Light-like Wilson Loops and Cusp Anomalous Dimensions in Non-conformal Gauge Theories
Leopoldo A. Pando Zayas, Daniel J. Phalen, Cesar A. Terrero-Escalante
aa r X i v : . [ h e p - t h ] J u l Light-like Wilson Loops and Cusp AnomalousDimensions in Nonconformal Gauge Theories
Leopoldo A. Pando Zayas ∗ , Daniel J. Phalen ∗ and César A.Terrero-Escalante † ∗ Michigan Center for Theoretical PhysicsRandall Laboratory of Physics, The University of MichiganAnn Arbor, MI 48109-1040 † Centro de Investigación y Estudios AvanzadosAv. Politécnico Nacional 2508, México D.F. 07360, México
Abstract.
We emphasize that nonconformal theories provide a natural playground for the ideas ofthe Maldacena conjecture opening the possibility of exploring properties that could potentially be inthe same universality class as QCD. In particular, we discuss in detail how light-like Wilson loops,an important ingredient in the prescription for scattering amplitudes, can be described in a numberof gravity duals of nonconformal gauge theories. We point out to a few universal properties and theprominent role of the strong scale.
Keywords:
Gauge/string duality, Strong-coupling expansions, Wilson loops
PACS:
INTRODUCTION
The idea that the description of gauge theories might effectively involve strings has along history starting from the dual models for hadronic resonances. In 1974 ‘t Hooft[1] showed that gauge theories admit a limit in which their perturbation theory can bethought of as coming from a string theory. Ten years ago, a precise formulation of thisequivalence was put forward for a particular gauge theory. Namely, it was argued that N = AdS × S [2, 3, 4]. In the ten years following the seminal work of Maldacena [2] thegauge/gravity correspondence has seen a period of maturity. In its current incarnationthe correspondence has turned into a very fruitful theoretical framework.Particularly exciting is the side of Maldacena’s conjecture relating the strongly cou-pled sector of the gauge theory with weakly coupled gravity. Many problems in gravityhave been solved during nearly a century of work. By turning strong coupling prob-lems of gauge theories into questions in classical gravity, the correspondence openeda window of opportunity into one of the long-standing puzzles of particle physics: theinfrared properties of gauge theories. Some of the questions to be addressed here areconfinement, the spectrum of hadrons and the properties of the strongly coupled quark-gluon plasma. However, important questions about nonprotected quantities lie beyondthe domain of classical gravity and should properly be tackled using string theory.There are two main obstacles in applying the gauge/gravity correspondence to phe-nomenologically interesting theories. The high amount of supersymmetry and conformalsymmetry is not present in nature. Although originally formulated as a duality betweentring theory on AdS × S and maximally SYM in four dimensions, it has quickly be-come clear that such dualities extend to more general situations. Significant results inunderstanding QCD-like theories have been achieved by constructing supergravity the-ories whose dual gauge theories contain N = SU ( N ) supersymmetric Yang-Mills[5, 6]. In a remarkable series of papers [7] Klebanov and collaborators carried out a pro-gram that concluded with a supergravity background that is dual to N = N = U ( ) R → Z N symmetry breaking, gluino condensationand spontaneous Z N → Z breaking, monopoles, domain walls, baryons and KK states.Moreover, a supergravity solution dual to the finite temperature warped conifold withcascading in the UV was found [9]. Evidence of a transition between this solution andthe KS solution was reported in [10] (see also [11]). Such transition could be interpretedas the gravity dual to the confinement/deconfinement transition in QCD-like theories.Another important direction in which the Maldacena conjecture has been expandedis in going beyond the lowest energy states. Much of the developments in the con-formal version regarding beyond the supergravity approximation were extended to thenonconformal case. For example, the BMN limit [12] was also understood in noncon-formal situations; the analogous structure describes hadronic states for which an exactstring Hamiltonian was presented [13]. Using semiclassical quantization in the contextof gauge/gravity correspondence, quantum corrections to Regge trajectories were calcu-lated [14], showing that quantum effects alter both the linearity of the trajectory and thevanishing classical intercept, J : = a ( t ) = a + a ′ t + b √ t .More recently, Alday and Maldacena proposed a prescription to calculate the gluonscattering amplitudes for N = IGURE 1.
Schematic representation of the Wilson loop.
WILSON LOOPS IN GAUGE THEORIES
The Wilson loop is a very important observable in gauge theories and in some ap-proaches it is treated as the central ingredient in the description of all gauge invariantquantities. Its description on the gravity side was proposed in [17, 18] where it wassuggested that the evaluation of the vacuum expectation value of a given Wilson loopon the contour C is given by the string worldsheet whose boundary coincides with C : h W ( C ) i = exp ( − S ( C )) .The Wilson loop provided the first example of the correspondence that goes beyondthe study of protected quantities at the supergravity level. Related developments allowedto understand that certain non-BPS operators can be described by classical string config-urations in a given supergravity background. For example, the large anomalous dimen-sion of operators of the form Tr X I (cid:209) ( m . . . (cid:209) m s ) X I , where X I are scalar fields in N = AdS , D − S = √ lp ln S , (1)where l = g Y M N is the ’t Hooft coupling. These operators are generalizations in N = F am (cid:209) m . . . (cid:209) m s − F a m s and ¯ yg m (cid:209) m . . . (cid:209) m s y . Soon after [19], a description ofthese operators was found in terms of light-like Wilson loops in the AdS/CFT correspon-dence [22, 23]. One important property of the anomalous dimensions of these operatorsis that it is conjectured to be in general of the form D − S = f ( l ) ln S , where f ( l ) is thecusp anomalous dimension and it is a function only of l .The Alday-Maldacena prescription for gluon scattering in N = AdS × S that end on light-like segments.These configurations are light-like Wilson loops. It is crucial that the space where theseilson loops live is the T-dual space to the original supergravity background. This con-struction motivates us to revisit and expand the analysis of worldsheet configurationsending on light-like segments in supergravity backgrounds dual to nonconformal theo-ries. Similar configurations have been considered before but with two main differencesfrom those we present here. First, they were considered in the original supergravity back-ground, that is, without performing a T-duality and second the light-like segments werealways placed in the asymptotic UV region. For example, the light-like Wilson loop onthe Klebanov-Strassler background was analyzed in [24].In this contribution we report on the situation where we perform a T-duality on the R , subspace where the field theory lives, and the light-like segments are placed in theregion corresponding to the IR of the field theory. Since we work on the T-dual space andplace the brane in an arbitrary position, the value of the cusp depends on the structure ofthe infrared region. This is a crucial difference with perturbative QCD where no accessto the IR region is possible other than on dimensional grounds. Cusps anomalous dimension
If a Wilson loop is evaluated over a closed contour that forms a light-like cusp withangle g in Minkowski space-time, then its expectation value is [25], W g ∼ (cid:18) L m (cid:19) − G cusp ( g ) , (2)where L is a UV cutoff, m is an IR cutoff, and G cusp measures the anomalous dimensionof the Wilson loop. The cusp anomalous dimension is also an important observable ingauge theories. It controls the scaling behavior of various gauge invariant quantities likethe logarithmic scaling of the anomalous dimension of higher-spin Wilson operators,double-log (Sudakov) asymptotics of elastic form factors in QCD, the gluon Reggetrajectory, infrared asymptotics of on-shell scattering amplitudes, and it is importantfor resumming the effects of soft gluon emission in the study of QCD at colliders [26].The cusp anomalous dimension depends only on the coupling constant and its expan-sion at weak coupling is known in QCD to three loops [27] and in N = N = R , × SL ( , R ) / U ( ) [31] and the noncritical solutionwith an infrared fixed point [32]. he general set up for the cusp Let us consider a general metric of the form: ds = q ( r ) dx m dx m + p ( r ) dr . Weperform a T-duality along the x m coordinates and arrive at a metric of the form ds = q ( r ) dy m dy m + p ( r ) dr . (3)We now consider embedding a string world sheet into this metric. For world sheetcoordinates ( t , s ) , this embedding is y = e t cosh s , y = e t sinh s , r = r ( t , s ) . Theaction is thus S = pa ′ Z d s d t e t q q − p q e − t (( ¶ t r ) − ( ¶ s r ) ) . (4)The methodology is then to derive the equation of motion for r , to solve it, substitutethe solution into (4) and integrate for the action. This integral typically diverges, so anUV cutoff must be chosen as upper limit of integration. Given that S = ln W , the cuspanomalous dimension can be obtained from (2) as, G cusp = − W ¶ W ¶ ln L = − ¶ S ¶ ln L . (5) The cusp in N = SYM
Here we present elements of a systematic analysis to the derivation of the solutionfor this most symmetric case. It might seem like an overkill, however, reproducinganalytical results for simple cases helps to adjust the numerical methods for handlingless symmetric situations. Here we study a building block for the configuration thatleads to the scattering amplitude.Let us consider the cusp as described in [15]. For
AdS we have that in (4) q = r / R and p = R / r . Moreover, if we consider an ansatz of the type r = e t w ( t ) , then we findthat the action is: S = R pa ′ Z d s d t w ( t ) q − ( w ( t ) + ˙ w ( t )) . (6)Following the methodology listed at the end of the previous subsection we derive thecorresponding equation of motion which, for the purpose of numerical integration, isbetter to write as the following system of first order ordinary differential equations,˙ w = v , ˙ v = w (cid:0) − vw − v − w + w + vw − v w − v w (cid:1) . (7)Here dot stands for derivative with respect to t . We solved this system using the seventh-eight order continuous Runge-Kutta method. Thanks to its adaptive scheme, this methodprovides a great control upon the output accuracy. Typical solutions are plotted in fig.2. IGURE 2.
Left: The flow defined by system (7). Center: The flow near the stable node. Right: Theflow near the saddle point.
An important point here is that numerical solutions are not necessarily black boxes.Qualitative and asymptotic analysis help to understand the numbers which in turn mightguide the analysis. For instance, in this case we can use the theory of dynamical systemsto understand the contents of figures 2. This theory includes a set of general techniquesfor the analysis of phase spaces that allows us to understand better the asymptotics ofsolutions and their dependence on the initial conditions. This is a natural way of tacklingproblems about classical solutions in string theory; we are not aware of previous worksemphasizing these systematic methods; most of the literature emphasizes exploitingsymmetries. While of paramount importance in
AdS × S , most symmetries are lackingin the context of supergravity backgrounds dual to nonconformal theories.Setting the vector field to zero in system (7), we see that there are two fixed points.They are respectively located at ( , ) and ( √ , ) . If these singular points were hyper-bolic, then the Hartman-Grobman theorem [33] ensures that, in a small neighborhood ofeach hyperbolic fixed point, the flow defined by the full nonlinear system (7) is topologi-cally equivalent to its linearized version. A singular point is hyperbolic if the eigenvaluesof the Jacobian of the linearization around the point have non-zero real parts. For the firstpoint we obtained l = − l = −
2. This indicates that in its neighbourhood theflow behaves like near a radial sink (stable node), as it is shown in the center of fig.2.For the second singular point l = − + √ l = − − √
3. So, in its neighborhoodthe flow behaves like near a saddle point, as shown to the right in fig.2.Having tested that the numerical and analytical results coincide, we can move forwardand integrate (6) numerically. Taking L = r max , we evaluate the action with a set ofdifferent upper limits t max which are defined by r max . Then using dd L = r max ddt max , weestimate the cusp anomalous dimension using (5). The outcome for this case nicelyreplicates the result in [15]. Because of lack of space, we refrain of presenting it here,and move forward to the nonconformal cases. USP ANOMALOUS DIMENSIONS IN NONCONFORMALGAUGE THEORIESWitten QCD
In this subsection we calculate the cusp anomalous dimension for the Witten back-ground [16]. With that aim we rewrite the metric in the relevant IR regime. The ten-dimensional string frame metric and dilaton of this model are ds = ( uR ) / ( h mn dx m dx n + R u f ( u ) d q ) + ( Ru ) / du f ( u ) + R / u / d W , f ( u ) = − u u , R = ( p Ng s ) a ′ , e F = g s u / R / . (8)The geometry consists of a warped, flat 4-d part, a radial direction u , a circle parame-terized by q with radius vanishing at the horizon u = u , and a four-sphere whose volumeis instead everywhere non-zero. It is non-singular at u = u . In the u → ¥ limit the dila-ton diverges: this implies that in this limit the completion of the present IIA model hasto be found in M-theory. The background is completed by a constant four-form fieldstrength F = R w , where w is the volume form of the transverse S .The main gauge theory parameter we will use in the following is the KK mass scale1 / R q = m /
2, where m = u / R . As can be read from the metric, m is also the typicalglueball mass scale, and its square is proportional to the ratio between the confiningstring tension T QCD and the UV ’t Hooft coupling l . The supergravity approximationworks in the regime opposite to that in which the KK degrees of freedom decouple fromthe low energy dynamics. Condition T QCD ≪ m implies l ≪
1, which is beyond thesupergravity regime of validity.Assuming that u ( t , s ) = r ( t ) , with q ( r ) = ( u / R ) / , p ( r ) = ( R / u ) / × / f , andintroducing the new coordinate t = e t , so that e − t ¶ / ¶t = ¶ / ¶ t , the action (4) is now: S = R / pa ′ Z d s Z dt t r / q − ( ¶ t r ) / f . (9)A convenient way to write the equation of motion is:¨ r = (cid:2) r ˙ r − tr ( r − r ) ˙ r − r ( r − r ) ˙ r + t ( r − r ) (cid:3) r t ( r − r ) , (10)where r and t are now measured in units of R , and a dot stands for derivative withrespect to t . Note that r = const = r is not properly a fixed point of the action. A typicalsolution of this second order non-linear non-autonomous equation is presented in theleft of fig.3. As with the rest of the examples plotted in this subsection, it was obtainedby setting r = t = − , r ( ) = + − and ˙ r ( ) =
0. To understand this result letus analyze the asymptotic regimes.
IGURE 3.
Left: a typical solution for r in the WQCD model. Center: error of the approximate solution(12) compared to the numerical solution. Right: phase space for very large t . Dr stands for the derivativeof r with respect to t . Asymptotic behavior of the solution
First, we look for a solution for ( t − t ) → e = r ( ) − r →
0. Balancing equation(10) the relevant terms are, ¨ r = r ˙ r − ( r − r ) ˙ rt ( r − r ) . (11)For this balance to be consistent it is required the absolute value of the second derivativeto be greater than the neglected terms. It can be readily tested that this condition isfulfilled in the limits of e and t we are considering.We inmediately note that any solution to (11) with initial condition ˙ r = t = r = ( r + e ) + e ( t − t ) + e ( r e + r e + r e + e − r − r e − e ) t ( r + r e + e ) . . . (12)It can be seen that, in the limit e → r actually acts as a fixed point of the system. Wetested this result numerically. In the center of fig.3 we can see that the difference between(12) and a corresponding numerical solution is negligible. Thus, the approximationworks in the range of t and e considered.Let us now consider the asymptotic behavior of the solution for t → ¥ Now theequation of motion (10) becomes¨ r +
32 ˙ r ( r − r )( r − r ) r − r − r r = . (13)With regards to the consistency of this balance, from fig.3 it can be assumed that the large t behavior of the solutions is linear. Then, the second derivative goes to zero and that isalso true for the neglected terms, ( r ˙ r ) / ( t ( r − r )) − ˙ r / t . For further verification weagain compare the analytical and numerical results. Note now that the second derivativeis zero when the first derivative is ± ( r − r )(( r − r r ) / r ) whose limit as r → ¥ IGURE 4.
Left: fit to 1 / t of the argument of the squared root in the WQCD action. Center: plot of theintegral as function of t max . Right: fit of C as function of z. gives ±
1. So, for any r , some solutions converge to a linear solution with slope 1 andsome diverge from a linear solution with slope −
1. In the right of fig.3 the phase-spaceof (13) is plotted. We see that, indeed, for any r , ˙ r = r = −
1. Now, recall that we are analyzing the behavior for t → ¥ and, according to the analysis of the solutions when t →
0, the interesting us region is r > r and ˙ r >
0. It means that the interesting us asymptotic behavior is given by thelinear solution with slope equal to 1.Summarizing, the analysis and the numerical results for the range of r tested ( r ∈ [ − , ] and t = − ) indicate that the generic behavior of the solutions is as follows: r acts as a fixed point of this system; with an initial condition near r , the solution startswith a regime where r ≈ r . Next there is a transient regime where the solution smoothlychanges until it finally sets into a linear behavior with unit slope. The closer the initialcondition to the horizon, the larger the length of the initial regime and the shorter thetransition regime. The length of the time intervals are to be considered relatively, sincethe figures show always the same behavior if the right lengths of the t interval are chosen. The cusp anomalous dimension
Next, following our methodology, we evaluate the numerical solution in the actionand integrate it numerically. To accurately compute the actions we used an adaptive 3-5Simpson’s quadrature. If numerical factors are not taken into account, the integrand is, tr / q r ( − ˙ r ) − r − . We have already seen that r → t while t → ¥ . Therefore, if we expectthis integral to diverge logarithmically, we need the argument of the squared root tobehave like 1 / t . Indeed, using a best fit to the numerical results, we have verified thatthis is the case, and we present an example in the left side of fig.4. In the center a typicalresult for the action as a function of t max (cid:181) r max = L is plotted.The coefficient of dS / dlnr max is A √ C , where A stands for the numerical factors infront of the integral, C = r ( bz p + b + b z + b z + b z ) , z = r ( ) / r − p = . , b = . , b = . , b = . , b = − . , b = . r r ’ rr -2 = 10 r Starting Condition
FIGURE 5.
There is a fixed point of the Klebanov-Strassler equations of motion at r =
0. This plot isfor e = − . The numbers were obtained by the best fit C ( z ) shown in the right of fig.4. The warped deformed conifold
The Klebanov-Strassler background [6] is very well known in the literature of gravityduals to confining gauge theories, a full presentation of its metrics and action, suited fornumerical calculations can be found in the appendix A of [10].Using coordinates where t = e t and setting r ( t , s ) = r ( t ) , the equation of motion forthis background is ddt − e / h / ( r ) ˙ r tK ( r ) q − e / t K ( r ) ˙ r = (14) th / ( r ) s − e / t K ( r ) ˙ r ¶ r h ( r ) h ( r ) − e / ˙ r ¶ r K ( r ) t K ( r )( − e / t K ( r ) ˙ r ) . This equation was solved by setting r ( ) = e → r ′ ( ) =
0, and then using a fourthorder Runge-Kutta method for the evolution of the solution as a function of t = e t . Thenumerical integration necessary for evaluating the warp factor h ( r ) was done using acomposite Simpson method. A typical solution is presented in fig.5. An analysis similarto the previous subsections shows that there is a fixed point at r =
0. The solutioninitially stays close to the fixed point, then there is a transition to a logarithmic lookingfunction. The region close to the fixed point is larger depending on how close to the fixedpoint the solution begins.To show that the solution grows logarithmically, we examine the form of the metric forlarge t . In this regime, r is also getting large, so we have K ( r ) → B exp ( − r / ) , where is a constant. The KS metric now has the form ds = h − / ( r ) dx + h / e r / d r . Substituting r = e r / and performing a T-duality on the x coordinates, the metric is ds = h / ( r )( dx + dr ) . (15)Now, in the large r (large r ) approximation h ( r ) → A ( r − / ) / r , where A is aconstant, and one gets ds = p r − / dx + dr r (16)which is AdS with a logarithmic prefactor. To a first approximation, we can ignore theprefactor which will change slowly, to the metric, to see that the cusp solution of AdS which is r = constant × t will hold, which means that r ∼ log t for large t .This way, the result for KS resembles very much the one obtained for WQCD. DISCUSSION AND OUTLOOK
In this contribution we have shown that light-like Wilson loops in nonconformal theoriesdisplay some rather universal features. We hope that this is the kind of question that canbe asked and whose answer should be trusted to be in the same universality class asQCD. The light-like Wilson loops we have considered here are building blocks of thescattering amplitudes. We hope, as is the case in the perturbative treatment of QCDamplitudes, that the main contribution to scattering amplitudes will come from the cuspregion and therefore, our calculations will be a crucial building block in the amplitudes.It is important to note that, although not included in this contribution, we haveperformed similar analysis for various theories and the behavior is rather similar tothe one observed in the two examples considered explicitly here. Basically, there is anatural infrared regulator which is characterized by the distance between the positionof the brane and the characteristic strong scale which in most supergravity solutions isgiven by the region where a cycle collapses ( r for WQCD and r = ACKNOWLEDGMENTS
First, we wish to thank the organizers of the workshop "Ten years of AdS/CFT" forthe invitation to present our results in this important meeting. CAT-E is grateful to theCEFIMAS and the MCTP for hospitality while part of this research was undertaken.LPZ and DJP are partially supported by DoE under grant DE-FG02-95ER40899.
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