Renormalization of the energy-momentum tensor in noncommutative complex scalar field theory

TL;DR

This paper investigates the renormalization process of the energy-momentum tensor and composite operators in noncommutative complex scalar field theory, highlighting differences from commutative and real scalar theories and proposing an improved energy-momentum tensor.

Contribution

It introduces a proper operator basis, calculates the mixing matrix at one-loop, and defines an improved energy-momentum tensor for noncommutative complex scalar fields.

Findings

The operator basis and mixing matrix differ from commutative theories.

The energy-momentum tensor requires improvement for finiteness.

The energy-momentum vector is unambiguous and conserved at one-loop.

Abstract

We study the renormalization of dimension four composite operators and the energy-momentum tensor in noncommutative complex scalar field theory. The proper operator basis is defined and it is proved that the bare composite operators are expressed via renormalized ones with the help of an appropriate mixing matrix which is calculated in the one-loop approximation. The number and form of the operators in the basis and the structure of the mixing matrix essentially differ from those in the corresponding commutative theory and in noncommutative real scalar field theory. We show that the energy-momentum tensor in the noncommutative complex scalar field theory is defined up to six arbitrary constants. The canonically defined energy-momentum tensor is not finite and must be replaced by the "improved" one, in order to provide finiteness. Suitable "improving" terms are found. Renormalization of dimension four composite operators at zero momentum transfer is also studied. It is shown that the mixing matrices are different for the cases of arbitrary and zero momentum transfer. The energy-momentum vector, unlike the energy-momentum tensor, is defined unambigously and does not require "improving", in order to be conserved and finite, at least in the one-loop approximation.