Abstract
We apply analytic conformal bootstrap ideas in Mellin space to conformal field theories with O(N) symmetry and cubic anisotropy. We write down the conditions arising from the consistency between the operator product expansion and crossing symmetry in Mellin space. We solve the constraint equations to compute the anomalous dimension and the OPE coefficients of all operators quadratic in the fields in the epsilon expansion. We reproduce known results and derive new results up to O(ϵ3). For the O(N) case, we also study the large N limit in general dimensions and reproduce known results at the leading order in 1/N.
Highlights
Introduction and summary of resultsWilson’s renormalization group approach to understanding critical phenomena [1] has led to profound insights over many years
We write down the conditions arising from the consistency between the operator product expansion and crossing symmetry in Mellin space
We find the anomalous dimensions and OPE coefficients of operators for the critical O(N ) model in d = 4 − at the Wilson-Fisher fized point, for general N
Summary
Wilson’s renormalization group approach to understanding critical phenomena [1] has led to profound insights over many years. In [55] it was shown how to make use of conformal symmetry of three point functions to get the leading order (in epsilon) anomalous dimension of a large class of scalar operators (see [56,57,58,59,60,61]). This approach depended indirectly on the equations of motion that follows from a lagrangian and leads to the question: how does one recover these results using the bootstrap and go further? This is expected from the free theory, the only nontrivial bit in this assumption is that it begins at O( ) rather than say O( 1/2)
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