Dyson-Schwinger equations with a parametrized metric

We construct and solve the Dyson-Schwinger equation of a quark propagator with a parameterized metric, which connects the Euclidean metric with the Minkowskian one. We show, in some models, that the Minkowskian vacuum is different from the Euclidean vacuum. The usual analytic continuation of the 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.