Considering each single requirement defined for the selection of K± → π0π0e±νe(¯νe) decays in Sect. 6.2, the total selection efficiency εe4tot0 can be re-expressed as the product of uncorrelated and independent terms:
εe4tot0 ≡ [εtrig· εf ilt· εK· εvtx· εclu]e40 · εac· εe4f it0 , (6.7) where
∗ ) εe4trig0 is the trigger [92] efficiency in Ke40 events;
∗ ) εe4f ilt0 is the efficiency of the event classification filter in Ke40 events;
∗ ) εe4K0 is the kaon reconstruction efficiency in Ke40 events;
∗ ) εe4vtx0 is the vertex reconstruction efficiency in Ke40 events, according to the requirements in Par. 6.2.1;
∗ ) εe4clu0 is the efficiency to observe and identify four on-time clusters in Ke40 events, according to all the cuts listed in Par. 6.2.2;
∗ ) εac is the track-to-cluster efficiency for the electron tracks reconstructed in Ke40 events;
∗ ) εe4f it0 is the efficiency of the kinematic fit for Ke40 events described in Par.
6.2.3.
The first five efficiencies listed above are defined in exactly the same way for the τ0 case and are discussed in Sect. 4.5; they appear in the definition of the total selection efficiency ετtot0 for the final τ0 sample:
ετtot0 ≡ [εtrig· εf ilt· εK· εvtx· εclu]τ0 · ε135· ετf it0 . (6.8) In the definition (6.8)
∗ ) ε135 is the efficiency of the cut (4.23) on the daughter track momentum in τ0 events;
∗ ) ετf it0 is the efficiency of the kinematic fit (exposed in Sect. 5.3) for τ0 events.
All the efficiencies defined above have to be considered as averages over the allowed values in the selected samples. The terms that, looking at event topologies, should be similar in (6.7) and in (6.8) are studied in Sect. 6.5, and their relative fractions are compared to the unity. The remaining terms are discussed in the following section.
6.4.1 Track-to-cluster efficiency
A large amount of events in the present analysis is rejected by the require-ment of an EMC cluster associated to the daughter track. The study of the track-to-cluster efficiency, εac, is performed directly on data as a function of the daughter track momentum pD.
Even if the association between tracks and clusters is basically a geomet-rical procedure [113] that doesn’t depend on the particle type, it is preferable to deal with electron tracks that topologically reproduce the Ke40 decay. One very interesting source of e± at low momenta (50÷ 150 MeV ) is given by Ke3±
decays: in fact, the vertex is spatially distributed as in a typical K± decay, and the corresponding absolute branching ratio is quite sizeable (Tab. 1.1).
While a standard selection filter for Ke3± decays [106] would require the pre-sence of the cluster associated to the daughter track, for the study of εac the track-to-cluster information has to be ignored.
The following criteria have been applied to select electron tracks from K±→ π0e±νe(¯νe) decays:
• an EMC-triggered event which has been filtered by any of the 5 KPM streaming algorithms;
• a 2-track vertex in the DC volume with transverse radius rV > 60 cm in which a K±track is involved, and such that max{d(KLH± , V ), d(π±F H, V )} <
7 cm is satisfied;
• a pair of neutral clusters with invariant mass 80 MeV < mγ1γ2 <
190 M eV ;
• the two clusters have total energy 165 MeV < E1+ E2 < 265 M eV ;
• the two clusters are on-time with respect to the vertex according to the condition |∆t012| < 5;
• the daughter track momentum and the reconstructed π0 momentum in the kaon rest frame don’t exceed 200 M eV and their sum is < 365 M eV ;
• the energies and momenta of the kaon (K), of the daughter (D) and of the di-photon (γγ) fulfill the relation for a ν missing energy-momentum:
(EK− ED − Eγγ)− |~pK− ~pD − ~pγγ|
< 20 M eV ;
Decays already identified as τ0, Kθ or Kµtags are filtered out. Since statistics is not a big problem in this case, many cuts have been hardened and new variables have been exploited, with ad hoc requirements, tuned by means of the MC simulation.
The purity reached in the obtained sample is 96.1%, the background com-ing essentially from Kµ3± and other K± decays. The efficiency εac has been defined as the ratio between the number of events in which the track-to-cluster algorithm has successfully associated a cluster to the daughter track, and the total number of selected Ke3± decays. The study has been performed for dif-ferent values of the e± momentum, pD: in the plot of Fig. 6.7, εac has been fitted according to a function
εac(pD) = P1− P2
pD(M eV ) (6.9)
in the region between 80 and 200 M eV , with P1 adimensional parameter and P2 expressed in M eV . As the momentum gets lower, the probability that the track reaches the EMC barrel decreases and the impact point is more likely to occur in the endcaps, where the cluster reconstruction efficiency is lower.
Unfortunately, there are no observed points in Fig. 6.7 covering momenta below∼ 70 MeV , therefore the fit (6.9) is needed to extrapolate the behaviour of εac at very low momenta 6. To the purpose of the Ke40 analysis, the
track-6 Although the fit function becomes negative below 60 M eV , it has to be considered null, meaning that when an electron is produced and reconsctructed with such a low momentum, the probability that the eventual cluster determined in the EMC endcaps is correctly found and associated to the track is quite negligible.
9.398 / 10
P1 1.211 0.3267E-01
P2 -73.40 5.208
Figure 6.7 Track-to-cluster efficiency εac for electrons, as extracted from real data on a selected sample of Ke3± decays. The efficiency is expressed as a function of the momentum of the electron track.
to-cluster efficiency in (6.7) has been extracted as the weighted average hεacie40 =
R εac(p)ϕe40(p)dp
R ϕe40(p)dp .
over the known electron momentum distribution ϕe40(p) in a Ke40 decay at the 4γLevel. In the resulting value,
εac = 0.359± 0.006stat± 0.009syst ,
the systematic error takes into account the uncertainties on the MC distribu-tion of ϕe40(p).
6.4.2 Kinematic fit efficiency
The study of the efficiency εe4f it0 is based on the Monte Carlo sample LP0P0NU.
In fact, there is no way to investigate how the kinematic fit works by playing directly on real data, as no physical environment can completely reproduce the Ke40 dynamics, apart from the simulated signal itself.
The fraction of Ke40 events which satisfy the two cuts on the χ2 functions defined for the signal and the τ0 background at the end of Par. 6.2.4, is given
by
εe4f it0 = 0.237± 0.006 , (6.10)
where the error is systematic and comprises the effects observed by moving the definitions of the MC σ’s in χ2e40 by an amount of 10%. Also the efficiency of the cut on the m3π invariant mass (∼ 98%) is included in (6.10). A study on the widths of the distributions shown in Fig. 6.4 can be performed on real data only for βD and δte/π, since these quantities can be studied in suitable samples (i.e. selecting Ke3± and τ0 decays and comparing the different responses between charged pions and electrons); a similar survey on m2m and on ∆E is not directly possible with analogous methods.
Concerning the efficiencies connected to the τ0 selection, the two terms ε135 and ετf it0 factorized in ετtot0 (6.8) have been already analyzed previously.
In particular, ε135is the efficiency corresponding to the cut|~pD| < 135 MeV or (4.23), which has been discussed in Par. 4.5.1 by indicating it as εpdau: the value
ε135 = 0.975± 0.002 ,
is extracted as the average of the estimates reported in Tab. 4.6 for the two selftags.
The efficiency of the kinematic fit in τ0 hypothesis has been discussed in Sect. 5.4: only the differential shape of ετf it0 was used for the extraction of the τ0 Dalitz plot parameters, whereas an absolute estimate is needed in this case.
This is extracted on real data as the ratio between the events selected after the fit and the initial number of events at the 4γLevel:
ετf it0 = 0.403± 0.002stat± 0.004syst .
In practice this definition is biased by the presence of non-τ0 background con-taminating the 4γLevel (∼ 1%) that is deemed as a source of systematics in the evaluation of ετf it0 .