In the exponential representation after spontaneous symmetry breaking of the SU(2)×SU(2) by a VEV, v, the non-linear sigma model can be written,
L=12[(∂μS)2−2μ2S2]+(v+S)24Tr(∂μΣ∂μΣ†)−λvS3−λ4S4+ˉψi∂μγμψ−g(v+S)(ˉψLΣψR+ˉψRΣψL)
where S and Σ are related to the linear representation of σ and →π by,
π=σ+i→τ⋅→π=(v+S)Σ,Σ=exp(i→τ⋅→π′v)
Spontaneous symmetry breaking lead to three Goldstone bosons, →π. These are typically identified as the pions. But where did this identification come from? We started with 4 unknown scalar fields and an SU(2)×SU(2) flavor symmetry. Why would we expect the Goldstone bosons of this broken symmetry give rise to composite particles made of the fermions?