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The covariance properties are verified with the chiral quark condensate and the pion decay constant in the chiral limit.<\/jats:p>","DOI":"10.3390\/sym17010110","type":"journal-article","created":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T04:01:52Z","timestamp":1736740912000},"page":"110","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Gauge Covariance of the Gap Equation: From the Rainbow Truncation to Gauge Symmetry Constraints"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7050-0148","authenticated-orcid":false,"given":"Bruno","family":"El-Bennich","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal de S\u00e3o Paulo, Rua S\u00e3o Nicolau 210, Diadema 09913-030, SP, Brazil"},{"name":"Instituto de F\u00edsica Te\u00f3rica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, S\u00e3o Paulo 01140-070, SP, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1088\/0253-6102\/58\/1\/16","article-title":"Collective perspective on advances in Dyson-Schwinger Equation QCD","volume":"58","author":"Bashir","year":"2012","journal-title":"Commun. 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B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"085005","DOI":"10.1103\/PhysRevD.101.085005","article-title":"Landau-Khalatnikov-Fradkin Transformations, Nielsen Identities, Their Equivalence and Implications for QCD","volume":"101","author":"Meerleer","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/j.nuclphysb.2004.12.010","article-title":"Dynamical fermion masses and constraints of gauge invariance in quenched QED(3)","volume":"709","author":"Bashir","year":"2005","journal-title":"Nucl. Phys. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"076001","DOI":"10.1103\/PhysRevD.93.076001","article-title":"Local Gauge Transformation for the Quark Propagator in an SU(N) Gauge Theory","volume":"93","author":"Aslam","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"074017","DOI":"10.1103\/PhysRevD.97.074017","article-title":"Fresh look at the Abelian and non-Abelian Landau-Khalatnikov-Fradkin transformations","volume":"97","author":"Meerleer","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF01090719","article-title":"Ward Identities in Gauge Theories","volume":"10","author":"Slavnov","year":"1972","journal-title":"Theor. Math. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1016\/0550-3213(71)90297-5","article-title":"Ward Identities and Charge Renormalization of the Yang-Mills Field","volume":"33","author":"Taylor","year":"1971","journal-title":"Nucl. Phys. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5651","DOI":"10.1142\/S0217751X97002978","article-title":"Transverse Ward-Takahashi identity, anomaly and Schwinger-Dyson equation","volume":"12","author":"Kondo","year":"1997","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/S0370-2693(00)00353-1","article-title":"Transverse Ward-Takahashi identity for the fermion boson vertex in gauge theories","volume":"480","author":"He","year":"2000","journal-title":"Phys. Lett. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1088\/0253-6102\/46\/1\/025","article-title":"Transverse vector vertex function and transverse Ward-Takahashi relations in QED","volume":"46","author":"He","year":"2006","journal-title":"Commun. Theor. 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D"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"114514","DOI":"10.1103\/PhysRevD.92.114514","article-title":"Lattice gluon propagator in renormalizable \u03be gauges","volume":"92","author":"Bicudo","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"085018","DOI":"10.1103\/PhysRevD.91.085018","article-title":"Gluon and ghost propagators in linear covariant gauges","volume":"91","author":"Huber","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"054003","DOI":"10.1103\/PhysRevD.104.054003","article-title":"Yang-Mills propagators in linear covariant gauges from Nielsen identities","volume":"104","author":"Napetschnig","year":"2021","journal-title":"Phys. Rev. D"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"094506","DOI":"10.1103\/PhysRevD.99.094506","article-title":"Quark propagator with two flavors of O(a)-improved Wilson fermions","volume":"99","author":"Oliveira","year":"2019","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/JHEP10(2013)193","article-title":"On the Quark-Gluon Vertex and Quark-Ghost Kernel: Combining Lattice Simulations with Dyson-Schwinger equations","volume":"10","author":"Rojas","year":"2013","journal-title":"JHEP"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"094027","DOI":"10.1103\/PhysRevD.99.094027","article-title":"Interplay of dynamical and explicit chiral symmetry breaking effects on a quark","volume":"99","author":"Serna","year":"2019","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2542","DOI":"10.1103\/PhysRevD.22.2542","article-title":"Analytic Properties of the Vertex Function in Gauge Theories. 1","volume":"22","author":"Ball","year":"1980","journal-title":"Phys. Rev. D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"014013","DOI":"10.1103\/PhysRevD.83.014013","article-title":"Chiral symmetry breaking with lattice propagators","volume":"83","author":"Aguilar","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"014029","DOI":"10.1103\/PhysRevD.96.014029","article-title":"Non-Abelian Ball-Chiu vertex for arbitrary Euclidean momenta","volume":"96","author":"Aguilar","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"014013","DOI":"10.1103\/PhysRevD.96.014013","article-title":"Charmed mesons with a symmetry-preserving contact interaction","volume":"96","author":"Serna","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1140\/epjc\/s10052-020-08517-3","article-title":"Distribution amplitudes of heavy mesons and quarkonia on the light front","volume":"80","author":"Serna","year":"2020","journal-title":"Eur. Phys. J. C"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"L091504","DOI":"10.1103\/PhysRevD.106.L091504","article-title":"D* and Ds* distribution amplitudes from Bethe-Salpeter wave functions","volume":"106","author":"Serna","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"034021","DOI":"10.1103\/PhysRevD.107.034021","article-title":"Strong two-meson decays of light and charmed vector mesons","volume":"107","author":"Serna","year":"2023","journal-title":"Phys. Rev. D"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"114033","DOI":"10.1103\/PhysRevD.110.114033","article-title":"Parton distribution functions and transverse momentum dependence of heavy mesons","volume":"110","author":"Serna","year":"2024","journal-title":"Phys. Rev. D"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"014022","DOI":"10.1103\/PhysRevD.63.014022","article-title":"Quark gluon vertex in arbitrary gauge and dimension","volume":"63","author":"Davydychev","year":"2001","journal-title":"Phys. Rev. D"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"014502","DOI":"10.1103\/PhysRevD.94.014502","article-title":"Lattice Gluon and Ghost Propagators, and the Strong Coupling in Pure SU(3) Yang-Mills Theory: Finite Lattice Spacing and Volume Effects","volume":"94","author":"Duarte","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"045205","DOI":"10.1103\/PhysRevC.85.045205","article-title":"Dynamical chiral symmetry breaking and the fermion\u2013gauge-boson vertex","volume":"85","author":"Bashir","year":"2012","journal-title":"Phys. Rev. C"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"034017","DOI":"10.1103\/PhysRevD.95.034017","article-title":"Schwinger mechanism in linear covariant gauges","volume":"95","author":"Aguilar","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/j.aop.2018.08.019","article-title":"High precision statistical Landau gauge lattice gluon propagator computation vs. the Gribov\u2013Zwanziger approach","volume":"397","author":"Dudal","year":"2018","journal-title":"Ann. Phys."},{"key":"ref_43","unstructured":"Cucchieri, A., Dudal, D., Mendes, T., Oliveira, O., Roelfs, M., and Silva, P.J. (2018, January 22\u201328). Lattice Computation of the Ghost Propagator in Linear Covariant Gauges. Proceedings of the 36th International Symposium on Lattice Field Theory, East Lansing, MI, USA."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/0375-9474(96)00174-1","article-title":"Electromagnetic pion form-factor and neutral pion decay width","volume":"605","author":"Roberts","year":"1996","journal-title":"Nucl. Phys. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1242","DOI":"10.1103\/PhysRevD.52.1242","article-title":"One loop QED vertex in any covariant gauge: Its complete analytic form","volume":"52","author":"Reenders","year":"1995","journal-title":"Phys. Rev. D"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/110\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:27:25Z","timestamp":1759919245000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/110"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,12]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010110"],"URL":"https:\/\/doi.org\/10.3390\/sym17010110","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,12]]}}}