Advertisement

The European Physical Journal C

, Volume 70, Issue 1–2, pp 155–175 | Cite as

MO analysis of the high statistics Belle results on γ γπ + π , π 0 π 0 with chiral constraints

  • R. García-Martín
  • B. Moussallam
Regular Article - Theoretical Physics

Abstract

We reconsider Muskhelishvili–Omnès (MO) dispersive representations of photon–photon scattering to two pions, motivated by the very high statistics results recently released by the Belle collaboration for charged as well as neutral pion pairs and also by recent progress in the determination of the low-energy π π scattering amplitude. Applicability of this formalism is extended beyond 1 GeV by taking into account inelasticity due to \(K\bar{K}\). A modified MO representation is derived which has the advantage that all polynomial ambiguities are collected into the subtraction constants and have simple relations to pion polarizabilities. It is obtained by treating differently the exactly known QED Born term and the other components of the left-hand cut. These components are approximated by a sum over resonances. All resonances up to spin two and masses up to ≃1.3 GeV are included. The tensor contributions to the left-hand cut are found to be numerically important. We perform fits to the data imposing chiral constraints, in particular, using a model independent sum-rule result on the p 6 chiral coupling c 34. Such theoretical constraints are necessary because the experimental errors are dominantly systematic. Results on further p 6 couplings and pion dipole and quadrupole polarizabilities are then derived from the fit. The relevance of the new data for distinguishing between two possible scenarios of isospin breaking in the f 0(980) region is discussed.

Keywords

Helicity Amplitude Charged Pion Neutral Pion Belle Collaboration Dispersive Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Bijnens, G. Colangelo, G. Ecker, Ann. Phys. 280, 100 (2000). arXiv:hep-ph/9907333 MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    J. Bijnens, G. Colangelo, G. Ecker, J. High Energy Phys. 9902, 020 (1999). arXiv:hep-ph/9902437 CrossRefADSGoogle Scholar
  3. 3.
    G. Colangelo, Nucl. Phys. A 827, 228C (2009) CrossRefADSGoogle Scholar
  4. 4.
    J. Ahrens et al., Eur. Phys. J. A 23, 113 (2005). arXiv:nucl-ex/0407011 ADSGoogle Scholar
  5. 5.
    A. Guskov (COMPASS Collaboration), Fizika B 17, 313 (2008) ADSGoogle Scholar
  6. 6.
    T. Mori et al. (Belle Collaboration), J. Phys. Soc. Jpn. 76, 074102 (2007). arXiv:0704.3538 [hep-ex] CrossRefADSGoogle Scholar
  7. 7.
    T. Mori et al. (Belle Collaboration), Phys. Rev. D 75, 051101 (2007). arXiv:hep-ex/0610038 CrossRefADSGoogle Scholar
  8. 8.
    S. Uehara et al. (Belle Collaboration), Phys. Rev. D 78, 052004 (2008). arXiv:0805.3387 [hep-ex] ADSGoogle Scholar
  9. 9.
    S. Uehara et al. (BELLE Collaboration), Phys. Rev. D 79, 052009 (2009). arXiv:0903.3697 [hep-ex] CrossRefADSGoogle Scholar
  10. 10.
    J.R. Batley et al. (NA48/2 Collaboration), Eur. Phys. J. C 54, 411 (2008) CrossRefADSGoogle Scholar
  11. 11.
    J.R. Batley et al. (NA48/2 Collaboration), Phys. Lett. B 633, 173 (2006). arXiv:hep-ex/0511056 ADSGoogle Scholar
  12. 12.
    J.R. Batley et al. Eur. Phys. J. C 64, 589 (2009) CrossRefADSGoogle Scholar
  13. 13.
    B. Adeva et al. (DIRAC Collaboration), Phys. Lett. B 619, 50 (2005). arXiv:hep-ex/0504044 CrossRefADSGoogle Scholar
  14. 14.
    S. Pislak et al. (BNL-E865 Collaboration), Phys. Rev. Lett. 87, 221801 (2001). arXiv:hep-ex/0106071 CrossRefADSGoogle Scholar
  15. 15.
    R. Oehme, Int. J. Mod. Phys. A 10, 1995 (1995). arXiv:hep-th/9412040 CrossRefADSGoogle Scholar
  16. 16.
    R. Omnès, Nuovo Cimento 8, 316 (1958) MATHCrossRefGoogle Scholar
  17. 17.
    N.I. Muskhelishvili, Singular Integral Equations (Noordhof, Groningen, 1953) MATHGoogle Scholar
  18. 18.
    M. Gourdin, A. Martin, Nuovo Cimento 17, 224 (1960) MATHCrossRefGoogle Scholar
  19. 19.
    R.L. Goble, J.L. Rosner, Phys. Rev. D 5, 2345 (1972) CrossRefADSGoogle Scholar
  20. 20.
    R.L. Goble, R. Rosenfeld, J.L. Rosner, Phys. Rev. D 39, 3264 (1989) CrossRefADSGoogle Scholar
  21. 21.
    D. Morgan, M.R. Pennington, Phys. Lett. B 192, 207 (1987) CrossRefADSGoogle Scholar
  22. 22.
    D. Morgan, M.R. Pennington, Phys. Lett. B 272, 134 (1991) CrossRefADSGoogle Scholar
  23. 23.
    J.F. Donoghue, B.R. Holstein, Phys. Rev. D 48, 137 (1993). arXiv:hep-ph/9302203 ADSGoogle Scholar
  24. 24.
    J.F. Donoghue, B.R. Holstein, Y.C. Lin, Phys. Rev. D 37, 2423 (1988) CrossRefADSGoogle Scholar
  25. 25.
    J. Bijnens, F. Cornet, Nucl. Phys. B 296, 557 (1988) CrossRefADSGoogle Scholar
  26. 26.
    K. Sundermeyer, A coupled channel analysis of the reactions gamma + Gamma → Pi + Pi And Gamma + Gamma → K + Anti-K. Preprint DESY 74/17 Google Scholar
  27. 27.
    O. Babelon, J.L. Basdevant, D. Caillerie, M. Gourdin, G. Mennessier, Nucl. Phys. B 114, 252 (1976) CrossRefADSGoogle Scholar
  28. 28.
    Y. Mao, X.G. Wang, O. Zhang, H.Q. Zheng, Z.Y. Zhou, Phys. Rev. D 79, 116008 (2009). arXiv:0904.1445 [hep-ph] CrossRefADSGoogle Scholar
  29. 29.
    G. Mennessier, Z. Phys. C 16, 241 (1983) CrossRefADSGoogle Scholar
  30. 30.
    M. Boglione, M.R. Pennington, Eur. Phys. J. C 9, 11 (1999). arXiv:hep-ph/9812258 ADSGoogle Scholar
  31. 31.
    J.A. Oller, L. Roca, C. Schat, Phys. Lett. B 659, 201 (2008). arXiv:0708.1659 [hep-ph] CrossRefADSGoogle Scholar
  32. 32.
    N.N. Achasov, G.N. Shestakov, Phys. Rev. D 77, 074020 (2008). arXiv:0712.0885 [hep-ph] CrossRefADSGoogle Scholar
  33. 33.
    G. Mennessier, S. Narison, W. Ochs, Phys. Lett. B 665, 205 (2008). arXiv:0804.4452 [hep-ph] CrossRefADSGoogle Scholar
  34. 34.
    A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1960) Google Scholar
  35. 35.
    M. Jacob, G.C. Wick, Ann. Phys. 7, 404 (1959) [Ann. Phys. 281, 774 (2000)] MATHCrossRefMathSciNetADSGoogle Scholar
  36. 36.
    J.F. Donoghue, J. Gasser, H. Leutwyler, Nucl. Phys. B 343, 341 (1990) CrossRefADSGoogle Scholar
  37. 37.
    B. Moussallam, Eur. Phys. J. C 14, 111 (2000). arXiv:hep-ph/9909292 ADSGoogle Scholar
  38. 38.
    F.E. Low, Phys. Rev. 96, 1428 (1954) MATHCrossRefMathSciNetADSGoogle Scholar
  39. 39.
    M. Gell-Mann, M.L. Goldberger, Phys. Rev. 96, 1433 (1954) MATHCrossRefMathSciNetADSGoogle Scholar
  40. 40.
    H.D.I. Abarbanel, M.L. Goldberger, Phys. Rev. 165, 1594 (1968) CrossRefADSGoogle Scholar
  41. 41.
    B. Hyams et al., Nucl. Phys. B 64, 134 (1973). AIP Conf. Proc. 13, 206 (1973) CrossRefADSGoogle Scholar
  42. 42.
    C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1 (2008) CrossRefADSGoogle Scholar
  43. 43.
    P. Ko, Phys. Rev. D 41, 1531 (1990) ADSGoogle Scholar
  44. 44.
    G. Ecker, J. Gasser, A. Pich, E. de Rafael, Nucl. Phys. B 321, 311 (1989) CrossRefADSGoogle Scholar
  45. 45.
    S. Godfrey, N. Isgur, Phys. Rev. D 32, 189 (1985) CrossRefADSGoogle Scholar
  46. 46.
    B. Collick et al., Phys. Rev. Lett. 53, 2374 (1984) CrossRefADSGoogle Scholar
  47. 47.
    M. Zielinski et al., Phys. Rev. D 30, 1855 (1984) CrossRefADSGoogle Scholar
  48. 48.
    G.T. Condo, T. Handler, W.M. Bugg, G.R. Blackett, M. Pisharody, K.A. Danyo, Phys. Rev. D 48, 3045 (1993) CrossRefADSGoogle Scholar
  49. 49.
    M. Nozar et al. (CLAS Collaboration), Phys. Rev. Lett. 102, 102002 (2009). arXiv:0805.4438 [hep-ex] CrossRefADSGoogle Scholar
  50. 50.
    H.J. Lipkin, Phys. Lett. B 72, 249 (1977) CrossRefADSGoogle Scholar
  51. 51.
    S. Bellucci, J. Gasser, M.E. Sainio, Nucl. Phys. B 423, 80 (1994) [Erratum: Nucl. Phys. B 431, 413 (1994)]. arXiv:hep-ph/9401206 CrossRefADSGoogle Scholar
  52. 52.
    U. Bürgi, Nucl. Phys. B 479, 392 (1996). arXiv:hep-ph/9602429 CrossRefADSGoogle Scholar
  53. 53.
    J. Gasser, M.A. Ivanov, M.E. Sainio, Nucl. Phys. B 728, 31 (2005). arXiv:hep-ph/0506265 CrossRefADSGoogle Scholar
  54. 54.
    J. Gasser, M.A. Ivanov, M.E. Sainio, Nucl. Phys. B 745, 84 (2006). arXiv:hep-ph/0602234 MATHCrossRefADSGoogle Scholar
  55. 55.
    K. Maltman, C.E. Wolfe, Phys. Rev. D 59, 096003 (1999). arXiv:hep-ph/9810441 CrossRefADSGoogle Scholar
  56. 56.
    S. Dürr, J. Kambor, Phys. Rev. D 61, 114025 (2000). arXiv:hep-ph/9907539 CrossRefADSGoogle Scholar
  57. 57.
    M. Knecht, B. Moussallam, J. Stern, Nucl. Phys. B 429, 125 (1994). arXiv:hep-ph/9402318 CrossRefADSGoogle Scholar
  58. 58.
    J. Gasser, C. Haefeli, M.A. Ivanov, M. Schmid, Phys. Lett. B 675, 49 (2009). arXiv:0903.0801 [hep-ph] CrossRefADSGoogle Scholar
  59. 59.
    G. Amorós, J. Bijnens, P. Talavera, Nucl. Phys. B 568, 319 (2000). arXiv:hep-ph/9907264 CrossRefADSGoogle Scholar
  60. 60.
    H. Terazawa, Phys. Rev. Lett. 26, 1207 (1971) CrossRefADSGoogle Scholar
  61. 61.
    B.R. Martin, D. Morgan, G. Shaw, Pion–Pion Interactions in Particle Physics (Academic Press, London, 1976) Google Scholar
  62. 62.
    S.M. Roy, Phys. Lett. B 36, 353 (1971) CrossRefADSGoogle Scholar
  63. 63.
    B. Ananthanarayan, G. Colangelo, J. Gasser, H. Leutwyler, Phys. Rep. 353, 207 (2001). arXiv:hep-ph/0005297 MATHCrossRefADSGoogle Scholar
  64. 64.
    D.V. Bugg, B.S. Zou, A.V. Sarantsev, Nucl. Phys. B 471, 59 (1996) CrossRefADSGoogle Scholar
  65. 65.
    R. Kamiński, L. Leśniak, K. Rybicki, Z. Phys. C 74, 79 (1997). arXiv:hep-ph/9606362 Google Scholar
  66. 66.
    R. Kamiński, J.R. Peláez, F.J. Ynduráin, Phys. Rev. D 77, 054015 (2008). arXiv:0710.1150 [hep-ph] CrossRefADSGoogle Scholar
  67. 67.
    R. García-Martín, R. Kamiński, J.R. Peláez, F.J. Ynduráin, arXiv:0906.5467 [hep-ph]
  68. 68.
    D.H. Cohen, D.S. Ayres, R. Diebold, S.L. Kramer, A.J. Pawlicki, A.B. Wicklund, Phys. Rev. D 22, 2595 (1980) CrossRefADSGoogle Scholar
  69. 69.
    A. Etkin et al., Phys. Rev. D 25, 1786 (1982) CrossRefADSGoogle Scholar
  70. 70.
    K.L. Au, D. Morgan, M.R. Pennington, Phys. Rev. D 35, 1633 (1987) CrossRefADSGoogle Scholar
  71. 71.
    P. Büttiker, S. Descotes-Genon, B. Moussallam, Eur. Phys. J. C 33, 409 (2004). arXiv:hep-ph/0310283 CrossRefADSGoogle Scholar
  72. 72.
    M.J. Losty et al., Nucl. Phys. B 69, 185 (1974) CrossRefADSGoogle Scholar
  73. 73.
    W. Hoogland et al., Nucl. Phys. B 126, 109 (1977) CrossRefADSGoogle Scholar
  74. 74.
    H. Marsiske et al. (Crystal Ball Collaboration), Phys. Rev. D 41, 3324 (1990) CrossRefADSGoogle Scholar
  75. 75.
    J. Boyer et al., Phys. Rev. D 42, 1350 (1990) CrossRefADSGoogle Scholar
  76. 76.
    H.J. Behrend et al. (CELLO Collaboration), Z. Phys. C 56, 381 (1992) CrossRefADSGoogle Scholar
  77. 77.
    G. Ecker, Acta Phys. Pol. B 38, 2753 (2007). arXiv:hep-ph/0702263 MathSciNetADSGoogle Scholar
  78. 78.
    M.R. Pennington, T. Mori, S. Uehara, Y. Watanabe, Eur. Phys. J. C 56, 1 (2008). arXiv:0803.3389 [hep-ph] CrossRefADSGoogle Scholar
  79. 79.
    N.N. Achasov, G.N. Shestakov, Phys. Rev. D 70, 074015 (2004). arXiv:hep-ph/0405129 CrossRefADSGoogle Scholar
  80. 80.
    C. Hanhart, B. Kubis, J.R. Peláez, Phys. Rev. D 76, 074028 (2007). arXiv:0707.0262 [hep-ph] CrossRefADSGoogle Scholar
  81. 81.
    J. Bijnens, J. Prades, Nucl. Phys. B 490, 239 (1997). arXiv:hep-ph/9610360 CrossRefADSGoogle Scholar
  82. 82.
    L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 73, 035210 (2006). arXiv:nucl-th/0512047 CrossRefADSGoogle Scholar
  83. 83.
    V. Bernard, D. Vautherin, Phys. Rev. D 40, 1615 (1989) CrossRefADSGoogle Scholar
  84. 84.
    B. Hiller, W. Broniowski, A.A. Osipov, A.H. Blin, Phys. Lett. B 681, 147 (2009). arXiv:0908.0159 [hep-ph] CrossRefADSGoogle Scholar
  85. 85.
    L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 72, 035211 (2005). arXiv:nucl-th/0505058 CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag / Società Italiana di Fisica 2010

Authors and Affiliations

  1. 1.Departamento de Física Teórica II, Facultad de Ciencias FísicasUniversidad Complutense de MadridMadridSpain
  2. 2.Groupe de Physique Théorique, Institut de Physique NucléaireUniversité Paris-Sud 11OrsayFrance

Personalised recommendations