Asymptotic freedom in the Hamiltonian approach to binding of color

TL;DR

This paper derives the property of asymptotic freedom in SU(3) Yang-Mills theory using a Hamiltonian approach with effective particles, providing a new perspective on the renormalization group flow of the coupling.

Contribution

It introduces a Hamiltonian framework with effective particles to derive asymptotic freedom and compute the beta function perturbatively up to third order.

Findings

Confirmed asymptotic freedom in the Hamiltonian formalism.

Calculated the SU(3) Yang-Mills beta function up to third order.

Established a relation between effective Hamiltonians and the canonical Hamiltonian.

Abstract

We derive asymptotic freedom and the $SU(3)$ Yang-Mills $\beta$-function using the renormalization group procedure for effective particles. In this procedure, the concept of effective particles of size $s$ is introduced. Effective particles in the Fock space build eigenstates of the effective Hamiltonian $H_s$, which is a matrix written in a basis that depend on the scale (or size) parameter $s$. The effective Hamiltonians $H_s$ and the (regularized) canonical Hamiltonian $H_{0}$ are related by a similarity transformation. We calculate the effective Hamiltonian by solving its renormalization-group equation perturbatively up to third order and calculate the running coupling from the three-gluon-vertex function in the effective Hamiltonian operator.