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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)?

Qmechanic
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Nirmalya Kajuri
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1 Answers1

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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.

CGH
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  • 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 – innisfree Aug 02 '16 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 02 '16 at 19:14
  • @innisfree: The conventional wisdom is that $\phi^4$ theory in 4d is trivial. – user1504 Aug 02 '16 at 19:27
  • Ah yes i agree, but i had in mind an effective theory, valid below eg Planck scale.... – innisfree Aug 03 '16 at 04:56
  • Are you saying there are no possible UV completions that don't introduce massive particles? – innisfree Aug 03 '16 at 04:59
  • Sorry, I don't understand. Let me rephrase my question. According to CMT, the biggest symmetry group of any field theory in Minkowski spacetime is a direct product of Poincare and internal symmetry groups. My question is, does this conclusion apply to theories with massless as well as massive particles? Does it apply to QED? Does it apply to perturbative Quantum Gravity?

    Naively it shouldn't, because these theories don't have mass gap, which the proof assumes.

     – Nirmalya Kajuri Aug 03 '16 at 14:17
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    @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 04 '16 at 13:00
  • @NirmalyaKajuri: PS: I think that now I see your problem. Your are confusing the term mass gap. Mass gap means energy gap (in flat space, $m^2=-p^2$.) To have a mass gap means that there are no rest massless particles, which have zero energy. – CGH Aug 04 '16 at 13:48
  • by perturbative quantum gravity, I mean just that - the quantum theory of a massless spin 2 particle in Minkwski space, which we know to be non renormalizable. Perhaps I'm confusing mass gap, but I don't see how exactly. QED does not have mass gap, according to the link you gave. CMT assumes there is a mass gap. I suppose you are saying that once the theory is IR regularized, there is no mass gap and CMT applies? – Nirmalya Kajuri Aug 04 '16 at 18:17
  • Dear @NirmalyaKajuri, I really fail to see your problem. QED has a mass gap when IR regularized (that's what says in the link, exactly the opposite of what you are stating). If there's no mass gap, then it's free (trivial). See Rychkov CFT lectures discussing this issue. Are you sure you understand what a mass gap is? Finally, I still don't understand your question regarding quantum gravity: if it is not regularized, how can we compute amplitudes? Is there a well defined S-matrix definition there?, I don't know about that. – CGH Aug 04 '16 at 19:51
  • Sorry there was a typo in my last line I meant 'the theory is IR regularized, there is a mass gap and CMT applies? I think that's what you are saying. – Nirmalya Kajuri Aug 05 '16 at 09:37
  • Then, I don't understand your question. If I'm saying "QED has a mass gap when IR regularized", why do you ask if "QED has a mass gap"? – CGH Aug 05 '16 at 12:46