Potential NRQCD: an effective theory for heavy quarkonium

TL;DR

Potential NRQCD (pNRQCD) provides a rigorous EFT framework to describe heavy quarkonium by sequentially integrating out hard and soft scales to yield a Schrödinger-type description with μ-dependent singlet and octet potentials and ultrasoft gluon interactions. The formalism cleanly separates perturbative and nonperturbative effects, incorporates renormalon cancellations, and gives a systematic procedure for computing leading and subleading contributions to the spectrum, including the static limit and gluonic excitations (gluelumps). It also extends to regimes where nonperturbative QCD effects dominate via pNRQCD' and hadronic degrees of freedom, outlining Schrodinger-equation potentials with nonperturbative corrections and Goldstone-boson dynamics in real QCD. Overall, pNRQCD provides a model-independent, QCD-based framework for computing heavy-quarkonium properties and for connecting perturbative calculations to nonperturbative inputs, with clear predictions for lattice tests and phenomenology.

Abstract

Within an effective field theory framework we study heavy-quark--antiquark systems with a typical distance between the heavy quark and the antiquark smaller than $1/Λ_{\rm QCD}$. A suitable definition of the potential is given within this framework, while non-potential (retardation) effects are taken into account in a systematic way. We explore different physical systems. Model-independent results on the short distance behavior of the energies of the gluonic excitations between static quarks are obtained. Finally, we show how infrared renormalons affecting the static potential get cancelled in the effective theory.

Figures (12)

• Figure 1: Propagators and vertices of the pNRQCD Lagrangian (\ref{['pnrqcd0']}). In perturbative calculations the octet propagator is understood without the gluonic string, using instead the Coulomb octet--octet vertex. Also the singlet--octet and the octet--octet vertices produce three diagrams each in perturbative calculations with the Coulomb gauge: one with a longitudinal gluon line, one with a transverse gluon line, and one with both a longitudinal and a transverse gluon line.
• Figure 2: A graphical representation of the static Wilson loop. We adopt the convention that time propagates from the left to the right. Therefore, the quarks trajectories are represented by horizontal lines and the equal-time endpoint Schwinger strings by shorter vertical lines.
• Figure 3: The matching of $V_s$ and $Z_s$ at next-to-leading order in the multipole expansion. On the left-hand side is the Wilson loop in NRQCD, on the right-hand side are the pNRQCD fields. The wavy line represents the ultrasoft gluon propagator.
• Figure 4: The matching of $V_o$ and $Z_o$ at next-to-leading order in the multipole expansion. On the left-hand side is the Wilson loop in NRQCD with colour matrices insertions, on the right-hand side are the pNRQCD fields.
• Figure 5: Graphs contributing to $\langle W_\Box \rangle$ giving rise to $T \alpha_{\rm s}^4\ln{r\,(V_o-V_s)}$ and to $\alpha_{\rm s}^3\ln{r\,(V_o-V_s)}$ terms. Dashed lines represent Coulomb exchanges.