The abstract you refer to seems not to actually be a real paper. It excited the blogosphere last april, but from what I can see it is not actually a real result. A CERN spokesperson talking to Nature pretty much quashed the rumour:
ATLAS’ spokeswoman Fabiola Gianotti stops short of disowning the leaked document, but tells Nature signals of the kind reported in the memo show up quite frequently in the course of data analysis and are later falsified after more detailed scrutiny. “Only official ATLAS results, i.e. results that have undergone all the necessary scientific checks by the Collaboration, should be taken seriously,” she says.
A $4\sigma$ result would certainly be something to be taken seriously, though still short of the $5\sigma$ level which is usually taken as a discovery. However, most of the bumps that appear tend to be more around the $2\sigma$ range, and with good reason. These kind of events are very rare, and it can take many years to build up enough recorded events to sufficiently decrease the sampling error. The reason for these kind of bumps often turns out to be sampling error. While naively one would think that a $2\sigma$ event is already a rare event and so should signal something, you have to keep in mind that they are searching across a large range, and so some small bumps are to be expected. These don't necessarily correspond to anything other than sampling error.
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