Communications in Number Theory and Physics

Volume 4 (2010)

Number 1

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

Pages: 1 – 47



Oliver Schnetz (Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany)


Perturbative quantum field theories frequently featurerational linear combinations of multiple zeta values(periods). In massless $\phi^4$-theory we show that theperiods originate from certain “primitive” vacuum graphs.Graphs with vertex connectivity 3 are reducible in thesense that they lead to products of periods with lower looporder. A new “twist” identity among periods is proved anda list of graphs (the census) with their periods, ifavailable, is given up to loop order 8.

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