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A sample of 1.69 × 10 7 fully reconstructed π 0 → γ e + e − decay candidates collected by the NA48/2 experiment at CERN in 2003-2004 is analyzed to search for the dark photon (A) production in the π 0 → γ A decay followed by the prompt A → e + e − decay. No signal is observed, and an exclusion region in the plane of the dark photon mass m A and mixing parameter ε 2 is established. The obtained upper limits on ε 2 are more stringent than the previous limits in the mass range 9 MeV/c 2 < m A < 70 MeV/c 2. The NA48/2 sensitivity to the dark photon production in the K ± → π ± A decay is also evaluated. 0. Introduction Kaons are a source of tagged neutral pion decays, and high intensity kaon experiments provide opportunities for precision π 0 decay measurements. The NA48/2 experiment at the CERN SPS collected a large sample of charged kaon (K ±) decays in flight, corresponding to about 2 × 10 11 K ± decays in the fiducial decay volume. This letter reports the search for a hypothetical dark pho-ton (DP, denoted A) using a large sample of tagged π 0 mesons from identified K ± → π ± π 0 and K ± → π 0 μ ± ν decays. In a rather general set of hidden sector models with an extra U (1) gauge symmetry [1], the interaction of the DP with the visible sector proceeds through kinetic mixing with the Standard Model (SM) hypercharge. Such scenarios with GeV-scale dark matter provide possible explanations to the observed rise in the cosmic-ray positron fraction with energy and the muon gyromag-netic ratio (g − 2) measurement [2]. The DP is characterized by two a priori unknown parameters, the mass m A and the mixing parameter ε 2. Its possible production in the π 0 decay and its subsequent decay proceed via the chain π 0 → γ A , A → e + e −. The expected branching fraction of the above π 0 decay is [3] B(π 0 → γ A) = 2ε 2 1 − m 2 A m 2 π 0 3 B(π 0 → γ γ), (1) which is kinematically suppressed as m A approaches m π 0. In the DP mass range 2m e < m A < m π 0 accessible in pion decays, the only allowed tree-level decay into SM fermions is A → e + e − , while the loop-induced SM decays (A → 3γ , A → ν ¯ ν) are highly suppressed. Therefore, for a DP decaying only into SM particles, B(A → e + e −) ≈ 1, and the expected total decay width is [3] A ≈ (A → e + e −) = 1 3 αε 2 m A 1 − 4m 2 e m 2 A 1 + 2m 2 e m 2 A. (2) It follows that, for 2m e m A < m π 0 , the DP mean proper lifetime τ A satisfies the relation cτ A = ¯ hc/ / A ≈ 0.8 μm × 10 −6 ε 2 × 100 MeV/c 2 m A. (3) This analysis is performed assuming that the DP decays at the production point (prompt decay), which is valid for sufficiently large values of m A and ε 2 , as quantified in Section 5. In this case, the DP production and decay signature is identical to that of the Dalitz decay π 0 D → e + e − γ , which therefore represents an irreducible but well controlled background and determines the sensitivity. The NA48/2 experiment provides pure π 0 D decay samples through the reconstruction of K ± → π ± π 0 and K ± → π 0 μ ± ν decays (denoted K 2π and K μ3). Additionally, the K ± → π ±...

We report results from the analysis of the K ± → π + π − e ± ν (K e4 ) decay by the NA48/2 collaboration at the CERN SPS, based on the total statistics of 1.13 million decays collected in [2003][2004]. The hadronic form factors in the S-and P-wave and their variation with energy are obtained. The phase difference between the S-and P-wave states of the ππ system is accurately measured and allows a precise determination of a 0 0 and a 2 0 , the I = 0 and I = 2 S-wave ππ scattering lengths: a 0 0 = 0.2220 ± 0.0128 stat ± 0.0050 syst ± 0.0037 th , a 2 0 = −0.0432 ± 0.0086 stat ± 0.0034 syst ± 0.0028 th . Combination of this result with the other NA48/2 measurement obtained in the study of K ± → π 0 π 0 π ± decays brings an improved determination of a 0 0 and the first precise experimental measurement of a 2 0 , providing a stringent test of Chiral Perturbation Theory predictions and lattice QCD calculations. Using constraints based on analyticity and chiral symmetry, even more precise values are obtained: a 0 0 = 0.2196 ± 0.0028 stat ± 0.0020 syst and a 2 0 = −0.0444 ± 0.0007 stat ± 0.0005 syst ± 0.0008 ChPT .

412The NA48/2 Collaboration: K e4 decay form factors and ππ scattering phase shifts Abstract. We report results from a new measurement of the K e4 decay K ± → π + π − e ± ν by the NA48/2 collaboration at the CERN SPS, based on a partial sample of more than 670 000 K e4 decays in both charged modes collected in 2003. The form factors of the hadronic current (F, G, H) and ππ phase difference (δ = δs − δp) have been measured in ten independent bins of the ππ mass spectrum to investigate their variation. A sizeable acceptance at large ππ mass, a low background and a very good resolution contribute to an improved experimental accuracy, a factor two better than in the previous measurement, when extracting the ππ scattering lengths a

The algebraic structure of Thermo Field Dynamics lies in the q-deformation of the algebra of creation and annihilation operators. Doubling of the degrees of freedom, tilde-conjugation rules, and Bogoliubov transformation for bosons and fermions are recognized as algebraic properties of h q (1) and of h q (1|1), respectively.