Abstract
The gluon condensate, \( \left\langle \frac{\alpha_s}{\pi }{G}^2\right\rangle \), i.e. the leading order power correction in the operator product expansion of current correlators in QCD at short distances, is determined from e+e− annihilation data in the charm-quark region. This determination is based on finite energy QCD sum rules, weighted by a suitable integration kernel to (i) account for potential quark-hadron duality violations, (ii) enhance the contribution of the well known first two narrow resonances, the J/ψ and the ψ(2S), while quenching substantially the data region beyond, and (iii) reinforce the role of the gluon condensate in the sum rules. By using a kernel exhibiting a singularity at the origin, the gluon condensate enters the Cauchy residue at the pole through the low energy QCD expansion of the vector current correlator. These features allow for a reasonably precise determination of the condensate, i.e. \( \left\langle \frac{\alpha_s}{\pi }{G}^2\right\rangle =0.037\pm 0.015 \) GeV4.
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M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and resonance physics. Theoretical foundations, Nucl. Phys. B 147 (1979) 385 [INSPIRE].
P. Colangelo and A. Khodjamirian, QCD sum rules, a modern perspective, in At the frontier of particle physics/Handbook of QCD, M.A. Shifman ed., World Scientific, Singapore (2001).
C.A. Dominguez and K. Schilcher, Is there evidence for dimension two corrections in QCD two point functions?, Phys. Rev. D 61 (2000) 114020 [hep-ph/9903483] [INSPIRE].
C.A. Dominguez and K. Schilcher, QCD vacuum condensates from τ -lepton decay data, JHEP 01 (2007) 093 [hep-ph/0611347] [INSPIRE].
A.A. Almasy, K. Schilcher and H. Spiesberger, QCD condensates of dimension D = 6 and D=8 from hadronic tau-decays, Phys. Lett. B 650 (2007) 179 [hep-ph/0612304] [INSPIRE].
A.A. Almasy, K. Schilcher and H. Spiesberger, Determination of QCD condensates from τ -decay data, Eur. Phys. J. C 55 (2008) 237 [arXiv:0802.0980] [INSPIRE].
C.A. Dominguez, L.A. Hernandez, K. Schilcher and H. Spiesberger, Chiral sum rules and vacuum condensates from τ -lepton decay data, JHEP 03 (2015) 053 [arXiv:1410.3779] [INSPIRE].
S. Bodenstein, C.A. Dominguez, S.I. Eidelman, H. Spiesberger and K. Schilcher, Confronting electron-positron annihilation into hadrons with QCD: an operator product expansion analysis, JHEP 01 (2012) 039 [arXiv:1110.2026] [INSPIRE].
G.S. Bali, C. Bauer and A. Pineda, Model-independent determination of the gluoncondensate in four-dimensional SU(3) gauge theory, Phys. Rev. Lett. 113 (2014) 092001 [arXiv:1403.6477] [INSPIRE].
T. Lee, Renormalon subtraction from the average plaquette and the gluon condensate, Phys. Rev. D 82 (2010) 114021 [arXiv:1003.0231] [INSPIRE].
V.I. Zakharov, Gluon condensate and beyond, Int. J. Mod. Phys. A 14 (1999) 4865 [hep-ph/9906264] [INSPIRE].
R. Horsley et al., Wilson loops to 20th order numerical stochastic perturbation theory, Phys. Rev. D 86 (2012) 054502 [arXiv:1205.1659] [INSPIRE].
B. Chakraborty et al., High-precision quark masses and QCD coupling from n f = 4 lattice QCD, Phys. Rev. D 91 (2015) 054508 [arXiv:1408.4169] [INSPIRE].
R. Shankar, Determination of the quark-gluon coupling constant, Phys. Rev. D 15 (1977) 755 [INSPIRE].
K. Schilcher and M.D. Tran, Duality in semileptonic τ decay, Phys. Rev. D 29 (1984) 570 [INSPIRE].
A.A. Pivovarov, Renormalization group analysis of the τ lepton decay within QCD, Z. Phys. C 53 (1992) 461 [hep-ph/0302003] [INSPIRE].
F. Le Diberder and A. Pich, The perturbative QCD prediction to R(τ) revisited, Phys. Lett. B 286 (1992) 147 [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.A. Dominguez and K. Schilcher, Chiral sum rules and duality in QCD, Phys. Lett. B 448 (1999) 93 [hep-ph/9811261] [INSPIRE].
C.A. Dominguez and K. Schilcher, Finite energy chiral sum rules in QCD, Phys. Lett. B 581 (2004) 193 [hep-ph/0309285] [INSPIRE].
M. Gonzalez-Alonso, A. Pich and J. Prades, Violation of quark-hadron duality and spectral chiral moments in QCD, Phys. Rev. D 81 (2010) 074007 [arXiv:1001.2269] [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].
O. Catà, M. Golterman and S. Peris, Duality violations and spectral sum rules, JHEP 08 (2005) 076 [hep-ph/0506004] [INSPIRE].
O. Catà, M. Golterman and S. Peris, Possible duality violations in tau decay and their impact on the determination of α s , Phys. Rev. D 79 (2009) 053002 [arXiv:0812.2285] [INSPIRE].
D. Boito, M. Golterman, M. Jamin, K. Maltman and S. Peris, Low-energy constants and condensates from the τ hadronic spectral functions, Phys. Rev. D 87 (2013) 094008 [arXiv:1212.4471] [INSPIRE].
S.I. Eidelman, L.M. Kurdadze and A.I. Vainshtein, e+e− annihilation into hadrons below 2 GeV. Test of QCD predictions, Phys. Lett. B 82 (1979) 278 [INSPIRE].
R.A. Bertlmann, C.A. Dominguez, M. Loewe, M. Perrottet and E. de Rafael, Determination of the gluon condensate and the four quark condensate via FESR, Z. Phys. C 39 (1988) 231 [INSPIRE].
C.A. Dominguez and J. Solà, Determination of quark and gluon vacuum condensates from τ lepton decay data, Z. Phys. C 40 (1988) 63 [INSPIRE].
B. Guberina, R. Meckbach, R.D. Peccei and R. Ruckl, Quarkonium sum rules: a critical reappraisal, Nucl. Phys. B 184 (1981) 476 [INSPIRE].
L.J. Reinders, H. Rubinstein and S. Yazaki, Hadron properties from QCD sum rules, Phys. Rept. 127 (1985) 1 [INSPIRE].
V. Giménez, J.A. Penarrocha and J. Bordes, QCD condensates from e+e− to hadrons data in the ϕ(1020) meson channel, Phys. Lett. B 214 (1988) 247 [INSPIRE].
E. Di Salvo and M. Pallavicini, Predictions in the pseudoscalar channel of charmonium by means of QCD sum rules, Nucl. Phys. B 427 (1994) 22 [INSPIRE].
G. Launer, Variation on finite energy sum rules: vacuum matrix elements, Z. Phys. C 32 (1986) 557 [INSPIRE].
M. Davier, A. Hocker and Z. Zhang, The physics of hadronic τ decays, Rev. Mod. Phys. 78 (2006) 1043 [hep-ph/0507078] [INSPIRE].
M. Davier, A. Höcker, B. Malaescu, C.-Z. Yuan and Z. Zhang, Update of the ALEPH non-strange spectral functions from hadronic τ decays, Eur. Phys. J. C 74 (2014) 2803 [arXiv:1312.1501] [INSPIRE].
J. Bordes, C.A. Dominguez, J. Penarrocha and K. Schilcher, Chiral condensates from τ decay: a critical reappraisal, JHEP 02 (2006) 037 [hep-ph/0511293] [INSPIRE].
J.H. Kühn, M. Steinhauser and C. Sturm, Heavy quark masses from sum rules in four-loop approximation, Nucl. Phys. B 778 (2007) 192 [hep-ph/0702103] [INSPIRE].
K.G. Chetyrkin, R. Harlander, J.H. Kuhn and M. Steinhauser, Mass corrections to the vector current correlator, Nucl. Phys. B 503 (1997) 339 [hep-ph/9704222] [INSPIRE].
P.A. Baikov, K. G. Chetyrkin and J.H. Kühn, R(s) and hadronic τ -decays in order α 4 s : technical aspects, Nucl. Phys. Proc. Suppl. B 189 (2009) 49 [arXiv:0906.2987] [INSPIRE].
K.G. Chetyrkin, R.V. Harlander and J.H. Kühn, Quartic mass corrections to Rhad at order α 3 s , Nucl. Phys. B 586 (2000) 56 [Erratum ibid. B 634 (2002) 413] [hep-ph/0005139] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Order α 4 s QCD corrections to Z and τ decays, Phys. Rev. Lett. 101 (2008) 012002 [arXiv:0801.1821] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Vacuum polarization in pQCD: first complete View the MathML source O(α 4 s ) result, Nucl. Phys. Proc. Suppl. B 135 (2004) 243.
R. Boughezal, M. Czakon and T. Schutzmeier, Charm and bottom quark masses from perturbative QCD, Phys. Rev. D 74 (2006) 074006 [hep-ph/0605023] [INSPIRE].
R. Boughezal, M. Czakon and T. Schutzmeier, Four-loop tadpoles: applications in QCD, Nucl. Phys. Proc. Suppl. B 160 (2006) 164.
A. Maier, P. Maierhofer and P. Marquard, Higher moments of heavy quark correlators in the low energy limit at O(α 2 s ), Nucl. Phys. B 797 (2008) 218 [arXiv:0711.2636] [INSPIRE].
A. Maier, P. Maierhofer and P. Marqaurd, The second physical moment of the heavy quark vector correlator at O(α 3 s ), Phys. Lett. B 669 (2008) 88 [arXiv:0806.3405] [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and C. Sturm, Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD, Eur. Phys. J. C 48 (2006) 107 [hep-ph/0604234] [INSPIRE].
A. Maier, P. Maierhofer, P. Marquard and A.V. Smirnov, Low energy moments of heavy quark current correlators at four loops, Nucl. Phys. B 824 (2010) 1 [arXiv:0907.2117] [INSPIRE].
D.J. Broadhurst, P.A. Baikov, V.A. Ilyin, J. Fleischer, O.V. Tarasov and V.A. Smirnov, Two loop gluon condensate contributions to heavy quark current correlators: Exact results and approximations, Phys. Lett. B 329 (1994) 103 [hep-ph/9403274] [INSPIRE].
C. McNeile, C.T.H. Davies, E. Follana, K. Hornbostel and G.P. Lepage, High-precision c and b masses and QCD coupling from current-current correlators in lattice and continuum QCD, Phys. Rev. D 82 (2010) 034512 [arXiv:1004.4285] [INSPIRE].
S. Bodenstein, J. Bordes, C.A. Dominguez, J. Penarrocha and K. Schilcher, QCD sum rule determination of the charm-quark mass, Phys. Rev. D 83 (2011) 074014 [arXiv:1102.3835] [INSPIRE].
K. Chetyrkin et al., Precise charm- and bottom-quark masses: theoretical and experimental uncertainties, Theor. Math. Phys. 170 (2012) 217 [arXiv:1010.6157] [INSPIRE].
K.G. Chetyrkin et al., Charm and bottom quark masses: an update, Phys. Rev. D 80 (2009) 074010 [arXiv:0907.2110] [INSPIRE].
Particle Data Group collaboration, J. Beringer et al., Review of particle physics, Phys. Rev. D 86 (2012) 010001 [INSPIRE].
A.H. Hoang and M. Jamin, MSBAR charm mass from charmonium sum rules with contour improvement, Phys. Lett. B 594 (2004) 127 [hep-ph/0403083] [INSPIRE].
CLEO collaboration, D. Cronin-Hennessy et al., Measurement of charm production cross sections in e+e− annihilation at energies between 3.97 and 4.26 GeV, Phys. Rev. D 80 (2009) 072001 [arXiv:0801.3418] [INSPIRE].
BES collaboration, J.Z. Bai et al., Measurements of the cross-section for e+e− → hadrons at center-of-mass energies from 2 GeV to 5-GeV, Phys. Rev. Lett. 88 (2002) 101802 [hep-ex/0102003] [INSPIRE].
J.Z. Bau et al., Measurements of the continuum R(uds) and R values in e+e− annihilation in the energy region between 3.650 and 3.872 GeV, Phys. Rev. Lett. 97 (2006) 262001 [hep-ex/0612054] [INSPIRE].
S. Eidelman, F. Jegerlehner, A.L. Kataev and O. Veretin, Testing nonperturbative strong interaction effects via the Adler function, Phys. Lett. B 454 (1999) 369 [hep-ph/9812521] [INSPIRE].
B.V. Geshkenbein, Calculation of gluon and four-quark condensates from the operator product expansion, Phys. Rev. D 70 (2004) 074027 [hep-ph/0309122] [INSPIRE].
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Dominguez, C.A., Hernandez, L.A. & Schilcher, K. Determination of the gluon condensate from data in the charm-quark region. J. High Energ. Phys. 2015, 110 (2015). https://doi.org/10.1007/JHEP07(2015)110
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DOI: https://doi.org/10.1007/JHEP07(2015)110