We construct an explicit solution of type-IIB supergravity describing the
strong coupling regime of a non-supersymmetric gauge theory. The latter has a
running coupling with an ultraviolet stable fixed point corresponding to the
N=4 SU(N) super-Yang-Mills theory at large N. The running coupling has a power
law behaviour, argued to be universal, that is consistent with holography.
Around the critical point, our solution defines an asymptotic expansion for the
gauge coupling beta-function. We also calculate the first correction to the
Coulombic quark-antiquark potential.Comment: 10 pages, latex, v2: Another form of the solution is given and a
  reference is added, v3: a reference is added, final form to appear in PL

A. Kehagias

Antoniadis

Bachas

Brandhuber

Gross

K. Sfetsos

Maldacena

Schwarz

Taylor

Physics Letters B

English

arXiv

We construct an explicit solution of type-IIB supergravity describing the strong coupling regime of a non-supersymmetric gauge theory. The latter has a running coupling with an ultraviolet stable fixed point corresponding to the N=4 SU(N) super-Yang-Mills theory at large N. The running coupling has a power law behaviour, argued to be universal, that is consistent with holography. Around the critical point, our solution defines an asymptotic expansion for the gauge coupling beta-function. We also calculate the first correction to the Coulombic quark-antiquark potential.We construct an explicit solution of type-IIB supergravity describing the strong coupling regime of a non-supersymmetric gauge theory. The latter has a running coupling with an ultraviolet stable fixed point corresponding to the N=4 SU(N) super-Yang-Mills theory at large N. The running coupling has a power law behaviour, argued to be universal, that is consistent with holography. Around the critical point, our solution defines an asymptotic expansion for the gauge coupling beta-function. We also calculate the first correction to the Coulombic quark-antiquark potential.We construct an explicit solution of type-IIB supergravity describing the strong coupling regime of a non-supersymmetric gauge theory. The latter has a running coupling with an ultraviolet stable fixed point corresponding to the N =4 SU ( N ) super-Yang–Mills theory at large N . The running coupling has a power law behaviour, argued to be universal, that is consistent with holography. Around the critical point, our solution defines an asymptotic expansion for the gauge coupling beta-function. We also calculate the first correction to the Coulombic quark–antiquark potential

Kehagias, A.

Sfetsos, K.

CERN Document Server

CERN-TH/99-37hep{th/9902125On running couplings in gauge theories fromtype-IIB supergravityA. Kehagias and K. SfetsosTheory Division, CERNCH-1211 Geneva 23, Switzerlandkehagias,sfetsos@mail.cern.chAbstractWe construct an explicit solution of type-IIB supergravity describing the strong couplingregime of a non-supersymmetric gauge theory. The latter has a running coupling withan ultraviolet stable xed point corresponding to the N = 4 SU(N) super-Yang{Millstheory at large N . The running coupling has a power law behaviour, argued to beuniversal, that is consistent with holography. Around the critical point, our solutiondenes an asymptotic expansion for the gauge coupling beta-function. We also calculatethe rst correction to the Coulombic quark{antiquark potential.CERN-TH/99-37February 19991 Introduction and computationsOne of the well-known vacua of the type-IIB supergravity theory is the AdS5  S5 one,rst described in [1]. The non-vanishing elds are the metric and the anti-self-dual ve-form F5. The latter is given by the Freund{Rubin-type ansatz, which is explicitly writtenasFµνρκλ = −p2µνρκλ ; ; ; : : : = 0; 1; : : : ; 4 ;Fijkpq =p2ijkpq ; i; j; : : : = 5; : : : ; 9 ; (1)and is clearly anti-self-dual. This background has received a lot of attention recentlybecause of its conjectural connection toN = 4 SU(N) super-Yang{Mills (SYM) theory atlarge N [2, 3]. The SYM coupling gYM is given, in terms of the dilaton , as g2YM = 4eΦ,and the ’t Hooft coupling is g2H = g2YMN , where N =∫S5 F5 is the flux of the ve-formthrough the S5. The dilaton is constant in this background, which is related to theniteness of the N = 4 SYM theory.In order to make contact with QCD, it is important to investigate deformationsof the SYM theory that break conformal invariance and supersymmetry. In this case,the couplings are running corresponding to a non-constant dilaton in the supergravityside. It is then clear that the background we are after is a perturbation of AdS5  S5.Attempts to nd supergravity backgrounds that allow a non-constant dilaton, and hencea running coupling of the YM theory, have been exploited within type-0 theories [4]{[6].Deformations of the N = 4 theory, which flow to an interacting conformal xed point,have been considered in [7].The purpose of the present paper is to show that running couplings are also possiblewithin the type-IIB string theory. We will study the \minimal" case, that is we willkeep the same F5 form as in (1) and turn on a non-constant dilaton. We will show thatsuch a solution, which breaks supersymmetry and conformal invariance, exists. We willassume for the metric four-dimensional Poincare invariance ISO(1; 3), since we wouldlike a gauge theory dened on Minkowski space-time. In addition, we will preserve theoriginal SO(6) symmetry of the AdS5S5. As a result, the ISO(1; 3)SO(6) invariantten-dimensional metric is of the form1ds2 = gµνdxµdxν + gijdxidxj ;wheregµνdxµdxν = Ω2()(d2 + dxαdxα) ;  = 0; 1; 2; 3 ; (2)and gij is the metric on S5. The dilaton, by ISO(1; 3) SO(6) invariance, can only bea function of . The supergravity equations turn out to beRµν = −gµν + 12@µ@ν ;1Supersymmetric solutions to type-IIB supergravity which, however, do not preserve the Poincareinvariance in the brane world-volume have been found in [8].11p−G@µ(p−GGµν@ν) = 0 ; (3)andRij = gij : (4)The above equation is automatically solved for a ve-sphere of radius 2=p and a rstintegral of the dilaton equation in (3) is@ρ = AΩ−3 ; (5)where A is a dimensionful integration constant. Moreover, the non-zero components ofthe Ricci tensor for the metric (2) areRρρ = −4@2ρ ln Ω ;Rαβ = −αβ(@2ρ ln Ω + 3(@ρ ln Ω)2); (6)and the rst equation in (3) reduces to solving(@ρ ln Ω)2 =A224Ω−6 +4Ω2 : (7)The solution of the above equation for Ω as a function of  is given implicitly, in termsof a hypergeometric function, byΩ3 F(12;38;118;−6A2Ω8)= √38jAj(− 0) ; (8)where 0 is another constant of integration. The dierent overall signs in the right-handside of (8) arise from taking the square root in (7). We impose the boundary conditionthat the space described by (2) becomes AdS5 when  ! 0+. That means that theconformal factor should assume the form Ω ’ R= for small , where R  (4gsN)1/4.Using well-known formulae for the hypergeometric functions, we see that this naturallyleads to the choice of the minus sign in (8) and also xes  = 4=R2. In addition, theconstants A and 0 are related byjAj = R3440;  √38Γ(3=8)Γ(1=8)243/8’ 1:87 : (9)Then (8) can be written as(Ω0R)3F(12;38;118;−248(Ω0R)8)=√384(1− 0): (10)We also nd thatΩ ’ (3=8)1/64/3 R0(1− 0)1/3; as ! −0 : (11)2We may solve the dilaton equation in (5) close to  = 0 and  = 0. The result for thestring coupling iseΦ = gs(1 + s44( 0)4+832( 0)8+ s11123456( 0)12+ : : :); as ! 0+ ; (12)where s = 1 = sign(A),2 while on the other side = 0 − s (8=3)1/2 ln(1− 0); as ! −0 : (13)The form of the above solution is dictated by the fact that, in the limit 0 ! 1, thedilaton should be eΦ = gs  eΦ0 and (12) and (13) should coincide. Now, at  = 0there is a singularity that may be easily seen by computing the Ricci scalar using (3).The latter is R = 12@µ@µ, which at  ’ 0 behaves as R  (0 − )−8/3. Hence, wemay consider our solution only as an asymptotic expansion around the AdS5 geometryat  = 0 of the formΩ() =R(1 +1∑n=1an( 0)8n);  < 0 ; (14)where the coecients an are computed using the series representation for the hypergeo-metric function in (10) for large Ω. The rst coecient of the expansion turns out to bea1 = −8=432 ’ −0:352. Hence, Ω() is given byΩ() ’ R(1− 8432( 0)8)(15)to a very good approximation, except when  takes values very close to 0. Then wemay use, instead of (15), (11) to a very good approximation as well, and the results areplotted in g. 1. Note that our analysis was done in the Einstein frame and it is notdicult to translate everything into the string frame by multiplying the Einstein metricby eΦ/2. Our solution is singular at  = 0 in both frames and can, indeed, be trustedaway from that point.2 Running CouplingIn the AdS/CFT scheme, the dependence of the bulk elds on the radial coordinate may be interpreted as energy dependence. In fact, it is a general feature in the AdS/CFTscheme that long (short) distances in the AdS space correspond to high (low) energiesin the CFT [2, 9]. In particular, if the dilaton in the supergravity side is a function of2The solutions corresponding to the two possible choices for s are related by an S-duality trans-formation and correspond to dierent gauge theories. This reflects the fact that except for ρ0 = 1,corresponding to the N = 4 SU(N) SYM at large N , our solution describes gauge theories that are notS-duality-invariant. Also in (12) we have used (14) below.3(2)(1)024681012141618200.2 0.4 0.6 0.8 1Figure 1: Plot of Ω()=R in units where 0 = 1. Curves (1) and (2) were plotted using(15) and (11) respectively. The curve corresponding to Ω()=R, obtained by numericallysolving (8), coincides with the union of these curves., then the ’t Hooft coupling of the boundary CFT has an energy dependence and canbe interpreted as the running coupling of the CFT. Running coupling means of coursethat we are away from conformality thus, backgrounds that admit non-constant dilatoncorrespond to non-conformal eld theories. As long as supersymmetry is unbroken, spin-sum rules for the AdS supersymmetry are expected to protect the ’t Hooft couplinggH of the boundary N = 4 YM theory against running. However, if supersymmetry isbroken, there are then no more cancellations between fermionic and bosonic contributionsleading to the running of gH. The specic background we found here clearly breakssupersymmetry, and =12γM@M =A2Ω3γρ ; M = DM (16)are the associated fermionic zero modes. If we now follow the correspondence betweenlong-distances/high-energies in the AdS/CFT scheme, we nd that the dual theory ofthe supergravity solution we obtained has a coupling with power-law running. Indeed,by changing the variable  = R2=U and interpreting U as the energy of the boundaryeld theory, we nd from (12) the running of gH :gH = gH(1 + s48R840U4+8128R1680U8+ s171227648R24120 U12+ : : :); (17)where gH = R2 is the UV value of the ’t Hooft coupling (the result in plotted in g. 2).From this expression, it may easily be found that the behavior of the beta-function for4s=–1s=+1–2024681 2 3 4UFigure 2: Plot of gH=gH as a function of U using (17).the ’t Hooft coupling around gH isUdgHdU= −4(gH − gH)− 2(gH − gH)2=gH −1427(gH − gH)3=g2H + O(gH − gH)4 : (18)However, the above equation does not specify the beta-function, but rather its derivativeat the gH = gH point. The reason is that our solution breaks down at energies U  R2=0.From (18) we see that 0(gH) = −4 ; (19)which means that gYM is a UV-stable xed point. We believe that (19) is universal,namely, that it is valid for all models that approach AdS5  S5 at some boundary. Thiscan be seen by recalling that, near AdS5  S5, the dilaton always satises (3) withΩ = R=. As a result,  will behave as eΦ−Φ0  4 where 0 is the value of  at  = 0.We also see that (18) determines the second and third derivatives of the coupling beta-function at the xed point, which, however, are not expected to be model-independent.Let us also note that there is no known perturbative eld theory with UV stable xedpoints. A behaviour of the form (17), namely power-law running of the couplings, wasalso found in type-0 theories (see second ref. in [6]) and extensively discussed in gauge-coupling unication in theories with large internal dimensions [10]{[13]. In this scenario,the internal dimensions are shown up in the four-dimensional theory as the massive KKmodes. These modes can run in the loops of the four-dimensional theory, giving rise toa power law running of the couplings. In particular, for d large extra dimensions and forenergies E above the infrared cuto, which is specied by the mass scale  of the extradimensions, we nd, just for dimensional reasons, that the running coupling constant ofthe eective four-dimensional theory is of the form g(4)−g(4)0  (=E)d/2, where g(4)0 is thebare coupling. Thus, in our case, since we have a four-dimensional theory coming fromten dimensions we should expect the coupling to run in the sixth power of E. Instead,5we nd here that the coupling depends on the fourth power of U = E, indicating thatwhen holography is involved, we get a softer running of the couplings.It is possible to identify those operators that are responsible for the running of thecoupling in the boundary eld theory. Since the dilaton approaches a constant value at = 0 and the asymptotic background is an AdS5 space, the corresponding boundary eldtheory is expected to be a deformation of the N = 4 SU(N) supersymmetric YM theory.The explicit form of the deformation may be specied by recalling that our solution stillhas an SO(6) symmetry. There are not many SO(6) singlets in the spectrum of the S5compactication. In fact, from the results in [14] we see that the only scalar singlets arethe complex scalar of type-IIB theory B, aijkl and hii, with masses m2B = 0; m2a = m2h = 32in AdS-mass units. Since we have not perturbed the ve-form F5, aijkl = 0 and thus theperturbations we have turned on are the real part of B and hii. From their masses we ndthat the former corresponds to marginal deformations of the type F 2µν , while the latter,to the dimension-eight operator F 4µν , which is irrelevant. However, it gives contributionsto the boundary eld theory since we have an IR cuto specied by 0. Sending 0 !1all bulk perturbations disappear and, similarly, the boundary eld theory turns out tobe the N = 4 large-N SU(N) SYM theory.3 The quark–antiquark potentialThe breaking of the superconformal invariance of the N = 4 theory by our solutionshould be apparent in the expression for the quark{antiquark potential, which we nowcompute along the lines of [15, 16]. We will nd corrections to the purely Coulombicbehaviour, which, on purely dimensional grounds, we expect to be in powers of AL4,where L is the quark{antiquark distance. We are eventually interested in the rst suchcorrection, which, as is apparent when comparing (14) with (12), is due to the dilaton,but for the moment we keep the formalism general. As usual, we have to minimize theNambu{Goto actionS =12∫dd√det (GMN@αXM@βXN) ; (20)where GMN is the target-space metric in the string frame. For the static congurationx0 =  , x1  x = , and x2; x3 as well as the coordinates of S5 held xed, we nd that(20) becomes (we use the notation of [15])S =T2∫dx eΘ/2√(@xU)2 + U4=R4 ; (21)where eΘ  eΦ4 and  = 1 + ∑1n=1 an( R2ρ0U )8n is the function multiplying R= in (14)(rewritten using  = R2=U). It is clear that any background that approaches AdS5 willalways have a  of the form (14), so that our analysis is quite general at this point. It iseasy to see that the solution is expressed asx =∫ UU0R2dUU2√eΘ−Θ0U4=U40 − 1=R4; (22)6where U0 is the smallest \distance" of the trajectory to the center and 0 is the valueof the function  evaluated at U0. We assume that one of the branes is taken out toU = 1 and that the string conguration starts and ends at this brane. The rest of thebranes are located at U = R2=0. Setting x = L=2 corresponds to U = 1. In turn, thisgives a condition that relates U0 to L asL = 2∫ 1U0R2dUU2√eΘ−Θ0U4=U40 − 1=R4: (23)Proceeding in a standard fashion, we substitute back our solution (22) into the action (21)and obtain an integral that is innity. This is because we have included into the potentialenergy the (equal) masses of the innitely heavy (in the supergravity approximation)quark and antiquark. In order to compute these masses, we assume that N -branes areat U = R2=0 and 1 at U = Umax that is assumed to be large but nite. The mass of asingle quark is computed if in (21) we consider a conguration with x0 =  , U =  andwith xed spatial world-volume coordinates xα,  = 1; 2; 3, as well as S5 coordinates.Then the self-energy of the quark isEself =12∫ UmaxR2/ρ0dUeΘ/2 : (24)Subtracting o this energy twice we obtain a nite result for the quark{antiquark poten-tial given byEqq¯ =1∫ UmaxU0dUeΘ/2 U2=U20√U4=U20 − eΘ0−Θ− 1 : (25)At this point we have to solve (23) for U0 as a function of L and substitute the resultback into (25) to obtain Eqq¯ as a function of L only. This can be done perturbativelyin powers of L=0, and we are interested in the rst correction to the Coulombic lawbehaviour of the potential. As explained, the correction due to the non-constant dilatonis dominant and we will use therefore Ω = R= which corresponds to AdS5  S5 for thestring metric, whereas for the string coupling, we will keep the rst two terms in (12).Then we nd thatU0 ’ 2R21L(1 +s8( 21)2( L0)4); 1 =1/2Γ(3=4)Γ(1=4)’ :599 ; (26)where the value of the numerical constant  has been given in (9) andEself ’ Umax2− R220(1− s424): (27)The result for the quark{antiquark potential isEqq¯ ’ −221R21L(1− s8( 21)4( L0)4)+R20(1− s424): (28)7We see that the Coulombic potential receives a correction proportional to L3 due to thebreaking of conformal invariance.3 This is remarkably similar to the potential obtainedin [17] for the quark{antiquark pair for the N = 4, at nite temperature, using thenear horizon supergravity solution for N coincident D3-branes. However, in that casesupersymmetry is broken by thermal eects, whereas in our case it is broken, by thepresence of a non-trivial dilaton, even at zero temperature. The last term in (28) doesnot depend on L and represents a constant shift of the potential energy.As a nal remark we note that the computation of the potential for the monopole{antimonopole pair proceeds along the same lines as that for the quark{antiquark pairwith the only dierence that we start, similarly to [18], with the action for a D-string.This means that the integrand in (21) should be multiplied by e−Φ. Hence, the functioneΘ entering into (21) is dened as eΘ = e−Φ4. Consequently, the rst correction to theCoulombic behaviour of the monopole{antimonopole potential is given byEmm¯ ’ −221R2gsL(1 +s8( 21)4( L0)4)+R20(1 + s424); (29)which is the same as (28) after we use the fact that under S-duality gs ! 1=gs and s! −s.Hence we see a screening (antiscreening) of the quark{antiquark pair for s = +1(−1) andexactly the opposite behaviour for the monopole{antimonopole pair.AcknowledgementsWe would like to thank K. Dienes, E. Dudas, T. Gherghetta and A. Zaaroni fordiscussions.References[1] J.H. Schwarz, Nucl. Phys. B226 (1983) 269.[2] J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200.[3] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Phys. Lett. B428 (1998)105, hep-th/9802109; E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150.[4] I.R. Klebanov and A.A. Tseytlin, D-branes and dual gauge theories in type 0 strings,hep-th/9811035, Asymptotic freedom and infrared behaviour in the type 0 stringapproach to gauge theories, hep-th/9812089; A non-supersymmetric large-N CFTfrom type 0 string theory, hep-th/9901101.[5] J.A. Minahan, Glueball mass spectra and other issues for supergravity duals of QCDmodels, hep-th/9811156; Asymptotic freedom and confinement from type 0 stringtheory, hep-th/9902074.3Supergravity is valid when ρ  ρ0, which means that U  U0  R2/ρ0. Using the leading term in(23), we deduce that L/ρ0  1, which is indeed the condition for the validity of (28).8[6] G. Ferretti and D. Martelli, On the construction of gauge theories from non-criticaltype 0 strings, hep-th/9811208; E. Alvarez and C. Gomez, Non-critical confiningstrings and the renormalization group, hep-th/9902012.[7] J. Distler and F. Zamora, Non-supersymmetric conformal field theories from stableAnti-de sitter spaces, hep-th/9810206; L. Girardello, M. 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On running couplings in gauge theories from type-IIB supergravity

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We construct an explicit solution of type-IIB supergravity describing the strong coupling regime of a non-supersymmetric gauge theory. The latter has a running coupling with an ultraviolet stable fixed point corresponding to the N=4 SU(N) super-Yang-Mills theory at large N. The running coupling has a power law behaviour, argued to be universal, that is consistent with holography. Around the critical point, our solution defines an asymptotic expansion for the gauge coupling beta-function. We also calculate the first correction to the Coulombic quark-antiquark potential

Kehagias, A

Sfetsos, K

Surrey Research Insight

On running couplings in gauge theories from type-IIB supergravity

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