Third-order correction to top-quark pair production near threshold I. Effective theory set-up and matching coefficients

TL;DR

This work develops a rigorous, three-loop-accurate framework for top-quark pair production near threshold in $e^+e^-$ annihilation using non-relativistic effective field theories. It establishes NRQCD and potential NRQCD as the appropriate hierarchies, computes the necessary hard, soft, and ultrasoft matching coefficients (including $O(\epsilon)$ terms), and derives the master formula for the third-order Green function $G(E)$ that governs the cross section. By detailing the Coulomb resummation through the Coulomb Green function, the spin- and color-projected heavy-quark potentials, and the current-matching structure up to NNNLO (with the ultrasoft contribution and non-Coulomb terms), the paper lays the groundwork for the Part II matrix-element computation and a precise top-quark mass extraction. Electroweak and non-resonant effects are discussed as known corrections to NNLO/N4LO, with the analysis indicating that non-resonant NNNLO effects are numerically small, while a full combination with non-resonant contributions will be addressed in the companion paper. Overall, the work advances the precision toolkit for top-threshold phenomenology and paves the way for high-accuracy top mass determinations at future $e^+e^-$ colliders.

Abstract

This is the first in a series of papers, in which we compute the third-order QCD corrections to top-antitop production near threshold in e+ e- collisions. The present paper provides a detailed outline of the strategy of computation in the framework of non-relativistic effective theory and the threshold expansion, applicable more generally to heavy-quark pair production near threshold. It summarizes matching coefficients and potentials relevant to the next-to-next-to-next-to-leading order and ends with the master formula for the computation of the third-order Green function. The master formula is evaluated in part II of the series.

Figures (13)

• Figure 1: Basic electroweak $t\bar{t}$ production processes (upper part of the figure) and production mechanisms ("$t\bar{t}$-radiation" and "singlet production") not contained in the heavy-quark correlation functions (lower part).
• Figure 2: Ladder diagrams.
• Figure 3: Diagram containing cuts not related to top quark production.
• Figure 4: NRQCD Feynman rules for two-quark vertices up to order $1/m^2$. Dashed (curly) lines denote the $A^0$ ($A^i$) gluon field. $q = p^\prime -p$.
• Figure 5: One-loop form factor diagrams: wave-function renormalization and vertex corrections.