We report the double-helicity asymmetry, A(LL)(J/Psi), in inclusive J/Psi production at forward rapidity as a function of transverse momentum p(T) and rapidity |y|. The data analyzed were taken during root 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/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 decay daughter muon pairs mu(+)mu(-) within the PHENIX muon spectrometers in the rapidity range 1.2 \u3c |y| \u3c 2.2. In this kinematic range, we measured the A(LL)(J/Psi) to be 0.012 +/- 0.010 (stat) +/- 0.003 (syst). The A(LL)(J.Psi) 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 approximate to 5 x 10(-2) where recent data of jet and pi(0) double helicity spin asymmetries have shown evidence for significant gluon polarization, and the other one covering the poorly known small-x region x approximate to 2 x 10(-3). Thus our new results could be used to further constrain the gluon polarization for x \u3c 5 x 10(-2)
