Eur. Phys. J. C (2013) 73: 2566
https://doi.org/10.1140/epjc/s10052-013-2566-8

Regular Article - Theoretical Physics

BPHZ renormalization and its application to non-commutative field theory

¹ Theory Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
² Institut de Physique Nucléaire, Université de Lyon, Université Lyon 1 and CNRS/IN2P3, Bat. P. Dirac, 4 rue Enrico Fermi, 69622, Villeurbanne, France
³ Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040, Vienna, Austria
⁴ Austro-Ukrainian Institute for Science and Technology, c/o TU Vienna, Wiedner Hauptstraße 8-10, 1040, Vienna, Austria

^* e-mail: dblaschke@lanl.gov

Received: 29 July 2013
Published online: 19 September 2013

Abstract

In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative ϕ ⁴-theory. In the present paper, we first review the BPHZ method and then we apply the modified BPHZ scheme as well as Zimmermann’s forest formula to the sunrise graph, i.e. a typical higher-loop graph involving overlapping divergences. Furthermore, we show that the application of the modified BPHZ scheme to the IR-singularities appearing in non-planar graphs (UV/IR mixing problem) leads to the introduction of a 1/p ² term and thereby to a renormalizable model. Finally, we address the application of this approach to gauge field theories.