Gauge coupling beta functions in the standard model to three loops

Abstract

In this Letter we compute the three-loop corrections to the beta functions of the three gauge couplings in the Standard Model of particle physics using the minimal subtraction scheme and taking into account Yukawa and Higgs self-couplings.PACS numbers: 11.10.Hi 11.15.BtRenormalization group functions are fundamental quantities of each quantum field theory and play an important role in various aspects. Besides controlling the energy dependence of parameters and fields they are also crucial for the resummation of large logarithms. Furthermore, renormalization group functions are important for the development of grand unified theories and the extrapolation of low-energy precision data to high energies, not accessible by collider experiments.As far as the strong interaction part of the Standard Model is concerned the corresponding gauge coupling beta function is known up to four-loop order [1][2][3][4][5][6][7][8][9][10] [21].) For a general theory based on a simple gauge group the three-loop corrections to the gauge coupling beta function have been calculated in Ref. [22]. In this Letter we provide results for the three-loop gauge coupling beta functions taking into account all sectors of the Standard Model, i.e., the gauge, Yukawa and Higgs boson self-couplings.Let us in a first step define the beta functions. We denote the three gauge couplings by α 1 , α 2 and α 3 and adopt a SU (5)-like normalization withwhere α is the fine structure constant, θ W the weak mixing angle and α s the strong coupling. In our calculation we consider in addition to the gauge couplings also the third-generation Yukawa couplings [34] α 4 = α t , α 5 = α b and α 6 = α τ , and the Higgs boson self-couplingwhere m x and M W are the fermion and W boson mass, respectively, and −λ(Φ † Φ) 2 is the part of the Lagrange density describing the quartic Higgs self interaction.The functions β i are obtained from the renormalization constants of the corresponding gauge couplings that are defined as g bare i = µ ǫ Z gi g i where α i = g 2 i /(4π). Exploiting the fact that the g bare i are µ-independent and taking into account that Z gi may depend on all seven couplings leads to the following formulawhere ǫ = (4 − d)/2 is the regulator of Dimensional Regularization with d being the space-time dimension used for the evaluation of the momentum integrals and the dependence of α i on the renormalization scale µ is suppressed. From Eq. (2) it is clear that the renormalization constants Z gi (i = 1, 2, 3) have to be computed up to three-loop order.In the modified minimal subtraction (MS) renormalization scheme the perturbative expansion of the gauge coupling beta functions can be written asIn this Letter we evaluate the three-loop terms (coefficients c ijk ) only for the gauge couplings (i.e. i = 1, 2, 3). For our calculation the beta functions for the Yukawa couplings are needed to the one-loop order and the treelevel expression [first term in Eq. (3)] is sufficient for β λ . In the MS scheme the beta functions are mass independent that allows us to use th...

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“…(1.2). The central value of the stability bound at NNLO on M h is shifted with respect to NLO computations (where the matching scale is fixed at µ = M t ) by about +0.5 GeV, whose main contributions can be decomposed as follows: + 0.6 GeV due to the QCD threshold corrections to λ (in agreement with [21]); + 0.2 GeV due to the Yukawa threshold corrections to λ; − 0.2 GeV from RG equation at 3 loops (from [19,20]); − 0.1 GeV from the effective potential at 2 loops.…”

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