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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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Efforts to describe nuclear structure and dynamics from first principles have advanced significantly in recent years. Exact methods for light nuclei are now able to include continuum degrees of freedom and treat structure and reactions on the same footing, and multiple approximate, computationally efficient many-body methods have been developed that can be routinely applied for medium-mass nuclei. This has made it possible to confront modern nuclear interactions from Chiral Effective Field Theory, that are rooted in Quantum Chromodynamics with a wealth of experimental data. Here, we discuss one of these efficient new many-body methods, the In-Medium Similarity Renormalization Group (IMSRG), and its applications in modern nuclear structure theory. The IMSRG evolves the nuclear many-body Hamiltonian in second-quantized form through continuous unitary transformations that can be implemented with polynomial computational effort. Through suitably chosen generators, we drive the matrix representation of the Hamiltonian in configuration space to specific shapes, e.g., to implement a decoupling of low- and high-energy scales, or to extract energy eigenvalues for a given nucleus. We present selected results from Multireference IMSRG (MR-IMSRG) calculations of open-shell nuclei, as well as proof-of-principle applications for intrinsically deformed medium-mass nuclei. We discuss the successes and prospects of merging the (MR-)IMSRG with many-body methods ranging from Configuration Interaction to the Density Matrix Renormalization Group, with the goal of achieving an efficient simultaneous description of dynamic and static correlations in atomic nuclei.
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.
We consider the $pipi$-scattering problem in the context of the Kadyshevsky equation. In this scheme, we introduce a momentum grid and provide an isospectral definition of the phase-shift based on the spectral shift of a Chebyshev angle. We address the problem of the unnatural high momentum tails present in the fitted interactions which reaches energies far beyond the maximal center-of-mass energy of $sqrt{s}=1.4$ GeV. It turns out that these tails can be integrated out by using a block-diagonal generator of the SRG.
We present for the first time a determination of the energy dependence of the isoscalar $pipi$ elastic scattering phase-shift within a first-principles numerical lattice approach to QCD. Hadronic correlation functions are computed including all required quark propagation diagrams, and from these the discrete spectrum of states in the finite volume defined by the lattice boundary is extracted. From the volume dependence of the spectrum we obtain the $S$-wave phase-shift up to the $Koverline{K}$ threshold. Calculations are performed at two values of the $u,d$ quark mass corresponding to $m_pi = 236, 391$ MeV and the resulting amplitudes are described in terms of a $sigma$ meson which evolves from a bound-state below $pipi$ threshold at the heavier quark mass, to a broad resonance at the lighter quark mass.
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