An alternative to diagrams for the critical O(N) model: dimensions and structure constants to order 1/N2

Abstract

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.

Analytic and numerical bootstrap of CFTs with $O(m)\times O(n)$ global symmetry in 3D

“…[13,14,18,[22][23][24][25][26], with the highest-loop study (six loops) performed recently in [27]. 1 Here we will use the analytic bootstrap method of large spin perturbation theory, introduced in [28,29] and developed further in [30][31][32], to confirm existing results in the literature and obtain some new large-n ones. The analytic bootstrap gives the same type of results as diagrammatic methods, but simplifies the computation of certain quantities, such as scaling dimensions of spinning operators, OPE coefficients and central charges.…”

“…At each order in the expansion, the whole double-discontinuity can be generated from an ansatz of contributions from a small set of crossed-channel operators, and the undetermined constants of this ansatz can later be fixed by consistency conditions, for instance conservation of symmetry currents. The method applies to a wide range of theories, and in particular it has been used to study the expansion for the Wilson-Fisher fixed point [30] and in the large-N expansion for the O(N ) model [32]. 2 In this paper we show how to generalize these implementations to critical φ 4 theories with any global symmetry group.…”

“…The equations of motion for S and W ab can be used to go back to (3). Below we will reproduce (8) and (9) and obtain more results at order 1/n following the analytic bootstrap logic of [32].…”

“…where the explicit form of the matrix M can be worked out from the tensor structures for the symmetry group under consideration. Here we choose normalizations in agreement with [32], so that the matrix in the O(N ) case takes the form…”

“…For certain global symmetry groups, the Hubbard-Stratonovich transformation can be generalized, where N corresponds to a specific group parameter. 4 More precisely, if certain bilinear operators φ 2 R have dimensions approaching 2, they should be promoted to auxiliary fields R. In large spin perturbation theory, these auxiliary fields will, together with 1, source the generalized 1/N expansion, and play the role same role as σ in the treatment of the O(N ) model in [32].…”