Radial coordinates for conformal blocks

Abstract

We develop the theory of conformal blocks in ${\mathrm{CFT}}_{d}$ expressing them as power series with Gegenbauer polynomial coefficients. Such series have a clear physical meaning when the conformal block is analyzed in radial quantization: individual terms describe contributions of descendants of a given spin. Convergence of these series can be optimized by a judicious choice of the radial quantization origin. We argue that the best choice is to insert the operators symmetrically. We analyze in detail the resulting ``$\ensuremath{\rho}$-series'' and show that it converges much more rapidly than for the commonly used variable $z$. We discuss how these conformal block representations can be used in the conformal bootstrap. In particular, we use them to derive analytically some bootstrap bounds whose existence was previously found numerically.

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