Elastic positivity vs extremal positivity bounds in SMEFT: a case study in transversal electroweak gauge-boson scatterings

TL;DR

This paper compares two positivity-bounds methods in SMEFT—elastic positivity (based on forward 2-to-2 amplitudes with all helicity superpositions) and extremal positivity (via the extremal representation of convex cones)—applied to transversal electroweak gauge-boson scattering. It develops both analytical and numerical frameworks, showing that extremal positivity yields the tightest, most complete bounds on dim-8 aQGC operators, and remains computationally efficient even when including multiple SM states. The study provides explicit bounds for the transversal quartic-gauge-boson couplings, illustrating that positivity excludes roughly 99.3% of the LHC-parameter space, and presents practical analytic inequalities along with numerical schemes. These results offer a robust, model-independent guide for SMEFT UV-completion tests and for interpreting VBS and tri-boson measurements, with a clear path to incorporate longitudinal modes and higher-order bounds in future work.

Abstract

The positivity bounds, derived from the axiomatic principles of quantum field theory (QFT), constrain the signs of Wilson coefficients and their linear combinations in the Standard Model Effective Field Theory (SMEFT). The precise determination of these bounds, however, can become increasingly difficult as more and more SM modes and operators are taken into account. We study two approaches that aim at obtaining the full set of bounds for a given set of SM fields: 1) the traditional elastic positivity approach, which exploits the elastic scattering amplitudes of states with arbitrarily superposed helicities as well as other quantum numbers, and 2) the newly proposed extremal positivity approach, which constructs the allowed coefficient space directly by using the extremal representation of convex cones. Considering the electroweak gauge-bosons as an example, we demonstrate how the best analytical and numerical positivity bounds can be obtained in several ways. We further compare the constraining power and the efficiency of various approaches, as well as their applicability to more complex problems. While the new extremal approach is more constraining by construction, we also find that it is analytically easier to use, numerically much faster than the elastic approach, and much more applicable when more SM particle states and operators are taken into account. As a byproduct, we provide the best positivity bounds on the transversal quartic-gauge-boson couplings, required by the axiomatic principles of QFT, and show that they exclude $\approx 99.3\%$ of the parameter space currently being searched at the LHC.

Figures (5)

• Figure 1: A comparison of elastic positivity bounds and extremal positivity bounds on the $F_{T,0}$-$F_{T,1}$ plane. All other coefficients are fixed to 0.
• Figure 2: Percentages of the parameter space, or solid angle $\Omega(\mathcal{C}^{el}_{A})$, of analytical elastic positivity bounds, Eqs. \ref{['eq:l1']}-\ref{['eq:nfbound3']}, computed from different samplings ($10^5, 10^6, 10^7$ or $10^8$ points). Each blue point is a sampling. The horizontal axis is the number of sampling points, while the vertical axis is the resulting $\Omega(\mathcal{C}^{el}_{A})$. The solid line (0.6937%) is the average of 15 largest samplings, each with $10^8$ points, while the dashed lines are 2$\sigma$ errors.
• Figure 3: Numerical results for the percentages of the parameter space that satisfy the full elastic positivity bounds. Each point in the plot represents a sampling, and points with the same color come from the same $10^8$ sampling. The horizontal axis is the number of points used for the sampling and the vertical axis is the percentage of these points satisfying the full elastic positivity bounds. The horizontal solid line (0.6937%) is the average of ten $10^8$-points samplings, while the dashed lines are 2$\sigma$ errors (square root of the sampling variance) for the samplings.
• Figure 4: A circular cone, $(1,\sqrt 2 r,r^2)$, and its inscribed octagonal cone, by taking $N=4$ in Eq. \ref{['eq:rsample']}.
• Figure 5: Solid angle $\Omega(\mathcal{C}_N)$ for different $N$ values.