My understanding was that a particle may have mass if there is a quadratic term in the fields without derivatives. For a single left-handed Weyl fermion, the following expression is lorentz invariant, hermitian, and quadratic in the fields:
$$\psi_L^T \epsilon \psi_L-\psi_L^\dagger \epsilon \psi_L^*$$ where $\epsilon = -i\sigma _y$, and $^T$ is the transpose. The whole thing works with $R \leftrightarrow L$ as well.
Why can't we throw an $m^2$ prefactor over the whole thing and call it a mass term? And if we can, then is there a reason I am missing why the sources I have read suggest that Weyl particles are massless?