Perturbative heavy quark-antiquark systems

TL;DR

This review surveys the development of perturbative heavy quark–antiquark systems within a systematic EFT framework, focusing on NRQCD, threshold expansion, and the potential NRQCD (PNRQCD) formalism to separate hard, soft, potential, and ultrasoft scales. It highlights how advanced heavy-quark mass definitions (PS, 1S, kinetic) improve perturbative convergence and reduce renormalon ambiguities, enabling NNLO predictions for quarkonium masses, leptonic decays, and top-quark pair production near threshold. The paper also covers NNLO sum rules for the b-quark mass, showing consistent mb determinations across different methods and tying these results to lattice and experimental data. Overall, the EFT approach yields a coherent, scale-aware description of perturbative heavy-quarkonia with significant improvements over traditional potential models, while also underscoring remaining large corrections and the need for higher-order calculations and refined non-perturbative treatments.

Abstract

In this review I cover recent developments concerning the construction of non-relativistic effective theories for perturbative heavy quark-antiquark systems and heavy quark mass definitions. I then discuss next-to-next-to-leading order results on quarkonium masses and decay, top quark pair production near threshold and QCD sum rules for $Υ$ mesons.

Figures (6)

• Figure 1: Decomposition of the planar, 2-loop vertex integral in the threshold expansion. Line coding: solid and curled -- hard quarks and gluons, respectively; double line and wavy -- soft quarks and gluons; long- and short-dashed -- potential quarks and gluons; zigzag -- ultrasoft gluons.
• Figure 2: (a) s-s-s region that gives rise to an infrared logarithm (left); p-p-us region which contains the corresponding ultraviolet logarithm. (b) In PNRQCD notation the two NRQCD graphs in (a) are interpreted as a (Coulomb) potential insertion (left) and an ultrasoft 1-loop diagram (right). The shaded bar represents the propagation of the $\bar{Q} Q$ according to the Coulomb Green function. Line coding as in figure \ref{['fig1']}.
• Figure 3: NRQCD graph that generates the mixed non-local/ultrasoft interaction in (\ref{['ultrasoft']}). Line coding as in figure \ref{['fig1']}.
• Figure 4: PNRQCD perturbative diagrams for the heavy quark current correlation function. At leading order the current generates a $Q\bar{Q}$ pair which propagates with the Coulomb Green function (first line). Black bars denote insertions of potentials. The last diagram contains an ultrasoft gluon exchange. Both diagrams in the last line are beyond NNLO.
• Figure 5: (a) [upper panel]: The normalized $t\bar{t}$ cross section (virtual photon contribution only) in LO (short-dashed), NLO (short-long-dashed) and NNLO (solid) as function of $E=\sqrt{s}-2 m_{t,\rm PS}(20\,\hbox{GeV})$ (PS scheme, $\mu_f=20\,$GeV). Input parameters: $m_{t,\rm PS}(20\,\hbox{GeV})=\mu_h=175\,$GeV, $\Gamma_t=1.40\,$GeV, $\alpha_s(m_Z)=0.118$. The three curves for each case refer to $\mu=\left\{15 (\hbox{upper}); 30 (\hbox{central}); 60 (\hbox{lower})\right\}\,$GeV. (b) [lower panel]: As in (a), but in the pole mass scheme. Hence $E=\sqrt{s}-2 m_t$. Other parameters as above with $m_{t,\rm PS}(20\,\hbox{GeV})\to m_t$. Plot from BSS99