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Multiple interactions at LHC: an exercise in elementary statistics February 13, 2008

Posted by dorigo in mathematics, physics, science.
7 comments

The LHC will start running towards the end of this year, at the design energy and with a bunch crossing time of 25 ns. That means 40 million intersections per second between two proton packets in the core of CMS and ATLAS [things are a bit more complicated -some bunches are empty- but this has no relevance for my point].

We expect that the beams will contain few protons in this initial phase: low luminosity, that is. That’s because a high energy beam requires a lot of tuning before it can accommodate a large number of particles. Charged particles, in fact, have the nasty tendency to repel each other, and squeezing them into a narrow corner of phase space -knee to knee, all together as a single man- is a tremendously hard task, requiring successive approximations. Moreover, as the beams travel through the LHC tunnel, each making over ten thousand turns a second, they generate strong induced currents on the machine’s hardware. This electromagnetic interplay is impossible to compute beforehand, and a trial and error procedure by the machinists is unavoidable.

Luminosity is a function of the number of protons spinning in the two directions. Basically one can compute it from the number of particles circulating in the two directions by taking their product N_1 N_2 and dividing it by the revolution frequency and the transverse section of the beam. One obtains a number whose units are inverse area (the beam size) times inverse time in seconds (the frequency). The LHC will start at L=10^{30} cm^{-2} s^{-1}, but we expect it to reach the design value of L=10^{34} in a couple of years.

Luminosity is not just a number with which machinists boast about their gadget. With it, you can compute the rate of production of any given process, if you know its cross section.

Cross section, a number labeled with the greek letter \sigma carrying units of area, basically tells you the effective area a proton must hit in another in order to give rise to a given reaction. The total cross section for proton-proton collisions at the LHC energy is \sigma_{tot} = 8 \times 10^{-26} cm^2: more or less like a circle with a radius of 1.6 millionths of a billionth of a meter - the “size” of a proton seen by another colliding with it head-on. But the total pp cross section is huge! Compare it with the cross section for producing a top quark pair: \sigma_{t \bar t} = 8 \times 10^{-34} cm^2, or a hundred million times smaller. It is like if the incoming proton had to hit the other one “just right there”, to produce a top pair.

With a knowledge of what a cross section is we can answer questions. What is the total rate of proton collisions at the LHC if the luminosity is L=10^{33} cm^{-2} s^{-1} - the one we will have in the “low luminosity” phase ? Simply,

N = \sigma_{tot} L.

With \sigma_{tot} as quoted above, we get a rate N=8 \times 10^7 of proton collisions: eighty million collisions per second! How many per bunch crossing ? Well, if all proton bunches contain the same number of particles, we get on average two interactions per bunch crossing, since the crossing rate is 4 \times 10^7. Easy, huh ?

Well, things in reality are just a bit more complicated. The probability of events that may come incoherently in integer numbers follows the rules of Poisson statistics. Poisson statistics allows us to compute the probability that a bunch crossing will contain no collisions, or one, or two, or N, given the average \mu=2 as computed above. The formula looks awful, but it is quite benign:

P(N) = \frac {\mu^N e^{-\mu}} {N!} ( the exclamation mark indicates taking the factorial of N).

We need a pocket calculator, but other than that there is nothing that should scare you out of this post. Keep reading if you want to use what you just learned to get some insight in the inner workings of the LHC experiments!

With the formula for the probability of N collisions, we have gained power - knowledge, they say, is just that. The power to make wonderful calculations. If I tell you that the cross section for producing an event with two energetic jets (say, energy above 30 GeV each) is \sigma_2 = 2 \times 10^{-28} cm^2, or 200 \mu b  (we prefer to use microbarns -labeled \mu b- for the area of 10^{-30} cm^2, a quite convenient unit),  how many such events will be produced in a single bunch crossing at the full LHC luminosity of 10^{34} cm^{-2} s^{-1}, on average ?

Easy. Use the formula N = \sigma L, and you get a rate N=2 \times 10^6 s^{-1}, that is,  2 MHz. Then, by dividing by 40 million bunch crossings per second, you get the rate per bunch crossing, 0.05: one in twenty. If instead we had taken the cross section for producing four energetic jets, \sigma_4 = 3 \mu b, we would have obtained a rate of 30 kHz, and a bunch crossing frequency of 7.5 in ten thousand. Mind you, the cross sections I quote are approximate - I estimated them with some back-of-the-envelope calculation. But let’s not be distracted by details and let me get to the point.

Those computed above are average rates. What happens if I ask you what is the chance that two, or more, separate proton collisions each producing two jets like the ones above in the same bunch crossing?

Now, that might sound like a weird question, devoid of any practical importance. Quite the contrary. Let me compute it for you before making my point. We use the Poisson probability formula, with \mu = 0.05 and N>=2. Instead of computing P(2), P(3), P(4)… and then adding them together, we use the fact that the sum of all P(N) is one: a nice property of probability, indeed! Here is the computation:

P(0) = e^{-\mu} = 0.95123,

P(1) = e^{-\mu} \mu^1 /1! = 0.04756, and so

P(>=2) = 1 - P(0) - P(1) = 1-0.95123-0.04756=0.00121.

Interesting! The chance of two distinct dijet events in a single bunch crossing is not that negligible… If we cannot distinguish where the jets come from (i.e., if the two proton collisions happen too close to each other), we will interpret the event as one with four energetic jets!

Now compare this 0.00121 with the number computed above with \sigma_4, the rate of collisions producing four jets in the final state from a single proton-proton interaction: we discover that at the LHC, there are instances when two separate collisions may conspire to mimic rarer processes! If I am looking for four-jet events, I will find 1.21 every thousand bunch crossings coming from two 2-jet “multiple interaction” events, while only an additional 0.75 every thousand will be genuine 4-jet events. I have a background to consider which lower luminosity machines would never have to care about!

The exercise is over. It is not an academic one: in the study of the very rare production of top pairs with higgs-strahlung, pp \to ttH, one gets to consider the collection of exceedingly rare events with up to eight energetic jets in the final state. The background from multiple interactions conjuring a multijet final state by adding different contributions is to be removed! We can do that by actually tracking the jets down to the space point where they were originated: we only keep events where all eight jets originated from the same spot, and we are ok. We can do it, since we have such a wonderful silicon tracker (see picture)…