When I looked at this picture of Easter Island and matched it to a recent picture of Peter Higgs the best fit was the first statue, but where does the Higgs Boson fit best on the search plots from the LHC?
It may be a little late now to try to analyse the latest public data from the LHC given that the collaborations themselves are now looking at 3 times the amount of integrated luminosity, but Tommaso Dorigo is claiming that the summer data best fits a Higgs boson at 119 GeV and Peter Woit is pressing the case for no Higgs at all. I have my doubts about either claim, so how can we see what really fits best?
To answer this we first have to think about what the familiar brazil band plots mean such as this one showing the recent Higgs combination for the summer data from the LHC.
If you look at the 140GeV point you will see that the observed CLs line is crossing the red line. The naive interpretation is that the probability for no Higgs boson at this mass is 0.95 so it is ruled out at the 95% confidence level. However, this is wrong. Such a probability can only be calculated when we plug in our prior probabilistic beliefs for the existence or not of a Higgs boson at that mass. The correct interpretation of the plot is that if there were a Standard Model Higgs boson at 140 GeV then the probability of getting a stronger signal than the one seen would be 0.95. This is a very different statement.
Looking at the plot again we see that there is also a nearly three sigma excess at the 140 GeV point. We tend to discount it because of the exclusion, but again this is the wrong thing to do. The excess tells us that if there were NOT a Higgs boson (SM or otherwise) at this point then the probability of getting a weaker signal than the one seen would be about 99% (roughly). So actually the signal indicating a Higgs boson at 140GeV is five times stronger than the one tending to exclude it. The symmetry between the signal and no signal possibility is best seen on this signal plot that uses the same information differently.
If we were being Bayesian, our prior probability for no Higgs at this point would probably be higher than the probability that one exists because there should be more places where it isn’t than where it is. If we favoured a light Higgs mass for theoretical reasons and discounted non-standard models we might assign a probability of 0.8 to no Higgs boson at around 140 GeV and 0.2 to a SM Higgs at 140GeV. In this case we would look at the 140GeV point on the plot and come down slightly in favour of the Higgs boson at that mass.
However, the plot contains much more information because it covers the whole mass range where a Standard Model Higgs might be. We can compare the probabilities for a Higgs boson at any mass in the range and see which one is favoured. For this we need to use our prior beliefs for where the Higgs might be over the whole range. For simplicity lets just assume that we believe in a single standard model Higgs boson and we favour equally each of the mass points where they plot a square on the graph. To apply this we need to know the width of the signal that a Higgs boson at a given mass would produce on the signal plot. The underlying decay width for a Higgs boson is predicted by the standard model as shown in this plot.
Below twice the mass of the W the width is very narrow and it is the resolution of the detectors that counts. This varies depending on the channel and the mass but I am going to assume that it is ±5 GeV at worst and fit to a bell curve on that assumption. If you think differently you may get a different result from me. The method is to overlay the bell curve on the signal plot with a peak at 1.0 where we think the mass of the Higgs may be and tending to zero either side. At each mass point we read the signal strength and use the observed data to tell us the conditional probability for that signal strength (assuming a flat normal PDF) . These probabilities are all multiplied together to give the conditional probability for the fit. We can then try all the curves for different Higgs masses we believe in and see which one has the best fit. Here is the result.
As you can see the best fit is actually at 141 GeV. Perhaps we should see how it works for the separate plots from ATLAS and CMS
ATLAS sees the Higgs at 144 GeV and CMS sees it at 141 GeV. That is pretty consistent given the resolution of the detectors. What about using different channel combinations. I will limit this to the three with the most data.
The best fits are 132 for WW (which has poor resolution), 143 GeV for ZZ and 139 GeV for diphoton. So it is a pretty consistent result.
I don’t think it is safe to conclude that the Higgs boson has mass around 140 GeV. All we can say is that the limited data published so far supports that as best fit. The summer data has not probed the 120 GeV region well enough so there could be something there with a stronger signal when we look at the 5/fb data for this winter. Rumours are that there is not much of a signal anywhere with 120 GeV being the best chance, but I am waiting until I have seen the data myself and repeated this objective analysis.