Abstract
We present a dispersion relation in conformal field theory which expresses the four point function as an integral over its single discontinuity. Exploiting the analytic properties of the OPE and crossing symmetry of the correlator, we show that in perturbative settings the correlator depends only on the spectrum of the theory, as well as the OPE coefficients of certain low twist operators, and can be reconstructed unambiguously. In contrast to the Lorentzian inversion formula, the validity of the dispersion relation does not assume Regge behavior and is not restricted to the exchange of spinning operators. As an application, the correlator 〈ϕϕϕϕ〉 in ϕ4 theory at the Wilson-Fisher fixed point is computed in closed form to order є2 in the E expansion.
Highlights
Wave expansion and makes analyticity in spin manifest
We present a dispersion relation in conformal field theory which expresses the four point function as an integral over its single discontinuity
While the fact that in perturbative settings the singularities are controlled only by the spectrum was already noticeable in the large spin perturbation theory approach [4], our formula makes it manifest that the CFT correlator is only sensitive to the operator dimensions and not the OPE coefficients
Summary
2.1 General setup Consider the four point function of a scalar operator φ(x) of scaling dimension ∆φ. Where the right hand side is expressed in terms of the usual cross ratios defined by We will refer to the three expressions as s-, t- and u-channel respectively. The function F (z, z) can be expanded in the (s-channel) conformal blocks. F (z, z) = (zz)−∆φ a∆, g∆(d,) (z, z)
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