  We continue our study of open string perturbation theory on the lightcone
worldsheet lattice, which is an $M\times N$ rectangular grid. Here $M$ is the
number of $P^+$ units and $N$ is the number of $ix^+$ units. We extend our
previous analysis to the bosonic open string one planar loop self-energy. We
find that, when all open string coordinates satisfy Neumann conditions, the
ultraviolet worldsheet divergences associated with the closed string tachyon
and boundary effects can be cancelled by renormalization of bulk ($AM^1$) and
boundary ($BM^0$) worldsheet "cosmological constants". The bulk divergence for
the open string matches that for the closed string. The open string tachyon
mass shift displays the dilaton logarithmic divergence with the correct
coefficient for its consistent absorption by renormalization of the string
tension. The ultraviolet contribution to the open string gluon mass shift
vanishes, in accord with its interpretation as a gauge particle. We also find
that when the bosonic string ends on a D-brane additional negative powers of
$\ln M$ multiply the bulk and boundary divergences. These can no longer be
cancelled by the "cosmological constants", perhaps pointing to the need, in the
presence of D-branes, for the cancellations of divergences provided by
supersymmetry.
