Orbital Angular Momentum Distribution of Gluons at Small-x

The proton spin crisis is a long-term issue in spin physics, implying that the quark only carries a fraction of the proton’s spin. To compensate for the proton’s spin, the spin sum rules are given. These rules state that in a longitudinally polarized proton, the spin carried by the quarks and gluons, as well as their orbital angular momentum (OAM), must add to the proton spin. This decomposition of the proton spin into a spin and OAM of both quarks and gluons is given by Jaffe-Manohar and Ji separately. In this work, we have derived a generalized form of the gluon orbital angular momentum operator at small-x for a longitudinally polarized proton. This expression reproduces the gluon OAM operator given by the Jaffe-Manohar and Ji in the appropriate combination of limits.