Camels and dromedaries – rapidity at a hadron collider May 12, 2008Posted by dorigo in physics, science.
Today we had our meeting of the CMS analysis group in Padova, a monthly recurrence where we get adjourned of the various efforts going on. It was my turn to chair the meeting (I am co-convener of the meeting with Ezio Torassa and we alternate), and I had put together a tightly packed agenda, which included updates on the global cosmic runs (weeks of data taking when muons from cosmic rays are collected and used to understand the detector response), the tracker checkout (issues with the final commissioning of the silicon tracker), the trigger studies for SLHC (or how to measure muon momenta accurately enough to prevent being overwhelmed by the huge rate of fake muons of low transverse momentum, when we will take data with CMS at a luminosity of ), plus analyses of the decay, ttH production, and dimuon mass spectra.
Ignazio Lazzizzera, from the associated group of Trento, presented some kinematical distributions of muon tracks extracted from minimum bias Monte Carlo that will be used for SLHC studies. Minimum bias is a jargon that particle physicists use to describe events that withstood no selection whatsoever: events which suffered the minimum possible bias by the fact of having been collected by the detector. Such a collection of events is useful to understand what our “priors” are: at the full LHC luminosity (just a factor 10 below SLHC ones), every 25 nanoseconds we will have 20 proton-proton collisions to deal with, and only very rarely these interactions originate a high-momentum muon, which tags a potentially very interesting event. We have to rely on these minimum bias simulations to understand how easy it is for a light hadron -a pion or a kaon- to fool our detection system and be identified as a muon by our trigger, if we want to understand our chances of tuning trigger cuts and select good muons with high efficiency without being drowned in impossibly high rates from fake muons.
As Ignazio showed the plot below, which is the distribution of rapidity of simulated muon tracks in minimum bias data, I jumped on my chair. What was going on ? The two-humped distribution resembled a camel’s back!
To let you understand why such a distribution is unphysical, I need to take a step back. When you collide protons with other protons at high energy, what you are actually doing is creating hard interactions about proton constituents: quarks and gluons. Each of these constituents of a high-energy proton carries a fraction of the proton momentum: the two streams of “partons” (i.e. quarks or gluons) travel together in the positive and negative direction along the z axis – the beam direction- inside each proton; but some carry a larger, and many a smaller fraction of the total protons momentum.
Because of the variable amount of momentum carried by each parton, the collision center-of-momentum reference frame is not at rest in the detector reference frame: if a 90mph truck hits a 50mph compact car head on the debris will fly away following the truck direction!
What governs the probability that quarks and gluons carry a certain momentum fraction of the proton containing them are some functions called “Parton Distribution Functions“. They are shown below for the different constituents of protons.
As you see, it is increasingly probable (in a measured described by the PDF xf(x)) to find a parton carrying a smaller and smaller momentum fraction x (forget the u-distribution, which has a local maximum due to valence quarks: we are discussing the low-x tail of these shapes, since we are discussing not-so-high-energy interactions which constitute the bulk of collisions). Is this enough to figure out what will be the distribution of the debris, and in particular, the motion of the most energetic particles produced in the collision in the detector frame ?
Well, basically yes. If we label the momentum fractions of the colliding partons (which can be assumed massless for all practical purposes at LHC), the center-of-mass energy will be their geometric average times the 14 TeV globally possessed by the colliding protons. The motion of the center-of-momentum frame in the detector frame will instead be described by rapidity – the quantity , which reduces to .
Rapidity is, for the muons, the quantity plotted in the two-humped histogram above. Can there be a hole at zero in this distribution ? Not really! It does not take complicated math to realize that if you pick at random two values from a monotonous function, their values are most likely to be close to each other, and so their ratio will be close to one more often than not. The logarithm of one is zero, and at zero there cannot be a minimum! The distribution has to have a single maximum at zero rapidity instead!
You might find the above reasoning rather complicated. It is. However, had you worked at a hadron collider for 16 years, you would not need the math at all: the rapidity distribution of any physics process is (with very few exceptions) a broad distribution with a maximum at zero, unless the data have been biased by selection cuts.
I could thus explain what was going on in the distribution Ignazio was showing: the data he was plotting had been stripped of events which could fire the CMS trigger -that is, events with high-Pt, central muons in our case. Take a dromedar, substract stuff in the middle (the muons which are central), and you are left with a camel!
It remains to be seen why the minimum bias Monte Carlo had been selected this way. I suppose one such sample is rather useless for trigger studies!