Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order g6
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 g4 and g6. 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.
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BibTeX
@misc{maniatis2026perturbative,
title = {{Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order $g^6$}},
author = {M. Maniatis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.02702},
eprint = {2606.02702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.02702},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order g^6
AU - M. Maniatis
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2606.02702
UR - https://arxiv.org/abs/2606.02702
AB - 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.
ER -