Open Wilson Lines and Generalized Star Product in Noncommutative Scalar Field Theories

TL;DR

This paper demonstrates that open Wilson lines and generalized star products are fundamental structures in noncommutative scalar field theories, with explicit calculations showing their emergence in the one-loop effective action, especially in the nonplanar sector.

Contribution

It reveals the presence of open Wilson lines and generalized star products in noncommutative scalar theories, extending their known role from gauge theories and providing explicit one-loop calculations.

Findings

Open Wilson lines appear in noncommutative scalar theories.

Generalized star products emerge in the nonplanar one-loop effective action.

At low-energy, large noncommutativity limit, the nonplanar part is expressed via scalar open Wilson lines.

Abstract

Open Wilson line operators and generalized star product have been studied extensively in noncommutative gauge theories. We show that they also show up in noncommutative scalar field theories as universal structures. We first point out that dipole picture of noncommutative geometry provides an intuitive argument for robustness of the open Wilson lines and generalized star products therein. We calculate one-loop effective action of noncommutative scalar field theory with cubic self-interaction and show explicitly that the generalized star products arise in the nonplanar part. It is shown that, at low-energy, large noncommutativity limit, the nonplanar part is expressible solely in terms of the {\sl scalar} open Wilson line operator and descendants.