  We present an initial design study for LDMX, the Light Dark Matter
Experiment, a small-scale accelerator experiment having broad sensitivity to
both direct dark matter and mediator particle production in the sub-GeV mass
region. LDMX employs missing momentum and energy techniques in multi-GeV
electro-nuclear fixed-target collisions to explore couplings to electrons in
uncharted regions that extend down to and below levels that are motivated by
direct thermal freeze-out mechanisms. LDMX would also be sensitive to a wide
range of visibly and invisibly decaying dark sector particles, thereby
addressing many of the science drivers highlighted in the 2017 US Cosmic
Visions New Ideas in Dark Matter Community Report. LDMX would achieve the
required sensitivity by leveraging existing and developing detector
technologies from the CMS, HPS and Mu2e experiments. In this paper, we present
our initial design concept, detailed GEANT-based studies of detector
performance, signal and background processes, and a preliminary analysis
approach. We demonstrate how a first phase of LDMX could expand sensitivity to
a variety of light dark matter, mediator, and millicharge particles by several
orders of magnitude in coupling over the broad sub-GeV mass range.
