  We apply analytic bootstrap techniques to the four-point correlator of
fundamental fields in the Wilson-Fisher model. In an $\epsilon$-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 $\epsilon$ 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.
