11institutetext: Jeremy W. Holt 22institutetext: Department of Physics and
Astronomy and Cyclotron Institute, Texas A&M University, College Station, TX
77843, 22email<EMAIL_ADDRESS>33institutetext: Taylor R. Whitehead
44institutetext: Facility for Rare Isotope Beams, Michigan State University,
East Lansing, MI 48824, 44email<EMAIL_ADDRESS>
# Modern approaches to optical potentials

Jeremy W. Holt and Taylor R. Whitehead corresponding author

###### Abstract

This chapter presents an overview of the optical model description of nucleon-
nucleus scattering and reactions based on fundamental nuclear two-body and
many-body forces. The chapter begins with a historical review followed by a
discussion of several commonly used theoretical techniques for deriving
nucleon-nucleus optical model potentials that account for antisymmetry, Pauli
blocking, and multiple scattering. This is followed by a summary of current
efforts to derive microscopic nuclear forces consistent with the fundamental
theory of strong interactions, quantum chromodynamics, and the use of such
interactions in the construction of microscopic optical potentials. The
chapter will also outline the dispersive optical potential approach, which
despite being primarily phenomenological, has its origins in formal Green’s
function theory. Finally, the results from modern optical potentials will be
benchmarked to experimental data.

##  1 Introduction

Nuclear reactions are fundamental for understanding the chemical evolution of
the universe as well as energy generation in stars. On Earth, nuclear
reactions find applications to national security, medical diagnostics and
therapy, and nuclear power. Nuclear reaction experiments are also used in
basic science research to probe the structure of atomic nuclei and the
properties of the strong nuclear force. In the coming decades, a major
research direction in the field of nuclear science will be to explore unknown
regions of the nuclear chart through next-generation rare-isotope beam
facilities. The masses, decays, and reactions of these exotic isotopes will
help elucidate subtle effects from proton-neutron asymmetries in nuclei and
nuclear matter with implications for neutron star physics as well as novel
nucleosynthesis in transient stellar phenomena such as core-collapse
supernovae and neutron star mergers.

The interpretation of reaction experiments involving exotic isotopes will
largely rely on theoretical models, rather than on ab initio many-body
calculations, since in most cases the exact solution of the quantum many-body
problem for a projectile nucleon incident on a target nucleus is intractable.
This is complicated by the numerous exit channels that may occur in a nucleon-
nucleus scattering experiment:

$\displaystyle A+N$ $\displaystyle\longrightarrow A+N$ Elastic scattering (1)
$\displaystyle A+N$ $\displaystyle\longrightarrow A^{*}+N^{*}$ Inelastic
scattering (2) $\displaystyle A+p$ $\displaystyle\longrightarrow A^{\prime}+n$
Charge exchange (3) $\displaystyle A+N$
$\displaystyle\longrightarrow(A+N)+\gamma$ Radiative capture (4)
$\displaystyle(B+x)+N$ $\displaystyle\longrightarrow B+(N+x)$ Transfer (5)
$\displaystyle(B+x)+N$ $\displaystyle\longrightarrow B+N+x$ Breakup (6)

where $A$ denotes an arbitrary nucleus, $N$ denotes an arbitrary nucleon,
$p/n$ denotes a proton/neutron, a star superscript denotes a change in energy,
and elements in parentheses denote the components of a nucleus.

The optical model is just one approach to analyze nucleon elastic scattering
and reaction cross sections. The virtue of the model is its simplicity:
although the nuclear force is characterized by complicated two-body, three-
body, four-body, etc. components, the interaction of a projectile nucleon with
a target nucleus can be surprisingly well approximated by potential scattering
from a complex, nonlocal, and energy-dependent one-body potential. This one-
body potential arises by averaging over the more fundamental nucleon-nucleon
and many-nucleon components of the nuclear force and by projecting out the
inelastic scattering channels. The optical model is therefore the positive-
energy analogue of the nuclear shell model, which similarly describes the
ground and excited (negative energy) states of a nucleus in terms of an
average one-body mean field potential with associated single-particle orbitals
filled by the nucleons. Despite its simplicity, the optical model also encodes
complicated quantum many-body effects including antisymmetry, Pauli blocking,
virtual excitations and correlations. However, the model does not resolve the
fluctuations in scattering cross sections due to individual states or
resonances in the combined nucleus, and therefore at low projectile energies
the model can only provide a description of energy-averaged (or low-resolution
experiment) cross sections. The optical model can also be extended to describe
inelastic nucleon-nucleus reactions by singling out specific exit channels
that are treated together with elastic scattering within a coupled channels
formalism .

The rest of this chapter will focus on recent developments in the construction
of nucleon-nucleus optical model potentials. Starting from a brief overview of
the phenomenological approach that has undergone continuous refinement over
many decades, the discussion turns to the different formal methods have been
developed to derive the effective interaction describing nucleon-nucleus
elastic scattering starting from quantum many-body theory incorporating
microscopic nuclear forces. These include Feshbach’s projection formalism ,
Watson’s multiple scattering theory  and its later extensions , and the
Green’s function approach . The motivation behind these modern approaches
based on microscopic many-body theory is to enable a more consistent
description of nuclear structure and reactions connected by fundamental
theories of the nuclear force. The dispersive optical model  will also be
discussed. This formalism is similarly motivated by the need for a more
consistent framework for understanding nuclear scattering and nuclear
structure data, though from a primarily phenomenological perspective. Given
the long history of the topic, there are numerous excellent review articles ;
; ; ;  devoted to the nuclear optical model and which can be referred to for
additional details and expanded discussions.

Original applications of the optical model grew out of the surprising
discovery that the total elastic and total reaction cross sections for
projectile nucleons on target nuclei are quite smooth as a function of
projectile energy and target mass number. This suggests a description in which
nucleons scatter from drops of nuclear matter with associated average one-body
scattering potential, whose range is roughly proportional to the size of the
liquid drop. Since the nucleon can either be scattered elastically or undergo
any number of different inelastic reaction processes, the potential should
contain both a real and imaginary component ,
$U(\vec{r})=V(\vec{r})+iW(\vec{r})$, much like the scattering and absorption
of light in a dielectric medium is represented by a complex index of
refraction. In the center-of-mass frame, the one-body wave equation then takes
the form

$\left(-\frac{1}{2\mu}\vec{\nabla}^{2}+U(\vec{r})-E\right)\Psi(\vec{r})=0,$
(7)

where $\mu$ is the reduced mass and $U$ is the complex optical potential. From
the time-dependent Schödinger equation and its complex conjugate

$\displaystyle\Psi^{*}(\vec{r},t)\left[i\frac{\partial\Psi(\vec{r},t)}{\partial
t}\right]$
$\displaystyle=\Psi^{*}(\vec{r},t)\left[\left(-\frac{1}{2\mu}\vec{\nabla}^{2}+V(\vec{r})+iW(\vec{r})\right)\Psi(\vec{r},t)\right]$
(8)
$\displaystyle\Psi(\vec{r},t)\left[-i\frac{\partial\Psi^{*}(\vec{r},t)}{\partial
t}\right]$
$\displaystyle=\Psi(\vec{r},t)\left[\left(-\frac{1}{2\mu}\vec{\nabla}^{2}+V(\vec{r})-iW(\vec{r})\right)\Psi^{*}(\vec{r},t)\right],$
(9)

one obtains the change in probability density

$\frac{\partial|\Psi(\vec{r},t)|^{2}}{\partial
t}=-\vec{\nabla}\cdot\vec{j}(\vec{r},t)+2W(\vec{r})|\Psi(\vec{r},t)|^{2},$
(10)

where the probability current is
$\vec{j}(\vec{r},t)=\frac{1}{2i\mu}(\Psi^{*}(\vec{r},t)\nabla\Psi(\vec{r},t)-\Psi(\vec{r},t)\nabla\Psi^{*}(\vec{r},t))$.
One observes that Eq. (10) is the usual continuity equation with the addition
of the imaginary part of the optical potential, which for $W(\vec{r})<0$ acts
as a sink that reduces the elastic scattering flux. In the top panel of Figure
1 is shown the experimental total cross section for neutron projectiles with
momentum $p$ (given in terms of the quantity $x=Rp$) on target isotopes with
mass number $A$. In the bottom panel of Figure 1 is shown the calculated total
cross section for neutron projectiles on target isotopes using a complex one-
body potential given by

$U=\begin{cases}-V_{0}-i\zeta V_{0}&r<R\\\ 0&r>R.\end{cases}$ (11)

It is seen that the choices $V_{0}=42$ MeV, $\zeta=0.05$, and $R=1.45\times
10^{-13}A^{1/3}$ cm produce an overall good reproduction of the experimental
data.

Figure 1: Top: experimental neutron total cross section as a function of the
mass number $A$ and projectile momentum $p$ expressed in terms of the
dimensionless quantity $x^{2}=(Rp)^{2}$. Bottom: same as top, except for the
neutron total cross section calculated from the complex single-particle
potential $U=-V_{0}-i\zeta V_{0}$ for $r<R$ and $U=0$ otherwise. Reprinted
figure with permission from Ref. , copyright (1954) by the American Physical
Society.

##  2 General form of phenomenological optical potentials

The complex square well potential in Eq. (11) in general gives rise to a much
lower reaction cross section compared to experiment ; , since the sharp
surface is responsible for greater reflection from the nuclear boundary. Early
optical model potentials therefore included a multiplicative Woods-Saxon shape
function:

$U(r)=(V+iW)f(r),$ (12)

where the parameters $V$ and $W$ are still constant (independent of energy),
and $f(r)$ is given by

$f(r)=\frac{1}{1+e^{(r-R)/a}}$ (13)

with $R$ the nuclear radius and $a$ the diffuseness. At low energies
($E\lesssim 50$ MeV), absorption takes place primarily on the nuclear surface,
and therefore it is common to replace the imaginary part of the optical
potential in Eq. (12) with a surface-peaked form

$W_{D}(r)=-4\,a\,w_{D}\frac{d}{dr}f(r),$ (14)

where the additional factor of $-4a$ is used to enforce the convenient
normalization $W_{D}(R)=w_{D}$. The inclusion of real and imaginary one-body
spin-orbit interactions

$U_{so}(r)=(V_{so}+i\,W_{so})\dfrac{1}{r}\dfrac{d}{dr}f(r)\vec{l}\cdot\vec{s},$
(15)

where $\vec{l}$ is the single-particle orbital angular momentum operator and
$\vec{s}$ is the single-particle spin angular momentum operator, allows for a
better reproduction of spin observables, such as the vector analyzing power

$A_{y}=\frac{1}{P_{y}}\frac{d\sigma_{\uparrow}-d\sigma_{\downarrow}}{d\sigma_{\uparrow}+d\sigma_{\downarrow}},$
(16)

where $P_{y}$ is the beam polarization perpendicular to the scattering plane,
$d\sigma_{\uparrow}$ is the differential cross section for spin-up
projectiles, and $d\sigma_{\downarrow}$ is the differential cross section for
spin-down projectiles. Finally, for proton scattering one also must include
the Coulomb interaction, where the charge distribution of the nucleus is
typically assumed to be uniform within the charge radius $R_{C}$.

Present-day phenomenological optical potentials have the basic structure
outlined above, except that the strength functions multiplying the Woods-Saxon
distributions can have more complicated energy dependences. For instance, the
modern and widely used phenomenological Koning-Delaroche  global optical
potential is written

$\displaystyle U(r,E)=V_{V}(r,E)+iW_{V}(r,E)+iW_{D}(r,E)$ (17)
$\displaystyle+V_{so}(r,E)\vec{l}\cdot\vec{s}+iW_{so}(r,E)\vec{l}\cdot\vec{s}+V_{C}(r),$

consisting of energy-dependent real volume, imaginary volume, imaginary
surface, real spin-orbit, and imaginary spin-orbit terms as well as the
central Coulomb interaction. One notes a few features of the Koning-Delaroche
global optical potential to be compared with microscopic optical potentials
discussed in subsequent sections. First, the dependence on the projectile
energy and on the geometric properties of the target are separated as follows:

$\displaystyle V_{V}(r,E)$ $\displaystyle=v_{V}(E)f(r,R_{V},a_{V})$ (18)
$\displaystyle W_{V}(r,E)$ $\displaystyle=w_{V}(E)f(r,R_{V},a_{V})$ (19)
$\displaystyle W_{D}(r,E)$
$\displaystyle=a_{D}w_{D}(E)\dfrac{d}{dr}f(r,R_{D},a_{D})$ (20) $\displaystyle
V_{so}(r,E)$
$\displaystyle=v_{so}(E)\dfrac{1}{r}\dfrac{d}{dr}f(r,R_{so},a_{so})$ (21)
$\displaystyle W_{so}(r,E)$
$\displaystyle=w_{so}(E)\dfrac{1}{r}\dfrac{d}{dr}f(r,R_{so},a_{so})$ (22)
$\displaystyle V_{C}(r)$
$\displaystyle=\begin{cases}\dfrac{Zze^{2}}{2R_{C}}\left(3-\dfrac{r^{2}}{R_{C}^{2}}\right)&r\leq
R_{C}\\\ \dfrac{Zze^{2}}{r}&r\geq R_{C},\end{cases}$ (23)

where $Z$ is the charge of the target and $z$ is the charge of the projectile.
The two central volume terms $V_{V}(r,E)$ and $W_{V}(r,E)$ are proportional to
Woods-Saxon distributions with identical radius and diffuseness parameters.
The real and imaginary spin-orbit interactions are proportional to the radial
derivative of a single Woods-Saxon distribution whose geometry parameters are
allowed to freely vary from those of the central interactions. The imaginary
surface contribution $W_{D}$ is also proportional to the derivative of a
Woods-Saxon distribution with independent geometry parameters. The Coulomb
interaction is that of a uniformly-charged sphere of fixed radius $R_{C}$. The
energy-dependent multiplicative strength functions for each term have only a
few adjustable parameters, and in total the Koning-Delaroche global
phenomenological optical potential has 35 parameters.

The first attempts ; ;  to model experimental scattering data via complex one-
body potentials were purely phenomenological and lacked a clear connection to
quantum many-body theory. In the 1950’s several different methods were used to
derive optical model potentials based on operator projection formalism as well
as second quantization techniques borrowed from quantum field theory. These
approaches complement calculations of nuclear ground states and excited states
using similar methods and have the advantage that they can employ high-
precision nuclear forces for a microscopic description of nuclear scattering
and reactions.

##  3 Microscopic nuclear forces

The description of nucleon-nucleus scattering in terms of the fundamental
theory of strong interactions, quantum chromodynamics (QCD), may one day
become feasible, however, it almost certainly will not be the most efficient
or practical approach for analyzing a wide range of experimental data.
Instead, the low-energy realization of QCD, chiral effective field theory
with its explicit nucleon and pion degrees of freedom, is more naturally
suited to describe low-energy nuclear phenomena while at the same time
incorporating the symmetries and symmetry breaking pattern of QCD. Most modern
approaches to constructing nucleon-nucleus optical potentials start from
microscopic nuclear forces derived from chiral effective field theory, which
includes realistic nuclear microphysics, such as multi-pion exchange processes
and three-body forces, that have been shown from ab initio calculations to be
crucial for the accurate description of nuclear structure and the saturation
phenomenon of nuclear matter. The short-distance details of the nuclear force
are not resolved at moderate projectile energies up to $E_{\rm max}\simeq
150-200$ MeV, and can therefore be encoded in a series of low-energy constants
that characterize the strengths of generalized contact interactions. The long-
range details of the nuclear force are governed by pion-exchange processes
constrained by chiral symmetry. In addition, effective field theory methods
allow for the order-by-order expansion of the nuclear force in powers of the
low-energy physical scale $q$ divided by the chiral symmetry breaking scale
$\Lambda\sim 1$ GeV. This allows one to quantify the effects of missing
physics beyond the truncated order in the effective field theory expansion ; .
The reader is referred to standard review articles ;  for further details.

One of the important goals of modern nuclear reaction theory methods is to
elucidate the role of three-body nuclear forces, which in practice can be
challenging to implement in microscopic and ab initio frameworks. Whereas two-
nucleon forces arise at leading order (LO), $(q/\Lambda)^{0}$, in the chiral
expansion, three-body forces arise at next-to-next-to-leading-order (N2LO),
$(q/\Lambda)^{3}$, in the chiral power counting without explicit
$\Delta$-isobar degrees of freedom and at one order lower, $(q/\Lambda)^{2}$,
in the power counting with explicit $\Delta$s. The technically challenging
task of including three-body forces in the formalism of scattering theory is
often simplified by the use of a normal-ordered Hamiltonian truncated to the
two-body level, as will now be demonstrated.

In second quantization, a general three-body force $V_{3N}$ is written

$V_{3N}=\frac{1}{36}\sum_{123456}\langle
123|\bar{V}_{3N}|456\rangle\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{2}\hat{a}^{\dagger}_{3}\hat{a}_{6}\hat{a}_{5}\hat{a}_{4}$
(24)

where $\bar{V}_{3N}$ is the antisymmetrized three-body force, and
$a_{i}^{\dagger}$ ($a_{i}$) is the creation (annihilation) operator for state
$|i\rangle$. Normal ordering the three-body force with respect to, e.g., the
ground state of the noninteracting many-body system yields

$\displaystyle V_{3N}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\sum_{ijk}\langle
ijk|\bar{V}_{3N}|ijk\rangle+\frac{1}{2}\sum_{ij14}\langle
ij1|\bar{V}_{3N}|ij4\rangle:\\!\hat{a}^{\dagger}_{1}\hat{a}_{4}\\!:$
$\displaystyle+$ $\displaystyle\frac{1}{4}\sum_{i1245}\langle
i12|\bar{V}_{3N}|i45\rangle:\\!\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{2}\hat{a}_{5}\hat{a}_{4}\\!:+{1\over
36}\sum_{123456}\langle
123|\bar{V}_{3N}|456\rangle:\\!\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{2}\hat{a}^{\dagger}_{3}\hat{a}_{6}\hat{a}_{5}\hat{a}_{4}\\!:,$

where the colons around an operator, $:\\!\mathcal{\hat{O}}\\!:$, denote
normal ordering. The in-medium two-body force

$\frac{1}{4}\sum_{i1245}\langle
i12|\bar{V}_{3N}|i45\rangle:\\!\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{2}\hat{a}_{5}\hat{a}_{4}\\!:$
(26)

is effectively obtained by summing the third particle over the filled states
in the noninteracting Fermi sea :

$\bar{V}_{\rm
med}=\sum_{s_{3}t_{3}}\int\frac{d^{3}k_{3}}{(2\pi)^{3}}\theta(k_{f}-k_{3})(1-P_{13}-P_{23})V_{3N},$
(27)

where $k_{f}$ is the Fermi momentum and $P_{ij}$ is the two-body
antisymmetrization operator. The operator $(1-P_{12})$ has been absorbed into
the definition of the antisymmetrized medium-dependent NN interaction
$\bar{V}_{\rm med}$. In several of the results shown below, chiral effective
field theory three-body forces have been implemented using this two-body
normal-ordered approximation.

##  4 Feshbach projection formalism

Feshbach’s projection operator method  was one of the first attempts to build
a formal connection between the optical model potential for elastic scattering
and quantum many-body theory, including the effects of inelastic scattering
channels. One starts from the Schödinger equation for the combined projectile
+ nucleus system:

$H\Psi=E\Psi,$ (28)

where

$H=H_{A}(\vec{r}_{1},\dots,\vec{r}_{A})+T_{0}+V(\vec{r}_{0},\vec{r}_{1},\dots,\vec{r}_{A})$
(29)

with $H_{A}$ the Hamiltonian for the $A$-particles of the nucleus, $T_{0}$ the
kinetic energy operator for the projectile, and $V$ the potential describing
the interaction of the projectile with the $A$ particles of the target
nucleus. The exact eigenvalues and eigenstates of the $A$-particle system

$H_{A}\Phi_{i}=\epsilon_{i}\Phi_{i},$ (30)

contain in addition to the nuclear ground and excited states also the
continuum states in which one or more nucleons are unbound. One then
constructs the total wavefunction of the combined system in the distorted wave
approximation as

$\Psi=\sum_{i=0}^{\infty}\phi_{i}(\vec{r}_{0})\Phi_{i}(\vec{r}_{1},\dots,\vec{r}_{A}).$
(31)

For simplicity, proper antisymmetrization between the projectile and target is
neglected, i.e.,

$\Psi={\cal
A}\sum_{i=0}^{\infty}\phi_{i}(\vec{r}_{0})\Phi_{i}(\vec{r}_{1},\dots,\vec{r}_{A}),$
(32)

where ${\cal A}$ is the antisymmetrization operator. This may be a reasonable
approximation for large projectile energies , where the wavefunction of the
incident nucleon overlaps weakly with those of the bound nucleons. Using the
orthonormality of the eigenfunctions $\\{\Phi_{i}\\}$, one can derive from Eq.
(31) a set of coupled equations for the amplitudes $\phi_{i}$:

$(T_{0}+V_{ii}+\epsilon_{i}-E)\phi_{i}=-\sum_{i\neq j}V_{ij}\phi_{j},$ (33)

where $V_{ij}(\vec{r}_{0})=\langle\Phi_{i}|V|\Phi_{j}\rangle$. The goal is to
isolate the elastic scattering amplitude $\phi_{0}$. From the definitions

$\displaystyle\vec{U}$ $\displaystyle=\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\vdots\end{pmatrix},$ (34) $\vec{V}_{0}=(V_{01},V_{02},\dots),$ (35)
$\vec{H}=H_{ij}=T_{0}\delta_{ij}+V_{ij}+\epsilon_{i}\delta_{ij}\hskip
7.22743pt\text{with}\hskip 7.22743pti,j>0,$ (36)

one can straightforwardly separate Eq. (33) into two channels

$\displaystyle(T_{0}+V_{00}-E)\phi_{0}$ $\displaystyle=-\vec{V}_{0}\vec{U}$
(37) $\displaystyle(\vec{H}-E)\vec{U}$
$\displaystyle=-\vec{V}_{0}^{\dagger}\phi_{0},$ (38)

where the nucleus ground state energy has been set to $\epsilon_{0}=0$ for
convenience. Formally solving Eq. (38) for $\vec{U}$ and substituting into Eq.
(37) yields

$\left(T_{0}+V_{00}+\vec{V}_{0}\frac{1}{E-\vec{H}+i\eta}\vec{V}_{0}^{\dagger}-E\right)\phi_{0}=0,$
(39)

where the additional $+i\eta$ is included to ensure outgoing waves. The
effective interaction

$V_{\rm op}=V_{00}+\vec{V}_{0}\frac{1}{E-\vec{H}+i\eta}\vec{V}_{0}^{\dagger}$
(40)

can therefore be identified as the optical model potential. Effectively, this
amounts to a Schrödinger equation for the wavefunction Eq. (31) projected onto
the ground state

$P\Psi(\vec{r}_{0},\vec{r}_{1},\dots,\vec{r}_{A})=\phi_{0}(\vec{r}_{0})\Phi_{0}(\vec{r}_{1},\dots,\vec{r}_{A}),$
(41)

where $P$ is the projection operator $P=|\Phi_{0}\rangle\langle\Phi_{0}|$.
Accounting for anti-symmetrization between the projectile and nucleons in the
target nucleus leads to a modified projection operator equation:

$P\Psi(\vec{r}_{0},\vec{r}_{1},\dots,\vec{r}_{A})={\cal
A}\phi_{0}(\vec{r}_{0})\Phi_{0}(\vec{r}_{1},\dots,\vec{r}_{A})$ (42)

with a more complicated associated projection operator. For additional details
the reader is referred to Ref. .

##  5 Optical potential at high energies: multiple scattering theory

At sufficiently high energies, the scattering of a projectile nucleon on a
target nucleus can be approximated by the superposition of scattering waves
generated by each of the constituent nucleons in the target acting
independently. In the lowest-order approximation, each of these scatterings is
therefore treated as a two-body collision, and more complicated many-body
effects such as Pauli blocking and three-body forces can be neglected. The
total transition amplitude is therefore written in terms of the individual
transition amplitudes for free-space nucleon-nucleon scattering, where one
also needs to specify the momentum distribution of nucleons in the nucleus.
The projectile wave incident on each of the constituent nucleons is then a
combination of a plane wave and all the waves generated from each of the
nucleons in the nucleus. One anticipates a modified Schrödinger equation of
the form

$\left(H_{0}-E+\int\rho(\vec{r})T(\vec{r})d\vec{r}\right)\langle\psi\rangle=0,$
(43)

where $\rho(\vec{r})$ is the density of scatters and $T(\vec{r})$ is the
$T$-matrix for two-body scattering of the incident projectile and scattering
nucleon located at position $\vec{r}$, and $\langle\psi\rangle$ is the average
total wave.

The goal will be to derive an equation for the optical model potential in
terms of two-body scattering amplitudes. As a necessary condition to apply
multiple scattering theory, the projectile wavelength $\lambda\sim 1/k$ should
be small compared to the interparticle separation distance $r_{0}$ between
nucleons in the nucleus:

$k\gg\frac{1}{r_{0}}.$ (44)

In this case, the wave propagating from one scatterer to a nearby scatter is
just the asymptotic form, namely an outgoing wave multiplied by the scattering
amplitude. Eq. (44) requires that the incident nucleon have sufficiently large
energy, and conservatively one may identify the applicability region to
projectile energies $E\gtrsim 70$ MeV (additional details and analyses using
chiral effective field theory interactions can be found in Refs. ; ).

One begins from the potential energy between projectile nucleon at
$\vec{r}_{0}$ and the $A$ target nucleons in the nucleus:

$V=\sum_{i=1}^{A}v(\vec{r}_{0},\vec{r}_{i}).$ (45)

From this one defines the transition matrix $T$ by

$T=V+V\frac{1}{E-H_{0}+i\eta}T.$ (46)

Formally this equation is similar to that of the free-space $T$-matrix but
includes bound and continuum states of the $A$-particle nucleus:

$\langle n^{\prime}\vec{k}^{\prime}|T|\vec{k}n\rangle=\langle
n^{\prime}\vec{k}^{\prime}|V|\vec{k}n\rangle+\sum_{m}\int\frac{\langle
n^{\prime}\vec{k}^{\prime}|V|\vec{p}m\rangle\langle
m\vec{p}|T|\vec{k}n\rangle}{E-E_{m}(\vec{p})+i\eta}d\vec{p},$ (47)

where the intermediate-state energy is given as the sum of the nucleus energy
$\epsilon_{m}$ and the kinetic energy $T_{0}(\vec{p})$ of the projectile. The
evaluation of Eq. (57) can be simplified by noting that for antisymmetrized
states of the nucleus, the $A$ independent terms in Eq. (45) are identical:

$T=Av\left(1+\frac{1}{E-H_{0}+i\eta}T\right).$ (48)

Note that antisymmetrization between the projectile and target nucleus has
been neglected.

From Eq. (48) one sees that the first-order approximation is just the sum of
the $A$ scattering amplitudes in the Born approximation. However, it is
desired to develop an expansion formalism in which the leading term is the sum
of the $A$ full scattering amplitudes. For this one introduces a new
scattering operator

$\tau=v\left(1+\frac{1}{E-H_{0}+i\eta}\tau\right)$ (49)

and seek to replace $v$ in Eq. (48) with $\tau$, which accounts for multiple
scattering. This can be achieved by solving Eq. (49) for $v$ and substituting
it into Eq. (48):

$\displaystyle v$
$\displaystyle=\left(1+\tau\frac{1}{E-H_{0}+i\eta}\right)^{-1}\tau$ (50)
$\displaystyle T$
$\displaystyle=\left(1-(A-1)\tau\frac{1}{E-H_{0}+i\eta}\right)^{-1}A\tau$ (51)
$\displaystyle T$ $\displaystyle=A\tau+(A-1)\tau\frac{1}{E-H_{0}+i\eta}T.$
(52)

Defining $T^{\prime}=(A-1)T/A$, one obtains

$T^{\prime}=(A-1)\tau\left(1+\frac{1}{E-H_{0}+i\eta}T^{\prime}\right),$ (53)

where the term $V^{(e)}\equiv(A-1)\tau$ is identified as the effective
potential. In fact, from this modified scattering matrix equation and
effective potential, one can rederive the optical potential using the Feshbach
projection formalism of the previous section simply replacing
$V\longrightarrow V^{(e)}$. One arrives at the result

$V_{\rm
op}^{(e)}=V_{00}^{(e)}+\vec{V}_{0}^{(e)}\frac{1}{E-\vec{H}^{(e)}+i\eta}\vec{V}_{0}^{(e)\dagger}.$
(54)

In practice one applies two further approximations. First, at high energies it
may be reasonable to neglect the effect from excited nuclear states on the
elastic scattering cross section:

$V_{\rm op}^{(e)}\simeq
V_{00}^{(e)}=(A-1)\langle\phi_{0}|\tau|\phi_{0}\rangle.$ (55)

In other words, only multiple scattering effects are taken into account.
Second, the many-body scattering matrix $\tau$ is replaced by the free-space
NN scattering matrix $T_{NN}$ (referred to as the impulse approximation):

$V_{\rm op}^{(e)}\simeq(A-1)\langle\phi_{0}|T_{NN}|\phi_{0}\rangle,$ (56)

which is nonlocal and depends on the energy associated to the $NN$ $T$-matrix.
Thus, one may write the optical potential in the form

$U_{i}(\vec{q},\vec{q}^{\prime})=\frac{A-1}{A}\sum_{sj}\\!\int
d\vec{p}\,\langle\vec{q}^{\prime},\vec{p}-\\!\frac{\vec{q}^{\prime}-\vec{q}}{2},s,j|T_{ij}(\epsilon)|\vec{q},\vec{p}+\\!\frac{\vec{q}^{\prime}-\vec{q}}{2},s,j\rangle\rho_{j}^{s}(\vec{q},\vec{q}^{\prime};\vec{p}),$
(57)

where $i$ ($j$) labels the isospin projection of the projectile (struck
nucleon in the target), $\rho_{j}^{s}(\vec{q},\vec{q}^{\prime};\vec{p})$ is
the one-body density matrix of the target associated with particles having
isospin projection $j$ and spin projection $s$. One may invoke several
approximations  to rewrite Eq. (57) in the form of the $NN$ $T$-matrix
multiplied by the nuclear density, e.g.,

$U_{i}(\vec{q},\vec{q}^{\prime};\epsilon)\sim\frac{A-1}{A}\sum_{j}T_{ij}(\vec{q},\vec{q}^{\prime};\epsilon)\rho(|\vec{q}^{\prime}-\vec{q}|).$
(58)

More recently, there have been efforts ;  to improve upon several of these
approximations, including the use of ab initio one-body densities as well as
explicit account of the spin of the struck nucleon in the target.

Figure 2:  (Upper two panels) Differential cross sections as a function of the
center-of-mass scattering angle for elastic proton scattering off 12C at 122
MeV and 200 MeV. The band shows the result obtained with the $t_{pN}$ matrix
computed with the $pN$ chiral interaction of Ref.  supplemented by a density-
dependent $NN$ interaction  (with $0.08$ fm${}^{-3}\leq\rho\leq 0.13$ fm-3)
and the one-body nonlocal density matrix computed with the NCSM method  using
$NN$ and $3N$ ;  chiral interactions. (Lower panel) Differential cross section
as a function of the center-of-mass scattering angle for elastic antiproton
scattering off 12C at 180 MeV. The result is obtained with the $t_{\bar{p}N}$
matrix computed with the $\bar{p}N$ chiral interaction of Ref.  and the one-
body nonlocal density matrix computed with the NCSM method using $NN$ and $3N$
;  chiral interactions.

In the top two panels of Figure 43 is shown the differential elastic
scattering cross sections for proton projectiles on a 12C target at energies
$E=122$ MeV and $200$ MeV computed within a multiple-scattering formalism
compared to experimental data. The proton-nucleon $T$-matrix, $t_{pN}$, is
derived from nucleon-nucleon interactions at N4LO in the chiral power counting
together with a density-dependent two-body force  derived from the chiral
effective field theory three-body force at N2LO. The target one-body nonlocal
density matrix entering into the derivation of the optical potential is
obtained from the same nucleon-nucleon interaction and the full N2LO chiral
three-body force ; . One finds overall quite good agreement with the measured
data across most scattering angles and for the two energies considered (see
also Refs. ;  for related work). In the lower panel of Fig. 43 is shown the
elastic scattering cross section for antiprotons on a 12C target at energy
$E=180$ MeV. The theory results are obtained using the antiproton-nucleon
$T$-matrix computed from the $\bar{p}N$ chiral interaction at N3LO  together
with the same 12C one-body nonlocal density matrix used in computing the
proton scattering cross sections above. Again, the multiple scattering theory
is able to predict well the measured cross sections.

##  6 Optical potential from Green’s function theory

While formally satisfying, the Feshbach construction of the effective
Hamiltonian operator that projects onto the elastic scattering channel (and
projects out the inelastic channels) can be difficult to implement in
practice. An alternative approach begins from a description of single-particle
motion based on the one-body Green’s function ; ; :

$G(\vec{r},t;\vec{r}^{\prime},t^{\prime})=-i\langle\Psi^{A}_{0}|\hat{T}[\hat{a}(\vec{r},t)_{H}\hat{a}^{\dagger}(\vec{r}^{\prime},t^{\prime})_{H}]|\Psi^{A}_{0}\rangle,$
(59)

which is the ground-state expectation value of the time-ordered product of
nucleon creation/annihilation operators in the Heisenberg representation
(denoted by the subscript $H$). The Green’s function characterizes the
propagation of particles or holes added to a many-body ground state. For
instance, in momentum space the particle part of the Green’s function
$G_{p}(\vec{k};t)$ describes the probability amplitude of finding a particle
at time $t>0$ with momentum $\vec{k}$ if it was added to the ground state of
the many-body system at time $t=0$ with the same momentum $\vec{k}$. For a
Hamiltonian invariant under time translation, one has the Fourier transformed
Green’s function

$\displaystyle G(\vec{r},\vec{r}^{\prime};E)$
$\displaystyle=\langle\Psi^{A}_{0}|a(\vec{r})\frac{1}{E-(H-E_{0}^{A})+i\eta}a^{\dagger}(\vec{r}^{\prime})|\Psi^{A}_{0}\rangle$
(60)
$\displaystyle+\langle\Psi^{A}_{0}|a^{\dagger}(\vec{r}^{\prime})\frac{1}{E+(H-E_{0}^{A})-i\eta}a(\vec{r})|\Psi^{A}_{0}\rangle,$
(61)

where $H|\Psi^{A}_{0}\rangle=E_{0}^{A}|\Psi^{A}_{0}\rangle$. Inserting a
complete set of eigenstates for the $A+1$ and $A-1$ many-body systems, one
obtains the so-called Lehmann representation

$G(\vec{r},\vec{r}^{\prime};E)=\sum_{n}\frac{f_{n}(\vec{r})f^{*}_{n}(\vec{r}^{\prime})}{E-(E_{n}^{A+1}-E_{0}^{A})+i\eta}+\sum_{n}\frac{g_{n}(\vec{r})g^{*}_{n}(\vec{r}^{\prime})}{E+(E_{n}^{A-1}-E_{0}^{A})-i\eta},$
(62)

where $f_{n}(\vec{r})=\langle\Psi^{A}_{0}|a(\vec{r})|\Psi^{A+1}_{n}\rangle$
and $g_{n}(\vec{r})=\langle\Psi^{A-1}_{n}|a(\vec{r})|\Psi^{A}_{0}\rangle$ are
overlap functions related to the probability for transition to bound or
scattering states of the $A-1$ or $A+1$ systems. One can show that the
particle-hole Green’s function satisfies the equation motion

$\left(E+\frac{\nabla_{r}^{2}}{2M}\right)G(\vec{r},\vec{r}^{\prime};E)-\int\Sigma(\vec{r},\vec{r}^{\prime\prime};E)G(\vec{r}^{\prime\prime},\vec{r}^{\prime};E)d\vec{r}^{\prime\prime}=\delta(\vec{r}-\vec{r}^{\prime}),$
(63)

where the nucleon self energy $\Sigma$ has been introduced and is defined by

$G(\vec{r},\vec{r}^{\prime};E)=G_{0}(\vec{r},\vec{r}^{\prime};E)+\int
d\vec{y}\int
d\vec{y}^{\prime}G_{0}(\vec{r},\vec{y};E)\Sigma(\vec{y},\vec{y}^{\prime};E)G(\vec{y}^{\prime},\vec{r}^{\prime};E).$
(64)

Substituting the Lehmann representation of the Green’s function into Eq. (63)
yields for the single-particle amplitudes

$\left((E_{n}^{A+1}-E_{0}^{A})+\frac{\nabla_{r}^{2}}{2M}\right)f_{n}(\vec{r})-\int\Sigma(\vec{r},\vec{r}^{\prime};E_{n})f_{n}(\vec{r}^{\prime})d\vec{r}^{\prime}=0.$
(65)

Therefore, one sees that the self energy
$\Sigma(\vec{r},\vec{r}^{\prime};E_{n})$ acts as an energy-dependent and non-
local single-particle potential in the wave equation Eq. (65) for the
amplitude $f_{n}$. Bell and Squires showed  that the spectrum of scattering
states for the Hamiltonian

$H(\vec{r},\vec{r}^{\prime})=-\frac{\nabla_{r}^{2}}{2M}\delta(\vec{r}-\vec{r}^{\prime})+\Sigma(\vec{r},\vec{r}^{\prime};E)$
(66)

is exactly equal to the overlap functions associated with the elastic
scattering channel, and hence $\Sigma(\vec{r},\vec{r}^{\prime};E)$ is
identified as the optical model potential. It is worth highlighting that the
above equations for the Green’s function and self energy are expressed in
terms of single-particle coordinates $\vec{r}$ and $\vec{r}^{\prime}$, rather
than relative coordinates with respect to the target’s center of mass. The
application of the Green’s function formalism in ab initio nuclear reaction
theory therefore requires additional approximations since in general the
center-of-mass motion is not completely suppressed .

Figure 3:  (Left panel) Differential elastic scattering cross section as a
function of the center-of-mass scattering angle for neutron scattering off 16O
(40Ca) at 3.286 (3.2) MeV from $NNLO_{\rm sat}$ (red curves) compared to data.
Reprinted figure with permission from Ref. , copyright (2019) by the American
Physical Society. (Right panel) Differential elastic scattering cross section
as a function of scattering angle for a neutron projectile at 5.17 MeV on a
40Ca target. Coupled-cluster calculations using the $NNLO_{\rm sat}$ chiral
potential are shown for different model spaces of the 2N force ($N_{max}$) and
3N force ($N_{3}$) and compared to data. Reprinted figure with permission from
Ref. , copyright (2018) by the American Physical Society.

The Green’s function approach has the advantage that a perturbation expansion
for the irreducible self energy $\Sigma(\vec{r},\vec{r}^{\prime};E)$ is
straightforward to develop. Additionally, antisymmetrization between the
nucleon projectile and target nucleus is naturally accounted for in
calculations of the self energy using the standard techniques of second
quantization. Note, however, that translational invariance may not be
automatically satisfied, and recent work ;  has sought to develop a modified
translationally-invariant second-quantized theory of optical model potentials.
Within the Green’s function formalism, the self energy for states below the
Fermi surface can also be linked to the shell model potential, however, to
include consideration of the imaginary part it is more convenient to define
the so-called “mass operator” ${\cal M}$ as

${\cal M}(k;E)=\begin{cases}\Sigma(k;E)&E>E_{F}\\\
\Sigma(k;E)^{*}&E<E_{F}.\end{cases}$ (67)

The mass operator has the property that it is an analytical function in the
upper-half plane.

Green’s function theory has been used in ab initio many-body calculations to
construct optical potentials and corresponding differential elastic scattering
cross sections for low-energy neutrons on selected doubly-magic nuclei. In the
left panel of Fig. 3 is shown as the red curves the elastic scattering cross
sections of 3.286 MeV neutrons on 16O and 3.2 MeV neutrons on 40Ca computed
from optical model potentials generated within self-consistent Green’s
function theory using the high-precision chiral 2N + 3N interaction
$NNLO_{sat}$ . One sees that in general the local minima are reproduced fairly
well, but overall the elastic scattering results are overestimated compared to
data (black dots), indicating that the absorptive strength of the ab initio
optical potential is too small. This is likely the result of missing
configurations, such as 3-particle–2-hole states, that at present cannot be
computed within the self-consistent Green’s function formalism and that would
open additional inelastic scattering channels. In the right panel of Fig. 3 is
shown the elastic scattering cross section of 5.17 MeV neutron projectiles on
40Ca as a function of scattering angle computed within a coupled cluster
Green’s function formalism  (colored curves) and compared to experimental data
(black dots). Results are shown for several different values of the model-
space parameter $N_{max}$ for which all shells with principal quantum number
$n$ and orbital angular momentum quantum number $l$ satisfying $2n+l\leq
N_{max}$ are included. In general, the parameter $N_{3}$ characterizing the
three-nucleon force model-space cutoff is taken the same ($N_{3}=N_{max}$),
except for the results shown by the red line for which $N_{max}=14$ and
$N_{3}=16$, highlighting the important role of three-body forces. One observes
that achieving converged results is challenging within the coupled cluster
Green’s function approach, though in general the theoretical cross section
predicts correctly the angles of the scattering local minima as well as the
scattering cross section at forward angles.

##  7 Green’s function theory for homogeneous nuclear matter

An alternative approach  to the Green’s function formalism outlined in the
previous section starts by calculating the nucleon self energy in homogeneous
infinite nuclear matter at varying isoscalar density $n=n_{n}+n_{p}$ and
isospin asymmetry $\delta_{np}=(n_{n}-n_{p})/(n_{n}+n_{p})$. In terms of the
proton and neutron Fermi momenta
$k_{f}^{p}=(3\pi^{2}(1-\delta_{np})n/2)^{1/3}$ and
$k_{f}^{n}=(3\pi^{2}(1+\delta_{np})n/2)^{1/3}$, the nucleon optical potential
in infinite homogeneous matter is then written:

$\displaystyle U_{p}(E;k_{f}^{p},k_{f}^{n})$ $\displaystyle=$ $\displaystyle
V_{p}(E;k_{f}^{p},k_{f}^{n})+iW_{p}(E;k_{f}^{p},k_{f}^{n}),$ $\displaystyle
U_{n}(E;k_{f}^{p},k_{f}^{n})$ $\displaystyle=$ $\displaystyle
V_{n}(E;k_{f}^{p},k_{f}^{n})+iW_{n}(E;k_{f}^{p},k_{f}^{n}).$ (68)

To obtain a nucleon-nucleus optical potential, the nucleon self energy in
homogeneous matter is folded with the target nucleus density distribution in
the local density approximation:

$V(E;r)+iW(E;r)=V(E;k_{f}^{p}(r),k_{f}^{n}(r))+iW(E;k_{f}^{p}(r),k_{f}^{n}(r)).$
(69)

The nucleon self-energy in an infinite system needs to be computed only once
and then can be applied to any finite system for which the density-
distribution is known. The nuclear matter approach is therefore ideal for
constructing microscopic global optical potentials capable of accommodating a
wide range of target nuclei and projectile energies.

Figure 4: First- and second-order perturbation theory diagrams contributing to
the nucleon self energy. The solid lines with up (down) arrows represent
particle (hole) states, and the wavy line represents the nucleon-nucleon
interaction.

In Figure 4 is shown the first- and second-order perturbative contributions to
the nucleon self energy, considering only the presence of two-body forces.
Upward (downward) going solid lines represent nucleon states with momenta
above (below) the Fermi surface, and the wavy line represents the
antisymmetrized nuclear two-body interaction $\bar{V}=(1-P_{12})V$ with
$P_{12}$ the permutation operator

$P_{12}=\left(\frac{1+\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}}{2}\right)\left(\frac{1+\vec{\tau}_{1}\cdot\vec{\tau}_{2}}{2}\right),\,\,\vec{k}_{1}\longleftrightarrow\vec{k}_{2}.$
(70)

Diagram (a) is the first-order (or Hartree-Fock) contribution, which in
isospin-symmetric nuclear matter is given by

$\Sigma^{(1)}_{2N}(k)=\sum_{1}\langle\vec{k}\>\vec{h}_{1}ss_{1}tt_{1}|\bar{V}_{2N}|\vec{k}\>\vec{h}_{1}ss_{1}tt_{1}\rangle
n_{1},$ (71)

where $\vec{k},s,t$ are the momentum, spin, and isospin of the projectile,
$n_{1}$ is the occupation probability $\theta(k_{f}-h_{1})$ for a filled state
with momentum $\vec{h}_{1}$ below the Fermi surface and the summation is over
intermediate-state momenta $\vec{h}_{1}$, spins $s_{1}$, and isospins $t_{1}$.
For example, if one considers a nucleon propagating in symmetric nuclear
matter at density $n=2k_{f}^{3}/3\pi^{2}$ and interacting through a scalar-
isoscalar boson exchange interaction

$V=\frac{g^{2}}{m^{2}+q^{2}},$ (72)

with $\vec{q}$ the momentum transfer, then the corresponding direct (D) and
exchange (E) terms are written

$\displaystyle\Sigma^{(1)}_{D}$
$\displaystyle=4\int\frac{d^{3}h_{1}}{(2\pi)^{3}}\frac{g^{2}}{m^{2}}\theta(k_{f}-h_{1})=\frac{2g^{2}}{3\pi^{2}}\frac{k_{f}^{3}}{m^{2}},$
(73) $\displaystyle\Sigma^{(1)}_{E}$
$\displaystyle=\int\frac{d^{3}h_{1}}{(2\pi)^{3}}\frac{g^{2}}{m^{2}+(\vec{k}-\vec{h}_{1})^{2}}\theta(k_{f}-h_{1})=\frac{g^{2}m}{4\pi^{2}}\left(\frac{k_{f}}{m}-\arctan{\left(\frac{k_{f}}{m}+\frac{k}{m}\right)}\right.$
(74)
$\displaystyle-\left.\arctan{\left(\frac{k_{f}}{m}-\frac{k}{m}\right)}+\frac{m^{2}+k_{f}^{2}-k^{2}}{4mk}\ln{\frac{m^{2}+(k_{f}+k)^{2}}{m^{2}+(k_{f}-k)^{2}}}\right).$
(75)

One sees that the direct term is just a constant and hence local in coordinate
space, whereas the exchange term depends on the magnitude of the projectile
momentum $k$. Moreover, both contributions have no imaginary part and do not
explicitly depend on the energy of the projectile. However, employing the
relation

$E(k)=\frac{k^{2}}{2M}+\Sigma^{(1)}(k),$ (76)

one can interchange the momentum-dependent (i.e., spatially nonlocal) mean
field with an energy-dependent local mean field.

The second-order perturbative contributions to the nucleon self energy in
symmetric nuclear matter read

$\Sigma^{(2a)}_{2N}(k;E)=\frac{1}{2}\sum_{123}\frac{|\langle\vec{p}_{1}\vec{p}_{3}s_{1}s_{3}t_{1}t_{3}|\bar{V}_{2N}|\vec{k}\vec{h}_{2}ss_{2}tt_{2}\rangle|^{2}}{E+\epsilon_{2}-\epsilon_{1}-\epsilon_{3}+i\eta}\bar{n}_{1}n_{2}\bar{n}_{3},$
(77)
$\Sigma^{(2b)}_{2N}(k;E)=\frac{1}{2}\sum_{123}\frac{|\langle\vec{h}_{1}\vec{h}_{3}s_{1}s_{3}t_{1}t_{3}|\bar{V}_{2N}|\vec{k}\vec{p}_{2}ss_{2}tt_{2}\rangle|^{2}}{E+\epsilon_{2}-\epsilon_{1}-\epsilon_{3}-i\eta}n_{1}\bar{n}_{2}n_{3},$
(78)

where the occupation probability for particle states above the Fermi momentum
is $\bar{n}_{i}=\theta(k_{i}-k_{f})$. Hence, one sees that the effective
optical potential for elastic scattering is influenced by the excited states
of the combined system, which serve as possible virtual excitation modes
contributing to the scattering amplitude. In contrast to the Hartree-Fock
contribution, the second-order contributions $\Sigma^{(2a)}$ and
$\Sigma^{(2b)}$ are in general energy-dependent and complex. Contribution
$\Sigma^{(2a)}$ is complex only for states above the Fermi surface, including
positive-energy scattering states, while contribution $\Sigma^{(2b)}$ is
complex only for states below the Fermi surface. In the latter case, the
imaginary part describes the width of the state produced in a knockout or
transfer reaction that leaves behind a hole vacancy in the target nucleus.

In Eqs. (77) and (78), the single-particle energies $E$ and $\epsilon$ should
be computed self-consistently according to

$E(k)=\frac{k^{2}}{2M}+\Sigma(k;E(k)).$ (79)

In the context of elastic scattering, the energy $E$ in Eq. (79) is real and
therefore the momentum $k=k_{r}+ik_{i}$ of the propagating nucleon is in
general complex, since $\Sigma(k;E(q))=V(k;E)+iW(k;E)$ is also complex. Since
the imaginary part is usually several times smaller than the real part, in
practice it is convenient to make the approximation:

$E=\frac{k_{0}^{2}}{2M}+V(k_{0};E),$ (80)

which defines $k_{0}$. Considering both $k_{i}/k_{r}$ and $W(k;E)/V(k;E)$ as
small parameters and expanding Eq. (79) to first order in both quantities, one
obtains

$E=\frac{k_{0}^{2}}{2M}+V(k_{0};E)+\left(\frac{k_{0}}{M}+\left.\frac{\partial
V}{\partial k}\right|_{k_{0}}\right)(k_{r}+ik_{i}-k_{0})+iW(k_{0};E)+\cdots.$
(81)

It follows that

$\displaystyle k_{0}=k_{r}$ $\displaystyle=\sqrt{2M(E-V(k_{r};E))}$ (82)
$\displaystyle k_{i}$
$\displaystyle=-W(k_{r};E)\left(\frac{k_{r}}{M}+\left.\frac{\partial
V}{\partial
k}\right|_{k_{r}}\right)^{-1}=-\frac{M}{k_{r}}W\left(1+\frac{M}{k_{r}}\left.\frac{\partial
V}{\partial k}\right|_{k_{r}}\right)^{-1}.$ (83)

On the other hand, from the Schrödinger equation (7) for a complex potential
$U(r)=V(r)+iW(r)$, one obtains the wavevector solution

$k=k_{r}+ik_{i}=\sqrt{2M(E-V-iW)}$ (84)

with real and imaginary components

$\displaystyle k_{r}$
$\displaystyle=\sqrt{M(E-V)+M\sqrt{(E-V)^{2}+W^{2}}}\simeq\sqrt{2M(E-V)}$ (85)
$\displaystyle k_{i}$ $\displaystyle=-\frac{M}{k_{r}}W.$ (86)

Comparing Eqs. (83) and (86), one finds that in order to compare a microscopic
optical potential derived under the assumption Eq. (80), one must include an
additional factor $\left(1+\frac{M}{k_{r}}\left.\frac{\partial V}{\partial
k}\right|_{k_{r}}\right)^{-1}$ for the imaginary part that arises from the
inherent spatial nonlocality (momentum dependence) of the optical potential ;
.

The nonlocality of the optical potential mean field is directly related to the
nucleon effective mass  in a nuclear medium:

$\frac{dE}{dk}=v=\frac{k}{M^{*}}=\frac{k}{M}+\frac{\partial V}{\partial
k}+\frac{\partial V}{\partial E}\frac{dE}{dk},$ (87)

where

$\frac{M^{*}}{M}=\left(1+\frac{M}{k}\frac{\partial V}{\partial
k}\right)^{-1}\left(1-\frac{\partial V}{\partial
E}\right)\equiv\frac{M_{k}^{*}}{M}\frac{M_{E}^{*}}{M}$ (88)

depends on both the spatial nonlocality and temporal nonlocality (energy
dependence) of the optical potential. The $k$-dependent effective mass is
known to be roughly $M_{k}^{*}/M\simeq 0.7$ , and therefore the imaginary
optical potential computed from microscopic many-body theory under the
assumption Eq. (80) should be reduced when accounting for nonlocality effects.
In addition, the nuclear mean free path $\lambda$ characterizes the distance
over which the flux of nucleons decreases by a factor $e$ and is related to
the imaginary wavevector $k_{i}$ via

$\lambda=\frac{1}{2k_{i}}=-\frac{k_{r}}{2M_{k}^{*}W(k_{r};E)},$ (89)

and therefore nonlocality effects increase the nucleon mean free path.

The nuclear matter approach as defined above has several obvious shortcomings.
First, the spin-orbit force is proportional to the gradient of the potential
and therefore vanishes in a homogeneous medium. In the past, it was common to
supplement the microscopic central components of the optical potential with a
phenomenological spin-orbit interaction. More recently, in Ref.  it was
suggested to use the density matrix expansion ;  to obtain the leading-order
Hartree-Fock approximation to the spin-orbit interaction in nuclei starting
from the same nuclear two-body and three-body forces used to compute the
nucleon self energy. Such a spin-orbit interaction is constructed at the Fermi
surface (rather than at positive scattering energies) and has no energy
dependence, but nevertheless has a magnitude similar to that from
phenomenology .

Figure 5: Radial dependence of the real, imaginary, and spin-orbit
contributions to the proton-40Ca optical potential at projectile energy 2.35
MeV. The green-dashed lines are from the phenomenological Koning-Delaroche
optical potential. The dotted lines are from homogeneous nuclear matter
calculations of the nucleon self energy from chiral effective field theory
two-body and three-body forces ;  combined with the local density
approximation (left) and improved local density approximation (right). The
solid lines are fits to the chiral effective field theory results using the
Koning-Delaroche global parameterization. Reprinted figure with permission
from Ref. , copyright (2019) by the American Physical Society.

Another problematic feature of the nuclear matter approach is that in the
local density approximation, Eq. (68), implies that a projectile nucleon only
begins to interact with the target nucleus once the two are in physical
contact. Since the range of the nuclear force is approximately 2 fm, the
surface diffuseness in the local density approximation is too small. This can
be improved by accounting for the finite range of the nuclear force using the
improved local density approximation (ILDA):

${V}(E;r)_{ILDA}=\frac{1}{(t\sqrt{\pi})^{3}}\int
V(E;r^{\prime})e^{\frac{-|\vec{r}-\vec{r}^{\prime}|^{2}}{t^{2}}}d^{3}r^{\prime},$
(90)

where a smearing function with adjustable length scale $t$ is introduced and
taken to be the typical range of the nuclear force. In Fig. 5 is shown the
difference between the real, imaginary, and spin-orbit contributions to the
proton-40Ca optical potential at projectile energy 2.35 MeV in the local
density approximation (left) and improved local density approximation (right).
The dotted lines are the results obtained from chiral effective field theory
calculations of the nucleon self energy in homogeneous nuclear matter folded
with the density distribution of 40Ca using a mean field model fitted to the
equation of state from chiral effective field theory . The solid lines show
the fits to the chiral effective field theory calculations using the
parameterized form of the Koning-Delaroche (KD) global optical potential , and
the dashed-green lines are calculated from the KD global optical potential.
First, one notices that the phenomenological parametrization accurately fits
the microscopic calculations. Second, it is evident that the main effect of
the improved local density approximation is to increase the diffuseness of the
optical potential surface. Finally, one sees that the microscopic imaginary
part is much less surface-peaked than phenomenology would suggest, while at
higher projectile energies the imaginary part becomes nearly a factor of two
larger than the phenomenological depth .

Figure 6:  The depth (left), radius (middle), and diffuseness (right)
parameters of the central real and imaginary terms in the WLH and KD global
optical potentials shown as a function of energy. The bands of the WLH
parameter distributions represent the 95% credible interval from 5000 random
parameter samples.

Systematic uncertainties in the construction of microscopic global optical
potentials based on the nuclear matter Green’s function approach have been
studied in Ref. . In this work, a set of five chiral N2LO and N3LO potentials
with varying momentum-space cutoff values was used to construct a global
optical potential, referred to as WLH, with associated theory uncertainties.
From a covariance matrix analysis, the optical potential parameters are
expressed in terms of a multivariate normal distribution. This distribution
can then be sampled many times to produce uncertainty estimates for optical
potential parameters and derived scattering observables. In Fig. 6 the 95%
probability contours for the WLH depth, radius, and diffuseness parameter
distributions for the central real and imaginary terms are shown as a function
of projectile energy. In general the predicted parameter values of WLH are
close to KD, however, one finds large uncertainties inherent to
microscopically derived optical potentials. The real depth of WLH is quite
similar to KD while the imaginary depth from the nuclear matter approach is
roughly a factor of two larger than that of KD. On average, the geometry
parameters of WLH are also quite similar to those from KD, though in the
microscopic approach there

Figure 7:  Proton and neutron elastic scattering cross sections predicted by
the WLH global optical potential compared to data. The bands of the WLH
results represent the 95% credible interval of 5000 random parameter samples.

can be significant differences between the geometry parameters for the real
and imaginary parts, in contrast to the KD parameterization which assumes the
real and imaginary volume geometry parameters are identical.

In Fig. 7 elastic differential cross sections for both proton and neutron
projectiles predicted by WLH are compared to experimental data for a wide
range of targets and energies. In general the WLH optical potential reproduces
elastic scattering well for projectile energies of $E\lesssim 150$ MeV. As the
projectile energy increases to $E\sim 200$ MeV, the predictions of WLH tend to
have more discrepancies with data and larger uncertainties. In Fig. 8,
analyzing powers from the WLH microscopic global optical potential for proton
and neutron projectiles are shown for a range of energies and target isotopes.
Again, the overall agreement with experimental appears satisfactory within the
theory uncertainties.

Figure 8:  Proton and neutron analyzing powers predicted by the WLH global
optical potential compared to data. The bands of the WLH results represent the
95% credible interval of 5000 random parameter samples.

##  8 Dispersive optical potentials

A great deal of experimental data exists to probe the nuclear mean field
(i.e., optical potential) at a continuum of positive energies. At negative
energies the nuclear mean field (i.e., shell model potential) affects the
single-particle level structure of nuclei and the bound-state eigenfunctions,
all at discrete values of the energy. Historically, the development of optical
potentials and shell model potentials evolved independently. The aim of the
so-called dispersive optical model  is to provide a phenomenological framework
to treat elastic scattering, reaction, and structure data all on an equal
footing connected by a nuclear mean field that is continuous from negative to
positive energies. The starting point is the dispersion relation linking the
real and imaginary parts of the optical potential through the requirement of
causality, where the scattering wave cannot be emitted before the arrival of
the incident wave. This gives rise to Kramers-Kronig relations between the
real and imaginary parts of the mean field:

$\displaystyle V(k;E)$ $\displaystyle=\frac{1}{\pi}{\cal
P}\int_{-\infty}^{\infty}\frac{W(k,E^{\prime})}{E^{\prime}-E}dE^{\prime}$ (91)
$\displaystyle W(k;E)$ $\displaystyle=-\frac{1}{\pi}{\cal
P}\int_{-\infty}^{\infty}\frac{V(k,E^{\prime})}{E^{\prime}-E}dE^{\prime},$
(92)

where ${\cal P}$ denotes the principal value integral. Thus, one sees that the
real nuclear mean field in one energy regime is governed by the absorptive
imaginary part across both regimes. Purely phenomenological optical potentials
fit separately the real and imaginary components without respecting their
mutual constraints through dispersion integrals. Such optical potentials
therefore fundamentally violate causality.

In the dispersive optical model, one starts by breaking up the real part of
the self energy into two parts

$V(r,E)=V_{HF}(r,E)+\Delta V(r;E).$ (93)

In principle, the energy dependence of the Hartree-Fock contribution
$V_{HF}(r,E)$ arises by replacing the initially energy-independent and
nonlocal part of the optical potential by its energy-dependent local
equivalent . In practice, however, the Hartree-Fock contribution in the
dispersive optical potential is often taken to be of the form in Eq. (93) from
the start. The energy-dependent (or “dispersive”) part of the optical
potential $\Delta V(r;E)$ that arises from temporal nonlocality is related to
the imaginary part of the self energy through either Eq. (91) or the
subtracted dispersion relation

$\Delta V(r;E)=\frac{1}{\pi}{\cal P}\int
W(r,E^{\prime})\left(\frac{1}{E^{\prime}-E}-\frac{1}{E^{\prime}-E_{F}}\right)dE^{\prime},$
(94)

which has several advantages, such as minimizing the magnitude of the
dispersive correction in the vicinity of the Fermi surface, $\Delta
V(r;E_{F})=0$, and taming the effects from large variations in the imaginary
part of the optical potential near the Fermi surface.

Since only the real dispersive part $\Delta V(r,E)$ of the optical potential
is fixed from the dispersion relation, a parameterized model for both the real
Hartree-Fock $V_{HF}(r,E)$ contribution and the imaginary $W(r,E)$
contribution is needed. There is considerable flexibility when choosing the
functional dependence on the energy $E$ and on the nuclear radius $R$ and
diffuseness $a$ parameters. The following discussion will focus only on those
features that are common to many of the different analyses. The Hartree-Fock
term is typically written in the form

$V(r,E)=V_{HF}^{vol}(E)f(r,r^{HF},a^{HF})+V_{HF}^{sur}\frac{d}{dr}f(r,r^{HF},a^{HF})+V_{C}(r)+V_{so}(r,E),$
(95)

consisting of volume, Coulomb, and spin-orbit contributions as well as an
optional surface real part. Most purely phenomenological optical potentials do
not include a surface real term, which in recent dispersive optical model
analyses  was found to be important to describe high-energy elastic
scattering. The imaginary component is parameterized as

$W(r,E)=W^{vol}(E)f(r,r^{vol},a^{vol})+4a^{sur}W^{sur}(E)\frac{d}{dr}f(r,r^{sur},a^{sur})+W_{so}(r,E),$
(96)

consisting of the usual volume, surface, and spin-orbit terms. The presence of
an imaginary spin-orbit contribution also implies a dispersive correction
$\Delta V^{so}(r,E)$ to the real spin-orbit interaction $V_{so}(r,E)$ in Eq.
(95).

While the dispersive optical model is primarily phenomenological, some of the
assumptions that go into defining the parametric form of the potential are
motivated by microscopic calculations. For instance, general arguments
indicate that the imaginary part of the self energy is symmetric about the
Fermi energy and in fact vanishes at $E=E_{F}$. This property is built into
the energy dependent parameters in Eq. (96). Far away from the Fermi energy,
however, the imaginary part of the dispersive optical potential is typically
allowed to break symmetry about $E_{F}$. In addition, microscopic calculations
of the Hartree-Fock contribution find that the real volume component is a very
smooth function from positive to negative energies, and hence one typically
takes a low-order polynomial in powers of $E-E_{F}$.

In practice, the parameters that define the energy dependence and shape
parameters of the real Hartree-Fock and imaginary components of the dispersive
optical potential can be fitted to a wide range of scattering and bound-state
data. For instance, single-particle energy levels, experimental one-particle
removal widths, binding energies, charge densities, charge radii, differential
elastic scattering cross sections, analyzing powers, reaction cross sections,
and total cross sections ;  have all been used in fitting the parameters of
modern dispersive optical potentials. In Fig. 9 is shown the results  of
fitted nucleon-nucleus elastic scattering cross sections for proton and
neutron projectiles at varying energy on calcium isotopes within the
dispersive optical model. One finds that it is possible to describe a large
amount of elastic scattering data within this phenomenological framework, and
similar quality of fits can also be obtained simultaneously for analyzing
powers and reaction cross sections.

Figure 9: Fitted differential elastic scattering cross sections for proton and
neutron projectiles at varying energies on calcium isotopes within the
dispersive optical model framework. Reprinted figure with permission from Ref.
, copyright (2007) by the American Physical Society.

In this review, some of the modern theoretical tools for constructing nucleon-
nucleus optical model potentials for nucleon elastic scattering and reactions
have been outlined. Ab initio methods, such as self-consistent Green’s
function theory and the coupled-cluster method, can at present be applied at
low projectile energies and for doubly-magic nuclei. In order to describe a
wider range of target nuclei and projectile energies, one must make
simplifying assumptions as in the nuclear matter Green’s function approach or
multiple scattering theory. These methods allow for the implementation of
realistic nuclear two-body and many-body forces, which nowadays is firmly
grounded in the low-energy realization of QCD, chiral effective field theory.
Ab initio and microscopic reaction theory models can be linked to the large
corpus of nuclear structure studies using the same underlying nuclear force
models. A phenomenological framework for consistently analyzing both positive-
energy scattering states and negative-energy bound states is provided by the
dispersive optical model, which in addition respects microscopic constraints
such as causal dispersion relations and the functional form of the imaginary
optical potential near the Fermi surface.

##  9 Acknowledgments

The authors thank M. Vorabbi for useful discussions and providing Figure 2.
The authors also thank C. Elster and R. C. Johnson for helpful feedback on our
manuscript. JWH is supported by the National Science Foundation under Grant
No. PHY1652199 and by the U.S. Department of Energy National Nuclear Security
Administration under Grant No. DE-NA0003841. TRW is supported in part by the
U.S. Department of Energy (Office of Science, Nuclear Physics) under grant DE-
SC0021422.

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