Baryons as Chiral Solitons

Abstract

In Sect. 3.5 we derived the Skyrme Lagrangian (3.98)

$$ \begin{gathered} \mathcal{L} = - \frac{1} {4}f_\pi ^2 tr(L_\mu L^\mu ) + \frac{1} {4}m_\pi ^2 f_\pi ^2 tr(U^\dag + U - 2) + \frac{1} {{32e^2 }}tr[L_\mu ,L_\nu ][L^\mu L^\nu ] \hfill \\ U = e^{2i\pi } , \pi = \pi ^i \frac{{\tau ^i }} {2}, L_\mu = U^\dag \partial _\mu U \hfill \\ \end{gathered} $$

(4.1)

from the non-local effective meson theory (3.26) which is equivalent to the Nambu-Jona-Lasinio model (3.1). We achieved this by using a covariant gradient expansion, the heat kernel method, and taking the limit of infinitely large vector and axialvector meson masses. We discussed already that the Skyrme model describes pions for small energies and amplitudes reasonably well. Allowing also large amplitudes one realizes that this model possesses special localized solutions: topological solitons. These are then interpreted as baryons.