This paper tests a simple idea: can the force between heavy quarks be split into two parts that add up — one coming from Abelian monopoles (a simpler, diagonal part of the field) and the other from the remaining nonabelian field with those monopoles removed? The authors study this question in SU(3) gluodynamics, the pure-glue part of the theory of the strong force, using numerical simulations on a spacetime lattice.

To do this they fix the Maximal Abelian Gauge (MAG). This is a way of choosing a gauge so that the field looks as diagonal (Abelian) as possible. After that they separate the lattice gauge field into an Abelian diagonal component and a non-diagonal remainder, and further split the Abelian angles into a singular “monopole” part and a regular “photon” part. From each field they compute the static potential V(r) — the energy between two static color charges at distance r — using standard lattice tools: Wilson loops and noise-reduction techniques called APE smearing and hypercubic blocking. The authors ran simulations at two couplings, β = 6.0 and 6.1, which correspond to lattice spacings a ≈ 0.93 fm and a ≈ 0.79 fm.

A key technical issue is the existence of many gauge-fixed versions of the same configuration, known as Gribov copies. The authors compare two ways to fix the gauge. One is simulated annealing (SA), an algorithm tuned to find very low values of the MAG functional. The other is relaxation with over-relaxation (RO), a simpler method that produces different Gribov copies. With the SA copies they confirm earlier findings: the sum V_mon(r)+V_mod(r) (monopole plus modified nonabelian piece) departs noticeably from the full V(r) at large distances. The mismatch comes from a small string tension in V_mon(r). In contrast, on the Gribov copies produced by the RO algorithm the decomposition V(r) ≈ V_mon(r)+V_mod(r) holds with good precision at all distances and for both lattice spacings. This difference is the main result of the paper.