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Introduction
Recent progress in strangeness S = −2 nuclear physics [1], in particular the observation of nuclear bound states of Ξ − − 14 N ( 15 Ξ C) [2][3][4] and possibly Ξ − − 11 B ( 12 Ξ Be) [5], and evidence from femtoscopic measurements for an attractive Ξ − p interaction [6,7] have again triggered considerable interest in studying Ξ hypernuclei theoretically [8][9][10][11][12][13] despite large uncertainties in the underlying ΞN interaction.The latter is due to the absence of direct hyperon-hyperon (YY) and ΞN scattering data, and the overall extremely scarce empirical information on S = −2 systems.There are several ΛΛ hypernuclei unambiguously determined in experiments, with the sshell 6  ΛΛ He [14,15] being the lightest one.Given huge challenges in experimental identifications of Ξ states, the possible existence of s-shell Ξ hypernuclei remains by and large an open question.Thus, theoretical predictions for Ξ hypernuclei especially for light systems are of great importance.Results from such studies can provide useful guidelines for experimentalists in searching for Ξ bound states [16,17].
In this work we explore the possible existence of light Ξ hypernuclei up to A = 7.For the lightest system, 3  Ξ H, several calculations can be found in the literature [18][19][20][21].Not surprisingly, the predictions strongly depend on the interaction models used.For example, Garcilazo et al. [18] and Hiyama et al. [20] both obtained a deeply bound 3  Ξ H (J π = 3/2 + , T = 1/2) state employing an effective ΞN potential that mimics the phase shifts of the Nijmegen ESC08c potential [22].On the other hand, for interactions derived within chiral effective field theory (EFT) [23] or for ΞN potentials deduced from lattice QCD simulations by the HAL QCD Collaboration [24], the system is found to be unbound [20,21].It has to be said that a strongly attractive ΞN force as suggested by that ESC08c potential, which even yields a ΞN two-body bound state [22], is not supported by the currently available empirical constraints [25] including the aforementioned femtoscopic measurements, and also not by lattice simulations at almost physical masses [24].Note that in the work by Miyagawa et al. [21] the original ΞN interactions have been used directly to obtain the t matrices entering the Faddeev equations for the three-baryon bound state.For the solution of these equations, however, only ΞNN channels are considered.The variational calculation of Hiyama et al. [20] is based on interactions where the various Y Y channels of the original potential models [22,24] are renormalized into an effective ΞN interaction and where the latter is then treated within the so-called Gaussian expansion method [26].Earlier studies of Ξ hypernuclei with A = 5 − 12 by Hiyama et al. were performed within a cluster model [27].
In the present study, we will employ the Jacobi nocore shell model (J-NCSM) in combination with microscopic nucleon-nucleon (NN) and YY-ΞN interactions derived within chiral EFT to investigate A = 4 − 7 Ξ hypernuclei.Chiral EFT [28] is a very powerful tool for precisely describing the NN interaction (see [29] and references therein) and allows for accurate calculations of nuclear observables [30][31][32][33].It has also been successfully utilized in studies of the ΛN and ΣN interactions by the Jülich-Bonn-Munich group [34][35][36].These chiral YN potentials have already been used to study Λ hypernuclei within the J-NCSM approach up to the p-shell [37,38].Likewise, the YY-ΞN potentials from chiral EFT up to NLO [23,25] yield promising results for s-shell ΛΛ hypernuclei [39].Therefore, it is very interesting to explore the predictions of chiral EFT for Ξ hypernuclei, in particular for light systems where a microscopic (ab initio) treatment is possible.
The paper is organized as follows: in the next section we describe the baryon-baryon (BB) interactions employed in this work focusing particularly on the S = −2 BB potentials.Section 3 contains a brief description of the J-NCSM and its application to Ξ hypernuclei, and of the method to extrapolate the binding and separation energies to infinite model spaces.In Section 4, our results for NNNΞ, 5  Ξ H and 7 Ξ H are discussed.Final conclusions are given in Section 5.

Baryon-baryon interactions for S = −2
For all calculations presented here, we employ BB interactions that are derived within chiral EFT [28].The high-order semilocal momentum-space regularized potential with a regulator of Λ N = 450 MeV (N 4 LO+(450)) [29] is used for the NN interaction.For the interaction in the ΞN channel, we employ the potential from Ref. [23].This interaction for the S = −2 sector has been constructed in agreement with empirical constraints on the ΛΛ S-wave scattering length and with published values and upper bounds for Ξ − p elastic and inelastic cross sections [25].Moreover, it yields a moderately attractive Ξ-nuclear interaction as suggested by experimental evidence for the existence of Ξ-hypernuclei [2,4].The value obtained for the Ξ single-particle potential U Ξ (k = 0) at nuclear matter saturation density is with around −9 MeV [40] noticeably smaller than the commonly cited potential depth of −14 MeV [41,42].An application of this single-particle potential to finite Ξ nuclei based on the local density approximation method [12,40] showed, however, that pertinent predictions for the aforementioned recently reported states [2][3][4] are quite in line with the energies observed in the experiments.
The considered ΞN interaction includes the coupling to other BB channels in the strangeness S = −2 sector (ΛΛ, ΛΣ, ΣΣ).However, in the actual calculation within the J-NCSM, it turned out that convergence of the eigenvalue iterations (that diagonalize the manybody Hamiltonian) to the lowest lying Ξ states is rather poor when the coupling of ΞN to ΛΛ is explicitly included.Thus, similar to what has been done by Hiyama et al. [20,27], the explicit coupling to ΛΛ is omitted.Instead, the contribution from the transition ΛΛ − ΞN, which anyway can occur only in the 1 S 0 partial wave with isospin I = 0, is incorporated effectively by readjusting the strength of the corresponding V ΞN−ΞN potential.The other channel couplings in the S = −2 sector, i.e.ΞN−ΛΣ −ΣΣ, are, however, still taken into account.In practice, the S-wave low-energy constants (LECs) in the I = 0 channel (to be concrete, those corresponding to the SU(3) singlet irrep.{1} [25,43]) are appropriately re-adjusted so that the real part of the ΞN scattering length remains practically the same after omitting the ΛΛ channel.
Nonetheless, there is a delicate issue connected with that step.The chiral EFT interaction at NLO predicts the existence of a virtual state in the I = 0, 1 S 0 wave extremely close to the ΞN threshold [23,25] -a possible remnant of the H-dibaryon [44].The virtual state is reflected in an impressive cusp structure in the ΛΛ phase shift and a large ΞN phase [25].The pertinent ΞN scattering length is large too, cf.Table 1 in Ref. [23], with an imaginary part in the order of 10 fm or more.Similar features are also seen in lattice QCD results [24].The large imaginary part somehow suggests an overall strong ΛΛ − ΞN coupling.However, its value is artificially enhanced by the near-by virtual state.This can be easily seen by performing the calculation in the particle basis and with physical masses.Then the cusp is strongly reduced [25], and the scattering lengths in the Ξ 0 n channel amount to a = (−1.30− i 0.07) fm, (−2.05 − i 0.27) fm, (−1.95 − i 0.25) fm, (−1.41 − i 0.09) fm, respectively, for the cutoffs Λ = 500, 550, 600, and 650 MeV in the regulator function considered in [23,25].Obviously, now the imaginary part is rather small and that means the actual ΛΛ − ΞN coupling is indeed fairly weak.The latter conclusion has also been drawn in Ref. [21] where this issue has been examined from a slightly different perspective.See also the discussion in Ref. [12]. 11S 0 31 S 0 13 S 1 33 S 1 Fig. 1 ΞN phase shifts predicted by the NLO(500) and HAL QCD potentials (left panel) compared to those of the Nijmegen ESC08c model (right panel).The NLO(500) results are shown by lines: 11 S 0 (dotted, red), 31 S 0 (dash-dotted, black), 13 S 1 (dashed, blue) and 33 S 1 (dash-double-dotted, green).The solid line indicates the 11 S 0 phase shift of the re-adjusted NLO(500) potential, see text.The HAL QCD and ESC08c results (values are taken from [20]) for 11 S 0 , 31 S 0 , 13 S 0 and 33 S 0 are indicated by crosses, circles, squares, and triangles, respectively.Note the different scales in the left and right panels.
For the present exploratory study, we chose the YY-ΞN NLO potential with the cutoff of Λ = 500 MeV, i.e. the interaction with the smallest imaginary part.Moreover, we adjusted the LECs to the Ξ 0 n scattering length, instead of the one for I = 0, in order to avoid any strong bias from the actual but unsettled location of the virtual state.After the re-adjustment the 11 S 0 potential, using the notation (2I+1)(2S+1) L J , yields a ΞN ( 11 S 0 ) = −7.00fm, to be compared with a ΞN ( 11 S 0 ) = −7.71− i 2.03 fm for the original potential where all coupled channels are considered [23].We believe that this procedure allows us to capture the essential features of the chiral ΞN interaction in the 11 S 0 channel reliably, while guaranteeing at the same time the applicability of the J-NCSM approach.Note that all other ΞN partial waves are not affected by this modification anyway and ΛΣ and ΣΣ components are included in the J-NCSM calculations.We, however, neglect YN interactions that are expected to give insignificant contributions but could potentially again induce ΛΛ components to the many-body state.We postpone a more thorough investigation on this issue and of the dependence of the Ξ binding energies on the chiral cutoff Λ Y Y to a future study.
Finally, to speed up the convergence of the J-NCSM calculations, the NN and YY interactions are evolved using the similarity renormalization group (SRG) [45].Thereby, we use an SRG flow parameter of λ N N = 1.6 fm -1 for the NN interaction.This value has already been used in Refs.[38,39] and is motivated by the observation that ordinary nuclei are bound fairly realistically even if three-nucleon forces are neglected for this The variations of the binding energies with respect to λ Y Y allow one to quantify the possible contribution of the omitted SRG-induced three-and more-body forces.Note that such contributions are remarkably small for ΛΛ hypernuclei [39].
It is worthwhile to compare the ΞN phase shifts of the employed EFT interaction NLO(500) with those predicted by the Nijmegen ESC08c [22] and the HAL QCD [24] potentials.As mentioned in the introduction, the latter two interactions have recently been considered in A = 3, 4 Ξ hypernuclear calculations by Hiyama et al. [20].The phase shifts for the four Swave states, namely 11 S 0 , 31 S 0 , 13 S 1 and 33 S 1 , are displayed in Fig. 1.As expected, the original NLO(500) interaction (cf. the dotted line) and the re-adjusted potential differ only slightly in the 11 S 0 phase shifts.Overall, the results by the NLO(500) and HAL QCD interactions are fairly similar to each other, but differ substantially from the Nijmegen ESC08c potential.The ESC08c is strongly attractive in the 33 S 1 channel (leading to a deuteron-like ΞN bound state), whereas the chiral NLO(500) (HAL QCD) interaction is only moderately (weakly) attractive in this channel.Moreover, while the 11 S 0 ΞN interaction is rather attractive in the HAL QCD and NLO(500) potentials, it is actually repulsive in the ESC08c model.Although the NLO(500) and HAL QCD ΞN phase shifts exhibit an overall rather similar trend, there are visible differences in all ΞN partial waves except for 13 S 1 .As we will dis-cuss later, such variations lead to qualitative differences in the predictions of the two interactions for light Ξ systems.

Jacobi NCSM for Ξ hypernuclei
The application of the Jacobi no-core shell model (J-NCSM) to Ξ hypernuclei follows very closely our J-NCSM formalism for ΛΛ systems described in [39].Here we also split the basis functions into two orthogonal sets: one set that involves two S = −1 hyperons, |α * (Y1Y2) , and the other that contains the doubly strange Ξ hyperon, |α * (Ξ) .The |α * (Ξ) states are exactly the same as constructed in [39], Since the ΛΛ−ΞN transition is absorbed into the strength of the V ΞN−ΞN potential and since we omit YN interactions, the basis states |α * (Y1Y2) can be restricted to respectively.As a consequence, the computed binding energies will depend on N max and on the HO frequency ω.To extract the converged results we will follow the two-step extrapolation procedure that has been successfully employed for nuclear and hypernuclear energy 1 To be consistent with [39] we use here and later on t, T for the isospins calculations [38,39,46].First, the energies E(N , ω) are computed for all accessible model spaces N max and for a wide range of ω.Then E N is determined for a given N max = N by minimizing the energies E(N , ω) with respect to ω.In the second step, an exponential fit is applied to E N in order to extrapolate to N → ∞.Furthermore, in order to write down the explicit form of the Hamiltonian, we also distinguish three parts of the Hamiltonian, namely H Y1Y2 , H Ξ and H S=−2 Y1Y2,ΞN , like for ΛΛ hypernuclei.As mentioned before, we do not take into account YN interactions in the S = −1 sector here.Hence, the H Y1Y2 , H S=−2 Y1Y2,ΞN and H Ξ can be written as with Here, m(t Y1 ), m(t Y2 ) , m Ξ and m N are the Y 1 , Y 2 , Ξ hyperon and nucleon rest masses, respectively.M denotes the total rest mass of the system, thus, are the ΞN and YY reduced masses, respectively.The rest mass differences within the nucleon-and hyperonisospin multiplets are neglected.V S=0 and V S=−2 are the two-body NN and YY (ΞN) potentials.Finally, the last term in Eq. ( 3) accounts for the difference in the rest masses of the hyperons arising due to particle conversions.
With the basis states defined in Eqs.(1,2), the matrix elements of the Hamiltonian Eqs.(3-5) can be evaluated analogously as done in [39].Likewise, the final Ξ wave functions and the corresponding binding energies are directly obtained via the Lanczos eigenvalue iterations.

Results and discussion
As it has been shown in [21], the 3 Ξ H hypernucleus is not bound with the chiral YY-ΞN NLO potential [23].Therefore, the lightest system that we study here is NNNΞ.A Ξ hyperon with isospin 1  2 can couple to the core nucleus 3 H/ 3 He in its ground state ( 1 2 + , 1  2 ) resulting in several NNNΞ states with (J π , T ) = (1 + , 0), (0 + , 1), (1 + , 1) and (0 + , 0).The first three states are found to be strongly bound in the work by Hiyama et al. when the Nijmegen ESC08c potential is used.The HAL QCD potential, however, supports only one weakly bound NNNΞ state, namely the (1 + , 0) [20].As discussed in Section 2, there are some differences in the predictions for the ΞN phase shifts by the NLO(500) and HAL QCD interactions.It is therefore interesting to calculate the A = 4 system based on the chiral potential in order to see whether such differences are manifest in the predictions for the NNNΞ binding energies.We also consider 5 Ξ H. Due to the strong binding of the α particle core, the mass difference between ΞN and ΛΛ is partly removed [47] which makes this light hypernucleus especially interesting.Likewise, theoretical predictions for 7  Ξ H are of importance since this system is expected to be investigated through the 7 Li(K − , K + ) reaction in upcoming experiments at J-PARC [17].Generally, we expect that a consistent study of hypernuclei for a range of mass numbers might provide constraints for the properties of Ξs in nuclear matter.In this work, we present such an investigation for Ξ hypernuclei up to A = 7.
In this exploratory study, we do not take the Ξ − -Ξ 0 mass difference of 6.85 MeV into account and assume isospin symmetry by assigning to each state a definite isospin.We believe that this is justified to identify states that are possibly bound but stress that the impact of the mass difference and possible isospin breaking by other contributions like the Coulomb interaction should be analysed in a future study.For the A = 3 and A = 5 Ξ hypernuclei, the isospins are well defined since the corresponding core nucleus is predominantly in its isospin zero state.For A = 7, we follow the choice of Hiyama et al. [20] and only consider the (1/2 + , 3/2) state.For this state, it is natural to expect that the Ξ − -6 He(0 + , 1) component is the dominant one.The situation is less clear for A = 4 since, a priori, none of the possible isospins is favored by the 3N core.Below we have therefore assumed isospin symmetry and give separate results for T = 0 and T = 1 states.Our results for the contribution of the Coulomb interaction suggest that this is a reasonable approximation since the binding is still predominately due to the strong interaction.However, a more careful analysis also taking the cascade mass difference into account should be performed in the future.
As mentioned earlier, in order to eliminate the effect of finite-basis truncation on the binding energies, we follow the two-step extrapolation procedure as explained in [38].The ω-and N -space extrapolations for the binding energy of the 4 Ξ H(1 + , 0) state, E(1 + , 0), are illustrated in panels (a) and (b) of Fig. 2, respectively.To obtain the converged Ξ separation energy B Ξ , we perform an analogous exponential fit on the quantities B Ξ (N ) = E N (1 + , 0)−E N ( 3 H), see also panel (c).Here E N (1 + , 0), E N ( 3 H) are the hypernuclear and nuclear binding energies, respectively, obtained at their optimal HO frequencies for a given model space N .For the separation energies, we cannot expect a monotonic convergences a priori.But for the results shown here, we observed that this is the case and that an exponential fit is appropriate.The same approach was used for the separation energies obtained in our earlier studies [38,39].Note that the error bars shown in panels (b)-(c) are given by the difference to the next model space.These error bars are not meant to provide a realistic uncertainty estimate, but rather to assign relative weights for the following extrapolation to N → ∞.Clearly, well-converged results for both E(1 + , 0) and B Ξ (1 + , 0) are achieved for model spaces up to N max = 30.For the NN interaction alone, the triton energy calculation is, however, converged already for model spaces N max = 24.Similar convergence patterns are also observed for the binding (separation) energies of the other states in NNNΞ and of the 5 2 ) hypernuclei.The convergence for A = 5 and A = 7 is generally faster since the separation energies are larger.Therefore, our current limits of N max = 16 and 12 for A = 5 and 7, respectively, still allow an accurate determination of the energies as can be seen from Table 1.
In order to minimize the contribution of the Coulomb interaction, we use the 4 Ξ n states for isospin T = 1.As can be seen in Table 1, both of these states, 4 Ξ n(0 + , 1) and 4 Ξ n(1 + , 1), are clearly bound for the chiral interaction.It turns out that their binding energies are comparable to that one for 4 Ξ H(1 + , 0).The 4 Ξ H(0 + , 0) state, on the other hand, is unbound for the chiral interactions at NLO.
The converged B Ξ (1 + , 0) computed for a wide range of SRG-flow parameter, 1.4 ≤ λ Y Y ≤ 3.0 fm -1 , are presented in panel (d) of Fig. 2. One sees that the overall variation of B Ξ (1 + , 0) is visible, about 190 ± 30 keV.It is also larger than the dependence of B ΛΛ on the SRGflow parameter, which was found to be of the order of 100 keV [38].This may be related to the fact that, unlike for the ΛΛ case where the coupling to the pion is suppressed by isospin conservation, π exchange contributes to the ΞN interaction at leading order.Longrange interactions are likely to be more strongly affected by the SRG evolution.However, the variation is much smaller than the one observed for single Λ hypernuclei (see e.g.[38]).For the A = 5 and 7 Ξ hypernuclei, we observed similarly large absolute variations, but still they are relatively smaller as compared to the estimated Ξ separation energies.Therefore, in all cases, the SRG dependence is small enough that it does not affect conclusions on the existence of bound states.Therefore, in the following discussion, we will present results for a specific flow parameter, namely λ Y Y = 1.6 fm -1 .
The predicted separation energies B Ξ for the A = 4 − 7 Ξ hypernuclei are listed in Table 1.We verified that all the bound states established here are predominantly due to the strong ΞN interaction.The Ξ − p Coulomb interaction contributes roughly 200, 600, and 400 keV to the binding energies of NNNΞ, 5  Ξ H and 7 Ξ H, respectively.Table 1 provides also an estimate of the corresponding decay width Γ .These widths have been evaluated perturbatively by adapting the procedure followed by Hiyama et al. [20,27].shell) ΞN T -matrix at the sub-threshold energy corresponding to the bound state.One can clearly see that the three states (1 + , 0), (0 + , 1) and (1 + , 1) in NNNΞ are only weakly bound, possessing quite similar B Ξ 's but substantially different decay widths.Interestingly, our result for B Ξ (NNNΞ(1 + , 0)) is close to that for the HAL QCD potential, reported in [20], although the (0 + , 1) and (1 + , 1) states are unbound for the HAL QCD interaction.There are substantial (but not surprising) differences between our separation energies B Ξ (NNNΞ) and the predictions [20] for the ESC08c potential [22].According to the discussion in Ref. [20], it is the strong attraction in the 33 S 1 and 13 S 1 channels that is responsible for the rather large binding energies predicted for that ΞN model in the (1 + , 0) and (1 + , 1) states.
In Appendix A, we summarize the relative weights of the different partial wave channels to the effective ΞN interaction in the s-shell Ξ hypernuclei.Although such an estimate is rather rough, it can nevertheless help to understand the pattern of different bound states found.
Our results for the 5 Ξ H separation energy and decay width are B Ξ ( 5 Ξ H) = 2.16±0.1 MeV and Γ ( 5 Ξ H) = 0.19 MeV, respectively.Oddly enough, these values agree roughly with the estimations by Myint and Akaishi [47] of 1.7 MeV and 0.2 MeV, respectively.However, given the differences in the underlying interactions and specifically in the employed approaches, this is certainly acci-  dental.We further note that in contrast to our finding where 5  Ξ H is bound primarily due to the strong ΞN interaction, the authors in [47] state that the binding energy of 1.7 MeV in 5  Ξ H largely comes from the 4 He-Ξ − Coulomb interaction.The mechanism for the narrow width of 5  Ξ H has been investigated in [48,49].Recently, Friedman and Gal, employing an optical potential, also reported a quite similar result for 5 Ξ H (B Ξ ( 5 Ξ H) = 2.0 MeV) [11].But also here the agreement seems to be more or less accidental given that the Ξ-nuclear interaction used as starting point in that work is with U Ξ −20 MeV significantly more attractive than the one predicted by the chiral ΞN potential employed in the present study which is only around U Ξ ≈ −9 MeV [40] as mentioned above.
The prediction of the chiral ΞN interaction for 7 Ξ H ( 1 2 + , 3  2 ), B Ξ ( 7 Ξ H) = 3.50 ± 0.39 MeV, is only slightly larger than the binding energy of 3.15 MeV reported by Fujioka et al. [17,50] for the HAL QCD interaction, based on a calculation within a four-body (αnnΞ) cluster model [27,51].An earlier study utilizing older S = −2 potentials from the Nijmegen group indicated somewhat smaller binding energies [16,27].Finally, as one can see from   1.Note that for each system all probabilities sum up to the probability of finding a Ξ hyperon in that system.
To shed light on the relation between the properties of the chiral ΞN potential and the binding of the A = 4 − 7 Ξ systems, we provide in Table 2 the contributions of different ΞN partial waves to the expectation value of the S = −2 potential V S=−2 .These results are computed at N = 28, ω = 10 MeV for NNNΞ, at N = 14, ω = 16 MeV for 5  Ξ H and at N = 10, ω = 16 MeV for 7 Ξ H.Here the second largest model space is chosen for each system in order to save computational resources.And, ω is the corresponding optimal HO frequency for the chosen model space.For completeness, the energy expectation values are also shown in the last column of Table 2. Clearly, in all the considered states except NNNΞ(0 + , 0) the attractive ΞN interaction in the 33 S 1 channel plays the most important role in binding the systems.It accounts for more than 50% of the expectation value V s=−2 .While the attraction in the 11 S 0 channel is essential as well for NNNΞ(1 + , 0) and (0 + , 1) (amounting to more than 30% of V S=−2 ), its contribution becomes less significant in other states.Furthermore, the ΞN repulsion in 31 S 0 contributes predominantly to the expectation value V S=−2 of NNNΞ(0 + , 0) (naturally with opposite sign), which causes the system to be unbound.The expectation value V S=−2 ( 31 S 0 ) is also sizable for 5  Ξ H and 7 Ξ H, however, its effect is largely canceled by the attraction in the 11 S 0 channel.Complementary to Table 2, the binding of the A = 4 − 7 hypernuclei can also be understood from Table 3, where probabilities of finding a ΞN pair, P ΞN , in different partial-wave states are listed.One clearly notices that, in most of the systems, a ΞN pair is predominantly found in those channels with J ≤ 1 and in particular in the 33 S 1 , except for the unbound 4 Ξ H(0 + , 0) state.In addition, the two extremely small probabilities P ΞN ( 11 S 0 ) = 0.02% and P ΞN ( 33 S 1 ) = 0.11% in 4 Ξ H(0 + , 0) are obvious manifestations of the small expectation values V S=−2 ( 11 S 0 ) = −0.002MeV and V S=−2 ( 33 S 1 ) = −0.006MeV listed in Table .2. Furthermore, the strong variation of P ΞN ( 11 S 0 ) in different states of the A = 4 − 7 hypernuclei clearly explains the large difference in the decay widths estimated for these systems, see Table 1.
As discussed in Section 2, we had to omit the ΛΛ − ΞN coupling in the J-NCSM application and we compensated that by a small modification of the ΞN potential strength in the 11 S 0 state.It is reassuring to see that the overall effect of this partial wave on the binding energies is not too large.Specifically, the existence of the predicted bound states does not depend on its precise contribution, as can be read off from Tables 2  and 3.In fact, the slightly more attractive 11 S 0 interaction predicted by the original ΞN potential, see Fig. 1, implies that all found Ξ hypernuclei could be simply minimally more bound.

Conclusions
In this work, we employed the Jacobi NCSM in combination with the chiral NLO(500) ΞN potential to explore A = 4 − 7 Ξ hypernuclei.Particle conversions like ΛΣ − ΞN−ΣΣ are fully taken into account, while the transition ΛΛ − ΞN is omitted and its contribution is incorporated effectively by re-adjusting the strength of the V ΞN potential appropriately.The latter approach facilitates a proper convergence of the energy calculations to the lowest lying Ξ states.Furthermore, to speed up the convergence, the ΞN potential is SRG-evolved to a wide range of flow parameters.The effect of SRG evolution on the Ξ separation energies is in general small, but, it is slightly larger than that observed for ΛΛ hypernuclei.We found three loosely bound states (1 + , 0), (0 + , 1) and (1 + , 1) for the NNNΞ system and more tightly bound 5 Ξ H, 7 Ξ H hypernuclei.These Ξ systems are bound predominantly due to the attraction of the chiral ΞN potential in the 33 S 1 channel.On the other hand, the repulsive nature in 31 S 0 prevents the binding of the NNNΞ(0 + , 0) state.All the investigated Ξ bound states are predicted to have very small decay widths.
In view of these results, which are based on an interaction that is fully consistent with presently available experimental constraints, and well in line with current lattice QCD results [24], it seems likely that light Ξ hypernuclei exist.Experimental confirmation is certainly challenging.However, theoretical estimates for yields of A = 4 hypernuclei [52] as well as actual measurements of 4 Λ H, 4 Λ He by the STAR Collaboration [53] raise hopes that NNNΞ bound states can be detected in heavy ion collisions in the not too far future.Also a bound 7 Ξ H system could be produced and studied in the 7 Li(K − , K + ) reaction, cf. the proposal P75 for J-PARC [16].Once these new experimental results are available, they will provide new insights into the properties of S = −2 BB interactions.The current manuscript sets up a framework that allows one to exploit these insights to constrain BB interactions in the future.

Fig. 2
Fig.2 (a-c): binding energy E and Ξ separation energy B Ξ for 4 Ξ H(1 + , 0) computed with the YY-ΞN interaction NLO(500), SRG-evolved to a flow parameter of λ Y Y = 3.0 fm -1 .For the NN interaction the SMS N 4 LO+(450) potential[29] with λ NN = 1.6 fm -1 is employed.B Ξ is measured with respect to the triton binding energy (which is E( 3 H) = −8.5 MeV for the used NN interaction).(a): Solid lines and symbols (with different colors) represent numerical results for different model spaces N = 14 − 30, from top to bottom.The dashed lines are obtained by using the ansatz Eq. (22) in[38].(b-c): Horizontal (red) lines with shaded areas indicate the converged results and the corresponding uncertainties.(d): Dependence of B Ξ ( 4 Ξ H(1 + , 0)) on the flow parameter λ Y Y .

Table 1 :
Hiyama et al. have used the imaginary part of the G matrix.Here, we employ the ΞN T -matrix in the 11 S 0 state from the original potential that includes the ΞN-ΛΛ coupling [23] instead.Schematically the width amounts to Γ −2 Im Ψ B Ξ |T ΞN −ΞN |Ψ B Ξ and involves the pertinent hypernuclear wave function Ψ B Ξ and the (off-B Ξ [MeV] Γ [MeV] Ξ separation energies B Ξ and estimated decay widths Γ for A = 4 − 7 Ξ hypernuclei.All calculations are based on the YY-ΞN interaction NLO(500) and the NN interaction SMS N 4 LO+(450).Both potentials are SRG-evolved to a flow parameter of λ NN = λ Y Y = 1.6 fm -1 .The values of B Ξ in NNNΞ, 5 Ξ H and 7 Ξ H are measured with respect to the binding energies of the core nuclei 3 H, 4 He and 6 He, respectively.

Table 2 :
Contributions of different partial waves to V S=−2 (first five columns), and the total binding energy (last column) for the A = 4 − 7 Ξ hypernuclei.The results are extracted at N = 28, ω = 10 MeV for NNNΞ, at N = 14, ω = 16 MeV for 5 Ξ H and at N = 10, ω = 16 MeV for7 Ξ H.All energies are given in MeV.Same interactions as in Table 1.Note that the calculated binding energy of 3 He( 3 H) is −7.79 (−8.50)MeV.

Table 3 :
Probabilities (in %) of finding a ΞN pair in different partial-wave states in the wave functions of A = 4 − 7 Ξ hypernuclei.Same interactions and model spaces as in Table