Cluster algebras for Feynman integrals

TL;DR

The paper provides evidence that cluster algebras, notably the finite $C_{2}$ algebra, govern the function space of certain Feynman integrals in dimensional regularization, specifically four-point scattering with one off-shell leg. By embedding $C_{2}$ into $A_{3}$, it links cluster-adjacency to the extended Steinmann relations, constraining the alphabet of admissible polylogarithmic functions and enabling a bootstrap approach. It also develops a practical procedure to match arbitrary alphabets to cluster algebras and presents multiple explicit identifications, including one-loop and finite higher-point cases mapped to $A_n$, $C_n$, and $D_n$ algebras, and connections to the $G(4,8)$ framework. The findings open pathways to generalizing bootstrap techniques beyond ${\cal N}=4$ sYM, with potential impact on Higgs-plus-jet, vector-boson-plus-jet, and form-factor computations, and suggest a broader role for cluster algebras in generic quantum field theories.

Abstract

We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a $C_{2}$ cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. By embedding $C_{2}$ inside the $A_3$ cluster algebra, we identify these adjacencies with the extended Steinmann relations for six-particle massless scattering. The cluster algebra connection we find restricts the functions space for vector boson or Higgs plus jet amplitudes, and for form factors recently considered in $\mathcal{N}=4$ super Yang-Mills. We explain general procedures for studying relationships between alphabets of generalized polylogarithmic functions and cluster algebras, and use them to provide various identifications of one-loop alphabets with cluster algebras. In particular, we show how one can obtain one-loop alphabets for five-particle scattering from a recently discussed dual conformal eight-particle alphabet related to the $G(4,8)$ cluster algebra.

Figures (5)

• Figure 1: The exchange graph of the $C_2$ cluster algebra, with cluster coordinates ordered as $a_i,a_{i+1}$.
• Figure 2: Examples of known two- and three-loop four-point integrals with one off-shell leg, $P^2 \neq 0$.
• Figure 3: The intersection of the exchange graph of the $A_3$ cluster algebra with the parity invariant plane (in pink) is shown in white. It can be identified with the $C_2$ cluster exchange graph shown in Fig. \ref{['B2C2ExchangeGraph']}.
• Figure 4: Momentum twistors in a dual conformal two-mass-hard hexagon configuration. This is equivalent to five-particle kinematics with one off-shell leg $P$.
• Figure 5: Two-mass-hard hexagon configuration. The dual coordinates $x_{i}$ denote region momenta.