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.
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References
J. Bijnens, G. Colangelo, G. Ecker, Ann. Phys. 280, 100 (2000). arXiv:hep-ph/9907333
J. Bijnens, G. Colangelo, G. Ecker, J. High Energy Phys. 9902, 020 (1999). arXiv:hep-ph/9902437
G. Colangelo, Nucl. Phys. A 827, 228C (2009)
J. Ahrens et al., Eur. Phys. J. A 23, 113 (2005). arXiv:nucl-ex/0407011
A. Guskov (COMPASS Collaboration), Fizika B 17, 313 (2008)
T. Mori et al. (Belle Collaboration), J. Phys. Soc. Jpn. 76, 074102 (2007). arXiv:0704.3538 [hep-ex]
T. Mori et al. (Belle Collaboration), Phys. Rev. D 75, 051101 (2007). arXiv:hep-ex/0610038
S. Uehara et al. (Belle Collaboration), Phys. Rev. D 78, 052004 (2008). arXiv:0805.3387 [hep-ex]
S. Uehara et al. (BELLE Collaboration), Phys. Rev. D 79, 052009 (2009). arXiv:0903.3697 [hep-ex]
J.R. Batley et al. (NA48/2 Collaboration), Eur. Phys. J. C 54, 411 (2008)
J.R. Batley et al. (NA48/2 Collaboration), Phys. Lett. B 633, 173 (2006). arXiv:hep-ex/0511056
J.R. Batley et al. Eur. Phys. J. C 64, 589 (2009)
B. Adeva et al. (DIRAC Collaboration), Phys. Lett. B 619, 50 (2005). arXiv:hep-ex/0504044
S. Pislak et al. (BNL-E865 Collaboration), Phys. Rev. Lett. 87, 221801 (2001). arXiv:hep-ex/0106071
R. Oehme, Int. J. Mod. Phys. A 10, 1995 (1995). arXiv:hep-th/9412040
R. Omnès, Nuovo Cimento 8, 316 (1958)
N.I. Muskhelishvili, Singular Integral Equations (Noordhof, Groningen, 1953)
M. Gourdin, A. Martin, Nuovo Cimento 17, 224 (1960)
R.L. Goble, J.L. Rosner, Phys. Rev. D 5, 2345 (1972)
R.L. Goble, R. Rosenfeld, J.L. Rosner, Phys. Rev. D 39, 3264 (1989)
D. Morgan, M.R. Pennington, Phys. Lett. B 192, 207 (1987)
D. Morgan, M.R. Pennington, Phys. Lett. B 272, 134 (1991)
J.F. Donoghue, B.R. Holstein, Phys. Rev. D 48, 137 (1993). arXiv:hep-ph/9302203
J.F. Donoghue, B.R. Holstein, Y.C. Lin, Phys. Rev. D 37, 2423 (1988)
J. Bijnens, F. Cornet, Nucl. Phys. B 296, 557 (1988)
K. Sundermeyer, A coupled channel analysis of the reactions gamma + Gamma → Pi + Pi And Gamma + Gamma → K + Anti-K. Preprint DESY 74/17
O. Babelon, J.L. Basdevant, D. Caillerie, M. Gourdin, G. Mennessier, Nucl. Phys. B 114, 252 (1976)
Y. Mao, X.G. Wang, O. Zhang, H.Q. Zheng, Z.Y. Zhou, Phys. Rev. D 79, 116008 (2009). arXiv:0904.1445 [hep-ph]
G. Mennessier, Z. Phys. C 16, 241 (1983)
M. Boglione, M.R. Pennington, Eur. Phys. J. C 9, 11 (1999). arXiv:hep-ph/9812258
J.A. Oller, L. Roca, C. Schat, Phys. Lett. B 659, 201 (2008). arXiv:0708.1659 [hep-ph]
N.N. Achasov, G.N. Shestakov, Phys. Rev. D 77, 074020 (2008). arXiv:0712.0885 [hep-ph]
G. Mennessier, S. Narison, W. Ochs, Phys. Lett. B 665, 205 (2008). arXiv:0804.4452 [hep-ph]
A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1960)
M. Jacob, G.C. Wick, Ann. Phys. 7, 404 (1959) [Ann. Phys. 281, 774 (2000)]
J.F. Donoghue, J. Gasser, H. Leutwyler, Nucl. Phys. B 343, 341 (1990)
B. Moussallam, Eur. Phys. J. C 14, 111 (2000). arXiv:hep-ph/9909292
F.E. Low, Phys. Rev. 96, 1428 (1954)
M. Gell-Mann, M.L. Goldberger, Phys. Rev. 96, 1433 (1954)
H.D.I. Abarbanel, M.L. Goldberger, Phys. Rev. 165, 1594 (1968)
B. Hyams et al., Nucl. Phys. B 64, 134 (1973). AIP Conf. Proc. 13, 206 (1973)
C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1 (2008)
P. Ko, Phys. Rev. D 41, 1531 (1990)
G. Ecker, J. Gasser, A. Pich, E. de Rafael, Nucl. Phys. B 321, 311 (1989)
S. Godfrey, N. Isgur, Phys. Rev. D 32, 189 (1985)
B. Collick et al., Phys. Rev. Lett. 53, 2374 (1984)
M. Zielinski et al., Phys. Rev. D 30, 1855 (1984)
G.T. Condo, T. Handler, W.M. Bugg, G.R. Blackett, M. Pisharody, K.A. Danyo, Phys. Rev. D 48, 3045 (1993)
M. Nozar et al. (CLAS Collaboration), Phys. Rev. Lett. 102, 102002 (2009). arXiv:0805.4438 [hep-ex]
H.J. Lipkin, Phys. Lett. B 72, 249 (1977)
S. Bellucci, J. Gasser, M.E. Sainio, Nucl. Phys. B 423, 80 (1994) [Erratum: Nucl. Phys. B 431, 413 (1994)]. arXiv:hep-ph/9401206
U. Bürgi, Nucl. Phys. B 479, 392 (1996). arXiv:hep-ph/9602429
J. Gasser, M.A. Ivanov, M.E. Sainio, Nucl. Phys. B 728, 31 (2005). arXiv:hep-ph/0506265
J. Gasser, M.A. Ivanov, M.E. Sainio, Nucl. Phys. B 745, 84 (2006). arXiv:hep-ph/0602234
K. Maltman, C.E. Wolfe, Phys. Rev. D 59, 096003 (1999). arXiv:hep-ph/9810441
S. Dürr, J. Kambor, Phys. Rev. D 61, 114025 (2000). arXiv:hep-ph/9907539
M. Knecht, B. Moussallam, J. Stern, Nucl. Phys. B 429, 125 (1994). arXiv:hep-ph/9402318
J. Gasser, C. Haefeli, M.A. Ivanov, M. Schmid, Phys. Lett. B 675, 49 (2009). arXiv:0903.0801 [hep-ph]
G. Amorós, J. Bijnens, P. Talavera, Nucl. Phys. B 568, 319 (2000). arXiv:hep-ph/9907264
H. Terazawa, Phys. Rev. Lett. 26, 1207 (1971)
B.R. Martin, D. Morgan, G. Shaw, Pion–Pion Interactions in Particle Physics (Academic Press, London, 1976)
S.M. Roy, Phys. Lett. B 36, 353 (1971)
B. Ananthanarayan, G. Colangelo, J. Gasser, H. Leutwyler, Phys. Rep. 353, 207 (2001). arXiv:hep-ph/0005297
D.V. Bugg, B.S. Zou, A.V. Sarantsev, Nucl. Phys. B 471, 59 (1996)
R. Kamiński, L. Leśniak, K. Rybicki, Z. Phys. C 74, 79 (1997). arXiv:hep-ph/9606362
R. Kamiński, J.R. Peláez, F.J. Ynduráin, Phys. Rev. D 77, 054015 (2008). arXiv:0710.1150 [hep-ph]
R. García-Martín, R. Kamiński, J.R. Peláez, F.J. Ynduráin, arXiv:0906.5467 [hep-ph]
D.H. Cohen, D.S. Ayres, R. Diebold, S.L. Kramer, A.J. Pawlicki, A.B. Wicklund, Phys. Rev. D 22, 2595 (1980)
A. Etkin et al., Phys. Rev. D 25, 1786 (1982)
K.L. Au, D. Morgan, M.R. Pennington, Phys. Rev. D 35, 1633 (1987)
P. Büttiker, S. Descotes-Genon, B. Moussallam, Eur. Phys. J. C 33, 409 (2004). arXiv:hep-ph/0310283
M.J. Losty et al., Nucl. Phys. B 69, 185 (1974)
W. Hoogland et al., Nucl. Phys. B 126, 109 (1977)
H. Marsiske et al. (Crystal Ball Collaboration), Phys. Rev. D 41, 3324 (1990)
J. Boyer et al., Phys. Rev. D 42, 1350 (1990)
H.J. Behrend et al. (CELLO Collaboration), Z. Phys. C 56, 381 (1992)
G. Ecker, Acta Phys. Pol. B 38, 2753 (2007). arXiv:hep-ph/0702263
M.R. Pennington, T. Mori, S. Uehara, Y. Watanabe, Eur. Phys. J. C 56, 1 (2008). arXiv:0803.3389 [hep-ph]
N.N. Achasov, G.N. Shestakov, Phys. Rev. D 70, 074015 (2004). arXiv:hep-ph/0405129
C. Hanhart, B. Kubis, J.R. Peláez, Phys. Rev. D 76, 074028 (2007). arXiv:0707.0262 [hep-ph]
J. Bijnens, J. Prades, Nucl. Phys. B 490, 239 (1997). arXiv:hep-ph/9610360
L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 73, 035210 (2006). arXiv:nucl-th/0512047
V. Bernard, D. Vautherin, Phys. Rev. D 40, 1615 (1989)
B. Hiller, W. Broniowski, A.A. Osipov, A.H. Blin, Phys. Lett. B 681, 147 (2009). arXiv:0908.0159 [hep-ph]
L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 72, 035211 (2005). arXiv:nucl-th/0505058
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García-Martín, R., Moussallam, B. MO analysis of the high statistics Belle results on γ γ→π + π −, π 0 π 0 with chiral constraints. Eur. Phys. J. C 70, 155–175 (2010). https://doi.org/10.1140/epjc/s10052-010-1471-7
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DOI: https://doi.org/10.1140/epjc/s10052-010-1471-7