Analytic Euclidean Bootstrap

We solve crossing equations analytically in the deep Euclidean regime. Large scaling dimension \(\Delta\) tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading \(1\over \Delta\) tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex \(\Delta\) plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We check our formulas in a few examples (for CFTs and scattering amplitudes) and find perfect agreement. Moreover, in these examples we observe that the large \(\Delta\) expansion works very well already for small \(\Delta \sim 1\). We make predictions for the 3d Ising model. Our analysis of dispersion relations via complex tauberian theorems is very general and could be useful in many other contexts.