# Coleman-Mandula theorem and mass gap

I had a couple of naive questions about Coleman-Mandula theorem.

1. One of the assumptions of the theorem is the non-existence of massless particles in the spectrum. Since we do have massless photons in the standard model, how is the theorem relevant?

2. Why aren't there examples of relativistic theories with hybrid symmetries and a massless particle in the spectrum (like some extension of QED)?

## 1 Answer

The Coleman-Mandula theorem (CMT) does not rule out theories with massless particles. What it rules out is a theory with only massless particles. If you only have massless particles you either:

• End up with a free theory. This theory has trivial S-matrix, thus, the CMT does not apply here
• Have conformal symmetry. If you have conformal symmetry you cannot strictly speak about particles (there is no localized state) and there is no S-matrix, since there is no asymptotic free states.

This is the reason why they add a mass gap. But you can have theories with massless particles and massive particles.

The second question can be answered as follows: Suppose you have a symmetry that is not a direct product of Poincaré and an internal symmetry. This means that you can apply to a particle state of mass $M_1$ at position $x_1$ your symmetry and transform it in a state of mass $M_2$ (or just a different particle) at position $x_2$. This looks like a long range force. The CMT assumes local forces, otherwise you cannot assume asymptotic free states.

• I don't understand your argument. Take $\phi^4$ theory. I can tune the scalar to be massless (pole mass is zero) by cancelling a bare mass with corrections. The theory has a massless particle but is neither conformal nor trivial Aug 2, 2016 at 18:48
• The theory is either massless or is it not. You cannot turn a massive theory into a massless one by any (re)normalization. Massless $\phi^4$ is conformal.
– CGH
Aug 2, 2016 at 19:14
• @innisfree: The conventional wisdom is that $\phi^4$ theory in 4d is trivial. Aug 2, 2016 at 19:27
• Ah yes i agree, but i had in mind an effective theory, valid below eg Planck scale.... Aug 3, 2016 at 4:56
• @NirmalyaKajuri: As I stated before, it applies to both massive and massless, but not for only massless theories. I gave the reasons for that. For a discussion on mass gap in QED, see this. The point is that, adding an IR regularization is equivalent to add a finite mass gap. About quantum gravity, I have no idea what is quantum gravity, other than string theory, which has its own regulator in the string length. In General Relativity, a black hole gives a mass gap to the theory.
– CGH
Aug 4, 2016 at 13:00