{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:49:30Z","timestamp":1760237370132,"version":"build-2065373602"},"reference-count":61,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,5,14]],"date-time":"2020-05-14T00:00:00Z","timestamp":1589414400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We review the emergence of hypergeometric structures (of F4 Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d &gt; 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to hypergeometric systems with four independent solutions. For symmetric correlators, they can be expressed in terms of a single 3K integral\u2014functions of quadratic ratios of momenta\u2014which is a parametric integral of three modified Bessel K functions. In the case of scalar 4-point functions, by requiring the correlator to be conformal invariant in coordinate space as well as in some dual variables (i.e., dual conformal invariant), its explicit expression is also given by a 3K integral, or as a linear combination of Appell functions which are now quartic ratios of momenta. Similar expressions have been obtained in the past in the computation of an infinite class of planar ladder (Feynman) diagrams in perturbation theory, which, however, do not share the same (dual conformal\/conformal) symmetry of our solutions. We then discuss some hypergeometric functions of 3 variables, which define 8 particular solutions of the CWIs and correspond to Lauricella functions. They can also be combined in terms of 4K integral and appear in an asymptotic description of the scalar 4-point function, in special kinematical limits.<\/jats:p>","DOI":"10.3390\/axioms9020054","type":"journal-article","created":{"date-parts":[[2020,5,14]],"date-time":"2020-05-14T10:27:19Z","timestamp":1589452039000},"page":"54","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The Generalized Hypergeometric Structure of the Ward Identities of CFT\u2019s in Momentum Space in d &gt; 2"],"prefix":"10.3390","volume":"9","author":[{"given":"Claudio","family":"Corian\u00f2","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica e Fisica \u201cEnnio De Giorgi\u201d, Universit\u00e0 del Salento and INFN-Lecce, Via Arnesano, 73100 Lecce, Italy"}]},{"given":"Matteo Maria","family":"Maglio","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Fisica \u201cEnnio De Giorgi\u201d, Universit\u00e0 del Salento and INFN-Lecce, Via Arnesano, 73100 Lecce, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1002\/andp.200852009-1005","article-title":"On the Advancements of Conformal Transformations and their Associated Symmetries in Geometry and Theoretical Physics","volume":"17","author":"Kastrup","year":"2008","journal-title":"Ann. 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