Positivity Bounds and the Massless Spin-2 Pole

TL;DR

This work probes whether standard positivity bounds survive when gravity is present, focusing on the massless $t$-channel spin-2 exchange and a proposed 3d compactification scheme. By analyzing a scalar QED toy model with a known partial UV completion, the authors show that compactified positivity bounds can conflict with UV data unless new UV physics enters at a surprisingly low scale $\Lambda \sim (M M_{\rm Pl})^{1/2}$, or unless a mass gap is introduced. They explore both renormalizable and non-renormalizable UV extensions and find that only low-scale derivative interactions can reconcile the bounds, at the cost of lowering the EFT cutoff and altering gravity–matter interactions. The paper also scrutinizes infrared issues in 3d gravity, showing that massless gravitons lead to ill-defined amplitudes and delta-function structures that undermine positivity arguments, while a mass gap can restore positivity. Collectively, these results cast doubt on the universal applicability of compactified positivity bounds in gravity and motivate a cautious reformulation or modification of the bound that does not rely on detailed UV completion.

Abstract

The presence of a massless spin-2 field in an effective field theory results in a $t$-channel pole in the scattering amplitudes that precludes the application of standard positivity bounds. Despite this, recent arguments based on compactification to three dimensions have suggested that positivity bounds may be applied to the $t$-channel pole subtracted amplitude. If correct this would have deep implications for UV physics and the Weak Gravity Conjecture. Within the context of a simple renormalizable field theory coupled to gravity we find that applying these arguments would constrain the low-energy coupling constants in a way which is incompatible with their actual values. This contradiction persists on deforming the theory. Further enforcing the $t$-channel pole subtracted positivity bounds on such generic renormalizable effective theories coupled to gravity would imply new physics at a scale parametrically smaller than expected, with far reaching implications. This suggests that generically the standard positivity bounds are inapplicable with gravity and we highlight a number of issues that impinge on the formulation of a three-dimensional amplitude which simultaneously satisfies the required properties of analyticity, positivity and crossing symmetry. We conjecture instead a modified bound that ought to be satisfied independently of the precise details of the high energy completion.

Figures (14)

• Figure 1: Non--gravitational contribution to the $\phi\phi\to\phi\phi$ scattering in the UV theory. Bold lines represent propagators of the heavy field $\psi$.
• Figure 2: Contributions to $\chi\phi\to\chi\phi$ scattering in the IR theory. The wiggly line represents the graviton propagator.
• Figure 3: Leading contributions to the $\chi\phi\to\chi\phi$ scattering as seen from the point of view of the partial UV theory. Wiggly lines stand for the graviton propagator while thick lines stand for the propagator of the heavy field $\psi$.
• Figure 4: Feynman diagrams contributing to $\chi\phi\to\chi\phi$ scattering from the new UV operators in \ref{['pUV_3']} up to one loop. Bold lines correspond to the propagators of the heavy field $\psi$.
• Figure 5: One-loop contribution to $\chi\phi\to\chi\phi$ scattering from cubic dimension 5 operators. The wiggly line represents the graviton propagator, while the bold line that of the heavy field $\psi$.