# Kahler potential with multiple gauge groups

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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?

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