# Double-folding potentials from chiral effective field theory

V. Durant<EMAIL_ADDRESS>Institut für Kernphysik,
Technische Universität Darmstadt, 64289 Darmstadt, Germany  ExtreMe Matter
Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291
Darmstadt, Germany  P. Capel<EMAIL_ADDRESS>Institut für Kernphysik,
Technische Universität Darmstadt, 64289 Darmstadt, Germany  ExtreMe Matter
Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291
Darmstadt, Germany  Physique Nucléaire et Physique Quantique (CP
229),Université libre de Bruxelles (ULB), B-1050 Brussels, Belgium  L. Huth
<EMAIL_ADDRESS>Institut für Kernphysik,
Technische Universität Darmstadt, 64289 Darmstadt, Germany  ExtreMe Matter
Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291
Darmstadt, Germany  A. B. Balantekin<EMAIL_ADDRESS>Department of
Physics, University of Wisconsin, Madison, WI 53706, USA  A. Schwenk
<EMAIL_ADDRESS>Institut für Kernphysik, Technische Universität
Darmstadt, 64289 Darmstadt, Germany  ExtreMe Matter Institute EMMI, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany  Max-
Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

###### Abstract

The determination of nucleus-nucleus potentials is important not only to
describe the properties of the colliding system, but also to extract nuclear-
structure information and for modelling nuclear reactions for astrophysics. We
present the first determination of double-folding potentials based on chiral
effective field theory at leading, next-to-leading, and next-to-next-to-
leading order. To this end, we construct new soft local chiral effective field
theory interactions. We benchmark this approach in the 16O–16O system, and
present results for cross sections computed for elastic scattering up to 700
MeV in energy, as well as for the astrophysical $S$-factor of the fusion
reaction.

##  I Introduction

Determining the interaction between two nuclei is a long-standing and
challenging problem Brandan and Satchler (1997). It constitutes an important
input in the modelling of nuclear reactions, which provide key information
about the structure of nuclei and are relevant for processes that take place
in stars. The interaction between two nuclei has been modelled by
phenomenological potentials, e.g., of Woods-Saxon form, whose parameters are
adjusted to reproduce elastic-scattering data. Numerical potentials have also
been obtained from inversion of scattering data Leeb  _et al._ (1985). Albeit
precise when experimental data exist, these potentials lack predictive
behavior and do not have controlled uncertainties. Alternatively, it has been
suggested to construct nucleus-nucleus potentials from the densities of the
colliding nuclei and a given nucleon-nucleon ($NN$) interaction using a
double-folding procedure Satchler and Love (1979). It is known that this
framework provides more realistic potentials for the nucleon-nucleus
interactions than for the nucleus-nucleus case Mahaux and Sartor (1991).
Nevertheless, it constitutes a first-order approximation to optical potentials
derived from Feshbach’s reaction theory Brandan and Satchler (1997).
Interesting results have been obtained in such a way, e.g., by considering
zero-range contact $NN$ interactions Chamon  _et al._ (2002); Pereira  _et
al._ (2009) or using a $G$-matrix approach, see, e.g., Refs. Furumoto  _et
al._ (2012); Minomo  _et al._ (2016) for recent work.

During the last decade, there have been great advances in nuclear structure
and nuclear reactions based on effective field theories (EFT). For example,
chiral EFT has become the standard method for developing systematic nuclear
forces rooted in the symmetries of quantum chromodynamics (see, e.g., Refs.
Epelbaum  _et al._ (2009); Machleidt and Entem (2011); Hammer  _et al._ (2013)
for recent reviews). Efforts have been made to derive nucleon-nucleus optical
potentials using chiral EFT interactions from many-body perturbation theory
Holt  _et al._ (2013); Vorabbi  _et al._ (2016) and self-consistent Green’s
function calculations Rotureau  _et al._ (2017); Idini  _et al._ (2017). In
this work, we focus on the derivation of nucleus-nucleus potentials from
chiral $NN$ interactions and nucleonic densities. In particular, we use local
chiral EFT interactions Gezerlis  _et al._ (2013, 2014); Lynn  _et al._
(2014); Tews  _et al._ (2016); Lynn  _et al._ (2016, 2017); Huth  _et al._
(2017), because this simplifies the double-folding calculation.

In this first study, we explore and test this idea for 16O–16O reactions,
comparing our calculations to elastic-scattering Stiliaris  _et al._ (1989);
Bohlen  _et al._ (1993); Sugiyama  _et al._ (1993); Kondō  _et al._ (1996);
Bartnitzky  _et al._ (1996); Nuoffer  _et al._ (1998); Nicoli  _et al._
(1999); Khoa  _et al._ (2000) and fusion data Tserruya  _et al._ (1978); Hulke
_et al._ (1980); Wu and Barnes (1984); Kuronen  _et al._ (1987); Duarte  _et
al._ (2015). For this system, phenomenological Woods-Saxon potentials Khoa
_et al._ (2000) and potentials obtained through inversion techniques Allen
_et al._ (1993); Bennett  _et al._ (1996) also exist. We show that the double-
folding potential and the reaction observables exhibit an order-by-order
behavior expected in EFT and observe that, for soft potentials, our
calculations have only a weak dependence on the regularization scale. The
comparison of our results with experiment leads us to suggest various
directions for improvements for constructing nucleus-nucleus potentials from
chiral EFT interactions with the double-folding method.

This paper is organized as follows. We start with a brief review of the
formalism for the double-folding potential in the following section. In Sec.
III, we discuss local chiral EFT interactions and the construction of new soft
local chiral $NN$ potentials. We then determine the double-folding potentials
at different chiral orders and apply these to 16O–16O elastic scattering in
Sec. IV and to the $S$-factor for 16O+16O fusion in Sec. V. Finally, we
summarize and give an outlook in Sec. VI.

##  II Double-folding potential: Formalism

We consider the potential between nucleus 1 (with atomic and mass numbers
$Z_{1}$ and $A_{1}$) and nucleus 2 (with $Z_{2}$ and $A_{2}$). In the double-
folding formalism, the nuclear part of the nucleus-nucleus potential
$V_{\text{F}}=V_{\text{D}}+V_{\text{Ex}}$ can be constructed from a given $NN$
interaction $v$ by double folding over the densities in the direct (D) channel
and the density matrices in the exchange (Ex) channel. The review of the
formalism for the double-folding potential in this section follows Ref.
Furumoto  _et al._ (2012). We will include only $NN$ interactions here and
leave the investigation of many-body contributions to future work.

In the direct channel, the double-folding potential is calculated by
integrating the $NN$ interaction over the neutron ($n$) and proton ($p$)
density distributions $\rho_{1}^{n,p}$ and $\rho_{2}^{n,p}$ of the colliding
nuclei,

$V_{\text{D}}({\bf r})=\sum_{i,j=n,p}\int\int\rho^{i}_{1}({\bf
r}_{1})\,v^{ij}_{\text{D}}({\bf s})\,\rho^{j}_{2}({\bf r}_{2})\,d^{3}{\bf
r}_{1}d^{3}{\bf r}_{2}\,,$ (1)

where ${\bf r}$ is the relative coordinate between the center of mass of the
nuclei, ${\bf r}_{1}$ and ${\bf r}_{2}$ are the coordinates from the center of
mass of each nucleus, ${\bf s}={\bf r}-{\bf r}_{1}+{\bf r}_{2}$ (the geometry
is shown in Fig. 1), and the sum $i,j$ is over neutrons and protons with their
respective densities.

Figure 1:  Coordinates of the nuclei involved in the double-folding
calculation.

To account for the antisymmetrization between nucleons, the double-folding
potential receives contributions also from the exchange channel,

$V_{\text{Ex}}({\bf r},E_{\text{cm}})=\sum_{i,j=n,p}\int\int\rho^{i}_{1}({\bf
r}_{1},{\bf r}_{1}+{\bf s})\,v^{ij}_{\text{Ex}}({\bf s})\\\
\times\rho^{j}_{2}({\bf r}_{2},{\bf r}_{2}-{\bf s})\exp\left[\frac{i{\bf
k}({\bf r})\cdot{\bf s}}{\mu/m_{N}}\right]\,d^{3}{\bf r}_{1}d^{3}{\bf
r}_{2}\,,$ (2)

where $\mu=m_{N}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass of the colliding
nuclei (with $m_{N}$ the nucleon mass) and the integral is over the density
matrices $\rho^{i}({\bf r},{\bf r}\pm{\bf s})$ of the nuclei. In the exchange
channel, there is an additional phase that renders the double-folding
potential dependent on the energy $E_{\text{cm}}$ in the center-of-mass
system. The momentum for the nucleus-nucleus relative motion ${\bf k}$ is
related to $E_{\text{cm}}$, the nuclear part of the double-folding potential,
and the double-folding Coulomb potential $V_{\text{Coul}}$ through

$k^{2}({\bf r})=2\mu\,\Bigl{[}E_{\text{cm}}-V_{\text{F}}({\bf
r},E_{\text{cm}})-V_{\text{Coul}}({\bf r})\Bigr{]}\,.$ (3)

As a result, $V_{\text{Ex}}$ has to be determined self-consistently. Note that
at our level of calculation the double-folding potential,
$V_{\text{F}}=V_{\text{D}}+V_{\text{Ex}}$, is real. The density matrices
entering in Eq. (2) are approximated using the density matrix expansion Negele
and Vautherin (1972) restricted to its leading term,

$\rho^{i}({\bf r},{\bf r}\pm{\bf s})=\frac{3}{s\,k^{i}_{\text{F}}({\bf
R})}\,j_{1}\bigr{(}s\,k^{i}_{\text{F}}({\bf R})\bigl{)}\,\rho^{i}({\bf R})\,,$
(4)

where ${\bf R}={\bf r}\pm{\bf s}/2$, $j_{1}$ is a spherical Bessel function of
the first kind, and we take the effective local Fermi momentum, which is an
arbitrary scale in the density-matrix expansion, as in Ref. Furumoto  _et al._
(2012):

$k^{i}_{\text{F}}=\biggl{[}(3\pi^{2}\rho^{i})^{2/3}+\frac{5(\nabla\rho^{i})^{2}}{12(\rho^{i})^{2}}+\frac{5\nabla^{2}\rho^{i}}{36\rho^{i}}\biggr{]}^{1/2}\,.$
(5)

In the case of spherical nuclei, the densities and the effective local Fermi
momenta depend only on the distance from the center of mass of the nucleus
($r_{i}$ or $R$).

For doubly closed-shell nuclei, the $NN$ interaction entering the double-
folding potential in the direct and exchange channels, $v_{\text{D}}$ and
$v_{\text{Ex}}$, respectively, at this level receive contributions only from
the central parts of nuclear forces. Then also the $NN$ interaction and the
double-folding potentials depend only on the relative distance ($s$ or $r$).
Writing the $NN$ interaction in terms of their two-body spin-isospin
components, $v^{ST}$, and distinguishing between proton-proton ($pp$), proton-
neutron ($pn,np$), and neutron-neutron ($nn$) interactions, $v_{\text{D}}$ and
$v_{\text{Ex}}$ read

$\displaystyle v^{pp,nn}_{\text{D,Ex}}(s)$
$\displaystyle=\frac{1}{4}\Bigl{[}v^{01}(s)\pm 3v^{11}(s)\Bigr{]}\,,$ (6)
$\displaystyle v^{pn,np}_{\rm D,Ex}(s)$ $\displaystyle=\frac{1}{8}\Bigl{[}\pm
v^{00}(s)+v^{01}(s)+3v^{10}(s)\pm 3v^{11}(s)\Bigr{]}\,,$ (7)

where the upper (lower) signs refer to the direct (exchange) term and we have
neglected the small isospin-symmetry-breaking corrections to $v$.

The densities of the colliding nuclei are an important input for the
calculation of the double-folding potential. In this first study based on
chiral EFT interactions, we adopt the two-parameter Fermi distributions
provided by the São Paulo group Chamon  _et al._ (2002) for the proton and
neutron densities, whose parameters were fitted to Dirac-Hartree-Bogoliubov
calculations

$\rho^{p,n}(r)=\frac{\rho_{0}}{1+\exp\left(\frac{r-R_{p,n}}{a_{p,n}}\right)}\,,$
(8)

where $\rho_{0}=0.091$ fm-3 and the radii $R_{p,n}$ and diffusenesses
$a_{p,n}$ depend on the proton and neutron numbers of the nucleus. Expressed
in fm, they are given by

$\displaystyle R_{p}$ $\displaystyle=1.81\,Z^{1/3}-1.12\,,\quad
a_{p}=0.47-0.00083\,Z\,,$ (9) $\displaystyle R_{n}$
$\displaystyle=1.49\,N^{1/3}-0.79\,,\quad a_{n}=0.47+0.00046\,N\,.$ (10)

##  III Local chiral EFT interactions

###  III.1 Nucleon-nucleon potentials

Chiral EFT provides a systematic expansion for nuclear forces using nucleons
and pions as degrees of freedom, which is connected to the underlying theory
of quantum chromodynamics Epelbaum  _et al._ (2009); Machleidt and Entem
(2011). The different contributions to $NN$ and many-nucleon interactions are
ordered according to a power counting scheme in powers of
$(Q/\Lambda_{b})^{\nu}$, where $Q$ is a typical momentum or the pion mass and
$\Lambda_{b}$ the breakdown scale of the theory of the order of 500 MeV. This
leads to a hierarchy of two- and many-nucleon interactions, with $NN$
interactions starting at leading order (LO, $\nu=0$) followed by a
contribution at next-to-leading order (NLO, $\nu=2$), whereas three-nucleon
interactions enter at next-to-next-to leading order (N2LO, $\nu=3$).

Because they facilitate the calculation of double-folding potentials, we use
local chiral $NN$ interactions, developed initially in Refs. Gezerlis  _et
al._ (2013, 2014), but construct new soft $NN$ interactions up to N2LO. As in
these references, the long- and short-range parts of the interaction are
regularized by

$f_{\text{long}}(r)=1-e^{-(r/R_{0})^{4}}\hskip 4.2679pt\text{and}\hskip
5.69054ptf_{\text{short}}(r)=\frac{e^{-(r/R_{0})^{4}}}{\pi\Gamma(3/4)R_{0}^{3}}\,,$
(11)

where $R_{0}$ is the coordinate-space cutoff in the $NN$ potentials used. The
long-range regulator is designed to remove the singularity at $r=0$ in the
pion exchanges, while it preserves its properties at large distances. The
short-range regulator smears out the $NN$ contact interactions. A second
cutoff $\tilde{\Lambda}$ is used in the spectral-function regularization of
the two-pion exchange, which enters first at NLO. In Refs. Gezerlis  _et al._
(2013, 2014) it was shown that the calculations are practically insensitive to
$\tilde{\Lambda}$ for local interactions; in the present work, we consider
$\tilde{\Lambda}=1000$ MeV.

Table 1: Low-energy constants (LECs) for the soft cutoffs $R_{0}=1.4$ fm and 1.6 fm at LO, NLO and N2LO. In the last row, the deuteron binding energy $E_{d}$, not used to constrain the LECs, is given in MeV; its experimental value is $E_{d}=2.224$ MeV. The LO LECs $C_{S}$ and $C_{T}$ are given in fm2, the others are in fm4. $R_{0}$ [fm] |  | 1.4 fm |  |  | 1.6 fm |
---|---|---|---|---|---|---
| LO | NLO | N2LO | LO | NLO | N2LO
$C_{S}$ | $-2.675$ | $-0.480$ | $1.331$ | $-3.590$ | $-0.988$ | $0.538$
$C_{T}$ | $-0.021$ | $0.723$ | $0.363$ | $-0.188$ | $0.660$ | $0.495$
$C_{1}$ |  | $0.124$ | $-0.104$ |  | $-0.098$ | $-0.206$
$C_{2}$ |  | $0.302$ | $0.188$ |  | $0.393$ | $0.368$
$C_{3}$ |  | $-0.224$ | $-0.217$ |  | $-0.311$ | $-0.278$
$C_{4}$ |  | $0.192$ | $0.166$ |  | $0.312$ | $0.267$
$C_{5}$ |  | $-2.268$ | $-2.083$ |  | $-2.416$ | $-2.282$
$C_{6}$ |  | $0.471$ | $0.355$ |  | $0.603$ | $0.536$
$C_{7}$ |  | $-0.578$ | $-0.529$ |  | $-0.798$ | $-0.790$
$E_{d}$ | $1.886$ | $2.151$ | $2.193$ | $2.043$ | $2.147$ | $2.178$
Figure 2: Phase shifts for $R_{0}=1.4$ fm in different partial waves as a
function of laboratory energy. Results are shown for the LO (blue), NLO (red),
and N2LO (grey) interactions compared to the Nijmegen partial wave analysis
(PWA) Stoks  _et al._ (1993). The bands at each order give the theoretical
uncertainty as discussed in the text.

As it turns out, the available local interactions from Refs. Gezerlis  _et
al._ (2013, 2014) with $R_{0}=1.0$ fm and 1.1 fm are too hard (see also Ref.
Hoppe  _et al._ (2017)) and, thus, not suitable for calculations of a nucleus-
nucleus potential at the simple “Hartree-Fock” level111The double-folding
potential is calculated at the Hartree-Fock level, but using phenomenological
densities, which would otherwise be deficient, when taking them from a
Hartree-Fock calculation based on $NN$ interactions. considered here, because
the resulting double-folding potentials are repulsive. Additional $NN$
attraction coming from beyond Hartree-Fock many-body contributions would solve
this behavior of the resulting double-folding potentials. To perform
calculations at the Hartree-Fock level, we can only use the existing
interaction with $R_{0}=1.2$ fm. In order to estimate the impact of the
regulator, we construct softer interactions with cutoffs $R_{0}=1.4$ fm and
1.6 fm. We determine the low-energy constants (LECs) by fitting to the $np$
phase shifts from the Nijmegen partial wave analysis (PWA) Stoks  _et al._
(1993). To this end, we minimize the following $\chi^{2}$

$\chi^{2}=\sum_{i}\frac{(\delta^{\mathrm{PWA}}_{i}-\delta^{\mathrm{theo}}_{i})^{2}}{\Delta\delta_{i}^{2}}\,,$
(12)

computed from the squared difference between the PWA phase shifts and the
calculated ones. The uncertainty $\Delta\delta_{i}^{2}$ is obtained from the
PWA, a model uncertainty, and a numerical error:

$\Delta\delta_{i}^{2}=(\Delta\delta_{i}^{\mathrm{PWA}})^{2}+(\Delta\delta_{i}^{\mathrm{mod}})^{2}+(\Delta\delta_{i}^{\mathrm{num}})^{2}\,.$
(13)

For the model uncertainty we use a relative uncertainty multiplied with a
constant value Carlsson  _et al._ (2016); Huth  _et al._ (2017),

$\displaystyle\Delta\delta_{i}^{\text{model, LO}}$
$\displaystyle=\left(\frac{Q}{\Lambda_{b}}\right)^{2}C\,,$ (14)
$\displaystyle\Delta\delta_{i}^{\text{model, }\nu}$
$\displaystyle=\left(\frac{Q}{\Lambda_{b}}\right)^{\nu+1}C\,,$ (15)

where $Q=\text{max}(m_{\pi},\,p=\sqrt{E^{\mathrm{lab}}_{i}m_{N}/2})$ and
$C=1^{\circ}$. For both cutoffs ($R_{0}=1.4$ fm and 1.6 fm), we take
$\Lambda_{b}=400$ MeV, which also roughly corresponds to a coordinate-space
cutoff $R_{0}=1.4$ fm to get a more conservative uncertainty estimate.

Our interactions are fit up to laboratory energies of 50 MeV at LO and up to
150 MeV at NLO and N2LO. In particular, we consider the energies 1, 5, 10, 25,
50, 100, and 150 MeV. The LO interaction is fit to the two $S$-wave channels,
while the NLO and N2LO interactions are also constrained by the four $P$-waves
and the ${}^{3}S_{1}$–${}^{3}D_{1}$ mixing angle $\varepsilon_{1}$. The LECs
and the deuteron binding energy obtained for each interaction are given in
Table 1. All other inputs and conventions for these softer local chiral $NN$
potentials are as in Refs. Gezerlis  _et al._ (2013, 2014). The phase shifts
for $R_{0}=1.4$ fm are shown in Fig. 2; we find similar results with
$R_{0}=1.6$ fm. The phase shift reproduction here is comparable to the
interactions from Refs. Gezerlis  _et al._ (2013, 2014).

###  III.2 Double-folding potential

To apply the double-folding method using local chiral $NN$ interactions, we
consider the 16O–16O system, where there are ample sets of data to which we
can compare our calculations. Elastic scattering has been accurately measured
at various energies Stiliaris  _et al._ (1989); Bohlen  _et al._ (1993);
Sugiyama  _et al._ (1993); Kondō  _et al._ (1996); Bartnitzky  _et al._
(1996); Nuoffer  _et al._ (1998); Khoa  _et al._ (2000); Nicoli  _et al._
(1999) and these data sets have been precisely analyzed with phenomenological
optical potentials Khoa  _et al._ (2000); Kondō  _et al._ (1990) or using
inversion techniques Allen  _et al._ (1993); Bennett  _et al._ (1996). This
enables us to compare our results with state-of-the-art phenomenological
calculations. At lower energy, the fusion of two 16O nuclei Tserruya  _et al._
(1978); Hulke  _et al._ (1980); Wu and Barnes (1984); Kuronen  _et al._
(1987); Duarte  _et al._ (2015) is another observable with which we can test
our double-folding potential. In this section, we present results for the
double-folding potential computed at different energies and we illustrate its
order-by-order behavior and the sensitivity to the cutoff scale.

Figure 3: Direct (upper panel) and exchange (lower panel) contributions to the
double-folding potential for the 16O–16O system based on the local chiral EFT
interaction at N2LO with $R_{0}=1.4$ fm. The direct contribution is energy
independent and we show results for different laboratory energies,
$E_{\text{lab}}$, in the exchange channel. Figure 4: Double-folding potential
for the 16O–16O system at $E_{\text{lab}}=350$ MeV. The results obtained at
LO, NLO, and N2LO (for $R_{0}=1.4$ fm) illustrate the order-by-order behavior,
while the shaded area at N2LO shows the sensitivity to the cutoff for
$R_{0}=1.2$, 1.4, and 1.6 fm.

Figure 3 shows the direct (upper panel) and exchange (lower panel)
contributions to the double-folding potential based on the local chiral N2LO
potential with $R_{0}=1.4$ fm. Since the $NN$ interaction is energy
independent, the direct contribution of the double-folding potential is also
energy independent [see Eq. (1)]. The exchange contribution given by Eq. (2),
however, includes an energy dependence through the relative momentum ${\bf k}$
in the exponential factor [see Eq. (3)]. The shape of this exchange
contribution does not vary significantly with energy, but its attractive
strength decreases with increasing energy, which can be understood by the
increasing variation of the exponential factor.

The final double-folding potential computed at different orders and with
different cutoffs is displayed for $E_{\text{lab}}=350$ MeV in Fig. 4. The
order-by-order behavior is similar to what is observed in Fig. 2. As explained
before, lower cutoffs ($R_{0}<1.2$ fm) provide harder $NN$ interactions, which
lead to repulsive double-folding potentials at LO and NLO. These interactions
require the additional attraction expected to come from many-body
contributions beyond the simple Hartree-Fock level considered here. At N2LO,
the calculations have been performed with three different $NN$ cutoffs:
$R_{0}=1.2$ fm (dotted line), 1.4 fm (solid line), and 1.6 fm (dashed line);
the lowest cutoff providing the less attractive potential. It is interesting
to notice that the sensitivity to the $NN$ cutoff $R_{0}$ decreases at larger
distance, where all three N2LO potentials present nearly identical
asymptotics. The range of the regularization cutoff, $R_{0}$, highlighted by
the shaded band in Fig. 4, will allow us to gauge the level of details needed
in $NN$ interactions to reproduce the physical observables in nucleus-nucleus
reactions.

##  IV Elastic scattering

The elastic scattering of medium to heavy nuclei can be described within the
optical model. In that model, the nuclear part of the interaction between the
colliding nuclei is described by a complex potential. Roughly speaking, the
real part corresponds to the attractive interaction between the nuclei,
whereas the imaginary part simulates the absorption of the incoming channel to
other open channels, such as inelastic scattering or transfer. Double-folding
potentials are often used for the real part of the optical potential. In this
first study, we follow the São Paulo group and assume the imaginary part of
the optical potential $U_{\rm F}$ to be proportional to its real part Pereira
_et al._ (2009)

$U_{\rm F}(r,E_{\text{cm}})=(1+i\,N_{W})\,V_{\rm F}(r,E_{\text{cm}})\,,$ (16)

where $V_{\rm F}$ is our double-folding potential and $N_{W}$ is a real
coefficient taken in the range 0.6–0.8.

Figure 5: Ratio of the cross section for elastic 16O–16O scattering to the
Mott cross section for laboratory energy $E_{\text{lab}}=350$ MeV. Results are
shown at LO, NLO, and N2LO for $R_{0}=1.4$ fm, and the sensitivity to
$R_{0}=1.2$–$1.6$ fm is illustrated at N2LO by the shaded area. In all cases,
we take for the imaginary part $N_{W}=0.8$ [see Eq. (16)]. The results are
compared to experimental data from Ref. Bohlen  _et al._ (1993).

The cross section for 16O–16O elastic scattering for laboratory energy $E_{\rm
lab}=350$ MeV is shown in Fig. 5 as a ratio to the Mott cross section. In
these calculations, we take for the imaginary part $N_{W}=0.8$, whereas we
study the sensitivity to $N_{W}$ later. Note that since 16O is a spinless
boson, the wave function for the 16O–16O relative motion needs to be properly
symmetrized.

Figure 6: Ratio of the cross section for elastic 16O–16O scattering to the
Mott cross section as a function of momentum transfer $q$ for various
laboratory energies (the different energy results are offset by a factor as
indicated). Results are shown at N2LO for $R_{0}=1.2$ fm (red), 1.4 fm (blue),
and 1.6 fm (green). For these cutoffs, the region between the results with
$N_{W}=0.6$ (upper limit) and $N_{W}=0.8$ (lower limit) is shaded. In the case
of $R_{0}=1.2$ fm and 1.4 fm, the upper line is shown as a dashed line. For
comparison, we also show the optical-potential results of Khoa et al. Khoa
_et al._ (2000) and the experimental data from Refs. Bohlen  _et al._ (1993);
Kondō  _et al._ (1996); Bartnitzky  _et al._ (1996); Nuoffer  _et al._ (1998);
Khoa  _et al._ (2000); Nicoli  _et al._ (1999).

As in Figs. 2 and 4, we observe a systematic order-by-order behavior. The
uncertainty related to the cutoff choice at N2LO (shaded area) is similar to
that observed in the double-folding potential itself (see Fig. 4). At forward
angles, i.e., up to $10^{\circ}$, the agreement of our calculations with
experiment is excellent, knowing in particular that there are no parameters
fitted to reproduce the data. At larger angles this agreement deteriorates.
Since the spread observed in the NN cutoff band remains small even at larger
angles, this discrepancy cannot be fully explained by the detail of the $NN$
interactions considered. It is likely due to the simple Hartree-Fock level of
the many-body calculation or to the choice of the 16O density, which could be
improved. In addition, it could also reflect the simple description of the
imaginary part. The elastic scattering cross sections computed at various
laboratory energies between 124 and 704 MeV are displayed in Fig. 6 as a ratio
to the Mott cross section. To compare the calculations performed at different
energies, we plot them as a function of the momentum transfer $q$. The bands
are delimited by results for the range $N_{W}=0.6-0.8$. Results generated by
the cutoffs $R_{0}=1.2$ fm, 1.4 fm, and 1.6 fm are displayed in red, blue, and
green, respectively. We find that the cutoff variation is less relevant than
the impact of the imaginary part coefficient $N_{W}$. As in Fig. 5, we observe
a general agreement between our calculations and the data, especially at
forward angles. At larger momentum transfer, the agreement is less good,
although the experimental points remain close to the spread obtained for the
$N_{W}$ range. This confirms that going beyond the simple description of the
imaginary part could improve our calculations.

For comparison, we also show the cross sections computed with the
phenomenological optical potential developed by Khoa et al. Khoa  _et al._
(2000) (dotted line in Fig. 6). This potential, containing nine adjustable
parameters that are modified at each energy, provides a near-perfect
reproduction of the data. Given that we do not include any adjustable
parameter to fit the data, our results with the double-folding potential based
on chiral EFT interactions are therefore very encouraging.

##  V Fusion reactions

The 16O+16O fusion reaction is another test for our double-folding potential.
This cross section $\sigma_{\text{fus}}$ has been measured at low energies to
study the role of intermediate resonances during fusion Tserruya  _et al._
(1978); Hulke  _et al._ (1980) and because this reaction takes place in
medium- to heavy-mass stars Hulke  _et al._ (1980); Kuronen  _et al._ (1987);
Wu and Barnes (1984); Duarte  _et al._ (2015). Oxygen fusion is crucial in
medium-mass nuclei burning chains, which provide the seeds to the synthesis of
heavy elements. At low energy, the reaction takes place through quantum
tunneling of the effective potential barrier that results from the combination
of the attractive strong interaction, the repulsive Coulomb interaction, and
the centrifugal term of the kinetic energy:

$V_{\text{eff}}(r,E_{\text{cm}})=V_{\text{F}}(r,E_{\text{cm}})+V_{\text{Coul}}(r)+\frac{l(l+1)}{2\mu
r^{2}}\,.$ (17)

Since the fusion reaction takes place at very low energies and involves light
spherical nuclei, we take the (real) double-folding potential as the nuclear
interaction for this reaction Hagino and Takigawa (2012). For light systems
like 16O+16O, the fusion barrier is at around 9 fm, well before the neck
formation, which justifies the use of the double-folding procedure. For the
code used in the computation of the fusion cross section, we approximate the
Coulomb interaction by a sphere-sphere potential of radius $R_{C}=2\times
4.39$ fm Baye and Pecher (1982). We do not expect this change from the double-
folding Coulomb term used in Eq. (3) to affect significantly our results.

The fusion cross section of 16O+16O can be obtained from the probability
$P_{l}$ to tunnel through the barrier in each of the partial waves Hagino and
Takigawa (2012)

$\sigma_{\text{fus}}(E_{\text{cm}})=\frac{\pi}{k^{2}}\sum_{l}(1+(-1)^{l})(2l+1)P_{l}(E_{\text{cm}})\,.$
(18)

The probabilities $P_{l}$ are determined using the incoming-wave boundary
condition detailed in Ref. Hagino and Takigawa (2012) and implemented in the
code CCFULL Hagino  _et al._ (1999), in which we have included the effects of
the symmetrization of the wave function for the fusing nuclei being identical
spinless bosons.

At low energy, the fusion process is strongly hindered by the Coulomb
repulsion. This effect is well accounted for by the Gamow factor, which is
usually factorized out of the cross section to define the astrophysical $S$
factor

$S(E_{\text{cm}})=E_{\text{cm}}\,e^{2\pi\eta}\,\sigma_{\text{fus}}(E_{\text{cm}})\,,$
(19)

where the Sommerfeld parameter is given by
$\eta=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}v)$, with $v$ the relative velocity
between the two nuclei.

Figure 7: Astrophysical $S$-factor for the fusion of 16O+16O as a function of
the energy $E_{\rm cm}$ in the center-of-mass system. Results are shown at LO,
NLO, and N2LO for $R_{0}=1.4$ fm, and the sensitivity to $R_{0}=1.2$–$1.6$ fm
at N2LO is illustrated by the shaded area. The results are compared to
experimental data from Refs. Tserruya  _et al._ (1978); Hulke  _et al._
(1980); Wu and Barnes (1984); Kuronen  _et al._ (1987); Duarte  _et al._
(2015).

The $S$ factor obtained at LO, NLO, and N2LO for $R_{0}=1.4$ fm and with
different cutoffs $R_{0}$ at N2LO is displayed in Fig. 7. Given the very weak
energy dependence of the double-folding potential observed at the relevant
energies, $V_{\text{Ex}}$ is taken at the center of the energy range,
$E_{\text{cm}}=12$ MeV. We have tested that taking a different energy in this
range leads to indistinguishable results from those in Fig. 7. It is
interesting to note that, due to the nearly cutoff-independent asymptotic
behavior of the nuclear folding potential, the spread between the results
obtained with different values of $R_{0}$ is small around the Coulomb barrier.
This leads to results at N2LO in Fig. 7 that are closer than what Fig. 4 would
suggest. Note also that the less attractive potentials (at NLO with
$R_{0}=1.4$ fm, and N2LO with $R_{0}=1.2$ fm) naturally lead to the lowest
fusion cross sections. The general agreement with the data is good, recalling
that there is no fitting parameter. As for elastic scattering, we observe that
the sensitivity to the details in the $NN$ interaction shown by the shaded
area can only partially explain the discrepancy with experiment. In future
work, we will explore how a better many-body calculation of the double-folding
potential and more realistic densities of the fusing nuclei may improve this
agreement.

##  VI Summary and outlook

We have presented a first study of constructing nucleus-nucleus potentials
from local chiral $NN$ interactions Gezerlis  _et al._ (2013, 2014); Huth  _et
al._ (2017) using the double-folding method applied to the 16O–16O system. Our
results show that for soft cutoffs, $R_{0}\gtrsim 1.4$ fm, the resulting
double-folding potential exhibits a systematic order-by-order behavior
expected in EFT and a weak cutoff dependence on the details of the $NN$
interactions used. These features carry through to the elastic scattering
cross section and the $S$-factor for the fusion reaction.

We have focused on the 16O–16O reactions, because these have been accurately
measured and are well studied theoretically Stiliaris  _et al._ (1989); Bohlen
_et al._ (1993); Sugiyama  _et al._ (1993); Kondō  _et al._ (1996); Bartnitzky
_et al._ (1996); Nuoffer  _et al._ (1998); Nicoli  _et al._ (1999); Khoa  _et
al._ (2000); Tserruya  _et al._ (1978); Hulke  _et al._ (1980); Wu and Barnes
(1984); Kuronen  _et al._ (1987); Duarte  _et al._ (2015); Kondō  _et al._
(1990). In all cases, a good agreement with the data has been obtained without
any fitting parameter. Our results thus suggest that the idea to derive
nucleus-nucleus potentials using the double-folding method based on local
chiral EFT interactions is very promising.

We consider this a first step in a more fundamental description of nucleus-
nucleus potentials, but there are several directions how the calculations can
be improved, both at the level of the input interactions and the many-body
folding method. First, the influence of the nucleon density of the colliding
nuclei needs to be evaluated. This can be done by using more realistic
densities, such as those obtained from electron-scattering measurements or
accurate nuclear-structure models. Second, we need to refine the imaginary
part of the potential. Assuming it to be proportional to the double-folding
potential provides a first estimate, but it is clear that this can be
improved. Comparisons with phenomenological potentials Khoa  _et al._ (2000)
and potentials built from inversion techniques Allen  _et al._ (1993); Bennett
_et al._ (1996) can also provide tests towards more realistic prescriptions.
In a calculation beyond Hartree-Fock, an imaginary part as well as nonlocal
contributions would arise (see, e.g., Refs. Feshbach (1958, 1962)). Moreover,
going beyond the level of the density-matrix expansion considered here, there
will be gradient corrections Negele and Vautherin (1972) (i.e., surface terms)
to the double-folding potential. Finally three-nucleon interactions need to be
investigated in this approach, as they also enter at N2LO.

In conclusion, coupling chiral EFT interactions with the double-folding method
provides nucleus-nucleus potentials that lead to very encouraging agreement
with elastic-scattering and fusion data in a broad range of energies. This
idea is thus a promising first step towards the construction of microscopic
optical potentials from first principles with control over uncertainty
estimates. Through the above future developments, we hope to improve this new
method to obtain a systematic way to build efficient optical potentials for
nuclear reactions.

## Acknowledgments

We thank J. E. Lynn and I. Tews for many helpful discussions, and in
particular for providing the code for the local chiral interactions as well as
for discussions on the softer local chiral interactions presented here. We
also thank D. Baye for helpful discussions on the reactions studied in this
work, and the International Atomic Energy Agency that provided the
experimental data through their web page www-nds.iaea.org. This work was
supported in part by the European Research Council Grant No. 307986 STRONGINT,
the GSI-TU Darmstadt cooperation, the European Union’s Horizon 2020 research
and innovation programme under Grant Agreement No. 654002, and the US National
Science Foundation Grant No. PHY-1514695.

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