Many-body effects in (p,pN) reactions within a unified approach
R. Crespo, A. Arriaga, R.B. Wiringa, E. Cravo, A. Mecca, A. Deltuva
aa r X i v : . [ nu c l - t h ] J un Many-body e ff ects in (p,pN) reactions within a unified approach R. Crespo a,b, ∗ , A. Arriaga c , R.B. Wiringa d , E. Cravo e,c , A. Mecca a,b , A. Deltuva f a Departamento de F´ısica, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001, Lisboa, Portugal b Centro de Ciˆencias e Tecnologias Nucleares, Universidade de Lisboa, Estrada Nacional 10, 2695-066 Bobadela, Portugal c Departamento de F´ısica, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal d Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA e Centro de F´ısica Te´orica e Computacional, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal f Institute of Theoretical Physics and Astronomy, Vilnius University, Saul˙etekio al. 3, LT-10257 Vilnius, Lithuania
Abstract
We study knockout reactions with proton probes within a theoretical framework where ab initio
Quantum MonteCarlo wave functions are combined with the Faddeev / Alt-Grassberger-Sandhas few-body reaction formalism. NewQuantum Monte Carlo wave functions are used to describe C, yielding, for the first time, results consistent with theexperimental point rms radii, electron scattering data and (p,2p) total cross sections data. Our results for A ≤
12 and( N − Z ) ≤ ab initio and Mean Field Approximation theoreticalcross sections, R σ , (ii) corresponding ratios between the spectroscopic factors, R Σ , summed over states below particleemission, depend moderately on the nucleon separation energy S N . These ratios are determined by a delicate interplaybetween the radii of the parent and the residual nuclei and the nucleon separation energy, and were found to be alwayssmaller for the knockout of the more correlated deficient species nucleon. In the case of the symmetric C nucleus,the theoretical ratios still appear to indicate that protons are more correlated than neutrons.
Keywords:
Many-body ab initio structure, few-body reaction, (p,pN) reactions, nucleon removal reactions, reduction factors.
1. Introduction
The mean field approach (MFA) to particle systemshas played an important role in atomic physics for de-scribing the periodic table of elements and in nuclearphysics for explaining many properties of nuclei, suchas the origin of the magic numbers leading to additionalstability.Nevertheless, one of the goals of Nuclear Physics isto describe simultaneously, and along the nuclear land-scape, nuclear binding, structure, electromagnetic andweak transitions, as well as reactions with electroweakand nuclear probes based on a microscopic descriptionof the interaction between individual nucleons.A formidable theoretical e ff ort has been performedin developing many body and cluster approaches to de-scribe nuclei and their application in the study of re-actions [1, 2, 3, 4, 5, 6, 7] . Strong deviations be-tween these models and the MFA indicate the presence ∗ Corresponding author
Email address: [email protected] (R. Crespo) of non-trivial many-body e ff ects, being interpreted asdue to nuclear correlations. Many body ab initio cal-culations of nuclear structure have demonstrated theneed to go beyond the simplified MFA and to considermodels with explicit nucleon-nucleon (NN) and three-nucleon (NNN) interactions and NN and NNN corre-lations [1, 2], in particular neutron-proton correlationsentirely absent in the MFA [8, 9].In parallel, for more than 30 years an extensive ex-perimental program, in particular nucleon knockout re-actions with electron and nuclear probes, has been de-voted to the study of the failure of the MFA [10, 11, 12,13, 14, 15, 16, 17, 18, 19, 20, 21, 22].In most calculations, the extracted information de-pends more or less strongly on the uncertainties ofthe reaction formalism and its associated interactions.Moreover, the variety of models, methods and energyregimes makes di ffi cult to extract a consistent explana-tion of the MFA inadequacy to describe nuclear struc-ture and to value the importance and nature of the cor-relations.The interpretation of the single nucleon knockout Preprint submitted to Physics Letters B June 26, 2019 rom an A-body nucleus has been relying on the one-nucleon spectroscopic overlap. This is defined as the in-ner product of the A parent nucleus wave function (WF)and the fully antisymmetrized A − ff erent nucleon angular momentum channels, ℓ j , satisfying the appropriate triangular relations [23].The spectroscopic factor (SF) for a given transition isgiven by the integral of the overlap function in each an-gular momentum channel.The analysis of earlier (e,e’p) knockout experimentshas been used to provide information on the one-nu-cleon spectroscopic overlaps at low momentum and forlow-lying energy states of the residual nucleus. The ex-perimentally extracted SFs were found to be reducedwith respect to the MFA ones [10, 11].The nucleon knockout for composite projectiles andtarget nuclei (called one-nucleon removal in the liter-ature) has been also analyzed extensively [16, 17] andrefs therein. The ratio between the inclusive experimen-tal and the MFA theoretical cross sections, R S , has beenfound to be smaller than unity and to have a strong de-pendence on the asymmetry parameter ∆ S , a measureof the asymmetry of the neutron and proton binding.This has been interpreted as additional correlations instrongly asymmetric (N-Z) systems.Concurrently, (p,pN) reactions on Oxygen, Carbonand Nitrogen isotopes with − ≤ ( N − Z ) ≤ He) on , , O [22] and of (p,d) on , Ar [21] have revealedsmall and nearly constant R S as a function of ∆ S .Theoretical calculations of one-nucleon spec-troscopic overlaps for asymmetric parent nuclei, , , , , O showed that SFs calculated with a micro-scopic coupled cluster model are quenched relativelyto the MFA ones, the quenching being particularly im-portant for the knockout of deficient species nucleonsin strongly asymmetric nuclei [7]. On the other hand,a small dependence of this quenching on the nucleonbinding was found [3].Conflicting results did follow from this vast theoret-ical and experimental work. A consistent analysis ofavailable experimental data, for all open reaction chan-nels as well as di ff erent probes, with state-of-the arttheory is lacking and of utmost importance for the un-derstanding of nuclear structure along the nuclear land-scape. Furthermore, it is essential to meet the challengesof new experimental developments and multiphysics re-search [25].In this letter our goal is to contribute to a unified the-oretical approach built on ab initio WFs, which can be used as a common input to transfer and nucleon knock-out reactions with electron and proton probes. We aimto shed light on the failure of the MFA to describe thistype of reactions and to provide an understanding of (i)the ratios R S , (ii) the ratios between the ab initio andMFA theoretical cross sections, R σ , (iii) their relationwith the corresponding ratios between the spectroscopicfactors, R Σ , and (iv) their behaviour as a function of theseparation energy of the knocked out nucleon. Our anal-ysis of (p,pN) reactions with light nuclei will contributeto the construction of a unified interpretation of nucleonknockout reactions along the nuclear landscape, includ-ing the (p,pN) experimental data collected at the R3B-LAND setup at GSI [18, 19, 26].To achieve the above goal, we use Quantum MonteCarlo (QMC) methods to solve the many body Schr¨o-dinger equation, which are state-of-the-art techniquesused in various subfields of physics, such as molecu-lar, atomic and nuclear physics. In particular, in the lat-ter, ab initio QMC calculations [1] have been used tointerpret sucessfully transfer reactions [20] and (e,e’p)experimental data [11]. We also aim to test, for thefirst time, the ability of QMC calculations in describing(p,pN) reactions.
2. Formalism
The main assumption of our model is that the knock-out / breakup operator does not act on the internal struc-ture of the residual nucleus, making the spectroscopicoverlap between the parent and residual nuclei the keynuclear structure input. These overlaps are calculatedfrom QMC many-body wave functions and then incor-porated in the state-of-the-art Faddeev / Alt-Grassberger-Sandhas (F / AGS) to solve the resulting three-body scat-tering problem [27, 28]. Our assumption is supported byrecent work [29], where it has been shown that dynami-cal core excitation e ff ects in (p,pN) reactions are small,to a good approximation validating the factorization ofthe cross section into the single-particle cross section,defined below, and the corresponding SF. This enablesdirect spectroscopic information from the comparisonbetween experimental and theoretical cross sections.The F / AGS allows a consistent and simultaneoustreatment of all open channels, providing an exact solu-tion of the three body scattering problem for an assumedthree-body Hamiltonian. This formalism includes allmultiple scattering terms, contrary to other scatteringframeworks that rely on assumed exact cancellations be-tween multiple scattering terms [30]. It has been usedrecently in several exploratory studies of (p,pN) reac-2ions [19, 30, 31, 32] and it is able to model the experi-mental transverse momentum distributions [30].We use the F / AGS in a nonrelativistic form since con-sistent treatment of relativistic kinematics and dynamicsin Refs. [33, 34] indicates only a small relativistic e ff ectfor the total three-body breakup cross section, less than10 % in our energy regime of interest.The reaction formalism requires three pair interac-tions. We take the realistic NN AV18 for the proton-nucleon pair. For the the N-C and p-C pair interac-tions we consider the Koning-Delaroche (KD) opticalparametrization [35] used in preliminary calculations[32] and the Cooper [36] for C, a global parametriza-tion developed for medium-heavy nuclei and in partic-ular for A =
12 that reproduce the elastic scattering data.From comparison with other parametrizations providedin [30] we estimate the uncertainty on the cross sectionsassociated with optical parametrizations of 15 %.In our approach, the spectroscopic overlaps are cal-culated from the QMC many-body WFs generated us-ing the NN Argonne V18 and the NNN Urbana X(AV18 + UX) potentials [2]. We consider variationalMonte Carlo (VMC) overlaps for p- and n-knockoutfrom Li, Be and C nuclei. For the nuclear struc-ture input we use new improved VMC WFs for C with[444] and [4431] spatial symmetries ( B with [443] and[4421]) as specified in Young diagram notation [37].Preliminary results with the Norfolk local chiral poten-tial NV2 + + UX. The GFMC SFsfor the Li parent and residual Li overlaps agree fairlywell with the VMC ones, supporting the use of the VMCoverlaps for the study of these reactions. We take theVMC and Green’s function Monte Carlo (GFMC) over-laps for the Li parent nucleus from Ref. [23], which areable to describe the (e,e’p) reaction [11].We have performed a convenient parametrization ofthe QMC overlaps using the procedure described inRef. [23], which incorporates the adequate asymptoticbehavior. With a correct tail by construction, these over-laps can be validated indirectly by comparing the exper-imental values of the point proton rms radii ( r p ) with theones obtained from the QMC WFs from which the over-laps are calculated. In Table 1, we present the VMC andthe experimental values for r p [40, 41] for all the studiednuclei, which exhibit a quite good agreement.We also consider an MFA where only the (A-1) + Nconfiguration is present in the parent nucleus statespace. The overlaps are then obtained as solutions ofthe one-body Schr¨odinger equation with an e ff ective av-erage interaction. We use a Woods-Saxon potential withstandard radius and depth adjusted to the separation en- ergy of the removed nucleon and no antisymmetrizationis considered. As discussed below, from the overlaps,the SFs are the crucial quantities in the cross section,and for this reason we take the SFs calculated fromthe more sophisticated e ff ective interaction of Cohen-Kurath (CK) with the well known center of mass (c.m.)correction, given by A / ( A − M ,(QMC and MFA) are denoted here as Z i ( M ), where i identifies the energy and the angular momentum of theresidual nucleus, as well as the nucleon angular momen-tum channels, with the sum Σ ( M ) = Σ i Z i ( M ).The theoretical inclusive cross section σ th ( M ) is ob-tained as the weighted sum σ th ( M ) = P i Z i ( M ) σ isp ( M )where the single-particle cross sections σ isp ( M ) arecomputed using the overlaps normalized to unity. -4 -3 -2 -1 k (fm -1 ) N ( k ) (f m ) ρ p ( C)- ρ p ( B)] = 4 Expt(err), MFA, VMC B( J - ) +p(p j ) SF = 2.96(28), 3.98, 3.33 B( / )+p(p ) SF = 2.43(28), 2.85, 2.36 B( / )+p(p ) SF = 0.29(03), 0.75, 0.87 B( / *)+p(p ) SF = 0.24(03), 0.38, 0.11 Figure 1: VMC overlaps in momentum space N ( k ) calculated for thelow-lying states of B. Also shown in black is the di ff erence of Cand B proton momentum distributions multiplied by 4 (the totalnumber of protons in a p-shell in the Independent Particle Model).
3. Results
We start by evaluating the one-nucleon spectroscopicoverlaps for the parent nucleus C, for which there are(p,2p) data [26]. The overlaps in momentum space arerepresented in Fig. 1, along with the di ff erence betweenthe VMC C and B proton momentum distributions.This di ff erence exhibits a significant high-momentumtail, where about 15% of the protons have momentaabove 1.4 fm − , unaccountable in any MFA. The resultfor C, shown here for the first time, is consistent withhigh momentum electron scattering analysis [12], sup-porting the VMC WF from which both the momentumdistributions and the spectroscopic overlaps are gener-ated and, therefore, corroborating our nuclear structuremodel. Interesting to say that the dominant source of3his high momentum tail is the NN tensor force, com-ing from the one-pion-exchange potential, with a furthersignificant contribution from the NNN force with itstwo-pion-exchange terms. The experimental SFs, Z i exp ,deduced from C (p,2p) at 400 MeV / u, are calculateddividing the experimental cross section of Ref. [26] bythe σ isp (VMC). These are shown in the inset of Fig. 1along with the theoretical VMC and MFA ones. Thetheoretical sums Σ ( M ) = P i Z i ( M ) and the experimen-tal one for all final states of the residual nucleus arealso shown. It is fair to say that, when compared tothe MFA ones, the VMC SFs lead to a significant im-provement. In addition, their sum agrees reasonablywell with the deduced experimental sum Σ exp (p,2p) = .
96 and moderately with those extracted from elec-tron scattering Σ exp (e,e’p) = . .
00) [44] andtransfer Σ exp (d, He) = . .
00) [14]. We also notethat Σ (MFA) is very close to the sum of particles inthe shell (before c.m. correction), the well known sumrule. We have obtained the total theoretical cross sec-tion σ th (QMC) = .
66 mb, close to the experimentalvalue of σ exp = . . ab initio VMCWFs combined with the Faddeev / AGS reaction formal-ism predict cross sections for (p,2p) from C that agreefairly well with the experimental data. In other words,this result shows the ability of QMC WFS to describe(p,pN) reactions, within a remnant uncertainy due to op-tical parametrizations and relativity.
Table 1: Radii, nucleon separation energies and SFs for the groundstates of the parent A X and residual nucleus A − Y. A X A − Y J π S N r p r n r m r p Z i VMC / Z i MFA (MeV) (fm) (fm) (fm) (fm) SFexp VMC VMC VMC exp g.s. Li 3 / + Li 1 + He 0 + Li 3 / − Li 2 + He 0 + Be 0 + Be 3 / + Li 3 / + C 0 + C 3 / − B 3 / − The calculated QMC and MFA SFs, together withtheir sums Σ and the ratios R Σ (including c.m. correc-tion) are shown in Table 2. Also shown are the results with the partial sum over the final states of the residualnucleus below its breakup threshold (called here belowparticle threshold, BPT). Table 2: Total and BPT sums of SFs, Σ , and ratios R Σ . The MFA ⋆ includes c.m. correction factors. A X A − Y Σ ( M ) R Σ MFA VMC VMC / MFA ⋆ Li Li BPT 1 .
016 0 . . .
999 1 . . He BPT 0 .
592 0 . . .
997 0 . . Li Li BPT 1 .
313 1 . . .
859 3 . . He BPT 0 .
847 0 . . .
000 0 . . Be Be BPT 2 .
356 2 . . .
990 3 . . Li BPT 1 .
990 1 . . .
990 1 . . C C BPT 3 .
603 3 . . .
980 3 . . B BPT 3 .
980 3 . . .
980 3 . . The ratios R Σ range from 0.6 to 0.8 being consistentwith Ref. [8]. This reduction is due to the fact that theMFA considers only the (A − + N partition for theparent nucleus wave function, therefore setting to unitythe probability of finding this configuration inside thenucleus. In contrast, the QMC overlaps are calculatedfrom fully microscopic WFs for parent and residual nu-clei, both normalized to unity. This means that manyother partitions are present in the parent nucleus WFleading to a probability associated with (A − + N con-figuration smaller than unity. Consequently, the MFASFs are necessarily larger than the QMC and the ex-perimental ones. This conclusion is independent of theinteraction models. The ratios calculated with partialsums over the final states of the residual nuclei, R Σ BPT,shown in the upper panel of Fig. 2, di ff er significantlyfrom those calculated with total sums, ranging from 0.5to 1. These results follow naturally from the fact that thespectroscopic strength is distributed among the statesdi ff erently in the VMC and MFA formalisms. The R Σ BPT exhibit a moderate dependance on S N , for the con-sidered small asymmetry ( N − Z ) ≤
3. Nevertheless, theratio R Σ is always smaller for the knockout of the morecorrelated deficient species nucleon, the proton in thesecases, in accordance with previous findings [7, 16].On the other hand, we have found that the overlaps,and consequently the SFs, are determined by a deli-cate interplay between the radii of the parent (A) and4he residual (A −
1) nuclei and the separation energy ofthe knockout nucleon. To further explore this interplay,we present in Table I, for the ground states of parentand residual nuclei, the separation energy S N , the pointproton and neutron rms radii, the matter radii, given by r m = q ( Zr + Nr ) / A, and the corresponding ratios ofVMC / MFA SFs for the ground state as well. It is evi-dent that there is no clear dependence of these ratios onthe separation energy, for instance they do not decreasenecessarily with increasing S N , reflecting the fact thatthe SFs do not probe exclusively the tail of the overlaps.In fact, these ratios are also determined by the proximitybetween the matter radii of parent and daughter nuclei,which tends to enhance the overlap. Being this inter-play dealt di ff erently in VMC and MFA calculations, itis clear the subsequence dependance of R Σ on it. R Σ S N (MeV) R σ , R s R σ , R s Li+n He+p Li+n He+p C+n B+p Li+p Be+n Li+n Li+n Be+n He+p Li+p Li+n Be+n Li+n He+p He+p B+p C+n Li+p
Removal(p,pN) B+p C+n
Figure 2: Theoretical ratios (QMC / MFA) of sums SFs (upper panel),(p,pN) (middle pannel) and removal (lower panel) cross sections re-stricted to final states below particle threshold (BPT) of the residualnucleus and quenching factors, R S (symbols filled with squares), asa function of the nucleon separation energy. The theoretical singleparticle removal cross sections are taken from Refs. [15, 17]. We also calculate the ratios between the QMC andMFA theoretical cross sections, but where the sum is re-stricted to states of the residual nucleus BPT. Since weare considering low-lying states of the residual nucleusand we found only weak dependence of the σ isp on S N [31], we expect these ratios to be nearly independent ofthe choice of the global parametrization and MFA de-tails. For the case of C we verified a model indepen-dence of these ratios since similar results are obtainedwith di ff erent parameterizations of the optical potential[36, 35] and of the MFA prescriptions [19, 42].The ratio BPT R σ = σ th (QMC) /σ th (MFA), that quantifies the importance of nuclear correlations in thedescription of the QFS reactions, is represented in themiddle panel of Fig. 2 and ranges from 0.6 to 1. Wehave found that the microscopic treatment of the over-laps has its biggest e ff ect on the evaluation of the the-oretical cross section through the SFs. We, therefore,expect R σ to be very close to R Σ , which is confirmedwhen comparing the middle and upper panels.The ratios R σ exhibit the same moderate dependenceon S N as the R Σ , the quenching being somewhat moresignificant for the knockout of the deficient species nu-cleon. For the symmetric case of C, the theoreticalratios still appear to indicate that protons are more cor-related than neutrons. Moroever, the ratios R σ are de-termined by the same delicate interplay between theradii of the parent and the residual nuclei and the nu-cleon separation energy. This is, in fact, a strong indica-tion that the reaction mechanism does not probe directlythe tail of the overlaps between the parent and residualnucleus. In addition, by the same physical argumentsdrawn for the SFs, we expect the quenching factors R S to be smaller than unity, independently of the interactionmodels. The ratio R S for (p,2p) from C, also repre-sented, is very close to R σ reflecting the ability of VMCto describe the data.We represent in the lower panel of Fig. 2 the ra-tios R σ and R S for nucleon removal from Li, Li and Be, at 80-120 Mev / u, and C measured at 250,1050,2100 Mev / u. The sp cross sections were taken fromRefs. [15, 17] and weighted by the QMC and CK SFs.The R S are consistently smaller than R σ . For C, thedeviation from R σ (which coincide for the 3 energies)appear to be dependant of the energy of the projectilefor p- removal. Also, R σ and R S appear to be morequenched for proton than neutron knockout, yet withthe exceptions of Be for the former and C for thelater. We point out, however, that in transfer reactionsand in nucleon knockout reactions where both projec-tile and target are composite nuclei, the reaction mech-anisms are substantially di ff erent from those of (p,pN)reactions, which prevents the conclusion that the crosssection factorizes into SFs and single particle cross sec-tions [29], being the factorization used merely by con-venience [17]. Consequently, the clean link betweenthe R Σ and R σ that exists for (p,pN) knockout reactions,shown in our work for the first time, is not expected tohold for the removal analysis of the MSU experimentaldata [17] and for transfer reactions. Accordingly, the be-haviour of the quenching factors with respect to a givenphysical quantity is not directly related to the behaviorof the ratio of the sums of the SFs with respect to thesame quantity, and may have a very intricate interpreta-5ion. This conclusion is supported by the comparison ofthe lower panel and upper panels of Fig. 2: the locationof the points in both cases is very di ff erent.
4. Conclusion
In conclusion, in the present letter we analyze, for thefirst time, (p,pN) reactions for A ≤
12 and ( N − Z ) ≤ / AGS reaction theory. New QMC WFs used todescribe C and B yield results consistent with ex-perimental data of point rms radius, electron scatteringdata analysis at high momentum and (p,2p) total crosssections at 400 MeV / u.We show the inadequacy of the MFA to describe(p,pN) reactions due to contributions in the parent nu-cleus wave function of many-body partitions beyond the(A − + N, the only present in the MFA. This leads nec-essarily to an overestimation of the cross sections, in-dependently of the interaction models. Further, nuclearcorrelations are a key ingredient and, therefore, struc-ture many-body e ff ects must be taken into account. Weshow that the ratio between the partial sums of QMCand MFA QFS cross sections is very close to the ra-tio of the partial sum of SFs. Hence, one expects thequenching ratios to be determined by the same delicateinterplay between the radii of the parent and the residualnuclei and the nucleon separation energy, as the ratioof the SFs. Last, R Σ and R σ show a moderate S N de-pendence and are smaller for the knockout of the morecorrelated deficient species nucleon. In the case of thesymmetric C, the theoretical ratios still appear to indi-cate that protons are more correlated than neutrons.A consistent experimental program of transfer andknockout (with light and heavier targets) with protonand electron probes for A ≤
12 nuclei will be very use-ful to get further insight on the inadequacy of the MFApicture, on the structure of light nuclei.
Acknowledgments
R.C., E.C., A.A., A.M. A.D. and R.B.W. are sup-ported by Fundac¸˜ao para a Ciˆencia e Tecnologia of Por-tugal, Grant No. PTD / FIS-NUC / / ffi ce of NuclearPhysics, contract No. DE-AC02-06CH11357 and theNUCLEI SciDAC program; computing time was pro-vided by the Laboratory Computing Resource Center at Argonne National Laboratory. We thank B. Jon-son, P. Diaz-Fernandez and A. Heinz for reading themanuscript. ReferencesReferences [1] R. B. Wiringa, et al. , Phys. Rev. C 89, 024305 (2014).[2] J. Carlson et al. , Rev. Mod. Phys. 87, 1067 (2015).[3] C. Barbieri, W.H. Dicko ff , Int. Journal Mod. Phys A24, 2060(2009)[4] S. Quaglioni and P. Navratil, Phys. Rev. Lett. 101, 092501(2008).[5] N. K. Timofeyuk, Phys Rev. C 88, 044315 (2013).[6] F. Barranco et al. , Phys. Rev. Lett. 119, 082501 (2017).[7] O. Jensen et al. , Phys. Rev. Lett. 107, 032501 (2011).[8] V. R. Pandharipande, I. Sick, P.K.A. de Witt Huberts, Rev. Mod.Phys. 69, 981 (1997).[9] The CLAS Collaboration, Nature, 560, 617 (2018).[10] L. L´apikas, Nucl. Phys. A 553,297 (1993).[11] L. L´apikas, J.Wesseling, R. B. Wiringa, Phys. Rev. Lett. 82,4404 (2009).[12] D. Rohe et al. , Phys. Rev. Lett. 93, 182501 (2004).[13] R. Subedi et al. , Science 320, 1475 (2009).[14] G. J. Kramer, H. P. Block, L. Lapikas, Nucl. Phys A 679, 267(2001).[15] B.A. Brown, et al. , Phys. Rev. C 65, 061601(R) (2002).[16] J.A. Tostevin and A. Gade, Phys. Rev. C 90, 057602 (2014).[17] G. F. Grinyer et al. , Phys. Rev. C 86, 024315 (2012).[18] L. Atar, et al. , Phys. Rev. Lett. 120, 052501 (2018).[19] P. D´ıaz Fern´andez et al. , Phys. Rev. C 97, 024311 (2018).[20] K. M. Nollett and R. B. Wiringa, Phys. Rev. C 83, 0410001(R)(2011).[21] J. Lee et al. , Phys. Rev. Lett. 104, 112701 (2010).[22] F. Flavigny et al. , Phys. Rev. Lett. 110, 122503 (2013).[23] I. Brida, Steven C. Pieper and R. B. Wiringa, Phys. Rev. C 84,024319 (2011).[24] M. G´omez-Ramos and A.M. Moro, Phys. Lett B, 785, 511(2018).[25] P. F. Bortignon and R. A. Broglia, Eur. Phys J. A 52, 64 (2016).[26] V. Panin, et al. , Phys. Lett. B 753, 204 (2016).[27] L. D. Faddeev. Zh. Eksp. Teor. Fiz. 39, 1459 (1960).[28] E. O. Alt, P. Grassberger, and W. Sandhas, Nucl. Phys. B2 , 167(1967).[29] A. Deltuva, Phys Rev C 99, 024613 (2019).[30] R. Crespo, E. Cravo, A. Deltuva, Phys Rev C 19, 054622 (2019).[31] E. Cravo, R. Crespo, A. Deltuva, Phys. Rev. C 93, 054612(2016).[32] R. Crespo et al. , J. Phys: Conf Series, 966, 012056 (2018).[33] H. Witala, J. Golak, R. Skibinski, Phys. Lett. B , 374 (2006).[34] H. Witala et al. , Few-Body Systems 49, 1-4, pp 61-64 (2011).[35] A. J. Koning and J. P. Delaroche, Nucl. Phys, A713, 231 (2003).[36] E. D. Cooper et al. , Phys. Rev. C 47, 297 (1993).[37] R. B. Wiringa, Phys. Rev. C 73, 034317 (2006).[38] M. Piarulli, et al. , Phys. Rev. Lett. 120, 052503 (2018).[39] M. Baroni, et al. , Phys. Rev. C 98, 044003 (2018).[40] H. De Vries, C. W. De Jager, C. De Vries, At. Data and Nucl.Data Tables , 495 (1987).[41] Z.-T. Lu et al. , Rev. Mod. Phys. , 1383 (2013).[42] S. Cohen, D. Kurath, Nucl. Phys A101, 1 (1967); D. Kurath(private communication).[43] A. E. L. Dieperink and T. de Forest, Jr. Phys. Rev. C 10, 543(1974).
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