{"id":"https://openalex.org/W7212145369","doi":"https://doi.org/10.48550/arxiv.2609.09262","title":"Level-Four Modular Symmetry Selected by the Harmonic Pattern of Quark and Lepton Masses","display_name":"Level-Four Modular Symmetry Selected by the Harmonic Pattern of Quark and Lepton Masses","publication_year":2026,"publication_date":"2026-09-08","ids":{"openalex":"https://openalex.org/W7212145369","doi":"https://doi.org/10.48550/arxiv.2609.09262"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.09262","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.09262","pdf_url":"https://arxiv.org/pdf/2609.09262","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.09262","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5061874111","display_name":"Vernon D. Barger","orcid":"https://orcid.org/0000-0003-1010-2150","observed_orcids":["https://orcid.org/0000-0003-1010-2150"]},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Barger, Vernon","raw_affiliation_strings":["Department of Physics , University of Wisconsin-Madison , Madison , WI 53706 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , University of Wisconsin-Madison , Madison , WI 53706 , USA","institution_ids":["https://openalex.org/I135310074"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5061874111"],"corresponding_institution_ids":["https://openalex.org/I135310074"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10048","display_name":"Particle physics theoretical and experimental studies","score":0.8817999958992004,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10048","display_name":"Particle physics theoretical and experimental studies","score":0.8817999958992004,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.11289999634027481,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10224","display_name":"Quantum Chromodynamics and Particle Interactions","score":0.0017999999690800905,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quark-mass-ratios","display_name":"quark mass ratios","score":0.7630000114440918},{"id":"https://openalex.org/keywords/charged-lepton-mass-ratios","display_name":"charged lepton mass ratios","score":0.8980000019073486},{"id":"https://openalex.org/keywords/modular-symmetry","display_name":"modular symmetry","score":0.8629999756813049},{"id":"https://openalex.org/keywords/theta-constants","display_name":"theta constants","score":0.628000020980835},{"id":"https://openalex.org/keywords/ckm-matrix","display_name":"CKM matrix","score":0.7680000066757202},{"id":"https://openalex.org/keywords/unitarity-triangle","display_name":"unitarity triangle","score":0.9279999732971191},{"id":"https://openalex.org/keywords/jarlskog-invariant","display_name":"Jarlskog invariant","score":0.8050000071525574},{"id":"https://openalex.org/keywords/neutrino-mass","display_name":"neutrino mass","score":0.6589999794960022},{"id":"https://openalex.org/keywords/neutrino-mixing-angles","display_name":"neutrino mixing angles","score":0.7210000157356262},{"id":"https://openalex.org/keywords/theta13","display_name":"theta13","score":0.5759999752044678},{"id":"https://openalex.org/keywords/cp-violation","display_name":"CP violation","score":0.6039999723434448},{"id":"https://openalex.org/keywords/electroweak-scale","display_name":"electroweak scale","score":0.597000002861023}],"concepts":[{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.8808000087738037},{"id":"https://openalex.org/C109214941","wikidata":"https://www.wikidata.org/wiki/Q18334","display_name":"Particle physics","level":1,"score":0.7113999724388123},{"id":"https://openalex.org/C179203047","wikidata":"https://www.wikidata.org/wiki/Q82586","display_name":"Lepton","level":3,"score":0.5760999917984009},{"id":"https://openalex.org/C7602139","wikidata":"https://www.wikidata.org/wiki/Q6718","display_name":"Quark","level":2,"score":0.5318999886512756},{"id":"https://openalex.org/C130295795","wikidata":"https://www.wikidata.org/wiki/Q3550684","display_name":"Unitarity","level":2,"score":0.42559999227523804},{"id":"https://openalex.org/C17863212","wikidata":"https://www.wikidata.org/wiki/Q466416","display_name":"Electroweak interaction","level":2,"score":0.4212000072002411},{"id":"https://openalex.org/C2778761060","wikidata":"https://www.wikidata.org/wiki/Q6714234","display_name":"MAJORANA","level":3,"score":0.3756999969482422},{"id":"https://openalex.org/C181633103","wikidata":"https://www.wikidata.org/wiki/Q1834342","display_name":"Modular group","level":2,"score":0.31940001249313354},{"id":"https://openalex.org/C188082385","wikidata":"https://www.wikidata.org/wiki/Q73792","display_name":"Charge (physics)","level":2,"score":0.30570000410079956},{"id":"https://openalex.org/C41002478","wikidata":"https://www.wikidata.org/wiki/Q12794416","display_name":"CP violation","level":2,"score":0.29030001163482666},{"id":"https://openalex.org/C205334942","wikidata":"https://www.wikidata.org/wiki/Q3151","display_name":"Muon","level":2,"score":0.28060001134872437},{"id":"https://openalex.org/C135585149","wikidata":"https://www.wikidata.org/wiki/Q339704","display_name":"Mass matrix","level":3,"score":0.2786000072956085},{"id":"https://openalex.org/C15580451","wikidata":"https://www.wikidata.org/wiki/Q3077640","display_name":"Mass formula","level":2,"score":0.2782999873161316},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.2775000035762787},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.2709999978542328},{"id":"https://openalex.org/C2776394503","wikidata":"https://www.wikidata.org/wiki/Q5282063","display_name":"Discrete symmetry","level":3,"score":0.26980000734329224},{"id":"https://openalex.org/C181830111","wikidata":"https://www.wikidata.org/wiki/Q214850","display_name":"Gauge theory","level":2,"score":0.2587999999523163},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.25040000677108765}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.09262","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.09262","pdf_url":"https://arxiv.org/pdf/2609.09262","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.09262","doi":"https://doi.org/10.48550/arxiv.2609.09262","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.09262","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.09262","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.09262","pdf_url":"https://arxiv.org/pdf/2609.09262","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G2066175637","display_name":null,"funder_award_id":"DE-SC0017647","funder_id":"https://openalex.org/F4320337511","funder_display_name":"High Energy Physics"},{"id":"https://openalex.org/G8718190532","display_name":"HIGH ENERGY PHYSICS RESEARCH: EXPERIMENTAL & THEORETICAL","funder_award_id":"DESC0017647","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"}],"funders":[{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320332359","display_name":"Office of Science","ror":"https://ror.org/00mmn6b08"},{"id":"https://openalex.org/F4320337511","display_name":"High Energy Physics","ror":"https://ror.org/035m6g344"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212145369.pdf","grobid_xml":"https://content.openalex.org/works/W7212145369.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3203147687","https://openalex.org/W4323044129","https://openalex.org/W3122359300","https://openalex.org/W4383556504","https://openalex.org/W4380374861","https://openalex.org/W2020009491","https://openalex.org/W4385951814","https://openalex.org/W3036043435","https://openalex.org/W7095002727","https://openalex.org/W4384665548"],"abstract_inverted_index":{"The":[0,13,40,93,169,184,229],"mass":[1,95],"ratios":[2,96],"of":[3,64],"the":[4,30,35,72,77,80,87,90,106,113,120,127,131,140,152,162,177,188,201,206,234,240],"quarks":[5],"and":[6,29,37,55,71,86,112,139,145,159,180,205,220],"charged":[7],"leptons":[8],"follow":[9],"a":[10,19,24,47,62,99,210,216],"simple":[11],"pattern.":[12],"logarithmic":[14],"generation":[15],"steps":[16],"come":[17],"in":[18,79,89,119],"two":[20],"to":[21,26,227,233],"one":[22,146],"ratio,":[23],"note":[25],"its":[27],"octave,":[28],"ordering":[31],"is":[32,50,60],"mirrored":[33],"between":[34],"up-down":[36],"down-lepton":[38],"comparisons.":[39],"pattern,":[41],"read":[42],"as":[43,110],"charge":[44],"counting":[45],"on":[46,98,126,200],"$Z_N$":[48],"clock,":[49],"realized":[51],"at":[52,56,82,102,198,225],"level":[53,58,69],"four":[54,70],"no":[57,166],"that":[59],"not":[61],"multiple":[63],"four,":[65],"which":[66,104,196],"motivates":[67],"modular":[68],"finite":[73],"group":[74],"$S_4$,":[75],"with":[76,165,173,213,238],"generations":[78],"triplet":[81],"$T$":[83],"charges":[84],"$(3,1,0)$":[85],"mirror":[88],"sign":[91],"singlet.":[92],"charged-lepton":[94],"lie":[97],"theta-dressed":[100],"lattice":[101],"$0.02\\%$,":[103],"sets":[105],"base":[107],"$\\varepsilon=0.18664$,":[108],"taken":[109],"$14/75$,":[111],"harmonic":[114,207],"modulus":[115],"$\u03c4=1.0685i$.":[116],"Five":[117],"postulates":[118],"weight":[121],"one-half":[122],"theta":[123,192],"constants,":[124],"identified":[125],"same":[128,185],"data,":[129,204],"reproduce":[130],"six":[132],"quark":[133],"masses":[134,138,223],"from":[135],"three":[136],"lepton":[137],"electroweak":[141],"scale.":[142],"Two":[143],"more":[144],"empirical":[147],"relation":[148],"for":[149],"$|V_{td}|$":[150],"close":[151],"CKM":[153],"matrix,":[154],"giving":[155],"$|V_{us}|=\\sqrt{m_d/m_s}=0.2255$,":[156],"$|V_{cb}|=0.04197$,":[157],"$|V_{ub}|=0.003721$,":[158],"$\u03b4=1.138$":[160],"against":[161,176,182],"measured":[163,178],"$1.139\\pm0.023$,":[164],"continuous":[167],"parameter.":[168],"unitarity":[170],"triangle":[171],"follows,":[172],"apex":[174],"$(0.161,0.348)$":[175],"$(0.161\\pm0.010,0.347\\pm0.010)$,":[179],"$J=3.11\\times10^{-5}$":[181],"$(3.09\\pm0.07)\\times10^{-5}$.":[183],"structure":[186],"carries":[187],"neutrinos.":[189],"A":[190],"single":[191],"insertion":[193],"gives":[194],"$\\sin\u03b8_{13}=\u03b8_2(\u03c4)m_2/m_3$,":[195],"holds":[197],"$0.7\u03c3$":[199],"207-day":[202],"JUNO":[203],"condition":[208],"completes":[209],"normal-ordered":[211],"spectrum":[212],"$m_1=0.25$~meV,":[214],"$\u03a3m_\u03bd=0.0589$~eV,":[215],"CP-conserving":[217],"phase":[218],"$\u03b4_{CP}=\u03c0$,":[219],"effective":[221],"Majorana":[222],"quantized":[224],"$1.3$":[226],"$3.9$~meV.":[228],"ultraviolet":[230],"completion":[231],"belongs":[232],"magnetized":[235],"torus":[236],"class,":[237],"Pati--Salam":[239],"natural":[241],"gauge":[242],"embedding.":[243]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-11T00:00:00"}
