Profiles , potential and string tension of the flux tube in SU ( 2 ) and SU ( 3 ) lattice gauge theories

In this work, we have studied SU(2) and SU(3) gauge theories explaining the colour interaction of a quark and an antiquark, and their identical and dissimilar properties. Using both gauge theories, we have performed simulations under similar conditions and have studied the differences in the results obtained. We have compared the transverse and longitudinal profiles of the chromoelectric and chromomagnetic components of the field strength, qqqq � potential, and temperature-dependent string tension of the flux tube. The potential between the quark and antiquark of the SU(3) theory was larger than that of the SU(2) under all temperatures. The string tension of SU(3) tends to stabilize starting from the critical temperature while that of SU(2) has a gradual decreasing feature.


INTRODUCTION
In Quantum chromodynamics, SU(2) and SU(3) gauge theories explain the colour interaction of a quark and an antiquark. SU (2) and SU(3) gauge groups are non-abelian gauge groups. SU(2) gauge group is represented by Pauli matrix with three generations, while SU(3) gauge group is represented by Gell-Mann matrix with eight generations. SU (2) gauge theory is used to describe more simple and general case of interaction between a quark and an antiquark. However, the theory based on this group can give us a clear and important information about the interaction of a quark and an antiquark. But, the theory based on the SU (3) gauge group can express the interaction between a quark and an antiquark more precisely and completely. There are eight types of gluons that carry the interaction and eight generators of SU(3) gauge group, which correspond to them. The two theories differ with the order of the phase transition from the confined to the deconfined phase. In other words, SU(2) gauge theory has the secondorder phase transition, while SU (3)

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In other words, physical observables that signal the phase transition exhibit different behaviour at critical point. Physical results from these theories are similar in terms of general dependency of the physical observables on the parameters, but are different in quantity.
In order to obtain a much more exact and precise information, we had to perform simulation using the SU(3) gauge theory. Proceeding from this premise, we converted the simulation programs based on SU(2) gauge theory that we had used earlier into the program based on SU(3) gauge theory. Then, in the ref.
[1], we checked our new program to assess the results generated for correctness. From this study, we observed that our new program works accurately and concluded that it can be used for more precise future studies. In the previous work [2], we analyzed and compared in detail the identical and dissimilar peculiarities of the gauge theories SU(2) and SU (3). Also, by comparing works carried out by other researchers, we analyzed the difference of physical results from these theories. In the present work, we have performed simulations with SU(3) program and sought to determine the differences in physical observables in the results from the new SU(3) program. For this reason, we made a comparison under the condition that the simulation parameters performed in this work were similar to those of the simulations performed in the previous work [3].
The rest of the paper is organized as follows. In Section 2, we provide information about the lattice definition of the observables to be measured, the details of the simulation program and the simulation parameter values. We have presented our results in Section 3, which has two subsections. The first subsection displays the comparative results of the profiles of the electric and magnetic components of the field strength in the flux tube. In the second subsection, we have computed � potential and temperature-dependent string tension in SU (2) and SU(3) gauge theories. Finally, out conclusions are listed in Section 4 followed by references.

Lattice measurement
Pure gauge theory with SU(3) gauge group and the standard Wilson action is used in our simulation. The measurements are performed on the lattices of size || × ⊥ × ⊥ × . We have performed simulations with the new program based on SU(3) gauge theory to see how both theories affect physical observables within flux tube. We have computed physical observables by extracting the chromoelectric and chromomagnetic components of the field strength in the flux tube from a correlation of a plaquette and the Polyakov loops as defined in the works referenced in [4,5].
There are two Polyakov loops, located at a distance from each other on the lattice and having oppositive directions, and they represent time propagation of the two static quarks sitting at a distance from each other. The plaquette variable, which is at ⃗ distance from the line connecting the � pair, with the orientation , . The six different combinations of and define the six chromoelectric and chromomagnetic components of the field strength [6]. Three space-space plaquettes correspond to the magnetic components and three space-time plaquettes correspond to the electric components.
Our simulation parameters and their values are summarized in Table 1.  (2) theory. This means that it is a coarser lattice.

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For the program based on the SU(2) gauge theory, the update algorithm was one heat bath [7][8][9] and four over-relaxation steps [10,11]. For noise reduction, we used the link integration method [12] and the reference point technique [13]. And for the program based on the SU(3) gauge theory, the update algorithm was one metropolis [14] and four overrelaxation steps [10,11]. Codes of both theories are written in C++ programming language.

Longitudinal and transverse profiles
In order to study how SU (2) (2) is approximately equal to those of SU(3). We used an error reduction method, which we called the "link integration", in our previous code based on SU(2) theory, to decrease statistical noise. But, during conversion of code, we could not include any error reduction method in the new simulation code based on the SU(3) theory since the error reduction technique did not work. Consequently, the error bars on the plots are wider. � potential and string tension For the next physical observable, we selected the � potential which depends on interquark distance linearly. The coloraveraged potential between a quark and an 1.13 -filled circle and 1.29 -triangle). From this figure, the general trends of the � potential which linearly increases with � separation and decreases with the temperature were same for both SU(2) and SU(3) theories. But it is clearly observable that the potential of the SU(2) is quantitatively smaller than that of the SU(3) in this figure. This results have been plotted in units of the temperature. After that, we computed the temperature-dependent string tension, ( ), by fitting following function on above data of the potential [15] ( , ) = 0 − + ( ) (2) with three free parameters. The potential includes a 1/ piece which accounts for a Coulomb type behaviour for small distances and the linear rising part. The results for the free parameters are listed in Table 2. The resulting string tension values in Table 2, normalized to their zero temperature values, are shown in Figure 3. At low temperatures, the values of string tension from SU(2) theory is slightly higher than those of SU (3). But, when the temperature increases, the difference becomes negligible. This is different from the results in ref. [16]. In this paper, the values of string tension from SU(3) theory is higher than those of SU(2) at low temperatures and this difference gets low with temperature and vanishes at the critical temperature. From this figure, we can see clearly that the string tension of SU(3) theory decreases to 0.98 and after that temperature, it stabilizes while string tension of SU(2) theory decreases to 1.13 and increases at 1.29 .

CONCLUSIONS
In this study, we have investigated the results of the two theories based on SU(2) and SU(3) gauge groups explaining the colour interaction of the quark and antiquark and compared their results. Simulation parameters were the same for both, namely, the lattice of size 24 × 12 2 × 6, temperatures 0.75 − 1.29 , number of measurement 20000, � separation 4 − 8 .
Longitudinal and transverse profiles of the chromoelectric and chromomagnetic components of the field strength in the flux tube, the potential between a quark and an antiquark and the temperature-dependent string tension are computed and compared.
For the profiles of the flux tube, all field strength components, except the parallel electric component, for SU(3) theory were smaller than those of the SU(2) theory. On the contrary, the potential between the quark and antiquark of the SU(3) theory was larger than that of the SU(2) at all temperatures. From the results, string tension of SU(3) tends to stabilize starting from the critical temperature while that of SU(2) has a gradual decreasing feature.