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Three-body scattering: ladders and resonances

  • M. MikhasenkoEmail author
  • Y. Wunderlich
  • A. Jackura
  • V. Mathieu
  • A. Pilloni
  • B. Ketzer
  • A.P. Szczepaniak
Open Access
Regular Article - Theoretical Physics
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Abstract

We discuss unitarity constraints on the dynamics of a system of three interacting particles. We show how the short-range interaction that describes three-body resonances can be separated from the long-range exchange processes, in particular the one-pion-exchange process. It is demonstrated that unitarity demands a specific functional form of the amplitude with a clear interpretation: the bare three-particle resonances are dressed by the initial- and final-state interaction, in a way that is consistent with the considered long-range forces. We postulate that the resonance kernel admits a factorization in the energy variables of the initial- and the final-state particles. The factorization assumption leads to an algebraic form for the unitarity equations, which is reminiscent of the well-known two-body-unitarity condition and approaches it in the limit of the narrow-resonance approximation.

Keywords

Phenomenological Models QCD Phenomenology 

Notes

Open Access

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References

  1. [1]
    R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge (1966).zbMATHGoogle Scholar
  2. [2]
    S.L. Olsen, T. Skwarnicki and D. Zieminska, Nonstandard heavy mesons and baryons: Experimental evidence, Rev. Mod. Phys. 90 (2018) 015003 [arXiv:1708.04012] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Esposito, A. Pilloni and A.D. Polosa, Multiquark Resonances, Phys. Rept. 668 (2017) 1 [arXiv:1611.07920] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    F.-K. Guo, C. Hanhart, U.-G. Meissner, Q. Wang, Q. Zhao and B.-S. Zou, Hadronic molecules, Rev. Mod. Phys. 90 (2018) 015004 [arXiv:1705.00141] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    COMPASS collaboration, Light isovector resonances in π p → π π π + p at 190 GeV/c, Phys. Rev. D 98 (2018) 092003 [arXiv:1802.05913] [INSPIRE].
  6. [6]
    COMPASS collaboration, Observation of a New Narrow Axial-Vector Meson a 1(1420), Phys. Rev. Lett. 115 (2015) 082001 [arXiv:1501.05732] [INSPIRE].
  7. [7]
    Hadron Spectrum collaboration, Energy dependence of the ρ resonance in ππ elastic scattering from lattice QCD, Phys. Rev. D 87 (2013) 034505 [Erratum ibid. D 90 (2014) 099902] [arXiv:1212.0830] [INSPIRE].
  8. [8]
    Hadron Spectrum collaboration, Resonances in coupled πK − ηK scattering from quantum chromodynamics, Phys. Rev. Lett. 113 (2014) 182001 [arXiv:1406.4158] [INSPIRE].
  9. [9]
    R.A. Briceño, J.J. Dudek, R.G. Edwards and D.J. Wilson, Isoscalar ππ scattering and the σ meson resonance from QCD, Phys. Rev. Lett. 118 (2017) 022002 [arXiv:1607.05900] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    R.A. Briceno, J.J. Dudek and R.D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90 (2018) 025001 [arXiv:1706.06223] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    D. Guo, A. Alexandru, R. Molina, M. Mai and M. Döring, Extraction of isoscalar ππ phase-shifts from lattice QCD, Phys. Rev. D 98 (2018) 014507 [arXiv:1803.02897] [INSPIRE].ADSGoogle Scholar
  12. [12]
    C.B. Lang, D. Mohler, S. Prelovsek and M. Vidmar, Coupled channel analysis of the rho meson decay in lattice QCD, Phys. Rev. D 84 (2011) 054503 [Erratum ibid. D 89 (2014) 059903] [arXiv:1105.5636] [INSPIRE].ADSGoogle Scholar
  13. [13]
    M.T. Hansen and S.R. Sharpe, Applying the relativistic quantization condition to a three-particle bound state in a periodic box, Phys. Rev. D 95 (2017) 034501 [arXiv:1609.04317] [INSPIRE].ADSMathSciNetGoogle Scholar
  14. [14]
    JPAC collaboration, Phenomenology of Relativistic 3 → 3 Reaction Amplitudes within the Isobar Approximation, Eur. Phys. J. C 79 (2019) 56 [arXiv:1809.10523] [INSPIRE].
  15. [15]
    M. Mai, B. Hu, M. Döring, A. Pilloni and A. Szczepaniak, Three-body Unitarity with Isobars Revisited, Eur. Phys. J. A 53 (2017) 177 [arXiv:1706.06118] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Mai and M. Döring, Three-body Unitarity in the Finite Volume, Eur. Phys. J. A 53 (2017) 240 [arXiv:1709.08222] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Mai and M. Döring, Finite-Volume Spectrum of π + π + and π + π + π + Systems, Phys. Rev. Lett. 122 (2019) 062503 [arXiv:1807.04746] [INSPIRE].
  18. [18]
    H.W. Hammer, J.Y. Pang and A. Rusetsky, Three particle quantization condition in a finite volume: 2. general formalism and the analysis of data, JHEP 10 (2017) 115 [arXiv:1707.02176] [INSPIRE].
  19. [19]
    H.-W. Hammer, J.-Y. Pang and A. Rusetsky, Three-particle quantization condition in a finite volume: 1. The role of the three-particle force, JHEP 09 (2017) 109 [arXiv:1706.07700] [INSPIRE].
  20. [20]
    COMPASS collaboration, First results from an extended freed-isobar analysis at COMPASS, PoS(Hadron2017) 034 (2018) [arXiv:1711.10828] [INSPIRE].
  21. [21]
    M. Mikhasenko, B. Ketzer and A. Sarantsev, Nature of the a 1(1420), Phys. Rev. D 91 (2015) 094015 [arXiv:1501.07023] [INSPIRE].ADSGoogle Scholar
  22. [22]
    F. Aceti, L.R. Dai and E. Oset, a 1(1420) peak as the πf 0(980) decay mode of the a 1(1260), Phys. Rev. D 94 (2016) 096015 [arXiv:1606.06893] [INSPIRE].ADSGoogle Scholar
  23. [23]
    D. Herndon, P. Soding and R.J. Cashmore, A generalized isobar model formalism, Phys. Rev. D 11 (1975) 3165 [INSPIRE].ADSGoogle Scholar
  24. [24]
    M.T. Grisaru, Three-particle contributions to elastic scattering, Phys. Rev. 146 (1966) 1098.ADSCrossRefGoogle Scholar
  25. [25]
    S. Mandelstam, J. Paton, R.F. Peierls and A. Sarker, Isobar approximation of production processes, Annals Phys. 18 (1962) 198.ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    D.R. Harrington, Two-particle approximation for the three-pion amplitude, Phys. Rev. 127 (1962) 2235.ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    G.N. Fleming, Recoupling Effects in the Isobar Model. 1. General Formalism for Three-Pion Scattering, Phys. Rev. 135 (1964) B551 [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    W.R. Frazer and D.Y. Wong, Width of three-pion resonances, Phys. Rev. 128 (1962) 1927.ADSCrossRefGoogle Scholar
  29. [29]
    W.J. Holman, Modified isobar approximation for π-ρ scattering, Phys. Rev. 138 (1965) B1286.ADSCrossRefGoogle Scholar
  30. [30]
    L.F. Cook and B.W. Lee, Unitarity and Production Amplitudes, Phys. Rev. 127 (1962) 283 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    J.S. Ball, W.R. Frazer and M. Nauenberg, Scattering and production amplitudes with unstable particles, in High-energy physics. Proceedings, 11th International Conference, ICHEP’62, Geneva, Switzerland, 4-11 July 1962 (1962), pp. 141-143 [INSPIRE].
  32. [32]
    R. C. Hwa, On the analytic structure of production amplitudes, Phys. Rev. 134 (1964) B1086.ADSCrossRefGoogle Scholar
  33. [33]
    J.D. Bjorken, Construction of Coupled Scattering and Production Amplitudes Satisfying Analyticity and Unitarity, Phys. Rev. Lett. 4 (1960) 473 [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    R. Blankenbecler and R. Sugar, Linear integral equations for relativistic multichannel scattering, Phys. Rev. 142 (1966) 1051 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    R.D. Amado, Minimal three-body equations with finite-range effects, Phys. Rev. C 12 (1975) 1354 [INSPIRE].ADSGoogle Scholar
  36. [36]
    R.D. Amado, Minimal three-body scattering theory, Phys. Rev. Lett. 33 (1974) 333 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    R. Aaron and R.D. Amado, Analysis of three-hadron final states, Phys. Rev. Lett. 31 (1973) 1157 [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    R. Aaron, A Relativistic Three-Body Theory, Top. Curr. Phys. 2 (1977) 139 [INSPIRE].CrossRefGoogle Scholar
  39. [39]
    R. Aaron, R.D. Amado and J.E. Young, Relativistic three-body theory with applications to pi-minus n scattering, Phys. Rev. 174 (1968) 2022 [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    N.N. Khuri and S.B. Treiman, Pion-Pion Scattering and K ± 3π Decay, Phys. Rev. 119 (1960) 1115 [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    R. Pasquier and J.Y. Pasquier, Khuri-Treiman-Type Equations for Three-Body Decay and Production Processes, Phys. Rev. 170 (1968) 1294 [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    R. Pasquier and J.Y. Pasquier, Khuri-treiman-type equations for three-body decay and production processes. 2, Phys. Rev. 177 (1969) 2482 [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    J.B. Bronzan and C. Kacser, Khuri-TreimanRepresentation and Perturbation Theory, Phys. Rev. 132 (1963) 2703 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    I.J.R. Aitchison, Dispersion Theory Model of Three-Body Production and Decay Processes, Phys. Rev. 137 (1965) B1070 [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    I.J.R. Aitchison and R. Pasquier, Three-Body Unitarity and Khuri-TreimanAmplitudes, Phys. Rev. 152 (1966) 1274 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    C. Kacser, The Discontinuities of the Triangle Graph as a Function of an Internal Mass. II, J. Math. Phys. 7 (1966) 2008.ADSCrossRefGoogle Scholar
  47. [47]
    I.J.R. Aitchison and J.J. Brehm, Unitary Analytic Isobar Model for the Reaction Nucleon-Meson to Nucleon-Meson Meson, Phys. Rev. D 17 (1978) 3072 [INSPIRE].ADSGoogle Scholar
  48. [48]
    I.J.R. Aitchison and J.J. Brehm, Are there important unitarity corrections to the isobar model?, Phys. Lett. 84B (1979) 349 [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    I.J.R. Aitchison and J.J. Brehm, Medium-energy N ππ dynamics. 1., Phys. Rev. D 20 (1979) 1119 [INSPIRE].ADSGoogle Scholar
  50. [50]
    L. Ray, G.W. Hoffmann and R.M. Thaler, Coulomb interaction in multiple scattering theory, Phys. Rev. C 22 (1980) 1454 [INSPIRE].ADSGoogle Scholar
  51. [51]
    K. Nakano, Two potential formalisms and the Coulomb-nuclear interference, Phys. Rev. C 26 (1982) 1123 [INSPIRE].ADSGoogle Scholar
  52. [52]
    S. Ropertz, C. Hanhart and B. Kubis, A new parametrization for the scalar pion form factors, Eur. Phys. J. C 78 (2018) 1000 [arXiv:1809.06867] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    C. Hanhart, A New Parameterization for the Pion Vector Form Factor, Phys. Lett. B 715 (2012) 170 [arXiv:1203.6839] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    D. Ronchen et al., Coupled-channel dynamics in the reactions πN → πN, ηN, KΛ, KΣ, Eur. Phys. J. A 49 (2013) 44 [arXiv:1211.6998] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    M. Döring, C. Hanhart, F. Huang, S. Krewald and U.G. Meissner, The Role of the background in the extraction of resonance contributions from meson-baryon scattering, Phys. Lett. B 681 (2009) 26 [arXiv:0903.1781] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    A. Matsuyama, T. Sato and T.S.H. Lee, Dynamical coupled-channel model of meson production reactions in the nucleon resonance region, Phys. Rept. 439 (2007) 193 [nucl-th/0608051] [INSPIRE].
  57. [57]
    H. Kamano, S.X. Nakamura, T.S.H. Lee and T. Sato, Unitary coupled-channels model for three-mesons decays of heavy mesons, Phys. Rev. D 84 (2011) 114019 [arXiv:1106.4523] [INSPIRE].ADSGoogle Scholar
  58. [58]
    C. Itzykson and J.B. Zuber, Quantum Field Theory, International Series In Pure and Applied Physics, McGraw-Hill, New York (1980).Google Scholar
  59. [59]
    H. Lehmann, K. Symanzik and W. Zimmermann, On the formulation of quantized field theories, Nuovo Cim. 1 (1955) 205 [INSPIRE].CrossRefzbMATHGoogle Scholar
  60. [60]
    R. Omnes, On the Solution of certain singular integral equations of quantum field theory, Nuovo Cim. 8 (1958) 316 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    E. Byckling and K. Kajantie, Particle Kinematics, University of Jyvaskyla, Jyvaskyla, Finland (1971).Google Scholar
  62. [62]
    T.W.B. Kibble, Kinematics of General Scattering Processes and the Mandelstam Representation, Phys. Rev. 117 (1960) 1159 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    Particle Data Group collaboration, Review of Particle Physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].
  64. [64]
    M.T. Hansen and S.R. Sharpe, Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude, Phys. Rev. D 92 (2015) 114509 [arXiv:1504.04248] [INSPIRE].ADSGoogle Scholar
  65. [65]
    I.J.R. Aitchison, K-matrix formalism for overlapping resonances, Nucl. Phys. A 189 (1972) 417 [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    A.D. Martin and T.D. Spearman, Elementary-particle theory, North-Holland, Amsterdam (1970).
  67. [67]
    JPAC collaboration, Pole position of the a 1(1260) from τ -decay, Phys. Rev. D 98 (2018) 096021 [arXiv:1810.00016] [INSPIRE].
  68. [68]
    J.L. Basdevant and E.L. Berger, Unstable Particle Scattering and an Analytic Quasiunitary Isobar Model, Phys. Rev. D 19 (1979) 239 [INSPIRE].ADSGoogle Scholar
  69. [69]
    M. Hoferichter, B. Kubis, S. Leupold, F. Niecknig and S.P. Schneider, Dispersive analysis of the pion transition form factor, Eur. Phys. J. C 74 (2014) 3180 [arXiv:1410.4691] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    D.I. Olive, Unitarity and the evaluation of discontinuities, Nuovo Cim. 26 (1962) 73.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    M. Albaladejo and B. Moussallam, Extended chiral Khuri-Treiman formalism for η → 3π and the role of the a 0(980), f 0(980) resonances, Eur. Phys. J. C 77 (2017) 508 [arXiv:1702.04931] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    F. Niecknig, B. Kubis and S.P. Schneider, Dispersive analysis of ω → 3π and φ → 3π decays, Eur. Phys. J. C 72 (2012) 2014 [arXiv:1203.2501] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    P. Guo, I.V. Danilkin, D. Schott, C. Fernández-Ram´ırez, V. Mathieu and A.P. Szczepaniak, Three-body final state interaction in η → 3π, Phys. Rev. D 92 (2015) 054016 [arXiv:1505.01715] [INSPIRE].ADSGoogle Scholar
  74. [74]
    I.V. Danilkin et al., Dispersive analysis of ω/φ → 3π, πγ , Phys. Rev. D 91 (2015) 094029 [arXiv:1409.7708] [INSPIRE].ADSGoogle Scholar
  75. [75]
    M. Mikhasenko, Three-pion dynamics at COMPASS: resonances, rescattering and non-resonant processes, Ph.D. Thesis, University of Bonn (2019).Google Scholar
  76. [76]
    I.J.R. Aitchison and C. Kacser, Singularities and discontinuities of the triangle graph, as a function of an internal mass, Nuovo Cim. A 40 (1965) 576.ADSMathSciNetCrossRefGoogle Scholar
  77. [77]
    F. Niecknig, Dispersive analysis of charmed meson decays, Ph.D. Thesis, Bonn U., HISKP (2016) [http://hss.ulb.uni-bonn.de/2016/4431/4431.htm].
  78. [78]
    COMPASS collaboration, Resonance Production and ππ S-wave in π + p → π π π + + p recoil at 190 GeV/c, Phys. Rev. D 95 (2017) 032004 [arXiv:1509.00992] [INSPIRE].
  79. [79]
    CLEO collaboration, Hadronic structure in the decay τ → ν τ π π 0 π 0 and the sign of the tau-neutrino helicity, Phys. Rev. D 61 (2000) 012002 [hep-ex/9902022] [INSPIRE].
  80. [80]
    M. Mikhasenko and B. Ketzer, Beyond the isobar model: Rescattering in the system of three particles, PoS(BORMIO2016) 024.Google Scholar
  81. [81]
    A. Martin and T. Spearman, Elementary Particle Theory, 1st edition, North-Holland Publishing Company (1970).Google Scholar
  82. [82]
    M. Jacob and G.C. Wick, On the general theory of collisions for particles with spin, Annals Phys. 7 (1959) 404 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    A.D. Martin and T.D. Spearman, Elementary Particle Theory, North-Holland, Amsterdam, and Elsevier, New York (1970).Google Scholar
  84. [84]
    K. Gottfried and J.D. Jackson, On the Connection between production mechanism and decay of resonances at high-energies, Nuovo Cim. 33 (1964) 309 [INSPIRE].ADSCrossRefGoogle Scholar
  85. [85]
    M.E. Peskin and D.V. Schroeder, An Introduction to quantum field theory, Addison-Wesley, Reading, U.S.A. (1995).Google Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • M. Mikhasenko
    • 1
    • 2
    Email author
  • Y. Wunderlich
    • 1
  • A. Jackura
    • 3
    • 4
  • V. Mathieu
    • 5
    • 6
  • A. Pilloni
    • 7
    • 8
  • B. Ketzer
    • 1
  • A.P. Szczepaniak
    • 3
    • 4
    • 5
  1. 1.Universität Bonn, Helmholtz-Institut für Strahlen- und KernphysikBonnGermany
  2. 2.CERNGeneva 23Switzerland
  3. 3.Center for Exploration of Energy and MatterIndiana UniversityBloomingtonU.S.A.
  4. 4.Physics DepartmentIndiana UniversityBloomingtonU.S.A.
  5. 5.Theory CenterThomas Jefferson National Accelerator FacilityNewport NewsU.S.A.
  6. 6.Departamento de Física TeóricaUniversidad Complutense de MadridMadridSpain
  7. 7.European Centre for Theoretical Studies in Nuclear Physics and related Areas (ECT *), and Fondazione Bruno KesslerVillazzano (Trento)Italy
  8. 8.INFN Sezione di GenovaGenovaItaly

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