Ising Spectroscopy II: Particles and poles at T>Tc

TL;DR

This work analyzes the Ising field theory in its high-temperature regime by tracking the evolution of the two-particle S-matrix poles as the scaling parameter varies, including real and pure-imaginary magnetic fields. By exploiting analyticity, crossing, unitarity, and exact results at integrable points, the study maps how stable particles, virtual states, and resonances reorganize from the integrable point to the free-particle limit and toward the Yang-Lee point, with TFFSA data supplying waypoint anchors. The paper identifies critical transition parameters (e.g., $\eta_3$, $\eta_{12}$, $\eta_{33}$, $\eta_2$) where poles cross between the physical and mirror strips, disappear into the continuum, or become resonances, and it discusses how higher resonances and possible weakly bound multi-particle states emerge and decay along these paths. Overall, the analysis provides a coherent, pole-based narrative for the real-pole sector of the Ising field theory across integrable and non-integrable regimes, offering predictions for resonance behavior and guiding future numerical checks via TFFSA.

Abstract

I discuss particle content of the Ising field theory (the scaling limit of the Ising model in a magnetic field), in particular the evolution of its mass spectrum under the change of the scaling parameter. I consider both real and pure imaginary magnetic field. Here I address the high-temperature regime, where the spectrum of stable particles is relatively simple (there are from one to three particles, depending on the parameter). My goal is to understand analytic continuations of the masses to the domain of the parameter where they no longer exist as the stable particles. I use the natural tool -- the $2\to 2$ elastic scattering amplitude, with its poles associated with the stable particles, virtual and resonance states in a standard manner. Concentrating attention on the "real" poles (those corresponding to stable and virtual states) I propose a scenario on how the pattern of the poles evolves from the integrable point $T=T_c,\ H\neq 0$ to the free particle point $T>T_c,\ H=0$, and then, along the pure imaginary $H$, to the Yang-Lee critical point. Waypoints along this evolution path are located using TFFSA data. I also speculate about likely behavior of some of the resonance poles.

Figures (7)

• Figure 1: The lowest masses $M_n$ as the functions of $\eta$. The dotted line shows the stability threshold $2 M_1$. As $\eta$ decreases the particles successively disappear from the spectrum, becoming virtual, and then resonance states (see Sect 3).
• Figure 2: Typical analytic structure of the two-particle scattering amplitude $S(i\alpha)$ in the complex $\alpha$-plane. The bold lines show the branch cuts associated with inelastic channels. The values of $S(i\alpha)$ 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 real $\alpha$-axis, within the physical strip $0 < \Re e \alpha < \pi$ are associated with the stable particles; complex poles on the mirror strip $-\pi<\Re e \alpha<0$ are interpreted as the resonance scattering states.
• Figure 3: (a) Physical characteristics of IFT (e.g. S-matrix) are assumed to admit analytic continuation onto the complex plane of $\xi^2$, with the branch cut from $-\infty$ to the Yang-Lee singularity $-\xi_{0}^2$. (b) Plot of the ratio $M_1/(-m)$ as the function of $\xi^2$, at real $\xi^2 > -\xi_{0}^2$ (The data is obtained via TFFSA zyl). The mass turns to zero at $-\xi_{0}^2$ according to \ref{['m1yl']}. At large positive $\xi^2$ the ratio $M_1/(-m)$ approaches asymptotic form $M_{1}^{(0)}\,(\xi^2)^\frac{4}{15}$, where $M_{1}^{(0)}$ is given in Appendix.
• Figure 4: Real poles $\bullet$ and zeros $\circ$ of $S(i\alpha)$ in the complex plane of the variable $\alpha=-i\theta$, at some values of $\eta$. (a) Integrable point $\eta=0.00$. The poles and zeros of \ref{['szero']} are shown (b) $\eta=-0.08$. The poles $\alpha_2$ and $\alpha_3$ move towards zero, but are still in the PS. At nonzero $\eta$ a number of complex poles associated with the resonances $A_4$, $A_5$, ... also appear (see Sect.4); in this and the subsequent drawings I ignore such poles. (c) $\eta=-0.27$. The pole $\alpha_3$ has crossed into the MS, the pole $\alpha_2$ has moved further towards zero. (d) $\eta=-0.49$. The pole $\alpha_2$ interchanges order with the pole $\pi-\alpha_1=\pi/3$. Simultaneously, the mirror zero at $-\alpha_3$ moves over the same point $\pi/3$.
• Figure 5: Poles $\bullet$ and zeros $\circ$ of $S(i\alpha)$ in the complex $\alpha$-plane, at some values of $\eta$. Except for the pair $\alpha_3, \alpha_{3}^*$, only real poles are shown. (a) $\eta = -0.94$. After colliding at $-\pi/2$ (which happens at $\eta_{33}$, Eq.\ref{['eta33']}) the poles at $\alpha_3$ and $-\pi-\alpha_3$ move away from the real $\alpha$-axis. (b) $\eta=-1.87$. The pole at $\alpha_2$ gets closer to zero (c) $\eta=-2.29$. The pole at $\alpha_2$ has left the PS. The only stable particle left is $A_1$. (d)$\eta=-4.35$. The pole at $\alpha_2$ approaches the fixed zero at $-\pi/3$, and its residue becomes small, Eq.\ref{['r1alpha']}. In the limit $\eta \to -\infty$ the zero cancels the pole, resulting in \ref{['sh0']}.