Zero-momentum correlator in SU ( 2 ) Landau gauge gluodynam-ics under various boundary conditions

Wemake new simulations of S U(2) zero spatial momentum (ZM) gluon correlator in Landau gauge gluodynamics paying special attention to possible lattice artefacts and Gribov copy effect. In particular we started investigation of the correlator dependence on the choice of boundary conditions (BCs) by comparing results for periodic and zerofield (ZF) boundary conditions at various β values. Time behaviour of the ZM correlator for ZF BCs corresponds to approximately constant in time effective gluon mass thus providing additional evidence in favour of decoupling behaviour of momentum-dependent gluon propagator in the IR region. We have found that at fixed lattice sizes and β values the ZM gluon correlator and effective gluon mass for periodic and ZF boundary conditions can differ considerably.


Introduction
The only solution which has been found in lattice studies of momentum-dependent gluon and ghost propagators within Landau gauge gluodynamics on sufficiently large lattices D ab μν = δ ab δ μν − q μ q ν q 2 Z(q 2 ) q 2 , G ab = δ ab J(q 2 ) q 2 is decoupling, or regular one, which is characterized by the finite nonzero gluon propagator D(p 2 ) and the finite ghost dressing function J(q 2 ) for q 2 → 0 (for details see [1][2][3]).This solution describes the "effectively massive" gluon and breaks the BRST symmetry contrary to the scaling, or conformal solution, corresponding to massless gluon, zero D(0) and infinite J(q 2 → 0).After thorough checking of possible lattice artefacts due to finite-volume, finite-size and Gribov copy effects the decoupling solution has been widely accepted by the "community of nonperturbative QCD studies" as the only physical solution.However up to now dependence of the lattice propagators on the choice of boundary conditions has not been reported, to our knowledge (note, that in most lattice studies of gluodynamics periodic boundary conditions (PBCs) are traditionally used).Here we present results of the comparative study of the ZM gluon correlator in S U(2) gluodynamics under periodic and zero-field (sometimes called "open") boundary conditions.

Definition of correlators
Lattice studies of the gauge-dependent "space", Tr A i (t, x) A i (0, x) , and "time" ZM gluon correlators in the Landau gauge was started in [4] and continued in [5][6][7].

Landau gauge fixing
To fix the Landau gauge, we apply to all links a gauge transformation , with the aim to maximize a gauge functional or, more exactly, to find the global maximum of F U [g] [8].Search for the global maximum in practice is a complicated problem which becomes exceedingly time consuming with increasing lattice volume.Starting from an initial random gauge transformation g x one generally arrives at one of many local maxima of F U [g].The corresponding gauge-fixed configurations are called Gribov copies; they all satisfy the differential gauge condition ∂ μ A μ = 0. To avoid appearance of lattice Gribov copies we successfully applied the Simulated Annealing (SA) method (see, e.g., [3] and references therein), finalized by overrelaxation (OR) iterations.The SA method -often called a "stochastic optimization method"-allows quasi-equilibrium tunneling through functional barriers, in the course of a "temperature" T decrease.It is proven that infinitely slow "cooling" of a system under consideration (decreasing of T ) allows to reach the global maximum of its underlying functional.In practice, the greater is the number of SA "cooling" steps, the closer is the value of F U [g] local maxima reached in simulations to its global maximum.In this work we for the first time applied the SA technique to the study of the ZM gluon correlator which allowed to diminish dispersion of the correlator S (t) values and thus raise accuracy of "effective" gluon mass calculation.
The typical number of MC configurations was of order 10 4 with one gauge fixed copy obtained for each MC configurations.For gauge fixing we use S A + OR method with rather slow cooling (large number of SA steps) thus reaching values of the gauge functional G F very close to its global maximum.We have checked that T (t) is constant in t with very high accuracy (in Landau gauge it should be t-independent [4]), which shows reliability of our simulations.
In the first series of simulations we use zero-field boundary conditions (ZF BCs).It means that link variables U x,μ are given by unit S U(2) matrix for all links out of the minimal hypercubic volume containing all L 3 s × L t lattice sites.The varying correlator S (t) obtained for β = 2.6 is plotted in figure 1.It worth noting that correlators S (t) found for ZF BCs at various lattice volumes are presented in the logarithmic scale by nearly straight lines which correspond to gluon "effective" masses independent of N t contrary to what was found for periodic boundary conditions (PBCs) in [4][5][6][7] (see also below).
In the second series of simulations we compare results obtained on the lattice L = 12 3 × 24 for 2 types of boundary conditions: periodic (which are traditionally used in lattice experiments, PBCs) and zero-field (ZF BCs) ones; the results are presented in figure 2. One can see that the S (t) correlators obtained for PBCs and ZF BCs can differ considerably.
Finite-volume effects (mainly) and insufficient statistics (number of MC configurations) can lead to distortion of awaited exponential decrease of S (T ) correlator; in particular, the decrease is in most cases smaller at small N t , and relative statistical errors become progressively larger when N t → L t /2 + 1.That is why the notion of N t -dependent "effective" mass is introduced.In figure 3 and figure 4 we compare dependencies of "effective" masses for PBCs and ZF BCs on the lattice L = 12 3 × 24 and various β = 2 g 2 (figure 3 for β = 2.2, 2.3, 2.4 and figure 4 for β = 2.5, 2.6).One can see the quick increase of M e f f (N t ) found under ZF BCs at larger β = 2.5, 2.6 for which correlation length becomes larger than the physical lattice extension L s × a(β) due to small values of a(β).One might assume that for larger β ZF BCs introduce stronger finite-volume effects, which however should become weaker with growing lattice extension L s .
Our simulations of ZM gluon correlators at β = 2.6 and various L = L 3 s × L t with L s = 2 L t allow to find out whether it is really the case.In figure 5 it is clearly seen that the effective mass curves for Effective mass, beta=2.6,various L t , L s zero-field vs periodic boundary conditions PBCs and ZF BCs mutually converge at growing lattice extensions, tending to some limiting curve which could be used for estimation of effective gluon mass being obtained from the ZM correlator S (t).
Note here, that mass estimates for larger β's are physically preferable because for these β's finitesize effects are considerably lower.Relying on the simulations of the ZM correlator with β = 2.6 one might propose the following bounds for gluon mass M g : 2MeV ≤ M g ≤ 3.5MeV.

Figure 5 .
Figure 5. Convergence of dependencies of effective masses at β = 2.6 for (i) PBCs and (ii) ZF BCs when increasing lattice extensions