Complementarity and chiral fermions in SU(2) gauge theories.

Complementarity the absence of a phase boundary separating the Higgs and confinement phases of a gauge theory can be violated by the addition of chiral fermions. We utilize chiral symmetry violating fermion correlators such as 〈ψ̄ψ〉 as order parameters to investigate this issue. Using inequalities similar to those of Vafa-Witten and Weingarten, we show that SU(2) gauge theories with Higgs and fermion fields in the fundamental representation exhibit chiral symmetry breaking in the confined phase and therefore do not lead to massless composite fermions. We discuss the implications for the Abbott-Farhi strongly interacting standard model. Junior Fellow, Harvard Society of Fellows. Email: [email protected], [email protected] 1 Complementarity and all that Certain gauge theories with scalars in the fundamental representation can be shown to exhibit a remarkable property known as complementarity [1, 2, 3]. Complementarity means that the Higgs phase (large vacuum expectation value v, small gauge coupling g) and confinement phase (small v, large g) are not separated by a phase boundary. (Here both g = g(Λ) and v are defined in terms of some lattice spacing Λ.) The result, proved by Fradkin and Shenker [2] using results of Osterwalder and Seiler [1] (see also Banks and Rabinovici [4] for a similar result for U(1) theories), consists of demonstrating that in a lattice formulation of the theory all correlators (ie free energy, n-point Greens functions) are analytic functions of g and v in a connected region which contains both the Higgs and confinement phases. (See figure 1 for a typical phase diagram.) Therefore, quantities such as the free energy of the theory vary smoothly without discontinuity as we interpolate between the two regions. The rigorous demonstration of complementarity coincided with observations by ’t Hooft [5] and Susskind (unpublished) that there exists a strong similarity between the spectrum of states in the standard Higgs picture and the confined picture of a gauge theory with scalars in the fundamental. This led ’t Hooft to remark that the question of confinement in this class of models could only be answered dynamically there being no fundamental difference between the confining and spontaneously broken phases. (Indeed, these remarks apply equally to QCD, despite its lack of fundamental colored scalars, because of composite fields which can be formed out of glue and fermions.) In this letter we wish to examine complementarity in gauge-Higgs models when chiral fermions are included. We will demonstrate in the SU(2) case that the addition of chiral fermions is capable of drastically altering the phase diagram of the theory. The models we study exhibit a phase transition associated with chiral symmetry breaking (χSB) as we move from the broken to confined phase. The above result was established previously by I.-H. Lee and R. E. Shrock [6] using analytical and numerical techniques on the lattice. Our analysis will be in the continuum, which makes it less rigorous from the viewpoint of constructive quantum field theory but perhaps easier to understand to theorists who work in the continuum. It is straightforward to argue that addition of chiral fermions to a purely bosonic theory can lead to a violation of complementarity. One has merely to consider the ’t Hooft anomaly matching conditions [5], which are necessary but not sufficient conditions for the existence of massless composite fermions. If the anomalies resulting from the fundamental fermion triangle graphs do not match those of the (putative) massless composite fermions, one can immediately conclude that there are no massless composites and chiral symmetries are broken. The effects of the anomaly are then reproduced by the Goldstone modes via the Wess-Zumino term [7] in the chiral lagrangian. Here we will examine a more interesting class