Anomalous Magnetic Moments and Quark Orbital Angular Momentum

Detailed measurements of the spin structure of the nucleon [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] have revealed that only a small fraction of the nucleon spin is carried by the quark spin. This result immediately raised the question, which degrees of freedom carry the rest. Unfortunately, both orbital angular momentum and the gluon spin are difficult to access experimentally, and therefore little rigorous information exists about quark orbital angular momentum. Meanwhile, many qualitative statements regarding orbital angular momentum have been made. For example, when one expresses the matrix element for the anomalous magnetic moment of the nucleon in terms of light-cone wave functions (summed over all Fock components), a nonzero anomalous magnetic moment can only result when there is a nonzero probability that the vector current flips the nucleon helicity [12, 13]. Since the same matrix element conserves the spin of the quarks it is evident that some orbital angular momentum must be transfered to the quarks. Hence a nonzero anomalous magnetic moment can only occur when the target wave function contains components with nonzero orbital angular momentum. While this argument is rigorous, it leaves open quantitative questions regarding the norm of those wave function components or perhaps the resulting net Lz. Within models, it has also been found that a point-like object cannot produce a nonzero anomalous magnetic moment [12, 14] and within this model one can even derive quantitative bounds. Similarly, the observation of a nonzero Sivers [15] effect by the HERMES collaboration [16] seems to indicate wavefunction components with nonzero Lz since the effect requires an interference between initial nucleon states that have opposite helicity. Furthermore, orbital angular momentum seems to play a central role in all models for the Sivers function [17, 18, 19, 20, 21]. The main purpose of this note is to make some of the statements regarding the anomalous magnetic moment more quantitative. In order to accomplish this goal, we start from the matrix element that yields the generalized parton distribution E and apply the Cauchy-Schwarz inequality which will then provide a lower bound on the norm of wave function components with nonzero orbital angular momentum.