  While collisionless cold dark matter models have been largely successful in
explaining a wide range of observational data, some tensions still exist, and
it remains possible that dark matter possesses a non-negligible level of self
interactions. In this paper, we investigate a possible observable consequence
of self-interacting dark matter: offsets between the central galaxy and the
center of mass of its parent halo. We examine 23 relaxed galaxy clusters in a
redshift range of 0.1 to 0.3 drawn from clusters in the Dark Energy Survey and
the Sloan Digital Sky Survey which have archival Chandra X-ray data of
sufficient depth for center and relaxation determination. We find that most
clusters in our sample show non-zero offsets between the X-ray center, taken to
be the centroid within the cluster core, and the central galaxy position. All
of the measured offsets are larger, typically by an order of magnitude, than
the uncertainty in the X-ray position due to Poisson noise. In all but six
clusters, the measured offsets are also larger than the estimated, combined
astrometric uncertainties in the X-ray and optical positions. A more
conservative cut on concentration to select relaxed clusters marginally reduces
but does not eliminate the observed offset. With our more conservative sample,
we find an estimated mean X-ray to central galaxy offset of $\mu = 5.5 \pm 1.0$
kpc. Comparing to recent simulations, this distribution of offsets is
consistent with some level of dark matter self interaction, though further
simulation work is needed to place constraints.
