[Paper Review] Fine Tuning in Lattice SU(2) Gluodynamics vs Continuum-Theory Constraints

В. И. Захаров|ArXiv.org|Jun 26, 2003

Quantum Chromodynamics and Particle Interactions5 references3 citations

TL;DR

This paper investigates the apparent fine-tuning in lattice SU(2) gluodynamics, where monopole and vortex contributions exhibit ultraviolet divergences despite asymptotic freedom in the continuum theory. It resolves the tension by showing that monopoles and vortices are confined to a two-dimensional subspace of spacetime, forming 'branes' that evade the constraints of asymptotic freedom, thus reconciling lattice data with continuum quantum field theory.

ABSTRACT

Recently, it has been observed that the non-Abelian action associated with lattice monopoles and vortices is ultraviolet divergent, at least at presently available lattices. On the other hand, the total length of the monopole trajectories and area of the vortices scale in physical units. Coexistence of the two different scales, infrared and ultraviolet, for the same vacuum fluctuations represents a fine tuning. To check consistency of the newly emerging picture of non--perturbative fluctuations we consider constraints from the continuum theory on the ultraviolet behaviour of the monopoles and vortices. The constraints turn to be satisfied by the data in a highly non-trivial way. Namely, it is crucial that the monopoles populate not the whole of the four dimensional space but a two-dimensional subspace of it.

Motivation & Objective

• To resolve the apparent fine-tuning conflict between UV-divergent monopole masses in lattice SU(2) gluodynamics and the asymptotic freedom of continuum Yang-Mills theory.
• To examine whether the observed ultraviolet divergence of monopole mass (scaling as 1/a) is consistent with continuum constraints.
• To investigate the geometric and topological structure of monopole and vortex configurations in the vacuum, particularly their spatial distribution.
• To explore the implications of monopoles and vortices populating a two-dimensional subspace for gauge-invariant descriptions of non-perturbative vacuum structure.
• To assess the viability of these configurations as physical degrees of freedom consistent with known continuum field theory principles.

Proposed method

• Analyzing lattice data on monopole and vortex contributions to the non-Abelian action, particularly their scaling with lattice spacing a.
• Applying constraints from continuum asymptotic freedom to rule out point-like or fully four-dimensional monopole distributions.
• Using the projection method to define monopoles and vortices via closest U(1) or Z₂ configurations on the lattice, ensuring gauge invariance of final observables.
• Deriving the monopole mass scaling ⟨M(a)⟩ ∼ 1/a from the excess of non-Abelian action associated with monopole currents.
• Examining the spatial distribution of monopoles and vortices to show they are confined to a two-dimensional subspace, not the full four-dimensional spacetime.
• Comparing lattice results with continuum field theory expectations, particularly regarding UV divergences and the absence of new elementary particles.

Experimental results

Research questions

• RQ1 Can the observed ultraviolet divergence of monopole mass in lattice SU(2) gluodynamics be reconciled with the asymptotic freedom of the continuum theory?
• RQ2 What geometric structure of monopole and vortex configurations allows them to evade the constraints of asymptotic freedom despite UV divergences?
• RQ3 Is the assumption that monopoles are confined to a two-dimensional subspace sufficient to resolve the fine-tuning problem?
• RQ4 How do the lattice results on monopole and vortex contributions compare with known continuum field theory expectations, such as the absence of new elementary particles?
• RQ5 Can gauge-invariant observables, such as the determinant of the color magnetic field or the Dirac operator spectrum, identify the same two-dimensional structures observed in the lattice projections?

Key findings

• The monopole mass scales as ⟨M(a)⟩ ∼ 1/a, indicating an ultraviolet divergence consistent with a point-like particle, yet this is not incompatible with asymptotic freedom.
• The UV divergence is resolved by the fact that monopoles and vortices are not distributed throughout four-dimensional spacetime but are confined to a two-dimensional subspace, effectively forming 'branes'.
• This two-dimensional localization allows the system to avoid the constraints of asymptotic freedom that would otherwise rule out such divergences.
• The contribution of vortices to the average plaquette action is estimated at ⟨1−P⟩_vort ≈ 0.1 GeV² a⁻², matching the scale of ultraviolet renormalons.
• The condensate ⟨|ϕ|²⟩ ≈ 0.8 fm⁻² is gauge-invariant and dimension-2, consistent with recent proposals for dimension-2 gluon condensates in terms of fundamental fields.
• The determinant of the color magnetic field D(x) = ||Hⁱᵃ(x)|| vanishes on two-dimensional surfaces, suggesting a gauge-invariant way to identify vortices and monopole cores.