On The S-Matrix of Ising Field Theory in Two Dimensions

TL;DR

The paper addresses the non-perturbative S-matrix of the two-dimensional Ising field theory, a prototypical non-integrable QFT, by combining truncated free-fermion space (TFFSA) with S-matrix bootstrap techniques to analytically continue the 2→2 S-matrix into the complex plane.Focusing on the high-temperature regime (η>0), they extract elastic scattering phases via Luscher’s method, determine 3-point couplings, and track the evolution of poles and resonances as η evolves from the integrable E8 point toward the free-fermion limit, while rigorously bounding possible unknown resonances.The authors introduce a novel error-bounding lemma for analytic continuation inside the z-disc, enabling robust bounds on the S-matrix interior given elastic data, and they validate their results against perturbative predictions, with good agreement for η>2.Overall, the work demonstrates how low-energy finite-size data can tightly constrain high-energy S-matrix structure in 2D and lays out directions for extensions to higher dimensions and more complex S-matrix elements.

Abstract

We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 to 2 S-matrix element to the complex s-plane. Focusing primarily on the "high temperature" regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.

Figures (21)

• Figure 1: In two-dimensional $2\to2$ scattering, energy-momentum conservation leaves the center of mass energy squared $s$ as the only kinematic invariant.
• Figure 2: A typical set of poles (black and red dots) and zeros (grey dots) of a $2\to 2$ S-matrix element on the $s$-plane (left) and on the $\theta$-plane (right). The poles due to the self-coupling of the particle are represented by red dots. Blue and green dashed lines correspond to physical scattering in $s$ and $t$ channels respectively. On the physical strip of the $\theta$-plane (corresponding to the first sheet of the $s$-plane), zeros are allowed everywhere but they must appear in complex conjugate and crossing symmetric quadruplets (or pairs if they are on the imaginary line or on the crossing symmetric line), whereas poles are allowed only on for purely imaginary $\theta$ values.
• Figure 3: Mapping from the $\theta$-plane (left) to the $z$-disc (right). The red dots, representing the self-coupling pole, are mapped to the center of the $z$-disc. Other typical poles are represented by black dots, whereas typical zeros are represented by grey dots. The particle production threshold is indicated by the black stubs.
• Figure 4: Level 19 result for energy levels as a function of $\eta$. The cylinder circumference is taken to be $2\pi R = 6\eta/m_f$, where $m_f$ is the free fermion mass. Here we have subtracted the ground state energy and normalized the energy levels with respect to the mass $m_1$ of the lightest particle. The parts of the curves for $E/m_1$ between 1 and 2 correspond to stable particles, whereas those with $E/m_1>2$ represent scattering states of two or more particles.
• Figure 5: A demonstration of finite size effects in the spectral sets, computed using TFFSA at truncation level $L=22$ for two sample values of $\eta$. The vertical axis is the approximation to the elastic scattering phase extracted from the Bohr-Sommerfeld quantization condition, and the horizontal axis is $\arg(z)$ (related to the energy via (\ref{['zmap']})). The value of the cylinder radius $R$ is indicated by the color scheme, from red (large radii) to violet (small radii). For $\eta=2.2$, in the range $-2.7<\arg{z}<-1.6$, the spectral set of the lowest 2-particle state at small radii and that of the second lowest 2-particle state at larger radii do not overlap, due to significant finite size effect in the former. To handle this problem, we discard a posterior the spectral set data for the lowest 2-particle state at $2\pi R\lessapprox10$. The finite size effects are significantly smaller for the higher 2-particle states, as seen from their approximate overlaps. For $\eta=0.4$, finite size effects plague higher level 2-particle states as well for $2\pi R\lessapprox 3$, again in a clearly visible manner.