Effective Field Theory for Ultrasoft Momenta in NRQCD and NRQED

TL;DR

The paper develops an effective field theory framework for ultrasoft momenta in heavy quarkonia by organizing a scale hierarchy (m, |p|, E) and introducing NRQCD and pNRQCD as successive EFTs. It provides one-loop matching results for four-quark operators in NRQCD/NRQED, including unequal-mass cases, and formulates pNRQCD with singlet and octet wavefunction fields whose dynamics are governed by a non-local-in-space, local-in-time potential. The authors clarify dimensional regularization and threshold matching to separate analytical and non-analytical mass-dependent pieces, and they demonstrate how the pNRQCD Lagrangian yields perturbative predictions such as the mα^5 energy corrections and Lamb shift in QED, while outlining the transition to non-perturbative regimes when Λ_QCD is relevant. Overall, this work establishes a systematic EFT pathway from QCD/QED to a potential-based description of heavy quark bound states, enabling controlled computations of bound-state energies and interactions at ultrasoft scales.

Abstract

We propose an effective field theory for heavy quark-antiquark bound states once the soft gluons have been integrated out. We also give new results for the matching between QCD (QED) and NRQCD (NRQED).