Double-folding potentials from chiral effective field theory

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 $^{16}$O-$^{16}$O 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.

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 16 O- 16 O 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 [1]. 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 [2]. 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 (N N ) interaction using a double-folding procedure [3]. It is known that this framework provides more realistic potentials for the nucleon-nucleus interactions than for the nucleusnucleus case [4]. Nevertheless, it constitutes a first-order approximation to optical potentials derived from Feshbach's reaction theory [1]. Interesting results have been obtained in such a way, e.g., by considering zero-range contact N N interactions [5,6] or using a G-matrix approach, see, e.g., Refs. [7,8] for recent work.
In this first study, we explore and test this idea for 16 O-16 O reactions, comparing our calculations to elasticscattering [23][24][25][26][27][28][29][30] and fusion data [31][32][33][34][35]. For this system, phenomenological Woods-Saxon potentials [30] and potentials obtained through inversion techniques [36,37] 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 doublefolding 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 N N potentials. We then determine the double-folding potentials at different chiral orders and apply these to 16  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 F = V D + V Ex can be constructed from a given N N 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. [7]. We will include only N N 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 N N interaction over the neutron (n) and proton (p) density distributions ρ n,p 1 and ρ n,p 2 of the colliding nuclei, where r is the relative coordinate between the center of mass of the nuclei, r 1 and r 2 are the coordinates from the center of mass of each nucleus, s = r − r 1 + r 2 (the geometry is shown in Fig. 1), and the sum i, j is over neutrons and protons with their respective densities.
To account for the antisymmetrization between nucleons, the double-folding potential receives contributions also from the exchange channel, where µ = 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 ρ i (r, r ± s) of the nuclei.
In the exchange channel, there is an additional phase that renders the double-folding potential dependent on the energy E cm in the center-of-mass system. The momentum for the nucleus-nucleus relative motion k is related to E cm , the nuclear part of the double-folding potential, and the double-folding Coulomb potential V Coul through As a result, V Ex has to be determined self-consistently. Note that at our level of calculation the double-folding potential, V F = V D + V Ex , is real. The density matrices entering in Eq. (2) are approximated using the density matrix expansion [38] restricted to its leading term, where R = r ± 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 densitymatrix expansion, as in Ref. [7]: 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 N N interaction entering the double-folding potential in the direct and exchange channels, v D and v Ex , respectively, at this level receive contributions only from the central parts of nuclear forces. Then also the N N interaction and the doublefolding potentials depend only on the relative distance (s or r). Writing the N N interaction in terms of their two-body spin-isospin components, v ST , and distinguishing between proton-proton (pp), proton-neutron (pn, np), where the upper (lower) signs refer to the direct (exchange) term and we have neglected the small isospinsymmetry-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 [5] for the proton and neutron densities, whose parameters were fitted to Dirac-Hartree-Bogoliubov calculations where ρ 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 R n = 1.49 N 1/3 − 0.79 , a n = 0.47 + 0.00046 N . (10)

III. LOCAL CHIRAL EFT INTERACTIONS
A. 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 [9,10]. The different contributions to N N and many-nucleon interactions are ordered according to a power counting scheme in powers of (Q/Λ b ) ν , where Q is a typical momentum or the pion mass and Λ 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 N N interactions starting at leading order (LO, ν = 0) followed by a contribution at next-to-leading order (NLO, ν = 2), whereas three-nucleon interactions enter at next-to-next-to leading order (N 2 LO, ν = 3). Because they facilitate the calculation of doublefolding potentials, we use local chiral N N interactions, developed initially in Refs. [16,17], but construct new soft N N interactions up to N 2 LO. As in these references, the long-and short-range parts of the interaction are regularized by where R 0 is the coordinate-space cutoff in the N N 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 shortrange regulator smears out the N N contact interactions. A second cutoffΛ is used in the spectral-function regularization of the two-pion exchange, which enters first at NLO. In Refs. [16,17] it was shown that the calculations are practically insensitive toΛ for local interactions; in the present work, we considerΛ = 1000 MeV.
As it turns out, the available local interactions from Refs. [16,17] with R 0 = 1.0 fm and 1.1 fm are too hard (see also Ref. [40]) and, thus, not suitable for calculations of a nucleus-nucleus potential at the simple  [39]. The bands at each order give the theoretical uncertainty as discussed in the text. "Hartree-Fock" level 1 considered here, because the resulting double-folding potentials are repulsive. Additional N N 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) [39]. To this end, we minimize the following χ 2 computed from the squared difference between the PWA phase shifts and the calculated ones. The uncertainty ∆δ 2 i is obtained from the PWA, a model uncertainty, and a numerical error: For the model uncertainty we use a relative uncertainty multiplied with a constant value [22,41], where Q = max(m π , p = E lab i m N /2) and C = 1 • . For both cutoffs (R 0 = 1.4 fm and 1.6 fm), we take Λ 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 N 2 LO. In particular, we consider the energies 1, 5, 10, 25, 50, 100, and 150 MeV. The LO interaction is fit to the two Swave channels, while the NLO and N 2 LO interactions are also constrained by the four P -waves and the 3 S 1 -3 D 1 mixing angle ε 1 . The LECs and the deuteron binding energy obtained for each interaction are given in Table I. All other inputs and conventions for these softer local chiral N N potentials are as in Refs. [16,17]. 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. [16,17].

B. Double-folding potential
To apply the double-folding method using local chiral N N interactions, we consider the 16 O- 16 O system, where there are ample sets of data to which we can compare our calculations. Elastic scattering has been accurately measured at various energies [23][24][25][26][27][28][29][30] and these data sets have been precisely analyzed with phenomenological optical potentials [30,42] or using inversion techniques [36,37]. This enables us to compare our results with state-of-theart phenomenological calculations. At lower energy, the fusion of two 16 O nuclei [31][32][33][34][35] 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 shows the direct (upper panel) and exchange (lower panel) contributions to the double-folding potential based on the local chiral N 2 LO potential with R 0 = 1.4 fm. Since the N N 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 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 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 N N 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 N 2 LO, the calculations have been performed with three different N N 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 N N cutoff R 0 decreases at larger distance, where all three N 2 LO 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 N N 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 F to be proportional to its real part [6] where V F is our double-folding potential and N W is a real coefficient taken in the range 0.6-0.8. The cross section for 16 O-16 O elastic scattering for laboratory energy E 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 16 O is a spinless boson, the wave function for the 16 O-16 O relative motion needs to be properly symmetrized.
As in Figs. 2 and 4, we observe a systematic orderby-order behavior. The uncertainty related to the cutoff choice at N 2 LO (shaded area) is similar to that observed in the double-folding potential itself (see Fig. 4). At forward angles, i.e., up to 10 • , 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 N N interactions considered. It is likely due to the simple Hartree-Fock level of the many-body calculation or to the choice of the 16 O 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. [30] (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. . For these cutoffs, the region between the results with NW = 0.6 (upper limit) and NW = 0.8 (lower limit) is shaded. In the case of R0 = 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. [30] and the experimental data from Refs. [24,[26][27][28][29][30].

V. FUSION REACTIONS
The 16 O+ 16 O fusion reaction is another test for our double-folding potential. This cross section σ fus has been measured at low energies to study the role of intermediate resonances during fusion [31,32] and because this reaction takes place in medium-to heavy-mass stars [32][33][34][35]. 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: 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 [43]. For light systems like 16   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 × 4.39 fm [44]. 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 16 O+ 16 O can be obtained from the probability P l to tunnel through the barrier in each of the partial waves [43] σ fus (E cm ) = π k 2 l (1 + (−1) l )(2l + 1)P l (E cm ) . (18) The probabilities P l are determined using the incomingwave boundary condition detailed in Ref. [43] and implemented in the code CCFULL [45], 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 where the Sommerfeld parameter is given by η = Z 1 Z 2 e 2 /(4πε 0 v), with v the relative velocity between the two nuclei. The S factor obtained at LO, NLO, and N 2 LO for R 0 = 1.4 fm and with different cutoffs R 0 at N 2 LO is displayed in Fig. 7. Given the very weak energy dependence of the double-folding potential observed at the relevant energies, V Ex is taken at the center of the energy range, E 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 N 2 LO 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 N 2 LO 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 N N 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 N N interactions [16,17,22] using the double-folding method applied to the 16 O- 16 O system. Our results show that for soft cutoffs, R 0 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 N N interactions used. These features carry through to the elastic scattering cross section and the Sfactor for the fusion reaction.
We have focused on the 16 O-16 O reactions, because these have been accurately measured and are well studied theoretically [23][24][25][26][27][28][29][30][31][32][33][34][35]42]. 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 manybody folding method. First, the influence of the nucleon density of the colliding nuclei needs to be evalu-ated. 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 [30] and potentials built from inversion techniques [36,37] 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. [46,47]). Moreover, going beyond the level of the density-matrix expansion considered here, there will be gradient corrections [38] (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 N 2 LO.
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.