{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T14:27:06Z","timestamp":1785421626000,"version":"3.56.0"},"reference-count":43,"publisher":"American Physical Society (APS)","issue":"25","license":[{"start":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T00:00:00Z","timestamp":1782259200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100021745","name":"Institut P\u00e9rim\u00e8tre de physique th\u00e9orique","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100021745","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000023","name":"Government of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000023","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011332","name":"Innovation, Science and Economic Development Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100011332","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100021784","name":"Ministry of Colleges and Universities","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100021784","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Province of Ontario"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. Lett."],"accepted":{"date-parts":[[2026,5,29]]},"abstract":"<jats:p>\n                    We propose a renormalization group flow equation for a functional that generates\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <a:mi>S<\/a:mi>\n                    <\/a:math>\n                    -matrix elements and which captures similarities to the well-known Wetterich and Polchinski equations. While the latter ones, respectively, involve the effective action and Schwinger functional, which are genuine off-shell objects, the presented flow equation has the advantage of working more directly with observables, i.e., scattering amplitudes. Compared to the Wetterich equation, our flow equation also greatly simplifies the notion of going on shell, in the sense of satisfying the quantum equations of motion. In addition, unlike the Wetterich equation, it is polynomial and does not require a Hessian inversion. The approach is a promising direction for nonperturbative quantum field theories, allowing one to work more directly with scattering amplitudes.\n                  <\/jats:p>","DOI":"10.1103\/jxcl-q2vt","type":"journal-article","created":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T16:59:15Z","timestamp":1780073955000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Nonperturbative\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <mml:mi>S<\/mml:mi>\n                    <\/mml:math>\n                    -Matrix Renormalization"],"prefix":"10.1103","volume":"136","author":[{"given":"Laurent","family":"Freidel","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2611-1038","authenticated-orcid":true,"given":"Jos\u00e9","family":"Padua-Arg\u00fcelles","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"},{"id":[{"id":"https:\/\/ror.org\/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Waterloo"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3724-0943","authenticated-orcid":true,"given":"Susanne","family":"Schander","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0778-4800","authenticated-orcid":true,"given":"Marc","family":"Schiffer","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/016xsfp80","id-type":"ROR","asserted-by":"publisher"}],"name":"Radboud University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,6,24]]},"reference":[{"key":"jxcl-q2vtCc1R1","doi-asserted-by":"publisher","DOI":"10.1016\/0029-5582(61)90056-6","type":"journal-article"},{"key":"jxcl-q2vtCc2R1","doi-asserted-by":"publisher","DOI":"10.1016\/0029-5582(61)90106-7","type":"journal-article"},{"key":"jxcl-q2vtCc3R1","doi-asserted-by":"publisher","DOI":"10.1016\/0370-2693(94)01419-D","type":"journal-article"},{"key":"jxcl-q2vtCc4R1","unstructured":"For early reference on the S-matrix and renormalization in string theory see Refs.\u00a0[5, 6].","type":"other"},{"key":"jxcl-q2vtCc5R1","doi-asserted-by":"publisher","DOI":"10.1016\/0550-3213(87)90605-5","type":"journal-article"},{"key":"jxcl-q2vtCc6R1","doi-asserted-by":"publisher","DOI":"10.1016\/0550-3213(89)90384-2","type":"journal-article"},{"key":"jxcl-q2vtCc7R1","doi-asserted-by":"publisher","DOI":"10.1016\/0550-3213(84)90287-6","type":"journal-article"},{"key":"jxcl-q2vtCc8R1","doi-asserted-by":"publisher","DOI":"10.1016\/0370-2693(93)90726-X","type":"journal-article"},{"key":"jxcl-q2vtCc9R1","doi-asserted-by":"publisher","DOI":"10.1142\/S0217751X94000972","type":"journal-article"},{"key":"jxcl-q2vtCc10R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01555911","type":"journal-article"},{"key":"jxcl-q2vtCc11R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2021.01.001","type":"journal-article"},{"key":"jxcl-q2vtCc12R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.14.4.069","type":"journal-article"},{"key":"jxcl-q2vtCc13R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.17.5.148","type":"journal-article"},{"key":"jxcl-q2vtCc14R1","doi-asserted-by":"publisher","DOI":"10.1103\/lkwg-697n","type":"journal-article"},{"key":"jxcl-q2vtCc15R1","doi-asserted-by":"publisher","DOI":"10.1103\/4sh5-w4yc","type":"journal-article"},{"key":"jxcl-q2vtCc16R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.181301","type":"journal-article"},{"key":"jxcl-q2vtCc17R1","doi-asserted-by":"publisher","DOI":"10.1007\/JHEP11(2020)136","type":"journal-article"},{"key":"jxcl-q2vtCc18R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.111.106005","type":"journal-article"},{"key":"jxcl-q2vtCc19R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.130.081501","type":"journal-article"},{"key":"jxcl-q2vtCc20R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01038094","type":"journal-article"},{"key":"jxcl-q2vtCc21R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.37.1485","type":"journal-article"},{"key":"jxcl-q2vtCc22R1","unstructured":"The boundary term ensures that on-shell variations of the action are compatible with Dirichlet boundary conditions.","type":"other"},{"key":"jxcl-q2vtCc23R1","unstructured":"More precisely for a functional F[\u03c6] we adopt the \u201ctensorial\u201d notation F(L)[\u03c6] to encode all \u201ccomponents\u201d F(L)[\u03c6]z1\u2026zL\u2254\u03b4\u03b4\u03c6(z1)\u22ef\u03b4\u03b4\u03c6(zL)F[\u03c6].As is customary, for small L, we may occasionally denote the number of derivatives with a corresponding number of primes. So, for example, F(2)[\u03c6]=F\u2032\u2032[\u03c6].\nWith this notation, index contractions correspond to spacetime integrals. E.g., Tr(F\u2032\u2032[\u03c60])=\u222b\u03b4\u03b4\u03c6(z)\u03b4\u03b4\u03c6(z)F[\u03c6]|\u03c6=\u03c60.","type":"other"},{"key":"jxcl-q2vtCc24R1","volume-title":"Quantum Field Theory. A Modern Perspective","author":"V.\u2009P. Nair","year":"2004","type":"book"},{"key":"jxcl-q2vtCc25R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/jxcl-q2vt for some details on the derivation of Eq.\u00a0(5), and for details on the relation to other functional equations.","type":"other"},{"key":"jxcl-q2vtCc26R1","unstructured":"We caution that the argument relies on the LSZ formalism, which might break down in the presence of long-range interactions [27].","type":"other"},{"key":"jxcl-q2vtCc28R1","doi-asserted-by":"publisher","DOI":"10.1007\/JHEP02(2026)151","type":"journal-article"},{"key":"jxcl-q2vtCc29R1","doi-asserted-by":"publisher","DOI":"10.1007\/JHEP10(2023)036","type":"journal-article"},{"key":"jxcl-q2vtCc30R1","doi-asserted-by":"publisher","DOI":"10.1007\/JHEP05(2012)021","type":"journal-article"},{"key":"jxcl-q2vtCc31R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.106.251302","type":"journal-article"},{"key":"jxcl-q2vtCc32R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.130.081501","type":"journal-article"},{"key":"jxcl-q2vtCc33R1","doi-asserted-by":"publisher","DOI":"10.1007\/s00023-023-01348-4","type":"journal-article"},{"key":"jxcl-q2vtCc34R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.109.066012","type":"journal-article"},{"key":"jxcl-q2vtCc35R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.111.106007","type":"journal-article"},{"key":"jxcl-q2vtCc36R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2007.01.007","type":"journal-article"},{"key":"jxcl-q2vtCc37R1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-27320-9","type":"journal-article"},{"key":"jxcl-q2vtCc38R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.15.2.074","type":"journal-article"},{"key":"jxcl-q2vtCc39R1","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3719\/7\/12\/004","type":"journal-article"},{"key":"jxcl-q2vtCc40R1","unstructured":"J. Pawlowski and S. Ray (to be published).","type":"other"},{"key":"jxcl-q2vtCc41R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1703676","type":"journal-article"},{"key":"jxcl-q2vtCc42R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.102.125001","type":"journal-article"},{"key":"jxcl-q2vtCc43R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.107.126016","type":"journal-article"},{"key":"jxcl-q2vtCc44R1","doi-asserted-by":"publisher","DOI":"10.1007\/JHEP09(2024)074","type":"journal-article"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/jxcl-q2vt","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/jxcl-q2vt\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T13:46:16Z","timestamp":1782308776000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/jxcl-q2vt"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,24]]},"references-count":43,"journal-issue":{"issue":"25","published-print":{"date-parts":[[2026,6]]}},"URL":"https:\/\/doi.org\/10.1103\/jxcl-q2vt","relation":{"has-preprint":[{"id-type":"doi","id":"10.48550\/arXiv.2509.00156","asserted-by":"subject"}]},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,24]]},"assertion":[{"value":"2026-03-03","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"Published by the American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2026","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92026 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"251602"}}