Complete Order alpha_s^3 Results for e^+ e^- to (gamma,Z) to Four Jets

TL;DR

This paper delivers complete $O(\alpha_s^3)$ predictions for $e^+e^- \rightarrow (\gamma,Z) \rightarrow$ four jets, including subleading-color and light-by-glue contributions previously neglected. It provides fixed-order results for multiple jet algorithms and demonstrates that subleading terms are small, while resummation of small-$y_{\rm cut}$ logarithms in the Durham algorithm, matched to fixed order, yields excellent agreement with $Z$-pole data. The work also highlights residual renormalization-scale uncertainties, indicating remaining higher-order effects, and discusses implications for precise determinations of $\alpha_s$ and potential constraints on light flavors. Overall, the combination of complete NLO calculations and resummed+matched predictions strengthens QCD tests in multi-jet e+e- annihilation and informs future angular-distribution studies.

Abstract

We present the next-to-leading order (O(alpha_s^3)) perturbative QCD predictions for e^+e^- annihilation into four jets. A previous calculation omitted the O(alpha_s^3) terms suppressed by one or more powers of 1/N_c^2, where N_c is the number of colors, and the `light-by-glue scattering' contributions. We find that all such terms are uniformly small, constituting less than 10% of the correction. For the Durham clustering algorithm, the leading and next-to-leading logarithms in the limit of small jet resolution parameter y_{cut} can be resummed. We match the resummed results to our fixed-order calculation in order to improve the small y_{cut} prediction.

Figures (5)

• Figure 1: Representative contributions of type (I), (II) and (III), as described in the text. The coupling of a quark to the $(\gamma,Z)$ vector boson is denoted by $\times$, with a $1$ ($\gamma_5$) for vector (axial vector) coupling. Dashed lines correspond to representative five-parton cuts; dotted lines to four-parton cuts.
• Figure 2: (a) Absolute value of the contributions of the different electroweak/color pieces to the four-jet fraction at $\sqrt{s} = M_Z$ for the E0 scheme, i.e.$(\alpha_s/2 \pi)^3 \, |C_4^{(x)}|/(1 + \frac{\alpha_s}{\pi})$ with $x \in \{ a,b,c,d,e,f,II,III \}$. We also show the Born and full one-loop prediction, and data from ref. SLDdata. (b) Dependence of the tree-level (dashed line) and one-loop (solid line) prediction on the renormalization scale $\mu$ for $y_{\rm cut} = 0.015$.
• Figure 3: NLO prediction for the four-jet rate using the Geneva algorithm for $N_f = 5$ and $N_f = 8$. The theoretical bands have been obtained by varying the renormalization scale from ${{{1}\over{2}}} \sqrt{s} < \mu < 2\sqrt{s}$ and from ${{{1}\over{3}}}\sqrt{s} < \mu < 3\sqrt{s}$. The data are from ref. SLDdata.
• Figure 4: The four-jet fraction for the Durham algorithm at $\sqrt{s}=M_Z$, illustrating the improvements to the Born term from adding successively the leading-color loop corrections, the subleading-color corrections, and the resummed corrections after matching. The data are from ref. SLDdata.
• Figure 5: Dependence on the renormalization scale of (a) the full one-loop prediction and (b) the resummed and matched result, for the Durham algorithm at $\sqrt{s}=M_Z$.