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The Large Hadron–Electron Collider (LHeC) is designed to move the field of deep inelastic scattering (DIS) to the energy and intensity frontier of particle physics. Exploiting energy-recovery technology, it collides a novel, intense electron beam with a proton or ion beam from the High-Luminosity Large Hadron Collider (HL-LHC). The accelerator and interaction region are designed for concurrent electron–proton and proton–proton operations. This report represents an update to the LHeC’s conceptual design report (CDR), published in 2012. It comprises new results on the parton structure of the proton and heavier nuclei, QCD dynamics, and electroweak and top-quark physics. It is shown how the LHeC will open a new chapter of nuclear particle physics by extending the accessible kinematic range of lepton–nucleus scattering by several orders of magnitude. Due to its enhanced luminosity and large energy and the cleanliness of the final hadronic states, the LHeC has a strong Higgs physics programme and its own discovery potential for new physics. Building on the 2012 CDR, this report contains a detailed updated design for the energy-recovery electron linac (ERL), including a new lattice, magnet and superconducting radio-frequency technology, and further components. Challenges of energy recovery are described, and the lower-energy, high-current, three-turn ERL facility, PERLE at Orsay, is presented, which uses the LHeC characteristics serving as a development facility for the design and operation of the LHeC. An updated detector design is presented corresponding to the acceptance, resolution, and calibration goals that arise from the Higgs and parton-density-function physics programmes. This paper also presents novel results for the Future Circular Collider in electron–hadron (FCC-eh) mode, which utilises the same ERL technology to further extend the reach of DIS to even higher centre-of-mass energies.

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We present accurate QCD predictions for the transverse momentum (p ⊥ ) spectrum of electroweak gauge bosons at the LHC for 13 TeV collisions, based on a consistent combination of a NNLO calculation at large p ⊥ and N 3 LL resummation in the small p ⊥ limit. The inclusion of higher order corrections leads to substantial changes in the shape of the differential distributions, and the residual perturbative uncertainties are reduced to the few percent level across the whole transverse momentum spectrum. We examine the ratio of p ⊥ distributions in charged-and neutral-current Drell-Yan production, and study different prescriptions for the estimate of perturbative uncertainties that rely on different degrees of correlation between these processes. We observe an excellent stability of the ratios with respect to the perturbative order, indicating a strong correlation between the corresponding QCD corrections.

The transverse momentum spectra of weak gauge bosons and their ratios probe the underlying dynamics and are crucial in testing our understanding of the standard model. They are an essential ingredient in precision measurements, such as the W boson mass extraction. To fully exploit the potential of the LHC data, we compute the second-order [next-to-next-to-leading-order (NNLO)] QCD corrections to the inclusive-p W T spectrum as well as to the ratios of spectra for W − =W þ and Z=W. We find that the inclusion of NNLO QCD corrections considerably improves the theoretical description of the experimental CMS data and results in a substantial reduction of the residual scale uncertainties.