Minkowski Conformal Blocks and the Regge Limit for SYK-like Models

We discuss scattering in a CFT via the conformal partial-wave analysis and the Regge limit. The focus of this paper is on understanding an OPE with Minkowski conformal blocks. Starting with a t-channel OPE, it leads to an expansion for an s-channel scattering amplitude in terms of t-channel exchanges. By contrasting with Euclidean conformal blocks we see a precise relationship between conformal blocks in the two limits without preforming an explicit analytic continuation. We discuss a generic feature for a CFT correlation function having singular $F^{(M)}(u,v)\sim {u}^{-\delta}\,$, $\delta>0$, in the limit $u \rightarrow 0$ and $v\rightarrow 1$. Here, $\delta=(\ell_{eff}-1)/2$, with $\ell_{eff}$ serving as an effective spin and it can be determined through an OPE. In particular, it is bounded from above, $\ell_{eff} \leq 2$, for all CFTs with a gravity dual, and it can be associated with string modes interpolating the graviton in AdS. This singularity is historically referred to as the Pomeron. This bound is nearly saturated by SYK-like effective $d=1$ CFT, and its stringy and thermal corrections have piqued current interests. Our analysis has been facilitated by dealing with Wightman functions. We provide a direct treatment in diagonalizing dynamical equations via harmonic analysis over physical scattering regions. As an example these methods are applied to the SYK model.

adopted an Euclidean treatment and an analytic continuation is then performed [14][15][16][17][18]. Recent interest in CFT in a Minkowski setting has increased to warrant a more systematic and direct treatment 1 . Such an approach provides a framework where one can directly treat scattering problems, for example, inclusive and exclusive high energy near-forward scattering 2 , among others. Many phenomenological applications to high energy physics at the LHC and HERA have been carried out with encouraging successes 3 . In this paper we demonstrate a new method for di- 1 In the context of holography there is a history of directly investigating Lorentz correlators of the gravity theory to learn about thermal properties of the strongly coupled CFT. We are not concerned with this approach in this paper, but rather focus on the CFT directly regardless of a gravity dual. 2 In literature this is often referred to as the "eikonal limit" or the "Regge limit" 3 Many holographic models have been used to successfully model collider physics. For a brief list of some applications that directly investigate conformal properties see [15-17, 19, 21-28].
rectly computing Minkowski conformal blocks (MCB) as well as elucidating details about the Minkowski conformal block expansion relevant for arbitrary dimension.

A. Overview
This paper deals with the intersection of three sometimes disparate subjects: (1) conformal field theory, (2) analytic scattering amplitudes, and (3) string-gauge duality. Because these three subjects often discuss similar methods, for example conformal block expansion vs partial wave expansion, using different formalism and notation-we outline here our approach to conformal scattering processes that best illustrates these intersections.
A conformal approach to scattering processes was initially developed through the AdS/CFT correspondence [11][12][13], but it can be presented entirely in a CFT language [14][15][16][17][18]. Both approaches are equivalent and each offer separate intuitive frameworks. In this paper, we closely follow the CFT approach 4 , but describe physical insights it can tell us about interpreting the gravity dual. A typical example of conformal scattering is the offshell photon process γ * (1) + γ * (3) → γ * (2) + γ * (4), (I. 1) which does not involve asymptotic states. The amplitude is related to a time-ordered (T ) four-point current correlator, 0|T (J 1 J 2 J 3 J 4 )|0 5 . Our convention refers to scattering from (1 + 3) to (2 + 4) as the s-channel. The t-channel OPE, J 1 J 2 = α c 12,α O α , can be expressed in terms of MCB, G (M ) (∆, ) , as in Eq. (I. 4). This ultimately leads to an expansion for s-channel scattering amplitudes in terms t-channel exchanges 6 . 4 We summarize briefly the AdS/CFT perspective in Appendix B. 5 In more intuitive notation that conforms with other literature, we will sometimes write 0|T (J 1 J 2 J 3 J 4 )|0 as T (R(x 1 )R(x 2 )L(x 3 )L(x 4 )) or T (R(1)R(2)L(3)L(4)) . 6 The word "channel" can refer to a scattering process or an OPE. See Appendix A 1 for our conventions.
Recently, high energy scattering in CFT has become important for holographic models with black holes [29][30][31] and the related SYK model [32][33][34][35][36][37][38]: understanding the bounds of chaotic behavior and elucidating the flow of information via thermodynamics. It has been explained in [30] that understanding this behavior is equivalent to examining high energy behavior of near-forward scattering through the AdS/CFT correspondence following the formalism introduced in [11][12][13]. High energy scattering, depicted in Fig. I.1, involving a timeordered four-point correlator, can address stringy and thermal corrections to scrambling times by calculating "out-of-time-ordered" thermo-correlation functions, W (t)V (0)W (t)V (0) β , with β the inverse temperature.
Near-forward scattering for a process like Eq. (I.1) involves a small momentum transfer between 1 and 2, with a small deviation in their directions of travel 7 . The process can best be illustrated by with x ij = x i −x j and x 2 ij defined with Lorentzian signature.
As is well-known [1][2][3], one can express the invariant function α (u, v) are MCB, each associated with a conformal primary O α entering into the tchannel OPE. Eq. (I.4) defines the t-channel Minkowski conformal block expansion. Most of our results will apply generically to CFT's of arbitrary dimension, but many techniques are motivated by previous analyses ofN = 4 SYM. Specifically for N = 4 SYM, we focus on contributions from single-trace conformal primaries. In general, conformal primaries can be organized according to their twists τ 0 . The dimension ∆ and spin , are related by the relation ∆ α ( ) = + γ α ( ) + τ 0 , (I. 5) with γ α ( ) the anomalous dimension. In the absence of interactions, γ( ) = 0. In this representation, the dynamics lies in knowing all the participating conformal primaries, O α , and the associated "partial-wave coefficients", a (12;34) α . The partial-wave coefficient is real and factorizable, a (12;34) α ∼ c 12,α c 34,α . An important focus of this paper is to demonstrate how the formal sum, Eq. (I.4), can be interpreted, through the use of a Sommerfeld-Watson transform, as the principal series for an unitary representation of non-compact groups, Eq. (I.14), contrasting Minkowski and Euclidean behavior.
We treat CFTs where F (M ) (u, v) can diverge at u → 0 but is polynomially bounded. Since G (M ) is constructed to be real, it follows that the contribution to Eq. (I.4) from each conformal primary is also real. However, as a scattering amplitude F (M ) (u, v) is in general complex. A complex phase can emerge as a consequence of summing over higher spins 9 . Therefore, as a scattering problem, it is equally important in addressing 9 Complex phases can also be generated through summing over multiple trace primaries of low spins. This can lead to eikonalization. See [12,18,39] for a discussion about eikonalization in CFTs. the issue of re-summation for OPEs in a Minkowski setting 10 . In this vein, Conformal invariance has historically also been applied to simplify the analysis for ladder-type integral equations as is commonly done in resummations leading to high energy Regge behavior [41][42][43][44][45][46][47][48][49][50][51]. By working with Wightman functions, the absorptive part of forward scattering amplitudes 11 , our treatment leads to a simpler diagonalization procedure for dynamical equations via the appropriate harmonic analysis.
We show in this paper that a generic feature for CFT correlation functions F (M ) (u, v) is its singular growth (I. 6) in the limit (DLC) 12 u → 0, v → 1, (I.7) 10 In CFT bootstrap program (reviewed in [40]), the OPE sums are typically truncated in all channels, thus the issue of re-summation does not arise. However, this is a separate issue from defining the region of convergence for conformal blocks via a series expansion. As noted in [3], the region of convergence for standard euclidean conformal blocks is restricted to √ u + √ v) ≤ 1. Continuation to the Minkowski region necessarily requires going beyond the region of convergence for euclidean OPE. Our treatment here avoids this cumbersome step. 11 To be precise, the absorptive part is a discontinuity. In a coordinate space treatment this corresponds to a vacuum expectation value of a double commutator, for example 0| [R(2), R(1)][L(4), L(3)] |0 , appropriate for Eq. (I.2) and non-zero only in the physical region. An explicit example is that for deep-inelastic scattering (DIS) which involves an OPE of currents. See [52] for a review and Appendix D for connection to this work. 12 This limit can be thought of as a Regge limit, which is normally formulated in the momentum space. With the momentum space description comes a long history of phenomena associated with Regge behavior. As illustrated in Fig. I.1, this limit can also be treated as a double light-cone limit in position space. However, we caution that some authors use this label to refer to broader limits, for example [20]. Our statement here will be made more precise in Sec. III and also in Appendix A via a Rindler-like parametrization [8,9]. with 1/2 > δ > 0. In Sec. III, this limiting behavior is shown to allow MCB to be directly calculated by considering the corresponding boundary conditions for solutions to a conformal casimir. The limit will be defined more precisely in Sec. II where it is formulated as a double light-cone limit 13 . There are various ways to map conformal cross ratios on to the light-cone. Some involve single lightcone limits-like u → 0 in [39,56], or various double light-cone limits correspoding to different physical regimes [57][58][59][60][61]. For example, the conformal bootstrap program examines, among other limits, crossing relations after taking a double light-cone limit where u → 0, v → 0 asymmetrically as in [58].
By working directly in a Minkowski limit, we show δ = ( ef f − 1)/2 with ef f serving as an effective spin. For all CFTs with a gravity dual, this effective spin obeys an upper bound [11], ef f ≤ 2, (I. 8) In the case of N = 4 SYM it can be associated with string modes interpolating the graviton in AdS [11,27]. (See Fig. I.2.) This singularity is historically referred to as the Pomeron in both the context of QCD and for strongly coupled gauge theories. For SYK-like 1-d effective CFT [32][33][34][35][36][37], ( ef f −1) drives the Lyapunov behavior for thermocorrelators, W (t)V (0)W (t)V (0) β , with Eq. (I. 6) becoming e −2π( ef f −1)t/β at t large. These models are nearly maximally chaotic with the stringy and thermal deviations being driven by this Regge limit effective spin.

B. Outline
This paper, including the major results, is organized as follows: In Sec. II we discuss the kinematics of near forward scattering in a CFT where the relevant regime can be described as a DLC. Understanding 13 A similar limit to ours is explored in [53][54][55]. for twist-2, (τ0 = 2), at weak (λ 1) and strong coupling (λ 1). This figure is similar to that from [11] where it was first introduced.
this limit and the physical scattering regions are essential to the analysis in Secs. III-V. The OPE in an Euclidean setting exploits dilatation invariance leading to a single-scale scaling, which specifies the relevant boundary conditions for ECB. In contrast, in a Minkowski setting there can be two scaling limits. It is useful to adopt a new parameterization where this scaling is easily expressed as Dilatation : σ → ∞, (I.9) Boost : w → ∞. (I.10) The dilatation limit is characterized by a scaling parameter σ, Eq. (II. 15), which singles out the conformal primary of leading twist. The second scaling parameter w, defined by Eq. (II.14), relates to a Lorentz boost, specified by a conformal rapidity y, as in Eq. (II.7), where From the s-channel scattering perspective, tchannel spin, , is conjugate to the rapidity [13]. In the scaling limit of large rapidity, conformal symmetry manifests itself through an effective spin as in Eq. (I.6). Eq. (I.6) is a generic feature for the CFT correlation function F (M ) (w, σ), in the limit w → ∞ with σ fixed. In Secs. II B-II C we discuss crossing. To contrast Minkowski t-channel OPE with s-channel OPE we use the w, σ parameterization where the s-channel physical region corresponds to 1 < w < ∞ and u-channel in −∞ < w < −1. Of particular importance, we clarify why the contribution from the stress-energy tensor in a t-channel OPE, as well as its stringy correction in a dual description, serves as the dominant contribution in the DLC limit.
In Sec. III we discuss conformal blocks themselves, directly deriving MCB, and looking at relations to ECB and their asymptotic behavior. More technical details are left to Appendix C. For a tchannel OPE, MCB obey boundary conditions (I.12) The limit u → 0 is to be taken first before v → 1, In contrast, for the corresponding limit of u → 0 and v → 1 in an Euclidean OPE, conformal blocks obey boundary conditions This direct approach shows that the G (M ) (∆, ) (u, v) are related to, but not directly given by the analytic continuation of G While the conventional asymmetrical limits of taking u → 0 first before v → 1 is useful to differentiate MCB from ECB, we show in Sec. III D that a more symmetric treatment, in terms of variable (w, σ), allows us to generalize our approach in treating conformal blocks for general dimension, d. It also helps elucidate their crossing properties, and allows a more explicit demonstration the connection in the high energy limit to Euclidean AdS d−1 bulk propagators. These generalizations allow a smooth transition to the interesting case of d = 1.
In Sec. IV, we provide a more precise treatment on how t-channel OPE, for s-channel scattering, should be interpreted. This involves identi- 14 The literature often refers to the fact that the Lorentzian version is proportional to the analytic continuation, but here we spell out an exact relation.
fying the principal series representation for a noncompact group via a standard harmonic analysis. Through a Sommerfeld-Watson resummation, it is shown that F (M ) (u, v) takes on a Mellin-like representation, Eq. (IV.4). It follows that, in the physical region, its imaginary part is given simply by By pushing the integration contour in Eq. (I.14) to the left, contributions from singularities in the complex-plane become dominant in the high energy limit. The leading contributions come from the family of conformal primaries which interpolate the stress-energy tensor, ∆ P ( ), with a branch-point singularity at ef f 2. For holographic theories, the deviation from = 2 can be understood in terms of stringy corrections for integrable theories and receives additional temperature corrections in thermal theories. We emphasize that Im F (M ) (u, v) has non-vanishing support only in the s-channel and u-channel physical region, 1 < |w|. In the region −1 < w < 1, the contour can be closed to the right, leading to vanishing Im F (M ) (u, v).
As a special application of this new approach, we turn in Sec. V to CFT scattering in d = 1 and SYK-like models. We use the above Mellin representation to formulate the relevant 4-pt correlator in Sec. V A and discuss the role of effective spin for SYK-like models in Sec. V B. By taking advantage of reparametrization invariance, an integral equation for Im Γ(w) 15 is constructed in Eq. (V.24). We stress that we formulate the model directly as a Minkowski scattering problem, leading to an equation involving Im Γ(w) in the physical region only. This integral equation can be diagonalized readily by exploiting the conformal boost invariance. Using this approach, the integral equations involved can be used to derive simpler algebraic relations as described in Sec. V B 1. For the case of SYK models, the partial-wave amplitude is given as an integral over a Legendre function of the second kind, Eq. (V.9). The effective spin shows up as the right-most singularity of the partial-wave amplitude, A( ), Eq. (V.9), a pole at = 2 and is analytic to the right.
We end in Sec. VI with a short summary and adding further discussion on the role of spectral curves. (See further discussion in Sec. B 2.) For the canonical AdS/CFT correspondence, conformal invariance leads to spectral curves, ∆( ), that are symmetric under Its importance for high energy scattering is discussed in Sec. IV C for d = 4 CFTs. This property can be seen in Fig. I.2, and plays an important role in determining the effective spin. The effective spin can be obtained by solving an equation involving anomalous dimension, [11,13] γ( ef f ) + ef f = 0 (I. 16) where γ( ) is the analytically continued anomalous dimension. More generally, non-thermal deviation from = 2 can be attributed to stringy corrections. Thermal theories receive additional temperature dependent corrections. For the graviton, Eq. (I. 15) can be thought of as coming from the AdS mass condition 16 , 16 It has been shown that for N = 4 SYM, ∆(d − ∆) is Borel summable while while ∆ is not [62,63]. In this case, where the theory is thought to be integrable, integrability techniques can be used to determine this sum to high order. For a review of the procedure and it's application to the Pomeron, see [19]. One should be careful when computing corrections to the spectral curve that the expansion is well defined.
with non-thermal stringy corrections respect this symmetry. The leading correction can be interpreted as introducing a spin dependent mass 2). For N = 4 SYM, it can be shown that m 2 ef f ( ) admits a systematic expansion about = 2, Each coefficient admits a strong coupling expan- [19,62,64,65]. (See Eqs. (3.14-3.15) of Ref. [19].) Including thermal corrections can introduce a new effective mass that breaks the symmetry in Eq. (I. 15). Nonetheless we believe one can still apply the analysis starting from Eq. (I. 16).
We have also included several appendices. These provide more details than is normally done since, in spite of the initial work of [11] more than a decade ago, scattering for CFT remain unfamiliar to most CFT practitioners. Additionally those interested in CFT scattering come from a variety of backgrounds so we have aimed to be as self contained as possible. In Appendix A we set channel conventions and provide kinematic relations between invariant cross ratios and position coordinates via a Rindler-like parametrization [8,9] appropriate for the DLC limit. Appendix B connects a conformal invariant 4-point function, F (M ) (u, v), to an ordinary momentum-space amplitude from the perspective of the AdS/CFT conjecture. In Appendix C we clarify in greater details the relation of MCB to the conventionally defined ECB. Appendix D focuses on the application of CFT scattering to DIS, focusing on exploiting the SO(1, 1) × SO(1, 1) symmetry and connecting DIS to conformal methods. Finally, in Appendix E a conventional Hilbert space treatment for d = 1 CFTs is carried out and extended to the case of non-square-integrable, but power-behaved like Eq. (I.6), functions. As explained in Secs. IV and V, this illustrates that, through the Sommefeld-Watson transform via complex angular momentum, the re-summed Minkowski OPE can be interpreted as a (deformed) harmonic analysis over non-compact group. We also outline the basics of the SYK theory needed for interpreting Sec. V. a. NOTE: Upon completing this work we were made aware of [66], extending the work of Caron-Huot [67], which has some overlap and similar conclusions as ours. Other related works include [38,[68][69][70][71]. It is useful to briefly comment on the relation of this study to that of [67] and [66]. The starting point of both [67] and [66] is CFT in an Euclidean setting. One impetus for the study of CFT in the Lorentzian limit is related to the question of chaos bound, e.g., [30] and SYK model. In [67], CFT scattering amplitude, M, is introduced by identifying it as a discontinuity of analytically continued Euclidean correlation function, (e.g., Eq. (2.13) in [67]), which in turn leads to a representation for its imaginary part, Im M, as a "double-discontinuity", (Eq. (2.14) of [67]), or, equivalently, a double-commutator. Recognizing the importance of the constraint imposed by the Regge asymptotics, the focus has been to find a relation between the partial-wave amplitude a( , ∆), analytically continued in complex , and Im M, leading to its key result, (Eq. (3.20) of [67]). One important feature is the asymptotic boundedness in the limit Re → ∞. The procedure adopted followed a traditional Regge analysis in introducing Froissart-Gribov representation.
Our direct study for Lorentzian CFT is motivated by that of [11] where conformal Regge behavior can be derived, and we discuss how a doublecommutator, as the discontinuity of a CFT scattering amplitude, can be related to a t-channel OPE through a principal series representation, e.g., Eq. (I.14). In a broader context, Eq. (I.14) itself can be derived from the unitary irreducible representation of the full non-compact SO(4, 2), Eq. (IV.8). In an Euclidean setting, this leads to the principal series representation for SO(5, 1), Eq. (IV.6). The importance of this relation has also been emphasized in [66], and also earlier in [72]. In a Regge context, it can be traced back to earlier work of M. Toller [73]. The key dynamical assumption in our approach is meromorphy in the complex ∆ − plane for the "partial-wave amplitude", a( , ∆), e.g., Eq.
(IV.7), which leads formally to a t-channel OPE via spectral curves.
Our study here complements that of [67]. Eq. (I.14) involves MCB, which can be introduced directly in a Minkowski setting, as discussed in Sec. III, thus avoiding the step of intricate analytic continuation. The ability to close various complex contours require specifying boundedness of a( , ∆ α ( )) for Re large. The necessary assumption involved is, in the end, equivalent to the assumption of polynomial boundedness in the DLC limit, Eq. (I.6), which is the main focus of our study. The close relation between these two approaches can be brought out more explicitly in the case of d = 1, e.g. Eqs. (V.6), (V.9) and (V.12). The issue of asymptotic boundedness can be analyzed explicitly in terms of an elementary Hilbert space treatment, which is carried out in Appendix E.

II. THE DLC LIMIT
In this section, we spell out more precisely how dilatations and Lorentz boosts can be related to the dependence of conformal correlators on invariant cross ratios in the limit u → 0 and v → 1, as in Eq. (I.7). The causal relationship described in the introduction is depicted in Fig. A.1 and defines the s-channel scattering region. This limit can of course be taken both in Euclidean and Minkowski signatures. For an Euclidean OPE, this limit involves only a single scale corresponding to a dilatation under SO(5, 1). The corresponding asymptotic boundary conditions for ECB are given by (I.13). In a Minkowski setting, however, because of the Lorentzian structure, the same limit can involve two scales, one for Lorentz boost, and the other for dilatation. As indicated in the introduction this limit involves particles being scaled along forward and backward light-cones and we we refer to this specific double light-cone limit as the DLC. The relevant scaling limit exploits the invariance under SO(1, 1) × SO(1, 1), a subgroup of the full conformal symmetry SO(4, 2).
In Sec. II A we discuss the DLC limit more explicitly. It is useful to adopt a Rindler-like parametrization, Eqs. (II.3-II.4), fixing the phase space of s-channel physical region. In Sec. II B, we discuss the related DLC limit under crossing. In place of (u, v), we introduce new sets of independent invariants, Eqs. (II.14) and (II.15), which are not only more useful for the DLC limit, but also simplify the description of s-u crossing. In Sec. II C, more explicit connection to the nearforward scattering is discussed. We consider the contribution from the stress-energy tensor, T µν , in a Minkowski setting, which in turn helps to motivate boundary conditions for MCB, (I.12).
Additional details and definitions can be found in Appendix A 2.
In a Minkowski setting, because of the Lorentzian structure, the same limit can involve two scales, one for Lorentz boost, and the other for dilatation. Consider light-cone coordinates, Focus first on the case d = 2. For each coordinate, let us define r = √ −x + x − > 0, which can parametrized by a parameter η i 17 , The allowed range 0 < r i < ∞ corresponds to −∞ < η i < ∞. We shall refer to r as "conformal virtuality" 18 and dilatation in light-cone components, x ± → λ x ± , corresponds to scaling in conformal virtuality. In terms of parametrization Eq.
To identify a Lorentz boost, we next introduce rapidity variable, 0 ≤ y < ∞, for each coordinate. Consider first the right-movers, (x 1 , x 2 ). For time ordering, we keep x + 1 < 0 and x + 2 > 0, and parametrize their light-cone components as with ε 1 = −1, ε 2 = +. Similarly, for the pair (x 3 , x 4 ), x ± j = ∓ε j r j e ∓yj , with ε 3 = −1 and ε 4 = +, so that x − 3 < 0 and x − 4 > 0. Sending all rapidities y i → ∞, with r i fixed, leads to Eq. (II.2). Therefore, one way to achieve this limit is to perform a global Lorentz boost. For our purpose, it is sufficient to fix a single global rapidity y for all four legs, with y = y 1 = y 2 = y 3 = y 4 .
In order to connect with Eqs. (II.1) and (II.2) for d = 2, it is now necessary to discuss the effect of transverse coordinates. To simplify the discussion, we will adopt a frame where x i,⊥ = x 2,⊥ and x 3,⊥ = x 4,⊥ and with b ⊥ = x 1,⊥ − x 3⊥ as the 17 The length scale µ 0 is introduced for 'visual" purpose, which can be set to unity. 18 We use this name as the quantity will play a similar role to conventional virtuality which is an energy like quantity defined as the off-shell energy of a particle. relative separation between (1,2) and (3,4) in the transverse impact space 19 . In terms of the global rapidity y and conformal virtuality r i for each coordinate, cross ratios u and v take on relatively simple forms, Eqs. (A.7-A.9). In the case of two pairs of equal conformal virtuality, r 1 = r 2 and r 3 = r 4 , where the transverse separation enters through with i = 1 and j = 3. The limit u → 0 can therefore be achieved either by y → ∞ or b 2 ⊥ → ∞ first. For near-forward scattering, or the DLC limit, we consider the first scenario of y → ∞ with b 2 ⊥ fixed. The limit u → 0 therefore exploits the scaling limit of Lorentz boost. In this limit, with conformal virtualities fixed, one also has v → 1. Together, they correspond to the DLC limit, Eq. (I.7), as promised.
For general unequal conformal virtualities, one finds, in the limit of large rapidity, where we have introduced a tentative set of variables, w 0 and σ 0 , and have also introduced joint conformal virtualities 20 , We adopt w 0 → ∞ and σ 0 → ∞ as two independent scaling limits, global boost and dilatation, which characterizes the DLC limit for CFT. For d = 2, the limit σ 0 → ∞ generically corresponds to b 2 ⊥ >> z 12 z 34 . For both d = 2 and d = 2 with b ⊥ fixed, large σ 0 corresponds to the limit of small conformal virtuality, z12 z34 → 0 or z34 z12 → 0. This is analogous to the "near massless" limit in a conventional scattering limit.

B. Physical Regions
To understand how CFT correlators can be used for scattering, we need to comment on the kinematics of physical regions and constraints due to s-u crossing for T (R(1)R(2)L(3)L(4)) . To simplify the discussion, we consider correlators for four identical scalar conformal primaries.
From Eqs. (II.5) and (II.15), more generally, from Eqs. (A.7-A.9), the s-channel physical region corresponds to 0 < u < 1 and 0 < v < 1. This is the causal structure of Fig. I.1. In terms of the previous section this is w 0 and σ 0 , 1 < w 0 < ∞ and 1 < σ 0 < ∞. It is in this regime we can examine a t-channel OPE.
A similar t-channel OPE applies to the uchannel, Fig. A.3, which can be found by interchanging either 1 ↔ 2 or 4 ↔ 3, leading to It is possible to adopt (u , v ) as an alternative choice for cross ratios. This alternative choice leaves the limit of interest (I.7) unchanged. Under such an exchange, we see that the u-channel physical region, 0 < u < 1 and 0 < v < 1, corresponds to 0 < u < ∞ and 1 < v < ∞. Therefore, variables (u, v) transform asymmetrically under su crossing, As shown in [3], it is useful to change variables from u and v to a pair of independent variables, (x,x), where In a Euclidean setting, x andx are complex and x * =x. In a Minkowski setting, the s-channel physical region corresponds to both x andx real and positive with 0 < x < 1 and 0 <x < 1. This From the u-channel perspective, the range for x andx are unbounded, (−∞, 0): s-u crossing remains asymmetrical.
In order to take a symmetric approach, it is useful to introduce a new set of variables (q,q), with the s-channel physical region corresponds to 1 < q,q < ∞, and u-channel physical region corresponds to −∞ < q,q < −1. Under s-u crossing, one simply has q → −q andq → −q. In terms of these, cross ratios are given by It will be useful to change variables one more time to (w, σ), which we will ultimately adopt in Sec. III D. These parameters can be defined in terms of (q,q) or more directly from (u, v), by The s-channel physical region now corresponds to 1 < w < ∞ and 1 < σ < ∞. Under s-u crossing, Therefore, the u-channel physical region corresponds to −∞ < w < −1 and 1 < σ < ∞. These variables, w and σ, serve as the most symmetric variables for describing scattering for CFT. In Eqs. (II.14-II. 15), approximate equalities hold in the DLC limit, i.e., w w 0 and σ σ 0 .
In what follows, we will use (u, v), (q,q) and (w, σ) as three equivalent sets of independent invariants for specifying Minkowski conformal blocks, and they can be used interchangeably.

C. Eikonal Scattering
It has been shown [11][12][13][14][15][16][17][18] that the connected part of the invariant function in a t-channel OPE, F (M ) conn (u, v), can be related to the scattering amplitude for high energy near-forward scattering at fixed virtualities 21 , T (s, t; z 12 , z 34 ). This correspondence can best be made through an eikonal phase, χ(s, b, z 12 , z 34 ), conn (u, v) expanded as a sum over t-channel single-trace conformal primaries 22 . A convenient way of introducing of this eikonal phase is through a "shock-wave" treatment for near-forward scattering [12,14,15,18]. (For early developments on this front, see [74][75][76][77][78][79][80].) At a large impact separation, χ(s, b, z 12 , z 34 ) is small and can be treated perturbatively, Here the eikonal is given in a mixed coordinate and momentum representation. In performing Fourier transforms for the light-cone components, large s receives contribution from small u region only, or, equivalently, large w region, and the relevant Fourier integrals can be dropped. This representation can also be interpreted holographically as scattering in the AdS bulk [11][12][13][14][15][16][17][18]. In (II.17), one has identified the small u limit with s large by the reciprocal relation where the scale µ 2 0 introduced in (II.3) can now be used as a global scale for scattering 23 . It is important to emphasize that the standard normalization we are adopting here corresponds to Therefore, the large w behavior for F (M ) conn differs from that for T (s, t; z 12 , z 34 ) by one power of w or, equivalently, one power of s. This extra factor of w corresponds to the relativistic state normalization, leading to the conventional Optical Theorem, σ total (1/s)Im T (s, 0; z 12 , z 34 ) 24 . As mentioned earlier, and motivated by the AdS/CFT convention, in what follows we will switch from (z 12 , z 34 ), to (z,z) for right-and left-movers for notational simplification.
A useful illustration is to consider the contribution from the stress-energy tensor, T µν , having ∆ = d and = 2.
Since it couples through conserved currents, the amplitude picks up a spin factor, s 2 , from the product of two large coupling terms, involving ∂ x − i ∂ x + j , i = 1, 2 and j = 3, 4. This factor reflects the effect of longitudinal Lorentz boost due to spin-2 exchange. This is consistent with the expectation 23 This identification can be done more formally, but it is sufficient for our present purpose to have qualitative understanding as follows. In terms of momenta, since ) as e −2y , corresponding precisely the limit of large s. To fix the scale, we identify s −1 with the average as µ 2 0 (r 1 + r 2 )(r 3 + r 4 )e −2y . 24 For a careful review on the derivation of this representation, Eq. (II. 19), see Secs. 3-4 of [30] where the formalism introduced in [11] is also discussed. This is also summarized briefly in Appendix B.

χ
T /s ∼ s −1 . In addition to this spin factor, at large separation, the amplitude is controlled by a scalar propagator, ϕ(x)ϕ(0) = 1/(x 2 ) ∆ . Since both particles are moving near their respective light-cones, the total amplitude, after integrating over x ± components [12], leads to where we have scaled the last expression on dimensional grounds, as in Eq. (II.15). Explicit conformal invariance can be achieved by expressing this as with = 2 and ∆ = d. This also agrees with what follows holographically for one-graviton exchange in the bulk based on AdS/CFT using Witten diagrams, as explained in Appendix B 25 . This stresstensor example also serves as the prototypical behavior expected in a Minkowski OPE expansion, as w → ∞ first and then σ → ∞. With Eqs.
(II.14) and (II.15), this corresponds to the boundary conditions for Minkowski conformal blocks, Eq. (I.12). Although this work is primarily concerned with the t-channel OPE, a similar eikonal form can be found from examining the s-channel OPE directly. Performing an OPE in the s-channel one finds (II.24) 25 See also [12]. It should be pointed out that the resulting eikonal from the stress-energy tensor is purely real. When stringy effect is taking into account, the eikonal turns complex, with Im χ > 0 by unitarity and with Im χ/Re χ constant in the limit w → ∞ first. This also necessitates in a broader discussion on the order in taking w and σ large. This issue has been discussed extensively in [13] and should be addressed also in application to SYK-like models. In this paper, we focus only on tree-graph contribution to the eikonal sum, and the limit is always taken with w → ∞ first.
with ∆ = h +h and J = h −h. In the DLC limit, after removing the identity conformal primary I, following [17,81] χ(s, b ⊥ , z,z) + O(χ 2 ). Although as shown above, this is most naturally interpreted as being dominated by t-channel stressenergy tensor exchange.

III. MINKOWSKI CONFORMAL BLOCKS
The conformal partial-wave expansion can be traced back to work of [82][83][84][85][86][87][88] and has been carried out more recently in a series of papers by Dolan and Osborn [1][2][3]. Application of partialwave expansions for non-compact group has had a long history [45,73,[89][90][91][92][93]. Recent works on these expansions have been carried out exclusively in an Euclidean framework, and Minkowski results are obtained via careful analytic continuation. In this section we detail explicitly how the MCB can be obtained directly by understanding the boundary conditions of the conformal block differential Casimir.

A. Definitions
Conformal Blocks are eigen-functions of a quadratic Casimir 26 , D, associated with an arbitration conforaml transformation acting on a scalar four-point function. The commutator algebra of generators involved can be realized as differential operators in terms of conformal cross ratios, There is a quartic Casimir that can be used to construct recurrence relations between Conformal blocks in different dimensions [3]. We use this below to relate results in even (respectively odd) dimensions.
where a = −∆ 12 /2, b = ∆ 34 /2, and ∆ ij = ∆ i −∆ j . We consider the case where ∆ 1 = ∆ 2 and ∆ 3 = ∆ 4 so that a = b = 0. Eigenvalues for the quadratic Casimir are then While most prior treatments have considered ECB, invariant under SO(d+1, 1), MCB, invariant under SO(d, 2), have been addressed recently by analytically continuing the Euclidean case. Here, we carry out a direct treatment of MCB by imposing DLC boundary conditions, Eq. (I.12). We first carry out a more traditional analysis in Sec. III C-III C, and, in particular, point out that MCB as defined are not given by a direct analytic continuation of ECB. In Sec. III D, an alternative, more symmetric treatment is carried out in terms of variables w and σ, (II.14) and (II.15).
In the Lorentzian setting, (∆, ) serve as representation labels for SO(d, 2). Since this is noncompact, ∆ and can be continuous. However, to make contact with the OPE, we shall consider initially the situation where ∆ is positive and real, and is a non-negative integer. We will return to the question of restrictions on ∆ and from the perspective of SO(d, 2) representation in Sec. IV.

B. Indicial Analysis
As a second order partial differential equation, Eq. (III.1) has removable singularities at u = 0, 1, ∞-and similarly for v. Independent solutions can be specified by their behavior at these singular points. Representations (III.4) and (III.5) also suggest possible different variable separation procedure in solving this partial differential equation.
(See Appendix C for more details.) As emphasized in the Introduction, and elaborated further in Sec. II, we are interested in the the solution space is at least 4-fold degenerate. The usual OPE restricts to be a non-negative integer, ∆ real, and ∆ > , which is equivalent to λ + ≥ λ − ≥ 0. A similar analysis has also be carried out in a related context in [67].
Anticipating Eqs. (I.12) and (I.13), we consider the following solutions: Note that (c) and (d) are reflections of (a) and (b), under ∆ ↔ d − ∆, leading to mirror, or shadow, conformal blocks. We will thus focus on case (a) and case (b) only. As boundary conditions, these solutions can be re-written as We see that case (a) is that appropriate for Euclidean conformal blocks, Eq. (I.13), and case (b) appropriate for the Minkowski setting, Eq. (I.12).
For both sets of boundary conditions, the limit √ u → 0 is taken first before v → 1. Equivalently, both (a) and (b) in Eq. (III.7) correspond to taking the limit √ u → 0 with 1−v √ u initially fixed, and then taking the limit 1−v √ u → ∞. Therefore, MCB and ECB are solutions to the same partial differential equation, defined with different boundary conditions, thus, in general, they cannot be expressed simply as analytic continuation of each other.

C. Explicit Construction of MCB
We next focus on treating case (b) directly. The standard procedure [1][2][3] is to change variables from (u, v) to (x,x), Eq. (II.11). In order to maintain explicit crossing symmetry, we prefer to work first directly with new variables (q,q), Eq. (II.12), where the differential operator for the Casimir can again be written as a sum of terms, (as the case for leading to second order ODE for Legendre functions. More importantly, D ε 1 = 0 at d = 2. (Relation to the more traditional use of (x,x) pair [3] is also provided in Appendix C.) Boundary conditions in the DLC limit are specified by taking u → 0 first before v → 1. In terms of q andq, two possible approaches can be adopted: (1) an asymmetrical limit taking either q → ∞ beforeq → ∞ or the opposite, and (2) a symmetrical treatment of w = √ qq → ∞ before taking σ = (q +q)/2 √ qq → ∞. We will follow the asymmetrical approach here and postpone the symmetrical treatment to Sec. III D.
In an asymmetrical limit, the boundary condition for MCB, Eq. (III.7) (b), becomes where q < is the smaller of the pair (q,q) and q > the other, as both q,q → ∞. To proceed, we will first treat the case of d = 2 and d = 4 before commenting on general d.
• d = 2: Here ε = 0 and the differential equations for q andq decouple. From Eq. (III.5), we choose with λ = λ ± , and similarly for D 0 (q). Each can be reduced to standard hypergeometric differential equations: the solution can be expressed as, for 1 < q < ∞, In the second line, Q ν (q) is simply the standard Legendre functions of the second kind.
It remains to properly implement the desired Minkowski boundary conditions, Eq. (III.9). If 1 < q <q → ∞, choose a = 0 for g(q < ) and b = 0 for g(q > ). If 1 <q < q → ∞ make the reverse choice. This leads to in agreement with Eq. (III.9).
• d = 4: It is possible to re-cast this case into one where q andq equations again decouple as done in [3] working with (x,x). The cor-responding solution is One can verify that Eq. (III.9) is again satisfied.
• General d: The d = 1 case is of special interest for SYK-like models, and will be discussed separately in Sec. III D 3. For general d, it is not possible to reduce the solution to a combination of simple products of hypergeometric functions. However, an iterative procedure will be discussed in Sec. III D by treating (q,q) symmetrically. One finds that the general structure for the leading behavior in u → 0 is shared for all d.
We stress here that the physical region of schannel scattering corresponds to 1 < q,q < ∞, where k 2λ (q), Eq. (III.12), is real. It follows that G is also real. However, since k 2λ (q) is defined as a real analytic function, it can be continued into the complex plane and acquire a phase for general . We will turn to this question in Sec. IV, as well as in Appendix C.
The boundary conditions for the Euclidean conformal blocks, Eq. (III.7) (a), can similarly be expressed as G The same approach as above then leads to 27 The connection to the canonically defined functions in terms of {x,x} is shown in Appendix C for the d = 2 case, and to 16) for the d = 4 case.
In an Euclidean treatment, q andq are complex conjugates, q * =q, thus not independent. It has been suggested that Minkowski conformal blocks are simply the analytic continuation of Euclidean conformal blocks, which changes the boundary conditions from Eq. (III.7) (a) to (b). Appendix C gives the detailed relation between G in terms of their analytic structure. We will demonstrate explicitly that Eqs. (III.13) and (III.14) are not given by a direct analytic continuation of the corresponding ECB, Eqs. (III.15) and (III. 16).
To state it more succinctly, in performing analytic continuation from Euclidean to Minkowski limit, the Euclidean boundary condition does not transform precisely into that for Minkowski limit.
and following normal path of continuation, one arrives in the Euclidean region where Conversely, an additional prescription is required in relating ECB to the desired MCB. This will be touched upon further in Sec. IV B. Let us end by examining the constraints on F (M ) (u.v) due to crossing symmetry. As mentioned earlier, for a t-channel OPE, we are interesting in s-u crossing, which corresponds to interchanging either 1 ↔ 2 or 4 ↔ 3. This leads to, as already discussed in Sec. II B, u → u/v and v → 1/v, or, Consider the OPE expansion (I.4), with integer. From Eq. (III.12), under s-u crossing, 19) As we show more explicitly in the next section, this pattern holds for all d.

D. Symmetric Treatment
We next turn to a symmetric construction of MCB for general d. This was first advocated for ECB in [3], starting with (x,x), by shifting to new variables u = xx and σ 0 = ( x/x + x/x)/2. This approach was pursued further in [94]. As explained in [3], this approach has the advantage of being able to extend the boundary condition, for ECB, Eq. (III.7) (a), to the region where x = O(x), and the desired boundary condition translates into that in the limit σ 0 → ∞.
We will begin with (q,q) and focus on the conformal blocks in the Minkowski limit. Recall that, for the DLC limit, we are interested in first tak- the resulting boundary condition specified at σ = ( q/q + q/q)/2 σ 0 → ∞. In what follows, we shall adopt (w, σ) as two independent variables with the physical region specified by 1 < w < ∞ and 1 < σ < ∞. 28 We will mainly work this asymptotic limiy, but show a higher order expansion can be obtained formally on an equal-footing for all d.
The differential operator D, Eq. (III.2), when expressed in terms of (w, σ), becomes a sum of three terms, 28 As we shall demonstrate below, for holographic CFTs, this allows a simpler representation connecting in the leading order to the Euclidean AdS d−1 bulk-to-bulk propagator. 29 See [94] for a related treatment in "radial quantization". and It thus suggests the following expansion for conformal blocks, Here, each g n (σ) also depends on , ∆ and d, which will be exhibited explicitly when necessary for clarity. They can be found recursively, with the leading order term satisfying a relatively simple D.E., This differential operator, L 0,σ , is of the same form as D 0 , (III.8), and it will appear repeatedly under several contexts, with solutions expressible in terms of hypergeometric functions. (However, each case may impose a different boundary condition. See Sec. III D 2 and Appendix C 1 for further discussion.) We will treat solutions to (III.24) with appropriate boundary conditions in Sec. III D 1. Before proceeding to solving for g 0 (σ), we mention first the ease of exhibiting crossing under this symmetric approach. As we shall demonstrate below, for Minkowski conformal blocks, this corresponds to the choice s = − 1. Under s-u crossing, w → −w and σ → σ, (III.23) leads to As expected, it is odd for even and even for odd.
A direct indicial analysis at w → ∞ leads to several degenerate possibilities, with s taking on (a) The corresponding solutions to Eq. (III.24), labelled by g 0a , g 0b , g 0c and g 0d respectively, are We will therefore concentrate on case (a) and (b). Of these two solutions, by switching back to u and v and comparing to Eq. (III.7), we find that case (a) is appropriate for Euclidean conformal blocks, and case (b) is appropriate for the Minkowski limit.
Observe that these differential equations are even under σ ↔ −σ. General solutions to these equations can be expressed in terms of hypergeometric functions and we need to impose the respective boundary conditions, Eq. (III.7) 30 . Let us focus here on Eq. (III.26) (b). With s b = − 1, the solution is with ∆. For case (a), sa = −∆, the solution is that appropriate for Euclidean conformal blocks, with coefficient b = 0. This leads to polynomial solutions for integral , e.g., for d = 4, Gegenbauer polynomials. As σ → ∞, G 28) in the limit w → ∞ and σ → ∞, as promised. Solutions for d = 4 and d = 2 can be expressed simply as Similarly, for d = 3 and d = 1, the solutions also simplify to Here Q ν (σ) is the Legendre function of the second kind. The case of d = 1 is discussed in more detail in Sec. III D 3, as it relates to SYK-like models.
We also point out that Eq. (III.27) corresponds precisely to an Euclidean bulk-to-bulk scalar propagator in AdS d−1 , or more precisely H d−1 , with conformal dimension ∆ − 1. Here Eq. (III.27) is derived purely from a CFT perspective, with σ − 1 plays the role of a chordal distance or, in an alternative mathematical usage, ξ = cosh −1 σ is the geodesic in H d−1 . (See also Appendix B.) This connection is meaningful physically only for d ≥ 2 but can be extended formally to all d. As such, g 0 (σ; ∆, d) is singular at σ = 1, consistent with with our procedure of fixing a vanishing boundary condition at σ = ∞.
Lastly, with σ = cosh ξ, Eq. (III.26) (b) can also be expressed as Introducing a reduced function f (ξ), with (III.34) This simplifies for d = 2 and d = 4, leading to This agrees with the desired result given above, and also serves as the leading term in an higher order expansion, (C.33) and (C.34) respectively. For general d, the solution can be expressed in terms of associated Legendre functions.

Higher Order Expansion:
As mentioned in the introductory remarks for this section, higher order expansion in (III.23) can be found iteratively. Once the leading index, s b = − 1, is identified and the corresponding solution g 0b (σ) is found, it is possible to solve each expansion function g n (σ) for 1 ≤ n. With (III.20) and (III.22), one finds I.e., given in terms of g n−1 (σ). Therefore, one can proceed iteratively. Focusing on 1 < σ < ∞, Eq. (III.36) can be solved formally via a standard Green's function procedure with For d = 2 and d = 4, since explicit solutions are already known. We will instead demonstrate that they can be re-expressed in the symmetric form, Eq. (III.23), in Appendix C. For general d, care must be exercised in defining appropriate boundary conditions. We will leave the case of general d to a future study, and will focus next on the case of d = 1.
3. The Case of d = 1: Due to the existence of a kinematic constraint, the leading order solution for d = 1 requires additional examination. From the perspective of Euclidean SO(d + 1, 1) invariance, the physical region is bounded by ( with the equality holding at the kinematical boundary [3]. This boundary also defines the d = 1 limit, which can be expressed more usefully as In terms of q andq, both cases lead to q =q and σ = cosh ξ = 1. It follows that there is only one independent variable, instead of two. Therefore for d = 1, conformal blocks, as functions of w and σ, must satisfy a constraint: ∂ σ G(w, σ) = 0. This means that care must be taken when re-interpreting the above results for the case of d = 1.
There are two possibilities.
• (a) The Euclidean option corresponds to keeping only = 0, 1, and going to higher order expansion in w, (III.39) It is easily shown that, with d = 1, g a/c,0 (σ) being a constant is a consistent solution to (III.26) for = 0, 1.
• (b) The Minkowski option corresponds to having ∆ = 0, 1, and (III.40) It is also easily shown that g b/d,0 (σ) being a constant is a consistent solution to (III.26), with ∆ = 0, 1. In Sec. V we show that this limit is more appropriate in treating scattering for SYK-like 1-d models.
For both cases it is necessary to go beyond the leading order to obtain the proper sums, Eqs. (III.39) and (III.40). In terms of cross ratio, w, the residual symmetry for both cases is O(1, 1), with dilation for case (a) and Lorentz boost for case (b).
Let us treat case (b) first. We denote G ) by acting on functions of w only. A bit of algebra then leads to which is of identical form as Eq. (III.21), with d = 3, and also with Eq. (III.8), i.e., it is again that for Legendre functions. It follows from Eq. (III.11), for w → ∞, there are in general two independent solutions, leading to where k 2 (w) is again given by Legendre function of second kind, Q (w), Eq. (III.12). In Secs. V A, we consider Minkowski scattering, with the physical region arranged to lie in the region 1 ≤ w ≤ ∞. Therefore, our choice corresponds to a = 1 and b = 0. The normalization corresponds to having conformal blocks We shall also restrict Re > 0, (thus Re > 1/2). From the identity one has P (z) = P − −1 (z), and Q − has poles at non-negative integers, = 0, 1, · · · . However, since where d n = π 1/2 2 1−n Γ(n) Γ(n−1/2) . Let us briefly return to case (a), appropriate for an Euclidean treatment. As pointed out in [3], the solution to the = 0 problem has a direct solution. Following the same approach as above, , leading to the same result as given in [3], with w replaced by x = 2/(w + 1). The solution can also be obtained from that for the case (b), with ∆ replacing .

IV. MINKOWSKI OPE AND SCATTERING
Let us now return to discuss how OPE in a Minkowski setting, Eqs. (IV.4) and (I.14), can be applied to high energy scattering. In this Section, we focus on certain formal steps necessary before Minkowski OPE can be applied. The emphasis will be on first developing a Mellin-like representation for the OPE sum so that it applies to the physical scattering region 31 . This formulation stresses the importance of a spectral curve, ∆( ), and its relation to effective spin, * . An equally important and related issue discussed is the relation between the t-channel OPE in a Minkowski setting to the 31 We emphasize that the discussion here focuses on Mellin amplitudes that are distinct from the Mellin representation discussed in [72,[95][96][97][98].
principal series for an unitary irreducible representation 32 of non-compact O(4, 2), leading to Eq. (IV.8). Applications of dimensional reductions are discussed in Appendix D for d = 2 and the case of DIS, and in Sec. V for d = 1 scattering for SYK-like models.

A. Kinematics
Before applying (I.4) to high energy nearforward scattering, it is important to address the issue of the phase of F (M ) (u, v). As pointed out in Sec. I, MCB, G (M ) are real valued functions over the physical region for s-channel scattering where 1 < w < ∞ and 1 < σ < ∞. It follows that the contribution to (I.4) from each conformal primary is also real. As a scattering amplitude, however, emerges as a consequence of re-summation. This can be carried via complex-utilizing the technique of Sommerfeld-Watson transform, which we turn to next. Through this procedure, one also allows an natural continuation in w from the s-channel where 1 < w < ∞ to the u-channel phyiscal region where −∞ < w < −1. it is therefore useful to have a closer examination of the symmetry involved under s-u crossing.
In the DLC limit, Feynman amplitudes and CFT correlators have opposite parity under crossing. A crossing even Feynman amplitude, T , corresponds to a CFT invariant function F (M ) which odd under s-u exchange. This is due to the presence of the extra factor of w in T, for example, Eq. (II.19). A general amplitude can have both components. A 4-point function for identical conformal primaries is s-u crossing symmetric. This corresponds to an even T , and thus F (M ) (u, v) will be crossing odd: 32 The case of d = 1 provides an explicit illustration. This will be carried out in Appendix E. A more detailed discussion for d ≥ 2 will be reported separately.
In this study, we are mostly dealing with scattering amplitudes T which are crossing even.
If we are more explicit in identifying s-channel and u-channel amplitudes separately by F In what follows, we will always work with s-channel amplitudes, F (M ) s , while dropping the subscript. When expressed in terms of variables (w, σ) and analytically continued, one has (IV.1)

B. Sommerfeld-Watson Transform
We begin by first re-grouping the OPE in Eq.
We have re-expressed the partial-wave amplitude as a (12), (34) ,α , switched the dependence on cross ratios from (u, v) to (w, σ) and have also re-grouped the sum in a form which allows a re-summation, leading to a representation for F (w, σ) valid for w → ∞.
Consider the case where scattering amplitude is even under crossing 33 , thus F (w, σ) odd in w. From (III.25), the sum is over even only. Sommerfeld-Watson transform corresponds to turning this discrete sum over into an integral over complex-plane and then opening up the contour into an integral along a vertical line. This vertical line is chosen initially for convergence, leading 33 In general, the sum can be separated into a sum of even spins or another over odd spins. The case of crossing odd has been treated in [19].
to a Mellin-like integral, i.e., We assume that F (w, σ) is polynomially bounded at w = ∞, i.e., |F (w, σ)| < O(w N ), thus N + 1 < L 0 . This allows one to represent F (M ) (w, σ) by The assumption of polynomial boundedness can also translate into having a In Eq. (IV.2), we have also separated the sum into a sum over families conformal primaries. This was described in the introduction in discussing anomalous dimensions for the leading twist conformal primaries. Conformal dimensions for each family can be interpolated by their spins continuously by ∆( ), i.e., leading to a spectral curve. There will be many families and each family is labeled by an index α 34 . The separation in Eq. (IV.4) into two terms is at first necessary due to possible existence of singularities for Re < L 0 . It is nevertheless interesting to note, since F 0 (w, σ) is real, one always has with L 0 sufficiently large. 34 In the absence of interactions, the label α is simply the twist, τ 0 ≡ ∆ − , and with additional index for other families of conformal primaries of the same twist. For simplicity, we shall keep in what follows only one family for each twist.
A brief discussion on various formal assumptions necessary in carrying the above analysis is in order. Here we summarize a few key points: (a) There exists a unique analytic continuation away from integral values for while satisfying the constraint of "Carlson's theorem" [99,100], with a This principal series combines a discrete sum in the spin and a Mellin transform in a complex ∆plane, with ∆ ≡ iν = ∆ − d/2. As stressed by Mack [72], this representation should serves as a more general starting point for CFT, with standard OPE a consequence of this representation, (See (d) and (e) below and also [72] for more discussions.) (c) Conformal Regge theory assumes a meromorphic representation in the ν − plane, with poles specified by the collection of allowed spectral curves, ∆ α ( ), e.g., a( , ν) = α r α ( ) (IV.7) The spectral curve associated with the energymomentum tensor plays the dominant role, with ∆ P (2) = 4, and on which the Pomeron singularity lies, as in Fig.I.2 35 .
(d) To recover the standard conformal block expansion, it is conventional to close the contour 35 For more discussion see [11,19,26,27].
(IV.8) 36 Due to conformal invariance, the integrand is even in ν, or, equivalently, symmetric in ∆ ↔ 4 − ∆. The contour can be closed either in the upper or the lower half νplane. The poles in the upper half ν-plane corresponds to "shadow" operators. 37 Such procedure was carried out in [11][12][13], based on a treatment equivalent to keeping Minkowski conformal block under the leading order approximation, Sec. III D 1. A simpler example for this separation is Eq. (III.44), appropriate in treating for d = 1 and also used in traditional Regge analysis. A related discussion has also been carried out recently [38] by identifying G (−) as corresponding to the shadow blocks. 38 This procedure was first discussed by M. Toller in the context of Lorentz symmetry, SO(3, 1), [73,90,91,93]. See also a related discussion for O(4, 2) in [72]. We have also simplified the analysis by keeping only crossing even contribution, even in Eq. (I.4). The contribution with odd leads to the so-called Odderon contribution, which has been discussed in [19,101,102].
This representation can formally be regarded as the principal series for a unitary irreducible representation for the non-compact group, SO(4, 2), with a( , ∆ i ( )) an analytic function of . The contour is along the imaginary axis, distorted to include all participating families of conformal blocks, appropriate for describing amplitudes which are bounded at u = 0 by a power (1/ √ u) N , or w N at w = ∞. In pushing the -contour to Re˜ = 0, or Re = −(d − 2)/2, the term F Regge (w, σ) corresponds to contribution coming from all singularities in d/2 < Re < L 0 . The contour distortion allows one to include all participating families of conformal blocks, appropriate for describing divergent amplitudes at w = ∞. (f) Finally, the above analysis also suggests a procedure to resolve the issue of the connection between MCB and ECB. A more desirable, top down approach is to first construct the principle series for a unitary irreducible representation of SO(d, 2), i.e. finding G( , ν; w, σ) in Eq. (IV.8). Then by reverse engineering, one arrives at MCB and ECB respectively. (See Sec. 2 of [19] for a related discussion.)

C. Spectral Curve
In Eq. (IV.4), the integration contour can be pushed further to the left, with pole contributions from the contour passing j = 2n cancelling that in F 0 (w, σ). This can be done until L 0 < 0 thus removing F 0 entirely. However, one has to pick up contributions from possible -plane singularities for 0 < Re < L 0 which might enter through a (12), (34) ( , ∆ α ( )) and also that from the conformal block through ∆ P ( ). These contributions will be collectively denote as A Regge (w, σ). In terms of = + (d − 2)/2, we can shift the contour until the integration path is over Re = 0, or Re = −(d − 2)/2, arriving at Eq. (IV.8). For d = 4, the continuum corresponds to an integral along the line Re = −1. (For d = 3, the path is along Re = −1/2, the traditional "background integral" in a Regge representation.) Let us turn next to Im F (w, σ). As one pushes the contour to the line along Re = 0, one finds that Here let us examine the limit w → ∞. With Re = −(d − 2)/2, the background integral is of the order O(w 1+(d−2)/2 ), and is bounded for 1 ≤ d. It follows that a divergent contribution to F (w, σ) occurs only if there exists a singularity in in the regions −(d − 2)/2 < Re < 2. This is entirely analogous to a conventional Regge theory where the sub-dominant contribution from the continuum is referred to as that from the background integral.
Without dynamical inputs, it is not possible to specify what singularities might exist in the region to the right of Re = −(d−2)/2 for a general CFT. Let us consider first N = 4 SYM at d = 4 and focus on the leading twist-two contribution, which interpolates the stress-energy tensor, with the associated spectral curve denoted ∆ P ( ). Based on weak-coupling perturbation analysis, there exists at least one singularity ef f , to the right of = 1 [46][47][48]. This enters through a branch point of ∆ P ( ). At strong coupling, the location of the corresponding singularity is bounded from above, ef f < 2. (See Fig. I.2 for a schematic representation.) The spectral curve in weak coupling can be found by solving the BFKL equation, Eq. (D.11) and, in strong coupling, via AdS/CFT [11,103]. This singularity is historically referred to as the Pomeron.
It can be shown that this singularity, in the immediate neighborhood of = 2 and at large λ, is of the square-root type [11,19,62,65]. Using the effective AdS mass introduced in (I.18), i.e., with B( , λ) analytic at = 2. It follows that the leading behavior as w → ∞ for the correlation functions F is As stressed in [11], for any d-dim CFT with a gravity dual, this singularity can be associated with string modes interpolating graviton in AdS. With m 2 (2) = 0, it follows that its location is bounded from above, ef f < 2. (IV.12) The deviation from ef f = 2, δ ≡ 2 − ef f > 0, can be attributed to stringy corrections.
To demonstrate the generality of (IV.10), let us recall that for flat-space string theory, massless graviton lies on a linear trajectory, (IV.13) with √ α = string providing a length scale. This can be understood by the on-shell condition for graviton, 1 = L 0 =L 0 = /2 − (α /4)t, where L 0 andL 0 are generators of dilation in a worldsheet treatment. Consider next string living on a curved background, e.g., AdS 5 . In the weakcurvature limit, 2 st /R 2 ads = 1/ √ λ << 1, one has 1 = L 0 =L 0 /2 − (α /4R 2 ads ) ∇ 2 ( ), with ∇ 2 ( ) dimensionless and ∇ 2 (2) = ∆(4 − ∆), with stressenergy tensor having conformal dimension ∆ = 4. This leads to a leading λ spectral curve condition (IV.10) can be interpreted as having an -dependent effective AdS mass. For deformed or thermal theories mass/momentum modes will be shifted. For general d and/or deformed AdS-background, it can be shown that where λ ef f may depend on temperature. We return to this point in Secs. V & VI.
The importance of the CFT 4-point function has appeared recently in the study of CFTs dual to AdS spaces with a black hole. This set up has become tantalizing for two main reasons: first, as a model of holography it is relatively simpler than the canonical N = 4 SYM duality and one might be able to demonstrate more general aspects of holography. Secondly, using the conformal symmetry on the boundary one might learn about the information loss paradox. In this section we apply Minkowski conformal blocks to d = 1 CFTs. The primary example of interest is the SYK theory [32][33][34], however, as most of our results apply to CFTs more generally, we refer to these as SYKlike models 39 . Here we address the issue of d = 1 CFT directly as a scattering problem where one explores the SO(1, 1) Lorentz boost symmetry. The related CFT invariant four point correlation function, expressed in terms of appropriate cross ratio, Γ(w), has a power behavior, as w → ∞, The precise definition for the variable w is given by Eq. (V.3), in analogy to that defined for d ≥ 2 in Eq. (II.12). When ef f → 2, this can be interpreted as corresponding to dual graviton contribution which saturates the chaos bound [29,30]. There are two challenging aspects involved. The first is to find what kind of CFT 39 For a brief description of the SYK theory relevant to this work see Appendix E leads to such power-behavior. An equally challenging task is, given a specific 1-d CFT, how best to identify the effective spin. In this section, we will focus on the latter aspect.
The relevant kinematics for a 1-d scattering is discussed in Sec. V A where we provide a Mellinlike representation for Γ(w), as well as Im Γ(w), analogous to Eq. (D.7). In Sec. V B we examing the Schwinger-Dyson equation for SYK-like models. This corresponds to treating a set of Wightman functions for which the integral equation simplifies since it involves only amplitudes over physical scattering regions. This integral equation can be diagonalized in , and the associated partialwave amplitude, A( ), can be then found algebraically. For these models, one can show that Γ(w) is power-behaved, as in Eq. (V.1), and the leading effective spin can be identified simply.
Our treatment parallels to that carried out in [35,37] but differs significantly in how the relevant spectral analysis is carried out. In working with Im Γ(w), we can in principle deal only with the Hilbert space of square-integral functions defined over half-line, 1 < w < ∞, instead of over the whole line, −∞ < w < ∞, as done in [35,37]. This leads to a significant simplification. In Appendix E, 1-d Green function for D w , Eq. (III.41), is discussed from a conventional spectral analysis, before generalizing to the case of functions which are polynomially bounded. Here, we follow more directly the procedure of re-summation performed in the last section for d ≥ 2, starting with the result of Sec. III D 3, which agrees with the result followed from a Hilbert space treatment, Eq. (E.12).

A. Kinematics
In analogy to the case of d ≥ 2, we consider a reduced invariant 4-point function, Γ, as a function of a cross ratio for the process 1+3 → 2+4. There are several options for cross ratios 40 , e.g., In 1-d, however, only one is independent due to the constraint |τ c | + |τ | = 1. Let us first adopt τ as the independent variable. For s-channel scattering, 1 + 3 → 2 + 4, we require t 4 , t 2 > t 3 , t 1 . Without loss of generality, we consider the causal limit of t 4 > t 2 > t 1 > t 3 . Reparametrization invariance allows one to keep three points fixed. We choose the ansatz t 4 = ∞, t 3 = 0, t 2 = 1, with t 1 ≡ t as the independent variable, thus 0 ≤ t ≤ 1. In terms of invariant cross ratios, τ = 1 − t and τ c = t, and the s-channel physical region corresponds to 0 ≤ τ ≤ 1, and τ + τ c = 1. The u-channel physical region, with 3 ↔ 4 interchanged, corresponds to τ → τ ≡ t12t34 t13t24 = τ /(τ − 1). The u-channel physical region, 0 ≤ τ ≤ 1, then leads to −∞ < τ < 0. In order to exhibit crossing more symmetrically, it is again convenient to adopt a new variable (similar to Eq. (II.12)) The s-channel physical region corresponds to with u-channel region given symmetrically by −∞ < w < −1.
Let us consider both the invariant amplitude, Γ(w), and its imaginary part in the s-channel physical region, Im Γ(w). We assume that Γ(w) is polynomial bounded, |Γ(w)| < w N , at w = ∞. As before, this invariant function has a factor of w removed from the 4-point correlator, as in Eq. (II. 19). For correlators which are crossing even, Γ(w) will be crossing odd, Γ(−w) = −Γ(w). As shown in Sec. III D 2, eigen-functions for the quadratic Casimir for 1-d scattering processes can 40 We have switched to using a timelike coordinates, t and τ , to emphasize that we are interested in the Lorentz boost symmetry and to conform with the more standard notation found in the literature. See also Appendix A 2 b. be chosen to be Legendre functions of the second kind, G 1d (w) =Q − (w) = c Q − (w), with normalization c specified in Eq. (III.43) 41 . In terms of the Mellin representation, described in Sec. IV B, reparametrization invariance leads to where N + 1 < L 0 . For w large, from Eq. (III.43), the Legendre function simplifies toQ − (w) For its imaginary part, defined for 1 < w < ∞, the sum over < L 0 does not contribute, and where we have also brought the representation into a more conventional form 42 , with tan π π Q − (w) regular in at positive integers. By taking advantage of the identity in Eq. (III.44), it is useful to simplify (V.6) further in terms of Legendre function, P (w), with complex and 1 < w < ∞, This follows from the fact that one can add terms under the integral in (V.6) which will then vanish by closing contour to the right. In applying 41 The utility of using Legendre functions of the second kind to describe Γ(w) has been noted by other authors studying the SYK model [104]. Also see [105][106][107]. 42 Our notation here conforms to that adopted in Appendix E 2, in particular, factoring out the term (2 − 1). A more natural choice would be shifting by one unit, leading to a more familiar factor of 2 + 1. We have not done so due to our choice of maintaining symmetry about Re˜ = 0. for d = 1, this corresponds to Re = (2 − d)/2. to 1-d scattering, it is this Mellin representation which will be particularly useful. As shown in Appendix E, this representation can also be arrived at through a Hilbert space treatment over 1 < w < ∞.
Making use of a standard orthogonality condition, one has the following inversion formula Conversely, we can examine the full amplitude,

B. Scattering for SYK-Like Models
In the low temperature limit of the SYK model there is a near conformal symmetry that is both explicitly and spontaneously broken. The theory is close to a 0+1 dimensional conformal theory. Within this approximation, the invariant function Γ can be obtained by solving a ladder-type Bethe-Salpeter equation 44 , which can be represented schematically as where Γ is a function of 4 coordinates: Γ(t 1 , t 2 , t 3 , t 4 ). We shall refer to models that admit such a representation as SYK-like models.
Reparametrization invariance allows diagonalization of this integral equation. This has indeed been carried out in [35][36][37], primarily in an Euclidean setting. To make contact with the chaos bound, an appropriate analytic continuation to the Minkowski setting has to be performed. As a scattering problem, Eq. (V.14) has the structure of a "box-diagram", and it can be expressed equivalently as an analogous integral equation for its absorptive part, the imaginary part of Γ in the physical region. Schematically, this can be expressed as with the convolution, ⊗ , is over the physical region only. We will discuss the integral equation, Eq. (V.14), as a scattering problem in real time, and 43 The final relation comes from the two-term recursion relation P n+1 (w) = wPn(w) − n 2 4n 2 −1 P n−1 (w) . The contributions from = 0 and = 2 require a more careful treatment. We defer to the comments after Eq. (V.40) 44 This type of integral equation has a long history of use in particle physics. For a different example see Appendix D 3 where it is discussed in the context of deep inelastic scattering. study its solution in the high energy limit. As such, it is analogous to the BFKL type integral equation when applied to DIS structure functions, Appendix D. We will spell out more precisely the nature of this integral equation after first discussing the simplification due to reparatrization invariance, allowing one to express Γ in terms of invariant cross ratios, Eq. (V.18). Reparametrization invariance leads to a simpler diagonalization procedure in arriving at the desired solution in identifying the leading effective spin, * which directly controls the large-time behavior for Γ.
It is also instructive to express the solution to Eq. (V.15) formally as an iterative sum, The process can be viewed as producing n-lumps of particles, with K 0 providing the relative probability of producing an additional "lump". Each lump is irreducible, and the allowed intermediate states consists of n such lumps, with n = 1, 2, 3, · · · . This interpretation is analogous to the case of d = 2 forward scattering discussed in Appendix D.
More precisely, Eq. (V.17) can be identified with the early work of Amati, Fubini and Stangelini [43,45] in constructing multiperipheral models (MPM) of particle production. Their construction is equivalent to a Bethe-Salpeter equation for the absorptive part of the forward scattering amplitudes. Here (Im Γ) n ∝ |T 2→n | 2 , where a factorizable 2-to-n production amplitude, T 2→n , leads to a recursive relation, Eq. (V.17). Furthermore, consistency requires that K 0 ≥ 0, over the allowed phase region. Given K 0 for a specific model, the challenge is to carry out the sum, Eq. (V.16). In exact analogy to the BFKL integral equation, Eq. (D.11), this integral equation can be solved by diagonalization due to SO(1, 1) symmetry.

Kinematics of Integral Equation for Im Γ:
In order to understand the kinematics of the integral equation for Im Γ, it is useful to first examine the Schwinger-Dyson equation in d = 1 for the full amplitude Γ. Consider a theory defined in such a way that it is given by a sum of ladder graphs, i.e. schematically it is analogous to that for BFKL integral equation, Eq. (D.12). For SYK-like models, the integral equation can be written for an amputated 4-point function, with labels corresponding to "scattering" in the s-channel of 1 + 3 → 2 + 4. Explicitly, Eq. (V.14) becomes Similarly to the BFKL case, the ladder sum is in the t-channel. The first term on the right corresponds to an inhomogeneous contribution. The second term corresponds to a convolution of the "2particle irreducible" kernel,, K 0 (t 2 , t 1 ; t 6 , t 5 ), with the connected full amplitude, Γ(t 6 , t 5 ; t 4 , t 3 ). Although the solution can also be expressed formally in an iterative sum, Eq. (V.16), it is important to emphasize that the integration is over all values of t 5 and t 6 . Some regions do not correspond to physical scattering, with K 0 taking on negative values. Reparametrization invariance dictates that K 0 (t 2 , t 1 ; t 6 , t 5 ) is a function of a cross ratio. For SYK like models, define where α 0 (q) = 2πq (q−1)(q−2) tan (π/q) , with 4 ≤ q < ∞. We have expressed K 0 , which is derived in an Euclidean treatment, such that K 0 > 0 over the physical region of 0 < τ k < 1. As mentioned above, it is necessary to treat different kinematic regions differently, as done in [35]. However, we are interested in treating this as a generic scattering problem for 1-d CFT, but will not address here the question of how this type of model can arise from a more fundamental perspective 45 .
The structure of the integral equation for Im Γ is the same as for the full amplitude Γ, except the integration region in Eq. (V.18) has to be restricted to the physical region. For 1-d, it is possible to express both Γ and Im Γ in terms of a single cross ratio. We will initially adopt τ , Eq. (V.2), as the independent variable and use the ansatz described at the beginning of Sec. V A. For the integrand in Eq. (V.18), we can construct invariants analogous to τ : τ k = t21t65 t25t61 and τ = t65t43 t63t45 . In terms of these invariants where the Jacobian is given by The restriction to the physical region, for Im Γ, corresponds to enforcing the constraint D ≥ 0. As discussed in Sec. V A, a more symmetric treatment for the physical scattering region can be carried out by working with variable w = 2−τ τ , with s-channel physical region 1 < w < ∞ and correspondingly −∞ < w < −1 for the u-channel. More directly, it can be shown that wheret ij = (t i + t j ). Again, for the integrand in Eq. (V.18), we can construct with 1 < w k < ∞ and 1 < w < ∞ for s-channel physical regions and −∞ < w k < −1 and −∞ < w < −1 for u-channel physical regions, for 1+5 → 45 In moving to a Minkowski setting, the kernel should acquire an additional phase. We will return to this question below.
2 + 6 and 5 + 3 → 6 + 4 processes respectively. The region will further be restricted by the requirement that the Jacobian of transformation from t 5 and t 6 to w k and w be real. It is not difficult to check that the Jacobian is given by , whereD(w, w k , w ) takes on a standard form of a "triangle function", 23) The condition of integrating over physical region corresponds toD(w, w k , w ) > 0, with 1 < |w |, |w k | < w. The condition for physical scattering, designated by θ + (D), further restricts 1 ≤ w k , w ≤ w, with contributions from u-channel, −∞ < w k , w < −1, folded into the 1 < w k , w < w by symmetry. Introducing Lorentz boost parameters, cosh β = w, cosh β k = w k and cosh β = w , the constraint due to the Jacobian can also simply be expressed as a triangle inequality, 0 < β k +β ≤ β, which respects the proper time-ordering. It follows that the integral equation for Im Γ(w), Eq. (V.16), in the s-channel physical region can be expressed as Im Γ(w) = Im Γ 1 (w) which is one of our key results. Let us turn next to the integral kernel K 0 (w k ) and its relation to the corresponding kernel K 0 (w k ) for the full amplitude. In higher dimensions, K 0 would be appropriate discontinuities of K 0 across various physical regions. For CFT, these discontinuities are all equal, up to a constant numerical factor. We will therefore replace K 0 in Eq. (V.24) by K 0 (w k ) = C K 0 (w k ). with K 0 real and positive, given by Eq. (V.19), in the s-channel scattering region of 1 < w k < ∞. For 1-d, to keep track of the contributing phases, we need to take account of time-ordering across two edges of the ladder. Denote the signs for t 53 and t 64 by ±. There are 4 distinct time orderings: (++), (−−), (+−), and (−+), which contribute to Im Γ(w) in the schannel 46 . For SKY-like models, the combined contribution becomes C = (e −iπ/q +e iπ/q )(e −iπ/q + e iπ/q ) * = 4 cos 2 (π/q), with C real and positive. This amounts to adopting an approach using a retarded propagator, as done in [37]. It follows that This shall serve as our ansatz for SYK-like models.

Diagonalization:
We proceed first to a general solution to this integral equation for ImΓ(w), Eq. (V.24), taking advantage of our analysis in Sec. V A. Just like the situation for d ≥ 2, we will consider the class of problems where Γ(−w) = −Γ(w) and Γ(w) is polynomially bounded as w → ∞. It follows that Im Γ(w) can be represented by an inverse Mellin transform, Eq. (V.7). A similar representation can be written for the kernel, (V.26) and also for the inhomogeneous term, Γ 1 . The inversion formula is given by Eq. (V.9), and, for the kernel and the inhomogeneous term, We emphasize again, for both A 1 ( ) and k( ), the integrals are over the physical regions only. With 46 In contrast, for the integral equation for the full amplitudes, depending on the counting schemes, there are many more configurations to consider in an Euclidean treatment [35,37]. k( ) given by a single integral, Eq. (V.27), this is a significant simplification when compared to the comparable treatment involving the full amplitude, Γ [35,37]. Let us apply the integral transform The lefthand side leads to A( ). With the help of the identity [108][109][110] (V.29) valid for Re > 1, the non-trivial term on the right can be diagonalized as k( ) A( ), thus leading to The ability to reduce the integral equation, Eq. (V.24), to an algebraic equation by a rather straight-forward analysis is another one of our key results. This process is analogous to that carried out in [108][109][110]. The treatment here corresponds to an harmonic analysis by non-unitary representation of SO(1, 1). From this diagonalized equation, the partialwave amplitude A( ) can be found simply, with k( ) given by Eq. (V.27). This in turn allows one to recover ImΓ via Eq. (V.7) and/or the full amplitude Γ more directly from via Eq. (V.5).
Since k( , q) is monotonic in Re and vanishing at Re = ∞, ef f = 2 is the leading singularity to the right of Re = 1. From Eq. (V.7), one has, as w → ∞, with γ given by residue of A( ) at = 2. Corrections to the leading order comes from a singularity at = 1 − 2/q, driven by the Born term, Im Γ 1 .
For the full amplitude, Eq. (V.38) corresponds to having a leading behavior with γ given by residue of ( −2)A( ) at = 2. The emergence of log is perhaps puzzling. It is mathematically necessary due to the fact that Im Γ(w) grows linearly with w. This is also related to the fact that Eq. (V.37) is singular at n = 0. To clarify this issue further, we turn next to a brief discussion on the continuation into the region |w| < 1, conformal symmetry breaking, and stringy corrections.

C. Analyticity and Corrections
Let us end this section with several additional comments.
a. Hilbert Space Treatment: Our treatment for Im Γ(w) can be framed in the context of an harmonic analysis over the non-compact group, SO(1, 1), as carried out in Appendix E. We begin first with a spectral analysis over the Hilbert space of square-integrable functions over the interval (1, ∞), Sec. E 1. The framework is next extended to allow functions which are polynomially bounded at w = ∞, in Sec. E 2, leading to the representation (V.6-V.7). As explained in [108][109][110], this can be regarded as an harmonic analysis by non-unitary representation of SO(1, 1).
Let us contrast our scattering treatment with other related Euclidean treatments, for example that of [35,37], which can also be framed in a Hilbert space treatment. In [37], one first considers the space of functions defined over 0 < τ < 2, which corresponds to 0 < w < ∞. It is then extended to the whole range in τ by symmetry. With w = (2 − τ )/τ , this corresponds to reflecting w ↔ −w, leading to the whole range −∞ < w < ∞. As a consequence, eigenfunctions contain log-singularities at τ = 1 (w = 1). In contrast, our eigenfunctions, (E.5)-(E.6), are defined over the interval (1, ∞). In order to extend their treatment to include functions which are more singular at τ = 1, e.g., (τ − 1) −N , N > 0, (∼ |w| N as |w| → ∞), it is implicit that an extension of the standard spectral analysis also has to be made. Therefore, our treatment here is in some sense no less general than that carried out in [35,37].
b. Continuation to Euclidean Region: Given Im Γ(w), for the region 1 < |w| < ∞, let us examine the specification of the full amplitude Γ(w). From Eq. (V.10), it is possible to extend Γ(w) to the complex w-plane, as a real analytic function with branch cuts for 1 < |w|. For the case of Eq. (V.38), one can initially choose 2 < L 0 < 4. In continuing to the region −1 < w < 1, the contour can be closed to the right, Eq. (V.11), arriving at In closing the contour, one finds Im Γ(w) = 0 in the region |w| < 1, as indicated earlier. In the first line on the right,Ā 2 a constant, is a priori unspecified. The second line is a re-write of the first, with¯ = − 1 and A¯ +1 summed only over¯ odd integers. This is precisely the ordinary Legendre expansion for square-integrable functions over (−1, 1) where Γ(w) = −Γ(−w).
Under normal circumstance, the sum over¯ can be extended to¯ = 1, withĀ 2 = 3A(2) given by the analytic continuation of A( ) to = 2. However, this is not a necessity 47 . For SYK-like models, the situation is more robust since A( ) has a pole at = 2. From Eq. (V.5), there is a double-pole at = 2, leading to logrithmic behavior. Lastly, we note that Eq. (V.40) can again be 47 In a traditional Regge treatment, if this were to happen, it would correspond to the theory not being uniquely defined by the analytic S-Matrix [99], A( ) would contain a Kronecker-delta term, and the theory would require "Castillejo-Dalitz-Dyson" (CDD) poles. See [111] for a historical discussion.
c. Conformal Symmetry Breaking: The existence of a pole at = 2, as explained in [35,37], is due conformal invariance, and it corresponds to the existence of a goldstone mode. For SYK models, this mode is unphysical and should be removed. There are several scenarios to consider. One possibility is simply to define the theory with this mode removed. For example, in Eq. (V.40), settingĀ 2 = 0, as is done in [35]. In this case, it is possible to reverse the procedure in re-summing Eq. (V.40) for the limit w → ∞, leading to Eqs. (V.38) and (V.39) with γ = 0. Analytically, this can be accomplished by introducing an extra zero to A( ), e.g., A( ) ∼ − 2, leading to γ = 0 and γ = 0. This will not alter the eigenvalue condition for k(2, δ) = 1. As a consequence, this "weak breaking" scenario does note alter the feature that ef f = 2; it leads to a situation where the log term in Eq. (V.39) is removed. A stronger modification to the kernel is required in order to change ef f from 2.
d. Stringy Corrections: On the other hand, the model can be embellished by considering stringy corrections as discussed in [37], and, more generally, in [30]. In this case, one has ef f < 2, which leads to as w → ∞. It follows that the log-term in Eq. (V.39) is again removed. In analogy with N = 4 YM, it is tempting to refer to this singularity as due to stringy corrections, as the 1-d Pomeron [112]. However, as discussed in the next section, thermal effects must be taken into account.

VI. DISCUSSION AND SUMMARY
In this paper we have focused on scattering in CFTs, for example off-shell photon-photon scattering, through an OPE with Minkowski conformal blocks. We review the major results here and discuss future connections and applications. This paper consists of three main components. The first part, directly defining and calculating MCB, is shown in Secs. III. MCB are solutions to the quadratic Casimir for the product of scalar conformal primaries. We show how "scattering along light-cones", also realized by taking a double lightcone limit as discussed in Sec. II, selects a natural basis for MCB satisfying a different set of boundary conditions from that for Euclidean conformal blocks. Due to the difference in boundary conditions, MCB are not given by a direct analytic continuation but only related to the ECB.
In an Euclidean setting, the DLC limit involves a single scale, dilatation, leading to a single scaling limit. In a Minkowski setting, there are two scaling limits: dilatation and boost. The dilatation limit is characterized by a parameter σ → ∞, and the second scaling limit, w → ∞, is analogous to a Lorentz boost. Dilatation leads to scaling dependent on the conformal dimension ∆, Eq. (II.23), while the Lorentz boost leads to a dependence on an effective spin ef f , Eq. (I. 16). When a 4-point correlator is expressed in terms of invariant cross ratios, for example F (u, v) in Eq. (I.2), these two scaling limits allow one to explore the consequence of the residual symmetry, O(1, 1) × O (1, 1).
The second part deals with the application of Minkowski OPE for formal scattering processes. We explain how a t-channel OPE leads to a description for s-channel scattering amplitudes in terms of contributions from t-channel conformal primaries. In the DLC limit, the leading contribution comes from the stress-energy tensor. For a realistic phenomenological treatment, stringy, and sometimes thermal, corrections must be included. This requires summing contributions from the tchannel OPE via the Sommerfeld-Watson transformation. This is discussed in Sec. IV, leading to a complex -plane representation, (I.14), as well as a double-Mellin representation in complex-∆ and complex-, Eq. (IV.8).
The third part turns to specific applications. The simplest phenomenological application, which directly explores the residual SO(1, 1) × SO(1, 1) symmetry, is Deep Inelastic Scattering (DIS). Both formal and phenomenological aspects, like the BFKL program, are summarized in Appendix D. In Sec. V we focus on scattering for 1-d SYK-like models. We show how MCB can be used to simplify the dynamics for these models and how the leading effective spin, associated with the chaos bound, can be identified.
We conclude here with a more detailed discussion on how stringy correction to ef f for SYK-like models can be framed through AdS/CFT along the formalism introduced in [11][12][13]. This issue has been previously addressed in [30] and [37,38]. We add further discussion by exploring the consequences of SO(1, 1) × SO(1, 1) symmetry via the spectral curve, ∆( ), for the leading twist conformal primaries interpolating the stress-energy tensor. This was explored in Sec. IV.
The importance of spectral curve for scattering in CFT can best be illustrated via DIS. It is well known that anomalous dimensions of the leading twist-2 conformal primaries of conformal dimension ∆ and spin , O ∆, , controls the large q 2 dependence for the moments of structure functions. For example, for F 2 (x, q 2 ), for q large, (D.9), for general d, leads to ∆( )) ≥ d/2. The second scaling is related to the limit x → 0 (E γ → ∞). Again, for F 2 , this leads to Eq. (D.14). The effective spin can be found by solving an eigenvalue condition Eq. (I.16) [11,13], which can be expressed more explicitly as ∆( ef f ) = 2, which saturates the positivity bound.
An alternative spectral representation in t can also be obtained in terms of regular Bessel function, (B 2). At integral , they correspond to momentum space representation of bulk-to-boundary 48 For N = 4 SYM, by taking advantage of integrabililty, Eq. (IV.10) represents a systematic expansion at strong coupling and low spin. Using this approach it has been possible to calculate the Pomeron and Odderon, associated with the anti-symmetric tensor, Bµν , intercepts to several high orders in 1/ √ λ [19].
propagators. In the DLC limit where p 1 − p 2 q ⊥ is asymptotically transverse, performing a 2-dim Fourier transform, one finds, up to a factor of (zz) 2 in reducing to scalar propopagator, with cosh ξ = (z 2 +z 2 + b 2 ⊥ )/2zz, i.e., a formal solution expressible in terms of geodesic ξ on H 3 . At = 2 and d = 5, other than the extra factor (zz) 2 , this is nothing but the scalar AdS 3 bulk-tobulk propagator.
Let us turn next to thermo-CFT-correlators and treat it similarly via AdS/CFT by considering a black-hole background. For d ≥ 3 we can write the metric as [113], where we have scaled the horizon to z = 1. A similar spectral analysis in t and in can be carried out, extending the treatment of [114]. Interestingly, the spectral in t is now discrete, with t = m 2 n > 0, n = 0, 1, · · · , and is analytic for t < 0, (VI.5) For = 2, these correspond to tensor glueballs calculated in [114]. There is a finite mass gap, with m 2 0 > 0. For t > 0, the spectrum in is also discrete for > 0, which, when combined with Eq. (VI.5), leads to Regge trajectories, n (t), n = 0, 1, · · · . However, for t < 0, the t-dependent term in the Laplacian turns repulsive, and the spectrum in is continuous. This leads to branch cut at (−∞, ef f ), where we have introduced an effective 't Hooft coupling, λ ef f ≡ (R ef f / string ) 4 . (This branch cut also persists for t > 0.) A similar expression for ef f has also been arrived in [30] and [37]. In particular, [37] finds that λ ef f should be temperature dependent. We defer to a future study on how our analysis can be framed accordingly [115].
We end by pointing out that we have focused in Sec. V, on the large w behavior for Im Γ, with the real part given by dispersion relation. However, in "lifting" the model to higher dimensions, the relation between the real part and the imaginary part becomes more complex. In [13], in the Regge limit, it was pointed out that the limit of large s and large impact parameter, b, do not commute. Our study here takes the limit s large before b is allowed to be large, corresponding to the limit u → 0 with (1 − v)/ √ u initially fixed. It is nevertheless interesting to examine the large s limit, but fixed, and taking b large. In that case, one regains the single gravity exchange, with a cutoff in impact space controlled by the lowest tensor glueball mass, m 0 , with m 0 scaled by the inverse temperature β [13]. It is equally important to recognize that this mass does not directly determine 49 the stringy correction to ef f . a. Acknowledgments: We would like to thank Antal Jevicki for useful discussions and for comments on an early draft of this paper. We would also like to thank the Brown High Energy Theory group for comments on an early version of this work presented in seminar. C-I. T. would like to thank J. C. Lee and Y. Yang of National Chiao-Tung University (NCTU), Taiwan, and its Yau Center, for hosting and discussion where this work began in earnest during recent visits. He is particularly indebted to J. Polchinski, as well as R. Brower and M. Strassler for collaborating on [11], which shaped his interest in scattering in AdS/CFT. He owes special thanks to R. Brower for collaboration and encouragement on issues relating to CFT in general and AdS/CFT in particular. T.R. is supported by the University of Kansas 49 For a possibly different perspective, see [30]. Here we provide more detail on many of the specific kinematic details, notations, and conventions, that are used throughout the main text.

Channels
An unfortunate side effect of a long history of literature is that similar syntax confusingly gets used to refer to different things. In an effort to clear up confusion, in this brief appendix we would like to highlight two different uses of the channel of a scattering process that are used in the text. We refer to the channel of a scattering process, for example s-channel scattering, as a reference to incoming and outgoing particles. Once a scattering channel is defined, the rest follow. This can be seen in Fig.A.1. A single scattering process can be written in terms of different OPE combinations as seen in Fig.A.2. Note that in a CFT, these contributions are not summed; a single OPE prescription describes the entire correlation function. Finally we note that the s-channel process and u-channel process can involve a similar t-channel OPE structure as in Fig.A.3.

Rindler Coordinates for the DLC Limit
In this Appendix, we review the Rindler-like hyperbolic coordinate parametrization which is used extensively in Sec. II. We consider here s-channel scattering approaching the DLC limit.
Consider LC coordinates, x = (x + , x − ; x ⊥ ), x ± = x 0 ± x L . Our convention for different time signatures is For the Minkowski case, x space-like implies where the time and longitudinal coordinates are explicitly, Sending all rapidities y i → ∞, with x i⊥ fixed, leads to (II.2).
a. Relating to Invariant Cross Ratios for General d: For invariants, we have, for general d, with transverse coordinates x ⊥,i kept for d > 2, u = where To simplify the discussion, we will adopt a frame where x i,⊥ = x 2,⊥ and x 3,⊥ = x 4,⊥ , i.e., b 12 = b 34 = 0, with b ⊥ = x 1,⊥ − x 3⊥ as the relative separation between (1, 2) and (3,4) in the transverse impact space.
In terms of the global rapidity y and virtuality r i for each coordinate, cross ratios u and v take on relatively simple forms, For all y i taking on a global value y, thus y ij = 0 andȳ ij = 2y,  where the transverse separation enters through R(i, j). In the case of two pairs of equal virtuality, r 1 = r 2 and r 3 = r 4 , these can further be simplified, leading to (II.5). The limit u → 0 can therefore be achieved either by y → ∞ or r i → 0, or by b 2 ⊥ → ∞ first. For near-forward scattering, or the DLC limit, we consider the first scenario of y → ∞ with r i and b 2 ⊥ fixed. The limit u → 0 therefore exploits the scaling limit of Lorentz boost.
It is straight forward to calculate the combination 1−v+u 2 √ u . The physical region is constraint to satisfy As an illustration, we consider the limit where r 1 = r 2 and r 3 = r 4 , where one finds It is easy to check that equality for (A.11) holds, Therefore, only one cross ratio is independent.

Impact Parameter Representation and Holography
Consider the Fourier transform of the connected correlation function defined in (I.2), The amplitude T (p j ) can be expressed as a function of Mandelstam invariants s, t, and p 2 j . The Regge limit corresponds to s large, which defines a light-cone direction, with t < 0 and p 2 j fixed. In this limit, the momentum transfer is asymptotically transverse, with t = (p 1 + p 2 ) 2 ≈ −q 2 ⊥ . In a coordinate representation, this corresponds to the DLC limit, discussed in Sec. II and explained further in Appendix A 2. Using conformal symmetry, it is possible to express the amplitude T (p j ) as where K(s, t, z,z) corresponds to a Pomeron-Regge kernel which in the Regge limit admits an impact parameter representation, [14][15][16][17] K(s, t, z,z) = (zz) 2 s The amplitude F(S, σ 0 ) encodes all dynamical information and, due to conformal symmetry, depends only on the variables S = zzs and σ 0 z 2 +z 2 +b 2 ⊥ 2zz , (II.8). It is important to note that the conformal representation (B.3) of the amplitude is valid for any value of the coupling constant, since it relies only on conformal invariance. The same representation was obtained through direct AdS/CFT considerations, [11][12][13], leading to an identical Regge kernel, K(s, b ⊥ , z, z ). Up to irrelevant constants, this kernel is related to T (S, σ 0 ) by The Regge limit is now S → ∞ with fixed σ 0 . We have therefore two representations of the correlation function in the Regge limit. One derived from the CFT analysis in position space F (M ) (u, v), given by Eqs. I.2) and (IV.4), and another from a computation in momentum space with a clear geometrical interpretation as a scattering process in AdS, given by Eq. (II.19). This establishes a dictionary, where, in the Regge limit, We will also identified, as done in Sec.
. More details can be found in [19]. It is also possible to carry out a more formal analysis in establishing this equivalence [16] and a useful more recent review can also be found in [30]. It suffices to emphasize the exact equivalence of the two approaches to identify the spectral curve, ∆( ) in Fig. I.2, which serves as the common link between them.

AdS d−1 Bulk-to-Bulk Propagator and the Pomeron Intercept
A more precise relation discussed above can best be illustrated by the well-known example of one graviton-exchange contribution through AdS/CFT. The contribution is proportional to the traceless-transverse bulk-to-bulk graviton propagator in AdS 5 , which can be identified with the leading contribution to the eikonal, (II.19). In the near forward limit, the momentum transfer is small and transverse to the LC, and one finds, the net contributions reduces to 51 , The transition fro AdS 5 to AdS 3 propagator can be understood best in a momentum treatment. See [12,13]. with = 2 and ∆(2) = 4 for graviton in d = 4. G ads3 is the Euclidean AdS 3 scalar propagator for conformal dimension ∆−1, with σ = cosh ξ related to the chordal distance. As explained in Sec. VI, one can begin by restricting the string propagator G = δ L 0 ,L 0 L0+L0−2 on AdS 5 to the graviton sector. In strong coupling, it can be reduced to that involving scalar AdS Laplacian . In a momentum-space representation, one has ∇ 2 0 = ∇ 2 0,radial − z 2 t and ∇ 2 0,radial = z 5 ∂ z z −3 ∂ z . A self-adjoint spectral analysis can be carried out in with t < 0 and also in t with > 2.
Finally in the DLC limit where p 1 − p 2 = q ⊥ is asymptotically transverse, performing a (d-2)-dim Fourier transform, one finds with G ads (d−1) (σ, , ∆) given earlier, i.e., a formal solution expressible in terms of geodesic ξ on H d−1 . At = 2 and d = 5, other than the factor (zz) 2 R −4 ads , this is nothing but the scalar AdS 3 bulk-to-bulk propagator. For general , it corresponds to having AdS mass, m 2 ( ), (I. 18).
Appendix C: Minkowski Conformal Blocks and Analytic Continuation

Useful Mathematical Facts
We summarize several useful facts relating to hypergeometric DE. First, differential equation for associated Legendre functions, P µ ν (q) or Q µ ν (q), is We will work here mostly with µ = 0, leading to either P ν (q) or Q ν (q), depending on appropriate boundary conditions. More generally, Eq. (III.26) (b), is of the form 2) It can be shown to correspond to the DE for AdS d−1 propagator, on H d−1 , with geodesics ξ = cosh −1 q and AdS mass, m. They can be related to hypergeometric DE by a change of variable. For instance, in terms of y = q 2 , it leads to the standard DE, ε(d) = (d − 2)/2, For general values of a, b, c, the regular solution at y = 0 is denoted by the standard notation y n Γ(n+1) . However, we are interested in the region 1 < y < ∞. Two independent solutions can be chosen as Since ab < 0, adopting the convention a > 0, the solution where P (q) vanishing at q → ∞ corresponds to F 1,∞ (q 2 ), In particular, for d = 3, this leads to Q ν (q), with ν = −1/2 + 1/4 + m 2 , Q ν (q) = π 1/2 Γ(ν + 1) Alternatively, in terms of z = (q + 1)/2, the resulting DE also take on the same form, with This leads to a useful identity in changing variable from z to q = 2z − 1, which, more generally, corresponds to the identity F (a, b, 2b, w) (C.9)

Standard Differential Equation for
Conformal Blocks The Casimir differential operator, D ε (a, b) G(u, v) = C ∆, G(u, v) , can be expressed, either in terms of (x,x) or (q,q), as a sum of terms [3], D ε (a, b) = D 0 (a, b) +D 0 (a, b) + D ε 1 . The first term involves x or q, and the same forD 0 (a, b) withx andq replacing x andxn respectively. The mixed term, D ε 1 , is where ε = (d − 2)/2. Here a and b stand for more general four point function where a = −∆ 12 and b = ∆ 34 /2. We will restrict in what follows to the case where a = b = 0. Solutions to D 0 (x)f (x) = λ(λ − 1)f can be expressed in terms of Hypergeometric functions, i.e., with f (x) = x λf (x), DE forf (x) becomes (C.14) i.e., again in the hypergeometric form, Eq. (C.3). A general solution can be expressed as The corresponding differential equation in terms of variable q is dg(q) dq + λ(λ − 1))g(q) = 0 (C.17) with boundary condition specified at q → ∞. This is precisely that for Legendre function of the second kind, Eq. (C.3). A general solution can also be expressed in terms hypergeometric funvtions, e.g., k 2λ (q), Eqs. (III.11-III.12). Eq. (C.3), valid for |q| > 1, is particularly useful in considering continuation from 1 < q < ∞ to −∞ < q < −1.

Comparison with Analytically Continued Euclidean Conformal Blocks
It has been suggested that Minkowski conformal blocks are simply an appropriate analytic continuation of Euclidean conformal blocks, which changes the boundary conditions in Eq. (III.7) from (a) to (b). We demonstrate below that Eq. (III.13) and Eq. (III.14) are not given by a direct analytic continuation of the corresponding ECB, Eq. (III.15) and Eq. (III.16).
The discussion for analytic continuation is normally framed in terms of variables x andx. One can easily transition from (q,q) to (x,x), with k 2λ (x) = k 2λ (q), (C. 18) due to the identity, Eq. (C.9), where z = x −1 . We examine more closely here the relation of between G (M ) (∆, ) (x,x) and G (E) (∆, ) (x,x) in terms of their analytic structure in x andx. It should be emphasized that, by treating x andx as independent complex variables, one necessarily extends beyond the Euclidean limit wherex = x * . For Minkowski limit both x andx are real but independent. To examine their possible connection, both x andx are to be treated as independent complex variables. In order to make this demonstration explicit, we will focus on the case of d = 2. The case d = 4 can also be dealt with explicitly.
It is sufficient to examine the analytic structure of k 2λ (x) as one circles around its branch point at x = 1, or equivalently, at x = ∞, while holdingx fixed. It is useful to expose square-root singularities at x = 1 and x = ∞ by a mapping This map exposes a double-sheet structure (See   (C.1)). In general ρ maps the first sheet of x, denoted as the physical sheet, to the region within the unit circle, |ρ| < 1, and the second sheet in x, analytically continued through the cut (1, ∞), to the region |ρ| > 1, outside of the unit circle. For x = 0, on the physical sheet sheet, ρ 0. On the other hand, when continued to the second sheet, the limit x → x II → 0 maps ρ 4/x II → ∞. This map exposes a double-sheet structure. In general ρ maps the first sheet of x, denoted as the physical sheet, to the region within the unit circle, |ρ| < 1, and the second sheet in x, analytically continued through the cut (1, ∞), to the region |ρ| > 1, outside of the unit circle. For x = 0, on the physical sheet sheet, ρ 0. On the other hand, when continued to the second sheet, the limit x → x II → 0 maps ρ 4/x II → ∞. Therefore, to understand the analytic continuation, it is simpler working directly with the variable ρ.
It is useful to again relate both MCB and ECB directly to hypergeometric functions. The differential operator D 0 simplifies in terms of ρ. For d = 2 it becomes and the relevant eigenvalue condition becomes It is convenient to introduce new notation, K α (ρ 2 ) = k 2α (x). From Eq. (C.21), one finds K α (ρ 2 ) = ρ α 2 F 1 (1/2, α; α + 1/2; ρ 2 ). In terms of K α (ρ 2 ), the Minkowski conformal block, for d = 2, is given by where ρ < is the small of (ρ,ρ), and ρ > the other. In contrast, for the Euclidean conformal block, We stress, for both Eq. (C.23) and Eq. (C.22), we are, for now, restricted to the region |ρ| < 1 and |ρ| < 1 so that x andx remain on the physical sheet. The advantage of using K α (ρ 2 ) over k 2α (x) lies in the fact that analytic continuation to the second sheet in x simply corresponds to moving outside of the unit circle, ρ = 1.
Let us focus on the analytic continuation of Eq. (III.15). For convenience, we will analytically continue x from the first sheet to the second sheet, x → x II → 0, withx → 0 on the first sheet. In terms of Eq. (C.23), this corresponds to taking ρ → ∞ andρ → 0. That is, in continuing x from first sheet to second sheet, x → x II → 0, The analytically continued ECB becomes This can be converted via the inversion identity, Eq. (V.9), to a simpler representation amenable to expansions about x II = 0 andx = 0. The continued conformal block consists of four groups of terms, each with unique small x andx behavior and G Of these, only the first term has the desired dependence, Eq. (III.9). Therefore, an analytically continued ECB does not lead to the desired Minkowski conformal block. As another check on the fact that G (M ) (x,x) and G (E) (x,x) are related but not directly given via analytic continuation, it is instructive to carry out the following exercise by starting with our G (M ) as defined on the "second sheet" and analytically continued it back to the first-sheet and then compare with G (E) (x,x). This can again be done by using ρ-representation by first continuing from ρ >II = 0 to ρ ∞ . Next making use of inversion formula, Eq. (V.9), to bring it back to a representation amenable to an expansion around

Symmetric Treatment
Once the leading index γ b = 1 − is identified, it is possible to solve each expansion function g n (σ) iteratively. We will not do it here in general except for the case of d = 1. For d = 2 and d = 4, since explicit solutions are already known, we will instead demonstrate that they can be re-expressed in the symmetric form, Eq. (III.23).
Let us begin with Eq. (III.13) and Eq. (III.14). With d = 2 and consider the caseq > q, expanding hypergeometric functions leads to an expansion with a n and b m given by the standard coefficient functions. By re-grouping, this can be re-expressed as c p e pξ (C.33) with c p given by a finite sum of products a r b s , with r + s = n. Turning next to the case of d = 4 and consider again the caseq > q. One finds, c p e pξ .
(C. 34) with coefficients c n again given by a finite sum.
As an integral over the U 2 , the imaginary part of T in the s-channel physical region, it also defines an analytic function 53 of , regular for 2 ≤ Re . Eq. (D.2) also corresponds to the Mellin transform of U 2 with respect to x −1 . It follows that, for 0 < x < 1, U 2 (x, q) can be recovered via an inverse Mellin transform, With u( , q 2 ) bounded for Re → ∞, the full amplitude can be represented for −1 < x < 1 as Correspondingly, (D.3) defines a distribution with U 2 (x, q 2 ) = 0 for 1 < |x| < ∞. Eq. (D 1) can be considered as the starting point of a "primitive" tchannel OPE. Therefore the Mellin-representation for T 2 (x, q 2 ) corresponds to a Sommerfeld-Watson re-summation introduced earlier, with Eq. (D.3) providing the imaginary part in the s-channel scattering region, 0 < x < 1.

Reduction to d = 2
Let us turn next to a CFT description. For the 4-point correlator, the forward limit, t = 0, corresponds to integrating F (M ) (w, σ), Eq. (IV.4) over impact space. Because of conformal invariance, b 2 ⊥ enters only through σ, it follows that, from Eq. (II.15), the amplitude at t = 0, with fixed conformal virtualities, is a total derivative. The contribution at b = ∞ vanishes, and the total contribution becomes 54 53 We assume that U 2 (x, q) < O(x −1 ) at x = 0, consistent with the requirement of energy-momentum conservation, i.e., the u(2, q) integral is finite. It follows, for the inverse transform, Eq. (D.3), L 0 can be chosen so that 2 − ε < L 0 < 2, with ε infinitesimal. For a related discussion, see [119]. 54 We also mention, for DIS, we have Ws having symmetry (−1) s−1 . More generally, the reduction of one power of s reverses the symmetry pattern for each.
In addition, since Eq. (D.7) is related to a cross section, a positivity constraint applies. In the Regge limit of w → ∞ and σ 2 large, we can keep the leading order for χ α (w, σ 2 ; ).
It is now clear that, in treating DIS as a CFT scattering process, several essential steps must be followed. To extract the DIS cross section, it is necessary (1) to approach the forward limit of t = 0 and (2) to take the imaginary part of a 4-point CFT amplitude.

BFKL-DGLAP Equation
SO(2, 2) invariance for QCD can be illustrated by a joint-integral-differential equation of BFKL-DGLAP [46][47][48][49][50][51] and its solution. These lead to a ∆ − spectral-curve from which the effective spin can be extracted. The BFKL program [46,47] demonstrates the inter-relation between the scaling behavior in longitudinal boost: the effective spin and the anomalous dimensions control the scaling under dilatation for moments of DIS structure functions as we see below.
Let's us focus on the DIS structure U 2 , introduced above. The BFKL-DGLAP integrodiferential equation corresponds to summing dominant contributions to the DIS cross section in the multiperipheral region. This summation is based on a series of approximations-"leading-log", "k Tfactorization", etc.-and, in its integral form, it can be expressed as We will refer to this as the BFKL-DGLAP equation. A corresponding (Bethe-Salpeter) integral equation can also be written for the full amplitude, H, symbolically expressed as = Im H (1) and R = Im R. One advantage of working with U is the fact that the integration in Eq. (D.11) is over physical region only. The solution, formally expressed as U 2 (x, q 2 ) = n U 2 (n) (x, q 2 ), corresponds to summing dominant contributions to the DIS cross section in the multiperipheral region 56 . Each term in the sum, U 2 (n) (x), can be associated with the cross section for the production of n gluons. This simplification becomes particularly useful when we discuss SYK-like 1-d CFT models in Sec. V B.
Approximate boost and dilatation invariance are reflected by the fact that the kernel is a function of x/x and q/q for x, x << 1 and q 0 << q, q . If one extends this to the whole physical range, the integral equation can be solved by a double 56 We have made use of gluon-dominance by dropping quark-contributions as well as other technical simplification in order to bring the equation into a manageable form. Therefore, Eq. (D.11) should be interpreted as a schematic representation. For more realistic discussion, see [46-51, 120, 121] Mellin-Fourier transform with respect to x −1 , as in Eq. (D.3), and η = log q respectively, leading to (D.13) The simplest approximation has R(x, x , q 2 , q 2 ) → R 0 (x/x ) constant, leading to u( , ν) = With u( , q) having a pole at ef f = 1 + λ, this leads to Regge behavior at small-x, (D.14) Furthermore, U 2 is q-independent, corresponding to naive Bjorken scaling, with vanishing anomalous dimension, γ n = 0. In a more refined treatment by taking into account q 2 -dependence properly, one finds [120,121] u( , ν) r( ) ν + iG( , ν) + regular terms (D. 15) Here we focus on the singularities in the lower-half ν-plane, appropriate for q large. At = n, u(n, ν) has a pole at ν = −iγ n , where the anomalous dimension is obtained by solving: γ n = G(n, iγ n ). There could multiple solutions and we will retain only the lowest solution for each . This leads to DGLAP-like evolution equation for M n (q) with n = 2, 4, · · · , − dM n (q) d log q γ n M n (q) → M n (q) ∼ q −γn (D.16) as q → ∞. This characterizes the dilatation symmetry as realized in DIS. In particular, γ 2 = 0, due to energy-momentum conservation.
Since enters as a continuous parameter, it is possible to consider γ( ) as an analytic function of . This defines a spectral curve, ∆( ) ≡ γ( ) − − d/2. If one shifts ν by i , one can formally express the singular part of Eq. (D.15) as u( , ν) r ( ) where ν → ν − i and ∆ = ∆ − 2. Note that we have endowed f ( , ν) certain analyticity and symmetry structure, similar to that in Eq. (IV.7). In particular, due to conformal invariance, after inversion to (∆), one has d (∆) d∆ = 0 at ∆ = 2. It follows that the spectral curve ∆( ) has a squareroot branch point at ef f , which can be found by solving ∆( ef f ) = 2.
(D. 17) Due to the presence of this singularity, one finds In weak coupling, one generically has [46][47][48]  can be found directly by the Wronskian method 57 . The desired Green's function is simply given by where ν + = −1/2 + m 2 + 1/4 and, as usual, w < = (w, w ) min and w > = (w, w ) max . P ν (w) and Q ν (w) are Legendre functions of the first and second kind discussed earlier.
It is also instructive to arrive at the same answer by a spectral analysis to illustrate the noncompactness involved. Consider first m 2 = 0. The eigenvalue problem can be expressed as with λ > 0. Eigenfunctions at w ∼ +∞ are oscillatory in ξ = cosh −1 w, P (w) ∼ e i±kξ , with wave-number k = λ − 1/4. With eigenfunctions also bounded at w = 1, the spectrum for −D w is positive and continuous, with λ = k 2 + 1/4, 0 < k < ∞. The corresponding eigenfunctions are Legendre functions, P −1/2+ik (w) = P −1/2−ik (w). With index taking on values −1/2 + ik, these are also known as toroidal or ring functions. It is worth noting that our analysis is comparable to that done in [35,37] in treating SKY model, but yet differs in a significant detail. In [37], the Hilbert space deals with functions, in terms of variable τ , Eq. (V.2), defined over (0, 2). (The range is extended to (−∞, ∞) by symmetry.) The range (0, 2) in τ corresponds to 0 < w < ∞. In our treatment, by restricting w to the range (1, ∞), the spectrum for D w is strictly positive and continuous. There is no accompanying discrete spectrum involved. Our ability to take advantage of this simplification is part due to our ability to deal with the absorptive part, Im Γ(w), for a scattering process. 57 Dw is the same as L 0,σ , Eq. (III.21), evaluated for d = 3, with solutions given by Legendre functions. A similar treatment can also be carried out for L 0,σ , d = 3, acting on reduced functionsf (w) = (w 2 − 1) (d−3)/2 f (w), leading to appropriate associated Legendre functions. We will treat this more general case elsewhere.

Mellin-Like Representation and Polynomial Boundedness
We now address the important case where F (w) is not square-integrable but polynomially bounded. This can be handled by either working with a reduced function, for exampleF (w) = w −L F (w), or adopting a deformed representation for Eq. (E.8). We will adopt the latter approach, which can be seen to correspond to a Sommerfeld-Watson resummation from the OPE context.
Let us now turn to functions which grow with w. To deal with functions which grow with w as a power, it is possible to enlarge the Hilbert space [91,93], and, for the class of functions which are polynomially bounded F (w) = O(w L0 ), the region of analyticity forf + ( ) gets pushed out to the right. In other words, L 0 < Re < ∞. It is possible to define f + ( ) as an analytic function by initially for L 0 < Re , and then analytically continue in the the region to the left of Re = L 0 . The function F (w) can be recovered by (E.14) This is precisely what we have arrived at earlier via Minkowski OPE analysis, (V.6). As also mentioned earlier, an equivalent representation, which we will make use of in Sec. V B, is Eq. (V.7). If f + ( ) contains a singularity at ef f where 1/2 < ef f < L, e.g., a pole, by pulling the contour in (E.14) to the left passing to pole, one finds F (w) diverges at w → ∞ as F (w) = O(w ef f −1 ) . (E.15)

AdS2/CF T1 Duality
In recent years, much work has been done to elucidate the duality between some string theory in AdS 2 and CFT in d = 1. Pure Einstein gravity in two dimensions has no propagating degrees of freedom, but if other fields are included in the theory, there can be interesting dynamics. Most of the interesting work has been aimed at resolving issues with the black hole information loss paradox. (For reviews see [122][123][124]) Maldacena [113] pointed out that d dimensional theories with eternal black holes in Anti-de Sitter space can have interesting consequences for the information loss paradox; the past boundary could be decomposed into two copies of the boundary CFT and initial states could be thought of as a thermal ensemble of two CFT states. The object of interest is often |E n CF T ⊗ E n | CF T , invariant, it was pointed out [125] that these transformations can affect the properties of entangled states as they fall into a black hole.
Of critical importance is to understand how information sent in from the boundary becomes "smeared" across the horizon as quanta falls into the black hole. Classical dynamics describes such a system as chaotic; a sensitive dependence upon initial conditions describes a situation in which final state information is "smeared" on a mathematically dense phase space. To this end, scrambling of infalling quanta can be defined via a maximal Lyapunov exponent, λ L . This can be determined by considering an out-of-time correlator 59 W † R (t)V † L (0)W L (t)V R (0) β ∼ 1 −αe λ L t + O(α) (E.17) for smallα. Here the index β indicates that it is a thermal correlation function, andα is a constant that encodes information. 60 The scrambling time can then be defined as the time when the exponential becomes order one, for an entropy S. The correlator Eq. (E.17) 61 probes chaotic behavior 62 . It can be calculated from the gravitational theory [29,30]. A key element to the calculation is that the operator W (t) can be seen to create a shock wave [116,117]. This shock wave can be interpreted as a boosting scattering particles by exp(2πt/β) to arbitrary high energies 63 . The result is that Eq. (E.17) can be described by high energy elastic eikonal scattering at fixed impact parameter b, similar to that described in [12,14,15,18,127,128] and shown in Eq. (II. 19).
Finally we emphasize that while the thermofield double approach, Eq. (E.16), to calculating Eq. (E.17) has been convenient in gravitational literature, it is not the only approach that can be used. 59 It was pointed out in [126] that this correlator can be considered a quantum variant on the Loschmidt echo. [126] then goes on to propose a cold-atom qubit set up that could be used to experimentally such correlations. 60α should be a function of the entropy. A common definition is forα ≡ 1/B, Where B is the number of bits of information. For holographic theories,α ∼ 1/N 2 . 61 Often in the literature a similar correlator is defined with different operator ordering and operators living on different initial state CFTs. These various 4 point correlators are all related via analytic continuation by continuing various operators by β/2 around the imaginary time thermal boundary S β . An example can be seen in Eq. (E.20). 62 It should be noted that this requires a chaotic system. For example, in integrable models, as in the d = 2 Ising model, the behavior of Eq. (E.17) will not be seen. [31] 63 In principle, for late enough times, this can boost particles to the string or Planck scales.
Consider a d = 1 CFT 64 . At finite temperature, a thermal correlator can be related to a vacuum correlator via a conformal transformation of the form f (t) =exp(2πt/β). The invariance of conformal correlation functions then leads to ...
× O 1 (f (t 1 ))O 2 (f (t 2 ))... β . (E. 19) a. SYK Theory Most of the discussion in this work applies generally to CFTs of arbitrary dimension. While these results are more general, without a specific theory the details of particular dynamic behavior can be hard to suss out. For integrable theories the canonical non-trivial example is that of the well known duality between string theory on AdS 5 × S 5 and N = 4 SY M , which is conjectured to be integrable 65 . For chaotic systems with black holes, holographic examples have been harder to come by.
In this vein, much recent attention has been given to the Sachdev-Ye-Kitaev (SYK) model, first proposed by Kitaev in a series of talks [32][33][34]. The details of this theory were quickly expanded upon in [35-38, 105-107, 130-135]. The boundary theory of this model is a many body system of Majorana fermions with an all-to-all four point interaction J . In the low temperature limit 66 the system is approximately conformal. The full 2pt function of the theory can be found using a Schwinger-Dyson equation. The four point function can found using a Bethe-Salpeter equation which involves a ladder like exchange process as in Eq. (V.14). In this theory, calculation of Eq. (E.17) leads to a Lyapunov exponent λ L = 2π/β with corrections coming from Regge string effects, longitudinal string spreading, and non-linear interactions. This calculation can be done from the perspective of the bulk string theory using the approach of [30]. The bulk process is described by an eikonal scattering who's dominant contribution can be traced to the BPST Pomeron. In both cases the ∆( ) spectral curve plays an important role as the conformal weight of the dominant OPE contribution for the conformal theory and as the Virasoro operator dimension for the bulk theory.
We should take a moment to emphasize that both of the above approaches in the literature follow one path: define the correlation function in the euclidean limit, calculate the correlator, then carefully analytically continue to the Minkowski region by adding an imaginary piece to the Euclidean time 67 . For a finite temperature conformal transformation f (t) =exp(2πt/β) as in Eq. (E. 19) this process can be outlined as correlator:

(E.20)
In the end we arrive at the time ordered process we are interested in in this paper (1,3)→(2,4) as in Eq. (I.2). The advantage of our approach is that, by taking advantage of boundary conditions, one can directly write down the Minkowski solution without having to carefully do the analytic continuation. 64 A similar procedure exists for d = 2 CFTs as described in [31]. Here a separate holomorphic and antiholomorphic conformal transformation exist for the two degrees of freedom. 65 This is backed up by a large library of literature. In the large N limit, some sectors of the theory have been shown to be exactly integrable. For a comprehensive review see [129]. 66 An effective coupling λ = βJ interpolations between a holographic λ >> 1 limit where the theory is nearly conformal and a thermal λ << 1 limit where the conformal symmetry is broken. 67 The type of analytic continuation was first spelled out in great detail in [14,15,18,127].