Abstract
Next to leading order corrections to the SU(3) × SU(3) Gell-Mann-Oakes-Renner relation (GMOR) are obtained using weighted QCD Finite Energy Sum Rules (FESR) involving the pseudoscalar current correlator. Two types of integration kernels in the FESR are used to suppress the contribution of the kaon radial excitations to the hadronic spectral function, one with local and the other with global constraints. The result for the pseudoscalar current correlator at zero momentum is ψ 5(0) = (2.8 ± 0.3) ×10-3 GeV4, leading to the chiral corrections to GMOR: δ K = (55 ± 5)%. The resulting uncertainties are mostly due to variations in the upper limit of integration in the FESR, within the stability regions, and to a much lesser extent due to the uncertainties in the strong coupling and the strange quark mass. Higher order quark mass corrections, vacuum condensates, and the hadronic resonance sector play a negligible role in this determination. These results confirm an independent determination from chiral perturbation theory giving also very large corrections, i.e. roughly an order of magnitude larger than the corresponding corrections in chiral SU(2) × SU(2). Combining these results with our previous determination of the corrections to GMOR in chiral SU(2) × SU(2), δ π , we are able to determine two low energy constants of chiral perturbation theory, i.e. \( L_8^r=\left( {1.0\pm 0.3} \right)\times {10^{-3 }} \), and \( H_2^r=-\left( {4.7\pm 0.6} \right)\times {10^{-3 }} \), both atthe scaleof the ρ-meson mass.
Similar content being viewed by others
References
H. Pagels, Departures from chiral symmetry, Phys. Rep. C 16 (1975) 219.
C. Dominguez, A. Ramlakan and K. Schilcher, Ratio of strange to nonstrange quark condensates in QCD, Phys. Lett. B 511 (2001) 59 [hep-ph/0104262] [INSPIRE].
M. Jamin and B.O. Lange, f (B) and f (Bs) from QCD sum rules, Phys. Rev. D 65 (2002) 056005 [hep-ph/0108135] [INSPIRE].
C.A. Dominguez, N.F. Nasrallah and K. Schilcher, Strange quark condensate from QCD sum rules to five loops, JHEP 02 (2008) 072 [arXiv:0711.3962] [INSPIRE].
P. Colangelo, A. Khodjamirian, QCD sum rules, a modern perspective, in At the frontier of particle physics. Handbook of QCD, M. Shifman ed., World Scientific, Singapore (2001) 1495.
M. Gell-Mann, R. Oakes and B. Renner, Behavior of current divergences under SU(3) × SU(3), Phys. Rev. 175 (1968) 2195 [INSPIRE].
Particle Data Group collaboration, J. Beringer et al., Review of particle physics (RPP), Phys. Rev. D 86 (2012) 010001 [INSPIRE].
J.L. Rosner and S. Stone, Leptonic decays of charged pseudoscalar mesons, arXiv:1002.1655 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral perturbation theory: expansions in the mass of the strange quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
S. Scherer, Introduction to chiral perturbation theory, Adv. Nucl. Phys. 27 (2003) 277 [hep-ph/0210398] [INSPIRE].
M. Jamin, Flavor symmetry breaking of the quark condensate and chiral corrections to the Gell-Mann-Oakes-Renner relation, Phys. Lett. B 538 (2002) 71 [hep-ph/0201174] [INSPIRE].
G. Amoros, J. Bijnens and P. Talavera, QCD isospin breaking in meson masses, decay constants and quark mass ratios, Nucl. Phys. B 602 (2001) 87 [hep-ph/0101127] [INSPIRE].
D.J. Broadhurst, A strong constraint on chiral symmetry breaking at short distances, Nucl. Phys. B 85 (1975) 189 [INSPIRE].
D.J. Broadhurst, Chiral symmetry breaking and perturbative QCD, Phys. Lett. B 101 (1981) 423 [INSPIRE].
D.J. Broadhurst and S.C. Generalis, Open university report OUT-4102-8 (1982) (unpublished).
V. Spiridonov and K. Chetyrkin, Nonleading mass corrections and renormalization of the operators \( M\;\bar{\psi}\psi \) and \( g_{{\mu \nu}}^2 \) ), Sov. J. Nucl. Phys. 47 (1988) 522 [INSPIRE].
M. Jamin and M. Münz, The strange quark mass from QCD sum rules, Z. Phys. C 66 (1995) 633 [hep-ph/9409335] [INSPIRE].
K. Chetyrkin, C. Dominguez, D. Pirjol and K. Schilcher, Mass singularities in light quark correlators: the strange quark case, Phys. Rev. D 51 (1995) 5090 [hep-ph/9409371] [INSPIRE].
J. Bordes, C. Dominguez, P. Moodley, J. Penarrocha and K. Schilcher, Chiral corrections to the SU(2) × SU(2) Gell-Mann-Oakes-Renner relation, JHEP 05 (2010) 064 [arXiv:1003.3358] [INSPIRE].
C. Dominguez, The Goldberger-Treiman relation: a probe of the chiral symmetries of quantum chromodynamics, Riv. Nuovo Cim. 8N6 (1985) 1 [INSPIRE].
C.A. Dominguez, N.F. Nasrallah, R. Röntsch and K. Schilcher, Strange quark mass from finite energy QCD sum rules to five loops, JHEP 05 (2008) 020 [arXiv:0712.0768] [INSPIRE].
C. Dominguez, N. Nasrallah, R. Röntsch and K. Schilcher, Up and down quark masses from finite energy QCD sum rules to five loops, Phys. Rev. D 79 (2009) 014009 [arXiv:0806.0467] [INSPIRE].
K. Chetyrkin, A. Kataev and F. Tkachov, Higher order corrections to Σ − t(e+e− → hadrons) in quantum chromodynamics, Phys. Lett. B 85 (1979) 277 [INSPIRE].
M. Dine and J. Sapirstein, Higher order QCD corrections in e+e− annihilation, Phys. Rev. Lett. 43 (1979) 668 [INSPIRE].
W. Celmaster and R.J. Gonsalves, An analytic calculation of higher order quantum chromodynamic corrections in e+e− annihilation, Phys. Rev. Lett. 44 (1980) 560 [INSPIRE].
S. Gorishnii, A. Kataev and S. Larin, The \( O\left( {\alpha_s^3} \right) \) corrections to σtot(e+e− → hadrons) and γ(τ − → τ -neutrino + hadrons) in QCD, Phys. Lett. B 259 (1991) 144 [INSPIRE].
L.R. Surguladze and M.A. Samuel, Total hadronic cross-section in e+e− annihilation at the four loop level of perturbative QCD, Phys. Rev. Lett. 66 (1991) 560 [Erratum ibid. 66 (1991) 2416] [INSPIRE].
T. van Ritbergen, J. Vermaseren and S. Larin, The four loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].
P. Baikov, K. Chetyrkin and J.H. Kuhn, Scalar correlator at \( O\left( {\alpha_s^4} \right) \) , Higgs decay into b-quarks and bounds on the light quark masses, Phys. Rev. Lett. 96 (2006) 012003 [hep-ph/0511063] [INSPIRE].
C. Dominguez, Hadronic corrections to QCD sum rules and light quark masses, Z. Phys. C 26 (1984) 269 [INSPIRE].
C. Dominguez and E. de Rafael, Light quark masses in QCD from local duality, Annals Phys. 174 (1987) 372 [INSPIRE].
N. Bilic, C.A. Dominguez and B. Guberina, QCD calculation of K0 anti-K0 mixing from three point function sum rules, Z. Phys. C 39 (1988) 351 [INSPIRE].
C. Dominguez, L. Pirovano and K. Schilcher, The strange quark mass from QCD sum rules in the pseudoscalar channel, Phys. Lett. B 425 (1998) 193 [hep-ph/9712369] [INSPIRE].
K. Maltman, Constraints on hadronic spectral functions from continuous families of finite energy sum rules, Phys. Lett. B 440 (1998) 367 [hep-ph/9901239] [INSPIRE].
C. Dominguez and K. Schilcher, Chiral sum rules and duality in QCD, Phys. Lett. B 448 (1999) 93 [hep-ph/9811261] [INSPIRE].
C. Dominguez and K. Schilcher, Finite energy chiral sum rules in QCD, Phys. Lett. B 581 (2004) 193 [hep-ph/0309285] [INSPIRE].
O. Catà, M. Golterman and S. Peris, Unraveling duality violations in hadronic τ decays, Phys. Rev. D 77 (2008) 093006 [arXiv:0803.0246] [INSPIRE].
M. Gonzalez-Alonso, A. Pich and J. Prades, Pinched weights and duality violation in QCD sum rules: a critical analysis, Phys. Rev. D 82 (2010) 014019 [arXiv:1004.4987] [INSPIRE].
C. Dominguez, N. Nasrallah and K. Schilcher, Confronting QCD with the experimental hadronic spectral functions from τ -decay, Phys. Rev. D 80 (2009) 054014 [arXiv:0903.3463] [INSPIRE].
S. Bethke, A.H. Hoang, S. Kluth, J. Schieck, I.W. Stewart, et al., Workshop on precision measurements of αs, arXiv:1110.0016 [INSPIRE].
C. Dominguez, Determination of light quark masses in QCD, Int. J. Mod. Phys. A 25 (2010) 5223 [arXiv:1005.3724] [INSPIRE].
C. Dominguez, Quark masses in QCD: a progress report, Mod. Phys. Lett. A 26 (2011) 691 [arXiv:1103.5864] [INSPIRE].
C. Dominguez and M. Loewe, Chiral and flavor SU(2) and SU(3) symmetry breaking in quantum chromodynamics, Phys. Rev. D 31 (1985) 2930 [INSPIRE].
G. Colangelo, S. Dürr, A. Juttner, L. Lellouch, H. Leutwyler, et al., Review of lattice results concerning low energy particle physics, Eur. Phys. J. C 71 (2011) 1695 [arXiv:1011.4408] [INSPIRE].
PACS-CS collaboration, S. Aoki et al., 2 + 1 flavor lattice QCD toward the physical point, Phys. Rev. D 79 (2009) 034503 [arXiv:0807.1661] [INSPIRE].
RBC-UKQCD collaboration, C. Allton et al., Physical results from 2 + 1 flavor domain wall QCD and SU(2) chiral perturbation theory, Phys. Rev. D 78 (2008) 114509 [arXiv:0804.0473] [INSPIRE].
MILC collaboration, C. Bernard et al., Status of the MILC light pseudoscalar meson project, PoS LAT2007 (2007) 090 [arXiv:0710.1118] [INSPIRE].
I. Rosell, J.J. Sanz-Cillero and A. Pich, Towards a determination of the chiral couplings at NLO in 1/NC : \( L_{\mu}^{r_8 } \) , JHEP 01 (2007) 039 [hep-ph/0610290] [INSPIRE].
J. Bijnens and I. Jemos, A new global fit of the \( L_i^r \) at next-to-next-to-leading order in chiral perturbation theory, Nucl. Phys. B 854 (2012) 631 [arXiv:1103.5945] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1208.1159
Rights and permissions
About this article
Cite this article
Bordes, J., Dominguez, C.A., Moodley, P. et al. Corrections to the SU(3) × SU(3) Gell-Mann-Oakes-Renner relation and chiral couplings \( L_8^r \) and \( H_2^r \) . J. High Energ. Phys. 2012, 102 (2012). https://doi.org/10.1007/JHEP10(2012)102
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2012)102