Assuming the existence of a local, analytic, unitary UV completion in a Poincaré invariant scalar field theory with a mass gap, we derive an infinite number of positivity requirements using the known properties of the amplitude at and away from the forward scattering limit. These take the form of bounds on combinations of the pole subtracted scattering amplitude and its derivatives. In turn, these positivity requirements act as constraints on the operator coefficients in the low energy effective theory. For certain theories these constraints can be used to place an upper bound on the mass of the next lightest state that must lie beyond the low energy effective theory if such a UV completion is to ever exist.The physical requirements of unitarity, locality, and crossing symmetry, are well known to provide powerful constraints on the scattering matrix of a Lorentz invariant theory, and were an integral part of the S-matrix program [1,2]. Relativistic locality and causality is encoded in the twin requirements of analyticity of the scattering amplitude, and polynomial boundedness. Taken together, these allow us to express the scattering amplitude in terms of dispersion relations with a finite number of subtractions, from which it is possible to infer bounds on the growth of the scattering amplitude at high energies.It is only more recently that these constraints have been used to infer properties of low energy effective field theories (LEEFT) [3]. In doing so we assume the existence of a (possibly unknown) local Lorentz invariant UV completion and use its properties to infer properties of the LEEFT. These typically come in the form of 'positivity bounds', i.e. bounds on the sign of coefficients in the Wilsonian effective action. For example, it is known that for analytic 2-to-2 scattering amplitudes in the forward scattering limit, an expansion in powers of the invariant mass s must have positive coefficients [3]. It has also been suggested that these can be pushed away from the forward limit [4][5][6][7][8].Exploiting unitarity, analyticity and crossing symmetry of the full (unknown) UV complete theory, we will use the known properties of the scattering amplitude of a scalar theory at and away from the forward limit to show that there are an infinite number of such bounds on the pole subtracted scattering amplitude B(s, t). These translate into bounds on the coefficients of every non-redundant (not removable by a field redefintion) operator that contributes to the 2-to-2 scattering amplitude at tree level. We first derive the bounds on the exact quantum scattering amplitude, and then show how they may be applied to the tree-level amplitudes in the LEEFT. In certain cases, we will show how these constraints lead to an upper bound of the mass of the first state that necessarily lies beyond the regime of validity of the LEEFT. Unitarity:The 2-to-2 scattering amplitude is best expressed in terms of the Mandelstam variables [9]: s, the center of mass energy, t, the momentum transfer, related to the scattering angle by c...

For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles, for a complete set of polarizations at and away from the forward scattering limit. This provides a set of powerful criteria which can be used to restrict the parameter space of any effective field theory, often considerably more so than its forward limit subset alone.

The recent direct detection of gravitational waves from a neutron star merger with optical counterpart has been used to severely constrain models of dark energy that typically predict a modification of the gravitational wave speed. However, the energy scales observed at LIGO, and the particular frequency of the neutron star event, lie very close to the strong coupling scale or cutoff associated with many dark energy models. While it is true that at very low energies one expects gravitational waves to travel at a speed different than light in these models, the same is no longer necessarily true as one reaches energy scales close to the cutoff. We show explicitly how this occurs in a simple model with a known partial UV completion. Within the context of Horndeski, we show how the operators that naturally lie at the cutoff scale can affect the speed of propagation of gravitational waves and bring it back to unity at LIGO scales. We discuss how further missions including LISA and PTAs could play an essential role in testing such models. Dark Energy after GW170817 and GRB170817A:The recent direct detections of gravitational waves (GWs) have had an unprecedented impact on our understanding of gravity at a fundamental level. The first event alone (GW150914 [1]) was already sufficient to put bounds on the graviton with better precision than what we know of the photon. Last year, the first detection of GWs from a neutron star merger (GW170817), some 10 15 light seconds away, which arrived within one second of an optical counterpart (GRB170817A), allowed us to constrain the GW speed with remarkable precision [2-4]with c T the GW phase velocity and c γ the speed of light.Such a constraint has had far-reaching consequences for models of dark energy. Within the context of the Effective Field Theory (EFT) for dark energy [5], it was rapidly pointed out that (1) was sufficient to suppress the EFT operators that predict non-luminal gravitational propagation [6][7][8][9][10][11][12][13][14]. In particular, within the framework of scalar-tensor theories of gravity, Horndeski [15] has played a major part in the past decade as a consistent ghost-free EFT in which the scalar degree of freedom could play the role of dark energy. Yet the interplay between the scalar and gravity typically implies that GWs would not travel luminally. The LIGO constraint on the GW speed only leaves out the generalization of the cubic Galileon [16], which is severely constrained by other observations. As a result the Horndeski EFT seems almost entirely ruled out as a dark energy candidate [17].Nevertheless, it should be noted that the recent LIGO bound applies to GWs at a frequency of 10 − 100Hz, while the EFT for dark energy is "constructed" as an effective field theory for describing cosmology on scales 20 orders of magnitude smaller. When it comes to constraining such EFT parameters, it is therefore

The EFT coefficients in any gapped, scalar, Lorentz invariant field theory must satisfy positivity requirements if there is to exist a local, analytic Wilsonian UV completion. We apply these bounds to the tree level scattering amplitudes for a massive Galileon. The addition of a mass term, which does not spoil the non-renormalization theorem of the Galileon and preserves the Galileon symmetry at loop level, is necessary to satisfy the lowest order positivity bound. We further show that a careful choice of successively higher derivative corrections are necessary to satisfy the higher order positivity bounds. There is then no obstruction to a local UV completion from considerations of tree level 2-to-2 scattering alone. To demonstrate this we give an explicit example of such a UV completion.

We apply the recently developed positivity bounds for particles with spin, applied away from the forward limit, to the low energy effective theories of massive spin-1 and spin-2 theories. For spin-1 theories, we consider the generic Proca EFT which arises at low energies from a heavy Higgs mechanism, and the special case of a charged Galileon for which the EFT is reorganized by the Galileon symmetry. For spin-2, we consider generic Λ 5 massive gravity theories and the special 'ghost-free' Λ 3 theories. Remarkably we find that at the level of 2-2 scattering, the positivity bounds applied to Λ 5 massive gravity theories, impose the special tunings which generate the Λ 3 structure. For Λ 3 massive gravity theories, the island of positivity derived in the forward limit appears relatively stable against further bounds.