Positivity Constraints on Interacting Spin-2 Fields

TL;DR

This work shows that forward-limit positivity bounds applied to EFTs with two interacting spin-2 fields, in both cycle and line formulations, impose surprisingly strong constraints on both mixed and self-interactions. The analysis reveals that including a second spin-2 field tightens bounds due to additional poles and more scattering channels, often shrinking the allowed parameter space for cubic and quartic couplings. In particular, cubic mixed couplings must be positive, and quartic cross-terms vanish if cubic mixings are absent; in Λ3-tuned cycle theories, positivity bounds align with the required ghost-free tunings that raise the EFT cutoff. Overall, the results indicate that demanding a standard UV completion can significantly restrict multi-spin-2 EFTs and naturally prefers couplings that raise the strong coupling scale, with implications for extending these methods to more fields or higher-spin sectors.

Abstract

The consistency of the EFT of two interacting spin-2 fields is checked by applying forward limit positivity bounds on the scattering amplitudes to exclude the region of parameter space devoid of a standard UV completion. We focus on two classes of theories that have the highest possible EFT cutoff, namely those theories modelled on ghost-free interacting theories of a single massive spin-2 field. We find that the very existence of interactions between the spin-2 fields implies more stringent bounds on all the parameters of the EFT, even on the spin-2 self-interactions. This arises for two reasons. First, with every new field included in the low-energy EFT, comes the `knowledge' of an extra pole to be subtracted, hence strengthening the positivity bounds. Second, while adding new fields increases the number of free parameters from the new interactions, this is rapidly overcome by the increased number of positivity bounds for different possible scattering processes. We also discuss how positivity bounds appear to favour relations between operators that effectively raise the cutoff of the EFT.

Figures (16)

• Figure 1: Analytic structure of the $2-2$ elastic scattering amplitude in the complex $s$ plane for spin-zero particles or for regularized combinations of general spin transversity amplitudes deRham:2017zjm. We show the analytic structure for arbitrary values of $t$ but calculate bounds in the forward limit, $t=0$.
• Figure 2: The allowed region of parameters obtained from the indefinite $hh\rightarrow hh$ scattering for different values of $c_1$ at $x=m_1/m_2=0.5$ (up) and $x=2$ (down). The results are presented in both $(c_3,d_5)$ plane (left) and $(\kappa_3,\kappa_4)$ plane (right). By increasing $c_1$ the island shrinks until it becomes a point at $c_1=c_{1\text{max}}$ shown by the black dot. For $x=0.5$ this point is reached at $c_{1\text{max}}=0.9$, and for $x=2$ at $c_{1\text{max}}=1.2$. The cross in all figures represents the minimal model with $c_3 = 1/6$ and $d_5 = -1/48$, or $\kappa_3=4/3$, $\kappa_4=1/2$.
• Figure 3: The allowed values of the cubic couplings, $c_1$ (blue) and $c_2$ (yellow), as a function of the mass ratio, $x$, obtained from $hh\rightarrow hh$ scattering. For a given value of $x$, the maximal allowed value, $c_1=c_{1\text{max}}$, is determined as the value at which the allowed $(c_3,d_5)$ island shrinks to a point.
• Figure 4: Comparison of the allowed region of cubic parameters $c_1, \kappa_3$ obtained from the indefinite $hh\rightarrow hh$ and $hf\rightarrow hf$ scatterings, for $x=m_1/m_2=1$, $c_2=c_1$, and vanishing quartic couplings $\lambda=\kappa_4=0$, $\gamma=1$. Each channel allows the removal of a different region of parameter space.
• Figure 5: Relation between the bounds from $hh\to hh$ and $hf\to hf$. Left: The allowed region of parameters $(\lambda,c_1)$ obtained from combining positivity bounds from both $hh\rightarrow hh$ and $hf\rightarrow hf$ scatterings in the $\mathbb{Z}_2$ symmetry case. The largest allowed parameter region is obtained from $\kappa_3=2.23$ and the smallest --- from $\kappa_3=0.74$ (shaded). In all cases we recover $\lambda=0$ when $c_1=0$. Right: The allowed region of $(\kappa_3,\kappa_4)$ obtained from the $hh\rightarrow hh$ scattering in the $\mathbb Z_2$ symmetric case. The island starts to shrink as $c_1$ increases until it reaches $c_{1\text{max}}^{\mathbb Z_2}=0.77$ (green). The bounds obtained from the $hf\rightarrow hf$ scattering amplitudes forbids the shrinking of the island to a point which would occur at $c_{1\text{max}}=1.23$. The minimal model with $\kappa_3=4/3$, $\kappa_4=1/2$ is depicted by a cross.