No abstract

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 the ^-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 the ^-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 the /^-functions become polynomials of the coupling parameters. The simplifications in case of supersymmetric models are discussed.

For gauge theories with confinement, the analytic structure of amplitudes is explored. It is shown that the analytic properties of physical amplitudes are the same as those obtained on the basis of an effective theory involving only the composite, physical fields. The corresponding proofs of dispersion relations remain valid. Anomalous thresholds are considered. They are related to the composite structure of particles. It is shown, that there are no such thresholds in physical amplitudes which are associated with confined constituents, like quarks and gluons in QCD. Unphysical amplitudes are considered briefly, using propagator functions as an example. For general, covariant, linear gauges, it is shown that these functions must have singularities at finite, real points, which may be associated with confined states.