by Sean Carroll
Probability of a Higgs boson with mass 125 GeV decaying into different particles. Numbers don’t add up to exactly 100 percent due to rounding.
We’re nowhere near done yet, however. Hearken back to our discussion of particle detectors in Chapter Six, where we saw how different layers of the experimental onion helped us pinpoint different outgoing particles: electrons, photons, muons, and hadrons. Then look back at this pie chart. More than 99 percent of the time, the Higgs decays into something that we don’t observe directly in our detector. Rather, the Higgs decays into something that then decays (or transforms) into something else, and that is what we end up detecting. This makes life harder, or more interesting, depending on your perspective.
About 70 percent of the time, the Higgs decays into quarks (bottom-antibottom or charm-anticharm) or gluons. These are colored particles, which aren’t found by themselves in the wild. When they are produced, the strong interactions kick in and create a cloud of quarks/antiquarks/gluons, which congeal into jets of hadrons. Those jets are what we detect in the calorimeters. The problem—and it’s a very big problem—is that jets are produced by all sorts of processes. Smash protons together at high energy and you’ll be making jets by the bushelful, and a tiny fraction of the total will be the result of decaying Higgs bosons. Experimentalists certainly do their best to fit this kind of signal to the data, but it’s not the easiest way to go about detecting the Higgs. In the first full year’s run of the LHC, it’s been estimated that more than 100,000 Higgs bosons were produced, but most of them decayed into jets that were lost in the cacophony of the strong interactions.
When the Higgs doesn’t decay directly into quarks or gluons, it usually decays into W bosons, Z bosons, or tau-antitau pairs. All of these are useful channels to look at, but the details depend on what these massive particles decay into themselves. When tau pairs are produced, they generally decay into a W boson of the appropriate charge plus a tau neutrino, so the analysis is somewhat similar to what happens when the Higgs decays into Ws directly. Often, the decaying W or Z will produce quarks, which lead to jets, which are hard to pick out from above the background. Not impossible—hadronic decays are looked at very seriously by the experimenters. But it’s not a clean result.
Some of the time, however, the W and Z bosons can decay purely into leptons. The W can decay into a charged lepton (electron or muon) and its associated neutrino, while the Z can decay directly into a charged lepton and its antiparticle. Without jets getting in the way, these signals are relatively clean, although quite rare. The Higgs decays to two charged leptons and two neutrinos about 1 percent of the time, and into four charged leptons about 0.01 percent of the time. When the W decays create neutrinos, the missing energy makes these events hard to pin down, but they’re still useful. The four-charged-lepton events from Z decays have no missing energy to confuse things, so they are absolutely golden, though so uncommon that they’re very hard to find.
Four promising decay modes for discovering a Higgs boson at 125 GeV. The Higgs can decay into two W bosons, which then (sometimes) decay into electrons or muons and their neutrinos. Or it can decay into two Z bosons, which then (sometimes) decay into electrons or muons and their antiparticles. Or it can decay into a tau-antitau pair, which then decays into neutrinos and other fermions. Or it can decay into some charged particle that then converts into two photons. These are all rare processes but relatively easy to pick out at LHC experiments.
And sometimes, through a bit of help from virtual particles with electric charge, the Higgs can decay into two photons. Because photons are massless they don’t couple directly to the Higgs, but the Higgs can first create a charged massive particle, and then that can transform into a pair of photons. This happens only about 0.2 percent of the time, but it ends up being the clearest signal we have for a Higgs near 125 GeV. The rate is just large enough that we can get a sufficient number of events, and the background is small enough that it’s possible to see the Higgs signal sticking out above the background. The best evidence we’ve gathered for the Higgs has come from two-photon events.
This whirlwind tour of the different ways a Higgs can decay is just a superficial overview of the tremendous amount of theoretical effort that has gone into understanding the properties of the Higgs boson. That project was launched in 1975 in a classic paper by John Ellis, Mary K. Gaillard, and Dimitri Nanopoulos, all of whom were working at CERN at the time. They investigated how one could produce Higgs bosons, as well as how to detect them. Since then a large number of works have reconsidered the subject, including an entire book called The Higgs Hunter’s Guide, by John Gunion, Howard Haber, Gordon Kane, and Sally Dawson, which has occupied a prominent place on the bookshelves of a generation of particle physicists.
In the early days, there was much we hadn’t figured out about the Higgs. Its mass was always a completely arbitrary number, which we have only homed in on through diligent experimental efforts. In their paper Ellis, Gaillard, and Nanopoulos gave a great deal of attention to masses of 10 GeV or less; had that been right, we would have found the Higgs ages ago, but nature was not so kind. And they couldn’t resist closing their paper with “an apology and a caution”:
We apologize to experimentalists for having no idea what is the mass of the Higgs boson . . . and for not being sure of its couplings to other particles, except that they are probably all very small. For these reasons, we do not want to encourage big experimental searches for the Higgs boson, but we do feel that people doing experiments vulnerable to the Higgs boson should know how it may turn up.
Fortunately, big experimental searches were eventually encouraged, although it took some time. And now they are paying off.
Achieving significance
Searching for the Higgs boson has frequently been compared to looking for a needle in a haystack, or for a needle in a large collection of many haystacks. David Britton, a Glasgow physicist, who has helped put together the LHC computing grid in the United Kingdom, has a better analogy: “It’s like looking for a bit of hay in a haystack. The difference being that if you look for a needle in a haystack you know the needle when you find it, it’s different from all the hay . . . the only way to do it is to take every bit of hay in that haystack, line them all up, and suddenly you’ll find there’s a whole bunch at one particular length, and this is exactly what we’re doing.”
That’s the challenge: Any individual decay of the Higgs boson, even into “nice” particles like two photons or four leptons, can also be produced (and will be, more often) by other processes that have nothing to do with the Higgs. You’re not simply looking for a particular kind of event, you’re looking for a slightly larger number of events of a certain kind. It’s like you have a haystack with stalks of hay in all different sizes, and what you’re looking for is a slight excess of stalks at one particular size. This is not going to be a matter of examining the individual bits of hay closely; you’re going to have to turn to statistics.
To wrap our heads around how statistics will help us, let’s start with a much simpler task. You have a coin, which you can flip to get heads or tails, and you want to figure out whether the coin is “fair”—i.e., if it comes up heads or tails with exactly fifty-fifty probability. That’s not a judgment you can possibly make by flipping the coin just two or three times—with so few trials, no possible outcome would be truly surprising. The more flips you do, the more accurate your understanding of the coin’s fairness is going to be.
So you start with a “null hypothesis,” which is a fancy way of saying “what you expect if nothing funny is going on.” For the coin, the null hypothesis is that each flip has a fifty-fifty chance of giving heads or tails. For the Higgs boson, the null hypothesis is that all of your data is produced as if there is no Higgs. Then we ask whether the actual data are consistent with the null hypothesis—whether there’s a reasonable chance we would have obtained these results with a fair coin, or with no Higgs lurking there.
Imagine that we flip the
coin one hundred times. (Really we should do a lot more than that, but we’re feeling lazy.) If the coin were perfectly fair, we would expect to get fifty heads and fifty tails, or something close to that. We wouldn’t be surprised to get, for example, fifty-two heads and forty-eight tails, but if we obtained ninety-three heads and only seven tails we’d be extremely suspicious. What we’d like to do is quantify exactly how suspicious we should be. In other words, how much deviation from the predicted fifty-fifty split would we need to conclude that we weren’t dealing with a fair coin?
There’s no hard and fast answer to this question. We could flip the coin a billion times and get heads every time, and in principle it’s possible that we were just really, really lucky. That’s how science works. We don’t “prove” results like we can in mathematics or logic; we simply add to their plausibility by accumulating more and more evidence. Once the data are sufficiently different from what we would expect under the null hypothesis, we reject it and move on with our lives, even if we haven’t attained metaphysical certitude.
Because we’re considering processes that are inherently probabilistic, and we look only at a finite number of events, it’s not surprising to get some deviation from the ideal result. We can actually calculate how much deviation we would typically expect, which is labeled with the Greek letter sigma, written as . This lets us conveniently express how big an observed deviation actually is—how much bigger is it than sigma? If the difference between the observed measurement and the ideal prediction is twice as big as the typical expected uncertainty, we say we have a “two-sigma result.”
Confidence intervals for flipping a coin 100 times, which has an expected value of 50 and an uncertainty of sigma = 5. The one-sigma interval stretches from 45 to 55, the three-sigma interval stretches from 35 to 65, and the five-sigma interval stretches from 25 to 75.
When we make a measurement, the variability in the predicted outcome often takes the form of a bell curve, as shown in the figure above. Here we are graphing the likelihood of obtaining different outcomes (in this example, the number of heads when we flip a coin a hundred times). The curve peaks at the most likely value, which in this case is fifty, but there is some natural spread around that value. This spread, the width of the bell curve, is the uncertainty in the prediction, or equivalently the value of sigma. For flipping a fair coin a hundred times, sigma = 5, so we would say, “We expect to get heads fifty times, plus or minus five.”
The nice thing about quoting sigma is that it translates into the probability that the actual result would be obtained (even though the explicit formula is a mess, and usually you just look it up). If you flip a coin one hundred times and get between forty-five and fifty-five heads, we say you are “within one sigma,” which happens 68 percent of the time. In other words, a deviation of more than one sigma happens about 32 percent of the time—which is quite often, so a one-sigma deviation isn’t anything to write home about. You wouldn’t judge the coin to be unfair just because you got fifty-five heads and forty-five tails in one hundred flips.
Greater sigmas correspond to increasingly unlikely results (if the null hypothesis is right). If you got sixty heads out of a hundred, that’s a two-sigma deviation, and such things happen only about 5 percent of the time. That seems unlikely but not completely implausible. It’s not enough to reject the null hypothesis, but maybe enough to raise some suspicion. Getting sixty-five heads would be a three-sigma deviation, which occurs about 0.3 percent of the time. That’s getting pretty rare, and now we have a legitimate reason to think something fishy is going on. If we had gotten seventy-five heads out of one hundred flips, that would be a five-sigma result, something that happens less than one in a million times. We are therefore justified in concluding that this was not just a statistical fluke, and the null hypothesis is not correct—the coin is not fair.
Signal and background
Particle physics, since it is powered by quantum mechanics, is a lot like coin flipping: The best we can do is predict probabilities. At the LHC, we smash protons together and predict the probability of different interactions occurring. For the particular case of the Higgs search, we consider different “channels,” each of which is specified by the particles that are captured by the detector: There’s the two-photon channel, the two-lepton channel, the four-lepton channel, the two-jets-plus-two-leptons channel, and so on. In each case, we add up the total energy of the outgoing particles, and the machinery of quantum field theory (aided by actual measurements) allows us to predict how many events we expect to see at every energy, typically forming a smooth curve.
That’s the null hypothesis—what we expect without any Higgs boson. If there is a Higgs at some specific mass, its main effect is to give a boost to the number of events we expect at the corresponding energy: A 125 GeV mass Higgs creates some extra particles with a total energy of 125 GeV, and so on. Creating a Higgs and letting it decay provides a mechanism (in addition to all the non-Higgs processes) to produce particles that typically have the same total energy as the Higgs mass, leading to a few additional events over the background. So we go “bump hunting”—is there a noticeable deviation from the smooth curve we would see if the Higgs wasn’t there?
Predicting what the expected background is supposed to be is by no means an easy task. We know the Standard Model, of course, but just because we know what the theory is doesn’t mean it’s easy to make a prediction. (The Standard Model also describes the earth’s atmosphere, but it’s not easy to predict the weather.) Powerful computer programs do their best to simulate the most likely outcomes of the proton collisions, and those results are run through a simulation of the detectors themselves. Even so, we readily admit that some rates are easier to measure than to predict. So it is often best to do a “blind” analysis—use some method to disguise the actual data of interest, by adding fake data to it or simply not looking at certain events, then making every effort to understand the boring data in other regions, and only once the best possible understanding is achieved do we “open the box” and look at the data where our particle might be lurking. A procedure like this helps to ensure that we don’t see things just because we want to see them; we only see them when they’re really there.
It wasn’t always so. In his book Nobel Dreams, journalist Gary Taubes tells the story of Carlo Rubbia’s work in the early 1980s that discovered the W and Z bosons and won him a Nobel Prize, as well as his less successful attempts to win a second Nobel by finding physics beyond the Standard Model. One of the tools that Rubbia’s team used in their analysis was the Megatek, a computer system that could display the data from particle collisions and let the user rotate the view in three dimensions by operating a joystick. Rubbia’s lieutenants, American James Rohlf and Englishman Steve Geer, became masters of the Megatek. They were able to glance at an event, twirl it a bit, pick out the important particle tracks, and declare with confidence that they were seeing a W or a Z or a tau. “You have all this computing,” in Rubbia’s words, “but the purpose of all this tremendous data analysis, the one fundamental bottom line, is to be able to let the human being give the final answer. It’s James Rohlf looking at the f***ing event who will decide whether this is a Z or not.” No longer—we have a lot more data now, but the only way to really understand what you’re seeing is to hand it over to the computer.
Whenever there is some excitement about a purported experimental result, your first instinct should be to ask, “How many sigma?” Within particle physics, an informal standard has arisen over the years, according to which a three-sigma deviation is considered “evidence for” something going on, while a five-sigma deviation is needed to claim “discovery of” that something. That might seem unduly demanding, since a three-sigma result is already something that only happens 0.3 percent of the time. But the right way to think about it is, if you look at three hundred different measurements, one of them is likely to be a three-sigma anomaly just by chance. So sticking to five sigma is a good idea.
At the
December 2011 seminars, the peak near 125 GeV had a significance of 3.6 sigma in the ATLAS data, and 2.6 sigma in the CMS data (which are completely independent). Suggestive, but not enough to claim a discovery. Speaking against the significance of the result was the “look-elsewhere effect”; the simple fact that, as we just alluded to, large deviations are likely if you look at many different possible measurements, which the two LHC experiments were certainly doing. But at the same time, the fact that the two experiments saw bumps in the same place was extremely suggestive. Taken all together, the sense of the community was that the experiments probably were on the right track, and we probably were seeing the first glimpses of the Higgs—but only more data would tell for sure.
When the predictions you are testing involve probabilities, the importance of collecting more data cannot be overemphasized. Think back to our coin-flipping example. If we had only flipped the coin five times instead of a hundred, the biggest possible deviation from the expected value would have been to get all heads (or all tails). But the chance of that happening is more than 6 percent. So even for a completely unfair coin, we wouldn’t be able to claim as much as a two-sigma deviation from fairness. On Cosmic Variance, a group blog I contribute to that is hosted by Discover magazine, I put up a post on the day before the CERN seminars, entitled “Not Being Announced Tomorrow: Discovery of the Higgs Boson.” It’s not that I had any inside information; it’s just that we all knew how much data the LHC had produced up to that time, and it simply wasn’t enough to claim a five-sigma discovery of the Higgs. That would have to wait for more data.