  Phosphorene, a single layer of black phosphorus, is a direct-band gap
two-dimensional semiconductor with promising charge and spin transport
properties. The electronic band structure of phosphorene is strongly affected
by the structural anisotropy of the underlying crystal lattice. We describe the
relevant conduction and valence bands close to the $\Gamma$ point by four- and
six-band (with spin) $k \cdot p$ models, including the previously overlooked
interband spin-orbit coupling which is essential for studying anisotropic
crystals. All the $k \cdot p$ parameters are obtained by a robust fit to {\it
ab initio} data, by taking into account the nominal band structure and the
$k$-dependence of the effective mass close to $\Gamma$-point. The inclusion of
interband spin-orbit coupling allows us to determine dipole transitions along
both armchair and zigzag directions. The interband coupling is also key to
determine the effective g-factors and Zeeman splittings of the Landau levels.
We predict the electron and hole g-factor correction of $\approx 0.03$ due to
the intrinsic contributions in phosphorene, which lies within the existing
range of experimental data. Furthermore, we investigate excitonic effects using
the $k \cdot p$ models and find exciton binding energy (0.81 eV) and exciton
diameters consistent with experiments and {\it ab initio} based calculations.
The proposed $k \cdot p$ Hamiltonians should be useful for investigating
magnetic, spin, transport, optical properties and many-body effects in
phosphorene.
