Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 094, 14 pages      arXiv:2103.01136      https://doi.org/10.3842/SIGMA.2021.094
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

Johannes Thürigen ab
a) Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
b) Institut für Physik/Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

Received February 28, 2021, in final form October 24, 2021; Published online October 27, 2021

Abstract
Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applies to any combinatorially non-local field theory which is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field theories of rank $r$>$2$.

Key words: non-local field theory; renormalization; Hopf algebras; multiple polylogarithms.

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