Born values for vector and tensor asymmetries in electron-deuteron scattering
Using the previously obtained analytic form factors for the deuteron wave function in the coordinate representation for the nucleon-nucleon potential Argonne v18, are calculated the Born values for vector AᴸB, AᵀB and tensor AᴸᴸB, AᵀᵀB, AᴸᵀB asymmetries, which are necessary for estimating radiative...
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Zhaba, V.I. 2023-11-27T14:16:05Z 2023-11-27T14:16:05Z 2020 Born values for vector and tensor asymmetries in electron-deuteron scattering / V.I. Zhaba // Problems of atomic science and tecnology. — 2020. — № 5. — С. 19-22. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 03.65.Nk, 13.40.Gp, 13.88.+e, 21.45.Bc https://nasplib.isofts.kiev.ua/handle/123456789/194559 Using the previously obtained analytic form factors for the deuteron wave function in the coordinate representation for the nucleon-nucleon potential Argonne v18, are calculated the Born values for vector AᴸB, AᵀB and tensor AᴸᴸB, AᵀᵀB, AᴸᵀB asymmetries, which are necessary for estimating radiative corrections to polarization observables in elastic electron-deuteron scattering in lepton variables. The momentum-angular dependence for vector and tensor asymmetries is illustrated in 3D format. Each component of the asymmetry has its own peculiarity of the form depending on the values of the scattering angle or momentum of the particle. З використанням раніше отриманих коефіцієнтів аналітичної форми хвильової функції дейтрона в координатному представленні для нуклон-нуклонного потенціалу Argonne v18 розраховані борнівські значення векторних AᴸB, AᵀB, і тензорних AᴸᴸB, AᵀᵀB, AᴸᵀB асиметрій, необхідні для оцінки радіаційних поправок до поляризаційних спостережуваних в пружному електрон-дейтронному розсіянні в лептонних змінних. Імпульсно-кутова залежність для векторних і тензорних асиметрій проілюстрована в форматі 3D. Кожна компонента асиметрії має свою особливість форми в залежності від значень кута розсіяння або імпульсу частинки. С использованием ранее полученных коэффициентов аналитической формы волновой функции дейтрона в координатном представлении для нуклон-нуклонного потенциала Argonne v18 рассчитаны борновские значения векторных AᴸB, AᵀB, и тензорных AᴸᴸB, AᵀᵀB, AᴸᵀB асимметрий, необходимые для оценки радиационных поправок к поляризационным наблюдаемым в упругом электрон-дейтронном рассеянии в лептонных переменных. Импульсно-угловая зависимость для векторных и тензорных асимметрий проиллюстрирована в формате 3D. Каждая компонента асимметрии имеет свою особенность формы в зависимости от значений угла рассеяния или импульса частицы. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Nuclear physics and elementary particles Born values for vector and tensor asymmetries in electron-deuteron scattering Борнівські значення векторних і тензорних асиметрій в електрон-дейтронному розсіянні Борновские значения векторных и тензорных асимметрий в электрон-дейтронном рассеянии Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Born values for vector and tensor asymmetries in electron-deuteron scattering |
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Born values for vector and tensor asymmetries in electron-deuteron scattering Zhaba, V.I. Nuclear physics and elementary particles |
| title_short |
Born values for vector and tensor asymmetries in electron-deuteron scattering |
| title_full |
Born values for vector and tensor asymmetries in electron-deuteron scattering |
| title_fullStr |
Born values for vector and tensor asymmetries in electron-deuteron scattering |
| title_full_unstemmed |
Born values for vector and tensor asymmetries in electron-deuteron scattering |
| title_sort |
born values for vector and tensor asymmetries in electron-deuteron scattering |
| author |
Zhaba, V.I. |
| author_facet |
Zhaba, V.I. |
| topic |
Nuclear physics and elementary particles |
| topic_facet |
Nuclear physics and elementary particles |
| publishDate |
2020 |
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English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Борнівські значення векторних і тензорних асиметрій в електрон-дейтронному розсіянні Борновские значения векторных и тензорных асимметрий в электрон-дейтронном рассеянии |
| description |
Using the previously obtained analytic form factors for the deuteron wave function in the coordinate representation for the nucleon-nucleon potential Argonne v18, are calculated the Born values for vector AᴸB, AᵀB and tensor AᴸᴸB, AᵀᵀB, AᴸᵀB asymmetries, which are necessary for estimating radiative corrections to polarization observables in elastic electron-deuteron scattering in lepton variables. The momentum-angular dependence for vector and tensor asymmetries is illustrated in 3D format. Each component of the asymmetry has its own peculiarity of the form depending on the values of the scattering angle or momentum of the particle.
З використанням раніше отриманих коефіцієнтів аналітичної форми хвильової функції дейтрона в координатному представленні для нуклон-нуклонного потенціалу Argonne v18 розраховані борнівські значення векторних AᴸB, AᵀB, і тензорних AᴸᴸB, AᵀᵀB, AᴸᵀB асиметрій, необхідні для оцінки радіаційних поправок до поляризаційних спостережуваних в пружному електрон-дейтронному розсіянні в лептонних змінних. Імпульсно-кутова залежність для векторних і тензорних асиметрій проілюстрована в форматі 3D. Кожна компонента асиметрії має свою особливість форми в залежності від значень кута розсіяння або імпульсу частинки.
С использованием ранее полученных коэффициентов аналитической формы волновой функции дейтрона в координатном представлении для нуклон-нуклонного потенциала Argonne v18 рассчитаны борновские значения векторных AᴸB, AᵀB, и тензорных AᴸᴸB, AᵀᵀB, AᴸᵀB асимметрий, необходимые для оценки радиационных поправок к поляризационным наблюдаемым в упругом электрон-дейтронном рассеянии в лептонных переменных. Импульсно-угловая зависимость для векторных и тензорных асимметрий проиллюстрирована в формате 3D. Каждая компонента асимметрии имеет свою особенность формы в зависимости от значений угла рассеяния или импульса частицы.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/194559 |
| citation_txt |
Born values for vector and tensor asymmetries in electron-deuteron scattering / V.I. Zhaba // Problems of atomic science and tecnology. — 2020. — № 5. — С. 19-22. — Бібліогр.: 12 назв. — англ. |
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2025-11-24T15:54:11Z |
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2025-11-24T15:54:11Z |
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| fulltext |
BORN VALUES FOR VECTOR AND TENSOR
ASYMMETRIES IN ELECTRON-DEUTERON SCATTERING
V. I. Zhaba∗
Uzhhorod National University, 88000 Uzhhorod, Ukraine, Voloshin Str. 54
(Received March 13, 2019)
Using the previously obtained analytic form factors for the deuteron wave function in the coordinate representation
for the nucleon-nucleon potential Argonne v18, are calculated the Born values for vector AL
B , A
T
B and tensor ALL
B ,
ATT
B , ALT
B asymmetries, which are necessary for estimating radiative corrections to polarization observables in elastic
electron-deuteron scattering in lepton variables. The momentum-angular dependence for vector and tensor asymme-
tries is illustrated in 3D format. Each component of the asymmetry has its own peculiarity of the form depending
on the values of the scattering angle or momentum of the particle.
PACS: 03.65.Nk, 13.40.Gp, 13.88.+e, 21.45.Bc
1. INTRODUCTION
A deuteron can be used as a target for an electron
beam or as a particle that is scattered on a proton
and nuclei. For example, in paper [1], results are
presented for spin-dependent electron scattering on
polarized protons and deuterons for the BLAST ex-
periment, carried out at the MIT-Bates Linear Accel-
erator Center. The paper [2] presents the results of a
study for spin observables in dp- scattering and test-
ing T -invariance in applying the modified Glauber
theory. In [3] the full set of deuteron analyzing pow-
ers in elastic dp-scattering at 190MeV/nucleon is in-
dicated. In [4] proton and deuteron analyzing powers
and 10 spin correlation coefficients were measured for
elastic p+d scattering at an energy of bombarding
protons of 135 and 200MeV . In this paper, analyti-
cal forms of DWF are used for theoretical calculations
of the Born values of vector and tensor asymmetries,
which are necessary for estimating radiative correc-
tions to polarization observables in elastic electron-
deuteron scattering in lepton variables [5]. For nu-
merical calculations used the realistic phenomenolog-
ical potential of the Argonne group – Argonne v18.
2. THE VECTOR AND TENSOR
ASYMMETRIES
The task of studying the lepton radiative corrections
in elastic electron-deuteron scattering remains rele-
vant in recent years [6].
In order to obtain formulas for radiative correc-
tions to the polarization observables for the reaction
e−(k1) + d(p1) → e−(k2) + d(p2), it is necessary to
parameterize the state of polarization of the target in
the definitions of the 4-moment of the particles in this
reaction [5]. The 4-vector sµ of deuteron polarization
and the quadrupole polarization tensor pµν describe
the polarization state of the target. For the polariza-
tion state, this parametrization depends on the di-
rections along which the longitudinal and transverse
polarization components of the deuteron in the fixed
frame are determined. The quantity sµ describes the
vector polarization of the deuteron.
Five Born values of vector AL
B , AT
B and tensor
ALL
B , ATT
B , ALT
B asymmetries were considered when
searching for radiative corrections to polarization ob-
servables in elastic ed- scattering in leptonic variables
[5].
The spin-dependent parts of the cross-section are
determined by the vector polarization of the initial
deuteron and by the longitudinal polarization of the
electron beam [5, 7]
dσL
B
dp2
= − πα2
4τV 2
2− ρ
ρ
√
ρ(4τ + ρ)G2
M ;
dσT
B
dp2
= −πα2
V p2
√
(4τ + ρ)c
τ
GMG,
where
G = 2GC +
2
3
ηGQ; c = 1− ρ− ρτ ;
η =
P 2
4M2
D
; ρ =
p2
V
; τ =
M2
D
V
.
In the laboratory system, these expressions for the
cross-sections allow one to write values for asymme-
tries (or the spin correlation coefficients) in elastic
ed-scattering in the Born approximation [7]
dσL
B
dp2
=
π
ε22
σNS
√
(1 + η)(1 + η sin2 φ) tanφ cscφG2
M ;
dσT
B
dp2
= 2
π
ε22
σNS
√
η(1 + η) tanφGM
(
GC +
η
3
GQ
)
,
∗Corresponding author E-mail address: victorzh@meta.ua
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2020, N5(129).
Series: Nuclear Physics Investigations (74), p.19-22.
19
where ε2 are the energy of the scattered electron;
GC(p), GQ(p), GM (p) are deuteron form factors;
φ = θe/2; θe is the electron scattering angle.
These two vector asymmetries are formed due to
the vector polarization of the deuteron target (respec-
tively, the longitudinal and transverse directions of
the spin 4-vectors) and the longitudinal polarization
of the electron beam [5]
AL
B = −η
√
(1 + η)(1 + η sin2 φ) tanφ cscφG2
MI−1
0 ; (1)
AT
B = −2
√
η(1 + η) tanφGM
(
GC +
η
3
GQ
)
I−1
0 , (2)
where
I0 = A+B tan2 φ.
The ratio between the vector longitudinal and
vector transverse polarization asymmetries is written
in the form [7]
AL
B
AT
B
=
dσL
B/dσB
dσT
B/dσB
=
2− ρ
4
√
ρ
cτ
GM
G
(3)
or in the laboratory system [5]
AL
B
AT
B
=
√
η(1 + η sin2 φ) cscφ
GM
G
. (4)
The 4-vector for a tensor-polarized deuteron tar-
get is written as
s(I)µ =
2εµλρσp1λk1ρk2σ
V
√
V cρ
at I = L, T, N.
In the Born approximation, part of the cross-
section depends on the tensor polarization of the
deuteron target [5, 7]
dσp
B
dp2
=
dσLL
B
dp2
RLL+
dσTT
B
dp2
(RTT −RNN )+
dσLT
B
dp2
RLT ,
where are the three components for this section:
dσLL
B
dp2
=
πα2
p4
2cη ×
×
{
8GCGQ +
8
3
ηG2
Q +
2c+ 4τρ+ ρ2
2c
G2
M
}
;
dσTT
B
dp2
=
πα2
p4
2cηG2
M ;
dσLT
B
dp2
=
πα2
p4
4η(2− ρ)
√
cρ
τ
GQGM .
In the laboratory system these expressions for
the cross-sections lead to the following three tensor
asymmetries (or analyzing capabilities) in elastic ed-
scattering, that were induced by tensor polarization
of the deuteron target and the unpolarized electron
beam [7]
dσp
B
dp2
=
π
ε22
σNS [SLLRLL+STT (RTT−RNN )+SLTRLT ],
or this formula is presented in [5] as
I0A
p
B = ALL
B RLL +ATT
B (RTT −RNN ) +ALT
B RLT ,
where ALL
B , ATT
B , ALT
B are the tensor polarizations
asymmetries:
ALL
B =
1
2
{
8ηGCGQ +
8
3
η2G2
Q+
+η
[
1 + 2(1 + η) tan2 φ
]
G2
M
}
I−1
0 ; (5)
ATT
B =
1
2
ηG2
MI−1
0 ; (6)
ALT
B = −4η
√
η + η2 sin2 φ secφGQGMI−1
0 . (7)
Between the components Sij and Aij
B there is the
following relationship (π/ε22)σNSSij = Aij
B/I0.
3. CALCULATIONS
In paper [8] for research of radiative corrections to
the polarization observed in elastic ed- scattering in
leptonic variables have been calculated the Born val-
ues of vector and tensor asymmetries. The deuteron
wave functions in coordinate representation for eight
nucleon-nucleon potentials (Nijm1, Nijm2, Nijm93,
Reid93, Argonne v18, OBEPC, MT and Paris) were
applied for numerical calculations of these asymme-
tries. The momentum-angular dependence of val-
ues vector Ai
B(p, θ) and tensor Aij
B(p, θ) asymme-
tries have been also evaluated in 3D format only for
Reid93 potentials. Due to the fact that in [8] the
values of the angular dependence of these five asym-
metries for the Argonne v18 potential are calculated,
therefore, in this paper, only the momentum-angular
dependence of the asymmetries will be calculated.
The deuteron wave function (DWF) in the coordi-
nate representation for the phenomenological realistic
nucleon-nucleon potential Argonne v18 in analytical
form [9] is used for numerical calculations
u(r) = r3/2
∑N
i=1 Ai exp(−air
3)
w(r) = r
∑N
i=1 Bi exp(−bir
3).
(8)
The coefficients of DWF for potential Argonne v18
are given in [9].
In the laboratory system, the Born values of
the vector (1), (2) and tensor (5)-(7) asymme-
tries are determined by the deuteron form factors.
In turn, the deuteron form factors depend on the
DWF in the coordinate representation (see [10, 11]).
20
Fig.1. The vector asymmetry
Fig.2. The vector asymmetry
Fig.3. The tensor asymmetries
Fig.4. The tensor asymmetries
Fig.5. The tensor asymmetries
The Figs.1-5 display momentum-angular depen-
dence in 3D format for vector Ai
B(p, θ) and tensor
Aij
B(p, θ) asymmetries, which are calculated for for
DWF (8) for Argonne v18 potential.
For vector asymmetry AL
B (see Fig.1), the charac-
teristic plane at small angles up to 70 degrees and a
rapid decrease at large scattering angles. According
to Fig.2 the vector asymmetry AT
B forms a kind of
”tray”.
As seen in Figs.3 and 4 for tensor asymmetries
ALL
B and ATT
B there is a hump (peak) near 3.7 fm-1
in the range of angles 0-180 degrees. For the tensor
asymmetry ALT
B (see Fig.5), on the contrary, there is
a pit.
4. CONCLUSIONS
The previously obtained coefficients of the analytical
form of the deuteron wave function (8) in the coordi-
nate representation for the phenomenological realis-
tic nucleon-nucleon potential Argonne v18 calculated
the set of Born values for vector AL
B, A
T
B and tensor
ALL
B , ATT
B , ALT
B asymmetries. These values for asym-
metries are necessary for the subsequent estimating
of radiative corrections to polarization observables in
elastic ed- scattering in lepton variables [5].
As shown in Figs.1-5 each component of the vector
and tensor asymmetries has its own peculiarity of the
form depending on the values of the scattering angle
or the momentum of the particle. The aforecited and
reviewed vector and tensor asymmetries (the spin cor-
relation coefficients) can be compared with two sets
of components for cross-sections [5]:
dσβ
B
dp2
= VβA(−θ)
dσA
B
dp2
atA = L, T andβ = l, t;
dσβ
B
dp2
= TβA(−θ)
dσA
B
dp2
atA = LL, TT andβ = ll, tt, lt
for polarization 4-vectors
s(l)µ =
2τk1µ − p1µ
MD
; s(n)µ = s(N)
µ ;
s(t)µ =
k2µ − (1− ρ− 2ρτ)k1µ − ρp1µ√
V cρ
.
In addition, further studies of polarization observ-
ables in elastic lepton-deuteron scattering, including
21
lepton masses [12] (and for the case of spin correla-
tion coefficients in the limit of zero lepton mass) are
promising.
References
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tering from Polarized Protons and Deuterons
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2. A.A.Temerbayev, Yu.N.Uzikov. Spin Observ-
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Invariance Test // Phys. Atom. Nucl. 2015, v.78,
p.35-42.
3. K. Sekiguchi et al. Complete set of deuteron
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p.064001.
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and deuteron wave function for different nucleon-
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tion in coordinate space and tensor polarization
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p.1650139.
10. F.Gross. Relativistic Calculation of the Deuteron
Electromagnetic Form Factor. II* // Phys. Rev.
1964, v.136, p.B140-B161.
11. R.Gilman, F.Gross. Electromagnetic structure
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R116.
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Polarization observables in lepton-deuteron elas-
tic scattering including the lepton mass // Phys.
Rev. C. 2014, v.90, p.064901.
ÁÎÐÍÎÂÑÊÈÅ ÇÍÀ×ÅÍÈß ÂÅÊÒÎÐÍÛÕ È ÒÅÍÇÎÐÍÛÕ ÀÑÈÌÌÅÒÐÈÉ
 ÝËÅÊÒÐÎÍ-ÄÅÉÒÐÎÍÍÎÌ ÐÀÑÑÅßÍÈÈ
Â.È.Æàáà
Ñ èñïîëüçîâàíèåì ðàíåå ïîëó÷åííûõ êîýôôèöèåíòîâ àíàëèòè÷åñêîé ôîðìû âîëíîâîé ôóíêöèè äåé-
òðîíà â êîîðäèíàòíîì ïðåäñòàâëåíèè äëÿ íóêëîí-íóêëîííîãî ïîòåíöèàëà Argonne v18 ðàññ÷èòàíû áîð-
íîâñêèå çíà÷åíèÿ âåêòîðíûõ AL
B , A
T
B è òåíçîðíûõ ALL
B , ATT
B , ALT
B àñèììåòðèé, íåîáõîäèìûå äëÿ îöåíêè
ðàäèàöèîííûõ ïîïðàâîê ê ïîëÿðèçàöèîííûì íàáëþäàåìûì â óïðóãîì ýëåêòðîí-äåéòðîííîì ðàññåÿíèè
â ëåïòîííûõ ïåðåìåííûõ. Èìïóëüñíî-óãëîâàÿ çàâèñèìîñòü äëÿ âåêòîðíûõ è òåíçîðíûõ àñèììåòðèé
ïðîèëëþñòðèðîâàíà â ôîðìàòå 3D. Êàæäàÿ êîìïîíåíòà àñèììåòðèè èìååò ñâîþ îñîáåííîñòü ôîðìû â
çàâèñèìîñòè îò çíà÷åíèé óãëà ðàññåÿíèÿ èëè èìïóëüñà ÷àñòèöû.
ÁÎÐÍIÂÑÜÊI ÇÍÀ×ÅÍÍß ÂÅÊÒÎÐÍÈÕ I ÒÅÍÇÎÐÍÈÕ ÀÑÈÌÅÒÐIÉ
 ÅËÅÊÒÐÎÍ-ÄÅÉÒÐÎÍÍÎÌÓ ÐÎÇÑIßÍÍI
Â. I.Æàáà
Ç âèêîðèñòàííÿì ðàíiøå îòðèìàíèõ êîåôiöi¹íòiâ àíàëiòè÷íî¨ ôîðìè õâèëüîâî¨ ôóíêöi¨ äåéòðîíà â
êîîðäèíàòíîìó ïðåäñòàâëåííi äëÿ íóêëîí-íóêëîííîãî ïîòåíöiàëó Argonne v18 ðîçðàõîâàíi áîðíiâñüêi
çíà÷åííÿ âåêòîðíèõ AL
B , A
T
B i òåíçîðíèõ ALL
B , ATT
B , ALT
B àñèìåòðié, íåîáõiäíi äëÿ îöiíêè ðàäiàöiéíèõ ïî-
ïðàâîê äî ïîëÿðèçàöiéíèõ ñïîñòåðåæóâàíèõ ó ïðóæíîìó åëåêòðîí-äåéòðîííîìó ðîçñiÿííi â ëåïòîííèõ
çìiííèõ. Iìïóëüñíî-êóòîâà çàëåæíiñòü äëÿ âåêòîðíèõ i òåíçîðíèõ àñèìåòðié ïðîiëþñòðîâàíà â ôîðìàòi
3D. Êîæíà êîìïîíåíòà àñèìåòði¨ ì๠ñâîþ îñîáëèâiñòü ôîðìè â çàëåæíîñòi âiä çíà÷åíü êóòà ðîçñiÿííÿ
àáî iìïóëüñó ÷àñòèíêè.
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