Color screening in ($2+1$)-flavor QCD

We study correlation functions of spatially separated static quark-antiquark pairs in ($2+1$)-flavor QCD in order to investigate onset and nature of color screening at high temperatures. We perform lattice calculations in a wide temperature range, $140\le T\le 5814\mathrm{MeV}$, using the highly improved staggered quark action and several lattice spacings to control discretization effects. By comparing at high temperatures our lattice results to weak-coupling calculations, as well as to the zero temperature result for the energy of a static quark-antiquark pair, we observe that color screening sets in at $rT\approx 0.3$. Furthermore, we also observe that in the range $0.3\lesssim rT\lesssim 0.6$ weak-coupling calculations in the framework of suitable effective field theories provide an adequate picture of color screening.

Figure 1

The continuum limit of the singlet free energy $F_S$ (left) and of the free energy $F_{Q\overline{Q}}$ (right). On the right and left, the black band and symbols show the $T=0$ static $Q\overline{Q}$ energy $V_S$; (right) the light gray band shows $F_S$ at high temperatures and very short distances, where finite temperature effects are smaller than the statistical errors. The horizontal bands correspond to the $r\to \infty$ limit of $F_S$ and $F_{Q\overline{Q}}$, i.e., $2F_Q$. The subtraction of $T\ln 9$ in the right panel is due to the normalization convention used for $F_{Q\overline{Q}}$ as discussed in the text.

Figure 2

The effective coupling obtained from the singlet free energy in the continuum limit and compared to the vacuum result (black band).

Figure 3

The subtracted free energy multiplied by $-r^{2}T$ at short distances in a log-log plot.

Figure 4

The effective coupling obtained from the free energy of a $Q\overline{Q}$ pair compared with the effective coupling obtainedom the $Q\overline{Q}$ static energy at zero temperature shown as a black band.

Figure 5

The logarithm of $-r{F}_W^{\mathrm{sub}}$ calculated from smeared and unsmeared Wilson loops. The black bursts show the result for the singlet correlator in the Coulomb gauge. (Upper) Correspond to $T=200\mathrm{MeV}$. (Lower) Correspond to $T=480\mathrm{MeV}$. (Left) Show the $N_{\tau }=8$ results. (Right) Show the $N_{\tau }=12$ result.

Figure 6

Continuum results for the difference $V_S-F_S$ in MeV (left) and in temperature units (right) as function of the distance $r$ between the static quark and antiquark.

Figure 7

Comparison of lattice and weak-coupling results for $V_S-F_S$. The weak-coupling results have been shifted by a small constant to match them to the lattice results at the shortest distance (see text). The dotted line corresponds to the renormalization scale $\mu =2\pi T$, while the band corresponds to its variation from $\pi T$ (solid) to $4\pi T$ (dashed).

Figure 8

Continuum results for $-r{F}_S^{\mathrm{sub}}$ as function of $rT$ at different temperatures. For visibility, we shifted the results for different temperatures. The uncertainty bands represent the leading-order (LO, hashed) and NLO (solid) results; the renormalization scale is varied from $\mu =\pi T$ (solid) to $2\pi T$ (dotted) and $4\pi T$ (dashed).

Figure 9

The octet contribution to the Polyakov loop correlator (left) and the octet free energy (right) at different temperatures as function of the distance $r$ calculated on $N_{\tau }=8$ lattices; see text.

Figure 10

The subtracted free energy multiplied by $-r^{2}T$ calculated on $N_{\tau }=12$ lattices (squares) and compared to the reconstruction based on the pNRQCD formula given by Eq. (30) (bands). (Left) The result for $T=172\mathrm{MeV}$ (Right) The result for $T=666\mathrm{MeV}$ (the fully reconstructed result is in magenta, while the blue band ignores Casimir scaling violating contributions to the octet potential and the green one ignores the octet contribution).

Figure 11

The $Q\overline{Q}$ free energy multiplied by $-9r^{2}T$ calculated for $T=1600\mathrm{MeV}$ in the continuum limit and compared with the NNNLO weak-coupling expression (lines). The upper line corresponds to $\mu =\pi T$, the middle one to $\mu =2\pi T$, and the lower line corresponds to $\mu =4\pi T$.

Figure 12

The free energy $F_{Q\overline{Q}}^{\mathrm{sub}}$ multiplied by $-9r^{2}T$ calculated on the lattice and in weak-coupling EQCD. The open symbols show lattice data for $N_{\tau }=4$ with aspect ratio $N_{\sigma }/N_{\tau }=6$ corrected for cutoff effects. The result for $T=1\mathrm{GeV}$ has $N_{\sigma }/N_{\tau }=4$. The bands represent the NLO (solid) results, where the scale has been varied as $\mu =\pi T$, $2\pi T$, and $4\pi T$ (solid, dotted, and dashed lines). The dash-dotted lines correspond to the LO result evaluated at $\mu =4\pi T$.

Figure 13

The subtracted free energy (upper) and the subtracted singlet free energy (lower) multiplied by $-r$ calculated for different temperatures and $N_{\tau }=4$.

Figure 14

The asymptotic screening mass in temperature units $m_S/T$ from the exponential fits of $-r{F}_S^{\mathrm{sub}}$. The lines correspond to the NLO Debye mass in temperature units calculated for $\mu =\pi T$, $2\pi T$, and $4\pi T$ (solid, dotted, and dashed) and multiplied by a constant to take into account nonperturbative effects. The open squares correspond to the screening masses determined using $N_{\tau }=4$ lattices with aspect ratio 6.

Figure 15

The asymptotic screening mass in temperature units $m_{Q\overline{Q}}/T$ from the exponential fits of $-r{F}_{Q\overline{Q}}^{\mathrm{sub}}$. The lines correspond to 2 times the NLO Debye mass in temperature units calculated for $\mu =\pi T$, $2\pi T$, and $4\pi T$ (solid, dotted, and dashed) and multiplied by a constant to take into account nonperturbative effects. The open squares correspond to the screening masses determined using $N_{\tau }=4$ lattices with aspect ratio 6. The horizontal band corresponds to the EQCD result for the screening mass obtained in 3D lattice calculations [79].

Figure 16

The ratio $\mathrm{\Delta }{f}_Q^{\mathrm{bare}}$ (left), the screening function $S_1$ for the subtracted singlet free energy $F_S^{\mathrm{sub}}$ (middle), and for the subtracted quark-antiquark free energy $F_{Q\overline{Q}}^{\mathrm{sub}}$ (right) are shown for different light sea quark masses. Results for different $\beta$ and $N_{\tau }$ have been shifted for reasons of visibility. The bands correspond to the ensembles with larger light sea quark masses. The observed deviation between ensembles with different masses can be understood in terms of larger statistical uncertainties of the lighter ensembles. For $N_{\tau }=4$, 6, and 12, the ensemble sizes of most lighter ensembles are sufficiently large such that results are in good agreement.

Figure 17

The screening function $S_1$ for the subtracted singlet free energy $F_S^{\mathrm{sub}}$ and for the subtracted free energy $F_{Q\overline{Q}}^{\mathrm{sub}}$. The asymptotic behavior exhibits a strong volume dependence ($N_{\tau }=4$ and $\beta =6.664$). Cancellation between quark-antiquark free energies and $2F_Q$ is better in the larger volume.

Figure 18

The continuum extrapolated entropy $S_Q$ in the high temperature region.

Figure 19

Cutoff effects in the tree-level lattice static energy with Lüscher-Weisz action are usually much smaller than for the Wilson gauge action. Both results are normalized by the corresponding continuum result.

Figure 20

The normalized deviation of the static energy from continuum estimates at different $\beta$ values. (Top) The results for $\beta =7.280$ (left) and $\beta =7.373$ (right). (Center) The results for $\beta =7.596$ (left) and $\beta =7.825$ (right). (Bottom) The results for $\beta =8.000$ (left) and $\beta =8.200$ (right). Points for different ${\beta }^{\mathrm{ref}}$ are displaced for better visibility. The averages (large open squares) are computed using the squared errors for each ${\beta }^{\mathrm{ref}}$ as weights.

Figure 21

Interpolation of the correction factors $⟨K⟩$. The points for “one ${\beta }^{\mathrm{ref}}$” have been obtained with ${\beta }^{\mathrm{ref}}=8.4$ for $\beta >7.373$ and with ${\beta }^{\mathrm{ref}}\ge 7.825$ for $\beta \le 7.373$.

Figure 22

Continuum interpolation of the correction factors $⟨K⟩$ with a shifted renormalization scheme. The enlarged errors in the range $(a/r_1{)}^2>0.1$ are not shown in the figure. The approach to the continuum limit is less steep in the shifted scheme.

Figure 23

Comparison of the correction factors $⟨K⟩$ from two different renormalization schemes. The results from both schemes are compatible with each other.

Figure 24

Improvement of $F_S$. We show results for $N_{\tau }=12$ for $\beta =9.67$ (left) and 8.2 (right) in the top row and for $\beta =7.373$ (left) and 6.664 (right) in the bottom row.

Figure 25

Improvement of $F_{Q\overline{Q}}$. We show results for $N_{\tau }=12$ for $\beta =9.67$ (left) and 8.2 (right) in the top row and for $\beta =7.373$ (left) and 6.664 (right) in the bottom row.

Figure 26

Comparison of local and global fits for $N_{\tau }=12$ as a log-log plot. We show the nonperturbatively corrected data entering the fits as large open symbols and the underlying data as small filled symbols. Both data sets are shown only for signal-to-noise ratio of at least five. The crosshatched bands represent local interpolations, and the diagonal-hatched bands represent the associated extrapolation up to $rT=0.75$. The solid bands represent global fits for low or high temperature ranges. In the overlap region, the shorter bands correspond to the low temperature global fit.

Figure 27

Scaling behavior of the subtracted free energy $F_{Q\overline{Q}}^{\mathrm{sub}}$ in units of the continuum result at various temperatures and at two representative separations (from top to bottom $rT=0.16$ and 0.40) as function of $N_{\tau }$. “CL” marks the continuum limit $(N_{\tau }\to \infty )$. Ratios for each temperature are shifted by constants for better visibility. The bands with filled pattern indicate fits with $b/N_{\tau }^2$, except for $rT=0.16$ and $T\in (200,400)\mathrm{MeV}$ or $T>1200\mathrm{MeV}$. The solid bands indicate fits with $c/N_{\tau }^4$. The large open (small filled) symbols indicate results from local (global) fits.

Figure 28

Scaling behavior of the subtracted singlet free energy $F_S^{\mathrm{sub}}$ in units of the continuum result at various temperatures and at two representative (from top to bottom $rT=0.16$ and 0.40) as function of $N_{\tau }$. “CL” marks the continuum limit $(N_{\tau }\to \infty )$. Results for each temperature are shifted by constants for better visibility. The bands with filled pattern indicate fits with $b/N_{\tau }^2$. The solid bands indicate fits with $c/N_{\tau }^4$. The large open (small filled) symbols indicate results from local (global) fits.

Figure 29

Scaling behavior of the effective coupling ${\alpha }_{Q\overline{Q}}[F_S]$ in units of the continuum result at various temperatures and at two representative separations (from top to bottom $rT=0.16$ and 0.40) as function of $N_{\tau }$. “CL” marks the continuum limit $(N_{\tau }\to \infty )$. Results for each temperature are shifted by constants for better visibility. The bands with filled pattern indicate fits with $b/N_{\tau }^2$. The solid bands indicate fits with $c/N_{\tau }^4$. The large open (small filled) symbols indicate results from local (global) fits.

Figure 30

Scaling behavior of $[V_S-F_S]/T$ in at various temperatures and at two representative separations (from top to bottom $rT=0.17$ and 0.35) as function of $N_{\tau }$. “CL” marks the continuum limit $(N_{\tau }\to \infty )$. Results for each temperature are shifted by constants for better visibility. The bands with filled pattern indicate fits with $b/N_{\tau }^2$. The solid bands indicate fits with $c/N_{\tau }^4$. The large open (small filled) symbols indicate results from local (global) fits.

Figure 31

Correction for finite volume [$\beta =6.664$, $N_{\tau }=4$, and aspect ratio $(\mathrm{AR})=N_{\sigma }/N_{\tau }$, equal to 4 and 6]. Uncorrected and finite-volume corrected data are shown with smaller filled and larger open symbols respectively.

Figure 32

Volume dependence of the screening mass $m_{Q\overline{Q}}$ from $F_{Q\overline{Q}}^{\mathrm{sub}}$. $\beta =6.664$ corresponds to a moderately high temperature ($T=421\mathrm{MeV}$), $\beta =8.200$ corresponds to a high temperature ($T=1687\mathrm{MeV}$), and $\beta =9.670$ corresponds to the highest temperature in the study ($T=5814\mathrm{MeV}$). We show the screening mass in units of the temperature, $m_{Q\overline{Q}}/T$. We shifted $m_{Q\overline{Q}}/T$ by a constant $\pm 1.5$ at the lowest or highest of the three temperatures for better visibility. The screening mass in the larger volume is shown with patterned bands for Polyakov loop point sources and with solid bands for Polyakov loop plane sources. The screening mass in the smaller volume is shown with filled symbols for Polyakov loop point sources and with open symbols for Polyakov loop plane sources. The triplets of horizontal lines indicate the naive weak-coupling expectation $2m_D/T$ for the scales $\mu =\pi T$, $2\pi T$, and $4\pi T$ (solid, dotted, and dashed) using the NLO result [Eq. (25)].

Figure 33

Volume dependence of the screening mass $m_S$ from $F_S^{\mathrm{sub}}$. $\beta =6.664$ corresponds to a moderately high temperature ($T=421\mathrm{MeV}$), $\beta =8.200$ corresponds to a high temperature ($T=1687\mathrm{MeV}$), and $\beta =9.670$ corresponds to the highest temperature in the study ($T=5814\mathrm{MeV}$). We show the screening mass in units of the temperature, $m_S/T$. The screening mass in the larger volume is shown with patterned bands, while the screening mass in the smaller volume is shown with filled symbols. The triplets of horizontal lines on the left side indicate the naive weak-coupling expectation $m_D/T$ for the scales $\mu =\pi T$, $2\pi T$, and $4\pi T$ (solid, dotted, and dashed) using the NLO result [Eq. (25)]. The triplets of horizontal lines on the right side indicate the naive weak-coupling expectation for exchange of a bound state $2m_D/T$ of two scalar modes, each with the mass $m_D/T$. The scales are the same. Lastly, the horizontal solid bands indicate the estimate of the asymptotic screening mass obtained by shifting the screening mass for $r{T}_{\min }=0.5$ by a temperature-independent constant.