[Paper Review] Wilson Line Integrals in the Unparticle Action

Alexis Licht|ArXiv.org|May 25, 2008

Black Holes and Theoretical Physics4 references3 citations

TL;DR

This paper demonstrates that the Mandelstam derivative—commonly used to derive vertexes from gauge-invariant unparticle actions—is mathematically inconsistent when applied in unparticle action integrals. It proposes a correct method based on ordinary derivatives of the Wilson line's explicit endpoint dependence and proves that only a straight-line path preserves both Poincaré and scale invariance in the unparticle action, resolving a key ambiguity in the formulation of gauge-invariant unparticle interactions.

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.

Motivation & Objective

• To identify and resolve mathematical inconsistencies in the use of the Mandelstam derivative for deriving unparticle-gauge field vertexes.
• To establish a mathematically consistent method for differentiating open Wilson lines in the context of unparticle field theory.
• To determine the unique path that preserves both Poincaré and scale invariance in the gauged unparticle action.
• To clarify the relationship between the Mandelstam derivative and functional derivatives of the path-dependent Wilson line.

Proposed method

• Derives the correct ordinary derivative of the Wilson line by explicitly differentiating its dependence on the endpoint x, using a parametrized path between x and y.
• Compares the results of partial integration using the ordinary derivative versus the Mandelstam derivative in a solvable abelian gauge model with a uniform magnetic field.
• Introduces a functional derivative formalism that generalizes the Wilson line dependence on the path function ζ(x,y,λ), showing the Mandelstam derivative as a limiting case.
• Applies scale and Lorentz invariance constraints to the path function ζ, deriving differential equations for the path's dependence on x, y, and λ.
• Uses scaling and translational symmetry to derive constraints on the path function, leading to the conclusion that only a straight-line path satisfies both symmetries.
• Performs explicit calculations in a 2D plane with Gaussian wavefunctions to compare results from ordinary derivative, Mandelstam derivative, and direct integration.

Experimental results

Research questions

• RQ1 Is the Mandelstam derivative mathematically consistent when used in integrals involving the unparticle action?
• RQ2 What is the correct way to differentiate the Wilson line in the unparticle action to preserve gauge invariance and consistency?
• RQ3 Which path in the Wilson line integral preserves both Poincaré and scale invariance in the unparticle action?
• RQ4 How does the Mandelstam derivative relate to the functional derivative of the path function ζ?
• RQ5 Can a gauge-invariant unparticle action be constructed without using a Wilson line integral?

Key findings

• The Mandelstam derivative leads to inconsistent results in the unparticle action integral, as shown by a discrepancy between partial integration using the Mandelstam derivative and direct integration in a solvable abelian model.
• The correct derivative of the Wilson line is the ordinary derivative with respect to the endpoint x, which yields consistent results when used in partial integration.
• The Mandelstam derivative is a special case of a functional derivative of the path function ζ, valid only in a limiting case and not suitable for evaluating the unparticle action.
• Only a straight-line path between x and y preserves both Poincaré and scale invariance in the unparticle action, as required by the symmetry constraints derived from scaling and translational invariance.
• The path function must satisfy f(u,v,w,λ)x^μ + g(u,v,w,λ)y^μ = λx^μ + (1−λ)y^μ, which implies a linear, straight-line parametrization.
• The requirement f+g=1 and the vanishing of partial derivatives with respect to u, v, w forces f and g to depend only on λ, confirming the straight-line path as the unique solution.