  The construction of non-Abelian Euclidean Yang-Mills theories in dimension
four, as scaling limits of lattice Yang-Mills theories or otherwise, is a
central open question of mathematical physics. This paper takes the following
small step towards this goal. In any dimension $d\ge 2$, we construct a scaling
limit of $\mathrm{SU}(2)$ lattice Yang-Mills theory coupled to a Higgs field
transforming in the fundamental representation of $\mathrm{SU}(2)$. The scaling
limit is obtained by sending the gauge coupling constant $g$ to zero and the
Higgs length $\alpha$ to infinity slower than $g^{-1}$, but faster than
$g^{-1+1/49d}$. After unitary gauge fixing and taking the lattice scaling to
zero as a constant multiple of $\alpha g$, a stereographic projection of the
gauge field is shown to converge to a scale-invariant massive Gaussian field.
This gives the first construction of a scaling limit of a non-Abelian lattice
Yang-Mills theory in a dimension higher than two, as well as the first rigorous
proof of mass generation by the Higgs mechanism in such a theory. Analogous
results are proved for $\mathrm{U}(1)$ theory as well. The question of
constructing a non-Gaussian scaling limit remains open.
