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We calculate the three-loop anomalous dimension of the leading operator Q_S2, the three-loop mixing of the current-current and penguin operators into Q_S2, and the corresponding two-loop matching conditions at the electroweak, the bottom-quark, and the charm-quark scale. As our final numerical result we obtain eta_ct = 0.496 +/- "},"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":"1007.0684","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-ph","submitted_at":"2010-07-05T14:09:28Z","cross_cats_sorted":[],"title_canon_sha256":"cf34dc3e718824f065b714d717ede583018fcd2321e6b50701d96c687dcaee6d","abstract_canon_sha256":"5ffe182adea143caa6ae10f266d20d24b65f3cc1c60166de5c8e91805b807584"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:17.956751Z","signature_b64":"6eqby+l4fjPSei8vqrXZu6+/bX2WyDJ5EU9UEJ7niuXzS9LoUXBgDrWUrVm7obJ4ojOO+XfL0d4fvp3Q77epAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4426f54523c99d2633967b90727198bbe2959c6fd481bee620d054fb8bc2f80","last_reissued_at":"2026-05-18T04:33:17.956029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:17.956029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Epsilon_K at Next-to-Next-to-Leading Order: The Charm-Top-Quark Contribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-ph","authors_text":"Joachim Brod, Martin Gorbahn","submitted_at":"2010-07-05T14:09:28Z","abstract_excerpt":"We perform a next-to-next-to-leading order (NNLO) QCD analysis of the charm-top-quark contribution eta_ct to the effective Delta S = 2 Hamiltonian in the Standard Model. eta_ct represents an important part of the short distance contribution to the parameter epsilon_K. We calculate the three-loop anomalous dimension of the leading operator Q_S2, the three-loop mixing of the current-current and penguin operators into Q_S2, and the corresponding two-loop matching conditions at the electroweak, the bottom-quark, and the charm-quark scale. 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