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


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*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? 

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


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*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.
