Positivity bounds for scalar field theories

Abstract

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...

Massive Galileon positivity bounds

“…While one might conclude from this that the non-scalar modes in Ghost-free massive gravity play an important role in satisfying the positivity requirements, based on the massive Galileon result we see that these requirements are already satisfied for the massive scalar sector alone. In fact, as we are able to apply our positivity bounds away from the forward limit [5], our requirements on the massive Galileon are stronger than equivalent bounds in the literature.…”

“…In [5], using the unitary and analytic properties of the scalar scattering amplitude, we have derived an infinite number of positivity bounds for the derivatives of pole subtracted amplitudes. In the following, we will review the main ingredients of the proof and results.…”

“…These bounds apply to any scalar EFT with a mass gap, which includes the massive Galileon on flat spacetime. The infinite number of bounds derived in [5] places constraints on the EFT coefficients, not only the leading Galileon operators, but also on all higher derivative operators that enter the massive Galileon LEEFT. 2 We point out that following the spirit of standard Wilsonian effective field theories, the notion of ghost-freeness is only meaningful till the scale of perturbative unitarity breaking.…”

“…[5]. These represent an infinite number of requirements that place constraints on all the coefficients of the scattering amplitude when Taylor expanded in terms of the Mandelstam variables.…”

“…massive) scalar theory to admit a standard Wilsonian UV completion, the 2-to-2 scattering amplitude must satisfy a number of so-called positivity bounds [4,5]. These are derived based on the cherished assumptions that the scattering amplitude is Lorentz invariant, unitary, polynomially bounded in momenta, crossing symmetric and is analytic in the complex energy plane modulo certain poles and branch cuts.…”