We report the double-helicity asymmetry, $A_{L L}^{J / ψ}$ , in inclusive $J / ψ$ production at forward rapidity as a function of transverse momentum $p_{T}$ and rapidity $| y |$ . The data analyzed were taken during $\sqrt{s} = 510 GeV$ longitudinally polarized $p + p$ collisions at the Relativistic Heavy Ion Collider in the 2013 run using the PHENIX detector. At this collision energy, $J / ψ$ particles are predominantly produced through gluon-gluon scatterings, thus $A_{L L}^{J / ψ}$ is sensitive to the gluon polarization inside the proton. We measured $A_{L L}^{J / ψ}$ by detecting the decay daughter muon pairs $μ^{+} μ^{-}$ within the PHENIX muon spectrometers in the rapidity range $1.2 < | y | < 2.2$ . In this kinematic range, we measured the $A_{L L}^{J / ψ}$ to be $0.012 \pm 0.010$ (stat) $\pm 0.003$ (syst). The $A_{L L}^{J / ψ}$ can be expressed to be proportional to the product of the gluon polarization distributions at two distinct ranges of Bjorken $x$ : one at moderate range $x \approx 5 \times 10^{- 2}$ where recent data of jet and $π^{0}$ double helicity spin asymmetries have shown evidence for significant gluon polarization, and the other one covering the poorly known small- $x$ region $x \approx 2 \times 10^{- 3}$ . Thus our new results could be used to further constrain the gluon polarization for $x < 5 \times 10^{- 2}$ .

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