Causality constraints on corrections to the graviton three-point coupling

JHEP02(2016)020 Published for SISSA by Springer Received:October 15, 2015 Accepted:December 28, 2015 Published:February 3, 2016 Causality constraints on corrections to the graviton three-point coupling Xi´an O. Camanho,aJos´e D. Edelstein,b,c Juan Maldacenadand Alexander Zhiboedove aMax-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut, Am M¨uhlenberg 1, D-14476 Golm, Germany bDepartment of Particle Physics and IGFAE, Universidade de Santiago de Compostela, E-15782, Santiago de Compostela, Spain cCentro de Estudios Cient´ıficos CECs, Casilla 1469, Valdivia, Chile dSchool of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, U.S.A. eDepartment of Physics, Princeton University, Princeton, NJ 08544, U.S.A. E-mail: [email protected],[email protected],[email protected], [email protected] Abstract: We consider higher derivative corrections to the graviton three-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond the one present in the Einstein theory. We argue that these are constrained by causality. We devise a thought experiment involving a high energy scattering process which leads to causality violation if the graviton three-point vertex contains the additional structures. This violation cannot be fixed by adding conventional particles with spins J≤2. But, it can be fixed by adding an infinite tower of extra massive particles with higher spins, J > 2. In AdS theories this implies a constraint on the conformal anomaly coefficients a−c c.1 ∆2 gap in terms of ∆gap, the dimension of the lightest single trace operator with spin J > 2. For inflation, or de Sitter-like solutions, it indicates the existence of massive higher spin particles if the gravity wave non-gaussianity deviates significantly from the one computed in the Einstein theory. Keywords: AdS-CFT Correspondence, Models of Quantum Gravity, Classical Theories of Gravity ArXiv ePrint: 1407.5597 Open Access,c The Authors. Article funded by SCOAP3.doi:10.1007/JHEP02(2016)020 JHEP02(2016)020 Contents 1 Introduction/motivation 1 2 Flat space causality and shock waves 4 2.1 Statement of flat space causality 4 2.2 Scattering through a plane wave in general relativity 5 2.3 Connection with the scattering amplitude computation 7 2.4 The effect of higher derivative interactions on particles with spin 8 3 General constrains on the on-shell three-point functions 10 3.1 The phase shift in impact parameter representation from three-point functions 10 3.2 The possible forms of three-point functions in various theories 14 3.3 Problems with higher derivative corrections to the three-point functions 15 3.4 Scattering of gravitons in D > 4 dimensions 18 3.5 Scattering of gravitons in four dimensions 20 4 Fixing the causality problem by adding massive particles 21 4.1 Massive spin two particles do not fix the problem in D= 4 21 4.2 Massive spin two particles do not fix the problem in D > 423 4.3 Exciting the graviton into new particles 24 4.4 Massive higher spin particles can solve the problem 26 4.5 Compositeness and the extra structures for graviton scattering 27 5 Anti-de Sitter discussion 28 5.1 Motivation: the emergence of bulk locality should happen in the classical theory 28 5.2 Statement of AdS causality 28 5.3 The effect of higher derivative interactions on particles with spin 30 5.4 Implications for a, c in theories with large operator dimensions 32 5.5 Implications for dimensions of double trace operators 32 6 Wormholes and time advances 35 7 Cosmological applications 36 8 Conclusions 37 8.1 Open problems 38 A Shapiro time delay 40 B Three-point amplitudes and their sums 41 B.1 Scattering of a scalar and a graviton 43 – i – JHEP02(2016)020 C Higher derivative terms from integrating out particles 44 C.1 QED case 44 C.2 Gravity case 45 D Causality and unitarity for a signal model 45 E Scattering in string theory 48 F Properties of the AdS shock wave 51 G Time advances and time machines 52 H Representations that couple to two gravitons 53 I The Weinberg-Witten theorem in Ddimensions 55 1 Introduction/motivation In this paper we consider weakly coupled gravity theories in the tree-level approximation. It is well-known that at long distances such theories should reduce to the Einstein gravity theory. However, at intermediate energies we can have higher derivative corrections. By intermediate energies we mean those that are low enough that the theory is still weakly coupled but high enough that we are sensitive to possible higher derivative corrections. An example of such a theory is weakly coupled string theory where the corrections appear at a length scale √α0, which is much larger than the Planck length, lp. The theory at energies comparable to 1/√α0is still weakly coupled. In this case, the higher derivative corrections are accompanied by extra massive higher spin particles which appear at the same scale. For higher energies the description is via a string theory which departs significantly from ordinary local quantum field theory. It is reasonable to expect that this is a generic feature. Namely, that higher derivative corrections only arise due to the presence of extra states with masses comparable to the scales where the higher derivative corrections become important. The objective of this paper is to sharpen this link for the simplest possible correction, that of the graviton three-point coupling. Due to the fact that the graviton has spin, the flat space on-shell three-point function is not uniquely specified. In general, it has three different possible structures. The most familiar is the one we get in the Einstein theory. The others can be viewed as arising from higher derivative terms in the gravitational action. The first new structure has two more derivatives. Relative to the size of the Einstein-Hilbert term, it scales like αp2, where αis a new quantity with dimensions of length squared which characterizes the relative importance of the new term. Here pis the typical scale of the momenta, it is not a Mandelstam invariant, since they all vanish for on-shell three-point functions. We work in a regime where both of the three-point vertices are small, so that gravity is weakly coupled. – 1 – JHEP02(2016)020 We find that new three-point vertices lead to a potential causality violation unless we get contributions from extra particles. This causality violation is occurring when the theory is still weakly coupled. It occurs in a high-energy, fixed impact parameter, scattering process at a center of mass (energy)2,s, which is large compared to 1/α but still small enough for the coupling to be weak. In General Relativity this scattering process leads to the wellknown Shapiro time delay [1], which is one of the classical tests of Einstein’s theory [2]. See also [3]. When the graviton three-point vertex is corrected, the new terms can lead to a time advance, depending on the spin of the scattered graviton. At short enough impact parameter this time advance can overwhelm Shapiro’s time delay and lead to a causality problem. This troublesome feature arises at an impact parameter of order b2∼α. At tree-level, this problem can only be fixed by introducing an infinite number of new massive particles with spin1J > 2 and m2comparable to α−1. In other words, it cannot be fixed by adding particles with spins J≤2, or by considering the existence of extra dimensions. These causality constraints are similar in spirit to those considered in [4,5] but they differ in two ways. First of all, here the problem will be shown to arise for small t/s, but large s. Second, the fact that the graviton has spin is crucial. On the other hand, in both cases we have locally Lorentz-invariant Lagrangians that nevertheless can lead to causality violations in non-trivial backgrounds. An example of a theory that is constrained by these considerations is given by the action S=l2−D pZdDx√gR+αRµνρσRµνρσ −4RµνRµν +R2,(1.1) where the second term is the Lanczos-Gauss-Bonnet term.2The constant αhas dimensions of length squared. For αl2 p, we will show that the theory is not causal. Furthermore, there is no way to make it causal by adding local higher curvature terms. In fact, our discussion refers to on-shell data, namely the three-point function, which reflects the real physical information and does not depend on the particular way that we write the Lagrangian. In other words, the discussion is invariant under field redefinitions. Note that if we view gravity as a low-energy effective theory with a UV cutoff of order the Planck mass, Mp, and we add higher derivative terms with dimensionless coefficients which are of order one, then we have nothing to say. The remarks in this paper only apply to theories where the coefficients of the higher derivative terms are much larger. The “natural” value for the coefficient αif we view (1.1) as an effective gravity theory is α∼l2 p. We will only constrain larger values of α. Of course, we are discussing this problem because it is indeed possible to have theories with αl2 p, for example a weakly coupled string theory. Another example where this discussion is relevant is the following. Imagine that we consider a large Ngauge theory. Such a gauge theory is expected to have a weakly coupled string dual. This theory will have a weakly coupled graviton corresponding to the stress 1In D > 4 dimensions by spin J > 2 particles we mean particles in representations of the little group SO(D−1) with both of the following two properties (see also appendix H): a) their maximal spin projection J+−≥2; b) their representations are labeled by Young tableaux with three or more boxes. 2It is the dimensional continuation of the four-dimensional Euler density. In four dimensions it is a topological term, while in higher dimensions it is not topological, in fact it contributes to the three-point coupling of the graviton. – 2 – JHEP02(2016)020 tensor operator [6–8]. However, we are not guaranteed that the dual will be an ordinary Einstein gravity theory. It might even be a Vasiliev-like gravity theory [9,10]. The field theory three-point functions of the stress tensor [11,12] determine the three-point functions of the gravity theory [13,14]. One can imagine a theory where the only light single trace operator is the stress tensor. It is natural to expect that this theory will have an Einstein gravity dual. It would be nice to prove that. For contact scalar interactions, it has been argued that there is a one-to-one correspondence between the solutions of crossing and bulk lagrangians [15], see also [16]. However, one can worry that the size of the vertices with higher derivatives might be comparable to the one in the Einstein theory. As a case in point, we can think about the graviton three-point coupling. This is constrained to be a linear combination of three structures, only one of which is the Einstein one. These extra structures necessarily lead to a causality problem unless we introduce new higher spin particles at the scale that appears in these new three-point functions. Thus we link the three-point function of the stress tensor to the operator spectrum of the theory. These three-point functions were constrained by causality in [14]. Here we get stronger constraints because we are making further assumptions about the operator spectrum of the theory. As another application we consider the possibility that gravity waves during inflation were generated by a theory that indeed had these higher derivative corrections with a size comparable to the Hubble scale. This is a possibility which is allowed by conformal invariance and would be realized if the dual description to inflation (in the spirit of dS/CFT [17–19]) was a weakly coupled theory or if inflation occurs in a string theory where the string scale is close to the Hubble scale. The theory is still weakly coupled, so that scalar and tensor fluctuations are small. In this case the gravity wave non-gaussianities would be different from the ones in the Einstein theory [20,21]. The observation of such gravity wave signals, combined with the arguments in this paper, would imply the existence of extra particles with spin J > 2 during inflation. Finally, as a further motivation we should mention the grand dream of deriving the most general weakly coupled consistent theory of gravity. It is quite likely that the only such theory is a string-like theory, broadly defined. We are certainly very far away from this dream, but hopefully our simple observation about three-point functions could be useful. In particular, this observation highlights the importance of spin. Spin is likely to grow in importance as we consider constraints on the four-point function, given what we got for the much simpler three-point function. In [22–24] various interesting constraints were derived by using crossing symmetry and the correct factorization on the pole singularities. The constraints discussed in this paper are additional constraints, not covered by their analysis. This paper is organized as follows. In section 2 we discuss the notion of asymptotic causality, both for asymptotically flat and asymptotically AdS spacetimes. We also discuss the propagation of particles and fields through a shock wave. The purpose of this discussion is to set the stage for the more general argument in the following section so that it becomes intuitively clearer. We present in section 3 the main thought experiment, which involves only on-shell amplitudes and does not refer explicitly to shock waves. In section 4 we discuss the effects of adding extra massive particles. We show that particles with spins two or less cannot solve the problem. We also discuss the appearance of these – 3 – JHEP02(2016)020 massive particles among the final states of the scattering process. In order to argue that the problem persists we offer an alternative presentation of the problem where we consider the analyticity properties of the Smatrix in impact parameter representation. We discuss in section 5 various aspects of the AdS version of the thought experiment and its implications for properties of the dual conformal field theory. In section 6 we briefly mention the implications of a possible time advance in the context of wormholes. In section 7 we discuss a cosmological application: we link the possible existence of new structures for the gravity wave non-gaussianity to the presence of higher spin massive particles during inflation. We present our conclusions in section 8. Finally, we include a number of appendices to share some important but rather technical details. 2 Flat space causality and shock waves In this section we consider the problem in asymptotically flat space, and start by discussing the causal structure. As a motivation for our later discussion we analyze the scattering of a probe graviton from the shock wave. We then present the argument for causality violation in purely on-shell terms. 2.1 Statement of flat space causality When we consider a gravity theory in asymptotically flat space we expect to be able to define scattering amplitudes. In particular, this presupposes that one can fix the asymptotic structure of the spacetime so that we can compare times between the past and the future asymptotic regions. In dimensions D > 4 we expect to be able to do this. Let us give a simple argument. Imagine we have a series of observers that sit at a large distance Lfrom each other, and also from the center. For simplicity, consider them at the vertices of a large spatial hypercube and moving along the time direction. We would like to argue that they can synchronize their clocks. If they were in flat space, they can just send signals to each other. On the other hand if there is an object of mass min the interior, then the metric components decay as 1/rD−3. This gives rise to a redshift of order Gm/LD−3at the position of the detectors. More importantly, as a signal travels from one detector to the other, staying at a distance of order Lfrom the center, it will get delayed by an amount δt =δL ∼Gm/LD−4(see appendix A). In D≥5, we can make this as small as we want by moving out to large enough L. For D= 4 this delay does not go to zero and we have a problem. This has been discussed in [25], and it seems closely related to the soft graviton issues that arise in perturbative attempts to define the gravitational S-matrix. This issue is not present in AdS4. In order to deal with the D= 4 case, Gao and Wald [26] have introduced another notion of causality saying that we cannot send signals faster than what is allowed by the asymptotic causal structure of the spacetime. In General Relativity, with the null energy condition, they argued that this notion of causality is respected; see [26] for a precise statement of the theorem. This holds in any number of dimensions. For D > 4 this becomes the more naive notion of causality introduced above. Note that this is an asymptotic notion and we are not relying on the locally defined lightcones as in [27]. – 4 – JHEP02(2016)020 u v ∆v P p u v Figure 1. A particle creates a shock wave localized at u= 0. A second probe particle propagates on the geometry and experiences a time delay, ∆v. The two particles are separated along the transverse directions, which are suppressed in this diagram. We expect that this notion of causality is actually a requirement for any theory of quantum gravity. For asymptotically AdS theories we expect that we should not be able to send signals through the bulk faster than through the boundary. For theories of gravity that are dual to a quantum field theory in the boundary, this is implied by causality in the boundary theory. Also, since we expect that quantum gravity in asymptotically flat space is Lorentz invariant, then these time delays can lead to acausality or closed time-like curves (see appendix G). Another reason to require this notion is to ensure that Lorentzian wormholes, such as the one obtained form the eternal Schwarzschild black hole and discussed in [28], do not lead to causality violations in the ambient spacetime. This is required by the so-called ER=EPR interpretation of such geometries [29–31]. In the rest of the paper we will assume this notion of causality and derive constraints on some higher derivative corrections. 2.2 Scattering through a plane wave in general relativity A well-known general relativistic effect is that light going near a massive body (e.g. the Sun) would suffer a time delay relative to the same propagation in flat space. This is known as the Shapiro time delay [1] and constitutes one of the classical tests of General Relativity (see appendix A). We will recall here the derivation of this time delay in the shock wave approximation, which will be all we need for our purposes. First we review the basic properties of shock wave solutions in gravitational theories [32]. This solution describes the gravitational field of an ultrarelativistic particle in a generic theory of gravity [33,34] and is directly relevant for the high-energy scattering. – 5 – JHEP02(2016)020 A generic shock wave solution in flat space can be written in the following form ds2=−dudv +h(u, xi)du2+ D−2 X i=1 (dxi)2.(2.1) This geometry admits a covariantly constant null Killing vector lµ∂µ=∂v. See figure 1. We will be interested in the case when this geometry is sourced by a particle that moves very fast in the vdirection. Classically, we can model such particle via the stress tensor [33] Tuu =−Puδ(u)δD−2(~x),(2.2) where r=qPD−2 i=1 x2 iand Pu<0 is the momentum of a particle. The Einstein equations then take the form ∂2 ⊥h(u, xi) = −16πG|Pu|δ(u)δD−2(~x).(2.3) The solution is h(u, xi) = 4Γ(D−4 2) πD−4 2 δ(u)G|Pu| rD−4.(2.4) Now we consider a probe particle that moves in the other light cone direction with momentum pvand is such that it crosses the shock with impact parameter b. In other words, the displacement in the transverse dimensions is r=b. The metric (2.1) is a bit peculiar because of the delta function δ(u) in h(u, xi). We can remove this delta function at the transverse position bby defining a new coordinate v=vnew +4Γ(D−4 2) πD−4 2 G|Pu| bD−4θ(u).(2.5) This then cancels the δ(u) term in the metric at r=b. Thus, the geodesic that goes through this point is continuous in the vnew coordinates. This means that in the original coordinates it suffers a shift ∆v=vAfter crossing −vBefore crossing =4Γ(D−4 2) πD−4 2 G|Pu| bD−4>0.(2.6) This represents the Shapiro time delay, see figure 1. In addition to this time delay, the trajectory is also subject to a deflection angle. We might worry that the deflection angle would hamper our ability to see a possible time delay or time advance from far away. In fact, we could consider two shocks in succession, but separated in the transverse direction. Considering the probe particle coming at r= 0, we set the shocks at r=bopposite to each other as shown in figure 2. In this case, the probe particle does not get a net deflection, but the time delays add. It is instructive to reproduce this formula for the time delay when we treat the probe as a quantum mechanical particle. The wave equation for a scalar field takes the form ∇2φ= 0 −→ ∂u∂vφ+h∂2 vφ−1 4∂2 iφ= 0 ,(2.7) – 6 – JHEP02(2016)020 Figure 2. Left: we consider a particle propagating through the superposition of two left moving shock waves localized at u= 0. The particle trajectory is given by the arrows. Right: in the transverse plane we separate shock waves by distance 2band send the particle between them so that the net deflection angle is zero. The time delay is the sum of the time delays due to each shock wave. Let us now consider the change in the value of φfrom u= 0−to u= 0+. Since the variation of the hterm is much faster than the variation in the other variables, we neglect the ∂iderivatives and write φ(u= 0+, v, xi) = e−R0+ 0−duh∂vφ(u= 0−, v, xi) =e−∆v ∂vφ(u= 0−, v, xi) = e−i∆v pvφ(u= 0−, v, xi),(2.8) where ∆vis given in (2.6). Thus we see that we reproduce the answer we got through the geodesic analysis. Note that pv=−i∂vis the generator of translations in v. In quantum field theory we would end the discussion here. Any time advance in v, relative to the background Minkowski metric would be a problem. In gravity, the situation is more subtle, in principle, we need to make observations from asymptotically far away. However, in that case, the puenergy also has a pvdependence3and it contributes positively to the time delay. Since pu=~q2/4pv, this is a small effect for large pv. However, if we multiply by the total u−time elapsed, it can add to a very big time delay. Here we will assume that we can sit far enough from the shock to be able to neglect the dynamics of spacetime, but close enough that we can neglect the v−time delay produced by the pu energy. This seems possible for small G. 2.3 Connection with the scattering amplitude computation Let us reproduce the computation above using scattering amplitudes [3]. It is well-known that the shock wave computation can be reproduced using the so-called eikonal approximation [35]. Consider the scattering amplitude for gravitating scalar particles. It is given by Atree(s, t) = −8πGs2 t.(2.9) The eikonal approximation resums a particular set of diagrams (horizontal ladders) in the deflectionless limit when t s→0. Under favorable circumstances,4the amplitude exponen3See section 3 for the relevant kinematic configuration. 4This particular point will be discussed later in more detail. – 7 – JHEP02(2016)020 3.2 The possible forms of three-point functions in various theories Given that the answer for the time delay depends on the on-shell three-point functions, it is useful to recall their possible structures. In four dimensions we can use the usual helicity basis. The Einstein-Hilbert gravity action gives rise to the + + −and −−+ three-point functions. We can also have +++ and −−−structures which could come from (Riemann)3 terms. There are two combinations, one of them being parity violating. In higher dimensions, D > 4, we have three possible structures for the graviton three-point functions, all parity preserving. They have the schematic structure, writing µν =µν, AR= (1.23.p1+1.32.p3+2.31.p2)2, AR2= (1.23.p1+1.32.p3+2.31.p2)1.p22.p33.p1,(3.7) AR3= (1.p22.p33.p1)2. The first is the usual one coming from Einstein gravity, while the second can arise from the Lanczos-Gauss-Bonnet term (2.17). The third one can arise from a (Riemann)3term.8 They can be viewed as products of the two possible structures that we can have for on-shell spin one particles. In a given theory, the total three-point function is given by a linear combination of these three answers Aggg =√32πG [AR+α2AR2+α4AR3],(3.8) where α2and α4are two parameters with dimension of (length)2and (length)4respectively. Notice that we have an overall coupling, given by G, which we take to be parametrically smaller than the other two parameters. In the high energy limit the three-point functions appearing in (3.6) simplify further and become A13I R= 2p2 u(~e1.~e3)2, A13I R2= 2p2 u(~e1.~e3)(~q.~e1)(~q.~e3),(3.9) A13I R3= 2p2 u(~q.~e1)2(~q.~e3)2, for the three terms in (3.7). We used (3.4). The amplitude AI24 has a similar expression with pu→pvand 1,3→2,4. The reader can express these in terms of transverse traceless two index tensors by using the replacement rule eiej→eij in (3.9) , where i, j are indices in the D−2 transverse directions. Note that the third structure in (3.7) or (3.9) is not allowed in a supersymmetric theory. This can be seen as follows. In D= 4 this structure gives rise to + + + and −−−amplitudes. However, by the methods described in [46] it is possible to show that 8(Riemann)3=RµνσδRσδργ Rµν ργ . In four dimensions we can replace one of the curvatures by Rµνσδ → e Rµνσδ =µνργ Rργ σδ to obtain the parity violating term. This parity violating term gives rise to the threepoint vertex µνδσpδ 1pσ 21,µµ02,νν0pµ0 2pν0 13,γρpρ 1pγ 2, which, despite appearances, is properly symmetric under exchange of any of the three particles. When this term is present the coefficient of the + + + amplitude is complex and the coefficient of the − − − amplitude is the complex conjugate. – 14 – JHEP02(2016)020 supersymmetry implies that this structure should be set to zero. This is actually true in all dimensions. The reason is that we can start with a theory in Ddimensions and consider external graviton three-point functions with four-dimensional kinematics. If we had a non-zero contribution from the third structure in Ddimensions, then it would lead to a contribution to the +++ or −−− amplitudes in four-dimensional kinematics. Since the arguments in [46] are purely kinematical, based on the symmetries of the theory, then they also force the amplitudes to vanish. In conclusion, supersymmetry implies that α4= 0. In the heterotic string we have a non-zero α2. And this is also true for compactifications of the string to Ddimensions. Thus α2is compatible with half maximal supersymmetry. With maximal supersymmetry, e.g. N= 8 in D= 4, we should also have α2= 0. This can be understood from the fact that the four point amplitude for the full supergravity multiplet is determined up to a unique function. Thus, there is no freedom to introduce the polarization dependent terms that would arise if we had the freedom to switch on α2. Of course, in standard maximal supergravity α2is not present, therefore the only value consistent with maximal supersymmetry is α2= 0. As an aside, notice that this is also related to the fact that in four-dimensional N= 4 superconformal theories, the three-point functions of the stress tensor are completely fixed by supersymmetry in terms of the two-point functions. Between a graviton and two photons the number of possible three-point functions is two A13I F2=I µν [pµ 1pν 3(1.3)−µ 1pν 3(3.p1)−µ 3pν 1(1.p3)] , A13I RF F =I µνpµ 1pν 3(1.p3)(3.p1),(3.10) so that Agγγ =√32πG [AF2+ ˆα2ARF F ]. One arising from the usual electrodynamics and the other from the second term in (2.13). Again this second structure is forbidden in a supersymmetric theory. As we did in (3.9) , in the high energy limit we can write them as A13I F2= 2p2 u(~e1.~e3), A13I RF F = 2p2 u(~q.~e1)(~q.~e3).(3.11) We emphasize that we work here with the on-shell three-point functions, independently of the precise way we write the Lagrangian. This discussion depends only on on-shell threepoint functions and not on other contact terms. Any contact four point interaction does not give rise (at tree level) to the long range force at a non-zero value of the impact parameter. 3.3 Problems with higher derivative corrections to the three-point functions We discussed above how the three-point functions give rise to the leading order expression for the phase shift δ(~ b, s) = sF(~ b). If this result were exponentiated, as eiδ, then we could get a time advance problem similar to what we found for the shock waves. Here we would like to explain how to get a time advance problem without using the particular non-linear structure of shock waves. The goal is to present the problem in a way that depends only on very general principles. First note that in order for time delay to be a problem we would like to find that the time delay ∆v=∂p2,v δis larger than the quantum mechanical uncertainty that is implicit – 15 – JHEP02(2016)020 Figure 5. We imagine a particle going through a set of successive scattering events. The intrinsic quantum uncertainty in vis ∆qv. We have drawn a situation where there is a final time advance after going through all the shocks that is larger than the quantum uncertainty. In this figure we have neglected the delay of the u-localized particles. in the definition of the wave packet for a particle of momentum p2,v. This uncertainty is of the order of ∆qv∼1/p2,v. Thus the figure of merit is ∆v ∆qv=p2,v∂p2,v δ=δ , (3.12) here we used that δis linear in s, and therefore linear in p2,v (s∝p1,up2,v). Thus in order to see a problem, we expect that δshould be greater than one. On the other hand, the validity of perturbation theory suggests that δshould be much less than one. In order to amplify the effect, we imagine that particle number two undergoes N successive instances of particle number one, see figure 5. Through each instance it gets a small phase shift which leads to a factor in the out state of the form (1 + iδ) with small δ. If we repeat this Ntimes we get (1 + iδ)N∼eiNδ , δ 1, N 1.(3.13) This is the total phase shift, and as explained in (3.12), we want Nδ to be of order one. This can be achieved by taking N∼1/δ. In addition, we would like to make sure that the approximations that we used remain valid. In particular, we have said that particle 2 remains localized at some distance bthrough the whole process. We can choose light-cone coordinates for the evolution of particle two so that uis time, then we have a non-relativistic problem with mass m∼pv. The spreading of the wavefunction during the time Uthat the whole process takes is ∆b∼pU/pv. The time Uthat the process takes is determined as follows. We want to separate the Ninstances of the scattering process from each other so that we can view them as independent and we can use (3.13). The best we can localize each of the Nparticles of type 1 is by an amount ∆qu∼1/pu. Thus U=N/puand then – 16 – JHEP02(2016)020 ∆b∼qN s. We want ∆bb. This translates into the condition Nsb2−→ 1 sb2δ , (3.14) where we used that N∼1/δ in order to have a problem. We see that the simultaneous validity of (3.14) and (3.13) can be achieved if sb21 and if scan also be increased so as to achieve 1 sb2δ1,(3.15) which can be done if δgrows with s. An additional issue is that we wanted to neglect the deflection angle. Then we can replace each of the 1 particles by a pair of particles localized in the transverse dimension at a distance bwith respect to particle number 2 on opposite sides of its trajectory, as shown in figure 2. For spin zero particles, notice that if we had a gφ3vertex, then the scalar exchange leads to δ∼1 s g2 bD−4. In this case we cannot obey the conditions (3.15) to obtain a possible causality problem. In fact, (3.14) implies that g2/bD−6>1, which means that we are at strong coupling. Thus we see that a crucial feature that we used is that the phase shift increases as a function of s. In the case that we exchange a spin one field, δis independent of sand there is also no causality issue. The reason is simple, the spin two field is effectively changing the metric and causal structure while the spin zero or one fields are not. The conclusion is that we have justified the exponentiation (3.13) and thus the derivation of the time delay problem in a way that does not depend on the details of the shock wave solution. Let us emphasize that the preceding shock wave discussion was motivational, but the thought experiment that we have set up in this section does not require us to rely on the non-linear structure of the shock wave. It was all derived from the on-shell three-point functions plus certain assumptions about the locality of the theory that allowed us to view each shock as an independent event. In fact, there are some cases where the shock wave computation does not give the right answer. For example, consider the pure gravity case, where δ=Gs/bD−4. If the energy is large enough to form a black hole, then the time delay computed from the shock wave is not the correct description for the physics. We expect to form a black hole when the centerof-mass energy is such that the associated Schwarzschild radius rD−3 S=√sG is larger than b[47–49]. It is easy to check that this never happens if (3.15) is obeyed. It is not obeyed even if we consider the energy of the Nparticles that we used for the argument (with N∼1/δ). There is still one more complication that we need to deal with in order to make the argument clearer. In the shock wave discussion of section two, the spin of the particle creating the shock did not matter. However, with the modified three-point functions, the spins of the scattered particles can change. The full interaction is a spin dependent force which acts of both spins. It acts on both the spin of the left and the right moving particles (particle one and two, see figure 3). I will be necessary for us to be able to fix the polarizations of particles one and three, see figure 3. This can be achieved by replacing particle one, by a coherent state of particles. In this case, due to the usual Bose enhancement factor, particle three will have a larger probability of remaining with the same – 17 – JHEP02(2016)020 polarization. In this set up we can set the spins, or polarization vectors, of particles one and three to be the same. Or, more precisely, 3=∗ 1. Since we are at weak coupling we can consider a coherent state with a mean occupation number which is large enough for us to be able to neglect the spin flips but small enough that the total scattering amplitude is still small. In other words, the use of coherent states allows us to effectively select the final state for particle 3. More explicitly, say that we form a coherent state for the oscillator mode created by a†,eλa†|0i. We could then have terms in the interaction Hamiltonian that leave this oscillator the same H1,int =h1a†aor that mix it with a second oscillator H2,int =h2b†a+h.c.. Here h1and h2can act on other degrees of freedom. Then, for a coherent state, the matrix elements where there is no change are enhanced, due to the usual Bose enhancement factors, relative to the ones where there is a change h0|e¯ λa H1,int eλa†|0i ∝ |λ|2hh1i, h0|e¯ λab H2,int eλa†|0i ∝ λhh2i.(3.16) We see that for large λthe first term is enhanced. We have not indicated explicitly the initial and final states on which h1and h2act, since they involve other oscillators. Since we are at weak coupling, we can choose λlarge enough so that the first term dominates relative to the second term, while still the whole process is in the weakly coupled approximation, or the total effect of the interaction Hamiltonian is small. Of course an alternative way to say this is that we are creating a classical background with the particles of type 1 in figure 3 the terms that have a non-zero expectation value in this classical background dominate over the others. We want a classical background with small enough amplitude that we can still trust the leading order perturbative expansion of the interaction Hamiltonian. The use of coherent states also allows us to select final states for particle 3 in figure 3 with a a small momentum pv. Namely, we form a coherent state out of a superposition of particles with large momenta p1,µ. We need a superposition since we need to localize this particles within the transverse plane to a location smaller than b. Thus we have some dispersion in the transverse momenta ~q. With a large pucomponent of the momentum, we then get a small momentum along the pv=~q 2 4pudirection. Since we have a coherent state, the particle 3 is also taken out of this superposition and has the same range of values for the momentum. Therefore the total momentum transfer in the t-channel along the vdirection is very small. Then the kinematics chosen in (3.2) is representative for the process in question. This still allows a possibly large amount of momentum transfer along the udirection. We will discuss this in subsection 4.3 . Another minor point, is that the phase shift δrepresents a time delay for both particles, it affects both particle 1 and particle 2. So far we have been focusing only on the effects on particle 2. These coherent states also allow us to effectively select a final state for particles 3, so that we can focus more clearly on the time delay with which particle four emerges, see figure 3. 3.4 Scattering of gravitons in D > 4 dimensions It is also easy to argue that α2and α4structures lead to causality problem in D > 4. To show this we consider the probe graviton that scatters off the coherent state. The phase – 18 – JHEP02(2016)020 shift takes in this case the following form (see appendix B for details) δ∼Gs(eij 1eij 3+α2eij 1eik 3∂bj∂bk+α4eij 1ekl 3∂bi∂bj∂bk∂bl)× ×(eij 2eij 4+α2eij 2eik 4∂bj∂bk+α4eij 2ekl 4∂bi∂bj∂bk∂bl)1 |~ b|D−4.(3.17) In order to find problems we will be choosing various polarizations for the particles. For example, let us firt consider particle 1 with polarization e1 xx =−e1 yy and the other components equal to zero. Here x, y represent to two directions in the transverse plane. We call this the ⊕polarization. We also choose e3=e1. We enforce this by sending a coherent state with this polarization. Now for particle 2 we can choose the same polarization or the crossed polarization, called ⊗, given by e2 xy =e2 yx = 1/√2 and all other components equal to zero. Then we find the following. If ~ bis along the ˆxdirection, then these two different polarizations for particle two do not mix as they go through the shock. They diagonalize the phase shift matrix. For small enough bthe α2 4terms dominate since they are the most singular in the small bexpansion. These terms have the form δ⊕⊕ ∼Gsα2 4O⊕O⊕ 1 bD−4=Gsα2 4 bD+4 (positive) , δ⊕⊗ ∼Gsα2 4O⊕O⊗ 1 bD−4=Gsα2 4 bD+4 (negative) ,(3.18) O⊕= (∂2 bx−∂2 by)2, O⊗= 4∂2 bx∂2 by, where we take the derivatives first and then set ~ b= (bx,0,··· ,0). Where the terms in parenthesis are polynomials in Dwhich are positive or negative definite.9Notice that the positivity of the first case is due to the following argument. If the polarizations of particle 2 and 4 are the same as those of 1 and 3, then the configuration is constrained by unitarity along the t-channel.10 Therefore in this case we should get a strictly positive answer for the time delay for the contribution of any particle with a non-zero coupling. Since we obtained a negative time delay for the second case in (3.18) , we conclude that α4should be set to zero unless new particles are present. Once α4has been set to zero, we can discuss α2. In that case we can still choose the ⊗xy polarization for particles 1 and 3 and the ⊗yz polarizations for particles 2 and four. We then focus on the terms proportional to α2 2since they are the dominant terms at small b(once we have set α4= 0). We then get δ⊗xy,⊗yz ∼Gsα2 2ˆ Oxy ˆ Oyz 1 bD−4=−Gsα2 2 bD2(D−4)(D−3)(D−2) , ˆ Oxy =−∂2 bx−∂2 by,ˆ Oyz =−∂2 by−∂2 bz.(3.19) Then we conclude that α2should also be set to zero unless new massive particles appear. 9For D > 4, (positive)= (D−4)(D2−4)D(336 + 128D+ 20D2+ 4D3+D4) and (negative)= −4(D− 4)(D2−4)D(D+ 3)(20 + 6D+D2). For D= 4 the derivatives act on −log band we get (positive) = -(negative) =80640. 10t-channel unitarity is the following statement. When the polarizations of 2 and 4 are related by conjugation and reflection along the ~ baxis to the polarizations of 1 and 3 respectively, then the residue of the t-channel pole is positive. – 19 – JHEP02(2016)020 3.5 Scattering of gravitons in four dimensions Let us now discuss in more detail the four-dimensional case, D= 4, which is a bit special. First, need to take into account that the Einstein term produces a log(L/b) time delay, where Lis an IR cutoff. Second we need to take into account the parity violating structure. The logarithm can be taken into account by modifying the causality criterion in the form suggested by Gao and Wald, who define it by comparing to the behavior of the same metric far away. In this way the log Lterm is eliminated and it is easy for a power law behavior produced by α4, which goes as 1/b4to overwhelm the logarithm. Also we will later repeat the computations for AdS4space and we will see that L→RAdS4. In conclusion, this is not a real issue. Note that in four dimensions the α2structure is identically zero. This is related to the fact that the Gauss-Bonnet term becomes topological in four dimensions. Thus, we have only the Einstein term structure and the α4one. With four-dimensional kinematics, we can consider the situation with coherent states of particles of type 1. Let us choose the spin of these particles to be plus. Due to the coherent state considerations, the spin of particle 3 also needs to be plus (in the outgoing notation, or minus in the incoming notation). In other words the amplitude does not have a spin flip. In fact, without a spin flip the α4 structure does not contribute in four dimensions. Thus in the vertex involving particles one, three, and the intermediate one, the only structure that contributes is the Einstein one. This contribution is effectively the same as the spin zero one. Then we can run an argument similar to the one above. Let us now discuss the parity violating structure, together with the parity preserving one. By considering the coherent state for particle one (see figure 3), with definite helicity (positive or negative) we get that only the Einstein structure contributes to the A13Ithreepoint amplitude (see figure 4). The we get the following matrix form for the phase shift for particle two δ= 2Gs 1+γ∂4 β|−ih+|+γ∗∂4 β∗|+ih−|(−log |β|) = 2Gs −1log |β|+3γ β4|−ih+|+3γ∗ β∗4|+ih−|,(3.20) where we introduced a complex variable β=b1+ib2in the two dimensional transverse space. We also used ∂β=1 2(∂b1−i∂b2) and e±∝ex∓iey. Here γis the coefficient of the + + + amplitude and γ∗the coefficient of the −−−one. Notice that these physically imply that the particle two undergoes a spin flip. We see that δis a two by two matrix in the space of helicities. In (3.20) , 1represents the identity matrix in this two dimensional space. The matrix in (3.20) can be diagonalized by choosing the polarization directions p1,2∝ |+i±sγ β∗4 γ∗β4|−i.(3.21) Then we find that δ1,2= 2Gs −log |β|±3|γ| |β|4.(3.22) Thus we see that we have a causality problem for small enough b=|β|. – 20 – JHEP02(2016)020 zz + _ + + ++ +− J =+4 zJ =0 z Figure 6. Consider the coupling of a massive spin two particle to two massless gravitons. Let us choose the kinematic configuration so that the massive particle decays into two massless gravitons along the ˆzaxis. The +−helicity configuration is impossible since the angular momentum along the zaxis would be +4. The ++ configuration is allowed. 4 Fixing the causality problem by adding massive particles Let us discuss how to evade the causality problem that we found above. This problem can be evaded by adding new particles at the scale α2or α4. We will discuss the case of a weakly coupled theory where the problem should be fixed at tree level. This is indeed what happens in string theory, see appendix E. For cases involving loops see appendix C. Let us first consider the corrections to the time delay due to new particles being exchanged in the t-channel. The new particles have to lead to a phase shift growing like s, or a higher power of s.Thus, we should add massive particles with spin J≥2. We will now argue that massive spin two particles do not help and that we need particles of higher spin. In particular, this will then rule out a solution involving Kaluza-Klein gravity, which would be a special example of the addition of massive spin two particles. For this reason we will analyze it in detail. 4.1 Massive spin two particles do not fix the problem in D= 4 Let us first discuss the four-dimensional case. Since the external states are massless spin two particles, the on-shell three-point vertices involve two massless particles and a massive spin two particle. In four dimensions, we can label the massless particles by their helicities. An important result is that, in all incoming notation, the only non-zero amplitudes involve ++ or −− helicities for the massless particles. In particular the +−combinations are zero. The argument is essentially the same as in the Weinberg Witten theorem [50], or the statement that gravity does not have a local stress tensor operator.11 Imagine that we have the massive spin two particle in its rest frame. We let it decay into two massless spin two particles. Let us suppose that the two decay products move in opposite directions along the ˆzaxis, see figure 6. In the +−configuration the total sum of the spins of the decay 11The matrix elements of the stress tensor operator between two on-shell graviton states is like the coupling to a massive spin two particle, where the square of the momentum of the stress tensor, q2corresponds to the mass of the massive spin two particle. – 21 – JHEP02(2016)020 products along the ˆzaxis is +4 or −4. However, the initial massive particle had spin at most ±2. Therefore a +−configuration is impossible. With a ++ or −− configuration there is not problem because the sum of the spins is zero. One can also write down explicitly the corresponding three-point amplitude eα4I µνpν 1pµ 3[(1.p3)(3.p1)−(1.3)(p1.p3)]2 →2eα4p2 u(~e1.~q)(~e3.~q) + m2 I 2(~e1.~e3)2 ,(4.1) where 1 µν =1 µ1 ν, and we used that the component of Iµν that contributes the largest factor of sin the sum over intermediate states is Iuu = 2. We have denoted the coupling by eα4 since it reduces to the α4structure in the massless limit. Here 1 and 3 are the massless particles. Of course, p1.p3is given by the mass of the massive particle. We see that this result is invariant under 1→1+p1, and so on. In the second line of (4.1) we have written the three-point amplitude including the leading terms in the high energy limit. This is written in terms of the purely transverse polarization vectors (or tensors) introduced in (3.4). In D= 4 there is also a parity violating structure which we will not need to write explicitly. If we now consider particles 1 and 3 in figure 3to be associated to a coherent state with definite spin, then we have no spin flip allowed and this coherent state does not couple to the massive spin two particles. Therefore in four dimensions the massive spin two particles cannot solve the problem, they simply do not couple to the type of source that we are considering. Note that it is important that the massless intermediate gravitons are still coupling to the 1-3 coherent state through the Einstein three-point function, and as discussed in section 3.5, it leads to a causality problem for particle two in figure 3. We can further show that the massive spin two particle with a coupling (4.1) by itself also leads to a causality problem and should therefore not be present. In fact, it will be useful for our later argument to understand this in more detail. For simplicity let us set to zero the parity violating massive structure. For the coherent state that involves particles one and three in figure 3we choose the ⊕polarization with e1 xx =−e1 yy and the other components equal to zero, and the same for e3. Here x, y represent the two directions in the transverse plane. Now for particle 2 we can choose the same polarization or the crossed polarization, called ⊗, given by e2 xy =e2 yx = 1/√2 and all other components equal to zero. Then we find the following. If ~ bis along the ˆxdirection, then these two different polarizations for particle two do not mix as they go through the shock. They diagonalize the phase shift matrix. If the polarizations of particle 2 and 4 are the conjugate to those of 1 and 3, and reflected along ~ b, then the configuration is constrained by unitarity along the t-channel to give a strictly positive answer for the contribution to the time delay of any particle with a non-zero coupling. On the other hand, if we average over all polarizations for particle 2, it is possible to see that the terms involving α4or eα4(the massive particle contributions) all vanish. Thus, the contribution from the crossed polarization has to have the opposite sign. In other words, unitarity fixes a plus sign for the time delay for one polarization and this implies a negative sign for the other. Indeed, it is possible to see this – 22 – JHEP02(2016)020 explicitly by computing the massive particle contribution to both answers, which are δ⊕,⊕= 4GsX meα2 4OmOmK0(mb), δ⊕,⊗=−4GsX meα2 4OmOmK0(mb),(4.2) Om≡∂2 bx∂2 by−m4 8, where the first subindex of δis the polarization of particles 1 and 3 and the second that of particles 2 and 4 in figure 3. By acting with this operator explicitly one can see that it gives a negative answer in the second case. This is independent of the sign of eα4. In fact, it is also negative for the contribution of the massless case when we have the α4structure on both sides. The full phase shift also has the General Relativity contribution. Once we have a single massive particle, it is possible to go to a small enough bso that we overwhelm the positive contributions from the General Relativity vertices. This shows that in a theory with up to spin two particles we cannot solve the causality problem that arises when α4is nonzero. In addition, we see any massive spin two particles, even if present, they should have eα4= 0 in order not to cause further causality problems. 4.2 Massive spin two particles do not fix the problem in D > 4 Now we now move on to a higher dimensional gravity theory, D > 4. The three-point amplitudes for two gravitons and a massive spin two particle now have two possible structures, first the one in (4.1), which can be multiplied by a coefficient that we will still call eα4. And a second one of the form eα2I µν [µ 1pν 3(3.p1) + µ 3pν 1(1.p3)−pµ 1pν 3(1.3)−µ 1ν 3(p1.p3)] × ×[(1.p3)(3.p1)−(1.3)(p1.p3)] (4.3) →2eα2p2 ue1 kiqie3 kjqj+m2 2e1 ije3 ij. where again I µν is the intermediate state polarization vector and we used that we only care about its Iuu = 2 component. We have introduced a new coefficient eα2. In the second line we have indicated the form that it takes in the high energy limit. In the last line the polarization tensors are purely in the transverse directions and qis the momentum transfer. We can first consider a setup with four-dimensional kinematics. Namely, we can consider particles 1 and 3 to be associated to a coherent state which is uniformly distributed along D−4 of the original dimensions. In this case the problem is essentially four-dimensional and the three-point amplitudes involving α2and eα2(both massless and massive) do not contribute. If we want to avoid causality problems, and without spin >2 particles, we conclude that both α4and eα4should be zero. The argument is the same as the one we presented in the four-dimensional discussion. Note that since we are getting to four dimensions by effectively dimensionally reducing the higher dimensional theory, then the parity violating four-dimensional structure does not arise. – 23 – JHEP02(2016)020 plane in flat space is played here by the transverse HD−2. It is convenient to think of the probe crossing this hyperbolic space at the center.16 The shock centered at (~y0, z0) has a number of Killing vectors that depend on f(u). For arbitrary f(u) the background has an obvious SO(D−3) symmetry that rotates ~y and a translation in v. For f(u) = δ(u) the geometry has extra Killing vectors which enhance the rotational symmetry to SO(D−2), as we had in flat space. In the original coordinates (5.2) , some of the extra symmetries involve special conformal generators. Since we are working in the high energy limit, we effectively have a delta function in u, so that we also expect to have this extra SO(D−2) symmetry. See [61] for a coordinate system that makes this manifest. General properties of the AdS shock wave and its different limits are considered in appendix F. In particular, when the center of the shock goes to the boundary z0→0 the problem becomes very similar to the one arising in the computation of energy correlators [14], whereas in the limit z0→ ∞ it reduces to the setup used in [62] to study causality. Our formulas will reduce to the ones considered before in those limits. 5.3 The effect of higher derivative interactions on particles with spin Now we would like to consider different type of probes and compute the time delay for them. We start with a simple example of a scalar probe and then move to the case of particles of spin one and two. If we consider a minimally coupled scalar in the shock wave background its equation of motion takes the form ∇2φ= 0 .(5.10) In our setup we are interested in corrections to this equation which are second order in derivatives.17 Considering terms yielding two derivative equations of motion for φwe have to consider terms like Hµν∇µ∇νφ , (5.11) where the tensor His made from the background metric, Riemann and covariant derivatives. However one can check that there is no two index symmetric tensor that is not vanishing on-shell [61]. Of course, this statement is equivalent to the uniqueness of the scalarscalar-graviton three-point vertex. In the high energy limit we get similar to flat space ∂u∂vφ+f(u)h(z, ~y)∂2 vφ= 0 ,(5.12) which produces the time delay ∆v=$(ρ) 1−ρ2,(5.13) reproducing the flat space computation for small ρ. We assumed that the perturbation crosses the shock at z= 1 and ~y = 0. 16From the CFT point of view we can create such a probe by acting with the operator of given energy and zero momentum in the AdS Poincare coordinates for which u= 0 is the future null infinity as explained in [14]. In [14] terms we are working here in the y-coordinates, while the operator with given momentum is inserted in the x-coordinates. In pure AdS case isometries of HD−2at fixed u= 0 correspond to the usual Lorentz symmetry group in the x-coordinates of [14]. 17Since we need derivatives to bring down large factors of momentum. – 30 – JHEP02(2016)020 For the gauge boson we imagine at the level of two derivative the following equation ∇µFµν +Hµαβ ν∇µFαβ = 0 ,(5.14) where His built out of the Riemann tensor and its covariant derivatives. Using the properties of the background discussed above (we defer the details to appendix F) one can check that the only term that we can have is a correction analogous to that appearing in the case of flat space ∇µFµν −ˆα2ˇ Rµαβ ν∇µFαβ = 0 ,(5.15) where ˇ Rreduces to the Weyl tensor on-shell (see appendix F for details). If we compute the time delay using the same action (5.15), considering that each mode corresponds to a different constant polarization, i, we get ∆v=$(ρ) 1−ρ21−ˆα21−ρ22$0(ρ)−ρ$00(ρ) 4ρ$(ρ).n2 . −1 D−2.(5.16) The final result is very similar to the one obtained in flat space. The only different is that the polarization dependent delay is slightly more complicated. The flat space result (2.16) is reproduced by considering ρ→0 limit, whereas the energy correlator constrained is recovered in the limit ρ→1. Similarly, in the case of gravity we are interested in the most general form of the second order equations. We choose to parameterize the equations of motions for perturbations as follows δRµν +α2ˇ Rραβ (µδRν)ραβ +α4 2[∇(µ∇ν)ˇ Rαβρσ]δRαβρσ = 0 ,(5.17) where the parameters αiare in units of the AdS radius RAdS that we set to one. In this case, even though there are several possible ways to contract the indices in each of the above terms, we may concentrate on the contributions to the transverse equations of motion that are the ones yielding the time delay. For that purpose (5.17) is the most general parametrization.18 The time delay for these equations of motion is then given by ∆v=$(ρ) 1−ρ21 + t2(ρ)(.n)2 . −1 D−2+t4(ρ)(.n)4 (.)2−2 D(D−2), t2(ρ) = 1−ρ22$0−ρ$00 4ρ $(ρ)−α2+α4 D(1 + ρ2)2−2ρ2 ρ2,(5.18) t4(ρ) = −Dα41−ρ22$0−ρ$00 4ρ $(ρ) D(1 + ρ2)2+ 2(1 + ρ4) 4ρ2, where ~n is a vector pointing from the center of the shock to the probe particle, and is the polarization of the probe particle. These time delays can become negative for small enough ρif α2or α4are non-zero. 18There is also the possibility of considering more than two derivatives acting on the perturbation as ∇nδR. These contributions in general change the number of degrees of freedom of the theory and will not be considered here. – 31 – JHEP02(2016)020 These results can be also reproduced using slightly different method of evaluating the on-shell action in an explicit gravitational theories in the shock wave background like Lovelock or quasi-topological theories (see e.g. [65,66]). In the limit ρ→0 these constraints reproduce the flat space analysis of section B.1 with ρ=b 2. In the ρ→1 limit the above result reproduces constraints discussed in the past. If we take the limit ρ→1 by taking the shock center to the boundary z0→0 we recover energy correlator computation [14]. If we on the other hand consider ρ∼1 by taking the shock center to the horizon z0→ ∞ we recover the shock wave discussed by Hofman [62]. 5.4 Implications for a, c in theories with large operator dimensions Imagine an abstract CFT4with large N1 and large gap ∆gap 1, where ∆gap is the dimension of the lightest higher spin single trace operator. This theory is described by a gravitational theory in the bulk with potentially some higher derivative corrections. String theory inspired intuition suggests that higher derivative corrections should be suppressed by the 1 ∆gap factor. Our argument shows that this is indeed the case for the simplest corrections, which are the ones affecting the three-point function of stress tensor. Indeed, as we showed in (5.18) , we run into a potential problems with causality at impact parameters ρc∼(α2)1/2,(α4)1/4. Since this can only be fixed by higher spin particles, we conclude that ∆gap has to be small enough so that it can start correcting the amplitude before we run into this problem. For ∆gap 1 then the relevant impact parameters are such that we can approximate the formulas by the flat space limit. Thus, we get the bound of the type (α2)1/2.1 ∆gap ,(α4)1/4.1 ∆gap ,(5.19) where .stands for some numerical coefficient that we cannot fix using our simple analysis. In the case of N= 1 superconformal theories, α4= 0 by supersymmetry and α2∝a−c c. Then we get  a−c c.1 ∆2 gap .(5.20) In the case of N= 4 SYM (5.19) is satisfied trivially since ∆gap ∼λ1/4and a=c. It will be very interesting to find an independent field theoretic argument that leads to the bounds of the type (5.19) or (5.20). It would also be nice to find the precise numerical coefficients in (5.19) and (5.20). In the supersymmetric case, a−cwas argued to control asymptotic density of BPS operators in [67]. It would be very interesting to understand if there is any relationship between their work and our analysis. 5.5 Implications for dimensions of double trace operators This computation of the time delay can be viewed as a special four-point correlation function with particular wave functions for external operators. We can use the OPE to expand the four-point function computation in terms of the time evolution eigenstates (or, equivalently, local operators). One can ask then the following question: what is the relation between the time delay and the OPE data? – 32 – JHEP02(2016)020 ρ Figure 8. We consider two energetic particles in AdS that oscillate back and forth with energy Eand angular momentum J. This models the high twist, high spin double trace operator in the dual CFT. This question was addressed in a nice series of papers [58,63,64] where the time delay was shown to be equal to anomalous dimension of double trace operators of the type O(∂2)n∂µ1. . . ∂µjOfor large nand large j δ(s, ρ) = −πγ(n, j), ρ=j j+ 2n,(5.21) s= 4n(j+n). where the relations between parameters on both sides of the equation are reviewed below. Intuitively, (5.21) follows from the fact that the phase shift of the correlator eiδ is given by e−i(∆∗−2∆O)∆τwhere ∆∗is the dimension of the operator that dominates the OPE, and ∆τ is the global AdS time that passed from the beginning to the end of the process. The time delay that we computed corresponds to particles starting at the boundary, getting close to each other at the center and then reaching the boundary again, see figure 8. It takes ∆τ=πfor this process to occur. From this fact (5.21) follows. Even though (5.21) was derived in General Relativity in the limit when δ1 it follows simply from the AdS graviton diagram exchanged. In the impact parameter representation, only the on-shell t-channel exchange diagram constributes. This diagram is fixed in terms of three-point function hOOTµνiin a generic gravitational theory with generic three-point couplings similar to our flat space analysis. Let us briefly review the results in [58,63,64]. The basic idea is the following: the state created by O(∂2)n∂µ1. . . ∂µjOfor large nand large jcan be thought of as two highly energetic particles that follow null geodesics in AdS, see figure 8. For two geodesics that are characterized by total energy Eand spin Jthe minimal separation is achieved in global coordinates at ρ=J E≃j j+ 2n.(5.22) – 33 – JHEP02(2016)020 where we matched energy and spin of the pair of particles to the ones of the double trace operator J=j,E= 2∆ + 2n+jand used that n, s 1, with ∆ also of order one. Thus, we see that probing distances much smaller than AdS radius (ρ1) corresponds to considering operators with nj. On the other hand njcorresponds to scattering at very large impact parameters. The Mandelstam variable sis given by s=E2−J2≃4n(j+n),(5.23) and the relation to the anomalous dimensions is that πγ(n, j) = pv∆v=−Gs $(ρ) 1−ρ2.(5.24) Let us consider different limits of this formula. First, consider very large impact parameters ρ∼1 or jn. In this limit we get γ(n, j)∼ −GnD−1 jD−3,(5.25) which is in agreement with the general results derived using the crossing equation [68–70]. In the opposite limitof small impact parameters scattering, njand j n>1 ∆gap , we have γ(n, j)∼ −Gn2n jD−4 .(5.26) We have several comments to add to this story. First, these results should be universal and applicable to generic CFTs with large Nand large gap. To write the answer in an abstract form we have to use the relation of Gto the two-point function of stress tensors which is well-known and is roughly G∼1 cT. It means that it should be possible to derive them using crossing equation which still be dominated by the stress tensor exchange. Probably the relevant limit is z→0, ¯z→1 with z 1−¯zbeing fixed. It would be nice to reproduce the formulas above using crossing equations. Second, we see that causality, or positivity of the time delay, implies the constraint γ(n, s)<0, which generalizes the ones that were previously known [69–71] for asymptotically large sn. Again it would be very interesting to understand how to prove these constraints purely from the field theory point of view. Third, we see that considering double trace operators of the type Tµν(∂2)n∂µ1. . . ∂µjTρσ we get new structures due to the dependence on polarization which potentially lead to causality violations and bounds (5.19), (5.20). Of course, the same is true about the double trace operators that involve the conserved current Jµ. It will be very interesting to understand them from the purely CFT viewpoint. Note that in the scattering process the polarizations of particles 3 and 4 in figure 3can change relative to those of particles 2 and 4, so that the phase shift is an operator that acts on this space of polarization tensors. Both t-channel unitarity, as well as our considerations, constrain only some matrix elements of this general matrix. While we leave the general case for the future, we note that, in some cases, we can ensure that the polarizations of 3 and 4 do not change by using conservation – 34 – JHEP02(2016)020 (a) (b) delay advance Figure 9. We consider a Lorentzian wormhole configuration that is described, near each wormhole, by the maximally extended Schwarzschild solution. We send a (green) particle from the left very close to the past horizon. We then send a (purple) particle from the right. (a) If this particle gets a time delay, it will fall into the singularity. (b) If the particle gets a time advance, then it can make it out of the other black hole and we would have a way of sending signals through the wormhole. Here the blue lines represent the average position of the horizon of the black hole, defined by null lines that are very far away from the two particles we send in. We assume that the impact parameter is much smaller than the Schwarzschild radius. of angular momentum along the directions orthogonal to the impact parameter direction. In these cases our considerations apply and we can say that the anomalous dimensions of the corresponding double trace operators should be negative. The positivity statement applies to the part of the phase shift that grows with the Mandelstam invariant s, which translate into the growth with nvia (5.23). However, in our case, this positivity requirement is not obvious from the CFT point of view. It would be nice to see whether this is a general requirement or is one that is present only in theories with a local bulk dual. 6 Wormholes and time advances General relativity has Lorentzian wormhole solutions that join far away points by short Einstein-Rosen bridges. The simplest configuration is the maximally extended Schwarzschild solution interpreted as an approximation to the metric of two distant black holes which share a single interior. As discussed in [28], these solutions do not lead to a violation of causality in the ambient space because it is not possible to send signals through the wormhole [72]. The inability to send signals through the wormhole depends crucially on the fact that we have a Shapiro time delay as opposed to a time advance. For example, if one sends a fast moving particle from the left side, then a particle send from the right will suffer a time delay that will make it go into the singularity, see e.g. [33,73]. However, if that particle were to suffer a time advance, as opposed to a time delay, then it would be able to go through the wormhole and we would have a violation of causality, see figure 9. Note that we can make the wormhole big enough that we can neglect the higher derivative corrections in the description of the background metric. It can also be big enough that we can neglect the backreaction of the two particles. So we are considering a situation where the impact – 35 – JHEP02(2016)020 parameter b(Schwarzschild radius). In the D= 4 case, the Schwarzschild radius acts at the IR cutoff of the logarithm. This impossibility of sending signals is crucial for interpreting the wormhole as an EPR state of two disconnected systems [29–31]. 7 Cosmological applications The gravity wave non-gaussianities produced by inflation are a direct measure of the graviton three-point vertex during inflation [17,20]. To leading order in the slow roll approximation we can do the computation in de Sitter space. The symmetries of de Sitter imply that only two different parity preserving structures are possible. These correspond to the two parity preserving structures that we have in four-dimensional flat space. One is the one produced by the Einstein action and the other can be produced by a term in the action of the form M2 pα4R3, where Ris a Riemann tensor (not the Ricci tensor). The relative size between the two types of gravity wave non-gaussianity is proportional to hhhhiR3 hhhhiEinstein ∝α4H4,(7.1) where His the Hubble scale during inflation. Of course, both are small compared to the two point function, h3pointi h2pointi3/2∼H Mp. See [20] for the explicit expressions. Thus, if the gravity wave three-point function was measured and it was found that this exotic new structure is present at a level comparable to the Einstein one, then one concludes that α4is of the order of the Hubble scale. The considerations in this paper imply that there should also be new particles with spins J > 2 with masses comparable to the Hubble scale during inflation. Thus, this would be an indirect evidence for string theory during inflation. Note that α4H4∼1 implies that supersymmetry had to be broken at the Planck scale and not at a lower scale, since the + + + and −−−structures are forbidden by supersymmetry.19 Let us be a bit more explicit about this point. If the short distance theory is supersymmetric, then the field theory Lagrangian does not contain the couplings giving rise to α4. Now, since supersymmetry is broken, this three-point function could arise from integrating out massive particles. These are expected to contribute to α4as α4∼1 M2 p1 m2 B−1 m2 F,−→ α4H4∼H2 M2 p1,(7.2) which is very small. Where, to maximize the effect, we assumed that the masses of the bosons and fermions, as well as their differences, are of order H. We then see from (7.1) and (7.2) that in this supersymmetric scenario the contributions are very small. Thus, in order for the right hand side of (7.1) to be of order unity (or say a few percent) the supersymmetry should be broken at the Planck scale (during inflation) so that the threepoint vertex is present in the original classical theory. Notice that most of the string inflation models do not predict a large α4since they are based on compactifications of the ten dimensional superstring. It should be a model where the string length is comparable 19We thank Nima Arkani-Hamed for emphasizing this to us. – 36 – JHEP02(2016)020 to the Hubble radius and with a very weak coupling to account for the small experimental upper bound for H/Mp[74]. Note that if one imagines that inflation is given a dual description in the spirit of dS/CFT and the dual field theory is weakly coupled, then one expects that α4H4∼1. This is what happens in the Vasiliev theory [75]. Of course, this theory also contains massless higher spin particles. It is also not suitable for building an inflationary model because the scalar does not appear to obey the slow roll conditions. If the gravity waves produced by inflation are as large as to explain the signal seen by BICEP2 [76], then probing the gravity wave three-point functions might be possible (with a lot of optimism!).20 8 Conclusions In this paper, we studied causality constraints on higher derivative corrections to the graviton three-point function. We considered a weakly coupled theory and studied higher derivative corrections which are important before the theory becomes strongly coupled. These are higher derivative corrections that arise in the classical regime of the theory. The constraints arise from a thought experiment where we scatter two gravitons at relatively high energy and fixed impact parameter. The energy is high compared to the inverse of the impact parameter but low compared to the scale where the theory becomes strongly coupled. More explicitly, we have the very small overall coupling Gand we consider corrections to the three-point functions which scale as powers of αp2relative to the Einstein one. The three-point amplitudes are very small because Gis very small. But αis a fixed quantity and we look at impact parameters of the order of b2∼α. We found that when the impact parameter b2∼α, then we see a causality violation. In this impact parameter representation, and in the field theory regime (without higher spin particles), the time delay comes from the singularities in the t-channel, which is simply a pole at t= 0 for the massless theory. More precisely it comes from the part that is quadratic in sof its residue. In other words, terms going like s2/t at t= 0. It is important that while t= 0, the momentum transfer itself is non-zero.21 The overall sign of the time delay, then depends on the contractions of the polarization tensors of the external particles with the momentum transfer in the t-channel. We have argued that this type of tree level causality violation can only be fixed, at tree level, by higher spin particles at a mass scale m2∼1/α. In string theory, this issue is fixed because the amplitude Reggeizes. Namely, it has a behavior s2+ α0t 2for large sand fixed t. This is due to extended strings being exchanged in the s-channel. If the amplitude Reggeizes, then corrections appear at a scale b2∼α0log(sα0). Due to the presence of the logarithm we did not find a sharp bound between the corrections to the graviton three point amplitude, α, and the Regge slope α0. 20At this time, there are alternative explanations for this signal [77,78], so we might have to wait till the dust settles. 21It is a null, non-zero momentum. – 37 – JHEP02(2016)020 We should stress that in this discussion we have assumed that we have a weakly coupled gravitational theory. We have also assumed the notion of asymptotic causality which says that the causal structured determined by the far away regions of spacetime cannot be violated by its interior regions. The analysis in this paper was also motivated by trying to understand better the AdS/CFT correspondence. In particular, if we consider large Ngauge theories we know that we have a weakly coupled theory in the bulk. However, we do not know under what conditions that weakly coupled theory is a local in the bulk. It is clear that the absence of light massive higher spin states is a requirement. Here we have tried to address the question of whether it is sufficient. We have only studied the simplest possible correction to gravity, its three-point function. We have argued that, as long as higher spin particles are very massive, there cannot be higher derivative corrections to the three-point functions. Previous discussions argued against such corrections by saying that they would make the theory strongly coupled at energies that are lower than the Planck scale [15,16], but still parametrically larger than the scale of the corrections.22 Here we have strengthened the bound by linking the scale of corrections to the appearance of new particles at the same scale. As a more concrete statement, we are linking the values of the constants appearing in the stress tensor three-point functions to the dimensions of the lightest particles with higher spins, J > 2. In other words, a−c c.1 ∆2 gap . Unfortunately, we could not determine the precise numerical constant in this inequality. Using the results in [58,63,64], we can link the time delay for a high energy scattering process in the bulk to the anomalous dimensions of certain double trace operators. These double trace operators have the rough form T∂j +(∂2)nT. They have both relative spin and relative radial excitations. The anomalous dimensions are γ(n, j)∼ −δ(s, b)/π, where the values of band sare given in terms of j, n in (5.22) (5.23). The requirement that the time delay is positive leads to the statement that the anomalous dimensions for some of these operators should be negative. For the de Sitter case, this analysis has potentially interesting phenomenological applications. If the gravity wave three-point functions were measured, we expect to see the structure predicted by Einstein theory. However a new structure is also possible. This new structure is the only one allowed by the approximate scale and conformal invariance of the inflationary phase. If such new structure was found with a strength comparable to the Einstein one, then it would be a direct signal of dramatically new physics at the Hubble scale: a tower of higher spin particles. Which is a rather drastic departure from ordinary field theory at the inflationary scales. It is not clear how likely this inflationary scenario is in the space of possible inflationary theories. 8.1 Open problems It would be nice to derive these constraints in a more direct way. If one understood directly the constraints of unitarity and causality at the level of the four point function, 22In [15,16], or in talks referring to those papers, they impose the bound α.1 G(∆gap)4(with G∼1/N2). This bound comes from demanding that the theory remains perturbatively unitary at the scale ∆gap where new particles appear. Here we argued for the stronger bound α.1 ∆2 gap . – 38 – JHEP02(2016)020 then one would not need to resort to the exponentiation argument we discussed in section 3.3. Furthermore, it might lead to sharper bounds that include numerical factors. Our discussion of massive intermediate particles in mixed representations was not complete.23 These are representations that have maximal spin two in the uv-plane, but have additional indices in the other directions (see appendix H). We suspect that a finite number of these cannot solve the causality problem, but we did not prove it. In the AdS case, it would be nice to derive the constraints from the conformal bootstrap point of view. This is in the spirit of [15], but it would involve the stress tensor as an external state. One of the main messages from this paper is the importance of spin in order to derive constraints. The structure constants (or three-point functions) of operators with spin are very numerous but, as we have shown, there can be powerful constraints on them. These constraints are not so easily seen when we scatter external operators with no spin. Notice that, even the simplest bounds for aand cdiscussed in [14], which should be valid for arbitrary CFTs, have not been derived from the conformal bootstrap approach. It would be nice to further constrain the interactions of all the higher spin particles and derive the general structure of the tree level theory. This is the program that was pursued in the sixties and that led to string theory. However, we would like to know how unique string theory is. Methods developed to tackle this problem might also be useful for analyzing large Ngauge theories such as large NQCD. It would also be nice to see whether in the de Sitter context there is a sharp bound for the gravity wave three-point correlators analogous to the one in [14] for AdS correlators. Our de Sitter discussion assumed the local gravity description and a locally flat space discussion in the bulk, so it applies most clearly when α4H4is somewhat smaller than one, but still of order one compared with H/Mp. Acknowledgments We would like to thank N. Arkani-Hamed, S. Caron-Huot, A. Dymarsky, S. Giddings, S. He, D. Hofman, Y. Huang, Z. Komargodski, D. McGady, R. Myers, J. Penedones, L. Rodina, D. Simmons-Duffin, E. Skvortsov, A. Strominger and R. Wald for discussions. X.C. and J.E. thank the Institute for Advanced Study for hospitality at the initial stages of this work. J.E. is supported in part by MINECO and FEDER (grant FPA2011-22594 and FPA201452218), by Xunta de Galicia (GRC2013-024), and by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042). J.M. is supported in part by U.S. Department of Energy grant DE-SC0009988. The Centro de Estudios Cient´ıficos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt. 23These are not present in D= 4, so that the existence of an infinite tower of higher spin states is clear in D= 4, or if α46= 0 in higher dimensions. In higher dimensional theories with only α4= 0 but non-zero α2, in order to establish that the tower is really infinite we need to rule out the possibility that the causality problem is fixed with a finite number of mixed representations. We leave this to the future. – 39 – JHEP02(2016)020 initial signal which is a function of time fin(t) and an out-signal fout(t) which, in Fourier space is given by fout(ω) = S(ω)fin(ω) or fout(t) = Zdt0ZdωS(ω)e−iω(t−t0)fin(t0).(D.1) Causality implies that if fin(t0) = 0 for t0<0, then fout(t) = 0 for t < 0. By unitarity we mean that the L2norm of the out-signal should be smaller than that of the in-signal Rdt|fout(t)|2≤Rdt|fin(t)|2. Now it is well known that the Fourier transform of a function which vanishes for t < 0 is analytic in the upper half ωplane. This follows directly from the explicit integral expression for the Fourier transform. Then if fin = 0 for t < 0 we find that both fin(ω) and fout(ω) are analytic in the upper half plane. This also implies that S(ω), which is given their ratio, is also analytic. One might worry that S(ω) could have poles at zeros of fin(ω). However, we can change the location of the zeros of fin(ω) by choosing different functions. Therefore S(ω) is analytic in the upper half plane. We will now prove that unitarity implies that |S(ω)| ≤ 1 in the upper half plane. With some foresight, we pick a particular fin(t) of the form fin(t) = e−γte−iω0tθ(t)p2γ , (D.2) with γ > 0 and ω0real. Note that ||fin||2=Rdt|fin(t)|2= 1. For Im(ω)>0 we can now write |fout(ω)|2≤Z∞ 0 dteiωtfout(t)≤Z∞ 0 dt|eiωt|2Z∞ 0 dt|fout(t)|2=1 2Im(ω)||fout||2 |fout(ω)|2≤1 2Im(ω)(D.3) |S(ω)|2=|fout(ω)|2 |fin(ω)|2≤1 2Im(ω) 1 |fin(ω)|2. Here we used the Cauchy-Schwartz inequality. Note that |eiωt|2=e−2Im(ω)t. We also used that ||fout||2≤ ||fin||2= 1. We can now set ω=ω0+iγ and find that fin(ω0+iγ)=1/√2γ for the specific function (D.2). Inserting this into (D.3) we then find that |S(ω0+iγ)| ≤ 1, which is what we wanted to prove, since ω0and γare arbitrary. In conclusion, we find that S(ω) should be analytic and bounded |S(ω)| ≤ 1 in the upper half plane. These are necessary and sufficient conditions.28 Let us now briefly mention how this is connected to the field theory situation. We consider light cone coordinates uand v. We consider a perturbation that is translation invariant in vbut is localized in the ucoordinate. We call this “the shock”. We expand the fields in the vcoordinate at some uin and then we expand them again at some uout after the shock. To make contact with the above discussion we call t=vand pv=−ω. We can expand the field as φ(t) = Z∞ 0 dω √ω(aωe−iωt +a† ωeiωt).(D.4) 28Note that some functions which are analytic in the upper half plane, such as S(ω) = eiω3are actually not causal. – 46 – JHEP02(2016)020 u v u uin out Figure 11. We consider a vindependent perturbation that is localized in the udirection, given here by the shaded region. We then consider signal propagating along the udirection, which are v dependent and demand causality. We can consider an Smatrix that connects the region before the perturbation to the region after the perturbation. We can do this for φin and φout in terms of ain and aout. These oscillators then are related by aω,out =S(ω)aω,in , a† ω,out =S(ω)∗a† ω,in .(D.5) This defines S(ω) for positive ω. For negative ωwe can define S(−ω) = S(ω)∗. Alternatively, we can define S(ω) = −Z∞ −∞ dteiωt[φout(t), i∂tφin(0)] = −Z∞ 0 dteiωt[φout(t), i∂tφin(0)] .(D.6) The commutation relations for the in and out oscillators require that |S(ω)|2= 1. In a case where there is particle mixing, but no particle creation, then the fields have indices φi in and φj out. Now Sis a matrix which obeys Sij(−ω) = Sij(ω)∗and Sij(ω)Skj(ω)∗=δik due to the commutation relations of the oscillators before and after the shock. In the preceding discussion we have neglected the transverse dimensions. We can now remedy that by including the momentum in the transverse dimensions as part of the indices we are discussing here. If we consider a signal that is made out of physical particles, one might correctly worry that the fact that ω > 0 will preclude us from localizing the signal in time. In order to avoid this issue we can consider a coherent state of the form |Ψi=eiRdtfin(t)φin(t)|0i,(D.7) with a real function fin. This is a state that could be produced by adding a hermitian term to the Hamiltonian at some early time uin. On this state we have the expectation values hΨ|∂tφin(t)|Ψi=fin(t),hΨ|∂tφout(t)|Ψi=fout(t),(D.8) where the functions are related as in the signal model.29 Here we assumed a linear relation between the inand out-signals. Furthermore, we can also consider the expectation values 29fout is real if fin is real when S(−ω) = S(ω)∗. – 47 – JHEP02(2016)020 of the normal ordered product Tvv =Ttt =: ∂tφ(t)∂tφ(t) :. When this is evaluated on the state (D.7) , and integrated over twe find that the answer is given by −Pin,out v=ZdvTvv =Zdt(fin,out(t))2=||fin,out||2.(D.9) Thus, the condition that the total light-cone momentum Pvshould not increase implies the norm condition ||fout||2≤ ||fin||2. More precisely, we can consider the signal fin exciting a mode involving a graviton with a given polarization. The signal fout is the same mode of the graviton. In addition, the initial graviton could go into other massive particles. Then the condition that the total Pv in the out-graviton mode should be no bigger than the initial Pv, which was all contained in the graviton mode, leads to the norm condition (or unitarity condition) for the signal model. In conclusion, the graviton-graviton matrix element Sgg(ω) obeys all the assumptions of the signal model. Therefore, it should be analytic and |Sgg(ω)| ≤ 1 in the upper half plane. Note that we have assumed here a perfect v-translation symmetry for the perturbation that creates the shock. In our scattering problem, see figure 3, particles 1 and 3 have small pvmomentum. Thus, in this discussion, we have neglected this small momentum. This is reasonable for sb21. E Scattering in string theory String theory in flat space is the simplest example of a theory that follows into the category of weakly coupled gravitational theories with higher derivative corrections that are subject of our analysis. As explained in the introduction, a motivation for this work was actually to argue that string theory is inevitable, at least, under certain assumptions. It is well known that effective gravitational actions in string theory contain higher derivative corrections at the string scale [84,85]. In particular graviton three-point amplitudes can contain the higher derivative terms that we constrained in this paper using causality. The potential problem is fixed by extra particles with string scale masses. Here we would like to understand how this is happening in detail. Let us first recall the form of the three-point gravity amplitudes in bosonic, heterotic and type II string theories [86] Aggg =√32πGµ1µ0 1 1µ2µ0 2 2µ3µ0 3 3Nµ1µ2µ3¯ Nµ0 1µ0 2µ0 3, Nµ1µ2µ3=kµ1 2ηµ2µ3+kµ2 3ηµ1µ3+kµ3 1ηµ1µ2+α0 2kµ1 2kµ2 3kµ3 1,(E.1) and we have bos = ¯bos =het = 1 and II = ¯II = ¯het = 0 . Translating it to our notations we get αbos 2= 2αhet 2=α0, αII 2= 0 , αbos 4=(α0)2 4, αhet 4=αII 4= 0 .(E.2) – 48 – JHEP02(2016)020 The vanishing of some of these corrections can be understood from supersymmetry, as explained in section 3.2. For the purposes of this paper, the type II case is not interesting since there are not corrections at all for the graviton three-point function. The high energy scattering problem in string theory was studied in a nice series of papers by Amati, Ciafaloni and Veneziano [51–55], see also e.g. [34,87]. Let us first review their picture. The scattering can be described in terms of a phase shift defined as δ(~ b, s) = [POL]δACV (~ b, s), δACV (~ b, s) = ZdD−2~q (2π)D−2ei~q.~ bC(s, t, u),(E.3) C(s, t, u) = Γ(−α0s 4)Γ(−α0t 4)Γ(−α0u 4) Γ(1 + α0s 4)Γ(1 + α0t 4)Γ(1 + α0u 4), where [POL] represents a factor that depends on the polarizations and is a polynomial in the momenta. We will only need its form in a specific limit. In the high energy limit C(s, t, u) has the celebrated Regge behavior C(s, t, u)∼Γ(−α0t 4) Γ(1 + α0t 4)−isα0 4−2+ tα0 2.(E.4) This Regge form is reflecting the creation of particles in the s-channel. The infinite sequence of s-channel poles is becoming a cut, the cut arising when s→se2πi. The creation of the massive s-channel states is also related to the fact that we get an imaginary part from the (−i)tα0 2factor in (E.4) is saying that the most likely process is to create a massive closed string, rather than scattering the gravitons. For large s, only small qwill contribute and we can approximate the prefactor in (E.4) by 1/t. Then the integral becomes δ[ACV ]∝ZdD−2~q 2(2π)D−2 ei~q.~ b−isα0 4−~q 2α0 2 ~q 2=1 (α0Y 2)D−4 2Z1 0 dρρD−6 2e−b2 2α0Yρ,(E.5) where Y= log(−isα0/4). We have two characteristic behaviors, depending on whether b2 is larger or smaller than α0log(sα0). For large bwe get the usual 1/bD−4behavior. For small bwe get the result in brackets in (4.8). Note that since the transition region occurs for a b2which is larger than α0by a log sfactor, we can, a posteriori, justify the fact that we have approximated the prefactor in (E.4) by 1/t. In our field theory discussion we had represented the phase shift as a sum over poles. We can wonder how this applies to the string theory discussion. Notice that from (E.4) we get a Gaussian integrand factor of the form ei~q.~ be−(~q)2α0 2log(−isα0/4) .(E.6) Let us assume that ~ b= (b, 0,··· ,0), so that it points along the first coordinates. When we do the integral over the first component of ~q, call it q1, we get a saddle point for (E.6) at qs=ib α0log(−isα0/4) .(E.7) – 49 – JHEP02(2016)020 x x x x x q1 Figure 12. We display the complex q1plane. We have displayed the poles in the t-channel by black crosses. The saddle point (E.7) of the Gaussian integral has been denoted by a red cross. We have shifted the original integration contour for q1, which was along the real axis, to the complex plane so that it passes through the saddle point (E.7). In the process we have picked up some of the poles in the t-channel. It is thus convenient to shift the contour to this location, (E.7) , where this saddle point contribution gives something of the order of e−b2 2α0log(−isα0/4) .(E.8) This is not the whole answer, since by shifting the contour to this location, we can pick up some poles from the prefactor (E.4) , see figure 12. We always pick up the pole at t= 0,30 which was the center of our gravity discussion, but we can also pick up the poles at t=4 α0n for n < −q2 s, where qsis given in (E.7). Notice that at these saddles the tdependent part of the exponent gives us factors of s2nas we expect for the corresponding spins,31 once we take into account that [POL] contains a factor of s4. When bis large b2α0log(sα0), we pick up many poles, but when bis small we only pick up the t= 0 pole, but, even then, the integral is more accurately computed using (E.5). Note that the factor [POL] in string theory phase simplifies at t= 0 and becomes the product of three-point amplitudes we discussed in the body of the paper and that we added as Pol in (4.8). In other words [POL]→Pol as t→0. In particular, note that the residues of poles associated to the massive states go as 1 (n!)2e−bq4n α0s2n. As a function of n, these contributions decrease and then start increasing again with a transition at a value of ncorresponding to the saddle point (E.7).32 We can ask: why don’t we include all poles in the t-channel?. If we were to include all poles in the t-channel, we would obtain the wrong answer. The reason it is wrong is that in string theory the argument that we can shift the contour is not correct because of contributions for large values of q. Such large values of qwere never meant to be included in the integral, since the kinematics of the process we consider restricts the real values of ~q 2to be much smaller than s. In deriving the physical picture, we certainly assumed that 30The location of this pole in the q1plane depends on ~qrest which we take to be real. 31For a closed string with NL=NR=n, the maximum spin is J= 2 + 2n. 32Here we are assuming that both b2and log sα0are large with a ratio larger than one but fixed, so that we can neglect the 1/(n!)2in this discussion. This last factor makes the sum converge for large n, but this convergent answer, as discussed below, is not the correct one. – 50 – JHEP02(2016)020 ~q 2s.33 Indeed, if we look at (E.4) we get a very small contribution from large real values of ~q 2. On the other hand, if we were to keep sfixed and we formally look at large real values of ~q 2in the original expression, (E.3) , then we would encounter the uchannel poles. The conclusion is that shifting the contour for the q1integral, while it can be done formally, it does not represent the real physical computation we want to do. Approximating the integrand using (E.4) , and then integrating gives the physically correct answer. One can qualitatively say that, for large b2/(α0log sα0), we get a contribution of some of the t-channel poles, as in figure 12, and then the rest of the poles are completely resummed via the saddle point integral in figure 12. Their contribution should be better thought of as coming from the creation of extended objects in the s-channel. Another remark we want to make is the following. It was shown in [34] that the plane wave solution is a solution to all orders in the α0expansion. This gravitational plane wave encodes the contribution from the t= 0 pole. However, we have seen that sometimes we get a subleading contribution from the other poles, due to massive states along the t-channel. These mean that the physical scattering process contains extra contributions not captured by the plane wave.34 In string theory we can also take into account the tidal excitations. In this case the phase shift can be viewed as an operator that maps the two initial gravitons to two final generic string states. Refs. [51–55] have shown that this operator has the remarkably simple expression ˆ δ∝Rdσdσ0δgrav(ˆ XL(σ)−ˆ XR(σ0)) where XLand XRare the transverse space positions of the string on the worldsheet and δgrav(b) is the ordinary gravity phase shift. This is valid for distances b2α0log(sα0). The effects of these tidal excitations do not help in resolving the causality issues discussed here and are unrelated to the appearance of closed strings in the s-channel discussed above. See [51–55] for further discussion. F Properties of the AdS shock wave Let us first examine the problem of higher derivative corrections for the AdS shock wave. As in the case of flat space the shock wave at hand continue to be an exact solution when arbitrary higher derivative corrections are included. The argument for this is identical to the one in section 5 of [61]. The vector lµ= ∂µu={1,0,0,0. . .}in coordinates u, v, yi, z. We can now compute the vector Vµthat the argument talks about. We find Vµ={0,··· ,0,−2/z}. In other words Vz=−2/z and the rest of the components are zero. This obeys that Vµlµ= 0 as required in their argument. In addition, one can also compute the substracted Riemann tensor ˇ Rµναβ =Rµναβ + 2gµ[αgβ]ν. It is indeed of the form stated in [61] with the symmetric tensor Kgiven by Kyi,yj=1 2(δij∂zh/z3−∂i∂jh/z2), 33Momentum conservation and on-shell conditions impose q2<s 4. 34This is not in contradiction with [34], since these extra terms can be viewed as a non-perturbative contribution in α0. – 51 – JHEP02(2016)020 Kz,yi=Kyi,z =−∂yi∂zh z2,(F.1) Kzz =1 2(∂zh/z3−∂2 zh/z2), with the rest of the components equal to zero. This Kobeys that Kµνlν= 0 as required by [61]. Notice that this form of Kalso leads to the equation of the form (5.3), when the Riemann tensor is used in Einstein’s equations, namely Kµ µ= 0. Once the former equation is imposed ˇ Rreduces to the Weyl tensor. Using this shock wave we can once again compute the time delay for different theories. It is convenient to evaluate Riemann tensor in the coordinates that make rotation symmetry manifest. The result is ˆ Ruiju|~y=0;z=1 =Kij =−f(u)1−ρ2$0(ρ)−ρ$00(ρ) 8ρninj−1 D−2δij,(F.2) where i= 1, . . . , D −2 so that we span ~y and zcomponents. In the bulk of the paper we consider propagation of different perturbations in the shock wave background described above. We first write the most general form of the second order equations of motion and then compute the time delay in the high energy limit. Let us consider several limits of the shock wave to make contact with previous investigations of similar type. First, we expect to recover the flat space shock wave for probes that come close enough to the center of the shock or, equivalently, for ρ→0. Indeed, it is easy to check that this asymptotic is correctly recovered (5.9). Let us consider a couple of other limits. We can fix ~y0and send z0→0 in which case we have a source at the boundary and the shock wave takes the form h∼zD−3 (z2+|~y −~y0|2)D−2,(F.3) which is exactly the shock wave considered in the energy correlator problem [14]. In the opposite limit z0→ ∞ we get h∼zD−3,(F.4) and the time delay in this background was computed, for example, in [62]. G Time advances and time machines In this appendix we would like to argue that a negative time delay enables us to build a time machine which leads to closed time-like curves. This is a standard argument [88,89] and the only thing we check is that the long range gravitational forces do not prevent us from setting it up. The setup we would like to consider is the following. We have two shock waves that correspond to energetic particles with momenta q1,u =√s 2and q2,v =√s 2 separated by distance rin the transverse plane. The first shock is localized around u= 0 and the second one around v= 0. We would like the separation to be such that rrS, namely we are in the regime where the black hole formation does not occur rD−3rD−3 S=G√s . (G.1) – 52 – JHEP02(2016)020 Figure 13. a) We imagine a background that consists of two shock waves located at u= 0 and v= 0 widely separated in the transverse directions which is not presented on the figure.The arrows show the motion of the probe massless particle projected on to the u, v plane. b) Same motion but projected on the transverse plane. The two background shocks are separated by rand the probe passes at a short distance bfrom each of them. The vertical region of the path in (a) corresponds to the horizontal motion in (b). We can build a closed time-like curve as depicted on the picture by crossing this pair of shocks if time advances are allowed. We need mirrors to reverse the motion in the transverse plane as we pass through the shocks. Next we would like to consider a test particle that propagates through both shocks in such a way that it ends up at the same position where it started, thus, forming a closed time-like curve. The situation is depicted on figure 13. When the test particle crosses each of the shocks it gets shifted by ∆v= ∆u∼G√s bD−4. Between the shock the particle travels the distance of order r. Thus, we want the time shift to be G√s bD−4=rS bD−3b∼rrSwhich becomes rS bD−41.(G.2) We also want berSwhere erSis the Schwarzschild radius for the shock wave-test particle pair, erD−3 S∼Gpp√s, where pis the energy of the probe particle. Together with (G.2) it implies √spwhere pis the energy of the probe particle. We also need that pb 1. The conclusion is that in D > 4 we can construct closed time-like curve using negative time delays by choosing b,rand sappropriately. For example, if we have a causality problem that appears at a scale b2∼α2, then we take this value for b. Since we are at weak coupling, we know that the Planck length lpb. We can then pick √slp∼X1+a, p lp∼X1−a0, with X= (b/lp)D−3and we can choose a > 0, a−a0<0, 1−a0+ 1/(D−3) >0 to ensure that rSb,erSband pb 1. We can achieve this with a0= 1/2 and a= 1/4, for example. H Representations that couple to two gravitons Here we would like to understand better what are the representations of little group SO(D− 1) of massive particles that can couple to two gravitons. We are interested only in bosonic fields since a single fermion does not couple to two gravitons. – 53 – JHEP02(2016)020 Let us consider the decay of a massive particle in its rest frame so that p2= (M,~ 0). Gravitons produced have ~p1=−~p3=~p. We characterize the original particle by some polarization tensor ei1...ikwhich has only spatial components and is traceless with respect to any pair of indices. We do not specify the symmetry properties of this tensor yet. We characterize gravitons by polarization tensors e1and e3such that ~e1.~p =~e3.~p = 0. We have three type of contractions (see also [79]) A1=ei1...ikpi1. . . pik(e1.e3)2, A2=ei1...ikei1 1ei2 3pi3. . . pik(e1.e3),(H.1) A3=ei1...ikei1 1ei2 1ei3 3ei4 3pi5. . . pik. The first amplitude A1exists only for particles in the symmetric traceless representations (Young tableau that consists of single horizontal row with kboxes). Actually all three amplitudes are allowed for symmetric representation and we discussed them in the bulk of the paper in detail. For the second and third amplitude we can add more rows to the Young diagram. Properties of these, so-called, mixed-symmetry tensors are nicely reviewed, for example, in appendix E of [90]. By thinking about the representation in terms of tensors which are manifestly anti-symmetric with respect to indices in a given column we can read off possible representations. In particular the fact that we have only three different vectors means that we can have at most three rows. Let us write all the amplitudes in a covariant manner. The general prescription is the following ei 1ej 3→Eµν 13 ≡µ 1pν 3(3.p1) + µ 3pν 1(1.p3)−pµ 1pν 3(1.3)−µ 1ν 3(p1.p3).(H.2) We then have for the amplitudes A1=µ1...µkpµ1 1. . . pµk 1[(1.3)(p1.p3)−(1.p3)(3.p1)]2, A2=µ1...µkEµ1µ2 13 pµ3 1. . . pµk 1[(1.3)(p1.p3)−(1.p3)(3.p1)] ,(H.3) A3=µ1...µkEµ1µ3 13 Eµ2µ4 13 pµ5 1. . . pµk 1. To compute the time delay we need to use the completeness relation X i ∗ ν1...νkµ1...µk= Πν1...νk|µ1...µk,(H.4) and contract both sides of the amplitude, where Π is a projector on to the space orthogonal to the intermediate momentum. As the next step we would like to focus on those diagrams that produce sawith a≥2 for the amplitude. Everything that is less is irrelevant for causality violation. In the language of Young tableaux it corresponds to having a≥2 boxes in the first row. It will be curious to understand if mixed symmetry fields with a= 2 can resolve the causality problem we observed in the bulk of the paper. As an example of such field would be (2,2) (a square with four boxes) or (2,2,2) fields (a vertical rectangle of 2 ×3 boxes). There is a short list of representations of this kind. – 54 – JHEP02(2016)020 One could wonder whether these particles alone (without the infinite tower of higher spin particles) could solve the causality problem. We think that the answer is no. We leave a full exploration of this question for the future. I The Weinberg-Witten theorem in Ddimensions The Weinberg-Witten theorem [50] constrains the matrix elements of a conserved stress tensor on single graviton states. They showed that the stress tensor cannot be Lorentz invariant, conserved, and such that it measures the energy of the gravitons. Their original argument is for D= 4. Here we extend it to arbitrary number of spacetime dimensions D. The matrix elements of a conserved stress tensor on single graviton states, h0, p0|Tµν(q)|p, i, has the Lorentz transformation properties of an on shell three point amplitude between two massless gravitons and a massive spin two particle. The “mass” is the same as the square of the four momentum of the stress tensor operator, q2.35 The generic three point amplitude for a massive spin two and two massless gravitons has the form Aggeg=ec2eµν [µ 1pν 3(3.p1) + µ 3pν 1(1.p3)−pµ 1pν 3(1.3)−µ 1ν 3(p1.p3)] [(1.3)(p1.p3)−(3.p1)(1.p3)]] (I.1) +ec4eµνpµ 1pν 3[(1.p3)(3.p1)−(1.3)(p1.p3)]2, where the indices one and three denote the massless gravitons. We are interested in the matrix element that corresponds to the measurement of particle’s energy lim p0→ph0, p0|Tµν|p, i=pµpν0. . (I.2) We approach the point p0=pfrom the space-like direction. In other words, the transferred momentum is space-like q2= (p0−p)2>0. Let us now consider p1=p= (pu,0,~ 0) and q= (0,0, ~q) so that p3=p0= (pu,~q2 4pu, ~q). We also consider the polarization tensor of the graviton to be purely transverse eij. Then for the outgoing graviton we have 0= (0,~q.~0 2pu,~0). To measure the energy in the limit of zero transfer we should have lim p0→phe0, p0|Tµν|p, ei=pupue0 ijeij .(I.3) Computing the three-point amplitude to leading order in puwe get Aggeg=−ec2(q2)pupu(1.3) [(1.3)(p1.p3)−(3.p1)(1.p3)]] +ec4(q2)pupu[(1.p3)(3.p1)−(1.3)(p1.p3)]2 =1 2ec2(~q2)~q2pupue0 ijeij −e0 ijeik qjqk ~q2 +1 4ec4(~q2)~q4pupue0 ijeij −2e0 ijeik qjqk ~q2+e0 ijekl qiqjqkql ~q4.(I.4) 35When q26= 0, the transversality condition on the polarization tensor, qµeµν = 0 corresponds to the conservation condition, qµTµν (q) = 0. – 55 –