Wolfhart Zimmermann

Reduction in the Number of Coupling Parameters

A method is developed for reducing the formulation of massless models with several independent couplings to a description in terms of a single coupling parameter. The original as well as the reduced system are supposed to be renormalizable and invariant under the renormalization group. For most models there are, if any, only a finite number of reductions possible including those which correspond to symmetries of the system. The reduction method leads to a consistent formulation of the reduced model in any order of perturbation theory even in cases where it is difficult to establish a symmetry in higher orders. An example where no symmetry seems to be involved is the interaction of a spinor field with a pseudoscalar field. For this model the reduction method determines the quartic coupling constant uniquely as a function of the Yukawa coupling constant. The Wess-Zumino model is an exceptional case for which the reduction method admits an infinite number of solutions besides the supersymmetric one.

Relation between effective couplings for asymptotically free models

For asymptotically free models with two independent couplings asymptotic expansions are constructed which express one effective coupling in terms of the other. The expansions involve powers (including fractional or irrational exponents) and logarithms. All orders of theβ-functions are taken into account. The expansions found are complete in the sense that they represent solutions (exact to any order) which generalize all the solutions obtained with theβ-functions approximated to second order. It is shown that higher orders are relevant since it is not possible in general to reparametrize the system such that theβ-functions become polynomials of the coupling parameters. The simplifications in case of supersymmetric models are discussed.

Higgs and top mass from reduction of couplings

Abstract We reduce the couplings in the standard model with one Higgs doublet and n generations and obtain for three generations 61 GeV and 81 GeV for the mass of the Higgs particle and the top quark respectively. The error is estimated to be about 10–15%.

New results in the reduction of the standard model

Abstract The reduction of couplings in the standard model is ameliorated and updated: a new method of calculating the electroweak corrections is developed, two-loop effects are estimated and recent experimental values of the strong coupling and the Weinberg angle are incorporated in the explicit calculation of the mass relations.

Cancellation of divergencies and reduction of couplings in the standard model

Abstract Cancellation of quadratical divergencies in the Higgs propagator is shown to be compatible with renormalization group invariance and the reduction of couplings. Requiring both - cancellation and reduction - fixes Higgs and top mass as a function of the strong coupling and the Weinberg angle.

Construction of gauge theories with a single coupling parameter for Yang-Mills and matter fields

Abstract Massless gauge theories are considered involving matter fields coupled invariantly to a Yang-Mills field. In general renormalization induces additional couplings with independent coefficients λ j . Consistent descriptions depending only upon the gauge coupling g may be constructed by determining the functions λ j ( g ) which are independent of the normalization point and allow for an asymptotic power series in g . Two examples with four couplings are shown to result in the pure N =2, N =4 supersymmetric Yang-Mills theories. In addition, one obtains some non-supersymmetric models. Stability properties are discussed using Lyapunovs theory.

Renormalization group equations with vanishing lowest order of the primary β-function

Abstract The evolution equations for the effective couplings are solved for massless models with two coupling and vanishing lowest order contribution of the primary β-function. The stability and asymptotic freedom of the solutions are analyzed. A specialization to supersymmetric theories with nonvanishing Yukawa couplings of superfields shows instability for asymptotically free models and for possible cases with identically vanishing β-functions.

Infrared Convergence of Feynman Integrals for the Massless a**4 Model

For the masslessA4-model it is proved that renormalization can be formulated such that each Feynman diagram yields an ultraviolet and infrared convergent contribution to the Greens functions.

The renormalization group of the model of

For the model ofA4-interaction the postulates of the renormalization group are stated within the abstract approach of quantum field theory. In the massive case these postulates follow if an on-shell formulation of the model is assumed to exist. For the massless model the postulates of the renormalization group imply that the propagator has a pole at momentum zero. Consequently there is no dynamic mass generation and the propagator is normalizable on the mass shell. It is shown that theS-matrix elements scale with canonical dimensions. A general method of rescaling parameter values is developed which takes into account the possibility of propagator zeros and stationary points of the effective coupling.

A^{4}

Abstract In the limit of vanishing electroweak interactions an asymptotic requirement is proposed which uniquely determines electroweak corrections to reduction solutions in the standard model. The precise form of these boundary conditions is worked out for the one- and two-loop approximation of QCD extended by the top and Higgs interactions. It is shown that the conditions are satisfied in Kubos two-loop computation of the top and Higgs mass.