Model independent analysis of polarization effects in elastic proton-electron scattering

Abstract

The experimental observables for the elastic reaction induced by protons on
electrons are calculated in the Born approximation. Model independent
expressions are derived for the differential cross section and polarization
observables. Numerical estimations are given for spin correlation coefficients,
polarization transfer coefficients and depolarization coefficients, in a wide
kinematical range. Specific attention is given to the kinematical conditions,
i.e., to the specific range of incident energy and transferred momentum.

Full text

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arXiv:1103.2540v1 [nucl-th] 13 Mar 2011
Model independent analysis of polarization effects in elastic
proton-electron scattering
G. I. Gakh
National Science Centre ”Kharkov Institute of Physics and Technology”
61108 Akademicheskaya 1, Kharkov, Ukraine
A. Dbeyssi,∗D. Marchand, and E. Tomasi-Gustafsson†
CNRS/IN2P3, Institut de Physique Nucl´ eaire, UMR 8608, 91405 Orsay, France
V. V. Bytev
Joint Institute for Nuclear Research, Dubna, Russia
Abstract
The experimental observables for the elastic reaction induced by protons on electrons are calcu-
lated in the Born approximation. Model independent expressions are derived for the differential
cross section and polarization observables. Numerical estimations are given for spin correlation
coefficients, polarization transfer coefficients and depolarization coefficients, in a wide kinematical
range. Specific attention is given to the kinematical conditions, i.e., to the specific range of incident
energy and transferred momentum.
PACS numbers:
∗Boursier du CNRS libanais/LNCSR Scholar
†E-mail: etomasi@cea.fr; Permanent address: CEA,IRFU,SPhN, Saclay, 91191 Gif-sur-Yvette Cedex,
France
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I.INTRODUCTION
The polarized and unpolarized scattering of electrons by protons has been widely stud-
ied, as it is considered the simpler way to access information on proton structure. Model
independent expressions, which relate the cross section and polarization observables to the
proton electromagnetic form factors were firstly derived in Ref. [1]. The scattering of proton
on electrons at rest (inverse kinematics) is more complicated, in principle, because approxi-
mations, such as neglecting the electron mass, do not hold anymore. Liquid hydrogen targets
are considered as proton targets, but any reaction on such targets involves also reactions
with atomic electrons, which we will assume to be at rest.
Large interest in inverse kinematics (proton projectile on electron target) aroused, due
to two possible applications: - the possibility to build beam polarimeters, for high energy
polarized proton beams, in the RHIC energy range [2], - the possibility to build polarized
antiprotons beams [3], which would open a wide domain of polarization studies at the FAIR
facility [4, 5]. Indeed, assuming C-invariance in electromagnetic interaction, the (elastic and
inelastic) reactions p + e−and ¯ p + e+are strictly equivalent.
Concerning the polarimetry of the high energy proton beams, in Ref. [2] analyzing powers
corresponding to polarized proton beam and electron target were numerically calculated for
the elastic proton-electron scattering, assuming one photon exchange mechanism and dipole
approximation for the proton form factors. It was shown that the analyzing powers, as
functions of the proton beam energy E, reach a maximum for forward scattering at E = 50
GeV, where the cross section is small. The authors concluded that the concept of such
polarimeter is realistic for longitudinal as well as transverse proton beam polarizations. On
the other hand, in that paper, explicit expressions for the analyzing powers were not given.
The possibility of polarizing a proton beam in a storage ring by circulating through a
polarized hydrogen target was reported in Ref. [6]. Possible explanations of the polarizing
mechanisms were published in a number of papers [7–9], and more recently in Refs. [10, 11].
In this work, we derive the cross section and the polarization observables for proton
electron elastic scattering, in a relativistic approach assuming Born approximation. We
derive relations connecting kinematical variables in direct and inverse kinematics. Three
types of polarization effects are studied: - the spin correlation, due to the polarization of
the proton beam and of the electron target, - the polarization transfer from the polarized
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electron target to the scattered proton, - and the depolarization coefficients which describe
the polarization of the scattered proton, depending on the polarization of the proton beam.
Numerical estimations of the polarization observables have been performed in a wide range
of the proton beam energy and for different values of the scattering angle.
We discuss model independent properties of the observables for proton–electron elastic
scattering and compare to the recent theoretical and experimental ongoing work related to
the production and the properties of high energy polarized (anti)proton beams.
II. FORMALISM
Let us consider the reaction (Fig. 1):
p(p1) + e(k1) → p(p2) + e(k2), (1)
where the particle momenta are indicated in parenthesis, and k = k1− k2 = p2− p1 is
the four momentum of the virtual photon. In the one photon exchange approximation, the
matrix element M of the reaction (1) can be written as:
M =e2
k2jµJµ, (2)
where jµ(Jµ) is the leptonic (hadronic) electromagnetic current:
jµ = ¯ u(k2)γµu(k1),
Jµ = ¯ u(p2)
= ¯ u(p2)?GM(k2)γµ− F2(k2)Pµ
?
F1(k2)γµ−
1
2MF2(k2)σµνkν
?
u(p1)
?u(p1). (3)
Here F1(k2) and F2(k2) are the Dirac and Pauli proton electromagnetic form factors (FFs),
GM(k2) = F1(k2) + F2(k2) is the Sachs proton magnetic FF, M is the proton mass, and
Pµ= (p1+ p2)µ/(2M). The matrix element squared is:
|M|2= 16π2α2
k4LµνWµν, with Lµν= jµj∗
ν, Wµν= JµJ∗
ν, (4)
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)
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)
1
p(p
(k)
*
γ
)
2
(k
−e
)
1
(k
−e
)
2
p(p
FIG. 1: Feynman diagram for the reaction p(p1)+e(k1) → p(p2)+e(k2). The transfer momentum
of the virtual photon is k = k1− k2= p2− p1.
The contribution to the electron tensor corresponding to a polarized electron target is
L(p)
µν= 2imǫµναβkαSβ, (6)
where Sβis the initial electron polarization four vector and m is the electron mass.
The hadronic tensor, W(0)
µν , for unpolarized initial and final protons can be written in the
standard form, through two unpolarized structure functions:
?
Averaging over the initial proton spin, the structure functions Wi, i = 1,2, can be expressed
W(0)
µν=−gµν+kµkν
k2
?
W1(k2) + PµPνW2(k2). (7)
in terms of the nucleon electromagnetic FFs as:
W1(k2) = −k2G2
W2(k2) = 4M2G2
M(k2),
E(k2) + τG2
1 + τ
M(k2)
, (8)
where GE(k2) = F1(k2) − τF2(k2) is the proton electric FF and τ = −k2/4M2.
The differential cross section is related to the matrix element squared (4) by
dσ =
(2π)4|M|2
4?(k1· p1)2− m2M2
d3?k2
(2π)32ǫ2
d3? p2
(2π)32E2δ4(k1+ p1− k2− p2), (9)
where p2(E2) is the momentum (energy) of the final proton, ǫ2is the energy of the scattered
electron. From this point, formulas will differ from the elastic electron-proton scattering, as
we introduce a reference system where the electron is at rest. In this system, the differential
cross section can be written as:
dσ
dǫ2
=
1
32π
|M|2
m? p2, (10)
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where ? p is the momentum of the proton beam. The average over the spins of the initial
particles has been included in the leptonic and hadronic tensors. Using the relation
k2= 2m(m − ǫ2) (11)
one can write
dσ
dk2=
1
64π
|M|2
m2? p2. (12)
The differential cross section over the solid angle can be written as:
dσ
dΩe
=
1
32π2
1
mp
?k3
−k2
2
|M|2
E + m,
(13)
where E is the proton beam energy and dΩe= 2πdcosθ (due to azimuthal symmetry). We
used the relation
dǫ2=
p
E + m
?k3
2
m(ǫ2− m)
dΩe
2π.
(14)
Let us focus here on three types of polarization observables, for elastic proton-electron
scattering
1. The polarization transfer coefficients which describe the polarization transfer from the
polarized electron target to the scattered proton, p +? e → ? p + e;
2. The spin correlation coefficients when both initial particles have arbitrary polarization,
? p +? e → p + e;
3. The depolarization coefficients which define the dependence of the scattered proton
polarization 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 polarize proton (antiproton) beams. The second case was considered in Ref. [2], in view
of using the polarized proton-electron scattering for the measurement of the longitudinal
and transverse polarizations of a high energy proton beams.
Let us calculate the hadronic tensor, when the initial or final proton is polarized. The
contribution of the proton polarization to the hadronic tensor is:
Wµν(ηj) = −2iGM(k2)?MGM(k2)ǫµναβkαηjβ+ F2(k2)(Pµǫναβγ− Pνǫµαβγ)p1αp2βηjγ
?, (15)
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where the four vector ηj (j = 1,2) stands for initial (final) proton polarization. One can
see that all the correlation coefficients in ? p? e collisions are proportional to the proton mag-
netic FF. This is a well known fact for ? e? p scattering [12]. The dependence of the different
polarization observables, namely, the spin correlation (the polarization transfer) coefficients
on the polarization four vector of the initial (scattered) proton is completely determined by
the spin dependent part of the hadronic tensor Wµν(ηj), j = 1 (j = 2).
The expression of the differential cross section for unpolarized proton-electron scattering,
in the coordinate system where the electron is at rest can be written as:
dσ
dk2=
πα2
2m2? p2
D
k4, (16)
with
D = k2(k2+ 2m2)G2
M(k2) + 2?k2M2+ 2mE?2mE + k2???F2
1(k2) + τF2
2(k2)?. (17)
It can be written in terms of the Sachs FFs as:
D = k2(k2+ 2m2)G2
M(k2) + 2
?
k2M2+
1
1 + τ
?
2mE +k2
2
?2?
?G2
E(k2) + τG2
M(k2)?. (18)
This expression is consistent with Ref. [2].
Note that the differential cross section diverges as k4when k2→ 0. This is expected from
the one photon exchange mechanism.
A. Polarization transfer coefficients, Tij, in the p +? e → ? p + e reaction
These polarization observables describe the polarization transfer from the polarized target
to the ejectile. The transfer coefficients are also called Ti00jin the notations from [13]. Here
the four subscripts denote, in the order, ejectile, recoil, projectile, target. The indexes i, j
correspond to n, t, ℓ, according to the direction of the polarization vectors of each particle.
The dependence of the scattered proton polarization on the polarization state of the
initial electron is obtained by contraction of the spin-dependent leptonic tensor L(p)
µν, Eq.
(6), and the spin-dependent hadronic tensor Wµν(η2), Eq. (15). The following formula hold
in any reference system and can be used to obtain the polarization transfer coefficients :
DT(S,η2) = 4mMGM(k2)?GE(k2)(k · Sk · η2− k2S · η2) − k2F2(k2)P · SP · η2
?. (19)
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In the frame where the initial electron is at rest, the polarization four vectors of the electron
Sµand of the scattered proton η2µhave the following components:
S ≡ (0,?ξ), η2≡
?
1
M? p2·?S2,?S2+
? p2(? p2·?S2)
M(E2+ M)
?
, (20)
where?ξ and?S2are the unit three-vectors of the initial electron and scattered proton polariza-
tions in their rest systems, respectively; ? p2(E2) is the momentum (energy) of the final proton.
In the laboratory system (inverse kinematics) one can write ? p =?k2+ ? p2, m + E = E2+ ǫ2,
where?k2(ǫ2) is the momentum (energy) of the scattered electron.
Using the P-invariance of the hadron electromagnetic interaction, one can parametrize
the dependence of the differential cross section on the polarizations of the electron target
and of the scattered proton as follows:
?dσ
where Tik, i,k = ℓ,t,n are the corresponding polarization transfer coefficients, with the
dσ
dk2(?ξ,?S2) =
dk2
?
un
[1 + TℓℓξℓS2ℓ+ TnnξnS2n+ TttξtS2t+ TℓtξℓS2t+ TtℓξtS2ℓ], (21)
following notations: ℓ is the component of the polarization vector along the momentum of
the initial proton, n is the component which is orthogonal to the momenta of the initial
proton and of the scattered electron, i.e., orthogonal to the scattering plane, and t is the
component which is orthogonal to the initial proton momentum and lies in the scattering
plane.
At high energy, the polarization transfer coefficients depend essentially on the direction
of the scattered proton polarization. Let us choose an orthogonal system with the z axis
directed along ? p,?k2lies in the xz plane (θ is the angle between the initial proton and the
final electron momenta) and the y axis is directed along the vector ? p ×?k2. Therefore, in
this system ℓ ? z, t ? x and n ? y. The explicit expressions for the polarization transfer
coefficients are given in Appendix A.
B.Polarization correlation coefficients, Cij, in the ? p +? e → p + e reaction
In the reaction involving polarized proton beam and polarized electron target, one can
derive explicit expressions for the spin correlation coefficients. These coefficients are also
called double analyzing powers and denoted A00ijin the notations from Ref. [13].
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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)?.
(22)
All spin correlation coefficients for the elastic ? p? e collisions can be obtained from this expres-
sion and they are, therefore, proportional to the proton magnetic FF.
In the considered frame, where the target electron is at rest, the polarization four vector
of the initial proton has the following components
η1=
?
? p ·?S1
M
,?S1+
? p(? p ·?S1)
M(E + M)
?
, (23)
where?S1is the unit vector describing the polarization of the initial proton in its rest system.
Applying the P-invariance of the hadron electromagnetic interaction, one can write the
following expression for the dependence of the differential cross section on the polarization
of the initial particles:
dσ
dk2(?ξ,?S1) =
?dσ
dk2
?
un
[1 + CℓℓξℓS1ℓ+ CttξtS1t+ CnnξnS1n+ CℓtξℓS1t+ CtℓξtS1ℓ], (24)
where Cik, i,k = ℓ,t,n are the corresponding spin correlation coefficients which characterize
? p? e scattering. Here also one expects large sensitivity of these observables to the direction
of the proton beam polarization. Small coefficients (in absolute value) are expected for the
transversal component of the beam polarization 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) + S1ℓ
?p
M,? pM
E
p
?
∼ S1ℓp1µ
M.
(25)
The effect of the transversal beam polarization appears 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.
The explicit expressions of the spin correlation coefficients, are given in Appendix B. One
can see that Cnn= Tnn.
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C. Depolarization coefficients, Dij, in the ? p + e → ? p + e reaction
In this section explicit expressions for the depolarization coefficients, (also denoted Di0j0
in the notations from Ref. [13]), which define the polarization transfer from the initial to
the final proton, are derived for the reaction ? p + e → ? p + e.
The part of the hadronic tensor, Wµν(η1,η2), which corresponds to polarized protons in
initial and final states can be written as:
Wµν(η1,η2) = A1? gµν+A2PµPν+A3(? η1µ? η2ν+? η1ν? η2µ)+A4(Pµ? η1ν+Pν? η1µ)+A5(Pµ? η2ν+Pν? η2µ),
where
? gµν= gµν−kµkν
and
(26)
k2, ? ηiµ= ηiµ−k · ηi
k2kµ, i = 1,2,
A1 =
G2
2
M
(2k · η1k · η2− k2η1· η2), A2= −η1· η22M2
1 + τ(G2
E(k2) + τG2
M(k2)),
A3 = G2
M(k2)k2
2, A4= −MGM(k2)GE(k2) + τGM(k2)
A5 = MGM(k2)GE(k2) + τGM(k2)
1 + τ
1 + τ
k · η2,
k · η1.
The dependence of the polarization of the scattered proton on the polarization state of
the proton beam is obtained by contraction of the spin independent leptonic tensor (not
averaged over the spin of the initial electron), i.e., 2L(0)
µν, Eq. (6), and the spin-dependent
hadronic tensor Wµν(η1,η2), Eq. (26).
One obtains the following formula which holds in any reference system:
DD(η1,η2) = 2(1 + τ)−1?
k · η1k · η2GM(k2)?k2?GM(k2) − GE(k2)?+ 2m2(1 + τ)GM(k2)?
+k2(1 + τ)G2
+4GM(k2)(k · η1k1· η2− k · η2k1· η1)?M2τ?GE(k2) − GM(k2)?
?G2
Applying the P-invariance of the hadron electromagnetic interaction, one can write the
M(k2)(2k1· η2k2· η1− m2η1· η2)
+mE?GE(k2) + τGM(k2)??
−η1· η2
E(k2) + τG2
M(k2)??k2(M2− 2mE) + 4m2E2??
. (27)
following expression for the dependence of the differential cross section on the polarization
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of the incident and scattered protons which participate in the reaction as:
?dσdσ
dk2(η1,η2) =
dk2
?
un
[1 + DttS1tS2t+ DnnS1nS2n+ DℓℓS1ℓS2ℓ+ DtℓS1tS2ℓ+ DℓtS1ℓS2t],
(28)
where Dik, i,k = ℓ,t,n are the corresponding spin depolarization coefficients which charac-
terize ? p + e → ? p + e scattering. The explicit expressions of the depolarization coefficients,
are given in Appendix C, in terms of the hadron form factors.
D.Kinematics
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 the colliding hadrons. Let us 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. The following formulas can be partly found in Ref. [12].
One can show 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. 2):
(−k2)max=
4m2? p2
M2+ 2mE + m2. (29)
Being proportional to the electron mass squared, the four momentum transfer squared is
restricted to very small values, where the proton can be considered point-like. Comparing
the expressions for the total energies in two reactions: sI= m2+M2+2mE, where E is the
proton energy in the elastic proton electron scattering, and sD= m2+ M2+ 2Mǫ, where ǫ
is the electron beam energy in the electron proton elastic scattering, one finds the following
relation between the proton beam energy and the electron beam energy, in order to reach
the same total energy sI= sD
E =M
mǫ ∼ 2000 ǫ. (30)
The four momentum transfer squared is expressed as a function of the energy of the scattered
electron, ǫ2, as:
k2= (k1− k2)2= 2m(m − ǫ2), (31)
where
ǫ2= m(E + m)2+ p2cos2θ
(E + m)2− p2cos2θ,
(32)
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E [GeV]
12345678 9 10
2
[GeV/c]
max
2
−k
0
0.02
0.04
0.06
0.08
0.1
0.12
−3
10
×
FIG. 2: Maximum four momentum transfer squared as a function of the proton beam energy.
and θ is the angle between the proton beam and the scattered electron momenta.
From energy and momentum conservation, one finds the following relation between the
angle and the energy of the scattered electron:
cosθ =(E + m)(ǫ2− m)
|? p|?(ǫ2
2− m2)
, (33)
which shows that cosθ ≥ 0. One can see from Eq. (32) that in the inverse kinematics, the
available kinematical region is reduced to small values of ǫ2:
ǫ2,max= m2E(E + m) + m2− M2
M2+ 2mE + m2
. (34)
From momentum conservation, on can find the following relation between the energy and
the angle of the scattered proton E2and θp:
E2=
(E + m)(M2+ mE) ± M|? p|2cosθp
?
m2
M2− sin2θp
(E + m)2− |? p|2cos2θp
. (35)
Let us introduce the invariant
ν = k · p1= E(m − ǫ2) + |?k2||? p|cosθ =k2
2m
?
E − |? p|cosθ
?
1 − 4m2
k2
?
. (36)
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The following relation holds: k2+ 2ν = 0.
For the angle between the initial and final hadron, it exists a maximum value which is
determined by the ratio of the electron and scattered hadron masses, sinθh,max = m/M.
One concludes that hadrons are scattered on atomic electrons at very small angles, and that
the largest is the hadron mass, the smaller is the available angular range for the scattered
hadron.
III.NUMERICAL RESULTS
A. Experimental observables
For a given proton beam energy E the observables are functions of only one kinematical
variable, that we chose as k2, as it is a kinematical invariant. Transformation to the scatter-
ing electron angle are straightforward. The proton structure is taken into account through
the parametrization of FFs. We took the dipole parametrization:
GE(k2) = GM(k2)/µp= [1 − k2/0.71]−2, (37)
where µpis the proton magnetic moment, and k2is expressed in GeV2. The normalization
to the static point is GE(0) = 1 and GM(0) = µp. The standard dipole parametrization
coincides with more recent descriptions for −k2< 1 GeV2. At higher k2, different choices
may affect the cross section and at a lesser extent, the polarization observables, but as
we showed above, the maximum value of k2which can be achieved in inverse kinematics,
justifies the choice of dipole parametrization, and even of constant FFs, where the constants
correspond to the static values.
The differential cross section, Eq. (16), is plotted as a function of (−k)2in Fig. 3. One
can see that it is monotonically decreasing as a function of k2up to a value of k2
maxaccording
to Eq. (29).
The polarization transfer coefficients, Eq. (A1), are shown in Fig. 6 as a function of
the incident energy for θ = 0 (black solid line), 10 mrad (red dashed line), 30 mrad (green
dash-dotted line), 50 mrad (blue dotted line). The spin correlation coefficients Eq. (B1) are
shown in Fig. 7. The spin depolarization coefficients, Eq. (C1), are shown in Fig. 8.
One can see that in collinear kinematics, in general, either polarization observables take
the maximal values or they vanish. An interesting kinematical region appears at E = 20
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10?10
10?9
10?8
10?7
10?6
?k² ?GeV²?
109
1011
1013
1015
1017
dΣ
dk²
?mb?GeV²?
FIG. 3: Differential cross section as a function of −k2for different incident energies: E=1 MeV
(black solid line), E=50 MeV (red dotted line), E=100 MeV (blue dashed line), E=1 GeV (green
thick line).
GeV, where a structure is present in agreement with the results of Ref. [2].
As shown in Section II, Eq. (16), the cross section diverges for k2→ 0. This condition
is obtained when the scattering angle is small (high energies, and large impact parameters),
or when the energy is small.
In the first case, one introduces a minimum scattering angle, which is related to the
impact parameter, which classical (c) and quantum expressions (q), are given by [14]:
θ(c)
min=2e2
pβb, θ(q)
min=?
pb,
(38)
where b is the impact parameter and β is the relative velocity. Let us take as characteristic
impact parameter, the Bohr radius, b = 0.519 · 105fm. We have shown above that there is
a maximum scattering angle for the proton, which does not depend on the energy, and a
corresponding maximum value for the transferred momentum k2. The condition kmin< kmax
from Eqs. (38), is obtained for E ≥ 1 MeV. When the relative energy is very low, the electron
and proton may be trapped in a bound system, and the present description based on one
photon exchange is not valid. The Born approximation corresponds to the first term of an
expansion in the parameter α/v which should be lower than unity. The condition α/v = 0.1c
is satisfied for E > 2.5 MeV.
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E [GeV]E [GeV]
00 50 50100 100 150150200 200
[mb/sterad]
e
Ω
/ d
σ
d
σ
d
11
10 10
22
10 10
33
10 10
[mb/sterad]
e
Ω
/ d
FIG. 4: Differential cross section as a function of the incident energy E for different angles: θ = 0
(black solid line), 10 mrad (red dashed line), 30 mrad (green dotted line), 50 mrad (blue dashed-
dotted line).
The description of Coulomb effects at low energies require approximations and it is outside
the purpose of this paper. We will apply the present calculation for E ≥ 3 MeV.
Al low energy, screening effects are introduced multiplying the cross section by the factor
χ =
χb
eχb+ 1, χb= −2πα
β.
(39)
Such factor is attractive for opposite charges and increases the cross section for the reaction
of interest here.
The total cross section has been calculated by integration from a value of k2
minextracted
from Eqs. (31,32), and it is given as a function of the incident proton kinetic energy T =
E − M in Fig. 5, for values of the proton kinetic energy in the MeV range.
Let us calculate the cross section for a non polarized proton beam colliding with a polar-
ized target:
σij=
?
NDTijPiPjdk2, N =
πα2
2m2p2k4. (40)
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T[MeV]
05 10 1520 25
[mb]
σ
-8
10
10
2
10
3
10
FIG. 5: Total unpolarized cross section as a function of the incident proton kinetic energy T.
Assuming Pi = Pj = 1, the values for different incident energies are reported in Table I
for the total polarized and unpolarized cross sections and in Table II for the corresponding
integrated polarization coefficients.
Tσunp
σtℓ
σℓt
σℓℓ
σtt
σnn
[GeV ] [mb][mb] [mb][mb] [mb][mb]
23 · 10−34.4 · 108
26 26.7
−125.3
−16.9
−139.3
50 · 10−3
2 · 108
11.5 12.2
−62.8
−7.4
−67
12.5 · 107
0.4 0.8-5.6-0.2 -2.9
101.9 · 1079.1 · 10−310.6 · 10−2
-1.01
−0.6 · 10−2
-0.09
501.8 · 1070.4 · 10−3
2.3 · 10−2
-0.2
−0.3 · 10−3−0.5 · 10−2
TABLE I: Unpolarized cross section and polarized transfer cross sections (in mb) for different
incident energies.
The spin transfer cross section σnnand σℓ= (σℓℓ+σℓt)/2, are illustrated in Fig. 9 in the
MeV range.
15