Direct and Indirect Loop Equations in Lattice Yang-Mills Theory
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911388594601984 |
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| author | Liu, Xizhe Yang, Gang |
| author_facet | Liu, Xizhe Yang, Gang |
| contents | The dynamics of Wilson loops are governed by an infinite set of Schwinger-Dyson equations and trace relations. In the context of the lattice positivity bootstrap, a central challenge is determining a dynamically independent basis of these operators within a truncated space. We present a systematic framework to address this problem, utilizing a geometric plaquette-cut and subloop-cut strategy to efficiently generate all (local) direct equations. Furthermore, we identify and analyze "indirect equations", which arise from the elimination of higher-length intermediate loops. We elucidate the origin of these subtle relations and propose a vertex-filtering strategy to construct them. Applying the above framework to SU(2) lattice Yang-Mills theory, we provide explicit counts of independent canonical loops and equations in 2, 3, and 4 dimensions, along with a statistical analysis of their asymptotic growth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_04316 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Direct and Indirect Loop Equations in Lattice Yang-Mills Theory Liu, Xizhe Yang, Gang High Energy Physics - Theory High Energy Physics - Lattice Mathematical Physics The dynamics of Wilson loops are governed by an infinite set of Schwinger-Dyson equations and trace relations. In the context of the lattice positivity bootstrap, a central challenge is determining a dynamically independent basis of these operators within a truncated space. We present a systematic framework to address this problem, utilizing a geometric plaquette-cut and subloop-cut strategy to efficiently generate all (local) direct equations. Furthermore, we identify and analyze "indirect equations", which arise from the elimination of higher-length intermediate loops. We elucidate the origin of these subtle relations and propose a vertex-filtering strategy to construct them. Applying the above framework to SU(2) lattice Yang-Mills theory, we provide explicit counts of independent canonical loops and equations in 2, 3, and 4 dimensions, along with a statistical analysis of their asymptotic growth. |
| title | Direct and Indirect Loop Equations in Lattice Yang-Mills Theory |
| topic | High Energy Physics - Theory High Energy Physics - Lattice Mathematical Physics |
| url | https://arxiv.org/abs/2601.04316 |