Possible quadrupole-first options with $\beta^* \leq$ 0.25M

A global model of an LHC insertion was prepared for the Arcidosso workshop. Its objective is to evaluate in a consistent way the potential of a class of solutions for the LHC low-β upgrades and help identifying what should be the R&D lines. The solutions studied are those of the present LHC baseline layout, namely quadrupole first and small crossing angle. Provision for dipole first solutions is included. The model tackles to the magnet layout, the beam optics, the beam-beam effect, the superconductor capabilities and the peak heat deposition in the coils. The approach is simplified to allow a gain by 6 orders of magnitude in the design/calculation time. The accuracy should be sufficient for the identification of the important features of the parameter space. The main results are: 1) a required quadrupole inner coil aperture larger than previously assumed, 2) the necessity to find simultaneously solutions for the upgrade of the triplet and for the minimization of the luminosity geometrical loss factor 3) a net global advantage of shifting the triplet by 4 meters towards the collision point, 4) the requirement to use the performance of the Nb$_{3}$Sn technology (or equivalent) to increase significantly the LHC luminosity. To contribute to 2), a new approach is proposed to reduce or suppress the consequence of the luminosity geometrical loss factor: the principle is to install an early separation scheme made of two small dipoles inside the detector as close as possible to the crossing point.