Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation

We continue the exploration of multipoint scalar comb channel blocks for conformal field theories in 3D and 4D. The central goal here is to construct novel comb channel cross ratios that are well adapted to perform projections onto all intermediate primary fields. More concretely, our new set of cross ratios includes three for each intermediate mixed symmetry tensor exchange. These variables are designed such that the associated power series expansion coincides with the sum over descendants. The leading term of this expansion is argued to factorise into a product of lower point blocks. We establish this remarkable factorisation property by studying the limiting behaviour of the Gaudin Hamiltonians that are used to characterise multipoint conformal blocks. For six points we can map the eigenvalue equations for the limiting Gaudin differential operators to Casimir equations of spinning four-point blocks.