Boson half-integer spin particles cannot be bosons and integer spin particles cannot be fermions. In large systems, the difference between bosonic and fermionic statistics is only apparent at large densities — when their wave functions overlap. At low densities, both types of statistics are well approximated by Maxwell–Boltzmann statistics, which is described by classical mechanics. All observed elementary particles are either fermions or bosons. The observed elementary bosons are all gauge bosons: photons, W and Z bosons, gluons, except the Higgs boson which is a scalar boson. Finally, many approaches to quantum gravity postulate a force carrier for gravity, the graviton,
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,