Axial Anomaly and Confinement in Two-dimensional QED: Singular Behavior of Vacuum Polarization at Threshold

Abstract

Performing the perturbative calculation of the vacuum polarization amplitude in QED1+1, one finds an anomalous axial vector Ward identity. We note that the photon self-energy function displays a singularity in the 1+1D case, in stark contrast to the 3+1D case. We discuss the nature of this singularity and its physical implications from the perspectives of axial anomaly and confinement in two-dimensional QED. Computing the total cross section of $e^+e^-\to μ^+μ^-$ and displaying the toy version of the R ratio in 1+1D with respect to the typical R ratio in 3+1D, we discuss the significance of using the dressed photon propagator in obtaining the finite cross section.

Figures (7)

• Figure 1: Feynman diagram for the photon self-energy at one-loop order.
• Figure 2: Plot of (a) $\text{Re}[T(q^2)]$ and (b) $\text{Im}[T(q^2)]$ in 1+1D case taking $m=1$ and $e=1$, and in 3+1D case taking $m=1$ and $\alpha=1$, where the dots represent results from the (subtracted) dispersion relation.
• Figure 3: Mass gap equation for the photon in $\text{QED}_{1+1}$.
• Figure 4: Plot of (a) $\text{Re}[\frac{1}{1+T(q^2)}]$ and (b) $\text{Im}[\frac{1}{1+T(q^2)}]$ in 1+1D and 3+1D, where the dots represent results from the subtracted dispersion relation.
• Figure 5: Total cross-section of $e^+e^-\rightarrow \mu^+\mu^-$.