Properties of $c_2$ invariants of Feynman graphs

Abstract

Abstract. The c2 invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the c2 invariant in momentum space and prove that it equals the c2 invariant in parametric space for overall log-divergent graphs. Then we show that the c2 invariant of a graph vanishes whenever it contains subdivergences. Finally, we investigate how the c2 invariant relates to identities such as the four-term relation in knot theory. IntroductionLet G be a connected graph. The graph polynomial of G is defined by associating a variable x e to every edge e of G and settingwhere the sum is over all spanning trees T of G. These polynomials first appeared in Kirchhoff's work on currents in electrical networks [16]. Let N G denote the number of edges of G, and let h G denote the number of independent cycles in G (the first Betti number). Of particular interest is the case when G is primitive and overall logarithmically divergent:N γ > 2h γ for all strict non-trivial subgraphs γ G .For such graphs, the corresponding Feynman integral (or residue) is independent of the choice of renormalization scheme and can be defined by the following convergent integral in parametric spaceThe numbers I G are notoriously difficult to calculate, and have been investigated intensively from the numerical [5,20] and algebro-geometric points of view [4,9]. For graphs in φ 4 theory with subdivergences, the renormalised Karen Yeats is supported by an NSERC discovery grant and would like to thank Samson Black for explaining knots. Francis Brown is partially supported by ERC grant 257638. Francis Brown and Oliver Schnetz thank Humboldt University, Berlin, for support as visiting guest scientists. All three authors thank Humboldt University for hospitality.

Modular forms in quantum field theory

“…Then the c 2 -invariant is constant: c 2 (G) q ≡ c mod q for some constant c ∈ Z. In [8] it was shown that any graph in φ 4 theory which is not primitive, i.e., containing a non-trivial subdivergence, has vanishing c 2 -invariant.…”

“…21) appear at loop order 8. In fact all four non quasi-constant graphs at loop order 8 are modular with respect to the newforms with weight and level equal to (3,7), (3,8), (4,5) and (6,3). The graphs are depicted in figure 5.…”

“…Furthermore, there is not a single graph with c 2 = −z 3 or −z 4 whose period is known. What we do have is the conjectured period of two completed primitive graphs at loop order 7 (graphs P 7,8 and P 7,9 in [20]) with c 2 -invariant −z 2 . They are weight 11 multiple zeta values, namely [4] To summarize the results for quasi-constant graphs, the 10 loop data are consistent with the following, rather surprising, conjecture.…”

“…Using the techniques of [8,9] Every completed primitive graph with at least five vertices has a c 2 -invariant and the first two examples are…”

A Special Case of Completion Invariance for thec2Invariant of a Graph

“…That this is well defined is proved in [24]. The same definition can be made for prime powers, but we will stick to primes p. The c 2 invariant has interesting properties [8,10,17,18], yields interesting sequences in p such as coefficient sequences of modular forms [9,20], and predicts properties of the Feynman period. A simple and striking example of the last of these is that the c (p) 2 (G) = 0 for all p corresponds to when the Feynman period apparently has a drop in transcendental weight relative to the size of the graph 3 .…”