  In the collapsing core of massive stars, the standing accretion shock
instability (SASI) can drive spiral modes that efficiently redistribute angular
momentum. This process can impart a spin to the forming neutron star even when
the progenitor star is non-rotating. Here we develop the first analytical
description of the angular momentum redistribution driven by a spiral mode of
the SASI. Our analysis, valid in the limit of small mode amplitude, shows that
the angular momentum separation is driven by the Reynolds stress generated by
the spiral mode. The resulting solutions compare favorably with previous
three-dimensional hydrodynamic simulations of the SASI in the linear and weakly
non-linear phases. Reasonable agreement is also found when extrapolating the
solutions into the fully non-linear phase. A Reynolds-decomposition of the flow
is performed in the saturated state of these simulations, showing that outward
angular momentum transport by the Reynolds stress and the fluctuating component
of the mass flux balance inward transport by advection. We derive an
approximate analytic expression for the maximum angular momentum deposited in
the neutron star as a function of the mass accretion rate, shock radius, shock
compression ratio, and amplitude of the spiral mode at the time of explosion.
Implications for the birth spin periods of neutron stars are discussed.
