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The failure of perturbation theory in describing strongly-coupled QCD is because it can't account for field configurations that are 'large'.

My questions is: from experience in lattice QCD, what kind of 'large' field configurations make important contributions to strong-coupling phenomena?

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    $\begingroup$ "The failure of perturbation theory in describing strongly-coupled QCD is because it can't account for field configurations that are 'large'." I don't really understand what this is supposed to mean: Perturbation theory fails for strongly-coupled theories because the whole assumption of perturbation theory is that the coupling parameter is small, which it isn't in strongly coupling theories. What do you mean by "field configurations that are 'large'" here, and what does that have to do with strong coupling? $\endgroup$
    – ACuriousMind ♦
    Commented Dec 29, 2022 at 21:06
  • $\begingroup$ The assumption of small coupling is really equivalent to the assumption of small field (i.e. field configurations close to zero everywhere). What non-perturbative calculations like lattice QCD manage to achieve is the sum over ALL configurations - even those that aren't small. So my question is: what kind of field configurations turn out to be important for strong-coupling phenomena? (and this is, most likely, only answerable from experience in lattice QCD) $\endgroup$
    – dennis
    Commented Dec 30, 2022 at 11:33

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The issue with perturbation theory in QCD is that it only works when the interaction strength is small compared to the energy scale you're considering. In terms of field configurations in a monte carlo simulation, a small interaction strength means that the probability distribution of field configurations is close to the distribution of the free theory. Perturbation theory breaks down when the distribution becomes substantially different from the free field theory. I don't think there's a meaningful way to point to specific configurations as the trouble makers (at least as far as perturbation theory is concerned). Certain types of field configurations can cause headaches for the lattice calculation though.

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  • In zero-temperature QCD it is generally BPST instantons. These are known to explain spontanteous breaking of $SU(N_f)_A$ chiral symmetry, which provides quarks with constituent mass. These gauge-configurations alone do not give rise to (i.e. "explain") confinement.

  • In non-zero termperature QCD with vanishing holonomy, it is KvBLL calorons.

  • With non-zero temperature and non-zero holonomy, it is the and instanton-dyons. These are suspected to possibly explain confinement.

The semi-classical approximation can be applied to all of these dominant configurations, which basically means approximating the path integral about these configurations via the saddle-point approximation. Despite the fact that this approximation integrates over the dominant contributions to the path integral, higher-order perturbative fluctations can (and usually do) play a crucial supplementary role (e.g. solving the large instanton problem). Their effects and often are implemented phenomenologically.

You can read about all this information in this lecture series by Edward Shuryak. [arxiv:1812.01509]

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Well, I guess the answer is: the saddle points.

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