Three-loop SM beta-functions for matrix Yukawa couplings. (English) Zbl 1317.81268

Summary: We present the extension of our previous results for three-loop Yukawa coupling beta-functions to the case of complex Yukawa matrices describing the flavour structure of the SM. The calculation is carried out in the context of unbroken phase of the SM with the help of the MINCER program in a general linear gauge and cross-checked by means of MATAD/BAMBA codes. In addition, ambiguities in Yukawa matrix beta-functions are studied.


81V22 Unified quantum theories
81T17 Renormalization group methods applied to problems in quantum field theory
Full Text: DOI arXiv


[1] Aad, G., Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B, 716, 1-29, (2012)
[2] Chatrchyan, S., Observation of a new boson at a mass of 125 gev with the CMS experiment at the LHC, Phys. Lett. B, 716, 30-61, (2012)
[3] Measurements of the properties of the Higgs-like boson in the two photon decay channel with the ATLAS detector using \(25 \text{fb}^{- 1}\) of proton-proton collision data.
[4] Combination of standard model Higgs boson searches and measurements of the properties of the new boson with a mass near 125 GeV.
[5] Gershon, T., Flavour physics in the LHC era
[6] Bilenky, S., Neutrino. history of a unique particle, Eur. Phys. J., H38, 345-404, (2013)
[7] Fritzsch, H.; Xing, Z.-z., Mass and flavor mixing schemes of quarks and leptons, Prog. Part. Nucl. Phys., 45, 1-81, (2000)
[8] Bednyakov, A.; Pikelner, A.; Velizhanin, V., Anomalous dimensions of gauge fields and gauge coupling beta-functions in the standard model at three loops, J. High Energy Phys., 1301, 017, (2013) · Zbl 1342.81336
[9] Bednyakov, A.; Pikelner, A.; Velizhanin, V., Three-loop Higgs self-coupling beta-function in the standard model with complex Yukawa matrices · Zbl 1284.81320
[10] Gross, D. J.; Wilczek, F., Ultraviolet behavior of nonabelian gauge theories, Phys. Rev. Lett., 30, 1343-1346, (1973)
[11] Politzer, H. D., Reliable perturbative results for strong interactions?, Phys. Rev. Lett., 30, 1346-1349, (1973)
[12] Jones, D., Two loop diagrams in Yang-Mills theory, Nucl. Phys. B, 75, 531, (1974)
[13] Tarasov, O.; Vladimirov, A., Two loop renormalization of the Yang-Mills theory in an arbitrary gauge, Sov. J. Nucl. Phys., 25, 585, (1977)
[14] Caswell, W. E., Asymptotic behavior of nonabelian gauge theories to two loop order, Phys. Rev. Lett., 33, 244, (1974)
[15] Egorian, E.; Tarasov, O., Two loop renormalization of the QCD in an arbitrary gauge, Teor. Mat. Fiz., 41, 26-32, (1979)
[16] Jones, D., The two loop beta function for a \(G_1 \times G_2\) gauge theory, Phys. Rev. D, 25, 581, (1982)
[17] Fischler, M. S.; Hill, C. T., Effects of large mass fermions on \(M_X\) and \(\sin^2 \theta_W\), Nucl. Phys. B, 193, 53, (1981)
[18] Machacek, M. E.; Vaughn, M. T., Two loop renormalization group equations in a general quantum field theory. 1. wave function renormalization, Nucl. Phys. B, 222, 83, (1983)
[19] Machacek, M. E.; Vaughn, M. T., Two loop renormalization group equations in a general quantum field theory. 2. Yukawa couplings, Nucl. Phys. B, 236, 221, (1984)
[20] Luo, M.-x.; Wang, H.-w.; Xiao, Y., Two loop renormalization group equations in general gauge field theories, Phys. Rev. D, 67, 065019, (2003)
[21] Jack, I.; Osborn, H., General background field calculations with fermion fields, Nucl. Phys. B, 249, 472, (1985)
[22] Gorishnii, S.; Kataev, A.; Larin, S., Two loop renormalization group calculations in theories with scalar quarks, Theor. Math. Phys., 70, 262-270, (1987)
[23] Arason, H.; Castano, D.; Keszthelyi, B.; Mikaelian, S.; Piard, E., Renormalization group study of the standard model and its extensions. 1. the standard model, Phys. Rev. D, 46, 3945-3965, (1992)
[24] Luo, M.-x.; Xiao, Y., Two loop renormalization group equations in the standard model, Phys. Rev. Lett., 90, 011601, (2003)
[25] Mihaila, L. N.; Salomon, J.; Steinhauser, M., Renormalization constants and beta functions for the gauge couplings of the standard model to three-loop order, Phys. Rev. D, 86, 096008, (2012)
[26] Chetyrkin, K.; Zoller, M.; Chetyrkin, K.; Zoller, M., β-function for the Higgs self-interaction in the standard model at three-loop level, J. High Energy Phys., J. High Energy Phys., 1309, 155, (2013), (Erratum)
[27] Bednyakov, A.; Pikelner, A.; Velizhanin, V., Higgs self-coupling beta-function in the standard model at three loops, Nucl. Phys. B, 875, 552-565, (2013) · Zbl 1331.81357
[28] Grisaru, M. T.; Siegel, W.; Rocek, M., Improved methods for supergraphs, Nucl. Phys. B, 159, 429, (1979)
[29] Jack, I.; Jones, D.; Kord, A., Snowmass benchmark points and three-loop running, Ann. Phys., 316, 213-233, (2005) · Zbl 1078.81531
[30] Hahn, T., Generating Feynman diagrams and amplitudes with feynarts 3, Comput. Phys. Commun., 140, 418-431, (2001) · Zbl 0994.81082
[31] Tentyukov, M.; Fleischer, J., A Feynman diagram analyzer DIANA, Comput. Phys. Commun., 132, 124-141, (2000) · Zbl 1073.81506
[32] Chetyrkin, K.; Zoller, M., Three-loop β-functions for top-Yukawa and the Higgs self-interaction in the standard model, J. High Energy Phys., 1206, 033, (2012)
[33] Bednyakov, A.; Pikelner, A.; Velizhanin, V., Yukawa coupling beta-functions in the standard model at three loops, Phys. Lett. B, 722, 336-340, (2013) · Zbl 1306.81390
[34] Vladimirov, A., Method for computing renormalization group functions in dimensional renormalization scheme, Theor. Math. Phys., 43, 417, (1980)
[35] Misiak, M.; Munz, M., Two loop mixing of dimension five flavor changing operators, Phys. Lett. B, 344, 308-318, (1995)
[36] Chetyrkin, K. G.; Misiak, M.; Munz, M., Beta functions and anomalous dimensions up to three loops, Nucl. Phys. B, 518, 473-494, (1998) · Zbl 0945.81064
[37] Gorishnii, S.; Larin, S.; Surguladze, L.; Tkachov, F., Mincer: program for multiloop calculations in quantum field theory for the schoonschip system, Comput. Phys. Commun., 55, 381-408, (1989)
[38] S. Larin, F. Tkachov, J. Vermaseren, The FORM version of MINCER.
[39] Steinhauser, M., MATAD: a program package for the computation of massive tadpoles, Comput. Phys. Commun., 134, 335-364, (2001) · Zbl 0978.81058
[40] van Ritbergen, T.; Schellekens, A.; Vermaseren, J., Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A, 14, 41-96, (1999) · Zbl 0924.22017
[41] Santamaria, A., Masses, mixings, Yukawa couplings and their symmetries, Phys. Lett. B, 305, 90-97, (1993)
[42] ’t Hooft, G.; Veltman, M., Regularization and renormalization of gauge fields, Nucl. Phys. B, 44, 189-213, (1972)
[43] Jarlskog, C., A basis independent formulation of the connection between quark mass matrices, CP violation and experiment, Z. Phys. C, 29, 491-497, (1985)
[44] Kazakov, D., Radiative corrections, divergences, regularization, renormalization, renormalization group and all that in examples in quantum field theory
[45] Grossman, Y., Introduction to flavor physics, (2010)
[46] Grojean, C.; Mulders, M., in: Proceedings, 2011 European School of High-Energy Physics (ESHEP 2011)
[47] Babu, K., Renormalization group analysis of the Kobayashi-maskawa matrix, Z. Phys. C, 35, 69, (1987)
[48] Naculich, S. G., Third generation effects on fermion mass predictions in supersymmetric grand unified theories, Phys. Rev. D, 48, 5293-5304, (1993)
[49] Balzereit, C.; Mannel, T.; Plumper, B., The renormalization group evolution of the CKM matrix, Eur. Phys. J. C, 9, 197-211, (1999)
[50] Jenkins, E. E.; Manohar, A. V., Algebraic structure of lepton and quark flavor invariants and CP violation, J. High Energy Phys., 0910, 094, (2009)
[51] Schwertfeger, S., Renormierungsgruppenfluss von flavourinvarianten, (2014), University of Siegen, Master’s thesis
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