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

We compute, using the method of large spin perturbation theory, the anomalous dimensions and OPE coefficients of all leading twist operators in the critical O(N ) model, to fourth order in the -expansion. This is done fully within a bootstrap framework, and generalizes a recent result for the CFT-data of the Wilson-Fisher model. The anomalous dimensions we obtain for the O(N ) singlet operators agree with the literature values, obtained by diagrammatic techniques, while the anomalous dimensions for operators in other representations, as well as all OPE coefficients, are new. From the results for the OPE coefficients, we derive the 4 corrections to the central charges C T and C J , which are found to be compatible with the known large N expansions. Predictions for the central charge in the strongly coupled 3d model, including the 3d Ising model, are made for various values of N , which compare favourably with numerical results and previous predictions. 34 , (2.2) and where G ijkl (z,z) satisfies crossing symmetry: 1 − z z ∆ϕ G ijkl (z,z) = z 1 −z ∆ϕ G kjil (1 −z, 1 − z).(2.3)

Motivated by applications to critical phenomena and open theoretical questions, we study conformal field theories with O(m)\times O(n)O(m)×O(n) global symmetry in d=3d=3 spacetime dimensions. We use both analytic and numerical bootstrap techniques. Using the analytic bootstrap, we calculate anomalous dimensions and OPE coefficients as power series in \varepsilon=4-dε=4−d and in 1/n1/n, with a method that generalizes to arbitrary global symmetry. Whenever comparison is possible, our results agree with earlier results obtained with diagrammatic methods in the literature. Using the numerical bootstrap, we obtain a wide variety of operator dimension bounds, and we find several islands (isolated allowed regions) in parameter space for O(2)\times O(n)O(2)×O(n) theories for various values of nn. Some of these islands can be attributed to fixed points predicted by perturbative methods like the \varepsilonε and large-nn expansions, while others appear to arise due to fixed points that have been claimed to exist in resummations of perturbative beta functions.

We apply the methods of modern analytic bootstrap to the critical O(N ) model in a 1/N expansion. At infinite N the model possesses higher spin symmetry which is weakly broken as we turn on 1/N . By studying consistency conditions for the correlator of four fundamental fields we derive the CFT-data for all the (broken) currents to order 1/N , and the CFT-data for the non-singlet currents to order 1/N 2 . To order 1/N our results are in perfect agreement with those in the literature. To order 1/N 2 we reproduce known results for anomalous dimensions and obtain a variety of new results for structure constants, including the global symmetry central charge C J to this order.

We apply the analytic conformal bootstrap method to study weakly coupled conformal gauge theories in four dimensions. We employ twist conformal blocks to find the most general form of the one-loop four-point correlation function of identical scalar operators, without any reference to Feynman calculations. The method relies only on symmetries of the model. In particular, it does not require introducing any regularisation and it is free from the redundancies usually associated with the Feynman approach. By supplementing the general solution with known data for a small number of operators, we recover explicit forms of oneloop correlation functions of four Konishi operators as well as of four half-BPS operators O 20 in N = 4 super Yang-Mills. arXiv:1710.06242v1 [hep-th]