The tuning of stochastic gradient algorithms (SGAs) for optimization and sampling is often based on heuristics and trial-and-error rather than generalizable theory.
We address this theory–practice gap by characterizing the large-sample statistical asymptotics of SGAs via a joint step-size–sample-size scaling limit.
We show that iterate averaging with a large fixed step size is robust to the choice of tuning parameters and asymptotically has covariance proportional to that of the MLE sampling distribution.
We also prove a Bernstein–von Mises-like theorem to guide tuning, including for generalized posteriors that are robust to model misspecification.
Numerical experiments validate our results and recommendations in realistic finite-sample regimes.

Our work lays the foundation for a systematic analysis of other stochastic gradient Markov chain Monte Carlo algorithms for a wide range of models.


figuret

#  Single- & double-strangeness hypernuclei up to $A=8$ within chiral
effective field theory

Hoai Le

###### Abstract

We investigate $S=-1$ and $-2$ hypernuclei with $A=4-8$ employing the Jacobi-
NCSM approach and in combination with baryon-baryon interactions derived
within the frame work of chiral effective field theory. The employed
interactions are transformed using the similarity renormalization group (SRG)
so that the low- and high-momentum states are decoupled, and, thereby,
convergence of the binding energies with respect to model space can be
significantly speeded up. Such an evolution is however only approximately
unitary when the so-called SRG induced higher-body forces are omitted. We
first explore the impact of the SRG evolution on the $\Lambda$ separation
energies $B_{\Lambda}$ in $A=3-5$ hypernuclei when only SRG-evolved two-body
and when both two- and three-body forces are included. For the latter
scenario, we thoroughly study predictions of the two almost phase-equivalent
NLO13 and NLO19 YN potentials for $A=4-7$ hypernuclei. The NLO19 interaction
yields separation energies that are comparable with experiment, whereas NLO13
underestimates all the systems considered. We further explore CSB splittings
in the $A=7,8$ multiplets employing the two NLO YN potentials that include
also the leading CSB potential in the $\Lambda$N channel, whose strength has
been fitted to the presently established CSB in $A=4$. Finally, we report on
our recent study for $\Xi$ hypernuclei based on the $\Xi$N interaction at NLO.

##  1  Introduction

Experimental information on hyperon-nucleon (YN), hyperon-hyperon (YY) and
$\Xi$ N scattering data is indispensable input for developing realistic YN and
YY interactions [1, 2, 3, 4, 5, 6]. However, despite significant progress over
the last several decades, scattering data in the strangeness $S=-1,-2$ sectors
remain extremely scarce and the situation is not expected to be substantially
improved in the near future. Yet, other important source of information is
hypernucler spectroscopy [7], in which not only ground-state energies but also
energy level splittings in single- and double-strangness systems can be
determined with high accuracy. This together with theoretical studies of
hypernuclei, especially with microscopic approaches are essential tools to
explore the underlying baryon-baryon (BB) interactions and thereby to improve
interactions models.

In this study, we investigate single-$\Lambda$ and $\Xi$ hypernuclear systems
with $A=4-8$ based on the “ab initio” Jacobi no-core shell model (NCSM) [8, 9]
and in combination with BB interactions derived within chiral effective field
theory (EFT). In this approach, potentials are established in terms of an
expansion in powers of low momenta in accordance with an appropriate power
counting scheme. The long-ranged parts of the interactions are fixed from
chiral symmetry, whereas unresolved short-distance dynamics are parameterized
in terms of the so-called low energy constants (LECs), whose strengths should
be determined via a fit to experiments. In the non-strangeness sector, thanks
to the wealth of NN scattering data, two- and three-body interactions have
been consistently derived up to high order, e.g. $\mathrm{N}^{4}\mathrm{LO}$
[10] and $\mathrm{N^{2}LO}$ [11], respectively, see also Ref. [12] for an
overview on the current status of the chiral NN and 3N interactions. In the
$S=-1,-2$ sectors, the situation is however less satisfactory due to, as
already said, the lack of the appropriate scattering data. Despite of that, YN
and YY potentials up to NLO have been derived [2, 3, 4, 5] and recently
extended to $\mathrm{N^{2}LO}$ [6]. Especially, at NLO, by pursuing different
strategies for fixing LECs, two realizations of the YN interaction, NLO13 [2]
and NLO19 [3], which yield similarly good description of all the available YN
scattering data, could be established. However, as discussed in [3], NLO19 is
characterized by a somewhat weaker $\Lambda\mathrm{N}-\Sigma$N transition
potential, and as a result, the two potentials yield slightly different
predictions for ${}^{4}_{\Lambda}\mathrm{He}$ and for nuclear matter. Also, in
[9] we have explored effects of NLO13 and NLO19 on light hypernuclei with
$A=4-7$, but only qualitatively. Below we revisit this topic and take a closer
look at the separation energies of these systems predicted by the two
potentials, and thereby, discuss the possible effect of chiral YNN forces. In
addition, we explore charge symmetry breaking (CSB) effects in the $A=7,8$
multiplets, employing the NLO13 and NLO19 interactions and with the inclusion
of the leading CSB potential derived in Ref. [16]. The latter involves
additional two LECs that have been determined via a fit to the separation
energies difference $\Delta B_{\Lambda}$ in the ground and excited states of
${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda}$H, with $\Delta
B_{\Lambda}(0^{+})=233\pm 92$ keV and $\Delta B_{\Lambda}(1^{+})=-83\pm 94$
[17], respectively. Finally, we discuss our recent work [13], where the
possible existence of $\Xi$ hypernuclei up to $A=7$ for the NLO $\Xi$N
potential [5] has been explored.

##  2  SRG evolution for YN interactions and impact on $B_{\Lambda}$ in
$A=3-5$ hypernuclei

BB interactions generally contain short-range and tensor correlations that
couple low- and high-momentum states, which make calculations expanded in
harmonic oscillator (HO) bases such as the NCSM converge slowly. A very large
number of HO functions is required in order to incorporate such strong
correlations and therefore well-converged results for systems with $A\geq 5$
are practically computationally inaccessible. So, in order soften the
interactions we will continuously evolve the Hamiltonian using the Similarity
Renormalization Group (SRG). The latter can be written in term of a flow
equation for potentials, which, in three-body YNN space, has following form

$\displaystyle\begin{split}\frac{dV_{s}}{ds}=[[T_{rel},V_{s}],H_{s}];\quad
H_{s}=T_{rel}+V_{s}+\Delta
M=T_{rel}+V^{\mathrm{NN}}_{12,s}+V^{\mathrm{YN}}_{32,s}+V^{\mathrm{YN}}_{31,s}+V^{\mathrm{YNN}}_{s}+\Delta{M}.\end{split}$
(1)

Here $T_{rel}$ is the intrinsic kinetic energy, $\Delta M$ is the rest mass
difference arising due to particle conversions such as
$\Lambda\mathrm{N}-\Sigma\mathrm{N}$, and s is a flow parameter that varies
from 0 to $\infty$ with $s=0$ meaning no evolution or bare interactions. For
characterising SRG-evolved interactions, we will however use the more
intuitive $\lambda$ parameter, $\lambda=(4\mu_{\mathrm{N}}^{2}/s)^{1/4}$ with
$\mu_{\mathcal{N}}$ being nucleon reduced mass, that provides a measure of the
width of the potentials in momentum space [18]. We solve the flow equation Eq.
(1) in momentum space and follow the same approach as done in [18, 19] that
explicitly separates the flow equations for two-body NN and YN, and three-body
YNN interactions by employing the following relation,

$\displaystyle\begin{split}\frac{dV_{s}}{ds}=\frac{dV^{\mathrm{NN}}_{12,s}}{ds}+\frac{dV^{\mathrm{YN}}_{32,s}}{ds}+\frac{dV^{\mathrm{YN}}_{31,s}}{ds}+\frac{dV^{\mathrm{YNN}}_{s}}{ds}.\end{split}$
(2)

In two-body NN and YN space, the flow equations for $V^{\mathrm{NN}}$ and
$V^{\mathrm{YN}}$ can be written as [9]

$\displaystyle\begin{split}\frac{dV^{\mathrm{NN}}_{12}}{ds}&=[[t^{\mathrm{NN}}_{12},V^{\mathrm{NN}}_{12}],\,t^{\mathrm{NN}}_{12}+V_{12}^{\mathrm{NN}}]\\\
\frac{dV^{\mathrm{YN}}_{32}}{ds}&=[[t^{\mathrm{YN}}_{32},V^{\mathrm{YN}}_{32}],\,t^{\mathrm{YN}}_{32}+V_{32}^{\mathrm{YN}}+\Delta
M];\qquad\frac{dV^{\mathrm{YN}}_{31}}{ds}=[[t^{\mathrm{YN}}_{31},V^{\mathrm{YN}}_{31}],\,t^{\mathrm{YN}}_{31}+V_{31}^{\mathrm{YN}}+\Delta
M],\end{split}$ (3)

hence, the evolution equation for $V^{\mathrm{YNN}}$ reads,

$\displaystyle\frac{dV^{\mathrm{YNN}}}{ds}=[[t^{\mathrm{NN}}_{12},V^{\mathrm{NN}}_{12}],\,V^{\mathrm{YN}}_{32}+V^{\mathrm{YN}}_{31}+V^{\mathrm{YNN}}]+[[t^{\mathrm{YN}}_{32},\,V^{\mathrm{YN}}_{32}],V^{\mathrm{NN}}_{12}+V^{\mathrm{YN}}_{31}+V^{\mathrm{YNN}}]$
(4)
$\displaystyle\qquad+[[t^{\mathrm{YN}}_{31},V^{\mathrm{YN}}_{31}],V^{\mathrm{YN}}_{32}+V^{\mathrm{NN}}_{12}+V^{\mathrm{YNN}}]+[[T_{rel},V^{\mathrm{YNN}}],T_{rel}+V^{\mathrm{NN}}_{12}+V^{\mathrm{YN}}_{32}+V^{\mathrm{YN}}_{31}+V^{\mathrm{YNN}}+\Delta
M]$

Note that now the Kronecker delta functions that appear when embedding two-
body interactions in a three-body YNN basis, see the right hand side of Eq.
(4), can be evaluated analytically [18, 19]. By projecting Eqs.(3,4) onto the
corresponding two- and three-body partial-decomposed momentum-space bases, we
obtain a system of coupled ordinary differential equations which will be
solved simultaneously employing the ARKsover from SUNDIAL. In the current
work, we do not include chiral YNN forces in the evolution, however, it should
be clear from Eq. (4) that the SRG evolution will eventually induce three-body
YNN interactions even if the initial (bare) potential $V^{\mathrm{YNN}}_{s=0}$
is set to zero, $V^{\mathrm{YNN}}_{s=0}=0$. Therefore, omitting those induced
higher-body forces, the SRG evolution will only be approximately unitary.
Indeed, as it can clearly be seen in Fig. 1, the binding energies (square blue
symbols) of the $A=3-5$ hypernuclei, computed for SRG-evolved NN and YN
interactions, exhibit a strong dependence on the $\lambda$ flow parameter.
With the inclusion of the induced YNN forces, we practically reproduce the
hypertrition binding energy $E(^{3}_{\Lambda}\mathrm{H})$ (dashed green line)
that was obtained when solving the Faddeev equation for the bare NN and YN
potentials, see panel (a) in Fig. 1. Note the large estimated errors for the
NCSM results because of the extremely small separation energy of
${}^{3}_{\Lambda}\mathrm{H}$. In panel (b), we clearly demonstrate that
$E(^{3}_{\Lambda}\mathrm{H})$ is indeed independent of the SRG parameter by
increasing the strength of the YN interaction, say, by a factor of 1.5, so
that the ${}^{3}_{\Lambda}\mathrm{H}$ hypernucleus is more tightly bound.
Similarly, with both SRG-induced YNN and 3N forces included, the SRG-
dependence of $B_{\Lambda}$ in the $A=4,5$ systems is significantly removed,
as clearly shown in panels (c-d). This is an important indication of likely
negligibly small contributions of SRG-induced higher-body forces to the
separation energies of these systems. However, for a more quantitative
estimate of those contributions, well-converged separation energies computed
for a wide range of flow parameter are required.

(a) (b) (c) (d)

Figure 1:  Dependence of the binding (separation) energies in $A=3-5$
hypernuclei on the SRG flow parameter $\lambda$. The results (blue circles,
panels (a, c,d)) are computed using the SMS $\mathrm{N^{4}LO^{+}450}$ [10] NN
potential SRG-evolved to $\lambda_{NN}=1.6$ fm-1 and the NLO13(500) YN
interaction (with CSB included) evolved to a wide range of flow parameter. In
panel (b) the strength of the original NLO13(500) potential is amplified by a
factor of 1.5 so that ${}^{3}_{\Lambda}\mathrm{H}$ is more tightly bound.
Black triangles are obtained with inclusion of SRG-induced YNN (panels (a-b))
and of the 3N forces [11] (panels (c-d)). Dashed green lines are binding
energies computed by solving the Faddeev equation for the bare NN and YN
interactions.

##  3 Effect of NLO YN interactions on $B_{\Lambda}$ in $A=4-7$ hypernuclei

In our work [9], using only SRG-evolved NN and YN forces, we have
qualitatively studied predictions of the two practically phase equivalent
NLO13 and NLO19 YN interactions for the separation energies in the $A=4-7$
systems. Overall, the NLO19 potential, that is characterized by a weaker
$\Lambda\mathrm{N}-\Sigma\mathrm{N}$ transition potential [3], yields
substantially larger $B_{\Lambda}$ in all systems considered except for
${}^{4}_{\Lambda}\mathrm{H}/^{4}_{\Lambda}\mathrm{He}(0^{+})$, see Fig. 10 in
Ref.[9]. As also discussed in that work, such a large discrepancy between the
NLO13 and NLO19 predictions could be attributed to possible contributions of
chiral YNN forces, which according to the underlying power counting will
appear at $\mathrm{N^{2}LO}$ [6, 20]. Now, taking into account both chiral and
SRG-induced 3N interactions as well as SRG-induced YNN forces, we can study
the actual separation energies predicted by the two YN potentials in more
detail. These results together with the experimental $B_{\Lambda}$ for the
$A=4-7$ hypernuclei are summarized in Table. 1. One sees that NLO13 and NLO19
yield very similar separation energies for the
${}^{4}_{\Lambda}\mathrm{H(0^{+})}$ state, with $B_{\Lambda}=1.551\pm 0.007$
MeV and $1.514\pm 0.007$ MeV, respectively, that are by roughly 0.6 MeV
smaller than the empirical value of $B_{\Lambda}=2.16\pm 0.08$ MeV [21].
Likewise, the separation energies for ${}^{4}_{\Lambda}\mathrm{H}(1^{+})$ and
the ground states in ${}^{5}_{\Lambda}\mathrm{He}$ and
${}^{7}_{\Lambda}\mathrm{Li}$ computed for NLO13 are $0.823\pm 0.003$ MeV,
$\,2.22\pm 0.06$ MeV and $5.28\pm 0.68$, respectively. In comparison with the
empirical $B_{\Lambda}$ of $1.07\pm 0.08$ MeV, $3.12\pm 0.02$ MeV and $5.58\pm
0.03$ MeV [21], respectively, the NLO13 potential clearly underestimates these
systems. In contrast, the predictions of NLO19 for these three states are
quite close to experiment, only slightly above. For example, the computed
separation energies for ${}^{4}_{\Lambda}\mathrm{H(1^{+})}$ and
${}^{5}_{\Lambda}\mathrm{He}$,
$B_{\Lambda}(^{4}_{\Lambda}\mathrm{H,1^{+})})=1.514\pm 0.007$ and
$B_{\Lambda}(^{5}_{\Lambda}\mathrm{He})=3.32\pm 0.03$, exceed the experiment
by less than 0.2 MeV. Interestingly, for the ground state of
${}^{7}_{\Lambda}\mathrm{Li}$ we obtained the value of $B_{\Lambda}=6.04\pm
0.30$ MeV, that is somewhat larger than the emulsion separation energy,
$B_{\Lambda}=5.58\pm 0.03$ [21], but it is in perfect agreement with the one
deduced from counter experiments, with $B_{\Lambda}=5.85\pm 0.13(10)$ [22]. In
conclusion, the discrepancies between $B_{\Lambda}$ predicted by NLO13 and
NLO19, and between the computed $B_{\Lambda}$ and experiment highlight the
importance of chiral YNN forces for properly describing light hypernuclei.

Table 1: $\Lambda$-separation energies for the $A=4-7$ hypernuclei computed for the YN potentials NLO13(500) and NLO19(500) including the CSB interaction. The SMS $\mathrm{N^{4}LO^{+}}(450)$ and $\mathrm{N^{2}LO(450)}$ are used to describe the 2N and 3N interactions, respectively. All the employed potentials are evolved to a SRG parameter of $\lambda=1.88$ fm-1. SRG-induced YNN forces are explicitly taken into account. All energies are given in MeV. | ${}^{4}_{\Lambda}\mathrm{H}$ | ${{}^{5}_{\Lambda}\mathrm{He}}$ | ${{}^{7}_{\Lambda}\mathrm{Li}}$
---|---|---|---
| $0^{+}$ | $1^{+}$ | ${1/2^{+}}$ | $(1/2^{+},0)$
NLO13(500) | ${1.551\pm 0.007}$ | $0.823\pm 0.003$ | $2.22\pm 0.06$ | $5.28\pm 0.68$
NLO19(500) | ${1.514\pm 0.007}$ | $1.27\pm 0.009$ | $3.32\pm 0.03$ | ${6.04\pm 0.30}$
Exp [21, 22] | $2.16\pm 0.08$ | $1.07\pm 0.08$ | $3.12\pm 0.02$ | $5.85\pm 0.13(10)$
|  |  | $5.58\pm 0.03$

##  4 CSB in $A=7,8$ isospin multiplets

Being able to accurately compute hypernuclear wave functions for hypernuclei
up to the p-shell, we can now employ those wave functions to explore CSB
effects in the $A=7,8$ multiplets. Here, we employ the NLO13(500) and
NLO19(500) potentials both without and with CSB components [16] as YN
interactions, while the SMS $\mathrm{N^{4}LO^{+}}(450)$ [10] and
$\mathrm{N^{2}LO}(450)$ [11] are used for NN and 3N interactions,
respectively. All the employed potentials are evolved to a SRG parameter of
$\lambda=1.88$ fm-1. In addition, both YNN and 3N forces that are generated
during the evolution are explicitly included in all calculations. Let us
further remark that, as one can also see in Table. 1, due to the limited
accessible model space, $B_{\Lambda}$ for systems with $A\geq 7$ are afflicted
with rather large uncertainties, which even exceed the experimentally found
CSB splittings in those systems. Therefore, it will be more meaningful to
extract the CSB results for the $A=7,\,8$ multiplets perturbatively based on
the ${}^{7}_{\Lambda}\mathrm{Li^{*}(T=1)}$ and
${}^{8}_{\Lambda}\mathrm{Li(T=1/2)}$ hypernuclear wave functions and the
${}^{6}\mathrm{Li^{*}(T=1)}$ and ${}^{7}\mathrm{Li(T=1/2)}$ nuclear wave
functions, respectively. Note that this perturbative treatment has been also
applied to study CSB effects in the $A=4$ systems in Ref. [16]. The employed
wave functions are computed for the largest computationally accessible model
space, namely $\mathcal{N}_{\mathrm{max}}=10\,(9)$ for $A=7\,(8)$, and for the
optimal HO frequency $\omega_{\mathrm{opt}}=16$ MeV that yields the lowest
binding energies. For estimating the uncertainty, we will consider only two-
body interactions and perform calculations using the model space
$\mathcal{N}=\mathcal{N}_{\mathrm{max}},\,\mathcal{N}_{\mathrm{\mathrm{max}}}+2$,
and at the HO frequencies $\omega=\omega_{\mathrm{opt}}\pm 2$ MeV. The
variation of the CSB results compared to the values obtained at
$\mathcal{N}_{\mathrm{max}}$ and $\omega_{\mathrm{opt}}$ will be assigned as
the uncertainty of our CSB estimations which amount to 30 (50) keV for
$A=7\,(8)$. Note that the latter is of the same magnitude as that for the
uncertainties of the CSB results deduced from experiment, therefore, a direct
comparison of our CSB predictions with experiment is possible.

Details of the computed CSB splittings for the mirror hypernuclei
$(^{7}_{\Lambda}\mathrm{Be}$-${}^{7}_{\Lambda}\mathrm{Li^{*}})$ and
$(^{8}_{\Lambda}\mathrm{Be}$-${}^{8}_{\Lambda}\mathrm{Li})$ are summarized in
Table. 2, where individual contributions of the kinetic energy $\Delta T$, of
the NN interaction $\Delta V_{\mathrm{NN}}$ and of the YN potential $\Delta
V_{\mathrm{YN}}$ to the total CSB are also provided. Note that the 3N and SRG-
induced YNN forces contributions are insignificantly small, they are therefore
omitted from the table. Furthermore, in order to understand the relation
between the CSB effect of the employed YN potentials and the $\Delta
V_{\mathrm{YN}}$ for larger systems, which as shown in [16] contributes
dominantly to the CSB splitting in the $A=4$ doublets, we also estimate the
${}^{1}S_{0}$ and ${}^{3}S_{1}$ partial-wave contributions to $\Delta
V_{\mathrm{YN}}$ separately.

| $\Delta T$ | $\Delta V_{\mathrm{NN}}$ | $\Delta V_{\mathrm{YN}}$ | $\Delta B_{\Lambda}$
---|---|---|---|---
|  |  | ${}^{1}S_{0}$ | ${}^{3}S_{1}$ | total |
${}^{7}_{\Lambda}$Be-${}^{7}_{\Lambda}$Li∗ | NLO13 | 7 | -24 | -1 | 0 | 0 | -17
NLO13-CSB | 8 | -24 | -49 | 26 | -24 | -40
NLO19 | 6 | -40 | -1 | 0 | 0 | -34
NLO19-CSB | 6 | -41 | -43 | 42 | 9 | -35
| Hiyama [24] |  | -70 |  |  | 200 | 150
| Gal [23] | 3 | -70 |  |  | 50 | -17
| experiment [17] |  |  |  |  |  | $-100\pm 90$
${}^{8}_{\Lambda}$Be-${}^{8}_{\Lambda}$Li | NLO13 | 12 | 8 | -2 | 0 | -4 | 16
NLO13-CSB | 12 | 7 | 100 | 56 | 159 | 178
NLO19 | 7 | -11 | -1 | 0 | -2 | -6
NLO19-CSB | 6 | -11 | 62 | 79 | 147 | 143
| Hiyama [24] |  | 40 |  |  |  | 160
| Gal [23] | 11 | -81 |  |  | 119 | 49
| experiment [25] |  |  |  |  |  | $40\pm 60$
Table 2:  Contributions to CSB in the $A=7$ and $8$ isospin multiplets, based
on the YN potentials NLO13(500) and NLO19(500) (including 3N and SRG-induced
YNN forces). The results are for the original potentials (i.e. without CSB
force) and for the scenario CSB1 [16]. The estimated uncertainties for CSB in
$A=7,8$ multiplets is 30 and 50 keV, respectively. All energies are in keV.

One clearly sees that, despite the substantial discrepancies in their
predicted $B_{\Lambda}$, the two YN potentials yield comparable CSB results in
the $A=7$ mutiplets. For both cases, with and without the CSB interactions
included, the overall CSB effects are small and consistent with the experiment
both in magnitude and in sign. It is further observed that the ${}^{1}S_{0}$
and ${}^{3}S_{1}$ states contribute with opposite signs to the total $\Delta
V_{\mathrm{YN}}$, like that for the $1^{+}$ states in the $A=4$ systems, which
in turns leads to a small total CSB in the $A=7$ isotriplet. Surprisingly, our
prediction for ${}^{7}_{\Lambda}\mathrm{Be}$-${}^{7}_{\Lambda}\mathrm{Li^{*}}$
for the original NLO13 potential (i.e. without CSB components), with $\Delta
B_{\Lambda}(\mathrm{NLO13})=-17$ keV, is identical to the value estimated by
Gal [23], for which a simple shell-model approach in combination with
effective potentials were employed. However, the individual contributions
$\Delta T$, $\Delta V_{\mathrm{NN}}$, and $\Delta V_{\mathrm{YN}}$ differ
substantially. For example, the original NLO13 yields a vanishing $\Delta
V_{\mathrm{YN}}$, whereas in Gal’s calculation this contribution amounts to
$50$ keV. That value is also of opposite sign and larger than our estimated
$\Delta V_{\mathrm{YN}}$ for the actual CSB potential. In addition, our NCSM
estimate for $\Delta V_{\mathrm{NN}}$ (that also includes the Coulomb effect)
is visibly smaller than the corresponding one used in Gal’s calculation. Note
that the latter is not computed consistently but rather taken from a cluster-
model study of Hiyama et al. [24].

CSB results for ${}^{8}_{\Lambda}\mathrm{Be}$-${}^{8}_{\Lambda}\mathrm{Li}$
are listed at the lower end of Table 2. Without the CSB components, the two
NLO13 and NLO19 potentials predict a negligibly small CSB for
${}^{8}_{\Lambda}\mathrm{Be}$-${}^{8}_{\Lambda}\mathrm{Li}$, with $\Delta
B_{\Lambda}=16\pm 50$ keV and $-6\pm 50$ keV, respectively, which are well in
line with the separation energies difference of $40\pm 60$ keV [25] deduced
from emulsion data. A similarly small $\Delta B_{\Lambda}$ was also predicted
by Gal in [23]. However, in contrast to the rather small $\Delta
V_{\mathrm{NN}}$ contribution in our calculation, e.g. $\Delta
V_{\mathrm{NN}}=-11$ keV for NLO19, Gal assigned a significantly larger value
to $\Delta V_{\mathrm{NN}}$, with $\Delta V_{\mathrm{NN}}=-81$ keV that was
taken from a shell model calculation by Millener [23]. Interestingly, with the
CSB potentials included, both NLO13 and NLO19 yield rather sizable CSB
results, $\Delta B_{\Lambda}=177\pm 50$ keV and $143\pm 50$ keV, respectively.
In this case, the ${}^{1}S_{0}$ and ${}^{3}S_{1}$ partial-wave contributions
are also large, and importantly, of the same sign, adding up to a pronounced
total CSB. We found that this exactly resembles the situation for the $0^{+}$
states of the $A=4$ mirror hypernuclei. One could therefore conclude that a
fairly large splitting in the $A=4(0^{+})$ states, as presently suggested by
experiments [17], implies automatically a likewise significant CSB splitting
in ${{}^{8}_{\Lambda}\mathrm{Be}}$–${{}^{8}_{\Lambda}\mathrm{Li}}$.
Interestingly, the predictions of NLO13 and NLO19 with CSB interaction are
comparable to the value of $\Delta B_{\Lambda}=160$ keV obtained in a
($\Lambda$+$\alpha$+3He/$t$) three-body cluster calculation by Hiyama et al.
[24]. However, it should be stressed that the phenomenological CSB YN
interaction employed in that work was fitted to an outdated CSB splittings for
the $A=4$ doublets, namely $\Delta B_{\Lambda}(0^{+})=350\pm 60$ keV and
$\Delta B_{\Lambda}(1^{+})=240\pm 60$ keV. Also it was estimated there that
the Coulomb interaction contributes about -80 keV to $\Delta B_{\Lambda}$,
which is substantially larger than our estimate of roughly 10 keV.

##  5 Results for $A=4-7$ $\Xi$ hypernuclei

In this section we discuss the results for $A=4-7$ $\Xi$ hypernuclei based on
our recent study in Ref. [13]. Here all the calculations have been computed
using the SMS $\mathrm{N^{4}LO^{+}}(450)$ NN [10] and NLO(500) $\Xi$N [5]
interactions. The original $\Xi$N potential includes coupling to all other
baryon-baryon channels in the $S=-2$ sector such as
$\Lambda\Lambda,\Lambda\Sigma$ and $\Sigma\Sigma$. However, in practice, the
coupling $\Lambda\Lambda-\Xi$N was omitted here and its contribution is
effectively incorporated by appropriately re-adjusting the strength of the
$V_{\Xi\mathrm{N}-\Xi\mathrm{N}}$ potential [13]. This facilitates a proper
convergence of the energy calculations to the lowest lying $\Xi$ state.
Furthermore, the employed NN and $\Xi$N potentials are evolved to a SRG flow
parameter of $\lambda_{\mathrm{NN}}=\lambda_{\Xi\mathrm{N}}=1.6$ fm-1. For
simplicity, the electromagnetic NN interaction [26] as well as the Coulomb
point-like interaction between $\Xi^{-}$ and a proton are only included after
the evolution. And, SRG-induced higher-body forces are not included in that
study. As a result, we observe a moderate dependence of the $\Xi$ separation
energies $B_{\Xi}$ on the $\lambda_{\Xi\mathrm{N}}$ parameter [13]. However,
the overall SRG dependence of $B_{\Xi}$ is relatively small as compared to the
actual separation energies, and therefore, it should not affect our conclusion
on the existence of the found bound state [13].

The computed separation energies $B_{\Xi}$ of $A=4-7$ $\Xi$ hypernuclei
together with their perturbatively estimated decay width $\Gamma$ are provided
in Table 3 . It is estimated that the $\Xi^{-}p$ Coulomb interaction
contributes only moderately to the total binding energies of the NNN$\Xi$,
${}^{5}_{\Xi}\mathrm{H}$ and ${}^{7}_{\Xi}\mathrm{H}$ systems, about 200, 600,
and 400 keV, respectively.

|  $B_{\Xi}$ [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 3:  $\Xi$ separation energies $B_{\Xi}$ and estimated decay widths
$\Gamma$ for $A=4-7$ $\Xi$ hypernuclei. All calculations are based on the
NLO(500) YY-$\Xi$N interaction and the SMS N4LO+(450) NN potential. Both
interactions are SRG-evolved to a flow parameter of
$\lambda_{\mathrm{NN}}=\lambda_{\Xi\mathrm{N}}=1.6$ 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.

Thus, the binding of these systems should be predominantly due to the strong
$\Xi$N interaction. Overall, we found three weakly bound states $(1^{+},0)$,
$(0^{+},1)$ and $(1^{+},1)$ in NNN$\Xi$, with quite similar $B_{\Xi}$’s but
substantially different decay widths. Interestingly, our result for the ${\rm
NNN}\Xi(1^{+},0)$ state, $B_{\Xi}=0.48\pm 0.01$ MeV, is close to the value of
$0.36\pm 0.16$ MeV obtained with the HAL QCD potential [27]. The latter
however does not support the binding of the two $(0^{+},1)$ and $(1^{+},1)$
states. In addition, there are substantial differences between our separation
energies $B_{\Xi}({\rm NNN}\Xi)$ and the results for the ESC08c potential [28]
also reported in [27]. Note that this version of the Nijmegen potential
predicts an unrealistically large strength for the $\Xi$N interaction in the
${}^{33}S_{1}$ state that even results in a bound $\Xi$N state.

Our computed separation energy and decay width for ${}^{5}_{\Xi}\mathrm{H}$
are $B_{\Xi}=2.16\pm 0.1$ MeV and $\Gamma=0.19$ MeV. Surprisingly, these
values agree roughly with the estimations by Myint and Akaishi [29] of $1.7$
MeV and $0.2$ MeV, respectively. However, in contrast to our finding that
${}^{5}_{\Xi}\mathrm{H}$ is bound primarily due to the strong $\Xi$N
interaction, the authors in [29] estimated that most of the $1.7$ MeV binding
energy comes from the ${}^{4}\mathrm{He}$-$\Xi^{-}$ Coulomb interaction. A
quite similar separation energy, $B_{\Xi}(^{5}_{\Xi}\mathrm{H})=$ 2.0 MeV, is
also reported in the work by Friedman and Gal, where an optical potential is
employed [30]. But, as discussed in [13], such an agreement could be more or
less accidental.

For the ${}^{7}_{\Xi}\mathrm{H}$ $(\frac{1}{2}^{+},\frac{3}{2})$ system, the
chiral $\Xi$N potential yields a separation energy of $B_{\Xi}=3.50\pm 0.39$
MeV and a narrow decay width, with $\Gamma=0.2$ MeV. The computed $B_{\Xi}$ is
only slightly larger than the value of 3.15 MeV reported by Fujioka et al.
[31] when the HAL QCD potentials is employed in combination with a four-body
($\alpha nn\Xi$) cluster model [32]. With an older $S=-2$ potential from the
Nijmegen group, the authors however obtained somewhat smaller binding energy,
namely $B_{\Xi}=1.8$ MeV [32].

Let us finally remark that in Ref. [13] we have also explored the relation
between properties of the employed chiral $\Xi$N interaction and the binding
of the $A=4-7$ $\Xi$ systems. We shown that the $\Xi$N attraction in the
${}^{33}S_{1}$ state is essential for the binding of all the systems
considered, which contributes to more than $50\%$ of the $\Xi$N potential
expectation value $\langle V_{\Xi\mathrm{N}}\rangle$.

##  6 Conclusion

In this work, we throughly studied single- and double-strangeness hypernuclei
with $A=4-8$ based on the Jacobi NCSM and in combination with the NN (3N), YN
and $\Xi$N interactions consistently derived within the chiral EFT approach.
In order to speed up the convergence of the NCSM with respect to model space,
all the employed two-body and 3N potentials are transformed using an SRG
evolution. We explicitly showed that, with the inclusion of the SRG-induced
three-body forces (3N and YNN), the strong dependence of $B_{\Lambda}$ on SRG
flow parameter is removed for ${}^{3}_{\Lambda}\mathrm{H}$ and significantly
reduced for the $A=4,5$ systems. After that, we studied the predictions of the
two practically phase-equivalent NLO13(500) and NLO19(500) YN interactions for
the separation energies in the $A=4-7$ hypernuclei. Both potentials
underestimate the ground state of ${}^{4}_{\Lambda}\mathrm{H(0^{+})}$. The
NLO19 potential yields $B_{\Lambda}$ for ${}^{4}_{\Lambda}\mathrm{H}(1^{+})$
and the ground states in ${}^{5}_{\Lambda}\mathrm{He}$ and
${}^{7}_{\Lambda}\mathrm{Li}$ that are comparable with the experiments (only
slightly above), whereas NLO13 substantially underestimates these systems. We
also investigated CSB splittings in the $A=7,\,8$ multiplets using the two NLO
YN potentials that also include the leading CSB interaction in the $\Lambda$N
channel derived in Ref. [16]. Overall, the predicted CSB for the $A=7$ systems
are small and agree with experiment while the results for the $A=8$ doublet
are somewhat larger than the available empirical value deduced based on
emulsion data. Finally, we discussed our recents results for $\Xi$ hypernuclei
with $A=4-7$ based on the NLO(500) $\Xi$N potential. We found three loosely
bound states $(1^{+},0)$, $(0^{+},1)$ and $(1^{+},1)$ for the NNN$\Xi$ system.
The ${}^{5}_{\Xi}\mathrm{H}$ and ${}^{7}_{\Xi}\mathrm{H}$
$(\frac{1}{2}^{+},\frac{3}{2})$ hypernuclei are more tightly bound. The decay
widths of these $\Xi$ bound states are expected to be very small which could
allow them to be observed in experiment.

Acknowledgements: I am very grateful to Ulf-G. Meißner, Johann Haidenbauer and
Andreas Nogga for the fruitful collaboration, for proofreading and providing
valuable suggestions to improve the manuscript. This work is supported by the
DFG and NSFC through the funds provided to the Sino-German Collaborative
Research Center TRR110 “Symmetries and the Emergence of Structure in QCD”.
Numerical calculations have been performed on JURECA and JURECA booster of the
JSC, Jülich, Germany.

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