Bootstrapping 2d $φ^4$ Theory with Hamiltonian Truncation Data

TL;DR

This paper develops a hybrid nonperturbative framework that blends Hamiltonian truncation data with a generalized S-matrix/form-factor bootstrap to study the 2d $\phi^4$ theory. By injecting LCT-derived two-particle stress-tensor form factors and spectral densities into the bootstrap, the authors compute the elastic 2-to-2 scattering amplitude and the $2$-particle form factor of $\Theta$ for the $\phi^4$ model, and they establish pure S-matrix bounds saturated by the sinh-Gordon and staircase models. Remarkably, the $\phi^4$ elastic S-matrix and related observables lie very close to these bounds, effectively matching the sinh-Gordon/staircase results in the elastic regime. The work demonstrates the power of combining truncation data with bootstrap constraints, revealing a surprising kinship between seemingly different 2d theories and outlining clear paths for extending the approach to more complex observables and higher dimensions.

Abstract

We combine the methods of Hamiltonian Truncation and the recently proposed generalisation of the S-matrix bootstrap that includes local operators to determine the two-particle scattering amplitude and the two-particle form factor of the stress tensor at $s>0$ in the 2d $φ^4$ theory. We use the form factor of the stress tensor at $s\le 0$ and its spectral density computed using Lightcone Conformal Truncation (LCT), and inject them into the generalized S-matrix bootstrap set-up. The obtained results for the scattering amplitude and the form factor are fully reliable only in the elastic regime. We independently construct the "pure" S-matrix bootstrap bounds (bootstrap without including matrix elements of local operators), and find that the sinh-Gordon model and its analytic continuation the "staircase model" saturate these bounds. Surprisingly, the $φ^4$ two-particle scattering amplitude also very nearly saturates these bounds, and moreover is extremely close to that of the sinh-Gordon/staircase model.

Figures (14)

• Figure 1: Bound on the parameters $\Lambda$ and $\Lambda^{(2)}$. The allowed region is depicted in blue.
• Figure 2: Real part of the scattering amplitude obtained by minimizing $\Lambda^{(2)}$ for various values of $\Lambda$.
• Figure 4: Lower and upper bounds on the non-perturbative quartic coupling constant $\Lambda$. Blue dots are the numerical data. The blue vertical lines indicate the allowed region for $\Lambda$ for each value of $\delta$. The horizontal dashed line indicates the analytic value of $\Lambda \approx 6.1538$, which the numerical lower and upper bounds are expected to converge to. Here we use the analytic large $N$ data of the 2d $O(N)$ model to mimic the LCT input data.
• Figure 7: Bound on the parameters $\Lambda$ and $\Lambda^{(2)}$. The allowed region is depicted in blue. The obtained numerical values for the $\phi^4$ model using the LCT data from table \ref{['tab:phi4_parameters']} are indicated by red and purple crosses. These crosses correspond to lower and upper bounds respectively.
• Figure 8: Real part of the interacting scattering amplitude in the $\phi^4$ theory computed using the LCT data as an input to the S-matrix/form factor bootstrap problem for various values of $\overline{\lambda}$. As a consistency check, we also plotted the real part of the perturbative two-loop scattering amplitude (equation (\ref{['eq:amp_pert']})) with $\overline{\lambda}=1$ (red dotted line).