Axial anomaly and confinement in two-dimensional QED: Singular behavior of vacuum polarization at threshold

Performing the one-loop calculation of the vacuum polarization amplitude in QED 1 + 1 , one finds that the axial-vector current exhibits an anomaly. 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 − → 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.