Interacting Open Wilson Lines in Noncommutative Field Theories

TL;DR

This paper demonstrates that in noncommutative field theories, the two-loop effective action reveals a purely geometrical cubic interaction among open Wilson lines, extending the understanding of their dynamics beyond one-loop effects.

Contribution

It introduces the computation of two-loop nonplanar contributions in noncommutative  theory, revealing a simple cubic interaction among open Wilson lines that is purely geometrical and noncommutative.

Findings

Two-loop effective action describes cubic interactions among open Wilson lines.

The interaction depends only on the sizes of the Wilson lines.

The contribution simplifies to a geometrical, noncommutative form.

Abstract

In noncommutative field theories, it was known that one-loop effective action describes propagation of non-interacting open Wilson lines, obeying the flying dipole's relation. We show that two-loop effective action describes cubic interaction among `closed string' states created by open Wilson lines. Taking d-dimensional noncommutative [\Phi^3] theory as the simplest setup, we compute nonplanar contribution at low-energy and large noncommutativity limit. We find that the contribution is expressible in a remarkably simple cubic interaction involving scalar open Wilson lines only and nothing else. We show that the interaction is purely geometrical and noncommutative in nature, depending only on sizes of each open Wilson line.