A scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills-Higgs theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918510495531008 |
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| author | Chatterjee, Sourav |
| author_facet | Chatterjee, Sourav |
| contents | 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 (under the degenerate potential) transforming in the fundamental representation of $\mathrm{SU}(2)$. After unitary gauge fixing and taking the lattice spacing $\varepsilon\to 0$, and simultaneously taking the gauge coupling constant $g\to 0$ and the Higgs length $α\to \infty$ in such a manner that $αg$ is always equal to $c\varepsilon$ for some fixed $c$ and $g= O(\varepsilon^{50d})$, 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10507 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills-Higgs theory Chatterjee, Sourav Probability High Energy Physics - Theory Mathematical Physics 70S15, 81T13, 81T25, 82B20 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 (under the degenerate potential) transforming in the fundamental representation of $\mathrm{SU}(2)$. After unitary gauge fixing and taking the lattice spacing $\varepsilon\to 0$, and simultaneously taking the gauge coupling constant $g\to 0$ and the Higgs length $α\to \infty$ in such a manner that $αg$ is always equal to $c\varepsilon$ for some fixed $c$ and $g= O(\varepsilon^{50d})$, 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 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. |
| title | A scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills-Higgs theory |
| topic | Probability High Energy Physics - Theory Mathematical Physics 70S15, 81T13, 81T25, 82B20 |
| url | https://arxiv.org/abs/2401.10507 |