Noncommutative GUT inspired theories with U(1) andSU(N)groups and their renormalizability

Abstract

We consider the grand unified theory (GUT)-compatible formulation of noncommutative QED, as well as noncommutative $\mathrm{SU}(N)$ 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 ${\ensuremath{\theta}}^{\ensuremath{\mu}\ensuremath{\nu}}$, 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 $O(\ensuremath{\theta})$ triple gauge-field interactions vanish, the divergences can be absorbed by means of multiplicative renormalizations and the inclusion of $\ensuremath{\theta}$-dependent counterterms that vanish on shell and are thus unphysical. For this to happen in the $\mathrm{SU}(N)$, $N>2$ case, the representations of the matter fields must have a common second Casimir; anomaly cancellation then requires the ordinary (commutative) matter content to be nonchiral. Together with the vanishing of the divergences of fermionic four point functions, this shows that GUT-inspired theories with U(1) and $\mathrm{SU}(N)$, $N>2$ gauge groups and ordinary vector matter content not only have a renormalizable matter sector, but are on-shell one-loop multiplicatively renormalizable at order one in $\ensuremath{\theta}$.