{
  "abstract": "We formulate a systematic on-shell construction of perturbative amplitudes by combining the Feynman-tree theorem with BCFW recursion. Amplitudes are built entirely from on-shell tree amplitudes generated by BCFW and are supplemented by additional unobservable state-conjugate on-shell pairs, called vacuum pairs, whose momenta are integrated over Lorentz-invariant phase space. At fixed coupling order, the contributing tree amplitudes are classified by polygon bookkeeping, while their products form forests whose relative signs are fixed by inclusion-exclusion. We show that these signs coincide with the signs assigned by the full Feynman-tree-theorem cut expansion, so that the construction reproduces the ordinary perturbative loop expansion order by order. The dynamical tree input consists of the allowed three-point on-shell amplitudes; the particle spectrum, couplings, and color data are taken as given, and no off-shell object is introduced at any stage. As a nontrivial test of this equivalence, we study the four-gluon all-plus amplitude through orders $g^4$ and $g^6$. The on-shell construction reproduces the finite rational one-loop primitive amplitude and the known bare planar and non-planar two-loop primitive integrands, including the bow-tie terms in the planar primitive.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.02702",
  "author_total": 1,
  "authors": [
    {
      "name": "M. Maniatis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ex0gfr7n1qsn4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.02702",
  "fulltext_url": null,
  "id": "ex0gfr7n1qsn4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.02702",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-ph"
    ],
    [
      "arxiv",
      "hep-th"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.02702",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ex0gfr7n1qsn4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2606.02702",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.02702"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-ph",
      "subfield": "high-energy"
    }
  ],
  "title": "Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order $g^6$",
  "title_text": "Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order g^6",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:22Z",
  "url": "https://research.draconitic.com/paper/ex0gfr7n1qsn4",
  "venue": "arXiv",
  "versions": []
}