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Copositivity criteria allow to derive analytic necessary and sufficient vacuum stability conditions for the matrix $\\lambda_{ab}$. We review the basic properties of copositive matrices and analytic criteria for copositivity. To illustrate these, we re-derive the vacuum stability conditions for the ine"},"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":"1205.3781","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-ph","submitted_at":"2012-05-16T20:00:02Z","cross_cats_sorted":[],"title_canon_sha256":"7e4e3977cd481c341eab9fb8c439240b607ca23b0e48c5598a2b2fef8da0d66b","abstract_canon_sha256":"d63a843e5ba8d8764647506f9635b95061d5d29426a5acf8f940ff7df259e908"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:57:31.222590Z","signature_b64":"d9WPkEOaSid1bHwESpi8uD7IZb2JtYZremqUI/86fIqxGQFFrytV8YrCGNujdTl/2PU5SOqWIRmhPmTIr/AyCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f31dde5a0d5c59b3e8ea362416e2318ccd0c14accfbfe843a893cf6256e82bf","last_reissued_at":"2026-05-18T01:57:31.222005Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:57:31.222005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vacuum Stability Conditions From Copositivity Criteria","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-ph","authors_text":"Kristjan Kannike","submitted_at":"2012-05-16T20:00:02Z","abstract_excerpt":"A scalar potential of the form $\\lambda_{ab} \\phi_a^2 \\phi_b^2$ is bounded from below if its matrix of quartic couplings $\\lambda_{ab}$ is copositive -- positive on non-negative vectors. 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