In addition to the cases which genneth has mentioned,

- Goldstone bosons: whose zero mass is protected by the broken symmetry
- Gauge invariance: where charged chiral fermions can't have a mass by gauge invariance (charge conservation)

I have to add the following cases, which make the complete list:

- Supersymmetry: (user1631's answer) This is the statement that a scalar particle is related by supersymmetry to a necessarily massless charged fermion. The fermion must be massless, the scalar (absent supersymmetry) doesn't have to be, but in order to have supersymmetry, the scalar ends up massless.
- Chiral symmetry enforcement: This is the only new one here.

The last one is a little confusing, because it isn't the same as the chiral symmetry breaking by the QCD quark-condensate that makes the pions light. The chiral symmetry breaking by the QCD quark-condensate is just an ordinary Goldstone thing, and the pions are Goldstone bosons.

The chiral symmetry enforcement is the statement that I require that the Lagrangian of a fermion be invariant under rotations of the left and right chiralities in opposite directions. The mass term of a Dirac equation is not invariant under this, because the mass term produces a left chirality from a right chirality with a definite phase (you fix this phase by making m real and positive).

So in order to make a Dirac equation (with two chiralities) massless, you impose chiral symmetry, which prevents the two chiralities from turning into each other. This symmetry can be a gauged symmetry, in which case this is an example of Gauge invariance--- charge conservation doesn't allow the two to flip into each other--- but it can also be a non-gauged symmetry, and then you are just forcing the mass to zero to impose a symmetry.

This is confusing because there is a tendency in high energy physics to write all fermions in Dirac form, because people often memorize Dirac matrix formulas, not Weyl formulas, and Dirac formulas generalize to higher dimension easier. Then any 2-component chiral fermion is represented as two components of a Dirac spinor, but with an extra chiral symmetry which decouples the other 2 components. The other 2 components are not physical--- they aren't there--- but the form of the Lagrangian makes the massless condition come from the chiral symmetry.

This is the sense in which the term "mass protected by a symmetry" comes up the most--- when mass terms for a (global or gauge) charge carrying Fermion are disallowed because of their 2-component nature. It's not a good way to say it, it is best to only have 2 components in your head, and consider it forbidden just by charge conservation, not by a chiral symmetry, but people unfortunately use the symmetry language in this case all the time anyway.

This post imported from StackExchange Physics at 2014-03-22 17:27 (UCT), posted by SE-user Ron Maimon