This interaction cannot be derived from QCD. It is also not quite correct.

1) QCD conserves parity (for $\theta=0$), and the pion field is a pseudoscalar, so $B$ must be zero.

2) The pion is an (approximate) Goldstone boson, so it is derivatively coupled
$$
{\cal L}=\frac{g}{f_\pi}\bar\psi\tau^a\gamma_5\gamma_\mu\partial^\mu\pi^a\psi
+ \ldots
$$
In chiral perturbation theory, this is the first term in an infinite tower of pion nucleon interactions, involving higher derivatives and higher powers of $\pi^a$. The value of $g$ has to be determined from experiment (or from lattice QCD). Chiral symmetry implies that $g$ is related to the axial-vector coupling of the nucleon, $g_A$, which can be measured using neutrino scattering or neutron beta decay.

3) The Skyrmion model is (as the name suggests) a model, not something that can be derived from QCD. First of all, the idea that the nucleon can be described by some kind of classical field can only be true in the large $N_c$ limit. Second, the nature of the classical field is not clear (why the chiral lagrangian?). Third, even if I accept the idea that the nucleon is a solitonic solution of the chiral lagrangian, its properties cannot be determined. For the soliton to be stable, the solution has to exist in the regime where all powers of the gradient expansion are of the same order, and there is no predicitive power.

Additional Remarks: In Witten's paper (http://inspirehep.net/record/140391?ln=en) the large $N_c$ limit is used to motivate a classical (mean field) picture of baryons, in which the nucleon might emerge as a soliton. If $N_c$ is not large, quantum corrections are $O(1)$ and the soliton picture makes no sense. Witten argues that the chiral lagrangian is a natural candidate for the mean field lagrangian (and that thanks to the WZ term the quantum numbers work out), but he does not claim to derive this. Finally, with or without vector mesons (why vectors? why not spin 5 mensons?) all terms in the gradient expansion are of the same order.

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Thomas