Decay of particles above threshold in the Ising field theory with magnetic field

TL;DR

This work studies the decay of heavy particles above threshold in the two-dimensional Ising field theory with a magnetic field by applying form-factor perturbation theory around the integrable point (critical Ising line). The authors construct three-particle form factors and compute mass corrections and decay widths when integrability is weakly broken by a small deviation in temperature, obtaining a universal lifetime ratio $t_4/t_5=0.23326$ and specific branching fractions for $A_5$ decays. They extract detailed FF data ($f_1,f_2,f_3,f_4$, $f_{411}$, $f_{511}$, $f_{512}$, and $\langle\varepsilon\rangle_{\tau=0}$) to determine mass shifts, imaginary parts, and decay channels, and validate the results with finite-volume numerical studies (TCSA) showing the expected level-crossing behavior and splittings. The analysis highlights the interplay between 2D phase-space constraints and integrability-driven dynamics, which together suppress decays and yield robust, universal quantities for unstable above-threshold states in this exactly solved QFT setting.

Abstract

The two-dimensional scaling Ising model in a magnetic field at critical temperature is integrable and possesses eight stable particles A_i (i=1,...,8) with different masses. The heaviest five lie above threshold and owe their stability to integrability. We use form factor perturbation theory to compute the decay widths of the first two particles above threshold when integrability is broken by a small deviation from the critical temperature. The lifetime ratio t_4/t_5 is found to be 0.233; the particle A_5 decays at 47% in the channel A_1A_1 and for the remaining fraction in the channel A_1A_2. The increase of the lifetime with the mass, a feature which can be expected in two dimensions from phase space considerations, is in this model further enhanced by the dynamics.

Figures (5)

• Figure 1: Poles and unitarity cuts for the scattering amplitudes $S_{11}$ and $S_{12}$ in the integrable case $\tau=0$, (a) and (b), respectively, and for $\tau$ slightly different from zero (c). In (c) the particle $A_4$ became unstable and the associated pole moved through the cut into the unphysical region.
• Figure 2: Diagrams determining the leading corrections to the real (a) and imaginary (b) parts of the masses at small $\tau$. In (b) also the intermediate particles are on shell and energy-momentum is conserved at each vertex. For $c>5$ also diagrams with more than two particles in the intermediate state contribute to the imaginary part.
• Figure 3: The first few trajectories which divide the $h$-$\tau$ plane into regions with a different number of stable particles. There are $n$ stable particles in between the trajectories labeled by $\eta_n$ and $\eta_{n+1}$. These trajectories densely fill the plane when the negative orizontal axis is approached ($\eta\to-\infty$).
• Figure 4: First eight energy levels of the finite volume Hamiltonian of the scaling Ising model with magnetic field at critical temperature as functions of the scaling variable $r =m_1R$. At $r=40$, starting from the bottom, the levels are identified as the ground state, the first three particle states $A_1$, $A_2$ and $A_3$, three scattering states $A_1A_1$, the particle above threshold $A_4$. Crossings between the line associated to the latter and the scattering states are visible around $r=18$, $r=25$ and $r=36$.
• Figure 5: First eight energy levels of the finite volume Hamiltonian of the scaling Ising model with magnetic field slightly away from the critical temperature. Observe the splitting of the crossings pointed out in the previous figure.