  We present the dominant two- and three-pion-exchange contributions to the
nucleon-nucleon interaction at sixth order
(next-to-next-to-next-to-next-to-next-to-leading order, N5LO) of chiral
perturbation theory. Phase shifts with orbital angular momentum L>=4 are given
parameter free at this order and allow for a systematic investigation of the
convergence of the chiral expansion. The N5LO contribution is prevailingly
repulsive and considerably smaller than the N4LO one, thus, establishing the
desired trend towards convergence. Using low-energy constants that were
extracted from an analysis of pi-N scattering at fourth order, the predictions
at N5LO are in excellent agreement with the empirical phase shifts of
peripheral partial waves.
