Positivity in electron-positron scattering: testing the axiomatic quantum field theory principles and probing the existence of UV states

TL;DR

This work derives the full set of positivity bounds for dimension-8, four-electron SMEFT operators and probes their phenomenology at future $e^+e^-$ colliders. By combining elastic forward-scattering bounds with a convex-cone, extremal-ray framework, it demonstrates how the sign and geometry of the dimension-8 coefficient space constrain or reveal UV states, enabling model-independent inferences or exclusions of new particles. The authors quantify potential positivity violations via a scale $oldsymbol{ riangle}$ and translate this into collider sensitivity reaches of order $1$–$10$ TeV for individual operators, while showing that even in the presence of dimension-6 effects the dimension-8 sector remains informative and testable. They also show that, if positivity holds, the positive geometry can non-trivially constrain UV completions and provide universal bounds on UV masses and couplings, offering a novel, model-independent route to confirm the Standard Model or to point toward specific classes of UV physics at high scales.

Abstract

We consider the positivity bounds on dimension-8 four-electron operators and study two related phenomenological aspects at future lepton colliders. First, if positivity is violated, probing such violations will revolutionize our understanding of the fundamental pillars of quantum field theory and the $S$-matrix theory. We observe that positivity violation at scales of 1--10 TeV can potentially be probed at future lepton colliders even if one assumes that dimension-6 operators are also present. Second, the positive nature of the dimension-8 parameter space often allows us to either directly infer the existence of UV-scale particles together with their quantum numbers or exclude them up to certain scales in a model-independent way. In particular, dimension-8 positivity plays an important role in the test of the Standard Model. If no deviations from the Standard Model are observed, it allows for simultaneous exclusion limits on all kinds of potential UV-complete models. Unlike the dimension-6 case, these limits apply regardless of the UV model setup and cannot be removed by possible cancellations among various UV contributions. This thus consists of a novel and universal test to confirm the Standard Model. We demonstrate with realistic examples how all the previously mentioned possibilities, including the test of positivity violation, can be achieved. Hence, we provide an important motivation for studying dimension-8 operators more comprehensively.

Figures (6)

• Figure 1: Amount of positivity violation in the studied dimension-8 Wilson coefficient parameter space. We consider several two-dimensional slices of the parameter space that we define by setting all Wilson coefficients but two to 0. Positivity violation is estimated through the quantity $\delta(\vec{C}^{(8)})$ of Eq. \ref{['eq:def3']}, so that the white areas correspond to regions compliant with positivity.
• Figure 2: Three-dimensional cross section of the convex cones ${\cal C}$ (yellow, bigger cone) and ${\cal C}_1$ (green). The cross section is taken to be perpendicular to the direction $(1,1,0,1)$. The three axes $x$, $y$, and $z$ are defined along the $(1,-1,0,0)$, $(0,0,1,0)$, and $(-1,-1,0,2)$ directions, respectively.
• Figure 3: Limits on the new physics characterization scale $\Lambda_c$ (in TeV) for the various considered future lepton colliders. "M" denotes marginalized limits (all other coefficients being floating) whereas "F" denotes individual limits (all other coefficients being vanishing). In addition, we represent by the darkest color the largest center-of-mass energy of each collider project.
• Figure 4: Bounds on the pairs of dimension-8 operator coefficients that are relevant for the benchmark points $B_3$ (upper) and $B_4$ (lower). In each case, the other two dimension-8 coefficients are set to 0, as in the benchmark scenario definitions from Eq. \ref{['eq:bench']}, whereas the dimension-6 coefficients are either marginalized over (M, solid) or fixed to 0 (F, dashed). We show the $2\sigma$ expectations for the reach of different future lepton collider projects as colored contours, and the light green area represents the parameter space region allowed by the positivity bounds. Outside this area, the gray isolines depict the dependence of the $\delta(\vec{C})= ({1\ \mathrm{TeV}}/{\Delta})^4$ quantity, which is thus used as an estimate for the amount of positivity violation, on the coefficients.
• Figure 5: Correlations between the amount of positivity violation ($\Delta^{-1}$) and the maximal sensititivity that can be reached at a given future lepton collider ($\Delta^{-1}_{\rm low}$). We present results for the CEPC (upper left), FCC-ee (upper right), ILC (center), and CLIC (lower) colliders, and each represented point has been obtained through a Monte Carlo sampling of the dimension-6 and dimension-8 Wilson coefficient parameter space.