The Vertex Algorithm

Presently, there are four ways of determining the vertex position along the beam axis $V_z$ in BRAHMS:

1. Using the ZDC time signal as described above. This is considered the least precise measurement, as the intrinsic time resolution $\approx 115ps$ [3] giving intrinsic $\sigma_{ZDC}(V_z) \approx 3.5cm$. However, current analysis gives $\sigma_{ZDC}(V_z) \approx 5cm$ [4].
2. Using the Beam-Beam time signal, in similar fashion to the ZDCs. However, the timing resolution is $\approx 70ps$ giving a much better resolution for $V_z$.
3. Using reconstructed hits or clusters in the first TPC of the mid rapidity spectrometer (TPM1). This is considered the most precise algorithm so far.
4. Using reconstructed tracks from TPM1 to point at a vertex. This method shows some promise, but isn't fully understood yet.

However, we can not choose the vertex as we see fit when we build up the distribution $\frac{d^2N}{dMdV_z}$, since that would introduce a bias in the data. We are therefore restricted to using the vertex measurement obtained from ZDCs, despite the non optimal resolution, since we have already required hits in the ZDCs by choosing the trigger 4 as our minimum bias trigger, and this obtain no further bias.

It is possible using the time signal from the two ZDC, $t_L$ and $t_R$, and assuming the particles that trigger the measurement in the ZDCs have $\beta \approx 1$, to obtain a very crude determination of the position of the primary vertex along the beam axis $Vz$:

$$V_z = c (t_R - t_0) - c (t_L - t_0) = c(t_R - t_L)$$ (5)

where $t_0$ is the, not measured, absolute time of the collision, and $t_R$ and $t_L$ is the average of the time signal from the first two modules in each ZDC.