Studies on the charged kaon decays K ± → π ± π 0 π 0

DOTTORATO DI RICERCA IN FISICA − XV CICLO

Tesi di Dottorato

Studies on the charged kaon decays K ^± → π ^± π ⁰ π ⁰

and K ^± → π ⁰ π ⁰ e ^± ν

¯

_e with the KLOE detector

Supervisors:

Prof. Edoardo Gorini

Dott. Margherita Primavera

Candidate:

Dott. Andrea Ventura

Lecce, September 2003

Introduction I

1 τ⁰ and K_e4⁰ decays 1

1.1 The branching ratio of the K^± → π^±π⁰π⁰ decay . . . 1

1.1.1 Review of experimental results on the τ⁰ branching ratio 2 1.2 The Dalitz plot of the K^± → π^±π⁰π⁰ decay . . . 5

1.2.1 K^±→ π^±π⁰π⁰ kinematics . . . 5

1.2.2 Review of experimental results on the Dalitz plot param- eters . . . 9

1.2.3 Present status and perspectives of direct CP violation measurements in K^± → (3π)^± processes . . . 15

1.3 The K^±→ π⁰π⁰e^±νe(¯νe) decay . . . 16

1.3.1 Kinematics of K_l4 decays . . . 17

1.3.2 Review of experimental results on the K_e4⁰ decay . . . 21

2 The KLOE experiment at DAΦNE 27 2.1 CP violation in the neutral kaon system . . . 27

2.2 The DAΦNE accelerator . . . 31

2.3 The KLOE experiment . . . 34

2.3.1 The drift chamber . . . 35

2.3.2 The electromagnetic calorimeter . . . 39

2.3.3 The trigger . . . 44

2.3.4 The data acquisition . . . 47

3 Data reconstruction and event classification 51 3.1 KLOE data taking . . . 51

3.2 Data reconstruction . . . 53

3.2.1 Clustering . . . 53 I

3.2.2 Cosmic and background filters . . . 56

3.2.3 Tracking . . . 59

3.2.4 Vertexing . . . 62

3.3 Event Classification . . . 62

3.3.1 Charged kaon events . . . 63

3.3.2 The KPM stream selection algorithms . . . 64

3.4 The Monte Carlo simulation . . . 68

3.4.1 Spurious DC hits . . . 69

3.4.2 Accidental EMC clusters . . . 70

4 Measurement of the τ⁰ branching ratio 73 4.1 The applied method . . . 74

4.1.1 Data and Monte Carlo samples . . . 76

4.2 The tagging strategy . . . 77

4.2.1 Preliminary vertex selection . . . 79

4.2.2 The Kθ-tag . . . 80

4.2.3 The Kµ-tag . . . 87

4.2.4 Self-triggering conditions . . . 88

4.2.5 Background in Kθ and Kµ samples . . . 89

4.3 Selection of K^± → π^±π⁰π⁰ decays . . . 90

4.4 Background in τ⁰ events . . . 93

4.4.1 Contamination from main K^± decays . . . 94

4.4.2 Contamination from K^± rare decays . . . 94

4.4.3 Contamination from K_l4⁰ decays . . . 95

4.4.4 Contamination from non K⁺K⁻ decays . . . 98

4.4.5 Comparison with data . . . 99

4.5 Efficiency evaluation . . . 101

4.5.1 Charged track and vertex efficiencies in τ⁰ events . . . . 102

4.5.2 Cluster efficiencies in τ⁰ events . . . 112

4.6 Systematics and residual effects . . . 123

4.6.1 Systematics on clustering . . . 123

4.6.2 Cosmic veto effect . . . 126

4.6.3 FILFO algorithm effect . . . 127

4.7 Final results . . . 127

5 Measurement of the τ⁰ Dalitz plot parameters 133

5.1 The measurement method . . . 134

5.2 Selection of τ⁰ decays . . . 136

5.2.1 Preselection cuts . . . 136

5.2.2 Further clustering requirements . . . 137

5.3 The kinematic fit . . . 139

5.3.1 Requirements and definition . . . 139

5.3.2 Fit performances . . . 142

5.3.3 Reconstructed Dalitz variables . . . 144

5.4 Differential efficiencies . . . 148

5.4.1 Background estimate . . . 150

5.5 Dalitz plot distribution and preliminary results . . . 152

5.5.1 A preliminary study on systematics . . . 154

5.5.2 Results on Dalitz plot parameters . . . 158

6 Measurement of the K_e4⁰ branching ratio 161 6.1 The method of measurement . . . 162

6.1.1 Data and Monte Carlo samples . . . 163

6.2 Selection of K^±→ π⁰π⁰e^±νe(¯νe) (K_e4⁰ ) decays . . . 163

6.2.1 Tracking and vertexing requirements . . . 166

6.2.2 Clustering requirements . . . 166

6.2.3 Further requirements for τ⁰ and K_e4⁰ . . . 168

6.2.4 Final requirements . . . 169

6.3 Background to K_e4⁰ events . . . 173

6.3.1 MC estimates of contamination . . . 174

6.3.2 Correction on data . . . 175

6.4 Selection efficiencies . . . 178

6.4.1 Track-to-cluster efficiency . . . 179

6.4.2 Kinematic fit efficiency . . . 181

6.5 Studies on efficiencies: a comparison between K_e4⁰ and τ⁰ decays 183 6.5.1 Trigger . . . 183

6.5.2 Event Classification . . . 185

6.5.3 Kaon identification and vertexing . . . 185

6.5.4 Clustering . . . 187

6.6 Results and discussion . . . 188

6.6.1 Branching ratio and form factor . . . 188

6.6.2 Differential spectra . . . 190

6.7 Outlook . . . 191

Conclusions 195

A Momentum distribution of charged pions in K^± rest frame 197 B Cluster time resolutions in charged kaon events 199 C Determination of the vertex position with 4 on-time clusters 201

Bibliography 205

Acknowledgments 213

The KLOE experiment has been collecting data since April 1999 at DAΦNE, the e⁺e⁻ collider of the Laboratori Nazionali di Frascati, operating at the φ resonance peak (√

s ' 1020 MeV ). The main goal of KLOE is to measure with an accuracy of∼ 10⁻⁴ the parameter<e(⁰/), which describes the direct CP violation in the neutral kaon system.

DAΦNE was designed to work at a luminosity of ∼ 5 × 10³²cm⁻²s⁻¹, but in the first three years of run it has reached a luminosity below this value (the peak was∼ 8×10³¹cm⁻²s⁻¹in the summer 2002) which, however, is estimated to increase of factor 2÷ 3 during the upcoming 2003-2004 run. The average luminosity per day has been improving continuosly with time, and the total integrated luminosity has grown year by year (2.5 pb⁻¹in 1999, 25 pb⁻¹ in 2000, 190 pb⁻¹in 2001 and 500 pb⁻¹in 2002). The total number of produced φ mesons corresponds to∼ 1.5 × 10⁹. This statistics is enough to perform a considerable number of interesting physics measurements, like studies on radiative φ decays and analyses of charged and neutral kaons copiously produced at DAΦNE.

Subject of this thesis is the study of some charged kaon decays and, in particular, three measurements: the absolute branching ratio and the Dalitz plot parameters of the K^± → π^±π⁰π⁰ decay and the ratio of decay rates

I

Γ(K^± → π⁰π⁰e^±νe(¯νe))/Γ(K^±→ π^±π⁰π⁰).

In the first chapter, a short review is given of the theoretical aspects con- nected to the kinematics and Dalitz plot distributions of the K^± in 3-pions decays (in particular K^± → π^±π⁰π⁰) and of the most relevant physics issues that can be studied by analyzing the Kl4 decays.

The second chapter is dedicated to a brief description of DAΦNE and of the KLOE detector. The trigger and the data acquisition systems are also pointed out.

The third chapter provides a general description of the offline data recon- struction procedures and of the streaming algorithms used for the event clas- sification (with more attention to the case of φ → K⁺K⁻ events). The role of the Monte Carlo simulation in the analysis is remarked, since the collected data have shown the presence of a considerably high background produced by DAΦNE, which had not been simulated when the KLOE data taking started.

The effects caused by machine background and electronic noise in the drift chamber, which are relevant in the case of K⁺K⁻ events, are described in detail, together with the procedure implemented in Monte Carlo to reproduce the background, to which part of the work described in this thesis has been devoted.

The fourth chapter gives an accurate description of the methods applied to measure BR(K^± → π^±π⁰π⁰). Two independent tagging strategies for K^±, based on the identification of K^∓ → µ^∓ν¯µ(νµ) or K^∓ → π^∓π⁰ decays in the event, are studied in detail, and the criteria adopted for selecting the K^± → π^±π⁰π⁰ signal are shown. A complete discussion of the procedures to compute the analysis efficiencies is provided: given the overconstrained kine- matics of this fully hadronic decay process, the redundance of information from the drift chamber and the calorimeter allows to select highly pure signal samples without using some of the variables, so that it is possible to extract the efficiency related to a cut on a specific physical quantity directly on a suit- able data sample, simply by hardening the cuts on the other quantities. The

Monte Carlo simulation is mainly used for a first estimate of the background in the final sample and to define the cuts in the analysis. Systematic errors are evaluated and checks on the stability of the measurement are performed respectively by varying the values of the cuts and by repeating the analysis over separate data subsamples.

The fifth chapter contains a study of the most relevant kinematic quanti- ties involved in K^± → π^±π⁰π⁰ decays and a detailed comparison between real and simulated data is reported. Then, in order to proceed in the analysis of the Dalitz plot distribution, a kinematic fit is carried out: the method and the results on the variables involved in the process are illustrated. A fine tuning of the simulation is then studied to better reproduce some data resolutions on invariant quantities. After discussing the efficiencies, determined through Monte Carlo studies, the final shape of the Dalitz plot is reconstructed and the parameters g, h and k are extracted from a fit to the theoretically predicted functional form of the distribution. The contributions to the total errors on these parameters are discussed and quantified.

The sixth chapter is devoted to the measurement of the ratio Γ(K^± → π⁰π⁰e^±νe(¯νe))/Γ(K^± → π^±π⁰π⁰). Firstly, the criteria used to select K^± → π⁰π⁰e^±νe(¯νe) (or K_e4⁰ ) decays are described and particular attention is paid in optimizing the rejection of the main sources of background, through the appli- cation of a kinematic fit, aiming at improving the knowledge of the dynamics of the decay, are then discussed. A very large Monte Carlo sample is used to tune the cuts applied in the analysis and to estimate the contamination.

The advantages of performing a measurement of BR(K → π⁰π⁰eνe) relative to the K → ππ⁰π⁰ decay are explained: in this way, an unprecedented sta- tistics of K_e4⁰ events can be collected (since no tag is required in the analysis), while keeping the systematic errors to an acceptable level, owing to the sub- stantial similarity concerning the event topologies and the selection algorithms of the two decays involved in the measurement. The ratio of decay rates is computed and the absolute branching ratio for the K_e4⁰ decay is derived, al- lowing a test of the validity of the ∆I = 1/2 rule by comparing the result obtained here with those of the other Ke4 decays. Finally, an estimate of the

form factor F00 is provided, and a preliminary study of the differential spectra with respect to a given set of kinematic variables is presented for the first time in this decay.

This chapter is focused on the theoretical and phenomenological aspects related to the two charged kaon decays examined in the present work: K^± → π^±π⁰π⁰ (or τ⁰) and K^± → π⁰π⁰e^±ν_e(¯ν_e) (or K_e4⁰ ). Particular attention is devoted to the implications of the direct CP violation that can be derived by observing the Dalitz plot distributions for both positive and negative kaons in the first decay, and to the knowledge of the Kl4 form factors in the second decay.

Brief descriptions of the experimental measurements performed in the past, concerning the branching ratio and the Dalitz plot parameters of the τ⁰ decay, and the branching ratio and the form factors of the K_e4⁰ decay, are reported.

1.1 The branching ratio of the K

→ π

π

π

decay

The K^± → π^±π⁰π⁰ decay mode, also indicated as τ⁰, represents an interest- ing process for testing directCP violation [1]. Since no oscillation is allowed by charge conservation, CP violation in K^± → (3π)^± decays can occur through the direct mode only.

1

CP invariance requires that partial rates have to be the same:

Γτ⁰(K⁺) = Γτ⁰(K⁻) ; consequently the asymmetry in the decay amplitudes

A = Γ⁺_τ⁰ − Γ⁻τ⁰

Γ⁺_τ⁰+ Γ⁻_τ⁰

constitutes a measurement of CP violation. While theoretical estimates [2, 3]

yield O(10⁻⁸) results, recent measurements [4] of the asymmetry have produced A = (0± 6) · 10⁻³ .

1.1.1 Review of experimental results on the τ⁰ branching ratio Among the six main K^± decays, listed in Tab. 1.1, the τ⁰ is the one which is known with the worst relative accuracy (more than 2%).

Charged kaon decay PDG value K^± → µ^±νµ(¯νµ) (63.43± 0.17)%

K^± → π^±π⁰ (21.13± 0.14)%

K^± → π^±π⁺π⁻ (5.576± 0.031)%

K^± → e^±π⁰νe(¯νe) (4.87± 0.06)%

K^± → µ^±π⁰νµ(¯νµ) (3.27± 0.06)%

K^± → π^±π⁰π⁰ (1.73± 0.04)%

Table 1.1 Branching ratios for the main K^± decay modes as reported by the PDG [4].

The experimental determinations of the branching ratio for the K^± → π^±π⁰π⁰ decay which are considered by the Particle Data Group [4] are shown in Tab. 1.2. Only the first four listed measurements have been taken into account in the PDG world fit:

BR(τ⁰) = (1.73± 0.04) · 10⁻² .

All the measurements date back to more than thirty years ago and are char-

Document ID Γ(π^±π⁰π⁰)/Γtotal (10⁻²) Events

Chiang 72 [5] 1.84± 0.06 1307

Pandoulas 70 [6] 1.53± 0.11 198

Shaklee 64 [7] 1.8± 0.2 108

Roe 61 [8] 1.7± 0.2

Taylor 59 [9] 1.5± 0.2 Alexander 57 [10] 2.2± 0.4 Birge 56 [11] 2.1± 0.5

Table 1.2 Experimental measurements of the branching ratio for the τ⁰. Only the first four results in the table are used by the PDG [4] for the fit.

acterized by rather large relative errors (3.5% in the best case); these reasons gave strong motivations to perform a new and independent measurement with larger statistics at KLOE.

The result obtained by Roe et al. [8] at the University of Michigan in 1961 is the oldest one. The measurement was based on a xenon bubble chamber experiment. 6300 positive kaons were produced in a 700 M eV /c beam at the Berkeley Bevatron and reduced in momentum down to ∼ 400 MeV/c by means of a moderator at the entry of the chamber where the K⁺ decayed (no magnetic field was used in the experimental setup). The scanned photographs were analyzed and classified in different categories depending on the various K⁺ decay modes; such a discrimination was carried out by recognizing the electron pairs produced by the conversion of the gamma rays from neutral pions and by observing the different radiative energy loss for the identified charged secondaries.

The determination of the efficiencies involved in the method of measure- ment consisted of two parts: computation of conversion probabilities in the chamber (through a Monte Carlo technique) and evaluation of the percentages of missed electron pairs due to scanning inefficiency (where the main system- atic effects were quoted). The branching ratios for the six main K⁺ decays were estimated, and about ∼ 100 events yielded a value of 0.017 ± 0.002 for

the K⁺ → π⁺π⁰π⁰ decay.

In 1964 a new measurement of the K⁺ meson decay modes was performed at the Bevatron by Shaklee et al. [7]. Even though experimentally both the ap- paratus and the technique used were the same as for the previously described measurement, a different method of analysis was used. Here the analyzed statistics was larger (21000 K⁺ mesons) and about 10600 events met prelim- inary requirements about fiducial volumes and kinematic cuts avoiding beam contamination of π⁺. The τ⁰decays were tagged as having three or four gamma rays, and were counted as 108± 10; the branching ratio was measured dividing this number by a Monte Carlo evaluated efficiency (about 57%), providing a final result of 0.018± 0.002.

The Pandoulas et al. [6] experiment was held in 1970 with a 400 M eV /c K⁺ beam at the Bevatron. In the measurement great remark was given to the study of partial rates and secondary energy spectra of the two decay modes in 3π of the charged kaon. The amount of analyzed data consisted of 12976 total K⁺mesons, 198 of which were classified as τ⁰by measuring the momentum and the energy loss of the secondary track. The obtained value of the branching ratio (0.0153± 0.0011) had an over-all uncertainty dominated by the quite poor statistics, while the main source of systematics was the misidentification of K⁺ primaries.

The most recent result for the K⁺→ π⁺π⁰π⁰ branching ratio was obtained by Chiang et al. [5] in 1972 with a 1.84 GeV /c K⁺meson beam at the Argonne Zero Gradient Synchrotron (ZGS). The K⁺flux was kept low and a 10:1 ratio of π⁺:K⁺ was highly reduced by means of a liquid differential Cherenkov counter:

a loose trigger ensured a very little bias in K⁺ mesons observed decaying in the spark chambers. The use of charged kaons in flight forced the decay products to be detected into a narrow forward cone and the well-collimated and energetic γ rays produced by π⁰’s decays resulted in a high efficiency for detecting τ⁰ events.

Event classification was based on the presence of a secondary charged parti- cle and on the number of visible γ ray conversions in the shower chamber. The

final scanning of the film yielded a total of 97967 K⁺ events. The distribution of the various decay modes was rearranged by performing some Monte Carlo calculations on the basis of a fit to the K⁺ → π⁺π⁰ hypothesis which could quantify the transferring of events from a classification category to another one and also help to estimate the background contribution. An amount of 1307 events selected as τ⁰ decays provided the result BR(τ⁰) = 0.0184± 0.0006, which is still the most precise estimate up to now.

All these four measurement of the τ⁰ branching ratio described so far have been performed on samples which contained only positive kaons, so that no information was reached about possible asymmetries in the K⁺ and the K⁻ decay rates andCP violation.

1.2 The Dalitz plot of the K

→ π

π

π

decay

Many theoretical issues [1] refer to the study of the K^± → π^±π⁰π⁰ and K^±→ π^±π⁺π⁻ decays (known as τ⁰ and τ , respectively), and in particular to the Dalitz plot parameters of these non-leptonic decays, since it can provide information on CP violation, validity of the ∆I = ¹₂ rule, etc. A precise measurement of the three “slope parameters” g, h and k can improve the knowledge of the contributions to the decay amplitude.

1.2.1 K^± → π^±π⁰π⁰ kinematics The definition

K^± → π1 π2 π3

can be applied to the case of the τ⁰ decay, where π₁ and π₂ are the two neutral pions and π3 is the charged pion.

The Lorentz-invariant quantities

si = (PK − Pⁱ)² = (mK− mⁱ)²− 2m^KTi , i = 1, 2, 3 , (1.1) being PK and Pi the kaon and the i^th pion quadrimomenta, and

s₀ = 1 3

X3 i=1

s_i = 1 3

m²_K+ m²₁+ m²₂+ m²₃^ (1.2)

can be introduced. Since s1+ s2+ s3 = 3s0, only two variables are independent in this system, and the Dalitz variables can be defined as:

X = s2− s¹

m²_π^± , (1.3)

Y = s₃− s0

m²_π^± . (1.4)

Figure 1.1 Dalitz plot for the K→ 3π decay. The kinetic energies Ti of the pions in the kaon rest frame are represented on the three axes at 120^◦. The centre of the diagram is the origin of the plot and represents the symmetric point in which T₁ = T₂ = T₃ = Q/3. If referred to this origin, any point in the plot has cartesian coordinates which are proportional to X/√

3 and Y , respectively.

The choice of dividing by the charged pion mass is used here to define X and Y as small and adimensional quantities. Starting from the kinetic energies Ti of the pions in the K^± rest frame, it is possible to define the constant quantity

Q = T₁ + T₂+ T₃ = m_K−

X3 i=1

m_i .

In order to easily evaluate phase space integrals, Ti can expressed in polar coordinates [12] considering the origin of the Dalitz plot (fig.1.1) as the pole:

T_1,2 = Q 3



1 + r cos

2 3π∓ ϕ



(1.5)

T₃ = Q

3 (1 + r cos ϕ) (1.6)

where r and ϕ are such that−π < ϕ ≤ π and 0 ≤ r ≤ r0, and r0 is a function of ϕ defining the contour of the Dalitz plot. Its expression is given by:

1− (1 + α) r²0− αr0³cos (3ϕ) = 0 (1.7) with α = 2Qm_K/(2m_K− Q)². The unit of length here is chosen so that the distance of a generic point in the Dalitz plot from the side which opposes the T_i axis is given by 3(T_i − mi)/Q. In the non-relativistic limit, α becomes negligible and r0 = 1 defines the circle inscribed in the triangle. Expressed in polar coordinates, the Dalitz variables are

X = 2

√3 mK

m²_π^±Qr sin ϕ (1.8)

Y = −2 3

mK

m²_π^±Qr cos ϕ . (1.9)

The decay rates can be expressed in polar variables and obtained integrating over the Dalitz plot:

Γ (K → 3π) = 1 (4π)³mK

√3 18Q²

Z Z

r|A (r, ϕ)|²drdϕ

where A (r, ϕ) represents the transition amplitude. In the kaon frame the kinetic energies allowed for the three pions are relatively small (at most 50 MeV [12]), so that a series expansion in X and Y is possible for the decay amplitude:

|A (K → 3π)|² ∝ 1 + gY + jX + hY²+ kX² . (1.10) The coefficient g measures the linear dependence of the s3 variable in the Dalitz plot, while h and k quantify the quadratic dependence on s3 and on (s2 − s¹), respectively. If CP is conserved, then the coefficient j must be 0 since it is related to the asymmetry of the Dalitz plot, and g, h and k must be the same for K⁺ and K⁻. In Tab. 1.3 the experimental values of the width and of the parameters g, h and k are listed for various K → 3π decay channels.

In the τ and in the τ⁰ decay cases, the widths and the linear slopes g are known at a few % level, while the quadratic coefficients are measured with a ∼ 30%

accuracy.

Channel K⁺→ π⁺π⁻π⁺ K⁺ → π⁺π⁰π⁰ KL→ π⁺π⁻π⁰ KL→ π⁰π⁰π⁰ Γ(10⁶s⁻¹) 4.52± 0.04 1.399± 0.032 2.43± 0.04 4.08± 0.06

g (10⁻¹) −2.154 ± 0.035 6.52± 0.31 6.78± 0.08 −

h (10⁻²) 1.2± 0.8 5.7± 1.8 7.6± 0.6 −0.33 ± 0.11 ± 0.07

k (10⁻²) −1.01 ± 0.34 ^1.97± 0.45 ± 0.29 0.99± 0.15 −

Table 1.3 Experimental values of widths and Dalitz plots parameters for the K⁺ and the K_L 3-pions decays [4].

The transition amplitudes can be expanded for processes with definite isospin selection rules (up to quadratic terms in X and Y ). For charged kaons, according to the transition rules ∆I = ¹₂ (index 1) and ∆I = ³₂ (index 3) only, and neglecting imaginary parts due to strong interactions, the result is given by [13, 14, 15]:

A(K⁺ → π⁺π⁺π⁻) = (2α1− α3) e^iδ1S +



β1 −1 2β3



e^iδ1M +√ 3γ3e^iδ2



Y (1.11) +2 (ζ1+ ζ3)



Y²+ 1 3X²



− (ξ1+ ξ2− ξ3⁰)



Y²− 1 3X²



A(K⁺ → π⁺π⁰π⁰) =−1

2(2α1− α3) e^iδ1S +



β1− 1 2β3



e^iδ1M −√ 3γ3e^iδ2



Y (1.12)

− (ζ1+ ζ3)



Y²+1 3X²



− (ξ1+ ξ2+ ξ₃⁰)



Y²− 1 3X²



where the coefficients α, β, γ, ζ, ξ are related to the three different states of isospin involved in the decay: |(3π)I=1, symm.i, |(3π)I=1, mixed symm.i and

|(3π)I=2i.

Both the equations (1.11) and (1.12) are valid only for small values of the variables X and Y , essentially nearby the central part of the Dalitz plot, and their general expression is quite more complicated. In the previous expansions, the phases δ due to strong interaction in the final state are contained, but both in non-relativistic approximation [16, 17] and in Chiral Perturbation Theory (χP T ) at first order [18], the relative phase at the center of the Dalitz plot is estimated to be rather small:

δ1S− δ1M ' 0.08 ,

as can be easily expected by considering the small phase space available in the decay.

IfCP is conserved, all the coefficients present in the expansions (1.11) and (1.12) must be real and the decay amplitudes must be equal for K⁺ and for K⁻ separately. In case of CP violation, the two different amplitudes could be studied through their asymmetry or, equivalently, by means of the asymmetry of the linear parameter g, which is orders of magnitude higher than the overall asymmetries [3] and therefore it’s easier to measure. For the two hadronic decays τ and τ⁰ the asymmetries of the coefficient g can be written as:

A⁺⁻_g = ∆g 2g

!

τ

=

= g (K⁺→ π⁺π⁺π⁻)− g (K⁻→ π⁻π⁺π⁻)

g (K⁺→ π⁺π⁺π⁻) + g (K⁻ → π⁻π⁺π⁻) = (1.13)

= Im^h(2α1− α³)^∗β1 −¹₂β3

sin(δ1S − δ^1M) + (2α1− α³)^∗√

3γ3sin(δ1S − δ²)ⁱ (2α₁− α3)^β₁− ¹₂β₃+√

3γ₃^

A⁰⁰_g = ∆g 2g

!

τ⁰

=

= g (K⁺ → π⁺π⁰π⁰)− g (K⁻→ π⁻π⁰π⁰)

g (K⁺→ π⁺π⁰π⁰) + g (K⁻ → π⁻π⁰π⁰) = (1.14)

= Im^h(2α1− α3)^∗β1− ¹₂β3

sin(δ1S− δ1M) + (2α1− α3)^∗√

3γ3sin(δ1S − δ2)ⁱ (2α₁− α3)^β₁− ¹₂β₃−√

3γ₃^

The predictions provided at first order by the Chiral Perturbation Theory and in the framework of the Standard Model [1, 19, 20] for A⁺⁻_g and for A⁰⁰_g range from 10⁻⁶ and 10⁻³. Recent theoretical errors have converged to values not exceeding ∼ 10⁻⁵ for the asymmetries [21]. A wide variety of supersym- metric models give larger values (∼ 10⁻⁴) [22].

1.2.2 Review of experimental results on the Dalitz plot parameters

In the central column of Tab. 1.3 the values of the parameters of the Dalitz distribution for the τ⁰ decay are shown, as extracted by fitting the published

results [4]. The estimate of g, though characterized by a 5% error, shows a low confidence level (0.001); moreover h is known with a large uncertainty, and only one measurement has been performed on k until now: then, besides the study of the asymmetry A⁰⁰_g , the parameters themselves are also worth to be investigated.

In the following, the seven measurements of g, h and k considered by the Particle Data Group will be shortly described. Other results have not been taken into account due to large errors [6, 23] or because only a linear fit in the Y variable was performed [24, 25, 26].

The first measurements of the Dalitz parameters for the τ⁰ were published by Davison et al. [27] in 1969. The experiment was held at the Lawrence Radiation Laboratory, Berkeley, and at the University of California, Riverside.

Two different methods were applied to measure the spectrum of the posi- tive pions produced by the K⁺decay: emulsions to scan the low kinetic energy region (from 1 to 21 M eV ), and a heavy-liquid bubble chamber to investigate the higher energy range (from 8 to 52 M eV ). Since the photons produced by the decay of the two π⁰’s were not detected in the experiment, the study of the kinematics only involved the Y variable. Both the emulsion and the bubble chamber spectra were separately normalized to the number of events observed in the respective energy region. For each energy bin a geometrical acceptance factor was estimated, and various corrections were also applied for the π⁺’s scattering, flight decays and other effects, mainly through Monte Carlo calculations.

The collected data (a total number of 4048 K⁺ → π⁺π⁰π⁰ events) were an- alyzed with six different fits, including a σ-resonance hypothesis: the quadratic fit in Y yielded the results g = 0.544± 0.048 and h = −0.026 ± 0.050.

The measurement by Aubert et al. [28] was the first one in which all the kinematic variables of the K⁺ → π⁺π⁰π⁰ decay were considered, since both the π⁺ and the π⁰ spectra were studied.

The data sample was collected at the X2 experiment [29] (CERN) and con- sisted of 1365 events, obtained by stopping 5· 10⁶ K⁺ in a heavy-liquid bubble chamber. Two separate analyses were carried on (Orsay - Paris and Brussels),

in which, respectively, the converted γ rays were and were not considered.

The contamination due to one spurious γ ray substituting a real photon was estimated by means of some calculations on a set of simulated τ⁰ decays. In both the two analyses the π⁺ energy spectrum was corrected for various ef- fects (flight decays, flight interactions, geometrical losses, contaminations due to K⁺ → µ⁺π⁰νµ decays with spurious γ rays), while the π⁰ energy spectrum was derived through the parametrization of the probability of revealing a γ ray as a function of its energy Eγ.

The hypothesis of a linear matrix element (i.e. with negligible second order terms) was tested both on a sample of 4 γ rays events and on one containing only 3 γ rays events with tighter kinematic cuts. After a fit of the results, the following values were found for the Dalitz plot parameters: g = 0.67± 0.06 and h =−0.01 ± 0.08, with a χ² probability of 39%.

In the measurement by Sheaff [30] the π⁺ energy spectrum in the τ⁰ decay was studied. The experiment, held in 1975 at the ANL in Michigan, made use of a heavy-liquid bubble chamber, with a high magnetic field and a long- radiation-length liquid which strongly reduced background.

Strict conditions were required during the scanning procedure, both on the K⁺ curvature and on the π⁺ momentum, dip angle and π− µ − e chain visibility: after accurate cuts, 5635 events were collected in the final analysis.

The cut efficiencies were estimated from the collected data and evaluated as functions of the kinetic energy interval of the charged pion.

The main source of physical background in the τ⁰ decays sample was sup- posed to come from the K_µ3^±: the estimate of such contamination was checked with two independent checks which gave consistent results.

Both the fits to a linear and to a quadratic matrix element were car- ried out, and gave no evidence for a value of h significantly different from 0:

g = 0.630±0.038 and h = 0.041±0.030 (χ² probability of 96.5%). In the same experiment also other tests were performed to check the theoretical predictions on the isospin selection rules in K → 3π decays.

In 1975 another measurement, by Smith et al. [31], gave, for the first time, results for both positive and negative kaons. The detector used in the

experiment [32] consisted of wire chambers and hodoscopes, but required a particular care to avoid sources of systematic errors in such measurement, for which the experiment itself hadn’t been specifically designed. First of all, only events with 4 γ were considered, so that the most important background, coming from the K⁺ semileptonic decays, could be reduced to less than 1%

and subtracted, leaving a sample of 27406 events, distributed as a function of the variable Y .

A Monte Carlo calculation was made to obtain the product of the appara- tus efficiency and the τ⁰ decay phase space: all the results were expressed for each interval in Y as averaged over all the allowed values of X. Much of the uncertainties came from the Monte Carlo model of the detector which didn’t perfectly reproduce the detection of the photons in the hodoscope, for which the efficiency was parametrized “ad hoc”.

The results of the Dalitz plot parameters presented in the paper after a quadratic fit were: g = 0.510± 0.060 and h = 0.009 ± 0.040. Furthermore, a comparison with previous results in the Dalitz plot for the τ decay by Ford et al. [33] led to the conclusion that transitions with ∆I = 3/2 or 5/2 are possible.

The authors attempted also a measurement of the slope asymmetry A⁰⁰_g : combining 3-γ and 4-γ events together and dividing the obtained Y plot in 4 bins, a weighted mean asymmetry was computed over all the statistics in each bin. Assuming no asymmetry in the quadratic coefficients and normalizing to the number of K^± → π^±π⁰ decays, a linear fit of the asymmetry provided an estimate compatible with 0: A⁰⁰_g = (0.19± 1.25)%.

A heavy-liquid bubble chamber filled with a propane-ethane mixture was used at CERN to study K⁺ → π⁺π⁰π⁰ decays by Braun et al. [34] in 1976.

Using 3263 fitted τ⁰ events, enough information was collected to analyze the Dalitz distribution up to the second-order parameters. A larger sample of unfitted events was also used to study the π⁺energy spectrum and to perform various independent cross checks. The background, due to Kµ3and to τ⁰ decays including wrongly reconstructed γ rays, was expected to be less than 1%, and possible scanning biases were avoided requiring at least 5 M eV for the π⁺ kinetic energy. The efficiencies for scanning and selection were evaluated as

practically 100%, and the problem of the wrong pairing of the four photons in the kinematic fit was verified to be completely neglibible.

A maximum likelihood analysis of the Dalitz plot was performed both with 3 and with 2 parameters and a clear agreement with a linear behaviour of the decay matrix element was found, as for the previous experiments. The quadratic fit for the π⁺ spectrum yielded the values g = 0.670± 0.054 and h = 0.152± 0.082.

In the measurement by Bolotov et al. [35] (1986), the K⁻ → π⁻π⁰π⁰ (or τ⁰⁻) decay was investigated at the ISTRA apparatus [36] of the Institute of Nuclear Physics of the USSR Academy of Sciences, which was specifically designed to study rare decays of negative pions and kaons in flight in the IHEP 70-GeV /c accelerator [37]. For this measurement a 25 GeV /c K⁻ beam was used. A total amount of 4.3· 10⁴ events were collected, following a number of cuts on the energy released in the Cherenkov spectrometer, and on the topology of the π⁻ track observed in the proportional chambers. Events having high χ² after the kinematic analysis were rejected.

Analyzing the event density on the Dalitz distribution by applying the maximum likelihood method, the quadratic term in X was observed to be compatible with 0 and the values g = 0.582± 0.021 and h = 0.037 ± 0.024 were obtained with χ² = 84.3 for 72 d.o.f.

The systematics involved possible effects as the event-selection conditions, non-reliability in Monte Carlo simulation of the detector, accuracy in γ ray energy determination, wrong assignment of the four γ’s. Compatible results within the limit of errors were also reached by analyzing the π⁻-meson energy spectrum, giving an ulterior proof of the presence in τ⁰ decays of transitions with ∆I > 1/2.

In the measurement by Batusov et al. [38] performed in 1998, a high statistics of K⁺ → π⁺π⁰π⁰ events was analyzed to get a precise estimate of the Dalitz plot parameters. The experiment took place at the HYPERON-2 spectrometer [39] at the Serpukhov accelerator. A K⁺ 10-GeV /c beam was extracted by three threshold gas Cherenkov counters, while the momentum could be measured with high precision (∆p/p = 0.5%) by a spectrometer in a

uniform-field magnet.

All the secondary charged particles were reconstructed with 1-3% relative momentum resolution; two electromagnetic calorimeters were used for detect- ing γ rays with∼ 10%/^qE(GeV ) energy resolution. The K⁺→ π⁺π⁰ decays, one of the principal sources of background, were used to calibrate the calorime- ters using the known π⁺ energy in the K⁺ rest frame. A suitable choice of the fiducial volume for the K⁺ decay and a well-defined set of kinematic cuts led to about 3.3· 10⁴ τ⁰ events, for which the background (∼ 0.5% as estimated by Monte Carlo).

Fitting the Dalitz plot to the function C(1+gY +hY²+kX²) the following results for the three parameters were obtained: g = 0.736± 0.014 ± 0.012, h = 0.128± 0.015 ± 0.024 and k = 0.0197 ± 0.0045 ± 0.0029.

Systematic errors included the stability over different periods of run and trigger conditions, the size of the bins, the behaviour at the edges of the distribution, the lower limit for γ ray energy, the possibility of incorrectly pair the photons and the “pulls” for each variable involved in the analysis.

It is worth to point out that the results quoted in this paper were in dis- agreement with the world average by the PDG with discrepancies going from 3 σ to more than 5 σ. The authors indicated as main motivation the fact that their experiment was the first and only electronic one where all the particle momenta involved in τ⁰ decays were measured, while the detectors in the other experiments were not able to close the complete kinematics of K^±→ π^±π⁰π⁰ events.

Besides the measurements considered by the PDG, also other results have been recently published.

In 2000 a kinematically complete measurement concerning positive τ⁰ de- cays was realized by Shin et al. [40] for the KEK-PS E246 Collaboration at the 12-GeV /c proton synchrotron in Tsukuba, Japan. The event selection relied on the information from the four-momentum conservation at the K decay ver- tex and time of flight of the π⁺. The bias due to the wrong pairing of photons was considered below 2%. Since a very narrow region in the Y interval could be investigated, the quadratic term h was not measured; the other parameters were determined as g = 0.518± 0.039 and k = 0.043 ± 0.020.

In 2002 Ajinenko et al. [41] obtained new precise results at the IHEP accel- erator [37] with “ISTRA+” (the upgrade of the ISTRA detector [36]). The se- lection criteria applied on a total of∼ 7·10⁸recorded events yielded an amount of about 250000 K⁻→ π⁻π⁰π⁰ decays. Monte Carlo was used to estimate and subtract the background contamination and to normalize the Dalitz distribu- tion. The parameters were obtained by means of a least squares fit according to the usual second order polynomial function: g = 0.627± 0.004 ± 0.010, h = 0.046± 0.004 ± 0.012 and k = 0.001 ± 0.001 ± 0.002.

As the slope parameters cannot be calculated theoretically, owing to the large uncertainties and the dependence on the particular model in the theo- retical calculations, the discrepancies in the experimental results can only be solved by new and more precise measurements.

KLOE offers the important chance to improve the knowledge of these pa- rameters and of their asymmetries, since it provides a very large statistics of completely reconstructed events with simultaneously positive and negative kaons decaying in the τ⁰ mode.

1.2.3 Present status and perspectives of direct CP violation mea- surements in K^± →(3π)^± processes

The study of the direct CP violation in the charged kaons 3π decays has produced only one relevant measurement: it was the work published by Ford et al. [33] in 1970 and involved only the three charged pions channel. The analysis of about 1.6·10⁶ τ^±events led to the result^∆g_2g^

τ = (−7.0 ± 5.3)·10⁻³ for the asymmetry of the linear slope parameter g.

Up to now no measurements have been performed on the A⁰⁰_g asymmetry in the K^± → π^±π⁰π⁰ decay described here. It has been possible only to give an estimate by matching together results coming from various experiments: the value 0.117 was found for the asymmetry [42], systematic uncertainties being very difficult to calculate.

HyperCP [43], or E-871, is an experiment now running at FNAL which is going to test the CP symmetry in hyperons and in charged kaons decays by

comparing the slopes of the Dalitz plot obtained for positive and negative par- ticles. In two runs in 1997 and 1999 5.5·10⁸ charged kaons have been revealed;

the collected K⁺ decays are about twice the K⁻ decays. Since the apparatus has been optimized for hyperon decays in particular, the present situation still requires to improve accuracy to understand the systematics that could affect the detection of kaons.

Finally, the upgrade of the NA48 experiment [44] (NA48/2) now running at CERN until September 2003 aims to measure both the asymmetries A⁺⁻_g and A⁰⁰_g in the charged kaons system. Symultaneous K⁺ and K⁻ beams are used and the experiment is supposed to collect 2· 10⁹ K^± → π^±π⁺π⁻ events and 1.2· 10⁸ K^±→ π^±π⁰π⁰ events in one year of SPS operation, which should provide an accuracy on the branching ratios of few parts in 10⁻⁴ and very precise estimates of the Dalitz plot parameters. Monte Carlo shows that the systematics should be at most of the order of 10⁻⁵ and the statistical uncer- tainties should then dominate with a contribution of less than 3.5· 10⁻⁴.

For charged kaons KLOE, running at a φ-factory like DAΦNE, represents a unique opportunity to study the asymmetries in the Dalitz plot parameters.

The exact cancellation [45] of many systematic effects in the evaluation of the efficiencies of positive and negative kaons in φ → K⁺K⁻ events and the collection of more than one million events (∼ 500 pb⁻¹ of data taken up to 2002), can ensure a precision of few 10⁻³ in the determination of A⁰⁰_g .

1.3 The K

→ π

π

e

ν

(¯ ν

) decay

The main theoretical interest in the study of K meson decays into two pions and a lepton pair is given by the possibility to extract information on the low-energy ππ interaction.

One relevant advantage shown by Kl4 decays is that only couplings to an external left-handed vector leptonic current are involved. Moreover, since the two pions can only be emitted in l = 0, 1 relative angular momentum states [46], and assuming the validity of the semileptonic ∆I = 1/2 rule, the only

possible quantum states allowed for the dipion system are (l = 0, I = 0) and (l = 1, I = 1): this implies that the K_l4 decays can be used to extract the ππ scattering phase-shift difference (δ₀⁰− δ¹1) as a function of the dipion invariant mass.

Furthermore, an accurate measurement of the form factors and of the branching ratios for the Kl4 decays can help in checking the validity of the

∆I = 1/2 rule and in testing the predictions of different theoretical models.

1.3.1 Kinematics of Kl4 decays The K_l4 semileptonic decays are:

K^± → π⁺π⁻l^±ν_l(¯ν_l) or K_l4^± K^± → π⁰π⁰l^±ν_l(¯ν_l) or K_l4⁰ K_L⁰ → π⁰π^∓l^±ν_l(¯ν_l) or K_l4⁰ ,

where the letter l stands for e or µ. In order to fully describe the kinematics of such decays, 5 independent variables are required. In their work on the angular correlations in K_e4 decays [47], Cabibbo and Maksymowicz consider three reference frames (see figure 1.2): the K rest frame (ΣK), the ππ center of mass system (Σππ) and the l^±νl center of mass system (Σlν). Given the four-momenta pK of the kaon, p1 and p2 of the two pions, and pl and pν of the two leptons, five quantities can be defined:

• s^π ≡ (p¹+p2)² ∈ [4m²π, (mK− m^l)²], invariant mass squared of the dipion system,

• sl ≡ (pl + p_ν)² ∈ ^hm²_l, (m_K−√s_π)²ⁱ, invariant mass squared of the dilepton system,

• θπ ∈ [0, π], angle of the “first” pion in Σππ with respect to the direction of flight of the dipion system in ΣK,

• θ^l∈ [0, π], angle of the charged lepton in Σ^lν with respect to the direction of flight of the dilepton system in Σ_K,

• φ ∈] − π, π], angle between the planes formed by the dipion and the dilepton systems in ΣK.

Figure 1.2 Kinematic quantities involved in a K_l4 decay. The angles φ, θ_π and θ_l are computed in Σ_K, Σ_ππ and Σ_lν, respectively.

In other terms, the kinematic approach of these decays analyzes the pro- duct as if they came out from two “resonances” (i.e. the dipion and the dilepton, having mass √sπ and √sl, respectively).

By introducing the four-vectors:

P ≡ p1+ p₂ , L≡ pl+ p_ν , Q≡ p1− p2 ,

the matrix elements of the hadronic axial-vector and vector currents can be expressed in the general form [48, 49]:

hππ|Jλ^A|Ki = F mK

Pλ+ G mK

Qλ+ R mK

Lλ ,

hππ|Jλ^V|Ki = iH

m³_KελµνσL^µP^νQ^σ ,

where the form factors F , G, H and R are dimensionless real analytic functions of p1p2, pKp1 and pKp2 or, equivalently, of sπ, sl and θπ.

Since the kaon’s and the pions’ states have opposite relative intrinsic pari- ties, the matrix element of the axial-vector current transforms as an ordinary vector, while hππ|Jλ^V|Ki transforms as an axial vector.

After the integration over all the variables on which the form factors don’t depend, the partial decay rate for the K_l4 can be written as:

dΓ = G²_F|Vus|²N (sπ, sl)J5(sπ, sl, θπ, θl, φ)dsπdsld(cos θπ)d(cos θl)dφ , (1.15) where J5 is expressed in terms of simple functions of θl and φ multiplying nine functions Ii(sπ, sl, θπ, F, G, H, R) [50], and the kinematic factor N (sπ, sl) is defined as

N (sπ, sl) = 1− m²l/s_l 2¹⁴π⁶m⁵_K

vu

ut 1− 4m²_π sπ

!

[(m²_K− sπ− sl)²− 4sπsl] . By integrating over θl and φ, the partial decay rate (1.15) becomes:

dΓ = G²_F|Vus|²N (sπ, sl)J3(sπ, sl, θπ)dsπdsld(cos θπ) , (1.16) J3 being explicitly defined as:

J₃(s_π, s_l, θ_π) = 4π

3 (1− z)



(2 + z)^h|F1|²+ (|F2|²+|F3|²) sin²θ_πⁱ+ 3z|F4|²



, with

z = m²_l/sl

F1 = ^q(P · L)²− sπsl· F +^q1− 4m²π/sπ(P · L) cos θπ · G F2 = ^qsl(sπ − 4m²π)

F3 = ^qsl(1− 4m²π/sπ)[(P · L)² − sπsl]· H/m²K

F4 = −(P · L)F −^q(1− 4m²π/sπ)[(P · L)²− s^πsl] cos θπ· G − s^lR . By exploiting the isospin symmetry connecting the current matrix elements after the decomposition of symmetric and antisymmetric parts under the ex- change (p1 ↔ p²) for all the Kl4 channels, the following isospin relation can be obtained for the decay rates:

Γ(K_l4^±) = 2Γ(K_l4⁰ ) + 1

2Γ(K_l4⁰) , (1.17) where the ∆I = 1/2 rule [51] has been applied.

Another result from the ∆I = 1/2 rule predicts that the form factor F has to be equal for the K_l4^± and the K_l4⁰ decays.

The form factors F , G, H and R can be described as partial wave ex- pansions in the variable θ_π, with amplitudes (f_l, g_l, h_l and r_l) which are real functions of sπ and sl and phases which are assumed to be the phase shifts δ_l^I involved in the elastic ππ scattering ¹.

According to the Pais-Treiman method [50], one relevant experimental test concerns the validity of the assumption of locality of the coupling of lepton pairs to hadrons, which implies that the dependence of the spectrum on the single quantities θl and φ (i.e. after that the integration over the other four variables has been performed) follows the expressions:

dΓ d cos θl

= a + b cos θl+ c cos 2θl , (1.18)

and dΓ

dφ = α + β cos φ + γ sin φ + δ cos 2φ +  sin 2φ .

If this test is satisfied, then the intensity distribution functions hIⁱi can be used as free parameters to fit the event distribution in the (θl, φ) plane, and to extract the phase shift difference (δ₀⁰− δ1¹) for each sπ bin from the relations:

tan(δ₀⁰− δ1¹) = 1 2

hI7i

hI⁴i , tan(δ⁰₀− δ1¹) = 2hI8i

hI⁵i . (1.19) Another test concerns the hypothesis of pion pairs produced exclusively in l = 0 and l = 1 states, which is more reasonable for low values of sπ: the form factors G and H are seen to be independent from θπ, while F and R are at most linear in cos θπ. Therefore, the intensity spectrum in the vari- able θπ, integrating over all the other four variables, behaves according to the expression:

dΓ d cos θπ

= A + B cos θπ + C cos 2θπ . (1.20) Finally, the isoscalar S-wave scattering length a⁰₀ can be extracted by us- ing a model based on solutions to the Roy equations [52], which has to be

1 Even if the phase shifts depend on the dipion invariant mass√sπ, δ^I_l are considered as constants, i.e. as if they were averaged over all the sπ spectrum.