The Advantages of Being Vectorlike

Consider an asymptotically free gauge theory with global symmetry group G. At short distances this gauge theory allows a perturbative definition. One fundamental question at long distances relates to how the symmetry group G is realized in the vacuum. Virtually no progress has been made in answering this question from first principles in QCD, the gauge theory responsible for all hadronic and nuclear phenomena. One might hope that ultimately lattice methods will provide insight. Unfortunately, there is at present no lattice formulation of gauge theory which preserves G if G is chiral, and therefore at present the lattice cannot reveal how QCD flavor symmetries are realized in the vacuum. For purposes of this paper, QCD has two massless flavors and global symmetry group G = SU(2)× SU(2)×U(1). This global symmetry group can be decomposed into H = SU(2)V ×U(1) which corresponds to isospin and baryon number and is vectorlike since it does not imply massless quarks, and SU(2)A which is chiral and therefore exact only in the limit of vanishing quark masses. No-go theorems formulated for QCD-like gauge theories imply that vectorlike symmetries are not spontaneously broken . Therefore, on theoretical grounds we expect that in the low-energy theory H will be unbroken, and SU(2)A can be unbroken, completely broken or partially broken by the vacuum. Observation reveals that SU(2)A is broken completely in the vacuum. In this paper we will motivate study of other gauge theories with QCD flavor symmetries by considering general properties of the nonlinear realization of G. We will show that the manifold possible G transformation properties of the hadronic mass-squared matrix suggest the existence of other gauge theories with precisely the same symmetry breaking pattern as QCD. We will construct