Semiclassical decay of topological defects

Perturbative estimates suggest that extended topological defects such as cosmic strings emit few particles, but numerical simulations of the fields from which they are constructed suggest the opposite. In this paper we study the decay of the two-dimensional prototype of strings, domain walls in a simple scalar theory, solving the underlying quantum field theory in the Hartree approximation. We conclude that including the quantum effects makes the picture clear: the defects do not directly transform into particles, but there is a nonperturbative channel to microscopic classical structures in the form of propagating waves and persistent localized oscillations, which operates over a huge separation of scales. When quantum effects are included, the microscopic classical structures can decay into particles.

Figure 1

Inverse domain wall density as a function of time. There is a perfect linear correspondence over more than 2 orders of magnitude ($t=10\dots 2000$), from the time of ending the damped evolution at $t=10$ to the end of the simulation, when the walls’ mean curvature radius is over ${10}^3$ times larger than their width.

Figure 2

Two snapshots of the lattice field configurations ($L=2000$). Blue and orange regions show the domains of the degenerate vacua. To the left, we show the configuration at $t=50$; we cropped the region $(0,L/2)\times (0,L/2)$ and scaled up by a factor of 2. To the right we show the uncropped lattice at $t=100$. There is no qualitative difference between the snapshots.

Figure 3

Scaling of the equal time correlation function. Notice that the function has an approximate Gaussian shape, which is due to the nature of spinodal instability that created the domains. (There are deviations at small distance.) ($L=1000$, 1024 runs averaged.)

Figure 4

The scaling of the power spectrum. The macroscopic part (domain walls) clearly follows the scaling law both in the classical (top) and in the quantum (bottom) case. The scaling breaks at the tail of the spectrum (small classical structures). ($L=500$, average of 16 000 and 192 runs for $\hslash =0$ and 1, respectively.)

Figure 5

The scaling of the correlation length (top) and the domain wall density (bottom).

Figure 6

Number of domains over lattice volume (top) in the classical and quantum framework. The classical scaling is counterintuitive. Dividing the difference of the total loop length by the number of loops (estimated by the number of domains) we get an average loop size (bottom).

Figure 7

Snapshot from a classical (left) and the corresponding quantum (middle) run. To the right the particle content is characterized by $⟨{\phi }^2(\overrightarrow{x},t){⟩}_E$. The darker points mean higher value. (These images were taken at $t=40$ on a $L=128$ lattice. We cropped out a piece of $75\times 75$.).