Physics:
The LHeC is designed to have a factor of 10-20 higher cms energy (s=4EeEp) and a factor of nearly 1000 higher luminosity (L near to 1 ab-1) than HERA. Therefore, the LHeC extends the kinematic range accessed with HERA from a maximum momentum transfer squared, Q2, of about 0.03 TeV2 to above 1 and from a maximum Bjorken x of about 0.6 to 0.9. Furthermore, the low x range extends down to 10-6. Main topics of the resulting physics program and their characteristic Q2-x values are illustrated in the plot below with highlights such as precision investigation of the Higgs boson or the clarification of the (non)existence of gluon saturation at low x. Details can be found in the 2012 CDR and its 2020 update.

The main physics deliverables of ep/eA collisions at the LHeC can be summarised in the following points:
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As the highest resolution microscope: discovery in QCD.
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Empowering the LHC/FCC programmes.
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Precision Higgs facility together with the HL-LHC/FCC-hh.
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Precision and discovery facility (top, EW, BSM).
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Unique nuclear physics facility.
Its contributions to the HL-LHC and FCC prgrammes are:
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Improving SM measurements.
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Searches for BSM.
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Flavor physics of heavy quarks and leptons.
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Higgs properties.
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QCD at high density/temperature.
Accelerator:
The LHeC could be realised either as a ring-ring (with a new lepton ring in the LHC tunnel) or as a linac-ring (based on a superconducting electron linac, configured as a recirculator) collider. In the CDR optics and beam dynamics studies are presented for both versions, along with technical design considerations on the interaction region, magnets including new dipole prototypes, cryogenics, RF, and further components. After careful consideration of installation issues and parameters of the electron beam, the linac-ring option has been chosen for the next phase of design as this is rather independent of the LHC.
The accelerator complex is of racetrack shape. A 500-MeV electron bunch coming from the injector is accelerated in each of the two 10-GeV SC linacs during three revolutions, after which it has obtained an energy of 60 GeV. The 60-GeV beam is focused and collided with the proton beam. It is then bent by 180° in the highest-energy arc beam line before it is sent back through the first linac, at a decelerating RF phase. After three revolutions with deceleration, re-converting the energy stored in the beam to RF energy, the beam energy is back at its original value of 500 MeV, and the beam is now disposed in a low-power 3.2-MW beam dump. A second, smaller (tune-up) dump could be installed behind the first linac.
Strictly speaking, with an injection energy into the first linac of 0.5 GeV, the energy gain in the two accelerating linacs need not be 10 GeV each, but about 9.92 GeV, in order to reach 60 GeV after three passages through each linac. Considering a rough value of 10 GeV means that we overestimate the electrical power required by about 1%. Each arc contains three separate beam lines at energies of 10, 30 and 50 GeV on one side, and 20, 40 and 60 GeV on the other. Except for the highest energy level of 60 GeV, at which there is only one beam, in each of the other arc beam lines there always co-exist a decelerating and an accelerating beam. The effective arc radius of curvature is 1 km, with a dipole bending radius of 764 m. The ERL configuration is depicted in the figure below.

While 60 GeV electrons is the default choice for the FCC-eh, for the LHeC cost studies led to considering smaller energies and racetrack sizes matching fraction of the LHC length. A compromise between cost and energy for the physics program led to choose as default option an electron energy of 50 GeV, 1/5 (around 5.4 km) of the LHC length, to be compared with 1/3 (around 9 km) for 60 GeV electrons, although the larger 1/3 length is also under consideration to lower energy consumption.
Detector:
The current default LHeC detector design is shown in the picture below. In this arrangement, the more energetic proton beam comes from the right and the electron beam from the left, with a dipole between the EMCAL and the HCAL to bend the electrons for head-on collisions. Due to the large energy imbalance there is a much denser and more energetic particle flow in the proton beam direction (forward), which determines the difference in inner tracking and forward/backward calorimetry. The detector as sketched here is about 14m long in z direction, parallel to the beam axis, and it has an approximate outer radius of 4.5m, which is achieved with a solenoid field of 3.5T. The detector components have presently been based on rather classic techniques such as Silicon pixel and strip tracking and Liquid Argon electromagnetic calorimetry, for example, but tracking based on cutting edge technologies like monolithic Si pixel sensors, are under consideration. The detector needs to be complemented by tagging devices in forward direction, for protons and neutrons, and in backward direction, for photons and low momentum transfer electrons.

More recently, it has been realised that a joint interaction region and detector able to study both ep/eA and pp/pA/AA collisions could be realised.
History:
The idea of an electron-proton (ep) collider in the LEP-LHC tunnel was discussed as early as 1984, at the first LHC workshop at Lausanne. This was the time when the first ever built ep collider, HERA, was approved by the German government. HERA was a machine of about 30 GeV electron beam energy and nearly 1 TeV proton beam energy, a combination of a warm dipole electron ring with a superconducting dipole proton ring, in a 6 km circumference tunnel. The machine started operation 8 years after its approval. It reached luminosities of 1031 cm-2s-1 in first phase of operation which were increased by about a factor of 4 in the subsequent, upgraded configuration. HERA never attempted to collide electrons with deuterons nor with ions.
The realisation of HERA at DESY had followed a number of attempts to realise ep interactions in collider mode: since the late 1960s, such machines had been considered and proposed to probe the proton's structure more deeply with an ep collider at DORIS, later at PETRA (PROPER) and subsequently at the SPS at CERN (CHEEP). Further ep collider studies were made for PEP, TRISTAN and also the Tevatron (CHEER).
In 1990, at a workshop at Aachen, the combination of LEP with the LHC was discussed, with studies on the luminosity, interaction region, a detector and the physics as seen with the knowledge of that time, before HERA. Following a request of the CERN Science Policy Committee (SPC), a brief study of the ring-ring ep collider in the LEP tunnel was performed leading to an estimated luminosity of about 1032 cm-2s-1 .
At the end of the eighties it had been anticipated that there was a possible end to the increase of the energy of ep colliders in the ring-ring configuration, because of the synchrotron radiation losses of an electron ring accelerator. The classic SLAC fixed target ep experiment had already used a 2 mile linac. For ep linac-ring collider configurations, two design sketches considering electron beam energies up to a few hundred GeV were published, in 1988 and in 1990. As part of the TESLA linear collider proposal, an option (THERA) was studied to collide electrons of a few hundred GeV energy with protons and ions from HERA. Later, in 2003, the possibility was evaluated to combine LHC protons with CLIC electrons. It was yet realised, that the bunch structures of the LHC and CLIC were not compliant with the need for high luminosities.
In September 2007, the SPC again asked whether one could realise an ep collider at CERN. Work in the year before had shown in detail, for the first time, that a luminosity of 1033 cm-2s-1 was achievable. This appeared possible in a ring-ring configuration based on the "ultimate" LHC beam, with 1.7 x 1011 protons in bunches 25 ns apart. Thanks to the small beam-beam tune-shift, it was found to be feasible to simultaneously operate pp in the LHC and ep in the new machine, which in 2005 was termed the Large Hadron Electron Collider (LHeC). Thus it appeared possible to realise an ep collider that was complementary to the LHC, just as HERA was to the Tevatron. The integrated luminosity was projected to be O(100) fb-1, a factor of a hundred more than HERA had collected over its lifetime of 15 years.
In the autumn of 2007, (r)ECFA and CERN invited to work out the LHeC concept to a degree, which would allow one to understand its physics programme, evaluate the accelerator options and their technical realisation. The detector design should be affordable and capable of realising a high precision, large acceptance experimental programme of deep inelastic scattering at the energy frontier. The electron beam energy range was set to be between about 50 - 150 GeV. The wall plug power consumed for the electron beam was limited to 100 MW.
For the installation of the LHC it had been decided to remove LEP from the tunnel and to re-use the injector chain. To realise an ep collider based on the LHC, a new electron accelerator has to be built. The following report details two solutions for the chosen default electron beam energy of Ee = 60 GeV. One option is to build and install a new ring, with modern magnet technology, on top of the LHC, using a new 10 GeV injector. Alternatively, one can build a "linac", actually two 10 GeV superconducting linacs in a racetrack configuration. By employing energy recovery techniques, this configuration could provide the equivalent of about 1 GW available power and reach 1033 cm-2s-1 luminosity. The LHeC linac would be of about the same length as the one used for the discovery of quarks at SLAC, but capable of probing parton interactions with a Q2 exceeding that of the 1969 machine by a factor of nearly 105.
It was agreed early on to devote a few years to the report, also because none of the people involved could work anything near to full time for this endeavour. Three workshops were held in 2008-2010, that annually assembled about a hundred experts on theory, experiment and accelerator to develop the LHeC design concepts. The project was presented annually to ECFA and in 2008 to ICFA. In view of the unique electron-ion scattering programme of the LHeC, the design became also supported by NuPECC, and the LHeC was part of the NuPECC roadmap for European nuclear physics as released in 2010. Following an intermediate report to the Science Policy Committee of CERN, in July 2010, the SPC considered the LHeC "an option for a future project at CERN".
In August 2011, a first complete draft of the conceptual design report was handed to more than twenty experts on various aspects of the physics and technology of the LHeC, which CERN had invited to referee the project and scrutinise its motivation and its design. The report has been completed following often close interactions with the referees and due consideration of their observations. The CDR was submitted for publication to Journal of Physics G prior to the discovery of the Higgs boson announced on July 4, actually on June 13, the day before the 2012 LHeC workshop began. The Higgs boson is produced dominantly in charged current scattering, radiated thus from the W (or Z in neutral currents). Its properties can be very well measured in ep. This was much studied in the design report but it is interesting that it did not make it to the prefaces of the report which are reproduced here as documents describing the past and a future of energy frontier deep inelastic scattering.
After the 2013 update of the European Strategy for Particle Physics, where the realisation of the HL-LHC was established as the top priority, and the proposal of the Future Circular Collider in 2014, this same year a new mandate to continue the studies for ep/eA collisions at CERN was given. Integral part of it was the study of the possibilities of Higgs studies, which demanded an increased luminosity 1034 cm-2s-1. This, together with demands on the minimal disruption for LHC operation and cost reduction compatible with the physics programme, led to the ERL racetrack configuration for 50 GeV electrons and positrons as the default choice for the LHeC (also for the FCC-eh with 60 GeV energy electrons). Also requirements for Higgs measurements led to several constraints on the detector. From all these considerations the development of an update of CDR followed, which was sent to arxiv on July 2020 and published the following year in Journal of Physics G.
After the 2020 update of the European Strategy for Particle Physics, where the explotation of the HL-LHC and the studies for an e+e- Higgs facility were set as top priorities, a new mandate was received from the CERN DG and DRC in October 2022 to further study the LHeC and the FCC-eh as potential options for the future and to provide input to the next Update of the European Strategy for Particle Physics, to happen in 2026. It was also realised in 2021 that a joint ep/pp detector and the possibility of both ep and pp (eA and AA) collision modes in the the same IP could be an option of interest for both the LHeC and the FCC-eh.