Basis independent neutrino masses in the $R_p$ violating MSSM

We calculate the neutrino mass matrix up to one loop order in the MSSM without R-parity, including the bilinears in the mass insertion approximation. This introduces additional diagrams usually neglected in the literature. We systematically consider the possible new diagrams, and find a few missing from our previous work. We provide analytic expressions for the mass matrix elements in the neutrino flavour basis, which are independent of the $H_d - L_i$ basis choice in the Lagrangian. We compare the contributions from different diagrams, and make ``Lagrangian-basis-independent'' estimates of when the new diagrams need to be included. We briefly discuss the phenomenology of a toy model of bilinear R-parity violation.