Abstract
The bottomonium spectrum up to n = 3 is studied within Non-Relativistic Quantum Chromodynamics up to N3LO. We consider finite charm quark mass effects both in the QCD potential and the \( \overline{\mathrm{MS}} \)-pole mass relation up to third order in the Y-scheme counting. The u = 1/2 renormalon of the static potential is canceled by expressing the bottom quark pole mass in terms of the MSR mass. A careful investigation of scale variation reveals that, while n = 1, 2 states are well behaved within perturbation theory, n = 3 bound states are no longer reliable. We carry out our analysis in the nℓ = 3 and nℓ = 4 schemes and conclude that, as long as finite mc effects are smoothly incorporated in the MSR mass definition, the difference between the two schemes is rather small. Performing a fit to \( b\overline{b} \) bound states we find \( {\overline{m}}_b\left({\overline{m}}_b\right) \) = 4.216 ± 0.039 GeV. We extend our analysis to the lowest lying charmonium states finding \( {\overline{m}}_c\left({\overline{m}}_c\right) \) = 1.273 ± 0.054 GeV. Finally, we perform simultaneous fits for \( {\overline{m}}_b \) and α s finding \( {\alpha}_s^{\left({n}_f=5\right)}\left({m}_Z\right)=0.1178\pm 0.0051 \). Additionally, using a modified version of the MSR mass with lighter massive quarks we are able to predict the uncalculated \( \mathcal{O}\left({\alpha}_s^4\right) \) virtual massive quark corrections to the relation between the \( \overline{\mathrm{MS}} \) and pole masses.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Godfrey and K. Moats, Bottomonium Mesons and Strategies for their Observation, Phys. Rev. D 92 (2015) 054034 [arXiv:1507.00024] [INSPIRE].
E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane and T.-M. Yan, Charmonium: The Model, Phys. Rev. D 17 (1978) 3090 [Erratum ibid. D 21 (1980) 313] [INSPIRE].
S. Godfrey and N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 32 (1985) 189 [INSPIRE].
J. Segovia, P.G. Ortega, D.R. Entem and F. Fernández, Bottomonium spectrum revisited, Phys. Rev. D 93 (2016) 074027 [arXiv:1601.05093] [INSPIRE].
J. Segovia, A.M. Yasser, D.R. Entem and F. Fernández, J P C = 1−− hidden charm resonances, Phys. Rev. D 78 (2008) 114033 [INSPIRE].
D. Mohler, Recent results on the meson and baryon spectrum from lattice QCD, EPJ Web Conf. 137 (2017) 05018 [arXiv:1701.05015] [INSPIRE].
S. Prelovsek, Lattice QCD review of charmonium and open-charm spectroscopy, arXiv:1310.4354 [INSPIRE].
Hadron Spectrum collaboration, L. Liu et al., Excited and exotic charmonium spectroscopy from lattice QCD, JHEP 07 (2012) 126 [arXiv:1204.5425] [INSPIRE].
R.J. Dowdall, C.T.H. Davies, T.C. Hammant and R.R. Horgan, Precise heavy-light meson masses and hyperfine splittings from lattice QCD including charm quarks in the sea, Phys. Rev. D 86 (2012) 094510 [arXiv:1207.5149] [INSPIRE].
W. Fischler, \( Q\overline{Q} \) Potential in QCD, Nucl. Phys. B 129 (1977) 157 [INSPIRE].
A. Billoire, How Heavy Must Be Quarks in Order to Build Coulombic \( q\overline{q} \) Bound States, Phys. Lett. B 92 (1980) 343 [INSPIRE].
Y. Schröder, The static potential in QCD to two loops, Phys. Lett. B 447 (1999) 321 [hep-ph/9812205] [INSPIRE].
A. Pineda and F.J. Yndurain, Calculation of quarkonium spectrum and m b , m c to order α 4 S , Phys. Rev. D 58 (1998) 094022 [hep-ph/9711287] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, The infrared behavior of the static potential in perturbative QCD, Phys. Rev. D 60 (1999) 091502 [hep-ph/9903355] [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 \( \mathcal{O}\left({\alpha}_s^5{m}_q\right) \) and bottom/top quark mass determination, Phys. Lett. B 538 (2002) 335 [hep-ph/0204290] [INSPIRE].
A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Fermionic contributions to the three-loop static potential, Phys. Lett. B 668 (2008) 293 [arXiv:0809.1927] [INSPIRE].
A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Three-loop static potential, Phys. Rev. Lett. 104 (2010) 112002 [arXiv:0911.4742] [INSPIRE].
C. Anzai, Y. Kiyo and Y. Sumino, Static QCD potential at three-loop order, Phys. Rev. Lett. 104 (2010) 112003 [arXiv:0911.4335] [INSPIRE].
M.E. Luke, A.V. Manohar and I.Z. Rothstein, Renormalization group scaling in nonrelativistic QCD, Phys. Rev. D 61 (2000) 074025 [hep-ph/9910209] [INSPIRE].
A. Pineda and J. Soto, Effective field theory for ultrasoft momenta in NRQCD and NRQED, Nucl. Phys. Proc. Suppl. 64 (1998) 428 [hep-ph/9707481] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Potential NRQCD: An effective theory for heavy quarkonium, Nucl. Phys. B 566 (2000) 275 [hep-ph/9907240] [INSPIRE].
Y. Kiyo and Y. Sumino, Full Formula for Heavy Quarkonium Energy Levels at Next-to-next-to-next-to-leading Order, Nucl. Phys. B 889 (2014) 156 [arXiv:1408.5590] [INSPIRE].
M. Peter, The static quark-anti-quark potential in QCD to three loops, Phys. Rev. Lett. 78 (1997) 602 [hep-ph/9610209] [INSPIRE].
R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Analytic three-loop static potential, Phys. Rev. D 94 (2016) 054029 [arXiv:1608.02603] [INSPIRE].
A. Pineda, Heavy quarkonium and nonrelativistic effective field theories, Ph.D. Thesis, Barcelona U. (1998) [INSPIRE].
A.H. Hoang, M.C. Smith, T. Stelzer and S. Willenbrock, Quarkonia and the pole mass, Phys. Rev. D 59 (1999) 114014 [hep-ph/9804227] [INSPIRE].
M. Beneke, A quark mass definition adequate for threshold problems, Phys. Lett. B 434 (1998) 115 [hep-ph/9804241] [INSPIRE].
N. Brambilla, Y. Sumino and A. Vairo, Quarkonium spectroscopy and perturbative QCD: A New perspective, Phys. Lett. B 513 (2001) 381 [hep-ph/0101305] [INSPIRE].
N. Brambilla, Y. Sumino and A. Vairo, Quarkonium spectroscopy and perturbative QCD: Massive quark loop effects, Phys. Rev. D 65 (2002) 034001 [hep-ph/0108084] [INSPIRE].
Y. Kiyo and Y. Sumino, Perturbative heavy quarkonium spectrum at next-to-next-to-next-to-leading order, Phys. Lett. B 730 (2014) 76 [arXiv:1309.6571] [INSPIRE].
A.H. Hoang, A. Jain, I. Scimemi and I.W. Stewart, Infrared Renormalization Group Flow for Heavy Quark Masses, Phys. Rev. Lett. 101 (2008) 151602 [arXiv:0803.4214] [INSPIRE].
A.H. Hoang et al., The MSR Mass and the \( \mathcal{O}\left({\varLambda}_{\mathrm{QCD}}\right) \) Renormalon Sum Rule, arXiv:1704.01580 [INSPIRE].
A. Pineda, Determination of the bottom quark mass from the ϒ(1S) system, JHEP 06 (2001) 022 [hep-ph/0105008] [INSPIRE].
A. Czarnecki, K. Melnikov and N. Uraltsev, NonAbelian dipole radiation and the heavy quark expansion, Phys. Rev. Lett. 80 (1998) 3189 [hep-ph/9708372] [INSPIRE].
A. Jain, I. Scimemi and I.W. Stewart, Two-loop Jet-Function and Jet-Mass for Top Quarks, Phys. Rev. D 77 (2008) 094008 [arXiv:0801.0743] [INSPIRE].
S. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart, Factorization approach for top mass reconstruction at high energies, eConf C 0705302 (2007) LOOP06 [arXiv:0710.4205] [INSPIRE].
A.H. Hoang, Z. Ligeti and A.V. Manohar, B decay and the Upsilon mass, Phys. Rev. Lett. 82 (1999) 277 [hep-ph/9809423] [INSPIRE].
A.H. Hoang, Z. Ligeti and A.V. Manohar, B decays in the upsilon expansion, Phys. Rev. D 59 (1999) 074017 [hep-ph/9811239] [INSPIRE].
A.H. Hoang, 1S and \( \overline{\mathrm{MS}} \) bottom quark masses from Upsilon sum rules, Phys. Rev. D 61 (2000) 034005 [hep-ph/9905550] [INSPIRE].
C. Ayala, G. Cvetič and A. Pineda, The bottom quark mass from the ϒ(1S) system at NNNLO, JHEP 09 (2014) 045 [arXiv:1407.2128] [INSPIRE].
A.H. Hoang, C. Lepenik and M. Preisser, On the Light Massive Flavor Dependence of the Large Order Asymptotic Behavior and the Ambiguity of the Pole Mass, JHEP 09 (2017) 099 [arXiv:1706.08526] [INSPIRE].
P. Marquard, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark Mass Relations to Four-Loop Order in Perturbative QCD, Phys. Rev. Lett. 114 (2015) 142002 [arXiv:1502.01030] [INSPIRE].
P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and D. Wellmann, \( \overline{\mathrm{MS}} \) -on-shell quark mass relation up to four loops in QCD and a general SU(N ) gauge group, Phys. Rev. D 94 (2016) 074025 [arXiv:1606.06754] [INSPIRE].
N. Gray, D.J. Broadhurst, W. Grafe and K. Schilcher, Three Loop Relation of Quark \( \overline{\mathrm{MS}} \) and Pole Masses, Z. Phys. C 48 (1990) 673 [INSPIRE].
S. Bekavac, A. Grozin, D. Seidel and M. Steinhauser, Light quark mass effects in the on-shell renormalization constants, JHEP 10 (2007) 006 [arXiv:0708.1729] [INSPIRE].
D. Eiras and J. Soto, Light fermion finite mass effects in non-relativistic bound states, Phys. Lett. B 491 (2000) 101 [hep-ph/0005066] [INSPIRE].
A.H. Hoang, Bottom quark mass from Upsilon mesons: Charm mass effects, hep-ph/0008102 [INSPIRE].
M. Butenschoen, B. Dehnadi, A.H. Hoang, V. Mateu, M. Preisser and I.W. Stewart, Top Quark Mass Calibration for Monte Carlo Event Generators, Phys. Rev. Lett. 117 (2016) 232001 [arXiv:1608.01318] [INSPIRE].
A.H. Hoang, S. Mantry, A. Pathak and I.W. Stewart, Extracting a Short Distance Top Mass with Light Grooming, arXiv:1708.02586 [INSPIRE].
M. Antonelli et al., Flavor Physics in the Quark Sector, Phys. Rept. 494 (2010) 197 [arXiv:0907.5386] [INSPIRE].
LHC Higgs Cross Section Working Group collaboration, J.R. Andersen et al., Handbook of LHC Higgs Cross Sections: 3. Higgs Properties, arXiv:1307.1347 [INSPIRE].
C. Ayala, G. Cvetič and A. Pineda, Mass of the bottom quark from ϒ(1S) at NNNLO: an update, J. Phys. Conf. Ser. 762 (2016) 012063 [arXiv:1606.01741] [INSPIRE].
Y. Kiyo, G. Mishima and Y. Sumino, Determination of m c and m b from quarkonium 1S energy levels in perturbative QCD, Phys. Lett. B 752 (2016) 122 [Erratum ibid. B 772 (2017) 878] [arXiv:1510.07072] [INSPIRE].
A. Pineda, Next-to-leading nonperturbative calculation in heavy quarkonium, Nucl. Phys. B 494 (1997) 213 [hep-ph/9611388] [INSPIRE].
M. Beneke, Y. Kiyo and K. Schuller, Third-order Coulomb corrections to the S-wave Green function, energy levels and wave functions at the origin, Nucl. Phys. B 714 (2005) 67 [hep-ph/0501289] [INSPIRE].
B. Dehnadi, A.H. Hoang, V. Mateu and S.M. Zebarjad, Charm Mass Determination from QCD Charmonium Sum Rules at Order α 3 s , JHEP 09 (2013) 103 [arXiv:1102.2264] [INSPIRE].
M. Beneke, A. Maier, J. Piclum and T. Rauh, The bottom-quark mass from non-relativistic sum rules at NNNLO, Nucl. Phys. B 891 (2015) 42 [arXiv:1411.3132] [INSPIRE].
G.S. Bali and A. Pineda, QCD phenomenology of static sources and gluonic excitations at short distances, Phys. Rev. D 69 (2004) 094001 [hep-ph/0310130] [INSPIRE].
P. Pietrulewicz, S. Gritschacher, A.H. Hoang, I. Jemos and V. Mateu, Variable Flavor Number Scheme for Final State Jets in Thrust, Phys. Rev. D 90 (2014) 114001 [arXiv:1405.4860] [INSPIRE].
I. Wolfram Research, Mathematica Edition: Version 10.0, Wolfram Research, Inc., Champaign, Illinois (2014).
GFortran, Gnu compiler collection (gcc), Version 6.3.0, Copyright © 2014 Free Software Foundation, Inc. (2014).
Particle Data Group collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE].
G. D’Agostini, On the use of the covariance matrix to fit correlated data, Nucl. Instrum. Meth. A 346 (1994) 306 [INSPIRE].
T.G. Kolda, R.M. Lewis and V. Torczon, Optimization by direct search: New perspectives on some classical and modern methods, SIAM Rev. 45 (2003) 385.
B. Colquhoun, R.J. Dowdall, C.T.H. Davies, K. Hornbostel and G.P. Lepage, ϒ and ϒ′ Leptonic Widths, a b μ and m b from full lattice QCD, Phys. Rev. D 91 (2015) 074514 [arXiv:1408.5768] [INSPIRE].
B. Dehnadi, A.H. Hoang and V. Mateu, Bottom and Charm Mass Determinations with a Convergence Test, JHEP 08 (2015) 155 [arXiv:1504.07638] [INSPIRE].
A. Hoang, P. Ruiz-Femenia and M. Stahlhofen, Renormalization Group Improved Bottom Mass from Upsilon Sum Rules at NNLL Order, JHEP 10 (2012) 188 [arXiv:1209.0450] [INSPIRE].
M. Beneke, A. Maier, J. Piclum and T. Rauh, NNNLO determination of the bottom-quark mass from non-relativistic sum rules, PoS(RADCOR2015)035 [arXiv:1601.02949] [INSPIRE].
Y. Maezawa and P. Petreczky, Quark masses and strong coupling constant in 2 + 1 flavor QCD, Phys. Rev. D 94 (2016) 034507 [arXiv:1606.08798] [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].
A. Pineda and J. Segovia, Improved determination of heavy quarkonium magnetic dipole transitions in potential nonrelativistic QCD, Phys. Rev. D 87 (2013) 074024 [arXiv:1302.3528] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1711.05755
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Mateu, V., Ortega, P.G. Bottom and charm mass determinations from global fits to \( Q\overline{Q} \) bound states at N3LO. J. High Energ. Phys. 2018, 122 (2018). https://doi.org/10.1007/JHEP01(2018)122
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2018)122