Breakdown of NRQCD Factorization in Processes Involving Two Quarkonia and its Cure

TL;DR

New types of operators whose quantum corrections absorb uncanceled infrared singularities are introduced to NRQCD effective field theory, inferring that such singularities will appear for double quarkonium hadroproduction at next-to-leading order.

Abstract

We study inclusive processes involving two heavy quarkonia in nonrelativisitic QCD (NRQCD) and demonstrate that, in the presence of two P-wave Fock states, NRQCD factorization breaks down, leaving uncanceled infrared singularities. As phenomenologically important examples, we consider the decay $Υ\to χ_{cJ}+X$ via $b\bar{b}({}^3P_{J_b}^{[8]})\to c\bar{c}({}^3P_J^{[1]})+gg$ and the production process $e^+e^-\to J/ψ+χ_{cJ}+X$ via $e^{+}e^{-}\to c\bar{c}({}^3P_{J_1}^{[8]})+c\bar{c}({}^3P_J^{[1]})+g$. We infer that such singularities will appear for double quarkonium hadroproduction at next-to-leading order. As a solution to this problem, we introduce to NRQCD effective field theory new types of operators whose quantum corrections absorb these singularities.

Figures (3)

• Figure 1: Typical Feynman diagrams for $b\bar{b}({}^3P_{J_b}^{[8]})\to c\bar{c}({}^3P_{J_c}^{[1]})+gg$ (upper panel), $b\bar{b}({}^3P_{J_b}^{[8]})\to c\bar{c}({}^3S_1^{[8]})+g$ (lower left panel), and $b\bar{b}({}^3S_1^{[8]})\to c\bar{c}({}^3P_{J_c}^{[1]})+g$ (lower right panel).
• Figure 2: Typical Feynman diagrams for $e^{+}e^{-}\to c\bar{c}({}^3P_{J_1}^{[8]})+c\bar{c}({}^3P_{J_2}^{[1]})+g$ (upper panel), $e^{+}e^{-}\to c\bar{c}({}^3P_{J_1}^{[8]})+c\bar{c}({}^3S_1^{[8]})$ (lower left panel), and $e^{+}e^{-}\to c\bar{c}({}^3S_{1}^{[1]})+c\bar{c}({}^3P_{J_2}^{[1]})$ (lower right panel).
• Figure 3: Sample diagrams for one-loop corrections to the annihilation or creation of two heavy-quark pairs $Q_1\bar{Q}_1$ and $Q_2\bar{Q}_2$. The solid dots represent four-quark vertices.