The motivation behind the work described in this chapter was the desire to know to high precision the CTDFLT efficiency across the whole of the accessible phase space. This is important for measurement of cross-sections as mentioned in the previous chapter. The naıve approach of simply generating large numbers of events in kinematic bins is not a suitable one since the constraints of available computer resources mean that the requisite precision cannot be obtained over all phase-space. For this reason, a method of simplifying the problem was searched for. For CC events, it is inherently plausible that the efficiency of the CTDFLT depends only on the polar angle of the current jet theta-jet. This hypothesis was shown to be consistent with the data by generating a large sample of events in small regions of phase space with fixed theta- jet.
The results for each angle were combined to produce high-precision efficiency data. These were then used to plot a map in x – Q2 space by assuming the same efficiency for all points in the phase space with the same jet angle. The method was also investigated with respect to NC events. As would be expected however, it was found to be unsatisfactory due to the scattered electron which plays an important part in triggering these events.
7.1.1 Special Jacquet-Blondel Kinematics
It is possible to manipulate the usual kinematic equations (see equation 1.26 in section 22.214.171.124) so that the theta-jet dependency becomes more explicit; in particular using half angle formulae and setting E_e = 30GeV gives equation 7.1:
So for a fixed jet angle, various different combinations of values of x and Q^2 are available for a given y. Contours of fixed y are shown in the x=theta-jet plane in figure 7.1 and for the Q^2 – theta-jet plane in figure 7.2.
Figure 7.1: Contours of fixed y in the x – theta-jet plane.
Figure 7.2: Contours of fixed y in the Q^2 – theta-jet plane.
We may define a SL polar angle such that a track from the nominal interaction point at this angle will leave the sensitive volume of the CTD at a position on the end-plate midway between where the two central wires are attached. The minimum angles for the instrumental SLs are 11. degrees, 18.9 degrees, 25.4 degrees for SL1, SL3 and SL5 respectively.
It is obvious that there will be no information from the CTDFLT concerning tracks with angles smaller than 11.6 degrees (or greater than 168.4 degrees). In fact, there will be some spread of tracks around the nominal jet angle so that some proportion of events have no tracks within the sensitive volume of the CTD. Clearly one expects this proportion to increase as the nominal jet angle is changed such that the tracks are expected to be closer to the beam-pipe.
7.2 Event Generation
A low-statistics pass across the whole of the angular range was made. Fifty events were generated in angular bins of two degrees. The information needed to produce bins in x and Q^2 corresponding to the required angular range is shown graphically in figure 7.1 and figure 7.2.
The events were generated with 10 degrees < theta-jet < 90 degrees. It was not necessary to generate any events with jet angles of larger than 90 degrees because symmetry means that eta(theta – 90 degrees) = eta(theta). Below 10 degrees there is not expected to be any activity in the detector. A similar sample was generated for NC events.
A selected set of five angles were chosen for high statistics runs. These angles were 13 degrees, 23degrees, 33degrees, 43degrees and 63 degrees. These were chosen with reference to the super-layer polar angles mentioned above. They correspond to the cases in which one expects the jet to pass through the one or two instrumental SLs for the two lowest angles and all three instrumented SLs for the remaining three angles.
Angular bins with a a range of one degree either side of the nominal value were defined for the low-statistics run. To measure the variation with respect to y from 0.1 to 0.9 were defined. Approximately one thousand events were generated in each bin so that in total 36250 events were used in this study. The CTDFLT simulation was run to find the efficiency.
The results for theta-jet = 63 degrees are shown in table 7.1, for theta-jet = 43 degrees in table 7.2, for theta-jet = 33 degrees in table 7.3, for theta-jet = 23 degrees in table 7.4 and for theta-jet = 13 degrees in table 7.5.
Table 7.1: CTDFLT efficiencies in the kinematic bins for theta-jet = 63 degrees +/- 1 degree.
Table 7.2: CTDFLT efficiencies in the kinematic bins for theta-jet = 43 degrees +/- 1 degree.
Table 7.3: CTDFLT efficiencies in the kinematic bins for theta-jet = 33 degrees +/- 1 degree.
Table 7.4: CTDFLT efficiencies in the kinematic bins for theta-jet = 23 degrees +/- 1 degree
Table 7.5: CTDFLT efficiencies in the kinematic bins for theta-jet = 13 degrees +/- 1 degree
Figure 7.3 shows that the results are consistent with the hypothesis that there is a smooth dependence of efficiency on jet angle. From the numbers in the tables it can be seen that efficiency is constant for a given angle independent of all other kinematic variables. Also the expected deterioration in efficiency is seen as the jet angle becomes closer to the beam-line.
Figure 7.3: Low statistics full angle pass for CC events.
For NC events however, figure 7.4 shows that the pattern does not show the same simple dependency on jet angle only. This is due to the presence of the scattered electron. It is unsafe therefore to attempt to proceed further with the method for this type of event.
Figure 7.4: Low statistics full angle pass for NC events.
Returning to the CC sample, it is now plausible to combine the various tables of results at the same jet angles to produce high-precision results. Since the results represent statistically independent measurements of the same quantity, they may be combined by taking the mean and dividing the error by root n where n is the number of entries in the relevant table. This yields the figures in table 7.6.
Table 7.6: Final combined figures for CTDFLT efficiency.
These figures may be used to generate contours of constant trigger efficiency in the x – Q^2 plane, remembering that symmetry allows the same efficiencies to be plotted for 180 degrees – theta also. This is shown in figure 7.5.
Figure 7.5: Efficiency for CC events.
It has been shown that CTDFLT CC efficiency is dependent on theta-jet only. Precise knowledge of the expected efficiency may now be obtained over a large part of the accessible phase space by deducing the jet angle from the kinematics of a given event if that event lies on or near one of the angles studied with high statistics. Otherwise, an interpolation may be made.