With estimates of mass constraints on magnetic monopoles, how likely is one to be found by the LHC (MoEDAL)? Fermilab seems to have ruled out monopoles with mass less than 850 GeV, but I have seen some estimates of the mass thought to be in the order of up to $10^{18}$ GeV, which, of course, would make them undetectable in any accelerators. By 2013, the LHC is scheduled to reach up to 14 TeV. The only disputed sighting of a cosmic ray produced magnetic monopole was in 1982 when Blas Cabrera reported discovering one (Valentine event monopole). This has never been duplicated. CERN has set up the Monopole and Exotics Detector MOEDAL. Other experiments set up to detect them are the Antarctic Impulsive Transient Antenna--ANITA, and the Antarctic Muon And Neutrino Detection Array, aka AMANDA. While both of these detected neutrinos, neither detected magnetic monopoles (looking for Cerenkov radiation produced by products of monopole interaction).
Another method of detection is to look for induced current in a superconducting ring when a monopole passes through.
Joseph Polchinski called the existence of monopoles “one of the safest bets that one can make about physics not yet seen.” This shows some optimism on his part that one will be seen. Is there any reason or way to limit the mass of the monopole on the upside to 14 TEV or less?
 A: Dear Gordon, Polchinski didn't imply anything about the lightness of the monopoles. You are overinterpreting what he said: his statement was an in-principle statement by a theorist. Just to be sure, the MoEDAL collaboration has overinterpreted him, too: he has never suggested that an experiment of their kind will find the beast. Still, it's very important to know whether such qualitatively different particles exist even if we think it's very likely that they won't be directly detected in the near (or far) future. Most of the top contemporary state-of-the-art theoretical physics deals with objects that will probably never be detected by direct experiments, for obvious financial and technological limitations.
Directly experimentally, the lower bound on the mass is just 500 GeV or so, but when some theory is wisely added, the mass is pretty much required to be exponentially higher than the TeV scale. In particular, some kind of unification seems necessary to allow the magnetic monopoles, and unification such as GUT can only appear at very high scales, such as the GUT scale of $10^{16}$ GeV: that's where the electromagnetic $U(1)$ may be embedded in a larger group that allows a topologically nontrivial solution (the $SU(2)$ of the weak interactions is not enough - it doesn't contain the hypercharge which has to be twisted as well). So it's unlikely that direct detection succeeds. Inflation has probably diluted the concentration of those massive magnetic monopoles to a very tiny number.
