Jan 28, 2019 - Single-particle energies of the Îc chamed baryon are obtained in several ... results for Îc-nuclei are compatible with those obtained...

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arXiv:1901.09644v1 [nucl-th] 28 Jan 2019

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I. Vida˜ na1 , A. Ramos2 and C. E. Jim´enez-Tejero3 Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Dipartimento di Fisica “Ettore Majorana”, Universit` a di Catania, Via Santa Sofia 64, I-95123 Catania, Italia 2 Departament de F´ısica Qu` antica i Astrof´ısica and Institut de Ci`encies del Cosmos (ICCUB), Facultat de F´ısica, Universitat de Barcelona, Mart´ı i Franqu`es 1, E-08028 Barcelona, Spain and 3 Barcelona Center for Subsurface Imaging, Institute of Marine Sciences, Spanish Research Council, Passeig Mar´ıtim de la Barceloneta 37, E-08003 Barcelona, Spain Single-particle energies of the Λc chamed baryon are obtained in several nuclei from the relevant self-energy constructed within the framework of a perturbative many-body approach. Results are presented for a charmed baryon-nucleon (Yc N ) potential based on a SU(4) extension of the meson˜ of the J¨ exchange hyperon-nucleon potential A ulich group. Three different models (A, B and C) of this interaction, that differ only on the values of the couplings of the scalar σ meson with the charmed baryons, are considered. Phase shifts, scattering lengths and effective ranges are computed for the three models and compared with those predicted by the Yc N interaction derived in Eur. Phys. A 54, 199 (2018) from the extrapolation to the physical pion mass of recent results of the HAL QCD Collaboration. Qualitative agreement is found for two of the models (B and C) considered. Our results for Λc -nuclei are compatible with those obtained by other authors based on different models and methods. We find a small spin-orbit splitting of the p−, d− and f −wave states as in the case of single Λ-hypernuclei. The level spacing of Λc single-particle energies is found to be smaller than that of the corresponding one for hypernuclei. The role of the Coulomb potential and the effect of the coupling of the Λc N and Σc N channels on the single-particle properties of Λc −nuclei are also analyzed. Our results show that, despite the Coulomb repulsion between the Λc and the protons, even the less attractive one of our Yc N models (model C) is able to bind the Λc in all the nuclei considered. The effect of the Λc N − Σc N coupling is found to be almost negligible due to the large mass difference of the Λc and Σc baryons. I.

INTRODUCTION

Soon after the discovery of charmed hadrons [1–6], the possible existence of charmed nuclei (bound systems composed of nucleons and charmed baryons) was proposed in analogy with hypernuclei (see e.g., Refs. [7–11]). This possibility motivated several authors to study the properties of these systems within different theoretical approaches, predicting a rich spectrum and a wide range of atomic numbers [12–18]. Production mechanisms of charmed nuclei by means of charm exchange or associate charm production reactions, analogous to the ones widely used in hypernuclear physics, were also proposed [19, 20]. However, the experimental production of charmed nuclei is difficult and, up to now, only three ambiguous candidates have been reported by an emulsion experiment carried out in Dubna in the mid-1970s [21–25]. Experimental difficulties arise mainly from (i) the kinematics of the production reactions: charmed particles are formed with large momentum making their capture by a targetnucleus improbable; and (ii) the short lifetimes of Dmeson beams, which makes necessary to place the target as close as possible to the D-meson production point. Such difficulties will be hopefully overcome at the future GSI–FAIR (Gesellschaft f¨ ur Schwerionenforschung– Facility for Antiproton and Ion Research) and JPARC (Japan Proton Accelerator Research Complex) facilities [26, 27]. The production of charmed particles in these facilities will be sufficiently large to make the study of charmed nuclei possible . Studies of p¯ reactions in nu-

clei at the conditions of the PANDA experiment predict forward differential cross sections for the formation of Λc hypernuclei in the range of a few µb/sr [28]. These future prospects have injected a renewed interest in this line of research [29]. In the last few years, theoretical estimations of the charmed baryons properties in nuclear matter and finite nuclei have been revisited using the quark-meson coupling model [30–33], a relativistic mean field approach [34], effective Lagrangians satisfying the heavy quark, chiral and hidden local symmetries [35], the quark cluster model [36], or a single-folding potential employing a Lattice QCD (LQCD) simulation of the Λc N interaction [37]. An extrapolation to the physical pion mass of the former LQCD Λc N interaction has recently become available [38]. In this work we study the single-particle properties of the Λc charmed baryon in several nuclei using a microscopic many-body approach. Our starting point is a nuclear matter G-matrix derived from a bare charmed baryon-nucleon (Yc N, Yc = Λc , Σc ) potential based on a SU(4) extension of the hyperon-nucleon (Y N ) potential A˜ of the J¨ ulich group [39]. This G-matrix is used to calculate the self-energy of the Λc in the finite nucleus including corrections up to the second order. Solving the Schr¨odinger equation with this self-energy we are able to determine the single-particle energies and the wave function of the bound Λc . Our approach also provides the real and imaginary parts of the Λc optical potential at positive energies, and therefore, allows one to study the Λc -nucleus scattering properties. This method was al-

2 ready used to study the properties in finite nuclei of the nucleon [40], the ∆ isobar [41], and the Λ and Σ hyperons [42–45]. The paper is organized in the following way. In Sec. II we present our model for the Yc N interaction. The method to obtain the Λc single-particle properties in finite nuclei is briefly described in Sec. III. Results for a variety of Λc −nuclei are shown in Sec. IV. Finally, a brief summary and some concluding remarks are given in Sec. V.

Λ

N

C

Λ

N

C

σ, ω

D, D *

N

Λ

N

Σ

C

Λ

N

C

Σ

C

σ, ω

C

N D, D *

π, ρ

II.

THE Yc N INTERACTION

Our model for the Yc N interaction is based on a generalization of the meson exchange Y N potential A˜ of the J¨ ulich group [39]. In analogy with that model, we describe the three different channels, Λc N → Λc N , Σc N → Σc N and Λc N ↔ Σc N , only as the sum of single scalar, pseudoscalar and vector meson exchange potentials shown in Fig. 1. As in the Y N J¨ ulich potential, the exchange of the effective scalar σ meson parametrizes the contribution of correlated 2π-exchange. The basic input of our model are the baryon-baryon-pseudoscalar (BBP) and the baryon-baryon-vector (BBV) vertices described, respectively, by the Lagrangian densities LBBP = gN N π (N † ~τ N ) · ~π ~ † · ~π Λc + Λ† Σ ~ c · ~π ] + gΛc Σc π [Σ c

c

~† ×Σ ~ c ) · ~π −i gΣc Σc π (Σ c † + gN Λc D [(N D)Λc + Λ†c (D† N )] ~c +Σ ~ †c (D† ~τ N )] + gN Σc D [(N † ~τ D) · Σ

and LBBV = gN N ρ (N † ~τ N ) · ρ ~ † ~ ~c ·ρ + gΛc Σc ρ [Σ · ρ ~Λc + Λ† Σ ~] c

c

~† ×Σ ~ c) · ρ −i gΣc Σc ρ (Σ ~ c

+ gN Λc D∗ [(N † D∗ )Λc + Λ†c (D∗† N )] ~c +Σ ~ † (D∗† ~τ N )] + gN Σc D∗ [(N † ~τ D∗ ) · Σ c

+ gN N ω N † N ω + gΛc Λc ω Λ†c Λc ω ~† ·Σ ~ cω . + gΣc Σc ω Σ c

We note that the isospin structure of these vertices is the same as that of their analogous strange ones. Similarly to the J¨ ulich Y N interaction, which is itself based on the Bonn NN one, the Yc N model presented here also neglects the contribution of the η and η ′ mesons. We use the SU(4) symmetry to derive the relations between the different coupling constants. Note, however, that this symmetry is strongly broken due to the use of the different physical masses of the various baryons and mesons, and that we employ it rather as a mathematical tool to get a handle on the various couplings of our model.

N

Σ

N

Σ

C

Σ

C

N

C

π, ρ

N

Σ

N

C

D, D *

Λ

C

N

Λ

C

FIG. 1: Single-meson exchange contributions included in our model for the Yc N interaction.

+

In particular, we are dealing with J P = 12 baryons and J P = 0− , 1− mesons belonging to 20′ - and 15-plet irreducible representations of SU(4), respectively. Since the baryon current can be reduced according to 20′ ⊗20′ = 1⊕151 ⊕152 ⊕20′′ ⊕45⊕45⊕84⊕175 , (1) there are two ways to obtain an SU(4)-scalar for the coupling 20′ ⊗ 20′ ⊗ 15 because the baryon current contains two distinct 15-plet representations, 151 and 152 . They couple to the meson 15-plets with strengh g151 and g152 , respectively. It is quite straightforward to relate these two couplings to the couplings gD and gF of the usual symmetric (“D-coupling”) and antisymmetric (“Fcoupling”) octet representations of the baryon current in SU(3). They read r √ 1 10 g8 (7 − 4α) g151 = (7gD + 5gF ) = 4 3 r √ 3 √ g152 = ( 5gD − 5gF ) = 40g8 (1 − 4α) , (2) 20 where in the last step we have written gD and gF in terms of the conventional SU(3) octet strengh coupling g8 , and the so-called F/(F + D) ratio α 40 gD = √ g8 (1 − α) , 30

√ gF = 4 6g8 α .

(3)

Let us consider first the coupling of the baryon current to the pseudoscalar mesons. The relations between all the relevant BBP coupling constants can be easily obtained

3 Model A,B,C A,B,C A,B,C

Vertex NNπ Λc Σc π Σc Σc π

√ √ ∗ gBBM / 4π fBBM / 4π ΛBBM (GeV) ments π → ρ, D → D , αp → αv in the above expressions. In addition, the couplings to the ω meson are 3.795 − 1.3 3.067 − 1.4 gN N ω = gN N ρ (4αv − 1) 2.277 − 1.2 gN N ρ

gΛc Λc ω =

A,B,C A,B,C

N Λc D N Σc D

−3.506 1.518

− −

2.5 2.5

A,B,C A,B,C A,B,C

NNρ Λc Σc ρ Σc Σc ρ

0.917 0.000 1.834

5.591 4.509 3.372

1.4 1.16 1.41

A,B,C A,B,C A,B,C

NNω Λc Λc ω Σc Σc ω

4.472 1.490 1.490

0.000 2.758 −2.907

1.5 2.0 2.0

A,B,C A,B,C

N Λc D∗ N Σc D∗

−1.588 −0.917

−5.175 2.219

2.5 2.5

A,B,C

NNσ

2.385

−

1.7

A A A

Λc Λc σ Σc Σc σ(I = 1/2) Σc Σc σ(I = 3/2)

2.138 3.061 3.102

− − −

1.0 1.0 1.12

B B B

Λc Λc σ Σc Σc σ(I = 1/2) Σc Σc σ(I = 3/2)

1.817 2.601 2.636

− − −

1.0 1.0 1.12

C C C

Λc Λc σ Σc Σc σ(I = 1/2) Σc Σc σ(I = 3/2)

1.710 2.448 2.481

− − −

1.0 1.0 1.12

TABLE I: Baryon-baryon-meson coupling constants gBBM , fBBM and cutoff masses ΛBBM for the models A, B and C of the Yc N interaction constructed and used in this work.

by using SU(4) Clebsch–Gordan cofficients [46] and the above relations. They read 2 gΛc Σc π = √ gN N π (1 − αp ) 3 gΣc Σc π = 2 gN N π αp 1 gN Λc D = − √ gN N π (1 + 2αp ) 3 gN Σc D = gN N π (1 − 2αp ) ,

(4)

where we have added the subindex p to the ratio α to specify that this is the ratio for the coupling of baryons with the pseudoscalar mesons and distinguish it from that for the vector ones used below. Similarly, the corresponding relations for the BBV couplings can be obtained by simply making the replace-

9

(6αv + 3)

gΣc Σc ω = gN N ρ (2αv − 1) ,

(5)

where we have assumed that the physical ω meson results from the ideal mixing of the mathematical members of the 15-plet ω8 and ω1 . The relations for the tensor coupling constants fBBM can be obtained by applying the corresponding SU(4) relations to the “magnetic” coupling GBBM = gBBM + fBBM . Thus, in the above relations gv has to be replaced simply by Gv and αv by αt . To determine the couplings of the scalar σ meson with the charmed baryons, we should remind that this meson is not a member of any SU(4) multiplet and, therefore, it is not possible to obtain these couplings by invoking the SU(4) symmetry as we did for the couplings with the pseudoscalar and vector mesons. This leaves us certain freedom to chose the values of the couplings gΛc Λc σ and gΣc Σc σ . To explore the sensitivity of our results to these couplings, in this work we consider three different sets of values for them that, together with those for the pseudoscalar and vector meson couplings, define three models for the Yc N interaction. From now on we will refer to these models simply as A, B and C. In model A the couplings of the σ meson with the charmed baryons are assumed to be equal to its couplings with the Λ and Σ hyperons, and their values are taken from the original Y N potential A˜ of the J¨ ulich group [39]. In models B and C these couplings are reduced by 15% and 20%, respectively, with respect to model A. The coupling gN N σ have been taken, for the three models, equal to that of the J¨ ulich A˜ Y N potential. Taking the values αP = 0.4, αv = 1 and αt = 0.4 employed in [39], we obtain the couplings reported in Table I where we also show the cutoff masses ΛBBM of the monopole form factors of the different vertices. We note that, to describe the nucleon-nucleon data quantitatively, the coupling gN N ω in the J¨ ulich A˜ Y N model was increased by a factor 1.626 with respect to its SU(3) value, gN N ω = 3gN N ρ , thereby accounting for missing shortrange correlations in an effective way. In the present work, we apply the same increasing factor to the gΛc Λc ω and gΣc Σc ω coupling constants of Eq. (5). We note also that the relation of these coupling constants to gN N ρ is a factor of two smaller than that obtained in the SU(3) sector, while the relations in Eq. (4), involving charmed baryons and the π, ρ, D and D∗ mesons, are the same as those involving their counterparts in the strange sector. The three Yc N interaction models have then been used to solve the coupled-channel (Λc N , Σc N ) Lipmann– Schwinger equation to obtain several scattering observables from the corresponding scattering amplitudes. The Λc N phase shifts in the 1 S0 and 3 S1 partial waves are shown as a function of the center-of-mass kinetic energy

4 70 Model A Model B Model C LQCD extrapolation at mπ=138 MeV

60 Phase shift δ [deg]

50

1

Model A Model B Model C Ref. [38] as −2.60 −1.11 −0.84 −0.85 · · · −1.00 rs 2.86 4.40 5.38 2.88 · · · 2.61

3

S0

S1

40

at −15.87 rt 1.64

30 20 10 0 -10

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 C. M. kinetic energy E [MeV] C. M. kinetic energy E [MeV]

-3

Diagonal matrix element <ΛcN|V|ΛcN> [MeV fm ]

FIG. 2: (color on-line) 1 S0 and 3 S1 Λc N phase shift as a function of the center-of-mass kinetic energy. Results are shown for models A, B and C. The band shows the extrapolation to the physical pion mass of the recent results of the HAL QCD Collaboration [37] made by Haidenbauer and Krein in Ref. [38].

15 3

1

S1

S0

10 5 0 -5

Model A Model B Model C

-10 -15 -20

0

10 5 -3 Relative momentum q [fm ]

15

0

10 5 -3 Relative momentum q [fm ]

FIG. 3: (color on-line) 1 S0 and 3 S1 Λc N → Λc N diagonal matrix element as a function of the relative momentum q. Results are shown for models A, B and C.

in Fig. 2 for the three models. The extrapolation to the physical pion mass of the recent results of the HAL QCD Collaboration [37] made by Haidenbauer and Krein in Ref. [38] is shown by the green band for comparison. One can clearly see from the phase shifts that model A predicts a more attractive Λc N interaction in the 1 S0 and 3 S1 partial waves than the one derived in Ref. [38]. The reduction of the gΛc Λc σ and gΣc Σc σ couplings in models B and C leads to a reduction of attraction in these two partial waves which translates into a qualitatively better agreement between the phase shifts predicted by these two models and those obtained from the interaction of Ref. [38], particularly in the low energy region. Note that the interaction derived in [38] predicts similar phase shifts for both partial waves since the corresponding 1 S0 and 3 S1 potentials are almost identical, a feature already noted by the HAL QCD Collaboration at different values of the pion mass (see Ref. [37]) that seems to persist when extrapolating the lattice results to the phys-

15

−1.52 2.79

−0.99 3.63

−0.81 · · · −0.98 3.50 · · · 3.15

TABLE II: Singlet and triplet Λc N scattering length and effective range predicted by the models A, B and C. The results of the extrapolation to the physical pion mass of the recent results of the HAL QCD Collaboration [37] made by Haidenbauer and Krein in Ref. [38] are shown in the last column. Units are given in fm.

ical point. This, however, is not the case of our models A, B and C which predict more overall attraction in the 3 S1 partial wave as it can be seen for example in Fig. 3 where we show the diagonal 1 S0 and 3 S1 matrix element in momentum space of the Λc N → Λc N channel. For completeness we report in Table II the singlet and triplet Λc N scattering length and the effective range predicted by the three models. The results obtained by Haidenbauer and Krein in Ref. [38] are shown for comparison in the last column of the table. There is a good agreement between model C and the result of [38] for both scattering lengths. However, is it pointed out in Ref. [38] that the scattering lengths at the physical pion mass could in fact be as larger as −1.3 fm if the uncertainty of ±0.2 fm, given by the HAL QCD Collaboration for their result at mπ = 410 MeV, is combined with the observation that variations in the scattering lengths of ±0.05 fm at this value of the pion mass amount to differences of about ±0.1 fm at mπ = 138 MeV. In this case, the prediction of model B would be in better agreement with the result of Haidenbauer and Krein than model C. Model A predicts a singlet effective range compatible with that obtained in Ref. [38] although a smaller triplet one. On the other hand, models B and C give a singlet effective range larger than that of [38] but their agreement is qualitatively better for the triplet one.

III.

Λc SINGLE-PARTICLE PROPERTIES IN FINITE NUCLEI

Here we briefly describe a method to obtain the Λc single-particle energies in a finite nucleus using an effective in-medium Yc N interaction derived from the bare Yc N potential presented in the previous section. The starting point of this method is the calculation of all the Yc N G-matrices, which describe the interaction between a charmed baryon (Yc = Λc , Σc ) and a nucleon in infinite nuclear matter. The G-matrices are obtained by solving the coupled-channel Bethe–Goldstone equation, written

5 schematically as GYc N →Yc′ N ′ (ω) = VYc N →Yc′ N ′ +

X

VYc N →Yc′′ N ′′

ΛC

The finite nucleus Λc self-energy can be obtained in the so-called Brueckner–Hartree–Fock approximation using

N

N

~

QYc′′ N ′′ × GY ′′ N ′′ →Yc′ N ′ (ω) , (6) ω − ǫYc′′ − ǫN ′′ + iη c

which involves the nuclear matter G-matrix and the difference between the finite nucleus and the nuclear matter propagators, written schematically as (Q/E)F N −(Q/E). This difference, which accounts for the relevant intermediate particle-particle states has been shown to be quite small (see Refs. [40–45]) and, therefore, in all practical calculations GF N can be well approximated by truncating the expansion (7) up second order in the nuclear matter G-matrix. Therefore, we have Q Q GF N ≈ G + G G. (8) − E FN E

G

N

Yc′′ N ′′

where V is the bare Yc N potential derived in the previous section, Q is the Pauli operator, that prevents the nucleon in the intermediate state Yc′′ N ′′ from being scattered below the Fermi momentum kFN , and ω is the nuclear matter starting energy which corresponds to the sum of the masses and the non-relativistic energies of the interacting charmed baryon and nucleon. We note that the Bethe–Goldstone equation has been solved in momentum space including partial waves up to a maximum value of the total angular momentum J = 4. We note also here that the so-called discontinuous prescription has been adopted, i.e., the single-particle energies ǫYc′′ and ǫN ′′ in the denominator of Eq. (6) are simply taken as the sum of the non-relativistic kinetic energy plus the mass of the corresponding baryon. The finite nucleus Yc N G-matrix, GF N , can be obtained, in principle, by solving the Bethe–Goldstone equation directly in the finite nucleus [47, 48]. The Bethe–Goldstone equation in finite nucleus is formally identical to Eq. (6), the only difference being the intermediate particle-particle propagator (i.e., Pauli operator and energy denominator), which corresponds to that in the finite nucleus. Alternatively, one can find the appropiate GF N by relating it to the nuclear matter Yc N G-matrix already obtained. Eliminating the bare interaction V in both finite nucleus and nuclear matter Bethe– Goldstone equations it is not difficult to write GF N in terms of G through the following integral equation: Q Q GF N = G + G GF N − E E F N Q Q = G+G G − E FN E Q Q Q Q + G G G − − E FN E E FN E + ··· , (7)

ΛC

ΛC

+

Λ ,Σ C

C

G

GFN ΛC

ΛC

G

ΛC (a)

(b)

(c)

FIG. 4: Brueckner–Hartree–Fock approximation to the finite nucleus Λc self-energy (diagram (a)), split into the sum of a first-order contribution (diagram (b)) and a second-order 2p1h correction (diagram (c)).

the GF N as an effective Yc N interaction, as it is shown in diagram (a) of Fig. 4. According to Eq. (8) it can be split into the sum of the diagram (b), which represents the first-order term on the right-hand side of Eq. (8), and the diagram (c), which stands for the so-called twoparticle-one-hole (2p1h) correction, where the intermediate particle-particle propagator has to be viewed as the difference of propagators appearing in Eq. (8). Schematically, it reads X X ΣBHF = hΛc N |GF N |Λc N i ≈ hΛc N |G|Λc N i N

+

X

Yc N

hΛc N |G|Yc N i

N

Q E

FN

−

Q E

hYc N |G|Λc N i . (9)

Detailed expressions for the first-order and the 2p1h contributions to ΣBHF can be derived in close analogy to those for the finite nucleus Λ self-energy given in Refs. [42–45] being, in fact, formally identical. The interested reader is referred to these works for details on the derivation and specific expressions of both contributions. Finally, the self-energy can then be used as an effective Λc –nucleus mean field potential in a Schr¨odinger equation in order to obtain the energies and wave functions of the bound states of the Λc in a finite nucleus. The Schr¨odinger equation is solved by diagonalizing the corresponding single-particle Hamiltonian in a complete basis within a spherical box following the procedure outlined in detail in Refs. [40–45]. Note that the Hamiltonian includes also the Coulomb potential since the Λc is a positively charged baryon. IV.

RESULTS

The energy of Λc single-particle bound states in 5Λc He, 209 91 41 , 17 Λc O, Λc Ca, Λc Zr and Λc Pb are shown in Table III for the three models considered. For comparison the

13 Λc C

6 5 Λc He

5 Λ He

13 Λc C

13 Λ C

17 Λc O

17 Λ O

Model A Model B Model C

JA˜

Model A Model B Model C

˜ JA

Model A Model B Model C

JA˜

1s1/2 −13.58 1p3/2 −1.74 1p1/2 −0.39 1d5/2 − 1d3/2 − 2s1/2

−

−3.24 − − − −

−1.05 − − − −

−

−

41 Λc Ca

Model A Model B Model C 1s1/2 1p3/2 1p1/2 1d5/2 1d3/2 1f7/2 1f5/2

−1.49 − − − −

−27.26 −14.91 −13.42 −4.10 −2.13

−

−3.59

41 Λ Ca

JA˜

−10.20 −2.13 −1.03 − −

−5.47 − − − −

−

−

91 Λc Zr

Model A Model B Model C

−7.84 − − − −

−31.76 −19.99 −18.79 −9.02 −6.96

−

−7.13

91 Λ Zr

˜ JA

−12.47 −4.32 −3.22 − −

−6.96 −0.51 − − −

−

−

209 Λc Pb

Model A Model B Model C

−10.04 −0.33 −0.35 − − −

209 Λ Pb

JA˜

−41.09 −32.39 −31.60 −23.10 −21.84 −13.54 −11.82

−16.89 −10.41 −9.67 −3.91 −2.74 − −

−9.60 −4.13 −3.42 − − − −

−17.33 −7.67 −7.78 − − − −

−44.76 −39.60 −39.24 −33.74 −33.17 −27.06 −26.29

−18.46 −14.27 −14.00 −9.63 −9.01 −4.65 −3.80

−10.51 −24.61 −52.52 −6.75 −17.66 −49.06 −6.49 −17.58 −48.84 −2.57 −9.12 −42.37 −1.95 −8.91 −41.97 − −1.35 −37.47 − −1.13 −37.07

−20.33 −18.28 −18.10 −12.94 −12.58 −9.11 −8.65

−10.32 −8.82 −8.64 −4.25 −3.88 −0.59 −0.10

−31.41 −27.59 −27.58 −19.24 −19.20 −10.51 −10.41

2s1/2 −20.47 2p3/2 −10.20 2p1/2 −9.24 2d5/2 −2.04 2d3/2 −0.95 − 2f7/2 − 2f5/2

−2.74 − − − − − −

− − − − − − −

− − − − − − −

−31.13 −22.81 −22.24 −14.62 −14.03 −7.90 −6.81

−8.05 −2.23 −1.45 − − − −

−1.29 − − − − − −

−6.60 −0.39 −0.38 − − − −

−40.53 −39.21 −38.95 −30.28 −29.83 −22.57 −22.10

−10.20 −9.28 −9.06 −5.36 −4.75 − −

−1.13 −0.03 − − − − −

−17.43 −7.68 −7.60 −4.85 −4.79 − −

3s1/2 3p3/2 3p1/2 3d5/2 3d3/2 3f7/2 3f5/2

−1.15 − − − − − −

− − − − − − −

− − − − − − −

− − − − − − −.

−13.41 −5.65 −5.61 − − − −

− − − − − − −

− − − − − − −

− − − − − − −

−23.80 −22.32 −21.95 −19.05 −18.33 −5.58 −5.02

−1.51 − − − − − −

− − − − − − −

−3.59 − − − − − −

4s1/2 4p3/2 4p1/2 4d5/2

− − − −

− − − −

− − − −

− − − −

− − − −

− − − −

− − − −

− − − −

−14.31 −1.19 −0.78 −0.68

− − − −

− − − −

− − − −

5s1/2

−

−

−

−

−

−

−

−

−0.52

−

−

−

TABLE III: Energy of Λc single-particle bound states of several charm nuclei from 5Λc He to 209 Λc Pb obtained for the three models considered. Results for the single-particle bound states of the Λ hyperon in the corresponding hypernuclei predicted by the ˜ Y N interaction are also shown for comparison. Units are given in MeV. original J¨ ulich A

energy of the single-particle bound states of the Λ hyperon in the corresponding hypernuclei, obtained with

the original J¨ ulich A˜ Y N interaction using the method described in the previous section, are also reported in the

7

40 Model A

-20

1s1/2 1p3/2 1p1/2 1d5/2 1d3/2

-40

2s1/2

20

Energy of the Λc single-particle bound state 1s/2 [MeV]

0

Model A Model B Model C JA˜ −31.54 −12.57 −7.11 −8.78 −19.69 −4.37 −0.58 − −18.45 −3.24 − − −8.71 − − − −6.62 − − − −7.02

−

−

−

-60 TABLE IV: Energy of Λc single-particle bound states of 17 Λc O when the coupling of the Λc N and the Σc N channels is switched off. Results for the Λ hyperon in 17 Λ O obtained with ˜ Y N interaction are also shown for comthe original J¨ ulich A parison. Units are given in MeV.

-80 Model B

20 0 -20 -40 -60 -80

Model C

20 0 -20 -40

Kinetic YcN interaction

-60

Coulomb Total

-80

0

50 100 150 Mass Number

200

FIG. 5: (color on-line) Contributions of the kinetic energy, the Yc N interaction and the Coulomb potential to the energy of the Λc single-particle bound state 1s1/2 as a function of the mass number of the Λc nuclei considered.

table. Note that all charmed nuclei (hypernuclei) considered consist of a closed shell nuclear core plus a Λc (Λ) sitting in a single-particle state. Model A predicts the most attractive Λc N interaction and, therefore, it predicts Λc single-particle states more bound than models B and C, and a larger number of them as it can be seen in the table. Note that in the lack of experimental data on Λc −nuclei we cannot say a priori which one of the three models is better. However, since models B and C predict, as we saw before, scattering observables in better agreement with those extrapolated from LQCD in Ref. [38], it would not be too risky to state that these two models are probably more realistic than model A. Looking now back at the table we observe (as in the

case of single Λ−hypernuclei) a small spin-orbit splitting of the p−, d− and f −wave states in all Λc −nuclei, specially in the case of the heavier ones where it is of the order of few tenths of MeV. In addition, we also note that, since the Λc is heavier than the Λ, the level spacing of the Λc single-particle energies is, for the three models, always smaller than the corresponding one for the hypernuclei. These observations are in agreement with the results previously obtained by in Tsushima and Khanna in Refs. [31–33] using the quark-meson coupling model and, later, by Tan and Ning in Ref. [34] within a relativistic mean field approach. Although there exist formal differences between our calculation and those of Refs. [31–34] that give rise to different predictions for the Λc single-particle bound states in finite nuclei, our results (particularly those for models B and C) are in general compatible with those of these works (see e.g., tables I and II of Ref. [31] and table I of Ref. [34]). It has been pointed in Refs. [31–34] and, more recently, also in Ref. [37] that the Coulomb interaction plays a nonnegligible role in Λc −nuclei, and that their existence is only possible if their binding energy is larger than the Coulomb repulsion between the Λc and the protons. To understand better the effect of the Coulomb force in our calculation, in Fig. 5 we explicitly show the contributions of the kinetic energy, the Yc N interaction and the Coulomb potential to the energy of the Λc single-particle bound state 1s1/2 as a function of the mass number (A = N + Z + 1, with N the neutron number and Z the atomic number) of the Λc −nuclei considered. Note that, while the Coulomb contribution increases because of the increase of the number of protons with the atomic number, those of the kinetic energy and the Yc N interaction decrease when going from light to heavy Λc −nuclei. The kinetic energy contribution decreases with the mass number because the wave function of the 1s1/2 state (see Fig. 6) becomes more and more spread due to the larger extension of the nuclear density over which the Λc wants to be distributed. The increase of the nuclear density with A leads to a more attractive Λc self-energy (see e.g., figures 2 and 3 of Ref. [45] for a detail discussion in the

1.5

1.5

1.5 13 C Λc

5 He Λc

Model A Model B Model C

1

17 O Λc

1

1

0.5

0.5

0.5

0

0

0

0.5

0.2

1

-3

Probability density [fm ]

-3

Probability density [fm ]

8

41 Ca Λc

0.8 0.6

0.3

0.4

0.2

0.2

0.1

209 Pb Λc

91 Zr Λc

0.4

0.1

0

0

1

2 3 r [fm]

4

5

0

0

1

2 3 r [fm]

4

5

0

0

1

2 3 r [fm]

4

5

FIG. 6: (color on-line) Λc probability density distribution for the 1s1/2 state in the six Λc −nuclei considered. Results are shown for the three models A, B and C of the Yc N interaction. Dashed lines show the result when the Coulomb interaction is artificially switched off.

case of single Λ−hypernuclei) that translates into a more negative contribution of the Yc N interaction. Note that, when adding the three contributions, the energy of the 1s1/2 bound state decreases by several MeV in the low mass number region and then it tends to saturate (being almost constant for model C) for heavier nuclei. This is due to a compensation between the attraction of the Yc N interaction and the repulsion of the Coulomb force. We note that this compensation leads, particularly in the case of model B, to values of the Λc single-particle bound state energies similar to those obtained for single Λ-hypernuclei with the J¨ ulich A˜ Y N potential (see Table III). We want to point out that even the less attractive one of our Yc N interactions (model C), despite the Coulomb repulsion, is able to bind the Λc in all the nuclei considered. This is in contrast with the recent results of the HAL QCD Collaboration [37] which suggest that only light or medium-mass Λc −nuclei could really exist. However, we note that this conclusion is based on results obtained for a value of the pion mass of 410 MeV which give rise to a Yc N interaction much less attractive than ours and the one derived in Ref. [38] when these lattice results are extrapolated to the physical pion mass (see figures 1 and 2 of Ref. [38]). Now we would like to focus the attention of the reader for a while on the effect of the coupling of the Λc N and Σc N channels. These two channels are located at approximately 3224 and 3394 MeV, respectively. Being separated by about 170 MeV it is expected, as it was already pointed out by Tsushima and Khanna (see e.g., [31, 32]), that the effect of their coupling on charmed nuclei will be less important than that of the ΛN and

ΣN channels (separated only by ∼ 80 MeV) on hypernuclei. This is illustrated in Table IV where we show as an example the energy of the Λc (Λ) single-particle 17 states bound states of 17 Λc O (Λ O) when the Λc N − Σc N (ΛN − ΣN ) coupling is switched off. Note that the differences between the levels obtained with the complete coupled-channel calculation for 17 Λc O (see Table III) and without the Λc N − Σc N coupling are almost negligible, being of the order of few tenths of MeV or less, whereas those for 17 Λ O are slightly larger than 1 MeV. Note also that the elimination of the coupling between the Λc N and Σc N channels leads, in the case of models B and C, to a bit more of attraction, contrary to what happens in the hypernuclei case where the Λ bound states become less bound when the ΛN − ΣN coupling is eliminated. We finish this section by showing in Fig. 6, for the three models, the probability density distribution (i.e., the square of the radial wave function) of the Λc in the 1s1/2 state for the six Λc −nuclei considered. The result when the Coulomb interaction is artificially schitwed off is also shown for comparison (dashed lines). Note that, due to the increase of the nuclear density, when moving from light to heavy nuclei the probability density of finding the Λc close to the center of the nucleus decreases, and it becomes more and more distributed over the whole nucleus. Note also that, as expected, due to the Coulomb repulsion the Λc is pushed away from the center of the nuclei. A similar discussion can be done for the probability densities of the the other Λc single-particle bound states.

9 V.

SUMMARY AND CONCLUSIONS

In this work we have determined the single-particle energies of the Λc charmed baryon in several nuclei. To such end, we have developed a charmed baryon-nucleon interaction based on a SU(4) extension of the meson-exchange hyperon-nucleon potential A˜ of the J¨ ulich group. We have considered three different models (A, B and C) of this interaction that differ only on the values of the couplings of the scalar σ meson with the charmed baryons. Several scattering observables have been computed with the three models and compared with those predicted by the Yc N interaction derived by Haidenbauer and Krein [38] from the extrapolation to the physical pion mass of the recent results of the HAL QCD Collaboration [37]. Qualitative agreement has been found between the predictions of our models B and C and those of the model by Haidenbauer and Krein [38]. The three models have then been used to obtain the self-energy of the Λc in finite nuclei by using a many-body approach that started with the construction of a nuclear matter Yc N G-matrix from which a finite nucleus one was derived through a perturbative expansion. Using the resulting Λc self-energy as an effective Λc −nucleus mean field potential in a Schr¨odinger equation we have finally obtained the energies and wave functions of the bound states of the Λc in the different nuclei. Our results (particularly those for models B and C) are compatible with those obtained by Tshushima and Khanna [31–33] and Tan and Ning [34], despite the formal differences between our calculation and those of these works based, respectively, on the quark-meson coupling model and the relativistic mean field approach. A small spin-orbit splitting of the p−, d− and f −wave states has been found as in the case of single Λ-hypernuclei. It has been also observed that level spacing of the Λc singleparticle energies is smaller than the corresponding one for hypernuclei.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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We have analyzed the role played by the Coulomb potential on the energies of the Λc single-particle bound states. This analysis has shown that the compensation between the Yc N interaction and the repulsion of the Coulomb force leads, particularly in the case of model B, to values of the Λc single-particle bound state energies similar to those obtained for the single Λ-hypernuclei with the original J¨ ulich A˜ Y N potential. The analysis has also shown that, despite the Coulomb repulsion, even the less attractive one of our Yc N interactions (model C) is able to bind the Λc in all the nuclei considered. This is in contrast with the recent results of the HAL QCD Collaboration [37] which suggest that only light or medium-mass Λc −nuclei could really exists. However, the conclusion of this work is based on results obtained for a value of the pion mass of 410 MeV which give rise to a Yc N interaction much less attractive than ours and the one derived in Ref. [38] when these lattice results are extrapolated to the physical pion mass. Finally, we have shown that the effect of the coupling of the Λc N and Σc N channels on the single-particle properties of charmed nuclei is much less important (being in fact almost negligible) than that of the ΛN and ΣN channels on the corresponding properties of single Λhypernuclei, due to the large mass difference of the Λc and Σc baryons of ∼ 170 MeV. Acknowledgments

The authors are very grateful to Johann Haidenbauer for his useful comments. This work has been partly supported by the COST Action CA16214 and by the Spanish Ministerio de Economia y Competitividad (MINECO) under the project MDM-2014-0369 of ICCUB (Unidad de Excelencia “Mar´ıa de Maeztu”), and, with additional European FEDER funds, under the project FIS2017-87534P.

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