Electroweak Absorptive Parts in NRQCD Matching Conditions

TL;DR

The paper develops a framework to incorporate electroweak absorptive corrections arising from top-quark instability into the NNLL treatment of the total top-pair threshold cross section in e+e− annihilation within NRQCD. By computing absorptive parts of electroweak matching conditions and analyzing their UV divergences via forward-scattering and time-ordered products, it shows how these corrections mix into (e+e−)(e+e−) operators and renormalize the cross section through an optical-theorem-based approach. The results indicate several-percent-level corrections with distinct energy dependence, slightly shifting the threshold line shape and peak position, and remaining comparable in size to NNLL QCD effects. Overall, the work demonstrates a novel NNLL electroweak effect that necessitates a complete treatment of electroweak corrections at NNLL for precision top-quark threshold studies.

Abstract

Electroweak corrections associated with the instability of the top quark to the next-to-next-to-leading logarithmic (NNLL) total top pair threshold cross section in e+e- annihilation are determined. Our method is based on absorptive parts in electroweak matching conditions of the NRQCD operators and the optical theorem. The corrections lead to ultraviolet phase space divergences that have to be renormalized and lead to NLL mixing effects. Numerically, the corrections can amount to several percent and are comparable to the known NNLL QCD corrections.

Figures (3)

• Figure 1: Full theory diagrams in Feynman gauge that have to be considered to determine the electroweak absorptive parts in the Wilson coefficients $C_A$ and $C_V$ related to the physical $bW^+$ and $\bar{b} W^-$ intermediate states. Only the $bW^+$ cut is drawn explicitly.
• Figure 2: Full theory Feynman diagrams describing the process $e^+e^-\to bW^+\bar{b} W^-$ with one or two intermediate top or antitop quark propagators. The circle in diagram (a) represents the QCD form factors for the $t\bar{t}$ vector and axial-vector currents.
• Figure 3: The corrections $\Delta\sigma_{\rm tot}^{\Gamma,1}$ and $\Delta\sigma_{\rm tot}^{\Gamma,2}$ in $pb$ for $M_{\rm 1S}=175$ GeV, $\alpha=1/125.7$, $s_w^2=0.23120$, $V_{tb}=1$, $M_W=80.425$ GeV, $\Gamma_t=1.43$ GeV and $\nu=0.1$ (solid curves), $0.2$ (dashed curves) and $0.3$ (dotted curves) in the energy range $346~\hbox{GeV} < \sqrt{s} < 354$ GeV. Panel (a) shows the sum of both corrections and panel (b) the individual size of $\Delta\sigma_{\rm tot}^{\Gamma,1}$ (energy-dependent lines) and $\Delta\sigma_{\rm tot}^{\Gamma,2}$ (straight lines).