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Register a new userKhuri-Treiman equations for $pipi$ scattering
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The Khuri-Treiman formalism models the partial-wave expansion of a scattering amplitude as a sum of three individual truncated series, capturing the low-energy dynamics of the direct and cross channels. We cast this formalism into dispersive equations to study $pipi$ scattering, and compare their expressions and numerical output to the Roy and GKPY equations. We prove that the Khuri-Treiman equations and Roy equations coincide when both are truncated to include only $S$- and $P$-waves. When higher partial waves are included, we find an excellent agreement between the Khuri-Treiman and the GKPY results. This lends credence to the notion that the Khuri-Treiman formalism is a reliable low-energy tool for studying hadronic reaction amplitudes.
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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.
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 $Q$.
The weak two-pion form factor $F_V^{pipi}$ is described as the product of a weak kernel $cal{K}_W$ by a strong function $Theta_{pipi}^P$, determined directly from $pipi$ scattering data. As the latter accounts at once for all effects associated with resonances, intermediate $Kbar{K}$ loops, and other possible inelasticities present in $pipi$ scattering, the need of modeling is restricted to $cal{K}_W$ only. The procedure proposed allows one to asses the weak kernel directly, which has a dominant cut beginning at the $Kbar{K}$ threshold. Even the simplest vector-meson-dominance choice for $cal{K}_W$ already yields a good qualitative description of $F_V^{pipi}$. The energy sector below $0.8$ GeV is quite well reproduced when a precise theoretical chiral perturbation $pipi$ amplitude is used as input, together with the single free parameter $F_V G_V/F^2=1.20$. The inclusion of kaon loops, along well established lines and using few parameters, produces a good description of the form factor in the entire energy range allowed by $tau$ decays. This indicates that the replacement of modeling by direct empirical scattering information can also be useful in the construction of theoretical tools to be used in analysesof hadronic heavy meson decay data.
A Wilsonian approach to $pipi$ scattering based in the Glazek-Wilson Similarity Renormalization Group (SRG) for Hamiltonians is analyzed in momentum space up to a maximal CM energy of $sqrt{s}=1.4$ GeV. To this end, we identify the corresponding relativistic Hamiltonian by means of the 3D reduction of the Bethe-Salpeter equation in the Kadyshevsky scheme, introduce a momentum grid and provide an isospectral definition of the phase-shift based on a spectral shift of a Chebyshev angle. We also propose a new method to integrate the SRG equations based on the Crank-Nicolson algorithm with a single step finite difference so that isospectrality is preserved at any step of the calculations. We discuss issues on the unnatural high momentum tails present in the fitted interactions and reaching far beyond the maximal CM energy of $sqrt{s}=1.4$ GeV and how these tails can be integrated out explicitly by using Block-Diagonal generators of the SRG.
A Wilsonian approach based on the Similarity Renormalization Group to $pipi$ scattering is analyzed in the $JI=$00, 11 and 02 channels in momentum space up to a maximal CM energy of $sqrt{s}=1.4$ GeV. We identify the Hamiltonian by means of the 3D reduction of the Bethe-Salpeter equation in the Kadyschevsky scheme. We propose a new method to integrate the SRG equations based in the Crank-Nicolson algorithm with a single step finite difference so that isospectrality is preserved at any step of the calculations. We discuss issues on the high momentum tails present in the fitted interactions hampering calculations.
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