The IR/UV Connection in the Non-Commutative Gauge Theories

TL;DR

This work analyzes one-loop corrections in noncommutative U(1) Yang–Mills, revealing ultraviolet modes can generate infrared anomalies via non-planar diagrams with phase factors $e^{i \tilde{p} l}$ ($\tilde{p}=\theta p$). It shows quadratic and linear poles in the noncommutativity scale appear in general gauge theories, while supersymmetric combinations cancel such poles; ${\cal N}=4$ SYM is expected to be free of even logarithmic theta-dependence, suggesting a smoother $\theta^{ij}\rightarrow 0$ limit. The findings demonstrate a nonanalytic, directionally nonlocal $\theta$-dependence that challenges the naive commutative limit and emphasizes the UV/IR connection intrinsic to noncommutative gauge theories. The results are consistent with gauge invariance, including in analyses of the S-matrix, and highlight the special role of SUSY in controlling infrared behavior.

Abstract

Quantum field theory on non-commutative spaces does not enjoy the usual ultraviolet-infrared decoupling that forms the basis for conventional renormalization. The high momentum contributions to loop integrations can lead to unfamiliar long distance behavior which can potentially undermine naive expectations for the IR behavior of the theory. These "anomalies" involve non-analytic behavior in the noncommutativity parameter Theta making the limit Theta goes to zero singular. In this paper we will analyze such effects in the one loop approximation to gauge theories on non-commutative space. We will see that contrary to expectations poles in Theta do occur and lead to large discrepancies between the expected and actual infrared behavior. We find that poles in Theta are absent in supersymmetric theories. The "anomalies" are generally still present, but only at the logarithmic level. A notable exception is non-commutative super Yang Mills theory with 16 real supercharges in which anomalous effects seem to be absent altogether.

Figures (7)

• Figure 1: Mass renormalization correction in $\phi^4$ thoery.
• Figure 2: Photon self energy: gauge loop corrections.
• Figure 3: Photon self energy: matter loop corrections.
• Figure 4: Vertex corrections: gauge sector.
• Figure 5: Vertex corrections: matter sector.