# Number of baryons in the Skyrme model

In QCD, the Eightfold Way organizes the number of baryons with respect to their flavor and color quantum numbers: for three light $(u,d,s)$ quark constituents, a spin-(1/2) baryon octet and a spin-(3/2) baryon decouplet emerge, because the antisymmetric baryonic wave function requires a symmetric spin-flavor part in the presence of the antisymmetric color wave function.

Alternatively, baryons can be modeled by topological objects called Skyrmions. Skyrmions arise if the third homotopy group satisfies $\pi_3(G/H)\neq 0$ in a symmetry breaking sequence $G\rightarrow H$: in QCD, the quark condensate spontaneously breaks the $SU(3)\times SU(3)\rightarrow SU(3)$ flavor symmetry yielding $\pi_3(SU(3))= Z$.

Now my first question is: since the existence of Skyrmions only requires spontaneous flavor symmetry breaking, can we infer the number of the different baryons in the Skyrmion model without any knowledge about color? In other words, could we infer from the symmetry breaking pattern itself (which yields $\pi_3(SU(3))= Z$) how many baryons exist in Nature?

And my second question is: could the number of baryons change for different flavor symmetry breaking patterns (which would not be allowed in QCD), such as $SU(3)\times SU(3)\rightarrow U(1)\times U(1)\times U(1)$ or $SU(3)\times SU(3)\rightarrow SU(2) \times SU(2)\times U(1)$?

• You need flavor, as in Adkins et al, but color barely enters ... (in the bland coeff of the WZW term.) – Cosmas Zachos Feb 12 '18 at 15:10
• Thanks a lot for the reference. How is your comment consistent with the argument I gave in the first paragraph, i.e., that the number of baryons would be different in the absence of the antisymmetric color wave function? And do you have an answer to the second question I added above? – Thomas Feb 19 '18 at 17:17
• Well, color provides antysmmetry in the combinatorics of quarks comprising wave functions, but such are absent in the Skyrme picture and antithetical to its spirit, so, basically, color "is not there". As indicated $N_c$ is a mere magical integer in the anomaly-summarizing WZW term. As for different nonstandard scheme chiral models, "daran scheitert meine ganze Kunst"; but I'm sure some serious review of Skyrmions might indulge in generalist alternative speculations. – Cosmas Zachos Feb 19 '18 at 18:06