The LHC collider

A new era of particle physics is about to begin with the construction of the Large Hadron Collider at CERN, Geneva, Switzerland. Hadrons can be described as combinations of quarks. Generally speaking, hadrons comprise all particles that are affected by the strong nuclear force[CD91, p.51]. Protons are one of about 100 different hadrons that are known today. Particles that are not affected by the strong force are called leptons (e.g. electron, muon). Protons are roughly 2000 times heavier than electrons, thus contain more energy and are more likely to produce new massive particles that theorists predict, but have not been ``seen'' yet. Furthermore $pp$collisions are essentially interactions between the enclosed quarks, thus allowing for very basic interactions. However, there are disadvantages with observing such interactions: as the collisions are not ``clean'' when bunches of protons hit each other, there is a lot of background information that has to be subtracted before the interesting event can be studied. This background consists of protons that did not collide as well as collision products of protons that did not hit each other with the full energy. Finally, as particle collisions will produce several different new products there is a huge amount of newly created ones, that are not wanted for a specific experiment. This makes the building of a data acquisition system much more difficult than for an experiment that has to observe clean lepton interactions.

In order to get an impression of the task that a data acquisition system has to carry out, let's have a closer look at the amounts of data that are fed into the system. The most interesting parameter is the rate at which particles collide. This reaction rate $\dot{N}$ is proportional to the particle flux $\Phi$ of beam $a$, to the cross section $\sigma_{pp}$ of the target particle and the number $N_b$ of particles in the target[MRRS88, p. 58] (i.e. the bunch of particles in the second beam $b$):

$$\dot{N} = \Phi_a \times N_b \times \sigma_{pp}$$ (1)

The expression $\Phi_a N_b$ is called luminosity and serves as a proportionality factor for the event rate[MRRS88, p. 59][PRSZ95, p. 44]. The flux $\Phi_a$ is given by the number of particles in a bunch and the frequency at which these bunches pass the interaction region divided by the target area[PRSZ95, p. 42]:

$$\Phi_a = f \frac{N_a}{4 \pi \sigma_x \sigma_y}$$ (2)

$N_a$ is the number of particles in a bunch of beam $a$, $f$ gives the frequency at which bunches pass the interaction region. $\sigma_x$ and $\sigma_y$ characterise the Gaussian transverse beam profiles in horizontal and vertical directions. For the luminosity ${\cal L}$ we get therefore the equation[ED98, p. 138]:

$${\cal L} = f \frac{N_a N_b}{4 \pi \sigma_x \sigma_y}$$ (3)

From the LHC key parameters in table 3 we can calculate ${\cal L}$:

$$\begin{eqnarray*} {\cal L} & = & f \frac{N_aN_b}{4 \pi \sigma_x \sigma_y}\\ & =... ...l L} & \approx & 1.2434 \times 10^{34}~{\rm cm}^{-2}{\rm s}^{-1} \end{eqnarray*}$$

The cross section $\sigma$ of a particle can be interpreted as the size[Gri87, p.190] that it represents to a stream of other incoming particles. When a particle is a solid sphere then its cross section will be the area given by its circumfence. For a ``soft'' particle, like a proton that manifests itself by a Coulomb potential that decreases with distance from the protons centre, the case is different: The distance from the centre of the particle at which another particle hits is called impact parameter. The angle at which a colliding particle is deflected at the target is called the scattering angle. The smaller the impact parameter, the larger the scattering angle. At a certain distance from the centre this angle will be close to zero and thus gives a limit to the effective area. The cross section of a particle depends on the kind of particles involved in the interaction and of the particles potentials, i.e. their energies[PRSZ95, pp.190]. Cross sections are given in barn (1 barn = $10^{-24}~{\rm cm}^2$).

For a $pp$ interaction we can look up the cross section for LHC energies in studies (see figure 6 for a fit to existing measurements). At a collision energy of 7 TeV per particle beam the cross section is about 100 mb. With this value we can continue our event rate calculation:

$$\dot{N} = {\cal L} \sigma_{pp} = 1.2434 \times 10^{34}~{\rm ... ...}{\rm s}^{-1} \cdot 100~{\rm mb} = 1.2434 \times 10^9~{\rm Hz}$$

As there are $25~ns$ time between two bunch crossings we have $\frac{1}{25 \times 10^{-9}} = 40 \times 10^6~{\rm Hz}$ bunch crossings per second. So, each bunch crossing contains roughly

$$\frac{1.2434 \times 10^9~{\rm Hz}}{40~{\rm MHz}} \approx 30$$

interactions.

For the $pp$ cross sections plotted against energy (figure 6) there are no real measurements yet, just fits. Therefore the event rate as well as the number of interactions per bunch crossing must not be seen as definitive values. They represent orders of magnitude for understanding the problems that arise when studying new physics processes. The high event rate of $10^9~{\rm Hz}$ forces us to reduce the amount of data and select the interesting events. Some tens of interactions per bunch crossing indicate that it is indeed difficult to find the created particles that shall be stored for later analysis. They are hidden by an enormous background of other particles and slower decaying debris from earlier interactions.