{"schema":"pubphys.bundle/1","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},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"f140e12f16b2c5e83ee6b9cf3e07fc7385e6fa519d90c5a2d70275c24e772239","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"6d3afbd62fa1dd12ac39306aa316e9b99d7244d127e6980ec174c2d7e4ed4813","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"BX5Y03U1inRixyxUBs_u16EwRaSaB-UTAUpQ5eM-q1-QHU8jlDHTPt1nCVIV1p6rNXd9XcmDzkWYOzG1BVfmDA"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIbTr71i-h3RKsOTBqoxbpuZ1yRNEn5pgOwXTC1-TtSBPxIG0oxdOb1D6F-bfQr07CcqItAygU38o-hLJhGpHghfTICPAgnCnusIDUPTJy4ky8565VS-HvbF70hpvDRgFUIq9YpoMI8SBadXb7MtqvBQ2rCJtxtjxauzEdTXUVlWv_tVSID847WwjxIA_dKXFv8PO7sDfgqE8wC8crjSv4joxRsze1rQK82IKbCPAgUmBzncrixG7TDnwHURz5xiwPIz2OdAbrkUa9jXf9DaEI8SD_5zFJHCmnJMVG3y8g48CODsGJwzCSeSjMybN-JiGepwjwICRMULf4Qk7B4Ixs-w5_AnqKLFYrm25Y_JDm1-0Q4j1tCPEgV483a5w_VdTK7my82rqJCPIXOvGhZNeRkwqfWifoP4MI8SBcP9hQb3hxUiLf8dFx-DuqZHA2xkmG-c_qCFt8QfQHCAjwIOv_x0B6HyZ6WMHKeIf_d_EUIvwPGdZeVY-hXN3RLDM5CPEgHd9u6OIGCvUV12A1B1xdiOprSJ-4w0cqwfe7CzVX5HsI8CDQo_sIWZiWXaTGRSQBLmU-lHJKskYKqsjO1r02DhB8LAj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEItrXY3ed2NqBTbjmUjutAi4iNm3X0qx-t8GlpFddjGA3xILajH2473A4P8-PZ2uGoFYbSS6AtKNn46hT3dW5OKR-eCPEg-lWeoa0Kn6eHkcn8qLGQX0mNNSM9r0pzFows24eER0MI8SAg-MYqI4Jq3ukCPffpjwFWdeJi_zarQ5uGlJMYcfS5MQjxIB0eHzw3zWdDP9SRtv4OYk2bpaKCpPqOMkXVYEmOLho6CPEg_c-UJMCql1K5vjusCC4VbVJns-NHj3-kgKZ7MfUAB1QI8CDuxny7d7Cv4pSdyJIKf1c5GX5jDtId83esEJK0hEHeFQjxIDwyhdMMcC9HUXvmnOqr-4hz0bthwNRePVXecoiwxma1CPAgsToia8UFTWcgAGCYERTkLASNsxtWD-53uc5ktQ4BIIgI8SDArwsGYVuomKH_DJiJGziPwECn7skqn20TGM58Ezco0wjwIIgwugvlgSNOmTWMhquqetKjeLf0x7d1a1MH4KU52JbFCPAg3mczMYDKOd7Nhf3ojA9ZvuTsN66Y1NnyYTp9j8GKQNwI8SCvAiiN-Hrv7IEejd8wdUihoozu5KOvpoBFJ0vg3UnGAQj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMm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