Cluster scattering in the non-local model

Cluster scattering in the non-local model N. K. Timofeyuk 1 * and M. Gómez-Ramos 2 1 Department of Physics, University of Surrey, Guildford, United Kingdom, 2 Departamento de FAMN, Universidad de Sevilla, Sevilla, Spain Scattering of weakly bound nuclei with a pronounced cluster structure is strongly affected by their breakup. Usually, this mechanism is accounted for in a threebody model with pairwise potentials. The interaction potentials between complex systems are non-local due to the existence of excitation channels and antisymmetrization. However, a common practice is to use local optical potentials in cluster scattering studies. To assess the validity of replacing nonlocal optical potentials by their local equivalents, we extend the local-equivalent continuum-discretized coupled-channel (LECDCC) approach proposed by us for deuteron scattering in [Phys. Rev. C98, 011601(R) (2018)] to the case of cluster scattering. We consider the case of 6 Li + 120 Sn at 27 and 60 MeV, and compare the angular distributions and reaction cross sections for elastic and breakup cross sections with those obtained in the standard continuum-discretized coupledchannel (CDCC) method with local equivalents of non-local potentials. We found that while elastic scattering is not significantly affected by non-locality, the breakup observables could be affected by up to 20% depending on kinematical conditions of their observation. KEYWORDS three-body model, elastic scattering, breakup, continuum-discretized coupled-channel method, non-localities in optical potential 1 Introduction The phenomenon of clustering has been known in nuclear physics for a long time. It is present in most light nuclei where two neutrons and two protons gain from gathering into an α-particle [1]. α-clustering also occurs in even–even N=Znuclei [2]. A special class of clustering, the halo nuclei, has been found around neutron and proton driplines [3]. More generally, clustering can occur in excited nuclear states close to the thresholds of particle emission channels. It influences cross sections of nuclear reactions involving them. Many such reactions are of importance for stellar nucleosynthesis [4], thus contributing much to shaping the world we live in. Scattering of weakly bound nuclei with a pronounced cluster structure is at the frontier of modern experimental nuclear research [5,6]. The information obtained from these studies is affected by the choice of theoretical models used to predict and/or analyze the measured observables. It has been known for a long time that theoretical models should include cluster breakup. To account for this important reaction mechanism, the scattering problem is often described in a three-body model with pairwise interactions, which are always chosen to be local. However, the general Feshbach theory states that interactions between complex systems are non-local [7]: they depend on the inter-nuclear separation both before and after collision. Non-locality comes both from removing the target excitation from consideration and from antisymmetrization between nucleons from both the target and projectile. It most naturally emerges in all ab initio calculations [8]. Therefore, an assessment OPEN ACCESS EDITED BY Valdir Guimaraes, University of São Paulo, Brazil REVIEWED BY Vladimir Zelevinsky, Michigan State University, United States Maria Colonna, Laboratori Nazionali del Sud (INFN), Italy Pierre Descouvemont, Université Libre de Bruxelles, Belgium *CORRESPONDENCE N. K. Timofeyuk, [email protected] RECEIVED 31 March 2023 ACCEPTED 25 May 2023 PUBLISHED 19 June 2023 CITATION Timofeyuk NK and Gómez-Ramos M (2023), Cluster scattering in the nonlocal model. Front. Phys. 11:1197726. doi: 10.3389/fphy.2023.1197726 COPYRIGHT © 2023 Timofeyuk and Gómez-Ramos. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Frontiers in Physics frontiersin.org01 TYPE Original Research PUBLISHED 19 June 2023 DOI 10.3389/fphy.2023.1197726 of validity of replacing non-local optical potentials by their local equivalents in cluster scattering calculations is a timely and vital task needed to correctly understand the uncertainties in observables due to optical potential choices. We first note that two-body scattering with non-local interactions is very well studied. Phenomenological non-local potentials are often considered in the Perey–Buck model [9], for which the localization procedure has been established [9,10]. Two-body scattering problems with non-local potentials of a general form can be treated easily by the R-matrix approach with aLagrangemesh[11]. Many other methods of solving the nonlocal Schrödinger equation exist, examples of which, as well as references to other works, could be found in [12–14]. The solutions of two-body non-local scattering problems can be used to model wave functions both in the entrance and exit channels of various nuclear reactionswhencalculatingtheircross sections. Examples of such calculations are reviewed in [15]for thecaseoftransferreactions. Three-body scattering problems with non-local potentials have been extensively studied for the case of three nucleons. They are usually solved either with the help of the Faddeev method [16,17]or by using hyperspherical harmonics expansions [18,19]. The Faddeev treatment of non-locality has also been extended to the case of deuteron scattering (d,d), transfer (d,p), and (p,d) reactions from heavy targets [20–22]. It was shown there that using localequivalent nucleon-target potentials does not reproduce exact nonlocal calculations. Moreover, including non-locality in nucleon–target potentials significantly improves the description of experimental differential cross sections of these reactions as compared to those obtained in studies with local-equivalent nucleon optical potentials. These findings, together with the rapid development of microscopic models for optical potentials, have motivated the present study. Modeling cluster scattering with non-local potentials requires a special theoretical technique. The Faddeev approach of [20–22] was applicable to (d,d)and(d,p) reactions because the two-body description of n−Aand p−Abound states for the carefully selected target Aworks well. When applied for cluster scattering, Faddeev approach’s requirement to achieve correct asymptotic conditions in all channels could be unachievable due to the complexity of the problem. In addition, there is no clarity in choosing two-body interaction potentials that would describe simultaneously bound and scattering states in the two-body subsystems. The normalization to unity of the two-body bound states in the Faddeev approach may not correspond to reality. For these reasons, we use the idea of [23], where the continuum-discretized coupled-channel (CDCC) approach [24, 25] was extended to provide an approximate three-body treatment of optical potentials’non-locality in the n+p+A problem. Following [23], we derive a leading-order localequivalent continuum-discretized coupled-channel (LECDCC) model for the case of scattering of nuclei with a well-pronounced two-cluster structure. We then compare the LECDCC predictions to those obtained in standard CDCC with localequivalent potentials. We perform calculations for a popular choice of the projectile, 6 Li, which is well represented by the α−dcluster structure. We start by formulating a three-body model for cluster scattering in Section 2, followed by developing LECDCC at the leading order in Section 3.InSection 4,weapplytheformalismto the case of 6 Li + 120 Sn scattering using non-local Sao Paulo potentials [26–28] comparing LECDCC with traditional CDCC, in which local-equivalent potentials are used. Section 5 summarizes the work undertaken and draws conclusion. Appendix provides additional mathematical details relevant to formalism development. 2 Three-body model for cluster scattering Let us consider scattering of the nucleus Ccomposed by two clusters C 1 and C 2 from the target A.Weassumethatthis problem can be well-described by the three-body wave function Ψ(r,R), which depends on coordinates rrC1−rC2 and RC1rC1+C2rC1 C1+C2−rA,showninFigure 1. We also assume that the interaction between any pair cof nuclei, chosen from C 1 ,C 2 ,andA, is described by a two-body Schrödinger equation: Tc+Vcoul crc () −Ec  ϕcrc () −drc ′Vcrc ′,rc  ϕcrc ′  (1) with non-local nuclear potential Vc(rc ′,rc)and the local Coulomb interaction Vcoul c(rc). The three-body Schrödinger equation in this case is written as follows (see Appendix for derivation): Tr+TR+Vcoul C1AR+γ1r  +Vcoul C2AR−γ2r  +Vcoul C1C2r ()−E  Ψr,R () −dr′VC1C2r′,r  Ψr′,R  −dR′J1VC1AR′+γ1r+γ1μC C1 R′−R  ,R+γ1r  Ψr+μC C1 R′−R  ,R′  +J2VC2AR′−γ2r+γ2μC C2 R′−R  ,R−γ2r  Ψr−μC C2 R′−R  ,R′  , (2) where T r and T R are the kinetic energy operators associated with the coordinate rand R, respectively, E=E 3 −E c.m. ,whereE 3 is the three-body energy in the laboratory system, while γ 1 =C 2 / (C 1 +C 2 ), γ 2 =C 1 /(C 1 +C 2 ), and μ c =CA/(C+A). The two Jacobians are J1C C1 A+C1 A+C  3 ,J2C C2 A+C2 A+C  3 .(3) We will assume for simplicity that the clusters and targets have zero spin. The three-body Schrödinger Eq. 2is solved within the CDCC framework using the expansion Ψr,R ()  nmax i1 χiR () ϕir () ,(4) where ϕ i is the C 1 −C 2 bound state wave function or a continuum bin, obtained in the non-local C 1 −C 2 model and χ i is the wave function of the relative motion of the bin iwith respect to the target A. It satisfies the equation TR+UC−Ei () χiR () − i′dR′Ui′iR′,R  χi′R′  ,(5) where U C is the local C−ACoulomb potential and Frontiers in Physics frontiersin.org02 Timofeyuk and Gómez-Ramos 10.3389/fphy.2023.1197726 Ui′iR′,R  drϕi′r ()J1VC1AR′+γ1r+γ1μC C1 R′−R  ,R+γ1r  ϕir+μC C1 R′−R  +J2VC2AR′−γ2r+γ2μC C2 R′−R  ,R−γ2r  ϕir−μC C2 R′−R  . (6) In this paper, we will consider phenomenological non-local potentials in each pair cchoosing them in the Perey–Buck form [9], Vr,r′  Hix () UiX () ,(7) where x=r−r′and X=(r+r′)/2, and Hx ()π−3/2β−3exp −x2 β2  ,(8) where βis the range of non-locality. Usually, the non-locality range βfor nucleon scattering is around 0.85–0.9 fm; for deuterons, it is ~ 0.54 fm, and for αparticles, it is ~ 0.25 fm. This is because in most cases, the clusters C 1 and C 2 are not the same, and different non-locality ranges are required in each pair c. Moreover, non-local potentials themselves could be represented by several terms of the Perey–Buck type with different non-locality ranges, such as in the non-local dispersive optical model (NLDOM) in [29]. Therefore, we consider pairwise potentials given by Vcrc ′,rc  j Hjxc () Uc () jXc () ,(9) where the index cis associated with a chosen pair of nuclei (clusters) and jis associated with a non-locality range β j . When dealing with potentials (9), we use the experience of handling NLDOM potentials in constructing the local-equivalent adiabatic model for (d,p) reactions from Ref. [30]. 3 Leading-order local-equivalent CDCC Our aim is to find an equivalent local potential model that gives the same scattering phase shifts as the non-local model (18) does. We will follow Refs. [23,31] to perform this. We will exploit the fact that the non-locality ranges are small. To simplify the R.H.S. of Eq. 6,we use the Perey–Buck presentation (7) of the non-local potentials VC1A and VC2A. After introducing a new variable, associated with cluster i, sC Ci A+Ci A+CR′−R  . (10) Equation 5can be rewritten as TR+UCR () −Ei () χiR () −jdsdrHjs () ×ϕi′r () UC1A () jR+γ1r+1 2s  ϕir+A A+C1 s  χi′α1s+R () +ϕi′r () UC2A () jR−γ2r+1 2s  ϕir−A A+C2 s  χi′α2s+R (), (11) where αiCi C A+C A+Ci . (12) In previous d+Astudies [30,31], the following step was the introduction of a new variable xr−2A A+2(R′−R)and neglecting s/(A+2)intheargumentsofU nA and U pA potentials. However, a similar strategy would result in the appearance of 1 2[(C2−C1)/C1· A/(A+C)−C/(A+C)]sin the arguments of these potentials. In the case of C 2 ≠C 1 , this term could be important, especially for scattering of one-nucleon halo nuclei such as 11 Be or 15 C. Therefore, a different strategy is adopted here. In the term associated with cluster C 1 , a new variable yγ1r+1 2sis introduced. In the term associated with cluster C 2 , the integration variable ris replaced by −r, followed by the introduction of yγ2r+1 2s.Itgives TR+UCR () −Ei () χiR () − i′cj πc () ii′dsdyHjs () ×ϕi′γ−1 cy−1 2γc s  Uc () jy+R  ϕiγ−1 cy−βcs  χi′αcs+R () , (13) where πc () ii′1 for cC1 − () li+li′for cC2,(14) βc1 2γc−A A+Cc . (15) Because of the short range of H j (s), the wave function χi′(αs+R)can be represented by the leading-order expansion that retains only spherical components in s(see [31]): χi′αcs+R () ≈ nmax n0 γnαcs () 2nTn Rχi′R () ,(16) in which γn− () n n!2n+1 () ‼ μ Z2  n ,(17) where μis the reduced mass of A+C(not to be confused with dimensionless reduced mass number μ C of the previous section). Then, the coupled system of equations takes the form TR+UCR () −Ei () χiR () − i′ nmax n0cj πc () ii′dyUc () jy+R  ×γnα2n cdss2nHjs () ϕiγ−1 cy−1 2γc s  ϕi′γ−1 cy−βcs  ×Tn Rχi′R () − i′ nmax n0 γnUn () ii′R () Tn Rχi′R () ,(18) where Un () ii′R () cj πc () ii′α2n cdyUc () jy+R  ×dss2nHjs () ϕiγ−1 cy−1 2γc s  ϕi′γ−1 cy−βcs  . (19) Equation 18 could be rewritten in the following form: TRχiR () − i′ Ei′−UCR ()() δii′− n γnUn () ii′R () Tn R ⎡⎣⎤⎦χi′R () . (20) Then, imposing the requirement that the local-equivalent coupling potentials Uloc ii′satisfy Frontiers in Physics frontiersin.org03 Timofeyuk and Gómez-Ramos 10.3389/fphy.2023.1197726 TR+UCR () −Ei () χiR () − i′ Uloc ii′R () χi′R () ,(21) we obtain a system of the transcendental matrix equations f0 () ii′−Ei−UC () δii′+ k f1 () ik +δik  Xki′+ kl f2 () ik XklXli′+/0, (22) for Xii′Ei′−UC () δii′−Uloc ii′,(23) in which f(n) ij γnU(n) ii′. The system of Eq. 22 could be solved using the Newton method, as was solved in [23], and then, Uloc ii′could be read into a CDCC reaction code such as FRESCO [32]. The continuum bins can have different values of orbital momentum. The partial wave decomposition of Uloc ii′for general orbital momentum for cluster scattering is presented in Appendix. 4 Application to 6 Li scattering We apply the aforementioned formalism to 6 Li+ 120 Sn scattering at two 6 Li incident energies, 27 MeV and 60 MeV, near and above the Coulomb barrier, respectively. 6 Li has a well-developed α−dcluster structure due to its small deuteron separation energy of 1.47 MeV. Since a non-local potential for α-dscattering is not available, to describe the continuum states of the 6 Li projectile, we used a local model with the same potential as in [33]togenerateα-dcontinuum bins. This model describes the α-ds-wave phase shifts and reproduces the experimental excitation energy and width of the 3 + resonance (E* = 2.186 MeV) of 6 Li. The 6 Li+ 120 Sn study around the Coulomb barrier in [33] has shown that this low-energy resonance in the d-wave continuum plays an important role. Therefore, the CDCC expansion considered a basis with l αd = 0 and 2. For the d-wave component, the continuum wave functions were weighted by the T-matrix in order to better describe the resonance [32]. The calculations have been performed, ignoring the spin of the deuteron. To calculate the couplings potentials U ii′ (R), we used six bins for each α-dpartial wave, which was deemed sufficient. The maximum α-denergy E αd was 30 MeV for the calculation at 60 MeV since the contribution of the last bin to the breakup cross section was very small. As for the calculation at 27 MeV, the maximum value of only 6 MeV was possible due to instability of calculations with higher E αd . Variations of the maximum of E αd lead to differences in the elastic cross section of less than 4% in the whole angular range. Convergence of the cross sections was reached by varying the matching radius R m and maximum total angular momentum J. The optimal value R m = 150 fm was obtained for both 6 Li incident energies, while the maximum total angular momentum Jwas 300 and 100 for 6 Li energies of 60 and 27 MeV, respectively. For α- 120 Sn and d- 120 Sn optical potentials, we used the Sao Paulo potential (SPP) [26–28]. This potential is given by Eqs 7,8with the non-locality parameter β=4Z/(μ C c), chosen the same as in [28]. The SPP uses for U i (X) the double-folding model both for real and imaginary parts, assuming that they are proportional. The two-point Fermi distribution [28] was chosen for model density of 120 Sn. The deuteron density was taken from the systematics found in [28], while that for the α-particle was a sum of Gaussians taken from [34]. For the imaginary part, we used the standard renormalization coefficient 0.78. The local-equivalent potential of SPP (LESPP) is obtained by solving the transcendental equation [28] VLE R, E () VFR () e−μβ2 2Z2E−VCR () −VLE R,E ()() . (24) The solution is energy-dependent. All three-body models employing optical potentials use them only at a chosen fixed energy equal to a certain fraction of the projectile energy E.For example, in d−Ascattering, this would be half the incoming deuteron energy for nucleon–target potential. For cluster scattering, the fixed energy of the C i −Apotential is proportional to its mass, EiCi C1+C2E.Wecomparethe standard CDCC calculations with LESPP to those obtained in LECDCC developed in this work. We made sure that for β→0, they coinside with standard local folding potential calculations when breakup is neglected. Figure 2 shows selected coupling potentials calculated in standard CDCC with LESPP and in LECDCC. The ULESPP gs−gs (R) and ULECDCC gs−gs (R)couplings, shown in Figures 2A, B for 6 Li incident energies of 27 and 60 MeV, respectively, are very similar in the surface region, around R≈8–12 fm. At R= 0 fm, their real parts differ by 4% at 27 MeV and by 19% at 60 MeV, but imaginary parts of ULECDCC gs−gs (0)are more than twice smaller than those of ULESPP gs−gs (0). A similar situation occurs for diagonal couplings ULECDCC ex−ex (R)between d-wave bin states with the first 2 + resonance (see Figure 2C). We have only plotted it for the 6 Li energy of 60 MeV because the differences there are expected to be higher. Non-diagonal couplings between these states, ULESPP ex−gs (R)and ULECDCC ex−gs (R), are also very similar at the surface. However, the difference in the internal region is more significant. Despite a significant difference in the coupling potentials U ii′ (R) within the short range of the optical potentials, the elastic scattering angular distributions calculated in LECDCC turned out to be very similar to those obtained in standard CDCC with LESPP. Therefore, we only show their ratio in Figure 3.Foran incident energy of 27 MeV, the difference between the two approaches does not exceed 4% in the whole angular range. At 60 MeV, numerical noise has not allowed us to unambiguously determine this ratio at angles larger than 90°.Below90 °,the difference between LECDCC and LESPP does not exceed 8%. We also observe some oscillations in the LECDCC/LESPP ratio, resulting from tiny oscillations in both these differential cross sections, which are slightly out of phase. These oscillations are FIGURE 1 Three-body C 1 +C 2 +Asystem associated with scattering of a two-cluster nucleus C=C 1 +C 2 from the target A. Frontiers in Physics frontiersin.org04 Timofeyuk and Gómez-Ramos 10.3389/fphy.2023.1197726 caused by a small near-far-side interference, which results in a small Fraunhofer diffraction pattern. The breakup cross sections are more strongly influenced by non-locality. The differential s-wave, d-wave, and total breakup cross sections are shown in Figure 4. One can see that these cross sections are dominated by the d-wave breakup and that treating non-locality at the leading order in CDCC increases these cross sections. At an incident energy of 27 MeV, this increase is most noticeable at the main peak around 40°,whileat60MeV,the increase occurs at most angles. The LECDCC/LESPP ratio for angle-integrated breakup cross sections is shown in Figure 5 as a function of bin energy. One can see that the non-locality effect can reach 20% in the breakup cross sections for both incident energies for higher energy bins. Since this effect is bin-dependent, it is possible that the non-locality effect can strongly depend on kinematic conditions of breakup observation. Finally, we found that the total breakup cross sections σ bu and reaction cross sections σ reac are less affected by non-local effects. These cross sections are collected in Table 1, showing about 0.1% difference in σ reac between LESPP and LECDCC calculations. For σ bu ,thisdifferenceis3%and5%for27and 60 MeV, respectively. The non-local effects in 6 Li elastic scattering turned to be much smaller than those associated with deuteron scattering —the simplest nucleus with a cluster structure. We have identified the reason why this happens, which is the Coulomb barrier (for smaller angles) and strong absorption that make the reaction mechanism less sensitive to the interactions in the nuclear interior. Since both LESPP and FIGURE 2 λ= 0 LECDCC diagonal (A–C) and non-diagonal (D) coupling potentials for 120 Sn+ 6 Li at an incident energy of 27 MeV (A) and 60 MeV (B–D). The excited state corresponds to the bin state with the first 2 + resonance. FIGURE 3 Ratio of the differential cross section for 6 Li + 120 Sn elastic scattering calculated in LECDCC to standard CDCC calculations with a local SPP potential at 27 and 60 MeV. TABLE 1 d+αbreakup and absorption cross sections for the 6 Li + 120 Sn reaction calculated in CDCC with the local-equivalent SPP (LESPP) and in LECDCC with the non-local SPP potential. σ bu (mb) 60 MeV σ reac (mb) 60 MeV σ bu (mb) 27 MeV σ reac (mb) 27 MeV LESPP 79.9 2,436 65.1 1,123 LECDCC 84.3 2,422 67.1 1,125 Frontiers in Physics frontiersin.org05 Timofeyuk and Gómez-Ramos 10.3389/fphy.2023.1197726 LECDCC potentials are very similar at the surface, the cross sections should be similar as well. We repeated both calculations reducing the imaginary part by 50% for an 6 Li incident energy of 60 MeV. The difference between LESPP and LECDCC has increased to 20% at 60° and become an order of magnitude greater after 100°. However, the elastic cross sections in this angular range are small. The difference in breakup cross sections exceeded 20% after E αd =5MeV.Theangular distribution for the breakup cross section also presents larger differences with a reduced absorption, of ~10%–20% for small angles and larger than 50% for angles larger than 60°.The difference in reaction cross sections remains small. 5 Summary and conclusion To assess the validity of a standard approach which replaces nonlocal optical potentials by their localequivalentsinclusterscattering calculations, we developed a leading-order method to include the nonlocality of pairwise optical potentials into a three-body Hamiltonian describing cluster scattering. This resulted in a coupled system of transcendental equations for local-equivalent coupling potentials that could be fed into standard CDCC equations. The application to 6 Li + 120 Sn scattering at the incident energy of 27 and 60 MeV has shown that elastic scattering is not much affected by non-locality, while breakup cross sections in some kinematical regions could increase the differential cross sections up to 20%. Both elastic and breakup observables are determined by scattering phase shifts in all CDCC channels, which determine asymptotics of the channel wave functions. The LECDCC has been constructed to provide phase-equivalent solutions to non-local problems. At small distances, the LECDCC local CDCC channel functions should differ more strongly from each other. Therefore, the cross sections of other reactions that are more sensitive to internal parts of the three-body wave function than the projectile breakup, such as transfer reactions, could be more affected by the non-locality. Using non-local optical potentials to describe various nuclear reactions induced by nuclei with the pronounced cluster structure requires further development of techniques to solve the three-body Schrödinger equation. For such reactions, beyond-leading-order effectscouldbeveryimportant.Itis FIGURE 4 Angular distribution for breakup using LECDCC and standard CDCC calculations with the local-equivalent SPP potential (LESPP). (A) Angular distribution for breakup in LECDCC (black curves) and standard CDCC calculations with local SPP potential (red curves) for 6 Li + 120 Sn scattering at 27 MeV (circles). Dashed, dot-dashed, and solid curves correspond to d-wave, s-wave, and total breakup, respectively. (B) Same for the 6 Li incident energy of 60 MeV. FIGURE 5 Ratio of the breakup cross section calculated in LECDCC to standard CDCC calculations with local-equivalent SPP potential (LESPP) for each bin. (A) Ratio of breakup cross sections for 6 Li + 120 Sn scattering at 27 MeV (circles). The ratios for s-andd-wave breakup are shown by triangles and squares, respectively. (B) Same for the 6 Li incident energy of 60 MeV. The s-wave ratio at the last point is equal to three. Frontiers in Physics frontiersin.org06 Timofeyuk and Gómez-Ramos 10.3389/fphy.2023.1197726 well-known that in the region of strong nuclear interaction, a two-body scattering wave function, obtained in a non-local model, is smaller than that obtained in the leading-order local-equivalent model by a quantity called the Perey factor [35]. The Perey factor also appears in the three-body model with velocity-dependent potentials [36]. However, incorporating the Perey effect into the LECDCC scheme is not easy. Therefore, it would make sense to try solving the non-local CDCC exactly. Theobstacleonthispathistheabsenceofformalismfornonlocal CDCC couplings. The easiest way to develop such formalism is to extend the non-local adiabatic model proposed for (d,p)in[37], but it would be applicable only to non-localities of the Perey–Buck type. Optical potentials coming from ab initio calculations have a more complicated structure, and including them into the CDCC approach requires a major effort, which will become an important task for future research. Finally, this paper considered only the processes in which the target Awas not excited. Many experiments with weakly bound projectiles involve nuclei, both the target and projectile, with lowlying excited states that play an important role in the reaction mechanism. It has just been demonstrated that using non-local optical potentials with core excitation achieves a considerably more successful description of the experimental data on (d,p) reactions as compared to previous studies with local potentials [38]. When nuclear excitations are explicitly included in coupled-channel calculations, the imaginary part that describes absorption is usually reduced. One can expect that non-local effects could be stronger in coupled-channel calculations with reduced absorption. The LECDCC introduced here could be readily extended to include core excitations to study these effects. Data availability statement The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author. Author contributions NT developed the reaction formalism presented in this work and wrote the article. MG-R developed the codes required for its implementation, run the calculations that produced the results shown, and contributed to writing. All authors contributed to the article and approved the submitted version. Funding This work was supported by the United Kingdom Science and Technology Facilities Council (STFC) under Grant No. ST/V001108/ 1 and by MCIN/AEI/10.13039/501100011033 (also known as Agencia Estatal de Investigacion) under Project PID2020-114687GB-I00; by MCIN/AEI/10.13039/501100011033 and “European Union Next- Generation EU/PRTR”under Grant: IJC2020-043878-I; and by Consejeria de Economia, Conocimiento, Empresas y Universidad, Junta de Andalucia, and “ERDF-A Way of Making Europe”under Project P20_01247. Conflict of interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Publisher’s note All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. 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Phys Lett B (2023) 840:137867. doi:10.1016/j. physletb.2023.137867 Frontiers in Physics frontiersin.org08 Timofeyuk and Gómez-Ramos 10.3389/fphy.2023.1197726 Appendix Three-body Schrödinger equation in laboratory and centre-of-mass frames In a fixed frame of reference where experimental cross-sections are measured, the laboratory frame, the Schrödinger equation for the C 1 +C 2 +Awave function Ψlab(rC1,rC2,rA)of the C 1 +C 2 +Asystem with non-local C 1 −A,C 2 −Aand C 1 −C 2 interactions is written as TC1+TC2+TA−E3  Ψlab rC1,rC2,rA  −drC1 ′drC2 ′drA ′ ×δrC2−C1rC1+ArA A+C1−rC2 ′+C1rC1 ′+ArA ′ A+C1  VC1ArC1 ′−rA ′,rC1−rA  +VC2ArC2 ′−rA ′,rC2−rA  δrC1−C2rC2+ArA A+C2−rC1 ′+C1rC2 ′+ArA ′ A+C2  +VC1C2rC1 ′−rC2 ′,rC1−rC2  δrA−C1rC1+C2rC2 C1+C2−rA ′+C1rC1 ′+C2rC2 ′ C1+C2  ×δC1rC1+C2rC2+ArA A+C1+C2−C1rC1 ′+C2rC2 ′+ArA ′ A+C1+C2  Ψlab rC1 ′,rC2 ′,rA ′  , (25) where E 3 is the three-body energy in the laboratory system. Separating the centre-of-mass motion, Ψlab rC1,rC2,rA  ϕcm C1rC1+C2rC2+ArA A+C1+C2  ΨrC1−rC2,rA−C1rC1+C2rC2 C1+C2  , (26) then introducing variables rrC1−rC2, RC1rC1+C2rC1 C1+C2−rA, Rcm C1rC1+C2rC2+ArA A+C1+C2 (27) and integrating over r′and Rcm ′in the r.h.s of Eq. (25) we get Eq. (2). Partial wave decomposition of coupling potentials U(n) ii′(R) In the case of non-zero orbital momenta of continuum bins we use, assuming zero spins for clusters C 1 and C 2 , we have ϕiay−bs   4π √ λ1λ2 ϕli λ1λ2ay, bs  Yλ1^ y ×Yλ2^ s () limi(28) Then for spherical optical potentials Un () ii′R () 4πcj πc () ii′α2n c λλ1λ2λ1 ′− () λYλμ ^ R ∞ 0 dy y2Uc () j,λy, R  ×∞ 0 ds s2+2nHjs () ϕli λ1λ2γ−1 cy, 1 2γc s  ϕli ′ λ1 ′λ2γ−1 cy, βcs  × μ1μ2μ λ1μ1λ2μ2|limi  λ1 ′μ1 ′λ2μ2|li ′mi ′  λ1−μ1λ1 ′μ1 ′|λμ  λ10λ1 ′0|λ0  ^ λ1^ λ1 ′  4π √^ λ− () μ1  4π √ λμ λμlimi|li ′mi ′  Yλμ ^ R c πc () ii′α2n c∞ 0 dy y2j Uc () j,λy, R  fn,λ j () lili ′y, R  ⎡⎢⎢⎣⎤⎥⎥⎦, (29) where fn,λ j () lili ′y, R   λ1λ2λ1 ′− () λ1 ′^ λ1^ λ^ liλ10λ1 ′0|λ0  Wλλ1li ′λ2;λ1 ′li  ×∞ 0 ds s2+2nHjs () ϕli λ1λ2γ−1 cy, 1 2γc s  ϕli ′ λ1 ′λ2γ−1 cy, βcs  . (30) Finally, the radial part of the partial wave decomposition of the continuum bin is ϕl λ1λ2x, y   Λlxly− () λ2 2l+1 () ! 2lx () !2ly  ! lx0Λ0|λ10 () ly0Λ0|λ20  Wλ1Λlly;lxλ2  xlxylyϕl () Λx, y (31) where l x +l y =land ϕl () Λx, y  2Λ+1 21 −1 dμPΛμ x2−2μxy +y2  −l/2ϕl x2−2μxy +y2   (32) where ϕ l (r) is the radial part of the continuum bin in the partial wave l. The integration variable s,chosenin(29), uses the shortrange nature of H(s), which makes accurate evaluation of (30) easy. Then the integral over dy is similar to a standard CDCC matrix element for which accurate numerical methods are in place. Frontiers in Physics frontiersin.org09 Timofeyuk and Gómez-Ramos 10.3389/fphy.2023.1197726