  This paper presents one of the case studies of the Gamma Factory initiative
-- a proposal of a new operation scheme of ion beams in the CERN accelerator
complex. Its goal is to extend the scope and precision of the LHC-based
research by complementing the proton-proton collision programme with the
high-luminosity nucleus-nucleus one. Its numerous physics highlights include
studies of the exclusive Higgs-boson production in photon-photon collisions and
precision measurements of the electroweak (EW) parameters. There are two
principal ways to increase the LHC luminosity which do not require an upgrade
of the CERN injectors: (1) modification of the beam-collision optics and (2)
reduction of the transverse emittance of the colliding beams. The former scheme
is employed by the ongoing high-luminosity (HL-LHC) project. The latter one,
applicable only to ion beams, is proposed in this paper. It is based on laser
cooling of bunches of partially stripped ions at the SPS flat-top energy. For
isoscalar calcium beams, which fulfil the present beam-operation constrains and
which are particularly attractive for the EW physics, the transverse beam
emittance can be reduced by a factor of $5$ within the $8$ seconds long cooling
phase. The predicted nucleon-nucleon luminosity of $L_{NN}= 4.2 \times
10^{34}\,$s$^{-1}$cm$^{-2}$ for collisions of the cooled calcium beams at the
LHC top energy is comparable to the levelled luminosity for the HL-LHC
proton-proton collisions, but with reduced pile-up background. The scheme
proposed in this paper, if confirmed by the future Gamma Factory
proof-of-principle experiment, could be implemented at CERN with minor
infrastructure investments.
