The LHC Safety Assessment Group, which prepared the report, has provided both a fully referenced analysis and a version targeted for the general public (both PDFs). Those looking for a really condensed reason not to worry can find reassurance in the fact that, "each collision of a pair of protons in the LHC will release an amount of energy comparable to that of two colliding mosquitoes."
Those looking for more sophisticated reasoning will find plenty of it. At its most basic level, the safety assessment boils down to a probability calculation. The collisions in the LHC will have an energy content that's equivalent to that of a cosmic ray hitting the earth with an energy of 1017eV. We've measured cosmic rays hitting the earth with energies of up to 1020eV, meaning that we're really doing nothing new.
Of course, it's possible that the frequency of collisions in the LHC is so high that we might unmask an extremely low-probability event. Based on the frequency of high energy cosmic rays, however, the authors calculate that, "nature has already conducted the equivalent of about a hundred thousand LHC experimental programmes on Earth already—and the planet still exists." Those numbers go up when one considers the sun, which is a much larger target for cosmic rays. Backing out even further, to the entire visible universe, the calculations suggest that every second, there are the equivalent of 3 x 1013 times the total number of collisions expected over the lifetime of the LHC. If any of these resulted in the sort of cataclysms people are worrying about, we'd be seeing the explosions.
The LHC's collider ring covers several miles near the Swiss Alps
The report also looks specifically at some of the exotic species that are proposed for bringing about global destruction. Patches of lower vacuum energy are ruled out by the cosmic ray analysis, as, were they possible, "bubbles of new vacuum would have expanded to consume large parts of the visible universe several billion years ago already." Magnetic monopoles have also been mentioned, but calculations show that the dangerous ones are too heavy to be created by the LHC; anything light enough to be within reach of the collider would only destroy a total of one microgram of matter before leaving the earth.
Microscopic black holes are another popular source of doom, but the report lists a whole lot of issues with them. For starters, Relativity suggests that gravity is simply not potent enough to overcome all the other forces acting on particles during these collisions, and even if that weren't the case, Hawking radiation should cause any black holes of this size to evaporate immediately. The only possible exception comes through some forms of String Theory, but even those have problems. In any versions of the theory with greater than seven dimensions, the black holes would draw in matter so slowly that we'd have several billion years to worry about them; less than seven, and we would already have observed neutron stars and white dwarfs exploding.
One last problem comes from quantum mechanics. Any object formed through quantum effects can decay through exactly the same pathway, meaning that these black holes aren't actually black in the traditional sense; they should decay back to the particles that formed them quite rapidly. All in all, black holes seem to be pretty seriously ruled out.
Last up in the report's list of things not to lose sleep over are strangelets, atoms that, instead of the normal up and down quarks, contain some strange quarks. A strange quark outside of a nucleus has a half life on the order of nanoseconds, but there are some hypothetical strange atoms that might be relatively stable. In a hypothetical once-removed, these might convert other normal atoms to strange matter, setting off a chain reaction. Fortunately, there is no way to get this to actually work in a collider.
Strange quarks are only produced at high energies, but atomic nuclei can't form at these energies. The time it takes to cool down to the point where they could form is longer than the lifetime of a strange quark. As a result, "the likelihood of strangelet production in relativistic heavy-ion collisions can be compared to the likelihood of producing an icecube in a furnace." The continued inability of Brookhaven's Relativistic Heavy Ion Collider to bring about the end of the earth is cited as further evidence.
Overall, it's hard to read this report and not wind up viewing the apocalyptic fears as simply being poorly thought through. It was striking how clearly the worries over the LHC have parallels to the fears over biotechnology, which came up during our recent interview with Carl Zimmer. There too, billions of years of natural experiments and decades' worth of scientific experiment should be informing our view of safety; for at least some segment of the public, that's not happening.