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The problem of absolute normalization

In nature, the probability for two nuclei to collide with impact parameter between $b$ and $b+\,db$ grows linearly with $b$, and majority of the collisions are peripheral. In the experiment, the events to be recorded are selected by a trigger which is sensitive to the event centrality via multiplicity of the emitted particles. A trigger can be described by its probability density of selecting an event of certain total multiplicity. In reality, this probability density will never have a sharp edge, and the poorer the equipment, the more smeared will the edge be. It is therefore best to characterize the centrality trigger by giving the functional form of the total multiplicity distribution it produces. If this form is identical for two triggers, we will say that they belong to the same centrality class.

The problem of absolute normalization arises when one wants to obtain the event averaged number of particles ($N$) emitted in the collision events of certain centrality (impact parameter) range, or probability density distributions of particle emission with respect to kinematical variables.

$$N = \frac{\mathcal{N}}{E}\frac{1}{\mathcal{A}},$$ (18)

where the desired average $N$ is calculated from the number $\mathcal{N}$ of detected particle tracks over $E$ collision events of given trigger-selected centrality class, using the apparatus which registers only $\mathcal{A} \le$1 fraction of such tracks.

$$E = B \times i \times c,$$ (19)

where the $E$ collisions are selected from the $B\times i$ inelastic beam-target interactions by means of the trigger. The trigger is represented by quantity $c$, which will be called trigger centrality. The interaction probability $i$ in eq. ( 4.17) is not measured precisely. Unknown in the eq. ( 4.17 ) is $i \times c$. The beam count $B$ is live-gated and measured by the Cherenkov beam counter (sections  3.5,  3.4.1,  [29]). In this section I quote $c$, assuming that $i=0.034$ (as discussed in section  3.3).

The $E$ includes collisions with and without tracks in the acceptance. Important here is that even though the $E$ includes collisions with no tracks in the acceptance, the requirement for them to be of the same centrality class as the trigger events (to ensure ``cleanliness'' of the average (4.16)) forces one to pay attention to all centrality-sensitive components of the trigger. The notion of the trigger-selected centrality class, or sample, and its characteristic $c$ becomes therefore the key issue of the normalization analysis.

$c$ is the fraction of inelastic interactions, which includes all of the following and only the following:

(a)

interactions that satisfy the trigger requirements

(b)

for every interaction of group (a), all interactions of the same total multiplicity that happened in reality during the run, but did not cause triggering due to absence of tracks in the spectrometer's acceptance, and only due to this reason.

The class of events just described will be also referred to as the ``normalization sample'' or ``$c$-sample''. This definition of $c$ guarantees that the $N$ in formula (4.16) is indeed the correct average multiplicity in the centrality class selected by the trigger.

One should bear in mind that there is also a group of events which do not belong to the same total multiplicity class as the trigger events, have acceptable T0 amplitude in the trigger and do not create tracks in the spectrometer. They must be excluded from the definition of the trigger-selected centrality class and be not confused with the events of group (b) ! (Otherwise the result will be an underestimate of the true $N$ !). This subtlety would not exist for a large acceptance experiment, due to absence of the group (b) itself. Among the mechanisms responsible for existence of the events with lower total multiplicity but acceptable T0 amplitude are $\delta$-electrons, noise and fluctuations associated with operation of T0, fluctuations in the distribution of charged tracks in space.

Let us list the centrality-sensitive components of the trigger and briefly discuss them.