TITLE: Why is conformal invariance only possible for massless theories?
QUESTION [34 upvotes]: I'm conscious that this isn't necessarily a research level question, but I've asked this question on mathstackexchange, and received no answer. So I'm trying it here.
A usual mantra in field theories is the assertion that only massless theories can be conformally invariant. By a theory I mean an action $$ S = \int \mathcal{L} \, \mathrm{dVol}, $$ where $\mathcal{L}$ is the Lagrangian density, and the integral is taken over a 4-dimensional Lorentzian manifold with metric $g$. By conformal invariance I mean the statement that under the conformal rescaling of the metric $$ \hat{g} = \Omega^2g, $$ the Lagrangian transforms as $\hat{\mathcal{L}} = \Omega^{-4} \mathcal{L}$. Then, as the volume form transforms as $\widehat{\mathrm{dVol}} = \Omega^{4} \mathrm{dVol}$, the action $S$ is invariant, and the theory is said to be conformally invariant.
The usual physics explanation given is that "if a theory is supposed to be conformally invariant, then there cannot exist an intrinsic scale to it, such as mass or a Compton wavelength". Of course, this is a load of hand waving. I guess I don't strictly know what I mean by a massless theory. Maxwell's equations, for example, are a massless conformally invariant theory. My guess would have been that the mass of a theory is its ADM mass, but as has been pointed out in the comments, this is a property of a solution to a theory, not the theory itself. So, if $m$ is the mass of a theory, whatever it stands for exactly, and $m \neq0$, why must conformal invariance fail?

REPLY [8 votes]: Why is "conformal invariance only possible for massless theories?" 
Actually, the answer is NO. 
Not only the massless, but also the infinite large massive theory can have "conformal invariance."  As long as the theory sets no scale in order to be scale invariance.
Some massless theories
(or gapless, which means the energy gap $\Delta_E$ for excitations away from the vacuum ground state is zero, $$\Delta_E\equiv E_{exc}- E_0 \to 0,$$ where $E_{exc}$ and $E_0$ are the energies of excited states and ground state respectively) theory can be "conformal invariance" or be a "conformal field theory (CFT)." The $\Delta_E=0$ sets no scale in the theory, thus it can be scale invariant.
The infinite large massive theory, or the infinite large gap theory can still be "conformal invariance" or be a "conformal field theory." The infinite large massive theory means there is an infinitely large mass $M\to \infty$, or say the energy gap $\Delta_E$  for excitations away from the vacuum ground state has 
$$\Delta_E\equiv E_{exc}- E_0 \to \infty.$$
So the length/time scale of the system (based on the dimensional analysis $\Delta_p \Delta_x \sim \Delta_E \Delta_t \sim \hbar$) $$\Delta_l \to 0$$
The well-known examples of such a massive $\Delta_E \to \infty$ is Topological Quantum Field Theory (TQFT). Indeed, Topological Quantum Field Theory at its ground state has no scales (thus scale invariance) and it is also Poincaré (translations + Lorentz) invariance, and also have a special conformal transformation (if needed). Thus TQFT at ground state is also conformal invariant. The $\Delta_E=\infty$ also sets no scale in the theory, thus it can be scale invariant.
TQFTs may be among the most simplest/trivial examples of CFT. But TQFTs are still complicated enough to have many different distinct classes.
The "conformal invariance" requires the scale invariance/dilations and the special conformal transformation, and Poincaré group (the semi direct product of translations group and Lorentz group).