Effective field theories for heavy quarkonium

TL;DR

This review shows how heavy quarkonium dynamics can be rigorously described within QCD by a hierarchy of effective field theories, NRQCD and pNRQCD, which separate hard, soft, and ultrasoft scales. It explains how Schrodinger-like equations emerge in both weak- and strong-coupling regimes through careful matching (diagrammatic and Wilson-loop based) and renormalization-group improvement, with potentials encoding QCD dynamics via Wilson loops and gluonic correlators. The authors detail spectroscopic and decay-width predictions, lattice comparisons, and renormalon issues, highlighting improved determinations of heavy-quark masses and αs, while also outlining remaining challenges in non-perturbative inputs and high-order corrections. The work emphasizes that EFTs convert complex QCD dynamics into a tractable, systematically improvable framework with clear paths for future advances in precision and scope, including top thresholds and finite-temperature extensions.

Abstract

We review recent theoretical developments in heavy quarkonium physics from the point of view of Effective Field Theories of QCD. We discuss Non-Relativistic QCD and concentrate on potential Non-Relativistic QCD. Our main goal will be to derive QCD Schrödinger-like equations that govern the heavy quarkonium physics in the weak and strong coupling regime. We also discuss a selected set of applications, which include spectroscopy, inclusive decays and electromagnetic threshold production.

Figures (26)

• Figure 1: Relevant one-loop diagrams for the matching of the 4-fermion operators at order ${\cal O}(1/m^2)$ for the case of unequal masses. The incoming and outgoing particles are on-shell and exactly at rest.
• Figure 2: Relevant diagrams to the matching for the 4-fermion operators at order ${\cal O}(1/m^2)$ and one loop that only appear for the equal mass case. The incoming and outgoing particles are on-shell and exactly at rest.
• Figure 3: Radial and orbital splittings in the $\Upsilon$ system from lattice QCD in the quenched approximation (open circles) and including dynamical $u, d$ and $s$ quarks (filled circles). The lattice spacing has been fixed on the radial splitting between $\Upsilon(2S)$ and $\Upsilon(1S)$. The $b$ quark mass has been tuned to get the $\Upsilon$ mass correct. Figure taken from Lepage:2004mq.
• Figure 4: Mass gap between the singlet and hybrid fields. From Bali:2000vr.
• Figure 5: Propagators and vertices of the pNRQCD Lagrangian (\ref{['Lpnrqcd']}). Dashed lines represent longitudinal gluons and curly lines transverse gluons. $P^\mu$ represents the gluon incoming momentum.