  In this Letter, we address the {long-range interaction} between kinks and
antikinks, as well as kinks and kinks, in $\varphi^{2n+4}$ field theories for
$n>1$. The kink-antikink interaction is generically attractive, while the
kink-kink interaction is generically repulsive. We find that the force of
interaction decays with the $(\frac{2n}{n-1})$th power of their separation, and
we identify the general prefactor for {\it arbitrary} $n$. Importantly, we test
the resulting mathematical prediction with detailed numerical simulations of
the dynamic field equation, and obtain good agreement between theory and
numerics for the cases of $n=2$ ($\varphi^8$ model), $n=3$ ($\varphi^{10}$
model) and $n=4$ ($\varphi^{12}$ model).
