  We prove that any scaling limit of a critical reflection positive Ising or
$\varphi^4$ model of effective dimension $d_{\text{eff}}$ at least four is
Gaussian. This extends the recent breakthrough work of Aizenman and
Duminil-Copin -- which demonstrates the corresponding result in the setup of
nearest-neighbour interactions in dimension four -- to the case of long-range
reflection positive interactions satisfying $d_{\text{eff}}=4$. The proof
relies on the random current representation which provides a geometric
interpretation of the deviation of the models' correlation functions from
Wick's law. When $d=4$, long-range interactions are handled with the derivation
of a criterion that relates the speed of decay of the interaction to two
different mechanisms that entail Gaussianity: interactions with a sufficiently
slow decay induce a faster decay at the level of the model's two-point
function, while sufficiently fast decaying interactions force a simpler
geometry on the currents which allows to extend nearest-neighbour arguments.
When $1\leq d\leq 3$ and $d_{\text{eff}}=4$, the phenomenology is different as
long-range effects play a prominent role.
