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The European Physical Journal C

, 71:1743 | Cite as

Roy–Steiner equations for γγππ

  • Martin HoferichterEmail author
  • Daniel R. Phillips
  • Carlos Schat
Regular Article - Theoretical Physics

Abstract

Starting from hyperbolic dispersion relations, we derive a system of Roy–Steiner equations for pion Compton scattering that respects analyticity, unitarity, gauge invariance, and crossing symmetry. It thus maintains all symmetries of the underlying quantum field theory. To suppress the dependence of observables on high-energy input, we also consider once- and twice-subtracted versions of the equations, and identify the subtraction constants with dipole and quadrupole pion polarizabilities. Based on the assumption of Mandelstam analyticity, we determine the kinematic range in which the equations are valid. As an application, we consider the resolution of the γγππ partial waves by a Muskhelishvili–Omnès representation with finite matching point. We find a sum rule for the isospin-two S-wave, which, together with chiral constraints, produces an improved prediction for the charged-pion quadrupole polarizability \((\alpha_{2}-\beta_{2})^{\pi^{\pm}}=(15.3\pm3.7)\times 10^{-4}\) fm5. We investigate the prediction of our dispersion relations for the two-photon coupling of the σ-resonance Γ σγγ . The twice-subtracted version predicts a correlation between this width and the isospin-zero pion polarizabilities, which is largely independent of the high-energy input used in the equations. Using this correlation, the chiral perturbation theory results for pion polarizabilities, and our new sum rule, we find Γ σγγ =(1.7±0.4) keV.

Keywords

Partial Wave Match Point Pion Polarizability Subtraction Constant Steiner Equation 
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.

References

  1. 1.
    I. Caprini, G. Colangelo, H. Leutwyler, Phys. Rev. Lett. 96, 132001 (2006). arXiv:hep-ph/0512364 ADSCrossRefGoogle Scholar
  2. 2.
    S.M. Roy, Phys. Lett. B 36, 353 (1971) ADSCrossRefGoogle Scholar
  3. 3.
    S. Weinberg, Physica A 96, 327 (1979) ADSCrossRefGoogle Scholar
  4. 4.
    J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984) MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    J. Gasser, H. Leutwyler, Nucl. Phys. B 250, 465 (1985) ADSCrossRefGoogle Scholar
  6. 6.
    H. Marsiske et al. (Crystal Ball Collaboration), Phys. Rev. D 41, 3324 (1990) ADSCrossRefGoogle Scholar
  7. 7.
    J. Boyer et al., Phys. Rev. D 42, 1350 (1990) ADSCrossRefGoogle Scholar
  8. 8.
    H.J. Behrend et al. (CELLO Collaboration), Z. Phys. C 56, 381 (1992) ADSCrossRefGoogle Scholar
  9. 9.
    T. Mori et al. (Belle Collaboration), J. Phys. Soc. Jpn. 76, 074102 (2007). arXiv:0704.3538 [hep-ex] ADSCrossRefGoogle Scholar
  10. 10.
    S. Uehara et al. (Belle Collaboration), Phys. Rev. D 78, 052004 (2008). arXiv:0805.3387 [hep-ex] ADSCrossRefGoogle Scholar
  11. 11.
    S. Uehara et al. (BELLE Collaboration), Phys. Rev. D 79, 052009 (2009). arXiv:0903.3697 [hep-ex] ADSCrossRefGoogle Scholar
  12. 12.
    A.I. L’vov, V.A. Petrun’kin, M. Schumacher, Phys. Rev. C 55, 359 (1997) ADSCrossRefGoogle Scholar
  13. 13.
    D. Drechsel, B. Pasquini, M. Vanderhaeghen, Phys. Rep. 378, 99 (2003). arXiv:hep-ph/0212124 ADSCrossRefGoogle Scholar
  14. 14.
    J. Bernabeu, J. Prades, Phys. Rev. Lett. 100, 241804 (2008). arXiv:0802.1830 [hep-ph] ADSCrossRefGoogle Scholar
  15. 15.
    M. Schumacher, Eur. Phys. J. C 67, 283 (2010). arXiv:1001.0500 [hep-ph] ADSCrossRefGoogle Scholar
  16. 16.
    M.R. Pennington, Phys. Rev. Lett. 97, 011601 (2006) ADSCrossRefGoogle Scholar
  17. 17.
    G. Mennessier, S. Narison, W. Ochs, Phys. Lett. B 665, 205 (2008). arXiv:0804.4452 [hep-ph] ADSCrossRefGoogle Scholar
  18. 18.
    G. Mennessier, S. Narison, X.G. Wang, Phys. Lett. B 696, 40 (2011). arXiv:1009.2773 [hep-ph] ADSCrossRefGoogle Scholar
  19. 19.
    N. Muskhelishvili, Singular Integral Equations (P. Noordhof, Groningen, 1953) zbMATHGoogle Scholar
  20. 20.
    R. Omnès, Nuovo Cimento 8, 316 (1958) zbMATHCrossRefGoogle Scholar
  21. 21.
    M. Gourdin, A. Martin, Nuovo Cimento 17, 224 (1960) zbMATHCrossRefGoogle Scholar
  22. 22.
    O. Babelon, J.L. Basdevant, D. Caillerie, M. Gourdin, G. Mennessier, Nucl. Phys. B 114, 252 (1976) ADSCrossRefGoogle Scholar
  23. 23.
    D. Morgan, M.R. Pennington, Z. Phys. C 37, 431 (1988). (Erratum-ibid. C 39 590 (1988)) ADSCrossRefGoogle Scholar
  24. 24.
    D. Morgan, M.R. Pennington, Phys. Lett. B 272, 134 (1991) ADSCrossRefGoogle Scholar
  25. 25.
    J.F. Donoghue, B.R. Holstein, Phys. Rev. D 48, 137 (1993). arXiv:hep-ph/9302203 ADSCrossRefGoogle Scholar
  26. 26.
    D. Drechsel, M. Gorchtein, B. Pasquini, M. Vanderhaeghen, Phys. Rev. C 61, 015204 (1999). arXiv:hep-ph/9904290 ADSCrossRefGoogle Scholar
  27. 27.
    L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 73, 035210 (2006). arXiv:nucl-th/0512047 ADSCrossRefGoogle Scholar
  28. 28.
    M.R. Pennington, T. Mori, S. Uehara, Y. Watanabe, Eur. Phys. J. C 56, 1 (2008). arXiv:0803.3389 [hep-ph] ADSCrossRefGoogle Scholar
  29. 29.
    J.A. Oller, L. Roca, C. Schat, Phys. Lett. B 659, 201 (2008). arXiv:0708.1659 [hep-ph] ADSCrossRefGoogle Scholar
  30. 30.
    J.A. Oller, L. Roca, Eur. Phys. J. A 37, 15 (2008). arXiv:0804.0309 [hep-ph] ADSCrossRefGoogle Scholar
  31. 31.
    Y. Mao, X.G. Wang, O. Zhang, H.Q. Zheng, Z.Y. Zhou, Phys. Rev. D 79, 116008 (2009). arXiv:0904.1445 [hep-ph] ADSCrossRefGoogle Scholar
  32. 32.
    R. García-Martín, B. Moussallam, Eur. Phys. J. C 70, 155 (2010). arXiv:1006.5373 [hep-ph] ADSCrossRefGoogle Scholar
  33. 33.
    Yu.M. Antipov et al., Phys. Lett. B 121, 445 (1983) ADSCrossRefGoogle Scholar
  34. 34.
    T.A. Aybergenov et al., Sov. Phys. Lebedev Inst. Rep. 6, 32 (1984) Google Scholar
  35. 35.
    T.A. Aybergenov et al., Czech. J. Phys. B 36, 948 (1986) ADSCrossRefGoogle Scholar
  36. 36.
    J. Ahrens et al., Eur. Phys. J. A 23, 113 (2005). arXiv:nucl-ex/0407011 ADSCrossRefGoogle Scholar
  37. 37.
    J. Bijnens, F. Cornet, Nucl. Phys. B 296, 557 (1988) ADSCrossRefGoogle Scholar
  38. 38.
    J.F. Donoghue, B.R. Holstein, Y.C. Lin, Phys. Rev. D 37, 2423 (1988) ADSCrossRefGoogle Scholar
  39. 39.
    S. Bellucci, J. Gasser, M.E. Sainio, Nucl. Phys. B 423, 80 (1994). (Erratum-ibid. B 431 413 (1994)) arXiv:hep-ph/9401206 ADSCrossRefGoogle Scholar
  40. 40.
    U. Bürgi, Phys. Lett. B 377, 147 (1996). arXiv:hep-ph/9602421 ADSCrossRefGoogle Scholar
  41. 41.
    U. Bürgi, Nucl. Phys. B 479, 392 (1996). arXiv:hep-ph/9602429 ADSCrossRefGoogle Scholar
  42. 42.
    J. Gasser, M.A. Ivanov, M.E. Sainio, Nucl. Phys. B 728, 31 (2005). arXiv:hep-ph/0506265 ADSCrossRefGoogle Scholar
  43. 43.
    J. Gasser, M.A. Ivanov, M.E. Sainio, Nucl. Phys. B 745, 84 (2006). arXiv:hep-ph/0602234 ADSCrossRefGoogle Scholar
  44. 44.
    A.V. Guskov, Phys. Part. Nucl. Lett. 7, 192 (2010) CrossRefGoogle Scholar
  45. 45.
    A. Guskov (COMPASS Collaboration), Nucl. Phys. B, Proc. Suppl. 198, 112 (2010) ADSCrossRefGoogle Scholar
  46. 46.
    A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1960) Google Scholar
  47. 47.
    M. Jacob, G.C. Wick, Ann. Phys. 7, 404 (1959). (Annals Phys. 281 774 (2000)). MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. 48.
    I. Guiaşu, E.E. Radescu, Ann. Phys. 120, 145 (1979) ADSCrossRefGoogle Scholar
  49. 49.
    B. Ananthanarayan, G. Colangelo, J. Gasser, H. Leutwyler, Phys. Rep. 353, 207 (2001). arXiv:hep-ph/0005297 ADSzbMATHCrossRefGoogle Scholar
  50. 50.
    G.E. Hite, F. Steiner, Nuovo Cimento A 18, 237 (1973) ADSCrossRefGoogle Scholar
  51. 51.
    B. Ananthanarayan, P. Büttiker, Eur. Phys. J. C 19, 517 (2001). arXiv:hep-ph/0012023 ADSCrossRefGoogle Scholar
  52. 52.
    P. Büttiker, S. Descotes-Genon, B. Moussallam, Eur. Phys. J. C 33, 409 (2004). arXiv:hep-ph/0310283 ADSCrossRefGoogle Scholar
  53. 53.
    F.E. Low, Phys. Rev. 96, 1428 (1954) MathSciNetADSzbMATHCrossRefGoogle Scholar
  54. 54.
    S. Mandelstam, Phys. Rev. 112, 1344 (1958) MathSciNetADSCrossRefGoogle Scholar
  55. 55.
    G. Höhler, in Landolt-Börnstein, vol. 9b2, ed. by H. Schopper (Springer, Berlin, 1983) Google Scholar
  56. 56.
    C. Itzykson, J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980) Google Scholar
  57. 57.
    H. Lehmann, Nuovo Cimento 10, 579 (1958) zbMATHCrossRefGoogle Scholar
  58. 58.
    K.M. Watson, Phys. Rev. 95, 228 (1954) ADSzbMATHCrossRefGoogle Scholar
  59. 59.
    L. Epele, G. Wanders, Phys. Lett. B 72, 390 (1978) ADSCrossRefGoogle Scholar
  60. 60.
    L. Epele, G. Wanders, Nucl. Phys. B 137, 521 (1978) ADSCrossRefGoogle Scholar
  61. 61.
    J. Gasser, G. Wanders, Eur. Phys. J. C 10, 159 (1999). arXiv:hep-ph/9903443 ADSGoogle Scholar
  62. 62.
    G. Wanders, Eur. Phys. J. C 17, 323 (2000). arXiv:hep-ph/0005042 ADSCrossRefGoogle Scholar
  63. 63.
    I. Caprini, G. Colangelo, H. Leutwyler, in preparation Google Scholar
  64. 64.
    G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603, 125 (2001). arXiv:hep-ph/0103088 ADSCrossRefGoogle Scholar
  65. 65.
    R. García-Martín, R. Kamiński, J.R. Peláez, J. Ruiz de Elvira, F.J. Ynduráin, Phys. Rev. D 83, 074004 (2011). arXiv:1102.2183 [hep-ph] ADSCrossRefGoogle Scholar
  66. 66.
    I. Caprini, private communication Google Scholar
  67. 67.
    R. García-Martín, R. Kamiński, J.R. Peláez, J. Ruiz de Elvira, Phys. Rev. Lett. 107, 072001 (2011). arXiv:1107.1635 [hep-ph] ADSCrossRefGoogle Scholar
  68. 68.
    J. Bijnens, J. Prades, Nucl. Phys. B 490, 239 (1997). arXiv:hep-ph/9610360 ADSCrossRefGoogle Scholar
  69. 69.
    K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010) ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag / Società Italiana di Fisica 2011

Authors and Affiliations

  • Martin Hoferichter
    • 1
    • 2
    Email author
  • Daniel R. Phillips
    • 2
  • Carlos Schat
    • 2
    • 3
  1. 1.Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical PhysicsUniversität BonnBonnGermany
  2. 2.Institute of Nuclear and Particle Physics and Department of Physics and AstronomyOhio UniversityAthensUSA
  3. 3.Instituto de Física de Buenos Aires, CONICET - Departamento de Física, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina

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