(* The solution is given in the form of five lists each containing 24 elements: 1) all 24 graphs considered in the paper (see Figure 5), described by a pair of two lists, vertices and edges, of the form: - vertices: {number, {connected edges}}, - edges: {type, number, from -> to, momentum flow} (use drawGraph function to produce an image of the graph in Mathematica), 2) numerators for each graph (use replaceRules to bring the numerators into a more general form), 3) color factors for each graph, 4) propagators for each graph, 5) symmetry factors for each graph, and, additionally, a list with expressions for the five relevant gravity numerators (eqs. (5.24)-(5.28)). *) (* Details on the notation: p[i]: external momenta l[i]: independent loop momenta s,t,u: Mandelstam variables (see replaceRules for exact definition) l1s,l1t,l1u,l2s,l2t,l2u: contractions of external and loop momenta (see replaceRules for exact definition) dot[ , ]: D-dimensional Minkowski dot product in signature (+ - - — . . . -) mu[l[i],l[j]]: two-dimensional Euclidean dot product for the extra-dimensional part of the loop momenta (note, we take mu[l[i],l[j]] to be positive before Wick rotation) e4: four-dimensional Levi-Civita invariant emu: two-dimensional Levi-Civita invariant for the extra-dimensional mu momenta f: gauge-group structure constants tr: trace over Lie algebra generators Ds: state counting parameter *) replaceRules = {l1s -> 2*(dot[l[1], p[1]] + dot[l[1], p[2]]), l1t -> 2*(dot[l[1], p[2]] + dot[l[1], p[3]]), l1u -> 2*(dot[l[1], p[1]] + dot[l[1], p[3]]), l2s -> 2*(dot[l[2], p[1]] + dot[l[2], p[2]]), l2t -> 2*(dot[l[2], p[2]] + dot[l[2], p[3]]), l2u -> 2*(dot[l[2], p[1]] + dot[l[2], p[3]]), l3s -> 2*(dot[l[1], p[1]] + dot[l[1], p[2]] + dot[l[2], p[1]] + dot[l[2], p[2]]), l3t -> 2*(dot[l[1], p[2]] + dot[l[1], p[3]] + dot[l[2], p[2]] + dot[l[2], p[3]]), l3u -> 2*(dot[l[1], p[1]] + dot[l[1], p[3]] + dot[l[2], p[1]] + dot[l[2], p[3]]), mu11 -> mu[l[1], l[1]], mu12 -> mu[l[1], l[2]], mu13 -> mu[l[1], l[1]] + mu[l[1], l[2]], mu22 -> mu[l[2], l[2]], mu23 -> mu[l[1], l[2]] + mu[l[2], l[2]], mu33 -> mu[l[1], l[1]] + 2*mu[l[1], l[2]] + mu[l[2], l[2]], emu -> emu[l[1], l[2]], s -> 2*dot[p[1], p[2]], t -> 2*dot[p[2], p[3]], u -> 2*dot[p[3], p[1]], l[3] -> l[1] + l[2]} drawGraph[graphData_] := DynamicModule[{T, vc, data = graphData}, data = {graphVertex /@ graphData[[1]], graphEdge /@ graphData[[2]]}; vc = (AbsoluteOptions[Graph[Sequence @@ (data /. {DirectedEdge -> UndirectedEdge})], VertexCoordinates] /. {HoldPattern[_ -> l_] :> l})[[1]]; LocatorPane[Dynamic[vc], Dynamic[Show[Graph[Sequence @@ data, VertexCoordinates -> vc], ImageSize -> Large]]]] graphVertex[{number_}] := number graphVertex[{number_, ordering_List}] := Tooltip[number, ordering] graphEdge[{type_, number_, from_ -> to_, tooltip_}] := Piecewise[{{Tooltip[Labeled[Style[DirectedEdge[from, to], Black, Dashing[{Small, Small}]], number], tooltip], type === "vector"}, {Tooltip[Labeled[Style[DirectedEdge[from, to], Black], number], tooltip], type === "hyper"}}, "Error"] graphs = {{{{1}, {2}, {3}, {4}, {5, {1, 5, -10}}, {6, {2, 6, -5}}, {7, {-6, -7, -11}}, {8, {7, 3, -8}}, {9, {8, 4, -9}}, {10, {9, 10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 8 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 8 -> 7, l[2] - p[3] - p[4]}, {"vector", 8, 9 -> 8, l[2] - p[4]}, {"vector", 9, 10 -> 9, l[2]}, {"vector", 10, 10 -> 5, l[1]}, {"vector", 11, 10 -> 7, -l[1] - l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 10}}, {6, {2, -6, 5}}, {7, {6, -7, -11}}, {8, {7, 3, -8}}, {9, {8, 4, -9}}, {10, {9, -10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 8 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 8 -> 7, l[2] - p[3] - p[4]}, {"hyper", 8, 9 -> 8, l[2] - p[4]}, {"hyper", 9, 10 -> 9, l[2]}, {"hyper", 10, 5 -> 10, -l[1]}, {"vector", 11, 10 -> 7, -l[1] - l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 10}}, {6, {2, -6, 5}}, {7, {6, -7, -11}}, {8, {7, 3, -8}}, {9, {8, 4, -9}}, {10, {9, -10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 8 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"vector", 7, 8 -> 7, l[2] - p[3] - p[4]}, {"vector", 8, 9 -> 8, l[2] - p[4]}, {"vector", 9, 10 -> 9, l[2]}, {"hyper", 10, 5 -> 10, -l[1]}, {"hyper", 11, 10 -> 7, -l[1] - l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -10}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {-7, -8, -11}}, {9, {4, -9, 8}}, {10, {9, 10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 9 -> 8, l[2] - p[4]}, {"vector", 9, 10 -> 9, l[2]}, {"vector", 10, 10 -> 5, l[1]}, {"vector", 11, 10 -> 8, -l[1] - l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 10}}, {6, {2, -6, 5}}, {7, {3, -7, 6}}, {8, {7, -8, -11}}, {9, {4, -9, 8}}, {10, {9, -10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 8 -> 7, -l[1] - p[4]}, {"hyper", 8, 9 -> 8, l[2] - p[4]}, {"hyper", 9, 10 -> 9, l[2]}, {"hyper", 10, 5 -> 10, -l[1]}, {"vector", 11, 10 -> 8, -l[1] - l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 10}}, {6, {2, -6, 5}}, {7, {3, -7, 6}}, {8, {7, -8, -11}}, {9, {4, -9, 8}}, {10, {9, -10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 8 -> 7, -l[1] - p[4]}, {"vector", 8, 9 -> 8, l[2] - p[4]}, {"vector", 9, 10 -> 9, l[2]}, {"hyper", 10, 5 -> 10, -l[1]}, {"hyper", 11, 10 -> 8, -l[1] - l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -10}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {-7, -8, 11}}, {9, {4, -9, 8}}, {10, {9, 10, -11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"hyper", 8, 9 -> 8, l[2] - p[4]}, {"hyper", 9, 10 -> 9, l[2]}, {"vector", 10, 10 -> 5, l[1]}, {"hyper", 11, 8 -> 10, l[1] + l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -9}}, {6, {2, 6, -5}}, {7, {-6, -7, -11}}, {8, {4, -8, 7}}, {9, {8, 9, 10}}, {10, {3, 11, -10}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 10 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 8 -> 7, l[2] - p[4]}, {"vector", 8, 9 -> 8, l[2]}, {"vector", 9, 9 -> 5, l[1]}, {"vector", 10, 9 -> 10, -l[1] - l[2]}, {"vector", 11, 10 -> 7, -l[1] - l[2] - p[3]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 9}}, {6, {2, -6, 5}}, {7, {6, -7, -11}}, {8, {4, -8, 7}}, {9, {8, -9, 10}}, {10, {3, 11, -10}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 10 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 8 -> 7, l[2] - p[4]}, {"hyper", 8, 9 -> 8, l[2]}, {"hyper", 9, 5 -> 9, -l[1]}, {"vector", 10, 9 -> 10, -l[1] - l[2]}, {"vector", 11, 10 -> 7, -l[1] - l[2] - p[3]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -9}}, {6, {2, 6, -5}}, {7, {-6, -7, 11}}, {8, {4, -8, 7}}, {9, {8, 9, -10}}, {10, {3, -11, 10}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 10 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"hyper", 7, 8 -> 7, l[2] - p[4]}, {"hyper", 8, 9 -> 8, l[2]}, {"vector", 9, 9 -> 5, l[1]}, {"hyper", 10, 10 -> 9, l[1] + l[2]}, {"hyper", 11, 7 -> 10, l[1] + l[2] + p[3]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -7}}, {6, {2, 6, -5}}, {7, {-8, 7, -6}}, {8, {8, -9, 11}}, {9, {3, -10, 9}}, {10, {4, -11, 10}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 9 -> 3, p[3]}, {"vector", 4, 10 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 5, l[1]}, {"vector", 8, 8 -> 7, p[1] + p[2]}, {"vector", 9, 9 -> 8, l[2] - p[3] - p[4]}, {"vector", 10, 10 -> 9, l[2] - p[4]}, {"vector", 11, 8 -> 10, l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 7}}, {6, {2, -6, 5}}, {7, {-8, -7, 6}}, {8, {8, -9, 11}}, {9, {3, -10, 9}}, {10, {4, -11, 10}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 9 -> 3, p[3]}, {"vector", 4, 10 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 5 -> 7, -l[1]}, {"vector", 8, 8 -> 7, p[1] + p[2]}, {"vector", 9, 9 -> 8, l[2] - p[3] - p[4]}, {"vector", 10, 10 -> 9, l[2] - p[4]}, {"vector", 11, 8 -> 10, l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 7}}, {6, {2, -6, 5}}, {7, {-8, -7, 6}}, {8, {8, -9, 11}}, {9, {3, -10, 9}}, {10, {4, -11, 10}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 9 -> 3, p[3]}, {"vector", 4, 10 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 5 -> 7, -l[1]}, {"vector", 8, 8 -> 7, p[1] + p[2]}, {"hyper", 9, 9 -> 8, l[2] - p[3] - p[4]}, {"hyper", 10, 10 -> 9, l[2] - p[4]}, {"hyper", 11, 8 -> 10, l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -11}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {4, 8, -7}}, {9, {-8, -9, 10}}, {10, {9, 11, -10}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 8 -> 9, l[1]}, {"vector", 9, 10 -> 9, l[2]}, {"vector", 10, 9 -> 10, l[1] + l[2]}, {"vector", 11, 10 -> 5, l[1]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 11}}, {6, {2, -6, 5}}, {7, {3, -7, 6}}, {8, {4, -8, 7}}, {9, {8, -9, 10}}, {10, {9, -10, -11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 8 -> 7, -l[1] - p[4]}, {"hyper", 8, 9 -> 8, -l[1]}, {"hyper", 9, 10 -> 9, l[2]}, {"vector", 10, 9 -> 10, l[1] + l[2]}, {"hyper", 11, 5 -> 10, -l[1]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -11}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {4, 8, -7}}, {9, {-8, -9, 10}}, {10, {9, 11, -10}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 8 -> 9, l[1]}, {"hyper", 9, 10 -> 9, l[2]}, {"hyper", 10, 9 -> 10, l[1] + l[2]}, {"vector", 11, 10 -> 5, l[1]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -8}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {-7, 9, 8}}, {9, {-9, -11, 10}}, {10, {4, -10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 10 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 8 -> 5, l[1]}, {"vector", 9, 8 -> 9, p[4]}, {"vector", 10, 9 -> 10, l[2]}, {"vector", 11, 10 -> 9, l[2] - p[4]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -8}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {-7, 9, 8}}, {9, {-9, -11, 10}}, {10, {4, -10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 10 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 8 -> 5, l[1]}, {"vector", 9, 8 -> 9, p[4]}, {"hyper", 10, 9 -> 10, l[2]}, {"hyper", 11, 10 -> 9, l[2] - p[4]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 8}}, {6, {2, -6, 5}}, {7, {3, -7, 6}}, {8, {7, 9, -8}}, {9, {-9, -11, 10}}, {10, {4, -10, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 10 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 8 -> 7, -l[1] - p[4]}, {"hyper", 8, 5 -> 8, -l[1]}, {"vector", 9, 8 -> 9, p[4]}, {"hyper", 10, 9 -> 10, l[2]}, {"hyper", 11, 10 -> 9, l[2] - p[4]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -9}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {4, 8, -7}}, {9, {-8, 10, 9}}, {10, {-10, -11, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 8 -> 9, l[1]}, {"vector", 9, 9 -> 5, l[1]}, {"vector", 10, 9 -> 10, 0}, {"vector", 11, 10 -> 10, l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -9}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {4, 8, -7}}, {9, {-8, 10, 9}}, {10, {-10, -11, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 8 -> 9, l[1]}, {"vector", 9, 9 -> 5, l[1]}, {"vector", 10, 9 -> 10, 0}, {"hyper", 11, 10 -> 10, l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, -5, 9}}, {6, {2, -6, 5}}, {7, {3, -7, 6}}, {8, {4, -8, 7}}, {9, {8, 10, -9}}, {10, {-10, -11, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 8 -> 4, p[4]}, {"hyper", 5, 6 -> 5, -l[1] + p[1]}, {"hyper", 6, 7 -> 6, -l[1] + p[1] + p[2]}, {"hyper", 7, 8 -> 7, -l[1] - p[4]}, {"hyper", 8, 9 -> 8, -l[1]}, {"hyper", 9, 5 -> 9, -l[1]}, {"vector", 10, 9 -> 10, 0}, {"hyper", 11, 10 -> 10, l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -8}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {-7, 9, 8}}, {9, {4, 10, -9}}, {10, {-10, -11, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 8 -> 5, l[1]}, {"vector", 9, 8 -> 9, p[4]}, {"vector", 10, 9 -> 10, 0}, {"vector", 11, 10 -> 10, l[2]}}}, {{{1}, {2}, {3}, {4}, {5, {1, 5, -8}}, {6, {2, 6, -5}}, {7, {3, 7, -6}}, {8, {-7, 9, 8}}, {9, {4, 10, -9}}, {10, {-10, -11, 11}}}, {{"vector", 1, 5 -> 1, p[1]}, {"vector", 2, 6 -> 2, p[2]}, {"vector", 3, 7 -> 3, p[3]}, {"vector", 4, 9 -> 4, p[4]}, {"vector", 5, 5 -> 6, l[1] - p[1]}, {"vector", 6, 6 -> 7, l[1] - p[1] - p[2]}, {"vector", 7, 7 -> 8, l[1] + p[4]}, {"vector", 8, 8 -> 5, l[1]}, {"vector", 9, 8 -> 9, p[4]}, {"vector", 10, 9 -> 10, 0}, {"hyper", 11, 10 -> 10, l[2]}}}} numerators = {((4*I)*(t*e4[p[1], p[2], l[1], l[2]] + l3t*e4[p[1], p[2], p[3], l[1]] + l2t*e4[p[1], p[2], p[3], l[2]])* (k[1, 4] - k[2, 3]))/t^2 + ((2*(l2t^2 + l1t*l3t)*s + (l2s^2 + l2t^2 - l2u^2 + l1s*l3s + l1t*l3t + l2u*l3t - l1u*l3u - l2t*l3u)*t + 2*s*t^2 - 4*s*t*(mu13 + mu22 + dot[l[1], l[3]] + dot[l[2], l[2]]))* (k[1, 4] + k[2, 3]))/(2*t^2) + ((4*I)*((l1u + 2*dot[p[3], l[2]])*e4[p[1], p[2], p[3], l[1]] + (l2u + 2*dot[p[1], l[1]])*e4[p[1], p[2], p[3], l[2]] + s*e4[p[1], p[3], l[1], l[2]])*(k[1, 3] - k[2, 4]))/u^2 + ((2*(l2u^2 + l1u*l3u)*s + (l2s^2 - l2t^2 + l2u^2 + l1s*l3s - l1t*l3t - l2u*l3t + l1u*l3u + l2t*l3u)*u + 2*s*u^2 - 4*s*u*(mu13 + mu22 + dot[l[1], l[3]] + dot[l[2], l[2]]))* (k[1, 3] + k[2, 4]))/(2*u^2) + (2*I)*emu*(k[1, 2] - k[3, 4]) + (-2*(mu13 + mu22) + s)*(k[1, 2] + k[3, 4]), ((2*I)*(t*e4[p[1], p[2], l[1], l[2]] + l2t*e4[p[1], p[2], p[3], l[1]])* (k[1, 4] - k[2, 3]))/t^2 + (((l1s*l2s + (l1t - l1u)*(l2t + l2u))*t + 2*s*(l1t*l2t - 2*t*(mu12 + dot[l[1], l[2]])))*(k[1, 4] + k[2, 3]))/ (4*t^2) + ((2*I)*(u*e4[p[1], p[2], l[1], l[2]] + l2u*e4[p[1], p[2], p[3], l[1]])*(k[1, 3] - k[2, 4]))/u^2 + (((l1s*l2s + (-l1t + l1u)*(l2t + l2u))*u + 2*s*(l1u*l2u - 2*u*(mu12 + dot[l[1], l[2]])))*(k[1, 3] + k[2, 4]))/ (4*u^2) + I*emu*(k[1, 2] - k[3, 4]) - mu12*(k[1, 2] + k[3, 4]), ((-2*I)*(t*e4[p[1], p[2], l[1], l[2]] + l3t*e4[p[1], p[2], p[3], l[1]])* (k[1, 4] - k[2, 3]))/t^2 - (((l1s*l3s + (l1t - l1u)*(l3t + l3u))*t + 2*s*(l1t*l3t - 2*t*(mu13 + dot[l[1], l[3]])))*(k[1, 4] + k[2, 3]))/ (4*t^2) - ((2*I)*(u*e4[p[1], p[2], l[1], l[2]] + l3u*e4[p[1], p[2], p[3], l[1]])*(k[1, 3] - k[2, 4]))/u^2 - (((l1s*l3s + (-l1t + l1u)*(l3t + l3u))*u + 2*s*(l1u*l3u - 2*u*(mu13 + dot[l[1], l[3]])))*(k[1, 3] + k[2, 4]))/ (4*u^2) - I*emu*(k[1, 2] - k[3, 4]) + mu13*(k[1, 2] + k[3, 4]), ((-((l1s + l1u)*l2t) + l1t*(l2s + l2u) + 4*t*dot[l[2], l[3]])* (k[1, 4] + k[2, 3]))/(2*t) + ((l1u*(l2s + l2t) - (l1s + l1t)*l2u + 4*u*dot[l[2], l[3]])* (k[1, 3] + k[2, 4]))/(2*u) + (4*I)*(-((e4[p[1], p[4], l[1], l[2]]*(k[1, 4] - k[2, 3]))/t) + (e4[p[2], p[4], l[1], l[2]]*(k[1, 3] - k[2, 4]))/u + (e4[p[3], p[4], l[1], l[2]]*(k[1, 2] - k[3, 4]))/s) + (2*I)*emu*(k[1, 2] + k[1, 3] - k[1, 4] + k[2, 3] - k[2, 4] - k[3, 4]) + ((-((l1t + l1u)*l2s) + l1s*(l2t + l2u) + 4*s*dot[l[2], l[3]])* (k[1, 2] + k[3, 4]))/(2*s), 0, 0, ((-2*I)*(-(l2t*e4[p[1], p[2], p[3], l[1]]) + l1t*e4[p[1], p[2], p[3], l[2]] + t*(2*e4[p[1], p[3], l[1], l[2]] + e4[p[2], p[3], l[1], l[2]]))*(k[1, 4] - k[2, 3]))/t^2 - ((-((l1s + l1u)*l2t) + l1t*(l2s + l2u))/(4*t) + dot[l[2], l[3]])* (k[1, 4] + k[2, 3]) - ((2*I)*e4[p[2], p[4], l[1], l[2]]* (k[1, 3] - k[2, 4]))/u - ((l1u*(l2s + l2t) - (l1s + l1t)*l2u)/(4*u) + dot[l[2], l[3]])*(k[1, 3] + k[2, 4]) - ((2*I)*(-(l2s*e4[p[1], p[2], p[3], l[1]]) + l1s*e4[p[1], p[2], p[3], l[2]] + s*(e4[p[1], p[2], l[1], l[2]] + 2*e4[p[1], p[3], l[1], l[2]]))*(k[1, 2] - k[3, 4]))/s^2 - I*emu*(k[1, 2] + k[1, 3] - k[1, 4] + k[2, 3] - k[2, 4] - k[3, 4]) + (-(-((l1t + l1u)*l2s) + l1s*(l2t + l2u))/(4*s) - dot[l[2], l[3]])* (k[1, 2] + k[3, 4]), ((4*I)*(l1t*e4[p[1], p[2], p[3], l[1]] + l3t*e4[p[1], p[2], p[3], l[2]] - t*e4[p[2], p[4], l[1], l[2]])* (k[1, 4] - k[2, 3]))/t^2 + (((l1s^2 + l1t^2 - l1u^2 - l1t*l2s + l1s*l2t + l2s*l3s + l2t*l3t - l2u*l3u)*t + 2*s*(l2t^2 + l1t*l3t + t^2 - 2*t*(mu13 + mu22 + dot[l[1], l[3]] + dot[l[2], l[2]])) - 4*t^2*dot[l[2], l[3]])*(k[1, 4] + k[2, 3]))/(2*t^2) + ((4*I)*((l1u + 2*dot[p[3], l[2]])*e4[p[1], p[2], p[3], l[1]] + (l2u + 2*dot[p[1], l[1]])*e4[p[1], p[2], p[3], l[2]] + s*e4[p[1], p[3], l[1], l[2]] - u*e4[p[2], p[4], l[1], l[2]])* (k[1, 3] - k[2, 4]))/u^2 - (2*I)*emu*(k[1, 3] - k[1, 4] + k[2, 3] - k[2, 4]) + (((l1s^2 - l1t^2 + l1u^2 - l1u*l2s + l1s*l2u + l2s*l3s - l2t*l3t + l2u*l3u)*u + 2*s*(l2u^2 + l1u*l3u + u^2 - 2*u*(mu13 + mu22 + dot[l[1], l[3]] + dot[l[2], l[2]])) - 4*u^2*dot[l[2], l[3]])*(k[1, 3] + k[2, 4]))/(2*u^2) - ((4*I)*e4[p[3], p[4], l[1], l[2]]*(k[1, 2] - k[3, 4]))/s + (((l1t + l1u)*l2s - l1s*(l2t + l2u) + 2*s^2 - 4*s*(mu13 + mu22 + dot[l[2], l[3]]))*(k[1, 2] + k[3, 4]))/(2*s), ((2*I)*(t*e4[p[1], p[2], l[1], l[2]] + l2t*e4[p[1], p[2], p[3], l[1]])* (k[1, 4] - k[2, 3]))/t^2 + (((l1s*l2s + (l1t - l1u)*(l2t + l2u))*t + 2*s*(l1t*l2t - 2*t*(mu12 + dot[l[1], l[2]])))*(k[1, 4] + k[2, 3]))/ (4*t^2) + ((2*I)*(u*e4[p[1], p[2], l[1], l[2]] + l2u*e4[p[1], p[2], p[3], l[1]])*(k[1, 3] - k[2, 4]))/u^2 + (((l1s*l2s + (-l1t + l1u)*(l2t + l2u))*u + 2*s*(l1u*l2u - 2*u*(mu12 + dot[l[1], l[2]])))*(k[1, 3] + k[2, 4]))/ (4*u^2) + I*emu*(k[1, 2] - k[3, 4]) - mu12*(k[1, 2] + k[3, 4]), -((I*emu + ((2*I)*(l3t*e4[p[1], p[2], p[3], l[2]] - t*e4[p[2], p[4], l[1], l[2]]))/t^2)*(k[1, 4] - k[2, 3])) - (((-(l1t*l2s) + l1s*l2t + l2s*l3s - l2u*l3u)*t + l2t*l3t*(s - u) + 4*t*(-(mu23*s) + u*dot[l[2], l[3]]))*(k[1, 4] + k[2, 3]))/(4*t^2) - ((-I)*emu + ((2*I)*(l2u*e4[p[1], p[2], p[3], l[3]] + u*e4[p[2], p[3], l[1], l[2]]))/u^2)*(k[1, 3] - k[2, 4]) - ((l2u*l3u*(s - t) + (-(l1u*l2s) + l1s*l2u + l2s*l3s - l2t*l3t)*u + 4*u*(-(mu23*s) + t*dot[l[2], l[3]]))*(k[1, 3] + k[2, 4]))/(4*u^2) - ((2*I)*(-(l2s*e4[p[1], p[2], p[3], l[1]]) + l1s*e4[p[1], p[2], p[3], l[2]] + s*(e4[p[1], p[2], l[1], l[2]] - 2*e4[p[2], p[3], l[1], l[2]]))*(k[1, 2] - k[3, 4]))/s^2 + ((-((l1t + l1u)*l2s) + l1s*(l2t + l2u) + 4*s*(mu23 + dot[l[2], l[3]]))* (k[1, 2] + k[3, 4]))/(4*s), ((4*I)*(t*e4[p[1], p[2], l[1], l[2]] + (l2t + 2*dot[l[2], p[3]])* e4[p[1], p[2], p[3], l[1]] + 2*dot[p[2], l[1]]*e4[p[1], p[2], p[3], l[2]] - s*e4[p[2], p[3], l[1], l[2]])*(k[1, 4] - k[2, 3]))/t^2 + (((l1s*l2s + (l1t - l1u)*(l2t + l2u))*t + 2*s*(l1t*l2t - 2*t*(mu12 + dot[l[1], l[2]])))*(k[1, 4] + k[2, 3]))/ t^2 + ((4*I)*(u*e4[p[1], p[2], l[1], l[2]] + (l2u + 2*dot[l[2], p[3]])* e4[p[1], p[2], p[3], l[1]] + 2*dot[p[1], l[1]]*e4[p[1], p[2], p[3], l[2]] + s*e4[p[1], p[3], l[1], l[2]])*(k[1, 3] - k[2, 4]))/u^2 + (((l1s*l2s + (-l1t + l1u)*(l2t + l2u))*u + 2*s*(l1u*l2u - 2*u*(mu12 + dot[l[1], l[2]])))*(k[1, 3] + k[2, 4]))/ u^2 + (4*I)*emu*(k[1, 2] - k[3, 4]) - 4*mu12*(k[1, 2] + k[3, 4]), ((-4*I)*(t*e4[p[1], p[2], l[1], l[2]] + l2t*e4[p[1], p[2], p[3], l[1]])* (k[1, 4] - k[2, 3]))/t^2 - (((l1s*l2s + (l1t - l1u)*(l2t + l2u))*t + 2*s*(l1t*l2t - 2*t*(mu12 + dot[l[1], l[2]])))*(k[1, 4] + k[2, 3]))/ (2*t^2) - ((4*I)*(u*e4[p[1], p[2], l[1], l[2]] + l2u*e4[p[1], p[2], p[3], l[1]])*(k[1, 3] - k[2, 4]))/u^2 - (((l1s*l2s + (-l1t + l1u)*(l2t + l2u))*u + 2*s*(l1u*l2u - 2*u*(mu12 + dot[l[1], l[2]])))*(k[1, 3] + k[2, 4]))/ (2*u^2) - (2*I)*emu*(k[1, 2] - k[3, 4]) + 2*mu12*(k[1, 2] + k[3, 4]), ((2*I)*(t*e4[p[1], p[2], l[1], l[2]] + l2t*e4[p[1], p[2], p[3], l[1]])* (k[1, 4] - k[2, 3]))/t^2 + (((l1s*l2s + (l1t - l1u)*(l2t + l2u))*t + 2*s*(l1t*l2t - 2*t*(mu12 + dot[l[1], l[2]])))*(k[1, 4] + k[2, 3]))/ (4*t^2) + ((2*I)*(u*e4[p[1], p[2], l[1], l[2]] + l2u*e4[p[1], p[2], p[3], l[1]])*(k[1, 3] - k[2, 4]))/u^2 + (((l1s*l2s + (-l1t + l1u)*(l2t + l2u))*u + 2*s*(l1u*l2u - 2*u*(mu12 + dot[l[1], l[2]])))*(k[1, 3] + k[2, 4]))/ (4*u^2) + I*emu*(k[1, 2] - k[3, 4]) - mu12*(k[1, 2] + k[3, 4]), 4*dot[l[2], l[3]]*(k[1, 2] + k[1, 3] + k[1, 4] + k[2, 3] + k[2, 4] + k[3, 4]), 0, -2*dot[l[2], l[3]]*(k[1, 2] + k[1, 3] + k[1, 4] + k[2, 3] + k[2, 4] + k[3, 4]), 4*dot[l[2], l[2] - p[4]]* (k[1, 2] + k[1, 3] + k[1, 4] + k[2, 3] + k[2, 4] + k[3, 4]), 2*dot[l[2], -l[2] + p[4]]*(k[1, 2] + k[1, 3] + k[1, 4] + k[2, 3] + k[2, 4] + k[3, 4]), 0, 8*dot[l[1], l[2]]*(k[1, 2] + k[1, 3] + k[1, 4] + k[2, 3] + k[2, 4] + k[3, 4]), -4*dot[l[1], l[2]]* (k[1, 2] + k[1, 3] + k[1, 4] + k[2, 3] + k[2, 4] + k[3, 4]), 0, -8*dot[p[4], l[2]]*(k[1, 2] + k[1, 3] + k[1, 4] + k[2, 3] + k[2, 4] + k[3, 4]), 4*dot[l[2], p[4]]*(k[1, 2] + k[1, 3] + k[1, 4] + k[2, 3] + k[2, 4] + k[3, 4])} colorFactors = {f[1, 5, 10]*f[2, 6, 5]*f[3, 8, 7]*f[4, 9, 8]*f[6, 7, 11]* f[9, 10, 11], tr[3, 4, 11, 1, 2, 11], f[3, 8, 7]*f[4, 9, 8]* tr[1, 2, 7, 9], f[1, 5, 10]*f[2, 6, 5]*f[3, 7, 6]*f[4, 9, 8]* f[7, 8, 11]*f[9, 10, 11], tr[1, 2, 3, 11, 4, 11], f[4, 9, 8]*tr[1, 2, 3, 8, 9], f[1, 5, 10]*f[2, 6, 5]*f[3, 7, 6]* tr[4, 10, 7], f[1, 5, 9]*f[2, 6, 5]*f[3, 11, 10]*f[4, 8, 7]*f[6, 7, 11]* f[8, 9, 10], f[3, 11, 10]*tr[1, 2, 11, 4, 10], f[1, 5, 9]*f[2, 6, 5]*tr[3, 6, 4, 9], f[1, 5, 7]*f[2, 6, 5]*f[3, 10, 9]* f[4, 11, 10]*f[6, 8, 7]*f[8, 9, 11], f[3, 10, 9]*f[4, 11, 10]* f[8, 9, 11]*tr[1, 2, 8], tr[1, 2, 8]*tr[3, 4, 8], f[1, 5, 11]*f[2, 6, 5]*f[3, 7, 6]*f[4, 8, 7]*f[8, 9, 10]*f[9, 11, 10], tr[1, 2, 3, 4, 10, 10], f[1, 5, 11]*f[2, 6, 5]*f[3, 7, 6]*f[4, 8, 7]* tr[8, 11], f[1, 5, 8]*f[2, 6, 5]*f[3, 7, 6]*f[4, 10, 11]*f[7, 9, 8]* f[9, 11, 10], f[1, 5, 8]*f[2, 6, 5]*f[3, 7, 6]*f[7, 9, 8]*tr[4, 9], tr[4, 9]*tr[1, 2, 3, 9], f[1, 5, 9]*f[2, 6, 5]*f[3, 7, 6]*f[4, 8, 7]* f[8, 10, 9]*f[10, 11, 11], f[1, 5, 9]*f[2, 6, 5]*f[3, 7, 6]*f[4, 8, 7]* f[8, 10, 9]*tr[10], tr[10]*tr[1, 2, 3, 4, 10], f[1, 5, 8]*f[2, 6, 5]*f[3, 7, 6]*f[4, 10, 9]*f[7, 9, 8]*f[10, 11, 11], f[1, 5, 8]*f[2, 6, 5]*f[3, 7, 6]*f[4, 10, 9]*f[7, 9, 8]*tr[10]} propagators = {{l[1] - p[1], l[1] - p[1] - p[2], l[2] - p[3] - p[4], l[2] - p[4], l[2], l[1], -l[1] - l[2]}, {-l[1] + p[1], -l[1] + p[1] + p[2], l[2] - p[3] - p[4], l[2] - p[4], l[2], -l[1], -l[1] - l[2]}, {-l[1] + p[1], -l[1] + p[1] + p[2], l[2] - p[3] - p[4], l[2] - p[4], l[2], -l[1], -l[1] - l[2]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[2] - p[4], l[2], l[1], -l[1] - l[2]}, {-l[1] + p[1], -l[1] + p[1] + p[2], -l[1] - p[4], l[2] - p[4], l[2], -l[1], -l[1] - l[2]}, {-l[1] + p[1], -l[1] + p[1] + p[2], -l[1] - p[4], l[2] - p[4], l[2], -l[1], -l[1] - l[2]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[2] - p[4], l[2], l[1], l[1] + l[2]}, {l[1] - p[1], l[1] - p[1] - p[2], l[2] - p[4], l[2], l[1], -l[1] - l[2], -l[1] - l[2] - p[3]}, {-l[1] + p[1], -l[1] + p[1] + p[2], l[2] - p[4], l[2], -l[1], -l[1] - l[2], -l[1] - l[2] - p[3]}, {l[1] - p[1], l[1] - p[1] - p[2], l[2] - p[4], l[2], l[1], l[1] + l[2], l[1] + l[2] + p[3]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1], p[1] + p[2], l[2] - p[3] - p[4], l[2] - p[4], l[2]}, {-l[1] + p[1], -l[1] + p[1] + p[2], -l[1], p[1] + p[2], l[2] - p[3] - p[4], l[2] - p[4], l[2]}, {-l[1] + p[1], -l[1] + p[1] + p[2], -l[1], p[1] + p[2], l[2] - p[3] - p[4], l[2] - p[4], l[2]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[1], l[2], l[1] + l[2], l[1]}, {-l[1] + p[1], -l[1] + p[1] + p[2], -l[1] - p[4], -l[1], l[2], l[1] + l[2], -l[1]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[1], l[2], l[1] + l[2], l[1]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[1], p[4], l[2], l[2] - p[4]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[1], p[4], l[2], l[2] - p[4]}, {-l[1] + p[1], -l[1] + p[1] + p[2], -l[1] - p[4], -l[1], p[4], l[2], l[2] - p[4]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[1], l[1], 0, l[2]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[1], l[1], 0, l[2]}, {-l[1] + p[1], -l[1] + p[1] + p[2], -l[1] - p[4], -l[1], -l[1], 0, l[2]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[1], p[4], 0, l[2]}, {l[1] - p[1], l[1] - p[1] - p[2], l[1] + p[4], l[1], p[4], 0, l[2]}} symmetryFactors = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1} gravityNumerators = {(n[1][1, 2, 3, 4, l[1], l[2]]^2 + (-6 + Ds)*(n[2][1, 2, 3, 4, l[1], l[2]]^2 + n[3][1, 2, 3, 4, l[1], l[2]]^2 + n[3][3, 4, 1, 2, -l[2] + p[3] + p[4], -l[1] + p[1] + p[2]]^2))/4, (n[8][1, 2, 3, 4, l[1], l[2]]^2 + (-6 + Ds)* (n[9][1, 2, 3, 4, l[1], l[2]]^2 + n[9][1, 2, 4, 3, l[1], -l[1] - l[2]] + n[10][1, 2, 3, 4, l[1], l[2]]^2))/2, (n[11][1, 2, 3, 4, l[1], l[2]]^2 + (-6 + Ds)* (n[12][1, 2, 3, 4, l[1], l[2]]^2 + n[12][3, 4, 1, 2, -l[2] + p[3] + p[4], -l[1] + p[1] + p[2]]^2) + (-6 + Ds)^2*n[13][1, 2, 3, 4, l[1], l[2]]^2)/8, (n[4][1, 2, 3, 4, l[1], l[2]]^2 + (-6 + Ds)* (n[5][1, 2, 3, 4, l[1], l[2]]^2 + n[6][1, 2, 3, 4, l[1], l[2]]^2 + n[7][1, 2, 3, 4, l[1], l[2]]^2))/2, (n[14][1, 2, 3, 4, l[1], l[2]]^2 + (-6 + Ds)* (n[15][1, 2, 3, 4, l[1], -l[1] - l[2]] + n[15][1, 2, 3, 4, l[1], l[2]] + n[16][1, 2, 3, 4, l[1], l[2]]))/4}