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

We consider a gauge theory on the 5D $\kappa$-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 Becchi-Rouet-Stora-Tyutin gauge-fixing with a Lorentz-type gauge, 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 nonvanishing tadpole for the time component of the gauge potential, while there is no nonvanishing 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. Possible consequences regarding the symmetries of the gauge model and the fate of the tadpole in other gauges of noncovariant type are discussed.