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Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case

Samuel Abreu, Ruth Britto, Claude Duhr, Einan Gardi

TL;DR

The paper introduces a diagrammatic coaction on cut and uncut one-loop Feynman graphs that mirrors the coaction on multiple polylogarithms (MPLs) when expanded in the dimensional regulator ε. By defining a deformation of the incidence coaction with a = 1/2, the authors formulate a precise conjecture: the MPL coaction of any one-loop integral equals a diagrammatic coaction on cut graphs, with cuts encoding the MPL residues and discontinuities. They validate the conjecture across a broad class of examples (tadpoles, bubbles, triangles, boxes) and show consistency with discontinuities, differential equations, and the symbol, while connecting to extended dual conformal invariance and to the master-integrand/master-contour framework. The work offers a new combinatorial and geometric perspective on the analytic structure of one-loop integrals, and lays groundwork for potential extensions to higher-loop integrals where MPLs still suffice. Overall, it provides a concrete, graph-based Hopf-algebra structure that reproduces known MPL behavior and yields practical tools for deriving differential equations and symbols recursively.

Abstract

We construct a diagrammatic coaction acting on one-loop Feynman graphs and their cuts. The graphs are naturally identified with the corresponding (cut) Feynman integrals in dimensional regularization, whose coefficients of the Laurent expansion in the dimensional regulator are multiple polylogarithms (MPLs). Our main result is the conjecture that this diagrammatic coaction reproduces the combinatorics of the coaction on MPLs order by order in the Laurent expansion. We show that our conjecture holds in a broad range of nontrivial one-loop integrals. We then explore its consequences for the study of discontinuities of Feynman integrals, and the differential equations that they satisfy. In particular, using the diagrammatic coaction along with information from cuts, we explicitly derive differential equations for any one-loop Feynman integral. We also explain how to construct the symbol of any one-loop Feynman integral recursively. Finally, we show that our diagrammatic coaction follows, in the special case of one-loop integrals, from a more general coaction proposed recently, which is constructed by pairing master integrands with corresponding master contours.

Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case

TL;DR

The paper introduces a diagrammatic coaction on cut and uncut one-loop Feynman graphs that mirrors the coaction on multiple polylogarithms (MPLs) when expanded in the dimensional regulator ε. By defining a deformation of the incidence coaction with a = 1/2, the authors formulate a precise conjecture: the MPL coaction of any one-loop integral equals a diagrammatic coaction on cut graphs, with cuts encoding the MPL residues and discontinuities. They validate the conjecture across a broad class of examples (tadpoles, bubbles, triangles, boxes) and show consistency with discontinuities, differential equations, and the symbol, while connecting to extended dual conformal invariance and to the master-integrand/master-contour framework. The work offers a new combinatorial and geometric perspective on the analytic structure of one-loop integrals, and lays groundwork for potential extensions to higher-loop integrals where MPLs still suffice. Overall, it provides a concrete, graph-based Hopf-algebra structure that reproduces known MPL behavior and yields practical tools for deriving differential equations and symbols recursively.

Abstract

We construct a diagrammatic coaction acting on one-loop Feynman graphs and their cuts. The graphs are naturally identified with the corresponding (cut) Feynman integrals in dimensional regularization, whose coefficients of the Laurent expansion in the dimensional regulator are multiple polylogarithms (MPLs). Our main result is the conjecture that this diagrammatic coaction reproduces the combinatorics of the coaction on MPLs order by order in the Laurent expansion. We show that our conjecture holds in a broad range of nontrivial one-loop integrals. We then explore its consequences for the study of discontinuities of Feynman integrals, and the differential equations that they satisfy. In particular, using the diagrammatic coaction along with information from cuts, we explicitly derive differential equations for any one-loop Feynman integral. We also explain how to construct the symbol of any one-loop Feynman integral recursively. Finally, we show that our diagrammatic coaction follows, in the special case of one-loop integrals, from a more general coaction proposed recently, which is constructed by pairing master integrands with corresponding master contours.

Paper Structure

This paper contains 68 sections, 265 equations, 3 figures.

Table of Contents

  1. Introduction
  2. One-loop Feynman integrals and their cuts
  3. One-loop Feynman integrals
  4. One-loop cut integrals
  5. Review of multiple polylogarithms
  6. First examples of the diagrammatic representation of the coaction
  7. The tadpole integral
  8. The massless bubble integral
  9. The triangle integral with three massive external legs
  10. The bubble integral with massive propagators
  11. A coaction on cut graphs
  12. The incidence coaction on cut graphs
  13. The deformed incidence coaction
  14. The main conjecture
  15. Dimensional regularization
  16. ...and 53 more sections

Figures (3)

  • Figure 1: (a) The integration path for $G(a_1,a_2,a_3,a_4;z)$. (b) The integration path for $G_{\vec{b}}(a_1,a_2,a_3,a_4;z)$ for $\vec{b}=(a_2,a_3,a_4)$. The path $\gamma_{\vec{b}}$ encircles each of the singularities $a_i\in \vec{b}$ counter-clockwise, in the order in which they appear in $\vec{b}$.
  • Figure 2: Notation for external and internal edges of three-point functions. Propagator $e_i$ has mass $m_i^2$, and external leg $\bm{i}$ has mass $p_i^2$.
  • Figure 3: Notation for external and internal edges of four-point functions. Propagator $e_i$ has mass $m_i^2$, and external leg $\bm{i}$ has mass $p_i^2$.