Revised 10 September 2009
Spatial string tension of finite temperature QCD matter in gauge/gravity duality
J. Alanenfirstname.lastname@example.org, K. Kajantie†††email@example.com, V. Suur-Uski‡‡‡firstname.lastname@example.org
Department of Physics, P.O.Box 64, FI-00014 University of Helsinki,
Helsinki Institute of Physics, P.O.Box 64, FI-00014 University of Helsinki, Finland
The computation of the spatial string tension of finite temperature QCD is discussed in QCD and in a gravity+scalar model of gauge/gravity duality.
The spatial string tension in finite temperature QCD is determined with lattice Monte Carlo techniques [1, 2, 3, 4, 5] by studying in asymmetric lattices expectation values of large Wilson loops with both sides in spatial directions. It is a nonperturbative quantity and thus not analytically calculable. However, if it is determined numerically in 3 dimensional (3d) SU(3) gauge theory , one can use the well established equations of perturbative high temperature dimensional reduction to compute it in full 4d finite QCD and compare it with the lattice result [7, 8]. The agreement is surprisingly good.
Since is so well under control in QCD, it is a good laboratory for testing various models for AdS/QCD [9, 10, 11]. The heart of QCD computations is conformal invariance breaking and running of the coupling constant while the heart of well established AdS/CFT duality is conformal invariance of CFT. The essential part of AdS/QCD models is thus how conformal invariance breaking is modelled. We shall in this article apply the AdS/QCD model of Kiritsis and coworkers [12, 13, 14, 15], approximating it in a form which permits an analytic solution.
2 QCD discussion
A first-principle method for determining the spatial string tension in hot QCD matter is to measure rectangular Wilson loops of size in the plane. A potential is then defined by:
and then finding from the large- behavior
Measurements are done with full 4d QCD action on a finite lattice and they can be done for pure SU(3) Yang-Mills theory or for QCD with dynamical quarks.
There is also a natural way to interpret the result and to derive it by splitting the problem in an analytic and a numerical nonperturbative part. The spatial string tension lives in the 3d spatial space and one can also determine it for 3d SU(3) Yang-Mills theory. Since the only dimensionful parameter in this theory is its coupling constant of dimension mass, one knows that the string tension must be a number times . In fact [6, 24],
On a schematic level, one knows that , where the 4d coupling constant has a simple logarithmic expansion. Thus it is natural to choose to plot
To obtain a controlled theoretical result one at least has to compute the in (2.3) and not fit it to data. To do this one can start from the nonperturbative magnetic sector 3d number (2.2) and work out backwards, first to the electric sector of hot QCD by restoring the electric mass by the 2-loop relation 
and finally the hard scale by the 2-loop relations  ()
Here is the running coupling in which we, for concreteness, optimized the scale so that the term in (2.5) vanishes, i.e., . A thorough analysis of the optimisation scale dependence is given in . Noting that  the logarithmic factors in could be written as , i.e., in (2.3) has been evaluated to be .
The reliable QCD prediction then is obtained by inserting to (2.2) from (2.4) with and as given by (2.5) and (2.6)111Note that in a 3d gauge theory the couplings are scale independent while has both a linear 1-loop and a logarithmic 2-loop divergence. It so happens that the -coefficient of the logarithmic divergence cancels for  and above is scale independent to the order shown, .. The prediction is shown as the continuous curve plotted over the range in Figs. 4 and 5 below. The lattice data, shown in the same figures, extends over the range .
The QCD computation is an entirely controlled perturbative computation, the nonperturbative part is isolated in the number (2.2). It works surprisingly well, perhaps even too well. The computation is, namely, based on integrating out the large scales and and one does not expect it to work down to . For the pressure a similar computation shows clear deviations from the lattice results for below .
Clearly, to assess the reliability of the result it would be useful to further compute the 3-loop contribution – a formidable task.
3 AdS/QCD model
Consider then the computation of the spatial string tension in AdS/QCD models. In bottom-up models conformal invariance can be broken by putting by hand some structure in the extra dimension : a hard wall or some soft function of the type . We shall use a model  the idea of which is to add a scalar field and to create this scale dynamically via the equations of motion. Related work on vacuum potentials is in .
3.1 Defining the AdS/QCD model
One starts from the gravity + scalar action (in the Einstein frame and in standard notation)
or, in the string frame, writing
One now assumes a metric ansatz
The four functions in the metric, the scalar field and the potential are then determined as the solutions of the three field equations following from (3.1):
(, etc.) and from a fourth equation,
where is the beta function of the field theory one is seeking the gravity dual for. This is the crucial assumption of the model. The logic here is that the energy scale of the coupling is identified by , so that () corresponds to the UV (IR). Any similar monotonic function would do and actually in the definition (3.8) one uses . Eq.(3.8) in the context of a black hole horizon at some is discussed below (see Fig. 1).
For given one can from the fourth equation solve , then from the third equation and finally from the second equation so that also and are determined, in terms of a number of integration constants. The role of the first equation then simply is to fix the scalar potential. Defining
the answer simply is
The normalisation follows from the requirement that the boundary be asymptotically AdS, .
In spite of appearances, the potential in Eq.(3.10) is expressible in terms of only. A thorough discussion of the choice of the potential has been given in -. The outcome is that SU(3) thermodynamics is well described by the above model with the potential222The numerical computation includes in (3.12) also two more terms which enforce QCD asymptotic freedom in the UV. These terms are totally irrelevant for how the model in  describes thermodynamics of hot SU(3) matter. In fact, they dominate for which, using (3.18), corresponds to
To supplement numerical computations it is very useful to have an analytic approximation. Clearly in the UV is small, but the outcome of the numerical computation is that even at is small, in fact . Assuming is small, , one has
One observes from (3.10) that a function
Thus the approximate form (3.15) with
gives the approximate potential in (3.14). We shall base our analysis on this approximate beta function.
Analytical solutions for the metric and the scalar for the model beta function (3.15) are given in the Appendix. Both small or large expansions and bulk thermodynamics can be derived from them. Physically, they are accurate enough for , but clearly the full potential (3.12) is required to describe the phase transition region.
A numerical solution [15, 21] of Eqs.(3.5) - (3.8) gives for the functions and the value of for each value of at the horizon: . Some properties of the functions computed are as follows (”numerics” refers to ):
when and, related to this, . In numerics .
There is a horizon at , . In numerics there is a different value of for each value of .
increases monotonically with from so that the leading term at small is
where (=1/174 in numerics) is a constant fixed by the normalisation (3.13).
The numerically computed solutions reproduce the vacuum beta function via the relation only when . In fact, if there is a horizon at some , the functions terminate at and thus also . The outcome of a computation of a ”thermal beta function” defined by the relation is shown in Figs. 1 and 2. One should emphasize that these are computed using the full potential (3.12). One also sees how well the approximate beta function (3.15) reproduces the beta function corresponding to the potential (3.12).
3.2 Deriving the spatial string tension
The equations giving the extremal string configurations and the corresponding potential are well known. There are two cases: the spatial string tension at finite is determined from a space-space Wilson loop, the usual string tension from a time-space Wilson loop at . We shall summarize the equations separately for these two cases. The end points of the string are always at and the string hangs in the fifth dimension so that the maximum depth it reaches at is . We abbreviate
1. Space-space loop:
The relation between and is given by
and the potential is
In (3.23) the first term separates a term proportional to , the second term is finite when and (one can put in it) and the third term contains the singularity when ; its divergence is cancelled when the energy of two independent (anti)quarks is included . Numerical examples are given in Fig. 3 for .
2. Time-space loop.
In this case the relations are ()
where the divergence in can be regulated as for an loop. If does not diverge, the equilibrium state is that of two separate quarks [23, 19]. However, taking and expanding near one finds that so that will diverge for that value of satisfying
From (3.28) the string tension then is given by
3.3 Evaluating the result for spatial string tension
The lattice results for in finite SU(3) Yang-Mills we want to compare with  are measured for . The QCD computation in Section 2 should be a good estimate of the data even at higher temperatures, the better the larger is, see Fig. 5.
It is rather automatic to evaluate the expression (3.25) numerically , but to control various effects contributing to it we want to see how far we can get with the approximate potential (3.14) and the approximate beta function in (3.15). The overall normalisation will be fixed from the string tension , measured as , using (3.30). This has the important property that the arbitrary normalisation of cancels.
The result (3.25) can be analytically approximated as follows:
where we first used the small- expansions in the Appendix on the leading log level, writing in the arguments of the logarithms , then introduced the explicit values used in the numerics  and finally and most subtly, used from the numerics.
To get the normalisation we shall use the lattice data for  and its prediction (3.30) in this model. To have an analytic approximation for we again use the beta function , although the accuracy deteriorates with increasing . Since implies one immediately obtains and then from Eq.(A.1) . Thus, with ,
With the lattice result , the numerical result , the value of and the correct numerical value of (reduction of the RHS of (3.36) by 0.00113/0.00146) this can be converted to
This result depends on the arbitrary normalisation of .
The QCD result (2.3) was given as a numerically computed -independent 3d quantity times a perturbatively computed dependence. In analogy, here the result is given as the numerically computed 4d quantity times a dependence computed from a gauge-gravity duality model.
Within the range where lattice data exists, it is fitted well by both by the QCD result and the AdS/QCD model discussed here. For larger the QCD prediction varies more rapidly with , , than the AdS/QCD model, . The exponent here is with , as followed from the small- expansion of the potential (3.12). In the extreme UV one meets with the QCD beta function with , corresponding to the result behaving . If the QCD-like behavior persists, the present AdS/QCD model will require some modification for .
We have in this article considered a particularly simple quantity in finite quarkless QCD, the spatial string tension, both in QCD and in a model for gauge/gravity duality, AdS/QCD. The QCD result is perfectly under control: after one nonperturbative number is determined numerically, the rest follows analytically via symbolic perturbative computations, which may be technically very demanding. For the AdS/QCD model the result needs solutions of Einstein’s gravity equations and a particular contribution of this article has been developing analytically tractable approximations to the model in .
There are some interesting parallels and differences in the QCD and AdS/QCD results. Both start from a independent nonperturbative quantity, for QCD the 3 dimensional Yang-Mills string tension, for AdS/QCD from a 4 dimensional computation of the string tension, expressed as . The dependence in QCD comes from perturbatively reintroducing the scales and , in AdS/QCD from the extra dimensional coordinate dependence of the metric and dilaton in the model used to break conformal invariance.
In the range where lattice data exists, Fig. 4, both calculations agree with the data within reasonable estimates of errors. At higher , Fig. 5, the AdS/QCD result, with used here, increases more slowly than the QCD result and it should be possible to distinguish between the two.
There are puzzling features in both approaches. The QCD approach adds perturbative corrections to a nonperturbative number and it is unexpected that it works so well down to . The AdS/QCD model fits very well and elegantly bulk thermodynamics  using the potential (3.12), but the power of is quite different from that in the QCD running coupling (2.7). This is not surprising near but one would expect that at least at some high the matter would probe distances small enough to see QCD asymptotic freedom.
The spatial string tension has also been measured for QCD . An interesting further project would be to study this in AdS/QCD by developing the AdS/QCD model used here to also include fundamental quarks.
Acknowledgements. We thank E. Kiritsis, Mikko Laine, York Schröder and Patta Yogendran for discussions and correspondence. We are particularly indebted to Francesco Nitti for the Mathematica code used to solve the Einstein equations of this model and for advice concerning its use.
Appendix A Appendix
where is a constant, the analogue of , the scale at which diverges. From (3.11)
The second equation can then be written in the form
from which by integration
Correct dimensions are here given by
Using (A.3) one can replace as a fifth coordinate by , or . For example,
If the potential satisfies the constraint (3.10), there is a horizon with the temperature
Using (A.5) one can derive various small- (, small , large ) approximations. One has, for example,
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