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Formulation of the problem

An offset of the event vertex with respect to the detector's symmetry axis results in a non-trivial functional dependence between the actual $\eta$ and $\zeta$ , and the $\eta^\prime$, $\zeta^\prime$ presumed based on the ``ideal'' geometry: $\eta = \eta(\eta^\prime,\zeta^\prime)$, $\zeta = \zeta(\eta^\prime,\zeta^\prime)$. This makes the observable multiplicity distribution $\,d^2N/\,d\eta^\prime\,d\zeta^\prime$ (in the presumed coordinates) differ from a simple function of $\eta^\prime$:

$$\frac{\,d^2N}{\,d\eta^\prime\,d\zeta^\prime} \not= \frac{1}{2\pi} \frac{\,dN}{\,d\eta^\prime}$$ (72)

In the true coordinates $\eta$ and $\zeta$, the inequality 6.19 becomes an equality. However, the detector's acceptance area in the true coordinates becomes distorted. Figures  6.2 and  6.3 show the pad multiplicity (gray level) in the misaligned and aligned coordinates, respectively.

Figure 6.2: Example of a monitoring plot used in the course of the analysis to understand the alignment procedure and the alignment quality. The color (or gray level) corresponds to the pad multiplicity. No misalignment correction is applied. The horizontal lines connect centers of the pads with $\Delta _{i,j}$ sufficiently small for the pairs to be used in formula 6.20 (compare with Fig. 6.3). Run 3192. The $\delta$-contaminated part of the detector is not shown.

Figure 6.3: Another example of a monitoring plot used in the course of the analysis to understand the alignment procedure and the alignment quality. The color (or gray level) corresponds to the pad occupancy. A misalignment correction is applied. One can see how both the acceptances of the pads and their (double differential !) multiplicities are modified. The horizontal lines connect centers of the pads with $\Delta _{i,j}$ sufficiently small for the pairs to be used in formula 6.20 (compare with Fig. 6.2). Run 3192. The $\delta$-contaminated part of the detector is not shown.

In the following we will refer to this as a ``Jacobian effect''. The Jacobian effect, obviously, contributes to the event textures, especially on the large scale, and needs to be evaluated and corrected for.