Playing the odds at the LHC: the luminosity frontier

We all like to play darts occasionally, and some others like buying lottery tickets. Although there is some skill that needs to be acquired when playing darts (we could call this the aim), playing the lottery requires no skill at all. In both of these situations a natural mathematical object comes to the fore. We usually call it the odds of winning. In some sense, the odds describe how you should bet money on the different possible outcomes of one of these games. If you do your analysis carefully you will find that betting on the lottery is always (statistically) a losing proposition, unless you could bet against winning.

Given that we are in the middle of a financial crisis, essentially because the odds of something happening were not calculated correctly, I thought this might be nice place to talk about the odds of stuff happening at another place: the Large Hadron Collider (I will use the standard acronym: LHC). Sadly, the odds were not in favor last week where there was a Helium leak.
Failures of these kinds are considered routine and they tend to happen more at the beginning, so there should be no alarm. It’s just that the schedule for collisions gets pushed back and the eager collective of particle physicists have to wait longer for new data. This gives us theoretical folk just a little more time to place bets as to what we will see come out of the LHC accelerator.

I want to describe the ‘bets’ that are made to find the missing link of the Standard Model: the Higgs particle. So if you want to hear about some odds, follow me through some reading of specs.

Unfortunately I will not have enough space to describe everything that goes into this analysis, but rather how one plays the odds at the LHC to try to see the Higgs. I thought it would also be a nice introduction as to how a physicist looks at LHC specs and tries to figure out real numbers from specs. By the way, don’t trust me. Trust the guys who work on the machine. I’m just trying to make sense of what they tell me.

In some sense this is also a post that should give you a sense of awe at how our experimental colleagues can pull a winning lottery ticket from a very noisy environment and how they perform truly outstanding feats to get this experiment to run. This is in case you are not awed already. Be prepared to deal with very small and very big numbers.

So what is the LHC? It is a proton-proton smasher. Two protons are accelerated to very large speeds (very very close to the speed of light) and they are made to collide. The experiment is aimed at looking at the debris that comes out of these collisions and to use this information to deduce the presence of new forms of matter that have not been observed by us yet. Some people have used the analogy of throwing watches at each other and seeing the bits and pieces fly out. I have been told that a better analogy is throwing watches at each other and seeing cars and cellphones shoot out. This is because we shoot light particles at each other, and produce very massive particles in return.

Ok, so far so good. The main notion that we need to describe collisions is that of a cross section. What is a cross section? Think of shooting arrows at a target. The area of the target that you have to hit is what one would call a cross section. So a cross section is the amount of area you have to hit to get something to happen.

The natural units of area in high energy collisions are called barns. A barn is equal to . Which is a really small target to hit. The typical proton-proton cross section is governed by the size of the proton. The cross section we are interested in is roughly . That is forty milli-barns (a barn is roughly the cross section to hit a uranium atom).
Talking about barns, my wife says that she never played basketball because she couldn’t hit the side of a barn. Seeing how barns are really tiny, this never surprised me much.

How well can one do with the collider? Well, if the collider works up to specifications, the beams of protons will be reduced to a cross section whose radius is roughly , which translates to an area of $1000\times 10^{-12} m^2$. The chances (odds) of one proton hitting another in such tight conditions is roughly

So, those chances are almost as bad as having to win a lottery three times in a row.
How does one overcome such pitifully small probabilities? Well, we buy as many lottery tickets as we can. As a matter of fact, the protons come in bunches of about . That is one hundred billion (this is smaller by a factor of ten than the roughly one trillion bailout program that the US government is considering as we speak, but these are just numbers of particles here, not dollars). So the total expected number of collisions that you get per crossing is about forty. Remember that you have two bunches of protons colliding each time, so each time there is about pairs of particles that could collide. If you read carefully the specs, their number is about one half the number I got above. This should be due to the beams not being exactly dead center on top of each other and other such things that reduce the efficiency.

Now each bunch goes around the ring 11245 times per second and there are 2808 bunches stored in the ring. If you multiply it all together, you get about 600.000.000 collisions per second on each of the detectors. The number of collisions per unit cross section that one can produce per second is called the luminosity. This is going to be the most luminous machine on earth, packing a huge number of collisions into an incredibly small area.

Each collision generates a huge amount of data. The current capabilities for transferring data to a hard drive ends up being about 100 data sets per second (I’m not sure of the number, but I’ve heard this from my colleagues, so I’ll just use it and hope to stand corrected afterwards). You would think that the other 599.999.000 collisions are wasted effort.

This is not so. Most of the collisions are really boring. This is because the proton is built of smaller pieces called quarks and gluons, and the interesting question is how the little quarks and gluons scatter with each other. Most of the times, the quarks and gluon bits scatter at relatively low energies, and only very seldom do the individual pieces of each proton carry enough stuff in them to make a really big collision.

For example, the typical cross section to produce a Higgs particle is roughly one to twenty picobarn (this is a number I got from chasing around the literature, and I think the twenty is for the nominal center of mass). Compared to the forty milibarns, this is about one to ten billion times less likely. Which means that one would be producing about one Higgs particle per ten seconds to a minute. Out of all those collisions, very few see the Higgs. Moreover, the Higgs has to decay to something that we can tell apart from the background. The preferred decay channel happens about once in a thousand times, because the other decays will be overwhelmed by other processes.

This means that there is about one nice well behaved Higgs particle per day. So if you were just taking chances, out of about 50 trillion collisions per day, one is good and the rest are ‘background’. If you look at those odds, it is not particularly good. You are more likely to win the lottery.

Fortunately, there is a solution to this problem. If you have to look for a needle in a haystack, use a magnet. Here, the way to look for a needle in a haystack is that you trigger on the interesting events. So a lot of the experimental effort being done is on designing and implementing hardware and software triggers to sift through the data and collect only those gold nuggets that are interesting and potentially relevant. That’s like rigging the lottery to get winning tickets. This is how the LHC really works: you have to know were to look.

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