A scaling limit of SU(2) lattice Yang–Mills–Higgstheory

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≥ 2, we construct a scaling limit of SU(2) lattice Yang-Mills theory coupled to a Higgs field (under the degenerate potential) transforming in the fundamental representation of SU(2). After unitary gauge fixing and taking the lattice spacing ε→ 0, and simultaneously taking the gauge coupling constant g→ 0 and the Higgs length α→ ∞ in such a manner that $αg$ is always equal to cε for some fixed c and g= O(ε⁵⁰ᵈ), 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 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 U(1) theory as well. The question of constructing a non-Gaussian scaling limit remains open.