$\pi\pi$ scattering in a renormalized Hamiltonian matrix

TL;DR

This paper applies a Wilsonian SRG approach to $ ext{p} ext{p}$ scattering, developing a relativistic Hamiltonian framework and a new numerical integration method to analyze phase shifts up to 1.4 GeV.

Contribution

It introduces a relativistic Hamiltonian for $ ext{p} ext{p}$ scattering using a 3D reduction and a novel Crank-Nicolson based SRG integration method that preserves isospectrality.

Findings

Effective Hamiltonian for $ ext{p} ext{p}$ scattering up to 1.4 GeV

New numerical method preserves spectral properties during SRG evolution

Insights into high momentum tails and their explicit integration

Abstract

A Wilsonian approach to $\pi\pi$ 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.