Validating the 2PI resummation: The Bloch-Nordsieck example

In this work, we provide a numerical method to obtain the Bloch–Nordsieck spectral function at finite temperature in the framework of the 2-particle-irreducible (2PI) approximation. We find that the 2PI results nicely agree with the exact one, provided we perform a coupling constant matching. In the paper, we present the resulting finite temperature running of the 2PI coupling constant. This result may apply for the finite temperature behavior of the coupling constant in QED, too.

Figure 1

The diagrammatic representation of the self-energy. The wavy line corresponds to the free (here also the exact) photon propagator with a loop momentum $k$, and the double solid line is for the exact fermion propagator with momentum $p-k$. Both the polarization and the Keldysh indices are shown.

Figure 2

The coupling constant dependence of the spectral function in the 2PI approximation (a) at fixed temperature $T=1$ and (b) at a fixed coupling value, $\alpha =0.5$. The curves widen with growing coupling and growing temperature.

Figure 3

Comparing the 2PI resummed spectral function to the exact one. The solid red line is obtained from the 2PI resummation, while the dashed blue line is the exact spectral function. Both of them are at $\mathrm{T}=1$, and the couplings are ${\alpha }_{\text{ex}}={\alpha }_{2\mathrm{PI}}=0.5$.

Figure 4

The fitting of the exact spectral function on the 2PI spectrum on (a) linear and (b) logarithmic plots. On the plot, the curves are normalized to the same height for better visibility. We can see an exact match at the peak and a small deviation in the asymptotics. The fit yields ${\alpha }_{2\mathrm{PI}}=0.5$ for ${\alpha }_{\text{ex}}=0.293$ at $T=1$.

Figure 5

The relation between the 2PI and the exact coupling at $u=0$ for temperatures (a) $T\in [0,12.03]$ and (b) $T\in (12.03,\infty )$, respectively. The dashed red line indicates the limiting function at $T=12.03$; for details, see the text.

Figure 6

The running of (a) $A_T$ and (b) $B_T/C_T$ with respect to the temperature. This latter quantity is the position of the pole [cf. Eq. (42)]. One can see the best matching on higher temperatures.

Figure 7

Finite-temperature running of ${\alpha }_{2\mathrm{PI}}$ for fixed ${\alpha }_{\text{ex}}=0.25$. One can observe the high-temperature (Landau) pole and the $T\to 0$ divergence.

Figure 8

The 2PI spectral functions with different rapidities [$\eta ={\tanh }^{-1}(v)$, where $v=u/u_0$] at fixed temperature $T=1$ and coupling $\alpha =0.5$. The shrinking of the width can be observed as the velocity grows, which is the same effect that we had for the exact solution [16].

Figure 9

The relation between the 2PI and the exact coupling for $u=\sqrt{3}$ at temperatures (a) $T\in [0,12.03]$ and (b) $T\in (12.03,\infty )$, respectively. The dashed red line indicates the limiting function at $T=12.03$; for details, see the text.