Comprehensive investigation of the symmetric space-star configuration in the nucleon-deuteron breakup
H. Witala, J. Golak, R. Skibinski, K. Topolnicki, E. Epelbaum, H. Krebs, P. Reinert
aa r X i v : . [ nu c l - t h ] M a r Comprehensive investigation of the symmetric space-starconfiguration in the nucleon-deuteron breakup
H. Wita la , ∗ J. Golak , R. Skibi´nski , K. Topolnicki ,E. Epelbaum , H. Krebs , and P. Reinert M. Smoluchowski Institute of Physics,Jagiellonian University, PL-30348 Krak´ow, Poland and Institut f¨ur Theoretische Physik II,Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany (Dated: March 3, 2021) bstract We examine a description of available cross section data for symmetric space star (SST) config-urations in the neutron-deuteron (nd) and proton-deuteron (pd) breakup reaction using numeri-cally exact solutions of the three-nucleon (3N) Faddeev equation based on two- and three-nucleon(semi)phenomenological and chiral forces. The predicted SST cross sections are very stable withrespect to the underlying dynamics for incoming nucleon laboratory energies below ≈
25 MeV. Wediscuss possible origins of the surprising discrepancies between theory and data found in low-energynd and pd SST breakup measurements.
PACS numbers: 21.30.-x, 21.45.-v, 24.10.-i, 24.70.+s
I. INTRODUCTION
Since the advent of numerically exact 3N continuum Faddeev calculations the elasticnucleon-deuteron (Nd) scattering and deuteron breakup reactions have become a powerfultool to test modern models of nuclear forces [1–3]. With the appearance of high precision(semi)phenomenological nucleon-nucleon (NN) potentials and first models of the 3N force(3NF) the question about the importance of the 3NF has become the main topic of studiesin the 3N system. That issue was given a new impetus from chiral perturbation theory(ChPT), which opened the possibility to employ consistent two- and many-body nuclearforces derived within this framework in 3N continuum calculations.First applications of (semi)phenomenological NN and 3N forces to elastic Nd scatteringand to the nucleon-induced deuteron breakup reactions revealed interesting cases of discrep-ancies between theoretical predictions based solely on two-nucleon (2N) potentials and data,indicating a possibility of large 3NF effects [4, 5]. The exclusive breakup reaction offers arich spectrum of kinematically complete geometries and the SST configuration from the verybeginning attracted attention as a possible candidate to reveal significant 3NF effects. Inthis kinematically complete geometry of Nd breakup the momenta of the three outgoingnucleons have the same magnitudes and they form a three-pointed ”Mercedes-Benz” starperpendicular to the beam direction in the 3N center-of-mass (c.m.) frame. Measurements ∗ [email protected]
2f the pd and nd breakup performed at low incoming nucleon energies in different labora-tories indeed revealed large discrepancies between predicted theoretical cross sections anddata for that geometry. The SST measurements for nd breakup have been performed at thefollowing energies: E = 10 .
25 MeV Bochum [6], E = 10 . E = 13 . E = 16 . E = 19 . E = 25 . E = 10 . E = 13 . E = 19 . E = 65 . π -exchange Tucson-Melbourne (TM99) [29] exhibited only moderate3NF effects [4, 5] at low energies.All analyses of the pd breakup performed by the Cracow-Bochum group have a permanentdrawback: they neglected the proton-proton (pp) long-range Coulomb interaction presentin the pd system and were based on nd calculations. The successful implementation of thelong-range proton-proton (pp) Coulomb force in the Faddeev formalism achieved in Ref. [30]permitted, for the first time, to perform exact calculations of pd breakup. It turned outthat the pp Coulomb interaction effects are practically negligible for the SST cross sections[30], which validated the results of analyses of that geometry in pd breakup based on ndcalculations.Recent progress in constructing nuclear forces within chiral effective field theory [31–33] resulted in several high precision NN potentials. Also, for the first time, a possibilityappeared of applying 2N and 3N forces derived consistently within the same formalism.Understanding of nuclear spectra and reactions based on these consistent chiral two- and3any-body forces has grown into a main topic of present day few-nucleon and many-nucleonstudies.In view of these new developments it is timely to check if it is possible to get new insightsabout the origin of the discrepancies between theory and data for the SST configuration byusing the newly developed chiral two- and three-body interactions.The paper is organized as follows: in Sec. II we present the available SST data togetherwith their description by (semi)phenomenological forces and additionally discuss sensitivityof the SST cross sections to particular NN force components. Results with chiral NN poten-tials alone are presented in Sec. III, while in Sec. IV the importance of 3NF is discussed. InSec. V we surmise on possible origins for the low-energy discrepancies between theoreticalpredictions and SST cross section data. We summarize and conclude in Sec. VI. II. RESULTS WITH (SEMI)PHENOMENOLOGICAL FORCES
Theoretical predictions which are shown in the present paper are obtained within the 3NFaddeev formalism using various 2N and 3N forces. The formalism itself and informationregarding our numerical performance were presented in numerous publications so for detailswe refer the reader to [1, 27, 34, 35].In Fig. 1 we show the available Nd SST cross section data and compare them to theorybased on the CD Bonn NN potential [25]. It is seen that at low laboratory energies E of theincoming nucleon, theoretical predictions clearly underestimate nd data by ≈ −
30% andsimultaneously overestimate pd data by ≈ E = 65 MeV theory describes the pd data.The predicted low-energy SST cross sections do not change when instead of the CD Bonnpotential another (semi)phenomenological interaction is used. The very narrow (red) darkshaded band at 13 MeV in Fig. 2, which comprises AV18 [24], CD Bonn [25], Nijm1 andNijm2 [26] predictions, reflects the astonishing stability of the low-energy SST cross sectionto the underlying dynamics. This stability is lost at 65 MeV as evidenced by broadening ofthe band showing the predictions with NN interactions only.The predicted low-energy SST cross sections are not only stable with respect to theunderlying NN (semi)phenomenological potentials. They are also insensitive to the standard2 π -exchange 3NF’s. In Fig. 2 we show also (cyan) light shaded bands containing predictions4ased on the AV18, CD Bonn, Nijm1 and Nijm2 interactions combined with the 2 π -exchangeTucson-Melbourne (TM99) 3NF [29]. The cut-off parameter Λ of that 3NF is adjusted foreach particular NN potential and TM99 3NF combination to reproduce the experimentaltriton binding energy [4]. In Fig. 2 the band of NN+3NF predictions contains also the crosssection for the combination of the AV18 potential and the Urbana IX 3NF [28]. At 13 MeV,effects of these 3NF’s are practically negligible, and the resulting band of predictions isvery narrow and overlaps with the band representing NN-only predictions. The astonishingstability with respect to the underlying dynamics present at low energies is lost at 65 MeV.Here, both bands broaden significantly and slightly move apart, indicating small effects ofthe 3NF. It is a region of energy where also in elastic Nd scattering 3NF effects start cominginto play [4, 36].The small effects of the 3NF and insensitivity to the underlying dynamics raise thequestion about the dominant NN force components, contributing to the SST cross section.It turns out that at low energies practically the whole input stems from the S and S NN force components only, with a dominating S contribution (see Fig. 3). These forcecomponents provide nearly the whole cross section at low energies as evidenced by nearlyoverlapping (red) solid (CD Bonn prediction) and (blue) dashed (CD Bonn restricted to S + S − D NN partial waves only) lines at 13 and 19 MeV in Fig. 3. With increasingenergy the contributions from the remaining partial waves become visible and at 65 MeVthey start to outweigh the S part.The low-energy dominance of the S and S contributions brings up the question towhat extent uncertainties of these NN force components could be responsible for the observeddiscrepancies between theory and data. To answer this question we investigated changesof the SST cross section caused by varying the strengths of the S and S NN forcecomponents. To this end we multiplied the corresponding potential matrix elements by afactor λ . It was shown in [37] that changes of the S interaction induced by the λ valuesin a vicinity of λ = 1 do not affect exclusive Nd elastic scattering observables and the totalcross sections significantly.In the first step, we investigated what changes of the S nn or pp interaction are requiredto get a proper description of the low-energy SST cross section data. It turned out that it wasnot possible to modify the S nn or pp potential in a way which would shift the cross sectionpredictions to the nd SST data. However, increasing the strength of S nn interaction by5 −
30% brings theoretical predictions close to the pd SST cross sections at low energies(see Fig. 4). Such a large increase of the strength ( λ = 1 .
21 or λ = 1 .
3) allows two neutronsto form a S bound state with the binding energy E λ =1 . b = − .
144 MeV or E λ =1 . b = − .
441 MeV. Existence of such a di-neutron state does not spoil the description of ndelastic scattering data [37] but it would have severe consequences for the H binding energy,increasing it from the CD Bonn value E CD Bonn H = − .
923 MeV to E λ =1 . H = − .
717 MeVor E λ =1 . H = − .
560 MeV. Admittedly, one could argue that an action of repulsive 3NF’scould provide again proper binding of that system. However, such a strong S force wouldalso spoil the description of nuclear structure. As can be seen in Fig. 4, such modificationsof the S nn force component would also lead to a significant overestimation of the pd SSTcross section data at E = 65 MeV but this could change upon including the 3NF tuned tothe H binding energy.In the case of the S component, a proper description of nd SST data would require areduction of its strength by ≈
5% ( λ = 0 .
95) (see Fig 5), what leads to a complete deterio-ration of the np data description and to a deuteron binding energy E λ =0 . d = − .
412 MeV,drastically different from the experimental value E expd = − . H would be bound by only E λ =0 . H = − .
338 MeV and a large effect of an attractive3NF would be required to regain the CD Bonn H binding. On the other hand, the SST pddata require only a 2% increase of S − D strength ( λ = 1 .
02) (see Fig. 5), what couldbe still tolerated by NN data. However, even such a small change of the strength wouldincrease the deuteron binding to E λ =1 . d = − .
592 MeV, in contradiction to the very preciseexperimental value. Also H would be bound stronger, with E λ =1 . H = − .
598 MeV.Summarizing, despite the fact that the SST cross sections are strongly dominated bythe S-wave NN force components, modifications of their strengths cannot serve to explaindifferences between theory and low-energy data for that configuration. Namely those NNforce components are very much restricted by available 2N and 3N data and their variationshave to be considered with great caution.
III. RESULTS WITH CHIRAL NN POTENTIALS
From the available chiral NN interactions we choose four of the most advanced potentialswhich provide a satisfactory description of NN data in a large energy range. One is an6lder set of Bochum forces [31, 32] developed up to fourth order (N LO) of chiral expan-sion. It reproduces experimental NN phase-shifts in a wide energy range with an almostcomparable accuracy as the high precision (semi)phenomenological NN potentials. We em-ploy five versions of that N LO chiral NN potential corresponding to different sets of cut-offparameters used to regularize the Lippmann-Schwinger equation and in spectral functionregularization, namely (450 , , , , , LO by the Bochum-Bonn [39, 40] and Idaho-Salamanca [41] groups. While inthe Idaho-Salamanca force, the nonlocal momentum-space regularization was applied witha cutoff parameter Λ, in the Bochum-Bonn potential the one-pion and two-pion exchangecontributions are regularized in coordinate space using the cutoff parameter R , and for thecontact interactions a simple Gaussian nonlocal momentum-space regulator with the cutoffΛ = 2 R − was used. The Idaho-Salamanca N LO force is available for three values of thecutoff parameter Λ = 450 , R = 0 . , . , . , . , and 1 . LO)of chiral expansion and augmented by an additional (the so-called N LO + ) version includ-ing some sixths-order terms (also the N LO Idaho-Salamanca potential is augmented bythe same Q contact terms). In this 2N chiral force, a new momentum-space regularizationscheme for the long-range contributions is employed, and a nonlocal Gaussian regulator forthe minimal set of independent contact interactions is introduced. These new features havealso been applied to the corresponding 3N forces at the N LO level [44]. This new familyof semilocal chiral 2N potentials provides an outstanding description of the NN data and isavailable up to N LO + for five values of the cutoff Λ = 350 , , , E = 13 and 65 MeV as bandscomprising the available range of cutoffs or regulator parameters for each of the four models.7t 13 MeV the bands are very narrow and practically overlap with each other. Similarlyto the (semi)phenomenological NN potentials, the chiral interactions also provide predic-tions for the SST cross sections at low energies that are very stable with respect to thetype of the underlying interaction and its parameters as far as they provide a satisfactorydescription of the NN data. The predictions of the chiral potentials agree with those of(semi)phenomenological forces, leading to the same disagreement with the low-energy SSTcross section data.As can be seen in Fig. 6 these bands are broadened at 65 MeV, especially for the olderBochum-Bonn potential (the (red) dark shaded band in Fig. 6), reflecting the increaseddependence of the predictions on potential parameters at that energy as well as the worsedescription of NN data by the older Bochum-Bonn potential. The bands of SCS and SMSinteractions are significantly constricted, in line with a good representation of the NN phase-shifts by these potentials.The applied chiral potentials differ not only in their regularization scheme. The olderBochum-Bonn potential leads to the deuteron wave function, which is quite different fromthe ones obtained with other chiral potentials [45]. Despite these differences the calculatedlow-energy SST cross sections are practically the same. IV. RESULTS WITH CHIRAL 3N-FORCES
First nonvanishing 3NF contributions appear at N LO [46, 47] and contain, in addition tothe 2 π -exchange term, two short-range contributions with the strength parameters c D and c E [48]. The latter two can be determined from the H binding energy and the Nd differentialcross section minimum at about E lab = 70 MeV, which is the energy at which effects of 3NFstart to appear in the Nd elastic scattering cross section [4, 36, 44, 49]. Specifically, first theso-called ( c D , c E ) correlation line is established, which for a particular chiral NN potentialcombined with a N LO 3NF yields values of ( c D , c E ) reproducing the H binding energy.Then, a fit to the experimental data for the elastic Nd cross section is performed and thevalues of both strengths, c D and c E , are uniquely determined.In Fig. 7a, b, and c we show predictions for the SST cross section at E = 13 MeV forthe SCS chiral potential with the regularization parameter R = 0 . LO, N LO,and N LO, respectively, combined with the N LO 3NF for four sets of the strength param-8ters taken from the corresponding correlation lines. Also, predictions of particular chiralpotentials are shown by (red) dashed line. All lines practically overlap showing that effectsof N LO 3NF on 13 MeV SST cross section are negligible. The predicted cross section isinsensitive to the order of the chiral NN potential used.Effects of the N LO 3NF start to appear at 65 MeV (see Fig. 7d). This is the energyregion, where effects of 3NF’s start to come into play also in elastic Nd scattering [4, 36].The overlapping predictions for four sets of strengths combinations from the correlation line( c D , c E ) shown in Fig. 7d indicates, that the magnitude of 3NF effects in this energy rangedoes not depend on strength values as far as they are taken from the correlation line.To investigate further how effects of the N LO 3NF depend on the strengths of thecontact terms we took the most precise chiral SMS potential at N LO + with the regulatorΛ = 450 MeV and combined it with the N LO 3NF [48]. In Fig. 8 we show predictions forSST cross sections at E = 13 and 65 MeV for eight combinations of strengths taken from thecorrelation line ( c D , c E ). Again, in spite of a very wide range of c D values, taken between c D = −
20 and c D = +20, the predicted 13 MeV cross sections are found to lie within arelatively narrow band. Contrary to that, at 65 MeV a very broad range of predictions isseen confirming the observation that in this energy region 3NF effects become important.One may now raise the question of the role of 3NF components from higher chiral ordersand their impact on the SST cross section. The necessary work to derive the chiral 3NFsat N LO has been done in [50, 51] using dimensional regularization. At that order, fivedifferent topologies contribute to the 3NF. Three of them are of long-range character [50]and are given by two-pion (2 π ) exchange graphs, by two-pion-one-pion (2 π − π ) exchangegraphs, and by the so-called ring diagrams. They are supplemented by the short-range one-pion-exchange-contact (1 π -contact) and two-pion-exchange-contact (2 π -contact) terms [51].The 3NF at N LO order does not involve any new unknown low-energy constants (LECs),see, however, a related discussion in [52], and depends only on two parameters, c D and c E ,that parameterize the leading one-pion-contact term and the 3N contact term appearingalready at N LO. Their values need to be fixed at a given order from a fit to few-nucleondata, as in the N LO case.In the first preliminary investigation of N LO 3NF effects [45] we considered the actionof the 3NF only in 3N states with the total 3N angular momenta J = 1 / / π -exchange-contact term was omitted in the short-range part of 3NF. The strengthsparameters c E and c D were determined at that time from the correlation line and the nddoublet scattering length a nd was used in addition to the H binding energy to uniquelydetermine both values.In Fig. 9a we show the SST cross sections at 13 MeV in form of a red band comprisingfive predictions of the N LO Bochum-Bonn potentials (versions 201-205). Combining thesepotentials with the N LO chiral 3NF gives the blue band. For the sake of comparisonalso the CD Bonn prediction is shown by the (orange) solid line. In b) the correspondingpredictions at N LO are also presented. It again turns out that the cross section for theSST configuration of the nd breakup is very stable with respect to the underlying dynamics.Not only (semi)phenomenological potentials, alone or combined with standard 3N forces,provide practically the same SST cross sections. Also the chiral 2N forces supplemented bythe N LO 3NF without relativistic 1/m corrections and short-range 2 π -contact term yieldsimilar predictions and cannot explain the discrepancy between the theory and the datafound for the low-energy SST configurations. Notice further that a consistent regularizationof the 3NF beyond N LO has not yet been achieved, see [53].Due to the restriction to the low total 3N angular momenta this result has to be confirmed.A systematic investigation of effects of the 3NF beyond N LO is the main aim of the LENPICcollaboration [49].At fifth order (N LO) the chiral 3NF comprises thirteen purely short-range operators[54] in addition to the long- and intermediate-range interactions generated by pion-exchangediagrams [55, 56]. In an exploratory study of [57], effects of these subleading short-rangeterms were investigated in pd scattering below E lab = 3 MeV within a hybrid approachbased on phenomenological two- and three-nucleon forces.To get insight into the expected 3NF effects on SST cross sections from these N LO short-range 3NF contributions we choose two out of the ten terms, namely the isoscalar central andspin-orbit interactions coming with strengths c E and c E , respectively [48, 54] and add themto the N LO 3NF. In Fig. 10 we show predictions for the SST cross section at E = 13 and65 MeV for the chiral SMS N LO + potential with regularization parameter Λ = 450 MeVcombined with that 3NF for a set of eight combinations of strengths from the correlationlines ( c D , c E , c E , c E ) for fixed values of c E and c E . For five of these combinations alsothe lines of predictions are drawn. It is seen that the inclusion of the c E - and c E -terms10as a negligible effect on the 13 MeV SST cross section and the predicted cross sectionsessentially coincide with each other and with the results of SCS N LO and SMS N LO + potentials alone. Again, at 65 MeV, a wide spread of predictions indicates significant effectsof the 3NF at this energy.The observed discrepancy between theoretical predictions and the nd and pd low-energySST cross section data thus indeed appears to be puzzling. Due to the observed strongstability of the low-energy space-star cross sections to the underlying dynamics it seemsvery unlikely that this puzzle can be resolved by the inclusion of omitted N LO terms orremaining N LO and higher order contributions to the 3NF. We checked for a combinationof SMS N LO + NN and N LO 3NF that even removing the requirement to reproduce Hbinding energy does not help to come into the vicinity of the 13 MeV nd SST cross sectiondata. Namely taking strengths c D = 2 . c E = 0 .
287 from the correlation line andincreasing c E to c E = 1 .
433 lowers the predicted cross sections and brings them close topd SST data (see (indigo) long dashed line in Fig. 10). For the combination of strengths c D = 2 . , c E = 1 . H is strongly bound with E H = − .
783 MeV. However, decreasingthe c E value and even changing its sign does not have any significant effect on the predictedcross section, which remains close to the stable region of predictions for strength values fromthe correlation line (see Fig. 10 and the (red) dash-double-dotted line, which is predictionfor c D = 2 . c E = − . c D = 2 . , c E = − . H is bound with E H = − .
792 MeV.
V. DISCUSSION OF THE LOW-ENERGY DISCREPANCY
The presented results support the conjecture that 3NF effects are not responsible for thediscrepancies between data and theory in the low-energy SST cross sections. One couldargue that, perhaps, modifications of the S and/or S − D NN force components, morerefined than a simple change of their strengths, would provide an explanation for at least pdSST low-energy cross section data. However, this seems to be difficult since there is no roomfor modifications of np and pp forces compatible with NN data [42, 58, 59]. In spite of thefact that the pd SST discrepancy is relatively small ( ≈
10 %) in comparison to the nd SSTone, since the theoretical predictions lie well outside statistical error bars and the systematicerrors are claimed to be small [19], it presents a significant discrepancy. A dedicated pd SST11easurement aimed to determine precise normalization of the SST cross sections wouldhelp to put some light on this discrepancy. Even larger is the discreapancy to nd SSTdata and between pd and nd data themselves. However, due to the strong insensitivity oflow-energy SST cross sections to the underlying dynamics it seems rather unlikely, that anyconceivable charge symmetry breaking mechanism in the NN and/or 3N force would be ableto explain the difference between pd and nd SST data and allow to describe nd data. Thissituation poses an interesting puzzle for theory and its solution has to be probably soughtin some exotic mechanism contributing to the nd breakup and irrelevant for the pd one.When looking for such a mechanism one could consider the contributions of hypotheticalbound state of two neutrons in the state S to a region of SST breakup phase-space.Such contributions could create in nd SST measurements additional background originatingfrom accidental coincidences between breakup neutrons and di-neutrons produced in ndscattering, increasing thus the measured cross section. That such a scenario is conceivablefollows from the fact that detection of neutrons in nd SST measurements was performedusing liquid scintillators with pulse-shape discrimination and their energy was determinedby the time-of-flight technique [12]. Such a detection system does not distinguish betweendi-neutrons and breakup neutrons. As a result, accidental coincidences between breakupneutrons and di-neutrons could appear in the region of SST phase-space which cannot bedistinguished from true events by the applied measurement technique. The kinetic energyassigned to di-neutron when using such an experimental arrangement, which is determinedby time of flight measurement of its velocity, will be in consequence twice as small as its realkinetic energy. To be specific, assuming that di-neutron is bound by E di − nb = − .
144 MeV(what corresponds to the factor λ S = 1 .
21) would lead at incoming neutron lab. energy E = 13 MeV to the energy of outgoing di-neutrons from d ( n, di − neutron ) p reaction atlaboratory angle of the SST configuration θ = 50 . o E di − nlab = 3 .
12 MeV and its energydetected by the TOF system will be 1 .
56 MeV. In Fig. 11, positions of breakup events(S-curve) in the plane of kinetic energies E − E of two outgoing neutrons detected incoincidence are shown together with positions of di-neutron energies determined by thedetection system for SST configurations at E = 13 , , and 65 MeV. Also the range of S-curvecovered by data is indicated by two circles. Since at 13 MeV di-neutrons would come nearestto the SST region, the data at that energy would be influenced most by the assumptivebackground of accidental coincidences. The intensity of accidental coincidences depends on12he number of neutrons or di-neutrons arriving at detectors, which is determined by theenergy spectra of outgoing neutrons in incomplete d ( n, n ) np breakup and by laboratoryangular distribution of di-neutrons from the d ( n, di − neutron ) p reaction. The predictionsfor these quantities based on solutions of the 3N Faddeev equation with a modified (in the S partial wave) CD Bonn potential [37] at the above considered three energies are shownin Fig. 12. Since the cross section for di-neutrons production is comparable to the crosssection for production of neutrons in incomplete nd breakup, it indeed seems plausible thataccidental coincidences could impact the measured low-energy SST cross sections. Rapiddiminishing of di-neutrons production with energy (see Fig. 12) would also explain why thediscrepancy between SST data and theory decreases with energy. New measurements of thelow-energy nd SST cross sections as well as a measurement of nd SST at E = 65 MeV, usinga detection system able to distinguish between neutrons and hypothetical di-neutrons, likethe one proposed in [60], would be very welcome. VI. SUMMARY AND CONCLUSIONS
In this investigation we performed a comprehensive analysis of the available SST Ndbreakup cross section data using high precision (semi)phenomenological NN potentials aloneor combined with the standard 3N forces as well as selected chiral forces. Four different chiralNN potentials including the most precise SMS N LO + of Ref. [43] have been applied aloneor in combination with chiral 3NF’s at different orders of chiral expansion. The main resultsare summarized as follows.- The available nd SST data cover the range of incoming neutron lab. energies between E = 10 −
25 MeV while the pd data were measured for proton energies in a region E = 10 −
65 MeV. The experiments were performed by different groups using differentexperimental arrangements or techniques. When at a particular energy several pd ornd data sets are available, the data from different measurements are consistent witheach other.- Using (semi)phenomenological NN potentials alone or accompanied by the TM99 orUrbana IX 3NF one is not able to explain the low-energy SST pd and nd SST data.All theoretical predictions practically overlap in nd and pd systems with pd data over-13stimated by ≈
10 % and nd data underestimated by ≈ −
30 %. The discrepanciesbetween theory and data diminish with increasing energy of the incoming nucleon.At E = 65 MeV NN force predictions agree with the pd SST cross sections, whileinclusion of 3NF provides a slight overestimation of the data.- Predicted low-energy SST cross sections based on different chiral NN potentials are in-dependent from the type of regularization used or from the regularization parameters,and are practically identical to predictions of (semi)phenomenological interactions. At E = 65 MeV that independence starts to be lost.- Adding the considered chiral 3NF at different orders of chiral expansion has no sig-nificant influence on the SST cross sections at low energy. Even a broader range ofstrengths from the correlation line ( c D , c E ) for N LO 3NF contact terms yields prac-tically the same low-energy SST cross sections. Again, at E = 65 MeV, this stabilitywith respect to changes of strengths vanishes and 3NF effects come into play.- The low-energy SST cross sections originate practically from S and S − D NNforce components. Changes introduced by a simple multiplication of the correspondingmatrix elements by a factor λ could explain the SST pd data but not nd ones. However,the required changes of the S and/or S − D are excluded by the NN data and/orby the H binding energy.- In view of the astonishing stability of the low-energy SST cross sections to the underly-ing dynamics it seems very unlikely that charge symmetry breaking mechanism of anyconceivable kind in 2N or 3N forces could be able to explain the low-energy nd SSTcross sections. The explanation of the nd data should be thus sought in some exoticphenomena such as e.g. the hypothetical bound state of two neutrons. Supposableexistence of the di-neutron would provide additional background in the region of theSST breakup phase-space, which could not be discerned in measurements performedso far.Further investigations and theoretical as well as experimental efforts are required tosolve that low-energy SST puzzle. From the experimental side, measurements of nd SSTcross sections at low-energies with experimental arrangement able to discern supposabledi-neutron background would be needed. Also dedicated pd SST measurement directed14o determine precise normalization of SST cross sections would be welcome. From thetheoretical side, efforts to fully include in 3N continuum calculations consistently regularizedN LO and N LO 3NF components are required. This is the aim of the LENPIC project.
ACKNOWLEDGMENTS
This study has been performed within Low Energy Nuclear Physics International Collab-oration (LENPIC) project and was supported by the Polish National Science Center underGrant No. 2016/22/M/ST2/00173, by Deutsche Forschungsgemeinschaft (DFG) and Natu-ral Science Foundation of China (NSFC) through funds provided to the Sino-German CRC110 “Symmetries and the Emergence of Structures in QCD” (NSFC Grant No. 11621131001,DFG Project-Id 196253076-TRR 110) and by Bundesministerium f¨ur Bildung und Forschung(BMBF), Grant No. 05P18PCFP1. The numerical calculations were performed on the su-percomputer cluster of the JSC, J¨ulich, Germany. We would like to thank other membersof the LENPIC Collaboration for interesting discussions and A. Nogga and K. Hebeler forproviding us with matrix elements of N LO chiral 3NF’s. [1] W. Gl¨ockle, H. Wita la, D. H¨uber, H. Kamada, J. Golak, Phys. Rep. , 107 (1996).[2] A. Kievsky, M. Viviani, S. Rosati, Phys. Rev. C , R15 (1993).[3] A. Deltuva, K. Chmielewski, and P. U. Sauer, Phys. Rev. C , 034001 (2003).[4] H. Wita la, W. Gl¨ockle, J. Golak, A. Nogga, H. Kamada, R. Skibi´nski and J. Kuro´s- ˙Zo lnierczuk,Phys. Rev. C , 024007 (2001), and references therein.[5] J. Kuro´s- ˙Zo lnierczuk, H. Wita la, J. Golak, H. Kamada, A. Nogga, R. Skibi´nski, W. Gl¨ockle,Phys. Rev. C , 024003 (2002).[6] M. Stephan, K. Bodek, J. Krug, W. L¨ubcke, S. Obermanns, H. R¨uhl, M. Steinke, D. Kamke,H. Wita la, Th. Cornelius and W. Gl¨ockle, Phys. Rev. C , 2133 (1989).[7] K. Gebhardt et al., Nucl. Phys. A , 232 (1993).[8] R. Macri, Ph.D. thesis, Duke University, 2004.[9] J. Strate, K. Geissd¨orfer, R. Lin, J. Cub, E. Finckh, K. Gebhardt, S. Schindler, H. Wita la,W. Gl¨ockle and T. Cornelius, J. Phys. G: Nucl. Phys. , L229 (1988).
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58] R. Navarro P´erez, J.E. Amaro, and E. Ruiz Arriola, Phys. Rev. C , 064002 (2013), ErratumPhys. Rev. C , 029901 (2015).[59] P. Reinert, H. Krebs, E. Epelbaum, arXiv:2006.15360 [nucl-th].[60] K. Bodek, Few-Body Systems , 713 (2014). d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=10.5 MeV θ =θ =49.4 ο φ =120 ο d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο d(n,nn)p SSTd(n,nn)p SSTE=19 MeVE=65 MeV θ =θ =52 ο φ =120 ο θ =θ =54 ο φ =120 ο d(n,nn)p SSTE=16 MeV θ =θ =51.5 ο φ =120 ο d(n,nn)p SSTE=25 MeV θ =θ =52.8 ο φ =120 ο FIG. 1. (Color online) The nd breakup five-fold differential cross section in the SST completegeometry at the incoming neutron laboratory energies E = 10 . , , , ,
25, and 65 MeV, shownas a function of arc-length of the S-curve. The solid (red) lines are predictions of the CD BonnNN potential. At E = 10 . . .
25 MeV Bochum ([6]), and 10 . . E = 13 MeV the (red) squares,the (maroon) circles, and the (green) squares are 13 MeV TUNL ([12]), 13 MeV TUNL ([8]), and13 MeV Erlangen ([9]) nd data. The (maroon) circles are 13 MeV K¨oln ([17]) and (violet) diamonds13 MeV Fukuoka ([18]) pd data. At E = 16 and 25 MeV the (red) squares are TUNL ([13]) andCIAE ([15]) nd data. The (maroon) cirles at E = 19 and 65 MeV are K¨oln ([20]) and PSI ([21])pd data. d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο E=65 MeVd(n,nn)p SST θ =θ =54.0 ο φ =120 ο FIG. 2. (Color online) The same as in Fig. 1 for the incoming neutron laboratory energies E = 13and 65 MeV. The (red) dark shaded and (cyan) light shaded bands comprise predictions of theAV18, CD Bonn, Nijm1 and Nijm2 NN potentials alone or combined with TM99 and for AV18also with Urbana IX 3NF, respectively. At E = 13 MeV the (red) squares are TUNL ([12]) nddata and the (maroon) circles are K¨oln ([20]) pd data. The (maroon) circles at E = 65 MeV arePSI ([21]) pd data. d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο E=19 MeVE=65 MeV d(n,nn)p SSTd(n,nn)p SST θ =θ =52.1 ο φ =120 ο θ =θ =54.0 ο φ =120 ο FIG. 3. (Color online) Contributions to the SST cross section at E = 13 ,
19 and 65 MeV fromdifferent partial waves. The (red) solid line is prediction of the CD Bonn NN potential. The(black) dotted and (green) dash-dotted lines show cross sections obtained with 3N partial wavesrestricted to these which contain only S and S - D NN partial waves, respectively. The resultobtained with only S + S - D is given by (blue) dashed lines and prediction with all partialwaves excluding S + S - D is shown by (brown) dash-double-dotted lines. The (maroon) circlesat E = 19 MeV are pd K¨oln data ([20]). For description of the data points at 13 and 65 MeV seeFig. 2. d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=10.5 MeV θ =θ =49.4 ο φ =120 ο d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο d(n,nn)p SST d(n,nn)p SSTE=19 MeV E=65 MeV θ =θ =52 ο φ =120 ο θ =θ =54 ο φ =120 ο FIG. 4. (Color online) Sensitivity of the SST cross section at E = 10 . , , , and 65 MeV tochanges of the S nn force component. The (red) solid line is prediction of the CD Bonn potential.The (black) dotted curve is the corresponding cross section when the strength of the nn S forcecomponent is reduced by multiplying its matrix elements with a factor λ = 0 .
9. The (orange)dashed and (magenta) dash-dotted lines shows the cross section when that strength is increasedby multiplying with factor λ = 1 .
21 and λ = 1 .
3, respectively. For description of the data pointssee Fig. 1. d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο E=19 MeVE=65 MeV d(n,nn)p SSTd(n,nn)p SST θ =θ =52.1 ο φ =120 ο θ =θ =54.0 ο φ =120 ο FIG. 5. (Color online) Sensitivity of the SST cross section at E = 13 , , and 65 MeV to changesof the S − D np force component. The (red) solid line is prediction of the CD Bonn potential.The (magenta) dashed curve is the corresponding cross section when the strength of that forcecomponent is reduced by 5 % by multiplying its matrix elements with a factor λ = 0 .
95. The(black) dotted line shows the cross section when that strength is increased by 2 % by multiplyingmatrix elements with factor λ = 1 .
02. For description of the data points see Fig. 4. d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο E=65 MeVd(n,nn)p SST θ =θ =54.0 ο φ =120 ο FIG. 6. (Color online) The predicted SST cross sections by chiral NN potentials at incomingneutron laboratory energies E = 13 and 65 MeV. The (red) shaded band comprises five predictionsof N LO chiral potentials (201, 202, 203, 204, 205) of Ref. [32] and the (magenta) band threepredictions of N LO potentials of Ref. [41] with regulator parameters Λ = 450 , ,
550 MeV.The (cyan) band covers five predictions of the SCS N LO potential of Ref. [39] with regulatorparameters R = 0 . , . , . , . , . LO + potential of Ref. [43] with regulators Λ = 400 , , ,
550 MeV. For description of the data pointssee Fig. 2. .00.51.01.5 0 5 10 0.00.51.01.50 5 10S [MeV]0.00.51.01.5 d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=65 MeV θ =θ =54.0 ο φ =120 ο d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο a) b)c) d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο d) FIG. 7. (Color online) Effects of N LO 3NF on SST cross section at E = 13 and 65 MeV whencombined with chiral SCS NN potential at different orders of chiral expansion. The dashed (red)lines are predictions of the SCS NN potential with the regulator R = 0 . LO (a), N LO(b), and N LO (c) and (d). Combining that potential with the N LO 3NF with four strengthsof the contact terms from the correlation lines ( c D , c E ) leads to results shown by different curves:solid (blue) (a: ( − . , . . , . . , − . − . , . . , − . . , − . . , − . . , − . . , − . . , . . , − . . , − . d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο E=65 MeVd(n,nn)p SST θ =θ =54.0 ο φ =120 ο FIG. 8. (Color online) SST cross sections at E = 13 and 65 MeV predicted by SMS chiral N LO + NN potential (the regulator parameter Λ = 450 MeV) alone (shown by dashed (red) lines) orcombined with consistently regularized N LO 3NF with different strengths of the contact termstaken from the correlation line ( c D , c E ) and shown by lines: dotted (blue) (2 . , . . , . . , . − . , − . . , . − . , . − . , − . d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο a) b) FIG. 9. (Color online) Effects of N LO 3NF, restricted to 3N total angular momenta J = 1 / /
2, with all long-range contributions with exception of 1 /m corrections and omitted 2 π -exchange-contact term in the short-range part on SST cross section at E = 13 MeV (a). The red band in(a) comprises predictions of five chiral N LO Bochum-Bonn potentials (201-205). Combining themwith N LO 3NF with strengths of the contact terms determined by the correlation line ( c D , c E )and requirement to reproduce a nd leads to the blue band. The (orange) solid line is predictionof the CD Bonn potential. In (b) the same is shown for chiral Bochum-Bonn potentials and 3NFat N LO. In this case the N LO 3NF acted up to J = 7 /
2. For description of the data points seeFig. 2. d σ / d S d Ω d Ω [ m b M e V - s r - ] d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο d(n,nn)p SSTE=65 MeV θ =θ =54 ο φ =120 ο a) b) FIG. 10. (Color online) SST cross sections at E = 13 and 65 MeV predicted by the chiral SMSN LO + NN potential with the regulator parameter Λ = 450 MeV alone (the long dashed (yellow)lines) or combined with 3NF comprising N LO and two out of ten N LO contact terms, withstrengths of the contact terms from the correlation line ( c D , c E , c E , c E ). The results with 3NF areshown by the following lines: dotted (orange) (2 . , . , . , . . , − . , − . , . − . , − . , . , . . , . , . , − . − . , − . , . , . . , . , − . , . . , . , . , − . . , . , . , . LO potential withthe regulator parameter R = 0 . c D , c E ) outside the correlation line, namely (indigo) long dashed:(2 . , . . , − . [MeV]012345 E [ M e V ] [MeV] 0480 5 10 15 20 25 30E [MeV]051015202530 E [ M e V ] d(n,nn)p SSTE=13 MeV θ =θ =50.5 ο φ =120 ο E=65 MeV d(n,nn)p SSTd(n,nn)p SST θ =θ =52.8 ο φ =120 ο θ =θ =54.0 ο φ =120 ο E=25 MeV
FIG. 11. (Color online) Position of d ( n, nn ) p breakup events (S curve) in the laboratory kineticenergy plane E − E of two detected in coincidence neutrons for SST configurations at E =13 , , and 65 MeV. The dashed lines show half of the energy of the outgoing di-neutron from d ( n, di − neutron ) p reaction at lab. angle of the corresponding SST geometry. The position on theS-curve where the SST condition is fulfilled is shown by square and the range of S-curve coveredby nd (pd) SST data is indicated by dots.
20 40 60 80 θ labdi-n [deg]0.11.010.0100.0 d σ / d Ω l a bd i - n [ m b s r - ] n [MeV]0246 d σ / d Ω n E n [ m b s r - M e V - ] d(n,di-n)p d(n,n)np θ n = θ SST a) b) c) a) b)c) FIG. 12. (Color online) The angular distributions of di-neutrons and the energy spectra of outgoingneutrons from incomplete breakup reaction d ( n, n ) np for lab. angle of the detected neutron equalto the SST configuration angle, predicted by the CD Bonn potential with a S nn force componentmodified by increasing its strength by a factor λ to get two neutrons bound. The (blue) dashedline corresponds to λ = 1 .
21 with di-neutron binding energy E di − nb = − .
144 MeV and the dashed-dotted line to λ = 1 . E di − nb = − .
441 MeV. The a), b) and c) are predictions for incomingneutron lab. energies E = 13 , , and 65 MeV, respectively.and 65 MeV, respectively.