Softness and amplitudes’ positivity for spinning particles

Abstract

We derive positivity bounds for scattering amplitudes of particles with arbitrary spin using unitarity, analyticity and crossing symmetry. The bounds imply the positivity of certain low-energy coefficients of the effective action that controls the dynamics of the light degrees of freedom. We show that low-energy amplitudes strictly softer than O(p 4 ) do not admit unitary ultraviolet completions unless the theory is free. This enforces a bound on the energy growth of scattering amplitudes in the region of validity of the effective theory. We discuss explicit examples including the Goldstino from spontaneous supersymmetry breaking, and the theory of a spin-1/2 fermion with a shift symmetry.

The other effective fermion compositeness

“…It is therefore crucial to understand how well can a framework of compositeness in the multi-TeV region (accessible at present or near-future colliders) fake the SM: how irrelevant can these new interactions be? From a low-energy perspective one would see no obstruction in arbitrarily soft interactions, but dispersion relations can be used to show that the basic requirement of unitarity of the microscopic (UV) theory, imposes certain positivity constraints that guarantee the existence of unsuppressed dimension-8 operators [12][13][14]. Given the SM field content and limiting our discussion to lepton and baryon number preserving effects, this implies that the leading BSM effects must be characterized either by operators of dimension-6 or by dimension-8, but not higher.…”

“…Since the kinetic terms can always be rescaled to a canonical form, the resulting Goldstini interactions are model independent, controlled by no free parameter except for the SUSY-breaking scale F (as the Goldstini self-interactions). 12 Calling X = ψ, A µ or φ any composite fermion, gauge boson or scalar, we construct their SUSY-invariant kinetic terms by making the replacements…”

“…This current can be found by noticing that in our geometrical construction Goldstini enter the action only through geometrical 12 In fact, this model independency is somewhat of a misnomer: the overall coefficient is controlled by F alone because we assumed full compositeness. Should some of these non-Goldstini states X = ψ, Aµ or φ be partially composite instead, the would-be model independent couplings would actually get rescaled by the degree of compositeness (squared), | X | 2 , which measures how large a fraction of the kinetic term arises from the SUSY-breaking sector.…”

“…usual positivity of the Goldstino amplitude [12]. However, a SUSY-preserving potential for φ must have vanishing vacuum energy, e.g.…”

“…Indeed, the basic requirement of unitarity in the underlying theory, together with analyticity of the 2 → 2 scattering amplitudes, implies that the Wilson coefficients of the operators in eqs. (5.1) and (5.8), (5.9), be strictly positive [12,14].…”