arXiv:hep-ph/0403295v1 30 Mar 2004 and fteeprils hi ssi,prt 2]adangular nature and determined. the [20] be However to parity yet been isospin, 19]. are their also momentum, 18, have , exotic [17, these structures SU(3) lattice of the the in and 4]. studied 3, 16], soliton chiral [2, 15, the model within 14, predicted three first [13, was a early anti-decuplet Jaffe the as since of or expected as works are multiquarks interpreted anti-quark discovered, Exotic an be be . and cannot to quark that a exotic numbers first quantum the with be may they aorS()teS()at-eult[,3 ]wihin- Θ to which discovered extends 4] recently 3, This [2, the anti-decuplet [1]. cludes SU(3) HERA the at SU(4) collaboration flavour H1 the by efis opt h asso l h osbes-wave possible the all of work that masses In p-wave the and MeV. compute -30 first of we energy binding with ∗ etqak r udeso e o ev oexplain to heavy too MeV Θ the of hundreds are hr eidct htteΘ the that indicate we where n Ξ and ihl on s-wave bound tightly a e eaigit a into decaying MeV prah h Θ the approach, oee nti hne efidaprl eusv exotic repulsive purely a find N we channel this in However nance d the add .6 e,idctn htti sa is state this -2 that = indicating Q GeV, -2, 1.862 S= the address Ξ also we [23], work recent isospin total and h Θ the a lcrncades [email protected] address: Electronic −− π − ercnl opee oko the on work a completed recently We nti ae td h anti-charmed the study I paper this In a , K icvrdb h A9eprmn ihams of mass a with experiment NA49 the by discovered , D + + −− ∗− adcr -aeitrcin hsecue,i our in excludes, This interaction. s-wave core hard D N eoac,ecp o the for except resonance, sabroentrebd -aebudtt of boundstate s-wave body three borromean a as π r h nicamdadbtoe etqak oeua he molecular pentaquarks bottomed and anti-charmed the Are ∗− 16)[0 1 2.TeΘ The 12]. 11, [10, (1860) p n a and 30)aeeteeyectn tts because states, exciting extremely are (3100) − p uudd I tnadqakmdlwt ur-niur niiainc annihilation the quark-antiquark excludes a repulsion that with model quark standard aina ea natcamdrsnnewsaraypredic Θ already resonance was S=1 resonance the anticharmed to An dedicated Hera. at ration π orma on-tt,alwrbnigeeg sepce i expected is energy binding Multiquark lower a bound-state, borromean sahpaur,euvln oa to equivalent heptaquark, a as N e.FıiaadCI,IsiuoSpro ´cio v Rov T´ecnico, Av. Superior Instituto CFIF, F´ısica and Dep. 30)(arwhdo eoac f3099 of resonance (narrow (3100) − .Ifidta the that find I 0. = td h charmed the study I , + D .INTRODUCTION I. π K s ¯ − sabr pentaquark bare a as I hnes norfaeokti tt shre obn hnt than bind to harder is state this framework our In channels. etqak,adw eiyta these that verify we and pentaquarks, 2,2,2] ihpstv aiy[27] parity positive with 26], 25, [21, .I htppr n navery a in and paper, that In 0. = K N and D − ∗− K N p eyrcnl discovered recently very ) + arwrsnne ethen We resonance. narrow − spoal a probably is I K N + 0, = N uudd 14)[,6 ,8 9] 8, 7, 6, [5, (1540) neatost study to interactions − K + ¯ − 14) Ξ (1540), π − D J D c ¯ uudd − ∗− N P resonance uudd D euso scneldb h trcineitn nthe in existing attraction the by cancelled is repulsion p − 1 = N , 30)a a as (3100) s uudd ¯ N + K K − ¯ tt ras or state 14) ocnr hs eetpeitos pl h same the apply I predictions, recent these confirm To (1540). / s ¯ −− − π 2 molecule − Θ + − π c ¯ (1860) π + D state. − D reso- − ∗− .Bicudo P. [21] ∗ B N iermlcl,wt oiieprt n oa isospin total and parity positive with molecule, linear ∗ p uudd and 30)vr eetydsoee yteH collabo- H1 the by discovered recently very (3100) c ¯ fsmlranti-charmed similar of MeV. -60 of energy binding a with otic rbt ocaiytentr ftepnaurs The pentaquarks. the of nature the state clarify con- certainly to will properties tribute its and [22], expected widely iur ¯ tiquark ie elcn the replacing sider ntenwpstv parity positive new the in o,bcuethe because son, n the and otescalar the to rhls n a osdrta ao ige quark- singlet flavor a that pair consider antiquark may one ertheless multipletH exotic the Θ just the in of not ) and (or pentaquarks pentaquarks) p-wave (or heptaquarks considering nryo h is the of energy ean ntesm aiyo h Θ the of family same the in remains ulctost the to publications fthe of Ξ hntersligheptaquark resulting the When arnmse u otebnigeeg.Ti princi- Θ This the of standard energy. mass these binding the qualitatively of the explains sum to ple exact due the masses than hadron heptaquark lower the slightly s-wave of two be in mass baryon The s-wave a . in rearranged naturally is tl arwsae n nti es the sense this is in threshold and above state, slightly narrow sys- located a a is still Nevertheless energy energy. which binding here tem negative because ones no previous is the from there thresh- differs above case MeV This +15 of old. energy an with molecule linear a arwrsnne iha nicamo anti-bottom or anti-charm an with resonances narrow lar emosadat-emos nti es h e hep- new the of sense parity this intrinsic In the taquark anti-. to and due fermions occurs parity reversed the tt iha poieprt oteoriginal the to parity opposite an with state a n,because And, N -aepnaur.Iepoethe explore I pentaquark. s-wave −− nrf 2]w locnld ugsigteexistence the suggesting conclude also we [21] ref. In oevrantrltertclmtvto xssfor exists motivation theoretical natural a Moreover nti ae xedtetcnqe sdi u first our in used techniques the extend I paper this In ∗ − ssuidadoecnldsta ti ntbe Nev- unstable. is it that concludes one and studied is uudd π D 2]adtemse ftenneoi multiquarks non-exotic the of masses the and [23] − D ∗− s + K s B gemn ihteH observation. H1 the with agreement n ntandb hrlsmer.Ifind I symmetry. chiral by onstrained H ¯ soPi,14-0 iba Portugal Lisboa, 1049-001 Pais, isco uoeta ie -aepnaur hadron pentaquark s-wave given a that Supose . elcdb ¯ a by replaced ¯ b p and ′ e,i eetpbiainmostly publication recent a in ted, − r lopredicted. also are arn.Teat-hre etqakwas pentaquark anti-charmed The hadrons. 30)myb iia oteΘ the to similar be may (3100) a ergre stecia ate of partner chiral the as regarded be can D D H D eΘ he u ∗ D s ′ u ¯ 22)adt h axial the to and (2320) s ∗ utqak r epcieycandidates respectively are multiquarks sepce ob poiaeysal,it stable, aproximately be to expected is D ∗− + ail [24]. familly K + ∗ p D d sas arwsae o instance For state. narrow a also is ecie ya by described eo ya by meson d 30)cnitn iha with consistent (3100) ¯ ∗− c or nti aei sntrlt con- to natural is it case this In . p uudd s 30)adt h te simi- other the to and (3100) s ¯ D scetdi h hadron the in created is s c ¯ eos 2]the [24] mesons, D H D n nibtoe ex- anti-bottomed and ptaquarks? N ∗− ′ k eo rb a by or meson − ean on,i is it bound, remains p + − (3100) π π n fteΞ the of and H − and ′ D + N sepce to expected is s ihtean- the with , + D 26) The (2460). ∗− 2]frthe for [21] D H ∗ p − where , D K ¯ (3100) π −− ∗ − − me- H H . N D . . 2 quark. A standard (QM) Hamiltonian is assumed, with a confining potential and a hyperfine term. Tp ✜ 1 ❭ ✚ 1 ❩ Moreover the Hamiltonian includes a quark-antiquark (a) ✜ ☛❄p ✄❈ p ✟❭ (b) ✚φ ☛❄p t ✄❈ p ✟φ❩ ✜ 2 ❈ ✄ ✻ ❭ ❩ A 2 ❈ ✄ ✻ A✚ φA φA ❩ ✚ annihilation term which is the result of spontaneous chi- ❭ ✡❄ ❈ ✄ ✠ ✜ ✏ ✡ ✄❈ ✠ ☛k k ✟✻ ❭ 3 ✄❈ ✜ T1P13 ✄ ❈ ral symmetry breaking. I start in this paper by reviewing ❭ ✡ ✠✜ h i = hEP14i = E× ✄ ❈ ψ ✚ 3 ✄ ❈ ❩ ✄ ❈ ✚φ ☛❄p p ✟φ❩ the QM, and the Resonating Group Method (RGM) [28] 4 ✄ ❈ ❩ B 4 ✻ B✚ ✚ ❩ ❩ ✡ ✠✚ which is adequate to study states where several ✚φ ☛❄p p ✟φ❩ ❩ B 5 ✻ B✚ ✑ ❩ ✡ ✠✚ overlap. Using the RGM, I show that the corresponding ✜ 1 ❭ ✜ ☛❄p ❉ ☎ p ✟❭ -meson short range s-wave interaction is re- 1 ✜φ 2 ❉ ☎ ✻φ ❭ ✚ ❩ (d) A ✡ ✠A (c) ✚φ ☛❄p q ✄❈ p ✟φ❩ ❭ ☛❄ ❉ ☎ ✟ ✜ pulsive in exotic channels and attractive in the channels ❩ A 2 ✻ A✚ k k ✻ ❩ ❈ ✄ ✚ ❭❭ 3 ❉ ☎ ✜✜ ✡ ′ ✄❈ ✠ ✡ ❉❉ ☎☎ ✠ with quark-antiquark annihilation. The short range re- V (p−p ) hA15i = hV13P13i = ✄ ❈ ✄✄r′❈ r❈ pulsion contradicts the existence of narrow pentaquarks ✚ 3 ✄ ❈ ❩ ✚ 4 ✄✄A(p−p )❈❈ ❩ ✚φ ☛❄p′ q p′ ✟φ❩ ✚φ ☛❄p′ ✄✄ ❈❈ p′ ✟φ❩ ❩ B 4 ✻ B✚ ❩ B 5 ✄✄ ❈❈ ✻ B✚ with an anti-charm or anti-. I proceed ❩ ✡ ✠✚ ❩ ✡ ✠✚ with the study of the linear molecules or heptaquarks D π N, D∗ π N, B π N and B∗ π N. In − − − − − − − − FIG. 1: Examples of RGM overlaps are depicted, in (a) the particular the total energy of these systems is discussed. norm overlap for the meson-baryon interaction, in (b) a ki- Finally I conclude interpreting the D∗−p (3100) and pre- netic overlap the meson-meson interaction, in (c) an interac- dicting related multiquarks. tion overlap the meson-meson interaction, in (d) the annihi- lation overlap for the meson-baryon interaction.

II. FRAMEWORK

and it must also be replaced by Vhyp MD∗ DK h Di ≃ − Our Hamiltonian is the standard QM Hamiltonian, . The quark-antiquark annihilation potential Ai¯j is also constrained when the quark model produces spontaneous H = Ti + Vij + Ai¯j (1) chiral symmetry breaking [33, 34]. The annihilation po- Xi Xi

+ + where each quark or antiquark has a kinetic energy Ti 2T + V A φ φ = Mπ (4) with a constituent quark mass, and the colour depen-  A 2T + V   φ−   φ−  − dent two-body interaction Vij includes the standard QM confining term and a hyperfine term, where the π is the only hadron with a large negative en- + ergy wave-function, φ− φ . In eq. (4) the annihilation 3 ≃ Vij = − ~λi ~λj Vconf (r)+ Vhyp(r)S~i S~j . (2) potential A cancels most of the kinetic energy and confin- 16 · h · i ing potential 2T +V . This is the reason why the has The QM of eq. (1) reproduces the meson and baryon a very small mass. From the hadron spectrum and using spectrum with quark and antiquark bound-states (from eq. (4) the matrix elements of the annihilation potential the heavy to the light pion mass). The RGM are determined, was first applied by Ribeiro [29] to show that in exotic 2 N N scattering, the quark-quark potential together 2T + V S=0 (2MN M∆) h i ≃ 3 − with− the Pauli repulsion of quarks explains the N N − 2 hard core repulsion. Deus and Ribeiro [30] also showed A S=0 (2MN M∆) , (5) that, in non-exotic channels, the quark-antiquark anni- ⇒ h i ≃ −3 − hilation could produce a short core attraction. Recently, where this result is correct for the annihilation of u or d addressing a system with the π π quantum − quarks. numbers, it was shown that the QM also fully complies The RGM [28] computes the effective multiquark en- with the chiral symmetry, including the Adler zero and ergy using the matrix elements of the microscopic quark- the Weinberg theorem [31, 32]. Therefore the QM is ad- quark interactions. Any multiquark state can be de- equate to address the anti-decuplet, which was predicted composed in combinations of simpler colour singlets, the [2, 3, 4] in an effective chiral model. baryons and mesons. The wave functions of quarks are For the purpose of this paper the details of the poten- arranged in anti-symmetrized overlaps of simple colour tials in eq. (1) are unimportant, only its matrix elements singlet hadrons. Once the internal energies EA and EB . The hadron spectrum constrains the hyperfine of the two hadronic clusters are accounted, potential, p φbφa H ( 1) P φaφb 4 h | p − | i p = Ea + Eb + Va b , (6) Vhyp (M∆ MN ) MK∗ MK . (3) φbφa P( 1) P φaφb h i≃ 3 − ≃ − h | p − | i P When a light quark is replaced by a heavy quark, say where ( 1)pP is the anti-symmetrizer, the remain- p − a charmed quark, the hyperfine interaction is decreased, ing energyP of the meson-baryon or meson-meson system 3 is computed with the overlap of the inter-cluster micro- for the different channels, are scopic potentials, 1 1 1 2 + 3 ~τD ~τN 2 VD N = 3 1 · Vhyp α− − 2 V bar A = φB φA (V14 + V15 +2V24 +2V25)3P14 4 3 ~τD ~τN h i N mes B h |− −1 1 · 1 + ~τD ~τN +3A15 φAφB / φB φA 1 3P14 φAφB 2 3 2 | i h | − | i + 3 1 · VhypD α− , 2 ~τD ~τN h i N V mes A = φB φA (1 + PAB)[ (V13 + V23 + V14 + V24) 4 3 mes B h | − − · 1+2√3 + 1+√3 ~τ ~τ P13 + A23 + A14] φAφB 8 3 D N 2 V D−N = · Vhyp α− × | i →D∗−N √ 1 5 h i N / φB φA (1 + PAB )(1 P13) φAφB , (7) 3+ 4 + 3 ~τD ~τN h | − | i 1 4 · 8 3 ~τD ~τN 2 + − − · Vhyp α− , where Pij stands for the exchange of i with parti- 1 5 D √3+ 4 + 3 ~τD ~τN h i N cle j, see Fig. 1. It is clear that quark exchange provides 7 · 1 2+ 3 ~τD ~τN 2 the necessary colour octets to match the Gell-Mann ma- ∗ VD N = 11 7 · Vhyp α− trices λi present in the potential Vij . This results in eq. − 2 + ~τD ~τN h i N 4 3 · (3) or eq. (5) times an algebraic colour spin flavour 1 5 × × 1 −2 + 3 ~τD ~τN 2 factor and a geometric momentum overlap [35]. + 11 7 · VhypD α− , 2 + ~τD ~τN h i N A good approximation for the wave-functions of the 4 3 · ground-state hadrons is the harmonic oscillator wave- 1 2 Vπ N = ~τπ ~τN A α− , function, − −3 · h i N 4 2 3 Vπ D = ~τπ ~τD A α− , 2 2 − −9 · h i N α 1 pρ α φ000(pρ)= α− exp 2 , α = , 4 2 N −2α  N 2√π  Vπ D∗ = ~τπ ~τD A α− , (9) (8) − −9 · h i N where the inverse hadronic radius α can not be estimated where ~τ are the isospin matrices, normalized with ~τ 2 = by -hadron scattering because it is masked by the τ(τ +1). The vanishing momentum case of eq. (9) is suf- dominance. α is the only free parameter in ficient to compute the scattering lengths with the Born this framework. In the case of vanishing external mo- approximation. However the study of binding needs the menta pA and pB, the momentum integral in eq. (7) is finite momentum case. In the exotic D N and D∗ N 2 − − 2 simply α− . channels, it can be proved that the geometric result − N α The annihilation potential only occurs in non-exotic is then replaced by the separable interaction φα Nφα . | 000ih 000| channels. Then it is clear from eq. (5) that the annihila- In the non-exoticπ ¯ N, π D and π D∗ channels the tion potential provides an attractive (negative) overlap. present state of the− art of− the RGM− does not allow a The quark-quark(antiquark) potential is dominated by precise determination of the finite momentum overlap. the hyperfine interaction of eq. (3), and in s-wave sys- Nevertheless I assume for simplicity the same separable tems with low spin this results in a repulsive interaction. interaction. This is a reasonable approximation, because These results are independent of the details of the quark the overlaps decrease when the relative momentum of model that one chooses to consider, provided it is chiral hadrons A and B increases. Moreover when the hadronic invariant. Therefore I arrive at the attraction/repulsion potential VAB is in a separable form v φ1 φ1 , the energy criterion, of the and the matrix element| ih of| the poten- - whenever the two interacting hadrons have quarks (or tial are simple to compute. A boundstate coincides with antiquarks) with a common flavour, the repulsion is in- a pole in the T matrix at a negative energy, creased by the Pauli principle, v - when the two interacting hadrons have a quark and an T = φ1 > < φ1 , | 1 g0 v | antiquark with the same flavour, the attraction is en- − 11 hanced by the quark-antiquark annihilation. 1 g0ij =< φi p2 φj > , (10) In the particular case of one interacting with |E + iǫ | − 2µ anti- and with kaons, this implies that the short therefore one just has to find the energy that cancels range exotic D N, D∗ N, B N and B∗ N in- − − − − 1 g0 v. This method also allows the computation of the teractions are repulsive. This shows that the I=0 s-wave − pentaquarks uuddc¯ and uudd¯b are certainly quite unsta- matrix element of the potential in the boundstate, 2 ble. Higher isospin or spin systems are certainly more g0 unstable in our framework because they have a higher < VAB >= v 2 . (11) g0 repulsion. On the other hand the short range π N, j j1 − P π D, π D∗, π B and π B∗ interactions can be When φ1 > is the harmonic oscillator state of eq. (8), − − − − attractive. This motivates the study of a linear molecule the necessary| condition for binding is, + with a N,a π and a D,ora D ,ora B,ora B∗. Quanti- 4µ v α2. (12) tatively [21, 32, 36, 37], the effective potentials computed − ≥ 4

¯ III. BINDING FLAVOUR uuddQ MULTIQUARKS VhypD << Vhyp . For example the strength of the re- hpulsiveiD hN potentiali is expected to decrease by 30 − The simplest pentaquarks are not expected to bind %. due to the attraction/repulsion criterion. For instance I now use an adiabatic Hartree method to study the stability of the linear N π D molecule and related the D∗−p (3100) cannot be the ground-state uuddc¯ pen- − − taquark because the elementary color singlets (uud) (dc¯) molecules with a D∗, a B or a B∗. Essentially the wave- or (udd) (uc¯) are repelled, since the elementary− color function of the pion is centered between the nucleon and singlets share− the same flavour u or d. This also implies the D, where the nucleon and the D don’t overlap with 0 each other. This results in a linear molecule. for simplic- that the D−p and D n systems are unbound. The uuddc¯ pentaquarks with spin, flavour or angular momentum ex- ity I use an averaged mass for the nucleon and D and citations will have larger masses and also large widths. for the pion interaction with these quark clusters. I solve Nevertheless thec ¯ pentaquarks are more subtle than the a Schr¨odinger equation for the nucleon in the potential produced by a pion placed at the origin and by the other s¯ ones, because the D∗ is quite stable when compared heavy-light meson placed at a distance a of the pion. with the K∗. Therefore one should also consider to ex- − cite the spin in the lc¯ cluster, and this amounts to study The potential of the pion is produced by the at the point a and the nucleon +a. This produces three D∗ N bound-states. Indeed the effective potential of − eq.− (9) is attractive in this case. However this state is binding energies ED Eπ, EN , and three wave-functions. coupled to the D N case also in eq. (9). Once the cou- In the Hartree method the total energy is the sum of these pled channel hamiltonian− is diagonalized, the energy of energies minus the matrix elements of the potential en- ergies, This is easily computed once the two Schr¨odinger the D∗ N is lifted and the attraction is essentially lost. Therefore− the simplest way to have attraction, together equations are solved, with eqs. (10), (11) and (12). The a with a low energy and with a narrow width, consists in total energy is a function of the distance , and I mini- adding at least one quark-antiquark pair to the system. mize it as a function of a. The same steps are repeated for the N π D∗, N π B and N π B∗ sys- Then this amounts to include a pion in the system, − − − − − − which can be attracted both by the N baryon and by the tems. At the point I am not yet able to bind these linear molecule systems, with a negative binding energy. The D or D∗ meson to produce a N π D or N π D∗ linear molecule. For instance the− flavour− includes− combi-− same happened for the N π K when we studied the Θ+. [21] Nevertheless the− picture− of a K π N with nations of terms like uud du¯ uc¯ where the anti-quark − − u¯ in the pion can be annihilated− − both by the u present in a binding energy of 30 MeV is still plausible because the the N and by the u present in the D. According to the medium range interaction remains to be used. Therefore binding or near-binding is also plausible in the N π D, attraction/repulsion criterion this produces an attractive − − N π D∗, N π B and N π B∗ systems. interaction. The quark d present in the nucleon cancels − − − − − − only part of the attraction to the pion. Incidently the pion-nucleon I = 1/2 attraction is fixed by chiral sym- metry, see reference [36]. IV. CONCLUSION AND OUTLOOK The proposed system N π D and N π D∗ are +− − − − similar to the model for Θ (1540) advocated in reference I find that N π D, N π D∗, N π B and − − − − − − [21], in the present case the anti-quarks ¯ is replaced by N π B∗ nearly bound I=0 linear s-wave molecules, a heavyc ¯ or ¯b. The increase of the quark mass does and− positive− parity are plausible. The absorption of a low not affect directly the attraction, where the Q¯ is just a energy pion and the resulting decay into a low energy p- spectator. However the size of the wave-functions 1/α is wave N D(D∗,B,B∗) results in a narrow decay width. − affected. For instance in an harmonic oscillator potential The N π D∗ and N π B∗ can also decay respectively 4 − − − − α is proportional to √µ, and the reduced mass µ doubles into the three body systems N π D and N π D, − − − − when one changes from a light-light meson to a heavy- but this is again narrow since the N D∗ and N B∗ − − light meson. This amounts to an increase of nearly 20% overlaps are suppressed. Moreover N D∗, and N B∗ of the α in the D or . Because the α parameter pure I=0 s-wave, negative parity, pentaquarks− are− not is increased only in one of the Jacobi coordinates, the completely ruled out, but they couple strongly with the average α in π D or π D or π B or π B is decay channels N D and N B (repulsive systems), ∗ ∗ − − only expected to− suffer a 10%.− This increase− of α−will de- and this results in a larger width. crease effectively the attractive interaction. for instance Comparing with the K π N and assuming that it is − − the condition for a 2-body binding in eq. (12) shows a bound system, I expect the N π D∗ and N π B∗ that this is equivalent to decrease the strength of the to be the most bound systems− of the− family studied− − in attractive potential by a factor of 20%. Similar results this paper, with a similar binding energy. Nevertheless are obtained in different models of confinement, say in the N π D∗ is less bound than the N π K, due − − − − the funnel interaction which is more adequate for heavy to a weaker π D(D∗,B,B∗) attraction when compared − quarks. In what concerns the repulsive D N potential, with the πK attraction. Thus an energy of 14 MeV above it is decreased in the same way. Moreover− the strength threshold, corresponding to a uuddc¯ mass of 3.099 GeV of the hyperfine potential is further decreased because as observed by the the H1 collaboration is plausible. The 5 corresponding mass of the multiquark with flavour uudd¯b interaction. The medium range interaction, which in nu- is of the order of 6.416 GeV. I what concerns the gound- clear physics is described by the sigma meson exchange, state N π D and N π B the binding energy is is equivalent to the coupling to channels with multiple predicted− to− be higher because− − in this case the nucleon- . heavy-light meson repulsion is larger. So these states would have an energy some MeV larger than respectively 2.957 GeV and 6.370 Gev. A more quantitative computation of the masses, sizes, Acknowledgments and decay rates of the proposed heptaquarks, includ- ing coupled channels and exact three body computations I thank Gon¸calo Marques for discussions on the al- will be done elsewhere. I expect that the most relevant gebraic computations of this paper. I am also grate- contributions that remain to be included in this frame- ful to Katerina Lipka and to Achim Geiser for discus- work are the solution of the full three body relativistic sions on the status of the experimental evidence of the Fadeev equations, and the inclusion of the medium range D∗ p(3100). −

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