Holographic models and the QCD trace anomaly
Abstract
Five dimensional dilaton models are considered as possible holographic duals of the pure gauge QCD vacuum. In the framework of these models, the QCD trace anomaly equation is considered. Each quantity appearing in that equation is computed by holographic means. Two exact solutions for different dilaton potentials corresponding to perturbative and nonperturbative functions are studied. It is shown that in the perturbative case, where the function is the QCD one at leading order, the resulting space is not asymptotically AdS. In the nonperturbative case, the model considered presents confinement of static quarks and leads to a nonvanishing gluon condensate, although it does not correspond to an asymptotically free theory. Calculating the NambuGoto action corresponding to a small circular Wilson loop, leads to an expression for the gluon condensate. The validity of the trace anomaly equation is considered for both models. It holds for the perturbative model and it does not hold for the nonperturbative one.
pacs:
11.15q, 11.15Tk, 11.25Tq, 12.38.Aw, 12.38.LgI Introduction
The relation between large N gauge theories and string theory tHooft () together with the antide Sitter/conformal field theory (AdS/CFT) correspondence Malda (); Gubser (); Witten (); MaldaReview () have opened new insights into strongly interacting gauge theories. The application of these ideas to QCD has received significant attention since those breakthroughs. From the phenomenological point of view, the so called AdS/QCD approach has produced very interesting results in spite of the strong assumptions involved in its formulation AdS/QCD (). It seems important to further proceed investigating these ideas and refining the current understanding of a possible QCD gravity dual. This endeavor has been followed in references GursoyKiritsis (). The aim of the present paper is to explore the simplest nonperturbative features of QCD. This is done in the framework of a holographic description of the pure YangMills (YM) QCD vacuum by means of 5dimensional dilaton gravity models.
At the basis of the AdS/CFT correspondence is the connection between scale transformations in the boundary field theory and isometries of the bulk gravitational theory. However, QCD is not a conformal field theory, as the scale symmetry is broken by the trace anomaly TraceAnomaly (). The trace anomaly equation describes the behavior of QCD under scale transformations. The question to be explored is how a holographic model can incorporate this behavior.
The trace anomaly equation TraceAnomaly () states that,
(1) 
where denotes the trace of the QCD energy momentum tensor (latin indices for spacetime), is the QCD function, is the t’Hooft coupling, is the QCD field strength tensor and the trace is taken in the fundamental representation of the SU(N) gauge group. In this respect it is important to note that holographic models can tell something about each of the three quantities involved in the trace anomaly equation, namely the vacuum expectation value (VEV) of the trace of the energy momentum tensor, the function and the VEV of .
According to the correspondence, evaluating the five dimensional action at a classical global solution gives information about the VEV of the trace of the energy momentum tensor. The function can be obtained in terms of the solutions to the 5dimensional equation of motion derived form the action in the bulk. Finally, there is a way of calculating the VEV of the Wilson loop by means of minimizing the NambuGoto (NG) action for a loop lying in the boundary space. This is known to work in the strictly AdS case, i.e. for a conformal boundary field theory, and its generalization to nonconformal cases is still an open important problem. In turn, the VEV defined by , known as the gluon condensate, can be determined from the coefficient of the area squared in the expansion of a small Wilson loop in powers of its area Banksetal (); DiGiacomoetal (); Rakow ().
The features and results of this work are summarized as follows,

Two exact solutions of 5dimensional dilaton gravity for different dilaton potentials are considered. The first model, to be referred to as perturbative model, has a function, which to leading order in the t’Hooft coupling is the same as the perturbative 1loop QCD function. The second model will be referred to as nonperturbative model (because its function is nonanalytic in ). This model, by choice of the parameter in the model, can be made to correspond asymptotically to the soft wall model often used in nondynamical models of holographic QCD. The model leads naturally to confinement in the sense of static quarks, and to a nonvanishing gluon condensate when tested with a Wilson loop. However, it does not lead to asymptotic freedom in the ultraviolet.

For the perturbative model the asymptotic behavior of the solutions in the ultraviolet is not AdS. In the language of the holographic renormalization group the difference with the AdS limit is produced by an irrelevant operator that flows away from the AdS fixed point. In the nonperturbative model considered, the function gives rise to an UV fixed point at finite and the metric is asymptotically AdS.

Using the correspondence, the VEV of the energy momentum tensor is obtained by evaluating the 5dimensional action in the corresponding exact solutions, regularizing by introducing an energy scale and subtracting. These subtractions are performed as proposed in hawhoro96 (), and employed in the holographic case in myers99 (). In the perturbative case, taking into account Eqn. (1), it is argued that the same solution should be subtracted, leading to a vanishing VEV for the energymomentum tensor. In the nonperturbative model, being asymptotically AdS, the AdS limit is subtracted.

In order to calculate the gluon condensate the VEV of a small circular Wilson loop is considered. This is carried out using the corresponding NG action. For the perturbative model this procedure leads to a vanishing gluon condensate, while a nonvanishing result is obtained in the nonperturbative case.

The validity of Eqn.(1) is considered for both models, and shown to hold in the perturbative one. In the nonperturbative model the dependence of the gluon condensate on the energy scale is not the one required by Eqn. (1). This is however not unexpected as this model does not give a consistent description of the QCD ultraviolet behavior.
The paper is organized as follows. Section II presents the 5dimensional dilatongravity model employed in what follows. Exact solutions of the dilaton model equations of motion and associated functions corresponding to the perturbative and nonperturbative models are studied in section III. Section IV deals with the evaluation, regularization and subtraction of the gravitational action evaluated in the above mentioned exact solutions. Section V discusses the relevant asymptotics of the solutions of section III, and gives the explicit result for the subtracted gravitational action for those solutions. Section VI presents a study of the VEV of a small circular Wilson loop by means of the minimization of the NG action. Section VII addresses the issue of validity of the trace anomaly equation in the models considered. A final section VIII presents conclusions and outlook.
Ii Dilaton Model
The model considered is that of a self interacting scalar field immersed in a dynamical gravitational field in dimensions (in the end the results are only valid at ). The action of the model is given by DilatonAction (),
(2) 
where is the Newton constant in dimensions (of dimension ), the metric tensor field, the scalar curvature, the dilaton field, and the dilaton potential. The last term is the GibbonsHawking term GibbonsHawking () where is the second fundamental form. This term is included to make the Lagrangian depend only on the first derivatives of the metric. The equations of motion derived from this action are,
(3) 
where the Einstein tensor reads: , and . Because here the focus is on the vacuum of the boundary field theory, only metrics and scalar fields having flat boundary space isometry invariance are considered, thus only solutions for the metric and scalar field of the following general form are considered,
(4) 
where is a flat metric, and the coordinates employed here are known as domain wall coordinates. The boundary of the space is at . The AdS metric corresponds to taking , where the coordinate is measured in units of the AdS radius . For this particular choice of fields which only depend on , the equations of motion are given by,
(5) 
where the prime denotes derivation with respect to . Introducing a superpotential according to:
(6)  
(7) 
the choice reduces the three equations in Eqn. (5) to the single equation:
(8) 
Since the intended realistic application to QCD is at , throughout could be replaced.
Iii functions in dilaton models
In the AdS/CFT correspondence the identification is made of the YM coupling with the dilaton profile according to MaldaReview ():
(9) 
The energy scale (measured in units of a scale , where is the length unit mentioned earlier) of the boundary theory is identified with the scale factor in domain wall coordinates: . These identifications give the function in the dilaton model GursoyKiritsis ():
(10) 
In the rest of this section two different and exactly soluble dilaton models are considered. These models are obtained according to the following scheme: a dilaton profile is given, where by expressing in terms of and employing Eqn. (7) the superpotential is obtained, followed by integrating Eqn. (6) to obtain , and finally from Eqn. (10) the function is obtained. The potential is determined from Eqn. (II).
The two models considered are extreme cases. One model corresponds at leading order in the gauge coupling to the perturbative QCD function, while the other one corresponds to a nonperturbative function, i.e., which is nonanalytic at small coupling and which leads to an UV fix point. These models are qualitatively different as the next sections show. The precise choice of dilaton profiles is made so as to be able to perform all the calculations analytically.
iii.1 Perturbative function
The following dilaton profile is considered,
(11) 
Note that this choice means that . Therefore, should be a quantity order . Using the procedure just described leads to:
(12) 
where for convenience the integration constants can be chosen in such a way that the leading asymptotic behavior be AdS, namely , and . Then, asymptotically . The resulting function reads:
(13) 
which to leading order in becomes:
(14) 
The choice reproduces the leading order term of the QCD function (see Fig. 1)
iii.2 Nonperturbative function
A function with nonperturbative behavior, i.e. nonanalytic in the coupling , is obtained from the following dilaton profile,
(15) 
where . In this case,
(16) 
giving an asymptotically AdS metric.
The resulting function is then given by,
(17) 
which is positive in the interval , leading to an UV fixed point at (see Fig. 1). Thus, this theory is not asymptotically free, and therefore is not related to a pure YM theory. The sign of the constant determines two phases of the theory: for the theory becomes free in the infrared, while for it becomes strongly coupled. Indeed this latter case describes a confining theory in the IR. In order to see this it is convenient to express the above result in the conformal coordinate , where asymptotically , Êand therefore . This matches the GürsoyKiritsis GursoyKiritsis () criterion for confinement ^{1}^{1}1For , differs from the one considered in AndreevZakharov () only by the sign of the term. The AZ model is not a dynamical one, in particular in that reference this same factor is employed in the calculation of the NambuGoto action, there is no string frame correction due to the dilaton and so the GürsoyKiritsis confinement criterium can not be applied to that model. The negative sign of the coefficient multiplying the term is crucial in two respects: it is necessary for the confinement criterion GursoyKiritsis () to be fulfilled and second, the behavior of the factor for is such that, , which as shown in the next section, makes the use of a infrared cutoff unnecessary in the evaluation of the 5dimensional action for this solution.
Iv The trace of the energymomentum tensor
According to the AdS/CFT conjecture, taking the metric as the source field of the energymomentum tensor of the boundary field theory, the VEV of the trace of the energy momentum tensor is evaluated by simply evaluating the action Eqn. (2) for the classical solutions of the previous section.
Taking the trace in the first Eqn. (3) gives,
(18) 
and the action for the classical solutions becomes:
(19) 
and using the first Eqn. (5),
(20) 
Noting that and leads to,
(21)  
where denotes the volume of the boundary ddimensional space. On the other hand, the classical GibbonsHawking boundary action is given by:
(22) 
where is the induced metric in the boundary , namely , and denotes a unit vector field orthogonal to the boundary of . In domain wall coordinates this vector field is simply , and therefore:
(23) 
which is just times the bulk action. For both exact solutions considered in the previous section there is no contribution from the infrared boundary. On the other hand, the ultraviolet boundary gives for both cases divergent contributions, as it happens in general for any holographic model. As proposed in Witten (), these contributions can be regularized by evaluating at a finite value . This leads finally to,
(24) 
It is important to note that for a boundary theory that is not quantum conformal invariant, as for example QCD, the regulator has a physical meaning. Indeed, as mentioned in the previous section, the energy scale at which the boundary theory is observed is related to , the boundary value of the domain wall coordinate .
As shown in hawhoro96 () and applied to holographic models in myers99 (), a well defined action can be obtained by subtracting from the regulated action an action corresponding to some background metric having the same asymptotic limit. That is,
(25) 
where denotes the action evaluated in a solution having the same asymptotic behavior as the classical one. The subtracted energymomentum tensor is obtained recalling that, according to the correspondence,
(26) 
leading to , where denotes the common asymptotic exponent. The choice of this background metric for the solutions considered in section III is discussed in the next section.
V The UV QCD fixed point
The perturbative model in subsection IIIA presents features the understanding of which leads to new insights. These are the following:

The model leads to a function that coincides at leading order with the perturbative QCD function.

The model is not asymptotically AdS. As Eqn. (12) shows, the deviation of from the AdS limit becomes .

As shown in the previous section, the action should be subtracted with the action evaluated in a background metric having the same asymptotic behavior as the one to be subtracted. Thus, it is not sufficient to perform a substruction with the AdS metric.

In the language of the holographic renormalization group RGE (), this correction corresponds to an irrelevant operator, that flows away from the AdS fixed point MaldaReview (). This can be seen from the fact that the dilaton field behaves as at the UV boundary.

Eqn. (1) implies that for QCD the trace of the energymomentum tensor should vanish in the UV. This can be independently seen in two ways. As shown in section VI for this model, the VEV of the Wilson loop, calculated via the NG action, does not have terms which are powers of its area, and therefore the gluon condensate must vanish. The other way is simply to recall that in perturbative QCD the log of the VEV of the Wilson loop follows a perimeter law.
All these points indicate that the UV fixed point of QCD does not correspond to AdS. It corresponds to another solution that is well approximated by the one in subsection IIIA in the UV, i.e. for large , and therefore the action evaluated in the same solution or one asymptotically equivalent must be subtracted, leading to a vanishing trace of the subtracted energymomentum tensor.
In the nonperturbative model the space is asymptotically AdS, and the subtracted action becomes:
(27) 
leading to,
(28) 
Vi Wilson loops
The VEV of the operator (gluon condensate) appearing in the trace anomaly is accessible through the power like behavior of small Wilson loops as a function of their size. In pure YM theory the expansion of a small smooth Wilson loop (e.g., square or circular) is expected to have the form given by Banksetal (); DiGiacomoetal (); AndreevZakharov (); Rakow ():
(29) 
where is the length of the loop, is its area, and where is the one loop function. It is argued in pure YM that the terms order vanish as these would require a gauge invariant dimension two condensate.
The connection between Wilson loops of the boundary conformal gauge theory and minimal surfaces was made in references MaldaWL (); ReyYee (). According to it, in a CFT such as SUSY YM, in the large limit and large ’tHooft coupling the VEV of the Wilson loop is determined by the minimal area surface in the AdS space subtended by the path of the loop . Specifically:
(30) 
where the minimal area is given by the NG action of a string whose ends run along the loop. Since for a loop located at the boundary diverges, it has to be regulated, and thus the proportionality factor above.
The extension of this identification to nonconformal YM theory is still an open problem, in particular because in that case, as discussed earlier, the theory cannot be obtained via a relevant deformation of a CFT MaldaReview (). This problem is closely related to the problem of finding the noncritical string action for QCD Polyakovbook (). An extension of the correspondence for Wilson loops to the nonconformal case has been proposed GursoyKiritsis (), in which the NG action is the one corresponding to the stringframe metric, namely: in dimensions.
For the present purpose a circular Wilson loop of radius is considered, for which the NG action turns out to be:
(31) 
where is the radial coordinate of the disk, and is the bulk coordinate in conformal coordinates. The equation of motion is:
(32) 
where the solution needed satisfies . In AdS limit it is , a half sphere.
The UV divergencies of the NG action result from the contributions to the integral for . Noticing that diverges as , one obtains:
(33) 
where can be interpreted as the location of the loop in the bulk coordinate (provided ). In dilaton models one readily obtains:
(34) 
where , which asymptotically for the models discussed . If the function is given as input to the model, the superpotential and are given by
(35) 
One readily checks the AdS case where and , giving .
The perturbative model asymptotically gives , and , where, without loss of generality, the constant of integration required for has been chosen to be . This leads to:
(36) 
This shows that, as one would expect from the fact that the metric is not asymptotically AdS, the UV divergence of the action is modified with respect to the AdS case by the second term in the exponent.
The nonperturbative model is asymptotically AdS and thus the expectation is that the UV divergence coincides with the AdS case. If the coefficient this is indeed the case as it is easily shown using Eqns. (15) to (17) for , which leads to:
(37) 
For a constant term remains, which corresponds to a term linear in in the UV divergence of or equivalently logarithmic in .
The UV divergencies stem from the fact that diverges at the boundary. Therefore, they must naturally be only proportional to the perimeter of the loop, i.e. proportional to . For this reason, the contributions of higher powers of , which are of interest here, are independent of the regularization of and unambiguous.
The central point of the discussion is the sufficient conditions for the presence of higher power terms in in . The simplest case is when the metric is asymptotically AdS and the UV divergence of corresponds as well to the AdS case. For small , , and expanding in leads to:
(38) 
where , and evaluation in the AdS limit solution has been performed. The first order approximation is adequate near the boundary only if the UV divergencies are strictly AdS. On the other hand, the dependencies of in powers of beyond the first power (perimeter terms) will stem primarily from the interior of the integration domain, where the approximation is expected to work. Thus, a sufficient condition for such power corrections is that contains terms which have power dependency in the argument. The contributions in Eqn. (38) are in general difficult to evaluate as they involve the corrections to the solution of the equation of motion (32) AndreevZakharov (). The arguments made here apply in particular to the nonperturbative model when .
When the metric is not asymptotically AdS, as is the case of the perturbative model, a more accurate evaluation is necessary. For sufficiently small the entire surface will lye near the boundary , and can be set to zero, thus , . Setting and evaluating with the asymptotic AdS solution leads to:
(39) 
It is readily checked that this has the UV divergence obtained earlier in Eqn. (36). Evidently the dependence of in is logarithmic, and therefore according to the evaluation of the Wilson loop in the perturbative model. A similar conclusion results if is in general analytic in . Therefore, in the present framework, this indicates that in order to obtain a nonvanishing gluon condensate, the function should include nonanalytic terms in .
As an illustration of the latter, where power corrections are obtained at small coupling as consequence of nonperturbative terms in the function, consider the asymptotically free theory with , which is found in certain SUSY gauge theories Seiberg () as the result of instanton contributions. Considering the nonperturbative piece as small (or expanding in ), asymptotically , , leading to:
(40) 
which as expected coincides asymptotically with the perturbative model. The evaluation of the finite pieces gives power terms in . Asymptotically, to first order in :
(41) 
where pert. indicates the case with discussed earlier. Using Eqn. (38) leads to:
(42) 
obtained after replacing in the evaluation. Note that the power correction in this case did not stem from the contribution to by the dilaton, but rather from the correction order to the metric itself. This model gives a nonvanishing if .
The nonperturbative model is now analyzed for , where , , and asymptotically . Applying Eqn. (39) leads to:
(43) 
where the term stems from the contribution to by the dilaton. Clearly, if the model gives a nonvanishing , namely . For it reproduces the additional logarithmic contribution in to the UV divergence in Eqn. (38). For the model is similar to the one analyzed in AndreevZakharov (). In that case, to obtain the power correction it is necessary to calculate to second order in the perturbation to the action, and therefore corrections to the solutions are to be calculated. As mentioned earlier, in QCD the power series in the area of the Wilson loop start at ; for there is however a nonvanishing term order AndreevZakharov ().
Vii The trace anomaly test
For the perturbative case the trace anomaly equation is clearly fulfilled. Indeed the subtraction to the 5dimensional action in section V was performed in order to match, through Eqn. (1), the vanishing of determined in the previous section for this model. On the other hand, for the nonperturbative case, it is shown below that it is not possible to match both sides of Eqn. (1).
vii.1 The trace anomaly equation for the nonperturbative case
Equations (17) and (28) for and lead asymptotically for to:
(44)  
(45)  
where the asymptotic relation between domain wall and conformal coordinates has been employed. If the trace anomaly equation Eqn. (1) would be fullfilled, replacing Eqns. (44) and (45) into Eqn. (1) would imply for the gluon condensate to vanish asymptotically as:
(46) 
vii.2 Wilson loop calculation of
The computation of using the Wilson loop calculations of the previous section involves a different choice of boundary conditions than the one employed in this section. This is because the Wilson loop should be situated at a finite value of the coordinate orthogonal to the boundary, corresponding to the finite value chosen in evaluating the 5dimensional action used to evaluate the trace of the energymomentum tensor. The boundary condition to be employed is,
(47) 
For the pure AdS case, a solution of the area minimization equation satisfying this boundary condition is given by, , which simply corresponds to a circle of radius that is the radius required to match the boundary condition (47). For the nonperturbative model the effect of the above mentioned change in boundary conditions is well aproximated by replacing the radius by the effective one corresponding to the AdS solution, i.e. . Making that replacement in Eqn. (43) shows that the coefficient of has a contribution, coming from the term proportional to ^{2}^{2}2Higher powers of also contribute to the coefficient of , giving contributions that vanish for . , which does not vanish for . In particular, the simplest case where gives a putative . This is however in contradiction with the dependence in Eqn. (46), which comes from assuming the validity of (1). Therefore, the trace anomaly equation is not fulfilled in this model, and this is so in general for .
Viii Conclusions and outlook
In this work the validity of the trace anomaly equation has been studied in the holographic framework. This was done by considering holographic evaluations of the VEV of the trace of the energymomentum tensor, the function and the gluon condensate . The function is directly related to the definition of the particular model under consideration. The VEV of the trace of the energy momentum tensor was evaluated according to the holographic correspondence, by evaluating the dimensional classical action of the dilaton model on the corresponding classical solution. The gluon condensate can be obtained in a YM theory from the VEV of the Wilson loop, which was here evaluated for the models studied by means of a NG action.
Two models were analyzed, which can be exactly solved and which have different qualitative characteristics. In the perturbative model, where , consistency is fulfilled as the evaluation of the classical action can be appropriately subtracted to give a vanishing trace for the energy momentum tensor. If indeed in QCD, this may already be a somewhat realistic model. On the other hand, the nonperturbative model shows an inconsistency for the trace anomaly equation. This is manifested by the fact that has different behavior in the scale in the two evaluations. Indeed, the evaluation of the action and Eqn. (1) give , while from the Wilson loop evaluation is nonvanishing in the limit . This inconsistency seems reasonable since the nonperturbative model fails to correctly describe the UV properties of QCD, being asymptotically AdS and not asymptotically free.
Various interesting conclusions can be drawn from these results. They indicate that, although a holographic model of the pure gauge QCD vacuum based on the AdS space is not feasible, they do not preclude a gravitational dual based on a dynamical 5dimensional Einstein gravitational theory. They also show that QCD Ward identities, as for example the trace anomaly equation, strongly restrict the possibilities. It is reasonable to expect that QCD symmetry restrictions can in principle lead to a more precise version of its putative gravitational dual. Such a dual should lead to a boundary theory having all the following properties: asymptotic freedom in the UV, confinement in the IR, (possibly) a nonvanishing gluon condensate, and consistency with the trace anomaly equation. As the examples considered have shown, it is not at all obvious how to obtain a consistent model with these properties. Work in this direction is in progress and will be reported in due course.
Among important fundamental nonperturbative effects in QCD, the existence of a nonvanishing gluon condensate was early on identified SumRules (). It has important manifestations in hadron phenomenology SumRules (); SumRulesReviews (), and there are indications of its nonvanishing from lattice QCD DiGiacomoetal (); Rakow (). Due to its importance, its further understanding in the framework of holographic models of QCD is going to play a key role in the development of such models, as it has been shown in this work.
Ix Appendix
This appendix presents an explicit calculation of the NG action in a case where the has deviations from the AdS limit which are integer powers of , namely,
(48) 
The equation of motion Eqn. (32) is solved using an asymptotic series:
(49) 
where the coefficients are obtained in a systematic fashion.
The evaluation presented here can be applied to the nonperturbative model discussed in the text. A straightforward but lengthy evaluation gives:
For instance, in a ”soft wall” model where only one obtains:
(51) 
and the resulting NG action becomes:
(52) 
For the nonperturbative model with , one keeps only the term with , and the NG action becomes:
(53) 
Acknowledgements
The authors wish to thank H. Casini, L. Da Rold, A. Rosabal and M. Schvellinger for useful discussions and insights. This work was supported by DOE Contract No. DEAC0506OR23177 under which JSA operates the Thomas Jefferson National Accelerator Facility, by the National Science Foundation (USA) through grants PHY0555559 and PHY0855789 (JLG), and by CONICET (Argentina) PIP N¼ 11220090101018(RCT). JLG thanks the Instituto Balseiro and the Centro Atómico Bariloche for hospitality during part of this project, the Programa Raices for sabbatical support through a Cesar Milstein Award. RCT thanks Jefferson Lab for hospitality during part of this project.
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