A Preliminary Explanation for the Pentaquark P found by LHCb

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1 A Preliminary Explanation for the Pentaquark P found by HCb Mario Everaldo de Souza Departamento de Físia, Universidade Federal de Sergipe, Av. Marehal Rondon, s/n, Rosa Elze, São Cristóvão, SE, Brazil Abstrat We propose that the two resonant states of the reently found pentaquark P with masses of 4380 MeV and 4450 MeV are two states of the hadroni moleule uud with similar properties to those of the Karliner-ipkin pentaquark. Applying the Morse moleular potential to the moleule its minimum size is estimated. If S states exist, the first two possible S states are suggested and their energies are estimated. It is shown that the oupling onstant is lose to that of harmonium, and this may mean Physis Beyond the Standard Model. I. INTRODUCTION The idea of the pentaquark was firstly proposed by Strottman in 1979 [1]. In 004 Karliner and ipkin proposed a very important model for a pentaquark in the desription of the []. They arrived at the onlusion that the bag model ommonly used for hadrons may not be adequate for the pentaquark. In their model they propose that the pentaquark system is omposed of two lusters, a diquark and a triquark, in a relative P-wave state, and the lusters an be separated by a distane larger than the range of the olor-magneti fore whih had been proposed by De Rujula, Georgi and Glashow [3]. It is important to reall that many authors have onsidered the possibility of the binding between J / and nulei. Among them we an ite [4,5,6,7,8,9,10,11]. They have raised some important observations onerning the interation suh as that meson exhange is suppressed by the OZI rule beause J / and nulei do not have quarks in ommon and also single gluon exhange is forbidden so that the interation should happen due to multiple-gluon exhange. The HCb Collaboration has reently [1] announed the disovery of 0 0 pentaquark-harmonium states whih resulted from the b exoti deay b PK, in whih P is a pentaquark with the quark ontent uud. The two resonant states have masses of about 4380 MeV and 4450 MeV in opposite parities with the preferred values for J 3/,5 / and with the orresponding s of MeV and MeV. All the data for the masses below were taken from the Partile Data Group [13].

2 II. A SIMPE MODE FOR THE HCb P We propose that the reently found HCb P is omposed of two olorless lusters, a meson and a baryon. The quark ontent of the P pentaquark, uud allows the two possibilities uud and u ud, but as the final deay is J / p we should rule out the seond possibility. The symbol is used to designate the binding between the two lusters. The two lusters should be weakly bound beause hadrons are olorless, but as they are bound, they form a hadroni moleule. The most famous hadroni moleule is the deuteron, also onstituted of olorless partiles. Beause the two partiles (nuleons) are olorless the binding is very weak, just about. MeV. But in the ase of the uud we do not know what kind of binding energy we have. Between two nuleon the nulear fore is a residual effet of the more fundamental fores of the olor field in the quark systems, but we still do not know the nature of the fore responsible for the binding between J / and the proton in the moleule uud. We an estimate below the minimum value for the binding energy. III. POSSIBIITIES FOR THE ANGUAR MOMENTA As the final deay is J / p we should onsider the two possibilities for the moleule: ( S 1) uud (S 1/ ) and ( S 1) uud (S 3/ ). ( S 1) is J/Psi and uud ( S 3/ ) is the resonane N(150) whose mass is atually about 1515 MeV. The sum of the masses 3096 MeV+1515 MeV = 4611 MeV is muh higher than 4380 MeV and 4450 MeV. Therefore, we should onsider only the first possibility. In order to reprodue the experimental values of J 3/,5 / we should have 1, for the moleule. Taking into aount the Karliner-ipkin model [] in whih the lusters are in a relative P-wave state, we an propose that the levels found by HCb for the P system are in relative P-wave and D-wave states. The possibility of S states is disussed below, in setion V. et us now analyze the angular momenta of the moleule ( S 1) uud (S 1/ ) and its omponents. Composing the two spins we obtain S 1/,3/. With 1 and S 1 we have J 1, 3, and for S 3 we have J 1, 3, 5. With and S 1 we have J 3, 5, and for S 3 we have J 1, 3, 5, 7. We see that the most favored values for J are 1, 3, 3, 5. IV. THE APPROXIMATE SIZE OF THE MOECUE We have no idea yet on the kind of effetive potential that exists in the P system. Assuming that there is a shallow potential well we may model the system through a moleular potential and obtain some important numbers. It is important to note that this system are heavy and, thus, we an use the Shrödinger equation for them. Reently, de

3 Souza [14,15] has used the Morse moleular potential in the desription of harmonium and bottomonium states. The Morse moleular potential an be expressed as [16] ( ) x x V r D e e (1) where D is the minimum of the well, a is the distane where V D, and x r a a. / For x 1 this potential an be expanded around the minimum up to order 3 in x and the expression is obtained where /a. 1 V ( x) D ka x ka x 3 3 () For this potential the solution of the Shrödinger equation yields the expression [16,17] 1 1 Enl A B( 1) 1 D ( 1) C ( 1)... for the vibrational and rotational levels above the minimum of the potential, where the quantum numbers, 0,1,,3,... In Eq. 3 the first term desribes harmoni vibrations, the seond term takes into aount the anharmoniity of the potential, the third term desribes rotations with onstant moment of inertia, the fourth term represents the entrifugal distortion and the fifth term represents the oupling between vibration and rotation. The onstant B is given by B /a where is the redued mass of the system. In the ase of P we only have two levels, one for =1 and another for = for 0 in Eq. 3 Therefore we an only determine two parameters of the above formula. The largest parameters in Eq. (3) are and B. In general A is not very small ompared to, but for 0 the nd term is A/4 while the first one is /, and thus, the first term has a muh larger ontribution than the nd term. Moreover, the inlusion of A would not hange the value of B in our fitting beause the two levels 4380 MeV and 4450 MeV have 0. The inlusion of A would inrease slightly the value of. The 4 th and 5 th terms are muh smaller than the 3 rd term. So, due to lak of data we an only have (3) 1 0 B B (4)

4 whih yield B MeV and MeV. Using the relation /a we find a 1.5 fm but with a large unertainty due to the large B unertainty of B. There are some available data for the radii of the proton and harmonium that allows us to have more information on the moleule uud. The size of ( J / ) is about fm [14] and the proton radius (harge radius) has been reported as being fm [18], fm [19], and fm [0,1]. Summing the radii of and the proton we obtain about 1. fm whih is very lose to the value of a. But in states with =1 and = is larger than 1.5 fm, of ourse, beause the wave P funtions of states ontain the term r. It is only worth making the alulation when we obtain a muh smaller value for the of the 4380 MeV level, and thus, as a onsequene, a more preise value for B. V. THE FIRST EVES OF THE MOECUE In our model above orresponds to n 1 in QCD models, that is, n= 1. Thus, the two states 4380 MeV and 4450 MeV are the states 1P and 1D, respetively. As we saw above we ould not alulate the value of A whose inlusion would inrease slightly the value of. This means that is a little larger than 9110 MeV. As B is very small ( B 17.5 MeV) and is, indeed, muh smaller than the of the level 4380 MeV whih is MeV, if the 1S level exists, it would have an energy around 4.3 GeV. This would explain why the level 4380 MeV is muh broader than the 4450 MeV level, and would mean that the 4380 MeV level is, atually, degenerate levels 1S and 1P. Of ourse, onsidering the spins of J / and the proton, the values of J for S states would be only 1 and 3. By the same token, if the S state ( 1) exists, it would have an energy around GeV. Thus we an propose that, if S states exist, the HCb pentaquark would have the levels shown in Table 1. evel ( nj ) Energy (MeV) 1S P D S Table 1. Possible levels of the pentaquark P if S states exist.

5 VI. THE BINDING ENERGY For a square well we have that in order to have a bound state (for an S wave) we should have a depth of D 8a (5) whih for a 1.5 fm and 70.1 MeV yields D 0.78 Mev. The oupling onstant is not small at all. From the value of MeV we obtain 3 k ( ) ( ) ( ) 10 GeV/fm whih is 3 almost the same as that of for harmonium, k GeV/fm [15]. Taking into aount the OZI rule, we an say that suh a strong oupling may be due to Physis Beyond the Standard Model. VII. CONCUSION It is proposed that the observed levels of the reently found pentaquark P are states of the moleule uud in the different relative angular momentum 1 and states, and is similar to the pentaquark system proposed by Karliner and ipkin []. It is shown how the observed angular momenta are generated, and an approximate minimum value for the size of the moleule is alulated. It is shown that the oupling onstant is not small and has the same order of magnitude of that of harmonium. This may mean Physis Beyond the Standard Model. Possible S states are proposed for the system. Although still preliminary, the work is relevant and shows a onsistent possibility for this reently disovered system of 5 quarks. More data is needed to improve the model and alulate other parameters of Eq. 3. It is important to find out, for example, if the system has or not exited states. This paper should be taking as one of the starting points for understanding this remarkable system. REFERENCES 1) D. Strottman, Multi-quarks baryons and the MIT bag model, Phys. Rev. D 0, , ) M. Karliner and H. J. ipkin, A Diquark-Triquark Model for the KN Pentaquark, Phys. ett. B 575, 49-55, ) A. De Rujula, H. Georgi and S.. Glashow, Hadron Masses in Gauge Theory, Phys. Rev. D 1, , ) M. uke, A. V. Manohar, and M. J. Savage, A QCD alulation of the interation of quarkonium with nulei, Phys. ett. B, 88, 355, ) D. Kharzeev and H. Satz, Quarkonium interations in hadroni matter, Phys. ett. B, 334, 155, 1994.

6 6) S. J. Brodsky, I Shmidt, and G. F. de Téramond, Nulear-bound quarkonium, Phys. Rev. ett. 64, 1011, ) S. J. Brodsky and G. A. Miller, Is J/Psi-nuleon sattering dominated by the gluoni van der Waals interation?, Phys. ett. B, 41, 15, ) K. Tsushina, Binding of D, D and J / mesons in nulei, arxiv ) I. Kawanai and S. Sasaki, Charmonium-nuleon Potential from attie QCD, Phys. Rev. ett. D, 8, ) G. Krein, Charmonium and D mesons inhadroni matter, AIP Conferene Proeedings, v. 165, p. 0-5, ) A. Yokota, E. Hiyama, and M. Oka, Possible existene of harmonium-lightnuleus bound states, XV International onferene on Hadron Spetrosopy- Hadron 013, Nara, Japan. 1) R. Aaij et al. (HCb Collaboration), Observation of J / with pentaquark states in 0 / deays, arxiv: b J K p p resonanes onsistent 13) K. A. Olive et al. (Partile Data Group) Chin. Phys. C, 38, (014). 14) M. E. de Souza, Bottomonium with an effetive moleular potential, Nul. Phys. B: Pro. Suppl., 37-40, ) M. E. de Souza, Charmonium with an effetive Morse moleular potential, J. Phys.: Conf. Ser. 556, , ) S. Flügge, Pratial Quantum Mehanis vol. I (New York: Springer-Verlag, New York), pp , ) MIT OpenCourseWare, eture # 3 Supplement Fall 008 Small-Moleule Spetrosopy and Dynamis. 18) I. Sik and D. Trautmann, On the rms Radius of the Deuteron, Nul. Phys. A 637, , ) A. Huber, Th. Udem, B. Gross, J. Reihert, M. Kourogi, K. Pahuki, M. Weitz, and T. W. Hänsh, Hydrogen-Deuterium 1S-S Isotope Shift and the Struture of the Deuteron, Phys. Rev. ett. 80, 468, ) J. Martorell, D. W.. Sprung and D. C. Zheng, Deuteron polarizability shifts and the deuteron matter radius, Phys. Rev. C 51, 117, ) A. Antognini et al. (CREMA:Muoni Hydrogen Collaboration), Proton Struture from the Measurement of S-P Transition Frequenies of Muoni Hydrogen, Siene 339, 417, 013.

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