Abstract
It is becoming more important to subtract renormalons efficiently from perturbative calculations, in order to achieve high precision QCD calculations. We propose a new framework “Dual Space Approach” for renormalon separation, which enables subtraction of multiple renormalons simultaneously. Using a dual transform which suppresses infrared renormalons, we derive a one-parameter integral representation of a general observable. We investigate systematically how renormalons emerge and get canceled in the entire operator product expansion (OPE) of an observable, by applying the expansion-by-regions (EBR) method to this one-parameter integral expression. In particular we investigate in detail OPEs in a solvable model, the 2-dimensional O(N) nonlinear σ model, by the dual space approach. A nontrivial mechanism of renormalon cancellation in this model can be understood from an integration identity on which the EBR method is founded. We demonstrate that the dual space approach can be useful by a simulation study imitating the QCD case. Application of this method to QCD calculations is also discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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?”, lectures given at the “Ettore Majorana” International School of Subnuclear Physics, Erice Italy (1977).
M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1 [hep-ph/9807443] [INSPIRE].
U. Aglietti and Z. Ligeti, Renormalons and confinement, Phys. Lett. B 364 (1995) 75 [hep-ph/9503209] [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].
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].
A. Pineda, Heavy Quarkonium And Nonrelativistic Effective Field Theories, Ph.D. Thesis, Universitat de Barcelona, Barcelona, Spain (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].
M. Neubert and C.T. Sachrajda, Cancellation of renormalon ambiguities in the heavy quark effective theory, Nucl. Phys. B 438 (1995) 235 [hep-ph/9407394] [INSPIRE].
P. Ball, M. Beneke and V.M. Braun, Resummation of running coupling effects in semileptonic B meson decays and extraction of |V (cb)|, Phys. Rev. D 52 (1995) 3929 [hep-ph/9503492] [INSPIRE].
A.A. Penin and N. Zerf, Bottom Quark Mass from ϒ Sum Rules to \( \mathcal{O}\left({\alpha}_s^3\right) \), JHEP 04 (2014) 120 [arXiv:1401.7035] [INSPIRE].
Y. Kiyo, G. Mishima and Y. Sumino, Determination of mc and mb from quarkonium 1S energy levels in perturbative QCD, Phys. Lett. B 752 (2016) 122 [arXiv:1510.07072] [Erratum ibid. 772 (2017) 878] [INSPIRE].
M. Beneke, A. Maier, J. Piclum and T. Rauh, NNNLO determination of the bottom-quark mass from non-relativistic sum rules, PoS RADCOR2015 (2016) 035 [arXiv:1601.02949] [INSPIRE].
Fermilab Lattice, MILC and TUMQCD collaborations, Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD, Phys. Rev. D 98 (2018) 054517 [arXiv:1802.04248] [INSPIRE].
C. Peset, A. Pineda and J. Segovia, The charm/bottom quark mass from heavy quarkonium at N3LO, JHEP 09 (2018) 167 [arXiv:1806.05197] [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, Z. Ligeti and A.V. Manohar, B decay and the Upsilon mass, Phys. Rev. Lett. 82 (1999) 277 [hep-ph/9809423] [INSPIRE].
A. Alberti, P. Gambino, K.J. Healey and S. Nandi, Precision Determination of the Cabibbo-Kobayashi-Maskawa Element Vcb, Phys. Rev. Lett. 114 (2015) 061802 [arXiv:1411.6560] [INSPIRE].
A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto and A. Vairo, Determination of αs from the QCD static energy, Phys. Rev. D 86 (2012) 114031 [arXiv:1205.6155] [INSPIRE].
K.G. Wilson, Nonlagrangian models of current algebra, Phys. Rev. 179 (1969) 1499 [INSPIRE].
A.H. Mueller, On the Structure of Infrared Renormalons in Physical Processes at High-Energies, Nucl. Phys. B 250 (1985) 327 [INSPIRE].
H. Takaura, T. Kaneko, Y. Kiyo and Y. Sumino, Determination of αs from static QCD potential with renormalon subtraction, Phys. Lett. B 789 (2019) 598 [arXiv:1808.01632] [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].
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].
G.S. Bali, C. Bauer and A. Pineda, Model-independent determination of the gluon condensate in four-dimensional SU(3) gauge theory, Phys. Rev. Lett. 113 (2014) 092001 [arXiv:1403.6477] [INSPIRE].
C. Ayala, X. Lobregat and A. Pineda, Hyperasymptotic approximation to the plaquette and determination of the gluon condensate, JHEP 12 (2020) 093 [arXiv:2009.01285] [INSPIRE].
C. Ayala, X. Lobregat and A. Pineda, Superasymptotic and hyperasymptotic approximation to the operator product expansion, Phys. Rev. D 99 (2019) 074019 [arXiv:1902.07736] [INSPIRE].
C. Ayala, X. Lobregat and A. Pineda, Hyperasymptotic approximation to the top, bottom and charm pole mass, Phys. Rev. D 101 (2020) 034002 [arXiv:1909.01370] [INSPIRE].
T. Lee, Surviving the renormalon in heavy quark potential, Phys. Rev. D 67 (2003) 014020 [hep-ph/0210032] [INSPIRE].
H. Takaura, Formulation for renormalon-free perturbative predictions beyond large-β0 approximation, JHEP 10 (2020) 039 [arXiv:2002.00428] [INSPIRE].
Y. Hayashi, Y. Sumino and H. Takaura, New method for renormalon subtraction using Fourier transform, Phys. Lett. B 819 (2021) 136414 [arXiv:2012.15670] [INSPIRE].
Y. Hayashi, Y. Sumino and H. Takaura, Renormalon subtraction in OPE using Fourier transform: formulation and application to various observables, JHEP 02 (2022) 016 [arXiv:2106.03687] [INSPIRE].
Y. Hayashi, Renormalon subtraction using Fourier transform: analyses of simplified models, JHEP 06 (2022) 157 [arXiv:2112.14408] [INSPIRE].
M.A. Benitez-Rathgeb, D. Boito, A.H. Hoang and M. Jamin, Reconciling the contour-improved and fixed-order approaches for τ hadronic spectral moments. Part I. Renormalon-free gluon condensate scheme, JHEP 07 (2022) 016 [arXiv:2202.10957] [INSPIRE].
M.A. Benitez-Rathgeb, D. Boito, A.H. Hoang and M. Jamin, Reconciling the contour-improved and fixed-order approaches for τ hadronic spectral moments. Part II. Renormalon norm and application in αs determinations, JHEP 09 (2022) 223 [arXiv:2207.01116] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
V.A. Smirnov, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1 [INSPIRE].
B. Jantzen, Foundation and generalization of the expansion by regions, JHEP 12 (2011) 076 [arXiv:1111.2589] [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].
W.A. Bardeen, B.W. Lee and R.E. Shrock, Phase Transition in the Nonlinear σ Model in 2 + ϵ Dimensional Continuum, Phys. Rev. D 14 (1976) 985 [INSPIRE].
F. David, Nonperturbative Effects and Infrared Renormalons Within the 1/N Expansion of the O(N) Nonlinear σ Model, Nucl. Phys. B 209 (1982) 433 [INSPIRE].
F. David, The Operator Product Expansion and Renormalons: A Comment, Nucl. Phys. B 263 (1986) 637 [INSPIRE].
V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Two-Dimensional Sigma Models: Modeling Nonperturbative Effects of Quantum Chromodynamics, Phys. Rept. 116 (1984) 103 [INSPIRE].
M. Beneke, V.M. Braun and N. Kivel, The Operator product expansion, nonperturbative couplings and the Landau pole: Lessons from the O(N) sigma model, Phys. Lett. B 443 (1998) 308 [hep-ph/9809287] [INSPIRE].
T. Lee, Nonperturbative effects from the resummation of perturbation theory, Phys. Rev. D 66 (2002) 034027 [hep-ph/0104306] [INSPIRE].
K. Ishikawa, O. Morikawa, A. Nakayama, K. Shibata, H. Suzuki and H. Takaura, Infrared renormalon in the supersymmetric ℂPN−1 model on ℝ × S1, PTEP 2020 (2020) 023B10 [arXiv:1908.00373] [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, Effective Field Theories for Heavy Quarkonium, Rev. Mod. Phys. 77 (2005) 1423 [hep-ph/0410047] [INSPIRE].
Acknowledgments
Y.H. acknowledges support from GP-PU at Tohoku University. This work was supported by JSPS KAKENHI Grant Numbers JP21J10226, JP20J00328, JP20K03923, JP19K14711, JP19H00689, and JP18H05542.
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: 2303.16392
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
Hayashi, Y., Mishima, G., Sumino, Y. et al. Renormalon subtraction in OPE by dual space approach: nonlinear sigma model and QCD. J. High Energ. Phys. 2023, 42 (2023). https://doi.org/10.1007/JHEP06(2023)042
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2023)042