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Higgs mechanism is known to give "mass" to gauge bosons, especially in electroweak theory where the gauge group is given by $SU(2) \times U(1)$.

However, as in this PE post or the statement of the mass gap problem, even a pure YM theory without a Higgs field is expected to have a mass gap.

Now, I have two related questions

It seems that the notion of "mass" appearing in Higgs mechanism just denotes the mass gap. Is this correct?

If a pure YM theory can have a mass gap on its own, it must be the case for electroweak theory as well, because the gauge group is $SU(2) \times U(1)$ and obviously non-Abelian. However, all standard literature in QFT seems to say just that $W$ and $Z$ bosons gain mass via the Higgs mechanism. So, I am confused here..

Could anyone please clarify for me?

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Here is a partial answer. There may be some deeper aspects that I don't fully understand yet.

The Higgs mechanism and the YM mass gap problem are associated with two different mechanisms in QFT. The former is fairly well understood as a spontaneous symmetry-breaking mechanism mediated by a scalar Higgs field. The latter is still much of a mystery and has to do with the appearance of the QCD scale and confinement.

In the appearance of the mass gap in YM theories, it is only the gauge field dynamics that is responsible for the phenomenon. There are no other fields, like some scalar fields that mediate the process. It is believed to be a consequence of the fact that the gauge coupling becomes strong at a lower energy scale due to asymptotic freedom. Eventually, the coupling becomes so strong that it confines all color charges into finite regions of space. The energy at which this phase transition occurs sets an energy scale that is called the QCD scale. This scale corresponds to the mass associated with the mass gap. Although we know that this happens from experimental observations, nobody has so far been able to give a satisfactory theoretical description of this process.

On the other hand, the process of spontaneous symmetry breaking has been described theoretically in a fairly satisfactory manner by many people, including Peter Higgs. The process is different though. It is primarily caused by a special kind of dynamics described by the potential of a scalar field that couples to the gauge field. The result is a non-zero vacuum expectation value for the scalar field, which causes a massive Higgs boson together with a Goldstone boson that in turn causes the gauge bosons to become heavy.

So why don't we see the YM mass gap in the weak force? Not sure, but I think the symmetry break spoils it. The electroweak scale is much higher than the QCD scale. So the symmetry breaking occurs at a scale where the gauge coupling is still too weak to produce the mass gap.

Hope this helps to clarify the issue.

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  • $\begingroup$ Weak interaction is $SU(2)$ gauge theory, and therefore falls within the scope of the mass gap problem. Are you saying that the mass gap problem is stated in a wrong way? $\endgroup$
    – Keith
    Commented May 28 at 6:16
  • $\begingroup$ Plus, without too many matter fields, we know that $SU(2)$ theory is asymptotically free, which is the case for the weak interaction. $\endgroup$
    – Keith
    Commented May 28 at 6:19
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    $\begingroup$ @Keith The problem is that the weak interaction comes from the breaking of the electroweak interaction, which has a $SU(2)\times U(1)$. In order to do this breaking, you need to introduce a new scalar field, $H$. In pure YM theory (which is what the problem considers), there is no such field! Apart, in the description of the problem they specify that the gauge group must be simple, which is not the case of $SU(2)\times U(1)$ $\endgroup$ Commented May 28 at 8:57
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    $\begingroup$ @GabrielYbarraMarcaida Ohh...I missed that detail. So, with just $SU(2)$, I guess that mass gap is expected without any Higgs mechanism and there is no contradiction with the standard model....Thank you!! $\endgroup$
    – Keith
    Commented May 28 at 10:51
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    $\begingroup$ @Keith Yeah, the main motivation for this problem is the "existence" (yet to be experimentally proven) of glueballs, which are massive excitations made out of gluons. These exist in QCD even without adding fermions to the theory (pure YM theory). Another big part of the problem is to prove this using axiomatic QCD... That's why it's a maths problem and not a physics one. Of course, proving it in a "non-axiomatic" way is a big Physics problem and probably worth a Nobel prize. $\endgroup$ Commented May 28 at 13:08

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