On the renormalization of non-commutative field theories

Abstract

This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). To start with, we discuss the Quantum Action Principle and provide evidence for its validity for noncommutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Z c [j] of connected Green functions makes sense (at least at one-loop level). Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field: Explicit computations are performed at one-loop order and the generalization to higher loops is commented upon. Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization.1 It is possible to remedy this problem by considering supersymmetry, e.g. the Wess-Zumino model on Moyal space [5]. However, in the present paper we restrict our attention to non-supersymmetric models.2 In this context, we should mention the so-called twisted approach to φ 4 -theory for which there has been some recent progress (see for instance reference [8]), but which we will not discuss here.3 In the Grosse-Wulkenhaar case, the initial proof was established by its authors by using the Wilson-Polchinski renormalization group approach in a matrix base. A later proof [10] relied on multi-scale analysis.

Gauge fields on non‐commutative spaces and renormalization

“…For the subtracted non-planar one-loop 2-point function of the model (1), we hence get a vanishing result (as one also does for the planar, UV-divergent diagram by virtue of the standard BPHZ subtraction scheme). For the subtracted non-planar one-loop 4-point function (4), one obtains a result which is regular inp, as shown in [1]. By proceeding in this way, the one-loop renormalization of the theory could be carried out for the (naïve) φ 4 -theory described by the action (1), as well as for this action supplemented by a 1/p 2 -term which is known to overcome the UV/IR mixing problem while maintaining the translation invariance of the model [8].…”

“…The quantization of this model and the renormalization of related scalar field models has been discussed over the last fifteen years, see for instance reference [1] for a brief review and list of references. In the latter work it was pointed out that the usual BPHZ momentum space subtraction scheme (which consists of subtracting appropriate polynomials in the external momentum from the integrand of divergent integrals) cannot be applied in noncommutative theories, e.g.…”

“…the appearance of an IR-singularity for small values of the external momentum p. Therefore, a modified subtraction scheme was proposed in Ref. [1]: It consists of considering p andp as independent variables (though satisfying pp = 0) when performing the subtraction: In particular, one subtracts from the integrand its Taylor series expansion with respect to the external momentum p around p = 0 up to the order of divergence of the graph, while maintaining the phase factors. Thus, for these non-planar graphs, our modified BPHZ subtraction amounts to an IR-subtraction rather than a UV-subtraction.…”

“…In this proceeding we review the discussion of the BPHZ renormalization of scalar and gauge models in non-commutative 4-dimensional Euclidean space initiated in References [1,2].…”

“…In the following section, we illustrate the main ideas behind the generalization of the BPHZ procedure to the non-commutative setting using a scalar φ 4 . We briefly discuss the results of [1,2] and describe how to apply the new method to a candidate for a renormalizable gauge field model on non-commutative space out forward in [14,15].…”