Probing for Instanton Quarks with epsilon-Cooling

TL;DR

This work investigates whether instantons in SU(2) gauge theory can be understood as composites of fractionally charged constituents (instanton quarks) in backgrounds with non-trivial holonomy. It develops and applies two lattice tools—$\varepsilon$-cooling, which tunes $O(a^2)$ lattice artifacts to control instanton size, and adiabatic cooling, which changes temperature via anisotropic couplings—to explore finite- and zero-temperature configurations, analyzing action/topological charge densities, Polyakov loops, and fermion zero-modes. The findings show well-separated fractional-charge constituents at finite temperature, and extended, overlapping structures at zero temperature, with boundary conditions (including twists) and cooling histories providing diagnostic signatures; zero-mode localization further informs the constituent picture. Overall, the results support the relevance of instanton quarks to the topological structure of the QCD vacuum and potentially to confinement, while acknowledging methodological limitations and the need for complementary analyses.

Abstract

We use epsilon-cooling, adjusting at will the order a^2 corrections to the lattice action, to study the parameter space of instantons in the background of non-trivial holonomy and to determine the presence and nature of constituents with fractional topological charge at finite and zero temperature for SU(2). As an additional tool, zero temperature configurations were generated from those at finite temperature with well-separated constituents. This is achieved by "adiabatically" adjusting the anisotropic coupling used to implement finite temperature on a symmetric lattice. The action and topological charge density, as well as the Polyakov loop and chiral zero-modes are used to analyse these configurations. We also show how cooling histories themselves can reveal the presence of constituents with fractional topological charge. We comment on the interpretation of recent fermion zero-mode studies for thermalized ensembles at small temperatures.

Figures (14)

• Figure 1: A charge 1 configuration on a $16^4$ lattice with periodic boundary conditions, generated from a Monte Carlo configuration in the confined phase, first being cooled with $\varepsilon=1$ to just above the one-instanton action, after which 500 sweeps of $\varepsilon=0$ (full), $-1$ (dashed) and $-10$ (dotted curves) cooling were applied. After interpolation of the lattice data we plot the action density (left) and Polyakov loop (right, in one of the directions only) along the line connecting its extrema. From the behaviour of the Polyakov loop we deduce that decreasing $\varepsilon$ pushes the constituents further apart.
• Figure 2: A charge 1 configuration on a $16^4$ lattice with the holonomies fixed to be trivial in one direction and maximally non-trivial in the other three directions, generated from a random start first being cooled with $\varepsilon =1$ to just above the one-instanton action, after which 80 sweeps of $\varepsilon=-1$ cooling were applied. We plot the Polyakov loop for two relevant directions, in a plane through the center of the instanton. In this plane the action density is shown in the middle.
• Figure 3: Starting from a continuum caloron solution with well-separated lumps, discretized on the anisotropic lattice and adjusted by 100 $\varepsilon=-10$ cooling sweeps, we performed the adiabatic cooling by reducing $\xi$ from 4 to 1, through $\xi=2\sqrt 2$, 2 and $\sqrt 2$, applying between each of the 4 steps 100 $\varepsilon=-10$ cooling sweeps. Shown is on the left the action density and on the right the Polyakov loop in the time direction along a line through the constituent locations. The dotted, dashed and full curves are for $\xi=4$, 2 and 1, respectively.
• Figure 4: Example of a charge 3 caloron solution with ${{{1\over 2}}}{\rm Tr}{{\cal P}_\infty}=-0.126$ on a $4\times 16^3$ lattice obtained from $\varepsilon=-1$ cooling. Shown are the surfaces where half the trace of the Polyakov loop takes on the values $0.5$ (light, red) and $-0.65$ (dark, blue), corresponding respectively to the constituent monopoles with positive and negative magnetic charge.
• Figure 5: A charge 2 caloron with ${\rm Tr}{{\cal P}_\infty}=0$ on a $4\times 16^3$ lattice obtained from a Monte Carlo generated configuration at $4/g^2=2.2$. We first went down to slightly above the 2 instanton action with $\varepsilon=1$. After that many thousands of $\varepsilon=-2$ cooling sweeps (followed by 500 with $\varepsilon=0$) were performed. This gives the finite volume modification of the so-called "rectangular" solution constructed in Ref. Us. Shown is a suitable surface of constant action density for the double doughnut structure, as well as (clockwise) the action density, the periodic zero-mode density and the Polyakov loop. The latter three are shown on a plane through the doughnut which supports the periodic zero-mode. The other doughnut seen by the action density and the anti-periodic zero-mode has the sign of the Polyakov loop inverted, but is not seen by the periodic zero-mode.