Bounds on OPE Coefficients from Interference Effects in the Conformal Collider
aa r X i v : . [ h e p - t h ] O c t Bounds on OPE Coefficients fromInterference Effects in the Conformal Collider
Clay C´ordova ∗ , Juan Maldacena † , and Gustavo J. Turiaci ‡ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA Physics Department, Princeton University, Princeton, NJ, USA
Abstract
We apply the average null energy condition to obtain upper bounds on the three-pointfunction coefficients of stress tensors and a scalar operator, h T T Oi , in general CFTs. Wealso constrain the gravitational anomaly of U (1) currents in four-dimensional CFTs, whichare encoded in three-point functions of the form h T T J i . In theories with a large N AdSdual we translate these bounds into constraints on the coefficient of a higher derivative bulkterm of the form R φ W . We speculate that these bounds also apply in de-Sitter. In thiscase our results constrain inflationary observables, such as the amplitude for chiral gravitywaves that originate from higher derivative terms in the Lagrangian of the form φ W W ∗ .October, 2017 ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] ontents T T O in d ≥ T T O in d = 3 T T J in d = 4 R -Current . . . . . . . . . . . . . . . . . . . . . . . 20 AdS
Effective Action 227 Constraints for de-Sitter and Inflation 25
A Absence of Positive Local Operators 27B Details of the Collider Calculation 28
B.1 Normalized States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B.2 Three-Point Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30B.3 Energy Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
C Free Scalar Correlators 36D h T T O i Parity-Odd Structures in d = 3 h T T J i Three-Point Function 38F Computing the Bound in the Gravity Theory 39
F.1 Four-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Introduction
In this paper we investigate some implications of the average null energy condition inconformal field theories. We consider the conformal collider physics experiment discussedin [1]. In that setup, we produce a localized excitation by acting with a smeared operatornear the origin of spacetime. Then we measure the energy flux at infinity per unit angle.Requiring that the energy flux is positive imposes constraints on the three-point functioncoefficients. This method was used to constrain three-point functions of the stress tensorin [1–3].In this paper we use this same method to constrain the three-point functions of twostress tensors and another operator h T T Oi . The new idea consists of creating the initialstate by a linear combination of a stress tensor operator and the operator O . The three-point function h T T Oi appears as a kind of interference term in the expression for the energy.Requiring that the total contribution to the energy flux is positive imposes a non-trivialupper bound on the absolute magnitude of this three-point correlator. We apply theseideas to general scalar operators O as well as conserved currents with spin one, J , wherewe use it to put bounds on the gravitational anomaly in d = 4 CFTs. Because the boundarises from quantum mechanical interference effects, these bounds are stronger than thoseobtained in states created by a single primary local operator and its descendants (thoughthe resulting bounds involve more OPE coefficients).This energy flux at infinity is given by an integral of the stress tensor. On the boundaryof Minkowski space this integral is simply the average null energy E = R dx − T −− . Wereview this in section 2. Physically, we expect that this energy should be positive forall angles. Recently, the average null energy condition was proven using entanglemententropy methods [4] as well as reflection positivity euclidean methods [5]. When we createa localized state using the stress tensor, this energy distribution is completely determinedby the three-point function of the stress tensor. Two of the insertions correspond to theinsertions creating the state in the bra and the ket. The third corresponds to the onemeasuring the energy flux at infinity. The resulting bounds could also be obtained byrequiring standard reflection positivity of the euclidean theory [6,7]. However, the conformalcollider calculations provide an efficient way to extract the results.One of our main results is a sum rule constraining the OPE coefficients of scalar primaryoperators O with the energy-momentum tensor T . In spacetime dimensions d ≥ h T T Oi three-point function. We find that thisdata is constrained as X Scalar Primaries O i | C T T O i | f (∆ i ) ≤ N B , (1.1)where N B is one of the three OPE coefficients in h T T T i (the one occurring in a theory of2ree bosons), and the non-negative function f (∆) is given explicitly by f (∆) = ( d − dπ d Γ (cid:0) d (cid:1) Γ( d + 1)Γ(∆)Γ (cid:0) ∆ − d − (cid:1) ( d − Γ (cid:0) ∆2 + 2 (cid:1) Γ (cid:0) d +∆2 (cid:1) Γ (cid:0) d − ∆2 (cid:1) . (1.2)This function arises by doing the integrals involved in smearing the operator as well asin computing the energy flux. We derive this bound in detail in section 3, and discusssome simple physical consequences such as its interpretation in free field theories, large N holographic systems, and general implications for the asymptotics of OPE coefficients.In section 4 we consider analogous results in spacetime dimension three. This caseis special because the three-point functions of interest admit both parity preserving andparity violating structures. The bounds we find generalize those recently obtained in [8].We apply our results to large N Chern-Simons matter theories, and further use them toobtain predictions on OPE coefficients C T T O for scalars in the Ising model using the recentresults of the conformal bootstrap [9]. For instance, we find that operator ε has an OPEcoefficient constrained as | C T T ε | ≤ . | C T T : φ : | , (1.3)where the right-hand side is the value in the free scalar theory based on the field φ .In section 5 we consider bounds in four-dimensional CFTs with a global symmetrycurrent J . We apply the same techniques to obtain universal constraints on the gravitationalanomaly of the current J. In section 6 we show that the h T T Oi correlator can be generated from a gravity theoryin AdS d +1 through a higher derivative term, R φW , in the bulk effective action. We matchthe coefficient of this term to the C T T O coefficient in the boundary theory by performingthe same collider experiment in the bulk, where it involves propagation through a shockwave. One interesting feature of this presentation is that the resulting bound is independentof the mass of φ. Thus, the ∆ dependence of (1.1) is purely kinematic and results fromtranslating the boundary three-point function coefficient to a bulk interaction. We use our
AdS presentation to show that α ′ corrections satisfy the bound.In section 7 we extrapolate the bounds we obtained in AdS to “quasi bounds” on thecoefficients of the effective action in de Sitter space. We call them “quasi-bounds” because,unfortunately, for de-Sitter we do not know how to prove a sharp bound. We can think ofthese as a good indication for where the bulk effective theory should break down. We applythese “quasi-bounds” to constrain the amplitude of chiral gravity waves, and to constrainthe violations of the inflationary “consistency condition” for the two-point function. Bothof these effect arise from higher curvature couplings of the form φW or φW W ∗ .In the appendices we include more explicit derivations of the material in the mainsections. 3 ANEC and the Conformal Collider
The null energy condition is a central assumption in many classical theorems of generalrelativity. These results allow us to exclude unphysical spacetimes where causality violation,naked singularities, or other physical pathologies occur [10].If we move beyond classical field theory, these results appear to be in doubt. Quantumeffects lead to fluctuations that prohibit any local operator from having a positive expec-tation value in every state [11]. (We review these ideas in appendix A.) In particular thelocal energy density and other components of the energy-momentum tensor have negativeexpectation value in some states.Deeper investigation reveals a potential resolution. While components of the energy-momentum tensor are pointwise non-positive, a weaker hypothesis, the so-called average nullenergy condition, is often sufficient to enforce causal behavior [12]. This condition statesthat the integral along a complete null geodesic of the null energy density is a positivedefinite operator E = Z ∞−∞ dx − T −− ≥ . (2.1)Recently there has been significant interest in understanding the average null energycondition (2.1) in the context of local quantum field theories. In [5], an argument was givenestablishing (2.1) in conformal field theories by examining the constraints of causality onthe light-cone operator product expansion. In [4], an alternative argument was given linkingthe average null energy operator to entanglement entropy, then establishing positivity usingstrong subadditivity. These information theoretic methods have also been extended toobtain new inequalities strengthening (2.1) [13].Given that the average null energy in quantum field theory is now a theorem, it isinteresting to take it as input and use it to constrain conformal field theory data. An efficient way to extract consequences of the average null energy condition in CFTs isto use the conformal collider setup of [1]. This technique is closely related to deep inelasticscattering experiments in conformal field theory [1, 14]. As we review, in the context ofAdS/CFT these bounds arise from demanding causality of the bulk theory in a shockwavebackground.The specific physical problem of interest is to create a disturbance in a conformal fieldtheory and then to measure the correlation of energy deposited at various angles at future4ull infinity (see Figure 1).(a) i + i J + (b)Figure 1: In the conformal collider experiment (a), the energy created by a localized exci-tation (blue) is measured far away by a calorimeter (red). (b) For a CFT, this is equivalentto measuring the energy at null infinity J + .The states in which we measure the energy are obtained by acting with local opera-tors O ( x ) on the Lorentzian vacuum | i . We further give these states definite timelikemomentum q . Thus we examine the state |O ( q, λ ) i = N Z d d x e − iqt λ · O ( x ) | i , (2.2)where λ is a polarization tensor accounting for the possible spin of O , and N is a normal-ization factor defined such that (2.2) has unit norm.We now measure the energy at null infinity in this state. In d dimensions null infinityis a sphere S d − and we parameterize it by a unit vector n . hE ( n ) i λ ·O = lim r →∞ r d − Z ∞−∞ dx − hO ( q, λ ) | T −− ( x − , rn ) |O ( q, λ ) i . (2.3)The average null energy condition implies that the resulting function is non-negative as afunction of the direction n .Since we are working in a conformal field theory this energy expectation value may beexplicitly evaluated. Indeed the object being integrated in (2.3) is a three-point function hO T Oi in Lorentzian signature with a prescribed operator ordering. Thus, the result of(2.3) is an explicit function of OPE coefficients. For technical reasons it is sometimes useful to create a localized wavepacket instead of an exact mo-mentum eigenstate. This subtlety will not affect our discussion. .2.1 External States Created by T Let us review the essential details of this calculation in the case where the external stateis created by an energy momentum tensor. In general in d ≥ h T T T i = N B h T T T i B + N F h T T T i F + N V h T T T i V , (2.4)where the various B, F, V structures are those that arise in a theory of respectively freebosons, fermions, or ( d − / Our conventions are such that for free fields, N B counts the number of real scalars, N F the total number of fermionic degrees of freedom(e.g. it is 2 ⌊ d/ ⌋ for a Dirac fermion), and N V counts the number of degrees of freedom in a( d − / d − / Γ( d/ ).A single linear combination of these coefficients is fixed by the conformal Ward identity,and related to the two-point function coefficient C T of energy momentum tensors (seeequation (B.7) for our conventions on the two-point function) C T = 1Ω d − (cid:18) dd − N B + d N F + d N V (cid:19) . (2.5)where Ω n is the area of a sphere S n . As another point of reference let us briefly specializeto the case of four-dimensional theories. In that case, the coefficients of the three-pointfunction are related to conformal anomalies a, c that parameterize the trace of the energy-momentum tensor in a general metric background h T µµ i [ g ] = c π W − a π E , (2.6)where W is the Weyl tensor and E is the Euler density. The coefficient c is proportional to C T , while a = 11440 (4 N B + 11 N F + 124 N V ) . (2.7)Returning to case of general dimensions we now investigate the null energy operatorusing these three-point functions. It is useful to organize the calculation using the relevantsymmetries, which are rotations on the null S d − . In addition, the three-point function of In odd d there is no free field associated to the structure parameterized by N V , but nevertheless thereis still a structure. See [3, 15] for details. Ω n − = 2 π n/ / Γ( n/ ’s is parity invariant. It follows that the most general expression for the null energy is hE ( n ) i λ · T = q Ω d − " t λ ∗ ij λ ik n j n k | λ | − d − ! + t λ ∗ ij λ kl n i n j n k n l | λ | − d − ! , (2.8)where the constants have been fixed so that the total energy of the state is q , and t and t are computable functions of N B , N F , N V .A useful way to understand the answer is to view the vector n as fixed and to decomposethe states (parameterized by their polarizations) under the remaining symmetry group SO ( d − ~n axis is λ ij ∝ (cid:18) n i n j − δ ij ( d − (cid:19) (2.9)In a similar way we can write polarization tensors that have spin one and spin two underrotation around the ~n axis. The energy flux in the direction n is the same for every statein a fixed SO ( d −
2) representation, and we denote them by qT i / Ω d − . Explicitly carryingout the integrals gives: T = (cid:18) − t d − − t d − (cid:19) + d − d − t + t ) = ρ ( d ) (cid:18) N B C T (cid:19) ,T = (cid:18) − t d − − t d − (cid:19) + t ρ ( d ) (cid:18) N F C T (cid:19) , (2.10) T = 1 − t d − − t d − ρ ( d ) (cid:18) N V C T (cid:19) , where the index labels the SO ( d −
2) charge and in the above ρ i ( d ) is a positive function thatdepends only on the spacetime dimension (and not the OPE coefficients). Their explicitform is given in equation (B.33).Additional symmetries imply constraints on the parameters above. In any superconfor-mal field theory we have t = 0. For holographic CFTs dual to Einstein gravity the parame-ters are t = t = 0, giving angle independent energy one-point functions T = T = T = 1.Returning to the general discussion, we can see from (2.10) that the average null energycondition implies the inequalities N B ≥ , N F ≥ , N V ≥ . (2.11)One significant remark concerning the bounds (2.11) is that they may clearly be satu-rated in free field theories. Conversely, it has been argued [16] that any theory that saturates In d = 3 the three-point function has a parity odd piece which we discuss in section 4. T T O in d ≥ We now turn to our main generalization of the conformal collider bounds reviewed insection 2.2. We explore the consequences of the average null energy condition in moregeneral states than those created by a single primary operator. Specifically in this sectionwe will investigate states which are obtained by a linear combination of primary operators.We will find that the average null energy condition in such states yields new inequalitieson OPE coefficients.In this section, the states we consider will be created by a linear combination of theenergy-momentum tensor and a general scalar hermitian operator O . We parameterize sucha state in terms of normalized coefficients v i | Ψ i = v | T ( q, λ ) i + v |O ( q ) i . (3.1)The energy one-point function in the collider experiment is now a matrix h Ψ |E ( n ) | Ψ i = v † (cid:18) h T ( q, λ ) |E ( n ) | T ( q, λ ) i h T ( q, λ ) |E ( n ) |O ( q ) ih T ( q, λ ) |E ( n ) |O ( q ) i ∗ hO ( q ) |E ( n ) |O ( q ) i (cid:19) v . (3.2)The average null energy condition implies that this matrix is positive definite. This is astronger condition than requiring that the diagonal entries are positive and will imply newinequalities on OPE coefficients.The majority of the entries in this matrix have already been computed. For instance, insection 2.2.1 we reviewed the portion of the matrix involving the energy expectation valuein states created by the energy momentum tensor. Even simpler is the entry involving theexpectation value in the scalar state which gives rise to a uniform energy distribution hO ( q ) |E ( n ) |O ( q ) i = q Ω d − . (3.3)It remains to determine the off-diagonal entries in the matrix. It is again useful toorganize the expected answer using the rotation group on the null sphere. Clearly we have h T ( q, λ ) |E ( n ) |O ( q ) i ∼ λ ij n i n j . (3.4)8herefore, the only polarization of the energy momentum tensor that participates in thenon-trivial interference terms is the scalar T aligned along the axis n (see equation (2.9)).To extract this matrix element we require the three-point function h T T Oi . In all d ≥ T imply that this correlator is fixed in terms of a singleOPE coefficient C T T O . We set conventions for our normalization of this OPE coefficientby examining a simple OPE channel. Specifically we restrict all operators to a two-plane,spanned by complex coordinates z, ¯ z . Then the OPE is T zz ( z ) T ¯ z ¯ z (0) ∼ C T T O | z | d − ∆ O (0) . (3.5)If we further assume that O is hermitian then the OPE coefficient C T T O is real. Additionaldetails of this correlator including the full d -dimensional Lorentz covariant OPE and relationto the spinning correlator formalism of [17] are given in appendix B.Based on these remarks, we can in general parameterize the energy flux in the direction n coming from the off-diagonal matrix element (3.4) as h T ( q, λ ) |E ( n ) |O ( q ) i = q Ω d − (cid:18) C T T O √ C T C O h (∆) (cid:19) , (3.6)where h (∆) is some universal function that may be extracted from the conformal collidercalculation, and the factors of C T and C O arise from normalizing the states. The relevantportion of the energy matrix (3.2) is two-by-two and takes the form q Ω d − T C TT O √ C T C O h (∆) C ∗ TT O √ C T C O h (∆) 1 ! . (3.7)Positivity of this matrix therefore leads to the constraint | C T T O | C T C O | h (∆) | ≤ T . (3.8)More generally we may instead consider the collider experiment in a state created by T plus a general linear combination of primary scalar operators. Positivity of the resultingenergy matrix is then equivalent to the following sum rule X Scalar Primaries O i | C T T O i | C T C O | h (∆ i ) | ≤ T . (3.9)In appendix B we explicitly compute the function h (∆) (see equation (B.48)). By combining9he result with the expression (2.10), we may reexpress the bound as X Scalar Primaries O i | C T T O i | C O f (∆ i ) ≤ N B , (3.10)where f (∆) is given as f (∆) = ( d − dπ d Γ (cid:0) d (cid:1) Γ( d + 1)Γ(∆)Γ (cid:0) ∆ − d − (cid:1) ( d − Γ (cid:0) ∆2 + 2 (cid:1) Γ (cid:0) d +∆2 (cid:1) Γ (cid:0) d − ∆2 (cid:1) . (3.11) We now turn to an analysis of the consequences of the general bound (3.10). The function f (∆) has a number of significant properties. • Expanded near the unitarity bound we find a first order pole: f (cid:18) d −
22 + x (cid:19) ∼ x . (3.12)Therefore in any family of theories, an operator O which is parametrically becomingfree (i.e. ∆ = ( d − / x with x tending to zero) must have | C T T O | vanish at leastas fast as √ x . • For large ∆ we find exponential growth f (∆) ∼ ∆ ∆ d +4 . (3.13)We may use this growth to approximate the sum in the bound for scalar operators oflarge ∆. Indeed, let ρ (∆) denote the asymptotic density of scalar primary operators.From convergence of the sum we then deduce that for large ∆ the spectral weightedOPE coefficients must decay exponentially fast ρ (∆) | C T T O | C O ≤ ∆ d +3 ∆ . (3.14)These estimates agree with those implied by convergence of the OPE expansion foundin [18] for scalar operators. • If ∆ is an even integer greater than or equal to 2 d we find that f (∆) vanishes. Wecan understand the necessity of this as follows. We can imagine a large N CFT dualto weakly coupled theory of gravity. In such theories we can consider the sequence10f operators O =: T AB ∂ n T AB :. At large N the dimensions of these operators arefixed to ∆ = 2 d + 2 n . Moreover, for these operators C TT O C O is of order C T . Thus,compatibility with the bound (3.10) for large C T , requires that f (∆) vanishes atthese locations.The above argument does not explain why f (∆) has double zeros. But the doublezeros imply that the bound may be obeyed at subleading order, where we include theanomalous dimensions of these operators which scale as 1 /C T , by truncating the sumon n . • The function f (∆) is non-zero for ∆ = d . Therefore the bound (3.10) may be appliedto marginal operators. In that context, it constrains the change in C T at leadingorder in conformal perturbation theory. Let us investigate the bound further in free field theories. These examples are interestingbecause the bound (3.10) is saturated.Consider first a theory of a free real boson φ in dimension d . There is a Z globalsymmetry under which φ is odd and the energy-momentum tensor T is even. Therefore weneed only consider scalars made from an even number of φ ’s. Since the explicit expressionfor T is quadratic in the free fields, the only possible scalars that may contribute to thebound are : φ : and : φ :.By a simple inspection of the Wick contractions we deduce that : φ : has vanishing T T O correlation function . Meanwhile : φ : has | C T T O | C O = ( d − Γ( d/ π d ( d − . (3.15)This exactly saturates the bound (3.10).We can also consider the bound applied to free fields of different spin. In d = 4 thetheory of free fermions or free gauge bosons have vanishing N B . Therefore the boundimplies that for all scalar operators O either C T T O vanishes, or O has dimension 2 d + 2 n for non-negative integer n .It is straightforward to directly verify this prediction. For instance consider the freevector. The gauge invariant field strength gives rise to two local operators F + µν and F − µν , which are respectively self-dual and anti-self-dual two-forms. Note that this free field theory We thank E. Perlmutter for comments on this point. The contractions imply that h T T : φ : i ∝ h T : φ : ih T : φ : i , which is zero since two-point functionsof different operators vanish. T (a) + T / / + T + O (d) + J / / + J φ φ (f) Figure 2: We consider operators with zero spatial momentum that create a pair of freeparticles. In (a,b,c) we consider a stress tensor operator. We examine the wavefunctionalong the direction specified by the long arrow and we decompose the stress tensor accordingto the spin around that axis. (a) The spin zero state is obtained for scalars, spin one forfermions (b) and spin two for vectors or self-dual forms (c). (d) is the state produced bya scalar operator with can interfere with (a). (e) is produced by a current with spin onealong the observation axis and can interfere with (b). Finally (f) is a current with spin zeroalong the observation axis in a theory of scalars. It produces two different real scalars inthe back to back configuration and cannot interfere with (a).enjoys a continuous electromagnetic duality symmetry under which F ± µν rotate with oppositecharge. The energy-momentum tensor T µν is neutral under this transformation, and hencea scalar operator O with non-vanishing C T T O must also be neutral. If we recall that F + µν F − µν vanishes identically, then we see that the lowest dimension neutral scalar operatoris ( F + µν F + µν )( F − αβ F − αβ ). Since this has dimension eight, the weight function f (∆) vanishes.Moreover all other scalar operators that are neutral have larger even integer dimension.Thus, the bound is obeyed.A more physical way to understand why the bound is saturated in the free scalar theoryis to visualize the state created by local operators.Let us consider the action of an operator with non-zero energy but zero spatial momen-tum. If the operator is a bilinear in the fields, such as the stress tensor in a free theory,then it will create a pair of particles with back to back spatial momenta. Of course, theoperator creates a quantum mechanical superposition of states where these momenta pointin various directions. For a scalar bilinear operator we get an s-wave superposition. For12he stress tensor we get a superposition determined by the polarization tensor.As in previous sections, we measure the energy in the angular direction n and hencecan focus on the properties of the wavefunction for the pair of particles in that particulardirection. As in section 2.2.1 it is convenient to decompose the polarization tensors of theoperators according to their angular momentum around the n axis. We can then easilycheck that a spin zero state T can be produced only in a theory of scalars, a spin one state T can be produced only in a theory of fermions, and T only in a theory of vectors (or d/ − O =: φ :, where φ is an elementary scalar, can alsoproduce a back to back combination of scalar particles, see Figure 2(d). Along the directionof observation this combination has the same form as the one produced by T , in Figure 2(a).It is clear that we can make a quantum mechanical superposition so that the wavefunctionfor the pair vanishes along that particular observation direction. This saturates the boundbecause we get zero energy along that direction. For that superposition of T and O theenergy along other directions is still non-zero.A similar argument helps us understand why we also saturate the h T T J i correlatorbound in the four dimensional theory of a Weyl fermion (see section 5). In that case wecan make a superposition of the state T in Figure 2(b) with the state J in 2(e). Noticethat we are using that J couples to a chiral fermion. If there was another fermion with thesame helicity but opposite charge, as it would be the case for a vector-like current, then wewould have an additional contribution to the state created by the current that will have arelative minus sign compared to the other charged particle pair. On the other hand, for thestate created by the stress tensor these two contributions have the same sign, therefore wecannot destructively interfere them.This highlights that the bound comes from a quantum mechanical interference effect.We saturate the bound through a destructive interference effect that prevents particles fromgoing into a particular direction. It is important to note that this is an interference for thepair of particles. For example, if we consider a theory of scalars with a U (1) symmetrygenerated by a current J , then in a basis of real scalars the current will create two differentscalars, say φ and φ . This cannot interfere with the state created by the stress tensorwhere we have the same scalar for the two particles indicated in Figure 2(a). T T O in d = 3 In this section we will consider the case of d = 3 separately. There are two reasons for doingthis. First, the stress-tensor three-point function has two parity even structures, insteadof three as in d ≥
4, and has a parity odd piece which is special to d = 3. Secondly, thecorrelation function h T T Oi also has an extra parity odd structure special to d = 3 [19].13irst we consider external states created by the stress-tensor. We parametrize the three-point function of energy-momentum tensors as h T T T i = N B h T T T i B + N F h T T T i F + N odd h T T T i odd , (4.1)where N B and N F already appeared in the d ≥ N odd parametrizes a newstructure. We use the same convention for the explicit expression for h T T T i odd as in [8] .In d ≥ SO ( d −
2) symmetryfor the calorimeter direction n . The linearly independent tensor polarizations are organizedas scalar, vectors or tensors with respect to this symmetry. In d = 3 the group becomes SO (1) and there are only two types of polarizations, which we take as λ = 1 √ (cid:18) − (cid:19) , λ = 1 √ (cid:18) (cid:19) . (4.2)The collider energy one-point function for an arbitrary polarization has the structure hE ( n ) i λ · T = q π " t (cid:18) | λ ij n i n j | | λ | − (cid:19) + d ε ij ( n i n m λ jm λ ∗ kp n k n p + n i n m λ ∗ jm λ kp n k n p )2 | λ | . (4.3)To obtain a bound on these parameters we can consider a state created by | Ψ i = v | T ( q, λ ) i + v | T ( q, λ ) i . The energy matrix becomes h Ψ |E ( n ) | Ψ i = q π v † (cid:18) T T odd T odd T (cid:19) v , (4.4)where T = 1 − t / T = 1 + t / T odd = d /
4. These parameters were computedin [8] in terms of the h T T T i parameters N B , N F and N odd obtaining C T T = 316 π N F , C T T = 316 π N B , C T T odd = 316 π N odd . (4.5)For supersymmetric CFTs t = 0 just as in the case d ≥
4. Also, CFTs dual to Einsteingravity have t = d = 0.The average null energy condition implies that the matrix (4.4) is positive definite.This implies t and d lie inside a circle t + d ≤ , or equivalently N B ≥ N F ≥
0, and N ≤ N B N F .Now we will generalize this construction along the same lines as presented in section 3. We identify our N odd with their π p T /
14e will consider a superposition between stress tensor and a scalar operator states | Ψ i = v | T ( q, λ ) i + v | T ( q, λ ) i + v |O ( q ) i . (4.6)As anticipated above, for d = 3 the correlation function h T T Oi is now determined by twoparameters h T T Oi = C even T T O h T T Oi even + C odd T T O h T T Oi odd , (4.7)where the even part is given by specializing the arbitrary d correlator d = 3 , and ourchoice of normalization for the odd part is given explicitly in appendix D. We can makeour conventions for this latter term as in (3.5) in the following way. We can define C odd T T O by the following OPE T zz ( z, ¯ z, y = 0) T zy (0) ∼ C odd T T O ¯ z | z | O (0) , (4.8)where the three spatial coordinates are ( z, ¯ z, y ).Using this normalization, the energy one-point function is given in terms of a three-by-three matrix as h Ψ |E ( n ) | Ψ i = q π v † T T odd C even TTO √ C T C O h even3 d (∆) T odd T C odd TTO √ C T C O h odd3 d (∆) C even TTO ∗ √ C T C O h even3 d (∆) C odd TTO ∗ √ C T C O h odd3 d (∆) 1 v , (4.9)where the functions h even3 d (∆) and h odd3 d (∆) can be obtained repeating the procedure reviewedin appendix B and we obtain h odd3 d (∆) = 12 √ π p Γ(2∆ − ∆+12 )Γ(∆ + 3) 1Γ( − ∆2 ) , (4.10) h even3 d (∆) = 12 √ π p Γ(2∆ − ∆2 )Γ(∆ + 3) 1Γ(3 − ∆2 ) . (4.11)Demanding positive definiteness of the energy matrix gives several types of constraintswhich involve the scalar OPE coefficients. Two of these bounds are easy to generalize toan arbitrary number of scalar operators X i | C even T T O i | C O i f even (∆ i ) ≤ N B , X i | C odd T T O i | C O i f odd (∆ i ) ≤ N F , (4.12)where we defined f odd / even = | h odd / even3 d | /
3. We can consider the positivity of the determi-nant of the 3 × h T T T i is sufficient for the positivity of the energy one-point function N B | C even T T O i | f even (∆ i ) C T C O i + N F | C odd T T O i | f odd (∆ i ) C T C O i − N odd Re [ C even T T O i p f even (∆ i ) C odd T T O i p f odd (∆ i )] C T C O i ≤ N B N F − N . (4.13)This bound can also be generalized to include an arbitrary number of scalar operators.However, as opposed to the situation in section 3, the bounds involving different numberof operators are independent. Their expressions in this case become more cumbersome andwe will omit them here.The (4.10) (4.11) have similar properties as the one appearing for the d ≥ / n (even) and7 + 2 n (odd) for integer n . The zeros in the even case were explained by the existence ofoperators with two stress tensors in theories that are dual to weakly coupled gravity, seethe last point in section 3.1. The odd ones have the same explanation, except that now thescalar operators have the structure ǫ ABC T AD ∂ n ∂ C T BD . In this section we apply the bounds derived to large N Chern Simons theories at level k coupled to fundamental matter. For definiteness we will consider fundamental fermions.We will denote the ’t Hooft coupling by θ = πN/ k . The elements of the energy matrixinvolving the stress tensor were computed in [8] using the explicit large N expressions forthe stress tensor three-point function [20]. The result is T = 2 cos θ , T = 2 sin θ , T odd = 2 sin θ cos θ . (4.14)Using the conventions in, for example, [21] we can compute the off-diagonal elements in-volving stress-tensor mixed with a scalar operator. In the fermionic theory we consider thescalar denoted by O ∼ ψ ¯ ψ has dimension ∆ = 2. The final result for the energy matrix is h Ψ |E ( n ) | Ψ i = q π v † θ θ cos θ √ θ θ cos θ θ √ θ √ θ √ θ v . (4.15)As a function of the ’t Hooft coupling, this matrix has the property that all the minors havevanishing determinant. This implies saturation for all types of superposition of states. Forthe case of the stress tensor this was noted in [8], but we find that this is a more generalfeature for states where we also act with O .16 − − − − − C T T ε /C free C TT ε ′ / C f r ee Figure 3: 3d Ising model allowed region for C T T ε and C T T ε ′ .Even though we do not have a concrete physical picture explaining this, we expect apicture along the lines of section 3.2, where the interaction with the Chern-Simons gaugefield has the effect of replacing free bosons or fermions by “free anyons”.This discussion can also be applied to the case of CS coupled to fundamental bosons.From [20] we know that the energy matrix, given in terms of CFT three-point functions,can be obtained from the fermionic theory by the replacement θ → θ + π when we considerthe operator O ∼ φ of dimension ∆ = 1. More generally we can consider the answer(4.15) as giving the energy matrix of a large N theory with a slightly broken higher spinsymmetry parametrized by θ . d Ising Model As another example, we can apply our bounds to obtain predictions for three-point coeffi-cients for the 3 d Ising model. First let us parameterize the three-point coefficients of theenergy-momentum tensor. Since this theory is parity preserving the coefficient N odd in (4.1)is necessarily zero. The remaining two structures in h T T T i have recently been computednumerically using the conformal bootstrap in [9, 22]. Explicitly N B ≈ . , N F ≈ . . (4.16)The Ising model has a Z global symmetry under which T is even. Therefore only Z evenscalars participate in the bound. The lightest Z even and parity even scalar is the operator In making these estimates we use a value of θ ≈ . whose dimension is known ∆ ε ≈ . . (4.17)Therefore, in a normalization where the two-point function coefficient of ε is one, we canevaluate (4.12) and find the bound | C T T ε | ≤ . . C free , (4.18)where in the last equation we normalized the answer by the expression (3.15) for the valueof the OPE coefficient in the free theory C free = | C T T : φ : | / p C : φ : . Note that although : φ :saturates the bound in the free field theory, the dimension of ε is larger than that of : φ :and hence the OPE coefficient C T T ε may be larger than C T T : φ : .We can obtain a stronger bound by including the operator ε ′ of dimension ∆ ε ′ ≈ . f even (∆) for these dimensions and nor-malizing by the T T : φ : OPE we obtain the constraint0 . | C T T ε | + 0 . | C T T ε ′ | ≤ C . (4.19)Since the operators ε and ε ′ are hermitian their OPE coefficients are real and the boundabove defines the allowed region of OPE coefficients as the interior of an ellipse shown inFigure 3. T T J in d = 4 As a final example, we consider states created by a linear combination of the energy-momentum tensor and a conserved vector current J in d = 4 spacetime dimensions. In thiscase the three-point function h T T J i is controlled by a single OPE coefficient C T T J and isparity violating. This three-point function is presented in detail in appendix E.One reason why this OPE coefficient is interesting is that it is equivalent to a non-trivialmixed anomaly between the flavor symmetry generated by J and the Lorentz symmetrygenerated by T [23]. In the presence of a background metric g , the current J is not conservedbut instead obeys [24–26] h∇ µ J µ i [ g ] = C T T J π ǫ µνρσ R µνδγ R δγρσ , (5.1)where R µνρσ is the Reimann tensor.In the above, our normalization is such that the coefficient C T T J may be expressed as18he net chirality of the charges of elementary Weyl fermions: C T T J = X Left Weyl i q i − X Right Weyl j q j . (5.2)In particular, for the theory of a single Weyl fermion C T T J is one. In an abstract CFTwithout a Lagrangian presentation our normalization of the OPE coefficient is defined asfollows. Fix complex coordinates ( z, w ) . Then the OPE of operators restricted to the w = 0plane is T ww ( z ) T ¯ w ¯ w (0) ∼ C T T J π | z | ( zJ ¯ z − ¯ zJ z ) . (5.3)We will also need the three-point function h T J J i . This correlator is controlled by twoindependent coefficients: h T J J i = Q CB h T J J i CB + Q W F h T J J i W F . (5.4)Here the structures CB and W F are those found for the U (1) current in a theory of freecomplex bosons ( CB ) or free Weyl fermions ( W F ). In a free field theory, these are expressedin terms of the charges of elementary fields as (see [15]) Q CB = X complex scalars i q i , Q W F = X Weyl fermions i q i . (5.5)In general, a single linear combination of these OPE coefficients is fixed by the Wardidentity. We have h J J i ∝ C J ≡ (cid:0) Q CB + 2 Q W F (cid:1) . (5.6)The two-point function coefficient C J can also be interpreted as a conformal anomaly.Indeed, in the presence of a non-trivial background gauge field that couples to J , theenergy-momentum tensor acquires an anomalous trace. In our conventions this is h T µµ i [ A ] = C J F αβ F αβ . (5.7)We can bound the anomaly coefficient C T T J using the same methods described in earliersections for scalar operators. We enforce positivity of the average null energy operator E in the state | Ψ i created by a linear combination of T and J | Ψ i = | T ( q, λ T ) i + | J ( q, λ J ) i . (5.8)The expectation values hE i λ T · T and hE i λ J · J have been computed in [1]. The matrix ofenergy expectation values in the states | Ψ i may again be decomposed in terms of the19 O (2) rotation symmetry about the vector n . The current operator J contributes states ofcharge − , , . As in the review of section 2.2.1 we may express the null energy expectationvalue as ( qJ i / π ) where i is the SO (2) charge. One then finds J ± = Q W F C J . (5.9)By repeating the collider calculation we find that the new off-diagonal matrix elementis given by h T ( q, λ T ) |E ( n ) | J ( q, λ J ) i = q π r π C T T J √ C T C J ε ijk λ ∗ T,im λ J,k n m n j ! . (5.10)Note that this structure is parity odd as expected. There are other allowed parity oddexpressions in terms of λ ij and n i , but they do not arise in the null-energy matrix element.An important feature of (5.10) is that only those states of SO (2) charge ± T defined in (2.10).Explicitly choosing appropriate polarization tensors we then find that positivity of thenull energy matrix E leads to a single constraint on these OPE coefficients: C T T J ≤ Q W F N W F , (5.11)where N W F = N F / h T T T i correlationfunction. This bound is saturated in the free field theory of Weyl fermions. This can beunderstood using the interference argument described in section 3.2. R -Current As in our analysis of scalar operators, we can generalize these results to states created bymultiple currents. This is particularly interesting in the case of supersymmetric theories.In supersymmetric theories, there is always a current J R contained in the same super-multiplet as T . In particular, since it resides in a different multiplet it can be distinguishedfrom an ordinary flavor current J F . We would like to improve our bound on the traceanomaly of J F to account for the fact that the R -current J R always exists. In order to dothis we consider the state created by | Ψ i = v | T ( q, λ T ) i + v | J R ( q, λ J ) i + v | J F ( q, λ J ) i . (5.12)The new ingredient appearing in the calculation of the energy matrix corresponding to20his state involves the three-point function h T J R J F i . Using superconformal invariancewe can fix this correlator completely. Since the details are not very illuminating we willoutline the procedure. The number of parity even structures, two of them, coincides withthe ones appearing in h T J J i , namely relaxing permutation symmetry does not add newstructures [17]. Moreover, using supersymmetric Ward identities [27] one can check that noparity odd structure is allowed for h T J R J F i . Out of the two OPE coefficients characterizing h T J R J F i , a linear combination of them is related to the two-point function h J R J F i , whichvanishes due to superconformal invariance. This leaves h T J R J F i fixed by a single OPEcoefficient. Finally, since J R lies in the same multiplet as the stress tensor we can relatethis number to C T T F , the mixed anomaly generated by the flavor current.Combining the results outlined in the previous paragraph, and the fact that there isno new structure involved in the collider calculation, it is straightforward to obtain theoff-diagonal matrix element h J R ( q, λ J ) |E | J F ( q, λ J ) i = q π r π C T T F √ C T C F ! , (5.13)where we chose n = (1 , ,
0) and λ J = (0 , , i ) for definiteness.We can express parameters related to the R -current in terms of a and c = C T π / C T by a supersymmetry Ward identity as C R = c .Its mixed anomaly is also fixed by supersymmetry to C T T R = 16( c − a ). Finally the energyone-point function is given by J R ± = ac [1, 7]. Supersymmetry also fixes this parameter forflavor currents as J F ± = 1. Taking these facts into account allows us to write down theenergy matrix as a function only of a , c , C T T F and C F . We obtain h Ψ |E | Ψ i = q π v † c − ac √ c − ac √ c C TTF √ C F √ c − ac ac √ c C TTF √ C F √ c C TTF √ C F √ c C TTF √ C F v , (5.14)where for definiteness we have chosen λ J = (0 , , i ) and a tensor polarization with the same SO (2) spin.Enforcing the positivity of this matrix yields several constraints. The leading two-by-twominor involving states | T ( q, λ T ) i + | J R ( q, λ J ) i gives the bound12 ≤ ac ≤ , (5.15)which coincides with those derived in [1]. This bound is saturated by a free chiral multiplet, This is not true for a three-point function of a stress tensor and two different conserved currents h T J J i in a generic theory. c = , or a free vector multiplet, ac = .To constrain the gravitational anomaly coefficient we evaluate the determinant of thefull three-by-three matrix (5.14). This gives the following bound on the mixed anomaly fora flavor current (cid:18) ac − (cid:19) (cid:18) c − a − C T T F C F (cid:19) ≥ . (5.16)For a free chiral multiplet the bound is automatically saturated, since the first term in theleft hand side vanishes independently of C T T F . Therefore we will assume that ac > . Thenwe obtain the following bound C T T F C F ≤ c − a , (5.17)which is stronger than the one derived in the previous section, without the use of super-symmetry. Note also that this is consistent with the free vector multiplet. In that case theright-hand-side vanishes, but there are also no flavor currents.To conclude this section, we can mention some contexts where such bound on the mixedanomaly is relevant. First of all, when we consider holographic CFT this anomaly is relatedto a 5d Chern-Simons term of the form R A ∧ R ∧ R , where A is the gauge field dual tothe current J (we will see in the next section how our bounds translate to bounds on thegravity couplings for the case of T T O ).Finally, in the context of hydrodynamics and transport, quantum anomalies induce aspecial type of transport coefficients, see [28] and, in particular, for the mixed anomaly[29–31]. The coefficient bounded in this section C T T J , is related to the mixed anomalyrecently observed experimentally in Weyl semimetals [32]. In the linear response regime,the mixed anomaly produces an energy current ~j given by [29–31] ~j = 24 C T T J T ~B , (5.18)where we denote the temperature by T and the system is placed in a fixed magnetic field ~B . This allows us to translate our results into concrete bounds for transport coefficients. AdS
Effective Action
If the d dimensional boundary theory has an AdS d +1 dual, then we would like to translatethe bounds on C T T O to bounds on the coefficients of the bulk effective action. We areimagining that the theory has a large N expansion. Then, to leading order, the bulk isgiven by a collection of free fields propagating on the AdS metric. The simplest interactionscorrespond to bulk three-point interactions. These lead to three-point functions in theboundary theory. For the case of gravitons we have a three-point interaction coming from22he Einstein Lagrangian, but it is also necessary to include higher derivative terms, of theform W and W , in order to get the most general structures for the tensor three-pointfunction. It is possible to match the coefficients of the new structures to the coefficients ofthese higher derivative terms in the Lagrangian [1, 3].Here we consider the same problem for the case of the h T T Oi correlator. The firstobservation is that in Einstein gravity this correlator is zero, since the action of any field,expanded around the minimum of its potential has an action without a linear term in thescalar field. Notice that this also implies that a massive scalar field cannot not decay intotwo gravitons. However, we can write the higher derivative term S = M d − pl α Z d d +1 x √ gχW (6.1)in the action, where we normalized the χ field to be dimensionless. This term enables thefield χ to decay into two gravitons. In flat space there is only one structure for the on shellthree-point function between a scalar and two gravitons, except in four dimensions wherethrere is also a parity odd one, as we discuss later. Therefore the vertex (6.1) representsthe general interaction that we can have in the theory. There can be other ways to writeit which give the same three-point function as (6.1). It is possible to check that (6.1) givesrise to a h T T Oi three-point function with the coefficient C T T O p f (∆) √ C T = 8 √ d ( d − π d/ √ d + 1 Γ( d/ αR AdS . (6.2)At first sight, it seems surprising that the function f (∆) appearing here is the same as theone that appears in the bound (3.10). This means that the ∆ dependence disappears whenwe express the bound in terms of α . This is easy to understand when we derive (6.2) asfollows.First we notice that integrating the stress tensor along a null line, as in the definitionof the energy measurement E = R dx − T −− ( x − , x + = 0 , ~y = 0), we produce a shock wavein the bulk that is localized at x + = 0. We can then imagine scattering a superpositionof χ and a graviton through this shock wave. This leads to a time delay that is given bya matrix mixing the graviton and the scalar. An important point is that the propagationthrough the shock wave is given by integrating the wave equation in a small interval beforeand after x + = 0. Only the shock wave contributes to this short integral over x + , butthe scalar mass term does not contribute. Therefore the time delay matrix is independentof the mass of the scalar. We can determine the precise coefficient in (6.2) by doing thisexplicit computation for Einstein gravity plus (6.1). We then get a bound on α by requiring Here M pl is the reduced Planck mass in d + 1 dimensions, defined so that the Einstein term is S = M d − pl R d d +1 x √ gR . Similarly, the action of the scalar field is S = M d − pl R [( ∇ χ ) − m χ ]. α if oneassumes that there is a gap to the higher spin particles, as was discussed in [33] for thecase of the graviton higher derivative interactions. A similar analysis can be done for the5d Chern-Simons term coupling dual to the mixed anomaly [34].In string theory, we expect that α is the order of α ′ , the inverse string tension. If gravityis a good approximation, α ′ ≪ R , then we find that the bound on (6.2) is far from beingsaturated. The bound is saturated only as the string length becomes of the order of theradius of AdS . In particular, this implies that the bound is satisfied, and far from beingsaturated, for the Konishi operator of N = 4 super Yang Mills at strong coupling. Thisoperator is the lightest non-protected single trace operator which has a dimension growinglike ∆ ∝ λ / at strong coupling, λ ≫ −− graviton helicities are Lorentz invariant by themselves. (The − + gravitonhelicities are forbidden by angular momentum conservation). We can then write the actionas S = M pl Z d x √ g (cid:20)
12 ( R − ∇ χ ) − m χ ] + Z α e χW + α o χW W ∗ (cid:21) , (6.3)where as above we have defined χ to be dimensionless. In this normalization α i hasdimensions of length squared. They can be related to the coefficients of the three-pointfunction as C even T T O h even (∆) √ C T = 24 √ α e R AdS , C odd T T O h odd (∆) √ C T = 24 √ α o R AdS . (6.4)The bounds in this case then read p α e + α o R AdS ≤ √ , (6.5)in the case that there are no purely gravitational corrections to Einstein gravity. Of course,if there are three-point functions that lead to corrections to Einstein gravity, then the boundis corrected to those given in section 4. We thank E. Perlmutter and D. Meltzer for discussions on this issue. We also define ( W ∗ ) µνρσ = ǫ µνδγ W δγρσ . Constraints for de-Sitter and Inflation
The physics of inflation might be our very best window into very high energy physics.The standard inflationary theory starts with a scalar field coupled to the Einstein actionand includes all two (or less) derivative interactions. The universe undergoes a period ofexpansion that is governed by a nearly de-Sitter solution, characterized by a Hubble scale H that is nearly constant. The effective coupling of the gravitational sector is of order H/M pl which is very small, less than 10 − . However, it is possible that there are correctionsto the two derivative action due to the presence of a light string scale. The value of thestring tension could be fairly low H . T . When the string tension becomes comparableto the Hubble scale, we expect significant corrections to the two derivative action. We donot have an explicit scenario where this happens. However, a similar situation happens in AdS space when we consider a gravity dual of a not so strongly coupled large N theory.Therefore it is natural to question whether something similar could happen in inflation andwe can look for signatures of such a low string scale. It is important to find signatures thatare as model independent as possible. Specially nice signatures are those that have a non-vanishing contribution in the de-Sitter approximation. These are not strongly suppressed byslow roll factors. In addition, their form is strongly constrained by the de-Sitter isometries.An example of such contributions are the three-point functions of gravity fluctuations,where the higher derivative corrections were discussed in [35]. Another interesting case arethe couplings of the form f e ( χ ) W or f ( χ ) W W ∗ . These two couplings are particularlyinteresting because their effects are visible at the two-point function level.Let us discuss first the parity odd coupling, which leads to chiral gravity waves [36, 37].Namely, we have different gravity wave two-point functions, hh L , hh R , for the left and righthanded circularly polarization. We can define the asymmetry A as A ≡ hh L − hh R hh L + hh R = 4 π ˙ f o ( χ ) H H = ± π √ ǫ (cid:18) ∂f∂χ (cid:19) H , χ = φM pl , (7.1)where χ is defined to be dimensionless and φ is the inflaton with canonical normalization.(The ± comes from going from ˙ χ to √ ǫ , since the derivative of the scalar can have eithersign). If we were in AdS we would have a sharp bound on the coefficients via the condition(6.5), after we identify α o = ∂f∂χ . It is reasonable to think that in the de-Sitter case too,there will be trouble is the bound is violated. Of course, we know that even near-saturationof the bound implies that the field theory approximation is breaking down.In the de-Sitter case we do not have a sharp derivation of a bound from boundary theoryreasoning. We do not have an analog of the null energy condition, discussed in section 4,for the boundary theory, since the boundary theory is purely spacelike. It would be niceto have a sharp derivation of a de-Sitter version of the bound. In de-Sitter, we can talk25f a “quasi-bound”, which we get by simply applying the same bound on the coefficientsof the action that we had in anti-de-Sitter. This quasi-bound should be viewed simplyas an educated guess, including numerical coefficients, for the validity of bulk effectivetheory. A near saturation of these quasi-bounds is a strong indication of a light string scalewhich could also have other manifestations such as indirect evidence of higher spin massiveparticles, etc [38]. In summary, in de-Sitter also we have a quasi-bound on the coefficientssimilar to (6.5), with 1 /R AdS → H s(cid:18) ∂f e ∂χ (cid:19) + (cid:18) ∂f o ∂χ (cid:19) = p α e + α o ≤ H √ . (7.2)This bound, then implies a quasi-bound on the asymmetry (7.1) of the form | A | ≤ π √ ǫ . (7.3)The allowed values by this quasi-bound seem to be smaller than the smallest possiblemeasurable value from the CMB B-modes as analyzed in [39]. Conversely, this means thatif chiral gravity waves through E-B mode correlators are measured, then we would need ahigher derivative coupling with a coefficient so large that it violates (7.2).Let us turn now to a discussion of the parity even coupling. This coupling gives riseto a violation of the consistency condition for the two-point function [40], even in the casethat the speed of sound is close to one, − n t r = 1 + 8 H Hd t f e ( d t χ ) = 1 ± H α e √ ǫ , (7.4)where we assumed that the speed of sound for the scalar is close to one. Here n t is the tensorspectral index and r the tensor to scalar ratio conventionally defined. Then the bound wehad in (7.2) translates into the following constraint on the violation of the consistencycondition (cid:12)(cid:12)(cid:12) − n t r − (cid:12)(cid:12)(cid:12) ≤ √ ǫ . (7.5) The φW higher derivative coupling between the scalar and the graviton also give rise tonew contributions to the scalar-tensor-tensor three-point function. This is a contribution,that is non-vanishing in the de-Sitter limit. More precisely, if we can approximate ∂ χ f e ( χ ( t ))by a constant, then we get a contribution even in de-Sitter space. The standard Einsteingravity contribution, [41], is suppressed by a slow roll factor √ ǫ , if we assume that ∂ χ f isof order one. Of course, our bound constrains the size of this three-point function because26t is constraining the size of the coefficient α e ∼ ∂ χ f e ( χ ( t )).The three-point function for the parity odd coupling f o ( χ ) W W ∗ was computed in [42],where it was found to be proportional to ∂ χ f . One might have naively expected a de-Sitterinvariant contribution proportional to α = ∂ χ f , when we approximate this by a constant.The explicit computation by [42] shows that there is no such contribution. This seemssurprising at first sight because this parity odd coupling does indeed give a non-vanishingcontribution to the three-point function in the AdS case. The reason it vanishes in de-Sitter is that it gives a contribution to the de-Sitter wavefunction that is a pure phase,which disappears when we take the absolute value squared of the wavefunction. The samehappens with the W W ∗ parity violating graviton three-point coupling [43]. The correlatorproportional to ∂ χ f found in [42] has an extra factor of ˙ φ and is not expected to be de-Sitterinvariant (though we did not check this explicitly).It should be noted that the correction to the two-point function consistency condition(7.4) has the right form so that the consistency condition involving the soft limit of thethree-point function [41, 44] is obeyed, though we have not explicitly checked the precisenumerical coefficients. A similar remark applies in the parity odd case; the correction tothe two-point function (7.1) is such that the soft limit of the three-point function in [42]obeys the consistency condition. Acknowledgements
We thank H. Casini, S. Giombi, R. Meyer, E. Perlmutter, D. Simmons-Duffin, and D.Stanford for discussions. We also thank E. Perlmutter for comments on a draft. C.C. issupported by the Marvin L. Goldberger Membership at the Institute for Advanced Study,and DOE grant de-sc0009988. J.M. is supported in part by U.S. Department of Energygrant de-sc0009988 and the Simons Foundation grant 385600.
A Absence of Positive Local Operators
Let us review the essential steps of [11] in a modern language. Let Φ be any Hermitianoperator and | i the Lorentz invariant vacuum state. We make two assumptions: • The one-point function h | Φ | i vanishes. • For all states | ψ i in the Hilbert space, the expectation value h ψ | Φ | ψ i is non-negative.27nder these assumptions we may prove that Φ annihilates the vacuum state, Φ | i = 0 . Indeed, for any positive operator, the Cauchy-Schwarz inequality implies that |h ψ | Φ | i| ≤ h ψ | Φ | ψ ih | Φ | i . (A.1)Since the right-hand side is zero by hypothesis, we conclude that Φ | i must vanish.If we now further assume that Φ is an operator localized within a compact region R ,we can deduce that Φ must vanish. To demonstrate this, consider operators localized in aregion R ′ that is spacelike separated from R . Let us denote by O R ′ a set (sum of products)of smeared operators in region R ′ , then we have0 = h | O R ′ O R ′ Φ( z ) | i = h | O R ′ Φ( z ) O R ′ | i , (A.2)where we have used that the operator O R ′ is spacelike separated from the region where Φis localized in order to move it to the right of Φ. However, according to the Reeh-Schliedertheorem [45], any state | ψ i may be approximated to arbitrary precision by acting with a(smeared) set of local operators in any open set in spacetime. Since the region of pointsthat are spacelike separated from a finite compact region is an open set, we may apply thisidea to the right-hand side above to conclude that for any sates | ψ i ih ψ | Φ | ψ i = 0 . (A.3)This implies that Φ vanishes as an operator.It is interesting to pinpoint exactly where this logic breaks down for non-local operatorssuch as the average null energy operator E . As long as the region that is spacelike separatedto Φ is open, one may repeat the Reeh-Schlieder argument and prove that Φ vanishes evenif it is non-local. The way the null energy operator E avoids this conclusion is that ithas support along a complete null line and hence the region of points that are spacelikeseparated to E is not open, since it consists just of the codimension one null plane containingthe null line. B Details of the Collider Calculation
In this appendix we will give more details on the calculation of the energy expectation valuefor a conformal collider experiment that we considered in this paper, giving a bound on C T T O in arbitrary dimensions. We that H. Casini for an enlightening discussion on this point. .1 Normalized States The states that we consider for the collider experiment are superposition of states of nor-malized wavepackets. Following [1] we take the state defined as |O ( q, λ ) i ≡ N Z d d x e − iqx exp (cid:20) − x + ~x σ (cid:21) λ · O ( x ) | i , q > qσ ≫
1. We find the normalization N by requiring the state to have unit norm inthis limit. We will give their values only for the operators and polarizations relevant tocomputing the bound on C T T O . Namely for a scalar operator and for the stress tensor withpolarization which is scalar with respect to the SO ( d −
2) symmetry perpendicular to n .We normalize the scalar operator such that its two-point function is hO ( x ) O (0) i = C O x − . Then the normalization condition for the state considered in the collider experi-ment is hO ( q ) |O ( q ) i = 1 ⇒ N − O = C O π d +22 Γ(∆)Γ(∆ − d + 1) (cid:16) q (cid:17) − d (B.2)where we used the following integral identity Z d d x e iqx x = 2 π d +22 Γ(∆)Γ(∆ − d + 1) (cid:16) q (cid:17) − d , q > | T ( q, λ ) i ≡ N T Z d d x e − iqx ( λ ) ij T ij ( x ) | i , (B.4)where we assumed the localized wavepacket limit. If we use conservation of the stress tensorwe can chose the polarization along spatial directions. The normalized scalar polarizationis ( λ ) ij = r d − d − (cid:20) n i n j − d − δ ij (cid:21) , (B.5)which satisfies Tr( λ ) = 0 and λ · λ = 1. In this case the normalization condition gives h T ( q, λ ) | T ( q, λ ) i = 1 ⇒ N − / T = C T d − π d +1 Γ (cid:0) d (cid:1) Γ( d + 2) (cid:16) q (cid:17) d , (B.6)where the normalization of the two-point function is h T µν ( x ) T ρσ (0) i = C T x d I µνρσ ( x ) , (B.7)29nd the tensor structure that appears derived in [15] is I µνρσ ( x ) = I µρ ( x ) I νσ ( x ) + I µσ ( x ) I νρ ( x )2 − d g µν g ρσ , (B.8) I µν ( x ) = g µν − x µ x ν x . (B.9)Of course, by SO ( d −
1) rotational symmetry, the normalizations for T and T are alsogiven by (B.6), once the polarizations are normalized to unity. Below we will perform theexperiment of [1] for linear superpositions of these normalized states. But first we willreview the form of the correlators we will need, mainly to fix notation and conventions. B.2 Three-Point Functions
The three-point functions we will need are h T T T i , h T OOi and h T T Oi . Their form werederived in [15] and the first two were studied in the context of the conformal collider infour dimensions in [1] and generalized to arbitrary dimensions in [3]. First, we will focus on h T T Oi which was not studied previously in the context of the conformal collider. The formconsistent with conformal symmetry and conservation of the stress-tensor found in [15] is h T µν ( x ) T ρσ ( x ) O ( x ) i = 1 x d − ∆12 x ∆23 x ∆31 I µν µ ′ ν ′ ( x ) I ρσρ ′ σ ′ ( x ) t µ ′ ν ′ ρ ′ σ ′ ( X ) , (B.10)where X = x x − x x and the tensor structure is a sum of three terms t µνρσ ( x ) ≡ ˆ a h µνρσ ( x ) + ˆ b h µνρσ ( x ) + ˆ c h µνρσ ( x ) , (B.11)where each h i is traceless and symmetric under µν ↔ ρσ , x → − xh µνρσ ( x ) = 1 x ( x µ x ν x ρ x σ + . . . ) , (B.12) h µνρσ ( x ) = 1 x ( x µ x ρ g νσ + . . . ) , (B.13) h µνρσ ( x ) = g µρ g νσ + . . . , (B.14)where dots represent terms needed for expressions to be traceless and symmetric and ineach line they involve a fixed number of factors of x . The main advantage of this approachis that it makes transparent the OPE limit x → x , or equivalently taking the x → ∞ limit T µν ( x ) T ρσ (0) ∼ x d − ∆ (ˆ a h µνρσ ( x ) + ˆ b h µνρσ ( x ) + ˆ c h µνρσ ( x )) O (0) . (B.15)30onservation can be imposed in this limit to the right hand side to the equation above,giving the two independent relationsˆ a + 4ˆ b −
12 ( d − ∆)( d − a + 4ˆ b ) − d ∆ˆ b = 0 , (B.16)ˆ a + 4ˆ b + d ( d − ∆)ˆ b + d (2 d − ∆)ˆ c = 0 . (B.17)This fixes the three point function to a single conserved structure up to an overall coefficient,which we can define as C T T O ≡ ˆ a + 8(ˆ b + ˆ c ) . (B.18)Another standard way of representing conformal three-point functions is given by thespinning correlator formalism of [17]. We will write the correlator in terms of the embeddingspace coordinate X i ∈ R d +1 , and the polarization Z i ∈ R d +1 , such that Z = 0. Thecorrelator we need is in appendix A of [17] and in terms of the usual conformal structures V i and H ij is given by h T ( X , Z ) T ( X , Z ) O ( X ) i = α V V + α H V V + α H ( − X · X ) d +2 − ∆2 ( − X · X ) ∆2 ( − X · X ) ∆2 , (B.19)where T ( X, Z ) ≡ Z A Z B T AB ( X ) (with the index running from 0 to d + 1) and the labels α corresponds to the subset of the 10 structures that h T T O ℓ i has when ℓ = 0. The buildingblocks are V = ( Z · X )( X · X ) − ( Z · X )( X · X ) X · X (B.20) V = ( Z · X )( X · X ) − ( Z · X )( X · X ) X · X (B.21) H = − Z · Z )( X · X ) − ( Z · X )( Z · X )) . (B.22)Starting from the expression in d + 2-dimensional embedding space we can obtain the d -dimensional correlator h T ( x , z ) T ( x , z ) O ( x ) i by using the replacements − X i · X j → x ij , Z i · Z j → z i · z j and X i · Z j → x ij · z j , where now in d -dimensions we define T ( x, z ) = z µ z ν T µν ( x ), with the index running from 0 to d −
1. Of course after these replacementsthe answer coincides with the Osborn-Petkou three-point function. We can match thecoefficients of the different representations by taking the OPE limit. The result gives α = ˆ a + 8(ˆ b + ˆ c ) , (B.23) α = 4(ˆ b + 2ˆ c ) , (B.24) α = 2ˆ c, (B.25)31he OPE coefficient (B.18) is now C T T O = ˆ a + 8(ˆ b + ˆ c ) = α . The conservation equationsin terms of these parameters are α (2 + ∆ − d (1 − d + ∆)) + α (cid:16) − − ∆ + d (cid:17) = 0 (B.26)2 α + 12 α ( − d − d ∆) + α d ∆ = 0 . (B.27)which we can solve in terms of C T T O .We defined in the main text the notation we will use for the stress-tensor three-pointfunction. Another correlator we need is h T µν ( x ) O ( x ) O ( x ) i = C T OO x d x − d x d I µνρσ ( x ) (cid:18) X ρ X σ X − d g ρσ (cid:19) , (B.28)which is fixed by a Ward identity to be C T OO = − C O d ∆( d − d − , with Ω d being the are of a S d sphere. B.3 Energy Matrix
As explained in the main text we want to consider states of the form | Ψ i = v | T ( q, λ ) i + v |O ( q ) i , (B.29)where we take v = ( v , v ) ∈ C such that | v | + | v | = 1. The energy one-point functionin the collider experiment is h Ψ |E ( n ) | Ψ i = v † (cid:18) h T ( q, λ ) |E ( n ) | T ( q, λ ) i h T ( q, λ ) |E ( n ) |O ( q ) ih T ( q, λ ) |E ( n ) |O ( q ) i ∗ hO ( q ) |E ( n ) |O ( q ) i (cid:19) v (B.30)In this section we will compute the entries of this matrix. The diagonal elements werealready computed in [1, 3] and are given by h T ( q, λ ) |E ( n ) | T ( q, λ ) i = q Ω d − (cid:18) − d − d − t − d ( d − − d − t (cid:19) , (B.31)where t = ( d −
1) (( d − N F − d − N V )( d − d − dN V + N F ) + 2 N B ) , t = ( d −
1) ( N B − N F + N V )2 N B + ( d − dN V + N F ) . (B.32)32sing these expressions we can find the parameters we called T , T and T in the maintext. They are given by equation (2.10) where the functions ρ i ( d ) are given by ρ ( d ) = 1Ω d − d ( d + 1)( d − d −
1) (B.33) ρ ( d ) = 1Ω d − d ( d + 1)4 (B.34) ρ ( d ) = 1Ω d − d ( d + 1)( d − d −
3) (B.35)The state created by a scalar operator gives hO ( q ) |E ( n ) |O ( q ) i = q Ω d − . (B.36)Now we will obtain the off-diagonal element of this matrix we has not been computed inthe literature. To perform the calculation in arbitrary dimensions it is convenient to use thespinning correlator formalism. Since we are computing an expectation value the correlatorwe need to consider is not time-ordered. The right iǫ prescription for this purpose wasexplained in [1] and [3], and we will omit it here to ease the notation. We start from the d -dimensional expression h T ( x , z ) T ( x , z ) O ( x ) i = α V V + α H V V + α H ( x ) d +2 − ∆2 ( x ) ∆2 ( x ) ∆2 (B.37)We will chose T ( x , z ) to be the insertion taken to infinity and giving E ( n ). First we take z = m = (1 , n ) (we chose the mostly plus convention for the metric in Minkowski space).Therefore T ( x , z ) → T −− ( x ). Then we take the limit x · ¯ m → ∞ , ¯ m = ( − , n ) . (B.38)To take this limit we can use the results of appendix F of [14], and obtainlim x · ¯ n →∞ h T ( x , z ) r d − T −− ( x ) O ( x ) i = α ˆ V ˆ V + α ˆ H ˆ V ˆ V + α ˆ H d ( x · m ) d +2 − ∆2 ( x ) ∆2 ( x · m ) ∆2 (B.39)where the structures in this limit areˆ V = z · m x − x · z x · mx · m , ˆ V = x · mx , H = − z · m. (B.40)We can get the correct polarization of the insertion T ( x , z ) by replacing z µ z ν → λ Tµν ,33ssuming λ is already traceless and symmetric which is true for expression (B.5). To simplifythe expressions we will choose n = (1 , , . . . ,
0) and write the positions as x = ( x + , x − , x ⊥ ),where x + = x · ¯ m = x + x , x − = x · m = x − x and x ⊥ corresponds to the d − G = lim x +2 →∞ ( x +2 / d − h λ · T ( x ) T −− ( x +2 , x − , O ( x = 0) i (B.41)which is given byˆ G = α λ T ( x − ) ( x − x x − x + x x ( x − ) − d − P i ⊥ ( x i ⊥ ) ( x − ) )2 d ( x − ) d +2 − ∆2 ( x ) ∆2 +2 ( x − ) ∆2 +2 + α λ T − d ( x − )( x x − − x )( x − ) d +2 − ∆2 ( x ) ∆2 +1 ( x − ) ∆2 +1 + α λ T − d ( x − ) d +2 − ∆2 ( x ) ∆2 ( x − ) ∆2 (B.42)First we do integral over x − , using the following identity Z dx − x − − iǫ ) b ( x − − iǫ ) a = 2 πi ( x − − iǫ ) a + b − Γ( a + b − a )Γ( b ) , (B.43)where we made explicit the pole prescription. Finally, to take the limit of the localizedwavepackets is equivalent to setting x → x with momentum ( q, , . . . , Z d d x e − iqx Z dx − ˆ G. (B.44)To do this we first integrate over the d − x ⊥ and then integrate overthe light-cone coordinates x ± . Because of SO ( d −
2) invariance, the integrand only dependson x +1 , x − and x ⊥ · x ⊥ . Then the integral can be written as Z d d x e − iqx Z dx − ˆ G = Ω d − Z dx +1 e − i q x +1 Z dx − e i q x − , × Z R d − dR F ( x +1 , x − , ( x ⊥ ) = R ) , (B.45)where we defined F = R dx − ˆ G to indicate the functional dependence explicitly. Afterperforming these integrals we use conservation conditions to write α and α in terms of α = C T T O using equations (B.16) and (B.23). Combining the three structures gives Z d d x e − iqx Z dx − ˆ G = C T T O − d ( d − π d +2 Γ( d + 1)( d − (cid:0) ∆2 + 2 (cid:1) Γ (cid:0) d − ∆2 (cid:1) Γ (cid:0) d +∆2 (cid:1) λ .n.n (cid:16) q (cid:17) ∆+1 (B.46)34n this expression we generalized the answer to arbitrary n by replacing λ → λ ij n i n j .Finally, we need to replace the specific value of the polarization tensor (B.5) and the propernormalization of the collider states (B.6) and (B.2). The final answer for the off-diagonalentry of the energy matrix is h T ( q, λ ) |E ( n ) |O ( q ) i = q Ω d − C T T O √ C T C O h (∆) , (B.47)where h (∆) ≡ π d +12 Γ( d + 1) q Γ (cid:0) d − (cid:1) Γ( d + 2)Γ(∆)Γ (cid:0) ∆ − d − (cid:1) d Γ (cid:0) d − (cid:1) Γ (cid:0) ∆2 + 2 (cid:1) Γ (cid:0) d +∆2 (cid:1) Γ (cid:0) d − ∆2 (cid:1) (B.48)Then the energy matrix that gives the expectation value for these superposition states is h Ψ |E ( n ) | Ψ i = v † T C TT O √ C T C O h (∆) C ∗ TT O √ C T C O h ∗ (∆) 1 ! v (B.49) T ≡ − d − d − t − d ( d − − d − t (B.50)Having computed the energy matrix the next step is to impose ANEC, which is equiva-lent to imposing positivity of the energy expectation value for the collider experiment. Thismeans that for all states | Ψ( v ) i = v | T ( q, λ ) i + v |O ( q ) i (B.51)we need to impose h Ψ( v ) |E ( n ) | Ψ( v ) i > , ∀ v ∈ C . (B.52)This constraint is equivalent to the positivity of all the leading principal minors of theenergy matrix. The first constraint is T = 1 − d − d − t − d ( d − − d − t = ρ ( d ) (cid:18) N B C T (cid:19) ≥ × | C T T O | C T C O | h (∆) | ≤ T (B.54)We can write T and C T in terms of the h T T T i structures N B , N F and N V . This gives theequivalent expression that we quoted in the introduction | C T T O | C O f (∆) ≤ N B (B.55)35here f (∆) = ( d − dπ d Γ (cid:0) d (cid:1) Γ( d + 1)Γ(∆)Γ (cid:0) ∆ − d − (cid:1) ( d − Γ (cid:0) ∆2 + 2 (cid:1) Γ (cid:0) d +∆2 (cid:1) Γ (cid:0) d − ∆2 (cid:1) (B.56)These two conditions (B.53) and (B.54) are necessary and sufficient for the energy to bepositive for any state of the form (B.51). For operators O that are not hermitian this bounddoes not have information about the phase of the OPE coefficient C T T O . C Free Scalar Correlators
In this section we will present some details on the calculation of the
T T O correlators for afree scalar that saturates the bound above. We use the normalization of [15] for h φ ( x ) φ (0) i = 1( d − d − x d − . (C.1)and the stress tensor is defined as T µν = : ∂ µ φ∂ ν φ : − d −
1) (( d − ∂ µ ∂ ν + g µν ∂ ) : φ : . (C.2)Since scalar operators have integer dimensions we only need to consider O such that ∆ < d .The first one is O ∼ φ . This one is predicted to vanish since f (∆ = d − ) → ∞ . This isindeed the case since an odd number of fields appear in h T T φ i . The next operator is O = : φ : of dimension ∆ = d −
2. The correct normalization of the two-point functiongives h : φ : ( x ) : φ : (0) i = 2( d − Ω d − x d − , C O = 2( d − Ω d − . (C.3)Using Wick contractions we can also compute h T T φ i . One can check that the answer hasthe conformal invariant structure (B.19) with C T T O = α = ( d − d d − d − , α = − d − α , α = 2( d − d α (C.4)Finally, the function appearing in the bound takes the value f (∆ = d −
2) = 8( d − π d ( d − Γ (cid:0) d + 1 (cid:1) (C.5)36utting everything together we find that the bound is saturated | C T T O | C O f ( d −
2) = 1 ≤ N B = 1 . (C.6)We have seen that the bound is saturated by a scalar field with O = : φ :. Neverthelessthere is one more primary scalar field we can make of dimension less than 2 d , namely O = : φ : which has dimension ∆ φ = 2( d − h T T : φ : i = 0. We can argue more generally that this is so. The form ofthe correlator (B.19) indicates that there is a x → x singularity whenever ∆ < d . Onthe other hand, to have a non-zero answer we can only take a Wick contraction which isbetween T and : φ : but not between the stress tensors. Therefore if this calculation wouldgive a non-zero answer, it will be finite when x → x . The only way this is consistent withthe form of the correlator fixed by conformal symmetry (B.19) is if it indeed vanishes. D h T T O i Parity-Odd Structures in d = 3 In dimensions d ≥ h T T O i has only a parity-even structureconsistent with permutation symmetry and conservation of the stress-tensors. The situationfor d = 3 is special since only for this number of dimensions a new parity-odd structureappears, that is also consistent with all the requirements. In this case the full correlator is h T T Oi = h T T Oi even + h T T Oi odd (D.1)where the parity-even part coincides with the answer for d ≥ h T ( X , Z ) T ( X , Z ) O ( X ) i even = α V V + α H V V + α H X − ∆2 X ∆2 X ∆2 , (D.2)and the new structure is h T ( X , Z ) T ( X , Z ) O ( X ) i odd = β V V + β H X − ∆2 X ∆2 X ∆2 ǫ ( Z , Z , X , X , X ) , (D.3)Since conservation put constraints independently for α , , and β , we can forget about theparity-even part and we get β (∆ − − β (∆ + 1) = 0. Therefore the parity-odd structureis also fixed by a single OPE coefficient which we denote C odd T T O , as opposed to the one inthe even part C even T T O = α . For completeness we present the same correlator in the Osborn37nd Petkou formalism h T µν ( x ) T ρσ ( x ) O ( x ) i = 1 x − ∆12 x ∆23 x ∆31 I µνµ ′ ν ′ ( x ) I ρσρ ′ σ ′ ( x ) t µ ′ ν ′ ρ ′ σ ′ ( X ) , (D.4)where t = ˆ d t + ˆ e t and we define t µνρσ ( X ) = x µ x ρ x γ x ε νσγ + . . . , (D.5) t µνρσ ( X ) = δ µρ x γ x ε νσγ + . . . , (D.6)where the dots represent other terms of the same form to make the answer symmetric,traceless and permutation symmetric. Conservation imposes ˆ e (∆ −
7) + ˆ d (∆ −
3) = 0.Either in the Osborn and Petkou formalism or in the spinning correlator formalism, wedefine the parity-odd OPE coefficient as C odd T T O ≡ ˆ d + ˆ e = ( β − β ) / E h T T J i Three-Point Function
In this appendix we will provide some details on the CFT three-point function controllingthe mixed gauge-gravitational anomaly h T T J i . Imposing permutation symmetry betweenthe stress-tensors and conservation, this correlator only involves an allowed parity-oddstructure. In the spinning correlator formalism it is given by h T ( X , Z ) T ( X , Z ) J ( X , Z ) i ∼ H − V V X X X ǫ ( Z , Z , Z , X , X , X ) , (E.1)where as usual the upper-case denote coordinates in embedding space and H ij and V i arethe usual structures defined in [17]. From this expression it is possible to deduce theconservation equation for the current when the CFT is placed on a curved background andgives the right normalization for C T T J [23]. Then the three-point function is h T ( X , Z ) T ( X , Z ) J ( X , Z ) i = C T T J π H − V V X X X ǫ ( Z , Z , Z , X , X , X ) (E.2)For completeness we can write this same correlator using the notation of Osborn andPetkou. This can be written as h T µν ( x ) T ρσ ( x ) J α ( x ) i = 1 x x x I µν µ ′ ν ′ ( x ) I ρσρ ′ σ ′ ( x ) t µ ′ ν ′ ρ ′ σ ′ α ( X ) (E.3)where t µνρσα ( x ) is the OPE structure T µν ( x ) T ρσ (0) ∼ | x | − t µνρσα J α (0) and X = x x − x x . The two structures possible, which are linear combinations of the H and V V , are38xplicitly t µνρσα ( x ) = x γ x µ x ρ | x | ǫ νσαγ + . . . , (E.4)and t µνρσα ( x ) = x γ δ µρ | x | ǫ νσαγ + . . . , (E.5)where the dots represent terms needed to add in ordered for the expression to be symmetric,traceless and permutation symmetric between the first two pair of indices. The most generalcase has t = ˆ at + ˆ bt . Imposing conservation and comparing with the spinning correlatorformalism we get ˆ a = − b = 3 C T T J /π . Using this information it is straightforward toapply the same procedure as was done for h T T O i to obtain the energy matrix elements inthe conformal collider experiment. F Computing the Bound in the Gravity Theory
In this appendix we relate the OPE coefficient C T T O to a coefficient, α , in the AdS D effectiveaction S = M D − pl (cid:20)Z √ g ( R − ∇ χ ) − m χ + 2 αχW (cid:21) , Λ = − ( D − D − R AdS , (F.1)where D is the dimension of AdS D . χ is defined to be dimensionless and α has dimensionsof length squared.In principle we can compute the relation between α and C T T O by computing the threepoint function between a scalar and the graviton produced by this cubic term in the La-grangian, using Witten diagrams. Instead, we will follow a different route. We will directlycompute the energy correlator in gravity and derive a bound on α by demanding its posi-tivity. We then relate α and C T T O by demanding that this gravity bound, in terms of α ,matches the bound we obtained in terms of C T T O in the field theory analysis.We will rely on [1, 3] where the energy correlators were computed in gravity. An im-portant point is that the insertion of T −− corresponds to a shock wave localized in a nullplane. Furthermore, an operator insertion at the origin with definite energy-momentumgives rise to an excitation that crosses this null plane at a localized point. For this reasonthe computation of the bound boils down to analyzing the propagation of an excitationthrough a suitable gravitational shock wave in flat space. The AdS D space is only relevantfor determining the transverse profile of the shock wave, as we will see below.For these reasons we consider a shock wave of the form ds = ds + ( dx + ) δ ( x + ) h ( y ) , ds = − dx + dx − + dy . (F.2)39dding gravitons we get ds = ds + ( dx + ) δ ( x + ) h ( y ) + dx µ dx ν ζ µ ζ ν e ip.x G ( p ) + dx µ dx ν ¯ ζ µ ¯ ζ ν e − ip.x ¯ G ( p ) , (F.3)with ζ = 0, ζ µ p µ = 0. Note that the graviton polarization is ζ µν = ζ µ ζ ν , and is normalizedto one ζ . ¯ ζ = 1. We can think of G ( p ) and ¯ G ( p ) as complex numbers, which in the quantumtheory will be related to a and a † . Inserting (F.3) into (F.1) we can derive the quadraticand cubic interaction terms. S = M D − pl Z dx + dx − d D − y (cid:8)(cid:2) p + p − + δ ( x + ) p − h (cid:3) (cid:2) G ( p ) ¯ G ( p ) + 4 p − p + χ ( p ) ¯ χ ( p ) (cid:3) ++8 p − αζ ij ∂ i ∂ j hδ ( x + ) G ( p ) ¯ χ ( p ) + c.c. (cid:9) , (F.4)where we only wrote the terms relevant for our computation, ignoring transverse derivativesin the kinetic terms. Momentarily setting the scalar field to zero, we see that we have thefollowing equation for the graviton as it crosses the shock wave∆ h µν ≡ h µν | x + =0 + − h µν | x + =0 − = ip − hh µν . (F.5)Exponentiating this, h µν ( x + = 0 + ) = e ip − h h µν ( x + = 0 − ), we see that the time delay issimply given by h . This is as expected from (F.2) since we can shift x − by h and makethe term involving h disappear if we ignore its y dependence. So far, we considered thecomputation in flat space. An insertion of the null energy integrated along a ray in theboundary theory gives rise to a shockwave in AdS D which is localized on a null direction.Its dependence on the transverse directions is the following. The transverse space is an H D − . This is easy to see in embedding coordinates where AdS D is ˜ W + ˜ W − + W µ W ν = − R AdS D = 1). The null plane is ˜ W + = 0. It contains the null direction parametrizedby ˜ W − as well as the transverse space W µ W µ = −
1. The profile of the wave is proportionalto h ∝ ( W − W i n i ) − D [1, 3], with a positive coefficient. Here ~n i is a vector on the sphereat infinity in the boundary Minkowski space. For (F.4) we need the derivatives at W i = 0,which are given by h → h , ∂ i ∂ j h = [(constant) δ ij + ( D − D − n i n j ] h R AdS D , (F.6)where the constant does not matter because the graviton is traceless. The relevant compo-nent of the graviton is the one with polarization along n i . This has the expression ζ ij = r D − D − (cid:20) n i n j − δ ij D − (cid:21) . (F.7)The expression for the time delay acting on a superposition of a graviton and a scalar is40ow a matrix proportional to (cid:18) γγ (cid:19) , γ ≡ D − p ( D − D − αR AdS D , (F.8)where the matrix is acting on a two dimensional space where one direction is the scalarand the other is the graviton with polarization (F.7). The unitarity bound comes from therestriction that the eigenvalues are non-negative, or | γ | ≤
1, which is | α | R AdS D ≤ D − p ( D − D −
2) = 14 d p ( d − d − , (F.9)where d is the dimension of the boundary. Comparing this with the bound obtained in(3.8), with the non-Einstein-gravity structures set to zero, we obtain (6.2). Of course, oncewe get the proportionality constant between α and C T T O for the Einstein gravity case, thesame constant holds also if we add the purely gravitational higher derivative terms thatgenerate the other tensor structures for h T T T i . We could add them to this computation,but we expect to reproduce the bounds we got in the general field theory analysis. F.1 Four-Dimensional Case
In the special case of the four dimensional theory, we actually have two couplings (6.3).This leads to a new interaction term in (F.4) of the form αζ ij ∂ i ∂ j h → α e ζ ij ∂ i ∂ j h + α o ζ il ǫ lj ∂ i ∂ j h , (F.10)where now ǫ ij is the two dimensional epsilon symbol. This means that the scalar can nowmix with the other graviton polarization component besides (F.7). Namely, defining (F.7)as ζ ⊕ , it can also mix with ζ ij ⊗ ≡ ǫ il ζ lj ⊕ . Now the time delay is a a three by three matrix γ βγ β , γ ≡ √ α e R AdS , β ≡ √ α o R AdS , (F.11)where the rows and columns correspond to the scalar and the two graviton polarizations.Now the bound is (6.5). Comparing this to (4.13), after setting the non-Einstein-gravitystructures to zero, we get the precise mapping to the C T T O coefficients (6.4).41 eferences [1] D. M. Hofman and J. Maldacena, “Conformal collider physics: Energy and chargecorrelations,” JHEP (2008) 012, arXiv:0803.1467 [hep-th] .[2] J. de Boer, M. Kulaxizi, and A. Parnachev, “AdS(7)/CFT(6), Gauss-Bonnet Gravity,and Viscosity Bound,” JHEP (2010) 087, arXiv:0910.5347 [hep-th] .[3] A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha, and M. Smolkin,“Holographic GB gravity in arbitrary dimensions,” JHEP (2010) 111, arXiv:0911.4257 [hep-th] .[4] T. Faulkner, R. G. Leigh, O. Parrikar, and H. Wang, “Modular Hamiltonians forDeformed Half-Spaces and the Averaged Null Energy Condition,” JHEP (2016)038, arXiv:1605.08072 [hep-th] .[5] T. Hartman, S. Kundu, and A. Tajdini, “Averaged Null Energy Condition fromCausality,” JHEP (2017) 066, arXiv:1610.05308 [hep-th] .[6] T. Hartman, S. Jain, and S. Kundu, “A New Spin on Causality Constraints,” JHEP (2016) 141, arXiv:1601.07904 [hep-th] .[7] D. M. Hofman, D. Li, D. Meltzer, D. Poland, and F. Rejon-Barrera, “A Proof of theConformal Collider Bounds,” JHEP (2016) 111, arXiv:1603.03771 [hep-th] .[8] S. D. Chowdhury, J. R. David, and S. Prakash, “Constraints on parity violatingconformal field theories in d = 3,” arXiv:1707.03007 [hep-th] .[9] A. Dymarsky, F. Kos, P. Kravchuk, D. Poland, and D. Simmons-Duffin, “The 3dStress-Tensor Bootstrap,” arXiv:1708.05718 [hep-th] .[10] R. W. Fuller and J. A. Wheeler, “Causality and Multiply Connected Space-Time,” Phys. Rev. (1962) 919–929.[11] H. Epstein, V. Glaser, and A. Jaffe, “Nonpositivity of energy density in Quantizedfield theories,”
Nuovo Cim. (1965) 1016.[12] J. L. Friedman, K. Schleich, and D. M. Witt, “Topological censorship,” Phys. Rev.Lett. (1993) 1486–1489, arXiv:gr-qc/9305017 [gr-qc] . [Erratum: Phys. Rev.Lett.75,1872(1995)].[13] S. Balakrishnan, T. Faulkner, Z. U. Khandker, and H. Wang, “A General Proof ofthe Quantum Null Energy Condition,” arXiv:1706.09432 [hep-th] .4214] Z. Komargodski, M. Kulaxizi, A. Parnachev, and A. Zhiboedov, “Conformal FieldTheories and Deep Inelastic Scattering,” Phys. Rev.
D95 no. 6, (2017) 065011, arXiv:1601.05453 [hep-th] .[15] H. Osborn and A. C. Petkou, “Implications of conformal invariance in field theoriesfor general dimensions,”
Annals Phys. (1994) 311–362, arXiv:hep-th/9307010[hep-th] .[16] A. Zhiboedov, “On Conformal Field Theories With Extremal a/c Values,”
JHEP (2014) 038, arXiv:1304.6075 [hep-th] .[17] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, “Spinning ConformalCorrelators,” JHEP (2011) 071, arXiv:1107.3554 [hep-th] .[18] D. Pappadopulo, S. Rychkov, J. Espin, and R. Rattazzi, “OPE Convergence inConformal Field Theory,” Phys. Rev.
D86 (2012) 105043, arXiv:1208.6449[hep-th] .[19] S. Giombi, S. Prakash, and X. Yin, “A Note on CFT Correlators in ThreeDimensions,”
JHEP (2013) 105, arXiv:1104.4317 [hep-th] .[20] J. Maldacena and A. Zhiboedov, “Constraining conformal field theories with aslightly broken higher spin symmetry,” Class. Quant. Grav. (2013) 104003, arXiv:1204.3882 [hep-th] .[21] E. Sezgin, E. D. Skvortsov, and Y. Zhu, “Chern-Simons Matter Theories and HigherSpin Gravity,” JHEP (2017) 133, arXiv:1705.03197 [hep-th] .[22] 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 II. c-Minimization andPrecise Critical Exponents,” J. Stat. Phys. (2014) 869, arXiv:1403.4545[hep-th] .[23] J. Erdmenger, “Gravitational axial anomaly for four-dimensional conformal fieldtheories,”
Nucl. Phys.
B562 (1999) 315–329, arXiv:hep-th/9905176 [hep-th] .[24] R. Delbourgo and A. Salam, “The gravitational correction to pcac,”
Phys. Lett. (1972) 381–382.[25] T. Eguchi and P. G. O. Freund, “Quantum Gravity and World Topology,”
Phys. Rev.Lett. (1976) 1251.[26] L. Alvarez-Gaume and E. Witten, “Gravitational Anomalies,” Nucl. Phys.
B234 (1984) 269. 4327] H. Osborn, “N=1 superconformal symmetry in four-dimensional quantum fieldtheory,”
Annals Phys. (1999) 243–294, arXiv:hep-th/9808041 [hep-th] .[28] D. T. Son and P. Surowka, “Hydrodynamics with Triangle Anomalies,”
Phys. Rev.Lett. (2009) 191601, arXiv:0906.5044 [hep-th] .[29] A. Vilenkin, “Parity Nonconservation and Rotating Black Holes,”
Phys. Rev. Lett. (1978) 1575–1577.[30] K. Landsteiner, E. Megias, and F. Pena-Benitez, “Gravitational Anomaly andTransport,” Phys. Rev. Lett. (2011) 021601, arXiv:1103.5006 [hep-ph] .[31] K. Landsteiner, E. Lopez, and G. Milans del Bosch, “Quenching the CME via thegravitational anomaly and holography,” arXiv:1709.08384 [hep-th] .[32] J. Gooth et al., arXiv:1703.10682 [cond-mat.mtrl-sci].[33] X. O. Camanho, J. D. Edelstein, J. Maldacena, and A. Zhiboedov, “CausalityConstraints on Corrections to the Graviton Three-Point Coupling,” arXiv:1407.5597 [hep-th] .[34] A. Bhattacharyya, L. Cheng, and L.-Y. Hung, “Relative Entropy, MixedGauge-Gravitational Anomaly and Causality,”
JHEP (2016) 121, arXiv:1605.02553 [hep-th] .[35] J. M. Maldacena and G. L. Pimentel, “On graviton non-Gaussianities duringinflation,” JHEP (2011) 045, arXiv:1104.2846 [hep-th] .[36] A. Lue, L.-M. Wang, and M. Kamionkowski, “Cosmological signature of new parityviolating interactions,” Phys. Rev. Lett. (1999) 1506–1509, arXiv:astro-ph/9812088 [astro-ph] .[37] S. Alexander and J. Martin, “Birefringent gravitational waves and the consistencycheck of inflation,” Phys. Rev.
D71 (2005) 063526, arXiv:hep-th/0410230[hep-th] .[38] N. Arkani-Hamed and J. Maldacena, “Cosmological Collider Physics,” arXiv:1503.08043 [hep-th] .[39] S. Saito, K. Ichiki, and A. Taruya, “Probing polarization states of primordialgravitational waves with CMB anisotropies,”
JCAP (2007) 002, arXiv:0705.3701 [astro-ph] .[40] D. Baumann, H. Lee, and G. L. Pimentel, “High-Scale Inflation and the Tensor Tilt,”
JHEP (2016) 101, arXiv:1507.07250 [hep-th] .4441] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single fieldinflationary models,” JHEP (2003) 013, arXiv:astro-ph/0210603 [astro-ph] .[42] N. Bartolo and G. Orlando, “Parity breaking signatures from a Chern-Simonscoupling during inflation: the case of non-Gaussian gravitational waves,” JCAP no. 07, (2017) 034, arXiv:1706.04627 [astro-ph.CO] .[43] J. Soda, H. Kodama, and M. Nozawa, “Parity Violation in GravitonNon-gaussianity,”
JHEP (2011) 067, arXiv:1106.3228 [hep-th] .[44] P. Creminelli and M. Zaldarriaga, “Single field consistency relation for the 3-pointfunction,” JCAP (2004) 006, arXiv:astro-ph/0407059 [astro-ph] .[45] H. Reeh and S. Schlieder, “Bemerkungen zur Unit¨ar¨aquivalenz vonLorentzinvarianten Felden,”
Nuovo Cim.22