The Problems

In figure 1 is shown $dN/dM$ for events with an inclusive trigger 4 i.e., all events for which trigger 4 fired, plus the requirement that there is at least one hit in the scintilating tiles array above threshold.

So far, much of the analysis has been done with centrality cuts as constant cuts in $M$.

One problem with this approach is that the acceptance of the tile array depends strongly on the position of the vertex, which means that $M$ is not only a function of the collision centrality, but also of where the collision occurs. The dependency of $M$ on the position of the primary collision vertex is illustrated in figure 2.

It can be seen from figure 2, that a global constant cut in $M$ does not correspond to a cut in centrality, applicable to all values of $V_z$.

One way to get around this problem is to make the $M$ distribution for a number of vertex bins and to determine centrality cuts for each bin. This is a kind of centrality calibration.

The Calibration Method

Having realized that the centrality is highly vertex dependent, we are forced to revisit our way of determining the centrality.

1. Starting from raw data, we look at minimum bias events (trigger 4 $\wedge 1 <$ tile hit above pedestal) with $\vert V_z\vert < 50cm$, where $V_z$ is determined by the ZDCs. This is to ensure that $V_z$ stays inside the coverage of the scintillator tiles, and to reject background from beam-gas, etc.
2. The multiplicity $M$ and the vertex $V_z$ are histogrammed to build up $\frac{d^2N}{dMdV_z}$, with 8 $V_z$-bins, corresponding to the 8 rings of tiles in the multiplicity array.
3. After having run through many events ( $\mathcal{O}(100,000)$), we select a vertex bin and plot $dN/dM$ for that vertex bin. See also figure 3
   
   
4. This distribution $dN/dM$ for a given vertex bin is then integrated in steps from $M_{max}$ to 1, and the centrality cut, $M_{cut}$, is determined.
5. Steps 3 and 4 are repeated for all vertex bins, and the values $M_{cut}$ are plotted as a function of $V_z$. This gives us series of pairs of numbers ($V_z$, $M_{cut}$), one series for each cut value.

   These points are then fitted with $3^{\mathrm{rd}}$-degree polynomial in $V_z$, to give us functions
   

   $$S_{cut}(V_z) = a_{cut,0} + a_{cut,1}V_z + a_{cut,2}V_z^2 + a_{cut,3}V_z^3$$ (1)

   
   
   See also figure 4.
   
   A third degree polynomial is used to give a reasonable fit with the fewest possible free parameters.

Centrality determination for a given event

For each subsequent event, one can, using $S_{cut}(V_z)$, now find the upper and lower limit for the collision centrality. For example, one could get the numbers 5 and 10 for upper and lower centrality limit - if the $S_{5\%}$ and $S_{10\%}$ functions are defined (in the calibration procedure) and the collision is in this centrality interval.