Abstract
In contrast to massless spinning particles, scalars are not heavily constrained by unitarity and locality. Off-shell, no gauge symmetries are required to write down manifestly local theories, while on-shell consistent factorisation is trivial. Instead a useful classification scheme for scalars is based on the symmetries they can non-linearly realise. Motivated by the breaking of Lorentz boosts in cosmology, in this paper we classify the possible symmetries of a shift-symmetric scalar that is assumed to non-linearly realise Lorentz boosts as, for example, in the EFT of inflation. Our classification method is algebraic; guided by the coset construction and inverse Higgs constraints. We rediscover some known phonon theories within the superfluid and galileid classes, and discover a new galileid theory which we call the extended galileid. Generic galileids correspond to the broken phase of galileon scalar EFTs and our extended galileids correspond to special subsets where each galileon coupling is fixed by an additional symmetry. We discuss the broken phase of theories that also admit a perturbation theory around Poincaré invariant vacua and we show that the so-called exceptional EFTs, the DBI scalar and special galileon, do not admit such a broken phase. Concentrating on DBI we provide a detailed account of this showing that the scattering amplitudes are secretly Poincaré invariant when the theory is expanded around the superfluid background used in the EFT of inflation. We point out that DBI is an exception to the common lore that the residue of the total energy pole of cosmological correlators is proportional to the amplitude. We also discuss the inevitability of poles in 2 → 2 scattering amplitudes when boost are spontaneously broken meaning that such theories do not admit Adler zeros and generalisations even in the presence of a shift symmetry.
Highlights
Derived purely using on-shell methods1 and without ever having to switch on a collider or think about a falling elevator
We discuss the broken phase of theories that admit a perturbation theory around Poincaré invariant vacua and we show that the so-called exceptional EFTs, the DBI scalar and special galileon, do not admit such a broken phase
We find an interesting example of a broken phase of the special galileon where a symmetry that was non-linearly realised in the Lorentz invariant phase becomes linearly realised on the phonon
Summary
In this paper we consider theories that live in flat spacetime, our classification is still useful for studying inflationary theories in their high energy limit, where the mixing with gravity is negligible and the background is effectively flat. Up to this point, Mn(t + π) are arbitrary functions, but in order to get a nearly scale invariant spectrum, one demands an approximate shift symmetry for π that in combination with dS dilatation enforces all the Mn’s to be constant [31, 32]. In the flat space limit gμν → ημν, the theory for π linearly realises time translations. Our results will have important applications for cosmological EFTs in their high energy limit, and this is manifest in the total energy pole of the cosmological correlators (see [38] for a recent study of exact linearly realised symmetries for cosmological correlators)
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