Higher Spin ANEC and the Space of CFTs
Abstract
We study the positivity properties of the leading Regge trajectory in higherdimensional, unitary, conformal field theories (CFTs). These conditions correspond to higher spin generalizations of the averaged null energy condition (ANEC). By studying higher spin ANEC, we will derive new bounds on the dimensions of charged, spinning operators and prove that if the HofmanMaldacena bounds are saturated, then the theory has a higher spin symmetry. We also derive new, general bounds on CFTs, with an emphasis on theories whose spectrum is close to that of a generalized free field theory. As an example, we consider the Ising CFT and show how the OPE structure of the leading Regge trajectory is constrained by causality. Finally, we use the analytic bootstrap to perform additional checks, in a large class of CFTs, that higher spin ANEC is obeyed at large and finite spin. In the process, we calculate corrections to large spin OPE coefficients to oneloop and higher in holographic CFTs.
CTWalter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, 91125 \institutionYaleDepartment of Physics, Yale University, New Haven, CT 06511
David Meltzer\worksat\CT,\Yale
Contents
 1 Introduction
 2 Review of the Lightcone OPE and HS ANEC
 3 Bounds on Higher Spin Couplings
 4 Spin2/Spin4 Mixed Systems
 5 Comparison to Analytic Bootstrap
 6 Conclusion
 A Integrals of ThreePoint Functions
 B Conservation and Ward Identities
 C Examples of Spin2 4 Matrix Elements
 D Sums of DoubleTwist Operators
1 Introduction
In this paper, we will study positivity conditions obeyed by the leading Regge trajectory in , Lorentzian conformal field theories (CFTs). The leading Regge trajectory is defined to be the set of operators with the smallest scaling dimension, , for each evenspin [Cornalba:2007fs, Costa:2012cb, CaronHuot:2017vep]. A universal operator which appears on this trajectory is the stressenergy tensor, , which has spin and saturates the unitarity bound, . In all QFTs the lightray integral of the stresstensor also obeys a positivity condition: the averaged null energy condition (ANEC). The ANEC states that the following operator is positive:
(1.1) 
where the integral is over a complete, null line.
This positivity condition was first studied extensively for CFTs in [Hofman:2008ar] to derive universal bounds on threepoint functions involving . The ANEC has since been proven via two different methods, through causality and OPE arguments in [Hartman:2016lgu] and through monotonicity of relative entropy in [Faulkner:2016mzt]. Here, we will be interested in exploring the results of [Hartman:2016lgu], where the proof of the ANEC also revealed an infinite set of new, higher spin positivity conditions^{1}^{1}1See [Komargodski:2016gci, Hofman:2016awc] for previous work on higher spin sum rules.. Higher spin ANEC, or HS ANEC, says the positivity of generalizes straightforwardly to the entire Regge trajectory. More precisely, the following operator is positive:
(1.2) 
where is the lightest spin operator in a reflection positive OPE. When this reduces to the ANEC operator.
Why might we be interested in studying the positivity properties of higher spin operators? The first, most basic, motivation is we want to use the fundamental principles of causality, unitarity and locality to map out the space of consistent quantum field theories. HS ANEC follows from the axioms of conformal field theory [Hartman:2016lgu] and gives new bounds on CFT data which have not been fully explored. Specifically, it singles out the operators with low twist, , which govern the lightcone OPE [light1, light2]. Understanding positivity conditions on the set of CFT data underlies the success of the conformal bootstrap program [Poland:2018epd, SimmonsDuffin:2016gjk], so it is natural to expect that this infinite set of positive operators will give a new analytic window into the space of CFTs.
As an illustrative case, by studying HS ANEC we can derive new bounds on how two scalar operators couple to the leading Regge trajectory. The corresponding ANEC bound is trivial by CFT Ward identities, while the HS ANEC bound is nontrivial for general CFTs. Therefore, while the first fourpoint function which is related to the ANEC in a nontrivial way involves spinning operators [Li:2015itl, krav, Dymarsky:2017xzb], HS ANEC can be related to a simpler fourpoint function consisting solely of scalars. These constraints, which have not been used thus far, can be straightforwardly applied to the study of mixed correlator systems.
In free CFTs, operators on the leading Regge trajectory also play an enhanced role as the generators of a higher spin symmetry. The presence of a single, conserved, higher spin current is enough to prove the existence of an infinitedimensional, higher spin symmetry, which in turn completely fixes the OPE of the higher spin currents [Maldacena:2011jn, Boulanger:2013zza, Alba:2015upa]. In addition, theories like ChernSimons vector models [Aharony:2011jz, Giombi:2011kc, Maldacena:2012sf] are also tightly constrained by a slightly broken, higher spin symmetry^{2}^{2}2See also [Aharony:2018npf, Turiaci:2018nua] for a bootstrap approach to these theories.. For weakly coupled CFTs, the HS ANEC operators are then natural objects to study as they are both manifestly positive and sensitive to the emergence of an infinitedimensional symmetry.
Finally, the Lorentzian inversion formula [CaronHuot:2017vep, ssw] also guarantees CFT data organizes nicely into analytic families parameterized by the scaling dimension and spin. Analyticity in spin follows from the fact that individual conformal blocks with spin diverge in the Regge limit, while fourpoint functions in a unitary CFT are bounded in this regime[Maldacena:2015waa, Hartman:2015lfa]. Therefore, we cannot independently vary the OPE data for a single, highspin operator without spoiling boundedness in the Regge limit. Moreover, in [Kravchuk:2018htv] it was shown that the analytic continuation in spin can also be done at the level of the lightray transformed operators themselves. These results imply that threepoint functions of operators on the leading Regge trajectory cannot be completely independent. As an example of this phenomena, we will use HS ANEC to put bounds on , , and in terms of . Using the AdS/CFT dictionary[Maldacena:1997re, Witten:1998qj, Gubser:1998bc], this corresponds to bounds on cubic interactions for AdS theories with many light, higher spin particles [Vasiliev:1990en, Vasiliev:1999ba, Sezgin:2002rt, Vasiliev:2003ev, Klebanov:2002ja]^{3}^{3}3The leading Regge trajectory we discuss here is the exact trajectory of the CFT and does not always correspond to the leading singletrace trajectory in large CFTs [Cornalba:2007fs, Costa:2012cb]..
1.1 Summary
This work is organized as follows. In section 2, we review the lightcone OPE for CFTs in , the proof of HS ANEC, the symmetry properties of lightray operators, and the behavior of the leading trajectory in CFTs. We will also establish notation and introduce the states used to derive the optimal bounds.
In section 3, we will present new constraints for two and three point functions from HS ANEC. To start, in section 3.1 we prove the twist of a charged, spin operator which appears in a reflection positive OPE is bounded below by the twist of the lightest, uncharged, spin operator that appears in the same OPE if and even. In other words, for generic CFTs, the leading Regge trajectory is necessarily composed of uncharged operators. In section 3.2, we consider simple examples of HS ANEC, with an emphasis on matrix elements involving a scalar operator. Here HS ANEC strongly constrains threepoint functions in theories whose spectrum is close to a generalized free field spectrum.
In section 4, we study HS ANEC in states created by the stresstensor and the lightest spin4 operator. In section 4.1, we prove that if the ANEC bounds for are saturated, then the CFT has a higher spin symmetry. We also show how saturation of ANEC implies saturation of HS ANEC. For practical applications, in section 4.2 we focus on CFTs with an Isinglike spectrum and derive bounds on threepoint functions involving the spin operator.
In section 5, we discuss the relation between HS ANEC and the analytic bootstrap. In CFTs, the higher spin positivity conditions bound OPE coefficients which are also computable using large spin expansions. We find that the exchange of isolated operators and towers of doubletwist operators in one channel always yield results consistent with HS ANEC in the dual channel, at finite and asymptotically large spin respectively. In the context of large CFTs, this implies AdS theories with only cubic interactions are consistent with HS ANEC at tree and oneloop level, with the corresponding restrictions on spin. We also consider examples where HS ANEC can naïvely be violated at finite spin if we do not include nonperturbative effects in the large spin expansion. Of independent interest, we also present new results for large spin OPE coefficients to all orders in for holographic CFTs. In the dual AdS theory, these OPE coefficients can be found through conformal block decompositions of ladder diagrams.
In section 6, we give a brief conclusion and discuss future directions. The appendices contain various technical details used throughout the paper. Appendix A describes the basis of conformally invariant threepoint structures and the integrals used to calculate HS ANEC matrix elements. Appendix B includes solutions to the conservation conditions and integrated Ward identities. In Appendix C, we give examples of relevant (HS) ANEC matrix elements. In Appendix D we give an example, relevant to section 5, where sums over doubletwist operators can yield negative corrections to large spin OPE coefficients.
2 Review of the Lightcone OPE and HS ANEC
In this section, we will give a brief overview of the lightcone OPE and how it leads to HS ANEC [Hartman:2016lgu]^{4}^{4}4See [Kravchuk:2018htv] for a generalization to continuous spin.. We will also review the behavior of the leading Regge trajectory in general CFTs and the properties of lightray operators.
For a CFT in flat space, we can always write down the OPE of identical scalars, , as:
(2.1) 
where is a symmetric, traceless tensor of spin and we have not used conformal symmetry to relate primaries and descendents.
As realized in [light1, light2], if we work in Lorentzian signature and take the limit , the dominant operator is the one with the minimal twist, . For a spin operator, , the leading contribution comes specifically from plus all descendents generated by acting with . At the level of the fourpoint function, , the conformal blocks reduce to a sum of blocks. Here is the group which leaves the lightray connecting the two, nullseparated operators invariant.
At the level of the OPE, we can write the contribution of the primary and all its minimal twist descendents as an integral over a null line. It was shown in [Hartman:2016lgu] that when , such that , the OPE becomes:
(2.2) 
where we have isolated the contribution of a given operator, , to the OPE and kept its normalization, , arbitrary. In this form, we also see how the operators with the minimal twist dominate the lightcone OPE. This is not the entire contribution of the multiplet to the OPE, but if we insert between two states, this integral captures the leading behavior in the above lightcone limit. For the remainder of this work, will always denote the operator with the smallest twist for a given spin. For OPE coefficients, we also use the label as a shorthand for .
In order to prove ANEC and its higher spin generalization, [Hartman:2016lgu] used the positivity properties of Rindler symmetric correlation functions in Minkowski space. The Rindler reflection for scalars is defined by:
(2.3) 
and maps operators in one wedge to the other. Rindler positivity^{5}^{5}5See also [Maldacena:2015waa, Casini:2010bf] for a derivation of Rindler positivity. states the following correlation function is positive:
(2.4) 
where the unbarred operators are inserted in the right Rindler wedge and Rindler reflection does not reverse the order of operators. To make a connection with ANEC, we consider the states:
(2.5) 
where and are defined on the right Rindler wedge and is real. Using Rindler positivity to define an inner product, the CauchySchwarz inequality implies:
(2.6) 
Using this inequality and analyticity properties of the fourpoint function, [Hartman:2016lgu] derived a sum rule for the normalized correlator:
(2.7) 
which is most clearly stated if we introduce the variables and . The sum rule is then:
(2.8) 
The integral runs over a semicircle of radius in the lower plane, just below the origin. They also take , such that the correlator on the arc is well approximated by the lightcone OPE. When we perform the OPE on the arc, the fact means we are projecting onto the minimaltwist operators, while the factor of projects onto the spin operators. The final result is:
(2.9)  
(2.10) 
where the right hand side of (2.9) is positive because the positive ordered correlation functions on the right hand side of (2.6) factorize at small [Hartman:2015lfa, Hartman:2016lgu]. In sum, they derived the positivity condition:
(2.11) 
When , this gives the averaged null energy condition (ANEC) since the stresstensor is always the lightest spintwo operator in any positive OPE^{6}^{6}6We will always assume our CFT does not have multiple decoupled sectors and the stresstensor is the unique, conserved, spintwo operator.. In the language of [Kravchuk:2018htv], is the lightray transform of a local operator on the leading Regge trajectory. In sections 3 and 4 we will assume the probe operator, , is chosen such that , while in section 5 it is more natural to keep the product of OPE coefficients explicit.
Next, we consider the behavior of the leading Regge trajectory in generic CFTs. From the work of [light1, light2], we know that for CFTs in , the large spin sector has the structure of a generalized free field theory. They showed that for any two operators, and , with twists , there always exists an infinite tower of doubletwist operators, , such that as the twists . In a generalized free field theory, these operators are given by [Heemskerk:2009pn]:
(2.12) 
Furthermore, it was shown in [light2, Komargodski:2016gci, Costa:2017twz] that the leading Regge trajectory in a reflection positive OPE, , is a monotonically increasing, convex function of the spin. Since every local CFT contains the stressenergy tensor, which has twist , we find the following bound for the twists, , of the leading trajectory:
(2.13) 
If there exists a light scalar, , with , then we can replace the upper bound with . We will generally assume the lower bound is not saturated for so the theory is not free [Maldacena:2011jn, Boulanger:2013zza, Alba:2015upa]. For generic CFTs, we expect that the leading Regge trajectory is composed of doubletwist operators, as is seen for example in the Ising CFT [SimmonsDuffin:2016wlq]. There are counterexamples, e.g. weakly coupled CFTs with a gauge theory description have low spin operators with twist and without having a low dimension scalar in the spectrum. To apply HS ANEC we do not need to make any assumptions on which scenario is realized, while when applying the analytic bootstrap we will assume the trajectory consists of doubletwist operators.
In deriving HS ANEC, it is important that the operator is the lightest, spin operator in a given positive OPE. It is not a priori clear that the same Regge trajectory gives the leading contribution in every positive OPE. In section 3.1, we will rule out a wide class of possible counterexamples by showing the leading trajectory must be in the singlet representation of any internal global symmetry.
Finally, we will review the structure of threepoint functions involving lightray operators. To derive the optimal bounds from the positivity of , our states will always be momentum eigenstates:
(2.14) 
We will set and use the mostly plus convention for the metric. It is also convenient to define a covariant version of the (HS) ANEC operators as [zhib]:
(2.15) 
We will always choose and such that .
We then want to calculate and impose positivity for all . To organize the bounds, we classify how transforms under the residual symmetry which leaves and fixed. If is a symmetric, traceless operator of spin , then can transform in the spin representations of .
In practice, it is convenient to construct the polarization tensors from the set of vectors , where and . For the spin bound we fix a set and consider all polarization tensors of the form:
(2.16) 
Then the general matrix becomes a block diagonal matrix, where for each we obtain a positive matrix:
(2.17) 
(2.18) 
For low spacetime dimensions or high external spin, some of these polarization choices are not possible, e.g. if we can not find vectors, , orthogonal to such that . Finally, if the external operators are conserved we can always eliminate , and we instead have linear bounds from . When it is clear from the context, we will drop the indices for the matrix .
If we set , then there are three linear bounds for and two linear bounds in . We can always write in terms of the tensor structures which appear in a free field theory:
(2.19) 
Here , , and refer to a free theory of scalars, fermions, and tensors^{7}^{7}7In even dimensions there exist free field theories of forms. For and odd, such free field theories do not exist, but the corresponding tensor structure still does., respectively. Then the ANEC yields:
(2.20)  
(2.21)  
(2.22) 
These are also known as the HofmanMaldacena bounds [Hofman:2008ar]. In we only have the first two bounds.
Generically, conservation of the stresstensor implies the ANEC bounds will be stronger than the corresponding HS ANEC bounds. For the threepoint function , conservation of the stresstensor at noncoincident points implies relations between the OPE coefficients, while the integrated CFT Ward identities relate this threepoint function to the twopoint function [Osborn:1993cr]. For interacting field theories, similar identities do not hold if we replace by a higher spin operator.
One related benefit of studying the ANEC operator is it makes solving the Ward identities simpler. From its definition we have the following property [Hofman:2008ar]:
(2.23) 
where we integrate over the sphere at infinity^{8}^{8}8The extra factor of is due to our unconventional normalization of .. We will use this equation to solve the Ward identity constraints on .
3 Bounds on Higher Spin Couplings
3.1 Bounds for Charged Operators
By studying CFT fourpoint functions and HS ANEC, we will prove a lower bound on the dimensions of spinning, charged operators in terms of the uncharged operators. More precisely, we show that if an operator satisfies the following properties:

has spin with and even,

appears in a reflection positive OPE, e.g. ,

transforms in a nontrivial representation, , of some internal, global symmetry group,
then where is the twist of the lightest, uncharged operator with spin. This implies that for generic CFTs, the leading Regge trajectory in any positive OPE must be composed of uncharged operators^{9}^{9}9For free CFTs with a global symmetry, we can have multiple, degenerate Regge trajectories with different global symmetry properties.. For , this is trivially satisfied since we assume there is a unique, conserved, spintwo operator. However, for all this statement is nontrivial and gives lower bounds on the dimensions of spinning, charged operators.
A simple way to motivate this bound is to consider a fourpoint function of scalar operators in the fundamental representation of . To make contact with [Hartman:2016lgu], we will also insert them symmetrically with respect to the Rindler wedges. The channel, or , conformal block decomposition now has the form:
where denotes the global symmetry representations of the exchanged operators. The points and will lie in the right Rindler wedge.
When we take the pair of operators to be lightlike separated, lightray operators will contribute to both and . Within the spin sector, the leading contribution to the lightcone OPE can come from either or , depending on the twists of the exchanged operators. In either case, the HS ANEC sum rule gives a sign constraint on the corresponding OPE coefficients.
If we assume the minimal twist, lightray operators are uncharged and appear in , the proof of HS ANEC is unchanged as the dependence on the polarizations is trivial. On the other hand, if we assume for some spin the lightest operator is in the adjoint representation, we see an immediate problem: the prefactor in front of is not signdefinite. The sum rule (2.9) now becomes:
(3.1) 
where the right hand side is still positive by the CauchySchwarz inequality and factorization.
By choosing different polarizations, the left hand side can take either sign and the HS ANEC sum rule implies . This is a contradiction since we assumed the lightest spin operator in the and OPEs was charged. Therefore, either the leading spin operator is in the singlet representation or we have two spin operators with degenerate twist, and , and the OPE coefficients for the charged operator are bounded in terms of the uncharged one. We can always derive the HS ANEC bound on the singlet operators alone by choosing the probe operator, , to be uncharged.
To prove this in general, we can use some simple properties of the ClebschGordon coefficients, or equivalently the 6jsymbol. We will consider a general fourpoint function of scalars:
(3.2) 
where we take to transform in the representation of the global symmetry. The operator can be smeared or inserted at a single point.
We will also assume a single operator, , where is the representation index, dominates the spin sector of the lightcone OPE, . When we isolate its contribution in this OPE and then calculate the threepoint function , we produce a product of ClebschGordon coefficients:
(3.3)  
(3.4) 
where we have dropped the overall spacetime dependence.
Next, we choose the polarization tensors to project onto a given representation , that is we choose . This produces another product of ClebschGordon coefficients, and we are left with the following group theory factors multiplying the OPE coefficients:
(3.5) 
where for brevity we suppressed various kinematical factors which are independent of the group representations. Here the is because we are studying the lightcone limit of the full fourpoint function, but the leading lightcone contribution is all we need for HS ANEC.
It is crucial that HS ANEC should hold for any choice of , since these correspond to different Rindler symmetric ways of creating our state and lightray operator, or different choices of and in (2.6). Our strategy will then be to sum over and use orthogonality properties of the ClebschGordon coefficients to show charged operators never give a signdefinite contribution in the lightcone OPE.
The orthogonality properties we need are:
(3.6)  
(3.7) 
If we sum (3.5) over we find:
(3.8) 
where in the last step the sums project onto the singlet representation. Put another way, by summing over we are averaging over all polarizations, so only operators in the singlet representation can appear in the channel.
Therefore if , the HS ANEC sum rule will fix the OPE coefficients to have either a positive or negative sign depending on how we choose the polarizations. This fixes the OPE coefficients to be zero, unless there is also an operator with the same or smaller twist in the singlet representation. This completes the proof that the twist of a charged operator in a positive OPE is bounded below by the twist of the lightest, uncharged operator with the same spin in the same OPE, for all and even. We have focused on the fourpoint function of scalar primaries, but the generalization to spinning operators or systems of fourpoint functions is straightforward.
It is also clear this bound cannot be improved for general CFTs. From the lightcone bootstrap, we know that if a CFT contains a light, charged scalar, , with , then the bound is saturated at infinite spin[Li:2015rfa]. By solving crossing for we can show there exists doubletwist operators for all global symmetry representations which can appear in the OPE and they all approach the same twist as . We also can use the results of [Li:2015rfa] to check this bound holds at asymptotically large spin.
In addition, if we assume the leading trajectory is composed of doubletwist operators , it is not hard to argue that the same leading trajectory appears in every positive OPE, for sufficiently large spin. Using the lightcone bootstrap [light1, light2], the coupling at large is nonzero in interacting CFTs and determined by the operators of minimal twist in the and OPEs. Given the results of [CaronHuot:2017vep], which proved the OPE data organizes into analytic families for , we also expect the entire trajectory to appear in the OPE. However, we cannot rule out the OPE coefficients having accidental zeros at finite spin.
3.2 HS ANEC Examples
We will now consider bounds from HS ANEC itself. The simplest bound is , for scalar , where we get a single positivity constraint on the OPE coefficient :
(3.9) 
Unitarity implies and convexity of the leading trajectory in combination with the lightcone bootstrap imply , so we have:
(3.10) 
One interesting case to study is when is the lightest scalar in the theory and . Then the leading Regge trajectory is and we can write . If is small, this matrix element is also small due to the factor of . This yields strong bounds on the offdiagonal, HS ANEC matrix elements when we consider a more general state. As an example, we can consider,
(3.11) 
where and are both scalars. Positivity of HS ANEC for this state gives the matrix condition:
(3.12) 
Keeping the twist, , generic we find:
(3.13)  
where . If and is small, the ratio of OPE coefficients scales like:
(3.14) 
and this bound becomes stronger as we increase [light1, light2]. This inequality already gives strong constraints for the Ising CFT where [ElShowk:2012ht] or in the Ising model where [Atanasov:2018kqw, Rong:2018okz]. Here it is important that the anomalous dimension, , is with respect to the generalized free field value, , and not the unitarity bound, .
The bound disappears when for integer, which is consistent with the structure of generalized free field theories. In such theories we can always construct the operator and the relevant coupling is not suppressed as . Finally, at large the bound becomes:
(3.15) 
which decays exponentially for large . This is consistent with results from OPE convergence [Pappadopulo:2012jk] and is similar to what was found using the ANEC [Cordova:2017zej] for other mixed systems. For this system, the ANEC bound is trivial since the stresstensor Ward identity implies for scalar . We will find similar bounds if we replace by a more general operator.
The next simplest case to consider is an external, conserved current . A similar calculation was also presented for and nonconserved vectors in [Hartman:2016lgu]. Based on symmetries, we find:
(3.16) 
which implies:
(3.17) 
The HS ANEC bound is identical in form to the ANEC bound [Hofman:2008ar]. One difference however is is related to by the Ward identity (2.23), but there is no such relation for . In general, when conservation can give additional relations between the coefficients, but here conservation for the external conserved currents is constraining enough that conservation of the stresstensor does not yield any additional relations [Costa:2011mg].
To find bounds on the underlying OPE coefficients, we will parametrize the three point function using the basis introduced in [Costa:2011mg]:
(3.18)  
The definitions of and can be found in appendix A.
For , permutation symmetry and conservation imply there are two independent OPE coefficients, which we will take to be and . The bounds then become:
(3.19)  
(3.20) 
which agrees with results found in [Hartman:2016lgu].
Unlike the case of scalar states, the HS ANEC matrix elements do not vanish in the limit . However, we now have the free parameter which can be tuned such that (HS) ANEC is saturated. It has been demonstrated in [zhib, Cordova:2017zej, Meltzer:2017rtf] that saturation of the ANEC bounds yields strong constraints on the CFT data, and it was conjectured in [zhib] that if ANEC is saturated in a state created by the stressenergy tensor, then the theory is free. In section 4.1 we will give a proof of this statement.
By requiring positivity for spin HS ANEC in the state , for scalar , we can also bound the threepoint function and find similar results as for a system of scalars. Assuming the external current is conserved, the threepoint function depends on a single OPE coefficient, , and the offdiagonal HS ANEC matrix element is:
(3.21) 
The HS ANEC bound then implies:
As before, if is the lightest operator in the theory, then and this bound becomes stronger as .
Finally, we will present bounds for systems involving the stresstensor in . Conservation, plus extra degeneracy conditions in , implies is a function of two OPE coefficients, and . We will only need the spin0 matrix element :
(3.22) 
After imposing conservation, is a function of a single OPE coefficient, , and we find: