Meson mass spectrum in Ising Field Theory

Abstract

We study the two-particle approximation of the Ising Field Theory (IFT), formulated in terms of the Bethe-Salpeter (BS) equation. Derived by Fonseca and Zamolodchikov as a systematic realization of the McCoy-Wu approach, this equation captures confinement by modeling ``mesons'' as bound states of two Majorana fermions (``quarks''), in a way analogous to the integral equation in the 't Hooft model. Despite its approximate nature, the BS equation provides remarkably accurate predictions for the mass spectrum of stable mesons across a wide range of parameters. Motivated by the striking structural similarity between the BS equation in IFT and the 't Hooft equation in two-dimensional QCD, we develop a new non-perturbative analytical framework inspired by the method of Fateev, Lukyanov, and Zamolodchikov (FLZ). Within this approach, we compute spectral sums and systematically derive the large-$n$ WKB expansion for the Bethe-Salpeter equation, which governs the spectrum of an infinite tower of mesons. We further examine how our analytical results capture the known behavior of the spectrum in well-studied asymptotic regimes, such as the $E_8$ limit and the free-fermion point, where exact solutions are available for comparison. Finally, we discuss how the obtained spectral data admit a natural analytic continuation to complex values of the parameters -- an extension that was one of the primary motivations for this work.