Quantum periods: A census of \phi^4-transcendentals

TL;DR

This paper catalogs the periods arising from primitive vacuum graphs in massless 4-theory, introduces a new identity among these periods, and provides a comprehensive list up to eight loops.

Contribution

It presents a detailed census of 4-transcendentals, proves a new twist identity among periods, and analyzes the reducibility of graphs based on vertex connectivity.

Findings

Periods originate from primitive vacuum graphs in 4-theory

Vertex connectivity 3 graphs are reducible to lower loop products

A new twist identity among periods is established

Abstract

Perturbative quantum field theories frequently feature rational linear combinations of multiple zeta values (periods). In massless \phi^4-theory we show that the periods originate from certain `primitive' vacuum graphs. Graphs with vertex connectivity 3 are reducible in the sense that they lead to products of periods with lower loop order. A new `twist' identity amongst periods is proved and a list of graphs (the census) with their periods, if available, is given up to loop order 8.