Enhanced Nonperturbative Effects in Z Decays to Hadrons

TL;DR

This paper develops a soft-collinear effective theory (SCET) framework to quantify enhanced nonperturbative QCD effects in Z decays to hadrons, especially in regions of phase space near jet endpoints. By factorizing the process into hard, collinear, and ultrasoft sectors, the authors express leading nonperturbative corrections through Wilson-line shape functions and a hierarchy of operators, with the jet-energy distribution near E_J ≈ M_Z/2 serving as a prime example. They extend the analysis to multiple event-shape observables (thrust, jet masses, jet broadening, C parameter, and energy-energy correlations), uncovering partial universality (notably between thrust and jet masses at leading order) and highlighting observables where no simple universal relation holds. The work emphasizes the limitations of simple shift Ansätze in the shape-function regime and provides a structured classification of observables by their ultrasoft inclusivity, offering a path toward more precise, model-informed comparisons with data. Overall, the SCET approach yields a principled description of nonperturbative effects in Z decays that informs both theory and experimental interpretation of precision QCD in high-energy jets.

Abstract

We use soft collinear effective field theory (SCET) to study nonperturbative strong interaction effects in Z decays to hadronic final states that are enhanced in corners of phase space. These occur, for example, in the jet energy distribution for two jet events near E_J=M_Z/2, the thrust distribution near unity and the jet invariant mass distribution near zero. The extent to which such nonperturbative effects for different observables are related is discussed.

Figures (2)

• Figure 1: Determination of the thrust axis. To the order we are working, the quark and antiquark have momenta $\left| \mathbf{p}_q \right| = \left| \mathbf{p}_{\bar{q}} \right| = M_Z/2$. The antiquark then makes an angle $\theta_{\bar{q}}= 2 \left| \mathbf{k}_{\bar{q}\perp} \right| /M_Z$ with the $z$-axis, and the thrust axis $\hat{\mathbf{}}{t}$ makes an angle $\theta_t = \left| \mathbf{k}_{\bar{q}\perp} \right| /M_Z$ with both the quark and antiquark.
• Figure 2: Plot of the function $M_1(f)$. The black solid curve shows the perturbative contributions only, while the red dashed line represents the moment including the nonperturbative contribution. The figure corresponds to $\beta=0.15$, $\delta=\pi/12$, $\langle 0|O_1|0\rangle = 0.5~{\rm GeV}$, and we have evaluated the strong coupling constant at the scale $\mu = \beta M_Z$.