On the analyticity of the lightest particle mass of Ising field theory in a magnetic field

TL;DR

This paper investigates the complex analyticity properties of the lightest particle mass in 2D Ising field theory under a magnetic field, proposing a new conjecture supported by numerical evidence.

Contribution

It introduces the 'extended analyticity' conjecture for the mass function in the complex plane and verifies it using numerical methods, extending previous high-temperature analyses.

Findings

Analytic structure includes Fisher-Langer's branch cut.

Discontinuity across the branch cut determines mass behavior.

Numerical verification supports the extended analyticity conjecture.

Abstract

We study the scaling functions associated with the lightest particle mass $M_1$ in 2d Ising field theory in external magnetic field. The scaling functions depend on the scaling parameter $\xi = h/|m|^{\frac{15}{8}}$, or related parameter $\eta = m / h^{\frac{8}{15}}$. Analytic properties of $M_1$ in the high-T domain were discussed in arXiv:2203.11262. In this work, we study analyticity of $M_1$ in the low-T domain. Important feature of this analytic structure is represented by the Fisher-Langer's branch cut. The discontinuity across this branch cut determines the behavior of $M_1$ at all complex $\xi$ via associated low-T dispersion relation. Also, we put forward the "extended analyticity" conjecture for $M_1$ in the complex $\eta$-plane, similar to the analyticity of the free energy density previously proposed in arXiv:hep-th/0112167. The extended analyticity implies the "extended dispersion relation", which we verify against the numerics from the Truncated Free Fermion Approach (TFFSA), giving strong support to the conjecture.