Flux tubes at Finite Temperature

TL;DR

This paper investigates the behavior of flux tubes in quark-antiquark and quark-quark pairs at finite temperature using SU(3) Lattice QCD, analyzing chromomagnetic and chromoelectric fields across phase transitions.

Contribution

It provides the first detailed lattice QCD analysis of flux tubes at finite temperature, including both chromomagnetic and chromoelectric fields.

Findings

Flux tubes are characterized at different temperatures.

Chromomagnetic and chromoelectric fields vary across the phase transition.

Results enhance understanding of confinement at finite temperature.

Abstract

In this work, we show the flux tubes of the quark-antiquark and quark-quark at finite temperature for SU(3) Lattice QCD. The chromomagnetic and chromoelectric fields are calculated above and below the phase transition.

Tables

Table 1. Table 1: Lattice simulations for a 48 3 × 8 superscript 48 3 8 48^{3}\times 8 volume. The lattice spacing was computed using the parametrization from [ 5 ] in units of the string tension at zero temperature. The ∗ means without configurations in the wrong phase transition.

$\beta$	$T/T_c$	$a\sqrt{\sigma }$	# config.
5.96	0.845	0.235023	5990
6.055	0.988	0.200931	5990/4775*
6.1237	1.100	0.180504	3669
6.2	1.233	0.161013	1868
6.338	1.501	0.132287	3688
6.5	1.868	0.106364	1868

Equations

$$f_{\mu \nu }(r,x)=\frac{\beta }{a^4}\left[\frac{⟨\mathcal{O}{\square }_{\mu \nu }(x)⟩}{⟨\mathcal{O}⟩}-⟨{\square }_{\mu \nu }(x)⟩\right]$$

$$f_{\mu \nu }(r,x)=\frac{\beta }{a^4}\left[\frac{⟨\mathcal{O}{\square }_{\mu \nu }(x)⟩-⟨\mathcal{O}{\square }_{\mu \nu }(x_R)⟩}{⟨\mathcal{O}⟩}\right]$$

$$f_{\mu \nu }\to \frac{1}{2}\left(-⟨B_x^2⟩,-⟨B_y^2⟩,-⟨B_z^2⟩,⟨E_x^2⟩,⟨E_y^2⟩,⟨E_z^2⟩\right)$$

$$U_4\to {\bar{U}}_4=\frac{\int {\left[\mathcal{D}{U}_4\right]}_{\Omega }{U}_4{e}^{\beta {\sum }_{\mu ,s}\text{Tr}\left[U_{\mu }(s)F^{†}(s)\right]}}{\int {\left[\mathcal{D}{U}_4\right]}_{\Omega }{e}^{\beta {\sum }_{\mu ,s}\text{Tr}\left[U_{\mu }(s)F^{†}(s)\right]}}$$

Full text

\PACS

11.15.Ha; 12.38.Gc

1 Introduction

The study of the chromo fields distributions inside the flux tubes formed $QQ$ and $Q\bar{Q}$ are presented in this study. How the flux tube evolves when the distance between quarks or the temperature increase beyond respective critical values are addressed in this paper. In section 2, we describe the lattice formulation. We briefly review the Polyakov loop for these systems and show how to compute the color fields as well as the Lagrangian distribution. In section 3, the numerical results are shown. Finally, we conclude in section 4.

2 Computation of the chromo-fields in the flux tube

The central observables that govern the event in the flux tube can be extracted from the correlation of a plaquette, $\square_{\mu\nu}$, with the Polyakov loops, $L$,

[TABLE]

where $\mathcal{O}=L(0)\,L^{\dagger}(r)$ for the $Q\bar{Q}$ system or $\mathcal{O}=L(0)\,L(r)$ for the $QQ$ system, $x$ denotes the distance of the plaquette from the line connecting quark sources, $r$ is the quark separation, $L(r)=\frac{1}{3}\Tr\Pi_{t=1}^{N_{t}}U_{4}(r,t)$ where $N_{t}$ is the number of time slices of the lattice and using the periodicity in the time direction for the plaquette, $\square_{\mu\nu}(x)=\frac{1}{N_{t}}\sum_{t=1}^{N_{t}}\square_{\mu\nu}(x,t)$, allows averaging over the time direction.

To reduce the fluctuations of the $\mathcal{O}\,\square_{\mu\nu}(x)$, we measure the following quantity, [1],

[TABLE]

where $x_{R}$ is the reference point placed far from the quark sources.

Therefore, using the plaquette orientation $(\mu,\nu)=(2,3),(1,3),(1,2),$ $(1,4),(2,4),(3,4)$, we can relate the six components in Eq. 2 to the components of the chromoelectric and chromomagnetic fields,

[TABLE]

and also calculate the total action (Lagrangian) density, $\Braket{\mathcal{L}}=\frac{1}{2}\left(\Braket{E^{2}}-\Braket{B^{2}}\right)$

In order to improve the signal over noise ratio, we use the multihit technique, [2, 3], replacing each temporal link by it’s thermal average, and the extended multihit technique, [4], which consists in replacing each temporal link by it’s thermal average with the first $N$ neighbors fixed. Instead of taking the thermal average of a temporal link with the first neighbors, we fix the higher order neighbors, and apply the heat-bath algorithm to all the links inside, averaging the central link,

[TABLE]

By using $N=2$ we are able to greatly improve the signal, when compared with the error reduction achieved with the simple multihit. Of course, this technique is more computer intensive than simple multihit, while being simpler to implement than multilevel. The only restriction is $R>2N$ for this technique to be valid.

3 Results

In this section, we present the results for different $\beta$ values suing a fixed lattice volume of $48^{3}\times 8$, Table 1. All the computations were done in NVIDIA GPUs using CUDA.

The $QQ$ and $Q\bar{Q}$ are located at $(0,-R/2,0)$ and $(0,R/2,0)$ for $R=4,6,8,10\text{ and }12$ lattice spacing units. In Figs. 2 and 4, we show the results for the $Q\bar{Q}$ system. As expected the strength of the fields decrease with the temperature. Also, in the confined phase the width in the middle of the flux tube increases with the distance between the sources, while above the phase transition the width decreases with the distance.

Just below the phase transition, we need to make sure that we don’t have contaminated configurations as already mentioned in [6]. By plotting the histogram of Polyakov loop history for $\beta=6.055$, Fig. 1, we were able to identify a second peak which then we were able to remove all the configurations that lie on the second peak. Therefore, in Table 1 the value with asterisk corresponds to the configurations after removing these contaminated configurations. In Figs. 3a and 3b, we show the results of this effect for the $QQ$ system below the phase transition.

4 Conclusions

As the distance increase between the sources, the field strength at the flux tube decreases as already seen in studies at zero temperature. Below the phase transition, the fields strength decreases as the temperature increases. However, above the phase transition the fields rapidly decrease to zero as the quarks are pulled apart. The width of the flux tube below the phase transition increases with the separation between the quark-antiquark, however above the phase transition the width seems to decrease.

Acknowledgments

Nuno Cardoso and Marco Cardoso are supported by FCT under the contracts SFRH/BPD/109443/2015 and SFRH/BPD/73140/2010 respectively. We also acknowledge the use of CPU and GPU servers of PtQCD, supported by NVIDIA, CFTP and FCT grant UID/FIS/00777/2013.