Second-Order Eikonal Corrections for A(e, e′p)
B. Van Overmeire, J. Ryckebusch
Department of Subatomic and Radiation Physics, Ghent University,
Proeftuinstraat 86, B-9000 Gent, Belgium
Abstract
The first-order eikonal approximation is frequently adopted in interpreting the re-
sults of A(e, e′p) measurements. Glauber calculations, for example, typically adopt
the first-order eikonal approximation. We present an extension of the relativis-
tic eikonal approach to A(e, e′p) which accounts for second-order eikonal correc-
tions. The numerical calculations are performed within the relativistic optical model
eikonal approximation. The nuclear transparency results indicate that the effect of
the second-order eikonal corrections is rather modest, even at Q2 ≈ 0.2 (GeV/c)2.
The same applies to polarization observables, left-right asymmetries, and differen-
tial cross sections at low missing momenta. At high missing momenta, however, the
second-order eikonal corrections are significant and bring the calculations in closer
agreement with the data and/or the exact results from models adopting partial-wave
expansions.
Key words: A(e, e′p) reactions, eikonal approximation, second-order corrections,
optical potentials
PACS: 11.80.Fv, 24.10.Ht, 24.10.Jv, 25.30.Dh
1 Introduction
The eikonal approximation [1–3] has a long history of successful results in
describing scattering processes like nucleon-nucleus scattering, heavy-ion col-
lisions, and electroinduced nucleon-knockout reactions. The latter class of re-
actions, usually denoted as A(e, e′p), provide access to a wide range of nuclear
phenomena like short- and long-range correlations, relativistic effects, the tran-
sition from hadronic to partonic degrees of freedom, and medium modifications
of nucleon properties. The interpretation of A(e, e′p) data heavily relies on an
Email address: Jan.Ryckebusch@UGent.be (J. Ryckebusch).
Preprint submitted to Elsevier 6 August 2021
http://arxiv.org/abs/0704.0705v1
accurate description of the effect of the final-state interactions (FSI), i.e., the
interactions of the ejected proton with the residual nucleus such as rescatter-
ing and/or absorption. The eikonal approximation has been widely used to
treat these distortions, either in combination with optical potentials [4–7], or
with Glauber theory, its multiple-scattering extension [8–15].
The eikonal scattering wave functions are derived by linearizing the continuum
wave equation for the ejected proton. Hence, the solution is only valid to first
order in 1/k, with k the proton’s momentum, and the eikonal approximation
is suited for the description of reactions at sufficiently high energies. To extend
the applicability to lower energies, Wallace [16] has developed systematic cor-
rections to the eikonal scattering amplitude. Several authors have investigated
the effect of higher-order eikonal corrections in elastic nuclear scattering by
protons, antiprotons, and α particles [17,18], heavy-ion collisions [19–22], and
inclusive electron-nucleus scattering [23]. The aim of this Letter is to deter-
mine the influence of higher-order eikonal corrections on A(e, e′p) observables.
To this purpose, we extend the relativistic optical model eikonal approxima-
tion (ROMEA) A(e, e′p) framework of Ref. [7]. Our formalism builds upon the
work of Baker [24], where an eikonal approximation for potential scattering
was derived to second order in 1/k. Here, this work is extended to include the
effect of the spin-orbit potential.
The outline of this Letter is as follows. In Section 2, the second-order eikonal
correction to the ROMEA model is derived. Section 3 presents the results of
the A(e, e′p) numerical calculations. We look into how the second-order eikonal
correction affects more inclusive quantities like the nuclear transparency, as
well as truly exclusive observables such as the induced normal polarization
Pn, the left-right asymmetry ALT , and the differential cross section. Finally,
in Section 4, we state our conclusions.
2 Formalism
For the description of the A(e, e′p) reaction, we adopt the impulse approxi-
mation (IA) and the independent-nucleon picture. Within this approach, the
basic quantity to be computed is the transition matrix element [25]
〈Jµ〉 =
~k,ms
(~r) Ĵµ(~r) ei~q·~r φα1(~r) . (1)
Here, φα1 and Ψ
~k,ms
are the relativistic bound-state and scattering wave func-
tions, with α1 the quantum numbers of the struck proton and ~k and ms the
momentum and spin of the ejected proton. The relativistic bound-state wave
function is obtained in the Hartree approximation to the σ − ω model [26]
with the W1 parametrization for the different field strengths [27]. The scat-
tering wave function Ψ
~k,ms
appears with incoming boundary conditions and is
related to Ψ
~k,ms
by time reversal. Furthermore, Ĵµ is the relativistic one-body
current operator. Throughout this Letter, we use the Coulomb gauge and the
CC2 form of Ĵµ [28].
We now turn our attention to the determination of the scattering wave func-
tion Ψ
~k,ms
. We start by considering the Dirac equation for a proton with
relativistic energy E =
k2 +M2N and spin state
subject to Lorentz
scalar and vector potentials Vs(r) and Vv(r). The Dirac equation for the four-
component spinor Ψ
~k,ms
(~r) is converted to a Schrödinger-like equation for the
upper component u
~k,ms
(~r) [7, 29]
+ Vc(r) + Vso(r) (~σ · ~L− i~r · ~̂p)
~k,ms
(~r) =
~k,ms
(~r) . (2)
The central Vc(r) and spin-orbit Vso(r) potentials are defined in terms of the
scalar and vector ones, Vs(r) and Vv(r). The lower component w
~k,ms
(~r) is
related to the upper one through
~k,ms
(~r) =
E +MN + Vs(r)− Vv(r)
~σ · ~̂p u
~k,ms
(~r) . (3)
When solving Eq. (2) in the eikonal approximation, a standard procedure is
to replace the momentum operator ~̂p by the asymptotic momentum ~k in the
spin-orbit (Vso(r)~σ · ~L) and Darwin (Vso(r) (−i~r · ~̂p)) terms, as well as in the
lower component (3). In literature, this is usually referred to as the effective
momentum approximation (EMA) [30]. For the upper component, one puts
forward a solution of the form
~k,ms
(~r) ≡ N η(~r) ei
~k·~r χ 1
, (4)
i.e., a plane wave modulated by an eikonal factor η(~r). Here, N is a normal-
ization factor.
In the ROMEA approach [7, 29], which adopts the first-order eikonal approx-
imation, Eq. (2) is linearized in ~̂p leading to a solution for the eikonal factor
of the form
ηROMEA(~r) = ηROMEA(~b, z) = exp
dz ′ Vopt(~b, z
 , (5)
where ~r ≡ (~b, z), the z axis lies along the momentum ~k of the proton, and
Vopt(~b, z) = Vc(~b, z) + Vso(~b, z) (~σ · ~b × ~k − ikz). Despite the fact that it is
written as an exponential phase, the solution (5) is only valid up to first order
in Vopt/k.
In what follows, we will derive an expression for the eikonal factor η(~r) that
is valid up to order Vopt/k
2. The momentum dependence in the spin-orbit
and Darwin terms makes that these terms are retained up to order Vso/k,
while central terms are included up to order Vc/k
2. Note that the expansion is
not expressed in terms of the Lorentz scalar and vector potentials Vs and Vv.
Looking for a solution of the form (4) for the Schrödinger-like equation (2),
Baker arrived at the following equation for the eikonal factor (see Eq. (14) of
Ref. [24]):
η(~b, z) = 1− i
dz ′ Vopt(~b, z
′) η(~b, z ′) +
Vopt(~b, z) η(~b, z)
dz ′ (z − z ′)
Vopt(~b, z
′) η(~b, z ′)
. (6)
Note that, apart from dropping contributions of order Vopt/k
3 and higher, no
additional assumptions were made when deriving Eq. (6). In Ref. [24], Eq. (6)
was subsequently solved for spherically symmetric potentials. The spin-orbit
and Darwin terms, however, break the spherical symmetry and a novel method
to solve Eq. (6) is needed. To that purpose, we assume that the derivative of
the function η is of higher order in 1/k than η itself (as is true for the ROMEA
solution (5)). This allows us to drop the ∂η/∂b contribution in the last term
of Eq. (6), as it is of order Vopt/k
3 or higher:
dz ′ (z − z ′)
Vopt(~b, z
′) η(~b, z ′)
dz ′ (z − z ′)
Vc(~b, z
′) + Vso(~b, z
′) (~σ ·~b× ~k − ikz ′)
η(~b, z ′) . (7)
Spherical symmetry implies that z ′ ∂Vc(~b, z
′)/∂b = b ∂Vc(~b, z
′)/∂z ′. Hence,
the z ′ ∂Vc/∂b term in Eq. (7) can be written as
dz ′ b
∂Vc(~b, z
η(~b, z ′)
dz ′ b
Vc(~b, z
′) η(~b, z ′)
2 + b
Vc(~b, z)
η(~b, z) . (8)
In the first step, we made use of the fact that the derivative ∂η/∂z ′ is of higher
order to turn the integrand into an exact differential. A similar reasoning,
followed by integration by parts, leads to
dz ′ (z − z ′)
∂Vso(~b, z
(−ikz ′) η(~b, z ′)
2 + b
Vso(~b, z)
η(~b, z ′) , (9)
for the Darwin term of Eq. (7). Inserting the expressions of Eqs. (8) and (9),
Eq. (6) adopts the form
η(~b, z) =
dz ′ Vopt(~b, z
′) η(~b, z ′)−
1 + b
Vc(~b, z)
η(~b, z)
1 + b
∂Vc(~b, z
η(~b, z ′)
Vso(~b, z) (~σ ·~b× ~k − ikz) η(~b, z)
1 + b
dz ′ (z − z ′)
Vso(~b, z
′)~σ ·~b× ~k
η(~b, z ′)
2 + b
Vso(~b, z)
η(~b, z ′) . (10)
We look for a solution of the form
η(~b, z) = f(~b, z) exp
dz ′ Vopt(~b, z
′) f(~b, z ′)
= f(~b, z) exp
i S(~b, z)
, (11)
which should reduce to the ROMEA result of Eq. (5) when terms of higher
order than Vopt/k are neglected. Accordingly, the function f(~b, z) should be of
the form f = 1+O(Vopt/k
2). Substituting (11) into Eq. (10) and multiplying
by e−i S(
~b,z) yields
f(~b, z) = 1−
1 + b
Vc(~b, z)
f(~b, z)
1 + b
∂Vc(~b, z
f(~b, z ′)
Vso(~b, z) (~σ ·~b× ~k − ikz) f(~b, z)
1 + b
dz ′ (z − z ′)
Vso(~b, z
′)~σ ·~b× ~k
f(~b, z ′)
2 + b
Vso(~b, z)
f(~b, z ′) . (12)
In deriving this equation, we set ei S(
~b,z ′) e−i S(
~b,z) equal to 1, since higher-order
terms are neglected. The difficulty in solving for f(~b, z) is that Eq. (12) is an
integral equation. An expression for f(~b, z) can, however, be readily obtained
by adding (1−f) terms, which introduce only higher-order terms, to the right-
hand side of Eq. (12). This is permitted since we seek for a solution up to order
Vopt/k
2. With this manipulation, the function f becomes
f(~b, z) = 1−
1 + b
Vc(~b, z) +
1 + b
∂Vc(~b, z
Vso(~b, z) (~σ ·~b× ~k − ikz)
1 + b
dz ′ (z − z ′)
Vso(~b, z
′)~σ ·~b× ~k
2 + b
Vso(~b, z) . (13)
The eikonal factor of Eq. (11) with f(~b, z) given by (13), is a solution of the
integral equation (6) to order Vopt/k
2 and reduces to the ROMEA result (5)
when truncated at order Vopt/k. Furthermore, it can be easily verified that the
derivative of η is of higher order in Vopt/k than η itself. Henceforth, calculations
performed with the eikonal factor of Eqs. (11) and (13), are dubbed as the
second-order relativistic optical model eikonal approximation (SOROMEA).
3 Results
One way to quantify the overall effect of FSI in A(e, e′p) processes is via
the nuclear transparency. The measurements are commonly performed under
quasielastic conditions [31–36]. We obtain the theoretical transparencies by
adopting similar expressions and cuts as in the experiments. Hence, the nuclear
transparency is defined as [37]
d~pmS
α(~pm, Em, ~k)
d~pmS
PWIA(~pm, Em)
. (14)
Here, Sα is the reduced cross section for knockout from the shell α
Sα(~pm, Em, ~k) =
dΩpdǫ′dΩǫ′
(e, e′p)
, (15)
where ~pm and Em are the missing momentum and energy, K is a kinemati-
cal factor and σep is the off-shell electron-proton cross section. S
PWIA is the
reduced cross section within the plane-wave impulse approximation (PWIA)
in the nonrelativistic limit. Further,
α extends over all occupied shells α in
the target nucleus. The phase-space volume in the missing momentum ∆3pm
is defined by the cut |pm| ≤ 300 MeV/c. The A-dependent factor cA corrects
in a phenomenological way for the effect of short-range correlations. We intro-
duce the cA in the denominator of Eq. (14) because the data have undergone
a rescaling with cA = 0.9 (
12C) and 0.82 (56Fe).
Transparencies have been computed for the nuclei 12C and 56Fe at planar
and constant (~q, ω) kinematics compatible with the phase space covered in
the experiments. For the optical potential, the EDAD1 parametrization of
Ref. [38] was used.
In Fig. 1 the ROMEA and SOROMEA results are displayed as a function
of the four-momentum transfer Q2 and compared to the data. Not surpris-
ingly, at high Q2, the ROMEA and SOROMEA predictions practically coin-
cide and the role of the second-order eikonal effects grows with decreasing Q2.
At Q2 = 1.7 (GeV/c)2, the ROMEA and SOROMEA transparencies agree to
within 1%; while at Q2 = 0.3 (GeV/c)2, the difference has risen to 3% for
56Fe and 5% for 12C. The enhancement of the nuclear transparency due to
the second-order eikonal corrections is modest, even for values of the four-
momentum transfer as low as Q2 = 0.2 (GeV/c)2. Both the ROMEA and
the SOROMEA predictions tend to slightly underestimate the measurements.
The second-order corrections move the predictions somewhat closer to the
Q2 = 0.34 (GeV/c)2 data point.
As the nuclear transparency involves integrations over missing momenta and
energies, it may hide subtleties in the theoretical treatment of the FSI mech-
anisms. Next, we focus on highly exclusive A(e, e′p) quantities and quantify
the role of second-order eikonal effects.
An observable that is particularly well suited to study FSI effects is the induced
normal polarization
d5σ (σn =↑)− d
5σ (σn =↓)
d5σ (σn =↑) + d5σ (σn =↓)
, (16)
where σn denotes the spin orientation of the ejectile in the direction orthogonal
to the reaction plane. Indeed, in the one-photon exchange approximation, Pn
vanishes in the absence of FSI.
Fig. 2 shows the missing momentum dependence of the induced normal polar-
ization for the kinematics of Ref. [39], corresponding with Q2 ≈ 0.5 (GeV/c)2.
The calculations are performed with the energy-dependent A-independent
(EDAI) potential of Ref. [38]. The ROMEA results are in line with the rela-
tivistic distorted-wave impulse approximation (RDWIA) calculations of Ref. [40].
The RDWIA framework was implemented by the Madrid-Sevilla group [41]
and relies on a partial-wave expansion of the exact scattering wave function.
It is similar to the (SO)ROMEA approach in that both models compute the
effect of the FSI with the aid of proton-nucleus optical potentials. Further,
the overall agreement with the data is excellent. The second-order eikonal
corrections are most pronounced for the 1s1/2 level. For missing momenta
pm > 125 MeV/c, they reduce the magnitude of the Pn for the 1s1/2 state
by roughly 20%, thereby resulting in a marginally better agreement with the
highest pm data point. For 1p3/2 knockout, on the other hand, the effect of the
second-order eikonal corrections is smaller than 5%.
The inclusion of the second-order eikonal effects is particularly visible at high
missing momentum, a region where also other mechanisms become impor-
tant. The qualitative behavior of the meson-exchange and ∆-isobar currents,
for instance, is alike [42]. At low missing momenta (pm ≤ 200 MeV/c), the in-
duced normal polarization Pn is relatively insensitive to the two-body currents;
whereas at higher missing momenta, sizable contributions from the meson-
exchange and isobar currents are predicted. The influence of the meson and
isobar degrees of freedom is also stronger for knockout from the 1s1/2 shell
than for 1p3/2 knockout.
In Fig. 2, also calculations neglecting the spin-orbit part Vso(~b, z)~σ ·~b×~k are
shown. They illustrate that the spin-orbit distortion is the largest source of
Pn. Hence, a correct inclusion of this term is essential. Moreover, Pn proves to
be rather sensitive to the choice of optical potential [40].
Another A(e, e′p) observable which has been the subject of many investigations
is the left-right asymmetry
ALT =
d5σ (φ = 0◦)− d5σ (φ = 180◦)
d5σ (φ = 0◦) + d5σ (φ = 180◦)
. (17)
The subscript LT indicates that this quantity is closely related to the longitudinal-
transverse response function.
Fig. 3 presents the ALT predictions for the removal of 1p-shell protons in
16O in the kinematics of Refs. [43, 44]. The FSI shift the dip in ALT , which
is located at pm ≈ 400 MeV/c in the relativistic PWIA (RPWIA), to lower
values of the missing momentum. This shift is essential to describe the data at
pm ≈ 350 MeV/c. The exact pm location and height of the ripple, however, are
affected by many ingredients of the calculations, such as the current operator,
bound-state wave function, and parametrization of the optical potential [44].
As can be inferred from Fig. 3, the second-order eikonal corrections affect the
height, but not the position of the ripple.
We also show the results of our SOROMEA calculations within the so-called
noSV approximation. In this approximation, the dynamical enhancement of
the lower component of the scattering wave (3) due to the Vs(r)− Vv(r) term
is omitted. As such, the SOROMEA-noSV calculations make the same set of
assumptions as the EMAf-noSV predictions by the Madrid-Sevilla group. The
EMAf-noSV approach is an RDWIA calculation which adopts the EMA in
combination with the noSV approximation. The second-order eikonal correc-
tions clearly increase the height of the oscillation in ALT and brings the eikonal
noSV calculations in excellent agreement with the corresponding partial-wave
prediction EMAf-noSV. Finally, the comparison between the SOROMEA and
the SOROMEA-noSV calculations demonstrates that the dynamical enhance-
ment plays a significant role in the description of the ALT data.
In Fig. 4, 16O(e, e′p) cross-section results are displayed for the kinematics
of Fig. 3. The spectroscopic factors, which normalize the calculations to the
data, were determined by performing a χ2 fit to the data and are summa-
rized in Table 1. The RDWIA spectroscopic factors are 5–10% higher than
the (SO)ROMEA ones. The second-order eikonal corrections hardly affect the
values of the extracted spectroscopic factors. Both our (SO)ROMEA calcula-
tions and the RDWIA predictions of the Madrid-Sevilla group do a very good
job of representing the data over the entire pm range. For missing momenta
|pm| ≤ 250 MeV/c, the (SO)ROMEA and RDWIA results are in excellent
agreement. The impact of the second-order eikonal corrections on the com-
puted differential cross sections is almost negligible for pm below the Fermi
momentum, but can be as large as 30% at high pm. The inclusion of the
second-order effects improves the agreement with the RDWIA calculations at
these high missing momenta. Results for the effective response functions RL,
RT , RLT , and RTT are not shown, but the effect of the second-order eikonal
corrections is similar to the effect on the differential cross section.
RPWIA ROMEA SOROMEA RDWIA
1p3/2 0.55 0.84 0.83 0.92
1p1/2 0.47 0.75 0.74 0.78
Table 1
The spectroscopic factors for the 16O(e, e′p) reaction of Ref. [43], as obtained with
a χ2 procedure.
4 Conclusions
We have developed a formalism to account for second-order corrections in the
eikonal approximation. Our model is relativistic and includes both the central
and spin-orbit parts of the optical potentials. The formalism has been applied
to A(e, e′p) processes. Our numerical calculations show that the effect of the
second-order eikonal corrections on A(e, e′p) observables is rather limited for
Q2 ≥ 0.2 (GeV/c)2. The nuclear transparency calculations confirm the ex-
pected energy dependence of the eikonal corrections: the effect decreases with
increasing Q2. Concerning the pm dependence of the A(e, e
′p) observables, the
effect of the second-order eikonal corrections is minor except at high missing
momenta. In this high-pm region, the eikonal corrections affect the observables
up to an order of 30%, thereby bringing the calculations closer to the data
and/or the RDWIA calculations. The robustness of the first-order eikonal ap-
proximation, which emerges from this study, can be invoked to explain the
success of the Glauber approach to A(e, e′p) down to relatively low kinetic
energies of 200 MeV.
Acknowledgements
This work was supported by the Fund for Scientific Research, Flanders (FWO).
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[44] K.G. Fissum, et al., Phys. Rev. C 70 (2004) 034606.
ROMEA
SOROMEA
Q2  (GeV2/c2)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fig. 1. Nuclear transparencies versus Q2 for A(e, e′p) reactions in quasielastic kine-
matics. The SOROMEA (dashed lines) are compared to the ROMEA (solid lines)
results. The EDAD1 potential [38] has been employed in both formalisms. Data
points are from Refs. [31] (open squares), [32, 33] (open triangles), [34, 35] (solid
triangles), and [36] (open diamonds).
1p3/2
ROMEA
SOROMEA
ROMEA-noSO
SOROMEA-noSO
1s1/2
pm (MeV/c)
0 50 100 150 200 250 300
Fig. 2. Induced normal polarization Pn for proton knockout from the 1p3/2 (upper
panel) and 1s1/2 (lower panel) shell in the
12C(e, e′~p) reaction. The kinematics is de-
termined by beam energy ǫ = 579 MeV, momentum transfer q = 760 MeV/c, energy
transfer ω = 292 MeV, and azimuthal angle φ = 180◦. The solid (dashed) curves
represent ROMEA (SOROMEA) calculations. The dot-dashed (dotted) curves re-
fer to predictions obtained within the ROMEA (SOROMEA) frameworks, with the
spin-orbit term Vso(~b, z)~σ ·~b× ~k turned off. The data are from Ref. [39].
1p1/2
ROMEA
SOROMEA
SOROMEA-noSV
EMAf-noSV
RPWIA
1p3/2
pm (MeV/c)
0 50 100 150 200 250 300 350 400 450
Fig. 3. The left-right asymmetry ALT for the
16O(e, e′p) experiment of [43].
The kinematics was ǫ = 2.442 GeV, q = 1 GeV/c, and ω = 445 MeV (i.e.,
Q2 = 0.8 (GeV/c)2). The red solid (green dashed) lines show the results of the
ROMEA (SOROMEA) calculations. The SOROMEA-noSV (orange long-dotted
curves) calculations differ from the SOROMEA calculations in that the dynamical
enhancement of the lower component of the scattering wave function is neglected.
The cyan short-dotted curves present the results from an RDWIA calculation where
the spinor distortions in the scattered wave are neglected. All calculations use the
EDAI version for the optical potentials [38]. The black short-dotted curves represent
the RPWIA results. The data points are from Ref. [43].
ROMEA
SOROMEA
RDWIA
RPWIA
1p3/2
1p1/2
pm (MeV/c)
-400 -300 -200 -100 0 100 200 300 400
Fig. 4. 16O(e, e′p) cross sections compared to ROMEA, SOROMEA, RDWIA, and
RPWIA calculations for the constant (~q, ω) kinematics of Fig. 3. The calculations
use the optical potential EDAI [38]. The data are from Ref. [43] and the RDWIA
results from Ref. [44]. The following convention is adopted: positive (negative) pm
corresponds to φ = 180◦ (φ = 0◦).
	Introduction
	Formalism
	Results
	Conclusions
	Acknowledgements
	References

  The first-order eikonal approximation is frequently adopted in interpreting
the results of $A(e,e'p)$ measurements. Glauber calculations, for example,
typically adopt the first-order eikonal approximation. We present an extension
of the relativistic eikonal approach to $A(e,e'p)$ which accounts for
second-order eikonal corrections. The numerical calculations are performed
within the relativistic optical model eikonal approximation. The nuclear
transparency results indicate that the effect of the second-order eikonal
corrections is rather modest, even at $Q^{2} \approx 0.2$ (GeV/c)$^2$. The same
applies to polarization observables, left-right asymmetries, and differential
cross sections at low missing momenta. At high missing momenta, however, the
second-order eikonal corrections are significant and bring the calculations in
closer agreement with the data and/or the exact results from models adopting
partial-wave expansions.

Introduction
The eikonal approximation [1–3] has a long history of successful results in
describing scattering processes like nucleon-nucleus scattering, heavy-ion col-
lisions, and electroinduced nucleon-knockout reactions. The latter class of re-
actions, usually denoted as A(e, e′p), provide access to a wide range of nuclear
phenomena like short- and long-range correlations, relativistic effects, the tran-
sition from hadronic to partonic degrees of freedom, and medium modifications
of nucleon properties. The interpretation of A(e, e′p) data heavily relies on an
Email address: Jan.Ryckebusch@UGent.be (J. Ryckebusch).
Preprint submitted to Elsevier 6 August 2021
http://arxiv.org/abs/0704.0705v1
accurate description of the effect of the final-state interactions (FSI), i.e., the
interactions of the ejected proton with the residual nucleus such as rescatter-
ing and/or absorption. The eikonal approximation has been widely used to
treat these distortions, either in combination with optical potentials [4–7], or
with Glauber theory, its multiple-scattering extension [8–15].
The eikonal scattering wave functions are derived by linearizing the continuum
wave equation for the ejected proton. Hence, the solution is only valid to first
order in 1/k, with k the proton’s momentum, and the eikonal approximation
is suited for the description of reactions at sufficiently high energies. To extend
the applicability to lower energies, Wallace [16] has developed systematic cor-
rections to the eikonal scattering amplitude. Several authors have investigated
the effect of higher-order eikonal corrections in elastic nuclear scattering by
protons, antiprotons, and α particles [17,18], heavy-ion collisions [19–22], and
inclusive electron-nucleus scattering [23]. The aim of this Letter is to deter-
mine the influence of higher-order eikonal corrections on A(e, e′p) observables.
To this purpose, we extend the relativistic optical model eikonal approxima-
tion (ROMEA) A(e, e′p) framework of Ref. [7]. Our formalism builds upon the
work of Baker [24], where an eikonal approximation for potential scattering
was derived to second order in 1/k. Here, this work is extended to include the
effect of the spin-orbit potential.
The outline of this Letter is as follows. In Section 2, the second-order eikonal
correction to the ROMEA model is derived. Section 3 presents the results of
the A(e, e′p) numerical calculations. We look into how the second-order eikonal
correction affects more inclusive quantities like the nuclear transparency, as
well as truly exclusive observables such as the induced normal polarization
Pn, the left-right asymmetry ALT , and the differential cross section. Finally,
in Section 4, we state our conclusions.
2 Formalism
For the description of the A(e, e′p) reaction, we adopt the impulse approxi-
mation (IA) and the independent-nucleon picture. Within this approach, the
basic quantity to be computed is the transition matrix element [25]
〈Jµ〉 =
~k,ms
(~r) Ĵµ(~r) ei~q·~r φα1(~r) . (1)
Here, φα1 and Ψ
~k,ms
are the relativistic bound-state and scattering wave func-
tions, with α1 the quantum numbers of the struck proton and ~k and ms the
momentum and spin of the ejected proton. The relativistic bound-state wave
function is obtained in the Hartree approximation to the σ − ω model [26]
with the W1 parametrization for the different field strengths [27]. The scat-
tering wave function Ψ
~k,ms
appears with incoming boundary conditions and is
related to Ψ
~k,ms
by time reversal. Furthermore, Ĵµ is the relativistic one-body
current operator. Throughout this Letter, we use the Coulomb gauge and the
CC2 form of Ĵµ [28].
We now turn our attention to the determination of the scattering wave func-
tion Ψ
~k,ms
. We start by considering the Dirac equation for a proton with
relativistic energy E =
k2 +M2N and spin state
subject to Lorentz
scalar and vector potentials Vs(r) and Vv(r). The Dirac equation for the four-
component spinor Ψ
~k,ms
(~r) is converted to a Schrödinger-like equation for the
upper component u
~k,ms
(~r) [7, 29]
+ Vc(r) + Vso(r) (~σ · ~L− i~r · ~̂p)
~k,ms
(~r) =
~k,ms
(~r) . (2)
The central Vc(r) and spin-orbit Vso(r) potentials are defined in terms of the
scalar and vector ones, Vs(r) and Vv(r). The lower component w
~k,ms
(~r) is
related to the upper one through
~k,ms
(~r) =
E +MN + Vs(r)− Vv(r)
~σ · ~̂p u
~k,ms
(~r) . (3)
When solving Eq. (2) in the eikonal approximation, a standard procedure is
to replace the momentum operator ~̂p by the asymptotic momentum ~k in the
spin-orbit (Vso(r)~σ · ~L) and Darwin (Vso(r) (−i~r · ~̂p)) terms, as well as in the
lower component (3). In literature, this is usually referred to as the effective
momentum approximation (EMA) [30]. For the upper component, one puts
forward a solution of the form
~k,ms
(~r) ≡ N η(~r) ei
~k·~r χ 1
, (4)
i.e., a plane wave modulated by an eikonal factor η(~r). Here, N is a normal-
ization factor.
In the ROMEA approach [7, 29], which adopts the first-order eikonal approx-
imation, Eq. (2) is linearized in ~̂p leading to a solution for the eikonal factor
of the form
ηROMEA(~r) = ηROMEA(~b, z) = exp
dz ′ Vopt(~b, z
 , (5)
where ~r ≡ (~b, z), the z axis lies along the momentum ~k of the proton, and
Vopt(~b, z) = Vc(~b, z) + Vso(~b, z) (~σ · ~b × ~k − ikz). Despite the fact that it is
written as an exponential phase, the solution (5) is only valid up to first order
in Vopt/k.
In what follows, we will derive an expression for the eikonal factor η(~r) that
is valid up to order Vopt/k
2. The momentum dependence in the spin-orbit
and Darwin terms makes that these terms are retained up to order Vso/k,
while central terms are included up to order Vc/k
2. Note that the expansion is
not expressed in terms of the Lorentz scalar and vector potentials Vs and Vv.
Looking for a solution of the form (4) for the Schrödinger-like equation (2),
Baker arrived at the following equation for the eikonal factor (see Eq. (14) of
Ref. [24]):
η(~b, z) = 1− i
dz ′ Vopt(~b, z
′) η(~b, z ′) +
Vopt(~b, z) η(~b, z)
dz ′ (z − z ′)
Vopt(~b, z
′) η(~b, z ′)
. (6)
Note that, apart from dropping contributions of order Vopt/k
3 and higher, no
additional assumptions were made when deriving Eq. (6). In Ref. [24], Eq. (6)
was subsequently solved for spherically symmetric potentials. The spin-orbit
and Darwin terms, however, break the spherical symmetry and a novel method
to solve Eq. (6) is needed. To that purpose, we assume that the derivative of
the function η is of higher order in 1/k than η itself (as is true for the ROMEA
solution (5)). This allows us to drop the ∂η/∂b contribution in the last term
of Eq. (6), as it is of order Vopt/k
3 or higher:
dz ′ (z − z ′)
Vopt(~b, z
′) η(~b, z ′)
dz ′ (z − z ′)
Vc(~b, z
′) + Vso(~b, z
′) (~σ ·~b× ~k − ikz ′)
η(~b, z ′) . (7)
Spherical symmetry implies that z ′ ∂Vc(~b, z
′)/∂b = b ∂Vc(~b, z
′)/∂z ′. Hence,
the z ′ ∂Vc/∂b term in Eq. (7) can be written as
dz ′ b
∂Vc(~b, z
η(~b, z ′)
dz ′ b
Vc(~b, z
′) η(~b, z ′)
2 + b
Vc(~b, z)
η(~b, z) . (8)
In the first step, we made use of the fact that the derivative ∂η/∂z ′ is of higher
order to turn the integrand into an exact differential. A similar reasoning,
followed by integration by parts, leads to
dz ′ (z − z ′)
∂Vso(~b, z
(−ikz ′) η(~b, z ′)
2 + b
Vso(~b, z)
η(~b, z ′) , (9)
for the Darwin term of Eq. (7). Inserting the expressions of Eqs. (8) and (9),
Eq. (6) adopts the form
η(~b, z) =
dz ′ Vopt(~b, z
′) η(~b, z ′)−
1 + b
Vc(~b, z)
η(~b, z)
1 + b
∂Vc(~b, z
η(~b, z ′)
Vso(~b, z) (~σ ·~b× ~k − ikz) η(~b, z)
1 + b
dz ′ (z − z ′)
Vso(~b, z
′)~σ ·~b× ~k
η(~b, z ′)
2 + b
Vso(~b, z)
η(~b, z ′) . (10)
We look for a solution of the form
η(~b, z) = f(~b, z) exp
dz ′ Vopt(~b, z
′) f(~b, z ′)
= f(~b, z) exp
i S(~b, z)
, (11)
which should reduce to the ROMEA result of Eq. (5) when terms of higher
order than Vopt/k are neglected. Accordingly, the function f(~b, z) should be of
the form f = 1+O(Vopt/k
2). Substituting (11) into Eq. (10) and multiplying
by e−i S(
~b,z) yields
f(~b, z) = 1−
1 + b
Vc(~b, z)
f(~b, z)
1 + b
∂Vc(~b, z
f(~b, z ′)
Vso(~b, z) (~σ ·~b× ~k − ikz) f(~b, z)
1 + b
dz ′ (z − z ′)
Vso(~b, z
′)~σ ·~b× ~k
f(~b, z ′)
2 + b
Vso(~b, z)
f(~b, z ′) . (12)
In deriving this equation, we set ei S(
~b,z ′) e−i S(
~b,z) equal to 1, since higher-order
terms are neglected. The difficulty in solving for f(~b, z) is that Eq. (12) is an
integral equation. An expression for f(~b, z) can, however, be readily obtained
by adding (1−f) terms, which introduce only higher-order terms, to the right-
hand side of Eq. (12). This is permitted since we seek for a solution up to order
Vopt/k
2. With this manipulation, the function f becomes
f(~b, z) = 1−
1 + b
Vc(~b, z) +
1 + b
∂Vc(~b, z
Vso(~b, z) (~σ ·~b× ~k − ikz)
1 + b
dz ′ (z − z ′)
Vso(~b, z
′)~σ ·~b× ~k
2 + b
Vso(~b, z) . (13)
The eikonal factor of Eq. (11) with f(~b, z) given by (13), is a solution of the
integral equation (6) to order Vopt/k
2 and reduces to the ROMEA result (5)
when truncated at order Vopt/k. Furthermore, it can be easily verified that the
derivative of η is of higher order in Vopt/k than η itself. Henceforth, calculations
performed with the eikonal factor of Eqs. (11) and (13), are dubbed as the
second-order relativistic optical model eikonal approximation (SOROMEA).
3 Results
One way to quantify the overall effect of FSI in A(e, e′p) processes is via
the nuclear transparency. The measurements are commonly performed under
quasielastic conditions [31–36]. We obtain the theoretical transparencies by
adopting similar expressions and cuts as in the experiments. Hence, the nuclear
transparency is defined as [37]
d~pmS
α(~pm, Em, ~k)
d~pmS
PWIA(~pm, Em)
. (14)
Here, Sα is the reduced cross section for knockout from the shell α
Sα(~pm, Em, ~k) =
dΩpdǫ′dΩǫ′
(e, e′p)
, (15)
where ~pm and Em are the missing momentum and energy, K is a kinemati-
cal factor and σep is the off-shell electron-proton cross section. S
PWIA is the
reduced cross section within the plane-wave impulse approximation (PWIA)
in the nonrelativistic limit. Further,
α extends over all occupied shells α in
the target nucleus. The phase-space volume in the missing momentum ∆3pm
is defined by the cut |pm| ≤ 300 MeV/c. The A-dependent factor cA corrects
in a phenomenological way for the effect of short-range correlations. We intro-
duce the cA in the denominator of Eq. (14) because the data have undergone
a rescaling with cA = 0.9 (
12C) and 0.82 (56Fe).
Transparencies have been computed for the nuclei 12C and 56Fe at planar
and constant (~q, ω) kinematics compatible with the phase space covered in
the experiments. For the optical potential, the EDAD1 parametrization of
Ref. [38] was used.
In Fig. 1 the ROMEA and SOROMEA results are displayed as a function
of the four-momentum transfer Q2 and compared to the data. Not surpris-
ingly, at high Q2, the ROMEA and SOROMEA predictions practically coin-
cide and the role of the second-order eikonal effects grows with decreasing Q2.
At Q2 = 1.7 (GeV/c)2, the ROMEA and SOROMEA transparencies agree to
within 1%; while at Q2 = 0.3 (GeV/c)2, the difference has risen to 3% for
56Fe and 5% for 12C. The enhancement of the nuclear transparency due to
the second-order eikonal corrections is modest, even for values of the four-
momentum transfer as low as Q2 = 0.2 (GeV/c)2. Both the ROMEA and
the SOROMEA predictions tend to slightly underestimate the measurements.
The second-order corrections move the predictions somewhat closer to the
Q2 = 0.34 (GeV/c)2 data point.
As the nuclear transparency involves integrations over missing momenta and
energies, it may hide subtleties in the theoretical treatment of the FSI mech-
anisms. Next, we focus on highly exclusive A(e, e′p) quantities and quantify
the role of second-order eikonal effects.
An observable that is particularly well suited to study FSI effects is the induced
normal polarization
d5σ (σn =↑)− d
5σ (σn =↓)
d5σ (σn =↑) + d5σ (σn =↓)
, (16)
where σn denotes the spin orientation of the ejectile in the direction orthogonal
to the reaction plane. Indeed, in the one-photon exchange approximation, Pn
vanishes in the absence of FSI.
Fig. 2 shows the missing momentum dependence of the induced normal polar-
ization for the kinematics of Ref. [39], corresponding with Q2 ≈ 0.5 (GeV/c)2.
The calculations are performed with the energy-dependent A-independent
(EDAI) potential of Ref. [38]. The ROMEA results are in line with the rela-
tivistic distorted-wave impulse approximation (RDWIA) calculations of Ref. [40].
The RDWIA framework was implemented by the Madrid-Sevilla group [41]
and relies on a partial-wave expansion of the exact scattering wave function.
It is similar to the (SO)ROMEA approach in that both models compute the
effect of the FSI with the aid of proton-nucleus optical potentials. Further,
the overall agreement with the data is excellent. The second-order eikonal
corrections are most pronounced for the 1s1/2 level. For missing momenta
pm > 125 MeV/c, they reduce the magnitude of the Pn for the 1s1/2 state
by roughly 20%, thereby resulting in a marginally better agreement with the
highest pm data point. For 1p3/2 knockout, on the other hand, the effect of the
second-order eikonal corrections is smaller than 5%.
The inclusion of the second-order eikonal effects is particularly visible at high
missing momentum, a region where also other mechanisms become impor-
tant. The qualitative behavior of the meson-exchange and ∆-isobar currents,
for instance, is alike [42]. At low missing momenta (pm ≤ 200 MeV/c), the in-
duced normal polarization Pn is relatively insensitive to the two-body currents;
whereas at higher missing momenta, sizable contributions from the meson-
exchange and isobar currents are predicted. The influence of the meson and
isobar degrees of freedom is also stronger for knockout from the 1s1/2 shell
than for 1p3/2 knockout.
In Fig. 2, also calculations neglecting the spin-orbit part Vso(~b, z)~σ ·~b×~k are
shown. They illustrate that the spin-orbit distortion is the largest source of
Pn. Hence, a correct inclusion of this term is essential. Moreover, Pn proves to
be rather sensitive to the choice of optical potential [40].
Another A(e, e′p) observable which has been the subject of many investigations
is the left-right asymmetry
ALT =
d5σ (φ = 0◦)− d5σ (φ = 180◦)
d5σ (φ = 0◦) + d5σ (φ = 180◦)
. (17)
The subscript LT indicates that this quantity is closely related to the longitudinal-
transverse response function.
Fig. 3 presents the ALT predictions for the removal of 1p-shell protons in
16O in the kinematics of Refs. [43, 44]. The FSI shift the dip in ALT , which
is located at pm ≈ 400 MeV/c in the relativistic PWIA (RPWIA), to lower
values of the missing momentum. This shift is essential to describe the data at
pm ≈ 350 MeV/c. The exact pm location and height of the ripple, however, are
affected by many ingredients of the calculations, such as the current operator,
bound-state wave function, and parametrization of the optical potential [44].
As can be inferred from Fig. 3, the second-order eikonal corrections affect the
height, but not the position of the ripple.
We also show the results of our SOROMEA calculations within the so-called
noSV approximation. In this approximation, the dynamical enhancement of
the lower component of the scattering wave (3) due to the Vs(r)− Vv(r) term
is omitted. As such, the SOROMEA-noSV calculations make the same set of
assumptions as the EMAf-noSV predictions by the Madrid-Sevilla group. The
EMAf-noSV approach is an RDWIA calculation which adopts the EMA in
combination with the noSV approximation. The second-order eikonal correc-
tions clearly increase the height of the oscillation in ALT and brings the eikonal
noSV calculations in excellent agreement with the corresponding partial-wave
prediction EMAf-noSV. Finally, the comparison between the SOROMEA and
the SOROMEA-noSV calculations demonstrates that the dynamical enhance-
ment plays a significant role in the description of the ALT data.
In Fig. 4, 16O(e, e′p) cross-section results are displayed for the kinematics
of Fig. 3. The spectroscopic factors, which normalize the calculations to the
data, were determined by performing a χ2 fit to the data and are summa-
rized in Table 1. The RDWIA spectroscopic factors are 5–10% higher than
the (SO)ROMEA ones. The second-order eikonal corrections hardly affect the
values of the extracted spectroscopic factors. Both our (SO)ROMEA calcula-
tions and the RDWIA predictions of the Madrid-Sevilla group do a very good
job of representing the data over the entire pm range. For missing momenta
|pm| ≤ 250 MeV/c, the (SO)ROMEA and RDWIA results are in excellent
agreement. The impact of the second-order eikonal corrections on the com-
puted differential cross sections is almost negligible for pm below the Fermi
momentum, but can be as large as 30% at high pm. The inclusion of the
second-order effects improves the agreement with the RDWIA calculations at
these high missing momenta. Results for the effective response functions RL,
RT , RLT , and RTT are not shown, but the effect of the second-order eikonal
corrections is similar to the effect on the differential cross section.
RPWIA ROMEA SOROMEA RDWIA
1p3/2 0.55 0.84 0.83 0.92
1p1/2 0.47 0.75 0.74 0.78
Table 1
The spectroscopic factors for the 16O(e, e′p) reaction of Ref. [43], as obtained with
a χ2 procedure.
4 Conclusions
We have developed a formalism to account for second-order corrections in the
eikonal approximation. Our model is relativistic and includes both the central
and spin-orbit parts of the optical potentials. The formalism has been applied
to A(e, e′p) processes. Our numerical calculations show that the effect of the
second-order eikonal corrections on A(e, e′p) observables is rather limited for
Q2 ≥ 0.2 (GeV/c)2. The nuclear transparency calculations confirm the ex-
pected energy dependence of the eikonal corrections: the effect decreases with
increasing Q2. Concerning the pm dependence of the A(e, e
′p) observables, the
effect of the second-order eikonal corrections is minor except at high missing
momenta. In this high-pm region, the eikonal corrections affect the observables
up to an order of 30%, thereby bringing the calculations closer to the data
and/or the RDWIA calculations. The robustness of the first-order eikonal ap-
proximation, which emerges from this study, can be invoked to explain the
success of the Glauber approach to A(e, e′p) down to relatively low kinetic
energies of 200 MeV.
Acknowledgements
This work was supported by the Fund for Scientific Research, Flanders (FWO).
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ROMEA
SOROMEA
Q2  (GeV2/c2)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fig. 1. Nuclear transparencies versus Q2 for A(e, e′p) reactions in quasielastic kine-
matics. The SOROMEA (dashed lines) are compared to the ROMEA (solid lines)
results. The EDAD1 potential [38] has been employed in both formalisms. Data
points are from Refs. [31] (open squares), [32, 33] (open triangles), [34, 35] (solid
triangles), and [36] (open diamonds).
1p3/2
ROMEA
SOROMEA
ROMEA-noSO
SOROMEA-noSO
1s1/2
pm (MeV/c)
0 50 100 150 200 250 300
Fig. 2. Induced normal polarization Pn for proton knockout from the 1p3/2 (upper
panel) and 1s1/2 (lower panel) shell in the
12C(e, e′~p) reaction. The kinematics is de-
termined by beam energy ǫ = 579 MeV, momentum transfer q = 760 MeV/c, energy
transfer ω = 292 MeV, and azimuthal angle φ = 180◦. The solid (dashed) curves
represent ROMEA (SOROMEA) calculations. The dot-dashed (dotted) curves re-
fer to predictions obtained within the ROMEA (SOROMEA) frameworks, with the
spin-orbit term Vso(~b, z)~σ ·~b× ~k turned off. The data are from Ref. [39].
1p1/2
ROMEA
SOROMEA
SOROMEA-noSV
EMAf-noSV
RPWIA
1p3/2
pm (MeV/c)
0 50 100 150 200 250 300 350 400 450
Fig. 3. The left-right asymmetry ALT for the
16O(e, e′p) experiment of [43].
The kinematics was ǫ = 2.442 GeV, q = 1 GeV/c, and ω = 445 MeV (i.e.,
Q2 = 0.8 (GeV/c)2). The red solid (green dashed) lines show the results of the
ROMEA (SOROMEA) calculations. The SOROMEA-noSV (orange long-dotted
curves) calculations differ from the SOROMEA calculations in that the dynamical
enhancement of the lower component of the scattering wave function is neglected.
The cyan short-dotted curves present the results from an RDWIA calculation where
the spinor distortions in the scattered wave are neglected. All calculations use the
EDAI version for the optical potentials [38]. The black short-dotted curves represent
the RPWIA results. The data points are from Ref. [43].
ROMEA
SOROMEA
RDWIA
RPWIA
1p3/2
1p1/2
pm (MeV/c)
-400 -300 -200 -100 0 100 200 300 400
Fig. 4. 16O(e, e′p) cross sections compared to ROMEA, SOROMEA, RDWIA, and
RPWIA calculations for the constant (~q, ω) kinematics of Fig. 3. The calculations
use the optical potential EDAI [38]. The data are from Ref. [43] and the RDWIA
results from Ref. [44]. The following convention is adopted: positive (negative) pm
corresponds to φ = 180◦ (φ = 0◦).
	Introduction
	Formalism
	Results
	Conclusions
	Acknowledgements
	References
