QED positivity bounds

TL;DR

The paper investigates whether positivity bounds, extended beyond the forward limit, are compatible with QED minimally coupled to gravity. Using improved dispersion relations that subtract known low-energy discontinuities, the authors show that insisting on positivity with the $t$-channel pole removed forces a low new-physics scale $\Lambda_{\rm new} \sim (e m M_{\rm Pl})^{1/2}$, regardless of the UV completion being weakly or strongly coupled. This bound arises from explicit one-loop calculations in both scalar and spinor QED, including gravitational corrections, and persists when considering higher-order graviton effects, unless one allows a controlled amount of negativity compatible with decoupling limits. The results imply positivity bounds in gravity can be significantly more constraining than weak-gravity conjecture expectations and motivate a nuanced view of positivity in gravitational contexts, including potential weakening to accommodate UV completions. The work thus constrains EFT validity and highlights subtle interplay between UV physics, gravity, and low-energy photon interactions.

Abstract

We apply positivity bounds directly to a $U(1)$ gauge theory with charged scalars and charged fermions, i.e. QED, minimally coupled to gravity. Assuming that the massless $t$-channel pole may be discarded, we show that the improved positivity bounds are violated unless new physics is introduced at the parametrically low scale $Λ_{\rm new} \sim (e m M_{\rm pl})^{1/2}$, consistent with similar results for scalar field theories, far lower than the scale implied by the weak gravity conjecture. This is sharply contrasted with previous treatments which focus on the application of positivity bounds to the low energy gravitational Euler-Heisenberg effective theory only. We emphasise that the low-cutoff is a consequence of applying the positivity bounds under the assumption that the pole may be discarded. We conjecture an alternative resolution that a small amount of negativity, consistent with decoupling limits, is allowed and not in conflict with standard UV completions, including weakly coupled ones.

Figures (6)

• Figure 1: The $AA\to AA$$t$-channel scattering in the gravitational Euler-Heisenberg theory. The wiggly line stands for the vector field $A_\mu$. The exchanged wavy line stands for the graviton $h_{\mu\nu}$.
• Figure 2: The $AA\to AA$ scattering in scalar QED due to non-gravitational interactions (first line) and gravitational interactions to order $1/M_{\rm Pl}^2$ (second line). The wiggly line stands for the vector field $A_\mu$ and the solid line stands for the scalar field $\phi$. The arrows depict the direction of the charge flow. We do not show all the crossed versions of the diagrams.
• Figure 3: The $AA\to AA$ scattering in spinor QED due to non-gravitational interactions (first line) and gravitational interactions to order $1/M_{\rm Pl}^2$ (second line). The wiggly line stands for the vector field $A_\mu$ and the solid line stands for the fermion $\psi$. The arrows depict the direction of the charge flow. We do not show all the crossed versions of the diagrams.
• Figure 4: Example gravitational contributions at order $1/M_{\rm Pl}^4$.
• Figure 5: One loop self-energy contribution for scalar and spinor QED.