 Leading pion mass from a two-flavour condensate

For condensate normalization $⟨{\bar{q}}_{Lj}{q}_{Ri}⟩=-v{\delta }_{ij}$, replacing the condensate matrix by $-vU$ maps the real quark mass matrix $M=\mathrm{diag}(m_u,m_d)$ to $v\mathrm{tr}[M(U+U^{†})]$. Since $(\mathbf{\pi }\cdot \mathbf{\tau }{)}^2=|\mathbf{\pi }{|}^{2}I$, expansion gives a constant minus $v(m_u+m_d)|\mathbf{\pi }{|}^2/F^2$. Comparison with $-m_{\pi }^2|\mathbf{\pi }{|}^2/2$ gives the displayed common leading mass. Strong isospin-breaking mass splitting enters beyond this leading action; electromagnetic effects are also excluded.