BPHZ renormalization and its application to non-commutative field theory

TL;DR

This paper reviews and extends a modified BPHZ renormalization scheme to complex non-commutative quantum field theories, demonstrating its effectiveness in handling divergences and UV/IR mixing, and exploring applications to gauge theories.

Contribution

It introduces a modified BPHZ scheme for non-commutative field theories and applies it to complex graphs, addressing UV/IR mixing and proposing a renormalizable model.

Findings

Modified BPHZ scheme successfully applied to higher-loop graphs.

Application to IR-singularities introduces a 1/p^2 term, aiding renormalization.

Potential extension to gauge field theories discussed.

Abstract

In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative phi^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.