= Leading pion mass from a two-flavour condensate
{title2=$m_\pi^2=2v(m_u+m_d)/F^2$}
For condensate normalization $\langle\bar q_{Lj}q_{Ri}\rangle=-v\delta_{ij}$, replacing the condensate matrix by $-vU$ maps the real quark mass matrix $M=\operatorname{diag}(m_u,m_d)$ to $v\operatorname{tr}[M(U+U^\dagger)]$. Since $(\boldsymbol\pi\cdot\boldsymbol\tau)^2=|\boldsymbol\pi|^2I$, expansion gives a constant minus $v(m_u+m_d)|\boldsymbol\pi|^2/F^2$. Comparison with $-m_\pi^2|\boldsymbol\pi|^2/2$ gives the displayed common leading mass. Strong isospin-breaking mass splitting enters beyond this leading action; electromagnetic effects are also excluded.
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