Rigorous QCD Analysis of Inclusive Annihilation and Production of Heavy Quarkonium

TL;DR

This work introduces NRQCD as a rigorous effective field theory to disentangle short-distance heavy-quark annihilation from long-distance quarkonium structure, enabling a double expansion in αs(M) and v. It derives factorization formulas for both annihilation rates and production cross sections, incorporating both color-singlet and color-octet contributions and systematically including relativistic corrections up to specified orders in v. The authors provide a comprehensive framework for matching, scale dependence, and relations between decay and production matrix elements, and demonstrate how lattice QCD can compute the nonperturbative inputs. The resulting formalism resolves infrared issues that plagued prior approaches (notably for P-waves), clarifies the role of higher Fock states, and lays the groundwork for quantitatively reliable predictions across many heavy-quarkonium processes with potential lattice validation. Overall, this NRQCD-based factorization framework marks a significant advance over the traditional color-singlet model by enabling controlled calculations and nonperturbative determinations of the long-distance factors governing quarkonium physics.

Abstract

A rigorous QCD analysis of the inclusive annihilation decay rates of heavy quarkonium states is presented. The effective-field-theory framework of nonrelativistic QCD is used to separate the short-distance scale of annihilation, which is set by the heavy quark mass $M$, from the longer-distance scales associated with quarkonium structure. The annihilation decay rates are expressed in terms of nonperturbative matrix elements of 4-fermion operators in nonrelativistic QCD, with coefficients that can be computed using perturbation theory in the coupling constant $α_s(M)$. The matrix elements are organized into a hierarchy according to their scaling with $v$, the typical velocity of the heavy quark. An analogous factorization formalism is developed for the production cross sections of heavy quarkonium in processes involving momentum transfers of order $M$ or larger. The factorization formulas are applied to the annihilation decay rates and production cross sections of S-wave states, up to corrections of relative order $v^3$, and of P-wave states, up to corrections of relative order $v^2$.

Figures (11)

• Figure 1: Example of a diagram that contributes to the quarkonium annihilation rate at order $\alpha_s^3$. The three cuts of the diagram participate in a KLN cancellation.
• Figure 2: Schematic representation of the topological factorization of the rate for quarkonium annihilation. The short distance part is represented by the circle labelled H. The quarkonium wavefunctions are represented by the shaded ovals. The wavefunctions can be connected by light partons, such as the two gluons that are shown explicitly. Soft gluon interactions between the light partons are represented by the circle labelled S.
• Figure 3: Example of a Feynman diagram for quarkonium annihilation at order $\alpha_s^2$. The shaded ovals represent the quarkonium wavefunctions.
• Figure 4: Examples of real-gluon emission in quarkonium decay at order $\alpha_s^3$. The shaded ovals represent the quarkonium wavefunctions.
• Figure 5: Examples of virtual-gluon emission in quarkonium decay at order $\alpha_s^3$. The shaded ovals represent the quarkonium wavefunctions.