The vertex reconstruction toolkit RAVE has been extended by an option for the inclusion of kinematic constraints, and embedded into the ILD analysis framework Marlin. The new tools have been tested with an exemplary reconstruction of WW and ZZ decays. The presented results show the improvements achieved in precision of the fitted masses, and demonstrate the usage and functionality of the toolkit.
2 Reconstruction of W + W -decays RAVE's kinematic capabilities were tested with a data sample consisting of 1376 events e + e -→ W + W -→ 4 jets at √ s = 500 GeV, generated by Pythia [5] and processed for the LDC 2007 layout [6]; the "true" W masses are plotted in Figure 1(a). Events with a mass |m W -80.32| > 6.5 GeV are eventually discarded, retaining 1008 events. To model the expected performance of jet reconstruction in ILC experiments, the following Gaussian errors have been applied to all four generated jets:
The direction resolution affects the θ (polar angle) and φ (azimuth) measurements.
A known general problem with the use of jets as virtual measurements is due to their compositeness: only direction and energy, but not an absolute momentum, are welldefined. However, an initialization of the jet’s momentum with its energy, followed by an appropriate inflation of the associated error, gives satisfactory results.
The constraints applied are those of energy and 3-momentum conservation. They are explicitly written as (the subscripts identify the four jets):
So far no distinction was made between the four jets: the applied kinematic constraints acted equally on all final states, and did not take into account which particles they could have originated from. Now one has the task of associating the four jets into two pairs (there are 3 possible combinations), with each pair originating from a W boson decay.
A trivial strategy is to assume that each combination is valid, and to calculate the two W masses accordingly. Figure 1(b) shows the results: each entry represents the W pair masses corresponding to one combination of one event. a a About 5% of the combinations yield unphysical mass values and do not enter the plot.
A comparison of Figure 1(b) with 1(a) suggests that, as a first step, dropping combinations where both W masses exceed a pre-defined cut limit would significantly improve the performance of the association. This cut is chosen at a value of 130 GeV. For a second step, several options are feasible. A simple “equal-mass hypothesis” introduces in the kinematic fit, as an additional constraint, the requirement of the two fitted masses to be equal (because they both belong to W bosons). However, such a requirement would strongly favour combinations along the 45 o diagonal over those parallel to the axes, in contradiction to the true distribution of the W pair masses as shown in Figure 1(a).
The most obvious improvement of such a hypothesis would be to model a selection requirement from two uncorrelated Breit-Wigner (BW) distributions. b But that is not possible in real-world scenarios, because the position parameter of the expected distribution is exactly the (unknown) value to be determined by this kinematic vertex fit. Therefore, a compromise is suggested between this idea and the “equal-mass hypothesis” above [8]:
The two W masses are known to not being equal, however, they are still picked from the same distribution. The likelihood of such a configuration is given by
Here, the first term holds the results of the kinematic vertex fit (parameters α c and covariances V c ). The second term represents the new “similar-mass hypothesis”; it models the distribution of the two masses dependent on the position parameter m, and it can be split into two symmetric terms if the masses are approximately uncorrelated. Although any p.d.f. can be used within the second term, a Gaussian model is certainly the easiest choice. Then, the objective function for the new fit can be written as
where σ t ≈ 9 GeV is a pre-defined scale (obtained by integration over the BW).
The objective function M is minimized w.r.t. α m and m and yields a pseudo-χ 2 , the probability of which is used as the new second step selection criterion for finding the best jet association (out of the 3 possible combinations).
c Note that the four kinematic constraints (equ. 3 and 4) and the similar-mass constraint (equ. 6) together do not suffice to account for the redundant degrees of freedom.
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