The Threshold t-tbar Cross Section at NNLL Order

TL;DR

This work develops a renormalization-group improved, NNLL-accurate calculation of the total cross section for $t\bar t$ production near threshold in $e^+e^-$ annihilation using the velocity NRQCD (vNRQCD) framework. It combines a consistent summation of velocity logarithms across hard and nonrelativistic scales with a Schrödinger-equation treatment of the nonrelativistic $t\bar t$ pair, including Coulomb and subleading potentials and production currents; the analysis leverages a 1S threshold mass to avoid renormalon ambiguities and achieves a remaining theoretical uncertainty of about $3\%$ in the cross section normalization. The results enable precision extraction of the strong coupling $\alpha_s$, the top width $\Gamma_t$, and the top Yukawa coupling $y_t$ from threshold scans, with the 1S mass stabilizing the peak structure and reducing scheme-related instabilities. The work also highlights the need for a complete electroweak treatment and non-resonant backgrounds for fully robust phenomenology, while showing that NNLL corrections are substantially smaller than fixed-order NNLO predictions and that the threshold region can yield high-precision top-quark parameters.

Abstract

The total cross section for top quark pair production close to threshold in e+e- annihilation is investigated. Details are given about the calculation at next-to-next-to-leading logarithmic order. The summation of logarithms leads to a convergent expansion for the normalization of the cross section, and small residual dependence on the subtraction parameter nu. A detailed analysis of the residual nu dependence is carried out. A conservative estimate for the remaining uncertainty in the normalization of the total cross section from QCD effects is $\lesssim \pm 3%$. This makes precise extractions of the strong coupling and top width feasible, and further studies of electroweak effects mandatory.

Figures (11)

• Figure 1: Examples of potential (a), soft (b), and ultrasoft (c) interactions in vNRQCD.
• Figure 2: Graphs with an ultrasoft gluon with ${\hbox{\boldmath $p$}}\cdot {\bf A}$ couplings which contribute to the running of $c_2(\nu)$.
• Figure 3: Results for the vector current $R$-ratio in Eq. (\ref{['Rveft']}) with a fixed pole mass, $m_t=175\,{\rm GeV}$. In (a) we reproduce the NNLO pole mass results (with matching values for Wilson coefficients). In (b) we display our NNLL pole mass results (with running coefficients). In both plots the dotted, dot-dashed, dashed, and solid curves correspond to $\nu=0.1, 0.15, 0.2$, and $0.4$ respectively.
• Figure 4: Pole mass results for the axial-vector current $R$-ratio in Eq. (\ref{['Rveft']}). The dotted, dashed, dot-dashed, and solid curves correspond to $\nu=0.1, 0.15, 0.2$, and $0.4$ respectively.
• Figure 5: Comparison of $Q_t^2 R^v$ with fixed $M_t^{\hbox{\scriptsize 1S}}$ mass for the fixed order and resummed expansions. The dotted, dashed, and solid curves in a) are LO, NLO, and NNLO, and in b) are LL, NLL, and NNLL order. For each order four curves are plotted for $\nu=0.1$, $0.125$, $0.2$, and $0.4$.