\section{Introduction}
\label{sec:intro}

The prospect of an ample production of baryons with charm offered by
facilities such as the LHC at CERN
\cite{Acharya:2018,Aaij:2018,Sirunyan:2019},
RHIC at BNL \cite{Adam:2019},
J-PARC and KEK in Japan
\cite{Noumi:2017,Niiyama:2018},
or FAIR in Germany
\cite{PANDA,Wiedner:2011,Friman:2011}
has led to a renewed interest in the in-medium properties of such
baryons \cite{Ohtani:2017,Carames:2018,Tsushima:2018,Yasui:2018}
and also in the question whether they, and notably the lightest
charmed baryon, the ${\Lambda}_c$(2286), could form bound states with ordinary matter
\cite{Liu:2012,Garcilazo:2015,Maeda:2016,vidana19,Wu:2020,Kopeliovich:2020}.
In fact, there is a long history of speculations
about possible bound systems involving the ${\Lambda}_c$  \cite{tyapkin75,Dover:1977,dover77b,iwao77,gatto78,bhamathi81,kolesnikov81,Bando:1982,bando83,Gibson:1983,Bando:1985,
bhamathi89,tsushima03b,tsushima04,tan04,Kopeliovich:2007}
that started soon after the discovery of charmed baryons \cite{cazzoli75,knapp76}
(see also the recent reviews \cite{Hosaka:2016,Krein:2017}). In
principle, charmed nuclei could be produced by means of {\it charm exchange}
or {\it associate charm production} reactions \cite{bressani89,bunyatov91},
in analogy to the ones widely used in hypernuclear physics. However, the experimental production of charmed nuclei is difficult due to the short lifetimes of $D$-meson beams which makes it necessary to place the target as close as possible to the $D$-meson production point, and due to the kinematics of the reactions: the charmed particles are formed with large momentum making their capture by a target-nucleus improbable. Because of these difficulties,
up to now, only three albeit controversial candidates have been reported
by an emulsion experiment carried out in Dubna in the mid-1970s \cite{batusov81a,batusov81b,batusov81d,lyukov89}.

The ${\Lambda}_c N$ forces employed in the past investigations
were predominantly derived within the meson-exchange framework,
see Refs.~\cite{Liu:2012,vidana19}
for recent examples, often utilizing SU(4) flavor symmetry in one
form or the other. Lately, also the constituent quark model \cite{Garcilazo:2019}
or a combination of meson-exchange and quark model \cite{Maeda:2016}
have been considered. Independently of that, in general, the resulting
potentials turned out to be fairly attractive.
 %
Interestingly, a rather different picture emerged from recent lattice
QCD (LQCD) simulations by the HAL QCD collaboration \cite{miyamoto18,Miyamoto:2018}.
Those studies, based on unphysical quark masses corresponding to pion masses
of $m_\pi = 410 - 700$ MeV, suggest that the ${\Lambda}_c N$ and ${\Sigma}_c N$ interactions
could be significantly less attractive than what had been proposed in the
phenomenological studies mentioned above.
In Ref.~\cite{Haidenbauer:2018}, an extrapolation of the HAL QCD results
to the physical point was presented, using chiral effective
field theory (EFT) \cite{Epelbaum:2003,Petschauer:2013} as guideline.
It revealed that the ${\Lambda}_c N$ interaction at the physical point is
expected to be somewhat stronger than for large pion masses, however,
still only moderately attractive and, specifically, considerably
less attractive than most of the phenomenological predictions for the
${\Lambda}_c N$ interaction.

In the present work, we use the ${\Lambda}_c N$ interaction from
Ref.~\cite{Haidenbauer:2018} to explore the binding energies of
charmed nuclei. In the literature different conventions for naming
${\Lambda}_c$ nuclei have been used in the past.
We adopt here the standard nomenclature
for nuclei with the proper generalization~\cite{Gibson:1983},
spelled out for hypernuclei in Sect. I.B of Ref.~\cite{Gal:2016}.
It takes into account that the characterizing letter(s) for the
nucleus depends on its {total} charge {and not just} on the number of protons.
For example, when adding the (uncharged) ${\Lambda}$ to hydrogen ($^2$H) one
gets the hypertriton ($^3_{{\Lambda}}$H) but adding the positively charged
${\Lambda}_c$ leads to $^{\ 3}_{{\Lambda}_c}$He.
Similarly, based on this convention, the bound state of $\Lambda_c$
and $^{208}$Pb is $^{209}_{{\Lambda}_c}$Bi.

The lightest nuclei considered, {the $A=3$ and $A=4$ systems
$^{\ 3}_{{\Lambda}_c}$He and $^{\ 4}_{{\Lambda}_c}$He }, are investigated by
solving corresponding Faddeev-Yakubovsky equations.
Indeed, for the three-body system, bound states have been reported in
Ref.~\cite{Maeda:2016} (with total angular momentum $J=1/2$ and $3/2$)
and Ref.~\cite{Garcilazo:2015} (for $J=3/2$). Note, however, that some of
the ${\Lambda}_c N$ interactions employed in the former works are so strongly
attractive that they even predict two-body bound states (in the
$^1S_0$ as well as in the $^3S_1$ partial wave) with binding
energies comparable to that of the deuteron.
For calculating heavier charmed nuclei, namely from $^{\ 5}_{\Lambda_c}$Li
onward to $^{209}_{\Lambda_c}$Bi, a perturbative many-body
approach is utilized that allows one to obtain the $\Lambda_c$
single-particle bound states in the
different nuclei from the corresponding $\Lambda_c$ self-energy.
Results for charmed nuclei computed within this framework have been
reported recently in Ref.~\cite{vidana19}, for a set of $Y_c N$
interactions deduced from an early version \cite{Reuber:1994} of the {hyperon-nucleon ($YN$)}
meson-exchange potential of the J\"ulich Group \cite{Haidenbauer:2005}
via SU(4) symmetry arguments. For these interactions,
the $^{\ 5}_{\Lambda_c}$Li system
{($^{\ 5}_{\Lambda_c}$He in the nomenclature used in \cite{vidana19})} turned out to be already bound.
In this context, let us mention that the HAL QCD collaboration has
likewise reported results for charmed nuclei~\cite{miyamoto18}.
The calculations were performed with the ${\Lambda}_c N$ potentials extracted
from the lattice simulations
at pion masses $410$-$700$ MeV, but using the physical masses of the ${\Lambda}_c$ and
the considered nuclei. Binding energies for ${\Lambda}_c$ bound to
$^{12}$C, $^{28}$Si, $^{40}$Ca, $^{58}$Na, $^{90}$Zr, and $^{208}$Pb were reported.

Since after the publication of the ${\Lambda}_c N$ interaction \cite{Haidenbauer:2018} new results from the HAL QCD collaboration became available that
include now the ${\Sigma}_c N$ channel \cite{Miyamoto:2018}, we also
revisit the ${\Lambda}_c N$ interaction in order to explore in how far the
inclusion of a direct interaction in the ${\Sigma}_c N$ channel modifies
the extrapolation of the ${\Lambda}_c N$ interaction from the results/masses
of the lattice simulations to the physical point, and in how far
it influences the predictions for charmed nuclei.
It turns out that adding/considering the interaction in the ${\Sigma}_c N$ channel
has very little influence on the ${\Lambda}_c N$ {scattering} results and also not on those for
${\Lambda}_c$ nuclei. May be this is not too surprising in view of the fact that the
thresholds of the two channels are separated by almost $170$ MeV. In any
case, for completeness, we report predictions for the ${\Sigma}_c N$ $S$-wave phase
shifts based on our extrapolation of the HAL QCD results \cite{Miyamoto:2018}.

Another extrapolation of the lattice results for ${\Sigma}_c N$,
performed in heavy baryon chiral perturbation theory and
taking into account heavy quark spin symmetry, has been
performed recently~\cite{Meng:2019}. Let us mention already now that
neither in that work nor in our calculation any signal for resonances
or bound states in the ${\Sigma}_c N$ channel are found.
The existence of such resonances has been suggested
in Ref.~\cite{Huang:2013}, where the ${\Lambda}_c N$-${\Sigma}_c N$-${\Sigma}_c^* N$
interaction was investigated within the framework of the quark
delocalization color screening model.
Bound states (and resonances) in the ${\Sigma}_c N$ $^3S_1$ partial wave
around the ${\Sigma}_c N$ and ${\Sigma}_c^* N$ thresholds were also predicted
in Ref.~\cite{Maeda:2018} based on a $Y_c N$ interaction
from meson exchange supplemented by short-range repulsion
from a quark exchange model.

The paper is structured in the following way: A summary of the main characteristics of the $\Lambda_c N$ and $\Sigma_c N$ potentials is presented in Sec.~\ref{sec:formalism} . Results of the properties of $\Lambda_c$ in infinite nuclear matter, and light and heavier nuclei are reported in Sec.~\ref{sec:nuc_prop}. Finally, a brief summary and some concluding
remarks are given in Sec.~\ref{sec:concl}.  The appendix summarizes results for ${\Sigma}_c N$ scattering.

\section{The ${\Lambda}_c N$ and ${\Sigma}_c N$ potentials}
\label{sec:formalism}

The ${\Lambda}_c N$ and ${\Sigma}_c N$ interactions are constructed by using chiral EFT
as guideline, following the scheme employed in our studies of the
$\Lambda N$ and $\Sigma N$ systems
\cite{Polinder,Haidenbauer:2013,Haidenbauer:2019,Petschauer:2020}.
We summarize the essentials below. More details of the approach can be found in
Ref.~\cite{Haidenbauer:2018}.
The $Y_c N$ potential consists of contact terms and contributions from pion exchange.
The former are given by
\begin{eqnarray}
V(^1S_0) = {\tilde{C}}_{^1S_0} + \tilde{D}_{^1S_0} m^2_\pi + ({C}_{^1S_0} + {D}_{^1S_0} m^2_\pi)\, ({p}^2+{p}'^2), &&\nonumber \\
V(^3S_1) = {\tilde{C}}_{^3S_1} + \tilde{D}_{^3S_1} m^2_\pi + ({C}_{^3S_1} + {D}_{^3S_1} m^2_\pi)\, ({p}^2+{p}'^2), &&\nonumber \\
V(^3D_1 -\, ^3S_1) = {C}_{\varepsilon_1}\, {p'}^2, \hskip 4.9cm && \nonumber \\
V(^3S_1 -\, ^3D_1) = {C}_{\varepsilon_1}\, {p}^2, \hskip 5.0cm &&
\label{LEC}
\end{eqnarray}
for the partial waves considered in the present study.
Here $p = |{\bf p}\,|$ and ${p}' = |{\bf p}\,'|$  are the initial and final
center-of-mass (c.m.) momenta in the ${\Lambda}_c N$ or ${\Sigma}_c N$ systems.
The quantities ${\tilde{C}}_{i}$, ${\tilde{D}}_{i}$, ${C}_{i}$, ${D}_{i}$ are
low-energy constants (LECs) that are fixed by a fit to lattice data (phase shifts) by
the HAL QCD collaboration at $m_\pi=410$ MeV and $570$ MeV.
The $m_\pi$ dependence in Eq.~(\ref{LEC}) is motivated by the
corresponding expression in the standard
Weinberg counting up to next-to-leading order (NLO) \cite{Epelbaum:2003,Petschauer:2013} but
differs from it by the term proportional
to $m^2_\pi\, ({p}^2+{p}'^2)$ which is nominally of higher order. Nevertheless,
we included that term in Ref.~\cite{Haidenbauer:2018} because it allowed us to obtain
an optimal description of the LQCD results at $m_\pi=410$ MeV as well as $570$ MeV.
We consider this as a prerequisite for a well constrained extrapolation to lower pion masses.
Without such a term the phase shifts by HAL QCD for $m_\pi=410$ MeV would be underestimated
at low energies, as exemplified by results shown in Figs.~6 and 7 of Ref.~\cite{Meng:2019}.

The contribution of pion exchange to the ${Y}_c N$ potential is given by
\begin{equation}
V^{OPE}_{{Y_c N\to Y_c N}} =-f_{{Y_c Y_c\pi}}f_{{NN\pi}}
\frac{\left({\bf \sigma}_1\cdot {\bf q} \right)
\left({\bf \sigma}_2\cdot {\bf q} \right)}{ {\bf q}^{\,2}+m_{\pi}^2} \ ,
\label{OPE}
\end{equation}
where ${\bf q}$ is the transferred momentum, ${\bf q} = {\bf p'} - {\bf p}$.
The coupling constants $f_{BB'\pi}$ are related to the axial-vector strength
via $f_{BB'\pi} = g_A^{BB'}/2F_\pi$ with $F_\pi$ being the pion decay constant ($F_\pi\approx 93$~MeV).
The coupling constant for the ${\Lambda}_c{\Sigma}_c\pi$ vertex can be
determined from the experimentally known ${\Sigma}_c \to {\Lambda}_c\pi$ decay rate, see
Refs.~\cite{Albertus:2005,Can:2016}. For the ${\Sigma}_c\Sc\pi$ coupling constant
lattice QCD results \cite{Alexandrou:2016} are employed.
To be concrete, $g_A^{{\Sigma}_c\Sc}=0.71$ \cite{Alexandrou:2016} and
$g_A^{{\Lambda}_c{\Sigma}_c}=0.74$ \cite{Albertus:2005,Can:2016} are used, together
with $g_A^{NN}=1.27$ \cite{PDG}.
Note that $f_{{\Lambda}_c\Lc\pi} \equiv 0$ under the assumption that isospin is
conserved.

Besides the coupling constants at the physical point, one needs also
their $m_\pi$ dependence:
\begin{equation}
f_{BB'\pi} (m^2_\pi) = \frac{g^{BB'}(m^2_\pi)}{2\,F_\pi (m^2_\pi) } \ .
\label{fpi}
\end{equation}
Results for the dependence of $F_\pi$ on $m^2_\pi$ are
available from lattice simulations \cite{Durr:2013}.
Based on that work the values
$F_\pi \approx 112$ MeV at $m_\pi = 410$ MeV and
$F_\pi \approx 129$ MeV at $m_\pi = 570$ MeV
were deduced and employed in Ref.~\cite{Haidenbauer:2018}.
With regard to the dependence of $g^{BB}_A$ on $m^2_\pi$
lattice QCD simulations indicate a rather small variation,
at least for $g^{{\Sigma}_c\Sc}_A$ and $g^{NN}_A$ where concrete
results are available \cite{Alexandrou:2014,Alexandrou:2016}.
Because of that the dependence of the $g_A$'s on $m_\pi$ was neglected
in Ref.~\cite{Haidenbauer:2018} and the values at the
physical point were used throughout.

There was no information from LQCD on the ${\Sigma}_c N$ interaction
at the time when the study in Ref.~\cite{Haidenbauer:2018} was performed,
and, thus, the interaction in the ${\Sigma}_c N$ channel was not considered.
Nonetheless, the coupling of ${\Lambda}_c N$ to ${\Sigma}_c N$ via pion exchange was already
included. Due to its long-range nature, this coupling plays an important role
in case of the $\Lambda N$ and $\Sigma N$ systems
\cite{Haidenbauer:2013,Haidenbauer:2019}.
Since $f_{{\Lambda}_c{\Sigma}_c\pi}$ is empirically known and only slightly smaller than
$f_{{\Lambda}{\Sigma}\pi}$ \cite{Haidenbauer:2013} the coupling between ${\Lambda}_c N$
and ${\Sigma}_c N$ via pion exchange should still be of relevance.
Indeed, the effective contribution to the ${\Lambda}_c N$ interaction,
$V_{{\Lambda}_c N \to {\Lambda}_c N} \sim V^{OPE}_{{\Lambda}_c N \to {\Sigma}_c N} \, G_{{\Sigma}_c N}\, V^{OPE}_{{\Sigma}_c N \to {\Lambda}_c N}$,
is expected to be smaller for energies around the ${\Lambda}_c N$
threshold as compared to the situation for ${\Lambda} N$, but just by a factor
$2-3$. The reduction is due to the larger
mass difference, $M_{{\Sigma}_c}-M_{{\Lambda}_c} \approx 167$ MeV versus $M_{\Sigma}-M_{\Lambda} \approx 78$ MeV,
that enters in the corresponding Green's functions
$G_{{\Sigma}_c N}$ or $G_{{\Sigma} N}$.
In any case, formally two-pion exchange contributions to the
${\Lambda}_c N$ interaction involving the ${\Sigma}_c$ do arise at NLO
\cite{Haidenbauer:2013}, and it can be expected that the piece with
a ${\Sigma}_c N$ intermediate state provides the dominant contribution \cite{Beane:2005}.
In the actual calculation, this (reducible) two-pion exchange contribution
to the ${\Lambda}_c N$ potential is generated by solving a coupled-channel
Lippmann-Schwinger {(LS) equation} (see below).
Further (irreducible) NLO contributions from two-pion exchange \cite{Haidenbauer:2013}
have been omitted in \cite{Haidenbauer:2018} for simplicity
reasons. It was assumed that those can be effectively absorbed
into the contact terms.

For the present study, we add a direct ${\Sigma}_c N$ interaction. The potential
is determined in the same way as the one for ${\Lambda}_c N$, by using results of
the HAL QCD collaboration for the corresponding $^3S_1$ phase shift at
$m_\pi=410$ MeV and $570$ MeV \cite{Miyamoto:2018}.
The main goal of this extension is to explore in how far the inclusion of a
direct ${\Sigma}_c N$ interaction modifies the ${\Lambda}_c N$ results achieved earlier.
Of particular interest is the question, whether it influences the results
for charmed nuclei that we are concerned with here.
Results for ${\Sigma}_c N$ scattering itself are discussed and summarized in the appendix.

The reaction amplitudes are obtained from the solution of a coupled-channel
LS equation for the interaction potentials.
After partial-wave projection \cite{Polinder}, the equation is given by
\begin{eqnarray}
T^{\nu'\nu,J}_{\rho'\rho}(p',p;\sqrt{s})&=&V^{\nu'\nu,J}_{\rho'\rho}(p',p) \nonumber\\
&+& \sum_{{\rho''},\nu''}\int_0^\infty \frac{dp''p''^2}{(2\pi)^3} \,
V^{\nu'\nu'',J}_{\rho'\rho''}(p',p'') \nonumber \\
&\times& \frac{2\mu_{\rho''}}{p^2_{\rho''}-p''^2+i\eta}
T^{\nu''\nu,J}_{\rho''\rho}(p'',p;\sqrt{s}) \ ,
\label{LS}
\end{eqnarray}
where the label $\rho$ indicates the channels (${\Lambda}_c N$, ${\Sigma}_c N$) and the label $\nu$
the partial wave. The quantity $\mu_\rho$ signifies the pertinent reduced mass.
The on-shell momentum in the intermediate state, $p_{\rho}$, is defined by
$\sqrt{s}=\sqrt{M^2_{B_{1,\rho}}+p_{\rho}^2}+\sqrt{M^2_{B_{2,\rho}}+p_{\rho}^2}$.

Since the integral in the LS equation (\ref{LS}) is divergent for the chiral
potentials specified above, a regularization scheme needs to be
introduced \cite{Epelbaum:2008,Machleidt:2011}.
For that purpose the potentials in the LS equation are cut off in
momentum space by multiplication with a regulator function
\cite{Polinder,Haidenbauer:2013},
$f(p',p) = \exp\left[-\left(p'^4+p^4\right)/\Lambda^4\right]$.
In Ref.~\cite{Haidenbauer:2018}, cutoff values $\Lambda =500$--$600$ MeV were
employed, in line with the range that yielded the best results in NLO studies of
the ${\Lambda} N$ and ${\Sigma} N$ interactions \cite{Haidenbauer:2013,Haidenbauer:2019}.
The variations of the results with the cutoff, reflecting uncertainties due to
the regularization, will be indicated by bands.

The baryon masses corresponding to the LQCD simulations at $m_\pi= 410$ and
$570$~MeV are taken from Ref.~\cite{miyamoto18}.
For the calculation at the physical point, we use the masses
$M_{N} = 938.92$ MeV,
$M_{{\Lambda}_c} = 2286.5$ MeV, and $M_{{\Sigma}_c} = 2455$ MeV.

\section{${\Lambda}_c$ nuclei and matter properties}
\label{sec:nuc_prop}

In this section, we report results on the properties of the ${\Lambda}_c$
in infinite nuclear matter and on charmed nuclei.
In the pertinent calculations, we employ $Y_c N$ interactions
extrapolated from results of lattice simulations by the HAL QCD
collaboration \cite{miyamoto18,Miyamoto:2018} to the
physical point. In particular, we use the
${\Lambda}_c N$ potential from Ref.~\cite{Haidenbauer:2018}, where
there is no direct ${\Sigma}_c N$ interaction, and the
two potentials $Y_c N$-A and $Y_c N$-B, introduced and described
in the appendix, which include a direct ${\Sigma}_c N$ interaction.

\begin{figure}[t]
\begin{center}
\includegraphics[width=85mm]{Vkfdep}
\includegraphics[width=85mm]{Vkdep}
\caption{${\Lambda}_c$ in symmetric nuclear matter.
Top: The ${\Lambda}_c$ s.p. potential $U_{{\Lambda}_c}(k_{{\Lambda}_c}=0)$ as a function of the
Fermi momentum $k_F$ in comparison to $U_{\Lambda}(k_{\Lambda}=0)$
\cite{Haidenbauer:2019}
(NLO13 with dash-dotted border; NLO19 with dashed border)
for cutoff variations ${\Lambda} =500$-$600$ MeV.
Bottom: Momentum dependence of $U_{{\Lambda}_c}$ (solid band) and $\varepsilon_{{\Lambda}_c}$ (hatched band)
at the Fermi momentum $k_F=1.35$ fm$^{-1}$.
}
\label{gm}
\end{center}
\end{figure}

\subsection{${\Lambda}_c$ in infinite nuclear matter}

In order to investigate the properties of the ${\Lambda}_c N$ interaction
in nuclear matter, we perform a Brueckner--Hartree--Fock calculation
where we adopt the so-called discontinuous prescription when solving
the Bethe--Goldstone equation.
We follow closely our corresponding
calculation for the ${\Lambda} N$ interaction \cite{Haidenbauer:2015}. In
that work and similar ones (see, {\it e.g.}, Refs.~\cite{schulze97,vidana00}),
the reader can find details how to solve the Bethe--Goldstone
equation and how the single-particle (s.p.) potential $U_{{\Lambda}_c}$
is determined self-consistently together with the G-matrices for a specific nuclear matter density $\rho$ (or
Fermi momentum $k_F$).

In Fig.~\ref{gm}(a), we present results for the dependence of $U_{{\Lambda}_c}(k_{{\Lambda}_c}=0)$
on the Fermi momentum, in comparison to those for the ${\Lambda}$ hyperon obtained
with the NLO interaction from Refs.~\cite{Haidenbauer:2013,Haidenbauer:2019}.
In Fig.~\ref{gm}(b), we display the dependence of $U_{{\Lambda}_c}(k_{{\Lambda}_c})$ and
the s.p. energy, $\varepsilon_{{\Lambda}_c}(k_{{\Lambda}_c})= k^2_{{\Lambda}_c}/2m_{{\Lambda}_c} + U_{{\Lambda}_c}(k_{{\Lambda}_c})$,
on the ${\Lambda}_c$ momentum at the Fermi momentum $k_F=1.35$ fm$^{-1}$,
{\it i.e.,} at nuclear matter saturation density.
The in-medium predictions for ${\Lambda}_c$ are based on the $Y_c N$-A potential
(cf. appendix) and the ${\Lambda}_c N$ potential from Ref.~\cite{Haidenbauer:2018}. Results for the ${\Lambda}_c$ properties for the alternative fit $Y_c N$-B,
considered in the appendix, practically coincide with the ones for
$Y_c N$-A and are therefore not shown.

\begin{table*}[h]
\renewcommand{\arraystretch}{1.4}
\centering
\caption{${\Lambda}_c N$ scattering lengths (in fm) and partial-wave contributions
to the s.p. potential $ U_{{\Lambda}_c} (k_{{\Lambda}_c} = 0)$ (in MeV)
at $k_F = 1.35 \ {\rm fm}^{-1}$.
Results are shown for the $Y_c N$-A potential which includes a direct
${\Sigma}_c N$ interaction (cf. appendix), and for the ${\Lambda}_c N$
interaction from Ref.~\cite{Haidenbauer:2018}.
The cutoff values used ($\Lambda = 500,\,600$~MeV) are indicated in brackets.
For comparison corresponding results for the $\Lambda N$ interactions
NLO13 \cite{Haidenbauer:2013} and NLO19 \cite{Haidenbauer:2019} are given.
}
\vskip 0.2cm
\begin{tabular}{ c|l||rr||rcrrrc|r }
\hline
\hline
& interaction & $a_{^1S_0}$ & $a_{^3S_1}$ & \multicolumn{7}{c }{$U_{{\Lambda}_c}$ ($U_{{\Lambda}}$)} \\
& &&& \ $^1S_0$ \ & \ $^3S_1+^3D_1$ \ & \ $^3P_0$ \ & \ $^1P_1$ \ & \ $^3P_1$ \ & \ $^3P_2+^3F_2$ \ & \ Total \\
\hline
${\Lambda}_c N$ & ${\Lambda}_c N$ (500) \cite{Haidenbauer:2018} &-0.85&-0.81&  $-$ 5.1 &  $-$13.7 & $-$0.4 &    0.0 & $-$0.3 &   $-$0.2 &  $-$19.8  \\
        & $Y_c N$-A(500)                        &-0.85&-0.79&  $-$ 5.1 &  $-$13.5 & $-$0.4 &    0.0 & $-$0.3 &   $-$0.2 &  $-$19.7  \\
        & ${\Lambda}_c N$ (600) \cite{Haidenbauer:2018} &-1.00&-0.98&  $-$ 5.5 &  $-$12.9 &    1.4 &    0.0 & $-$0.5 &   $-$0.4 &  $-$18.0  \\
        & $Y_c N$-A(600)                        &-1.00&-0.91&  $-$ 5.5 &  $-$12.4 &    1.4 &    0.0 & $-$0.5 &   $-$0.4 &  $-$17.6  \\
\hline
$\Lambda N$ & NLO13 (500) &-2.91&-1.61&  $-$15.3 &  $-$14.6 &    1.1 &    0.3 &  1.8 &   $-$1.3 &  $-$28.3  \\
            & NLO19 (500) &-2.91&-1.52&  $-$12.5 &  $-$28.0 &    1.1 &    0.3 &  1.8 &   $-$1.2 &  $-$39.3  \\
            & NLO13 (600) &-2.91&-1.54&  $-$12.3 &  $-$10.9 &    0.9 &    0.3 &  1.7 &   $-$1.1 &  $-$21.6  \\
            & NLO19 (600) &-2.91&-1.41&  $-$11.2 &  $-$22.8 &    0.9 &    0.4 &  1.7 &   $-$1.1 &  $-$32.6  \\
\hline
\hline
\end{tabular}
\label{tab:La}
\renewcommand{\arraystretch}{1.0}
\end{table*}

Partial-wave contributions to $ U_{{\Lambda}_c} (k_{{\Lambda}_c} = 0)$
at $k_F=1.35$ fm$^{-1}$ are listed in Table~\ref{tab:La}.
Note that the contributions of the $P$ waves come solely
from two-pion exchange involving the ${\Sigma}_c N$ intermediate state.
The total potential depth amounts
to around {$-20\div-18$} MeV and is quite insensitive to whether a direct ${\Sigma}_c N$ interaction is included or not.
As a reminder, the ``empirical'' value in case of the ${\Lambda}$ hyperon
is {$-30\div-27$} MeV \cite{Gal:2016}.
Comparing the ${\Lambda}_c$ results with the ${\Lambda}$ case in detail, one can see
that the contribution in the $^1S_0$ partial wave is reduced by roughly
a factor three. This is well in line with the corresponding interaction
strengths; the ${\Lambda}_c N$ scattering length is also about a factor three
smaller than that for ${\Lambda} N$, see Table~\ref{tab:La}.
For the $^3S_1$ partial wave the ${\Lambda}_c N$ and ${\Lambda} N$ contributions (for
NLO13 \cite{Haidenbauer:2013}) are of comparable magnitude, despite of
the fact that the ${\Lambda}_c N$ interaction is less attractive as reflected in
the corresponding scattering lengths which is about 30-50\,\% smaller
than that for ${\Lambda} N$ scattering.
Obviously, for ${\Lambda}_c N$ the dispersive effects, which play an important
role for the contribution of that partial wave in case of the
${\Lambda}$ \cite{Haidenbauer:2019},
are smaller because the ${\Lambda}_c N$-${\Sigma}_c N$ coupling is weaker due to a
weaker transition potential and/or due to the larger threshold separation.
Apparently, that reduced effect compensates for the somewhat less
attractive ${\Lambda}_c N$ interaction.
When comparing with the results for the NLO interaction from 2019 \cite{Haidenbauer:2019},
where the ${\Lambda} N$-${\Sigma} N$ transition potential is noticeably weaker,
one sees a clear correlation between the smaller $^3S_1$ scattering
length and the reduced contribution to $U_{{\Lambda}_c}$
(cf. Table~\ref{tab:La}).

Finally, let us compare our nuclear matter results with other predictions
for $U_{{\Lambda}_c}$ found in the literature.
Ref.~\cite{Bando:1985} contains some results for $U_{{\Lambda}_c}$ based on an ${Y}_c N$
potential that is adapted from one of the $YN$ meson-exchange potentials by the
Nijmegen Group by imposing SU(4) flavor symmetry. In that work, a value of
$ U_{{\Lambda}_c} (k_{{\Lambda}_c} = 0)\approx -25$ MeV at nuclear matter saturation density
has been found. However, note the large contributions from $P$ waves in that
study. The two $S$ wave alone yield only around $-3.5$~MeV.
A study utilizing parity-projected QCD sum rules \cite{Ohtani:2017}
reports a potential depth of {$\sim -20$} MeV for ${\Lambda}_c$ at nuclear matter
saturation density.
Yasui, in a perturbative approach based on a heavy-quark effective theory,
finds a ${\Lambda}_c$ binding energy of around {$-25\div-20$} MeV in nuclear matter
\cite{Yasui:2018}.
Though to some extent surprising, it is interesting to see that the achieved
results are all fairly similar, despite of the different interactions
and approaches employed.

\subsection{$^{\ 3}_{{\Lambda}_c}$He and $^{\ 4}_{{\Lambda}_c}$He systems}

For $A=3$ and $4$ charmed nuclei, Faddeev-Yakubovsky calculations
are performed in the same way as in former studies of hypernuclei
\cite{Miyagawa:1995,Nogga:2002,Haidenbauer:2019}.
As discussed in Ref.~\cite{Haidenbauer:2018},
based on the results for ${\Lambda}$-hypernuclei and the relative strengths
of the ${\Lambda}_c N$ and ${\Lambda} N$ interactions, one can guess which light
charmed nuclei could be bound. For that the pertinent mixtures of
the spin-singlet and spin-triplet ${\Lambda}_c N$ (${\Lambda} N$) interaction for
$s$-shell nuclei are relevant \cite{Haidenbauer:2019,Gibson:1994}
and, of course, the reduction of the kinetic energy associated with
the ${\Lambda}_c$ as a consequence of its larger mass \cite{Gibson:1983}.
In view of the fact that, for the considered $Y_c N$ interactions,
the ${\Lambda}_c N$ $^1S_0$ scattering length is only one third of
the one for ${\Lambda} N$, while there is somewhat less difference in
the $^3S_1$ state (cf. Table~\ref{tab:La}), binding of light
systems is expected to be mainly possible for charmed nuclei with a
dominating spin-triplet ${\Lambda}_c N$ contribution, i.e. $^{\, 3}_{{\Lambda}_c}$He
($J=\frac{3}{2}^+$), $^{\, 4}_{{\Lambda}_c}$He ($1^+$),
and $^{\, 5}_{{\Lambda}_c}$Li \cite{Haidenbauer:2019,Miyagawa:1995,Gibson:1994}.

Additionally, whereas the Coulomb interaction is of less importance
for $\Lambda$ separation energies of hypernuclei, its contribution to
${\Lambda}_c$ separation energies of charmed nuclei is often decisive for binding
\cite{Gibson:1983}. For the solution of the Faddeev-Yakubovsky
equations here, we take the Coulomb interactions fully into account
as described in \cite{Nogga:2000uu}.

First, in order to benchmark our few-body calculations, we devised
${\Lambda}_c N$ interactions that mimic the effective range parameters
predicted by a ${\Lambda}_c N$ potential obtained in the constituent quark
model by Garcilazo et al.~\cite{Garcilazo:2019}.
In that model, the triplet interaction is much stronger
($a_{^3S_1} = -2.31$ fm) than the one in the singlet channel
($a_{^1S_0} = -0.86$ fm), cf. Table~3 in that reference.
Consequently, and in line with the above
arguments and the explorations in Ref.~\cite{Miyagawa:1995},
a bound state for the $J=\frac{3}{2}$ state of $^{\ 3}_{{\Lambda}_c}$He
has been reported, with a ${\Lambda}_c$ separation energy of approximately
$140$~keV including Coulomb~\cite{Garcilazo:2015}.
Since our interactions do not reproduce the phase shifts of the
quark model perfectly over a larger energy region, we use
two different realizations with cutoff $600$-$700$~MeV. We find
separation energies between $60$~keV and $264$~keV. This includes
a variation of $60$ keV due to different $NN$ interactions.
The uncertainty due to different cutoffs of the ${\Lambda}_c N$
interaction is larger than the one due to different $NN$
interactions. We also found that no bound state exists for that
interaction for $^{\ 3}_{{\Lambda}_c}$He with $J=\frac{1}{2}$.
This result confirms the earlier calculations of Garcilazo et al.

We then performed calculations for the ${\Lambda}_c N$ potentials from
Ref.~\cite{Haidenbauer:2018}, and the interactions $Y_c N$-A
and $Y_c N$-B of this work. Since all of these potentials predict
a considerably weaker interaction in the $^3$S$_1$ partial wave,
none of the charmed $A=3$ nuclei are found to be bound.
This remains even true when the
Coulomb interaction is not taken into account.

Note that model 4 employed by Gibson et al.~\cite{Gibson:1983}
has ${\Lambda}_c N$ $^1S_0$ and $^3S_1$ scattering lengths close to
those of the $Y_c N$ interactions considered by us. No bound state
for $A=3$ was found in that work, but a bound
$^{\, 4}_{{\Lambda}_c}$He with a ${\Lambda}_c$ separation energy of approximately
$1.25$~MeV was predicted, after including an estimate
for the contribution of the Coulomb interaction.

\begin{table*}[tbp]
\renewcommand{\arraystretch}{1.4}
    \centering
    \caption{${\Lambda}_c$ separation energies $E_{{\Lambda}_c}$ and binding energies with respect to breakup up into four baryons $E$
    for the $J=1^+$ state of $^{\ 4}_{{\Lambda}_c}$He. The $NN$ interaction at order N$^4$LO+ with a cutoff of 450~MeV of Ref.~\cite{Reinert:2018}
    was used leading to $E(^3{\rm H})=-8.141$~MeV. Expectation values for the kinetic energy $\langle T \rangle$, the
    $NN$ potential energy $\langle V_{NN} \rangle$, and the $Y_c N$ potential energy $\langle V_{Y_c N} \rangle$ are also given.
    The probability that the four-baryon have orbital angular momentum zero and two ( $P(S)$ and  $P(D)$ ) are
    listed together with the probability $P({\Sigma}_c)$ for the ${\Sigma}_c$ component. Energies are given in MeV and probabilities in \%.
    }
    \label{tab:4lamcHe1}
\begin{tabular}{l|r|r|rrrr|rrr}
\hline
\hline
interaction     &   $E_{{\Lambda}_c}$ &     $E$ &   $\langle H \rangle$ & $\langle T \rangle$ & $\langle V_{NN} \rangle$ &  $\langle V_{Y_c N} \rangle$ & $P(S)$ &  $P(D)$ &  $P({\Sigma}_c)$ \\
\hline
    ${\Lambda}_c N$(500) \cite{Haidenbauer:2018} & 0.13 &    -8.27 &   -8.26 &  36.82 &    -42.51 &      -2.58 &   93.21 & 6.75 & 0.21       \\
    $Y_c N$-A(500) & 0.11 &    -8.25 &   -8.25 &  36.65 &    -42.50 &      -2.39 &   93.22 & 6.73 & 0.19      \\
   $Y_c N$-B(500)& 0.11 &    -8.25 &   -8.25 &  36.65 &    -42.50 &      -2.39 &   93.22 & 6.73 & 0.19       \\
\hline
    ${\Lambda}_c N$(600)  \cite{Haidenbauer:2018} & 0.37 &    -8.51 &   -8.50 &  39.71 &    -42.52 &      -5.69 &   92.81 & 7.13 & 0.46       \\
    $Y_c N$-A(600) & 0.30 &    -8.44 &   -8.43 &  39.15 &    -42.51 &      -5.07 &   92.88 & 7.06 & 0.39      \\
   $Y_c N$-B(600)& 0.30 &    -8.44 &   -8.43 &  39.15 &    -42.51 &      -5.07 &   92.88 & 7.06 & 0.39       \\
 \hline
 \hline
\end{tabular}
\renewcommand{\arraystretch}{1.0}
\end{table*}

\begin{table*}[tbp]
\renewcommand{\arraystretch}{1.4}
    \centering
    \caption{Same as Table~\ref{tab:4lamcHe1} for the $J=0^+$ state of $^{\ 4}_{{\Lambda}_c}$He.}
    \label{tab:4lamcHe0}
\begin{tabular}{l|r|r|rrrr|rrr}
\hline
\hline
interaction     &   $E_{{\Lambda}_c}$ &     $E$ &   $\langle H \rangle$ & $\langle T \rangle$ & $\langle V_{NN} \rangle$ &  $\langle V_{Y_c N} \rangle$ &  $P(S)$ &  $P(D)$&  $P({\Sigma}_c)$ \\
\hline
    ${\Lambda}_c N$(500) \cite{Haidenbauer:2018} &   --  &    --  & -8.08 &   34.54 &    -42.51 &  -0.11 &   93.39 &  6.57 & 0.01       \\
    $Y_c N$-A(500) &   --  &    --  & -8.12 &   35.30 &    -42.55 & -0.86 &   93.33 &  6.63 & 0.09      \\
    $Y_c N$-B(500)&   --  &    --  & -8.12 &   35.30 &    -42.55 &  -0.86 &   93.33 &  6.63 & 0.09       \\
\hline
    ${\Lambda}_c N$(600) \cite{Haidenbauer:2018} & 0.18 & -8.32 & -8.30 &   39.21 &    -42.67 &
    -4.84 &   92.99 &  6.95 & 0.47       \\
    $Y_c N$-A(600) & 0.10 & -8.25 & -8.23 &   38.17 &    -42.57 &      -3.83 &   93.09 &  6.85 & 0.34     \\
    $Y_c N$-B(600)& 0.10 & -8.25 & -8.23 &   38.17 &    -42.57 &
    -3.83 &   93.09 &  6.85 & 0.34       \\
 \hline
 \hline
    \end{tabular}
    \renewcommand{\arraystretch}{1.0}
\end{table*}

The $A=4$ results for the interactions considered in this work are
summarized in Tables~\ref{tab:4lamcHe1} and \ref{tab:4lamcHe0}.
In all of our calculations, we found that the $J=1^+$ state is more
bound than the $J=0^+$ state. As obvious from Table~\ref{tab:4lamcHe1},
the results are very independent on whether the direct ${\Sigma}_c N$ interaction
has been included or omitted. The cutoff of the $Y_c N$ interactions has
a much larger effect on the energies than the inclusion of the direct interaction. Even more striking is the observation that the results for
$A=4$ are identical for the potentials $Y_c N$-A and $Y_c N$-B.
This shows unmistakably the insensitivity of the predicted bound-state
properties on the ${\Sigma}_c N$ channel.
The separation energies for the $J=1^+$ state are between
$100$~keV and $370$~keV, a clear evidence that $A=4$ charmed
nuclei could be bound.
The binding energies are, however, somewhat
smaller than predicted in \cite{Gibson:1983}.
We believe that this is partly due to omitting tensor interactions
in the $Y_c N$ in \cite{Gibson:1983} which is fully taken into account
in our calculations.
For comparison, we have also used the interactions that simulate the quark-model potential of Ref. \cite{Garcilazo:2019}. This interaction
clearly provides stronger binding
leading to $1.2$ to $2.1$~MeV separation energy depending on the cutoff used.

A few properties of the resulting wave functions are summarized in
Tables~\ref{tab:4lamcHe1} and \ref{tab:4lamcHe0}, too.
First of all, it is interesting to compare
the expectation value of the Hamiltonian to the energy.
For the numerical calculations, we need to restrict the
number of partial waves. The most significant restriction
is that the algebraic sum of all orbital angular momenta is less or equal 8. We checked that the solution of the Yakubovsky equations is converged such that the energy $E$ is accurate to approximately 10 keV. The expection values differ at most by
20 keV. The slightly larger differences is due to a slower
convergence of the wave functions compared to the Yakubovsky components. The good agreement of both numbers is a confirmation of the consistency of the numerical calculation. We also show separate expectation values of the kinetic
energy, the $NN$ potential energy and the $Y_c N$ potential
energy. It sticks out that the $NN$ potential energy is very similar for all considered bound states. Clearly, the
nuclear core is not very much distorted by the presence of
the charmed hyperon. The expectation value of the
$Y_c N$ interaction is mainly dependent on the cutoff
of the interaction and less dependent on the $\Sigma_c N$
contribution as can be seen from the similarity of the results of
$\Lambda_c N$ \cite{Haidenbauer:2018}, $Y_c N$-A and $Y_c N$-B.

Finally, we give probabilities for the total orbital
angular momentum of 0 and 2 $P(S)$ and $P(D)$. $P$-waves
and $F$-waves only give a negligible contribution. Obviously, the tensor components of the $NN$ and $Y_c N$ interactions
induce a $D$-wave contribution of approximately 7\%.
The probability to find a $\Sigma_c$ is, similar to the
$\Sigma$ in ordinary hypernuclei, small and depends strongly
on the cutoff of the $Y_c N$ interaction.

The outcome for the $J=0^+$ state is compiled in
Table~\ref{tab:4lamcHe0}. In this case, we do not find
a bound $^4_{\Lambda_c}$He for the interactions with a cutoff
of 500~MeV. Such a state is however close to being bound as can
be seen from the expectation values based on an approximate solution
of the Yakubovsky equation, cf. Table~\ref{tab:4lamcHe0}.
Also for the larger cutoff, the $\Lambda_c$ separation energy is only $100$-$180$~keV. Therefore, for our interactions, we can neither confirm
nor exclude the existence of a $0^+$ bound state.
We note that we do find a bound $0^+$ state for the interactions that
simulate the quark model potential of Ref.~\cite{Garcilazo:2019}.
In that case the separation energy is between $130$~keV and $470$~keV
depending on the cutoff used.

Given that the dependence on the $NN$ interaction was smaller than the
dependence on the employed regulator in the $Y_c N$ interaction for $A=3$,
we refrain from repeating the computationally very expensive $A=4$
calculations for different $NN$ interactions.
We do not expect that the results will be significantly different for other
choices.

\subsection{Heavier charmed nuclei}

\begin{table*}[ht!]
\caption{Energy of $\Lambda_c$ single-particle bound states (in MeV) of
several charmed nuclei from $^{\ 5}_{\Lambda_c}$Li to
$^{209}_{\Lambda_c}$Bi. For
convenience the corresponding core nucleus is indicated in brackets.
Results are shown for two values of the cutoff $\Lambda=500,\, 600$ MeV.}
\renewcommand{\arraystretch}{1.6}
\begin{center}
\begin{tabular}{c|cc|cccc}
\hline
\hline
  & \multicolumn{2}{c|}{${\Lambda}_c N$ \cite{Haidenbauer:2018}}
  &  \multicolumn{4}{c}{$Y_cN$} \cr
& (500) & (600)
& A(500) & B(500) & A(600) & B(600)    \\
\hline
$^{\ 5}_{{\Lambda}_c}$Li [$^{4}$He] & $$  & $$ & $$ & $$ & $$ & $$   \\
$1 s_{1/2}$ & $-0.86$  &  $-0.59$ &  $-0.82$ & $-0.81$ & $-0.47$ & $-0.45$  \\
\hline
$^{13}_{{\Lambda}_c}$N [$^{12}$C] & $$  & $$ & $$ & $$ & $$ & $$   \\
$1 s_{1/2}$ & $-3.71$  &  $-2.78$ &  $-3.61$ & $-3.59$ & $-2.53$ & $-2.49$  \\
\hline
$^{17}_{{\Lambda}_c}$F  [$^{16}$O] & $$  & $$ & $$ & $$ & $$ & $$   \\
 $1 s_{1/2}$ & $-5.18$  & $-3.70$ &  $-5.07$ & $-5.04$ & $-3.41$ & $-3.36$  \\
\hline
$^{41}_{{\Lambda}_c}$Sc  [$^{40}$Ca]  & $$  & $$ & $$ & $$ & $$ & $$   \\
$1 s_{1/2}$ & $-6.36$  & $-4.35$ &  $-6.21$ & $-6.18$ & $-3.94$ & $-3.87$  \\
$1 p_{3/2}$ & $-0.93$  & $-$ &  $-0.79$ & $-0.77$ & $-$ & $-$  \\
$1 p_{1/2}$ & $-1.01$  & $-$ &  $-0.96$ & $-0.94$ & $-$ & $-$  \\
\hline
$^{91}_{{\Lambda}_c}$Nb  [$^{90}$Zr] & $$  & $$ & $$ & $$ & $$ & $$   \\
$1 s_{1/2}$ & $-6.15$  & $-4.39$ &  $-5.95$ & $-5.91$ & $-3.88$ & $-3.79$  \\
$1 p_{3/2}$ & $-2.65$  & $-1.13$ &  $-2.46$ & $-2.42$ & $-0.65$ & $-0.57$  \\
$1 p_{1/2}$ & $-2.60$  & $-0.76$ &  $-2.41$ & $-2.37$ & $-0.29$ & $-0.21$  \\
\hline
$^{209}_{{\Lambda}_c}$Bi  [$^{208}$Pb] & $$  & $$ & $$ & $$ & $$ & $$   \\
$1 s_{1/2}$ & $-4.89$  & $-2.15$ &  $-4.64$ & $-4.59$ & $-1.51$ & $-1.40$  \\
$1 p_{3/2}$ & $-4.37$  & $-1.73$ &  $-4.12$ & $-4.07$ & $-1.11$ & $-1.01$  \\
$1 p_{1/2}$ & $-4.36$  & $-1.52$ &  $-4.11$ & $-4.06$ & $-0.91$ & $-0.80$  \\
$1 d_{5/2}$ & $-0.18$  & $-$ &  $-$ & $-$ & $-$ & $-$  \\
$1 d_{3/2}$ & $-0.30$  & $-$ &  $-0.12$ & $-0.08$ & $-$ & $-$  \\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:bound}
\renewcommand{\arraystretch}{1.0}
\end{table*}
Now we consider the energy of the $\Lambda_c$ single-particle bound states in heavier nuclei.
To such end, we follow a perturbative many-body approach whose starting point is a nuclear matter $G$-matrix derived from the bare $Y_c N$ interactions described in Sec.~\ref{sec:formalism} and the appendix.
This $G$-matrix is then used to calculate the self-energy
of the $\Lambda_c$ in the finite nucleus. Solving the Schr\"{o}dinger equation with this self-energy, finally, we are able
to determine the energies of all the single-particle bound states of the $\Lambda_c$ in the nucleus. This approach also provides the
real and imaginary parts of the $\Lambda_c$ optical potential at positive energies and, therefore, allows one to study
the $\Lambda_c$-nucleus  scattering properties. This method was already used to study the properties of the nucleon \cite{borromeo92},
the $\Delta$ isobar \cite{morten94} and the $\Lambda$ and $\Sigma$ hyperons \cite{morten96,vidana98,vidana17}
in finite nuclei, and very recently also those of the $\Lambda_c$ using a meson-exchange $Y_cN$ interaction \cite{vidana19}.
A comprehensive description of the method can be found in these works and the interested reader is referred to any of them for details.

Results for  $^{\ 5}_{{\Lambda}_c}$Li, $^{13}_{{\Lambda}_c}$N , $^{17}_{{\Lambda}_c}$F, $^{41}_{{\Lambda}_c}$Sc, $^{91}_{{\Lambda}_c}$Nb
and $^{209}_{{\Lambda}_c}$Bi are summarized in Table \ref{tab:bound} for the
${\Lambda}_c N$ interaction from Ref.~\cite{Haidenbauer:2018} and the two
potentials $Y_c N$-A and $Y_cN$-B with inclusion of a direct ${\Sigma}_c N$
interaction. We note that all charmed nuclei considered consist of a
closed-shell nuclear core plus a $\Lambda_c$ sitting in a single-particle state. We note also that, although the $\Lambda_cN$ interaction with the
cutoff $600$ MeV is more attractive, the $\Lambda_c$ single-particle
bound states predicted in this case are actually less bound.
This is due to dispersive effects \cite{nogami70,bodmer71,dabrowski73} in
the calculation of the $G$-matrix which suppress the contribution from
the $\Lambda_cN\rightarrow\Sigma_cN\rightarrow\Lambda_cN$ coupling.
That coupling is significantly larger for the $600$ MeV cutoff and, accordingly, likewise the reduction of the overall attraction.
Before analyzing the results, we would like to point out that, as discussed in Ref.~\cite{Haidenbauer:2019V},
the approach followed tends to underestimate the energies of
the $\Lambda$ hyperon single-particle bound states for light hypernuclei such as $^{\,5}_{\Lambda}$He.
Accordingly, we expect $^{\ 5}_{{\Lambda}_c}$Li to be somewhat more strongly bound than what is suggested
by the values given in Table \ref{tab:bound}.

It is interesting to observe that, contrary to single-$\Lambda$ hypernuclei where the $\Lambda$ is more and more bound when
going from light to heavy nuclei,  the binding energy of the $\Lambda_c$ increases from $^{\ 5}_{\Lambda_c}$Li to $^{41}_{\Lambda_c}$Sc
and then it decreases. This is due to the Coulomb repulsion between the $\Lambda_c$ and the protons of the nuclear core, which together
with the kinetic energy of the $\Lambda_c$, compensates most of the attraction of the $\Lambda_cN$ interaction. The possible existence
of $\Lambda_c$ nuclei is, therefore, subject to a delicate balance between the $\Lambda_cN$ interaction, the kinetic energy and the
Coulomb force as it has been already pointed out in Refs.\ \cite{vidana19,tsushima03,tsushima03b,tsushima04,miyamoto18}. In particular,
in Ref.\ \cite{miyamoto18} it was suggested that only light- or medium-mass $\Lambda_c$ nuclei could really exists whereas, for
instance, in Ref.\ \cite{vidana19} it was found that even the heavier $\Lambda_c$ nucleus considered in that work, namely
$^{209}_{\Lambda_c}$Bi, could exist, as in the present work. A small spin-orbit splitting of the $p-$ and $d-$wave states of the order
of a few tenths of MeV is observed in all $\Lambda_c$ nuclei in agreement with the results obtained in
Refs.\ \cite{vidana19,tsushima03,tsushima03b,tsushima04,tan04}. In addition,
we note also that the level spacing of the $\Lambda_c$
single-particle states is smaller than those for the corresponding hypernuclei (see {\it e.g.} Table I of Ref.\ \cite{vidana17}).
This is simply due to the fact that the mass of the $\Lambda_c$ is larger than that of the $\Lambda$ hyperon.

\begin{figure}[t]
\begin{center}
\vskip 1.2cm
\includegraphics[width=85mm]{fig_coul}
\caption{Contributions of the kinetic energy, the $Y_cN$ interaction and the Coulomb potential to the energy
of the $\Lambda_c$ single-particle bound state $1s_{1/2}$ as a function of the mass number of the $\Lambda_c$ nuclei
considered. Results are shown for the potential $Y_c N$-A with a cutoff of $500$ MeV.}
\label{fig:coul}
\end{center}
\end{figure}

To understand better the role of the Coulomb force in our calculation, in Fig.\ \ref{fig:coul} we show the separate
contributions of the kinetic energy, of the $Y_c N$ interaction, and of the Coulomb potential to the energy of the $\Lambda_c$
single-particle bound state $1s_{1/2}$ for the different charmed nuclei considered in this work as function of the mass number
($A=N+Z$, with $N$ and $Z$ being the neutron and atomic numbers,
respectively, of the specific nucleus). When going
from light to heavy $\Lambda_c$ nuclei, the Coulomb contribution increases because of the increase of the atomic number
whereas those of the kinetic energy and of the $Y_cN$ interaction decrease. The contribution of the kinetic energy decreases
with the mass number because the wave function of the $1s_{1/2}$ state becomes more and more spread due to the larger
extension of the nuclear density over which the $\Lambda_c$ wants to be distributed (see Fig.\ \ref{fig:wf}). The increase of the mass number leads
to a more attractive $\Lambda_c$ self-energy (see, {\it e.g.,} Figs.\ 2 and 3 of Ref.\ \cite{vidana17} for a detailed
discussion in the case of single-$\Lambda$ hypernuclei) that translates into a more negative contribution of the
$Y_cN$ interaction.  Note that, when adding the three contributions they compensate in such a way that
the energy of the $1s_{1/2}$ decreases only by about 5 MeV from  $^{\ 5}_{\Lambda_c}$Li to $^{17}_{\Lambda_c}$F
and then it increases very smoothly from $^{41}_{\Lambda_c}$Sc to $^{209}_{\Lambda_c}$Bi.

\begin{figure}[t]
\begin{center}
\includegraphics[width=85mm]{wave_fun}
\caption{${\Lambda}_c$ probability density distribution for the $1s_{1/2}$ state in the six ${\Lambda}_c$ nuclei considered. Results are presented for the $Y_c N$-A potential. The dependence on the cut-off is indicated by bands. The red bands show the results when the Coulomb interaction is artificially switched off.}
\label{fig:wf}
\end{center}
\end{figure}

To end this section, we display in Fig.\ \ref{fig:wf} the probability density distribution ({\it i.e.,} the square of the radial wave function) of the $\Lambda_c$ in the $1s_{1/2}$ state for the six $\Lambda_c$ nuclei considered,
for the $Y_c N$-A. The cut-off dependence is indicated by bands. Results
for the ${\Lambda}_c N$ interaction from Ref.~\cite{Haidenbauer:2018} and
for $Y_c N$-B are not shown since the differences in the probability
density are so small that they cannot be resolved in the plot.
Note that, when moving from light to heavy $\Lambda_c$ nuclei, due to the increase of the size of the nuclear core, the probability of finding the $\Lambda_c$ close to the center of the nucleus decreases, and it becomes more and more distributed over the whole nucleus.
The probability density distribution when the Coulomb interaction is artificially switched off is also shown for comparison.
Obviously, and as expected, the Coulomb repulsion pushes the $\Lambda_c$ away from the center of the nuclei. A similar effect is observed for the probability densities of the other $\Lambda_c$ single-particle bound states.

\section{Summary and Conclusions}
\label{sec:concl}

In the present work, we have investigated the binding energies
of charmed nuclei. As input we used ${\Lambda}_c N$ and ${\Sigma}_c N$
interactions that have been extrapolated from lattice QCD simulations
by the HAL QCD collaboration~\cite{miyamoto18,Miyamoto:2018} at
quark masses corresponding to $m_\pi=410 - 570$~MeV to the physical point.
For this extrapolation, we used a framework based on chiral effective
field theory~\cite{Haidenbauer:2018,Epelbaum:2003,Petschauer:2013}.
The ${\Lambda}_c N$ interaction
established in this way is significantly weaker than what has
been employed in most of the studies of charmed nuclei in the
literature so far.
The bound state calculations for light charmed nuclei have been
carried out within the Faddeev-Yakubovsky framework.
The results for heavier nuclei are from calculations
of the energies of ${\Lambda}_c$ single-particle bound states,
performed within a perturbative many-body approach, which allows one
to determine the finite nuclei ${\Lambda}_c$ self-energy from which
the energies of the different bound states can be obtained.

Our results indicate that even for a weak ${\Lambda}_c N$ interaction as
suggested by the lattice simulations of the HAL QCD collaboration
already $A=4$ charmed nuclei are likely to exist.
Only the lightest nucleus considered, a charmed helium $^{\ 3}_{{\Lambda}_c}$He,
turned out to be unbound, in contrast to conjectures reported in
Refs.~\cite{Maeda:2016,Garcilazo:2015}.

An additional aspect considered in the present work is the effect of
the ${\Sigma}_c N$ interaction. Some results from lattice simulations for
this channel have become available recently \cite{Miyamoto:2018}.
There is admittedly a sizable uncertainty in the extrapolation
of the HAL QCD results to the physical point, not least due to
missing information on the behavior in the ${\Sigma}_c^* N$ channel,
closely connected to the former by heavy quark spin symmetry.
This makes reliable predictions for ${\Sigma}_c N$ observables rather
difficult at the moment.
On the other hand, we found that the uncertainties due to the present
situation in the ${\Sigma}_c N$ channel do not affect the conclusions
on the properties of the ${\Lambda}_c N$ interaction at low energies, relevant
for the quest of charmed nuclei. Specifically, taking into account
the coupling of ${\Lambda}_c N$ to ${\Sigma}_c N$ and the direct ${\Sigma}_c N$ interaction
as suggested by the HAL QCD results has very little influence on the
existence of such bound states.
Indeed, for the ${\Lambda}_c N$ interaction, the extrapolation of
the lattice results to the physical point seems to be fairly reliable
and stable and, therefore, we believe that robust predictions for the
properties of the ${\Lambda}_c$ in finite and infinite nuclear matter can be
given based on the ${\Lambda}_c N$ potentials established in
Ref.~\cite{Haidenbauer:2018} and in this work.

Prospects for detecting charmed nuclei at J-PARC have been discussed
at various occasions, see, e.g., Ref.~\cite{Tsunemi:2008}.
Corresponding opportunities by the CBM experiment at FAIR are
considered in Ref.~\cite{Steinheimer:2017}.
The option for discovering charmed nuclei with neutrino beams is
addressed in Ref.~\cite{Imai:2019}.
An alternative on a different scope is offered by high-energy experiments
such as the $pp$- and/or heavy-ion collisions \cite{Cho:2017}
presently pursued by the
ALICE collaboration \cite{Braun:2018,Donigus:2019} at the LHC/CERN or
the STAR collaboration at RHIC/BNL
\cite{Agakishiev:2011,TheSTAR:2016}.
Here ``exotic'' nuclei such as the anti-hypertriton or the
$^4\overline{\rm He}$ were already produced and detected, and
the lightest charmed nuclei might be within reach - now or in
the near future - should they indeed exist.
That said, one should be aware that there are tremendous experimental challenges for producing and detecting charmed nuclei, as has been summarized in Ref.~\cite{vidana19} but also indicated in the introduction to the present work.

\begin{acknowledgement}
We acknowledge helpful communications with Vadim Baru, Benjamin D\"onigus,
and Hirokazu Tamura.
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 COST Action CA16214.
The numerical calculations were performed on JURECA, the JURECA-Booster of the J\"ulich Supercomputing Centre, J\"ulich, Germany,
and the Centro di Calcolo of the INFN Sezione di Catania, Catania, Italy.
\end{acknowledgement}

\section*{Appendix: ${\Sigma}_c N$ scattering}
\label{sec:results}

In this appendix, we discuss the results for ${\Sigma}_c N$ scattering. However, let us
emphasize from the beginning that these have to be interpreted with caution.
For charmed baryons, heavy quark spin symmetry plays a role \cite{Meng:2019,Lu:2017}
and, thus, one should include not only the ${\Sigma}_c N$ channel but also ${{\Sigma}_c^*} N$.
Indeed the ${\Sigma}_c N$ and ${{\Sigma}_c^*} N$ thresholds are just about $65$ MeV apart so
that the coupling between those systems should be important. Unfortunately,
there are no results for the ${{\Sigma}_c^*} N$ interaction from lattice simulations
and, therefore, it cannot be explicitly included in the analysis.
For that reason, it was omitted in our earlier study \cite{Haidenbauer:2018} and
it was assumed that any effect of the ${{\Sigma}_c^*} N$ interaction can be effectively
absorbed into the ${\Lambda}_c N$ LECs. After all, since $M_{{\Sigma}_c^*}-M_{{\Lambda}_c} = {234}$ MeV,
there should be little influence on the low-energy ${\Lambda}_c N$ amplitude anyway.
In case of the ${\Sigma}_c N$ channel, such an assumption is questionable.
In Ref.~\cite{Meng:2019} heavy quark spin symmetry was taken into account in
the derivation of the potential. But also in that work, the actual coupling of
the ${\Sigma}_c N$ and ${{\Sigma}_c^*} N$ channels was ignored in the evaluation of the
scattering amplitude.

\begin{figure}[t]
\begin{center}
\includegraphics[width=85mm]{S1E}
\includegraphics[width=85mm]{S1Eb}
\caption{${\Sigma}_c N$ $^3S_1$ phase shifts as function of the c.m.
kinetic energy.
Lattice QCD results~\cite{Miyamoto:2018} for $m_\pi=570$~MeV
(open circles) and $410$~MeV (filled circles) are shown together with
our fits (blue and green narrow bands).
The broader (red) band is the prediction for
$m_\pi=138$~MeV. Two scenarios, $Y_c N$-A (top) and $Y_c N$-B (bottom)
are considered, see text.
The bands represent the cutoff variation $\Lambda = 500$-$600$ MeV.
}
\label{ph3s1S}
\end{center}
\end{figure}

Another complicacy comes from the fact that the mass difference
$M_{{\Sigma}_c}-M_{{\Lambda}_c}$ is larger than the pion mass. Because of that the use of
the static approximation for the pion exchange (see Eq.~(\ref{OPE})) in the
${\Lambda}_c N \to {\Sigma}_c N$ transition potential is rather problematic for
energies near and/or above the ${\Sigma}_c N$ threshold where the exchanged
pion can go on-shell. In principle, one should
take into account that there is a ${\Lambda}_c N\pi$ three-body cut, see
Refs.~\cite{Baru:2011,Baru:2013,Baru:2015D} for a discussion on this issue for
the analogous $D\bar D^*$ - $D\bar D\pi$ case in the context of the $X(3872)$.
As a matter of fact, for the unphysical pion masses of the LQCD calculation,
this problem does not arise. However, it could have a noticeable influence on
the extrapolation of the ${\Sigma}_c N$ results to the physical point.
This aspect is ignored in our calculation, and it is also not addressed
in the ${\Sigma}_c N$ study of Meng et al.~\cite{Meng:2019}.
We emphasize that, for energies around the ${\Lambda}_c N$ threshold, the
static approximation is well justified.

LQCD results for the ${\Sigma}_c N$ $^3S_1$ phase shifts are available
for $m_\pi = 410,\, 570,\, 700$ MeV \cite{Miyamoto:2018}.
We determine the LECs of the contact interaction, cf. Eq.~(\ref{LEC}),
by a fit to the lattice data at the two lower pion masses. It
allows us to determine the LECs $\tilde C_i$ and $C_i$ as well
as $\tilde D_i$ and $D_i$, \textit{i.e.}, the ones which encode
the pion-mass dependence of the contact interaction.
The fits are done to the central values of the phase shifts as given
in Fig.~2 of Ref.~\cite{Miyamoto:2018}, for energies
up to $30$~MeV, cf. upper panel of Fig.~\ref{ph3s1S}.
Alternative fits with particular emphasis on the near-threshold behavior
of the HAL QCD results were performed, too, cf. the lower panel.
Of course, in both cases, we made sure that we produce larger near-threshold
phase shifts for $m_\pi = 410$ MeV than for $570$ MeV,
as suggested by the lattice simulation.
It should be said that trying to fit to the HAL QCD results at somewhat
higher energies is not very meaningful
in view of the fact that the ${{\Sigma}_c^*} N$ channel is not explicitly included.
Its threshold is around $85$~MeV for the HAL QCD calculations \cite{Miyamoto:2018}
and, as said above, at roughly $65$~MeV for physical masses.

\begin{table}[t]
\renewcommand{\arraystretch}{1.5}
\centering
\caption{${\Lambda}_c N$ and ${\Sigma}_c N$ ($I=1/2$) scattering lengths in
the $^3S_1$ partial wave (in fm), for the $Y_c N$
potentials described in the text. In case of ${\Sigma}_c N$ real and
imaginary parts are given. ${\Lambda}_c N$ results for the interaction
from Ref.~\cite{Haidenbauer:2018} are listed as well.
}
\vskip 0.2cm
\begin{tabular}{l|c|c}
\hline
\hline
& ${\Lambda}_c N$ & ${\Sigma}_c N$ \\
\hline
${\Lambda}_c N$ (500) \cite{Haidenbauer:2018} & -0.81 & \\
$Y_c N$-A (500) & -0.79 & (-1.56, -1.35) \\
$Y_c N$-B (500) & -0.78 & (-2.09, -1.70) \\
${\Lambda}_c N$ (600) \cite{Haidenbauer:2018} & -0.98 & \\
$Y_c N$-A (600) & -0.91 & (-0.08, -1.97) \\
$Y_c N$-B (600) & -0.90 & (-0.21, -2.50) \\
\hline
\hline
\end{tabular}
\label{tab:ERE}
\renewcommand{\arraystretch}{1.0}
\end{table}

A combined fit to the ${\Lambda}_c N$ and ${\Sigma}_c N$ $^3S_1$ phase shifts turned out to
be unnecessary because the inclusion of a direct ${\Sigma}_c N$ interaction
had practically no effect on the ${\Lambda}_c N$ results reported in \cite{Haidenbauer:2018},
at least for the energies considered there.
We did not attempt to reproduce the inelasticity parameter,
given in Ref.~\cite{Miyamoto:2018} in terms of the ${\Lambda}_c N$ $S$-matrix,
$\eta = |S_{{\Lambda}_c N,{\Lambda}_c N}|$. In the lattice simulation it is with
values around $0.995$ basically compatible with $1$, which suggests that there
is practically no channel coupling. However, we believe that this could be an
artifact of the way how the analysis by the HAL QCD collaboration is done.
In the
${\Lambda} N - {\Sigma} N$ systems, the strong channel coupling arises primarily from
the tensor force mediated by pion exchange, and that leads to a strong
coupling
of the $^3S_1$ to the $^3D_1$ partial wave, see, e.g., Fig.~7 in
Ref.~\cite{Haidenbauer:2013}.
Indeed, for $^3S_1$$\to$$^3S_1$ transitions (as well as for $^1S_0$$\to$$^1S_0$),
the expectation value of the tensor operator is zero.
However, in the analysis of the HAL QCD collaboration, $D$ waves are not
considered so that this component of the $Y_c N$ force is only effectively
included in the $S$-wave interactions.
In our calculation, we include the full one-pion exchange and that means that
there is a coupling to the $^3D_1$ partial wave - at the physical point
and also for the pion masses of the lattice simulation.
Since the pion mass is known and the ${\Lambda}_c{\Sigma}_c\pi$ coupling constant is known,
too (at least at the physical point), one can consider our result as genuine
prediction for the strength of the channel coupling and, thus, we did
not impose any further constraints on it.

Results for the $^3S_1$ phase shift are presented in Fig.~\ref{ph3s1S}
for pion masses of $570$~MeV and $410$~MeV, together with the extrapolation to
$138$~MeV. The bands represent the dependence of the results on
variations of the cutoff $\Lambda$.
One can see that the lattice results at $m_\pi=410$~MeV and $570$~MeV are
reproduced quantitatively by our potential ($Y_c N$-A) up
to c.m. energies of around $50$~MeV, cf. upper panel of Fig.~\ref{ph3s1S}.
If we require a quantitative reproduction of the low-energy behavior
($Y_c N$-B; lower panel), then there is
agreement with the lattice results only up to around $25$~MeV.
The cutoff dependence of the fits to the HAL QCD results is fairly small
so that the (hatched) bands are barely visible. The phase shift
obtained from the interaction at the physical point (solid bands)
do exhibit a noticeable but still moderate cutoff dependence.
That said, there is a sizable uncertainty in the extrapolation of the two
fitting scenarios considered. When emphasis is put on the very low-energy
results by HAL QCD, then the phase shifts at the physical point are larger
and actually close to the lattice results for unphysical pion masses
(Fig.~\ref{ph3s1S}; lower panel) whereas the fit over a larger energy
region yields perceptibly smaller results.

\begin{figure}[t]
\begin{center}
\includegraphics[width=85mm]{VA1E}
\caption{Results for the ${\Lambda}_c N$ $^3S_1$ phase shift for the interaction
$Y_c N$-A.
Same description of curves and symbols as in Fig.~\ref{ph3s1S}.
Lattice QCD results are from Ref.~\cite{miyamoto18}.
}
\label{ph3s1L}
\end{center}
\end{figure}

For convenience, we compiled the scattering lengths for $Y_c N$-A and
B in Table.~\ref{tab:ERE}.
Obviously, with the ${\Sigma}_c N$ interaction included there is a small
reduction in the ${\Lambda}_c N$ $^3S_1$ scattering length (for A and for B)
as conpared to the results from Ref.~\cite{Haidenbauer:2018}.
However, the variation is rather moderate and stays within the
uncertainty due to the regulator dependence. Also the variation in the
corresponding ${\Lambda}_c N$ $^3S_1$ phase shift is very small. Actually,
the ${\Lambda}_c N$ results with inclusion of a direct ${\Sigma}_c N$ interaction,
cf. Fig.~\ref{ph3s1L} for $Y_c N$-A, can be hardly distinguished from
those presented in Fig.~2 in Ref.~\cite{Haidenbauer:2018}.
For the ${\Sigma}_c N$ channel the situation is rather different.
First there is a sizable difference in the scattering length obtained
for the interactions A and B. In addition, and more disturbing, there
is a fairly drastic regulator dependence. One can certainly
say that there is no indication for a near-by ${\Sigma}_c N$ bound state,
a conclusion already drawn in Ref.~\cite{Meng:2019}.
(The existence of ${\Sigma}_c N$ bound states or resonances has been suggested
in some studies in the past \cite{Huang:2013,Maeda:2018}).
On the other hand, more quantitative conclusions are difficult to draw.
Definitely, there is a significant coupling to the ${\Lambda}_c N$ channel,
mediated by one-pion exchange, which gives rise to an appreciable imaginary
part of the scattering length in our calculation. Moreover, one should not
forget that the static approximation is used by us for
simplicity reasons. In the real world another channel is open,
namely ${\Lambda}_c N\pi$, which contributes likewise to the inelasticity.
An appropriate treatment is desirable but technically demanding and,
thus, postponed to the future when hopefully more information from lattice
simulations will be available.
Fortunately, these uncertainties have basically no effect on
the predictions for the properties of the ${\Lambda}_c N$ interaction at
low energies, cf. Table.~\ref{tab:ERE} and Fig.~\ref{ph3s1L},
and also not on the binding energies of ${\Lambda}_c$ nuclei, as discussed
in the main part of this paper.