Instanton modifications of the bound state singularity in the Schwinger Model

TL;DR

This paper analyzes how instanton effects modify the bound state singularities in the Schwinger Model's quark-antiquark Green's function, revealing non-factorizable contributions and additional branch point singularities.

Contribution

It provides an explicit calculation of instanton contributions to the bound state pole and branch point singularities in the Schwinger Model's Green's function, highlighting their impact on factorization properties.

Findings

Instanton sectors alter the residue of the bound state pole.

Nonzero instanton contributions break the factorization property.

Additional logarithmic and dilogarithmic branch point singularities are identified.

Abstract

We consider the quark-antiquark Green's function in the Schwinger Model with instanton contributions taken into account. Thanks to the fact that this function may analytically be found, we draw out singular terms, which arise due to the formation of the bound state in the theory -- the massive Schwinger boson. The principal term has a pole character. The residue in this pole contains contributions from various instanton sectors: $0,\pm 1, \pm 2$. It is shown, that the nonzero ones change the factorizability property. The formula for the residue is compared to the Bethe-Salpeter wave function found as a field amplitude. Next, it is demonstrated, that apart from polar part, there appears in the Green's function also the weak branch point singularity of the logarithmic and dilogarithmic nature. These results are not in variance with the universally adopted $S$-matrix factorization.