Low energy spectrum of SU(2) lattice gauge theory

Indrakshi Raychowdhury

doi:10.1140/epjc/s10052-019-6753-0

Abstract

Prepotential formulation of gauge theories on honeycomb lattice yields local loop states, which are exact and orthonormal being free from any spurious loop degrees of freedom. We illustrate that, the dynamics of orthonormal loop states are exactly same in both the square and honeycomb lattices. We further extend this construction to arbitrary dimensions. Utilizing this result, we make a mean field ansatz for loop configurations for SU(2) lattice gauge theory in 2+1 dimension contributing to the low energy sector of the spectrum. Using variational analysis, we show that, this type of mean loop configurations has two distinct phases in the strong and weak coupling regime and shows a first order transition at g=1. We also propose a reduced Hamiltonian to describe the dynamics of the theory within the mean field ansatz. We further work with the mean loop configuration obtained towards the weak coupling limit and analytically calculate the spectrum of the reduced Hamiltonian. The spectrum matches with that of the existing literature in this regime, establishing our ansatz to be a valid alternate one which is far more easier to handle for computation.

Highlights

Understanding the low energy behaviour of gauge theories is one of the most important problem of particle physics
In this work, we have proposed a general technique of constructing explicit orthonormal loop states for SU(2) lattice gauge theory in any arbitrary dimension
We make a mean field ansatz that only an average loop configuration contribute to the low energy spectrum of SU(2) lattice gauge theory. We show that this average loop configuration has two distinct phases at the weak and strong coupling regime of the theory and shows a first order phase transition at g = 1 denoting two distinct vacuum at two regimes of the theory

Summary

Introduction

Understanding the low energy behaviour of gauge theories is one of the most important problem of particle physics. The magnetic Hamiltonian consists of 26 = 64 terms, each of which is a set of six local loop operator at each of the six vertices of the hexagon, the action of which on respective local loop states are computed following the Table 2. At this point we compare the Hamiltonian dynamics on hexagonal lattice with that on the square lattice numerically. This numerical study proves that the dynamics of loop states on a square plaquette is identical to that on an hexagonal plaquette as long as one is interested only in orthonormal loop states, which are relevant for exact physical degrees of freedom

Point Splitting and virtual hexagonal lattice

The Hamiltonian and Average loop configurations

Mean Field Ansatz

The reduced Hamiltonian and its spectrum

Summary and future directions