account and performs a statistical sampling of all possible spin, flavor, and color configurations. However, in case of an intermediate τ lepton decay, i.e. (Z →ττ) j j events, this decay is not
described by VECBOS and needs to be modeled with the τ transfer functions which yield the
probability for an observed lepton to carry a given fraction of theτlepton energy.
In the reconstruction of the final state, the transverse momentum of theZ j jsystem is assumed
to be zero. While developing the method two additional procedures have been tested. In the first one, the transverse momentum of the Z j j system was not constrained at all, while in the
second one, the final probability was multiplied with the a priori probability for the corresponding transverse momentum. As in the case of the top pair momentum, this probability was derived using Monte Carlo simulated events. Since the assumption of zero transverse momentum yields the best separation of signal and background, the computation is significantly faster, and the effect is less important than in the calculation of the signal likelihood, this assumption is used to calculate the background likelihood.
To assess the(Z→ττ) j jlikelihood, the directions of theτleptons are assumed to be identical
to those of the final-state electron and muon, which is a good approximation because of the large boost of the τ leptons from Z decay. The jet and lepton directions are directly taken from the
detector measurements, while the two jet energies are obtained from random numbers distributed according to jet transfer functions under the assumption of light-flavor jets. For each event, at least 10000 random numbers are drawn. If needed, up to a maximum of 50000 are drawn, until the standard deviation of the individual values is no larger than 10% of the average. The absolute values of the twoτmomenta are determined from the constraint of zero event transverse momen- tum, pZ j jT = 0. Then, the sum of the matrix elements times PDF factors is calculated as in the (Z →``)j jcase using VECBOS, and multiplied with the phase space factor and the jet transfer
function factors. To account for theτlepton decay, theτtransfer functions are included yielding the likelihood to observe the lepton energy for the givenτenergy.
5.8
Normalization and Performance of the Background Likelihood
In principle, the normalization of the background likelihoods can be calculated in the same way as the signal likelihoods are normalized. However, the computation of the integral given in Equation (5.9) would be very computing intensive, and a different approach is chosen. It makes use of the fact that only the correct background-to-signal normalization reproduces the correct signal frac- tion. Thus, the following procedure is used to determine the relative background normalization.
• For each channel and each top mass, one large ensemble has been built using tt¯ and (Z→ττ) +jets Monte Carlo simulated events.
• The signal fraction is fitted in the sample and the normalization of the background likelihood
is adjusted, until the estimate yields the true signal fraction withinε=0.005.
• Since the normalization of the background likelihood cannot depend on the top quark mass,
the above steps are applied to each availablett¯Monte Carlo sample. The mean of all results
is taken as the signal to background normalization scale.
To demonstrate the good performance of the background likelihoods, Figure 5.12 shows the separation oftt¯signal and(Z→ττ) +jets background events in the electron+muon channel using
62 5. The Matrix Element Method signal and background likelihoods. The top mass hypotheses of the signal likelihoods have been chosen such that they correspond to the generated top masses of 160 GeV in the upper, 170 GeV in the middle and 185 GeV in the lower plot. Both signal and background likelihoods are normalized as described above. To make use of signal events where the requirement of zeroZ j j transverse
momentum cannot be fulfilled, the background likelihoods are set to 1·10−31. This explains the
vertical band in the plots. 4% of the(Z→ττ) +jets events and 40% of the signal events fail this
5.8. Normalization and Performance of the Background Likelihood 63 ) nrm ) jj τ τ (Z−> log (L −70 −60 −50 −40 −30 ) nrm tt log (L −60 −50 −40 (160GeV) t t ) jj τ τ (Z−> ) nrm ) jj τ τ (Z−> log (L −70 −60 −50 −40 −30 ) nrm tt log (L −60 −50 −40 (170 GeV) t t ) jj τ τ (Z−> ) nrm ) jj τ τ (Z−> log (L −70 −60 −50 −40 −30 ) nrm tt log (L −60 −50 −40 (185 GeV) t t ) jj τ τ (Z−>
Figure 5.12:Normalized signal likelihoods compared to normalized background likelihoods for both signal
t¯tand background(Z→ττ) +jets events. The mass hypothesis corresponds to the generated mass of the
64 5. The Matrix Element Method