Inelastic scattering and elastic amplitude in Ising field theory in a weak magnetic field at T>T_c. Perturbative analysis

TL;DR

This work develops a perturbative analysis of two-particle scattering in Ising field theory at $T>T_c$ with a weak magnetic field $h$, expanding in $h^2$ to compute the $2\to 3$ cross-section and its dominance of inelastic processes across energies. Using a dispersion-relation approach anchored by the optical theorem, the authors extract the high-energy behavior of the elastic amplitude, revealing a logarithmic growth in the inelastic cross-section controlled by the third moment $G_3$ of the Euclidean spin-spin correlator. They identify a direct-channel pole in the leading elastic amplitude and show that the dominant inelastic channel at order $h^2$ is $2\to 3$, with a closed-form expression and asymptotics for its cross-section. The high-energy limit is analyzed through the sinh-Gordon/Lax formalism, linking correlation functions to exact form-factor structures and leading to a resummation picture where leading logarithms exponentiate, producing a power-like decay of the elastic amplitude and a universal high-energy suppression of $2\to 2$ scattering. The results illuminate the interplay between near-integrable perturbations, analytic structure of the S-matrix, and the role of $G_3$ in governing inelastic scattering in 2D quantum field theory.

Abstract

Two-particle scattering in Ising field theory in a weak magnetic field h is studied in the regime T>T_c, using perturbation theory in h^2. We calculate explicitly the cross-section of the process 2->3 to the order h^2. To this order, the corresponding cross-section dominates the total cross-section (the probability of all inelastic processes) at all energies E. We show that at high energies the h^2 term in the total cross-section grows as 16 G_3 h^2 log(E) where G_3 is exactly the third moment of the Euclidean spin-spin correlation function. Going beyond the leading order, we argue that at small h^2 the probability of the 2->2 process decays as E^(-16G_3 h^2) as E->infinity.

Figures (3)

• Figure 1: Analytic structure of the two-particle scattering amplitude $S(\theta)$ in the complex $\theta$-plane. The solid lines represent the branch cuts associated with inelastic channels. The values of $S(\theta)$ at different edges of the branch cuts represent physical $S$-matrix element $S$, its complex conjugate $S^*$, and the inverse values. The bullets $\bullet$ and circles $\circ$ indicate possible positions of poles and zeroes, respectively. Poles located on the imaginary axis, within the physical strip $0 < \Im m\, \theta < \pi$ correspond to stable particles; poles on the strip $-\pi<\Im m\,\theta<0$ are associated with resonance scattering states.
• Figure 2: Energy dependence of the partial cross-section $\sigma^{(2)}_{2\to 3}(E)$ (solid line). The dashed line shows the asymptotic form \ref{['sigma23ass']}, which is seen to approximate $\sigma^{(2)}_{2\to 3}$ very closely starting from relatively low energies.
• Figure 3: Relative contributions of $A_p(w)$ (solid line) and $A_\sigma(w)$ (dashed line) to the amplitude $A(w)$ at low values of $w$.