Reduction of Coupling Parameters and Duality

TL;DR

This paper discusses a general method for reducing multiple coupling parameters in theories to a single parameter using renormalization group invariance, with applications to supersymmetric gauge theories and dualities.

Contribution

It introduces a broad reduction method for coupling parameters, including solutions with non-integer powers, and applies it to supersymmetric gauge theories with explicit solutions.

Findings

Reduced theories often have specific symmetries or none at all.

Unique power series solutions relate Yukawa and gauge couplings.

Explicit coefficients depend on the number of colors and flavors.

Abstract

The general method of the reduction in the number of coupling parameters is discussed. Using renormalization group invariance, theories with several independent couplings are related to a set of theories with a single coupling parameter. The reduced theories may have particular symmetries, or they may not be related to any known symmetry. The method is more general than the imposition of invariance properties. Usually, there are only a few reduced theories with an asymptotic power series expansion corresponding to a renormalizable Lagrangian. There also exist `general' solutions containing non-integer powers and sometimes logarithmic factors. As an example for the use of the reduction method, the dual magnetic theories associated with certain supersymmetric gauge theories are discussed. They have a superpotential with a Yukawa coupling parameter. This parameter is expressed as a function of the gauge coupling. Given some standard conditions, a unique, isolated power series solution of the reduction equations is obtained. After reparametrization, the Yukawa coupling is proportional to the square of the gauge coupling parameter. The coefficient is given explicitly in terms of the numbers of colors and flavors. `General' solutions with non-integer powers are also discussed. A brief list is given of other applications of the reduction method.