Ising Spectroscopy I: Mesons at T < T_c

TL;DR

This paper develops and tests a Bethe‑Salpeter framework for Ising Field Theory in a magnetic field at low temperature, modeling stable mesons as two‑quark bound states under confinement. By deriving the BS equation in the infinite‑momentum limit and including perturbative multi‑quark corrections (encoded as self‑energy and string‑tension renormalizations), the authors obtain weak‑coupling expansions and a robust numerical solution that matches TFFSA data for real parameters across a wide range of $\eta$. The work clarifies the limitations near the stability thresholds and outlines how to extend the approach to resonances and complex parameter regimes, as well as its relation to integrable points and analytic properties. Overall, the BS framework provides quantitative insight into the mass spectrum and offers a controlled path to study confinement and resonance phenomena in 2D QFTs with significant implications for related nonperturbative methods.

Abstract

This paper is our progress report on the project "Ising spectroscopy", devoted to systematic study of the mass spectrum of particles in 2D Ising Field Theory in a magnetic field. Here we address the low-temperature regime, and develop quantitative approach based on the idea (originally due to McCoy and Wu) of particles being the "mesons", consisting predominantly of two quarks confined by a long-range force. Systematic implementation of this idea leads to a version of the Bethe-Salpeter equation, which yields infinite sequence of meson masses. The Bethe-Salpeter spectrum becomes exact in the limit when the magnetic field is small, and we develop the corresponding weak-coupling expansions of the meson masses. The Bethe-Salpeter equation ignores the contributions from the multi-quark components of the meson's states, but we discuss how it can be improved by treating these components perturbatively, and in particular by incorporating the radiative corrections to the quark mass and the coupling parameter (the "string tension"). The approach fails to properly treat the mesons above the stability threshold, where they are expected to become resonance states, but it is shown to yield very good approximation for the masses of all stable particles, at all real values of the IFT parameters in the low-temperature regime. We briefly discuss how the Bethe-Salpeter approximation can be used to address the case of complex parameters, which was the main motivation of this work.

Figures (8)

• Figure 1: The mass spectrum $M_n(\eta)$ of IFT particles at positive (and some negative) $\eta$. The solid lines represent numerical data obtained using the TFFSA. The dotted line shows the stability threshold $2 M_1$; after crossing this line the particles become unstable, and their masses (not shown) develop imaginary parts. The exception is the point $\eta=0$, where there are eight stable particles, whose masses are indicated by bullets $\bullet$ . The dashed lines show the masses obtained from the Bethe-Salpeter equation \ref{['bs']}, with the renormalized string tension $f$ defined as in Eq.\ref{['fre']} (see Section 8 for explanations). The circles $\circ$ indicate positions of the higher thresholds $M_1 + M_2$, $M_1 + M_3$, etc, at $\eta=0$.
• Figure 2: Possible world lines of quarks in a meson. (a) Both quarks propagate forward in time. (b) Creation an annihilation of virtual pairs leads to the presence of more then two quarks in the intermediate state.
• Figure 3: Diagrams of the perturbation theory in $h$. (a) The "ladder" diagrams. The blobs represent the matrix elements \ref{['matrix']} . These diagrams are summed up by the Eq.\ref{['inteq']}. (b) The diagram with four intermediate quarks. (c) Example of a disconnected part of the diagram in (b).
• Figure 4: The quark self-energy: (a) The simplest self-energy diagram of the order $\sim h^2$; its contribution $\Sigma_{2}^{(3)}(p)$ appears in the Eq.\ref{['disconnected']}. (b), (c) Examples of more complicated self-energy diagrams.
• Figure 5: One-particle reducible self-energy diagrams. Parts (a) and (b) show two types of the order $h^2$ diagrams, with one and three quarks in the intermediate state, respectively. Part (c) shows an example of the higher-order diagrams which contribute to $\varepsilon(p)$.