# Theory for quarkonium: from NRQCD factorization to soft gluon factorization

• We demonstrate that the recently proposed soft gluon factorization (SGF) is equivalent to the nonrelativistic QCD (NRQCD) factorization for heavy quarkonium production or decay, which means that, for any given process, these two factorization theories are either both valid or both violated. We use two methods to arrive at this conclusion. In the first method, we apply the two factorization theories to the physical process $J/\psi \to e^+e^-$. Our explicit calculation shows that both SGF and NRQCD can correctly reproduce the low energy physics of full QCD, and the two factorizations are thus equivalent. In the second method, by using equations of motion, we successfully deduce SGF from NRQCD effective field theory. By identifying SGF with NRQCD factorization, we establish relations between the two factorization theories and prove the generalized Gremm-Kapustin relation as a byproduct. Compared with the NRQCD factorization, the advantage of SGF is that it resums the series of relativistic corrections originating from kinematic effects to all powers, yielding better convergence of the relativistic expansion.
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An-Ping Chen and Yan-Qing Ma. Theory for quarkonium: from NRQCD factorization to soft gluon factorization[J]. Chinese Physics C. doi: 10.1088/1674-1137/abc683
An-Ping Chen and Yan-Qing Ma. Theory for quarkonium: from NRQCD factorization to soft gluon factorization[J]. Chinese Physics C.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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## Theory for quarkonium: from NRQCD factorization to soft gluon factorization

###### Corresponding author: Yan-Qing Ma, yqma@pku.edu.cn
• 1. School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
• 2. Center for High Energy physics, Peking University, Beijing 100871, China
• 3. College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China
• 4. Collaborative Innovation Center of Quantum Matter, Beijing 100871, China

Abstract: We demonstrate that the recently proposed soft gluon factorization (SGF) is equivalent to the nonrelativistic QCD (NRQCD) factorization for heavy quarkonium production or decay, which means that, for any given process, these two factorization theories are either both valid or both violated. We use two methods to arrive at this conclusion. In the first method, we apply the two factorization theories to the physical process $J/\psi \to e^+e^-$. Our explicit calculation shows that both SGF and NRQCD can correctly reproduce the low energy physics of full QCD, and the two factorizations are thus equivalent. In the second method, by using equations of motion, we successfully deduce SGF from NRQCD effective field theory. By identifying SGF with NRQCD factorization, we establish relations between the two factorization theories and prove the generalized Gremm-Kapustin relation as a byproduct. Compared with the NRQCD factorization, the advantage of SGF is that it resums the series of relativistic corrections originating from kinematic effects to all powers, yielding better convergence of the relativistic expansion.

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