Abstract
We compute anomalous dimensions of higher spin operators in Conformal Field Theory at arbitrary space-time dimension by using the OPE inversion formula of [1], both from the position space representation as well as from the integral viz. Mellin representation of the conformal blocks. The Mellin space is advantageous over the position space not only in allowing to write expressions agnostic to the space-time dimension, but also in that it replaces tedious recursion relations in terms of simple sums which are easy to perform. We evaluate the contributions of scalar and spin exchanges in the t-channel exactly, in terms of higher order Hypergeometric functions. These relate to a particular exchange of conformal spin β = Δ + J in the s-channel through the inversion formula. Our results reproduce the special cases for large spin anomalous dimension and OPE coefficients obtained previously in the literature.
Highlights
Of large spin operators [10, 11]
This inversion formula is our main tool in this paper to compute the anomalous dimension of the large spin double-twist operators at large but still finite values of the spin, or in other words we show that the inversion formula resums the large spin expansion of the anomalous dimension
We want to show here that from the previous expressions computed in Mellin space we can recover the coefficients obtained from the conformal blocks in position space
Summary
We would like to consider the correlator of four conformal primary scalar operators, which by conformal invariance, is only a function of cross ratios, O4(x4) · · · O1(x1). The conformal blocks can be expressed in a closed form in terms of products of hypergeometric functions. They are very well known in two and four dimensions and are given respectively by GJ,∆(z, z) k∆−J (z)k∆+J (z) + k∆+J (z)k∆−J (z) 1 + δJ,0. The contour integral pick up the physical poles associated to the exchange of operators in a OPE expansion and are contained in the function c(J, ∆). In the remaining of this paper we are mainly interested on an equal-dimensions scalar four-point function In such case several comments are in order: the operator exchanges are limited to even spins J. We would like to consider the z → 0 limit in which the conformal blocks dependence on z splits into a singular contribution containing a log(z) factor and a regular power contribution
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