Injecting the UV into the Bootstrap: Ising Field Theory

TL;DR

This work develops a dual, linear S-matrix/bootstrap framework in $d=2$ that integrates scattering amplitudes, form factors, and spectral densities with UV input via the central charge $c_{UV}$ using the $c$-sum rule. By solving a SDPB-amenable optimization, the authors bound IR observables for gapped QFTs and, in particular, prove a rigorous lower bound on $c_{UV}$ for a class of $\mathbb{Z}_2$-symmetric theories, including the $\phi^4$-like case, while also targeting Ising Field Theory (IFT) by constraining the UV data and exploring the IR parameter space. They show that, for fixed quartic coupling $\Lambda$, the minimal $c_{UV}$ runs from the free-boson value $c_{UV}=1$ down to the Ising value $c_{UV}=1/2$, with results matching analytic bootstrap bounds. In the IFT sector, they extract bounds on the cubic coupling and one-particle form factor at $c_{UV}=1/2$ and map a 3D pyramid in the IR-UV coupling space, demonstrating improved convergence over primal methods and outlining clear paths for incorporating additional UV input and extending to more complex spectra and higher dimensions.

Abstract

We merge together recent developments in the S-matrix bootstrap program to develop a dual setup in 2 space-time dimensions incorporating scattering amplitudes of massive particles and matrix elements of local operators. In particular, the stress energy tensor allows us to input UV constraints on IR observables in terms of the central charge $c_{UV}$ of the UV Conformal Field Theory. We consider two applications: (1) We establish a rigorous lower bound on $c_{UV}$ of a class of $\mathbb{Z}_2$ symmetric scalar theories in the IR (including $φ^4$); (2) We target Ising Field Theory by, first, minimizing $c_{UV}$ for different values of the magnetic field and, secondly, by determining the allowed range of cubic coupling and one-particle form-factor for fixed $c_{UV} = 1/2$ and magnetic field.

Figures (29)

• Figure 2: In this pyramid lies Ising Field Theory
• Figure 3: Rigorous lower bound on the UV central charge in $\mathbb{Z}_2$ symmetric QFTs with a single stable particle, which is $\mathbb{Z}_2$ odd. The parameter $\Lambda$ is defined as (minus) the value of the amplitude at the crossing-symmetric point $s = 2m^2$. The lower bound on $c_{UV}$ goes from $c_{UV}^{(min)} = 1$ at $\Lambda = 0$, where the S-matrix, form factor and spectral density become that of a free massive boson, to $c_{UV}^{(min)} = 1/2$ at the other end $\Lambda = 8$, where the matrix elements are that of a free massive fermion.
• Figure 4: Complex planes of the two particles form factor $\mathcal{F}_2^\mathcal{O}(s)$ (right) and the scattering amplitude $S(s)$ (left) in the presence of an asymptotic particle of mass $m = 1$. The red crosses symbolize the presence of a pole and the red lines represent the branch cuts.
• Figure 5: Allowed region in the $(\Lambda, \Lambda^{(2)}$) plane. The boundary is obtained from the dual bootstrap problem and matches perfectly the analytical bounds, with the color gradient corresponing to the value of the minimal central charge discussed below. Using the non linear dual approach given by eq.\ref{['nonlindual1']} to \ref{['nonlindual2']}, the numerics for the points on the boundary were simple enough to be done on Mathematica with only $N=2$ parameters in the Ansatz.
• Figure 6: Allowed region in the $(\Lambda, \Lambda^{(2)}, c_{UV})$ space from different angles. We used $N=50$. We find perfect agreement with the analytical bootstrap result which assumes that the optimal S-matrix minimizes $\Lambda^{(4)} = \lim_{s\to 2} \frac{\partial^4 }{\partial s^4} \mathcal{T}(s)$ (see appendix \ref{['sec:analytic_bootstrap']}).