aa r X i v : . [ h e p - t h ] O c t Comments on operators with large spin
Luis F. Alday a,b and Juan Maldacena ba Institute for Theoretical Physics and Spinoza Institute
Utrecht University, 3508 TD Utrecht, The Netherlands b School of Natural Sciences, Institute for Advanced StudyPrinceton, NJ 08540, USA
We consider high spin operators. We give a general argument for the logarithmic scalingof their anomalous dimensions which is based on the symmetries of the problem. By ananalytic continuation we can also see the origin of the double logarithmic divergence in theSudakov factor. We show that the cusp anomalous dimension is the energy density for aflux configuration of the gauge theory on
AdS × S . We then focus on operators in N = 4super Yang Mills which carry large spin and SO(6) charge and show that in a particularlimit their properties are described in terms of a bosonic O(6) sigma model. This can beused to make certain all loop computations in the string theory. . Introduction In this paper we focus on two issues. First we discuss how the so called “cusp anoma-lous dimension”, f ( λ ), appears in various computations. Namely in the dimension of highspin operators and in lightlike Wilson loops with a cusp. These are well known relationsand our only objective here is to give a different perspective on these issues. First wegive a general argument for the logarithmic behavior of the anomalous dimension of highspin operators ∆ − S = f ( λ ) log S [1,2] which is based on the symmetries of the problem.Then we argue that the Sudakov form factor for two light-like particles has a behavior e − f ( λ )4 (log µ ) as a function of the IR cutoff [3,4,5,6] (for a review see [7]). This factor givesthe leading IR behavior when we consider the exclusive scattering of colored particles andit is an important ingredient in the computation of amplitudes [8]. Both of these propertiesfollow from symmetries of the theory plus the fact that we have gauge fluxes.For the particular case of planar N = 4 super Yang Mills, exact results were derivedusing integrability. In particular, an exact integral equation was written whose solutiongives the cusp anomalous dimension for arbitrary λ [9]. This equation was analyzed furtherin [10].In the second part of this paper we consider the question of high spin operators in N = 4 super Yang Mills that carry spin S and one of the SO(6) charges, J , in the large S, J limit such that J/ (log S ) is kept finite. In that case one can show that the anomalousdimension continues to have a logarithmic scaling∆ − S = (cid:20) f ( λ ) + ǫ ( λ, J log S ) (cid:21) log S (1 . ǫ was com-puted up to one loop in the strong coupling expansion. We derive an exact expression for ǫ in a suitable limit. We do this by noticing that the full AdS × S sigma model reducesto the O (6) bosonic sigma model in a suitable limit. In the O (6) sigma model we have aconfiguration with finite charge density whose free energy can be computed as a solutionof an integral equation [13]. The decoupling limit that gives the O (6) sigma model in-volves taking a strong ’t Hooft coupling and a small J/ (log S ) in such a way that quantumcorrections remain finite. Interestingly, the massive excitations of the O (6) sigma model,which are in the vector of SO(6), can be interpreted as insertions of the fundamental scalarfields φ I . 1his provides a way to compute higher loop corrections in the string theory sidewithout too much effort. These results could then be compared to a suitable generalizationof the BES equation [9] for finite J/ (log S ), which will probably be sensitive to higher ordercorrections of the phase of the fundamental magnon S matrix. (We do not perform thiscomputation here).This paper is organized as follows. In section two we explain the log S scaling of highspin operators using the symmetries of a conformal gauge theory. We then discuss therelated issue of the double logarithmic infrared divergencies in the Sudakov form factors.We also discuss how to simplify the strong coupling computation of f ( λ ) in N = 4 superYang Mills at tree level and one loop by using the symmetries we mentioned above. (Thesecomputations were originally performed in [14] and [11]).In section three we discuss high spin operators in N = 4 super Yang Mills and thereduction to an O(6) sigma model.Finally, in section four we present come conclusions.
2. High spin operators and Sudakov factors in conformal gauge theories log S scaling In this section we would like to offer a geometric argument for the logarithmic behaviorof the anomalous dimensions of high spin operators in gauge theories. Namely, we consideroperators with very high spin S → ∞ keeping the twist finite. For simplicity, consider firstoperators of the schematic form O S = ¯ q ( D ↔ ) S q (2 . q is in the fundamental representation. These operators have conformal dimensionsof the form [1,2] ∆ − S = f ( λ )2 log S (2 . S . The factor of 1/2 is a convention. Here we are considering a theory with alarge number of colors and we are disregarding the mixing between operators with differentnumbers of quarks. Alternatively, we could be considering the weakly coupled theory inwhich this mixing is suppressed in perturbation theory for the lowest twist operators.We can also consider an operator of a similar form in a theory with only adjoint fields,such as N = 4 super Yang Mills. In that case we consider a single trace operator of theschematic form T r [ φ I ( D ↔ ) S φ I ]. In the planar limit the high spin anomalous dimension2oes as ∆ − S = f ( λ ) log S , which is twice the value we had in (2.2) because, at large N we can view an adjoint particle a quark and an antiquark, each of which gives rise to (2.2).We now present an argument that explains the log S behavior in (2.2). A previousargument can be found in [2]. In a conformal field theory, the anomalous dimension ofan operator is equal to the energy of the corresponding state of the field theory on thecylinder R × S . On the cylinder, a high spin operator consists of two particles (or groupof particles) that are moving very rapidly along a great circle of S . These particles arecolored and the color field lines go between the two particles. See figure 1 . τ ϕ Fig. 1:
Quark and antiquark moving very fast on opposite sides of the cylinder.They become localized and can be replaced by light-like Wilson lines.
For simplicity, let us first consider the case of a quark and an antiquark moving veryfast along a great circle of the S , with color gauge fields joining them. Parametrizing thecylinder as ds R × S = − dτ + cos θdϕ + dθ + sin θdψ (2 . ϕ = τ and the antiquark at ϕ = τ + π ,both at θ = 0.Notice that if we had a color neutral object that is moving fast on the sphere, then itsenergy would go like S , namely ∆ − S ∼ finite, since we can get a particle which is movingfast along the equator from a particle that is at rest by applying conformal transformations.3n other words from an operator O , we can consider its descendent ∂ S O . In our case wehave a pair of particles and each particle carries color indices. In this case we have a largecontribution to ∆ − S from the color electric field lines emanating from the particles. Inorder to evaluate the effects of these fields, it is convenient to replace the quark and theanti-quark by a Wilson line. Thus, we first consider a Wilson line, which corresponds to S = ∞ , and then we go back to finite S . We consider a lightlike Wilson line at θ = 0 and ϕ = τ and an oppositely oriented line along ϕ = τ + π , see figure 1 . This configuration isclearly invariant under τ → τ + c, ϕ → ϕ + c where c is a constant. Less obvious is thefact that these Wilson lines are also invariant under a second symmetry, which acts as aconformal transformation which is not an isometry of R × S . As we will see, the log S behavior is associated to this second symmetry. In order to exhibit this symmetry moreclearly we can make a Weyl transformation of the R × S metric (2.3) to AdS × S bywriting ds R × S = sin θ (cid:20) − dτ + cos θdϕ + dθ sin θ + dψ (cid:21) = sin θds AdS × S ds AdS = − cosh ρdτ + sinh ρdϕ + dρ , sinh ρ = 1tan θ (2 . AdS × S . AdS × S is the space where thefield theory is defined and it should not be confused with the AdS space that will appearlater when we consider the gravity dual of the field theory. For the moment we are makingan argument purely in the context of the field theory and is valid regardless of the value ofthe coupling or whether the theory has a gravity dual or not. Thus we are considering thefour dimensional field theory on a four dimensional space which happens to be AdS × S .The Wilson lines, which sit at θ = 0, are mapped to a pair of lines along the boundary of AdS at ρ = ∞ . It is now convenient to introduce new coordinates where the AdS metrictakes the form ds AdS = − du + dχ − σdudχ + dσ (2 . AdS as the SL (2 , R ) group manifold parametrized as g = e iuσ e σσ e χσ (2 . A CFT can have Weyl anomalies that are local and thus should not affect the results forthe non-local part of the Wilson loop expectation value that we are considering. σ i are the usual Pauli matrices. In contrast, to get the metric in (2.4) we set g = e iσ ( τ + ϕ − π ) e ρσ e iσ ( τ − ϕ + π ) (2 . σ = − sin( τ − ϕ ) sinh 2 ρe iu = e i ( τ + ϕ ) cos( τ − ϕ ) + i cosh 2 ρ sin( τ − ϕ )cos( τ − ϕ ) − i cosh 2 ρ sin( τ − ϕ )sinh 2 χ = cos( τ − ϕ ) sinh 2 ρ q ( τ − ϕ ) sinh ρ (2 . χ → ±∞ and at σ = 0.Thus in the new coordinates (2.5), the two commuting non-compact symmetries of theproblem correspond to explicit isometries. Namely, to shifts in u and χ . In particular, wehave that the Hamiltonian in the new coordinates corresponds to ∆ − S = i∂ u . Note thatthe SL (2) L × SL (2) R isometries of AdS act on g as left and right matrix multiplication.The two commuting isometries corresponding to u and χ translations are embedded in SL (2) L and SL (2) R respectively.Since the Wilson loop is at the boundary, we end up with a configuration where wehave some color electric flux in the u, χ directions. It turns out that the flux is localized inthe direction σ due to the warp factor in (2.5). Thus the flux leads to a constant energydensity per unit χ and the energy is extensive in χ .Let us now explain in more detail why the energy is confined in the direction σ . Notethat for large σ the sinh 2 σ term in (2.5) dominates. For very large and positive σ wehave that ds ∼ e σ dudχ . Thus we can view u and χ as lightcone coordinates of a twodimensional space with a large warp factor or gravitational potential. Thus the flux ispushed towards smaller values of σ . For very large and negative σ we can make a similarargument. The conclusion is that the flux is concentrated around σ ∼
0. Note that thedirection ψ in (2.4) is compact, so that the flux cannot dissipate in this direction either. Inappendix A we consider explicitly the case of a U (1) gauge field and we show that indeedthe flux is confined to the region around σ ∼
0. The computation in appendix A alsoprovides a derivation for the one loop computation for the energy density and f ( λ ).The conclusion of this discussion is that the expectation value of the Wilson loop isdivergent because of the infinite extent of the χ direction, but it has a finite energy densityper unit distance in the χ direction, due to the fact that the flux does not dissipate due5o the gravitational potential in the σ direction which leads to a confining potential. Moreprecisely the energy, ∆ − S , for the configuration is∆ − S = 12 f ( λ )∆ χ (2 . / χ to S . First notice that the spin is an isometry of AdS in (2.4). The spin generator written in terms the coordinates in (2.5) has terms which golike e ± χ (see the explicit expressions in appendix A). In order to see this, note that 2 χ translations are conjugate to the generator σ (2.6), while the other SL (2) R generatorshave charges plus or minus one under the action of this generator. This implies that wehave exponentials of the form e ± χ . In the case that we have finite spin, we expect thatthe quark and the anti-quark are sitting around ± χ respectively. This would lead to aspin of the form S ∼ e | χ | , or ∆ χ = 2 χ = log S (2 . χ of a factor that grows like e χ . So, if the configuration has spin S we need to cut off this integral around e | χ | ∼ S . Then for a quark-antiquark high spinoperator we get ∆ − S = 12 f ( λ ) log S (2 . − S = f ( λ ) log S (2 . f ( λ ) / u, χ in the coordinates (2.5).We should also mention that one can consider an operator containing n fast movingpartons. In the planar limit, the flux joins neighboring partons and we have a contributiongoing like ∆ − S = n f log S if the spins of all partons are equal. This type of configurationswere studied in [15,16]. 6 .2. Finite N and relation to two dimensional QCD Let us make now some remarks about the finite N case. If we have dynamical quarks,or if we consider the adjoint case (2.12) then we see that we can nucleate colored particlesthat can screen the flux. In that case the energy no longer scales like log S . Of course, thisis energetically convenient only once f ( λ ) log S (or f ( λ )∆ χ ) becomes of order one. Thus,within the context of perturbation theory, where f ( λ ) is very small, we can ignore thisissue and argue that we have the log S scaling also for finite N . Note that the nucleationprobability goes as e − πf ( λ ) and it is very small as long as λ is small. However, for strongenough coupling the leading twist operators are not single trace operators. This is relevantfor deep inelastic scattering processes in strongly coupled field theories [17]. On the otherhand, in N = 4 super Yang Mills, we can consider the lightlike Wilson loop operator forfundamental external quarks. In this case, the flux configuration is protected by a Z N symmetry, the center of the gauge group. Thus in N = 4 super Yang mills we have awell defined problem in terms of which we can define f ( λ, N ) /
2, for arbitrary values of thearguments. We can also consider the strong coupling limit and relate this to a ’t Hooftloop. We can also consider the function f for higher representations. In the large N limitthe result is simply n times the result for the fundamental, where n is the number of boxesand anti-boxes for the representation. On the other hand, for finite N the result dependsonly on the N -ality (charge under Z N ) of the representation. Such configurations wereconsidered at strong coupling in [18] using D5 branes in AdS × S which are very similarto the ones appearing in the discussion of 1/2 BPS Wilson loops [19].In fact, it is interesting to consider the question of computing f for different rep-resentations at weak coupling. Once we view the problem in the coordinates we haveproposed(2.4), (2.5), we see that we can do a Kaluza Klein reduction on to the directionsparametrized by u and χ . The u -energies of various modes are given by the values of∆ − S . Thus we see that all modes in the field theory are massive except for the gaugefield along the u, χ direction. Thus we have a reduction to a 2d QCD problem. This leadsto an effective low energy action S = − g Z dudχT r [ F ] + · · · (2 . If N is finite the coefficient for the adjoint operator is not equal to twice the coefficient forquark-anti-quark operator. g = π g (cid:0) const ) g N + · · · (cid:1) (2 . g is the four dimensional coupling. In principle this can include planar as well asnon planar contributions .Notice that this effective field theory description is correct as long as there is largeseparation between the Kaluza Klein scale which is of order 1 and the 2d QCD scale whichis of the order of g N . This will be the case as long as we are at weak coupling. If the2d QCD lagrangian were the full description, then we would conclude that for a generalrepresentation R we have Casimir scaling for f , since that is what we get in 2d QCD [20] f g C ( R ) (2 . F . These operators were represented as dots in (2.13) . The firstoperators we can write down which are consistent with the symmetries of the problem are S = − g Z dudχT r [ F ] + cN Z T r [ F ] + c ′ Z T r [ F ] T r [ F ] (2 . c, c ′ are numerical constants. Such operators lead to a violation of Casimir scalingat four loops. Thus we have the prediction that in any theory we would get Casimir scalingup to three loops, and then at four loops we will get a violation of Casimir scaling. Wealso see that this effective field theory description breaks down when the 2d QCD scalegets to be of the order one, namely, when g N ∼
1. In that case we should consider thefull theory.
The argument given here for the logarithmic scaling of anomalous dimensions of highspin operators is completely geometrical and can also be generalized to other dimensionswere we can have conformal field theories with a gauge symmetry. For a field theoryin D dimensions we can do a Weyl transformation of the metric to write the metric as AdS × S D − and repeat the above arguments. In the case of D = 3 we get two copiesof AdS which are connected through the boundary conditions for the fields. In the twodimensional case, it might also be possible to find an argument (see [21] for a relatedproblem) but we leave this for the future. Note that if we start with a U ( N ) theory we have to distinguish between the g for SU ( N )and the one for U (1). .4. High spin limit of double trace operators Notice that a crucial part of the argument leading to the logarithmic scaling is thepresence of a conserved flux. In cases where we do not have a conserved flux we do not havea log S scaling. As an example, we can consider the behavior of double trace operators ina conformal theory, such as N = 4 super Yang Mills. Here we consider operators of theschematic form O d = O s ( D ↔ ) S O s , where O s are gauge invariant single trace operators.To leading order in N the dimension of these operators goes like ∆ − S = 2∆ s , where ∆ s is the dimension of the single trace operator. Here we consider the 1 /N correction to thisresult.The discussion we had above regarding the symmetries applies to this case too. How-ever, in this case we do not have a color flux along χ . Instead we are exchanging colorneutral states between the states created by the two operators O s . Let us first study whichstates can be exchanged. Our first task is to determine the energies of the possible states.The u -energies are simply given by ǫ = ∆ − S , where ∆ is the energy of the state in thecylinder. Thus we expect to find that we get a potential of the form V ∼ e − (∆ e − S e )∆ χ ,where ∆ e , S e are the conformal dimensions and spin of the exchanged particles . Using(2.10) we find that for large spin S we get∆ − S = 2∆ s + const N S (∆ e − S e ) (2 . e − S e . Notice that∆ e and S e are not large. In general the possible exchanged states are subject to selectionrules due to the symmetries of the operator O s , thus we cannot simply take the operatorwith lowest ∆ e − S e in the spectrum of the theory. However, we can always exchange thestress tensor operator, which has twist ∆ e − S e = 2. Thus, the power of S in (2.17) isbounded by 1 /S . Notice that this result is valid whether or not the gauge theory has an AdS dual or not. Our argument is purely field theoretic and it is valid for any conformaltheory, including non-gauge theories.The result (2.17) agrees with the more detailed analysis performed in [22,23] whichused the gravity description . The more detailed analysis of [23] makes it possible to In performing this argument we have implicitly assumed that the operator that performs χ translations has eigenvalues which are related to the ones of the operator performing u translations.This can be understood from analytic continuation in the metric (2.5). See section 4.6 in [23] and take h ∼ S , ¯ h ∼ small, ∆ → ∆ e , j → S e . S for large spin is fixedby this argument .As we will see below, these results also explain why do not get double log Sudakovdivergences in theories without conserved fluxes, such as the φ theory in six dimensionsconsidered in [7].In summary, the reason for the log S is simply that the configuration develops anadditional symmetry in the infinite spin limit. This symmetry becomes manifest whenwe do a Weyl rescaling of the metric. We then see that the energy is extensive along thecoordinate conjugate to this symmetry generator. For finite spin, we have only a finiterange for this coordinate, a range proportional to log S . Another issue that can be understood by performing Weyl transformations in themetric and coordinate redefinitions is the behavior of soft divergences in scattering ampli-tudes. It is well known that exclusive scattering amplitudes of colored (or charged) haveinfrared divergencies due to the emission of low energy gluons (or photons). These IR di-vergences disappear when we consider a physical observable (see [8] for example), but aresometimes replaced by explicit dependence on detector resolutions or parameters enteringin the definition of jet observables. For this reason a great deal of effort was devoted tounderstanding the structure of these divergences. These divergences can be resumed intoan expression of the form [3,4,7,5,6,24]
A ∼ e − h ( λ )(log µ IR ) − h ′ ( λ ) log µ IR (2 . h is some function of the coupling. In a planar gauge theory the color of each gluonis correlated with the anticolor of the next and so on. For each consecutive pair we geta factor of the form (2.18), with a function h given by f /
4. These double logarithmicdivergences can be computed by replacing the hard gluons by Wilson loops [4] (see also[25] for a more systematic discussion). In particular, we have light-like Wilson loops witha cusp. We would like to see that this cuspy Wilson line in the fundamental representationhas a behavior of the form h W i ∼ e − f ( λ )4 (log µ IR /µ UV ) (2 . Maybe is possible to make a general field theory argument for all the operators consideredin [23]. µ IR and µ UV are the UV and IR cutoffs. This behavior depends on how we introducethe UV cutoff. Here we are introducing it in the way that arises when we consider scatteringamplitudes, where the UV cutoff appears when we can no longer replace the hard particlesby Wilson lines . In other words, imagine we have two gluons coming out of an interactionregion around the origin with momenta ( k − ,
0) and (0 , k + ). We can then replace the gluonwith momentum k − with a Wilson line along x + = t + x > x − given by ∆ x − ∼ / | k − | . the other gluon gives rise to a Wilson line along x − = t − x > ds = ds AdS × S = − dt + dx + dr r + dψ = ds R , r (2 . t ± xr = sin αe ± γ , r = cos αe − τ ds AdS = − dα + sin αdγ + cos αdτ (2 . τ = −∞ and χ → ±∞ . These coordinates cover only aportion of AdS , namely the region 0 < t − x < r . We can cover other regions by goingthrough α ∼ α = π/ α → iα ′ , we go to the region t − x < α → iα ′′ + π/ t − x − r > t = x = r = 0 in AdS .The region parametrized by (2.21) intersects the boundary along t = ± x >
0, whichcoincides with the cuspy Wilson loop we want to consider. This is a time dependentbackground and we consider first the ordinary
AdS vacuum which leads to a particularstate at both the future and past horizons in (2.21). Then, starting and ending with sucha vacuum, we insert a Wilson line operator at α = π/ τ = −∞ and χ → ±∞ . We thenhave flux lines joining these two asymptotic regions. We now note that we can get themetric in (2.21) by performing an analytic continuation of the metric in (2.5) α = iσ + π/ , τ = iu − χ, γ = iu + χ (2 . If we were to choose a boost invariant UV and IR regulator we would get a divergence dueto the boost invariance. The UV regulator we choose here is not boost invariant. σ = 0 corresponds to α = π/
4. This implies that the fluxconfiguration that appears in the Sudakov computation leads to a factor A = e − f ( λ )4 R dτdγ (2 . f is the same as the function that appeared above (2.9). The factor of four in(2.23) is due to the change in measure R dudχ → − i R dτ dγ . This analytic continuationis also responsible for the fact that we get a real exponent in (2.23). Thus, this analyticcontinuation explains the connection between the f that appears in the Sudakov factorand the function f that appears for high spin operators [4]. A similar analytic continuationwas also used in [26] to argue, from the AdS side, that the same function appears on bothcalculations.Now we need to relate the range of τ and γ in (2.23) to the IR and UV cutoffs. Weexpect the IR cutoff to be boost invariant so that we get τ < − log µ IR . If we want torelate this computation to the computation of gluon scattering amplitudes, then the UVcutoff is not boost invariant since the approximation of replacing a hard gluon by a Wilsonline fails if we do a very high boost since the gluon ceases to be hard. Thus, we get aconstraint of the form τ UV ≡ − log µ UV < τ ± γ (2 . µ UV ∼ − s . Thus we are integrating overa triangular shaped region. The regions near each boundary in (2.24) correspond to thecollinear regions. Near these boundaries we can replace only one of the gluons by a Wilsonline but not both of them. Thus, we are integrating over a domain of the form Z dτ dγ = Z τ IR τ UV dτ Z τ − τ UV − ( τ − τ UV ) dγ = (cid:20) log µ IR µ UV (cid:21) A ∼ exp ( − f (cid:20) log µ IR µ UV (cid:21) ) , τ UV = − log µ UV , τ IR = − log µ IR (2 . τ . Thus weget an additional term of the form [5]exp (cid:26) g ( λ ) Z τ IR τ UV dτ (cid:27) (2 . We have the cutoffs ∆ x ± ∼ / | k ± | . Since the final answer will be boost invariant we cantake ∆ x ± ∼ /µ UV , with µ UV = − s .
12e should also note that using similar reasoning, but now considering the matchingbetween different regions of
AdS in (2.20) that we discussed above, we could consider anon-lightlike cusp Wilson line with a boost angle γ . For a large angle γ , then we expectan expression of the form h W i ∼ e − f ∆ χ R dτ ∼ e − f γ log µUVµIR , ∆ χ = γ (2 . γ for large γ . The reader should not be confused by thefact that one calls f a “cusp anomalous dimension”, though the scaling with the cutoff isdifferent for a light-like cusp (2.19) than for the original, non-lightlike, cusp discussed in[27], which has a single log. It is also possible to understand the above behavior in non-conformal theories. Herethe new element is that the gauge theory has a scale, which we parametrize with Λ.We can think of this scale as the UV cutoff, where we set the coupling constant. Thecoupling can then be evolved to a lower scale via the RG equation. Then, as we do aWeyl transformations of the metric by some function Ω( x ), ds = Ω ( x ) ds ′ we find thatthe new theory in the new metric, ds ′ , has a scale Λ ′ = ΛΩ( x ). This means that in thetheory on the new space ds ′ we are setting the coupling to a constant at the scale Λ ′ which depends on the position. If we now were to choose a constant cutoff Λ ′′ in the newtheory we would find that the coupling on the new cutoff scale Λ ′′ is not constant but x dependent. This dependence can be obtained by solving the RG equation to relate theconstant coupling at scale Λ ′ ( x ) to the scale Λ ′′ . As long as the starting Λ is sufficientlylarge, the value of Λ ′ is large enough so that the dependence on x is be very slow and wecan use the ordinary RG equation for a constant coupling. In appendix B we discuss moreexplicitly the simpler case where we map a non-conformal theory between the plane, R ,and the cylinder R × S .In our case, in (2.20) we find that Ω ∼ r and thus the new scale isΛ ′ = Λ r = Λ e τ cos α (2 . α dependence is not too important for us. The τ dependence implies that the Hamil-tonian generating τ translations is τ dependent and the value of the Wilson loop involves13n integral over τ . When we do that integral, it is important to remember that the rangeof γ also depends on τ and the UV and IR cutoffs. Thus, when we repeat the abovecomputations we find that g and f are the τ dependent eigenvalues of the τ -Hamiltonian.They depend on the coupling, which itself depends on τ through the β function equation.If we use dimensional regularization in order to cutoff the IR divergencies, then we shoulduse the β function in 4 + ǫ dimensions to determine the τ dependence of the couplingconstant. The bottom line is that we obtain an expression similar to the above one but thefunction f ( λ ( τ )) depend explicitly on the “time” τ and should placed inside the integrals.These expressions agree with the expressions in [5,24].If we concentrate on the leading dependence on the IR cutoff, then we are integratingin τ up to τ = − log µ IR and for each value of τ the range of γ is ∆ γ ∼ µ UV − τ ).Thus we have h W i ∼ e − R − log µIR − log µUV dτ f ( λ ( τ ))4 ∆ χ ( τ ) ∼ e − R − log µIR − log µUV dτ f ( λ ( τ ))4 µ UV − τ ) (2 . AdS/CF T one can consider the computation of Wilson loops with cuspsin non-conformal theories. This was done for the Klebanov-Strassler cascading theory in[28]and in 4 + ǫ dimensions in [29]. We now consider N = 4 super Yang Mills at strong coupling and analyze it using thegravitational dual. From our general discussion we concluded that f ( λ ) can be computedin terms of a light-like Wilson loop. It is convenient to slice AdS in coordinates wherethe boundary is manifestly AdS × S ds = cosh ζds AdS + sinh ζdψ + dζ == cosh ζ (cid:2) − du + dχ − σdudχ + dσ (cid:3) + sinh ζdψ + dζ (2 . AdS and the boundary sits at ζ → ∞ .In the gauge theory we considered a configuration with flux in the uχ direction (2.5).This gives rise to a string extended along the uχ directions of AdS . The warp factor inthe ζ direction pushes the string to ζ = 0, which is a U (1) ψ symmetric point. In additionthe warp factor in the σ direction pushes the string to σ = 0.14here are some interesting features of this string. First, its tension gives us the energydensity. Thus, simply the tension of the string gives us the strong coupling behavior forthe cusp [14,30] f T πα ′ = R πα ′ = √ λ π (2 . u, χ directions. This is not a symmetry of the full problem, but it is a symmetry of the theoryat the quadratic level. We can easily find the bosonic excitations and we can computetheir masses. We find that there are five massless goldstone bosons associated to themotion on S . The oscillations in the σ direction are described by a massive goldstonefield with mass m = 4 that comes from the SL (2) L symmetries that the string breaks .In other words, the creation and annihilation operators for the modes of the σ field on theworldsheet come from J ± L in SL (2) L , recall that 2 J L = i∂ u is the energy. This correspondsto oscillations in the σ direction inside AdS . Then there are two bosons of m = 2associated to motion in ζe iψ , these are not obviously Goldstone bosons. Nevertheless onecan view them as Goldstone bosons according the following heuristic argument. The fulltheory has conformal symmetries which are not isometries of AdS × S . In particularwe have conformal generators in the spin ( , , ±
1) under SL (2) L × SL (2) R × U (1) S .These symmetries are broken by the string. They create modes with wavefunctions ofthe form e − iu ± χ . We see that these are exponentially growing in the χ direction andthus would carry momentum p = ± i . Thus, the dispersion relation should be such that ǫ ( p = ± i ) = 1. If the dispersion relation is relativistic, then we get m = 2. Thisargument is heuristic because we are talking about non-normalizable modes. Now let usturn to the fermions. All the fermions have the same mass since they have to transformunder the spinor representation of SO (6) and the lowest dimensional representations havefour complex dimensions, corresponding to eight real fermions. They all have m = 1 whichcan be viewed again as goldstone fermions. Their mass is fixed by the transformationproperties of the supercharges under SL (2) L , where SL (2) L are the left isometries of AdS .The advantage of viewing them as Goldstone bosons or fermions is that their energies stay Goldstone bosons can be massive when the broken symmetry does not commute with theHamiltonian. . ¿Fromthis spectrum of masses it is straightforward to compute the vacuum energy and we obtainthe one loop contribution [11] f Z ∞−∞ dp π h | p | + p p + 4 + 2 p p + 2 − p p + 1 i = − π (2 .
3. The O(6) sigma model from string theory
In this section we consider further the worldsheet theory describing the string as-sociated with highly spinning operators or lightlike Wilson loops. We consider a stringstretched along the u, χ coordinates in (2.30), and we work in static gauge. Notice thatthe theory is not invariant under boosts in the χ and u directions, but it does becomeboost invariant at low energies. In fact, the spectrum we discussed above is precisely boostinvariant. Let us now imagine taking a low energy limit where we look at the system atdistances (in χ ) much larger than one, which is the mass of the fermions and the orderof magnitude of the mass of the massive bosons. In this case only the massless excitationssurvive. In two dimensions, massless fields have large fluctuations which lead to interestingdynamics in the IR. In our case, the massless fields describe an S . In other words, theydescribe the O (6) sigma model. This is a model where the coupling becomes strong in theIR and the theory develops a mass gap. Moreover, this is an exactly solvable theory [31].The scale set by the mass of the fermions (and massive bosons) acts as a UV cutoff for theO(6) sigma model, where the O(6) theory merges into the full AdS × S sigma model.When we compute the cusp anomalous dimension at strong coupling we are computingthe vacuum energy of this theory. Of course, the vacuum energy in the O (6) sigma modelis UV divergent. In our case, this UV divergence is cut off at the scale where the fermionsstart contributing. We can see this explicitly in the one loop result (2.32) . Thus thevacuum energy does not seem to have a clear contribution that comes purely from the O (6) sigma model. A two loop computation was attempted in [32]. We will see that the 5 massless modes will actually get a mass non-perturbatively in the α ′ expansion and the SO(6) symmetry will be restored. O (6) model. First, notethat the massive excitations of the theory transform in the vector representation of O (6),thus it is natural to identify them with the fundamental scalars φ I of the gauge theory. Itlooks like such excitations should appear naturally as we compute the spectrum of chargedoperators around the lowest twist high spin operator.This relation to the O(6) sigma model allows us to perform all orders computationsin AdS by focusing on the right observable. We choose an observable which receives mostof its contribution from the low energy region described by the O (6) sigma model. Thereis an interesting concrete set of operators that has been studied recently [11,12] which hasa limit that can be explored in terms of the O (6) theory. These are operators which carrylarge spin S and also large charge J , where J scales like log S . In other words, considersingle trace operators in planar N = 4 super Yang Mills with S, J → ∞ , j ≡ J S = fixed (3 . S to the extension of the string (or corresponding field theory configura-tion) in the χ direction we see that in this limit we have a configuration with finite currentdensity along the χ direction on the worldsheet. We then conclude that the anomalousdimensions scale as log S and thatlim S →∞ ∆ − S log S = f ( λ ) + 2 ǫ ( λ, j ) (3 . f ( λ ) is the cusp anomalous dimension and 2 ǫ ( λ, j ) is the additional energy due tothe SO(6) charge density j . Note that ǫ ( j = 0) = 0. The argument of the previous sectionimplies that this is the right scaling for all values of λ and j .Note that at strong coupling we can consider the same string we discussed above,which is stretched along the u and χ directions. The string carries a current densityproportional to j since 2 log S = ∆ χ is the length of the folded string corresponding to asingle trace operator of spin S . Similarly the factor of 2 in (3.2) is chosen so that ǫ ( j ) isthe energy density along a single string stretched along the χ direction.We can compute ǫ ( j ) using the O (6) sigma model in the regime where the characteristictime variations of the angular coordinates are much smaller than the mass of the fermions.We have j = √ λ π ˙ ϕ (3 . ϕ ≪
1, thus we want j ≪ √ λ for starting to trust the O (6) results. Theclassical sigma model result is ǫ ( j ) = √ λ π ˙ ϕ = π √ λ j (3 . m = kλ / e − √ λ [1 + o ( 1 √ λ )] (3 . k is a constant that depends on the details of how the O (6) model is embedded inthe full AdS × S string sigma model. This formula is valid at large √ λ ≫ O (6) sigma model can be suitably decoupled from the rest.Let us specify more precisely the decoupling limit that gives the O(6) model. We takethe limit S → ∞ with j fixed (3.2). We then take the limit λ → ∞ , j → , with jm = jk − λ − / e √ λ = fixed (3 . ǫ ( j ) = j E ( j/m ) (3 . E can be determined purely in terms ofthe O (6) sigma model [13]. For large j/m this function can also be expanded using O (6)perturbation theory. Thus, we can use the O (6) results to compute the α ′ expansion ofthis observable.The problem of computing the energy of a configuration with constant current densitywas considered in [13]. These authors derived an integral equation determining the energyas a function of the chemical potential h for the charge j . They found that F ( h ) = h F ( h/m ) = − h (cid:18) − β log( h/m ) + β β log log( h/m ) + c + ˜ c log log( h/m )log( h/m ) + ... (cid:19) (3 . β and β can be related to the one and two loop beta functioncoefficients in the O(6) theory. Of course the first two terms can also be easily computedusing perturbation theory in the sigma model. But the computation of c amounts toa computation of the mass gap which is more involved [13]. In appendix C we review[13] and give the values of these coefficients for an O(N) model. In appendix C we alsoshow that the structure of the perturbative series can be used to fix the coefficients of the18ogarithmic terms (for example ˜ c ). The energy density discussed above can be computedby first computing j from (3.8) and then performing the Legendre transform to get ǫ ( j ) = F ( h ) + jh . In this fashion we can obtain the function E in (3.7).This gives a precise prediction that could be used to test the BES/BHL [33,9] guess forthe S-matrix to all orders in the α ′ expansion, and probably in an exact way. Hopefully, allthat one needs to perform this comparison is to have a closer look at the Bethe equationsin the SL(2) subsector and repeat the steps in [34,9,35] keeping J/ log S constant. O (6) free energy and comparison to string theory One loop string theory computations on a related regime have been considered byFrolov, Tirziu and Tseytlin in [11]. They considered closed string configurations with y ≡ π √ λ j (3 . . As a check of what we have been saying we show that the limit of small j , y of the formulas in [11], match precisely the expectation from the O(6) side. In addition,using their computation we can fix the coefficient k in (3.5), which is sensitive to thresholdcorrections.In the small j limit their tree level and one loop results read ǫ ( j ) = ∆ − S − f ( λ ) log S S = y √ λ π − π log y + 34 π ! (3 . j , j/m ≫ j ≪ √ λ , so that the O (6) description isvalid), and using the relations (3.5), (3.8) and (3.9), we can expand the energy density inpowers of √ λ . We find ǫ ( λ, y ) = y " √ λ π + β y ) + k , k = − c − β log k + β β log( − πβ )(3 . y term whose coefficient depends on β . We see that for thecorrect O(6) value, β = − /π , we obtain agreement with (3.10) [11]. The constant piece, k , depends on k in (3.5). This is a quantity that we cannot determine by purely O(6) They wrote their results in terms of x , with x = 1 /y . k = 2 / Γ(5 /
4) (3 . y fixed and arbitrary, while herewe are just focusing on the region y ≪ O (6)theory. Indeed, the coefficient of the log y was determined by O (6) but not the constantpart. Here we consider higher loop predictions for the energy density. At higher loopsand very small y , the most important terms in the expansion are the logarithmic terms.Such terms are determined by the renormalization group equations in terms of lower orderterms. The expression (3.8) is the expansion of the free energy for the first two ordersin perturbation theory. This expression also determines the β function at one and twoloops. We thus expect that using the renormalization group we could determine the twoleading logs at each order in perturbation theory. Of course, we could just blindly expandthe expression (3.8) and directly see that the two leading logs are determined. Of course,these logs are renormalizing the coupling from the UV scale where it is given in terms of √ λ down to the scale set by y . However, since the coupling constant expansion in the fullsigma model is 1 / √ λ we might still wish to do an expansion in powers of 1 / √ λ . In thatcase we can give the expression for the first two leading log corrections at each order (afterwe used (3.12)), ( ǫ = P ∞ k =0 ǫ k ) ǫ ( y ) = y √ λπ (cid:0) y − y + · · · (cid:1) ǫ ( y ) = y πλ (cid:0) −
16 log y + 6 log y + · · · (cid:1) ǫ ( y ) l +1 = y πλ l/ (cid:16) ( − l +1 l log l +1 y + ( − l l − l + 1)(1 − h ( l )) + l ] log l y + · · · (cid:17) (3 . h ( l ) = P ln =1 /n is a harmonic sum.It would be nice to see whether these terms, which are easily computed, contain anyinformation about the higher order corrections for the dressing phase [36,33,9] when one20omputes this energy using a suitable generalization of the techniques in [34,9,35]. If thatis the case, then one could test the higher order terms in the dressing phase. If the leadinglogs do not depend on the higher order corrections to the phase, then one would be forcedto consider higher order corrections in the O(6) sigma model, which are still much easierto compute than higher order corrections in the full AdS × S string sigma model.We should note that at each order in the 1 / √ λ expansion there are also terms whicharise from higher order threshold corrections in (3.5). Such terms are not calculable in thepurely O (6) theory. These higher order corrections disappear if we take the decouplinglimit (3.6). Only the constant k in (3.6) appears, and we have already fixed it in (3.12)using the results in [11]. j limit Note that our discussion makes sense even non-perturbatively in the O (6) coupling.Thus, we can consider extremely small values of j . In this regime we have a very lowcharge density, so we have well separated massive particles in the O (6) theory thus makethe simple prediction that ǫ ( j ) ∼ mj , j ≪ m (3 . m is given in (3.5). This is valid for λ ≫
1. Thus we have particles that transform inthe vector representation of O (6). This is reminiscent of what happens at weak coupling, λ ≪
1, where for low j we also have an answer linear in j , ǫ ( j ) ∼ j . This is simply thestatement that the scalar fields φ I , which carry the SO (6) charge contribute one unit tothe twist (∆ − S ). We see that, for very small j , the functional dependence on j is thesame at weak and strong coupling. However, the coefficient is very different. The smallvalue of the coefficient at strong coupling signals the existence of the region where thephysics is described by the O(6) sigma model, since there is large difference between themass gap of the O (6) particles an the mass of the fermions, which sets the scale where the O (6) theory breaks down. At weak coupling, λ ≪
1, there is no such large separation andit would be wrong to use O (6) formulas to compute the energy.It would be interesting to see if one can get similar reductions to an O(4) or O(3)sigma models by considering strings in AdS × S or AdS × S .21 . Conclusions In this article we have presented a simple picture for the cusp anomalous dimension.This is a quantity that appears in various computations in gauge theories. We found itconvenient to perform a Weyl transformation of the metric from R , to AdS × S , whichsimplifies the action of the symmetries that determine the form of the results. The cuspanomalous dimension becomes the energy density of a certain flux configuration of thegauge theory on AdS × S . It is a flux configuration that is invariant under two non-compact translation symmetries. These symmetries explain the logarithmic behavior ofcertain quantities. For example, the logarithmic behavior of the dimension of high spinoperators, ∆ − S = f ( λ ) log S , arises when the configuration has a finite extent along thecoordinate conjugate to one of the translation symmetries. Similarly, the flux configurationassociated to the Sudakov factor arises after performing an analytic continuation and thedouble logarithmic behavior arises from imposing a finite range for both of the coordinatesconjugate to non-compact translation symmetries. We have also discussed how to obtainthe spin dependence of anomalous dimensions of double trace operators and we found thatit is given by a power determined by the twist of the lowest twist operator that couples toeach single trace operator (2.17). We also explained how the weak coupling computationof the cusp anomalous dimension can be reduced to an effective two dimensional QCDproblem and argued that the cusp anomalous dimension for arbitrary representations dis-plays Casimir scaling up to three loops. The arguments are made on the field theory sideand are valid for any conformal gauge theory regardless of the value of the coupling. Wehave discussed also the extension to the non-conformal case for the case of the Sudakovfactor.We then considered operators with high spin and charge in N = 4 super Yang Millsand argued that in the limit where J/ log S is finite we get ∆ − S = [ f + ǫ ( J/ log S )] log S [11]. We showed that when J/ log S is suitably small the computation of the function ǫ ( J/ log S ) reduces to a computation in the bosonic O(6) sigma model. This relation givesus a way of obtaining exact results for the worldsheet string theory. These can, hopefully,be used to test the BHL/BES [33,9] prediction for the phase of the S-matrix at higherloops. Acknowledgments
We would like to thank S. Ellis, G. Korchemsky, A. Manohar, G. Sterman, E.Sokatchev and E. Witten for discussions. 22his work was supported in part by U.S. Department of Energy grant
Note Added:
After this paper appeared, [37] computed the strong coupling expansion of the cuspanomalous dimension from the BES equation [9], finding exponentially small correctionsthat precisely agree with m , with m given by eqn. (3.5). Since the cusp anomalousdimension has the interpretation of an energy density, we expect this kind of correctionsbesides the perturbative series in 1 / √ λ .
5. Appendix A: One loop computation for the cusp anomalous dimension
In this appendix we outline the one loop computation of the cusp anomalous dimensionusing the coordinates we introduced above. Of course, this is a well known computationthat has been done in many ways. We perform the computation by considering a Wilsonloop in the coordinates (2.5). We consider first a U (1) theory, and start with a configurationwith an electric flux which is non-zero only in the u, χ directions, F uχ , which is constantdue to the Bianchi identity. We can write the action as S act = − g Z d x √− gF µν F µν = | F uχ | g Z dψdudχdσ cosh 2 σ = π g Z dudχ | F uχ | (5 . ψ in (2.20). The total energydensity is f π | F uχ | g (5 . F can be obtained in the standard way after we say thatfundamental charges couple as e i R A and compute the amount of flux that this chargegenerates. This gives F uχ = g π . We then obtain f g π (5 . U ( N ) theory. In that case we get a similarresult except that we get an additional factor of N in (5.3) and there is an additional factorof two that comes from the conventional definition of the coupling. Alternatively, we canreduce the four dimensional theory to two dimensions along ψ and σ . This leads to a twodimensional theory with a coupling g = g /π . Then we can use the two dimensionalQCD result (2.15) with C = N/ f / g N π . 23 .1. Spin generators in the new coordinates As we see in the metric (2.4) the generator that measures spin, − i∂ ϕ corresponds toan isometry in AdS . We can now compute the form of this Killing vector in the other AdS coordinates (2.5)∆ = 12 (cid:18) χ cosh 2 σ (cid:19) i∂ u + 12 cosh 2 χ tanh 2 σi∂ χ + cosh χ sinh χi∂ σ S = 12 (cid:18) − χ cosh 2 σ (cid:19) i∂ u + 12 cosh 2 χ tanh 2 σi∂ χ + cosh χ sinh χi∂ σ (5 . − S = i∂ u . We see from theseexpressions that if we have a configuration which goes up to some distance χ , then itsspin scales as S ∼ e | χ | . We can understand this better if we compute the contribution tothe spin of the constant flux configuration discussed above. Given a general Killing vectorof the form V = iξ µ ∂ µ the conserved current associated to it is given by contracting ξ µ with the stress tensor. Using the expression for the stress tensor for a gauge field we find,after integrating over ψ in (2.5),∆ = πg Z dσdχ F uχ cosh 2 σ (cid:18)
12 + cosh 2 χ σ (cid:19) S = πg Z dσdχ F uχ cosh 2 σ (cid:18) −
12 + cosh 2 χ σ (cid:19) (5 . χ ∼ χ . However, we argue that the dynamical particles that we put at χ ∼ χ still have a spin of order S ∼ e | χ | . In this fashion we connect the range of χ to the spin,via ∆ χ = log S .
6. Appendix B: Renormalization group and evolution on the cylinder
A conformal field theory on the plane is equivalent to a conformal field theory on thecylinder and the spectrum of anomalous dimensions of operators on the plane correspondsto the energy spectrum for the theory on the cylinder. Now, suppose that we have a non-conformal field theory on the plane. For simplicity, imagine we have a theory with a singlecoupling that runs g ( µ ). Then we would like to understand what type of theory we geton the cylinder. 24or simplicity, consider the Euclidean theory. Then the plane and the cylinder arerelated by the following Weyl transformation ds R = r (cid:0) dτ + d Ω (cid:1) = r ds R × S , r = e τ (6 . ds = Ω ds ′ , Ω = r = e τ . Let us imagine regularizingthe field theory on the plane with a cutoff Λ so that the value of the coupling at the cutoff isconstant. On the cylinder this leads to a field theory where the cutoff Λ ′ = Ω( x )Λ dependson position. In our case, this leads to a cutoff Λ ′ = e τ Λ which depends on the Euclideantime direction along the cylinder. The coupling constant on the cylinder is a constant atscale Λ ′ . This can be related to the more conventional way of defining the theory on thecylinder which uses a fixed (time independent) cutoff Λ ′′ . We can obtain the value of thecoupling at Λ ′′ by using the renormalization group equation to evolve the coupling fromthe scale Λ ′ ( τ ) to the scale Λ ′′ . We can apply the ordinary flat space renormalizationgroup equation as long as the coupling is varying slowly at the scale of the cutoff. Thiscondition reads ∂ τ Λ ′ / Λ ′ ≪ Λ ′ . In our case this requires that e τ Λ ≫
1. This says that thecutoff Λ ′ should be bigger than the inverse radius of the S . Thus, if we want to explorethe theory at τ → −∞ (or r → τ . The time dependence of the coupling constant can be computedexactly if we know the exact β function. Since we have a time dependent theory on thecylinder we have a time dependent Hamiltonian. We can nevertheless diagonalize thisHamiltonian at each time and this leads to the scale dependent anomalous dimensions wehave in a non-conformal theory ∆( g ( τ )).For theories that have a gravity dual, it is useful to understand this also from thegravity perspective. Let us start with a five dimensional metric and scalar field ds = w ( z ) dz + dx z , φ ( z ) (6 . AdS case, which is1 z = e − τ cosh ρ , rz = sinh ρ (6 . ds = w (cid:18) e τ cosh ρ (cid:19) (cid:2) cosh ρdτ + dρ + sinh ρd Ω (cid:3) , φ (cid:18) e τ cosh ρ (cid:19) (6 . z = 1 / Λ.This cutoff would then correspond to e − τ cosh ρ c ( τ ) = Λ, where ρ c ( τ ) is determined bythis equation. From the point of view of the new theory, we would say that the cutoff isat Λ ′ ∼ e ρ c ( τ ) / ∼ Λ e τ if ρ c is large. Thus we get Λ ′ = e τ Λ as we had in the generaldiscussion. Notice that the condition that the time variation was slow, translates intothe condition that ρ c ≫
1. If we fix the cutoff at a time independent value Λ ′′ = e ρ ′′ c / φ c ∼ φ (2 e τ − ρ ′′ c ).
7. Appendix C: O(N) sigma model
In this appendix we recall some results for the O ( N ) non linear sigma model. Weconsider the O ( N ) sigma model in the presence of a chemical potential h coupled to oneof the conserved charges (an SO (2) ⊂ SO (6)) and we compute the free energy f ( h ) = min j [ ǫ ( j ) − jh ]. By dimensional analysis f = h F ( h/m ), where m is the mass gap.Given the two particle S − matrix for the O ( N ) σ − model S ( θ ) = − Γ(1 + x )Γ(1 / − x )Γ(1 / x )Γ(∆ − x )Γ(1 − x )Γ(1 / x )Γ(1 / − x )Γ(∆ + x ) x = iθ π ∆ = 1 N − . f ( h ) = − m π Z B − B cosh( θ ) ρ ( θ ) dθ (7 . ρ ( θ ) satisfies the following integral equation with the following boundary condition ρ ( θ ) − Z B − B K ( θ − θ ′ ) ρ ( θ ′ ) dθ ′ = h − m cosh( θ ) , ρ ( ± B ) = 0 (7 . K ( θ ) = πi ddθ log S ( θ ).In order to make a comparison with computations from the string sigma model, oneneeds to consider the regime h/m ≫ f ( h ) = − h (cid:18) − β log( h/m ) + β β log log( h/m ) + c + ˜ c log log( h/m )log( h/m ) + ... (cid:19) β = − N − π , β = − N − π , c = N − π log "(cid:18) e (cid:19) N − e − / Γ(1 + N − ) (7 . β , turn out to be the one and twoloop beta functions for the O ( N ) sigma model. Next, we perform a Legendre transformand express ǫ = f ( h ) + jh = j E ( j/m ) in terms of the charge density j ≡ − f ′ ( h ). Startingfrom (7.4) it is straightforward to iteratively solve for E ( j/m ) E ( j/m ) = 1 − β log j/m + 1 − β log j/m (cid:0) ( β + β ) log log j/m + β c + β log ( − β ) (cid:1) + 1 − β log j/m (cid:2) β + β ) (log log j/m ) − k ′ log log j/m + const (cid:3) ++ O ( 1log j/m ) k ′ ≡ β (cid:0) β + 2 β β − β c − β (2 c + ˜ c ) − β ( β + β ) log ( − β ) (cid:1) (7 . j/m depends on a higher order term, of the form1 / log( h/m ), in (7.4) which we have not computed. In order to make a comparison withthe results of [11] we need to express our expansion parameters j and m in terms of y andthe coupling constant λ using (3.9), (3.5), or m = kλ − β β e √ λ πβ (7 . ǫ ( λ, x ) as a power series on λ we obtain ǫ ( λ, x ) = y " √ λ π + β y ) + k ++ π √ λβ (cid:0) β log ( y/ (2 πk )) + k log y + (2 β + 2 β ˜ c ) log( λ ) + const (cid:1) + ... (7 . k and k entering at one and two loops are2 k = − c − β log k + β β log( − πβ ) k =4 β (cid:2) β + β − β c + 2 β log(2 π ) + 2 β log( − πβ ) (cid:3) (7 . F in (7.4). This function has astructure of the form F ( t ) = f /h = − − β t + β β log t + c + ∞ X n =1 n X m =0 a nm (log t ) m t n ! (7 . a nm with m >
0, are determined in termsof lower order coefficients, a n ′ m with n ′ < n . In particular, the constant ˜ c is determinedby the structure of perturbation theory ˜ c = − β β (7 . c the term involving a log λ in (7.7) disappears. In general,the logarithmic terms in (7.9) are fixed by demanding that we can express the answer interms of a power series expansion in terms of the effective coupling ¯ g ( µ ) of the sigmamodel, with the additional condition that the dependence of the coupling constant ¯ g ( µ )on the scale µ is described by the Callan-Symanzik equation µ∂ µ ¯ g ( µ ) = β ¯ g ( µ ) + β ¯ g ( µ ) + ... = β (¯ g ( µ )) (7 . β ( g ) also has a power series expansion in ¯ g . This beta function equation can besolved as 1¯ g = − β log( µ/ Λ) + β β log(log µ/ Λ) + · · · (7 . µ = h ,then we can think of log( h/ Λ) = t +constant. In that case we can solve the equation as t + const = 1 − β ¯ g + β β log(1 / ¯ g ) + o (¯ g ) (7 . g on a small circle around the origin ¯ g → ¯ g e πi , then t performsa circle around infinity, but in addition we get a shift, t → te − πi − β β πi . When we saythat F has a power series expansion in ¯ g we are saying that each term is invariant underthis shift. However, t is not invariant and this one way to see that we need the logarithmicterms in (7.9), with coefficients determined by the lower order terms. All the logarithmicterms in (7.9) would vanish if β were zero. 28ow let us turn to the question of determining the leading logarithmic terms ateach loop order in the 1 / √ λ expansion. One can determine such terms by performingmanipulations similar to the ones performed above. However, it is also nice to see moredirectly how they are determined by using the renormalization group equations. For thatpurpose we imagine that we compute the tree level and one loop expressions for F usingperturbation theory. The simplest answer is obtained by choosing µ = h in which case weget f = h F , F = − (cid:20) g ( h ) + a + o (¯ g ) (cid:21) (7 . a is a constant that we will fix momentarily. We now run the coupling from thescale h to the scale 1 corresponding to the UV cutoff where the coupling is ¯ g = ¯ g (1) = π √ λ (1 + ˆ c √ λ ). The coupling ¯ g O (6) theory andconstant ˆ c is an unknown threshold correction. We can use the solution (7.13). We thensee that we can express the coupling at scale h as1¯ g ( h ) = 1¯ g (cid:20) z + ¯ g β β log(1 + ¯ z ) (cid:21) , ¯ z ≡ − β ¯ g log h (7 . g but kept all orders in ¯ z . We now compute j to find j = − ∂f∂h = h g (cid:20) z + ¯ g β β log(1 + ¯ z ) + ¯ g a − ¯ g β (cid:21) (7 . h as a function of ¯ y ≡ ¯ g j . This gives h = ¯ y z + ¯ g β log(1 + ˆ z ) + ¯ g β β log(1 + ˆ z ) + a ¯ g − ¯ g β , ˆ z ≡ − β ¯ g log ¯ y (7 . ǫ ( y ) = ¯ y g z ) − a ¯ g + ¯ g β (1 + β β ) log(1 + ˆ z )(1 + ˆ z ) (7 . y = π √ λ j . Then ¯ y = y (1+ ˆ c √ λ ). Similarly we can define z = − β π √ λ log y ,so that ˆ z = z (1 + ˆ c √ λ ). We can then write (7.18) as ǫ ( y ) = y √ λ π z ) − k π √ λ + π √ λ β (1 + β β ) log(1 + z )(1 + z ) (7 . k is a combination of the unknown parameters. k can be fixed by performing theone loop expansion of (7.19) and matching to the results in [11]. The one loop expansionof (7.19) is ǫ = y " √ λ π + β y − k (7 . − k / / (4 π ). Inserting this in (7.18) and usingthe values of β , β for the O(6) model (7.4), we find ǫ = y √ λ π (cid:20) z ) + 3 √ λ z )(1 + z ) (cid:21) , z = 4 √ λ log y (7 . g ( µ ), at some scale µf ( h ) = h (cid:18) K ( h/µ )¯ g ( µ ) + K ( h/µ ) + K ( h/µ )¯ g ( µ ) + ... (cid:19) (7 . K i are determined by the RG equations up to a constant.The relation between the mass gap m and the scale defined in the M S scheme via theequation (7.12) , Λ MS , was computed in [13] m = (cid:18) e (cid:19) N − N − ) Λ MS (7 . eferences [1] D. J. Gross and F. Wilczek, Phys. Rev. D , 980 (1974). H. Georgi and H. D. Politzer,Phys. Rev. D , 416 (1974).[2] G. P. Korchemsky, Mod. Phys. Lett. A , 1257 (1989). G. P. Korchemsky andG. Marchesini, Nucl. Phys. B , 225 (1993) [arXiv:hep-ph/9210281].[3] V. Sudakov, Sov. Phys. JETP , 65 (1956). R. Jackiw, Ann. Phys. (N.Y.) , 292(1968). A. H. Mueller, Phys. Rev. D , 2037 (1979). J. C. Collins, Phys. Rev. D ,1478 (1980). A. Sen, Phys. Rev. D , 3281 (1981).[4] G. P. Korchemsky and A. V. Radyushkin, Phys. Lett. B , 459 (1986); G. P.Korchemsky, and A. V. Radyushkin, Nucl. Phys. B283 , 342 (1987); G. P. Korchemsky,Phys. Lett. B , 629 (1989). S. V. Ivanov, G. P. Korchemsky and A. V. Radyushkin,Yad. Fiz. , 230 (1986) [Sov. J. Nucl. Phys. , 145 (1986)].[5] L. Magnea and G. Sterman, Phys. Rev. D , 4222 (1990). G. Sterman andM. E. Tejeda-Yeomans, Phys. Lett. B , 48 (2003) [arXiv:hep-ph/0210130].[6] S. Catani, Phys. Lett. B , 161 (1998) [arXiv:hep-ph/9802439].[7] J. C. Collins, “Sudakov form factors,” Adv. Ser. Direct. High Energy Phys. , 573(1989) [arXiv:hep-ph/0312336].[8] J. C. Collins, D. E. Soper and G. Sterman, Adv. Ser. Direct. High Energy Phys. , 1(1988) [arXiv:hep-ph/0409313].[9] N. Beisert, B. Eden and M. Staudacher, J. Stat. Mech. , P021 (2007) [arXiv:hep-th/0610251].[10] M. K. Benna, S. Benvenuti, I. R. Klebanov and A. Scardicchio, Phys. Rev. Lett. ,131603 (2007) [arXiv:hep-th/0611135]. A. V. Kotikov and L. N. Lipatov, Nucl. Phys.B , 217 (2007) [arXiv:hep-th/0611204]. L. F. Alday, G. Arutyunov, M. K. Benna,B. Eden and I. R. Klebanov, JHEP , 082 (2007) [arXiv:hep-th/0702028]. I. Kos-tov, D. Serban and D. Volin, arXiv:hep-th/0703031.[11] S. Frolov, A. Tirziu and A. A. Tseytlin, Nucl. Phys. B , 232 (2007) [arXiv:hep-th/0611269].[12] A. V. Belitsky, A. S. Gorsky and G. P. Korchemsky, Nucl. Phys. B , 24 (2006)[arXiv:hep-th/0601112].[13] P. Hasenfratz and F. Niedermayer, Phys. Lett. B , 529 (1990). See also P. Hasen-fratz, M. Maggiore and F. Niedermayer, Phys. Lett. B , 522 (1990).[14] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Nucl. Phys. B , 99 (2002)[arXiv:hep-th/0204051].[15] A. V. Belitsky, A. S. Gorsky and G. P. Korchemsky, Nucl. Phys. B , 3 (2003)[arXiv:hep-th/0304028].[16] M. Kruczenski, JHEP , 014 (2005) [arXiv:hep-th/0410226].[17] J. Polchinski and M. J. Strassler, JHEP , 012 (2003) [arXiv:hep-th/0209211].3118] A. Armoni, JHEP , 009 (2006) [arXiv:hep-th/0608026].[19] N. Drukker and B. Fiol, JHEP , 010 (2005) [arXiv:hep-th/0501109]. S. Yam-aguchi, JHEP , 037 (2006) [arXiv:hep-th/0603208]. S. A. Hartnoll and S. PremKumar, Phys. Rev. D , 026001 (2006) [arXiv:hep-th/0603190]. J. Gomis andF. Passerini, JHEP , 074 (2006) [arXiv:hep-th/0604007].[20] A. Migdal, Zh. Eksp. Teor. Fiz. , 810 (1975) (Sov. Phys. JETP. A5 , 693 (1990); V. Kazakov, Zh. Eksp. Teor. Fiz. , 1887 (1983)(Sov. Phys. JETP. 58 1096); I. Kostov, Nucl. Phys. B265 , 223 (1986); D. J. Grossand W. I. Taylor, Nucl. Phys. B , 181 (1993) [arXiv:hep-th/9301068].[21] C. Bachas, J. de Boer, R. Dijkgraaf and H. Ooguri, JHEP , 027 (2002) [arXiv:hep-th/0111210].[22] L. Cornalba, M. S. Costa, J. Penedones and R. Schiappa, Nucl. Phys. B , 327(2007) [arXiv:hep-th/0611123], arXiv:hep-th/0611122.[23] L. Cornalba, M. S. Costa and J. Penedones, arXiv:0707.0120 [hep-th].[24] Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D , 085001 (2005) [arXiv:hep-th/0505205].[25] C. W. Bauer, S. Fleming and M. E. Luke, Phys. Rev. D , 014006 (2001) [arXiv:hep-ph/0005275]. C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D , 114020 (2001) [arXiv:hep-ph/0011336]. C. W. Bauer, D. Pirjol and I. W. Stewart,Phys. Rev. Lett. , 201806 (2001) [arXiv:hep-ph/0107002]. C. W. Bauer, D. Pir-jol and I. W. Stewart, Phys. Rev. D , 054022 (2002) [arXiv:hep-ph/0109045].A. V. Manohar, Phys. Rev. D , 114019 (2003) [arXiv:hep-ph/0309176].[26] M. Kruczenski, R. Roiban, A. Tirziu and A. A. Tseytlin, arXiv:0707.4254 [hep-th].[27] A. M. Polyakov, Nucl. Phys. B , 171 (1980).[28] M. Kruczenski, Phys. Rev. D , 106002 (2004) [arXiv:hep-th/0310030].[29] L. F. Alday and J. Maldacena, JHEP , 064 (2007) [arXiv:0705.0303 [hep-th]].[30] M. Kruczenski, JHEP , 024 (2002) [arXiv:hep-th/0210115].[31] A. B. Zamolodchikov and A. B. Zamolodchikov, Annals Phys. , 253 (1979).A. B. Zamolodchikov and A. B. Zamolodchikov, Nucl. Phys. B , 525 (1978) [JETPLett. , 457 (1977)].[32] R. Roiban, A. Tirziu and A. A. Tseytlin, arXiv:0704.3638 [hep-th].[33] N. Beisert, R. Hernandez and E. Lopez, JHEP , 070 (2006) [arXiv:hep-th/0609044].[34] B. Eden and M. Staudacher, J. Stat. Mech. , P014 (2006) [arXiv:hep-th/0603157].[35] P. Y. Casteill and C. Kristjansen, arXiv:0705.0890 [hep-th].[36] G. Arutyunov, S. Frolov and M. Staudacher, JHEP0410