As a reliability test of the simulation, the distributions of X and Y can be compared between MC and data at the final level of selection. As shown in Fig. 5.9, these two normalized MC distributions are found in agreement with the corresponding plots of the real data sample 7. The satisfactory agreement
Figure 5.9 Normalized distributions of |X| (left) and Y (right) at the final level of selection for real (blue) and for simulated (red) data.
between data and MC ensures that it is possible to use the simulation for the evaluation of the differential total efficiency εtot(X, Y ). This is evaluated on MC according to the definition (5.1), as the ratio between the histogram of the reconstructed Dalitz plot GM C(X, Y ) (left panel of Fig. 5.10) and the generated plot ΨM C(X, Y ). The resulting normalized plot is shown in the right panel of Fig. 5.10 in a three-dimensional view. The statistical uncertainty associated to each bin content typically ranges from 3% in the central region to 10% near the contour. As a measure of the systematic effects induced by the smearing on X and Y over the whole Dalitz plot, the number of entries lying where ΨM C is not defined (especially outside the physical interval allowed for X) is below the order of percent. The two peaks in the right plot of Fig.
7 The quality of the simulation on the final sample has been also checked by comparing more elementary quantities, such as the energy spectrum of the selected γ clusters, the π± and the π0’s momenta and the si variables.
Figure 5.10Left: a box view of the reconstructed Monte Carlo distribution G(X, Y ).
Right: a three-dimensional representation of the differential efficiency εtot(X, Y ).
5.10 concern two bins (symmetric with respect to the X = 0 axis) nearby the contour with very small physically allowed area, in which the smearing coming from the close bins has produced an unbalanced increase of entries.
Such effect appears quite randomly also in other peripheral regions of the plot, if the binning is slightly changed.
The total efficiency, averaged over the entire Dalitz plot, can be measured on the simulated data sample as hεtoti = 0.0612 ± 0.0008 . It can be regarded as the efficiency ετsel0 (described in Sect. 4.5 and summarized in Tab. 4.9) multiplied by the efficiency of the kinematic fit ετf it0 and of the cut on the three quantities (sAi − sBi )/si defined previously. It is important to stress that the absolute value of εtot does not enter the extraction of the Dalitz plot parameters, since only the shape of the differential efficiency is relevant for this analysis.
The projections of εtot(X, Y ) on the X and on the Y axis (i.e. obtained by integrating over Y and X, respectively) are plotted in Fig. 5.11 and fitted as polynomials with coefficients An.
The efficiency εtot and its projections on the X and the Y axes have been obtained by averaging the results from two different subsamples of the PP0P0 generated events: in about 14 of the Monte Carlo statistics the dynamics of
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Figure 5.11 Integrated efficiencies, obtained as the projections of the differential efficiency εtot(X, Y ) on the X axis (left) and on the Y axis (right).
the τ0 decays follows the parametrization reported in the PDG [4], while the remainder has been generated according to arbitrary values of g, h and k. If separately considered, the integrated efficiencies compare satisfactorily within the errors.
While εtot appears to behave quite independently of |X|, it decreases in an approximately linear way as Y grows in the allowed interval. This can be explained through kinematic arguments. The X variable is proportional to a difference of invariants which behave similarly (s1 and s2) and its modulus is therefore weakly correlated to the tracking/clustering efficiencies correspond-ing to the reconstruction of the charged/neutral pions. In the average, a large value for Y means a high-momentum π± in the decay: this allows a slightly better reconstructed daughter track but the lower total energy at disposal of the two π0’s significantly reduces the probability to correctly reconstruct and identify the four final state photons.
Before extracting the values of the Dalitz plot parameters, the contribution of the background in G(X, Y ) has to be taken into account.
5.4.1 Background estimate
The main contribution of non-τ0 events in the finally selected MC sam-ple comes from the other charged kaon decays. They amount to 0.4% after
performing the kinematic fit and populate rather uniformly the (X, Y ) plane.
Their relatively low statistics on Monte Carlo does not allow to clearly define a shape of the physical background in terms of the Dalitz variables.
On the other hand, the wrongly reconstructed τ0 decays should be con-sidered as a source of background as far as the determination of the Dalitz plot parameters is concerned, and their presence in the final distribution has to be quantified and subtracted. From Monte Carlo, those τ0 decays in which one or both of the Dalitz variables have been reconstructed with values dif-fering from the true values by more than half of the dimension of the bins (δX× δY = 0.6 × 0.2) have been considered as background.
In general, the background can be parametrized as a function ξ(X, Y ), defined as the sum of the two above-mentioned sources (non-τ0 and badly reconstructed τ0 decays). Since no evidence of dependence on X has been found, the background function has been expressed simply as a function of Y (Fig. 5.12). A linear fit of ξ(Y ) demonstrates that the background increases at low values of Y : the main reason for this comes from the functional form of the Dalitz distribution itself, which grows as Y increases (see the generated plot in Fig. 5.8), while the background is more or less flatly distributed in the (X, Y ) plane, and is therefore relatively more frequent in the low-Y region of the Dalitz plot.
Figure 5.12 Normalized dis-tribution of the background ξ as a function of Y . A linear fit shows a larger background for low values of Y , while no sig-nificant dependence on X is ob-served on MC.
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A0 0.1613E-01 0.6864E-03 A1 -0.3136E-02 0.1083E-02
The average fraction of background events in GM C(X, Y ) amounts to 0.0162± 0.0006, mainly to be attributed to the category of the badly recon-structed τ0 decays. For each bin in the plot in Fig. 5.12, ξ(Y ) is normalized to the total number of selected decays in that bin, and the uncertainties derive from the MC statistics only. As a correction on data, ξ(Y ) is then scaled by the multiplicative factor k defined and used in Par. 4.4.5 to take into account the difference between real and simulated events.