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 A00g , 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 Kµ3±: 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 A00g : 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: A00g = (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