Abstract
We study interacting critical UV regime of the long-range O(N) vector model with quartic coupling. Analyzing CFT data within the scope of ϵ- and 1/N-expansion, we collect evidence for the equivalence of this model and the critical IR limit of the cubic model coupled to a generalized free field O(N) vector multiplet.
Highlights
The scaling dimension of the field φ is fixed exactly by the bi-local kinetic term in the action (1.1) to be [2]
The main statement of our paper is that the fixed point (3.2) is described by a CFT that is equivalent to the CFT in the UV critical regime of the non-local O(N ) vector model (1.5)
While the fixed point (3.2) was determined perturbatively near d = 6 and s = d/2 − 1, we suggest that such an IR stable fixed point exists for the range d/3 < s < min(d/2, s⋆), and that it is described by a CFT that is equivalent to the critical UV regime of the long-range O(N ) vector model (1.5)
Summary
We will perform our calculations in the model (1.14) perturbatively in the couplings g, h near the free UV fixed point, working at the linear order in ǫ1,2. (2.3) , where we took advantage of the simple behavior of loops in position space, discussed in appendix A, and used the propagator merging relation (A.1) and the function U (a, b, c), given by (A.2). It is given by a tree-level diagram, defined by the Feynman rule for the vertex corresponding to the coupling constant g. After one of the three vertices is integrated over, via the uniqueness relation (A.4), producing the factor of U (2, 2, 2)|d=6 = π3, we will encounter a logarithmically divergent integral, that we regulate using the UV cutoff μ0, d6x |x|6 Such divergences are removed with the δg and δh counterterms (that we keep mostly implicit in our calculation) proportional to − log(μ0/μ), at the expense of introducing an arbitrary RG scale μ. −3 h3 + N g2 (h − 4 g) + O(ǫ21,2)
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