Asymmetry for tensor $t_{2j}$ and vector $t_{1i}$ polarizations with   taking into account the deuteron wave function in coordinate space

V.I. Zhaba

arXiv:1907.09853·nucl-th·July 24, 2019

Asymmetry for tensor $t_{2j}$ and vector $t_{1i}$ polarizations with taking into account the deuteron wave function in coordinate space

V.I. Zhaba

PDF

Open Access

TL;DR

This paper uses deuteron wave functions and various nucleon-nucleon potentials to theoretically calculate and analyze tensor and vector polarization asymmetries across a range of momenta and angles, with implications for scattering processes.

Contribution

It provides detailed calculations of deuteron polarization asymmetries using realistic wave functions and potentials, extending understanding of angular-momentum dependence in scattering experiments.

Findings

01

Calculated asymmetries for momenta 0-7 fm$^{-1}$

02

Analyzed polarization dependencies on scattering angles

03

Compared results across different nucleon-nucleon potentials

Abstract

The analytic forms of deuteron wave function in coordinate space were applied for theoretical calculations of full set asymmetries of tensor $t_{2 j}$ and vector $t_{1 i}$ polarizations. Nucleon-nucleon realistic phenomenological potentials of Nijmegen group (Nijm1, Nijm2, Nijm93, Reid93) and Argonne group (Argonne v18) as well as other widely used and popular potentials (OBEPC, MT, Paris) are used for numerical calculations. The angular asymmetry is calculated in the range of momentums 0-7 fm $^{- 1}$ . The values of the asymmetry $t_{ij,}$ by polarization type and with each other, are analysed. This is the angular-momentum dependence of values vector $t_{1 i} (p, θ_{e})$ and tensor $t_{2 i} (p, θ_{e})$ polarizations in 3D format at momentums 0-7 fm $^{- 1}$ and scattering angles 1-180 degrees were calculated and compared for Reid93 potential. The perspectives of further application of…

Equations40

\left\{{\begin{array}[]{l}u\left(r\right)=\sum\limits_{j=1}^{N}{C_{j}\exp\left({-m_{j}r}\right),}\\ w\left(r\right)=\sum\limits_{j=1}^{N}{D_{j}\exp\left({-m_{j}r}\right)\left[{1+\frac{3}{m_{j}r}+\frac{3}{\left({m_{j}r}\right)^{2}}}\right],}\\ \end{array}}\right.

\left\{{\begin{array}[]{l}u\left(r\right)=\sum\limits_{j=1}^{N}{C_{j}\exp\left({-m_{j}r}\right),}\\ w\left(r\right)=\sum\limits_{j=1}^{N}{D_{j}\exp\left({-m_{j}r}\right)\left[{1+\frac{3}{m_{j}r}+\frac{3}{\left({m_{j}r}\right)^{2}}}\right],}\\ \end{array}}\right.

u (r) \to r; w (r) \to r^{3} .

u (r) \to r; w (r) \to r^{3} .

\left\{{\begin{array}[]{l}u(r)\sim A_{S}\exp(-\beta r),\\ w(r)\sim A_{D}\exp(-\beta r)\left[{1+\frac{3}{\beta r}+\frac{3}{(\beta r)^{2}}}\right],\\ \end{array}}\right.

\left\{{\begin{array}[]{l}u(r)\sim A_{S}\exp(-\beta r),\\ w(r)\sim A_{D}\exp(-\beta r)\left[{1+\frac{3}{\beta r}+\frac{3}{(\beta r)^{2}}}\right],\\ \end{array}}\right.

\left\{{\begin{array}[]{l}u(r)=r^{3/2}\sum\limits_{i=1}^{N}{A_{i}\exp(-a_{i}r^{3}),}\\ w(r)=r\sum\limits_{i=1}^{N}{B_{i}\exp(-b_{i}r^{3}).}\\ \end{array}}\right.

\left\{{\begin{array}[]{l}u(r)=r^{3/2}\sum\limits_{i=1}^{N}{A_{i}\exp(-a_{i}r^{3}),}\\ w(r)=r\sum\limits_{i=1}^{N}{B_{i}\exp(-b_{i}r^{3}).}\\ \end{array}}\right.

\left\{{\begin{array}[]{l}u(r)=r\sum\limits_{i=1}^{N}{A_{i}\exp(-a_{i}r^{2}),}\\ w(r)=r^{3}\sum\limits_{i=1}^{N}{B_{i}\exp(-b_{i}r^{2}).}\\ \end{array}}\right.

\left\{{\begin{array}[]{l}u(r)=r\sum\limits_{i=1}^{N}{A_{i}\exp(-a_{i}r^{2}),}\\ w(r)=r^{3}\sum\limits_{i=1}^{N}{B_{i}\exp(-b_{i}r^{2}).}\\ \end{array}}\right.

\left\{{\begin{array}[]{l}u(r)=r\sum\limits_{i=1}^{N}{A_{i}\exp(-a_{i}r^{2}),}\\ w(r)=r\sum\limits_{i=1}^{N}{B_{i}\exp(-b_{i}r^{2}).}\\ \end{array}}\right.

\left\{{\begin{array}[]{l}u(r)=r\sum\limits_{i=1}^{N}{A_{i}\exp(-a_{i}r^{2}),}\\ w(r)=r\sum\limits_{i=1}^{N}{B_{i}\exp(-b_{i}r^{2}).}\\ \end{array}}\right.

u_{l} (r) = i^{l} \frac{2}{π} 0 \int \infty u_{l} (p) j_{l} (p r) d p,

u_{l} (r) = i^{l} \frac{2}{π} 0 \int \infty u_{l} (p) j_{l} (p r) d p,

σ = σ_{0} [1 + \frac{1}{2} P_{z z} A_{d}^{T} + \frac{3}{2} P_{e} P_{z} A_{e d}^{V}],

σ = σ_{0} [1 + \frac{1}{2} P_{z z} A_{d}^{T} + \frac{3}{2} P_{e} P_{z} A_{e d}^{V}],

A_{d}^{T} = \frac{3 cos ^{2} θ ^{*} - 1}{2} t_{20} (p, θ_{e}) - \frac{3}{2} sin (2 θ^{*}) cos ϕ^{*} t_{21} (p, θ_{e}) + \frac{3}{2} sin^{2} θ^{*} cos (2 ϕ^{*}) t_{22} (p, θ_{e});

A_{d}^{T} = \frac{3 cos ^{2} θ ^{*} - 1}{2} t_{20} (p, θ_{e}) - \frac{3}{2} sin (2 θ^{*}) cos ϕ^{*} t_{21} (p, θ_{e}) + \frac{3}{2} sin^{2} θ^{*} cos (2 ϕ^{*}) t_{22} (p, θ_{e});

A_{e d}^{V} = 3 [\frac{1}{2} cos θ^{*} t_{10} (p, θ_{e}) - sin θ^{*} cos ϕ^{*} t_{11} (p, θ_{e})];

A_{e d}^{V} = 3 [\frac{1}{2} cos θ^{*} t_{10} (p, θ_{e}) - sin θ^{*} cos ϕ^{*} t_{11} (p, θ_{e})];

\frac{d σ}{d Ω d Ω _{2}} = \frac{d σ}{d Ω d Ω _{2}}_{0} [1 + \frac{3}{2} h t_{11} A_{y} sin φ_{2} + \frac{1}{2} t_{20} A_{z z} - \frac{2}{3} t_{21} A_{x z} cos φ_{2} + \frac{1}{3} t_{22} (A_{xx} - A_{y y}) cos 2 φ_{2}],

\frac{d σ}{d Ω d Ω _{2}} = \frac{d σ}{d Ω d Ω _{2}}_{0} [1 + \frac{3}{2} h t_{11} A_{y} sin φ_{2} + \frac{1}{2} t_{20} A_{z z} - \frac{2}{3} t_{21} A_{x z} cos φ_{2} + \frac{1}{3} t_{22} (A_{xx} - A_{y y}) cos 2 φ_{2}],

\frac{d σ}{d Ω _{e}} (h; p_{z}, p_{z z}) = Σ (θ, φ) + h Δ (θ, φ),

\frac{d σ}{d Ω _{e}} (h; p_{z}, p_{z z}) = Σ (θ, φ) + h Δ (θ, φ),

Σ (θ, φ) = \frac{d σ _{0}}{d Ω _{e}} [1 + Γ (θ, φ)],

Σ (θ, φ) = \frac{d σ _{0}}{d Ω _{e}} [1 + Γ (θ, φ)],

\begin{array}[]{l}\Gamma(\theta,\varphi)=p_{zz}\left[{\frac{1}{\sqrt{2}}}\right.P_{2}^{0}(\cos\theta)T_{20}(p^{2},\theta_{e})-\frac{1}{\sqrt{3}}P_{2}^{1}(\cos\theta)\cos\varphi T_{21}(p^{2},\theta_{e})+\\ +\left.{\frac{1}{2\sqrt{3}}P_{2}^{2}(\cos\theta)\cos(2\varphi)T_{22}(p^{2},\theta_{e})}\right].\\ \end{array}

\begin{array}[]{l}\Gamma(\theta,\varphi)=p_{zz}\left[{\frac{1}{\sqrt{2}}}\right.P_{2}^{0}(\cos\theta)T_{20}(p^{2},\theta_{e})-\frac{1}{\sqrt{3}}P_{2}^{1}(\cos\theta)\cos\varphi T_{21}(p^{2},\theta_{e})+\\ +\left.{\frac{1}{2\sqrt{3}}P_{2}^{2}(\cos\theta)\cos(2\varphi)T_{22}(p^{2},\theta_{e})}\right].\\ \end{array}

h Δ (θ, φ) = h \frac{d σ _{0}}{d Ω} p_{z} [\frac{3}{2} P_{1} (cos θ) T_{10} (p^{2}, θ_{e}) - 3 P_{1}^{1} (cos θ) cos φ T_{11} (p^{2}, θ_{e})] .

h Δ (θ, φ) = h \frac{d σ _{0}}{d Ω} p_{z} [\frac{3}{2} P_{1} (cos θ) T_{10} (p^{2}, θ_{e}) - 3 P_{1}^{1} (cos θ) cos φ T_{11} (p^{2}, θ_{e})] .

t_{10} (p, θ_{e}) = - \frac{2}{3} \frac{η}{S} (1 + η) (1 + η sin^{2} (\frac{θ _{e}}{2})) G_{M}^{2} (p) \mbox t g (\frac{θ _{e}}{2}) sec (\frac{θ _{e}}{2}),

t_{10} (p, θ_{e}) = - \frac{2}{3} \frac{η}{S} (1 + η) (1 + η sin^{2} (\frac{θ _{e}}{2})) G_{M}^{2} (p) \mbox t g (\frac{θ _{e}}{2}) sec (\frac{θ _{e}}{2}),

t_{11} (p, θ_{e}) = \frac{2}{3 S} η (1 + η) G_{M} (p) [G_{C} (p) + \frac{η}{3} G_{Q} (p)] \mbox t g (\frac{θ _{e}}{2}),

t_{11} (p, θ_{e}) = \frac{2}{3 S} η (1 + η) G_{M} (p) [G_{C} (p) + \frac{η}{3} G_{Q} (p)] \mbox t g (\frac{θ _{e}}{2}),

t_{20} (p, θ_{e}) = - \frac{1}{2 S} (\frac{8}{3} η G_{C} (p) G_{Q} (p) + \frac{8}{9} η^{2} G_{Q}^{2} (p) + \frac{1}{3} η [1 + 2 (1 + η) \mbox t g^{2} (\frac{θ _{e}}{2})] G_{M}^{2} (p)),

t_{20} (p, θ_{e}) = - \frac{1}{2 S} (\frac{8}{3} η G_{C} (p) G_{Q} (p) + \frac{8}{9} η^{2} G_{Q}^{2} (p) + \frac{1}{3} η [1 + 2 (1 + η) \mbox t g^{2} (\frac{θ _{e}}{2})] G_{M}^{2} (p)),

t_{21} (p, θ_{e}) = \frac{2}{3 S cos ( \frac{θ _{e}}{2} )} η η + η^{2} sin^{2} (\frac{θ _{e}}{2}) G_{M} (p) G_{Q} (p),

t_{21} (p, θ_{e}) = \frac{2}{3 S cos ( \frac{θ _{e}}{2} )} η η + η^{2} sin^{2} (\frac{θ _{e}}{2}) G_{M} (p) G_{Q} (p),

t_{22} (p, θ_{e}) = - \frac{1}{2 3 S} η G_{M}^{2} (p),

t_{22} (p, θ_{e}) = - \frac{1}{2 3 S} η G_{M}^{2} (p),

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Taxonomy

TopicsNuclear physics research studies · Superconducting Materials and Applications · Geophysics and Gravity Measurements

Full text

Asymmetry for tensor t2j and vector t1i polarizations with taking into account the deuteron wave function in coordinate space

V. I. Zhaba

Uzhgorod National University, Department of Theoretical Physics,

54, Voloshyna St., Uzhgorod, UA-88000, Ukraine

Abstract

The analytic forms of deuteron wave function in coordinate space were applied for theoretical calculations of full set asymmetries of tensor $t_{2j}$ and vector $t_{1i}$ polarizations. Nucleon-nucleon realistic phenomenological potentials of Nijmegen group (Nijm1, Nijm2, Nijm93, Reid93) and Argonne group (Argonne v18) as well as other widely used and popular potentials (OBEPC, MT, Paris) are used for numerical calculations. The angular asymmetry is calculated in the range of momentums 0-7 fm*-1*. The values of the asymmetry $t_{ij,}$ by polarization type and with each other, are analysed. This is the angular-momentum dependence of values vector $t_{1i}(p,\theta_{e})$ and tensor $t_{2i}(p,\theta_{e})$ polarizations in 3D format at momentums 0-7 fm*-1* and scattering angles 1-180 degrees were calculated and compared for Reid93 potential. The perspectives of further application of the obtained results for calculating the values of the cross-sections, asymmetries and other characteristics of processes with the participation of a deuteron are discussed.

Keywords: wave function, deuteron, spin observables, electron scattering, tensor polarization, vector polarization, asymmetry.

PACS 03.65.Nk, 13.40.Gp, 13.88.+e, 21.45.Bc

1. Introduction

Deuteron is the simplest core, which consists of the two elementary particles both a proton and a neutron. The simplicity and clarity of the deuteron structure always serves as a convenient laboratory for evident simulation and structural analysing nucleon-nucleon forces [1].

Deuteron is the possible target for an electron beam or the particle that dissipates on a proton or atomic nuclei. Some examples could be presented. The paper [27] presents recent results for spin-dependent scattering of electrons on polarized protons and deuterons for BLAST experiment in MIT-Bates. The radiative corrections are investigated for polarization observed in elastic ed- scattering in leptonic variables by [3]. Global analysis of quasi-elastic ed-scattering data’s in the weak-binding approximation was conducted in [4].

The experiments were performed on the separate measurement of deuteron form factors in elastic ed-scattering in the interval of transmitted momentums $p^{2}$ =8-15 fm*-2* on the electron positron storage VEPP-3 [5]. The results of measuring the components of the analyzing powers $T_{2i}$ in the photodisintegration reaction of a tensor-polarized deuteron are presented.

The authors [5] first measured the analyzing powers for the coherent photoproduction of a neutral pion $\pi^{0}$ on a tensor-polarized deuteron. According to [6], tensor target spin asymmetries are calculated in coherent $\pi^{0}$ - photoproduction on the deuteron, including an intermediate $\eta$ NN- interaction in the three-particle approach.

Spin observables in dp-scattering and T-invariance test were studied in the application of the modified Glauber theory by [7]. A complete set of deuteron analyzing powers in elastic dp-scattering at 190 MeV/nucleon is given in [8]. The proton and deuteron analyzing powers and 10 spin correlation coefficients were measured for elastic p+d scattering with the energy for bombarding protons 135 and 200 MeV [9]. Experimental data’s are compared with Fadeev’s calculations for CD-Bonn and AV18 potentials. Vector $A_{y}$ and tensor $A_{xx}$ , $A_{yy}$ analyzing powers in elastic dp-scattering at deuteron kinetic energy $T_{d}$ =1.2 and 2.27 GeV are obtained using the ANKE spectrometer on the COSY storage ring [10]. The results are compared with other experimental data’s and predictions within the framework of the multiple scattering of Glauber theory. The authors of [11] investigated the effects of polarization observed in elastic lepton-deuteron scattering including the lepton masses.

The angular dependence of the tensor and vector analyzing powers in inelastic (d,d’)-scattering of deuterons with a momentum at 9.0 GeV/c on hydrogen and carbon in [12] and of 5.0 GeV/c on beryllium in [13] is measured. Further theoretical calculations and prospects for the process (d,d’) are analysed in [1].

Generally speaking, in general for the theoretical study of mechanisms, changes and characteristics for the vast majority of these processes with the participation of a deuteron it is necessary to know the deuteron wave function (DWF) in the coordinate or momentum space, as well as the deuteron form factors.

In the last detailed review [14], the static parameters of deuteron obtained from DWF for different nucleon-nucleon potentials and models are systematized, as well as a review, list and characteristics for analytical forms of DWF in the coordinate space have been reviewed.

In this paper we use the analytic forms of DWF in coordinate space for theoretical calculations of set asymmetries of tensor $t_{2j}$ and vector $t_{1i}$ polarizations. Nucleon-nucleon realistic phenomenological potentials of Nijmegen group (NijmI, NijmII, Nijm93, Reid93) and Argonne group (Argonne v18) as well as other widely used and popular potentials (OBEPC, MT, Paris) are used for numerical calculations.

2. DWF in Coordinate Space

Among the large and varied list of analytical forms for DWF in the coordinate space, elementary parametrization of DWF for the Paris potential [15] is worth highlighting

[TABLE]

where $m_{j}=\beta+(j-1)m_{0}$ ; $\beta=\sqrt{ME_{d}}$ , $m_{0}$ =0.9 fm*-1*; $M$ – nucleon mass; $E_{d}$ – binding energy of the deuteron. The asymptotics of the deuteron wave function (1) at $r\to 0$ :

[TABLE]

The asymptotics of the deuteron wave function (1) for values $r\to\infty$

[TABLE]

where $A_{S}$ and $A_{D}$ are the asymptotics of $S$ - and $D$ - state normalizations accordingly.

Usually, coefficients $C_{j}$ , $D_{j}$ are indicated in the tables in most cases for specific potentials, except for the last values of the coefficients $C_{n}$ , $D_{n-2}$ , $D_{n-1}$ , $D_{n}$

The search for the coefficients $C_{j}$ , $D_{j}$ of the analytical form (1) was done for the Paris [15] and the Bonn (OBEPC [16] and charge-dependent Bonn (CD-Bonn) [17]) potentials and the fss2 model (with the Coulomb exchange kernel [18]), with $N$ =13, 11 and 11 respectively. The formula (1) was also applied to MT model [19], where for radial component of DWF for both the S-states and D- states of number made $N_{S}$ =16 and $N_{D}$ =12 accordingly.

In addition to the DWF (1), the next analytical form was suggested in paper [20] for Nijmegen group potentials (NijmI, NijmII and Nijm93) and further was used in paper [21] for approximation of DWFs for the Reid93 and Argonne v18 potentials:

[TABLE]

A complete set of coefficients for DWF (2) for these five potentials is given in [20, 21].

Coefficients of DWF for the dressed dibaryon model (DDM) [22] were obtained for the analytical form [23]:

[TABLE]

The new analytical DWFs were also proposed in coordinate space in such a simple form [24] in 2016:

[TABLE]

DWF in the coordinate representation $u_{l}(r)$ can be obtained from the DWF in the momentum representation $u_{l}(p)$ by means of the Hankel transform [18]:

[TABLE]

where $j_{0}$ (pr) and $j_{2}$ (pr) are Bessel functions of zero and second order. It was for PEST [25] that the Hankel transform (5) can be used the known momentum parameterisation of DWF.

Certov-Mathelitsch-Moravcsik (CMM) DWF [26] (the variant of the potential model in the original paper is designated as $\psi_{LS}^{6B})$ have been parameterised by formulas (1) with $N$ =7.

Obtaining method, approximation interval and knots for DWF are presented in Table 1.

Table 1. DWF for nucleon-nucleon interaction potentials

[TABLE]

3. Asymmetry for Tensor t2j and Vector t1i Polarizations

If the deuteron target is tensor and vector polarized, then the polarized cross-section is written as follows [27]:

[TABLE]

where target tensor $A_{d}^{T}$ and beam-target vector asymmetries $A_{ed}^{V}$ :

[TABLE]

where the angles $\theta$ ∗ and $\phi$ ∗ define the polarization direction in a frame; $P_{e}$ , $P_{z}$ , $P_{zz}$ are the longitudinal polarization of the electron, the deuteron vector and tensor polarizations.

The polarization of the outgoing (reflected) deuteron can be measured if the scattering process is analysed in detail. Differential cross section for the double scattering process is the following [28, 29]:

[TABLE]

where $h=\pm$ 1/2 - a polarization of the incoming electron beam; $\phi$ 2 - the angle between the two scattering planes; $A_{y}$ and $A_{ij}$ - the vector and tensor analyzing powers of the second scattering.

The differential cross section for elastic scattering of a polarized electron beam from a polarized deuteron target is given by an expression in a laboratory system [30, 31]

[TABLE]

where $h$ is a helicity of the incident electron beam; $p_{z}$ and $p_{zz}$ determine the degree of vector and tensor polarizations of the deuteron target; the angles $\theta$ and $\phi$ determine the polarization direction of the deuteron in the frame.

The first part on the right in a formula (10) defines a cross-section for an unpolarized electron but a polarized target deuteron

[TABLE]

where $\frac{d\sigma_{0}}{d\Omega_{e}}$ - is an unpolarized differential cross-section. The value $\Gamma$ contains tensor deuteron analyzing powers $T_{2j}$ :

[TABLE]

The second part on the right in a formula (10) describes a helicity-dependent differential cross-section for a polarized electron beam and a polarized deuteron target and contains vector deuteron analyzing powers of $T_{10}$ and $T_{11}$ [31]:

[TABLE]

The formulas (12) and (13) $P_{l}(x)$ and $P_{l}^{m}(x)$ determine Legendre polynomials and associated Legendre polynomials respectively.

The values of the tensor $T_{20}$ , $T_{21}$ , $T_{22}$ and vector $T_{10}$ , $T_{11}$ of deuteron analyzing powers are determined formulas (7)-(9), (12) and (13) through form factors such as [32, 31, 27] (in equivalent more widely used terms [33, 34, 28] tensor $t_{2j}$ and vector $t_{1i}$ polarizations):

[TABLE]

where the factor $S(p,\theta_{e})=A(p)+B(p)\mbox{tg}^{2}\left({\frac{\theta_{e}}{2}}\right)$ is determined by the structure functions $A,B$ and the scattering angle $\theta$ ; the charge $G_{C}(p)$ , quadrupole $G_{Q}(p)$ and magnetic $G_{M}(p)$ form factors contain information about the electromagnetic properties of the deuteron. The values of the tensor $t_{20}$ and vector $t_{11}$ polarization are determined by the form factors $G_{C}(p)$ , $G_{Q}(p)$ , $G_{M}(p)$ and scattering angle $\theta$ , and $t_{21}$ – by $G_{Q}(p)$ , $G_{M}(p)$ and $\theta$ . The values of $t_{22}$ and $t_{10}$ depend only on the form factors of the $G_{M}(p)$ and the scattering angle $\theta$ .

4. Results of calculations

The values of the angular asymmetry for the components of the vector $t_{1i}$ and the tensor $t_{2j}$ polarizations for the first eight potentials in Table 1 (Nijm1, Nijm2, Nijm93, Reid93, Argonne v18, OBEPC, MT, Paris) are given on Fig. 1-5. These values at angles 100, 400, 700, 900, 1200 calculated by DWF for CD-Bonn, Nijm-93, OSBEP, AV18 and Paris potentials in [31, 35]. The comparison of theoretical results and experimental data for vector $t_{1i}$ and tensor $t_{2j}$ polarization has already been indicated by [21] for Argonne v18 and in [21, 36] for Reid93 potentials. Therefore, in this paper, the the theory and experiment are not compared.

\pdfximage

width 100mm t10.jpg\pdfrefximage

Fig. 1. Angular asymmetry of vector polarization $t_{10}$

\pdfximage

width 100mm t11.jpg\pdfrefximage

Fig. 2. Angular asymmetry of vector polarization $t_{11}$

\pdfximage

width 100mm t20.jpg\pdfrefximage

Fig. 3. Angular asymmetry of tensor polarization $t_{20}$

\pdfximage

width 100mm t21.jpg\pdfrefximage

Fig. 4. Angular asymmetry of tensor polarization $t_{21}$

\pdfximage

width 100mm t22.jpg\pdfrefximage

Fig. 5. Angular asymmetry of tensor polarization $t_{22}$

\pdfximage

width 90mm t10R933D.jpg\pdfrefximage

Fig. 6. Vector polarization $t_{10}$ for Reid93 potential

\pdfximage

width 90mm t11R933D.jpg\pdfrefximage

Fig. 7. Vector polarization $t_{11}$ for Reid93 potential

\pdfximage

width 90mm t20R933D.jpg\pdfrefximage

Fig. 8. Tensor polarization $t_{20}$ for Reid93 potential

\pdfximage

width 90mm t21R933D.jpg\pdfrefximage

Fig. 9. Tensor polarization $t_{21}$ for Reid93 potential

\pdfximage

width 90mm t22R933D.jpg\pdfrefximage

Fig. 10. Tensor polarization $t_{22}$ for Reid93 potential

The angle and impulse asymmetry of vector and tensor polarization for DWF (2) for the Argonne v18 potential in [37] and for the Reid93 potential in [36] has been partially studied. These data are quoted here in Figs. 1-5 for the specified scattering angles. For other potentials, results are obtained for the first time.

Now let’s analyse the calculated components of vector and tensor polarizations. As it can be seen in Figs. 1 and 2, vector polarization $t_{1i}$ strongly depend on the scattering angle $\theta$ e. So, there is an angular asymmetry for both vector polarization components $t_{10}$ and $t_{11}$ . In addition, at $p$ ¿2.5 fm*-1*, they also depend on the choice of the potential for nucleon-nucleon interaction. The minimum values for $t_{10}$ and $t_{11}$ are located in the area $p$ =4 6 fm*-1* respectively. The comparison of vector polarizations indicates that the asymmetry $t_{10}$ is less than the asymmetry of $t_{11}$ at the same angles.

In contrast to the vector polarizations $t_{1i}$ , the deuteron tensor polarization $t_{20}$ (Fig. 3) weakly depends on the scattering angle. That is, the angular asymmetry for $t_{20}$ will be slightly intense and weakly sensitive to the scattering angle. The value of $t_{20}$ will have a constant limit value of 0.3-0.7 in the absence of the angle asymmetry and independence from the deuteron form factors. The minimum value for $t_{20}$ is in the pulse region $p$ =3-4 fm*-1*.

Due to this behaviour of polarizations $t_{20}$ and $t_{11}$ , the terms ”single and double spin asymmetries” are used [35].

Tensor polarizations $t_{21}$ and $t_{22}$ (Figs. 4 and 5) are characterized by angular asymmetry, which is more pronounced for $t_{22}$ at large angles. Asymmetries $t_{21}$ and $t_{22}$ with increasing scattering angle are described by curves with progressive convexity and concavity with maximum and minimum at $p\approx$ 4 fm*-1*, respectively. The values of $t_{20}$ and $t_{21}$ give a greater contribution to the absolute value of the cross-section with the same scattering angles, and $t_{22}$ will be an order of magnitude smaller.

In next Figs. 6-10 are given calculated values $t_{ij}$ in the $p\theta_{e}$ plane for Reid93 potential. This is the angular-momentum dependence of values vector $t_{10}(p,\theta_{e})$ and tensor $t_{2i}(p,\theta_{e})$ polarizations in 3D format. The values $t_{10}(p,\theta_{e})$ and $t_{11}(p,\theta_{e})$ are typical of flat forms at small angles and asymmetric momentum dependences at large angles. There is a hump (peak) at 4 fm*-1* in the range of angles 0-180 degrees for tensor polarization $t_{21}$ in 3D format here. There is a pit for the tensor polarizations $t_{20}$ and $t_{22}$ , on the contrary.

5.Conclusions

The novelty of this paper is the complete presentation for angular and angular-momentum asymmetries for all components of tensor and vector polarizations asymmetries. And the resulting angular-momentum asymmetry in the 3D format is original for nucleon-nucleon potentials for different models.

In general, it is necessary to detailed and exact study the influence of the form DWF on the behaviour and changes for characteristics of processes with the participation of deuteron. By the way, results of influence of a choice of the analytical form in coordinate space on calculations of density distribution in the deuteron and transition density for Reid93 and Moscow potentials are given in paper [38].

And further, the obtained results for components of vector and tensor polarizations can be used to calculate the values of the cross-sections (6), (8), asymmetries (7), (8) and other characteristics of processes with the participation of a deuteron.

Prospective are further studies of the angular-momentum dependence of values vector $A_{B}^{i}(p,\theta)$ and tensor $A_{B}^{ij}(p,\theta)$ asymmetries in 3D format [3]. Moreover, it is of interest to conduct the further study of polarization observables in elastic lepton-deuteron scattering [11] (first of all for the case of spin correlation limit of zero lepton mass).

It can be added that the obtained results allow us to estimate the integral picture for the behavior of tensor and vector polarizations in a wider angular, pulse and angular momentum scaling for various types of nucleon-nucleon potentials. In addition, the obtained forms for tensor and vector polarizations in 3D format (the results of this paper) can be systematized together with the angular-momentum dependence for vector and tensor asymmetries in elastic electron-deuteron scattering [3], as well as with spin correlation coefficients and tensor asymmetries in lepton-deuteron elastic scattering [11]. To do this, you need to use a set of different types of nucleon-nucleon potentials and models.

Bibliography38

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] V. I. Zhaba, Mod. Phys. Lett. A 33 , 1850160 (2018).
2[2] D. K. Hasell et al., Annu. Rev. Nucl. Part. Sci. 61 , 409 (2011).
3[3] G. I. Gakh et al., J. Exp. Theor. Phys. 115 , 212 (2012).
4[4] J. J. Ethier et al., Phys. Rev. C 89 , 065203 (2014).
5[5] D. M. Nikolenko et al., Phys. Atom. Nucl. 73 , 1322 (2010).
6[6] E. M. Darwish et al., Braz. J. Phys. 48 , 168 (2018).
7[7] A. A. Temerbayev and Yu. N. Uzikov, Phys. Atom. Nucl. 78 , 35 (2015).
8[8] K. Sekiguchi et al., Phys. Rev. C 96 , 064001 (2017).