Spin polarization for massive fermion in a shear flow: complete results at $O(\partial)$

Motivated by the key role of shear induced polarization in understanding the local spin polarization puzzle of $\Lambda$ hyperons in heavy ion collisions, we perform a complete analysis of spin polarization of massive fermion in a quantum electrodynamic plasma with shear flow. Apart from the well-known spin-shear coupling in free theory, we include two more collision dependent contributions: one is a non-dynamical contribution fixed by shifted spin-averaged distribution in steady state; the other is a dynamical contribution following from spin evolution. Despite of the dependencies on collision, we find the dependencies on coupling drop out in the final results. These contributions can lead to significant enhancement of the spin-shear coupling in phenomenologically interesting regime.