{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PRUMQALPEUP3E4SKDXZ7EJPKKI","short_pith_number":"pith:PRUMQALP","canonical_record":{"source":{"id":"1507.03814","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-ph","submitted_at":"2015-07-14T11:35:06Z","cross_cats_sorted":["hep-th"],"title_canon_sha256":"df1b0fd5867f3885f27886b6937c12eb8fac4ef9c28178a43ec9a6cdf3e5598c","abstract_canon_sha256":"e70703e057729d58be913435deddbc96943a02a9bbb2944ee77ea8131c831f4f"},"schema_version":"1.0"},"canonical_sha256":"7c68c8016f251fb2724a1df3f225ea5234c992716bd6ede3a8051404d97ae3ea","source":{"kind":"arxiv","id":"1507.03814","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.03814","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"arxiv_version","alias_value":"1507.03814v2","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03814","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"pith_short_12","alias_value":"PRUMQALPEUP3","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PRUMQALPEUP3E4SK","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PRUMQALP","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PRUMQALPEUP3E4SKDXZ7EJPKKI","target":"record","payload":{"canonical_record":{"source":{"id":"1507.03814","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-ph","submitted_at":"2015-07-14T11:35:06Z","cross_cats_sorted":["hep-th"],"title_canon_sha256":"df1b0fd5867f3885f27886b6937c12eb8fac4ef9c28178a43ec9a6cdf3e5598c","abstract_canon_sha256":"e70703e057729d58be913435deddbc96943a02a9bbb2944ee77ea8131c831f4f"},"schema_version":"1.0"},"canonical_sha256":"7c68c8016f251fb2724a1df3f225ea5234c992716bd6ede3a8051404d97ae3ea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:13.492113Z","signature_b64":"K8SrmOgVJLFP+yLVeRZFksGPrRoe045g6i2AzkIj9l5tnGwuE/hDEMe7nXeG+55+s2e/Yh8kh8uZrvNML4tSBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c68c8016f251fb2724a1df3f225ea5234c992716bd6ede3a8051404d97ae3ea","last_reissued_at":"2026-05-18T01:31:13.491362Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:13.491362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.03814","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A40SHjjPvkpjhphkzQeRARLytu7d+fTDWkP1AeDawAC2NOHe9otfaI64Gv2plm259OZc5mWHMHJ0r5dZg9lTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T09:55:56.093283Z"},"content_sha256":"c6f7e6bf4c61ed9c2bbd01736fceb5e2fa261a3f4293c92c58d07e66fc002f5e","schema_version":"1.0","event_id":"sha256:c6f7e6bf4c61ed9c2bbd01736fceb5e2fa261a3f4293c92c58d07e66fc002f5e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PRUMQALPEUP3E4SKDXZ7EJPKKI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Decomposing color structure into multiplet bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th"],"primary_cat":"hep-ph","authors_text":"Johan Thoren, Malin Sjodahl","submitted_at":"2015-07-14T11:35:06Z","abstract_excerpt":"We illustrate how QCD color structure elegantly can be decomposed into orthogonal multiplet bases corresponding to irreducible representations of SU(Nc) with the aid of Wigner 3j and 6j coefficients. We also show how to calculate the relevant 3j and 6j coefficients using multiplet bases and birdtrack techniques and argue that only a relatively small number of Wigner 3j and 6j coefficients are required. For up to six gluons plus quark-antiquark pairs we explicitly calculate all 6j coefficients required for up to NLO calculations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03814","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BimwFIycMqjrWe05QOR7NDYnNxB+4kvgcppNklUBq/GhgOnICyPmahn5I1k/c1ZBkAvrJXdOMG9hZiNf2b91Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T09:55:56.094224Z"},"content_sha256":"1edc9f3afa493c430c76761deb9f0faa7a9b8a331d6fd1e2f238807855521a07","schema_version":"1.0","event_id":"sha256:1edc9f3afa493c430c76761deb9f0faa7a9b8a331d6fd1e2f238807855521a07"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PRUMQALPEUP3E4SKDXZ7EJPKKI/bundle.json","state_url":"https://pith.science/pith/PRUMQALPEUP3E4SKDXZ7EJPKKI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PRUMQALPEUP3E4SKDXZ7EJPKKI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T09:55:56Z","links":{"resolver":"https://pith.science/pith/PRUMQALPEUP3E4SKDXZ7EJPKKI","bundle":"https://pith.science/pith/PRUMQALPEUP3E4SKDXZ7EJPKKI/bundle.json","state":"https://pith.science/pith/PRUMQALPEUP3E4SKDXZ7EJPKKI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PRUMQALPEUP3E4SKDXZ7EJPKKI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PRUMQALPEUP3E4SKDXZ7EJPKKI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e70703e057729d58be913435deddbc96943a02a9bbb2944ee77ea8131c831f4f","cross_cats_sorted":["hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-ph","submitted_at":"2015-07-14T11:35:06Z","title_canon_sha256":"df1b0fd5867f3885f27886b6937c12eb8fac4ef9c28178a43ec9a6cdf3e5598c"},"schema_version":"1.0","source":{"id":"1507.03814","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.03814","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"arxiv_version","alias_value":"1507.03814v2","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03814","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"pith_short_12","alias_value":"PRUMQALPEUP3","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PRUMQALPEUP3E4SK","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PRUMQALP","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:1edc9f3afa493c430c76761deb9f0faa7a9b8a331d6fd1e2f238807855521a07","target":"graph","created_at":"2026-05-18T01:31:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We illustrate how QCD color structure elegantly can be decomposed into orthogonal multiplet bases corresponding to irreducible representations of SU(Nc) with the aid of Wigner 3j and 6j coefficients. We also show how to calculate the relevant 3j and 6j coefficients using multiplet bases and birdtrack techniques and argue that only a relatively small number of Wigner 3j and 6j coefficients are required. For up to six gluons plus quark-antiquark pairs we explicitly calculate all 6j coefficients required for up to NLO calculations.","authors_text":"Johan Thoren, Malin Sjodahl","cross_cats":["hep-th"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-ph","submitted_at":"2015-07-14T11:35:06Z","title":"Decomposing color structure into multiplet bases"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03814","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c6f7e6bf4c61ed9c2bbd01736fceb5e2fa261a3f4293c92c58d07e66fc002f5e","target":"record","created_at":"2026-05-18T01:31:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e70703e057729d58be913435deddbc96943a02a9bbb2944ee77ea8131c831f4f","cross_cats_sorted":["hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-ph","submitted_at":"2015-07-14T11:35:06Z","title_canon_sha256":"df1b0fd5867f3885f27886b6937c12eb8fac4ef9c28178a43ec9a6cdf3e5598c"},"schema_version":"1.0","source":{"id":"1507.03814","kind":"arxiv","version":2}},"canonical_sha256":"7c68c8016f251fb2724a1df3f225ea5234c992716bd6ede3a8051404d97ae3ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c68c8016f251fb2724a1df3f225ea5234c992716bd6ede3a8051404d97ae3ea","first_computed_at":"2026-05-18T01:31:13.491362Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:13.491362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K8SrmOgVJLFP+yLVeRZFksGPrRoe045g6i2AzkIj9l5tnGwuE/hDEMe7nXeG+55+s2e/Yh8kh8uZrvNML4tSBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:13.492113Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.03814","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6f7e6bf4c61ed9c2bbd01736fceb5e2fa261a3f4293c92c58d07e66fc002f5e","sha256:1edc9f3afa493c430c76761deb9f0faa7a9b8a331d6fd1e2f238807855521a07"],"state_sha256":"23aef7c5f76e68cd005b7412431e338b322530a484531eaa08cd8787786bcd48"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8fHyLieT50X0JV1BwyAyeqxJMotVtDrVK+c25gE/O1BCsWR15JxskO7IShblHzDyNELAkgJXyHE8WKXHFLagCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T09:55:56.098816Z","bundle_sha256":"2a02000b30424b309e1e27d95d4881de22b8ea65396773c12f00ea0957ad84c2"}}