Abstract
We study massless fermions interacting through a particular four fermion term in four dimensions. Exact symmetries prevent the generation of bilinear fermion mass terms. We determine the structure of the low energy effective action for the auxiliary field needed to generate the four fermion term and find it has an novel structure that admits topologically non-trivial defects with non-zero Hopf invariant. We show that fermions propagating in such a background pick up a mass without breaking symmetries. Furthermore pairs of such defects experience a logarithmic interaction. We argue that a phase transition separates a phase where these defects proliferate from a broken phase where they are bound tightly. We conjecture that by tuning one additional operator the broken phase can be eliminated with a single BKT-like phase transition separating the massless from massive phases.
Highlights
In this paper, we construct a continuum theory of strongly interacting fermions in four dimensions in which exact symmetries prohibit the appearance of mass terms
We argue that the fermions acquire masses at strong coupling by virtue of their interactions with a nontrivial vacuum corresponding to a symmetric fourfermion condensate
Our work points out the existence of new classes of theories of strongly interacting fermions which may be important in the search for candidate theories of BSM physics
Summary
We construct a continuum theory of strongly interacting fermions in four dimensions in which exact symmetries prohibit the appearance of mass terms. Progress in understanding the nature of this phase diagram was given recently in [8] In four dimensions, it appears that a very narrow symmetry broken phase emerges between the massless and massive phases. The ingredients of the theory are somewhat unusual; the fermions appear as components of a (reduced) KählerDirac field and as a consequence the theory is invariant only under a diagonal subgroup of the Lorentz and flavor symmetries together with an additional SOð4Þ symmetry. It is this reduced symmetry, which is enforced by the structure of the four-fermion term, that plays a key role in prohibiting conventional Dirac mass terms. Our paper offers a way to understand the structure of the four-dimensional models from a continuum perspective where we will see that topological features of the continuum theory can play an important role
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