In the following, we attempt to understand the resolution in the direction along the padrow, so as to correctly estimate the position uncertainty hit-by-hit.
The first term in Equation 9 does not depend on crossing angles and only on the signal:noise and signal shape. To evaluate whether the hitfinder estimates this component of the uncertainty correctly, we generated low-track-density toy events, in which tracks cross the padrow with zero crossing angle. The field was turned off (in this simulation, this did not change the transverse diffusion from its field-on value) so that highly ionizing particles as well as m.i.p's would provide straight tracks. Then, the electronics noise in the simulator was varied and the resolution studied. ``100%'' noise was the nominal noise: , . RMS resolutions and uncertainties are shown in Figure 22 as a function of noise. There, it is seen that the hitfinder slightly overestimates the uncertainty from the first term of Equation 9. This is understood to be the consequence of using Equation 8, which is based on an idealized assumption of infinite shaping time, to calculate the uncertainty on pad projections.
Figure 22: Simulated =0 tracks passed through the slow simulator with varying levels of noise are used to study resolution and calculated uncertainties. The amount of noise is quantified in terms of the nominal electronics noise in the default simulator. Note that resolution and uncertainty do not vanish at ``% noise''=0, because the effective noise introduced by digitization is not removed.
Figure 23 shows that, for each noise value, the resolution scales as expected with signal size: resolution increases linearly with the inverse of the signal size, .
Figure 23: The dependence of the resolution (RMS of the residual distribution) is seen to show a linear dependence on the inverse of the hit size for simulated =0 tracks, for all values of noise. Note the scale change for resolution as the noise increases.
Having established that the resolution due to the first term in Equation 9 is understood and well-estimated by the hitfinder code, we turn to tracks with finite crossing angle. The slow simulator currently does not incorporate finite electron effects, so the second term in Equation 9 is not expected to contribute to the resolution (=0) . The third term, which arises from fluctuations in energy loss as a track crosses the padrow at non-zero angle, will increase the resolution strongly. The resolution in the outer sector from a Au-Au event is shown in Figure 24 as a function of crossing angle .
Figure 24: Resolution as a function of padrow crossing angle is shown for the outer sectors in a simulated Au+Au event.
A series of low-multiplicity events were run through the slow simulator, with varying vertex position (in order to study seperately effects of crossing angle and drift length L), and varying electronic noise. The RMS of the residual distribution was calculated as a function of , dE/dx, L, , and noise level. No dependence of the padrow spatial resolution was seen as a function of or L, as expected since =0 (see above). The dependence of the RMS on and signal size is shown in the case of zero electronics noise in Figure 25. The worsening of the resolution with crossing angle is clear, as is the fact that the rate of this worsening is greater for large hits (large dE/dx) .
Figure 25: Observed resolution (RMS of residual distribution) for low-multiplicity events with zero electronics noise is shown as a function of the crossing angle alpha, with cuts on dE/dx of the hit.
Equation 9 contains two constants, and , which depend only on the chamber geometry and gas properties . (The primes on the constants are in part to distinguish them from similar ``constants'' used elsewhere [6,7,10] which depend on hit properties.) These constants must be determined empirically. We find using P-10 gas for the outer sector and for the inner sector . will be determined after TSS is updated. We note that the values for may also change if, for example, the energy loss gets calculated differently.
The tracker, with and in hand, and a hypothesis of , can accurately estimate the uncertainty of the hit position, as is shown in Figure 26, where the uncertainties estimated by a tracker using the form 9 are compared for the range of hit classes to the rms of the residual distribution for that hit class.
As numerical examples of the resolution in the x- and y- directions, we note that, averaging over all particles in a Au+Au event, the Gaussian widths of the residual distribution , whereas for those hits whose parent tracks had . It is this small resolution that scales as signal:noise.
Figure 26: Estimated uncertainty in the position along the padrow that would be calculated by a tracker is compared to the position resolution for inner and outer sector, for two values of electronic noise. lines are drawn for reference.