More Analytic Bootstrap: Nonperturbative Effects and Fermions

TL;DR

The paper advances analytic bootstrap in three dimensions by incorporating nonperturbative finite-spin effects through the Lorentzian inversion formula and by extending the framework to fermionic correlators. It shows that exponentially suppressed finite-spin contributions are essential for achieving high-precision predictions in the 3d Ising and $O(2)$ CFTs, including a near-perfect reconstruction of the stress-tensor twist in Ising and a prediction of charge-2 operator couplings in $O(2)$. The authors develop fermion blocks via embedding-space differential operators, derive leading and subleading large-spin corrections for fermion double-twist trajectories, and analyze both parity-even and parity-odd exchanges, including stress-tensor and scalar contributions. These results deepen analytical control over CFT spectra at finite spin, provide cross-checks for numerical bootstrap, and lay groundwork for integrating analytic and numerical bootstrap methods and for applications to fermionic and gauge-theoretic 3d CFTs. The work thus enhances precision, consistency, and reach of the analytic bootstrap in both scalar and fermion sectors, with potential implications for holography and beyond.

Abstract

We develop the analytic bootstrap in several directions. First, we discuss the appearance of nonperturbative effects in the Lorentzian inversion formula, which are exponentially suppressed at large spin but important at finite spin. We show that these effects are important for precision applications of the analytic bootstrap in the context of the 3d Ising and O(2) models. In the former they allow us to reproduce the spin-2 stress tensor with error at the $10^{-5}$ level while in the latter requiring that we reproduce the stress tensor allows us to predict the coupling to the leading charge-2 operator. We also extend perturbative calculations in the lightcone bootstrap to fermion 4-point functions in 3d, predicting the leading and subleading asymptotic behavior for the double-twist operators built out of two fermions.

Figures (4)

• Figure 1: Spectrum for $[\sigma\sigma]_{0}$ in the Ising CFT derived using the inversion formula, asymptotic lightcone expansion, and numerical bootstrap. Numerical data is taken from Simmons-Duffin:2016wlq. The curves in this and later plots are obtained by matching at $z=.1$.
• Figure 2: OPE coefficients $f_{\sigma\sigma[\sigma\sigma]_{0}}$ in the Ising CFT. Numerical data is taken from Simmons-Duffin:2016wlq and the OPE coefficients are normalized by dividing by the mean field theory OPE coefficients.
• Figure 3: Spectrum of $[\phi\phi]_{0}^{(I)}$ and $[\phi\phi]_{0}^{(A)}$ in the O(2) model. The black dots corresponds to the stress tensor and conserved current which have twist one.
• Figure 4: OPE coefficients for $f_{\phi\phi[\phi\phi]^{(I)}_{0}}$ and $f_{\phi\phi[\phi\phi]^{(A)}_{0}}$ in the O(2) model. Numerical data is taken from Kos:2013tga and the OPE coefficients are normalized by dividing by the mean field theory OPE coefficients.