A study on prefixes of $c_2$ invariants

Abstract

This paper begins by reviewing recent progress that has been made by taking a combinatorial perspective on the c 2 invariant, an arithmetic graph invariant with connections to Feynman integrals. Then it proceeds to report on some recent calculations of c 2 invariants for two families of circulant graphs at small primes. These calculations support the idea that all possible finite sequences appear as initial segments of c 2 invariants, in contrast to their apparent sparsity on small graphs.Thanks to Oliver Schnetz for goading me into doing the prefix calculation, and for his continued excitement about the c 2 invariant. Thanks to Iain Crump and Freddy Cachazo for discussions. Thanks to NSERC and the Humboldt foundation for support. Thanks to the organizers of the Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) conference for their excellent conference.

Further investigations into the graph theory of $\phi^4$-periods and the $c_2$ invariant

“…In our results, we show that the ratio of unique c 2 invariants to prime ancestors does not change much between different loop orders. A related observation about the distribution of c 2 invariants appeared in [37]. One of us in [37] showed that the distribution of decompleted c 2 prefixes for the circulant graphs C n .1; 3/ and C n .2; 3/ is very uniform as we increase n. This uniformity, even within the same family of graphs, gave rise to the idea that maybe all finite prefixes show up in 4 c 2 invariants if the loop order is high enough.…”

“…A related observation about the distribution of c 2 invariants appeared in [37]. One of us in [37] showed that the distribution of decompleted c 2 prefixes for the circulant graphs C n .1; 3/ and C n .2; 3/ is very uniform as we increase n. This uniformity, even within the same family of graphs, gave rise to the idea that maybe all finite prefixes show up in 4 c 2 invariants if the loop order is high enough. This line of thought seems to be supported by the evidence of the present calculations.…”

“…However, 11 loops remained out of reach. Another of us alone and with Chorney [12,34,37] has used a different approach only for very small primes though applicable to the entirety of certain special families of graphs. Part of this approach can also be applied to individual graphs and is more tractable for large graphs than the previous approach, though larger primes are less accessible as the complexity growth in the size of the prime is worse.…”

c₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction

“…The c 2 has been studied quite deeply in the context of ϕ 4 quantum field theory (Section 2.7). The focus of these studies can either be the general mathematical structure of the c 2 [11,15,31] or the zoology of the geometries identified by c 2 s [9,10,13,27,30,32]. The nature of this article is more in the latter direction, particularly when we analyze the c 2 s of small kernels in Section 4.…”

“…In a certain sense these hourglass chains can be considered as 'telescopes' that enable us to look into geometries of Feynman graphs at very high loop order (i.e., the number of independent cycles in L − v). This has never been achieved before: all previous techniques were either restricted to the analysis of small graphs, or they worked in a way which was fundamentally prime-by-prime [13,30,32] and hence did not lead to non-trivial graph families with the same underlying geometries.…”