Partonic Contributions to the Proton Spin in Lattice QCD

Abstract

In Feynman's parton picture, the proton spin can be understood as sum of the contributions from the spin and orbital angular momentum of the quark and gluon partons. However, in gauge theories, there is no local gauge-invariant notion of the spin or orbital angular momentum of the gauge particles. It is shown that in the infinite momentum frame of the proton, the gluons can be equivalent to free radiation, which is analogous to the Weizsaecker-Williams approximation in electrodynamics, and therefore one can talk about gluon helicity and longitudinal orbital angular momentum. We will justify the physical meaning of the Jaffe-Manohar sum rule for the longitudinal proton spin which uses the free-field expression of the QCD angular momentum operator in the light-cone gauge. Furthermore, it is discovered that each term in the Jaffe-Manohar sum rule can be related to the matrix element of a gauge-invariant, but frame-dependent operator through a factorization formula in large-momentum effective field theory. This provides a new approach for the nonperturbative calculation of the proton spin content in lattice QCD, and can be applied to the other parton observables as well. We present all the matching coefficients for the proton spin sum rule and non-singlet quark distributions at one-loop order in perturbation theory. These results will be useful for a first direct lattice calculation of the corresponding parton properties, especially the gluon helicity and parton orbital angular momentum.