Abstract
For arbitrary four-dimensional quantum field theories with scalars and fermions, renormalisation group equations in the \( \overline{\mathrm{MS}} \) scheme are investigated at three-loop order in perturbation theory. Collecting literature results, general expressions are obtained for field anomalous dimensions, Yukawa interactions, as well as fermion masses. The renormalisation group evolution of scalar quartic, cubic and mass terms is determined up to a few unknown coefficients. The combined results are applied to compute the renormalisation group evolution of the gaugeless Litim-Sannino model.
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M. E. Machacek and M. T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 1. Wave function renormalization, Nucl. Phys. B 222 (1983) 83 [INSPIRE].
M. E. Machacek and M. T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 2. Yukawa couplings, Nucl. Phys. B 236 (1984) 221 [INSPIRE].
M. E. Machacek and M. T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 3. Scalar quartic couplings, Nucl. Phys. B 249 (1985) 70 [INSPIRE].
M.-X. Luo, H.-W. Wang and Y. Xiao, Two loop renormalization group equations in general gauge field theories, Phys. Rev. D 67 (2003) 065019 [hep-ph/0211440] [INSPIRE].
I. Schienbein, F. Staub, T. Steudtner and K. Svirina, Revisiting RGEs for general gauge theories, Nucl. Phys. B 939 (2019) 1 [Erratum ibid. 966 (2021) 115339] [arXiv:1809.06797] [INSPIRE].
M. Sperling, D. Stöckinger and A. Voigt, Renormalization of vacuum expectation values in spontaneously broken gauge theories, JHEP 07 (2013) 132 [arXiv:1305.1548] [INSPIRE].
M. Sperling, D. Stöckinger and A. Voigt, Renormalization of vacuum expectation values in spontaneously broken gauge theories: two-loop results, JHEP 01 (2014) 068 [arXiv:1310.7629] [INSPIRE].
L. Sartore, General renormalization group equations for dimensionful couplings in the \( \overline{\mathrm{MS}} \) scheme, Phys. Rev. D 102 (2020) 076002 [arXiv:2006.12307] [INSPIRE].
A. G. M. Pickering, J. A. Gracey and D. R. T. Jones, Three loop gauge β-function for the most general single gauge coupling theory, Phys. Lett. B 510 (2001) 347 [Erratum ibid. 535 (2002) 377] [hep-ph/0104247] [INSPIRE].
L. N. Mihaila, J. Salomon and M. Steinhauser, Renormalization constants and β-functions for the gauge couplings of the Standard Model to three-loop order, Phys. Rev. D 86 (2012) 096008 [arXiv:1208.3357] [INSPIRE].
L. Mihaila, Three-loop gauge β-function in non-simple gauge groups, PoS(RADCOR2013)060 (2013) [INSPIRE].
C. Poole and A. E. Thomsen, Constraints on 3- and 4-loop β-functions in a general four-dimensional quantum field theory, JHEP 09 (2019) 055 [arXiv:1906.04625] [INSPIRE].
F. Staub, SARAH 4: a tool for (not only SUSY) model builders, Comput. Phys. Commun. 185 (2014) 1773 [arXiv:1309.7223] [INSPIRE].
L. Sartore and I. Schienbein, PyR@TE 3, Comput. Phys. Commun. 261 (2021) 107819 [arXiv:2007.12700] [INSPIRE].
D. F. Litim and T. Steudtner, ARGES — Advanced Renormalisation Group Equation Simplifier, arXiv:2012.12955 [INSPIRE].
A. E. Thomsen, RGBeta: a Mathematica package for the evaluation of renormalization group β-functions, arXiv:2101.08265 [INSPIRE].
F. Jegerlehner, Facts of life with γ5, Eur. Phys. J. C 18 (2001) 673 [hep-th/0005255] [INSPIRE].
I. Jack and C. Poole, The a-function for gauge theories, JHEP 01 (2015) 138 [arXiv:1411.1301] [INSPIRE].
C. Poole and A. E. Thomsen, Weyl consistency conditions and γ5, Phys. Rev. Lett. 123 (2019) 041602 [arXiv:1901.02749] [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
I. Jack and H. Osborn, Constraints on RG flow for four dimensional quantum field theories, Nucl. Phys. B 883 (2014) 425 [arXiv:1312.0428] [INSPIRE].
I. Jack and C. Poole, Scheme invariants in ϕ4 theory in four dimensions, Phys. Rev. D 98 (2018) 065011 [arXiv:1806.08598] [INSPIRE].
T. Steudtner, General scalar renormalisation group equations at three-loop order, JHEP 12 (2020) 012 [arXiv:2007.06591] [INSPIRE].
J. Davies, F. Herren, C. Poole, M. Steinhauser and A. E. Thomsen, Gauge coupling β functions to four-loop order in the Standard Model, Phys. Rev. Lett. 124 (2020) 071803 [arXiv:1912.07624] [INSPIRE].
I. Jack and H. Osborn, Analogs for the c theorem for four-dimensional renormalizable field theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
A. Bednyakov and A. Pikelner, Six-loop β-functions in general scalar theory, JHEP 04 (2021) 233 [arXiv:2102.12832] [INSPIRE].
F. Herren, L. Mihaila and M. Steinhauser, Gauge and Yukawa coupling β-functions of two-Higgs-doublet models to three-loop order, Phys. Rev. D 97 (2018) 015016 [Erratum ibid. 101 (2020) 079903] [arXiv:1712.06614] [INSPIRE].
K. G. Chetyrkin and M. F. Zoller, Three-loop β-functions for top-Yukawa and the Higgs self-interaction in the Standard Model, JHEP 06 (2012) 033 [arXiv:1205.2892] [INSPIRE].
A. V. Bednyakov, A. F. Pikelner and V. N. Velizhanin, Higgs self-coupling β-function in the Standard Model at three loops, Nucl. Phys. B 875 (2013) 552 [arXiv:1303.4364] [INSPIRE].
K. G. Chetyrkin and M. F. Zoller, β-function for the Higgs self-interaction in the Standard Model at three-loop level, JHEP 04 (2013) 091 [Erratum ibid. 09 (2013) 155] [arXiv:1303.2890] [INSPIRE].
A. V. Bednyakov, A. F. Pikelner and V. N. Velizhanin, Three-loop Higgs self-coupling β-function in the Standard Model with complex Yukawa matrices, Nucl. Phys. B 879 (2014) 256 [arXiv:1310.3806] [INSPIRE].
A. V. Bednyakov, A. F. Pikelner and V. N. Velizhanin, Three-loop SM β-functions for matrix Yukawa couplings, Phys. Lett. B 737 (2014) 129 [arXiv:1406.7171] [INSPIRE].
N. Zerf, L. N. Mihaila, P. Marquard, I. F. Herbut and M. M. Scherer, Four-loop critical exponents for the Gross-Neveu-Yukawa models, Phys. Rev. D 96 (2017) 096010 [arXiv:1709.05057] [INSPIRE].
L. N. Mihaila, N. Zerf, B. Ihrig, I. F. Herbut and M. M. Scherer, Gross-Neveu-Yukawa model at three loops and Ising critical behavior of Dirac systems, Phys. Rev. B 96 (2017) 165133 [arXiv:1703.08801] [INSPIRE].
I. Jack, D. R. T. Jones and C. G. North, N = 1 supersymmetry and the three loop anomalous dimension for the chiral superfield, Nucl. Phys. B 473 (1996) 308 [hep-ph/9603386] [INSPIRE].
A. J. Parkes, Three loop finiteness conditions in N = 1 super Yang-Mills, Phys. Lett. B 156 (1985) 73 [INSPIRE].
C. G. Bollini and J. J. Giambiagi, Lowest order “divergent” graphs in ν-dimensional space, Phys. Lett. B 40 (1972) 566 [INSPIRE].
C. G. Bollini and J. J. Giambiagi, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].
G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].
W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Deep inelastic scattering beyond the leading order in asymptotically free gauge theories, Phys. Rev. D 18 (1978) 3998 [INSPIRE].
S. P. Martin and M. T. Vaughn, Two loop renormalization group equations for soft supersymmetry breaking couplings, Phys. Rev. D 50 (1994) 2282 [Erratum ibid. 78 (2008) 039903] [hep-ph/9311340] [INSPIRE].
G. ’t Hooft and M. J. G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
H. Bélusca-Maïto, A. Ilakovac, M. Mađor-Božinović and D. Stöckinger, Dimensional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme for γ5 applied to chiral YM theories: full one-loop counterterm and RGE structure, JHEP 08 (2020) 024 [arXiv:2004.14398] [INSPIRE].
A. V. Bednyakov and A. F. Pikelner, Four-loop strong coupling β-function in the Standard Model, Phys. Lett. B 762 (2016) 151 [arXiv:1508.02680] [INSPIRE].
M. F. Zoller, Top-Yukawa effects on the β-function of the strong coupling in the SM at four-loop level, JHEP 02 (2016) 095 [arXiv:1508.03624] [INSPIRE].
A. V. Bednyakov, A. F. Pikelner and V. N. Velizhanin, Yukawa coupling β-functions in the Standard Model at three loops, Phys. Lett. B 722 (2013) 336 [arXiv:1212.6829] [INSPIRE].
I. Jack and H. Osborn, Scheme dependence and multiple couplings, arXiv:1606.02571 [INSPIRE].
A. V. Bednyakov, On three-loop RGE for the Higgs sector of 2HDM, JHEP 11 (2018) 154 [arXiv:1809.04527] [INSPIRE].
D. F. Litim and F. Sannino, Asymptotic safety guaranteed, JHEP 12 (2014) 178 [arXiv:1406.2337] [INSPIRE].
D. F. Litim, M. Mojaza and F. Sannino, Vacuum stability of asymptotically safe gauge-Yukawa theories, JHEP 01 (2016) 081 [arXiv:1501.03061] [INSPIRE].
A. D. Bond, D. F. Litim, G. Medina Vazquez and T. Steudtner, UV conformal window for asymptotic safety, Phys. Rev. D 97 (2018) 036019 [arXiv:1710.07615] [INSPIRE].
A. Salam and J. A. Strathdee, On superfields and Fermi-Bose symmetry, Phys. Rev. D 11 (1975) 1521 [INSPIRE].
M. T. Grisaru, W. Siegel and M. Roček, Improved methods for supergraphs, Nucl. Phys. B 159 (1979) 429 [INSPIRE].
W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. B 84 (1979) 193 [INSPIRE].
D. M. Capper, D. R. T. Jones and P. van Nieuwenhuizen, Regularization by dimensional reduction of supersymmetric and nonsupersymmetric gauge theories, Nucl. Phys. B 167 (1980) 479 [INSPIRE].
C. Gnendiger et al., To d, or not to d: recent developments and comparisons of regularization schemes, Eur. Phys. J. C 77 (2017) 471 [arXiv:1705.01827] [INSPIRE].
S. P. Martin and M. T. Vaughn, Regularization dependence of running couplings in softly broken supersymmetry, Phys. Lett. B 318 (1993) 331 [hep-ph/9308222] [INSPIRE].
S. P. Martin, A supersymmetry primer, Adv. Ser. Direct. High Energy Phys. 18 (1998) 1 [Adv. Ser. Direct. High Energy Phys. 21 (2010) 1] [hep-ph/9709356] [INSPIRE].
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Steudtner, T. Towards general scalar-Yukawa renormalisation group equations at three-loop order. J. High Energ. Phys. 2021, 60 (2021). https://doi.org/10.1007/JHEP05(2021)060
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DOI: https://doi.org/10.1007/JHEP05(2021)060