Potential NRQCD and Heavy-Quarkonium Spectrum at Next-to-Next-to-Next-to-Leading Order

TL;DR

This work advances the perturbative understanding of heavy-quarkonium at next-to-next-to-next-to-leading order by deriving the complete $N^3LO$ nonrelativistic Hamiltonian within the NRQCD/pNRQCD framework and performing the full matching to ultrasoft dynamical gluons via threshold expansion. It provides explicit one- and two-loop operator corrections, plus the leading ultrasoft retardation effects, and expresses the heavy-quarkonium spectrum corrections up to $O(\alpha_s^3)$ with QCD Bethe logarithms $L^E_n$. The results demonstrate sizeable $N^3LO$ contributions for both top-antitop threshold and bottomonium, while showing encouraging convergence in some cases and highlighting the need for an exact $a_3$ coefficient and nonlogarithmic $\beta$-dependent terms. By delivering the Hamiltonian and ultrasoft matching at this order, the paper lays a solid foundation for a full $N^3LO$ analysis of heavy-quarkonium production, annihilation, and spectrum, with direct implications for precise determinations of $\alpha_s$ and heavy-quark masses. Future work will complete the remaining nonlogarithmic $\beta$-dependent terms and compute $a_3$ exactly, enabling refined predictions and potential resummations of logarithms in $\alpha_s$ and $v$.

Abstract

The next-to-next-to-next-to-leading order (N^3LO) Hamiltonian of potential nonrelativistic QCD is derived. The complete matching of the Hamiltonian and the contribution from the ultrasoft dynamical gluons relevant for perturbative bound-state calculations is performed including one-, two-, and three-loop contributions. The threshold expansion is used to disentangle and match contributions of different scales in the effective-theory calculations. As a physical application, the heavy-quarkonium spectrum is obtained at N^3LO for the case of vanishing QCD beta function. Our results set the stage for a full N^3LO analysis of the heavy-quarkonium system.

Figures (4)

• Figure 1: Examples of two-particle-irreducible two-loop diagrams. The standard quark-gluon vertex represents the leading Coulomb interaction. The black circles correspond to the three types of ${\cal O}(1/m_q)$ terms generated by the quark covariant-derivative term.
• Figure 2: Example of a two-particle-reducible two-loop diagram. $B$ stands a general one-loop two-particle-irreducible subgraph. The threshold expansion of this diagram is discussed in the text.
• Figure 3: Feynman diagram giving rise to the ultrasoft contribution at N$^3$LO. The shaded and light double lines stand for the singlet and octet Green functions, respectively. The loopy line represents the ultrasoft-gluon propagator in the Coulomb gauge, and the black circles correspond to the chromoelectric dipole interaction.
• Figure 4: Examples of two- and three-loop diagrams encoded in the diagram of Fig. \ref{['fig:three']}, which require additional matching to bring Eq. (\ref{['Hamus']}) in agreement with the threshold expansion. The dashed and loopy lines represent the potential (Coulomb) and ultrasoft (transverse) gluon propagators in the Coulomb gauge, respectively. The black circles correspond to the interaction generated by the quark covariant-derivative term.