[Paper Review] On The S-Matrix of Ising Field Theory in Two Dimensions

Barak Gabai, Xi Yin|arXiv (Cornell University)|May 2, 2019

Quantum Chromodynamics and Particle Interactions22 references4 citations

TL;DR

This paper develops a numerical method to compute the analytically continued S-matrix of two-dimensional Ising Field Theory using truncated free fermion space approach (TFFSA) and L"uscher's method, enabling precise determination of scattering phases and resonance locations. It establishes rigorous error bounds on the S-matrix in the complex energy plane, allowing identification of on-shell three-point couplings and resonances, with validated results across varying coupling regimes.

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.

Motivation & Objective

• To compute the non-perturbative S-matrix of two-dimensional Ising Field Theory (IFT) with rigorous error control.
• To extract scattering phases and resonance locations using finite-size spectrum data from TFFSA.
• To validate the analytic continuation of the S-matrix using unitarity and analyticity constraints.
• To constrain inelastic cross sections and detect unknown resonances in moderate energy ranges.
• To establish a framework for inferring high-energy physics from low-energy experimental data in 2D QFTs.

Proposed method

• Uses truncated free fermion space approach (TFFSA) to compute the finite-size energy spectrum of IFT on a circle as a function of radius R.
• Applies L"uscher's method to extract elastic scattering phases from the finite-size spectrum below inelastic thresholds.
• Employs a two-step fitting procedure: first, fit scattering phases to a CDD-like ansatz with polynomial approximation of Q(x); second, use analytic continuation via unitarity and analyticity to extend S(z) to the complex z-plane.
• Implements two numerical methods: Method I (direct construction of χ(z) function with constraints on the unit circle) and Method II (parametrized fitting with CDD zeros and exponential factor).
• Computes error bounds using a discrete Cramér-von Mises criterion and a bound on the difference between finite-level and extrapolated data, ensuring reliability.
• Validates results via consistency checks between Method I and Method II, and by testing sensitivity to assumptions about CDD zeros and resonance locations.

Experimental results

Research questions

• RQ1 Can the analytically continued S-matrix of non-integrable 2D IFT be computed numerically with rigorous error bounds?
• RQ2 What are the locations of resonances (zeros of S(z)) and on-shell three-point couplings in the non-integrable regime of IFT?
• RQ3 To what extent can high-energy physics be inferred from low-energy scattering data in 2D QFT?
• RQ4 How do finite-size effects and truncation errors in TFFSA affect the accuracy of S-matrix reconstruction?
• RQ5 Can the presence of unknown resonances be excluded over a moderate energy range using this method?

Key findings

• The analytically continued S-matrix elements of IFT are computed numerically with rigorous error bounds, valid across different values of the coupling ratio η.
• The method successfully identifies the location of resonances via zeros of S(z) on the first sheet, including a pair of complex conjugate resonances near z ≈ 0.5 ± 0.5i.
• On-shell three-point couplings of the first and second lightest particles are extracted, which are inaccessible via L"uscher’s method alone due to unphysical energy regimes.
• Error bounds from Method I and Method II agree reasonably well, with discrepancies attributed to finite-size and truncation errors in the input scattering data.
• The method excludes the existence of additional resonances with sufficiently large width over a moderate energy range, based on consistency with unitarity and analyticity.
• The quality of the extrapolation is validated via convergence tests: for small η, the Cramér-von Mises criterion ensures convergence; for large η, the difference between extrapolated and finite-level data remains small.