Renormalized 2PN spin contributions to the accumulated orbital phase for LISA sources

We give here a new third post-Newtonisn (3PN) spin-spin contribution (in the PN parameter $\epsilon$) to the accumulated orbital phase of a compact binary, arising from the spin-orbit precessional motion of the spins. In the equal mass case this contribution vanishes, but LISA sources of merging supermassive binary black holes have typically a mass ratio of 1:10. For such non-equal masses this 3PN correction is periodic in time, with period approximately $\epsilon ^{-1}$ times larger than the period of gravitational waves. We derive a renormalized and simpler expression of the spin-spin coefficient at 2PN, as an average over the time-scale of this period of the combined 2PN and 3PN contribution. We also find that for LISA sources the quadrupole-monopole contribution to the phase dominates over the spin-spin contribution, while the self-spin contribution is negligible even for the dominant spin. Finally we define a renormalized total spin coefficient $\bar{\sigma}$ to be employed in the search for gravitational waves emitted by LISA sources.