[Paper Review] Third-order correction to top-quark pair production near threshold II. Potential contributions

Μ. Beneke, Y. Kiyo|arXiv (Cornell University)|Sep 9, 2024

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper computes third-order quantum corrections to top-quark pair production near threshold using potential non-relativistic QCD (PNRQCD), focusing on non-Coulombic potentials such as $1/r^2$, delta, and contact interactions. It completes the NNNLO calculation by deriving finite, renormalized expressions for single and double insertions of these potentials, with key results including the full analytic structure of the S-wave Green function and detailed numerical analysis of scale dependence and mass scheme effects.

ABSTRACT

We provide a detailed account of the methods and calculations for the third-order corrections to the S-wave Green function from heavy-quark potentials other than the Coulomb potential. The results of this paper are relevant to the top-antitop threshold production process in next-to-next-to-next-to-leading (NNNLO) order and to the determination of the bottom-quark mass from high-moment sum rules, and have been employed in corresponding previous publications. Further to the third-order calculation, we discuss in detail three refinements necessary to obtain reliable third-order results for the top threshold: finite-width effects, pole resummation, and the implementation of the potential-subtracted mass scheme. A detailed numerical analysis of residual scale dependence and the size of various contributions to the top production cross section is provided. The S-wave energy levels and wave functions at the origin for heavy quarkonium states of arbitrary principal quantum number $n$ are collected in an appendix.

Motivation & Objective

• To complete the third-order calculation of the S-wave Green function in PNRQCD for top-quark pair production near threshold, beyond the leading Coulomb potential.
• To systematically compute and renormalize finite, ultraviolet-divergent contributions from higher-order potentials such as $1/r^2$, delta, and contact interactions.
• To incorporate and analyze three critical refinements: finite-width effects, pole resummation, and the potential-subtracted mass scheme for reliable NNNLO predictions.
• To provide a detailed numerical analysis of residual scale dependence and the relative size of individual contributions to the top-quark production cross section.
• To tabulate exact corrections to S-wave energy levels and wave functions at the origin for arbitrary principal quantum number $n$.

Proposed method

• Employ potential non-relativistic QCD (PNRQCD) to treat the strong Coulomb interaction as an unperturbed Hamiltonian, with higher-order corrections from non-Coulomb potentials.
• Use dimensional regularization with $d = 4 - 2\epsilon$ to regulate ultraviolet divergences arising from singular potentials like $1/r^2$ and delta functions.
• Compute single and double insertions of non-Coulomb potentials via Feynman-like diagrams in momentum space, solving the resulting integrals analytically and extracting divergent parts.
• Apply renormalization procedures to remove $\epsilon$-poles, including matching to the potential-subtracted (PS) mass scheme and handling $O(\epsilon)$ terms in matching coefficients.
• Implement pole resummation to resum $\alpha_s/v$-enhanced logarithms and improve convergence in the threshold region.
• Derive and include finite-width corrections via $d$-dimensional top decay width and $O(\epsilon)$ hard matching coefficients to improve accuracy in the on-shell regime.

Experimental results

Research questions

• RQ1 What is the analytic structure of the third-order correction to the S-wave Green function from non-Coulomb potentials in top-quark pair production near threshold?
• RQ2 How do finite-width effects and the choice of mass scheme (e.g., pole vs. potential-subtracted) affect the scale dependence and reliability of NNNLO predictions?
• RQ3 What is the relative numerical significance of non-Coulomb contributions (e.g., $1/r^2$, delta, contact) compared to pure Coulomb corrections in the total top-quark pair production cross section?
• RQ4 How do the corrections to the S-wave energy levels and wave functions at the origin depend on the principal quantum number $n$?
• RQ5 What is the impact of pole resummation and $O(\epsilon)$ corrections on the final result and its scale invariance?

Key findings

• The paper provides the complete analytic expression for the third-order correction $\delta_3 G(E)$ in PNRQCD, including single and double insertions of non-Coulomb potentials such as $1/r^2$, delta, and contact interactions.
• The non-logarithmic ultrasoft correction $\delta^{us}_n$ is computed numerically for $n = 1$ to $6$, with values ranging from 353.06 to 187.16, showing significant suppression with increasing $n$.
• The constant term $c_{\psi,3}^{nC}/(64\pi^2)$ in the non-Coulomb correction is derived analytically, including $S_1$, $S_2$, $S_3$, and harmonic sum $H_n$ contributions, with explicit dependence on $C_F$, $C_A$, $T_F$, and $n$.
• The scale dependence of the $-\frac{3}{4}C_A^3 L_m$ term in the ultrasoft correction cancels exactly with the corresponding term in the third-order Coulomb contribution, ensuring renormalization scale invariance.
• The inclusion of $O(\epsilon)$ terms in the $1/m^2$ potential matching coefficients and the new $O(\epsilon)$ term $b_2^{(\epsilon)}$ for the two-loop $1/r^2$ potential improves the accuracy of the final result.
• The potential-subtracted mass scheme is implemented via both insertion and shift methods, with consistent results, and pole resummation is shown to significantly reduce scale dependence in the cross section prediction.