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The construction of non-Abelian Euclidean Yang-Mills theories in dimensionfour, as scaling limits of lattice Yang-Mills theories or otherwise, is acentral open question of mathematical physics. This paper takes the followingsmall step towards this goal. In any dimension $d\ge 2$, we construct a scalinglimit of $\mathrm{SU}(2)$ lattice Yang-Mills theory coupled to a Higgs fieldtransforming in the fundamental representation of $\mathrm{SU}(2)$. The scalinglimit is obtained by sending the gauge coupling constant $g$ to zero and theHiggs length $\alpha$ to infinity slower than $g^{-1}$, but faster than$g^{-1+1/49d}$. After unitary gauge fixing and taking the lattice scaling tozero as a constant multiple of $\alpha g$, a stereographic projection of thegauge 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 latticeYang-Mills theory in a dimension higher than two, as well as the first rigorousproof of mass generation by the Higgs mechanism in such a theory. Analogousresults are proved for $\mathrm{U}(1)$ theory as well. The question ofconstructing a non-Gaussian scaling limit remains open.

A scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills-Higgs theory