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Abstract
We calculate various CFT data for the O(N) vector model with the long-range interaction, working at the next-to-leading order in the 1/N expansion. Our results provide additional evidence for the existence of conformal symmetry at the long-range fixed point, as well as the continuity of the CFT data at the long-range to short-range crossover point s* of the exponent parameter s. We also develop the N > 1 generalization of the recently proposed IR duality between the long-range and the deformed short-range models, providing further evidence for its non-perturbative validity in the entire region d/2 < s < s*.
Highlights
Where J > 0 governs the strength of interaction and si = ±1 are the Ising spins
Our results provide additional evidence for the existence of conformal symmetry at the long-range fixed point, as well as the continuity of the CFT data at the long-range to short-range crossover point s of the exponent parameter s
While the dimension of φ in the long range CFT is (d − s)/2, its corresponding dimension in the short-range CFT is d/2−1+γφ, with γφbeing the anomalous dimension of the short-range field φ
Summary
We review the basic setup of the long-range critical O(N ) vector model that we will be studying in this paper. When s < d/2, the quartic coupling g is irrelevant, and the theory flows to a fixed point in the UV limit, albeit the resulting model suffers from instabilities. This situation is analogous to the Wilson-Fisher fixed point of the short-range O(N ) vector model in 2 < d < 4 and 4 < d < 6 dimensions, respectively. In the effective action (2.13) we omitted higher-order vertices for σ, which are suppressed in 1/N We will encounter these vertices in our calculation of CFT data in this paper, where they will be explicitly represented diagrammatically via polygon graphs with internal φ propagator lines. In appendix A we collect some well-known identifies for conformal graphs in position space, that have been used for calculations in this paper
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