Wilson Loops in 2D Noncommutative Euclidean Gauge Theory: 2. 1/\theta Expansion

TL;DR

This paper investigates the 1/θ and 1/N expansions of Wilson loop averages in 2D noncommutative U(θ)(N) gauge theory, deriving integral representations and analyzing asymptotic behaviors to understand their decay and implications for string interpretations.

Contribution

It provides a non-perturbative integral representation for the next-to-leading term in the 1/θ expansion and analyzes its asymptotic behavior, revealing limitations for string interpretations.

Findings

Next-to-leading term exhibits power-like decay for large areas.

Large θ asymptote matches the next-to-leading term in 1/θ expansion.

Subleading terms hinder a stringy interpretation similar to the commutative case.

Abstract

We analyze the $1/\theta$ and 1/N expansions of the Wilson loop averages $<W(C)>_{U_\theta (N)}$ in the two-dimensional noncommutative $U_\theta (N)$ gauge theory with the parameter of noncommutativity $\theta$. For a generic rectangular contour $C$, a concise integral representation is derived (non-perturbatively both in the coupling constant $g^{2}$ and in $\theta$) for the next-to-leading term of the $1/\theta$ expansion. In turn, in the limit when ${\theta}$ is much larger than the area $A(C)$ of the surface bounded by $C$, the large $\theta$ asymptote of this representation is argued to yield the next-to-leading term of the $1/\theta$ series. For both of the expansions, the next-to-leading contribution exhibits only a power-like decay for areas $A(C)>>\sigma^{-1}$ (but $A(C)<<{\theta}$) much larger than the inverse of the string tension $\sigma$ defining the range of the exponential decay of the leading term. Consequently, for large $\theta$, it hinders a direct stringy interpretation of the subleading terms of the 1/N expansion in the spirit of Gross-Taylor proposal for the $\theta=0$ commutative D=2 gauge theory.