Dyson-Schwinger Equations with a Parameterized Metric

TL;DR

This paper develops a method to solve Dyson-Schwinger equations with a parameterized metric connecting Euclidean and Minkowskian spaces, revealing differences in vacua and enabling meaningful analytic continuation.

Contribution

It introduces a novel algorithm and framework for solving DSEs with a parameterized metric, preserving Ward-Takahashi identity and analyzing vacuum differences.

Findings

Minkowskian and Euclidean vacua can differ in certain models.

Analytic continuation is valid when the vacuum remains unchanged during metric evolution.

The proposed method enables meaningful analytic continuation of Green functions.

Abstract

We construct and solve the Dyson-Schwinger equation (DSE) of quark propagator with a parameterized metric, which connects the Euclidean metric with the Minkowskian one. We show, in some models, the Minkowskian vacuum is different from the Euclidean vacuum. The usual analytic continuation of Green function does not make sense in these cases. While with the algorithm we proposed and the quark-gluon vertex ansatz which preserves the Ward-Takahashi identity, the vacuum keeps being unchanged in the evolution of the metric. In this case, analytic continuation becomes meaningful and can be fully carried out.