Basis independent neutrino masses in the R_p violating minimal supersymmetric standard model

We calculate the neutrino mass matrix up to one loop order in the minimal supersymmetric standard model 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 flavor 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.