BPHZ renormalization and its application to non-commutative field theory

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.