The measurements of the branching ratios of the Kl4 decays reported by the Particle Data Group [4] are listed in Tab. 1.4.
The branching ratio of Ke40 shown in Tab. 1.4 is obtained by fitting three in-dependent experimental results, in which the absolute branching ratio and/or relative fractions involving decay rates of other channels have been studied.
2 The kinematic dependence on sπ, sland θπ has never been resolved experimentally for this decay.
Decay channel i Γi/Γtotal (·10−5) K+ → π+π−e+νe 3.91± 0.17 K+ → π0π0e+νe 2.1± 0.4 KL0 → π0π±e∓ν¯e(νe) 5.18± 0.29 K+ → π+π−µ+νµ 1.4± 0.9
Table 1.4 Experimental measurements of the branching ratios for the Kl4 decays, as reported by the PDG [4].
The large relative error (∼ 20%) associated to ΓKe40 /Γtotal is due to the low statistics reached so far by experiments in this channel. The isospin relation (1.17) connecting the first three decay modes in Tab. 1.4 is verified within the errors.
The first experimental analysis on the Ke40 decay dates back to 1971, when F. Romano et al. [54] used a large amount of semileptonic decays from the X2 experiment [29] at CERN to measure the relative branching ratio Γ(K+ → π0e+νeγ)/Γ(K+ → π0e+νe). As no evidence of events compatible with the Ke40 kinematics was found, an upper limit at the 90% confidence level was determined for the absolute branching ratio: BR(K0+e4) < 1.8· 10−4.
In 1973 D. Ljung and D. Cline [55] studied K+ → π0π0e+νe among some rare K+ decay modes. The analyzed data set was collected over a period of three years at Argonne National Laboratory. K+ mesons of ∼ 500 MeV/c at the Zero Gradient Synchrotron (ZGS) when entering a bubble chamber filled with heavy freon in a 46-kG magnetic field.
The scanning of 674200 stopped K+ decays produced the selection of 148 candidates for the Ke40 mode, in which an electron track and 4 converted γ’s pointing at the decay vertex were observed. Then, by applying a cut on the χ2 probability of a three-constraint fit, and on the event topology to reject background, the sample was reduced to 9 events.
In the last step, only 2 decays were selected, as a low χ2 fit was obtained in the K+ → π+π0π0 hypothesis for the other seven decays. The final result,
BR(K0+e4) = (1.8+2.4−0.6)·10−5, was obtained from the ratio with 22952 Ke3+ decays found in the same initial sample, which yielded Γ(K+ → π0π0e+νe)/Γ(K+ → π0e+νe) = 3.8· 10−4.
The work by V. N. Bolotov et al. [56] (1986) studied two rare K− decay modes, i.e. the radiative Ke3− and the K0−e4. The experimental measurement was held with the ISTRA apparatus [36] on a 25-GeV energy beam of π- and K-mesons from the IHEP accelerator [37]. Both these two analyses were per-formed after a calibration process based on the identification of about 170000 Ke3− decays.
The selection of Ke40 events was carried out through two stages. As a first requirement, 5 or 6 showers had to be found in the spectrometer and the distances between their centers had to be at least 10 cm. Subsequently, a 99% C. L. cut was applied on a 6-constraint χ2 fit under the K → ππ0π0 (τ0) hypothesis, which had to reject the “main background process” (i.e. τ0−
decays in which the π− emits a δ electron, or decays into e−ν¯e, or follows the chain: π → µν µ → eν ¯ν). Also charge exchange and other processes were considered as sources of contamination.
The surviving candidates were then analyzed by a 4-constraint fit under the K → π0π0e¯ν hypothesis, and a 99% C. L. cut was designed to defini-tively select K0−e4 decays. The estimate of the amount of background was made by measuring the different counting efficiencies for pions and electrons.
From the 25 events resulting after the background subtraction, it was pos-sible to produce the final result BR(K0−e4) = (2.0+0.5−0.4)· 10−5 from the ratio:
Γ(K−→ π0π0e−ν¯e)/Γ(K− → π0e−ν¯e) = (4.2+1.0−0.9)· 10−4.
Two years later, Barmin et al. [57] performed an absolute branching ratio measurement on a small amount of positive Ke40 decays. The work was done with photographs obtained in a xenon bubble chamber exposed to a 0.85-GeV /c K+ meson beam extracted from the proton synchrotron of ITEP. The total statistics included about 6· 109 K+ mesons observed in the acceptance region of the chamber. Both K+ decays in flight and at rest were used, defi-ning two separate groups of events (which were called primary and secondary, respectively).
The topology for Ke40 events consisted in a track connected to the K+ and accompanied by an electromagnetic shower, and other four showers pointing to the supposed kaon decay vertex. The main source of background came from the K+ → π+π0π0 decay with the π+ misidentified as a e+. A Monte Carlo method was used to improve the purity of the sample, and the total depth of the positron shower was the most powerful feature for rejecting the τ0 background.
A ∼ 20% background was estimated in the secondary decays, among which 4.9± 2.7 events were counted as signal, while the 5 events observed in the primary decays were considered as uncontaminated, and affected by the poissonian uncertainty only. After the corrections due to the various ef-ficiences (scanning, checking, final identification, etc.), the combinations of the results from the two sets of data samples gave the final measurement:
BR(K0+e4) = (2.54± 0.89) · 10−5, in agreement with the previous experimental results.
Form factors
Despite the very low statistics collected for the measurement of the Ke40 branching ratio in all the experiments described so far, the relation (1.22) allows to obtain in a straightforward way an estimate of the modulus of the form factor F in this decay mode.
Some published papers on the Ke40 decay mode express F in a different notation by means of the parameter f1; this can be transcribed according to the convention on the form factors used in this work: F ≡ f1/Vus. The value
|Vus| ≈ 0.22 is considered; both theoretical and experimental uncertainties on
|Vus| [4] are negligible with respect to the precision reached for F .
In Tab. 1.5 the values of |F | are reported 3 together with the estimates BR(Ke40 ) corresponding to each of the experiments mentioned in the previous paragraph.
3 Actually only the average value of F have been considered in the various measurements, since the poor statistics couldn’t resolve kinematics satisfactorily.
Document ID K± |F | ΓKe40 /Γtotal (·10−5) Norm. Events
Romano 71 [54] K+ − < 18 Ke3+ 0
Ljung 73 [55] K+ 4.4+2.2−0.9 1.8+2.4−0.6 Ke3+ 2 Bolotov 86 [56] K− 6.4+0.8−0.6 2.0+0.5−0.4 Ke3− 25 Barmin 88 [57] K+ 7.3± 1.3 2.54± 0.89 K+→ all 10
Table 1.5Experimental determinations for the modulus of the form factor F and for the absolute branching ratio of the Ke40 decay. Also the charge of the kaons collected in each measurement is reported, and the normalization decay mode.
A possible parametrization for the form factor F in the Ke40 channel is given by
F = F0(1 + λq2)eiδ00 , (1.23) where q2 = (sπ − 4m2π)/4m2π and λ is set to 0.08 [58]. The resulting value of the amplitude, F0 = 5.72+0.57−0.49 [49] agrees very well with the analogous measurement performed by Rosselet et al. at the Geneva-Saclay experiment [58] on a sizeable sample of 30000 Ke4+ decays (F0 = 5.59± 0.14), so confirming the isospin prediction on the equality of F in Ke4± and Ke40 decays.
Another measurement has been performed in 2003 by E865 [59] at the Brookhaven Alternate Gradient Synchrotron (AGS) on a large statistics of Ke4+ events (400000): according to the parametrization (1.23), the values F0 = 5.83± 0.08 and λ = 0.079 ± 0.015 have been measured.
Also the agreement with the various theoretical predictions on F in the Ke40 decay is remarkable (see for example [60, 61, 62, 63, 64, 65]), although a higher statistics in experiments would be helpful in improving the knowledge of this form factor.
at DAΦNE
The KLOE experiment [66] has been designed to perform precision tests on CP violation in the system of neutral kaons, in particular the primary goal was to measure <e(0/) with an accuracy of 10−4. The experiment is located at the Frascati φ-factory DAΦNE [68, 69], a double ring e+e− collider at the energy of the φ meson resonance (mφ = (1019.456±0.020)MeV [4]). The peak cross section is σ(e+e− → φ) ' 3.1 µb.
A short description of the CP violation phenomenology will be given in Sec. 2.1. The DAΦNE collider will be described in Sec. 2.2 and the KLOE experiment in Sec. 2.3.
2.1 CP violation in the neutral kaon system
The largest part of φ mesons decay into pairs of charged or neutral kaons, as can be observed in Tab. 2.1.
In 1955 Gell-Mann and Pais [70] pointed out that K0 and ¯K0 mesons are different particles in the strong interactions, but they can transform into each other by means of second-order weak interactions. The quantum state
27
φ decay mode Branching ratio
Table 2.1 φ meson decays and their branching ratios [4].
for a neutral kaon can be described by a linear combination of|K0i and | ¯K0i with coefficients α and β, whose evolution is described by an effective 2× 2 Hamiltonian:
where the matrices M and Γ are hermitian. By imposing CPT conservation, the masses and the decay rates have to coincide for K0 and ¯K0:
M11= M22 , Γ11= Γ22 ,
whileCP conservation the matrix elements to be invariant under the exchange of the indices 1 and 2 if the phase convention such that CP|K0i = | ¯K0i is observed, then adding:
M12 = M21 = M12∗ , Γ12 = Γ21 = Γ∗12
The CP invariance in the neutral kaon system can be likewise expressed inde-pendently of the phase convention as:
IfCP was conserved, the eigenstates of the Hamiltonian should be the masses (MS and ML) and the widths (ΓS and ΓL) of the KS and the KL physical states. In such hypothesis, mass andCP eigenstates should be the same:
CP|K1,2i = ±|K1,2i , |K1,2i = 1
√2[|K0i ± | ¯K0i] ,
where the|K1i is allowed to decay into 2-pion final states and |K2i into 3-pion states, which are respectively even and oddCP eigenstates.
In 1964 CP violation was observed [71] by measuring a fraction ∼ 2 · 10−3 of KL mesons (supposed to be exactly the |K2i state) decaying into π+π−. Therefore, mass eigenstates are obtained by mixing the |K1i and |K2i CP eigenstates:
|KS,Li =h|K1,2i + ˜|K2,1ii.q2(1 +|˜|2) with the parameter ˜ defined as:
1 + ˜ 1− ˜=
vu
utM12− iΓ12/2 M12∗ − iΓ∗12/2 .
This is the so-called “indirect CP violation”, which has to be ascribed to KL↔ KS oscillations with strangeness S undergoing into ∆S = 2 transitions.
A second possible CP-violating effect can be produced by ∆S = 1 weak transitions (“directCP violation”) from the odd CP eigenstate |K2i to a 2-pion final state.
Direct CP violation in ππ final states can be parametrized by decomposing K0 and ¯K0 amplitudes in isospin (I) components:
AIeiδI ≡ hI|Hweak|K0i ,
where δI is the phase shift between the |ππi physical state and the isospin eigenstate|Ii. CP simmetry allows to obtain
AIeiδI =hI|CP−1HweakCP| ¯K0i = A∗IeiδI ,
so that A0 can be set to a real value and only the difference between the weak phases, δ2− δ0, becomes relevant forCP violation.
By introducing the KS and KL decay widths η+−= hπ+π−|Hweak|KLi
hπ+π−|Hweak|KSi , η00= hπ0π0|Hweak|KLi hπ+π−|Hweak|KSi
and using the experimental observation (∆I = 1/2 rule) that the final state
|I = 0i is favoured with respect to |I = 2i (or equivalently <eA2 A0), η+−
and η00 can be expressed as
η+−' + 0 , η00 ' − 20
represents a powerful experimental method to extract information on the direct CP violation parameter 0 by means of the relation:
R' 1 + 6<e(0/) .
=m(0/) is expected to be small since the phases of and 0 are almost the same [4], therefore implying<e(0/)≈ 0/.
The statistical accuracy in the measurement of R is mainly driven by the KL → ππ decay statistics (about 3 orders of magnitude less than KL → 3π decays):
All measurements of <e(0/) have been performed at fixed target exper-iment (E731 [72], NA31 [73], KTeV [74], NA48 [75]), unambiguosly showing
the presence of direct CP violation.
DAΦNE allows to produce φ → K0K¯0 decays, corresponding to coherent superpositions of KLKS states; since the kaons are produced with a well-defined momentum (∼ 110 MeV ) 1 in the φ-meson rest frame, the detection of a KS (KL) tags the presence of a KL(KS) emitted in the opposite direction, so that very clear neutral kaon beams are produced. As shown in the following sections, given the KSKL rates at DAΦNE and the present KLOE detector performances, an accuracy of 10−4 on 0/ requires to collect 4.4× 104 pb−1.
The accelerator was designed to reach an instantaneous luminosity of 5× 1032cm−2s−1, but this goal has not been achieved up to now. The peak luminos-ity delivered by DAΦNE during years 2001-2002 has been∼ 8 × 1031cm−2s−1, with a total integrated luminosity of∼ 500 pb−1. However, this amount of data set is not enough to perform a measurement of<e(0/) with the projected ac-curacy, the KS part of the double ratio has been studied [76] at a 10−3 level of precision and further analyses and tests have been carried out concerning the KL part [77].
Apart from the study of 0/, the statistics collected by KLOE so far is enough to explore many other interesting physics issues and to perform a sizeable number of measurements with unprecedented precision, involving ra-diative φ decays [78, 79, 80], the hadronic cross section, neutral and charged kaons [81, 82].