  The chiral nucleon-nucleon ($NN$) interaction at high cutoffs has been
plagued by the presence of spurious bound states. In this work, the chiral $NN$
interaction at N$^3$LO is studied by the local projection method as the cutoff
increases. The evolution of short-range behaviors of pion-exchange interactions
and contact interactions is intuitively demonstrated. The $P$-channel
potentials toward high cutoffs appear to be erratic at short ranges to
compromise with phase shifts, while such erratic behaviors can be avoided in
$S$ and $D$ channels. Furthermore, a chiral $NN$ interaction at N$^3$LO is
studied at a cutoff of 700 MeV. The properties of deuteron and triton are
testified with this interaction. Such a hard interaction is expected to provide
an alternative choice for studies of short-range correlations and high density
nuclear matter.
