Two-loop renormalization group restrictions on the standard model and the fourth chiral family

In the framework of the two-loop renormalization group, the restrictions on the Higgs mass from the electroweak vacuum stability and from the absence of the strong coupling are refined, while the more precise value of the top mass is taken into account. When the SM cutoff is equal to the Planck scale, the Higgs mass must be GeV and GeV, where the corridor is the theoretical one and the errors are due to the top-mass uncertainty. The SM two-loop functions are generalized to the case with massive neutrinos from extra families. The requirement of self-consistency of the perturbative SM as an underlying theory up to the Planck scale excludes a fourth chiral family. Under the precision-experiment restriction GeV, the fourth chiral family, if alone, is excluded even when the SM is regarded as an effective theory. Nevertheless a pair of chiral families constituting a vector-like one could exist. Received: 2 September 1998 / Revised version: 4 January 1999 / Published online: 28 September 1999     

Two-loop renormalization group equations in a general quantum field theory: (III). Scalar quartic couplings

The two-loop β-functions for the scalar quartic couplings are computed in a general renormalizable quantum field theory with scalar, $\text{spin}\text{-}\text{1}\text{2}$, and (vector) gauge fields associated with a general gauge group G, using dimensional regularization and modified minimal subtraction (^?MS). A more explicit form is given for the two-loop β-function of the quartic coupling of the Higgs doublet in the minimal QCD electroweak theory based on SU(3) × SU(2) × U(1).     

Anomalous dimensions and the renormalization group equations

An interpretation is given of scale and anomalous dimensions in the framework of the renormalization group, and the renormalization group equations are derived which are regarded to represent the conservation of these scale dimensions. By the use of continuous dimensional regularization all coefficient functions appearing in these equations and in the Callan-Symanzik equations are explicitly expressed in terms of the residues of the single poles at n = 4 as well as the finite part of renormalization counter-terms.     

The equations of Wilson's renormalization group and analytic renormalization

In the present series of two papers we solve exactly Wilson's equations for a long-range effective hamiltonian. These equations arise when one seeks a fixed point of the Wilson's renormalization group transformations in the formulation of perturbation theory. The first paper has a general character. Wilson's renormalization transformation and its modifications are defined and the group property for them is established. Some topological aspects of the renormalization transformations are discussed. A space of projection hamiltonians is introduced and a theorem on the invariance of this space with respect to the renormalization transformations is proved.     

On several parameter renormalization group equations and renormalization scheme specifications

An example of the explicit realization of the full renormalization group for physical quantities is investigated. It introduces two-index beta functions β_ij(g_j): (i) the first index specifying the renormalization scheme chosen (including the scale μ); (ii) the second index specifying the coupling of which the appropriate beta function is the differential transformation. The treated example is presented in a 4 — ? dimensional O(n) massless (?·?)² theory with emphasis on the calculation of the fixed point.     

Two-loop renormalization of three-quark operators in QCD

Renormalization of composite three-quark operators in dimensional regularization is complicated by the mixing of physical and unphysical (evanescent) operators. This mixing must be taken into account in a consistent subtraction scheme. In this work we propose a particular scheme that allows one to avoid the necessity of additional finite renormalization and is convenient in QCD applications. As an illustration, we calculate the two-loop anomalous dimensions of local three-quark operators in this scheme.     

Analytical solutions of exact renormalization group equations

We study exact renormalization group equations in the framework of the effective average action. We present analytical solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of physical behaviour such as fixed points governing the universal behaviour near second-order phase transitions, critical exponents, first-order transitions (some of which are radiatively induced) and tricritical behaviour.     

The equations of Wilson's renormalization group in dimension 4 and analytic renormalization

Wilson's renormalization group equations are introduced and investigated in the framework of perturbation theory with respect to the deviation of the renormalization exponent from its bifurcation value. We consider the case when the dimension is equal to 4. An exact solution of these equations is constructed using analytic renormalization of the projection Hamiltonians.     

Quarkonia from charmonium and renormalization group equations

A prediction of the upsilon and strangeonium spectra is made from the charmonium spectrum by solving the Salpeter equation using an identical potential to that used in charmonium. Effective quark masses and coupling parameters α_s are functions of the inter-quark distance according to the renormalization group equations. The use of the Fermi-Breit-Hamiltonian for obtaining the charmonium hyperfine splitting is criticized.     

A new method to solve the non-perturbative renormalization group equations

We propose a method to solve the non-perturbative renormalization group equations for the n-point functions. In leading order, it consists in solving the equations obtained by closing the infinite hierarchy of equations for the n-point functions. This is achieved: (i) by exploiting the decoupling of modes and the analyticity of the n  -point functions at small momenta: this allows us to neglect some momentum dependence of the vertices entering the flow equations; (ii) by relating vertices at zero momenta to derivatives of lower order vertices with respect to a constant background field. Although the approximation is not controlled by a small parameter, its accuracy can be systematically improved. When it is applied to the O(N)$O(N)$ model, its leading order is exact in the large-N limit; in this case, one recovers known results in a simple and direct way, i.e., without introducing an auxiliary field.     

Masses of the right-handed gauge bosons from the renormalization group equations

Lower bounds on the masses of the right-handed gauge bosons are obtained from the renormalization group analysis. The perturbative approach breaks down unless M_WR≳10⁷ GeV.     

Scaling laws and surface exponents from renormalization group equations

The critical behaviour of a semi-infinite system with O(n) spin symmetry is studied. Renormalization group equations are derived whose solutions yield scaling laws for surface exponents. Results to order ² are given which disprove the conjecture η_; = ν⁻¹ by Bray and Moore.     

Restrictions on realistic superstring models from renormalization group equations

The renormalization group equations for supersymmetric models are re-examined considering the normalization factor of the hypercharge as a free parameter. This possibility has recently been raised in the context of superstring theories. Very important (and simple) restrictions for the phenomenological viability of the models are obtained. Several examples (some of them with great physical relevance) are given in order to illustrate how the analysis of an important class of SU(3)×SU(2)×U(1)_Ysuperstring constructions can be carried out. Many models can be ruled out in this way, which is welcome for the predictive power of the theory.