Exact Solution of Schrödinger Equation for Pentaquark Systems

Document Type : Original Scientific Paper


Department of Physics, University of Kashan, Kashan, I. R. Iran



In this paper we present an exact analytical solution for five interacting quarks. We solve Schrödinger equation for pentaquarks in the framework of five-body and two-body problems. For this purpose, we utilize Yukawa potential in Jacobi coordinates. Also finding the relation between the reduced masses and coupling constants of pentaquarks, we obtain the coupling constant of Yukawa potential for pentaquark systems. We calculate the energy of these systems in their ground state. The results are well consistent with the theoretical results. Our procedure to obtain these results is appropriate for other potentials and n-body systems.


Main Subjects

[1] C. Alt, T. Anticic, B. Baatar, D. Barna, J. Bartke, L. Betev, H. Biakowska, A. Billmeier, C. Blume, B. Boimska and M. Botje, J. Bracinik, R. Bramm, R. Brun, P. Buncic, V. Cerny, P. Christakoglou, O. Chvala, JG. Cramer, P. Csato, N. Darmenov, A. Dimitrov, P. Dinkelaker, V. Eckardt, G. Farantatos, P. Filip, D. Flierl, Z. Fodor, P. Foka, P. Freund, V. Friese, J. Gal, M. Gazdzicki, G. Georgopoulos, E. Gladysz, S. Hegyi, C. Hohne, K. Kadija, A. Karev, S. Kniege, V. I. Kolesnikov, T. Kollegger, R. Korus, M. Kowalski, I. Kraus, M. Kreps, M. van Leeuwen, P. Levai, L. Litov, M. Makariev, A. I. Malakhov, C. Markert, M. Mateev, BW. Mayes, G. L. Melkumov, C. Meurer, A. Mischke, M. Mitrovski, J. Molnar, S. Mrowczynski, G. Palla, A. D. Panagiotou, D. Panayotov, K. Perl, A. Petridis, M. Pikna, L. Pinsky, F. Puhlhofer, J. G. Reid, R. Renfordt, W. Retyk, C. Roland, G. Roland, M. Rybczynski, A. Rybicki, A. Sandoval, H. Sann, N. Schmitz, P. Seyboth, F. Sikler, B. Sitar, E. Skrzypczak, G. Stefanek, R. Stock, H. Strobele, T. Susa, I. Szentpetery, J. Sziklai, T. A. Trainor, D. Varga, M. Vassiliou, G. I. Veres, G. Vesztergombi, D. Vranic, A. Wetzler, Z. Wlodarczyk, I. K. Yoo, J. Zaranek, J. Zimanyi, Evidence for an Exotic S = −2, Q = −2 Baryon Resonance in Proton-Proton Collisions at the CERN SPS, Phys. Rev. Lett. 92 (4) (2004) 042003.
[2] K. Azizi, Y. Sarac and H. Sundu, Analysis of P+c (4380) and P+c(4450) as pentaquark states in the molecular picture with QCD sum rules, Phys. Rev. D 95 (9) (2017) 094016.
[3] M. Karakoc and I. Boztosun, Accurate iterative and perturbative solutions of the Yukawa potential, Int. J. Mod. Phys. E 15 (06) (2006) 1253 − 1262.
[4] D. Diakonov, V. Petrov and M. V. Polyakov, Exotic anti-decuplet of baryons: Prediction from chiral solitons, Z. Phys. A 359 (3) (1997) 305 − 314.
[5] A. Doté, T. Hyodo and W. Weise, Variational calculation of the system ppKbased on chiral SU(3) dynamics, Phys. Rev. C 79 (1) (2009) 014003.
[6] E. Santopinto, F. Iachello and M. M. Giannini, Exactly solvable models of baryon spectroscopy, Nucl. Phys. A 623 (1-2) (1997) 100 − 109.
[7] J. Stein, Akiva Ron, I. B. Goldberg and R. H. Pratt, Generalized WKB approximation to nonrelativistic normalizations and phase shifts in a screened Coulomb potential, Phys. Rev. A.: Gen. Phys. 36 (12) (1987) 5523 − 5529.
[8] B. Gönül, K. Köksal and E. Bakir, An alternative treatment for Yukawa-type potentials, Phys. Scr. 73 (3) (2006) 279.
[9] F. J. Rogers, H. C. Graboske, Jr. and D. J. Harwood, Bound eigenstates of the static screened Coulomb potential, Phys. Rev. A 1 (6) (1970) 1577−1586.
[10] R. Jaffe and F. Wilczek, Diquarks and exotic spectroscopy, Phys. Rev. Lett. 91 (23) (2003) 232003.
[11] B. K. Jennings and K. Maltman, Z ∗ resonances: Phenomenology and models, Phys. Rev. D 69 (2004) 094020.
[12] T. Hyodo, D. Jido and T. Kunihiro, Nature of the σ meson as revealed by its softening process, Nucl. Phys. A 848 (2010) 341.
[13] T. Hyodo and W. Weise, Effective¯KN interaction based on chiral SU(3) dynamics, Phys. Rev. C 77 (2008) 035204.
[14] A. Khalaghi, M. Monemzadeh and N. Tazimi, Investigation of binding energy and potential in mesons, Commun. Theor. Phys. 71 (7) (2019) 813.
[15] S. H. Lee, S. Yasui, W. Liu and C. Ko, Charmed exotics in heavy ion collisions, Eur. Phys. J. C 54 (2) (2008) 259 − 265.
[16] M. Monemzadeh, P. Sadeghi and N. Tazimi, Tetraquarks as diquark-antidiquark bound systems, Phys. Lett. B 741 (2015) 124 − 127.
[17] P. Sadeghi Alavijeh, N. Tazimi and M. Monemzadeh, Calculation of the Top-Quark Yukawa coupling constant, Int J. Mod. Phys. A 36 (19) (2021) 2150124.
[18] F. Okiharu, T. Doi, H. Ichie, H. Iida, Noriyoshi Ishii, M. Oka, H. Suganuma and T. T. Takahashi, Tetraquark and pentaquark systems in lattice QCD, J. Modern Phys. 7 (2016) 774 − 789.
[19] M. Radin, Sh. Babaghodrat and M. Monemzadeh, Estimation of heavy baryon masses Ω++ccc and Ωbbb by solving the Faddeev equation in a three-dimensional approach, Phys. Rev. D 90 (2014) 047701.
[20] M. Radin and N. Tazimi, Quark-antiquark bound state in momentum-helicity representation, Phys. Rev. D 90 (2014) 085020.
[21] Ö. Süleyman and M. Simsek, Exact solutions of the radial Schrödinger equation for inverse-power potentials, Phys. Lett. A 152 (3-4) (1991) 145 − 150.
[22] I. Schmidt and M. Siddikov, Production of pentaquarks in pA-collisions, Phys. Rev. D 93 (9) (2016) 094005.
[23] FI. Stancu, Positive parity pentaquarks in a Goldstone boson exchange model, Phys. Rev. D 58 (11) (1998) 111501(R).
[24] C. Stubbins, Bound states of the Hulthén and Yukawa potentials, Phys. Rev. A 48 (1993) 220 − 227.
[25] S. Yasui and K. Sudoh, Exotic nuclei with open heavy flavor mesons, Phys. Rev. D 80 (3) (2009) 034008.
[26] M. Karliner, Quark masses, hyperfine interactions and exotic baryons, Pentaquark Workshop, Jlab, hep-ph/0307243 and hep-ph/0307343 Nov. 7, 2003.
[27] M. Karliner and H. J. Lipkin, The anticharmed exotic baryon Θ c and its relatives hep-ph/0307343.
[28] Y. Sarac, H. Kimb and S. H. Lee, QCD sum rules for the anticharmed pentaquark, Phys. Rev. D 73 (1) (2006) 014009.
[29] H. Yukawa, On the interaction of elementary particles, Proc. Phys. Math. Soc. Jap. 17 (1935) 48 − 57.
[30] R. Zhu, X. Liu, H. Huang and C. -F. Qiao, Analyzing doubly heavy tetra-and penta-quark states by variational method, Phys. Lett. B 797 (2019) 134869.