Question

Do descendant fields always die off quicker than the slowest primary?

Answer 1

By translation and dilatation invariance, the two-point function of two scalar fields with conformal dimensions Δ1 and Δ2 is

⟨O1(x1)O2(x2)⟩∝|x1−x2|−Δ1−Δ2

This holds whenever the two fields are eigenvalues of the dilatation operator, which is true in particular for descendents and primaries. (If the fields are primary you have additional conformal Ward identities that say that the two-point function vanishes unless Δ1=Δ2.)

By definition, descendents are obtained from primaries by acting with creation operators. Such operators increase conformal dimensions. So if you assume that all your fields are primary or descendent, and that conformal dimensions are bounded from below, then the field with the lowest conformal dimension must be primary.

Answer 2

One of the best-known examples of anevent horizon derives from general relativity's description of a black hole, a celestial object so massive that no nearby matter or radiation can escape its gravitational field. ... The Schwarzschild radius of an object is proportional to its mass.