QFT with massless particles

Question

EDIT: Often in textbooks one discusses QFT with massless particles, e.g. massless fermions or scalar particles. What mass vanishes: bare or physical? or both?

Answer 1

When textbooks specifically mention massless particles, they always mean renormalized mass. Unless protected by some type of symmetry, if the bare mass vanishes, the renormalized mass does not (due to loop corrections).

Examples of cases where renormalized mass vanishes are:

1. Goldstone bosons (spin-0): Goldstone bosons arise due to spontaneous symmetry breaking. They have a non-linear symmetry transformation of the form $\phi \to \phi + a$. The mass term $m^2 \phi^2$ is not invariant under such a shift symmetry, so the renormalized mass vanishes.

2. Chiral fermions (spin-1/2): The fermion mass term $m {\bar \psi} \psi$ is not invariant under chiral symmetry transformations, so chiral fermions are massless. Note, however, that a chiral symmetry is often anomalous, in which case the renormalized mass is not expected to be zero.

3. Gauge symmetry (spin-1): The mass term $m^2 A^\mu A_\mu$ is not gauge-invariant so in this case, the renormalized mass is also zero.

4. Diffeomorphism symmetry (spin-2): Same as above.