Conformal blocks for highly disparate scaling dimensions

TL;DR

The paper analyzes conformal blocks in the regime where an external pair of scaling dimensions are highly disparate, showing that the spacetime dimension essentially contributes only through an overall normalization. By solving the Casimir differential equation in the large-$a$ limit with $a = -\tfrac{1}{2}\Delta_{12}$, the leading block behavior becomes a simple monomial in the cross-ratios, up to a dimension-dependent constant $C^{(d)}_{\mathcal{O}}$. The authors derive this constant via a spinless base case and an inductive recurrence argument, confirming agreement with exact blocks in even dimensions and extending universality across $d$. These results suggest a tractable route to incorporate heavy operators in the conformal bootstrap and offer a Mellin-space perspective aligned with AdS/CFT intuition. The work highlights both practical simplifications for bootstrap analyses and deeper connections between conformal blocks, bulk duals, and high-dimension operator dynamics.

Abstract

We show that conformal blocks simplify greatly when there is a large difference between two of the scaling dimensions for external operators. In particular the spacetime dimension only appears in an overall constant which we determine via recurrence relations. Connections to the conformal bootstrap program and the AdS / CFT correspondence are also discussed.

Figures (4)

• Figure 1: This simple shape has exactly one boundary point of each type. The left and right boundary points as we are defining them are circled with dotted red lines. The top and bottom boundary points are circled with dashed blue lines.
• Figure 2: The solution in four dimensions corresponds to the point at $(\lambda_1, \lambda_2 - 1)$. In six dimensions, this becomes 5 points centred at $(\lambda_1, \lambda_2 - 2)$. If one were to solve for conformal blocks in eight dimensions, this would most likely involve the 13 points centred at $(\lambda_1, \lambda_2 - 3)$.
• Figure 3: Each boundary point for this square has a boundary point across from it that is three units away. However, this is not a sufficient shape for the coefficients in five dimensions because some of the boundary points are adjacent.
• Figure 4: For three different pairs $\left ( \Delta, \Delta_{34} \right )$, we compare the four dimensional conformal block (black line) to the asmyptotic version of it (red dotted line). The top plot has $x \in (0.1, 0.5)$, while the bottom plot has $x \in (0.5, 0.9)$. For each of these plots $z = \frac{1}{2}$, $\Delta_{12} = -40$ and $\ell = 1$.