Differential observables for the Higgs-strahlung process to all orders in EFT

TL;DR

This work presents an all-orders framework to map the full differential amplitude of $f\bar{f}\to Z(\ell\ell) h$ onto EFT Wilson coefficients, using a fixed-$J$ decomposition to relate partial waves to higher-dimension operators. By introducing principal amplitudes and their descendants, the authors build an EFT basis that isolates fixed-$J$ contributions and reveal how angular observables, via angular moments, encode the EFT structure across invariant-mass bins. They show that at high energy the principal operators provide as many independent directions as independent partial waves, while at lower orders EFT induces correlations that manifest as helicity-directions, which map to specific higher-dim operators. The framework is then interpreted in SMEFT up to dimension-8 and extended to differential observables, enabling systematic extractions of Wilson coefficients from angular-moment analyses in current and future collider data. Overall, the method provides a general, all-orders procedure to connect differential collider measurements to high-scale physics across a broad class of 2-to-2 processes, with practical steps for experimental implementation using angular moments and energy binning.

Abstract

We develop methods to obtain the fully differential cross-section for the $f \bar{f} \to Z(\ell\ell)\,h$ process to any desired order in effective field theory (EFT). To achieve this, we first derive a mapping between the partial wave expansion and the EFT expansion to all orders. We find that at lower orders, EFT predicts correlations between the different partial wave coefficients. This allows us to construct linear combinations of partial wave coefficients that get their leading contributions from a higher dimension EFT operator. We then introduce experimental observables, the so called angular moments -- that probe these linear combinations of partial wave coefficients -- and can be determined from a fully differential analysis of the angular distribution of the leptons arising from the $Z$ decay. We show that analysing the dependence of these angular moments on the $Zh$ invariant mass allows us to systematically probe all higher dimension EFT operators contributing to this process. While we take the Higgs-strahlung process as an example, the methods developed here are completely general and can be applied to other 2-to-2 collider processes.

Figures (3)

• Figure 1: Mapping between partial waves and EFT contact terms in the CCLL basis of eq. (\ref{['ccll']}), and in our EFT basis defined in eq. (\ref{['basisdef']}). Our basis in eq. (\ref{['basisdef']}) has been designed to simplify this mapping (see text for more details).
• Figure 2: We show the three angles, $\left(\Theta, {\theta}, {\phi}\right)$, in the figure above. The angle, $\Theta$ is the scattering angle of the $f \bar{f} \to Zh$ process. The cartesian axes, $\{x,y,z\}$, are defined as follows: the $z$ direction is identified as the direction of the $Z$-boson; $y$ is defined to be normal to the plane of $Z$, i.e. $\hat{y}=\hat{z}\times \hat{f}$, $\hat{f}$ being the direction of the fermion shown by the red arrow; $x$ is defined such that it completes the right-handed set. The angles, $\theta$ and $\phi$ ($\hat{\theta}$ and $\hat{\phi}$), are the polar and azimuthal angles of the positively charged (positive helicity) lepton in the $Z$ rest frame in the spherical polar coordinate system defined with respect to the above cartesian system.
• Figure 3: A schematic summary of the steps we have taken to arrive at our final results. See the text for more details.