Decomposition of the static potential in the Maximal Abelian gauge

Decomposition of SU(2) gauge field into the monopole and monopoleless components is studied in the Maximal Abelian gauge using Monte-Carlo simulations in lattice SU(2) gluodynamics as well as in two-color QCD with both zero and nonzero quark chemical potential. The interaction potential between static charges is calculated for each component and their sum is compared with the non-Abelian static potential. A good agreement is found in the confinement phase. Implications of this result are discussed.


Introduction
We study the decomposition of the non-Abelian gauge field in the Maximal Abelian gauge (MAG) [1,2] into the sum of the monopole component and the monopoleless component. For this purpose we employ the SU(2) lattice gauge theory.
In terms of vector potentials A µ (x), the decomposition has the form where A mon µ (x) is the monopole component and A mod µ (x) is the monopoleless component defined below and referred to as the modified gauge field.
In the MAG, the Abelian dominance for the string tension has long been known [3][4][5][6][7] (for a review see e.g. [8,9]). Moreover, it was found [5,10,11] that the properly determined monopole component of the gauge field produces the string tension close to its exact value in agreement with conjecture that the monopole degrees of freedom are responsible for confinement [12,13].
In Refs. [14,15] it was shown that the topological charge, chiral condensate and effects of chiral symmetry breaking in quenched light hadron spectrum disappear after removal of the monopole contribution from the relevant operators. Similar computations were made within the scope of the Z 2 projection studies [16]. In particular it was shown that after removal of Pvortices the confinement property disappears. We perform a similar removal of monopoles. We consider the following types of the static potential: V mod (r) obtained from the Wilson loops of the modified gauge field U mod µ (x), V mon (r) obtained from the Wilson loops of the monopole gauge field u mon µ (x) and the sum of these two static potentials.
For one value of lattice spacing it was shown [17] that V mod (r) can be well approximated by purely Coulomb fit function and the sum V mod (r) + V mon (r) provides a good approximation to the original non-Abelian static potential V(r) at all distances.
Here we study this phenomenon at three lattice spacings using the Wilson action and thus we can draw conclusions about the continuum limit. We also present the results for one lattice spacing obtained with the improved lattice field action thus checking the universality. Furthermore, we present results for the SU(2) theory with dynamical quarks, i.e. for QC 2 D. These results were partially presented in [18].

Details of simulations
We study the SU(2) lattice gauge theory. Vector potentials of the gauge field can be defined in terms of the link variables by the formula where a is the lattice spacing. Up to terms of the order O(a 2 ), the decomposition (1) can be rearranged to the form which furnishes the subject of our research.
To fix the MAG we use the simulated annealing algorithm [5] with one gauge copy per configuration. Usually after fixing MAG the following decomposition of the non-Abelian lattice gauge field U µ (x) is made where u µ (x) is the Abelian field and C µ (x) is the non-Abelian coset field. The Abelian gauge field u µ (x) is further decomposed [19] into the monopole (singular) part u mon µ (x) and the photon (regular) part u ph µ (x): Then it follows from eq. (1) that Note that u ph (x) is the Abelian projection of U mod µ (x) and involves no monopoles. We need to compute the usual Wilson loops the monopole Wilson loops and the non-Abelian Wilson loops with removed monopole contribution It is known that MAG fixing leaves U(1) gauge symmetry unbroken. The monopole Wilson loop W mon (C) is invariant under respective residual gauge transformations. This is not true for W mod (C) [17]. Thus we need to fix the Landau U(1) gauge by finding the maximum of the gauge-fixing functional, where ω ∈ U(1) is the gauge transformation. To imrove the noise-to-signal ratio for the static potential we use the APE smearing [20] in computations of the Wilson loops. We generate 100 statistically independent gauge field configurations with the Wilson lattice action at β = 2.4, 2.5 on the 24 4 lattice and at β = 2.6 on the 32 4 lattice. The respective values of lattice spacing are a = 0.118, 0.085 and 0.062 fm, which are determined from a fit to the lattice data [21] on the string tension, whose "experimental" value is set to √ σ = 440 MeV. As a check of universality the computations were done with the tadpole improved action at β = 3.4 on 24 4 lattices. Additionally we present our results [22] obtained in QC 2 D on 32 4 lattice at zero and nonzero quark chemical potential µ q .
It is worth to note that another decomposition, namely eq. (4) was investigated in Ref. [7] in the case of SU(3) gluodynamics. Good agreement between the static potenial V(r) and the sum V abel (r) + V o f f (r) was found. We believe that this decomposition also deserves further study.

Results
In Fig. 1 we show the usual non-Abelian static potential V(r) denoted as 'full' and compare it with the sum V sum (r) = V mon (r) + V mod (r). One can see that approximate equality  is satisfied for all three lattice spacings and the approximation improves toward the continuum limit. To give further support to this statement we plot in Fig. 2 the relative deviation ∆(r) defined as It is clear that ∆(r) decreases with decreasing lattice spacing. We also studied the universality of the decomposition of the static potential eq. (11). The simulations were made with the tadpole improved action at β = 3.4 with lattice spacing approximately equal to that of the Wilson action at β = 2.5. The results of our computations are presented in Fig. 3. It is seen that the agreement between V(r) and V sum (r) is nearly as good as in Fig. 1 for β = 2.5.
Furthermore, we completed the same computations in QC 2 D on 32 4 lattices with small lattice spacing at zero and nonzero quark chemical potential µ q (details of simulations in QC 2 D can be found, e.g., in [22]). The results of these computations are presented in Fig. 4 for µ q = 0 and in Fig. 5 for aµ q = 0.19. It can be seen clearly that approximate decomposition (11) is fulfilled with rather high both at zero and nonzero µ q .

Conclusions
We have studied the decomposition of the static potential into the linear term produced by the monopole (Abelian) gauge field U mon (x) and the Coulomb term produced by the monopoleless non-Abelian gauge field U mod (x). In the case of Wilson action we have presented the results (Figs. 1 and 2) for three values of lattice spacing to demonstrate that the agreement becomes better with a decrease of lattice spacing. Thus our results imply that the relation (1) becomes exact in the continuum limit. Further work is needed to provide more evidence for this conclusion. Next, we have demonstrated that the decomposition (1) holds true also in the case of improved lattice action (see Fig. 3). Furthermore, we have found that it works also in QC 2 D for both zero and nonzero quark chemical potential. It should be noticed that in Ref. [18] we also presented results for the static potential in the adjoint representation and found that the decomposition (1) works quite well in this case although not so well as in the case of the fundamental representation. It is of course interesting to verify how decomposition (1) works for other observables, e.g. action density and energy density of the flux tube. Also the decomposition should be checked in the case of SU(3) gauge group.
One can draw the following conclusions from the decomposition (1). The monopole part of the gauge field U mon (x) is responsible for the classical part of the energy of the hadronic string, whereas the monopoleless part U mod (x) is associated with the fluctuating part of its energy, i.e. U mod (x) reproduces perturbative results at small distances and contributes to the nonperturbative physics at large distances.