# Nucleon-meson interaction

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Suppose interaction lagrangian between neutron-proton doublet and $\pi$-mesons: $$\tag 1 L_{\pi pn} = \bar{\Psi}\pi_{a}\tau_{a}(A\gamma_{5} + B)\Psi , \quad \Psi = \begin{pmatrix} p \\ n\end{pmatrix}$$ Is it possible to derive it from the first principles? I realize that proton and neutron aren't pseudogoldstone bosons, like pions, and thus their lagrangian cannot be derived simply. But, maybe, it is possible to derive directly $(1)$.

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

edited Sep 24, 2016

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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
answered Feb 23, 2016 by (310 points)

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Name YYY
"...First of all, the idea that the nucleon can be described by some kind of classical field can only be true in the large Nc limit..." Since spin statistics and baryon number doesn't depend on the number $N_{c}$, then skyrmions in $SU_{c}(3)$ theory are fermions which have baryon number 1. In my understanding, we may identify them with hadrons, but we need to discuss their phenomenology for completeness of identification.

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Name YYY
"...Second, the nature of the classical field is not clear (why the chiral lagrangian?)..." I realize that chiral perturbation theory can't be derived from $N_{c} = 3$ QCD. But, if I understand correctly, the chiral perturbation theory is the only theory which may arise as effective theory of QCD which describes scalar mesons.

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Name YYY
"...Third, even if I accept the idea that the nucleon is a solitonic solution of the chiral lagrangian, its properties cannot be determined..." Yes, since skyrmions in "minimal" chiral perturbation theory are shrinked, we may stabilize them only by adding higher derivative terms, which in general makes theory unpredictable. However, if we introduce vector mesons in chiral perturbation theory (by the method of hidden broken symmetry), they stabilize the skyrmion solution, and we may discuss their properties.

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Name YYY
I've also seen text about $SU(2)$ skyrmions (embedded into $SU(3)$) phenomenology which contains derivation of effective skyrmion-pion coupling at zero momenta. An axial coupling was derived.

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Name YYY
Thank you one more time. But why, despite the fact that fluctuations are large, Witten et. al. continued to discuss phenomenological implications of $SU(2)$ Skyrme model in the context of the role of skyrmions as the nucleons (for example, inspirehep.net/record/190174?ln=ru). Also, it I understand correctly, fluctuations are of order $\frac{1}{N_{c}}$, but for $N_{c} = 3$ $\frac{1}{3} < 1$, so in bad approximation we may neglect them, don't we?

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Name YYY
@NameYYY 1) You do what you can (this was at a time when Witten was still doing some phenomenological work). 2) Well, 30% error is not that great (no of way knowing that it's not a 100%). Also, using a specific lagrangian to find the Skyrmion is an additional approximation beyond large N.

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Thomas
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Now I know an answer, so I draw it here.

Direct derivation of nucleon-meson interaction is possible from chiral perturbation theory, which arises from the QCD spontaneous symmetry breaking. We look for finite classical field configurations which leaves chiral action finite. Since homotopic group $\pi_{3}(SU(3)) = Z$ is nontrivial, then such configurations exist. They are called skyrmions. Corresponding Maurer-Cartan invariant, which defines skyrmion winding number, coincides with anomalous baryon charge which can be obtained formally from gauging baryon number symmetry in Wess-Zumino term. Also, the spin of winding number one skyrmion is one half. We may therefore try to identify skyrmions with proton and neutron.

The next steps are straightforward. Defining proton and neutron fields through skyrmion solution, we may, due to general logic of perturbation field theory, treat proton-neutron state as the ground state of theory, and calculate perturbations near it. Perturbations in chiral perturbation theory are defined in terms of goldstone bosons. Considering the pion sector and calculating low-energy matrix elements $\langle \psi_{n}| A_{\mu}^{a}|\psi_{m}\rangle$, where $\psi_{n}$ is proton-neutron doublet and $A_{\mu}^{a}$ is axial urrent in terms of skyrmion field, we may obtain low-energy theorems which contain axial couplings from my question.

This post imported from StackExchange Physics at 2016-09-24 12:20 (UTC), posted by SE-user Name YYY
answered Feb 22, 2016 by anonymous

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