Abstract
We analyze the renormalization group equations for the Standard Model at the one and two loops levels. At one loop level we find an exact constant of evolution built from the product of the quark masses and the gauge couplings g1 and g3 of the U(1) and SU(3) groups. For leptons at one loop level we find that the ratio of the charged lepton mass and the power of g1 varies ≃4×10−5 in the whole energy range. At the two loop level we have found two relations between the quark masses and the gauge couplings that vary ≃4% and ≃1%, respectively. For leptons at the two loop level we have derived a relation between the charged lepton mass and the gauge couplings g1 and g2 that varies ≃0.1%. This analysis significantly simplifies the picture of the renormalization group evolution of the Standard Model and establishes new important relations between its parameters. There is also included a discussion of the gauge invariance of our relations and its possible relation to the reduction of couplings method.
Highlights
In particle physics the renormalization group is used for the study of the asymptotic properties of the theory [1, 2]
The renormalization group equations (RGE) for the Standard Model [3,4,5,6,7,8,9,10,11,12,13,14] is a set of coupled nonlinear differential equations, derived perturbatively, for the parameters of the theory
The K1 relation at the two loop solution of the renormalization group equations is not constant, what mathematically is expected, because K1 was derived from the one loop equations
Summary
In particle physics the renormalization group is used for the study of the asymptotic properties of the theory [1, 2]. Equations for the gauge couplings decouple and are solved exactly, but at two loops this is not the case Another approach, is to use the hierarchy of the parameters of the Standard model, keeping only certain powers of the quark and lepton masses and of λCKM ≈ 0.21 of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [17]. In such a way one obtains the exact solutions of the approximate equations [18]. We include an Appendix, where we derive an equation needed in our analysis
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