  In addition to the familiar contribution from a holomorphic function $\FF$,
the K\"ahler potential of the scalars in the nonabelian $N=2$ vector multiplet
receives contributions from a real function $\HH$. We determine the latter at
the one-loop level, taking into account both supersymmetric matter and gauge
loops. The function $\HH$ characterizes the four-point coupling of the $N=2$
vector-multiplet vectors for constant values of their scalar superpartners. We
discuss the consequences of our results.
