Polarization effects in elastic proton-electron scattering
Proton elastic scattering from electrons at rest is calculated in the Born approximation. The interest of this reaction is related to the possibility of polarizing high energy antiproton beam and to high energy proton polarimetry. The differential cross section and polarization observables have been...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Polarization effects in elastic proton-electron scattering / G.I. Gakh, A.Dbeyssi, D. Marchand, V.V.Bytev, E.Tomasi-Gustafsson // Вопросы атомной науки и техники. — 2012. — № 1. — С. 79-83. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860017905563860992 |
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| author | Gakh, G.I. Dbeyssi, A. Marchand, D. Bytev, V.V. Tomasi-Gustafsson, E. |
| author_facet | Gakh, G.I. Dbeyssi, A. Marchand, D. Bytev, V.V. Tomasi-Gustafsson, E. |
| citation_txt | Polarization effects in elastic proton-electron scattering / G.I. Gakh, A.Dbeyssi, D. Marchand, V.V.Bytev, E.Tomasi-Gustafsson // Вопросы атомной науки и техники. — 2012. — № 1. — С. 79-83. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Proton elastic scattering from electrons at rest is calculated in the Born approximation. The interest of this reaction is related to the possibility of polarizing high energy antiproton beam and to high energy proton polarimetry. The differential cross section and polarization observables have been derived assuming one photon exchange. Numerical estimates are given for the cross section and the spin correlation coefficients in a wide kinematical range.
В Борновском приближении вычислено упругое рассеяние протонов на покоящихся электронах. Интерес к этой реакции обусловлен возможностью поляризации высокоэнергетического антипротонного пучка и поляриметрией протонов высоких энергий. В предположении однофотонного обмена получены выражения для дифференциального сечения и поляризационных наблюдаемых. Выполнены численные оценки сечения и коэффициентов корреляции спина в широкой кинематической области.
У Борнівському наближенні обчислено пружне розсіювання протонів на електронах, які знаходяться у стані спокою. Інтерес до цієї реакції обумовлено можливістю поляризації високоенергетичного антипротонного пучка та поляриметрією протонів високих енергій. У передбаченні однофотонного обміну одержані вирази для диференційного перерізу та поляризаційних спостережуваних. Виконані чисельні оцінки перерізу та коефіцієнтів кореляції спінів у широкій кінематичній області.
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| first_indexed | 2025-12-07T16:46:11Z |
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| fulltext |
POLARIZATION EFFECTS IN ELASTIC
PROTON-ELECTRON SCATTERING
G.I. Gakh 1∗, A. Dbeyssi 2, D. Marchand 2, V.V. Bytev 3,
and E. Tomasi-Gustafsson 2,4
1National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
2CNRS/IN2P3, IPNO, UMR 8608, Univ. Paris-Sud, 91405, Orsay, France
3BLTP, Joint Institute for Nuclear Research, Dubna, Russia
4CEA,IRFU,SPhN, Saclay, 91191, Gif-sur-Yvette, France
(Received October 12, 2011)
Proton elastic scattering from electrons at rest is calculated in the Born approximation. The interest of this reaction
is related to the possibility of polarizing high energy antiproton beam and to high energy proton polarimetry. The
differential cross section and polarization observables have been derived assuming one photon exchange. Numerical
estimates are given for the cross section and the spin correlation coefficients in a wide kinematical range.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION
The polarized and unpolarized scattering of electrons
by protons has been widely studied, as it is consid-
ered the simplest way to access information on pro-
ton structure. The expressions which relate the po-
larization observables to the proton electromagnetic
form factors were firstly derived in Ref. [1], assuming
that the interaction occurs through the exchange of
a virtual photon. In the scattering of proton from
electrons at rest (inverse kinematics) approximations
such as neglecting the electron mass no longer hold.
Liquid hydrogen targets are considered as proton tar-
gets, but any reaction with such targets also involves
reactions with atomic electrons, which we will assume
to be at rest.
A large interest in inverse kinematics (proton pro-
jectile on electron target) has been aroused due to
two possible applications: the possibility to build
beam polarimeters, for high-energy polarized proton
beams, in the relativistic heavy-ion collider (RHIC)
energy range [2] and the possibility to build polar-
ized antiprotons beams [3], which would open a wide
domain of polarization studies at the GSI facility for
Antiproton and Ion Research (FAIR) at Darmstadt
(Germany).
Concerning the polarimetry of high-energy proton
beams, in Ref. [2] analyzing powers corresponding to
polarized proton beam and electron target were nu-
merically calculated for elastic proton-electron scat-
tering.
The possibility of polarizing a proton beam in a
storage ring by circulating through a polarized hy-
drogen target would be extremely interesting. Such
polarization was indeed observed [4]. Possible expla-
nations of the polarizing mechanisms were published
in a number of papers (see [5] and references therein).
Two mechanism could be responsible for the polar-
ization: ‘spin-filtering’, where the proton-proton in-
teraction would scatter preferentially at large angles
protons with one spin component which would be lost
from recirculating in the beam, and ‘spin-flip’ where
the reaction proton-electron at very small scatter-
ing angles would have very large analyzing powers.
The second explanation is extremely interesting as
one could polarize antiproton beams without losses
of particles.
We calculated the cross section and the polariza-
tion observables for proton electron elastic scattering,
in a relativistic approach assuming the Born approx-
imation with particular attention to the kinematics
which is very specific for this reaction. Three types of
polarization effects were studied: – the spin correla-
tion, due to the polarization of the proton beam and
of the electron target, – the polarization transfer from
the polarized electron target to the scattered proton,
– and the depolarization coefficients which describe
the polarization of the scattered proton which de-
pends on the polarization of the proton beam. Nu-
merical estimations of the polarization observables
have been performed over a wide range of proton-
beam energy and for different values of scattering
angle. Our results show that polarization effects are
sizable in the high energy domain.
2. GENERAL FORMALISM
Let us consider the reaction p(p1) + e(k1) → p(p2) +
e(k2), where particle momenta are indicated in paren-
∗Corresponding author E-mail address: gakh@kipt.kharkov.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 79-83.
79
theses, and k = k1 − k2 = p2 − p1 is the four-
momentum of the virtual photon.
A general characteristic of all reactions of elastic
and inelastic hadron scattering by atomic electrons
(which can be considered at rest) is the small value
of the transfer momentum squared, even for relatively
large energies of colliding hadrons. Let us first give
details of the order of magnitude and the range which
is accessible to the kinematic variables, as they are
very specific for this reaction, and then derive the
spin structure of the matrix element and the unpo-
larized and polarized observables.
Kinematics
The following formulas can be partly found in
Ref. [6]. One can show [6] that, for a given energy
of the proton beam, the maximum value of the four-
momentum transfer squared, in the scattering on the
electron at rest, is: (Fig. 1):
(−k2)max =
4m2(E2 − M2)
M2 + 2mE + m2
, (1)
where m(M) is the electron (proton) mass. Be-
ing proportional to the electron mass squared, the
four momentum transfer squared is restricted to very
small values.
E [GeV]
1 2 3 4 5 6 7 8 9 10
2
[
G
eV
/c
]
m
ax
2
−k
0
0.02
0.04
0.06
0.08
0.1
0.12
−310×
Fig. 1. Maximum four-momentum squared as
a function of the proton beam energy
In order to have the same total energy s in proton-
electron and electron-proton scattering the proton en-
ergy should be 2000 times the electron energy. The
electron can never be scattered backward. For one
proton angle there may be two values of the proton
energy, (and two corresponding values for the recoil-
electron energy and angle, and for the transferred
momentum k2). The two solutions coincide when
the angle between the initial and final hadron takes
its maximum value, sin θh,max = m/M . Protons are
scattered from atomic electrons at very small angles.
In the one-photon-exchange approximation, the
matrix element M of reaction p + e → p + e can be
written as:
M =
e2
k2
jμJμ, (2)
where jμ(Jμ) is the leptonic (hadronic) electromag-
netic current:
jμ = ū(k2)γμu(k1),
Jμ = ū(p2)
[
F1(k2)γμ − 1
2M
F2(k2)σμνkν
]
u(p1)
= ū(p2)
[
GM (k2)γμ − F2(k2)Pμ
]
u(p1). (3)
Here F1(k2) and F2(k2) are the Dirac and Pauli pro-
ton electromagnetic form factors (FFs), GM (k2) =
F1(k2) + F2(k2) is the Sachs proton magnetic FF,
and Pμ = (p1 + p2)μ/(2M).
The matrix element squared is:
|M|2 = 16π2 α2
k4
LμνWμν , (4)
with Lμν = jμj∗ν , Wμν = JμJ∗
ν , where α = 1/137
is the electromagnetic fine structure constant. The
leptonic tensor, L
(0)
μν , for unpolarized initial and final
electrons (averaging over the initial electron spin) has
the form:
L(0)
μν = k2gμν + 2(k1μk2ν + k1νk2μ). (5)
The contribution to the electron tensor corresponding
to a polarized electron target is
L(p)
μν = 2imεμναβkαSβ , (6)
where Sβ is the initial electron polarization four-
vector.
The hadronic tensor, W
(0)
μν , for unpolarized ini-
tial and final protons can be written in the standard
form, through two unpolarized structure functions:
W (0)
μν =
(
−gμν +
kμkν
k2
)
W1(k2)+PμPνW2(k2). (7)
Averaging over the initial proton spin, the structure
functions Wi, i = 1, 2, can be expressed in terms of
the nucleon electromagnetic FFs as:
W1(k2) = −k2G2
M (k2),
W2(k2) = 4M2 G2
E(k2) + τG2
M (k2)
1 + τ
, (8)
where GE(k2) = F1(k2)− τF2(k2) is the proton elec-
tric FF and τ = −k2/(4M2).
The differential cross section over the solid angle
can be written as:
dσ
dΩe
=
1
32π2
1
m| p|
k3
2
(−k2)
|M|2
E + m
, (9)
where dΩe = 2πd cos θe (due to azimuthal
symmetry), p is the three-momentum of the proton
beam. The expression of the differential cross section
80
for unpolarized proton-electron scattering, in the co-
ordinate system where the electron is at rest, can be
written as:
dσ
dk2
=
πα2
2m2 p2
1
D
k4
, (10)
D = k2(k2 + 2m2)G2
M (k2) + 2
[
k2M2
+ 2mE
(
2mE + k2
)] [
F 2
1 (k2) + τF 2
2 (k2)
]
.
The differential cross section diverges as k4 when
k2 → 0. This is a well known result, which is a con-
sequence of the one photon exchange mechanism.
The proton structure is taken into account
through the parametrization of FFs. However, due to
the small maximum value of k2 which can be achieved
in inverse kinematics any FFs parametrization, and
even of constant FFs, where the constants correspond
to the static values would give the same results.
E [GeV]
0 50 100 150 200
[
m
b
/s
te
ra
d
]
e
Ω
/
d
σd
1
10
210
310
E [GeV]
0 50 100 150 200
[
m
b
/s
te
ra
d
]
e
Ω
/
d
σd
1
10
210
310
Fig. 2. Differential cross section as a function of
incident energy E for different angles: θe = 0 (solid
line), 10 mrad (dashed line), 30 mrad (dotted line),
50 mrad (dash-dotted line)
The energy dependence of the cross section for
different angles: θe = 0 (solid line), 10 mrad (dashed
line), 30 mrad (dotted line), 50 mrad (dash-dotted
line) is given in Fig. 2. The unpolarized differential
cross section is divergent at small values of energy; it
has an angle dependent minimum and then increases
smoothly up to large energies.
3. POLARIZATION OBSERVABLES
Let us focus here on three types of polarization ob-
servables, for elastic proton-electron scattering:
1. The polarization transfer coefficients which de-
scribe the polarization transfer from the po-
larized electron target to the scattered proton,
p + e → p + e.
2. The spin correlation coefficients when both ini-
tial particles have arbitrary polarization, p +
e → p + e.
3. The depolarization coefficients which define the
dependence of the scattered proton polariza-
tion on the polarization of the proton beam,
p + e → p + e. In our knowledge, this case was
not previously considered in the literature.
The first case is the object of a number of recent
papers [3] in connection with the possibility to po-
larize proton (antiproton) beams. The second case
was considered in Ref. [2], in view of using polarized
proton-electron scattering to measure the longitudi-
nal and transverse polarizations of high-energy pro-
ton beams.
The explicit expressions of the polarization ob-
servables can be found in Ref. [5]. At high energy,
the polarization transfer coefficients depend essen-
tially on the direction of the scattered proton polar-
ization.
Let us give, for illustration, the correlation coeffi-
cients when the incident proton and the target elec-
tron are polarized.
The contraction of the spin-dependent leptonic
L
(p)
μν and hadronic Wμν(η1) tensors, in an arbitrary
reference frame, gives:
DC(S, η1) = 8mMGM (k2)[(k · Sk · η1 − k2S · η1)
×GE(k2) + τk · η1(k · S + 2p1 · S)F2(k2)].
All spin correlation coefficients for the elastic p e col-
lisions can be obtained from this expression and are,
therefore, proportional to the proton magnetic FF.
This is a well known fact for e p scattering [6].
In the considered frame, where the target electron
is at rest, the four-vector of the proton beam polar-
ization has the following components:
η1 =
(
p · S1
M
, S1 +
p( p · S1)
M(E + M)
)
, (11)
where S1 is the unit vector describing the polarization
of the initial proton in its rest system.
Applying the P-invariance of the hadron electro-
magnetic interaction, one can write the following ex-
pression for the dependence of the differential cross
section on the polarization of the initial particles:
dσ
dk2
( ξ, S1) =
(
dσ
dk2
)
un
[1 + C��ξ�S1� + CttξtS1t
+ CnnξnS1n + C�tξ�S1t + Ct�ξtS1�] ,
where Cik, i, k = �, t, n are the corresponding spin
correlation coefficients which characterize p e scatter-
ing. Small coefficients (in absolute value) are ex-
pected for the transverse component of the beam po-
larization at high energies. This can be seen from
the expression of the components of the proton-beam-
polarization four-vector at large energies, E � M :
η1μ = (0, S1t) + S1�
( | p|
M
,
p
M
E
p
)
∼ S1�
p1μ
M
. (12)
81
The effect of the transverse beam polarization ap-
pears to be smaller by a factor 1/γ, γ = E/M � 1.
This is a consequence of the relativistic description
of the spin of the fermion at large energies.
n
n
C
-0.6
-0.4
-0.2
0n
n
C
-0.6
-0.4
-0.2
0
E [GeV]
0 50 100 150
lt
10
C
-0.1
0
tt
C
-0.6
-0.4
-0.2
0tt
C
-0.6
-0.4
-0.2
0
E [GeV]
0 50 100 150
tl
C
-0.2
-0.1
0
E [GeV]
0 50 100 150
ll
C
-1
-0.5
0
Fig. 3. Spin correlation coefficients as a function
of E for different angles. Notations are the same as
in Fig. 2
The spin correlation coefficients are shown in Fig. 3.
In collinear kinematics, in general, either polar-
ization observables take the maximal values or they
vanish. An interesting kinematic region appears at
E = 20 GeV, where a structure is present in various
observables.
It appears that polarization coefficients are in gen-
eral quite large, except at low energy. Proton electron
scattering can be used, in principle, to measure the
polarization of high-energy beams. Let us calculate
the figure of merit, for measuring the polarization of a
secondary proton beam, after scattering from atomic
electrons.
The differential figure of merit is defined as
F2(θp) = ε(θp)A2
ij(θp),
where Aij stands for a generic polarization coefficient
and ε(θp) = Nf (θp)/Ni is the number of useful events
over the number of the incident events in an interval
Δθp around θp. Because it is related to the inverse
of the statistical error on the polarization measure-
ment, this quantity is useful for a proton with degree
of polarization P :(
ΔP (θp)
P
)2
=
2
Ni(θp)F2(θp)P 2
(13)
=
2
Ltm(dσ/dΩ)dΩA2
ij(θp)P 2
,
where tm is the time of measurement.
The integrated quantity, calculated for a trans-
verse polarized proton beam scattering from a longi-
tudinally polarized electron target ( p + e → p + e)
F 2 =
∫
dσ
dk2
C2
�t(k
2)dk2 (14)
is shown in Fig. 4 as a function of the incident energy.
In Refs. [7] it was suggested to use the scattering of
a transverse polarized proton beam from a longitu-
dinally polarized electron target. From Fig. 4, one
can see that F 2 takes its maximum value for T � 10
GeV. Assuming a luminosity of 1032 cm−2 s−1 for an
ideal detector with an acceptance and efficiency of
100%, one could measure the beam polarization with
an error of 1% in a time interval of 3 min.
T[GeV]
0 50 100 150 200
2 tl
F4
10
3
4
5
6
7
8
9
10
11
Fig. 4. Variation of the quantity F 2 [a.u.] as a
function of proton kinetic energy T for a transverse
polarized proton beam scattering from a longitudi-
nally polarized electron target ( p + e → p + e)
4. CONCLUSIONS
The elastic scattering of protons from electrons at
rest was investigated in a relativistic approach in the
one-photon-exchange (Born) approximation. This re-
action, where the target is three orders of magnitude
lighter than the projectile, has specific kinematical
features due to the “inverse kinematics” (i.e., the pro-
jectile is heavier than the target). For example, the
proton is scattered at very small angles and the al-
lowed momentum transfer does not exceed the MeV2
scale, even when the proton incident energy is of the
order of GeV. The differential cross section and var-
ious double spin polarization observables have been
calculated in terms of the nucleon electromagnetic
FFs. However, for the values of transferred momen-
tum involved, any parametrization of FFs is equiva-
lent and is very near to the static values. The spin
transfer coefficients to a polarized scattered proton
were calculated when the proton beam is polarized or
when the electron target is polarized. The correlation
spin coefficients when the proton beam and the elec-
tron target are both polarized were also calculated.
Numerical estimates showed that polarization effects
may be sizable in the GeV range, and that the po-
larization transfer coefficients for p + e → p+ e could
be used to measure the polarization of high energy
82
proton beams. The calculated values of the scattered
proton polarization for the reaction p + e → p + e at
proton-beam energies lower then a few tens of MeV
show that it is not possible to obtain sizable polariza-
tion of the antiproton beam in an experimental setup
where antiprotons and electrons collide with small
relative velocities. The present results confirm that
the polarization of the scattered proton has large val-
ues at high proton-beam energies. Thus, one could
consider an experimental setup where high-energy
protons collide with a polarized electron target at
rest. The low values of momentum transfer which
are involved ensure that the cross section is sizable.
References
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| id | nasplib_isofts_kiev_ua-123456789-106987 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:46:11Z |
| publishDate | 2012 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Gakh, G.I. Dbeyssi, A. Marchand, D. Bytev, V.V. Tomasi-Gustafsson, E. 2016-10-10T15:55:35Z 2016-10-10T15:55:35Z 2012 Polarization effects in elastic proton-electron scattering / G.I. Gakh, A.Dbeyssi, D. Marchand, V.V.Bytev, E.Tomasi-Gustafsson // Вопросы атомной науки и техники. — 2012. — № 1. — С. 79-83. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk https://nasplib.isofts.kiev.ua/handle/123456789/106987 Proton elastic scattering from electrons at rest is calculated in the Born approximation. The interest of this reaction is related to the possibility of polarizing high energy antiproton beam and to high energy proton polarimetry. The differential cross section and polarization observables have been derived assuming one photon exchange. Numerical estimates are given for the cross section and the spin correlation coefficients in a wide kinematical range. В Борновском приближении вычислено упругое рассеяние протонов на покоящихся электронах. Интерес к этой реакции обусловлен возможностью поляризации высокоэнергетического антипротонного пучка и поляриметрией протонов высоких энергий. В предположении однофотонного обмена получены выражения для дифференциального сечения и поляризационных наблюдаемых. Выполнены численные оценки сечения и коэффициентов корреляции спина в широкой кинематической области. У Борнівському наближенні обчислено пружне розсіювання протонів на електронах, які знаходяться у стані спокою. Інтерес до цієї реакції обумовлено можливістю поляризації високоенергетичного антипротонного пучка та поляриметрією протонів високих енергій. У передбаченні однофотонного обміну одержані вирази для диференційного перерізу та поляризаційних спостережуваних. Виконані чисельні оцінки перерізу та коефіцієнтів кореляції спінів у широкій кінематичній області. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section B. QED Processes at High Energies Polarization effects in elastic proton-electron scattering Поляризационные эффекты в упругом протон-электронном рассеянии Поляризаційні ефекти в пружному електрон-протонному розсіюванні Article published earlier |
| spellingShingle | Polarization effects in elastic proton-electron scattering Gakh, G.I. Dbeyssi, A. Marchand, D. Bytev, V.V. Tomasi-Gustafsson, E. Section B. QED Processes at High Energies |
| title | Polarization effects in elastic proton-electron scattering |
| title_alt | Поляризационные эффекты в упругом протон-электронном рассеянии Поляризаційні ефекти в пружному електрон-протонному розсіюванні |
| title_full | Polarization effects in elastic proton-electron scattering |
| title_fullStr | Polarization effects in elastic proton-electron scattering |
| title_full_unstemmed | Polarization effects in elastic proton-electron scattering |
| title_short | Polarization effects in elastic proton-electron scattering |
| title_sort | polarization effects in elastic proton-electron scattering |
| topic | Section B. QED Processes at High Energies |
| topic_facet | Section B. QED Processes at High Energies |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106987 |
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