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Skyrmions with vector mesons in the hidden local symmetry approach

The roles of light $ρ$ and $ω$ vector mesons in the Skyrmion are investigated in a chiral Lagrangian derived from the hidden local symmetry (HLS) up to $O(p^4)$ including the homogeneous Wess-Zumino (hWZ) terms. We write a general "master formula" that allows us to determine the parameters of the HLS Lagrangian from a class of holographic QCD models valid at large $N_c$ and $λ$ ('t Hooft constant) limit by integrating out the infinite towers of vector and axial-vector mesons other than the lowest $ρ$ and $ω$ mesons. Within this approach we find that the physical properties of the Skyrmion as the solitonic description of baryons are \textit{independent} of the HLS parameter $a$. Therefore the only parameters of the model are the pion decay constant and the vector meson mass. Once determined in the meson sector, we have a totally parameter-free theory that allows us to study unequivocally the role of light vector mesons in the Skyrmion structure. We find, as suggested by Sutcliffe, that inclusion of the $ρ$ meson reduces the soliton mass, which makes the Skyrmion come closer to the Bogomol'nyi-Prasad-Sommerfield (BPS) soliton, but the role of the $ω$ meson is found to increase the soliton mass. In a stark contrast, the $Δ$-$N$ mass difference, which is determined by the moment of inertia in adiabatic collective quantization of the Skyrmion, is increased by the $ρ$ vector meson, while it is reduced by the inclusion of the $ω$ meson. All these observations show the importance of the $ω$ meson in the properties of the nucleon and nuclear matter in the Skyrme model.