# Ab initio calculation of charge symmetry breaking in $A=7$ and 8
$\Lambda$-hypernuclei

Hoai Le<EMAIL_ADDRESS>Institut für Kernphysik, Institute for Advanced
Simulation and Jülich Center for Hadron Physics, Forschungszentrum Jülich,
D-52425 Jülich, Germany  Johann Haidenbauer<EMAIL_ADDRESS>Institut für Kernphysik, Institute for Advanced Simulation and Jülich Center
for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany  Ulf-G.
Meißner<EMAIL_ADDRESS>Helmholtz-Institut für Strahlen- und
Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, D-53115
Bonn, Germany  Institut für Kernphysik, Institute for Advanced Simulation and
Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich,
Germany  Tbilisi State University, 0186 Tbilisi, Georgia  CASA,
Forschungszentrum Jülich, D-52425 Jülich, Germany  Andreas Nogga  a.nogga@fz-
juelich.de Institut für Kernphysik, Institute for Advanced Simulation and
Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich,
Germany  CASA, Forschungszentrum Jülich, D-52425 Jülich, Germany

(October 2022)

###### Abstract

The $\Lambda$ separation energies of the isospin triplet ${{}^{7}_{\Lambda}\rm
He}$, ${{}^{7}_{\Lambda}\rm Li}^{*}$, ${{}^{7}_{\Lambda}\rm Be}$, and the
$T=1/2$ doublet ${{}^{8}_{\Lambda}\rm Li}$, ${{}^{8}_{\Lambda}\rm Be}$ are
investigated within the no-core shell model. Calculations are performed based
on a hyperon-nucleon potential derived from chiral effective field theory at
next-to-leading order. The potential includes the leading charge-symmetry
breaking (CSB) interaction in the $\Lambda$N channel, whose strength has been
fixed to the experimentally known difference of the $\Lambda$ separation
energies of the mirror hypernuclei ${{}^{4}_{\Lambda}\rm He}$ and
${{}^{4}_{\Lambda}\rm H}$. It turns out that the CSB predicted for the $A=7$
systems is small and agrees with the splittings deduced from the empirical
binding energies within the experimental uncertainty. In case of the $A=8$
doublet, the computed CSB is somewhat larger than the available experimental
value.

Hyperon-nucleon interaction, Effective field theory, Hypernuclei, Charge-
symmetry breaking

##  I Introduction

Charge symmetry breaking (CSB) in $\Lambda$ hypernuclei has been
experimentally established for many decades. The first and probably most
pronounced evidence came from the difference of the ${\Lambda}$-separation
energies of the mirror nuclei ${{}^{4}_{\Lambda}\rm He}$ and
${{}^{4}_{\Lambda}\rm H}$ [1, 2], eventually followed by data on other
$\Lambda$-hypernuclei isospin multiplets up to $A=16$ [3, 4, 5, 6], see also
[7, 8]. However, a solid theoretical understanding of the CSB effects has been
lacking for a long time. Certainly, one of the possible CSB mechanisms, namely
${\Lambda}-{\Sigma}^{0}$ mixing, had already been identified and investigated
at an early stage [1]. That mechanism facilitates pion exchange between the
${\Lambda}$ and the nucleons, otherwise forbidden by isospin conservation, and
thus yields a long-ranged CSB force. However, with ${\Lambda}-{\Sigma}^{0}$
mixing alone, commonly included in elaborate hyperon-nucleon (YN) potentials
like those of the Nijmegen group [9], no quantitative description of the
observed CSB in the ground ($0^{+}$) and excited ($1^{+}$) states of
${{}^{4}_{\Lambda}\rm He}$-${{}^{4}_{\Lambda}\rm H}$ could be achieved [10].
One could attribute that to the fact that the separation-energy difference
$\Delta B_{\Lambda}(0^{+})=B^{0^{+}}_{\Lambda}(^{4}_{\Lambda}{\rm
He})-B^{0^{+}}_{\Lambda}(^{4}_{\Lambda}{\rm H})$ of $340$ keV [3] and $\Delta
B_{\Lambda}(1^{+})=240$ keV [11] accepted at that time are exceptionally large
when compared to those found for, say, the mirror nuclei ${{}^{3}\rm H}$ and
${{}^{3}\rm He}$ of about 80 keV after the Coulomb-energy correction [12].
Indeed, they were also large when compared to the CSB effects found for
heavier $\Lambda$ hypernuclei with $A=7$ and $A=8$. In fact, cluster model
calculations for $A=7-10$ mirror hypernuclei [13, 14, 15], which implemented
phenomenological ${\Lambda}$N CSB forces that were tuned to the splittings
found for ${{}^{4}_{\Lambda}\rm He}$-${{}^{4}_{\Lambda}\rm H}$, overestimated
the CSB splittings for the heavier systems and/or predicted shifts in the
wrong direction.

In this work, we present a calculation of the binding energies for the
isotriplet ${{}^{7}_{\Lambda}\rm He}$, ${{}^{7}_{\Lambda}{\rm Li}^{*}}$,
${{}^{7}_{\Lambda}\rm Be}$ (${{}^{7}_{\Lambda}{\rm Li}^{*}}$ denotes the
excited state of ${{}^{7}_{\Lambda}{\rm Li}}$ with isospin $T=1$), as well as
of the A=8 doublet ${{}^{8}_{\Lambda}\rm Li}$, ${{}^{8}_{\Lambda}{\rm Be}}$.
The study is motivated by the significant experimental and theoretical
progress that has been made since the last extended calculation by Hiyama et
al. [13]. On the experimental side, there has been a reliable determination of
the binding energy of the ${{}^{7}_{\Lambda}\rm He}$ hypernucleus [16].
Moreover, and more importantly, there has been a re-evaluation of CSB in the
${{}^{4}_{\Lambda}\rm He}$-${{}^{4}_{\Lambda}\rm H}$ systems. Refined data
from experiments at J-PARC [17] and Mainz [18, 19] that became available in
the years 2015/16 established the splittings to be $\Delta
B_{\Lambda}(0^{+})=233\pm 92$ keV and $\Delta B_{\Lambda}(1^{+})=-83\pm 94$
keV [5, 20]. Thus, there is a sizable reduction of the CSB effect in the
$0^{+}$ state as compared to the former value. In the $1^{+}$ state there is
even a change in the sign, and the new value is practically compatible with
zero.

With regard to theory, a consistent description of the charge-symmetry
preserving and CSB components of the $\Lambda N$ interaction has been achieved
within chiral effective field theory (EFT) applying an appropriate power
counting. The resulting potentials yield an excellent description of the
available low-energy ${\Lambda}p$ and $\Sigma$N data [21, 22]. Earlier studies
usually omitted the CSB contact interactions leading to significant dependence
of the predictions of the CSB for A=4 hypernuclei on details of the
interactions [20]. This problem could be resolved by taking the CSB contact
interactions into account and fixing them using the $A=4$
$\Lambda$-hypernuclei data [23]. In addition, microscopic “ab initio”
calculations of hypernuclei up to $A=8$ and beyond are feasible now, say,
within the no-core shell model (NCSM) [24, 25, 26, 27]. As input elementary YN
interactions can be used, together with sophisticated nucleon-nucleon (NN) and
three-nucleon (3N) forces. Specifically, the important coupling between the
${\Lambda}$N and ${\Sigma}$N systems can be fully taken into account and, of
course, CSB which induces differences in the ${\Lambda}p$ and ${\Lambda}n$
interactions.

The paper is structured in the following way: in Sect. II, we give a brief
account of the employed YN interactions. Specifically, we explain how the CSB
part is determined from the separation-energy differences in the $0^{+}$ and
$1^{+}$ states of the $A=4$ hypernuclei ${{}^{4}_{\Lambda}\rm He}$ and
${{}^{4}_{\Lambda}\rm H}$. In Sect. III, we summarize the treatment of the
$A=4-8$ hypernuclei within the Jacobi no-core shell model. Our results and a
discussion of the CSB effects are presented in Sect. IV. Some further details
are relegated to the appendix. The paper ends with a brief summary.

##  II Hyperon-nucleon interaction including CSB

For the present study, we utilize the YN interactions from Refs. [21, 22],
derived within SU(3) chiral EFT at next-to-leading order (NLO). At that order
of the chiral expansion, the YN potential consists of contributions from one-
and two-pseudoscalar-meson exchange diagrams (involving the Goldstone boson
octet $\pi$, $\eta$, $K$) and from four-baryon contact terms without and with
two derivatives. The two YN interactions are the result of pursuing different
strategies for fixing the low-energy constants (LECs) that determine the
strength of the contact interactions. In the YN interaction from 2013 [21],
denoted by NLO13 in the following, all LECs have been fixed exclusively by a
fit to the available $\Lambda$N and $\Sigma$N data. The other potential [22]
(NLO19) has been guided by the objective to reduce the number of LECs that
need to be fixed from the YN data by inferring some of them from the NN sector
via the underlying (though broken) SU(3) flavor symmetry. A thorough
comparison of the two versions for a range of cutoffs can be found in Ref.
[22], where one can see that the two YN interactions yield essentially
equivalent results in the two-body sector.

The YN potentials NLO13 and NLO19 do not include any explicit CSB
contributions. However, in Ref. [23], we derived the leading CSB interaction
within chiral EFT and added it to those YN interactions. At the order
considered, CSB contributions arise from a non-zero ${\Lambda}{\Lambda}\pi$
coupling constant which is estimated from $\Lambda-\Sigma^{0}$ and
$\pi^{0}-\eta$ mixing, the mass difference between $K^{\pm}$ and $K^{0}$, and
from two contact terms that represent short-ranged CSB forces. In the actual
calculation, the two arising CSB low-energy constants (LECs) were fixed by
considering the known differences in the energy levels of the $0^{+}$ and
$1^{+}$ states of the aforementioned $A=4$ hypernuclei. Then, by construction,
the resulting interaction describes all low-energy $\Lambda p$ and
$\Sigma\mathrm{N}$ scattering data, the hypertriton and the CSB in
${{}^{4}_{\Lambda}\rm He}$ and ${{}^{4}_{\Lambda}\rm H}$ accurately.

For a detailed discussion of the CSB effects we refer the reader to [23]. As
main outcome, it turned out that the reproduction of the splittings of $\Delta
B_{\Lambda}(0^{+})=233\pm 92$ keV and $\Delta B_{\Lambda}(1^{+})=-83\pm 94$
keV [5] (scenario CSB1 in [23]) requires a sizable difference between the
strengths of the ${\Lambda}p$ and ${\Lambda}n$ interactions in the
${}^{1}S_{0}$ state, whereas the modifications in the ${}^{3}S_{1}$ partial
wave are much smaller. The effects go also in opposite directions, i.e. while
for ${}^{1}S_{0}$ the ${\Lambda}p$ interaction is found to be noticeably less
attractive than that for ${\Lambda}n$, in case of ${}^{3}S_{1}$ it is slightly
more attractive. In terms of the difference in the scattering lengths, $\Delta
a^{CSB}=a_{{\Lambda}p}-a_{{\Lambda}n}$, a value of $0.62\pm 0.08$ fm has been
predicted for the ${}^{1}S_{0}$ partial wave and $-0.10\pm 0.02$ fm for the
${}^{3}S_{1}$ [23].

Recently, the STAR collaboration has reported a new measurement for the $A=4$
systems, which suggests somewhat different CSB splittings of the $0^{+}$ and
$1^{+}$ states [28]. Their results are $\Delta B_{\Lambda}(0^{+})=160\pm
140({\rm stat})\pm 100({\rm syst})$ keV and $\Delta B_{\Lambda}(1^{+})=-160\pm
140({\rm stat})\pm 100({\rm syst})$ keV. Of course, considering the sizable
statistical and systematic uncertainties, those values are compatible with the
ones cited above, so that quantitative conclusions cannot be drawn at present.
Nevertheless, it is interesting to explore the implication of such a possible
modification of the CSB in the $A=4$ $\Lambda$ hypernuclei for that in the
$A=7$ and $8$ systems, though, in view of the uncertainties, we refrain from
doing more elaborate calculations at present. Thus, we only re-adjust the two
CSB LECs for the NLO19 potential in order to reproduce the central values of
the STAR results. We find that the difference in the $\Lambda p$ and $\Lambda
n$ scattering lengths is somewhat reduced in the ${}^{3}S_{1}$ partial wave,
$\Delta a^{\mathrm{CSB}}(^{3}S_{1})=-0.05$ fm, whereas it slightly increases
in the ${}^{1}S_{0}$ state, $\Delta a^{\mathrm{CSB}}(^{1}S_{0})=0.71$ fm. This
new set of CSB LECs will be referred to as CSB∗ and will be employed to
explore the impact on the splittings in the $A=7$ and $8$ isospin multiplets.
For a recent and detailed overview on the experimental situation regarding the
CSB splittings in the ${{}^{4}_{\Lambda}\rm He}$ and ${{}^{4}_{\Lambda}\rm H}$
systems see Ref. [6].

As shown in previous bound-state calculations, the ${\Lambda}$ separation
energies of light hypernuclei are not very sensitive to the employed NN
interaction [10, 22]. Therefore, we use in all of the calculations presented
here the same state-of-the-art chiral NN interaction, namely the semi-local
momentum-space-regularized (SMS) NN potential of Ref. [29] at order N4LO+ with
cutoff ${\Lambda}_{\mathrm{N}}=450$ MeV. Indeed, the variation of the
separation energy for N4LO+ potentials with other cutoffs is of the order of
$100$ keV for ${}^{4}_{\Lambda}\mathrm{He}/^{4}_{\Lambda}\mathrm{H}$ [22] and
within the range expected from calculations based on phenomenological
interactions [10]. A recent dedicated study performed within the NCSM, using
however only N2LO NN potentials and a LO $YN$ interaction, reported
uncertainties of around $100$ keV for $A=4$ and of around $400$ keV for A=5
hypernuclei [30]. Earlier, Wirth and Roth [31] found uncertainties of $\approx
200$ keV and $\approx 400$ keV for ${}^{7}_{\Lambda}$Li and
${}^{9}_{\Lambda}$Be, respectively, utilizing N3LO and N4LO NN potentials but
also only LO for the $YN$ interaction. In preliminary calculations, we
observed that the sensitivity of the $\Lambda$ separation energies to the
employed NN interactions depends also on the YN interaction itself. E.g., for
${}^{3}_{\Lambda}$H, we found variations of the separation energy of 18 keV
when using the recent SMS N2LO interactions and 13 keV when using the SMS
N4LO+ interactions in conjunction with NLO19(650). With the same NN
interactions the variation is 60 keV and 23 keV for the LO(650) YN
interaction. The surprisingly large dependence of the variation of separation
energies on the order of the chiral NN interaction and on the order of the YN
interaction might in part explain why our estimate of the dependence of the
separation energies on the NN interaction is smaller than other values
available in the literature. A more detailed study on this issue is in
progress but beyond the scope of this work. We also note that Ref. [30] found
that the NN force dependence of the CSB in $A=4$ hypernuclei is anyhow smaller
due to correlations.

In order to accurately describe the parent nuclei, the chiral 3N interaction
at order N2LO with the regulator of ${\Lambda}_{\mathrm{N}}=450$ [32] is also
included. Note that such a combination of the NN and 3N forces gives a fairly
good description for the binding energies of light and medium mass nuclei
[32]. It should, however, be stressed that although 3N forces contribute
moderately to the nuclear and hypernuclear binding energies, their overall
effect on the $\Lambda$ separation energies and, in particular, on the CSB
splittings is expected to be rather small for light and ground states of
$p$-shell hypernuclei [33, 22, 23]. The inclusion of 3N forces can improve the
description of excited states that are linked to an excited core nucleus.

##  III Jacobi no-core shell model

We apply the Jacobi NCSM for calculating the $\Lambda$-separation (binding)
energies of the $A=4-8$ hypernuclei. A detailed description of the formalism
and of the procedure to extract the binding (separation) energies can be found
in Ref. [27]. In that reference, and in [34], one can also find results for
${}^{7}_{\Lambda}$Li based on the YN interactions NLO13 and NLO19 without CSB
contribution. As already mentioned, for the current study, we shall employ
chiral NN, 3N and YN potentials to describe the interactions among the
nucleons and between a nucleon and a hyperon, respectively. In all
calculations, contributions of the NN(YN) potentials in partial waves up to
$J=6(5)$ are included, while for the 3N interaction all partial waves with
total angular momentum $J_{\mathrm{3N}}\leq 9/2$ are taken into account. It
has been checked that higher partial waves only contribute negligibly compared
to the HO model space uncertainties. In order to speed up the convergence of
the NCSM with respect to the model space, all the employed NN, 3N and YN
potentials are SRG-evolved to a flow parameter of $\lambda=1.88$ fm-1, see
[27] and references therein. The latter is commonly used in nuclear
calculations, which, on the one hand yields rather well-converged nuclear
binding energies, and on the other hand, minimizes the possible contribution
of SRG-induced 4N and higher-body forces [32]. Furthermore, in most of the
calculations, the SRG-induced YNN interaction with the total angular momentum
$J_{\mathrm{YNN}}\leq 5/2$ is also explicitly included. Based on the
contributions of $J_{\mathrm{YNN}}\leq 1/2$, $3/2$ and $5/2$, the contribution
from higher partial waves $J_{\mathrm{YNN}}\geq 7/2$ is estimated to be
negligibly small and therefore is omitted from the calculations. With the
proper inclusion of these SRG-induced three-body forces, the otherwise strong
dependence of the $\Lambda$ separation energies $B_{\Lambda}$ on the SRG-flow
parameter [24, 27] is largely removed [31, 35].

It should be further noted that, for large systems like $A=7$ and $8$, the
extrapolated NCSM separation energies are afflicted with appreciable
uncertainties, see [27] and also Table 3, which even exceed the experimentally
found CSB splittings in these systems. Therefore, it is not advisable to
estimate CSB based on the extrapolated separation energies. Instead, one can
compute the CSB effects directly from the corresponding nuclear and
hypernuclear energy expectation values for each model space
$\mathcal{N}_{max}$ and HO frequency $\omega$. It has been observed that
because of the correlations between those binding energies the directly
extracted $\Delta B_{\Lambda}(\omega,\mathcal{N}_{max})$ converges
significantly faster with respect to $\mathcal{N}_{max}$ and $\omega$ than the
individual binding energies and to some extent the separation energies, so
that a direct comparison with experiment is possible. Accordingly, the
separation energies difference, say, for $A=4$ systems, can be computed as

$\displaystyle\Delta B_{\Lambda}$ $\displaystyle=$ $\displaystyle
B_{\Lambda}(^{4}_{\Lambda}{\rm He})-B_{\Lambda}(^{4}_{\Lambda}{\rm H})$ (1)
$\displaystyle=$ $\displaystyle E(^{3}{\rm He})-E(^{3}{\rm
H})-\big{(}\,E(^{4}_{\Lambda}{\rm He})-E(^{4}_{\Lambda}{\rm H})\,\big{)}.$ (2)

Let us further separate contributions from the kinetic energy, and from the NN
and YN interactions to the total binding energies. This decomposition is
justified by the observation that the contributions due to three-body forces
are negligibly small. Hence, the CSB splitting in Eq. (1) can finally be
expressed as follows

$\displaystyle\Delta B_{\Lambda}$ $\displaystyle=$ $\displaystyle T(^{3}{\rm
He})-T(^{3}{\rm H})-\big{(}T(^{4}_{\Lambda}{\rm He})-T(^{4}_{\Lambda}{\rm
H})\big{)}+V_{\mathrm{NN}}(^{3}{\rm He})-V_{\mathrm{NN}}(^{3}{\rm H})$ (3)
$\displaystyle\qquad-$ $\displaystyle\big{(}V_{\mathrm{NN}}(^{4}_{\Lambda}{\rm
He})-V_{\mathrm{NN}}(^{4}_{\Lambda}{\rm
H})\big{)}-\big{(}V_{\mathrm{YN}}(^{4}_{\Lambda}{\rm
He})-V_{\mathrm{YN}}(^{4}_{\Lambda}{\rm H})\big{)}$ $\displaystyle=$
$\displaystyle\Delta T+\Delta V_{\mathrm{NN}}+\Delta V_{\mathrm{YN}}\ .$

Note that the operators $V_{NN}$ and $V_{YN}$ employed in Eq. (3) are also
SRG-evolved, like the full Hamiltonian. Furthermore, we will follow the
approach in [23, NoggaPhD:2001] to estimate the individual contributions
$\Delta T,\,\Delta V_{NN}$ and $\Delta V_{YN}$ perturbatively based on the two
(hyper)nuclear wave functions of ${}^{4}_{\Lambda}{\rm He}$ and ${}^{3}{\rm
He}$ (or ${}^{7}_{\Lambda}{\rm Li^{*}}$ and ${}^{6}{\rm Li}$, and
${}^{8}_{\Lambda}{\rm Li}$ and ${}^{7}{\rm Li}$ in cases of $A=7$ and 8
systems, respectively). The former is computed for the YN interactions that
also include the CSB components. Using the wave functions that include CSB
effects is strictly speaking a deviation from first order perturbation theory.
However, the deviation is of second order and therefore not relevant here. As
it has been shown in [23, NoggaPhD:2001] and will be discussed in the
following section, such a perturbative estimate of $\Delta B_{\Lambda}$ is a
good approximation to the exact calculations. The wave functions for $A=4,7$
and 8 hypernuclear ($A=3,6$ and 7 for nuclear) systems are generated using the
largest computationally accessible model spaces, namely
$\mathcal{N}_{max}=24,10$ and 9, respectively, and at the optimal
$\omega_{opt}=16$ MeV that is (or very close to) the variational minimum.

For estimating numerical uncertainties due to the model space truncation, we
have performed the same calculations for two-body interactions at the same
$\mathcal{N}_{max}$ and for $\mathcal{N}_{max}+2$ and with the same
$\omega_{opt}$ and $\omega_{opt}\pm 2$ MeV. The variation of these
calculations gave our uncertainty estimate of 10, 30 and 50 keV for the $A=4$,
$7$, and $8$ isospin multiplets, respectively. Note the larger uncertainty for
the $A=8$ doublet because of the smaller accessible model space.

As already said, we will employ the high-order SMS NN interaction with
$\Lambda_{\mathrm{N}}=450$ MeV (SMS N4LO+(450)) [29] and the N2LO 3N force
with the same chiral cutoff [32]. Two chiral potentials at next-to-leading
order, namely NLO13 and NLO19 [21, 22] with a regulator of $\Lambda_{Y}=500$
MeV, are chosen for the YN interaction. We know by experience that the SRG
evolution for such low cutoff values converges very quickly thanks to the
overall small YN potential matrix elements. For larger cutoffs and especially
for the NLO13 interaction, which contains sizable off-diagonal potential
matrix elements, the ordinary differential equations (ODE) solver used for the
SRG evolution demands an extremely small time step for achieving an accurate
solution which requires prohibitively large computing resources. The predicted
difference in the $\Lambda$p and $\Lambda$n scattering lengths, i.e. $\Delta
a^{CSB}$, has been found to be basically the same for all cutoffs and for the
two realizations of the YN interaction [23]. Obviously, the regulator
dependence is efficiently absorbed by the contact terms of the CSB component
of the YN potentials, when fixing the pertinent LECs from the $A=4$ CSB level
splittings. Therefore, we expect that the CSB splittings for $A=7$ and $8$
hypernuclei based on those interactions exhibit likewise a fairly weak or even
a negligible cutoff dependence. Finally, chiral $Y$NN forces are not
considered in the current study, only those from the SRG evolution. In the
following “3N forces” stands for (the inclusion of) chiral as well as SRG-
induced 3N forces, unless explicitly stated otherwise.

##  IV Results

| ${}^{4}_{\Lambda}\mathrm{H}(0^{+})$ | ${}^{4}_{\Lambda}\mathrm{H}(1^{+})$ | ${}^{5}_{\Lambda}\mathrm{He}$
---|---|---|---
NLO13-CSB | full | $1.551\pm 0.007$ | $0.823\pm 0.003$ | $2.22\pm 0.06$
$\lambda=0.765$ | $1.29\pm 0.005$ | $0.779\pm 0.02$ | $2.22\pm 0.04$
FY | $1.513$ | $0.813$ |
NLO19-CSB | full | $1.514\pm 0.007$ | $1.27\pm 0.009$ | $3.32\pm 0.03$
$\lambda=0.823$ | $1.41\pm 0.003$ | $1.131\pm 0.01$ | $3.35\pm 0.02$
FY | $1.511$ | $1.268$ |
experiment |  $2.16\pm 0.08$ [19] |  $1.07\pm 0.08$ [19] |  $3.12\pm 0.02$ [3]
Table 1:  $\Lambda$-separation energies for the
${}^{4}_{\Lambda}\mathrm{H}(0^{+},1^{+})$ states and for
${}^{5}_{\Lambda}\mathrm{He}$, computed for the YN potentials NLO13(500) and
NLO19(500) including the CSB interaction. Listed are our full results, with
inclusion of the corresponding SRG-induced YNN forces, and those with the YN
potentials SRG-evolved to the magic flow parameter, $\lambda_{magic}=0.765$
and $0.823$ fm-1, respectively. The $B_{\Lambda}$ values are obtained by
performing the two-step $\omega$\- and $\mathcal{N}$-space extrapolation, see
[27] for more details. The FY calculations are performed with the bare NN, 3N
and YN potentials. Energies are given in MeV.

###  IV.1 Charge symmetry breaking in the $A=4$ systems

The hypernuclei ${}^{4}_{\Lambda}$H and ${}^{4}_{\Lambda}$He constitute an
important test case for our calculations, because here we can directly compare
the NCSM predictions with results obtained from solutions of the Faddeev-
Yakubovsky (FY) equations [23]. Table 1 shows the comparison of the separation
energies obtained within the two methods. The NLO13 and NLO19 potentials with
chiral cutoff of $500$ MeV have been employed to describe the YN interaction,
while the standard combination of the SMS $\mathrm{N^{4}LO}^{+}(450)$ NN and
$\mathrm{N^{2}LO}(450)$ 3N interactions is used [32] for the nucleons. For the
NCSM calculations, the employed NN, 3N and YN potentials are SRG-evolved to a
flow parameter of $\lambda=1.88\,\mathrm{fm}^{-1}$. Furthermore, the SRG-
induced YNN interaction is taken into account so that the separation energies
are practically independent of the SRG-flow parameter [35]. For the FY
calculations, the bare NN, 3N and YN interactions have been employed. The
small discrepancy between the FY results listed in Table 1 and those provided
in [23] is essentially due to the contribution of the 3N force, neglected in
the latter work, which clearly amounts to less than $50$ keV. It is reassuring
to observe that for NLO19 the actual separation energies computed within the
NCSM approach agree perfectly with the results of the FY equations, for the
ground state as well as for the excited state. The extremely small difference
could be an indication that the contribution of SRG-induced YNNN forces to the
separation energies in the $A=4$ systems are negligibly small. However, for a
more quantitative estimate, well-converged calculations using a wide range of
values for the SRG flow parameter are still necessary. For NLO13, the
difference of the FY result and the full calculations is more visible and of
the order of $40$ keV indicating larger contributions of the missing SRG-
induced YNNN forces in this case which are probably related to the larger
$\Lambda$-$\Sigma$ transition matrix elements [22]. We stress that the
agreement of the FY and full calculations are still excellent.

Additionally, the table contains our NCSM results for the $\Lambda$ separation
energies of ${}^{5}_{\Lambda}$He, which are
$B_{\Lambda}(^{5}_{\Lambda}\mathrm{He})=2.22\pm 0.06$ MeV and $3.32\pm 0.03$
MeV [35], respectively. Evidently, NLO13 significantly underestimates the
${}^{5}_{\Lambda}\mathrm{He}$ separation energy, while the result for the
NLO19 potential is rather close to and only slightly above the experimental
value of $B_{\Lambda}(^{5}_{\Lambda}\mathrm{He})=3.12$ MeV. The discrepancy
between the two NLO13 and NLO19 predictions signals the need for including
proper chiral ${\Lambda}\mathrm{NN}$ and ${\Sigma}\mathrm{NN}$ three-body
forces [36], given that the ${\Lambda}$p and ${\Sigma}$N results of those
potentials are practically identical. Indeed, three-body forces, with a
distinct spin-isospin dependence might also be needed to bring the $A=4$
results in a better agreement with the experiment.

Finally, we include in Table 1 results of NCSM calculations where only the NN
potential, SRG-evolved to $\lambda_{\mathrm{NN}}=1.6$ fm-1, and the two YN
potentials, SRG-evolved to the “magic” flow parameters,
$\lambda_{magic}(\mathrm{NLO19})=0.823$ fm-1 and
$\lambda_{magic}(\mathrm{NLO13})=0.765$ fm-1, are employed. The values of
$\lambda_{magic}$ for NLO19 and NLO13 are chosen in such a way that the
pertinent full NCSM results for the ${}^{5}_{\Lambda}$He separation energy are
reproduced. Let us remark that our way of fixing $\lambda_{magic}$ here
slightly differs from the strategy in [27] where the experimental value of
${}^{5}_{\Lambda}\mathrm{He}$ has been used as benchmark. Note that at the SRG
parameter of $\lambda_{\mathrm{NN}}=1.6$ fm-1, the parent nuclear cores can be
fairly well described even when 3N forces are omitted [27]. With
$\lambda_{magic}$ fixed to the actual $B_{\Lambda}(^{5}_{\Lambda}\mathrm{He})$
for NLO13 and NLO19, we observe a fair to good agreement between the $A=4$
separation energies from the full NCSM calculations and those computed at
$\lambda_{magic}$, as can be seen in Table 1. The small discrepancy between
the two results, up to around $200$ keV for the $0^{+}$ state and in the order
of $100$ keV for $1^{+}$, can be attributed again to possible contributions
from YNN forces [27, 36].

| $\mathrm{YN}$ | $\Delta T$ | $\Delta V_{\mathrm{NN}}$ | $\Delta V_{\mathrm{YN}}$ | $\Delta B_{\Lambda}$ | $\Delta B_{\Lambda}(FY)$
---|---|---|---|---|---|---
|  |  |  | ${}^{1}S_{0}$ | ${}^{3}S_{1}$ | total |  |
$(0^{+})$ | NLO13 | 17 | -12 | -3 | 0 | -3 | 3 | 43
NLO13-CSB | 18 | -13 | 152 | 76 | 224 | 229 | 252
NLO19 | 9 | -15 | -1 | 0 | -1 | -7 | 10
NLO19-CSB | 9 | -16 | 126 | 118 | 245 | 238 | 238
$(1^{+})$ | NLO13 | 6 | -5 | 0 | 0 | -1 | 0 | -9
NLO13-CSB | 6 | -5 | -114 | 19 | -95 | -94 | -75
NLO19 | 5 | -15 | 0 | 0 | -1 | -11 | 5
NLO19-CSB | 5 | -15 | -114 | 36 | -76 | -85 | -85
Table 2:  Contributions to CSB for ${}^{4}_{\Lambda}$He and
${}^{4}_{\Lambda}$H in the $0^{+}$ and $1^{+}$ states, based on the YN
potentials NLO13 and NLO19 (including 3N forces and SRG-induced YNN forces)
with cutoff of $\Lambda=500$ MeV. The results are for the original potentials
(without CSB force) and for the scenario CSB1 of Ref. [23]. FY indicates the
exact CSB results extracted from Faddeev-Yakubovsky calculations which employ
the bare NN, 3N and YN interactions. All results are in keV. The estimated
uncertainty from the NCSM and FY calculations are $10$ and $20$ keV,
respectively. The experimental reference values are $\Delta
B_{\Lambda}(0^{+})=233\pm 92$ and $\Delta B_{\Lambda}(1^{+})=-83\pm 94$ keV.

In Table 2, we analyse the CSB in the $A=4$ isodoublet ${}^{4}_{\Lambda}$He
and ${}^{4}_{\Lambda}$H in detail. The results are based on NLO13(500) and
NLO19(500) as published originally [21, 22] and including a CSB interaction
[23] that was adjusted to the experimental splittings $\Delta
B_{\Lambda}(0^{+})=233\pm 92$ keV and $\Delta B_{\Lambda}(1^{+})=-83\pm 94$
keV (CSB1 of Ref. [23]). Similarly to [23], we break down the different
contributions to the total CSB splitting $\Delta B_{\Lambda}$, due to the
kinetic energy $\Delta T$, the NN interaction ($\Delta V_{\mathrm{NN}}$) and
the YN interaction ($\Delta V_{\mathrm{YN}}$), see Eq. (3). The perturbatively
estimated contributions of the 3N and YNN forces are negligibly small and,
therefore, omitted in the table. The CSB contribution $\Delta T$ is also small
when using chiral interactions, but contributes with positive sign to the
total CSB. The contribution of the nuclear core $\Delta V_{\mathrm{NN}}$,
mostly due to the point Coulomb interaction between the protons, is of similar
magnitude as $\Delta T$ but comes with a negative sign. As expected, $\Delta
V_{\mathrm{YN}}$ for the original YN potentials is insignificant. However,
when the CSB interaction [23] is included, $\Delta V_{\mathrm{YN}}$ becomes
sizable and, by construction, the total CSB results for the $0^{+}$ state as
well as for $1^{+}$ are in line with the aforementioned empirical information.

Also for the CSB splittings, we can compare our NCSM results with those
obtained by solving the FY equations, cf. the last column in Table 2. Again,
there is good agreement between the two calculations within the estimated
uncertainties. Note that the FY values are from an exact solution of the
equations. The comparison of perturbative and exact CSB results in Tables 6
and 7 of Ref. [23] reveals that there is very little difference. In addition,
for $A=7$ systems, we have also explicitly studied and observed a discrepancy
of only less than ten keV between the perturbativly estimated CSB and the CSB
results that are computed based on the expectation values of the $T$, $V_{NN}$
and $V_{YN}$ operators estimated with respect to the corresponding
hyper(nuclear) wavefunctions ${}^{7}_{\Lambda}\mathrm{Be}(^{6}\mathrm{Be})$
and ${}^{7}_{\Lambda}\mathrm{Li^{*}}(^{6}\mathrm{Li})$. Let us again stress
that due to the large uncertainties of the extrapolated separation energies
for the $A\geq 7$ systems, see Table 3, a direct extraction of CSB splittings
based on those separation energies is not useful. One could also calculate the
CSB differences for each model space separately and study the model space and
$\omega$ dependence more carefully. For $A=4$ and $A=7$ hypernuclei, our
results for this approach were also consistent with the perturbative estimate,
but there was still a visible dependence on the model space size and HO
frequencies which made the extraction of an uncertainty rather difficult. We
therefore favor the perturbative approach which is robust and computationally
less demanding and use it for obtaining the CSB effects in the $A=7$ and $8$
$\Lambda$ hypernuclei below.

Let us now have a closer look at the different contributions of the
${}^{1}S_{0}$ and ${}^{3}S_{1}$ partial waves, $\Delta
V_{\mathrm{YN}(^{1}S_{0})}$ and $\Delta V_{\mathrm{YN}(^{3}S_{1})}$, to the
total $\Delta V_{\mathrm{YN}}$. From the fourth column in Table 2, it follows
that $\Delta V_{\mathrm{YN}(^{1}S_{0})}$ and $\Delta
V_{\mathrm{YN}(^{3}S_{1})}$ are sizable and of the same sign in the $0^{+}$
state, resulting in a large $\Delta V_{\mathrm{YN}(0^{+})}$. In the excited
state, the two contributions are, however, smaller and of opposite sign so
that there is some cancellation. The signs of the two contributions $\Delta
V_{\mathrm{YN}(^{1}S_{0})}$ and $\Delta V_{\mathrm{YN}(^{3}S_{1})}$ are
directly related to the different strengths of the $\Lambda n$ and $\Lambda p$
interactions in the singlet and triplet states, as manifested by the
respective scattering lengths, and to the relative weights of the $\Lambda n$
and $\Lambda p$ components in those spin states. More details are given in the
appendix.

| NLO19 | NLO13 | experiment
---|---|---|---
| full | $\lambda=0.823$ | full | $\lambda=0.765$ |  |
${}^{7}_{\Lambda}$Be | $5.54\pm 0.22$ | $5.44\pm 0.03$ | $4.30\pm 0.47$ | $4.53\pm 0.34$ | $5.16\pm 0.08$ |
${}^{7}_{\Lambda}$Li∗ | $5.64\pm 0.28$ | $5.49\pm 0.04$ | $4.42\pm 0.58$ | $4.59\pm 0.34$ | $5.26\pm 0.03$ | $5.53\pm 0.13$
${}^{7}_{\Lambda}$He | $5.64\pm 0.27$ | $5.43\pm 0.06$ | $4.39\pm 0.54$ | $4.45\pm 0.35$ |  | $5.55\pm 0.1\phantom{0}$
${}^{8}_{\Lambda}$Be |  | $7.15\pm 0.10$ |  | $5.56\pm 0.25$ | $6.84\pm 0.05$ |
${}^{8}_{\Lambda}$Li | $7.33\pm 1.15$ | $7.17\pm 0.10$ | $5.75\pm 1.08$ | $5.57\pm 0.30$ | $6.80\pm 0.03$ |
Table 3:  $\Lambda$ separation energies for the $A=7$ and $8$ systems,
computed for NLO13(500) and NLO19(500) including the SRG-induced YNN forces
(full), and at the magic flow parameters (third and fifth columns). Note that
the separation energies of $A=7(8)$ for NLO19 at $\lambda=0.823$ fm-1 have
been computed with model spaces up to $\mathcal{N}_{max}=12(11)$, whereas the
other calculations are performed with $\mathcal{N}_{max}=10(9)$. The listed
$B_{\Lambda}$ values are obtained by performing the two-step $\omega$\- and
$\mathcal{N}$-space extrapolation, see [27] for more details. Values from
emulsion (left) and counter (right) experiments are taken from the compilation
in Ref. [4]. Energies are given in MeV.

###  IV.2 Charge symmetry breaking in the $A=7$ and 8 systems

We now employ the NLO13(500) and NLO19(500) potentials to study CSB in the
$A=7$ isotriplet and the $A=8$ isodoublet. Predictions for the separation
energies of the $(1/2^{+},1)$ mirror hypernuclei ${}^{7}_{\Lambda}$He,
${}^{7}_{\Lambda}$Li∗, ${}^{7}_{\Lambda}$Be without CSB terms are provided in
Table 3. The values listed in the second and fourth columns have been obtained
with inclusion of both the chiral and SRG-induced 3N forces as well as of the
SRG-induced YNN interactions, whereas $B_{\Lambda}$ displayed in the third and
fifth columns is computed at the corresponding $\lambda_{magic}$. Obviously,
there is a fairly good agreement between the separation energies extracted
from the full calculations and the one at $\lambda_{magic}$. This confirms our
observation in [34, 27] that the magic SRG-flow parameters can be utilized to
simplify the calculation of light hypernuclear systems. The NLO13 interaction
predicts separation energies of $B_{\Lambda}=4.30\pm 0.47$, $4.42\pm 0.58$ and
$4.39\pm 0.54$ MeV for ${}^{7}_{\Lambda}\mathrm{Be}$,
${}^{7}_{\Lambda}\mathrm{Li^{*}}$, and ${}^{7}_{\Lambda}\mathrm{He}$,
respectively, and, thus, underestimates the empirical values by about $1$ MeV.
On the other hand, the results based on NLO19 are rather close to experiment.
In particular, the obtained separation energies for the $T_{3}=0$ and
$T_{3}=-1$ members $B_{\Lambda}(^{7}_{\Lambda}\mathrm{Li^{*}})=5.64\pm 0.28$
MeV and $B_{\Lambda}(^{7}_{\Lambda}\mathrm{He})=5.64\pm 0.27$ MeV are
perfectly in line with the values of
$B_{\Lambda}(^{7}_{\Lambda}\mathrm{Li^{*}})=5.53\pm 0.13$ and
$B_{\Lambda}(^{7}_{\Lambda}\mathrm{He})=5.55\pm 0.13$ MeV, extracted from
counter experiments with an absolute energy calibration [4]. For the
${}^{7}_{\Lambda}\mathrm{Be}$ hypernucleus, we obtain a separation energy of
$B_{\Lambda}(^{7}_{\Lambda}\mathrm{Be})=5.54\pm 0.22$ MeV, which exceeds the
emulsion result of $B_{\Lambda}(^{7}_{\Lambda}\mathrm{Be})=5.16\pm 0.08$
MeV[4]. However, considering the unresolved difference of $270\pm 170$ keV
between the $B_{\Lambda}(^{7}_{\Lambda}\mathrm{Li^{*}})$ determinations in
counter and emulsion experiments, cf. Table 3, the actual discrepancy for
$B_{\Lambda}(^{7}_{\Lambda}\mathrm{Be})$ could be much smaller. Hopefully,
future counter experiments will settle this issue.

The separation energies for the $A=8$ systems are likewise summarized in Table
3. The results for ${}^{8}_{\Lambda}\mathrm{Li}$ with both 3N forces and SRG-
induced YNN interactions included are obtained from the full calculations with
model space up to $\mathcal{N}_{max}=9$. Extending the calculation for model
spaces up to $\mathcal{N}_{max}=11$ will definitely help to reduce the
estimated errors. Unfortunately, such a calculation is very CPU-time consuming
and we need to postpone it to a future study. Nevertheless, in spite of the
large uncertainty, it clearly follows from Table 3 that the separation energy
for ${}^{8}_{\Lambda}\mathrm{Li}$ for the NLO13 potential is substantially too
low whereas the prediction for NLO19,
$B_{\Lambda}(^{8}_{\Lambda}\mathrm{Li})=7.33\pm 1.15$ MeV, exceeds the
empirical value of $B_{\Lambda}=6.80\pm 0.03$ MeV [4] only moderately. Again,
the difference in the predictions of NLO13 and NLO19 can be attributed to
possible contributions of chiral YNN forces [22, 27, 36]. Although full
calculations for ${}^{8}_{\Lambda}\mathrm{Be}$ have not been performed yet, a
result for $B_{\Lambda}(^{8}_{\Lambda}\mathrm{Be})$ very similar to that for
the ${}^{8}_{\Lambda}\mathrm{Li}$ hypernucleus can be expected. $B_{\Lambda}$
values for ${}^{8}_{\Lambda}\mathrm{Be}$ and ${}^{8}_{\Lambda}\mathrm{Li}$
computed at the magic SRG-flow parameters are given in the third and fifth
columns of Table 3. Evidently, the obtained separation energies, e.g.
$B_{\Lambda}(^{8}_{\Lambda}\mathrm{Li},\lambda_{magic})=7.17\pm 0.10$ MeV, is
close to the result of $B_{\Lambda}(^{8}_{\Lambda}\mathrm{Li})=7.33\pm 1.15$
MeV of the full calculations. This is not too surprising in view of what we
had already observed in the pertinent comparison for the $A=4$ and $7$
systems. Note that $B_{\Lambda}(^{8}_{\Lambda}\mathrm{Li})$ based on
$\lambda_{\rm magic}$ exceeds the value from the emulsion experiment only by
$0.37\pm 0.13$ MeV. Anyway, in view of the rather good agreement of our
predictions for the $A=7$ systems with the separation energies from counter
experiments, corresponding measurements for $A=8$ hypernuclei, that could
either confirm or revise the emulsion results, are desirable.

| $\Delta T$ | $\Delta V_{\mathrm{NN}}$ | $\Delta V_{\mathrm{YN}}$ | $\Delta B_{\Lambda}$
---|---|---|---|---
|  |  | ${}^{1}S_{0}$ | ${}^{3}S_{1}$ | 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 [13] |  | -70 |  |  | 200 | 150
| Gal [37] | 3 | -70 |  |  | 50 | -17
| experiment [6] |  |  |  |  |  | $-100\pm 90$
${}^{7}_{\Lambda}$Li∗-${}^{7}_{\Lambda}$He | NLO13 | 8 | -13 | 0 | 0 | 0 | -5
NLO13-CSB | 7 | -14 | -49 | 26 | -24 | -31
NLO19 | 5 | -22 | -43 | 42 | 0 | -17
NLO19-CSB | 5 | -21 | -38 | 37 | -1 | -16
| Hiyama [13] |  | -80 |  |  | 200 | 130
| Gal [38] | 2 | -80 |  |  | 50 | -28
| experiment [6] |  |  |  |  |  |  $-20\pm 230$111The difference between BΛ(${}^{7}_{\Lambda}$Li∗) and BΛ(${}^{7}_{\Lambda}$He) is $-20\pm 230$ keV for the FINUDA and JLab results, but $-50\pm 190$ keV when the revised SKS and JLab results are used [6].
|  |  |  |  |  |  |  $-50\pm 190$
${}^{8}_{\Lambda}$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 [13] |  | 40 |  |  |  | 160
| Gal [37] | 11 | -81 |  |  | 119 | 49
| experiment [4] |  |  |  |  |  | $40\pm 60$
Table 4:  Contributions to CSB in the $A=7$ and $8$ isospin multiplets, based
on the YN potentials NLO13(500) and NLO19(500) (including 3N forces and SRG-
induced YNN interactions). The results are for the original potentials
(without CSB force) and for the scenario CSB1, see text. Results by Gal [37]
and by Hiyama et al. [13] are included for the ease of comparison. All
energies are in keV. The estimated uncertainties for $A=7$ and $8$ systems are
30 and 50 keV, respectively.

Table 4 provides a detailed view on the CSB splittings for the three members
of the $A=7$ isotriplet, by comparing
${}^{7}_{\Lambda}\mathrm{Be}$-${}^{7}_{\Lambda}\mathrm{Li}^{*}$ and
${}^{7}_{\Lambda}\mathrm{Li}^{*}$-${}^{7}_{\Lambda}\mathrm{He}$, computed for
NLO13 and NLO19 without and with CSB interaction. The 3N forces and the SRG-
induced YNN interactions are explicitly taken into account. One sees that,
despite the substantial discrepancy in the predicted $\Lambda$ separation
energies, the two potentials yield comparable CSB results in the $A=7$
systems. The overall CSB effect is rather small, with as well as without the
CSB part of the potentials, and consistent with the experiment, both in
magnitude and sign. It is also interesting to note that, like in the $1^{+}$
state of the $A=4$ systems, the ${}^{1}S_{0}$ and ${}^{3}S_{1}$ states
contribute with opposite signs to the total $\Delta V_{\mathrm{YN}}$, which,
in turn, leads to a small total CSB for the $A=7$ isotriplet.

We include also results of former studies for the ease of comparison. Those of
Gal [37, 38] were computed by employing a shell-model approach in combination
with an effective ${\Lambda}{\Sigma}$ coupling model. The $A=7$ calculation by
Hiyama et al. [13] is done within a ($\Lambda+\mathrm{N+N}+\alpha$) four-body
cluster model. Surprisingly, our prediction for
${}^{7}_{\Lambda}\mathrm{Be}$-${}^{7}_{\Lambda}\mathrm{Li^{*}}$ for the
original NLO13 potential (without CSB interaction), $\Delta
B_{\Lambda}(\mathrm{NLO13})=-17$ keV, is identical to the CSB estimated by Gal
[37]. However, the individual contributions $\Delta T$, $\Delta
V_{\mathrm{NN}}$, and $\Delta V_{\mathrm{YN}}$ differ substantially. For
example, the NLO13 potential yields a vanishing $\Delta V_{\mathrm{YN}}$
(because, as said, there is no CSB part), whereas in Gal’s calculation this
contribution amounts to $50$ keV. $\Delta V_{\mathrm{YN}}$ evaluated for the
actual chiral CSB interaction is of opposite sign and smaller. There is also a
large difference in $\Delta V_{\mathrm{NN}}$ (that quantity includes also the
Coulomb effect). Note that $\Delta V_{\mathrm{NN}}$ used by Gal is taken from
the cluster-model study of Hiyama et al. [13] whereas our value is calculated
consistently within the NCSM.

CSB results for the two $A=8$ mirror nuclei are listed at the lower end of
Table 4. When using the potentials NLO13 and NLO19 without the CSB
interaction, a negligibly small CSB is predicted for
${}^{8}_{\Lambda}\mathrm{Be}$-${}^{8}_{\Lambda}\mathrm{Li}$, namely $\Delta
B_{\Lambda}=16\pm 50$ keV and $-6\pm 50$ keV, respectively. This is, however,
well in line with the empirical CSB of $40\pm 60$ keV [4] based on the
separation energies determined in emulsion experiments. A similarly small
$\Delta B_{\Lambda}$ was also predicted by Gal in [37], but, 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 $\Delta V_{\mathrm{NN}}$, namely $\Delta V_{\mathrm{NN}}=-81$
keV. The latter was not computed directly but taken from the shell model
calculation by Millener [37].

With the CSB interaction included, both the NLO13 and NLO19 potentials yield
rather sizable CSB results, $\Delta B_{\Lambda}(\mathrm{NLO13})=177\pm 50$ keV
and $\Delta B_{\Lambda}(\mathrm{NLO19})=143\pm 50$ keV. In this case, the
${}^{1}S_{0}$ and ${}^{3}S_{1}$ partial-wave contributions are large, and more
importantly, are of the same sign, and, therefore, add up to a pronounced
total CSB. This exactly resembles the situation for the $0^{+}$ states of the
$A=4$ mirror hypernuclei discussed in Sect. IV.1. Indeed, it is conceivable
that a fairly large splitting in the $0^{+}$ state, as presently established,
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.
[13, 39]. However, it should be noted that the phenomenological CSB YN
interaction used in Ref. [13] was fitted to an outdated CSB splitting in the
$A=4$ systems, namely $\Delta B_{\Lambda}(0^{+})=350\pm 60$ keV and $\Delta
B_{\Lambda}(1^{+})=240\pm 60$ keV. Also, it should be said that, when using
only the charge symmetric phenomenological interactions adjusted so that the
experimental value of $B_{\Lambda}(^{8}_{\Lambda}\mathrm{Li})=6.80$ MeV is
reproduced, Hiyama et al. obtained a separation energy of $B_{\Lambda}=6.72$
MeV for ${}^{8}_{\Lambda}\mathrm{Be}$. The difference of $-80$ keV between
$B_{\Lambda}(^{8}_{\Lambda}\mathrm{Be})$ and
$B_{\Lambda}(^{8}_{\Lambda}\mathrm{Li})$ was then attributed to the difference
of the Coulomb interaction [13], which only amounts to about $10$ keV in our
calculations.

|  | ${{}^{4}_{\Lambda}\mathrm{He}}-{{}^{4}_{\Lambda}\mathrm{H}}$ | ${{}^{7}_{\Lambda}\mathrm{Be}}-{{}^{7}_{\Lambda}\mathrm{Li^{*}}}$ | ${{}^{7}_{\Lambda}\mathrm{Li^{*}}}-{{}^{7}_{\Lambda}\mathrm{He}}$ | ${{}^{8}_{\Lambda}\mathrm{Be}}-{{}^{8}_{\Lambda}\mathrm{Li}}$
---|---|---|---|---|---
|  | ${0^{+}}$ | ${1^{+}}$ |  |  |
NLO13-CSB | full | 229 | -94 | -40 | -5 | 178
NLO13-CSB | $\lambda=0.765$ | 213 | -80 | -10 | 0 | 204
NLO19-CSB | full | 238 | -85 | -35 | -16 | 143
NLO19-CSB | $\lambda=0.823$ | 210 | -71 | -26 | -3 | 135
NLO19-CSB∗ | $\lambda=0.823$ | 130 | -135 | -83 | -62 | 74
Table 5:  CSB splittings $\Delta B_{\Lambda}$ (in keV) for $A=4-8$ systems.
Results for the full calculation and those based on the YN potentials NLO13
and NLO19, SRG-evolved to $\lambda_{magic}=0.765$ and $\lambda_{magic}=0.823$
fm-1, respectively, are compared. CSB∗ corresponds to a CSB interaction
adjusted to the new STAR data [28], see text. The estimated uncertainties for
$A=4,7$ and 8 systems are 10, 30 and 50 keV, respectively.

Finally, for illustration, we compare in Table 5 CSB results based on
calculations with the magic flow parameter $\lambda_{magic}$, i.e. of
calculations without 3N forces and without the SRG-induced YNN interaction,
with the full results. Furthermore, we discuss the implications of a somewhat
different CSB splitting in the $A=4$ system, as suggested by a recent STAR
measurement [28]. For the latter aspect a new scenario is introduced, called
CSB∗, where the LECs of the CSB interaction have been re-adjusted to match the
CSB splittings reported by STAR, namely $160\pm 160$ keV for the $0^{+}$ state
and $-160\pm 140$ keV for $1^{+}$.

It is reassuring though not surprising that the full $A=4$ CSB results differ
from the values computed at $\lambda_{magic}$ by at most $30$ keV. The CSB
splittings for the $A=7$ systems, computed at the magic SRG-flow parameters,
are likewise in rather good agreement with the results extracted from the full
calculations. The same is also true for the $A=8$ isodoublet. Apparently, the
magic SRG-flow parameter is a fairly reliable starting point for studying the
separation energies as well as CSB effects in light hypernuclei. This
important observation could help to significantly save computational
resources.

Regarding the new STAR data, it clearly sticks out from Table 5 that the
corresponding scenario CSB∗ yields somewhat larger CSB for
${}^{7}_{\Lambda}\mathrm{Be}$-${}^{7}_{\Lambda}\mathrm{Li}^{*}$ and
${}^{7}_{\Lambda}\mathrm{Li}^{*}$-${}^{7}_{\Lambda}\mathrm{He}$, $\Delta
B_{\Lambda}=-83\pm 30$ keV and $\Delta B_{\Lambda}=-62\pm 30$ keV,
respectively, as compared to the values of $\Delta B_{\Lambda}=-26\pm 30$ keV
and $\Delta B_{\Lambda}=-3\pm 30$ keV, predicted by the standard CSB
interaction. However, overall, both the CSB∗ and CSB results are still
consistent with the experimental values of $\Delta
B_{\Lambda}(^{7}_{\Lambda}\mathrm{Be}$-${}^{7}_{\Lambda}\mathrm{Li^{*}})=-100\pm
90$ keV and $\Delta
B_{\Lambda}(^{7}_{\Lambda}\mathrm{Li}^{*}$-${}^{7}_{\Lambda}\mathrm{He})=-20\pm
230$ keV [6]. Also in case of the A=8 isodoublet the splitting of $74\pm 50$
keV predicted for the scenario CSB∗ is well in line with the experimental
value of $40\pm 60$ keV [4].

##  V Summary

In this work, we have presented results for the separation energies of the
isospin triplet ${{}^{7}_{\Lambda}\rm He}$, ${{}^{7}_{\Lambda}\rm Li}^{*}$,
${{}^{7}_{\Lambda}\rm Be}$, and the $T=1/2$ doublet ${{}^{8}_{\Lambda}\rm
Li}$, ${{}^{8}_{\Lambda}\rm Be}$, calculated within the NCSM. The underlying
YN interactions, taken from Refs. [21, 22, 23], are derived from chiral
effective field theory at NLO. The potentials include the leading CSB
interaction in the $\Lambda\mathrm{N}$ channel, whose strength has been fixed
to the experimental difference of the $\Lambda$ separation energies of the
mirror hypernuclei ${{}^{4}_{\Lambda}\rm He}$ and ${{}^{4}_{\Lambda}\rm H}$ as
established by the J-PARC and Mainz data [17, 18, 19]. In order to speed up
the convergence of the NCSM with respect to the model space, all included
interactions are SRG-evolved and the arising SRG-induced three-body forces are
taken into account.

We have found that the YN potential NLO13 [21] produces too low separation
energies for the $A=7$ and $8$ systems considered in the present work.
However, the predictions for the YN potential from 2019 (NLO19) [22] agree
quite well with the experimental values for ${{}^{7}_{\Lambda}\rm He}$ and
${{}^{7}_{\Lambda}\rm Li}^{*}$, deduced from counter experiments. On the other
hand, separation energies obtained from emulsion experiments for
${{}^{7}_{\Lambda}\rm Be}$ and for the $A=8$ hypernuclei ${{}^{8}_{\Lambda}\rm
Be}$ and ${{}^{8}_{\Lambda}\rm Li}$ are overestimated. For either potentials
the discrepancies between theory and experiments and the differences of NLO13
and NLO19 might be a signal for the necessity of chiral $\Lambda$NN and
$\Sigma$NN three-body forces [36], which have been not included so far in our
calculations. At the same time, one has to keep in mind that the experimental
situation for the hypernuclei studied in the present work is not yet settled,
specifically concerning the emulsion data, see the discussion in Refs. [4, 5,
6].

With regard to CSB, the predicted values for the $A=7$ systems are small and
agree with the splittings deduced from the empirical binding energies within
the experimental uncertainty. In case of the $A=8$ doublet, the computed CSB
is somewhat larger than the available experimental value. We stress that
possible YNN three-body forces should have only a minor influence on the
calculated CSB splittings. In view of the still uncertain experimental
situation, we also considered a scenario motivated by recent data from the
STAR collaboration for $A=4$. We found slightly increased values for the CSB
in $A=7$ and a significant reduction in $A=8$. The different effects in $A=7$
and 8 hypernuclei are related to contributions of different sign in the 1S0
and 3S1 partial waves. Accurate experimental data in these systems will
therefore allow one to independently check the CSB deduced from $A=4$
hypernuclei.

We have also explored in detail the possibility to use the so-called magic
flow parameter of the SRG evolution in the actual NCSM computations. In this
case, in contrast to the full calculation which includes 3N forces and the
SRG-induced YNN interaction, only two-body interactions are taken into
account. In such a scenario, one can save a significant amount of
computational resources. But then, as a consequence, the results depend on the
actual value of the SRG-flow parameter. We consider the option to fix its
value by requiring that the same ${}^{5}_{\Lambda}$He separation energies are
obtained as in the full NCSM calculation. It turned out that the separation
energies obtained with that choice of the flow parameter are fairly close to
the full results. This suggests that the “magic” SRG-flow parameter is a
fairly reliable starting point for studying the separation energies as well as
CSB effects in light hypernuclei in an “inexpensive” way.

## Acknowledgements

We thank Stefan Petschauer for his collaboration in the early stage of this
work. This project is part of the ERC Advanced Grant “EXOTIC” supported the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 101018170). This work
is further supported in part by the DFG and the NSFC through funds provided to
the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD”
(DFG Project-ID 196253076 - TRR 110), the VolkswagenStiftung (grant no.
93562), and by the MKW NRW under the funding code NW21-024-A. The work of UGM
was supported in part by The Chinese Academy of Sciences (CAS) President’s
International Fellowship Initiative (PIFI) (grant no. 2018DM0034). We also
acknowledge support of the THEIA net-working activity of the Strong 2020
Project. The numerical calculations were performed on JURECA and the JURECA-
Booster of the Jülich Supercomputing Centre, Jülich, Germany.

##  Appendix A Contribution of ${}^{1}S_{0}$ and ${}^{3}S_{1}$ partial waves
to $\Delta V_{YN}$

| ${}^{1}S_{0}$ | ${}^{3}S_{1}$ | $\langle V_{\mathrm{YN}}\rangle$
---|---|---|---
| $\Lambda p$ | $\Lambda n$ | $\Lambda p$ | $\Lambda n$ | ${}^{1}S_{0}$ | ${}^{3}S_{1}$
${}^{4}_{\Lambda}\mathrm{He}(0^{+})$ | 13.92 | 27.60 | 44.54 | 0.42 | -4.383 | -3.916
${}^{4}_{\Lambda}\mathrm{H}(0^{+})$ | 27.1 | 13.66 | 0.41 | 43.79 | -4.257 | -3.797
${}^{4}_{\Lambda}\mathrm{He}(1^{+})$ | 14.48 | 0.13 | 42.47 | 27.07 | -1.383 | -5.743
${}^{4}_{\Lambda}\mathrm{H}(1^{+})$ | 0.128 | 14.48 | 27.16 | 42.48 | -1.423 | -5.8685
${}^{7}_{\Lambda}\mathrm{Be}$ | 11.13 | 7.22 | 33.25 | 21.67 | -3.733 | -9.364
${}^{7}_{\Lambda}\mathrm{Li^{*}}$ | 9.17 | 9.17 | 27.44 | 27.44 | -3.768 | -9.321
${}^{7}_{\Lambda}\mathrm{He}$ | 7.22 | 11.10 | 21.65 | 33.13 | -3.802 | -9.278
${}^{8}_{\Lambda}\mathrm{Be}$ | 9.49 | 12.24 | 28.67 | 19.33 | -5.315 | -9.959
${}^{8}_{\Lambda}\mathrm{Li}$ | 11.71 | 9.5 | 19.84 | 28.58 | -5.254 | -9.876
Table 6:  Probability (in $\%$) of finding $\Lambda p$ and $\Lambda n$ pairs
in the $A=4\,-\,8$ wave functions, and the contributions of the ${}^{1}S_{0}$
and ${}^{3}S_{1}$ $\Lambda\mathrm{N}$ partial waves to the expectation value
$\langle V_{\mathrm{YN}}\rangle$ (in MeV). The calculations are based on the
NLO19(500) YN potential. The SRG-induced YNN interaction is also included in
the calculations for
${}^{4}_{\Lambda}\mathrm{He}$-${}^{4}_{\Lambda}\mathrm{H}$ whereas the $A=7$
and $8$ wave functions were computed at the magic SRG-flow parameter of
$\lambda_{magic}=0.823$ fm-1. The singlet and triplet $\Lambda p\,(\Lambda n)$
scattering lengths predicted by the NLO19(500) are $a_{{}^{1}{\rm
S}_{0}}=-2.649(-3.202)$ fm and $a_{{}^{3}{\rm S}_{1}}=-1.580(-1.467)$ fm.

In this appendix, we provide a brief summary of the contributions from the
${}^{1}S_{0}$ and ${}^{3}S_{1}$ $\Lambda\mathrm{N}$ partial waves to the
expectation value of the corresponding YN potentials for the considered
$A=4\,-\,8$ $\Lambda$-hypernuclei, see 6. The weights of the respective
${\Lambda}p$ and ${\Lambda}n$ components are listed, too, which differ, of
course, for the mirror hypernuclei in question. Those weights, in combination
with the different strengths of the $\Lambda n$ and $\Lambda p$ interactions
in the singlet and triplet states as manifested by the respective scattering
lengths, see Table 2 in Ref. [23], determine the value for $\langle
V_{\mathrm{YN}}\rangle$ and, in turn, also the values for $\Delta
V_{\mathrm{YN(^{1}S_{0})}}$ and $\Delta V_{\mathrm{YN(^{3}S_{1})}}$ that are
listed in Tables 2 and 4. Clearly, the signs of the two contributions $\Delta
V_{\mathrm{YN(^{1}S_{0})}}$ and $\Delta V_{\mathrm{YN(^{3}S_{1})}}$ can be the
same or the opposite, depending on the concrete interplay realized in a
specific mirror hypernucleus.

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