on the D-state component of the DWF was investigated in Refs. [60,61], respectively. It was found that the D-wave contribution becomes visible at backward scattering angles.

We would like to point out that similar characterizations for $A_{xx}^{(0)}$, $A_{xz}^{(0)}$, and $A_{zz}^{(0)}$ are observed in Refs. [31,32] in the zero lepton mass limit. The present estimations agree well with the calculations of tensor SAs in the same kinematical range. From the figures presented in the present work it appears that the sensitivity of the estimated tensor SAs to realistic DWF is large at muon scattering angles greater than 30${}^{\circ }$, in particular at muon beam energies greater than 0.6 GeV. Unfortunately, measurements for these tensor SAs are not available in the literature. Thus, it would be very desirable to have experimental data for SAs in elastic lepton scattering on the deuteron in the relevant kinematical region.

The lepton mass cannot always be neglected because some of the polarization observables contain contributions which are proportional to the lepton mass. In presence of lepton mass, the contributions of tensor SAs which are related to the lepton mass become important at low incident beam energies and backward scattering angles and the lepton mass should be explicitly considered. It was shown in Refs. [31,32] that the relative effect of the mass is about 10% on $A_{xx}$ and $A_{xz}$ and can reach 50% on $A_{zz}$.

4. Conclusions and outlook

We reported theoretical estimations for tensor SAs in the elastic scattering of leptons on the deuteron ignoring the lepton mass. Numerical results for the SAs $A_{xx}^{(0)}$, $A_{xz}^{(0)}$, and $A_{zz}^{(0)}$ with unpolarized lepton beam and tensor deuteron target in two- and three-dimensional plots are given. As an application, the elastic scattering of muon on the deuteron in the laboratory system is shown. The sensitivity of the estimated results for tensor SAs to the realistic DWF of modern NN potential is studied for the first time. In our estimations, we used four realistic NN potentials, which are the Nijmegen-II [35], Argonne v18 [39], CD-Bonn [40], and Bonn-Q [41] potentials. For the proton and neutron form factors, the standard dipole fit [42] is used.

We found that the tensor SAs $A_{xx}^{(0)}$, $A_{xz}^{(0)}$, and $A_{zz}^{(0)}$ vanish at zero muon scattering angle and small values of muon beam energy. By increasing the scattering angle and the incident energy of the muon, the tensor spin asymmetries become sizable. It is also shown that the results for the SAs obtained using various realistic DWFs are comparable at extreme forward muon scattering angles and muon beam energies less than 0.6 GeV. When the energy of the muon beam and the scattering angle of the muon increase, theoretical discrepancies among the results for $A_{xx}^{(0)}$, $A_{xz}^{(0)}$, and $A_{zz}^{(0)}$ using various realistic DWFs are obtained, which maybe due to the tensor force between two nucleons.

More theoretical and experimental investigation on SAs in the elastic scattering of leptons on the deuteron are needed. For instance, numerical estimations of physical observables in the elastic scattering of leptons on the deuteron with and without the lepton mass are useful and promising. This makes it possible to compare between the results for physical observables with and without the lepton mass. The issue of taking lepton mass effects into account is relevant for experimental observables in elastic lepton-deuteron scattering, in particular for the extraction of the magnetic form factor of the deuteron $G_M$ from the cross section of elastic muon-deuteron scattering at backward muon angles. This subject is also relevant for the proton-antiproton annihilation experiment PANDA at GSI facility in Darmstadt [62] (see also Ref. [63] for a theoretical overview). It is also interesting to estimate results for the tensor SAs with the lepton mass. On the experimental side, measurements for spin observables in lepton-deuteron elastic scattering in the relevant kinematical region are needed.

Acknowledgments

The authors acknowledge with thanks the University of Jeddah for technical and financial support. We are very grateful to the anonymous referees for the evaluation of our manuscript and for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia [grant number UJ-21-DR-64].