Propagator from nonperturbative worldline dynamics

We use the worldline representation for correlation functions together with numerical path integral methods to extract nonperturbative information about the propagator to all orders in the coupling in the quenched limit (small-$N_{\mathrm{f}}$ expansion). Specifically, we consider a simple two-scalar field theory with cubic interaction ($\mathrm{S}{}^2\mathrm{Q}\mathrm{ED}$) in four dimensions as a toy model for QED-like theories. Using a worldline regularization technique, we are able to analyze the divergence structure of all-order diagrams and to perform the renormalization of the model nonperturbatively. Our method gives us access to a wide range of couplings and coordinate distances. We compute the pole mass of the $\mathrm{S}{}^2\mathrm{Q}\mathrm{ED}$ electron and observe sizable nonperturbative effects in the strong-coupling regime arising from the full photon dressing. We also find indications for the existence of a critical coupling where the photon dressing compensates the bare mass such that the electron mass vanishes. The short distance behavior remains unaffected by the photon dressing in accordance with the power-counting structure of the model.

Figure 1

Mean value $⟨V⟩$ of the potential as a function of ${\log }_{2}N$, the base two logarithm of the number $N$ of point per loops, for lines whose endpoints distance are $\mathrm{\Delta }y=1$ (left) and $\mathrm{\Delta }y=14$ (right). The exact analytic expression (solid red line) and the fitting with a straight line (dashed green line) are also shown.

Figure 2

Fit parameters $a_V$ (left) and $b_V$ (right) of the fit Eq. (28) to the expectation value of the interaction potential (light blue points), as functions of the distance $\mathrm{\Delta }y$. The orange solid lines show the exact analytical prefactor $(\ln 2)/2$ of the log divergence (left) and the remainder function of Eq. (20) after subtraction of the log-divergence (right).

Figure 3

Probability distributions $\mathcal{P}$ and the corresponding fits $P$ for $\mathrm{\Delta }y=0$, $N=5$ (left) and $\mathrm{\Delta }y=1$, $N=3,4,\dots ,16$ (right), with $N$ increasing from left to right in the right plot.

Figure 4

The $\alpha$ (left) and $\beta$ (right) parameters for $\mathrm{\Delta }y=1$ as a function of $N$. The blue dots correspond to the data coming from the fit of our Ansatz $P(v,\mathrm{\Delta }y)$ to $\mathcal{P}$ for different values of $N$, while the green solid line is the constant fit in the large $N$ region.

Figure 5

Left: $N$ dependence of the fit parameter $v_0$ for $\mathrm{\Delta }y=1$; the blue dots show the results from fitting $P(v,\mathrm{\Delta }y)$ to $\mathcal{P}$ for different values of $N$, and the green solid line represents the linear ${\log }_{2}N$ fit in the large $N$ region. Right: self-consistency check using the regularized analytical result [Eq. (c4), solid orange line] and the regularized numerical one [Eq. (35) using the large-$N$ fit values of $\alpha$, $\beta$ and $b_{v_0}$, blue dots] for $T=1$.

Figure 6

PDF fit parameters $\alpha$ (left) and $\beta$ (right) as a function of the distance $\mathrm{\Delta }y$. The blue dots corresponds to the large $N$ fits of the $\alpha$ and $\beta$ parameters, while the green solid line is a third order polynomial fit.

Figure 7

PDF fit parameters $a_{v_0}$ (left) and and $b_{v_0}$ (right) representing the fit parameter $v_0$ via Eq. (33) as a function of the distance $\mathrm{\Delta }y$. The blue dots corresponds to the large $N$ fits of the $v_0$ parameter. In the left panel, the green solid line is a constant fit, whereas in the rigth panel it shows the estimate for $b_{v_0}$ using Eq. (37) as described in the text.

Figure 8

Left: peak-normalized integrand $\mathcal{G}$ of Eq. (48), for $\overline{g}=0.5$ and distances $\mathrm{\Delta }\overline{x}=1$, 3, 5 (solid violet, dashed blue and dotted cyan lines respectively). Right: density plot of the proper-time peak position $T_{\max }$ of $\mathcal{G}$ as a function of $\overline{g}$ and $\mathrm{\Delta }\overline{x}$.

Figure 9

Left panel: behavior of the propagator ${\overline{G}}_P(·)$ as a function of the distance for $\overline{g}=0.1$ (dashed red line), $\overline{g}=0.4$ (dot-dahed blue line) and $\overline{g}=0.7$ (dotted green line). Right panel: propagator (dot-dashed red line) and the one-loop propagator $G_{1\text{−}\mathrm{loop},\mathrm{WR}}$ in the worldline regularization (dashed green line) together with their large distance fits (solid blue and violet line, for the full and one-loop propagator, respectively).

Figure 10

Pole mass $m^{\star }$ as a function of the coupling constant for various estimates: within worldline regularization, the analytical one-loop result is shown as orange squares, exhibiting satisfactory agreement with the same result extracted from a fit procedure (green triangles). The nonperturbative worldline result using the same fit procedure is shown as red circles. For comparison, we show the one-loop pole mass extracted from the propagator using cutoff regularization (purple diamonds).

Figure 11

Probability distribution function for the action $S_W$ considering $\mathrm{\Delta }y=10$, $D=4$ and $N=2^5$. The histogram contains 100 bins and corresponds to an ensemble of $10^4$ lines, whereas the solid green line is given by Eq. (a18).