QED Radiative Corrections to $Z$ Boson Production and the Forward Backward Asymmetry at Hadron Colliders

TL;DR

This work provides a comprehensive ${\cal O}(\alpha)$ QED treatment of di-lepton production at hadron colliders, including initial- and final-state radiation and their interference with lepton-mass effects. The analysis shows that final-state radiation, enhanced by mass-singular logarithms, dominates shape distortions of the $m(\ell^+\ell^-)$ distribution and the forward-backward asymmetry, while initial-state corrections are comparatively modest. Detector effects and lepton-identification strategies can significantly mitigate these corrections, though they remain sizable in key observables such as $A_{FB}$ below the $Z$ peak; the study also quantifies the impact on the extracted $Z$ mass and explores the potential for high-precision $\sin^2\theta^{\ell}_{\text{eff}}$ measurements at the LHC. Together, these results provide essential guidance for precision $W/Z$ physics and luminosity determinations at hadron colliders and highlight the importance of incorporating complete electroweak radiative corrections in analyses.

Abstract

The ${\cal O}(α)$ radiative corrections to the process $p\,p\hskip-7pt\hbox{$^{^{(\!-\!)}}$} \rightarrow γ^*, \, Z \rightarrow \ell^+ \ell^-$ ($\ell=e,\,μ$) are calculated. Factorizing the collinear singularity associated with initial state photon bremsstrahlung into the parton distribution functions, we find that initial state corrections have a much smaller effect than final state radiative corrections. Due to mass singular logarithmic terms associated with photons emitted collinear with one of the final state leptons, QED radiative corrections strongly affect the shape of the di-lepton invariant mass distribution, the lepton transverse momentum spectrum, and the forward backward asymmetry, $A_{FB}$. They lead to a sizeable shift in the $Z$ boson mass extracted from data, decrease the di-lepton cross section by up to 10\%, and increase the integrated forward backward asymmetry in the $Z$ peak region by about 7\% at the Tevatron. We also investigate how experimental lepton identification requirements modify the effect of the QED corrections, and study the prospects for a high precision measurement of $\sin^2θ^{lept}_{eff}$ using the forward backward asymmetry at the Large Hadron Collider (LHC).

Figures (14)

• Figure 1: The $p\bar{p}\to\ell^+\ell^-(\gamma)$, ($\ell=e,\,\mu$) cross section for ${\sqrt{s}=1.8}$ TeV and $75~{\rm GeV}<m(\ell^+\ell^-)$$<105$ GeV as a function of a) $\delta_c$ for $\delta_s=0.01$, and b) $\delta_s$ for $\delta_c=0.0005$, including initial state radiation corrections only. Shown are $\sigma(2\to 2) -\sigma({\rm Born})$, $\sigma(2\to 3)$, and $\sigma({\rm NLO)}-\sigma({\rm Born})$. $\sigma({\rm NLO})$ denotes the ${\cal O}(\alpha^3)$ cross section.
• Figure 2: The cross section a) $\sigma(p\bar{p}\to e^+e^-(\gamma))$ and b) $\sigma(p\bar{p}\to\mu^+\mu^-(\gamma))$ as a function of $\delta_s$, including final state radiation corrections only, for ${\sqrt{s}=1.8}$ TeV and $75~{\rm GeV}<m(\ell^+\ell^-) <105$ GeV. Shown are the $2\to 2$ and $2\to 3$ contributions, and the total ${\cal O}(\alpha^3)$ cross section. The solid line represents the Born cross section.
• Figure 3: The lepton pair invariant mass distribution for $p\bar{p}\to\ell^+\ell^-(\gamma)$ at ${\sqrt{s}=1.8}$ TeV in the vicinity of the $Z$ peak. The solid (dotted) line shows $d\sigma/dm(\ell^+\ell^-)$ for electron (muon) final states including ${\cal O}(\alpha)$ QED corrections. The dashed lines gives the ${\ell^+\ell^-}$ Born cross section.
• Figure 4: Ratio of the ${\cal O}(\alpha^3)$ and lowest order differential cross sections as a function of the di-lepton invariant mass for $p\bar{p}\to\ell^+\ell^-(\gamma)$ at ${\sqrt{s}=1.8}$ TeV. The solid line shows the result obtained for final state electrons, whereas the dashed line displays the cross section ratio for muons.
• Figure 5: Ratio of the ${\cal O}(\alpha^3)$ cross section and the cross section obtained in the fragmentation function approach ($\sigma^{FF}$) as a function of the di-lepton invariant mass for $p\bar{p}\to\ell^+\ell^-X$ at ${\sqrt{s}=1.8}$ TeV. The solid line shows the result obtained for final state electrons, whereas the dashed line displays the cross section ratio for muons. In the fragmentation function approach, only final state corrections are taken into account.