The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT

TL;DR

The paper develops a comprehensive program combining numerical bootstrap (extremal functional method with SDPB) and analytic Lorentzian inversion to study the 3d O(2) CFT. By analyzing four-point functions of scalars with charges 0, 1, and 2, it derives high-precision low-twist OPE data across charge sectors, validates analytic predictions against numerics, and provides evidence that certain scalars lie on double-twist Regge trajectories with estimated Regge intercepts. Key technical innovations include dimensional reduction and SL2 expansions of 3d blocks, double-twist improvement (DTI), and a twist Hamiltonian to resolve mixing among near-degenerate towers. The work not only corroborates the inversion formula’s predictive power but also charts directions for improving numerical and analytic bootstrap methods and applying the data to broader CFT contexts and experimental probes.

Abstract

We study the spectrum and OPE coefficients of the three-dimensional critical O(2) model, using four-point functions of the leading scalars with charges 0, 1, and 2 ($s$, $φ$, and $t$). We obtain numerical predictions for low-twist OPE data in several charge sectors using the extremal functional method. We compare the results to analytical estimates using the Lorentzian inversion formula and a small amount of numerical input. We find agreement between the analytic and numerical predictions. We also give evidence that certain scalar operators lie on double-twist Regge trajectories and obtain estimates for the leading Regge intercepts of the O(2) model.

Figures (34)

• Figure 1: We plot the Euclidean four-point function $\langle\phi\phi\phi\phi\rangle$ projected onto $0^+$ exchange in the $s$-channel and normalized by the corresponding MFT four-point function. We include both the isolated and double-twist operators and expand to 5$^{\text{th}}$ order in dimensional reduction. The regions around $z=0$ and $z=1$ are computed using the $s$ and $t$-channel, respectively.
• Figure 2: Dimension of the leading charge 4 scalar operator (above the gap) as one changes the imposed gap in the charge 4 sector. Here we have chosen a single primal point and computed the extremal spectra by optimizing the upper bound on the OPE coefficient $f_{\phi\phi s}$. This plot gives a clear illustration of the sharing effect, where when we impose gaps closer to the target operators, we get less accurate results from the extremal functional method in some cases.
• Figure 3: Analytical and numerical predictions for the twists $\tau=\Delta-\ell$ of the leading-twist Regge trajectories $[\phi\phi]^{0^{\pm},2}_{0}$ in the charge sectors $0^{\pm},2$ as a function of spin $\ell$. The lowest trajectory corresponds to charge $0^+$ (the trivial representation of O(2)). Note that both the spin-2 operator in the charge $0^+$ sector and the spin-1 operator in the charge $0^-$ sector have twist 1, corresponding to the stress tensor and O(2) current, respectively.
• Figure 4: Analytical and numerical predictions for the OPE coefficients of the leading-twist Regge trajectories $[\phi\phi]^{0^{\pm},2}_{0}$ in the charge sectors $0^{\pm},2$, as a function of spin $\ell$. We divide each coefficient by the corresponding coefficient in Mean Field Theory (MFT).
• Figure 5: Analytic and numerical predictions for the spectrum of the $[\phi t]^{3}_{0}$ Regge trajectories. The upper (blue) curve represents even spin operators, while the lower (orange) curve represents odd-spin operators.