Third-order Coulomb corrections to the S-wave Green function, energy levels and wave functions at the origin

TL;DR

The paper addresses large higher-order corrections in heavy-quarkonium and top-quark threshold physics by computing third-order (NNNLO) Coulomb corrections to S-wave energy levels, wave functions at the origin, and the Green function within a nonrelativistic QCD framework. It provides analytic expressions for the Coulomb contributions e_i^C and f_i^C (and non-Coulomb pieces e_i^{nC}, f_i^{nC}) for arbitrary n, including new terms such as c_E,3 and c_ψ,3, with Bethe logarithms L_E(n) entering the energy corrections. The results enable a complete S-wave spectrum at order α_s^5, improve the perturbative treatment of bottomonium masses through the PS-mass scheme (e.g., m_b,PS(2 GeV) = 4.57 GeV), and offer controlled predictions for top-quark pair production near threshold with residual Coulomb uncertainties around a few percent. This work lays the groundwork for precise quarkonium spectroscopy and top-threshold phenomenology by clarifying the convergence and scale- dependence of perturbative Coulomb corrections.

Abstract

We obtain analytic expressions for the third-order corrections due to the strong interaction Coulomb potential to the S-wave Green function, energy levels and wave functions at the origin for arbitrary principal quantum number n. Together with the known non-Coulomb correction this results in the complete spectrum of S-states up to order alpha_s^5. The numerical impact of these corrections on the Upsilon spectrum and the top quark pair production cross section near threshold is estimated.

Figures (4)

• Figure 1: The bottom PS mass, $m_{b,\rm PS}(2\,\hbox{GeV})$, extracted from the experimental value $M_{\Upsilon(1S)}=9.460\,$GeV as a function of renormalization scale $\mu$ at LO (long dashes, black), NLO (long-short dashes, red), NNLO (short dashes, green) and NNNLO (solid, blue).
• Figure 2: Predicted masses of the $\Upsilon(\rm{2S})$ and $\Upsilon(\rm{3S})$ as a function of the renormalization scale $\mu$. The lines refer to LO (long dashes, black), NLO (long-short dashes, red), NNLO (short dashes, green) and NNNLO (solid, blue). The widths of the bands for the experimental mass values are exaggerated.
• Figure 3: The Coulomb wave function at the origin squared for the ground state ($n=1$) normalized by $|\Psi_1^{(0)}(0)|^2$ at $\mu_B=32.6\,$GeV is shown as a function of the renormalization scale $\mu$. The input parameters are $m_{t,\rm PS}(20\,\hbox{GeV})=175\,$GeV, $\nu=m_{t,\rm PS} C_F \alpha_s(\mu)$. The lines refer to LO (long dashes, black), NLO (long-short dashes, red), NNLO (short dashes, green) and NNNLO (solid, blue).
• Figure 4: Top quark pair production cross section (Coulomb corrections only) for $m_{t,\rm PS}=175\,$GeV, $\Gamma_t=1.5\,$GeV. Upper panel: successive approximations up to the third order for $\mu=30\,$GeV. Lower panel: Scale dependence of the third-order approximation. See text for further explanation.