$\Xi$ hyper-nuclear states predicted by NLO chiral baryon-baryon interactions

The $\Xi$ single-particle potential obtained in nuclear matter with the next-to-leading order baryon-baryon interactions in chiral effective field theory is applied to finite nuclei by an improved local-density approximation method. As a premise, phase shifts of $\Xi N$ elastic scattering and the results of Faddeev calculations for the $\Xi NN$ bound state problem are presented to show the properties of the $\Xi N$ interactions in the present parametrization. First, the $\Xi$ states in $^{14}$N are revisited because of the recent experimental progress, including the discussion on the $\Xi N$ spin-orbit interaction that is relevant to the location of the $p$-state. Then the $\Xi$ levels in $^{56}$Fe are calculated. In particular, the level shift which is expected to be measured experimentally in the near future is predicted. The smallness of the imaginary part of the $\Xi$ single-particle potential is explicitly demonstrated.

The Ξ single-particle potential obtained in nuclear matter with the next-to-leading order baryonbaryon interactions in chiral effective field theory is applied to finite nuclei by an improved localdensity approximation method.As a premise, phase shifts of ΞN elastic scattering and the results of Faddeev calculations for the ΞN N bound state problem are presented to show the properties of the ΞN interactions in the present parametrization.First, the Ξ states in 14 N are revisited because of the recent experimental progress, including the discussion on the ΞN spin-orbit interaction that is relevant to the location of the p-state.Then the Ξ levels in 56 Fe are calculated.In particular, the level shift which is expected to be measured experimentally in the near future is predicted.The smallness of the imaginary part of the Ξ single-particle potential is explicitly demonstrated.

I. INTRODUCTION
New experimental information on the Ξ-nucleus interaction is increasing from the analyses of experiments at J-PARC.The first observation of twin single-Λ hypernuclei in the experiment at J-PARC identified a Ξ − -14 N bound state with the binding energy B Ξ = 1.27 ± 0.21 MeV [1].The energy is close to the candidate of the Ξ − -14 N state observed in the previous KEK E373 experiment [2] with B Ξ = 1.03 ± 0.18 MeV.In the near future, further observation of Ξ bound states in nuclei is expected.The inclusive spectra of (K − , K + ) reactions on nuclei [3] should provide the properties of Ξ-nucleus potential in a wide energy range.Another ongoing experiment to detect Ξ atomic level shifts by measuring electromagnetic transition spectra [4] is also valuable to inform the Ξnucleus potential in the surface region.
On the theoretical side, the construction of baryonbaryon interactions in the strangeness S = −2 sector has been developed in the framework of chiral effective field theory (ChEFT) [5][6][7].The lattice QCD method by the HAL-QCD group also provides the parametrization of the S = −2 interactions [8,9].Both descriptions are based on the QCD, namely the underlying theory of hadrons and their interactions, either in a direct way or by way of low-energy chiral effective field theory.ΞN interactions of these two methods are, interestingly, are resembling even at the quantitative level, as is shown in the following section.The S = −2 sector of the octet baryon-baryon interactions is the middle of the possible strangeness contents from S = 0 to −4 and therefore all the combinations of the flavor SU(3) bases participate in the feature of the interactions.
Because it is not feasible in the near future to measure directly Ξ-nucleon scattering, the information on the Ξ bound states is the chief source for the ΞN interactions.It is hard, however, to find detailed spin and isospin structure of the ΞN interaction from the analysis of the bound state data in itself.It is necessary to compare the experimental data with the results of microscopic calculations using theoretical interactions as reliable as possible.
One of the present authors reported, in Ref. [10], the properties of the Ξ-nucleus single-particle potential which are obtained on the basis of G-matrix calculations in symmetric nuclear matter with next-to-leading order (NLO) ChEFT potentials constructed by the Jülich-Bonn-München group [6,7].There, Ξ potentials in light nuclei such as 9 Be, 12 C, and 14 N are predicted through the translation of the potential in infinite matter to that in a finite nucleus by an improved local-density approximation (ILDA) method.In view of the current experimental efforts, it is meaningful to revisit the case of 14 N and extend the calculation of the chiral Ξ potential to heavy nuclei such as 56 Fe.
In Sec. 2, the basic properties of the chiral NLO ΞN interactions are elucidated by presenting ΞN phase shifts, and the results of the Faddeev calculations for searching a ΞN N bound state.The Ξ − single-particle potentials in heavier nuclei are studied in Sec.III.First, the Ξ − states predicted on 14 N are revisited.The probable 0p Ξ − state experimentally observed [1] is conducive to the discussion of the Ξ spin-orbit single-particle potential.Next, the potential for 56 Fe is presented, for which the measurement of the level shift of a certain atomic level is aimed in the J-PARC experiment.The very small imaginary part of the Ξ single-particle potential is demonstrated.Summary follows in Sec.IV.
It is basic to evaluate phase shifts of elastic scattering to elucidate the properties of the ΞN interaction in each spin and isospin channel.The s-wave phase shifts calculated with an updated version of the chiral NLO interactions [7] are shown in Fig. 1 by the solid curves.Calculations are in the isospin base.That is, the average masses are assigned for the N, Σ, and Ξ baryons, respectively.The phase shifts with the interactions parametrized on the basis of HAL-QCD calculations are also included  for comparison.There are two sets of parametrization based on the same HAL-QCD calculations.The dashed curves represent the results of the potential by Inoue et al. [8] in which the baryon-channel coupling components are explicitly parametrized as a local function.The dotted curves are the results of the potential of the fit for t/a = 12 by Sasaki et al. [9] in which the effects of the tensor coupling and the baryon-channel coupling except for ΛΛ are simulated by a local ΞN potential in coordinate space.It is seen that three potentials predict qualitatively similar behavior of the phase shifts in all spin and isospin states.The interaction in the isospin T = 1 and 1 S 0 state is repulsive, and the interactions in the remaining three states are attractive.Nevertheless, some quantitative differences are remarked.The repulsion of the T = 1 1 S 0 part of the Sasaki potential is very weak.The attraction in the T = 1 3 S 1 state of the HAL-QCD parametrization is smaller than that of ChEFT.The attraction in the T = 0 3 S 1 state, in which no baryon-channel coupling is present, is weak.The T = 0 1 S 0 state is most attractive, although no bound state exists.This attraction originates from the coupling to the ΛΛ as well as ΣΣ states, though the effect of the ΞN -ΛΛ coupling is smaller than that of the ΞN -ΣΣ coupling in the HAL-QCD potentials.The attractive character in the T = 1 3 S 1 state is not so prominent as in the T = 0 1 S 0 state but plays an important role to generate an attractive Ξ single-particle potential in a nucleus because of the spin-isospin weight factor of (2S + 1)(2T + 1) = 9.
The uncertainties in the ChEFT parametrization of p-waves are larger than those in the s-waves [6].The anti-symmetric spin-orbit interactions, which couples the spin-single and triplet states with the same total spin J, are absent in the present chiral NLO interactions [6].
Still, it is meaningful to present p-wave phase shifts for inferring the effects of the p-waves on the Ξ-nucleus potential.The p-wave phase shifts calculated with the chiral NLO interactions [6] are shown in Fig. 2. The phase shifts are rather small, except for in the T = 1 3 P 2 state, the attraction of which grows with increasing energy.The corresponding attractive contribution to the Ξ single-particle potential in symmetric nuclear matter was presented in Fig. 2 of Ref. [10].It is also seen in that figure that the contributions to the Ξ single-particle from other p-states are small and tend to cancel each other among them.

B. ΞN N three-body system
It is important to figure out whether the chiral NLO interaction can support a ΞN N three-body bound state.The results of the Faddeev calculation for the ΞN N bound state problem are recapitulated in this section.In the present calculations, two-body ΞN T -matrices are first prepared by solving a baryon-channel coupled Lippmann-Schwinger equation in momentum space.In the isospin T = 0 case, the ΞN -ΛΛ-ΣΣ coupling is present in the 1 S 0 state, while no baryon-channel coupling exists in the 3 S 1 -3 D 1 tensor correlated state.In the T = 1 case, the ΞN -ΛΣ coupling is present in the 1 S 0 state, and the ΞN -ΛΣ-ΣΣ coupling takes place in the 3 S 1 -3 D 1 tensor correlated state.Then, the evaluated T -matrices are used in the Faddeev equation for the ΞN N bound-state problem: Ψ (12) =G 0 T ΞN (Ψ (23) − P 23 Ψ (12) ), where G 0 is a three-particle propagator, Ψ (ij) is the Faddeev component, and P 23 is the transposition operator for the 2-3 pair with assigning the number 1 to Ξ and the remaining 2 and 3 to the nucleons.This procedure means that while the pairwise correlation is fully solved, the entire three-baryon coupling is not considered.The interactions are also restricted to the s-wave except for the tensor-coupled d-wave.The Coulomb force is also not taken into account.Still, the calculation is an important attempt for a realistic description of the ΞN N system.
In the literature, possible ΞN N bound states have been reported [11,12], using the s-wave single-channel ΞN potential simulating the Nijmegen ESC08c model [13] for the ΞN interaction and the central s-wave Malfliet-Tjon N N potential [15] for the N N interaction.The Faddeev calculations in Ref. [12] without the Coulomb force show that the bound state exists at the binding energy B = 17.2 MeV in the spin-isospin (S, T ) = (3/2, 1/2) state and B = 2.9 MeV in the (S, T ) = (1/2, 3/2) state.These results are reproduced in our momentum-space Faddeev calculations.It is noted, however, that a substantial revision was made for ESC08c to construct the new version as ESC16 by those authors [14].
The situation is different when the chiral NLO S = −2 interactions are used together with the N 3 LO N N interactions [16].The results of our Faddeev calculations show that no hyper-nuclear bound ΞN N system is expected in every possible spin-isospin channel.It is also ascertained that even if the repulsive T=1 1 S 0 ΞN interaction is omitted, the ΞN N system is not bound.The details are reported in a separate paper [17].

III. Ξ BOUND STATES IN FINITE NUCLEI
Although the ΞN N system is not bound with the present NLO chiral interactions, the Ξ hyperon can be bound in heavier nuclei due to the attraction in the T = 1 3 S 1 channel with the statistical factor of (2S +1)(2T +1).As shown in Ref. [10], the Ξ − single-particle potentials predicted for 9 Be, 12 C, and 14 N by the ILDA method using the G-matrices in symmetric nuclear matter with the NLO chiral interactions are rather shallow but enough attractive to support hyper-nuclear bound states.In this section, first, the calculated Ξ − -14 N bound states are revisited concerning the recent experimental data and the possible effect of the Ξ spin-orbit potential for the pstate.Next, Ξ − bound states in 56 Fe are presented.In particular, the atomic level shift of Ξ − in 56 Fe is focused, for which the X-ray spectroscopy experiment to detect it ongoing at J-PARC [4].The very small imaginary part of the Ξ potential is exemplified, which was not included in Ref. [10].
A. Ξ − -14 N bound states and Ξ spin-orbit potential After the prediction for Ξ − bound states in 14 N was reported in Ref. [10] based on the NLO ChEFT S = −2 interactions, new experimental information [1] was obtained through the first observation of twin single-Λ hypernuclei: Ξ − + 14 N → 10 Λ Be + 5 Λ He.The state is a probably 0p level at 1.27 ± 0.21 MeV.Further observation of Ξ − states in 14 N was also reported [18] from the analyses of the data of KEK and J-PARC experiments.That is, three candidates for the 0s state are at 8.00 ± 0.77, 4.96 ± 0.77, and 6.27 ± 0.27 MeV, respectively, and a possible 0p state is at 0.90 ± 0.62 MeV.These energies are shown on the left side of Fig. 4.
Observing that the calculated energies in Ref. [10] correspond reasonably well to these experimental data, additional calculations are given in this subsection.First, if the experimental energy of 6.27 ± 0.27 MeV is taken seriously, the calculated Ξ − 0s energy of −5.40 MeV in Table II in Ref. [10] is slightly short.To reproduce the range of the empirical energy, −6.00 ∼ −6.54 MeV, it is needed to multiply a factor of 1.10 ∼ 1.19 to the calculated Ξ-14 N potential.This factor appears within the uncertainties of the G-matrix calculations and the ILDA method.The calculated Ξ-14 N single-particle potential is shown in Fig. 3.The potential by the ILDA method is energy-dependent.U Ξ (r; E = −5MeV) is employed for the 0s state, and U Ξ (r; E = 0MeV) is for the 0p and 0d states.The evaluated single-particle energies are presented in Fig. 4 both for U Ξ and 1.15 × U Ξ .The position of the 0p level is reasonable.It is noted that the Coulomb 0d state is hardly affected by the addition of the hypernuclear potential U Ξ (r; E = 0MeV).
Another subject to discuss here is the effect of a Ξnucleus spin-orbit potential.If the Ξ spin-orbit interaction is not negligible, the location of the 0p state does not simply imply the strength of the Ξ central single-particle FIG.3: Energy-dependent Ξ-14 N single-particle potential by the ILDA method, using ΞN G-matrices in symmetric nuclear matter with the NLO ChEFT interactions [7].The potentials enhanced by a factor of 1.15 are also shown.U Coulomb is the potential of uniform charge distribution with a radius of RC = 1.15A 1/3 fm.potential.Although the ground state of 14 N is not simply shell-closed, it is instructive to estimate how the Ξ 0p level in 14 N is affected by the possible spin-orbit potential in a mean-field consideration; that is, without considering the detailed structure of the 14 N ground state.
The interesting feature of the Ξ spin-orbit singleparticle potential in nuclei is that the potential may be repulsive in contrast to the attractive nucleon spin-orbit potential.Various theoretical studies have predicted a repulsive Ξ spin-orbit mean-field, though the strength is considerably smaller than that of the nucleon.When the spin-orbit potential is repulsive, the downward shift of the single-particle level with j < = − 1 2 is twice as large as that of the level with j > = + 1  2 for the attractive one.In a relativistic mean-field description [19][20][21], the repulsive spin-orbit mean-field is brought about by an ωmeson exchange with the tensor coupling.The repulsive character is also predicted in a microscopic description based on two-body ΞN interactions constructed in a nonrelativistic SU(6) quark model [22], in which the contribution from the ordinary spin-orbit component of the ΞN interaction is attractive, while the anti-symmetric spin-orbit component contributes opposite and the net spin-orbit single-particle potential becomes repulsive.In all these estimations, the repulsive strength is one-fifth of the attractive strength of the nucleon spin-orbit potential or less.
The effective spin-orbit strength generated by the bare baryon-baryon interaction is properly measured by the Scheerbaum factor [23] calculated in nuclear matter.The expression for the nucleon case was extended to the hyperon cases in Ref. [22].The Scheerbaum factor for Ξ in symmetric nuclear matter with the Fermi momentum k F reads Here, ζ ≡ m N /m Ξ , q max = 1 2 (k F + q) and the weight factor W (q, q) is defined by where θ(k F − q) is a step function.In Eq. 3, G JT 1 ,1 is the abbreviation of the momentum-space diagonal G-matrix element in the spin-triplet channel with the total spin J and total isospin T .
The above definition of S Ξ is different from the original constant in Ref. [23] by a factor of − 2π 3 .Then, S Ξ can be identified with the strength W 0 of the δ-type effective two-body spin-orbit interaction customarily used in Skyrme-Hartree-Fock calculations [24]: In the present NLO ChEFT interaction, the antisymmetric spin-orbit term is not included by putting the pertinent low-energy constant to zero.Therefore, the spinorbit interaction in the nuclear medium is expected to be attractive.The actual G-matrix calculation in symmetric nuclear matter gives S Ξ 21.5 MeV•fm 5 at q ≈ 0.7 fm −1 that is the value prescribed by Scheerbaum on the basis of the wavelength of the density distribution.This value is about one-fifth of S N = 102 MeV•fm 5 with the same sign [25].It is noted that S N = 102 MeV•fm 5 is somewhat smaller than the typical value of S N = 120 MeV•fm 5 used in Skyrme-Hartree-Fock calculations in the literature.
Changes of the Ξ − 0p level in 14 N of the potential 1.15 × U Ξ (E = 0M eV ) depending on the spin-orbit parameter W 0 from −25 to 25 MeV•fm 5 are depicted in Fig. 4. The negative sign of W 0 means the repulsive spin-orbit potential.If the spin-orbit potential strength is about one-fifth of that of the nucleon, the energy splitting of the shallow 0p 1/2 and 0p 3/2 states in 14 N is about 0.4 MeV.

B. Ξ bound states in 56 Fe and atomic level shifts
In applying the ILDA method [10] to generate the Ξ single-particle potential in 56 Fe using G-matrices eval- uated in symmetric nuclear matter, one has to beware that the 56 Fe nucleus is asymmetric in the proton and neutron density distributions.At present, however, it is very demanding to perform Brueckner self-consistent calculations of the Ξ potential at various asymmetric nuclear matter, because all single-particle potentials of the 8 octet baryons (n, p, Λ, Σ − , Σ 0 , Σ + , Ξ − , and Ξ 0 ) have to be determined self-consistently.Fortunately, the asymmetric effect can be expected to be small as explained in the following.If the neutron and proton contributions are individually written, the Ξ potential is obtained by the sum of the proton and neutron contributions, which is written in an abbreviated notation as where ρ n and ρ p are neutron and proton density distributions, respectively, and G 2T +1 represents the contribution of the ΞN G-matrix in the isospin T channel.Introducing the asymmetry parameter α as α ≡ (ρ n − ρ p )/(ρ n + ρ p ) ≡ (ρ n − ρ p )/ρ, the potential U Ξ − is written as The profile of the neutron and proton density distributions of the density-dependent Hartree-Fock calculation with the G-0 force of Sprung and Banerjee [26], which is used in the present ILDA calculations of the Ξ potential in 56 Fe, is shown in Fig. 5.The asymmetry of 56 Fe is seen to be about α ≈ 0.01 0.15 = 1 15 .The additional factor (G 3 − G 1 )/(3G 3 + G 1 ) is smaller than 1  3 as inferred from the properties of the ΞN interaction presented in Sec.II.Therefore, the contribution of the second term in Eq. ( 7) is estimated to be at most 2 % of the first term.This indicates that the estimation of the Ξ − potential based on G-matrices in symmetric nuclear matter is reliable.
The Ξ-56 Fe single-particle potential calculated by the ILDA method with the Gaussian smearing range of β = 1.0 fm is shown in Fig. 6.The solid and dashed curves represent the real and imaginary parts, respectively.The potential in symmetric nuclear matter is energy-dependent.The energy is set to be 0 MeV because the shallow Ξ level is mainly concerned to discuss the Coulomb energy level shift.The potential shape is well simulated by a standard Woods-Saxon form both in the real and imaginary parts.The fitted strength and geometry parameters are V R = −8.39MeV, R 0,R = 5.01 fm, and a R = 0.499 fm for the real part, and V I = −0.247MeV, R 0,I = 5.32 fm, and a I = 0.303 fm for the real part.These Woods-Saxon potentials are shown by the dotted curves in Fig. 6.The depth of the real part of about 8 MeV corresponds to that in nuclear matter.The imaginary potential, which mainly originates from the energy-conserving ΞN → ΛΛ process, turns out to be very small.Note that the imaginary part is scaled up by a factor of 10 in Fig. 6.In this transition process, the kaon exchange has to be involved and therefore the interaction is short-ranged.The smallness of the ΞN -ΛΛ coupling potential is also pointed out in the HAL-QCD calculations [9].Another factor of the smallness is the spin-isospin structure.The ΞN ↔ ΛΛ conversion is possible only in the isospin T = 0 1 S 0 channel.The statistical factor (2S + 1)(2T + 1) suggests that the contribution from the T = 1 1 S 0 channel is comparably suppressed, namely 1/16 in all spin-isospin combinations of the ΞN pair.Ξ − single-particle energies evaluated by the ILDA potential of Fig. 6  of R c = 1.15A 1/3 fm are also included for comparison.
The level position e real and the width Γ of the Ξ state correspond to the complex eigenvalue of the Schrödinger equation: The inclusion of U Ξ appreciably lowers the Coulomb levels with the angular momentum ≤ 3. Reflecting the small imaginary part, the width of the level is at most 0.5 MeV.The level with = 4 is the main target to experimentally detect the atomic level shift by the Ξ-56 Fe hyper-nuclear potential [4].Table 1 tabulates energies and predicted shifts of the = 4 and adjacent levels.These energies are not affected by the inclusion of the Ξ spin-orbit potential argued in the preceding subsection.The smallness of the width of the = 4 level is remarkable, though uncertainties are kept in mind in the various stage of the present calculation.

IV. SUMMARY
Ξ hyper-nuclear single-particle states predicted by the Ξ-nucleus potential derived from the chiral NLO ΞN interactions by the Jülich-Bonn-München group [6,7] are presented.To learn the basic spin-isospin structure of the present ΞN interactions, ΞN phase shifts are discussed and compared with those of the two sets of the parametrization based on the HAL-QCD calculations Fe.The level shift given at the right end is ∆e ≡ eC − e real .Entries for the energies eC , e real , Γ, and ∆e are in keV.The unit for the root-meansquare-radius r 2 is fm.[8,9].It is also pointed out by Faddeev calculations that no ΞN N bound state is expected in every spinisospin state.First, the Ξ states in 14 N are revisited.Considering the experimental observation of a probable Ξ − p-state in 14 N, the discussion is included about the ΞN spin-orbit interactions which are relevant to the location of the p-state.Then, the Ξ single-particle states in 56 Fe are calculated.In particular, the atomic level shift which is expected to be measured experimentally in the near future is predicted.The smallness of the imaginary part of the Ξ single-particle potential is demonstrated.The smallness is due to the small transition interaction between ΞN and ΛΛ, in addition to the fact that the transition to the ΛΛ state is possible only in the isospin T = 0 1 S 0 channel.The parametrization of the baryon-baryon interactions in the S = −2 sector seems to be still in an exploratory stage due to the scarce and less-accurate experimental scattering data.Although Ξ hyper-nuclear data is valuable, it is difficult to deduce spin-isospin properties of the ΞN interactions by phenomenological analyses of the experimental data of Ξ states in nuclei because there are 4 spin-isospin channels and various baryon-channel couplings are involved.Therefore, studies based on the microscopic baryon-baryon interactions as much reliable as possible are important.Experimental data in the near future and theoretical microscopic studies should improve our understanding of baryon-baryon interactions in strangeness sectors.

FIG. 4 :
FIG.4: Ξ − states in14 N. Experimental data on the left side are taken from the compilation in Ref.[18].The Coulomb attraction is treated by the potential of uniform charge distribution with a radius of Rc = 1.15A 1/3 fm.The ILDA potential is energy-dependent.UΞ(r; E = −5MeV) is employed for the 0s state and UΞ(r; E = 0MeV) for the 0p and 0d states.It is demonstrated that an enhancement factor of 1.15 is needed to fit the 0s energy at 6.27 MeV.The shifts of the 0p 3/2 and 0p 1/2 energies due to the addition of the ΞN spin-orbit potential is shown on the right side, as a function of the strength W0 of Eq. 5.

FIG. 6 :
FIG.6: Ξ potential obtained in the ILDA method, based on the Ξ potential in symmetric nuclear matter calculated with the NLO ChEFT interactions[6].The imaginary part shown by the dashed curve is scaled up by a factor of 10.The dotted curves are the potential fitted in a Woods-Saxon form, the depth and the geometry parameters are given in the text.U Coulomb depicts the Coulomb potential of the uniform charge distribution with the radius with RC = 1.15A 1/3 fm.

FIG. 7 :
FIG.7: Ξ single-particle levels in 56 Fe obtained by the potential U Coulomb + UΞ(E = 0).The width Γ is indicated by the error bar in the case the error bar is larger than that of the symbol.Energy levels of the pure Coulomb potential U Coulomb of the uniform charge distribution with the radius of RC = 1.15A 1/3 fm are also shown with dashed connecting lines.