Convexity and Liberation at Large Spin
aa r X i v : . [ h e p - t h ] O c t PUPT-2433WIS/21/12-DEC-DPPA
Convexity and Liberation at Large Spin
Zohar Komargodski , and Alexander Zhiboedov Weizmann Institute of Science, Rehovot 76100, Israel Institute for Advanced Study, Princeton, NJ 08540, USA Department of Physics, Princeton University, Princeton, NJ 08544, USA
We consider several aspects of unitary higher-dimensional conformal field theo-ries (CFTs). We first study massive deformations that trigger a flow to a gapped phase.Deep inelastic scattering in the gapped phase leads to a convexity property of dimensionsof spinning operators of the original CFT. We further investigate the dimensions of spin-ning operators via the crossing equations in the light-cone limit. We find that, in a sense,CFTs become free at large spin and 1 /s is a weak coupling parameter. The spectrum ofCFTs enjoys additivity: if two twists τ , τ appear in the spectrum, there are operatorswhose twists are arbitrarily close to τ + τ . We characterize how τ + τ is approachedat large spin by solving the crossing equations analytically. We find the precise form ofthe leading correction, including the prefactor. We compare with examples where theseobservables were computed in perturbation theory, or via gauge-gravity duality, and findcomplete agreement. The crossing equations show that certain operators have a convexspectrum in twist space. We also observe a connection between convexity and the ratioof dimension to charge. Applications include the 3d Ising model, theories with a gravitydual, SCFTs, and patterns of higher spin symmetry breaking.December 2012 ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1. General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2. Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153. Crossing Symmetry and the Structure of the OPE . . . . . . . . . . . . . . . . . 173.1. The Duals of the Unit Operator . . . . . . . . . . . . . . . . . . . . . . . 183.2. An Argument Using the Light-cone OPE . . . . . . . . . . . . . . . . . . . 213.3. Relation to Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4. The Leading Finite Spin Correction . . . . . . . . . . . . . . . . . . . . . 253.5. An Argument in the Spirit of Alday-Maldacena . . . . . . . . . . . . . . . . 294. Examples and Applications of (3.17) . . . . . . . . . . . . . . . . . . . . . . . 304.1. Correction due to the Stress Tensor and Theories with Gravity Duals . . . . . . . 304.2. The large ∆ Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3. The Case of Parametrically Small ∆ − τ min . . . . . . . . . . . . . . . . . . 324.4. The Charge to Mass Ratio is Related to Convexity . . . . . . . . . . . . . . . 345. Applications of Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1. Free Theories and Weakly Coupled Theories . . . . . . . . . . . . . . . . . . 375.2. The Critical O ( N ) Models . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3. Strongly Coupled Theories and AdS/CFT . . . . . . . . . . . . . . . . . . . 415.4. Higher Spin Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . 426. Conclusions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . 43Appendix A. Normalizations and Conventions . . . . . . . . . . . . . . . . . . . . 45Appendix B. Relating Spin to σ . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.1. A Naive Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.2. A Systematic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 48B.3. The Coefficient c τ min in (3.12) . . . . . . . . . . . . . . . . . . . . . . . . 49B.4. The Most General (Scalar) Case . . . . . . . . . . . . . . . . . . . . . . . 50Appendix C. OPE of Chiral Primary Operators in SCFT . . . . . . . . . . . . . . . 52Appendix D. DIS for Traceless Symmetric Representations . . . . . . . . . . . . . . 54Appendix E. The s Correction in the Critical O ( N ) Model . . . . . . . . . . . . . . 55
1. Introduction
Even though conformal field theories (CFTs) have been studied for many years, verylittle is known about the general structure of their operator spectrum and three-pointfunctions. The interest in conformal field theories stems from their role in the description of1hase transitions [1] and from their relation to renormalization group flows. In addition,significant motivation to study conformal field theories is derived from the connectionbetween quantum gravity in AdS and conformal field theories [9,10,11].A classic result is that, in unitarity CFTs, operators obey the so-called unitaritybounds [12] (see also [13]). For example, the dimension of primary symmetric tracelesstensors of spin s ≥ ≥ d − s . (1 . d is the dimension of space-time.)Primary operators that saturate the bound are known to be conserved, namely, theysatisfy the equation ∂ µ J µ... = 0 inside any correlation function. Similarly, scalar operators( s = 0) are known to satisfy ∆ ≥ d − . If this inequality is saturated, a second orderdifferential equation holds true O = 0.The unitarity bounds above are traditionally derived from reflection positivity of Eu-clidean correlation functions. It has been known for a while that considering processes inLorentz signature can lead to further constraints on quantum field theory, and often suchconstraints do not seem to admit a straightforward derivation in Euclidean space. Simpleexamples based on an application of the optical theorem are discussed in [14,15,16,17].In this note we will use some properties of quantum field theory that are particularlyeasy to understand in Minkowski space in order to derive various constraints. In addition,we will find it useful to embed our conformal field theories in renormalization group flowsthat lead to a gapped phase.Before we plunge into a detailed summary of our results, we would like to reviewquickly some basic notions in conformal field theory, mostly in order to set the terminologywe will be using throughout this note. Given scalar operators O ( x ), O ( x ) we can considertheir OPE O ( x ) O (0) ∼ X s α ( s ) X k =1 C ks x − ∆ − ∆ +∆ ks − s x µ x µ · · · x µ s O kµ ,µ ,...,µ s (0) . (1 . A priori there may also be renormalization group flows where one finds scale invariant theoriesat the end points without the special conformal generators. The feasibility of this scenario hasbeen discussed recently, for example, in [2-8]. See also references therein. To date, scale invariantbut non-conformal theories have not been constructed. s is the sum over spin and the sum over k is the sum over all the operators thathave spin s . The sum over k is a finite sum. One can further re-package the sum above interms of representations of the conformal group by combining the contributions of all theoperators that are derivatives of some primary operator. (The coefficients C ks are fixed byconformal symmetry in terms of the coefficients of primary operators.)In Euclidean space it is natural to consider x → x µ go to zero. In this case the terms that dominate the sum (1.2) are the operators O kµ ,...,µ s with the smallest dimensions ∆ ks .In Minkowski signature we can send x → x − , x + , and set x i = 0. Then we consider the light-cone limit x + → x − is finite. The operators thatdominate then are O − , − ,..., − and their contribution to the OPE goes like x − ∆ − ∆ +∆ ks − s ,hence the strength with which they contribute is dictated by the twist τ ks ≡ ∆ ks − s . Wedenote τ ∗ s ≡ min k ( τ ks ), i.e. τ ∗ s is the minimal twist that appears in the OPE for a givenspin. Another useful notation is τ min ≡ min s τ ∗ s , that is, the overall minimal twist thatappears in the OPE, excluding the unit operator , which has twist zero.The twists of conformal primaries are constrained by (1.1). Traceless symmetric con-formal primaries of spin s ≥ d − d −
1. The unit operator has twist zero. The energy momentumtensor and conserved currents have twist d − SL (2 , R ) preserves the correspond-ing light-ray. It is therefore natural to consider representations of it. The representationsare classified by collinear primaries of dimension ∆ and spin s + − in the two dimensionalplane. We can act on this collinear primary any number of times with ∂ − in order togenerate the complete collinear representation. The twist of a collinear primary is definedas τ coll ≡ ∆ − s + − .A conformal primary O µ ,...,µ n can be decomposed into (generally infinitely many)collinear primaries. In this decomposition, the collinear primary with the smallest collineartwist is clearly O − ,..., − , and this minimal collinear twist coincides with the conformal twist.It would be useful for our purposes to rewrite the OPE (1.2) in terms of collinearrepresentations. From the discussion above we see that it takes the form as x + → O ( x + , x − ) O (0 , ∼ ( x + ) − (∆ +∆ ) X τ coll ,s + − ,... e C ks ( x + ) τ coll F ( x − , ∂ − ) O k − , − , ..., − | {z }
12 ( s + s + − ) , + , + ..., + | {z }
12 ( s − s + − ) (0) . (1 . F are determined from representation theory, see [18] for a review. Theimportant message in (1.3) is that the approach to the light cone is controlled by thecollinear twists.Having introduced the basic terminology that we will employ, let us now summarizethe constraints we discuss in this note.First, we merely extend the observations of Nachtmann [19] about the QCD sum rulesto an argument pertaining to arbitrary CFTs. In [19] it was pointed out that the DeepInelastic Scattering (DIS) QCD sum rules imply certain convexity properties of the highenergy limit of QCD. We discuss such constraints on the operators in the OPE in thecontext of a generic CFTs. We emphasize the assumptions involved in this argument.To this end, we consider an RG flow where the CFT of interest is the UV theoryand in the IR we have a gapped phase. We consider an experiment with the operatorof interest, O ( x ), playing the role of the usual electromagnetic current in DIS. AssumingRegge asymptotics of the amplitude in the gapped phase (namely, the amplitude is boundedby some power of s ) lim s →∞ A ( s, t = 0) ≤ s N − (1 . τ ∗ s of operators appearing in the O ( x ) O † (0) OPE τ ∗ s − τ ∗ s s − s ≤ τ ∗ s − τ ∗ s s − s , s > s > s ≥ s c . (1 . s c is the minimal spin above which the convexity property starts. From (1.4) we knowthat s c is finite and certainly does not exceed N . The argument leading to the convexityproperty (1.5) depends on the following three assumptions:a) unitarity;b) the CFT can flow to a gapped phase;c) polynomial boundedness of the DIS cross section in the gapped phase (1.4).Assumption c) may seem as the least innocuous. However, there are formal argumentsconnecting it to causality and other basic properties of QFT. See for example [20], anda more recent discussion in [21]. In addition, we do not know of field theories violatingassumption b) (such theories are known to exist in the realm of critical phenomena, but, toour knowledge, there are no concrete examples of unitary Lorentz invariant theories thatcannot be deformed to a gapped phase). Hence, convexity of the function τ ∗ s is establishedunder very general assumptions. Moreover, in all the examples we checked, the convexity4bove holds true starting from s c = 2. We do not prove here that this is the case in mostgenerality.A second topic we study here is the crossing symmetry (i.e. the bootstrap equations)in Lorentzian signature. The bootstrap equations were introduced long ago by [22,23,24].It has been demonstrated recently, starting from [25], that one can go about and sys-tematically bound numerically solutions to these formidable equations. Sometimes thesenumerical bounds seem to be saturated by actual CFTs, as has been demonstrated, forexample, in [26]. These methods of bounding solutions to the bootstrap equations areeasily described in Euclidean space, and the consistency of the procedure follows from thefact that the contribution of the operators with high dimension to the OPE in Euclideanspace is exponentially suppressed [27].By contrast, as we recalled above, the structure of the operator product expansionin Minkowski space is such that we can probe directly operators with very high spin andsmall twist. We therefore use the bootstrap methods in Lorentzian signature to constrainoperators with high dimension and high spin, but with low twist. Studying the light-cone OPE and bootstrap equations in the appropriate limit, weconclude that starting from any two primaries with twists τ and τ , operators with twistsarbitrarily close to τ + τ are necessarily present. This can be regarded as an additiv-ity property of the twist spectrum of general CFT. In the derivation of this result, it isimportant to assume that d >
2. Indeed, this additivity property is definitely violated intwo-dimensional models, such as the minimal models. The condition d > O with any other operator O are bounded from above as follows (in d > s →∞ τ ∗ s ≤ τ O + τ O . (1 . τ ∗ s > τ O + τ O . This could be useful to bear in mind when trying to bound solutions tothe crossing equations along the lines of [25]. Different applications of the bootstrap equations in Lorentzian signature were discussedin [28,29,30]. In these papers the properties of operators with both high spin and high twistwere analyzed. τ O + τ O . We refer to these operators as double-twist operators. We alsodenote these operators symbolically as O ∂ s O . Similar argument shows that operatorswith twists asymptoting to τ O + τ O + 2 n for any integer n are present as well. Thosecan be denoted symbolically as O ∂ s n O . We do not discuss the case n = 0 in as muchdetail as n = 0. In addition, it also makes sense to talk about multi-twist operators.Given that their twists approach τ O + τ O for large enough spin, it is interesting toask how precisely this limit of τ O + τ O is reached. It has been argued in [31] that forsufficiently large spin we can parameterize the twists of these operators as follows τ ( s ) = τ O + τ O − c τ min s τ min + · · · . (1 . τ min is the twist of O min which is the operator of smallest twist exchanged by O and O . (As in the definition before, we exclude from this the unit operator.) In many caseswe would expect this to be the energy momentum tensor, thus τ min = d − AdS × S d − background. By choosing special coordinates in AdS one can think about the four-point functions above as being described by a two-dimensional gapped theory embedded in AdS . The energy in this theory is the twist,and (1.7) corresponds to the leading interaction between separated localized excitationswhere the separation is governed by the spin.Here we arrive at the same conclusion (1.7) by studying the crossing equation inflat Minkowski space. Furthermore, by using the bootstrap equations, we compute thecoefficient c τ min in (1.7) analytically in every d > O ( x ) O † (0). Then thecoefficient governing the asymptotic twists of double-twist operators is c τ min = Γ( τ min + 2 s min )2 s min − Γ (cid:0) τ min +2 s min (cid:1) Γ(∆ O ) Γ(∆ O − τ min ) f ,f = C OO † O τ min C OO C O † O † C O τ min O τ min , (1 . s min and τ min are the spin and the twist of the minimal twist operator (other thanthe unit operator) appearing in the OPE of O with its conjugate. Hence, c τ min is fixed By this we mean it appears both in the OPE of O O † and O O † .
6y some universal factors and the two- and three-point functions of the operators O and O τ min . As we remarked, in many models one would expect O τ min to be the stress tenors,in which case the three-point function above is fixed in terms of ∆ O and the two-pointfunction of the stress tensor (which is some measure of the number of degrees of freedomof the theory).Note that in unitary CFTs we have τ min ≤ O , and an equality can only be reachedfor free fields. In general, there could be several operators of the same minimal twist τ min .In this case c is the sum over all of these operators. It is worth mentioning that (1.8)provides another point of view on convexity. Since all the leading contributions to c are manifestly positive, the spectrum of double-twist operators at large enough spins isconvex. This statement is slightly different from the one outlined in (1.5), which is aboutthe minimal twists in reflection positive OPEs. Here the statement is about double-twistoperators in reflection positive OPEs.We also consider non-reflection positive OPEs, such as those of a charged operator(with charge q under some U (1) global symmetry, with the generalization to non-Abeliansymmetries being straightforward) O with itself, O ( x ) O (0). The double-twist operators O ∂ s O with twists that approach 2∆ O are necessarily present. Here we can compute thecorrections to the twists at large spin τ s − O and we find a formula very similar to (1.8)(we will describe it in detail in the text) with one small but important difference, a factorof ( − s . Hence, the conserved current and the energy momentum tensor, both of whichhave twist d − OO † . The spectrum is non-concave at large enough spin if and onlyif ∆ ≥ d − | q | , (1 . U (1) current that we define carefullyin the main text. Hence convexity at large spin of such operators is related to whether ornot the operator O satisfies a BPS-like bound.In SUSY theories, for chiral primaries, (1.9) is saturated (the charge q is that of thesuperconformal U (1) R symmetry) and the leading corrections to the anomalous twistsvanish. For chiral operators that are not superconformal primaries, (1.9) is satisfied inunitary theories. Hence, the spectrum of twists of the double-twist operators O ∂ s O isconvex. It would be interesting to understand more generally whether convexity at large In this note when we say “reflection positive OPE” we mean an OPE of the type O ( x ) O (0) † . If we combine (1.5) and (1.6) (i.e. convexity and additivity) we are led to ratherstringent predictions concerning interesting models. In particular, we predict that in the3d critical O ( N ) models (which are highly relevant for 3d phase transitions) there is a setof almost conserved currents of spin s ≥ s = s + τ ∗ s with1 ≤ τ ∗ s ≤ σ , (1 . σ is the dimension of the spin field σ i . Furthermore, we can argue using additivitythat lim s →∞ τ ∗ s = 2∆ σ , (1 . τ ∗ s is monotonically rising and convex as a function of the spin.In the case of the 3d Ising model, it is known that there is a spin 4 operator with thetwist τ ∗ ∼ .
02, which agrees with our expectations. Using convexity, this allows us tofurther narrow down the possible window for higher spin currents and predict that in the3d Ising model (which describes, among others, boiling water)1 . ≤ τ ∗ s ≤ . , s ≥ , lim s →∞ τ ∗ s = 2∆ σ ≈ . . (1 . τ ∗ s is again monotonic and convex as a function of the spin. In addition, theupper bound above is in fact saturated for s → ∞ (this follows from additivity and thefact that the set of operators of the Ising model is continuously connected to free fieldtheory). We will present several other examples in the text.We also discuss the relation of some of our results to small breaking of higher spinsymmetry and, through AdS/CFT, to quantum gravity in AdS.For the parity preserving Vasiliev theory in AdS [33] with higher spin breaking bound-ary conditions, (1.11) implies for the masses of the higher spin bosons in the bulk m s ≈ γ σ s + O (1) , s ≫ . Throughout the paper, convexity is to be understood as non-concavity. γ σ is the anomalous dimension of the spin field in a non-singlet O ( N ) model and theresult is exact in N .The outline of the paper is as follows. In section 2 we discuss DIS and estab-lish (1.5). In section 3 we consider the bootstrap equations in Lorentz signature andderive (1.6),(1.7),(1.8). We also review the argument of [31] in this section. In sections 4,5we consider examples and applications. In section 6 we describe some open problems andconclude. Five appendices complement the main text with technical details. Note: After this work had been completed, we became aware of a related work byA. Liam Fitzpatrick, Jared Kaplan, David Poland, and David Simmons-Duffin [34].
2. Deep Inelastic Scattering
In this section we explain how one can derive inequalities about operator dimensionsin CFT by considering RG flows that start in the ultraviolet from the CFT and end up inthe gapped phase. The basic ingredients are as follows.
Fig. 1:
A relevant perturbation of a CFT that leads to a gapped phase.
Take any CFT (in any number of space-time dimensions, d ) and assume it can beperturbed by a relevant deformation to a gapped phase (see fig. 1). Then we take thelightest particle in the Hilbert space, and denote this particle by | P i . Its mass is denotedby M .There may exist CFTs which have no relevant operators whatsoever (“self-organizing”CFTs), or CFTs which have relevant operators but can never flow to a gapped phase.We are not aware of such examples, and we will henceforth assume that flowing to a9apped phase is possible. In the critical O ( N ) models one can make the following heuristicargument to support this scenario.The critical O ( N ) models can be reached if one tunes appropriately the relevantperturbations of free N bosons. On the other hand, if the mass parameter in the UV islarge enough, the model clearly flows to a gapped phase. Now we can decrease the massgradually until the RG flow hovers very close to the critical O ( N ) model. In this case wecan describe the late part of the flow as a perturbation of the O ( N ) model by the energyoperator, R d xǫ ( x ). Indeed, it is a primary relevant operator in the critical O ( N ) models. If there are no phase transitions as a function of this mass, then it means that the critical O ( N ) models, perturbed by the energy operator, flow to a gapped phase. This is alsoconsistent with the fact that the energy operator corresponds to dialing the temperatureaway from the critical temperature. Fig. 2:
When we add a large mass m in the ultraviolet, the model clearly flows toa gapped phase. We can gradually reduce the mass and tune the flow such that itpasses close to the critical model. The flow after this stage can be interpreted as aperturbation of the critical O ( N ) model by the energy operator, ǫ ( x ) . This heuristic argument is summarized in fig. 2. In the following we discuss generalCFTs, assuming they can be deformed to a gapped phase by adding relevant operators(and maintaining unitarity). This is known to be certainly true at large N and at some small values of N . It seems naturalthat this would be indeed the case for any N . .1. General Theory Deep Inelastic Scattering (DIS) allows to probe the internal structure of matter. Onebombards some state with very energetic (virtual) particles and examines the debris. Ofparticular interest is the total cross section for this process, as a function of various kine-matical variables. l l * qP Fig. 3:
A lepton emits a virtual, space-like, photon that strikes a hadron. As aresult, the hadron generally breaks up into a complicated final state.
Traditionally, the most common setup to consider is that of fig. 3. A lepton emits avirtual, space-like, photon that strikes a hadron.To study the inner structure of | P i we can try to shoot various particles at it. If thetheory (i.e. the CFT and the relevant deformations) preserves a global symmetry, we cancouple the conserved currents to external gauge fields and repeat the experiment of fig. 3.(We can introduce arbitrarily weakly interacting charged matter particles and make thegauge field dynamical in order to have a perfect analog of fig. 3.) A more universal probethat exists in any local QFT is the EM tensor. We can couple it to a background gravitonand consider the same experiment. We can also perform DIS with any other operatorin the theory. Let us start by reviewing the kinematics of DIS in the case of a scalarbackground field J ( x ). In other words, a background field that is coupled to some scalaroperator in the theory O ( x ) via R d d xJ ( x ) O ( x ). ( J ( x ) and O ( x ) are taken to be real.)11 qP ~ q −qP −P I m * *
Fig. 4:
The total inclusive cross section can be extracted from the imaginary partof a Compton-like scattering where the momenta of the outgoing states are identicalto the momenta of the incoming states.
The optical theorem allows to extract the inclusive amplitude for DIS via the analogof Compton scattering (see fig. 4). The amplitude in our “scalar DIS” setup is A ( q µ , P µ ) ≡ Z d d ye iqy h P | T ( O ( y ) O (0)) | P i , (2 . | P i by P µ . We will only discuss thecase of q <
0, i.e. space-like momentum for the virtual particle. The above amplitudeobviously depends on the mass scales of the theory (we set P = 1 for convenience), andthe two invariants q , ν ≡ q · P . Fig. 5:
The analytic structure in the ν plane.
12e can promote ν to a complex variable. Since we have assumed the particle | P i isthe lightest in the theory, the above amplitude has a branch cut for ν ≥ − q and ν ≤ q .The ν plane is depicted in fig. 5. The optical theorem connects the discontinuity along thebranch cut of (2.1) to the square of the DIS amplitude. (There are no other cuts since q is space-like.)To get a handle on (2.1), we can invoke the OPE expansion. In the case that O ( x ) isreal, only even spins contribute. (Odd spins would be allowed if O + = O .) O ( y ) O (0) = X s =0 , , ,... X α ∈I s f ( α ) s ( y ) y µ ...y µ s O ( α ) µ ...µ s (0) . (2 . s labels the spins, and we only write primary fields and ignore descendants since theydo not contribute to the correlation function under consideration. The set I s correspondsto the collection of operators of spin s .Conformal symmetry at small y fixes the OPE coefficients in (2.2) to be f ( α ) s ( y ) = C ( α ) s y τ ( α ) s − O (1 + · · · ) , (2 . τ ( α ) s = ∆ O ( α ) µ ...µs − s is the twist of the operator. The · · · denote corrections sup-pressed by positive powers of y . These corrections depend on the relevant operators.Parameterizing these corrections would be unnecessary for our purposes.The expectation values of O αµ ...µ s in the state P are parameterized as follows h P |O αµ ...µ s (0) | P i = A ( α ) s ( P µ P µ · · · P µ s − traces) , (2 . A ( α ) s some dimensionful coefficients.We can now insert the expansion (2.2) into (2.1) and find A ( q µ , P µ ) ≡ X s =0 , , ,... X α ∈I s A ( α ) s (cid:18)(cid:18) P · ∂∂q (cid:19) s − traces (cid:19) e f ( α ) s ( q ) . (2 . P (cid:16) P · ∂∂q (cid:17) s − (cid:16) ∂∂q · ∂∂q (cid:17) etc.At this point it is natural to switch to the variable x = − q /ν , and write the answerin terms of x, q . The OPE allows to control easily the leading terms as − q → ∞ . Hence,for any given power of x we keep only those terms that are leading in the limit of large − q .The leading terms take the form A ( x, q ) ≡ X s =0 , , ,... A ∗ s C ∗ n x − s ( q ) − τ ∗ s +∆ O − d/ . (2 . s , I s , we select the one that has the smallesttwist and denote the corresponding coefficients and twists with an asterisk. In the limit oflarge − q the “traces” appearing in (2.5) can be dropped. We see that the OPE expansion is useful for finite ν and large q (i.e. large x andlarge q ). The physically relevant configuration is, however, large q and x ∈ [0 , x to a complex variable and recall that the branch cut extendsover x ∈ [ − , x ∼
0, we writesum rules by following the usual trick of pulling the contour from infinite x to the branchcut. We define µ s ( q ) as the s-th moment µ s ( q ) = Z dxx s − Im A ( x, q ) , (2 . − q µ s ( q ) → ( q ) − τ ∗ s +∆ O − d/ A ∗ s C ∗ s . (2 . x behavior. Equivalently, itdepends on the behavior of the amplitude for fixed q and large ν . This is the Reggelimit. We would like to take a conservative approach to the problem of determining the The reason is as follows. The nature of the expansion is that we keep all powers of x andfor each power we only retain the most dominant power of q . The trace terms are negligible,because, for instance, the first “trace term” scales as x − s +2 ( q ) − τ ∗ s +∆ O − d/ − . This has to becompared to the contribution from the spin s − x − s +2 ( q ) − τ ∗ s − +∆ O − d/ . We seethat the “trace terms” can be indeed consistently neglected, for example, if the smallest twistof a spin n operator, τ ∗ s , is monotonically increasing as a function of the spin. We will justifythis assumption self-consistently later. This subtlety is apparently overlooked in various places.Another way to avoid this issue is to use the spin projection trick [19]. lim x → A ( x, q ) ≤ x − N +1 , (2 . N . Polynomial boundedness is discussed in [20]. A recent discussion andmore references to the original literature where polynomial boundedness is discussed canbe found in [21]. In this case the contour manipulation leading to (2.7) can be justifiedonly for s ≥ N . This has to be borne in mind in the following, where we derive someconsequences of (2.7) using convexity properties. The first place we are aware of whereconvexity properties appear in this context is [19]. We will discuss various simple inequalities that the moments µ s ( q ) have to satisfy.These properties follow simply from unitarityIm A ( x, q ) ≥ , (2 . A ( x, q ) must be nonzero at least around some points, otherwise, the scatteringis trivial.The general inequalities stated below can be proven with the method we are usingonly for s ≥ N with some finite N . However, they may or may not be true for smallerspins as well. We will denote the spin from which these inequalities become true by s c .We see that s c is finite and it is at most N .From (2.7) it is clear (because of (2.10)) that µ s > µ s +1 , which together with the sumrule (2.8) leads to τ ∗ s ≤ τ ∗ s +1 . (2 . For on-shell scattering processes, we have the bound of Froissart-Martin [35,36] σ ≤ C (log( s )) , where C is some dimensionful coefficient. In deep inelastic scattering we are dealingwith an amplitude involving an off-shell particle (for example, a virtual photon), so the argumentof Froissart-Martin does not carry over. In the context of QCD, one can use various phenomeno-logical approaches to model the small x behavior. For example, see the review [37]. The result ofthese phenomenological models is that the amplitude grows only as a power of a logarithm with1 /x . This may be more general than QCD, for example, it would be interesting to understand thesmall x asymptotics in N = 4 [38]. We thank M. Lublinsky and A. Schwimmer for discussionsabout the Regge limit. s < s < s , then (cid:18) µ s µ s (cid:19) s − s ≥ (cid:18) µ s µ s (cid:19) s − s . (2 . τ ∗ s − τ ∗ s s − s ≤ τ ∗ s − τ ∗ s s − s . (2 . We remind again that these inequalities start from some finite spin, s c . In all the exampleswe checked they are in fact true for all spins above two (at spin two we have the energymomentum tensor, which has the minimal possible twist allowed by unitarity, d − Fig. 6:
The generic structure of the spectrum of minimal twists in a CFT. Let us first prove this in the case s = s + 2 = s + 4, and we denote s ≡ s . In thiscase, (2.12) is true due to the simple identity0 < Z ( x,y ) ∈ [0 , dxdyx s − y s − ( x − y ) Im A ( x, q )Im A ( y, q ) = µ s µ s +4 − µ s +2 . This establishes local convexity, namely, τ ∗ s +4 − τ ∗ s ≤ τ ∗ s +2 − τ ∗ s ) . From this, global convexity (2.13) follows trivially. One can also show that without furtherassumptions on Im A ( x, q ) (or some extra input) there is no stronger inequality that followsfrom (2.7),(2.8),(2.10). d >
2. The situation is slightly subtler in d = 2. Intwo-dimensional CFTs there are infinitely many operators with vanishing twist (these arefound in the Verma module of the unit operator). Hence, the minimal twist is zero for allspins. This is however consistent with our analysis here, as we have only shown that τ ∗ s isnon-concave. (When all the minimal twists are identical, unitarily (2.10) simply translatesto inequalities between the coefficients appearing in (2.8).)So far we have established convexity and monotonicity (starting from some finite spin).It is also natural to ask what is the asymptotic limit of the minimal twist as the spin goesto infinity. We will see that τ ∗ s has a finite and determined limit as s → ∞ . This is thetopic of the next section.
3. Crossing Symmetry and the Structure of the OPE
In this section we analyze the crossing equations in Minkowski space. Using a con-formal transformation we can always put any four points on a plane. On this plane it isnatural to introduce light-like coordinates, such that the metric on R d − , takes the form ds = dx + dx − + ( dx i ) . We study the crossing equations in the limit when one of thepoints approaches the light cone of another point. We have explained in the introductionthat one should distinguish between the collinear twist ∆ − s + − and the conformal twist∆ − s .Our analysis consists of two parts. First, we analyze how the unit operator constrainsgeneral CFTs via the crossing equations. We conclude that in any unitary CFT in d > τ coll1 and τ coll2 , we either have in the theoryoperators with collinear twist precisely τ coll1 + τ coll2 , or there exist operators whose twistsare arbitrarily close to τ coll1 + τ coll2 .Combined with convexity (which holds in reflection positive OPEs), this implies thatin any unitary CFT the minimal twist obeys the following inequality lim s →∞ τ ∗ s ≤ τ O . (3 . In some sense, a predecessor of this statement is the so-called Callan-Gross theorem [39].The discussion of [39] is in the context of weakly coupled Lagrangian theories. given any two operators of twists τ and τ , there will be an operator with twistarbitrary close to τ + τ . This sharper statement motivates the symbolic notation O ∂ s O introduced above.We present several arguments for this stronger statement, and we also emphasize thatit can be understood in the analysis of Alday and Maldacena [31]. The arguments we giveand the discussion in [31] provide very compelling reasons for making this stronger claim.We leave the task of constructing a detailed proof of this statement to the future. Thus, the spectrum of any CFT has an additivity property in twist space. It makessense to talk about double- and multi-twist operators. We denote the double-twist op-erators appearing in the OPE of O and O symbolically as O ∂ s O . The three-pointfunctions hO O ( O ∂ s O ) i are given by the generalized free fields values to leading orderin s . By analyzing terms that are sub-leading in the small z limit we conclude that oper-ators which can be symbolically denoted as O ∂ s n O appear as well, and for large spintheir twist asymptotes to τ + τ +2 n . Again the three-point functions hO O ( O ∂ s n O ) i approach the generalized free fields values to leading order in s . In this paper we do notdiscuss the case n = 0 in detail.We then include in the crossing equations the operator with the minimal twist after theunit operator. Using the crossing equations we characterize how the limiting twist τ + τ is approached as the spin is taken to infinity. The corrections to the twist, τ s − τ − τ ,go to zero at large s as some power of s that we characterize in great detail below. Forsimplicity, we mostly concentrate on the case where O and O are scalar operators. As a warm-up, consider any large N vector model, and a four-point function of someflavor-neutral operators. (With minor differences the same holds for adjoint large N the-ories.) We decompose the four-point function into its disconnected and connected contri-butions hO ( x ) O ( x ) O ( x ) O ( x ) i = hO ( x ) O ( x ) ihO ( x ) O ( x ) i + hO ( x ) O ( x ) ihO ( x ) O ( x ) i + hO ( x ) O ( x ) ihO ( x ) O ( x ) i + hO ( x ) O ( x ) O ( x ) O ( x ) i conn . (3 . In the setup of [31] this would amount to understanding better some analytic continuations.In our setup this presumably amounts to understanding in more detail the structure of the de-composition of a conformal primary into collinear primaries. N , this correlator is dominated by thedisconnected pieces. The connected contribution is suppressed by N compared to thedisconnected pieces. This fact by itself guarantees the existence of the double trace oper-ators, which, as a matter of definition, we can denote O ∂ s O , with the scaling dimensions∆ s = 2∆ O + s + O ( N ).To see that these operators are present and that their dimensions are as above, wesimply expand the disconnected pieces in a specific channel. This exercise is reviewed inappendix B. Of course, it is well known that such operators indeed exist in all the large N theories, and the notation O ∂ s O makes sense because the dimensions of these operatorsare very close to the sum of dimensions of the “constituents.”We would like to prove that a similar structure exists in arbitrary d > τ + τ .This universal property of higher-dimensional CFTs can be seen from crossing sym-metry and the presence of a twist gap. By the twist gap we mean the non-zero twistdifference between the unit operator that is always present in reflection positive OPEs andany other operator in the theory. This fact follows from unitarity.To establish this we consider a four-point correlation function in the Lorentzian do-main. The main idea is to consider the consequences in the s-channel of the existenceof the unit operator in the t-channel. In Euclidean space, the consequences of the unitoperator in the dual channel were recently discussed in [27]. It turns out to constrain thehigh energy asymptotics of the integrated spectral density. Here we would like to exploitthe presence of the unit operator in the Lorentzian domain.For simplicity we consider the case of four identical real scalar operators O of dimen-sion ∆ and then generalize the discussion. The argument in the most general case goesthrough essentially verbatim.Consider the four-point function hO ( x ) O ( x ) O ( x ) O ( x ) i = F ( z, z )( x x ) ∆ ,u = x x x x = zz , v = x x x x = (1 − z )(1 − z ) . (3 . We thank Juan Maldacena for many useful discussions on this topic. ig. 7: We consider all four points to lie on a plane, with the following light cone ( z, z ) coordinates: x = (0 , , x = ( z, z ) , x = (1 , , x = ( ∞ , ∞ ) . We considerthe light-cone OPE in the small z fixed z channel and then explore its asymptoticas z → . Let us introduce two variables which will parameterize the approach to different lightcones σ, β ∈ (0 , ∞ ). In this domain all points are space-like separated. The cross ratiosare related to σ and β as follows z = e − β ,z = 1 − e − σ . (3 . z and z are two independent real numbers. The reason for the notation weare using is that in Euclidean signature z is a complex number and z is its conjugate. Aconvenient choice of a four-point function realizing the notation above is depicted in fig. 7.Below we present two versions of the argument. In the first version we exploit crossingsymmetry in Minkowski space. We use some basic properties of the light-cone OPE andshow how crossing symmetry can lead to certain constraints on the operators spectrumof the theory. We find that crossing symmetry leads to a certain “duality” between fastspinning operators with very large dimensions and the low-lying operators of the theory.Here we compute the first two orders in the large spin expansion that we introduce. Thesecond version of the argument relies on the picture of [31] for fast spinning operators. Inthis nice, intuitive, picture many facts that we derive from the bootstrap equations havea straightforward interpretation in terms of particles that interact weakly with each otherin some gapped theory. We begin by presenting the approach based on the bootstrapequations, and then re-interpret some of the facts we find using the ideas of [31].20 .2. An Argument Using the Light-cone OPE We would like to consider the s-channel light-cone OPE expansion z → + and z fixed. As we have reviewed in the introduction, this expansion is governed by the twists ofthe operators. More precisely, it is governed by the collinear twist ∆ − s − + . Any primaryoperator can be decomposed into irreducible representations of the collinear conformalgroup and one can easily see that, for any operator, the minimal collinear twist coincideswith the usual conformal twist ∆ − s . An operator with collinear twist τ coll contributes F ( z, z ) ∼ z τ coll2 F ( z ). The function F ( z ) is a partial wave of the collinear conformal group.We quote some properties of it when needed. As usual, by re-summing the light-cone OPEexpansion we can recover the correlation function at any fixed z and z .In the notation we defined after (3.3), the light-cone OPE expansion in the s-channelcorresponds to a large β , fixed σ , expansion F ( β, σ ) = X i e − τ colli β f i ( σ ) = Z ∞ dτ e − τ coll β f ( σ, τ ) . (3 . σ, β ∈ (0 , ∞ ).The s-channel unit operator gives the most important contribution for finite σ andlarge β . Let us now consider a different regime. We keep β fixed and start increasing σ . Inthis way our coordinate z approaches 1 and hence gradually becomes light-like separatedfrom the insertion x . If we take σ ≫ β the four-point function is dominated by the t-channel OPE, and the most dominant term comes from the unit operator in the t-channel.We can estimate the behavior of (3.5) in this limit: F ( β, σ ) ∼ e − τ O β e τ O σ (cid:0) O ( e − σ ) (cid:1) . (3 . d > τ coll = 2 τ O . This is, however, not necessarily true. Imagine we have a series of operatorswith twists 2 τ O + a i such that a i → i → ∞ . Then for arbitrary large σ there willbe a i < e − τ min σ such that their twist will be effectively 2 τ O to the precision we are probingthe correlation function. Thus, equation (3.6) only implies that the spectral density f ( σ, τ )is non-zero in any neighborhood of τ coll = 2 τ O . This shows that there are operators withcollinear twist in any neighborhood of τ coll = 2 τ O .21he discussion above concerned with the collinear twists. As mentioned in the intro-duction to this section, we claim a slightly stronger result – that there are operators whoseconformal twists are arbitrarily close to τ = 2 τ O .Let us explain the motivations for this stronger claim. First of all, as we will reviewlater, this follows from the construction of [31]. Second, if τ = 2 τ O had not been the min-imal collinear twist in the decomposition of a conformal primary into collinear primaries,then the conformal twist of the corresponding conformal primary would have been 2 τ O − n for some nonzero integer n . That conflicts with unitarity for some choices of O . Finally,a heuristic argument: different collinear primaries that follow from a single conformal pri-mary are obtained by applying ∂ z to the collinear primary with the lowest collinear twist.However, since we are interested in the behavior for small 1 − z (this is the e τ O σ in (3.6)),higher collinear twists are expected to produce the same (or weaker) singularity as theyroughly transport O in the z direction.We conclude that there must be operators with twists arbitrarily close to 2 τ O . Thisfollows from the presence of the unit operator in the t-channel. We can say more aboutthese operators. An easy lemma to prove is that we must have infinitely many operatorswith twists 2 τ O (or arbitrarily close to it). The argument is simply that conformal blocksbehave logarithmically as z → z ∆ (1 − z ) ∆ in terms of s-channel collinear conformal blocks. This canbe done using the generalized free fields solution of the crossing equations. The z → z → hO O ( O ∂ s O ) i should coincide with the ones ofgeneralized free fields to the leading order in s .So far we analyzed the consequences of matching the z ∆ (1 − z ) ∆ piece from the unitoperator contribution to the s-channel OPE. However, we can consider the complete de-pendence on z coming from the unit operator in the t-channel, z ∆ (1 − z ) ∆ (1 − z ) ∆ . Clearly, theprimary operators discussed so far are not enough. Again the theory of generalized freefields reproduces the necessary function by means of additional primary operators of theform O ∂ s n O . By taking the large spin limit of this solution (equivalently, z →
1) weconclude that operators with twists 2 τ O + 2 n and three-point functions approaching thoseof generalized free fields are always present in the spectrum, of any CFT.22ll these facts naturally combine into the following picture (that we will henceforthuse): CFTs are free at large spin – they are given by generalized free fields. More precisely,in the OPE of two arbitrary operators O ( x ) and O ( x ), at large enough spin, there areoperators whose conformal twists are arbitrarily close to τ + τ . We denote these operatorssymbolically by O ∂ s O . Their twists behave as τ + τ + O ( s α ) where the power correctionwill be discussed in the next subsection ( α is some positive number). These operators, O ∂ s O , are referred to as double-twist operators . Their three-point functions asymptotethe ones of the theory of generalized free fields. Similar remarks apply to O ∂ s n O .In principle, one can have either of the following two scenarios:a) Operators with the twist τ = τ + τ are present in the spectrum.b) The point τ = τ + τ is a limiting point of the spectrum at infinite spin.By option b) we mean that in a general CFT there is a set of operators with largeenough spin such that their twist is τ + τ − O ( s α ) with α >
0. In this way, for arbitrary ǫ >
0, there will be operators X with twist τ + τ − τ X < ǫ . They will be responsible forthe behavior (3.6). Strictly speaking, so far we only discussed scalar external operators. However, thegeneralization to operators with spin is trivial in this case. Indeed, the usual compli-cation of having many different structures in three point functions for operators withspin is irrelevant for us. The reason is that our problem is effectively two dimensional,and by considering, for example, operators with all the indices along the z direction hO z...z e O z...z e O z...z O z...z i the argument above goes through.Indeed, in the case of four identical real operators, by keeping only unit operators inthe s- and t-channel we get hO z...z O z...z O z...z O z...z i = z s ( zz ) ∆+ s + (1 − z ) s [(1 − z )(1 − z )] ∆+ s = z s ( zz ) ∆+ s " (cid:18) z − z (cid:19) ∆ − s (cid:18) z − z (cid:19) ∆+ s . (3 . z ∆ − s corresponds to the τ coll = 2 τ O = 2(∆ − s ) operators in the s-channel. The crucial simplification is that our perturbation theory is effectively 2d and,thus, all the usual complications of three-point functions of operators with spin are absent. At one loop this was observed, for example, in [40],[41]. Here we see this is a general propertyof any CFT above two dimensions. d = 2. There are many 2 d theories where the double-twist operators are absent. As we have mentioned above, this isdue to the fact that in two dimensions there is no twist gap above the unit operator.This is also related to why some theories in 2 d can be very simple: the absence ofthe twist gap provides a way out from this additivity property. Then we can have a verysimple twist spectrum. For example, in minimal models the twist spectrum takes the form τ i + n , where n is an integer that corresponds to the presence of the Virasoro descendants,and τ i is some finite collection of real numbers. Such a spectrum is generally inconsistentin higher dimensions.As a simple illustration of how this result is evaded in two dimensions, we can considerthe four-point function of spin fields in the 2d Ising model [24]. In this case the onlyVirasoro primaries have quantum numbers (0 , , ), and ( , ) for the spin field.One can check explicitly that the leading piece in the z → z τ with τ = 0, consistently with the known spectrum. The spectral densityaround 2 τ σ vanishes. Fig. 8:
The general form of the minimal twist spectrum in an arbitrary CFT in d > . The discussion above allows us to put an upper bound on the minimal twist of thelarge spin operators appearing in the OPE of any operator with its Hermitian conjugatein d > τ O . This leads to a compelling picture ofthe spectrum of general CFTs if we combine this observation with the main claims ofsection 2, (2.11),(2.13). τ ∗ s is thus a nondecreasing convex function that asymptotes toa finite number, not exceeding 2 τ O . See fig. 8. In addition, as we emphasized above,there must be some operators populating the region of twists around 2 τ O . These may beoperators of non-minimal twist in the OPE.24n interesting direct application of these results is for the Euclidean bootstrap al-gorithm of [25]. If we assume that s c = 2 (which is consistent with all the examples weknow), then we see that imposing the gap condition for operators with s > , τ ∗ s > τ + τ should not have any solutions (in unitary conformal field theories) . Let us, for simplicity, focus on the case of four scalar operators of the type hOO † OO † i where the dimension of O is ∆. (The case of non-identical scalar operators is consideredin appendix B.) The crossing equation in this case takes the form F ( z, z ) = (cid:18) zz (1 − z )(1 − z ) (cid:19) ∆ F (1 − z, − z ) . (3 . z → z finite. We have taken into account the contribution of the unit operator in this limit. Letus now consider the next operator that contributes in this limit. This operator has thesmallest twist among the operators appearing in the OPE (excluding the unit operator).We denote this smallest twist by τ min . We denote the dimension and spin of this operatorby ∆ min , s min such that ∆ min − s min = τ min . The twist gap τ min > e − στ min compared to the unit operator contribution.It would be very important for our purpose to determine the contribution of thisoperator in more detail. For this we need the conformal block corresponding to thisoperator. The general conformal block entering the s-channel is denoted by g ∆ ,s ( z, z ).Since in this case we are doing it for the t-channel we need to evaluate it with the arguments g ∆ ,s (1 − z, − z ).One useful property of conformal blocks that we need is their behavior near the light-cone. In the limit z → z the conformal block is given by z τ F ( z ), where F is simply related to the hypergeometric function F . This is true in any number ofdimensions. We can consider F ( z ) as z approaches 1. We find the following leadingasymptotic form [42] g ∆ ,s ( z, z ) → − Γ( τ + 2 s )( − s Γ (cid:0) τ +2 s (cid:1) z τ (log (1 − z ) + O (1)) . (3 . z → z approach 1.25ombining all the factors we arrive at the following result for (3.5) in the limit σ ≫ β F ( z, z ) = (cid:18) zz (1 − z )(1 − z ) (cid:19) ∆ O − e f Γ( τ min + 2 s min )( − s min Γ (cid:0) τ min +2 s min (cid:1) (1 − z ) τ min2 log ( z ) ! + · · · . (3 . · · · stand for operators with higher twist than τ min and for various subleading con-tributions from the O min conformal block. The contributions from operators with highertwist than τ min are further suppressed by powers of 1 − z (this translates to exponentialsuppression in σ in the limit σ ≫ β fixed). f is the usual coefficient appearing inthe conformal block decomposition. It is fixed by the three point function hO min OO † i andthe two-point functions of these operators.We rewrite (3.10) in terms of β and σ . We get F ( β, σ ) = (cid:18) e − β (1 − e − σ )(1 − e − β ) e − σ (cid:19) ∆ O β e f Γ( τ min + 2 s min )( − s min Γ (cid:0) τ min +2 s min (cid:1) e − τ min σ ! + · · · . (3 . s dom = σ + log (2∆ − (2∆ + 1)
4∆ + O ( e − σ ) . (3 . → ∞ .(In this case the formula simplifies to log s dom = σ + log ∆ + O ( e − σ ).) Otherwise, thesaddle point is not localized and its precise location is not very meaningful. It is howevermeaningful that when σ becomes large the dominant spins are large approximately as (3.12)dictates, s dom ∼ e σ . Below we will “re-sum” all the contributions around the saddle pointin order to get various detailed predictions.Thus, we are building a perturbation theory around the point s = ∞ , which is dom-inated by the unit operator in the t-channel and double-twist operators in the s-channel.Note that this perturbation theory is very different from the one in [43], where the ex-pansion is for large N theories. The expansion we are considering is valid in any CFT in d >
2. 26n (3.11) we have included the contribution of the first leading operator in the t-channel, after the unit operator. Unitarity guarantees that τ min < τ min ≤ d − ≥ d − . Thus, equality is only possible for free fields. This is importantbecause it means that this contribution to F ( β, σ ) from τ min is rising exponentially. Hence,it cannot be accounted for by adding finitely many operators in the s-channel, or changingfinitely many three-point functions. The saddle point (3.12) suggests that this contributionfrom τ min should be accounted for by introducing small corrections to operators with verylarge spins.Taking into account the exponentially suppressed correction proportional to e − τ min σ in (3.11), the twists of the operators propagating in the s-channel are now modified. Themodification is very small for large enough spin because of the relation (3.12) between thespin and σ . Treating this term proportional to e − τ min σ as a perturbation, to leading orderwe can thus replace e − τ min σ by 1 /s τ min , where s is the spin. Then we can interpret thisperturbation as a correction to the twist, δτ s , of such high spin operators by matching thedependence on β . We find that for large sδτ s = − c τ min s τ min + O (cid:18) s τ min + ǫ (cid:19) , (3 . c is some constant independent of the spin and ǫ >
0. If these double-twist operatorsare also the minimal twist operators in the OPE (as we will discuss in some examples),then we know from monotonicity of the twist that c τ min >
0. We see that the limiting twist2 τ O is approached in a manner that is completely fixed by the operator with the lowestlying nontrivial twist in the problem. For example, it could be some low dimension scalaroperator, or the energy momentum tensor (for which τ min = d − c τ min . Here we compute it for the caseof external scalar operators of dimension ∆. As explained in appendix B, the relevantcontribution from the s-channel takes the form −
12 log z z ∆ × lim z → ∞ X s =Λ c τ min s τ min c s z ∆+ s F (∆ + s, ∆ + s, s + 2∆ , z ) . (3 . In other words, we use e − βτ − βδτ = e − βτ (1 − βδτ + · · · ). For the small spin cut-off we chose Λ. Since the sum is dominated by large spin operators inthe limit we are considering, nothing will depend on Λ in an important way. P ∞ s =0 → R ds and also make a change ofvariables motivated by the saddle point analysis of appendix B, s → s √ − z . Denoting ǫ = 1 − z and using for the ǫ → F ( h √ ǫ , h √ ǫ , h √ ǫ , − ǫ ) ∼ h √ ǫ √ hK (2 h ) √ πǫ , (3 . ǫ limit) that (3.14) is equal to c τ min (1 − z ) ∆ − τ min2 Z ∞ ds s − τ min − K (2 s )= c τ min (1 − z ) ∆ − τ min2 Γ(∆ − τ min ) Γ(∆) . (3 . c τ min Γ(∆ − τ min ) Γ(∆) = 2 e f Γ( τ min + 2 s min )( − s min Γ (cid:0) τ min +2 s min (cid:1) , (3 . c τ min to get c τ min = Γ( τ min + 2 s min )2 s min − Γ (cid:0) τ min +2 s min (cid:1) Γ(∆) Γ(∆ − τ min ) f ,f = C OO † O τ min C OO C O † O † C O τ min O τ min = ( − s min e f . (3 . − s min in the second line of (3.18) comes about because C OO † O τ min differs from C O † OO τ min by a minus sign for odd spins. This will be important below.For convenience, above we have included the expression connecting f with vari-ous two- and three-point functions (our conventions for two- and three-point functionsare summarized in appendix A). Thus, we have computed the coefficient that controlsthe leading correction to the anomalous dimension of double-twist operators in any CFT τ O ∂ s O † = 2 τ O − c τ min s τ min + · · · . Let us emphasize several features of this formula:- It is manifestly invariant under redefinitions of the normalizations of operators;- The formula is valid for any d > O ∂ s O † is clearly convex for large enough spin, in any CFT.- If the minimal twist operator O min is not unique, we just have to sum the differentcontributions c = P c i ;- From unitarity we know that τ min < τ min ≤ d −
2. In section 5 we will seean example where all the contributions with τ min ≤
2∆ cancel out. Then the leadingcontribution to the asymptotic anomalous twist will come from τ min > The general results discussed in the previous subsections can be naturally understoodat the qualitative level using the ideas of [31]. Such ideas were also applied in [45].One can think about the four-point correlation function in a d -dimensional CFT as afour-point function in a two-dimensional gapped theory. Denote the coordinates on thistwo-dimensional space by ( u, v ). The gap in the spectrum is the twist gap of the underlying d -dimensional CFT. More generally, the evolution in the u direction is dictated by the twistof the underlying d -dimensional CFT. The operators could be thought as being insertedat ( ± u , ± v ).When operators are largely separated in the v direction the interaction is weak andcorrections to the free propagation e − τ O u are small. The corrections are governed by theseparation v . Since the theory is gapped, the correction to the energy due to the exchangeof a particle of “mass” τ exch takes the form e − τ exch v , which could be made arbitrarily smallby taking large enough v .Since ∂ u measures the twist we see that in this limit of large v we have a propagatingstate with the twist arbitrarily close to 2 τ O . By the state-operator correspondence we haveto identify it with some operators in the theory. This is the additivity property alluded toabove. 29oreover, [31] identified how the v separation is related to the spin of the opera-tors. They suggested the relation log s dom ∼ v . That allows to interpret the correctionpotential energy due to the leading interaction, e − τ exch v , in terms of a correction to thetwist of high spin operators, δτ ∼ cs τexch . For large v we the most important exchangedparticle is the one with the smallest twist, hence τ exch is identified with τ min . This is thestatement of (3.13). The fact that at large spin we approach the generalized free fieldpicture corresponds to the fact that at large separations excitations do not interact andpropagate freely. (In other words, locality in the v coordinate.)We see that the free propagation in the Alday-Maldacena picture corresponds to theunit operator dominance in the OPE picture. Corrections due to the exchange of massiveparticles in the Alday-Maldacena picture correspond to the inclusion of the next termsin the light-cone OPE. Also in this picture the state-operator correspondence plays animportant role.The relation log s dom ∼ v of [31] is qualitatively correct, but there are very importantcorrections to it that we could determine precisely in our formalism (3.12). (And as weexplained, the saddle point itself is wide unless ∆ → ∞ , so one needs to re-sum thecorrections around it. This is what we have essentially done in (3.15),(3.16).) Thosecorrections play a central role in the determination of the precise shift in the twists of fastspinning operators. In the picture of [31] those corrections should result from taking intoaccount the finite width of the wave function in the v coordinate. This would be interestingto understand better, perhaps along the lines of [45].
4. Examples and Applications of (3.17)
In this section we study the correction formula (3.18) in different regimes and circum-stances, comparing it to known results when possible.
It is well known that in the case of the stress tensor the coupling of it to other operatorsis universal and depends only on the dimension of the operator and two-point function ofthe stress tensor. More precisely, in (3.18) f = d ∆ ( d − c T so that we get c stress = d Γ( d + 2)2 c T ( d − Γ( d +22 ) ∆ Γ(∆) Γ(∆ − d − ) . (4 . c T = d + 1 d − L d − πG ( d +1) N Γ( d + 1) π d Γ( d ) , (4 . c stress = 4Γ( d ) π d − ( d − G ( d +1) N L d − Γ(∆ + 1) Γ(∆ − d − ) . (4 . N = 4 SYM theory. Wesubstitute d = 4 and G ( d +1) N L d − = π N so that c T = 40 N and we get that the correction dueto the stress tensor exchange is given by c N =4 stress = 2∆ (∆ − N . (4 . τ : O ∂ s O : = 8 − s + 1)( s + 6) 1 N . (4 . O stands for the operator dual to the dilaton. The only twist 2operator in the t-channel is the stress tensor. By plugging ∆ = 4 into our formula wecorrectly reproduce N !For other computations of the anomalous dimensions of double-trace operators intheories with gravity duals see [48,49]. In all those cases we found that the sign of thecorrection is consistent with convexity, and the leading power of s is as predicted. However,often more than one operator of twist 2 is exchanged in the t-channel and (4.4) will not bethe complete answer. To get the complete answer we just have to sum over finitely manycontributions. ∆ Limit
It is curious to consider large ∆ limit of (3.18). The factor
Γ(∆ − τ min2 ) Γ(∆) becomes ∆ τ min and, thus, c τ min has a simple dependence on ∆, c τ min ∼ ∆ τ min f where f is the combinationof three- and two-point functions written in (3.18). The limit of large ∆ is also nice in31hat the saddle point described in appendix B becomes exact and simplifies as describedafter (3.12).The point we would like to make in this subsection is that since for large ∆ the saddlepoint (3.12) becomes exact, the idea of [31], which we outlined in subsection 3.5, can bemade precise.According to the picture we reviewed in subsection 3.5, the scaling dimensions ofthe double-twist operators are related to the total energy of two static charges in sometwo-dimensional space interacting through a Yukawa-like potential∆ + ∆ + gq q e − mv , (4 . v is the separation between the particles. We assume for simplicity ∆ = ∆ ≡ ∆.Now we use the fact that the saddle point is exact if ∆ is large and the relation at thesaddle point is v = log s ∆ . Additionally, from our formula for the leading correction to thedimensions of double-twist operators we read gq q ∼ C OO†O τ min C OO C O†O† C O τ min O τ min . The mass m appearing in (4.6) corresponds to the minimal twist that we exchange (this is directlyrelated to the fact that the lightest particle that we exchange leads to the longest rangeinteractions).These identifications are natural, for example, the sign of the correction has a verysimple interpretation. Gravity is always an attractive force, thus, the correction fromthe potential energy is always negative. This is reflected by the fact that for the energymomentum tensor we have g ∼ − c T and q ∼ ∆. However, if we have a U (1) current theforce could be either attractive or repulsive depending on the signs of the charges.The identification v = log s ∆ and the relation between gq q and various two- andthree-point functions all receive nontrivial corrections in ∆ − . We have accounted for allof them precisely in the previous section. ∆ − τ min The formula (3.18) goes to zero when ∆ = τ min . This is consistent because such anequality could only be realized in free field theory (or, if one abandons unitarity it couldbe realized in models such as generalized free fields). And indeed, in free field theory(or generalized free fields) the answer is zero; there are no corrections to the twists ofdouble-twist operators. 32owever, the case of small ∆ − τ min is in fact more subtle. For example, consider someweakly coupled CFT. Then if we choose the external state appropriately, the leading twistoperators in the OPE would have τ min very close to 2∆. But there would be infinitelymany such operators. In order to get the right answer for the anomalous dimensions offast spinning operators we would generally have to sum them all and be careful aboutperforming the perturbation theory consistently.An example is the φ theory in 4 − ǫ dimensions. We consider the φ ( x ) φ (0) OPE,which includes many operators whose twists are very close to 2. In order to obtain theright anomalous twists of fast spinning operators of the type φ∂ s φ as discussed in section3, we have to sum all of those operators up. In this case ǫ breaks a higher spin symmetryand so the infinitely many operators with twists close to 2 can be regarded as due to aslightly broken higher spin symmetry.More generally, imagine we have a CFT which contains a small parameter ǫ . Imaginewe also know the twist τ O ∂ s O ( s, ǫ ) of some double-twist operators exactly as a function of∆ and s . From the discussion above we know that at sufficiently large spin we expect itto have the form τ O ∂ s O ( s, ǫ ) = 2∆ − c ( ǫ ) s d − − c ( ǫ ) s d − γ ( ǫ ) − ... (4 . γ ( ǫ ) is the anomalous dimension of the minimal twist spin-four operator. In theabove we have assumed that the minimal twists are realized by the energy-momentumtensor, spin-four operators etc. Of course it could also be that O has a twist smaller than d − ǫ is small, the expansion (4.7) is only usefulfor spins much larger than any other parameter in the theory (including e γ ǫ ) that wouldnaturally arise in a situation like (4.7)).If one decides to fix the spin and take ǫ arbitrarily small, then all the operatorsin (4.7) become important. In this regime, the problem we are discussing is essentiallythat of solving perturbative CFTs via the bootstrap program. This is not our goal here,and we will not comment on it further. The interested reader should consult [43,50] forinteresting recent progress in this direction.In any given CFT, the results (3.13),(3.18) always apply for sufficiently large spin.For example, in the Ising model τ d Isingσ∂ s σ ∼ . − . s + ... (4 . s in order for the next terms to be suppressed (the nextterm is controlled by the anomalous twist of the spin 4 operator, which is 0.02, and thusthe formula (4.8) is useful only for exponentially large spins). In the next section we willdiscuss some results about the Ising model that are more powerful from the practical pointof view. As we explained above, the correction c τ min in (3.13) could have either sign dependingon which operators propagates in the t-channel and depending on the precise form of thefour-point function we study. We demonstrate this point here by considering four pointfunctions of the form hO i ( x ) O j ( x ) O † i ( x ) O † j ( x ) i where O i and O i are operators chargedunder global symmetries. In the expansion in the s-channel we necessarily encounterdouble-twist operators of the type O ∂ s O which are charged under some global symmetries.We cannot directly apply the results of section 2 for such operators since we do not havepositivity of the cross section argument. It is thus interesting to ask whether such operatorsapproach the limiting twist 2∆ in a convex, flat, or concave manner. Here we will explainthat under some very general circumstances this is, roughly speaking, fixed by the chargeto dimension ratio of O . Let us set up the framework more precisely.Imagine we have N f flavor currents J µK . And imagine also we have a set of scalaroperators O i charged under this symmetry so that[ Q, O i ] = − ( T K ) ji O j (4 . J µK ( x ) O i (0) ∼ − iS d x µ x d ( T K ) ji O j (0) . (4 . S d is the surface area of a d − hO i ( x ) O † i (0) i = g ii x , h J µI ( x ) J µJ (0) i = τ IJ S d I µν x d − . (4 . hO i ( x ) O † j ( x ) J µI ( x ) i = − iS d ( T I ) ji g jj Z µ x − ( d − x d − x d − (4 . Z µ = x µ x − x µ x .Now we consider the four-point function hO i ( x ) O j ( x ) O † i ( x ) O † j ( x ) i . Using (3.18)we can account for the contributions of the conserved currents in the t-channel to δτ s . Wefind ( δτ s ) currents ∼ − X I,J C O i O † i J I C O j O † j J J C O i O † i C O j O † j C J I J J = − X I,J ( T I ) ki g ki ( T J ) mj g mj τ IJ g ii g jj . (4 . − s ) we get for the sum of the stress-tensor andrepulsive flavor corrections (here we have assumed that the minimal twists that appear inthe t-channel are the EM tensor and conserved currents) δτ s ∼ − s d − d ∆ d − c T Γ( d + 2)Γ( d +22 ) − Γ( d )2Γ( d ) X I,J ( T I ) ki g ki ( T J ) mj g mj τ IJ g ii g jj . (4 . δτ s is convex, flat, or concave for large spinis determined by the ratio of ∆ /c T compared to, roughly speaking, q /τ IJ where q is thecharge of the operators (more generally, the representation). Therefore, the question ofconvexity at large spin is determined by whether O satisfies a BPS-like bound. The BPS-like bound that controls convexity for the simple case of stress tensor and a single U (1)symmetry is ∆ √ c T ≥ √ d − p d ( d + 1) | q |√ τ . (4 . ig. 9: The correction due to the U (1) current exchange has the opposite sign forthe anomalous twists of O ∂ s O and O ∂ s O † . At the level of the bootstrap equations,it corresponds to the fact that the three-point function hOO † J s i is odd under apermutation of O and O † when s is odd. (Here J s is an operator of spin s .) Let us now mention that if we consider instead the correlation function in the order-ing hO i ( x ) O † i ( x ) O j ( x ) O † j ( x ) i and study the s-channel expansion, then we encounterdouble-twist operators of the type O ∂ s O † . In this case, the t-channel expansions gives adifferent results from before. Because we flipped two operators (see fig. 9), all the evenspins still contribute positively to c but now the odd spins contribute positively as well.Hence, this essentially proves convexity in a new way (without alluding to an RG flow) forlarge enough spins! In fact, in some sense, it proves a slightly different result than the onein section 2, since here we do not need to assume that the double-twist operators are theminimal twist operators to get convexity – the convexity of section 2 is always about theminimal twist operators.As an example of the discussion above let us consider an N = 1 SCFT in four dimen-sions. In the case of SCFT we have a U (1) R global symmetry. The two-point function ofthe R-current is related via supersymmetry to the two-point function of the stress tensor(see, for example, formula (1.11) in [51] ). The relation in our conventions is c T τ = d ( d + 1)2 . (4 . Our conventions are related to theirs by τ here S d = d − d − π ) d τ there and c T ; here S d = c T ; there . hO ( x ) O ( x ) O † ( x ) O † ( x ) i where O is a chiralprimary and O † is an anti-chiral primary. We can use the formula (4.14) to evaluatethe asymptotic corrections to the operators O ∂ s O appearing in the chiral × chiral OPE.Using the relation (4.16) and the fact that for chiral primaries ∆ = r where r is the R -symmetry charge of O , we find that the correction (4.14) precisely vanishes. Furthermore,if O carries some global symmetry quantum numbers other than the R -symmetry, weshould include the contributions to δτ s from the exchanges in the t-channel of the flavorsupermultiplet (this is a linear multiplet of N = 1). We find that this contribution againvanishes identically.One could ask why these corrections to δτ s vanish identically for chiral primaries. Inthe OPE of two chiral primaries one finds short representations with twists precisely 2∆and long representations with twists larger than 2∆. (There are no smaller twists.) Thecancelation above may be related to the presence of infinitely many short representations inthe OPE. It would be nice to understand whether the OPE of two chiral primaries containsinfinitely many protected operators, as the computations above hint. For completeness,the structure of the OPE of two chiral primaries is reviewed in appendix D.Another curiosity that we would like to note is that if O is chiral but not a chiralprimary, then ∆ > r and the exchange of the stress tensor dominates over the R -current.Thus, we have convexity again. We see that the general connection between the BPS-likebound (4.15) and convexity is realized very naturally in SUSY.
5. Applications of Convexity
In this section we consider different known examples and verify that the convexity andasymptotic form of the twists hold true. Moreover, we will see that in all known examplesthe convexity starts from spin 2 (thus s c = 2). It is easy to see that the propositions made in the previous sections are true in freetheories. In this case we have an infinite set of conserved currents j s with the minimalpossible twist τ = d − O we expect the three-point functions hOO † j s i to be non-zero.This is suggested by the way higher spin symmetry is realized in the sector of conservedcurrents as well as from the way it acts on the fundamental field.37 ig. 10: The form of the minimal twist spectrum in free theories.
The conserved currents will be the leading twist operators. In this way convexity with s c = 2 is realized in the case of free theories. See fig. 10.It is also trivial to see that there exist operators with twist arbitrarily close to 2 τ O .Indeed, for any composite operator O in a free theory, in the OPE of this operator withitself there will appear the operators O ∂ s O † (written schematically). Those have twistprecisely 2 τ O . If the operator O is the elementary free field itself, then these operatorsare the higher spin currents. Their twist is precisely d −
2, which coincides with twice thedimension (and hence twice the twist) of the elementary scalar field.Weakly coupled CFTs inherit some of their OPE structure from their higher spinsymmetric parents. (The OPE coefficients and the dimensions change a little. Variousshort multiplets could combine, for example, the exactly conserved higher-spin currentsundergo a “Higgs mechanism” and acquire a small anomalous twist. Also new operatorsthat were not present in the free field theory could appear.) If the CFT is sufficientlyweakly coupled, the almost conserved currents will still be the minimal twist operatorsin the OPE of any operator with itself. This means that in this case our statement fromsection 2 boils down to the convexity of twists of almost conserved currents. The examplesbelow are of this type.Notice that in the case when we have a flux (e.g. in gauge theories) the situation isdifferent. The almost conserved currents universally receive a correction to the twist of theform log s . Thus, for large enough spin, their twist could be arbitrary large so that theycease to be the minimal twist operators. In this case some other operators (the doubletwist operators) ensure additivity and convexity.38 .2. The Critical O ( N ) Models
Let us consider the O ( N ) critical point in 4 − ǫ dimensions. We can use the resultsof [52] for the anomalous dimensions. We have δτ σ i ∂ s σ i − γ σ = − ǫ N + 22( N + 8) s ( s + 1) ,δτ σ ( i ∂ s σ j ) − γ σ = − ǫ N + 22( N + 8) N + 6)( N + 2) s ( s + 1) , (5 . γ σ is the anomalous dimension of the spin field. The operators σ i ∂ s σ i are singletsunder the global symmetry O ( N ), and σ ( i ∂ s σ j ) are symmetric traceless representations of O ( N ).Let us make several comments about the formulae above. First of all, both formulaehave a well-defined large s → ∞ limit that implies δτ s − γ σ → This means thatthe prediction (3.1) is saturated, as it must in weakly coupled CFTs where we consideroperators that are bilinear in the fundamental field. Second, the leading correction aroundinfinite spin in both cases is of the form s . This is what we expect due to the presence ofthe stress tensor and other currents which are conserved at this order. (To this order also σ has twist d − s = 2.At higher orders in ǫ , the higher spin-currents will lead to different typical exponents atlarge spin, s τ min( ǫ ) . The dominant exponent at very large spin will still be due to the stresstensor, s − ǫ . (The reason for that is that higher spin currents have a larger anomaloustwist and the operator σ appears to have dimension, and thus twist, larger than d − σ is suggested from the epsilon expansion, from the knownresults about low N critical O ( N ) models, and the results of large N .)Alternatively, we can consider the case of large N with the number of space-timedimensions being d . In this case we can use the results of the large N expansion [53],[54] δτ σ i ∂ s σ i = 2 γ σ d + 2 s − d + 2 s − (cid:20) ( d + s − s − −
12 Γ[ d + 1]Γ[ s + 1]2( d − d + s − (cid:21) ,δτ σ ( i ∂ s σ j ) = 2 γ σ d + s − s − d + 2 s − d + 2 s − ,γ σ = 2 N sin π d π Γ( d − d − d + 1) . (5 . To leading order 2 γ σ = N +22( N +8) ǫ .
39n the large s limit these formulae become δτ σ i ∂ s σ i − γ σ γ σ ∼ − Γ( d + 1)2( d −
1) 1 s d − − d ( d − s ,δτ σ ( i ∂ s σ j ) − γ σ γ σ ∼ − d ( d − s . (5 . N we have conservedcurrents operators with the twist τ = d −
2, and we also see 1 /s coming from the operator σ . The difference in the coefficients of s d − for the operators σ i ∂ s σ i and σ ( i ∂ s σ j ) comesfrom the fact that different conserved higher spin currents contribute. We see that the sumover all the higher spin currents (each contributing to 1 /s d − ) actually vanishes for thesymmetric combination. However, the s contribution is governed solely by the three pointfunction h σ i σ i σ i and, thus, the coefficient of 1 /s is the same for both operators σ i ∂ s σ i and σ ( i ∂ s σ j ) . We can easily reproduce the coefficient of 1 /s using our formula (3.18)and the known two- and three-point functions in this model (at large N ). We find aprecise match between the perturbative calculation (5.3) and our general methods usingthe bootstrap equations. The necessary details are collected in appendix E.Now we can turn to predictions using convexity. We consider arbitrary N and ǫ .Of particular interest are three-dimensional small N models (for which there is no smallparameter in the problem). They describe many second order phase transitions. Forexample, in the case of N = 1 we have the 3d Ising model which describes the vapor-liquidcritical point of water.All these theories are known to contain an almost free scalar operator (the spin field)∆ σ ∼ .
5, thus, the prediction is that all these theories contain operators of spin 4 , , ..., ∞ with twists 1 < τ s < σ . (5 . The cancelation of 1 /s d − in the second line of (5.3) is easily understood in our language.Consider the four-point function of spin fields h σ σ σ σ i . The only operator in the t-channelthat can propagate is the interacting field σ . The higher spin currents cannot propagate becausethey behave essentially like free fields and so contribute zero to the t-channel diagrams. Thisshows that the operators σ ∂ s σ do not have a piece that goes like 1 /s d − at large spin. σ = 0 . < τ s < . τ ∼ . s = 2. Fig. 11:
The form of the minimal twist spectrum implied by convexity in the 3DIsing model. The shaded region describes the allowed values for almost conservedcurrents of spin s ≥ . Using monotonicity and the known twist of the spin four operator, we can furthermake the prediction that there is an infinite set of almost conserved currents in the 3DIsing model starting from s = 6 with the twists1 . < τ s < . , s = 6 , , ..., ∞ . (5 . For theories with gravity duals, convexity could be tested via AdS/CFT. One may beable to think about convexity as some hidden constraint on the low-energy effective actionsin AdS in the spirit of [15]. We leave the exploration of this question for the future. Atypical characteristic of theories with gravity duals is that these theories possess a largegap in the spectrum. See [43] for a systematic approach to analyzing such theories.
Fig. 12:
The form of the minimal twist spectrum implied by convexity in a theorydescribed by supergravity.
41n this context, convexity is at statement about the anomalous dimensions of variousdouble-trace operators, which are the double-twist operators in this case. For very largespins convexity was proved using the crossing equations in section 3 while for spins oforder O (1) we can use the argument of section 2. In the bulk gravity duals, this correspondsto the energy of various spinning two-particle states. Hence, convexity implies that thebinding energy of fast spinning operators in the bulk is always negative (i.e. the particleshave net attraction) and the binding energy approaches zero in a power law fashion. Thisis depicted schematically in fig. 12, where 2 τ O is the twist of the two-particle state at verylarge spin (hence, the binding energy goes to zero). We emphasize that all these bindingenergies are of order 1 /N or less. Hence, the properties delineated above are about various1 /N corrections that come from tree-level diagrams in supergravity. Recently there was some progress in understanding the higher spin symmetric phaseof the AdS/CFT correspondence [55,56,57]. On the CFT side one can notice that suchsymmetries fix correlations functions up to a number [58]. However, in the case when higher spin symmetry is broken the situation is much lessclear. On the boundary sometimes it is possible to use the constraints to get some generalpredictions [60,61,62]. But they are very limited and are not applicable in the generalsituation. In the bulk the situation is even less clear. It would be very interesting tounderstand higher spin symmetry breaking in the bulk better.Convexity is a generic constraint on any possible higher spin symmetry breaking.On the boundary, slightly broken higher spin symmetry currents are the minimal twistoperators discussed before. In the bulk, this maps to a statement about the masses ofHiggsed gauge bosons.The scaling dimensions of operators with spin s are mapped to the masses of the dualstates in the bulk [63] m = C (∆ , s ) − C ( d − s, s ) = δτ s ( δτ s + 2 s + d − . (5 . C is the quadratic Casimir of SO ( d,
2) and δτ s is the anomalous twist (when it iszero the particle is massless). The expression for n -point correlation functions of currents was recently found in [59]. AdS with higherspin breaking boundary conditions. We know from additivity that for large spins τ s = d − γ σ + O ( s − d ) so we can approximate the masses of the higher spin bosons in thebulk as m s ≈ γ σ s + O (1) . (5 . γ σ is the anomalous dimension of the spin field in the O ( N ) model (the singlet O ( N )model was first considered in [55]).In the singlet O ( N ) model, σ is not part of the spectrum, but we can neverthelessdefine its anomalous dimension: γ σ = lim s →∞ ∂m s ∂s = lim s →∞
12 ( τ s − . (5 .
6. Conclusions and Open Problems
In this paper we investigated unitary CFTs in the Lorentzian domain. By flowing tothe gapped phase and using positivity of a certain cross section, we established a convexityproperty of the minimal twists in the O ( x ) O † (0) OPE (2.13). Studying the crossingequations in the light-cone limit we found that there is an additivity property in thetwist space and that the inequality (3.1) has to hold. Additivity means that in d > s . We reformulated their picture in terms of the OPE and that allowedus to extend the analysis of [31] and compute the leading spin corrections exactly.43e proceeded by studying various examples and applications of these general prop-erties. We have found that convexity is a recurring theme at large spin. We checked thatthe general picture that we obtained is indeed realized in known theories (mostly weaklycoupled theories or theories with gravity duals).We concluded that there should not be solutions of the crossing equations with τ ∗ s > τ + τ and we checked our picture against the available numerical results [26].We emphasized some predictions for the 3d critical O ( N ) models that are relevant forphase transitions. One can set bounds on an infinite set of operators in the critical O ( N )models (5.4). In the case of the 3d Ising model one can do better (5.5).Another example we considered is the correction to the twists of chiral operators in N = 1 SCFTs. We found that to the order we computed the twist spectrum of chiralprimaries remained flat at large spin (in principle it could have been concave). If weconsider chiral operators that are not chiral primaries, then, as we have explained insection 4, the usual superconformal unitarity bound leads to convexity at large spin.For a generic CFT we found that there is a relation between the convexity/concavity ofthe large spin twists of double-twist operators and the mass/charge ratio. More precisely,assuming that the minimal twist operators are the stress tensor and a U (1) current wefound that O ∂ s O convexity ↔ ∆ ≥ ( d − | q | , (6 . c T τ = d ( d +1)2 (we can alwaysdo this for Abelian currents. For concavity the inequality works in the opposite direction.)Hence, by studying the dimensions and charges of arbitrary primary operators invarious CFTs one can conclude about convexity/concavity of the large spin anomalousdimensions.Our results are also relevant for the high spin symmetry breaking in CFTs and throughAdS/CFT to theories of quantum gravity in the bulk. As higher spin symmetry breaks,higher spin gauge bosons become massive through the Higgs mechanism. The minimaltwists discussed above correspond to almost conserved higher spin currents. For the caseof parity preserving Vasiliev theory, from the property of additivity of the twists we inferthat the masses of these gauge bosons at high spin follow a Regge-like formula, with theprefactor fixed by the dimension of the spin field.There are several concrete open questions which would be interesting to address.44ne obvious generalization would be to compute the next, subleading, orders in the s expansion. Another concrete problem that would be nice to address is the computation ofthe coefficients controlling the power law corrections for the operators O ∂ s n O . Finally,it would be interesting to derive convexity using the ideas of [65]. It was noticed in [65]that the Regge limit of the DIS amplitude we considered in section 2 is related througha conformal transformation to the problem of measuring energy distributions in the finalstate created by some operator. Moreover, the fact that the energy distributions areintegrable may constrain the Regge asymptotics of graviton deep inelastic scattering. Itwould be interesting to understand whether this is indeed the case. Acknowledgments:
We would like to thank Thomas Dumitrescu, Tom Hartman, JuanMaldacena, Nathan Seiberg and David Simmons-Duffin for useful discussions. ZK wassupported by NSF grant PHY-0969448, a research grant from Peter and Patricia GruberAwards, a grant from the Robert Rees Fund for Applied Research, and by the Israel ScienceFoundation under grant number 884/11. ZK would also like to thank the United States-Israel Binational Science Foundation (BSF) for support under grant number 2010/629.Any opinions, findings, and conclusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect the views of the funding agencies.
Appendix A. Normalizations and Conventions
Here we collect our normalization and conventions for two- and three-point functionsas well as the normalizations of conformal blocks. We mostly follow the conventions of [66] r ij = ( x i − x j ) ,I µν ( x ) = δ µν − x µ x ν x ,Z µ = x µ r − x µ r . (A.1)Then for the two- and three-point functions contracted with polarization tensors C µ wehave h O l · C ( x ) O l · C ( x ) i = c O l O l ( C µ I µν C ν ) l r ∆12 , hO ∆ ( x ) O ∆ ( x ) O l · C ( x ) i = c OO O l ( Z · C ) l r (∆ +∆ − ∆)12 r (∆ +∆ − ∆ )13 r (∆ +∆ − ∆ )23 . (A.2)45or the conformal block we have the following asymptotic behavior [67] G d, ∆ ,s ( z, z ) ∼ z → ( −
12 ) s z ∆ − s z ∆+ s F (∆ + s, ∆ + s, s + 2∆ , z ) . (A.3)This leading behavior is independent of the number of space-time dimensions.In the text various notions of twist, minimal twist etc. appeared. For convenience,we summarize here some of the notation and terminology we used.- Conformal twist. It is given by τ = ∆ − s and governs the DIS experiment weconsidered. Unless stated otherwise, by twist we always mean the conformal twist.- Collinear twist. It is given by τ coll = ∆ − s + − and the light-cone OPE is arrangedaccording to collinear twists of operators.- By τ ∗ s we denote the conformal twist of the spin s minimal conformal twist operatorthat appears in the OPE of some operator with its conjugate O ( x ) O † (0). This isagain relevant for DIS.- By τ min we denote the minimal twist among the operators different from the unitoperator that appear in the t-channel of various four point functions. The spin of thecorresponding operator could be arbitrary.- By τ s we denote the twists of double-twist operators that appear in the s-channel andcorrespond via the bootstrap equations to low twist operators in the t-channel. Appendix B. Relating Spin to σ B.1. A Naive Approach
Consider the disconnected part of the four-point function of real scalar operators withdimension ∆. In our notation for the cross ratios, the four point function takes the form F ( z, z ) = 1 + ( zz ) ∆ + (cid:20) zz (1 − z )(1 − z ) (cid:21) ∆ . (B.1)As in the main text, we take z to be small and consider the piece z ∆ only. This correspondsto focusing on the contributions of the leading collinear twist “double-trace” operators.This leading piece in the four-point function takes the form z ∆ f ( z ) where f ( z ) = z ∆ (cid:20) − z ) ∆ (cid:21) . (B.2)46e can decompose this function in terms of collinear conformal blocks f ( z ) = ∞ X s =0 c s z ∆+ s F (∆ + s, ∆ + s, s + 2∆ , z ) , (B.3)where the c s are known to be (see e.g. [43]) c s = (1 + ( − s ) Γ(∆ + s ) Γ(2∆ + s − s + 1)Γ(∆) Γ(2∆ + 2 s − . (B.4)This expansion is valid in any number of dimensions even though the full conformal blocksfor generic d are not known. The simplification is due to the fact we consider small z andthus collinear conformal blocks, which are much simpler than the full conformal blocks.Let us now substitute 1 − z = e − σ . We would like to study where does the contributioncome from in the sum (B.3) for some given σ . In other words, we would like to find thetypical spin of operators that dominate the contribution for some given σ . It is helpful toreplace the sum over spins by an integral and also utilize the integral representation of thehypergeometric function. We get for f ( z ) Z dt Z ∞ ∆ d b s b s − b s − Γ(1 + b s − ∆) h (1 − e − σ ) t (1 − t )1 − (1 − e − σ ) t ib s t (1 − t ) , (B.5)We expect the large σ ≫ b s b s − b s − Γ(1 + b s − ∆) = 4 b s − Γ(∆) + O ( 1 b s ) , (B.6)so that we get for the integral4Γ(∆) Z dt Z ∞ ∆ d b s e (2∆ −
1) log b s + b s log h (1 − e − σ ) t (1 − t )1 − (1 − e − σ ) t i t (1 − t ) . (B.7)We perform the integral over b s by the saddle point approximation. We get for the extremumvalue b s b s = − − h (1 − e − σ ) t (1 − t )1 − (1 − e − σ ) t i . (B.8)After evaluating the integral we get4 e − √ π (2∆ − − √ − Z dt (cid:16) − log ( e σ − − t ) te σ (1 − t )+ t (cid:17) − t (1 − t ) . (B.9)47ext we use the saddle point approximation to evaluate the t integral. The extremal valuefor t is t = 1 − (cid:18) −
12∆ + 1 (cid:19) e − σ + O ( e − σ ) . (B.10)Plugging it back to (B.8) we see that the dominant spin, s dom , islog s dom = σ + log (2∆ − (2∆ + 1)
4∆ + O ( e − σ ) . (B.11)The leading term in the relation (B.11) coincides with the identification of [31]. Noticethat for ∆ ≫ s dom ≈ σ + log ∆ . (B.12)It is important to remark that the saddle point above is only localized when ∆ ≫ z → s dom ≈ σ .In the next subsection we will present a more rigorous approach to the problem. Thisapproach allows us to control the sum (B.3) efficiently. The following discussion is crucialfor reproducing the main results of section 3. B.2. A Systematic Approach
In the previous subsection we learned that the spins dominating the s-channel expan-sion which reproduces the unit operator in the t-channel are given by s ∼ − z ) . Wedenote ǫ = 1 − z and plug this into the hypergeometric function. Below we will discuss theleading terms are ǫ →
0. From the integral representation of the hypergeometric functionwe see that it behaves as follows in this scaling limit ( A is an arbitrary coefficient thatstays finite as ǫ → F (cid:18) A √ ǫ , A √ ǫ , A √ ǫ , − ǫ (cid:19) = Γ( A √ ǫ )Γ ( A √ ǫ ) Z dt (1 − t ) A √ ǫ − t A √ ǫ − (1 − t + ǫt ) − A √ ǫ . (B.13)We approximate the pre-factor using the Stirling formula as q A π ǫ / e A log(2) √ ǫ and so F (cid:18) A √ ǫ , A √ ǫ , A √ ǫ , − ǫ (cid:19) = r A π ǫ / A √ ǫ Z dt (1 − t ) A √ ǫ − t A √ ǫ − (1 − t + ǫt ) − A √ ǫ . (B.14)48ow we need to take the ǫ → − t = λ √ ǫ and find F (cid:18) A √ ǫ , A √ ǫ , A √ ǫ , − ǫ (cid:19) = r A π ǫ / A √ ǫ Z / √ ǫ dλλ (1 − λ √ ǫ ) A √ ǫ − (1 + √ ǫλ ) − A √ ǫ . (B.15)We take the ǫ → F (cid:18) A √ ǫ , A √ ǫ , A √ ǫ , − ǫ (cid:19) = r A π ǫ / A √ ǫ Z ∞ dλλ e − λA − Aλ . (B.16)This is a nicely convergent integral, given by the modified Bessel function of the secondkind. Thus F (cid:18) A √ ǫ , A √ ǫ , A √ ǫ , − ǫ (cid:19) → r Aπ A √ ǫ ǫ / K (2 A ) . (B.17)We can now use this limiting expression and reconsider (B.3) using this approximationfor the hypergeometric function. We find f ( z ) = 4Γ (∆) ǫ ∆ Z ∞ dAA − K (2 A ) . (B.18)It is encouraging to see that the dependence on ǫ is correct in the small ǫ limit. Nowlet us consider the pre-factor. This would be another test for the fact that the procedureabove indeed captures the important contributions in the small ǫ limit. We use the integral R ∞ dAA − K (2 A ) = Γ (∆). Plugging this back into (B.18) we find f ( z ) = 1 ǫ ∆ (1 + O ( ǫ )) . (B.19)This is precisely the correct asymptotics for small ǫ , including the pre-factor. B.3. The Coefficient c τ min in (3.12) In the main text in section 3 we have seen that since the correction to the anomalousdimension is of the form e − τ min and log s dom = σ , we expect the anomalous twist at large s to take the form c τ min s τ min . Here we would like to compute c τ min exactly. The idea is thatthe small corrections to the anomalous twists in the s-channel reproduce the contributionof the operator O min in the t-channel. This contribution in the t-channel is given by − Γ( τ min + 2 s min )( − s min Γ (cid:0) τ min +2 s min (cid:1) C OOO τ min C OO C O τ min O τ min log z e (2∆ − τ min ) σ . (B.20)49t is easy to match this in the s-channel because of the log z . We reproduce thislogarithm by summing the corrections to the twists in the s-channel z ∆+ δτs ∼ δτ s z ∆ log z .Setting δτ s = c τ min s τ min , we get the basic equation that determines c τ min − c τ min Γ( τ min + 2 s min )( − s min Γ (cid:0) τ min +2 s min (cid:1) C OOO τ min C OO C O τ min O τ min e (2∆ − τ min ) σ = ∞ X s =0 c s s τ min z ∆+ s F (∆ + s, ∆ + s, s + 2∆ , z ) , (B.21)where z = 1 − e − σ and the equality should be understood in terms of leading pieces in σ .Solving for c τ min we get c τ min = − τ min + 2 s τ min )( − s τ min Γ (cid:16) τ min +2 s τ min (cid:17) C OOO τ min C OO C O τ min O τ min f ( τ min , ∆) ,f ( τ min , ∆) = lim σ →∞ e (2∆ − τ min ) σ P ∞ s =Λ c s s τ min z ∆+ s F (∆ + s, ∆ + s, s + 2∆ , z ) . (B.22)and since the sum is dominated by the large spins we expect the result to be independentof Λ.We compute f ( τ min , ∆) by performing the sum in the denominator carefully. We firstswitch to an integral and repeat the procedure of the previous subsection, including thedouble scaling limit for the hypergeometric function (B.17). Doing the integral we finallyfind f ( τ min , ∆) = Γ(∆) Γ(∆ − τ min ) . (B.23) B.4. The Most General (Scalar) Case
We are interested in generalizing the computation of the correction c τ min to the casewhen not all the operators are identical and ∆ = ∆ . This is relevant for a variety ofapplications in the main body of the text. More precisely, we are considering the followingcorrelation function hO ( x ) O ( x ) O ( x ) O ( x ) i . What we call the s-channel is the OPE12 →
34. What we call the t-channel is the OPE 23 →
14. Let us write the OPE expansionin each of the channels.For the s-channel OPE we have hO ( x ) O ( x ) O † ( x ) O † ( x ) i = (cid:18) x x x (cid:19) ∆ − ∆ X ∆ ,s p ∆ ,s g ∆ ,s ( z, z ) x ∆ +∆ x ∆ +∆ . (B.24)50nd for the t-channel OPE we have hO † ( x ) O ( x ) O ( x ) O † ( x ) i = X ∆ ,s e p ∆ ,s g ∆ ,s (1 − z, − z ) x x . (B.25)The crossing equation takes the form X ∆ ,s p ∆ ,s g ∆ ,s ( z, z ) = z ∆1+∆22 z ∆1+∆22 (1 − z ) ∆ (1 − z ) ∆ X ∆ ,s e p ∆ ,s g ∆ ,s (1 − z, − z ) . (B.26)As before we focus on the unit operator in the t-channel. It is dominant when z → z ∆1+∆22 (1 − z ) ∆2 where we again keep only theleading piece in the z expansion. This is so that we only have to deal with the collinearconformal blocks instead of the full ones.The collinear conformal blocks in this case are slightly different from the case ∆ = ∆ ,so that we get the following equationlim z → ∞ X s =Λ c s z s F ( s + ∆ , s + ∆ , s + ∆ + ∆ ) = 1(1 − z ) ∆ . (B.27)We know that the coefficients of the theory of generalized free fields solve this problem.They could be easily found using the results of [68] c s = Γ(∆ + s )Γ(∆ + s )Γ(∆ + ∆ + s − s + 1)Γ(∆ )Γ(∆ )Γ(∆ + ∆ + 2 s − . (B.28)The limit z → F ( s √ ǫ + ∆ , s √ ǫ + ∆ , s √ ǫ + ∆ + ∆ ) ∼ ∆ +∆ ǫ ∆1 − ∆22 s √ ǫ √ sK ∆ − ∆ (2 s ) √ πǫ . (B.29)Switching from the sum (B.27) to an integral we get4(1 − z ) − ∆ Γ(∆ )Γ(∆ ) Z ∞ ds s ∆ +∆ − K ∆ − ∆ (2 s ) = 1(1 − z ) ∆ . (B.30)This is analogous to (B.18) and (B.19). It serves to check that we have understood correctlyhow to resum the hypergeomtric functions on the LHS of (B.27) in this subtle limit.To compute the correction to anomalous dimension we need a slightly different integral4Γ(∆ )Γ(∆ ) Z ∞ ds s ∆ +∆ − τ min − K ∆ − ∆ (2 s ) = Γ(∆ − τ min )Γ(∆ ) Γ(∆ − τ min )Γ(∆ ) . (B.31)51o that the crossing equation becomes c τ min Γ(∆ − τ min )Γ(∆ ) Γ(∆ − τ min )Γ(∆ ) = 2 f Γ( τ min + 2 s min )( − s min Γ (cid:0) τ min +2 s min (cid:1) , (B.32)and we have for the correction to the twist of O ∂ s O in this case c τ min = 2 f Γ( τ min + 2 s min )( − s min Γ (cid:0) τ min +2 s min (cid:1) Γ(∆ )Γ(∆ − τ min ) Γ(∆ )Γ(∆ − τ min ) ,f = C O O † O τ min C O † O O τ min C O O † C O O † C O τ min O τ min . (B.33)If we take O = O † , the coefficient c τ min is positive for any intermediate operator O min . Appendix C. OPE of Chiral Primary Operators in SCFT
Here we discuss d = 4 N = 1 SCFTs (see also [69] and references therein). Thesuperconformal algebra contains as a bosonic sub-algebra the conformal algebra times U (1) R . Performing radial quantization, we label representations by the quantum numbers(∆ , j , j , R ), where ∆ is the dimension of the operator.Superconformal primaries are annihilated by S α and S ˙ α . This implies that they areannihilated by special conformal transformations as well, hence, they are primaries in theusual sense. The complete representation can be constructed by acting with Q and Q onthe superconformal primary.The quantum numbers of Q α are (1 / , / , , −
1) and of Q ˙ α are (1 / , , / , , j , j , R ), we can act on it with Q α andfind states in the representation (∆+ , j − , j , R − ⊕ (∆+ , j + , j , R − Q ˙ α and find states in (∆ + , j , j − , R + 1) ⊕ (∆ + , j , j + , R + 1).In both cases the irreducible representation with the smaller norm is then one withthe smaller spins. Demanding therefore that the states above have non-negative norm wefind two inequalities ∆ + 2 δ j , − (cid:18) j − r (cid:19) ≥ , ∆ + 2 δ j , − (cid:18) j + 3 r (cid:19) ≥ . (C.1)If one of the inequalities above is saturated, the superconformal primary is annihilated bysome supercharges. 52n the special case that either j = 0 or j = 0 (or both), there is more information thatcan be extracted by considering states at level 2. (This is similar to the case j = j = 0in the ordinary conformal group.) Let us take for example j = 0. Then states at level 2transform in (∆ + 1 , , j , R − − r < ∆ < − r excluded . The norm of this state at level 2 precisely vanishes when ∆ = − r/ − r/
2. Similarly, if j = 0, the disallowed range is r < ∆ < r .Chiral primaries are annihilated by all the Q ˙ α , which means they carry j = 0, andthus satisfy ∆ = 3 r/ x )Φ(0). It is well known that there are no singularities in this OPE. Letus examine all the possible primaries that can appear in this OPE. (The operators thatappear on the right hand side are not necessarily superconformal primaries.) We denote R (Φ) ≡ R Φ = 2∆ Φ / A Chiral primaries, i.e. states with j = 0. Those must have R = 2 R Φ and thus∆ = 2∆ Φ . So this is just the operator Φ in the OPE. It has j = j = 0.The other operators in the OPE must still be chiral (because the LHS is chiral) butthey cannot be chiral primaries. So we have to write general supeconformal descendantsthat are chiral (annihilated by Q ) and are primaries of the usual conformal group. So wecan have B Q ( ˙ α O ˙ α ... ˙ α l ) , ( α ,...,α l ) with O a superconfromal primary. This is a conformal pri-mary, and to make sure it is chiral we need the superconformal primary to obey Q ˙ α O ( ˙ α... ˙ α l ) , ( α ,...,α l ) = 0. This first order condition guarantees that Q ( ˙ α O ˙ α ... ˙ α l ) , ( α ,...,α l ) is a chiral field. By matching the R -charge we also find R O = 2 R Φ −
1. Theshortening condition and the R -charge imply immediately from (C.1) that ∆ O =2 + 2 j + 3 R Φ − /
2. We dropped the term δ j , because we know that O has tohave half-integer spin in the ˙ α indices. (In other words, l is even.) Finally, we sub-stitute j = l/ − / O = l + 2∆ Φ − /
2, and hence the dimension of Q ( ˙ α O ˙ α ... ˙ α l ) , ( α ,...,α l ) is l + 2∆ Φ . This means that the twist of these operators isprecisely 2∆. Another closely related class of operators one could think of would be obtained by antisym-metrizing the spinor index of Q with some superconformal primary O and imposing a first orderequation that the symmetrization gives zero. (This equation is necessary for chirality.) This isclearly inconsistent, since, as we explained, among the descendants, the state of lowest norm isalways the state of smallest spin. We can have Q O ( ˙ α ... ˙ α l ) , ( α ,...,α l ) for O superconformal primary. We immediately findthat R O = 2 R Φ − Q O ) ≥ l + 3 R Φ − l + 2∆ and ∆( Q O ) ≥ l − R Φ + 3 = l + 6 − Appendix D. DIS for Traceless Symmetric Representations
Consider a symmetric traceless operator of spin s , O µ ...µ s ( y ). We can contract itwith a light like complex polarization vector ζ µ , ζ = 0. We denote the result as O ( ζ, y ).The object we are interested in is A ( ν, q , ζ ) = iπ Z d d ye iqy h P | T ( O ( ζ ∗ , y ) O ( ζ, | P i , ν = 2 q · P . (D.1)The imaginary part of A is related to the total cross section for DIS.Let us consider the most general possible form for the operator product expansion O ( ζ ∗ , y ) O ( ζ,
0) = X s =0 , , ... X α ∈I s f ( α ) s ( y, ζ, ζ ∗ ) µ ...µ s O ( α ) µ ...µ s (0) , (D.2)where the symbols s, I s represent the same objects as in section 2. If the operator O µ ...µ s ( x ) is conserved that would constrain the allowed structures in (D.2). In theOPE (D.2) we retain only primary symmetric traceless operators on the right hand side.The other operators are discarded because they do not contribute to DIS as long as thetarget particle | P i is a scalar. (We will henceforth assume the target is a scalar particlefor simplicity.)We substitute the OPE expansion (D.2) in (D.1) and evaluate the expectation valuesof the operators according to h P |O αµ ...µ s (0) | P i = A ( α ) n ( P µ P µ · · · P µ s − traces ) . (D.3)The kinematics is a little more complicated than in the scalar case of section 2 because ofthe polarization vector ζ . However, there is a simple choice for ζ which makes the problemvirtually isomorphic to the scalar case. We can always pick ζ · P = 0 . (D.4) One can in fact argue the twist must be strictly larger than 2∆. Suppose the twist were 2∆.Then the second inequality in (C.1) would have been saturated. In this case Q ˙ α O ( ˙ α... ˙ α l ) , ( α ,...,α l ) =0. But in this case Q O ( ˙ α ... ˙ α l ) , ( α ,...,α l ) = 0 and thus there is no such operator in the OPE.
54n this case, the contribution from the P µ P µ · · · P µ s term in (D.3) automatically selectsthe same kinematic structure as in the scalar case, and all the arguments from section 2 gothrough. One finds the same sum rules and obtains convexity under the same assump-tions as in section 2.
Appendix E. The s Correction in the Critical O ( N ) Model
Here we would like to show how one can use the general formula for the coefficient c τ min in order to compute the s tem in (5.3). The correction 1 /s arises from the exchangeof the σ operator in the t-channel.To contrast (5.3) with our formula (3.18) we need to compute the three-point function h σ i ( x ) σ j ( x ) σ ( x ) i . For the two-point functions we have [57] h σ i ( x ) σ j (0) i = δ ij N Γ( d − π d x d − = γ s N x d − , h σ ( x ) σ (0) i = γ φ N x , γ φ = 2 d +2 sin (cid:0) πd (cid:1) Γ( d − ) π Γ( d − . (E.1) Fig. 13:
The diagram for the three-point function h σ i ( x ) σ i ( x ) σ ( x ) i . Again, we need to be careful that the “trace” terms in (D.3) do not overwhelm contributionscoming from lower spins (with a different power of ν ). In section 2 we have already analyzedthis for the case f ( α ) s ( y, ζ, ζ ∗ ) µ ...µ s ∼ y µ · · · y µ s . Here we could obtain new structures, such as f ( α ) s ( y, ζ, ζ ∗ ) µ ...µ s ∼ ζ µ ( ζ ∗ ) µ y µ · · · y µ s . Due to (D.4), this could contribute only when dottedinto the “trace” terms, e.g. g µ µ P µ · · · P µ s . A simple calculation shows that this would scalelike (at large − q ) x − s +2 ( q ) − τ ∗ s +∆ O − d/ . Comparing this to the already existing contributionfrom spin s − x − s +2 ( q ) − τ ∗ s − +∆ O − d/ , we see that again it is sufficient that the twists τ ∗ s arenondecreasing, as found in (2.11). h σ i ( x ) σ i ( x ) σ ( x ) i at leading order is given by the dia-gram fig. 13, where the interaction vertex is − N σ i . So that we get h σ i ( x ) σ i ( x ) σ ( x ) i = − γ σ γ s N Z d d x x d − x d − x = − γ σ γ s N π d Γ( d − d − ) x d − x x . (E.2)(We used the standard formula [70], and there is no summation over i ). Thus, we get C σσσ C σ σ C σσ = 1 N γ σ γ s π d Γ( d − Γ( d − ) . (E.3)Now we have everything we need to apply (3.18). (Notice that in this case τ min > s min = 0, τ min = 2 and ∆ = d − to get c σ = 2 d − Γ( d − ) sin( πd ) N π / Γ( d − , (E.4)which can be easily checked to reproduce the result (5.3) precisely.56 eferences [1] A. M. Polyakov, “Conformal symmetry of critical fluctuations,” JETP Lett. , 381(1970), [Pisma Zh. Eksp. Teor. Fiz. , 538 (1970)].[2] I. Antoniadis and M. Buican, “On R-symmetric Fixed Points and Superconformality,”Phys. Rev. D , 105011 (2011). [arXiv:1102.2294 [hep-th]].[3] M. A. Luty, J. Polchinski and R. Rattazzi, “The a -theorem and the Asymptotics of4D Quantum Field Theory,” [arXiv:1204.5221 [hep-th]].[4] J. -F. Fortin, B. Grinstein and A. Stergiou, “Scale without Conformal Invariance inFour Dimensions,” [arXiv:1206.2921 [hep-th]].[5] J. -F. Fortin, B. Grinstein and A. Stergiou, “A generalized c-theorem and the consis-tency of scale without conformal invariance,” [arXiv:1208.3674 [hep-th]].[6] Y. Nakayama, “Supercurrent, Supervirial and Superimprovement,” [arXiv:1208.4726[hep-th]].[7] J. -F. Fortin, B. Grinstein, C. W. Murphy and A. Stergiou, “On Limit Cycles inSupersymmetric Theories,” [arXiv:1210.2718 [hep-th]].[8] Y. Nakayama, “Is boundary conformal in CFT?,” [arXiv:1210.6439 [hep-th]].[9] J. M. Maldacena, “The Large N limit of superconformal field theories and supergrav-ity,” Adv. Theor. Math. Phys. , 231 (1998). [hep-th/9711200].[10] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators fromnoncritical string theory,” Phys. Lett. B , 105 (1998). [hep-th/9802109].[11] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. , 253(1998). [hep-th/9802150].[12] G. Mack, “All Unitary Ray Representations of the Conformal Group SU(2,2) withPositive Energy,” Commun. Math. Phys. , 1 (1977)..[13] B. Grinstein, K. A. Intriligator and I. Z. Rothstein, “Comments on Unparticles,” Phys.Lett. B , 367 (2008). [arXiv:0801.1140 [hep-ph]].[14] T. N. Pham and T. N. Truong, “Evaluation Of The Derivative Quartic Terms Of TheMeson Chiral Lagrangian From Forward Dispersion Relation,” Phys. Rev. D , 3027(1985)..[15] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, “Causality,analyticity and an IR obstruction to UV completion,” JHEP , 014 (2006). [hep-th/0602178].[16] Z. Komargodski and A. Schwimmer, “On Renormalization Group Flows in Four Di-mensions,” JHEP , 099 (2011). [arXiv:1107.3987 [hep-th]].[17] Z. Komargodski, “The Constraints of Conformal Symmetry on RG Flows,” JHEP , 069 (2012). [arXiv:1112.4538 [hep-th]].[18] V. M. Braun, G. P. Korchemsky and D. Mueller, “The Uses of conformal symmetryin QCD,” Prog. Part. Nucl. Phys. , 311 (2003). [hep-ph/0306057].5719] O. Nachtmann, “Positivity constraints for anomalous dimensions,” Nucl. Phys. B ,237 (1973).[20] H. Epstein, V. Glaser and A. Martin, “Polynomial behaviour of scattering amplitudesat fixed momentum transfer in theories with local observables,” Commun. Math. Phys. , 257 (1969)..[21] S. B. Giddings and R. A. Porto, “The Gravitational S-matrix,” Phys. Rev. D ,025002 (2010). [arXiv:0908.0004 [hep-th]].[22] S. Ferrara, A. F. Grillo and R. Gatto, “Tensor representations of conformal algebraand conformally covariant operator product expansion,” Annals Phys. , 161 (1973).[23] A. M. Polyakov, “Nonhamiltonian approach to conformal quantum field theory,” Zh.Eksp. Teor. Fiz. , 23 (1974).[24] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, “Infinite Conformal Sym-metry in Two-Dimensional Quantum Field Theory,” Nucl. Phys. B , 333 (1984).[25] R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi, “Bounding scalar operator dimen-sions in 4D CFT,” JHEP , 031 (2008). [arXiv:0807.0004 [hep-th]].[26] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi,“Solving the 3D Ising Model with the Conformal Bootstrap,” [arXiv:1203.6064 [hep-th]].[27] D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, “OPE Convergence in Con-formal Field Theory,” [arXiv:1208.6449 [hep-th]].[28] L. Cornalba, M. S. Costa, J. Penedones and R. Schiappa, “Eikonal Approximationin AdS/CFT: From Shock Waves to Four-Point Functions,” JHEP , 019 (2007).[hep-th/0611122].[29] L. Cornalba, M. S. Costa, J. Penedones and R. Schiappa, “Eikonal Approximation inAdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions,” Nucl. Phys.B , 327 (2007). [hep-th/0611123].[30] L. Cornalba, M. S. Costa and J. Penedones, “Eikonal approximation in AdS/CFT: Re-summing the gravitational loop expansion,” JHEP , 037 (2007). [arXiv:0707.0120[hep-th]].[31] L. F. Alday and J. M. Maldacena, “Comments on operators with large spin,” JHEP , 019 (2007). [arXiv:0708.0672 [hep-th]].[32] A. Pelissetto and E. Vicari, “Critical phenomena and renormalization group theory,”Phys. Rept. , 549 (2002). [cond-mat/0012164].[33] M. A. Vasiliev, “Higher spin gauge theories: Star product and AdS space,” In *Shif-man, M.A. (ed.): The many faces of the superworld* 533-610. [hep-th/9910096].[34] A. L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, “The Analytic Boot-strap and AdS Superhorizon Locality,” [arXiv:1212.3616 [hep-th]].[35] M. Froissart, “Asymptotic behavior and subtractions in the Mandelstam representa-tion,” Phys. Rev. , 1053 (1961).. 5836] A. Martin, “Unitarity and high-energy behavior of scattering amplitudes,” Phys. Rev. , 1432 (1963)..[37] L. Frankfurt, M. Strikman and C. Weiss, “Small-x physics: From HERA to LHC andbeyond,” Ann. Rev. Nucl. Part. Sci. , 403 (2005). [hep-ph/0507286].[38] J. Polchinski and M. J. Strassler, “Deep inelastic scattering and gauge / string dual-ity,” JHEP , 012 (2003). [hep-th/0209211].[39] C. G. Callan, Jr. and D. J. Gross, “Bjorken scaling in quantum field theory,” Phys.Rev. D , 4383 (1973)..[40] S. E. Derkachov and A. N. Manashov, “Generic scaling relation in the scalar phi**4model,” J. Phys. A , 8011 (1996). [hep-th/9604173].[41] S. K. Kehrein, “The Spectrum of critical exponents in phi**2 in two-dimensions theoryin D = (4-epsilon)-dimensions: Resolution of degeneracies and hierarchical structures,”Nucl. Phys. B , 777 (1995). [hep-th/9507044].[42] F. A. Dolan and H. Osborn, “Conformal Partial Waves: Further Mathematical Re-sults,” [arXiv:1108.6194 [hep-th]].[43] I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, “Holography from ConformalField Theory,” JHEP , 079 (2009). [arXiv:0907.0151 [hep-th]].[44] H. Osborn, “Conformal Blocks for Arbitrary Spins in Two Dimensions,” Phys. Lett.B , 169 (2012). [arXiv:1205.1941 [hep-th]].[45] L. F. Alday, B. Eden, G. P. Korchemsky, J. Maldacena and E. Sokatchev, “Fromcorrelation functions to Wilson loops,” JHEP , 123 (2011). [arXiv:1007.3243[hep-th]].[46] P. Kovtun and A. Ritz, “Black holes and universality classes of critical points,” Phys.Rev. Lett. , 171606 (2008). [arXiv:0801.2785 [hep-th]].[47] L. Hoffmann, L. Mesref and W. Ruhl, “Conformal partial wave analysis of AdS am-plitudes for dilaton axion four point functions,” Nucl. Phys. B , 177 (2001). [hep-th/0012153].[48] F. A. Dolan and H. Osborn, “Superconformal symmetry, correlation functions and theoperator product expansion,” Nucl. Phys. B , 3 (2002). [hep-th/0112251].[49] P. J. Heslop, “Aspects of superconformal field theories in six dimensions,” JHEP ,056 (2004). [hep-th/0405245].[50] P. Liendo, L. Rastelli and B. C. van Rees, “The Bootstrap Program for Boundary CF T d ,” [arXiv:1210.4258 [hep-th]].[51] E. Barnes, E. Gorbatov, K. A. Intriligator, M. Sudano and J. Wright, “The Exactsuperconformal R-symmetry minimizes tau(RR),” Nucl. Phys. B , 210 (2005).[hep-th/0507137].[52] K. G. Wilson and J. B. Kogut, “The Renormalization group and the epsilon expan-sion,” Phys. Rept. , 75 (1974).. 5953] K. Lang and W. Ruhl, “Critical O(N) vector nonlinear sigma models: A Resume oftheir field structure,” [hep-th/9311046].[54] K. Lang and W. Ruhl, “The Critical O(N) sigma model at dimensions 2 ¡ d ¡ 4: Fusioncoefficients and anomalous dimensions,” Nucl. Phys. B , 597 (1993)..[55] I. R. Klebanov and A. M. Polyakov, “AdS dual of the critical O(N) vector model,”Phys. Lett. B , 213 (2002). [hep-th/0210114].[56] E. Sezgin and P. Sundell, “Massless higher spins and holography,” Nucl. Phys. B ,303 (2002), [Erratum-ibid. B , 403 (2003)]. [hep-th/0205131].[57] S. Giombi and X. Yin, “Higher Spin Gauge Theory and Holography: The Three-PointFunctions,” JHEP , 115 (2010). [arXiv:0912.3462 [hep-th]].[58] J. Maldacena and A. Zhiboedov, “Constraining Conformal Field Theories with AHigher Spin Symmetry,” [arXiv:1112.1016 [hep-th]].[59] V. E. Didenko and E. D. Skvortsov, “Exact higher-spin symmetry in CFT: all corre-lators in unbroken Vasiliev theory,” [arXiv:1210.7963 [hep-th]].[60] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia and X. Yin, “Chern-Simons Theory with Vector Fermion Matter,” Eur. Phys. J. C , 2112 (2012).[arXiv:1110.4386 [hep-th]].[61] O. Aharony, G. Gur-Ari and R. Yacoby, “d=3 Bosonic Vector Models Coupled toChern-Simons Gauge Theories,” JHEP , 037 (2012). [arXiv:1110.4382 [hep-th]].[62] J. Maldacena and A. Zhiboedov, “Constraining conformal field theories with a slightlybroken higher spin symmetry,” [arXiv:1204.3882 [hep-th]].[63] S. Ferrara, C. Fronsdal and A. Zaffaroni, “On N=8 supergravity on AdS(5) and N=4superconformal Yang-Mills theory,” Nucl. Phys. B , 153 (1998). [hep-th/9802203].[64] B. Sundborg, “Stringy gravity, interacting tensionless strings and massless higherspins,” Nucl. Phys. Proc. Suppl. , 113 (2001). [hep-th/0103247].[65] D. M. Hofman and J. Maldacena, “Conformal collider physics: Energy and chargecorrelations,” JHEP , 012 (2008). [arXiv:0803.1467 [hep-th]].[66] F. A. Dolan and H. Osborn, “Conformal four point functions and the operator productexpansion,” Nucl. Phys. B , 459 (2001). [hep-th/0011040].[67] M. S. Costa, J. Penedones, D. Poland and S. Rychkov, “Spinning Conformal Blocks,”JHEP , 154 (2011). [arXiv:1109.6321 [hep-th]].[68] A. L. Fitzpatrick and J. Kaplan, “Unitarity and the Holographic S-Matrix,” JHEP , 032 (2012). [arXiv:1112.4845 [hep-th]].[69] D. Poland and D. Simmons-Duffin, “Bounds on 4D Conformal and SuperconformalField Theories,” JHEP , 017 (2011). [arXiv:1009.2087 [hep-th]].[70] D. I. Kazakov, “The Method Of Uniqueness, A New Powerful Technique For MultiloopCalculations,” Phys. Lett. B133