QCD Thermodynamics with Improved Actions

TL;DR

This paper investigates SU(3) gauge theory thermodynamics using improved lattice actions, demonstrating significant reduction in finite cut-off effects and presenting results for pressure, critical temperature, surface tension, and latent heat.

Contribution

It introduces the use of improved Symanzik actions to reduce lattice artifacts in QCD thermodynamics calculations, showing enhanced accuracy over standard Wilson actions.

Findings

Improved actions drastically reduce finite cut-off effects.

Results for pressure, T_c, surface tension, and latent heat are provided.

First thermodynamics results for four-flavour QCD with improved staggered action.

Abstract

The thermodynamics of the SU(3) gauge theory has been analyzed with tree level and tadpole improved Symanzik actions. A comparison with the continuum extrapolated results for the standard Wilson action shows that improved actions lead to a drastic reduction of finite cut-off effects already on lattices with temporal extent $N_\tau=4$. Results for the pressure, the critical temperature, surface tension and latent heat are presented. First results for the thermodynamics of four-flavour QCD with an improved staggered action are also presented. They indicate similarly large improvement factors for bulk thermodynamics.

Tables

Table 1. Table 1: Critical temperature in units of σ 𝜎 \sqrt{\sigma} on lattices with temporal extent N τ = 4 subscript 𝑁 𝜏 4 N_{\tau}=4 . Infinite volume extrapolations of β c subscript 𝛽 𝑐 \beta_{c} for the Symanzik actions are based on lattices with N σ = 16 subscript 𝑁 𝜎 16 N_{\sigma}=16 , 24 and 32.

action	$N_{\sigma }$	${\beta }_c$	$T_c/\sqrt{\sigma }$
standard Wilson	$\infty$	5.69254 (24)	$0.5983\left(30\right)$
(2,2) Symanzik (tree level)	24	4.3995 (2)	$0.624\left(4\right)$
(1,2) Symanzik (tree level)	$\infty$	4.07297 (28)	$0.631\left(3\right)$
(1,2) Symanzik (tadpole)	$\infty$	4.35228 (39)	$0.635\left(3\right)$

Table 2. Table 2: Surface tension and latent heat for the three improved actions and the Wilson action. Results for the Wilson action are based on data from [9] using the non-perturbative β 𝛽 \beta -function calculated in [5].

action	$V_{\sigma }$	$N_{\tau }$	${\sigma }_I/T_c^3$	$\Delta ϵ/T_c^4$	${\left(\Delta ϵ/T_c^4\right)}_{\mathrm{pert}}$
standard Wilson	${24}^2\times 36$	4	0.0300 (16)	2.27 (5)	4.06 (8)
	${36}^2\times 48$	6	0.0164 (26)	1.53 (4)	2.39 (6)
(1,2) Symanzik (tree level)	${32}^3$	4	0.0116 (23)	1.57 (12)	2.28 (8)
(1,2) Symanzik (tadpole)	${32}^3$	4	0.0125 (17)	1.40 (9)	1.94 (8)

Equations

$$\frac{ϵ(N_{\tau })}{{ϵ}_{\mathrm{SB}}}$$

$$S^{(1,2)}$$

$$S^{(2,2)}$$

$$P(|L|)\sim\exp\bigl{(}-\bigl{[}f_{1}V_{1}+f_{2}V_{2}+2\sigma_{I}A\bigr{]}/T\bigr{)}$$

$$\frac{\Delta ϵ}{T_c^4}$$

$$M[U{]}_{ij}$$

$$\mathrm{with}$$

$$A[U{]}_{ij}$$

$$B[U{]}_{ij}$$

Full text

1 Why improved actions are expected to improve the QCD thermodynamics

Although the plasma phase of QCD does in many respects show strong non-perturbative properties (screening masses, poor convergence of perturbation theory,…), bulk thermodynamic observables like energy density and pressure do approach the non-interacting ideal gas limit at high temperatures ($T$) where they are expected to receive large contributions from high momentum modes, $\langle{\rm avg.momentum}\rangle\sim T$. In lattice calculations $T$ is determined through the temporal extent $N_{\tau}$ of the lattice and the cut-off $a^{-1}$, i.e. $T\equiv 1/N_{\tau}a$. In the ideal gas limit the relevant momenta are therefore of the order of the cut-off, where lattice and continuum dispersion relations differ strongly from each other. Indeed this is the origin of the well-known discrepancy between the energy density of an ideal gas calculated on a finite lattice ($\epsilon(N_{\tau})$) and in the continuum ($\epsilon_{\rm SB}$),

[TABLE]

The standard Wilson action leads to nearly 50% corrections on a lattice with temporal extent $N_{\tau}=4$ while the cut-off dependence is drastically reduced in the case of Symanzik improved actions which eliminate the leading ${\cal O}(N_{\tau}^{-2})\equiv{\cal O}((aT)^{2})$ corrections. In the case of the (1,2)-Symanzik action this is less than 2% for $N_{\tau}=4$, i.e. $c_{I}=0.044$. This clearly demonstrates the significance of improved actions in the ideal gas limit ($T\rightarrow\infty$).

In a first exploratory analysis we have shown that the improvement found analytically in the ideal gas limit also persists at finite temperature and does seem to be important even close to $T_{c}$ [1]. Here we will present further evidence that tree level and tadpole improved actions do lead to a significant reduction of finite cut-off effects even at $T_{c}$ [2, 3, 4].

2 SU(3) Thermodynamics

We have analyzed the thermodynamics of the SU(3) gauge theory using ${\cal O}(a^{2})$ improved tree level and tadpole improved actions, which have been constructed by adding to the Wilson plaquette action, $S^{(1,1)}$, an appropriately weighted contribution from planar (1,2) or (2,2) loops,

[TABLE]

where the factor $u_{0}$ appearing in the definition of the (1,2)-Symanzik action denotes the tadpole improvement factor defined in terms of the self-consistently determined plaquette expectation value [7].

2.1 Bulk thermodynamics

Bulk thermodynamic quantities like the energy density ($\epsilon$) or pressure ($p$) calculated with the Wilson action on lattices of size $N_{\sigma}^{3}\times N_{\tau}$ with $N_{\tau}=4$, 6 and 8 [5] show a strong cut-off dependence in the plasma phase. A first analysis with the (2,2)-Symanzik action [1] has shown that this cut-off dependence gets drastically reduced already on a $N_{\tau}=4$ lattice. The work presented in Ref. [1] has been extended in several ways. On lattices with temporal extent $N_{\tau}=4$ we have analyzed now also the tree-level improved (1,2)-Symanzik action, which shows even smaller cut-off distortions in the high temperature limit than the (2,2) action. In addition we have studied the influence of tadpole improvement in the case of the (1,2)-Symanzik action. In all cases we have determined the temperature scale non-perturbatively through the calculation of the string tension on symmetric ($N_{\sigma}^{4}$) lattices, $T/T_{c}\equiv\sqrt{\sigma}a(\beta_{c})/\sqrt{\sigma}a(\beta)$. From this calculation we also obtain $T_{c}$ in units of $\sqrt{\sigma}$ for the different actions. These are given in Table 1 together with results for the Wilson action. In that case $T_{c}/\sqrt{\sigma}$ has also been calculated on lattices with $N_{\tau}$ varying between 4 and 12 and extrapolated to the continuum limit. This gave $T_{c}/\sqrt{\sigma}=0.629\pm 0.003$ [5]. On $N_{\tau}=4$ lattices the results obtained with improved actions are thus much closer to this extrapolated value than those obtained with the Wilson action.

The pressure has been calculated by integrating differences of the action densities obtained on $16^{4}$ and $16^{3}\times 4$ lattices (in the case of the (2,2) action we used $24^{4}$ and $24^{3}\times 4$ lattices) following the standard procedures [1, 5]. Results are shown in Figure 1.

2.2 Surface tension and latent heat

The success of improved actions for the calculation of bulk thermodynamics even at temperatures close to $T_{c}$ naturally leads to the question whether these actions also do lead to an improvement at $T_{c}$. This is, of course, a highly non-perturbative regime. However, observables like the latent heat ($\Delta\epsilon$) and the surface tension ($\sigma_{I}$), which characterize the discontinuities at the first order deconfinement phase transition in a $SU(3)$ gauge theory, do depend on properties of the low as well as the high temperature phase. As the latter is largely controlled by high momentum modes it may be expected that some improvement does result even from tree level improved actions.

At the SU(3) deconfinement transition $\Delta\epsilon$ and $\sigma_{I}$ have been studied on lattices up to temporal extent $N_{\tau}=6$ [8, 9]. A strong cut-off dependence has been found when comparing calculations for $N_{\tau}=4$ and 6. For this reason an extrapolation to the continuum limit has so far not been possible for these observables.

We have extracted $\sigma_{I}$ [3] from the probability distribution of the absolute value of the Polyakov loop following the analysis presented in Ref. [9]. The probability distribution at the minimum is proportional to

[TABLE]

where $f_{i}$ denotes the free energy in the phase $i$, and $V_{i}$ is the volume occupied by that phase and $A$ denotes the interface area of the finite system. From the depth of the minimum one thus can extract the surface tension. The distribution functions for the tadpole improved actions for three different lattice sizes are shown in Figure 2.

In Table 2 we give results for $\sigma_{I}$ on the largest lattices considered. Clearly the surface tension extracted from simulations with improved actions on lattices with temporal extent $N_{\tau}=4$ are substantially smaller than corresponding results for the Wilson action. In fact, they are compatible with the $N_{\tau}=6$ results for the Wilson action.

The latent heat is calculated from the discontinuity in ($\epsilon-3P)$. This in turn is obtained from the discontinuity in the various Wilson loops entering the definition of the improved actions,

[TABLE]

with $\tilde{S}\equiv S-{\rm d}S/{\rm d}\beta$.

The difference of action expectation values at $\beta_{c}$ is obtained by calculating these in the two coexisting phases at $\beta_{c}$. In the time histories of the Polyakov loop values we have cuted out the transition regions in order to classify configurations belonging to either of the two phases [8].

In order to extract the latent heat one does still need the $\beta$-function entering the definition of $\Delta\epsilon/T_{c}^{4}$ in Eq. 4. The necessary relation $a(\beta)$ has been obtained from a calculation of $\sqrt{\sigma}a$ (improved actions) or a determination of $T_{c}a$ (Wilson action). Results for $\Delta\epsilon/T_{c}^{4}$ are summarized in Table 2. Here we also give the result obtained with the 1-loop $\beta$-function.

3 Four-flavour QCD with an improved staggered action

In the fermionic sector of QCD the influence of a finite cut-off on bulk thermodynamic observables is known to be even larger than in the pure gauge sector. For instance, in the staggered formulation the energy density of an ideal fermi gas differs by more than 70% from the continuum value on a lattice with temporal extent $N_{\tau}=4$ and approaches the continuum value only very slowly with increasing $N_{\tau}$. This cut-off dependence can drastically be reduced with an $O(a^{2})$ improved staggered action. We use an improved action, $S^{I}[U]=S^{(1,2)}+\bar{\psi}M\psi$, where a higher order difference scheme (one-link and three-link terms), is used to improve the fermionic part [10]. The improved fermion matrix reads

[TABLE]

With this action the overall cut-off distortion of the ideal gas limit on a $16^{3}\times 4$ lattices reduces to about 20%. We have performed simulations for two quark masses, $ma=0.05$ and 0.1 [4]. Like in the pure gauge sector the improvement is visible already close to $T_{c}$. For instance, the energy density stays close to the ideal gas limit. Although we observe an overshooting of the ideal gas limit close to $T_{c}$ for the non-zero quark masses considered by us this feature does seem to disappear in the ”chiral limit” [4].

4 Conclusions

Thermodynamic observables of SU(3) gauge theory and QCD studied with improved gauge and fermion actions show a drastic reduction of the cut-off dependence in the high temperature limit as well as at $T_{c}$. The major improvement effect is already obtained with tree level improved actions. Tadpole improvement plays a minor role even close to $T_{c}$.