e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>e4e-mail<EMAIL_ADDRESS>
11institutetext: IAS-4, IKP-3 and JHCP, Forschungszentrum Jülich, D-52428
Jülich, Germany 22institutetext: HISKP and BCTP, Universität Bonn, D-53115
Bonn, Germany 33institutetext: Tbilisi State University, 0186 Tbilisi, Georgia

#  $A=4-7$ $\Xi$ hypernuclei based on interactions from chiral effective field
theory

Hoai Leaddr1,e1  Johann Haidenbaueraddr1,e2  Ulf-G.
Meißneraddr2,addr1,addr3,e3  Andreas Noggaaddr1,e4

(March 14, 2021)

###### Abstract

We investigate the existence of bound $\Xi$ states in systems with $A=4-7$
baryons using the Jacobi NCSM approach in combination with chiral NN and
$\Xi$N interactions. We find three shallow bound states for the NNN$\Xi$
system (with $(J^{\pi},T)=(1^{+},0)$, $(0^{+},1)$ and $(1^{+},1)$) with quite
similar binding energies. The
${}^{5}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{1}{2})$ and
${}^{7}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{3}{2})$ hypernuclei are also
clearly bound with respect to the thresholds ${}^{4}\mathrm{He}+\Xi$ and
${}^{6}\mathrm{He}+\Xi$, respectively. The binding of all these $\Xi$ systems
is predominantly due to the attraction of the chiral $\Xi$N potential in the
${}^{33}S_{1}$ channel. A perturbative estimation suggests that the decay
widths of all the observed states could be rather small.

###### Keywords:

Hyperon-Hyperon interactions $\Xi$-Hypernuclei Forces in hadronic systems and
effective interactions Shell model

###### pacs:

13.75.Ev 21.80.+a 21.30.Fe 21.60.Cs

††journal: Eur. Phys. J. A

##  1 Introduction

Recent progress in strangeness $S=-2$ nuclear physics Hiyama:2018lgs , in
particular the observation of nuclear bound states of
$\Xi^{-}-^{14}\mathrm{N}\,(^{15}_{\Xi}\mathrm{C})$ Nakazawa:2015joa ;
PhysRevLett.126.062501 ; Yoshimoto:2021ljs  and possibly
$\Xi^{-}-^{11}\mathrm{B}\,(^{12}_{\Xi}\mathrm{Be})$ Nagae:2019uzt , and
evidence from femtoscopic measurements for an attractive $\Xi^{-}p$
interaction ALICE:2019hdt ; ALICE:2020mfd  have again triggered considerable
interest in studying $\Xi$ hypernuclei theoretically Sun:2016tuf ; Jin:2019sqc
; Shyam:2019laf ; Friedman:2021rhu ; Kohno:2021str ; Hu:2021ttg  despite large
uncertainties in the underlying $\Xi$N interaction. The latter is due to the
absence of direct hyperon-hyperon (YY) and $\Xi$N scattering data, and the
overall extremely scarce empirical information on $S=-2$ systems. There are
several $\Lambda\Lambda$ hypernuclei unambiguously determined in experiments,
with the s-shell ${}^{\text{ }\text{ }\text{
}\text{}6}_{\Lambda\Lambda}\text{He}$ PhysRevLett.87.212502 ; Nakazawa:2010zza
being the lightest one. Given huge challenges in experimental identifications
of $\Xi$ states, the possible existence of s-shell $\Xi$ hypernuclei remains
by and large an open question. Thus, theoretical predictions for $\Xi$
hypernuclei especially for light systems are of great importance. Results from
such studies can provide useful guidelines for experimentalists in searching
for $\Xi$ bound states Ajimura:2019 ; Fujioka:2021S .

In this work we explore the possible existence of light $\Xi$ hypernuclei up
to $A=7$. For the lightest system, ${}^{3}_{\Xi}\mathrm{H}$, several
calculations can be found in the literature Garcilazo:2016ylj ;
Filikhin:2017fog ; Hiyama:2019kpw ; Miyagawa:2021krh . Not surprisingly, the
predictions strongly depend on the interaction models used. For example,
Garcilazo et al.Garcilazo:2016ylj  and Hiyama et al.Hiyama:2019kpw  both
obtained a deeply bound ${}^{3}_{\Xi}\mathrm{H}\,(J^{\pi}=3/2^{+},T=1/2)$
state employing an effective $\Xi$N potential that mimics the phase shifts of
the Nijmegen ESC08c potential Nagels:2015dia . On the other hand, for
interactions derived within chiral effective field theory (EFT)
Haidenbauer:2018gvg  or for $\Xi$N potentials deduced from lattice QCD
simulations by the HAL QCD Collaboration HALQCD:2019wsz , the system is found
to be unbound Miyagawa:2021krh ; Hiyama:2019kpw . It has to be said that a
strongly attractive $\Xi$N force as suggested by that ESC08c potential, which
even yields a $\Xi$N two-body bound state Nagels:2015dia , is not supported by
the currently available empirical constraints Haidenbauer:2015zqb  including
the aforementioned femtoscopic measurements, and also not by lattice
simulations at almost physical masses HALQCD:2019wsz . Note that in the work
by Miyagawa et al.Miyagawa:2021krh  the original $\Xi$N 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 $\Xi$NN channels are considered. The variational calculation of Hiyama et
al.Hiyama:2019kpw  is based on interactions where the various $YY$ channels of
the original potential models Nagels:2015dia ; HALQCD:2019wsz  are
renormalized into an effective $\Xi$N interaction and where the latter is then
treated within the so-called Gaussian expansion method Hiyama:2012sma .
Earlier studies of $\Xi$ hypernuclei with $A=5-12$ by Hiyama et al.were
performed within a cluster model Hiyama:2008fq .

In the present study, we will employ the Jacobi no-core shell model (J-NCSM)
in combination with microscopic nucleon-nucleon (NN) and YY-$\Xi$N
interactions derived within chiral EFT to investigate $A=4-7$ $\Xi$
hypernuclei. Chiral EFT Epelbaum:2008ga  is a very powerful tool for precisely
describing the NN interaction (see Reinert:2017usi  and references therein)
and allows for accurate calculations of nuclear observables Epelbaum:2018ogq ;
Piarulli:2017dwd ; Epelbaum:2019zqc ; Maris:2020qne . It has also been
successfully utilized in studies of the $\Lambda$N and $\Sigma$N interactions
by the Jülich-Bonn-Munich group Polinder:2006eq ; Haidenbauer:2013oca ;
Haidenbauer:2019boi . These chiral YN potentials have already been used to
study $\Lambda$ hypernuclei within the J-NCSM approach up to the $p$-shell
Le:2019gjp ; Le:2020zdu . Likewise, the YY-$\Xi$N potentials from chiral EFT
up to NLO Haidenbauer:2015zqb ; Haidenbauer:2018gvg  yield promising results
for s-shell $\Lambda\Lambda$ hypernuclei Le:2021wwz . Therefore, it is very
interesting to explore the predictions of chiral EFT for $\Xi$ 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 $\Xi$ hypernuclei, and of the method to extrapolate the
binding and separation energies to infinite model spaces. In Section 4, our
results for NNN$\Xi$, ${}^{5}_{\Xi}\mathrm{H}$ and ${}^{7}_{\Xi}\mathrm{H}$
are discussed. Final conclusions are given in Section 5.

##  2 Baryon-baryon interactions for $S=-2$

For all calculations presented here, we employ BB interactions that are
derived within chiral EFT Epelbaum:2008ga . The high-order semilocal momentum-
space regularized potential with a regulator of $\Lambda_{N}=450$ MeV
(N4LO+(450)) Reinert:2017usi  is used for the NN interaction. For the
interaction in the $\Xi N$ channel, we employ the potential from Ref.
Haidenbauer:2018gvg . This interaction for the $S=-2$ sector has been
constructed in agreement with empirical constraints on the $\Lambda\Lambda$
$S$-wave scattering length and with published values and upper bounds for
$\Xi^{-}p$ elastic and inelastic cross sections Haidenbauer:2015zqb .
Moreover, it yields a moderately attractive $\Xi$-nuclear interaction as
suggested by experimental evidence for the existence of $\Xi$-hypernuclei
Nakazawa:2015joa ; Yoshimoto:2021ljs . The value obtained for the $\Xi$
single-particle potential $U_{\Xi}(k=0)$ at nuclear matter saturation density
is with around $-9$ MeV Kohno:2019oyw  noticeably smaller than the commonly
cited potential depth of $-14$ MeV AGSE885:1999erv ; Gal:2016boi . An
application of this single-particle potential to finite $\Xi$ nuclei based on
the local density approximation method Kohno:2019oyw ; Kohno:2021str  showed,
however, that pertinent predictions for the aforementioned recently reported
states Nakazawa:2015joa ; PhysRevLett.126.062501 ; Yoshimoto:2021ljs  are
quite in line with the energies observed in the experiments.

The considered $\Xi$N interaction includes the coupling to other BB channels
in the strangeness $S=-2$ sector ($\Lambda\Lambda$, $\Lambda\Sigma$,
$\Sigma\Sigma$). However, in the actual calculation within the J-NCSM, it
turned out that convergence of the eigenvalue iterations (that diagonalize the
many-body Hamiltonian) to the lowest lying $\Xi$ states is rather poor when
the coupling of $\Xi$N to $\Lambda\Lambda$ is explicitly included. Thus,
similar to what has been done by Hiyama et al.Hiyama:2008fq ; Hiyama:2019kpw ,
the explicit coupling to $\Lambda\Lambda$ is omitted. Instead, the
contribution from the transition $\Lambda\Lambda-\Xi$N, which anyway can occur
only in the ${}^{1}S_{0}$ partial wave with isospin $I=0$, is incorporated
effectively by re-adjusting the strength of the corresponding
$V_{\Xi\mathrm{N}-\Xi\mathrm{N}}$ potential. The other channel couplings in
the $S=-2$ sector, i.e. $\Xi\mathrm{N}-\Lambda\Sigma-\Sigma\Sigma$, 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} Polinder:2007mp ; Haidenbauer:2015zqb ) are
appropriately re-adjusted so that the real part of the $\Xi$N scattering
length remains practically the same after omitting the $\Lambda\Lambda$
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 $\Xi$N threshold Haidenbauer:2015zqb
; Haidenbauer:2018gvg  – a possible remnant of the $H$-dibaryon
Haidenbauer:2011ah . The virtual state is reflected in an impressive cusp
structure in the $\Lambda\Lambda$ phase shift and a large $\Xi$N phase
Haidenbauer:2015zqb . The pertinent $\Xi$N scattering length is large too, cf.
Table 1 in Ref. Haidenbauer:2018gvg , with an imaginary part in the order of
$10$ fm or more. Similar features are also seen in lattice QCD results
HALQCD:2019wsz . The large imaginary part somehow suggests an overall strong
$\Lambda\Lambda-\Xi$N 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 Haidenbauer:2015zqb , and the scattering lengths in the
$\Xi^{0}$n channel amount to $a=(-1.30-{\rm i}\,0.07)$ fm, $(-2.05-{\rm
i}\,0.27)$ fm, $(-1.95-{\rm i}\,0.25)$ fm, $(-1.41-{\rm i}\,0.09)$ fm,
respectively, for the cutoffs $\Lambda=500$, $550$, $600$, and $650$ MeV in
the regulator function considered in Haidenbauer:2015zqb ; Haidenbauer:2018gvg
. Obviously, now the imaginary part is rather small and that means the actual
$\Lambda\Lambda-\Xi$N coupling is indeed fairly weak. The latter conclusion
has also been drawn in Ref. Miyagawa:2021krh  where this issue has been
examined from a slightly different perspective. See also the discussion in
Ref. Kohno:2021str .

Figure 1: $\Xi N$ 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 Hiyama:2019kpw ) 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-$\Xi$N NLO potential with
the cutoff of $\Lambda=500$ MeV, i.e. the interaction with the smallest
imaginary part. Moreover, we adjusted the LECs to the $\Xi^{0}$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 $a_{\Xi N}({{}^{11}S_{0}})=-7.00$ fm, to be
compared with $a_{\Xi\mathrm{N}}({{}^{11}S_{0}})=-7.71-{\rm i}\,2.03$ fm for
the original potential where all coupled channels are considered
Haidenbauer:2018gvg . We believe that this procedure allows us to capture the
essential features of the chiral $\Xi$N 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 $\Xi$N partial waves are not affected by
this modification anyway and $\Lambda\Sigma$ and $\Sigma\Sigma$ 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 $\Lambda\Lambda$ components to the many-body state. We postpone a more
thorough investigation on this issue and of the dependence of the $\Xi$
binding energies on the chiral cutoff $\Lambda_{YY}$ 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)
Bogner:2007hn . Thereby, we use an SRG flow parameter of $\lambda_{NN}=1.6$
fm-1 for the NN interaction. This value has already been used in Refs.
Le:2020zdu ; Le:2021wwz  and is motivated by the observation that ordinary
nuclei are bound fairly realistically even if three-nucleon forces are
neglected for this $\lambda_{NN}$. The $S=-2$ potential is SRG-evolved to a
wide range of SRG flow parameter (denoted generically by $\lambda_{YY}$),
namely $1.4\leq\lambda_{YY}\leq 3.0$ fm-1. The variations of the binding
energies with respect to $\lambda_{YY}$ 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 $\Lambda\Lambda$ hypernuclei
Le:2021wwz .

It is worthwhile to compare the $\Xi$N phase shifts of the employed EFT
interaction NLO(500) with those predicted by the Nijmegen ESC08c
Nagels:2015dia  and the HAL QCD HALQCD:2019wsz  potentials. As mentioned in
the introduction, the latter two interactions have recently been considered in
$A=3,4$ $\Xi$ hypernuclear calculations by Hiyama et al.Hiyama:2019kpw . The
phase shifts for the four S-wave 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 $\Xi$N bound state), whereas the chiral NLO(500)
(HAL QCD) interaction is only moderately (weakly) attractive in this channel.
Moreover, while the ${}^{11}S_{0}$ $\Xi$N 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 $\Xi$N phase shifts exhibit an
overall rather similar trend, there are visible differences in all $\Xi N$
partial waves except for ${}^{13}S_{1}$. As we will discuss later, such
variations lead to qualitative differences in the predictions of the two
interactions for light $\Xi$ systems.

##  3 Jacobi NCSM for $\Xi$ hypernuclei

The application of the Jacobi no-core shell model (J-NCSM) to $\Xi$
hypernuclei follows very closely our J-NCSM formalism for $\Lambda\Lambda$
systems described in Le:2021wwz . Here we also split the basis functions into
two orthogonal sets: one set that involves two $S=-1$ hyperons,
$|\alpha^{*(Y_{1}Y_{2})}\rangle$, and the other that contains the doubly
strange $\Xi$ hyperon, $|\alpha^{*(\Xi)}\rangle$. The
$|\alpha^{*(\Xi)}\rangle$ states are exactly the same as constructed in
Le:2021wwz ,

$\displaystyle|\alpha^{*(\Xi)}\rangle$
$\displaystyle=|\alpha_{(A-1)N}\rangle\otimes|\Xi\rangle$ (1)
$\displaystyle=|\mathcal{N}{J}{T},\alpha_{(A-1)N}\,n_{\Xi}\,I_{\Xi}\,t_{\Xi};$
$\displaystyle\quad(J_{A-1}(l_{\Xi}\,s_{\Xi})\,I_{\Xi}){J},(T_{A-1}\,t_{\Xi}){T}\rangle\,.$

Since the $\Lambda\Lambda-\Xi$N transition is absorbed into the strength of
the $V_{\Xi\mathrm{N}-\Xi N}$ potential and since we omit YN interactions, the
basis states $|\alpha^{*(Y_{1}Y_{2})}\rangle$ can be restricted to

$\displaystyle|\alpha^{*(Y_{1}Y_{2})}\rangle$
$\displaystyle=|\alpha_{(A-2)N}\rangle\otimes|Y_{1}Y_{2}\rangle$ (2)
$\displaystyle=|\mathcal{N}{J}{T},\alpha_{(A-2)N}\,\alpha_{Y_{1}Y_{2}}\,n_{\lambda}\lambda;$
$\displaystyle\qquad((l_{Y_{1}Y_{2}}(s_{Y_{1}}s_{Y_{2}})S_{Y_{1}Y_{2}})J_{Y_{1}Y_{2}}(\lambda
J_{A-2})I_{\lambda})J,$
$\displaystyle\qquad((t_{Y_{1}}t_{Y_{2}})T_{Y_{1}Y_{2}}T_{A-2})T\rangle,$

with $|Y_{1}Y_{2}\rangle=|\Lambda\Sigma\rangle$ or $|\Sigma\Sigma\rangle$. The
notations used in Eqs. (1,2) are the same as in Le:2021wwz 111To be consistent
with Le:2021wwz  we use here and later on $t,\,T$ for the isospins. For
example, the symbol $\alpha_{(A-2)\mathrm{N}}$ stands for all quantum numbers
characterizing the antisymmetrized states of $A-2$ nucleons: the total number
of oscillator quanta $\mathcal{N}_{A-2}$, total angular momentum $J_{A-2}$,
isospin $T_{A-2}$ and state index $\zeta_{A-2}$ as well. Similarly,
$\alpha_{Y_{1}Y_{2}}$ stands for a complete set of quantum numbers describing
the subcluster of two hyperons $Y_{1}$ and $Y_{2}$. The principal quantum
number $n_{\lambda}$ of the harmonic oscillator (HO) together with the orbital
angular $\lambda$ describe the relative motion of the $(A-2)$N core with
respect to the center-of-mass (C.M.) of the $Y_{1}Y_{2}$ subcluster. The
orders, in which these quantum numbers are coupled, are shown after the
semicolon. For practical calculations, we truncate the model space by limiting
the total HO energy quantum number $\mathcal{N}$ in Eqs. (1,2) to
$\mathcal{N}\leq\mathcal{N}_{max}$ where
$\mathcal{N}=\mathcal{N}_{A-2}+2n_{\lambda}+l_{\lambda}+\mathcal{N}_{Y_{1}Y_{2}}$
or $\mathcal{N}=\mathcal{N}_{A-1}+l_{\Xi}+2n_{\Xi}$, respectively. As a
consequence, the computed binding energies will depend on $\mathcal{N}_{max}$
and on the HO frequency $\omega$. To extract the converged results we will
follow the two-step extrapolation procedure that has been successfully
employed for nuclear and hypernuclear energy calculations Liebig:2015kwa ;
Le:2020zdu ; Le:2021wwz . First, the energies $E(\mathcal{N},\omega)$ are
computed for all accessible model spaces $\mathcal{N}_{max}$ and for a wide
range of $\omega$. Then $E_{\mathcal{N}}$ is determined for a given
$\mathcal{N}_{max}=\mathcal{N}$ by minimizing the energies
$E(\mathcal{N},\omega)$ with respect to $\omega$. In the second step, an
exponential fit is applied to $E_{\mathcal{N}}$ in order to extrapolate to
$\mathcal{N}\to\infty$.

Furthermore, in order to write down the explicit form of the Hamiltonian, we
also distinguish three parts of the Hamiltonian, namely
$H_{Y_{1}Y_{2}},H_{\Xi}$ and $H^{S=-2}_{Y_{1}Y_{2},\Xi\mathrm{N}}$, like for
$\Lambda\Lambda$ hypernuclei. As mentioned before, we do not take into account
YN interactions in the $S=-1$ sector here. Hence, the $H_{Y_{1}Y_{2}}$,
$H^{S=-2}_{Y_{1}Y_{2},\Xi\mathrm{N}}$ and $H_{\Xi}$ can be written as

$\displaystyle
H_{Y_{1}Y_{2}}=\sum_{i<j=1}^{A-2}\Big{(}\frac{2p^{2}_{ij}}{M(t_{Y_{1}},t_{Y_{2}})}\,+V^{S=0}_{ij}\Big{)}$
$\displaystyle\qquad\qquad+\frac{m(t_{Y_{1}})+m(t_{Y_{2}})}{M(t_{Y_{1}},{t_{Y2}})}\,\frac{p^{2}_{Y_{1}Y_{2}}}{2\mu_{Y_{1}Y_{2}}}\,+V^{S=-2}_{Y_{1}Y_{2}}$
$\displaystyle\qquad\qquad+\big{(}m(t_{Y_{1}})+m({t_{Y_{2}}})-m_{\Xi}-m_{N}\big{)},$
(3) $\displaystyle\begin{split}H^{S=-2}_{Y_{1}Y_{2},\Xi
N}=\sum_{i=1}^{A-1}V^{S=-2}_{Y_{1}Y_{2},\Xi i}\,,\end{split}$ (4)
$\displaystyle H_{\Xi}=H^{S=0}_{\Xi}\,+\,H^{S=-2}_{\Xi}$ (5)
$\displaystyle\qquad=\sum_{i<j=1}^{A-1}\Big{(}\frac{2p^{2}_{ij}}{M({\Xi})}\,+V^{S=0}_{ij}\Big{)}$
$\displaystyle\qquad\,\,+\sum_{i=1}^{A-1}\Big{(}\frac{m_{N}+m_{\Xi}}{M({\Xi})}\,\frac{p^{2}_{\Xi
i}}{2\mu_{\Xi i}}\,+V^{S=-2}_{\Xi i}\Big{)},$

with $|Y_{1}Y_{2}\rangle=|\Lambda\Sigma\rangle$ or $|\Sigma\Sigma\rangle$.
Here, $m({t_{Y_{1}}}),m(t_{Y_{2}})$ , $m_{\Xi}$ and $m_{N}$ are the $Y_{1}$,
$Y_{2}$, $\Xi$ hyperon and nucleon rest masses, respectively. $M$ denotes the
total rest mass of the system, thus,
$M(t_{Y_{1}},t_{Y_{2}})=m(t_{Y_{1}})+m(t_{Y_{2}})+(A-2)m_{N}$ and
$M(\Xi)=m_{\Xi}+(A-1)m_{N}$. $\mu_{i\Xi}$ and $\mu_{Y_{1}Y_{2}}$ are the
$\Xi$N and YY reduced masses, respectively. The rest mass differences within
the nucleon- and hyperon-isospin multiplets are neglected. $V^{S=0}$ and
$V^{S=-2}$ are the two-body NN and YY ($\Xi$N) 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 Le:2021wwz .
Likewise, the final $\Xi$ wave functions and the corresponding binding
energies are directly obtained via the Lanczos eigenvalue iterations.

##  4 Results and discussion

(a) $E_{\mathcal{N}}(^{4}_{\Xi}\mathrm{H}(1^{+},0))$ as a function of
$\omega$. (b) $E(^{4}_{\Xi}\mathrm{H}(1^{+},0))$ as a function of
$\mathcal{N}$. (c) $B_{\Xi}(^{4}_{\Xi}\mathrm{H}(1^{+},0))$ as a function of
$\mathcal{N}$. (d) $B_{\Xi}(^{4}_{\Xi}\mathrm{H}(1^{+},0))$ as a function of
$\lambda_{YY}$.

Figure 2:  (a-c): binding energy $E$ and $\Xi$ separation energy $B_{\Xi}$ for
${}^{4}_{\Xi}\mathrm{H}(1^{+},0)$ computed with the YY-$\Xi$N interaction
NLO(500), SRG-evolved to a flow parameter of $\lambda_{YY}=3.0$ fm-1. For the
NN interaction the SMS N4LO+(450) potential Reinert:2017usi  with
$\lambda_{\mathrm{NN}}=1.6$ fm-1 is employed. $B_{\Xi}$ is measured with
respect to the triton binding energy (which is $E(^{3}\mathrm{H})=-8.5$ MeV
for the used NN interaction). (a): Solid lines and symbols (with different
colors) represent numerical results for different model spaces
$\mathcal{N}=14-30$, from top to bottom. The dashed lines are obtained by
using the ansatz Eq. (22) in Le:2020zdu . (b-c): Horizontal (red) lines with
shaded areas indicate the converged results and the corresponding
uncertainties. (d): Dependence of $B_{\Xi}(^{4}_{\Xi}\mathrm{H}(1^{+},0))$ on
the flow parameter $\lambda_{YY}$.

As it has been shown in Miyagawa:2021krh , the ${}^{3}_{\Xi}\mathrm{H}$
hypernucleus is not bound with the chiral YY-$\Xi$N NLO potential
Haidenbauer:2018gvg . Therefore, the lightest system that we study here is
NNN$\Xi$. A $\Xi$ hyperon with isospin $\frac{1}{2}$ can couple to the core
nucleus ${}^{3}\mathrm{H}/^{3}\mathrm{He}$ in its ground state
$(\frac{1}{2}^{+},\frac{1}{2})$ resulting in several NNN$\Xi$ states with
$(J^{\pi},T)=(1^{+},0)$, $(0^{+},1)$, $(1^{+},1)$ 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 NNN$\Xi$ state, namely the $(1^{+},0)$
Hiyama:2019kpw . As discussed in Section 2, there are some differences in the
predictions for the $\Xi$N 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 NNN$\Xi$ binding energies. We also consider
${}^{5}_{\Xi}\mathrm{H}$. Due to the strong binding of the $\alpha$ particle
core, the mass difference between $\Xi$N and $\Lambda\Lambda$ is partly
removed Myint:1994qv  which makes this light hypernucleus especially
interesting. Likewise, theoretical predictions for ${}^{7}_{\Xi}\mathrm{H}$
are of importance since this system is expected to be investigated through the
${}^{7}{\mathrm{Li}}(K^{-},K^{+})$ reaction in upcoming experiments at J-PARC
Fujioka:2021S . Generally, we expect that a consistent study of hypernuclei
for a range of mass numbers might provide constraints for the properties of
$\Xi$s in nuclear matter. In this work, we present such an investigation for
$\Xi$ hypernuclei up to $A=7$.

In this exploratory study, we do not take the $\Xi^{-}$-$\Xi^{0}$ 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
$A=5$ $\Xi$ 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.Hiyama:2019kpw  and only consider the
$(1/2^{+},\,3/2)$ state. For this state, it is natural to expect that the
$\Xi^{-}$-${}^{6}\mathrm{He}(0^{+},1)$ 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 Le:2020zdu . The $\omega$\- and $\mathcal{N}$-space
extrapolations for the binding energy of the ${}^{4}_{\Xi}{\rm H}(1^{+},0)$
state, $E(1^{+},0)$, are illustrated in panels (a) and (b) of Fig. 2,
respectively. To obtain the converged $\Xi$ separation energy $B_{\Xi}$, we
perform an analogous exponential fit on the quantities
$B_{\Xi}(\mathcal{N})=E_{\mathcal{N}}(1^{+},0)-E_{\mathcal{N}}(^{3}\mathrm{H})$,
see also panel (c). Here $E_{\mathcal{N}}(1^{+},0)$,
$E_{\mathcal{N}}(^{3}\mathrm{H})$ are the hypernuclear and nuclear binding
energies, respectively, obtained at their optimal HO frequencies for a given
model space $\mathcal{N}$. 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
Le:2020zdu ; Le:2021wwz . 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
$\mathcal{N}\rightarrow\infty$. Clearly, well-converged results for both
$E(1^{+},0)$ and $B_{\Xi}(1^{+},0)$ are achieved for model spaces up to
$\mathcal{N}_{max}=30$. For the NN interaction alone, the triton energy
calculation is, however, converged already for model spaces
$\mathcal{N}_{max}=24$. Similar convergence patterns are also observed for the
binding (separation) energies of the other states in NNN$\Xi$ and of the
${}^{5}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{1}{2})$,
${}^{7}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{3}{2})$ hypernuclei. The
convergence for $A=5$ and $A=7$ is generally faster since the separation
energies are larger. Therefore, our current limits of $\mathcal{N}_{max}=16$
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
${}^{4}_{\Xi}{\rm n}$ states for isospin $T=1$. As can be seen in Table 1,
both of these states, ${}^{4}_{\Xi}{\rm n}(0^{+},1)$ and
${}^{4}_{\Xi}\rm{n}(1^{+},1)$, are clearly bound for the chiral interaction.
It turns out that their binding energies are comparable to that one for
${}^{4}_{\Xi}{\rm H}(1^{+},0)$. The ${}^{4}_{\Xi}{\rm H}(0^{+},0)$ state, on
the other hand, is unbound for the chiral interactions at NLO.

The converged $B_{\Xi}(1^{+},0)$ computed for a wide range of SRG-flow
parameter, $1.4\leq\lambda_{YY}\leq 3.0$ fm-1, are presented in panel (d) of
Fig. 2. One sees that the overall variation of $B_{\Xi}(1^{+},0)$ is visible,
about $190\pm 30$ keV. It is also larger than the dependence of
$B_{\Lambda\Lambda}$ on the SRG-flow parameter, which was found to be of the
order of 100 keV Le:2020zdu . This may be related to the fact that, unlike for
the $\Lambda\Lambda$ case where the coupling to the pion is suppressed by
isospin conservation, $\pi$ exchange contributes to the $\Xi$N interaction at
leading order. Long-range interactions are likely to be more strongly affected
by the SRG evolution. However, the variation is much smaller than the one
observed for single $\Lambda$ hypernuclei (see e.g. Le:2020zdu ). For the
$A=5$ and 7 $\Xi$ hypernuclei, we observed similarly large absolute
variations, but still they are relatively smaller as compared to the estimated
$\Xi$ 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 $\lambda_{YY}=1.6$ fm-1.

|  $B_{\Xi}$ [MeV] |  $\Gamma$ [MeV]
---|---|---
${}^{4}_{\Xi}\mathrm{H}(1^{+},0)$ | $0.48\pm 0.01$ | 0.74
${}^{4}_{\Xi}\mathrm{n}(0^{+},1)$ | $0.71\pm 0.08$ | 0.2
${}^{4}_{\Xi}\mathrm{n}(1^{+},1)$ | $0.64\pm 0.11$ | 0.01
${}^{4}_{\Xi}\mathrm{H}(0^{+},0)$ | - | -
${}^{5}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{1}{2})$ | $2.16\pm 0.10$ | 0.19
${}^{7}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{3}{2})$ | $3.50\pm 0.39$ | 0.2
Table 1:  $\Xi$ separation energies $B_{\Xi}$ and estimated decay widths
$\Gamma$ for $A=4-7$ $\Xi$ hypernuclei. All calculations are based on the
YY-$\Xi$N interaction NLO(500) and the NN interaction SMS N4LO+(450). Both
potentials are SRG-evolved to a flow parameter of
$\lambda_{\mathrm{NN}}=\lambda_{YY}=1.6$ fm-1. The values of $B_{\Xi}$ in
NNN$\Xi$, ${}^{5}_{\Xi}\mathrm{H}$ and ${}^{7}_{\Xi}\mathrm{H}$ are measured
with respect to the binding energies of the core nuclei ${}^{3}\mathrm{H}$,
${}^{4}{\mathrm{He}}$ and ${}^{6}{\mathrm{He}}$, respectively.

The predicted separation energies $B_{\Xi}$ for the $A=4-7$ $\Xi$ hypernuclei
are listed in Table 1. We verified that all the bound states established here
are predominantly due to the strong $\Xi$N interaction. The $\Xi^{-}p$ Coulomb
interaction contributes roughly 200, 600, and 400 keV to the binding energies
of NNN$\Xi$, ${}^{5}_{\Xi}\mathrm{H}$ and ${}^{7}_{\Xi}\mathrm{H}$,
respectively. Table 1 provides also an estimate of the corresponding decay
width $\Gamma$. These widths have been evaluated perturbatively by adapting
the procedure followed by Hiyama et al.Hiyama:2008fq ; Hiyama:2019kpw . Hiyama
et al.have used the imaginary part of the $G$ matrix. Here, we employ the
$\Xi$N $T$-matrix in the ${}^{11}S_{0}$ state from the original potential that
includes the $\Xi$N-$\Lambda\Lambda$ coupling Haidenbauer:2018gvg  instead.
Schematically the width amounts to $\Gamma\simeq-2\,{\rm
Im}\,\langle\Psi_{B_{\Xi}}|T_{\Xi N-\Xi N}|\Psi_{B_{\Xi}}\rangle$ and involves
the pertinent hypernuclear wave function $\Psi_{B_{\Xi}}$ and the (off-shell)
$\Xi$N $T$-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 NNN$\Xi$ are only weakly bound, possessing quite similar
$B_{\Xi}$’s but substantially different decay widths. Interestingly, our
result for $B_{\Xi}({\rm NNN}\Xi(1^{+},0))$ is close to that for the HAL QCD
potential, reported in Hiyama:2019kpw , 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
$B_{\Xi}({\rm NNN}\Xi)$ and the predictions Hiyama:2019kpw  for the ESC08c
potential Nagels:2015dia . According to the discussion in Ref. Hiyama:2019kpw
, 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 $\Xi$N model in the $(1^{+},0)$ and $(1^{+},1)$ states.

In A, we summarize the relative weights of the different partial wave channels
to the effective $\Xi$N interaction in the $s$-shell $\Xi$ 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 ${}^{5}_{\Xi}\mathrm{H}$ separation energy and decay width
are $B_{\Xi}(^{5}_{\Xi}\mathrm{H})=2.16\pm 0.1$ MeV and
$\Gamma(^{5}_{\Xi}\mathrm{H})=0.19$ MeV, respectively. Oddly enough, these
values agree roughly with the estimations by Myint and Akaishi Myint:1994qv
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 accidental. We further note that in contrast to our finding where
${}^{5}_{\Xi}\mathrm{H}$ is bound primarily due to the strong $\Xi$N
interaction, the authors in Myint:1994qv  state that the binding energy of
$1.7$ MeV in ${}^{5}_{\Xi}\mathrm{H}$ largely comes from the
${}^{4}\mathrm{He}$-$\Xi^{-}$ Coulomb interaction. The mechanism for the
narrow width of ${}^{5}_{\Xi}\mathrm{H}$ has been investigated in Dover:1994qs
; Kumagai-Fuse:1995zeb . Recently, Friedman and Gal, employing an optical
potential, also reported a quite similar result for ${}^{5}_{\Xi}\mathrm{H}$
($B_{\Xi}(^{5}_{\Xi}\mathrm{H})=$
2.0 MeV) Friedman:2021rhu . But also here the agreement seems to be more or
less accidental given that the $\Xi$-nuclear interaction used as starting
point in that work is with $U_{\Xi}\lesssim-20$ MeV significantly more
attractive than the one predicted by the chiral $\Xi$N potential employed in
the present study which is only around $U_{\Xi}\approx-9$ MeV Kohno:2019oyw
as mentioned above.

The prediction of the chiral $\Xi$N interaction for ${}^{7}_{\Xi}\mathrm{H}$
$(\frac{1}{2}^{+},\frac{3}{2})$, $B_{\Xi}(^{7}_{\Xi}\mathrm{H})=3.50\pm 0.39$
MeV, is only slightly larger than the binding energy of 3.15 MeV reported by
Fujioka et al.Fujioka:2021S ; Fujioka:2021T  for the HAL QCD interaction,
based on a calculation within a four-body ($\alpha nn\Xi$) cluster model
Hiyama:2008fq ; Hiyama:2021P . An earlier study utilizing older $S=-2$
potentials from the Nijmegen group indicated somewhat smaller binding energies
Ajimura:2019 ; Hiyama:2008fq . Finally, as one can see from Table 1, the
${}^{7}_{\Xi}\mathrm{H}\,(\frac{1}{2}^{+},\frac{3}{2})$ state is also very
narrow, with a width of $\Gamma=0.2$ MeV.

| $V^{S=-2}$ | E
---|---|---
| ${}^{11}S_{0}$ | ${}^{31}S_{0}$ | ${}^{13}S_{1}$ | ${}^{33}S_{1}$ | total |
${}^{4}_{\Xi}\mathrm{H}(1^{+},0)$ | -1.95 | 0.02 | -0.7 | -2.31 | -5.21 | -8.97
${}^{4}_{\Xi}\mathrm{n}(0^{+},1)$ | -0.6 | 0.25 | -0.004 | -0.74 | -1.37 | -9.07
${}^{4}_{\Xi}\mathrm{n}(1^{+},1)$ | -0.02 | 0.16 | -0.13 | -1.14 | -1.30 | -9.0
${}^{4}_{\Xi}\mathrm{H}(0^{+},0)$ | -0.002 | 0.08 | -0.01 | -0.006 | -0.11 | -6.94
${}^{5}_{\Xi}\mathrm{H}(1/2^{+},1/2)$ | -0.96 | 0.94 | -0.58 | -3.63 | -4.88 | -31.43
${}^{7}_{\Xi}\mathrm{H}(1/2^{+},3/2)$ | -1.23 | 1.79 | -0.79 | -6.74 | -8.04 | -33.22
Table 2: Contributions of different partial waves to $\langle V^{S=-2}\rangle$
(first five columns), and the total binding energy (last column) for the
$A=4-7$ $\Xi$ hypernuclei. The results are extracted at $\mathcal{N}=28$,
$\omega=10$ MeV for NNN$\Xi$, at $\mathcal{N}=14$, $\omega=16$ MeV for
${}^{5}_{\Xi}\mathrm{H}$ and at $\mathcal{N}=10$, $\omega=16$ MeV for
${}^{7}_{\Xi}\mathrm{H}$. All energies are given in MeV. Same interactions as
in Table 1. Note that the calculated binding energy of
${}^{3}\mathrm{He}(^{3}\mathrm{H})$ is $-7.79\,(-8.50)$ MeV.

To shed light on the relation between the properties of the chiral $\Xi$N
potential and the binding of the $A=4-7$ $\Xi$ systems, we provide in Table 2
the contributions of different $\Xi$N partial waves to the expectation value
of the $S=-2$ potential $\langle V^{S=-2}\rangle$. These results are computed
at $\mathcal{N}=28,\omega=10$ MeV for NNN$\Xi$, at $\mathcal{N}=14,\omega=16$
MeV for ${}^{5}_{\Xi}\mathrm{H}$ and at $\mathcal{N}=10,\omega=16$ MeV for
${}^{7}_{\Xi}\mathrm{H}$. Here the second largest model space is chosen for
each system in order to save computational resources. And, $\omega$ 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 NNN$\Xi(0^{+},0)$ the
attractive $\Xi$N 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 $\langle V^{s=-2}\rangle$. While the attraction in the
${}^{11}S_{0}$ channel is essential as well for NNN$\Xi(1^{+},0)$ and
$(0^{+},1)$ (amounting to more than $30\%$ of $\langle V^{S=-2}\rangle$), its
contribution becomes less significant in other states. Furthermore, the $\Xi
N$ repulsion in ${}^{31}S_{0}$ contributes predominantly to the expectation
value $\langle V^{S=-2}\rangle$ of NNN$\Xi(0^{+},0)$ (naturally with opposite
sign), which causes the system to be unbound. The expectation value $\langle
V^{S=-2}(^{31}S_{0})\rangle$ is also sizable for ${}^{5}_{\Xi}\mathrm{H}$ and
${}^{7}_{\Xi}\mathrm{H}$, however, its effect is largely canceled by the
attraction in the ${}^{11}S_{0}$ channel.

| $|\Xi N\rangle$
---|---
| $|^{11}S_{0}\rangle$ | $|^{31}S_{0}\rangle$ | $|^{13}S_{1}\rangle$ | $|^{33}S_{1}\rangle$ | $J\geq 2$
${}^{4}_{\Xi}\mathrm{H}(1^{+},0)$ | 12.88 | 0.18 | 25.91 | 35.72 | 24.80
${}^{4}_{\Xi}\mathrm{n}(0^{+},1)$ | 8.24 | 13.32 | 0.23 | 23.29 | 54.73
${}^{4}_{\Xi}\mathrm{n}(1^{+},1)$ | 0.14 | 9.22 | 9.83 | 33.08 | 47.56
${}^{4}_{\Xi}\mathrm{H}(0^{+},0)$ | 0.02 | 11.87 | 14.65 | 0.11 | 73.33
${}^{5}_{\Xi}\mathrm{H}(1/2^{+},1/2)$ | 4.82 | 12.18 | 14.37 | 35.53 | 32.59
${}^{7}_{\Xi}\mathrm{H}(1/2^{+},3/2)$ | 3.71 | 12.92 | 11.11 | 38.36 | 32.94
Table 3:  Probabilities (in %) of finding a $\Xi$N pair in different partial-
wave states in the wave functions of $A=4-7$ $\Xi$ hypernuclei. Same
interactions and model spaces as in Table 1. Note that for each system all
probabilities sum up to the probability of finding a $\Xi$ hyperon in that
system.

Complementary to Table 2, the binding of the $A=4-7$ hypernuclei can also be
understood from Table 3, where probabilities of finding a $\Xi$N pair,
$P_{\Xi\mathrm{N}}$, in different partial-wave states are listed. One clearly
notices that, in most of the systems, a $\Xi$N pair is predominantly found in
those channels with $J\leq 1$ and in particular in the ${}^{33}S_{1}$, except
for the unbound ${}^{4}_{\Xi}\mathrm{H}(0^{+},0)$ state. In addition, the two
extremely small probabilities $P_{\Xi\mathrm{N}}(^{11}S_{0})=0.02\%$ and
$P_{\Xi N}(^{33}S_{1})=0.11\%$ in ${}^{4}_{\Xi}\mathrm{H}(0^{+},0)$ are
obvious manifestations of the small expectation values
$V^{S=-2}(^{11}S_{0})=-0.002$ MeV and $V^{S=-2}(^{33}S_{1})=-0.006$ MeV listed
in Table. 2. Furthermore, the strong variation of
$P_{\Xi\mathrm{N}}(^{11}S_{0})$ 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 $\Lambda\Lambda-\Xi$N coupling
in the J-NCSM application and we compensated that by a small modification of
the $\Xi$N 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 $\Xi$N potential, see Fig. 1, implies that all found $\Xi$
hypernuclei could be simply minimally more bound.

##  5 Conclusions

In this work, we employed the Jacobi NCSM in combination with the chiral
NLO(500) $\Xi$N potential to explore $A=4-7$ $\Xi$ hypernuclei. Particle
conversions like $\Lambda\Sigma-\Xi$N$-\Sigma\Sigma$ are fully taken into
account, while the transition $\Lambda\Lambda-\Xi$N is omitted and its
contribution is incorporated effectively by re-adjusting the strength of the
$V_{\Xi N}$ potential appropriately. The latter approach facilitates a proper
convergence of the energy calculations to the lowest lying $\Xi$ states.
Furthermore, to speed up the convergence, the $\Xi$N potential is SRG-evolved
to a wide range of flow parameters. The effect of SRG evolution on the $\Xi$
separation energies is in general small, but, it is slightly larger than that
observed for $\Lambda\Lambda$ hypernuclei. We found three loosely bound states
$(1^{+},0)$, $(0^{+},1)$ and $(1^{+},1)$ for the NNN$\Xi$ system and more
tightly bound ${}^{5}_{\Xi}\mathrm{H}$, ${}^{7}_{\Xi}\mathrm{H}$ hypernuclei.
These $\Xi$ systems are bound predominantly due to the attraction of the
chiral $\Xi$N potential in the ${}^{33}S_{1}$ channel. On the other hand, the
repulsive nature in ${}^{31}S_{0}$ prevents the binding of the
NNN$\Xi(0^{+},0)$ state. All the investigated $\Xi$ 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 HALQCD:2019wsz , it seems likely that light
$\Xi$ hypernuclei exist. Experimental confirmation is certainly challenging.
However, theoretical estimates for yields of $A=4$ hypernuclei
Steinheimer:2012tb  as well as actual measurements of
${}^{4}_{\Lambda}\mathrm{H}$, ${}^{4}_{\Lambda}\mathrm{He}$ by the STAR
Collaboration Leung:2021  raise hopes that NNN$\Xi$ bound states can be
detected in heavy ion collisions in the not too far future. Also a bound
${}^{7}_{\Xi}\mathrm{H}$ system could be produced and studied in the
7Li$(K^{-},K^{+})$ reaction, cf. the proposal P75 for J-PARC Ajimura:2019 .
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.

Acknowledgements: This work is supported in part by the NSFC and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the funds
provided to the Sino-German Collaborative Research Center TRR110 “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076 - TRR 110). We also acknowledge support of the THEIA net-
working activity of the Strong 2020 Project. The numerical calculations have
been performed on JURECA and the JURECA booster of the JSC, Jülich, Germany.
The work of UGM was supported in part by the Chinese Academy of Sciences (CAS)
President’s International Fellowship Initiative (PIFI) (Grant No. 2018DM0034)
and by VolkswagenStiftung (Grant No. 93562).

##  Appendix A Estimate of partial wave contributions

In this appendix, we summarize approximate partial wave contributions to
$s$-shell $\Xi$ hypernuclei. The relations are similarly derived as the ones
for $\Lambda$ hypernuclei and the $\Lambda N$ potential Gibson:1994yp . For
these rough estimates it is assumed that there is no particle conversion
contributing. Additionally, the $\Xi$ hypernucleus exhibits a clear core-$\Xi$
structure. Both, the core nucleons and the $\Xi$ are in $s$-wave states. To
justify these assumptions, we provide in Tables 4 and 5 probabilities of
finding the nucleons in certain angular momentum and isospin states and
together with $\Xi^{-}$ or $\Xi^{0}$. For the isospin zero states of $A=4$,
the hypernucleus seems to be dominated by the 3He/3H component together with
$\Xi^{-}$ and $\Xi^{0}$, respectively. The choice of $T=0$ enforces that both
parts contribute equally. For the other isospin and $A=5$ and 7, the total
charge of the systems is chosen such that the $\Xi^{-}$ contribution dominates
in conjunction with the expected core nuclei. In the approximation that the
hypernucleus only contains these dominant components, one obtains for the
effective interactions

${}^{3}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{1}{2}):\tilde{V}_{\Xi N}$
$\displaystyle\approx\frac{3}{16}V_{\Xi N}^{{}^{11}S_{0}}+\frac{9}{16}V_{\Xi
N}^{{}^{31}S_{0}}\,+\,\frac{1}{16}V_{\Xi N}^{{}^{13}S_{1}}$ (6)
$\displaystyle\quad+\,\frac{3}{16}V_{\Xi N}^{{}^{33}S_{1}}$

${}^{3}_{\Xi}\mathrm{H}(\frac{3}{2}^{+},\frac{1}{2}):\tilde{V}_{\Xi N}$
$\displaystyle\approx\frac{1}{4}V_{\Xi N}^{{}^{13}S_{1}}+\frac{3}{4}V_{\Xi
N}^{{}^{33}S_{1}}$ (7)

${}^{4}_{\Xi}\mathrm{H}(1^{+},0):\tilde{V}_{\Xi
N}\approx\frac{1}{6}V^{{}^{11}S_{0}}_{\Xi
N}\,+\,\frac{1}{3}V^{{}^{13}S_{1}}_{\Xi
N}\,+\,\frac{1}{2}V^{{}^{33}S_{1}}_{\Xi N}$ (8)

${}^{4}_{\Xi}\mathrm{H}(0^{+},1):\tilde{V}_{\Xi N}$
$\displaystyle\approx\frac{1}{6}V^{{}^{11}S_{0}}_{\Xi
N}\,+\,\frac{1}{3}V^{{}^{31}S_{0}}_{\Xi
N}\,+\,\frac{1}{2}V^{{}^{33}S_{1}}_{\Xi N}$ (9)

${}^{4}_{\Xi}\mathrm{H}(1^{+},1):\tilde{V}_{\Xi N}$
$\displaystyle\approx\frac{1}{6}V^{{}^{31}S_{0}}_{\Xi
N}\,+\,\frac{1}{6}V^{{}^{13}S_{1}}_{\Xi
N}\,+\,\frac{2}{3}V^{{}^{33}S_{1}}_{\Xi N}$ (10)

${}^{4}_{\Xi}\mathrm{H}(0^{+},0):\tilde{V}_{\Xi
N}\approx\frac{1}{2}V^{{}^{31}S_{0}}_{\Xi
N}\,+\,\frac{1}{2}V^{{}^{13}S_{1}}_{\Xi N}$ (11)

${}^{5}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{1}{2}):\tilde{V}_{\Xi N}$
$\displaystyle\approx\frac{1}{16}V^{{}^{11}S_{0}}_{\Xi
N}\,+\,\frac{3}{16}V^{{}^{31}S_{0}}_{\Xi
N}\,+\,\frac{3}{16}V^{{}^{13}S_{1}}_{\Xi N}$ (12)
$\displaystyle\quad+\,\frac{9}{16}V^{{}^{33}S_{1}}_{\Xi N}.$

It clearly follows from Eqs. (6-12) that the ${}^{33}S_{1}$ $\Xi$N interaction
strength dominates the $(\frac{3}{2}^{+},\frac{1}{2})$ state in
${}^{3}_{\Xi}\mathrm{H}$, the $(1^{+},0)$, $(0^{+},1)$ and $(1^{+},1)$ states
in NNN$\Xi$ and the ${}^{5}_{\Xi}\mathrm{H}$ hypernucleus. The repulsive
${}^{31}S_{0}$ and weakly attractive ${}^{13}S_{1}$ potentials contribute
practically with equal weight to NNN$\Xi\,(0^{+},0)$, whereas the contribution
from the ${}^{31}S_{0}$ channel is dominant in
${}^{3}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{1}{2})$.

$(J_{core},T_{core},m^{t}_{\Xi})$ | ${}^{4}_{\Xi}H(1^{+},0)$ | ${}^{4}_{\Xi}n(0^{+},1)$ | ${}^{4}_{\Xi}n(1^{+},1)$ | ${}^{4}_{\Xi}\mathrm{H}(0^{+},0)$
---|---|---|---|---
$(\frac{1}{2},\frac{1}{2},-\frac{1}{2})$ | 49.66 | 97.48 | 97.44 | 49.98
$(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | 49.66 | – | – | 49.98
$(\frac{1}{2},\frac{3}{2},\frac{1}{2})$ | – | 0.54 | 0.55 | –
$(\frac{1}{2},\frac{3}{2},-\frac{1}{2})$ | – | 1.6 | 1.6 | –
others | 0.16 | 0.17 | 0.22 | 0.02
Table 4:  Probabilities (in %) of finding the nucleons in a total $J_{core}$
and $T_{core}$ angular momentum and isospin state and with a $\Xi$ hyperon
with third component of isospin $m^{t}_{\Xi}$ in the wave functions of $A=4$
$\Xi$ hypernuclei.

$(J_{core},T_{core},m^{t}_{\Xi})$ | ${}^{5}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{1}{2})$
---|---
$(0,0,-\frac{1}{2})$ | 96.03
$(0,0,\frac{1}{2})$ | 1.1
$(0,1,\frac{1}{2})$ | 2.1
others | 0.3

$(J_{core},T_{core},m^{t}_{\Xi})$ | ${}^{7}_{\Xi}\mathrm{H}(\frac{1}{2}^{+},\frac{3}{2})$
---|---
$(0,1,-\frac{1}{2})$ | 94.44
$(0,2,-\frac{1}{2})$ | 0.7
$(0,2,\frac{1}{2})$ | 2.8
others | 1.13

Table 5:  Same as Table 4 for $A=5$ and 7 $\Xi$ hypernuclei.

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