φ³ Graph Atlas tensor structures of βijk and γφ
Conventions: d = 6 − ε, λ0 = με/2(λ + Z-poles), so β = −(ε/2)λ + β and an L-loop 1/ε residue a1 gives βL = L·a1, γL = L·(residue of Zφ1/2); the loop factor is absorbed into the real coupling λ (Schnetz's convention; arXiv:2507.20761 uses iλ, a relative (−1)L), so the one-loop β is the triangle λiabλjbcλkca with coefficient 1 (V1.1) and γφ = −(1/12) λiabλjab (S1.1). S = 1/|Aut| is the symmetry factor, automorphisms counted with the external legs held fixed and parallel lines included. |stab| counts the permutations of ijk (ij for a propagator) leaving the tensor unchanged, orbit = 6/|stab| (2/|stab|) its distinct terms; every coefficient multiplies their sum, so βL = L·(1/ε residue of Zλ) and γφ,L = −L·(1/ε residue of Zφ−½). The percolation factor is Schnetz's group factor of one term for the n-state Potts theory (percolation is n = 1; the theory has n − 1 fields), and a structure contributes coefficient × orbit × percolation factor in that reduction.
Graph ordering and the six-loop input data follow O. Schnetz's
HyperlogProcedures 0.8; the structures and every coefficient shown here were extracted and recomputed by tensorial minimal subtraction (MS-bar) with Claude Fable 5.1. The full set of 35208 structures is available as
phi3_atlas.json (36.9 MB). Provenance, citation and licence (CC BY 4.0):
github.com/andstergiou/phi3-graph-atlas.