The isospin-symmetry-breaking effects in <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">e</Emphasis>4</Subscript> decays

The final-state interaction of pions in K _e4 decay allows to obtain the value of the isospin and angular-momentum-zero ππ scattering length a ₀⁰.We take into account the electromagnetic interaction of pions and isospin-symmetry-breaking effects caused by different masses of neutral and charged pions and estimate the impact of these effects on the procedure of scattering-length extraction from K _e4 decays.     

The electromagnetic effects in <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">e</Emphasis>4</Subscript> decay

The final-state interaction of pions in K _e4 decay allows to obtain the value of the isospin and angular-momentum-zero ππ scattering length a ₀ ⁰ .We take into account the electromagnetic interaction of pions and isospin-symmetry-breaking effects caused by different masses of neutral and charged pions and estimate the impact of these effects on the procedure of scattering-length extraction from K _e4 decays.     

Isospin breaking in the phases of the K e4 form factors

Isospin breaking in the K _?4 form factors induced by the difference between charged and neutral pion masses is studied. Starting from suitably subtracted dispersion representations, the form factors are constructed in an iterative way up to two loops in the low-energy expansion by implementing analyticity, crossing, and unitarity due to two-meson intermediate states. Analytical expressions for the phases of the two-loop form factors of the K ^±→π ⁺ π ^? e ^± ν _e channel are given, allowing one to connect the difference of form-factor phase shifts measured experimentally (out of the isospin limit) and the difference of S- and P-wave ππ phase shifts studied theoretically (in the isospin limit). The isospin-breaking correction consists of the sum of a universal part, involving only ππ rescattering, and a process-dependent contribution, involving the form factors in the coupled channels. The dependence on the two S-wave scattering lengths $a_{0}^{0}$ and $a_{0}^{2}$ in the isospin limit is worked out in a general way, in contrast to previous analyses based on one-loop chiral perturbation theory. The latter is used only to assess the subtraction constants involved in the dispersive approach. The two-loop universal and process-dependent contributions are estimated and cancel partially to yield an isospin-breaking correction close to the one-loop case. The recent results on the phases of K ^±→π ⁺ π ^? e ^± ν _e form factors obtained by the NA48/2 collaboration at the CERN SPS are reanalysed including this isospin-breaking correction to extract values for the scattering lengths $a_{0}^{0}$ and $a_{0}^{2}$ , as well as for low-energy constants and order parameters of two-flavour χPT.     

Pion-pion phase shift and K2π decay

We remark that, using the Cabibbo transformation properties for the weak Hamiltonian we can obtain Δ₀ = δ₀⁽⁰⁾ (s = M_K⁽²⁾ (s = M_K²) from the experimental decay rates for K_s → 2π, K⁺ → π⁰π⁺, obtaining δ₀ = 58.0° ± 4.6°. This result implies in particular a value of ≈ 0.6 for the S-wave isospin zero scattering length, in accordance with K_e4 results but in violent disagreement with Weinberg's calculation.     

The strong interaction shift and width of the ground state of pionic hydrogen

The 3p-1s transition in pionic hydrogen was investigated with a high-resolution crystal spectrometer system. From the precisely measured transition energy, together with the (calculated) electromagnetic energy, the strong interaction shift of the 1s state was obtained as ϵ_1s = −7.127 ± 0.028(stat.)± 0.036(syst.) eV (attractive). From the natural line width, measured for the first time, we determine the decaywidth of the 1s state: Γ_1s^(decay) = 0.97 ± 0.10(stat.)± 0.05(syst.) eV. With the recently calculated electromagnetic corrections the s-wave scattering lengths of an isospin symmetric strong interaction are deduced. The scattering length for elastic scattering of a negative pion on a proton is a_{π⁻p→π⁻p}^h = 0.0885±0.00003(stat.)±0.0006(syst.)m_π⁻¹. The scattering lengthe for single charge exchange is found to be a_{π⁻p→π⁰n}^h = −0.136 ± 0.007(stat.) ± 0.003(syst.)m_π⁻¹.The experiment was performed at the Paul Scherrer Institute (PSI) in Switzerland. A focussing crystal spectrometer with an array of bent crystals, the cyclotron trap (a magnetic system designed to increase the particle stop density) and a CCD (charge-coupled device) detector system were employed. The results from the pionic hydrogen experiment — together with those from the pionic deuterium experiment — were used to test the isospin symmetry of the strong interaction. The present data are still consistent with isospin sysmmetry.     

Isospin violation in the standard model and the decay F+ → π+π0

The decay F⁺ → π⁺π⁰ can proceed via isospin violating mechanisms similarly to $\text{ψ′ →}\text{J}\text{ψπ}{}^0$. I estimate that π⁰-η-′ mixing together with symmetry breaking in the amplitude lead to Γ(F⁺ → π⁺π⁰)/Γ(F⁺ → π⁺η) = (2.2 ± 0.7) × 10^?3 or (1.0 ± 0.3) × 10^?3, if annihilation processes dominate nonleptonic charmed meson decays. These values correspond to different assumptions for two-meson production in F decays.     

Measurement of the slope of the π0-electromagnetic form factor

Using a magnetic spectrometer, we have studied 30 000 π⁰ → e⁺e^?γ events coming from K_π2⁺ decays in flight. We find a positive value for the slope of the neutral-pion electromagnetic form factor: a = +0.10 ± 0.03 (or a = +0.05 ± 0.03 without radiative corrections). This value is higher than the predictions of vector meson models.     

The process e + e − → ωπ0 near the Φ resonance

The reaction e ⁺ e ^?→ωπ⁰ near a Φ resonance was studied with a spherical neutral detector at VEPP-2M e ⁺ e ^? collider. Both main modes of decay of a ω meson were investigated: ω→π⁺π^?π⁰ and ω→π ⁰γ. The probability of decay Φ→ωπ⁰ was obtained from the magnitude of the interference wave in the cross section for the reaction e ⁺ e ^?→ωπ⁰→ π⁺π^?π⁰π⁰: B(Φ→ωπ⁰) = (5.5 _?1.4 ^+1.6 ±0.3) ×10^?5. The ratio of the partial widths of the ω meson was obtained from the ratio of the cross sections for the two modes: Γ(ω→π⁰γ )/Γ(ω→π⁺π^?π⁰) = 0.0994 ± 0.0036 ± 0.0038.     

Measurement of elastic ω photoproduction at HERA ZEUS Collaboration

The reaction γp →ωp(ω → π⁺π^?π⁰ and π⁰ → γ γ) has been studied in ep interactions using the ZEUS detector at photon-proton centre-of-mass energies between 70 and 90GeV and ∣t∣ < 0.6GeV², where t is the squared four momentum transferred at the proton vertex. The elastic ω photoproduction cross section has been measured to be σ_{γp → ωp} = 1.21 ±0.12 ±0.23 μb. The differential cross section dσ_γp→ωp/d¦t¦ has an exponential shape e^?b∣t∣ with a slope b =10.0 ± 1.2 ± 1.3 GeV^?2. The angular distributions of the decay pions are consistent with s-channel helicity conservation. When compared to low energy data, the features of ω photoproduction as measured at HERA energies are in agreement with those of a soft diffractive process. Previous measurements of the ρ⁰ and ? photoproduction cross sections at HERA show a similar behaviour.     

Observation of the π0 → e+e− decay

We have observed for the first time the very rare π⁰ → e⁺e^? decay. The tagged π⁰'s come from K⁺ → π⁺π⁰ decays in flight. We find the branching ratio to be (2.23_?1.1^+2.4) × 10^?7 (90% C.L.). This value is well above the QED unitary limit and comes in rough agreement with a second-order electromagnetic process mediated by vector mesons.     

Structure and experimental consequences of chiral SU3 × SU3 breaking

Chiral SU₂ × SU₂ breaking, Cabibbo angle, non-electromagnetic isospin breaking, η → 3π decay rate, and pion and kaon mass differences are related to one another in a nonlinear Lagrangian model of broken chiral SU₃ × SU₃ symmetry. The agreement with experiment is very good.     

The e + e −→π0π0γ process below 1.0 GeV

The e ⁺ e ^?→π⁰π⁰ γreaction was studied in the energy range 0.36–0.97 GeV on a VEPP-2M e ⁺ e ^? collider with an SND detector. The decay probabilities were found to be B(ω→ π⁰π⁰γ)= (7.8 ±2.7±2.0)×10^?5 and B(π→ π⁰π⁰γ)= (4.8 _?1.8 ^+3.4 ±0.2)×10^?5.     

A study of the decay KL0 → π±e∓νe

In a study of charged K_L⁰ three-body decays a sample of 6668 K_L⁰ → π^±e^?ν candidates has been obtained. The Dalitz plot distribution is in agreement with V ? A theory, and limits are presented for scalar and tensot contributions to the weak current. Using a conventional expansion for the form factor f₊ we find λ₊ = 0.055 ± 0.010 with systematic effects estimated at ± 0.01.     

Isospin breaking in chiral perturbation theory and the decaysη →πℓν andτ →ηπν

Violation of isospin due tom _u ≠m _d and electromagnetism is investigated within the framework of the standard model. We discuss the electromagnetic contributions to the masses of the pseudoscalar mesons and theK _?3 amplitudes. The decay η→π?ν is calculated at the one-loop level in chiral perturbation theory including an estimate of the electromagnetic contributions. Adding all four decay channels, we obtain 2.0·10^?13 as a rather accurate upper bound for the branching ratio. We determine the leading meson resonance contributions to τ→ηπν predicting a branching ratio of 1.2·10^?5.     

Comment on the η→3π, η→2γ decay rates

The main decay modes of the eta meson are discussed with emphasis on large symmetry breaking effects induced by the mass of the strange quark. It is argued that these SU3-breaking effects amount to corrections involving factors of 2 to 3 in rate in the above decay modes. This is in line with the well-known large correction factors in the nonleptonic decays K_S →2π and D⁰→K⁺K^?, π⁺π^?.     

New high statistics measurement of Ke4 decay form factors and ππ scattering phase shifts

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=δ_s-δ_p) 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 a₀ ⁰ and a₀ ². Under the assumption of isospin symmetry and using numerical solutions of the Roy equations, the following values are obtained in the plane (a₀ ⁰,a₀ ²): a₀ ⁰=0.233±0.016stat±0.007syst,a₀ ²=-0.0471±0.011stat±0.004syst. The presence of potentially large isospin effects is also considered and will allow comparison with precise predictions from Chiral Perturbation Theory. PACS 13.20.Eb; 13.75.Lb     

Radiative Kaon decays and CP violation in chiral perturbation theory

Chiral perturbation theory is a very useful framework for testing the standard model in processes where long-distance effects are expected to play an essential rôle. We analyze the rare K decays K⁰→γℓ⁺ℓ⁻, K⁺→π⁺γ3 and K_L→π⁰ℓ⁺ℓ⁻ in the effective chiral formulation of the standard model. These processes, like the decays K⁰→γγ, K⁺→π⁺ℓ⁺ℓ⁻, K_S→π⁰ℓ⁺ℓ⁻ and K⁰→π⁰γγ discussed in previous work, have the property that the corresponding amplitudes vanish to lowest order in chiral perturbation theory. Precise predictions for decay rates and spectra are made in terms of a few coupling constants not restricted by softly broken chiral symmetry alone. Special consideration is given to various possible tests of CP noninvariance in these decays, in particular to effects due to intrinsic CP violation in the amplitudes. We find that chiral perturbation theory correlates different CP-violating observables such as the charge asymmetries in K^±→π^±γγ and K^±→π^±ℓ⁺ℓ⁻, the one-photon exchange contribution to K_L→π⁰e⁺e⁻ and the transverse polarization in K_L →π⁰μ⁺μ⁻. Detailed numerical results are shown.     

Pion-pion scattering lengths from Veneziano formula and η→3π,K→3π decays

Theη→ 3π decay amplitude is factorized into (a) the transition matrix element fromη toπ ⁰ and (b) Lovelace's expression of the Veneziano type for the pion-pion scattering amplitude. (a) is related with the electromagnetic mass splitting betweenK ⁺ andK ⁰. The absolute value of (b) is then obtained by fitting theη decay width. The pion-pion scattering lengths from (b) are reasonable. Since the analogous treatment ofK→ 3π decay together with a soft-pion determination of theK→π transition matrix element yield very small scattering lengths we suggest that theK→2π transition matrix element depends strongly on the pion masses.     

Study of some rare decays of the K+ meson

In a search for very rare decays of the K⁺ meson we observed the K⁺→μ⁺νe⁺e^? and K⁺→e⁺ν e⁺e^? modes with branching ratios of (10±3) ×10^?7 and (2⁺²_?2)×10^?7 respectively. We also put new limits on a decay allowed by the alternate scheme of lepton conservation K⁺→π^?μ⁺e⁺, on a decay with double charged weak current K⁺→π^? e⁺e⁺ and on other forbidden decays K⁺→μ^?νe⁺e⁺ and K⁺→π⁺μ^±e^±.     

Measurement of the relative probability of the decay φ → η′γ in the channel η′ → ηπ0π0, η → γγ

The process e ⁺ e ^? → η′γ, η′ → η π⁰π⁰, η → γγ is investigated by an SND detector in experiment on a VEPP-2M equipment with colliding electron-positron beams. The analysis of experimental data yields a value of B(? → η′γ)=(6.7 _?4.2 ^+5.0 ±1.5)×10^?5 for the decay probability. Taking into account a previous measurement of this quantity by an SND detector in the decay channel η′ → π⁺π^?η, one finally obtains B(?→η ′γ)=(6.7 _?2.4 ^+2.8 ±0.8)×10^?5.