Finite Temperature Spacelike Gluon Propagators in Lattice Momentum Space

TL;DR

This paper investigates the behavior of gluon propagators in lattice momentum space for SU(2) gauge theory at finite temperature, revealing a specific magnetic mass value and comparing results in three dimensions.

Contribution

It provides a detailed analysis of gluon propagators at finite temperature and introduces the measurement of the magnetic mass in this context.

Findings

Magnetic mass is 0.26 g^2(T) T

Results are consistent with three-dimensional calculations

Enhances understanding of gluon behavior at finite temperature

Abstract

We study the behaviour of lattice momentum-space gluon propagators for a pure $SU(2)$ gauge theory at finite temperature. We find out that the magnetic mass is $0.26 g^2(T) T$; we have repeated the same calculations in three dimensions.

Equations

$$m_e=\sqrt{\frac{2N_c+N_f}{6}}g(T)T,$$

$$P_{\mu \nu }^T={\delta }_{\mu i}({\delta }_{ij}-\frac{k_i{k}_j}{{\mathbf{k}}^2}){\delta }_{j\nu }$$

$$P_{\mu \nu }^L=({\delta }_{\mu 4}-\frac{k_4{k}_{\mu }}{k^2})\frac{k^2}{{\mathbf{k}}^2}({\delta }_{\nu 4}-\frac{k_4{k}_{\nu }}{k^2})$$

$$P_{\mu \nu }^G=\frac{k_{\mu }{k}_{\nu }}{k^2}$$

$$P_{\mu \nu }^S=\frac{1}{\sqrt{2{\mathbf{k}}^2}}(k_{\mu }({\delta }_{\nu 4}-\frac{k_4{k}_{\nu }}{k^2})+(k_{\nu }({\delta }_{\mu 4}-\frac{k_4{k}_{\mu }}{k^2}))$$

$${\Pi }_{\mu \nu }^{ab}(k_4,\mathbf{k})={\mathbf{\Pi }}_L^{ab}{P}_{\mu \nu }^L+{\mathbf{\Pi }}_T^{ab}{P}_{\mu \nu }^T+{\mathbf{\Pi }}_S^{ab}{P}_{\mu \nu }^S+{\mathbf{\Pi }}_G^{ab}{P}_{\mu \nu }^G.$$

$$\begin{array}[]{lll}G_{\mu\nu}^{ab}({\bf k},k_{4})&\equiv<A_{\mu}^{a}A_{\nu}^{b}>\\ &=\delta^{ab}({{1}\over{k_{4}^{2}+{\bf k}^{2}+\Pi_{L}({\bf k},k_{4})}}P^{L}_{\mu\nu}+{{1}\over{k_{4}^{2}+{\bf k}^{2}+\Pi_{T}({\bf k},k_{4})}}P^{T}_{\mu\nu})\end{array}$$

$$C_j(\mathbf{k})\equiv \frac{1}{2}\sum _{a,b=1}^2{\delta }_{ab}{G}_{jj}^{ab}(\mathbf{k},k_4=0)\equiv G(\mathbf{k},k_4=0)P_{jj}^T(\mathbf{k},k_4=0),$$

$$G(\mathbf{k},k_4=0)=\frac{1}{{\mathbf{k}}^2+{\Pi }_T(\mathbf{k},k_4=0)}.$$

$$\Pi ({\mathbf{k}}^2)=\frac{m^2(g,\mu ,T)}{{\mathbf{k}}^2}+\frac{b^4(g,\mu ,T)}{({\mathbf{k}}^2{)}^2}+\dots$$

$$G(k)=\frac{{\mathbf{k}}^2}{({\mathbf{k}}^2{)}^2+b^4},$$

$$A_{\mu }(x)\equiv \frac{U_{\mu }(n)-U_{\mu }^{†}(n)}{2i},\mu =1,2,3,4.$$

$$\Sigma [\Lambda ]\equiv \sum _{n,\mu }Tr[\Lambda (n)U_{\mu }(n){\Lambda }^{†}(n+\mu )]$$

$$\begin{array}[]{ll}c(N_{t}=2)=0.20(3),&\lambda(N_{t}=2)=0.55(15)\\ c(N_{t}=3)=0.26(2),&\lambda(N_{t}=3)=0.26(8).\end{array}$$

$$m(N_t=2)=1.9T,m(N_t=3)=1.6T$$

$$m=0.49(3)g_3^2.$$

Full text

Finite Temperature Spacelike Gluon Propagators

in Lattice Momentum Space

G. Koutsoumbas

Physics Department

National Technical University

157 80 Zografou, Athens, Greece

We study the behaviour of lattice momentum-space gluon propagators for a pure $SU(2)$ gauge theory at finite temperature. We find out that the magnetic mass is $0.26g^{2}(T)T$; we have repeated the same calculations in three dimensions.

NTUA 49/96January 1996

1 Introduction

The subject of the finite temperature behaviour of the gluon propagators has attracted attention due to its relevance in understanding processes in the early universe. Very early approaches [1, 2] have identified the two different kinds of masses arising when considering longitudinal or transverse degrees of freedom, called electric and magnetic masses respectively. It has been argued [3] that they are gauge invariant, thus measurable quantities.

The pole for the electric gluon propagator may be obtained in perturbation theory, with the well-known one-loop result:

[TABLE]

for $N_{f}$ massless quark flavours and the $SU(N_{c})$ gauge group. This has been measured numerically using the heavy quark potential and the correlations of Polyakov lines [4, 5].

The magnetic gluon propagator has no (non-zero) pole at the one-loop level, however at two loops there is a contribution to the magnetic mass proportional to $g^{2}(T)T$. Higher loops contribute multiples of the same quantity, leaving no possibility for perturbative calculations. A non-zero value for this non-perturbative quantity may soften the severe infrared problems of finite temperature perturbative calculations, since it will act as an infrared cut-off. The subject of the magnetic mass has been attacked since the eighties, using various methods. On the lattice, it has been measured using the effects of twisted boundary conditions on bulk quantities [6, 7] and yielded the value $0.24g^{2}(T)T$. The next approach [8] has been more straightforward, in the sense that it studied the correlators of (gauge-variant) gluon operators. It found effective masses increasing with distance, being thus in conflict with the Källen-Lehmann representation. The interpretation was that this odd behaviour can be acceptable for a confined state (such as the gluon), although it should be rejected for a physical particle.

Restricting ourselves to pure gauge theories, we have three scales that enter the game: the temperature T, the scale of the electric mass, $g(T)T$, and the scale of the magnetic mass, $g^{2}(T)T$. At sufficiently high temperatures one expects that $g(T)$ will be very small, therefore the electric mass will be much smaller than the temperature scale and the magnetic mass even smaller. This is not the case however in realistic lattice simulations, since at high temperatures the finite-size-effects become big, thus restricting the investigation to relatively low temperatures.

There have been two recent determinations of the electric and magnetic masses; the first one [9] measured correlators of gauge invariant objects in QCD, yielding $m_{mag}\approx(2.9\pm 0.2)~{}T,~{}~{}m_{el}\approx(1.4\pm 0.2)~{}T$. The second one [10] is based on the gluon propagators themselves in $SU(2)$ gauge theory and yields $m_{mag}=0.466(15)~{}g^{2}(T)~{}T,~{}~{}m_{el}=2.484(52)~{}T$.

On the other hand, there has been a different interpretation [11] of the results of [9], claiming that the gluon propagators do not really behave as decreasing exponentials, but have instead a power law behaviour at small distances, which would mean that the results of [9] are compatible even with a vanishing value for the magnetic mass.

An approach that could shed some light on the space-time dependence of the gluon propagator is the study of momentum space propagators ([12, 13]). This has the advantage that one studies the whole momentum space propagator, rather than the few numbers which survive the sum over the hyperplanes. Moreover it opens the possibility to study the behaviour in each momentum region separately and relate the momentum under study to the scales that enter the problem, which is not possible in configuration space studies. Finally, as shown in [12], there is the very important technical advantage of a much better behaved covariance matrix. In particular, the covariance matrix for configuration space propagators is singular for the lattice sizes presently used; this is not the case in momentum space, offering the possibility to perform fully correlated chi-squared fits.

2 The finite temperature gluon propagator

At finite temperature any second rank tensor may be expanded in the basis of the following four tensors:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In the Landau gauge the free gluon propagator is of the form: $D_{\mu\nu}(k)={{-1}\over{k_{4}^{2}+{\bf k}^{2}}}(P_{\mu\nu}^{L}+P_{\mu\nu}^{T}).$ If one expands the self energy tensor $\Pi_{\mu\nu}$ on the above basis:

[TABLE]

Taking into account that ${\bf\Pi}_{J}^{ab}({\bf k},k_{4})=\delta^{ab}\Pi_{J}({\bf k},k_{4})$, where the index $J$ may be $L$, $T$, $S$ or $G$, one may show that the full gluon propagator at finite temperature is given by the expression:

[TABLE]

Now we observe that $P_{4\mu}^{T}(k)=P_{\mu 4}^{T}(k)=0,~{}\mu=1,...4$ for any value of k, while $P^{L}_{ij}(k)=k_{4}^{2}{{k_{i}k_{j}}\over{k^{2}~{}{\bf k}^{2}}},~{}i,j=1,2,3$, which vanishes for all i, j when $k_{4}=0$. This fact, that the static spatial self-energy is always transverse, may be shown [1] using the Ward identities satisfied by $G_{\mu\nu}^{ab}$ making no appeal to a specific gauge. In the sequel we stick to the static case, $k_{4}=0$. The limits of $\Pi_{L}({\bf k},k_{4}=0)$ and $\Pi_{T}({\bf k},k_{4}=0)$ as ${\bf k}$ goes to zero are the electric and magnetic masses (squared) respectively. In this work we are going to measure the (lattice versions of the) quantities

[TABLE]

where j runs from 1 to 3 and no sum over j is implied. In fact, we do not fit $C_{j}({\bf k})$, but the quantities

[TABLE]

The infrared non-perturbative behaviour of the gluon propagator has been studied using a variety of methods, with conflicting conclusions. Other works [16] predict for the propagator a very singular behaviour at zero k, like $\frac{1}{({\bf k}^{2})^{2}}$ and a confining property is conjectured; on the other side, there are claims [17] that the tree-level pole at ${\bf k}^{2}=0$ is removed, due to non-perturbative effects and the Green’s function vanishes at zero momentum. It is a fact that in the infrared region there should appear a dynamically generated mass $M(g,\mu)$. However, the generation of this mass cannot be seen in perturbation theory, since it must behave like $M(g,\mu)\approx\mu e^{-\frac{constant}{g^{2}}}$ for very small $g$, that is it should exhibit an essential singularity at zero gauge coupling. It is expected that non-perturbative effects may yield negative powers of the momentum in the vacuum polarization function of the gluon, introducing various mass scales:

[TABLE]

g is the coupling constant and $m(g,\mu,T)$, $b(g,\mu,T)$ have dimensions of mass and depend non-analytically on g. We have only shown the first two terms of an expansion in $\frac{1}{{\bf k}^{2}}$, since they correspond to well-known suggestions about the gluon propagators. In particular, if $b(g,\mu,T)=0$, a non-zero $m(g,\mu,T)$ gives rise to a mass pole in the gluon propagator. On the other hand, if $m(g,\mu,T)=0$, a non-zero $b(g,\mu,T)$ will give rise to a propagator of the form

[TABLE]

which has been proposed by Gribov [15].

3 The method

We measured correlations of the gauge potential, defined through the relation:

[TABLE]

The various expectation values should be calculated in a lattice version of the Landau gauge, implemented by making the quantity

[TABLE]

a global maximum with respect to the gauge transformations $\Lambda(n)$. This guarantees not only the satisfaction of the lattice version of the condition $\partial_{\mu}A^{\mu}=0$, but also the additional constraint imposed by Gribov. Let us note that the lattice Faddeev-Popov operator is just the Hessian matrix of $\Sigma[\Lambda]$ with respect to $\Lambda[n]$. The maximization condition fixes the sign of the Hessian to be the correct one. Moreover, the maximization condition also enforces the smoothness of the continuum limit. Of course, one cannot say whether this proposal really gets rid of all of the Gribov copies; however, it is easily seen that the Gribov copies of the trivial configuration are eliminated, since the latter is the only constant configuration which maximizes $\Sigma[\Lambda]$. Thus, it may be expected that also the problem in its general form may be less severe.

In order to calculate the gauge dependent correlators, we updated with the usual Wilson action and transformed the resulting configurations to the Landau gauge before taking measurements. This can be proved [8] to incorporate the effects of the Faddeev-Popov determinant. For SU(2), which we are considering, we sweep through the lattice and at each lattice site we calculate analytically the gauge transformation that maximizes the sum of the links beginning or ending at this site. Of course, this gauge transformation will disturb the gauge condition on the neighboring sites, so we expect that the algorithm relaxes to the global maximum we are looking for after several sweeps. A good gauge fixing is very important for the reliability of the results. A straightforward check is to calculate the $<A_{3}(x)A_{3}(0)>$ correlator in configuration space, summed over the directions 1, 2 and 4: this quantity should be constant, as a consequence of the gauge fixing. We have considered the gauge fixing as good enough if the variation of this correlator was not larger than 0.1 percent. This in turn dictated that the quantity $\frac{1}{N^{3}N_{t}}\sum_{k,n,\mu}(\partial_{\mu}A^{\mu}_{k}(n))^{2}$ (which must vanish in the Landau gauge) should be less than about $10^{-5}$. This last requirement has been practically used as the criterion to stop the gauge fixing iterations. Let us note that the number of measurements done varied between 3000 and 5000 for each point; we also mention that a number of overrelaxation sweeps ( 5) was performed between successive Monte Carlo steps. In figure 1 we show the configuration space correlators $<A_{1}(x)A_{1}(0)>$ and $<A_{3}(x)A_{3}(0)>$ versus $x$. The first correlator decays as usual, while the second one is constant, because of the gauge fixing, as just explained.

4 Results and conclusions

Let us first explain the way of analyzing our results. We have measured the correlators of link variables $A_{j}~{}~{}(j=1,2,3)$ on the lattice and then performed the Fourier transform, considering only the case $k_{4}=0$. The lattice versions of the propagators have been considered in the fit function $G({\bf k}^{2})=\frac{C}{\hat{\bf k}^{2}+\mu^{2}}$ ($\mu\equiv m\alpha$), which means that we used everywhere the dimensionless lattice version $\hat{\bf k}$ of the three-momentum rather than the usual one ${\bf k}$. Let us recall the relevant definitions: $\hat{p}_{i}a\equiv\hat{k}_{i}\equiv 2sin\frac{k_{i}a}{2}$ where $k_{i}a=\frac{2\pi}{N}n_{i},n_{i}=-\frac{N}{2}+1,\dots,\frac{N}{2}$ for the even N we have been using. We have considered lattices with two different temporal extents ($N_{t}=2$ and $N_{t}=3$) to get some feeling about the finite size effects due to the smallest of the lattice dimensions. Of course the corresponding momentum $k_{4}$ has been set to zero, as an external momentum, however it also appears as an internal momentum in the self-energy graphs and may very well influence the results. In most simulations the spatial dimension $N$ of the lattice has been $10$, however we have also used $N=12$ and $N=14$ in some cases.

We have chosen to measure the momentum $\hat{p}$ ($\hat{p}\equiv|\hat{\bf p}|$, $\hat{k}\equiv|\hat{\bf k}|$) in units of the temperature $T$ of each lattice, since this is one of the dominant scales of the problem (along with $g(T)T$ and $g^{2}(T)T$). We note that $\frac{\hat{p}_{max}}{T}=N_{t}\hat{k}_{max}$, so the range of the quantity $\frac{\hat{p}}{T}$ explored by each lattice is proportional to its “temporal” extent, provided $N$ is kept constant. Taking this fact into account, the (lattice) momentum range $[0,\hat{k}_{max}]$ has been divided into $N_{t}$ parts and the gluon propagators have been considered over the intervals: $\left[\hat{k}_{max}\frac{n-1}{N_{t}},\hat{k}_{max}\frac{n}{N_{t}}\right],n=1,\dots,N_{t}$, for the quantity $\hat{k}$, corresponding to the intervals $\left[(n-1)\hat{k}_{max},n\hat{k}_{max}\right],n=1,\dots,N_{t}$ for the quantity $\frac{\hat{p}}{T}$. We note that this division of the intervals is arbitrary; one might divide the intervals in a different number of subintervals (bigger or smaller than $N_{t}$). We observe that a specific value of n yields the same values of $\frac{\hat{p}}{T}$ no matter which lattice we are considering. In particular, the value n=1 corresponds to the results we are going to depict in the figures and has to do with the interval $0\leq\frac{\hat{p}}{T}\leq\hat{k}_{max}$. Note that for the lattices $10^{3}\times 2$, at most ten momenta may be considered in the fits, while for the lattices $10^{3}\times 3$ the corresponding number of momenta is four. In particular for the $N_{t}=2$ case we observed that the qualitative results we refer to in the following do not change if we consider shorter intervals within the ones corresponding to the various values of n; there are minor changes in the values of the mass and one gets smaller values for the chis squared. There is a serious exception, though, for $n=1$: the point $\hat{\bf k}={\bf 0}$ should be included in the fit in any case, since the second smallest momentum possible ($\frac{2\pi}{N\alpha}$) is already of the order of $T$.

We have measured the propagators for several values of $\frac{T}{T_{c}}$, ranging from $0.60$ to $6.0$ (the values greater than $4.0$ could only be reached on the lattices with $N_{t}=2$; for $N_{t}=3$ the required $\beta_{g}$ would be too large, given the spatial dimensions of the lattices we have been using).

In figures 2a, 2b, we show the results for the spacelike propagators (magnetic sector) as a function of the temperature for $N_{t}=2,~{}3$ respectively, at “low” momenta (n=1). We plot the output for $\frac{T}{m}=\frac{1}{N_{t}\mu}$ versus $\frac{T}{T_{c}}$, as well as curves fitting these data . We have also included in the figures the data for $\frac{T}{T_{c}}<1$. These have nothing to do with the magnetic mass, of course, however it is worth noting that no dramatic effects occur in the values of these screening masses at the phase transition point. One may observe that lattices with $N_{t}=2$ and $N_{t}=3$ give values quite close for equal $\frac{T}{T_{c}}$ (with some small discrepancies around $\frac{T}{T_{c}}=2$).

To construct the fitting functions we invoke the expected behaviour $m=cg^{2}(T)T$ of the magnetic mass along with the ansatz $g^{-2}(T)=\frac{11}{12\pi^{2}}log(\frac{T}{\lambda T_{c}})$, inspired by (the leading order in) the renormalization group equation for the gauge coupling. In practice we have fitted $\frac{T}{m}$ to the form $A~{}log(\frac{T}{T_{c}})+B$ and determined $c,\lambda$ through $c=\frac{11}{12\pi^{2}A},\lambda=e^{-\frac{B}{A}}$. The fit has taken into account only the data with $\frac{T}{T_{c}}>1.6$. It turns out that

[TABLE]

These values for $c$ are more or less in agreement to the early publications [6] on the subject, where $c$ has been found $0.24$.

Let us note that using the above expression for $g(T)$ and the just mentioned values of $\lambda$, we may find the ratio of the magnetic mass to the temperature:

[TABLE]

For even larger momenta ($n=2,~{}n=3$) the mass is zero in all cases. Thus, at high momenta the correlators tend to free field ones.

It is of interest to check what happens to the quantities measured above in the case we have a three-dimensional lattice, which would correspond to the “infinite temperature” limit. In figure 3 we show the results for the resulting masses for a three - dimensional ($12^{3}$) lattice. On dimensional grounds one expects that any mass should be proportional to $g_{3}^{2}$, the square of the three-dimensional gauge coupling. We find that

[TABLE]

If we make the (crude) identification of the (bare) $g_{3}^{2}$ to $g^{2}(T)T$, we see that the three-dimensional result would imply a much larger value for the quantity $c$ than the one found from the four-dimensional lattices; this result is, in fact, close to the result given in [10].

Let us note that, if we also treat $2\times 2$ plaquette correlators in the same way as the link correlators, we get $m_{pl}=2.3(5)g^{2}_{3}$ (in agreement with [14]), that is a much larger factor than the one extracted from the link variables.

Acknowledgements: The author wants to thank the DESY theory group for its warm hospitality during the early stages of this work. In particular, I thank I.Montvay for giving me his simulation code for $SU(2)$ gauge fields, R.Sommer for providing an efficient random generator and W.Buchmueller, Z.Fodor, M.Reuter and K.Farakos for useful discussions. A very enlightening discussion with F.Karsch is gratefully acknowledged.