5σ seems like an unnecessarily high standard to this non-physicist; what's the rationale for that? At 5σ we could publish 1,000 major discoveries a year and have E[false discovery accepted] = 1,740 years.
Also more generally known as the "multiple testing" problem, fwiw (not sure why it has a different name in physics, unless I'm missing a subtlety).
It's a major problem in "big data" also, where people just data-dredge thousands of possible parameter choices and pairwise correlations, and then report the p<0.01 results that came up, even though you'd expect several false positives just by chance with that methodology.
I'm kind of joking - most other scientists don't collect enough data to have to worry about 1 in 10,000 events happening by chance. In medicine, though, I'm not joking at all - those guys publish absolute statistical garbage all the time, I hesitate to even consider it a science because the data dredging is so bad. I can "prove" just about anything if the publication standard is 95% significance...