Using the bound state version of the topological soliton model for the baryons we show that the existence of a bound (or quasi-bound) D-soliton state leads to the possibility of having hidden charm pentaquarks with quantum numbers and masses, which are compatible with those of the candidates recently reported by the LHCb experiment. The implications of heavy quark symmetry are elaborated.Fil: Scoccola, Norberto Nerio. Universidad Favaloro. Facultad de Ingeniería y Ciencias Exactas y Naturales; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Investigación y Aplicaciones no Nucleares. Gerencia Física (CAC). Grupo de Física Teórica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Riska, D. O.. Finnish Society Of Science And Letters; FinlandiaFil: Rho, María Gabriela. Centre D'etudes de Saclay; Franci

Rho, María Gabriela

Riska, D. O.

Scoccola, Norberto Nerio

Physical Review D

English

CONICET Digital

arXiv:1508.01172v1  [hep-ph]  5 Aug 2015On the pentaquark candidates P+c(4380) and P+c(4450) within the soliton picture ofbaryonsN.N. Scoccolaa,b,c, D.O. Riskad and Mannque Rhoea Department of Theoretical Physics, Comisio´n Nacional de Energ´ıa Ato´mica,Av.Libertador 8250, 1429 Buenos Aires, Argentinab CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentinac Universidad Favaloro, Sol´ıs 453, 1078 Buenos Aires, Argentinad Finnish Society of Science and Letters, Fabianinkatu 4 B 16, 00130 Helsinki, FinlandeInstitut de Physique The´orique, CEA Saclay, 91191 Gif-sur-Yvette, FranceUsing the bound state version of the topological soliton model for the baryons we show thatthe existence of a bound (or quasi-bound) D¯-soliton state leads to the possibility of having hiddencharm pentaquarks with quantum numbers and masses, which are compatible with those of thecandidates recently reported by the LHCb experiment. The implications of heavy quark symmetryare elaborated.PACS numbers: 12.39.Dc,12.39.Hg,14.20.PtI. INTRODUCTIONThe LHCb collaboration at CERN recently reported the discovery of two states, which in the quark model correspondto “pentaquarks” with hidden charm Ref.[1]. The first one has the mass (width) 4380± 8± 29 (205± 18± 86 ) MeVand the second the mass (width) 4449.8± 1.7± 2.5 ( 39± 5± 19 ) MeV. The preferred quantum number assignmentsare Jπ = 3/2−, 5/2+, respectively, although acceptable solutions are also found for additional cases with oppositeparity. Note that their decay channel is J/ψ p+ which implies I = 1/2. After this announcement several articles thatconsider possible theoretical interpretations of the observed states have appeared. Some of these [2–5] suggest, withsome variations, that the observed states may be interpreted as anticharmed meson (D¯ or D¯∗)-hyperon (Σ or Σ∗)molecular states, while others base their description on diquark models [6–8]. The possibility that at least one of theobserved peaks might be only a kinematical effect was discussed in Refs.[9–11]. Further suggestions have been putforward in Refs.[12, 13]. Moreover, the production and formation of the hidden-charm pentaquarks in γ - nucleoncollisions have been discussed in Refs.[14, 15].The purpose of the present note is to show that it is also possible to account for the quantum numbers and masses ofthese observed pentaquark candidates within the topological soliton picture of baryons. The possible existence of stablepentaquarks with negative charm in that framework was already considered many years ago [16]. In this approach theheavy flavor hyperons were described as bound states of heavy flavour mesons and a topological soliton [16–18] in theextension to heavy flavor of the bound state approximation [19–21] to the Skyrme model, supplemented with suitablesymmetry breaking terms. This “Naive Skyrme Model” (NSM) formulation does however only approximately [22]incorporate heavy quark symmetry (HQS) according to which in the heavy quark limit the heavy pseudoscalar andvector fields become degenerate and, therefore should be treated on an equal footing [23]. An improved way to proceedis therefore to apply the bound state approach to the heavy meson effective lagrangian [24–27], which simultaneouslyincorporates chiral symmetry and heavy quark symmetry (for details see Ref.[28] and refs. therein). The possibleexistence of a C = −1 meson bound state in the context of a model consistent with HQS was first discussed inRefs.[30, 31]. There, however, only pseudoscalar mesons as described by the Skyrme model were considered in thelight sector. In what follows we will refer to this as SMHQS formulation. It was later pointed out [32] that theinclusion of light vector mesons in the corresponding effective lagrangian tends to push this state into the continuum.On other hand, the calculation in Ref.[33], which incorporates the center of mass corrections in a more consistent way,still leaves the possibility of a loosely bound state. Additional arguments for the existence of C = −2bound state have been given in Ref.[34].In the NSM approach it is the Wess-Zumino term in the lagrangian, which is responsible for the difference in theinteraction of the soliton and the mesons with opposite massive flavor quantum number. This term is repulsive in thecase of mesons with massive antiflavor quantum number. In the case of S = +1 kaons, this repulsion, in combinationwith the repulsive effect of the meson kinetic energy term, pushes them into the continuum [35, 36]. (It should here bementioned that the indications for the existence of the conjectured strange pentaquark “θ(1540)” [37] have hithertonot been experimentally confirmed[38]). As the repulsive effect of the kinetic energy term weakens with increasingmeson mass and the strength of the Wess-Zumino term is smaller for heavy flavors, the existence of anticharm (and afortiori antibottom) meson-soliton bound states becomes possible. Below we show that the existence of a bound (orquasi-bound state) D¯-soliton state naturally leads to the possibility of having some hidden charm pentaquarks withquantum numbers and masses which are compatible with those of the candidates proposed in Ref.[1]II. BOUND STATE DESCRIPTION OF HIDDEN HEAVY FLAVOURED PENTAQUARKSSince the pentaquark candidates reported in Ref.[1] have no net charm quantum number we propose to describethem as bound states of a soliton and two pseudoscalar mesons (one charm and the other anticharm) in the presentpicture. We recall that in the bound state approximation, while a bound C = +1 meson behaves as a quark [17–21],a D¯-meson corresponds to a antiheavy-light quark pair [16, 30, 31]. To determine the possible quantum numbers andestimate the values of the associated masses we need to know which bound states of charm and anticharm mesonsare possible. In fact, both the NSM and the SMHQS models lead to a number of meson bound states. Those boundstates can be labelled by kπ, where ~k = ~i + ~ℓ and π = (−1)ℓ+1. Here, ℓ and i = 1/2 are the angular momentum andisospin of the bound pseudoscalar meson. In Table I the corresponding quantum numbers, binding energies b andhyperfine splittings constants c are given . The latter are needed for calculation of the nonadiabatic corrections to bediscussed below. These results were obtained using the standard Skyrme model parameters fπ = 64.5 MeV, e = 5.45,which lead to the empirical masses for the nucleon and the ∆ resonance. The values of the binding energies b andhyperfine splittings c associated with the C = ±1 mesons in the NSM scheme have been calculated in Refs.[16, 18].Those of the SMHQS scheme have been extracted from Ref.[29] (Set 5) except for the binding of the C = −1 state,which is taken from Ref.[33]. Note that most of the associated hyperfine splittings have not been given in those works,however. The results corresponding to the NSM have been obtained using the decay constant ratio fD/fπ = 1.8. Thisvalue has been updated in recent years. The current estimate is fD/fπ = 1.57 [41]. While the use of this value hardlyaffects the predictions for the hyperfine splitting constants, one obtains an enhancement of about 100 MeV for all theC = +1 binding energies. Since the results for the binding energies obtained with fD/fπ = 1.8 are closer to those ofthe SMHQS formulation we will assume that this overbinding is a consequence of the simplicity inherent to the NSMformulation.NSM SMHQSl kpi b(MeV ) c b(MeV ) cC = +1 1 1/2+ 568 0.20 518 0.150 1/2− 355 0.52 239 0.302 3/2− 243 0.15 2121 3/2+ 140 0.28 491 1/2+ 118 0.03 65C = −1 1 1/2+ 38 0.16 54Table I: Quantum numbers of the meson bound states and associated binding energies and hyperfine splitting constants.The single meson spectra that follow from Table I are illustrated in Fig.1. It can be seen that the meson spectrumobtained in the NSM formulation is qualitatively similar to that obtained using the SMHQS formulation. Actually,this fact was already noticed long ago [22].3-1.0-0.8-0.6-0.4-0.20.01/2+  C= +1                      C= -13/2+1/2+3/2-1/2-1/2+C= +1                      C= -1    NSM                                                  SMHQS   Binding Energy [GeV] 1/2+1/2+3/2+3/2-1/2-1/2+Figure 1: Binding energies and quantum numbers corresponding to possible soliton-(anti)charmed mesons. The left panelcorresponds to the spectrum obtained using the NSM formulation while the right panel to the SMHQS one. The lowestJpi = 3/2− configuration is indicated by the two black circles and the Jpi = 5/2+ by the open ones.Clearly, the C = −1 meson has to be bound in the kπ−− = 1/2+ state. Here, the subindex stands for the charmquantum number of the meson. For the C = +1 one, we have, however, several possibilities. The preferred quantumnumber assignments of the pentaquark candidates are Jπ = 3/2−, 5/2+, respectively. Since, obviously, π = π− π+ itfollows that π+ = −(+) for the 3/2 (5/2) pentaquarks. To determine k+ we note that, according to the usual rules,the total grand spin ~K = ~k++~k− satisfies ~K = ~J+ ~I where J and I are the total spin and isopin of the bound system.Noting that the observed states are isospin doublets, it follows that 1/2 ≤ k+ ≤ 5/2 for the state with J = 3/2 and3/2 ≤ k+ ≤ 7/2 for that with J = 5/2. From Fig.1 it is clear that the lowest lying meson configurations (kπ++ , kπ−− )that satisfy all the requirements are (1/2−, 1/2+) for the Jπ = 3/2− pentaquark candidate and (3/2+, 1/2+) for theJπ = 5/2+ one. In the figure the first configuration is indicated by the two black circles and the second by the openones. In principle many other states can be obtained by populating the alternative C = +1 bound states. The fulllist is given in Table II together with the corresponding masses up to non-adiabatic corrections. These have beenobtained usingM =Msol + 2mD − b+ − b−, (1)where Msol = 866 MeV is the soliton mass, mD the pseudoscalar charm meson mass that we take to be mD = 1867MeV and b± are the meson binding energies given in Table I.It is seen that three low lying 3/2− with masses (4.21-4.32, 4.31-4.33) GeV for (NSM,SMHQS), respectively, arepredicted in both schemes. The rather large width found in the experiment might be explained by these 3 close lyingstates. One the other hand only one single 5/2+ state at (4.42, 4.50) GeV is predicted.So far we have not included the non-adiabatic contributions to the masses. To first order perturbation theory therotational corrections can be obtained by considering12Ω< (I(k+k−)K)J |(~R − ~Θ)2|(I(k+k−)K)J >, (2)where Ω is the moment of inertia of the soliton and Θ the total isospin of the meson bound system. Moreover, ~R hasthe role of spin of the light quark system (basically the spin of the rotating soliton, which coincides with its isospin).For a system of a C = +1 and a C = −1 bound mesons one has ~Θ = ~θ+ + ~θ−. The calculation of the matrix element4I kpi++ kpi−−K Jpi NSM SMHQS1/2 1/2+ 1/2+ 0 1/2+ 4.00 4.031 1/2+1 3/2+1/2 1/2− 1/2+ 0 1/2− 4.21 4.311 1/2−1 3/2−1/2 3/2− 1/2+ 1 1/2− 4.32 4.331 3/2−2 3/2−2 5/2−1/2 3/2+ 1/2+ 1 1/2+ 4.42 4.501 3/2+2 3/2+2 5/2+1/2 1/2+ 1/2+ 0 1/2+ 4.44 4.481 3/2+1 3/2+Table II: Quantum numbers and O(N0c ) masses (in GeV).in Eq.(2) requires some assumptions and/or approximations. The usual procedure in the bound state approach (BSA)to the Skyrme model [17–21] is to employ the approximation < θ2i >= c2i ki(ki + 1). We denote this as BSA option.Here, ci is the corresponding hyperfine splitting constant. As noted in Ref.[39] this approximation does not hold inthe HQS limit, where it can be shown that < θ2i >= 3/4, since for all the cases considered the bound mesons haveisospin 1/2. We denote this as HQS option. Since it is not yet settled which the best way to proceed at the charmmass scale is, both cases will be considered here in order to determine the related uncertainty. The second one has todo with the way in which < ~R · ~Θ > is calculated (see Ref.[40]). In fact, this term might induce mixings between stateswith different K and/or k+. This effect will be neglected in what follows. The resulting formula for the rotationalcorrections to the mass of a system composed by a soliton and two bound mesons (one in a state with k+ and theother with k−) is thenMrot(I, J, k+, k−,K) =12Ω{I(I + 1) + c+c− [K(K + 1)− k+(k+ + 1)− k−(k− + 1)] + δ[J(J + 1)−K(K + 1)− I(I + 1)][c+ + c−2+c+ − c−2k+(k+ + 1)− k−(k− + 1)K(K + 1)]}(3)whereδ ={ c2+ k+(k+ + 1) + c2− k−(k− + 1) BSA3/2 HQS(4)Using the BSA option one recovers the mass formula given in Refs.[17, 18]. Using the parameters of Table I therotational corrections can then be calculated. Note that in the case of SMHQS scheme most of the hyperfine splittingsneeded for the calculation of the rotational corrections have not been given in the literature. Thus, only the predictionsas obtained in the NSM scheme are reported in Table III. The quoted values correspond to the average between theresults obtained using each of the two options for the calculation of the rotational corrections. The correspondinguncertainty is considered to be half of the difference between these two values, which turns out to be about 70 MeV.From this table we see that the prediction for the mass of the lowest 3/2− is in the range 4.33− 4.47 GeV. In the caseof the 5/2+ the predicted mass is in the range 4.57− 4.71 GeV.5Jpi Mass [GeV]1/2+ 4.11, 4.14, 4.51, 4.57, 4.593/2+ 4.16, 4.52, 4.60, 4.601/2− 4.30, 4.35, 4.443/2− 4.40, 4.43, 4.485/2− 4.505/2+ 4.64Table III: Masses including rotational corrections for the NSM scheme. All quoted values have an uncertainty of about 70 MeVdue to the ambiguities in the formula for the rotational corrections. The lower value of this uncertainty range corresponds tothe BSA option for the calculation of the rotational corrections while the upper to the HQS one.III. CONCLUSIONSThe utility of the bound state interpretation described above is of course hinged on the existence of a C = −1meson bound state. The calculations made within the framework of the NSM approximation indicate that such astate does exist. The SMHQS approach is less definite on this issue, but most of the calculations point towards theexistence of a loosely bound state in the C = −1 channel. It should be noted that even if it is unbound but lies closeto threshold, one can still argue that an attractive D¯D interaction can make the whole soliton-D¯-D system bound.Given the assumption of the existence of a loosely C = −1 bound state, the existence of a pentaquark-type statewith quantum numbers (I, Jπ) = (1/2, 3/2−) and mass in a region compatible with the LHCb observation followsnaturally. Note that in the present picture at least one of the components of this state comes from populating thekπ++ = 1/2− meson state. In this sense it appears as akin to the Λ(1405). Note also that according to HQS sucha meson bound state should be degenerate with a kπ++ = 3/2− state (see Fig.1). This probably then explains theexistence of the other two 3/2− state close by. The situation concerning the (I, Jπ) = (1/2, 5/2+) state is lessclear. The model does predict such state but the associated mass lies about 4.6 GeV, which is somewhat too high ascompared with the observed value. It has in fact been suggested that the observed peak at 4450 MeV might even bea kinematical effect [9]. In any case it is important to recall that, given the approximations made in the calculationof the masses, the quoted values have to be viewed only as first estimates.The present model obviously predicts the existence of several other states. In particular, it predicts two 1/2+ andone 3/2+ states with masses in the range ≃ 4.1 − 4.2 GeV, which arise by putting the C = +1 meson in the lowestkπ++ = 1/2+ bound state. Whether those states are too wide to be discriminated by the experiment remains to beseen.This bound state approach picture can be fairly straightforwardly extended to the bottom sector, where it wouldimply the existence also of hidden bottom pentaquarks in view of the large mass of the bottom mesons. This wouldprobably require a full calculation within the SMHQS scheme, which is anyhow required to obtain more accuratepredictions for the properties of the states discussed in the present work.AcknowledgmentsThis work was partially supported by CONICET (Argentina) under grant PIP 00682 and by ANPCyT (Argentina)under grant PICT-2011-0113.[1] R. Aaij et al. [LHCb Collaboration], arXiv:1507.03414 [hep-ex].[2] R. Chen, X. Liu, X. Q. Li and S. L. Zhu, arXiv:1507.03704 [hep-ph].[3] H. X. Chen, W. Chen, X. Liu, T. G. Steele and S. L. Zhu, arXiv:1507.03717 [hep-ph].[4] L. Roca, J. Nieves and E. Oset, arXiv:1507.04249 [hep-ph].[5] J. He, arXiv:1507.05200 [hep-ph].[6] L. Maiani, A. D. Polosa and V. Riquer, arXiv:1507.04980 [hep-ph].[7] R. F. Lebed, arXiv:1507.05867 [hep-ph].[8] V. V. Anisovich, M. A. Matveev, J. Nyiri, A. V. Sarantsev and A. N. Semenova, arXiv:1507.07652 [hep-ph].[9] F. K. Guo, U. G. Meißner, W. Wang and Z. Yang, arXiv:1507.04950 [hep-ph];[10] X. H. Liu, Q. Wang and Q. Zhao, arXiv:1507.05359 [hep-ph].6[11] M. Mikhasenko, arXiv:1507.06552 [hep-ph].[12] A. Mironov and A. Morozov, arXiv:1507.04694 [hep-ph].[13] U. G. Meißner and J. A. Oller, arXiv:1507.07478 [hep-ph].[14] Q. Wang, X. H. Liu and Q. Zhao, arXiv:1508.00339 [hep-ph].[15] V. Kubarovsky and M. B. Voloshin, arXiv:1508.00888 [hep-ph].[16] D. O. Riska and N. N. Scoccola, Phys. Lett. B 299, 338 (1993).[17] M. Rho, D. O. Riska and N. N. Scoccola, Z. Phys. A 341, 343 (1992).[18] D. O. Riska and N. N. Scoccola, Phys. Lett. B 265, 188 (1991).[19] C. G. Callan, Jr. and I. R. Klebanov, Nucl. Phys. B 262, 365 (1985).[20] C. G. Callan, Jr., K. Hornbostel and I. R. Klebanov, Phys. Lett. B 202, 269 (1988).[21] N. N. Scoccola, H. Nadeau, M. A. Nowak and M. Rho, Phys. Lett. B 201, 425 (1988) [Phys. Lett. B 220, 658 (1989)].[22] M. Bjornberg, K. Dannbom, D. O. Riska and N. N. Scoccola, Nucl. Phys. A 539, 662 (1992).[23] See e.g. the following monographs where references to the original work can be found: A. V. Manohar and M. B. Wise,Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1 (2000); A. G. Grozin, Springer Tracts Mod. Phys. 201, 1 (2004).[24] M.B. Wise, Phys. Rev. 45, R2188 (1992).[25] G. Burdman and J.F. Donoghue, Phys. Lett. B280, 287 (1992).[26] T.-M. Yan, H.-Y. Cheng, C.-Y.Cheung, G.-L. Lin, Y.C. Lin and H.-L. Yu, Phys. Rev. D 46, 1148 (1992).[27] J. L. Goity, Phys. Rev. D 46, 3929 (1992).[28] N. N. Scoccola, in The Multifaceted Skyrmion ed. G.E. Brown and M. Rho (World Scientific, Singapore, 2010),arXiv:0905.2722 [hep-ph].[29] Y. Oh and B. Y. Park, Phys. Rev. D 53 1605 (1996) [hep-ph/9510268].[30] Y. Oh, B. Y. Park and D. P. Min, Phys. Lett. B 331 362 (1994) [hep-ph/9405297].[31] Y. Oh and B. Y. Park, Phys. Rev. D 51 5016 (1995) [hep-ph/9501356].[32] J. Schechter, A. Subbaraman, S. Vaidya and H. Weigel, Nucl. Phys. A 590 655 (1995) [Nucl. Phys. A 598 583 (1996)][hep-ph/9503307].[33] Y. Oh and B. Y. Park, Z. Phys. A 359, 83 (1997) [hep-ph/9703219].[34] M. Harada and Y. L. Ma, Phys. Rev. D 87, no. 5, 056007 (2013) [arXiv:1212.5079 [hep-ph]].[35] N. N. Scoccola, Phys. Lett. B 236, 245 (1990).[36] N. Itzhaki, I. R. Klebanov, P. Ouyang and L. Rastelli, Nucl. Phys. B 684, 264 (2004) [hep-ph/0309305].[37] D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359, 305 (1997) [hep-ph/9703373].[38] K. H. Hicks, Eur. Phys. J. H 37, 1 (2012)[39] Y. Oh, B. Y. Park and D. P. Min, Phys. Rev. D 50, 3350 (1994) [hep-ph/9407214].[40] Y. Oh, Phys. Rev. D 75, 074002 (2007) [hep-ph/0702126].[41] K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C 38, 090001 (2014).

On the pentaquark candidates p+c (4380) and p+c (4450) within the soliton picture of baryons

N.N. Scoccola

D.O. Riska

Mannque Rho

Crossref

On the Pentaquark Candidates Pc+(4380) and Pc+(4450) within the Soliton Picture of Baryons

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

arXiv:1508.01172v1  [hep-ph]  5 Aug 2015On the pentaquark candidates P+c(4380) and P+c(4450) within the soliton picture ofbaryonsN.N. Scoccolaa,b,c, D.O. Riskad and Mannque Rhoea Department of Theoretical Physics, Comisión Nacional de Enerǵıa Atómica,Av.Libertador 8250, 1429 Buenos Aires, Argentinab CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentinac Universidad Favaloro, Soĺıs 453, 1078 Buenos Aires, Argentinad Finnish Society of Science and Letters, Fabianinkatu 4 B 16, 00130 Helsinki, FinlandeInstitut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, FranceUsing the bound state version of the topological soliton model for the baryons we show thatthe existence of a bound (or quasi-bound) D̄-soliton state leads to the possibility of having hiddencharm pentaquarks with quantum numbers and masses, which are compatible with those of thecandidates recently reported by the LHCb experiment. The implications of heavy quark symmetryare elaborated.PACS numbers: 12.39.Dc,12.39.Hg,14.20.PtI. INTRODUCTIONThe LHCb collaboration at CERN recently reported the discovery of two states, which in the quark model correspondto “pentaquarks” with hidden charm Ref.[1]. The first one has the mass (width) 4380± 8± 29 (205± 18± 86 ) MeVand the second the mass (width) 4449.8± 1.7± 2.5 ( 39± 5± 19 ) MeV. The preferred quantum number assignmentsare Jπ = 3/2−, 5/2+, respectively, although acceptable solutions are also found for additional cases with oppositeparity. Note that their decay channel is J/ψ p+ which implies I = 1/2. After this announcement several articles thatconsider possible theoretical interpretations of the observed states have appeared. Some of these [2–5] suggest, withsome variations, that the observed states may be interpreted as anticharmed meson (D̄ or D̄∗)-hyperon (Σ or Σ∗)molecular states, while others base their description on diquark models [6–8]. The possibility that at least one of theobserved peaks might be only a kinematical effect was discussed in Refs.[9–11]. Further suggestions have been putforward in Refs.[12, 13]. Moreover, the production and formation of the hidden-charm pentaquarks in γ - nucleoncollisions have been discussed in Refs.[14, 15].The purpose of the present note is to show that it is also possible to account for the quantum numbers and masses ofthese observed pentaquark candidates within the topological soliton picture of baryons. The possible existence of stablepentaquarks with negative charm in that framework was already considered many years ago [16]. In this approach theheavy flavor hyperons were described as bound states of heavy flavour mesons and a topological soliton [16–18] in theextension to heavy flavor of the bound state approximation [19–21] to the Skyrme model, supplemented with suitablesymmetry breaking terms. This “Naive Skyrme Model” (NSM) formulation does however only approximately [22]incorporate heavy quark symmetry (HQS) according to which in the heavy quark limit the heavy pseudoscalar andvector fields become degenerate and, therefore should be treated on an equal footing [23]. An improved way to proceedis therefore to apply the bound state approach to the heavy meson effective lagrangian [24–27], which simultaneouslyincorporates chiral symmetry and heavy quark symmetry (for details see Ref.[28] and refs. therein). The possibleexistence of a C = −1 meson bound state in the context of a model consistent with HQS was first discussed inRefs.[30, 31]. There, however, only pseudoscalar mesons as described by the Skyrme model were considered in thelight sector. In what follows we will refer to this as SMHQS formulation. It was later pointed out [32] that theinclusion of light vector mesons in the corresponding effective lagrangian tends to push this state into the continuum.On other hand, the calculation in Ref.[33], which incorporates the center of mass corrections in a more consistent way,still leaves the possibility of a loosely bound state. Additional arguments for the existence of C = −1 meson-soliton2bound state have been given in Ref.[34].In the NSM approach it is the Wess-Zumino term in the lagrangian, which is responsible for the difference in theinteraction of the soliton and the mesons with opposite massive flavor quantum number. This term is repulsive in thecase of mesons with massive antiflavor quantum number. In the case of S = +1 kaons, this repulsion, in combinationwith the repulsive effect of the meson kinetic energy term, pushes them into the continuum [35, 36]. (It should here bementioned that the indications for the existence of the conjectured strange pentaquark “θ(1540)” [37] have hithertonot been experimentally confirmed[38]). As the repulsive effect of the kinetic energy term weakens with increasingmeson mass and the strength of the Wess-Zumino term is smaller for heavy flavors, the existence of anticharm (and afortiori antibottom) meson-soliton bound states becomes possible. Below we show that the existence of a bound (orquasi-bound state) D̄-soliton state naturally leads to the possibility of having some hidden charm pentaquarks withquantum numbers and masses which are compatible with those of the candidates proposed in Ref.[1]II. BOUND STATE DESCRIPTION OF HIDDEN HEAVY FLAVOURED PENTAQUARKSSince the pentaquark candidates reported in Ref.[1] have no net charm quantum number we propose to describethem as bound states of a soliton and two pseudoscalar mesons (one charm and the other anticharm) in the presentpicture. We recall that in the bound state approximation, while a bound C = +1 meson behaves as a quark [17–21],a D̄-meson corresponds to a antiheavy-light quark pair [16, 30, 31]. To determine the possible quantum numbers andestimate the values of the associated masses we need to know which bound states of charm and anticharm mesonsare possible. In fact, both the NSM and the SMHQS models lead to a number of meson bound states. Those boundstates can be labelled by kπ, where ~k = ~i + ~ℓ and π = (−1)ℓ+1. Here, ℓ and i = 1/2 are the angular momentum andisospin of the bound pseudoscalar meson. In Table I the corresponding quantum numbers, binding energies b andhyperfine splittings constants c are given . The latter are needed for calculation of the nonadiabatic corrections to bediscussed below. These results were obtained using the standard Skyrme model parameters fπ = 64.5 MeV, e = 5.45,which lead to the empirical masses for the nucleon and the ∆ resonance. The values of the binding energies b andhyperfine splittings c associated with the C = ±1 mesons in the NSM scheme have been calculated in Refs.[16, 18].Those of the SMHQS scheme have been extracted from Ref.[29] (Set 5) except for the binding of the C = −1 state,which is taken from Ref.[33]. Note that most of the associated hyperfine splittings have not been given in those works,however. The results corresponding to the NSM have been obtained using the decay constant ratio fD/fπ = 1.8. Thisvalue has been updated in recent years. The current estimate is fD/fπ = 1.57 [41]. While the use of this value hardlyaffects the predictions for the hyperfine splitting constants, one obtains an enhancement of about 100 MeV for all theC = +1 binding energies. Since the results for the binding energies obtained with fD/fπ = 1.8 are closer to those ofthe SMHQS formulation we will assume that this overbinding is a consequence of the simplicity inherent to the NSMformulation.NSM SMHQSl kπ b(MeV ) c b(MeV ) cC = +1 1 1/2+ 568 0.20 518 0.150 1/2− 355 0.52 239 0.302 3/2− 243 0.15 2121 3/2+ 140 0.28 491 1/2+ 118 0.03 65C = −1 1 1/2+ 38 0.16 54Table I: Quantum numbers of the meson bound states and associated binding energies and hyperfine splitting constants.The single meson spectra that follow from Table I are illustrated in Fig.1. It can be seen that the meson spectrumobtained in the NSM formulation is qualitatively similar to that obtained using the SMHQS formulation. Actually,this fact was already noticed long ago [22].3-1.0-0.8-0.6-0.4-0.20.01/2+  C= +1                      C= -13/2+1/2+3/2-1/2-1/2+C= +1                      C= -1    NSM                                                  SMHQS   Binding Energy [GeV] 1/2+1/2+3/2+3/2-1/2-1/2+Figure 1: Binding energies and quantum numbers corresponding to possible soliton-(anti)charmed mesons. The left panelcorresponds to the spectrum obtained using the NSM formulation while the right panel to the SMHQS one. The lowestJπ = 3/2− configuration is indicated by the two black circles and the Jπ = 5/2+ by the open ones.Clearly, the C = −1 meson has to be bound in the kπ−− = 1/2+ state. Here, the subindex stands for the charmquantum number of the meson. For the C = +1 one, we have, however, several possibilities. The preferred quantumnumber assignments of the pentaquark candidates are Jπ = 3/2−, 5/2+, respectively. Since, obviously, π = π− π+ itfollows that π+ = −(+) for the 3/2 (5/2) pentaquarks. To determine k+ we note that, according to the usual rules,the total grand spin ~K = ~k++~k− satisfies ~K = ~J+ ~I where J and I are the total spin and isopin of the bound system.Noting that the observed states are isospin doublets, it follows that 1/2 ≤ k+ ≤ 5/2 for the state with J = 3/2 and3/2 ≤ k+ ≤ 7/2 for that with J = 5/2. From Fig.1 it is clear that the lowest lying meson configurations (kπ++ , kπ−− )that satisfy all the requirements are (1/2−, 1/2+) for the Jπ = 3/2− pentaquark candidate and (3/2+, 1/2+) for theJπ = 5/2+ one. In the figure the first configuration is indicated by the two black circles and the second by the openones. In principle many other states can be obtained by populating the alternative C = +1 bound states. The fulllist is given in Table II together with the corresponding masses up to non-adiabatic corrections. These have beenobtained usingM =Msol + 2mD − b+ − b−, (1)where Msol = 866 MeV is the soliton mass, mD the pseudoscalar charm meson mass that we take to be mD = 1867MeV and b± are the meson binding energies given in Table I.It is seen that three low lying 3/2− with masses (4.21-4.32, 4.31-4.33) GeV for (NSM,SMHQS), respectively, arepredicted in both schemes. The rather large width found in the experiment might be explained by these 3 close lyingstates. One the other hand only one single 5/2+ state at (4.42, 4.50) GeV is predicted.So far we have not included the non-adiabatic contributions to the masses. To first order perturbation theory therotational corrections can be obtained by considering12Ω< (I(k+k−)K)J |(~R − ~Θ)2|(I(k+k−)K)J >, (2)where Ω is the moment of inertia of the soliton and Θ the total isospin of the meson bound system. Moreover, ~R hasthe role of spin of the light quark system (basically the spin of the rotating soliton, which coincides with its isospin).For a system of a C = +1 and a C = −1 bound mesons one has ~Θ = ~θ+ + ~θ−. The calculation of the matrix element4I kπ++ kπ−−K Jπ NSM SMHQS1/2 1/2+ 1/2+ 0 1/2+ 4.00 4.031 1/2+1 3/2+1/2 1/2− 1/2+ 0 1/2− 4.21 4.311 1/2−1 3/2−1/2 3/2− 1/2+ 1 1/2− 4.32 4.331 3/2−2 3/2−2 5/2−1/2 3/2+ 1/2+ 1 1/2+ 4.42 4.501 3/2+2 3/2+2 5/2+1/2 1/2+ 1/2+ 0 1/2+ 4.44 4.481 3/2+1 3/2+Table II: Quantum numbers and O(N0c ) masses (in GeV).in Eq.(2) requires some assumptions and/or approximations. The usual procedure in the bound state approach (BSA)to the Skyrme model [17–21] is to employ the approximation < θ2i >= c2i ki(ki + 1). We denote this as BSA option.Here, ci is the corresponding hyperfine splitting constant. As noted in Ref.[39] this approximation does not hold inthe HQS limit, where it can be shown that < θ2i >= 3/4, since for all the cases considered the bound mesons haveisospin 1/2. We denote this as HQS option. Since it is not yet settled which the best way to proceed at the charmmass scale is, both cases will be considered here in order to determine the related uncertainty. The second one has todo with the way in which < ~R · ~Θ > is calculated (see Ref.[40]). In fact, this term might induce mixings between stateswith different K and/or k+. This effect will be neglected in what follows. The resulting formula for the rotationalcorrections to the mass of a system composed by a soliton and two bound mesons (one in a state with k+ and theother with k−) is thenMrot(I, J, k+, k−,K) =12Ω{I(I + 1) + c+c− [K(K + 1)− k+(k+ + 1)− k−(k− + 1)] + δ[J(J + 1)−K(K + 1)− I(I + 1)][c+ + c−2+c+ − c−2k+(k+ + 1)− k−(k− + 1)K(K + 1)]}(3)whereδ ={c2+ k+(k+ + 1) + c2− k−(k− + 1) BSA3/2 HQS(4)Using the BSA option one recovers the mass formula given in Refs.[17, 18]. Using the parameters of Table I therotational corrections can then be calculated. Note that in the case of SMHQS scheme most of the hyperfine splittingsneeded for the calculation of the rotational corrections have not been given in the literature. Thus, only the predictionsas obtained in the NSM scheme are reported in Table III. The quoted values correspond to the average between theresults obtained using each of the two options for the calculation of the rotational corrections. The correspondinguncertainty is considered to be half of the difference between these two values, which turns out to be about 70 MeV.From this table we see that the prediction for the mass of the lowest 3/2− is in the range 4.33− 4.47 GeV. In the caseof the 5/2+ the predicted mass is in the range 4.57− 4.71 GeV.5Jπ Mass [GeV]1/2+ 4.11, 4.14, 4.51, 4.57, 4.593/2+ 4.16, 4.52, 4.60, 4.601/2− 4.30, 4.35, 4.443/2− 4.40, 4.43, 4.485/2− 4.505/2+ 4.64Table III: Masses including rotational corrections for the NSM scheme. All quoted values have an uncertainty of about 70 MeVdue to the ambiguities in the formula for the rotational corrections. The lower value of this uncertainty range corresponds tothe BSA option for the calculation of the rotational corrections while the upper to the HQS one.III. CONCLUSIONSThe utility of the bound state interpretation described above is of course hinged on the existence of a C = −1meson bound state. The calculations made within the framework of the NSM approximation indicate that such astate does exist. The SMHQS approach is less definite on this issue, but most of the calculations point towards theexistence of a loosely bound state in the C = −1 channel. It should be noted that even if it is unbound but lies closeto threshold, one can still argue that an attractive D̄D interaction can make the whole soliton-D̄-D system bound.Given the assumption of the existence of a loosely C = −1 bound state, the existence of a pentaquark-type statewith quantum numbers (I, Jπ) = (1/2, 3/2−) and mass in a region compatible with the LHCb observation followsnaturally. Note that in the present picture at least one of the components of this state comes from populating thekπ++ = 1/2− meson state. In this sense it appears as akin to the Λ(1405). Note also that according to HQS sucha meson bound state should be degenerate with a kπ++ = 3/2− state (see Fig.1). This probably then explains theexistence of the other two 3/2− state close by. The situation concerning the (I, Jπ) = (1/2, 5/2+) state is lessclear. The model does predict such state but the associated mass lies about 4.6 GeV, which is somewhat too high ascompared with the observed value. It has in fact been suggested that the observed peak at 4450 MeV might even bea kinematical effect [9]. In any case it is important to recall that, given the approximations made in the calculationof the masses, the quoted values have to be viewed only as first estimates.The present model obviously predicts the existence of several other states. In particular, it predicts two 1/2+ andone 3/2+ states with masses in the range ≃ 4.1 − 4.2 GeV, which arise by putting the C = +1 meson in the lowestkπ++ = 1/2+ bound state. 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On the pentaquark candidates p+c (4380) and p+c (4450) within the soliton picture of baryons

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