Massive Higher Spins: Effective Theory and Consistency

TL;DR

Massive higher-spin resonances in flat space can be described by an EFT that cleanly separates transverse (massless) and longitudinal (Goldstone) sectors and uses a Higgs-like mixing to generate a consistent massive multiplet. Positivity bounds on forward scattering and beyond-positivity analyses force the cutoff scale $\Lambda$ to lie parametrically near the mass $m$ (i.e., $\epsilon=m/\Lambda$ is typically $\mathcal{O}(1)$ unless couplings are tiny) and strongly constrain longitudinal self-interactions, often favoring more derivative, suppressed operators. A detailed spin-3 case demonstrates that leading $\Phi^4$-type self-interactions are inconsistent with positivity, indicating HS self-interactions are extremely constrained and dominated by higher-derivative terms. The paper also maps out couplings to matter and gauge fields, showing how positivity tightens these couplings and suggesting implications for cosmology, colliders, and dark matter. Overall, the study clarifies the UV consistency landscape for massive HS EFTs and highlights symmetry structures, decoupling limits, and tuning as central features.

Abstract

We construct the effective field theory for a single massive higher-spin particle in flat spacetime. Positivity bounds of the S-matrix force the cutoff of the theory to be well below the naive strong-coupling scale, forbid any potential and make therefore higher-derivative operators important even at low energy. As interesting application, we discuss in detail the massive spin-3 theory and show that an extended Galileon-like symmetry of the longitudinal modes, even with spin, emerges at high energy.

Figures (2)

• Figure 1: LEFT: Different strong-coupling scales for $s=3$ as function of $\sqrt{\lambda_L}$ in Eq. (\ref{['Potential']}) or Eq. (\ref{['generalLagrangian']}). For $E\gtrsim m$ the interaction strength increases at high energy; the $S(V)$ polarizations are strongly coupled at $\Lambda^\text{sc}_{12}(\Lambda^\text{sc}_{8})$ generically (solid lines) or at $\Lambda^\text{sc}_{10}(\Lambda^\text{sc}_{6})$ for the tuned theory (dashed lines). Beyond-positivity bounds for the $\lambda_L\Phi^4$ interaction (dotted) and $\lambda_L\partial^4\Phi^4/\Lambda^4$ (dot-dashed): this represents the maximal cutoff $\Lambda$ of the theory, well below the strong-coupling scales. RIGHT: similar energy scales and beyond-positivity bounds for the $\mathcal{R}^4$ interaction as function of $m$ (solid line for $s$=3, dotted for $s\to\infty$).
• Figure 2: Allowed regions of the coefficients $\lambda_{1},\lambda_{2}$ as function of the sign of $\lambda_3$. Blue: result from scattering of linear-definite polarizations Eq. (\ref{['pos1']}). Red: result from $\Sigma^{(2)}_\text{IR}>0$ by scattering different choices of linear combination of polarizations. Yellow: result from $\Sigma^{(4)}_\text{IR}>0$ by scattering only linear-definite polarizations. The dotted line correspond to the tuning $\lambda_1 =\lambda_2= -\lambda_3$ in Eq. (\ref{['tunedPotential']}).