An on-shell construction using BCFW trees plus vacuum-pair phase-space integrals with inclusion-exclusion signs reproduces the known one- and two-loop all-plus four-gluon amplitudes.
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Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order $\boldsymbol{g}^{\boldsymbol{6}}$
An on-shell construction using BCFW trees plus vacuum-pair phase-space integrals with inclusion-exclusion signs reproduces the known one- and two-loop all-plus four-gluon amplitudes.