Neutron-{}^{19}\mathrm{C} scattering: Towards including realistic interactions
aa r X i v : . [ nu c l - t h ] A ug Neutron- C scattering: Towards including realistic interactions
A. Deltuva
Institute of Theoretical Physics and Astronomy, Vilnius University, Saul˙etekio al. 3, LT-10257 Vilnius, Lithuania
Abstract
Low-energy neutron- C scattering is studied in the three-body n + n + C model using a realistic nn potential and a numberof shallow and deep n - C potentials, the latter supporting deeply-bound Pauli-forbidden states that are projected out. ExactFaddeev-type three-body scattering equations for transition operators including two- and three-body forces are solved in themomentum-space partial-wave framework. Phase shift, inelasticity parameter, and cross sections are calculated. For the elastic n - C scattering in the J Π = 0 + partial wave the signatures of the Efimov physics, i.e., the pole in the effective-range expansionand the elastic cross section minimum, are confirmed for both shallow and deep models, but with clear quantitative differencesbetween them, indicating the importance of a proper treatment of deeply-bound Pauli-forbidden states. In contrast, the inelasticityparameter is mostly correlated with the asymptotic normalization coefficient of the C bound state. Finally, in the regime of veryweak C binding and near-threshold (bound or virtual) excited C state the standard Efimovian behaviour of the n - C scatteringlength and cross section was confirmed, resolving the discrepancies between earlier studies by other authors [I. Mazumdar, A. R. P.Rau, V. S. Bhasin, Phys. Rev. Lett. 97 (2006) 062503; M. T. Yamashita, T. Frederico, L. Tomio, Phys. Rev. Lett. 99 (2007) 269201].
Key words:
Three-body scattering, Efimov physics, Faddeev equations, Pauli-forbidden states
1. Introduction
Few-particle systems whose two-particle ( ij ) subsystemsare characterized by large s -wave scattering lengths a ij exhibit universal properties. Their systematic theoreticalstudy was pioneered by V. Efimov almost 50 years ago [1]but the experimental confirmation came many years later[2–5]. It was achieved in cold-atom systems where the two-atom scattering length in the vicinity of the Feshbach res-onance can be controlled by an external magnetic field andthereby tuned to a large value significantly exceeding theinteraction range, a condition needed for the manifestationof the so-called universal or Efimov physics. The possibil-ity of tuning the scattering length is not available in thenuclear physics. Nevertheless, some nuclear systems havequite large two-particle scattering lengths and qualitativelyshow some properties characteristic for Efimov physics. Thesimplest case is the three-nucleon system [1,6–11]. Furtherexamples are systems consisting of a nuclear core ( A ) andtwo neutrons ( n ) provided that there is a weakly boundor virtual s -wave state in the ( A + n ) subsystem [12–16].Among them the C + n + n system has been studied in anumber of works (see Refs. [15,16] for a review) hoping to Email address: [email protected] (A. Deltuva). establish the existence of a C excited Efimov state assum-ing that C has the binding energy of only S n = 0 .
16 MeV[17] while the ground state of C is bound with S n = 3 . C+ n + n threshold). However, morerecent experiments have not confirmed such a weak bind-ing of C and presently accepted value is S n = 0 . C as a real bound state. Nevertheless,the C + n + n system in the low-energy n + C scatter-ing process is expected to exhibit some universal proper-ties that have been studied theoretically both in zero-rangeand finite-range models [19–22]. However, there is no con-sensus on the fate of the C excited Efimov state as Cbinding increases towards its physical value. Refs. [21–23]predict that it becomes a virtual state leading to a pole inthe effective-range expansion for n + C scattering simi-lar to the neutron-deuteron case [24] while Refs. [19,20,25]claim that the C excited state turns into a continuumresonance seen as a pronounced peak in the n + C elas-tic cross section around 1.5 keV center-of-mass (c.m.) en-ergy. One of the conclusions drawn in Ref. [25] was that“there is a need to undertake a detailed investigation usingrealistic interaction”. Indeed, all the calculations for the n + C scattering so far have been performed using simplerank-one separable potentials with Yamaguchi form factors
Preprint submitted to Elsevier 8 October 2018 or n - n and n - C pairs. Although in the ideally universalregime the predictions for observables should be indepen-dent of the interaction details, some remnant dependenceis expected for realistic systems whose universal behaviouris modified by finite-range corrections.The goal of the present work is to study the low-energy n + C scattering using more realistic interactions, possi-bly establishing shortcomings of rank-one separable mod-els, and sort out the differences between findings of Refs.[19,20,25] and [21–23]. The improvement of the interac-tion models is threefold: (i) For the n - n pair a realistichigh-precision charge-dependent (CD) Bonn potential [26]is used. (ii) A rank-one separable potential can support atmost one bound state, thus, it misses deeply-bound core-neutron states. These states are occupied by internal neu-trons in the core (not treated explicitly) and therefore arePauli-forbidden and must be projected out. In this way theidentity of external neutrons and those within the core isapproximately taken into account. A proper treatment ofPauli-forbidden states was found to be important in scat-tering processes, see e.g. Ref. [27] for α -deuteron collisions.Using a n - C potential supporting the 2 s state with the0.58 MeV experimental value [18] of the C binding en-ergy and projecting out the deep 1 s state will enable tostudy the importance of the Pauli-forbidden state for the n + C scattering and its impact on the Efimov physics.(iii) Depending on the chosen two-particle potentials, anadditional three-body force (3BF) may be needed to fixthe C ground-state binding energy that must be includedalso in the n + C scattering calculations.Thus, for the desired study of the n + C scatteringan accurate theoretical description of three-particle scat-tering process including general form of potentials and3BF is needed; the separable quasi-particle formulationof Refs. [19–22] is not applicable. In the present work thedescription is obtained by a combination and extensionof momentum-space techniques from Refs. [27,28]. Bothare based on exact Faddeev three-body theory [29] in theintegral form for transition operators as proposed by Alt,Grassberger, and Sandhas (AGS) [30], but either neglect-ing the 3BF [27] or limited to the three-nucleon system[28].The employed potentials are described in Sec. 2 and thethree-body scattering equations with 3BF in Sec. 3. Resultsare presented in Sec. 4, and a summary is given in Sec. 5.
2. Potentials
The system of two neutrons and C core is considered asa three-body problem. Particle masses m n = 1 . m N and m A = 18 m N are given in units of the average nucleonmass m N = 938 .
919 MeV. The dynamics of the system isdetermined by two-particle potentials v nn and v nA actingwithin the nn pair and two nA pairs, and, eventually, byan additional 3BF. Unless explicitly stated otherwise, v nn is taken to be the high-precision CD Bonn potential [26]; it is allowed to act in the s , p and d waves thereby ensur-ing a perfect convergence of the nn partial-wave expansion.Given the uncertainty in the v nA , a number of models willbe used. A very common choice is the Woods-Saxon poten-tial, in the coordinate space defined as¯ v nA ( r ) = − V c [1 + exp(( r − R v ) /a v )] − . (1)In the present work this kind of potentials is numericallytransformed into the momentum-space representation andthen used in two- and three-particle equations. The poten-tials parameters are the strength V c , radius R v = r v A / ,and diffuseness a v with standard values being around r v =1 . a v = 0 . r v = 0 . r v = 0 .
332 fm keeping the ratio a v /r v = 1 /
2. In allthree cases the strength V c is adjusted such that the excited s -wave state 2 s has the experimental C binding energyvalue ε s = S n = 0 .
58 MeV. The ground state is deeplybound with the energy ε s and the wave function | φ s i de-pending on the chosen r v , but it is Pauli-forbidden andtherefore must be projected out. This is achieved [31,32,27]by taking the neutron-core s -wave potential as v snA = ¯ v nA + | φ s i Γ h φ s | (2)where formally Γ → ∞ but in practice Γ must be largeenough such that the results for the three-body boundstate(s) and scattering in the considered energy regime be-come independent of it; in the present work Γ = 50 GeVwas proven to be sufficiently large. Simultaneously this en-sures also the absence of deeply-bound three-body states.The potential models projecting out deeply-bound Pauli-forbidden (DP) states will be denoted as DPa, DPb, andDPc; their parameters are collected in table 1. The pre-dictions for the nA scattering length a nA and the effec-tive range r nA are presented as well. As the potentials ofRefs. [19–22], they are restricted to act in the s -wave only.The manifestation of the Efimov physics is governed byresonant s -wave interactions but in realistic systems alsohigher partial waves contribute. To estimate their effect,one more model, labeled DPp, is introduced that uses ¯ v nA parameters from DPa but is allowed to act in p -waves aswell. It turns out that ¯ v nA supports a deeply-bound 1 p state | φ p i for C with ε p = 11 .
54 MeV that is Pauli-forbiddenas well and must be projected out in the same way, i.e., v pnA = ¯ v nA + | φ p i Γ h φ p | . (3)Furthermore, as in Refs. [21,22] the energy of the three-body bound state, i.e., the two-neutron separation energyof C, is fixed at its experimental value of S n = 3 . R v was adjusted toreproduce S n , in general case the pairwise nn and nA in-teractions are insufficient for S n and an additional 3BF isneeded. In fact, when the 3BF is not included, S n = 2 . able 1Parameters of the employed nA force models DPa, DPb, DPc andDPp with projected-out deeply-bound Pauli-forbidden states and3BF. In the last line the parameters of the shallow WS potential aregiven. R v (fm) V c (MeV) Λ(fm − ) W c (fm MeV) ε s (MeV)DPa 3.145 44.569 1.0 312.40 25.52DPb 2.097 95.995 1.5 43.34 53.98DPc 0.870 530.745 0.00 292.00DPp 3.145 44.569 1.0 287.50 25.52WS 3.119 8.317 0.00 0.58 [32] or hypermomentum [33,34]. The latter choice is obvi-ously more convenient in the momentum-space frameworkand therefore is used in the present calculations. The three-body bound-state Faddeev equations, their solution tech-nique, and the form of the 3BF is taken over from Ref. [34].The latter is h p α q α | W | p ′ α q ′ α i = − (4 π ) − W c g ( K ) g ( K ′ ) (4)with the hypermomentum K = m N ( p α /µ α + q α /M α ) ex-pressed in terms of Jacobi momenta p α for the pair and q α for the spectator and the associated reduced masses µ α and M α . Note that K / m N is the internal motion kineticenergy, thus, the 3BF has the same form in any Jacobi con-figuration labeled by the spectator particle α in the odd-man-out notation (see next section for more details). Theform factor g ( K ) = exp ( −K / ) is chosen as a Gaus-sian. The cutoff parameter Λ is related to the interactionrange R w roughly as Λ R w ∼ √
2; for each two-body model R w /R v ∼ √ / (Λ R v ) < / v nA . The strength W c is adjusted to re-produce the desired three-body binding energy. These pa-rameters are collected in table 1 as well.To isolate the effect of deeply-bound Pauli-forbiddenstates, several models without those states are used, i.e.,they support only one C bound state 1 s with the bind-ing energy ε s = 0 .
58 MeV. The model WS uses a shallowWoods-Saxon potential (1) with r v = 1 .
19 fm, while themodel labeled Y is a rank-one separable potential withYamaguchi form factor as in Ref. [22] with the momentum-range parameter β nc = 0 . − . Finally, to get theinsight on the importance of a realistic nn interaction, in-stead of the CD Bonn the rank-one separable nn potentialwith Yamaguchi form factor from Ref. [22] is used; its com-bination with the nA potential of the same type as Y butwith β nc = 0 . − will be labeled YY in the follow-ing. The above choices of the range parameter values forWS, Y, and YY models ensure the desired binding energyof C without the 3BF. The WS, Y, and YY potentialswill be referred in the following as shallow , in contrast tothe deep ones DPa, DPb, DPc, and DPp.To get an insight into the correlations between the inter-action models and physical properties of C and C, intable 2 the predictions for the n - C scattering length a nA and effective range r nA , the asymptotic normalization co-efficient (ANC) of the C bound state, and the C ground
Table 2Predictions of the employed force models for the n - C scatteringlength a nA and effective range r nA , ANC of the C nucleus, andthe internal kinetic energy expectation value ¯ K b of the C nucleuswith the binding energy of S n = 3 . a nA (fm) r nA (fm) ANC(fm − / ) ¯ K b (MeV)DPa 9.23 4.32 0.948 32.83DPb 8.22 3.19 0.802 47.00DPc 7.01 1.54 0.657 86.82DPp 9.23 4.32 0.948 33.12WS 8.12 3.04 0.792 9.87Y 8.67 3.66 0.871 10.64YY 8.68 3.67 0.872 11.14 state internal kinetic energy expectation value ¯ K b are col-lected. Within the group of deep models one may easilynotice the well-known feature that a longer-range potentialleads to larger values of the ANC, effective range, and, to alesser extent, scattering length. However, comparing DPaand WS that have almost the same R , one can conclude thatdeeply-bound Pauli-forbidden states cause larger a nA , r nA ,and ANC values, and significantly higher ¯ K b . Within thegroup of DP models the kinetic energy expectation valuedepends also strongly on the range R , but in all cases itconsiderably exceeds the predictions of shallow potentials.Thus, deep potentials strongly enhance high-momentumcomponents in the C ground state.
3. Neutron- C scattering equations including3BF
The momentum-space formulation of the three-bodyscattering theory is convenient when the underlying po-tentials have nonlocal terms such as those in the deep nA potentials projecting out Pauli-forbidden states. Thepresent work is based on Faddeev equations for the multi-channel transition operators U βα in the version derived byAlt, Grassberger, and Sandhas (AGS) [30] but extended toinclude also the 3BF. Such extensions have been proposedin a number of works, e.g., [35,28], but their practical appli-cations mostly were limited to the symmetrized version inthe three-nucleon system. The general form of three-bodyequations from Ref. [28] is taken for the present study ofthe n - C scattering, i.e., U βα = ¯ δ βα G − + u α + X γ =1 ¯ δ βγ T γ G U γα + X γ =1 u γ G (1 + T γ G ) U γα , (5)with ¯ δ βα = 1 − δ βα , the free resolvent G = ( E + i − H ) − at the available energy E , the free Hamiltonian for theinternal motion H , the two-body transition matrix T γ = v γ + v γ G T γ , (6)3nd the 3BF arbitrarily decomposed into three components W = X α =1 u α . (7)The odd-man-out notation is used, i.e., the channel α cor-responds to the configuration where the particle α is thespectator and the remaining two are the pair. The decom-position of the 3BF into three symmetric parts (7) is essen-tial for the symmetrization of three-nucleon equations [28]but is not needed in the present work. Labeling the parti-cles n, A, n as 1 , , u α = δ α W , the system ofthe AGS equations (5) for the n - C scattering simplifies to U β = ¯ δ β G − + X γ =1 ¯ δ βγ T γ G U γ + W G (1 + T G ) U (8)with β = 1 , ,
3. The above system of integral equa-tions is solved in the momentum-space partial-waverepresentation employing three sets of base functions | p α q α ( l α { [ L α ( s β s γ ) S α ] j α s α }S α ) JM i with ( α, β, γ ) beingcyclic permutations of (1 , , p α and q α are mag-nitudes of Jacobi momenta for the corresponding pair andspectator, while L α and l α are the associated orbital an-gular momenta, respectively. Together with the particlespins s α , s β , s γ they are coupled, through the intermediatesubsystem spins S α , j α and S α , to the total angular mo-mentum J with the projection M . Only the basis α = 2 isantisymmetric with respect to the permutation of the twoneutrons; this is achieved by considering only even L + S states. However, the neutron identity is accounted for bytaking the antisymmetrized elastic scattering amplitude f ν ′ ν ( k ′ , k ) = − (2 π ) M [ h Φ ν ′ ( k ′ ) | U | Φ ν ( k ) i− h Φ ν ′ ( k ′ ) | U | Φ ν ( k ) i ] . (9)Here | Φ να ( k ) i is the asymptotic state in the channel α ; itis given by the product of the bound state wave functionfor the pair and the plane wave with the on-shell momen-tum k for the relative motion between the bound pair andspectator α satisfying E = − S n + k / M ; the spin quan-tum numbers are abbreviated by ν . In the normalizationof Eq. (9) the n - C elastic differential cross section for the ν k → ν ′ k ′ transition is simply dσ/d Ω = | f ν ′ ν ( k ′ , k ) | .
4. Results
The Efimov physics manifests itself in the states dom-inated by the s -wave components L α = l α = 0 for all α ; this condition is satisfied only for J Π = 0 + whereΠ = ( − L α + l α is the total parity. For the notationalbrevity suppressing the dependence on the on-shell mo-mentum k , the S -matrix and the amplitude in the 0 + state are parametrized as s = e iδ and f = e iδ sin δ/k =( k cot δ − ik ) − , respectively. The phase shift δ is realbelow the inelastic threshold, i.e., at c.m. kinetic ener-gies E k = k / M ≤ .
58 MeV, but becomes complex
Table 3Parameters of the n - C effective-range expansion for the employedinteraction models together with r nA for n - C. a (fm) b (fm − MeV − ) c (fm − MeV − ) E (MeV) r nA (fm)DPa -6.299 1.176 -0.2726 0.20626 4.32DPb -6.103 1.078 -0.1538 0.26235 3.19DPc -5.369 1.033 -0.0511 0.33770 1.54DPp -6.310 1.176 -0.2744 0.20644 4.32WS -9.419 0.8194 -0.0496 0.39663 3.04Y -9.802 0.8520 -0.0838 0.34710 3.66YY -9.653 0.8591 -0.0828 0.34413 3.67 above this value due to the open breakup channel whoseimportance is parametrized by the inelasticity parameter η = | e iδ | ≤ k cot δ ≈ − a − + bE k + cE k − E k /E , (10)where a is the n - C singlet scattering length and E is theposition of the pole. The values for the parameters a , b , c , and E obtained fitting the n - C phase shift results at E k ≤ .
58 MeV for all employed interaction models are col-lected in table 3 while the corresponding reduced effective-range functions (1 − E k /E ) k cot δ are plotted in Fig. 1. Itturns out that Eq. (10) yields a very good approximation- the quantities calculated directly and from the fitted pa-rameters are indistinguishable in the plot. One notices im-mediately that (1 − E k /E ) k cot δ predictions for the groupsof the shallow (YY,Y,WS) and deep (DPa,DPb,DPc,DPp)potentials clearly separate. A closer inspection of the ta-ble 3 reveals that this is mostly due to the differences inthe n - C scattering length a and, to a lesser extent, in theparameter b . Within each group one can see qualitativelythe same trend in correlations between the n - C effectiverange r nA and n - C parameters, i.e., | a | , b and | c | increasewith increasing r nA while E decreases. However, it turnsout that the presence of deep Pauli-forbidden states is moredecisive for a and b than the correlation with r nA , whilefor c and E these two effects are of comparable impor-tance. The parameters c and E show a broad spread ofvalues, especially in the group of deep potentials. However,if one disregards the DPc model as being of unrealisticallyshort range, one can see again some trend, i.e., larger | c | and smaller E for deep potentials as compared to shallowones. The parameters of DPa and DPp stay very close in-dicating that the n - C p -wave interaction is indeed irrele-vant in the present context. The deviations between Y andYY for all parameters are insignificant as well, thus, therank-one separable s -wave nn potential is able to capturerelevant physics for the n - C low-energy J Π = 0 + elasticscattering.The differences in a and E are clearly reflected in the J Π = 0 + total elastic cross section σ + for the n - C scat-tering shown in Fig. 2: a determines σ + at E k = 0 while4 .00.20.40.60.0 0.2 0.4 ( - E k / E ) k c o t δ ( f m - ) E k (MeV) YYYWSDPaDPbDPcDPp
Fig. 1. (Color online) n - C reduced effective-range functions(1 − E k /E ) k cot δ in the J Π = 0 + partial wave for the interactionmodels YY(double-dashed-dotted), Y (dotted), WS (double-dotted–dashed), DPa (solid), DPb (dashed-dotted), DPc (dashed), and DPp(bullets). E k = E corresponds to the minimum of σ + . However,this minimum is only clearly seen when the initial state n and C spins are anti-parallel, such that the total channelspin S = 0 couples with l = 0 to J Π = 0 + . If the initialstate is not polarized, one has to take into account also the n + C triplet state ( S = 1 , l = 0) J Π = 1 + whose crosssection σ + is also shown in Fig. 2. In fact, σ + yields byfar the most sizable contribution to the unpolarized low-energy cross section, given (neglecting l = 1 and higherwaves) as the spin-weighted average σ = ( σ + + 3 σ + ) / S ( nn ) configuration is not allowed in the J Π =1 + state, σ + is governed by the nA interaction. In fact, forall models with the nn CD Bonn potential the n - C tripletscattering length a + ≈ a nA + 0 .
02 fm is simply related tothe n - C scattering length.The results in Fig. 2 extend above the breakup threshold;in that regime σ + depends on E k only weakly, with thedeep models (except for DPc) providing higher cross sectionthan the shallow ones, although the spread within eachgroup is comparable to the difference between groups. TheDPa-DPp and Y-YY similarities remain valid also over thebroader regime.However, the situation is quite different for the inelas-ticity parameter η studied in Fig. 3. It exhibits some DPa-DPp and Y-YY deviations but shows no trend for the dif-ferences between shallow and deep potentials, the spreadfor the latter being very broad. Looking back to the modelproperties in table 2, one may notice the correlations be-tween the ANC (or a nA , or r nA ) and η . To make it moreevident, the inelasticity parameter at E k = 1 .
14 and 1.90MeV for all force models is plotted in Fig. 4 against thecorresponding ANC value. The dependence is roughly lin-ear with deviations by YY and DPp models, that eitherhave a different nn force (YY) from all the others, or havean additional nA p -wave dynamics (DPp). This is not sur-prising since one can expect the n - C breakup reaction to -4 -3 -2 -1 σ J + ( b ) E k (MeV) YYYWSDPaDPbDPcDPpDPp(1 + ) Fig. 2. (Color online) n - C total elastic cross section σ + in the J Π = 0 + partial wave as a function of the c.m. kinetic energy E k fordifferent interaction models. In addition, the J Π = 1 + wave crosssection σ + is shown for the DPp model as the upper triple-dotted–dashed curve, other curves are as in Fig. 1. η E k (MeV) YYYWsDPaDPbDPcDPp
Fig. 3. (Color online) n - C inelasticity parameter η in the J Π = 0 + partial wave as a function of the c.m. kinetic energy E k for differentinteraction models. Curves are as in Fig. 1. be peripheral at these low energies and dominated by themechanism of the n - n knockout. In fact, even neglectingthe 3BF for DPa, DPb and DPp models leads to changesof η that are significantly smaller than the spread of pre-dictions in Fig. 3. Thus, the breakup and inelasticity pa-rameter η is mostly governed by the properties of the Cbound state and the nn force, i.e., by the two-body physicswithout clear evidence for the three-body Efimov physics.Finally I turn to the disagreement between Refs.[19,20,25] and [21–23] near the regime where the boundexcited C Efimov state disappears. To reach that regimethe strength of the nA potential V c is reduced withoutchanges in other force parameters; this leads to the varia-tions of C and C binding energies and n - C scatteringobservables. The appearance of the bound excited Cstate, depending on the potential, takes place when S n isreduced to 0.07 - 0.09 MeV and S n to 1.4 - 1.9 MeV. Thisis different from the strategy of Ref. [22] where S n wasfixed at 3.5 MeV, but nevertheless the present results sup-port the conclusions of Refs. [21,22] that the excited Cstate at the n + C threshold corresponds to a pole in the n - C scattering length, i.e., a → ±∞ , with +( − ) for S n below (above) the critical value. This behaviour is shown5 .91.0 0.7 0.8 0.9 η ANC (fm -1/2 ) E k = 1.14 MeVE k = 1.90 MeV Fig. 4. (Color online) n - C inelasticity parameter η in the J Π = 0 + partial wave at E k = 1 .
14 (boxes) and 1.90 MeV (circles). Thesymbols from left to right correspond to the interaction models DPc,WS, DPb, YY, Y, DPa, and DPp. The lines are for guiding the eyeonly. -1500-1000-500050010001500 0.2 0.4 a ( f m ) S n (MeV) YDPbDPc
Fig. 5. (Color online) Dependence of the n - C singlet scatteringlength a on the C binding energy S n for the evolved models Y(dotted), DPb (filled boxes) and DPc (dashed). in Fig. 5 for selected potential models, but is characteristicfor all of them. In contrast, the authors of Refs. [19,20,25]claim that the n - C scattering length remains positivealso for S n above the critical value while the low-energyelastic n - C cross section exhibits a resonance around E k = 1 . n - C scattering in this regimeperformed in the present work excludes such a behaviour:the cross section rapidly and monotonically decreases withincreasing energy without any signs of resonant peaks. Asexample the J Π = 0 + elastic cross section calculated usingthe evolved DPb model is shown in Fig. 6.
5. Summary and conclusions
Low-energy neutron- C scattering was studied in thethree-body C + n + n model. Realistic nn CD Bonn po-tential and a number of shallow and deep n - C potentialsof different range were used. All deep potentials supportdeeply-bound Pauli-forbidden states that were projectedout thereby accounting for the identity of external neutronsand those within the C core in an approximate way, whileshallow models ignore this aspect. For all models the po- σ + ( b ) E k (keV) S n = 84.0 keVS n = 108.9 keVS n = 136.8 keV Fig. 6. (Color online) n - C total elastic cross section σ + in the J Π = 0 + partial wave as a function of the c.m. kinetic energy E k forthe DPb model evolved to have the C binding energy S n of 84.0keV (dotted), 108.9 keV (dashed-dotted) and 136.8 keV (solid). The C bound excited state exists below S n = 71 .
66 keV. tential parameters were adjusted to reproduce the experi-mental binding of the C ground state; most of the deepmodels had to be supplemented by a 3BF to achieve thisgoal. Exact three-body Faddeev-type scattering theory inthe AGS version for transition operators, extended to in-clude also the 3BF, was implemented in the momentum-space partial-wave framework yielding numerically accu-rate results for the n - C scattering both below and abovebreakup threshold.Given the weak binding of C and large nn scatteringlength, the C + n + n system in the J Π = 0 + partial waveexhibits some features characteristic for Efimov physics. Inparticular, the presence of an excited C Efimov state asa virtual state leads to a pole in the J Π = 0 + n - C effec-tive range expansion. The reduced effective range functions(1 − E k /E ) k cot δ clearly separate for shallow and deepmodels, indicating the importance of a proper treatmentfor deeply-bound Pauli-forbidden states. For some observ-ables like the n - C singlet scattering length the presenceof deep Pauli-forbidden states appears to be more decisivethan the correlation with the n - C effective range. On theother hand, the observed differences between the groups ofshallow and deep models are of comparable size as the fi-nite range effects found in Ref. [22], and therefore do notinvalidate the concept of the Efimov physics being inde-pendent of the short-range interaction details. However,the present work shows that deeply-bound Pauli-forbiddenstates may lead to systematic shifts within the limits offinite-range corrections. The effect is even more importantfor non-observable quantities like the expectation value ofthe C internal kinetic energy.For the elastic n - C scattering in the J Π = 0 + partialwave the signature of the Efimov physics, i.e., the pres-ence of the cross section minimum, was confirmed for bothshallow and deep models. It was also shown that withoutthe initial antiparallel n - C polarization this minimum is,however, hidden to a large extent due to the dominatingcontribution of the J Π = 1 + partial wave.In the hypothetical situation of very weak C binding6nd near-threshold (bound or virtual) excited C statethe standard Efimovian behaviour of the n - C scatteringlength and cross section was confirmed as well, clearlysupporting Refs. [21–23] and excluding the possibility ofnear-threshold resonances predicted in Refs. [19,20,25]. Asboth groups have solved Faddeev equations with rank-one s -wave potentials, a possible explanation for this differ-ence could be inaccurate numerical implementation inRefs. [19,20,25].In contrast to the elastic n - C scattering, the breakupreaction is dominated by two-body physics. The inelasticityparameter in the J Π = 0 + partial wave is mostly correlatedwith the ANC of the C bound state; this suggests a simple nn -knockout picture for the reaction mechanism.Although the present work demonstrated the impor-tance of the deeply-bound Pauli-forbidden states in thelow-energy elastic n - C scattering, further changes can beexpected given the low excitation energy of the C core[18]. This would lead to the d -wave admixture in the Cground state and possibly to d -wave excited states or reso-nances, thereby bringing d -wave corrections to the s -wavedominated Efimov physics of the C + n + n system. Forexample, significant d -wave effects have been found in thestudy of cold atom systems with van der Waals interactions[36].This work was supported by Lietuvos Mokslo Taryba(Research Council of Lithuania) under Contract No. MIP-094/2015 and by Alexander von Humboldt-Stiftung. 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