Abstract
We study the role of the lattice artifacts associated with the Coulomb binding effects in the analysis of the heavy quarkonium within lattice NRQCD. We find that a “na¨ıve” perturbative matching generates spurious linear Coulomb artifacts, which result in a large systematic error in the lattice predictions for the heavy quarkonium spectrum. This effect is responsible, in particular, for the discrepancy between the recent determinations of the bottomonium hyperfine splitting in the radiatively improved lattice NRQCD [1, 2]. We show that the correct matching procedure which provides full control over discretization errors is based on the asymptotic expansion of the lattice theory about the continuum limit, which gives M Y(1S) − M ηb (1S) = 52.9 ± 5.5 MeV [1].
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References
M. Baker, A.A. Penin, D. Seidel and N. Zerf, Bottomonium Hyperfine Splitting on the Lattice and in the Continuum, Phys. Rev. D 92 (2015) 054502 [arXiv:1504.05979] [INSPIRE].
HPQCD collaboration, R.J. Dowdall, C.T.H. Davies, T. Hammant and R.R. Horgan, Bottomonium hyperfine splittings from lattice nonrelativistic QCD including radiative and relativistic corrections, Phys. Rev. D 89 (2014) 031502 [Erratum ibid. D 92 (2015) 039904] [arXiv:1309.5797] [INSPIRE].
W.E. Caswell and G.P. Lepage, Effective Lagrangians for Bound State Problems in QED, QCD and Other Field Theories, Phys. Lett. B 167 (1986) 437 [INSPIRE].
G.T. Bodwin, E. Braaten and G.P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D 51 (1995) 1125 [Erratum ibid. D 55 (1997) 5853] [hep-ph/9407339] [INSPIRE].
HPQCD collaboration, R.J. Dowdall et al., The Upsilon spectrum and the determination of the lattice spacing from lattice QCD including charm quarks in the sea, Phys. Rev. D 85 (2012) 054509 [arXiv:1110.6887] [INSPIRE].
BaBar collaboration, B. Aubert et al., Observation of the bottomonium ground state in the decay υ 3S → γeta b , Phys. Rev. Lett. 101 (2008) 071801 [Erratum ibid. 102 (2009) 029901] [arXiv:0807.1086] [INSPIRE].
B.A. Kniehl, A.A. Penin, A. Pineda, V.A. Smirnov and M. Steinhauser, Mass of the η b and α s from nonrelativistic renormalization group, Phys. Rev. Lett. 92 (2004) 242001 [Erratum ibid. 104 (2010) 199901] [hep-ph/0312086] [INSPIRE].
BaBar collaboration, B. Aubert et al., Evidence for the η b (1S) Meson in Radiative Y(2S) Decay, Phys. Rev. Lett. 103 (2009) 161801 [arXiv:0903.1124] [INSPIRE].
CLEO collaboration, G. Bonvicini et al., Measurement of the η b (1S) mass and the branching fraction for Y(3S) → γη b (1S), Phys. Rev. D 81 (2010) 031104 [arXiv:0909.5474] [INSPIRE].
S. Dobbs, Z. Metreveli, K.K. Seth, A. Tomaradze and T. Xiao, Observation of η b (2S) in Y(2S) → γη b (2S), η b (2S) → hadrons and Confirmation of η b (1S), Phys. Rev. Lett. 109 (2012) 082001 [arXiv:1204.4205] [INSPIRE].
Belle collaboration, R. Mizuk et al., Evidence for the η b (2S) and observation of h b (1P ) → η b (1S)γ and h b (2P ) → η b (1S)γ, Phys. Rev. Lett. 109 (2012) 232002 [arXiv:1205.6351] [INSPIRE].
Particle Data Group collaboration, J. Beringer et al., Review of Particle Physics (RPP), Phys. Rev. D 86 (2012) 010001 [INSPIRE].
T.C. Hammant, A.G. Hart, G.M. von Hippel, R.R. Horgan and C.J. Monahan, Radiative improvement of the lattice NRQCD action using the background field method and application to the hyperfine splitting of quarkonium states, Phys. Rev. Lett. 107 (2011) 112002 [Erratum ibid. 115 (2015) 039901] [arXiv:1105.5309] [INSPIRE].
T.C. Hammant, A.G. Hart, G.M. von Hippel, R.R. Horgan and C.J. Monahan, Radiative improvement of the lattice nonrelativistic QCD action using the background field method with applications to quarkonium spectroscopy, Phys. Rev. D 88 (2013) 014505 [arXiv:1303.3234] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, The Heavy quarkonium spectrum at order mα 5 s ln α s , Phys. Lett. B 470 (1999) 215 [hep-ph/9910238] [INSPIRE].
B.A. Kniehl, A.A. Penin, V.A. Smirnov and M. Steinhauser, Potential NRQCD and heavy quarkonium spectrum at next-to-next-to-next-to-leading order, Nucl. Phys. B 635 (2002) 357 [hep-ph/0203166] [INSPIRE].
A.A. Penin and M. Steinhauser, Heavy quarkonium spectrum at O(alpha**5(s) m(q)) and bottom/top quark mass determination, Phys. Lett. B 538 (2002) 335 [hep-ph/0204290] [INSPIRE].
B.A. Thacker and G.P. Lepage, Heavy quark bound states in lattice QCD, Phys. Rev. D 43 (1991) 196 [INSPIRE].
G.P. Lepage, L. Magnea, C. Nakhleh, U. Magnea and K. Hornbostel, Improved nonrelativistic QCD for heavy quark physics, Phys. Rev. D 46 (1992) 4052 [hep-lat/9205007] [INSPIRE].
A. Pineda and J. Soto, Matching at one loop for the four quark operators in NRQCD, Phys. Rev. D 58 (1998) 114011 [hep-ph/9802365] [INSPIRE].
A. Pineda and J. Soto, Potential NRQED: The Positronium case, Phys. Rev. D 59 (1999) 016005 [hep-ph/9805424] [INSPIRE].
A.A. Penin, A. Pineda, V.A. Smirnov and M. Steinhauser, M(B ∗ c ) − M(B c ) splitting from nonrelativistic renormalization group, Phys. Lett. B 593 (2004) 124 [Erratum ibid. 677 (2009) 343] [Erratum ibid. 683 (2010) 358] [hep-ph/0403080] [INSPIRE].
A.A. Penin, A. Pineda, V.A. Smirnov and M. Steinhauser, Spin dependence of heavy quarkonium production and annihilation rates: Complete next-to-next-to-leading logarithmic result, Nucl. Phys. B 699 (2004) 183 [Erratum ibid. B 829 (2010) 398] [hep-ph/0406175] [INSPIRE].
T. Becher and K. Melnikov, The Asymptotic expansion of lattice loop integrals around the continuum limit, Phys. Rev. D 66 (2002) 074508 [hep-ph/0207201] [INSPIRE].
E. Hairer, S.P. Nørsett and G. Wanner, ‘Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin Heidelberg, Germany (2008).
G.S. Bali and P. Boyle, A Lattice potential investigation of quark mass and volume dependence of the Upsilon spectrum, Phys. Rev. D 59 (1999) 114504 [hep-lat/9809180] [INSPIRE].
R.J. Hill and G.P. Lepage, \( \mathcal{O}\left({\alpha}^2\varGamma, {\alpha}^3\varGamma \right) \) binding effects in orthopositronium decay, Phys. Rev. D 62 (2000) 111301 [hep-ph/0003277] [INSPIRE].
HPQCD collaboration, C.T.H. Davies et al., Bottomonium and B results from full lattice QCD, PoS(LATTICE 2013)438 [arXiv:1312.5874] [INSPIRE].
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ArXiv ePrint: 1609.07151
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Liu, T., Penin, A.A. & Rayyan, A. Coulomb artifacts and bottomonium hyperfine splitting in lattice NRQCD. J. High Energ. Phys. 2017, 84 (2017). https://doi.org/10.1007/JHEP02(2017)084
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DOI: https://doi.org/10.1007/JHEP02(2017)084