Abstract
We discuss the {β}-expansion for renormalization group invariant quantities tracing this expansion to the different contractions of the corresponding incomplete BPHZ R-operation. All of the coupling renormalizations, which follow from these contractions, should be taken into account for the {β}-expansion. We illustrate this feature considering the nonsinglet Adler function D NS in the third order of perturbation. We propose a generalization of the {β}-expansion for the renormalization group covariant quantities — the {β, γ}-expansion.
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S.J. Brodsky, G.P. Lepage and P.B. Mackenzie, On the elimination of scale ambiguities in perturbative quantum chromodynamics, Phys. Rev. D 28 (1983) 228 [INSPIRE].
S.V. Mikhailov, Generalization of BLM procedure and its scales in any order of pQCD: a practical approach, JHEP 06 (2007) 009 [hep-ph/0411397] [INSPIRE].
X.-G. Wu, S.J. Brodsky and M. Mojaza, The renormalization scale-setting problem in QCD, Prog. Part. Nucl. Phys. 72 (2013) 44 [arXiv:1302.0599] [INSPIRE].
T. Gehrmann, N. Häfliger and P.F. Monni, BLM scale fixing in event shape distributions, Eur. Phys. J. C 74 (2014) 2896 [arXiv:1401.6809] [INSPIRE].
A.L. Kataev and S.V. Mikhailov, Generalization of the Brodsky-Lepage-Mackenzie optimization within the {β}-expansion and the principle of maximal conformality, Phys. Rev. D 91 (2015) 014007 [arXiv:1408.0122] [INSPIRE].
A.L. Kataev and S.V. Mikhailov, New perturbation theory representation of the conformal symmetry breaking effects in gauge quantum field theory models, Theor. Math. Phys. 170 (2012) 139 [arXiv:1011.5248] [INSPIRE].
A.L. Kataev and S.V. Mikhailov, {β}-expansion in QCD, its conformal symmetry limit: theory + applications, Nucl. Part. Phys. Proc. 258-259 (2015) 45 [arXiv:1410.0554] [INSPIRE].
A.L. Kataev, The generalized BLM approach to fix scale-dependence in QCD: the current status of investigations, J. Phys. Conf. Ser. 608 (2015) 012078 [arXiv:1411.2257] [INSPIRE].
A.P. Bakulev, S.V. Mikhailov and N.G. Stefanis, Higher-order QCD perturbation theory in different schemes: from FOPT to CIPT to FAPT, JHEP 06 (2010) 085 [arXiv:1004.4125] [INSPIRE].
S.J. Brodsky and X.-G. Wu, Scale setting using the extended renormalization group and the principle of maximum conformality: the QCD coupling constant at four loops, Phys. Rev. D 85 (2012) 034038 [Erratum ibid. D 86 (2012) 079903] [arXiv:1111.6175] [INSPIRE].
M. Mojaza, S.J. Brodsky and X.-G. Wu, Systematic all-orders method to eliminate renormalization-scale and scheme ambiguities in perturbative QCD, Phys. Rev. Lett. 110 (2013) 192001 [arXiv:1212.0049] [INSPIRE].
S.J. Brodsky, M. Mojaza and X.-G. Wu, Systematic scale-setting to all orders: the principle of maximum conformality and commensurate scale relations, Phys. Rev. D 89 (2014) 014027 [arXiv:1304.4631] [INSPIRE].
H.-H. Ma, X.-G. Wu, Y. Ma, S.J. Brodsky and M. Mojaza, Setting the renormalization scale in perturbative QCD: comparisons of the principle of maximum conformality with the sequential extended Brodsky-Lepage-Mackenzie approach, Phys. Rev. D 91 (2015) 094028 [arXiv:1504.01260] [INSPIRE].
N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields, IV edition, in Collection of scientific works in 12 volumes, N.N. Bogoliubov, volume 10, section 30, Nauka, Moscow Russia (2008) [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, Computation of the α 2 s correction σ tot (e + e − → hadrons) in QCD, preprint IYaI-P-0170, (1980) [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, New approach to evaluation of multiloop Feynman integrals: the Gegenbauer polynomial x space technique, Nucl. Phys. B 174 (1980) 345 [INSPIRE].
P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn and J. Rittinger, Vector correlator in massless QCD at order O(α 4 s ) and the QED β-function at five loop, JHEP 07 (2012) 017 [arXiv:1206.1284] [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 [Erratum ibid. 66 (1991) 2416] [INSPIRE].
K.G. Chetyrkin, Corrections of order α 3 s to R had in pQCD with light gluinos, Phys. Lett. B 391 (1997) 402 [hep-ph/9608480] [INSPIRE].
S.L. Adler, Some simple vacuum polarization phenomenology: e + e − → hadrons: the μ-mesic atom X-ray discrepancy and (g − 2) of the muon, Phys. Rev. D 10 (1974) 3714 [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, Higher order corrections to σ tot (e + e − → hadrons) in quantum chromodynamics, Phys. Lett. B 85 (1979) 277 [INSPIRE].
S.G. Gorishnii, A.L. Kataev and S.A. Larin, The three loop QED contributions to the photon vacuum polarization function in the MS scheme and the four loop corrections to the QED β-function in the on-shell scheme, Phys. Lett. B 273 (1991) 141 [Erratum ibid. B 275 (1992) 512] [Erratum ibid. B 341 (1995) 448] [INSPIRE].
A.A. Vladimirov, Methods of multiloop calculations and the renormalization group analysis of ϕ 4 theory, Theor. Math. Phys. 36 (1979) 732 [Teor. Mat. Fiz. 36 (1978) 271] [INSPIRE].
O.V. Tarasov and A.A. Vladimirov, Three loop calculations in non-Abelian gauge theories, preprint JINR-E2-80-483, (1980) [Phys. Part. Nucl. 44 (2013) 791] [arXiv:1301.5645] [INSPIRE].
S.-Q. Wang, X.-G. Wu, X.-C. Zheng, J.-M. Shen and Q.-L. Zhang, The Higgs boson inclusive decay channels \( H\to b\overline{b} \) and H → gg up to four-loop level, Eur. Phys. J. C 74 (2014) 2825 [arXiv:1308.6364] [INSPIRE].
A.A. Petrov, S. Pokorski, J.D. Wells and Z. Zhang, Role of low-energy observables in precision Higgs boson analyses, Phys. Rev. D 91 (2015) 073001 [arXiv:1501.02803] [INSPIRE].
G. Cvetič and A.L. Kataev, Adler function and Bjorken polarized sum rule: perturbation expansions in powers of the SU(N c ) conformal anomaly and studies of the conformal symmetry limit, Phys. Rev. D 94 (2016) 014006 [arXiv:1604.00509] [INSPIRE].
L. Clavelli, P.W. Coulter and L.R. Surguladze, Gluino contribution to the three loop β-function in the minimal supersymmetric Standard Model, Phys. Rev. D 55 (1997) 4268 [hep-ph/9611355] [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].
A.V. Bednyakov and A.F. Pikelner, On the four-loop strong coupling β-function in the SM, EPJ Web Conf. 125 (2016) 04008 [arXiv:1609.02597] [INSPIRE].
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Kataev, A.L., Mikhailov, S.V. The {β}-expansion formalism in perturbative QCD and its extension. J. High Energ. Phys. 2016, 79 (2016). https://doi.org/10.1007/JHEP11(2016)079
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DOI: https://doi.org/10.1007/JHEP11(2016)079