Abstract
We provide a detailed description and analysis of a low-scale short-distance mass scheme, called the MSR mass, that is useful for high-precision top quark mass determinations, but can be applied for any heavy quark Q. In contrast to earlier low-scale short-distance mass schemes, the MSR scheme has a direct connection to the well known \( \overline{\mathrm{MS}} \) mass commonly used for high-energy applications, and is determined by heavy quark on-shell self-energy Feynman diagrams. Indeed, the MSR mass scheme can be viewed as the simplest extension of the \( \overline{\mathrm{MS}} \) mass concept to renormalization scales ≪ mQ. The MSR mass depends on a scale R that can be chosen freely, and its renormalization group evolution has a linear dependence on R, which is known as R-evolution. Using R-evolution for the MSR mass we provide details of the derivation of an analytic expression for the normalization of the \( \mathcal{O}\left({\varLambda}_{\mathrm{QCD}}\right) \) renormalon asymptotic behavior of the pole mass in perturbation theory. This is referred to as the \( \mathcal{O}\left({\varLambda}_{\mathrm{QCD}}\right) \) renormalon sum rule, and can be applied to any perturbative series. The relations of the MSR mass scheme to other low-scale short-distance masses are analyzed as well.
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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].
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].
S. Bodenstein, J. Bordes, C.A. Dominguez, J. Penarrocha and K. Schilcher, Bottom-quark mass from finite energy QCD sum rules, Phys. Rev. D 85 (2012) 034003 [arXiv:1111.5742] [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].
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].
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].
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].
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].
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].
J. Erler, P. Masjuan and H. Spiesberger, Charm Quark Mass with Calibrated Uncertainty, Eur. Phys. J. C 77 (2017) 99 [arXiv:1610.08531] [INSPIRE].
A.H. Hoang et al., Top-anti-top pair production close to threshold: Synopsis of recent NNLO results, Eur. Phys. J. direct 2 (2000) 3 [hep-ph/0001286] [INSPIRE].
A.H. Hoang and M. Stahlhofen, The Top-Antitop Threshold at the ILC: NNLL QCD Uncertainties, JHEP 05 (2014) 121 [arXiv:1309.6323] [INSPIRE].
M. Beneke, Y. Kiyo, P. Marquard, A. Penin, J. Piclum and M. Steinhauser, Next-to-Next-to-Next-to-Leading Order QCD Prediction for the Top Antitop S-Wave Pair Production Cross Section Near Threshold in e + e − Annihilation, Phys. Rev. Lett. 115 (2015) 192001 [arXiv:1506.06864] [INSPIRE].
M. Czakon, P. Fiedler and A. Mitov, Total Top-Quark Pair-Production Cross Section at Hadron Colliders Through O(α 4 s ), Phys. Rev. Lett. 110 (2013) 252004 [arXiv:1303.6254] [INSPIRE].
CMS collaboration, Measurement of the t-tbar production cross section in the e-mu channel in proton-proton collisions at \( \sqrt{s} = 7 \) and 8 TeV, JHEP 08 (2016) 029 [arXiv:1603.02303] [INSPIRE].
ATLAS collaboration, Measurement of the tt production cross-section using eμ events with b-tagged jets in pp collisions at \( \sqrt{s}=13 \) TeV with the ATLAS detector, Phys. Lett. B 761 (2016) 136 [Erratum ibid. B 772 (2017) 879] [arXiv:1606.02699] [INSPIRE].
M. Czakon, P. Fiedler, D. Heymes and A. Mitov, NNLO QCD predictions for fully-differential top-quark pair production at the Tevatron, JHEP 05 (2016) 034 [arXiv:1601.05375] [INSPIRE].
S. Alioli et al., A new observable to measure the top-quark mass at hadron colliders, Eur. Phys. J. C 73 (2013) 2438 [arXiv:1303.6415] [INSPIRE].
S. Frixione and A. Mitov, Determination of the top quark mass from leptonic observables, JHEP 09 (2014) 012 [arXiv:1407.2763] [INSPIRE].
CMS collaboration, Measurement of masses in the tt system by kinematic endpoints in pp collisions at \( \sqrt{s}=7 \) TeV, Eur. Phys. J. C 73 (2013) 2494 [arXiv:1304.5783] [INSPIRE].
A. Kharchilava, Top mass determination in leptonic final states with J/ψ, Phys. Lett. B 476 (2000) 73 [hep-ph/9912320] [INSPIRE].
I.I.Y. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein, The pole mass of the heavy quark. Perturbation theory and beyond, Phys. Rev. D 50 (1994) 2234 [hep-ph/9402360] [INSPIRE].
M. Beneke and V.M. Braun, Heavy quark effective theory beyond perturbation theory: Renormalons, the pole mass and the residual mass term, Nucl. Phys. B 426 (1994) 301 [hep-ph/9402364] [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].
M. Beneke, A Quark mass definition adequate for threshold problems, Phys. Lett. B 434 (1998) 115 [hep-ph/9804241] [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 MS-bar bottom quark masses from Upsilon sum rules, Phys. Rev. D 61 (2000) 034005 [hep-ph/9905550] [INSPIRE].
A. Pineda, Determination of the bottom quark mass from the ϒ(1S) system, JHEP 06 (2001) 022 [hep-ph/0105008] [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, Jets from massive unstable particles: Top-mass determination, Phys. Rev. D 77 (2008) 074010 [hep-ph/0703207] [INSPIRE].
K. Seidel, F. Simon, M. Tesar and S. Poss, Top quark mass measurements at and above threshold at CLIC, Eur. Phys. J. C 73 (2013) 2530 [arXiv:1303.3758] [INSPIRE].
T. Horiguchi et al., Study of top quark pair production near threshold at the ILC, arXiv:1310.0563 [INSPIRE].
M. Vos et al., Top physics at high-energy lepton colliders, arXiv:1604.08122 [INSPIRE].
MS collaboration, Measurement of the top quark mass using proton-proton data at \( \sqrt{(s)} = 7 \) and 8 TeV, Phys. Rev. D 93 (2016) 072004 [arXiv:1509.04044] [INSPIRE].
ATLAS collaboration, Measurement of the top quark mass in the \( t\overline{t} \) → dilepton channel from \( \sqrt{s}=8 \) TeV ATLAS data, Phys. Lett. B 761 (2016) 350 [arXiv:1606.02179] [INSPIRE].
CDF, D0 collaboration, T.E.W. Group, Combination of CDF and D0 results on the mass of the top quark using up to 9.7 fb −1 at the Tevatron, arXiv:1407.2682 [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].
A.H. Hoang and I.W. Stewart, Top Mass Measurements from Jets and the Tevatron Top-Quark Mass, Nucl. Phys. Proc. Suppl. 185 (2008) 220 [arXiv:0808.0222] [INSPIRE].
A.H. Hoang, The Top Mass: Interpretation and Theoretical Uncertainties, in Proceedings, 7th International Workshop on Top Quark Physics (TOP2014), Cannes, France, September 28-October 3, 2014 [arXiv:1412.3649] [INSPIRE].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, N-Jettiness: An Inclusive Event Shap to Veto Jets, Phys. Rev. Lett. 105 (2010) 092002 [arXiv:1004.2489] [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 and I.W. Stewart, Designing gapped soft functions for jet production, Phys. Lett. B 660 (2008) 483 [arXiv:0709.3519] [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].
V. Mateu and P.G. Ortega, Bottom and Charm Mass determinations from global fits to \( Q\overline{Q} \) bound states at N 3 LO, JHEP 01 (2018) 122 [arXiv:1711.05755] [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].
R. Tarrach, The Pole Mass in Perturbative QCD, Nucl. Phys. B 183 (1981) 384 [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].
K. Melnikov and T.v. Ritbergen, The Three loop relation between the \( \overline{\mathrm{MS}} \) and the pole quark masses, Phys. Lett. B 482 (2000) 99 [hep-ph/9912391] [INSPIRE].
K.G. Chetyrkin and M. Steinhauser, Short distance mass of a heavy quark at order α s, Phys. Rev. Lett. 83 (1999) 4001 [hep-ph/9907509] [INSPIRE].
K.G. Chetyrkin and M. Steinhauser, The Relation between the \( \overline{\mathrm{MS}} \) and the on-shell quark mass at order α 3 s , Nucl. Phys. B 573 (2000) 617 [hep-ph/9911434] [INSPIRE].
P. Marquard, L. Mihaila, J.H. Piclum and M. Steinhauser, Relation between the pole and the minimally subtracted mass in dimensional regularization and dimensional reduction to three-loop order, Nucl. Phys. B 773 (2007) 1 [hep-ph/0702185] [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].
M.B. Voloshin, ‘Optical’ sum rule for form-factors of heavy mesons, Phys. Rev. D 46 (1992) 3062 [INSPIRE].
I.I.Y. Bigi, M.A. Shifman and N. Uraltsev, Aspects of heavy quark theory, Ann. Rev. Nucl. Part. Sci. 47 (1997) 591 [hep-ph/9703290] [INSPIRE].
A.H. Hoang, A. Jain, I. Scimemi and I.W. Stewart, R-evolution: Improving perturbative QCD, Phys. Rev. D 82 (2010) 011501 [arXiv:0908.3189] [INSPIRE].
R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Thrust at N 3 LL with Power Corrections and a Precision Global Fit for α s(m Z), Phys. Rev. D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].
R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Precision Thrust Cumulant Moments at N 3 LL, Phys. Rev. D 86 (2012) 094002 [arXiv:1204.5746] [INSPIRE].
A.H. Hoang, D.W. Kolodrubetz, V. Mateu and I.W. Stewart, C-parameter distribution at N 3 LL ′ including power corrections, Phys. Rev. D 91 (2015) 094017 [arXiv:1411.6633] [INSPIRE].
A.H. Hoang, D.W. Kolodrubetz, V. Mateu and I.W. Stewart, Precise determination of α s from the C-parameter distribution, Phys. Rev. D 91 (2015) 094018 [arXiv:1501.04111] [INSPIRE].
S. Gritschacher, A. Hoang, I. Jemos and P. Pietrulewicz, Two loop soft function for secondary massive quarks, Phys. Rev. D 89 (2014) 014035 [arXiv:1309.6251] [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].
V. Mateu, I.W. Stewart and J. Thaler, Power Corrections to Event Shapes with Mass-Dependent Operators, Phys. Rev. D 87 (2013) 014025 [arXiv:1209.3781] [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].
F. Campanario and A. Pineda, Fit to the Bjorken, Ellis-Jaffe and Gross-Llewellyn-Smith sum rules in a renormalon based approach, Phys. Rev. D 72 (2005) 056008 [hep-ph/0508217] [INSPIRE].
T. Lee, Renormalons beyond one loop, Phys. Rev. D 56 (1997) 1091 [hep-th/9611010] [INSPIRE].
G.S. Bali, C. Bauer, A. Pineda and C. Torrero, Perturbative expansion of the energy of static sources at large orders in four-dimensional SU(3) gauge theory, Phys. Rev. D 87 (2013) 094517 [arXiv:1303.3279] [INSPIRE].
M. Beneke, P. Marquard, P. Nason and M. Steinhauser, On the ultimate uncertainty of the top quark pole mass, Phys. Lett. B 775 (2017) 63 [arXiv:1605.03609] [INSPIRE].
A.S. Kronfeld, The Perturbative pole mass in QCD, Phys. Rev. D 58 (1998) 051501 [hep-ph/9805215] [INSPIRE].
A.L. Kataev and V.S. Molokoedov, On the flavour dependence of the \( \mathcal{O}\left({\alpha}_s^4\right) \) correction to the relation between running and pole heavy quark masses, Eur. Phys. J. Plus 131 (2016) 271 [arXiv:1511.06898] [INSPIRE].
M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1 [hep-ph/9807443] [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].
A.H. Hoang, Bottom quark mass from Upsilon mesons: Charm mass effects, hep-ph/0008102 [INSPIRE].
K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser, Decoupling relations to O (alpha-S 3 ) and their connection to low-energy theorems, Nucl. Phys. B 510 (1998) 61 [hep-ph/9708255] [INSPIRE].
M. Beneke, More on ambiguities in the pole mass, Phys. Lett. B 344 (1995) 341 [hep-ph/9408380] [INSPIRE].
A. Jain, Heavy quarks in effective field theories, Ph.D. Thesis, Massachusetts Institute of Technology (2009).
J. Komijani, A discussion on leading renormalon in the pole mass, JHEP 08 (2017) 062 [arXiv:1701.00347] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Five-Loop Running of the QCD coupling constant, Phys. Rev. Lett. 118 (2017) 082002 [arXiv:1606.08659] [INSPIRE].
M. Beneke and V.M. Braun, Naive nonAbelianization and resummation of fermion bubble chains, Phys. Lett. B 348 (1995) 513 [hep-ph/9411229] [INSPIRE].
K.G. Chetyrkin, B.A. Kniehl and A. Sirlin, Estimations of order α 3 s and α 4 s corrections to mass dependent observables, Phys. Lett. B 402 (1997) 359 [hep-ph/9703226] [INSPIRE].
A.L. Kataev and V.T. Kim, Peculiar features of the relations between pole and running heavy quark masses and estimates of the O(α 4 s ) contributions, Phys. Part. Nucl. 41 (2010) 946 [arXiv:1001.4207] [INSPIRE].
Y. Sumino, Estimate of 4-loop Pole-MSbar Mass Relation from Static QCD Potential, Phys. Lett. B 728 (2014) 73 [arXiv:1309.5436] [INSPIRE].
G.S. Bali, C. Bauer and A. Pineda, The static quark self-energy at O(α 20) in perturbation theory, PoS(LATTICE 2013)371 [arXiv:1311.0114] [INSPIRE].
A.O.G. Kallen and A. Sabry, Fourth order vacuum polarization, Kong. Dan. Vid. Sel. Mat. Fys. Med. 29 (1955) 1 [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, Heavy quark vacuum polarization to three loops, Phys. Lett. B 371 (1996) 93 [hep-ph/9511430] [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, Three loop polarization function and \( \mathcal{O}\left({\alpha}_s^4\right) \) corrections to the production of heavy quarks, Nucl. Phys. B 482 (1996) 213 [hep-ph/9606230] [INSPIRE].
R. Boughezal, M. Czakon and T. Schutzmeier, Four-Loop Tadpoles: Applications in QCD, Nucl. Phys. Proc. Suppl. 160 (2006) 160 [hep-ph/0607141] [INSPIRE].
M. Czakon and T. Schutzmeier, Double fermionic contributions to the heavy-quark vacuum polarization, JHEP 07 (2008) 001 [arXiv:0712.2762] [INSPIRE].
A. Maier, P. Maierhofer and P. Marquard, Higher Moments of Heavy Quark Correlators in the Low Energy Limit at \( \mathcal{O}\left({\alpha}_s^2\right) \), Nucl. Phys. B 797 (2008) 218 [arXiv:0711.2636] [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].
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].
C. Sturm, Moments of Heavy Quark Current Correlators at Four-Loop Order in Perturbative QCD, JHEP 09 (2008) 075 [arXiv:0805.3358] [INSPIRE].
A. Maier, P. Maierhofer and P. Marqaurd, The Second physical moment of the heavy quark vector correlator at \( \mathcal{O}\left({\alpha}_s^3\right) \), Phys. Lett. B 669 (2008) 88 [arXiv:0806.3405] [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].
T. Appelquist, M. Dine and I.J. Muzinich, The Static Potential in Quantum Chromodynamics, Phys. Lett. B 69 (1977) 231 [INSPIRE].
T. Appelquist, M. Dine and I.J. Muzinich, The Static Limit of Quantum Chromodynamics, Phys. Rev. D 17 (1978) 2074 [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].
C. Anzai, Y. Kiyo and Y. Sumino, Static QCD potential at three-loop order, Phys. Rev. Lett. 104 (2010) 112003 [arXiv:0911.4335] [INSPIRE].
A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Three-loop static potential, Phys. Rev. Lett. 104 (2010) 112002 [arXiv:0911.4742] [INSPIRE].
Y. Kiyo, G. Mishima and Y. Sumino, Strong IR Cancellation in Heavy Quarkonium and Precise Top Mass Determination, JHEP 11 (2015) 084 [arXiv:1506.06542] [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, Higher Order Corrections to σ t(e + e − → Hadrons) in Quantum Chromodynamics, Phys. Lett. B 85 (1979) 277 [INSPIRE].
S.G. Gorishnii, A.L. Kataev and S.A. Larin, The O(α 3 s )-corrections to σ tot(e + e − → hadrons) and Γ(τ − → ν τ + 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 [INSPIRE].
L.R. Surguladze and M.A. Samuel, Erratum: Total hadronic cross-section in e + e − annihilation at the four loop level of perturbative QCD, Phys. Rev. Lett. 66 (1991) 2461.
P.A. Baikov, K.G. Chetyrkin and J.H. Kuhn, Order α 4 s QCD Corrections to Z and tau Decays, Phys. Rev. Lett. 101 (2008) 012002 [arXiv:0801.1821] [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and Resonance Physics. Theoretical Foundations, Nucl. Phys. B 147 (1979) 385 [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and Resonance Physics: Applications, Nucl. Phys. B 147 (1979) 448 [INSPIRE].
F. Simon, Impact of Theory Uncertainties on the Precision of the Top Quark Mass in a Threshold Scan at Future e + e − Colliders, PoS(ICHEP2016)872 [arXiv:1611.03399] [INSPIRE].
D. d’Enterria, Physics at the FCC-ee, in Proceedings, 17th Lomonosov Conference on Elementary Particle Physics, Moscow, Russia, August 20-26, 2015, pp. 182-191 (2017) [DOI:https://doi.org/10.1142/9789813224568_0028] [arXiv:1602.05043] [INSPIRE].
D. d’Enterria et al. eds., Proceedings, High-Precision α s Measurements from LHC to FCC-ee, CERN, Geneva, Switzerland (2015) [arXiv:1512.05194] [INSPIRE].
A. Pineda, Comment on “The MSR Mass and the \( \mathcal{O}\left({\varLambda}_{\mathrm{QCD}}\right) \) Renormalon Sum Rule”, arXiv:1704.05095 [INSPIRE].
O.V. Tarasov, A.A. Vladimirov and A.Yu. Zharkov, The Gell-Mann-Low Function of QCD in the Three Loop Approximation, Phys. Lett. B 93 (1980) 429 [INSPIRE].
S.A. Larin and J.A.M. Vermaseren, The Three loop QCD β-function and anomalous dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208] [INSPIRE].
T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The Four loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The three loop splitting functions in QCD: The Nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
M. Czakon, The four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].
W. Fischler, Quark-anti-Quark 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].
M. Peter, The Static quark-anti-quark potential in QCD to three loops, Phys. Rev. Lett. 78 (1997) 602 [hep-ph/9610209] [INSPIRE].
Y. Schröder, The static potential in QCD to two loops, Phys. Lett. B 447 (1999) 321 [hep-ph/9812205] [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.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].
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].
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Hoang, A.H., Jain, A., Lepenik, C. et al. The MSR mass and the \( \mathcal{O}\left({\Lambda}_{\mathrm{QCD}}\right) \) renormalon sum rule. J. High Energ. Phys. 2018, 3 (2018). https://doi.org/10.1007/JHEP04(2018)003
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DOI: https://doi.org/10.1007/JHEP04(2018)003