The Continuum Spectrum of Hypernuclear Trios

The spectrum of hypernuclear trios composed of a $\Lambda$ baryon and two nucleons is the subject of an ongoing experimental campaign, aiming to study the interaction of the $\Lambda$ particle with a neutron, and the 3-body $\Lambda$-nucleon-nucleon force. In this manuscript we utilize baryonic effective field theory at leading order, constrained to reproduce the available low energy light hypernuclear data, to study the continuum spectrum of such hypernuclear trios. Using the complex scaling method and the inverse analytic continuation in the coupling constant method we find the existence of a virtual state in the $\Lambda n p$ $J^{\pi}=\frac{3}{2}^{+}$ channel, leading to cross-section enhancement near threshold. For the $\Lambda n n$ $J^{\pi}=\frac{1}{2}^{+}$ channel we predict a resonance state. Depending, however, on the value of the $\Lambda N$ scattering length, the resonance pole moves from the physical to the unphysical complex energy sheet within the experimental bounds.


I. INTRODUCTION
Understanding the interaction between nucleons and a Λ hyperon is the subject of an ongoing experimental and theoretical campaign [1]. In the last few years much effort is dedicated to the study of hypernuclear trios (ΛN N ) aiming to determine the unknown Λ-neutron (Λn) interaction, and the ΛN N 3-body force. The latter is known to have a crucial effect in the nuclear equation of state at high density, and therefore on our understanding of neutron stars.
The Λ-nucleon interaction is not strong enough to bind a ΛN pair, making the hypertriton 3 Λ H(I = 0, J π = 1/2 + ) the lightest hypernuclei. It is weakly bound with a Λ separation energy B Λ = 0.13 ± 0.05 MeV [2]. The experimental search for other bound hypernuclear trios has found no evidence for the hypertriton state 3 Λ H * , 3 Λ H(I = 0, J π = 3/2 + ), indicating that the singlet s = 0 ΛN interaction is somewhat stronger than the triplet s = 1 interaction. Recently, the HypHI collaboration [3] has claimed evidence for a bound Λnn state, 3 Λ n(I = 1, J π = 1/2 + ). However, this observation contradicts theoretical analyses demonstrating that such a bound state cannot exist. Since the first calculation by Dalitz and Downs [4], numerous theoretical studies of I = 0, 1 and J = 1/2, 3/2 ΛN N states have been performed, confirming the observation that no bound Λnn and 3 Λ H(I = 0, J π = 3/2 + ) exist within Faddeev calculations for separable potentials [5,6], chiral constituent quark model of Y N interactions [7,8] or the Nijmegen hyperon-nucleon potentials [9]. The same conclusion was drawn in [10] within variational calculations using Y N model, simulating the realistic Nijmegen interaction. The Λnn system was also studied within a baryonic (pionless) effective field theory (/ πEFT ) [11,12], however, due to uncertainty in fixing the three-body Λnn force no firm predictions of its stability could be made.
In spite of the theoretical consensus regarding a bound Λnn, the nature of hypernuclear ΛN N trios remains a subject of an ongoing discussion [13]. Specifically, the search for the Λnn system is a goal of the JLab E12-17-003 experiment [14], and the study of the 3 Λ H(I = 0, J π = 3/2 + ) state is part of the JLab proposal P12-19-002 [15].
Regardless the apparent interest, the possible existence of Λnn and 3 Λ H * hypernuclear continuum states has been directly addressed in only few theoretical works. Calculating zeros of the three-body Jost function, Belyaev et al. found a very wide, near-threshold, Λnn resonance [16]. Afnan and Gibson [17] using Faddeev calculation and separable potentials, fitted to reproduce ΛN and N N scattering length and effective range, concluded that the Λnn state exists as a sub-threshold resonance. They also found that a small increase of the ΛN interaction strength shifts the resonance position above threshold and thus yields an observable resonance. We are not aware of any direct calculation of the 3 Λ H * continuum state, however, as Garcilazo et al. concluded, there is a hint of near-threshold pole which gives rise to large Λd scattering length in S = 3/2 channel [7].
The aforementioned continuum studies [7,8,16,17] were limited to A = 3 systems. Therefore, the predictive power of their interaction models was not verified against the available experimental B Λ data in e.g. 4-body or 5-body s-shell hypernuclei. In fact, applying a gaussian potential mimicking the low energy behavior of the separable potential of [17] we find substantial overbinding in these systems. Given the relatively poorly known ΛN scattering parameters, and the precise B Λ data, such comprehensive study is called for.
Motivated by the debate regarding the nature of the hypernuclear 3-body states, and the soon to be published JLab E12-17-003 Λnn results [14], in the present work we report on precise few-body calculations of the hypernuclear ΛN N bound and continuum spectrum, using Hamiltonians constructed at leading order (LO) in / πEFT [18]. This / πEFT is an extension, including Λ hyperons, of the n, p nuclear / πEFT Hamiltonian, first reported in [19,20] and more recently used to study lattice-nuclei in [21][22][23][24]. At LO / πEFT contains both 2-body and 3-body contact interactions. The theory's parameters, i.e. the 2-and 3-body low-energy constants (LECs), were fitted to reproduce the ΛN, N N scattering lengths, 3 H binding energy, and the available 3,4body B Λ data [18]. The predictive power of the theory was tested against the measured 5 Λ He separation energy [18,49]. The / πEFT breakup scale can be associated with 2-pion exchange 2m π , or the threshold value for exciting ΣN pair. These two values are remarkably close. Assuming a typical energy scale E Λ of about 1 MeV, the momentum scale Q ≈ 2M Λ E Λ = 47 MeV/c, suggesting a / πEFT expansion parameter (Q/2m π ) ≈ 0.2. This implies a / πEFT LO accuracy of order (Q/2m π ) 2 ≈ 4%. The 3-body calculations were performed with the Stochastic Variational Method (SVM) expanding the wave function on a correlated gaussian basis [25,26], the continuum states were located using the Complex Scaling Method (CSM) [34], or the Inverse Analytic Continuation in the Coupling Constant (IACCC) Method [48].
Our main findings are: (a) The possible existence of a bound Λnn, or 3 Λ H * state is ruled out, confirming findings of previous theoretical studies [4-10, 16, 17]. (b) The excited state of hypertriton, 3 Λ H * (J π = 3/2 + ), is a virtual state. (c) The Λnn state is a resonance pole near the three-body threshold in a complex energy plane. The position of this pole depends on the value of the ΛN scattering length. Within the current bounds on the ΛN scattering length it can either be a real resonance or a sub-threshold resonance.

II. CALCULATIONAL DETAILS
A. Hypernuclear / πEFT at LO At LO the / πEFT of neutrons, protons and Λ-hyperons is given by the Lagrangian density (1) where N and Λ are nucleon and Λ-hyperon fields, respectively, and L 2B , L 3B are 2-body, and 3-body, s-wave contact interactions, with no derivatives. These contact interactions are regularized by introducing a local gaussian regulator with momentum cutoff λ, see e.g. [27], that smears the Dirac delta appearing in the contact terms over distances ∼ λ −1 . This procedure yields Hamiltonian containing two-body V 2 and three-body V 3 interactions where P I,S ij and Q I,S ijk are the 2-and 3-body projection operators into an s-wave isospin-spin (I, S) channels. The cutoff λ dependent parameters C I,S λ , and D I,S λ are the 2-and 3-body LECs, fixed for each λ by the appropriate renormalization condition. For λ higher than the breakup scale of the theory (λ > 2m π ), observables posses residual cutoff dependence, at LO O(Q/λ), suppressed with λ approaching the renormalization group invariant limit λ → ∞ [18].
In are not well constrained by experiment, we consider different values both as given by direct analysis of experimental data [28], or as predicted by several ΛN interaction models [29][30][31], see Table I. For the particular values of the LECs see [18]. The A-body Schrödinger equation is solved expanding the wave function Ψ in correlated gaussians basis [25] whereÂ stands for the antisymmetrization operator over nucleons, x = (x 1 , ..., x A−1 ) denotes a set of Jacobi vectors, and χ i SM S (ξ i IM I ) is the spin (isospin) part. The information about interparticle correlations is contained in the (A − 1) dimensional positive-definite symmetric matrix A i . Once we fix all basis functions ψ i , both energies and coefficients c i are obtained through diagonalization of the Hamiltonian matrix. The A(A − 1)/2 nonlinear variational parameters contained in each A i matrix are determined using the Stochastic Variational Method (SVM) [25,26].
Unlike bound states, continuum wave functions are not square-integrable. Therefore, resonances or virtual states can not be directly described using an L 2 basis set of correlated gaussians. Techniques such as CSM or IACCC have to be used to study such states with a correlated gaussians. Below we discuss in some detail the techniques we applied in our study.

C. The Complex Scaling Method
The CSM [34] is a reliable tool to study few-body resonances [35]. The basic idea in the CSM is to locate resonances introducing complex rotation of coordinates and momenta that transforms the continuum states into integrable L 2 states. This transformation rotate continuum state energies by 2θ uncovering a section of the second energy plane between the real axis and a ray defined by |argE| = 2θ, exposing resonances with argument θ r = arctan(Γ/2E r )/2 smaller than θ. Using gaussian regulator (3) the rotation angle is restricted to be θ < π 4 , to prevent divergence of the rotated gaussian, limiting the scope of the CSM.
The SVM method use the variational principle as a tool to optimize the nonlinear basis parameters A i (4), minimizing the basis size. This do not apply to resonance states, making it a highly non trivial problem to choose the appropriate basis. Here, we present a new efficient procedure to determine the basis set for an accurate description of resonance states. To optimize the basis, we supplement the Hamiltonian H with an additional harmonic oscillator (HO) trap where m is an arbitrary mass scale, and b is the HO trap length. The potential V HO (b) gives rise to a HO spectrum of the ground and excited states which is affected by the presence of a resonance in the Hamiltonian H [42]. For a given trap length b we select basis states ψ i (4) using the SVM, optimizing the variational parameters for the ground state energy and then subsequently for excited states energies up to E max > E r + Γ/2. The SVM procedure prefers basis states which promote interparticle distances r jk in a specific region given by the trap length b. Increasing b we enlarge the typical radius of the correlated gaussians ψ i . For large enough b, the CSM resonance solution for the Hamiltonian H starts to stabilize and both the short range and the suppressed long range asymptotic parts of a resonance wave function are described sufficiently well. In order to further enhance the accuracy of our CSM solution, we use a grid {b k }, of a HO trap lengths, and for each grid point we independently select correlated gaussians basis. Then we merge basis states determined for each b k into a larger basis while ensuring linear independence and numerical stability of the overlap matrix. We have found that this procedure works well for both narrow, and broad resonances.

D. Inverse Analytic Continuation in the Coupling Constant Method
The Analytic Continuation in the Coupling Constant (ACCC) method [43] has been successfully applied in various calculations of few-body resonances and virtual states [44,45]. Moreover, it was pointed out that the ACCC method provides rather convenient way how to extend applicability of the SVM into the continuum region [44,46]. We consider a few-body Hamiltonian consisting of the physical part H and an auxiliary attractive potential V aux which introduces a bound state for a certain value of α, but ensures that the physical dissociation thresholds for the various subsystems remain unaffected. By decreasing the strength α the bound state moves closer to the threshold and for a certain α 0 it turns into a resonance or virtual state. It has been demonstrated for a two-body system that in the vicinity of the branching point α 0 the square root of an energy k = √ E behaves as k ≈ (α−α 0 ) for s-wave (l = 0) and k ≈ √ α − α 0 for l > 0 [43]. Defining new variable x = √ α − α 0 one obtains two branches k(x) and k(−x) where the former one describes motion of the S-matrix pole assigned to a bound state on a positive imaginary k-axis to the third quadrant of a k-plane. Using analyticity of the function k(x) one can continue from a bound region α > α 0 to a resonance region α < α 0 . In practice this is done by constructing a Padé approximant for the function k(x) using M + N + 1 bound state solutions {(x j , k j ); j = 1, . . . , M + N + 1} for different values of α > α 0 . The evaluation of the Padé approximant (8) at x = √ −α 0 yields complex k which is assigned to the physical resonance solution k 2 = E r − iΓ/2 corresponding to the Hamiltonian H. For more details regarding the ACCC method see [47].
The ACCC method suffers from two drawbacks which are predominantly of numerical nature. The first issue is high sensitivity of the numerical solution to precise determination of the branching point value α 0 [43]. The second obstacle appears with increasing orders M and N of the Padé approximant (8) when the numerical solution starts to deteriorate.
Rather recently Horáček et al. [48] have introduced a modified version of the ACCC method called the Inverse Analytic Continuation in the Coupling Constant (IACCC) method which provides more robust numerical stability. Starting in the same manner as in the ACCC case, we consider the Hamiltonian (7) and calculate series of bound states for different values of α > α 0 . Next, we construct a Padé approximant of a function α(κ), where κ = −ik, using a relevant set of bound state solutions The parameters of the physical resonance or virtual state pole are then readily obtained by setting α = 0 as the physical root of a simple polynomial equation P M (κ) = 0.
To ensure that the properties of the 2-body part of the Hamiltonian, such as scattering lengths or deuteron binding energy, remain unaffected, we choose the auxiliary potential to be an attractive 3-body force. The natural choice is to select it to have the same form as the / πEFT 3-body potential (3), where the amplitude d I,S λ defines its strength, corresponding to the parameter α in Eq. (7), and is negative for an attractive auxiliary potential.
The accuracy of our IACCC resonance solutions in the fourth quadrant of the complex energy plane, Re(E) > 0, Im(E) < 0, are better then ≈ 10 −3 MeV. These results compare very well with the CSM calculations in their region of applicability θ < π/4.

III. RESULTS
Using / πEFT at LO with the LECs fitted to the available data as described earlier [18], we find no bound Λnn or 3 Λ H * states. Further examining the hypothetical existence of these states, we found that they are incompatible with the well measured A = 4, 5 hypernuclear spectrum.

A. A Λnn resonance?
We start our study of three-body hypernuclear continuum states with the Λnn system. To understand the cutoff dependence of our theory we present, in Fig. 1 , the resonance poles move along a circular trajectory in the complex energy plane starting from the Λ + n + n threshold to the physical end point where d I=1,S=1/2 λ = 0. From the figure it can be seen that the the trajectories E Λnn (d I=1,S=1/2 λ , λ) and the physical end points converge with increasing cutoff, and already at λ = 2.5 fm −1 we reach stabilized results.
Repeating the same calculations for all sets of scatter-ing lengths given in Table I, we find that regardless the cutoff value, the imaginary part of the physical solution Im(E λ Λnn ) lies in the interval −1.32 ≤ Im(E λ Λnn ) ≤ −0.58 MeV for all a ΛN s sets. In contrast, the real part Re(E λ Λnn ) exhibit large cutoff dependence. As shown in Fig. 1 for the NSC97f case, the pole moves with increasing λ from the unphysical part of the Riemann sheet (Re(E) < 0, Im(E) < 0; third quadrant) towards the physical one (Re(E) > 0, Im(E) < 0; fourth quadrant).   Table I, at cutoff λ = 4 fm −1 . From the figure, we can deduce that the existence of a physically observable Λnn resonance is very sensitive to the ΛN interaction. The latter must be strong enough to ensure the pole's location in the fourth quadrant of a complex energy plane. The figure and Table I show that with increasing size of the spin-averaged scattering lengthā ΛN = 3/4a ΛN 1 + 1/4a ΛN 0 the Λnn pole trajectories move closer to the Λ + n + n threshold. Moreover, by increasing the cutoff λ the physical Λnn pole is shifted closer to or into the fourth quadrant. In this sense the pole position in the renormalization group invariant limit λ → ∞ could be considered as the most favorable to the existence of an observable resonance. Nevertheless, in the λ → ∞ limit only two sets of a ΛN s -NSC97f and χEFT(NLO) undoubtedly predict a physical resonance. From the results shown in Fig.2  is the minimal value for the Λnn pole to enter the fourth quadrant, becoming a physical resonance. It should be noted that though the size ofā ΛN plays a dominant role, one should take into account also the effect of the three-body force which might introduce more complicated dependence on a ΛN 0 and a ΛN 1 .
B. The hypertriton excited state 3 Λ H * (J π = 3/2 + ) The excited state of the hypertriton 3 Λ H * (J π = 3/2 + ) might be considered as a good candidate for a nearthreshold resonance. Indeed, several works demonstrated an emergence of a bound state by increasing rather moderately the ΛN interaction strength. Applying the IACCC method we follow the pole trajectory given by the amplitude of auxiliary 3-body force d I=0,S=3/2 λ from a bound region to its physical position in a Λ+deuteron (Λ+d) continuum. In Fig. 3 , as a function of d I=0,S=3/2 λ for Alexander B ΛN scattering lengths and λ = 6 fm −1 . We observe that with a decreasing auxiliary attraction the imaginary part of the momentum Im(k) decreases from positive value (bound state) to a negative value (unbound state) whereas the real part Re(k) remains equal to zero. This behavior is regarded as definition of a virtual state [50].
Repeating the calculations for various cutoffs and different ΛN scattering lengths, Table I, we find 3 Λ H * to be a virtual state in all considered cases. As we have seen in the Λnn calculations, the energy of the virtual state E v is stabilized at cutoffs λ ≥ 4 fm −1 .
The existence of the 3 Λ H * virtual state is further confirmed by the CSM. We do not see any sign of resonance for all sets of ΛN scattering lengths, cutoffs, or auxiliary 3-body force values d I=0,S=3/2 λ . Odsuren et al. [51] have showed that the rotated discretized CSM continuum spectra reflect phenomena such as near-threshold virtual states, although one would naively assume that virtual states having |argE| = π/2 are beyond the reach of the CSM. From continuum level density they have extracted the scattering phase shifts which revealed enhancement due to the vicinity of a the pole [51,52]. Following this approach we calculated the Λd s-wave phase shifts δ Λd The scattering length a Λd 3/2 and effective range r Λd 3/2 extracted from the Λd phase shifts reveal through their sign, negative a Λd 3/2 and positive r Λd 3/2 , the existence of a virtual state [53]. Using a Λd 3/2 , r Λd 3/2 the virtual state binding mo- In is undermined by the 3-body force in the (I, S) = (0, 3/2) channel, fixed by B Λ ( 4 Λ H * ). Comparing the IACCC and CSM results, one clearly see that both approaches are in mutual agreement, they exhibit the same dependence on the ΛN interaction strength, though, the CSM yields larger estimates for |E v |. It is a well known drawback of the CSM that eigenvalues in a vicinity of the threshold start to be affected by inaccuracies caused by complex arithmetic.
Concluding this section, we see that at LO / πEFT firmly predicts the excited state of hypertriton 3 Λ H * (J π = 3/2 + ) to be a virtual state in the vicinity of the Λ − d threshold. This result has important implications for prospective experimental search of this state. Experimental observation of 3 Λ H * as a resonance state seems to be highly unlikely. Instead, there is a nearthreshold virtual state which should be seen through the enhancement of s-wave Λd phase shifts in the J = 3/2 channel as demonstrated in Fig. 4.

IV. CONCLUSIONS
In this work we have presented the first comprehensive / πEFT study of continuum hypernuclear ΛN N trios. The underlying nucleon and hyperon interactions were described within a / πEFT at LO, with the LECs fixed by 2-body low energy observables and experimental input from 3-and 4-body s-shell systems. The Λnn and 3 Λ H * energies were then obtained as predictions of the theory. In view of poor low energy ΛN scattering data we considered several sets of ΛN scattering lengths, whereas the N N interaction remained constrained by experiment [18].
Few-body wave functions were described within a correlated gaussians basis. Bound state solutions were obtained using the SVM. The continuum region was studied employing two independent methods -the IACCC method and CSM.
The / πEFT predicts that both Λnn and 3 Λ H * are unbound. Tuning the 3-body LECs to put the Λnn or 3 Λ H * binding energy on threshold, yielded considerable discrepancy between the calculated and measured B Λ in the A = 4, 5 hypernuclei. Our findings further strengthen the conclusions of previous theoretical studies that both states are unbound [4-10, 16, 17].
Our LO / πEFT calculations predict Λnn and 3 Λ H * to be near-threshold continuum states. We thus anticipate that the EFT truncation error is small due to low characteristic momenta and thus higher order corrections would not change our results qualitatively. We conclude that position of the Λnn pole depends strongly on the spin independent scattering lengthā ΛN . Forā ΛN ≥ 1.7 fm −1 the Λnn pole becomes a physical resonance close to threshold with E r ≤ 0.3 MeV, and a large width most likely in the range 1.16 ≤ Γ ≤ 2.00 MeV. If observed, the position of the Λnn resonance can yield tight constraints on the ΛN scattering length. We note, owever, that the exact position of the Λnn depends on both on a ΛN 0 and a ΛN 1 , and also on subleading / πEFT terms neglected here. The excited state of hypertriton 3 Λ H * was firmly predicted to be a near-threshold virtual state regardless of the value of a ΛN s . We have demonstrated that this virtual state has a strong effect on the Λd s-wave phase shifts in J π = 3/2 + channel.