I faintly remember the rule of thumb that only the up, down and strange quarks can form condensates because their mass is below the QCD scale $\Lambda_\text{QCD}$. But why is that? Where‘s the connection between condensates and the QCD scale?

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    $\begingroup$ Free theories don’t tend to condense. Unless I am mistaken, for processes which appreciably involve the heavy quarks (above the QCD scale) the theory is asymptotically free/weakly coupled. $\endgroup$ – hulsey May 18 at 8:26

@hulsey essentially answered your question, qualitatively. Perhaps he should just post his comment as an answer. For quark masses larger than QCD's Λ~200MeV, weakly-coupled gluons cannot effect something as drastic as condensing the respective fermions. Chiral condensation is an intrinsically non-perturbative phenomenon.

For quark masses smaller than the scale of the quark condensate, ~250MeV, (our world's u, s, d: 2, 5, 95MeV), one may consider them as a perturbation around a χ-symmetric massless limit; that is, QCD may act to SSBreak the extant background χ-symmetry and condense, thus realizing it in the Goldstone-Nambu mode.

One may then perturb around this ideal chiral symmetry limit by m/(condensation-scale), and obtain realistic results. Lots of Lattice QCD simulation effort has gone into investigating the mechanism, and a proper answer should involve lattice reviews of these investigations. I believe there have been lattice verifications of Dashen's formula (sometimes referred to as GMOR) breakdown, confirming that heavy-quark pseudoscalar-meson masses do not go as the square-root of the quark (>Λ) masses, as if they were underlain by chiral condensation.

So, if you did have a heavy quark condensate, how would you estimate it? How would you dare use it in Dashen’s formula? How would you know?

The qualitative argument has been fleshed out by ultra technical Lattice simulations, I believe, without good review recommendations... To assay when chiral perturbation theory breaks down, as discussed above, look at Edwards, Heller, & Narayanan (1998), "Spectral flow, condensate and topology in lattice QCD", Nucl Phys B535 (1-2) 403-422.

You might enjoy this: Jamin (2002), "Flavour-symmetry breaking of the quark condensate and chiral corrections to the Gell-Mann–Oakes–Renner relation", Phys Lett B538 (1-2) 71-76.


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