Feynman diagrams as a weight system: four-loop test of a four-term relation

Abstract

At four loops there first occurs a test of the four-term relation derived by the second author in the course of investigating whether counterterms from subdivergencefree diagrams form a weight system. This test relates counterterms in a four-dimensional field theory with Yukawa and φ 4 interactions, where no such relation was previously suspected. Using integration by parts, we reduce each counterterm to massless two-loop two-point integrals. The four-term relation is verified, with G 1 − G 2 + G 3 − G 4 = 0 − 3ζ 3 + 6ζ 3 − 3ζ 3 = 0, demonstrating non-trivial cancellation of the trefoil knot and thus supporting the emerging connection between knots and counterterms, via transcendental numbers assigned by four-dimensional field theories to chord diagrams. Restrictions to scalar couplings and renormalizable interactions are found to be necessary for the existence of a pure four-term relation. Strong indications of richer structure are given at five loops.

Weight Systems from Feynman Diagrams

“…Also, in a parallel paper, we will present an explicit 4-loop calculation which establishes the 4TR as proposed here, and demonstrates its failure in cases when our derivation is not applicable [15].…”

“…An example is when I 3,2 provides a subdivergence at s 2 = 0, while I 8,2 does only so at s 2 = 0. This asymmetry provides us finally with an extra contribution which diverges with λ, which can be shown to have the form of a finite part of a two-loop integral of master topology [15]. A more detailed analysis will be given elsewhere.…”

Properties of $c_2$ invariants of Feynman graphs

“…In this paper, we investigated the 4-term relation for chord diagrams, which was shown to hold in some cases in [6], but found no such relation on the level of c 2 invariants in φ 4 theory. To our surprise, however, we found that the 4-term identity actually holds true on the level of the denominator polynomials D 7 G .…”

“…If we subtract 2h [1], and the theorem follows. (6) and N G/ /e = 0 or N G/ /e ≥ h G/ /e + 1, hence q|[Ψ G/ /e ] q . Again, the theorem follows.…”

A dilogarithmic 3-dimensional ising tetrahedron

“…We shall show that the dispersive methods of [4,5] enable a reduction of C T et , as for any assignment of masses, to single integrals of logarithms. Then we shall describe how the lattice algorithm PSLQ [6] achieved a very simple reduction of C T et to a Clausen integral, which gives an exponentially convergent sum that reveals a new feature of the distinctive mapping [7] of diagrams [5,8,9,10,11,12] to numbers [13,14,15,16] provided by quantum field theory.…”