QCD Corrections to the $e^+e^-$ Cross Section and the $Z$ Boson Decay Rate

TL;DR

The paper provides a comprehensive, technique-driven review of QCD corrections to the e+e- hadronic cross section and the Z hadronic width, with emphasis on practical formulas and mass effects. It combines operator product expansion, heavy-mass expansion, and decoupling/matching to treat light and heavy quarks across energy scales, delivering exact O(α_s) results and extensive mass-expansion results up to α_s^3 in various channels. The work distinguishes singlet and nonsinglet contributions, vector and axial-vector currents, and offers detailed numerical analyses for LEP-scale observables as well as low-energy regions around 10 GeV. It furnishes explicit prescriptions for partial-rate decompositions, assesses theoretical uncertainties, and provides compact formulas and appendices to support precise α_s extractions and Standard Model tests from hadronic Z- decays and low-energy cross sections.

Abstract

QCD corrections to the electron positron annihilation cross-section into hadrons and to the hadronic $Z$ boson decay rate are reviewed. Formal developments are introduced in a form particularly suited for practical applications. These include the operator product expansion, the heavy mass expansion, the decoupling of heavy quarks and matching conditions. Exact results for the quark mass depen- dence are presented whenever available, and formulae valid in the limit of small bottom mass ($m_{b}^2\ll s$) or of large top mass ($m_{t}^2\gg s$) are presen- ted. The differences between vector and axial vector induced rates as well the classification of singlet and nonsinglet rates are discussed. Handy formulae for all contributions are collected and their numerical relevance is investi- gated. Prescriptions for the separation of the total rate into partial rates are formulated. The applicability of the results in the low energy region, relevant for measurements around 10 GeV and below, is investigated and numerical predictions are collected for this energy region.

Figures (20)

• Figure 1: Contour integral.
• Figure 2: The absorptive part of a current correlator is obtained by cutting the diagram in all possible ways.
• Figure 3: Singlet contribution of order ${\cal O}(\alpha_s^2)$ and ${\cal O}(\alpha_s^3)$.
• Figure 4: (a) Lowest order contribution to the correlator $\Pi"(Q^2)$. (b) Vacuum diagram contributing to the perturbative VEV of the operator $\bar{q} q$.
• Figure 5: Hard Mass Procedure.