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Register a new userSolving integral equations in $etato 3pi$
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A dispersive analysis of $etato 3pi$ decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: i) The angular averages of the amplitudes need to be performed along a complicated path in the complex plane. ii) The averaged amplitudes develop singularities along the path of integration in the dispersive representation of the full amplitudes. It is a delicate affair to handle these singularities properly, and independent checks of the obtained solutions are demanding and time consuming. In the present article, we propose a solution method that avoids these difficulties. It is based on a simple deformation of the path of integration in the dispersive representation (not in the angular average). Numerical solutions are then obtained rather straightforwardly. We expect that the method also works for $omegato 3pi$.
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A reliable determination of the isospin breaking double quark mass ratio from precise experimental data on $etato 3pi$ decays should be based on the chiral expansion of the amplitude supplemented with a Khuri-Treiman type dispersive treatment of the final-state interactions. We discuss an extension of this formalism which allows to estimate the effects of the $a_0(980)$ and $f_0(980)$ resonances and their mixing on the $etato 3pi$ amplitudes. Matrix generalisations of the equations describing elastic $pipi$ rescattering with $I=0,,2$ are introduced which accomodate both $pipi/Kbar{K}$ and $etapi/Kbar{K}$ coupled-channel rescattering. Isospin violation induced by the physical $K^+-K^0$ mass difference and by direct $u-d$ mass difference effects are both accounted for in the dispersive integrals. Numerical solutions are constructed which illustrate how the large resonance effects at 1 GeV propagate down to low energies. They remain small in the physical region of the decay, due to the matching constraints with the NLO chiral amplitude, but they are not negligible and go in the sense of further improving the agreement with experiment for the Dalitz plot parameters.
We present a measurement of the slope parameter $alpha$ for the $etato 3pi^{0}$ decay, with the KLOE experiment at the DA$Phi$NE $phi$-factory, based on a background free sample of $sim$ 17 millions $eta$ mesons produced in $phi$ radiative decays. By fitting the event density in the Dalitz plot we determine $alpha = -0.0301 pm 0.0035,stat;_{-0.0035}^{+0.0022},syst,$. The result is in agreement with recent measurements from hadro- and photo-production experiments.
We report a preliminary measurement of the slope parameter $alpha$ for the $etato 3piz$ decay carried out with KLOE at DA$Phi$NE; where $alpha$ is the parameter describing the energy dependence of the square of the matrix element for this decay. By fitting the event density in the Dalitz plot with a collected statistic of 420 pb$^{-1}$ we determine $alpha = -0.027 pm 0.004 (stat) ^{+0.004}_{-0.006} (syst)$. This result is consistent with current chiral perturbation theory calculations within the unitary approach.
Recent experiments on $etato 3pi$ decays have provided an extremely precise knowledge of the amplitudes across the Dalitz region which represent stringent constraints on theoretical descriptions. We reconsider an approach in which the low-energy chiral expansion is assumed to be optimally convergent in an unphysical region surrounding the Adler zero, and the amplitude in the physical region is uniquely deduced by an analyticity-based extrapolation using the Khuri-Treiman dispersive formalism. We present an extension of the usual formalism which implements the leading inelastic effects from the $Kbar{K}$ channel in the final-state $pipi$ interaction as well as in the initial-state $etapi$ interaction. The constructed amplitude has an enlarged region of validity and accounts in a realistic way for the influence of the two light scalar resonances $f_0(980)$ and $a_0(980)$ in the dispersive integrals. It is shown that the effect of these resonances in the low energy region of the $eta to 3pi$ decay is not negligible, in particular for the $3pi^0$ mode, and improves the description of the energy variation across the Dalitz plot. Some remarks are made on the scale dependence and the value of the double quark mass ratio $Q$.
Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a family of appropriate Hilbert scales so that the operator is ill-posed of order a in the scale. We provide weak regularity assumptions on the kernel underlying the operator for the above to hold true. Our construction leads to a well-defined regularisation strategy by Tikhonov regularisation in Hilbert scales. We thereby generalise the results of Gorenflo and Yamamoto for a < 1 to arbitrary a > 0 and more general kernels. Thanks to tools from interpolation theory, we also show that the a priori associated to the Hilbert scale formulates in terms of smoothness in usual Sobolev spaces up to boundary conditions, and that the regularisation term actually amounts to penalising derivatives. Finally, following the theoretical construction, we develop a comprehensive numerical approach, where the a priori is encoded in a single parameter rather than in a full operator. Several numerical examples are shown, both confirming the theoretical convergence rates and showing the general applicability of the method.
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