During the recent EPS conference when some new Higgs Exclusion plots were unveiled I has a stab at putting together some combinations of the plots using some basic formulas. Despite the broad caveats I gave them the plots got a surprising amount of attention. At a plenary session during EPS Bill Murray referred to my plots as “nonsense based on absolutely nothing” (which is not too far from the truth). Then at the Higgs Hunting workshop that followed EPS, John Ellis showed my “bloggers conbinations” saying that they were garbage but in the absence of anything better he would use them anyway. I hope this all added to everyone’s amusement and excitement as all the great new results were shown and discussed.
The formulas I used in those combinations were just quick guesses but they worked quite well for the Tevatron combination of CDF and Dzero Higgs results. In two or three weeks the LHC will reveal their combination for ATLAS and CMS at the Lepton-Photon conference in Mumbai so we will see how well my combination for that worked too.
Now that there has been a little more time to think about it I have looked at the basic statistic theory behind the plots to see why my formulas worked (so far). As a result I have come up with some improvements so I want to show some new plots that I think will be more accurate. There will be many more plots to combine in the near future as the LHC and Tevatron continue to churn out more data, so if they do work even approximately they may have some real use.
First some theory. Imagine you are looking for a signal of new physics in some decay channel. The standard model (without Higgs) will predict a certain background cross-section in a given mass bin. The new process (such as a Higgs decay) will add a signal cross section to give a total cross-section . After gathering lots of integrated luminosity you may see events with the required signal so you calculate the observed cross-section . Now you are interested in whether corresponds to the background or the background plus signal . In practice you can’t be sure so you have to look at the uncertainty.
To make things even simpler I am going to assume that the signal is smaller than the background but there are plenty of events . For a Higgs search this is a better approximation for low mass than for bigger mass but there are lots of other things we are going to ignore so why not start here?
Our estimate of the cross section has an uncertainty which we can write as . One thing we can say is that with 95% confidence the cross section is less than a limit . We calculate the limit minus the background over the expected signal
If this is less than one it means that the cross-section is less than the signal required for the Higgs boson with 95% confidence. This is roughly what the experiments plot against the Higgs mass. They also look at background uncertainty, trial error and combine different channels in a non-trivial way, but let’s ignore those things and see what happens. The expected value if there is no signal is just what we would get for if . This is also added to the plot as a function of mass with the familiar green and yellow uncertainty bands.
Now imagine that there are two experiments measuring the same quantity. They have different amounts of luminosity recorded and may be working at different energies and they will surely see different number of events. For now let’s pretend the background and signal are the same for each. This would be roughly true for two experiments at the same collider, but since the actual values of these numbers will not enter into the final formula we can try and use it even for different colliders.
For experiment 1 the observed value of is
and for the expected value it is
Similarly for experiment 2 with observed and expected , and . If we combine the two sets of events we will have events in total, and total Luminosity . This combination of luminosities can be substituted into the formula for excpected to derive the following combination law
This is exactly the formula I used before, so far so good. However I used the same formula to combine the observed $CL_s $, this was not quite correct. The excess is given by
Using the large approximation this reduces . If you dont like this approximation and you know the signal to background ratio you can improve it. I found that this does not make much difference in practice.
The observed cross-sections combine with weights given by the luminosities
Which implies a similar combination law for . Using the relationship between the expected and the luminosity this reduces to
This allows us to combine the observed and expected without knowing the background cross-sections.
Here is what it does for the combination of CDF and Dzero. This is slightly better than my previous attempt when compared with the official combination shown at EPS.
Next here is the new result for the LHC combination that has not yet been shown officially.
As you can see this gives much more significant excesses than my earlier combination. It is even a little above the upper limit of the grey uncertainty area I drew before. The broad excess around 140 GeV is well over three sigma so it can be claimed as an “observation” of a candidate Higgs if this is how the official plot looks. The excess at 120 GeV is also hard to ignore at over 2 sigma and even the limit at the high end near 600 GeV cannot be ruled out. I hope that CERN will decide to extend the plot to higher masses so that we can see this a little better if it appears on their plot.
To look at this in another way we can plot just the size of the excess as seen on the logarithmic graph. In doing so it would be useful to know the expected size of the excess when there is a Higgs boson rather than when there is not as shown on the plot above. I can approximate this by adding 1 to the expected and showing it with the excess. I also hope CERN will decide to do an accurate version of this or something like it. It is fine to show expected values for no Higgs boson when you are just excluding, but as soon as a signal appears you need to know what a signal is expected to look like with the boson.
This plot is less familiar so let me explain what it is telling us. The black line shows the observed excess in numbers of sigma. There is a broad region of excess above two sigma for masses from 112 GeV to 172 GeV, but this is below the red exclusion line above 149 GeV. It lies within the bands for an expected Higgs boson signal between 110 GeV and 144 GeV. 144 GeV is also where we see the maximum excess at 3.4 sigma, but there is also a minor peak at 119 GeV where the signal reaches 2.6 sigma. Finally there is also a less significant peak at 580 GeV of 1.7 sigma. Although the plot does not exclude a signal for a small window around 250 GeV this is lower than the excess expected for a Higgs boson.
That is not the end of the story because we also have the full Tevatron combination and we can add that in as well to produce a global Higgs combination plot. Nothing changes above 200 GeV so here is a closeup of the low mass window
The excess at 120 GeV is a little reduced, but otherwise the message is similar.
With twice as much data now recorded by ATLAS and CMS we can expect some clarification on what this is telling us quite soon. Until then the conclusions are uncertain and you are free to speculate.