Trace anomaly for a conformal 2D vector field model

TL;DR

The paper analyzes the trace (Weyl) anomaly for a conformal 2D vector field, addressing the classical singularity of the $D\to2$ limit by employing dimensional regularization with an auxiliary scalar to preserve conformal and gauge invariance. It derives the one-loop divergences for scalars, fermions, and the 2D vector model, showing that in 2D the vector case produces a nontrivial anomaly depending on both the metric and the auxiliary scalar, with an explicit anomaly structure $\mathcal{T}=aR + \omega (\nabla\phi)^2 + b \Box\phi$ where $a=b=-1/(2\pi)$ and $\omega=1/\pi$. The authors construct the corresponding anomaly-induced effective action, which can be nonlocal or localized with two auxiliary fields and may include an arbitrary conformal invariant functional $S_c$, highlighting a richer 2D anomaly landscape akin to higher dimensions. The results extend the conventional 2D anomaly framework to include vector fields and external scalars, providing a groundwork for comparing scalars, fermions, and vectors in anomaly-induced actions in low dimensions.

Abstract

The trace anomaly and anomaly-induced action are evaluated for the two-dimensional $2D$ vector theory with classical conformal symmetry. Implementing local conformal symmetry while preserving the gauge invariance requires either giving up locality of the classical action or, equivalently, introducing an auxiliary scalar field. The two-dimensional limit in such a theory is singular. However, in the dimensional regularization, the limit $D \to 2$ in the one-loop divergence is smooth. As a result, we arrive at the expression for anomaly, which has a rich general structure, typical for the dimensions $D \geq 4$. For comparison and completeness, we also evaluate anomalies for conformal scalar and fermions, also in the presence of auxiliary external scalars.