Abstract
We solve crossing equations analytically in the deep Euclidean regime. Large scaling dimension ∆ 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 frac{1}{varDelta } 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 ∆ 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 ∆ expansion works very well already for small ∆ ∼ 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.
Highlights
Crossing equations express associativity of the operator product expansion (OPE) [1, 2].They are nonperturbative consistency conditions on the CFT data
We argue that at large ∆ away from the real axis both cJ (∆) and the contribution of kinematic poles can be computed by the OPE in the crossed channel via the Lorentzian inversion formula [22]
We show that for meromorphic amplitudes a rigorous way to use finite energy sum rules (FESR) is via complex tauberian theorems
Summary
Crossing equations express associativity of the operator product expansion (OPE) [1, 2]. It was shown in [16] that F (E) has a universal asymptotic2 This rigorous result follows from unitarity and the leading contribution of the unit operator in the crossed channel Euclidean OPE via the so-called Hardy-Littlewood tauberian theorem. In this paper we point out that the situation changes if we note that the OPE expansion is valid in a complex domain of the corresponding cross ratios In this case one can apply more powerful complex tauberian theorems to the problem at hand [18,19,20,21]. Our analysis grew out of an attempt to understand FESR for meromorphic amplitudes
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