The Spin Budget of the Proton at NNLO and Beyond

We revisit the scale evolution of the quark and gluon spin contributions to the proton spin, $\frac{1}{2}\Delta \Sigma$ and $\Delta G$, using the three-loop results for the spin-dependent evolution kernels available in the literature. We argue that the evolution of the quark spin contribution may actually be extended to four-loop order, and that to all orders a single anomalous dimension governs the evolution of both $\Delta \Sigma$ and $\Delta G$. We present analytical solutions of the evolution equations for $\Delta \Sigma$ and $\Delta G$ and investigate their scale dependence both to large and down to lower "hadronic" scales. We find that the solutions remain perturbatively stable even to low scales, where they come closer to simple quark model expectations. We discuss a curious scenario for the proton spin, in which even the gluon spin contribution is essentially scale independent and has a finite asymptotic value as the scale becomes large. We finally also show that perturbative three-loop evolution leads to a larger spin contribution of strange anti-quarks than of strange quarks.