The $P^\prime$-operator, the $Q^\prime$-curvature, and the CR tractor calculus

We establish an algorithm which computes formulae for the CR GJMS operators, the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the $P^\prime$-operator, and shows that the $Q^\prime$-curvature is constant, with the constant explicitly given in terms of the Webster scalar curvature. We also use our algorithm to derive local formulae for the $P^\prime$-operator and $Q^\prime$-curvature of a five-dimensional pseudo-Einstein manifold. Comparison with Marugame's formulation of the Burns--Epstein invariant as the integral of a pseudohermitian invariant yields new insights into the class of local pseudohermitian invariants for which the total integral is independent of the choice of pseudo-Einstein contact form.