The EFT Bootstrap at Finite $M_{PL}$

TL;DR

Beadle et al. analyze how one-loop IR effects modify positivity bounds in EFTs of massless scalars coupled to gravity. They develop and compare fixed-$t$ and crossing-symmetric dispersion relations with momentum-space smearing to tame forward-limit divergences, showing that loop corrections, while generally small, can meaningfully alter tree-level bounds and induce running of leading EFT coefficients depending on spacetime dimension. A key result is that gravity-induced non-analyticities and the necessity of IR-safe smearing constrain the range of admissible EFTs and require a consistent perturbative framework, especially near extremal (saturated) bounds. The work bridges tree-level positivity results with non-perturbative S-matrix bootstrap ideas and informs how finite-coupling UV completions might be constrained in quantum gravity contexts.

Abstract

We explore the impact of loop effects on positivity in effective field theories emerging in the infrared from unitary and causal microscopic dynamics. Focusing on massless particles coupled to gravity, we address the treatment of forward-limit divergences from loop discontinuities and establish necessary conditions for maintaining computational control in perturbation theory. While loop effects remain small, ensuring consistency in our approach leads to a significant impact on bounds, even at tree level.

Figures (15)

• Figure 1: Non-trivial cuts used for the matching in Eq. (\ref{['eqap:PVreduction']}). LEFT: cuts for $1$-loop diagrams with scalar propagators in the '$t$-channel'. RIGHT: cuts for $1$-loop diagrams with gravitons. The grey blobs denote all possible tree-level interactions associated with either graviton exchange or insertion of an EFT $4$-point interaction.
• Figure 2: One of many diagrams contributing to the $1$-loop amplitude at order $\kappa^4$.
• Figure 3: Example of one-loop diagram contributing to the cusp anomalous dimension in an $N$-point scattering amplitude. The graviton internal line is taken to have soft momentum. External particles are taken to be on-shell.
• Figure 4: The analyticity structure in the complex $s\in\mathbb{C}$ plane for fixed-t amplitudes. The integral along the semicircle at infinity vanishes, implying that the IR contour integral is equal and opposite the integral along the UV discontinuity.
• Figure 5: Solid lines: 1-loop contributions to the fixed-$t$ arc $a_0^{\text{FT}}$ from the $O(\kappa^4)$ terms, for various values of $N$ in Eq. (\ref{['eq:geralimp']}), in $d=6$. The dashed vertical line shows the radius of convergence of our expresions Eq. (\ref{['eq:tmaxUB']}). The dashed blue line shows the same contribution to the crossing-symmetric arc $a_0^{\text{CS}}$, where we identify $-t=p^2$. Despite the discrepancy at large $p$, the two methods give bounds on Wilson coefficients that are in agreement with eachother, see Fig. \ref{['fig:gsandman']}