The Kahler potential for a gauge theory is given by

\begin{equation}

\Phi ^\dagger e ^{ V} \Phi

\end{equation}

where $ V $ is the vector superfield corresponding to the gauge group.

This doesn't seem to extend as smoothly to multiple gauge groups as I would have thought. Suppose we have two gauge superfields, $ V $ and $ V ' $. Since they live in different spaces I would think we would just have,

\begin{equation}

\Phi ^\dagger e ^{ V} e ^{ V ' } \Phi

\end{equation}

If we work in the Wess-Zumino gauge,

\begin{equation}

V = ( \theta \sigma ^\mu \bar{\theta} ) V _\mu ^a T ^a

\end{equation}

then we have,

\begin{equation}

{\cal L} \supset \int \,d^4\theta \Phi ^\dagger \left( 1 + V + \frac{1}{2} V ^2 \right) \left( 1 + V ' + \frac{1}{2}V ^{ \prime 2} \right) \Phi

\end{equation}

Since the generators in $ V $ and $ V ' $ act on different spaces typically the exponentials are independent of one another as you'd expect.

However, there is one term that I wouldn't have expected which couples the different gauge fields,

\begin{equation}

{\cal L} \supset \int \,d^4\theta \Phi ^\dagger V V' \Phi = \phi ^\dagger t _a t _b ' V _\mu ^a V ^{ \prime \, \mu b } \phi

\end{equation}

where $ t _a , t _b ' $ are the generators and $ \phi $ are the scalar components of the chiral superfields.

I can't think of such an analogue of such terms in non-supersymmetric theories. Are these special to supersymmetric gauge theories or do they not exist for some reason?