Critical O(N) model to orderϵ⁴from analytic bootstrap

Abstract

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)

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

“…The new results for OPE coefficients in this paper make it possible to perform similar consistency checks for the currents J ( ) R to order 4 /N 2 in all representations and for all spins. Using the 4− expansions computed in [38] we find perfect agreement. Unfortunately this does not give any constraint on the function ρ 1 (h, µ), since it does not appear until order 5 /N 2 , which can be seen from the explicit prefactor (γ (1) ϕ ) 2 /(µ − 2) 2 ∼ 2 in (F.1) and the additional factor (µ − 2) 3 ∼ 3 in the series (3.35).…”

“…A conformal block decomposition of this can be performed using the subleading corrections to the conformal blocks G τ, (z,z). The functional form of these block corrections is known [14], and by imposing the Casimir equation, they can be computed systematically order by order in z; for a short summary, see [38]. We have computed the OPE coefficients order by order, and found that they satisfy…”

“…where the a GFF n, | ∆=2 are the generalized free field OPE coefficients (normalized as in eq. (2.10) of [38]) of a free scalar of dimension 2, where J 2 2n+4, = (n + + 2)(n + + 1) is the conformal spin of the operator [σ, σ] n, , and where K(µ) = µ(µ−1)…”

“…19 Note that there is a typo in the definition of the parameter α in [17]; as pointed out in [15] Table 2: Evaluation of the truncated series in d = 3 for C J given in (3.54). The column 1/N 2 contains the new results of this paper and show great improvement when compared to results from numeric bootstrap [47] and Padé approximants (we use the Padé [3,3] of [38] which uses both the 4 − and the 2 + expansions). For N = 2, C J /C J,free is related to the universal conductivity σ ∞ of the 3d O(2) model [48] and has been estimated using Monte Carlo simulations to 0.914(10) [49].…”

“…Some operators with low twist and their names in the different regimes. using large spin perturbation theory[38] 18 .…”