Abstract
We apply analytic bootstrap techniques to the four-point correlator of fundamental fields in the Wilson-Fisher model. In an ϵ-expansion crossing symmetry fixes the double discontinuity of the correlator in terms of CFT data at lower orders. Large spin perturbation theory, or equivalently the recently proposed Froissart-Gribov inversion integral, then allows one to reconstruct the CFT data of intermediate operators of any spin. We use this method to compute the anomalous dimensions and OPE coefficients of leading twist operators. To cubic order in ϵ the double discontinuity arises solely from the identity operator and the scalar bilinear operator, making the computation straightforward. At higher orders the double discontinuity receives contributions from infinite towers of higher spin operators. At fourth order, the structure of perturbation theory leads to a proposal in terms of functions of certain degree of transcendentality, which can then be fixed by symmetries. This leads to the full determination of the CFT data for leading twist operators to fourth order.
Highlights
An algebraic machinery to compute the CFT data as a series in inverse powers of the spin was developed
In an -expansion crossing symmetry fixes the double discontinuity of the correlator in terms of CFT data at lower orders
The inversion formula of [13] can be regarded as the resummed version of large spin perturbation theory, but it proves that its results do resum into analytic functions of the spin
Summary
An algebraic machinery to compute the CFT data as a series in inverse powers of the spin was developed. In a remarkable paper [10] it was shown that this is the case even in a non-perturbative context.1 All this was put on firmer ground in a beautiful paper [13] which proved that the CFT data is an analytic function of the spin and arises solely from the singularities of the correlator. A remarkable feature of these operators, together with intuition from perturbation theory, makes it possible to guess their contribution to the divergence, and to determine the CFT data of weakly broken currents to fourth order. The results for the anomalous dimensions agree with those in the literature, computed by Feynman techniques, while the OPE coefficients are a new result From the latter we deduce the central charge of the WF model to fourth order in the -expansion: Cfree. This introduces a correction to the scaling dimensions and OPE coefficients of the leading-twist operators
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