[Paper Review] The Gluon Propagator in Lattice Landau Gauge with twisted boundary conditions

T. Tok, Kurt Langfeld+2 more|ArXiv.org|Sep 26, 2005

Quantum Chromodynamics and Particle Interactions5 citations

TL;DR

This study investigates the infrared behavior of the gluon propagator in lattice Landau gauge using twisted boundary conditions to suppress zero-momentum fluctuations. By comparing with periodic boundary conditions, the authors find that twisted boundaries reduce large zero-mode fluctuations, leading to a smaller gluon propagator that approaches zero at low momentum—supporting the non-compact space-time prediction over the torus-based non-zero limit.

ABSTRACT

We investigate the infrared behaviour of the gluon propagator in Landau gauge on a lattice with twisted boundary conditions. Analytic calculations using Dyson-Schwinger equations, exact renormalization group and stochastic quantization show that the gluon propagator in Landau gauge approaches zero for small momentum. On the other hand lattice calculations and calculations on a four-torus seem to give rise to a non-zero limit. One possible reason for this difference is the existence of zero-momentum fluctuation modes which potentially give a massive contribution to the gluon propagator. Our simulations show that with twisted boundary conditions these zero-momentum modes are suppressed and the gluon propagator becomes smaller than in a periodic ensemble.

Motivation & Objective

• To resolve the discrepancy between Dyson-Schwinger equation predictions (vanishing gluon propagator at p→0) and lattice simulations on a torus (non-zero limit) in Landau gauge.
• To examine how zero-momentum fluctuation modes on compact manifolds like the four-torus affect the gluon propagator's infrared behavior.
• To test whether twisted boundary conditions suppress these zero modes and lead to a lower bound on the gluon propagator compared to periodic boundary conditions.
• To clarify whether the non-zero infrared limit observed in periodic lattice simulations is an artifact of finite-size effects and zero-mode contributions.

Proposed method

• Implementation of twisted boundary conditions via constant transition functions Ωμ(x) = iσμ for spatial directions, ensuring anti-periodicity in specific color components.
• Use of SU(2) gauge group with fixed transition functions that preserve gauge invariance and allow consistent Landau gauge fixing.
• Numerical simulation of the gluon propagator D(p²) at β=2.15 on various lattice sizes (e.g., 6⁴) with both twisted and periodic boundary conditions.
• Comparison of the gluon propagator in twisted vs. periodic ensembles to assess finite-size and zero-mode effects.
• Fourier transformation of the gauge potential Aμ(x) to extract the zero-momentum component Cμ = ˜Aμ(p=0), monitoring its fluctuations.
• Adoption of the vacuum configuration Uμ(x) ≡ 1 as a reference, compatible with twisted boundary conditions, to maintain translational invariance.

Experimental results

Research questions

• RQ1 Does the presence of zero-momentum fluctuation modes on a compact four-torus explain the non-zero infrared limit of the gluon propagator observed in lattice simulations?
• RQ2 How do twisted boundary conditions affect the magnitude and stability of the gluon propagator in the infrared regime?
• RQ3 Can twisted boundary conditions suppress zero-mode contributions and lead to a gluon propagator that vanishes at p→0, aligning with Dyson-Schwinger equation results on R⁴?
• RQ4 Is the difference in the infrared behavior of the gluon propagator between compact and non-compact space-times intrinsic, or an artifact of finite-size effects and boundary conditions?

Key findings

• Twisted boundary conditions significantly suppress fluctuations in the zero-momentum component Cμ of the gauge potential, as shown in simulation histories on a 6⁴ lattice.
• The gluon propagator D(p²) in the twisted ensemble is consistently smaller than in the periodic ensemble across all lattice sizes and momenta.
• As lattice spatial extension increases, the difference between the twisted and periodic gluon propagators decreases, indicating diminishing finite-size effects from the twist.
• The gluon propagator in the twisted ensemble approaches zero at low momentum, supporting the Dyson-Schwinger equation prediction D(p²) ∝ (p²)^(2κ−1) with κ≈0.6.
• The results suggest that the non-zero infrared limit observed in periodic lattice simulations arises from zero-mode contributions, which are effectively removed by twisted boundary conditions.
• The twisted boundary condition setup provides a lower bound for the gluon propagator in periodic simulations, indicating that the non-zero limit may not reflect the true infrared behavior on R⁴.