The Galois coaction on $\phi^4$ periods

Abstract

We report on calculations of Feynman periods of primitive logdivergent 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. P φ 4 := lin Q P (G) : G primitive log-divergent and φ 4 ⊆ P log := lin Q {P (G) : G primitive log-divergent} ⊂ P denote the Q-vector spaces spanned by primitive log-divergent periods. They are subspaces of the Q-algebra P of periods in the sense of Kontsevich and Zagier [44]. 1 We obtain finite-dimensional subspaces if we restrict the loop order of the graphs, P •,≤n := lin Q {P (G) : h G ≤ n} (1.4)for φ 4 graphs or general primitive log-divergent graphs G, respectively.

Renormalization group functions of $\phi^4$ theory in the MS-scheme to six loops

“…Our approach can in principle also be used at 7 loops, but the calculation using graphical functions promises to be much more efficient and is already underway [59]. The contributions from 7-loop graphs without subdivergences (these are expected to give the most complicated transcendental numbers) have been completed already [52].…”

“…Of the many recent advances made in the evaluation of Feynman integrals, parametric integration is one of the most powerful methods for p-integrals; surpassed only by the position-space approach of graphical functions [58] and the combination [34] of both techniques used in [52,59].…”

From positive geometries to a coaction on hypergeometric functions

“…Φϕ 13 = 1 + O( ) . (7.8) 6 We chose this configuration so that the intersection of the hyperplanes H3 and H4 is inside the first quadrant of the (u, v) plane. This leads to a pole associated with the factor 1 − xu − yv inside the standard integration region, γ123.…”

Origin of the Six-Gluon Amplitude in Planar N=4 Supersymmetric Yang-Mills Theory

“…with ρ = ρ(g 2 ) a function of the coupling constant. This function was determined iteratively in [13,33] by demanding that the spaces of functions in which the perturbative amplitudes live obey a coaction principle associated to a cosmic Galois group [48][49][50]. The implementation of this requirement fixes ρ order by order in perturbation theory, ln ρ = 8ζ 2 3 g 6 − 160ζ 3 ζ 5 g 8 + 16(−2ζ 4 ζ 2 3 + 57ζ 2 5 + 105ζ 3 ζ 7 )g 10 + .…”