Proton elastic scattering on calcium isotopes from chiral nuclear optical potentials

T. R. Whitehead, Y. Lim, and J. W. HoltCyclotron Institute, Texas A&M University, College Station, TX 77843, USA and

Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA

We formulate microscopic optical potentials for nucleon-nucleus scattering from chiral two- andthree-nucleon forces. The real and imaginary central terms of the optical potentials are obtained fromthe nucleon self energy in infinite nuclear matter at a given density and isospin asymmetry, calculatedself-consistently to second order in many-body perturbation theory. The real spin-orbit term isextracted from the same chiral potential using an improved density matrix expansion. The density-dependent optical potential is then folded with the nuclear density distributions of 40,42,44,48Ca fromwhich we study proton-nucleus elastic scattering and total reaction cross sections using the reactioncode TALYS. We compare the results of the microscopic calculations to those of phenomenologicalmodels and experimental data up to projectile energies of E = 180 MeV. While overall satisfactoryagreement with the available experimental data is obtained, we find that the elastic scattering andtotal reaction cross sections can be significantly improved with a weaker imaginary optical potential,particularly for larger projectile energies.

I. INTRODUCTION

Nucleon-nucleus optical potentials are a valuable toolfor predicting a wide range of scattering and reactionprocesses by replacing the complicated many-body dy-namics of nucleons interacting through two- and three-body forces with an average complex, energy-dependentsingle-particle potential. Global phenomenological opti-cal potentials [1, 2] have been constructed by fitting toexperimental data spanning a large range of projectileenergies across many nuclei. Although phenomenolog-ical optical potentials are very successful at describingscattering processes involving nuclei near stability, mi-croscopic optical potentials are not tuned to experimen-tal data and therefore may have greater predictive powerfor reactions involving exotic isotopes.

Semi-microscopic global optical potentials [3, 4] de-rived in the 1970’s from high-precision one-boson-exchange nucleon-nucleon interactions are still widelyused today [5–9]. Whereas the density and isospin-asymmetry dependence is computed microscopically, theoverall strengths of the real and imaginary volumeterms are often adjusted with energy-dependent empiri-cal strength factors. The effects of three-body forces weregenerally neglected in these early works, and within theoriginal nuclear matter approach no spin-orbit optical po-tential could be derived. More recently, three-body forceshave been implemented [10–12] in calculations of thereal and imaginary central terms, while phenomenologi-cal spin-orbit optical potentials have been added [7, 8, 13]in order to better describe analyzing powers and differ-ential cross sections at large scattering angles. An alter-native approach [14–21] to constructing microscopic op-tical potentials is based on multiple scattering theory in-volving the nucleon-nucleon T -matrix. Such an approachnaturally generates a spin-orbit contribution, but the im-plementation of medium effects [22, 23] and three-bodyforces remains challenging, which in practice often limitsthe theory to large scattering energies E & 200 MeV.Other current approaches to deriving predictive opti-

cal potentials include the self-consistent Green’s functionmethod [24] and the dispersive optical model [25].

Recently there has been much interest in the develop-ment of microscopic optical potentials [10, 12, 21, 26–30]based on chiral effective field theory (EFT) [31–33]. Themain motivation is to implement more realistic micro-physics involving multi-pion exchange contributions tothe nuclear force, three-body interactions, and theoret-ical uncertainty estimates. Chiral optical potentials arewell suited to describe low-energy scattering processesbut are expected to break down for energies approach-ing the relevant momentum-space cutoff employed. Inpractice, the presence of the cutoff constrains nucleonprojectile energies to lie below E . 200 MeV.

In the present study, we aim to lay the groundworkfor a revised nuclear matter description of the globalnucleon-nucleus optical potential based on chiral EFT.Ultimately the goal will be to develop a theory fornucleon-nucleus scattering across a large range of iso-topes, including those off stability, at energies up to 200MeV. As a starting point we consider differential elas-tic and total reaction cross sections for proton-nucleusscattering along a chain of calcium isotopes, 40,42,44,48Ca,at energies ranging from 2-160 MeV where experimentaldata are available. We also compare to the global phe-nomenological optical potential of Koning and Delaroche[2] and investigate to what extent modern phenomeno-logical parametrizations of the optical potential are con-sistent with microscopic analyses. Our calculations areperformed within the TALYS [34] reaction code for whichwe have developed an implementation of our microscopicoptical potential.

We take as a starting point for the calculation a partic-ular high-precision 2N + 3N chiral nuclear potential withmomentum-space cutoff Λ = 450 MeV. The low-energyconstants of the potential are fitted to nucleon-nucleonscattering phase shifts, deuteron properties, and in thecase of three-body contact terms also the triton bindingenergy and lifetime. The nucleon-nucleon interaction istaken at next-to-next-to-next-to-leading order (N3LO),

arX

iv:1

812.

0872

5v2

[nu

cl-t

h] 2

3 O

ct 2

019

2

while only the N2LO three-nucleon force is included. Theinconsistent treatment of two- and three-body forces atthe level of the chiral expansion is undesirable, but worktoward fully consistent two- and many-body forces is inprogress [35–37]. We note that the chiral nuclear po-tential employed in the present work exhibits good nu-clear matter properties (saturation energy and density[38], thermodynamics [39, 40], and Fermi liquid param-eters [41]) when calculated at least to second order inmany-body perturbation theory. In the future we planto perform calculations of the nucleon-nucleus optical po-tential from a wider range of high-precision chiral nuclearforces in order to better assess theoretical uncertainties.

In quantum many-body theory, the optical potentialfor scattering states is identified with the energy- andmomentum-dependent single-particle self energy [42].We first compute the nucleon self energy in homoge-neous nuclear matter at arbitrary density and compo-sition (proton fraction) from chiral two- and three-bodyforces to second order in many-body perturbation the-ory. We next compute nuclear density distributions forselected calcium isotopes (40Ca, 42Ca, 44Ca, 48Ca) frommean field theory employing recently derived [43] Skyrmeeffective interactions constrained by chiral effective fieldtheory. In the local density approximation (LDA) thenucleon-nucleus optical potential is computed [44] byfolding the nucleon self-energy in homogeneous matterwith the derived density distributions. Since the LDA isknown [44] to underestimate the surface diffuseness of theoptical potential in finite nuclei, we employ the improvedlocal density approximation (ILDA) described in Refs.[7, 44] to account for the non-zero range of the nuclearforce.

The method outlined thus far is versatile since it can beused to produce optical potentials for a very wide rangeof nuclei. However, the LDA nuclear matter approachcannot capture the physics of collective surface modes,shell effects [45], and surface-peaked spin-orbit opticalpotentials. The latter are particularly important for spinobservables and elastic scattering cross sections at largeangles. In the present work we therefore construct a spin-orbit optical potential from the improved density ma-trix expansion [46–48], which improves the description ofthe spin-dependent part of the energy density functionalcompared to the standard density matrix expansion ofNegele and Vautherin [49]. We then benchmark our ap-proach to experimental data for proton elastic scatteringand total reaction cross sections on the calcium isotopes40Ca, 42Ca, 44Ca, 48Ca.

The paper is organized as follows. In Section II we de-scribe details of the microscopic calculation of the opticalpotential in nuclear matter from chiral EFT. We then cal-culate nuclear density distributions from mean field the-ory and employ the ILDA to construct nucleon-nucleusoptical potentials for calcium isotopes. The microscopicoptical potentials are parametrized in the form of theKoning-Delaroche (KD) [2] phenomenological optical po-tential in order to implement them into the reaction code

(a) (b) (c)

FIG. 1. Diagrammatic representations of the first and secondorder contributions to the self energy. The solid lines repre-sent nucleon propagators and the wavy lines represent the inmedium two-nucleon interaction.

TALYS. In Section III we compute proton-nucleus elasticdifferential scattering cross sections up to projectile en-ergy E = 160 MeV and total reaction cross sections up toE = 180 MeV. These results are compared to empiricaldata and predictions from the KD phenomenological op-tical potential. We end with a summary and conclusions.

II. OPTICAL POTENTIAL FROM CHIRALEFFECTIVE FIELD THEORY

A. Real and imaginary central terms

In recent work [10, 12] the nucleon self energy in homo-geneous nuclear matter has been computed employing aset of nuclear potentials derived from chiral effective fieldtheory. The first- and second-order perturbative contri-butions to the nucleon self energy are shown graphicallyin Fig. 1 and given quantitatively by

Σ(1)2N (q; kf ) =

∑1

〈~q ~h1ss1tt1|V eff2N |~q ~h1ss1tt1〉n1, (1)

Σ(2a)2N (q, ω; kf ) (2)

=1

2

∑123

|〈~p1~p3s1s3t1t3|V eff2N |~q~h2ss2tt2〉|2

ω + ε2 − ε1 − ε3 + iηn1n2n3,

Σ(2b)2N (q, ω; kf ) (3)

=1

2

∑123

|〈~h1~h3s1s3t1t3|V eff

2N |~q~p2ss2tt2〉|2

ω + ε2 − ε1 − ε3 − iηn1n2n3,

where ni is the occupation probability θ(kf − ki) for

a filled state with momentum ~ki below the Fermi sur-face, the occupation probability for particle states isni = θ(ki−kf ), the summation is over intermediate-state

momenta for particles ~pi and holes ~hi, their spins si, andisospins ti. The nuclear potential V eff

2N represents the anti-symmetrized two-body interaction consisting of the barenucleon-nucleon (NN) potential VNN together with an ef-fective, medium-dependent NN interaction V med

NN derivedfrom the N2LO chiral three-nucleon force by averagingone particle over the filled Fermi sea of noninteracting

3

nucleons [50–52]. In the first-order Hartree-Fock con-tribution, Eq. (1), the effective interaction is given byV eff

2N = VNN + 12V

medNN , while for the higher-order con-

tributions, Eqs. (2) and (3), the effective interaction isgiven by V eff

2N = VNN + V medNN . The Hartree-Fock contri-

bution is nonlocal, energy-independent, and purely real,while the second-order contributions are in general nonlo-cal, energy-dependent, and complex. The single-particleenergies in the denominators of Eqs. (2) and (3) are com-puted self-consistently according to

ε(q) =q2

2M+ ReΣ(q, ε(q)), (4)

where M is the free-space nucleon mass.In the present work the self energy is computed for

arbitrary isospin-asymmetry, δnp = (ρn − ρp)/(ρn + ρp),which is essential for an accurate description of nuclei forwhich N 6= Z. The resulting optical potentials for nu-cleons propagating in homogeneous matter characterizedby its proton and neutron Fermi momenta kpf and knf aregiven by

Up(E; kpf , knf ) = Vp(E; kpf , k

nf ) + iWp(E; kpf , k

nf ),

Un(E; kpf , knf ) = Vn(E; kpf , k

nf ) + iWn(E; kpf , k

nf ) (5)

with

Vi(E; kpf , knf ) = ReΣi(q, E(q); kpf , k

nf ), (6)

Wi(E; kpf , knf ) =

Mk∗i

MImΣi(q, E(q); kpf , k

nf ), (7)

where the subscript i denotes a propagating proton orneutron. In relating the physical imaginary part of theoptical potential to the imaginary part of the nucleonself-energy we have multiplied [53, 54] by the effectivek-mass Mk∗

i defined by

Mk∗i

M=

(1 +

M

k

∂

∂kVi(k,E(k)

)−1

, (8)

in order to account for the non-locality of the opticalpotential.

B. Spin-orbit optical potential

The effective one-body spin-orbit interaction vanishesin homogeneous nuclear matter and therefore cannot becomputed within the framework described above. In-stead we employ an improved density matrix expan-sion [47, 48, 55] to construct the one-body spin-orbit in-teraction from chiral two- and three-body forces. Theimproved density matrix expansion takes advantage ofphase space averaging to derive a more accurate spin-dependent energy density functional compared to thestandard density matrix expansion of Negele-Vautherin[49].

40Ca42Ca44Ca48Ca

2 4 6 8

0.05

0.10

0.15

2 4 6 8

0.05

0.10

0.15

r (fm)

ρ(fm

-3)

FIG. 2. The nucleon density distributions for 40,42,44,48Ca cal-culated in mean field theory from the Skyrme Skχ450 effectiveinteraction constrained by chiral effective field theory.

From the definition of the density matrix

ρ(~r1σ1τ1;~r2σ2τ2) =∑α

Ψ∗α(~r2σ2τ2)Ψα(~r1σ1τ1), (9)

where Ψα are energy eigenfunctions associated with oc-cupied orbitals of the non-relativistic many-body system,the energy density functional for N = Z even-even nucleiin the Hartree-Fock approximation can be expanded upto second order in spatial gradients as

E [ρ, τ, ~J ] = ρ E(ρ) +

[τ − 3

5ρk2f

][1

2MN+ Fτ (ρ)

](10)

+(~∇ρ)2 F∇(ρ) + ~∇ρ · ~J FSO(ρ) + ~J 2 FJ(ρ) ,

where ρ(~r ) = 2k3f (~r )/3π2 =

∑α Ψ†α(~r )Ψα(~r ) de-

fines the local density with kf (~r ) the local Fermi mo-

mentum, τ(~r ) =∑α~∇Ψ†α(~r ) · ~∇Ψα(~r ) is the ki-

netic energy density, and ~J(~r ) = i∑α~Ψ†α(~r )~σ ×

~∇Ψα(~r ) is the spin-orbit density. These terms aremultiplied by the density-dependent strength func-tions E(ρ), Fτ (ρ), F∇(ρ), FSO(ρ), FJ(ρ), of which we arepresently only interested in the spin-orbit term FSO(ρ).In effect, the spin-orbit optical potential is therefore cal-culated for N = Z nuclei to first order in many-bodyperturbation theory. In the future, higher-order pertur-bative contributions [56] to the microscopic nuclear en-ergy density functional may be investigated. We notethat we do not include the isovector part [57] of the spin-orbit interaction for N 6= Z nuclei in this study since itis known to be small compared to the isoscalar part [48].

C. Improved local density approximation

We employ the improved local density approximation(ILDA) to construct the nucleon-nucleus optical potentialfor finite nuclei. The density dependent optical potential

4

●

● ● ●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

KD Global

Chiral Fit

● Chiral Data

1 2 3 4 5 6 7

-2.0

-1.5

-1.0

-0.5

0

r (fm)

VSO(MeV

)

R2=0.9998

● ● ● ● ● ● ● ● ● ● ● ● ● ●●

●●

●

●

●

●

●

●

●

●

●

●●

● ● ● ● ● ● ● ● ● ● ●

KD Global

Chiral Fit

● Chiral Data-15

-10

-5

0

WV+WD(MeV

)

R2=0.9998

● ● ● ● ● ● ● ● ● ● ● ● ●●

●●

●

●

●

●

●

●

●

●

●

●

●●

● ● ● ● ● ● ● ● ● ● ●

KD Global

Chiral Fit

● Chiral Data-40

-30

-20

-10

01 2 3 4 5 6 7

VV(MeV

)

R2=0.9999

KD Global

Chiral ILDA

1 2 3 4 5 6 7

-2.0

-1.5

-1.0

-0.5

0

r (fm)

KD Global

Chiral ILDA-15

-10

-5

0

KD Global

Chiral ILDA

1 2 3 4 5 6 7

-40

-30

-20

-10

0E = 35 MeV

FIG. 3. Left (right): The real, imaginary, and spin-orbit terms of the microscopic optical potential for proton-40Ca scatteringat E = 2.35 MeV before (after) applying the improved local density approximation. In the left panel the dots represent rawresults from chiral EFT, while the solid black lines represent fits to the Koning-Delaroche (KD) form. We also compare to theglobal KD phenomenological optical potential, shown as the dashed green line, at the same energy.

(both central and spin-orbit parts) is folded with the ra-dial density distribution of a target nucleus. The nucleardensity distributions are calculated within mean field the-ory from the Skχ450 Skyrme interaction [43], which fitsboth finite nuclei properties as well as theoretical cal-culations of the asymmetric nuclear matter equation ofstate from the N3LO Λ = 450 MeV chiral potential usedin the calculation of the self energy. In Fig. 2 we showthe resulting nucleon density distributions for each of thecalcium isotopes 40Ca, 42Ca, 44Ca, 48Ca.

In the standard local density approximation, thestrength of the nucleon-nucleus optical potential at agiven radial distance r is evaluated as

V (E; r) + iW (E; r) = V (E; kpf (r), knf (r)) (11)

+iW (E; kpf (r), knf (r)),

where kpf (r) and knf (r) are the local proton and neu-tron Fermi momenta. This approximation is strictlyvalid only for zero-range nuclear forces, and when ap-plied to nucleon-nucleus optical potentials it is knownto underestimate the surface diffuseness [44, 58]. Conse-quently, such an approach is inadequate for an accuratedescription of nuclear elastic scattering and reaction pro-cesses. The improved local density approximation applies

a Gaussian smearing

V (E; r)ILDA =1

(t√π)3

∫V (E; r′)e

−|~r−~r′|2

t2 d3r′ (12)

characterized by an adjustable length scale t associatedwith the non-zero range of the nuclear force. In the lim-

iting case of t → 0, a factor of δ(|~r − ~r′|) replaces theGaussian, giving VILDA(E; r) → V (E; r). In Ref. [7]it is found that for the central part of the interactiontC = 1.2 fm gives the best fit to experimental reactioncross sections for 10 MeV < E < 200 MeV and targetsranging from 40Ca to 208Pb. In the present work wevary the range parameter 1.15 fm < tC < 1.25 fm. Thisvariation is used to estimate the theoretical uncertaintyassociated with our choice of the length scale tC . Forthe spin-orbit part of the optical potential, we estimatethe range parameter tSO from the root mean square ra-dius of the Argonne v18 spin-orbit nucleon-nucleon poten-tial [59]. We found tSO = 1.07 fm and took two values,tSO = 1.0, 1.1 fm to estimate the uncertainty.

In Figs. 3 and 4 we show the comparison between thecentral and spin-orbit optical potentials in the LDA (leftpanels) and ILDA (right panels) for the proton-40Ca op-tical potential at projectile energies E = 2.35 MeV and

5

●

● ● ●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

KD Global

Chiral Fit

● Chiral Data

1 2 3 4 5 6 7-2.5

-2.0

-1.5

-1.0

-0.5

0

r (fm)

VSO(MeV

)

R2=0.9998

● ●●

●●

●● ● ● ● ●

●●

●●

● ● ●●

●

●

●

●

●

●

●●

● ● ● ● ● ● ● ● ● ● ● ●

KD Global

Chiral Fit

● Chiral Data-6

-4

-2

0

WV+WD(MeV

)

R2=0.9998

● ● ● ● ● ● ● ● ● ● ● ● ● ●●

●

●

●

●

●

●

●

●

●

●

●

●●

●● ● ● ● ● ● ● ● ● ●

KD Global

Chiral Fit

● Chiral Data-50

-40

-30

-20

-10

01 2 3 4 5 6 7

VV(MeV

)

R2=0.9999

KD Global

Chiral ILDA

1 2 3 4 5 6 7-2.5

-2.0

-1.5

-1.0

-0.5

0

r (fm)

KD Global

Chiral ILDA-6

-4

-2

0

KD Global

Chiral ILDA

1 2 3 4 5 6 7

-50

-40

-30

-20

-10

0E = 2.35 MeV

FIG. 4. Same as Fig. 3, except that the projectile energy is E = 35 MeV.

E = 35 MeV respectively. In the left panels, the dots in-dicate the results from chiral effective field theory, and forcomparison we show as the green dashed lines the phe-nomenological optical potentials from Koning and De-laroche [2]. In particular, the real central part of themicroscopic optical potential has a much smaller surfacediffuseness compared to phenomenology. When the im-proved local density approximation is employed in theright panels (solid blue bands), the comparison to phe-nomenology is much improved. In addition, the overallstrength of the real central part of the optical potential isin very good agreement with the Koning-Delaroche phe-nomenological optical potential. Varying the ILDA rangeparameter tC between 1.15 fm and 1.25 fm yields only asmall change in the overall shape of the real central part,which suggests that the theoretical predictions for scat-tering cross sections computed in the next section willnot be especially sensitive to the precise choice of tC .

In the middle panels of Figs. 3 and 4 we plot the imagi-nary part of the microscopic optical potential for proton-40Ca scattering at E = 2.35 MeV and E = 35 MeV.This is again compared to the phenomenological opti-cal potential of Koning and Delaroche, which is writtenas the sum of a volume imaginary part WV and a sur-face imaginary part WD. In the microscopic descriptionof the imaginary part, there is no distinction betweenthese two components. We observe that in contrast to the

real central part of the optical potential, the microscopicimaginary part exhibits large qualitative differences com-pared to phenomenology. The most striking differenceis a much smaller surface peak, which only appears atlow projectile energies in the microscopic calculation butpersists to much higher energies (E ' 100 MeV) in thephenomenological optical potential. For instance, at thescattering energy E = 35 MeV, the surface peak has es-sentially vanished in the microscopic calculation, andthe remaining “volume” imaginary part is large com-pared to phenomenology. In fact, this is a common fea-ture [4, 6, 60, 61] in microscopic optical potentials com-puted in the nuclear matter approach. Modern semi-microscopic optical potentials therefore include empiricalenergy-dependent strength factors multiplying the realand imaginary central parts [7, 9].

In the bottom panels of Figs. 3 and 4 we show the realspin-orbit part of the microscopic optical potential com-pared to the Koning-Delaroche phenomenological opticalpotential for proton-40Ca scattering at E = 2.35 MeVand E = 35 MeV. The radial shape of the microscopicspin-orbit optical potential is found to be very similarto that of the Koning-Delaroche optical potential, how-ever, the strength of the microscopic potential is larger.Indeed, the density matrix expansion carried out at theHartree-Fock level is known [48, 62] to produce a strongerspin-orbit interaction, by about 20-50%, than is required

6

TABLE I. Shape parameters for the proton-40Ca microscopicoptical potential at the three energies E = 2.35, 35, 100 MeV.Also shown are the corresponding shape parameters for theKoning-Delaroche (KD) optical potential that are indepen-dent of energy.

E = 2.35 MeV VV WV VSO

r (fm) 1.213 1.344 1.015

a (fm) 0.723 0.583 0.706

E = 35 MeV VV WV VSO

r (fm) 1.183 1.204 1.015

a (fm) 0.730 0.751 0.706

E = 100 MeV VV WV VSO

r (fm) 1.173 0.846 1.015

a (fm) 0.713 0.702 0.706

KD VV WV VSO

r (fm) 1.185 1.185 0.996

a (fm) 0.672 0.672 0.590

from traditional mean field theory studies of finite nuclei.Higher-order perturbative contributions are expected toremedy this feature. In particular, multi-pion-exchangeprocesses have been shown [63] to reduce the strength ofthe one-body spin-orbit interaction in finite nuclei. Thisprovides additional motivation for including G-matrixcorrelations in the density matrix expansion as outlinedin [56]. As in the case of the central components of theoptical potential, we find relatively small differences be-tween spin-orbit potentials produced with two choices ofthe ILDA length scale tSO = 1.0, 1.1 fm.

D. Parameterization of the chiral optical potential

In order to facilitate the implementation of our micro-scopic optical potential into standard nuclear reactioncodes, such as TALYS, we fit our optical potential to thephenomenological form of Koning and Delaroche. Even-tually our aim is to construct a global microscopic opticalpotential and make it available in a convenient form fornuclear reaction practitioners. This exercise may alsohelp to reveal any deficiencies in the assumed form ofphenomenological optical potentials. We recall that inthe phenomenological description, the optical potentialtakes the form

U(r, E) = VV (r, E) + iWV (r, E) + iWD(r, E) (13)

+VSO(r, E)~ · ~s+ iWSO(r, E)~ · ~s+ VC(r),

consisting of a real volume term, an imaginary volumeand surface term, a real and imaginary spin-orbit term,and finally the central Coulomb interaction. In Eq. (13),~ and ~s are the single-particle orbital angular momen-tum and spin angular momentum operators, respectively.Since the phenomenological imaginary spin-orbit term isvery small and cannot be extracted within the present

microscopic approach, we neglect it in the rest of thediscussion. The energy and radial dependence of the dif-ferent terms in the phenomenological optical potentialare assumed to factorize according to

VV (r, E) = VV (E)f(r; rV , aV ), (14)

WV (r, E) =WV (E)f(r; rW , aW ), (15)

WD(r, E) = −4aDWD(E)d

drf(r; rD, aD), (16)

VSO(r, E) = VSO(E)1

m2π

1

r

d

drf(r; rSO, aSO), (17)

where

f(r; ri, ai) =1

1 + e(r−A1/3ri)/ai(18)

is of the Woods-Saxon form with A the mass numberand {ri,ai} the energy-independent geometry parame-ters that encode the size and diffuseness of a given targetnucleus respectively. In phenomenological optical poten-tials, these shape parameters vary weakly with the targetnucleus. The energy-dependent strength functions in theKD parametrization have the form

VV (E) = v1(1− v2E + v3E2 − v4E

3), (19)

WV (E) = w1E2

E2 + w22

, (20)

WD(E) = d1E2 e−d2E

E2 + d23

, (21)

VSO(E) = vSO1e−vSO2E , (22)

where E = E−EF is the projectile energy relative to theFermi energy EF .

In the left panels of Figs. 3 and 4 we show as the solidblack lines the best fit functions of the form Eqs. (14)-(22) to the microscopic calculations. We see that overallthe phenomenological form can reproduce well the radialdependence of the microscopic optical potential. In thepresent study we have isolated the optical potential atlow energy, where there is a defined surface imaginarypeak, and fitted to the phenomenological form separately.At larger energies E > 50 MeV, we have also fitted tothe phenomenological form separately. To show that nocrucial features of the chiral potential are lost in thisparameterization, we also display in Figs. 3 and 4 theaccompanying coefficient of determination defined by

R2 = 1−∑i(yi − f(ri))

2∑i(yi − y)2

, (23)

7

TABLE II. Volume integrals for the real central VC , imagi-nary central WC , and real spin-orbit VSO parts of the proton-40Ca optical potential. Results are shown for the microscopicchiral optical potential and for the phenomenological Koning-Delaroche (KD) optical potential.

E = 2.35 MeV Chiral (MeV fm3) KD (MeV fm3)

VC 524 480

WC 64 80

VSO 19 13

E = 35 MeV Chiral (MeV fm3) KD (MeV fm3)

VC 413 374

WC 120 113

VSO 17 11

E = 100 MeV Chiral (MeV fm3) KD (MeV fm3)

VC 236 220

WC 163 109

VSO 13 9

where yi represents the value of the potential from chiralEFT at location ri, f(ri) is the value of the fitted func-tion, and y is the mean of the chiral optical potentialvalues. In the right panels of Figs. 3 and 4, the ILDAresults are obtained from the parametrized form of thecorresponding parts of the optical potentials.

For the global KD phenomenological optical potentialswe note that the real and imaginary volume terms haveidentical Woods-Saxon shape functions. From the mi-croscopic perspective there is little justification for thisassumption. In fact, we find that the shape parametersof the real and imaginary central optical potentials haveto be fitted separately in order to achieve a good fit tothe microscopic results. In Table I we show the values ofall shape parameters for the three energy windows overwhich we fit to the phenomenological form together withthose of the phenomenological Koning-Delaroche opti-cal potential, whose shape parameters are independentof energy. For these results we have chosen the valuestC = 1.15 fm and tSO = 1.0 fm in the ILDA. We see thatthere is not a very large energy dependence in the shapeparameters of the real volume part of the microscopic op-tical potential, but there is a significant difference amongthe real volume, imaginary volume, and real spin-orbitparts.

Finally, we note that the microscopic real spin-orbitoptical potential calculated from the density matrix ex-pansion has no energy dependence. The phenomenolog-ical energy dependence used in TALYS (vSO2 = 0.004)is constant across all nuclei, and in fact since vSO2 issmall, the real spin-orbit term does not strongly dependon the energy. We have therefore incorporated this phe-nomenological energy dependence into our parametriza-tion of the spin-orbit optical potential. As mentionedabove, the imaginary spin-orbit part of the optical po-tential is neglected since its magnitude for the relevant

●

●

●

●

ChiralKD Global

● Experiment

80 100 120 140 160

150

25035045080 100 120 140 160

150

250350450

θ(deg)

dσ

dΩ(mb

sr)

40Ca E=2.35 MeV

FIG. 5. Differential elastic scattering cross sections forproton-40Ca at projectile energy E = 2.35 MeV. The mi-croscopic cross section is given by the blue band. The KDphenomenological cross section is given by the green dashedcurve, and experimental data are represented by red circles.

energy range is ∼ 0.1 MeV and has been shown to havea negligible effect on elastic scattering cross sections atrelatively low energies [64].

E. Volume integrals of the real and imaginaryparts of the optical potential

We end this section by comparing the volume inte-grals of the various components of the microscopic opti-cal potential to those from phenomenology. It has beendemonstrated [3] that physical scattering observables canremain unchanged even if the various parameters of anoptical potential are allowed to vary, provided that thevolume integrals, defined by

J

A=

1

A

∫U(r)d3r, (24)

remain roughly constant. In Table II we show the vol-ume integrals for each term of the microscopic and phe-nomenological optical potentials for the proton-40Ca sys-tem at the three energies E = 2.35, 35, 100 MeV. We seethat the microscopic real volume and spin-orbit terms areboth slightly larger than their phenomenological coun-terparts for all three energies considered. The centralimaginary term features a volume part that grows withenergy and a surface peak that diminishes with energy.The imaginary term of the chiral optical potential over-estimates the volume component and underestimates thesurface component. For E = 35 MeV these compet-ing effects nearly cancel out and the chiral volume in-tegral is close to the phenomenological volume integral.At E = 2.35 MeV, the volume integral for the chiralimaginary term is smaller than the KD model since itssurface peak is at a smaller r value. For higher energies,the microscopic imaginary term becomes larger than thephenomenological imaginary term.

8

III. RESULTS

As a first test of the microscopic optical poten-tials constructed in the present work, we consider pro-ton scattering on calcium isotopes. Both the differen-tial elastic scattering cross sections and total reactioncross sections are calculated for selected calcium iso-topes at energies for which there are available experi-mental data. In particular, we compute differential elas-tic scattering cross sections for 40,42,44,48Ca targets atE = 25, 35, 45 MeV projectile energies. For 40Ca, differ-ential elastic scattering cross sections are also calculatedat E = 2.35, 55, 65, 80, 135, 160 MeV. Experimental dataare taken from Refs. [65–69]. The total reaction cross sec-tions for proton scattering on 40,42,44,48Ca are calculatedand compared to experimental data [70–73]. Energiesexceeding 200 MeV are not considered, since the chiralexpansion is expected to breakdown around that energyscale.

The TALYS reaction code is used to calculate the crosssections in the different reaction channels. In all cases weemploy the microscopic optical potential parametrized tothe KD form implemented in the ILDA. In the presentwork, the only theoretical uncertainties considered arethose for the ILDA length scales tC and tSO. In thefuture, we will consider a wider class of chiral nuclearpotentials in order to more accurately assess the com-plete theoretical uncertainty. We also benchmark againstresults from the KD global phenomenological optical po-tential [2].

A. Microscopic optical potential at low energy

Low-energy nuclear reactions are important for a widerange of astrophysical applications. One of the primarymotivations for the construction of new global micro-scopic optical potentials is to reduce the uncertainty incalculated radiative neutron capture reaction rates on ex-otic, neutron-rich isotopes. These reactions play an im-portant role in r-process nucleosynthesis [74, 75], espe-cially in cold r-process environments such as neutron starmergers where freeze-out is achieved rapidly and neu-tron capture plays an enhanced role. Neutron-capturerates are included in most modern r-process reaction net-work codes, and the neutron-nucleus optical potential(together with level densities and γ transition strengthfunctions) is one of the key ingredients for the theoreti-cal calculations. Most relevant is the imaginary part ofthe optical potential at low energies [9].

In Fig. 5 we show the differential elastic scatteringcross sections for proton projectiles on a 40Ca targetat E = 2.35 MeV. The red circles indicate experimen-tal data [65–69], the green dashed curve is the result ofthe global phenomenological optical potential from Kon-ing and Delaroche, while the blue band is the prediction

from the microscopic optical potential constructed in thepresent work. Interestingly, there is very little differencebetween the phenomenological optical potential predic-tions and those from chiral effective field theory. Bothcalculations agree well with experimental data at scat-tering angles up to θ ' 120◦, but overpredict the crosssection at large angles.

B. Microscopic optical potential at medium energy

In Fig. 6 we plot the differential elastic scatteringcross sections for protons on 40,42,44,48Ca targets atE = 24, 35, 45 MeV. For scattering angles in the range0◦ < θ < 80◦, the microscopic optical potential yieldscross sections that are consistent with experiment andoften more accurate than predictions based on the phe-nomenological KD optical potential. However, at largerscattering angles the microscopic calculations of the crosssections exhibit a weaker interference pattern, which per-sists as the energy increases. Overall, the microscopicelastic scattering cross sections are larger than experi-ment at high scattering angles.

From Fig. 4, we suspect that the underlying cause ofthese discrepancies may be due to the imaginary part ofthe microscopic optical potential. At these intermediateprojectile energies, the imaginary volume integral is closeto phenomenology according to Table II. However, themicroscopic surface imaginary peak is too small, as canbe seen in Fig. 4, which leads to larger elastic scatteringcross sections. In contrast the imaginary volume part,probed at higher projectile energies, is much larger thanphenomenology.

In order to investigate this conjecture, we substitutethe phenomenological imaginary term into the micro-scopic optical potential. This replacement is meant tobe a simple way of showing the possible improvements inthe chiral optical potential and should not be interpretedas a substitute for proper microscopic modeling. In Fig.7 we show the differential elastic scattering cross sectionsfor protons on 40,42,44,48Ca targets at E = 24, 35, 45 MeVwith this phenomenological replacement. Indeed we findthat the calculated cross sections are much improved atlarge angles across all isotopes. The enhanced surfaceimaginary part leads to stronger interferences and anoverall decrease in the elastic scattering cross section.Hence, there is a strong motivation for future work aimedat improving the microscopic description of the imagi-nary part of the optical potential.

C. Microscopic optical potential at high energy

To test the chiral optical potential at higher energies,we calculate proton-40Ca differential elastic scatteringcross sections at E = 55, 65, 80, 135, 160 MeV. In Fig. 8we plot the results from the chiral optical potential andthe KD phenomenological optical potential together with

9

●●●●●

●

●

●●●●●●●●

●●●●●●●●●●●●

●●

● ● ● ●●

●●

● ● ● ● ●● ● ● ● ●

●

ChiralKD Global

● Experiment50 100 150

0.010.101101001000

θ(deg)

40Ca E = 45 MeV

●

●●●●

●

●

●

●●●●●●●

●●●●●●●●●●●●●●

●

●

●● ● ● ●

●

●

●

● ●

●● ● ● ●

●

1

10

100

1000

dσ

dΩ(mb

sr)

40Ca E = 35 MeV

●

●

●

●

●

●

●

●

●●

●

●●●●●●●

●●●

●●●●●●● ● ●

●●

●●

●●

● ● ●●

● ● ● ● ●●

1

10

100

1000

50 100 150

40Ca E = 25 MeV

●●●●

●

●

●

●●●●●●●

●●●●●●●●●●●●

●●●

● ● ●●

●

●● ● ● ● ●

●● ● ● ● ●

●

ChiralKD Global

● Experiment50 100 150

θ(deg)

42Ca E = 45 MeV

●

●●●

●

●

●

●

●●

●●●●●●

●●●●

●●●●●●●●●

●

●

● ● ● ●●

●●

●● ●

●● ● ●

●●

42Ca E = 35 MeV

●

●

●

●

●

●

●

●

●●

●

●●●●●

●●●●●

●●●●●●● ● ●

●

●

●●

●●

● ● ● ● ● ● ● ● ● ●

50 100 150

42Ca E = 25 MeV

●●●●

●

●

●

●●●●●●●

●●●●●●●●●●●

●●●●

● ● ● ●●

●● ● ● ●

●●

●● ● ● ●

●

ChiralKD Global

● Experiment50 100 150

θ(deg)

44Ca E = 45 MeV

●

●●●

●

●

●

●

●●●●●●●

●●●

●

●●●●●●●●●

●

●

●● ● ● ●

●●

●●

● ●● ● ● ●

●●

44Ca E = 35 MeV

●

●

●

●

●

●

●

●

●●

●

●●●●●

●●●●

●

●

●●●●● ● ●

●

●

●● ● ● ● ● ● ● ● ● ● ● ● ●

50 100 150

44Ca E = 25 MeV

●●●●

●

●

●

●●●●●●

●●●●●●●●●●●

●●●●●●●●● ●

●

●● ● ● ● ●

●●

● ●●

● ●●

ChiralKD Global

● Experiment50 100 150

0.010.101101001000

θ(deg)

48Ca E = 45 MeV

●

●●

●

●

●

●

●●

●●●

●●●●●

●

●●●●●●●●●

●●●●●● ● ● ●

●●

●●

● ● ● ● ● ● ●●

●

1

10

100

100048Ca E = 35 MeV

●

●

●

●

●

●

●

●●

●

●●●●●

●●●●

●

●●●●

●●●●●●●●

●

●●

●● ●

●●

●● ●

●● ● ● ●

50 100 150

1

10

100

100048Ca E = 25 MeV

FIG. 6. Differential elastic scattering cross sections for proton projectiles on calcium targets at the energies E = 25, 35, 45 MeV.The microscopic cross sections are shown as the blue band. The KD phenomenological cross sections are given by the greendashed curves and experimental data are represented by red circles.

●●●●●

●

●

●●●●●●●●

●●●●●●●●●●●●

●●

● ● ● ●●

●●

● ● ● ● ●● ● ● ● ●

●

Chiral+KD Im.KD Global

● Experiment50 100 150

0.010.101101001000

θ(deg)

40Ca E = 45 MeV

●●●●●

●

●

●

●●●●●●●●●

●●●●●●●●●●●●

●

●●

● ● ● ●●

●

●

● ●

●● ● ● ●

●

0.1

1

10

100

1000

dσ

dΩ(mb

sr)

40Ca E = 35 MeV

●

●

●

●

●

●

●

●

●●

●

●●●●●●●

●●●

●●●●●●● ● ●

●●

●●

●●

● ● ●●

● ● ● ● ●●

1

10

100

1000

50 100 150

40Ca E = 25 MeV

●●●●

●

●

●

●●●●●●●

●●●●●●●●●●●●

●●●

● ● ●●

●

●● ● ● ● ●

●● ● ● ● ●

●

Chiral+KD Im.KD Global

● Experiment50 100 150

θ(deg)

42Ca E = 45 MeV

●

●●●●

●

●

●

●●●●●●●●

●●●●●●●●●●●●●

●

●

● ● ● ●●

●●

●● ●

●● ● ●

●●

42Ca E = 35 MeV

●

●

●

●

●

●

●

●

●●

●

●●●●●

●●●●●

●●●●●●● ● ●

●

●

●●

●●

● ● ● ● ● ● ● ● ● ●

50 100 150

42Ca E = 25 MeV

●●●●

●

●

●

●●●●●●●

●●●●●●●●●●●

●●●●

● ● ● ●●

●● ● ● ●

●●

●● ● ● ●

●

Chiral+KD Im.KD Global

● Experiment50 100 150

θ(deg)

44Ca E = 45 MeV

●●●●

●

●

●

●

●●●●●●●

●●●●●●●●●●●●●

●

●

●● ● ● ●

●●

●● ● ● ● ● ● ●

●●

44Ca E = 35 MeV

●

●

●

●

●

●

●

●

●●

●

●●●●●

●●●●

●

●

●●●●● ● ●

●

●

●● ● ● ● ● ● ● ● ● ● ● ● ●

50 100 150

44Ca E = 25 MeV

●●●●

●

●

●

●●●●●●

●●●●●●●●●●●

●●●●●●●●● ●

●

●● ● ● ● ●

●●

● ●●

● ●●

Chiral+KD Im.KD Global

● Experiment50 100 150

0.010.101101001000

θ(deg)

48Ca E = 45 MeV

●

●●●

●

●

●

●●

●●●

●●●●●●●●●●●●●●●

●●●●●● ● ● ●

●●

●● ● ● ● ● ● ● ● ●

●

0.1

1

10

100

100048Ca E = 35 MeV

●

●

●

●

●

●

●

●●

●

●●●●●

●●●●

●

●●●●

●●●●●●●●

●

●●

●● ●

●●

●● ●

●● ● ● ●

50 100 150

1

10

100

100048Ca E = 25 MeV

FIG. 7. Differential elastic scattering cross sections for proton projectiles on calcium targets at the energies E = 25, 35, 45 MeV.In comparison to Fig. 6, the black bands show the cross sections that result from replacing the microscopic imaginary part inthe chiral optical potential by the Koning-Delaroche phenomenological imaginary part.

experimental data from Refs. [65–69]. The cross sectionsfrom the chiral optical potential stay close to phenomeno-logical and experimental results for E = 55, 65 MeV butbegin to deviate strongly for E > 80 MeV. The micro-scopic imaginary term becomes much more absorptivefor E > 80 MeV, as the large volume contribution fromthe chiral optical potential becomes more relevant. Theeffect of this can be seen especially in the lower threeplots of Fig. 8, where the cross sections exhibit large in-

terference oscillations. Since there are more open inelas-tic channels at higher energy, a stronger imaginary partin general corresponds to a lower elastic scattering crosssection.

In order to assess the quality of the microscopic imag-inary part of the optical potential, we again substitutethe KD phenomenological imaginary part into the chi-ral optical potential. The results for proton-40Ca elasticscattering cross sections at E = 55, 65, 80, 135, 160 MeV

10

● ●●

●●

●

●● ● ● ● ●

●●

●●

● ●●

●● ● ● ●

●●

●●

● ● ● ● ●●

●

●

●● ● ● ●

ChiralChiral+KD Im.KD Global

● Experiment20 40 60 80

0.001

0.100

10

1000

0.001

0.100

10

1000

θ(deg)

40Ca E =160 MeV

●●

●

●

●

●●

● ● ●●

●●

●● ● ●

●●

● ● ● ● ●●

●●

●● ● ● ● ●

● ●●

●

●0.001

0.100

10

1000

0.001

0.100

10

100040Ca E =135 MeV

●

● ● ●●

●

●

●

●

●● ●

● ● ●●

●●

●●

● ● ● ● ●●

●●

●● ● ● ● ● ● ●

●●

●●0.1

1101001000

0.11101001000

dσ

dΩ(mb

sr)

40Ca E = 80 MeV

●●●●●●●●●

●●●●●

●● ● ●

●

●

●

● ● ●● ●

●

● ●1

10

100

1000

1

10

100

100040Ca E = 65 MeV

●

●●●

●

●

●

●

●

●

●●

●

●

●

●

●1

10

100

1000

20 40 60 80

1

10

100

100040Ca E = 55 MeV

FIG. 8. Differential elastic scattering cross sections forproton projectiles on a 40Ca target at the energies E =55, 65, 80, 135, 160 MeV. Full microscopic cross sections areshown as the blue bands, microscopic real optical potentialplus phenomenological imaginary optical potential are shownby the black bands, the KD phenomenological cross sectionsare given by the green dashed curves, and experimental dataare represented by red circles.

are shown in Fig. 8. Again, we find that the replacementof the large microscopic imaginary optical potential bythe KD phenomenological imaginary part leads to signif-icant improvements in the elastic scattering cross sectionsacross all energies. For E = 135 and 160 MeV, the purelyphenomenological cross sections are still more accurate,but the microscopic optical potential with phenomeno-logical imaginary part gives a quality description of thedata.

●●●

●●●

●

●●●

●●

●●

●

●

●●

●●●●●

●

●●

● ● ●ChiralChiral+KD Im.KD Global

● Experiment50 100 150

200

400

600

800

100050 100 150

200

400

600

800

1000

E (MeV)

σR(mb)

p+40Ca

FIG. 9. The proton-40Ca total reaction cross section as a func-tion of energy calculated from the microscopic optical poten-tial (blue band), the phenomenological KD optical potential(dashed green curve), and the microscopic optical potentialwith the phenomenological KD imaginary part (black band).Experimental data are shown as red circles.

● ●

●

●●

●●

ChiralChiral+KD Im.KD Global

● Experiment

25 30 35 40 45

800

900

1000

1100

800

900

1000

1100

E (MeV)

p+48Ca

● ●

●

●

●

● ●900

950

1000

1050

1100

900

950

1000

1050

1100

σR(mb)

p+44Ca

● ●

●

●

●

●●

85090095010001050

25 30 35 40 45

85090095010001050p+42Ca

FIG. 10. The proton-42,44,48Ca total reaction cross sectionsfrom 20 MeV < E < 50 MeV calculated from the microscopicoptical potential (blue bands), the phenomenological KD op-tical potential (dashed green curves), and the microscopic op-tical potential with the phenomenological KD imaginary part(black bands). Experimental data are shown as red circles.

11

D. Total reaction cross section

In Figs. 9 and 10 we plot the total reaction cross sec-tions for proton scattering on 40,42,44,48Ca from our mi-croscopic optical potential and the KD phenomenologicaloptical potential. The chiral EFT results are shown asthe blue band, while the KD predictions are shown asdashed green lines. Experimental data [70–73] are shownwith red circles. We see that for all energies the purelymicroscopic optical potential predicts a total reactioncross section that is too large, while the phenomenolog-ical potential gives an overall good description for mostisotopes and energies. However, for proton-48Ca, the KDphenomenological potential also gives a larger total reac-tion cross section compared to experiment. This suggeststhat there might be room for improvement in the phe-nomenological description of the global isovector opticalpotential probed in this neutron-rich nucleus.

We also show in Figs. 9 and 10 the results for the to-tal reaction cross sections (black solid bands) when themicroscopic imaginary part is replaced by the KD phe-nomenological imaginary potential. For intermediate en-ergies (20 MeV < E < 50 MeV), there is not a substan-tial improvement in the comparison to experimental data.However, beyond energies of E = 50 MeV that are shownin Fig. 9, the replacement of the phenomenological imag-inary part again leads to a significant improvement inthe description of the total reaction cross section. Nev-ertheless, the modified microscopic optical potential stilloverestimates the total reaction cross section for all en-ergies due to the real volume and real spin-orbit termshaving slightly larger depths than their phenomenologi-cal counterparts.

IV. CONCLUSIONS

We have calculated a microscopic optical potentialfrom chiral two- and three-body forces for proton scatter-ing on calcium isotopes. We started from a self-consistentsecond-order calculation of the proton and neutron selfenergies in isospin-asymmetric nuclear matter from whichwe derived the central real and imaginary parts of theoptical potential in finite nuclei within the framework

of the improved local density approximation. The realspin-orbit potential was constructed from the improveddensity matrix expansion using the same chiral two- andthree-body forces.

We found that chiral nucleon-nucleus optical poten-tials describe low-energy (E . 5 MeV) scattering pro-cesses rather well, due in part to a well-defined sur-face peak in the imaginary part of the optical potential.At all energies, the real central term is consistent withphenomenological modeling, while the microscopic spin-orbit strength is larger by ∼ 20%. At moderate energies(E & 35 MeV), the imaginary part of the chiral opticalpotential develops a large volume term without a signif-icant surface peak. This leads to discrepancies betweenour theoretical calculations and experimental data, espe-cially at large scattering angles. At the highest energies(E ' 100−160 MeV) considered in the present work, thelarge imaginary term leads to over suppression of the elas-tic scattering cross section. We have shown that substi-tuting the microscopic imaginary part with the KD phe-nomenological optical potential leads to excellent agree-ment with elastic scattering and total reaction cross sec-tions for nearly all isotopes and projectile energies inves-tigated. In the future we plan to compute higher-orderperturbative contributions that may improve the descrip-tion of the imaginary part of the optical potential and theoverall spin-orbit strength. We will also explore a widerrange [76] of chiral nuclear potentials in order to providea more comprehensive estimate of the theoretical uncer-tainties.

ACKNOWLEDGMENTS

We thank Carlos Bertulani for useful discussions andL. Huff for comments on the manuscript. Work sup-ported by the National Science Foundation under GrantNo. PHY1652199 and by the U.S. Department of EnergyNational Nuclear Security Administration under GrantNo. DE-NA0003841. Portions of this research were con-ducted with the advanced computing resources providedby Texas A&M High Performance Research Computing.

[1] R. L. Varner, W. J. Thompson, T. L. McAbee, E. J.Ludwig, and T. B. Clegg, Phys. Rep. 201, 57 (1991).

[2] A. J. Koning and J. P. Delaroche, Nucl. Phys. A713, 231(2003).

[3] J. P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rep.25, 83 (1976).

[4] A. Lejeune and P. E. Hodgson, Nucl. Phys. A295, 301(1978).

[5] L. F. Hansen, F. S. Dietrich, B. A. Pohl, C. H. Poppe,and C. Wong, Phys. Rev. C 31, 111 (1985).

[6] J. S. Petler, M. S. Islam, R. W. Finlay, and F. S. Dietrich,Phys. Rev. C 32, 673 (1985).

[7] E. Bauge, J. P. Delaroche, and M. Girod, Phys. Rev. C58, 1118 (1998).

[8] E. Bauge, J. P. Delaroche, and M. Girod, Phys. Rev. C63, 024607 (2001).

[9] S. Goriely and J.-P. Delaroche, Phys. Lett. B653, 178(2007).

[10] J. W. Holt, N. Kaiser, G. A. Miller, and W. Weise, Phys.Rev. C 88, 024614 (2013).

12

[11] M. Toyokawa, K. Minomo, M. Kohno, and M. Yahiro,J. Phys. G 42, 025104 (2015).

[12] J. W. Holt, N. Kaiser, and G. A. Miller, Phys. Rev. C93, 064603 (2016).

[13] H. S. Camarda, F. S. Dietrich, and T. W. Phillips, Phys.Rev. C 39, 1725 (1989).

[14] A. K. Kerman, H. McManus, and R. Thaler, Ann. Phys.(N.Y.) 8, 551 (1959).

[15] F. A. Brieva and J. R. Rook, Nucl. Phys. A291, 299(1977).

[16] L. Ray and G. W. Hoffmann, Phys. Rev. C 31, 538(1985).

[17] C. Elster, T. Cheon, E. F. Redish, and P. C. Tandy,Phys. Rev. C 41, 814 (1990).

[18] H. F. Arellano, F. A. Brieva, and W. G. Love, Phys.Rev. C 41, 2188 (1990).

[19] H. F. Arellano, F. A. Brieva, and W. G. Love, Phys.Rev. C 52, 301 (1995).

[20] M. Vorabbi, P. Finelli, and C. Giusti, Phys. Rev. C 93,034619 (2016).

[21] M. Vorabbi, P. Finelli, and C. Giusti, Phys. Rev. C 98,064602 (2018).

[22] C. R. Chinn, C. Elster, and R. M. Thaler, Phys. Rev. C48, 2956 (1993).

[23] R. Crespo, R. C. Johnson, and J. A. Tostevin, Phys.Rev. C 48, 351 (1993).

[24] S. J. Waldecker, C. Barbieri, and W. H. Dickhoff, Phys.Rev. C 84, 034616 (2011).

[25] R. J. Charity, J. M. Mueller, L. G. Sobotka, and W. H.Dickhoff, Phys. Rev. C 76, 044314 (2007).

[26] K. Egashira, K. Minomo, M. Toyokawa, T. Matsumoto,and M. Yahiro, Phys. Rev. C 89, 064611 (2014).

[27] J. Rotureau, P. Danielewicz, G. Hagen, G. R. Jansen,and F. M. Nunes, (2018), arXiv:1808.04535 [nucl-th].

[28] J. Rotureau, P. Danielewicz, G. Hagen, F. M. Nunes,and T. Papenbrock, Phys. Rev. C 95, 024315 (2017).

[29] M. Toyokawa, M. Yahiro, T. Matsumoto, K. Minomo,K. Ogata, and M. Kohno, Phys. Rev. C 92, 024618(2015).

[30] V. Durant, P. Capel, L. Huth, A. Balantekin, andA. Schwenk, Phys. Lett. B782, 668 (2018).

[31] S. Weinberg, Physica A 96, 327 (1979).[32] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev.

Mod. Phys. 81, 1773 (2009).[33] R. Machleidt and D. R. Entem, Phys. Rep. 503, 1 (2011).[34] A. J. Koning, S. Hilaire, and M. C. Duijvestijn, Proc. of

the Int. Conf. on Nucl. Data for Science and Technology,EDP Sciences (2008).

[35] I. Tews, T. Kruger, K. Hebeler, and A. Schwenk, Phys.Rev. Lett. 110, 032504 (2013).

[36] C. Drischler, A. Carbone, K. Hebeler, and A. Schwenk,Phys. Rev. C 94, 054307 (2016).

[37] N. Kaiser and V. Niessner, Phys. Rev. C 98, 054002(2018).

[38] L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt, L. E.Marcucci, and F. Sammarruca, Phys. Rev. C 89, 044321(2014).

[39] C. Wellenhofer, J. W. Holt, N. Kaiser, and W. Weise,Phys. Rev. C 89, 064009 (2014).

[40] C. Wellenhofer, J. W. Holt, and N. Kaiser, Phys. Rev.C 92, 015801 (2015).

[41] J. W. Holt, N. Kaiser, and T. R. Whitehead, Phys. Rev.C 97, 054325 (2018).

[42] J. S. Bell and E. J. Squires, Phys. Rev. Lett. 3, 96 (1959).

[43] Y. Lim and J. W. Holt, Phys. Rev. C 95, 065805 (2017).[44] J. P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rev.

C 16, 80 (1977).[45] S. Kosugi and H. Yoshida, Nucl. Phys. A373, 349 (1982).[46] S. K. Bogner, R. J. Furnstahl, and L. Platter, Eur. Phys.

J. A 39, 219 (2009).[47] B. Gebremariam, T. Duguet, and S. K. Bogner, Phys.

Rev. C 82, 014305 (2010).[48] J. W. Holt, N. Kaiser, and W. Weise, Eur. Phys. J. A

47 (2011).[49] J. W. Negele and D. Vautherin, Phys. Rev. C 5, 1472

(1972).[50] S. K. Bogner, A. Schwenk, R. J. Furnstahl, and

A. Nogga, Nucl. Phys. A763, 59 (2005).[51] J. W. Holt, N. Kaiser, and W. Weise, Phys. Rev. C 79,

054331 (2009).[52] K. Hebeler and A. Schwenk, Phys. Rev. C 82, 014314

(2010).[53] J. W. Negele and K. Yazaki, Phys. Rev. Lett. 47, 71

(1981).[54] S. Fantoni, B. L. Friman, and V. R. Pandharipande,

Phys. Lett. B104, 89 (1981).[55] B. Gebremariam, S. K. Bogner, and T. Duguet, Nucl.

Phys. A851, 17 (2011).[56] Y. N. Zhang, S. K. Bogner, and R. J. Furnstahl, Phys.

Rev. C 98, 064306 (2018).[57] N. Kaiser, Eur. Phys. J. A 48, 36 (2012).[58] F. A. Brieva and J. R. Rook, Nucl. Phys. A291, 317

(1977).[59] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys.

Rev. C 51, 38 (1995).[60] C. Lagrange and A. Lejeune, Phys. Rev. C 25, 2278

(1982).[61] M. Kohno, D. W. L. Sprung, S. Nagata, and N. Yam-

aguchi, Phys. Lett. B137, 10 (1984).[62] J. W. Holt, M. Rho, and W. Weise, Phys. Rept. 621, 2

(2016).[63] N. Kaiser and W. Weise, Nucl. Phys. A836, 256 (2010).[64] F. A. Brieva and J. R. Rook, Nucl. Phys. A297, 206

(1978).[65] R. H. McCamis, T. N. Nasr, J. Birchall, N. E. Davison,

W. T. H. van Oers, P. J. T. Verheijen, R. F. Carlson,A. J. Cox, B. C. Clark, E. D. Cooper, S. Hama, andR. L. Mercer, Phys. Rev. C 33, 1624 (1986).

[66] K. Yagi, H. Ejiri, M. Furukawa, Y. Ishizaki, M. Koike,K. Matsuda, Y. Nakajima, I. Nonaka, Y. Saji, E. Tanaka,and G. Satchler, Phys. Lett. 10, 186 (1964).

[67] H. Sakaguchi, M. Nakamura, K. Hatanaka, A. Goto,T. Noro, F. Ohtani, H. Sakamoto, and S. Kobayashi,Phys. Lett. B89, 40 (1979).

[68] A. Nadasen, P. Schwandt, P. P. Singh, W. W. Jacobs,A. D. Bacher, P. T. Debevec, M. D. Kaitchuck, andJ. T. Meek, Phys. Rev. C 23, 1023 (1981).

[69] E. Koltay, L. Mesko, and L. Vegh, Nucl. Phys. A249,173 (1975).

[70] A. Auce, A. Ingemarsson, R. Johansson, M. Lantz,G. Tibell, R. F. Carlson, M. J. Shachno, A. A. Cow-ley, G. C. Hillhouse, N. M. Jacobs, J. A. Stander, J. J.vanZyl, S. V. Fortsch, J. J. Lawrie, F. D. Smit, andG. F. Steyn, Phys. Rev. C 71, 064606 (2005).

[71] J. F. Dicello and G. Igo, Phys. Rev. C 2, 488 (1970).[72] R. F. Carlson, A. J. Cox, J. R. Nimmo, N. E. Davison,

S. A. Elbakr, J. L. Horton, A. Houdayer, A. M. Sourkes,W. T. H. van Oers, and D. J. Margaziotis, Phys. Rev.

13

C 12, 1167 (1975).[73] R. F. Carlson, A. J. Cox, N. E. Davison, T. Eliyakut-

Roshko, R. H. McCamis, and W. T. H. van Oers, Phys.Rev. C 49, 3090 (1994).

[74] M. R. Mumpower, R. Surman, G. C. McLaughlin, andA. Aprahamian, Prog. Part. Nucl. Phys. 86, 86 (2016).

[75] C. J. Horowitz et al., arXiv:1805.04637 (2018).[76] F. Sammarruca, L. E. Marcucci, L. Coraggio, J. W. Holt,

N. Itaco, and R. Machleidt, arXiv:1807.06640 (2018).