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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.

Now that you have all you need to read these plots you can go check out the limits yourself! Here are the talks from the main experiments:

Electroweak fit

On an unrelated note, Happy Pi Approximation Day!

## Errata

This post initially incorrectly credited the plot to A. Baroncelli. This plot was presented at EPS by Kyle Cranmer. Apologies to Kyle!

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### 13 Responses to “Life at the limit”

1. John says:

Don’t you technically need to be using Bayesian credible intervals, not frequentist confidence intervals, to be able to say you are 95% confident that the value X is in a range?

2. The whole Bayesian vs frequentist debate is a bit of a holy war in particle physics. Personally I much prefer an honest Bayesian limit, (By honest I mean we state our results and how we got from them to the limit.) whereas quite a few of my colleagues favor frequentist limits. When I asked one of them how to obtain such a limit they said “Well you could assume you have three signal events in the sample and see what branching fraction you’d get”, to which I replied “huh?”. I suppose there are advantages and disadvantages to both methods. I think you’re right, but I’m not a statistics expert.

3. BongoBern says:

It’s such a shame that I can read the same lines over and over and not understand a word. I’ve never been good at learning from reading without a teacher to break it down. Might have something to do with my ADD. Anyway, green or yellow lines crossing 1 with 95% confidence may not mean anything to me, but I will be delighted when someone finds the Higgs. So carry on without me, but let us all know when you you are at least 95% sure you’ve found it!

Aidan, that is so funny: I consider frequentist limits honest, since they make for hypothesis tests. While bayesian methods makes for some of those hypotheses to test. (This is from biology, where the latter makes for good phylogenetic trees.)

I can see how you could think the latter substitutes for some of the former, but not all of it.

Btw, I don’t understand the first comment. Don’t that mean “95 % credible”? How can you replace confidence and still claim confidence?

5. Hi Torbjörn! Thanks for your comment- it’s a real thought provoker. When we have a high statistics sample and we don’t need to set a limit I definitely prefer the frequentist approach (I think everyone does.) When it comes to very low statistics things get very tricky. For example, suppose we expect to see 0.2 +/- 0.3 events in the background, and we see 1 event in the data, how can we make a frequestist statement about a limit? There are certainly ways to do it, but they all make me feel a little uneasy. At least with a Bayesian approach we come right and state our assumptions (preferably before we’ve looked at the data!)

The definition of a 95% confidence limit is usually along the lines of “The probability that these limits enclose the true value is 95%”. I suppose we shouldn’t be claiming confidence at all, and should remain dispassionate about what the data are telling us, but we’re stuck with the word “confidence”, so that’s what we have to use. If you want, you can replace “certain” with “confident” and “confidence” with “credibility” in my post if that makes you feel more comfortable about it. Or maybe I’ve missed your point. The semantics aren’t trivial!

6. Hi Bongo! If I get time I’ll see if I can explain it in a video post, but the coming week is going to be a very busy one…

7. John says:

Torbjörn:

The words ‘credible’ vs ‘confidence’ are not the issue here, as Aidan points out. Frequentist and Bayesian statistics have fundamentally different methods for creating intervals around datapoints to allow for inferences: Bayesians simply call their method ‘credible intervals’, while frequentists call their method ‘confidence intervals’. The crucial difference between the two is that only Bayesian methods, with their incorporation of an estimation of the prior probability distribution, allow for a direct computation of the probability of a certain value occurring within the constructed range. Frequentist confidence intervals, properly understood, do not allow such a direct statement of the probability of the value occurring in the constructed range because the value is either in the range or not and therefore has a probability of 0 or 1. This is because with frequentism, population values are unknown but fixed, while said values can vary with Bayesianism.

8. [...] Diaries Survivor, Resonaances, Not Even Wrong, ArcadianPseudofunctor, Of Particular Significance, US LHC Blog, Quantum Diaries, TDG Diary Eco World Content From Across The Internet. Featured on [...]

9. Stephen Brooks says:

LEP was “a collection of four experiments”? Think you’ll find there was a 100GeV electron and positron synchrotron joining them together.

10. Hi Stephen! Okay, to be a bit more precise, LEP was the accelerator upon which four experiments’ detectors were located. But that’s just not as catchy, right?

Actually, that whole paragraph is a little sloppy. I’ll fix that now. Thanks for your comment!

11. [...] Aidan Randle-Conde said at Quantum Diaries, “Like anyone else, physicists have issues with confidence,” by which he means [...]

12. [...] For more details on how to read these plots, please see Aidan Randle-Conde’s post about how to interpret Higgs limit plots from EPS and if you want to learn even more, I recommend Ben Kilminster’s lecture [...]

13. [...] how to read one of these plots (I don’t blame you!) there is an excellent explanation in the Quantum Diaries. The basic message is that if a Higgs particle exists we will see a peak, measured by the distance [...]