Wavefunction coefficients from Amplitubes

TL;DR

This work develops a unified graph-theoretic framework that connects wavefunction coefficients $\Psi_G$ (edge-centric binary tubes) with amplitudes $A_G$ (vertex-centric unary tubes) through cut tubings and decorated orientations. It derives a central edge-cut expansion $\Psi_G = \sum_{\mathbf{e}\subset E_G} (-1)^{|\mathbf{e}|} \prod_{b\in \mathbf{b}_e} A_b$ and shows how substituting directed-graph expansions yields decorated amplitudes consistent with partial-fraction representations of cosmological time integrals. By introducing cut tubings, the authors link acyclic decorated orientations $\text{aDec}(G)$ to the basis functions in the kinematic flow, with counts expressible via Tutte-polynomial data and connections to graph associahedra and cosmological polytopes. The results illuminate a deep combinatorial structure behind flat-space wavefunctions and amplitudes, offering a path to systematic counting and representation through graph-theoretic invariants and decorated orientations.

Abstract

Given a graph its set of connected subgraphs (tubes) can be defined in two ways: either by considering subsets of edges, or by considering subsets of vertices. We refer to these as binary tubes and unary tubes respectively. Both notions come with a natural compatibility condition between tubes which differ by a simple adjacency constraint. Compatible sets of tubes are refered to as tubings. By considering the set of binary tubes, and summing over all maximal binary-tubings, one is lead to an expression for the flat space wavefunction coefficients relevant for computing cosmological correlators. On the other hand, considering the set of unary tubes, and summing over all maximal unary-tubings, one is lead to expressions recently referred to as amplitubes which resemble the scattering amplitudes of $\text{tr}(φ^3)$ theory. In this paper we study the two definitions of tubing in order to provide a new formula for the flat space wavefunction coefficient for a single graph as a sum over products of amplitubes. Motivated by our rewriting of the wavefunction coefficient we introduce a new definition of tubing which makes use of both the binary and unary tubes which we refer to as cut tubings. We explain how each cut tubing induces a decorated orientation of the underlying graph satisfying an acyclic condition and demonstrate how the set of all acyclic decorated orientations for a given graph count the number of basis functions appearing in the kinematic flow.

Figures (4)

• Figure 1: The $22$ acyclic decorated orientations of the two-loop sunrise.
• Figure 2: The $50$ acyclic decorated orientations of the three cycle.
• Figure 3: The $40$ acyclic decorated orientations of the one-loop frying pan.
• Figure 4: The $14$ decorated orientations of the three cycle that fail to be acyclic.