RG flow, Wilson loops and the dilaton tadpole

1 June 2000

Ž .Physics Letters B 482 2000 329–336

RG flow, Wilson loops and the dilaton tadpole

Carlo Angelantonj a,1, Adi Armoni b,2

a ´ 3Laboratoire de Physique Theorique de l’Ecole Normale Superieure , 24, rue Lhomond, F-75231 Paris Cedex 05, France´ ´b ´ 4Centre de Physique Theorique de l’Ecole Polytechnique , F-91128 Palaiseau Cedex, France´

Received 10 March 2000; accepted 6 April 2000Editor: L. Alvarez-Gaume

Abstract

We discuss the role of the dilaton tadpole in the holographic description of non-supersymmetric gauge theories that areconformal in the planar limit. On the string theory side, the presence of the dilaton tadpole modifies the AdS backgroundinducing a logarithmic running for the radius and the dilaton. Using the holographic prescription we compute the Wilsonloop on the gravity side and find a smooth interpolating potential between asymptotic freedom and confinement, as expectedfrom field theory. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

The AdSrCFT correspondence has improvedconsiderably our understanding of strongly coupled

w xlarge N gauge theories 1 , showing that the naturaldescription of 4d theories in this regime is in termsof 5d gravity, where the extra dimension is related tothe 4d energy scale. So far, strong evidence for thevalidity of the conjecture has come from the descrip-tion of NNs4 SYM in terms of IIB supergravity,while limited efforts have been devoted to the holo-graphic description of non-supersymmetric non-con-formal field theories. Moreover, the few examples

Žanalyzed were mostly based on the tachyonic ori-. w xented 0B string 2,3 , where the presence of the

1 E-mail: [email protected] E-mail: [email protected] Unite mixte du CNRS et de l’ENS, UMR 8549.´4 Unite mixte du CNRS et de l’EP, UMR 7644.´

w xtachyon complicates the gravity description 4,5 andleads to instabilities in the dual strongly coupled

w xfield theory 6 .The first attempt to find a non-tachyonic holo-

graphic description of non-supersymmetric gaugew xtheories was recently made in 7 . The analysis was

still performed within the framework of 0B strings,w xbut resorting to a particular orientifold projection 8

to eliminate the tachyon from the bulk and to avoidproblems with the doubling of R-R sectors. Actually,

Žthe gauge theories that naturally emerge from orien-.tifolds of type 0 models have the peculiar property

of becoming conformal in the planar limit, and there-fore several results can be mutuated from NNs

w x Ž .4 9,7 . In particular, the leading planar geometry isknown to be of the form AdS =X with a constant5 5

dilaton field. For finite N, however, the gauge theoryis no longer conformal and new features are ex-pected in the gravity description. Indeed, a generalproperty of non-supersymmetric theories seems to bethe presence of a non-vanishing dilaton tadpole at

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00475-5

( )C. Angelantonj, A. ArmonirPhysics Letters B 482 2000 329–336330

w xgenus one-half 10 . On the other hand, supersymme-try relates R-R tadpoles to NS-NS ones, and thusdemanding neutral configurations of orientifoldplanes and D-branes automatically ensures the van-ishing of all massless tadpoles. However, wheneversupersymmetry is not present one can find configura-tions for which the R-R and NS-NS tadpole condi-

w xtions are incompatible 8,10,11 . While uncancelledR-R tadpoles manifest themselves in pathologies of

w xthe string theory 12 , NS-NS tadpoles are less prob-lematic. They correspond to potential terms in thelow-energy action and their main effect is to modifythe vacuum. In our case this is precisely what oneneeds! Indeed, studying D3 branes in non-tachyonic

Ž Xorientifolds of type 0B with O 7-planes and D7-branes, where a prime indicates the non-tachyonic

. w xinvolution we showed explicitly 7 how a genusone-half dilaton tadpole in the bulk theory doesreproduce qualitatively and quantitatively the ex-

Žpected logarithmic gauge theory RG flow. Recently,a logarithmic running of the gauge coupling was alsofound in the gravitational description of NNs

Ž . Ž . w x2 SU N =SU NqM gauge theory 13 . See re-w x .lated discussion on dilaton-induced RG flow in 14 .

The purpose of this letter is to pursue this pro-gram further, studying the back-reaction on the met-ric induced by the presence of a dilaton tadpole. Inparticular, we compute the Wilson loop in stringtheory and find a quark anti-quark potential interpo-

Ž .lating between a logarithmically running Coulombphase and a confining phase in the IR, as expectedfrom the field theory analysis. To the best of ourknowledge, the interpolating solution that we find isthe only known example in which a single solutiondescribes the expected behavior in the entire energyregime. Although our analysis is quite general andapplies to a vast class of non-supersymmetric non-conformal theories for which a non-vanishing dilaton

w xtadpole is expected to develop 10 , the simplestmodel one may conceive originates from D3-branesand OX3-planes in type 0B. In the large N limit the

5 w xnear horizon geometry is AdS =RP 15 , where5

RP5 sS5rZ , with Z the non-tachyonic involution2 2w x X 58 associated to O 3. Since the Z acts freely on S ,2

no open sector is present in the bulk and the spec-trum of the model consists of all the Z -invariant2

harmonics of the 0B fields. In particular, the tachyonis projected out. Moreover, since two-cycles of RP5

are unorientable, as for instance RP2 ;RP5, newinteractions in the low-energy action are expected tooriginate from unoriented world-sheet topologies.

Ž .These are weighted by generic not only even pow-ers of the dilaton, and in particular a dilaton tadpolecan be generated. Unfortunately, an explicit compu-tation of one-point amplitudes for the dilaton in thecurved AdS =RP5 geometry seems out of reach5

Ž .within the actual perturbative definition of stringtheory, although such a term is expected from theabove arguments, from our experience with non-su-persymmetric theories, and from the results we aregoing to present here.

The organization of this letter is as follows: inSection 2 we introduce the class of field theories wewant to study, and present an explicit example fromŽ .unoriented D-branes in the 0B string. In Section 3we turn to the gravity description and find an explicitsolution for the dilaton and the metric up to next-to-leading order in 1rN expansion. In Section 4 weextract the holographic quark anti-quark potentialand compare it with the field theory analysis ofSection 2. Finally, Section 5 contains our final com-ments about the validity of the gravity description.

2. Field theory considerations

One of the most remarkable results in field theoryis the running of coupling constants

da uŽ .syb a , 1Ž . Ž .

dlogu

Ž . 2 Ž .where a u sg u N is the ’t Hooft coupling.YM

The precise expression of the beta-function is modeldependent and encodes the quantum corrections to

Ž .the classical behavior a u sa . Whenever a quan-0

tum theory does not depend on any scale and thus isexactly conformally invariant, as for instance NNs

Ž .4 SYM, b a vanishes identically and the couplingconstants take their classical values for all energyscales. Combining these two arguments, it is thenevident that theories that are conformally invariant inthe planar limit are governed by a RG equation

da u 1 1Ž .sy b a y b a q . . . , 2Ž . Ž . Ž .1 22dlogu N N

( )C. Angelantonj, A. ArmonirPhysics Letters B 482 2000 329–336 331

Ž .so that one recovers the expected a u sa result0Ž . thfor N™`. Here, b a do not represent the i -loopi

contribution to the RG equation. Rather, they encodeall the contributions at a given order in 1rN.

To be concrete, the simplest prototype one canŽ .conceive is an SU N gauge theory with six real

scalars in the adjoint and 4 Weyl spinors in theantisymmetric and antisymmetric-conjugate repre-sentations, arising from the non-tachyonic orientifold

w xprojection on type 0B D3 branes 8,11 . This theoryŽ .was shown to converge to the SU N NNs4 SYM

w xin the planar limit 7 , and to have a one-loop16Ž .beta-function suppressed by 1rN with b s , and1 3

Ž .a two-loop beta-function contributing both to b a1Ž . Ž w xand b a in 2 3 .2

One can extract some interesting results from theŽ . Ž .RG equation 2 . Since in 2 there is a natural small

parameter, 1rN, it is natural to solve this equationby iterations

1a u sa q a q . . . . 3Ž . Ž .0 1N

The solution

1a u sa y b a logu 4Ž . Ž . Ž .0 1 0N

that, to this order, can be equivalently written as

1 1 1 b aŽ .1 0s q logu , 5Ž .2a u a N aŽ . 0 0

leads then to a quark anti-quark potential

a L aŽ . 0E L s s , 6Ž . Ž .

1 b aŽ .L 1 0L 1y log Lž /N a0

as expected for asymptotically free theories. Actu-ally, since for large N the beta-function varies slowly,

Ž .the logarithmic behavior 5 persists beyond the per-turbative regime, up to energy scales such that

Ž .1 b a1 0 logu <1. Note that the expansion is notN2a0

Ž .valid in the extreme IR.Ž .The description of the non-perturbative IR

regime is less straightforward. Although for the spe-

cific model we introduced previously the scalars dow xnot acquire a potential at the one-loop level 3,7 , in

general one would not expect to find a moduli spaceof vacua for non-supersymmetric theories. Thus, apotential for the scalars should develop: the scalarsmight become massive, or tachyonic, and might alsoacquire a non-vanishing vacuum expectation value.Although the specific shape of the potential is modeldependent, in the following we shall assume that thescalars become massive without acquiring any ÕeÕ.As we shall see, the string theory description sup-ports this assumption.

Once the massive scalars decouple, the IR degreesof freedom will consist of Yang-Mills theory with

Ž .four massless spinors a QCD like theory , and aconfining potential is expected

E L ss L , 7Ž . Ž .

where ssL2 is the QCD string tension. For pureQCDŽ .large N Yang-Mills theory one expects a constantŽ .finite string tension s . In the present case, how-0

ever, one expects a different behavior. Since thetheories we are describing are conformal in theplanar limit, the string tension should to vanishaccordingly. To estimate the N dependence of the

Ž .string tension, one can integrate the RG equation 2Žfrom an arbitrary weak coupling scale ;u in the0

. Ž .UV to a strong coupling one ;L :QCD

dayNHL su e , 8Ž . Ž .b aQCD 0 1

that translates into an exponentially suppressed stringtension

s;eyN . 9Ž .

3. The gravity solution

We now turn to a dual gravitational description ofthe gauge theories introduced in the previous section,

w xin the spirit of the AdSrCFT correspondence 1 . Asw xdiscussed in 7 , the dual gravity description involves

the ten-dimensional metric, the dilaton and the R-R


Ž .four-form potential with self-dual field strength ,whose dynamics is essentially encoded in the action 5

110 y2F r'Ss d x yg e Rq4E FE FŽ .H r4X

aŽ .1 4 2X1yF < <q Ce y a F , 10Ž . Ž .X 54a

or in its equations of motion2y2F 2 yFe 8 =F y8= Fy2 R yCe s0 , 11Ž . Ž .Ž .

1y2F yFe R q2= = F q g C eŽ .mn m n mn4

1 1rslt rslthy F F y g F F s0 .Ž .mrslt n mn rslth96 10

12Ž .

The term CeyF sC gy1 corresponds to the dilatons

tadpole expected from half-genus in string perturba-tion theory. Although the direct computation of thisterm is difficult, there are plausible arguments infavor of its presence, as discussed in the introduc-tion.

In the spirit of AdSrCFT correspondence, wemake for the metric the ansatz

X Ž . Ž .2 2 2 l t ab 2n t 2ds sa dt qe h dx dx qe dV ,Ž .a b 5

13Ž .

compatible with four-dimensional Poincare invari-´Ž .ance and SU 4 flavor symmetry. Moreover, the

1five-form field strength has Ny units of flux2

induced by the N D3 branes and the OX3 plane. Inthe following we shall ignore the contribution of theorientifold plane, manifestly suppressed in the pres-ence of a large number of branes.

Ž .Inserting the above ansatz in the Eqs. 11 andŽ .12 and allowing for a t dependence of the variousfields one obtains

12 2 2 2Fy10n¨ ¨ ˙2Fy4ly5ny4l y5n q N e¨ ˙ 2

1 Fy Ce s0 , 14Ž .4

1 2 2Fy10n¨ ˙ ˙ ˙ly 2Fy4ly5n ly N e˙Ž . 2

1 Fq Ce s0 , 15Ž .4

5 w xTo be more precise one should use the PST action 16 , sincewe are dealing with a self-dual 5-form F .5

1y2 n 2 2Fy10n˙ ˙ny 2Fy4ly5n ny4e q N e¨ ˙ ˙Ž . 2

1 Fq Ce s0 , 16Ž .4

˙2 2 ˙ 2 ˙ ˙ ˙ ˙y12l y20n y4F q20Fnq16Fly40ln˙ ˙ ˙q20ey2 n yN 2e2Fy10n qCeF s0 . 17Ž .

As usual in General Relativity, the tt componentŽ .17 of Einstein’s equations is not independent, andtranslates into a vanishing energy constraint for theassociated mechanical system.

In order to better appreciate the role of the differ-ent contributions in the low-energy action, it is use-ful to redefine F™Fy log N, in order that eF be

Ždirectly related to the ’t Hooft coupling. The inde-.pendent field equations then read

C2 F¨ ˙ ˙ ˙ ˙2Fy4F q8Flq10Fnq3 e s0 , 18Ž .˙

N1 2Fy10n¨ ˙ ˙ ˙ly 2Fy4ly5n ly e˙Ž . 2

C1 Fq e s0 , 19Ž .4 N

1y2 n 2Fy10n˙ ˙ny 2Fy4ly5n ny4e q e¨ ˙ ˙Ž . 2

C1 Fq e s0 . 20Ž .4 N

From these one can neatly understand the role of thedilaton tadpole: it represents a 1rN correction to theleading type IIB supergravity equations, and thusinduces 1rN corrections to the AdS =RP5 leading5

geometry, in agreement with field theory expecta-tions. Moreover, 1rN is a natural small parameter inthe theory and suggests to solve the equations byiterations

1F t sF t q F t q . . . , 21Ž . Ž . Ž . Ž .0 1N

1l t sl t q l t q . . . , 22Ž . Ž . Ž . Ž .0 1N

1n t sn t q n t q . . . . 23Ž . Ž . Ž . Ž .0 1N

The leading order equations

¨ ˙ 2 ˙ ˙ ˙2F y4F q8F l q10F n s0 , 24Ž .˙0 0 0 0 0 0

1 2F y10n0 0¨ ˙ ˙ ˙l y 2F y4l y5n l y e s0 ,˙ž /0 0 0 0 0 2

25Ž .


˙ ˙ y2 n 0n y 2F y4l y5n n y4e¨ ˙ ˙ž /0 0 0 0 0

1 2F y10n0 0q e s0 , 26Ž .2

independent of the dilaton tadpole, are consistentlyŽ . 5solved by the expected leading AdS =RP geom-5

etryF sw , 27Ž .0

8 1y w4'l s 8 e t , 28Ž .01 1

n sy log8q w . 29Ž .0 8 4

The next-to-leading order equations then reduce to8 1 3y w w4¨ ˙'F q4 8 e F q Ce s0 , 30Ž .1 1 2

8 1y w4¨ ˙ ˙'l q 8 e y2F q8l q5nž /1 1 1 1

4 1 1y w w2'y8 8 e F y5n q Ce s0 , 31Ž . Ž .1 1 4

8 81 1y w y w4 2' 'n q4 8 e n q8 8 e F y5nŽ .¨ ˙1 1 1 11 wq Ce s0 , 32Ž .4

and are solved by5w3 4F sy Ce t , 33Ž .1

8'8 853 w 2 w15 4l s Ce t y Ce t , 34Ž .1 64

8'256 85 5 31w y w3 4 4 2n sy Ce ty 8 Ce . 35Ž .1 32

8'32 8

In principle one should also consider the higherorder iterations. However, we shall stop here since inthe low energy action we are already neglectingcontributions of higher order in 1rN, as for examplea cosmological constant.

Collecting the various contributions, one finds thefollowing expressions for the dilaton

5C w

3 41Fswy e t , 36Ž .88 N'8

and for the metric tensor1

2dsXa

C8 1 32 y w w 2 a b4'sdt qexp 2 8 e tq e t h dx dxab32 N

C 51 1 w 23 4qexp y log8q wy e t dV .54 28 N'16 8

37Ž .

It is not hard to realize that the introduction of adilaton tadpole results into a running dilaton andrunning radii for the AdS and RP5 spaces. This can5

be better appreciated if one parametrizes the metricas

1 du2 u22 2 a bds sR u q h dx dxŽ .X ab2 2a u R uŽ .

˜2 2qR u dV , 38Ž . Ž .5

in terms of the energy scale

us elŽt . dt , 39Ž .Hwith the radii

dtn Žt Žu..˜R u s , R u se . 40Ž . Ž . Ž .

dlogu

Collecting our previous results, we find1 3 Cw w 21 4ts e loguy e log u , 41Ž .Ž .64 N

8'8

1w1 2e

4'82R u s , 42Ž . Ž .C3 w1q e logu16 N

1w3 2e

4'82R u s , 43Ž . Ž .C 2 w31q e loguN4'16 8

CyF yw w3e se q e logu . 44Ž .N4'8 8

At this point, before we interpret these results,various comments are in order. First, in the aboveexpressions we are repeatedly ignoring 1rN 2 sub-leading terms. Moreover, the value of ew can not beinterpreted any longer as the ’t Hooft coupling, sincenow the field theory is not conformal and the gauge

Ž . Ž .coupling runs. In addition, from Eqs. 42 and 43˜one might argue that R and R develop singularities

Žat some point in the IR notice that for our model Cis positive and then singularity presents itself for

.logu-0 . This, however, is not the case. The singu-larities are at points beyond the validity of the 1rNapproximation and are an artifact of our choice of


Ž .coordinates. Note, in fact, that the metric 37 is welldefined for every energy scale. Finally, contrary tothe familiar NNs4 super-conformal case the radii R

˜and R have now different values. This is expectedsince at order 1rN the dilaton tadpole behaves like

Žan effective cosmological constant, and therefore at.that order the overall curvature of the ten-dimen-

sional space is no longer vanishing.Ž .We can now turn to the interpretation of Eqs. 42

Ž .and 44 , in the spirit of AdSrCFT correspondence.The behavior of the dilaton is consistent with the factthat, in the dual gauge theory, no longer conformal,the coupling constant is expected to run. More pre-cisely, once the dilaton is identified with the ’t Hooft

F Žu. Ž . Ž .coupling, e ;a u , Eq. 44 reproduces pre-Ž .cisely the logarithmic behavior 5 expected from the

w xgauge theory side, as already shown in 7 . More-over, in the large N limit the running of the couplingconstant is suppressed, in agreement with the factthat in this limit the theory is conformal. Followingw x7 , it would be interesting to compute also in thiscase the numerical value of the dilaton tadpole C andcompare it with the one-loop beta-function coeffi-

16cient b s . To this end, however, one should1 3

compute the contributions from closed strings in theAdS =RP5 background with unoriented world-sheet5

wrapping the RP5. Unfortunately, computations ingenuinely curved backgrounds are still an open prob-lem in string perturbation theory.

w xFollowing 18 the running of the AdS radiusplays a natural role in the computation of the quarkanti-quark potential, the subject of the next section.

4. The Wilson loop

We can now use the ‘‘running’’ AdS =RP55

geometry to compute the quark anti-quark potentialŽ w xusing the holographic Wilson loop see 17 for a

. w xrecent review . In 19,18 it was suggested that anatural description of the Wilson loop in terms ofgravity is

² : ySW CC ;e , 45Ž . Ž .where, in the limit of large ’t Hooft coupling, S isthe proper area of a string world-sheet

1m nSs dt ds det g E X E X 46Ž .(HX mn a b2pa

describing the loop CC on the boundary of AdS. Thequark anti-quark potential can then be extractedchoosing the standard rectangular loop with sides Land T , with T™`.

Ž .Using the parametrization 38 for the metric g ,mn

the Nambu-Goto action reads

2 4 4(SrTsEs dx E u qu rR u , 47Ž . Ž . Ž .H x

and thus the quark anti-quark separation and thew xinteraction energy are 18

R2 dy0L; , 48Ž .H

2 4 4 4u (y y yR rR yuŽ .0 0 0

and

y2dyE;u , 49Ž .H0 4 4 4(y yR rR yuŽ .0 0

Ž .with R sR u .0 0

If the energy u is within the range of validity of01 4 4Ž .the expansion, logu <1, the ratio R rR yu0 0 0N

can be approximated by

3 C4 wR 1q e log yu0 08 N; 3 C w4 1q e loguR yuŽ . 08 N0

3 C w;1q e log y . 50Ž .8 N

Thus, one has

R20

L; , and E;u , 51Ž .0u0

and the potential

1 1w2e

42 2R R 1rL 'Ž . 80E; ; s CL L 3 wL 1y e log L16ž /N

52Ž .

can be expressed in terms of the running AdS radius,w xsimilarly to 4 .

Ž .Eq. 52 reproduces precisely the behavior ex-pected from the gauge theory side: at finite N theCoulomb phase is characterized by a quark anti-quark

Ž . Ž .potential 6 with a logarithmically running gauge


coupling. The fact that the two prescriptions, thedilaton running

a L eF Ž L. ewŽ .Es s ;

L L C2 w3L 1y e log L

4ž /N'8 8

53Ž .

Ž . Žand the Wilson loop 52 , yield similar results inw x.contrast to tachyonic type 0 models 5,4 is encour-

aging, although different powers of ew are still in-Ž .volved in the two cases. Note that 53 is a perturba-

Ž .tive prescription in contrast to 52 that should, apriori, be applicable to the non-perturbative regime.

Finally, let us describe the IR behavior of thegauge theory. To this end, it is simpler to work with

Ž .the metric in the form 37 . The quark anti-quarkenergy is then

2lŽt . 2 lŽt .(Es dx e E t qe , 54Ž . Ž .H x

and, since

12l t satq bt , 55Ž . Ž .

N

with a and b positive constants, el has a minimumat t syaNr2b , that corresponds to the IR,min

w ximplying a confining long distance potential 20

E;elmin Lss L . 56Ž .

The reason behind this result is rather simple tounderstand. For large enough separation, the stringwill prefer to stay at the minimum of l, thus mini-

Ž .mizing the action 54 . Therefore the energy will besimply proportional to the separation on the AdS

Žboundary. Moreover, the QCD string tension mea-X. w xsured in units of 1ra 20

a 2Nssexpl sexp ymin ž /4b

4 316 y w y12'sexp y 8 e C N 57Ž .3ž /is exponentially suppressed with N, in agreement

Ž .with field theory analysis 9 .

5. Comments

In conclusion, let us make a few comments aboutthe validity of the holographic description that wehave just presented. The holographic conjecture re-

Žlates 4d strongly coupled gauge theories to 5d su-.per gravity, and is thus valid, in principle, only

within a certain range of energies. Moreover, theprocedure of solving the gravitational equations byiterations further restricts the solution. First, we aretaking into account only the next-to-leading contribu-

Ž .tion associated to the dilaton tadpole a 1rN effect ,and therefore we need N41 to be sure that higherorder string loops be negligible. Moreover, the pro-cedure of solving the differential equations by itera-tions is valid only when the expansion parameter issmall. Since the expansion parameter is e

31 w2' e logu, the solution is valid only in someN

Ž .regions of the u , N space. Note, however, that for1w4any given energy u and initial AdS radius e , it is

always possible to find an N such that e is small,compatibly with the previous requirement. Finally,the curvature of the space should be small enough toallow one to trust classical gravity and to neglect a

X

corrections. Once the previous requirements are met,1w4this last constraint simply translates into large e , as

in the NNs4 case.Let us now see whether our solution meets these

requirements. It is not hard to see that large ew andsmall expansion parameter e translate into large ’tHooft coupling. But, from the field theory considera-tions, one expects the logarithmic behavior for small

Ž .and intermediate coupling 5 . One should assumenon-large ew and therefore a

X corrections should notbe neglected, thus invalidating our solution. How-ever, this is not the case. A more careful analysisreveals that our results depend only on the existenceof an AdS =RP5 solution at infinite N. Now, since5

it is a common lore that aX corrections do not

w x Žchange the geometry of the space 21 namely, thatNNs4 is described by an AdS =S5 metric at any5

.coupling , the logarithmic running should not beaffected. Moreover, there seem to be problems in thecomputation of the short-distance Wilson loop. First,the direct Wilson loop computation results into avalue for the one-loop beta-function different fromthat obtained by the running dilaton recipe. This isactually related to a

X corrections that, as in the


w xNNs4 case 21 , are expected to affect the parame-ters, but not the qualitative picture. In addition, thestring world-sheet ‘‘invades’’ also regions where e

is not small. Namely, the holographic computation ofthe Wilson loop for a given quark anti-quark separa-tion L requires that the metric be known everywhere

Ž .throughout the range u;1rL , ` . In order to avoidthis problem and thus be able to trust our solution,the world-sheet should extend deeply inside AdS,

Ž .and thus the potential 52 applies only to not tooŽsmall quark anti-quark separations i.e. our results

.are not valid in the extreme UV . Finally, the confin-ing phase for long distances relies on the existence

Ž .of a minimum for l t . However, at tst themin

contribution of next-to-leading term l is of the1

same order as l . Thus, we can not exclude that0

higher order contributions could eliminate the mini-mum, but hopefully the only effect of higher correc-tions could be simply to shift t . However, amin

definite answer to this problem requires an exactsolution to the exact low-energy action.

Acknowledgements

We are grateful to E. Gardi, B. Kol, A. Sagnotti,D. Seminara and especially to G. Grunberg and J.Sonnenschein for useful comments. A.A. thanks theDepartment of Applied Mathematics and TheoreticalPhysics at Cambridge University for the warm hospi-tality while this work was being completed. Thisresearch was supported in part by EEC under TMRcontract ERBFMRX-CT96-0090.

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RG flow, Wilson loops and the dilaton tadpole