I Focused proton beam waist diameter

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The minimal waist diameter is roughly determined by emittance (size of the bunches in phase space) and the spread in flight direction at the smallest point.

The emittance is given by the proton source - how uniform the protons are in position and velocity. After that the emittance can only grow, at best you keep it nearly the same the whole time.
Electron-positron accelerators can reduce their emittance via synchrotron radiation, some proton accelerators can use electron cooling or stochastic cooling, but these would be impractical at the LHC.

A larger spread in flight direction can lead to a smaller spread in position, but the beams still have to fit into the beam pipe - to get a large spread inside the experiments but not in the rest of the accelerator you need strong focusing magnets very close to the interaction point and there are simply limits how much you can do there.
 
Electron cooling seems useful but won't there be significant losses due to the recombination of electrons, or is there a way of increasing the probability of electron scattering such that recombination losses become negligible?
 
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You lose some protons that form hydrogen atoms but there are conditions where that rate is small.
With antiprotons you don't have that mechanism, of course.
 

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