11institutetext: Johann Haidenbauer 22institutetext: IAS-4, IKP-3 and JHCP,
Forschungszentrum Jülich, D-52428 Jülich, Germany
22email<EMAIL_ADDRESS>33institutetext: Ulf-G. Meißner
44institutetext: HISKP and BCTP, Universität Bonn, D-53115 Bonn, Germany
IAS-4, IKP-3 and JHCP, Forschungszentrum Jülich, D-52428 Jülich, Germany
Tbilisi State University, 0186 Tbilisi, Georgia
44email<EMAIL_ADDRESS>55institutetext: Andreas Nogga
66institutetext: IAS-4, IKP-3 and JHCP, Forschungszentrum Jülich, D-52428
Jülich, Germany
66email<EMAIL_ADDRESS>
# Constraints on the $\Lambda$-neutron interaction from charge symmetry
breaking in the $\mathbf{{}^{4}_{\Lambda}\rm\bf He}$ \-
$\mathbf{{}^{4}_{\Lambda}\rm\bf H}$ hypernuclei

Johann Haidenbauer  Ulf-G. Meißner  Andreas Nogga

(draft: July 1, 2021)

###### Abstract

We utilize the experimentally known difference of the $\Lambda$ separation
energies of the mirror hypernuclei ${{}^{4}_{\Lambda}\rm He}$ and
${{}^{4}_{\Lambda}\rm H}$ to constrain the $\Lambda$-neutron interaction. We
include the leading charge-symmetry breaking (CSB) interaction into our
hyperon-nucleon interaction derived within chiral effective field theory at
next-to-leading order. In particular, we determine the strength of the two
arising CSB contact terms by a fit to the differences of the separation
energies of these hypernuclei in the $0^{+}$ and $1^{+}$ states, respectively.
By construction, the resulting interaction describes all low energy hyperon-
nucleon scattering data, the hypertriton and the CSB in ${{}^{4}_{\Lambda}\rm
He}$-${{}^{4}_{\Lambda}\rm H}$ accurately. This allows us to provide first
predictions for the $\Lambda$n scattering lengths, based solely on available
hypernuclear data.

###### Keywords:

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

###### pacs:

13.75.Ev,12.39.Fe, 14.20.Jn

††journal: Few-Body Syst

##  1 Introduction

The large charge symmetry breaking (CSB), manifested in the differences of the
${\Lambda}$-separation energies of the mirror nuclei ${{}^{4}_{\Lambda}\rm
He}$ and ${{}^{4}_{\Lambda}\rm H}$, is one of the mysteries of hypernuclear
physics. Already experimentally established in the early 1960s Dalitz:1964fu ;
Raymund:1964an , for the ground ($0^{+}$) state, there is still no plausible
and generally accepted explanation of it despite of numerous investigations
Gibson:1973zza ; Coon:1979si ; Bodmer:1985km ; Coon:1998jd ; Nogga:2001ef ;
Gal:2015hh ; Gazda:2015qyt ; Gazda:2016ir . Indeed, the separation-energy
difference $\Delta E(0^{+})$ of $340$ keV Juric:1973zq , benchmark for many
decades, is about half of the corresponding difference in the mirror nuclei
${{}^{3}\rm H}$ and ${{}^{3}\rm He}$ which amounts to $764$ keV Miller:2006tv
. However, while in the latter case about 90 % of the difference is due to the
Coulomb force, its effect is rather small for the $A=4$ hypernuclei and,
moreover, it goes into the wrong direction Bodmer:1985km ; Nogga:2001ef .
Thus, most of the CSB seen in the $A=4$ hypernuclei must come from the strong
interaction.

The separation-energy difference for the excited $(1^{+})$ state was
established with a measurement from 1979 Bedjidian:1979ih  and found to be
$\Delta E(1^{+})=240$ keV. Thus, at that time, it looked as if CSB effects are
practically spin (state) independent. The situation changed considerably
around 2015-2016 when new and more refined data from experiments at J-PARC
Yamamoto:2015avw  and Mainz Esser:2015fm ; Schulz:2016dt  became available.
These led to the presently accepted values of $\Delta E(0^{+})=233\pm 92$ keV
and $\Delta E(1^{+})=-83\pm 94$ keV Gazda:2016ir .

As already indicated above, initial calculations of the ${{}^{4}_{\Lambda}\rm
He}$-${{}^{4}_{\Lambda}\rm H}$ binding energy difference based on a two-body
model Dalitz:1964fu ; Coon:1979si  failed to describe the data. The principle
CSB mechanism considered in those studies consisted of
${\Lambda}-{\Sigma}^{0}$ mixing. It facilitates pion exchange between the
${\Lambda}$ and the nucleons Dalitz:1964fu , which is otherwise forbidden by
isospin conservation. In addition, contributions from $\eta-\pi^{0}$,
$\omega-\rho^{0}$, etc., mixing were taken into account. The situation did not
improve with first more elaborate studies that employed four-body wave
functions from variational Monte Carlo calculations Coon:1998jd . And it
remained also unchanged when the first full-fledged four-body calculations
based on the Faddeev-Yakubovsky approach became available Nogga:2001ef . The
coupled-channel ${\Lambda}N$-${\Sigma}N$ interactions employed in the latter
study, constructed by the Nijmegen group Maessen:1989hm ; Rijken:1999fc , all
include ${\Lambda}-{\Sigma}^{0}$ mixing as essential source of CSB. In
addition, further sources of CSB such as the Coulomb interaction in the $NN$
and ${\Sigma}N$ subsystems and the mass differences between ${\Sigma}^{-}$,
${\Sigma}^{0}$ and ${\Sigma}^{+}$ were taken into account in Ref. Nogga:2001ef
. But these calculations could only explain a fraction of the experimentally
found CSB in $A=4$ hypernuclei. Very recently four-body calculations within
the no-core shell model were presented by Gal and Gazda Gazda:2015qyt ;
Gazda:2016ir  which promised, finally, a solution to the CSB “puzzle”.
However, the CSB mechanism is somewhat unorthodox and rests on the assumption
that the CSB part of the ${\Lambda}N$ interaction can be entirely and uniquely
fixed by the ${\Lambda}N\to{\Sigma}N$ transition potential Gal:2015hh .

In the present work, we study CSB in the hyperon-nucleon ($YN$) interaction
within SU(3) chiral effective field theory (EFT) Polinder:2006eq ;
Haidenbauer:2013oca ; Haidenbauer:2019boi ; Petschauer:2020urh , which is an
extension of Weinberg’s idea suggested for nuclear forces Weinberg:1990bf  to
systems involving baryons with strangeness. In this approach, the long-range
part of the interaction (due to exchange of pseudoscalar mesons) is fixed by
chiral symmetry. The short-distance part is not resolved and effectively
described by contact terms whose strengths, encoded in low-energy constants
(LECs), need to be determined by a fit to data Weinberg:1990bf ;
Epelbaum:2008ga ; Machleidt:2011gh . This notion applies to the charge-
symmetry conserving as well as to the charge-symmetry breaking part of the
interaction Walzl:2000cx ; Friar:2003yv ; Epelbaum:2005ci . Accordingly, in
our investigation, we do not follow the aforementioned procedure applied by
Gal and Gazda. Rather, we fix the CSB part of the ${\Lambda}N$ potential from
the $A=4$ separation energies and then predict CSB effects for the elementary
${\Lambda}p$ and ${\Lambda}n$ interactions. We do not share the view of Gazda
and Gal who consider this procedure basically as a tautology Gazda:2016ir  but
actually as an excellent tool to pin down the ${\Lambda}n$ interaction,
relying on and being consistent with available hypernuclear data.

The paper is structured in the following way: in the subsequent section, we
give a detailed account of the CSB part of the ${\Lambda}N$ interaction. Here
we follow closely the arguments from analogous studies of the nucleon-nucleon
($NN$) system. Technical details of the treatment of the three- and four-body
systems are summarized in Sec. 3. In Sec. 4, we explain how the CSB part is
determined from the separation-energy differences in the $0^{+}$ and $1^{+}$
states of the $A=4$ hypernuclei. Specifically, considering those differences
allows us to fix the low-energy constants of corresponding CSB contact terms
that arise at next-to-leading order in the chiral expansion Epelbaum:2005ci .
Once those are established, predictions for ${\Lambda}p$ and ${\Lambda}n$
scattering lengths are presented. The paper ends with some concluding remarks.

##  2 Hyperon-nucleon interaction

###  2.1 $YN$ interaction in chiral EFT

For the present study, we utilize the $YN$ interactions from Refs.
Haidenbauer:2013oca ; Haidenbauer:2019boi , derived within SU(3) chiral EFT up
to 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 Haidenbauer:2013oca , 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
Haidenbauer:2019boi  (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. Haidenbauer:2019boi , where one can see that the two $YN$
interactions yield essentially equivalent results in the two-body sector. Note
that there is no explicit CSB in the ${\Lambda}N$ potential of the published
$YN$ interactions. However, since the scattering amplitude in Refs.
Haidenbauer:2013oca ; Haidenbauer:2019boi  is obtained from solving a coupled-
channel Lippmann-Schwinger equation in the particle basis, the mass
differences between ${\Sigma}^{-}$, ${\Sigma}^{0}$, and ${\Sigma}^{+}$ enter
and likewise the Coulomb interaction in the ${\Sigma}^{-}p$ channel. Because
of that isospin symmetry is broken and the results for ${\Lambda}p$ and
${\Lambda}n$ scattering are (slightly) different.

###  2.2 CSB in chiral EFT

As noted by Dalitz and von Hippel many decades ago Dalitz:1964fu ,
${\Lambda}-{\Sigma}^{0}$ mixing leads to a long-ranged CSB contribution to the
${\Lambda}N$ interaction due to pion exchange, see Fig. 1. The strength of the
potential can be estimated from the electromagnetic mass matrices,

$\displaystyle\langle{\Sigma}^{0}|\delta m|{\Lambda}\rangle$ $\displaystyle=$
$\displaystyle[m_{{\Sigma}^{0}}-m_{{\Sigma}^{+}}+m_{p}-m_{n}]/\sqrt{3},$
$\displaystyle\langle\pi^{0}|\delta M^{2}|\eta\rangle$ $\displaystyle=$
$\displaystyle[M^{2}_{\pi^{0}}-M^{2}_{\pi^{+}}+M^{2}_{K^{+}}-M^{2}_{K^{0}}]/\sqrt{3}$
(1)

and subsumed in terms of an effective ${\Lambda}{\Lambda}\pi$ coupling
constant

$f_{{\Lambda}{\Lambda}\pi}=\left[-2\frac{\langle{\Sigma}^{0}|\delta
m|{\Lambda}\rangle}{m_{{\Sigma}^{0}}-m_{{\Lambda}}}+\frac{\langle\pi^{0}|\delta
M^{2}|\eta\rangle}{M^{2}_{\eta}-M^{2}_{\pi^{0}}}\right]\,f_{{\Lambda}{\Sigma}\pi}\
.$ (2)

Based on the latest PDG mass valuesZyla:2020zbs , one obtains

$f_{{\Lambda}{\Lambda}\pi}=f^{({\Lambda}-{\Sigma}^{0})}_{{\Lambda}{\Lambda}\pi}+f^{(\eta-\pi^{0})}_{{\Lambda}{\Lambda}\pi}\approx(-0.0297-0.0106)\,f_{{\Lambda}{\Sigma}\pi}\
.$ (3)

In this context, let us mention that there are also lattice QCD calculations
of ${\Lambda}-{\Sigma}^{0}$ mixing Kordov:2019oer ; Horsley:2014koa ;
Gal:2015iha ; Horsley:2015lha .

Figure 1: CSB contributions involving pion exchange, according to Dalitz and
von Hippel Dalitz:1964fu , due to ${\Lambda}-{\Sigma}^{0}$ mixing (left two
diagrams) and $\pi^{0}-\eta$ mixing (right diagram).

Figure 2: CSB contributions from $K^{\pm}$/$K^{0}$ exchange (left) and from
contact terms (right).

In our implementation of CSB within chiral EFT, we follow closely the
arguments given in pertinent studies of isospin-breaking effects in the
nucleon-nucleon ($NN$) system, see Refs. Walzl:2000cx ; Friar:2003yv ;
Epelbaum:2005ci . According to Ref. Friar:2003yv , the CSB contributions at
leading order are characterized by the parameter $\epsilon
M^{2}_{\pi}/\Lambda^{2}\sim 10^{-2}$, where
$\epsilon\equiv\frac{m_{d}-m_{u}}{m_{d}+m_{u}}\sim 0.3$ and $\Lambda\sim
M_{\rho}$. In particular, one expects a potential strength of $V^{\rm
CSB}_{BB}\sim(\epsilon M^{2}_{\pi}/\Lambda^{2})\,V_{BB}$. At order $n=2$ (NLØ
in the notation of Ref. Epelbaum:2005ci ), there are contributions from
isospin violation in the pion-baryon coupling constant, which in the
${\Lambda}N$ case arise from the aforementioned $\Sigma^{0}-\Lambda$ mixing as
well as from $\pi^{0}-\eta$ mixing. In addition, there are contributions from
short range forces (arising from $\rho^{0}-\omega$ mixing, etc.). In chiral
EFT, such forces are simply represented by contact terms involving LECs (Fig.
2 right) that need to be fixed by a fit to data. Contributions at $n=1$ (LØ)
are due to a possible Coulomb interaction between the baryons in question and
due to mass differences between $M_{\pi^{\pm}}$ and $M_{\pi^{0}}$. Such
contributions do not arise in the $\Lambda N$ system. However, in the
extension to SU(3), there is CSB induced by the $M_{K^{\pm}}$-$M_{K^{0}}$ mass
difference, see left side of Fig. 2. We take that into account in our
calculation, since it is formally at leading order. But because the kaon mass
is rather large compared to the mass difference, its effect is actually very
small. For a general overview, we refer the reader to Table I in Ref.
Epelbaum:2005ci .

The CSB part of the ${\Lambda}N$ potential at NLØ is given by

$\displaystyle V^{CSB}_{{\Lambda}N\to{\Lambda}N}$ $\displaystyle=$
$\displaystyle\Bigg{[}-f^{({\Lambda}-{\Sigma}^{0})}_{{\Lambda}{\Lambda}\pi}f_{NN\pi}\frac{\left(\mbox{\boldmath$\sigma$}_{1}\cdot{\bf
q}\right)\left(\mbox{\boldmath$\sigma$}_{2}\cdot{\bf q}\right)}{{\bf
q}^{2}+M^{2}_{\pi^{0}}}$ (4) $\displaystyle-$ $\displaystyle
f^{(\eta-\pi^{0})}_{{\Lambda}{\Lambda}\pi}f_{NN\pi}\left(\mbox{\boldmath$\sigma$}_{1}\cdot{\bf
q}\right)\left(\mbox{\boldmath$\sigma$}_{2}\cdot{\bf
q}\right)\left(\frac{1}{{\bf q}^{2}+M^{2}_{\pi^{0}}}-\frac{1}{{\bf
q}^{2}+M^{2}_{\eta}}\right)$ $\displaystyle+$
$\displaystyle\frac{1}{4}(1-\mbox{\boldmath$\sigma$}_{1}\cdot\mbox{\boldmath$\sigma$}_{2})\,C^{CSB}_{s}\,+\,\frac{1}{4}(3+\mbox{\boldmath$\sigma$}_{1}\cdot\mbox{\boldmath$\sigma$}_{2})\,C^{CSB}_{t}\Bigg{]}\
\tau_{N}\ ,$

where $C^{CSB}_{s}$ and $C^{CSB}_{t}$ are charge-symmetry breaking contact
terms in the spin-singlet (${}^{1}S_{0}$) and triplet (${}^{3}S_{1}$) partial
waves, respectively, and $\tau_{p}=1$ and $\tau_{n}=-1$. In the treatment of
the contribution from $\pi^{0}-\eta$ mixing, we follow Ref. Coon:1979si .

Besides the CSB in the ${\Lambda}N$ potential, there is also some effect due
to the coupling to the ${\Sigma}N$ channel. Specifically, the coupled-channel
Lippmann-Schwinger (LS) equation is solved in the particle basis and the
physical masses of the ${\Sigma}$’s and $p$, $n$ are used. In addition, the
Coulomb interaction in the ${\Sigma}^{-}p$ channel that couples to
${\Lambda}n$ is taken into account. As mentioned above, these effects are
already considered in our standard calculation Haidenbauer:2013oca ;
Haidenbauer:2019boi  and lead to a small but noticeable CSB breaking in the
${\Lambda}N$ results but also in case of ${{}^{4}_{\Lambda}\rm He}$ /
${{}^{4}_{\Lambda}\rm H}$ Haidenbauer:2007ra ; Nogga:2013pwa ; Nogga:2019bwo .
CSB contributions to the ${\Lambda}N$ potential from (irreducible) two-pion
exchange, which in our case not only involves the $p$-$n$ mass difference but
also the one for ${\Sigma}^{+}$-${\Sigma}^{0}$-${\Sigma}^{-}$, are expected to
be small Friar:2003yv  and, therefore, omitted in the present study. Note that
the corresponding two-pion exchange diagrams involve either $\pi^{\pm}$ or
$\pi^{0}$ so that the pion-mass difference does not enter.

We note already now that the value for $f_{{\Lambda}{\Lambda}\pi}$ given in
Eq. (3) as well as the CSB LECs used in the actual calculation are in line
with the aforementioned order-of-magnitude estimate by Friar et al.
Friar:2003yv .

##  3 Faddeev-Yakubovsky equations

Our predictions for $A=3$ and $A=4$ systems are based on solutions of Faddeev-
Yakubovsky equations in momentum space Nogga:2001ef ; NoggaPhD:2001 ;
Nogga:2013ey . Here, we just briefly summarize our way of solving the
Yakubovsky equations for $A=4$ hypernuclei. For one hyperon and three
identical nucleons, the Schrödinger equation can be rewritten in a set of five
coupled Yakubovsky equations

$\displaystyle|\psi_{1A}\rangle$ $\displaystyle=$ $\displaystyle
G_{0}t_{12}(P_{13}P_{23}+P_{12}P_{23})$
$\displaystyle\qquad\qquad\left[|\psi_{1A}\rangle+|\psi_{1B}\rangle+|\psi_{2A}\rangle\right]$
$\displaystyle|\psi_{1B}\rangle$ $\displaystyle=$ $\displaystyle
G_{0}t_{12}\left[(1-P_{12})(1-P_{23})|\psi_{1C}\rangle\right.$
$\displaystyle\qquad\qquad\left.+(P_{13}P_{23}+P_{12}P_{23})|\psi_{2B}\rangle\right]$
$\displaystyle|\psi_{1C}\rangle$ $\displaystyle=$ $\displaystyle
G_{0}t_{14}\left[|\psi_{1A}\rangle+|\psi_{1B}\rangle+|\psi_{2A}\rangle-
P_{12}|\psi_{1C}\rangle\right.$
$\displaystyle\qquad\qquad\left.+P_{13}P_{23}|\psi_{1C}\rangle+P_{12}P_{23}|\psi_{2B}\rangle\right]$
$\displaystyle|\psi_{2A}\rangle$ $\displaystyle=$ $\displaystyle
G_{0}t_{12}\left[(P_{12}-1)P_{13}|\psi_{1C}\rangle+|\psi_{2B}\rangle\right]$
$\displaystyle|\psi_{2B}\rangle$ $\displaystyle=$ $\displaystyle
G_{0}t_{34}\left[|\psi_{1A}\rangle+|\psi_{1B}\rangle+|\psi_{2A}\rangle\right]\
.$ (5)

For the solution of this problem, we distinguish the proton and neutron and
the $\Sigma$ masses in the free propagator $G_{0}$ and for the solution of the
Lippmann-Schwinger equations for the $t$-matrices $t_{ij}$ that are embedded
into the four-baryon system. The reduction to only five Yakubovsky equations
is possible because of the identity of the nucleons which allows to relate
different Yakubovsky components to each other using permutation operators
$P_{ij}$ that interchange the quantum numbers of nucleon $i$ and $j$.

The five remaining Yakobovsky components $|\psi_{1A}\rangle$,
$|\psi_{1B}\rangle$, $|\psi_{1C}\rangle$, $|\psi_{2A}\rangle$ and
$|\psi_{2B}\rangle$ are expanded in terms of their natural Jacobi coordinate

$\displaystyle\textstyle|p_{12}p_{3}q_{4}\alpha_{A}\rangle$ $\displaystyle=$
$\displaystyle\textstyle\left|p_{12}p_{3}q_{4}\left[\left[(l_{12}s_{12})j_{12}\left(l_{3}\frac{1}{2}\right)I_{3}\right]j_{123}\left(l_{4}\frac{1}{2}\right)I_{4}\right]J\left[(t_{12}\frac{1}{2})\tau_{123}t_{Y}\right]TM_{T}\right\rangle\
$ $\displaystyle\textstyle|p_{12}p_{4}q_{3}\alpha_{B}\rangle$ $\displaystyle=$
$\displaystyle\textstyle\left|p_{12}p_{4}q_{3}\left[\left[(l_{12}s_{12})j_{12}\left(l_{4}\frac{1}{2}\right)I_{4}\right]j_{124}\left(l_{3}\frac{1}{2}\right)I_{3}\right]J\left[(t_{12}t_{Y})\tau_{124}\frac{1}{2}\right]TM_{T}\right\rangle\
$ $\displaystyle\textstyle|p_{14}p_{2}q_{3}\alpha_{C}\rangle$ $\displaystyle=$
$\displaystyle\textstyle\left|p_{14}p_{3}q_{4}\left[\left[(l_{14}s_{14})j_{14}\left(l_{2}\frac{1}{2}\right)I_{2}\right]j_{124}\left(l_{3}\frac{1}{2}\right)I_{3}\right]J\left[(t_{14}\frac{1}{2})\tau_{124}\frac{1}{2}\right]TM_{T}\right\rangle\
$ $\displaystyle\textstyle|p_{12}p_{34}q\beta_{A}\rangle$ $\displaystyle=$
$\displaystyle\textstyle\left|p_{12}p_{34}q\left[\left[(l_{12}s_{12})j_{12}\lambda\right]I\
(l_{34}s_{34})j_{34}\right]J\ (t_{12}t_{34})TM_{T}\right\rangle\ $
$\displaystyle\textstyle|p_{34}p_{12}q\beta_{B}\rangle$ $\displaystyle=$
$\displaystyle\textstyle\left|p_{34}p_{12}q\left[\left[(l_{34}s_{34})j_{34}\lambda\right]I\
(l_{12}s_{12})j_{12}\right]J\ (t_{34}t_{12})TM_{T}\right\rangle\ .$ (6)

There are two types of Jacobi coordinates required. The first three basis sets
are of the “3+1” type. Here, three momenta $p_{ij}$, $p_{k}$ and $q_{l}$ are
required that are relative momenta within the pair $ij$, of particle $k$ with
respect to pair $ij$ and of particle $l$ with respect to the three-body
subsytem $ijk$. Corresponding orbital angular momenta $l_{ij}$, $l_{j}$ and
$l_{k}$ are used to expand angular dependencies. $s_{ij}$ and $j_{ij}$ are the
spin and total angular momentum of the two-body subsystem. We also introduce
$j_{ijk}$ and $\tau_{ijk}$ for the total angular momentum and isospin of the
three-body subsystem. $J$, $T$ and $M_{T}$ are the total angular momentum,
isospin and third component of isospin of the four-baryon system. We have
omitted the spins and isospins of the two baryons in the inner most subsystem
since only $t_{4}=t_{Y}$ differs from ${1}/{2}$. The last two basis sets are
of the “2+2” type. Here, relative momenta of two two-body subsytems $p_{ij}$
and $p_{kl}$ are introduced together with angular momenta and isospins for
these subsystems. Additionally, the relative momentum of the two pairs $q$ and
its angular momentum $\lambda$ is required. In order to finally define the
total four-body angular momentum in this case, an additional intermediate
angular momentum $I$ is required and coupled to the other angular momenta as
shown in Eq. (3).

Once the Yakubovsky components are found, we obtain the wave function by

$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle(1+P_{13}P_{23}+P_{12}P_{23})|\psi_{1A}\rangle$ (7)
$\displaystyle+(1+P_{13}P_{23}+P_{12}P_{23})|\psi_{1B}\rangle\rangle$
$\displaystyle+(1-P_{12})(1+P_{13}P_{23}+P_{12}P_{23})|\psi_{1C}\rangle$
$\displaystyle+(1+P_{13}P_{23}+P_{12}P_{23})|\psi_{2A}\rangle\rangle$
$\displaystyle+(1+P_{13}P_{23}+P_{12}P_{23})|\psi_{2B}\rangle\ .$

For the solution of the Yakubovsky equations, the permutation operators
$P_{ij}$ and transformations between different Jacobi coordinates Eq. (3) need
to be evaluated. For these parts of the code, we use averages of the nucleon
and $\Sigma$ masses. A comparison of the resulting energies and expectation
values of the Hamiltonian shows that this approximation does not alter the
results.

For the numerical solution, the partial wave states have to be constrained.
Here, we choose $j_{ij}\leq 4$, $l_{i}\leq 4$, $\lambda\leq 4$. Additionally,
we restrict $l_{ij}+l_{k}+l_{l}\leq 10$ and $l_{ij}+l_{kl}+\lambda\leq 10$. We
also only take the dominant isospin state $T=1/2$ into account.

We checked carefully that, for the chiral interactions employed here, these
constraints ensure that the numerical accuracy is better than 10 keV for the
energies entering the Yakubovsky equations (3) and 20 keV for expectation
values of the energy. It turns out that the additional isospin components with
$T=3/2$ and $T=5/2$ induced by isospin breaking effects which we do not take
into account here lead to changes of the energy in this order and contribute
most to this uncertainty.

For the solution of this bound state problem, the Coulomb interaction in $YN$
and $NN$ is included. As discussed above, the Coulomb interactions only
contribute a few keV to the separation energies. The same is true for the
$n$-$p$ mass difference as has been shown in Ref. Nogga:2002qp  for 3H-3He. We
therefore do not distinguish between contributions due to the $n$-$p$ mass
difference and the one of the $\Sigma$’s in our results. In fact, the $n$-$p$
mass difference also contributes to the CSB of the core nucleus which we also
do not separate from the other CSB contributions of the core.

Table 1:  ${}^{3}_{\Lambda}$H, ${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda}$H separation energies for NLO13 and NLO19 for various cutoffs in combination with SMS N4LO+ (450) Reinert:2018ip  and different cutoffs for NLO13 and/or NLO19. No explicit CSB is included in the $YN$ potentials. Energies are in MeV.  interaction | $E_{\Lambda}(^{3}_{\Lambda}{\rm H})$ | $E_{\Lambda}(^{4}_{\Lambda}{\rm He})$ | $E_{\Lambda}(^{4}_{\Lambda}{\rm H})$
---|---|---|---
|  | $J^{\pi}=0^{+}$ | $J^{\pi}=1^{+}$ | $J^{\pi}=0^{+}$ | $J^{\pi}=1^{+}$
NLO13(500) | 0.13 | 1.71 | 0.80 | 1.66 | 0.78
NLO13(550) | 0.09 | 1.51 | 0.59 | 1.45 | 0.57
NLO13(600) | 0.09 | 1.48 | 0.59 | 1.43 | 0.56
NLO13(650) | 0.08 | 1.50 | 0.62 | 1.45 | 0.60
NLO19(500) | 0.10 | 1.65 | 1.23 | 1.63 | 1.23
NLO19(550) | 0.09 | 1.55 | 1.25 | 1.53 | 1.24
NLO19(600) | 0.10 | 1.47 | 1.06 | 1.44 | 1.05
NLO19(650) | 0.09 | 1.54 | 0.92 | 1.50 | 0.91
Expt. |  $0.13(5)$ Juric:1973zq  | 2.39(3)Yamamoto:2015avw  | 0.98(3)Yamamoto:2015avw  | 2.16(8)Schulz:2016dt  | 1.07(8) Schulz:2016dt

As shown in previous calculations, the ${\Lambda}$ separation energies of
light hypernuclei are only mildly dependent on the underlying $NN$ interaction
Nogga:2001ef ; Haidenbauer:2019boi . Therefore, we employ in all of the
calculations presented here the same chiral semi-local momentum-space-
regularized (SMS) $NN$ interaction of Ref. Reinert:2018ip  at order N4LO+ for
a cutoff of ${\Lambda}=450$ MeV. We have also used the Idaho interaction
Entem:2003ft  for these calculations and have not found any significant
changes of the CSB predictions. We note that at LO, we find somewhat larger
separation energies for $A=4$ hypernuclei for N4LO+ compared to Idaho.
Apparently, the missing repulsion at short distances at LO increases the
sensitivity to configuration of the nucleons in the core. At NLO, the $NN$
force dependence of the separation energies is of the order of 100 keV and
within the range expected from calculations based on phenomenological
interactions Nogga:2001ef . To define a baseline, we summarize our results for
the separation energies of $A=3$ and 4 hypernuclei for the original NLO13 and
NLO19 interactions in Table 1.

Below we will present two kinds of results for the CSB. First, we will perform
complete calculations for ${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda}$H. This
will allow us to obtain the CSB of the separation energies directly. But it
does not allow for an easy separation of the different contributions to CSB.
Second, we will use the wave function and Yakubovsky components for
${}^{4}_{\Lambda}$He of the original interactions to evaluate CSB
perturbatively. For this, we calculate the expectation values of the
differences of the kinetic energy and the $NN$ and $YN$ potentials when $n$
and $p$ and $\Sigma^{+}$ and $\Sigma^{-}$ are interchanged. The total CSB of
both calculations agrees to better than 10 % for our standard calculations.

##  4 Results

As argued in Refs. Haidenbauer:2019boi ; Le:2019gjp , the contribution of
three-body forces (3BFs) is probably negligible for $A=3$, but very likely
becomes relevant for the more strongly bound $A=4$ system. The dependence of
the separation energies on the regulator (cutoff) is an effect of next-to-
next-to-leading order (N2LO) which includes also 3BFs Hammer:2012id . As can
be seen from the results in Table 1, the variation is negligible for the
hypertriton but can be as large as $200$–$300$ keV for $A=4$. Even larger is
the difference between the two different realizations of $YN$ interactions:
NLO19 and NLO13. For the $1^{+}$ state, the predictions of these two
essentially equivalent realizations of the $YN$ interaction can differ by as
much as $500$ keV. These variations in the predicted $A=4$ separation energies
have to be kept in mind and, ultimately, should be explained by similarly
large 3BF contributions.

However, such 3BFs should have only a minor influence on the observed
splittings between the ${{}^{4}_{\Lambda}\rm He}$ and ${{}^{4}_{\Lambda}\rm
H}$ states, which are primarily due to CSB two-baryon forces. This is the
basic assumption in the strategy pursued below.

###  4.1 CSB in $A=4$ hypernuclei

To start with, we need to fix the CSB-breaking LECs $C^{CSB}_{s}$ and
$C^{CSB}_{t}$ in the ${\Lambda}N$ interaction, cf. Eq. (4). We do this by
considering the observed CSB splittings in the $A=4$ hypernuclei, defined in
the usual way in terms of the separation energies

$\displaystyle\Delta E(0^{+})$ $\displaystyle=$ $\displaystyle
E^{0^{+}}_{{\Lambda}}({{}^{4}_{\Lambda}\rm
He})-E^{0^{+}}_{{\Lambda}}({{}^{4}_{\Lambda}\rm H}),$ $\displaystyle\Delta
E(1^{+})$ $\displaystyle=$ $\displaystyle
E^{1^{+}}_{{\Lambda}}({{}^{4}_{\Lambda}\rm
He})-E^{1^{+}}_{{\Lambda}}({{}^{4}_{\Lambda}\rm H})\ .$ (8)

In our principal results, we aim at a reproduction of the present experimental
situation, based on the recent measurements of the ${{}^{4}_{\Lambda}\rm H}$
$0^{+}$ state in Mainz Schulz:2016dt  and the one of the ${{}^{4}_{\Lambda}\rm
He}$ $1^{+}$-$0^{+}$ splitting at J-PARC Yamamoto:2015avw , i.e. $\Delta
E(0^{+})=233\pm 92$ keV and $\Delta E(1^{+})=-83\pm 94$ keV. It is the same
scenario as considered by Gazda and Gal in Ref. Gazda:2016ir . Below, we will
refer to this choice as CSB1. In order to illustrate the effect of CSB in the
$A=4$ hypernuclei on the underlying ${\Lambda}N$ interaction, we consider also
two other scenarios. One (CSB3) corresponds to the situation after the
publication of the J-PARC experiment Yamamoto:2015avw  but before the final
results from Mainz became available: $\Delta E(0^{+})=350\pm 50$ keV and
$\Delta E(1^{+})=30\pm 50$ keV. It is the status considered by Gazda and Gal
in Ref. Gazda:2015qyt  and discussed in the review Gal:2016boi . In addition,
we look at the situation up to 2014 (which will be labeled CSB2), namely
$\Delta E(0^{+})=350\pm 50$ keV and $\Delta E(1^{+})=240\pm 80$ keV
Bedjidian:1979ih . It is the one discussed by Gal in Ref. Gal:2015hh  and, of
course, in all pre-2014 studies of CSB in the $A=4$ hypernuclei. Note that the
CSB splitting in the $1^{+}$ states in the scenarios CSB1 and CSB3 is
compatible with zero, given the present experimental uncertainty.

Table 2: Comparison of different CSB scenarios, based on the $YN$ interactions NLO13 and NLO19 with cutoff $\Lambda=600$ MeV. Results are shown for the original NLO interactions, with addition of OBE contribution to CSB, and for the scenarios CSB1, CSB2, CSB3 with added CSB contact terms. CSB1 corresponds to the present experimental status. Note that the $\chi^{2}$ for the NLO interactions differs slightly from the ones given in Refs. Haidenbauer:2013oca ; Haidenbauer:2019boi  because there the small differences between ${\Lambda}p$ and ${\Lambda}n$ have not been taken into account. Small deviations of the CSB from values of the three scenarios are due to using perturbation theory for fitting and using a smaller number of partial waves for fitting.  | $a^{\Lambda p}_{s}$ | $a^{\Lambda p}_{t}$ | $a^{\Lambda n}_{s}$ | $a^{\Lambda n}_{t}$ | $\chi^{2}(\Lambda p)$ | $\chi^{2}(\Sigma N)$ | $\chi^{2}({\rm total})$ | $\Delta E(0^{+})$ | $\Delta E(1^{+})$
---|---|---|---|---|---|---|---|---|---
NLO13 | -2.906 | -1.541 | -2.907 | -1.517 | 4.47 | 12.34 | 16.81 | 58 | 24
CSB-OBE | -2.881 | -1.547 | -2.933 | -1.513 | 4.39 | 12.43 | 16.83 | 57 | 20
CSB1 | -2.588 | -1.573 | -3.291 | -1.487 | 3.43 | 12.38 | 15.81 | 256 | -53
CSB2 | -3.983 | -1.281 | -2.814 | -0.948 | 4.51 | 12.31 | 16.82 | 299 | 161
CSB3 | -2.792 | -1.666 | -3.027 | -1.407 | 9.52 | 12.41 | 21.93 | 370 | 56
NLO19 | -2.906 | -1.423 | -2.907 | -1.409 | 3.58 | 12.70 | 16.28 | 34 | 10
CSB-OBE | -2.877 | -1.415 | -2.937 | -1.419 | 3.30 | 13.01 | 16.31 | -6 | -7
CSB1 | -2.632 | -1.473 | -3.227 | -1.362 | 3.45 | 12.68 | 16.13 | 243 | -67
CSB2 | -3.618 | -1.339 | -3.013 | -1.117 | 4.02 | 12.09 | 16.12 | 218 | 129
CSB3 | -2.758 | -1.546 | -3.066 | -1.300 | 7.49 | 12.64 | 20.14 | 359 | 45
Table 3: Singlet ($s$) and triplet ($t$) $S$-wave scattering lengths and $\chi^{2}$ values for the fits to the present experimental CSB splittings of $\Delta E(0^{+})=233$ keV and $\Delta E(1^{+})=-83$ keV (CSB1), based on the $YN$ interactions NLO13 and NLO19.  | $a^{\Lambda p}_{s}$ | $a^{\Lambda p}_{t}$ | $a^{\Lambda n}_{s}$ | $a^{\Lambda n}_{t}$ | $\chi^{2}(\Lambda p)$ | $\chi^{2}(\Sigma N)$ | $\chi^{2}({\rm total})$
---|---|---|---|---|---|---|---
NLO13(500) | -2.604 | -1.647 | -3.267 | -1.561 | 4.47 | 12.13 | 16.60
NLO13(550) | -2.586 | -1.551 | -3.291 | -1.469 | 3.46 | 12.03 | 15.49
NLO13(600) | -2.588 | -1.573 | -3.291 | -1.487 | 3.43 | 12.38 | 15.81
NLO13(650) | -2.592 | -1.538 | -3.271 | -1.452 | 3.70 | 12.57 | 16.27
NLO19(500) | -2.649 | -1.580 | -3.202 | -1.467 | 3.51 | 14.69 | 18.20
NLO19(550) | -2.640 | -1.524 | -3.205 | -1.407 | 3.23 | 14.19 | 17.42
NLO19(600) | -2.632 | -1.473 | -3.227 | -1.362 | 3.45 | 12.68 | 16.13
NLO19(650) | -2.620 | -1.464 | -3.225 | -1.365 | 3.28 | 12.76 | 16.04

We determine the CSB LECs from perturbative calculations of the CSB
contribution to the ${}^{4}_{\Lambda}$H-${}^{4}_{\Lambda}$He splittings for
the three scenarios CSB1-3. Table 2 provides a comparison of the results for
the different scenarios with those of the initial (NLO13 and NLO19) $YN$
potentials, for a regulator with cutoff ${\Lambda}=600$ MeV, cf. Ref.
Haidenbauer:2013oca  for details. The total $\chi^{2}$ for the NLO13 and NLO19
potentials is from a global fit to $36$ ${\Lambda}N$ and ${\Sigma}N$ data
points Haidenbauer:2019boi  while the $\chi^{2}$ for ${\Lambda}p$ includes
$12$ data points SechiZorn:1969hk ; Alexander:1969cx . In case of the
scenarios CSB1 and CSB3, the CSB contributions are just added to the NLO13 and
NLO19 potentials as published in Refs. Haidenbauer:2013oca  and
Haidenbauer:2019boi , respectively. However, for CSB2 the required CSB changes
the overall $YN$ results considerably and the total $\chi^{2}$ increases to
values around $40-50$. Here we had to re-fit the charge-symmetry conserving
part in order to achieve a description of the $YN$ data that is comparable to
those in the other scenarios, cf. the $\chi^{2}$ values in Table 2. After that
for CSB2 the scattering length of the singlet state produced by the charge-
symmetric part amounts to $a_{s}=-3.3$ fm, as compared to $-2.9$ fm for the
other two scenarios. We want to emphasize that the predicted binding energies
of the hypertriton remain practically unchanged for the different considered
scenarios for CSB. The variations are in the order of at most $30$ keV, and
thus remain well within the experimental uncertainty. This is expected for a
$T=0$ $\Lambda np$ state where the $\Lambda p$ and $\Lambda n$ interactions
are basically averaged.

Results for the principal scenario CSB1 are summarized in Table 3, based on
NLO13 and NLO19 and for the standard range of cutoffs $\Lambda=500-650$ MeV
Haidenbauer:2013oca ; Haidenbauer:2019boi . Finally, the actual values for the
short range CSB counter terms for that principal scenario are listed in Table
4. As one can see, those LECs are indeed much smaller than the ones of the
regular (charge-symmetry conserving) contact terms (cf. Table 3 in
Haidenbauer:2013oca ) and in line with the expectations from power counting,
see Sect. 2.2.

Table 4: The CSB LECs for the ${}^{1}S_{0}$ (s) and ${}^{3}S_{1}$ (t) partial waves, cf. Eq. (4), fixed from the present experimental splittings $\Delta E(0^{+})=233$ keV and $\Delta E(1^{+})=-83$ keV (CSB1).  $\Lambda$ | NLO13 | NLO19
---|---|---
|  $C_{s}^{CSB}$[MeV-2] |  $C_{t}^{CSB}$[MeV-2] |  $C_{s}^{CSB}$[MeV-2] |  $C_{t}^{CSB}$[MeV-2]
500 | 4.691×10-3 | -9.294×10-4 | 5.590×10-3 | -9.505×10-4
550 | 6.724×10-3 | -8.625×10-4 | 6.863×10-3 | -1.260×10-3
600 | 9.960×10-3 | -9.870×10-4 | 9.217×10-3 | -1.305×10-3
650 | 1.500×10-2 | -1.142×10-3 | 1.240×10-2 | -1.395×10-3

Let us start with the comparison of the various scenarios based on the
calculations for the cutoff $600$ MeV, listed in Table 2, and focus first on
the $\Lambda n$ scattering lengths, Obviously there is a sizable splitting
between the $\Lambda n$ and $\Lambda p$ results, depending on the CSB scenario
for $A=4$ hypernuclei. In particular, for CSB2 the ${\Lambda}n$ interaction
becomes significantly less attractive as compared to $\Lambda p$ and, in the
triplet state, also as compared to the case without CSB forces. In the other
two scenarios, the singlet interaction in $\Lambda n$ is more attractive than
that in $\Lambda p$. Furthermore, there are noticeably smaller changes for the
triplet ${\Lambda}n$ scattering length in those two scenarios. In particular,
for CSB1 the values for $\Lambda n$ and $\Lambda p$ are fairly close to that
without CSB.

Table 2 also provides the results of the full (non-perturbative) calculation
of the CSB splittings of the $0^{+}$ and $1^{+}$ states for $A=4$ hypernuclei
for all three CSB scenarios. In addition, the predictions for the original
$YN$ potentials, without any explicit CSB force, and for the case where only
the one-boson-exchange CSB contributions (CSB-OBE) (${\Lambda}-{\Sigma}^{0}$
mixing, $\eta-\pi^{0}$ mixing, $K^{\pm}/K^{0}$ exchange) are added. For CSB1
and CSB3, the CSB of the separation energy agrees within experimental
uncertainties with the values mentioned above. For CSB2, there are some
deviations to the pre-2014 situation. Given that this is an outdated scenario
anyway and that CSB2 required a complete refit of the $YN$ interaction, we
refrained from further improving the description of CSB. The obtained
splittings without CSB contact terms confirm the conclusion from earlier
studies Nogga:2001ef ; Haidenbauer:2007ra ; Nogga:2013pwa  that the standard
mechanisms can only explain a very small fraction of the experimentally found
CSB in $A=4$ hypernuclei. In particular, because of cancellations between the
OBE contributions, once $\eta-\pi^{0}$ mixing is treated properly Coon:1979si
, the overall results do not really improve when including those. In addition,
the large variation between the NLO13 and NLO19 results is a clear signal for
the missing CSB contact terms.

Now we analyze in more detail the results for scenario CSB1, the one which is
in line with the present experimental situation. Corresponding results are
summarized in Table 3. There is a clear and universal trend for a sizable
splitting between the ${\Lambda}p$ and ${\Lambda}n$ scattering length in the
singlet state, once we impose the reproduction of $\Delta E(0^{+})$ and
$\Delta E(1^{+})$. The splitting in the triplet state is much smaller and
actually goes into the opposite direction. In particular, for reproducing the
experimentally observed CSB splitting in the $A=4$ hypernuclei, in the
${}^{1}S_{0}$ state the ${\Lambda}n$ interaction is required to be more
attractive than for ${\Lambda}p$, whereas for ${}^{3}S_{1}$ the ${\Lambda}n$
interaction is slightly less attractive than that for ${\Lambda}p$.

With regard to the $\Lambda n$ scattering lengths the results for the singlet
channel are quite robust. The predictions are in the narrow range of $-3.2$ to
$-3.3$ fm and practically independent on the cutoff and whether NLO13 or NLO19
is used. There is more variation in case of the triplet state which, however,
is simply a reflection of the situation observed already in the calculation
without CSB forces. One very interesting aspect is that, adding the CSB
interaction to our NLO potentials established in Refs. Haidenbauer:2013oca ;
Haidenbauer:2019boi , improves also the overall description of the $\Lambda p$
data as quantified by the $\chi^{2}$ value – without any refit, see Table 2.
It is due to the noticeable reduction of the strength of the $\Lambda p$
interaction in the singlet channel by the needed CSB force, cf. the pertinent
scattering lengths in the table. In fact, one could interpret this as sign for
a consistency of the available $\Lambda p$ data with the present values of the
CSB level splittings in the $A=4$ hypernuclei. In this context we want to
mention that a recent measurement of the $\Lambda p$ momentum correlation
function in $pp$ collisions at $13$ TeV Acharya:2021fdf  likewise indicates
that a slightly less attractive $\Lambda p$ interaction is favored by the
data.

Finally, note that $\Delta a_{1S0}^{CSB}\equiv a_{{\Lambda}p}-a_{{\Lambda}n}$
is $\approx 0.62\pm 0.08$ fm for the ${}^{1}S_{0}$ partial wave, which is
comparable to but noticeably smaller than the CSB effects in the $pp$ and $nn$
scattering lengths where it amounts to $\Delta a^{CSB}=a_{pp}-a_{nn}=1.5\pm
0.5$ fm Miller:2006tv . On the other hand, in case of the triplet state, the
prediction is with $\Delta a_{3S1}^{CSB}\approx-0.10\pm 0.02$ fm significantly
smaller and of opposite sign.

###  4.2 Relation of CSB in the separation energies to the expectation values

Table 5:  ${}^{3}_{\Lambda}$H, ${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda}$H separation energies for scenario CSB1 for NLO13 and NLO19 for various cutoffs in combination with SMS N4LO+ (450) Reinert:2018ip  and different different cutoffs for NLO13 and/or NLO19. Energies are in MeV.  interaction | $E_{\Lambda}(^{3}_{\Lambda}{\rm H})$ | $E_{\Lambda}(^{4}_{\Lambda}{\rm He})$ | $E_{\Lambda}(^{4}_{\Lambda}{\rm H})$
---|---|---|---
|  | $J^{\pi}=0^{+}$ | $J^{\pi}=1^{+}$ | $J^{\pi}=0^{+}$ | $J^{\pi}=1^{+}$
NLO13(500) | 0.14 | 1.82 | 0.76 | 1.56 | 0.82
NLO13(550) | 0.10 | 1.62 | 0.56 | 1.36 | 0.61
NLO13(600) | 0.09 | 1.59 | 0.55 | 1.34 | 0.60
NLO13(650) | 0.09 | 1.61 | 0.59 | 1.36 | 0.64
NLO19(500) | 0.10 | 1.77 | 1.19 | 1.52 | 1.27
NLO19(550) | 0.10 | 1.67 | 1.21 | 1.42 | 1.28
NLO19(600) | 0.09 | 1.58 | 1.03 | 1.34 | 1.09
NLO19(650) | 0.10 | 1.65 | 0.89 | 1.40 | 0.96
Expt. |  $0.13(5)$ Juric:1973zq  | 2.39(3)Yamamoto:2015avw  | 0.98(3)Yamamoto:2015avw  | 2.16(8)Schulz:2016dt  | 1.07(8) Schulz:2016dt

This section is a short summary of the relation between the expectation values
obtained and the CSB in the separation energies of
${}^{4}_{\Lambda}$He-${}^{4}_{\Lambda}$H for CSB1. We start with a summary of
the $A=3$ and $A=4$ results for the separation energies in Table 5. These
results are qualitatively quite similar to the ones of Table 1. We will
discuss the CSB contributions for $A=4$ in more detail below. Here, we just
remark that we observed small changes of the separation energy of
${}^{3}_{\Lambda}$H of the order of 3 keV since we take isospin $T=1$ and
$T=2$ components into account. The effect of CSB is calculated here
perturbatively based on the wave function for ${}^{4}_{\Lambda}$He for the
orginal interaction. The results are given as differences of expectation
values

$\langle H\rangle_{{\rm CSB}}\equiv\langle H\rangle_{{}^{4}_{\Lambda}{\rm
He}}-\langle H\rangle_{{}^{4}_{\Lambda}{\rm H}}\ .$ (9)

For the separation energies, one therefore finds

$\displaystyle\Delta E_{\Lambda}$ $\displaystyle=$ $\displaystyle
E_{\Lambda}(^{4}_{\Lambda}{\rm He})-E_{\Lambda}(^{4}_{\Lambda}{\rm H})$ (10)
$\displaystyle=$ $\displaystyle E(^{3}{\rm He})-E(^{3}{\rm
H})-\left(\,E(^{4}_{\Lambda}{\rm He})-E(^{4}_{\Lambda}{\rm H})\,\right)\ \ ,$

where then the last contribution is approximated by the expectation values. We
separate

$\displaystyle E(^{4}_{\Lambda}{\rm He})-E(^{4}_{\Lambda}{\rm
H})\approx\langle H\rangle_{{\rm CSB}}$ $\displaystyle\qquad=\langle
T\rangle_{{\rm CSB}}+\langle V_{NN}\rangle_{{\rm CSB}}+\langle
V_{YN}\rangle_{{\rm CSB}}\ \ .$ (11)

It turns out that $E(^{3}{\rm He})-E(^{3}{\rm H})$ is similar to $\langle
V_{NN}\rangle_{{\rm CSB}}$. Therefore, the contributions largely cancel. For
example, one finds for the SMS $NN$ interaction at order N4LO+ Reinert:2018ip
with cutoff $450$ MeV $E(^{3}{\rm He})-E(^{3}{\rm H})=751$ keV. Based on the
scenario CSB1 for NLO19 (600), $\langle V_{NN}\rangle_{{\rm CSB}}=740$ keV.
Thereby, $E(^{3}{\rm He})-E(^{3}{\rm H})$ contains approximately $10$ keV due
to the mass difference of proton and neutron. The remaining $741$ keV are
mostly due to the point proton Coulomb interaction but also include CSB of the
$NN$ interaction. We note that this combined effect in most cases contribute
positively to the CSB whereas the Coulomb interaction alone is expected to
give a negative contribution Bodmer:1985km ; Nogga:2001ef .

Table 6:  Perturbative estimate of different contributions to the CSB of ${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda}$H for the $0^{+}$ state based on ${}^{4}_{\Lambda}$He wave functions for scenario CSB1. The SMS N4LO+ (450) $NN$ interaction Reinert:2018ip  was used in all cases. The contributions of the kinetic energy $\langle T\rangle_{{\rm CSB}}$, the $YN$ interaction $\langle V_{YN}\rangle_{{\rm CSB}}$ and the contribution of the nuclear core $V_{NN}^{\rm CSB}=\langle V_{NN}\rangle_{{\rm CSB}}-E(^{3}{\rm He})+E(^{3}{\rm H})$ are separated and combined to the total CSB $\Delta E_{\Lambda}^{pert}$. The direct comparison of separation energies for full calculations of ${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda}$H, $\Delta E_{\Lambda}$, is also given. All energies are in keV.  interaction | $\langle T\rangle_{{\rm CSB}}$ | $\langle V_{YN}\rangle_{{\rm CSB}}$ | $V_{NN}^{\rm CSB}$ | $\Delta E_{\Lambda}^{pert}$ | $\Delta E_{\Lambda}$
---|---|---|---|---|---
NLO13(500) | 44 | 200 | 16 | 261 | 265
NLO13(550) | 46 | 191 | 20 | 257 | 261
NLO13(600) | 44 | 187 | 20 | 252 | 256
NLO13(650) | 38 | 189 | 18 | 245 | 249
NLO19(500) | 14 | 224 | 5 | 243 | 249
NLO19(550) | 14 | 226 | 7 | 247 | 252
NLO19(600) | 22 | 204 | 12 | 238 | 243
NLO19(650) | 26 | 207 | 12 | 245 | 250
Table 7:  Perturbative estimate of different contributions to the CSB of ${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda}$H for the $1^{+}$ state based on ${}^{4}_{\Lambda}$He wave functions for scenario CSB1. Same interactions and notations as in Table 6.  interaction | $\langle T\rangle_{{\rm CSB}}$ | $\langle V_{YN}\rangle_{{\rm CSB}}$ | $V_{NN}^{\rm CSB}$ | $\Delta E_{\Lambda}^{pert}$ | $\Delta E_{\Lambda}$
---|---|---|---|---|---
NLO13(500) | 5 | -90 | 15 | -71 | -66
NLO13(550) | 5 | -86 | 18 | -63 | -56
NLO13(600) | 4 | -83 | 19 | -59 | -53
NLO13(650) | 3 | -80 | 17 | -59 | -55
NLO19(500) | 1 | -84 | 3 | -80 | -75
NLO19(550) | 2 | -81 | 2 | -77 | -72
NLO19(600) | 4 | -82 | 6 | -71 | -67
NLO19(650) | 4 | -79 | 9 | -66 | -69

In Tables 6 and 7, we summarize the different contributions to CSB for the
scenario CSB1 for different $YN$ potentials. It can be seen that the
contribution of the kinetic energy (mostly due to the mass difference within
the $\Sigma$ multiplet) is strongly dependent on the chosen interaction. After
properly including the CSB LECs, the $YN$ potential provides the by far
largest contribution to the CSB. The total CSB is by construction fairly
independent of the $YN$ interaction. The comparison of the perturbative
estimate to the direct result for the CSB $\Delta E_{\Lambda}$ shows that both
calculations agree well with each other. We note that this is also so because
we chose ${}^{4}_{\Lambda}$He wave functions for the evaluation of the
expectation values. Results for ${}^{4}_{\Lambda}$H reproduce the full
calculation with slightly lower accuracy.

As already seen in Table 3, also the predictions for the $\Lambda p$ and
$\Lambda n$ scattering lengths are largely independent of the interaction. The
latter property is not trivial and suggests that the CSB of the scattering
lengths can be indeed determined using $A=4$ data.

###  4.3 The prescription of CSB employed by Gazda and Gal

In the calculations by Gazda and Gal Gazda:2015qyt ; Gazda:2016ir , a global
prescription for the CSB potential was employed, suggested in Ref. Gal:2015hh
:

$\langle
N{\Lambda}|V^{CSB}_{{\Lambda}N}|N{\Lambda}\rangle=-0.0297\tau_{N}\frac{1}{\sqrt{3}}\langle
N{\Sigma}|V|N{\Lambda}\rangle\ .$ (12)

Obviously here the CSB contribution to the ${\Lambda}N$ interaction is equated
with the full ${\Lambda}N\to{\Sigma}N$ transition potential, appropriately
scaled with the ${\Lambda}-{\Sigma}^{0}$ mixing matrix element. Indeed, this
is appropriate for isovector mesons whose direct contribution to the
${\Lambda}N$ potential is forbidden when isospin is conserved, but becomes
non-zero via $\Lambda-\Sigma^{0}$ mixing. For example, in the Nijmegen $YN$
potentials such as NSC97 Rijken:1999fc  this aspect is implemented and
${\Lambda}-{\Sigma}^{0}$ mixing is taken into account for all isovector mesons
($\pi$, $\rho$, $a_{0}$, $a_{2}$) that contribute to the
${\Lambda}N\to{\Sigma}N$ transition potential. However, with such a global
prescription CSB contributions are also attributed to $K$ and/or $K^{*}$
exchange - though strange mesons contribute anyway directly to the charge-
symmetry conserving part of the ${\Lambda}N$ potential. If there is CSB from
say $K$ exchange it should arise directly from the ${\Lambda}N\to{\Lambda}N$
potential, cf. Sect. 2.

Apart from that, the prescription Eq. (12) dismisses other CSB contributions
not related to ${\Lambda}-{\Sigma}^{0}$ mixing, e.g. the ones from
$\eta-\pi^{0}$, $\rho^{0}-\omega$, etc. mixing. According to the literature,
$\rho^{0}-\omega$ mixing provides the main contribution to the CSB observed
between the $pp$ and $nn$ systems Miller:2006tv . Though pertinent studies for
${}^{\,4}_{\Lambda}\rm He$ and ${}^{\,4}_{\Lambda}\rm H$ are inconclusive
Coon:1979si ; Coon:1998jd , they certainly reveal a strong model dependence.
This is a clear signal that corresponding contact terms representing short-
range physics should be and have to be included. In fact, also in case of
Gazda and Gal, the result for the CSB splitting of the $0^{+}$ state based on
the LO interaction exhibits a large cutoff dependence Gazda:2016ir . For the
$1^{+}$ state, the energy levels themselves show a sizable cutoff dependence,
while the CSB splitting itself appears to be too large as compared to the
present experimental value. In order to quantify the uncertainty of the
prescription in Eq. (12) by ourselves, we have performed calculations for the
$0^{+}$ state using the LO, NLO13 and NLO19 interactions. We reproduce the CSB
of Ref. Gazda:2016ir  for LO fairly well ($36$-$309$ keV). Using the same
prescription, we find $461$-$2266$ keV for NLO13 and $86$-$458$ keV for NLO19.
Evidently, the results at NLO are likewise strongly cutoff dependent and there
are also strong variations between the two variants NLO13 and NLO19, which
otherwise yield practically identical results for ${\Lambda}N$ and ${\Sigma}N$
scattering. The discussion in Section 4 of Ref. Gazda:2016ir  suggests that
the cutoff dependence and the interplay between contact terms and
(unregularized) pion exchange was indeed a concern for the authors and an
attempt was made to stabilize the outcome within an ad hoc procedure. However,
adding genuine CSB contact terms as done in the present work is the
appropriate remedy – and anyway required in a consistent application of chiral
EFT. Finally, we note that this again stresses that the strength of the
$\Lambda N$-$\Sigma N$ transition potential is not observable and intimately
correlated with consistently defined three-baryon interactions
Haidenbauer:2019boi ; Wirth:2016iwn .

##  5 Conclusions

In the present work, we have studied effects from CSB in the $YN$ interaction.
Specifically, we have utilized the experimentally known difference of the
$\Lambda$ separation energies in the mirror nuclei ${{}^{4}_{\Lambda}\rm He}$
and ${{}^{4}_{\Lambda}\rm H}$ to constrain the $\Lambda$-neutron interaction.
For that purpose, we derived the contributions of the leading CSB interaction
within chiral effective field theory and added them to our NLO chiral hyperon-
nucleon interactions Haidenbauer:2013oca ; Haidenbauer:2019boi . CSB
contributions arise from a non-zero ${\Lambda}{\Lambda}\pi$ coupling constant
which is estimated from $\Lambda-\Sigma^{0}$ 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 are 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 hyperon-nucleon scattering data, the hypertriton and the CSB in
${{}^{4}_{\Lambda}\rm He}$ and ${{}^{4}_{\Lambda}\rm H}$ accurately.

It turned out that the reproduction of the presently established splittings of
$\Delta E(0^{+})=233\pm 92$ keV and $\Delta E(1^{+})=-83\pm 94$ keV requires a
sizable difference between the strength 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 ${\Lambda}n$, in case of ${}^{3}S_{1}$
it is slightly more attractive. In terms of the pertinent scattering lengths
we predict for $\Delta a^{CSB}=a_{{\Lambda}p}-a_{{\Lambda}n}$ a value of
$0.62\pm 0.08$ fm for the ${}^{1}S_{0}$ partial wave and $-0.10\pm 0.02$ fm
for ${}^{3}S_{1}$.

The required CSB implies a significantly stronger ${\Lambda}n$ interaction in
the ${}^{1}S_{0}$ partial wave and the pertinent scattering length of our NLO
potentials Haidenbauer:2013oca ; Haidenbauer:2019boi  increases from $-2.9$ fm
to around $a^{\Lambda n}_{s}=-3.2$ fm. Therefore, it is worthwhile to explore
in how far this has consequences for the possible existence of ${\Lambda}nn$
resonances Garcilazo:2014lva ; Gal:2014efa ; Hiyama:2014cua ; Richard:2014pwa
; Ando:2015fsa ; Afnan:2015ahc ; Kamada:2016ozg ; Gibson:2019occ ;
Gibson:2020npg ; Schafer:2020vzl ; Schafer:2020jnb . It will be also
interesting to utilize the CSB forces established in the present work in
calculations of $p$-shell hypernuclei. Indeed, there are several
experimentally established mirror hypernuclei in the $p$-shell region with
$A=7-10$ Botta:2017eg ; Davis:2005npa ; Botta:2019has  which have been already
studied within phenomenological approaches in the past Hiyama:2009ki ;
Hiyama:2012sq ; Hiyama:2013owa . However, now those systems can be also
explored by more systematic microscopic approaches such as ab-initio
calculations based on the no-core shell model Wirth:2017bpw ; Wirth:2019cpp ;
Le:2020zdu .

Interestingly, adding the CSB interaction to our NLO potentials established in
Refs. Haidenbauer:2013oca ; Haidenbauer:2019boi , improves slightly the
overall description of the $YN$ data as quantified by the $\chi^{2}$ value –
without any refit. This could be interpreted as sign for a consistency of the
available $YN$ data with the present values of the CSB level splittings in the
$A=4$ hypernuclei. In any case, with regard to the latter new but still
preliminary results by FINUDA for the ground-state binding energy of
${}^{\,4}_{\Lambda}\rm H$ have been reported already Botta:2019has  and
further measurements of the splitting of the $0^{+}$ and $1^{+}$ states of
${}^{\,4}_{\Lambda}\rm H$ are planned at J-PARC Akazawa:2016 ;
Yamamoto:2019dcf .

###### Acknowledgements.

We acknowledge stimulating discussions with Avraham Gal and Josef Pochodzalla.
This work is 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 grant. no. TRR 110) and the VolkswagenStiftung (grant no. 93562). 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.

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