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It was noticed in the 1960s that chiral perturbation theory, describing the goldstone-bosons (pions) of the breaking $SU(N_f)_L \times SU(N_f)_R \to SU(N_f)_V$ of the chiral symmetry, has solitonic solutions (skyrmions), so long as you include four-derivative terms in your chiral lagrangian (the original proposal just used one of these terms). As I understand, there are various reasons to believe this skyrmion can be identified with nucleons such as the proton/neutron, delta baryons, etc:

  • Their mass has the correct linear in $N_c$ scaling as expected from large $N$ calculations.
  • The conserved topological current describing the winding number of the solitonic solution can be identified as the $U(1)$-baryon current - if you gauge your chiral perturbation theory with weak interactions, you can see that you recover the $U(1)_\text{Baryon} SU(2)^2$ anomaly
  • With a bit of work, you see exactly the right excitations, in terms of isospin and spin representations

When I look in the literature, I mainly see papers from the 1980s hyping all of this up, and it seems very solid. What I am confused about is (to quote Witten) to what extent the skyrmions actually are baryons. To be concrete:

Are there any predictions from Skyrmionic theory of baryons that are not compatible with our current measurements/understanding of baryon phenomenology?

The original papers include calculations of various physical quantities which match up to ~30%, which seems fine to me given that the original Skyrme model only contains one of the four-derivative terms of the chiral-lagrangian, while it seems to me the "proper" thing to do would be to include all terms of the same order in chi-PT with coefficients extracted from experiment/lattice, and then perform skyrmionic calculations. It's believable to me that going to higher order will fix these numerical issues

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I think there are two basic reasons why there has not been much work on skyrmions in QCD over the past $\sim$ 25 years.

  1. The model is not systematically improvable: In the Skyrmion model, stability of the soliton comes from a balance between two-derivative and four derivative terms in the chiral Lagrangian. This means that the four-derivative contribution is not parametrically small, and higher derivative (six, eight, $\ldots$) are not small either. This is contrast to applications of the chiral lagrangian in the meson sector, where higher derivative terms are suppressed, and systematic calculations are possible.

  2. Lattice QCD provides accurate (few per cent level) predictions of properties of the protons. These are based on correlation functions of local three-quark operators. It is not clear how one would use these results to determine in what sense the skyrmion is an approximate description of the proton in the real world. Indeed, I would have expected that if the proton is a soliton of an emergent collective field, then the coupling to local $qqq$ operators is small (this is not the case).

There has been some recent work on skyrmions in connection with the AdS/CFT correspondence, and AdS/QCD models, see here.

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  • $\begingroup$ Thanks for the answer! Indeed the four-derivative terms are important for stability of the skyrmion, I had naively thought however that six-derivative and higher terms have suppressed effects for the skyrmion - is this not the case? Perhaps what I had misunderstood is that the effects of higher derivative operators pick up factors of (m_proton/lambda_QCD) instead of (momentum/lambda_QCD) that you usually get from chiral perturbation theory, so the series is not good? $\endgroup$
    – QCD_IS_GOOD
    Jun 2, 2022 at 2:00
  • $\begingroup$ For the second point - I understand that the well-established, systematically improvable way to get nonperturbative information is via lattice-QCD, however it seems to me still very attractive that skyrmionic models can get nonperturbative information very quickly/easily, with much less computational time required than lattice QCD methods (for example, I think the early papers gave predictions for radial charge distribution of proton which would normally be a very difficult lattice computation) $\endgroup$
    – QCD_IS_GOOD
    Jun 2, 2022 at 2:02
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    $\begingroup$ Indeed, in the skyrmion model the nucleon mass is $M_p\sim f_\pi$ (really, $\sqrt{N_c}f_\pi$), and all terms in the derivative expansion contribute at the same order. $\endgroup$
    – Thomas
    Jun 2, 2022 at 3:09
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    $\begingroup$ I think the issue is that there are many models, the non-relativistic quark model, the MIT bag model, the skyrmion, etc. They all "work" in the sense of making $O(30\%)$ predictions, but nobody has found a way to rigorously check their assumptions. How would you tell, looking at lattice or real world observables, that the skyrmion is better than the NR quark model? $\endgroup$
    – Thomas
    Jun 2, 2022 at 3:11

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