K -> 3 pi Final State Interactions at NLO in CHPT and Cabibbo's Proposal to Measure a_0-a_2

Abstract

We present the analytical results for the K -> 3 pi final state interactions at next-to-leading order (NLO) in CHPT. We also study the recent Cabibbo's proposal to measure the pi-pi scattering lenghts combination a_0-a_2 from the cusp effect in the pi^0-pi^0 energy spectrum at threshold for K^+ -> pi^0 pi^0 pi^+ and K_L -> pi^0 pi^0 pi^0$, and give the relevant formulas to describe it at NLO. For that, we use the NLO CHPT expression to fit the real part of K -> 3 pi to data while the pi-pi scattering lenghts are treated non-perturbatively. Using them, we make a quantitative estimate of the theoretical uncertaintity of the a_0-a_2 determination at NLO in our approach and obtain that it is not smaller than 5 % if added quadratically and 7 % if linearly for K^+ -> pi^0 pi^0 pi^+. One gets similar theoretical uncertainties if the neutral K_L -> pi^0 pi^0 pi^0 decay data below threshold are used instead. For this decay, there are very large theoretical uncertainties above threshold due to cancellations and data above threshold cannot be used to get the scattering lenghts. All the numbers we present are in the isospin limit apart of two-pion phase space factors which are physical. We compare our results for the cusp effect with Cabibbo and Isidori's results and discuss the differences and agreements. We also comment on the apperance of the singularity at the K -> 3 pi pseudo-threshold s=(m_K-m_pi)^2 in the discontinuity that defines the cusp.

arXiv comment31 pages, 8 figures. v2=v3 Added the full contributions to the cusp from the real part of the discontinuity. v4 Improved text. Matches published version

References in corpus (22)

• Big Bang Nucleosynthesis (in "The Review of Particle Properties" 2004)
• ππscattering
• Chiral perturbation theory
• Chiral Perturbation Theory
• Dijet Cross Sections and Parton Densities in Diffractive DIS at HERA
• The pi pi S-wave scattering lengths
• Virtual Photons in Low Energy pi-pi Scattering
• Cusps in K --> 3 pi decays
• The pion-pion scattering amplitude
• Virtual photons in SU(2) chiral perturbation theory and electromagnetic corrections to $\pi \pi$ scattering
• Determination of the a0-a2 pion scattering length from K->pi+ pi0 pi0 decay
• Pion-pion scattering and the K->3pi decay amplitudes
• Observation of a cusp-like structure in the pizero-pizero invariant mass distribution from K+- ==> pi+- pizero pizero decay and determination of the pi-pi scattering lengths
• $K\to 3\pi$ Decays in Chiral Perturbation Theory
• Isospin Breaking in $K\to 3\pi$ Decays III: Bremsstrahlung and Fit to Experiment
• Charged Kaon K \to 3 pi CP Violating Asymmetries at NLO in CHPT
• Strong rescattering in K-> 3pi decays and low-energy meson dynamics
• Isospin Breaking in $K\to 3\pi$ Decays II: Radiative Corrections
• Isospin Breaking in $K\to 3\pi$ Decays I: Strong Isospin Breaking
• Generating Functional for Strong and Nonleptonic Weak Interactions
• Virtual Photon Correction to the $K^{+}\to {\pi }^{+}{\pi }^0{\pi }^0$ Decay
• Three-body dispersion-relation N/D equations for the coupled decay channels ppbar (J^{PC}=0^{-+}) --> pi^0 pi^0 pi^0, eta pi^0 pi^0, eta eta pi^0, K Kbar pi^0

Cited by in corpus (29)

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• Cusps in K --> 3 pi decays
• Electromagnetic corrections in eta --> 3 pi decays
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• Cusps in K_L --> 3 pi decays
• Cusps in K --> 3pi decays: a theoretical framework
• Measurement of the Matrix Elements for the Decays $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $\eta /{\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$
• The cusp effect in eta' --> eta pi pi decays
• Measurement of the eta->3pi0 Dalitz Plot Distribution with the WASA Detector at COSY
• Predictions for the cusp in eta->3pi0 decay
• Soft-photon corrections in multi-body meson decays
• Measurement of the matrix elements for the decays ${\eta }^{\prime }\to \eta {\pi }^{+}{\pi }^{-}$ and ${\eta }^{\prime }\to \eta {\pi }^0{\pi }^0$
• Low-energy pi-pi and pi-K scatterings revisited in three-flavour resummed chiral perturbation theory
• Extraction of $ND$ scattering lengths from the ${\Lambda }_b\to {\pi }^{-}p{D}^0$ decay and properties of the ${\Sigma }_c(2800{)}^{+}$
• Hadronic Atoms
• Dispersive construction of two-loop $P\to \pi \pi \pi$ $(P=K,\eta )$ amplitudes
• Two-loop representations of low-energy pion form factors and pi-pi scattering phases in the presence of isospin breaking
• Dispersive representation of K -> 3 pi amplitudes and cusps
• Theoretical progress on cusp effect and Kl4 decays
• An introduction to chiral perturbation theory
• Measurement of the decay ${\eta }^{\prime }\to {\pi }^0{\pi }^0\eta$ at MAMI
• Evidence for the Cusp Effect in ${\eta }^{\prime }$ Decays into $\eta {\pi }^0{\pi }^0$
• Workshop on exotic hadronic atoms, deeply bound kaonic nuclear states and antihydrogen: present results, future challenges
• Final state interaction in kaons decays
• KAON 2007: Conference Summary
• Status of chiral perturbation theory
• Finite-temperature effects on the threshold cusps in $\pi \pi$ and $D{\bar{D}}^{\ast }$ scatterings from relativistic heavy-ion collisions
• Measuring the a0-a2 pion scattering lengths through K to 3 pi decays