The five-twist identity for Feynman periods

TL;DR

This work introduces a new five-twist identity for Feynman periods, acting on five-vertex cuts of completed primitive graphs, and proves its independence from the known twist, Fourier identity/split, and planar duality in $\\phi^4$ theory. The authors develop a general framework for internally completed four-point integrals using graphical functions and dual planar graphs, showing that double transpositions of external legs yield invariances under suitable degree and planarity conditions. They then derive the five-twist by inserting a planar dual $G_1^*$ and performing a sequence of decomplete–dualize–twist operations, establishing a robust mechanism for relating subgraph insertions without altering the overall period. Applied to $\\phi^4$ theory, the five-twist generates loop-ordered identities up to eleven loops, including several relations between $\\phi^4$ and non-$\\phi^4$ periods and evidence of independence from established identities (e.g., via $P_{9,103}$), suggesting a richer combinatorial structure of periods and potential extensions to broader QFT contexts.

Abstract

We prove a new identity for Feynman periods that acts on five-vertex cuts of completed primitive Feynman graphs. It is shown that in $φ^4$ theory this identity is independent from existing identities which are the twist, the Fourier identity and the Fourier split.

Figures (6)

• Figure 1: The bubble and the tetrahedron are the smallest primitive graphs in $\phi^4$ theory.
• Figure 2: The five-twist on the completed graph (left) and as reflections along diagonals of a square in the decompleted graph (right). The shaded areas stand for any subgraphs. Only if the graph $G_1$ has specific properties, the five-twist becomes an identity for Feynman periods.
• Figure 3: The smallest nontrivial five-twist links $P_{7,2}$ to the non-$\phi^4$ period $P^{\mathrm{non}\,\phi^4}_{7,17}$. The dashed edge on the right hand side has weight $-1$ (a numerator edge), all other edges have weight $1$.
• Figure 4: Examples of insertions $G_1$ in the five-twist.
• Figure 5: The only four vertex split of the $\phi^4$ graph $P_{9,103}$ is at the gray vertices. The twist at these vertices differs from the five-twist along the subgraph that is depicted at the left of Figure \ref{['fig:G1']}. The vertex $\infty$ is the center of the small square.

Theorems & Definitions (15)

• Proposition 2
• Definition 3
• Proposition 4
• proof
• Proposition 5
• proof
• Corollary 6
• proof
• Remark 7
• Proposition 8