Abstract
We formally extend the CFT techniques introduced in arXiv: 1505.00963, to $$ \phi \frac{2{d}_0}{d_0-2} $$ theory in d = d 0 − ϵ dimensions and use it to compute anomalous dimensions near d 0 = 3, 4 in a unified manner. We also do a similar analysis of the O(N) model in three dimensions by developing a recursive combinatorial approach for OPE contractions. Our results match precisely with low loop perturbative computations. Finally, using 3-point correlators in the CFT, we comment on why the ϕ 3 theory in d 0 = 6 is qualitatively different.
Highlights
Our results are based purely on constraints from three point functions
Using 3-point correlators in the CFT, we comment on why the φ3 theory in d0 = 6 is qualitatively different
One goal of this paper is to present the discussion in a somewhat unified manner — we will see that the CFT formalism goes through without hitch for the d0 = 3 case as well
Summary
The net result for the number of contractions is nCn−r This quantity is equal to ρf because f = (n + 1)! One might hope to generalize the conclusions to generic r by re-writing the factorials in terms of Gamma functions, but the meaning of such an operation is unclear This is because the arguments for the contractions were combinatorial. For d0 = 6 were r = 1/2, we will see that the situation is qualitatively different This is the first indication from the CFT approach that the d0 = 6 case where r is no longer integral is bound to have a conceptually different -expansion compared to the d0 = 3, 4 cases. We will see that the latter theories have an anomalous dimension γφ that starts at O( 2) while the six dimensional theory it starts at O( )
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