{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"95ee9ec19933d7cd42af53914f7422744df5adf643e0f0f92427c8b92b56f538","created":"2026-10-03T07:18:09Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"5f123c1045413ae88c187f5a8a7a8a71ea9e0358a35dda9489f545c9daadfafb","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":"1","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":null,"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":"aa2c9297ee0f9c5637717e0362dc66e39fae921c636c489893e101ec3f0ff180"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"8383c54977822e9fa9a6744bf14a05cd55f4084ac834e6212a67c1f772be4aec","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"05bff71b168d1ce176f8a8da5ae77f349ffafba0adfdc45b32978c87eb2f00f6","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"BWn9I9CdOkSFnyQGQDM7uWbQGwZXzvdqLLoNB5tA6miaSBDbqlmSyPJTpIPWaGzvY_mW7fytejLVyZUAHzOpBg"},"schema":"pubphys.envelope/1"},"record_hash":"8383c54977822e9fa9a6744bf14a05cd55f4084ac834e6212a67c1f772be4aec","leaf_index":1947}