Quantum periods: A census of φ^4-transcendentals

TL;DR

The article investigates the origin and structure of $\phi^4$-periods in massless four-dimensional theory, showing that primitive vacuum graphs generate these periods and that certain graph-theoretic reductions (e.g., vertex connectivity $3$) yield products of lower-order periods. It introduces a twist identity and leverages Fourier-type identities to reduce the catalog of irreducible periods up to loop order $8$, producing a census with 73 primitive graphs, 60 irreducible before reductions, 48 irreducible after reductions, and 31 periods identified as integer combinations of multiple zeta values, while 17 remain inaccessible. The work situates these periods as algebraic periods in the Kontsevich–Zagier sense and discusses connections to mixed Tate motives, the Hopf-algebra of renormalization, and potential implications for high-precision QFT calculations. Overall, it provides a rigorous framework and computational results that link perturbative QFT, number theory, and algebraic geometry, offering a benchmark for evaluating advanced calculational techniques.

Abstract

Perturbative quantum field theories frequently feature rational linear combinations of multiple zeta values (periods). In massless φ^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.

Figures (2)

• Figure 1: A primitive graph with one loop (a) and one with three loops (b).
• Figure :

Theorems & Definitions (2)

• proof
• Example 2.2