Quantum stability of gauge theories on $\kappa$-Minkowski space

We consider a gauge theory on the 5-d $\kappa$-Minkowski which can be viewed as the noncommutative analog of a $U(1)$ gauge theory. We show that the Hermiticity condition obeyed by the gauge potential $A_\mu$ is necessarily twisted. Performing a BRST gauge-fixing, we carry out a first exploration of the one loop quantum properties of this gauge theory. We find that the gauge-fixed theory gives rise to a non vanishing tadpole for the time component of the gauge potential, while there is no non-vanishing tadpole 1-point function for the spatial components of $A_\mu$. This signals that the classical vacuum of the theory is not stable against quantum fluctuations. This result is briefly discussed.