The U(1)s in the Finite N Limit of Orbifold Field Theories

TL;DR

This paper investigates the behavior of U(1) gauge symmetries in orbifolded N=4 super Yang-Mills theories at finite N, revealing that while non-abelian couplings remain conformal, abelian U(1) couplings do not, affecting the theories' conformality.

Contribution

It demonstrates that the beta functions of non-abelian SU(N) gauge couplings vanish up to three loops at finite N when considering U(1) factors, highlighting the non-conformality of abelian U(1) couplings.

Findings

Non-abelian SU(N) beta functions vanish up to three loops at finite N.

Abelian U(1) gauge couplings have non-zero beta functions.

Theories are not conformal at finite N due to U(1) running.

Abstract

We study theories generated by orbifolding the {\cal N}=4 super conformal U(N) Yang Mills theory with finite N, focusing on the r\^ole of the remnant U(1) gauge symmetries of the orbifold process. It is well known that the one loop beta functions of the non abelian SU(N) gauge couplings vanish in these theories. It is also known that in the large N limit the beta functions vanish to all order in perturbation theory. We show that the beta functions of the non abelian SU(N) gauge couplings vanish to two and three loop order even for finite N. This is the result of taking the abelian U(1) of U(N)=SU(N)xU(1) into account. However, the abelian U(1) gauge couplings have a non vanishing beta function. Hence, those theories are not conformal for finite N. We analyze the renormalization group flow of the orbifold theories, discuss the suppression of the cosmological constant and tackle the hierarchy problem in the non supersymmetric models.