A condition for the reduction of couplings in the $P = \frac{1}{3}Q$ supersymmetric theories

TL;DR

This paper establishes a condition linking anomalous dimensions and renormalization group invariance in certain supersymmetric theories, providing explicit verification at low orders and proposing a higher-order extension.

Contribution

It introduces a simple relation between anomalous dimensions that ensures the RG invariance of coupling ratios in $P=\frac{1}{3}Q$ supersymmetric theories, valid at all perturbation orders.

Findings

Explicit verification at one- and two-loop levels.

Proposed higher-order relation for planar supergraphs.

Reformulation of the anomalous dimension equation.

Abstract

We demonstrate that in the $P=\frac{1}{3}Q$ supersymmetric theories the renormalization group invariance of the ratio $\lambda^{ijk}/e$ (of the Yukawa couplings to the gauge coupling) is equivalent to a simple relation between the anomalous dimensions of the quantum gauge superfield, of the Faddeev--Popov ghosts, and of the matter superfields, which should be valid in each order of the perturbation theory. In the one- and two-loop approximations it is verified explicitly. Presumably, in higher orders this relation can be satisfied for the planar supergraphs under a certain renormalization prescription. Assuming that it is valid we rewrite the exact equation for the (corresponding contribution to the) anomalous dimension of the matter superfields in the theories under consideration in a different (but equivalent) form.