Frontiers of Physics

, 13:132106 | Cite as

Study of various few-body systems using Gaussian expansion method (GEM)

  • Emiko Hiyama
  • Masayasu Kamimura
Review Article


We review our calculation method, Gaussian expansion method (GEM), to solve accurately the Schrödinger equations for bound, resonant and scattering states of few-body systems. Use is made of the Rayleigh-Ritz variational method for bound states, the complex-scaling method for resonant states and the Kohn-type variational principle to S-matrix for scattering states. GEM was proposed 30 years ago and has been applied to a variety of subjects in few-body (3- to 5-body) systems, such as 1) few-nucleon systems, 2) few-body structure of hypernuclei, 3) clustering structure of light nuclei and unstable nuclei, 4) exotic atoms/molecules, 5) cold atoms, 6) nuclear astrophysics and 7) structure of exotic hadrons. Showing examples in our published papers, we explain i) high accuracy of GEM calculations and its reason, ii) wide applicability of GEM to various few-body systems, iii) successful predictions by GEM calculations before measurements. The total bound-state wave function is expanded in terms of few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates. Gaussians with ranges in geometric progression work very well both for shortrange and long-range behavior of the few-body wave functions. Use of Gaussians with complex ranges gives much more accurate solution than in the case of real-range Gaussians, especially, when the wave function has many nodes (oscillations). These basis functions can well be applied to calculations using the complex-scaling method for resonances. For the few-body scattering states, the amplitude of the interaction region is expanded in terms of those few-body Gaussian basis functions.


few-body problems Gaussian expansion method Gaussian ranges in geometric progression 



It is our great pleasure to submit this invited review paper to the international symposium in honor of Professor Akito Arima to the celebration of his 88th birthday. We are very grateful for his continuous encouragement on our work. We would like to thank Professor Y. Kino for valuable discussions on GEM and its applications. The writing of this review was partially supported by the Japan Society for the Promotion of Science under grants 16H03995 and 16H02180 and by the RIKEN Interdisciplinary Theoretical Science Research Group project.


  1. 1.
    M. Kamimura, Nonadiabatic coupled-rearrangementchannel approach to muonic molecules, Phys. Rev. A 38(2), 621 (1988)ADSGoogle Scholar
  2. 2.
    H. Kameyama, M. Kamimura, and Y. Fukushima, Coupled-rearrangement-channel Gaussian-basis variational method for trinucleon bound states, Phys. Rev. C 40(2), 974 (1989)ADSGoogle Scholar
  3. 3.
    E. Hiyama, Y. Kino, and M. Kamimura, Gaussian expansion method for few-body systems, Prog. Part. Nucl. Phys. 51(1), 223 (2003)ADSGoogle Scholar
  4. 4.
    E. Hiyama, Few-body aspects of hypernuclear physics, Few-Body Syst. 53(3–4), 189 (2012)ADSGoogle Scholar
  5. 5.
    E. Hiyama, Gaussian expansion method for few-body systems and its applications to atomic and nuclear physics, Prog. Theor. Exp. Phys. 2012(1), 01A204 (2012)MathSciNetGoogle Scholar
  6. 6.
    D. E. Groom, et al. (Particle Data Group), Reviews, tables, and plots in the 2000 review of particle physics, Eur. Phys. J. C 15, 1 (2000)Google Scholar
  7. 7.
    H. A. Torii, R. S. Hayano, M. Hori, T. Ishikawa, N. Morita, et al., Laser measurements of the density shifts of resonance lines in antiprotonic helium atoms and stringent constraint on the antiproton charge and mass, Phys. Rev. A 59(1), 223 (1999)ADSGoogle Scholar
  8. 8.
    Y. Kino, M. Kamimura, and H. Kudo, High-accuracy 3-body coupled-channel calculation of metastable states of antiprotonic helium atoms, Nucl. Phys. A 631, 649 (1998)ADSGoogle Scholar
  9. 9.
    Y. Kino, M. Kamimura, and H. Kudo, Non-adiabatic high-precision calculation of antiprotonic helium atomcules, Hyperfine Interact. 119(1/4), 201 (1999)ADSGoogle Scholar
  10. 10.
    E. Hiyama, RCNP Physics Report (Research Center for Nuclear Physics, Osaka Univ.), RCNP-P 132, 35 (1994)Google Scholar
  11. 11.
    E. Hiyama, Proceedings of International Workshop on the 4-Body Problems, Uppsala, 1995 (Uppsala Univ., 1996), p. 28Google Scholar
  12. 12.
    S. Aoyama, T. Myo, K. Kato, and K. Ikeda, The complex scaling method for many-body resonances and its applications to three-body resonances, Prog. Theor. Phys. 116(1), 1 (2006)ADSGoogle Scholar
  13. 13.
    M. Kamimura, A coupled channel variational method for microscopic study of reactions between complex nuclei, Prog. Theor. Phys. Suppl. 62, 236 (1977)ADSGoogle Scholar
  14. 14.
    E. Hiyama and M. Kamimura, Variational calculation of 4He tetramer ground and excited states using a realistic pair potential, Phys. Rev. A 85(2), 022502 (2012)ADSGoogle Scholar
  15. 15.
    E. Hiyama and M. Kamimura, Linear correlations between 4He trimer and tetramer energies calculated with various realistic 4He potentials, Phys. Rev. A 85(6), 062505 (2012)ADSGoogle Scholar
  16. 16.
    E. Hiyama and M. Kamimura, Universality in Efimovassociated tetramers in 4He, Phys. Rev. A 90(5), 052514 (2014)ADSGoogle Scholar
  17. 17.
    S. Ohtsubo, Y. Fukushima, M. Kamimura, and E. Hiyama, Complex-scaling calculation of three-body resonances using complex-range Gaussian basis functions: Application to 3α resonances in 12C, Prog. Theor. Exp. Phys. 2013(7), 073D02 (2013)Google Scholar
  18. 18.
    T. Matsumoto, T. Kamizato, K. Ogata, Y. Iseri, E. Hiyama, M. Kamimura, and M. Yahiro, New treatment of breakup continuum in the method of continuum discretized coupled channels, Phys. Rev. C 68(6), 064607 (2003)ADSGoogle Scholar
  19. 19.
    T. Matsumoto, E. Hiyama, K. Ogata, Y. Iseri, M. Kamimura, S. Chiba, and M. Yahiro, Continuumdiscretized coupled-channels method for four-body nuclear breakup in 6He + 12C scattering, Phys. Rev. C 70(6), 061601 (2004)ADSGoogle Scholar
  20. 20.
    M. Kamimura, E. Hiyama, and Y. Kino, Big bang nucleosynthesis reactions catalyzed by a long lived negatively charged leptonic particle, Prog. Theor. Phys. 121(5), 1059 (2009)ADSzbMATHGoogle Scholar
  21. 21.
    H. Preuss, Bemerkungen zum Self-consistent-field-Verfahren und zur Methode der Konfigurationenwechselwirkung in der Quantenchemie, Z. Naturforsch 11a, 823 (1956)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    J. L. Whitten, Gaussian expansion of hydrogen-atom wavefunctions, J. Chem. Phys. 39(2), 349 (1963)ADSGoogle Scholar
  23. 23.
    H. Sambe, Use of 1 s Gaussian wavefunctions for molecular calculations (I): The hydrogen atom and the hydrogen molecule ion, J. Chem. Phys. 42(5), 1732 (1965)ADSGoogle Scholar
  24. 24.
    J. F. Harrison, On the Gaussian-Lobe representation of atomic orbitals, J. Chem. Phys. 46(3), 1115 (1967)ADSGoogle Scholar
  25. 25.
    A. A. Frost, Floating spherical gaussian orbital model of molecular structure (I): Computational procedure, LiH as an example, J. Chem. Phys. 47(10), 3707 (1967)ADSGoogle Scholar
  26. 26.
    K. Nagamine and M. Kamimura, Muon catalyzed fusion: Interplay between nuclear and atomic physics, Advance in Nuclear Physics 24, 151 (1998)Google Scholar
  27. 27.
    V. I. Korobov, I. V. Puzynin, and S. I. Vinitsky, A variational calculation of weakly bound rotationalvibrational states of the mesic molecules ddμ and dtμ, Phys. Lett. B 196(3), 272 (1987)ADSGoogle Scholar
  28. 28.
    S. A. Alexander and H. J. Monkhorst, High-accuracy calculation of muonic molecules using random-tempered basis sets, Phys. Rev. 38(1), 26 (1988)ADSGoogle Scholar
  29. 29.
    R. B. Wiringa, R. A. Smith, and T. A. Ainsworth, Nucleon-nucleon potentials with and without D(1232) degrees of freedom, Phys. Rev. C 29(4), 1207 (1984)ADSGoogle Scholar
  30. 30.
    M. Kamimura and H. Kameyama, Coupled rearrangement channel calculations of muonic molecules and A = 3 nuclei, Nucl. Phys. A 508, 17 (1990)ADSGoogle Scholar
  31. 31.
    S. A. Coon, M. D. Scadron, P. C. McNamee, B. R. Barrett, D. W. E. Blatt, and B. H. J. McKellar, The two-pion-exchange three-nucleon potential and nuclear matter, Nucl. Phys. A 317(1), 242 (1979)ADSGoogle Scholar
  32. 32.
    C. R. Chen, G. L. Payne, J. L. Frier, and B. F. Gibson, Convergence of Faddeev partial-wave series for triton ground state, Phys. Rev. C 31(6), 2266 (1985)ADSGoogle Scholar
  33. 33.
    S. Ishikawa and T. Sasakawa, Faddeev partial-wave calculations with a three-nucleon potential for the triton ground state, Few-Body Syst. 1(3), 143 (1986)ADSGoogle Scholar
  34. 34.
    T. Sasakawa, in: Proceedings of the Workshop on Electron Nucleus Scattering, Elba International physics Center, Italy, 1988Google Scholar
  35. 35.
    Y. Wu, S. Ishikawa, and T. Sasakawa, Private communications (1989)Google Scholar
  36. 36.
    G. L. Payne and B. F. Gibson, Variational aspects of Faddeev calculations, Few-Body Syst. 14(3), 117 (1993)ADSGoogle Scholar
  37. 37.
    H. Kamada, A. Nogga, W. Glöckle, E. Hiyama, M. Kamimura, et al., Leidemann, and G. Orlandini, Benchmark test calculation of a four-nucleon bound state, Phys. Rev. C 64(4), 044001 (2001)ADSGoogle Scholar
  38. 38.
    B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C. Pieper, and R. B. Wiringa, Quantum Monte Carlo calculations of nuclei with A ≤ 7, Phys. Rev. C 56(4), 1720 (1997)ADSGoogle Scholar
  39. 39.
    E. Hiyama, B. F. Gibson and M. Kamimura, Fourbody calculation of the first excited state of 4He using a realistic NN interaction: 4He (e, e′)4He (0+ 2) and the monopole sum rule, Phys. Rev. C 70, 031001(R) (2004)ADSGoogle Scholar
  40. 40.
    W. Horiuchi and Y. Suzuki, Excited states and strength functions of 4He in correlated gaussians, Few-Body Syst. 54(12), 2407 (2013)ADSGoogle Scholar
  41. 41.
    C. Caso, et al. (Particle Data Group), Review of Particle Physics, Eur. Phys. J. C 3(1–4), 1 (1998)Google Scholar
  42. 42.
    Y. Kino, M. Kamimura, and H. Kudo, High-precision calculation of antiprotonic helium atomcules and antiproton mass, Few-Body Syst. Suppl. 12, 40 (2000)Google Scholar
  43. 43.
    Y. Kino, N. Yamanaka, M. Kamimura, P. Froelich, and H. Kudo, High-precision calculation of the energy levels and auger decay rates of the metastable states of the antiprotonic helium atoms, Hyperfine Interact. 138(1/4), 179 (2001)ADSGoogle Scholar
  44. 44.
    E. Braaten and H. W. Hammer, Universality in fewbody systems with large scattering length, Phys. Rep. 428(5–6), 259 (2006)ADSMathSciNetGoogle Scholar
  45. 45.
    R. A. Aziz and M. J. Slaman, An examination of ab initio results for the helium potential energy curve, J. Chem. Phys. 94(12), 8047 (1991)ADSGoogle Scholar
  46. 46.
    R. Lazauskas and J. Carbonell, Description of He4 tetramer bound and scattering states, Phys. Rev. A 73(6), 062717 (2006)ADSGoogle Scholar
  47. 47.
    M. Hori, J. Eades, R. S. Hayano, T. Ishikawa, J. Sakaguchi, E. Widmann, H. Yamaguchi, H. A. Torii, B. Juhász, D. Horváth, and T. Yamazaki, Sub-ppm laser spectroscopy of antiprotonic helium and a CPTviolation limit on the antiprotonic charge and mass, Phys. Rev. Lett. 87, 093401 (2001)ADSGoogle Scholar
  48. 48.
    K. Hagiwara, et al. (Particle Data Group), Review of particle properties, Phys. Rev. D 66(1), 010001 (2002)ADSGoogle Scholar
  49. 49.
    T. Motoba, H. Bando, and K. Ikeda, Light p-shellhypernuclei by the microscopic three-cluster model, Prog. Theor. Phys. 70(1), 189 (1983)ADSGoogle Scholar
  50. 50.
    T. Motoba, H. Bando, K. Ikeda, and T. Yamada, Production, structure an decay of light p-shell L-hypernuclei, Prog. Theor. Phys. Suppl. 81, 42 (1985)ADSGoogle Scholar
  51. 51.
    E. Hiyama, M. Kamimura, K. Miyazaki, and T. Motoba, γ transitions in A = 7 hypernuclei and a possible derivation of hypernuclear size, Phys. Rev. C 59(5), 2351 (1999)ADSGoogle Scholar
  52. 52.
    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Three-body model study of A = 6–7 hypernuclei: Halo and skin structures, Phys. Rev. C 53(5), 2075 (1996)ADSGoogle Scholar
  53. 53.
    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Three-and four-body structure of light hypernuclei, Nucl. Phys. A 684(1–4), 227 (2001)ADSGoogle Scholar
  54. 54.
    K. Tanida, H. Tamura, D. Abe, H. Akikawa, K. Araki, et al., Measurement of the B(E2) of 6 ΛLi and shrinkage of the hypernuclear size, Phys. Rev. Lett. 86(10), 1982 (2001)ADSGoogle Scholar
  55. 55.
    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Three-and four-body cluster models of hypernuclei using the G-matrix N interaction: 9Be, 13C, 6He and 10Be, Prog. Theor. Phys. 97(6), 881 (1997)ADSGoogle Scholar
  56. 56.
    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, LN spin-orbit splittings in 9 ΛBe and 13 Λ C studied with one-boson-exchange LN interactions, Phys. Rev. Lett. 85(2), 270 (2000)ADSGoogle Scholar
  57. 57.
    H. Akikawa, S. Ajimura, R. E. Chrien, P. M. Eugenio, G. B. Franklin, et al., Hypernuclear fine structure in 9 Be, Phys. Rev. Lett. 88(8), 082501 (2002)ADSGoogle Scholar
  58. 58.
    S. Ajimura, H. Hayakawa, T. Kishimoto, H. Kohri, K. Matsuoka, et al., Observation of spin-orbit splitting in L single-particle states, Phys. Rev. Lett. 86(19), 4255 (2001)ADSGoogle Scholar
  59. 59.
    M. M. Nagels, T. A. Rijken, and J. J. de Swart, Baryonbaryon scattering in a one-boson-exchange-potential approach (I): Nucleon-nucleon scattering, Phys. Rev. D 12, 744 (1975)ADSGoogle Scholar
  60. 60.
    M. M. Nagels, T. A. Rijken, and J. J. de Swart, Baryonbaryon scattering in a one-boson-exchange-potential approach (II): Hyperon-nucleon scattering, Phys. Rev. D 15, 2547 (1977)ADSGoogle Scholar
  61. 61.
    M. M. Nagels, T. A. Rijken, and J. J. de Swart, Baryonbaryon scattering in a one-boson-exchange-potential approach (III): A nucleon-nucleon and hyperon-nucleon analysis including contributions of a nonet of scalar mesons, Phys. Rev. D 20, 1633 (1979)ADSGoogle Scholar
  62. 62.
    T. A. Rijken, V. G. J. Stoks, and Y. Yamamoto, Softcore hyperon-nucleon potentials, Phys. Rev. C 59(1), 21 (1999)ADSGoogle Scholar
  63. 63.
    O. Morimatsu, S. Ohta, K. Shimizu, and K. Yazaki, Baryon-baryon spin-orbit interaction in a quark model, Nucl. Phys. A 420(3), 573 (1984)ADSGoogle Scholar
  64. 64.
    Y. Fujiwara, C. Nakamoto, and Y. Suzuki, Unified description of NN and YN interactions in a quark model with effective meson-exchange potentials, Phys. Rev. Lett. 76(13), 2242 (1996)ADSGoogle Scholar
  65. 65.
    E. Hiyama, Y. Yamamoto, T. Motoba, and M. Kamimura, Structure of A = 7 iso-triplet Λ hypernuclei studied with the four-body cluster model, Phys. Rev. C 80(5), 054321 (2009)ADSGoogle Scholar
  66. 66.
    S. N. Nakamura, A. Matsumura, Y. Okayasu, T. Seva, V. M. Rodriguez, et al., Observation of the 7 ΛHe hypernucleus by the (e, e′ K +) reaction, Phys. Rev. Lett. 110(1), 012502 (2013)ADSGoogle Scholar
  67. 67.
    T. Gogami, Ph.D thesis, Tohoku University, 2014Google Scholar
  68. 68.
    E. Hiyama, M. Isaka, M. Kamimura, T. Myo, and T. Motoba, Resonant states of the neutron-rich Λ hypernucleus 7 ΛHe, Phys. Rev. C 91(5), 054316 (2015)ADSGoogle Scholar
  69. 69.
    J. K. Ahn, et al., in: Hadron and Nuclei, AIP Conf. Proc. No. 594, Ed. II-Tong Cheon, et al., AIP, Meville, NY, 2001, page 180Google Scholar
  70. 70.
    A. Ichikawa, Ph.D. thesis, Kyoto University, 2001Google Scholar
  71. 71.
    K. Nakazawa, et al., Double-Λ hypernuclei via the Ξ- hyperon capture at rest reaction in a hybrid emulsion, Nucl. Phys. A 835(1–4), 207 (2010) (The proceedings on the 10th International Conference on Hypernuclear and Strange Particle Physics (Hyp X), Tokai, Sept. 14–18, 2009)ADSGoogle Scholar
  72. 72.
    J. K. Ahn, H. Akikawa, S. Aoki, K. Arai, S. Y. Bahk, et al., Double-Λ hypernuclei observed in a hybrid emulsion experiment, Phys. Rev. C 88(1), 014003 (2013)ADSGoogle Scholar
  73. 73.
    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Four-body cluster structure of A = 7–10 double-Λ hypernuclei, Phys. Rev. C 66(2), 024007 (2002)ADSGoogle Scholar
  74. 74.
    E. Hiyama, M. Kamimura, Y. Yamamoto, and T. Motoba, Five-body cluster structure of the double-Λ hypernucleus 11 ΛΛBe, Phys. Rev. Lett. 104(21), 212502 (2010)ADSGoogle Scholar
  75. 75.
    E. Hiyama and T. Yamada, Structure of light hypernuclei, Prog. Part. Nucl. Phys. 63(2), 339 (2009)ADSGoogle Scholar
  76. 76.
    E. Hiyama, M. Kamimura, Y. Yamamoto, T. Motoba, and T. A. Rijken, S =–1 hypernuclear structure, Prog. Theor. Phys. Suppl. 185, 106 (2010)ADSzbMATHGoogle Scholar
  77. 77.
    E. Hiyama, M. Kamimura, Y. Yamamoto, T. Motoba, and T. A. Rijken, S =–2 hypernuclear structure, Prog. Theor. Phys. Suppl. 185, 152 (2010)ADSzbMATHGoogle Scholar
  78. 78.
    J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Commun. Math. Phys. 22(4), 269 (1971)ADSzbMATHGoogle Scholar
  79. 79.
    E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilatationanalytic interactions, Commun. Math. Phys. 22(4), 280 (1971)ADSzbMATHGoogle Scholar
  80. 80.
    B. Simon, Quadratic form techniques and the Balslev–Combes theorem, Commun. Math. Phys. 27(1), 1 (1972)ADSMathSciNetzbMATHGoogle Scholar
  81. 81.
    Y. K. Ho, The method of complex coordinate rotation and its applications to atomic collision processes, Phys. Rep. 99(1), 1 (1983)ADSGoogle Scholar
  82. 82.
    N. Moiseyev, Quantum theory of resonances: Calculating energies, widths and cross-sections by complex scaling, Phys. Rep. 302(5–6), 212 (1998)Google Scholar
  83. 83.
    E. Hiyama, R. Lazauskas, J. Carbonell, and M. Kamimura, Possibility of generating a 4-neutron resonance with a T = 3/2 isospin 3-neutron force, Phys. Rev. C 93(4), 044004 (2016)ADSGoogle Scholar
  84. 84.
    K. Kisamori, S. Shimoura, H. Miya, S. Michimasa, S. Ota, et al., Candidate resonant tetraneutron state populated by the 4He (8He, 8Be) reaction, Phys. Rev. Lett. 116(5), 052501 (2016)ADSGoogle Scholar
  85. 85.
    C. Kurokawa and K. Katō, New broad 0+ state in 12C, Phys. Rev. C 71, 021301(R) (2005)ADSGoogle Scholar
  86. 86.
    M. Kusakabe, G. J. Mathews, T. Kajino, and M.K. Cheoun, Review on effects of long-lived negatively charged massive particles on Big Bang Nucleosynthesis, Int. J. Mod. Phys. E 26(08), 1741004 (2017)ADSGoogle Scholar
  87. 87.
    F. Iocco, G. Mangano, G. Miele, O. Pisanti, and P. D. Serpico, Primordial nucleosynthesis: From precision cosmology to fundamental physics, Phys. Rep. 472(1–6), 1 (2009)ADSGoogle Scholar
  88. 88.
    M. Kamimura, Non-adiabatic coupled-rearrangementchannels approach to muonic molecules and muon transfer reactions, Muon Catal. Fusion 3, 335 (1988)Google Scholar
  89. 89.
    Y. Kino and M. Kamimura, Non-adiabatic calculation of muonic atom-nucleus collisions, Hyperfine Interactions 82(1–4), 45 (1993)ADSGoogle Scholar
  90. 90.
    J. S. Cohen and M. C. Struensee, Improved adiabatic calculation of muonic-hydrogen-atom cross sections (I): Isotopic exchange and elastic scattering in asymmetric collisions, Phys. Rev. A 43, 3460 (1991)ADSGoogle Scholar
  91. 91.
    C. Chiccoli, V. I. Korobov, V. S. Melezhik, P. Pasini, L. I. Ponomarev, and J. Wozniak, The atlas of the cross sections of mesic atomic processes (III): The process +(d, t), +(p, t) and +(p, d), Muon Catal. Fusion 7, 87 (1992)Google Scholar
  92. 92.
    O. I. Tolstikhin, and C. Namba, Hyperspherical calculations of low-energy rearrangement processes in dtμ, Phys. Rev. A 60(6), 5111 (1999)ADSGoogle Scholar
  93. 93.
    K. Hamaguchi, T. Hatsuda, M. Kamimura, Y. Kino, and T. T. Yanagida, Stau-catalyzed 6Li production in big-bang nucleosynthesis, Phys. Lett. B 650(4), 268 (2007)ADSGoogle Scholar
  94. 94.
    M. Kubo, J. Sato, T. Shimomura, Y. Takanishi, and M. Yamanaka, Big-bang nucleosynthesis and leptogenesis in the CMSSM, Phys. Rev. D 97(11), 115013 (2018)ADSGoogle Scholar
  95. 95.
    M. Kusakabe, K. S. Kim, M. K. Cheoun, T. Kajino, and Y. Kino, 7Be charge exchange between 7Be3+ ion and an exotic long-lived negatively charged massive particle in big bang nucleosynthesis, Phys. Rev. D 88(6), 063514 (2013)ADSGoogle Scholar
  96. 96.
    S. Bailly, K. Jedamzik, and G. Moultaka, Gravitino dark matter and the cosmic lithium abundances, Phys. Rev. D 80(6), 063509 (2009)ADSGoogle Scholar
  97. 97.
    E. Hiyama, M. Kamimura, A. Hosaka, H. Toki, and M. Yahiro, Five-body calculation of resonance and scattering states of pentaquark system, Phys. Lett. B 633(2–3), 237 (2006)ADSGoogle Scholar
  98. 98.
    T. Nakano, et al. (LEPS Collaboration), Evidence for a narrow S = +1 baryon resonance in photoproduction from the neutron, Phys. Rev. Lett. 91(1), 012002 (2003)ADSGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsKyushu UniversityFukuokaJapan
  2. 2.Nishina CenterRIKENWakoJapan

Personalised recommendations