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 ansätz 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.

Figure 1

Calculated result of the analytical structure of the mass function in our parameterized metric and the comparison with those given in the standard perturbative procedure, where $p^2=\rho e^{-i\epsilon }$. The solid line is that given by perturbative theory in the exact Minkowskian space (with $\epsilon =\pi$ in the expression of $p^2$), the dot line is also that calculated in perturbative theory but with $ϵ=0.95\pi$, the dash line is that obtained with our parameterized metric at $\epsilon =0.95\pi$.

Figure 2

Schematics of the area and the contour of the complex momentum $p^2$ (lower panel) and $p^{2*}$ (upper panel) with the parameterized metric $g^{\mu \nu }(\epsilon )$.

Figure 3

Calculated variation behaviors of $A(\rho e^{-i\epsilon })$ and $B(\rho e^{-i\epsilon })$ with respect to $\rho$ at $\epsilon =3\pi /4$.

Figure 4

The comparison between the data generated by our algorithm and those obtained directly in the Euclidean space.

Figure 5

Calculated variation behavior of the scalar part of the quark propagator $|{\sigma }_B|$ in the whole momentum area in the case of $\epsilon =3\pi /4$ (upper panel) and that at two special values of $\theta$ (lower panel).

Figure 6

Calculated results of the real part of the functions $A$ and $B$ in the quark propagator [$\mathrm{Re}(A)$ in upper panel, $\mathrm{Re}(B)$ in middle panel] with several values of parameter $\epsilon$ and those of the real and the imaginary parts of the functions $A$ and $B$ at $\epsilon =0.95\pi$ (lower panel).

Figure 7

Calculated result of the variation behavior of the function $|{\sigma }_B|$ in the whole momentum area in the case of $\epsilon =0.95\pi$.