The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop
António Antunes, Miguel S. Costa, Tobias Hansen, Aaditya Salgarkar, Sourav Sarkar
PPrepared for submission to JHEP
The perturbative CFT optical theorem andhigh-energy string scattering in AdS at one loop
Ant´onio Antunes † , Miguel Costa † , Tobias Hansen ♦ , Aaditya Salgarkar † , Sourav Sarkar † † Centro de F´ısica do Porto, Departamento de F´ısica e AstronomiaFaculdade de Ciˆencias da Universidade do PortoRua do Campo Alegre 687, 4169–007 Porto, Portugal ♦ Mathematical Institute, University of Oxford, Andrew Wiles Building,Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK ♦ Department of Physics and Astronomy, Uppsala University,Box 516, SE-751 20 Uppsala, Sweden
E-mail: [email protected], [email protected],[email protected], [email protected],[email protected]
Abstract:
We derive an optical theorem for perturbative CFTs which computes thedouble discontinuity of conformal correlators from the single discontinuities of lower ordercorrelators, in analogy with the optical theorem for flat space scattering amplitudes. Thetheorem takes a purely multiplicative form in the CFT impact parameter representationused to describe high-energy scattering in the dual AdS theory. We use this result to studyfour-point correlation functions that are dominated in the Regge limit by the exchange ofthe graviton Regge trajectory (Pomeron) in the dual theory. At one-loop the scattering isdominated by double Pomeron exchange and receives contributions from tidal excitationsof the scattering states which are efficiently described by an AdS vertex function, in closeanalogy with the known Regge limit result for one-loop string scattering in flat space atfinite string tension. We compare the flat space limit of the conformal correlator to theflat space results and thus derive constraints on the one-loop vertex function for type IIBstrings in AdS and also on general spinning tree level type IIB amplitudes in AdS.
Keywords:
CFT, Regge theory, AdS/CFT a r X i v : . [ h e p - t h ] D ec ontents N expansion . . . . . . . . . . . . . . . . . . . . 14 s -channel discontinuities in the Regge limit . . . . . . . . . . . . . . . . . . 314.4 Spinning particles and the vertex function . . . . . . . . . . . . . . . . . . . 34 gap limit . . . . . . . . . . . . . . . . . . . . . 375.2 Extracting t-channel CFT data . . . . . . . . . . . . . . . . . . . . . . . . . 38 A.1 Chiral Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.2 Closed string amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B Tensor products for projectors 57C Branching relations for projectors 59
C.1 All 5d closed string amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 62– 1 –
Introduction
In recent years it has been shown that powerful analytical results for scattering amplitudesin quantum field theory, namely the Froissart-Gribov formula and dispersion relations,have equally powerful CFT analogues in the Lorentzian inversion formula [1–4] and thetwo-variable CFT dispersion relation [5, 6]. Dispersion relations reconstruct a scatteringamplitude from the discontinuity of the amplitude, while the Froissart-Gribov formulaextracts the partial wave coefficients from the discontinuity and makes their analyticity inspin manifest. The utility of these methods as computational tools for scattering amplitudesstems from the fact that the discontinuity of an amplitude (or that of its integrand) inperturbation theory is determined in terms of lower-loop data by the optical theorem,which in turn is a direct consequence of unitarity.The CFT analogue of the discontinuities of amplitudes, which contain the dispersivedata and are of central importance in the aforementioned analytical results, is the dou-ble discontinuity (dDisc) of CFT four-point functions. The Lorentzian inversion formulacomputes OPE data (anomalous dimensions and OPE coefficients) from the dDisc of four-point functions and establishes the analyticity in spin of OPE data. The CFT dispersionrelation, much like its QFT inspiration, directly reconstructs the full correlator from thedDisc. There also exist simpler single-variable dispersion relations in terms of a singlediscontinuity (Disc) of the correlation function that determine only the OPE coefficientswhile the anomalous dimensions are required as inputs [7].The unitarity based methods to compute amplitudes inspire the development of similarunitarity methods for CFT, in particular, for the dDisc of four-point functions one gainsa loop or leg order for free. It was first noticed in large spin expansions [8–10] and laterunderstood more generally in terms of the Lorentzian inversion formula that OPE data atone-loop can be obtained from tree-level data [11, 12]. Generically, in perturbative CFTcalculations the dDisc at a given order only depends on OPE data from lower order orlower-point correlators. More recently, in the context of the AdS/CFT correspondence[13–15], these unitarity methods for CFT have been related to cutting rules for computingthe dDisc of one-loop Witten diagrams from tree-level diagrams [16, 17]. See also the earlierwork of [18].However, so far we have been missing a direct adaptation of the optical theorem toCFT correlation functions. More concretely, we still lack the ability to express the dDiscof a perturbative correlator, at a given order in the perturbative parameter, in terms oflower order correlators, without the detour via the OPE data and without making explicitreference to AdS Witten diagrams. In this paper we provide a direct CFT derivationof such unitarity relations. In particular we present an optical theorem for 1-loop four-point functions wherein the dDisc is fixed in terms of single discontinuities of lower-loopcorrelators.Let us briefly describe the logic that underlies the perturbative CFT optical theorem.Throughout this paper we will consider the correlator A ( y i ) = (cid:104)O ( y ) O ( y ) O ( y ) O ( y ) (cid:105) . (1.1)– 2 –e begin by expanding the dDisc of this correlator in t -channel conformal blocks. We maydo this by expanding in conformal partial waves and then projecting out the contributionof the exchange of the shadow operator ˜ O . The advantage of this procedure is that whenwriting the partial waves as an integrated product of three-points functions, the dDiscoperation factorizes as a product of discontinuities,dDisc t A ( y i ) = − (cid:88) O (cid:90) dydy (cid:48) Disc (cid:104)O O O ( y ) (cid:105)(cid:104) ˜ O ( y ) ˜ O ( y (cid:48) ) (cid:105) Disc (cid:104)O O O ( y (cid:48) ) (cid:105) (cid:12)(cid:12)(cid:12) O , (1.2)where we use the shorthand notation d d y ≡ dy . Notice that the sum runs over all operatorsin the theory. We give the precise definitions of the double and single discontinuities of thecorrelator in section 2.Next, let us assume the correlator admits an expansion in a small parameter aroundmean field theory (MFT). The example we have in mind is the 1 /N expansion, A = A MFT + 1 N A tree + 1 N A + · · · . (1.3)We can then separate the sum over intermediate operators O into single-, double-, andhigher-trace operators, and rewrite the multi-trace contributions as higher-point functionsof single-trace operators. The contribution of single-trace operators to the t -channel ex-pansion of dDisc in (1.2) is left unchanged and is still given in terms of discontinuities ofthree-point functionsdDisc t A ( y i ) (cid:12)(cid:12)(cid:12) s.t. = − (cid:88) O∈ s.t. (cid:90) dydy (cid:48) Disc (cid:104)O O O ( y ) (cid:105)(cid:104) ˜ O ( y ) ˜ O ( y (cid:48) ) (cid:105) Disc (cid:104)O O O ( y (cid:48) ) (cid:105) (cid:12)(cid:12)(cid:12) O . (1.4)Here no simplifications occur, however this contribution is already simple as loop correctionscome from corrections to the three-point functions of single-trace operators.The essential simplification that we call the perturbative optical theorem arises forthe contributions of double-trace operators to (1.2), which are now expressed in terms ofdiscontinuities of four-point functions of single-trace operatorsdDisc t A − loop ( y i ) (cid:12)(cid:12)(cid:12) d.t. = − (cid:88) O , O ∈ s.t. (cid:90) dy dy Disc A ( y k ) S S Disc A ( y k ) (cid:12)(cid:12)(cid:12) [ O O ] . (1.5)Here and henceforth, we shall use the notation A abcd ( y k ) = (cid:104)O a O b O c O d (cid:105) to denote thecorrelator of a set of operators other than (cid:104)O O O O (cid:105) , which we denote simply as A ( y i ). S S A is defined as the shadow transform of A with respect to the operators O and O . The operators O and O are summed over all single-trace operators for which thetree-level correlators exist. These may have spin, in which case the indices are contractedbetween the two tree-level correlators. Importantly, in this case dDisc is of order 1 /N andcan be computed from the product of the discontinuities of tree-level four-point functions,each of order 1 /N .Together equations (1.4) and (1.5) compute the full double discontinuity at one-loop inlarge N CFTs, since the contributions from higher traces will start at higher loops. Their– 3 –Disc t
12 43 ∼ (cid:88) O , O (cid:90) dy , Disc
52 63 Disc (cid:101) (cid:101) Figure 1 . In the Regge limit the dDisc of the genus one closed string amplitude in AdS is givenby the perturbative CFT optical theorem in terms of genus zero amplitudes. analogue is of course the optical theorem for amplitudes which computes discontinuitiesof one-loop amplitudes in terms of two- and one-line cuts. Note that although we use thenotation A tree and A , these refer to conformal correlation functions and in generalare not Witten diagrams. The notation with the terms “one-loop” and “tree” for thecorrelators is used only because we always refer to a perturbative expansion. The resultis valid for CFTs with an expansion in a small parameter around MFT. The fact that itnaturally handles cuts of spinning particles gives an advantage over previous CFT unitaritymethods that work in terms of OPE data.In the second part of the paper, we employ the perturbative CFT optical theorem inthe context of the AdS/CFT correspondence [13–15] to study high-energy scattering ofstrings in AdS, which is governed by the CFT Regge limit [19, 20]. This is illustrated infigure 1. High-energy string scattering in flat space has been of interest for a long time, bothin the fixed angle case [21, 22] and in the fixed momentum transfer Regge regime [23–25].This second set of works studied the effects of the finite string size on the exponentiationof the phase shift (eikonalization) in the Regge limit. In particular, it was shown thatthe amplitudes indeed eikonalize provided we allow the phase shift to become an operatoracting on the string Hilbert space, whose matrix elements account for the possibility of theexternal particles becoming intermediate excited string states, known as tidal excitations.The phase shift δ ( s, b ), which depends on the Mandelstam s and on the impact parameter b , is obtained by Fourier transforming the amplitude with respect to momentum transferin the directions transverse to the scattering plane. This gives a multiplicative opticaltheorem of the formIm δ ( s, b ) = 12 (cid:88) m ,ρ ,(cid:15) m ,ρ ,(cid:15) δ ( s, − b ) ∗ δ ( s, b ) , (1.6)where the sum is over all possible exchanged particles, characterized by their mass m i and Little group representation ρ i , and their polarization tensors (cid:15) i . In [24] the one-loop amplitude for four-graviton scattering in type IIB string theory was presented in aparticularly nice form, where the tidal excitations, which constitute a complicated sum in(1.6), are packaged into a single explicit scalar function, the so-called vertex function.To study the analogous process in AdS we derive an AdS/CFT analogue of (1.6) bytransforming the correlators in the CFT optical theorem (1.5) to AdS impact parameter– 4 –pace [19, 20]. This gives the following multiplicative optical theorem for CFTs − Re B ( p, p ) (cid:12)(cid:12)(cid:12) d.t. = 12 (cid:88) O , O ∈ s.t. B ( − p, − p ) ∗ B ( p, p ) (cid:12)(cid:12)(cid:12) [ O O ] . (1.7)Here B denotes the impact parameter transform of A . These transforms depend on twocross ratios S and L , respectively interpreted as the square of the energy and as theimpact parameter of the AdS scattering process, that can be expressed in terms of two d -dimensional vectors p and p , as will be detailed below. When O or O have spin, B hastensor structures that depend on p and p . Equations (1.7) and (1.6) are related through theflat space limit for the impact parameter representation, where the radius of AdS is sentto infinity and where B ( p, p ) is mapped to iδ ( s, b ). In this way, each of the infinite numberof tree-level correlators with spinning particles 5 and 6 that appear on the right hand sideof (1.7) is partially fixed by the corresponding flat space phase shift. Moreover, we will beable to efficiently describe the summed result in terms of an AdS vertex function, which isin turn constrained by the one-loop flat space vertex function, as constructed for examplefor type IIB strings in [24].For neutral scalar operators of dimension four in d = 4, the four-point function con-sidered here is dual to the scattering of four dilatons in the bulk of AdS . There are twoexpansion parameters that we need to consider, the loop order parameter 1 /N , and thet’Hooft coupling λ . The large λ limit is given by supergravity in AdS. In this limit the tree-level four-point function is dominated by graviton exchange [19, 20] and beyond tree-levelone can safely resum the 1 /N expansion by exponentiating the single graviton exchange[26, 27]. For finite λ , string effects are included at tree-level via Pomeron exchange [28]and can be described using conformal Regge theory [29, 30]. A very non-trivial questionwe address in this paper is the inclusion of string effects beyond tree-level.To account for such effects in the Regge limit, the earlier works [29, 31, 32] conjecturedthe exponentiation of the tree-level Pomeron phase shift, assuming stringy tidal excitationsto be negligible [33]. More recently [34], the loop effects of Pomeron exchange were system-atically taken into account from the CFT side in the AdS high-energy limit S (cid:29) λ (cid:29) λ (cid:29)
1, in agreement with [29, 31, 32]. In the present work, we take finite λ (or α (cid:48) )and include all tidal or stringy corrections. This is made possible because the perturbativeCFT optical theorem is able to describe cuts involving spinning operators, so we can takeinto account intermediate massive string excitations that are exchanged in the t-channel.This paper has the following structure. In section 2 we first motivate how (1.2) fordouble-trace operators leads to the perturbative CFT optical theorem (1.5) using the tech-nique of “conglomeration” [18], and then give a detailed derivation of (1.5) using tools fromharmonic analysis of the conformal group. Then in section 3 we review some importantideas from flat space scattering, including impact parameter space, unitarity cuts and thevertex function, both to guide the AdS version and to serve as a target for the flat spacelimit. We subsequently move to the holographic case in section 4, where we transform thecorrelator to CFT impact parameter space to write a multiplicative optical theorem for– 5 –hase shifts. We use conformal Regge theory in the case of arbitrary spinning operatorsleading to the derivation of the AdS vertex function. In section 5 we recover the results forthe one-loop correlator in the large λ limit [34] and also derive new t-channel constraintson CFT data at finite λ . We give the details of the flat space limit prescription in section 6,and consider the specific four-dilaton amplitude of type IIB strings in section 7, constrain-ing several spinning tree-level correlators of the dual N = 4 SYM theory. We concludeand briefly discuss some generalizations and applications of our work in section 8. Manytechnical details and additional considerations about spinning amplitudes are relegated tothe appendices. In this section we will give a derivation for the perturbative CFT optical theorem in (1.5)using results from harmonic analysis of the conformal group following [35], but first let usmotivate (1.5) and (1.4) using the conglomeration of operators [18].Unitarity in CFT can be formulated as completeness of the set of states correspondingto local operators 1 = (cid:88) O |O| . (2.1)The right hand side is a sum over projectors associated to a primary operator O . Suchprojectors can be formulated in terms of a conformally invariant pairing known as theshadow integral [36, 37] |O| = (cid:90) dy |O ( y ) (cid:105)(cid:104) S [ O ]( y ) | (cid:12)(cid:12)(cid:12) O , (2.2)which defines the projector to the conformal family with primary operator O , automati-cally taking into account the contribution of descendants of O . Here we used the shadowtransform, defined by S [ O ]( y ) = 1 N O (cid:90) dx (cid:104) (cid:101) O ( y ) (cid:101) O † ( x ) (cid:105)O ( x ) , (2.3)with an index contraction implied for spinning operators. We normalize the two-pointfunctions to unity and N O = π d (∆ − | ρ | ( d − ∆ − | ρ | Γ (cid:0) ∆ − d (cid:1) Γ (cid:0) d − ∆ (cid:1) Γ( d − ∆ + | ρ | )Γ(∆ + | ρ | ) . (2.4)Note that with this normalization of S [ O ], S is 1 /N O times the identity map. | ρ | is thenumber of indices of the operator O . The shadow transform is a map from the operator O to (cid:101) O , where (cid:101) O is in the representation labeled by ( (cid:101) ∆ = d − ∆ , ρ ). O † is an operator withscaling dimension ∆ but transforming in the dual SO ( d ) representation ρ ∗ .Inserting the projector (2.2) into a four-point function, one finds the contribution ofthe t-channel conformal partial wave Ψ O to the four-point function (cid:104)O O |O|O O (cid:105) ∝ Ψ O . (2.5)– 6 –he conformal partial wave is a linear combination of the conformal blocks for exchangeof O and its shadow (cid:101) O . This explains the notation | O adopted in (2.2), since we need toproject onto the contribution from O and discard that of (cid:101) O .In the large N expansion of CFTs, there exists a complete basis of states spanned bythe multi-trace operators. In a one-loop four-point function of single trace operators, withan expansion as shown in (1.3), only single- and double-trace operators appear A ( y i ) = (cid:88) O∈O s.t. , O d.t. (cid:104)O O |O|O O (cid:105) . (2.6)The right hand side involves three-point functions with single- and double-trace operators.The double-trace operators are composite operators of the schematic form[ O O ] n,(cid:96) ∼ O ∂ n ∂ µ . . . ∂ µ (cid:96) O , (2.7)and have conformal dimensions∆ + ∆ + 2 n + (cid:96) + O (cid:0) /N (cid:1) . (2.8)Below we often omit the n and (cid:96) labels when talking about a family of double-trace op-erators. To obtain an optical theorem resembling the one in flat space, we would like toproject onto states created by products of single-trace operators |O ( y ) O ( y ) (cid:105) , ratherthan the often infinite sum over n and (cid:96) of the double-trace operators | [ O O ] n,(cid:96) ( y ) (cid:105) . Thiscan be achieved by relating these two states using the technique of conglomeration [18],which amounts to using the formula | [ O O ] n,(cid:96) ( y ) (cid:105) = (cid:90) dy dy |O ( y ) O ( y ) (cid:105)(cid:104) S [ O ]( y ) S [ O ]( y )[ O O ] n,(cid:96) ( y ) (cid:105) . (2.9)This shows that we can define a projector onto double-trace operators in terms of a doubleshadow integral |O O | = (cid:90) dy dy |O ( y ) O ( y ) (cid:105)(cid:104) S [ O ]( y ) S [ O ]( y ) | (cid:12)(cid:12)(cid:12) [ O O ] , (2.10)and thus (2.9) is just the projection |O O | [ O O ] n,(cid:96) (cid:105) = | [ O O ] n,(cid:96) (cid:105) . (2.11)The notation | [ O O ] means that we project onto the contributions from the double-tracesof the physical operators and discard contributions coming from the shadows, which, aswe will discuss below, can be generated when using this bi-local projector. Using thisprojector, together with (2.2) for the single-traces, we can write the one-loop four-pointfunction in (2.6) as A ( y i ) = (cid:88) O∈O s.t. (cid:104)O O |O|O O (cid:105) + (cid:88) O , O ∈O s.t. (cid:104)O O |O O |O O (cid:105) . (2.12)The important step in (2.12) is that we replaced the sum over double-trace operators witha double sum over the corresponding single-trace operators. This is already close to thesingle- and double-line cuts that appear in the flat-space optical theorem at one-loop.– 7 –he main difference of (2.12) with the flat space optical theorem is that in flat spaceone needs to sum only over cuts of internal lines, while if we express (2.12) in terms ofWitten diagrams it would also contain contributions from external line cuts. Anotherway to see this is that even the disconnected correlator for O = O and O = O hascontributions of the form (cid:104)O O |O O |O O (cid:105) , (2.13)while internal double line cuts in a diagram can only appear starting at one-loop. Thisproblem is resolved by acting on (2.12) with the double discontinuity. This procedureshifts the contributions of external double-traces to a higher order in N . In the contextof (1.5) that we propose for conformal correlation functions (and not for Witten diagramsspecifically), taking the double discontinuity suppresses the contributions of the externaldouble-trace operators [ O O ] and [ O O ]. We will expand on this further in section 2.3.We will make the definitions of the double discontinuity and the single discontinuitiesmore precise in sec. 2.3 but for now, let us mention that the double discontinuity can bewritten in the following factorized formdDisc t A ( y i ) = −
12 Disc Disc A ( y i ) . (2.14)The discontinuities on the right hand side are defined in terms of analytic continuations ofthe distances y and y to the negative real axis,Disc jk A ( y i ) = A ( y i ) | y jk → y jk e πi − A ( y i ) | y jk → y jk e − πi . (2.15)Note that each term in this discontinuity is defined through a Wick rotation of the twocoordinates y j and y k while we hold the other points Euclidean (or spacelike separated).The result (1.4) for the exchange of single-trace operators comes from the first termon the right hand side of (2.12) with the double discontinuity taken on both sides. Thisare simply the single-trace terms in the conformal block expansion of the correlator. Forthe more interesting result (1.5), let us use the explicit form of the projector (2.10) in thesecond term on the right hand side of (2.12). This gives (cid:88) O , O ∈O s.t. (cid:90) dy dy (cid:104)O O |O ( y ) O ( y ) (cid:105)(cid:104) S [ O ]( y ) S [ O ]( y ) |O O (cid:105)| (cid:12)(cid:12)(cid:12) [ O O ] . (2.16)We can now take the double discontinuity on the left hand side using (2.14), while onthe right hand side we can take Disc on the first correlator and Disc on the second.This gives the result (1.5). In the next subsections we provide a detailed proof of thisperturbative CFT optical theorem using results from harmonic analysis of the conformalgroup [35]. A conformal correlator can be expanded in s -channel conformal blocks as follows, A ( y i ) = T ( y i ) A ( z, z ) , A ( z, z ) = (cid:88) O c O c O g O ( z, z ) , (2.17)– 8 –ith the kinematical prefactor T ( y i ) = 1 y ∆ +∆ y ∆ +∆ (cid:18) y y (cid:19) ∆212 (cid:18) y y (cid:19) ∆342 , (2.18)where ∆ ij = ∆ i − ∆ j and the cross-ratios are defined as zz = y y y y , (1 − z )(1 − z ) = y y y y . (2.19)The t -channel OPE is obtained by exchanging the labels 1 and 3, thus A ( y i ) = T ( y i ) A ( z, z ) , A ( z, z ) = (cid:88) O c O c O g O (1 − z, − z ) , (2.20)Note that although A jklm ( y i ) is invariant under permutations of the jklm labels, the or-dering of the labels is meaningful in A jklm ( z, z ) because of the pre-factor T jklm ( y i ). Forthe conformal blocks we will also use the notation G O ( y k ) = T ( y k ) g O ( z, z ) , (2.21)and similarly for t -channel blocks.In order to perform harmonic analysis of the conformal group, one expands the four-point function not in conformal blocks but in conformal partial waves of principal seriesrepresentations ∆ = d + iν , ν ∈ R + [38]. A conformal correlator can be expanded in termsof s -channel conformal partial waves as follows A ( y i ) = (cid:88) ρ (cid:90) d + i ∞ d d ∆2 πi I ab (∆ , ρ )Ψ ab ) O ( y i ) + discrete , (2.22)where the operator O is labeled by the scaling dimension ∆ and a finite dimensionalirreducible representation ρ of SO ( d ), which we take to be bosonic. I ab is the spectralfunction carrying the OPE data, and it can be extracted from the correlator using theEuclidean inversion formula. We will assume that there are no discrete contributions. Theconformal partial waves are defined as a pairing of three-point structuresΨ ab ) O ( y i ) = (cid:90) dy (cid:104)O O O ( y ) (cid:105) ( a ) (cid:104)O O (cid:101) O † ( y ) (cid:105) ( b ) , (2.23)where a and b label different tensor structures in case the external operators have spin.The conformal partial wave Ψ ab ) O is related to the conformal block G ab ) O and to theblock for the exchange of the shadow byΨ ab ) O = S ( O O [ (cid:101) O † ]) bc G ac ) O + S ( O O [ O ]) ac G cb ) (cid:101) O . (2.24)The matrices S ( O i O j [ O k ]) ab are part of the action of the shadow transform (2.3) on three-point functions, (cid:104)O O S [ O ] (cid:105) ( a ) = S ( O O [ O ]) ab N O (cid:104)O O (cid:101) O (cid:105) ( b ) , (2.25)– 9 –ith N O as defined in (2.4). Acting with the shadow transform on an operator within athree-point structure also rotates into a different basis of tensor structures. The shadowcoefficients/matrices S act as a map between the two bases. Note that the inverse of S ( O O [ O ]) ab is (1 /N O ) S ( O O [ (cid:101) O ]) ab .The usual conformal block expansion (2.17) can be obtained from (2.22) by inserting(2.24) and using the identity [2] I ab (∆ , ρ ) S ( O O [ (cid:101) O † ]) bc = I bc (cid:0) (cid:101) ∆ , ρ (cid:1) S ( O O [ (cid:101) O ]) ba , (2.26)to replace the contribution of the shadow block with an extension of the integration regionto d − i ∞ , A ( y i ) = (cid:88) ρ (cid:90) d + i ∞ d − i ∞ d ∆2 πi C ab (∆ , ρ ) G ab ) O ( y i ) , (2.27)where C ab (∆ , ρ ) = I ac (∆ , ρ ) S ( O O [ (cid:101) O † ]) cb . (2.28)The conformal block decays for large real ∆ >
0, so the contour can be closed to the rightand the integral is the sum of residues − Res ∆ → ∆ ∗ C ab (∆ , ρ ∗ ) = (cid:88) I c I O ∗ ,a c I O ∗ ,b . (2.29)The sum over I in (2.29) is over degenerate operators with the quantum numbers (∆ ∗ , ρ ∗ ).Degeneracies among multi-trace operators are natural in expansions around mean fieldtheory. In section 2.2 we will use the partial wave expansion of the shadow transformed four-point function. To obtain it let us now apply the shadow transform in (2.3) to O and O on both sides of the partial wave expansion (2.22). Using (2.23) this gives (cid:104) S [ O ] S [ O ] O O (cid:105) = (cid:88) ρ (cid:90) d + i ∞ d d ∆2 πi I ab (∆ , ρ ) (cid:90) dy (cid:104) S [ O ] S [ O ] O ( y ) (cid:105) ( a ) (cid:104)O O (cid:101) O † ( y ) (cid:105) ( b ) . (2.30)From (2.25), we thus obtain the partial wave expansion of the shadow transformed corre-lator (cid:104) S [ O ] S [ O ] O O (cid:105) = (cid:88) ρ (cid:90) d + i ∞ d d ∆2 πi I S [1] S [2]34 ab (∆ , ρ ) Ψ (cid:101) (cid:101) ab ) O ( y i ) , (2.31) A simple example built with spin 1 operators are the families O µ (cid:3) n O ,µ and O µ ∂ µ ∂ ν (cid:3) n − O ν , whichwe wrote schematically. Both these sets of operators have quantum numbers ∆ = ∆ + ∆ + 2 n and ρ = • . – 10 –here I S [1] S [2]34 ab = I mb (∆ , ρ ) S ( O [ O ] O ) mn N O S ([ O ] (cid:101) O O ) na N O , Ψ (cid:101) (cid:101) ab ) O ( y i ) = (cid:90) dy (cid:104) (cid:101) O (cid:101) O O ( y ) (cid:105) ( a ) (cid:104)O O (cid:101) O † ( y ) (cid:105) ( b ) . (2.32)There are examples of the S coefficients computed in [35] which tell us that they have theappropriate zeroes to kill the double-trace poles in I and replace them with the polesfor the double-traces of the shadows, as would be appropriate for I S [1] S [2]34 . We are ready to begin the derivation of the perturbative CFT optical theorem (1.5). S S A ( y i ) in (1.5) is the coefficient of 1 /N in the correlator (cid:104) S [ O ] † S [ O ] † O O (cid:105) , and A ( y i ) is the coefficient of 1 /N in (cid:104)O O O O (cid:105) . Consider the following conformallyinvariant pairing of two four-point functions (cid:90) dy dy (cid:104)O O O O (cid:105)(cid:104) S [ O ] † S [ O ] † O O (cid:105) = (2.33) (cid:88) ρ,ρ (cid:48) (cid:90) d + i ∞ d d ∆2 πi d ∆ (cid:48) πi I ab (∆ , ρ ) I S [6] S [5]14 cd (∆ (cid:48) , ρ (cid:48) ) (cid:90) dy dy Ψ ab ) O ( y i ) Ψ (cid:101) (cid:101) cd ) O (cid:48) ( y i ) . To compute the y and y integrals, we use (2.23) and the following result for the pairingof the three-point structures by two legs, which is known as the bubble integral, (cid:90) dy dy (cid:104)O O O ( y ) (cid:105) ( a ) (cid:104) (cid:101) O † (cid:101) O † (cid:101) O (cid:48) † ( y (cid:48) ) (cid:105) ( b ) = (cid:16) (cid:104)O O O(cid:105) ( a ) , (cid:104) (cid:101) O † (cid:101) O † (cid:101) O † (cid:105) ( b ) (cid:17) µ (∆ , ρ ) yy (cid:48) δ OO (cid:48) , (2.34)with δ OO (cid:48) ≡ πδ ( s − s (cid:48) ) δ ρρ (cid:48) . Here µ (∆ , ρ ) is the Plancherel measure and the bracketsdenote a conformally invariant pairing of 3-point functions, given by (cid:16) (cid:104)O O O (cid:105) , (cid:104) (cid:101) O † (cid:101) O † (cid:101) O † (cid:105) (cid:17) = (cid:90) dy dy dy volSO( d + 1 , (cid:104)O O O (cid:105)(cid:104) (cid:101) O † (cid:101) O † (cid:101) O † (cid:105) . (2.35)Using (2.23) and the bubble integral in (2.34) we find (cid:90) dy dy Ψ ab ) O ( y i )Ψ (cid:101) (cid:101) cd ) O (cid:48) ( y i ) = (cid:16) (cid:104)O O (cid:101) O † (cid:105) ( b ) , (cid:104) (cid:101) O † (cid:101) O † O(cid:105) ( c ) (cid:17) µ (∆ , ρ ) δ OO (cid:48) Ψ ad ) O ( y i ) . (2.36)We can now plug (2.36) into (2.33) which gives (cid:90) dy dy (cid:104)O O O O (cid:105)(cid:104) S [ O ] † S [ O ] † O O (cid:105) = (2.37) (cid:88) ρ (cid:90) d + i ∞ d d ∆2 πi I ab (∆ , ρ ) I S [6] S [5]14 cd (∆ , ρ ) (cid:16) (cid:104)O O (cid:101) O † (cid:105) ( b ) , (cid:104) (cid:101) O † (cid:101) O † O(cid:105) ( c ) (cid:17) µ (∆ , ρ ) Ψ ad ) O ( y i ) . – 11 –n the next steps we will show that the factor (cid:0) (cid:104)O O (cid:101) O † (cid:105) , (cid:104) (cid:101) O † (cid:101) O † O(cid:105) (cid:1) in (2.37) , alongwith the various shadow coefficients, will cancel the contribution of the OPE coefficients c MFT56[56] in the spectral functions I and I S [6] S [5]14 . In the simple case where at least one ofthe spectral functions in (2.37) belong to scalar MFT correlators (which requires pairwiseequal operators) this is particularly easy to see, since [35] I MFT (∆ , ρ ) = µ (∆ , ρ ) (cid:16) (cid:104)O O (cid:101) O † (cid:105) , (cid:104) (cid:101) O † (cid:101) O † O(cid:105) (cid:17) S ([ (cid:101) O ] (cid:101) O O ) S ( O [ (cid:101) O ] O ) , (2.38)so that the pairing of three-point functions can be canceled directly with one of the spectralfunctions. The general case is less obvious because the cancellation happens on the levelof OPE coefficients, not spectral functions. Here we use (2.31) in (2.37), and extend therange of the principal series integral as in (2.27) by repeated use of (2.26). This gives (cid:90) dy dy (cid:104)O O O O (cid:105)(cid:104) S [ O ] † S [ O ] † O O (cid:105) = (cid:88) ρ (cid:90) d + i ∞ d − i ∞ d ∆2 πi I ab I md S ( O O [ (cid:101) O † ]) dl S ( O [ O ] O ) mn N O S ([ O ] (cid:101) O O ) nc N O (cid:16) (cid:104)O O (cid:101) O † (cid:105) ( b ) , (cid:104) (cid:101) O † (cid:101) O † O(cid:105) ( c ) (cid:17) µ (∆ , ρ ) G al ) O ( y i ) . (2.39)Using (2.28) we can express (2.39) as (cid:90) dy dy (cid:104)O O O O (cid:105)(cid:104) S [ O ] † S [ O ] † O O (cid:105) = (cid:88) ρ (cid:90) d + i ∞ d − i ∞ d ∆2 πi C ak C md Q km O G ad ) O , (2.40)where, Q km O = S ( O [ O ] O ) mn N O S ([ O ] (cid:101) O O ) nc N O (cid:16) (cid:104)O O (cid:101) O † (cid:105) ( b ) , (cid:104) (cid:101) O † (cid:101) O † O(cid:105) ( c ) (cid:17) µ (∆ , ρ ) S ( O O [ O † ]) kb N O . (2.41)Next we analyze the pole structure of the spectral function in (2.40) and close the integra-tion contour to obtain the block expansion. First let us consider the simple poles at thedimensions of the double-trace operators O [56] in each of C and C . We will showthat Q O (∆ , ρ ) has a zero at each of these dimensions, canceling one of the two polesfrom C and C . This ensures that in the MFT limit the spectral function in (2.40)has a simple pole for each double-trace dimension. This can be seen explicitly in specificexamples for the S coefficients computed in [35], but in general let us note the followingidentity, which can be derived by applying Euclidean inversion on the expansion (2.22) forthe MFT correlator [35] I , MFT ab (∆ , ρ ) µ (∆ , ρ ) (cid:16) (cid:104)O † O † (cid:101) O † (cid:105) ( b ) , (cid:104) (cid:101) O (cid:101) O O(cid:105) ( c ) (cid:17) = S ([ (cid:101) O ] (cid:101) O O ) cl S ( O [ (cid:101) O ] O ) la . (2.42)Since all operators are bosonic, (2.42) can be expressed as( − J I , MFT ab (∆ , ρ ) µ (∆ , ρ ) S ( O [ O ] O ) mn N O S ([ O ] (cid:101) O O ) nc N O (cid:16) (cid:104)O O (cid:101) O † (cid:105) ( b ) , (cid:104) (cid:101) O † (cid:101) O † O(cid:105) ( c ) (cid:17) = δ ma . (2.43)– 12 –sing (2.27) and (2.41), we rephrase (2.43) as C , MFT ak (∆ , ρ ) Q km O (∆ , ρ ) = δ ma . (2.44)Let (∆ , ρ ) be (∆ ∗ , ρ ∗ ) for the double-trace operators O I [56] ∗ , where I labels degenerate oper-ators, as discussed previously. The coefficient C , MFT ak has a simple pole at this locationand therefore (2.44) implies that Q km O (∆ , ρ ) is its inverse matrix and has a correspondingzero at this value. Evaluated at ∆ = ∆ ∗ , (2.44) takes the form (cid:32)(cid:88) I c MF T,I ∗ ,a c MF T,I ∗ ,k (cid:33) q km = δ ma , (2.45)where c MF T,I ∗ ,a c MF T,I ∗ ,k is the contribution to the residue of C , MFT ak corresponding to O I [56] ∗ and q km is the coefficient of the first order zero of Q km O at ∆ ∗ .Note that the matrix of OPE coefficients c MF T,I ∗ ,a c MF T,I ∗ ,k for a specific double-traceoperator is singular. In general, (2.44) and (2.45) imply that there are sufficiently manydegenerate double-trace families so the matrix obtained by summing over all of them is notsingular. In the case where there is a unique tensor structure, such as when O and O arescalars, the 1 × c MF T,J ∗ ,m , we obtain c MF T,I ∗ ,k q km c MF T,J ∗ ,m = δ IJ . (2.46)Finally, using (2.29) and (2.46) we obtain the contribution of the (∆ ∗ , ρ ∗ ) pole to thespectral integral in (2.40) − Res ∆ → ∆ ∗ C ak C md Q km O G ad ) O (cid:12)(cid:12) ρ → ρ ∗ = (cid:88) I c I ∗ ,a c I ∗ ,d G ad )[56] ∗ ( y i ) . (2.47)Given that this is precisely the contribution of the double-trace operators [ O O ] to thecorrelator A ( y i ), this shows that the conformally invariant pairing we started with in(2.33) computes precisely this contribution, to leading order in 1 /N because we used MFTexpressions along the way. Thus (cid:16) O (cid:0) /N (cid:1)(cid:17) A ( y k ) (cid:12)(cid:12) [ O O ] = (cid:90) dy dy A ( y k ) S S A ( y k ) (cid:12)(cid:12)(cid:12) [ O O ] . (2.48)In the context of the the projector defined in the previous section in (2.10), this result canbe phrased as (cid:88) n,(cid:96),I (cid:12)(cid:12)(cid:12) [ O O ] In,(cid:96) (cid:12)(cid:12)(cid:12) = |O O | + O (cid:0) /N (cid:1) . (2.49)The labels n, (cid:96) sum over the double-trace operators with different dimensions and spins,while I sums over degenerate operators. The projection | [ O O ] appears on the two sidesof (2.48) for different reasons. On the left hand side it selects one family of double-traceoperators among all the operators appearing in the OPE, while on the right hand side– 13 –t serves to discard poles from shadow operators that we would pick up when we closethe contour in (2.40). For example, it is evident from the first equation in (2.32) that Q (∆ , ρ ) has poles at the double-traces O I [ (cid:101) (cid:101) composed of (cid:101) O and (cid:101) O and we pick up thesecontributions too. Let us take for simplicity the case with O and O scalars and O withinteger spin (cid:96) in 4 dimensions. The corresponding three-point function has only one tensorstructure and the expressions for S ( O [ O ] O ) and S ([ O ] (cid:101) O O ) are known [35] S ( O [ O ] O ) ∼ Γ (cid:16) ∆ + (cid:101) ∆ − ∆+ (cid:96) (cid:17) Γ (cid:16) ∆ +∆ − ∆+ (cid:96) (cid:17) , S ([ O ] (cid:101) O O ) ∼ Γ (cid:16) (cid:101) ∆ + (cid:101) ∆ − ∆+ (cid:96) (cid:17) Γ (cid:16) ∆ + (cid:101) ∆ − ∆+ (cid:96) (cid:17) . (2.50)Therefore, the product has poles at the double-traces [ (cid:101) O (cid:101) O ] (and zeroes at the double-traces [ O O ]). To determine such contributions in the same way as above, we shouldexpress I in terms of I S [5] S [6] by inverting (2.31) at (2.37) in the derivation above.We can follow the remaining steps and use an identity for the MFT spectral function similarto (2.42) (see [35]). This gives the contribution from the double-traces of shadows O I [ (cid:101) (cid:101) tobe of the same form as in (2.47). Note that in the case of scalar MFT correlators, thesepoles in Q (∆ , ρ ) are canceled by zeros in the MFT spectral function (2.38) and hence wedo not have these contributions from the double-traces of shadows. N expansion Equation (2.48) by itself is not very useful because of the O ( N ) error term. External doubletraces contribute already at O ( N ) so that their contributions at O ( N ) are already notattainable by (2.48). This problem is solved by taking the double discontinuity of (2.48),which will ensure that both sides of the equation are valid to O ( N ) for all double traces[ O O ], both external and internal.The discontinuities are given by commutators in Lorentzian signature, hence we ana-lytically continue the correlators to Lorentzian signature and take the difference of differentoperator orderings. Euclidean correlators can be continued to Wightman functions usingthe following prescription [39] (cid:104)O ( t , (cid:126)x ) O ( t , (cid:126)x ) · · · O n ( t n , (cid:126)x n ) (cid:105) = lim (cid:15) i → (cid:104)O ( t − i(cid:15) , (cid:126)x ) · · · O n ( t n − i(cid:15) n , (cid:126)x n ) (cid:105) , (2.51)with τ i = it i where τ is Euclidean and t Lorentzian time. The limits are taken assuming (cid:15) > (cid:15) > · · · > (cid:15) n .Let us assume without loss of generality that O is in the future of O , that O isin the future of O and that all other pairs of operators are spacelike from each other.Now we apply the epsilon prescription to (cid:104)O O O O (cid:105) with (cid:15) > (cid:15) and (cid:15) > (cid:15) . Therelative ordering of epsilons is unimportant for the spacelike separated pairs. This givesthe Lorentzian correlator A (cid:9) = (cid:104)O O O O (cid:105) , which is equal to the time ordered correlatorfor the assumed kinematics. Similarly, we obtain A (cid:8) = (cid:104)O O O O (cid:105) from the ordering (cid:15) < (cid:15) , (cid:15) < (cid:15) . The Euclidean configurations A Euc correspond to the mixed orderings (cid:15) > (cid:15) , (cid:15) < (cid:15) and (cid:15) < (cid:15) , (cid:15) > (cid:15) . We can then relate the dDisc t to these four– 14 –onfigurations bydDisc t A ( y i ) = A Euc ( y i ) − (cid:0) A (cid:9) ( y i ) + A (cid:8) ( y i ) (cid:1) = − (cid:104) [ O , O ] [ O , O ] (cid:105) . (2.52)Using (2.17) this gives the conventional definition of the double discontinuity [1]dDisc t A ( y i ) = T ( y i ) (cid:20) cos (cid:0) π ( a + b ) (cid:1) A ( z, z ) −− (cid:16) e iπ ( a + b ) A ( z, z (cid:9) ) + e − iπ ( a + b ) A ( z, z (cid:8) ) (cid:17) (cid:21) , (2.53)where a = ∆ / b = ∆ / z (cid:9) and z (cid:8) denote that z is analytically continued by afull circle counter-clockwise and clockwise around z = 1, respectively. The gluing of correlators on the right hand side in (2.48), with the shadow integralsnow written explicitly, is a sum of terms of the form1 N O N O (cid:90) dy dy dy dy (cid:104)O O O O (cid:105) (cid:104) (cid:101) O (cid:101) O † (cid:105) (cid:104) (cid:101) O (cid:101) O † (cid:105) (cid:104)O O O O (cid:105) . (2.54)Note that O = O and O = O but we have used the different labels to denote the inser-tion points. We can apply the same (cid:15) -prescriptions on (2.54) while we hold y , y , y , y to be Euclidean. Taking the same combinations as in (2.52) we arrive at1 N O N O (cid:90) dy dy dy dy (cid:104) [ O , O ] O O (cid:105) (cid:104) (cid:101) O (cid:101) O † (cid:105) (cid:104) (cid:101) O (cid:101) O † (cid:105) (cid:104)O O [ O , O ] (cid:105) . (2.55)The commutators in (2.55) give discontinuities in the correlator as defined in (2.15).We will now show that taking the dDisc of (2.33) ensures that the external double-traces [ O O ] and [ O O ] which usually appear at O ( N ) are suppressed in 1 /N so thatthey appear at the same order as other double trace operators. To this end, let us brieflydiscuss the 1 /N expansion of correlators and associated CFT data. The leading contri-bution is A MFT , which is simply the disconnected correlator if the external operators arepairwise equal and is absent otherwise. Because of this, the only operators that appear at O ( N ) are the ones appearing in the disconnected correlator, c ij [ O i O j ] n,(cid:96) = c MFT ij [ O i O j ] n,(cid:96) + 1 N c (1) ij [ O i O j ] n,(cid:96) + · · · , ∆ [ O i O j ] n,(cid:96) = ∆ i + ∆ j + 2 n + (cid:96) + 1 N γ [ O i O j ] n,(cid:96) + · · · . (2.56)Other double-trace operators can only appear at higher orders in the OPE, therefore c ij [ O k O l ] n,(cid:96) = 1 N c (1) ij [ O k O l ] n,(cid:96) + · · · , i, j (cid:54) = k, l . (2.57)The analytic continuation of a t -channel conformal block to the Regge sheet is given bythe following simple expression g O (cid:0) − z, (1 − z ) e iβ (cid:1) = e iβ τ O g O (1 − z, − z ) . (2.58) The relation between (2.52) and (2.53) can be obtained by assigning the phases y ij → y ij e ± iπ to thetimelike distances y and y . – 15 –s a result, the action of the single and double discontinuities on the t -channel blockexpansion in (2.20) is given byDisc A ij ( y k ) = (cid:88) O i sin (cid:0) π ( τ O − ∆ − ∆ ) (cid:1) c ij O c O G ij O ( y k ) , Disc A ji ( y k ) = (cid:88) O i sin (cid:0) π ( τ O − ∆ − ∆ ) (cid:1) c O c ij O G ij O ( y k ) , (2.59)dDisc t A ( y k ) = (cid:88) O (cid:0) π ( τ O − ∆ − ∆ ) (cid:1) sin (cid:0) π ( τ O − ∆ − ∆ ) (cid:1) c O c O G O ( y k ) . The sines in the expansions are responsible for suppressing the contribution of externaldouble-traces. Therefore, using (2.59), (2.56) and (2.57), the leading contribution to thediscontinuity of a correlator is O (1 /N )Disc A ( y i ) = 1 N Disc A tree ( y i ) + O (1 /N ) = (2.60) (cid:88) O =[ O O ] iπ γ O N c MFT14 O c O G O + (cid:88) O(cid:54) =[ O O ] i sin (cid:0) π ( τ O − ∆ − ∆ ) (cid:1) c (1)14 O N c O G O , and similarly the leading contribution to the double discontinuity is O (1 /N )dDisc t A ( y i ) = 1 N dDisc t A ( y i ) + O (1 /N ) . (2.61)In particular, when acting with (2.14) on the left hand side of (2.33) we haveDisc A = (cid:88) O =[ O O ] iπ γ O N c MFT32 O c O G O + (cid:88) O(cid:54) =[ O O ] i sin (cid:0) π ( τ O − ∆ − ∆ ) (cid:1) c (1)32 O N c O G O , Disc A (cid:101) (cid:101) = (cid:88) O =[ O O ] iπ γ O N c (cid:101) (cid:101) O c MFT14 O G (cid:101) (cid:101) O + (cid:88) O(cid:54) =[ O O ] i sin (cid:0) π ( τ O − ∆ − ∆ ) (cid:1) c (cid:101) (cid:101) O c (1)14 O N G (cid:101) (cid:101) O . (2.62)Since every term is these expansions already has an explicit factor of 1 /N , the only oper-ators that can contribute at this order are the ones with c O = O ( N ) or c (cid:101) (cid:101) O = O ( N ),which are the double-traces O = [ O O ] and O = [ (cid:101) O (cid:101) O ]. Applying the discontinuities toboth sides of (2.48) leaves us with one of our main results, the perturbative optical theo-rem for the contributions of double-trace operators to the 1-loop dDisc of the correlator,as stated in the introductiondDisc t A − loop ( y i ) (cid:12)(cid:12)(cid:12) d.t. = − (cid:88) O , O ∈ s.t. (cid:90) dy dy Disc A ( y k ) S S Disc A ( y k ) (cid:12)(cid:12)(cid:12) [ O O ] . (2.63)The integrals in this formula are over Euclidean space.– 16 – Review of flat space amplitudes
In this section we review the Regge limit in D -dimensional flat space. Then we reviewthe optical theorem in impact parameter space and explain how the notion of a one-loopvertex function arises. Not only does this serve as a hopefully more familiar introductionbefore discussing the same concepts in AdS, but it also provides the results we need laterwhen we take the flat space limit of our AdS results and match them to known flat spaceexpressions. To mimic the 1 /N expansion in the CFT, it will be convenient to define anexpansion in G N for the flat space scattering amplitude A ( s, t ) = 2 G N π A tree ( s, t ) + (cid:18) G N π (cid:19) A ( s, t ) + . . . , (3.1)and we use an identical expansion for the phase shifts δ ( s, b ) defined below. We start by introducing the impact parameter representation, following [40]. Let us con-sider a tree-level scattering process with incoming momenta k and k that have largemomenta along different lightcone directions. For simplicity we assume for now that allexternal particles are massless scalars. This process is dominated by t -channel exchangediagrams of the type k k k k q (3.2)and the amplitude can be expressed in terms of the Mandelstam variables s = − ( k + k ) , t = − ( k − k ) . (3.3)The amplitude now depends only on s and the momentum exchange q in the transversedirections, because we are considering the following configuration of null momenta, writtenin light-cone coordinates p = ( p u , p v , p ⊥ ) k µ = (cid:18) k u , q k u , q (cid:19) , k µ = (cid:18) q k v , k v , − q (cid:19) ,k µ = (cid:18) k u , q k u , − q (cid:19) , k µ = (cid:18) q k v , k v , q (cid:19) . (3.4)Notice that we reserve the letter q for ( D − k u ∼ k v → ∞ the Mandelstams are given by s ≈ k u k v , t = − q . (3.5)– 17 –he tensor structures in such amplitudes are fixed in terms of the on-shell three-pointamplitudes. For the case with two external scalars and an intermediate particle (labeled I ) with spin J there is only one possible tensor structure for the three-point amplitudesgiven by ˜ A I = a J ( (cid:15) I · k ) J , ˜ A I = a J ( (cid:15) I · k ) J , (3.6)where we encode traceless and transverse polarization tensors in terms of vectors satisfying (cid:15) i = (cid:15) i · k i = 0. We can then write the four-point amplitude as A ( m,J ) ( s, t ) = (cid:80) (cid:15) I ˜ A I ˜ A I t − m ≈ a J s J t − m , (3.7)where we used that for large s the sum over polarizations is dominated by (cid:15) Iu k u ∼ k u and (cid:15) Iv k v ∼ k v . The s J behavior is naively problematic at high energies, especially ifthe spectrum contains particles of large spin, as is the case in string theory. However,boundedness of the amplitude in the Regge limit means there is a delicate balance betweenthe infinitely many contributions in the sum over spin. The precise framework to describethis phenomenon is Regge theory [41], which was reviewed for flat space in [30, 42].In the Regge limit one has to consider the particle with the maximum spin j ( m ) foreach mass. The function j ( m ) is called the leading Regge trajectory and the contributionsfrom these particles get resummed into an effective particle with continuous spin j ( t ). Inthis work we will focus on the leading trajectory with vacuum quantum numbers known asthe Pomeron. At tree-level the amplitude for Pomeron exchange factorizes into three-pointamplitudes involving a Pomeron and the universal Pomeron propagator β ( t ). For example,in the case of 4-dilaton scattering in type IIB strings we have A tree ( s, t ) = 8 α (cid:48) A P β ( t ) A P (cid:18) α (cid:48) s (cid:19) j ( t ) , (3.8)with β ( t ) = 2 π Γ( − α (cid:48) t )Γ(1 + α (cid:48) t ) e − iπα (cid:48) t . (3.9) A ijP are the three-point amplitudes between the external scalars and the Pomeron withthe s -dependence factored out and normalized such that in the case of 4-dilaton scatter-ing A ijP = 1. This is convenient since later on, when we consider more general stringstates with spin, the string three-point amplitudes defined this way will contain just tensorstructures. Diagrammatically we can write (3.8) as12 34 P = 12 P × P × G N π α (cid:48) β ( t ) (cid:18) α (cid:48) s (cid:19) j ( t ) . (3.10) The couplings a J are dimensionful, [ a J ] = 3 − D/ − J , and accommodate for higher derivatives in thecouplings to higher spin fields. In string theory the dimensionful scale is α (cid:48) and the dimensionless couplingsare all proportional to the string coupling g s . – 18 –mplitudes involving a Pomeron can be computed in string theory using the Pomeronvertex operator [28, 43, 44]. The factorization into three-point functions and a Pomeronpropagator holds for general external string states [28, 45]. Next we consider the expression for the two-line cut of the one-loop amplitude in the impactparameter representation, which will be given in terms of the tree-level pieces we havediscussed so far. The two-line cut receives a contribution from two-Pomeron exchange,which is the leading term in the Regge limit of the one-loop amplitude. Consider thefollowing configuration of momenta k k k k l l k k . (3.11)The external momenta are again in the configuration (3.4) with Mandelstams (3.5). Theoptical theorem tells us to cut the internal lines of the diagram, putting the correspondinglegs on-shell. This implies the following equation for the discontinuity of the amplitude2 Im A ( s, q ) = (cid:88) m ,ρ ,(cid:15) m ,ρ ,(cid:15) (cid:90) dl (2 π ) D πiδ ( k + m ) 2 πiδ ( k + m ) A ( s, l ) ∗ A ( s, l ) , (3.12)where one sums over all possible particles 5 and 6 with masses m , in Little group repre-sentations ρ and with polarization tensors (cid:15) . The sums over polarizations can be evaluatedusing completeness relations. In order to remove the delta functions we express k and k in terms of l and the external momenta (3.4). Then we write the loop momentumas l µ = ( l u , l v , q ) and use the delta-functions to fix the forward components of the loopmomentum l u and l v to l u = m + q + q · q k v , l v = − m + q − q · q k u , (3.13)leaving only the transverse integration over q . We arrive at the equationIm A ( s, q ) = (cid:88) m ,ρ ,(cid:15) m ,ρ ,(cid:15) (cid:90) dq dq (2 π ) D − δ ( q − q − q )4 s A ( s, q ) ∗ A ( s, q ) , (3.14)where we introduced the transverse momentum q = q − q to write the expression in amore symmetrical way. Using that the tree-level amplitudes are given in the Regge limit– 19 –2 34 P P ∼ (cid:88) m ,ρ ,(cid:15) m ,ρ ,(cid:15) (cid:90)
15 64 P ×
52 36 P Figure 2 . Optical theorem in the Regge limit in terms of Feynman diagrams. The tree-levelcorrelators are dominated by s -channel Pomeron exchange. The ellipses on the l.h.s. indicate thatall string excitations are taken into account. by Pomeron exchange, we can write (3.14) diagrammatically as in figure 2. The opticaltheorem can be simplified even further by transforming it to impact parameter space. Tothis end the amplitude is expressed in terms of the impact parameter b , which is a vectorin the transverse impact parameter space R D − , using the following transformation δ ( s, b ) = 12 s (cid:90) dq (2 π ) D − e iq · b A ( s, t ) . (3.15)We can use this definition together with (3.14) to computeIm δ ( s, b ) = 12 (cid:88) m ,ρ ,(cid:15) m ,ρ ,(cid:15) δ ( s, − b ) ∗ δ ( s, b ) . (3.16)We conclude that the impact parameter representation absorbs the remaining phase spaceintegrals in the optical theorem, resulting in a purely multiplicative formula. In fact, in thecase where the particles on the left and right of the diagram do not change (i.e. 1,5,2 and3,6,4 are identical particles), such a statement holds to all-loops, leading to exponentiationof the tree-level phase shift, which is the basis for the famous eikonal approximation. Another notion we will use is that of the vertex function, which arises when combiningthe optical theorem (3.14) with the factorization of the tree-level amplitudes (3.8) intothree-point amplitudes. By combining the two results one sees that the sums over particlesand their polarizations factorize into separate sums for particles 5 and 6, which we call thevertex function
V V ( q , q ) ≡ (cid:88) m ,ρ ,(cid:15) A P ( q ) A P ( q ) . (3.17)Moreover, such a sum over representations and polarizations for each mass is given bytree-level unitarity as the residue of the four-point amplitudes with two external PomeronsRes k = − m A P P ( k , q , q ) = (cid:88) ρ ,(cid:15) A P ( q ) A P ( q ) . (3.18)– 20 –n terms of diagrams this reads V ( q , q ) ≡ (cid:88) m ,ρ ,(cid:15) P × P = (cid:88) m Res k = − m P P . (3.19)The vertex function combines all information about the exchanges of possibly spinningparticles 5 and 6 into a single scalar function. In terms of the vertex function the opticaltheorem (3.14) in the Regge limit becomesIm A ( s, q ) = − s (cid:90) dq dq (2 π ) D − δ ( q − q − q ) (3.20) (cid:18) α (cid:48) (cid:19) β ( t ) ∗ β ( t ) V ( q , q ) (cid:18) α (cid:48) s (cid:19) j ( t )+ j ( t ) , where t i = − q i . Since it will be important later to compare tensor structures in AdS and flat space, wewill provide here some more details on the tensor structures of the three-point amplitudesthat appear in the unitarity cut of the four-point amplitude A P P discussed above. Theexternal momentum k and the exchanged momentum l , with light-cone components givenin the Regge limit by (3.13), fix the momentum k = k − l as shown in the figure below.We may, however, change frame such that k has no transverse momentum [45]. Suchchange of frame does not alter the fact that the light-cone components of l are subleading.The same applies to l . Thus in the Regge limit we can safely write k k l l k k ≈ (cid:18) k u , m k u , (cid:19) ,l ≈ (0 , , q ) , k = k − l ,l ≈ (0 , , q ) , k = k + l . (3.21)We focus on the three-point amplitude A P ( q ), which is related to the four-point ampli-tude via the tree-level unitarity (3.18). In this relation there appears a sum over a basisof possible polarizations (cid:15) , which can be evaluated using completeness relations, e.g. formassive bosons [46] (cid:88) (cid:15) (cid:15) µ ...µ | ρ | (cid:15) ν ...ν | ρ | = P µ γ . . . P µ ρ γ ρ π γ ...γ | ρ | ; σ ...σ | ρ | ρ P ν σ . . . P ν ρ σ ρ , (3.22)where π ρ is the projector to the irreducible SO ( D −
1) representation ρ and P µ ν = δ µν − k µ k ν k , (3.23)– 21 –s a projector to the space transverse to k . We will always absorb the projectors P µ ν into the three-point amplitudes, i.e. consider amplitudes in a transverse configuration.That means that the indices corresponding to particle 5 have to be constructed from theprojected momenta of the other particles, which are identical P µ ν l µ = P µ ν k µ . (3.24)Apart from that, massive particles can also have a longitudinal polarization v which satisfies v · k = 0 , v = 1 , (3.25)and is given in this frame explicitly by v µ = 1 m (cid:18) k u , − m k u , (cid:19) . (3.26)For the case that particle 1 is a scalar, we can then construct A P in terms of the followingmanifestly transverse tensor structures A Pm ,ρ , µ = | ρ | (cid:88) k =0 a km ,ρ ( t ) i | ρ | √ α (cid:48)| ρ |− k v µ . . . v µ k q µ k +1 . . . q µ | ρ | , (3.27)where we introduced boldface indices µ as multi-indices that stand for the | ρ | indices forthe irrep ρ . By abuse of language we defined the vector q ≡ (0 , , q ), since q is transverse.If particle 1 carries spin as well, as will be the case for the gravitons considered later on,we construct the polarization tensors out of the vector ξ = ( ξ u , ξ v , (cid:15) ). In this case, againdefining (cid:15) ≡ (0 , , (cid:15) ), the amplitudes take the following form in the Regge limit A Pm ,ρ , µ = (cid:96) (cid:88) n =0 | ρ |− n (cid:88) k =0 a k,nm ,ρ ( t ) i | ρ | √ α (cid:48)| ρ | + (cid:96) − n − k ( (cid:15) · q ) (cid:96) − n (cid:15) µ . . . (cid:15) µ n v µ n +1 . . . v µ n + k q µ n + k +1 . . . q µ | ρ | , (3.28)as can be checked by comparing with the explicit amplitudes computed in [45]. Thesechoices for the tensor structures are particularly convenient since q · v = (cid:15) · v = 0.Contact with the momentum frame used in the previous subsections is made by identifyingthe Lorentz invariant A P P . Our goal in this section is to compute the Regge limit of a scalar four-point function in aperturbative large N CFT at one-loop and finite ∆ gap . At finite ∆ gap we have to considerthe t -channel exchange of all possible double-trace operators and also single-trace operators,which are respectively dual to tidal excitations of the external scattering states and tolong-string creation in the string theory context. It was shown in [34] that the exchangeof single-trace operators dual to the long-string creation effects is subleading in the Regge– 22 – y y y y + y − Figure 3 . Kinematics in the central Poincar´e patch with coordinates y i . Time is on the verticalaxis, transverse directions are suppressed. limit. Therefore we only need to consider the exchange of double-trace operators. This iswhere the new perturbative CFT optical theorem (1.5) takes a central role, as it allows usto compute the contributions of double-trace operators to the correlator starting from thecorresponding tree-level correlators. The contribution of the leading Regge trajectory tothe scalar tree-level correlators is known to leading order in the Regge limit [29, 30].In this section we will therefore study (1.5) in the Regge limit, and this time we expandthe tree-level correlators in the s -channel. In the Regge limit the four external points are inLorentzian kinematics as depicted in figure 3. In this configuration all distances betweenpoints are spacelike except for y , y <
0. The Regge limit is reached by sending thefour-points to infinity along the light cones y +1 → −∞ , y +2 → + ∞ , y − → −∞ , y − → + ∞ . (4.1)The Regge limit can be directly applied to the left hand side of (1.5). The terms onthe Regge sheets A (cid:9) ( y i ) and A (cid:8) ( y i ) are dominant over the Euclidean terms in this limit.However, we cannot apply the Regge limit directly to the right hand side of (1.5) as theshadow integrals range over Euclidean configurations. Hence we will apply Wick rotationson the points y , y , y , y to obtain a gluing of the discontinuities of Lorentzian correlators.We will assume that in the Regge limit the dominant contribution to the gluing formulacomes from the domain where the individual tree-level correlators are in the Regge limitthemselves. We do not provide a proof of this assumption but we justify it in section 4.1.When each four-point function in (2.54) is in the Regge limit, the points y , y , y , y are placed in the same positions as y , y , y , y in fig. 3, respectively. Thus y is in thefuture of y and this pair is spacelike from y , y , y , y . Similarly, y is in the future of y and is spacelike from y , y , y , y . For the chosen kinematics we put the pair y , y inanti-time order using the epsilon prescription of (2.51) with (cid:15) > (cid:15) , and the pair y , y intime order using (cid:15) > (cid:15) . Applying this on (2.63) gives the following formula for the Regge– 23 –imit of the double discontinuitydDisc t A ( y k ) (cid:12)(cid:12)(cid:12) d.t. = − (cid:88) O , O N O N O (cid:90) dy dy dy dy (cid:104) [ O , O ] O O (cid:105) tree (cid:104) (cid:101) O (cid:101) O † (cid:105)(cid:104) (cid:101) O (cid:101) O † (cid:105)(cid:104)O O [ O , O ] (cid:105) tree (cid:12)(cid:12)(cid:12) [ O O ] . (4.2)The relative ordering between (cid:15) , (cid:15) and (cid:15) , (cid:15) is irrelevant as the pairs, appearing in thetwo-point functions on the right hand side of (4.2), are spacelike separated in the Reggeconfiguration.In this section we define the discontinuities as the commutators inserted into the fullyLorentzian correlatorsDisc A ( y i ) := (cid:104)O O [ O , O ] (cid:105) = A (cid:9) ( y i ) − A ( y i ) , Disc A ( y i ) := (cid:104) [ O , O ] O O (cid:105) = A ( y i ) − A (cid:8) ( y i ) . (4.3)This definition differs slightly from the one in (2.15). Stripping out the appropriate pre-factors from (4.3), one can check that these single discontinuities can be equivalently definedas Disc A ( z, z ) := e iπ ( a + b ) A ( z, z (cid:9) ) − e − iπ ( a + b ) A ( z, z ) , Disc A ( z, z ) := A ( z, z ) − A ( z, z (cid:8) ) , Disc A ( z, z ) = e − iπ ( a + b ) A ( z, z ) − e iπ ( a + b ) A ( z, z (cid:8) ) . (4.4)Starting from the discontinuity defined in (2.15), these expressions result from continuinganother half circle in z , so that the different terms are either evaluated at the originalposition or continued a full circle around 1. The extra phase comes from the additionalWick rotations. The final result matches the definition of the discontinuity in [42]. Notethat z is continued an extra half circle in opposite directions for the first two lines in (4.4),so that with these definitions the relation to the dDisc in (2.14) remains valid. Thereforethe optical theorem in the Regge limit can still be expressed asdDisc t A − loop ( y i ) (cid:12)(cid:12)(cid:12) d.t. = − (cid:88) O , O ∈ s.t. (cid:90) dy dy Disc A ( y k ) S S Disc A ( y k ) (cid:12)(cid:12)(cid:12) [ O O ] , (4.5)with the discontinuities as defined in the first and third lines of (4.4), and the gluing andshadow integrals now ranging over Minkowski space. This formula is also depicted in figure4 in terms of Witten diagrams.We should also note that for real z, z , Disc in the third line of (4.4) is related toDisc in the first line byDisc A ( z, z ) = − (cid:0) Disc A ( z, z ) (cid:1) ∗ (cid:12)(cid:12)(cid:12) ( a,b ) → ( − b, − a ) , < z, z < . (4.6)Applied to the correlators appearing in (4.5) this readsDisc A ( z, z ) = − (cid:0) Disc A ( z, z ) (cid:1) ∗ (cid:12)(cid:12)(cid:12) → . (4.7) For t -channel blocks, the new definitions for the discontinuities in (4.4) are related to the old definitionin (2.15) by a phase, for example, for Disc the relative phase is e iπτ O / . – 24 –Disc t
12 43 P P ∼ (cid:88) O , O (cid:90) Disc
52 63 P Disc (cid:101) (cid:101) P Figure 4 . Optical theorem in the Regge limit in terms of Witten diagrams. The tree-level cor-relators are dominated by s -channel Pomeron exchange. The external operators are scalars, while O and O are summed over all states that couple to the external scalars and the Pomeron (tidalexcitations). The ellipses on the l.h.s. indicate that all string excitations are taken into account. To benefit from this useful relation, we will always strip out a pre-factor such that we obtainthe correlator A ( z, z ) on the right hand side of (4.5). Otherwise we would have to usethe second line of (4.4) for Disc . Finally, in the Regge limit the analytically continuedcorrelators are dominant over the Euclidean contributions so that we haveDisc A ( z, z ) ≈ e iπ ( a + b ) A ( z, z (cid:9) ) , Disc A ( z, z ) ≈ − e iπ ( a + b ) A ( z, z (cid:8) ) , dDisc t A ( z, z ) ≈ − (cid:16) e iπ ( a + b ) A ( z, z (cid:9) ) + e − iπ ( a + b ) A ( z, z (cid:8) ) (cid:17) , (4.8)where the ≈ sign means we took the Regge limit. In order to account for the tidal excita-tions, the operators O and O can carry spin, in which case their indices are contractedwith the ones of (cid:101) O and (cid:101) O and sums over tensor structures are implied. In subsections4.1, 4.2 and 4.3 below we will mostly suppress the aspect of spinning correlators. We willcome back to this issue in subsection 4.4. To obtain the impact parameter representation, we first change the coordinate systemplacing each point on a different Poincar´e patch as shown in figure 5. We use the followingcoordinate transformations x i = (cid:0) x + i , x − i , x i ⊥ (cid:1) = − y + i (cid:0) , y i , y i ⊥ (cid:1) , i = 1 , , , ,x i = (cid:0) x + i , x − i , x i ⊥ (cid:1) = − y − i (cid:0) , y i , y i ⊥ (cid:1) , i = 3 , , , . (4.9)In the new x i coordinates, the Regge limit corresponds to placing the four external pointsat the origin of their respective Poincar´e patches, x , x , x , x → . (4.10)However, x to x are integrated over in the CFT optical theorem.– 25 – x x x P P P P Figure 5 . The external operators at coordinates x i in their respective Poincar´e patches P i . Theblack dotted lines are identified when the Poincar´e patches are wrapped on the boundary of theglobal AdS cylinder. Conformal correlators transform covariantly under the transformation (4.9). In thescalar case we have A ( y i ) = ( − y +1 ) − ∆ ( y +2 ) − ∆ ( − y − ) − ∆ ( y − ) − ∆ A ( x i ) . (4.11)In the spinning case, one must additionally account for the Jacobian matrix ∂y a /∂x m . Next we use conformal symmetry to express the correlator in terms of two vectors.This is similar to expressing the correlator in terms of two scalar cross-ratios, with thedifference that here we fix two, instead of the customary three, positions using translationsand special conformal transformations to express the correlator in terms of the remainingtwo position vectors. We can follow [33] and use a translation to send x to 0 which,due to the different transformations in (4.9) (see also [47]), will act as a special conformaltransformation on the Poincar´e patches for x and x , x → , x → x − x , x , → x , − x , x − x , · x + x , x . (4.12)Next we implement a translation on the x and x patches (acting as special conformaltransformation on x , ) to also map x to 0 in its own patch and find that the correlatoras a function of the Poincar´e patch coordinates A ( x i ), as defined in (4.11), can always beexpressed as A ( x , x , x , x ) ≈ A (0 , − x, x/x , ≡ A ( x, x ) , (4.13) For external spinning operators, the conformal transformations have a non-trivial rotation matrix ∂y a /∂x m . Conformal covariance of the correlators gives, in the representative example of two vectorsand two scalars [33], A ab ( y i ) = (cid:0) − y +1 y +2 (cid:1) − − ∆ V (cid:0) − y − y − (cid:1) − ∆ S ∂y a ∂x m ∂y b ∂x n A mn ( x i ). These matrices ensurethat the inversion tensors are correctly mapped from y i to x i variables, preserving their form. – 26 –ith x ≈ x − x , x ≈ x − x , (4.14)in the Regge limit (4.10).It is further convenient to implement the coordinate change using embedding spacecoordinates P M ∈ R ,d P M = (cid:0) P + , P − , P m (cid:1) , P · P = − P + P − + η mn P m P n . (4.15)These are related to the coordinates y m ∈ R ,d − of physical Minkowski space by [33] P M = (cid:0) y + , y − , , y , y ⊥ (cid:1) ⇒ P ij ≡ − P i · P j = ( y i − y j ) , (4.16)and to the coordinates x i by P M = − y +1 (cid:0) − , − x , x m (cid:1) , P M = y +2 (cid:0) − , − x , x m (cid:1) ,P M = − y − (cid:0) − x , − , x m (cid:1) , P M = y − (cid:0) − x , − , x m (cid:1) . (4.17)One can easily show that the cross-ratios (2.19) are given in terms of x and x as zz = x x , (1 − z )(1 − z ) = 1 + x x + 2 x · x , (4.18)and the kinematic prefactor (2.18) becomes T = ( − y +1 ) − ∆ ( y +2 ) − ∆ ( − y − ) − ∆ ( y − ) − ∆ x ∆ +∆ x ∆ +∆ . (4.19)When combining (2.17) and (4.11), the numerator of the last expression cancels the Jaco-bian prefactor in (4.11) to give, A ( x i ) = A ( z, z ) x ∆ +∆ x ∆ +∆ . (4.20)If we now study the correlator A ( x i ), a priori we have to take into account that only x and x are affected by the Regge limit. However, we will assume that the integrationwill be dominated by the region where the integration points are also boosted. Using theembedding space coordinates P M = − y +5 (cid:0) − , − x , x m (cid:1) , P M = y − (cid:0) − x , − , x m (cid:1) , (4.21)we find x (cid:48) ≈ x − x , x (cid:48) ≈ x − x . (4.22)Where the primed variables are meant to emphasize that the points 1 , , t A ( x , x ) = − (cid:88) O , O (cid:90) dx dx Disc A ( x , x ) S S Disc A ( x , x ) (cid:12)(cid:12)(cid:12) [ O O ] , (4.23)– 27 –here we stop explicitly mentioning that we are dealing with only the contribution ofdouble-trace operators as single-trace contributions are subleading in the limit considered.Let us stress that in order to write the correlators on the right hand side in terms oftwo differences, we assumed that each of the individual tree-level correlators are in theRegge limit themselves. The easiest way to justify this is in Fourier space using the impactparameter transform defined below. Each tree-level position space correlator is dominatedby a power σ − j ( ν ) in the Regge limit, which maps to a power of the AdS center of massenergy S j ( ν ) − in impact parameter space. Since the optical theorem is multiplicativein impact parameter space, subleading Regge trajectories or kinematical corrections fromthe conformal block at finite boost get mapped to smaller powers of S , and thereforedo not contribute to the leading behavior. The eikonal approximation in AdS [26] givesadditional intuition for this, since it means that even in AdS, the particles remain essentiallyundeflected, scattering forward each time they exchange a Pomeron. Furthermore, we willshow in section 5 that this configuration reproduces the behavior at one-loop derived in[34]. Let us now consider the two-point functions (cid:104) (cid:101) O (cid:101) O (cid:105) and (cid:104) (cid:101) O (cid:101) O (cid:105) for the shadow transformsin (4.23). In the Regge configuration, x is the patch of x , x is the patch of x , x isthe patch of x and x is the patch of x . As explained in [3, 26], the two-point functionbetween two coordinates on adjacent Poincar´e patches has an additional phase factor e iπ ∆ and this phase can be accounted for by switching from the i(cid:15) prescription of a Feynmanpropagator to that of a Wightman propagator (see [26]). The normalization of the shadowtransform in (2.3) and (2.4) is obtained from the Fourier transform of a two-point functionas the shadow transform acts multiplicatively in Fourier space (see section 3.2 of [35]).The normalization in (2.4) is obtained from the Fourier transform of a Euclidean two-point function, which matches the one of a Lorentzian two-point function with Feynman i(cid:15) prescription. The Wightman propagator in momentum space however has support only onthe future lightcone and the coefficient of the Fourier transform is different (see AppendixB of [26] and section 2.1 of [48]) (cid:90) dx e − iq · x [ − ( x − i(cid:15) ) + (cid:126)x ] ∆ = M O Θ( q ) Θ( − q ) (cid:0) − q (cid:1) ∆ − d , (4.24)with M O = 2 π d +1 Γ(∆)Γ (cid:0) ∆ − d + 1 (cid:1) . (4.25)Consequently we change the normalization N O of the shadow transform in (2.3) to (forscalar operators) N O = M O M (cid:101) O . (4.26)– 28 –ext we define, following [29], the impact parameter representation as the Fouriertransform of the discontinuity of the correlator in the two remaining vectors Disc A jk ( x j , x k ) = (cid:90) dp dp e − ip · x j − ip · x k B jk ( p, p ) , (4.27)where the function B ( p, p ) has support only on the future Milne wedge of p and p . Using(4.7) on (4.27) we get the Fourier transform of Disc .Disc A kj ( x k , x j ) = − (cid:90) dp dp e − ip · x k − ip · x j B kj ( − p, − p ) ∗ . (4.28)The causal relations and thus the i(cid:15) prescription in (4.28) are opposite to those in (4.27)and the complex conjugation prescribed in (4.7) compensates for that. Inserting (4.27)into (4.23) and using that A is a double shadow transform of A we obtain, uponusing (4.7) for the Disc ,dDisc t A ( x , x ) = 12 (cid:88) O , O (cid:90) dx dx dx dx (cid:90) dp dp dp (cid:48) dp (cid:48) × e − i ( p (cid:48) · x + p (cid:48) · x + p · x + p · x ) B ( − p (cid:48) , − p (cid:48) ) ∗ B ( p, p ) × T ( ρ ) ( x ) N O (cid:2) − ( x − i(cid:15) ) + (cid:126)x (cid:3) d − ∆ T ( ρ ) ( x ) N O (cid:2) − ( x − i(cid:15) ) + (cid:126)x (cid:3) d − ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ O O ] . (4.29) T ( ρ ) ( x ij ) is the tensor structure for the two-point function of an operator with SO ( d )quantum number ρ . For example it is the familiar inversion tensor η µν − x µ x ν x for spin 1operators. Note that we have B instead of B , as the superscripts now only indicatethe dimensions of the corresponding operators and ∆ = ∆ , ∆ = ∆ .We can now express the two-point functions from the shadow transforms in Fourierspace by inverting (4.24) T ( ρ ) ( x )[ − ( x − i(cid:15) ) + (cid:126)x ] d − ∆ = M (cid:101) O π d (cid:90) M dq e − iq · x (cid:98) T ( ρ ) ( q ) ( − q ) d − ∆ . (4.30)The Fourier space integral is over the future Milne wedge M as the Fourier transform hassupport only on this domain. (cid:98) T ( ρ ) ( q ) is the tensor structure of the two-point function inFourier space, which has been discussed for example in [48]. It is a tensor composed of q µ and η µν that can be factorized into a product of new tensors t ( ρ ) ( q ) as follows (cid:98) T ( ρ ) ( q ) µ ...µ | ρ | ν ...ν | ρ | = t ( ρ ) ( q ) µ ...µ | ρ | σ ...σ | ρ | t ( ρ ) ( q ) σ ...σ | ρ | ν ...ν | ρ | . (4.31) The AdS impact parameter representation was previously defined in [20, 29, 31] as the Fourier transformof the correlator on the second sheet A ( z, z (cid:9) ). Since this contribution dominates in the Regge limit, it isindistinguishable from the discontinuity of the correlator in this limit. However, in the t -channel the twonotions are clearly different and it was necessary to take the discontinuity to derive (1.5). In the nextsection we will see that the discontinuity is the better choice also in the s -channel. – 29 –sing (4.30) in (4.29) we end up with four position integrals over x , x , x , x and sixintegrals over q, q (from the four-point functions) and p, p, p (cid:48) , p (cid:48) . The four position integralsgive four Dirac delta functions with which we can eliminate the q, q, p (cid:48) , p (cid:48) integrals to obtaindDisc t A ( x , x ) = π d (cid:88) O , O M O M O (cid:90) dp dp e − i ( p · x + p · x ) B ( − p, − p ) ∗ (cid:98) T ( ρ ) ( p ) (cid:98) T ( ρ ) ( p ) B ( p, p )( − p ) ∆ − d ( − p ) ∆ − d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ O O ] , (4.32)with an implicit index contraction between B and B and the tensor structures (cid:98) T ( ρ ) .At this point we use (4.31) and absorb the t ( ρ ) ( q ) tensors into the definition of the phaseshifts B tree . This means that one needs to take it into account if one wants to relate tensorstructures of CFT correlators and phase shifts, but we will not need to do such a basischange explicitly in this work. Using (4.8) we see that the double discontinuity correspondsto the quantity − Re B ( p, p ) in impact parameter space. Thus we find the following gluingformula for the impact parameter representation, which is purely multiplicative, − Re B ( p, p ) = π d (cid:88) O , O M O M O B ( − p, − p ) ∗ B ( p, p )( − p ) ∆ − d ( − p ) ∆ − d (cid:12)(cid:12)(cid:12) [ O O ] . (4.33)Let us consider the case when O = O and O = O . In this case, it is useful to strip outa scale factor similar to that in (4.20) from the impact parameter representation B jjkk ( p, p ) = M O j M O k B jjkk ( p, p )( − p ) d − ∆ j (cid:0) − p (cid:1) d − ∆ k . (4.34)Using (4.24) one sees that with this choice of normalization the impact parameter repre-sentation of the MFT correlator is B jjkk MFT = 1, which is necessary for the eikonalization ofthe phase shift in AdS gravity. Inspired by this fact we choose the normalization B ijkl ( p, p ) = (cid:112) M O i M O j M O k M O l B ijkl ( p, p )( − p ) d − ∆ j (cid:0) − p (cid:1) d − ∆ k , (4.35)for the general case. This gives the following compact form for the optical theorem inimpact parameter space − Re B ( p, p ) = 12 (cid:88) O , O B ( − p, − p ) ∗ B ( p, p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ O O ] . (4.36)In [34], the Regge limit of a one-loop four-point function of scalars was studied in thelarge λ regime with S (cid:29) λ (cid:29)
1. The contribution of tidal excitations to the correlator issuppressed in this regime. It corresponds to just one term in the sum on the right handside of (4.36) i.e. with O = O and O = O . We show in section 5.1 that this termfrom our formula (4.36) reproduces the result from [34] in the large λ or equivalently thelarge ∆ limit. We do not need to discard any shadow double-trace contributions for thismatch. This motivates us to assume that the only non-zero contributions to the gluing oftree-level correlators in the Regge limit is from the physical double-traces [ O O ], and wewill drop the explicit projections henceforth.– 30 – .3 s -channel discontinuities in the Regge limit Next we have to analyze the discontinuities on the right hand side of the optical theorem(4.5). The discontinuity of the scalar s -channel block was recently computed in generalwithout taking the Regge limit in [42] and we will review it here. The generalization toexternal spinning operators is done in section 4.4, after taking the Regge limit. Let us takethe conformal partial wave expansion (2.22) in the s channel and use the symmetry of theintegrand to extend the integration region at the cost of a factor 1 / A ( z, z ) = 12 (cid:88) J (cid:90) d + i ∞ d − i ∞ d ∆2 πi I (∆ , J ) ψ , O ( z, z ) . (4.37)Let ψ ( z, z ) be the partial wave Ψ ( y i ) with the prefactor T stripped off. ψ , O ( z, z )is the conformal partial wave with an additional term that vanishes for integer spin due toits favorable properties for non-integer spin [42]. The new partial wave is given by ψ , O ( z, z ) = ψ O ( z, z ) + 2 π S ( O O [ (cid:101) O † ]) K J + d − , − ∆ ξ ( a,b )∆ ,J g J + d − , − ∆ ( z, z ) , (4.38)where g ,J ( z, z ) is the usual conformal block, the constants a, b are defined below (2.53)and ξ ( a,b )∆ ,J = (cid:16) s ( a,b )∆+ J − s ( a,b )∆+2 − d − J (cid:17) Γ (cid:0) − J − d − (cid:1) Γ( − J ) ,s ( a,b ) β = sin (cid:0) π ( a + β/ (cid:1) sin (cid:0) π ( b + β/ (cid:1) sin( πβ ) ,K ∆ ,J = Γ(∆ − (cid:0) ∆ − d (cid:1) κ ( a,b )∆+ J ,κ ( a,b ) β = Γ (cid:0) β − a (cid:1) Γ (cid:0) β + a (cid:1) Γ (cid:0) β − b (cid:1) Γ (cid:0) β + b (cid:1) π Γ( β − β ) . (4.39)With this conformal partial wave it is possible to compute the discontinuity exactly [42]Disc ψ , O ( z, z ) S ( O O [ (cid:101) O † ]) = R O ( z, z ) πiκ ( a,b )∆ ,J . (4.40)Here R is the so-called Regge block R O ( z, z ) = g − J, − ∆ − κ (cid:48) ( a,b )∆+ J g ,J − Γ( d − ∆ − (cid:0) ∆ − d (cid:1) Γ(∆ − (cid:0) d − ∆ (cid:1) κ (cid:48) ( a,b ) d − ∆+ J g d − ∆ ,J ++ Γ( J + d − (cid:0) − J − d − (cid:1) Γ (cid:0) J + d − (cid:1) Γ( − J ) κ (cid:48) ( a,b )∆+ J κ (cid:48) ( a,b ) d − ∆+ J g J + d − , − ∆ , (4.41)with κ (cid:48) ( a,b ) β defined as κ (cid:48) ( a,b ) β = r ( a,b ) β r ( a,b )2 − β , r ( a,b ) β = Γ( β + a )Γ( β + b )Γ( β ) . (4.42)– 31 –isc in the 3412 OPE channel can be obtained by using (4.6) on (4.40)Disc ψ , O ( z, z ) S ( O O [ (cid:101) O † ]) = R O ( z, z ) πiκ ( − b, − a )∆ ,J , (4.43)which was the reason to consider this channel for the correlators on the right hand side of(4.5). The Regge block is dominated in the Regge limit by [1, 29, 42] g − J, − ∆ ( z, z ) = 4 π d Γ(∆ − d )Γ(∆ − σ − J (cid:16) Ω ∆ − d ( ρ ) + O ( σ ) (cid:17) . (4.44)The σ, ρ cross-ratios introduced here are defined as σ = √ zz = √ x x , cosh( ρ ) = z + z √ zz = − x · x √ x x . (4.45)Ω iν ( ρ ) is the harmonic function on d − H d − transverse tothe scattering plane in AdS d +1 [19]Ω iν ( ρ ) = − iν sin( πiν )Γ( h − iν )Γ( h − − iν )2 h − π h + Γ (cid:0) h − (cid:1) F (cid:18) h − iν, h − − iν, h − , − cosh( ρ )2 (cid:19) . (4.46)Inserting everything into (4.37), we find the following expression for the discontinuity ofthe correlator in the Regge limitDisc A ( z, z ) = 2 πi (cid:88) J ∞ (cid:90) −∞ dν α ( ν, J ) σ − J Ω iν ( ρ ) , (4.47)with α ( ν, J ) = − π d − S (cid:0) O O (cid:2)(cid:0) − iν − d (cid:1) † (cid:3)(cid:1) Γ( iν )2 πκ ( a,b ) iν + d ,J Γ (cid:0) iν + d − (cid:1) I (cid:18) iν + d , J (cid:19) . (4.48)As in flat space the sum in (4.47) is dominated by the large J contributions in the Reggelimit and only finite due to a conspiration of the coefficients to ensure Regge boundedness.The next step is therefore to perform a Sommerfeld-Watson resummation over J to evaluate(4.47). Also, note that the spectral function, as given by the Lorentzian inversion formula[1], is of the form I ( ν ) = I ,t ( ν ) + ( − J I ,u ( ν ) . (4.49)Let us first consider the case of a correlator with pairwise equal external operators i.e. a = b = 0, I ,t = I ,u , where only even spins are exchanged. Now for the resummationwe replace the sum by an integral,2 (cid:88) J even → (cid:90) C dJ e iπJ − e iπJ . (4.50)– 32 –he contour C encloses all poles on the positive real axis (at even integers) in a clockwisedirection. The leading Regge trajectory is given by the operators with the lowest dimension∆( J ) for every even spin J and α ( ν, J ) has poles at iν = ± (∆( J ) − d/ j = j ( ν ) of the spectral function ∆( J ) by ν + (cid:0) ∆( j ( ν )) − d (cid:1) = 0 , (4.51)we see that the poles in ν translate into a single pole at J = j ( ν ). By deforming the J contour to the left one sees that the J integral is given by the residue at J = j ( ν ), i.e.Disc A ( z, z ) = 2 πi ∞ (cid:90) −∞ dν α ( ν ) σ − j ( ν ) Ω iν ( ρ ) , (4.52)where α ( ν ) = − Res J = j ( ν ) i e iπJ − e − iπJ π d − S (cid:0) O O (cid:2)(cid:0) − iν − d (cid:1) † (cid:3)(cid:1) Γ( iν ) κ ( a,b ) iν + d ,J Γ (cid:0) iν + d − (cid:1) I ,t (cid:18) iν + d , J (cid:19) . (4.53)When the operators are not pairwise equal, the even and odd spin operators organize intotwo analytic families as evident from the Lorentzian inversion formula [1]. To obtain thecontribution of the leading Regge trajectory we still sum over the even spin exchanges.The result is of the same form as in (4.52) with I ,t replaced by I in α ( ν ). For theother correlator we can use (4.6) to see that we get an analogous result with the complexconjugate spectral functionDisc A ( z, z ) = 2 πi ∞ (cid:90) −∞ dν α ( ν ) ∗ σ − j ( ν ) Ω iν ( ρ ) . (4.54)One can show that the corresponding impact parameter representation is given in generalby the same spectral function times a multiplicative factor which cancels poles for theexternal double-trace operators [20] B ( p, p ) = 2 πi ∞ (cid:90) −∞ dν β ( ν ) S j ( ν ) − Ω iν ( L ) , (4.55)where β ( ν ) = 4 π − d ( (cid:112) M O M O M O M O ) − α ( ν ) χ j ( ν ) ( ν ) χ j ( ν ) ( − ν ) , (4.56)with the definition χ j ( ν ) ( ν ) = Γ (cid:18) ∆ + ∆ + j ( ν ) − d/ iν (cid:19) Γ (cid:18) ∆ + ∆ + j ( ν ) − d/ iν (cid:19) . (4.57)The impact parameter space cross-ratios, analogous to (4.45), are S = (cid:112) p p , cosh L = − p · p (cid:112) p p . (4.58)In the dual AdS scattering process these cross ratios are interpreted as the squared of theenergy with respect to global time and as the impact parameter in the transverse space H d − . – 33 – .4 Spinning particles and the vertex function In this section we will introduce concrete expressions for the tree amplitudes with spin-ning external legs and show that the contributions of the contracted spinning legs canbe expressed in terms of a scalar function of three spectral parameters which we call thevertex function, analogous to (3.17) in flat space. We construct tensor structures in termsof differential operators, which are a Regge limit version of weight-shifting operators thatgenerate spinning conformal blocks from the scalar ones [49, 50]. It is convenient to workwith tensor structures which are homogeneous in p and p , i.e. independent of the cross-ratio S in (4.58), such that all tensor structures have the same large S behavior in the Reggelimit. These differential operators can be constructed from the covariant derivative on thehyperboloid H d − and from (cid:98) p = p/ | p | , (cid:98) p = p/ | p | [33, 51]. The possible differential operatorsthat generate spin for a single particle are D ρ,k m ( p ) = (cid:98) p m . . . (cid:98) p m k ∇ pm k +1 . . . ∇ pm | ρ | , k = 0 , . . . , | ρ | . (4.59)Tree diagrams for exchange of the Pomeron then have the form B (∆ ,ρ ) , (∆ ,ρ ) mn ( p, p ) = 2 πi ∞ (cid:90) −∞ dν S j ( ν ) − D (∆ ,ρ ) , (∆ ,ρ ) mn ( ν ) Ω iν ( L ) . (4.60)Here B (∆ ,ρ ) , (∆ ,ρ ) mn ( p, p ) is defined just as in (4.27) and (4.35), but with tensor structuresconstructed from (cid:98) p and (cid:98) p . In (4.60) we introduced the following definition for the combi-nation of spectral functions β ( ν ) and differential operators that generate different tensorstructures D (∆ ,ρ ) , (∆ ,ρ ) mn ( ν ) = | ρ | (cid:88) k =0 | ρ | (cid:88) k =0 β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) D ρ ,k m ( p ) D ρ ,k n ( p ) . (4.61)Notice that, in contrast to flat space, we do not impose a full factorization into three-point structures but rather allow for a separate spectral function for each combination ofthree-point structures.The next step is to derive the general functional form of (4.36) after the contractionsand sums have been done. We begin by inserting (4.60) into (4.36), − Re B ( p, p ) = 2 π (cid:88) ∆ , ∆ ,ρ ,ρ ∞ (cid:90) −∞ dν dν S j ( ν )+ j ( ν ) − (4.62) D (∆ ,ρ ) , (∆ ,ρ ) mn ( ν ) ∗ Ω iν ( L ) π m ; p ρ π n ; q ρ D (∆ ,ρ ) , (∆ ,ρ ) pq ( ν ) Ω iν ( L ) . Here π ρ is the projector to the irreducible representation ρ of SO ( d ), which is necessarybecause the operators (4.59) do not ensure the properties of irreducible representationssuch as tracelessness and Young symmetrization. Next we will show how one can replacethe contractions and derivatives in the previous equation by spectral parameters. Note firstthat due to p · ∇ p = 0, all contractions involving (cid:98) p or (cid:98) p give factors of their norm −
1. The– 34 –emaining contractions involve only covariant derivatives. These contracted derivatives canall be replaced by functions of the spectral parameters by using the Laplace equation forthe harmonic function (cid:16) ∇ H d − + ν + ( d/ − (cid:17) Ω iν ( L ) = 0 . (4.63)Using this equation, factors of ∇ p can directly be replaced. To evaluate contractionsbetween derivatives acting on different harmonic functions we expand the product of twoscalar harmonic functions as follows,Ω iν ( L )Ω iν ( L ) = ∞ (cid:90) −∞ dν Φ( ν , ν , ν )Ω iν ( L ) , (4.64)where Φ( ν , ν , ν ) was computed (for the similar case of harmonic functions on AdS d +1 ) inappendix D of [52]. By acting repeatedly with (4.63) on this equation, one can determinethe function W k that appears in ∇ pm . . . ∇ pm k Ω iν ( L ) ∇ m p . . . ∇ m k p Ω iν ( L ) = ∞ (cid:90) −∞ dν W k (cid:0) ν , ν , ν (cid:1) Φ( ν , ν , ν ) Ω iν ( L ) . (4.65) W k is a fixed kinematical polynomial of maximal degree k in its arguments. For example,the first non-trivial case is ∞ (cid:90) −∞ dν Φ( ν , ν , ν ) ν Ω iν ( L ) = (cid:0) ν + ν + ( d − (cid:1) Ω iν ( L ) Ω iν ( L ) − ∇ µ Ω iν ( L ) ∇ µ Ω iν ( L ) , (4.66)from which one can read off W and W to be W (cid:0) ν , ν , ν (cid:1) = 1 , W (cid:0) ν , ν , ν (cid:1) = 12 (cid:16) ν + ν − ν + ( d/ − (cid:17) . (4.67)More generally, by acting with the Laplacian on both sides of (4.65) one can derive arecursion relation of the form (cid:90) dν W k +1 ( ν i ) Φ( ν i ) Ω iν ( L ) = (cid:90) dν W k ( ν i ) W ( ν i ) Φ( ν i ) Ω iν ( L )+ 12 (cid:0) [ ∇ , ∇ m . . . ∇ m k ]Ω iν ( L ) ∇ m . . . ∇ m k Ω iν ( L ) + ( ν ↔ ν ) (cid:1) . (4.68)The terms with commutators, which will vanish in the flat space limit, can be evaluatedusing the fact that the commutators of covariant derivatives can be replaced by Riemanntensors, which for the hyperboloid can be written in terms of the metric. This means thatthese terms have two derivatives less than the other terms, and will therefore produce lessthan maximal powers of ν i . This shows that the maximal power of ν i in W k is just givenby repeatedly multiplying W . Therefore we have W k (cid:0) ν , ν , ν (cid:1) = (cid:18) ν + ν − ν (cid:19) k + O (cid:16) ν k − i (cid:17) . (4.69) Note that Φ here Ω iν (0) = Φ there . – 35 –aving shown that all derivatives can be replaced by polynomials of the spectral parame-ters, we can define D (∆ ,ρ ) , (∆ ,ρ ) mn ( ν ) ∗ Ω iν ( L ) π m ; p ρ π n ; q ρ D (∆ ,ρ ) , (∆ ,ρ ) pq ( ν ) Ω iν ( L )= ∞ (cid:90) −∞ dν W (∆ ,ρ ) , (∆ ,ρ ) (cid:0) ν , ν , ν (cid:1) Φ( ν , ν , ν ) Ω iν ( L ) . (4.70)This gives the contribution of a given pair of intermediate states labeled by (∆ , ρ ) and(∆ , ρ ) to − Re B ( p, p ). Now we can define the vertex function V ( ν , ν , ν ), which iseven in all its arguments, in analogy to (3.17) as the sum over all such contributions in(4.62) (cid:88) ∆ , ∆ ,ρ ,ρ , W (∆ ,ρ ) , (∆ ,ρ ) (cid:0) ν , ν , ν (cid:1) = β ( ν ) ∗ β ( ν ) V ( ν , ν , ν ) , (4.71)and reach the following representation for the 1-loop amplitude − Re B ( p, p ) = 2 π ∞ (cid:90) −∞ dνdν dν β ( ν ) ∗ β ( ν ) V ( ν , ν , ν ) S j ( ν )+ j ( ν ) − Φ( ν , ν , ν ) Ω iν ( L ) . (4.72)All the information about the spinning tree-level correlators and their contractions is en-coded in the vertex function V ( ν , ν , ν ) which mirrors the role of its flat space analogue.However, in order to compute the full impact parameter representation rather thanjust its real part, we have to go through a detour via the Lorentzian inversion formula, asdescribed in [34]. We first Fourier transform back to dDisc t A from which we obtainthe s -channel OPE coefficients. Then we can compute Disc A which we can finallyFourier transform to obtain B ( p, p ). Since in the Regge limit the difference betweendDisc t A and Disc A is just a phase factor (see [34]), the same happens for theimpact parameter representation B ( p, p ) = − π ∞ (cid:90) −∞ dνdν dν e − iπ ( j ( ν )+ j ( ν ) − − e − πi ( j ( ν )+ j ( ν ) − β ( ν ) ∗ β ( ν ) V ( ν , ν , ν ) S j ( ν )+ j ( ν ) − Φ( ν , ν , ν ) Ω iν ( L ) . (4.73)It is important to emphasize that this provides a finite ∆ gap description for the one-loopcorrelator in the Regge limit up to the knowledge of the vertex function V ( ν , ν , ν ) . ForCFTs that admit a flat space limit, we will see in sections 6 and 7 how one can fix part ofthis vertex function from the knowledge of its flat space analogue. In section 5 below, wemake a comparison with the large ∆ gap limit studied in reference [34], and also describethe implications of (4.72) for t-channel CFT data.– 36 – Constraints on CFT data ∆ gap limit In [34] the Regge limit of the one-loop four-point correlator of pairwise identical scalars wasstudied in an expansion in 1 /N in the limit of large ∆ gap . The specific limit consideredwas S (cid:29) ∆ (cid:29)
1, so the result is sensitive to all the higher spin interactions in theleading Regge trajectory, but tidal excitations are ignored. Since we have kept ∆ gap finite,we should be able to obtain a match between the result for the one-loop correlator in [34]with our result (4.36) after dropping the tidal excitations O (cid:54) = O and O (cid:54) = O .We pick the term ∆ = ∆ and ∆ = ∆ that is the sole contribution to (4.36) in thelarge ∆ gap limit, and use (4.55) to obtain − Re B ( S, L ) = 2 π (cid:90) dν dν dν β ∗ ( ν ) β ( ν ) Φ( ν , ν , ν ) S j ( ν )+ j ( ν ) − Ω iν ( L ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ O O ] . (5.1)Now let us extract the corresponding result from [34]. Equation (3.15) of [34] gives thedouble discontinuity of the one-loop correlator G (2) as follows (with φ = ∆ and ψ = ∆ )dDisc t [ G (2) ( z, z )] = π (cid:90) dν dν dν χ j ( ν )+ j ( ν ) − ( ν ) χ j ( ν )+ j ( ν ) − ( − ν ) N (cid:98) γ (1) ( ν ) (cid:98) γ (1) ( ν ) Φ( ν , ν , ν )( zz ) − j ( ν − j ( ν Ω iν (cid:18)
12 log( z/z ) (cid:19) , (5.2)where χ j ( ν )+ j ( ν ) − ( ν ) is defined in (4.57) but with j ( ν ) replaced by j ( ν ) + j ( ν ) − = ∆ and ∆ = ∆ . It accounts for the double-trace exchanges [ O O ] and[ O O ] analytically continued to spin j ( ν ) + j ( ν ) −
1. The operators contributing to the t -channel expansion in the large ∆ gap limit are the double-traces [ O O ] n,(cid:96) with dimensionsand OPE coefficients given by∆ h,h = ∆ (0) h,h + 1 N γ (1) h,h + 1 N γ (2) h,h + · · · , ∆ (0) h,h = ∆ + ∆ + 2 n + (cid:96) ,P h,h = P MF Th,h (cid:18) N δP (1) h,h + 1 N δP (2) h,h + · · · (cid:19) , (5.3)where h, h = ∆ ∓ (cid:96) . The tree-level anomalous dimensions γ (1) h,h and tree-level corrections toOPE coefficients δP (1) h,h can be extracted respectively from (cid:98) γ (1) ( ν ) and (cid:99) δP (1) ( ν ) by γ (1) h,h ≈ ∞ (cid:90) −∞ dν (cid:98) γ (1) ( ν ) ( hh ) j ( ν ) − Ω iν (cid:0) log( h/h ) (cid:1) ,δP (1) h,h ≈ ∞ (cid:90) −∞ dν (cid:99) δP (1) ( ν ) ( hh ) j ( ν ) − Ω iν (cid:0) log( h/h ) (cid:1) . (5.4)– 37 – γ (1) ( ν ) and (cid:99) δP (1) ( ν ) can be obtained respectively from the real and imaginary parts of thephase shift, and are related to β by (cid:98) γ (1) ( ν ) = 2 Re β ( ν ) , (cid:99) δP (1) ( ν ) = − π Im β ( ν ) . (5.5)Taking the Fourier transform to impact parameter space on (5.2) and then using (5.5)gives − Re B ( S, L ) = 2 π (cid:90) dν dν dν Re β ( ν ) Re β ( ν ) Φ( ν , ν , ν ) S j ( ν )+ j ( ν ) − Ω iν ( L ) . (5.6)We need to compare (5.1) with (5.6). The only difference are the real parts in (5.6),however Im β in (5.1) is related to tree-level corrections to the OPE coefficients and theseare suppressed at large ∆ gap [34]. This can be seen for example from (5.5) and using in itthe explicit expression for α ( ν ) from [34]. The result is (cid:99) δP (1) ( ν ) = − π Im (cid:16) ie iπj ( ν ) − e iπj ( ν ) (cid:17) Re (cid:16) ie iπj ( ν ) − e iπj ( ν ) (cid:17) (cid:98) γ (1) ( ν ) = − π tan (cid:16) π j ( ν ) (cid:17) (cid:98) γ (1) ( ν ) . (5.7)The suppression is due to the tan factor, since for large N theories it is known that [28–30] j ( ν ) = 2 − ν + ( d/ ∆ + O (cid:0) ∆ − (cid:1) . (5.8)The anomalous dimensions γ (1) h,h are order 1, while the the corrections to the OPE coefficients δP (1) h,h are at order ∆ − .Thus we have matched our result for the Regge limit of the dDisc of a one-loop correla-tor at large ∆ gap to that of [34]. Note that we managed to reproduce the result without theneed for any projections to the physical double-traces. Therefore it is reasonable to assumethat the gluing of tree-level correlators in the Regge limit does not receive contributionsfrom the double-traces of shadows and we can use the optical theorem (4.36) without theprojections onto [ O O ]. Next we shall see how we can extract the CFT data for the double-trace operators ex-changed in the t -channel to order 1 /N from the vertex function V ( ν , ν , ν ). To thisend we follow section 3.2 of [34] and extend the results there in by including tidal ex-citations, which make our statements valid at finite ∆ gap . As discussed in the previoussection, the only operators contributing to the t -channel expansion in the large ∆ gap limitare the double-traces [ O O ]. The three-point function of these double-traces with theirconstituent operators O and O has the large N behavior (cid:104)O O [ O O ] (cid:105) ∼ . (5.9) Note that due to difference in conventions, N β for us is equal to β , as defined in [34]. In our conventions the Fourier transform takes dDisc t [ G (2) ( z, z )] to −N Re B upto scaling factors. – 38 –y including tidal excitations we have to include also double-traces [ O O ] correspondingto additional double-traces coupling to O and O . These satisfy (cid:104)O O [ O O ] (cid:105) ∼ N , [ O O ] (cid:54) = [ O O ] , (5.10)so that only their classical dimension and leading OPE coefficient squared ∆ h (cid:48) ,h (cid:48) = ∆ O + ∆ O + 2 n + (cid:96) , P h (cid:48) ,h (cid:48) = 1 N P MFT h (cid:48) ,h (cid:48) δP (56) h (cid:48) ,h (cid:48) + . . . , (5.11)appear in the one-loop correlator. This is compatible with the large N behavior for single-trace exchange in the direct channel, (cid:104)O O O ∆( J ) (cid:105) ∼ N , (cid:104)O ∆( J ) O O (cid:105) ∼ N , (5.12)which justifies that the OPE coefficients c O O [ O O ] start at order 1 /N . As explained in[34], the cross channel expansion of the correlator is then dominated by the terms A (cid:9) ( z, z )( zz ) ∆ φ ≈ (cid:88) h,h P MF Th,h (cid:20) iπγ (2) h,h + δP (2) h,h + iπγ (1) h,h δP (1) h,h − π (cid:16) γ (1) h,h (cid:17) (cid:21) g h,h (1 − z, − z )+ (cid:88) h (cid:48) ,h (cid:48) P MF Th (cid:48) ,h (cid:48) δP (56) h (cid:48) ,h (cid:48) g h (cid:48) ,h (cid:48) (1 − z, − z ) . (5.13)We shall now compare with our result for the one-loop correlator in the Regge limitand use it in the light of (5.13) to extract CFT data. We start with the dDisc of thecorrelator in the impact parameter representation as in (4.72). Doing an inverse Fouriertransform on this and taking out the appropriate scale factors gives us the dDisc of theone-loop correlator in the Regge limitdDisc t A ( z, z ) = π N ∞ (cid:90) −∞ dνdν dν (cid:0) β ∗ ( ν ) β ( ν ) + β ( ν ) β ∗ ( ν ) (cid:1) V ( ν , ν , ν ) Φ( ν , ν , ν ) χ j ( ν )+ j ( ν ) − ( ν ) χ j ( ν )+ j ( ν ) − ( − ν ) σ − j ( ν ) − j ( ν ) Ω iν ( ρ ) , (5.14)where we symmetrized the product of β ’s by using the symmetry of the expression under ν ↔ ν . We can now use the Lorentzian inversion formula [1] on (5.14), as shown in [34],to obtain the one-loop correlator in the Regge limit, and then use (5.5) to express it as A (cid:9) ( z, z ) ≈ − π N ∞ (cid:90) −∞ dν dν dν e − iπ ( j ( ν )+ j ( ν ) − − e − πi ( j ( ν )+ j ( ν ) − V ( ν , ν , ν ) Φ( ν , ν , ν ) χ j ( ν )+ j ( ν ) − ( ν ) χ j ( ν )+ j ( ν ) − ( − ν ) σ − j ( ν ) − j ( ν ) Ω iν ( ρ ) (cid:20)(cid:98) γ (1) ( ν ) (cid:98) γ (1) ( ν ) + 1 π (cid:99) δP (1) ( ν ) (cid:99) δP (1) ( ν ) (cid:21) . (5.15) We are free to insert P MFT here, defining δP accordingly. This will be useful below in (5.18). – 39 –e now take the t -channel expansion (5.13), and use in it (5.4), (4.64), and the followingansatz, γ (2) h,h ≈ ∞ (cid:90) −∞ dν dν dν (cid:98) γ (2) ( ν , ν , ν ) ( hh ) j ( ν )+ j ( ν ) − Ω iν (cid:0) log( h/h ) (cid:1) ,δP (2) / (56) h,h ≈ ∞ (cid:90) −∞ dν dν dν (cid:99) δP (2) / (56) ( ν , ν , ν ) ( hh ) j ( ν )+ j ( ν ) − Ω iν (cid:0) log( h/h ) (cid:1) , (5.16)to obtain( zz ) − ∆ φ A (cid:9) ( z, z ) ≈ ∞ (cid:90) −∞ dν dν dν (cid:34) (cid:88) h,h ( hh ) j ( ν )+ j ( ν ) − Ω iν (log h/h ) P MF Th,h (5.17) (cid:20) iπ (cid:98) γ (2) ( ν , ν , ν ) + iπ (cid:18)(cid:98) γ (1) ( ν ) (cid:99) δP (1) ( ν ) + (cid:98) γ (1) ( ν ) (cid:99) δP (1) ( ν ) (cid:19) Φ( ν , ν , ν ) − π (cid:98) γ (1) ( ν ) (cid:98) γ (1) ( ν )Φ( ν , ν , ν ) + (cid:99) δP (2) ( ν , ν , ν ) (cid:21) g h,h (1 − z, − z )+ (cid:88) h (cid:48) ,h (cid:48) P MF Th (cid:48) ,h (cid:48) (cid:99) δP (56) ( ν , ν , ν )( h (cid:48) h (cid:48) ) j ( ν )+ j ( ν ) − Ω iν (log h (cid:48) /h (cid:48) ) g h (cid:48) ,h (cid:48) (1 − z, − z ) (cid:35) . We can approximate the h, h and h (cid:48) , h (cid:48) sums with integrals, (cid:80) h,h → (cid:82) ∞ dh dh , andevaluate them using Bessel function integrals (see section 2.2 of [34]) to arrive at the result A (cid:9) ( z, z ) ≈ π N ∞ (cid:90) −∞ dν dν dν χ j ( ν )+ j ( ν ) − ( ν ) χ j ( ν )+ j ( ν ) − ( − ν ) σ − j ( ν ) − j ( ν ) Ω iν ( ρ ) (cid:20) iπ (cid:98) γ (2) ( ν , ν , ν ) − π (cid:98) γ (1) ( ν ) (cid:98) γ (1) ( ν ) Φ( ν , ν , ν ) + (cid:99) δP (2) ( ν , ν , ν ) (5.18)+ iπ (cid:18)(cid:98) γ (1) ( ν ) (cid:99) δP (1) ( ν ) + (cid:98) γ (1) ( ν ) (cid:99) δP (1) ( ν ) (cid:19) Φ( ν , ν , ν ) + (cid:99) δP (56) ( ν , ν , ν ) (cid:21) . Comparing the real parts of the coefficient of χ ( ν ) χ ( − ν ) σ − j ( ν ) − j ( ν ) Ω iν ( ρ ) in the inte-grands of (5.15) and (5.18), and using1 + e − iπ ( j ( ν )+ j ( ν ) − − e − πi ( j ( ν )+ j ( ν ) − = 12 + i (cid:16) π (cid:0) j ( ν ) + j ( ν ) (cid:1)(cid:17) , (5.19)we conclude that (cid:99) δP (2) ( ν , ν ; ν ) + (cid:99) δP (56) ( ν , ν ; ν ) = − (cid:20) π (cid:16) V ( ν , ν , ν ) − (cid:17)(cid:98) γ (1) ( ν ) (cid:98) γ (1) ( ν ) (5.20)+ V ( ν , ν , ν ) (cid:99) δP (1) ( ν ) (cid:99) δP (1) ( ν ) (cid:21) Φ( ν , ν , ν ) . This is the general result for fixed ∆ gap that extracts OPE data from the AdS vertexfunction. Let us now take the large ∆ gap limit to make contact with [34]. In this limit,– 40 – ( ν , ν , ν ) = 1 and (cid:99) δP (1) , (cid:99) δP (56) are suppressed with respect to (cid:98) γ (1) . Therefore δP (2) h,h = 0,as was obtained in [34].Similarly, comparing the imaginary parts we have (cid:98) γ (2) ( ν , ν ; ν ) = − (cid:20) (cid:18)(cid:98) γ (1) ( ν ) (cid:99) δP (1) ( ν ) + (cid:99) δP (1) ( ν ) (cid:98) γ (1) ( ν ) (cid:19) + π tan (cid:16) π j ( ν ) + j ( ν )) (cid:17) V ( ν , ν , ν ) (cid:16)(cid:98) γ (1) ( ν ) (cid:98) γ (1) ( ν ) + 1 π (cid:99) δP (1) ( ν ) (cid:99) δP (1) ( ν ) (cid:17)(cid:21) Φ( ν , ν , ν ) . (5.21)The term (cid:99) δP (1) ( ν ) (cid:99) δP (1) ( ν ) is suppressed by ∆ − with respect to the other terms. Atleading order in ∆ gap , we can discard this term and set V ( ν , ν , ν ) = 1. We can then use(5.7) to simplify the expression to, (cid:98) γ (2) ( ν , ν ; ν ) = − π tan (cid:18) πj ( ν ) (cid:19) tan (cid:18) πj ( ν ) (cid:19) tan (cid:18) π ( j ( ν ) + j ( ν )) (cid:19) × (cid:98) γ (1) ( ν ) (cid:98) γ (1) ( ν )Φ( ν , ν , ν ) . (5.22)This is the same result as obtained in [34] for (cid:98) γ (2) ( ν , ν ; ν ). More generally, knowledge ofthe vertex function V ( ν , ν , ν ) and of the (cid:104)O O [ O O ] (cid:105) OPE coefficients gives additionalinformation about the one-loop CFT data of the [ O O ] double-trace operators. It wouldbe interesting to analyze these equations order by order in the 1 / ∆ expansion. Having fixed the general form of the impact parameter representation of the one-loopcorrelator from first principles in section 4, we now want to fix part of the dynamical databy taking the flat space limit, which relates it to the known flat space amplitudes. Theprescription to achieve this was discovered in [29], where it was applied to scalar tree-levelamplitudes. This limit is taken by sending the AdS radius R to infinity while scaling therelevant quantities in order to match them to flat space quantities in a sensible way. Thedimensionless quantities S and L are sent to dimensionless combinations of R with the flatspace center of mass energy s and impact parameter b as S = R s , L = bR . (6.1)Note that L is the AdS impact parameter, as it describes the geodesic distance on H d − between the impact points in transverse space. If we impose the identification of Casimireigenvalues ∆(∆ − d ) = R m , (6.2)for the states on the leading Regge trajectory and take this equation off-shell, it becomes ν + (cid:18) d (cid:19) = R q , (6.3)so for large R we further impose ν = R q . (6.4)– 41 –ur expressions in AdS are integrals in ν , while in flat space we have vector integrals in q , where we recall that q is a vector in the transverse space R D − . In order to comparethe expressions, it is instructive to do the flat space angular integrals and keep only theintegral over the modulus | q | . In this way, the exponential is replaced by the harmonicfunction ω q ( b ) according to (cid:90) R D − dq (2 π ) D − e ib · q = 2 ∞ (cid:90) d | q | ω q ( b ) , (6.5)so that [53] ω q ( b ) = q (cid:90) R D − dp (2 π ) D − e ib · p δ ( p − q ) = 12(2 π ) D − | q | D − | b | D − J D − ( | q || b | ) , (6.6)where J denotes the Bessel J -function and we recall that ω q ( b ) only depends on the modulusof the vectors q and b . One can check that the flat space limit of the H d − harmonic function(4.46) yields the flat space harmonic function R − D Ω iν ( L ) → ω q ( b ) , ν ≥ . (6.7)For even d this can be checked directly, while for general d it is convenient to use anintegral representation for the hypergeometric function which under the limit is related toan integral representation for the Bessel function [54]. For even d the relation is also validfor ν < λ which isexpressed in terms of α (cid:48) and R as √ λ = R α (cid:48) , (6.8)meaning we can also express S as S = √ λ α (cid:48) s . (6.9)To summarize, the flat space limit is taken by sending the AdS radius R to infinity whilereplacing S = √ λα (cid:48) s , L = bR , ν = R q , ν = R q , ν = R q , √ λ = R α (cid:48) , (6.10)and impact parameter representations can be compared by using (6.7). We can also usethese relations to relate ∆ gap to λ taking as reference a string state of mass m = 4 /α (cid:48) ,therefore ∆ = 4 R α (cid:48) = 4 √ λ . (6.11)– 42 – .1 Matching in impact parameter space Let us now see what we can learn when we apply the flat space limit to the impact parameterrepresentations studied in section 4. We begin with the tree-level correlator of four scalarsfor which the limit was originally imposed in [29]. The flat space limit of the AdS result B in (4.55) should match the flat space impact parameter representation iδ from (3.15) ofthe amplitude (3.8) B tree ( p, p ) = 4 πi ∞ (cid:90) dν β ( ν ) S j ( ν ) − Ω iν ( L ) → iδ tree ( s, b ) = 2 i ∞ (cid:90) d | q | β ( t ) (cid:18) α (cid:48) s (cid:19) j ( t ) − ω q ( b ) . (6.12)Here and below we do not always write the overall factors of R as in (6.7), but they dowork out correctly when including the expansion parameters from (1.3) and (3.1) and usingthe relation 1 N = 1 R D − G N π . (6.13)From (6.7) and (6.9) we see that this does indeed match, provided the flat space limit ofthe AdS Regge trajectory and spectral function are sent to the flat space Regge trajectoryand Pomeron propagator j ( ν ) → j ( t ) , λ j ( ν ) − β ( ν ) → π β ( t ) . (6.14)The power of λ in the relation of β ’s is necessary to cancel the powers of λ in the relationbetween S and α (cid:48) s . It is compatible with the expectation that each derivative in thecouplings of the spin J operators forming the Pomeron comes at least with a power of λ − .Next we consider the optical theorem in AdS (4.36) and flat space (3.16) − Re B = 12 (cid:88) ∆ ,ρ ∆ ,ρ B ∗ tree B → Im δ ( b ) = 12 (cid:88) m ,ρ ,(cid:15) m ,ρ ,(cid:15) δ ∗ tree δ . (6.15)The similarity is striking, however we have to make sure the sums and summands are infact related by the flat space limit. The additional sums over polarizations can be evaluatedusing completeness relations such as (3.22), which evaluate to contractions just as in theAdS equation. We also have to make sure that the labels ρ on both sides are irreduciblerepresentations of the same group SO ( d ). This is indeed the case for massive particles ifwe consider the flat space limit AdS d +1 → R ,d , which has the massive Little group SO ( d ).The next step is to match the tree-level correlators (4.60) and amplitudes (3.8) that The relative factor i in B → iδ can be determined by matching the exponents in the eikonal approxi-mation for λ → ∞ . j ( ν ), j ( t ) and β ( ν ), β ( t ) are different functions and not the same function with different arguments. – 43 –nvolve spinning particles 5 and 6. In this case the flat space limit gives B (∆ ,ρ ) , (∆ ,ρ ) mn ( p, p ) = 4 πi ∞ (cid:90) dν S j ( ν ) − D (∆ ,ρ ) , (∆ ,ρ ) mn ( ν ) Ω iν ( L ) → (6.16) → iδ ( m ,ρ ) , ( m ,ρ ) mn ( s, b ) = i (cid:90) R D − dq (2 π ) D − (cid:18) α (cid:48) s (cid:19) j ( t ) − A Pm ,ρ , m ( q, v ) β ( t ) A Pm ,ρ , n ( q, v ) e iq · b = 2 i ∞ (cid:90) d | q | (cid:18) α (cid:48) s (cid:19) j ( t ) − A Pm ,ρ , m ( − i∂ b , v ) β ( t ) A Pm ,ρ , n ( − i∂ b , v ) ω q ( b ) , where the derivative ∂ b is with respect to the components of the transverse vector b . Thedifference compared to (6.12) is that in AdS we have differential operators that generatetensor structures, while in flat space the tensor structures are the ones of the on-shell three-point amplitudes. As discussed in section 3.4, these three-point amplitudes are given interms of the Pomeron momentum q and, for massive particles, the longitudinal polarizationvector v , which is transverse to q . We will now study the relation of these two kinds oftensor structures to the flat space limit.We begin with the covariant derivatives and will argue that they become derivativesin impact parameter in flat space, i.e. ∇ mp Ω iν ( L ) → R∂ mb e ib · q = Riq m e ib · q . (6.17)In order to show this, we will act with two contracted covariant derivatives either on asingle harmonic function or on two different ones, covering all situations that can occur.Acting on a single harmonic function we obtain, from (4.63) and (6.10),1 √ λ ∇ p Ω iν , ( L ) → − α (cid:48) q , ω iν , ( b ) . (6.18)The action of contracted covariant derivatives on two different harmonic functions is cap-tured by the functions W k in (4.65), which is given in the flat space limit by the leadingterm (4.69) W k (cid:0) ν , ν , ν (cid:1)(cid:0) √ λ (cid:1) k → (cid:18) ν + ν − ν √ λ (cid:19) k → (cid:18) α (cid:48) q + q − q (cid:19) k = ( α (cid:48) ) k ( − q · q ) k , (6.19)where we used that q = q + q . This implies that the flat space limit of (4.65) is1 (cid:0) √ λ (cid:1) k ∇ pm . . . ∇ pm k Ω iν ( L ) ∇ m p . . . ∇ m k p Ω iν ( L ) → (6.20) → ( − α (cid:48) ) k q m . . . q m k e ib · q q m . . . q m k e ib · q . (6.21)We conclude that both (6.18) and (6.21) are compatible with (6.17). Apart from thecovariant derivative, tensor structures depend also on the direction (cid:98) p , which is normal tothe transverse space H d − and satisfies (cid:98) p = −
1. In flat space the only possible direction– 44 –or polarizations that is normal to the transverse space is the unit vector v , hence we haveto require that in the flat space limit (cid:98) p m → iv m . (6.22)With the identifications (6.17) and (6.22), the matching in (6.16) works provided thatthe spectral functions β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) in (4.61) are such that λ j ( ν ) − D (∆ ,ρ ) , (∆ ,ρ ) mn ( ν ) Ω iν ( L ) → π A Pm ,ρ , m ( − i∂ b , v ) β ( t ) A Pm ,ρ , n ( − i∂ b , v ) ω q ( b ) . (6.23)Using the explicit tensor structures for three-point amplitudes in (3.27), the matching(6.23) can also be expressed for the spectral function for any given tensor structure λ j ( ν ) − λ | ρ |− k λ | ρ |− k β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) → π a k m ,ρ ( t ) a k m ,ρ ( t ) β ( t ) . (6.24)The powers of λ are again compatible with a factor of λ − in β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) for eachderivative in the coupling. Such a scaling is expected from the general arguments of [51, 55].We have now shown that all tree-level phase shifts appearing in the AdS and flat spaceoptical theorems can be related by the flat space limit.Let us also compare the vertex functions that appear in both AdS and flat space impactparameter representations of the one-loop amplitudes. The flat space limit of (4.72) is givenby the impact parameter transform of (3.20), i.e. − Re B ( p, p ) → Im δ ( s, b ) , (6.25)becomes2 π ∞ (cid:90) −∞ dνdν dν β ( ν ) ∗ β ( ν ) V ( ν , ν , ν ) S j ( ν )+ j ( ν ) − Φ( ν , ν , ν ) Ω iν ( L ) →→ (cid:90) R D − dqdq dq (2 π ) D − β ( t ) ∗ β ( t ) V ( q , q ) (cid:18) α (cid:48) s (cid:19) j ( t )+ j ( t ) − δ ( q − q − q ) e iq · b . (6.26)In this case we can use the delta function to write all the other scalar functions in termsof q , q and q , however we need to do the angular integral over the delta function itself.To this end we can define (cid:90) R D − dq dq (2 π ) D − δ ( q − q − q ) = 4 ∞ (cid:90) d | q | d | q | φ ( q , q , q ) . (6.27)Using this and (6.5), we can compute the angular integrals in (cid:90) R D − dq dq (2 π ) D − e ib · ( q + q ) = (cid:90) R D − dq dq dq (2 π ) D − δ ( q − q − q ) e ib · q , (6.28)– 45 –o find the flat space version of (4.64) ω q ( b ) ω q ( b ) = 2 ∞ (cid:90) d | q | φ ( q , q , q ) ω q ( b ) . (6.29)Using the explicit expressions for Φ and φ (which can be found for instance in appendix Eof [52]), one can further check that under the flat space limit R − D Φ( ν , ν , ν ) → φ ( q , q , q ) . (6.30)With this relation is clear that the expressions in (6.26) are indeed related by the flat spacelimit provided that the vertex functions are related by V ( ν , ν , ν ) → V ( q , q ) = V ( t , t , t ) . (6.31) We saw above that all elements of the impact parameter optical theorems in AdS and flatspace are related by the flat space limit provided that j ( ν ), β ( ν ), β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) and V (cid:0) ν , ν , ν (cid:1) are given by their flat space counterparts in the limit. In this subsection, webriefly review how the limit actually constrains these functions. All of these objects dependon two dimensionless quantities, the spectral parameters ν and the t’Hooft coupling λ . Letus discuss this for a generic function f ( ν ) that is required to satisfy the flat space limit f ( ν, λ ) → f ( t ) . (6.32)Existence of the gravity limit requires the function to have an expansion in negative powersof √ λ of the form f ( ν, λ ) = ∞ (cid:88) n =0 f n ( ν ) λ n/ . (6.33)In order for the flat space limit (6.10) of the function to be finite, the functions f n ( ν ) musthave an expansion in large ν with leading power not larger than 2 n , f n ( ν ) = a n,n ν n + a n,n − ν n − + a n,n − ν n − + . . . , (6.34)which ensures finiteness of the limit order by order in the large λ expansion. The flat spacelimit of f ( ν, λ ) is then f ( ν, λ ) → ∞ (cid:88) n =0 a n,n (cid:18) ν √ λ (cid:19) n → ∞ (cid:88) n =0 a n,n ( α (cid:48) q ) n . (6.35)At every order in 1 / √ λ the leading power of ν survives and is fixed by the flat space limit,while all the other powers are subleading, and cannot be determined from this condition.These considerations hold for j ( ν ), β ( ν ), β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) and V ( ν , ν , ν ), fixing part ofthese functions. These facts have been explored in detail for the functions j ( ν ) and β ( ν )in [30, 31]. – 46 – Relating type IIB string theory in AdS and flat space
Let us now apply our general ideas to a concrete example, the scattering of four dilatons intype IIB superstring theory on
AdS × S . In the flat space limit this is related to type IIBsuperstring theory on 10-dimensional flat space where the kinematics is restricted to thefive dimensions arising from AdS. This happens since both the dilatons and the Pomeronsare R -symmetry singlets, meaning the tidal excitations they couple to also have to besinglets. As a consequence, the Regge limit does not probe the 10 dimensional nature ofthe string scattering process, as we consider only states with the vacuum quantum numbersassociated to the compact manifold S . For this case the discontinuity of the (finite α (cid:48) )one-loop amplitude in the Regge limit was computed in [24] and is precisely of the form(3.20) with D = 5. The regime of validity of this description was discussed in detail in[24]. All we need to specify are the four dynamic quantities that we already discussed inthe previous section. For the Regge trajectory and Pomeron propagator we have j ( t ) = 2 + α (cid:48) t , β ( t ) = 2 π Γ (cid:0) − α (cid:48) t (cid:1) Γ (cid:0) α (cid:48) t (cid:1) e − iπα (cid:48) t . (7.1)As discussed in section 3.3 the vertex function can be obtained from the scattering am-plitude of two dilatons and two Pomerons. This amplitude was computed in [24] andreads A P P ( k, q , q ) = − Γ(1 + α (cid:48) q /
2) Γ (cid:0) − α (cid:48) k − α (cid:48) q / (cid:1) Γ (cid:0) α (cid:48) k (cid:1) − α (cid:48) q /
2) Γ (cid:0) α (cid:48) q / α (cid:48) k (cid:1) Γ (cid:0) − α (cid:48) k (cid:1) , (7.2)where q = q · q . This amplitude has poles at the masses ( m = 4 n/α (cid:48) ) of the stringstates with residuesRes k = − n/α (cid:48) A P P ( k, q , q ) = (cid:18) ( − α (cid:48) q / n n ! (cid:19) , n = 0 , , , . . . , (7.3)and the resulting vertex function is given by (3.19) V ( q , q ) = ∞ (cid:88) n =0 (cid:18) ( − α (cid:48) q / n n ! (cid:19) = Γ (cid:0) α (cid:48) t + t − t (cid:1) Γ (cid:0) α (cid:48) t + t − t (cid:1) . (7.4)Using the reasoning of section 6.2, this immediately fixes the leading terms in ν, ν i of theAdS vertex function at every order in λV ( ν , ν , ν ) = Γ (cid:16) − ν + ν − ν √ λ (cid:17) Γ (cid:16) − ν + ν − ν √ λ (cid:17) + vanishing in flat space limit . (7.5)Thus, the first two corrections from expanding (7.5) at large λ are V ( ν , ν , ν ) = 1 + (cid:16) · (cid:0) ν + ν − ν (cid:1) + c , (cid:17) √ λ (7.6)+ (cid:18) π (cid:0) ν + ν − ν (cid:1) + c , (cid:0) ν + ν (cid:1) + c (cid:48) , ν + c , (cid:19) λ + . . . , – 47 –here we also included the constants that are not fixed by the flat space limit. We notethat the constants multiplying the leading power of ν at order (cid:0) √ λ (cid:1) − n have a uniformtranscendentality of weight n , which can be seen by explicitly expanding (7.5). It wouldbe interesting to understand the relation of this property with features of maximal tran-scendentality in N = 4 SYM [56, 57].Finally, all the spectral functions β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) of tree-level correlators that con-tribute to the optical theorem are constrained by the flat space limit. Their flat space limit(6.24) is parameterized by on-shell three-point amplitudes in flat space. These amplitudesare in principle encoded in the result (7.3), which separates the contributions of particleswith different masses, but not the ones of particles in different representations ρ . Theattempt to expand (7.3) into products of three-point amplitudes for different ρ and tensorstructures k using (3.18) shows that this does not fully fix the a km ,ρ ( t ) in (3.27) becausethe equations are quadratic. However the three-point amplitudes can of course be com-puted in string theory, which is what the next subsection is about. We will start with the10D open superstring amplitudes of a massless vector, a Pomeron and an open string stateup to mass level 2 which were computed in [45] by studying string-brane scattering. Theseamplitudes have to be squared to obtain closed string amplitudes. Then the irreduciblerepresentations of the 10D massive Little group SO (9) have to be branched into irreduciblerepresentations of SO (4) to match with the CFT irreps and account for the fact that wehave five non-compact dimensions. Now we will discuss the flat space three-point amplitudes that take part in the process andthat will fix part of the tree-level correlators with external spinning legs in AdS via the flatspace limit. The goal is to derive the three-point amplitudes that appear in the unitaritycut (7.3) of the four-point amplitude of two dilatons and two Pomerons (7.2). It will beconvenient to consider the more general case of two gravitons instead of dilatons, withpolarizations (cid:15) µνi = (cid:15) µi (cid:15) νi , and obtain the dilaton amplitudes in the very end by replacing (cid:15) µνi with η µν . By using explicitly transverse three-point amplitudes and the completenessrelation (3.22), we can write tree-level unitarity (3.18) in the formRes k = − n/α (cid:48) A P P ( k, q ) = (cid:18) ( − α (cid:48) q / n n ! (cid:19) ( (cid:15) · (cid:15) ) = (cid:88) ρ,i A P n,ρ,i, m π m , n ρ A P n,ρ,i, n , (7.7)where for the massive levels, on which we will mostly focus, ρ is summed over irreduciblerepresentations of SO (4) and i is summed over degenerate states in the same representation.Our starting point will be the open string three-point amplitudes of a massless vector,a Pomeron and an arbitrary massive state up to mass level 2 (we give some simpler explicitexamples in Appendix A). These amplitudes were computed in [45] by studying string-brane scattering. Since in flat space there is no interaction between the left- and right-moving string modes, the closed string amplitudes factorize into products of open stringamplitudes. We can indeed check that the square root of the residues (7.7) matches the– 48 –xpansion in terms of the open string three-point amplitudes of [45] (cid:114) Res k = − n/α (cid:48) A P P ( k, q ) = ( − α (cid:48) q / n n ! ( (cid:15) · (cid:15) ) = (cid:88) ρ L A P n,ρ L , α π α , γ ρ L A P n,ρ L , γ . (7.8)We did this consistency check for the first three mass levels, for which ρ L is summed overthe bosonic part (NS sector) of the chiral superstring spectrum in 10 dimensions, given by[58] n = 0 : ,n = 1 : ⊕ ,n = 2 : ⊕ ⊕ ⊕ ⊕ . (7.9)In order to obtain three-point amplitudes for closed strings in 10 D , we need to square (7.8)and expand again in irreducible representations. The first step is trivialRes k = − n/α (cid:48) A P P ( k, q ) = (cid:88) ρ L ,ρ R A P n,ρ L , α A P n,ρ R , β π α , γ ρ L π β , δ ρ R A P n,ρ L , γ A P n,ρ R , δ , (7.10)however expanding this into irreducible representations requires some more work. On anabstract level this is easily done in terms of the tensor product ρ L ⊗ ρ R = (cid:77) ρ C ρ C , (7.11)which can be computed explicitly in terms of characters using e.g. the WeylCharacterRingimplementation in SageMath [59]. For example, the closed string spectrum for the firsttwo mass levels is n = 0 : ⊗ = ⊕ ⊕ • , (7.12) n = 1 : (cid:32) ⊕ (cid:33) = ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ • . To use the tensor product in explicit calculations requires considerably more work andcan be done by formulating (7.11) as an equation in terms of projectors to irreduciblerepresentations π α ; γ ρ L π β ; δ ρ R = (cid:88) ρ C ⊂ ρ L ⊗ ρ R p αβ ρ L ⊗ ρ R → ρ C , µ π µ ; ν ρ C p γδ ρ L ⊗ ρ R → ρ C , ν . (7.13)– 49 –he tensors p ρ L ⊗ ρ R → ρ C are constructed from Kronecker deltas and are uniquely determinedby this equation. By inserting (7.13) into (7.10) we find the expansion of the residueRes k = − n/α (cid:48) A P P ( k, q ) = (cid:88) ρ L ,ρ R ,ρ C A P n,ρ L ⊗ ρ R → ρ C , µ π µ , ν ρ C A P n,ρ L ⊗ ρ R → ρ C , ν , (7.14)in terms of the closed string amplitudes A P n,ρ L ⊗ ρ R → ρ C , µ = A P n,ρ L , α A P n,ρ R , β p αβ ρ L ⊗ ρ R → ρ C , µ . (7.15)The final step is to restrict the indices of the amplitudes to five dimensions and expandonce again into irreducible representations, this times for the massive Little group SO (4).In terms of representation theory, this is done by using branching rules to expand the SO (9) representations in terms of irreps of the product SO (4) × SO (5), ρ C = (cid:77) ( ρ,σ ) ⊂ ρ C ( ρ, σ ) , (7.16)where, for massive levels, ρ is an irreducible representation of SO (4) and σ of SO (5). Sincewe consider Pomeron exchange, which caries the vacuum quantum numbers, we projectonto the singlets of SO (5) ρ C | • = (cid:77) ( ρ, • ) ⊂ ρ C ( ρ, • ) . (7.17)This step is also abstractly implemented in SageMath. For example, we have = (cid:16) , • (cid:17) ⊕ (cid:16) , (cid:17) ⊕ (cid:16) • , (cid:17) ⊕ (cid:16) • , • (cid:17) . (7.18)and after projection to SO (5) singlets (cid:12)(cid:12)(cid:12) • = ⊕ • . (7.19)In this way we find the SO (5) singlets for the closed string spectrum in terms of SO (3) or SO (4) irreps for the first two levels n = 0 : ⊕ ⊕ • ,n = 1 : ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ • . (7.20)As for the tensor product, we can rephrase (7.17) as an equation in terms of projectors.In this case we get an equation for the SO (9) projector with indices restricted to the SO (4)directions a = 1 , . . . , π a ; b ρ C = (cid:88) ρ ⊂ ρ C | • b a ρ C → ρ, m π m ; n ρ b b ρ C → ρ, n , (7.21)where the tensors b ρ C → ρ are uniquely determined by this equation and can be expressed interms of Kronecker deltas and the SO (4) Levi-Civita symbol. Since we are assuming the– 50 –at space limit kinematics to be restricted to five dimensions, we can simply insert thisinto (7.14) and obtain the residue in the anticipated form (7.7)Res k = − n/α (cid:48) A P P ( k, q ) = (cid:88) ρ L ,ρ R ,ρ C ,ρ A P n,ρ L ⊗ ρ R → ρ C → ρ, m π m , n ρ A P n,ρ L ⊗ ρ R → ρ C → ρ, n , (7.22)with the 5 D closed string amplitudes given by A P n,ρ L ⊗ ρ R → ρ C → ρ, m = A P n,ρ L ⊗ ρ R → ρ C , a b a ρ C → ρ, m . (7.23) Let us now give a specific example of the procedure outlined above. We will consider thefollowing chain of expansions at mass level 1, starting from the product of two open stringmassive spin 2 fields that give rise to a 5 D scalar ⊗ → → • . (7.24)In this example we discard the 5 D massive spin 2 field that also appears in the projection(7.19). We alert the reader that whenever we write explicit amplitudes they are neitherappropriately symmetrized nor traceless in order to write them more compactly. All explicitamplitudes should be understood as objects to be contracted with the projector for theassociated representation.We start with the open string amplitude for the state (1 , ) from [45] A P , [2] ,α α = − (cid:114) α (cid:48) (cid:18) (cid:15) α q α + 12 ( q · (cid:15) ) v α v α (cid:19) . (7.25)Squaring this amplitude produces the following closed string states ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ • . (7.26)To construct the relation (7.13) we need the projectors for all the representations in thislist. They can be found in [60, 61], however here we just remind the reader of two of themost familiar ones π µ µ ,ν ν [2] d = 12 (cid:0) η µ ν η µ ν + η µ ν η µ ν (cid:1) − d η µ µ η ν ν , π • = 1 , (7.27)and state that the closed string state comes in (7.13) with the tensor p α α β β [2] ⊗ [2] → [2] ,µ µ = (cid:114) π α α ; γ γ [2] π β β ; δ δ [2] η γ δ η γ µ η δ µ , (7.28)where we introduced additional projectors contracted with metrics in order to have thecorrect index properties. This determines the following closed string amplitude via (7.15) A P , [2] ⊗ [2] → [2] ,µ µ = α (cid:48) √ (cid:104)(cid:0) q ,µ ( (cid:15) ,µ q · (cid:15) + 3 q ,µ )+ (cid:15) ,µ ( q ,µ q · (cid:15) − t (cid:15) ,µ ) + v µ v µ ( q · (cid:15) ) (cid:1)(cid:105) . (7.29)– 51 –he branching rule for was already considered in (7.19). Using the projectors (7.27)it is easy to see that we can write explicitly π a a ,b b [2] = δ a m δ a m π m m ,n n [2] δ b n δ b n + 536 δ a a π • δ b b , (7.30)from which we read off b a a [2] → [2] ,m m = δ a m δ a m , b a a [2] →• = (cid:114) δ a a . (7.31)Finally, inserting (7.29) and (7.31) into (7.23) we compute the 5 D closed string amplitude A P , [2] ⊗ [2] → [2] →• = 18 (cid:114) α (cid:48) (cid:0) ( q · (cid:15) ) − t (cid:1) (7.32)We derive the complete list of such level 1 three-point amplitudes in appendices B and C. In this section we use the flat-space string amplitudes to constrain the high-energy, tree-level AdS amplitudes with two dilatons and two spinning operators (cid:104) φφ O O (cid:105) . Since theoperators in question are of stringy nature (i.e. the bulk fields have m ∼ n/α (cid:48) for avery large AdS radius), their dimensions grow with the ’t Hooft coupling (∆ ∼ λ / ) andtransform in the SO (4) representations discussed above in the flat space case. This meansthat generically O and O are in bosonic mixed-symmetry representations.As discussed in section 6.1, the spectral functions that determine the spinning AdScorrelators (4.60) via (4.61) are determined in the flat space limit by the three-point am-plitudes (3.27) λ j ( ν ) − λ | ρ |− k λ | ρ |− k β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) → π a k m ,ρ ( t ) a k m ,ρ ( t ) β ( t ) . (7.33)In other words, the leading term in ν at each order in λ is fixed by the flat space expression(see section 6.2) λ j ( ν ) − λ | ρ |− k λ | ρ |− k β k ,k (∆ ,ρ ) , (∆ ,ρ ) ( ν ) = 12 π a k m ,ρ ( t ) a k m ,ρ ( t ) β ( t ) (cid:12)(cid:12)(cid:12) α (cid:48) t = − ν √ λ + . . . , (7.34)where . . . are terms that vanish in the flat space limit. Let us now use this to studysome specific examples. We begin with the example of section 7.1.1 where we computed anamplitude involving a graviton, a Pomeron and a particular scalar at mass level 1 in (7.32).In this result (cid:15) ,µ is such that (cid:15) µν = (cid:15) ,µ (cid:15) ,ν parametrizes a general graviton polarizationwhich must be replaced by the metric (cid:15) µν → η µν to obtain the dilaton amplitude A D P , [2] ⊗ [2] → [2] →• = a /α (cid:48) , [2] ⊗ [2] → [2] →• ( t ) = − (cid:114) α (cid:48) t , (7.35)where we used that ( q · (cid:15) ) = − t for the dilaton case. Thus, for the AdS correlator ofthis particular scalar and three dilatons we have β , λ + ..., • ) , (4 , • ) ( ν ) = 38 (cid:114) ν √ λ β ( ν ) + vanishing in the flat space limit . (7.36)– 52 –ote that with respect to the case of four dilaton scattering we have an extra power of1 / √ λ , so the term of order λ is absent, confirming that tidal excitations are suppressedat large λ , which in turn agrees with the considerations of [34] (we verified this fact forall amplitudes at level 1). In particular, this is consistent with the large λ suppressionof c φ φ j ( ν ) for non-identical scalars, since our stringy mode is certainly different from thedilaton. Such a suppression is not a priori obvious from writing a bulk interaction betweentwo different scalars and a spin J field (to be Sommerfeld-Watson transformed into aPomeron), which makes this a non-trivial realization of the bounds derived in [51, 55].This example is particularly simple, since there is a unique three-point structure in thecase of the scalar.More generally, we can consider amplitudes with several tensor structures constructedfrom v a ’s and q a ’s (equivalently, (cid:98) p and ∇ p in AdS). Let us take as a representative example,the case of the spin 4 operator at level 1. This operator is typically used to define ∆ gap andsits in the leading Regge trajectory. The corresponding graviton-Pomeron-spin 4 amplitudewas worked out in Appendices B and C, and reads A a a a a [2] ⊗ [2] → [4] → [4] ( (cid:15) ) = 18 α (cid:48) (cid:0) (cid:15) ,a q ,a + v a v a q · (cid:15) (cid:1)(cid:0) (cid:15) ,a q ,a + v a v a q · (cid:15) (cid:1) . (7.37)We again emphasize that it is understood that the amplitude should be contracted with π α ; β [4] . Furthermore, we are using the simplifying transverse kinematics discussed above.This means that upon doing the dilaton replacement, we have A a a a a [2] ⊗ [2] → [4] → [4] = α (cid:48) (cid:0) − t v a v a v a v a + 4 v a v a q ,a q ,a (cid:1) , (7.38)where we used the symmetry of the indices and note that transverse kinematics ensures thatthe term proportional to (cid:15) ,a (cid:15) ,a gets mapped to a transverse metric which is annihilatedby the projector to [4] . In this case we can directly use (6.23) to match both tensorstructures at once D (2 λ + ..., [4] ) , (4 , • ) a a a a ( ν ) = β ( ν )8 √ λ (cid:0) ν (cid:98) p a (cid:98) p a (cid:98) p a (cid:98) p a + 4 (cid:98) p a (cid:98) p a ∇ p a ∇ p a (cid:1) + vanishing in flat space limit . (7.39)We again note that these corrections are suppressed at large λ . In this work, we derived a perturbative CFT optical theorem which computes the dDisc of acorrelator in the 1 /N expansion in terms of single discontinuities of lower order correlators.Notably, this allows the determination of double-trace contributions to a given one-loopholographic correlator, even when the intermediate fields have spin, which makes themmuch harder to handle using unitarity formulas in terms of the CFT data. This alsoclarifies the underlying CFT principles behind cutting formulas for AdS Witten diagrams,which so far used bulk quantities [16, 17]. – 53 –sing the perturbative CFT optical theorem we fixed the form of the AdS one-loopfour-point scattering amplitude in the high-energy limit, accounting for the physical effectof tidal excitations. This corresponds to box Witten diagrams with two-Pomeron exchangeand general string fields as intermediate states. To do this, we transformed the opticaltheorem to CFT impact parameter space, in which the loop level phase shift is obtained asa contraction of tree-level phase shifts. Using the general structure of spinning correlatorsin the s-channel Regge limit, we rewrote all the tidal excitations in terms of a single scalarfunction, the AdS vertex function.For the case of N = 4 SYM, dual to type IIB strings, we fixed part of the answer byrelating our expression to the flat space results of [23–25] for high energy string scattering,requiring consistency with the flat space limit in impact parameter space. This procedurefixes part of the AdS vertex function and therefore also part of the CFT correlation functionat one-loop in the Regge limit. Additionally, interpreting the previous result in terms ofunitarity, we used the flat-space behavior to constrain the spectral function for certainspinning CFT correlators at tree level in the Regge limit.There are several open directions and applications of this work. First, we emphasizethat the CFT optical theorem is quite general and does not rely on AdS ingredients.Moreover, it works directly at the level of correlators instead of having to extract the CFTdata, which is very difficult to resum into correlators. It would be interesting to test and usethis formula for more general holographic correlators and, since the expansion parameterdoes not necessarily need to be 1 /N , in weakly coupled CFTs such as φ theory at theWilson-Fisher fixed point in 4 − (cid:15) dimensions.Another playground to apply our gluing formula is N = 4 SYM at weak t’Hooftcoupling in the Regge limit. One could try to derive the order 1 /N correlator explicitlyat leading order in λ , using the techniques introduced in [31]. The corresponding doublediscontinuity should be the square of order 1 /N correlators with impact factors thatinclude the intermediate states.In the Regge limit there are kinematical conditions in the CFT optical theorem thatsimplified the integrations over Lorentzian configurations. An interesting generalizationwould be to systematically study kinematic corrections to the Regge limit. In fact, in therecent work [42] the authors derived a Regge expansion for the correlator valid for anyboost. It would be interesting to see how to incorporate this into our analysis, both ina general structural way, but also potentially to impose specific constraints from the flatspace limit in a more general kinematic setup. More generally, it would be interestingto understand the Regge limit integrations in terms of light-ray operators [3], and to usethe more general Lorentzian machinery of [2, 3] to write an intrinsically Lorentzian gluingformula in general kinematics.A possible extension of this work is to consider a higher number of bulk loops. Thiswas analyzed in the large ∆ gap limit in [34]. Let us give a few concrete ideas for the stringygeneralization of that analysis. The leading contribution in the Regge limit at k − k -Pomeron exchange, related to a k -fold product of tree-level phase shifts.By repeatedly using (4.64) one can define a generalization of the function Φ for such aproduct, so we expect that the contribution of intermediate states can again be expressed– 54 –y a vertex function − Re B k − ( S, L ) = ∞ (cid:90) −∞ dν (cid:32) k (cid:89) n =1 dν n β ( ∗ ) ( ν n ) (cid:33) V ( ν , . . . , ν k , ν ) S (cid:80) m j ( ν m ) − k Φ( ν , . . . , ν k , ν ) Ω iν ( L ) , (8.1)where the product of β ( ν n ) must be real, which means that the answer is slightly differentdepending on whether the number of loops is even or odd [34]. In order to find the flatspace limit of this ( k − V k ( q , . . . , q k ) = (cid:90) k (cid:89) i =1 dσ i π (cid:89) ≤ j A Additional examples of string amplitudes In this appendix we provide some additional examples and comments on the chiral andclosed string amplitudes. This corresponds to eikonalization in the operator sense of [24] where the phase shift is an operator inthe string Hilbert space, with matrix elements between all possible string states. – 55 – .1 Chiral Amplitudes In the chiral case it is trivial to reproduce level 0. Here we have only the massless particleswith residue (cid:114) Res k =0 A P P ( k, q ) = (cid:15) µ A P µα π α ; β [1] A P βν (cid:15) ν = (cid:15) µ η µα η αβ η βν (cid:15) ν = (cid:15) · (cid:15) , (A.1)where we have used A P µα = η µα and π α ; β [1] = η αβ . At higher levels we will have non-trivialthree-point functions. It will be convenient to absorb the external polarization into theamplitude (cid:15) µ A P n, µ,ρ , ν ≡ A P n,ρ , ν , (A.2)to be more compact in writing the amplitudes (we are using the integer n to label the masslevel of the state).From the spectrum described above, we will have two amplitudes at level 1 which are A P , [1 , , , α and A P , [2] , α .Here we will keep in mind the Young diagrams explained above, along with the indexsymmetrization that comes with them, packaged in our boldface multi-index notation.The explicit level 1 three point amplitudes in the IIB superstring are A P , [1 , , , α = √ m (cid:15) α q α v α , A P , [2] , α = − (cid:114) α (cid:48) (cid:18) (cid:15) α q α + 12 ( q · (cid:15) ) v α v α (cid:19) , (A.3)with m = 2 / √ α (cid:48) being the mass at level 1, and q is the transverse momentum carriedby the Pomeron P (similarly for q and the Pomeron P ). All the relative factors betweenthe different tensor structures and the overall normalization are fixed by computing thethree-point amplitudes with the correctly normalized vertex operators for the excited NSstates in IIB super string theory [45]. We can now contract the three-point amplitudes oneach side using the projector for the appropriate representation and check the residue (cid:114) Res k = − /α (cid:48) A P P ( k, q ) = A P , α π α ; β [1 , , A P , β + A P , α π α ; β [2] A P , β = α (cid:48) − q · q )( (cid:15) · (cid:15) ) , (A.4)where we refrained from writing the representation labels in the amplitudes since they arecontracted with a projector with the appropriate label. This matches what we extractedfrom A α α ( q ), or equivalently from the vertex function. We can continue this procedureto the second level, where mixed symmetry tensors appear for the first time. For example,the [2 , , tensor has the amplitude A P , [2 , , , α = (cid:114) (cid:114) α (cid:48) (cid:18) m q α + 2 v α (cid:19) (cid:15) α q α v α , (A.5)where m = (cid:112) /α (cid:48) is the mass at level 2. It is important to emphasize that the level 2amplitude contains a term with more powers of α (cid:48) than any of the level 1 amplitudes. Thiswould lead to further suppression in 1 / √ λ in the AdS theory. Clearly, states with higher spin, which can only appear at higher levels, can have higher powers of α (cid:48) leading to a spin-dependent suppression of couplings, as is expected from the general arguments of [51, 55]. – 56 –he remaining amplitudes can be found in section 5 of [45]. For our purposes it is justimportant to know that the amplitudes satisfy (cid:114) Res k = − /α (cid:48) A P P ( k, q ) = (cid:88) ρ ∈ S A P ,ρ, α π α ; β ρ A P ,ρ, β = ( − α (cid:48) q · q ) 2! ( (cid:15) · (cid:15) ) ,S = { [3] , [2 , , , [2 , , [1 , , [1] } , (A.6)which we explicitly checked. More generally, we can conclude that the square root of theresidue at mass level n of A P P ( k, q ) can be recovered by unitarity if we account forall the covariant SO(9) representations corresponding to the massive NS states. This givesa microscopic interpretation for the vertex function at a given mass level. As alreadymentioned, summing over all these mass levels reconstructs the full vertex function. A.2 Closed string amplitudes Here we consider the simple but instructive level 0 case for the closed string amplitudes,where the little group is SO(8). The square of the residue reads A P L ,α A P R ,β (cid:16) π α ; γ [1] π β ; δ [1] (cid:17) A P L ,γ A P R δ (A.7)= (cid:88) ρ C =[2] , [1 , , • A P L ,α A P R ,β ( p αβ [1] ⊗ [1] → ρ C ,µ µ π µ µ ; ν ν ρ C p γδ [1] ⊗ [1] → ρ C ,ν ν ) A P L ,γ A P R δ = ( (cid:15) · (cid:15) ) , where we used the group theoretical tensor product identity for projectors π α ; γ [1] π β ; δ [1] = (cid:88) ρ C =[2] , [1 , , • p αβ [1] ⊗ [1] → ρ C ,µ µ π µ µ ; ν ν ρ C p γδ [1] ⊗ [1] → ρ C ,ν ν . (A.8)We can solve this equation for the tensors p by contracting with polarization vectors forthe left and right modes on both sides of the projector ( z L , z R and z L , z R ) and equatingthe polynomials in scalar products of z ’s. In practice we will always use this procedure, ora similar one where we contract with amplitudes to fix coefficients. In this case it is trivialto directly check that π α ; γ [1] π β ; δ [1] = η αγ η βδ ≡ π αβ ; γδ [2] + π αβ ; γδ [1 , + 18 η αβ η γδ (A.9)= (cid:18) 12 ( η αγ η βδ + η αδ η βγ ) − η αβ η γδ (cid:19) + 12 ( η αγ η βδ − η αδ η βγ ) + 18 ( η αβ η γδ ) , where, obviously p αβ [1] ⊗ [1] → [2] ,µ µ = δ αµ δ βµ , p αβ [1] ⊗ [1] → [1 , ,µ µ = δ αµ δ βµ and p αβ [1] ⊗ [1] →• = (cid:113) η αβ extracts traces, thereby projecting to a singlet state. B Tensor products for projectors In this appendix we explain how to realize the tensor product of open string states intoclosed string states in terms of the corresponding projectors/tensors. We will consider masslevel n = 1. The chiral spectrum at this level is n = 1 : ⊕ . (B.1)– 57 –e square the irreps using the tensor product as in the main text and analyze the de-composition term by term. For example, taking ρ L = ρ R = [2] corresponds to the tensorproduct ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ • , (B.2)which we want to write in terms of SO(9) tensors as π α ; γ [2] π β ; δ [2] = (cid:88) ρ C ∈ S p αβ [2] ⊗ [2] → ρ C , µ π µ ; ν ρ C p γδ [2] ⊗ [2] → ρ C , ν ,S = { [4] , [3 , , [2 , , [2] , [1 , , • } . (B.3)It will be useful to manifestly symmetrize the α and β indices of the tensors p , in order towrite down these tensors more compactly. Therefore, we will use p αβ ρ L ⊗ ρ R → ρ C , µ ≡ π α ρ L ; α (cid:48) π β ρ R ; β (cid:48) ˜ p α (cid:48) β (cid:48) ρ L ⊗ ρ R → ρ C , µ , (B.4)and we will present the simpler trial tensors ˜ p for each case. In this case ρ L = ρ R = [2] and we used the trial tensors ˜ p α α β β [2] ⊗ [2] → [4] ,µ µ µ µ = δ α µ δ α µ δ β µ δ β µ ≡ δ αβµ , ˜ p αβ [2] ⊗ [2] → [3 , , µ = δ αβµ , (B.5)˜ p αβ [2] ⊗ [2] → [2 , , µ = √ δ αβµ , ˜ p α α β β [2] ⊗ [2] → [2] ,µ µ = (cid:114) δ α µ δ β µ η α β , ˜ p α α β β [2] ⊗ [2] → [1 , ,µ µ = (cid:114) δ α µ δ β µ η α β , ˜ p α α β β [2] ⊗ [2] →• = (cid:114) η α β η α β . The remaining tensor products have some additional subtleties. Taking the cross term inthe tensor product ⊗ = ⊕ ⊕ ⊕ , (B.6)we note that there is a 4 row tensor appearing. When contracted with the amplitudes, thiscontribution will vanish, because we have only 3 independent vectors. However, from thepoint of view of the projector equation, we must still have π α ; γ [2] π β ; δ [1 , , = (cid:88) ρ C ∈ S (cid:48) p αβ [2] ⊗ [1 , , → ρ C , µ π µ ; ν ρ C p γδ [2] ⊗ [1 , , → ρ C , ν ,S (cid:48) = { [3 , , , [1 , , , [2 , , [2 , , , } . (B.7)with a non-vanishing p αβ [2] ⊗ [1 , , → [2 , , , , µ . However, by contracting directly with theamplitudes A α [2] A β [1 , , A γ [2] A δ [1 , , we automatically eliminate the contribution of this For irreps with the same number of indices as the tensor product we will always take ˜ p αβµ ∝ δ αβµ . – 58 –ensor (this also avoids the computation of a complicated 4 row projector). With this inmind, we use again (B.4), with the trial projectors˜ p αβ [2] ⊗ [1 , , → [3 , , , µ = (cid:114) δ α µ δ α µ δ β µ δ β µ δ β µ ≡ (cid:114) δ αβµ , ˜ p αβ [2] ⊗ [1 , , → [1 , , , µ = (cid:114) η α β δ α µ δ β µ δ β µ , ˜ p αβ [2] ⊗ [1 , , → [2 , , µ = η α β δ α µ δ β µ δ β µ . (B.8)With these tensors and setting p αβ [2] ⊗ [1 , , =[2 , , , , µ → 0, which suffices for our purposes,we have that the identity (B.7) holds, but only when inserted between the amplitudes.The remaining tensor product ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ • , (B.9)can be dealt with similarly, by contracting with only 3 independent polarizations instead ofsix, eliminating the contributions of the complicated 4 row tensors. The only non-vanishingcontributions to projectors turn out to be π α ; γ [1 , , π β ; δ [1 , , = (cid:88) ρ C ∈ S (cid:48)(cid:48) p αβ [1 , , ⊗ [1 , , → ρ C , µ π µ ; ν ρ C p γδ [1 , , ⊗ [1 , , → ρ C , ν ,S (cid:48)(cid:48) = { [2 , , , [2 , , [2] , • } . (B.10)where we used (B.4) and the trial projectors˜ p αβ [1 , , ⊗ [1 , , → [2 , , , µ = √ δ α µ δ α µ δ α µ δ β µ δ β µ δ β µ ≡ √ δ αβµ , ˜ p αβ [1 , , ⊗ [1 , , → [2 , , µ = (cid:114) η α β δ α µ δ β µ δ α µ δ β µ , ˜ p αβ [1 , , ⊗ [1 , , → [2] , µ = (cid:114) η α β η α β δ α µ δ β µ , ˜ p αβ [1 , , ⊗ [1 , , →• = (cid:114) η α β η α β η α β , (B.11)where the unusual index ordering is to ensure that the resulting tensor doesn’t vanish whenwe act with π [1 , , on the trial projectors ˜ p . This turns (B.10) into an identity which holdsfor two identical [1 , , tensors, as will be the case for our amplitudes. C Branching relations for projectors In this Appendix we provide a detailed account of all the branching relations for closedstring state projectors utilized in section 7.1. Let us start by recalling the SO(9) closed– 59 –tring states at level 1 (cid:32) ⊕ (cid:33) = ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ • . (C.1)We recall that states with more than 3 columns will not contribute as we only have 3different vectors to anti-symmetrize. We are going to perform the branchingSO(9) → SO(4) × SO(5) | • , (C.2)where we denote the projection to singlets of SO(5) by | • . Note that certain representationscan naturally be restricted to SO(4) by simply taking the 5d subset of 10d indices. It isobvious that → , • → • . (C.3)Additionally, the projector for these representations is identical for SO(9) and SO(4) (upto restriction of the indices α → a , β → b , µ → m , . . . ) π a a ; b b [1 , = π a a ; b b [1 , = 12 (cid:16) η a b η a b − η a b η a b (cid:17) . (C.4)Other representations admit a direct restriction, but can also give additional irreps, bythe creation of SO(5) singlets, through the contraction of indices with legs on the compactmanifold. For example the spin 2 states branch as → ⊕ • . (C.5)The spin 2 on the RHS is interpreted as the restriction of indices to the SO(4) and thesinglet as a trace over the compact space indices. In terms of projectors the statement issimply π a a ; b b [2] = π a a ; b b [2] + 536 η a a η b b . (C.6)Similarly, for the spin 4 case → ⊕ ⊕ • , (C.7)and the projector equation is π a ; b [4] = (cid:88) ρ =[4] , [2] , • b a [4] → ρ, m π m ; n ρ b b [4] → ρ, n . (C.8)It will be again convenient to manifestly symmetrize the tensors, in order to present themmore compactly. We define b a ρ C → ρ, m ≡ π a ρ C , a (cid:48) ˜ b a (cid:48) ρ C → ρ, m , (C.9)– 60 –nd then present a list of the simpler ˜ b . In this case we have˜ b a a a a [4] → [4] ,m m m m = δ a m δ a m δ a m δ a m ≡ δ am ˜ b a a a a [4] → [2] ,m m = (cid:114) δ a m δ a m η a a , ˜ b a a a a [4] →• = (cid:114) η a a η a a . (C.10)The fact that the direct restriction of the irrep [4] → [4] comes with coefficient 1 is anon-trivial consistency check of the previous procedure.There are other irreps that don’t admit a direct restriction, because they have morethan two columns (SO(4) Young tableaux have at most two columns, and traces can vanishby antisymmetry). For this we need to use the SO(4) Levi-Civita tensor. We will simplywrite it as ε a a a a . The simplest case is the 3-form → , (C.11)and the corresponding projector equation is π a ; b [1 , , = 16 ε a a a m π m ; n [1] ε b b b n = 4 π a m ; b n [1 , , , π [1] ,m ; n , (C.12)From the first equation we can read off b a a a [1 , , → [1] ,m = (cid:114) ε a a a m . (C.13)For the second equality in (C.12) we used the standard identity (cid:15) a a a a (cid:15) b b b b = 4! π a ; b [1 , , , . (C.14)This is convenient to square the three-point amplitudes when computing the residue of thefour-point function A P P . Using trace subtractions and products of epsilon tensors wecan now derive branching identities for all the relevant irreps. Let us list the remainingidentities, where we write the trace subtractions and the Levi-Civita tensors using the trialprojectors ˜ b , but then appropriately symmetrize them through (C.9) → ⊕ , ˜ b a a a a [2 , , → [2] ,m m = (cid:114) ε a a a m δ a m , ˜ b a a a a [2 , , → [1 , ,m m = (cid:114) ε a a a m δ a m , → ⊕ , ˜ b a [2 , → [2 , , m = δ am , ˜ b a a a [2 , → [1] ,m = (cid:114) δ a m η a a , (C.15)– 61 – → ⊕ ⊕ , ˜ b a [3 , , → [3] , m = (cid:114) ε a a a m δ a m δ a m , ˜ b a [3 , , → [2 , , m = (cid:114) ε a a a m δ a m δ a m , ˜ b a a a a a [3 , , → [1] ,m = (cid:114) ε a a a m η a a , → ⊕ ⊕ , ˜ b a [3 , → [3 , , m = δ am , ˜ b a [3 , → [1 , , m = (cid:114) δ a m η a a δ a m , ˜ b a [3 , → [2] , m = (cid:114) δ a m η a a δ a m , → ⊕ ⊕ • , ˜ b a [2 , → [2 , , m = δ am , ˜ b a [2 , → [2] , m = (cid:114) δ a m η a a δ a m , ˜ b a [2 , →• = (cid:114) η a a η a a , → ⊕ • , ˜ b a [2 , , → [2] ,m m = √ ε a a a m ε a a a m , ˜ b a [2 , , →• = √ ε a a a m ε a a a m . (C.16)Note that for the last diagram, which has more than 2 boxes in both columns, we are forcedto use two pairs of epsilon tensors. Tensors with more than three rows aren’t allowed bythe 10d kinematics, but even if they were, their branchings do not contain singlets of SO(5)so we can simply discard them. C.1 All 5d closed string amplitudes Let us first write down in generality the 5d amplitudes using the relations derived in themain text. We have A P n,ρ L ⊗ ρ R → ρ C → ρ, m = A P ρ L , α A P ρ R , β p αβ ρ L ⊗ ρ R → ρ C , a b a ρ C → ρ, m . (C.17)With all the group theory identities established, we can enumerate all the amplitudes usedin the main text to reproduce the residue at the cut with mass level 1. However, we againemphasize that we have not explicitly symmetrized the amplitudes by contracting with therespective projector, in order to maintain some compactness of the tables below. Namely,all amplitudes are to be contracted with the projector to the SO(4) irrep, and furthermore,amplitudes where rank( ρ ) = rank( ρ L ) + rank( ρ R ) are also not explicitly symmetrized withrespect to ρ L and ρ R as in equation (B.4). Additionally, for amplitudes where the b tensorscontain a Levi-Civita symbol, we write the square of the amplitude (cid:0) A ρ L ⊗ ρ R → ρ C → ρ m (cid:1) ≡ A ρ L ⊗ ρ R → ρ C → ρ m π m ; n ρ A ρ L ⊗ ρ R → ρ C → ρ n , (C.18)since the amplitude itself cannot be written nicely in terms of v α , q α , (cid:15) α . With these caveatsin mind, we list the amplitudes starting by the ones with the most indices– 62 – A [2] ⊗ [2] → [4] → [4] m m m m = 18 α (cid:48) (2 (cid:15) ,m q ,m + v m v m q · (cid:15) ) × (2 (cid:15) ,m q ,m + v m v m q · (cid:15) ) , A [2] ⊗ [2] → [2 , → [2 , m m m m = 18 α (cid:48) (2 (cid:15) ,m q ,m + v m v m q · (cid:15) ) × (2 (cid:15) ,m q ,m + v m v m q · (cid:15) ) ,A [1 , , ⊗ [1 , , → [2 , → [2 , m m m m = α (cid:48) √ 15 [( v m (cid:15) ,m − v m (cid:15) ,m ) ( q · (cid:15) ( v m q ,m − v m q ,m )+ t ( v m (cid:15) ,m − v m (cid:15) ,m ))+ q ,m ( (cid:15) ,m ( (cid:15) ,m q ,m − (cid:15) ,m q ,m )+ v m ( v m ( (cid:15) ,m q · (cid:15) − q ,m ) + v m ( q ,m − (cid:15) ,m q · (cid:15) )))+ q ,m ( (cid:15) ,m ( (cid:15) ,m q ,m − (cid:15) ,m q ,m )+ v m ( v m ( q ,m − (cid:15) ,m q · (cid:15) ) + v m ( (cid:15) ,m q · (cid:15) − q ,m )))] , (cid:16) A [2] ⊗ [111] → [311] → [3] m (cid:17) = − ( α (cid:48) ) (cid:0) 48 ( q · (cid:15) ) (cid:0) q · (cid:15) ) + t (cid:1) + 48 t (cid:0) ( q · (cid:15) ) + t ( (cid:15) · (cid:15) ) (cid:1) + 48 ( q · q ) (cid:0) − (cid:15) · (cid:15) ) (cid:1) + 115( q · q )( (cid:15) · (cid:15) )( q · (cid:15) )( q · (cid:15) ) − q · (cid:15) )( q · (cid:15) )( q · (cid:15) )( q · (cid:15) )) , (cid:16) A [2] ⊗ [111] → [311] → [1] m (cid:17) = − 625 ( α (cid:48) ) q · (cid:15) )( q · (cid:15) ) × (( q · (cid:15) )( q · (cid:15) ) − ( q · q )( (cid:15) · (cid:15) )) , (cid:16) A [2] ⊗ [111] → [111] → [1] m (cid:17) = 65 ( α (cid:48) ) q · (cid:15) )( q · (cid:15) ) × (( q · q )( (cid:15) · (cid:15) ) − ( q · (cid:15) )( q · (cid:15) )) . (C.19)– 63 –nd 2 A [2] ⊗ [111] → [21] → [21] m m m = α (cid:48) √ q ,m ( v m (2 (cid:15) ,m q · (cid:15) − q ,m )+ v m (3 q ,m − (cid:15) ,m q · (cid:15) ))+ 2 q · (cid:15) ( q ,m ( v m (cid:15) ,m − v m (cid:15) ,m )+ q ,m ( v m (cid:15) ,m − v m (cid:15) ,m ))+ 3 t (cid:15) ,m ( v m (cid:15) ,m − v m (cid:15) ,α ))] , (cid:16) A [2] ⊗ [111] → [311] → [21] m (cid:17) = 5 ( α (cid:48) ) (cid:0) t ( q · (cid:15) ) + 6 t (cid:0) ( q · (cid:15) ) + t ( (cid:15) · (cid:15) ) (cid:1) + ( q · q )( (cid:15) · (cid:15) )( q · (cid:15) )( q · (cid:15) ) − ( q · (cid:15) )( q · (cid:15) )( q · (cid:15) )( q · (cid:15) ) + 6 ( q · q ) (cid:1) , (C.20)We also have the sixfold degenerate spin 2 states6 A [2] ⊗ [2] → [4] → [2] m m = α (cid:48) (cid:114) 539 [ q ,m (5 (cid:15) ,m ( q · (cid:15) ) + 2 q ,m )+ (cid:15) ,m (5 q ,m ( q · (cid:15) ) − t (cid:15) ,m )+5 v m v m ( q · (cid:15) ) (cid:3) ,A [2] ⊗ [2] → [2 , → [2] m m = α (cid:48) (cid:114) 542 [ − q ,m ( q ,m − (cid:15) ,m ( q · (cid:15) ))+ (cid:15) ,m (2 q ,m ( q · (cid:15) ) + t (cid:15) ,m )+ 2 v m v m ( q · (cid:15) ) (cid:3) ,A [2] ⊗ [2] → [2] → [2] m m = α (cid:48) √ 91 [( q ,m ( (cid:15) ,m ( q · (cid:15) ) + 3 q ,m )+ (cid:15) ,m ( q ,m ( q · (cid:15) ) − t (cid:15) ,m )+ v m v m ( q · (cid:15) ) (cid:1) ] , (C.21)– 64 – [1 , , ⊗ [1 , , → [2 , → [2] m m = α (cid:48) √ 14 [( q ,m ( q ,m − (cid:15) ,m ( q · (cid:15) ))+ (cid:15) ,m ( − q ,m ( q · (cid:15) ) − t (cid:15) ,m )+ v m v m (cid:0) − ( q · (cid:15) ) − t (cid:1)(cid:1) ] ,A [1 , , ⊗ [1 , , → [2] → [2] m m = α (cid:48) √ 21 [( q ,m ( (cid:15) ,m ( q · (cid:15) ) − q ,m )+ (cid:15) ,m ( q ,m ( q · (cid:15) ) + t (cid:15) ,m )+ v m v m (cid:0) ( q · (cid:15) ) + t (cid:1)(cid:1) ] , (cid:16) A [1 , , ⊗ [1 , , → [2 , , → [2] m (cid:17) = ( α (cid:48) ) (cid:0) t ( q · (cid:15) ) − t ( (cid:15) · (cid:15) )( q · (cid:15) )( q · (cid:15) )+17 ( q · (cid:15) ) (cid:0) ( q · (cid:15) ) + t (cid:1) + 17 t (cid:0) ( q · (cid:15) ) + t (cid:1) + ( q · (cid:15) ) (cid:0) 117 ( q · (cid:15) ) + 37 t (cid:1) − q · (cid:15) )( q · (cid:15) ) (( q · (cid:15) )( q · (cid:15) ) + t ( (cid:15) · (cid:15) ))+ ( q · q ) (cid:0) 117 ( (cid:15) · (cid:15) ) − (cid:1) + q · q (2 q · (cid:15) (37 q · (cid:15) − (cid:15) · (cid:15) q · (cid:15) )+74 q · (cid:15) ( q · (cid:15) − (cid:15) · (cid:15) q · (cid:15) )) − t t ( (cid:15) · (cid:15) ) (cid:1) , (C.22)and 8-fold degenerate spin 0 states8 • A [2] ⊗ [2] → [4] →• = 148 (cid:114) α (cid:48) (cid:0) t − 15 ( q · (cid:15) ) (cid:1) ,A [2] ⊗ [2] → [2 , →• = 124 (cid:114) α (cid:48) (cid:0) q · (cid:15) ) + t (cid:1) ,A [2] ⊗ [2] → [2] →• = 18 (cid:114) α (cid:48) (cid:0) ( q · (cid:15) ) − t (cid:1) ,A [2] ⊗ [2] →• →• = α (cid:48) √ (cid:0) ( q · (cid:15) ) − t (cid:1) ,A [1 , , ⊗ [1 , , → [2 , →• = 18 (cid:114) α (cid:48) (cid:0) − ( q · (cid:15) ) − t (cid:1) , (C.23)– 65 – [1 , , ⊗ [1 , , → [2] →• = 18 (cid:114) α (cid:48) (cid:0) ( q · (cid:15) ) + t (cid:1) ,A [1 , , ⊗ [1 , , →• →• = α (cid:48) √ (cid:0) ( q · (cid:15) ) + t (cid:1) , (cid:0) A [1 , , ⊗ [1 , , → [2 , , →• (cid:1) = ( α (cid:48) ) (cid:0) − t ( q · (cid:15) ) + 98 t ( (cid:15) · (cid:15) )( q · (cid:15) )( q · (cid:15) ) − 49 ( q · (cid:15) ) (cid:0) ( q · (cid:15) ) + t (cid:1) − 29 ( q · (cid:15) ) (cid:0) ( q · (cid:15) ) + t (cid:1) − t (cid:0) ( q · (cid:15) ) + t (cid:1) − 49 ( q · q ) (cid:0) ( (cid:15) · (cid:15) ) − (cid:1) + 98( q · (cid:15) )( q · (cid:15) ) (( q · (cid:15) )( q · (cid:15) ) + t ( (cid:15) · (cid:15) )) − q · q ( q · (cid:15) ( q · (cid:15) − ( (cid:15) · (cid:15) )( q · (cid:15) ))+ q · (cid:15) ( q · (cid:15) − ( (cid:15) · (cid:15) )( q · (cid:15) ))) + 49 t t ( (cid:15) · (cid:15) ) (cid:1) . 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