Low-energy limit of Yang–Mills with massless adjoint quarks: chiral Lagrangian and Skyrmions

If the fundamental quarks of QCD are replaced by massless adjoint quarks, the pattern of the chiral symmetry breaking drastically changes compared to the standard one. It becomes SU (Nf) -> SO (Nf). While for Nf=2 the chiral Lagrangian describing the 'pion" dynamics is well-known, this is not the case at Nf>2. We outline a general strategy for deriving chiral Lagrangians for the coset spaces Mₖ=SU (k) /SO (k), and study in detail the case of Nf=k=3. We obtain two- and four-derivatives terms in the chiral Lagrangian on the coset space M₃ = SU (3) /SO (3), as well as the Wess-Zumino-Novikov-Witten term, in terms of an explicit parameterization of the quotient manifold. Then we discuss stable topological solitons supported by this Lagrangian. Aspects of relevant topological considerations scattered in the literature are reviewed. The same analysis applies to SO (N) gauge theories with Nf Weyl flavors in the vector representation.