Abstract
A method, known as “minimal renormalon subtraction” [Phys. Rev. D 97 (2018) 034503, JHEP 08 (2017) 62], relates the factorial growth of a perturbative series (in QCD) to the power p of a power correction Λp/Qp. (Λ is the QCD scale, Q some hard scale.) Here, the derivation is simplified and generalized to any p, more than one such correction, and cases with anomalous dimensions. Strikingly, the well-known factorial growth is seen to emerge already at low or medium orders, as a consequence of constraints on the Q dependence from the renormalization group. The effectiveness of the method is studied with the gluonic energy between a static quark and static antiquark (the “static energy”). Truncation uncertainties are found to be under control after next-to-leading order, despite the small exponent of the power correction (p = 1) and associated rapid growth seen in the first four coefficients of the perturbative series.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Fermilab Lattice et al. collaborations, Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD, Phys. Rev. D 98 (2018) 054517 [arXiv:1802.04248] [INSPIRE].
TUMQCD collaboration, Relations between Heavy-light Meson and Quark Masses, Phys. Rev. D 97 (2018) 034503 [arXiv:1712.04983] [INSPIRE].
J. Komijani, A discussion on leading renormalon in the pole mass, JHEP 08 (2017) 062 [arXiv:1701.00347] [INSPIRE].
C.M. Bender and T.T. WU, Large order behavior of Perturbation theory, Phys. Rev. Lett. 27 (1971) 461 [INSPIRE].
C.M. Bender and T.T. Wu, Anharmonic oscillator. II: A study of perturbation theory in large order, Phys. Rev. D 7 (1973) 1620 [INSPIRE].
D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].
B.E. Lautrup, On High Order Estimates in QED, Phys. Lett. B 69 (1977) 109 [INSPIRE].
G. ’t Hooft, Can We Make Sense Out of Quantum Chromodynamics?, in the proceedings of the 15th Erice School of Subnuclear Physics: The Why’s of Subnuclear Physics, Erice, Italy, July 23 – August 10 (1977), p. 943–982 [INSPIRE].
M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1 [hep-ph/9807443] [INSPIRE].
T. Lee, Renormalons beyond one loop, Phys. Rev. D 56 (1997) 1091 [hep-th/9611010] [INSPIRE].
T. Lee, Normalization constants of large order behavior, Phys. Lett. B 462 (1999) 1 [hep-ph/9908225] [INSPIRE].
A. Pineda, Determination of the bottom quark mass from the Υ(1S) system, JHEP 06 (2001) 022 [hep-ph/0105008] [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].
L.S. Brown, L.G. Yaffe and C.-X. Zhai, Large order perturbation theory for the electromagnetic current current correlation function, Phys. Rev. D 46 (1992) 4712 [hep-ph/9205213] [INSPIRE].
A.F. Falk and M. Neubert, Second order power corrections in the heavy quark effective theory. I. Formalism and meson form-factors, Phys. Rev. D 47 (1993) 2965 [hep-ph/9209268] [INSPIRE].
A.F. Falk, M. Neubert and M.E. Luke, The residual mass term in the heavy quark effective theory, Nucl. Phys. B 388 (1992) 363 [hep-ph/9204229] [INSPIRE].
T. Mannel, Higher order 1/m corrections at zero recoil, Phys. Rev. D 50 (1994) 428 [hep-ph/9403249] [INSPIRE].
A.S. Kronfeld, The perturbative pole mass in QCD, Phys. Rev. D 58 (1998) 051501 [hep-ph/9805215] [INSPIRE].
I.I.Y. Bigi and N.G. Uraltsev, Anathematizing the Guralnik-Manohar bound for Λ, Phys. Lett. B 321 (1994) 412 [hep-ph/9311337] [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].
M. Beneke, More on ambiguities in the pole mass, Phys. Lett. B 344 (1995) 341 [hep-ph/9408380] [INSPIRE].
M.E. Luke, A.V. Manohar and M.J. Savage, Renormalons in effective field theories, Phys. Rev. D 51 (1995) 4924 [hep-ph/9407407] [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, U.S.A. (1972).
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].
R.M. Corless et al., On the LambertW function, Adv. Comput. Math. 5 (1996) 329 [INSPIRE].
K.G. Wilson, Confinement of Quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].
TUMQCD collaboration, Static energy in (2 + 1 + 1)-flavor lattice QCD: Scale setting and charm effects, Phys. Rev. D 107 (2023) 074503 [arXiv:2206.03156] [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 anti-q Bound States, Phys. Lett. B 92 (1980) 343 [INSPIRE].
M. Peter, The static potential in QCD: A full two loop calculation, Nucl. Phys. B 501 (1997) 471 [hep-ph/9702245] [INSPIRE].
Y. Schroder, The static potential in QCD to two loops, Phys. Lett. B 447 (1999) 321 [hep-ph/9812205] [INSPIRE].
B.A. Kniehl, A.A. Penin, M. Steinhauser and V.A. Smirnov, Non-Abelian \( {\alpha}_s^3(s)/\left({m}_q{r}^2\right) \) heavy-quark–antiquark potential, Phys. Rev. D 65 (2002) 091503 [hep-ph/0106135] [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].
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].
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].
D.J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys. Rev. Lett. 30 (1973) 1343 [INSPIRE].
H.D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30 (1973) 1346 [INSPIRE].
D.R.T. Jones, Two Loop Diagrams in Yang-Mills Theory, Nucl. Phys. B 75 (1974) 531 [INSPIRE].
W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].
O.V. Tarasov, A.A. Vladimirov and A.Y. 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].
M. Czakon, The four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].
M.F. Zoller, Four-loop QCD β-function with different fermion representations of the gauge group, JHEP 10 (2016) 118 [arXiv:1608.08982] [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].
F. Herzog et al., The five-loop beta function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [arXiv:1701.01404] [INSPIRE].
T. Luthe, A. Maier, P. Marquard and Y. Schroder, Complete renormalization of QCD at five loops, JHEP 03 (2017) 020 [arXiv:1701.07068] [INSPIRE].
T. Appelquist, M. Dine and I.J. Muzinich, The Static Limit of Quantum Chromodynamics, Phys. Rev. D 17 (1978) 2074 [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].
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].
B.A. Kniehl and A.A. Penin, Ultrasoft effects in heavy quarkonium physics, Nucl. Phys. B 563 (1999) 200 [hep-ph/9907489] [INSPIRE].
I.X.G. Tormo, Review on the determination of αs from the QCD static energy, Mod. Phys. Lett. A 28 (2013) 1330028 [arXiv:1307.2238] [INSPIRE].
W.R. Inc., Mathematica, Version 13.2.1, Champaign, Illinois (2022). [https://www.wolfram.com/mathematica/].
C. Ayala, X. Lobregat and A. Pineda, Determination of α(Mz) from an hyperasymptotic approximation to the energy of a static quark-antiquark pair, JHEP 09 (2020) 016 [arXiv:2005.12301] [INSPIRE].
A. Pineda and J. Soto, The renormalization group improvement of the QCD static potentials, Phys. Lett. B 495 (2000) 323 [hep-ph/0007197] [INSPIRE].
N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo, The logarithmic contribution to the QCD static energy at N**4 LO, Phys. Lett. B 647 (2007) 185 [hep-ph/0610143] [INSPIRE].
N. Brambilla, A. Vairo, X. Garcia i Tormo and J. Soto, The QCD static energy at NNNLL, Phys. Rev. D 80 (2009) 034016 [arXiv:0906.1390] [INSPIRE].
H. Takaura, T. Kaneko, Y. Kiyo and Y. Sumino, Determination of αs from static QCD potential: OPE with renormalon subtraction and lattice QCD, JHEP 04 (2019) 155 [arXiv:1808.01643] [INSPIRE].
TUMQCD collaboration, Determination of the QCD coupling from the static energy and the free energy, Phys. Rev. D 100 (2019) 114511 [arXiv:1907.11747] [INSPIRE].
B. Ananthanarayan, D. Das and M.S.A. Alam Khan, QCD static energy using optimal renormalization and asymptotic Padé-approximant methods, Phys. Rev. D 102 (2020) 076008 [arXiv:2007.10775] [INSPIRE].
N. Brambilla et al., Lattice gauge theory computation of the static force, Phys. Rev. D 105 (2022) 054514 [arXiv:2106.01794] [INSPIRE].
Y. Sumino and H. Takaura, On renormalons of static QCD potential at u = 1/2 and 3/2, JHEP 05 (2020) 116 [arXiv:2001.00770] [INSPIRE].
A.H. Hoang, Bottom quark mass from Υ mesons: Charm mass effects, hep-ph/0008102 [INSPIRE].
Acknowledgments
This work is supported in part by the Technical University of Munich, Institute for Advanced Study, funded by the German Excellence Initiative. Fermilab is managed by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2310.15137
Rights and permissions
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.
About this article
Cite this article
Kronfeld, A.S. Factorial growth at low orders in perturbative QCD: control over truncation uncertainties. J. High Energ. Phys. 2023, 108 (2023). https://doi.org/10.1007/JHEP12(2023)108
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2023)108