QCD tests with Kaons

Final results from an analysis of about 400 K± → π±γγ rare decay candidates collected by the NA48/2 and NA62 experiments at CERN are presented. The results include a model-independent decay rate measurement and fits to Chiral Perturbation Theory (ChPT) description. The data support the ChPT prediction for a cusp in the di-photon invariant mass spectrum at the two pion threshold. The NA48/2 Collaboration at CERN has accumulated unprecedented statistics of rare kaon decays in the Ke4 modes K±e4 → π+π− eν and K00e4 → π0π0eν with one percent background contamination. The detailed study of the form factors brings new inputs to low energy QCD description and crucial tests of predictions from ChPT and lattice QCD calculations.


The NA48/2 and NA62-RK experiments
At an early stage the NA62 [1] experiment collected data, with modified beam conditions, using the NA48/2 detector, aiming to measure the ratio R K of the rates of the leptonic kaon decays. The NA48/2 beam line has been designed to deliver simultaneous narrow momentum band K + and K − beams derived from the 400 GeV/c protons extracted from the CERN SPS with central momenta of 60 GeV/c (for NA48/2) or 74 GeV/c (for NA62-RK). The beam kaons decayed in a fiducial volume contained in a 114 m long cylindrical vacuum tank. The momenta of charged decay products were measured in a magnetic spectrometer consisting of four drift chambers (DCHs), two upstream and two downstream of a dipole magnet. The dipole provided a horizontal transverse momentum kick of 120 MeV/c (for NA48/2) or 265 MeV/c (for NA62-RK) to charged particles. Each DCH was composed of eight planes of sense wires. A plastic scintillator hodoscope producing fast trigger signals and providing precise time measurements of charged particles was placed after the spectrometer. Further downstream was a liquid krypton electromagnetic calorimeter (LKr), an almost homogeneous ionization chamber with an active volume of 7 m 3 of liquid krypton, 27X 0 deep, segmented transversally into 13248 projective 2x2 cm 2 cells and with no longitudinal segmentation. The LKr information is used for photon measurements and charged particle identification. An iron/scintillator hadronic calorimeter and muon detectors were located further downstream. A description of the detector can be found in [2].
2. The K ± → π ± γγ decay Measurements of radiative non-leptonic kaon decays provide crucial tests for the ability of the Chiral Perturbation Theory (ChPT) to explain weak low energy processes. In the ChPT framework, the K ± (p) → π ± (p 3 )γ(q 1 )γ(q 2 ) differential decay rate can be parametrized as [3]: where z = m 2 γγ /m 2 K and y = p · (q 1 − q 2 )/m 2 K . The dominant contribution at O(p 4 ) comes from the loop term A(z,ĉ), including the pion and kaon loop amplitudes, the B and D terms are zero and the Wess-Zumino-Witten term C accounts for ∼10% of the total decay rate. Higher order unitarity corrections from K ± → 3π ± decays, including the main O(p 6 ) contribution as well as those beyond O(p 6 ), have been found to contribute significantly to the branching ratio (BR) (up to 30-40%) and to the shape of M γγ spectrum [3]. The total decay rate is predicted to be BR(K ± → π ± γγ) ∼ 10 −6 . The only K ± → π ± γγ experimental observation published so far is by the BNL E787 experiment [4] measuring the BR andĉ: based on 31 K + decay candidates in the kinematic region 100 MeV/c< p π < 180 MeV/c (p π is the π + momentum in the K + rest frame). The value of the BR quoted in (2) refers to the entire kinematical range and is obtained with a ChPT extrapolation using the value 1.8 forĉ.

The data analysis
The measurements described here have been performed using minimum bias data sets collected during a 3-day special NA48/2 run in 2004 with 60 GeV/c K ± beams, and a 3-month NA62 run in 2007 with 74 GeV/c K ± beams. The effective kaon fluxes collected are similar, but the background conditions and resolution on kinematic variables differ significantly. Signal events are selected in the region z = (m γγ /m K ) 2 > 0.2 to reject the K ± → π ± π 0 background peaking at z = 0.075. In total 149 (232) decays candidates are observed by NA48/2 (NA62), with backgrounds contaminations of 15.5 ± 0.7 (17.4 ± 1.1) dominated by K ± → π ± π 0 (π 0 )(γ) decays with merged photon clusters in the LKr calorimeter. The data spectra of the z kinematic variable, together with the signal and background expectations, are displayed in Figure 1: they support the ChPT prediction of a cusp at the two-pion threshold. The values of theĉ parameter in the framework of the ChPT O(p 4 ) and O(p 6 ) parameterizations according to [3] have been measured by performing likelihood fits to the data. The uncertainties are dominated 2 the polynomial contribution terms are η 1 = 2.06, η 2 = 0.24 and η 3 = -0.26 as suggested in [3], while the K ± → 3π ± amplitude parameters come from a fit to the experimental data [5]. Along with the separate 2004 and 2007 results, the combined NA48/2 and NA62 results are also presented in Table 1. The measured BR, in agreement with the E787 one, improves the 1.37 ± 0.36 1.41 ± 0.40 0.877 ± 0.087 stat ± 0.017 syst 0.910 ± 0.072 stat ± 0.022 syst NA62 [8] 1.93 ± 0.27 2.10 ± 0.33 1.088 ± 0.093 stat ± 0.027 syst 1.058 ± 0.066 stat ± 0.044 syst Combined [8] 1.72 ± 0.21 1.86 ± 0.25 0.965 ± 0.061 stat ± 0.014 syst 1.003 ± 0.051 stat ± 0.024 syst precision of the measurement by a factor ∼5. The obtained value ofĉ, for both the O(p 4 ) and O(p 6 ) fits, is in very good agreement with the previous measurement by E787 [4]. The same parameter has been measured by NA48/2 using the decay K ± → π ± e + e − γ, with a compatible value:ĉ = 0.90 ± 0.45 [6].

The K e4 decay formalism
Four-body final state decays are described by five kinematic variables historically called, for K e4 decays, Cabibbo-Maksymowicz variables [9]: two invariant masses S π = M 2 ππ and S e = M 2 eν and three angles ϑ π , ϑ e and ϕ. The hadronic current is described by form factors which can be developed in a partial wave expansion as suggested in [10] Limiting the expansion to S-and P-waves and considering a unique phase δ p for all P-wave form factors, two axial (F, G) and one vector (H) complex form factors contribute to the transition amplitude: F = F s e iδs + F p e iδp cosθ π , G = G p e iδp , H = H p e iδp . From the differential rate study in the 5-dimensional space, four real form factors (F s , F p , G p and H p ) and a single phase difference (δ = δ s − δ p ) have been measured, together with their energy variation with S π and S e , by the NA48/2 experiment [11]. In the neutral pion mode (K 00 e4 ), the differential rate depends on a single hadronic form factor F s whose variation with (S π ,S e ) is unknown and will be studied. No such study is available so far in the literature.

K 00
e4 form factor measurement The form factor (FF) study requires a sample free of large radiative effects which can pollute the original kaon decay amplitude. The event density in the plane (S π ,S e ) is proportional to F 2 s . The fit procedure minimizes a χ 2 expression in the two-dimensional space describing F s by a polynomial expansion in q 2 = (S π /4m 2 π+ − 1) and y 2 =(S e /4m 2 π+ ): The term C(q 2 ) = |q 2 /(1 + q 2 )| parameterizes the cusp-like function. The results in Table 2 are in good agreement with those obtained in a high statistics measurement of the corresponding form factor of the K ± e4 [11] mode see Figure 2b). Below S π = 4m 2 π+ , the observed deficit of events it is well described by a cusp-like function Figure 2a).

Measurement of the K 00
e4 branching ratio The K 00 e4 rate is measured relative to the abundant K 00 3π normalization channel. As the topologies of the two modes are similar the two samples are collected concurrently using the same trigger  Figure 2. a) Ratio of the two q 2 distributions in equal population bins. b) FF description in 2-parameter planes obtained in K 00 e4 [12] (red) and K ± e4 [11] (black). logic and a common selection is employed as far as possible. This leads to partial cancellation of the systematic effects and avoids relying on the absolute kaon flux measurement. After the selection described in [12] a sample of 65210 candidate is identified with 1% background dominated by K 00 3π with a misidentified pion as an electron. The BR measurement [12], using the value BR(K 00 3π ) = (1.761 ± 0.022)% as normalization, is obtained: BR(K 00 e4 ) = (2.552 ± 0.010 stat ± 0.010 syst ± 0.032 ext ) · 10 −5 This measurement improves the current world average precision by one order of magnitude. Both total rate and form factor description are used to obtain the absolute FF value at S π = 4m 2 π+ , S e = 0 (q 2 = 0, y 2 = 0) [12]: f s = 6.079 ± 0.012 stat ± 0.027 syst ± 0.046 ext , where the dominating external error comes from uncertainties on the normalization mode K 00 3π branching ratio, on the mean kaon life time and on |V us |. This value shows some tension with the corresponding form factor of the K ± e4 mode f ± s = 5.705 ± 0.003 stat ± 0.017 syst ± 0.031 ext [11].