{"paper":{"title":"Modular symmetry at level 6 and a new route towards finite modular groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"hep-ph","authors_text":"Cai-Chang Li, Gui-Jun Ding, Xiang-Gan Liu","submitted_at":"2021-08-04T17:14:13Z","abstract_excerpt":"We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as $\\Gamma(N')/\\Gamma(N\")$, and the modular group $SL(2,\\mathbb{Z})$ is extended to a principal congruence subgroup $\\Gamma(N')$. The original modular invariant theory is reproduced when $N'=1$. We perform a comprehensive study of $\\Gamma'_6$ modular symmetry corresponding to $N'=1$ and $N\"=6$, five types of models for lepton masses and mixing with $\\Gamma'_6$ modular symmetry are discussed and some example models are studied numerically. The case of $N'=2$ and $N\"=6$ is considered, the fi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.02181","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.02181/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}