Abstract
The fermion spectrum in the Standard Model (SM) exhibits hierarchical structures between the eigenvalues of the Yukawa matrices which determine the fermion masses, as well as certain hierarchical patterns in the mixing matrix that describes weak transitions between different fermion generations. A basis-independent description of the SM flavour structure can be given in terms of a complete set of flavour invariants. In this paper, we construct a convenient set of such invariants, and discuss the general form of the renormalization-group equations. We also discuss the simplifications that arise from exploiting hierarchies in Yukwawa couplings and mixings which are present in the SM or its minimal-flavour violating extensions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
B. Grzadkowski and M. Lindner, Nonlinear evolution of Yukawa couplings, Phys. Lett. B 193 (1987) 71 [INSPIRE].
B. Grzadkowski, M. Lindner and S. Theisen, Nonlinear evolution of Yukawa couplings in the double Higgs and supersymmetric extensions of the Standard Model, Phys. Lett. B 198 (1987) 64 [INSPIRE].
H. Arason et al., Renormalization group study of the Standard Model and its extensions. 1. The Standard Model, Phys. Rev. D 46 (1992) 3945 [INSPIRE].
V.D. Barger, M.S. Berger and P. Ohmann, Universal evolution of CKM matrix elements, Phys. Rev. D 47 (1993) 2038 [hep-ph/9210260] [INSPIRE].
C. Balzereit, T. Mannel and B. Plümper, The renormalization group evolution of the CKM matrix, Eur. Phys. J. C 9 (1999) 197 [hep-ph/9210260] [INSPIRE].
C.R. Das and M.K. Parida, New formulas and predictions for running fermion masses at higher scales in SM, 2HDM and MSSM, Eur. Phys. J. C 20 (2001) 121 [hep-ph/0010004] [INSPIRE].
A. Crivellin and U. Nierste, Supersymmetric renormalisation of the CKM matrix and new constraints on the squark mass matrices, Phys. Rev. D 79 (2009) 035018 [arXiv:0810.1613] [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization group evolution of the Standard Model dimension six operators II: Yukawa dependence, JHEP 01 (2014) 035 [arXiv:1310.4838] [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].
E.E. Jenkins and A.V. Manohar, Algebraic structure of lepton and quark flavor invariants and CP-violation, JHEP 10 (2009) 094 [arXiv:0907.4763] [INSPIRE].
G.C. Branco and L. Lavoura, Rephasing invariant parametrization of the quark mixing matrix, Phys. Lett. B 208 (1988) 123 [INSPIRE].
G. Colangelo, E. Nikolidakis and C. Smith, Supersymmetric models with minimal flavour violation and their running, Eur. Phys. J. C 59 (2009) 75 [arXiv:0807.0801] [INSPIRE].
G. D’Ambrosio, G.F. Giudice, G. Isidori and A. Strumia, Minimal flavor violation: an effective field theory approach, Nucl. Phys. B 645 (2002) 155 [hep-ph/0207036] [INSPIRE].
T. Feldmann, M. Jung and T. Mannel, Sequential flavour symmetry breaking, Phys. Rev. D 80 (2009) 033003 [arXiv:0906.1523] [INSPIRE].
R. Alonso, M.B. Gavela, L. Merlo and S. Rigolin, On the scalar potential of minimal flavour violation, JHEP 07 (2011) 012 [arXiv:1103.2915] [INSPIRE].
E. Nardi, Naturally large Yukawa hierarchies, Phys. Rev. D 84 (2011) 036008 [arXiv:1105.1770] [INSPIRE].
J.R. Espinosa, C.S. Fong and E. Nardi, Yukawa hierarchies from spontaneous breaking of the SU(3) L × SU(3) R flavour symmetry?, JHEP 02 (2013) 137 [arXiv:1211.6428] [INSPIRE].
C.S. Fong and E. Nardi, Quark masses, mixings and CP-violation from spontaneous breaking of flavor SU(3)3, Phys. Rev. D 89 (2014) 036008 [arXiv:1307.4412] [INSPIRE].
T. Feldmann and T. Mannel, Minimal flavour violation and beyond, JHEP 02 (2007) 067 [hep-ph/0611095] [INSPIRE].
G. Isidori and D.M. Straub, Minimal flavour violation and beyond, Eur. Phys. J. C 72 (2012) 2103 [arXiv:1202.0464] [INSPIRE].
W. Buchmüller and D. Wyler, Effective Lagrangian analysis of new interactions and flavor conservation, Nucl. Phys. B 268 (1986) 621 [INSPIRE].
B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-six terms in the Standard Model Lagrangian, JHEP 10 (2010) 085 [arXiv:1008.4884] [INSPIRE].
S.R. Juarez Wysozka, S.F. Herrera, H., P. Kielanowski and G. Mora, Scale dependence of the quark masses and mixings: leading order, Phys. Rev. D 66 (2002) 116007 [hep-ph/0206243] [INSPIRE].
T. Feldmann, T. Mannel and S. Schwertfeger, Flavour invariants and residual flavour symmetries, in preparation.
R. Alonso, Dynamical Yukawa couplings, arXiv:1307.1904 [INSPIRE].
T. Feldmann and T. Mannel, Large top mass and non-linear representation of flavour symmetry, Phys. Rev. Lett. 100 (2008) 171601 [arXiv:0801.1802] [INSPIRE].
A.L. Kagan, G. Perez, T. Volansky and J. Zupan, General minimal flavor violation, Phys. Rev. D 80 (2009) 076002 [arXiv:0903.1794] [INSPIRE].
H. Fritzsch, Quark masses and flavor mixing, Nucl. Phys. B 155 (1979) 189 [INSPIRE].
C. Jarlskog, Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP-violation, Phys. Rev. Lett. 55 (1985) 1039 [INSPIRE].
C.D. Froggatt and H.B. Nielsen, Hierarchy of quark masses, Cabibbo angles and CP-violation, Nucl. Phys. B 147 (1979) 277 [INSPIRE].
L.-X. Liu, Renormalization invariants and quark flavor mixings, Int. J. Mod. Phys. A 25 (2010) 4975 [arXiv:0910.1326] [INSPIRE].
P.F. Harrison, R. Krishnan and W.G. Scott, Exact one-loop evolution invariants in the Standard Model, Phys. Rev. D 82 (2010) 096004 [arXiv:1007.3810] [INSPIRE].
M. Antonelli et al., Flavor physics in the quark sector, Phys. Rept. 494 (2010) 197 [arXiv:0907.5386] [INSPIRE].
LHCb collaboration, Implications of LHCb measurements and future prospects, Eur. Phys. J. C 73 (2013) 2373 [arXiv:1208.3355] [INSPIRE].
A.J. Buras and J. Girrbach, Towards the identification of new physics through quark flavour violating processes, Rept. Prog. Phys. 77 (2014) 086201 [arXiv:1306.3775] [INSPIRE].
A.J. Buras and J. Girrbach, BSM models facing the recent LHCb data: a first look, Acta Phys. Polon. B 43 (2012) 1427 [arXiv:1204.5064] [INSPIRE].
A.J. Buras, Minimal flavour violation and beyond: towards a flavour code for short distance dynamics, Acta Phys. Polon. B 41 (2010) 2487 [arXiv:1012.1447] [INSPIRE].
M.E. Albrecht, T. Feldmann and T. Mannel, Goldstone bosons in effective theories with spontaneously broken flavour symmetry, JHEP 10 (2010) 089 [arXiv:1002.4798] [INSPIRE].
B. Grinstein, M. Redi and G. Villadoro, Low scale flavor gauge symmetries, JHEP 11 (2010) 067 [arXiv:1009.2049] [INSPIRE].
T. Feldmann, F. Hartmann, W. Kilian and C. Luhn, Combining Pati-Salam and flavour symmetries, accepted for publication in JHEP [arXiv:1506.00782] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1507.00328
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Feldmann, T., Mannel, T. & Schwertfeger, S. Renormalization group evolution of flavour invariants. J. High Energ. Phys. 2015, 7 (2015). https://doi.org/10.1007/JHEP10(2015)007
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2015)007