In principle, all the efficiencies required for measuring Γ(Ke40 )/Γ(τ0) have been already discussed in Sect. 6.4. Anyway the ratio ετtot0 /εe4tot0 in the expres-sions (6.2) and (6.3) includes a set of similar terms− put in evidence in (6.8) and in (6.7) −, that in principle could not coincide for τ0 and Ke40 . Suitable techniques have been implemented and discussed in this section to quantify this effect.
6.5.1 Trigger
Due to the high cluster multiplicity in the event, the EMC trigger efficiency εtrig for the π±π0π0 and the π0π0e±νe(¯νe) final states assumes the highest values among the observed K± decays. Rather than studying the ratio Rtrig ≡ ετtrig0 /εe4trig0 , it is possible to analyze the analogous ratio Rself trig between the self-trigger efficiencies (i.e. the probabilities that the set of EMC clusters belonging to the “signal” hemisphere satisfies the conditions described in Par.
4.2.4). In fact, for any decay channel the trigger efficiency is higher than the self-trigger efficiency 7, which means that it is closer to 1 than εself trig: this implies that the following relation holds between the uncertainties of the ratios defined above:
δRtrig ≡ |Rtrig− 1| < |Rself trig− 1| ≡ δRself trig .
Put differently, by evaluating Rself trig and the corresponding error, a very con-servative estimate of the discrepancy between ετtrig0 /εe4trig0 and 1 can be extracted.
Moreover, this can be done directly on real data by defining suitable classes of events for the two decay modes:
• ) 4γ + π: events in which four photons/clusters have been found on-time with a charged vertex and, when present, also the cluster associated to
7 The condition εtrig > εself trig is always verified as an obvious consequence of the trigger logic: if a restricted set of clusters is enough to trigger an event with a given probability, the additional presence of clusters from the other hemisphere can only increase such probability.
the π± track (only τ0 decays);
• ) 4γ + x: a subclass of τ0 events where, besides the four on-time clusters, a fifth cluster (different from the one associated to the charged pion) in the endcaps has been reconstructed. More than one case can be classified for a given τ0 decay.
The two classes are subdivided in bins (20 M eV -wide) of energy E5of the fifth cluster (from 0 to 220 M eV , which is the typical energy range for the charged particle in the decays of interest).
Figure 6.8 Self-trigger effi-ciencies for the “4γ+π” sam-ple (upper panel) and for the
“4γ +x” sample (lower panel) as functions of the energy of the fifth cluster (see the text for a more detailed explana-tion). The linear fits for both plots provide compatible re-sults.
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A0 0.8947 0.9175E-02
A1 0.3178E-03 0.9065E-04
19.72 / 8
A0 0.8837 0.4281E-02
A1 0.4030E-03 0.4100E-04
In Fig. 6.8 the efficiencies εself trig for the “4γ + π” and “4γ + x” samples are shown, as functions of E5 on a small 2002 data sample. In both samples the probability to self-trigger an event increases as the fifth cluster becomes more and more energetic, and a linear fit seems satisfactory to explain such increase.
By comparing the two fits, and adapting the E5 distribution of the “4γ +x”
sample to the expected energy spectrum of the electron cluster in Ke40 events, the estimate of the ratio between the two considered trigger efficiencies can be
finally expressed as
ετtrig0 /εe4trig0 = 1.02± 0.02syst± 0.01stat (6.11) which is compatible with 1. This result implies that− to the required precision level − the EMC trigger is independent of the nature of the cluster (e/π in this case).
6.5.2 Event Classification
Other differences between the Ke40 and the τ0 selection algorithms could in principle concern the criteria adopted at the Event Classification level. These effects have been analyzed on Monte Carlo by evaluating the efficiency εf ilt
as the ratio between the number of selected events and the total number of triggered events. In particular, the ratio between the efficiencies for the τ0 and the Ke40 modes, respectively, is measured as:
ετf ilt0 /εe4f ilt0 = 1.006± 0.0011 , (6.12) in agreement with the hypothesis that the streaming algorithms treat the two decay channels in the same way.
6.5.3 Kaon identification and vertexing
To study how the DC variables could be differently treated in the two ana-lyses, the MC simulation has been used to understand the effects on the ef-ficiencies εK and εvtx, concerning the reconstruction/identification of the K± track and the reconstruction of the vertex, respectively.
In the left panel of Fig. 6.9 εK is plotted as a function of the transverse radius of the generated decay point (no matter if the kaon track and/or its vertex have been found); from the superposition of εKfor Ke40 (in black) and for τ0(in red), a very good agreement is evident. The probability to reconstruct the charged kaon track increases as the time of flight gets higher until a maximum around∼ 100 cm is reached; in general, the shape of εK depends on the cuts applied at the level of Event Classification (Par. 3.3.1). The average ratio between the values of the efficiencies to reconstruct a kaon track is found to
be
ετK0/εe4K0 = 0.992± 0.0013 . (6.13)
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A0 0.6957 0.1922E-02
91.57 / 49
A0 0.7070 0.1912E-02
Figure 6.9 Left: Monte Carlo shape of εK as a function of the transverse radius rV of the K± decay point, for Ke40 (black) and for τ0 (red). Right: vertex efficiency εvtx as a function of the cosine of the angle between the trimomenta of the kaon and of the daughter particle in the laboratory frame, for Ke40 (upper black plot) and for τ0 (lower red plot).
The vertex efficiency εvtx can slightly differ between the Ke40 and the τ0 decays for kinematic reasons, since the first one is a four-body and the second is a three-body decay. This can imply two possible orders of effect: (1) the electron in the Ke40 has a lower momentum than the charged pion in a τ0 decay but a longer tail up to ∼ 200 MeV , and (2) a different angular distribution for the emitted charged particle characterizes the two decay channels.
In the right panel of Fig. 6.9 the efficiency εvtx of the cuts on the vertex is displayed for Ke40 (τ0) in black (in red) as a function of the cosine of the angle ϑKD between the kaon and the daughter tracks at the vertex in the laboratory frame. The MC also shows that the efficiency is slightly lower for backward-emitted K±’s products (i.e. for cos ϑKD → −1), but it is essentially constant over all the possigle angular configurations. The value
ετvtx0 /εe4vtx0 = 1.016± 0.005 . (6.14) is obtained by fitting to a constant and comparing the two MC distributions.
6.5.4 Clustering
As far as the clustering and the corresponding cuts in the analysis are concerned, the observed behaviour of the two neutral pions in both Ke40 and τ0 is in practice the same. The most significant difference between the photons from a Ke40 decay and those coming from a τ0 is in the energy spectrum, which reaches a larger tail at higher values in the case of the K± → π0π0e±νe(¯νe) decay.
Supposing that the geometrical acceptance for a single neutral cluster has the same dependence on the energy Aclu(E) as the one parametrized in (4.39), this effect can be studied by checking on MC how the average value of hA4clui changes between the Ke40 and the τ0 distributions. As can be seen in Fig. 6.10, the two MC distributions are very similar if the average values are compared;
the ratio of the εclu efficiencies can be obtained as
ετclu0 /εe4clu0 = 1.007± 0.007 , (6.15) where the error includes also the small discrepancies related to the efficiency of the on-time condition (4.25).
Figure 6.10 Monte Carlo distributions of hA4clui for Ke40 (in black) and for τ0 (in red).
Although dominantly based on Monte Carlo, the investigation on the ef-ficiencies corresponding to the analysis cuts performed until the 4γLevel has shown that both τ0 and Ke40 selections reduce the initial signal samples by roughly the same amounts 8.