Detector Simulation and Lepton Response

The way that GOPAL [90] simulates the detector so that the Monte Carlo can be compared directly to the data is an integral part of the analysis. It is very difficult to measure this as any differences between the data and Monte Carlo could be due to physical effects, such as anomalous TGCs, rather than poor modelling of the detector.

The most important part of detecting a ${\rm q}\bar{\rm q}\ell\bar{\nu}_{\ell}$ event is identification of the lepton. Parameters of the lepton which are less sensitive to the TGC value must be compared. For the lepton, the polar angle in the lab frame, $\cos\theta_{\ell}$, and the lepton energy are chosen. A comparison using these two variables between the data and fully simulated Monte Carlo are made, and then lines are fitted to the distributions. These lines are used to weight the Monte Carlo. Then a comparison between the SDM elements and $\cos\theta_{\rm W}$ distributions, for the TGC fit, and the helicity proportions for the helicity studies, before and after the weighting is made. As this test is limited by statistics and also TGC dependent, no correction is made to the overall result. However, a systematic uncertainty is assigned due to each of the tests. For the SDM elements and $\cos\theta_{\rm W}$ these can be seen in tables 10.5 and 10.6.

Table 10.5: The systematic uncertainty due to lepton identification as a function of the lepton polar angle in the lab frame. The numbers represent the size of the error as a fraction of the statistical error from the data sample on each bin of each variable used in the TGC fits.

	1	2	3	4	5	6	7	8
$\cos\theta_{\rm W}$	0.01	0.05	0.15	0.06	0.06	0.05	0.15	0.03
$\rho_{++}$	0.01	0.01	0.01	0.09	0.10	0.05	0.01	0.13
$\rho_{--}$	0.02	0.01	0.10	0.11	0.03	0.01	0.02	0.02
$\rho_{00}$	0.01	0.01	0.05	0.13	0.09	0.02	0.01	0.07
Re($\rho_{+-}$)	0.01	0.05	0.05	0.07	0.01	0.01	0.08	0.01
Re($\rho_{+0}$)	0.13	0.33	0.02	0.01	0.01	0.04	0.19	0.13
Re($\rho_{-0}$)	0.01	0.29	0.01	0.02	0.03	0.01	0.20	0.20
Im($\rho_{+-}$)	0.06	0.11	0.06	0.11	0.04	0.14	0.06	0.11
Im($\rho_{+0}$)	0.16	0.15	0.04	0.06	0.05	0.01	0.09	0.02
Im($\rho_{-0}$)	0.21	0.07	0.02	0.09	0.01	0.01	0.01	0.08

Table 10.6: The systematic uncertainty due to lepton identification as a function of the lepton energy. The numbers represent the size of the error as a fraction of the statistical error from the data sample on each bin of each variable used in the TGC fits.

	1	2	3	4	5	6	7	8
$\cos\theta_{\rm W}$	0.13	0.15	0.11	0.09	0.02	0.04	0.11	0.13
$\rho_{++}$	0.19	0.25	0.21	0.23	0.19	0.21	0.20	0.17
$\rho_{--}$	0.05	0.10	0.16	0.21	0.28	0.32	0.44	0.44
$\rho_{00}$	0.09	0.11	0.05	0.02	0.03	0.10	0.20	0.24
Re($\rho_{+-}$)	0.01	0.02	0.02	0.01	0.02	0.03	0.01	0.01
Re($\rho_{+0}$)	0.01	0.02	0.05	0.07	0.09	0.14	0.13	0.02
Re($\rho_{-0}$)	0.02	0.08	0.11	0.10	0.11	0.11	0.05	0.10
Im($\rho_{+-}$)	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01
Im($\rho_{+0}$)	0.00	0.01	0.01	0.01	0.01	0.01	0.01	0.02
Im($\rho_{-0}$)	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01

The overall uncertainty due to the lepton identification on the helicity proportions is shown in table 10.9. However, for the calculation of the helicity fractions, a further systematic uncertainty is assigned due to the detector simulation. In correcting for detector effects Standard Model Monte Carlo is used, however this could cause a bias towards the Standard Model. This is tested by comparing the calculated helicity fractions for non-Standard Model samples of Monte Carlo at generator level, to those from the same sample after full detector simulation and detector correction. Samples with anomalous couplings, $\Delta\kappa_{\gamma}$, $\Delta g^{z}_{1}$, and $\lambda$ set at $\pm$1 are used. The largest variation from the six tests is taken as the systematic uncertainty. These are shown in table 10.9.

Uncertainties on the measured lepton energy are estimated to be less than 0.3%. Any effects this would have are tested by shifting the lepton energy by this amount. In all cases this is found to have a negligible effect.

Differences in the charge misassignment between the Monte Carlo and data could have a significant effect on all the results, especially as the wrong value of $\cos\theta_{\rm W}$ is assigned when the charge is incorrectly identified. Misidentification most commonly occurs in the higher momentum leptons where there is little bending of the lepton track in the magnetic field. A test of the effect was performed by randomly doubling the amount of expected misassigned charges in a large Monte Carlo sample from 0.8% to 1.5%. The effect this had was found to be small.

Any charge misassignment can also cause a problem in the measured lepton momentum. This was accounted for by varying the resolution in Q/p$_{t}$ by 10% in the Monte Carlo. Where Q is the lepton charge and p$_{t}$ is the transverse momentum of the lepton. Differences before and after the change are taken as the systematic uncertainty. The uncertainties for the TGC fit can be seen in table 10.7 and the uncertainties on the measured helicity fractions are shown in table 10.9.

Table 10.7: The systematic uncertainty due to lepton charge/momentum uncertainty. The numbers represent the size of the error as a fraction of the statistical error from the data sample on each bin of each variable used in the TGC fits.

	1	2	3	4	5	6	7	8
$\cos\theta_{\rm W}$	0.05	0.11	0.01	0.00	0.02	0.03	0.07	0.23
$\rho_{++}$	0.07	0.11	0.09	0.08	0.05	0.10	0.09	0.16
$\rho_{--}$	0.08	0.12	0.08	0.06	0.01	0.01	0.00	0.02
$\rho_{00}$	0.09	0.15	0.11	0.09	0.04	0.06	0.05	0.08
Re($\rho_{+-}$)	0.16	0.16	0.09	0.08	0.09	0.08	0.08	0.15
Re($\rho_{+0}$)	0.00	0.03	0.06	0.04	0.02	0.02	0.03	0.04
Re($\rho_{-0}$)	0.08	0.01	0.06	0.04	0.01	0.00	0.02	0.03
Im($\rho_{+-}$)	0.02	0.05	0.00	0.03	0.01	0.02	0.01	0.02
Im($\rho_{+0}$)	0.03	0.01	0.02	0.01	0.02	0.02	0.05	0.04
Im($\rho_{-0}$)	0.03	0.03	0.01	0.00	0.01	0.02	0.01	0.02