Corrections of Order α_s^2 to the Forward-Backward Asymmetry

TL;DR

This paper delivers the first full $O(\alpha_S^2)$ calculation of the non-singlet component of the forward-backward asymmetry $A_{FB}$ in $e^+e^-$ annihilation, addressing discrepancies between earlier massless results and extending the analysis to the experimentally favored thrust-axis definition. By decomposing NNLO corrections into non-singlet, singlet, and triangle contributions and handling four-quark final states with a tagging weight, the authors reveal a logarithmic enhancement from quark-mass effects that renders the massless $A_{FB}$ ill-defined in perturbation theory. They show that the antisymmetric piece is finite in the massless limit and agrees with Ravindran and van Neerven, while the symmetric piece contains a problematic logarithmic term and differs from Altarelli and Lampe; the thrust-axis calculation is performed for the first time, showing an ~0.8% axis-definition effect at typical couplings. The results influence precision electroweak fits and the inferred Higgs mass, and they highlight remaining theoretical uncertainties from higher orders, mass effects, and non-perturbative fragmentation.

Abstract

We calculate the second-order QCD corrections to the forward-backward asymmetry in $e^+e^-$ annihilation. Using the quark axis definition, we do not agree with either existing calculation, but the difference relative to one of them is small and understood. In particular, we point out that the forward-backward asymmetry of massive quarks is enhanced by logarithms of the quark mass. This implies that the forward-backward asymmetry of massless quarks is not computable in QCD perturbation theory and affected by non-power-suppressed corrections coming from the non-perturbative fragmentation functions. We also calculate the second-order corrections using the experimentally-preferred thrust axis definition for the first time.

Figures (6)

• Figure 1: Some of the diagrams contributing to the $b\bar{b}$ final state up to $\mathcal{O}(\alpha_{\mathrm{S}}^2)$. The dashed line represents either the axial or vector current, the thick line the $b$ and the thin line another quark $q$, which must be summed over flavours, including the $b$- and $t$-quark contributions.
• Figure 2: Some of the diagrams contributing to the $b\bar{b}g$ final state up to $\mathcal{O}(\alpha_{\mathrm{S}}^2)$. The dashed line represents either the axial or vector current, the thick line the $b$ and the thin line another quark $q$, which must be summed over flavours, including the $b$- and $t$-quark contributions.
• Figure 3: One of the diagrams contributing to the $b\bar{b}gg$ final state at $\mathcal{O}(\alpha_{\mathrm{S}}^2)$. The dashed line represents either the axial or vector current.
• Figure 4: Some of the diagrams contributing to the $b\bar{b}q\bar{q}$ final state at $\mathcal{O}(\alpha_{\mathrm{S}}^2)$. The dashed line represents either the axial or vector current, the thick line the $b$ and the thin line some other quark flavour $q$, with $q \neq b$.
• Figure 5: One of the diagrams contributing to the $b\bar{b}b\bar{b}$ final state at $\mathcal{O}(\alpha_{\mathrm{S}}^2)$. The dashed line represents either the axial or vector current. The cross indicates which of the two $b$ quarks is triggered on: both contributions must be summed.