Axions: Past, Present, and Future

Consider quantum chromodynamics (QCD) with color group SU(3) and three flavors of massless quarks. The lagrangian is invariant under an additional global flavor group, U(3)L × U(3)R. Although we will use it anyway, this “left–right” terminology is actually somewhat misleading; in four dimensions, a right-handed fermion field is the hermitian conjugate of a left-handed fermion field, and so we could (and will) adopt the convention that all fundamental fermion fields are left-handed (with, of course, right-handed hermitian conjugates). In this case, there would be three lefthanded fields transforming as 3’s of SU(3)color, and three left-handed fields transforming as 3’s of SU(3)color; thus the flavor group would be better named U(3)3 × U(3)3. [We note in passing that if quarks were in a real or pseudo-real representation of the color group, such as 3 of O(3)color or 2 of SU(2)color, the lagrangian flavor group would be U(6), since in this case there is no distinction between left-handed quarks and left-handed antiquarks.] For now we ignore the effects of the quantum anomaly, which will of course play a key role later. QCD exhibits dynamical breaking of the flavor symmetry; this can be understood as the formation of a quark condensate,