[Paper Review] Gauge Invariant Monopoles in Lattice SU(2) Gluodynamics

F. V. Gubarev|ArXiv.org|Apr 12, 2002

Quantum Chromodynamics and Particle Interactions1 references4 citations

TL;DR

This paper presents a lattice implementation of a gauge-invariant monopole charge definition in SU(2) gluodynamics, using interpolated spin coherent states to construct conserved monopole currents. Despite lattice artifacts and an effective Ising model complicating the extraction of physical monopoles, the density of physical monopoles scales correctly toward the continuum limit, with an estimated density of approximately (760 MeV)³ at β = 2.6.

ABSTRACT

We consider lattice implementation of the recently proposed gauge invariant definition of the monopole charge. Because of the lattice discretization the algorithm gives rise to specific lattice artifacts and an effective Ising model. The Ising-model problem might in principle be solved and we discuss the role of the Maximal Abelian gauge in this respect. The lattice artifacts are much more difficult to deal with since they are mixed up with monopoles thus obscuring the physical observables. Nevertheless, it is possible to extract the density of physical monopoles which seems to scale correctly towards the continuum limit.

Motivation & Objective

• To implement a gauge-invariant monopole charge definition on the lattice, avoiding gauge-fixing ambiguities.
• To address the challenge of lattice artifacts that mix with physical monopoles and distort observables.
• To test whether the physical monopole density scales correctly toward the continuum limit.
• To investigate the role of the Maximal Abelian gauge in stabilizing the monopole solution and reducing Ising model frustration.
• To separate physical monopoles from lattice artifacts using a dipole-based assumption, enabling quantitative estimation of physical monopole density.

Proposed method

• Use of spin coherent states |n(t)⟩ to define a gauge-invariant phase evolution along Wilson loops, ensuring consistency with the continuum monopole definition.
• Interpolation of spin states between lattice sites to construct a continuous vector field n on the dual lattice, enabling integer-valued monopole currents.
• Application of the gauge-invariant monopole charge formula (4) on the interpolated lattice, leading to conserved monopole currents.
• Identification of lattice artifacts arising from the modified geometry of the dual lattice, which contribute to apparent monopole density but vanish in the continuum limit.
• Use of simulated annealing and Maximal Abelian gauge preconditioning to minimize the effective Ising model action and resolve sign ambiguities.
• Assumption of monopole-artifact dipole structure in 4D SU(2) LGT to subtract artifacts and extract physical monopole density.

Experimental results

Research questions

• RQ1 Can a gauge-invariant monopole charge be consistently defined and implemented on the lattice without gauge-fixing ambiguities?
• RQ2 How do lattice artifacts affect the measured monopole density, and can they be systematically separated from physical monopoles?
• RQ3 Does the density of physical monopoles extracted from lattice simulations scale correctly toward the continuum limit?
• RQ4 To what extent does the Maximal Abelian gauge improve the stability and accuracy of monopole detection on the lattice?
• RQ5 Is the observed scaling of the physical monopole density consistent with the expected renormalization group behavior?

Key findings

• The lattice implementation successfully produces integer-valued, conserved monopole currents using interpolated spin coherent states.
• Lattice artifacts, which are not physical, exhibit a nearly constant density across the β range, confirming their actionless nature.
• The physical monopole density ρ_phys = ρ_monopoles − ρ_artifacts follows the expected renormalization group scaling behavior in the weak coupling regime.
• The estimated physical monopole density at β = 2.6 is approximately (760 MeV)³, using the string tension σ = (440 MeV)².
• The Maximal Abelian gauge significantly improves the convergence of monopole detection, even without explicit gauge fixing, suggesting a privileged role in monopole physics.
• Despite the lack of rigorous artifact subtraction, the dipole assumption enables a consistent and scaling physical monopole density estimate.