{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:SXRTKLN2ETGQYSVZ6I4LVPXHRL","short_pith_number":"pith:SXRTKLN2","canonical_record":{"source":{"id":"2106.06967","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2021-06-13T11:42:16Z","cross_cats_sorted":[],"title_canon_sha256":"de8bc7d2e034bbed76577488e0a85e672b92c9e9fd15958631dd296f73c17fca","abstract_canon_sha256":"f457b59e376cc693a5e35143b7b3b8d69b4d3580fe146033560669c2faa97093"},"schema_version":"1.0"},"canonical_sha256":"95e3352dba24cd0c4ab9f238babee78ae190829022bb97bfcc7f0561fde8045f","source":{"kind":"arxiv","id":"2106.06967","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.06967","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"arxiv_version","alias_value":"2106.06967v2","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.06967","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"pith_short_12","alias_value":"SXRTKLN2ETGQ","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"pith_short_16","alias_value":"SXRTKLN2ETGQYSVZ","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"pith_short_8","alias_value":"SXRTKLN2","created_at":"2026-07-05T03:19:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:SXRTKLN2ETGQYSVZ6I4LVPXHRL","target":"record","payload":{"canonical_record":{"source":{"id":"2106.06967","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2021-06-13T11:42:16Z","cross_cats_sorted":[],"title_canon_sha256":"de8bc7d2e034bbed76577488e0a85e672b92c9e9fd15958631dd296f73c17fca","abstract_canon_sha256":"f457b59e376cc693a5e35143b7b3b8d69b4d3580fe146033560669c2faa97093"},"schema_version":"1.0"},"canonical_sha256":"95e3352dba24cd0c4ab9f238babee78ae190829022bb97bfcc7f0561fde8045f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:19:13.777649Z","signature_b64":"JO68vGCdQibEdUlfLOiIV8Kb5MPEbtBS2oZcHUXt+DDKKu14/1Jsl/gI+yWANlG/G4FVqRzuSvX0jyH6Sw+2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95e3352dba24cd0c4ab9f238babee78ae190829022bb97bfcc7f0561fde8045f","last_reissued_at":"2026-07-05T03:19:13.777209Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:19:13.777209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2106.06967","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":"2026-07-05T03:19:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZafTFptuZ22SXE8Tqc0ex30R70g/QghRFn914kvh0SusyBCbEJmJwDEK+B1R5LBBoXVNLy3F7V/HyCmwmGBWBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T00:06:52.728353Z"},"content_sha256":"198ca88de55bbab5c9a2a66ac0a18dfc189b168ec2756428e452bf770b3b07a0","schema_version":"1.0","event_id":"sha256:198ca88de55bbab5c9a2a66ac0a18dfc189b168ec2756428e452bf770b3b07a0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:SXRTKLN2ETGQYSVZ6I4LVPXHRL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiplicity distribution of dipoles in QCD from Le, Mueller and Munier equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"hep-ph","authors_text":"Eugene Levin (Tel Aviv U./UTFSM)","submitted_at":"2021-06-13T11:42:16Z","abstract_excerpt":"In this paper we derived in QCD the BFKL linear, inhomogeneous equation for the factorial moments of multiplicity distribution($M_k$) from LMM equation. In particular, the equation for the average multiplicity of the color-singlet dipoles($N$) turns out to be the homogeneous BFKL while $M_k \\propto N^k$ at small $x$. Second, using the diffusion approximation for the BFKL kernel we show that the factorial moments are equal to: $M_k=k!N( N-1)^{k-1}$ which leads to the multiplicity distribution:$ \\frac{\\sigma_n}{\\sigma_{in}}=\\frac{1}{N} ( \\frac{N\\,-\\,1}{N})^{n - 1}$. We also suggest a procedure f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.06967","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.06967/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:19:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/w98iytPyWuDkFRT7njusyCkC8DlJ+iDZwcQdnEtYc4ADlGrEHSgGBM7mMfMUDww3rzwOrUGdis3UcQzhC23DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T00:06:52.728882Z"},"content_sha256":"2907d05d43ae9d017a1241661951e829bc43ed1d83202e614e0e5c87fb941295","schema_version":"1.0","event_id":"sha256:2907d05d43ae9d017a1241661951e829bc43ed1d83202e614e0e5c87fb941295"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SXRTKLN2ETGQYSVZ6I4LVPXHRL/bundle.json","state_url":"https://pith.science/pith/SXRTKLN2ETGQYSVZ6I4LVPXHRL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SXRTKLN2ETGQYSVZ6I4LVPXHRL/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":"2026-08-04T00:06:52Z","links":{"resolver":"https://pith.science/pith/SXRTKLN2ETGQYSVZ6I4LVPXHRL","bundle":"https://pith.science/pith/SXRTKLN2ETGQYSVZ6I4LVPXHRL/bundle.json","state":"https://pith.science/pith/SXRTKLN2ETGQYSVZ6I4LVPXHRL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SXRTKLN2ETGQYSVZ6I4LVPXHRL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:SXRTKLN2ETGQYSVZ6I4LVPXHRL","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":"f457b59e376cc693a5e35143b7b3b8d69b4d3580fe146033560669c2faa97093","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2021-06-13T11:42:16Z","title_canon_sha256":"de8bc7d2e034bbed76577488e0a85e672b92c9e9fd15958631dd296f73c17fca"},"schema_version":"1.0","source":{"id":"2106.06967","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.06967","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"arxiv_version","alias_value":"2106.06967v2","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.06967","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"pith_short_12","alias_value":"SXRTKLN2ETGQ","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"pith_short_16","alias_value":"SXRTKLN2ETGQYSVZ","created_at":"2026-07-05T03:19:13Z"},{"alias_kind":"pith_short_8","alias_value":"SXRTKLN2","created_at":"2026-07-05T03:19:13Z"}],"graph_snapshots":[{"event_id":"sha256:2907d05d43ae9d017a1241661951e829bc43ed1d83202e614e0e5c87fb941295","target":"graph","created_at":"2026-07-05T03:19:13Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2106.06967/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we derived in QCD the BFKL linear, inhomogeneous equation for the factorial moments of multiplicity distribution($M_k$) from LMM equation. In particular, the equation for the average multiplicity of the color-singlet dipoles($N$) turns out to be the homogeneous BFKL while $M_k \\propto N^k$ at small $x$. Second, using the diffusion approximation for the BFKL kernel we show that the factorial moments are equal to: $M_k=k!N( N-1)^{k-1}$ which leads to the multiplicity distribution:$ \\frac{\\sigma_n}{\\sigma_{in}}=\\frac{1}{N} ( \\frac{N\\,-\\,1}{N})^{n - 1}$. We also suggest a procedure f","authors_text":"Eugene Levin (Tel Aviv U./UTFSM)","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2021-06-13T11:42:16Z","title":"Multiplicity distribution of dipoles in QCD from Le, Mueller and Munier equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.06967","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":"sha256:198ca88de55bbab5c9a2a66ac0a18dfc189b168ec2756428e452bf770b3b07a0","target":"record","created_at":"2026-07-05T03:19:13Z","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":"f457b59e376cc693a5e35143b7b3b8d69b4d3580fe146033560669c2faa97093","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2021-06-13T11:42:16Z","title_canon_sha256":"de8bc7d2e034bbed76577488e0a85e672b92c9e9fd15958631dd296f73c17fca"},"schema_version":"1.0","source":{"id":"2106.06967","kind":"arxiv","version":2}},"canonical_sha256":"95e3352dba24cd0c4ab9f238babee78ae190829022bb97bfcc7f0561fde8045f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95e3352dba24cd0c4ab9f238babee78ae190829022bb97bfcc7f0561fde8045f","first_computed_at":"2026-07-05T03:19:13.777209Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:19:13.777209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JO68vGCdQibEdUlfLOiIV8Kb5MPEbtBS2oZcHUXt+DDKKu14/1Jsl/gI+yWANlG/G4FVqRzuSvX0jyH6Sw+2CQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:19:13.777649Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.06967","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:198ca88de55bbab5c9a2a66ac0a18dfc189b168ec2756428e452bf770b3b07a0","sha256:2907d05d43ae9d017a1241661951e829bc43ed1d83202e614e0e5c87fb941295"],"state_sha256":"e86a4886c2c4ece0728298cafefc737aa88cc3f9bbb4e5c325ea8f73a69baa1c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JI65h5CzC1c4RcM/4Tt1bTpHkgjyGVH6pRHbtfXHPd2V3wwUin+uU9B71jy5t/9wN0Awff8aUL1KrIbEO6n2BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T00:06:52.733777Z","bundle_sha256":"2c889fa2a30eed2c71380079b083035e4248ebb10a05a0d3430089d3322bed69"}}