Quantum chromodynamics observed particles make isospin and SU(3) multiplets. The approximate flavor symmetries do have associated gauge bosons, observed particles like the rho and the omega, but these particles are nothing like the gluons and they are not massless. They are emergent gauge bosons in an approximate string description of QCD. The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The gauge invariant QCD Lagrangian is where formula_1 is the quark field, a dynamical function of spacetime, in the fundamental representation of the SU(3) gauge group, indexed by formula_2; formula_3 is the gauge covariant derivative; the γ
Chirality (physics) to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons. For more "light" quark species, flavors in general, the corresponding chiral symmetries are "U(N)×U(N)", decomposing into and exhibiting a very analogous chiral symmetry breaking pattern. Most usually, = 3 is taken, the "u, d", and "s" quarks taken to be light (the Eightfold way (physics)), so then approximately massless for the symmetry to be meaningful to a lowest order, while the other three quarks are sufficiently heavy to barely have a residual chiral symmetry be visible for practical purposes. In theoretical physics,