Diagonal Limit for Conformal Blocks in d Dimensions

Abstract

Conformal blocks in any number of dimensions depend on two variables z, zbar. Here we study their restrictions to the special "diagonal" kinematics z = zbar, previously found useful as a starting point for the conformal bootstrap analysis. We show that conformal blocks on the diagonal satisfy ordinary differential equations, third-order for spin zero and fourth-order for the general case. These ODEs determine the blocks uniquely and lead to an efficient numerical evaluation algorithm. For equal external operator dimensions, we find closed-form solutions in terms of finite sums of 3F2 functions.

21 pages; v2: misprints in Eqs. 2.17 and 4.9 corrected; v3: misprints in A.5 and A.6 corrected

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• Bounds on 4D Conformal and Superconformal Field Theories
• The $(2,0)$ superconformal bootstrap
• Bounds in 4D Conformal Field Theories with Global Symmetry
• Central Charge Bounds in 4D Conformal Field Theory
• Conformal Blocks in the Large D Limit

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