Advertisement

Roy-Steiner equations for pion-nucleon scattering

  • C. DitscheEmail author
  • M. Hoferichter
  • B. Kubis
  • U.-G. Meißner
Article

Abstract

Starting from hyperbolic dispersion relations, we derive a closed system of Roy-Steiner equations for pion-nucleon scattering that respects analyticity, unitarity, and crossing symmetry. We work out analytically all kernel functions and unitarity relations required for the lowest partial waves. In order to suppress the dependence on the high energy regime we also consider once- and twice-subtracted versions of the equations, where we identify the subtraction constants with subthreshold parameters. Assuming Mandelstam analyticity we determine the maximal range of validity of these equations. As a first step towards the solution of the full system we cast the equations for the \(\pi \pi \to \overline N N\) partial waves into the form of a Muskhelishvili-Omnès problem with finite matching point, which we solve numerically in the single-channel approximation. We investigate in detail the role of individual contributions to our solutions and discuss some consequences for the spectral functions of the nucleon electromagnetic form factors.

Keywords

Chiral Lagrangians QCD 

References

  1. [1]
    D. Gotta et al., Pionic hydrogen, in Precision Physics of Simple Atoms and Molecules, Lect. Notes Phys. 745 (2008) 165.ADSCrossRefGoogle Scholar
  2. [2]
    T. Strauch et al., Pionic deuterium, Eur. Phys. J. A 47 (2011) 88 [arXiv:1011.2415] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga and D.R. Phillips, Precision calculation of the π − d scattering length and its impact on threshold πN scattering, Phys. Lett. B 694 (2011) 473 [arXiv:1003.4444] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga and D.R. Phillips, Precision calculation of threshold π d scattering, πN scattering lengths and the GMO sum rule, Nucl. Phys. A 872 (2011) 69 [arXiv:1107.5509] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    J. Gasser, Hadron Masses and Sigma Commutator in the Light of Chiral Perturbation Theory, Annals Phys. 136 (1981) 62 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    A. Bottino, F. Donato, N. Fornengo and S. Scopel, Size of the neutralino-nucleon cross-section in the light of a new determination of the pion-nucleon sigma term, Astropart. Phys. 18 (2002) 205 [hep-ph/0111229] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    J.R. Ellis, K.A. Olive and C. Savage, Hadronic Uncertainties in the Elastic Scattering of Supersymmetric Dark Matter, Phys. Rev. D 77 (2008) 065026 [arXiv:0801.3656] [INSPIRE].ADSGoogle Scholar
  8. [8]
    K.A. Olive, The impact of XENON100 and the LHC on Supersymmetric Dark Matter, arXiv:1202.2324 [INSPIRE].
  9. [9]
    A. Walker-Loud, Evidence for non-analytic light quark mass dependence in the baryon spectrum, arXiv:1112.2658 [INSPIRE].
  10. [10]
    T. Cheng and R.F. Dashen, Is SU(2) × SU(2) a better symmetry than SU(3)?, Phys. Rev. Lett. 26 (1971) 594 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    S. Roy, Exact integral equation for pion-pion scattering involving only physical region partial waves, Phys. Lett. B 36 (1971) 353 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    B. Ananthanarayan, G. Colangelo, J. Gasser and H. Leutwyler, Roy equation analysis of ππ scattering, Phys. Rept. 353 (2001) 207 [hep-ph/0005297] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  13. [13]
    S. Descotes-Genon, N. Fuchs, L. Girlanda and J. Stern, Analysis and interpretation of new low-energy ππ scattering data, Eur. Phys. J. C 24 (2002) 469 [hep-ph/0112088] [INSPIRE].CrossRefGoogle Scholar
  14. [14]
    R. García-Martín, R. Kaminski, J. Peláez, J. Ruiz de Elvira and F. Ynduráin, The Pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV, Phys. Rev. D 83 (2011) 074004 [arXiv:1102.2183] [INSPIRE].ADSGoogle Scholar
  15. [15]
    I. Caprini, G. Colangelo and H. Leutwyler, Regge analysis of the ππ scattering amplitude, Eur. Phys. J. C 72 (2012) 1860 [arXiv:1111.7160] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    I. Caprini, G. Colangelo and H. Leutwyler, in preparation.Google Scholar
  17. [17]
    B. Moussallam, Couplings of light I = 0 scalar mesons to simple operators in the complex plane, Eur. Phys. J. C 71 (2011) 1814 [arXiv:1110.6074] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    B. Ananthanarayan and P. Büttiker, Comparison of pion-kaon scattering in SU(3) chiral perturbation theory and dispersion relations, Eur. Phys. J. C 19 (2001) 517 [hep-ph/0012023] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    P. Büttiker, S. Descotes-Genon and B. Moussallam, A new analysis of πK scattering from Roy and Steiner type equations, Eur. Phys. J. C 33 (2004) 409 [hep-ph/0310283] [INSPIRE].ADSGoogle Scholar
  20. [20]
    T. Becher and H. Leutwyler, Low energy analysis of πN → πN, JHEP 06 (2001) 017 [hep-ph/0103263] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    G. Hite and F. Steiner, New dispersion relations and their application to partial-wave amplitudes, Nuovo Cim. A 18 (1973) 237 [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    R. Koch, A New Determination of the πN Sigma Term Using Hyperbolic Dispersion Relations in the (ν 2 , t) Plane, Z. Phys. C 15 (1982) 161 [INSPIRE].ADSGoogle Scholar
  23. [23]
    G. Höhler, Determinations of the πN Sigma term, PiN Newslett. 15 (1999) 123.Google Scholar
  24. [24]
    J. Stahov, The subthreshold expansion of the πN invariant amplitudes in dispersion theory, PiN Newslett. 15 (1999) 13.Google Scholar
  25. [25]
    J. Stahov, Calculation of πN partial waves from hyperbolic dispersion relations, PiN Newslett. 16 (2002) 116.Google Scholar
  26. [26]
    N.I. Muskhelishvili, Singular Integral Equations, Wolters-Noordhoff Publishing, Groningen (1953) [Dover Publications, 2nd edition (2008)].zbMATHGoogle Scholar
  27. [27]
    R. Omnès, On the Solution of certain singular integral equations of quantum field theory, Nuovo Cim. 8 (1958) 316 [INSPIRE].CrossRefzbMATHGoogle Scholar
  28. [28]
    R. Koch and E. Pietarinen, Low-Energy πN Partial Wave Analysis, Nucl. Phys. A 336 (1980) 331 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    G. Höhler, Pion-Nukleon-Streuung: Methoden und Ergebnisse, in Landolt-Börnstein: Numerical Data and Functional Relationships in Science and TechnologyNew Series / Elementary Particles, Nuclei and Atoms 9b2, H. Schopper ed., Springer Verlag, Berlin (1983).Google Scholar
  30. [30]
    J. Stahov, Determination of πN low-energy parameters from forward dispersion relations, PiN Newslett. 13 (1997) 174.Google Scholar
  31. [31]
    M. Hoferichter, D.R. Phillips and C. Schat, Roy-Steiner equations for γγ → ππ, Eur. Phys. J. C 71 (2011) 1743 [arXiv:1106.4147] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    J. Gasser and G. Wanders, One-channel Roy equations revisited, Eur. Phys. J. C 10 (1999) 159 [hep-ph/9903443] [INSPIRE].ADSGoogle Scholar
  33. [33]
    G. Wanders, The Role of the input in Roys equations for ππ scattering, Eur. Phys. J. C 17 (2000) 323 [hep-ph/0005042] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    T.W.B. Kibble, Kinematics of General Scattering Processes and the Mandelstam Representation, Phys. Rev. 117 (1960) 1159 [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Döring, C. Hanhart, F. Huang, S. Krewald and U.-G. Meißner, Analytic properties of the scattering amplitude and resonances parameters in a meson exchange model, Nucl. Phys. A 829 (2009)170 [arXiv:0903.4337] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    W.B. Kaufmann and G.E. Hite, Tests of current algebra and partially conserved axial-vector current in the subthreshold region of the pion-nucleon system, Phys. Rev. C 60 (1999) 055204 [INSPIRE].ADSGoogle Scholar
  38. [38]
    L.S. Brown, W. Pardee and R. Peccei, Adler-Weisberger theorem reexamined, Phys. Rev. D 4 (1971) 2801 [INSPIRE].ADSGoogle Scholar
  39. [39]
    V. Bernard, N. Kaiser and U.-G. Meißner, On the analysis of the pion-nucleon sigma term: The Size of the remainder at the Cheng-Dashen point, Phys. Lett. B 389 (1996) 144 [hep-ph/9607245] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    G.E. Hite, W.B. Kaufmann and R.J. Jacob, New evaluation of the πN Sigma term, Phys. Rev. C 71 (2005) 065201 [INSPIRE].ADSGoogle Scholar
  41. [41]
    D. Bugg, A. Carter and J. Carter, New values of pion-nucleon scattering lengths and F 2, Phys. Lett. B 44 (1973) 278 [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    J. de Swart, M. Rentmeester and R. Timmermans, The Status of the pion-nucleon coupling constant, PiN Newslett. 13 (1997) 96 [nucl-th/9802084] [INSPIRE].Google Scholar
  43. [43]
    R. Arndt, W. Briscoe, I. Strakovsky and R. Workman, Extended partial-wave analysis of πN scattering data, Phys. Rev. C 74 (2006) 045205 [nucl-th/0605082] [INSPIRE].ADSGoogle Scholar
  44. [44]
    W.R. Frazer and J.R. Fulco, Partial-Wave Dispersion Relations for Pion-Nucleon Scattering, Phys. Rev. 119 (1960) 1420 [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  45. [45]
    S.W. MacDowell, Analytic Properties of Partial Amplitudes in Meson-Nucleon Scattering, Phys. Rev. 116 (1959) 774 [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  46. [46]
    J. Baacke and F. Steiner, πN partial wave relations from fixed-t dispersion relations, Fortsch. Phys. 18 (1970) 67 [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    F. Steiner, On the generalized πN potentiala new representation from fixed-t dispersion relations, Fortsch. Phys. 18 (1970) 43 [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    F. Steiner, Partial wave crossing relations for meson-baryon scattering, Fortsch. Phys. 19 (1971) 115 [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    W.R. Frazer and J.R. Fulco, Partial-Wave Dispersion Relations for \(\pi \pi \to N\overline N\), Phys. Rev. 117 (1960) 1603 [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  50. [50]
    M. Jacob and G.C. Wick, On the general theory of collisions for particles with spin, Annals Phys. 7 (1959) 404 [Annals Phys. 281 (2000) 774] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  51. [51]
    D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, Quantum Theory of Angular Momentum, World-Scientific Publishing, Singapore (1988).CrossRefGoogle Scholar
  52. [52]
    K.M. Watson, Some general relations between the photoproduction and scattering of π mesons, Phys. Rev. 95 (1954) 228 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  53. [53]
    B. Ananthanarayan, I. Caprini, G. Colangelo, J. Gasser and H. Leutwyler, Scalar form-factors of light mesons, Phys. Lett. B 602 (2004) 218 [hep-ph/0409222] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    M.J. Musolf, H.-W. Hammer and D. Drechsel, Nucleon strangeness and unitarity, Phys. Rev. D 55 (1997) 2741 [Erratum ibid. D 62 (2000) 079901] [hep-ph/9610402] [INSPIRE].ADSGoogle Scholar
  55. [55]
    W.R. Frazer and J.R. Fulco, Effect of a Pion-Pion Scattering Resonance on Nucleon Structure. II, Phys. Rev. 117 (1960) 1609 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    E. Pietarinen, A calculation of \(\pi \pi \to N\overline N\) amplitudes in the pseudophysical region, Preprint Series in Theoretical Physics HU-TFT-17-77, Helsinki University, unpublished.Google Scholar
  57. [57]
    M. Froissart, Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053 [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    A. Martin, Unitarity and high-energy behavior of scattering amplitudes, Phys. Rev. 129 (1963) 1432 [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  59. [59]
    P. Büttiker and U.-G. Meißner, Pion-nucleon scattering inside the Mandelstam triangle, Nucl. Phys. A 668 (2000) 97 [hep-ph/9908247] [INSPIRE].CrossRefGoogle Scholar
  60. [60]
    A. Gasparyan and M.F.M. Lutz, Photon- and pion-nucleon interactions in a unitary and causal effective field theory based on the chiral Lagrangian, Nucl. Phys. A 848 (2010) 126 [arXiv:1003.3426] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    J.L. Basdevant, J.C. Le Guillou and H. Navelet, Crossing and physical partial-wave amplitudes, Nuovo Cim. A 7 (1972) 363 [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    A. Schenk, Absorption and dispersion of pions at finite temperature, Nucl. Phys. B 363 (1991) 97 [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    C.D. Froggatt and J.L. Petersen, Phase-shift analysis of π+ π scattering between 1.0 GeV and 1.8 GeV based on fixed momentum transfer analyticity. 2., Nucl. Phys. B 129 (1977) 89 [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    E. Pietarinen, Dispersion relations and experimental data, Nuovo Cim. A 12 (1972) 522 [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    R. Koch, Improved πN Partial Waves, Consistent With Analyticity And Unitarity, Z. Phys. C 29 (1985) 597 [INSPIRE].ADSGoogle Scholar
  66. [66]
    R. Koch, A Calculation of Low-Energy πN Partial Waves Based on Fixed-t Analyticity, Nucl. Phys. A 448 (1986) 707 [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    R.A. Arndt, R.L. Workman, I.I. Strakovsky and M.M. Pavan, πN elastic scattering analyses and dispersion relation constraints, nucl-th/9807087 [INSPIRE].
  68. [68]
    R.A. Arndt, W.J. Briscoe, I.I. Strakovsky and R.L. Workman, Partial-wave analysis and baryon spectroscopy, Eur. Phys. J. A 35 (2008) 311 [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
  70. [70]
    A. Anisovich et al., Partial-wave analysis of \(\overline p p \to {\pi^{-} }{\pi^{+} },{\pi^0}{\pi^0},\eta \eta\) and ηη′, Nucl. Phys. A 662 (2000)319 [arXiv:1109.1188] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    M.E. Sainio, Analyticity constrained pion-nucleon analysis, PoS(CD09)013.
  72. [72]
    P. Metsä, Forward analysis of πN scattering with an expansion method, Eur. Phys. J. A 33 (2007) 349 [arXiv:0705.4528] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    F. Huang, A. Sibirtsev, J. Haidenbauer, S. Krewald and U.-G. Meißner, Backward pion-nucleon scattering, Eur. Phys. J. A 44 (2010) 81 [arXiv:0910.4275] [INSPIRE].ADSCrossRefGoogle Scholar
  74. [74]
    T.N. Pham and T.N. Truong, Muskhelishvili-Omnès Integral Equation with Inelastic Unitarity: Single- and Coupled-Channel Equations, Phys. Rev. D 16 (1977) 896 [INSPIRE].MathSciNetADSGoogle Scholar
  75. [75]
    I. Caprini, Omnès representations with inelastic effects for hadronic form factors, Rom. J. Phys. 50 (2005) 7.Google Scholar
  76. [76]
    S.M. Flatté, Coupled-Channel Analysis of the πη and \(K\overline K\) Systems Near \(K\overline K\) Threshold, Phys. Lett. B 63 (1976) 224 [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    R. García-Martín, R. Kaminski, J.R. Peláez and J. Ruiz de Elvira, Precise determination of the f 0(600) and f 0(980) pole parameters from a dispersive data analysis, Phys. Rev. Lett. 107 (2011) 072001 [arXiv:1107.1635] [INSPIRE].ADSCrossRefGoogle Scholar
  78. [78]
    M.M. Nagels, T.A. Rijken and J.J. de Swart, A Low-Energy Nucleon-Nucleon Potential from Regge Pole Theory, Phys. Rev. D 17 (1978) 768 [INSPIRE].ADSGoogle Scholar
  79. [79]
    P.M.M. Maessen, T.A. Rijken and J.J. de Swart, Soft Core Baryon Baryon One Boson Exchange Models. 2. Hyperon-Nucleon Potential, Phys. Rev. C 40 (1989) 2226 [INSPIRE].ADSGoogle Scholar
  80. [80]
    V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen and J.J. de Swart, Construction of high quality N N potential models, Phys. Rev. C 49 (1994) 2950 [nucl-th/9406039] [INSPIRE].ADSGoogle Scholar
  81. [81]
    T.A. Rijken, H. Polinder and J. Nagata, Extended-soft-core NN potentials in momentum space. 2. Meson-pair exchange potentials, Phys. Rev. C 66 (2002) 044009 [nucl-th/0201020] [INSPIRE].ADSGoogle Scholar
  82. [82]
    M. Hoferichter, C. Ditsche, B. Kubis and U.-G. Meißner, Dispersive analysis of the scalar form factor of the nucleon, arXiv:12046251, accepted for publication in JHEP.
  83. [83]
    G.C. Oades, Finite contour dispersion relations and the subthreshold expansion coefficients of the πN invariant amplitudes, PiN Newslett. 15 (1999) 307.Google Scholar
  84. [84]
    B.R. Martin and G.C. Oades, Threshold and subthreshold πN scattering amplitudes: Comparison with chiral perturbation theory predictions, PiN Newslett. 16 (2002) 133.Google Scholar
  85. [85]
    N. Fettes, Pion-nucleon physics in Chiral Perturbation Theory, Thesis, University of Bonn (2000).Google Scholar
  86. [86]
    M.M. Pavan, R.A. Arndt, I.I. Strakovsky and R.L. Workman, Determination of the πNN coupling constant in the VPI/GW πNN partial wave and dispersion relation analysis, PiN Newslett. 15 (1999) 171 [Phys. Scripta 87 (2000) 65 ] [nucl-th/9910040] [INSPIRE].Google Scholar
  87. [87]
    G. Höhler, Some results on πN phenomenology, PiN Newslett. 15 (1999) 7.Google Scholar
  88. [88]
    N. Fettes, U.-G. Meißner and S. Steininger, Pion-nucleon scattering in chiral perturbation theory. 1. Isospin symmetric case, Nucl. Phys. A 640 (1998) 199 [hep-ph/9803266] [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    N. Fettes and U.-G. Meißner, Pion-nucleon scattering in chiral perturbation theory. 2.: Fourth order calculation, Nucl. Phys. A 676 (2000) 311 [hep-ph/0002162] [INSPIRE].ADSCrossRefGoogle Scholar
  90. [90]
    G. Höhler and E. Pietarinen, Electromagnetic Radii of Nucleon and Pion, Phys. Lett. B 53 (1975) 471 [INSPIRE].ADSCrossRefGoogle Scholar
  91. [91]
    M.A. Belushkin, H.-W. Hammer and U.-G. Meißner, Dispersion analysis of the nucleon form-factors including meson continua, Phys. Rev. C 75 (2007) 035202 [hep-ph/0608337] [INSPIRE].ADSGoogle Scholar
  92. [92]
    J. Gasser, H. Leutwyler and M.E. Sainio, Form-factor of the sigma term, Phys. Lett. B 253 (1991) 260 [INSPIRE].ADSCrossRefGoogle Scholar
  93. [93]
    J.F. Donoghue, J. Gasser and H. Leutwyler, The decay of a light Higgs boson, Nucl. Phys. B 343 (1990) 341 [INSPIRE].ADSCrossRefGoogle Scholar
  94. [94]
    G. Colangelo, Hadronic contributions to a μ below one GeV, Nucl. Phys. Proc. Suppl. 131 (2004) 185 [hep-ph/0312017] [INSPIRE].ADSCrossRefGoogle Scholar
  95. [95]
    F.-K. Guo, C. Hanhart, F.J. Llanes-Estrada and U.-G. Meißner, Quark mass dependence of the pion vector form factor, Phys. Lett. B 678 (2009) 90 [arXiv:0812.3270] [INSPIRE].ADSCrossRefGoogle Scholar
  96. [96]
    Bateman Manuscript Project, Higher Transcendental Functions 1, A. Erdélyi ed., McGraw-Hill, New York (1953).Google Scholar
  97. [97]
    G.F. Chew, M.L. Goldberger, F.E. Low and Y. Nambu, Application of Dispersion Relations to Low-Energy Meson-Nucleon Scattering, Phys. Rev. 106 (1957) 1337 [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  98. [98]
    S. Descotes-Genon and B. Moussallam, The \(K_0^*\) (800) scalar resonance from Roy-Steiner representations of πK scattering, Eur. Phys. J. C 48 (2006) 553 [hep-ph/0607133] [INSPIRE].ADSCrossRefGoogle Scholar
  99. [99]
    S. Mandelstam, Determination of the pion-nucleon scattering amplitude from dispersion relations and unitarity. General theory, Phys. Rev. 112 (1958) 1344 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  100. [100]
    S. Mandelstam, Analytic properties of transition amplitudes in perturbation theory, Phys. Rev. 115 (1959) 1741 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  101. [101]
    S. Mandelstam, Construction of the perturbation series for transition amplitudes from their analyticity and unitarity properties, Phys. Rev. 115 (1959) 1752.MathSciNetADSCrossRefGoogle Scholar
  102. [102]
    A. Martin, Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity - I., Nuovo Cim. A 42 (1965) 930 [INSPIRE].ADSGoogle Scholar
  103. [103]
    A. Martin, Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity - II., Nuovo Cim. A 44 (1966) 1219 .ADSCrossRefGoogle Scholar
  104. [104]
    S.W. MacDowell, Analytic continuation of reduced pion-nucleon partial-wave amplitudes, Phys. Rev. D 6 (1972) 3512 [INSPIRE].ADSGoogle Scholar
  105. [105]
    F.F.K. Cheung and F.S. Chen-Cheung, Uniqueness of amplitudes satisfying the Mandelstam representation, Phys. Rev. D 5 (1972) 970 [INSPIRE].ADSGoogle Scholar
  106. [106]
    H. Lehmann, Analytic properties of scattering amplitudes as functions of momentum transfer, Nuovo Cim. 10 (1958) 579 .CrossRefzbMATHGoogle Scholar
  107. [107]
    J. Stahov, Dispersion relations on hyperbolas and higher pion-nucleon partial waves (in Croatian), Thesis, University of Zagreb (1983).Google Scholar
  108. [108]
    T. Regge, Introduction to complex orbital momenta, Nuovo Cim. 14 (1959) 951 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  109. [109]
    P.D.B. Collins, An introduction to Regge theory and high energy physics, Cambridge University Press, Cambridge (1977).CrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • C. Ditsche
    • 1
    Email author
  • M. Hoferichter
    • 1
  • B. Kubis
    • 1
  • U.-G. Meißner
    • 1
    • 2
  1. 1.Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical PhysicsUniversität BonnBonnGermany
  2. 2.Institut für Kernphysik, Institute for Advanced Simulation, and Jülich Center for Hadron PhysicsForschungszentrum JülichJülichGermany

Personalised recommendations