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.

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非相对论性QCD因式分解在涉及两个Quarkonia的过程中的及其解决方法

我们研究了非相对论性QCD（NRQCD）中涉及两个重夸克族的包容性过程，并证明了在存在两个P波福克状态的情况下，NRQCD分解会分解，留下未消除的红外奇点。作为现象学上重要的示例，我们考虑通过$ b \ bar {b}（^ 3P_ {J_b} ^ {[[8]}）\\到c \ bar {c }（^ 3P_J ^ {[[1]}）+ gg $和生产过程$ e ^ + e ^-\通过$ e ^ {+} e ^ {-到J / \ psi + \ chi_ {cJ} + X $ } \到c \ bar {c}（^ 3P_ {J_1} ^ {[8]}）+ c \ bar {c}（^ 3P_J ^ {[1]}）+ g $。我们推断，这种奇异性将以接近领先的顺序出现在双夸克hadro生产中。为解决此问题，我们向NRQCD有效场论介绍了量子校正吸收了这些奇点的新型算子。