Einstein, but results from quantum mechanical experiments show a difference in the behavior of left-chiral versus right-chiral subatomic particles. Example: and quarks in QCD Consider quantum chromodynamics (QCD) with two massless quarks and (massive fermions do not exhibit chiral symmetry). The Lagrangian reads In terms of left-handed and right-handed spinors, it reads (Here, is the imaginary unit and the Dirac operator.) Defining it can be written as The Lagrangian is unchanged under a rotation of qL by any 2×2 unitary matrix , and qR by any 2×2 unitary matrix . This symmetry of the Lagrangian is called flavor chiral symmetry, and denoted as . It decomposes into The singlet vector symmetry, , acts as and thus invariant under gauge symmetry. This corresponds to baryon number conservation. The singlet axial group transforms as the following global transformation However, it does not correspond to a conserved quantity, because the associated axial current is not conserved. It is explicitly violated by a quantum anomaly. The remaining chiral symmetry turns out to be spontaneously broken by a quark condensate formed through nonperturbative action of QCD gluons, into the diagonal vector subgroup SU(2)V known as isospin. The Goldstone bosons corresponding to the three broken generators are the three pions. As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this chiral symmetry breaking induces the bulk of hadron masses, such as those for the nucleons — in effect, the bulk of the mass of all visible matter. In the real world, because of the nonvanishing and differing masses of the quarks, SU(2)L × SU(2)R is only an approximate symmetry to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons. More flavors For more "light" quark species, flavors in general, the corresponding chiral symmetries are U(N)L × U(N)R, 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. An application in particle physics In theoretical physics, the electroweak model breaks parity maximally. All its fermions are