Massless theories can be conformally invariant

[Joker93]

Homework Statement

The exercise needs us to first show that $P^2$ (with $P_\mu=i\partial_\mu$) is not a Casimir invariant of the Conformal group. From this, it wants us to deduce that only massless theories could be conformally invariant.

Homework Equations

The Attempt at a Solution

I have shown that $P^2$ is not a Casimir operator; that is, it does not commute with all the generators of the conformal group. Specifically, I have found that:
$[P^2,D]=2iP^2$
$[P^2,K_\nu]=2i\ \{P_\nu,D\}+2i\{L_{\mu\nu},P^\mu\}$
and the commutators between $P^2$ and the remaining generators vanish. This shows that $P^2$ is a Casimir operator of the Poincare group (that does not contain $D$ and $K_\mu$).
Now, for the last part on how this shows us that only massless theories can be conformally invariant, I have no idea on how to show this. It might just be that the lecturer needs a heuristic argument though. We could do it using Lagrangians that contain a mass term and show that the mass term is not invariant under, say, scalings $x^\mu\rightarrow \alpha x^\mu$, but this way does not follow from the fact that $P^2$ is not a Casimir operator.

 

[king vitamin]

What is the eigenvalue of $P^2$? That is - (1) what is the physical interpretation of this eigenvalue, and (2) what value must it take for the conformal algebra to hold?