Asymptotic Freedom and Bound States in Hamiltonian Dynamics

TL;DR

This paper investigates a model of asymptotically free theories with bound states using the similarity renormalization group, demonstrating effective Hamiltonian approximations and methods to improve bound state energy calculations.

Contribution

It introduces a novel approach to approximate effective Hamiltonians in asymptotically free theories and improves bound state energy calculations using expansion in the running coupling.

Findings

Effective Hamiltonians can be approximated over a large width range.

Bound state energies are accurately obtained with a few expansion terms.

Approximation accuracy decreases at small widths but can be improved.

Abstract

We study a model of asymptotically free theories with bound states using the similarity renormalization group for hamiltonians. We find that the renormalized effective hamiltonians can be approximated in a large range of widths by introducing similarity factors and the running coupling constant. This approximation loses accuracy for the small widths on the order of the bound state energy and it is improved by using the expansion in powers of the running coupling constant. The coupling constant for small widths is order 1. The small width effective hamiltonian is projected on a small subset of the effective basis states. The resulting small matrix is diagonalized and the exact bound state energy is obtained with accuracy of the order of 10% using the first three terms in the expansion. We briefly describe options for improving the accuracy.