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The final result for $\\Delta(J;D)$ in the (resummed) $\\epsilon$-expansion, valid when $J\\gg 1/\\epsilon \\gg 1$, turns out to be \\begin{equation*} \\Delta(J;D)=\\left[\\frac{2(D-1)}{3(D-2)}\\left(\\frac{9(D-2)\\pi}{5D}\\right)^{\\frac{D}{2(D-1)}}\\left[\\frac{5\\Gamma\\left(\\frac{D}{2}\\right)}{24\\pi^2}\\right]^{\\frac{1}{D-1}} \\epsilon^{\\frac{D-2}{2(D-1)}}\\right]\\times J^{\\frac{D}{D-1}}+O\\left(J^{\\frac{D-2}{D-1}}\\right)"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1909.01337","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2019-09-03T17:59:50Z","cross_cats_sorted":[],"title_canon_sha256":"d56908f5ccf230d3a459abb0b6fda703bcf35379b6f787ed0fafb2a399cfbde2","abstract_canon_sha256":"93e8e7b892a6cb94a46b32785e43bb7f4c311fa54098bff002d1493c7e7ffa45"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:01:17.972001Z","signature_b64":"Dkkd9YO/bOj8dLYJ+weG4FNgrd3ZOhbAzy5DvpoKCbbqOq0ZQFzZiCwt72R4q5odlFv2BizA451haLp6yN5ODw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4aa55cf3d8c214f3bebfcac7eb2e238345834c730629a7b15c0af950be5f88b6","last_reissued_at":"2026-07-05T00:01:17.971528Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:01:17.971528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Accessing Large Global Charge via the $\\epsilon$-Expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"Masataka Watanabe","submitted_at":"2019-09-03T17:59:50Z","abstract_excerpt":"We compute the lowest operator dimension $\\Delta(J;D)$ at large global charge $J$ in the $O(2)$ Wilson-Fisher model in $D=4-\\epsilon$ dimensions, to leading order in both $1/J$ and $\\epsilon$. The final result for $\\Delta(J;D)$ in the (resummed) $\\epsilon$-expansion, valid when $J\\gg 1/\\epsilon \\gg 1$, turns out to be \\begin{equation*} \\Delta(J;D)=\\left[\\frac{2(D-1)}{3(D-2)}\\left(\\frac{9(D-2)\\pi}{5D}\\right)^{\\frac{D}{2(D-1)}}\\left[\\frac{5\\Gamma\\left(\\frac{D}{2}\\right)}{24\\pi^2}\\right]^{\\frac{1}{D-1}} \\epsilon^{\\frac{D-2}{2(D-1)}}\\right]\\times J^{\\frac{D}{D-1}}+O\\left(J^{\\frac{D-2}{D-1}}\\right)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.01337","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/1909.01337/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1909.01337","created_at":"2026-07-05T00:01:17.971587+00:00"},{"alias_kind":"arxiv_version","alias_value":"1909.01337v1","created_at":"2026-07-05T00:01:17.971587+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.01337","created_at":"2026-07-05T00:01:17.971587+00:00"},{"alias_kind":"pith_short_12","alias_value":"JKSVZ46YYIKP","created_at":"2026-07-05T00:01:17.971587+00:00"},{"alias_kind":"pith_short_16","alias_value":"JKSVZ46YYIKPHPV7","created_at":"2026-07-05T00:01:17.971587+00:00"},{"alias_kind":"pith_short_8","alias_value":"JKSVZ46Y","created_at":"2026-07-05T00:01:17.971587+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2406.19441","citing_title":"Moduli Spaces in CFT: Large Charge Operators","ref_index":43,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN","json":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN.json","graph_json":"https://pith.science/api/pith-number/JKSVZ46YYIKPHPV7ZLD6WLRDQN/graph.json","events_json":"https://pith.science/api/pith-number/JKSVZ46YYIKPHPV7ZLD6WLRDQN/events.json","paper":"https://pith.science/paper/JKSVZ46Y"},"agent_actions":{"view_html":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN","download_json":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN.json","view_paper":"https://pith.science/paper/JKSVZ46Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1909.01337&json=true","fetch_graph":"https://pith.science/api/pith-number/JKSVZ46YYIKPHPV7ZLD6WLRDQN/graph.json","fetch_events":"https://pith.science/api/pith-number/JKSVZ46YYIKPHPV7ZLD6WLRDQN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN/action/storage_attestation","attest_author":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN/action/author_attestation","sign_citation":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN/action/citation_signature","submit_replication":"https://pith.science/pith/JKSVZ46YYIKPHPV7ZLD6WLRDQN/action/replication_record"}},"created_at":"2026-07-05T00:01:17.971587+00:00","updated_at":"2026-07-05T00:01:17.971587+00:00"}