I'm trying to learn conformal field theory and getting rather frustrated, because I can't find any source that gives decent examples or straightforward logic.
In most sources I have found, conformal invariance for a classical theory is established by showing that the theory is scale invariant, then performing some handwaving. However, this reasoning can't be remotely correct, as every theory with only dimensionless couplings is scale invariant, as long as we give each field a scaling dimension equal to its usual mass dimension.
For example, under this reasoning, the Standard Model without a Higgs mass term is a conformal field theory at the classical level, but it can't possibly be, or else I would have heard about that already. Similarly, massless $\phi^4$ theory in $d = 4$ would be conformally invariant under this logic, but it isn't, as its stress-energy tensor isn't traceless. Yet another example is a free scalar field in a curved spacetime background, which requires a non-minimal coupling term $- R \phi^2 / 12$ to achieve conformal invariance.
What's going on? Are all CFT sources just being incredibly sloppy, and if so, why are they being this sloppy? Is there some reason that this fuzzy logic actually works in most examples they consider, e.g. is the situation different in $d = 2$, perhaps? And what's the real way to show a theory is classically conformally invariant?