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The linear sigma model packages scalar and pseudoscalar fields into a linearly transforming multiplet. A symmetry-breaking potential gives a massive radial sigma mode and massless pions before explicit breaking is added.

Introduce the linear sigma model fieldA suitable renormalizable potential iswith a small symmetry-breaking source proportional to the quark-mass matrix when desired. A vacuum $⟨\Sigma ⟩=vI$ is fixed precisely by $L=R$, so the unbroken group is the diagonal isospin $SU(2{)}_V$. The three broken axial generators produce the pion triplet ${\pi }^{\pm },{\pi }^0$, which become pseudo-Goldstone bosons when $m_u,m_d$ are restored.