Dynamical chiral symmetry breaking in unquenched ${\mathrm{Q}\mathrm{E}\mathrm{D}}_3$

We investigate dynamical chiral symmetry breaking in unquenched ${\mathrm{Q}\mathrm{E}\mathrm{D}}_3$ using the coupled set of Dyson-Schwinger equations for the fermion and photon propagators. For the fermion-photon interaction we employ an ansatz which satisfies its Ward-Green-Takahashi identity. We present self-consistent analytical solutions in the infrared as well as numerical results for all momenta. In Landau gauge, we find a phase transition at a critical number of flavors of $N_f^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\approx 4$. In the chirally symmetric phase the infrared behavior of the propagators is described by power laws with interrelated exponents. For $N_f=1$ and $N_f=2$ we find small values for the chiral condensate in accordance with bounds from recent lattice calculations. We investigate the Dyson-Schwinger equations in other linear covariant gauges as well. A comparison of their solutions to the accordingly transformed Landau gauge solutions shows that the quenched solutions are approximately gauge covariant, but reveals a significant amount of violation of gauge covariance for the unquenched solutions.

Figure 1

The Dyson-Schwinger equations of the photon and fermion propagators in diagrammatic notation.

Figure 2

Here we display the anomalous dimension of the fermion vector dressing function obtained from our infrared analysis, $\kappa (N_f)$, compared with the conjecture from perturbation theory, $\eta (N_f)$.

Figure 3

Shown are the variation of the mass function, the wave function renormalization and the polarization with the number of flavors in Landau gauge, $\xi =0$. On the left-hand side we employed the first term of the BC-vertex (1BC) in both the photon and the fermion DSEs. On the right-hand side we display the same functions calculated with the CP-vertex in the fermion DSE and the BC-vertex in the photon DSE. The scale is set by choosing $e^2=1$.

Figure 4

The phase transition: Analytical vs numerical results for the scalar fermion dressing function, $B(p^2=0)$, as function of $N_f$. Shown are results for four different truncation schemes: The $1/N_f$-expansion of Ref. 8, and three different vertex truncations of the coupled photon and fermion DSEs. The scale is set by choosing $e^2=1$.

Figure 5

Shown are our results for the dressing functions for three different values of the gauge parameter $\xi$ and $N_f=1$. The scale is set by choosing $e^2=1$.

Figure 6

Shown are the variation of the mass function, the wave function renormalization and the polarization with the number of flavors in Feynman gauge, $\xi =1$. The scale is set by choosing $e^2=1$.

Figure 7

Here we display the fermion mass function, $M(p^2)$, and the wave function renormalization, $Z_f(p^2)=1/A(p^2)$ in quenched approximation for three different values of the gauge parameter $\xi$. The scale is set by choosing $e^2=1$.

Figure 8

Here we display the photon polarization $\Pi (p^2)$ as calculated from the photon DSE using $N_f=1$ and the quenched fermion propagator functions without back-coupling for three different values of the gauge parameter $\xi$. The scale is set by choosing $e^2=1$.