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I am curious as to precisely why one can't introduce masses for gluons in Yang-Mills by a Higgs-type mechanism as in electroweak theory. Is it because then one would end up with an unwanted massless Goldstone boson? Or is there a different reason?

Thanks for any insight!

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Solving the mass gap problem for a field theory, means proving that there is an energy gap between the ground state and the first excited state. (The alternative situation is where there is a continuum of excited states with energies approaching arbitrarily close to zero.) For example, if you could prove that glueballs are the lightest bound state in QCD, and have a mass greater than zero.

In the electroweak theory, the Higgs field is something extra that you introduce, in addition to the gauge fields, in order to indirectly give the gauge bosons mass. (The alternative here would be for the gauge fields to be defined as massive from the beginning, by having a mass term in their Lagrangian.)

So it's the difference between proving that a theory already has mass gap, and modifying it so that it has a mass gap as a result of the modification. You actually could add a Higgs field with color charge to QCD, and thereby give the gluons mass, but that's not the problem.

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  • $\begingroup$ Thank you! I agree with everything you said here...so I guess the question can perhaps be rephrased like this: In endowing gauge bosons with mass, why is the Higgs mechanism (which as you point out adds something extra - we couple on the Higgs field) an acceptable solution in the case of electroweak theory, but for QCD we are not allowed to add anything extra and must prove that the excited spectrum is bounded away from zero in the first place? $\endgroup$ – Idempotent Jan 3 '16 at 13:19
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    $\begingroup$ Physically, the motivating question was "why are nuclear forces short-ranged". For the weak force, the answer is "because the gauge bosons are massive", and the mechanism is Higgs. For the strong force, the answer is believed to be, "because color charge is confined", so it's not that the individual gluons are massive, but rather that they cannot travel freely beyond the confinement distance. $\endgroup$ – Mitchell Porter Jan 3 '16 at 14:34
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    $\begingroup$ Confinement for quarks would explain a lot about mesons and baryons, but isn't proven. Proving confinement for gluons is a first step and would be implied by a mass gap for the gluon field. $\endgroup$ – Mitchell Porter Jan 3 '16 at 14:49
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You can prove the existence of a mass gap for the strong interaction without adding a Higgs-like correction term in the Yang-Mills Lagrangian to account for the limited range of the strong interaction, as it is the case for the weak interaction.

Simone Farinelli

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