Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order $\boldsymbol{g}^{\boldsymbol{6}}$

Abstract

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. As a test case, we study the color-ordered four-gluon all-plus amplitude through orders $g^4$ and $g^6$, and compare the resulting signed phase-space sums with the standard one- and two-loop contributions. The fixed-order bookkeeping of the tree amplitudes is organized in terms of polygons. At order $g^4$ the construction reproduces the finite rational one-loop result. At order $g^6$ the non-vanishing polygon sectors are the octagon, hexagon-quadrilateral, two-pentagon, and three-quadrilateral sectors. Taken together, they reproduce the known planar, non-planar, and bow-tie expressions.