Strongly-coupled unquenched ${\mathrm{QED}}_4$ propagators using Schwinger-Dyson equations

We study unquenched QED in four dimensions using renormalized Schwinger-Dyson equations and focus on the behaviour of the fermion and photon propagators. For this purpose we use an improved Kızılersü-Pennington (KP) vertex which respects gauge invariance, multiplicative renormalizability for the massless case, agrees with perturbation theory in the weak coupling regime, and is free of kinematic singularities. We find that the KP vertex performs very well as expected especially in comparison with other vertex choices. We find that the Landau pole problem familiar from perturbative QED persists in the nonperturbative case with the renormalized inverse photon propagator having zero crossing.

Figure 1

The Schwinger-Dyson equations for the fermion and photon propagators in QED.

Figure 2

Fermion Schwinger-Dyson equation.

Figure 3

Photon Schwinger-Dyson equation.

Figure 4

Massless fermion and photon wave-function renormalizations versus $p^2$ using bare, BC, CP, mod. CP, and KP vertices for $\alpha =0.05$ and $\xi =0.5$. The lines except CP and BC in above plot and CP in below plot overlaps.

Figure 5

Vertex comparison of massless and massive $\alpha =0.2$ solutions in Landau gauge. The massive solutions have $m_{\mu }=400$.

Figure 6

Vertex comparison of the massive $\alpha =0.2$, $\xi =0.5$ solutions for different $m_{\mu }$ values.

Figure 7

Percentage relative differences between the KP-type2 and KP-type3 in fermion, photon wave-function renormalizations, and mass function versus $p^2$ using vertices for $\alpha =0.6$ and $\xi =0$, 0.5, 1.0.

Figure 8

Massive bare, BC, modified CP, and KP vertex comparisons: $\alpha =0.2$, $\xi =0$ solutions for different $N_F$, $m_{\mu }=4$.

Figure 9

Massive bare, BC, modified CP, and KP vertex comparisons: $\alpha =0.2$, $\xi =0.5$ solutions for different $N_F$, $m_{\mu }=4$.

Figure 10

Asymptotic tail comparison of numerical solutions for $\alpha =0.6$, $\xi =0$, $m_{\mu }=4$ (solid line) and their power-law fit (dashed line).

Figure 11

Massive bare, BC, modified CP, and KP vertex comparisons: $\alpha =0.2$ solutions for different $\xi$, $m_{\mu }=4$.

Figure 12

Massive bare, BC, modified CP, and KP vertex comparisons: $\alpha =0.2$, $\xi =0$ solutions for different ${\Lambda }^2$, $m_{\mu }=4$.

Figure 13

Test of multiplicative renormalizability on $\alpha =0.6$, $\xi =0$, $m_{\mu }=4$ solutions.

Figure 14

Massive bare, BC, modified CP, and KP vertex comparisons: $\xi =0$ solutions for different $\alpha$, $m_{\mu }=4$.

Figure 15

Massless bare, modified CP, and KP vertex comparisons: $\xi =0$ solutions for different $\alpha$, ${\mu }^2={\Lambda }^2$.

Figure 16

Massless high $\alpha$ solutions for KP vertex.

Figure 17

A comparison of the bare, modified CP, and KP vertices for $\xi =0,{\mu }^2={\Lambda }^2$ for different $\alpha$’s over a range of $p^2$ values.

Figure 18

Cycle 1 solutions for low and high mass from the numerical and analytic calculations.

Figure 19

High alpha high mass decreasing solutions $\alpha =1.1$, $\xi =0$, $m_{\mu }={10}^6\to 5\times {10}^5$.