Charmed and beautiful baryonic systems (hypernuclei) in chiral soliton models
aa r X i v : . [ nu c l - t h ] J a n Charmed and beautiful baryonic systems (hypernuclei)in chiral soliton models
V.B. Kopeliovich a,b ∗ and D.E. Lanskoy c † a) Institute for Nuclear Research of RAS, Moscow 117312, Russiab) Moscow Institute of Physics and Technology (MIPT), Dolgoprudny, Moscow district, Russiac) Department of Physics, Moscow State University, Moscow, Russia Abstract
The binding energies of baryonic systems with quantum number charm or beauty and neutronexcess (or large isospin) are estimated within the bound state version of the chiral topologicalsoliton model. The procedure of the skyrmion rescaling is applied which is of importance for largeenough flavor excitation energies (for heavy flavors). Inclusion of the flavor excitation energies intothe minimization of the total mass of the baryonic state naturally leads to its decrease, and thiseffect increases with increasing flavor mass. Our results concerning the binding energies turned outto be partly negative.
The nature of baryonic states (nuclear fragments) with very unusual properties, in particular, withheavy flavors - charm and/or beauty, is of interest not only for the nuclear physics itself, but incosmology and astrophysics as well, because production and subsequent decay of such states couldplay an important role at the early stages of the Universe evolution, which are not well understood sofar. Studies of charmed nuclei are the unique tool to understand the low-energy interactions ofcharmed particles. Generally speaking, the Nature arrangement is based on 6 quark flavors obeyingunitary SU(6) symmetry, though strongly violated. On the other hand, our knowledge of baryonicinteractions is drastically asymmetric in this sense and mainly corresponds only to the SU(2) subgroup(nucleons). Strange hypernuclei are known to give the unique possibility of extension our knowledge tothe strange sector (for reviews, see [1] and other references). Then, studies of charmed and beautifulnuclei is the way to extend the picture of the low-energy hadronic interactions to the SU(4) and theSU(5) world. The earlier theoretical treatment of the heavy flavored hypernuclei properties can befound in [2] - [16] mostly within various potential approaches.The chiral (topological) soliton models provide conceptionally different approach to this prob-lem which has both some advantages and certain disadvantages. The attempt to estimate the totalbinding energies of heavy flavored hypernuclei within the chiral soliton approach (CSA) was made in[34] where the total binding energies of states with flavor up to | F | = 2 have been roughly estimatedfor baryon numbers up to B = 4, and somewhat later in [36] where estimates have been extended upto B = 8 using the rational map ansatz by Houghton, Manton and Sutcliffe [37]. The tendency of ∗ e-mail : [email protected] † e-mail : [email protected] A possibility of existence of the Λ c nuclei was first discussed in [2] soon after the discovery of charm.Then a number of theoretical papers appeared [3]-[11] studying properties of charmed nuclei. Mostly,meson-exchange Λ c N interactions or schematic approaches adopting some model Λ c A potentials wereutilized. This stage of investigations was summarized in [12]. Then the theoretical activity becomespoor, probably due to lack of empirical information. In the more recent years, few studies werepublished [13, 14, 15, 16].So far, charmed nuclei have not been observed confidently. Some candidate events were pro-posed in the emulsion experiment [17]. On the other hand, negative results were reported by threeother experimental groups [18, 19, 20], which used nuclear emulsions too. Other ways to producecharmed and bottomed nuclei were discussed in [12, 21, 22].Theoretical studies of Λ c nuclei naturally use the rich experience achieved in the field of strangehypernuclei. On the other hand, Λ c -nucleus dynamics differs from the Λ s one in some substantialpoints. Below, we briefly discuss main qualitative features of charmed hypernuclei, following Ref. [12]to some extent, emphasizing mostly properties, which are different from those of strange hypernuclei.Of course, quantitative predictions unlikely can be reliable in this stage.1. Seemingly all theoretical considerations predict attraction between Λ c and nuclei. The intensityof this attraction varies, however, in different approaches. When coupling constants Λ s Λ s m andΛ c Λ c m for nonstrange (and noncharmed) mesons m (or nonrelativistic Λ s N and Λ c N potentials)are supposed to be equal to each other (e.g., [3, 6]), Λ c in light systems appears to be bounddeeper by the strong interaction than Λ s due to the greater mass and corresponding reductionof the kinetic energy (but see discussion of the Coulomb potential below). On the other hand,it is possible that the Λ c N interaction is weaker than the Λ s N one ([12] and references therein).In this case, the nuclear well forΛ c is shallower and its depth can be, for instance, only a half of that for Λ s . In the first case(comparable Λ s N and Λ c N interaction), the charmed deuterons are probably unbound, but thecharmed tritons can exist [3, 10, 16]. In the second case (a shallow well) charmed nuclei becomebound only at relatively large A (maybe, at A >
10 [12, 14]).2. The important role in the Λ s N interaction is played by kaonic exchange, which leads to Majorana(space-exchange) interaction. This implies the strong orbital moment dependence of the inter-action: The considerable attraction in the s wave and relatively weak interaction (maybe evenslight repulsion) in the p wave [26]. For Λ c hyperon, D meson exchange is extremely short-rangeand probably negligible. So one may expect comparable s and p wave attractions [9, 11]. The2ituation is similar to that for the double-strange Ξ hypernuclei, where the well depth probablygrows substantially with A [23]. Probably, the nuclear potential for Λ c also deepens with A .Some dependence of the well depth on A is seen in [9], but this dependence was not studied sofar.3. The important role of Coulomb Λ + c A repulsion was shown first in [6]. It appears that, contraryto strange hypernuclei, Λ c binding energy B Λ c does not saturate, but rather reaches a maximumin the vicinity of Ca ( A ∼
40) and then falls. In the case of a deep well, charmed nuclei remainbound up to heaviest ones. But for a shallow well, they exist only in the finite range of A (island,in terms of [12]). Even for a relatively deep well, 20 MeV, bound states exist only at A < A dependence was studied only with constant well depths. If the well deepenswith A , it can provide a competitive effect.4. In the early stage of strange hypernuclear physics, production and decay of Λ s hypernuclei wereinvestigated in emulsion experiments [24]. Identification of hypernuclei was performed by theirweak decay. A hypernucleus can be produced in the ground or some excited state. But adiscrete excited state deexcites by γ emission (nonobserved in emulsions) well before the weakdecay. Therefore, only hypernuclear ground states were available for study. Kolesnikov et al.pointed out [6] that discrete excited states of Λ c hypernuclei probably decay weekly rather thanelectromagnetically since the Λ c lifetime is by few orders of magnitude smaller than that of Λ s .So, one may expect that not only ground state can be observed in an emulsion experiment.To our knowledge, nobody calculated so far rates of electromagnetic transitions between Λ c hypernuclear levels (they depend strongly on unknown excitation energies), but evidently theelectromagnetic lifetimes are greater than the Λ c lifetime (2 · − s) often, if not always.5. As for heavier charmed hyperons, Σ c can convert (Σ c N → Λ c N ) quickly in a nucleus similarlyto Σ s . However, Ξ c ( C = +1, S = − usc or dsc ) hypernuclei can be interesting. The keypoint [12] is that the energy release from the conversion Ξ c + N → Λ c + Λ is very small, namely,5–6 MeV [25]. It is possible even that Ξ c hypernuclei are stable with respect to the stronginteraction (Ξ c cannot convert) due to nuclear binding. Even if this is not the case, the widthsare expectedly rather small. Strange charmed nuclei have been considered only very briefly[3, 12] without taking into account Ξ c N − Λ c Λ mixing. In view of current studies of double-strangeness hypernuclei [27, 28], one can say that there exists very strong baryonic mixing instrange charmed nuclei. To our knowledge, nobody studied the mixing of baryonic states in C = +1, S = − c states, but rather superpositions of Ξ c N and Λ c Λ states.6. Λ b nuclei were considered [6, 12, 13, 14] in similar lines. Differences between Λ c and Λ b nucleioriginate evidently from the greater mass and zero electric charge of Λ b . The starting point of the CSA, as well as of the chiral perturbation theory, is the effective chirallagrangian written in terms of the chiral fields incorporated into the unitary matrix U ∈ SU (2) inthe original variant of the model [47, 48], U = cos f + isin f ~τ~n , n z = cosα, n x = sinα cosβ, n y = sinα sin β , where functions f (the profile of the skyrmion), and angular functions α, ; β in general caseare the functions of 3 coordinates x, y, z . To get the states with flavor S, c or b we make extension ofthe basic U ∈ SU (2) to U ∈ SU (3) with ( u, d, s ), ( u, d, c ) or (( u, d, b ).3t is convenient to write the lagrangian density of the model in terms of left (or right) chiralderivative l µ = ∂ µ U U † = − U ∂ µ U † (3 . L = − F π l ρ l ρ + 1 e [ l ρ l τ ] + F π m π T r ( U + U † −
2) (3 . δM are due to the term in the lagrangian L M ≃ − ˜ m D Γ s ν λ , (3 . s ν = sin ( ν ), ν is the angle of rotation into ”strange” direction, ˜ m K = F D m D /F π − m π includes the SU (3)-symmetry violation in flavor decay constants, the quantity Γ, proportional to the sigma - termΓ( λ ) ≃ F π Z (1 − c f ) λ d r. (3 . B = 1 configuration, Γ ∼ Gev − moments of inertia Θ π ∼ (5 − Gev − , Θ K ∼ (2 − Gev − , see [ ? , 42] and references here. All moments of inertia Θ ∼ N c .The advantage of the CSA consists in the possibility to consider baryonic states with differentflavors - strange, charmed or beautiful - and with different atomic (baryon) numbers from uniquepoint of view, using one and the same set of the model parameters. The properties of the systemare evaluated as a function of external quantum numbers which characterize the system as a whole,whereas the hadronic content of the state plays a secondary role. This is in close correspondence withstandard experimental situation where e.g. in the missing mass experiments the spectrum of statesis measured at fixed external quantum numbers - strangeness or other flavor, isospin, etc. The socalled deeply bound antikaon-nuclei states have been considered from this point of view in [44] not incondratiction with data (this is probably one of most striking examples).Remarkably that the moments of inertia of multiskyrmions carry information about theirinteractions. Probably, the first example how it works are the moments of inertia of the toroidal B = 2 biskyrmion. The orbital moment of inertia θ J is greater than the isotopic moment of inertia θ I ,as a result, the quantized state with the isospin I = 0 and spin J = 1 (analogue of the deuteron) hassmaller energy than the state with I = 1 , J = 0 (quasi-deuteron, or nucleon-nucleon scattering state),in qualitative agreement with experimental observation that deuteron is bound stronger [45, 46].The total binding energies of strangeness S = − ∼
32 and isospin up to ∼ − ∼
10) [41]. The variant of the model with the 6-th order term in chiral derivatives in the lagrangiandensity has been included, but flavors strangeness, charm or beauty have not been involved in thisconsideration.Recently a variant of the model with the 6-th order stabilizing term in the lagrangian attractedmuch attention [29] - [32], and it has been noted that the binding energies of heavy nuclei are describedbetter in this variant than in the original variant with the Skyrme stabilizing term. The contribution of the chiral and flavor symmetry breaking mass terms into the baryon mass equals to δM SB = m π m K C S Γ , where the first part may be interpreted as sigma-term, Σ = m π Γ /
2. For Γ = 6
Gev − we get Σ ≃ MeV .
4n view of this moderate success we can hope that further studies of baryonic states withdifferent quantum numbers in framework of the CSA, including states with unusual properties, maybe of interest and useful.Previously estimates of the flavor excitation energies were made mostly in perturbation theory,i.e. the flavor excitation energy has been simply added to the skyrmion energy. This is not justified,however, when the flavor excitation energy is large. Here we include this energy into simplifiedminimization procedure which is made by means of the change of the soliton dimension (rescalingof the soliton). This procedure takes into account the main degree of freedom of skyrmions givenby the rational map anzatz [37] and leads to considerable decrease of the energy of states. Thismodification of the skyrmion was made, in particular, by B.Schwesinger et al [33] to improve thedescription of strange dibaryon configurations.
In this section we present some static properties of multiskyrmions which are necessary to performthe procedure of the SU (3) quantization and to obtain the spectrum of states with definite quantumnumbers.The flavored moment of inertia equals within the rational map approximation for the originalvariant of the model with the 4-th order in chiral derivatives term as the soliton stabilizer (the SK λ to make evident the behaviourunder the rescaling procedure r → rλ )Θ SK F = λf + λ f (0)3 F D F π = θ (0) F + λ f (0)3 F D F π − ! (4 . f = π e Z (1 − c F ) (cid:18) f ′ + 2 B s f r (cid:19) r dr ; f (0)3 = π F π Z (1 − c F ) r dr. (4 . λ . In Table 1 we present numerical values for f and f .There is simple connection between total moment of inertia in the SK θ SK F and the sigma-term: θ tot,SK F = F D F π Γ + θ SK F = F D F π f (0)3 + f . (4 . t ( SK f ( SK t ( SK f ( SK
4) Γ( SK t ∗ ( SK f ∗ ( SK t ∗ ( SK f ∗ ( SK .
64 0 .
85 2 .
92 1 .
20 4 .
80 6 .
67 2 .
14 6 .
13 2 .
522 6 .
69 1 .
84 4 .
81 2 .
34 9 .
35 13 .
57 4 .
64 10 .
73 5 .
223 8 .
14 2 .
84 6 .
26 3 .
50 14 . .
53 7 .
18 14 .
17 7 .
924 9 .
37 3 .
77 7 .
43 4 .
50 18 . .
35 9 .
38 16 .
55 10 .
025 14 .
67 4 .
85 8 .
83 5 .
95 23 . .
76 12 . .
74 13 .
36 15 .
39 5 .
85 10 .
01 7 .
25 29 . .
27 14 . .
33 16 .
187 18 .
19 6 .
62 10 .
71 8 .
08 32 . .
57 16 . .
03 18 .
128 21 .
41 7 .
68 11 .
99 9 .
72 38 . .
96 19 . .
94 21 .
859 24 .
28 9 .
02 13 .
52 11 .
58 46 . .
41 21 . .
61 25 . .
85 10 . .
55 13 . . .
79 24 . .
74 28 . .
54 10 .
98 15 .
66 14 .
62 58 . .
26 27 . .
74 31.512 31 .
91 11 .
98 16 .
59 16 .
02 64 . .
28 29 . .
72 34.513 34 .
54 12 .
95 17 .
56 17 .
55 70 . .
23 31 . .
77 37.75
Table 1.
The Skyrme term contribution to the isotopic moment of inertia of multiskyrmions Θ I (SK4),the ”flavor” inertia Θ F (SK4) and the sigma term Γ in the SK4 variant of 2 the model with e = 4 . e = 3 (columns 7 — 10), in GeV − .Similarly, the isotopic moment of inertia θ I within the rational map aproximation can bewritten as Θ SK I = λ t + λ t (4 . t = 4 π Z s F e F ′ + B s F r ! r dr, t = 2 π F π Z s F r dr. (4 . SK F ( SK
6) = f λ + f /λ, (4 . I ( SK
6) = t λ + t /λ, (4 . t = 18 Z (1 − c f )2 c s f (cid:18) Bf ′ + I s f r (cid:19) dr. (4 . SK θ tot,SK F = F D F π Γ + θ SK F . (4 . θ F ( λ ) = π Z (1 − c f ) " λ F D + λe (cid:18) f ′ + 2 B s f r (cid:19) + 2 c λ s f r (cid:18) Bf ′ + I s f r (cid:19) r dr (4 . F π we used analytical approachdeveloped in [35]. Under scale transformation we have θ SK F ∼ λ − . B Θ I (SK6) Θ F (SK6) Γ(SK6) Θ I (SK6 ∗ ) Θ F (SK6 ∗ ) ∗ Γ(SK6 ∗ ) ∗ .
13 0 .
76 6 .
08 14 . .
38 15 .
32 9 .
26 1 .
44 14 . . .
62 35 .
93 12 . .
18 20 . . .
92 53 .
94 15 . .
80 24 . . .
85 64 .
65 18 . .
60 32 . . .
35 86 .
26 21 . .
28 39 . . .
65 1037 23 . .
88 42 . . . . .
60 51 . . . . .
32 59 . . . . .
05 65 . . . . .
70 73 . . . .
32 79 . . . .
02 87 . . Table 2.
Same as in Table 1, for the SK6 variant of the model. e ′ = 4 .
11 and the rescaledvariant of the model, e ′ = 2 .
84 .Here θ SK F scales like 1 /λ . The total binding energies are estimated using the double subtraction procedure.We shall use the following mass formula for the quantized state which allowed to estimate thebinding energies of hypernuclei in [39] M ( B, F, I, J ) = M cl + | F | ω F,B + 12 θ F,B [ c F I r ( I r + 1) + (1 − c F ) I ( I + 1) + (¯ c F − c F ) I F ( I F + 1)] ++ J ( J + 1)2 θ J , (5 . | F | = 1 we have in present paper I F = 1 / ω c (SK4) ω b (SK4) ω C (SK4 ∗ ) ω b (SK4 ∗ ) ω c (SK6) ω b (SK6) ω c (SK6 ∗ ) ω b (SK6 ∗ )1 1 .
54 4 .
80 1 .
55 4 .
77 1 .
61 4 .
93 1 .
62 4 .
892 1 .
52 4 .
77 1 .
54 4 .
75 1 .
64 4 .
98 1 .
66 4 .
953 1 .
51 4 .
76 1 .
54 4 .
74 1 .
64 4 .
98 1 .
66 4 .
954 1 .
50 4 .
74 1 .
52 4 .
72 1 .
62 4 .
92 1 .
64 4 .
935 1 .
51 4 .
75 1 .
53 4 .
74 1 .
63 4 .
96 1 .
65 4 .
946 1 .
51 4 .
76 1 .
54 4 .
74 1 .
634 4 .
96 1 .
65 4 .
947 1 .
50 4 .
74 1 .
53 4 .
73 1 .
623 4 .
95 1 .
64 4 .
938 1 .
51 4 .
76 1 .
54 4 .
75 1 .
63 4 .
96 1 .
65 4 .
949 1 .
52 4 .
77 1 .
54 4 .
76 1 .
63 4 .
97 1 .
65 4 . .
52 4 .
78 1 .
558 4 .
76 1 .
63 4 .
97 1 .
65 4 . .
53 4 .
79 1 .
55 4 .
77 1 .
63 4 .
97 1 .
65 4 . .
53 4 .
79 1 .
55 4 .
77 1 .
63 4 .
97 1 .
65 4 . .
53 4 .
79 1 .
55 4 .
77 1 .
63 4 .
98 1 .
65 4 . Table 3.
Flavor excitation energies for charm and beauty, for the SK SK F D /F π = 1 .
576 and F B /F π ≃ .
44 according to [49].The flavor excitation energy is ω F,B = 3 B θ F,B ( µ F,B −
1) (5 . µ F,B = " m D Γ B θ F,B B / At large enough m D the expansion can be made µ F,B ≃ m D (Γ B θ F,B ) / B + 3 B m D Γ B θ F,B , (5 . ω F,B ≃
12 ¯ m D Γ B θ F,B ! / − B θ F,B . (5 . The correction to the energy of states which is formally of the 1 /N c order in the number of flavors hasbeen obtained previously in [ ? , ? ]∆ E /N c = 12 θ I [ c F I r ( I r + 1) + (1 − c F ) I ( I + 1) + (¯ c F − c F ) I F ( I F + 1)] (6 . B is omitted for the sake of brevity, I is the isospin of the state, I F is the isospin carriedby flavored meson ( K, D, or B − meson, for unit flavor I F = 1 / I r can be interpreted as ”right”isospin, or isospin of basic non-flavored configuration. The hyperfine splitting constants c F = 1 − θ I ( µ F − θ F µ F , ¯ c F = 1 − θ I ( µ F − θ F µ F , (6 . I is large.In our previous calculations we used also the following expression for the difference of energiesbetween the state with flavor | F | isospin I and the state with zero flavor F , isospi I r which belongsto same SU (3) multiplet ( p, q ):∆ E ( B, F ) = | F | ω F + µ F − µ F θ F [ I ( I + 1) − I r ( I r + 1)] + ( µ F − µ F − µ F θ F I F ( I F + 1) (6 . The flavor excitation energy and hyperfine splitting correction have been considered previously assmall corrections to the energy of the state. Such an approach can be, however, not justified when theflavor excitation is not so small, as for heavy flavors, charm and beauty.In these cases it is reasonable to include into consideration the overall scale of the soliton andto perform further minimization of the energy as a function of this scaling parameter.Let us consider several examples of interest. B -number, | F | = 1 For the case of even baryon number, not very large, the ground state of the nucleus has zero isospin,and it belongs to the SU (3) multiplet ( p, q ) = (0 , B/ I = 0 , I r = 0 , I F = 0 . (7 . E ( B, , , ,
0) = M clB . (7 . | F | = 1 we should take I r = 0 , I = I F = 12 (7 . E ( B, | F | = 1 , I F = 1 / , I r = 0 , I = 1 /
2) = M clB + ω F + 3( µ F − θ F µ F . (7 . x and to findminimal energy E min ( x min .The next step is to estimate the change of the binding energy of state, when substitution ofthe nucleon by the Λ − hyperon was made. 9 .2 Odd B -numbers, | F | = 1 In this case for the ground state we have I F = 0 , I = I r = 1 /
2. The energy of these states E ( B, , , / , /
2) = M clB + 38 θ I . (7 . | F | = 1 , I F = 1 / , I r = 1 / , I = 0 its energy is E ( B, , / , / ,
0) = M clB + ω F + 38 θ I − µ F − θ F µ F (7 . B = 1 case should be considered in similar way. The nucleon mass, I = I r = 1 / , I F = 0: M N = M cl + 38 θ I, (7 . − hyperon mass M Λ = M cl + ω F, + 3¯ c F, θ I, = M cl + ω F, + 38 θ I, − µ F, − µ F, θ F, . (7 . M N and M Λ should be minimized separately with own scaling parameter x .We do not pretend to calculate the binding energies, but we can estimate the changes in bindingenergies of baryonic systems with flavor (hypernuclei) in comparison with nonflavored baryonic system:∆ ǫ ( B, F ) = − E B,F + M B + M Λ − M N . (7 . We have estimated the total binding energies of baryonic states (hypernuclei) with quantum numberscharm or beauty, and some neutron excess, or high isotopic spin.The rescaling procedure is important for several values of baryon number, and with increasingB-numbers it becomes less important.We thank Yura Ivanov for important help in numerical computations.
The rational map approximation for multiskyrmions, proposed in [37], allows to get analytical expres-sions describing characteristics of multiskyrmions (masses, moments of inertia, sigma term) [35], validwith an acuracy of several percents.Starting point of the anlytical treatment is parametrization of the multiskyrmion profile func-tion in the form φ = cos F = ( r/r ) b − r/r )) b + 1 , ( A . cos f (0) = − cos f ( ∞ ) = 1, and the constants r ( B ) - dimen-sion of the skyrmion, b ( B ) - the effective power, depending on the baryon number of multiskyrmion,10an be found by the static mass minimization procedure [35]. The integrals over 3-dimensional spacewhich appear in calculation of the skyrmion mass, as well as other skyrmions properties are theEuler-type integrals which can be evaluated in general enough form Z ∞ ( r/r ) c drβ + ( r/r ) b = β (1+ c ) /b − π r b sin [ π (1 + c ) /b ] . ( A . r min and b min were found for the SU (2) model with the Skyrme term as the skyrmionstabilizer to be [35] r min ≃ [2 √I / / , b min ≃ I / . ( A . Z (1 − cos F ) d r = 8 π r b sin (3 π/b ) . ( A . (2) F = Z sin F d r = 48 π r b sin (3 π/b ) , ( A .
10 Appendix 2. The classical mass of the skyrmion, and corrections
The classical mass of the soliton can be written in form (original SK M cl = λm + m /λ + λ m . ( A . θ F = λ f + λ f , ( A . θ I = t = λ t + λ t . ( A . F = o, I F = 0 , I = I r = 1 / λ ; E gr ( odd ) = M cl + 38Θ I = λm + m /λ + λ m + 3 λt + λ t ( A . | F | = 1 , I F = 1 / , I = 0) can be eathearly written.11 eferences
1. H. Band¯o, T. Motoba, J. ˘Zofka. Int.J.Mod.Phys. A5(1990)4021; O. Hashimoto,H. Tamura. Prog.Part.Nucl.Phys. 57(2006)564; A. Gal, E.V. Hungerford, D.J. Mil-lener. Rev.Mod.Phys. 88(2016)035004.2. A.A. Tyapkin, Sov.J.Nucl.Phys. 22 (1975) 89 [Yad.Fiz. 22 (1975) 181]3. C.B. Dover and S.H. Kahana, Phys.Rev.Lett. 39 (1977) 15064. S. Iwao. Lett.Nuov.Cim. (1977)647.5. R. Gatto, F. Paccanoni. Nuov.Cim. A46 (1978)313.6. N.N. Kolesnikov et al. Yad.Fiz. (1981)957.7. G. Bhamathi. Phys.Rev. C24 (1981)1816.8. H. Band¯o, M. Band¯o. Phys.Lett.
B109 (1982)164.9. H. Band¯o, S.Nagata. Prog.Theor.Phys. (1983)557; H. Band¯o. Prog.Theor.Phys.Suppl. (1985)197.10. B.F. Gibson et al. Phys.Rev. C27 (1983)2085.11. Y. Yamamoto. Prog.Theor.Phys. (1986)639.12. N.I. Starkov, V.A.Tsarev. Proc. of 1986 INS Int. Symp. on Hypernuclear Physics,ed. by H. Band¯o, O. Hashimoto, K.Ogawa, p.247; Nucl.Phys. A450 (1986) 507;S.A.Bunyatov et al. Sov.J.Part.Nucl. (1992)581.13. K. Tsushima, F.C. Khanna. Phys.Rev. C67 (2003)015211; Prog.Theor.Phys.Suppl. (2003)160.14. Y.-H. Tan et al. Commun.Theor.Phys. (2003)473; Phys.Rev. C70 (2004)054306.15. B. Julia-D´ıaz, D.O. Riska. Nucl.Phys. A755(2005)431.16. H. Garcilazo, A. Valcarce, T.F. Carames. Phys.Rev.
C92 (2015)024006.17. Yu.A. Batusov et al., JETP Lett. 33(1981)52; V.V.Lyukov. Nuov.Cim. A102(1989)583.18. G. Coremans-Bertrand et al., Phys.Lett. B 65 (1976) 480.19. W. Bozzoli et al., Lett.Nuov.Cim. 19(1977)32.20. G. ¨Oneng¨ut et al., Nucl.Phys. B718(2005)35.21. T. Bressani, F.Iazzi. Nuov.Cim. A102(1989)597.22. A. Feliciello. Nucl.Phys. A881 (2012)78.23. Y. Yamamoto. Few-Body Syst.Suppl. (1995)145.24. D.H. Davis, J.Pniewski. Contemp.Phys. 27(1986)91.25. C. Patrignani et al. (Particle Data Group). Chin.Phys. C40(2016)100001.26. D.E. Lanskoy, Y. Yamamoto. Phys.Rev. C55 (1997) 233027. K.S. Myint, Y. Akaishi. Prog,Theor.Phys.Suppl. 117(1994)25128. D.E. Lanskoy, Y. Yamamoto. Phys.Rev. C69 (2004) 01430329. Eric Bonenfant, Luc Marleau, Phys.Rev. D82 (2010) 054023; arXiv:1007.1396 [hep-ph]30. C. Adam, C. Naya, J. Sanchez-Guillen, A. Wereszczynski, Phys.Rev.Lett. 111 (2013)23, 232501; C. Adam, C. Naya, J. Sanchez-Guillen, A. Wereszczynski, Phys.Rev. C88(2013) 5, 05431331. Marc-Olivier Beaudoin, Luc Marleau, Nucl.Phys. B883 (2014) 328; e-Print:arXiv:1305.4944 [hep-ph]32. C. Adam, C. Naya, J. Sanchez-Guillen, J.M. Speight, R. Vazquez, A. Wereszczynski,Conference: C14-07-02 ;e-Print: arXiv:1412.1487 [hep-th]33. V.B. Kopeliovich, B. Schwesinger, and B.E. Stern, Phys. Lett. B242 (1990) 14534. V.B. Kopeliovich, JETP Lett. 67 (1998) 896 [Pisma Zh.Eksp.Teor.Fiz. 67 (1998) 854]
5. V.B. Kopeliovich, J. Phys. G: Nucl. Part. Phys. 28 (2002) 10336. V.B. Kopeliovich and W.J. Zakrzewski, JETP Lett. 69 (1999) 721 [PismaZh.Eksp.Teor.Fiz. 69 (1999) 675]; Eur.Phys.J. C18 (2000) 36937. C. Houghton, N. Manton, and P. Sutcliffe, Nucl. Phys. B 510 (1998) 50738. C.L. Schat and N.N. Scoccola. Multibaryons with heavy flavors in the Skyrme modelPhys.Rev. D61 (2000) 03400839. V.B. Kopeliovich, J.Exp.Theor.Phys. 96 (2003) 782 [Zh.Eksp.Teor.Fiz. 123 (2003) 891]40. Martin Schvellinger and Norberto N. Scoccola. Phys.Lett. B430 (1998) 3241. V.B. Kopeliovich, A.M. Shunderuk, and G.K. Matushko, Phys.Atom.Nucl. 69 (2006)120 [Yad.Fiz. 69 (2006) 124]42. V.B. Kopeliovich and A.M. Shunderuk, J.Exp.Theor.Phys. 100 (2005) 929[Zh.Eksp.Teor.Fiz. 127 (2005) 1055]43. V.B. Kopeliovich, JETP Lett. 96 (2012) 210 [Pisma Zh.Eksp.Teor.Fiz. 96 (2012) 226]44. V.B. Kopeliovich and I.K. Potashnikova, Phys.Rev. C83 (2011) 06430245. V.B. Kopeliovich, Sov. J. Nucl. Phys. 47 (1988) 949 [Yad. Fiz. 47 (1988) 1495]46. E. Braaten and L. Carson, Phys. Rev. D38 (1988) 352547. T.H.R. Skyrme, Proc.R.Soc. London, A260 (1961) 127; Nucl.Phys. 31 (1962) 55648. E. Witten, Nucl.Phys. B 223 (1983) 422, 433 ; G. Adkins, C. Nappi, and E. Witten,Nucl.Phys. B228 (1983) 55249. H. Na, C. Monahan, C. Davies et al, arXiv: 1212.0586 [hep-lat]; PoS LATTICE2012(2012) 1025. V.B. Kopeliovich, J. Phys. G: Nucl. Part. Phys. 28 (2002) 10336. V.B. Kopeliovich and W.J. Zakrzewski, JETP Lett. 69 (1999) 721 [PismaZh.Eksp.Teor.Fiz. 69 (1999) 675]; Eur.Phys.J. C18 (2000) 36937. C. Houghton, N. Manton, and P. Sutcliffe, Nucl. Phys. B 510 (1998) 50738. C.L. Schat and N.N. Scoccola. Multibaryons with heavy flavors in the Skyrme modelPhys.Rev. D61 (2000) 03400839. V.B. Kopeliovich, J.Exp.Theor.Phys. 96 (2003) 782 [Zh.Eksp.Teor.Fiz. 123 (2003) 891]40. Martin Schvellinger and Norberto N. Scoccola. Phys.Lett. B430 (1998) 3241. V.B. Kopeliovich, A.M. Shunderuk, and G.K. Matushko, Phys.Atom.Nucl. 69 (2006)120 [Yad.Fiz. 69 (2006) 124]42. V.B. Kopeliovich and A.M. Shunderuk, J.Exp.Theor.Phys. 100 (2005) 929[Zh.Eksp.Teor.Fiz. 127 (2005) 1055]43. V.B. Kopeliovich, JETP Lett. 96 (2012) 210 [Pisma Zh.Eksp.Teor.Fiz. 96 (2012) 226]44. V.B. Kopeliovich and I.K. Potashnikova, Phys.Rev. C83 (2011) 06430245. V.B. Kopeliovich, Sov. J. Nucl. Phys. 47 (1988) 949 [Yad. Fiz. 47 (1988) 1495]46. E. Braaten and L. Carson, Phys. Rev. D38 (1988) 352547. T.H.R. Skyrme, Proc.R.Soc. London, A260 (1961) 127; Nucl.Phys. 31 (1962) 55648. E. Witten, Nucl.Phys. B 223 (1983) 422, 433 ; G. Adkins, C. Nappi, and E. Witten,Nucl.Phys. B228 (1983) 55249. H. Na, C. Monahan, C. Davies et al, arXiv: 1212.0586 [hep-lat]; PoS LATTICE2012(2012) 102