Bounds on Regge growth of flat space scattering from bounds on chaos
Deeksha Chandorkar, Subham Dutta Chowdhury, Suman Kundu, Shiraz Minwalla
PPrepared for submission to JHEP
TIFR/TH/21-2
Bounds on Regge growth of flat space scatteringfrom bounds on chaos
Deeksha Chandorkar, b, Subham Dutta Chowdhury, a, Suman Kundu, a, ShirazMinwalla a, a Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Rd,Mumbai 400005, India b Department of Physics, Jai Hind College (Autonomous), A Rd, Churchgate, Mumbai 400020,India
Abstract:
We study four-point functions of scalars, conserved currents, and stress tensorsin a conformal field theory, generated by a local contact term in the bulk dual description, intwo different causal configurations. The first of these is the standard Regge configurationin which the chaos bound applies. The second is the ‘causally scattering configuration’in which the correlator develops a bulk point singularity. We find an expression for thecoefficient of the bulk point singularity in terms of the bulk S matrix of the bulk dualmetric, gauge fields and scalars, and use it to determine the Regge scaling of the correlatoron the causally scattering sheet in terms of the Regge growth of this S matrix. We thendemonstrate that the Regge scaling on this sheet is governed by the same power as in thestandard Regge configuration, and so is constrained by the chaos bound, which turns out tobe violated unless the bulk flat space S matrix grows no faster than s in the Regge limit.It follows that in the context of the AdS/CFT correspondence, the chaos bound applied tothe boundary field theory implies that the S matrices of the dual bulk scalars, gauge fields,and gravitons obey the Classical Regge Growth (CRG) conjecture. [email protected] [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] F e b ontents τ limit 82.3.2 The Regge limit 92.3.3 Overlap between small τ and Regge 92.4 A path in cross ratio space 102.5 The same path on the complex cross ratio sheets 122.5.1 The neighbourhood of ¯ z = 1 z = 0 τ limit 17 σ → Regge limit 17 θ a e ρ ρ → ( τ → ) limit 23 φ correlator in AdS AdS AdS D +1 CRG conjecture from the chaos bound 39 A (cid:48) ≥ A ≥ A (cid:48) B.1 Definition of
AdS D +1 Q (cid:48) points in embedding space 52B.5 Conformal cross ratios and intersection lightcones for four boundary points 53B.5.1 u and v σ and ρ z and z bar 54B.5.4 z and z bar in terms of σ and ρ σ and ρ R , R , C Exploration of the neighbourhood of the Regge point 63
C.1 Relationship to global coordinates 63C.2 Causal relations 64C.3 R p,q z and ¯ z in this neighbourhood 68 D i(cid:15) in position space 71 D.1 The i(cid:15) prescription 71D.2 Relation to Euclidean space 71– ii –
Mapping into the ρ plane 72 E.1 The ρ frame 72E.2 Mapping into the ρ plane: qualitative comments 74E.3 Mapping into the ρ plane : quantitative formulae 75E.4 Summary of the mapping 76 F Regge expansion of bulk integrand 77G Example of bulk Regge scaling 77
G.1 Exact result for every value of n when a > n = 1 a H Singularities of holographic correlators from contact interactions 82
H.1 End point singularities 82H.2 Pinch singularities 83 I ρ = 0 branch cuts and UV softening 84 I.1 D = 2 D > J Massive higher spin particles 87
J.1 Multiple powers of ρ It has recently been conjectured that the tree level S matrix of any ‘consistent’ classicaltheory never grows faster with s than s in the Regge limit. Arguments supporting thisso-called Classical Regge Growth (CRG) conjecture were advanced in the CEMZ paper [5],the chaos bound paper [1] and also in discussions of the inversion formula [6] in large N theories. (see e.g. [7]).The authors of [2] demonstrated that the classical Einstein S matrix is the only CRGconsistent local tree-level gravitational S matrix in D ≤ . This striking result suggeststhat if the CRG conjecture indeed holds then the set of consistent classical gravitational That is a causal classical theory whose energy is bounded from below (and perhaps is also required toobey other constraints of a similar general nature). See [2, 3] for a discussion and [4] for a generalizationto gluons. That is an S matrix that is the sum of a polynomial in momenta plus a finite number of physical particleexchange poles. – 1 – matrices is highly constrained by simple general considerations , and motivates a moredetailed investigation of the CRG conjecture.We are aware of three arguments in support of the CRG conjecture. The first (andweakest) of these arguments is the simple observation that the CRG conjecture is indeedobeyed in every classical theory that is known for certain to be consistent. For instance,all two derivative classical theories always obey the CRG conjecture [2]. Moreover, theclassical Einstein S matrix saturates CRG growth and the Type II and Heterotic analogsof the Virasoro Shapiro amplitude of string theory temper this growth to a power that isstrictly less than two at every physical finite value of α (cid:48) t (see [2] for some more details).A second reason to believe the CRG conjecture is contained in the analysis of [5]. Theauthors of [5] used a ‘signal model’ (see Appendix D of that paper) to argue that the thefunction T ( δ, s ) = 1 + i S ( δ, s ) s (1.1)should obey the inequality |T ( δ, s ) | ≤ (1.2)everywhere in the upper half complex s plane for every nonzero physical value of δ . Here S ( δ, s ) is the transition amplitude - atleast roughly speaking- for a particle passing through ashock at impact parameter δ . Roughly, S ( δ, s ) can be thought of as the invariant amplitudefor the scattering in impact parameter space (see [8] for related discussions). In the classicallimit the tree amplitude is parametrically small and (1.2) reduces to the condition Im (cid:18) S ( δ, s ) s (cid:19) ≥ (1.3)If the large | s | fixed δ expression for the classical S matrix is S ( δ, s ) = B ( δ ) s A ( δ ) and s = | s | e iφ then (1.3) implies T ( δ ) sin (( A ( δ ) − φ ) ≥ , ≤ φ ≤ π (1.4)When A ( δ ) > the LHS of (1.4) switches sign as φ varies over the range [0 , π ] so (1.4)cannot be obeyed for all values of φ in this range. . It follows that the condition (1.2) canonly be satisfied if A ( δ ) ≤ , i.e if the CRG conjecture is obeyed. As the passage from theimpact space S matrix S ( δ, s ) to the usual momentum space S matrix S ( t, s ) is, roughlyspeaking a Fourier Transform in t at fixed s , this argument suggests that S ( t, s ) obeysthe CRG conjecture. The weakness of this beautiful argument lies in the fact it containselements that are intuitive rather than completely precise. For instance the transition from See in particular the introduction of [2], and in particular Conjectures 1-3 presented in that introduction,for a detailed discussion. Demonstrating the impossibility of a tighter than s bound on Regge growth. This is why loops are sub dominant compared to trees. When | A ( δ ) | ≤ , the sin function that appears in (1.4) is always positive, and (1.4) is obeyed when B ( δ ) > . – 2 –impact parameter space’ to ‘momentum space’ may have subtleties related to δ functionterms in impact parameter space The third (and possibly the strongest) reason to believe in the correctness of the CRGconjecture is its connection to the chaos bound in holographic theories. A relationshipbetween the chaos bound and the CRG conjecture was already suggested in the originalpaper [1] (see Appendix A for a discussion and of the relevant comments in that paper)and has been somewhat elaborated upon since (see e.g. [7]). The key strength of the chaos-bound based argument for the CRG conjecture is that the starting point for this argumentis a theorem, at least by physicists’ standards. One weakness of this argument lies in thefact that the connection between the chaos bound and the CRG conjecture has never (toour knowledge) been carefully argued through (see Appendix A for some discussion). The goal of this paper is to fill in this gap for the case that bulk particles are either scalars,gauge fields, or the metric (i.e. all particles relevant to the analysis of [2]).In particular, in this paper we demonstrate that the four-point function generated holo-graphically by a local bulk contact term for scalars, gauge fields, or the metric necessarilyviolates the chaos bound whenever the flat space S matrix generated by the same contactbound violates the CRG conjecture. In other words, the classical holographic dual of aconsistent unitary boundary field theory necessarily obeys the CRG conjecture.Our argument for a sharp connection between the chaos bound and the CRG conjectureconsists largely of stitching together well known results from the now classic papers [29]and [30] (see also [31–34] ). In section 2 study time-ordered four-point functions in aholographic conformal field theory of four operators of arbitrary spin inserted on the two-parameter family of points (2.1). Our four-point function is normalized by dividingwith a product of two two-point functions (see (2.21)). As the insertion parameters θ and τ run over the range of study (2.2), the conformal cross ratios σ and ρ (or equivalently z and ¯ z -see around (2.6) for definitions) range over three different sheets in the complexcross ratio space corresponding to three distinct causal configurations (2.3) of the boundarypoints. The first of these is the principal sheet which we refer to as the Causally Euclideansheet through this paper. The second sheet -which we refer to as the Causally Regge sheetthrough this paper - is reached starting from the Causally Euclidean sheet by circling thebranch point at ¯ z = 1 in a counter-clockwise manner. The third sheet - which we refer toas the Causally Scattering sheet through this paper- is obtained starting from the CausallyRegge sheet by circling counter-clockwise around the branch point at z = 0 .Our argument proceeds by focusing attention on two special one parameter limits ofour two-parameter set of insertion points (2.1). The first of these is the much-studied We thank S. Caron Huot for very useful discussions on this point. See however the papers [8–15] which put use the chaos bound (often the sign constraint rather thanthe growth constraint part of this bound) or closely related causality constraints [16–24, 24–28] to obtainmany interesting results that are in the same broad universality class as this paper. Another weakness is the fact that it uses an elaborate theoretical framework - namely that of holographyand consistency of the quantum structure of the boundary field theory - to establish a fact about scatteringin the simpler theoretical structure of classical field theories. See [35, 36] for recent progress regarding external massive scalar states. The same kinematic configurations were studied in section 6.1 of [30]. – 3 –Regge limit’ (2.9) in which σ is taken to be small but ρ is allowed to be arbitrary. Hereand throughout this paper, ρ and σ are conformal cross ratios, related by the more familiarcross ratios z and ¯ z by the relations z = σe ρ and ¯ z = σe − ρ . In several studies of conformalfield theory, the Regge limit is studied on the Causally Regge sheet (2.3) of cross ratiospace. The Regge limit studied in section 3 of this paper, however, straddles across boththe Causally Regge and the Causally scattering configuration In section 3 below we useuse a small variant of the analysis of section 5.2 of [29] to demonstrate that to leading orderin this limit (i.e. to leading order in small σ ) the σ and ρ dependence of our correlatortakes the form g CS ( e ρ ) σ A (cid:48) − and g CR ( e ρ ) σ A (cid:48) − (1.5)respectively in the Causally Scattering and Causally Regge regimes, where A (cid:48) is a fixedbut as yet unknown number. g CS ( e ρ ) and g CR ( e ρ ) are as yet unknown functions of thecross ratio ρ . The crucial point here, however, is that they are not completely independentof each other. There exists a function ˜ H ( z ) which is analytic apart away from the branchcut at z = 0 . g CR ( e ρ ) equals ˜ H ( z ) evaluated on z = e ρ on one sheet of this function,while g CR ( e ρ ) equals ˜ H ( z ) evaluated on z = e ρ on a second sheet. This fact makes itimpossible for g ( e ρ ) to vanish identically in the Causally Regge branch if it is nontrivialon the Causally Scattering branch. It follows, in other words, that in the small σ limit,correlators scale with the same power of σ in the Causally Scattering and Causally Reggesheets of cross ratio space.In section (4) we turn to the study the limit ρ → on the Causally Scattering sheet;this is the bulk point limit of [31] [30]. In section (4) we generalize the discussion of section6 of [30] to demonstrate that the four point function is given, to leading order in this limit,by the expression G sing = − (cid:112) σ (1 − σ ) (cid:90) H D − d D − X (cid:90) dωω ∆ − e iωP.X S (1.6)where S is the classical flat space S matrix of four specified waves with momenta (4.21),polarizations (4.68), ω is the energy of each of these waves, ∆ is the sum of the scaling di-mensions of the four operators and the H D − is the part of AdS D +1 space that is orthogonalto the four boundary points P i at τ = 0 (see subsection 2.6). Whenever the bulk S matrix is generated by a local contact term, S grows like a non-negative power of ω . In this situation the integral over ω in (1.6) receives its dominantcontributions from large values of ω . The phase factor e iωP.X cuts off a would-be large ω One reason for this is that our ‘coordinate patch’ (2.1) is more flexible than the ρ coordinate patch (E.2)often employed in CFT studies. As we explain in Appendix E, that part of the two-parameter insertionspace (2.1) which lies on the Causally Scattering sheet has no image into time-ordered correlators on the ρ plane (E.2). On the configurations studied in this paper e ρ is real and lies in the interval (0 , . The point τ = θ in the configuration (2.1) maps to e ρ = z = 0 , the branch point of the correlator.As a consequence the functions g CS ( ρ ) and g CR ( e ρ ) are not directly smoothly connected but are moreindirectly related, as explained in this paragraph. In other words it consists of the points X in the embedding space R D, which obey the equations P i .X = 0 for all i together with X = − . See Appendix B.3 for details. – 4 –ivergence in this integral and turns it into a power-law singularity, of the form ρ a , in ρ ;this is the famous bulk point singularity of [30](see Section 4 for details). (1.6) allows one tocompute the coefficient of this singularity; we find that it is proportional to a simple knownfunction of σ times an integral of flat space S matrix over H D − , with the scattering angle θ and scattering polarizations determined in terms of boundary cross ratios and polarizationin a simple way. Our final result for the coefficient of the singularity /ρ a in terms ofthe flat space S matrix, presented in (4.69), is a generalization of the results of [31] to thestudy of holographic correlators of arbitrary spin, and reduces to the results of [31] in thespecial case that all particles have spin-zero.Recall that (1.5) applies when σ is small while (1.6) applies when ρ is small. We willnow extract information from the fact that these two expressions must hold simultaneouslywhen ρ and σ are both small. When specialize the small ρ discussion ((1.6) and surrounding)to small values of σ it turns out we find the following simple universal result. If the flatspace S matrix S scales at large s but fixed t like S ∝ s A (1.7)then the small σ limit of the ρ → limit of the Greens function - normalized as in (2.21) -is proportional to G ∝ ρ a σ A − (1.8)(1.8) tells us how fast our correlator grows with σ at small ρ . On the other hand (1.5)captures the fastest growth of the correlator at any value of ρ . It follows immediately that A (cid:48) ≥ A (1.9)Recall that the chaos bound theorem of [1] implies (see the Appendix of that paper andsection 5 for a brief review) that the correlator for a well behaved (unitary etc) boundarytheory cannot grow faster than σ in the small σ limit on the causally Regge sheet. As A (cid:48) in (1.5) equals A even on the causally Regge sheet, it follows as a consequence of the chaosbound as A ≤ . (1.10)In words, the flat space S matrix cannot grow faster than s in the Regge limit. Restated,the chaos bound applied to boundary correlators of a unitary theory implies that the bulkdual of that theory obeys the CRG conjecture. When S grows like ω r , a = ∆ + r − . Specifically, θ is identified with the conformal cross ratio σ according sin θ = σ , while the transversepolarizations are identified with the parallel transport or boundary polarizations to the bulk point X on H D − along the unique null geodesic that connects the boundary points P a to X . – 5 – t = τP t = πtP P θ Figure 1 : Insertion points in global AdS
In this paper we study the four point function of boundary operators in a holographicfield theory. Following section 6 of [30], we study correlators of operators inserted at thefollowing two parameter set of boundary points of
AdS D +1 : P = (cos τ, sin τ, , ,(cid:126) P = (cos τ, sin τ, − , ,(cid:126) P = ( − , , − cos θ, − sin θ,(cid:126) P = ( − , , cos θ, sin θ,(cid:126) (2.1)where we parameterize boundary points in AdS D +1 by null rays in embedding spacewith the first two coordinates timelike and the remaining D coordinates space-like (seeAppendix (B) for notations and conventions and a brief review of the embedding spaceformalism), (cid:126) is the zero vector in D − dimensional Euclidean space. Points P and P are inserted at global time τ , and points P and P are inserted at global time π . We willfocus our attention on the range of parameters ≤ τ ≤ π, ≤ θ ≤ π (2.2) The restriction to θ in the range ≤ θ ≤ π rather than ≤ θ ≤ π is a matter of convenience; the flip ↔ maps the smaller to the bigger range. – 6 – .2 Causal relations and cross ratio sheets The causal relations between the points P a are given by τ > π − θ Causally Euclidean ,π − θ > τ > θ Causally Regge ( P > P , and P > P ) τ < θ Causally Scattering ( P , P ) > ( P , P ) (2.3)where we have used the notation A > B to denote that A is in the causal future of B and it is understood that two points are space-like separated with respect to each other iftheir causal ordering is not specified
20 21
Note that the three distinct causal relations (2.3)correspond to three distinct sheets in complex cross ratio space.We end this subsection with an aside. We see from (2.3) that when τ > θ , τ = θ or τ < θ , the pairs of points ( P , P ) and ( P , P ) are, respectively both space-like, nullor time-like separated. The fact that the switch of causal relations for the pairs ( P , P ) and ( P , P ) happens simultaneously indicates that the two parameter set of coordinates(2.1) are rather special especially in the neighbourhood of τ = θ . If we allowed for generalvariations of all four boundary points, in the neighbourhood of the configuration (2.1) with τ = θ we would find points with P > P but ( P , P ) space-like and vice versa. Theconfigurations (2.1) do not include any such causal configurations, and so are non genericin the neighbourhood of τ = θ . See Appendix C for a detailed analysis in the case that τ and θ are both small. See around Fig. 8 of[30]. The argument is as follows. The pairs ( P , P ) and ( P , P ) are each insertedat equal global times; it follows that elements of the same pair are always space-like separated from eachother. The time difference between the insertion points of the second and first pair is π − τ . The angularseparation between and (or and ) is π − θ , and the angular separation between and (or and ) is θ . Using the fact that two points are time-like separated only if the time difference between them exceedstheir angular separation gives (2.3). For complete clarity, all four points are space-like separated from each other in the Euclidean configu-ration, P is in the future light-cone of P , P is the the future light-cone of P while each of the pair P , P is space-like separated with each of the pair P , P in the Regge configuration. On the other hand P and P are space-like separated with respect to each other, P , and P are also space-like separated with respectto each other, but each of P , P lie in the future light-cones of each of P , P in the causally scatteringregions. We use the name Causally Regge because this is the causal relationship between points in the wellstudied Regge limit of CFT on the cross ratio sheet on which the chaos bound constrains the small σ behavior of correlators. – 7 – .3 Conformal cross ratios The conformal cross ratios (see Appendix B.5) associated with these four points are givenby z ¯ z ≡ ( P .P )( P .P )( P .P )( P .P ) = (cos τ − cos θ ) ≡ A (1 − z )(1 − ¯ z ) ≡ ( P .P )( P .P )( P .P )( P .P ) = (cos τ + cos θ ) ⇒ z + ¯ z = 1 − cos τ cos θ ≡ Bz − Bz + A = 0 z = B ± (cid:112) B − Az = 1 − cos ( τ − θ )2 , ¯ z = 1 − cos ( τ + θ )2 , (2.4)Note that both z and ¯ z lie in the interval z ∈ [0 . , ¯ z ∈ [0 , (2.5)As usual we define the cross ratios σ and ρ by the relations z = σe ρ , ¯ z = σe − ρ (2.6)With these definitions σ = (cos θ − cos τ ) ρ = sin θ sin τ (cos θ − cos τ ) (2.7)To end this subsection we specialize (2.4) in the two parametric limits that are ofparticular interest in this paper. τ limit In the small τ limit at fixed θ , (2.4) simplifies to z = sin θ (cid:18) sin θ − τ cos θ (cid:19) + O ( τ )¯ z = sin θ (cid:18) sin θ τ cos θ (cid:19) + O ( τ ) σ = sin θ O ( τ ) ρ = − τ cot θ O ( τ ) (2.8)Note, in particular, that ρ approaches zero (from the negative side) as τ → . See Appendix B.5 for more about various cross ratios. – 8 – .3.2 The Regge limit
On the other hand in the Regge limit τ → , θ → , τθ = a = fixed (2.9)(2.15) simplifies to z = ( θ − τ − i(cid:15) ) θ − a − i(cid:15) ) + O ( θ )¯ z = ( θ + τ + i(cid:15) ) θ a + i(cid:15) ) + O ( θ ) σ = θ (1 − a )16 + O ( θ ) e ρ = (cid:18) − a − i(cid:15) a + i(cid:15) (cid:19) + O ( θ ) (2.10)(in (2.9) we have presented i(cid:15) corrected formulae using the method of subsection 2.5 below).Note that the Regge limit explores the neighbourhood of the boundary point (2.1)obtained by setting τ = θ = 0 , i.e. the point P = (1 , , , ,(cid:126) P = (1 , , − , ,(cid:126) P = ( − , , − , ,(cid:126) P = ( − , , , ,(cid:126) (2.11)In this paper we study only that part of the neighbourhood of this point that we can bereached by turning on small values of θ and τ in (2.1). As an aside we note that this twoparameter set of points do not give a complete cover of the neighbourhood of (2.11) moduloconformal transformations. In addition to the points obtained from (2.1) at small valuesof θ and τ , there are additional infinitesimal deformations of (2.11) in which the insertionpoints enjoy different causal relations from any of those listed in (2.3). This fact (which wewill never use anywhere else in this paper) is explained in some detail in Appendix C. τ and Regge Note that the small θ limit of the small τ limit (2.8) overlaps with the small a limit of theRegge limit (2.10). In particular if we expand the RHS of (2.8) at leading order in θ weobtain σ = θ ρ = − τθ (2.12)On the other hand if we expand (2.10) to leading order in a we obtain σ = θ ρ = − a (2.13)– 9 –t follows from (2.9) that (2.10) and (2.8) are equivalent.Notice that the τ → limit places us in the Causally Scattering regime of (2.3). Onthe other hand, the Regge limit lies in the causally scattering regime when a < but inthe Causally Regge regime for a > . The fact that the Regge limit straddles two distinctcausal regimes will be of central importance to this paper. It is useful to track the evolution of z , ¯ z , σ and ρ as we keep θ fixed and adiabaticallydecrease τ from τ = π to τ = 0 . When τ = π , z = ¯ z = θ and our configuration isEuclidean. As we decrease τ , z decreases while ¯ z increases. ¯ z reaches its maximum value,namely unity when τ = π − θ (i.e. at the boundary between the Causally Euclidean andCausally Regge regimes) . As τ is further decreased, z continues to decrease, but now ¯ z also begins to decrease. Once τ reaches θ (the boundary between the causally Regge andcausally scattering regime) z reaches its minimum value namely zero. As τ is decreasedeven further ¯ z continues to decrease but z now starts to increase. Finally, at τ = 0 we have z = ¯ z = − cos θ . In the previous paragraph, we have described a one parameter path in configurationspace. The evolutions of various conformal cross ratios along this path is depicted in thegraphs Figs 2, 3 and 4 below in which we have displayed graphs of the cross ratios z , ¯ z , σ ρ versus τ (note the arrows in those graphs track the journey from Causally Euclideanconfigurations, starting at τ = π , to Causally Scattering configurations, ending at τ = 0 .) Figure 2 : The evolution of z and ¯ z as τ is decreased from π down to at fixed θ . Note ¯ z touches its maximum value unity at τ = π − θ and z touches its minimum value, z = 0 at τ = θ . It follows (see (B.27)) that the starting point τ = π and the end point τ = 0 of our paths each have ρ = 0 . – 10 – igure 3 : The evolution of the cross ratio σ as τ is decreased from π down to at fixed θ .Note that σ touches its minimum value, σ = 0 at τ = θ , when z vanishes. Figure 4 : The evolution of the cross ratio e ρ = z ¯ z as τ is decreased from π down to atfixed θ . Note that e ρ touches its minimum value, e ρ = 0 at τ = θ , when z vanishes.We end this subsection with a remark. Several investigations of CFT work in theso-called ‘ ρ plane’. In this plane the four CFT operators are inserted in R , at thelocations (E.2). A question that might occur to the reader is the following: how doesthe path described in this subsection (fixed θ , τ lowered from π to zero) map to the ρ plane? This question is addressed in detail in Appendix E. Here we only make a simplequalitative point. In the ρ coordinate system (E.2) the insertion points P and P are fixedand unmoving; in particular, these two points are spacelike separated from each other.It follows immediately that the part of the trajectory of this subsection that lies in theCausally scattering region ( τ < θ ) has no image in the ρ plane. On the other hand, thepart of the trajectory of this subsection that lies on the Causally Euclidean and CausallyRegge sheets has a faithful map onto the ρ plane, as we describe in Appendix E. To forestall confusion we emphasize that in this paragraph and in parts of Appendix E - but nowhereelse in this paper - ρ refers to the insertion coordinate in (E.2) and not to the conformal cross ratio pairedwith σ . – 11 – .5 The same path on the complex cross ratio sheets As conformal correlators have branch points at ¯ z = 1 and z = 0 , and as the trajectoriesdescribed in the previous subsection all ‘touch’ these branch points, it follows that thedescription of the paths presented in the previous subsection is ambiguous and needs tobe improved. The true paths traversed in cross ratio space are given by making thereplacements τ → τ − i(cid:15)τ (see Appendix (D.2)). The i(cid:15) corrected insertion points are P = (cos( τ − i(cid:15)τ ) , sin( τ − i(cid:15)τ ) , , P = (cos( τ − i(cid:15)τ ) , sin( τ − i(cid:15)τ ) , − , P = (cos( π − iπ(cid:15) ) , sin( π − iπ(cid:15) ) , − cos θ, − sin θ ) P = (cos( π − iπ(cid:15) ) , sin( π − iπ(cid:15) ) , cos θ, sin θ ) (2.14)with (cid:15) > . The i(cid:15) corrected cross ratios are given by z ¯ z ≡ ( P .P )( P .P )( P .P )( P .P ) = (cos θ − cos [ τ + i(cid:15) ( π − τ )]) ≡ A (1 − z )(1 − ¯ z ) ≡ ( P .P )( P .P )( P .P )( P .P ) = (cos θ + cos [ τ + i(cid:15) ( π − τ )]) ⇒ z + ¯ z = 1 − cos θ cos [ τ + i(cid:15) ( π − τ )] ≡ Bz − Bz + A = 0 z = B ± (cid:112) B − Az = 12 (1 − cos( θ − τ − i ( π − τ ) (cid:15) )) , ¯ z = 12 (1 − cos( θ + τ + i ( π − τ ) (cid:15) )) ,σ = (cos θ − cos[ τ + i(cid:15) ( π − τ )]) ρ = sin θ sin[ τ + i(cid:15) ( π − τ )] (cos θ − cos[ τ + i(cid:15) ( π − τ )]) (2.15) ¯ z = 1 To examine the route traced by the path of subsection 2.4 in complex cross ratio space, wefirst examine (2.15) in the neighbourhood of ¯ z = 1 by setting τ = π − θ + δτ , assuming δτ = O ( (cid:15) ) and expanding to second order in (cid:15) . We find that the formula for ¯ z reduces to ¯ z − − ( δτ + i ˜ (cid:15) ) | ¯ z − | = δτ + ˜ (cid:15) z −
1) = − π + 2 tan − (cid:18) ˜ (cid:15)δτ (cid:19) , tan − ( x ) ∈ [0 , π ) (2.16) Along the path of the previous subsection, for instance, ¯ z increases to unity and then retreats back. Weneed to know whether the retreat occurs on a different sheet from the onward trajectory, and if so whichone. A similar question arises for the part of the trajectory that touches z = 0 . – 12 –here ˜ (cid:15) = ( π − τ ) (cid:15) It follows immediately from (2.16) that as τ is lowered from just above π − θ to justbelow π − θ , our path in cross ratio space circles around the branch point at ¯ z = 1 ina counter-clockwise manner. The much-studied passage from the Euclidean to standard‘Regge’ behavior - involves traversing precisely the same path in cross ratio space (see e.g.subsection 5.1 of [37] ). It follows that the Causally Regge sheet of (2.3) is the sheetencountered in studies of the CFT in the lightcone or Regge limit (see e.g. [6, 9, 37, 38],i.e. the sheet on which CFT correlators are constrained by the chaos bound of [1]. z = 0 In the similar fashion we examine the behaviour of our path in the neighbourhood of z = 0 by setting τ = θ + δτ . Once again we take O ( δτ ) = O ( (cid:15) ) and work to second order in (cid:15) toobtain z = ( δτ + i ˜ (cid:15) ) | z | = δτ + ˜ (cid:15) z ) = 2 tan − (cid:18) ˜ (cid:15)δτ (cid:19) , tan − ( x ) ∈ [0 , π ) (2.17)It follows that as τ is lowered from just above θ to just below θ , our path circles round thebranch point at z = 0 in a counter-clockwise manner (note ˜ (cid:15) > . In summary, as we lower τ from π to zero at constant θ we move along the trajectories in(complex) cross ratio space depicted schematically in Fig 5 (the vertical scale in these graphsis highly exaggerated; all curves hug the real axis except when ¯ z is in the neighbourhoodof unity or when z is in the neighbourhood of zero). In other words, our path starts on theprincipal or Causally Euclidean sheet when τ = π . As τ is lowered below π − θ our pathcircles around the branch cut at ¯ z = 1 counter-clockwise, bringing us onto the CausallyRegge sheet. As τ is further lowered past θ , the path circles counter-clockwise around thebranch cut at z = 0 , taking us to the Causally Scattering sheet.– 13 – igure 5 : The path traversed in the complex plane by the variables z (purple) and ¯ z (green) as we lower τ from π to at fixed θ . The vertical scale in these graphs is greatlyexaggerated to make them visible. The actual curves should be thought of as hugging thereal axis except in the neighbourhood of the branch points which they circle in the mannershown in this Figure. When do the lightcones emanating out of the four points P , P , P and P have a commonintersection point in the bulk? For τ (cid:54) = 0 the four vectors P , P , P and P span a fourdimensional subspace of R D, and this subspace is metrically R , . It follows from theanalysis of Appendix B.4 that the lightcones that emanate out of these four points haveno common intersection for τ (cid:54) = 0 . When τ = 0 , on the other hand (i.e. at the edge of the parameter range (2.2), i.e.thelimit (2.8) in which ρ → ) the vectors P , P , P and P are linearly dependent. Inparticular, at τ = 0 P + P + P + P = 0 (2.18)In this case these vectors span a three dimensional subspace of R D, . This subspace ismetrically R , . In the language Appendx B.4, (2.1) is a Case 2 configuration of boundarypoints with ˆ Q = 3 . The lightcones that emanate out of these four points intersect on In the language Appendx B.4(2.1) is a Case 1 configuration of boundary points with ˆ Q = 4 . In this generic situation the subalgebra of the conformal algebra that stabilizes the collection of points P , P , P and P - and so the vector space R , of embedding space vectors spanned by P . . . P - is SO ( D − . There are actually two distinct limits of the insertions (2.1) in which ρ → . The first of these is thelimit τ → of interest to this paper. The second is the simpler limit τ → π in which, once again, ρ → .These two limits are physically distinct because they lie on different sheets on the z and ¯ z complex plane. Inthe second simpler limit the correlators lie on the principal or Euclidean sheet (i.e. the Causally Euclideanregion of (2.3)). Correlation functions at nonzero θ are manifestly non-singular in this second ρ → limit(see under equation 6.21 of [30]). This non-singular limit is of no interest to this paper and will never againbe considered here. – 14 –n H D − , and the subalgebra of the conformal algebra that stabilizes the four point is SO ( D − , .In the special case that τ = 0 and θ = 0 (i.e. the Regge limit (2.9)), it is easy to checkthat (2.18) breaks up into two equations (see (2.11)) P + P = 0 P + P = 0 (2.19)It follows that the subspace of R D, spanned by the vectors P , P , P , P in this limit istwo dimensional; infact it is the R , spanned by ( a, , b, , . . . for arbitrary a and b . The space orthogonal to this R , is R D − , spanned by coordinatesof the form (0 , y , , y , y i ) (2.20)and the subgroup of the conformal group that stabilizes the collection of points P , P , P , P is SO ( D − , . Although this fact will have no implications for the main flow of this paper, we noteas an aside that the neighbourhood of the Regge point (2.11) includes points whose spanin embedding space is an R , (in addition to the points obtained from small values of θ and τ in (2.1) whose span in embedding space make up an R , or an R , ); this point isexplained in some detail in Appendix C. In this paper, we will be interested in studying the normalized four-point function of oper-ators inserted at the positions P . . . P defined by G norm = GG G (2.21)where G is the simple four-point function, and G and G are two-point functions. Wewill be particularly interested in the scaling of the correlator (2.21) in the Regge and small τ limits. In the subsequent two sections, we present a study of the four-point function - thenumerator of (2.21) - in these two scaling limits. In this brief subsection, we present themuch simpler analysis of the scaling of the denominator of (2.21) in these limits.To end this section we study the scaling of the two-point function of spin J operatorsinserted at the locations P and P in the Regge and τ → limits. Moving beyond the consideration of the specific boundary configurations studied in this paper, thereader may wonder there exists a general interplay between the R p,q classification of boundary points, theirpossible causal structures and the nature of their cross ratios. The answer to this general question (that hasno application to the current paper) is clearly in the affirmative and we present a preliminary investigationof this point in Appendix B.7. For example we demonstrate that whenever the four boundary points are inan R , configuration (this can happen for a number of different causal relations between the points), the ρ cross ratio corresponding to these points is always imaginary. On the other hand ρ could be either realor imaginary when the four points span an R , . – 15 –s we have reviewed in Appendix B.10, the boundary to boundary two point functionfor a spin J field is given by G ij = C ∆ ,J ( Z i .Z j P i .P j − Z i .P j Z j .P i ) J ( P i .P j ) ∆+ J (2.22)where Z i is the boundary polarization vector for the i th particle (see under (B.15) for adiscussion of boundary tangent vectors in the embedding space formalism) and C ∆ ,J = ( J + ∆ − π d/ (∆ − − h ) (2.23)Note that on the configurations (2.1) P .P = P .P = cos τ − cos θ (2.24) In this limit (2.24) simplifies to P .P = P .P = θ − τ θ − a ) (2.25)Also Z .P = Z . ( P + P ) = θ (cid:16) Z . (0 , a, , − ,(cid:126) (cid:17) + O ( θ ) ≤ O ( θ ) Z .P = Z . ( P + P ) = θ (cid:16) Z . (0 , a, , − ,(cid:126) (cid:17) + O ( θ ) ≤ O ( θ ) Z .Z ≤ O (1) (2.26)where we have used Z .P = Z .P = 0 (see around (B.15)). It follows from (2.26) that ( Z .Z P .P − Z .P Z .P ) J ≤ O ( θ J ) (2.27)Generic polarizations saturate the inequality (2.27). Through this paper and we restrictattention to such correlators; In other words in this paper we restrict attention to genericpolarizations - those that obey ( Z .Z P .P − Z .P Z .P ) J = O ( θ J ) (2.28)(similar comments apply to G .) It follows immediately that in the Regge limit, the two point function G scales like G = 1 σ ∆ G = 1 σ ∆ (2.29) This restriction allows for greater simplicity of presentation and does not really result in loss of gen-erality. Our assumption is maximally violated when the LHS of (2.28) vanishes, in which situation (2.21).See section 5 for a discussion of how the omission of such correlators does not result in a lack of generality. – 16 – .7.2 Small τ limit In the limit in which τ is taken to zero P .P = P .P = 1 − cos θ (2.30)If we now also take θ to be small P .P = P .P = θ (2.31)in agreement with (2.25) at a = 0 .At generic θ we generically have ( Z .Z P .P − Z .P Z .P ) J = O (1) (2.32)When we take θ small, however, we recover the a → limit of (2.28). While at generic θ the G and G are of order unity, at small θ (2.29) applies. σ → Regge limit
Consider a correlation function with four operators O , O , O , O inserted at the points P , P , P , P given in (2.1) In all the concrete formulae presented in this paper we willassume that O i are primary operators of the conformal group SO ( D, that have somedimension ∆ i = w i + J i and transform in the traceless symmetric representation with J i indices of SO ( D ) .In this section we study the correlator described above holographically, and in theRegge limit (2.9), i.e. the limit in which the conformal cross ratios are given by (2.10). Weemphasize that, in the terminology of this paper, the Regge limit straddles the CausallyRegge and Causally Scattering sheets (see the discussion around (2.10)). The discussion inthis section is very similar to that of section 5.2 of [29]; the slight novelty of our presentationis our emphasis on the fact that our analysis applies both to the Causally Regge as wellas the Causally Scattering sheets, and in fact allows for smooth interpolation between thetwo. θ We wish to study a four point function generated by a bulk contact term in the Regge limit.Any such four point function is given by the sum over expressions of the schematic form (cid:90) d D +1 X N ( − P .X + i(cid:15) ) ˜ a ( − P .X + i(cid:15) ) ˜ a ( − P .X + i(cid:15) ) ˜ a ( − P .X + i(cid:15) ) ˜ a (3.1)where N is a numerator function; N = N ( Z i , X ) , Z i are boundary polarization vectors and ˜ a i are positive numbers (not necessarily integers). Bellow we will have use for the symbol B = (cid:88) i ˜ a i . The extension of this discussion to general representations of SO ( D ) seems conceptually straightforwardbut is not considered here as it is notationally cumbersome. – 17 –s we have explained around (2.19), in the strict Regge limit the four points P . . . P span an R , . The space orthogonal to this R , is R D − , spanned by vectors of the form(2.20). It is thus useful to parameterize the general bulk point in this limit as X = ( u + v , y , v − u , y , y i ) (3.2)with − ( y µ ) + uv = 1 (3.3)When u = v = 0 and in the strict Regge limit the points (3.2) are orthogonal to all P a ,and so all denominators in (3.1) vanish when u , v , θ , τ all equal zero. Recall that in theRegge limit θ and τ are of the same order of smallness. It will turn out that the the regionof the integral in u and v that contributes maximally at small σ is u and v of order θ . It isthus useful to expand the integrand in (3.1) in a power series expansion in the four smallvariables ( θ, τ, u, v ) with all treated as being of the same order in smallness. Let ussuppose that the numerator N is of order θ M plus subleading in this expansion. If we thendefine B = ˜2 B − M (3.4)then the integrand of (3.1) scales like θ B in the limit under consideration.At leading order − P .X = u + τ y − P .X = − u + θ y − P .X = v − τ y − P .X = − v − θ y (3.5)where y = (cid:113) y i − uv → (cid:113) y i . By Taylor expanding both the numerator and thedenominator, it is fairly obvious (and we check carefully in Appendix F) that the integrandin (3.1) can be rewritten in the form (cid:88) i N n i ( y i , Z i )( u + τ y + i(cid:15) ) a i ( − u + θ y + i(cid:15) ) a i ( v + τ y + i(cid:15) ) a i ( − v − θ y + i(cid:15) ) a i (3.6)where a im = ˜ a im − n i (3.7)where n i are integers (positive or negative) subject to the inequality (cid:88) i =1 n i ≥ M (3.8)so that (cid:88) i a i ≤ B (3.9) To do this we use (3.3) to solve for y in a power series expansion in uv . The coefficients of the powerseries expansion in θ, τ, u, v are all functions of y i and Z i . y i are the coordinates on the H D − below. – 18 –3.6), (3.7) (3.8) and (3.9) are simply a long-winded way of saying that our originalexpression at small θ is of order θ B − M = θ B (the M arises from the scaling of the nu-merators) and that there are additional power series corrections to this leading scalingbehaviour.We now turn to the bulk integral over u and v . It is convenient to break up this integralinto two regions; the first R being disk of radius (say) A in the u, v plane and the second, R , which is the complement of R . A is a fixed number, independent of θ , and is chosento be smaller than the radius of convergence of the power series expansions for u and v . The point of this split is the following; in order to generate a σ expansion of our result, itis useful to use the Taylor expansion (3.6). However this expansion is only valid at smallenough u , v . , We are allowed to use this expansion only in the region R but not in R .Indeed, the appropriate Taylor expansion of the integrand in R is in a power seriesin θ and τ , but with u , v being treated of order unity, i.e. u and v dependence beingdealt with exactly and not in expansion.Performing this expansion we immediately findthat the integral (3.1) in R is of order unity (in θ , τ smallness or smaller. This will beparametrically smaller than the θ dependence we will obtain from region R , and so theintegral over R will be irrelevant for the determination of the behavior of the integral atleading order in θ , and so will be ignored in the rest of this section.For the integral over R we employ the expansion (3.6) and perform the integral termby term. Upto corrections of order unity or smaller (that come from the fact that theregion of integration in R is bounded) we can perform the integral as follows. We makethe variable change u = θU and v = θV to obtain (cid:88) i θ − ( (cid:80) m ( a im ) − ) N n i ( y µ , Z i ) f { a im } ( a, y , y ) (3.10)where f { α i } ( a, y , y i ) = (cid:90) dU dV ( U + a y + i(cid:15) ) α ( − U + y + i(cid:15) ) α ( V + a y + i(cid:15) ) α ( − V − y + i(cid:15) ) α (3.11)Recall a = τθ . The integration (3.11) is performed over the full real line for U and V ; thecorrection from the finite integration range is of order unity or smaller, as we have explainedabove. In the integral above y = (cid:113) y i . The integrals over in (3.11) are easily evaluatedusing Schwinger parameters; one obtains f { α i } ( a, y , y i ) = C α ,α ,α ,α ( a y + y + i(cid:15) ) α + α − ( a y − y + i(cid:15) ) α + α − C α ,α ,α ,α = Γ ( α + α −
1) Γ ( α + α − α ) Γ ( α ) Γ ( α ) Γ ( α ) (3.12) This expansion has a finite radius of convergence because it arises from the expansion of simple rationalfunctions of u and v . In particular if we illegally used the expansion at large u and v , terms that appear at high enoughorders in this integrand would have divergent integrals over u and v . – 19 –n the small θ limit the dominant term in (3.10) is of order θ B − ∼ σ B − . Thecontribution of this term to the full integral (3.1) is θ − B +2 H ( a, Z i ) (3.13)where H ( a, Z ) = (cid:90) H D − (cid:88) i N i ( y i , Z i ) f i ( a, (cid:113) y i , y ) (3.14)where the summation in (3.14) runs only over those terms for which (cid:80) i a im = 2 B .The integral in (3.14) is taken over the hyperboloid y − y i = 1 , more precisely on itsbranch in which y = + (cid:113) y i , ≤ τ ≤ π (3.15) aH ( a, Z i ) is a function of a as well as the boundary polarizations Z i . In what follows weallow these polarizations to be a function of a , but demand that this function is chosen tobe analytic. For any such choice H ( a, Z i ) = ˜ H ( a ) and our four-point function in the Reggelimit takes the form θ − B +2 ˜ H ( a ) (3.16)While the function ˜ H ( a ) is in general complicated, in this paper, we are concerned onlywith its analytic properties, which are easy to understand. ˜ H ( a ) has two singularities; the power-law ‘bulk point’ singularity at a = 0 and alightcone singularity at a = 1 . In the next section, we study the bulk point singularity ingreat detail. In this section, we focus our attention on the lightcone singularity at a = 1 .The main concern we address in this section is the following. Given that ˜ H ( a ) hasa singularity at a = 1 , one might worry that it is possible for ˜ H ( a ) ∝ θ (1 − a ) . If ˜ H ( a ) behaved in this manner then the effective value of B in (3.13) could be smaller for a > than it is for a < . In this subsection we explain that this cannot happen.Our result follows almost immediately once we analytically continue the function ˜ H ( a ) to complex values of a . The fact that such an analytic continuation is clear from the integralrepresentations (3.14) and (3.12) (see Appendix G for a class of worked examples). It is alsoexpected on general grounds; the continuation to complex a is effectively a continuation tocomplex cross ratios (see (2.10)), and conformal correlators are, of course, famously analyticfunction of cross ratios with branch cuts at lightcone singularities (the branch cut natureof the complex singularity is explicitly displayed in a class of simple examples in AppendixG). Elsewhere the correlators are analytic functions (see Appendix (H) for a review). In embedding space the curve y − y i = 1 has two branches y = ± (cid:113) y i Each of these branches maps into an infinite number of branches in covering space. The branch (3.15) isthe one of relevance to this paper. This is the branch on which the bulk point singularity lies, and is alsothe branch on which the i(cid:15) assignment in (3.11) is correct. – 20 –ndeed the i(cid:15) prescription effectively tells us that we need to perform an analyticcontinuation even to work at physical values of parameters. In particular, we see from(3.12) , the ˜ H ( a ) is actually a function of the combination of variables ˜ a = a + i ˜ (cid:15) (3.17)for an appropriate definition of ˜ (cid:15) > . It is useful to view ˜ H as a function of ˜ a .With this convention, the function ˜ H has its branch cut singularity exactly at ˜ a = 1 . .Physics instructs us to work at real values of a , and so at slightly complex values of ˜ a , moreparticularly to evaluate the function ˜ H (˜ a ) on a contour that passes just above the real axis.Restated, both for a > and a < , ˜ H ( a ) is the restriction of the analytic function ˜ H (˜ a ) on the real axis, with the limit to the real axis being taken from above (i.e. frompositive values of Im(˜ a ) ). It follows that ˜ H ( a ) for a < is related to ˜ H ( a ) for a > via analytic continuation, the continuation being taken in the upper half complex ˜ a plane.Despite the singularity at a = 1 , therefore, it follows that ˜ H ( a ) for a > and a < areanalytic continuations of each other . In particular if ˜ H ( a ) = 0 for all a > then it followsthat ˜ H ( a ) also vanish for all a < , and so in the limit a → . Restated, if H ( a ) does notvanish at small a , it cannot vanish identically all a > . Figure 6 : The path traversed in the complex plane by the cross ratio e ρ as a moves fromgreater to one to less than one. The vertical scale in these graphs are greatly exaggeratedin order to make it visible. The actual curves should be thought of as hugging the real axisexcept in the neighbourhood of the branch point at zero. The path followed by e ρ circlesround the branch point at zero in a counter-clockwise e ρ The discussion above about the analytic properties of the function H ( a ) can also be un-derstood in what may be more familiar terms when worded in terms of cross ratio function And using the fact that y > everywhere on the integration domain in (3.14). The analytic function ˜ H ˜ (cid:15) ( a ) is most conveniently defined to have a branch cut running from a = 1 down to a = 0 . When we take (cid:15) → – 21 – ρ = z ¯ z . Recall from (2.10) that e ρ = (cid:18) − a − i(cid:15) a + i(cid:15) (cid:19) (3.18)Or in other words e ρ = (cid:18) ˜ a −
11 + ˜ a (cid:19) (3.19)with ˜ a given in (3.17). If we ensure that ˜ a has a small positive imaginary part and we takethe real part of ˜ a from greater than one less than one then the variable ζ = e ρ effectivelycircles counter-clockwise round the branch cut at ζ = 0 In other words, the passagefrom a > to a < (roughly speaking a π rotation in the complex a plane) correspondsto winding counter-clockwise around the branch point at zero of the cross ratio variable e ρ (roughly speaking performing a π rotation in the counter clockwise direction in the e ρ plane). The analytic continuation that takes us from a > to a < is simply theanalytic continuation that takes us from just above to just below the branch cut of thecorrelator when viewed as a function of the cross ratio variable e ρ . The impossibilityof the function ˜ H ( a ) being nontrivial for a < but vanishing for a > now follows as inthe previous subsection. Figure 7 : The path traversed in the complex plane by the cross ratio e ρ as a moves fromgreater to one to less than one. The vertical scale in these graphs are greatly exaggeratedin order to make it visible. The actual curves should be thought of as hugging the real axisexcept in the neighbourhood of the branch point at zero. The path followed by e ρ circlesround the branch point at zero in a counter-clockwise To see this note that that as we pass from
Re(˜ a ) > to Re(˜ a < with Im( a ) = (cid:15) > , the argument of ˜ a − changes from to π . It follows, therefore, that when this happens the argument of (˜ a − increasefrom to π . Noting that z = ¯ ze ρ and recalling that ¯ z is fixed and finite at a = 1 we see that this is basically thesame as the fact, established in the previous section, that passing from τ > θ to τ < θ corresponds tocircling counter-clockwise round the branch cut singularity at z = 0 . – 22 –ll the analytic properties described above are explicitly illustrated in Appendix Hin the context of a simple example; the correlator generated by a φ interaction for fouridentical bulk scalars dual to boundary operators of dimension ∆ in Appendix G. In this paper our interest lies mainly in the normalized correlator (2.21) (because thiscorrelator is constrained by the chaos bound). Recall this normalized correlator is onlydefined when ∆ = ∆ and ∆ = ∆ . Defining A (cid:48) = B − ∆ − ∆ (3.20)it follows immediately from (3.16) and (2.29) that the leading behavior of normalized fourpoint function (2.21) in the Regge limit takes the form H ( a ) θ A (cid:48) − (3.21)The fact that (3.21) holds uniformly - i.e. with the same value of A both for a > and a < , i.e. with the same value of A in both the Causally Scattering and the CausallyRegge sheets - is the main result of this section. ρ → ( τ → ) limit As we have mentioned in the introduction, it is possible to show on very general groundsthat the correlation function studied in this paper is an analytic function of the insertionpoints away from τ = 1 (see the previous section) and τ = 0 (see Appendix H following[30]). In this section, we will study the singularity at τ = 0 more closely, and in particular,determine its precise structure including its coefficient. The analysis of this section followssection 6 of [30], and also overlaps with [31]. Recall that every boundary to bulk propagator is a polynomial times − P.X + i(cid:15) ) A (4.1)for some value of A . In this section we adopt the following strategy. We choose an arbitraryfunction f ( a ) with the following properties f A ( a ) = 1 − a A P ( a ) , a (cid:28) P ( a ) = ∞ (cid:88) m =0 c m a m lim a →∞ f A ( a ) = 0 (4.2)– 23 –n words, the function f vanishes at large values of its argument. It tends to unity at a = 0 ,and the deviations from unity take a very particular form. An example of an f functionwith these properties is f A ( a ) = 1 − a A (1 + a ) A (4.3)Given any such f A function we next use the trivial identity x + i(cid:15) ) A = 1 − f A ( xL )( x + i(cid:15) ) A + f A ( xL )( x + i(cid:15) ) A (4.4)to split every propagator into two pieces. At the moment L is an arbitrary constant. Wewill, choose its value so that τ (cid:28) L (cid:28) (4.5)(this is possible in the small τ limit, the regime of interest to this section).A key point is that the first term on the RHS of (4.4) is non-singular at x = 0 ; wecall this the smooth piece. On the other hand, the second term in (4.4) continues to besingular at x = 0 ; indeed this term (which we call the modulated term) is essentiallyindistinguishable from (4.1) for x (cid:28) L .Now bulk contact interaction contribution to any correlator is given by the productof four boundaries to bulk propagators (to the bulk point X ) sewn together by the bulkinteraction and integrated over the bulk. Our strategy is to replace each of the four bulkto boundary propagators by the sum of two terms (the regular part of the propagatorand the singular part of the propagator) in the integrand above using the identity (4.4).The integrand now breaks up into = 16 terms, depending on whether we retain theregular or modulated part of each propagator. Upon performing the integral over X , itfollows immediately from the analysis of Appendix H that the integral of any term in whicheven one of the propagators is replaced by its regular part evaluates to an expression thatis non-singular at τ = 0 . The singularity in our correlator - the object of study of thecurrent section - comes entirely from the integrand with every propagator replaced by thecorresponding modulated propagator.Let us summarize. We have demonstrated that the singular term in the correlation isunaffected if we make the replacement − P.X + i(cid:15) ) A → f A ( − P.XL )( − P.X + i(cid:15) ) A (4.6)in every propagator that appears in the integrand of our correlation functions, i.e. if wereplace every propagator by a modulated propagator.All through this section, we will make the replacement (4.6) on every propagator in ourintegrand. The utility of this maneuver is the following. As the envelope function, f A ( P.X ) decays away from the lightcone of the boundary point P , once we make the replacement(4.6), it follows immediately that the singularity of our correlator receives contributionsonly from an envelope of width L around the common intersections of the lightcones of theboundary points P i , a fact that simplifies our analysis below.– 24 – .2 Wave representation for propagators The next step in our procedure is to express the modulated propagators as a sum overpropagating waves (a sort of Fourier transform. This may be accomplished ([30]) using theidentity − P.X + i(cid:15) ) A = e − iπA A Γ( A ) (cid:90) ∞ dω ω A − e iω ( − P.X + i(cid:15) ) (4.7)It follows that the modulated spin J propagator can be written as G J ( P, Z, X, W ) = C D ∆ ,J e − iπ (∆+ J )2 ∆ Γ(∆ + J ) f ( P.XL ) ( − Z.W P.X + Z.X W.P ) J (cid:90) dω ω ∆+ J − e iω ( − P.X + i(cid:15) ) (4.8)where C D ∆ ,J = ( J + ∆ − π D (∆ − − D + 1) (4.9)(4.8) represents the propagator as a sum over waves of momenta ωP M . These plane wavesare multiplied by the envelope function f polynomial dressings. However these factors (aswell as the curvature of the underlying AdS background) all vary on scales ranging from L to unity. On the other hand the singularity of our correlator will turn out to have itsorigin in ω of order τ (cid:29) L (cid:29) . At distance scales τ the multiplying factors are effectivelyconstant, and AdS D +1 space is effectively flat, so at these scales (4.8) is literally a FourierTransform representation of the propagator.Following [30] we will now explain how the wave representation of propagators, (4.8),can be used to find an expression for the singularity in τ of the correlators. We begin byconsidering very simple situations but building up to more complicated ones. φ correlator in AdS Let us first study the correlation function induced by a φ φ φ φ bulk interaction between4 scalars of dimension ∆ , ∆ , ∆ and ∆ in D = 2 (i.e. 2 boundary or 3 bulk dimensions).We need to evaluate the integral (4.8) is G sing = (cid:32)(cid:89) a C ∆ a e − iπ ∆ a ∆ a Γ(∆ a ) (cid:33) (cid:90) dX (cid:32)(cid:89) a dω a ω ∆ a − a f ∆ a ( P a .X/L ) (cid:33) e − iX. ( (cid:80) a ω a P a ) (4.10)where C ∆ a is short notation for C ∆ a ,J =0 .Recall that there are no bulk points X that solve the equation P i .X = 0 for all i exceptwhen τ = 0 . When τ = 0 , the unique bulk point that obeys this equation in D = 2 (andalso has global time between and π ) is X = X = (0 , , , . (4.11)It follows that when τ (cid:29) L the envelopes around the four lightcones (the lightcones of P , P , P , P ) never intersect and the integral in (4.10) is very small. On the other hand– 25 –hen τ (cid:28) L the four envelope functions overlap in a spacetime region in AdS of ‘size’ L (volume L ) centred around X . We refer to this region as ‘the elevator’.It is useful to set X = X + x (4.12)As only values of x with x less than or of order L lie in the ‘elevator’ (and so contributesignificantly to the integral) and as L (cid:28) , x effectively a tangent space coordinate aboutthe point X in AdS . It follows that the space parameterized by the coordinate x iseffectively flat. Now P MNX = η MN + X M X N (4.13)is the projector in R D, that projects vectors in this space orthogonal to X . Expandingthe equation ( X + x ) = − to first order in x , we conclude that P MNX x M = x N (4.14)Now the integral (4.10) evaluates to G sing = (cid:32)(cid:89) a C ∆ a e − iπ ∆ a ∆ a Γ(∆ a ) (cid:33) (cid:90) (cid:32)(cid:89) a dω a ω ∆ a − a (cid:33) e − i (cid:80) a ω a P a .X (2 π ) ˜ δ ( (cid:88) a k a ) (4.15)where k Ma = ω a ( P X ) MN P Na ˜ δ is a δ function broadened in a Gaussian manner over scale of order δω ∼ L . In terms of the vector P = (cid:88) a ω a P a (4.16)(4.15) can be recast as G sing = (2 π ) (cid:32)(cid:89) a C ∆ a e − iπ ∆ a ∆ a Γ(∆ a ) (cid:33) (cid:90) (cid:32)(cid:89) a dω a ω ∆ a − a (cid:33) e − iP.X ˜ δ (cid:0) ( P X ) MN P N (cid:1) (4.17)The ˜ δ function in (4.17) clicks when ( P X ) MN P N vanishes. This happens on a one-parameter set of ω a . The values of ω a at which this happens is easy to deduce. Expandingto first order in τ we find P + P + P + P = 2 τ X (4.18) The precise form of this function can be systematically evaluated in a saddle point approximation.For our purposes (i.e. leading order computations in the small τ limit) , the only thing about about thisfunction that is relevant is that it integrates to unity, and is nonzero only over a range of of order δL of itsarguments. The counting is the following. The space in which the vectors k Ma varies is the tangent space of thepoint X and so is 3 dimensional. Any four vectors in 3-dimensional space are linearly dependent and soobey an equation of the form ζ a k a = 0 . It follows that the δ function in (4.17) clicks for ω a = ωζ a . Noteall this is true at generic τ , despite the fact that the four 4 dimensional vectors P a are linearly independentunless τ = 0 . – 26 –o that (cid:88) a k a = 0 (4.19)It follows that the ˜ δ function is nonzero if and only if for every value of aω a = ω (4.20) With these values of ω a we find k = ω (1 , , , k = ω (1 , , − , k = − ω (1 , , cos θ, sin θ ) k = − ω (1 , , − cos θ, − sin θ ) (4.21)Now it is easy to check that (cid:90) dω dω dω dω ˜ δ (cid:0) ( P X ) MN P N (cid:1) = −
12 sin θ (cid:90) dω (4.22)(where ω is defined in (4.20)) and so (4.17) reduces to G sing = − π e − iπ ∆2 sin θ (cid:32)(cid:89) a C ∆ a ∆ a Γ(∆ a ) (cid:33) (cid:90) ∞ dωω ∆ − e − iωτ (4.23)where, ∆ = (cid:88) a ∆ a Note that (4.23) would have diverged at large ω in the absence of the phase factor e − iωτ . The phase factor regulates this divergence at ω ∼ τ , converting it into a singularityat τ = 0 . As promised the singularity has its origin in waves with ω ∼ /τ and so is a UV effect at small τ . The precise form of the singularity is G sing = − π e − iπ ∆2 sin θ (cid:32)(cid:89) a C ∆ a ∆ a Γ(∆ a ) (cid:33) (cid:18) − ie − iπ ∆2 Γ(∆ −
3) 1(2 τ ) ∆ − (cid:19) = 4 iπ e − iπ ∆ sin θ (cid:32) Γ(∆ − (cid:89) a C ∆ a ∆ a Γ(∆ a ) (cid:33) τ ) ∆ − (4.24) As the delta function has a spread of order /L , the solution (4.20) also has the same fuzz. This fuzzwill not be important in what follows and we ignore it. The final result (4.24) is correct only at leading order in the limit τ → . The corrections to this answercome from the fact that the ˜ δ function is not a completely sharp but has a width δw . It follows, for instance,that the integrand in (4.23) is actually a power series in ω of the schematic form ω a → ω a + ω a − δω + ω a − δω + . . . As every factor of ω is effectively of order τ while δω is independent of τ , these correction give correctionsto the singularity in (4.24) of order O (1 /τ ∆ +∆ +∆ +∆ − m ) where m is an integer. – 27 – .4 General scalar contact correlator in AdS The generalization of the discussion of the previous subsubsection to the study of thesingularity of a scalar correlator resulting from a more general bulk contact interaction than φ (i.e. some number of derivatives acting on the φ fields) is completely straightforward.The analogue of (4.15) in this case is G sing = (cid:32)(cid:89) a ˜ C ∆ a (cid:90) dω a ω ∆ a − a (cid:33) S ( k a ) e − i (cid:80) a ω a P a .X (2 π ) ˜ δ ( (cid:88) a k a ) (4.25)where, ˜ C ∆ ,J = C ∆ ,J e − iπ (∆+ J )2 ∆ Γ(∆ + J ) and, ˜ C ∆ = ˜ C ∆ , (4.26)and, S ( k a ) is the contact interaction evaluated on the waves k a = ωP a . Evaluating theintegral over ω a , as in the previous subsection, gives the analogue of (4.23) G sing = − π sin θ (cid:32)(cid:89) a ˜ C ∆ a (cid:33) (cid:90) dωω ∆ − S ( ω ) e − iωτ (4.27)where S ( ω ) is now the interaction contact term evaluated on the waves (4.28), i.e. theinvariant transition amplitude of the S matrix generated in flat space by the contact termin question for the scattering momenta (4.28).The vectors on which S ( ω ) is evaluated are (4.21) where they are displayed as momentain the embedding space R , . All the vectors in (4.21), however, are orthogonal to X andso lie in its tangent space. This tangent space consists of the collection of R , vectors withsecond component set to zero. Effectively these vectors lie in the R , obtained by deletingthe second component of all vectors in (4.21). With this convention k = ω (1 , , k = ω (1 , − , k = − ω (1 , cos θ, sin θ ) k = − ω (1 , − cos θ, − sin θ ) (4.28) Note that k and k are 3 momenta with positive energy, and so particles and are initial states. On the other hand the momenta k and k have negative energy, so thatparticles and are the final states. The 3 momenta (4.28) are conserved (this is, of course,a consequence of the delta function in (4.25)). And S ( ω ) in (4.27) is simply the S matrix For complete clarity we reiterate that the first coordinate of each of the vectors here is timelike, thesecond and third coordinates are spacelike. The vectors in (4.28) are obtained from those in (4.21) bydeleting the zero in the second component of all vectors in (4.21). – 28 – for the scattering process with the momenta (4.21). The Mandlestam variables for thisscattering process are given by s ≡ − ( k + k ) = 4 ω ,t ≡ − ( k + k ) = − ω (1 − cos θ ) u ≡ ( k + k ) = − ω (1 + cos θ ) (4.29)From (2.8) we see that sin θ = 2 (cid:112) σ (1 − σ ) so that (4.27) can be rewritten more invari-antly as G sing = − π (cid:112) σ (1 − σ ) (cid:32)(cid:89) a ˜ C ∆ a (cid:33) (cid:90) dωω ∆ − S ( ω ) e − iωτ (4.30)In the case of the scattering of four scalars, the S matrix is a function only of s and t .Let the invariant amplitude as a function of s and t be denoted by T ( s, t ) . It follows that S ( ω ) = iT (cid:0) ω , − ω (1 − cos θ ) (cid:1) (4.31)It follows that (4.27) can be rewritten in terms of the invariant transition amplitude T ( s, t ) generated by the contact interaction as G sing = − i π (cid:112) σ (1 − σ ) (cid:32)(cid:89) a ˜ C ∆ a (cid:33) (cid:90) dωω ∆ − e − iωτ T (cid:0) ω , − ω (1 − cos θ ) (cid:1) (4.32)Let us suppose that the contact term in question is of r th order in spacetime derivatives.Then T (cid:0) ω , − ω (1 − cos θ ) (cid:1) = ω r T (4 , − − cos θ )) (4.33)so that G sing = − π (cid:112) σ (1 − σ ) (cid:32)(cid:89) a ˜ C ∆ a (cid:33) (cid:90) dω ω ∆+ r − e − iωτ ( iT (4 , − − cos θ )))= − π ( iT (4 , − − cos θ ))) (cid:112) σ (1 − σ ) (cid:32)(cid:89) a ˜ C ∆ a (cid:33) (cid:90) dω ω ∆+ r − e − iωτ = − π ( iT (4 , − − cos θ ))) (cid:112) σ (1 − σ ) (cid:32)(cid:89) a ˜ C ∆ a (cid:33) (cid:18) − ie − iπ (∆+ r )2 Γ(∆ + r −
3) 1(2 τ ) ∆+ r − (cid:19) (4.34)which simplifies to, G sing = 2 iπ e − iπ (∆+ r )2 (cid:112) σ (1 − σ ) (cid:32)(cid:89) a ˜ C ∆ a (cid:33) Γ(∆ + r −
3) ( iT (4 , − − cos θ )))(2 τ ) ∆+ r − (4.35)Using the relationship (2.8), (4.35) can be rewritten as Or more precisely the invariant transition amplitude, the S matrix with the momentum conservingdelta function deleted. – 29 – sing = 2 iπ e − iπ (∆+ r )2 ∆+ r − (cid:32)(cid:89) a ˜ C ∆ a (cid:33) Γ(∆ + r − √ − σ (∆+ r − ( iT (4 , − σ )) σ ∆+ r − ( − ρ ) ∆+ r − (4.36)(4.36) is a completely explicit expression for the leading singularity at small ρ of the fourpoint function generated by a particular contact term in AdS in terms of the flat spaceS matrix generated by the same contact term. (4.45) agrees perfectly with the expressionfor the S-matrix in terms of the coefficient of singularity of the CFT four point function asderived using a slightly different method (consisting of producing localized bulk waves bysmearing boundary insertion points) in [31] (see eq (3.37)).As an aside let us highlight an initially puzzling aspect of (4.36). As ∆ is not necessarilyan integer, the correlator (4.36) at fixed σ has a branch cut around ρ = 0 in the complex ρ plane. This is puzzling, and branch cuts in correlation functions in conformal field theoriesare usually associated with operator ordering ambiguities across lightcones; however, thereis no causal significance to the point ρ = 0 (the boundary separation between no two pointschanges from spacelike to timelike across ρ = 0 ). We present a brief discussion of thisquestion in Appendix I. AdS D +1 The new element in this story when
D > is that the boundary lightcones intersect over amanifold ( H D − ) rather than at a point.The integrand in the analogue of the expression (4.10) is now nonzero everywhere in anenvelope around an H D − . To describe this hyperboloid it is useful to choose coordinatesso that X M = ( V , Y , V , V , Y i ) , i = 1 . . . D − (4.37) The hyperboloid over which boundary lightcones intersect is located at V µ = 0 . Pointson the hyperboloid are parameterized by the coordinates Y µ , subject to the relation Y = − . It is useful to proceed as follows. We break up the H D − into little cells of volume V D − . Like the variable L earlier in this section, these cells are chosen to be large comparedto τ but small compared to unity. Restricting ourselves to a particular cell and performingthe integral over x we find G sing = (cid:32)(cid:89) a ˜ C ∆ a (cid:33) (cid:90) (cid:32)(cid:89) a dω a ω ∆ a − a (cid:33) S ( k a ) e i (cid:80) a ω a P a .X (2 π ) ( D +1) ˜ δ D +1 ( (cid:88) a k a ) (4.38)Note that (4.38) is the same as (4.25) except for the replacement ˜ δ (cid:0) ( P X ) MN P N ) (cid:1) → ˜ δ D +1 (cid:0) ( P X ) MN P N ) (cid:1) (4.39) Let us recall that the space spanned by the P a is R , (preserving the symmetry group SO ( D − )when τ (cid:54) = 0 , but is R , (preserving the symmetry group SO ( D − , ) when τ = 0 . The SO ( D − , is the subgroup that rotates the Y µ coordinates into each other in the usual Minkowskian fashion. The SO ( D − rotates the Y i into each other. – 30 –he manipulations leading up (4.32) continue to work, except that the integral over 3 ofthe four ω a variables (see (4.15)) are sufficient to use up only 3 of the D − δ functions onthe RHS of (4.39). The analogue of (4.22) is (2 π ) ( D +1) (cid:90) dω dω dω dω ˜ δ D +1 (cid:0) ( P X ) MN P N (cid:1) = − (2 π ) ( D +1) δ D − (0)2 sin θ (cid:90) dω = − (2 π ) V D − θ (cid:90) dω (4.40)We now need to sum the results (4.40) over all the cells in H D − . Note that thissummation is weighted by the volume of those cells. As the size of each cell is smallcompared to the AdS radius (unity in our units) the summation is well approximated byan integral and (4.32) turns into G sing = − π (cid:16)(cid:81) a ˜ C ∆ a (cid:17)(cid:112) σ (1 − σ ) (cid:90) H D − √ g D − d D − X (cid:90) dωω ∆ − e iωP.X S ( ω ) (4.41)where √ g D − d D − X is the volume element on H D − where S ( k ) is the S matrix (moreprecisely invariant transition amplitude) for four scalars with momenta k = ω (1 , , , ,(cid:126) k = ω (1 , , − , ,(cid:126) k = − ω (1 , , cos θ, sin θ,(cid:126) k = − ω (1 , , − cos θ, − sin θ,(cid:126) (4.42)Note that all the vectors k Ma have vanishing components in all Y M directions. It followsthat the vectors k a lie in the tangent space of the H D − at every value of Y µ . It follows that(4.29) and (4.31) apply without modification to this D + 1 dimensional scattering process.Note, in particular, that in this case S ( k ) is independent of the point Y µ on the hyperboloid.For this reason the integral over the hyperboloid in (4.41) is easily performed (the samewill not be true for scattering of spinning particles, as we will see in the next section). Todo this is is useful to parameterize points on H D − by Y i = sinh ζ ˆ n i , Y = cosh ζ (4.43)where ˆ n i is a unit vector in R D − . It is easily verified that d D − X = dζ sinh D − ζ dω D − . It follows from (4.18) that X.P = − τ cosh ζ The integral over angles produce the volume of the unit D − sphere, Ω D − . It remains toperform the integrals over ζ and ω . While we can perform these integrals in any order, itis easier to perform the (elementary) integral over ω first. Assuming that it is of r th orderin derivatives, we find G sing = i (2 π ) (cid:32)(cid:89) a ˜ C ∆ a (cid:33) Ω D − e − i (∆+ r )2 Γ(∆ + r − ∆+ r − √ − σ (∆+ r − ( iT (4 , − σ )) σ ∆+ r − ( − ρ ) ∆+ r − (cid:90) dζ sinh D − ζ cosh ζ ∆+ r − (4.44)– 31 –he integral over ζ is now simply a finite number. Our final answer is G sing = (cid:34) i (2 π ) N D, ∆ (cid:32)(cid:89) a ˜ C ∆ a (cid:33) Ω D − e − i (∆+ r )2 Γ(∆ + r − ∆+ r − (cid:35) √ − σ (∆+ r − ( iT (4 , − σ )) σ ∆+ r − ( − ρ ) ∆+ r − ,N D, ∆ = (cid:90) ∞ dζ sinh D − ζ cosh ζ ∆+ r − = Γ (cid:0) D − (cid:1) Γ (cid:0) (∆ + r − D ) (cid:1) (cid:0) (∆ + r − (cid:1) , Ω n = 2 π n +12 Γ (cid:0) n +12 (cid:1) (4.45)As in the case D = 2 (4.45) agrees exactly with the expression for the S-matrix in terms ofthe coefficient of singularity of the CFT four point function derived in [31] (see eq (3.37))using slightly different methods. The method of derivation employed in this paper willallow for easy generalization to the scattering of massless spinning particles in the rest ofthis section.As in the previous subsection, it might seem odd to the reader that (4.45) displays abranch cut type singularity at ρ = 0 , a point at which no causal relations get changed. Weinvestigate this question in Appendix I.2. The analysis of the previous subsection goes through when studying photons (or non-abeliangauge bosons) and gravitons with a few extra twists. From the point of view of the analysisof this section, the main difference between gauge bosons, gravitons, and scalars lies in theirboundary to bulk propagators.
Let us denote the propagator of a spin J particle of dimension ∆ by ( G ∆ ) M ...M J A ...A J where A i are the boundary indices and M J are the bulk indices. Then (see Appendix B.9 and [39]) Z A . . . Z A J ( G ∆ ) M ...M J A ...A J W M . . . W M J = C ∆ ,J e − iπJ ∆ ( Z.W P.X − Z.X W.P ) J ( − P.X ) ∆+ J (4.46)where Z A and W M , respectively, are arbitrary boundary and bulk tangent vectors thatobey Z A Z A = W M and the explicit form of C ∆ ,J is given by (4.9). When J = 1 , the bulkgauge field at position X , sourced by ‘current’ with polarization Z M , is given by Z A ( G ∆ ) AM = −C ∆ , ( D − − ∆+1 (cid:18) Z M ( P.X ) ∆ − ( Z.X ) P M ( P.X ) ∆+1 (cid:19) (4.47)where Z A is anHere A is a bulk index which is understood to be projected orthogonal to X A . It isuseful to make the projection more explicit. Let ( P X ) MN ≡ δ MN + X M X N (4.48)denote the projector orthogonal to X . Given any vector ‘field A M let us also define A ⊥ M ≡ ( P X ) MN A N (4.49)– 32 –t follows that the gauge field at the point X is given by Z A ( G ∆ ) AM = −C ∆ , ( D − − ∆+1 (cid:18) Z ⊥ M ( P.X ) ∆ − ( Z.X ) P ⊥ M ( P.X ) ∆+1 (cid:19) (4.50)This expression can further be manipulated to Z A ( G ∆ ) AM = −C ∆ , ( D − − ∆+1 (cid:18)(cid:18) − (cid:19) Z ⊥ M ( P.X ) ∆ + ∇ M (cid:18) Z.X ∆( P.X ) ∆ (cid:19)(cid:19) (4.51)where the AdS D +1 covariant derivative is defined by ∇ M = ( P X ) NM ∂ N . It follows that ( G ∆ ) AM = ( G ) AM + ( G ) AM (4.52)with ( G ) AM = −C ∆ , ( D − − ∆+1 (cid:18) − (cid:19) η AM + X A X M ( P.X ) ∆ , ( G ) AM = ∇ M ( ζ ∆ ) A , ζ A ∆ = −C ∆ , ( D − − ∆+1 X A ∆( P.X ) ∆ , (4.53)It is easily verified (see Appendix B.11) that ∇ A ( G ) AM = 0 (4.54) It follows that (4.52) decomposes the boundary propagator into two pieces, the first ofwhich is conserved (divergenceless) on the boundary while the second of which is a gradientin the bulk. So far we have worked at arbitrary values of the dimension, ∆ of our spin one operator.In this paper, we will focus our attention on the special case ∆ = D − . In this case,the bulk gauge field enjoys invariance under bulk gauge transformations. In this case, the The boundary tangent vector Z A has D independent components, which is the same as the number ofdegrees of freedom in a massive vector field propagating in D + 1 bulk dimensions. (4.51) makes clear that -roughly speaking - the D − components in Z ⊥ parameterize the transverse degrees of freedom while Z.X labels the longitudinal degree of freedom. On the other hand ∇ A (cid:0) G AM (cid:1) = ∇ M ∇ A ζ A = ∇ M (cid:18) P.X ) ∆+1 (cid:19) (4.55) It follows that the contribution of G AM to Witten diagrams yields a term in the boundary correlatorthat is identically conserved. On the other hand the contribution of G AM to Witten diagrams yields aterm in the bulk correlator that is generically non conserved. However this second contribution sometimesvanishes. This happens, for instance, when all bulk interactions happen to be ‘gauge invariant’ - i.e. builtonly out of field strengths of the bulk vector field and their derivatives. For general ∆ there is no reason forbulk interactions to be gauge invariant. The case of ∆ = D − is special and will be dealt with in detailbelow. – 33 –econd terms in (4.51) and (4.52) are pure gauge and can be dropped (as their contributionto any correlator vanishes). It follows in this case that, effectively, Z A ( G D − ) AM = C D − , ( D − D (cid:18) D − D − (cid:19) Z ⊥ M ( − P.X ) D − ( G D − ) AM = C D − , ( D − D (cid:18) D − D − (cid:19) η AM + X A X M ( − P.X ) D − (4.56)The ‘plane wave’ representation of this propagator is Z A ( G D − ) AM = C D − , ( D − D i D − Γ( D − (cid:90) ∞ dω ω D − Z ⊥ M e − iωP.X − (cid:15)ω (4.57) In the case of the spin two propagator, similar manipulations to those performed in theprevious subsubsection yield the analogue of (4.50) Z A Z A ( G ∆ ) A A M M = C ∆ , ( D − − ∆ (cid:32) Z ⊥ M Z ⊥ M ( P.X ) ∆ − ( Z.X )( P ⊥ M Z ⊥ M + Z ⊥ M P ⊥ M )( P.X ) ∆+1 + ( Z.X ) ( P.X ) ∆+2 P ⊥ M P ⊥ M (cid:19) (4.58)and of (4.51) Z A Z A ( G ∆ ) A A M M = C ∆ , ( − ∆ (cid:32) − (cid:0) D − (cid:1) ( X.P ) − ∆ (cid:0) ( X.Z ) ( η M M + X M X M ) − (∆ − Z ⊥ M Z ⊥ M (cid:1) (cid:0) D − (cid:1) (cid:16) ∇ M ξ ⊥ M + ∇ M ξ ⊥ M (cid:17)(cid:17) (4.59)where, ξ ⊥ A = 12(∆ + 1) Z.X ( P.X ) ∆ (cid:18) − Z ⊥ A + 12 Z.XP.X P ⊥ A (cid:19) (4.60)Note that when ∆ = D , the second term in (4.59) is pure gauge. As in the previoussubsection it follows that the propagator can be decomposed into two pieces Z A Z A ( G ) A A M M = C ∆ , ( − ∆ (cid:32) − (cid:0) D − (cid:1) ( X.P ) − ∆ (cid:0) ( X.Z ) ( η M M + X M X M ) − (∆ − Z ⊥ M Z ⊥ M (cid:1) (cid:33) Z A Z A ( G ) A A M M = C ∆ , ( − ∆ (cid:16)(cid:0) D − (cid:1) (cid:16) ∇ M ξ ⊥ M + ∇ M ξ ⊥ M (cid:17)(cid:17) (4.61)It is then easily verified (see Appendix B.11) that for ∆ = D ∇ A Z A ( G D ) A A M M = 0 (4.62)We will focus on the special case ∆ = D . In this case G D is pure gauge and can be ignored.Effectively Z A Z A ( G D ) A A M M = − C ∆ , ( D − − D ( X.P ) D (cid:16) ( X.Z ) ( η M M + X M X M ) − ( D − Z ⊥ M Z ⊥ M (cid:17) (4.63)– 34 –he ‘plane wave’ representation of this propagator is Z A Z A ( G D ) A A M M = C D, i D Γ( D ) (cid:18) ( D − D +1 (cid:19) (cid:90) ∞ dω ω D − (cid:18) Z ⊥ M Z ⊥ M − ( X.Z ) ( η M M + X M X M ) D − (cid:19) e − iωP.X − (cid:15)ω (4.64) It follows immediately from (4.57) and (4.64) that the singular part of correlator involvingboundary scalars, conserved current and conserved stress tensor operators is given by thefollowing generalization of (4.41) G sing = − π (cid:16)(cid:81) a ˜ C ∆ a ,J (cid:17)(cid:112) σ (1 − σ ) (cid:90) H D − √ g D − d D − X (cid:90) dωω ∆ − e iωP.X S X ( ω ) (4.65)with ˜ C ∆ , = C ∆ , e − iπ (∆)2 ∆ Γ(∆) , ˜ C D − , = C D − , ( D − D i D − Γ( D − , ˜ C D, = C D, i D Γ( D ) (cid:18) ( D − D +1 (cid:19) (4.66)for scalars, photons and gravitons respectively and S X is the S matrix for the scattering ofthe waves φ = e ik.x ScalarA M = Z ⊥ M e ik.x V ectorh MN = (cid:18) Z ⊥ M Z ⊥ M − ( X.Z ) ( η M M + X M X M ) D − (cid:19) Graviton (4.67)Here Z M is the boundary polarization of the current or stress tensor operator, and k is themomentum for the appropriate particle listed in (4.42)).Notice that while all four scattering momenta, given by (4.42), are independent of thepoint X , the polarizations listed in (4.67) depend on the point X on the hyperboloid (overwhich the integral in (4.65) is performed) because Z ⊥ M = ( P X ) NM Z N (4.68)(see (4.48) for the definition of the projector ( P X ) ). It follows that the S matrix S X ( ω ) that appears in (4.65) depends on X , and so we cannot trivially perform the integral overthe S D − in (4.65) (as we were able to do in subsection (4.5)).Performing the integral over ω in (4.65) we find the analogue of (4.44): G sing = i (cid:18) π (cid:16) ˜ C ∆ ,J (cid:17) (cid:19) Γ(∆+ r − e − i (∆+ r )2 √ − σ (∆+ r − σ ∆+ r − ρ ∆+ r − (cid:90) d Ω D − dζ sinh D − ζ cosh ζ ∆+ r − (cid:18) S X ( ω ) ω r (cid:19) (4.69)– 35 –here we have assumed, as in the subsection (4.5), that the interaction term is of order r in derivatives so that the S matrix scales with overall energy scale like S ( ω ) ∼ ω r so thatthe quantity ˜ S ( ω ) = (cid:18) S X ( ω ) ω r (cid:19) (4.70)that appears in (4.69) is actually independent of ω . While the formulae presented in the previous subsection are all accurate, the flat spacegraviton polarization that appears in (4.67) is presented in an unusual gauge. In thissubsection, we will explain this fact and also gauge transform to a more standard gauge.A linearized onshell graviton in flat space always obeys the massless onshell condition k = 0 . In addition, the linearized Einstein equations also impose the following conditionon h MN k M h MN − k N h = 0 (4.71)where h = h MM is the trace of h MN .When studying gravitational scattering, it is conventional to work in a Lorentz typegauge in which k M h MN = 0 (4.72)If we impose this condition, it follows immediately from (4.71) that h = 0 (4.73)Let us now turn to the graviton wave presented in (4.67). It is easy to verify that h MN that appears in (4.67) obeys k M h MN = − k N ( X.Z ) D − , h = − X.Z ) D − (4.74)It follows that h MN listed in (4.67) obeys (4.71), even though it does not obey the equations(4.72) or (4.73) individually.It is, of course, possible to gauge transform h MN listed in (4.67) to ensure that it obeys(4.72) (and so, automatically, (4.73)). Under an infinitesimal gauge transformation ˜ h MN = h MN + ζ M k N + ζ N k M (4.75)It follows from (4.74) that the transfrormed ˜ h MN obeys both (4.72) and (4.73) provided k.ζ = ( X.Z ) D − (4.76) Note that the RHS of (4.69) involves an integral over the hyperboloid H D − . ζ and the angles on the D − sphere Ω D − are coordinates on this hyperboloid. d Ω D − dζ sinh D − ζ is the usual volume element onthe hyperboloid. The factor of cosh ζ ∆+ r − in the denominator is a result of doing the ω integral, and is aconsequence of the fact that the phase factor e iP.X breaks the SO ( D − , isometry of the hyperboloid. – 36 –f course (4.76) is one condition on D + 1 variables, and so does not completely determine ζ . If we want a concrete particular formula for the transformed h MN we need to impose D additional conditions; these conditions are arbitrary and can be chosen as per convenience. A physically natural additional gauge condition - the one we will choose to adopt - isto demand that all gravitons polarizations are transverse to the centre of mass momentum k + k = − ( k + k ) . Since we have already demanded that h aMN is transverse to k a , thisadditional requirement requires us only to impose the conditions ( k ) M h MN = ( k ) M h MN = ( k ) M h MN = ( k ) M h MN = 0 (4.77)Despite first appearances, (4.77) imposes D (rather than D + 1 ) conditions on each gravitonas one of each of the four groups of D + 1 equations in (4.77) is automatic from (4.72). If we choose to impose both (4.72) as well as (4.77) on our scattering gravitons, thescattering graviton waves reported in (4.67) are modified to h MN = (cid:18) Z ⊥ M Z ⊥ N − ( X.Z ) ( η MN + X M X N ) D − (cid:19) + ( X.Z ) (cid:0) k M k N + k M k N (cid:1) ( D − k .k − k .Z (cid:0) k N Z ⊥ M + k M Z ⊥ N (cid:1) k .k + k M k N (cid:0) k .Z (cid:1) ( k .k ) h MN = (cid:18) Z ⊥ M Z ⊥ N − ( X.Z ) ( η MN + X M X N ) D − (cid:19) + ( X.Z ) (cid:0) k M k N + k M k N (cid:1) ( D − k .k − k .Z (cid:0) k N Z ⊥ M + k M Z ⊥ N (cid:1) k .k + k M k N (cid:0) k .Z (cid:1) ( k .k ) h MN = (cid:18) Z ⊥ M Z ⊥ N − ( X.Z ) ( η MN + X M X N ) D − (cid:19) + ( X.Z ) (cid:0) k M k N + k M k N (cid:1) ( D − k .k − k .Z (cid:0) k N Z ⊥ M + k M Z ⊥ N (cid:1) k .k + k M k N (cid:0) k .Z (cid:1) ( k .k ) h MN = (cid:18) Z ⊥ M Z ⊥ N − ( X.Z ) ( η MN + X M X N ) D − (cid:19) + ( X.Z ) (cid:0) k M k N + k M k N (cid:1) ( D − k .k − k .Z (cid:0) k N Z ⊥ M + k M Z ⊥ N (cid:1) k .k + k M k N (cid:0) k .Z (cid:1) ( k .k ) (4.78)In final summary, the S matrix that appears in (4.69) is the S matrix for the scalarsand vectors reported in (4.67) and either the gravitons reported in (4.67) or the gravitons The reason for this ambiguity is the following. In D + 1 spacetime dimensions the number of metriccomponents is ( D +2)( D +1)2 while the number of independent graviton polarizations is D ( D − − . Thedifference between these two numbers is D + 1) . If we impose the D + 1 conditions (4.72), we get the oneadditional condition (4.73) free. This still leaves us with D more parameters in h MN than the number ofphysical gravitons. We need to fix these additional parameters by imposing an additional (arbitrary) gaugecondition; these are the D undetermined parameters in ζ . The Maxwell analogue of the discussion of thisfootnote is simply the fact that Lorentz gauge does not completely fix photon polarizations; the remainingambiguity is (cid:15) M → (cid:15) M + k M . For the graviton inserted at the point P , for instance, the condition k M h MN k N is already ensured by(4.72). – 37 –eported in (4.78). The two sets of gravitons are gauge related and have equal S matrices.The advantage of expressions presented in (4.67) is that the polarization for the i th particledoes not refer to any other particle. Its disadvantage is that it appears in an unfamiliargauge. These advantages and disadvantages are reversed in the polarizations (4.78). Thereader is free to choose either of these gauges (or any other) according to her convenience. In (4.69), the small ρ behavior of the four-point function is expressed in terms of the integralover flat space S matrices over an H D − . As we have emphasized above, all the S matricesthat appear in this formula have the same effective scattering momenta, but the effectivescattering polarizations depend on the scattering point X .The Regge scaling of the S matrices S X that appear in (4.69) is not necessarily thesame for all values of X . Recall, however, that the dependence of the S matrix on X is verysimple; it arises entirely through the dependence on the scattering polarizations on X . Thedependence of the S matrix is very simple. In the case of photons, it is a linear function ofpolarizations (separately in the polarization of every particle) while in the case of gravitons,it is a bilinear function of polarizations. It follows from this fact that if the S matrix displaysa certain Regge growth at some point X on the hyperboloid, it must grow at least as fastat all but possibly (measure zero sets of) isolated points on the hyperboloid. Ignoring theseisolated points that make no significant contribution to the integral, it follows that all theS matrices that appear in (4.69) scale in the same manner in the Regge limit.Let us suppose that in the small angle limit, this generic S matrix behaves like ˆ S ∝ θ r − A (4.79)Using the fact that our S matrix scales with energy like ω r , and that in the small anglelimit t ∼ θ ω , it follows from (4.79) that the Regge (i.e. fixed t ) scaling of S is S ∼ ω A ∼ s A (4.80)Plugging this estimate into (4.69) we find that in the small σ limit G sing ∝ σ ∆2 × σ A − (4.81)Note in particular that the scaling (4.81) is independent of r . In applications related to the chaos bound it is useful to study the normalized four pointcorrelator (2.21). Recall that in order for the denominator of (2.21) to be nonvanishing, ∆ = ∆ and ∆ = ∆ . It follows that ∆ = 2∆ + 2∆ Combining (2.29) and (4.81), it follows that G normsing ∝ σ A − (4.82)– 38 – .11 Massive higher spin scattering In this subsection, we have restricted our attention to the study of the massless spin one andspin two fields (dual to the correlators of conserved currents and the stress tensor on theboundary). While the generalization to the study of massive higher spin particles shouldcertainly be possible, such a generalization involves new complications, which will be brieflyoutlined in this subsection (and discuss in much greater detail in Appendix J).Consider the case of a massive vector field. Because the bulk interactions of such a fieldare not necessarily ‘gauge invariant’, in this case we are forced to deal with the full propa-gator (4.51) rather than a simplified propagator analogous to (4.56). As a consequence, thescattering modes include an additional polarization; a longitudinal polarization (which ispure gauge in the massless theory). In equations, at leading order, the wave representationof his propagator, tells us that the effective scattering wave of the massive spin one particleis given by A iM = (cid:15) iM e ik i .x (cid:15) iM = (cid:18) − i (cid:19) ( Z ⊥ i ) M + ik iM ∆ i ( Z i .X ) (4.83)( k iM are as listed in (4.28)). The second term in (4.83) is the new longitudinal polarization.Notice that the coefficient of the longitudinal mode in (4.83) is of order ω , while thatof the transverse mode is of order unity. Recall also that increasing the number of powersof ω in the S matrix increases the degree of singularity in ρ (in formulae like (4.69)). Itfollows that the analog of the formula (4.69) involves multiple terms on the RHS withdifferent inverse powers of ρ . The coefficient of the maximally singular term in ρ is theS matrix involving the maximal number of longitudinal polarizations. The coefficients ofsubleading singularities in ρ include the S matrices of modes with less than the maximalnumber of longitudinal polarizations - but also receive contributions from subleading effectsfrom, e.g., the scattering of the maximal number of longitudinal modes. These subleadingeffects - which presumably involve corrections to the simple flat space S matrix (resultingfrom the fact that the relevant propagators are not precisely plane waves and from thecurvature of AdS D +1 ) - complicate the analysis. While we believe that these complicationsare tractable, and the main result of this paper - namely that scattering amplitudes thatviolate the CRG conjecture lead to correlators that violate the chaos bound - likely alsoholds for these massive modes, the proof of this claim needs more care in these cases, andwe leave this to future work. See Appendix J for more discussion of this case. A (cid:48) ≥ A The normalized correlation function G norm (see (2.21)) under study in this paper is a func-tion of the cross ratios e ρ and σ . In section 3 we demonstrated that to leading order in thesmall σ limit (equivalently the Regge limit, see (2.10)), the normalized four point function– 39 –nder study in this paper takes the form (see (3.21) ) g CS ( e ρ ) σ A (cid:48) − and g CR ( e ρ ) σ A (cid:48) − (5.1)on the Causally Scattering and the Causally Regge sheets respectively. We also demon-strated that g CS ( e ρ ) and g CR ( e ρ ) are both nontrivial (neither of them vanishes identi-cally).In section (4), on the other hand, we computed the expansion of the normalized cor-relator at leading order in the small ρ expansion. Our explicit final result was presented in(4.69). The coefficient of the leading small ρ expansion of is a function of σ . We demon-strated that at leading order in the small σ limit, this coefficient function scales like σ A − (5.2)(see (4.81)) where A characterizes the leading order large s fixed t scaling of the flat spaceS matrix of the corresponding contact term (see (4.80)).From these two results, what can we say about the relationship between the two num-bers A and A (cid:48) ? Clearly, the simplest way the facts reviewed so far in this subsection couldboth be true is if A = A (cid:48) . This equality does not, however, necessarily follow just from thefacts reviewed above . To see this consider the following simple function that obeys allthe properties reviewed above σ A ρ a + 1 σ A (cid:48) ρ a (cid:48) (5.3)with A (cid:48) ≥ A but a ≥ a (cid:48) . At leading order in the small ρ limit this function reduces to σ A ρ a and so scales like σ A as required, whereas at leading order in the small σ limit thisfunction reduces to σ A (cid:48) ρ a (cid:48) and so scales like σ A (cid:48) as also required. Thus the example listedin (5.3) obeys all the properties listed earlier in this section, even though A (cid:48) and A are notnecessarily equal.Note, however, that in the example function above A (cid:48) ≥ A (5.4)Indeed it is obvious that this inequality must always hold. A (cid:48) controls the coefficient ofthe most singular scaling of the Greens function with σ at any possible value of ρ . On theother hand A controls the scaling of the correlator with σ in the small ρ limit. (5.4) thusfollows just from definitions. ≥ A (cid:48) In their celebrated chaos bound paper, the authors of [1] demonstrated that out of timeorder thermal four-point function in a large N theory cannot grow faster with time than e πT t where T is the temperature of the ensemble. In Appendix A of the same paper, the In the context of the current paper we believe this fact is indeed true, i.e. it is indeed the case that A = A (cid:48) . Proving this statement would, however, take more work and is not needed to establish the mainresult of this paper, so we do not attempt to construct a careful argument for this fact here. – 40 –uthors explained that, in the special case of conformal large N field theories, their boundalso constrains the growth of ordinary time-ordered correlators in the Regge limit on theCausally Regge sheet. We present a brief summary of this connection.Consider the large N CFT first in Euclidean space, and consider the insertion of fouroperators in a particular plane. Now consider this theory in ‘angular quantization’, i.e.with the angular coordinate θ of the plane being regarded as Euclidean time and the radialcoordinate r thought of as space. As the angular coordinate θ is periodic with periodicity π , the theory in this quantization is effectively thermal with T = π . Figure 8
With this choice of quantization we study the out of time ordered correlator depictedin Fig 8. In this correlator Operator
III is inserted at θ = 0 and r = 1 . Operator II isinserted at r = x ( x < ) and θ = (cid:15) + iτ . Operator IV is inserted at θ = π and r = 1 .Operator II at r = x < and θ = iτ and operator I at r = 1 and θ = π + (cid:15) + iτ .After dividing by the appropriate normalization factor, the path integral depicted in Fig.8 computes the following operator expectation value (in the Hilbert space obtained fromangular quantization) (cid:104) O O O O (cid:105)(cid:104) O O (cid:105)(cid:104) O O (cid:105) (5.5) The chaos bound theorem of [1] asserts that the correlator (5.5) grows no faster with τ than e τ .As explained in Appendix A of [1], the correlators in (5.5) have a simple representationin the quantization of the same theory in usual Minkowski time (in the plane R , obtainedby starting with the plane R and performing the usual analytic continuation to go to R , ).In the Hilbert space obtained by the usual quantization of Minkowski space, the normalizedversion of the path integral of Fig 8 (or equivalently the operator expression (5.5)) has the The ordering of operators in (5.6) is simply the order in which the operators are inserted along thecontour in Fig 8. Note also that this ordering of operators also follows directly from the time assignmentsof operators in this paragraph, as operators inserted at different Euclidean times have only one consistentordering - that in which Euclidean time increases moving from right to left. In other words, the contourdepicted in 8 is the only one consistent with the time assignments of this paragraph, as path integralcontours are only allowed to move forward in Euclidean time. – 41 –ollowing representation. (cid:10) B ( τ ) O ( x, x ) B − ( τ ) O ( − , − O (1 , B ( τ ) O ( − x, − x ) B − ( τ ) (cid:11) (cid:104) B ( τ ) O ( x, x ) B − ( τ ) B ( τ ) O ( − x, − x ) B − ( τ ) (cid:105) (cid:104) O ( − , − O (1 , (cid:105) (5.6)The locations of operator insertions in (5.6) is given by O ( w, ¯ w ) where w and ¯ w are thelightcone coordinates defined in (E.1). B ( τ ) is the boost operator by rapidity τ . Note thatthe operators in (5.6) are time ordered in Minkowski time (i.e. the earliest insertions arefurthest to the right).The equality of (5.6) and (5.5) is a consequence of the fact that time evolution inanalytically continued angular time (i.e. Rindler time) is simply a boost in Minkowskispace. The change in ordering of operators between (5.5) and (5.6) is a consequence of thefact that angular time and regular time run in opposite directions in the left half of theMinkowskian plane. This has the following implication. Operators O and O which areboth inserted at Rindler time τ − i(cid:15) are respectively inserted at Minkowski time t = ± x sinh( τ − i(cid:15) ) ≈ ± x (sinh τ − i(cid:15) cosh τ ) (5.7)In other words, while O is inserted at Minkowski time x sinh τ − i ˜ (cid:15) and so at a positive valueof Euclidean time, O is inserted at Minkowski time − x sinh τ + i ˜ (cid:15) and so at a negativevalue of Euclidean time. The fact that operator insertions must always be ordered sothat Euclidean time increases from right to left then determines the ordering of operatorinsertions in (5.6).We can simplify (5.6) as follows. Let us assume that the operator O m has weight λ m under boosts. It follows that B ( τ ) O ( x, x ) B − ( τ ) = e λ τ O ( xe − τ , xe τ ) B ( τ ) O ( − x, − x ) B − ( τ ) = e λ τ O ( − xe − τ , − xe τ ) (5.8)Inserting (5.8) into (5.6) and asserting the chaos bound, we find that the expression in thatequation simplifies to (cid:104) O ( e − τ x, e τ x ) O ( − , − O (1 , O ( − e − τ x, − e τ x ) (cid:105)(cid:104) O ( e − τ x, e τ x ) O ( − e − τ x, − e τ x ) (cid:105) (cid:104) O ( − , − O (1 , (cid:105) (5.9)Note that the factors of e λ τ and e λ τ have cancelled between the numerator and denomi-nator.When τ is large enough operator insertions (5.9) lie on the Causally Regge sheet. Theconformal cross ratios associated with the insertions (5.9) may be computed using (E.2); inthe large τ limit we find z = 4 xe − τ , ¯ z = 4 x e − τ , σ = 4 e − τ , e ρ = x (5.10)The chaos bound theorem, which asserts that (5.9) grows no faster with τ than e τ , tells usthat the correlator (5.9) can grow no faster in the small σ limit than σ at any fixed ρ . Inother words the Chaos bound asserts that A (cid:48) ≤ .Although the chaos bound holds only on the Causally Regge sheet, the fact that A (cid:48) isthe same on the Causally Regge and Causally Scattering sheets tells us that A (cid:48) ≤ even onthe Causally Scattering sheet in the situation studied in this paper (i.e. correlators inducedby a local bulk contact term). – 42 – .3 The CRG bound from the chaos bound Putting together the results of the previous two subsections it follows that A ≤ A (cid:48) ≤ , (5.11)Now using (4.69) it is easy to convince oneself that there is always a choice of boundarypolarizations for which any given bulk polarization appears on the scattering H D − . Itfollows that (1.10) must hold for every choice of bulk polarizations. In other words, thechaos bound implies the CRG conjecture. The argument presented earlier in this section made important use of the normalized four-point function. As this object is undefined if either G or G vanish, it may at first seemthat the argument of this paper is restricted to the scattering of particles that are ‘equalin pairs’. This is not the case. Suppose we are given four particles that are created by theHermitian boundary operators O , O , O and O such that (cid:104) O O (cid:105) = (cid:104) O O (cid:105) = 0 . Theargument presented in this paper applies unmodified to the scattering created by the fourcorrelators (cid:104) O O O O (cid:105) , (cid:104) O O O O (cid:105) , (cid:104) O O O O (cid:105) , (cid:104) O O O O (cid:105) and (cid:104) O O O O (cid:105) . Thistells us that the scattering processes → , → , → , → all obey the CRG conjecture. We can now consider the argument of this paper to thecorrelator (cid:104) ( O + αO )( O + αO ) O O (cid:105) . The argument of this paper tells us that a linearcombination of the scattering processes → , → , → obey the CRG conjecture. But as we already know the first two processes obey this con-jecture, this tells us that the same is also true for → Similar arguments establish that all scattering processes in which only one of the productparticles is different from one of the reactant particles obey the CRG conjecture. Finally,we study the correlator (cid:104) ( O + αO )( O + αO )( O + βO )( O + βO ) (cid:105) . The argumentof this paper tells us that a linear combination of several scattering amplitudes obeys theCRG conjecture. But we have already established that all of these amplitudes except → obey CRG scaling. It thus follows that this last scattering amplitude also obeys the CRGconjecture . Similar arguments have been used in [40] to constrain the growth of OTOC of mixed correlators usingchaos bound. – 43 –
Discussion and conclusions
In this paper, we have demonstrated the following. Consider a conformal field theory thathas a local bulk AdS/CFT dual. If the contact interaction terms in this bulk dual lead toa flat space S matrix that grows faster than s in the Regge limit, then the correspondingfour-point functions of the boundary conformal field theory violate the chaos bound.We have established the result of the previous paragraph by studying a two-parameterfamily of CFT correlators with insertions at the points (2.1) that interpolate between theCausally Regge sheet and the Causally Scattering sheet (see (2.3)). Our argument proceedsby first demonstrating that on each of these sheets and in the limit that the conformal crossratio σ → , our correlators respectively take the form g CS ( e ρ ) σ A (cid:48) − and g CR ( e ρ ) σ A (cid:48) − (6.1)(here ρ is the second cross ratio). The key point here is that the exponent A (cid:48) is thesame on the Causally Regge and Causally Scattering sheets. We then used the fact thatthe correlator has a singularity - the so-called bulk point singularity - at ρ = 0 on theCausally scattering sheet. The coefficient of this singularity is the flat space S matrix ofthe corresponding bulk modes which we assume to scale like s A at fixed t . This connectionallows us to demonstrate that A (cid:48) ≥ A . However, the chaos bound applied to the CausallyRegge sheet tells us that A (cid:48) ≤ . Putting these results demonstrates that A ≤ , i.e. impliesthe CRG conjecture.The fact the bulk duals to ‘good’ conformal field theories - i.e. CFTs that obey thechaos bound - always obey the CRG conjecture (see the introduction for terminology) seemsto us to be very strong evidence for the correctness of the CRG conjecture, and thereforeof the recent results of [2].While a link between the chaos bound and the allowed Regge scaling of S matrices hasbeen suspected for some time (see e.g. [1, 2, 6, 7], and also Appendix A), to the best ofour knowledge this connection has never previously been made precise, especially in thecontext of the scattering of particles with spin (like the photons and gravitons studied inthis paper).At the technical level, one of the accomplishments of this paper is the generalizationof the results of [31] - which determined the coefficient of the bulk point singularity ofa four-point function of scalar operators in terms of the coefficient of the S matrix ofthe corresponding bulk scalar waves - to a similar result for conserved vectors and theconserved stress tensor (the corresponding S matrices are those of scalars, photons, andgravitons). While the techniques employed in this paper can also be used to study four-point functions of non-conserved spinning operators corresponding to massive spinning bulkparticles, the argument linking the singularity in these correlators to the Regge scaling offlat space S matrices is complicated by the enhanced high energy behavior of the scatteringof longitudinal polarizations in the bulk (see subsection 4.11 and Appendix J). It wouldbe interesting (and should not prove too difficult) to work through these complicationsand generalize the tight connection between the CRG conjecture and the chaos bound toparticles of arbitrary mass and spin. – 44 –s a slight aside from the main flow of this paper, we highlight an issue that we donot understand. To set the context for this question, let us first recall that the weakenedversion of the bulk point singularity (I.13) was conjectured by the authors of [30] to occur insuitable correlators of scalar operators in all non-interacting theories, even at finite N . Oneexplanation for this conjecture, presented in [30], is the observation that the boundary con-figurations that have a bulk point singularity are stabilized by a non-compact SO ( D − , subgroup of the conformal group (see Appendix B.4). Now while boundary configura-tions that have a bulk point singularity are co-dimension in cross ratio space and so arespecial, there exist a less special collection of boundary insertion points for scalar operatorsthat span an R , subspace of embedding space (see Appendix B.4). These points are co-dimension zero or generic in cross ratio space. The subgroup of the conformal symmetrythat is preserved by these R , configurations, SO ( D − , , is also non-compact when D ≥ , suggesting that these correlators are also ill defined (infinite) unless the coefficientof this divergence vanishes for some unknown reason. This conclusion seems to us to beunphysical. It is presumably possible to define the correlator in such ‘ R , configurations’by analytically continuing the answer from the better behaved ‘ R , configurations’ butwe do not understand how such correlators can be computed directly, without resortingto an analytic continuation, even in the simple context of a holographic computation atleading order in large N in a theory with a local bulk dual. As we have mentioned above, inthis context the integral of the integration point over foliation hyperboloids H D − appearsto give an infinite result. While this issue has no bearing on the current paper, it would benice to clear it up.In this paper, we have focussed our attention on correlators generated by contact di-agrams in the AdS bulk. While we have not thought the issue through very carefully, webelieve that the generalization of our paper to the study of bulk exchange diagrams is likelyto be straightforward, and the final result of this paper is likely to apply without modi-fication to this case. More ambitiously, in this paper we have focussed our attention oncorrelation functions generated by classical dynamics in the bulk. However the logical flowof the argument presented in this paper would appear, atleast at first sight, to go througheven accounting for quantum effects in the bulk. As the boundary dual to a quantum bulk If we imagine computing a boundary correlator in conformal perturbation theory, the integration of the‘interaction points’ over the orbit of this symmetry would thus appear give an infinite answer, leading tothe bulk point singularity. While the two parameter set of insertions (2.1) that we have focussed on in this paper are never of thisform - indeed this is part of the reason we chose to study the special set of insertions (2.1) - points withthis property occur in the ‘neighbourhood’ of the Regge point τ = θ = 0 of (2.1) (see Appendix C). Indeedin Appendix C it is the set of ‘bulk point singular’ R , configurations that separate those configurationsthat are R , from the ‘good’ R , configurations like all of (2.1) at τ (cid:54) = 0 . In the context of a holographic computation for a classical bulk dual, such configurations do not giverise to a pinch singularity of the form that we get for R , configurations, see around (H.8). The originof this apparent divergence is more elementary; it is simply the infinite volume of the orbits of the H D − foliations of AdS D +1 that are generated by the preserved symmetry subgroup SO ( D − , . We thank J.Penedones for discussions on this point. The fact that conformal blocks are well defined in ‘ R , configurations’ (though they diverge in ‘ R , configurations’ as pointed out in [30]) suggests this should be possible. – 45 –heory is a CFT at finite N , it may be possible to use the ‘finite N chaos bound’ i.e. thefinite N Cauchy Schwarz inequality (which forms the starting point of the large N analysisin [1]) to obtain a stronger bound for the Regge growth of bulk quantum S matrices thanthe s bound we have derived for classical S matrices in our paper. We think this is a veryinteresting direction for future work. While the results of this paper may be taken to be strong evidence for the correctness ofthe CRG conjecture, the argument presented here is very indirect. It relies on the AdS CFT,the flat space limit of
AdS , and a theorem (the chaos bound) on quantum field theories toconstrain the growth of classical bulk scattering amplitudes. It should be possible to givea simple general - possibly classical - argument for the CRG conjecture that does not relyon all these bells and whistles. Given the results of this paper, such an argument couldalso allow one to ‘derive’ the chaos bound directly from the bulk. We think this is a veryinteresting problem for the future. Finally, just as the chaos bound only constrains four-point functions in a conformal fieldtheory, the CRG conjecture only constrains the growth of → scattering amplitudes. Itwould be very interesting to find nontrivial generalizations of both the chaos bound andthe CRG conjecture to high point correlators and multi-particle S matrices. We leave thisto the future. Acknowledgments
We would like to thank Soumangsu Chakraborty for the initial collaboration and A. Gadde,I. Haldar, M Mezei, and S. Stanford for useful discussions. We would like to thank S. Caron-Huot, T. Hartman, S. Kundu, J.Penedones and A.Sinha for comments on a preliminearyversion of this manuscript. Part of this work was presented in ‘Recent Developments inS-matrix theory’ program (code: ICTS/rdst2020/07) in ICTS, Bengaluru, and also in theYITP workshop YITP-W-20-03 on "Strings and Fields 2020". The work of all authors wassupported by the Infosys Endowment for the study of the Quantum Structure of Spacetimeand by the J C Bose Fellowship JCB/2019/000052. We would all also like to acknowledgeour debt to the people of India for their steady support to the study of the basic sciences.
A Discussion of remarks in [1] relevant to the CRG conjecture
The main theorem of the remarkable paper [1] is the starting point of the analysis of thecurrent paper.In a discussion of their results, the authors of [1] also anticipated the connection betweenthe chaos bound and the CRG conjecture (see the last paragraph of Section 3 of [1]).Somewhat confusingly, however, in a separate discussion, (the second paragraph of Section3), the authors of [1] assert that no finite set of higher derivative corrections to Einstein’sequations affect the fact that Einstein gravity saturates the chaos bound. As individual We thank A. Gadde and S. Caron Huot for related discussion. S. Zhiboedov already has some very interesting results on this question. We thank him for discussionson this point. – 46 –igher derivative corrections to Einstein’s equations certainly violate the CRG conjecture(see [2]) this assertion apparently contradicts the connection between the chaos bound andthe CRG conjecture.We believe that the resolution to this apparent contradiction is that the claim thathigher derivative corrections do not modify the chaos scaling of Einstein gravity is incorrect.The authors of [1] appears to have based their claim on the expectation that scatteringamplitudes that involve only spin two particles always scale like s in the Regge limit.As explained around 1.5.2 in [2], while this is correct for pole exchange diagrams in the t channel, it is not correct either for pole exchange diagrams in the s and u channels or forcontact diagrams.All diagrams that violate the ‘spin two implies s ’ intuition (diagrams whose contribu-tions to S matrices potentially grow faster than s in the Regge limit) are polynomials in t .It follows that these diagrams contribute to bulk scattering only at zero impact parameter. Despite this fact, these diagrams contribute to correlators in the Regge limit at genericvalues of the cross ratio ρ and not only at very special values of ρ . This follows formallyfrom the fact that correlators are analytic functions of ρ , and more physically from thespreading of waves between boundary and bulk.In summary, we agree with the expectation (expressed in the last paragraph of Section 3of [1]) for a tight connection between the chaos bound and CRG scaling. Indeed this papermay be thought of as an attempt to establish this connection more clearly. We believe,however, that the atleast naively contradictory claim of universality of the saturation of thechaos bound in a class of higher derivative gravitational theories is incorrect.Happily, the remarks about universality were made only in a motivational context in [1];their validity or otherwise does not affect any of the actual conclusions of that remarkablepaper.We thank M. Mezei and D. Stanford for discussion related to this Appendix. B Review of the embedding space formalism
B.1 Definition of
AdS D +1 Through this paper we work with
AdS D +1 using the so called embedding space formalism,within which AdS D +1 is thought of as the ‘unfolding’ or universal cover of the sub-manifold Y.Y ≡ η MN Y M Y N = − (B.1)in the space R D, with line element ds = η MN dY M dY N (B.2)( η MN has eigenvalues ( − , − , . . . . Recall that the transformation from t to impact parameters is, roughly, a Fourier transformation. – 47 – .1.1 Global coordinates For some purposes it is convenient to choose an arbitrary decomposition of R D, as R D, = R , ⊗ R D, Let Y − and Y be Cartesian coordinates on R , and Y a ( a = 1 , . . . , D ) beCartesian coordinates on R D, so the line element on R D, is ds = − ( dY − ) − ( dY ) + dY a (B.4)and the AdS manifold is given by the equation − Y − − Y + Y M = − (B.5)The very natural ‘global AdS’ coordinate system associated with any such split parametrizespoints on AdS D +1 according to the formulae Y − = cosh ζ cos( τ ) Y = cosh ζ sin( τ ) Y a = sinh ζ (cid:126)n a (B.6)where (cid:126)n a is a unit vector on a unit D − sphere The metric in these coordinates is givenby ds = dζ − cosh ζdτ + sinh ζd Ω D − (B.7) B.1.2 Poincare coordinates
The construction of the Poincare Patch begins with the choice of an R , hyperplane (of R D, ) that passes through the origin. We label the two lightlike directions of R , as Y + and Y − . These are chosen so that the metric on R , is ds = − dY + dY − . These conditions fix Y + and Y − : upto the Z ambiguity of interchanging Y + and Y − .One choice for the R , is the space spanned by Y − and Y D ; once we have made thischoice one of the (two possible) choices for the coordinates Y + and Y − are Y + = Y − + Y D , Y + = Y − + Y D . (B.9)With these choices the equation (B.5) and the metric in the embedding space can be rewrit-ten as − Y + Y − + Y µ Y µ = − , ds = − dY + dY − + dX µ dX µ (B.10) The inequivalent ways of making this decomposition are labelled by elements of the D parameter coset SO ( D, SO ( D ) × SO (2) . (B.3). Clearly the set of such hyperplanes are parameterized by the D dimensional coset SO ( D, / ( SO ( D − , × SO (1 , (B.8)It follows that we have D inequivalent Poincare patches. – 48 –here µ is an index in D dimensional Minkowski space with a mostly positive metric.We can obtain an explicit parameterization of the half of space with Y − > by usingthe first of (B.10) to solve for Y + and plugging the solution back into (B.10). In this processit is useful to perform the redefinition Y µ = Y − x µ . Renaming Y − as u we now have ( Y + , Y − , Y µ ) = ( ux + 1 u , u, ux µ ) (B.11)We find ds = duu + u dx µ dx µ (B.12) u is the usual Maldacena (energy scale) coordinate for the Poincare patch metric. Notethat this metric in this coordinate becomes singular when u = Y − = 0 .The region of negative Y − is a second Poincare Patch. This two Poincare patchestogether cover all of the manifold (B.5).It is not difficult to visualize the half of AdS D +1 that has Y − > and so is containedin a single Poincare patch. We work in global coordinates, (B.6) and choose coordinates onthe sphere so that Y D = sinh ζ cos φ . φ is the ‘angle with the Y D axis’. φ = 0 is the ‘northpole’ of the sphere, while φ = π is the ‘south pole’ of the sphere. Note ≤ φ ≤ π . Let usfirst characterize the intersection of the Poincare patch with the boundary of AdS D +1 . Todo this we take limit ζ → ∞ where Y + ∝ (cos τ + cos φ ) , Y − ∝ (cos τ − cos φ ) The intersection of the Poincare patch with the boundary consists of points with φ > | τ | . The region above may be characterized invariantly as follows. Consider the north poleat τ = 0 . . The intersection of the Poincare patch and the boundary is the complementof the past and future boundary lightcones of this point.The description of the previous paragraph applies with very little modification in thebulk as well. The full Poincare patch is the complement of the (past and future) bulklightcones of this distinguished boundary point. Using dY + = − dY − Y − + 2( Y − ) x µ dx µ + dY − x µ x µ . so that − dY + dY − + dX µ dX µ = + dY − Y − + − Y − ) x µ dx µ dY − + ( dY − ) x µ x µ + d ( Y − x µ ) d ( Y − x µ ) And so includes all of the boundary sphere at τ = 0 , progressively less of this sphere (a region aroundthe north pole is excluded) as | τ | increases, and only a neighbourhood around the south pole at | τ | → π . In the labelling of boundary points (B.14) and (B.15)), this is the point Y + = 0 – 49 – .2 General analysis in embedding space While the ‘global’ coordinate system (B.6) is familiar and useful for many purposes, andthe Poincare coordinates above are convenient for other purposes, each of them suffers fromdefects. For example, any choice of global coordinates involves an arbitrary splitting of R D, into R D ⊗ R and so obscures the SO ( D, invariance of our space. Poincare coordinatesshare a similar (and more severe) defect of this sort. They also cover only half of AdS space. It is often convenient not to tie ourselves to any particular coordinate system, but toemploy a more intrinsically geometrical view, regarding (B.5) and (B.4) as the fundamentalcoordinate independent definitions of our space. In this subsection, we describe some detailsof this approach.The geodesic distance, d ( U, V ) between two points U and V on (B.4) is given by cos ( d ( U, V )) =
U.V (B.13)where
U.V is the standard dot product in the embedding space R D, . In particular, U and V are null related (i.e. lie on the same null geodesic) if and only if U.V = 0 .Within the embedding space formalism the boundary of
AdS D +1 is the collection ofnull rays in R D, , i.e. by points P in R D, such that P = 0 (B.14)subject to the equivalence relationship P ∼ λ ( P ) P, λ ( P ) > (B.15)It follows that the tangent space to the boundary point P given by the set of vectors δP orthogonal to P (i.e. P.δP = 0 ) subject to the equivalence relation δP ∼ δP + aP where a is any real number.If we wish to parameterize boundary points by particular null vectors P rather thanequivalence classes (B.15) of such vectors we need to choose a ‘gauge’ - say of the schematicform χ ( P ) = 0 , χ ( P ) ∼ χ ( P ) + P χ (cid:48) ( P ) (B.16)to fix the ambiguity (B.15) (the equivalence relationship in (B.16) follows because χ shouldbe evaluated only at the boundary). The (gauge dependent) one-form field n = dχ, ( n → n + 2 χ (cid:48) P ) (B.17) allows us to define the tangent space of the boundary in the gauge slicing (B.16); theallowed class of δP are those vectors that are orthogonal to the plane generated by n and This formula is an analytic continuation of a familiar fact of Euclidean geometry. Recall that thegeodesic distance between two points on the Euclidean unit sphere is the angle between them. It followsthat the cosine of the geodesic distance is the dot product of the unit vectors from the center of the sphereto the two points. The bracketed equation in (B.17) displays how n transforms under the ‘gauge’ transformations (secondo (B.16)). – 50 – . This two plane is typically an R , - and so δP is constrained to lie in the orthogonal R D − , . Note that unlike n itself this two-plane is gauge invariant under χ (cid:48) shifts (B.16).Any such choice of gauge leads to a metric on the boundary given by ds = ( δP ) (B.18)Under the shift (B.15) induces the shift ds → λ ( P ) ( δP ) (upto terms of third order in infinitesimals which we ignore). It follows that λ in (B.15) isa Weyl factor for an effective Weyl transformation.One example of a gauge choice which fixes the choice of λ and hence of Weyl frame is χ ( P ) = P − + P − . (B.19)With this choice the boundary is parametrized, in (B.6), by the points (cos τ, sin τ, ˆ n ) (where ˆ n is a unit vector on R D ). The boundary metric with this choice and with thesecoordinates is ds − dτ + dω D − , i.e the metric on a unit sphere times time. Note that with this choice of coordinates − P .P = 2 cos( τ − τ ) − n . ˆ n (B.20)Another choice of Weyl frame is χ ( P ) = ( Y − −
1) = 0 (B.21)With this choice the boundary is parameterized, in the coordinates of (B.11) by ( x , , x µ ) The boundary metric with this choice of Weyl frame and these coordinates is dx µ dx µ the usual metric on Minkowski space. With this choice of coordinates − P .P = ( x − x ) (B.22)– 51 – .3 Light-cones emanating out of a boundary point in embedding space Consider the boundary point P . The light sheet that emanates out of the boundary point P is given by the set of points X.P = 0 and X = − . This is a D dimensional nullsub-manifold of AdS D +1 ; the normal one form to this manifold is P . The non-degeneratesections of this manifold - parameterized by the equivalence classes X ∼ X + P - are labeledby the distinct null geodesics that generate this manifold. The tangent along every pointalong each of these geodesics is given by the vector P ; note that this embedding spacevector also lies in the tangent space of AdS D +1 because P.X = 0 .The set of all equivalence classes of vectors in R D, that are orthogonal to a null vector P and are subject to the identification V ∼ V + aP (for any a ) is an R D − , . The space ofgeodesics (the space of equivalence classes of the previous paragraph) is the set of points inthis R D − , that obey the AdS D +1 equation X = − . It follows that the set of geodesicsis the space H D − .The space of polarizations along a geodesic parameterized by the point X in H D − isgiven by the set of vectors (cid:15) such that (cid:15).P = (cid:15).X = 0 , modulo the shifts (cid:15) ∼ (cid:15) + P . Thisspace of polarizations spans an R D − . This R D − can be thought of as the tangent spaceof H D − . Equivalently it is the subspace of the tangent space of P (i.e. vectors that areorthogonal to P modulo shifts of P ) that is orthogonal to the point X .The full H D − family of distinct geodesics ‘meet’ at the boundary point P . It followsthat the set of allowed polarizations at the boundary point P - the vector space of associatedspanned by the full set of the polarizations above over all allowed values of X - is simplythe tangent space of P . B.4 Overlap of boundary lightcones of Q (cid:48) points in embedding space Consider a set of Q (cid:48) boundary points P Ma ( a = 1 . . . Q (cid:48) ) . The overlap of the lightcones ofthese Q (cid:48) points is given by the set of simultaneous solutions to the equations P Ma X M = 0 (B.23)and the AdS D +1 condition X = − . Let us suppose ˜ Q of these equations are linearly independent, i.e. the set of points P Ma spana ˜ Q dimensional subspace. Clearly ˜ Q ≤ D + 2 (the dimensionality of embedding space). If ˜ Q = D + 2 these lightcones never intersect. . For the rest of this subsection we assumethat ˜ Q ≤ D + 1 . The space spanned by the vectors P Ma is either1. R ˜ Q − , R ˜ Q − , The first condition is the Lorentz gauge condition ∂.A = 0 . The second condition asserts that ourpolarization lies in the
AdS D +1 manifold. The last condition is the residual gauge invariance on onshellconfigurations after we have imposed the Lorentz gauge; (cid:15) ∼ (cid:15) + αk . We see this as follows. In this situation the only solution of (B.23) is X M = 0 for all M . But thissolution does not obey X = − . – 52 –. Null with non-degenerate sections R ˜ Q − ,
4. Null with non-degenerate sections R ˜ Q − ,
5. Doubly null with non-degenerate sections R ˜ Q − , .For future convenience we define Q = D − ˜ Q + 1 Note Q ≥ .In case (1) above the the isometry group that stabilizes all P Ma is SO ( Q + 1) . Thespace of solutions to (B.23) is an R Q +1 , . This solution set has no intersection with the AdS equation X = − . It follows that the lightcones of corresponding four points neverintersect.In case (2) above the isometry subgroup that stabilizes the points P Ma is SO ( Q, .The space of solutions of (B.23) is R Q, . The intersection of this space with X = − is H Q , a Q dimensional (Euclidean) hyperboloid.In case (3) above the space of solutions of (B.23) is a null manifold with non-degeneratesections R Q, . None of the solutions obey X = − and so the light cones do not intersect.In case (4) above the space of solutions of (B.23) is a null manifold with spatial sections R Q − , . In this case, the intersection of lightcones is a null sub-manifold, whose non-degenerate sections are an H Q − (the null combination of P Ma is the normal to this sub-manifold).In case (5) above the space of solutions of (B.23) is a doubly null sub-manifold with non-degenerate sections are R Q − , . None of these solutions obey X = − so the light-conesnever intersect in this case.Note that in cases (2) and (4), where the intersections of light-cones is nontrivial, theintersection manifold is a homogeneous space; any point on the intersection manifold canbe mapped to any other by the action of the isometry subgroup that preserves all P Ma .In this subsection, we have presented a complete classification of all possible subspacespreserved spanned by a collection of P a . In the special case of interest to this paper -namely the insertions (2.1), we only encounter Case 1 with ˜ Q = 4 (when τ (cid:54) = 0 ) or Case1 with ˜ Q = 4 (when τ (cid:54) = 0 ). In our exploration of the neighbourhood of the Regge pointwe also encounter Case 2 with ˜ Q = 4 (this happens inside the ellipse depicted in Fig. 9).Nowhere in this paper do we ever encounter Cases 3, 4, or 5. B.5 Conformal cross ratios and intersection lightcones for four boundary points
In this subsection we study the insertion of four boundary points on
AdS D +1 .Any SO ( D, invariant expression in the four points P a that is separate of homogeneityzero in each of the four points (so that the expression is ‘gauge’ invariant under the gaugescalings P a → λ ( P ) P a ) is a conformal cross ratio. In this section we briefly recall thedefinitions of commonly used cross ratios. – 53 – .5.1 u and v The conformal cross ratios u and v are defined by u = ( P .P )( P .P )( P .P )( P .P ) v = ( P .P )( P .P )( P .P )( P .P ) (B.24) B.5.2 σ and ρ The conformal cross ratios σ and ρ are defined by σ = u − σ cosh ρ + σ = v (B.25) B.5.3 z and z bar The conformal cross ratios z and ¯ z are defined by the relations u = z ¯ zv = (1 − z )(1 − ¯ z ) (B.26) B.5.4 z and z bar in terms of σ and ρ It follows that z and ¯ z are the two solutions to the quadratic equation x − xσ cosh ρ + σ = 0 (B.27)The two solutions to this equation are x = σ (cosh ρ ± sinh ρ ) (B.28)In particular our conventions are z = σe ρ , ¯ z = σe − ρ (B.29)Note that z and ¯ z are real and independent when ρ is real (i.e. when cosh ρ > ) butare complex conjugates of each other when ρ is imaginary (i.e. when cosh ρ < ).In the second case it is convenient to set ρ = iφ in terms of which z = σe iφ , ¯ z = σe − iφ (B.30) B.5.5 More symmetric expression for σ and ρ Let us adopt the notation P i .P j = p ij (B.31)Using σ = p p p p , σ − σ cosh ρ = p p p p , (B.32)– 54 –t follows that σ = p p p p , cosh ρ = ( p p − p p − p p ) p p p p , (B.33)or with a slight rearrangement σ = p p p p , sinh ρ = p p + p p + p p − p p p p + p p p p + p p p p )4 p p p p (B.34) For any × matrix p ij Det( p ij ) = p p + p p + p p − p p p p + p p p p + p p p p ) (B.36)and so (B.33) can be rewritten as (see eq. (3.20) of [31]) σ = p p p p , sinh ρ = Det( p ij )4 p p p p (B.37)For the special case of the insertions (2.1) we have − P .P = − P .P = 2 − P .P = − P .P = cos θ − cos τ − P .P = − P .P = − cos τ − cos θ (B.38) Inserting (B.38) into (B.37) reproduces (2.7).
B.6 Classes of cross ratios
As we have seen above, the conformal cross ratios fall into two broad classes. Cross ratiosof Type I are those for which z and ¯ z are complex conjugates of each other. Cross ratios ofType II are those for which z and ¯ z are real and independent of each other. The algebra in the last step is cosh ρ − p p − p p − p p ) − p p p p p p p p = p p + p p + p p − p p p p + p p p p + p p p p )4 p p p p . (B.35)The equality of the first and second lines can be seen by expanding the square in the first line. As a quick qualitative check of (B.38), note that they (together with the fact that P a and P b arespace/time like separated in embedding space if − P a .P b is positive/negative see around (B.13)) are consis-tent with the causal relations (2.3). – 55 –ross ratios of Type I take the form (B.30) with σ and φ both real. For such crossratios the conformal cross ratio σ is real while ρ = iφ is imaginary. In this case sinh ρ = i sin φ, sinh ρ = − sin φ ≤ (B.39)cross ratios of Type II are of four sorts. Type IIa are those for which z and ¯ z are bothpositive. Type IIb are those for which z and ¯ z are both negative. Type IIc are those for z is positive and ¯ z is negative. Type IId is the reverse; those for which z is negative and ¯ z ispositive.In the case of Type IIa configurations, σ > and ρ is real. In this case sinh ρ isreal and positive. The configurations (2.1) are all of Type IIa. In the case of Type IIbconfigurations σ < and ρ is real. Once again, in this case sinh ρ is real and positive.configurations of Type IIc are those for σ = − iα with α > and ρ = ζ + i π with ζ real. Inthis case σ = − α and sinh ρ = − cosh ζ . Finally configurations of Type IId are thosefor which σ = iα with α > and ρ = ζ + i π . Once again in this case σ = − α and sinh ρ = − cosh ζ . Note that sinh ρ is positive (and varies in the range (0 , ∞ ) for configurations of typeIIa and IIb). On the other hand, sinh ρ and has a modulus less than or equal to unity inthe case of configurations of Type I. Finally, for configurations o Type IIc and IId, sinh ρ is negative and of modulus greater than unity. B.7 Lightcones and cross ratios
In this subsection, we will initiate an investigation into the relationship between the causalproperties of the four boundary points, the reality (or otherwise) of ρ and the R p,q classifi-cation of these points (see section B.4).All through this subsection, we work on the manifold (B.1), not on its universal cover.By ‘causal relations’ we mean only the following: given a pair of points, is the separationbetween them spacelike or timelike on the manifold (B.1) (recall this manifold has a compacttime circle)? B.7.1 R , Let us first assume that the vectors P i span an R , . In this case Det( p ij ) - the determinantof the metric in a coordinate system oriented towards P a - is negative. Recall the vectors P a are all null; each of these vectors is either past or future directed. Let us define the variable (cid:15) a = 1 when P a is future directed but (cid:15) a = − when P a is past directed. Then p ab has thesign of − (cid:15) a (cid:15) b . As each P a appears twice in the expression p p p p , it follows that this In the case of configurations of type IIc we could also have set σ = iα with α > and ρ = ζ + − i π . Inthe case of configurations of type IId we could also have set σ = − iα with α > and ρ = ζ + i π . Whichchoice we make is unimportant; in particular σ and sinh ρ are left unaffected. The relationship between this spacelike/ timelike dichotomy and causality in the covering space is thefollowing. If two points are timelike separated in embedding space manifold (B.1) then their pre images arenecessarily timelike separated in the covering space. However, points that are spacelike separated in theembedding space manifold (B.1) may be either spacelike or timelike separated in covering space. – 56 –xpression is positive. We conclude from (B.37) that in this case sinh ρ is negative and so ρ is imaginary.What are the boundary ‘causal’ relations between points in this situation? With thisclarification, two points P a and P b are spacelike separated in embedding space if − P a .P b = (cid:15) a (cid:15) b is positive, and are timelike separated if this quantity is negative.With this terminology, an R , situation is consistent with three distinct ‘causal’ con-figurations (upto permutations of particle labels).• When all (cid:15) a have the same sign all points spacelike separated from each other.• When - say - (cid:15) has a different sign from (cid:15) i ( i = 2 , , ) is a configuration in which P , P and P are mutually spacelike, but are timelike separated from P .• When - say - (cid:15) and (cid:15) have the same sign, but this sign is different from the sign of (cid:15) and (cid:15) , P and P are mutually spacelike, P and P are mutually spacelike, butthe pair ( P , P ) are timelike separated from ( P , P ) We reiterate that as long as the P a span an R , , the cross ratio ρ is imaginary in eachof these causal configurations. As ρ is imaginary, the cross ratios must, in the classificationof the previous subsection, be either of Type 1 ( z and ¯ z complex conjugates of each other)or of Type IIc or Type IId ( z and ¯ z both real, but one positive and the other negative). B.7.2 R , When the four points P a span an R , Det( p ij ) is positive, so it follows from (B.37) that ρ is real if an even number of the pairs of the tuples (12) , (13) (24) (34) are timelikeseparated, but ρ is imaginary if an odd number of these pairs are timelike separated. Inother words the reality or otherwise of ρ is determined by the causal relations betweenpoints. We now turn to examining the possibilities for these causal relations.Note that, in this case, the vectors P a can each (by scaling) be put in the form P a = (cos τ a , sin τ a , cos θ a , sin θ a ) (B.40)in a Cartesian coordinate system with signature R , ( τ a and θ a are both angle valued). Itfollows that − p ab = cos( τ a − τ b ) − cos( θ a − θ b ) Now suppose that all τ a are concentrated around a given point while the four θ a are widelyspread on the circle. Then all − p ab are positive, so all four points are spacelike separatedfrom each other. In this case it follows from (B.37) that ρ is real. In the opposite configu-ration - when all θ a are near to each other and all τ a widely separated - then all points aremutually timelike separated, and ρ is still real.There are many other possible configurations. For instance let the times of P , P , P be clumped around a given point, the time of P be near the opposite end of the time circle This is a ‘causal’ relation similar to the scattering configuration we study in this paper, even thoughthe P a in the scattering configuration span an R , not an R , . – 57 – and all angle separated at distances large compared to the first time differences but smallcompared to π . Then P is timelike separated from all other points - while the rest aremutually spacelike separated. This configuration also has real ρ . (A similar configurationwith the role of τ and θ flipped - with P , P , P mutually timelike separated but allspacelike separated from P - also have real ρ ).Similarly if the times of P and P are near to each other the times of P and P arealso near to each other but the times of each pair are separated by a larger amount, and theangular separation between all particles is large compared to the separation between timeseparations within a pair, but small compared to the time separation between pairs then ( P , P ) are mutually spacelike separated, and ( P , P ) are mutually spacelike separated,but the two pairs are mutually timelike separated. Once again ρ is real in this configuration.Now consider yet another configuration - one in which τ = τ = τ = 0 , τ = a , θ = 0 , θ = a , θ = 2 a and θ = 3 a . In this configuration − p is negative (so P and P are spacelike separated) while all other points are timelike separated. In this configuration ρ is imaginary.Let us summarize. While we have not attempted a complete careful classification ofthe allowed causal configurations for R , , it appears at first sight that these configurationsallow for virtually any causal combinations; some of these causal configurations have real ρ while other have imaginary ρ , as spelled out by the rule enunciated at the beginning ofthis subsubsection.As explained in the previous subsection, configurations with real ρ are either of TypeIIa or IIb, ( z and ¯ z both real and either both positive or both negative). In the case that ρ isimaginary cross ratios must be either of Type 1 ( z and ¯ z complex conjugates of each other)or of Type IIc or Type IId ( z and ¯ z both real, but one positive and the other negative). B.7.3 Null subspaces
Finally if the P i span a three or lower dimensional subspace of R D, or if the four dimen-sional manifold spanned by these vectors is null, Det( p ij ) , and so ρ vanishes. We have notcarefully investigated the question of which causal configurations are consistent with thesenull configurations. We leave it to the interested reader to fill this gap. B.8 Weyl weights and scaling dimensions
Recall that the combination of a conformal diffeomorphism and a compensating Weyl trans-formation leaves the metric invariant. A CFT is a theory that is Weyl covariant. In sucha theory the correlators of a primary operator O A ...A n on a space with metric g AB are thesame as the correlators of a primary operator e wφ O A ...A n on the space with metric e φ g AB .In a schematic equation ( O A ...A n , g AB ) = ( e wφ O A ...A n , e φ g AB ) (B.41)– 58 – We call the real number w the Weyl weight of the primary operator O . Note that if theoperator O A has weight w by this definition then ( O A , g CD ) = ( e wφ O A , e φ g CD )= ⇒ ( g BA O A , g CD ) = ( e wφ g BA O A , e φ g CD )= ⇒ ( g BA O A , g CD ) = ( e ( w +2) φ ( e − φ g BA ) O A , e φ g CD )= ⇒ ( O B , g CD ) = ( e ( w +2) φ O B , e φ g CD ) (B.42)and so it follows that the operator O A has weight w + 2 . The general rule is that raisingthe index of an operator increases its Weyl weight by two.Let us now specialize this discussion to the space R D (or R D − , ). Let X A representCartesian coordinates in this space. and let the metric in X A coordinates be η AB . Wefirst perform a Weyl transformation on this space with a constant φ (and use the variable e φ = λ ) and then perform the coordinate transformation X A = ˜ X A λ . (B.43)After the Weyl transformation, the metric on our space is λ η AB . It follows that after thisWeyl transformation ds = λ η AB dX A dX B = η AB d ˜ X A d ˜ X B It follows that ( O B ...B m A ...A n ( X ) , η AB ) = ( λ w O B ...B m A ...A n ( X ) , λ η AB ) = ( λ w + n − m O B ...B m A ...A n ( λX ) , η AB ) (B.44)The = in this equation means ‘has the same correlation functions as’. In the first equalityin (B.44) we have performed a Weyl transformation with the constant Weyl factor λ . Inthe second equality, we have made the variable change (B.43). Note that if an operator isinserted at the point X in the X coordinate system, it is inserted at the point ˜ X = λX inthe ˜ X coordinate system. This is why the argument of insertions in the third bracket in(B.44) is λX . .Comparing the first and the third brackets, setting and suppressing the metric (as it is η AB on both sides) we conclude in summary that O B ...B M A ...A n ( λX ) = O B ...B m A ...A n ( X ) λ w + n − m (B.45) This definition is chosen to ensure that a scalar operator of Weyl weight w has scaling dimension w .We can see this as follows. Let e φ = λ be a constant. Consider the two point function of a scalar operator O . If λ is small then the action of scaling on the metric reduces the proper distance between the insertionsby a factor of λ , and so increases the 2 point function by a factor of λ where ∆ is the scaling dimensionof the operator. The equality (B.41) thus tells us that λ w − ∆ = 1 so that w = ∆ . We will figure out theconnection between the Weyl weight and the scaling dimension for tensor operators below. This is just a complicated way of saying that g AB = e φ ˜ g AB . We emphasize that in any of the brackets above, the value of the argument is simply the location - inthe coordinate system relevant to that bracket - of the operator insertion – 59 –equality means has the same correlators as) so that the field O A ...A n is of scaling weight ∆ = w + n − m. (B.46)Note, in particular, that raising and lowering indices leaves ∆ invariant (because w , n and m all change in a coordinated manner to leave ∆ invariant).In summary the scaling dimension of a primary operator with n lower indices and m upper indices is its Weyl weight plus n - m .
81 82
In particular, we know that the scalingdimension of a conserved current is D − . It follows that the Weyl weight of the corre-sponding operator with an upper index is D , while the Weyl weight of the correspondingoperator with a lower index is D − . Similarly the Weyl weight of the stress tensor (scalingdimension D ) with both lower indices is D − , with one upper and one lower index D , andwith both upper indices D + 2 .Now let’s say we parameterize boundary points by the real projective coordinates P A ,and compute the correlators of insertions with polarizations Z A . The final expression forthe correlator will be a function of the P A and Z A coordinates of all of the operators. Wewish to find a rule that connects the scaling and Weyl dimensions of all operators withthe homogeneity of the final correlators in its arguments. The appropriate rules are thefollowing:• If an expression scales like λ − ∆ when we replace P i by λP i , the expression has scalingdimension ∆ .• If the RHS scales like λ − w when we make the replacements P iM → λP iM , Z Mi → λZ Mi ,then the i th operator (viewed as an object with lower indices) has Weyl weight w .To see how these rules work it is useful to consider an example. Consider, for instance,a one-form vector field A M of definite scaling dimension and Weyl weight. The two pointfunction of such a field could, for instance, have terms of the form Z .A M ( P ) Z .A N ( P ) = a Z .Z ( − P .P ) w +1 + b Z .P Z .P ( − P .P ) w +2 (B.47)which means A M ( P ) A N ( P ) = a η MN ( − P .P ) w +1 + b ( P ) M ( P ) N ( − P .P ) w +2 (B.48) We can understand this intuitively as follows. Consider the two point function < O µ ...µ m ν ...ν n O α ...α m β ...β n > . If the operator has Dimension ∆ , one possible term in the two point function of this operator is ( η µ α . . . η µ n α n ) (cid:0) η ν β . . . η ν m β m (cid:1) ( η θφ x θ x φ ) ∆ . Under a Weyl transformation O → λ w O , η ab → λ η ab and η ab → η ab λ so it follows that the expressionabove picks up a factor of λ w − − m +2 n . It follows ∆ = w + n − m as deduced above. One example of this rule is provided by the world sheet theory of the bosonic string. In this theory b = b z and c = c zz both have Weyl weight. This is why their scaling dimensions are − and respectively. – 60 –here − P .P = ( P − P ) = ( P − P ) P ( P − P ) Q η P Q and η P Q is the flat metric in embedding space.With the choice of Weyl frame (and coordinate system) (B.11), this two point functionreduces to A µ ( x ) A ν ( x ) = a η µν x w +1)12 + b x µ x ν x w +2)12 (B.49)We see immediately from (B.49) that the operators in question have scaling dimension w + 1 , in agreement with the first rule (see the expressions (B.48) and (B.47)). We nowturn to the Weyl weights. Taking into account the x = ( x − x ) µ ( x − x ) ν η µν , and that x µ = η µν x ν we see that the Weyl scaling of the RHS of (B.49) is e − wφ ( x ) − wφ ( x ) , so that(in order that the RHS be Weyl invariant), the two operators must each have Weyl weight w , in agreement with the second rule (see the expression (B.47)). B.9 Bulk to boundary propagators
Consider the bulk field T ( X, W ) = T M ...M J W M . . . W M J (B.50)where W M is a constant vector field. This field corresponds to a boundary operator O M ...M J of scaling dimension ∆ . The Weyl weight of O M ...M J (note we have taken it to have alllower indices) is, then w = ∆ − J (B.51)In brief subsection we determine the bulk to boundary propagator of this field - up toan overall constant - from general considerations.The bulk to boundary propagator is a function of a boundary vector Z M ( Z A has upperindices), a boundary point P M , a bulk point X M and a bulk polarization vector W M . Thepropagator is a homogeneous polynomial of degree J separately in Z and W . Moreoverwe have seen in the previous subsection that it is of homogeneity ∆ in P . It must alsobe invariant under the shift Z → Z + αP for any α (see the discussion around (B.17)).Finally, it must be invariant under SO ( D, transformations, and so must be made up ofdot products of the four vectors that form the data of this correlator.It follows from the discussion of the last paragraph that the propagator takes the generalform F ( − P.X ) ∆ where F is a polynomial in Z.X , Z.W and
W.P ( P.X ) . The fact that F is of degree J separatelyin Z and W tells us that it is a polynomial of degree J in the two variables ( Z.W ) and (cid:0) Z.X W.PP.X (cid:1) J − m . However this expression must also be invariant under the shift δZ = P .It is easy to check that the unique combination of these two monomials that is invariantunder this shift is ( Z.W − Z.X W.PP.X ) – 61 –t follows that the spin J bulk to boundary propagator is proportional to [39] (cid:0) Z.W − Z.X W.PP.X (cid:1) J ( − P.X ) ∆ = ( Z.W P.X − Z.X W.P ) J e − iπJ ∆ ( − P.X ) ∆+ J (B.52)Even though we did not explicitly put in this requirement, note that our propagator isinvariant under the shift δW = X . It is satisfying, as the bulk gauge field is A M W M . Theinvariance tells us that the gauge field read off in this fashion is automatically orthogonalto the AdS D +1 sub-manifold.Note that the propagator is singular precisely on the light front of the point P , i.e.on the sub-manifold spanned by light rays emanating out of P . As explained above, thegeodesics that make up this light front each have tangent vectors proportional to P M ; i.e.the various light rays each move in the direction P M . B.10 Boundary to boundary correlators
The boundary to boundary correlator is given by the expression (B.52) upon replacing X by P (cid:48) and W by Z (cid:48) . So, in particular, the field theory two point function for spin J operatorslocated at P i , Z i and P j , Z j is equal to [41] G ij = C ∆ ,J ( − Z i .Z j P i .P j + 2 Z i .P j Z j .P i ) J ( − P i .P j ) ∆+ J (B.53)where, C ∆ ,J = ( J + ∆ − π d/ (∆ − − h ) (B.54) B.11 Boundary calculus in embedding space
In the D dimensional boundary CFT, a generic traceless symmetric polynomial tensor isencoded in the embedding space by a ( D + 2) dimensional polynomial. It is a (polynomial)function of the position P and polarisation vector Z subject to the condition P = Z = P · Z = 0 . More specifically, we can encode a traceless symmetric tensor of spin- l , in thefollowing way [41], T ( P, Z ) = T A A A ··· A l Z A Z A · · · Z A l (B.55)Conservation condition of this spin- l tensor implies [41, 42], ( ∂ · D Z ) T ( P, Z ) = 0 ,∂ · D Z = ∂∂P M (cid:18)(cid:18) D − Z · ∂∂Z (cid:19) ∂∂Z M − Z M ∂ ∂Z · ∂Z (cid:19) (B.56)– 62 – Exploration of the neighbourhood of the Regge point
Consider the neighbourhood of the Regge configuration (2.11) P = (cid:18)(cid:18) a (cid:19) , a , (cid:18) − a (cid:19) , a i (cid:19) P = (cid:18)(cid:18) a (cid:19) , a , − (cid:18) − a (cid:19) , a i (cid:19) P = (cid:18) − (cid:18) a (cid:19) , a , − (cid:18) − a (cid:19) , a i (cid:19) P = (cid:18) − (cid:18) a (cid:19) , a , (cid:18) − a (cid:19) , a i (cid:19) (C.1)where a iµ = ( a i , a ji ) are vectors in R D − , ( j = 1 . . . D − ). In the parameterization (C.1)we have fixed the scale symmetry of each P i in a convenient manner (for instance we havefixed the scale symmetry of P by the requirement that the sum of the first and thirdcomponents of P equals ).It follows from (C.1) that − P .P = − ( a + a ) , − P .P = − ( a + a ) (C.2)It follows that if ( a + a ) < then P and P are timelike related on the manifold (B.5)and so also that P > P in the full covering space AdS. If ( a + a ) < , on the other hand,the two points are spacelike separated on the manifold (B.5). They may be either spacelikeor timelike separated on the covering space AdS - we will discover the correct rule below. C.1 Relationship to global coordinates
Let δτ i represent the deviation in global coordinates (B.6) from the τ value of the point P i from the Regge point. Similarly let δθ i represent the deviation in global coordinatesfrom the θ value of the point P i . It follows that the embedding space coordinates of theboundary points in the gauge (B.19) - working to second order in smallness - are P = (cid:18) − δτ , δτ , − ( b i ) + δθ , δθ , b i (cid:19) P = (cid:18) − δτ , δτ , − (cid:18) − ( b i ) + δθ (cid:19) , − δθ , b i (cid:19) P = (cid:18) − (cid:18) − δτ (cid:19) , − δτ , − (cid:18) − ( b i ) + δθ (cid:19) , − δθ , b i (cid:19) P = (cid:18) − (cid:18) − δτ (cid:19) , − δτ , (cid:18) − ( b i ) + δθ (cid:19) , δθ , b i (cid:19) (C.3)– 63 –t follows that (cid:16) a , a , a j (cid:17) = (cid:16) δτ , δθ , b j (cid:17)(cid:16) a , a , a j (cid:17) = (cid:16) δτ , − δθ , b j (cid:17)(cid:16) a , a , a j (cid:17) = (cid:16) − δτ , − δθ , b j (cid:17)(cid:16) a , a , a j (cid:17) = (cid:16) − δτ , δθ , b j (cid:17) (C.4)(The points (C.1) with (C.4) agree with (C.3) upto a scaling for each P M )Below we will find it useful to define α = P + P = a + a , α = P + P = a + a (C.5)Finally note that in the special kinematical configuration (2.1) ( α , α , α j ) = ( δτ, − δθ, α , α , α j ) = ( δτ, δθ, (C.6) C.2 Causal relations
The global time difference (see (B.6)) between points 2 and 1 is π + δτ − δτ . The spatialangular difference is computed from P .P in (C.3) and is given by π − (cid:112) ( δθ − δθ ) + ( b + b ) As two points are timelike/ spacelike separated depending on whether their separation inglobal time is larger or smaller than their angular difference, it follows that P > P iff δτ − δτ < (cid:112) ( δθ − δθ ) + ( b + b ) i . e .P > P iff δτ − δτ < (cid:112) ( δθ − δθ ) + ( b + b ) i . e . (C.7)In other words P > P iff α < (cid:113) ( α i ) P > P iff α < (cid:113) ( α i ) (C.8)As a consistency check on this answer we see from (C.7) that wheneever P and P aretimelike separated on the manifold (B.1) (i.e. when − P .P < i.e. when ( α i ) > ( α ) ) )then P > P in the covering AdS space as expected on general grounds. On the otherhand when P and P are spacelike separated on the manifold (B.1) then they are eitherspacelike or timelike separated in global AdS , depending on whether α is positive ornegative. Identical remarks hold for the points and .On the special configuration (2.1), both conditions (C.8) hold whenever δτ < | δθ | and neither apply if this relation does not hold. This is as we have seen before in the maintext. – 64 – .3 R p,q If all the a i are small (as we assume) it follows that the subspace generated by the fourpoints P i is the R , plus the subspace generated by α and α . The signature of this spaceis determined by the sign of the determinant D = α α − ( α .α ) (C.9)The space is R , if this determinant is negative and R , if the determinant is positive.Note, in particular, that if one or both of α i are timelike then the combination in (C.9)is necessarily negative On the special configuration (C.6), the quantity (C.9) evaluates to − δτ δθ and so is always negative, consistent with the fact that the configuration (2.1) is always R , . C.4 Cross ratios
The conformal cross ratios for these points, at leading order in smallness of the a i is givenby z = 14 (cid:18) − α .α − (cid:113) ( α .α ) − α α (cid:19) ¯ z = 14 (cid:18) − α .α + (cid:113) ( α .α ) − α α (cid:19) sinh ρ = ( α .α ) − α α α α σ = 116 α α (C.10)For the special configuration (C.6) , using (C.6), the above cross ratios evaluates to, z = 14 ( δτ − δθ ) ¯ z = 14 ( δτ + δθ ) e ρ = (cid:18) δθ − δτδθ + δτ (cid:19) σ = 116 (cid:0) δτ − δθ (cid:1) (C.11)in perfect agreement with (2.10). When both vectors are timelike, this follows because the dot product of two timelike vectors - in a spacewith only one timelike direction - is necessarily larger in magnitude than the product of norms of vectors. – 65 – .5 A diagram of the neighbourhood of the Regge point
In order to understand the neighbourhood of the Regge point it is useful to vary α and α keeping α i and α i fixed. Let us define x = α | (cid:126)α | , y = α | (cid:126)α | , θ = (cid:126)α .(cid:126)α | (cid:126)α || (cid:126)α | , w = yx (C.12)Note that | θ | ≤ (C.13)and also that θ = − (and so (C.13) is saturated) on the special configuration (C.6).Using (C.10) we find z = | (cid:126)α || (cid:126)α | (cid:18) ( xy − θ ) − (cid:113) ( θ − xy ) − ( x − y − (cid:19) ¯ z = | (cid:126)α || (cid:126)α | (cid:18) ( xy − θ ) + (cid:113) ( θ − xy ) − ( x − y − (cid:19) z ¯ z = ( xy − θ ) − (cid:113) ( θ − xy ) − ( x − y − xy − θ ) + (cid:113) ( θ − xy ) − ( x − y − (C.14) D in (C.9) is given by | (cid:126)α | | (cid:126)α | times − x + 2 xyθ − y + 1 − θ = − x (cid:0) w − θw + 1 (cid:1) + (cid:0) − θ (cid:1) (C.15)(this is the negative of the quantity in the square root in (C.14)). It follows that D vanisheswhenever x = 1 − θ ( w − θw + 1) (C.16)Note that (cid:0) w − θw + 1 (cid:1) = ( w − θ ) + (cid:0) − θ (cid:1) (C.17)It follows from(C.13) that the RHS of (C.16) is positive for every value of w . This tells usthat (C.16) has a real solution for x solution for every value of w . It follow the curve ρ = 0 is a closed curve that surrounds the origin (infact it is an ellipse in the x , y plane).– 66 – igure 9 : A plot of the causal and R p,q structure of the neighbourhood of the Regge point.The red line in this plot represents the boundary between R , and R , for a typical valueof θ . R , region is represented by the deep blue shaded region. P i > P j , P i ≯ P j meansthat P i is in the causal future of P j or not respectively. The two dashed lines divide theplane into four distinct causal configurations.Note that the precise shape of the ellipse depicted in Fig. 9 depends on the value of θ . The slope of the major axis of the ellipse has the same sign as θ (Fig. 9 is plotted with θ = − . ). At θ = 0 the ellipse becomes a circle, and at positive θ it begins to ‘tilt to theright’ (see the yellow-green ellipse at θ = 0 . plotted in Fig. 10). Focussing on negativevalues of θ for a moment, the thickness of ellipse decreases as θ reduces to its minimumvalue, θ = − . In particular when θ is precisely − the ellipse degenerates to a line withslope -1 (see Fig. 10. ) – 67 – igure 10 : Various colored ellipses (Brown for θ = − , Red for θ = − . , Yellow for θ = − . and Green for θ = 0 . ) represents the contour for ρ = 0 for various θ includinga special case θ = 0 . Blue line represents parameter space for configuration (2.1).Recall that θ = − on the special configuration (C.6) that we have focussed attentionon in the main text of this paper. In fact the special configuration (C.6) lies on the blueline (with slope unity) of Fig 10. The fact that the ellipse degenerates to a line at θ = − explain why the configurations (2.1) are always either R , or R , but never R , . C.6 The sign and ratio of z and ¯ z in this neighbourhood In the previous subsection we have presented a detailed two parameter blow up of theneighbourhood of the Regge point. The small θ and τ limit of (2.1) can also be thought ofas a restricted blow up of the Regge point. In this subsection we will analyze how genericthe configurations (2.1) (more accurately (C.6)) are - upto conformal transformations - inthe neighbourhood of the Regge point.Let us first recall that, as depicted in Fig. 9, the neighbourhood of the Regge pointincludes points with four different causal configurations. The top right quadrant of Fig. 9( x > , y > ) has the same causal structure as the Causally Regge sheet of (2.3). Thebottom left quadrant of the same figure, ( x < , y < ), has the same causal structure asthe Causally Scattering sheet of (2.3). On the other hand the top left ( x < , y > ) andbottom right ( x > , y < ) quadrants of Fig. 9 display causal relations that never occur inour special configurations (2.1). It follows that the special configuration (2.1) lie entirely– 68 – igure 11 : Positivity of z . Figure 12 : Positivity of ¯ z . Figure 13 : In this plot we have shown the positivity of z and ¯ z in the ( x, y ) plane by thedark shaded region. In the light shaded region they are negative. White part is excludedsince z and ¯ z are imaginary in that region.in the top right and bottom left quadrants of Fig 9; the remaining quadrants of that figureare not covered by (2.1).What parts of the top right and bottom left quadrants of Fig. 9 are covered by thesmall θ, τ limits of (2.1) (i.e. by (C.6))? To answer this question we first recall that in thespecial configuration (2.1)• The spanning space of the P a is either R , or R , • z > and ¯ z > (i.e. are of Type IIa, see Appendix B.6).• ≤ z ¯ z ≤ The first point above immediately tells us that points in the interior of the ellipse in Fig.9 are not symmetry related (conformally related) to any of the points (C.6). The seconditemized point above tells us that no point in Fig 9 for which either z or ¯ z is negative isconformally related to any of the points (C.6). .Using (C.14) it is easy to convince oneself that the regions in which z and ¯ z are positiveare the dark shaded regions in Figs 13. In particular z and ¯ z are both positive if and only if xy − θ is positive, and x and y are both either greater than unity or both less than unity. In the language of Appendix B.6, this means that no point of Type IIb, IIc or IId is conformallyequivalent to any of the points (C.6) – 69 –n this case ( xy − θ ) > (cid:113) ( θ − xy ) − ( x − y − > . and so it follows immediatelyfrom (C.14) that in this region z ¯ z < so that the third item above is also met.It follows all points that are dark shaded in the first of Fig. 13 are symmetry relatedto one of the points (C.6). None of the light-shaded points - or the points in the interior ofthe ellipse- are symmetry related to the points (C.6). Figure 14 : The dark shaded represents the region inside the R , elipse and the lightshaded represents the type IId region, where, ¯ z is negative and z is positive.As an aside note that ρ is imaginary at every point in Fig. 9 which either lies insidethe R , ellipse (these are points of Type I in the language of Appendix B.6) or when ¯ z isnegative and z is positive (these are configurations of Type IId in the language of AppendixB.6, and are the shaded regions in Fig 14.) We have carefully checked that in each of theshaded regions in Fig 14, one of the pairs of points P and P are spacelike related on the(compact time) manifold (B.1) while the other pair of points is timelike related on the samemanifold, so that the fact that these points have imaginary ρ is consistent with the analysisof Appendix B.7.2. We re emphasize that points that this is a distinct question from the causal relation of points in thefull covering space
AdS D +1 . Points that are timelike related on the manifold (B.1) are necessarily timelikerelated in the full embedding space, but points that are spacelike related in (B.1) could be either spacelikeor timelike separated in the full embedding space. This dichotomy explains the apparent inconsistencybetween the statements of this paragraph (which refer to the timelike and spacelike separation betweenpoints on the space (B.1)) and causal relations asserted in Fig. 9, which refer to causal relations in the fullcovering AdS D +1 space. – 70 –n summary we conclude that the special configuration (2.1) completely covers theCausally Regge sheet of correlators, gives a partial cover of the Causally Scattering sheet(it covers the part of the neighbourhood of the Regge point contained in the dark shadedpoints in (2.1)). D i(cid:15) in position space D.1 The i(cid:15) prescription
Consider a quantum theory with a Hamiltonian H whose spectrum is bounded from below.A wave function may be evolved either forward or backward in time by the time evolutionoperation | ψ ( t − t ) (cid:105) = e − iH ( t − t ) | ψ ( t ) (cid:105) . (D.1)The expression on the RHS of (D.1) consists of a sum of terms with many different frequen-cies. As the energy of a quantum system is typically unbounded from above, the frequenciesthat appear in (D.1) have no bound, and the expression (D.1) is potentially ill defined. Ifwe are interested in evolving only forward in time then we can improve the situation. (D.1)continues to be well defined under the replacement t → te − i(cid:15) ≈ t − i(cid:15)t (D.2)with (cid:15) > . Once we make this replacement, extremely high energy components of (D.1)are damped out, and (D.1) is well defined. This is the i(cid:15) prescription that is used to givedefinite meaning to expressions that are otherwise ill defined in Lorentzian space. D.2 Relation to Euclidean space
More generally, (D.1) continues to be well defined under the replacement t = e − iα t (cid:48) with t (cid:48) real and < α < π . The variable t (cid:48) at α = π is the so called Euclidean time τ . Inother words t = − iτ (D.3)The evolution of the wave function in Euclidean time is given by | ψ ( τ − τ ) (cid:105) = e − ( τ − τ ) H | ψ ( τ ) (cid:105) . (D.4)As usual, (D.4) is well defined only for τ > τ . (D.4) is, of course, extremely well behavedas it is a sum of exponentially decaying rather than oscillating terms.One point of view that is sometimes useful to take is the following. The wave function(D.1) is defined starting with the manifestly well defined object (D.4) and then making thereplacement τ E = ite − i(cid:15) (D.5)(D.3) and (D.5) together, of course, reduce to the replacement rule (D.2).– 71 – Mapping into the ρ plane In this appendix we attempt to map the configurations studied in this paper into a possiblymore familiar conformal coordinate system - the so called ρ coordinate frame. E.1 The ρ frame Consider the coordinates z and ¯ z in R , defined by w = x − t, ¯ w = x + t (E.1) t is Lorentzian time.
86 87
Now consider the following standard insertions (insertions are specified by ( w, ¯ w ) , shownin Fig. (15)) and their corresponding cross ratios I : ( − ρ, − ¯ ρ ) II : ( ρ, ¯ ρ ) III : (1 , IV : ( − , − z ( ρ ) = z z z z = 4 ρ ( ρ + 1) , ¯ z ( ¯ ρ ) = ¯ z ¯ z ¯ z ¯ z = 4 ¯ ρ ( ¯ ρ + 1) (E.2)These insertions are depicted schematically in the Figure (15). Figure 15 : A schematic of the four insertion points on the ρ plane. We have drawn boththe x and t as well as the w and ¯ w axes on this graph. The values of coordinates are givenin w and ¯ w . w and ¯ w are the standard σ + and σ − configurations of a 2 D CFT. After continuation to Euclidean time, t = − iτ find w = x + iτ, ¯ z = w − iτ ; note that w and ¯ w are now complex conjugates of each other. – 72 –we will consider the i(cid:15) corrected cross ratios shortly).Note that the ρ plane is a double cover of the z plane; similarly the ¯ ρ plane is a doublecover of the ¯ z plane. Any given value of z corresponds to two distinct values of ρ . Infact if ρ is one solution to the equation ρ ( ρ + 1) = z (E.3)then ρ is a second solution to the same equation. Identical remarks apply to ¯ ρ and ¯ z .In what follows we will need the i(cid:15) corrections to (E.2). This is achieved by makingthe following replacements in (E.2) ρ → ( ρ + ¯ ρ )2 − ( ¯ ρ − ρ )2 e − i(cid:15) , ¯ ρ → ( ρ + ¯ ρ )2 + ( ¯ ρ − ρ )2 e − i(cid:15) (E.4)((E.4) simply implements (D.2)). Figure 16 : Specification of
E, R, and TR regions in ( t, x ) plane.From a causal point of view the ( ρ, ¯ ρ ) plane has 16 inequivalent regions. This comesabout as follows. As depicted in Fig. 16, the ρ plane has 3 interesting lightcone lines withslope unity. As the operator II goes through these lines (moving from down to up), it cutsthese lines - the right-moving lightcones of operators III , I and IV . In a similar way theplane has 3 interesting lightcone lines of slope − (again see Fig. 16). As the operator IImoves from bottom to top, it cuts these lines - the lightcones of operators IV , I and III .This grid of lightcones divides the plane into a criss cross of (3 + 1) × (3 + 1) = 16 distinctcausal regions, as depicted in Fig. 16. Except z = 1 , see below – 73 –n this section we will only explore 3 of these causal regions. The first of these is thediamond marked E (for Euclidean) in Fig. 4.33 defined by the conditions < ρ < and < ¯ ρ < . In the region E all operators are spacelike separated with respect to each other.In other words the causal relations between operators in the region E is the same as thatin the Causally Euclidean configurations of (2.3).The second region we consider is the half strip marked R (for Regge) in Fig. 4.33defined by the conditions < ρ < together with ¯ ρ > . The causal relations betweenoperators in this region is the same as that of the Causally Regge configurations in (2.3).Finally, the third region we consider is the half strip marked TR (for Timelike Regge)in Fig. 4.33 defined by the conditions − < ρ < together with ¯ ρ > . The causal relationsbetween operators in this region is in between that of the Causally Regge and the Causallyscattering sheets (it is not identical to that of the Causally Scattering sheet because theoperators III and IV are spacelike separated with respect to each other in the region TR unlike in the Causally Scattering region). E.2 Mapping into the ρ plane: qualitative comments Provided ρ > it from (E.3) that < z < . Similarly whenever ¯ ρ > < ¯ z < . Itfollows that both in the regions E and the region R < z < , < ¯ z < (E.5)This is precisely the range over which the cross ratios in the Causally Euclidean and CausallyRegge regions of (2.1) vary (see (2.5)). So it seems very plausible that the Causally Eu-clidean and Causally Regge regions of (2.1) map into the regions E and R of (E.2).In the region TR , on the other hand we have z < , < ¯ z < (E.6)However the z cross ratio for the insertions (2.1) is never negative. This is another demon-stration of the fact that the Causally Scattering region of (2.1) cannot map into the region TR of (E.2) (this follows more elementarily, of course, from causal considerations, as wehave explained above). In fact the causally scattering region simply has no analogue in thecoordinate charge (E.2). If it is indeed the case that the Causally Euclidean region maps to E while the CausallyRegge region maps to R then it should also be the case that the transition from E to R involves circling counterclockwise around the branch cut at ¯ z = 1 , as was the case for (2.1)(see around (2.16)). It is easy to directly verify that this is indeed the case. Indeed in theneighbourhood of the transition region the Minkowski time is positive which implies that ρ → ρ + i(cid:15), ¯ ρ → ¯ ρ − i(cid:15) A very rough analogy might be the following. If (2.1) is like the ingoing Eddington Finklestein co-ordinate system in a black hole, (E.2) is the analogue of the outgoing Eddington Finklestein coordinatesystem. Just like the two EF coordinates agree on the exterior of the event horizon but continue to differentregions of spacetime for r < r H , the two conformal coordinate patches (2.1) and (E.2) agree on the CausallyEuclidean i.e. E and Causally Regge i.e. R regions, but have different continuations beyond z = 0 . – 74 –n particular ¯ z = 4( ¯ ρ − i(cid:15) )( ¯ ρ − i(cid:15) + 1) , ¯ ρ = ρ e τ (E.7)An analysis very similar to that around (2.16) will convince the reader that the transitionfrom the region ¯ E to ¯ R - which is the transition from ¯ ρ < to ¯ ρ > takes us along a pathin cross ratio space that circles counter clockwise around the branch cut at ¯ z = 1 . E.3 Mapping into the ρ plane : quantitative formulae The quantitative map from (2.1) to (E.2) is obtained by equating the z and ¯ z cross ratiosof the two configurations, i.e. by imposing the equations ρ ( ρ + 1) = 1 − cos( θ − τ )24 ¯ ρ ( ¯ ρ + 1) = 1 − cos( θ + τ )2 (E.8)In the E region the solution to these equations is given by ρ = tan (cid:18) τ − θ (cid:19) ¯ ρ = cot (cid:18) τ + θ (cid:19) (E.9)It is interesting to investigate how the equivalent insertion points in the ρ , ¯ ρ planeevolve as we move along the path in cross ratio space described in (2.4). Recall that thispath is obtained by varying τ from π to at fixed t . As depicted in Fig 17, at τ = π thecorresponding insertion for operator II in the ( ρ, ¯ ρ ) plane starts out at the point on the x axis, x = cot (cid:0) π + θ (cid:1) , t = 0 . This point lies in the E region of Fig 4.33. Figure 17 : The image of the part of the path of subsection 2.4 that has a map in the ρ , ¯ ρ plane. The right black curve depicts the insertion position of opertor II. The trajectory isone of decreasing τ ; it starts at τ = π and ends at τ = θ .– 75 –s τ decreases the insertion point follows the path depicted on the rightmost curve inFig. 17. This path cuts the left-moving lightcone of the operator III at ¯ ρ = 1 , ρ = − sin θ θ .It then moves into the region R of Fig. 4.33. As τ decreases further the path approachesnearer and nearer to the right moving lightcone of operator I, cutting it when τ = θ at thepoint x = t = cot (cid:0) θ (cid:1) .In Fig 18 we depict the image of the analogous trajectory at a smaller value of θ . Notethat the path starts much nearer to the point x = 1 on the x axis and then runs along thelightcone of the origin for a much longer ‘time’, intersecting it only at late times (and fardistances) x = t ≈ θ . Figure 18 : The image of the trajectory of subsection 2.4 - with a small fixed value of θ -onto the ρ plane. E.4 Summary of the mapping
Let us summarize. The point τ = π in (2.1) maps to a point on the x axis ( x =tan (cid:0) π − θ (cid:1) , t = 0) in the ρ plane. Note that the x coordinate of this insertion is lessthan unity. As τ decreases, the our ρ point leaves the x axis, moving to positive values ofthe time t in the ρ plane (see the trajectory of the point II in Fig. 17 ). When τ decreasesto π − θ , the trajectory in the ρ plane cuts the lightcone ¯ ρ = 1 (the red line which is orientedat angle − π to the x axis in Fig 17). As τ further increased the trajectory approaches thelightcone ρ = 0 (the red line oriented at an angle π with the x axis in Fig. 17), hitting it at τ = θ . As we have pointed out, the subsequent evolution of the trajectory of this subsec-tion (the part between τ = θ and τ = 0 , i.e. the part of our trajectory on the ConformallyScattering sheet) has no image in the ρ plane.Let us now return to the part of the fixed θ trajectory of this subsection that lies onthe Causally Regge sheet. As we have seen above this part does have an image on the ρ plane. When we choose the fixed value of θ to be small, the image of this trajectory on the ρ plane is qualitatively (though not quantitatively) similar to a familiar trajectory on this– 76 –lane, namely the path traced out by boosting a point on the x axis, i.e. the trajectoryoften discussed in the study of the Regge or chaos limits. F Regge expansion of bulk integrand
In this brief appendix we check that the integrand in (3.1) can be recast in the form (3.10).The numerator N is a polynomial in Z i and X and so can be expanded in a powerseries in the small quantities. It is useful to use the quantities that appear on the RHS of(3.5) as basis for this expansion, so that the expansion takes the form (cid:88) n i a n ,n ,n ,n ( Z i , y i )( u + τ y ) n ( − u + θ y ) n ( v − τ y ) n ( − v − θ y ) n where M is the smallest homogeneity (i.e. smallest value of n + n + n + n ) that appearsin this expression)In a similar manner each expression in the denominator can also be expanded. Forinstance − P .X + i(cid:15) ) ˜ a = ∞ (cid:88) n =1 E n (F.1)where E n can be takes the form E n = (cid:88) Q n ,n ,n ,n ( Z i , y i )( u + τ y ) ˜ a + n ( − u + θ y ) − n ( v − τ y ) − n ( − v − θ y ) − n (F.2)where all n i are integers, n , n , n are all positive, n is either positive or negative and n + n + n − n = n Putting the two expansions together, it follows that the integrand in (3.1) admits theexpansion (3.10).
G Example of bulk Regge scaling
In this Appendix we explicitly evaluate the bulk integral (3.1) in the Regge limit in thespecial case that N is a constant (let’s take it to be unity).At leading order in the small θ limit the bulk integral simplifies to I = (cid:90) H D − d D − yf i ( a, y , y i ) (G.1)where f i ( a, y , y i ) = (cid:90) dU dV ( U + a y + i(cid:15) ) a ( − U + y + i(cid:15) ) a ( V + a y + i(cid:15) ) a ( − V − y + i(cid:15) ) a (G.2)As we have explained in the main text, the integral over U and V is easily evaluated. Usinga change of variable, U + a y = ˜ u and V + a y = ˜ v , the above integral becomes, f i ( a, y , y i ) = (cid:90) d ˜ u d ˜ v (˜ u + i(cid:15) ) a ( − ˜ u + a y + y + i(cid:15) ) a (˜ v + i(cid:15) ) a ( − ˜ v + a y − y + i(cid:15) ) a (G.3)– 77 –sing a Schwinger parameter representation of the denominators we find that f i ( a, y , y i ) = C a ,a ,a ,a ( a y + y + i(cid:15) ) a + a − ( a y − y + i(cid:15) ) a + a − (G.4)where C a ,a ,a ,a = Γ ( a + a −
1) Γ ( a + a − a ) Γ ( a ) Γ ( a ) Γ ( a ) (G.5)Plugging (3.12) into (G.1), we find that I = (cid:90) H D − d D − y C a ,a ,a ,a ( a y + y + i(cid:15) ) a + a − ( a y − y + i(cid:15) ) a + a − (G.6)In order to perform the integral over H D − it is convenient to use the following coordinates: y µ = ( y , y , y i ) = (cosh r cosh θ, cosh r sinh θ, sinh r ˆ n i ( φ a )) (G.7)in terms of which I = (cid:90) sinh D − r (cosh r ) − (cid:80) a i dr dθ d Ω D − C a ,a ,a ,a ( a cosh θ + sinh θ + i(cid:15) ) a + a − ( a cosh θ − sinh θ + i(cid:15) ) a + a − = C a ,a ,a ,a Ω D − (cid:90) dr sinh D − r (cosh r ) − (cid:80) a i (cid:90) dθ ( a cosh θ + sinh θ + i(cid:15) ) a + a − ( a cosh θ − sinh θ + i(cid:15) ) a + a − (G.8)The integral over r above is a number independent of a . Although it will play no role inwhat follows, for completeness we record the value of this constant. M ≡ (cid:90) ∞ dr sinh D − r cosh − r = Γ (cid:0) D − (cid:1) Γ (cid:0) − D (cid:1) − (G.9)where, ∆ = (cid:80) i =0 a i .It is convenient to club all the constants together, i.e. to define N = M Ω D − C a ,a ,a ,a (G.10)in terms of which I = N a cosh θ + sinh θ + i(cid:15) ) a + a − ( a cosh θ − sinh θ + i(cid:15) ) a + a − (G.11)For ease the rest of this Appendix we will specialize to the case a + a = a + a = 2∆ (this case is relevant, for instance, to the evaluation of the four point function of 4 operators,each of dimension ∆ , caused by a bulk φ interaction). With this specialization (G.11)simplifies to I = N (cid:90) ∞−∞ dθ (cid:0) ( a + i(cid:15) ) cosh θ − sinh θ (cid:1) − (G.12)– 78 –hen a > our integrand has no singularities on the real axis. In this case the i(cid:15) in (G.12)makes no difference to the integral and can be dropped. When a < , on the other hand,the integrand in (G.12) has two poles on the real axis, located at tanh θ = ± ( a + i(cid:15) ) In this case the i(cid:15) is crucial to the definition of the integral in (G.12).In the main text we have argued that the integral I has a branch cut singularity at a = 1 . Moreover we have argued that if we evaluate I for a > and then analyticallycontinue this result to a < via the upper half of the complex a plane, then we will obtainthe correct result for I for a < . In the rest of this Appendix we will directly verify theseclaims by explicitly evaluating I separately for a > and a < . G.1 Exact result for every value of n when a > In this case, as we have explained above, we can ignore the i(cid:15) in the integrand. For thisreason Mathematica is able to evaluate the integral; we find (cid:90) dθ (cid:0) a cosh θ − sinh θ (cid:1) − = 4 − F (cid:16) −
1; 2∆ − , −
1; 2∆; − aa +1 , a +11 − a (cid:17) (2∆ −
1) ( a − − (G.13)Where F is the Appell function. While the function F may be unfamiliar, it is not par-ticularly complicated, atleast for the values of parameters of relevance to our computation.To illustrate this in (1) below we have listed the specific functional form of F for smallinteger values of n = 2∆ − in terms of elementary functions n n n ( a − ) n F (cid:0) n ; n , n ; n + ; − aa + , a + − a (cid:1) a log (cid:0) a +1 a − (cid:1) a (cid:2)(cid:0) a + 1 (cid:1) log (cid:0) a +1 a − (cid:1) − a (cid:3) a (cid:2)(cid:0) a + 2 a + 3 (cid:1) log (cid:0) a +1 a − (cid:1) − a (cid:0) a + 1 (cid:1)(cid:3) ... ... Table 1 : Functional form of the integral (G.13) for few integer n .As we see from the examples listed in Table 1, the integral I does indeed always have abranch cut singularity at a = 1 as anticipated on general grounds. Indeed the fact that thisbranch cut is logarithmic in nature is true at (atleast) all integer values of n = (2∆ − aswe see from the formula lim a → + n F (cid:16) n ; n, n ; n + 1; − aa +1 , a +11 − a (cid:17) n ( a − n = log (cid:18) a − (cid:19) + O ( a − . ∀ n ∈ Z + (G.14)We will now check that the analytic continuation of this result - taken through the upperhalf complex a plane - correctly reproduces the result for I ( a ) for a < . For simplicity we– 79 –estrict attention in this part of the Appendix to the especially simple case n = 1 , thoughwe do not think it would be too difficult to generalize the computations presented in therest of this Appendix to (at least) arbitrary integer values of n . G.2 Complete analytic structure in a special case n = 1 We will now completely analyse I in the case n = 1 . We do this by re evaluating the integralover θ , first for the case a > , but in a manner that easily allows us to generalize to a < . I = N (cid:90) ∞−∞ dθ ( a cosh θ + sinh θ + i(cid:15) ) ( a cosh θ − sinh θ + i(cid:15) )=2 N (cid:90) ∞ dθ ( a cosh θ + sinh θ + i(cid:15) ) ( a cosh θ − sinh θ + i(cid:15) ) (G.15)Now we will do a change of variable, e θ = w , which gives, I = 2 N (cid:90) ∞ dww (cid:0) a (cid:0) w + w (cid:1) + (cid:0) w − w (cid:1) + i(cid:15) (cid:1) (cid:0) a (cid:0) w + w (cid:1) − (cid:0) w − w (cid:1) + i(cid:15) (cid:1) = 2 N (cid:90) ∞ w dw (( a + 1) w + a − i(cid:15) ) (( a − w + a + 1 + i(cid:15) )= N (cid:90) ∞ dz (( a + 1) z + a − i(cid:15) ) (( a − z + a + 1 + i(cid:15) ) (G.16)In going from the second to the third line in (G.16) we have made the variable change z = w .(G.16) applies both to the cases a > and a < . In the case a > the integral (G.16)is elementary because none of the poles lie on the integration axis. In this case (cid:15) can simplybe set to zero and we obtain I > N = 1 a log (cid:18) a + 1 a − (cid:19) = − a log (cid:18) a − a + 1 (cid:19) (G.17)As the analytic continuation of ln( a − to a < via the upper half complex plane is ln( a − → ln(1 − a ) + iπ it follows that the analytic continuation of I > to a < via the upper half plane, I < , isgiven by I (cid:8) < N = 1 a (cid:20) − iπ − log (cid:18) − a a (cid:19)(cid:21) = 1 a (cid:20) − iπ + log (cid:18) a − a (cid:19)(cid:21) (G.18)We will now directly evaluate the integral (G.16) for the case a < and verify that weindeed obtain the result (G.18). When a < , the integrand in (G.16) has two poles thatlie (approximately) on the integration contour (i.e. the positive real axis). These two polesoccur at z = z ± where z + = 1 + a − a + i(cid:15) − a , and, z − = 1 − aa + 1 − i(cid:15) a . (G.19) This is because the argument of a − changes continuously from to π as we go from the the real axiswith a > to the real axis with a < via the upper half of the complex plane. – 80 –he integral (G.16) may be evaluated by rotating the contour counter-clockwise by anangle π , i.e. changing the integral from zero to ∞ along the positive real axis to zero to −∞ along the negative real axis. In performing this integral we cut the pole at z = z + . Asthe integrand decays like / | z | at infinity, the contribution to the integral from the arc atinfinity vanishes, and so it follows that the integral in (G.16) equals the value of the sameintegral evaluated along the negative real axis plus the contribution of the pole at z = z + .We now evaluate these two contributions separately.The integral along the negative real axis can be evaluated by setting z = − y in (G.16).rotated path has no pole in its path (recall a > ) so can be integrated in an elementarymanner (and in particular dropping the i(cid:15) ) and yields I ← N = (cid:90) ∞ dy ( − ( a + 1) y + a −
1) (( a − − y + a + 1)= 1 a log (cid:18) a − a (cid:19) . (G.20)The contribution of the pole at z + is given by I (cid:12) N = 2 πi Res (cid:20) a + 1) z + a −
1) (( a − z + a + 1) (cid:21) z = z + = − iπa . (G.21)It follows that the final answer for I at a < is I < N = I ← N + I (cid:12) N = 1 a (cid:20) − iπ + log (cid:18) a − a (cid:19)(cid:21) = I (cid:9) < . (G.22)We have thus verified in this special example that the appropriate analytic continuation ofthe a > result for I does indeed reproduce I at a < , as predicted on general grounds inthe main text. G.3 Integral at small a In this subsection we evaluate (G.11) at leading order in the small a expansion. For small a , major contribution to the integral (G.11) comes θ ∼ a . So the last integral in (G.11)becomes, (cid:90) dθ ( a + θ + i(cid:15) ) a + a − ( a − θ + i(cid:15) ) a + a − = (cid:90) dθ ( θ + i(cid:15) ) a + a − (2 a − θ + i(cid:15) ) a + a − = Γ( a + a + a + a − a + a − a + a −
2) 1(2 a ) − (cid:80) i a i (G.23)– 81 – Singularities of holographic correlators from contact interactions
In this subsection we will demonstrate that the classical holographic correlator with inser-tion points (2.1) - generated by a bulk local contact interaction - is an analytic function ofof its parameters ( τ and θ ) away from the ‘bulk point singularity’ line τ = 0 and also the‘lightcone singularity lines’ τ = θ and τ = π − θ . The result of this section is a simplespecialization of the general results of sections 2 and 3 of [30] to the particular case of (2.1).Any correlator obtained from a local contact interaction is a sum of terms of the form C = (cid:90) AdS D +1 d D +1 X Q ( Z i , P i , X ) (cid:81) i =1 ( − P i .X ) q i (H.1)Here P i are the the boundary insertion points, Z i are the boundary polarizations, X positionof the interaction vertex in AdS D +1 and q i = ∆ i + n i where ∆ i is the dimension of theoperator inserted at P i and n i is a positive integer. Because of potential singularities from the denominator, the expression (H.1) is not yetcompletely unambiguous; it needs to be i(cid:15) corrected. Note that − P i .X = 2 cosh r cos( τ − τ i ) + ... where τ i is the global time of the boundary point P i and τ is the global point of the bulkinteraction point X . The i(cid:15) replacement rule (D.2) instructs us to make the replacement − P i .X → − P i .X + i sin( τ − τ i )( τ − τ i ) (cid:15) (H.2)In particular when | τ − τ i | < π (H.3)(H.2) simplifies to − P i .X → − P i .X + i(cid:15) (H.4)For X such that (H.3) is obeyed, (H.1) is modified to C = (cid:90) AdS D +1 d D +1 X Q ( Z i , P i , X ) (cid:81) i =1 ( − P i .X + i(cid:15) ) q i (H.5)The integral over X in (H.5) is potentially singular when one or more of the fourdenominator factors in (H.5) vanish - and the contour of integration over X µ cannot bemodified in the complex X M plane (in a manner consistent with Cauchy’s theorem, i.e.without crossing a pole) to avoid all singular points. H.1 End point singularities
The boundary of the integration contour for the integral over the
AdS lies at the boundaryof
AdS . The integral is potentially singular when the integrand of (H.5) has a pole at a In the case that the operator O i is traceless symmetric with J i indices, n i ≥ J i . We get terms with n i > J i when derivatives - from the bulk interaction vertex - hit the propagator. – 82 –oundary point. This happens at four points on the boundary, namely X = P , X = P , X = P and X = P .For generic values of P a these potentially problematic points do not in fact lead to asingularity in the integral. Let us, for instance, consider the pole at X = P . The i(cid:15) in (H.5)ensures that the singularity does not really lie on the integration contour, the potentiallydangerous contribution from integral in the neighbourhood of this point is proportionalto the non-singular residue of this ‘pole’. This residue is singular only if P a .P = 0 for a = 2 , , . This happens only when the points P and P a are lightlike separated on theboundary (i.e), and leads to the usual ‘lightcone’ singularities familiar from the study ofconformal field theories. As is familiar - and as we have explained earlier in this section -these light cone singularities lead to branch cuts in correlators, and the correlation functioncontinues to be analytic on the branched cover of cross ratio space.In this paper we are interested in the correlator at the points (2.1). Lightcone singular-ities occur at τ = π − θ (at which point the pairs of points ( P , P ) and ( P , P ) are lightlikeseparated, and the conformal cross ratio ¯ z = 1 ) and also at τ = θ (at which point ( P , P ) and ( P , P ) are lightlike separated and the conformal cross ratio z = 1 ). At exactly thepoints described, (namely τ = π − θ and τ = θ ) we land exactly at the branch points of thecorrelator which is thus singular. As we have explained, however, we can continue past thissingularity by going around it; and so the value of the correlator at in the ranges τ ∈ (0 , θ ) , τ ∈ ( θ, π − θ ) , ( π − θ, π ) are different ‘boundary values’ (across cuts) of the same analyticfunction. H.2 Pinch singularities
As explained in [30], a potential singularity at X = X (where some subset { P a } = S obey P a .X = 0 ) can be avoided provided we can find a small vector δQ so that on the newcontour X → X + iδQ and δQ.X = 0 , and − P a .δQ > , for all P a in S, (H.6)The first equation is needed to ensure that ( X + iδQ ) = − (H.7)(i.e. the modified contour is on the AdS D +1 sub-manifold - note we work to first order inthe small modification δQ ). The second condition is needed to ensure that the modificationdoes not pass through any of the poles in (H.1). To see why this is the case, consider the single variable integral (cid:82) dzz + i(cid:15) along the real axis. The pole inthe integrand lies at z = i(cid:15) . Hence we are free to move the contour of integration in the upper half plane, i.e.to give z a positive imaginary part. In other words the + i(cid:15) in the integrand tells us that we are allowed tochange the integration contour so that Im( z ) is positive when Re( z ) = 0 ; however the reverse modificationis not allowed. In the same way the integral contour in (H.5) can be modified (without changing the valueof the integral) if we ensure that at all X that obey P a .X = 0 , P a . Im( X ) > . – 83 –he first condition (H.6) tells us that δQ is orthogonal to X , and so forces δQ to liein an R D, in R D, . As each P a obeys P a .X = 0 , each P a lies in this R D, . Each P a isassociated with a co-dimension one hyperplane in R D, that passes through the origin ( P a is the one-form normal to this plane - this is the plane, whose intersection with X = − gives the light sheet that emanates out of P a ). The condition P a .δQ > tells us that δQ lies on one side of this hyperplane in R D, .Let us first consider a point X at which the equation P a .X = 0 is obeyed for m valuesof a . Since no collection of three or fewer P a in (2.1) are linearly dependent (and since D − ≥ ) the hyperplanes associated with the m P a slice up the R D, into m distinctsectors. The positivity condition in (H.6) is met provided we choose δQ to lie within one ofthese (the all positive) sector. It follows that (H.6) admits an infinite number of solutions,and the integral (H.5) receives regular contributions from all such points.If τ (cid:54) = 0 then the four P i span an R , and there are no solutions (in AdS D +1 ) to thesimultaneous equation P i .X = 0 .It follows that the integral (H.5) can only be singular at τ = 0 . At this value of τ thefour P i span an R , and the simultaneous equations P i .X = 0 are obeyed on the H D − X = (0 , cosh r, , , (sinh r ) (cid:126)m ) where (cid:126)m is any unit vector in R D − . When τ = 0 P + P + P + P = 0 (H.8)It follows from (2.18) that P .δQ + P .δQ + P .δQ + P .δQ = 0 (H.9)As (2.18) is clearly inconsistent with the condition P a .δQ > for all a [30], it follows that(H.6) cannot be obeyed, and so it is not possible to deform the integration contour awayfrom the singular hyperboloid and the contribution to (H.5) is singular.In summary, we have demonstrated that the holographic correlator with insertions at(2.1) has a pinch singularity at τ = 0 . It also has endpoint singularities at τ = θ and τ = π − θ , but these are branch cut singularities that can be continued around, as we haveexplained earlier in this section. I ρ = 0 branch cuts and UV softening I.1 D = 2 The branch cut singularity in (4.36) had its origin in the integral (cid:90) ∞ dω (cid:0) ω ∆ − ω r e − iωτ (cid:1) (I.1)(see the middle equation in (4.32)). The factor of ω r in the integrand in (I.1) had its originin the fact that the bulk S matrix for the corresponding contact term grew like ω r . Of– 84 –ourse this is an approximation. In any ‘real’ bulk theory we expect this power law growthof the S matrix to be ameliorated by stringy and quantum effects. As a crude model forthe stringy softening, we can follow [30] and make the replacement ω r → ω r e − α (cid:48) ω .Consider the function I ( τ ) = l ∆ − rs (cid:90) ∞ dω (cid:16) ω ∆ − r e − α (cid:48) ω − iωτ (cid:17) (I.2)(where we have added the normalization factor for future convenience). The change ofvariables ω = xl s and y = τl s transforms (I.2) to I ( y ) = (cid:90) ∞ dx (cid:16) x ∆ − r e − x − ixy (cid:17) (I.3)Note that τ , ρ and y are all proportional to each other at fixed σ , and so the analyticstructure of I ( y ) around y = 0 is the same as the analytic structure of the RHS of (4.36).In the rest of this subsection we analyze the function I ( y ) .First note that I ( y ) is a manifestly single valued function of y that is analytic every-where in the y complex plane. It is instructive to study this function in various differentregimes.For | y | (cid:28) the term ixy is a small perturbation of the argument of the exponent in(I.3). It follows that the integral in (I.3) admits a power series expansion in y , which canbe evaluated by first expanding the integrand in a power series and performing the integralterm by term. We find I ( y ) = (cid:88) n a n y n a n = ( − i ) n n ! Γ (cid:18) ∆ − r + n (cid:19) (I.4)At large values of n a n ≈ (cid:32) − i (cid:114) en (cid:33) n (I.5)From which it follows that the expansion (I.4) is convergent, with an infinite radius of con-vergence. Of course the truncation of (I.4) to the first few terms gives a good approximationto I ( y ) only for all complex values such that | y | (cid:28) or | τ | (cid:28) l s (I.6)When | y | (cid:29) , on the other hand, I ( y ) behaves very differently depending on whether Im( y ) > or Im( y ) < . When Im( y ) > the damping and oscillations from the factor e ixy cuts off the integral before the factor e − x becomes important at all when y is large.In this case the integral is well approximated by dropping the factor e − x and so is wellapproximated by I ( y ) = Γ (∆ − r )(2 i ) (∆ − r ) y ∆ − r (I.7)Note that y ∝ ρ , so (I.7) reproduces the singularity visible in (4.36). Corrections to (I.7)can be systematically computed by expanding the factor of e − x in a power seies in x and– 85 –hen integrating term by term. This is reasonable as the integral receives its dominantcontribution from small values of x when Imy is large and positive. We find I ( y ) = ∞ (cid:88) n =0 ( − n n ! Γ(∆ + r + 2 n − i ) ∆+ r +2 n − y ∆+ r +2 n − (I.8)In contrast with the expansion (I.4), (I.8) is an asymptotic rather than a convergent ex-pansion. This mathematical fact reflects the physical fact that the truncation of (I.8) to afew terms gives us a good approximation to the actual behaviour of the function I ( y ) onlywhen Im( y ) is positive rather than negative. When | y | (cid:29) but Im( y ) < , I ( y ) actually behaves completely differently from thecase just examined above. In this case the factor e ixy exponentially enhances (instead ofcutting off) the integrand, which continues to grow until it is eventually cut off at of orderunity by the factor e − x (which now plays a crucial role; without this factor the integralwould have divergent and so ill defined). In this regime, very, very approximately, | I ( y ) | ∼ e − Im( y ) (I.9)(moreover we expect the phase of I ( y ) to oscillate rapidly with the real part of y ).We now have a good qualitative picture of the behaviour of I ( y ) on the complex y plane. This function is well approximated by the first few terms of (I.4) when | y | (cid:28) , iswell approximated by the first few terms of (I.8) when | y | (cid:29) and Im( y ) > , but is harderto control (very approximately given by (I.9) when | y | (cid:29) and Im( y ) < .In summary, the softening of the integral induced by the factor e − x in I ( y ) impacts thefunctional dependence of (4.36) in the complex ρ differently at different values of ρ (recall ρ is proportional to y ). First it modifies the function at small values of ρ to smoothen outthe ρ = 0 ( y = 0 ) singularity in (4.36). Second it has a negligible impact on the value ofthe function at large values of | ρ | provided Im( ρ ) > ; in this regime (4.36) continues to bea good approximation to a regulated function. When Im( ρ ) < , on the other hand, I ( y ) isvery different from (4.36) even at large values of | y | (the function behaves rather wildly inthis region, growing very large in modulus and also oscillating very rapidly). It follows that(4.36) cannot be used to reliably compute the discontinuity around ρ = 0 even at large ρ .In fact I ( y ) is an everywhere analytic and single valued function of y , that does not have abranch cut. I.2
D > When
D > the singular part of (4.44) is generated by the integral J ( y ) = (cid:90) ∞ dζ sinh D − ζ (cid:90) ∞ dx (cid:16) x ∆ − r e − x − ixy cosh ζ (cid:17) . (I.10) An analogy is the following. The perturbation series for the energy spectrum of an harmonic oscillatoris asymptotic rather than convergent, reflecting the fact that the first few terms of the expansion give agood approximation to the spectrum of the theory in the case that the quartic term in the potential ispositive, but a very bad approximation to the (ill defined, unbounded) spectrum of the theory in the casethat the quartic term is negative. – 86 –erforming the integral over x we obtain J ( y ) = (cid:90) ∞ dζ sinh D − ζ I ( y cosh ζ ) (I.11)It is easy to approximately perform the integral in (I.11) in two limits. When | y | (cid:29) and Im( y ) > the argument of the function I in (I.11) shares these two properties, so we canuse the approximation (I.7) for I (cosh ζy ) to find J ( y ) ≈ Γ (∆ − r )(2 i ) (∆ − r ) y ∆ − r (cid:90) ∞ dζ sinh D − ζ cosh ∆ − r ζ = N D, ∆ Γ (∆ − r )(2 i ) (∆ − r ) y ∆ − r (I.12)where N D, ∆ is defined in (4.45). The approximation (I.11) is easily improved by by inserting(I.8) rather than (I.7) into (I.11) but we will not bother to do so here.At small | y | , on the other hand, the dominant contribution to the integral in (I.11)comes from ζ such that y cosh ζ less than or of order unity. Working in a very crudeapproximation one can replace I ( y cosh ζ ) by I (0) θ (1 − y cosh ζ ) . Making this replacementwe find that for | y | (cid:28) J ( y ) ∼ D −
3) Γ (cid:18) ∆ − r (cid:19) y D − (I.13)The approximation (I.13) is very crude (even the overall coefficient on the RHS is not reliablebut the form of the y dependence is). We will not attempt to improve this approximationhere, but leave this as an exercise for the interested reader.The main qualitative point is that the function J ( y ) roughly similar to I ( y ) in theprevious section with one key difference. While I ( y ) interpolates from a constant to a rapiddecay ∝ y ∆ − r as y increases from zero to infinity, J ( y ) interpolates from the weaker power y D − [30] to the stronger power (more rapid decay) y ∆ − r as y varies over the same range.Most of the comments in the previous subsection about the analytic properties of I ( y ) alsohold for J ( y ) with small modifications. In particular it is a single valued function of y everywhere in the complex plane. J Massive higher spin particles
J.1 Multiple powers of ρ In this Appendix we outline some of the complications that arise when attempting togeneralize the analysis of this paper to massive higher spin particles. We focus on thesimplest case, namely that of massive vector particles. The complications that arise all havetheir origin in a familiar fact; namely that the S matrices of ‘longitudinal’ polarizations ofmassive particles grow faster with energy in the high energy limit than those of transversepolarizations. – 87 –uantitatively, the complications arise as follows. As we have explained in the maintext ((4.51)) the bulk to boundary propagator for a vector operator of dimension ∆ is givenby (cid:18) − (cid:19) Z ⊥ A ( P.X ) ∆ + ∇ A (cid:18) Z.X ∆( P.X ) ∆ (cid:19) (J.1)Applying (4.7) on (J.1) and working to leading order we find the wave form (4.83).Let us first proceed by supposing (4.65) (which we reproduce here for convenience) G sing = − π (cid:16)(cid:81) a ˜ C ∆ a , (cid:17)(cid:112) σ (1 − σ ) (cid:90) H D − √ g D − d D − X (cid:90) dωω ∆ − e iωP.X S ( ω )∆ = (cid:88) i ∆ i ˜ C ∆ a , = C ∆ a , ∆+1 i ∆ Γ(∆) (J.2)continues to capture all relevant singularities (we will see later this is untrue) where S ( ω ) is the flat space S matrix for the waves (J.1).Let us suppose that the bulk interaction term is of r th order in derivatives. Let usdecompose the S matrix in (J.2) as follows S ( (cid:15) i , k i ) = (cid:88) m =0 S m ( (cid:15) i , k i ) (J.3)where S n is the S matrix of n ‘longitudinal’ polarization (i.e. the polarization proportionalto k i ) in (4.83) and − n transverse polarizations (i.e. the polarizations proportional to Z ⊥ i ). . When the scattering matrix results from an r derivative bulk interaction term,the quantity S m scales with the overall energy scale of the scattering momenta (4.21) like S m ∼ ω r + m (J.5)(the dependence of the RHS of (J.5) on m follows from the extra factor of momentumin (4.83)). Note that the scattering waves (4.83) that produce S n are not canonicallynormalized (more about this in the next subsection) . S n can be defined more formally as follows. We formally modify (4.83) to included a new countingvariable θ (which is set to unity at the end of the computation) as (cid:15) M = (cid:18) − i (cid:19) ( Z ⊥ i ) M + iθk iM ( Z i .X ) (J.4) S n is the part of the S matrix which scales like θ n . The fact that these waves do not have standard normalization is of no qualitative importance forthe transverse wave - whose normalization factor is independent of ω . However this point is of crucialimportance for the longitudinal mode. As we discuss in more detail in the next subsection, the norm ofthis mode in (4.83) is ω dependent, and goes to zero in the limit ω → , cancelling the singular behaviourof longitudinal mode scattering amplitudes at high energy. – 88 –lugging (J.3) into (J.2) we find that one contribution to the singularities in ρ of thecorrelator is given by G sing = (cid:88) m =0 G m sing G m sing = i (cid:18) π (cid:16) ˜ C ∆ a , (cid:17) (cid:19) Γ(∆ + r + m − e − i (∆+ r + m )2 √ − σ (∆+ r + m − σ ∆+ r + m − ρ ∆+ r + m − × (cid:90) dω D − dζ sinh D − ζ cosh ∆+ r + m − ζ ˆ S m ( X )ˆ S m = S m ω r + m X = (0 , cosh ζ, , , sinh ζ ˆ n i ) , i = 1 . . . D − (J.6)and S m is the part of the S matrix of the un normalized waves (4.83) involving m longitudinaland − m transverse modes, where the scattering takes place for the waves (4.83), withthe scattering momenta given by (4.21) with ω set to unity. The dependence of the powerof the ρ singularity on m in (J.6) is a consequence of (J.5).Had (J.6) accurately captured the coefficients of all the ρ singularities that appear inthat formula, we could have used it to establish that if any of the S matrices that appear ascoefficients of different powers of ρ grow faster than s in the Regge limit then the correlatorcontinued to the Causally Regge sheet would violate the chaos bound. Unfortunately (J.6)is not complete. The problem is that the term with, say, m = 4 in (J.3) has its origin inthe overlap of four longitudinal modes. The m = 4 term in (J.3) does indeed capture themost singular term that arises from this overlap. However, the overlap of four longitudinalmodes could also produce lower order singularities (the coefficients of these singularitieswould ‘see’ the fact that the bulk to boundary propagator does not quite yield a planewave, and that the elevator is not quite a flat space). These lower order singularities can, inprinciple, modify the coefficients of ρ ∆+ r + m − for m ≤ . Similarly, subleading correctionsfrom the overlap of 3 longitudinal and one transverse polarization can modify the coefficientsof ρ ∆+ r + m − for m ≤ . Unlikely as it seems, these correction terms could, in principle,cancel that rapid Regge growth of the S matrix from a particular contact term, allowing itto give rise to a correlator that obeys the chaos bound even though the S matrix in questionviolates the CRG conjecture. While it seems very likely that this will in fact happen a clear argument for the connection between the chaos bound and massive higher spinscattering requires further work. The reader might hope that the contamination from subleading corrections could be ‘quarantined’ in aclass of index structures, allowing us to read off the scattering of less than a maximal number of longitudinalpolarizations from the other index structures. We were, however, unable to show this will always be thecase. Note that the fact that the terms in G m sing listed in (H) obeys the equations ∇ M ∇ M . . . ∇ M m +1 G M M ...M = 0 ∇ M ∇ M . . . ∇ M m G M M ...M (cid:54) = 0 (J.7)does not help us, as the nothing we can see prevents the subleading corrections to flat space scattering fromobeying the same equations. – 89 – .2 Normalizations and connections to scattering We have indicated above that the additional power of ω in the longitudinal waves is es-sentially a reflection of the enhanced ω scaling of the scattering of longitudinal waves ascompared to transverse waves. In this brief subsection we explain this connection moreclearly.Through this paper we have been slightly imprecise in discussing the flat space S matrixof the wave (4.83) in the case of massive particles (even massive scalars). For massiveparticles (4.83) are free solutions of the relevant flat space bulk wave equations only in thestrict ω → ∞ limit (because the momenta of the scattering waves obey k = 0 rather than k = − m ). As the singularity in the flat space correlator arises from the ω → ∞ part offrequency space, this imprecision was unimportant in the case of scalars.In the case of massive vectors the imprecision extends also to the polarizations. Apolarization proportional to k M is not quite transverse and so not quite allowed except inthe ω → ∞ limit. In order to see this more clearly we could, for instance, arbitrarily replacethe scattering waves (4.83) with A iM = (cid:18)(cid:18) − i (cid:19) ( Z ⊥ i ) M + im i ∆ i (cid:15) (cid:107) i ( Z i .X ) (cid:19) e i ˜ k i .x (J.8)where ˜ k = ( ω , , p , (cid:15) (cid:107) = 1 m ( p , , ω , k = ( ω , , − p , (cid:15) (cid:107) = 1 m ( p , , − ω , k = − ( ω , , p cos θ, p sin θ ) (cid:15) (cid:107) = 1 m ( p , , ω cos θ, ω sin θ )˜ k = ( − ω , , − p cos θ, − p sin θ ) (cid:15) (cid:107) = 1 m ( p , , − ω cos θ, − ω sin θ ) p = p = p, p = p = p (cid:48) ω i = (cid:113) m i + p i (cid:113) m + p + (cid:113) m + p = (cid:113) m + ( p (cid:48) ) + (cid:113) m + ( p (cid:48) ) (J.9)Note that the waves (J.8) are identical to the waves (4.83) in the large ω limit that is relevantfor generating the singularity of the correlators. Unlike the waves (4.83), however, (J.8) aregenuine solutions of the free flat space massive massive vector equation with mass m , and so the S matrix of these waves is genuinely well defined. The quantity S that enters In (J.9) we have corrected the momenta to ensure that they obey the true mass shell condition k = − m , and have simultaneously modified the expression for (cid:15) (cid:107) i to make sure that the equation (cid:15) (cid:107) .k = 0 continues to be obeyed. Note that we have chosen our modifications to ensure that scattering continues totake place in the centre of mass frame, as was the case in for the momenta (4.21). We have also ensuredthat the normalization vectors (cid:15) (cid:107) i are all normalized. This is what results in the factor m i in the expressionsfor (cid:15) (cid:107) i in (J.9), and the compensating factors of m i in the coefficient of (cid:15) (cid:107) i in (J.8). Note, in particular, thatthe normalization of the part of the scattering wave proportional to (cid:15) (cid:107) in (J.8) is proportional to m i . Thisis what allows the scattering amplitudes of longitudinal polarizations (which come with a single factor of m i for each factor of m i ) to be well defined even in the limit m i → . – 90 –ormulae such as (J.6) should really be thought of as the S matrix of the true scatteringwaves (J.8).The point we want to make is the following. In the expression (4.83) the extra factor of ω in the longitudinal polarization comes from the fact that the polarization is proportionalto k µ . The reader might have suspected that this extra scaling with ω is a consequence ofusing an ω dependent normalization for this wave. This is not the case. In (J.9) makesclear that the polarization that appears in (4.83) differs from the properly normalizedpolarization (cid:15) (cid:107) by the ω independent factor m i . 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