Full positivity bounds for anomalous quartic gauge couplings in SMEFT

Abstract

Electroweak boson scattering at the LHC provides a crucial avenue for probing physics beyond the Standard Model, particularly regarding deviations in quartic gauge couplings. We derive the complete set of positivity bounds for the $22$ dimension-$8$ anomalous quartic gauge coupling (aQGC) coefficients within the Standard Model Effective Field Theory (SMEFT). Moving beyond previous studies limited to transverse vector bosons, our analysis incorporates all electroweak boson modes, explicitly constructing the extremal rays (ERs) of the positivity cone through a group theoretic framework. We utilize two independent methods--direct construction and Casimir operator analysis--to determine these rays, addressing complexities such as parity-violating operators and continuous parameter degeneracies. Our results indicate that the positivity bounds impose severe constraints, restricting the physically viable parameter space to approximately $0.0313\%$ of the naive total space. Furthermore, we derive linear analytical bounds for various operator combinations and provide an easy-to-use Python package, {\tt SMEFTaQGC}, which implements algorithms to numerically verify positivity and compute the optimized positivity bounds for general aQGC configurations.

Figures (5)

• Figure 1: Cartoon representation of the positivity (convex) cone within the 22D SMEFT aQGC parameter space. Imposing the fundamental principles of the S-matrix, we find that the theoretically consistent aQGC region, i.e., the positivity cone, occupies only $0.0313\%$ of the total naive parameter space.
• Figure 2: Contours to derive dispersion relations for the pole-subtracted, forward amplitude $\bar{A}_{ijkl}(s)$. The red branch cuts encode possible UV states.
• Figure 3: Example of extra linear bounds from mixing different types of operators. The light blue region represents the allowed parameter space carved out by $F_{S0}\geq0,F_{T7}\geq0$, and $-F_{M8}\geq0$. The mixing introduces another linear bound $16F_{S0}+2F_{T7}+F_{M8}\geq0$ (space above the brown plane), which chops off the triangular cone below the brown plane.
• Figure 4: Optimal bound on a hyperplane section. The hyperplane section is defined by $\vec{c}\cdot\vec{F}=1$ ($\vec{c}$ being the normal vector to the hyperplane), and the objective function is $L=\vec{v}\cdot\vec{F}$ ($\vec{v}$ being the normal vector to the blue hyperplane). Varying $\vec{F}$ within the section is equivalent to parallel-translating the hyperplane $\vec{v}\cdot\vec{F}=L$ along the $\vec{v}$ direction. The optimal value of $L$ is attained on the boundary of the cone on the hyperplane section.
• Figure 5: Percentage of positivity and convergence of the discretization scheme. The vertical axis shows the fraction of the positivity cone, measured by its solid angle in the higher-dimensional aQGC sphere. Each blue point is a sampling with the number of sampling points denoted on the lower horizontal axis ($10^5$, $10^6$, $10^7$ or $10^8$ points), after discretizing the continuous parameters with $N=200$ according to Eq. \ref{['eq:disScheme']}. The dashed line $(0.0313\%)$ denotes the averaged percentage of the positivity region for $15$ samplings with $10^8$ points. Each maroon point denotes the average of $15$ samplings, each with $10^7$ points, evaluated with different discretizations $N=25,50,100,150,200,250$.