standard model - From $U(3)$ to $SU(3)times U(1)$ Color symmetry. There is a "gluon" photon-like?

Suppose that $U(3)$ was the gauge group. We can decompose this as $U(3)=U(1)\times SU(3)$, which implies that in addition to the $SU(3)$ that has eight generators corresponding to eight gluons, there would be an additional generator for $U(1)$.

Would that generator behave like a photon? (no self interaction and no color)