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The string tension of spacelike Wilson loops is computed in SU(3) lattice gauge theory in the high-temperature deconfined phase. Its physical value is extracted following the continuum limit. An upper bound for the length scale where nonperturbative fluctuations become dominant is computed. The value of the string tension and the nonperturbative length scale are very close to their zero-temperature counterparts.
©1987 The American Physical Society.
PACS: 12.38.Gc 11.15.Ha
The investigations of strong interactions at high temperature were initiated by the observation that the effective coupling constant g_{roman}eff^{2}(T)_ becomes small at high temperature. In fact, the temperature T plays a similar role to the momentum scale in the standard manipulations with the renormalization group equations. It was hoped that QCD becomes a weak coupling theory at high enough temperature and we shall have better chances to identify the elementary constituents of the strong interactions. It was encouraging too when analytical^{1}and numerical^{2}evidence was found for the existence of the deconfined phase at high temperature. It was soon realized that the infrared modes ( p vec -> 0) spoil the weak coupling expansion even at arbitrarily high temperature.^{3}The real expansion parameter in summing up finite temperature Feynman graphs is g^{2}T / m_{m_,}where m_{m_}is the mass gap in the chromomagnetic propagator, which is at most m_{m}= c^{-1}g^{2}T_. Thus one finds an expansion parameter c whose actual value is inaccessible in the weak coupling expansion.
There were attempts to estimate a mass scale related to the magnetic field by means of numerical simulations.^{4}But in order to make statements about the high temperature weak coupling expansion we need control over finite size effects and the verification of the proper continuum behavior on the lattice. In addition, one has to be careful in referring to (lattice) weak coupling graphs when g_{roman}eff^{2}(T) > g_{roman}eff^{2}(T/N_{0}) = g_{roman}lattice^{2}app 1_ (N_{0_}is the lattice size in the timelike direction). Furthermore, it is not obvious how the mass scale associated with the Z_{n_}magnetic fluxes^{4}is related to m_{m_.}The latter controls the infrared behavior of small fluctuations of the magnetic field.
Special care is needed when interpreting the numerical results for the internal energy density at high temperature.^{5}The not uncommon view that we have evidence for a gas of weakly interacting gluons at high temperature is not necessarily correct because of finite size effects occurring in the calculations done so far. All that one can say is that the results are compatible with having eight light, weakly interacting quasiparticles in quarkless QCD. We believe that this conclusion survives the thermodynamic limit and that the quasiparticles are plane waves of gluons locally (asymptotic freedom) and become distorted beyond a certain length scale.^{6}The aim of this Letter is to estimate this length scale which signals the onset of nonperturbative effects. Such a calculation can be done numerically without our making assumptions about the details of the dynamics of the quasiparticles. The discussion of this dynamics and how these distorted gluon waves form a weakly interacting gas requires a nonperturbative framework^{7,8}and is beyond the scope of the present work.
First it is worthwhile to refer to an exact result of high temperature lattice gauge theory. For a finite value of the lattice spacing and sufficiently high temperature the spacelike Wilson loops follow area behavior.^{9}The proof is a formal elaboration on the point that high temperature corresponds to strong coupling in spacelike directions. In fact, when we control the temperature by introducing an anisotropic lattice, xi = a_{t}/ a_{s}!= 1_, in order to separate dependences on time and spacelike extents of the system, the lattice action becomes
S = 6 over {g^{2}}[ xi ^ sum from \(sq ^ \(sq_{{}sp"-" sp} + xi^{-1}^ sum from \(sq ^ \(sq_{{}sp"-" t} right ] ,
where \(sq_{{}sp"-" sp}_ and \(sq_{{}sp"-" t}_ denote contributions to the action from spacelike and timelike plaquettes, respectively. At high temperature xi << 1 and the spacelike links behave as in a strong coupling model, i.e., the leading contribution to the spacelike Wilson loop is obtained by tiling up the area of the loop with spacelike plaquettes. Unfortunately such simple considerations can tell us nothing about the cutoff dependence of the spacelike string tension and thus about continuum QCD.
A more systematic way to describe nonperturbative features of thermal Green's functions of the continuum theory is to consider the effective theory of the gluon fields at a fixed time slice t = t_{0_,}
exp lcurl - S_{roman}eff [ A tilde_{mu}( x vec ) ] rcurl = int D [ A_{mu}( x vec , t ) ] exp lcurl - S [ A_{mu}( x vec , t ) ] rcurl ^ prod from x vec ^ delta (A_{mu}( x vec , t_{0}) - A tilde_{mu}( x vec ) ).
In fact, the spacelike Wilson loops are given in terms of A_{mu}( x vec , t_{0})_ only and are thus governed by S_{roman}eff [ A tilde_{mu}( x vec ) ]_. It is known that S_{roman}eff [ A tilde_{mu}( x vec ) ]_ describes a three dimensional Yang Mills Higgs system with A tilde_{0}( x vec )_ as the matter field in the adjoint representation. In addition at sufficiently high temperature, where the static modes dominate the path integral, S_{roman}eff_ is of the form
S_{roman}eff = - ( 1 / 4g^{2}T ) F tilde_{ij}F tilde_{ij}- D_{i}A tilde_{0^{dag}D}sub i A tilde_{0}- V ( A tilde_{0}) ,
i,j = 1,2,3 . R
F tilde_{ij_}is the antisymmetric tensor of the spatial field components. The coupling constants of the manifestly gauge invariant local potential V( A tilde_{0})_ are functions of g^{2}T. It is known that the three dimensional Yang Mills theory is confining.^{10}Moreover, all the nonperturbative dynamics of high temperature QCD come from this effective theory.
A Monte Carlo calculation was carried out to determine the actual value of the string tension sigma_{s_}of spatial Wilson loops and to study its continuum limit. We obtained an upper bound for the length scale where nonperturbative effects dominate thermal Green's functions as a by product of the calculation in the following way: Consider V(b,c) = - ln [ W (b,c) ], where W(b,c) is the expectation value of a spacelike Wilson loop with sizes b and c. The ln [ W(b,c)] selects the connected parts of the correlation function. It involves the contributions of modes with length scale xi < b or c. In order to select the contributions of modes with xi app b we form Z (b,c) = partial V (b,c) / partial b. Confinement of the three dimensional theory guarantees that the nonperturbative tail Z(b,c) = sigma_{s}c_, as b tends to infinity. We define the length scale xi_{0_,}that at which Z( xi_{0}, c) = 2 sigma_{s}c_. It is clear that for b > xi_{0_}the nonperturbative contributions dominate in Z. This discussion is similar to the one in the case of the force acting between static charges. In that case, one considers timelike Wilson loops and takes the derivative with respect to the spatial separation in order to obtain the force. It is obvious that the force is the relevant quantity to look into as opposed to the potential. This way of defining xi_{0_}is rather arbitrary, but the important point is to use the same definition when we compare zero and finite temperature.
In the numerical part, the spacelike Wilson loop W(b,c) was replaced by the correlation function of the Polyakov lines wrapping around the periodic lattice in spatial direction. We define
W_{{}mu j } (b,L) = langle trOMEGA_{mu}(0) trOMEGA_{mu^{dag}(}b e vec_{j}) rangle ,
OMEGA_{mu}( n_{{}j != mu } ) = prod from { n_{{}mu =1 } } to { N_{mu}} ^ U_{mu}( n_{{}j != mu } , n_{mu}) ,
L = a N_{mu}, mu = 0,1,2,3, j = 1,2,3, mu != j .
The difference between the Wilson loops and W_{{mu}j}_ is obvious when the z direction is interpreted as the Euclidean time. In that case W includes the contribution of the color singlet and octet states of the static quark antiquark system. The Wilson loop corresponds to the singlet case only. Consequently W_{ij_}and the Wilson loop agree when both follow area behavior and L > b.
We may define a static ``potential'' from the spacelike correlation function as V_{s^{{}(}N_{0}) } (b) = - [ ln W_{ij}(b,L) ] / L _ + const. The V_{s_}computed on a 10^{3}times 6 lattice at beta == 6/g^{2}= 6 is shown in Fig. 1 (squares). The physical static potential V_{t^{{}(}N_{0}) } (b) = - [ ln W_{0j}(b,c) ] / c + const_,measured (by timelike Wilson loops) on confined (zero temperature) lattice at the same value of beta, taken from D'Hoker,^{10}is plotted as well (dashed line). The constants in the previous expressions are chosen such that case 1 over 2 [ V_{s}(2a) + V_{s}(4a) ] = case 1 over 2 [ V_{t}(2a) + V_{t}(4a ) ] = 0_. The physical static potential at finite temperature can be obtained from the correlation function of the timelike Polyakov lines as measured on our 10^{3}times 6 lattice. This potential, V_{t^{{(N}sub}0 )} (b) = - [ ln W_{0j}(b, beta ) ] / beta + const_,is also plotted in Fig. 1 (dash dotted line). Since at T=0, W_{ij}= W_{0j_,}the obvious lesson of this calculation is that the correlation function of the spacelike components of the gauge field at the present T app 2T_{roman}dec_ is very close to its zero temperature confined counterpart. The act of deconfinement appearing in the behavior of the timelike component of the gauge field does not influence the spatial components in a substantial way.^{12}
The lattice spacing a was left unchanged in the previous discussion. In order to make contact with the continuum theory one has to follow the limit a -> 0. This program has been carried out quite exhaustively for the physical zero temperature static potential and the result is consistent with having finite string tension in the continuum limit.^{13}The results of Fig. 1 and the standard flux tube picture of the linearly rising part of the static potential suggest that the slope of the function V_{s}(b)_ is insensitive to the dynamics of the shorter timelike direction. Thus it seems likely that the slope of V_{s}(b)_ remains finite in the continuum limit.
To verify this conjecture, V_{s}(b)_ was calculated on two lattices with lattice spacings a and a^{sprime}with a^{sprime}= 2a, in the following way: The ratio of the lattice spacings a^{sprime}/ a = 2 is achieved on lattices N_{s^{3}times}N_{t_}and (2N_{s})^{3}times 2N_{t_}by the tuning of g^{2}( a^{sprime}) and g^{2}(a), so that the finite part of the correlation functions V_{t^{{(N}sub}t ) } (na^{sprime})_ and V_{t^{{(2N}sub}t ) } (2na)_ computed at the corresponding g^{2}agree. It is advantageous to use V_{t}(b)_ to find the values of the coupling constant g^{2}(a) and g^{2}( a^{sprime}) which correspond to the same temperature,^{14}because it has strong temperature dependence.
Because of limitation of computer power we had to choose lattices of the size 10^{3}times 6 and 6^{3}times 3 in the present calculation. The subtracted potential
V_{t^{{}(}N_{0}) } (b) = - [ ln W_{0j}(b, beta ) - 2 ^ ln langle trOMEGA_{0}(0) rangle ] / beta
is plotted in Fig. 2. The best matching between the two lattices 10^{3}times 6 and 6^{3}times 3 was found at 6/g^{2}equal to 6 and 5.5615, respectively. So DELTA beta which realizes the halving of the lattice spacing is 0 for beta = 6. DELTA beta has been calculated with the use of much larger lattices and was found around 0.3^^{14}or 0.35.^{11,15}This discrepancy is related to finite size effects. In fact, the spatial size of the two lattices was almost the same in our case. So our DELTA beta corresponds to the theory placed in a finite quantization box with size L app 1 fm which tends to increase DELTA beta. The spatial correlation function V_{s^{(3)}(b)_}is plotted in Fig. 1 (circles). The constant term was determined on the smaller lattice by requiring case 1 over 2 [V_{s^{(3)}(a)}+ V_{s^{(3)}(2a)}] = 0 _. The stability of the potential V_{s}(b)_, as far as this change a -> 2a is concerned, is supported by this result.
The static potential can be approximated by the function V(b) = - alpha / b + sigma b + c . In the case of the correlation function of spacelike Polyakov lines we find alpha = 0.184 +- 0.02, sqrt sigma = 0 +- 0.03. The corresponding zero temperature values are alpha = pi /12 and sqrt sigma = 0.22 +- 0.02.^{11}
The numerical results presented above support the picture that spacelike Wilson loops follow area behavior even in the high temperature deconfined phase. Moreover it was found that at the temperature considered, T app 2T_{roman}dec_, the value of the string tension of the spacelike Wilson loop is very close to those of the zero tem\%perature theory. Considering the path integral as the partition function of a four dimensional classical system, one can say that the compactification of one (time) direction does not influence the asymptotic behavior of the correlation function of the gauge field components of the other (spacelike) directions. We find this behavior surprising, since the system undergoes a phase transition at T = T_{roman}dec_ which changes the dynamics of the time component of the gauge field completely.
The physical string tension at finite temperature becomes zero at T = T_{roman}dec_.^{16}It remains to be seen how the spacelike string tension behaves in the vicinity of T_{roman}dec_. Spacelike Wilson loops are controlled by the three dimensional Yang Mills Higgs system of the static modes at sufficiently high temperature and by the complete four dimensional theory at low T. There is no obvious reason to believe that the spacelike string theory would be even approximately the same for T < T_{roman}dec_ and T > T_{roman}dec_. Thus the temperature dependence of the spacelike string tension is a proper test of our understanding of the confinement mechanism. As an example one may consider the suggestion that chromomagnetic monopoles are present in QCD.^{6}This scenario involves the condensate at T < T_{roman}dec_ and the gas of localized monopoles for T > T_{roman}dec_. Although no self consistent analytical approximation is known to sum up the contributions of such objects in the path integral, it seems plausible that the spacelike string tension originates ultimately from the condensate by dual Meissner effect^{17}at low temperature^{18}and from the fluctuations around localized monopoles^{19}at high temperature. The monopole density becomes zero at the transition temperature and increases as a singular function of T - T_{roman}dec_. If correct, this description should explain why the spacelike string tension depends weakly on the monopole density and happens to be similar in both cases.
The actual form of V_{s}(b)_ allows us to give an upper bound for the length scale xi_{0_}characterizing the appearance of nonperturbative contributions in thermostatic correlation functions. We find xi_{0}app 2 app 0.20_ fm at T app 2T_{roman}dec_ as opposed to xi_{0}app 2 app 0.23_ fm at T =0. Although the string tension appears to be almost the same, the short distance behavior differs at the two values of the temperature studied. The coefficient of the 1/b term in V_{s}(b)_ is consistent with the value obtained from the two transverse fluctuations of the flux tube at T=0, alpha = pi / 12.^{20}At T app 2T_{roman}dec_ this coefficient becomes smaller and lies between the values corresponding to two or one transverse mode, pi / 24 < alpha < pi / 12. In general, alpha should approach pi / 24 as the temperature increases. Since the correlation function V_{s_}is less steep at short distances for high temperature, xi_{0}( 2T_{roman}dec ) < xi_{0}(0)_. But xi_{0}(T)_ is an upper estimate for the nonperturbative length scale only. The applicability of the perturbation expansion must be even more restricted in the vicinity of the critical temperature. xi_{0_}defined by V_{t}(b)_ diverges in this region since the physical string tension vanishes at T = T_{roman}dec_. The upper bound xi_{0_,}based on the generation of a linear term in static potentials, is clearly inadequate in this case. One needs an insight into the dynamics at T app T_{roman}dec_ to select another nonperturbatively generated term for our purpose. What one can say at the present stage is that it does not help to increase the temperature if we intend to eliminate the nonperturbative aspects of the strong interactions. Any description of the dynamics of high temperature QCD, which involves length scales larger than 0.2 fm, must rely substantially on nonperturbative effects.
We thank J. Smit for a useful discussion. The numerical calculations were done at the VAX system at the Laboratory for Nuclear Science of Massachusetts Institute of Technology and the Cray XMP machine of the Lawrence Livermore National Laboratory. This work was supported in part by funds provided by the U.S. Department of Energy under Contract No. DE AC02 76ER03069.
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