Theories with fundamental quarks experience spontaneous chiral symmetry breaking:

$$ SU_L(N_f) \times SU_R(N_f) \rightarrow SU_A(N_f)$$

($N_f$ is the number of flavors)

(This is the observed approximate symmetry breaking in nature where the pions are the approximate Goldstone bosons).

In contrast, theories with adjoint quarks experience the chiral symmetry breaking pattern:

$$ SU(N_f) \rightarrow SO(N_f)$$

(modulo discrete groups). Please see for example the following article Auzzi, Bolognesi, and Shifman.

The reasoning is that since the adjoint representation is real, it has only one copy of $SU(N_f)$ flavor symmetry and $SO(N_f) $ is am maximal subgroup of $SU(N_f)$, thus any symmetry breaking will start in this pattern.

The Goldstone Boson manifold will be:

$$\mathcal{M} = SU(N_f)/SO(N_f)$$

The topology of the Goldstone Boson manifold determines the existence of t’ Hooft-Polyakov monopoles, since an non trivial homotopy group $\pi_2(\mathcal{M} )$ is required for a stable monopole solution to exist. This happens in our case when $N_f =2$, in this:

$$\mathcal{M} = SU(2)/SO(2) = S^2$$

Thus $\pi_2(\mathcal{M} ) = \mathbb{Z}$ and monopoles exist.

In addition for any number of flavors there will be Skyrmions in $\mathcal{M}$ as elaborated in , Bolognesi, and Shifman's article.

This post imported from StackExchange Physics at 2014-08-22 05:03 (UCT), posted by SE-user David Bar Moshe