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The data analyzed were taken during $\\sqrt{s}=510$ GeV longitudinally polarized $p$$+$$p$ collisions at the Relativistic Heavy Ion Collider (RHIC) in the 2013 run using the PHENIX detector. At this collision energy, $J/\\psi$ particles are predominantly produced through gluon-gluon scatterings, thus $A_{LL}^{J/\\psi}$ is sensitive to the gluon polarization inside the proton. We measured $A_{LL}^{J/\\psi}$ by detecting the d","authors_text":"A. Adare, A. Bazilevsky, A. Berdnikov, A.C. Mignerey, A. Datta, A. Denisov, A. Deshpande, A.D. Frawley, A. Dion, A. Drees, A. Durum, A. Enokizono, A. Franz, A. Glenn, A. Iordanova, A.J. Miller, A. Khanzadeev, A.K. Mohanty, A. Lebedev, A. Manion, A. Meles, A. Milov, A. Mwai, A. Oskarsson, A. Sen, A. Sexton, A. Sickles, A.S. Nyanin, A. Sukhanov, A.S. White, A. Takahara, A. Taketani, A. Taranenko, A. Timilsina, A.V. Kazantsev, A. Yanovich, A. Zelenski, B. Azmoun, B. Bannier, B. Fadem, B. Hong, B.J. Ramson, B. Kimelman, B. Komkov, B.K. Schmoll, B.K. 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