= Quark chiral condensate
{title2=$\langle\bar q q\rangle\ne0$}
A nonzero vacuum value of the <scalar> <quark> density breaks the massless nonsinglet <chiral symmetry> to its diagonal vector subgroup. Indeed, the <nonsinglet axial charge algebra of quark densities> gives $\langle[Q_{5,a},P_b]\rangle=-i\delta_{ab}\langle\bar q q\rangle$, whereas invariance of the vacuum under every conserved <axial charge> would make these expectations vanish. The three broken generators produce three massless <Goldstone bosons> in the exact chiral limit. Small equal <quark masses> explicitly break that <symmetry> and turn the <pions> into <pseudo-Goldstone bosons>, with squared masses linear in the <quark> <mass> at leading order.
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