Generalized elastic positivity bounds on interacting massive spin-2 theories

TL;DR

This work develops generalized elastic positivity bounds that leverage inelastic scatterings among multiple spin-2 fields to constrain the Wilson coefficients of multi-field massive spin-2 EFTs. By regularizing the kinematic singularities inherent in inelastic amplitudes and applying forward-dispersion relations, the authors derive positivity conditions that hold for arbitrary real linear combinations of external species. Applied to bi-field pseudo-linear spin-2, cycle spin-2, and line spin-2 theories, the approach completely rules out bi-field pseudo-linear models and confines the cycle and line theories to finite regions in their parameter spaces, with quantitative volume estimates of the allowed regions. The analysis extends to generalizations with more than two fields, showing that inelastic information can yield new cross-field constraints, thereby tightening the viable parameter space for multi-field massive gravity setups and informing UV completion prospects. Overall, the results demonstrate that generalized elastic positivity bounds are a powerful tool for testing the consistency of high-cutoff spin-2 EFTs and for guiding model-building in multi-field gravity scenarios.

Abstract

We use generalized elastic positivity bounds to constrain the parameter space of multi-field spin-2 effective field theories. These generalized bounds involve inelastic scattering amplitudes between particles with different masses, which contain kinematic singularities even in the $t=0$ limit. We apply these bounds to the pseudo-linear spin-2 theory, the cycle spin-2 theory and the line spin-2 theory respectively. For the pseudo-linear theory, we exclude the remaining operators that are unconstrained by the usual elastic positivity bounds, thus excluding all the leading (or highest cutoff) interacting operators in the theory. For the cycle and line theory, our approach also provides new bounds on the Wilson coefficients previously unconstrained, bounding the parameter space in both theories to be a finite region ({\it i.e.}, every Wilson coefficient being constrained from both sides). To help visualize these finite regions, we sample various cross sections of them and estimate the total volumes.

Figures (14)

• Figure 1: Positive regions in the $(\kappa_3,\kappa_4)$-plane for different $c$, ${\lambda}$ and $d$ (the regions within the solid lines are allowed by generalized elastic positivity bounds) in the $\mathbb{Z}_2$ symmetric case. The black line (the largest region) corresponds to the elastic positivity bounds in dRGT gravity. The size of the cross section of this region largely depends on $c$, while $d$ can dramatically change the shape of this region. At $d=d_{min}\simeq -0.5$ or $d=d_{max}\simeq 1.4$, this region shrinks to a point.
• Figure 2: Positive regions in the $(c,{\lambda})$-plane for different $d$ with $\kappa_3=1.4$, $\kappa_4=0.36$ (the regions within the solid lines are allowed by generalized elastic positivity bounds) in the $\mathbb{Z}_2$ symmetric case. The green dashed line is the positivity bound from elastic scattering $hf\rightarrow hf$. We see that the positive cross section changes dramatically with $d$ in this plane.
• Figure 3: Positive regions in the $({\lambda},d)$-plane with different $\kappa_3$, $\kappa_4$ and $c$ (the regions within the solid lines are allowed by generalized elastic positivity bounds) in the $\mathbb{Z}_2$ symmetric case. The black line is when parameters other than ${\lambda}$ and $d$ take their approximate central values in Eq. (\ref{['centralV']}). The top left, top right and bottom plot show the influence of changing $\kappa_3$, $\kappa_4$, and $c$ respectively.
• Figure 4: Positive regions in the $(c,d)$-plane with different $\kappa_3$, $\kappa_4$ and ${\lambda}$ (the regions within the solid lines are allowed by generalized elastic positivity bounds) in the $\mathbb{Z}_2$ symmetric case. The black line is when parameters other than $c$ and $d$ take their approximate central values in Eq. (\ref{['centralV']}). The top left, top right and bottom plot show the influence of changing $\kappa_3$, $\kappa_4$, and ${\lambda}$ respectively.
• Figure 5: Positive regions in the ($c,d$)-plane with different $x$ (the regions within the solid lines are allowed by generalized elastic positivity bounds) in the $\mathbb{Z}_2$ symmetric but $x\neq 1$ case. We choose $\kappa_3=1.4, \kappa_4=0.38, \lambda=0.2$ and $1/2<x<2$.