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Register a new userTwo-Loop Renormalization in the Standard Model Part III: Renormalization Equations and their Solutions
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In part I and II of this series of papers all elements have been introduced to extend, to two loops, the set of renormalization procedures which are needed in describing the properties of a spontaneously broken gauge theory. In this paper, the final step is undertaken and finite renormalization is discussed. Two-loop renormalization equations are introduced and their solutions discussed within the context of the minimal standard model of fundamental interactions. These equations relate renormalized Lagrangian parameters (couplings and masses) to some input parameter set containing physical (pseudo-)observables. Complex poles for unstable gauge and Higgs bosons are used and a consistent setup is constructed for extending the predictivity of the theory from the Lep1 Z-boson scale (or the Lep2 WW scale) to regions of interest for LHC and ILC physics.
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In part I general aspects of the renormalization of a spontaneously broken gauge theory have been introduced. Here, in part II, two-loop renormalization is introduced and discussed within the context of the minimal Standard Model. Therefore, this paper deals with the transition between bare parameters and fields to renormalized ones. The full list of one- and two-loop counterterms is shown and it is proven that, by a suitable extension of the formalism already introduced at the one-loop level, two-point functions suffice in renormalizing the model. The problem of overlapping ultraviolet divergencies is analyzed and it is shown that all counterterms are local and of polynomial nature. The original program of t Hooft and Veltman is at work. Finite parts are written in a way that allows for a fast and reliable numerical integration with all collinear logarithms extracted analytically. Finite renormalization, the transition between renormalized parameters and physical (pseudo-)observables, will be discussed in part III where numerical results, e.g. for the complex poles of the unstable gauge bosons, will be shown. An attempt will be made to define the running of the electromagnetic coupling constant at the two-loop level.
In this paper the building blocks for the two-loop renormalization of the Standard Model are introduced with a comprehensive discussion of the special vertices induced in the Lagrangian by a particular diagonalization of the neutral sector and by two alternative treatments of the Higgs tadpoles. Dyson resummed propagators for the gauge bosons are derived, and two-loop Ward-Slavnov-Taylor identities are discussed. In part II, the complete set of counterterms needed for the two-loop renormalization will be derived. In part III, a renormalization scheme will be introduced, connecting the renormalized quantities to an input parameter set of (pseudo-)experimental data, critically discussing renormalization of a gauge theory with unstable particles.
We analize the impact of two-loop renormalization group equations on the $SU(3)_ctimes SU(2)_wtimes U(1)_Y$ gauge couplings unification in various supersymmetric theories. In general the presence of superfields in higher representation than the doublet spoil the gauge couplings unification at one-loop. The situation is more interesting when the renormalization group equations are calculated at two-loop. In this case we show that the unification of the gauge couplings can be achieved for models with triplet superfield(s). In the analysis of the models with triplet superfield(s) we show that the dimensionless couplings do not have a Landau pole in their evolution at high energies but they run to a nontrivial ultraviolet fixed point.
We present the full two-loop $beta$-functions for the minimal supersymmetric standard model couplings, extended to include R-parity violating couplings through explicit R-parity violation.
Motivated by models for neutrino masses and lepton mixing, we consider the renormalization of the lepton sector of a general multi-Higgs-doublet Standard Model with an arbitrary number of right-handed neutrino singlets. We propose to make the theory finite by $overline{mbox{MS}}$ renormalization of the parameters of the unbroken theory. However, using a general $R_xi$ gauge, in the explicit one-loop computations of one-point and two-point functions it becomes clear that---in addition---a renormalization of the vacuum expectation values (VEVs) is necessary. Moreover, in order to ensure vanishing one-point functions of the physical scalar mass eigenfields, finite shifts of the tree-level VEVs, induced by the finite parts of the tadpole diagrams, are required. As a consequence of our renormalization scheme, physical masses are functions of the renormalized parameters and VEVs and thus derived quantities. Applying our scheme to one-loop corrections of lepton masses, we perform a thorough discussion of finiteness and $xi$-independence. In the latter context, the tadpole contributions figure prominently.
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