Cluster efficiencies in τ 0 events

The efficiency εclu to find the four clusters from the two π⁰’s in the τ⁰ decay has been factorized as a product of two independent terms:

εclu=hA⁴clui · ε4onT ,

where Aclu represents the single cluster acceptance for each of the four τ⁰ final state photons and ε4onT is the efficiency of the cuts (4.24) and (4.25).

Estimate of Aclu

Aclu has been parametrized as a function of the cluster energy Eγ and has been estimated for photons produced in K^± → π^±π⁰ decays, in which the photon energy spectrum has been weighted to reproduce the one from τ⁰.

From a geometrical point of view, photons from K^± → π^±π⁰ and K^± → π^±π⁰π⁰ are produced with the same spatial distribution at the K^± decay ver-tex, and propagate with a uniform angular distribution before impacting on the calorimeter. No dependence on the total cluster multiplicity is hypothesized in Aclu(Eγ).

The SoN used for the evaluation of A_clu, called Kp+1, is selected from data and MC according to the drift chamber requirements for the Kθ sample de-scribed in sections 4.2.1 and 4.2.2. The cluster associated to the pion daughter track is required in accordance with the track-to-cluster algorithm. Let dπ be the length of the flight path of the charged pion and βπ its velocity, computed as pπ/Eπ: a neutral cluster (γ1 in Fig. 4.19) with energy E1 and distance d1

from the vertex is requested in the event, such that the time difference ∆tπ1

satisfies:

|∆t_q^π1− (d¹+ dπ/βπ)/c| σt(E1)²+ σt(Eπ)² < 3 ,

and |E1− ˜E1|

E˜1

< 5%

qE˜1(GeV ) ,

Figure 4.19 Sketch of a typical K⁺→ π⁺π⁰ event selected in the Kp+1 sample.

where ˜E1is the true energy of the neutral cluster estimated from the knowledge of the versor along the vertex-cluster direction, ~c₁:

E˜₁ = m²_π⁰

2 [(EK − Eπ)− (~pK− ~pπ)· ~c1] .

The purity of Kp+1 is defined as the percentage of events in which the selected neutral cluster actually comes from a photon of a K^± → π^±π⁰ decay, and is estimated from MC to be as high as 0.9956± 0.0006.

By closing the kinematics at the vertex, the four-momentum (Eγ2, ~pγ2) of the other photon (γ2 in Fig. 4.19) and its flight direction and the expected impact point P2 on the EMC can be determined. In order not to suffer by systematic uncertainties in the estimate of the coordinates of P2, in the Kp+1 sample it has been required that the expected flight direction does not inter-sect the inner cylindrical region of the DC, so that the photon γ₂ gets to the calorimeter surface without interacting through any material before. This

re-Figure 4.20 Monte Carlo

quest does not imply any relevant change on the acceptance, because photons propagating after an interaction in the detector’s volume still behave in the same way from the geometrical viewpoint, despite small effects may take place in case they lose a consistent amount of their energy.

The accuracy in the determination of E_γ (≡ Eγ²) has been studied in the Monte Carlo Kp+1 sample. The distribution is fitted with the sum of two gaussians (see fit in Fig. 4.20); it has been chosen to subdivide the energy spectrum for the τ⁰ decay (i.e. 0÷ 200 MeV ) in 10 MeV -wide bins. Then, for each Eγ bin, the cluster acceptance Aclu has been evaluated as the probability to find in a Kp+1 event a cluster j within a spherical space region of radius

∆c from P2 14 that satisfies extremely loose time and energy cuts, according to what expected for γ2 15.

14 It has been preferred to define a spherical space region of given radius ∆c around P2

rather than setting an acceptance cone of a fixed semiaperture, since the area that it would intercept would depend on the cylindrical geometry of the EMC and mainly on the position of the vertex, whereas the resolution on the cluster position is approximately constant in the barrel as in the endcaps.

15 By asking for ∆t⁰(γ1, γj) < 10 and for|E^j−E²|/E²< 10·5%/p

E2(GeV ), the fraction of background clusters observed nearby the expected position P2 is strongly reduced, without losing signal.

20.93 / 11

Figure 4.21 Cluster acceptance for photons in K → ππ⁰ events as a function of their expected energy E_γ: the plot on the left refers to data, while the one on the right is obtained from MC. In each case the plateau has been fitted to a constant.

In Fig. 4.21 the plot of A_clu(E_γ) is shown for data (left) and for MC (right) when ∆c is fixed to 100 cm. As the curves reach the plateau (at ∼ 80 M eV ), the acceptance becomes purely geometrical. Since the Monte Carlo sample ALLPHI correctly reproduces the data, it has been used to compute the systematics on Aclu to be applied on the data-extracted curve.

If no correction had to be applied, the acceptance Aclushould be considered as the ratio between the number of events with a cluster γ2inside the ∆cregion, NKp+1(∆c), and the total number of events in Kp+1 (NKp+1). In practice, a multiplicative factor ζ(∆c), obtained from Monte Carlo, has to be introduced to take into account the cases in which a spurious cluster enters the acceptance region (∆c region) instead of γ2 or γ2 is found not where expected, etc. The corrected acceptance then becomes:

Aclu = NKp+1(∆c)

N_Kp+1 · ζ(∆^c) . (4.37)

Through a more detailed study of the MC Kp+1 sample, five different background cathegories can be observed and classified:

(a) correct γ₂, but outside the ∆_c region,

(b) γ2 wrongly not found while it should be found in the EMC,

∆c= 80 cm ∆c= 100 cm ∆c = 120 cm ζ(∆c) 1.0056± 0.0012 1.0045 ± 0.0011 1.0042 ± 0.0011

Table 4.7 Dependence of ζ(∆_c) from ∆_c. The values are averaged over E_γ.

(c) spurious cluster in the ∆c region while no γ2 is in the EMC, (d) spurious cluster outside the ∆c region when correct γ2 is found, (e) not a K^±→ π^±π⁰ decay; a fake γ₂ found (e1) or not found (e2) inside the ∆c region.

As a matter of fact, ζ(∆_c) can be expressed as:

ζ(∆c) = 1 + ξ^(a) + ξ^(b)− ξ^(c)+ ξ^(d)− ξ^(e1)

1− ξ^(e) (4.38)

where the various ξ^(·) are the fractions of events belonging to each of the cathegories listed above. They slightly depend on ∆_c, showing fluctuations of the order of 0.2% when ∆c is moved around 100 cm by ±20 cm (which is the typical minimum distance between two distinct clusters in the EMC), and an even smaller net effect on ζ(∆c), as can be noticed from Tab. 4.7 and from the fit in Fig. 4.22 (left panel).

Then the corrected acceptance (4.37) is computed and reported in the right panel of Fig. 4.22

Finally, Aclu has been parametrized according to the function:

Aclu(E) = P1− P₂ E− P3

, (4.39)

with the results of the fit reported in Fig. 4.22, and hA⁴clui on the final sample of τ⁰ decays has been obtained from data as:

hA⁴clui = 1

where the systematic error derives from the uncertainties of the parameters in the fit of A_clu(E) and the statistical error has been lowered by avoiding the selftag requirement.

13.66 / 13

P1 1.005 0.1237E-02

33.81 / 16

P1 0.9623 0.1910E-02

P2 -1.081 0.1438

P3 10.34 0.7906

Figure 4.22Left: corrective multiplicative factor for A_cludue to the various sources of background in the MC Kp+1 SoN. Right: Cluster acceptance A_clu(E_γ) after the corrections discussed in the text; the fit refers to the function (4.39), with P₁ adi-mensional parameter and P2, P3 expressed in M eV .

Estimate of ε4onT

The efficiency to find on-time the four accepted clusters produced in a τ⁰ decay, ε4onT, cannot be easily factorized as a product of more elementary prob-abilities. Indeed, a SoN built up in the same fashion as the Kp+1 sample could only solve the problem of one single π⁰ → γγ, and even if this efficiency was correctly determined, the result couldn’t be extrapolated to the 4 γ’s case, since the six resulting on-time relations (4.25) would be heavily correlated, depending on the peculiar topology of the decay.

A suitable SoN, referred to as K+d, has been used firstly requiring the drift chamber preselection (Sect. 4.3) in tagged events. Then, a set of clusters in the event is selected, coherently with what done in the analysis, having energy greater than 15 M eV and not involved in the tag hemisphere, nor associated to any of the kaon daughter tracks. A very loose cut is subsequently applied on the times of the clusters left: once defined dm(dM) as the minimum (maximum) distance between the vertex and the EMC surface, the quantity

δt= dM − dm+ 2ρ^max_V

c ⊕ 2δt^maxc

is defined, where ρ^max_V and δt^max_c are, respectively, the typical maximum

dis-tance reconstructed-true vertex position and the maximum time spread for a generic EMC cluster. The values chosen from the MC (ρ^max_V = 100 cm and δt^max_c = 1 ns) ensure that 100% of the τ⁰ quartets of clusters have absolute times which are contained within a δt time window.

A K^±decay vertex is finally selected as K+d if at least one of the quartets extracted from the possible multiplets of clusters found in the event satisfies the additional conditions:

• 20 MeV < Ei < 220 M eV , ∀ i = 1, . . . , 4

• a permutation (i^k)k=1,...,4 is such that

135 M eV < (E_i1 + E_i2), (E_i3 + E_i4) < 195 M eV

• at most 2 clusters with Eⁱ > 115 M eV

• at most 2 clusters with Eⁱ < 60 M eV .

Except for the very safe request on times, only information on the cluster ener-gies has been used for the selection; no other variables like di-photon invariant masses, four-momentum conservation at vertex, etc., have been considered here, in order not to involve possible effects due to tracking and vertexing in the selected sample ¹⁶.

From the results reported in Tab. 4.8 for both Kθ and Kµ tagged K+d samples (the data sample B has been used), it is evident that the MC is not able to reproduce data and that ε4onT has to be separately calculated for the two tags, providing two different estimates of the branching ratio (4.5), to be suitably weighted for the final result.

The statistical and systematic errors on the estimates of ε_4onT come from the statistics collected in the SoN, and from the uncertainty of background subtraction, respectively.

16 Though apparently ε4onT is a quantity depending only on clustering, it actually includes some effects due to tracking and vertexing, since when the vertex is badly reconstructed (i.e.

far from its true position, normally not farer than δt^max_c ), the condition (4.25) may not be fulfilled even if the 4 clusters were effectively on-time with the true vertex.

Data MC ε4onT,θ 0.697± 0.004 0.824 ± 0.006 ε4onT,µ 0.745± 0.004 0.876 ± 0.005

Table 4.8 Comparison between data and MC on efficiency ε_4onT for K_θ and K_µ tagged K+d samples.

This discrepancy between data and Monte Carlo in the result of ε4onT is the most evident among all terms involved in ε^τ_sel⁰ . There are two possible, non-exclusive contributions that go in the direction of decreasing ε4onT in the data:

(1) the time of one (or more) of the 4 clusters may be badly reconstructed, so that the condition (4.25) is no longer satisfied, and/or (2) the resolution on the reconstructed vertex is worse in the data than in MC, so that the four clusters cannot be found on-time with it. The second hypothesis can be studied on MC by looking at the distance between the reconstructed and the true decay vertex in case of τ⁰(see Fig. 4.23): whereas∼ 91% of the vertices are reconstructed within 5 cm from its true position, in agreement with the known spatial resolution of the vertexing algorithm [99], the remainder populates a long and randomly uniform tail, reaching even very high values (> 200 cm).

It is not possible to disentangle these two contributions on data, but an independent method has been developed in order to comprehend which of them is the main responsible for the above-mentioned discrepancy.

Indeed, 4 clusters selected as those in K+d provide a sufficient number of equations to determine analitically the position of the decay vertex from which the respective photons have been originated. The method used for finding such vertex is discussed in Appendix C. The SoN constituted by those events in which this method has succeeded in the search of an acceptable vertex is called K+4s.

Hence, three different points can be considered in the intersection sample:

K+d∩K+4s:

• Vrec: the vertex position as reconstructed from the vertexing procedure;

• Vclu: the vertex position found by making use of the clusters’ times and

spatial coordinates (see Appendix C);

• Vtrue: the “true” position of the decay vertex (known on MC only).

Since Vrec and Vclu are obtained by means of totally uncorrelated methods, statistically the following relation holds:

hd(Vrec, V_clu)i =^qhδ(Vrec)i²+hδ(Vclu)i² (4.40) where δ(Vrec) and δ(Vclu) stand for the distances d(Vrec, Vtrue) and d(Vclu, Vtrue), respectively.

An approximated evaluation of the contribution in 1− ε4onT due to faults in the charged vertex can be carried out from d(Vrec, Vclu), for which the distri-bution can be observed on data as well as on MC; this quantity can be thought as the result of a convolution of δ(Vrec) and δ(Vclu) according to (4.40).

Figure 4.23Left: Monte Carlo distributions of δ(V_clu) (solid line) and d(V_rec, V_clu) (dashed line). Right: Monte Carlo distribution of δ(V_rec).

The MC distribution of δ(Vclu) is represented by the solid line of the left plot in Fig. 4.23: after the convolution, the resulting distribution d(Vrec, Vclu) is given by the dashed line. Since δ(Vrec) (see right plot in Fig. 4.23) peaks to very small values and has a quite flat behaviour at higher distances, the effect in passing from δ(Vclu) to d(Vrec, Vclu) is a light spread of its distribution and

a smearing mainly toward larger values that makes the population in the tail increase.

If only the region δ(Vrec) ∈ [6, 20] cm region is considered in the plots of Fig. 4.23, qualitatively three situations can be verified:

1. δ(Vrec) is so small ( 5 cm) that d(V^rec, Vclu)≈ δ(V^clu) and no smearing takes place;

2. δ(Vrec) is so large ( 20 cm) that δ(V^clu) shifts to some value of d(Vrec, Vclu) far outside from [6, 20] cm;

3. δ(Vrec) is comparable with δ(Vclu), which is moved to some other bin in the [6, 20] cm interval.

The last case occurs in K+d∩K+4s with probability smaller than 5%. Sup-posing that the smearing in that case is so uniform in [6, 20] cm that the mean value of the bin contents is not altered ¹⁷, the inefficiency contribution to 1− ε4onT coming from the vertexing can be regarded as the average frac-tion RV V (in the interested bins) between the number of vertices found in K+d∩K+4s after and before the requirements (4.24) and (4.25).

The fit to a constant shown in Fig. 4.24 demonstrates a quite good agree-ment between real and simulated data, at least within the uncertainties caused by the smearing effect described above. One can then conclude that the differ-ence in ε_4onT between data and Monte Carlo is determined by some possible effect involving the timing and/or the identification of the clusters in the data, which is not perfectly reproduced in the simulation. An evidence is also given by the higher value of ε4onT in the Kµ-tag hypothesis than in the Kθ-tag one (Tab. 4.8), the only relevant difference being in a lower cluster multiplicity per event.

Spurious clusters in K⁺K⁻ events

The probability Pwγ for a “spurious” neutral cluster to be found on-time with the γ’s from a π⁰ decay is studied from a data sample of K_θ⁺ K_θ⁻ events.

17 A 2 cm-binning is chosen for d(Vrec, Vclu).

Figure 4.24Fraction R_{V V} between the number of vertices in K+d∩K+4s after and before the cuts (4.24) and (4.25) applied in the analysis. Fits to a constant are performed in the interval between 6 and 20 cm for MC (left) and for data (right).

In these events the association of the two cluster pairs to the respective π⁰’s is estimated from MC to be the correct one ∼ 99.5% of the times.

Given a Kθ vertex and the clusters/photons produced by its π⁰, Pwγ is then evaluated as the probability for any other generic cluster i in the event to satisfy the on-time cut (|∆t⁰ij| < 4, j = 1, 2), as applied in the analysis for τ⁰ decays.

Figure 4.25 Energy distribu-tions of neutral clusters: green histogram represents all neutral clusters, red histogram indicates the fraction of those which are wrongly found on-time with a correct pair of π⁰’s clusters.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0 25 50 75 100 125 150 175 200 225 250 275 300

The energy distribution for these clusters after the requirement E > 15 M eV

is plotted in Fig. 4.25 and compared with the energy spectrum of all the neu-tral clusters: the ratio between the two integrals yields the average estimate

Pwγ = 0.0786± 0.0014 , which is assumed to be valid also in Kµ tagged events.

Estimate of εEtot

The efficiency to select a τ⁰ according to the cut (4.26) on the total energy of the neutral clusters given by MC is:

εEtot(M C) = 0.9970± 0.0006stat± 0.0007syst .

After the correction taking into account the larger contamination in the data, estimated from the distribution of the missing mass at vertex (see Par. 4.4.5), the efficiency becomes:

ε_Etot = 0.9947± 0.0006stat± 0.0012syst ,

where the error is dominated by the finite statistics of the MC sample and by the systematics coming from the difference between Kθ- and Kµ- tags.

Nel documento Studies on the charged kaon decays K ± → π ± π 0 π 0 (pagine 122-133)