CCoupled-cluster computations of opticalpotential for medium-mass nuclei
J. Rotureau
NSCL/FRIB Laboratory, Michigan State University, East Lansing, Michigan 48824,USA
Correspondence*:J. [email protected]
ABSTRACT
Recent progress in the numerical solution of the nuclear many-body problem and in thedevelopment of nuclear Hamiltonians rooted in Quantum Chromodynamics, has opened the doorto first-principle computations of nuclear reactions. In this article, we discuss the current statusof ab initio calculations of nucleon-nucleus optical potentials for medium-mass systems, with afocus on results obtained with the coupled-cluster method.
Keywords: nuclear reactions, nuclear structure, optical potential, ab-initio method, Green’s function, chiral effective field theory.
Understanding the structure and dynamics of atomic nuclei in terms of nucleons and their mutualinteractions is one of the main goals of nuclear physics. At the typical energy scale of nuclear phenomena,the quarks and gluons degrees of freedom are not resolved. As a consequence, in this context, nucleonscan be treated as point-like particles and the nuclear problem with protons and neutrons can be viewedas a low-energy effective approximation to QCD. Within the framework of Effective Field Theory (EFT),inter-nucleon interactions consistent with the chiral symmetry can nowadays be derived systematically interms of nucleon-nucleon, three-nucleon, and higher many-nucleon forces [1, 2, 3, 4, 5, 6]. Starting with agiven Hamiltonian, ab initio calculations of nuclei aim at solving the many-body Schr¨odinger equationwithout any uncontrolled approximations. Within the last decades, the increase in computing power andthe development of powerful many-body methods, combined with the use of chiral-EFT interactions, haveenabled a quantitative description of light and medium-mass nuclei ab initio [7, 8, 9, 10, 11, 12]. With theinclusion of continuum effects in many-body methods, ab-initio calculations have also reached parts ofthe nuclear chart far from stability where the coupling to continuum states and decay channels plays animportant part in the structure of nuclei [13, 14, 15, 16, 17, 18, 19, 20, 21]).A lot of progress has been made as well in the development of ab initio methods for nuclear reactions.The No-Core Shell Model with the Resonating Group Method (NCSM/RGM) or with continuum (NCSMC)have successfully described scattering and transfer reactions for light targets [22, 23, 24], the Green’sFunction Monte Carlo [25, 26] has recently been applied to nucleon-alpha scattering using chiral NN, 3Nforces [27], and lattice-EFT computations of alpha-alpha scattering have recently been reported [28]. Formedium-mass nuclei, nucleon-nucleus optical potentials and elastic scattering cross sections have beencomputed with chiral forces within the Self Consistent Green’s Function (SCGF) approach [9, 29, 30, 31]and the coupled-cluster method [32, 33, 34]. a r X i v : . [ nu c l - t h ] J u l . Rotureau Coupled-cluster computations of optical potential for medium-mass nuclei
The optical potential plays an important role in reaction theory. It is usual (and practical) in thiscontext to reduce the many-body problem into a few-body one where only the most relevant degreesof freedom are retained[35]. Correspondingly, the many-body Hamiltonian is replaced by a few-bodyHamiltonian expressed in terms of optical potentials i.e. effective interactions between the particlesconsidered at the few-body level. Traditionally, optical potentials have been constructed by fitting to data,particularly data on β -stable isotopes [36, 37]. For instance, global phenomenological nucleon-nucleuspotentials enable the description of scattering processes for a large range of nuclei and projectile energies.However, extrapolation of these phenomenological potentials to exotic regions of the nuclear chart areunreliable and have uncontrolled uncertainties. Moreover, since fitting to two-body elastic scattering data(as it is most often done) does not constrain the off-shell behavior of potentials , a dependence on thechoice of potentials may arise in transfer reactions observables (and other reactions) as shown in e.g. [38, 39, 40]. It is then critical, in order to advance the field of nuclear reactions and notably for reactionswith exotic nuclei undertaken at rare-isotope-beam facilities[41, 42], to connect the optical potentialsto an underlying microscopic theory of nuclei. Since potentials derived from ab initio approaches arebuilt up from fundamental nuclear interactions without tuning to data, they may have a greater predictivepower in regions of the nuclear chart that are unexplored experimentally. Furthermore, they can guide newparametrization of phenomenological potentials by providing insights on form factors, energy-dependenceand dependence on the isospin-asymmetry of the target.It is useful for pedagogical purpose and the introduction of key concepts, to start with the derivationof the optical potential within the Feshbach projection formalism [43, 44]. Let us consider the processof scattering of a nucleon on a target A . One can partition the Hilbert space for this A + 1 system into P the subspace of elastic scattering states and Q the complementary subspace. Denoting P and Q theprojectors operators on respectively P and Q , by construction one has P + Q = Id . We introduce H theHamiltonian of the system and E its energy. The optical potential describing the elastic scattering processcan be identified with the effective Hamiltonian H effP ( E ) acting in P , which by construction, reproducesthe eigenvalues of H with a model wavefunction in P . One can show that H effP ( E ) = H P P + H P Q E − H QQ + iη H QP (1)where H P P ≡ P HP , H P Q ≡ P HQ, . . . and η → + . The optical potential H effP ( E ) is non-local andfrom Eq. 1, it is clear that it is also energy-dependent and complex. The imaginary (absorptive) componentof the potential represents the loss of flux in the elastic channel due to the opening of other channels, forinstance, the excitation of the target to a state of energy E Ai for E > E Ai or breakup channels. By addingthe Hilbert space of the A − system (hole states in the target) in the formalism, it has been shown thatthe resulting optical potential corresponds to the self-energy defined in Green’s function theory [45]. Theparticle part of the self-energy is equivalent to the optical potential (1), whereas the hole part describesthe structure of the target. By including information on both the ( A + 1) - and ( A − -system in theformalism, the Green’s function approach, which will be used in this paper, provides a consistent treatmentof scattering and structure.In this article, we present some recent results for the ab-initio computation of nucleon-nucleus opticalpotential for medium-mass nuclei, constructed by combining the Green’s function approach with thecoupled-cluster method [46]. The coupled-cluster method is an efficient tool for the computation of ground- Two phase-equivalent potentials will reproduce the same elastic two-body scattering data but may have different off-shell behavior.
Frontiers . Rotureau Coupled-cluster computations of optical potential for medium-mass nuclei and low-lying excited states in nuclei with a closed (sub-)shell structure and in their neighbors with ± nucleons. By including complex continuum basis states in the formalism, it also provides a versatileframework to consistently compute bound, resonant states and scattering processes [13, 15, 16, 17, 32]. Inour approach, the optical potential is obtained by solving the Dyson equation after a direct computation ofthe Green’s function with the coupled-cluster method. As we will see in Sec. 2, the inclusion of complexcontinuum basis states enables also a precise computation of Green’s functions and optical potentials.We want to point out here that there has been a lot of work over the years to compute optical potentialsfrom various microscopic approaches. In the following, we mention some of the most recent worksdedicated to that goal (for a more exhaustive review we refer the reader to e.g. [47]). The authors in [48]have computed optical potentials for neutron and proton elastic scattering on Ca based on the applicationof the self-consistent Hartree-Fock and Random-Phase Approximations to account for collective statesin the target. Using the phenomenological Gogny interaction, a good reproduction of data for scatteringat E ≤ MeV has been reported in [48]. In [49, 50], nucleon-nucleus potentials are computed for finitenuclei from a folding of optical potentials obtained by many-body perturbation theory calculations innuclear matter with chiral forces. In these papers, several calcium isotopes are considered and an overallsatisfactory agreement with data is achieved. For the scattering of nucleons at intermediate and high energy(E (cid:38)
100 MeV) optical potentials can be derived within the multiple scattering formalism [51, 52] wherethe optical potential is obtained based on the folding of the nucleon-nucleon T-matrix or G-matrix with thenuclear density [53, 54, 55]. Recent applications of this approach, in which the nucleon-nucleus T-matrixand the density are computed consistently starting from the same chiral-EFT interaction, have been reportedand shown a successful reproduction of data [56, 57]. In the Dispersive Optical Model [47, 58, 59, 60], a(semi-) phenomenological potential is constructed by exploiting formal properties of the Green’s function,such as the dispersion relation, which connects the real part and imaginary part of the potential [61].Applications of this data-driven approach have been made using local and non-local form factors of thepotential for Ca and Pb isotopes.This paper is organized as follows. In Sec. 2, we will briefly review the formalism to construct opticalpotentials by combining the Green’s function approach and the coupled-cluster method. In Sec. 3, recentresults for neutron- , Ca optical potentials at negative and positive energies are presented. In Sec. 4, wewill discuss challenges and possible solutions for the construction of fully predictive optical potentials withthe coupled-cluster method. Finally, we will conclude in Sec. 5.
In this part, we will briefly review the formalism for deriving ab-initio nucleon-nucleus optical potentialsby combining the Green’s function approach and the coupled-cluster method. We start first by introducingbelow, key quantities of the Green’s function formalism.
Given a single-particle basis {| α (cid:105) , | β (cid:105) , . . . } , the Green’s function [62] of a nucleus A has matrix elements G ( α, β, E ) = (cid:104) Ψ | a α E − ( H − E Ags ) + iη a † β | Ψ (cid:105) + (cid:104) Ψ | a † β E − ( E Ags − H ) − iη a α | Ψ (cid:105) . (2)Here, H is the Hamiltonian and | Ψ (cid:105) the ground state of A with the energy E Ags and by definition η → + .The operators a † α and a β create and annihilate a fermion in the single-particle state α and β , respectively. α Frontiers 3 . Rotureau
Coupled-cluster computations of optical potential for medium-mass nuclei is shorthand for the quantum numbers α = ( n, l, j, j z , τ z ) . By inserting completeness relations expressedwith the eigenstates of the A ± systems in (2), one obtains the Lehmann representation of the Green’sfunction: G ( α, β, E ) = (cid:88) i (cid:104) Ψ | a α | Ψ A +1 i (cid:105)(cid:104) Ψ A +1 i | a † β | Ψ (cid:105) E − ( E A +1 i − E Ags ) + iη + (cid:88) j (cid:104) Ψ | a † β | Ψ A − j (cid:105)(cid:104) Ψ A − j | a α | Ψ (cid:105) E − ( E Ags − E A − j ) − iη , (3)where | Ψ A +1 i (cid:105) ( | Ψ A − j (cid:105) ) is an eigenstate of H for the A + 1 ( A − ) system with energy E A +1 i ( E A − j ). Tosimplify the notation, the completeness relations are written in (3) as discrete summations over the states inthe A ± systems. The Lehmann representation has the merit to reveal somewhat more clearly some of theinformation content of the Green’s Function. As one can see from (3), the poles of the Green’s functioncorrespond to the energies of the eigenstates of H in the A ± systems.The Green’s function fulfills the Dyson equation G ( E ) = G (0) ( E ) + G ( E )Σ ∗ ( E ) G ( E ) , (4)where G ( E ) is the Green’s function associated with a single-particle potential U and Σ ∗ ( E ) the irreducibleself energy. The optical potential is given by V opt ( E ) ≡ Σ ∗ ( E ) + U. (5)The potential U is usually taken as the Hartree-Fock (HF) potential since the corresponding Green’sfunction is a first-order approximation to G ( E ) in eq. (4). In our approach, since the Green’s function isdirectly computed with the coupled-cluster method and is input of Eq. 4, the resulting optical potential isindependent of the choice of U .For E + ≡ E − E Ags ≥ , V opt ( E ) corresponds to the optical potential for the elastic scattering fromthe A -nucleon ground state[62]. In other words, the scattering amplitude ξ E + ( r ) = (cid:104) Ψ | a r | Ψ E + (cid:105) (here | Ψ E + (cid:105) is the elastic scattering state of a nucleon on the target with the energy E + and a r is the annihilationoperator of a particle at the position r ) fulfills the Schr¨odinger equation − (cid:126) µ ∇ ξ ( r ) + (cid:90) d r (cid:48) V opt ( r , r (cid:48) , E ) ξ ( r (cid:48) ) = E + ξ ( r ) , (6)where µ is the reduced mass of the nucleus-nucleon system. For simplicity, we have suppressed any spinand isospin labels. The optical potential is non-local, energy-dependent and complex [62] and for E + ≥ ,its imaginary component describes the loss of flux in the elastic channels to other channels. For E + < ,Eq. 6 admits a discrete number of physical solution at E n = E A +1 n − E Ags , which corresponds to the boundstates energies in A+1. In that case, the solutions are given by the overlap ξ n ( r ) = (cid:104) Ψ | a r | Ψ A +1 n (cid:105) where | Ψ A +1 n (cid:105) is a bound state of energy E A +1 n in the A + 1 system .In the following section, we present the main steps involved in the computation of the Green’s functionwith the coupled-cluster method. n, l, j, j z , τ z label the radial quantum number, the orbital angular momentum, the total orbital momentum, its projection on the z-axis, and the isospinprojection, respectively Similarly, for E = E Ags − E A − n , the solution of the optical potential V opt ( E ) are the radial overlap ξ − n ( r ) = (cid:104) Ψ | a † r | Ψ A − n (cid:105) [62]. Frontiers . Rotureau Coupled-cluster computations of optical potential for medium-mass nuclei
We start with the computation of the ground state | Ψ (cid:105) of the A -nucleon system. Working in the laboratoryframe, the intrinsic Hamiltonian reads H = A (cid:88) i =1 (cid:126)p i m − (cid:126)P mA + (cid:88) i
H, T ] , T ] + . . . (12)denotes the similarity transformed Hamiltonian, which is computed by making use of the Baker-Campbell-Hausdorff expansion [46]. For two-body forces and in the CCSD approximation, this expansion terminatesat fourfold nested commutators . The CCSD equations (11) show that the CCSD ground state is an The NF s component V ijk of the Hamiltonian in (8) is truncated at the normal-ordered two-body level in the HF basis (see Sec. 3). Frontiers 5 . Rotureau
Coupled-cluster computations of optical potential for medium-mass nuclei eigenstate of the similarity-transformed Hamiltonian ¯ H = e − T He T in the space of p − h , p − h , p − h configurations. The operator e T being not unitary, ¯ H is not Hermitian. As a consequence, its left-and right-eigenvectors form a bi-orthonormal set [46].Denoting (cid:104) Φ ,L | the left eigenvector for the ground state of A , we can now write the matrix elements ofthe coupled cluster Green’s function G cc as G CC ( α, β, E ) ≡ (cid:104) Φ ,L | a α E − ( H − E Ags ) + iη a † β | Φ (cid:105) + (cid:104) Φ ,L | a † β E − ( E Ags − H ) − iη a α | Φ (cid:105) . (13)Here, a α = e − T a α e T and a † β = e − T a † β e T are the similarity-transformed annihilation and creation operators,respectively. These are computed with the Baker-Campbell-Hausdorff expansion (12).In principle, the Green’s function could be computed from the Lehman decomposition (3) with thesolutions of the particle-attached equation of-motion (PA-EOM) and particle-removed equation-of motion(PR-EOM) for the A + 1 and A − sytems, respectively [46]. However, as the sum over all states in Eq. (3)involves also eigenstates in the continuum, this approach is difficult to pursue in practice. Instead, wemake use of the Lanczos continued fraction technique, which allows for an efficient and numerically stablecomputation of the Green’s function [33, 63, 64, 65, 66, 67].By definition of the Green’s function, the parameter η in the matrix elements (2) is such that η → + .However, in this limit, because of the appearance of poles at energies E = ( E A +1 i − E Ags ) in the Green’sfunction (see Eq. (3)), the calculation of optical potential for elastic scattering becomes numericallyunstable. In order to resolve this issue, we compute an analytic continuation of the Green’s function in thecomplex-energy plane by working in a Berggren basis [68, 69, 70, 71, 72, 73, 74, 75] (generated by the HFpotential) that includes bound, resonant, and complex-continuum states. The solutions of the (PA-EOM)and (PR-EOM) in the Berggren basis, i.e the eigenstates of the A ± systems, are either bound, resonantor complex-scattering states. In other words, the poles of the analytically continued Green’s function arelocated either at negative real or complex energy. As a result, the Green’s function matrix elements for E ≥ smoothly converge to a finite value as η → + (this is illustrated below in Fig. 1).The scattering states entering the Berggren basis are defined along a contour L + in the fourth quadrant ofthe complex momentum plane, below the resonant single-particle states. According to the Cauchy theorem,the shape of the contour L + is not important, under the condition that all resonant states lie between thecontour and the real momentum axis. The Berggren completeness reads (cid:88) i | u i (cid:105)(cid:104) ˜ u i | + (cid:90) L + dk | u ( k ) (cid:105)(cid:104) ˜ u ( k ) | = ˆ1 , (14)where | u i (cid:105) are discrete states corresponding to bound and resonant solutions of the single-particle potential,and | u ( k ) (cid:105) are complex-energy scattering states along the complex-contour L + . In practice, the integralalong the complex continuum is discretized yielding a finite discrete basis set.In Fig. 1, we illustrate the numerical stability provided by the use of the Berggren basis for the computationof the Green’s function. We are interested in the level density [76, 77] ρ lj ( E ) = − π T r (cid:104) Im ( G lj ( E ) − G (0) lj ( E )) (cid:105) , (15) Frontiers . Rotureau Coupled-cluster computations of optical potential for medium-mass nuclei M e V O Figure 1.
Computed level densities in O. For the J π = 3 / + level density, results are shown for severalvalues of the parameter η to illustrate the smooth convergence pattern for η → . The inset shows theenergies of the ground state, first excited and / + resonant states in O calculated at the PA-EOM-CCSDtruncation level (see text for details).where G lj ( E ) and G lj ( E ) are respectively the component of the Green’s functions and the HF Green’sfunction in the ( l, j ) partial wave . We show in Fig. 1, the J π = 3 / + level density in O calculatedwith the
NNLO sat interaction. The ground state in O is computed at the CCSD level while the Green’sfunction is computed with the PA-EOM and PR-EOM Lanczos vectors truncated at the p − h and p − h excitation level, respectively (other details of the calculation are also the same as in Sec. 3). As η approaches 0, the level density smoothly converges, and the position of the peak at η = 0 corresponds,as expected, to the position of the J π = 3 / + resonance in O (see inset in Fig. 1, which shows thePA-EOM-CCSD energies in O). For completeness, we also show the J π = 5 / + , / + level densities. Inthese cases, the level density at negative energies are equal to a Dirac delta function peaked at respectivelythe ground state and first excited state energies in O (see inset in Fig. 1). For purpose of illustrationin Fig. 1, we have used a finite value of η for the J π = 5 / + , / + densities and set the height of thecorresponding peaks to 1. We now show in this section a few results of the computation of neutron optical potentials for thedouble-magic nuclei Ca and Ca .All calculations presented here are performed using the NNLO sat chiral interaction [5], which reproducesthe binding energy and charge radius of both systems [78, 79]. We want to point out here that a properreproduction of the distribution of nuclear matter, and, more specifically, nuclear radii is critical in orderto obtain an accurate account of reactions observables. All results are obtained from coupled-clustercalculations truncated at the CCSD level, while the Lanczos vectors in the PA-EOM (PR-EOM) havebeen truncated at the p − h ( p − h ) excitation level. Since the computation of the Green’s function is Since the Green’s functions are here defined by adding (and removing) a nucleon from the + ground state in the target A , the quantum number ( l, j ) areconserved. Frontiers 7 . Rotureau
Coupled-cluster computations of optical potential for medium-mass nuclei performed using the laboratory coordinates (the Hamiltonian H in Eq. (8) is defined with these coordinates),the calculated optical potential is identified with the optical potential in the relative coordinates of the n − A Ca system. This identification will result in a small error, which is a decreasing function of the targetmass number A [33, 34] (see also Sec. 4).The HF calculations are performed in a mixed basis of harmonic oscillator and Berggren states, dependingon the partial wave. The NNLO sat interaction contains two-body and three-body terms. Denoting N and N the cutoffs in the harmonic oscillator (HO) basis of respectively, the two-body and three-body part ofthe interaction, we set N = N = N max except for the most extensive calculations where N = 14 and N = 16 . Finally, we truncate the three-nucleon forces at the normal-ordered two-body level in the HFbasis. This approximation has been shown to work well in light- and medium mass nuclei [80, 81]. Theharmonic oscillator frequency is kept fixed at (cid:126) ω =
16 MeV (for more details see [33, 34]). N max E(7/2 - ) E(3/2 - ) E(1/2 - ) 12 -7.35 -3.47 -1.3114 -7.62 -3.87 -1.8014/16 -7.84 -4.07 -2.15E exp -8.36 -6.42 -4.74 Ca Figure 2.
Diagonal part of the n + Ca optical potential for the bound states in Ca computed with the NNLO sat interaction. Results are shown for several values of N max and the corresponding bound stateenergies (with respect to the Ca ground state) are shown in the table (in MeV). The components of theHF potential in the associated partial waves are shown for ( N max , N ) = (14 , (see text for details).We start with the computation of the n + Ca optical potentials associated with the bound states in Ca.At the PA-EOM-CCSD level of truncation considered here, there are only three bound states supported bythe
NNLO sat
Hamiltonian. In order to show the convergence pattern of the potentials, we present in Fig. (2)results at several values of N max with the corresponding bound state energies. We present the diagonal partof the potentials, and for comparison the HF potential (for ( N max , N ) = (14 , ) in each partial wave isalso shown in Fig. (2). The energies are shown in the table in Fig. (2) along with the experimental values.As expected, the convergence of energies is slower for higher-energy states. The difference between the Ca energies at ( N max , N ) =(14,14) and (14,16) is ∼
220 keV in the case of the ground-state, whereas itis ∼
350 keV in the case of the J π = 1 / − second excited state. Even though the absolute binding energyis underestimated in the CCSD approximation, when compared to experiment (the CCSD binding energyof Ca is 299.28 MeV for ( N max , N ) = (14, 16), whereas the experimental value is 342.05 MeV), the Frontiers . Rotureau Coupled-cluster computations of optical potential for medium-mass nuclei neutron separation energies are consistently within 600 keV of the experimental values for , Ca . Theeigenenergies of these potentials are equal, by construction, to the bound states energies when using theeffective mass mA/ ( A − instead of the actual reduced mass. This can be traced to Eq. (8) where theeffective mass associated with the one-body kinetic operator is equal to mA/ ( A − (see also Sec. 4).We now consider the neutron elastic scattering on Ca and Ca . The phase shift is computed in eachpartial wave with the optical potential calculated in the largest space ( N max , N ) = (14 , . The angulardistributions are then obtained by summing the contributions from each partial wave. Figure 3 showsthe resulting differential elastic cross section for Ca(n , n) Ca at 5.2 MeV and Ca(n , n) Ca at 7.8MeV. We find that at these energies the inclusion of partial waves with angular momentum L ≤ and L ≤ is sufficient for Ca and Ca respectively, the contribution of partial waves with higher L beingnegligible (see also the computations of elastic scattering on , Ca at other energies in [34]). The angulardistributions obtained with the phenomenological Koning Delaroche (KD) potential [36] and the measuredcross sections are also shown in Fig. 3 for comparison. As Fig. 3 indicates, the data at small angle wherethe cross section is larger, are well reproduced for Ca whereas the computed cross section is slightlyabove the data for Ca . Overall, the shape of the experimental cross sections and the positions of theminima are well reproduced for both nuclei, as expected from the correct reproduction of matter densitiesin , Ca by the
NNLO sat interaction. d σ / d Ω [ m b / s t r] CCSDdataKoning-Delaroche Θ [deg] d σ / d Ω [ m b / s t r] Ca(n,n) Ca @ 5.2 MeV Ca(n,n) Ca @ 7.8 MeV
Figure 3.
Differential elastic cross section for Ca(n , n) Ca at 5.2 MeV (top) and Ca(n , n) Ca at7.8 MeV (bottom) calculated with the NNLO sat interaction. Results obtained with the phenomenologicalKoning-Delaroche potential potential are shown (dashed line) for comparison. Data points are taken from[36] (errors on the data are smaller than the symbols). By including both perturbative triple excitations and perturbative estimates for the neglected residual 3NFs (3NF terms beyond the normal-ordered two-bodyapproximation), a good agreement with experimental binding energies can be obtained for , Ca[78]
Frontiers 9 . Rotureau
Coupled-cluster computations of optical potential for medium-mass nuclei
The experimental energy of the first two excited-states in Ca, namely E (0 + ) =3.35 MeV and E (3 − ) =3.74MeV are below the scattering energy E scat =5.2 MeV of the elastic process Ca(n , n) Ca shown in Fig. 3.In other words, the channels for excitation of the Ca target are open at this scattering energy. This shouldresult in a loss of flux in the initial elastic channel and the corresponding occurrence of an absorptiveimaginary part in the phase shifts. The first excited + state, which has a strong p − h components,cannot be properly reproduced at the truncation level considered here: its computed energy, solution ofthe EOM-CCSD equations, is ∼
16 MeV above the ground state. On the other hand, the − excited stateis well reproduced with E EOM − CCSD (3 − ) =3.94 MeV. Nevertheless, we have found that the computedabsorption is practically negligible and none of the computed phase shifts at E scat =5.2 MeV have asignificant imaginary part. A similar pattern happens for Ca(n , n) Ca at 7.8 MeV: in that case, the firstexcited state E( + )=3.83 MeV is fairly well reproduced, the computed value is E EOM − CCSD (2 + ) =4.65MeV, but again the absorption in that case is negligible too.Although some excited states below the scattering energy are reproduced by the EOM-CCSD calculations,the absorption is negligible in both situations. This suggests that at the level of truncation considered here,namely p − h above the CCSD ground state, the computed wavefunctions are not correlated enough(in the perturbative expansion of the Dyson equation Eq. (4), the absorption appears at second-order,beyond the HF contribution [62]). In other words, at these energies, the computed level density (15) inthe n + A Ca system is too small. We have observed that only at higher energy E (cid:38)
20 MeV the absorptionstarts to increase significantly (a similar pattern can be seen in Fig. 4 of [33] for the CCSD computation of n + O optical potential). It is possible to increase artificially the absorption by using a finite value of η inEq. (13). This amounts to increasing the correlations content of the coupled-cluster wavefunctions and asshown in [33, 34], the computed elastic cross section in that case will decrease. In Sec.4, we will return tothis lack of absorption in the computed potential.We should emphasize here that the computation of the optical potential with the coupled-cluster method iscarried out without any free parameter. It is then not surprising that it does not allow for the same quality ofreproduction of data as a phenomenological potential such as the KD interaction (see Fig. 3). But still, sincemicroscopic optical potentials are built up from fundamental nuclear interactions without tuning to data,they may yield guidance for parameterizations of phenomenological potential, by providing informationon the form factor, energy dependence and dependence on the isospin asymmetry of the targets. A recentseries of studies has shown that non-locality can affect transfer reaction observables (e.g. [38, 39, 40])and it is expected that it can equally affect other reaction channels. Microscopic potential can provideguidance on this aspect of the optical potential. Keeping in mind that a potential is not an observable andis not uniquely defined (for a given potential, it is possible to modify its high-energy component with aunitary transformation without affecting experimental predictions [82, 83]), we focus in the following onthe non-locality of the CCSD optical potential.We plot in Fig. 4, the n + , Ca potentials in several partial waves, at a fixed value of R = ( r + r (cid:48) ) / and as a function of r − r (cid:48) . We fix R to be equal to the charge radius in both nuclei, namely 3.48 fmand 3.46 fm for respectively Ca and Ca [5]. We consider the same energy as previously, namely 5.2MeV for Ca and 7.8 MeV for Ca. A fit of the potential using a Gaussian form factor, is also shownin Fig. 4. As one can see, the shape of potentials in Fig. 4 are well reproduced by the fit. For Ca, thevalues of the range β of the fitted Gaussian somehow varies slightly with the partial wave: we obtain β = 1 . , . , . fm for the f / , p / and p / component of the potentials, respectively. For Ca, β = 1 . , . , . fm for the f / , p / and and d / partial waves, respectively. We have observed evensmaller variations of the range with the energies although a more exhaustive study would be required to Frontiers . Rotureau Coupled-cluster computations of optical potential for medium-mass nuclei -40-30-20-100 f p p -6 -4 -2 0 2 4 6(r-r’ ) [fm]-40-30-20-100 R e [r r’ V l (r-r’)] [ M e V / f m ] f p d n+ Can+ Ca Figure 4.
Real part of the neutron potential in several partial waves for , Ca at respectively 5.2 and 7.8MeV. The potentials are shown at fixed values of R (equal to the charge radius in both nuclei) and as afunction of r − r (cid:48) . Symbols corresponds to the calculated potentials and the lines are the results of a fitwith a Gaussian form factor (see text for details).draw definitive conclusion about the dependence of β on the value of R and the energy. Nevertheless, in allcases, the non-local pattern of the optical potential display a Gaussian dependence, which correspondsto the choice made for the non-local form factor in the phenomenological potentials by Perey and Buck[84]. Note that due to the non-hermiticity of the Coupled Cluster Hamiltonian (see Sec. 2.2) the potential isslightly non symmetric in r and r (cid:48) . However since this effect is small [33, 34], it is hardly noticeable inFig. 4. In this section, we discuss some challenges and possible solutions for the development of fully predictiveab-initio optical potentials with the coupled-cluster method.We saw in the previous section that with the ab-initio optical potentials computed at the CCSD level, onecan arrive at an overall fair reproduction of data for medium-mass nuclei. However, the absorptive part ofthe potential was shown to be negligible at low energy. This lack of absorption was linked to neglectedconfigurations in the computed Green’s function.Currently, ab-initio computation of optical potentials for medium-mass nuclei using chiral
N N and3NFs, have only been performed with the coupled-cluster method and the Self Consistent Green’s Function(SCGF) method [31]. The SCGF is based on an iterative solution of the Dyson equation performed until aself-consistency between the input Green’s function and the result of the Dyson equation has been reached[9]. In [31], the authors compute neutron optical potential for O and Ca with the
NNLO sat interactionand include up to p − h configurations in the Green’s function. In that work, the minima in the elasticcross sections are well reproduced for both systems, and as in the CCSD computation of the potential, anoverall lack of absorption was observed and attributed to neglected configurations in the model space. Frontiers 11 . Rotureau
Coupled-cluster computations of optical potential for medium-mass nuclei
The natural next step to address the lack of absorption at the CCSD level would be to include higher-ordercorrelations in the Green’s function by considering next order excitations in the coupled-cluster calculations,namely triple corrections. One should expect in that case an increased level density in the A + 1 systemand as a result, a larger absorptive part of the optical potential. Coupled-cluster calculations with triplecorrections are routinely used for nuclear spectroscopy [46] and have recently been implemented in thecomputation of the dipole polarizability of Ca [85]. In that paper, the authors show that by including p − h excitations in the computation of the nuclear response function to an electromagnetic probe (theGreen’s function is a similar object since it is the response function to the addition/removal of a nucleon),the results improve over previous computations at CCSD.For most nuclei, and particularly for heavier systems, there are many compound-nucleus resonancesabove the particle threshold. Since these states consist of a high number of particle-hole excitations theycannot be reproduced accurately by ab-initio methods and are usually best described by a stochasticapproach [86]. In order to account for the formation of the compound nucleus and the resulting loss offlux in the elastic channel, one could add a polarization term to the ab-initio potential. A possible way tocompute this term would be to use Random Matrix Theory to generate an effective Hamiltonian belongingto a Gaussian Orthogonal Ensemble [87].Since the coupled-cluster Green’s function is computed in the laboratory frame, the optical potentialsolution of the Dyson equation is defined with respect to the origin of that frame O . As mentioned in Sec. 3,we have identified this potential with the potential in the relative n − A coordinate. For the medium-massnuclei considered here, this prescription creates a small error, which decreases with A [33, 34, 88]. Forlight systems, a correction to the optical potential becomes necessary to account for the identificationbetween laboratory and relative coordinates. It has been demonstrated that the coupled cluster wavefunctionfactorizes to a very good approximation into a product of an intrinsic wave function and a Gaussian inthe center-of-mass coordinate [89]. Since both the potential and the center-of-mass wavefunction of thetarget are computed in the laboratory frame, it seems reasonable to suggest that such a correction could beintroduced in the form of a folding of the potential with the center-of-mass wavefunction (nevertheless,such a prescription would have to be worked out and checked). Another possible way to introduce acorrection of the potential could be to use the integral method utilized in the GFMC approach ( see e.g [90]) for computation of overlap functions (see also e.g. [91, 92]). In this article, we have presented recent developments in the computation of nucleon-nucleus opticalpotential constructed by combining the Green’s function and the coupled-cluster method. A key element inthis approach is the use of the Berggren basis, which enables a consistent description of bound, resonantstates and scattering process of the (nucleon-target) system and at the same time, allows to properly dealwith the poles of the Green’s function on the real energy axis.We have shown results for optical potentials at negative and positive energy for the double magicsystems Ca and Ca using a chiral
N N and 3NFs that reproduces the binding energy and charge radiiin both systems. We pointed out that a proper reproduction of the distribution of nuclear matter, and,more specifically, nuclear radii, by the Hamiltonian, is essential to give an accurate account of reactionobservables. At the truncation level considered here, namely p − h and p − h / h − p in thecomputation of the target and the Green’s function, respectively, an overall fair agreement with data wasobtained. Nevertheless, in that case, the optical potential at positive energy suffers from a lack of absorption, Frontiers . Rotureau Coupled-cluster computations of optical potential for medium-mass nuclei which stems from the neglect of higher-order configurations. In (near) future development, higher-orderexcitations in the coupled-cluster expansion will be included to address this issue.In the future, the Green’s function formalism and coupled-cluster method could be combined forapplications to other reaction channels such as transfer, capture, breakup and charge-exchange. Anotherpossible approach toward the ab-initio computation of transfer reactions with medium-mass nuclei is theGreen’s Function Transfer (GFT) method [93]. Using the optical potential and Green’s function computedwith the coupled-cluster method as input of the GFT equations, as well as phenomenological ingredients,a very good reproduction of data for populating the ground states in , Ca was obtained with thisapproach. Although the current implementations of the GFT method require phenomenological inputs,future extensions of the formalism should allow ab-initio computation of transfer reactions [93].
ACKNOWLEDGMENTS
The author would like to thank his collaborators P. Danielewicz, G. Hagen, G. R. Jansen, F. M. Nunes, andT. Papenbrock for their contributions to the studies presented in this work.
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