Four point functions in CFT’s with slightly broken higher spin symmetry

We compute spinning four point functions in the quasi-fermionic three dimensional conformal field theory with slightly broken higher spin symmetry at finite t’Hooft coupling. More concretely, we obtain a formula for jsj0˜j0˜j0˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\langle {j}_s{j}_{\tilde{0}}{j}_{\tilde{0}}{j}_{\tilde{0}}\right\rangle $$\end{document}, where js is a higher spin current and j0˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {j}_{\tilde{0}} $$\end{document} is the scalar single trace operator. Our procedure consists in writing a plausible ansatz in Mellin space and using crossing, pseudo-conservation and Regge boundedness to fix all undetermined coefficients. Our method can potentially be generalised to compute all spinning four point functions in these theories.


Introduction and summary of results
The dualities between conformal field theories and higher spin gravity theories in AdS are one of the most intriguing topics in the AdS/CFT correspondence. Potentially, these dualities should allow for an improved understanding of the AdS/CFT correspondence, since both sides of the duality are simple, at least when compared to the more standard case of N = 4 SYM and type IIB superstring theory. 1 Of particular interest are CFT's with slightly broken higher spin symmetry, that were studied most notably in the paper by

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Maldacena and Zhiboedov [5], where all three point functions of single trace operators at the planar level were computed at finite t'Hooft coupling. In our paper, we compute some four point functions of spinning single trace operators at the planar level at finite t'Hooft coupling. The formulas we obtain are very simple and our formalism, which is based on pure CFT arguments in which Mellin space plays an important role, potentially paves the way for the computation of all spinning four point functions.
CFT's with slightly broken higher spin symmetry are large N CFT's where higher spin symmetry is broken by 1/N effects. There are two such theories, the quasi-boson theory and the quasi-fermion theory, which are defined in 3 dimensions. We will focus on the quasi-fermion theory. This theory depends on two parameters,Ñ andλ (we follow the notation of [5]). We will study the theory at the planar level, i.e. at leading order inÑ . In that case the theory interpolates between the free fermion theory atλ = 0 and the critical point of the O(N ) model (critical boson) atλ = ∞.
Being a large N theory, the spectrum of the quasi-fermion theory organises into single and multitrace primary operators. Let us describe the single trace operators. There is one single trace operator for each even spin s = 0, 2, . . .. The scalar primary, which we will denote by j0, has dimension 2 + O 1 N [6]. The spin 2 primary j 2 is exactly conserved. A higher spin primary j s of spin s > 2 has dimension s+1 and acquires anomalous dimensions of O 1 N [7,8]. This theory is believed to be solvable in the planar limit. In [5] three point functions of single trace operators were computed at the planar level and for finiteλ through the use of slightly broken higher spin Ward identities. 2 In [10] four point functions of scalar operators were computed using the Lorentzian inversion formula and Schwinger-Dyson equations. In [11] the four point function j 2 j0j0j0 was computed using the pseudo-conservation equations. 3 We obtain a formula for j s j0j0j0 for generic spin s ≥ 4: where j s j0j0j0 f f is the correlator in the free fermion theory (which is fully known) and j s j0j0j0 cb is the corresponding correlator in the critical boson theory. The critical boson theory is the IR fixed point of the theory ofÑ free real scalar fields perturbed by (φ i φ i ) 2 . This result agrees with the 3d bosonization picture advanced in [5], where it is proposed that the quasi-fermionic theory interpolates between a tridimensional theory ofÑ free fermions and the critical theory ofÑ bosons, in the limitsλ → 0 andλ → ∞ respectively.
We obtain that where V (i; j, k) is a conformal structure (see (2.3)) and u and v are the usual conformal cross ratios. M (γ 12 , γ 14 ; s, k) is equal to M (γ 12 , γ 14 ; s, k) = Γ(−k + γ 14 − 1)Γ −k + γ 14 where p(γ 12 , γ 14 ; s, k) is a polynomial in γ 12 and γ 14 . This polynomial is fully determined by crossing, pseudo-conservation and Regge boundedness, see equations (2.8) and (2.9), see (2.11) and see also (3.16), (3.17) and (3.18). We explain in section 2 how formula (1.1) solves the crossing and pseudo-conservation equations and correctly accounts for the exchange of single trace operators with the OPE coefficients derived in [5]. In section 3 we show that formula (1.1) is the unique solution to the pseudo-conservation and crossing equations which is consistent with the bound on chaos. In particular we analyse AdS contact diagrams for j s j0j0j0 and we conclude that such diagrams violate the bound on chaos, provided s ≥ 4. In section 4 we discuss open directions. In appendix A we study the bulk point limit of j s j0j0j0 . In appendix B we calculate j s j0j0j0 in position space for spins s = 2, . . . , 14. This calculation agrees with the Mellin space result. In appendix C we recompute j 2 j0j0j0 by solving the higher spin Ward identities.

The bootstrap of j s j0j0j0
We will compute j s j0j0j0 . Let us start by examining theÑ andλ dependence. It is expected that the quasi-fermion theory interpolates between a theory ofÑ free fermions atλ = 0 and the critical boson theory atλ = ∞.
We will work in a normalization where j s j s ∼ 1, i.e. two point functions of single trace operators do not depend onÑ orλ. We use the ∼ sign to mean that we do not keep track of numerical factors, but we do keep track of theÑ andλ dependence. Thus, j s j0j0j0 ∼ 1 N . At this order, we can only have exchanges of single trace operators or double trace operators [j0, j0] or [j s , j0].
Let us consider exchanges of single trace operators. The relevant three point functions are j s j0j s and j s j0j0 , with s ≥ 2. Note that j0j0j0 = 0 [5]. From [5] we see that j s j0j0 ∼ 1 √Ñ . There are two possible structures for j s j0j s , the fermion and the odd structure. We have that j s j0j s fermion ∼ 1 √Ñ√ 1+λ 2 and j s j0j s odd ∼λ √Ñ√ Based on this we propose the following ansatz where j s j0j0j0 f f is the four point function in the free fermion theory, whose form can be read in [13]. To the best of our knowledge, j s j0j0j0 cb has not yet been computed and it will be the subject of this section to do precisely that. We attached the subscript cb since it is expected that it corresponds to a four point function in the critical boson theory. The reader might be confused about the factor of 1 N . In our normalization, Given that the quasifermionic theory interpolates between a theory ofÑ free fermions and the critical theory ofÑ bosons, the reader might be confused about why there is a factor of 1 N . There are two things happening in this context. First, when we write j s j0j0j0 f f and j s j0j0j0 cb we have decided to factor out the dependence onÑ . Second, the reader might thus have expected to encounter j s j0j0j0 ∼Ñ , but this is only true when two point functions are normalized such that j s j s ∼ j0j0 ∼Ñ , whereas we are using different normalizations, namely j s j s ∼ j0j0 ∼ 1. This justifies why does j s j0j0j0 ∼ 1 N . We can write parity even and parity odd structures for the correlator j s j0j0j0 . The parity odd structures are realised in the free fermion theory. This is because j0 is parity odd in the free fermion theory. The parity even structures are realised in the quasi-boson theory. Thus, we write where V (i; j, k) is a conformal structure which is given in embedding space [14] by is a function of the distances between the points, with appropriate weights on each of the points. We find it advantageous to consider the Mellin representation Eq. (2.2) can be rewritten as We will callM (γ 12 , γ 14 ; s, k) the Mellin amplitude. 4 4 Spinning Mellin amplitudes are analysed in [15][16][17]. The definitions slightly differ among these works, but at least concerning conformal four point functions the basic idea is to decompose the correlator in a basis of spinning structures and take the Mellin transform with respect to each function of the positions multiplying each structure. Up to now all works use the embedding space formalism, which has the serious drawback of involving many degeneracies for arbitrary spinning correlators. For generic spinning correlators, we think it would be interesting to define Mellin amplitudes with the conformal frame formalism [18,19], which does not have the problem of degeneracies. We think that it is an interesting problem to work out the poles and residues of the Mellin amplitude for spinning correlators using the conformal frame formalism.

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The location of the poles of the Mellin amplitude is related to the operator product expansion of the external operators. Let us make this point explicitly. Consider two external operators O 1 , O 2 of dimensions ∆ 1 , ∆ 2 and spins s 1 , s 2 and suppose they exchange an operator of dimension ∆ and spin s. Then the most singular term in the lightcone OPE is where τ = ∆ − s. From this logic we expect the Mellin amplitude to have poles at where γ 13 = 2s + 1 − γ 12 − γ 14 . The Γ functions contain all the poles implied by the OPE. For this reason we assume that p(γ 12 , γ 14 ; s, k) is a polynomial in the Mellin variables.
The bound on chaos [20] bounds the degree of the polynomial p(γ 12 , γ 14 ; s, k). This is worked out in section 3, see (3.16), (3.17) and (3.18) for the precise formulas. Furthermore, j s j0j0j0 is constrained by invariance under interchange of points 2 ↔ 3 and 2 ↔ 4. This crossing symmetry implies the equations j s j0j0j0 is constrained by pseudoconservation of j s . We implement this condition in embedding space. The differential operator for conservation is ∂ Since ∂ · j s is a primary operator of spin s − 1 and dimension s + 2, then ∂ · j s j0j0j0 is a conformal four point function of primary operators. ∂ · j s j0j0j0 factorizes into products of a two point function times a three point function. Such a four point function is made up of powers of u and of v and so its Mellin amplitude vanishes.
Four point functions of scalars with vanishing Mellin amplitudes were analysed in [21], see in particular section E.E.1. A similar analysis can be performed for the spinning case, though we will not pursue it here. The important conclusion is that in Mellin space pseudoconservation is the same as conservation. In other words, ∂·j s j0j0j0 has a vanishing Mellin amplitude.

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Pseudoconservation implies the equation (2.11) The coefficients are written in the appendix D, see formula (D.1). The crossing equations (2.8), (2.9), the pseudoconservation equation (2.11) and Regge boundedness (3.16), (3.17) and (3.18) determine p(γ 12 , γ 14 ; s, k) up to a multiplicative constant. This has to do with the fact that we have not picked a normalization for the higher spin current j s . It is simple to solve this set of equations in a computer algebra system for each spin s. We find that the solution always has the form
We write this correlator in position space in appendix D, see formula (D.2). In appendix B we implement an algorithm to compute j s j0j0j0 in position space. We managed to determine j s j0j0j0 in position space for spins 2, . . . , 14 using this algorithm. We write the formulas for the correlators in position space in an accompanying notebook. Taking the Mellin transform we get precisely the same as we get with the procedure in Mellin space. The advantage of Mellin space is that it allows to write equations (2.8), (2.9) and (2.11) that determine the solution for generic s.
Let us mention some checks on our solution. One such check is compatibility of the pseudo-conservation equations with conformal symmetry. ∂ · j s is a conformal primary at leading order in 1 N . ∂ · j s can have contributions coming from [j s 1 , j0] and [j s 1 , j s 2 ]. Only

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the former matter since we are interested in j s j0j0j0 . More precisely, The coefficients c m are fixed by conformal symmetry (see formula (B.7)). When we run our algorithm in position space we do not need to input the values of c m , we prefer to keep them unknown. It turns out that our algorithm fixes c m in agreement with (B.7). This is an important check on our results. We also checked that the short distance limit of our expression for j s j0j0j0 cb agrees with the correct three point structures for the exchange of higher spin currents. Let us take s = 4 for concreteness. The short distance limit u → 0 captures the exchange of the higher spin currents in the s-channel. If afterwards we take v → 1, we find that the correlator behaves as The ∼ sign means that we just keep track of the conformal structure that appears, but we do not keep track of numerical coefficients. Eq. (2.16) is matched by the behaviour of conformal blocks of higher spin currents in the same limit. Formula (1.1) correctly accounts for the exchange of single trace operators in j s j0j0j0 . However, it is not obvious that it correctly accounts for the exchange of double trace operators. Indeed, one can imagine adding to (1.1) AdS contact diagrams, which are solutions to crossing that only involve the exchange of double trace operators. By taking linear combinations of AdS contact diagrams one can furthermore obtain solutions to the conservation equations. However, in the next section we consider such linear combinations and show that they always violate the bound on chaos. For this reason, it is not legal to add them to (1.1).

Bound on chaos for j s j0j0j0
The bound on chaos [22] constrains the Regge limit of j s j0j0j0 . In this section we review the bound on chaos and derive its consequences for j s j0j0j0 . There are two possible structures one can write for j s j0j0j0 . One structure involves the tensor and the other one does not. We examine the two cases separately in sections (3.2) and (3.4) and derive bounds on the Regge growth of the Mellin amplitude for both of these cases.
Solutions to crossing that only involve the exchange of double twist operators are given by AdS contact diagrams. This was proven in [23], for the special case of four point functions of external scalars. We will assume that such a result holds for any n-point function of spinning conformal primaries. We study AdS contact diagrams in sections (3.3) and (3.4). Our main conclusion is that AdS contact diagrams for j s j0j0j0 are incompatible with the bound on chaos, provided s ≥ 4. For s = 2 we construct the contact diagrams that are compatible with the bound on chaos, see formulas (3.37) and (3.52). This completes the proof of formula (1.1).

Review of the bound on chaos and Rindler positivity
Conformal field theories are constrained by the Regge behaviour of Lorentzian correlators. For nonperturbative CFT's, correlators in the Regge limit are bounded by the Euclidean OPE in the first sheet. For large N CFT's one needs to use the bound on chaos to bound correlators in the Regge limit. In this subsection we review the bound on chaos [22].
We will consider the following kinematics for a four point function, in which we set all four points on the same plane (x ± = t ± x) The bound on chaos applies for systems at finite temperature with a large number of degrees of freedom. For the case of a large N conformal field theory, a correlation function of single trace primaries V ( where the Lyapunov exponent λ L obeys the bound λ L ≤ 2πT , where T is the temperature of the system. The proportionality constant α does not depend on t. The bound on chaos can be applied to large N CFT's in Minkowski space, in which case we should consider the temperature T = 1 2π of the Rindler horizon. We cannot apply directly (3.2) to j s j0j0j0 . However, we can use Rindler positivity [24] to bound j s j0j0j0 by j s j s j0j0 and j0j0j0j0 and use the bound on chaos to bound the latter two quantities, as we will explain next.
The Rindler conjugateŌ of an operator O is defined asŌ µ,ν..
where A and B are operators (that might be composite) defined on a single Rindler wedge.
. Then, the time-ordered correlation function in the configuration (3.1) is given by The bound on chaos on the r.h.s. of the previous expression implies a bound on j s j0j0j0 .

Consequences for j s j0j0j0 cb
Let us work out the consequences of the bound on chaos for the Mellin amplitudes of j s j0j0j0 . In the critical boson theory, where V (i; j, k) was defined in (2.3) and We call M (γ 12 , γ 14 ; s, k) a Mellin amplitude. The arguments of the Γ functions are just the Mellin variables defined in (2.4).
In the limit t → ∞ of the kinematics (3.1), the conformal cross-ratio v acquires a monodromy v → ve 2πi . Furthermore (3.8)

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The polynomial growth of the Mellin amplitude is related to the Regge limit, in a manner that we explain next, following appendix C of [25]. Let us consider the limit The factor e −2πiγ 14 becomes very large in the regime γ 14 → i∞. This is cancelled by the exponential decay of the Γ functions. Let us suppose that the Mellin amplitude grows polynomially as γ , when γ 14 is large and imaginary and γ 12 is fixed. In this regime we can rewrite (3.9) as where M 1 is an irrelevant large number. If we substitute m 1 → m 1 σ we get that the integral (3.9) scales like σ 1−∆ 1 −∆ 2 −α(s,k) . In order to compare (3.6) with (3.5), we should furthermore take into account the prefactor and the structures in (3.6), which scale with σ.
We can use the crossing symmetry equationŝ M (γ 12 , γ 14 ; s, k) =M (γ 14 , γ 12 ; s, s − k). (3.12) to derive the following bounds on the polynomial growth of the Mellin amplitude We can apply these bounds to the ansatz (2.7). We conclude that The solution that we found respects this bound.

The Regge limit of AdS contact diagrams for the parity even structure in j s j0j0j0
We will study the Regge limit of a generic AdS contact diagram for j s j0j0j0 (see figure 2), using the methods of [26]. We use vectors P i and Z i in embedding space to describe the position and polarization vectors of an operator O i defined on the boundary of AdS. For tensor fields defined on the bulk of AdS, we use vectors X i and W i to denote the position and the polarization. The following identities are obeyed: We denote the bulk to boundary propagator of a dimension ∆ and spin J field by Π ∆,J (X, P ; W, Z). Its formula is where C ∆,J is a proportionality constant (whose value will not be relevant for us). An important class of contact diagrams contributing to the parity even structure in j s j0j0j0 is given by where s 1 = s 2 + s 3 . There are other contact diagrams one can write by contracting more derivatives among the propagators, but such diagrams will diverge more in the Regge limit, which is the issue we wish to discuss here. The covariant derivative is given by

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The operator K is given by where for our purposes d = 3.
The following identity . is useful for us. D ij is an operator that only acts on the external points. It increases the spin at position i by 1 and it decreases the conformal dimension at position j by 1.
is a constant of proportionality, which will not be relevant for us. The precise definition of D ij is We confirmed the identity (3.24) for a few values of the external spins using Mathematica. So, with the help of identity (3.24) we can perform the integration in (3.21) using only scalar propagators and afterwards we act with the differential operators D 12 and D 13 . The AdS integral with only scalar propagators corresponds to a contact quartic scalar diagram, whose Mellin amplitude is a constant. Afterwards we act with the differential operators and obtain an expression in the form of (2.5).
Let us exemplify what we mean for the case of j 2 j0j0j0 . Let us take s 2 = 1 and s 3 = 1 in (3.21). Up to a proportionality constant, the contact diagram is given by x 34 where the ∼ symbol means that we neglected a numerical factor. We now act with the differential operators D 12 and D 13 and reorganise the result into the form (3.6), (3.7). 5 For this contact diagram, we conclude that M (γ 12 , γ 14 , s = 2, k = 2) = γ 12 (3 − 8γ 12 + 4γ 2 12 ) (−4 + γ 12 + γ 14 )

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This contact diagram obeys the chaos bounds (3.13), (3.14) and (3.15). We found that contact diagrams of the type (3.21) obey the bound on chaos for spin 2, but violate the bound on chaos for spin s ≥ 4.
Our goal is to investigate if there are extra solutions to crossing, conservation and Regge boundedness for j s j0j0j0 . AdS contact diagrams are solutions to the crossing equations, however they are not necessarily conserved, nor Regge bounded. To see that contact diagrams are not necessarily conserved, let us consider a generic contact diagram where we denoted by J(X, P i , W, Z i ) the dependence on the other AdS fields. It turns out that the action of the conservation operator (2.10) on Π ∆=s+1,s gives a pure gauge expression This vanishes only if J(X, P i , K, Z i ) is conserved in the bulk of AdS, i.e. a contact diagram involving a bulk to boundary propagator is conserved only when the bulk to boundary propagator is coupled to a conserved current. Clearly, this is not the case for a generic contact diagram (3.21). So, we consider instead linear combinations of AdS contact diagrams. The most economical way of doing this is to notice that the Mellin transform of any contact diagram, or any linear combination of contact diagrams, can be written aŝ where p dt (γ 12 , γ 14 ; s, k) is a polynomial. Let us explain this important formula. If we act with the differential operators on the scalar contact diagram, they will shift the arguments of the Γ functions by integers. So, the Mellin transform of an AdS contact diagram will involve 6 Γ functions times a polynomial. The arguments of the Γ functions are related to the operators that appear in the OPE of the external operators. Thus, we arrive at (3.33). Notice that p dt (γ 12 , γ 14 ; s, k) will eventually have zeros.

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The chaos bound for p dt (γ 12 , γ 14  More explicitly, the spin 2 parity even contact term in position space is given by where p dt (γ 12 , γ 14 ; s = 2, k) is given by (3.37). 6 There is some arbitrariness in the choice of the contour. What is important is that it passes to the right of the poles in γ12, γ13 and γ14, see [21].

The Regge limit of AdS contact diagrams for the parity odd structure in j s j0j0j0
The parity odd structure is We define the Mellin amplitude M odd (γ 12 , γ 14 ; s, k) in the following manner The following equations encapsulate crossing symmetry: Let us use the bound on chaos to derive a bound on the polynomial growth of the Mellin amplitude. Let us define the exponent α(s; k) such that lim β→∞ M (γ 12 , βγ 14 ; s, k) ∼ β α(s;k) . In the Regge limit, the Mellin integral goes as σ −2s−3−α(s;k) . The prefactor times the structure goes as σ 3+2s−k . So, (3.41) behaves as σ −k−α(s;k) . By comparing with the bound on chaos (3.5) and using (3.43), (3.44) we conclude that The Mellin amplitude of an AdS contact diagram of the type (3.41), or of a linear combination of contact diagrams, is given bŷ
However, for s = 2 there is one solution that respects the bound on chaos. This solution is

Open directions
The methods developed in this paper potentially pave the way to compute all four point functions in conformal field theories with slightly broken higher spin symmetry. We believe that the next steps in this program are the following: 1. Compute j s j 0 j 0 j 0 in the quasi-boson theory. The conformal structures involved are the same as in this paper, so the calculation should be very similar.

2.
Demonstrate that AdS contact diagrams are not present in j0j0j0j0 and j 2 j0j0j0 in the quasi-fermion theory using pure CFT arguments. The chaos bound allows for contact diagrams in j0j0j0j0 and j 2 j0j0j0 . Their absence for j0j0j0j0 was demonstrated in [10] using Feynman diagrams. It should be possible to give a pure CFT demonstration of this fact. The idea is to write down the higher spin Ward identity that connects j0j0j0j0 and j 2 j0j0j0 , plug the AdS contact diagrams multiplied by arbitrary functions of the t'Hooft coupling and obtain that the only way for the Ward identity to be satisfied is if such functions vanish.
Let us mention some more ambitious problems: 1. Develop a code that computes all spinning four point functions in CFT's with slightly broken higher spin symmetry. Such a code should: • generate the structures involved for a given four point function • generate an ansatz for the Mellin transform, which should be a product of 6 Gamma functions (whose arguments are determined by the lightcone OPE, which is known) times polynomials • impose crossing, pseudo-conservation and Regge boundedness to fix all the undetermined coefficients in the polynomials.

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What differs from what we did here is that for generic spins we should not use embedding space, since the conformal structures in embedding space are generically linearly dependent on each other. It is best to use conformal frame techniques instead. Concretely, one would need the 3 dimensional version of [19] (see also [18]).

2.
Demonstrate that AdS contact diagrams are not present in four point functions in CFT's with slightly broken higher spin symmetry. As above, the hurdle should be in adapting our formalism to use the 3d conformal frame.
Recently, a new formalism for correlators of conserved currents was proposed in [27]. The idea is to write the conformal structures in a helicity basis. It would be very interesting to apply this idea to correlators in CFT's with slightly broken higher spin symmetry.
Ultimately, one would like to understand higher spin symmetry from the point of view of the bulk of AdS. We hope that our CFT computations can be of some utility for this ultimate goal.

Acknowledgments
I am very grateful to Joao Penedones and Alexander Zhiboedov for suggesting me this problem and for all the help they provided me. All the errors are of course mine. Furthermore, I would like to thank discussions with Aditya Hebbar, Evgeny Skvortsov, Subham Chowdhury and Vasco Gonçalves. This work was partially supported by a grant from the Simons Foundation (Simons Collaboration on the Nonperturbative Bootstrap: 488649) and by the Swiss National Science Foundation through the project 200021-169132. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement number 949077).

A Bulk point limit
Correlation functions of conformal field theories in Lorentzian signature may diverge even when none of the distances between the points vanish. At the moment a full classification of the singularity structure of correlation functions in conformal field theories does not exist.
One such singularity is the so called "bulk point singularity". In terms of cross ratios, we can obtain such a singularity in the following manner. In Lorentzian signature z andz are independent real numbers. The four point function has branch points. When z andz go around the branch points the four point function may develop a divergence when z =z. More specifically, suppose z goes around the branch point at 1,z goes around ∞ and now take z →z. We generically expect the four point function to diverge in this limit. A detailed examination of the bulk point limit for a four point function of equal scalars was carried out in [20].
In the bulk point limit a d dimensional conformal block where the external operators are scalars diverges as 1 (z−z) d−3 [20]. For this reason it is expected that a generic JHEP05(2021)097 nonperturbative four point function of scalars diverges as However, when the CFT has a local bulk dual, then we expect the divergence to be more severe. For example, a contact quartic diagram in AdS diverges as The plan for this section is the following. In A.1 we calculate the bulk point singularity of an AdS contact diagram for a scalar four point function of unequal primaries. The result is a trivial generalisation of (A.2), however to our knowledge its derivation had not appeared before in the literature. We need such a result in order to calculate the bulk point singularity of an AdS contact diagram for j s j0j0j0 , which we do in appendix A.2. Finally, in appendix A.3 we calculate the expected bulk point divergence of j s j0j0j0 in CFT's with slightly broken higher spin symmetry. We assume that j s j0j0j0 does not diverge more than conformal blocks in the bulk point limit. We conclude that AdS contact diagrams diverge more severely in the bulk point limit than what is expected for j s j0j0j0 for s ≥ 2 in CFT's with slightly broken higher spin symmetry. Thus, bulk point softness implies that we cannot add AdS contact diagrams to the solution to the pseudo-conservation equations that we found in section 2.
Let us add a caveat. Our result for j s j0j0j0 does not rely on assuming bulk point softness and is independent of it. Nevertheless, we choose to keep this appendix, because it was useful for us to think in terms of the bulk point limit in the early stages of our work, and maybe this can be of use to someone else.

A.1 Bulk point singularity of an AdS contact diagram for a scalar four point function of unequal primaries
A quartic contact diagram has a Mellin amplitude equal to 1. We will use this to compute the bulk point divergence, proceeding similarly to section 7.5.1 in [21]. Upon analytic continuation, the diagram is given by where a ij = 2(∆ i + ∆ j ) − k ∆ k and γ 13 = ∆ 1 − γ 12 − γ 14 . The integral diverges when γ 12 and γ 14 have a very big and positive imaginary part. We can use Stirling's approximation

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for the Γ functions. Indeed suppose we take γ 12 = isβ and γ 14 = is (1 − β). Then for very large s we have where f (β) is a function of β that will not play any role. The integral has a saddle point for In that case the exponential dependence of the integrand becomes e is ( The integral in β is Gaussian and can be readily evaluated. Furthermore, the phase is stationary when (A.5)

A.2 Bulk point singularity of AdS contact diagrams for j s j0j0j0
Identity (3.24) allows us to obtain spinning contact AdS diagrams from scalar contact AdS diagrams. So, with the help of identity (3.24) we can perform the integration in (3.21) using only scalar propagators and afterwards we act with the differential operators D 12 and D 13 . The scalar propagators cause a divergence like 1 (z−z) i ∆ i −3+s , see formula (A.5). After acting with the differential operators, we find that the bulk point divergence of the integral (3.21) is

A.3 Bulk point singularity of j s j0j0j0 in CFT's with slightly broken higher spin symmetry
Conformal field theories with slightly broken higher spin symmetry have an infinite number of light single trace operators. For this reason, they are not expected to be dual to a local theory in AdS. Thus, their bulk point singularity should not be enhanced with respect to that of an individual conformal block. We want to calculate the bulk point divergence of j s j0j0j0 . For our discussion, it is useful to introduce the operator where we used embedding space coordinates [14]. This operator acts on conformal blocks where the operator exchanged is symmetric and traceless. It increases the spin of the operator in position 1 by 1 and it decreases its conformal dimension by 1 also. It turns out that d s 11 (z −z) a ∼ (z −z) a−2s , i.e. the action of d s 11 increases the divergence by a power of 2s. For this reason, we expect the divergence of j s j0j0j0 to be

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since the scalar conformal block diverges logarithmically. We could have picked other differential operators than d 11 to create spin from the scalar conformal block. Since such operators only contain first derivatives of P i (and not higher derivatives), they lead to the same divergence (A.7).

B Algorithm for computing j s j0j0j0 in position space
We will implement an algorithm in position space to calculate j s j0j0j0 cb . The results match with the Mellin space calculation. j s j0j0j0 cb is constrained by conformal symmetry, crossing, consistency with OPE and the pseudo-conservation equation that j s obeys. Conformal symmetry implies that x 2 jk and we use the notation ( The indices are symmetric and traceless. f j (u, v) is a function of the cross ratios not determined by conformal symmetry. We write the following ansatz.
where c n j ,m j are parameters that will be fixed by crossing and the pseudo-conservation equation. The values of a(j), b(j), M (j) and N (j) will follow from consistency with the operator product expansion. Let us motivate the preceding ansatz. The spinning four point functions are related to the scalar four point functions by slightly broken higher spin Ward identities. The scalar four point function is a linear combination of powers of u and of v. So, it is natural that f j (u, v) is made up of powers of u and of v.
We will see below that the contribution to the operator product expansion of a certain operator goes as ∼ u τ 2 , where τ is the twist, which is defined as the conformal dimension minus the spin. Since all operator dimensions are integers, it is natural that the ansatz involves semi-integer powers of u and of v. The denominator For j s j0j0j0 the primary operators exchanged can have twist 1 (higher spin currents), 3 + 2n (double traces [j s , j0]) and 4 + 2n (double traces [j0, j0]), where n is a nonnegative integer. There is no primary operator of twist 2 being exchanged. This is an important condition that we impose in our algorithm.
More explicitly The final ingredient is compatibility with pseudo-conservation. ∂ · j s can have contributions coming from [j s 1 , j0] and [j s 1 , j s 2 ]. Only the former matter since we are interested in j s j0j0j0 . More precisely, Since the right-hand side must be a conformal primary, this implies [8] Thus ∂ · j s j0j0j0 is a linear combination of terms of type ∂ n 1 j0j0 ∂ n 2 j s 1 j0j0 . Crossing and compatibility with pseudo-conservation fix all coefficients in (B.3) up to a number. This number is related to the normalizaton of j s . In fact we did not even need to input formula (B.7), we kept the coefficients c m as unknowns and our algorithm correctly returns (B.7). This serves as a check on our results. We checked that the algorithm fixes the solution for s = 2, . . . , 14. Afterwards the computation becomes heavy for our laptop.

C Mixed Fourier transform
We will solve the higher spin Ward identities to compute j 2 j0j0j0 . This is a rederivation of the main result of [11]. Our method involves the use of a mixed Fourier transform, see [27] and [28].
We use the metric ds 2 = −dx − dx + + dy 2 . We will take all indices lowered and in the minus component. We will study the action of the charge

JHEP05(2021)097
on the four point function j0j0j0j0 . We make use of equations [5,10] ∂ · j 4 = αλ √Ñ 1 +λ 2 : ∂ − j0j 2 : − 2 5 : j0∂ − j 2 : , α, α 4 and β are numerical coefficients that can be obtained from solving Ward identities at the level of three point functions. 7 We will not need their precise value in what follows. The scalar four point function obeys the slightly broken spin 4 Ward identity where by . . . we mean the permutations (12), (13), (14). Note that where j0j0j0j0 f f denotes the connected piece in the free fermion theory and j0j0j0j0 disc denotes the disconnected piece. The disconnected piece obeys where we summed over all permutations. For this reason the disconnected piece drops out of (C.4). Using our ansatz (2.1) we conclude that From the Ward identities in the free fermion theory this becomes Using (C.2) in the right-hand side of (C.4) we get Ñ α 4 d 3 x ∂ · j 4 (x)j0j0j0j0 = αα 4λ 1 +λ 2 d 3 x ∂ − j0(x)j0 j 2 (x)j0j0j0 (C.9) We use the decomposition (2.1) to obtain that (C.9) is equal to j0(x)j0 ∂ − j 2 (x)j0j0j0 cb + . . .
We solved (C.11) and (C.12) using a mixed Fourier transform. We define the mixed Fourier transform of a four point function
The position space correlator j 4 j0j0j0 cb is given by (2.2), with