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The Galois coaction on $φ^4$ periods

Erik Panzer, Oliver Schnetz

TL;DR

This work investigates the Galois coaction on motivic φ^4 periods, compiling extensive computations up to eleven loops and formulating strong conjectures that φ^4 periods close under the coaction. It leverages motivic periods, the $f$-alphabet, and a parity basis to express periods and their conjugates, revealing a highly constrained structure tied to the $c_2$-invariant and to Deligne-type bases. The results distinguish φ^4 periods from general primitive log-divergent periods, illustrate weight-drop phenomena, and identify key graphs (e.g., ladders and $K_5$-based ancestors) driving the observed patterns. The data, including explicit representations and a complete dataset for up to eleven loops, provide a valuable resource for validating the coaction conjecture and guiding future explorations beyond multiple polylogarithms.

Abstract

We report on calculations of Feynman periods of primitive log-divergent $φ^4$ graphs up to eleven loops. The structure of $φ^4$ periods is described by a series of conjectures. In particular, we discuss the possibility that $φ^4$ periods are a comodule under the Galois coaction. Finally, we compare the results with the periods of primitive log-divergent non-$φ^4$ graphs up to eight loops and find remarkable differences to $φ^4$ periods. Explicit results for all periods we could compute are provided in ancillary files.

The Galois coaction on $φ^4$ periods

TL;DR

This work investigates the Galois coaction on motivic φ^4 periods, compiling extensive computations up to eleven loops and formulating strong conjectures that φ^4 periods close under the coaction. It leverages motivic periods, the -alphabet, and a parity basis to express periods and their conjugates, revealing a highly constrained structure tied to the -invariant and to Deligne-type bases. The results distinguish φ^4 periods from general primitive log-divergent periods, illustrate weight-drop phenomena, and identify key graphs (e.g., ladders and -based ancestors) driving the observed patterns. The data, including explicit representations and a complete dataset for up to eleven loops, provide a valuable resource for validating the coaction conjecture and guiding future explorations beyond multiple polylogarithms.

Abstract

We report on calculations of Feynman periods of primitive log-divergent graphs up to eleven loops. The structure of periods is described by a series of conjectures. In particular, we discuss the possibility that periods are a comodule under the Galois coaction. Finally, we compare the results with the periods of primitive log-divergent non- graphs up to eight loops and find remarkable differences to periods. Explicit results for all periods we could compute are provided in ancillary files.

Paper Structure

This paper contains 29 sections, 16 theorems, 77 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Feynman periods
  3. Galois theory for periods
  4. Motivic Feynman periods
  5. The $c_2$-invariant
  6. quasi-constants
  7. modular forms
  8. Results
  9. Iterated integrals and multiple zeta values
  10. Iterated integrals
  11. Motivic iterated integrals
  12. The Deligne basis
  13. The $f$-alphabet
  14. The derivation $\delta_m$
  15. The parity basis
  16. ...and 14 more sections

Key Result

Theorem 1.1

For every integer $n \geq 3$, the period of the zig-zag graph $Z_{n}$ is given by

Figures (7)

  • Figure 1: The zig-zag graphs with five and with six loops.
  • Figure 2: The completion (see Definition \ref{['def:completion']}) of the graph $P_{7,11}$ is the circulant $C^9_{1,3}$. Its period \ref{['eq:p711']} evaluates to multiple polylogarithms at sixth roots of unity.
  • Figure 3: The completions of the zig-zag graphs $Z_{n}$ of Figure \ref{['fig:zig-zags']} are circulants $\overline{Z_{n}}=C_{1,2}^{\,n+2}$.
  • Figure 4: A vertex cut of size three factorizes the period of a graph.
  • Figure 5: Double triangle reduction: Replace the joint vertex of two attached triangles by a crossing. Vertices $v$ and $w$ are not allowed to be identical.
  • ...and 2 more figures

Theorems & Definitions (48)

  • Theorem 1.1: F. Brown and O. Schnetz BrownSchnetz:ZigZag, conjectured in BK:KnotsAndNumbers
  • Conjecture 1.2
  • Conjecture 1.3
  • Definition 1.4
  • Theorem 1.5: small graphs principle, Brown:FeynmanAmplitudesGalois
  • Remark 1.6
  • Example 2.1
  • Theorem 2.2: P. Deligne Deligne:GroupeFondamental
  • Remark 2.3
  • Corollary 2.4
  • ...and 38 more