{"paper":{"title":"Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order $\\boldsymbol{g}^{\\boldsymbol{6}}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["hep-ph"],"primary_cat":"hep-th","authors_text":"M. Maniatis","submitted_at":"2026-06-01T18:00:01Z","abstract_excerpt":"We formulate a fixed-order perturbative on-shell construction of amplitudes. The basic input is the particle spectrum together with the allowed on-shell three-point amplitudes. The construction is formulated in terms of tree amplitudes generated by BCFW recursion, supplemented by additional unobservable state-conjugate on-shell pairs, called vacuum pairs, and integrated over the Lorentz-invariant phase space of these pairs. The relative signs are assigned as inclusion-exclusion signs for repeated phase-space ranges in the on-shell construction.\n  As a test case, we study the color-ordered four"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02702","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.02702/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}