  The Principle of Maximum Conformality (PMC) has been suggested in the
literature to eliminate conventional renormalization scheme-and-scale
ambiguities in precision test of the QCD theory. It has been shown that by
applying the PMC single-scale-setting approach (PMCs), all non-conformal
$\{\beta_i\}$-terms of the perturbative QCD series will be resummed into strong
coupling constant $\alpha_s$, which then sets the correct magnitude of
$\alpha_s$ and results in a scale-fixed and scheme-invariant prediction. In
this paper, by applying the PMCs, we present an improved analysis of the
top-quark pair production via the process $e^{+}e^{-}\to \gamma^{*}\to
t\bar{t}$ near the threshold region up to next-to-next-to-next-to-leading order
(N$^3$LO) QCD corrections. Near the threshold region, the top-quark velocity
$v$ tends to zero, leading to Coulomb singularity. For the purpose, we first
reconstruct the analytical expression for the Coulomb-terms up to N$^{3}$LO
accuracy by using the PSLQ algorithm. We then apply the PMCs to deal with the
Coulomb and non-Coulomb parts separately, and two effective coupling are fixed
accordingly. After that, the Coulomb part is resummed into a
Sommerfeld-Gamow-Sakharov factor, which leads to a more reasonable behavior
near the threshold region. We thus achieve a more precise and a better
understanding of the top-quark pair production near the threshold region.

High Energy Physics - Phenomenology