Help interpreting "five sigma" standard? So, I am coming from a math/stats background.  I was just wondering about this in the abstract, and tried Googling around, and came across this article which says the following regarding some experiments undertaken at CERN:

it is the probability that if the particle does not exist, the data that CERN scientists collected in Geneva, Switzerland, would be at least as extreme as what they observed.

But, "does not exist" doesn't seem to me to be a very well-defined hypothesis to test:
In my understanding of frequentist hypothesis testing, tests are always designed with the intent to provide evidence against a particular hypothesis, in a very Popperian sort of epistemology.  It just so happens that in a lot of the toy examples used in stats classes, and also in many real-life instances, the negation of the hypothesis one sets out to prove wrong is itself an interesting hypothesis.  E.g. ACME corp hypothesizes that their ACME brand bird seed will attract >90% of roadrunners passing within 5m of a box of it.  W.E. Coyote hypothesizes the negation.  Either can set about gathering data to provide evidence against the hypothesis of the other, and because the hypotheses are logical negations of one another, evidence against ACME is evidence for W.E.C. and vice versa.
In the quote above, they attempt to frame one hypothesis as "yes Higgs' Boson" and it's negation as "no Higgs' Boson". It seems that if the intent is to provide evidence for "yes Higgs' Boson", then in normal frequentist methodology,  one gathers evidence against "no Higgs' Boson" and can quantify that evidence into a p-value or just a number of standard errors of whatever quantity predicted by the theory we happen to be investigating.  But this seems to me to be silly, since the negation of the physical model that includes the Higgs' is an infinite space of models.  OTOH, this is the only context in which the "five sigma" p-value surrogate seems to make any sense.
In fact, this was my original thought when I set out Googling: the five sigma standard implies that we are gathering evidence against something, but modern physics theories seem to encompass such a breadth, and are yet so specific, that gathering evidence against their bare negation is nonsense.
What am I missing here?  What does "five sigma" evidence for the Higgs hypothesis (or other physics hypotheses) mean in this context?
 A: The Higgs-discovery experiment is a particle-counting experiment. Lots of particles are produced by collisions in the accelerator, and appear in its various detectors. Information about those particles is stored for later: when they appeared, the direction they were traveling, their kinetic energy, their charge, what other particles appeared elsewhere in the detector at the same time. Then you can reconstruct “events,” group them in different ways, and look at them in a histogram, like this one:

[Mea culpa: I remember this image, and others like it, from the Higgs discovery announcement, but I found it from an image search and I don’t have a proper source link.]
These are simultaneous detections of two photons (“diphotons”), grouped by the “equivalent mass” $m_{\gamma\gamma}$ of the pair. There are tons and tons and tons of photons rattling around these collisions, and directional tracking for photons is not very good, so most of these “pairs” are just random coincidences, unrelated photons that happened to reach different parts of the detector at the same time.
Because each collision is independent of all the others, the filling of each little bin is subject to Poisson statistics: a bin with $N$ events in it has an intrinsic “one-sigma” statistical uncertainty of $\pm\sqrt N$.  You can see the error bars in the total-minus-fit plot in the bottom panel: on the left side, where $N\approx 6000$ events per interval in the top figure, the error bars are roughly $\sqrt{6000}\approx 80$ events; on the right side, where there is less signal, the error bars are appropriately smaller.
The “one-sigma” confidence limit is 68%. Therefore, if those data were really independently generated by a Poissonian process whose average behavior were described by the fit line, you would expect the data points to be equally distributed above and below the fit, with about 68% of the error bars crossing the fit line. The other third-ish will miss the fit line, just from ordinary noise.
In this plot we have thirty points, and about ten of them have error bars that don’t cross the fit line: totally reasonable.
On average one point in twenty should be, randomly, two or more error bars away from the prediction (or, “two sigma” corresponds to a 95% confidence limit).
There are two remarkable bins in this histogram, centered on 125 GeV and 127 GeV, which are different from the background fit by (reading by eye) approximately $180\pm60$ and $260\pm60$ events.  The “null hypothesis” is that these two differences, roughly $3\sigma$ and $4\sigma$, are both statistical flukes, just like the low bin at 143 GeV is probably a statistical fluke.
You can see that this null hypothesis is strongly disfavored, relative to the hypothesis that “in some collisions, an object with mass near 125 GeV decays into two photons.”
This diphoton plot by itself doesn’t get you to a five-sigma discovery: that required data in multiple different Higgs decay channels, combined from both of the big CERN experiments, which required a great deal of statistical sophistication.  An important part of the discovery was combining the data from all channels to determine the best estimate for the Higgs’s mass, charge, and spin.  Another important result out of the discover was the relative intensities of the different decay modes.  As another answer says, it helped a lot that we already had a prediction there might be a particle with this mass. But I think this data set shows the null hypothesis nicely: most of ATLAS’s photon pairs come from a well-defined continuum background of accidental coincidences, and the null hypothesis is that there’s nothing special about any of the photon pairs which happen to have an equivalent mass of 125 GeV.
A: I think this question may arise from a difference between somewhat rough layman's-terms presentations and the more careful statistics which goes on in the actual labs. But even after a given body of data has been analyzed to death, there is no formal way to capture in full the evidence underlying the way knowledge of physics grows. The evidence surrounding the Higgs mechanism, for example, would not be nearly as convincing if the Higgs mechanism itself were not an elegant combination of ideas which already find their place in a coherent whole.
The hypothesis that one is gathering evidence against is always the hypothesis that we are mistaken as to how a given body of data (such as a peak in a spectrum) came about. The mistake could be quite simple, as for example when in fact the underlying distribution is flat and the peak is an artifact of random noise. But usually one has to consider the possibility that the peak is there but is owing to something else than the mechanism under study.
The hypothesis one is testing in the strict sense---the sense of ruling out at some level of confidence---is the set of all other ways we have thought of yet as to how the data could arise. In this set of ways we only need to consider ways that reflect known physics and known amounts of noise etc. in the apparatus.
I think what the community of physicists do is a bit like Sherlock Holmes: we try to think of plausible other ways the data could arise, and then give reasons as to why those other ways can be ruled out. The final step, where we proceed to the claim that the leading candidate explanation is what really happened, is not a step that can be quantified by any statistical measure. This is because it relies not only on a given data set, but also on a judgement about the quality of the theory under consideration.
A: The null hypothesis here is that the data was generated by physics which obeys the effective field theory describing all the Standard Model particles except the Higgs.  This model doesn't usually have a name, but could reasonably be called the 'Standard Model without Higgs'.  It's a perfectly good effective field theory.  It's predictions are barely different from the usual Standard Model (with Higgs).
Asking for a 5 sigma rejection of the null hypothesis in this case means accumulating a lot of data which is incompatible with the 'Standard Model without Higgs".  Enough data that a couple of 1 sigma experimental errors don't ruin the result.
A: Refresher on hypothesis testing
In (frequentist) hypothesis testing one always have (at least) two hypotheses: the null hypothesis, and the alternative hypothesis. Then p-value is the probability of observing certain dataset, given that the null hypothesis is true, whereas the power of the test is the probability that the alternative hypothesis is true, given the observed data.
If p-value is smaller than a pre-defined threshold (significance level), one rejects the null hypothesis as improbable. In the example given in the OP the data is supposed to follow the Gaussian/normal distribution, and five sigma determines the significance level in terms of this distribution (a rather stringent one).
What does it have to do with Popper?
From statistical viewpoint, Popperian epistemiology simply means that designing a test to reject a hypothesis and calculating its p-value is usually easier than calculating the test power (which typically requires some ad-hoc assumptions about the underlying probability distributions). In other words, disproving the null hypothesis is easier than proving that the alternative hypothesis is correct. One then chooses the null hypothesis in such a way that it can be disproved, rather than trying to prove it. Choosing whether the particle exists as the null hypothesis and particle does not exist as the alternative one, or vice versa, depends not on the philosophical meaning of either statement, but on our ability to disprove it.
Remark
In my opinion the chapter on statistical testing published by the Partciel Data Group is one of the best crash courses on statistics for physicists.
