Automated One-loop Computation in Quarkonium Process within NRQCD Framework

Abstract

In last decades, it has been realized that the next-to-leading order corrections may become very important, and sometimes requisite, for some processes involving quarkoinum production or decay, e.g., $e^+e^- \to J/\psi + \eta_c$ and $J/\psi \to 3\gamma$. In this article, we review some basic steps to perform automated one-loop computations in quarkonium process within the Non-relativistic Quantum Chromodynamics (NRQCD) factorization framework, and we give an introduction to some related public tools or packages and their usages in each step. We start from generating Feynman diagrams and amplitudes with \textsc{FeynArts} for the quarkonium process, performing Dirac- and Color- algebras simplifications using \textsc{FeynCalc} and \textsc{FeynCalcFormLink}, and then to doing partial fractions on the linear-dependent propagators by \textsc{APart}, and finally to reducing the Tensor Integrals (TI) into Scalar Integrals (SI) or Master Integrals (MI) using Integration-By-Parts (IBP) method with the help of \textsc{Fire}. We will use a simple concrete example to demonstrate the basic usages of the corresponding packages or tools in each step.