Nature of the Λnn ( J π =1/ 2 + ,I=1) and 3 Λ H ∗ ( J π =3/ 2 + ,I=0) states
NNature of the Λ nn ( J π = 1 / + , I = 1) and H ∗ ( J π = 3 / + , I = 0) states M. Sch¨afer ∗ Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering,Bˇrehov´a 7, 11519 Prague 1, Czech Republic andNuclear Physics Institute of the Czech Academy of Sciences, 25069 ˇReˇz, Czech Republic
B. Bazak † and N. Barnea ‡ Racah Institute of Physics,The Hebrew University, Jerusalem 91904, Israel
J. Mareˇs § Nuclear Physics Institute of the Czech Academy of Sciences, 25069 ˇReˇz, Czech Republic (Dated: July 21, 2020)The nature of the Λ nn and H ∗ ( J π = 3 / + , I = 0) states is investigated within a pionless effectivefield theory at leading order, constrained by the low energy Λ N scattering data and hypernuclear3- and 4-body data. Bound state solutions are obtained using the stochastic variational method,the continuum region is studied by employing two independent methods - the inverse analyticcontinuation in the coupling constant method and the complex scaling method. Our calculationsyield both the Λ nn and H ∗ states unbound. We conclude that the excited state H ∗ is a virtualstate and the Λ nn pole located close to the three-body threshold in a complex energy plane couldconvert to a true resonance with Re( E ) > N interactions. Finally, thestability of resonance solutions is discussed and limits of the accuracy of performed calculations areassessed. I. INTRODUCTION
The s -shell Λ hypernuclei play an important role in thestudy of baryon-baryon interactions in the strangenesssector. In view of scarce hyperon-nucleon scattering datathey provide a unique test ground for the underlying in-teraction models thanks to reliable few-body techniques.In particular, experimental values of the Λ separation en-ergies in A = 3 , H ∗ and He ∗ excitation energies represent quite stringent constraints(see [1] and references therein).The hypertriton H ( J π = 1 / + , I = 0) is the light-est known hypernucleus, with the Λ separation energy B Λ = 0 . ± .
05 MeV [2]. In view of the small valueof B Λ in the hypertriton ground state, it is likely thatthe excited state H ∗ ( J π = 3 / + , I = 0) is located justabove the Λ + d threshold, however, its physical nature isnot yet known. Moreover, since the isospin-triplet N N state is unbound, it is highly unlikely that there existsa bound state in the I = 1 Λ nn system. A thoroughstudy of the A = 3 hypernuclear systems with differentspin and isospin, addressing the question whether theyare bound or continuum states, provides invaluable in-formation about the spin and isospin dependence of theΛ N interaction, as well as dynamical effects in these few-body systems caused by a Λ hyperon. Moreover, the issueof the Λ nn and also ΛΛ nn states as possible candidates ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] for widely discussed bound neutral nuclear systems hasattracted increased attention recently in connection withthe experimental evidence for the bound Λ nn state re-ported by the HypHI collaboration [3].The first variational calculation demonstrating thatthe Λ nn system is unbound was performed by Dalitz andDowns more than 50 years ago [4]. Later, this conclu-sion was further supported by Garcilazo using Faddeevapproach with separable potentials [5]. Following, moredetailed, studies of both Λ nn and H ∗ systems withinFaddeev approach using either Nijmegen Y N potential[6] or chiral constituent quark model of
Y N interactions[7] confirmed that both systems are indeed unbound. Inaddition, these calculations revealed that with increas-ing
Y N attraction the binding of H ∗ comes first. Theinvestigation of the Λ d scattering length in J π = 3 / + channel indicated existence of a pole in the vicinity ofthe Λ + d threshold. Continuum calculations of the un-bound Λ nn system were performed by Belyaev et al. us-ing a phenomenological Λ N potential [8]. This neutralhypernuclear system was found to form a very wide, near-threshold resonance.In view of the above theoretical calculations, theclaimed evidence of the Λ nn reported by HypHI Collab-oration was quite surprising and it stimulated renewedinterest in the nature of the 3-body hypernuclear states.The HypHI conclusions were seriously challenged by suc-ceeding calculations [9, 10], demonstrating inconsistencyof the existence of the Λ nn bound state with Λ N scat-tering as well as 3- and 4-body hypernuclear data. Fur-thermore, the renewed analysis of the BNL-AGS-E906experiment [11] led to conclusion that the formation of abound Λ nn nucleus is highly unlikely. In addition, ratherrecently Gal and Garzilazo [12] made a rough but solid a r X i v : . [ nu c l - t h ] J u l estimate of Λ nn lifetime which, if bound, is considerablylonger than the one of free Λ hyperon τ Λ . This result isin disagreement with the shorter Λ nn lifetime with re-spect to τ Λ extracted from the HypHI events assigned tothis system. The Λ nn was also explored within pionlesseffective field theory ( /π EFT ) [13, 14].In spite of the apparent interest the Λ nn and H ∗ con-tinuum states have been investigated in only few theoret-ical works[8, 15, 16]. Afnan and Gibson [15] performedFaddeev calculations of Λ nn using two-body separablepotentials fitted to reproduce N N and Λ N scatteringlengths and effective ranges. They pointed out that whileΛ nn pole appears in the subthreshold region (Re( E ) < N interaction strength pro-duces a Λ nn resonance (Re( E ) > nn system in the JLab E12-17-003 experiment [17].In this work, we performed few-body calculationsof the Λ nn and H ∗ hypernuclear systems within LO /π EFT , both in the bound and continuous region, ex-ploring thoroughly their nature. The first selected resultshave been reported in Ref. [16]. As demonstrated in thatwork the virtual state H ∗ pole position close to the Λ+ d threshold strongly affects the Λ d s -wave phase shifts in J π = 3 / + channel. The calculated Λ d scattering lengthsand effective ranges from this work were further employedby Haidenbauer in the study of Λ d correlation functionswithin the Lednicky-Lyuboshits formalism [18]. It is tobe noted that the nature of the H ∗ state is a subject ofthe JLab proposal P12-19-002 [19].The /π EFT approach was applied to s -shell Λ hyper-nuclei and, among others, the experimental value ofthe Λ separation energy B Λ in He was successfullyreproduced [20]. The /π EFT was further extended to S = − /π EFT is applied to the study of continuumstates in 3-body hypernuclear systems. Bound statecalculations are performed using the Stochastic Vari-ational Method (SVM), the continuum states are de-scribed within the Inverse Analytic Continuation in theCoupling Constant (IACCC) Method and the ComplexScaling Method (CSM). The IACCC calculations arebenchmarked against the CSM and the stability of reso-nance solutions is discussed. The CSM is in addition usedto set limits of the accuracy of performed calculations.The paper is organized as follows: In Section II, wefirst give a brief description of the /π EFT approach andthe SVM method applied in the calculations of few-bodyhypernuclear systems. Then, we introduce the CSM andIACCC method used to describe continuum states andpole movement in a complex energy plane. In SectionIII, we present results of our study of the Λ nn and H ∗ systems. We discuss in more detail the relation betweenthe applied LO /π EFT approach and phenomenologicalmodels and, in particular, the stability and numerical ac-curacy of our /π EFT calculations. Finally, we summarizeour findings in Section V.
II. MODEL AND METHODOLOGY
Hypernuclear systems studied in this work are de-scribed within the /π EFT at LO which was introducedin detail in [20]. In this section we present only basicingredients of the theory. The LO /π EFT contains 2- and3-body s -wave contact interaction terms, each of themassociated with corresponding isospin-spin channel. Thecontact terms are then regularized by applying a Gaus-sian regulator with momentum cutoff λ . This procedureyields two-body V and three-body V potentials whichtogether with the kinetic energy T k enter the total Hamil-tonian H : H = T k + V + V , (1)where V = (cid:88) I,S C I,Sλ (cid:88) i N N ) LECs. Nuclear LECs C I =0 ,S =1 λ , C I =1 ,S =0 λ , and D I =1 / ,S =1 / λ are fitted to the deuteronbinding energy, N N spin-singlet scattering length a NN ,and to the triton binding energy, respectively. Hyper-nuclear two-body LECs C I =1 / ,S =0 λ and C I =1 / ,S =1 λ arefixed by the Λ N scattering length in a spin-singlet a Λ N and spin-triplet a Λ N channel. Three-body hypernuclearLECs D I =0 ,S =1 / λ , D I =1 ,S =1 / λ , and D I =0 ,S =3 / λ are fit-ted to the experimental values of Λ separation energies B Λ ( H), B Λ ( H) and the excitation energy E exc ( H ∗ ).Since a Λ N and a Λ N are not constrained sufficientlywell by experiment, we use their values given by directanalysis of scattering data [22] or predicted by severalmodels of Λ N interaction [23–25]. Considered a Λ N and a Λ N together with the data used to fix N N spin-singlet S and spin-triplet S LECs are given in Table. I. The /π EFT approach was applied to s-shell Λ hypernuclei and,among others, the experimental value of the Λ separa-tion energy B Λ in He was successfully reproduced [20]as demonstrated in the last column of Table I.The calculation of A = 3 , , s -shell Λ hyper-nuclear systems are performed within finite basis set of TABLE I. Values of spin-singlet a Λ N and spin-triplet a Λ N scattering lengths a used to fit hypernuclear 2-body LECs to-gether with effective ranges r Λ N and r Λ N (in fm). Corre-sponding Λ separation energies B Λ ( He; ∞ ) (in MeV), pre-dicted within /π EFT for λ → ∞ [20] are to be compared withthe experimental value B Λ ( He) = 3 . a Λ N r Λ N a Λ N r Λ N B Λ ( He; ∞ )Alexander B [22] -1.80 2.80 -1.60 3.30 3.01(10)NSC97f [23] -2.60 3.05 -1.71 3.33 2.74(11) χ EFT(LO) [24] -1.91 1.40 -1.23 2.20 3.96(08) χ EFT(NLO) [25] -2.91 2.78 -1.54 2.27 3.01(06) NN [26, 27] -18.63 2.75 E B ( H) = − . a We use the effective range expansion sign convention defined as k cotg( δ ) = − a s + r s k + · · · . correlated Gaussians [28] ψ i = ˆ A exp (cid:18) − x T A i x (cid:19) χ iSM S ξ iIM I , (4)where the operator ˆ A ensures antisymmetrization be-tween nucleons, x T = ( x , . . . , x A − ) is a set of Jacobicoordinates, and χ iSM S and ξ iIM I stand for correspond-ing spin and isospin parts, respectively. Each ψ i includes A ( A − / A − 1) dimensional positive-definite symmetric matrix A i plus 2 discreet parameters which represent differentspin and isospin configuration in χ iSM S and ξ iIM I , respec-tively.In order to choose ψ i with the most appropriate nonlin-ear parameters we use the Stochastic Variational Method(SVM) [29] which was proved to provide systematic pro-cedure to optimize the finite basis set, thus reachinghighly accurate bound state description.Resonances and virtual states, predominantly inter-preted as poles of S -matrix [30, 31], can not be addresseddirectly using the SVM with the finite basis set. Conse-quently, in order to study hypernuclear continuum weapply the Inverse Analytic Continuation in the CouplingConstant (IACCC) method [32] which was proposed asnumerically more stable alternative to the Analytic Con-tinuation in the Coupling Constant [33].Following the spirit of analytical continuation tech-niques we supplement the Hamiltonian H (1) by an aux-iliary 3-body attractive potential V IACCC3 = d I,Sλ (cid:88) i I, S )three-body channel - (1 , ) for Λ nn or (0 , ) for H ∗ .If not explicitly mentioned λ in V IACCC3 is equal to the /π EFT cutoff λ . In principle one can use a rather large class of 2- or 3-body attractive auxiliary potentials whichfulfill certain criteria imposed by analytic continuation[31]. Using V IACCC3 (5) ensures that the properties of2-body part of the /π EFT Hamiltonian (1) such as scat-tering lengths or deuteron binding energy remain unaf-fected. Its form is selected to be the same as of the /π EFT 3-body potential (1).With increasing attractive strength of d I,Sλ the res-onance or virtual state S -matrix pole described by H starts to move towards the bound state region and atcertain d I,S , λ becomes a bound state. The other wayaround, studying bound state energy E B as a functionof d I,Sλ < d I,S , λ we can perform an analytic continuationof the pole position from the bound region back into thecontinuum ( d I,Sλ > d I,S , λ ) up to the point of its physicalposition with no auxiliary force ( d I,Sλ = 0).In practice, we apply the SVM to calculate a set of M + N + 1 bound state energies for different values of thecoupling constant { E i B ( d i ) ; d i < d ; i = 1 , . . . , M + N +1 } , where d i = d I,Si, λ . Next, using this set we construct thePad´e approximant of degree ( M , N ) P ( M,N ) of function d ( κ ) P ( M,N ) ( κ ) = (cid:80) Mj =0 b j κ j (cid:80) Nj =1 c j κ j ≈ d ( κ ) , (6)where b j and c j are real parameters of the P ( M,N ) . The κ is defined as κ = − i k = − i √ E with E standing fora bound state energy with respect to the nearest dis-sociation threshold. The position of the S -matrix polecorresponding to H is calculated setting d = 0 in Eq. (6)which leads to the the simple polynomial equation M (cid:88) j =0 b j κ j = 0 . (7)The resonance or virtual state energy with respect to thenearest threshold is then obtained as E = (i κ ) , where κ now corresponds to the physical root of Eq. (7). Here,for complex resonance energy, we use the notation E = E r − iΓ / 2, where E r = Re( E ) is the position of theresonance and Γ = − E ) stands for the resonancewidth.Using the IACCC method we study the whole poletrajectory E ( d ) in the continuum region d ∈ (cid:104) d ; 0 (cid:105) (see Fig. 4). For a given set of bound state energies { E i B ( d i ) ; d i < d ; i = 1 , . . . , M + N + 1 } , we shift d i → d − d i in the E i B ( d i ) set, construct new Pad´e ap-proximant (6), and obtain E ( d ) as a corresponding rootof Eq. (7).The specific choice of V IACCC3 (5) provides clear phys-ical interpretation for any d I,Sλ solution. By varying d I,Sλ the Λ nn or H ∗ pole moves along its trajectory E ( d I,Sλ , λ )which is defined purely by the underlying 2-body inter-actions and cutoff λ . Supplementing the physical Hamil-tonian (1) by V IACCC3 might be understood as a shift ofthe three-body LEC constant D I,Sλ → D I,Sλ + d I,Sλ . Since D I =1 ,S =1 / λ and D I =0 ,S =3 / λ have been fitted for each λ to the experimental value of B Λ ( H) and E exc ( H ∗ ), re-spectively [20], one could assign the parts of trajectoriesfor d I,Sλ < a Λ N and cutoff λ the trajec-tory E ( d I =1 ,S =1 / λ , λ ) of Λnn pole positions correspondsto different values of B Λ ( H) and similarly the trajec-tory E ( d I =0 ,S =3 / λ , λ ) of H ∗ pole positions correspondsto different values of E exc ( H ∗ ).For each IACCC resonance calculation we benchmarkpart of the corresponding pole trajectory against theComplex Scaling Method (CSM) [34]. The main ingre-dient of the CSM is a transformation U ( θ ) of relativecoordinates r and their conjugate momenta k U ( θ ) r = r e i θ , U ( θ ) k = k e − i θ , (8)where θ is a real positive scaling angle. Applying thistransformation to the Schr¨odinger equation one obtainsits complex scaled version H ( θ )Ψ( θ ) = E ( θ )Ψ( θ ) , (9)where H ( θ ) = U ( θ ) HU − ( θ ) is the complex scaledHamiltonian and Ψ( θ ) = U ( θ )Ψ is the correspondingwave function. For large enough θ , the divergent asymp-totic part of the resonance wave function is suppressedand Ψ( θ ) is normalizable - possible resonant states canthen be obtained as discrete solutions of Eq. (9) [35]. Inorder to prevent divergence of the complex scaled Gaus-sian potential (1) the scaling angle is limited to θ < π .A mathematically rigorous formulation of the CSM fora two-body system results in the ABC theorem [34] whichprovides description of the behavior of a complex scaledenergy E ( θ ) with respect to θ : (i) Bound state ener-gies remain unaffected (ii) The continuum spectrum ro-tates clockwise in a complex energy plane by angle 2 θ from the real axis with its center of rotation at the cor-responding threshold (iii) For θ > θ r = arctan (cid:16) Γ2 E r (cid:17) corresponding to the resonance energy E r and width Γ,the resonance is described by a square-integrable func-tion and its energy and width are given by a complexenergy E ( θ ) = E r − iΓ / θ .In this work, we expand Ψ( θ ) in a finite basis of corre-lated Gaussians (4)Ψ( θ ) = N (cid:88) i =1 c i ( θ ) ψ i . (10)Both resonance energies E ( θ ) and corresponding coeffi-cients c i ( θ ) are then obtained using the c -variational prin-ciple [36] as a solution of generalized eigenvalue problem N (cid:88) j =1 ( ψ i | H ( θ ) | ψ j ) c αj ( θ ) = E α ( θ ) N (cid:88) j =1 ( ψ i | ψ j ) c αj ( θ ) , (11) where ( | ) stands for the c -product (bi-orthogonal prod-uct) [35, 37]. In the case of real ψ i , the c-product inEq. (11) is equivalent to the inner product < | > . Itwas proved that the solutions of Eq. (11) are station-ary in the complex variational space, and for N → ∞ they are equal to exact solutions of the complex scaledSchr¨odinger equation (9) [36]. Nevertheless, with increas-ing number of basis states the solution stabilizes andthere is no upper or lower bound to an exact resonancesolution [38].In fact, due to a finite dimension of the basis set theresonance energy E ( θ ) (11) moves with increasing scal-ing angle along the θ -trajectory even for θ > θ r , featuringresidual θ dependence [35, 39]. It was demonstrated thatfollowing the generalized virial theorem [36, 40] the bestestimate of a resonance energy is given by the most sta-tionary point of the θ -trajectory, i.e. such E ( θ opt ) forwhich the residual θ dependence is minimal but not nec-essarily equal to zero (cid:12)(cid:12)(cid:12)(cid:12) d E ( θ )d θ (cid:12)(cid:12)(cid:12)(cid:12) θ opt ≈ . (12)A real scaling angle θ is frequently used in finite basisCSM calculations with satisfactory results [39, 41, 42].However, identifying the resonance energy with E ( θ opt )using the θ -trajectory (Im( θ ) = 0, Re( θ ) changing) isstill approximate. As pointed out by Moiseyev [38] theresonance stationary condition requires exact equality inEq. (12), which can be achieved in a finite basis set byconsidering complex θ opt . Consequently, taking θ real in-troduces certain theoretical error and it is problematic toquantify how much the result obtained using θ -trajectorytechnique deviates from the true CSM resonance solution(zero derivative in Eq. (12)).Following Aoyama et al. [35] we use both θ -trajectoryand β -trajectories (Re( θ ) fixed, Im( θ ) changing) to lo-cate the position of the true CSM solution. In the abovework it was numerically demonstrated that for certainRe( θ opt ) the θ -trajectory approaches the stationary pointand then starts to move away. On the other hand, the β -trajectories are roughly circles with decreasing radius asthe corresponding Re( θ ) approaches Re( θ opt ). In view oforthogonality of the θ - and β -trajectories at given scalingangle θ , the true CSM solution is then located inside anarea given by circular β -trajectories. More specifically,it is identified as the center of the circular β -trajectorywith the smallest radius where the CSM error is given bythe size of this radius [35].Another non-trivial task is to determine an appropriateyet not excessively large correlated Gaussian basis whichyields stable CSM resonance solution. In this work, weapply the HO trap technique [16] which introduces sys-tematic algorithm how to select such basis. First, weplace a resonant system described by the Hamiltonian H into a harmonic oscillator (HO) trap H trap ( b ) = H + V HO ( b ) , V HO ( b ) = ¯ h mb (cid:88) j We applied the LO /π EFT approach with 2- and 3-body regulated contact terms defined in Eq. (1) tothe study of the s -shell Λ hypernuclei, the Λ nn and H ∗ ( J π = 3 / + , I = 0) systems in particular. In thissection, we present results of the calculations and pro-vide comparison of the results obtained within our LO /π EFT approach and phenomenological models. In a sep-arate subsection, we discuss in detail stability and nu-merical accuracy of the presented SVM and IACCC res-onance solutions.The additional auxiliary 3-body potential V IACCC3 (5)introduced to study continuum states allows us to varythe amount of attraction and thus explore different sce-narios, as demonstrated in Fig. 1. Here, the Λ nn and H ∗ bound state energies E B are plotted as a functionof the strength d I,Sλ of the auxiliary force normalized tothe strength D I,Sλ of the 3-body Λ N N potential of the /π EFT . In the limiting case d I,Sλ /D I,Sλ = − 1, the 3-bodyrepulsion is completely canceled and the systems undergoThomas collapse [43] in the limit of λ → ∞ . For suitablychosen values of d I,Sλ /D I,Sλ between -1 and 0, both Λ nn and H ∗ are bound and one can study implications forthe 4- and 5-body s -shell hypernuclei as will be shownbelow where we tune d I,Sλ to get either Λ nn or H ∗ just d I , S / D I , S [-]5432101 E B [ M e V ] B o un d nn B o un d H * E B ( nn)=-0.001 MeV + n + nB ( H * )=0.001 MeV+ d FIG. 1. The Λ nn and H ∗ bound state energies E B as afunction of d I,Sλ normalized to D I,Sλ for I = 1 , S = 1 / I = 0 , S = 3 / 2, respectively. The calculation is performed forthe Alexander B set of Λ N scattering lengths and λ = 6 fm − . bound by 0.001 MeV. Finally, for the zero auxiliary force d I,Sλ /D I,Sλ = 0 one gets physical solutions, namely con-tinuum states of Λ nn and H ∗ (either resonant or virtualstates). The figure suggests that the value of d I,Sλ /D I,Sλ considerably closer to 0, i.e. much less additional attrac-tion, is needed to get H ∗ bound then in the case of Λ nn .We will now demonstrate that such Λ interactionstuned to bind Λ nn and/or H ∗ are inconsistent with Λseparation energies in A = 4 and 5 hypernuclei. We keep2- and 3-body LECs fixed and fit the attractive strengthof the auxiliary 3-body force, either d I =0 ,S =3 / λ to Λ sep-aration energy B Λ ( H ∗ ) = 0 . 001 MeV or d I =1 ,S =1 / λ tobound state energy E B (Λ nn ) = − . 001 MeV.Consequences of such tuning are illustrated in Fig. 2.Here, we present Λ separation energies B Λ in s -shellhypernuclei, calculated for selected Λ N scatteringlengths and cutoff λ = 6 fm − which already exhibitspartial renormalization group invariance. Variations of d I =0 ,S =3 / λ or d I =1 ,S =1 / λ do not affect the I, S = (cid:0) , (cid:1) three-body channel, consequently, the Λ separationenergy of the hypertriton ground state remains unaf-fected and is not shown in the figure. In order to getthe H ∗ system just bound (left panel), the amountof repulsion in the (cid:0) , (cid:1) three-body channel mustdecrease, which leads in return to overbinding of boththe H ∗ excited state and the He hypernucleus. Thewave function of the H ground state does not includethe (cid:0) , (cid:1) component and thus its B Λ remains intact.As was already noted and demonstrated in Fig. 1, the B [ M e V ] Bound H * ( B ( H * )=0.001 MeV)NSC97f EFT(NLO) EFT(LO) Alex. B Bound nn ( E B ( nn)=-0.001 MeV)NSC97f EFT(NLO) EFT(LO) Alex. B B ( H) B ( H * ) B ( He) B ( H ) exp. B ( H * ) exp. B ( He) exp. FIG. 2. Λ separation energies B Λ from SVM calculations using cutoff λ = 6 fm − and several sets of Λ N scattering lengths fortwo cases - just bound H ∗ (left) and just bound Λ nn (right). Horizontal dotted lines mark experimental values of B Λ . binding of the Λ nn system requires a larger change inthe corresponding auxiliary three-body force. Indeed,decreasing amount of repulsion in the (cid:0) , (cid:1) three-bodychannel induces even more severe overbinding than inthe H ∗ case - B Λ s are more than twice larger thanexperimental values (right panel). We might deduce thatby varying the strength of Λ interactions, it is harderto get Λnn bound - the bound H ∗ state appears morelikely first. This result is in agreement with previousworks [6, 7].In Fig. 3 we show the physical solutions (with no aux-iliary force) corresponding to the /π EFT Hamiltonian H (1). Here, the real Re( E ) and imaginary Im( E ) partsof the Λ nn resonance energy (left panel) and the energy E v of the virtual state H ∗ (right panel) are plotted asa function of the cutoff λ for the Λ N scattering lengthversions listed in Table I. The calculated energies in theboth hypernuclear systems depend strongly on the in-put Λ N interaction strength. In the case of H ∗ , weobtain for all considered Λ N scattering lengths a vir-tual state solution. Namely, in accord with the definitionof a virtual state [30], the imaginary part of the H ∗ pole momentum Im( k ) decreases from a positive value(bound state) to a negative value (unbound state) witha decreasing auxiliary attraction whereas the real partRe( k ) remains equal to zero [30] (as was demonstratedin ref. [16]). On the other hand, in the case of the Λ nn system the /π EFT predicts a resonant state. Moreover,only the NSC97f and χ EFT(NLO) yield Λ N interactionstrong enough to ensure for λ ≥ − the Λ nn pole po-sition in the fourth quadrant of a complex energy plane(Re( E ) > 0, Im( E ) < nn resonance.In Fig. 3 we also demonstrate stability of the solu-tions with respect to the cutoff λ . The calculated en-ergies vary smoothly beyond the value λ = 2 fm − andalready at λ = 4 fm − they stabilize within extrapo-lation uncertainties at an asymptotic value correspond-ing to the renormalization scale invariance limit λ → ∞ .This is illustrated in the right panel, where we presentfor the Alexander B case the extrapolation function andthe asymptotic value including the extrapolation error forthe energy E v of the H ∗ virtual state. It is to be notedthat one might naively expect clear dependence on thestrength of the Λ N spin-triplet interaction which solelyenters the H ∗ hypernuclear part on a two-body level.However, the dominance of the spin-triplet interaction isundermined by 3-body force in the (cid:0) , (cid:1) channel com-pensating the size of the spin-singlet scattering length a Λ N , being fixed by the B Λ ( H ∗ ) experimental value.One could argue that considering different values of a Λ Ns , Λ N N three-body forces or an effect of non-zero ef-fective range r s would open a possibility to locate theΛ nn resonance in the fourth quadrant closer to the realaxis and thus decrease its width Γ. This would certainlyfacilitate its experimental observation. However, Λ N N forces are fixed by experimental B Λ s of 3- and 4-bodyhypernuclear systems. Considering unusually large val-ues of a Λ Ns would allow Λ nn pole position closer to thethreshold but Λ N interactions would have to be rec-onciled again with remaining s -shell systems. At LO /π EFT we would be constrained by a possibility of bound H ∗ and by the experimental value of B Λ ( He).Incorporation of a non-zero effective range for λ → ∞ is restricted by the Wigner bound [44] and leads to per- ]1.501.251.000.750.500.250.000.250.50 R e ( E ) , I m ( E ) [ M e V ] Alexander BNSC97fEFT(LO)EFT(NLO) ]0.250.200.150.100.050.000.05 E v [ M e V ] Alexander BNSC97fEFT(LO)EFT(NLO) FIG. 3. Real Re( E ) (full symbols) and imaginary Im( E ) (empty symbols) parts of the Λ nn resonance energy (left) and energy E v of the H ∗ virtual state (right) as a function of cutoff λ calculated using the IACCC method for several Λ N interactionstrengths. For H ∗ virtual state (right) and Alexander B we perform extrapolation for λ → ∞ . The red dashed line is theextrapolation function, the solid red line and shaded area mark the contact limit and the extrapolation error. turbative inclusion of NLO term [45]. Assuming thatthe total energy of the Λ nn resonance is below 1 MeVone might estimate maximal Λ ( N ) typical momentumas p Λ ≈ √ M Λ E = 47 MeV ( p N ≈ √ M N E = 43 MeV)which yields truncation error of of the LO /π EFT of or-der ( Q m π ) ≈ 3% (( Qm π ) ≈ nn and H ∗ hypernuclear systems in a continuum. There-fore, we find it appropriate to discuss difference of ourapproach with respect to the previous calculations of theΛ nn resonance performed by Afnan and Gibson usinga phenomenological approach [15]. Following their workwe neglect three-body force but instead of separable non-local two-body potentials we employ one range Gaussians V ( r ) = (cid:88) I,S ˆ P I,S C I,S exp (cid:32) − λ I,S r (cid:33) (14)to describe s -wave interaction in nuclear I, S =(0 , , (1 , 0) and hypernuclear I, S = (1 / , , (1 / , P I,S is the projection oper-ator. The parameters C I,S and λ I,S are fitted to thevalues of a s and r s listed in [15]. Moreover, we tookinto account a Λ Ns and r Λ Ns related to Alexander B and χ EFT(LO) given in Table I.The calculated Λ nn pole trajectories for the Phen-2Bpotential (14) are presented in Fig. 4, left panel. Theauxiliary interaction is in a form of three-body force (5)with cutoff λ = 1 fm − . We observe that calculatedphysical pole positions (filled larger symbols) are in good agreement with those presented in [15] (empty symbols).Indeed, as might be expected the position of the near-threshold Λnn resonance is predominantly given by low-momentum characteristics of an interaction - a s and r s which are the same in both cases.In order to reveal the relation between theLO /π EFT and phenomenological approaches discussedabove, one can consider the finite cutoff λ s which givesroughly the same values of r s as used in the above phe-nomenological calculations. Such a value, λ s ≈ . 25 fm − for NSC97f and χ EFT(NLO), yields in addition B Λ ( He)remarkably close to experiment [46]. As explained bythe authors one might understand that λ s absorbs intoLECs NLO contributions of the theory which are likelyto increase its precision, however, success of this proce-dure is not in general guaranteed for all systems. In-deed, higher orders above NLO which behave as powersof ( Q/λ ) are induced as well and are not suppressed by λ → ∞ . In Fig. 4, right panel, we present Λ nn poletrajectories calculated using the /π EFT for this specific λ s value and several Λ N interaction strengths. One no-tices very close positions of the Λ nn resonance calculatedfor χ EFT(NLO) and NSC97f using the Phen-2B poten-tial (left panel) and the /π EFT (right panel). The LO /π EFT for λ = 1 . 25 fm − could thus be considered as asuitable phenomenological model which yields good pre-dictions for 4- and 5- body hypernuclei and hypertriton[20, 46].In addition, in both panels of Fig. 4 we comparethe Λ nn pole positions calculated within the CSM andIACCC method for the same values of d I =1 ,S =1 / λ locatedin the area reachable by the CSM. We might see re-markable agreement between IACCC (dots) and CSM E ) [MeV]1.41.21.00.80.60.40.20.00.2 I m ( E ) [ M e V ] EFT ( = 1.25 fm ) Alexander BNSC97fEFT(LO)EFT(NLO) E ) [MeV]1.41.21.00.80.60.40.20.00.2 I m ( E ) [ M e V ] Phen 2B Alexander BNSC97fEFT(LO)EFT(NLO)Julich04Mod D AG NSC97fEFT(NLO)Julich04Mod D FIG. 4. Trajectories of the Λ nn resonance pole in a complex energy plane determined by a decreasing attractive strength d I =1 ,S =1 / λ for several Λ N interaction strengths. Left panel: Calculations using Λ N and NN phenomenological potential Phen-2B (14). Larger full symbols stand for the physical position of the Λ nn pole ( d I =1 ,S =1 / λ = 0), empty symbols (left panel) markcorresponding solutions obtained by Afnan and Gibson (AG) [15] for the same scattering lengths and effective ranges used tofix potential Phen-2B (14). Right panel: /π EFT calculations for cut-off λ = 1 . 25 fm − . In a region accessible by the CSMwe also show for each IACCC solution (dots) the one obtained by the CSM (crosses) for the same amplitude of the auxiliarythree-body force. (crosses) solutions, which provides benchmark of the cal-culations and demonstrates high precision of our results.In Fig. 5, we show B Λ of remaining s -shell hypernu-clear systems, calculated using the Phen-2B potential(14). The hypertriton ground state H is in most casesoverbound, calculated B Λ ( H) are consistent with thoseobtained by Afnan and Gibson using separable non-localpotentials fitted to the same Λ N interaction strengths[15]. The excited state of hypertriton H ∗ turns to bebound, which is in disagreement with previous theoreti-cal calculations [6, 10]. Heavier s -shell systems are con-siderably overbound as well, regardless of which specificset of a Λ Ns and r Λ Ns is fitted. Overbinding of s -shell hy-pernuclear systems brought about by the Phen-2B inter-action (14) clearly indicates a missing piece which wouldintroduce necessary repulsion. This could be providedby introducing a Λ N N three-body force. In fact, Afnanand Gibson stated that more detailed study of the Λ nn resonance including three-body forces should be consid-ered [15]. In /π EFT additional repulsion is included rightthrough the Λ N N force fitted for each cutoff λ to exper-imental values of B Λ in 3- and 4-body hypernuclei. As aresult, though both the Phen-2B (as well as AG) inter-action and the /π EFT for λ = 1 . 25 fm − yield close posi-tions of the Λ nn resonance (see Fig. 4), the interplay be-tween three-body forces in the /π EFT exhibits large effectwhich completely removes overbinding presented for thePhen-2B interaction in Fig. 5, yielding correct B Λ ( He),exact B Λ ( H), B Λ ( H), and E exc ( H ∗ ) plus unbound H ∗ as presented in Fig. 3. This suggests that the sen- sitivity of the Λ nn system to the three-body Λ N N forceseems to be relatively small. A. Stability and error of continuum solutions In this subsection, we demonstrate stability and ac-curacy of our CSM and IACCC resonance solutions fora particular point of the Λ nn pole trajectory. Moreprecisely, we use the χ EFT(LO) /π EFT interaction with λ = 1 . 25 fm − and the strength of auxiliary three-bodyinteraction d I =1 ,S =1 / λ = − 24 MeV. This specific choicewas motivated by large θ r = arctan( E/ / χ EFT(LO) CSMsolution in the right panel of Fig. 4).Using the CSM in a finite basis we make sure that ourresonant solution is stable and does not change with anincreasing number of basis states. Here, we apply theharmonic oscillator (HO) trap technique [16] with massscale m = 939 MeV (13) which provides us with an ef-ficient algorithm to select an appropriate, yet not exces-sively large CSM basis. For a chosen HO trap length b (13), this procedure yields stochastically optimized ba-sis of correlated Gaussians with a maximal typical ra-dius R max which gets larger as the trap becomes morebroad. We choose a grid of increasing trap lengths b i ranging from 20 fm to 80 fm with 2 fm step and usingHO trap technique for each b i , we prepare 31 different B [ M e V ] B ( H) AG B ( H) Phen-2B B ( H * ) Phen-2B B ( H) exp. B [ M e V ] Julich04 Mod D NSC97f EFT(NLO) EFT(LO) Alex. B B ( He) Phen-2B B ( H) Phen-2B B ( H * ) Phen-2B B ( He) exp. B ( H) exp. B ( H * ) exp. FIG. 5. Λ separation energies B Λ from SVM calculations using various Λ N interaction strengths of the Phen-2B interaction(14). The nuclear part is given by the same form of a phenomenological potential. Experimental values of B Λ are marked bydashed horizontal lines. basis sets. In the next step, we build the CSM basis forour resonance calculation in the following way : First,we fix correlated Gaussian states obtained for the low-est b = 20 fm trap length. Second, we take the basisstates for b = 22 fm leaving out the states which arenearly linear dependent to any of already fixed b corre-lated Gaussians and we merge b and b basis sets. Next,in the same way, we add correlated Gaussians from the b = 24 fm basis set to already fixed b and b states. Wecontinue this procedure for all b i up to certain b max andconstruct our final CSM basis set.The stability of the CSM solution with respect to HOtrap length b is illustrated in Fig. 6. Here, we present cal-culated real and imaginary parts of the Λ nn resonanceenergy using different CSM bases obtained combiningHO trap sets up to a certain b max . Black dots stand forthe most stationary point of the resonance θ -trajectory E CSMΛ nn ( θ opt ) for which (cid:12)(cid:12) d E d θ (cid:12)(cid:12) θ opt is minimal. Shaded ar-eas then show the spread of resonance energy E CSMΛ nn ( θ )within the θ opt ± ◦ range (darker shaded area) and the θ opt ± ◦ range (lighter shaded area) thus indicating thelevel of the CSM resonance energy dependence on thescaling angle θ (8). Calculated Λ nn resonance energystabilizes already using the CSM basis constructed for b max = 36 fm. It is clearly visible that considering higher b max and thus including more basis states does not affectthe CSM solution.In Fig. 7 we show the calculated θ -trajectory and sev-eral β -trajectories for two different CSM bases whichwere obtained for b max = 24 fm (left panel) and for b max = 80 fm (right panel). For b max = 24 fm we can 20 30 40 50 60 70 800.3000.3250.3500.375 R e ( E ) [ M e V ] 20 30 40 50 60 70 80 b max [fm]0.720.700.680.66 I m ( E ) [ M e V ] FIG. 6. Stability of the Λ nn CSM resonant solution E ( θ ) = Re( E ( θ )) + iIm( E ( θ )) as a function of increasing HOtrap length b max . Black dots show the most stationary pointof the θ -trajectory E ( θ opt ). Darker shaded area shows un-certainty of E ( θ ) within θ opt ± ◦ range, lighter shaded areashows the same within θ opt ± ◦ range. The particular poleposition was calculated for /π EFT interaction with χ EFT(LO)Λ N scattering lengths and λ = 1 . 25 fm − , strength of auxil-iary three-body force was set to d I =1 ,S =1 / λ = − 24 MeV. clearly see that β -trajectories are not circular and man-ifest highly unstable behaviour due to poor quality ofthe employed basis set. In fact, we have already pointedout in Fig. 6 that the Λ nn resonance solution stabilizesat least for b max = 36 fm. Using the CSM basis for0 E ) [MeV]0.710.700.690.680.670.660.650.640.63 I m ( E ) [ M e V ] 40 41 4345 363738394041 42 43 44 HO b max = fm -trajectory-trajectory E ) [MeV]0.710.700.690.680.670.660.650.640.63 I m ( E ) [ M e V ] HO b max = fm -trajectory-trajectory FIG. 7. Λ nn resonance θ -trajectory (Im( θ )=0; black solid line) and β -trajectories (colored dotted lines) showing movementof corresponding E ( θ ) as a function of θ in the complex energy plane. Trajectories are calculated for two different CSM basissets which were obtained combining HO trap sets up to b max = 24 fm (left panel) and up to b max = 80 fm (right panel). β -trajectories are presented for several different Re( θ ) changing Im( θ ) from 0 to 0.44 radians with 0.01 step. Black cross inthe left panel indicates estimated Λ nn resonance position of the true CSM solution satisfying Eq. (12). Shaded gray area thenshows corresponding CSM error. Λ nn calculation is performed using the same interaction as in Fig. 6. b max = 80 fm (right panel) our results are stable show-ing almost circular β -trajectories characterised by theirdecreasing radius as the corresponding Re( θ ) approachesRe( θ opt ) ≈ ◦ . The β -trajectory for Re( θ ) = 41 ◦ ex-hibits oscillatory behaviour within a small region aroundthe true CSM solution. We assume that this effect isrelated to a finite dimension of our CSM basis set andcorresponding circular trajectory would be recovered byconsidering more basis states. The most probable Λ nn resonance energy E CSMΛ nn is in the center of the grey shadedcircle while its radius defines the error of our true CSMsolution. In this particular case, the Λ nn resonance en-ergy is E CSMΛ nn = 0 . − i 0 . nn resonance energies E IACCCΛ nn using different degrees ( M, N ) of the Pad´e ap-proximant P ( M,N ) (6). As expected, calculated E IACCCΛnn start to stabilize with increasing ( M, N ). The IACCCsolution saturates already for (7,7) and does not improvedramatically with further increase of ( M, N ). This ispredominantly explained by finite precision of our SVMbound state energies which are used to fix the parame-ters of P ( M,N ) and by numerical instabilities which slowlystart to affect our IACCC solution at higher degrees ofthe approximant. Comparing saturated IACCC solutionobtained with different ( M, N ) ranging from (7,7) up to(13,13) we estimate for this specific example the E IACCCΛ nn accuracy ∼ × − MeV. Despite considerable differ-ence between IACCC and CSM, both approaches predictremarkably consistent Λ nn resonance energies. In fact, all presented IACCC energies starting from the Pad´e ap-proximant of degree (7,7) and higher lie within the errorsof the corresponding CSM prediction.Dependence of our IACCC calculations of the H ∗ virtual state energy E IACCCv , H ∗ on different degrees ofthe Pad´e approximant is demonstrated in Table II aswell. In this particular case we use as an example the /π EFT interaction with the χ EFT(LO) Λ N scatteringlengths, cut-off λ = 1 . 25 fm − , and no auxiliary inter-action. We see that the H ∗ solution starts to stabilizealready for P (4 , and it is approximately by two ordersmore accurate than the solutions for the Λ nn resonance.The reason is that the H ∗ virtual state lies in the vicin-ity of the Λ + d threshold, analytical continuation fromthe bound region is thus not performed far into the con-tinuum, which enhances the IACCC precision.The uncertainty of our IACCC resonance solutions inthe fourth quadrant of a complex energy plane (Re( E ) > 0, Im( E ) < 0) does not exceed ≈ × − MeV. AllIACCC results are crosschecked by the CSM in a regionof its applicability determined by the maximal resonanceangle θ r ≈ ◦ for which our complex scaling results arestill reliable. Up to this point the CSM solution possessesthe same minimal accuracy as the IACCC solution, how-ever, for higher θ r approaching the limiting value 45 ◦ theCSM solution quickly starts to deteriorate due to numer-ical instabilities.Subthreshold resonance positions are calculated withinthe IACCC method. For poles residing deeper in this re-gion of a complex energy plane (Re( E ) < 0, Im( E ) < TABLE II. Stability of the Λ nn resonance energy E IACCCΛ nn and H ∗ virtual state energy with respect to the Λ + d threshold E IACCCv , H ∗ calculated within the IACCC for increasing degree ( M, N ) of the Pad´e approximant. Λ nn calculation is performedusing the same interaction as in Fig. 6. Position of the H ∗ virtual state is determined for /π EFT interaction with χ EFT(LO)Λ N scattering lengths and λ = 1 . 25 fm − with no auxiliary three-body force, i.e. d I =0 ,S =3 / λ = 0 MeV. E diff stands forthe difference between absolute values of IACCC solution calculated for two neighbouring Pad´e approximants E ( M,N )diff = | E ( M,N ) | − | E ( M − ,N − | . All energies are given in MeV.( M, N ) E IACCCΛ nn | E IACCCΛ nn | E diff (Λ nn ) E IACCCv , H ∗ E diff ( H ∗ )(3,3) -0.0588 - i0.5605 0.5636 -0.04216(4,4) 0.3367 - i0.7041 0.7805 0.2169 -0.05192 0.00976(5,5) 0.2965 - i0.6559 0.7198 -0.0652 -0.05154 -0.00038(6,6) 0.2941 - i0.6770 0.7381 0.0183 -0.05161 0.00007(7,7) 0.3003 - i0.6796 0.7430 0.0050 -0.05160 -0.00001(8,8) 0.2997 - i0.6796 0.7427 -0.0003 -0.05160 < − (9,9) 0.3001 - i0.6796 0.7429 0.0002 -0.05156 -0.00004(10,10) 0.3014 - i0.6791 0.7430 0.0001 -0.05159 0.00003(11,11) 0.3012 - i0.6795 0.7433 0.0003 0.05160 0.00001(12,12) 0.3020 - i0.6757 0.7401 -0.0032 -0.05160 < − (13,13) 0.3026 - i0.6765 0.7411 0.0010 -0.05161 0.00001 the precision of our results, predominantly of the imagi-nary part Im( E ), decreases. For Re( E ) ∈ ( − . , 0) MeVthe maximal error of Im( E ) is ≈ × − MeV, forRe( E ) ∈ ( − . , − . 25) MeV it is ≈ . 03 MeV, and forRe( E ) ∈ ( − . , − . 5) MeV it is ≈ . ≈ − MeV in all considered cases. IV. CONCLUSIONS In the present work, we have studied few-body hy-pernuclear systems Λ nn and H ∗ ( J π = 3 / + , I = 0)within a LO /π EFT with 2- and 3-body regulated con-tact terms. The Λ N LECs were associated with Λ N scattering lengths given by various interaction modelsand the Λ N N LECs were fitted to known Λ separationenergies B Λ in A ≤ E exc ( H ∗ ). Few-body wave functions were de-scribed within a correlated Gaussians basis. Bound statesolutions were obtained using the SVM. The continuumregion was studied by employing two independent meth-ods - the IACCC and CSM. Our LO /π EFT approach,which accounts for known s -shell hypernuclear data, rep-resents a unique tool to describe within a unified inter-action model 3-, 4-, 5- and 6-body hypernuclar systems– single- and double-Λ hypernuclei including continuumstates. In that it differs from other similar studies whichfocused solely on few particular hypernuclei. Moreover, the /π EFT approach allows us to develop systematicallyhigher orders corrections, assess reliably precision of cal-culations and evaluate errors of their solutions.The additional auxiliary 3-body potential introducedto study Λ nn and H ∗ continuum states allows us to ex-plore different scenarios. Fixing the attractive strengthof the auxiliary force in order to get these systems justbound yields considerable discrepancy between calcu-lated and experimental B Λ s of 4- and 5-body s -shell hy-pernuclei. Our conclusions thus ruled out the possibilityfor the existence of bound Λ nn and H ∗ states, which isin accord with conclusions of previous theoretical stud-ies [4–7, 9, 10]. Moreover, we found that by increasingthe strength of the Λ attraction, the onset of the H ∗ comes before the Λ nn binding. The experimental evi-dence for the bound Λ nn state reported by the HypHIcollaboration [3] would thus imply existence of the boundstate H ∗ .On the basis of our /π EFT calculations with the auxil-iary force set to zero, we firmly conclude that the excitedstate H ∗ is a virtual state. On the other hand, the Λ nn pole located close to the three-body threshold in a com-plex energy plane could convert to a true resonance withRe( E ) > N interactions [e.g.,for NSC97f and χ EFT(NLO)] but most likely does notexceed E r ≈ . . ≤ Γ ≤ . 00 MeV. Even larger width wouldbe obtained for a rather weak Λ N interaction strengthbut it does not yield experimentally observable Λnn pole.On the contrary, the observation of a sharp resonancewould definitely attract considerable attention since itwould signal that the Λ N interaction at low-momenta isstronger than most Λ N interaction models suggest.Besides the model dependence of our calculations we2explored the stability of solutions with respect to thecutoff parameter λ . We demonstrated that already for λ = 4 fm − the calculated energies stabilize close to theasymptotic value corresponding to the renormalizationscale invariance limit λ → ∞ . We anticipate that thetruncation error, describing effects of higher order cor-rections, is small due to low typical momenta and doesnot change our results qualitatively. In a region acces-sible by the CSM we performed comparison of the CSMwith IACCC method, which yielded highly consistent so-lutions, hence proving reliability of our results. More-over, we verified that our CSM solutions for Λ nn arestable with respect to the considered number of basisstates. Exploring both the θ and β trajectories of theΛ nn pole for one particular case we set the true CSMsolution including its error. The stability of the IACCCmethod with respect to the degree of the employed Pad´eapproximant was investigated and the uncertainty of thecalculations was assessed.A rather different situation occurs when we considerjust 2-body phenomenological interactions fitted to N N and Λ N scattering lengths and effective ranges. We thenobtain subthreshold Λ nn pole positions close to those ofAfnan and Gibson [15]. However, these interactions failto describe other few-body Λ hypernuclei. The predictedoverbinding of the s -shell hypernuclei induced by thesephenomenological 2-body interactions indicates a miss-ing repulsive part of the Λ interaction. In the /π EFT ,it is provided by an additional Λ N N /π EFT calculations revealed thatthe results of Afnan and Gibson could be reproducedfor the finite cutoff value λ s ≈ . 25 fm − . However,thanks to the repulsive Λ N N force the s -shell hyper- nuclear data are now described successfully. The LO /π EFT with λ s ≈ . 25 fm − could thus be considered asa suitable phenomenological model.Our method presented here can be directly applied tothe double-Λ hypernuclear continuum using the recentlyintroduced ΛΛ extension of a LO /π EFT [21]. It is highlydesirable to explore possible resonances in the neutralΛΛ n and ΛΛ nn systems or in the H hypernucleus,where a consistent theoretical continuum study has notbeen performed yet. Indeed, an example of its impor-tance is the continuing ambiguity in interpretation of theAGS-E906 experiment [47] referred to as the E906 puz-zle. 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