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

TL;DR

This paper demonstrates the breakdown of NRQCD factorization in processes involving two P-wave quarkonia due to uncanceled infrared singularities and proposes a new operator framework to resolve this issue.

Contribution

It identifies a fundamental limitation of NRQCD factorization with two P-wave states and introduces new operators to absorb the resulting singularities, improving the theory's consistency.

Findings

NRQCD factorization breaks down with two P-wave quarkonia due to infrared singularities.

Singularities are present in specific decay and production processes involving P-wave states.

New operators are proposed to absorb the infrared singularities and restore factorization.

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 $\Upsilon \to \chi_{cJ}+X$ via $b\bar{b}({}^3P_{J_b}^{[8]})\to c\bar{c}({}^3P_J^{[1]})+gg$ and the production process $e^+e^-\to J/\psi+\chi_{cJ}+X$ 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.