4.2 The tagging strategy
4.2.2 The K θ -tag
In a K± → π±π0 decay at KLOE, if the kaon reaches the DC volume, two charged tracks joining in a vertex are expected in the chamber and up to three clusters can be found in the calorimeter (two neutral clusters from the π0 → γγ decay and one from the π± or from the charged particle coming out from the possible π-µ-e chain).
It is important to notice that, besides the tagging task, K → ππ0 decays can be successfully used for the study of efficiencies and resolutions directly from data, by exploiting the overconstrained knowledge of the kinematics at the decay vertex (see Par. 4.5.2).
The charged pion coming out from a K± → π±π0 vertex has a momentum
~
pπ which is fixed to 205 M eV in the K±rest frame (see Appendix A). Starting from the knowledge of the pion momentum in the laboratory frame ~pD, a Lorentz transformation in the K± rest frame yields:
~pπ = ~pD + γKβ~K
β~K = ~pK/EK
ED =qp2D+ m2π± . (4.9) The distribution of pπ are shown in Fig. 4.1 for MC and for data. The two
0 0.005 0.01 0.015 0.02 0.025 0.03
180 190 200 210 220 230 240 250 260 270
Figure 4.1 Momentum distribution of the daughter particle (charged pion mass hypothesis) in the K± rest frame for data and Monte Carlo K+K− events, after the preliminary 2-track vertex selection.
peaks correspond to the two 2-body decay modes: K → ππ0 at ∼ 205 MeV and K → µν at ∼ 236 MeV . It has to be remarked that the µν peak is not centered at the right value and is distorted by the wrong mass hypothesis in the definition (4.8). The shape of the spectrum at lower momenta is due to the semileptonic and the τ0 decays 6.
By fitting to a gaussian function the ππ0 peak of the distribution in Fig.
4.1, a kinematic cut on the daughter track momentum is chosen 7 for selecting Kθ decays:
•) 200 M eV < pπ < 210 M eV . (4.10)
6 Also unrejected K±→ π±π+π− decays or any other badly reconstructed charged kaon decay can be included in the low-momentum region, at this level.
7 The width of this cut (and similarly for the following ones) has been chosen as 3σ of the corresponding distribution, and has been opportunely defined for data and MC.
After applying this cut, the remaining sources of background on Kθ are the K → µν decays (with daughter track momentum underestimated in the track-ing procedure) and the Ke3 decays with high momentum e±. Then, to reduce these contaminations, the missing mass mm, defined as
mm =
q
(EK− ED)2− |~pK− ~pD|2 , has been required to satisfy:
•) 122 M eV < mm < 148 M eV . (4.11) In Fig. 4.2 the distributions of mm after the cut (4.10) are shown for Monte Carlo (left panel) and for real data (right panel); in the MC plot, the background distribution is superimposed. The mean values of the mm distri-bution are in perfect agreement with the known π0 mass (mπ0 = 134.9766± 0.0006 M eV [4]).
124 126 128 130 132 134 136 138 140 142 144 146
Constant 0.1770E+06
124 126 128 130 132 134 136 138 140 142 144 146
Figure 4.2 Distributions of the neutral missing mass mm, after the cut on the daughter track momentum, for MC (left) and for data (right). In the left panel both the distributions for K → ππ0 signal and background are represented on the ALLPHI sample.
Up to now only drift chamber information has been used to select K± → π±π0 decays; the two cuts described above are correlated, but they bring to a
∼ 98% purity 8 of the Kθ sample, with still no use of EMC variables.
Assuming that the π0 coming from the K± → π±π0 decay vertex produces two γ’s, a pair of neutral (i.e. not associated to any reconstructed track [113]) clusters has to be found in the EMC. Owing to the very short mean life of the π0 meson (∼ 10−16 s), the γ’s from neutral pions are practically originated at the K± vertex point.
For each cluster i reconstructed in the EMC, the energy Ei, the absolute time ti (depending on the event global time offset T 0) and the position vector
~ridescribing the coordinates of the cluster centroid are given. The definition of T 0 in charged kaons events is performed according to the complete reconstruc-tion of a tagged K± decay in the event (as already mentioned in Par. 3.2.1), from the IP to the point in the EMC where its charged decay product stops;
the correspondence with a bunch crossing in DAΦNE is taken into account to correctly determine T 0. It is important to notice that in the present anal-ysis no absolute time measurement is used, and the times of clusters ti
only appear in expressions as differences (e.g. ti− tj), so that all time offsets in the event cancel out.
Moreover, the electomagnetic calorimeter allows a very good time resolu-tion (see Par. 2.3.2), even for very low-energy clusters (down to 20÷30 MeV ).
Actually this resolution for the calorimeter has been evaluated on photons coming out from φ radiative decays, which significantly differ from those pro-duced in charged kaons events, both for their higher energy range and for their origin vertex. In fact, γ’s coming from K+K− events can be emitted uniformly in the DC volume, so their time of flight and their passage through the detector materials are more complicated to study than the ones for those generated in the origin. As far as charged kaon events are concerned, the EMC time reso-lution is presented in Appendix B, extracted from photons in K → ππ0decays.
As already discussed, accidental clusters represent the most relevant source
8 As for all the samples described in the present analysis, the “purity” is defined, for a Monte Carlo sample, as the fraction of selected decay vertices in which the K±decay mode is the one considered as “signal”.
of background in reconstructing neutral pions. What concerns the present analysis has mainly two features: (1) these clusters are randomly distributed and not related to physics, so are usually out of the vertex time and (2) their energy distribution peaks to rather low values (as can be seen from the left plot of Fig. 4.3).
Figure 4.3 Left: Energy distribution of all the neutral clusters for data and MC.
Right: Neutral clusters multiplicity in charged kaons events for data and MC. The MC sample has a lower population of accidentals than the real data.
The neutral clusters multiplicity in charged kaons events is shown in the right plot of Fig. 4.3, in which a discrepancy between data and Monte Carlo is observable.
Other background sources on neutral clusters are: low-energy cosmic rays that haven’t been rejected at trigger level or inefficiencies in the track-to-cluster algorithm (e.g. in the case of clusters due to charged pions/muons impinging on the calorimeter and producing large electromagnetic showers to which no track has been associated).
Additional neutral clusters are due to splitting, i.e. the fragmentation of the energy deposit of a particle impinging on the EMC that produces a wrong estimate of the position and of the energy of the originary clusters.
The scatter plot in Fig. 4.4 shows, for each possible pair of reconstructed clusters in a suitable time window, the tridimensional distance rij versus the
Figure 4.4Plot of the distances (in cm) of all the possible clusters pairs versus the minimum energy (in M eV ) in the pair. The line at 15 M eV refers to the minimum energy cut applied in the analy-sis.
energy Emin of the less energetic cluster in the pair. Both data and MC are characterized by a fraction of pairs of short-distanced clusters having a low Emin: this is easily explained as a little cluster which has been split from another nearby cluster with higher energy. A “splitting recovery” algorithm is applied before the analysis, merging pairs such that
rij < r0
Emin(M eV ) and Emin < Esup , (4.12) (where r0 = 1000 cm and Esup= 40 M eV ) and subsequently redefining times, energies and positions of the clusters. The effect of the algorithm, and specifi-cally the condition (4.12), has been considered as a possible source of system-atic error in the branching ratio (see Sect. 4.6).
Afterwards, to select a Kθ decay and reject part of both accidental and split clusters, a cut on the energy of the neutral clusters is performed:
•) Ei > 15 M eV . (4.13)
This cut has been tuned by looking at the energy spectrum of the γ’s involved in the K± decays studied in the present analysis.
A fundamental requirement is to find two neutral clusters j and k on-time with the previously found vertex V , i.e. the time interval defined as
∆tjk = (tj − tk)− 1
c · (|~rV j| − |~rV k|) (4.14)
has to be near to 0. In equation (4.14) ~rV irepresents the vector from the vertex point V to the centroid of the cluster i, along which direction the photons are assumed to travel at the speed of light c with no energy loss. Since the time resolution depends on the cluster energy, for each possible clusters pair, the on-time condition is required by considering the adimensional quantity:
∆t0jk = ∆tjk
qσt(Ej)2+ σt(Ek)2 , (4.15)
and asking for
•) |∆t0jk| < 5 . (4.16)
Accidental clusters surviving the cut on energy that may belong to these pairs of clusters (j, k) are powerfully rejected by applying this cut.
In the next step the invariant mass mγγ is studied for each selected pair of on-time neutral clusters,
mγjγk =q2EjEk(1− cos θjk) , (4.17) where θjk is the angle between ~rV j and ~rV k. Only the pairs that satisfy the condition
•) 85 M eV < mγγ < 185 M eV (4.18) are considered.
Finally, Kθ decays are selected by requiring the four-momentum conserva-tion in the K → ππ0 vertex, according to the cut:
•) |∆Pππ0| < 50 MeV , (4.19) where
|∆Pππ0|2 ≡ (EK − ED− Eγγ)2+|~pK− ~pD− ~pγγ|2 .
and ~pγγ and Eγγ are the di-photon momentum and energy, respectively.
The two plots in Fig. 4.5 show the distributions of mγγ for MC (left) and data (right) Kθ samples; also the results to a gaussian fit in the central re-gion of the plots are reported. The shift in the peak of the distributions with
Constant 0.1322E+05
Figure 4.5Distributions of the clusters pair invariant mass mγγ, for MC (left) and for data (right) after the Kθ sample selection.
respect to the known π0 mass [4] originates from the non-optimization of the EMC calibration for low-energy clusters [114], as those involved in this par-ticular analysis; moreover, a non perfectly linear response of the calorimeter in the region below 100 M eV can cause a systematic shift of the π0 mass peak.
In the MC simulation, the background to K± → π±π0 events is seen to come essentially from the other main K± decay modes. The detailed estimate of such background is provided in Par. 4.2.5.