New positivity bounds from full crossing symmetry

TL;DR

This work develops a comprehensive set of nonlinear positivity bounds for scalar EFTs by combining fixed-$t$ dispersion relations, partial-wave unitarity, and full crossing symmetry, including triple crossing. The four bound families—$PQ$, $D^{\rm su}$, $D^{\rm stu}$, and $\bar{D}^{\rm stu}$—provide tighter, often complementary constraints on Wilson coefficients than traditional linear bounds, and they constrain amplitudes away from the forward limit. Applied to weakly broken Galileons, the bounds rule out standard local UV completions for soft amplitudes, while also yielding stronger, universal constraints in chiral perturbation theory. The results connect with recent crossing-symmetry analyses and demonstrate substantial reductions in viable EFT parameter space, with broad implications for EFT viability and phenomenology. Overall, the paper significantly strengthens the theoretical toolkit for constraining EFTs using fundamental S-matrix principles.

Abstract

Positivity bounds are powerful tools to constrain effective field theories. Utilizing the partial wave expansion in the dispersion relation and the full crossing symmetry of the scattering amplitude, we derive several sets of generically nonlinear positivity bounds for a generic scalar effective field theory: We refer to these as the $PQ$, $D^{\rm su}$, $D^{\rm stu}$ and $\bar{D}^{\rm stu}$ bounds. While the $PQ$ bounds and $D^{\rm su}$ bounds only make use of the $s\leftrightarrow u$ dispersion relation, the $D^{\rm stu}$ and $\bar{D}^{\rm stu}$ bounds are obtained by further imposing the $s\leftrightarrow t$ crossing symmetry. In contradistinction to the linear positivity for scalars, these inequalities can be applied to put upper and lower bounds on Wilson coefficients, and are much more constraining as shown in the lowest orders. In particular we are able to exclude theories with soft amplitude behaviour such as weakly broken Galileon theories from admitting a standard UV completion. We also apply these bounds to chiral perturbation theory and we find these bounds are stronger than the previous bounds in constraining its Wilson coefficients.

Figures (8)

• Figure 1: Convex hull of $\bar{D}^{\rm stu}_{2,1}(\eta,\kappa)$ in 4D for different $\eta=\ell(\ell+1),~\ell=0,2,4, ...$. This convex hull gives $T^\kappa_{2,1}={\rm min}_\kappa \bar{D}^{\rm stu}_{2,1}(\eta,\kappa)$. The maximum of $T^\kappa_{2,1}$ over positive $\kappa$ gives $T_{2,1}$ in the positivity bound (\ref{['boundood34']}), which is at the intersection point of line $\bar{D}^{\rm stu}_{2,1}(12,\kappa)$ and line $\bar{D}^{\rm stu}_{2,1}(20,\kappa)$.
• Figure 2: Positivity bounds on $c_{3,3}/\sqrt{c_{4,0}c_{5,0}}$ and $c_{4,1}/\sqrt{c_{4,0}c_{5,0}}$. The red (blue) lines are the ${D}^{\rm stu}_{3,3}$ ($\bar{D}^{\rm stu}_{3,3}$) bounds with different choices of $k$. The enclosed region pentagon is the region allowed by the optimal positivity bounds. Non-optimal positivity bounds are also plotted with equal interval choices of $k$.
• Figure 3: Positivity bounds on $c_{4,4}/c_{6,0}$ and $c_{5,2}/c_{6,0}$. The red (blue) lines are the ${D}^{\rm stu}_{4,4}$ ($\bar{D}^{\rm stu}_{4,4}$) bounds with different choices of $k$. The enclosed region hexagon is the region allowed by the optimal positivity bounds.
• Figure 4: Comparison of the $Y$ bounds, $PQ$ bounds and $D$ bounds. The total parameter space is a 6-dimensional sphere. The percentages denoted in the corresponding areas are the percentages of the total solid angle. The blue disk schematically represents the solid angle of the parameter space that satisfies the $Y$ bounds. The red disk represents the solid angle that satisfies the 3 types of $PQ$ bounds, excluding the bounds that are already in the $Y$ bounds. The green circle represents the solid angle that satisfies the $D$ bounds (plus $a_{n,0}>0$, because of the square roots in the $D$ bounds). We assume that the EFT is valid up to the cutoff, so settting $\epsilon=1$ and $\sigma=1$.
• Figure 5: Comparison of the $D^{\rm stu}$ and $\bar{D}^{\rm stu}$ bounds and the nonlinear $PQ$ bounds.