Better Higgs Combinations

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 b in a given mass bin. The new process (such as a Higgs decay) will add a signal cross section s to give a total cross-section b+s . After gathering lots of integrated luminosity L you may see N events with the required signal so you calculate the observed cross-section x = \frac{N}{L} . Now you are interested in whether x corresponds to the background b  or the background plus signal b+s . 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 N . 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 x has an uncertainty which we can write as \sigma = \pm \frac{x}{\sqrt{N}} . One thing we can say is that with 95% confidence the cross section is less than a limit x + 2 \sigma . We calculate the limit minus the background over the expected signal

CL_s = \frac{x -b +2 \sigma}{s}

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 CL_s if x = b . 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 CL_s is

c_1 = \frac{x_1 (1 + \frac{2}{\sqrt{N_1}}) - b}{s}

and for the expected value it is

e_1 = \frac{2 b}{ s \sqrt{ b L_1} }

Similarly for experiment 2 with observed and expected CL_s , c_2 and e_2. If we combine the two sets of events we will have N = N_1 + N_2 events in total, and total Luminosity L = L_1 + L_2 . This combination of luminosities can be substituted into the formula for excpected CL_s to derive the following combination law

e = \frac{1}{\sqrt{\frac{1}{e_1^2 } + \frac{1}{e_2^2 }}}

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 \Delta = c - e is given by

s \Delta = (x - b) (1 + \frac{2}{\sqrt{N}} )

Using the large N approximation this reduces s \Delta = x - b . 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

x L = x_1 L_1 + x_2 L_2

Which implies a similar combination law for \Delta . Using the relationship between the expected CL_s and the luminosity this reduces to

\frac{\Delta}{e^2} = \frac{\Delta_1}{e_1^2} + \frac{\Delta_2}{e_2^2}

This allows us to combine the observed and expected CL_s 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 CL_s 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.

32 Responses to Better Higgs Combinations

  1. Luboš Motl says:

    Dear Phil,

    there have been some problems with your calculations but it’s an obvious piece of work that the individual professionals didn’t do – perhaps because they’re lazy – but the insults against your work are clearly meant just as a protection of the (hypocritical) speakers against the obvious expectation (or stereotype) of their colleagues: “Blogosphere is pure garbage. Why would you associate yourself with it?”

    Now, 99.99% of the physics blogosphere is garbage, indeed. Still, one may sometimes find things that are at least as sensible as what the professionals would do if they were not lazy.


    • Bill K says:

      The guys at ATLAS and CMS did a tremendous job of processing and analyzing 1/fb of data in just a few weeks, under a great deal of pressure. They certainly don’t deserve to be called lazy.

    • Luboš Motl says:

      Dear Bill,
      they’re great, I love them, and blah blah blah.

      But I am not quite sure whether you realize and appreciate that ATLAS is composed of 3,000 people, CMS – including everything – is composed of 3,600 people, and those 6,000+ people are running methods and algorithms that have been developed and prepared in advance for the LHC collider by tens of thousands of people – experimenters, phenomenologists, theorists, and others – during the last 10-20 years or so when they had nothing more “tangible” to do.

      So before you establish a new religious cult, please divide the work that has been done by something like 6,000-20,000. Thank you very much.

      It’s exactly in the somewhat irregular things – such as the combination of experiments – in which one sees things that were not pre-planned and pre-prepared in years of work. And in this sense, it’s just slow. And it’s not just the experimenters in the collaborations themselves who didn’t do the obvious – a combination of the data. Maybe many of them believe that it’s not possible to combine graphs at all. Well, in that case, they’re surely wrong.


    • Bill K says:

      There are obvious political reasons why a group would benefit by publishing its own results first, and not want to immediately combine them with those of a second group. They promise to do so at the next meeting.

      I didn’t say it was impossible to combine the results. I said you lose something when you do so. There’s a loss of information involved. Or if not, explain to me how to reverse the process. The two original graphs contain more information than the combined one. But primarily there’s a big payoff in confidence to be able to hold two graphs side by side and say that independent groups have both arrived at similar results. Which is the reason for having two experiments in the first place.

    • Luboš Motl says:

      Two graphs combined to one graph obviously “lose information”. But that’s the very purpose of doing the combination.

      I don’t care about someone’s political explanations because they have no room in proper science. Moreover, this is not just about members of collaborations. This is about phenomenologists as well.

      Anyone in the hep-ex, hep-ph fields could have made the combinations Phil did – without the mistakes, with all the proper optimized statistical methods. Whether they like it or not, he’s the leader in this business. If they were not able to do a better job, they have no right to say that Phil’s work is just junk.

      I have already commented on the “value of having two experiments”. These were long discussions. It’s just stupid to create a religious value – or a “scientific necessity” – out of this completely irrelevant technicality that there are two big detectors at the collider.

      It was just sensible to spend more than 10% for detectors – relatively to the ring – so they built two big detectors to make the investments more balanced – and to use more space around the ring. The comments about the virtues of the “independent confirmations” are just a posteriori ideological justifications of something that had different, financial, more important reasons.

      Science doesn’t need this kind of confirmations. Moreover, much of the work of the two collaborations isn’t really that independent because aside from a very small set of their individual “proprietary” methods, they use the same whole literature of previous methods, expectations, and maybe even wrong expectations and misconceptions. It’s just not true that the two detectors’ results are “completely independent”.

      Getting 1/fb from ATLAS and 1/fb from CMS brings pretty much the same information as we would have if we got 2/fb in a single detector. The separation of this 2/fb of collisions into two halves is completely arbitrary – that’s why your demands that the combinations should be reversible are completely irrational. Only the overall number of events matters in the most accurate calculations one can do.

  2. ervin goldfain says:


    The global Higgs combination plot shows also a broad excess between about 145 GeV and about 172 GeV that falls below the red exclusion line. How do you interpret it? Is there a hint on a possible systematic error creeping in from 125 GeV to 172 GeV?



    • A thousand different kinds of higgs crounding their space!

    • Philip Gibbs says:

      The broad signal is very odd. For now I’d just say that it is a clear signal of something in that mass range but it is too soon to say what. I hope they show some simulations to see what kind of particle or multiplets can account for it.

      The unparticle theory is nice but other ideas may work too. It is hard to see how it could agree with a single particle. The gfitter group were also quoting quite a low p-value for fitting to a lone 140 GeV Higgs (17% I think, at Higgs Hunting 2011) so a non-standard model outcome should be favoured by now, but Standard Model Higgs can still fight back.

      I hope they start a new analysis at the next technical stop. Three times the data will make a big difference but if the outcome is complicated it may take a lot more data to resolve it clearly. Good thing the LHC is going so well. It’s developping into a dramatic story.

  3. [...] presentaban un pequeño sesgo en su figura de 28 de julio. Nos cuenta cómo lo ha hecho en “Better Higgs Combinations,” viXra log, August 6th, 2011. Merece la pena leer su entrada. Como puedes ver, la línea [...]

  4. ervin goldfain says:


    Here is a fairly well-written introduction to unparticle theory:

    Also, refer to the last entry of:

    for two representative papers on the concept of unhiggs.

    It can be shown that unparticle physics follows from the onset of non-equilibrium dynamics in QFT above the electroweak scale. If you wish to get into a more detailed discussion, we can talk offline.



  5. Dear Phil,

    than you for a nice explanation about how a man from street;-) can combine these data.

    I remember from discussions with some people in CERN that individuals in these groups are like ants in ant nest. A group member might privately combine various type of data in this manner but whether the publication of the result is even considered for publication is highly improbable. As individuals they cannot publish anything: this was the situation more than decade ago.

    These large ant nest like collaborations demonstrate the enormous power of group intelligence but there is also the free intelligence of individuals who have passionately devoted to the mission to understand. These people remain for obvious reasons outside the collaborations and academic world: they are like dissidents in Soviet Union. They can however use completely freely their own brains and certainly do so! I think that these big research institutions should try to utilize this anarchistic form of intelligence. Free intelligence is especially valuable source of Nature at the criticality for a paradigm change.

    By the way, my own bet is still that the “Higgs signal” below 150 GeV corresponds to 145 GeV CDF bump and has therefore nothing to do with Higgs. CDF, D0 and ATLAS should do analysis about the causes of discrepancy (I hope that this is politically possible;-)).

  6. kevin says:

    here is a preliminary one by CERN
    [link deleted]

    • Philip Gibbs says:

      It’s nice that they are putting the combinations in a well known public place as they go along :) . I think we are not meant to discuss them too much yet but they do look quite similar to the crude combination above.

      • kevin says:

        Your crude combination is indeed quite precise, the first limit is exact and the second one is within 1 GeV. It seems that they are working on the combination with one team from CMS and the other from Atlas, starting with the one I posted on monday, and meeting each morning to confront their updates. Anyway there is a clear presence of an higgs around 140 GeV. As of now they have only shown exclusion plots. With the new amount of data they may present discovery plots in autumn, with 3 times more data.

      • Philip Gibbs says:

        The signal plots are very good. For a Higgs it is at one on the y-axis and for no Higgs it is at zero.

        They may get a discovery signal but they may have to wait for individual channels to be clear before they can say what they actually have, unless the 144 GeV Higgs becomes the dominant signal.

      • Dilaton says:

        Dear Phil

        Is the possible higgs at 144 GeV now the only one so far, or are there still other (higgs) signals worth discussion, when You look at the combinations in the well known public place?

        (It is their fault if they have learned nothing from the past and leave the plots again at the same place, ha ha :-P …)


      • Philip Gibbs says:

        I wont talk about their plots yet because I don’t know if they have included everything yet. The signal in my crude combination peaks at 144 GeV but is very broad so it is too early to say what it is. There has also been talk of the bump at 120 GeV but it is not nearly as significant. The excess at 600 GeV only comes from one of the expriments. It may even be a single exceptional event. When there is a second it could make a three sigma signal :)

      • Dilaton says:

        Thanks Phil,
        so I must be careful to not explode with impatience to see the final outcome :-)

      • Luboš Motl says:

        Dear Phil, the below-120-GeV signal is not that significant but a less significant signal – around 2 sigma – is exactly what is expected after 1/fb, see this graph:

        It’s easier to find a Higgs at 144 GeV and indeed, the excess looks more significant, but it’s also expected and the mass around 144 GeV seems closer to being excluded as the “only” Higgs mass for the Standard Model.

        So the evidence for a 116 and 144 GeV Higgs is comparably strong – and of course, chances are that both of these Higgs bosons exist, like in MSSM.

        Cheers, LM

      • Philip Gibbs says:

        I like the two light Higgs theory too, but if there is one at 116 GeV then the Tevatron was very unlucky not to get any signal there. I like three Higgs even more 120 GeV, 144 GeV and 600 GeV. :)

  7. Evil String Theorist says:

    Is it just me, or does the excess between ~125-145 GeV just seem to mirror the “expected” curve, just at a higher level. So, is it possible that the peak around 119 GeV is the real-one, but the excess between 125-145 GeV is just a larger-than-expected background?

    • Philip Gibbs says:

      A Higgs signal is one plus the expected line but it is surprising that it does this over such a wide mass range. I think the background should be well enough understood but maybe you are right.

      It will be interesting to see how CERN spin this. A press release in advance of the conference seems likely

  8. Marc Sher says:

    Phil—in the Sphicus talk at the Higgs Hunting workshop two weeks ago, there is a plot showing the expected results if a 120,130,140 or 150 GeV Higgs signal is inserted. For 140, the result is, as you say, one plus the expected line, but over a range from 120 to 145 or so. Now, the plot is only for the WW -> 2 lepton plus two nu signal, and I haven’t seen one for all modes, but it should be similar. It is so broad due to difficulties in energy resolution, I think..

    • Luboš Motl says:

      Dear Marc, indeed, the channels with neutrinos – which look like missing energy – are the worst ones when it comes to the measurement of the energy of the decaying particles, so if this WW channel is what leads to the “Higgs excess”, it’s not surprising that the excess is broad.

    • Philip Gibbs says:

      Thanks Marc, I remember seeing that but then I could not find it again. It does look like you could just about account for the whole broad excess with one particle plus some fluctuations.

  9. [...] already been leaked but rather than show results that may change I am just going to discuss my own unofficial combinations that are not very different. So here again is my combined plot for CMS, ATLAS and the [...]

  10. [...] previous Combo after EPS was consistent with a standard model Higgs somewhere between 125 GeV and 145 GeV, or a more complex [...]


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