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### Life at the limit

At the moment anyone who has even a passing interest in particle physics is thinking about the results being presented that European Physical Society High Energy Physics 2011 conference (referred to as simply “EPS”.) It is at EPS that we hear news about the search the Higgs boson, and the news is tantalizing! The talks are all publicly available, and to understand them fully you need to know a little bit about how limits work.

## What is a limit?

A limit is an upper or lower bound for a physical quantity, and we place limits when we don’t have enough information to estimate the value accurately or precisely. When we say something like “The lower limit for the mass of the Higgs boson is 114GeV” what we mean is that given the data we have had access to we can be confident that the mass of the Higgs boson is at least 114GeV.

One of the most interesting parts of the search for the Higgs is that we have several limits at the moment, so one experiment might say “The mass is at least 114GeV” and another experiment will say “It’s not between 155 and 190GeV”, and experts in electroweak physics will say “Don’t bother to look above 300GeV”. Each mass region requires a slightly different search, and that’s why some limits appear before others.

## Confidence problems

Like anyone else, physicists have issues with “confidence”. To a physicist, “confidence” means the extent to which they trust a measurement, so it’s an important concept to get right! Our data are statistically limited, so we can never be 100% certain in any of our measurements. What we usually do is say something like “We’re 95% certain that the Higgs mass is not in the region 157-174GeV”. To understand what that really means you need to think backwards. We’ve got some data and the probability that we would get this data, given that the Higgs mass in the region 157-174GeV is 5%, or 1 in 20.

You can probably see why this gives us confidence problems… if we have 20 data points that show us measurements with 95% confidence then we expect 1 of them to be incorrect. As we look across one of our plots we can see lots of data points (generally, every time there’s a kink in the plot there’s another data point.) How do we know when we’ve got it right? The answer is that we don’t know, and the fluctuations can take the distributions up and down. At first this seems like a minor irritation, but it has serious implications.

A lot of the talks at EPS contain plots like this:

ATLAS limit on Higgs mass (K. Cranmer, on behalf of ATLAS. EPS HEP 2011)

They look pretty, but they don’t look simple. The green and yellow bands show us the expected confidence bounds for some number, and that’s what we should look at first to get a feeling for what the plot is telling us. The line at the center of these bands shows us the expected limit. The “Observed” line shows us what we actually see in the experiment. If the “Observed” line stays within the bands then our expectations are about right.

The y-axis shows the production cross section of the Higgs boson, multiplied by the branching fraction to the final state, and some other factors. These numbers all vary as the mass of the Higgs boson, which is one of the reasons why the graphs look so wiggly. The exciting part is the horizontal line at 1. This is the line where we would expect to see the Higgs boson being produced. If the “Observed” line crosses the line at 1 then we can conclude that the Higgs boson probably does not exist at that mass, because our limit is already at 1 times the Standard Model. As the upper yellow band passes under the line at 1 we can be almost certain that the Higgs doesn’t exist there. (Remember the definition of the confidence: “At this mass point, we’re 95% sure that the Higgs production cross section is less than what the Standard Model predicts.”)

## What the plots tell us

Exciting things start to happen when the limits change! As we gather more data the limits improve and we exclude more mass points. On the plot, we would see the green and yellow bands move down. If the Higgs doesn’t exist in a particular mass region then the “Observed” line would move down as well. But, if we see the bands move down and the “Observed” line get left behind then that’s a hint that the Higgs boson mass is in that region!

This is cause for major excitement for some physicists and skepticism for others. Remember the confidence problem of fluctuations and you can see that this kind of fluctuation would happen very often. When does a “fluctuation” turn into “evidence”? It’s a topic that’s not very well defined, but we’ve chosen to say three standard deviations (imagine a third colored band on the plot) is a good indication of evidence, and five standard deviations (a veritable rainbow of confidence!) is proof of new physics. When we see a fluctuation the answer is to add more data and see if it remains. If it stays there while the bands move down around it then there’s probably a particle there.

## LEP? What is LEP?

In these talks you’ll often see “LEP” on the plots. This refers to a collection of four high-precision experiments that operated at CERN before the era of the LHC. LEP was an electron-positron collider and the LHC now occupies the tunnels where LEP were. The four experiments (ALEPH, Delphi, L3 and Opal) searched for the Higgs boson directly via a process known as Higgsstrahlung. They excluded the Higgs boson mass up to 114GeV, and their contribution to the hunt, although it looks small on these plots, is certainly substantial. (In fact, you can see how the limits at the LHC experiments get poorer and poorer at lower masses. The LEP experiments were better suited to these regions by design, and by design the LHC experiments are better at higher mass regions.)

The Higgs boson is just one of many particles, and since all the other particles interact with each other we expect the Higgs boson to interact as well. If it does it will affect all kinds of processes, most notably the electroweak processes, which are finely constrained. This allows us to identify the most likely place where the Higgs boson would be found, so if you see anything about electroweak fits and exclusions it’s usually the result of fitting a huge number of electroweak parameters in an attempt to limit the mass of the Higgs boson.