Abstract

The construction of nonabelian 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 $\mathrm{SU}<!--FUNCTION\; APPLICATION-->(2)$ lattice Yang–Mills theory coupled to a Higgs field (under the degenerate potential) transforming in the fundamental representation of $\mathrm{SU}<!--FUNCTION\; APPLICATION-->(2)$. After unitary gauge fixing and taking the lattice spacing $𝜀\to 0$, and simultaneously taking the gauge coupling constant $g\to 0$ and the Higgs length $\alpha \to \infty$ in such a manner that $\alpha g$ is always equal to $c𝜀$ for some fixed $c$ and $g=O({𝜀}^{50d})$, a stereographic projection of the gauge field is shown to converge to a massive Gaussian field. This gives the first construction of a scaling limit of a nonabelian 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 $U<!--FUNCTION\; APPLICATION-->(1)$ theory as well. The question of constructing a non-Gaussian scaling limit remains open.

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