Abstract
We discuss the role of higher dimensional operators in the spontaneous breaking of internal symmetry and scale invariance, in the context of the Lorentz invariant scalar field theory. Using the $\varepsilon$-expansion we determine phase diagrams and demonstrate that (un)stable RG flows computed with a certain basis of dimension 6 operators in the Lagrangian, map to (un)stable RG flows of another basis related to the first by field redefinitions. Crucial is the presence of reparametrization ghosts if Ostrogradsky ghosts appear.
Highlights
The Higgs potential responsible for the spontaneous breaking of gauge symmetry breaks scale invariance via an explicit mass term, already at the classical level
We analyzed the one-loop relation between renormalization group (RG) flows related by field redefinitions in the scalar field theory, up to operator dimension 6
In particular we showed how a basis that contains an Ostrogradsky ghost can be consistently defined with the addition of an extra sector of ghostlike fields
Summary
The Higgs potential responsible for the spontaneous breaking of gauge symmetry breaks scale invariance via an explicit mass term, already at the classical level. Without the mass term this model does not have a phase where the Z2 symmetry is broken neither in the classical limit nor at the quantum level, as far as perturbation theory is concerned. Scale invariance on the other hand generically does break even in the massless limit by the coupling developing nonzero β function Another generic consequence of quantization is the appearance of classically irrelevant, higher dimensional operators (HDOs) in the action, suppressed by appropriate powers of a dimensionful scale Λ. If the phase diagram develops a Z2 broken phase at the quantum level, perturbation theory should be able to detect it via the presence of these HDOs. There is a interesting interpretation of this picture, revealed by the observation that field redefinitions leave the S-matrix invariant. The reason is that in these cases the phase diagram may contain Wilson-Fisher (WF) fixed points where scale invariance is restored, to the order that
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