  This Conceptual Design Report describes LUXE (Laser Und XFEL Experiment), an
experimental campaign that aims to combine the high-quality and high-energy
electron beam of the European XFEL with a powerful laser to explore the
uncharted terrain of quantum electrodynamics characterised by both high energy
and high intensity. We will reach this hitherto inaccessible regime of quantum
physics by analysing high-energy electron-photon and photon-photon interactions
in the extreme environment provided by an intense laser focus. The physics
background and its relevance are presented in the science case which in turn
leads to, and justifies, the ensuing plan for all aspects of the experiment:
Our choice of experimental parameters allows (i) effective field strengths to
be probed at and beyond the Schwinger limit and (ii) a precision to be achieved
that permits a detailed comparison of the measured data with calculations. In
addition, the high photon flux predicted will enable a sensitive search for new
physics beyond the Standard Model. The initial phase of the experiment will
employ an existing 40 TW laser, whereas the second phase will utilise an
upgraded laser power of 350 TW. All expectations regarding the performance of
the experimental set-up as well as the expected physics results are based on
detailed numerical simulations throughout.
