Centrality

Naïve Approach

In a first attempt to define the centrality $C'$ as dependent on the measured multiplicity $M$ deposited in the scintillating tiles, we write

$$C'\left(M_m\right) \equiv 1 - \frac{\int_0^{M_m} dM\, \frac{dN}{dM}}{\int_0^{M_{max}} dM\, \frac{dN}{dM}}\quad,$$ (2)

where $M_{max}$ is the maximum observed deposited energy in the tiles, and $dN$ is the number of events with a given deposited energy $dM$.

The distribution $dN/dM$ should be measured over all collisions, since the measured multiplicity $M$ is most likely to be correlated with other conditions one might impose.

However, as is most often the case, we do not have a clear, unambiguous signal for a collision; hence we must rely on some sort of measurement to tell us that a collision has occurred.

Minimum Bias

Hence the need for a clear minimum bias trigger. The signal from the two Zero Degree Calorimeters (ZDCs) being closely spaced in time (trigger 4) and the requirement at least one tile has a ADC over pedestal, is generally considered to be a good minimum bias trigger [2]. This is because trigger 4 is the most inclusive⁴ of the available triggers in BRAHMS, and the the requirement that at least one tile fires, vetoes pure Columb interactions between the colliding nuclei, and other kinds of events that may be considered as `bad' events.

Taking the Primary Vertex Position into Account

For minimum bias events, we plot $\frac{d^2N}{dMdV_z}$ (see figure 2), and find that

$$N\vert _{M_m} = \int_0^{M_{m}}dM \, \frac{dN}{dM }$$

is not constant over $V_z$ for a given $E$. This can be expressed be re-writing equation 2:

$$\frac{dC(M_m)}{dV_z} \equiv 1 - \frac{\int_0^{M_m}dM \, \frac{d^2N}{dM dV_z}}{\int_0^{M_{max}}dM \,\frac{d^2N}{dM dV_z}}$$ (3)