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I had heard statements that Skyrmion of pions (pseudo-scalar mesons) cannot be an object like baryon, however, Skyrmion of vectors mesons may indeed form an object like baryon. For example, see this Ref and the work of this author and Ref 2.

For example, some said "baryon must be a skyrmion in 4D with an infinite tower of vector mesons, but not just of pseudo-scalar mesons." What is the intuition behind this and the distinction Skyrmion of pseudo-scalar mesons and vector mesons? And their relations to baryons or other hadrons?

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What is the intuition behind this and the distinction Skyrmion of pseudo-scalar mesons and vector mesons?

I'm not sure, but the authors may just state the problem of non-stability of skyrmions in "minimal" chiral perturbation theory.

The skyrmion itself is the non-point-like object characterized by some radius $r$. It can be shown by using the simple scaling arguments that in chiral perturbation theory with the lagrangian $$ L = f_{\pi}^{2}\text{tr}\left[|\partial_{\mu}U|^{2}\right] $$ the skyrmions are unstable, i.e., the radius $r$ tends to zero even if initially it is non-zero. Therefore the baryons can't be described by the skyrmion solutions in this theory.

There are two ways to solve this problem. The first one is to assume the non-minimal lagrangian $$ L = f_{\pi}^{2}\text{tr}\left[|\partial_{\mu}U|^{2}\right] + a \text{Tr}[|\partial_{\mu}U|^{4}]+... $$ Within this theory, the skyrmion solutions are stable.

The second one is to introduce background massive vector fields $V_{\mu}$ minimally coupled to $U$. The lagrangian now is $$ L = f_{\pi}^{2}\text{tr}[|D_{\mu}U|^{2}] - \frac{1}{4}V_{\mu\nu}V^{\mu\nu}, \quad D_{\mu} = \partial_{\mu} - igV_{\mu} $$ These vector fields are identified as the vector mesons. This approach sometimes is called "$\rho$-stabilized skyrmions", by the name of the lowest mass vector meson, the $\rho$-meson.

If you need, I can add the clarifying details in the answer.

And their relations to baryons or other hadrons?

Starting from the QCD (with the $SU_{L}(3)\times SU_{R}(3)$ global symmetry) and constructing the chiral perturbation theory, the skyrmion can be shown to have half-integer spin (1/2, 3/2,...), non-zero mass and integer baryon number (in dependence on the corresponding topological number). Next, the skyrmion is non-point like object with finite radius. Finally, starting from the skyrmion solution $U_{\text{skyrmion}}$, introducing the perturbations around this solution (which are parametrized in terms of the pions), it can be shown that the skyrmion-pion vertices are similar to the couplings of the nucleons to the pions.

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