The European Physical Journal A

, Volume 41, Issue 1, pp 7–11 | Cite as

Improved bounds on the radius and curvature of the K\( \pi\) scalar form factor and implications to low-energy theorems

Open Access


We obtain stringent bounds in the 〈r2\( \rangle_{S}^{{K\pi}}\) -c plane where these are the scalar radius and the curvature parameters of the scalar K\( \pi\) form factor, respectively, using analyticity and dispersion relation constraints, the knowledge of the form factor from the well-known Callan-Treiman point \(\mathrm{\ensuremath m_K^2-m_\pi^2}\) , as well as at \(\mathrm{\ensuremath m_\pi^2-m_K^2}\) , which we call the second Callan-Treiman point. The central values of these parameters from a recent determination are accomodated in the allowed region provided the higher loop corrections to the value of the form factor at the second Callan-Treiman point reduce the one-loop result by about 3% with \(\mathrm{\ensuremath F_K/F_\pi=1.21}\) . Such a variation in magnitude at the second Callan-Treiman point yields 0.12 fm2\( \lesssim\)r2\( \rangle_{S}^{{K\pi}}\)\( \lesssim\) 0.21 fm2and 0.56 GeV-4\( \lesssim\)c\( \lesssim\) 1.47 GeV-4and a strong correlation between them. A smaller value of \(\mathrm{\ensuremath F_K/F_\pi}\) shifts both bounds to lower values.


11.55.Fv Dispersion relations 12.39.Fe Chiral Lagrangians 


  1. 1.
    FlaviaNet Working Group on Kaon Decays (M. Antonelli), arXiv:0801.1817 [hep-ph].Google Scholar
  2. 2.
    J. Gasser, H. Leutwyler, Nucl. Phys. B 250, 517 (1985).Google Scholar
  3. 3.
    P. Post, K. Schilcher, Eur. Phys. J. C 25, 427 (2002) (arXiv:hep-ph/0112352).Google Scholar
  4. 4.
    J. Bijnens, P. Talavera, Nucl. Phys. B 669, 341 (2003) (arXiv:hep-ph/0303103).Google Scholar
  5. 5.
    M. Jamin, J.A. Oller, A. Pich, JHEP 0402, 047 (2004) (arXiv:hep-ph/0401080).Google Scholar
  6. 6.
    C.G. Callan, S.B. Treiman, Phys. Rev. Lett. 16, 153 (1966).Google Scholar
  7. 7.
    R.F. Dashen, M. Weinstein, Phys. Rev. Lett. 22, 1337 (1969).Google Scholar
  8. 8.
    J. Bijnens, K. Ghorbani, arXiv:0711.0148 [hep-ph].Google Scholar
  9. 9.
    A. Kastner, H. Neufeld, Eur. Phys. J. C 57, 541 (2008) (arXiv:0805.2222 [hep-ph]).Google Scholar
  10. 10.
    V. Cirigliano, G. Ecker, M. Eidemuller, R. Kaiser, A. Pich, J. Portoles, JHEP 0504, 006 (2005) (arXiv:hep-ph/ 0503108).Google Scholar
  11. 11.
    V. Bernard, M. Oertel, E. Passemar, J. Stern, arXiv: 0903.1654 [hep-ph].Google Scholar
  12. 12.
    R. Oheme, Phys. Rev. Lett. 16, 215 (1966).Google Scholar
  13. 13.
    C. Bourrely, I. Caprini, Nucl. Phys. B 722, 149 (2005) (arXiv:hep-ph/0504016).Google Scholar
  14. 14.
    C. Bourrely, B. Machet, E. de Rafael, Nucl. Phys. B 189, 157 (1981).Google Scholar
  15. 15.
    I. Caprini, Eur. Phys. J. C 13, 471 (2000) (arXiv:hep-ph/9907227).Google Scholar
  16. 16.
    B. Ananthanarayan, S. Ramanan, Eur. Phys. J. C 54, 461 (2008) (arXiv:0801.2023 [hep-ph]).Google Scholar
  17. 17.
    HPQCD Collaboration and MILC Collaboration and UKQCD Collaboration (C. Aubin), Phys. Rev. D 70, 031504 (2004) (arXiv:hep-lat/0405022).Google Scholar
  18. 18.
    QCDSF Collaboration and UKQCD Collaboration (M. Gockeler, R. Horsley, A.C. Irving, D. Pleiter, P.E.L. Rakow, G. Schierholz, H. Stuben), Phys. Lett. B 639, 307 (2006) (arXiv:hep-ph/0409312).Google Scholar
  19. 19.
    European Twisted Mass Collaboration (B. Blossier), JHEP 0804, 020 (2008) (arXiv:0709.4574 [hep-lat]).Google Scholar
  20. 20.
    V. Bernard, M. Oertel, E. Passemar, J. Stern, JHEP 0801, 015 (2008) (arXiv:0707.4194 [hep-ph]).Google Scholar

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© The Author(s) 2009

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Authors and Affiliations

  1. 1.Centre for High Energy PhysicsIndian Institute of ScienceBangaloreIndia

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