Radial Coordinates for Conformal Blocks

TL;DR

The paper develops a radial-coordinate framework for conformal blocks in $d$-dimensional CFTs, introducing the symmetric $\rho$-coordinate to map the block’s regularity domain to the unit disk and yield rapidly convergent power-series representations with nonnegative, dimension- and spin-bounded coefficients. By deriving recursion relations from the Casimir equation in both the Dolan–Osborn ($z$) and $\rho$ coordinates, the authors show that $\rho$-series coefficients remain uniformly bounded in the exchanged dimension $\Delta$ and spin $l$, in stark contrast to the $z$-series growth. The $\rho$-series converges faster than the traditional $z$-series, enabling inexpensive derivatives for all $\Delta$ and $l$, and it admits analytic toy bootstrap constructions that illuminate how bootstrap bounds arise; a truncated bootstrap with explicit error estimates is also proposed. Together, these results provide a practical and analytic toolkit to accelerate and extend the conformal bootstrap, including potential extensions to unequal external dimensions and non-scalar exchanges.

Abstract

We develop the theory of conformal blocks in 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 "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.

Figures (9)

• Figure 1: Any CFT is characterized by conformal data---primary operator dimensions and spins $(\Delta_i,l_i)$ and the OPE coefficients $f_{ijk}$. Using the OPE, the four point functions can be expanded into conformal partial waves, fixed by conformal symmetry in terms of the operator quantum numbers, times the products of the OPE coefficients. That the different expansions agree is a nontrivial constraint on the conformal data.
• Figure 2: By conformal symmetry, three operators can be put at $x_1=0$, $x_3=(1,0,\ldots,0)$, $x_4\to \infty$, with the fourth point $x_2$ somewhere in the (12)-plane parametrized by the complex coordinate $z$.
• Figure 3: Using a Weyl transformation, the configuration in Fig. \ref{['fig:z']} is mapped onto a cylinder matrix element with operators inserted as shown.
• Figure 4: Elastic center-of-mass scattering of two scalar particles. When a spin-$j$ resonance dominates the scattering process, the amplitude is proportional to $P_j(\cos \theta)$.
• Figure 5: This more symmetric configuration of operation insertions can be obtained from the one in Fig. \ref{['fig:z']} by a global conformal transformation.