Noncommutative GUT inspired theories with U(1), SU(N) groups and their renormalisability

TL;DR

This paper demonstrates that GUT-inspired noncommutative gauge theories with U(1) and SU(N) groups, involving fermionic matter and no scalars, are one-loop multiplicatively renormalisable at first order in the noncommutativity parameter, under specific conditions.

Contribution

It provides the first analysis of renormalisability for noncommutative GUT-inspired theories with fermionic matter and no scalars, identifying conditions for divergence absorption.

Findings

Divergences can be absorbed via multiplicative renormalisation with specific trace choices.

Anomaly cancellation requires non-chiral matter content in SU(N) cases.

Theories are on-shell one-loop multiplicatively renormalisable at order theta.

Abstract

We consider the GUT compatible formulation of noncommutative QED, as well as noncommutative SU(N) GUTs, for N>2, with no scalars but with fermionic matter in an arbitrary, anomaly-free representation, in the enveloping algebra approach. We compute, to first order in the noncommutativity parameters theta, the UV divergent part of the one-loop background-field effective action involving at most two fermion fields and an arbitrary number of gauge fields. It turns out that, for special choices of the ambiguous trace over the gauge degrees of freedom, for which the O(theta) triple gauge-field interactions vanish, the divergences can be absorbed by means of multiplicative renormalisations and the inclusion of theta-dependent counterterms that vanish on-shell and are thus unphysical. For this to happen in the SU(N), N>2 case, the representations of the matter fields must have a common second Casimir; anomaly cancellation then requires the ordinary (commutative) matter content to be non-chiral. Together with the vanishing of the divergences of fermionic four point functions, this shows that GUT inspired theories with U(1) and SU(N), N>2 gauge groups and ordinary vector matter content not only have a renormalisable matter sector, but are on-shell one-loop multiplicatively renormalisable at order one in theta.