Wilson Line Integrals in the Unparticle Action

TL;DR

This paper examines the gauge invariance of the unparticle action involving Wilson lines, identifies inconsistencies in the Mandelstam derivative, and proposes a consistent differentiation method, emphasizing the straight line as the only Poincare and scale invariant path.

Contribution

It clarifies the correct way to differentiate Wilson lines in unparticle actions and establishes the straight line as the unique path preserving invariance.

Findings

Mandelstam derivative is mathematically inconsistent.

Differentiating the explicit endpoint dependence is consistent.

Only straight lines preserve Poincare and scale invariance.

Abstract

We consider the unparticle action that is made gauge invariant by inclusion of an open Wilson line factor. In deriving vertexes from such an action it has been customary to use a form of differentiating the Wilson line originally proposed by Mandelstam. Using a simple example, we show that the Mandelstam derivative is mathematically inconsistent. We show that there are two ways to define differentiation of the Wilson line. The mathematically consistent method is to differentiate the explicit dependence of the line on the endpoint. The other method is a functional derivative and corresponds in a limiting case to the Mandelstam derivative. We also show that the only path that can be used in the Wilson line integral that leaves the unparticle action both Poincare and scale invariant is the straight line.