{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"9148cd7256529c694a505198a99d4fbd7e7af9be21f2fc68b449b2071d571a61","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["8383c54977822e9fa9a6744bf14a05cd55f4084ac834e6212a67c1f772be4aec","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"709caf2c42d3cbe6021a153e8a1f125d19d351d969e949088d32db3182ef48ef","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"qft.rg-irreversibility.strong-a-gradient","kind":"well-posed","literature_status":"partially-resolved","n":"2","parents":[],"plain":"In four dimensions a certain count of degrees of freedom is proven to be lower at long distances than at short ones. A stronger claim, that the couplings always move steadily downhill along a height function given by this count, is checked only in perturbation theory.","posed_since":"1990","precise":"Jack and Osborn (1990) showed perturbatively that the beta functions $\\beta^{j}$ of a 4D renormalizable QFT satisfy $d_{i} a\\sim = \\chi_{ij}\\beta^{j} + (d_{i} w_{j} - d_{j} w_{i})\\beta^{j}$ for a function $a\\sim(g)$ equal to the a-anomaly at fixed points. Determine whether, beyond perturbation theory, $\\chi_{ij}$ is positive definite so that $a\\sim$ decreases monotonically along every flow (strong a-theorem). The weak version $a_{\\mathrm{UV}} > a_{\\mathrm{IR}}$ between fixed points is proven (Komargodski and Schwimmer 2011).","problem_ref":"8383c54977822e9fa9a6744bf14a05cd55f4084ac834e6212a67c1f772be4aec","references":"","settled_by":"A nonperturbative proof that $\\chi_{ij}$ is positive definite in unitary 4D QFT, or a unitary flow along which $a\\sim$ increases.","status_note":"$\\chi_{ij}$ is positive at leading orders and $a\\sim$ has been constructed to high loop order for general gauge theories (Jack and Poole 2015); the nonperturbative statement is open.","title":"Is four-dimensional RG flow a gradient flow of the a-function?","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"f140e12f16b2c5e83ee6b9cf3e07fc7385e6fa519d90c5a2d70275c24e772239","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"6d3afbd62fa1dd12ac39306aa316e9b99d7244d127e6980ec174c2d7e4ed4813","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"BX5Y03U1inRixyxUBs_u16EwRaSaB-UTAUpQ5eM-q1-QHU8jlDHTPt1nCVIV1p6rNXd9XcmDzkWYOzG1BVfmDA"},"schema":"pubphys.envelope/1"},"record_hash":"f140e12f16b2c5e83ee6b9cf3e07fc7385e6fa519d90c5a2d70275c24e772239","leaf_index":3674}