Form Factors and Spectral Densities from Lightcone Conformal Truncation

TL;DR

This work develops nonperturbative, LCT-based methods to compute dynamical observables in a strongly coupled two-dimensional QFT, focusing on the φ^4 model and the stress-tensor trace Θ. By extracting form factors from one-particle LCT eigenstates and employing crossing symmetry, Padé approximants, and analyticity-guided rational fits, the authors obtain both two-particle form factors F_{2,0}^Θ(s) (via analytic continuation from F_{1,1}^Θ) and spectral densities ρ_Θ(s) with improved smoothness and consistency across the unbroken phase. The approach yields accurate perturbative checks at small coupling, provides smooth spectral densities from a discretized spectrum, and demonstrates robust analytic continuation to s>0, setting the stage for S-matrix/bootstrap input via a companion study. The results offer a practical nonperturbative handle on dynamical observables in LCT and point to broad applicability and future extensions to other operators and truncation schemes.

Abstract

We use the method of Lightcone Conformal Truncation (LCT) to obtain form factors and spectral densities of local operators $\mathcal{O}$ in $φ^4$ theory in two dimensions. We show how to use the Hamiltonian eigenstates from LCT to obtain form factors that are matrix elements of a local operator $\mathcal{O}$ between single-particle bra and ket states, and we develop methods that significantly reduce errors resulting from the finite truncation of the Hilbert space. We extrapolate these form factors as a function of momentum to the regime where, by crossing symmetry, they are form factors of $\mathcal{O}$ between the vacuum and a two-particle asymptotic scattering state. We also compute the momentum-space time-ordered two-point functions of local operators in LCT. These converge quickly at momenta away from branch cuts, allowing us to indirectly obtain the time-ordered correlator and the spectral density at the branch cuts. We focus on the case where the local operator $\mathcal{O}$ is the trace $Θ$ of the stress tensor.

Figures (16)

• Figure 1: Imaginary part of the time-ordered correlator $\mathbf{\Delta}_\Theta(s)$ from LCT using the spectral representation (\ref{['eq:TOCandSD']}) at $\Delta_{\rm max}=12$ ( red, solid) compared to the exact result ( black, dashed) in the free scalar theory (\ref{['eq:free2pt']}). Because the truncated Hamiltonian has a discrete spectrum, the resulting time-ordered correlator has a series of discrete poles at $s>4m^2$, whereas the continuum limit should give a smooth function as shown. The real part from LCT is not shown because it is a sum of $\delta$ functions.
• Figure 2: Real ( left) and imaginary ( right) parts of the time-ordered correlator $\mathbf{\Delta}_\Theta(s)$ in the free scalar theory, comparing the exact result ( black, dashed) to the LCT result at $\Delta_{\rm max}=12$ ( red, solid) after using Padé approximants as described in the text.
• Figure 3: Convergence of the Taylor coefficients $c_n$ of $\mathbf{\Delta}_\Theta({\varrho}(s))=\sum_{n=0}^\infty c_n {\varrho}^n$ at ${\varrho}=0$. For a given $\overline{\lambda}$, we computed the coefficients $c_n$ for each $\Delta_\text{max}$ up to $\Delta_\text{max}=40$, and then extrapolated them to $\Delta_\text{max}=\infty$ by fitting them as a function of $x=1/\Delta_\text{max}$. The function we used to fit these coefficients is $a+ b x^2$, and the solid lines are the results of the fits.
• Figure 4: Spectral densities of the trace of the stress tensor $\rho_\Theta$ in the $\phi^4$ model computed for various values of $\overline{\lambda}$ from LCT (with the inset showing more details near $s=4m^2$). These plots are obtained by taking the imaginary part of the two-point Padé approximant of the trace of the stress tensor two-point function as described in the main text. As a comparison, we also plotted the perturbative spectral density for $\overline{\lambda}=1$ (red dotted line) computed via $\left|\mathcal{F}_{2,0}^{\Theta}(s)\right|^{2}/(2 \pi \mathcal{N}_{2})$, where $\mathcal{F}_{2,0}^{\Theta}(s)$ is the two-loop form factor given in equation (\ref{['eq:FF_pert']}). One can see that it agrees with the LCT non-perturbative $\overline{\lambda}=1$ result very well at large $s$ as expected, since $\overline{\lambda}=1\ll4\pi$ is in the perturbative regime. This provides a consistency check for our procedure for computing the non-perturbative spectral density from LCT. However, one can also see the difference near $s=4m^2$, where perturbative theory breaks down. Especially, the perturbative result has a singularity at $s=4m^2$, while the non-perturbative LCT result is regular there.
• Figure 5: The $C$-functions for various coupling constants. The piecewise continuous lines are computed from the truncation data with $\Delta_\text{max}=40$ directly (that is, by integrating the $\delta$ functions in equation (\ref{['eq:SDdefn']})), while the dashed lines are computed from integrating the Padé approximant of the spectral density of $\Theta$ (i.e., real part of the Padé approximant of the $\langle\Theta\Theta\rangle$ two-point function). One can see that they agree with each other fairly well, but the Padé approximant smooths out the unphysical steps in the raw truncation result.