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The mass-gap existence in quantum YM theory is the statement that the spectrum is bounded from below by some positive value $\delta$.

The spectrum should be independent of the energy at which you probe the theory. In other words, $\delta$, for one, should not "run" with the renormalization group scale.

At high energies, YM is described well by free massless gluons due to asymptotic freedom. In this description, there is no mass-gap because massless gluons can have arbitrarily small energy. So my question is: if $\delta$ doesn't run, then why does it appear as though $\delta=0$ at very large energies?

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Because $\delta$ has the dimensions of energy, and compared to very large energy values it is negligible?

Constants only “run” when you’re considering a family of approximating theories with cutoffs. Real QFT constants aren’t running couplings, they are constants just like any other. It’s the relationship between these real constants and our naive “renormalized interaction couplings” from the (renormalized) classical action that runs with scale.

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  • $\begingroup$ That's an attractive solution. So you're saying that the mass-gap is swamped by the energy of the process...That somehow, the gluons are massive but their mass is small compared to the energy of the process? I have to disagree with your statements about running couplings though. There is a sense in which couplings run even after the cutoff is removed. You only have to lookup the beta function to see that. $\endgroup$
    – dennis
    Commented Oct 15, 2023 at 11:18
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    $\begingroup$ @dennis there’s no such thing as a gluon. It’s a perturbative approximation that’s only useful at high energies, when the (relatively) small value of the mass gap is negligible. $\endgroup$ Commented Oct 15, 2023 at 12:27

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