Positivity Bounds for Massive Spin-1 and Spin-2 Fields

TL;DR

This work extends positivity bounds for spinning particles away from the forward limit to the low-energy EFTs of massive spin-1 and spin-2 fields, under a weakly coupled UV completion. Using the transversity formalism, it derives new nonforward bounds that orthogonally constrain Wilson coefficients in Proca, Charged Galileon, and massive gravity theories, revealing that Λ5 massive gravity is compelled toward the ghost-free Λ3 structure at tree level, while Λ3 theories occupy a relatively stable positivity island. The results highlight the diagnostic power of nonforward positivity bounds for assessing UV completions and potential Higgs-like mechanisms for spin-2, offering a detailed map of allowed parameter spaces and necessary tunings. Overall, the paper provides a concrete framework for testing the consistency of spinning EFTs with analytic, unitary UV completions and informs the viability of proposed UV completions and symmetry-breaking patterns.

Abstract

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.

Figures (2)

• Figure 1: Parameter space for charged Galileon theory constrained by analyticity. A suppressed $a_0 \sim m^2/\Lambda_{\phi}^2$ is required to recover a Galileon symmetry in the decoupling limit. The forward limit bound from scattering four scalar modes was previously found in Bonifacio:2016wcb, which excluded the red region. By looking at scattering superpositions of modes, one use forward limit bounds to further restrict the parameter space to a semi-infinite strip (the white region plus the light green region) in the parameter space spanned by $( \mu_3 , \bar{a}_t + 3\bar{a}_0 )$. Utilizing the first $\partial_t$ bound (i.e., going away from the limited forward limit formalism), we reduce this semi-infinite strip even further, potentially greatly, depending on the sign of $3 \bar{a}_0 = 3 a_0 - a_5 - C_1 - 4 C_2$. In terms of the EFT coefficients in \ref{['eqn:ProcaLeading']}, we have $2 \bar{a}_t = a_3 + a_4 - 2 a_5,\; \mu_3 = a_4 - a_5 - a_3/2 - C_1/2 - 2C_2$. The $SSSS$ bound corresponds to $\bar{a}_t + 3 \bar{a}_0 - 2 \mu_3 > 0$.
• Figure 2: Parameter space of massive gravity constrained by analyticity. The forward limit bounds on $\Lambda_5$ massive gravity (left) gradually constricts as $\Delta d$ is made more negative. However, going beyond the forward limit \ref{['eqn:L5dtbound']} rules out $\Delta d \neq 0$ completely. In $\Lambda_3$ massive gravity (right), the higher $t$ derivative bounds marginally restrict the forward limit bounds. This lends further evidence to the idea that $\Lambda_3$ massive gravity may admit a Wilsonian UV completion within this narrow island.