Conformal collider physics from the lightcone bootstrap

Abstract

Abstract

We analytically study the lightcone limit of the conformal bootstrap equations for 4-point functions containing global symmetry currents and the stress tensor in 3d CFTs. We show that the contribution of the stress tensor to the anomalous dimensions of large spin double-twist states is negative if and only if the conformal collider physics bounds are satisfied. In the context of AdS/CFT these results indicate a relation between the attractiveness of AdS gravity and positivity of the CFT energy flux. We also study the contribution of non-Abelian conserved currents to the anomalous dimensions of double- twist operators, corresponding to the gauge binding energy of 2-particle states in AdS. We show that the representation of the double-twist state determines the sign of the gauge binding energy if and only if the coefficients appearing in the current 3-point function satisfies a similar bound, which is equivalent to an upper bound on the charge flux asymmetry of the CFT.

Published for SISSA by Springer

Received: December 10, 2015

Accepted: February 1, 2016

Published: February 22, 2016

Daliang Li,a,b David Meltzera and David Polanda,c

aDepartment of Physics, Yale University,

New Haven, CT 06511, U.S.A.

bDepartment of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, U.S.A.

cSchool of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We analytically study the lightcone limit of the conformal bootstrap equations for 4-point functions containing global symmetry currents and the stress tensor in 3d CFTs. We show that the contribution of the stress tensor to the anomalous dimensions of large spin double-twist states is negative if and only if the conformal collider physics bounds are satised. In the context of AdS/CFT these results indicate a relation between the attractiveness of AdS gravity and positivity of the CFT energy ux. We also study the contribution of non-Abelian conserved currents to the anomalous dimensions of double-twist operators, corresponding to the gauge binding energy of 2-particle states in AdS. We show that the representation of the double-twist state determines the sign of the gauge binding energy if and only if the coe cients appearing in the current 3-point function satises a similar bound, which is equivalent to an upper bound on the charge ux asymmetry of the CFT.

Keywords: Conformal and W Symmetry, AdS-CFT Correspondence, Global Symmetries, Higher Spin Symmetry

ArXiv ePrint: 1511.08025

Open Access, c

The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP02(2016)143

Web End =10.1007/JHEP02(2016)143

Conformal collider physics from the lightcone bootstrap

JHEP02(2016)143

Contents

1 Introduction 1

2 Correlation functions 22.1 The embedding formalism 22.2 3-point functions 42.3 4-point functions 6

3 Lightcone bootstrap 83.1 Review: scalar 4-point functions 83.2 Spinning operators 113.3 Bootstrap with SU(N) adjoint operators 12

4 Mixed current-scalar 4-point functions 144.1 Identity matching 144.1.1 U(1) 144.1.2 SU(N) 154.2 Stress tensor and current matching 164.2.1 U(1) 164.2.2 SU(N) 18

5 Current 4-point functions 205.1 Identity matching 205.1.1 U(1) 205.1.2 SU(N) 225.2 Stress tensor and current matching 235.2.1 U(1) 235.2.2 SU(N) 245.3 Higher spin symmetry at large N 25

6 Mixed stress tensor-scalar 4-point functions 266.1 Identity matching 266.2 Stress tensor matching 27

7 Superconformal eld theories 28

8 Discussion 32

A Collinear conformal blocks at large spin 34

B Singularities in direct and crossed channel 34

C Degeneracy equations 36

i

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D 3-point functions and di erential operators 38D.1 hJi and hT i 38

D.2 hJJJi and hJJT i 39

D.3 Di erential operators for hT T T i and hT [T ]i 40

E Charge 1-point function 43

F Other technical details 44F.1 Change of basis for hT T T i 44

F.2 SU(N) adjoint crossing matrix 45

1 Introduction

The conformal bootstrap program [13] has seen a marked revival in recent years for CFTs in d > 2 spacetime dimensions. Using only unitarity and associativity of the operator product expansion (OPE), it was found in [4] that one could derive numerical bounds on the spectrum of an arbitrary CFT by studying the four point function of identical scalars. The numerical work has been extended greatly to include global symmetries [514], supersymmetry [1533], and correlation functions of non-identical scalars [34]. In particular, there has been remarkable progress in numerically solving the 3d Ising [3437] and critical O(N) models [9, 10]. Furthermore, it was found that there do exist limits where the conformal bootstrap equations can be solved analytically. The pertinent example for this paper is the lightcone limit, rst studied in [38, 39] and extended in [4048]. In this limit the expansion parameter is the twist of the exchanged operator, as opposed to its dimension. In a unitary, interacting CFT in d > 2 dimensions there exist a nite number of operators with very low twist, namely the identity operator, conserved operators, and possibly some low dimension scalars. This fact is crucial in solving the lightcone bootstrap equations.

With the exception of recent work on the four point function of 3d fermions [49], the bootstrap equations have primarily been studied for four external scalars. In this work we will focus instead on four point functions of 3d CFTs that include external conserved spin 1 and spin 2 operators, i.e. conserved currents J and the stress-energy tensor T.

The restriction to 3d is technical, as in this dimension all the relevant conformal blocks are known [50]. We will study these equations analytically in the lightcone limit and solve for the anomalous dimensions of a wide variety of double-twist states. To be specic, we will study the correlation functions hJJi, hJJJJi, and hT T i where J is a conserved

current for a global U(1) or SU(N) symmetry, T is the stress energy tensor, and is a scalar of arbitrary dimension. The s-channel conformal block expansion is dominated by the contribution from low twist operators such as the identity, the conserved currents J, the stress tensor T , and possibly some scalars with small dimensions. As in [38, 39], we show that large spin double-twist operators must exist to satisfy the crossing equations. Their anomalous dimensions are determined by the OPE coe cients of the lower twist

1

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operators exchanged in the s-channel. In an AdS dual description, the contributions to the anomalous dimensions from J and T correspond to the binding energies of well separated 2-particle states arising from gauge and gravitational interactions.

An important feature of our results will be that the contributions of J or T exchange to the anomalous dimensions of double-twist states ip signs if the relevant s-channel OPE coe cients do not lie between the free boson and free fermion values. For the exchange of T , this means requiring that gravity in the bulk be attractive yields the Hofman-Maldacena conformal collider physics bounds [51] on coe cients appearing in hJJT i and hT T T i (ex

tended to general dimensions in [52, 53]), which were originally discovered by requiring that the integrated energy ux produced by a localized perturbation in Minkowski space is positive.

Unlike the energy ux, it is not obvious if the integrated charge ux should have a denite sign. Consequently it is not clear if analogous bounds on the current three point function coe cients should hold. However, we will see that when these coe cients do not lie between the free eld theory values, the contributions of J to the anomalous dimensions would have counter-intuitive signs, motivating us to speculate that analogous bounds on the coe cients appearing in hJJJi might hold. A corollary of our results is that

when these OPE coe cients saturate their free eld theory values, some of the anomalous dimension asymptotics vanish, possibly indicating that subsectors or the entirety of the theory are free [54].

The organization of this paper is as follows. In section 2 we review the construction of correlation functions for operators with spin using an embedding formalism. By constructing a di erential representation of the 3-point functions we can calculate the relevant conformal blocks. In section 3 we review the lightcone bootstrap for four external scalars and generalize it to include external operators with spin. In section 4 we consider hJJi

with J being either a U(1) or a SU(N) conserved current. We rst solve the crossing equation at leading order in the lightcone limit, where large spin double-twist operators must contribute in order to reproduce the identity contribution. We then solve the equation at the rst subleading order, where the exchange of a conserved current and stress energy tensor in the s-channel is reproduced by the anomalous dimensions of the aforementioned double-twist operators. This procedure is repeated in sections 5 and 6 for hJJJJi and hT T i respectively. We also generalize a su cient condition for the existence of higher

spin symmetry in the limit of large global symmetry groups discovered in [47]. In section 7 we discuss some applications of our results to superconformal eld theories (SCFTs) and in section 8 we discuss some implications of our results and possible future work. In the appendices we collect technical details referenced in the paper.

2 Correlation functions

2.1 The embedding formalism

We will rst review the embedding space formalism for CFTs developed in [50, 55]. The idea, rst noted by Dirac [56], is that the conformal group in d Euclidean dimensions, SO(d+1, 1), can be realized linearly in an embedding space Md+2 as the group of isometries.

2

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The constraints on correlation functions of primary operators simplies to the constraints of Lorentz symmetry once we lift the elds to the embedding space. In this paper we will be interested in CFTs living in a d-dimensional Minkowski spacetime with conformal group SO(d, 2), but we can always Wick rotate between the two pictures.

The lift of Rd to Md+2 is accomplished by identifying points x in Rd with null rays in Md+2 as

P A = (1, x2, xa),

R, P A

Md+2, (2.1)

where P A is written in the lightcone basis

P A = (P +, P , P a), (2.2)

with the metric given by

P P ABP AP B = P +P + abP aP b. (2.3)

A linear SO(d + 1, 1) transformation maps null rays to null rays and therefore denes a transformation of the physical space onto itself. It can be shown that any SO(d + 1, 1) transformation of M induces a conformal transformation on R and that every conformal transformation can be obtained in this way.

We now need to give the correspondence between elds in the physical space and those in the embedding space. This was done using an index-free notation for symmetric traceless tensor elds in [50, 55] and has also been generalized to arbitrary tensor elds [57], spinors in three [49] and four dimensions [5860], and various situations with supersymmetry, e.g. [25, 6166] and references therein. Three dimensions is special because the only irreducible tensor representations of SO(3) are the totally symmetric and traceless representations. We only study 3d CFTs in this paper so we will restrict our review to these representations.

The mapping is as follows. Consider a eld FA1...A(P ), a tensor of SO(d + 1, 1), with the following properties:

1. Dened on the cone P 2=0,

2. Homogeneous of degree : FA1...A(P ) = FA1...A(P ), > 0,

3. Symmetric and traceless,

4. Transverse: (P F )A2...A P AFAA2...A = 0.

Now we dene the Poincar section as

P Ax = (1, x2, xa), x

Rd. (2.4)

Due to the homogeneity property, once F is known on the Poincar section it is known everywhere on the lightcone. The projection to the Poincar section denes a symmetric traceless eld on Rd,

fa1...a(x) = P A1

xa1 . . .

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P A

xa FA1...A(Px). (2.5)

3

To encode the symmetric traceless tensors in terms of a polynomial we introduce an auxiliary complex polarization vector za, contract it with the tensor eld, and restrict to the submanifold dened by z2 = 0,

fa1...a symmetric and traceless f(z)

z2=0. (2.6)

We do not lose any information with this condition since our tensors will be traceless. Two polynomials that di er by terms that vanish when z2 = 0 correspond to the same tensor. There is in fact a one to one correspondence between symmetric traceless tensors fa1...a

and polynomials f(z)

z2=0. The same idea is applied for tensors in the embedding space and we have that

FA1...A(P ) symmetric and traceless F (P, Z)

Z2=0,ZP =0. (2.7)

Once again we restrict to Z2 = 0 since the tensor will be traceless and Z P = 0 since it

is transverse. That is, the polynomial is invariant under Z Z + P . Any polynomial

that di ers from F (P, Z) by such terms corresponds to the same underlying tensor eld. Dening Zz,x (0, 2x z, z), the correspondence between the polynomials can be stated as f(x, z) = F (Px, Zx,z). (2.8)

2.2 3-point functions

In embedding space the classication of 3-point functions simplies. Conformal symmetry xes the basic building blocks for symmetric, traceless elds to be:

Hij 2 (Zi Zj)(Pi Pj) (Zi Pj)(Zj Pi)

. (2.10)

To simplify notation we dene Pij = 2Pi Pj. When projected to the Poincar section we

have Pij x2ij with xij xi xj. The 3-point function can be written as

G1,2,3({Pi; Zi}) =

where i = i + i. Dening

V1 V1,23, V2 V2,31, V3 V3,12, (2.12)

then Q can be written as a linear combination of structures of the form

Yi

V mii

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, (2.9)

Vi,jk

(Zi Pj)(Pi Pk) (Zi Pk)(Pi Pj)

(Pj Pk)

Q1,2,3({Pi, Zi}) (P12)

2+31

2 (P31)

2 (P23)

1+23

1+32 2

, (2.11)

Yi<j

Hnijij, (2.13)

where the homogeneity properties of the operators imply

mi +

Xj6=inij = i. (2.14)

4

In three dimensions some of these tensor structures are degenerate. In particular,

(V1H23 + V2H13 + V3H12 + 2V1V2V3)2 = 2H12H13H23 + O({Z2i, Zi Pi}). (2.15)

We discuss these degeneracy conditions in more detail in appendix C.

There is an alternative way to represent the 3-point functions that will be useful for constructing conformal blocks. When the 3rd operator is symmetric and traceless, the spinning 3-point function can be expressed in terms of di erential operators acting on an appropriate scalar 3-point function:

h{a}1(x1){b}2(x2)O{e}(x3)i = Da,bx1,x2h1(x1)2(x2)O{e}(x3)i. (2.16)

Explicit construction of these operators can be done in the embedding space, where they satisfy the consistency conditions DO(P 2i, Pi Zi, Z2i) = O(P 2i, Pi Zi, Z2i). Such operators can be built out of the building blocks

D11 (P1 P2)

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Z1 P2

(Z1 P2)

P1 P2

(Z1 Z2)

P1 Z2

+ (P1 Z2)

Z1 Z2

,

D12 (P1 P2)

Z1 P1 (Z1 P2)

P1 P1 + (Z1 P2)

Z1 Z1

. (2.17)

There are two more operators D22 and D21 which can be found by permuting 1 2.

Dij acts to increase the spin at point i by 1 and decreases the dimension at point j by1. A fth operator is multiplication by H12 which increases the spin and decreases the dimension at both points by one. The most general parity even spinning 3-point function can be constructed with the following basis:1

Hn1212Dn1012Dn2021Dm111Dm222 m1+n20+n12,m2+n10+n12h1(P1)2(P2)O(P3, Z3)i, (2.18)

where m1 +n10 +n12 = 1 and m2 +n20 +n12 = 2. The a,b operators shift the dimensions by 1 1 + a and 2 2 + b. We call this the di erential basis. The transformation

to the standard basis, (2.11) and (2.13), is computed in [55].

In three dimensions we also have parity odd structures, which are given by the parity even tensor structures above multiplied by the epsilon tensor. In the standard basis we have the 3-point function structure

ij Pij(Zi, Zj, P1, P2, P3), (2.19)

where on the right hand side we have used the 5d epsilon tensor. We could also consider the structure formed with three Z vectors and two P vectors, but these can always be solved for in terms of ij. Similarly we can always solve for 12 in terms of 13 and 23 [50, 55].

Therefore, we only need to use 13 and 23 to construct parity odd 3-point functions. Note that when multiplying by ij the scaling dimensions must be shifted to preserve the desired scaling properties.

1The operators can be reordered, keeping in mind two pairs do not commute: [D11, D22] 6= 0 and

[D12, D21] 6= 0. All other di erential operators commute with each other.

5

The corresponding parity odd di erential operators are:2

~D1

Z1, P1,

P1 , P2,

P2

+

Z1, P1, P1 , Z2, Z2

, (2.20)

~D2

Z2, P2,

P2 , P1,

. (2.21)

~Di increases the spin at point i by 1. To construct a parity odd 3-point function we act with a single odd di erential operator, ~D1 or ~D2, and then the parity even operators.

Finally, we will be interested in studying correlation functions involving conserved currents. As explained in [50, 55], requiring that a 3-point function be conserved at point Pi is equivalent to requiring that Pi DZi vanish when acting on the embedding space

correlation function, where

P DZ

P1

+

Z2, P2, P2 , Z1, Z1

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PM

d

2 1 + Z

Z

ZM

1

2ZM

2 Z Z

. (2.22)

2.3 4-point functions

In this section we will review the structure of conformal blocks that appear in the 4-point functions of scalars as well as how to construct the conformal blocks for external operators with spin. First we start with four distinct scalars with dimensions i, so the four point function is xed by conformal invariance to be of the form

h1(x1)2(x2)3(x3)4(x4)i =

12 12

x224 x214

x214 x213

12 34 G(u, v)(x212)12 ( 1+ 2)(x234)12 ( 3+ 4) . (2.23)

Here G(u, v) is an arbitrary function of the conformal cross ratios

u = x212x234 x213x224

, v = x214x223 x213x224

. (2.24)

Next we note that the OPE of two scalars takes the general form

1(x1)2(x2) =

XO12OC(x12, x2)e1...eOe1...e(x2). (2.25)

The sum runs over all primary operators which can appear in this OPE. The contribution of all the descendants is given by the kinematical function C(x12, x2), which can be found using the 3-point functions by multiplying both sides with Of1...f(x3) and taking the

vacuum expectation value. The OPE coe cients, 12O, are related to the coe cient of the

3-point functions and are not determined by kinematics. Applying the OPE for 12 and

2There are two more di erential operators, ~D121 = [parenleftBig]Z1, Z2, P1, P2,

P1

[parenrightBig] and ~D122 =

Z1, Z2, P1, P2, P2

. For this paper, they can be ignored since their action on the scalar 3-point functions can be re-expressed in terms of the rst two operators.

6

34 the contribution of a single irreducible representation is given by a conformal block gO(u, v) or equivalently the conformal partial wave WO(x1, x2, x3, x4) [6769],

h1(x1)2(x2)3(x3)4(x4)i =XO12O34OWO(x1, x2, x3, x4), (2.26)

WO(x1, x2, x3, x4) = C(x12, x2)e1...eC(x34, x4)f1...fhOe1...e(x2)Of1...f(x4)i,

=

12 12

where gO and WO also depend on the external scaling dimensions.3

To repeat the same idea for operators with spin we need to consider the OPE

{a}1(x1){b}2(x2) =

XO12OC(x12, x2){a,b,e}O{e}, (2.27)

where we have used the shorthand {a} a1 . . . a. As described in [50], the OPE structures

for spinning operators can be found by acting with di erential operators on the scalar structures

C(x12, x2){a,b,e} = Da,bx1,x2C(x12, x2){e}. (2.28)

Da,b is a di erential operator constructed via the methods of section 2.2. The conformal partial waves are then given by

W {a,b,c,d}O(x1, x2, x3, x4) = Da,bx1,x2Dc,dx3,x4WO(x1, x2, x3, x4). (2.29)

To be more explicit, the 4-point function of spinning operators is:

h (P1, Z1) (P2, Z2) (P3, Z3) (P4, Z4)i = Xs({Pi}) Xk

Gsk(u, v)Q(k)1234({Pi; Zi}), (2.30)

Xs({Pi}) =

where i denotes the representation of the operator and Q(k) denote independent tensor structures. Xs is the universal prefactor appearing in the s-channel expansion. The coe -cient functions, Gsk(u, v), depend only on the conformal cross ratios. The conformal block decomposition is:

Gsk(u, v) =

XO,i,j

(i) 1 2O(j) 3 4Og12,34,(ij)O,k(u, v). (2.32)

Note that an exchanged operator can generically give rise to di erent tensor structures. Rewriting (2.29) in terms of conformal blocks, we get:

g(12,34),(ij)O(u, v) = X1sDs,(i)LDs,(j)RW { 1, 2, 3, 4}O(P1, P2, P3, P4), (2.33) g12,34,(ij)O,k(u, v) = g12,34,(ij)O(u, v)

k, (2.34)

3In this paper we will work with a slightly di erent normalization than the one found using the above method, namely our conformal blocks will have an extra factor of (2): g(here)O = (2)gOPEO. An extra factor of (1/2) will then appear multiplying 12O34O in the conformal block expansion.

7

x224 x214

x214 x213

12 34 gO(u, v)(x212)12 ( 1+ 2)(x234)12 ( 3+ 4) ,

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P24

P14

122 P14

P13

34 2

(P12)1+22 (P34)3+42, (2.31)

1 2

where Ds,(i)L and Ds,(j)R give the s-channel di erential operators and

(i) 1 2O(j) 3 4O = 1 2

k means we project onto the k-th tensor structure. We will construct the di erential operators case by case explicitly.

We have two extra indices, i and j, which label the independent 3-point function tensor structures for each operator. That is, the 3-point function h 1 2Oi may have multiple,

linearly independent tensor structures with unxed relative coe cients. For example, in three dimensions, hT T T i has three structures, two parity even and one parity odd. The

parity even structures can be identied with the structures found in a theory of free bosons or free fermions while the odd structure can only appear in an interacting theory. The superscript labels the OPE channels under consideration. To simplify notation later we will write expressions in terms of the conformal block coe cients

P 12,34;(ij)O

1 2

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(i) 1 2O(j) 4 3O. (2.35)

In all cases under investigation this matrix is diagonal in the di erential basis (at leading order in 1/ where we will be working), and positive denite.

Everything we have said corresponds to doing a conformal partial wave expansion in the (12)(34) channel or s-channel. The partial wave expansion in the (14)(32) channel, or t-channel, can be found by exchanging 2 4 in all of these expressions. We will denote

these t-channel di erential operators with a superscript t to distinguish them from the s-channel di erential operators.

3 Lightcone bootstrap

We will now review the lightcone bootstrap when looking at the correlation function of four scalars (see [38, 39] for a more thorough analysis). The important result is that for any CFT in d > 2 spacetime dimensions, the large spin spectrum contains multi-twist states with a Fock space structure whose anomalous dimension asymptotics are determined by the minimal twist sector of the theory. The twist of an operator is dened as the di erence between its conformal dimension and spin, = . Analogous results hold when we

consider correlation functions involving the stress-energy tensor and conserved currents, except the contributions of T and J to the anomalous dimension of these double-twist states can have either sign. Requiring that it be non-positive for the stress-energy tensor yields the d = 3 conformal collider bounds. We will also see interesting behavior when the contribution of J changes sign, but we are not aware of any pre-existing bounds on the relevant OPE coe cient.

3.1 Review: scalar 4-point functions

We start by reviewing the basic results of [38, 39] and establishing some notation. Given a 4-point function of scalars, h1(x1)1(x2)2(x3)2(x4)i, we can perform the OPE inside the correlation function in three di erent ways, corresponding to three distinct channels.

Requiring that the resulting sums of conformal blocks agree yields the bootstrap equations. For our purpose, we only need the bootstrap equations from the (12)-(34) and the (14)-(32)

8

channels,

XO1,21,2P 11,22Og11,22,(u, v) = u 2v12 ( 1+ 2)

XO12P 12,21Og12,21,(v, u), (3.1)

where the coe cients P ij,klO are related to the OPE coe cients as in (2.35), we label the

conformal blocks gij,kl,(u, v) by the twist and spin of the exchanged operator, and we work in a normalization such that g,(u, v) u/2(1 v) when u 0 and then v 1.

In the eikonal (or lightcone) limit of u v 1, the conformal blocks in (12)-(34)

channel are proportional to u/2. Therefore the l.h.s. of (3.1) is dominated by the low twist operators: the identity with = 0, conserved currents with = d 2 and scalars with low

dimensions. In spacetime dimensions d > 2, the leading u contribution comes exclusively from the identity operator, yielding the following crossing equation:

u 2v

1

2 ( 1+ 2) =

X,

P 12,21,g12,21,(v, u). (3.2)

One puzzle is that the left hand side has the power law singularity u 2 while the crossed channel partial waves can have at most a u 1 2 divergence, for generic i. This problem is even more dramatic if 1 = 2, in which case the right hand side has at most a log(u) divergence. The resolution discovered in [38, 39] is that the correct power law singularity can only be reproduced on the r.h.s. by the innite sum over large spin operators with twist 1 + 2 with the OPE coe cients given by the generalized free eld theory. Solving (3.2) at leading order in u and all orders in v reveals the existence of large spin operators with twists 1 + 2 + 2n, where n is a non-negative integer. We refer to them as double-twist operators. Solving (3.1) to the next leading order in u includes contributions to the l.h.s. from conserved currents and low dimensional scalars, which are reproduced on the r.h.s. by large- suppressed anomalous dimensions of the double-twist operators correcting the canonical twists given above.

To see this explicitly we need an approximate form of the conformal blocks in this limit. Generalizing for the moment, we will start with the conformal block in the (14)-(32) channel when all four operators are distinct scalars with dimensions i. Then in the limit u v < 1 with u [lessorsimilar] O(1) we can use the approximation

g12,34,(v, u)

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2 K12 ( 1+ 2 3 4)(2u), (3.3)

where Kn(x) is the modied Bessel function of the second kind. Details about the derivation of this equation can be found in appendix A.

The n = 0 operators, i.e. those with twist 1+ 2, are required to match the v

1

2 ( 1+ 2)

2+2

u

14 ( 1+ 2 3 4)v

term on the left hand side. The generalized free eld theory OPE coe cients squared in the large spin limit are given by

4

P0,

( 1) ( 2)20+2 1+ 2

32 . (3.4)

9

After setting 2 = 1 and 4 = 3 in the above formula for the conformal block and approximating the sum over as an integral we obtain

X,

P 12,21,g12,21,(v, u) 4 ( 1) ( 2)

Z

d( 1+ 21)K 1 2(2u). (3.5)

Using the integral

Z

daKb() = 2a1

12(1 + a b)

12(1 + a + b)

, (3.6)

we reproduce exactly u 2v

12 ( 1+ 2). Considering a general CFT in d > 2 dimensions, where we have isolated the identity by taking the u 0 limit, this illustrates that at large

spin there exist double-twist states of the schematic form 11 . . . 2,4 whose twists are approximately 1 + 2. However, if we only have this tower of operators, the crossing equations cannot be solved because their conformal blocks gives higher order contributions in v that do not appear on the left hand side. To cancel these we must include operators with twist = 1 + 2 + 2n. These correspond to the 1(2)n1 . . . 2 operators.

Going to higher order in u in (3.1), we must include non-identity operators Om with

small twist m in the s-channel:

1 +

2

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X,

P 12,21,g12,21,(v, u). (3.7)

Using this equation, the anomalous dimensions and correction to the OPE coe cients of the double-twist states can be solved in terms of m, m and P 11,22m. For the moment we will assume that for each n = 1 + 2 + 2n there exists a unique operators at each spin , with twist approaching n as . We will relax this assumption later when considering

4-point functions of conserved currents.

In a unitary theory the stress energy tensor will always be present in the s-channel with = d 2. There is also the possibility of conserved currents with = d 2 and

scalars with d22 < d 2. Higher spin conserved currents also have = d 2, but the

existence of a single higher spin current in a theory with a nite central charge CT would imply the theory is free [70]. Therefore we will restrict the sum on the l.h.s. to m 2.

When u 1 the conformal blocks in the s-channel have the following behavior [69]

g12,34,(u, v) u

XmP 11,22mg11,22m,m(u, v) u 2v12 ( 1+ 2)

2 (1 v) 2F1

1

2( + 2 1) + ,

1

2( + 3 4) + , + 2; 1 v

.

(3.8)

When 3 = 4 and 1 = 2,

2F1(, , 2, 1 v) =

(2)

2()

Xn=0

()n n!

2 vn

2((n + 1) ()) log(v)

, (3.9)

4This form is schematic since we also have to symmetrize the Lorentz indices, remove the traces and the descendant contributions to construct a conformal primary operator. The exact form of these primaries will not be important to us.

10

(x) . At leading order in v, the log(v) singularity from the 2F1 is reproduced by the anomalous dimensions of the double-twist operators on the right hand side: 1 + 2 + (n, ), where (n, ) is power-law suppressed at large .

The log(v) piece arises from expanding g12,21,(v, u) v

where (x)n = (x+n) (x) and (x) =

(x)

2 to leading order in the anomalous

dimension, (n,)

2 log(v). The power law singularity in u is then reproduced from the t-channel expansion via the innite sum over spins. Approximating the anomalous dimension at large as (n, ) = n and matching the u divergence from the s-channel yields = m.

Matching the v0 log(v) term in the s-channel yields the coe cient [3840]

0 =

2 ( 1) ( 2)

( 1 m2 ) ( 2 m2 ) X

m

Pm (m + 2m)

2(m2 + m)

2 v

. (3.10)

The case of matching 0 is particularly simple since we just need to multiply each 0th order OPE coe cient by 12

0m . The t-channel OPE coe cient receives a correction of the form P, cnm, where the rst coe cient c0 is proportional to 0 and can be found

by matching the v0 term. Recently there has been success in nding n for general n in arbitrary dimension [41, 45], but here we will restrict ourselves to the 0 terms.

3.2 Spinning operators

The case of external operators with spin is complicated by the presence of multiple tensor structures appearing in the 4-point function. Each independent tensor structure yields an independent crossing equation. For the 4-point function of two pairs of identical operators

h 1(P1, Z1) 1(P2, Z2) 2(P3, Z3) 2(P4, Z4)i, crossing symmetry becomes u2v

1+2

JHEP02(2016)143

2 Gsk(u, v) = Gtk(v, u) k, (3.11)

Gsk(u, v) =

XO,i,jP 11,22;(ij)Og11,22,(ij)O,k(u, v), (3.12)

Gtk(v, u) =

XO,i,jP 12,21;(ij)Og12,21,(ij)O,k(v, u), (3.13)

where k runs over the allowed 4-point tensor structures and i = i + i.

The strategy for solving these equations in the lightcone limit mimics the scalar case. We consider the limit u v < 1 and u [lessorsimilar] O(1). The s-channel is dominated in

this limit by the operators with minimal twist, which is the identity for d > 2. The identity contribution has a power law divergence of u2, while all the conformal blocks in the t-channel have a weaker power law singularity in u. We will show that the identity contribution in the s-channel is reproduced in the t-channel via an innite sum over spins for multiple families of double-twist states. The simplest such states have the schematic form 1,(1...1 1 . . . k 2,1...2) and have twist = 1+2 and spin 1+2+k. But there

are other families of double-twist operators arising from contractions between the elds, derivatives, and/or the epsilon tensor. In particular, to reproduce the full tensor structure of the identity exchange in the s-channel, we need to include a few double-twist families with non-minimal twist in the t-channel. The matching will x their OPE coe cients at leading order in 1/.

11

Unlike in the scalar case we do not know the closed form expressions for the generalized free eld theory OPE coe cients. Instead we will make the ansatz that the modied OPE coe cients squared for the double-twist states, the P (ij) terms, have the form AB22 at large . Here A and B are determined by matching the t-channel expansion to the identity contribution. There are a few justications for this form, besides the fact that it gives the right answer. One is that we computed the exact generalized free theory OPE coe cients for the [J]n, double-twist states using conglomeration [71], and their large spin limit is of this form. Alternatively, if we follow the analysis of [46], we can decompose operators with spin into representations of the lightcone (collinear) subgroup of the conformal group. The problem then reduces to a 2d CFT problem where a single correlation function containing an operator with spin can be rewritten in terms of multiple correlation functions containing the lightcone primaries. Then the lightcone bootstrap equations can be solved in the usual way with the scalar collinear blocks and the OPE coe cients take the above form.

With the results from the identity matching, we can expand the crossing equations to the next leading order in u and solve for the large asymptotics of the double-twist anomalous dimensions. In the s-channel, this includes the next-to-minimal twist operators, which we will assume to be conserved spin 1 and 2 operators, whose conformal blocks have an additional log(v) divergence. This logarithm is reproduced in the t-channel in the same way as the scalar case via the anomalous dimensions: v

2

, (3.14)

where is xed by reproducing the u dependence in the s-channel. We will nd = 1, which in 3 dimensions is the twist of conserved currents. The coe cient n can be determined in terms of the s-channel OPE coe cients using the crossing equations.

In our analysis we will only consider contributions in the s-channel from 3-point structures of even parity. In a 3d CFT with a parity symmetry, conserved currents and the stress tensor have even parity so these are guaranteed to be the only contributions. The e ect of parity odd contributions, which would arise in theories without parity such as Chern-Simons-matter theories, as well as the e ects of scalar exchange, will be considered in future work.

3.3 Bootstrap with SU(N) adjoint operators

In this subsection, we briey review the structure of the bootstrap equations when all four operators transform under a global symmetry. In general, the 4-point function will contain several di erent index structures. Matching their coe cients in di erent OPE channels generically leads to independent crossing equations.

We take the adjoint representation of SU(N) as an example. We will rst discuss the scalar 4-point function habcdi before generalizing to the spinning case. See [27, 47] for

more thorough reviews of the conformal bootstrap with adjoint scalars.

For N 4, there are 7 representations that can appear in the product of two ad-

joints (using the notation of [27]): I , Adja , Adjs , (S,)a (A,

. The

v

1

2 (+(n,))

12n,v

2 log(v).

JHEP02(2016)143

The anomalous dimension asymptotics take the form

(n, ) = n

S)a , (A,)s , (S, S)s

12

subscript s or a denotes whether the operator appears in the symmetric or antisymmetric combination of the two adjoints. The notation (A, S) means the tensor is antisymmetric in the two fundamental indices and symmetric in the anti-fundamental indices. (A, S)a and (S,)a are complex conjugate and appear together in 4-point functions of real operators.

The 4-point function can then be decomposed into 6 independent index structures corresponding to these representations. If we construct them using the tensor product in the s-channel, we obtain:

ha(x1)b(x2)c(x3)d(x4)i = XsXr

Gsr(u, v)(Isr)abcd, (3.15)

where r = (I , Adja , Adjs , (S,)a(A,

S)a , (A,)s , (S, S)s) runs over the 6 representations and Isr gives the corresponding index structures. We can further decompose each Gsr(u, v) in terms of conformal blocks,

Gsr(u, v) =

XO()r

POg O,O(u, v), PO |O|22O . (3.16)

We can do a similar decomposition in the t-channel and require that the results agree. Matching the coe cients of the 6 index structures gives rise to 6 crossing equations:

u v

Gtr(v, u) = MrrGsr(u, v). (3.17)

The explicit matrix Mrr is given in appendix F.

We can generalize this discussion to 4-point functions of operators with spin. Each di erent choice of global index structure and tensor structure, (r, k), gives rise to a crossing equation:

u2v

Gsr,k(u, v) =

Gtr,k(u, v) =

JHEP02(2016)143

1

2 (1+2)Gtr,k(v, u) = MrrGsr

,k(u, v), (3.18)

XO()r,i,j(1)rP 12,34;(ij)Og(ij) O,O,k(u, v), (3.19)

XO()r,i,j(1)rP 14,32;(ij)Og(ij) O,O,k(u, v), (3.20)

where r keeps track of the extra minus signs due to the exchange symmetries of the global index structure. It is 0 for representations that appear in the symmetric product of adjoints and 1 for those that appear in the antisymmetric product of adjoints. Note that here the coe cients P 12,34;(ij) also contain a factor of (1) which in some cases cancels against the

(1)r (this implicitly occurred in (3.16)).

For N = 2 and 3 some representations do not appear, but the large OPE coe cients and anomalous dimensions can still be found by dropping these representations and setting N to the appropriate values. For N = 3 the (A,)s representation does not appear and for N = 2 the (S,)a, (Adj)s, and (A,)s representations do not appear.

13

4 Mixed current-scalar 4-point functions

In this section we will investigate correlation functions of the form hJJi, where J is a

conserved spin one current and is a real scalar operator of arbitrary dimension. The current has dimension J = d1 and corresponds to either a U(1) or SU(N) global symmetry.

For the U(1) case we will take the scalars to be uncharged under the U(1) symmetry, while for the SU(N) case we will assume they transform in the adjoint representation. There is no loss in generality for the U(1) case since the 3-point function of identical U(1) currents,

hJJJi, vanishes for a 3d CFT [72].

4.1 Identity matching

4.1.1 U(1)

At leading order in the small u limit, the 4-point function is given by:

hJ(P1; Z1)J(P2; Z2)(P3)(P4)i = CJ

H12

(P12)d(P34) + . . . . (4.1)

This is the result of the identity exchange in the s-channel. In other words, the 4-point function factorizes at this order and is equal to the generalized free eld theory result, even when we are not assuming a large N limit. CJ is the current central charge and describes the normalization of the current 2-point function

hJ(P1; Z1)J(P2; Z2)i = CJ

JHEP02(2016)143

H12

(P12)d . (4.2)

In this subsection we reproduce this contribution from the t-channel conformal block expansion. We will show that this requires the t-channel to receive contributions from two families of double-twist operators, the parity even ones [J]n, J(2)n1 . . . 1 with twist n, = J ++2n, as well as the parity odd ones [

fJ]n, J(2)n1 . . . 1 with twist J + + 2n + 1. We will solve the crossing equations at leading order in u and v, where only the lowest twist states (n = 0) from both families contribute.

To construct the spinning conformal blocks in the t-channel, we use the fact that the

hJ[J]i 3-point function can be represented in terms of di erential operators acting on

the scalar correlator. After imposing the conservation conditions, for [J]0, we have

hJ(P1; Z1)(P2)[J]0,(P3; Z3)i

2 d

= + 1

D11 1,0L + D12 0,1L

J[J]0,V 3

(P12)

1

2

(P13)d+

32 (P23) +

1

2

, (4.3)

where

J[J]0, is an arbitrary coe cient. For the parity odd double-twist states, we have

hJ(P1; Z1)(P2)[

fJ]0,(P3; Z3)i =~D1

J[tildewidest]

[J]0,V 3

(P12)

1

2

(P13)d+

32 (P23) +

1

2

. (4.4)

14

This correlation function is automatically conserved. The t-channel di erential operators that generate the spinning blocks are then5

Dt[J]0, =

1 + 1Dt11 1,0L,t + Dt14 0,1L,t

1 + 1

Dt22 1,0R,t + Dt23 0,1R,t

, (4.5)

Dt[tildewidest][J]0, = ~Dt1 ~Dt2, (4.6)

where we have set d = 3. The crossing equation at leading order in u is

CJ H12(P12)3(P34) =

Pn, P[J]n,Dt[J]n,W t[J]n, + P[[tildewider]J]n,Dt[tildewidest][J]n, W t[tildewidest][J]n, , (4.7)

where P[J]n, and P[[tildewider]J]n, are positive squares of OPE coe cients as normalized in (2.35). W tO denotes the t-channel conformal partial wave with scalars, which is (2.26) with 2 4

exchanged.

We now solve (4.7) at the lowest order in v. Thus we set n = 0 and restrict the sum to be over .6 As mentioned in section 3.2, we make the ansatz that as we have

Pi AiBi22. The bootstrap equation is now straightforward solve, we act with the

di erential operators on the large spin conformal blocks in (3.3), which produces terms of a similar form, aKb(2u). The sum is then approximated as an integral and evaluated

using (3.6). Computationally, it is more e cient to compute the integral rst and then act with the di erential operators. The result gives the OPE coe cients at leading order in 1/:

JHEP02(2016)143

P[J]0,

CJ

22+ 1 ( )

1

2 (2 1), P[[tildewider]J]0,

CJ

22+ +1 ( )

1

2 (2 7). (4.8)

4.1.2 SU(N)

At leading order in u, the 4-point function is dominated by the identity exchange:

hJa(P1; Z1)Jb(P2; Z2)c(P3)d(P4)i = CJabcd

H12

(P12)d(P34) + . . . . (4.9)

As explained in section 3.3, there are 6 independent bootstrap equations corresponding to the 6 index structures. They are given in eq. (3.19) with the matrix dened in appendix F. For each equation the analysis is the same as the U(1) case. Matching the identity contribution shows that double-twist operators in all representations appear in the t-channel with the following coe cients:

P[J]0, = CJ

2 +1+2 ( )

1

2 (2 1)P, P[[tildewider]J]0, =

CJ

2 +3+2 ( )

1

2 (2 7)P, (4.10)

where we have dened the vector P = 4

N21 , 2N ,

2NN24 , 2, 1, 1

using the basis of representa-

. One interesting point is that the OPE coe cients for the singlet representations decay like 1/N2, the (Adj)s and (Adj)a

coe cients like 1/N, and all others approach a constant as N , showing that the

former states decouple at large N.

5We explicitly label the di erential operators by t as a reminder that these are the di erential operators for the t-channel. In other words, in our original formulas we must let 2 4. In these expressions a,bL shifts 1 by a and 4 by b, while a,bR shifts 2 by a and 3 by b.

6The crossing equation only depends on the cross ratios u, v even when this is not manifest in (4.7).

tions I , Adja , Adjs , (S,)a (A,

S)a , (A,)s , (S, S)s

15

4.2 Stress tensor and current matching

4.2.1 U(1)

We now solve the crossing equation at the next leading order in u. In the U(1) case, the s-channel contains the contribution of the stress-energy tensor.7 The spinning conformal block for stress-tensor exchange can be obtained by acting with a di erential operator DL,T

on the scalar partial wave, which is xed by the condition DL,T hJJT i = hJJT i, where in general J is a real scalar with dimension d 1. Conformal symmetry implies

hJ(P1; Z1)J(P2; Z2)T (P3; Z3)i =

V1V2V 23+(H13V2 + H23V1)V3+H12V 23 + H13H23

(P12)

d

2 1(P13)

d

2 +1(P23)

JHEP02(2016)143

d

2 +1 .

(4.11)

After imposing conservation conditions and the Ward identity, this 3-point function is xed in terms of two coe cients, JJT and CJ [73]:

= dCJ(d2 4) 2JJT dSd

2Sd , = 2JJT , (4.12)

= dJJT +

d(d 2)CJ

2Sd , = 2JJT +

dCJ

Sd , (4.13)

where CJ is the current central charge and Sd is the volume of a d 1-dimensional sphere,

Sd = 2

d2

( d2 ) . The coe cient JJT is arbitrary and was denoted as c in [73] and in [53]. We can reproduce this 3-point function with the following di erential operator:

DL,T =

2JJT

d(d2)CJ

(d1)Sd

D11D22+

2JJT d2CJ (d1)Sd

D12D212JJT H12

1,1L,

(4.14)

where a,bL shifts the dimensions of the rst two operators.

In the 4-point function, the identity contribution is corrected by the stress tensor exchange in the s-channel, which is suppressed by an extra factor of u:

hJ(P1; Z1)J(P2; Z2)(P3)(P4)i = CJ

H12

(P12)d(P34) + T

14CT DL,T W sT({Pi}) + . . . ,

(4.15)

where T is also xed by the Ward identity to be

T =

1CT . (4.16)

In this rst correction, the leading contribution at small v is a logarithmic singularity. This log(v) is matched in the t-channel by expanding the conformal blocks in the anomalous

7To simplify the analysis, we are assuming that there are no scalars with 12 < < 1. Their contributions can also be included with the methods described here.

16

d

(d 1)Sd

dimensions of double-twist operators. The crossing equation then takes the form8

T

4CT DL,T W sT =Xn,

P[J]n,[J]n,nDt[J]n,W t[J]n, +P[tildewidest][J]n,[tildewidest][J]n,~nDt[tildewidest][J]n, W t[tildewidest][J]n, ,

(4.17)

where we implicitly restrict to terms multiplying log(v). This equation can be solved with the same technique as before. The anomalous dimensions are necessarily 1/ suppressed at large for their e ect to appear at the correct order in u. With the notation On, On/, the n = 0 results are:

[J]0 =

4 (3CJ 8JJT ) ( )

7/2CT CJ 12

JHEP02(2016)143

, [tildewidest]

[J]0 =

8 (16JJT 3CJ) ( )

7/2CT CJ 12

. (4.18)

In an AdS bulk description, these anomalous dimensions correspond to the gravitational binding energies for well-separated 2-particle states [38]. Since gravity is expected to be attractive, we expect the anomalous dimensions to be negative. Assuming that the CFT is unitary, so CT > 0 and CJ > 0, and that the scalar is not free, or equivalently > 12, we see that the anomalous dimensions are non-positive if and only if the relevant 3d conformal collider bounds [53] are satised:

3CJ

16 JJT

3CJ

8 . (4.19)

These bounds were originally discovered by requiring the integrated energy ux at spatial innity due to a localized perturbation in the CFT to be positive. Our results suggest that the positivity of this energy ux is equivalent to the attractiveness of bulk gravity at long distances. We could also turn this around and conclude that our analysis combined with the conformal collider bounds gives an argument for the attractive nature of bulk gravity at long distances, using entirely properties of the eld theory. The negativity of the anomalous dimensions is also related to bulk causality in large N theories [74]. We hope to explore this connection in more detail in future work.

Note that the two boundary values, 3CJ

8 and

3CJ

16 , correspond to the values found in a theory of free fermions and free bosons respectively. When JJT saturates one of the bounds, one of the asymptotic anomalous dimensions vanishes. This could be an indication that certain sectors of the theory are decoupling (see [54] for related work in 4d). It would be interesting to extend this analysis to higher order in n or to see if this behavior continues to hold. The fact that the other anomalous dimensions remains non-zero at the boundary free values indicates that our analysis is incomplete for free theories, which contain an innite number of higher spin conserved currents. We have not included them in the s-channel analysis since we focused on interacting theories which have a twist gap separating the spin 1, 2 conserved currents from the other operators. After summing up all the contributions from higher spin conserved currents, we expect the logarithm to disappear and the anomalous dimensions of the double-twist states to vanish.

8The division by CT comes from how we normalize the stress energy tensor. A division by CJ will also appear for current exchange. See appendix D for our conventions.

17

4.2.2 SU(N)

For a 4-point function of SU(N) adjoint operators, the SU(N) conserved currents can appear in the OPE decomposition. This gives rise to another contribution at the same order compared to the stress tensor exchange. The spinning conformal block for the current exchange can be obtained by acting with a di erential operator DL,J on the scalar partial

wave, which is xed by the condition DL,JhJJJi = hJJJi, where J is a real scalar with dimension d 1. After imposing the conservation conditions and the Ward identity, we see

that this 3-point function is xed by two coe cients, CJ and JJJ [73]:

hJa(P1, Z1)Jb(P2, Z2)Jc(P3, Z3)i

= fabc (CJdSd (2 + d)JJJ)V1V2V3 JJJ(H12V3 + H13V2 + H23V1)(P12)

, (4.20)

where fabc are the structure constants, corresponding to the (Adj)a representation in the tensor product. In [73] the coe cient JJJ was called b.

The associated di erential operator that generates this 3-point function is given by

DL,J = dCJ(d2)Sd

(D12D22 0,2L+ D11D21 2,0L D12D21 1,1L) +

JHEP02(2016)143

d

2 (P13)

d

2 (P23)

d

2

4JJJSd dCJ (d 2)Sd

D11D22 1,1L.

(4.21)

This is all we need to construct the spinning conformal block. After solving the crossing equation at the next-to-leading order in u, we nd that the t-channel anomalous dimensions receive separate contributions from T and J exchange. The anomalous dimensions asymptote to On, = On1 at large . The n = 0 coe cients are given by:

[J]0 =

2(CJ 4JJJ) ( )

7/2C2J 12

J

4 (3CJ 8JJT ) ( )

7/2CT CJ 12

T ,

[[tildewider]J]0 =

2(8JJJ CJ) ( )

7/2C2J 12

J

8 (16JJT 3CJ) ( )

7/2CT CJ 12

T , (4.22)

J = (2N, N, N, 0, 2, 2), T = (1, 1, 1, 1, 1, 1),where J and T give the results for double-twist operators in di erent representations of SU(N): I , Adja , Adjs , (S,)a (A,

. The second terms in these expressions are the corrections to the dimension due to the stress tensor exchange in the s-channel. As in the U(1) case, they correspond to the gravitational binding energies between well separated 2-particle states in AdS. The fact that they are the same for di erent representations is consistent with the universality of gravity. From the CFT perspective this is due to the fact that T appears in the singlet representation of SU(N). Since gravity at long distance is expected to be attractive, we expect these anomalous dimensions to be negative. Once again, we nd that this holds if and only if the same conformal collider bounds (4.19) are satised.

The anomalous dimensions from current exchange are given by the rst terms in (4.22). In a dual AdS description, they correspond to the binding energy from non-Abelian gauge

18

S)a , (A,)s , (S, S)s

interactions for well-separated 2-particle states. We nd that this interaction is attractive for the neutral 2-particle states, or the singlet, if and only if

CJ

8 JJJ

CJ

8 while in

a theory of free fermions JJJ = CJ4 [73]. Just as in the case of T -exchange, when hJJJi

saturates the free boson or free fermion structures some of the asymptotic anomalous dimensions vanish.

The inequalities (4.23) are intimately related to conformal collider physics. By acting with a properly chosen non-Abelian current on the vacuum, we can create a localized state with a positive charge under a U(1) SU(N) at the origin.9 This local perturbation

propagates and the charge ux at innity can be measured. We show in appendix E that the expectation value of the charge ux at innity is positive if and only if (4.23) is satised. In contrary to the energy ux, the charge ux in any single event may trivially have di erent signs at di erent angles, as expected from a showering of charged particles. But this doesnt imply that (4.23) is generically violated. Indeed, to make the expectation value of the charge ux negative, one needs a large charge ux asymmetry. This is the much more non-trivial behavior that is forbidden by (4.23).

From the perspective of gauge interactions in the bulk, the regime violating (4.23) seems rather strange. Eq. (4.23) is equivalent to the statement that the gauge representation of a well separated 2-particle state determines the sign of its gauge binding energy. In particular, this sign will not depend on the spin of the particles or the parity of the state. This follows from comparing (4.22) to the corresponding result in scalar 4-point functions [47], where this sign is uniquely determined by the representation. However if, for example, JJJ > CJ4 , then all the parity-even 2-particle states consisting of a scalar and a gauge boson [J] will have binding energies with opposite signs compared to the scalar-scalar state [] or the parity odd states [

fJ]. These behaviors seems counter-intuitive. For example, the singlet 2-particle state [J] which intuitively holds the least energy in the gauge eld congurations will become the most energetic one. We do not have a rigorous way to forbid this situation, but it is tempting to conjecture that the bound (4.23) holds in all unitary CFTs.

It would be interesting to see if there could exist consistent CFTs or theories in AdS that violate (4.23). We are not aware of any examples. Some holographic constraints on massive triple vector boson couplings were found in [75], but their analysis does not apply to this case. For all superconformal theories this bound holds. Supersymmetry xes the parity even 3-point functions of conserved global symmetry currents up to an overall coe cient [76]. Therefore, JJJ can be calculated in a free theory and the result holds for all SCFTs since the positivity of the number of bosons and fermions in the free theory implies (4.23).

9Although we only analyzed the case of SU(N), we expect similar features to appear more generally. In particular, from the analysis of [47] we expect them to show up in 4-point functions of operators in other representations of SU(N) as well as in CFTs with other global symmetries such as O(N).

19

CJ

4 , (4.23)

where JJJ and CJ parameterize hJJJi. In a theory of free bosons JJJ =

JHEP02(2016)143

Furthermore, using the constraints of slightly broken higher spin symmetry, Maldacena and Zhiboedov [77] found the correlation functions of all currents to leading order in N in two classes of CFTs parametrized by an e ective t Hooft coupling , which they called the quasi-bosonic and quasi-fermionic theories. Large N Chern-Simons theories with fundamental matter fall into this category and include as special cases the critical O(N) model and UV Gross-Neveu O(N) model. Specically they found that hJ(s1)J(s2)J(s3)i is parametrized by three structures hibos, hifer, hiodd, which refer to correlators found in a theory of free bosons, free fermions, and a structure that only shows up in an interacting theory. The coe cients in front of the bosonic and fermionic structures are always positive semidenite, so the conjectured bound on JJJ holds for CFTs with slightly broken higher spin symmetries.

Finally, let us consider the dependence of the anomalous dimensions with respect to N. We expect that CT scales with some positive power of N, while the behavior of CJ is less clear.10 The bounds (4.19) and (4.23), if true, indicate that JJT CJ and JJJ CJ.

At large N we see that the contribution of T to the anomalous dimension becomes small for all the operators. This is consistent with bulk gravity turning o . If CJ stays constant as N , the anomalous dimensions of double-twist states in the singlet and adjoint

channels can start to decrease like N/. In this case our results should only hold when

N , otherwise we cannot treat the anomalous dimensions as perturbative parameters. If

CJ also scales with some positive power of N, our results may have a wider range of validity.

5 Current 4-point functions

In this section we will generalize the above analysis to 4-point functions of currents hJJJJi.

As before, we will rst match the identity contribution in the s-channel to an innite sum over large spin double-twist states. Then we will match the current and stress-tensor contributions to compute the anomalous dimensions of these states.

5.1 Identity matching

5.1.1 U(1)

At the leading order in the u v 1 limit, the 4-point function factorizes:

hJ(P1, Z1)J(P2, Z2)J(P3, Z3)J(P4, Z4)i = C2J

H12H34

(P12)d(P34)d + . . . . (5.1)

This corresponds to the contribution from the identity exchange in the s-channel.11 We

will see that to reproduce all polarizations of (5.1) at leading order in the lightcone limit from the t-channel conformal block decomposition, we would need to include contributions

10The scaling properties of CJ can be determined if we are close to a free eld theory description. For example, the contribution of a free eld in representation r of the global symmetry group to CJ scales like the index of the representation C(r), which is dened as Trr(T aT b) = C(r)ab [73]. For the fundamental

representation this is a constant, but for the adjoint representation, this grows like N.

11There are also identity exchanges in the t- and u-channel, but these contributions are subleading in the small u limit.

JHEP02(2016)143

20

operator twist parity spin constructions

[JJ]0, 2 even even J1 . . . 2J[JJ]1, 4 even even J21 . . . 2J, J1 . . . J

[

fJJ]0, 3 odd even & odd J1 . . . 2J, J1 . . . 1J

Table 1. T-channel double-twist operators that reproduce the s-channel identity exchange in

hJJJJi. From the bootstrap perspective, we cannot distinguish di erent constructions with the

same twist and parity, with the exception that the second construction of [

fJJ]0, exists only for even . Our solutions represent the sum of their OPE coe cients and the average of their anomalous dimensions, e.g., PO

Pi POi and O

JHEP02(2016)143

[summationtext]i POi Oi

POi .

from the three families of large spin double-twist operators given in table 1. It is perhaps surprising that the twist 4 states [JJ]1, should contribute at leading order. One of the polarizations in (5.1), (Z1 Z2)(Z3 Z4), receives a contribution at leading order from this

state when we take into account degeneracies among the four point function structures (see appendix C).

We will start by constructing the spinning conformal blocks associated with these operators. The conformal blocks for exchanging a general spin- operator in hJJJJi can

be written in terms of di erential operators acting on the known blocks for scalar 4-point functions. In the conformal partial wave expansion, the contribution from a primary O to

hJJJJi can be written asP (ij)ODL,iDR,jW tO, (5.2)

where W tO is the scalar conformal partial wave (2.26) with (2 4) permuted. P (ij)O are

products of OPE coe cients in the normalization of (2.35). Our goal in this subsection is to solve for them in the large regime with the crossing equations. The t-channel di erential operators are [50]

DtL,1 =

2 + ( 1)( 3)( + 2)( + ) C ,

Dt11Dt44 1,1L

( 1)( + )(Dt41Dt11 2,0L + Dt14Dt44 0,2L) + C ,Dt14Dt41 1,1L, (5.3) DtL,2 = 4Dt11Dt44 1,1L + C ,H14 1,1L, (5.4)

~DtL,+ = (Dt41 ~Dt1 1,0L+ Dt14 ~D4 0,1L) +

(3 ) + (1+)

C , (Dt44

~Dt1 0,1L + Dt11 ~Dt4 1,0L), (5.5)

~DtL, = (Dt41

~Dt1 1,0L Dt14

~Dt4 0,1L) +

(3 ) + (1 + ) 4

C , (Dt44

~Dt1 0,1L Dt11

~Dt4 1,0L),

(5.6)

where C , = ( d) + ( + d 2) is the quadratic Casimir and , refer to the

scaling dimension and spin of the exchanged operator. Note that each DtL,i respects the

21

conservation conditions of the external currents. The rst two are parity even and appear in the conformal blocks of [JJ], while the last two are parity odd and appear with [

fJJ].

DtR,i is obtained by permuting (1 3, 4 2) in DtL,i.

At leading order in u v 1, the crossing equation becomesC2J H12H34

(P12)d(P34)d =X,i,j

hP (ij)[JJ]0,DtL,iDtR,jW t[JJ]0, + P (ij)[JJ]1,DtL,iDtR,jW t[JJ]1,

+ P (ij)[[tildewider]JJ]0, ~DtL,i ~DtR,jW t[tildewidest][JJ]0,

i. (5.7)

As explained in appendix C, there are two degeneracy conditions among the four point function tensor structures. In practice, it is simpler to match particular dot products appearing in the four point function tensor structures, taking into account the aforementioned degeneracies.

We nd a few simplications when solving (5.7). First, both the parity-even di erential operators and the parity-odd di erential operator ~DL,+ are symmetric under the exchange of 1 4. Therefore they only appear when is even.

~DtL,, on the other hand, is

odd under this exchange and appears when is odd. Second, for any coe cient matrix P (ij) constructed from (2.35), the cross terms, DtL,1DtR,2 and DtL,1DtR,2, give sub-dominant contributions compared to DtL,1DtR,1 and DtL,2DtR,2, and can be ignored. Finally, we nd from identity matching that to leading order in , P (22)[JJ]0, = 0 and P (11)[JJ]1, = 0.

To summarize, we nd that at leading order, the twist-2 parity even states only contribute through DtL,1DtR,1, and the twist-4 ones only contribute through DtL,2DtR,2. This is a nice simplication as we do not have to worry about a matrix of OPE coe cients. The di erential operators for each double trace state are given by:

Dt[JJ]0, DtL,1DtR,1

=2+,, (5.8)

JHEP02(2016)143

Dt[JJ]1, DtL,2DtR,2

=4+,,

Dt[[tildewider]JJ]0,

1 4

(1+(1))~DtL,++(1+(1)+1)~DtL, (1+(1))~DtR,++(1+(1)+1) ~DtR,

,

with a corresponding OPE coe cient PO. For the odd di erential operators we have

grouped the even and odd spin di erential operators together. In practice we should separate these contributions, split the sum over even and odd spins, approximate as an integral, and then solve. However, we nd that parity odd states of even and odd spin yield contributions of the same form, so we can only determine the sum of their OPE coe cients, which is denoted by P[[tildewider]

JJ]0,.

Matching all the dot products that appear in H12H34, we nd the OPE coe cients of

double-twist states at leading order in 1/ to be

P[JJ]0, =

C2

J

22+2

72 , P[JJ]1, =

C2

J

22+6

32 , P[[tildewider]

JJ]0, =

C2

J

22+2

52 . (5.9)

5.1.2 SU(N)

The SU(N) case is similar to the U(1) case with extra structures from the global symmetry indices. At leading order in the lightcone limit, the s-channel decomposition is dominated

22

by the identity exchange and the 4-point function factorizes. This contribution is reproduced in the t-channel by large spin double-twist states. As in section 4.1.2, two conserved currents can form 6 types of double-twist states corresponding to di erent representations of SU(N). For each type, there is a crossing equation similar to (5.7) that requires 3 families of double-twist operators with di erent twists as given in table 1.

For the parity even operators the same selection rules hold as in the purely scalar case, operators of (odd) even spin appear in representations (anti)symmetric under the exchange of adjoint indices. There are no such selection rules for the parity odd sector and we will need to keep track of extra minus signs relative to the scalar case when operators of odd spin appear in representations symmetric with respect to the adjoint indices and vice versa. This is the origin of the factor (1)r that appears multiplying P (ij)O in eq. (3.19).

Exchange symmetry for the parity odd operators implies we need the following di erential operator

Dt[tildewidest][JJ],r,n, =

1 4

(1 + (1)r)~DtL,+ + (1 + (1)r+1)~DtL,

(1 + (1)r)~DtR,+ + (1 + (1)r+1) ~DtR,

, (5.10)

where r is 0 or 1 for representations that appear in the symmetric or antisymmetric product of adjoints. As in the U(1) case, we choose to group together operators of even and odd spin for each representation because their contributions have the same form.

Using the crossing symmetry equations for SU(N) adjoints, the crossing equation at leading order in u v 1 can be solved to nd the OPE coe cients of double-twist

states at leading order in 1/:

P[JJ]0, = C2J

JHEP02(2016)143

24+2

72 P, P[JJ]1, = C2J

28+2

32 P, P[[tildewider]

JJ]0, = C2J

24+2

52 P, (5.11)

with P = 4

N21 , 2N ,

giving the result for each double-twist state in di erent representations under SU(N), which are I , Adja , Adjs , (S,)a(A,

S)a , (A,)s , (S, S)s

2NN24 , 2, 1, 1

.

5.2 Stress tensor and current matching

We now solve the crossing equations at the next-to-leading order. We include in the s-channel the contribution of the exchange of the stress tensor T , as well as the conserved currents Ja in the SU(N) case. These contributions are suppressed by a factor of u relative to the identity contribution. The log(v) singularity in the conformal blocks of T and J are reproduced on the right hand side by the anomalous dimensions of double-twist operators, n, = n/. The power of is determined by matching the extra u suppression in the s-channel. We will only focus on the n = 0 and n = 0, 1 case for the parity odd and even double-twist operators, respectively.

5.2.1 U(1)

Including the exchange of T and yields the following equations for the anomalous dimensions:

14DL,T DR,T W sT =

X,OOPOODtOW tO, (5.12)

23

where we have implicitly restricted to terms proportional to log(v). The sum for O runs over

the 3 families of operators [JJ]0,, [JJ]1,, and [

fJJ]0, as given in table 1. The di erential operators DO are given in (5.8), DL,T is given in (4.14), and DR,T is obtained by permuting

1 3, 2 4. The OPE coe cients PO are solutions to the leading order problem and are

given in (5.9).

Solving (5.12), we obtain the anomalous dimensions On, = On/ of operators in table 1 at leading order in 1/:

[JJ]0 =

16(3CJ 8JJT )2

34CT C2J

, (5.13)

JHEP02(2016)143

[JJ]1 =

64(3CJ 16JJT )2

34CT C2J

, (5.14)

[[tildewider]

JJ]0 =

32(3CJ 8JJT )(16JJT 3CJ)

34CT C2J

. (5.15)

We see the parity even double-twist states cannot have positive anomalous dimensions while the parity odd anomalous dimensions are not sign denite. Requiring that they be negative semidenite yields the conformal collider bounds (4.19). The fact that the negativity conditions of (5.15) agrees with that of (4.18) provides a non-trivial consistency check for our calculations. The results from the hJJJJi analysis is more general because

it does not assume the existence of scalar operators in the spectrum.

5.2.2 SU(N)

In the non-Abelian case the s-channel includes, in addition to the stress tensor, a conserved current in the Adja representation. At leading order in 1/, the resulting anomalous dimensions of the double-twist operators again take the form On, On/, where

[JJ]0 =

8(CJ 4JJJ)2

4C3J

J

16(3CJ 8JJT )2

34CT C2J

T , (5.16)

[JJ]1 =

8(8JJJ CJ)2 4C3J

J

64(16JJT 3CJ)2 34CT C2J

T , (5.17)

[[tildewider]

JJ]0 =

8(CJ 4JJJ)(8JJJ CJ)

4C3J

J

32(3CJ 8JJT )(16JJT 3CJ)

34CT C2J

T ,

(5.18)

with J = (2N, N, N, 0, 2, 2), T = (1, 1, 1, 1, 1, 1). They give the result for double-twist

operators in di erent representations I , Adja , Adjs , (S,)a (A,

S)a , (A,)s , (S, S)s

.

The second terms in (5.16)(5.18) are corrections due to the stress tensor exchange in the s-channel and correspond to the gravitational binding energies in AdS between well separated 2-particle states. The fact that they are the same for di erent representations is consistent with the universality of gravity. Once again, we nd that these anomalous dimensions are negative, or gravity in AdS is attractive at long distances, if and only if the same conformal collider bounds (4.19) are satised.

24

The corrections to the dimensions from the current exchange are given in the rst terms in (5.16)(5.18). In the dual AdS theory, they correspond to the binding energy from non-Abelian interactions for well separated 2-particle states. For the parity even states, we nd that the sign of the binding energy only depends on the SU(N) representation of the double-twist state. For a given family, it is the most negative for the singlet double-twist state and only positive for the symmetric representation (S, S)s. This matches with our intuition and also agrees with the signs found in the anomalous dimensions of scalar-scalar 2-particle states [47]. However, for the parity odd state this is not true a priori. The parity odd SU(N) singlet 2-particle state has non-positive gauge binding energy if and only if the 3-point function coe cients in hJJJi satisfy the bounds (4.23). This is also equivalent to

demanding the sign of the gauge binding energy to be independent of the parity of the bound state. As noted in section 4.2.2, the bounds (4.23) also imply that the charge ux one-point function does not change sign at di erent angles.

5.3 Higher spin symmetry at large N

As discussed in the hJJi case, if CJ does not grow with N, then the leading anomalous

dimension we computed holds only when N/ 1. Even with the freedom of choosing

JJJ, the anomalous dimensions of at least one family of double-twist operators would grow with N. The N regime is subtle because the large expansion is not separated

from the large N expansion. In this subsection we will focus on the opposite regime of N , where we will not be able to establish the presence of the double-twist operators.

However, the crossing equations and unitarity imply that if CJ and hJJJJi stay nite in

the N limit, then the theory must contain an innite number of higher spin currents

at innite N. We will show this by assuming the higher spin currents do not exist and deriving a contradiction.

We focus on the rst two crossing equations in (F.2), where we rst take N and

then go to the lightcone limit. These crossing equations become:

u v

3GtI = Gs(S,) + Gs(A,) + Gs(S,S), (5.19)

JHEP02(2016)143

u v

GsAdja + GsAdjs + Gs(A,) Gs(S,S) , (5.20)

where Gs,tr denotes the part of the 4-point function corresponding to the representation r in either the s- or t-channel. These functions implicitly depend on the polarization vectors. Since the absence of higher spin currents is assumed, we only need to consider the exchange of the identity and twist 1 operators of spin 2 in the s-channel. Note

that GsI, which includes the contribution from identity and T exchange, drops out in this limit. GsAdja includes the contribution from J exchange. This contribution is non-zero if the OPE coe cients for hJJJi are not suppressed, or equivalently, if CJ stays nite as

N . We do not make any assumptions on the twist 1 operators exchanged in the

other representations except for the absence of higher spin currents.

The contribution from the global symmetry current to GsAdja contains a log(v) term at leading order in u. In addition, we have shown in [47] that it is impossible to reproduce such

25

3GtAdja = 1 2

a term when each primary operator in the t-channel contributes with the same sign. The same problem shows up here when we consider the terms multiplying the (Z1 Z2)(Z3 Z4)

and (Z1 P2)(Z2 P1)(Z3 P4)(Z4 P3) structures in both equations. Furthermore, the log(v)

terms in the s-channel Gsr multiplying these structures also contribute with the same sign. In particular, if one tries to cancel the logarithm in the r.h.s. of (5.20) by allowing for the exchange of twist 1 operators in the (S, S) representation, then this exchange will induce a log(v) term on the r.h.s. of (5.19), leading to another contradiction. The analysis then reduces to that in [47]: the exchange of a nite number of higher spin currents in the s-channel cannot remove the logarithms, rather we need to sum over an innite tower of higher spin currents. These higher spin currents necessarily transform non-trivially under the global symmetry group of the CFT.

Let us now consider the exchange of scalars with 12 < < 1. In fact, it is easy to see that they cannot appear in the s-channel with O(1) coe cients at N in a unitary

CFT. Such scalars would contribute a log(v) term that could not be cancelled by a sum over higher spin operators. The reason is that, if such higher spin operators existed, they would necessarily violate the unitarity bound. Thus, there are no nite contributions from scalars with 1/2 < < 1 to Gsr6=I at N .

The argument presented here for the existence of higher spin currents at N

(assuming CJ does not grow with N) is more general than the one made in [47], because we did not need to assume the existence of scalar operators in the spectrum.

6 Mixed stress tensor-scalar 4-point functions

In this section, we study the correlation functions of the form hT T i where T is the stress-energy tensor and is a scalar operator of arbitrary dimension. Since T is conserved it saturates the unitarity bound and has dimension d. At leading order in u 1, the 4-

point function will be dominated by the identity contribution in the s-channel. We will assume that the next leading order correction comes from stress-energy tensor exchange. Note that the correlator hT T Ji vanishes in three dimensional CFTs [72]. We will also not

consider the corrections due to the exchange of a light scalar.

6.1 Identity matching

At leading order in u, the 4-point function is approximately given by the factorized form found in generalized free theories. In the t-channel this is reproduced by two families of double-twist operators with even or odd parity, of the schematic form:

[T ]n, = T(2)n1 . . . 2,

g[T ]n, = T(2)n1 . . . 2. (6.1)

The parity-even operators [T ]n, have twist 1 + 2n + , while the parity-odd operators [

fT ]n, have twist 2 + 2n + . The crossing equation at leading order in u is given by

CT H212(P12)3(P34) =

26

JHEP02(2016)143

Pn, P[T]n,Dt[T]n,W t[T]n, + P [tildewidest][T ]n,Dt[tildewidest][T]n, W t[tildewidest][T]n, . (6.2)

We solve this equation at leading order in v, which restricts the t-channel operators to have n = 0. The t-channel di erential operators are constructed in appendix D.3, and take the form

Dt[T]0, =

6( + 2)( + 1)

(Dt11)2 2,0L

4 + 1

Dt14Dt11 1,1L + (Dt14)2 0,2L

6( + 2)( + 1)

(Dt22)2 2,0R

4 + 1

Dt23Dt22 1,1L + (Dt23)2 0,2R

,

(6.3)

Dt[tildewidest][T]0, =

3 +1

Dt11 ~Dt1 1,0L+ Dt14 ~Dt1 0,1L

3 +1

Dt22 ~Dt1 1,0R+ Dt23 ~Dt1 0,1R

.

(6.4)

We match all dot products appearing in H212 in a basis of independent tensor structures. In particular, we matched the coe cient of three structures: (Z1 P2)2(Z2 P1)2, (Z1 Z2)2, and (Z1 Z2)(Z1 P2)(Z2 P1). This results in a linearly dependent set of equations that

can be solved for the OPE coe cient at leading order in 1/:

JHEP02(2016)143

P[T]0, CT

2

223 ( )

1

2 , P [tildewidest]

[T ]0, CT

2

223 ( )

72 . (6.5)

6.2 Stress tensor matching

At the next leading order in u we include the exchange of T in the s-channel. This contribution is suppressed by a factor of u compared to the identity. This implies that the anomalous dimensions are 1/ suppressed at large . At leading order in v and next-to-leading order in u, the crossing equation takes the form

T

4CT DT W sT =

X[T]0,P[T]0,0Dt[T]0,W t[T]0, + [tildewidest][T ]0,P [tildewidest][T ]0,~0Dt[tildewidest][T]0, W t[tildewidest][T]0, ,

(6.6)

where we project onto the log(v) terms. The di erential operator DT is constructed in ap

pendix D.3 by matching to the 3-point function of the stress tensor. This 3-point function depends on two coe cients, CT and TT T , where CT is the central charge and TT T is dened explicitly in terms of the 3-point function structures in appendix D.3.12 Solving (6.6) then gives the anomalous dimensions of the leading double-twist states On, = O0/ with coe cients

[T]0 =

16 (3CT 16TT T ) ( )

7/2C2T 12

, (6.7)

8 (128TT T 21CT ) ( )

7/2C2T 12

. (6.8)

In an AdS bulk description, these anomalous dimensions corresponds to the correction to the energy of well separated graviton-scalar two particle states from gravitational interactions. We expect gravity to be attractive at large distances and requiring the anomalous

12It is related to the coe cient t4 used in [52] by the relation TTT = 3CT (60t4)210.

27

[[tildewider]

T ]0 =

dimensions to be negative semidenite yields the following constraints

CT > 0, 21CT128 TT T

3CT

16 . (6.9)

In a theory of free bosons and fermions in three dimensions we have [73]

CT = 3(2n + n)

322 , TT T =

9(16n + 7n)

40963 . (6.10)

It follows that free theories saturate the bounds:

TT T

CT

n=0

= 3

TT T

CT

128 . (6.11)

Moreover, the bounds on TT T above correspond to n 0 and n 0, which match the

conformal collider bounds found in [52], eq. (3.43).

7 Superconformal eld theories

It can be shown that every 3d superconformal eld theory (SCFT) trivially satises the conformal collider bounds on JJT , TT T , and the conjectured bound on JJJ. For the moment we will assume the conserved current corresponds to a avor (non-R) symmetry. Working in N = 1 superspace, it was found in [76] that the parity even 3-point functions

of conserved operators is xed up to an overall coe cient.13

Since these correlation functions are xed up to an overall coe cient we can calculate JJJ, JJT , and TT T in free supersymmetric theories in terms of the central charges using section 5 of [73]. For general free eld theories in three dimensions we have

JJJ

CJ = P

JHEP02(2016)143

16 ,

n=0

= 21

i 2C(ri)mf + C(ri)s

Pi 8(C(ri)mf + C(ri)s), (7.1)

JJT

CJ = P

i 3(2C(ri)mf + C(ri)s)

Pi 16(C(ri)mf + C(ri)s), (7.2)

TT T

3(8nmf + 7ns)

128(nmf + ns). (7.3)

Here C(ri) is the index of the representation and the subscripts mf and s stand for Majorana fermions and real scalars respectively. Finally, nmf and ns give the total number of real Majorana fermions and real scalars. In a free supersymmetric theory there are an equal number of Majorana fermions and real scalars, in total and in a given representation of the avor symmetry, so we have

JJJ

CJ =

3 16 ,

CT =

JJT

TT T

45256 . (7.4)

13In [76] any multiplet containing conserved operators is referred to as a supercurrent, but we will follow the terminology of [78, 79] and refer to only the supermultiplet containing the stress-energy tensor as the supercurrent.

CJ =

9 32 ,

CT =

28

Although we computed these using the free theory, these results holds for any 3d supercon-formal eld theory. We also see that JJT /CJ and TT T /CT satisfy the conformal collider bounds on hJJT i (4.19) and hT T T i (6.9). These values lead to a uniform integrated,

energy distribution measured at spatial innity after a local perturbation is created by a conserved current ([53], eq. (6.14)) or the stress-energy tensor ([52], eqs. (3.8) and (3.43)). We also note that the ratio JJJ/CJ satises the conjectured bounds on hJJJi in (4.23).

We have computed the charge correlator in appendix E in terms of JJJ/CJ, where we nd that the charge ux is also uniform for the supersymmetric value given above.

We will now move on to the case of R-symmetry currents. We will start with N = 2

supersymmetric theories which have a U(1)R symmetry. For clarity we can make the replacement C(ri)mf,s (qmf,si)2, where qi denotes the charge under the U(1)R symmetry.

As shown in [79], the three point function of the supercurrent is xed up to an overall constant, so once again we can calculate JJT /CJ and TT T /CT using a free eld theory of chiral multiplets (JJJ does not appear since we have three U(1) currents). A free N = 2

chiral multiplet consists of a scalar with R-charge 1/2 and a fermion of R-charge -1/2, so the results for JJT and TT T found in (7.4) still hold.

Next we consider theories with N = 3 SUSY, which have an SO(3) R-symmetry. Once

again we have a one parameter family of free eld theories, this time with an equal number of complex scalars and fermions in the spinor representation of the R-symmetry group [76]. The three point function of the supercurrent J is xed up to an overall constant [79], so

we will nd the same ratios as before.Finally, we will study theories with N = 4 SUSY, which were extensively studied

in [78]. What makes these theories special is that the R-symmetry group, SO(4), is locally isomorphic to SU(2)L SU(2)R. Therefore, we can consider a two-parameter family of free

eld theories, consisting of hypermultiplets qi and q in the (2,1) and (1,2) representation of the R-symmetry group, respectively. Here we label representations by their dimension and note that each hypermultiplet consists of four real scalars and four Majorana fermions. The supercurrent is given by a real scalar supereld J and its three point function has

two linearly independent tensor structures parametrized by dN=4 and ~dN=4. This is in

contrast to theories with less supersymmetry, where the three point function is xed up to an overall constant. The resolution found in [78] was that the N = 4 supermultiplet

contains two N = 3 supermultiplets, S and J, the latter being the N = 3 supercurrent,

and that only one of the tensor structures contributes to hJJJi.

We can now repeat the above analysis for a free theory with m left hypermultiplets and n right hypermultiplets, specializing to study the SU(2)L R-current, J,L. The correlation functions hJL,JL,Ti and hJL,JL,JL,i are once again xed up to an overall coe -cient and our results for JJJ and JJT hold as before with CJ CJ,L. The analogous

substitution will also have to be made for the SU(2)R current.

Using these results, we can determine the large spin spectrum of double-twist states involving an R-symmetry current. We will consider only the exchange of the R-current itself and the stress-energy tensor in the s-channel and not the exchange of light scalars or other conserved currents. For N = 2, hJRJRJRi vanishes so all double-twist states formed

from two R-currents or a R-current and a scalar will have negative anomalous dimensions.

29

JHEP02(2016)143

Moving on to theories with N = 3 SUSY, we rst need to nd the ratio CT /CJ. The

R-current and stress-energy tensor lie in the same supermultiplet, so we can calculate this ratio either by expanding the supercurrent two point function as in [80] or by calculating it in a free theory. Using eqs. (5.5) and (5.6) in [73], we nd that CT /CJ = 3. We then nd that the coe cient of anomalous dimensions for double-twist states formed from the R-current and a scalar in the adjoint representation of the R-symmetry group become

[JR]0 =

3( + 2) ( ) 7/2CT 12

,

3( + 1) ( ) 7/2CT 12

,

3( 1) ( )

7/2CT 12

!, (7.5)

JHEP02(2016)143

[ [tildewidest]

JR]0 =

12( + 1) ( ) 7/2CT 12

,

6(2 + 1) ( ) 7/2CT ( 12)

, 6(1 2 ) ( )

7/2CT ( 12)!

, (7.6)

where we have used that SO(3) SU(2) and expressed our results in the basis

I , Adja, (S, S)

. Note that all the anomalous dimensions are non-positive if 1. This bound is nothing more than the unitarity bound for 3d scalars in the adjoint representation of the SO(3) R-symmetry group [81]. Furthermore, scalars which saturate this bound belong to a short representation of the superconformal algebra and their leading anomalous dimension asymptotic for the parity even (S, S) double-twist state vanishes. Finally, for double-twist states formed from two R-currents, we have

[JRJR]0 =

94CT , 64CT , 0 , (7.7)

[JRJR]1 =

724CT , 604CT , 36 4CT

, (7.8)

. (7.9)

Once again, all the anomalous dimensions either vanish or are negative.Finally, we will consider theories with N = 4 SUSY and focus on the SU(2)L R-current.

For the moment we only consider the e ect of the R-currents and T in the s-channel. The supercurrent multiplet also contains a dimension 1 scalar which will contribute to the anomalous dimensions at the same order which we will consider later.

If we consider a double-twist state formed from the SU(2)L R-current and a scalar in the same representation (i.e., adjoint of SU(2)L and singlet of SU(2)R), we nd that the contribution of the R-current and stress energy tensor to the anomalous dimension asymptotics is given by

[JL] = ( )(2CT 3CJ,L )

7/2CT CJ,L 12

[[tildewider]

JRJR]0 =

244CT , 184CT , 6 4CT

,

( )(CT + 3CJ,L ) 7/2CT CJ,L 12

, ( )(CT 3CJ,L ) 7/2CT CJ,L 12

!,

(7.10)

[tildewider][J

L] =

4 ( )(CT +3CJ,L ) 7/2CT CJ,L 12

,

2 ( )(CT +6CJ,L ) 7/2CT CJ,L 12

, 2 ( )(CT 6CJ,L )

7/2CT CJ,L 12

!.

(7.11)

30

So the contribution is non-positive for all double trace states if and only if

CT 3CJ,L .

When this inequality is saturated the parity even (S, S) term vanishes to leading order.

For the double-twist states formed from two SU(2)L R-currents, the contribution of J,L and T to the anomalous dimensions becomes

, (7.12)

, (7.13)

. (7.14)

The above quantities are all negative if CT 3CJ,L. When this inequality is saturated

the contribution of T and J,L vanishes for twist two, parity even double-twist states in the (S, S) representation. Similar results can be found for the SU(2)R current by letting CJ,L CJ,R.

If these inequalities are not satised then for some double trace states the contribution of the R-current is greater than the contribution from the stress energy tensor. This may be related to a non-Abelian version of the weak gravity conjecture [82].

To be complete we would also have to include the dimension 1 scalar superconformal primary of the supercurrent multiplet, J. One might wonder whether, after taking it into account, we will obtain a convex spectrum once the relevant unitarity bounds are satised. There is a simple way to see this cannot be the case. As mentioned earlier, upon reduction from N = 4 superspace to N = 3 superspace, the supercurrent J splits into two superelds,

J, which contains the N = 3 R-symmetry currents, and S which contains the dimension

1 scalar and the missing N = 4 R-currents. In [78] they found that both hJJSi and

hSSSi are determined by a single parameter

~dN=4, which in a free theory is proportional

to m n. Therefore if the theory has m = n, or

~dN=4=0, the scalar makes no contribution

and some of the double trace states will still have a concave spectrum in twist space at large spin.

The e ect of the dimension 1 scalar on the double twist states of two JL currents is to shift the anomalous dimension of only the parity even, twist two state by

[JJ]0 =

diag

SU(2), given by Jad = JaL,+JaR,. We have re-introduced the adjoint indices for the currents to emphasize we are considering the diagonal subgroup. The analysis then exactly mimics the N = 3 case, assuming that when studying hJJi the scalar also transforms in the

adjoint representation of the diagonal subgroup.

31

[JLJL]0 =

2CT + 3CJ,L4CT CJ,L ,

CT + 3CJ,L4CT CJ,L ,CT 3CJ,L 4CT CJ,L

[JLJL]1 =

8(CT + 6CJ,L)4CT CJ,L , 4(CT + 12CJ,L)4CT CJ,L ,4(CT 12CJ,L) 4CT CJ,L

JHEP02(2016)143

[tildewider]

[JLJL]0 =

4(CT + 3CJ,L)4CT CJ,L , 2(CT + 6CJ,L)4CT CJ,L ,2(CT 6CJ,L) 4CT CJ,L

4(CT 6CJ,L)2 34CT C2J,L

1, 1, 1

. (7.15)

One question that arises is how to to reproduce the N = 3 results, where convexity was

automatically satised, from our N = 4 results. The natural choice is to identify the N = 3

R-symmetry currents as the generators of the diagonal subgroup SU(2)L SU(2)R

8 Discussion

By studying the conformal bootstrap equations in the lightcone limit, we have generalized the CFT argument for the cluster decomposition principle to operators with spin. In doing so we have derived the existence of large spin double-twist conformal primaries constructed from spinning operators. We computed their anomalous dimensions and showed that they turn o as . In an AdS dual description, a large spin double-twist operator cor

responds to a two particle state with a separation log(). The anomalous dimension

then describes the binding energy induced by the exchange of light particles such as gauge bosons and gravitons.

In this work we discovered a connection between the signs of the anomalous dimensions and the conformal collider bounds (4.19) and (6.9) in parity-symmetric 3d CFTs. In all cases under consideration, the anomalous dimensions due to stress-energy tensor exchange are negative semi-denite if and only if the conformal collider bounds are satised. These anomalous dimensions are expected to be non-positive from the bulk point of view, since we expect gravity to be attractive at large distances. We can turn the logic around and conclude that the conformal collider bounds, combined with our analysis, provide a pure CFT argument for the attractiveness of bulk gravity at long distances, which does not require a large N limit and holds for all unitary theories.

It would be interesting to see if the same bounds can be derived from more basic axioms such as unitarity or causality. The connection to unitarity and deep inelastic scattering (DIS) arguments were explored in [39, 83]. In [84] it was also seen that the conformal collider bounds on hT T T i in 4d can be derived from unitarity if the stress tensor is the

only spin-2 conserved operator in the T T OPE that can get a vacuum expectation value at nite temperature.

For classes of large N CFTs it has been shown that causality is related to energy ux positivity [52, 8591]. Causality of bulk gravity is also related to the negativity of the anomalous dimensions due to the exchange of T in the direct channel. The anomalous dimensions of double-twist states of large spin and twist formed from scalars in large N CFTs were found to be related to Shapiro time delay in the bulk [9294]. This result was generalized to arbitrary double-twist states formed from scalars in large N CFTs [74]. At least in large N theories, there is an intimate relation between the negativity of the anomalous dimensions or the attractiveness of gravity at long distances, causality in the bulk, and positivity of integrated energy one point functions in the Lorentzian CFT. Our work provides the direct link between the anomalous dimensions and the energy positivity conditions, and also extends the discussion beyond the large N limit to generic, nonperturbative CFTs. In this regime, it may also be possible to establish the connection to causality along the lines of [95].

Furthermore, we speculated that a new conformal collider-like bound (4.23) may exist for hJJJi, that its undetermined coe cient must lie in between the free fermion and

free boson values. This is equivalent to the requirement that the signs of the anomalous dimensions due to J exchange only depend on the global symmetry representations of the double-twist states and not on their spin or parity. In the conformal collider set-up, we computed the charge 1-point function in terms of hJJJi. We nd that the same bound

32

JHEP02(2016)143

implies that the expectation value of the integrated charge ux is positive at all angles after a positive amount of charge is injected with a local perturbation created by J, thus putting constraints on the charge ux asymmetry. In all cases we are aware of in three dimensions, this bound holds and it would be interesting to see if there exists a proof or explicit counterexamples.

We also applied our results to the study of 3d superconformal eld theories. The conformal collider bounds and conjectural bound for hJJJi are found to be satised for

a theory with any amount of SUSY. The value of the corresponding 3-point functions result in uniform energy/charge ux distributions at innity after a local perturbation. In addition, we nd that for SCFTs with N = 2, 3 SUSY the exchange of the supercurrent

multiplet induces non-positive anomalous dimensions for several families of double-twist operators formed by two R-currents or one R-current and one scalar in the adjoint of the R-symmetry. This does not seem to hold for theories with N = 4 symmetry. The

distinguishing characteristic of N = 4 theories in comparison to CFTs with less SUSY in

three dimensions is that the R-symmetry group is locally isomorphic to a product of groups.

We have restricted ourselves to 3d CFTs since this is the only case where all the conformal blocks are currently known. Given recent progress in calculating conformal blocks for hJJi [96], it should be straightforward to extend the arguments for hJJT i

to higher dimensions. Generalizing our study of hT T i and hJJJJi to 4d will require

more work. Another straightforward generalization will be to include the e ects of parity-violating couplings in 3d. We have also not yet studied hT T T T i in three dimensions as

incorporating all possible degeneracy equations requires an intricate analysis [97] and it is not required to probe the conformal collider bounds.

Our work is just a rst step in analytically solving the bootstrap equations for spinning operators. Some simple extensions would be to include external fermions, operators of higher spin, or non-conserved spin 1 and 2 operators. By studying anomalous dimensions of double-twist states with twist comparable to or much greater than their spin we can also hope to derive the more stringent bounds of [74] on corrections to Einstein gravity in AdS4.

Finally, we should note these correlation functions have not yet been studied with the numerical bootstrap. It will be exciting to see if these bounds can be derived there. Studying the 4-point functions of these conserved operators, both analytically and numerically, is a key step in mapping out the space of consistent CFTs.

Acknowledgments

We thank Tom Hartman, Jared Kaplan, Zuhair Khandker, Filip Kos, Petr Kravchuk, Juan Maldacena, David Simmons-Du n, Fernando Rejon-Barrera, Andreas Stergiou, Junpu Wang, Matthew Walters and Sasha Zhiboedov for discussions. This work is supported by NSF grant 1350180. DP and DL thank the Aspen Center for Physics for its hospitality during the completion of this work, supported by NSF Grant 1066293. DL thanks the Simons Center for Geometry and Physics at Stony Brook University for its hospitality during the completion of this work. DP receives additional support as a Martin. A and Helen Chooljian Founders Circle Member at IAS. DL and DM also thank the Institute for Advanced Study for its hospitality during the completion of this work.

33

JHEP02(2016)143

A Collinear conformal blocks at large spin

In the small v limit the t-channel conformal blocks become [69]

g{ i},(v, u) = v

1

2 (1 u) 2F1

1

2( + 2) + a,

1

2( + 2) + b; + 2; 1 u

, (A.1)

1

2 ( 3 2). The

form is same for the s-channel blocks in small u limit but with u v, a = 12( 2 1), and

b = 1

2 ( 3 4). This approximation is su cient in the s-channel since there we have a

nite number of blocks, but we will need to make further approximations in the t-channel. Using the integral representation of the hypergeometric function, we rewrite this as

g{ i},(v, u) = v

1

2 (1u)Z

where we use {, } in place of { , }. Here a =

1

2 ( 4 1) and b =

JHEP02(2016)143

1 (2 + )(1 t)b++

2 1tb++

2 1(t(u 1) + 1)a

2

.

(A.2)

We want to expand the above expression at large , where we keep y u2 [lessorsimilar] O(1). Dening s

ty(1t) and expanding in this limit yields

g{ i},(v, u) v

1

2

(b + + 2) (b + + 2)

22+1e

Z

0 ds

s2+ys (1)ab(ys )ab s

1

2 22+(1)aby

1

2 (ab)Ka+b(2y)

. (A.3)

Plugging in our values for a, b, and y yields our nal expression for crossed channel blocks in the limit with u2 [lessorsimilar] O(1):

g{ i},(v, u) v

1

2 22+u

= v

14 ( 1+ 2 3 4)K12 ( 1 2+ 3+ 4)(2u)

. (A.4)

This approximation breaks down when u2 1, but all of our sums are dominated by

regions of xed u2.

B Singularities in direct and crossed channel

A key result of this work is that to reproduce the identity block in the s-channel an innite number of double-twist states are required in the t-channel. In the case of four identical scalars hi, this can be explained by the fact that the identity block is power law

divergent in u while the t-channel blocks have a log(u) divergence (see (3.11), which reduces to this case with k=1 and 2 = 1 = ). Thus, any nite sum of the t-channel blocks cannot reproduce the s-channel contribution.

In the spinning case, the t-channel spinning conformal blocks are obtained by acting derivatives on the scalar blocks, which produces power law singularities in u that can potentially become comparable to the s-channel divergences. If this were the case, then

34

there may exist a solution to the crossing equation with a nite number of t-channel blocks. In this appendix we will rule out this possibility.

First we need to look at the small u limit of the collinear t-channel conformal block, given by

v 2 (1 u) 2F1

1 (12( 1 2 + 3 + 4 + 2)) (12( 1 4 + 2 + )) (12( 2 3 + 2 + ))

1

2 ( 1+ 2 3 4)

Clearly only the second term can lead to a singular behavior in a single spinning conformal block that matches the identity contribution. In the following calculations this will be the only term kept.

We start with hJJi and the parity even double-twist states. Looking at the

P12(Z1 Z2) structure in the even channel we get a contribution of order u2 , which is less

singular then the u identity contribution. Similarly, the contribution to (Z1P2)(Z2P1)

is of order u3 . For the parity odd blocks the contribution to the P12(Z1 Z2) structure

starts at order u +2, while the contribution to (Z1 P2)(Z2 P1) starts at order u +3.

Since we have to match all the dot products appearing in the identity piece, we only need to look at one structure and see that it is subleading for all the double-twist states to conclude that we cannot match the identity contribution with a nite number of blocks.

For hT T i, the even and odd double-twist states contribute to (Z1 P2)2(Z2 P1)2 starting at order u5 . The identity contribution has a power law singularity of order u in comparison, so we cannot match this with a nite number of spinning blocks.

Finally, we need to look at hJJJJi. For simplicity we restrict to the U(1) case, but

the symmetry group will not a ect our results. Furthermore, we will need to be more careful with our approximation of the collinear block due to logarithmic singularities that can arise for special values of the dimensions, e.g. if they are all equal. To take into account these singularities we start with the hypergeometric form of the collinear blocks and do not expand in u until after we act with the derivatives. The result is that both the twist 2 and 4 double-twist states contribute to the (Z1 P2)(Z2 P1)(Z3 P4)(Z4 P3) structure

at order log(u). The contribution of the twist 3 parity odd double-twist states to this structure vanishes at order u1, so we cannot match the u3 power law singularity from the identity channel.

To conclude, no t-channel block is singular enough at small u to match the identity contribution. Therefore we need an innite number of states, which as we showed in the body of the text, has the spectrum of the double-twist states.

35

1

2( + 2) +

1

2( + 2) +

3 2

4 1

2 ,

2 ; + 2; 1 u

(B.1)

v/2 (2 + ) csc

1

2( 1 + 2 3 4)

(12( 1+ 2 3 4+2)) (12( 1 + 4 + 2 + )) (12( 2 + 3 + 2 + ))

JHEP02(2016)143

u

.

C Degeneracy equations

Here we will review the degeneracy equations that appear in 3- and 4-point functions.

We start by deriving the degeneracy relations among tensor structures appearing in the 3-point functions of spinning operators in three dimensions. These degeneracies arise because the 3-point structures depend on 6 vectors {Pi, Zi}, which cannot be linearly

independent in the 5-dimensional embedding space. The relation between the structures are found to be [50, 55]:

(V1H23 + V2H13 + V3H12 + 2V1V2V3)2 = 2H12H13H23 + O({Z2i, Zi Pi}). (C.1) To prove this we embed the vectors in a 6d space so that they lie on the x6 = 0 surface. It follows that (Z1, Z2, Z3, P1, P2, P3) = 0, or that the contraction of the vectors with the 6d epsilon tensor vanishes. Squaring this expression and using the identity

(Z1, Z2, Z3, Z4, Z5, Z6)(W1, W2, W3, W4, W5, W6) = det1i,j6(Zi Wj), (C.2) we obtain the above degeneracy. We will use this identity repeatedly to derive degeneracy conditions for the 4-point functions of spinning operators.

Conformal invariance required that the 4-point tensor structures have the following properties:

Q(k)1234({iPi; iZi}) = Q(k)1234({Pi; Zi})YI(ii)i. (C.3)

The Q(k)(u, v) structures will be polynomials in the Hij and Vi,jk tensor structures. There are additional degeneracy equations for the four point function. The rst is that in general dimensions there are two independent Vi,jk for each i. For example when i = 1 we have

P23P14V1,23 + P24P13V1,42 + P34P12V1,34 = 0, (C.4)

with related identities for i = 2, 3, 4 related by permutation. Note that these conditions do not depend on the spacetime dimension. For d < 6 there are more degeneracies among the tensor structures. The four point function depends on 8 vectors, the four pairs of position and polarization vectors, while the embedding space, if d < 6, is at most 7-dimensional. We will only focus on d = 3 here with the embedding space being 5d. We will use (C.2) with di erent vectors to derive the 4-point degeneracies.

For hJJi there are no linear relations among the tensor structures. This is easy

to see, because the only nontrivial contraction of the vectors with the 6d epsilon tensor is (P1, P2, P3, P4, Z1, Z2), which must vanish. The only degeneracy conditions apart from (C.4) is then found from (P1, P2, P3, P4, Z1, Z2)2 = 0, and rewriting it in terms of dot products. This constraint is quadratic in Z1 and Z2, while the four point function

hJJi is linear in both. Therefore, there are no degeneracies among the relevant tensor

structures.

We now will consider possible degeneracies for the four point function tensor structures in hJJJJi. The basic structures are

{V1,23, V1,24, V2,34, V2,31, V3,41, V3,42, V4,12, V4,13, H12, H13, H14, H23, H24, H34}, (C.5) out of which one can construct 43 distinct structures.

36

JHEP02(2016)143

There are three degeneracy equations, linear in each Zi, which follow from the fact that we have six 5d vectors which cannot be linearly independent:

(P1, P2, P3, P4, Z1, Z2)(P1, P2, P3, P4, Z3, Z4) = 0, (C.6)

(P1, P2, P3, P4, Z1, Z3)(P1, P2, P3, P4, Z2, Z4) = 0, (C.7)

(P1, P2, P3, P4, Z1, Z4)(P1, P2, P3, P4, Z2, Z3) = 0. (C.8)

Each individual contraction with the epsilon tensor vanishes and the product of two yields the degeneracy equations for the H and V structures. The three equations are not linearly independent; solving two implies the third. We will choose to solve for the latter two. Converting to the standard basis yields:

H12v(H34(2(u+1)v + (u1)2 + v2) + 2uV4,12(V3,41(uv1) + 2V3,42) 2V4,13(V3,41(u + v 1) + V3,42(u v + 1))) + H14u(H23(u2 2u(v + 1) + (v 1)2) + 2vV3,41(V2,34(u v + 1) 2V2,31)

+ 2V3,42(V2,31(u+v+1) + V2,34(u + v1))) + 2(uV4,12(H23vV1,23(uv+ 1) + H23V1,24(u+v1)

2vV3,41(V2,31(V1,23 + V1,24) V2,34(uV1,23 vV1,23 + V1,24)) + 2V3,42(vV1,23(V2,31 + V2,34) + V1,24(V2,31 V2,34))) + V4,13(H23u(uV1,24 + v(V1,24 2V1,23) + V1,24)

+ 2vV3,41(V1,23 V1,24)(V2,31 uV2,34) 2V3,42(v(uV1,23V2,34 + V1,23V2,31 V1,24V2,31) + uV1,24(V2,31 V2,34))) + H34v(V1,23V2,31((u + v 1)) + uV1,23V2,34(u v 1)

+ V1,24V2,31(u + v 1) + 2uV1,24V2,34)) = 0, (C.9)

H12v(H34(u2 2u(v + 1) + (v 1)2) + 2uV4,12(V3,41(u v 1) + 2V3,42) 2V4,13(V3,41(u+v1) + V3,42(uv+1)))+H13u(H24v(u22u(v+1)+(v1)2)+2uV4,12(V2,34((2u+1)v+(u1)u+v2)

+ V2,31(u + v + 1)) + 2V4,13(V2,31(u + v 1) + uV2,34(u + v + 1)))+ 2(H24v(u2vV1,23V3,41 + uvV3,41(2vV1,23 V1,23 + V1,24) + uV3,42(vV1,23 + V1,24)

+ (v1)(vV1,23 V1,24)(vV3,41 V3,42)) + H34v(V1,23V2,31((u + v 1)) + uV1,23V2,34(uv1) + V1,24V2,31(u + v 1) + 2uV1,24V2,34) 2(u3vV1,23V2,34V3,41V4,12+ u2(2v2V1,23V2,34V3,41V4,12 v(V4,12(V1,23V3,41(V2,31 + V2,34)

V1,23V2,34V3,42 V1,24V2,34V3,41) + V1,23V2,34V3,41V4,13) + V1,24V2,34V3,42V4,12)+ uV4,12(vV3,41(vV2,34(vV1,23 V1,24) + (v + 1)V1,23V2,31) V3,42(V2,31(vV1,23 + V1,24)

+ vV2,34(vV1,23V1,24)))+uV4,13(vV3,41(V1,23(vV2,34+V2,31+V2,34)V1,24V2,34) V1,24V2,34V3,42) + (v 1)V2,31V4,13(vV1,23V3,41 V1,24V3,42))) = 0. (C.10)

Using (C.9) we can solve for V1,23V2,34V3,41V4,12 and using (C.10) we can solve for H13V2,34V4,12. The reason for choosing these structures is as follows. For each equation we would like to solve for the tensor structure that will yield the most singular contribution to H12H34. That is, we want to take into account the behavior of the tensor structures themselves in the lightcone limit when solving the degeneracy equations. In practice, we then solve the above equations in terms of the dot products (Z1 P3)(Z3 P1)(Z2 P4)(Z4 P2) and (Z1 Z3)(Z2 P3)(Z4 P2) in the respective equations.

Solving (C.9) for (Z1 P3)(Z3 P1)(Z2 P4)(Z4 P2) will a ect the large spin cross channel

37

JHEP02(2016)143

results, but not the s-channel. Solving (C.10) for (Z1 Z3)(Z2 P3)(Z4 P2) will not a ect

either channel in the lightcone limit.

For hT T i, we choose the basis of structures to be {V1,23, V1,24, V2,31, V2,34, H12}, from which one can construct the 14 four-point function structures:

{H212, H12V1,24V2,31, H12V1,24V2,34, V 21,24V 22,31, V 21,24V2,31V2,34, V 21,24V 22,34, H12V1,23V2,31,

H12V1,23V2,34, V1,23V1,24V 22,31, V1,23V1,24V2,31V2,34, V1,23V1,24V 22,34, V 21,23V 22,31,

V 21,23V2,31V2,34, V 21,23V 22,34}. (C.11)

There is a single degeneracy equation following from (P1, P2, P3, P4, Z1, Z2)2 = 0, which is

H212(u2 2u(v + 1) + (v 1)2) 4H12V2,31(V1,23(u + v 1) + V1,24(u v + 1))

+ 4H12uV2,34(V1,23(u v 1) + 2V1,24) + 4(uV1,23V2,34 V1,23V2,31 + V1,24V2,31)2 = 0.(C.12)

Following the same logic as for hJJJJi, we want to solve for the most singular tensor

structure in the lightcone limit. Since the above equation must hold for all congurations and polarizations, we see that this structure must be V 21,23V 22,34. The degeneracy equation says V 21,23V 22,34 = 14u2H212 + (. . .). Equivalently, we can expand the above equation in

terms of the dot products and solve for (Z1 P2)2(Z2 P4)2 to nd (Z1 P2)2(Z2 P4)2 =

14u2 (2(P1 P2)(Z1 Z2))2 + O(u).

D 3-point functions and di erential operators

In this appendix we provide more details about the structure of various 3-point functions in our analysis and the construction of the corresponding di erential operators.

D.1 hJi and hT i

For the scalars in the adjoint representation we have the general form

ha(P1)b(P2)Jc(P3; Z3)i =

The Ward identity implies

J = 1Sd . Given our normalization of the current, what

appears in the conformal partial wave expansion is

J =

d2

( d2 ) . CJ is the current central charge and describes the normalization of the current 2-point function,

hJ(P1; Z1)J(P2; Z2)i = CJ

38

JHEP02(2016)143

Jfabc V3(P12) d/2(P13)d/2(P23)d/2 . (D.1)

1

SdCJ , (D.2)

where Sd gives the volume of d1 dimensional sphere, Sd =

2

J

CJ =

H12

(P12)d . (D.3)

Similarly, for 3-point functions between scalars and the stress tensor we have

h(P1)(P2)T (P3; Z3)i =

T V 23 (P12) 1

d

2 (P13)

d

2 +1(P23)

d

2 +1 ,

T =

d

(d 1)Sd

, (D.4)

where we use the normalization

H212(P12)d+2 , (D.5)

where CT is the central charge. The term appearing in the conformal partial wave expansion has an extra division by CT ,

T =

hT (P1; Z1)T (P2; Z2)i = CT

d

(d 1)Sd

1CT . (D.6)

D.2 hJJJi and hJJT iWe now present the di erential representation of the parity preserving three point functions for hJJJi and hJJT i.

The parity preserving 3-point function for hJaJbJci in embedding space is

hJa(P1, Z1)Jb(P2, Z2)Jc(P3, Z3)i = fabc

JHEP02(2016)143

a1V1V2V3 + a2H12V3 + a3H13V2 + a4H23V1

(P12)

d

2 (P13)

, (D.7)

where fabc are the structure constants. Conservation imposes a2 = a3 = a4. The relation to the parametrization found in [73], eq. (3.10), is a1 = a 2b and a2 = b. The Ward

identity further imposes that

Sd

1da + b

d

2 (P23)

d

2

= CJ, (D.8)

We have labelled the OPE coe cient b as JJJ. The correct di erential operator that reproduces (4.20) when acting on a scalar-scalar-current 3-point function is:

DL,J = dCJ(d2)Sd

(D12D22 0,2L+D11D21 2,0L D12D21 1,1L) +

4JJJSd dCJ Sd(d 2)

D11D22 1,1L.

(D.9)

We now proceed to study hJJT i. Without loss of generality we can now restrict to the

case where J is a U(1) current. Conformal invariance and symmetry under 1 2 implies

hJ(P1; Z1)J(P2; Z2)T (P3; Z3)i =

V1V2V 23 + (H13V2 + H23V1)V3 + H12V 23 + H13H23

(P12)

d

d

2 1(P13)

d

2 +1(P23)

2 +1 .

(D.10)

Conservation implies

d + (2 + d) = 0, 2 + 2 + (2 d) = 0. (D.11)

39

The implications of conservation for hJJT i was rst solved in [73], see eqs. (3.11)(3.14).

The relations between our parametrization and theirs is given by

= 2e, = 2c, = a

bd

4cd , = 2a + b

1 2 d

8cd , (D.12)

where the parameters a and c used to parametrize hJJT i are unrelated to those used in hJJJi. Furthermore they solved the Ward identities to nd2Sd(c + e) = dCJ. (D.13)

So hJJT i is xed up to one OPE coe cient and CJ. We labelled the OPE coe cient c as

JJT in the body of the paper. The required di erential operator is then found to be

DL,T =

JHEP02(2016)143

2JJT

CJd(d2) (d 1)Sd

D11D22+

2JJT + CJd2 Sd(1d)

D12D212JJT H12

1,1L.

(D.14)

D.3 Di erential operators for hT T T i and hT [T ]iWe will start by analyzing hT T T i in the standard basis and then the di erential basis.

Restricting to parity-preserving correlators, the allowed tensor structures are

Q1 = V 21V 22V 23, (D.15)

Q2 = H23V 21V2V3 + H13V1V 22V3, (D.16)

Q3 = H12V1V2V 23, (D.17)

Q4 = H12H13V2V3 + H12H23V1V3, (D.18)

Q5 = H13H23V1V2, (D.19)

Q6 = H212V 23, (D.20)

Q7 = H213V 22 + H223V 21, (D.21)

Q8 = H12H13H23. (D.22)

The H12H13H23 structure is not linearly independent in three dimensions, as follows from eq. (2.15). Above we only required symmetry under interchange between 1 2. We could

have also required symmetry under 2 3, but the above basis is simpler when comparing

the results to [73] where the latter symmetry was obscured.

The constraints of conservation were solved in [73] for general dimensions where they parametrized the correlation function in terms of 8 variables: a, b, b, c, c, e, e, and f.

These parameters are unrelated to those appearing in the hJJJi and hJJT i correlation

functions. Labelling the coe cients of Qi by xi, the mapping between the bases is given by

x1 = 8(c + e) + f, x2 = 4(4b + e), x3 = 4(2c + e), (D.23)

x4 = 8b, x5 = 8b + 16a, x6 = 2c, (D.24) x7 = 2c, x8 = 8a. (D.25)

40

Conservation at P1 or P2 imposes that

x1 = 2x2 + 14(d2 + 2d 8)x4

1

2d(2 + d)x7, x8 =

x2

d2 + 1

x4 + 2dx7

, (D.26)

x2 = x3, x4 = x5, x6 = x7, (D.27)

which is consistent with the conservation conditions found in [73].Furthermore, they found that the Ward identity constraints are given by

4Sd (d 2)(d + 3)a 2b (d + 1)c

d(d + 2) = CT . (D.28)

Imposing the Ward identity and using the degeneracy equation (2.15), we nd in d = 3

x1 = 147TT T

d2 2

2

JHEP02(2016)143

405CT

16 , (D.29)

x2 = x3 = 52TT T

75CT

8 , (D.30)

15CT

x4 = x5 = 16TT T

4 , (D.31) x6 = x7 = 2TT T , (D.32)

x8 = 0, (D.33)

where TT T = 2a c. As expected, we nd that in three dimensions the parity-even part

of hT T T i has two linearly independent forms, which we parametrize with TT T and CT .

Note that in [52] the extra parameter was called t4. Our parametrization is related to theirs by the relation TT T = 3CT (60t4)210.

We now need to nd the mapping between the standard basis and the di erential basis. An over-complete di erential basis symmetric under 1 2 is given by:

W1 = D211D222 2,2L, (D.34)

W2 = H12D11D22 2,2L, (D.35)

W3 = D21D211D22 3,1L + D12D222D11 1,3L, (D.36)

W4 = H12(D21D11 3,1L + D12D22 1,3L), (D.37)

W5 = D12D21D11D22 2,2L, (D.38)

W6 = H212 2,2L, (D.39)

W7 = D221D211 4,0L + D212D222 0,4L, (D.40)

W8 = H12D12D21 2,2L, (D.41)

W9 = D212D221 2,2L, (D.42)

W10 = D12D221D11 3,1L + D21D212D22 1,3L. (D.43)

Although there are 10 possible di erential operators, only the rst 8 are required to express

hT T T i in terms of di erential operators acting on a scalar structure. That is, we can nd

41

an invertible matrix such that

Wi =

8

Xj=1aijQj and Qi =

8

Xj=1(a1)ijWj. (D.44)

The matrix (a1)ij in general dimensions is given in appendix F. The di erential representation of hT T T i is then given by

DT =

Xijxi(a1)ijWj. (D.45)

Taking into account our normalization for hT T i, what appears in the conformal block

expansion is then 1

CT DT .

Let us consider the 3-point functions hT O()i where O() are double-twist states.

The two possible operators take the schematic form [T ]n, = T(2)n1 . . . 2 and

g[T ]n, = T(2)n1 . . . 2, with twists 2+2n and 3+2n respectively. Below we will restrict to the n = 0 operators.

Starting with the parity even di erential operators, the most general operator is

f1D211 2,0L + f2D12D11 1,1L + f3D212 0,2L. (D.46)

Imposing conservation yields

f1 = 6f3( + 2)( + 1)

JHEP02(2016)143

, f2 =

4f3 + 1

. (D.47)

For the parity odd states, the di erential operator has the form

t1D11 ~D1 1,0L + t2D12 ~D1 0,1L. (D.48)

Conservation implies

t1 =

3t2 + 1

. (D.49)

The t-channel left di erential operators are constructed in the usual way by letting 2 4

in the denition of the di erential building blocks. The right di erential operators are then constructed from the left operators by letting 1 2 and 3 4. The end result is

Dt[T]0, =

6( + 2)( + 1)

(Dt11)2 2,0L

4 + 1

Dt14Dt11 1,1L + (Dt14)2 0,2L

6( + 2)( + 1)

(Dt22)2 2,0R

4 + 1

Dt23Dt22 1,1L + (Dt23)2 0,2R

,

(D.50)

Dt[tildewidest][T]0, =

3 +1

Dt11 ~Dt1 1,0L + Dt14 ~Dt1 0,1L

3 +1

Dt22 ~Dt1 1,0R + Dt23 ~Dt1 0,1R

.

(D.51)

42

E Charge 1-point function

In this appendix, we compute the charge ux 1-point function in general dimensions. The charge ux 1-point function was dened in [51]. In a CFT with a non-Abelian global symmetry G, we inject a unit amount of charge with a local perturbation iJ+i at the origin, where the + indicates that the operator carries charge +1 under a chosen U(1) G.

The perturbation propagates and carries the charge away to innity. A charge detector at spatial innity along the direction of a unit vector will detect an integrated charge ux given by

hQ(n)i =

1 Sd

JHEP02(2016)143

1 +2

| n|2

||2

1d 1

. (E.1)

The second piece integrates to zero and characterizes the asymmetry in the charge ux.2 is a coe cient determined by the microscopic theory. In this appendix, we determine2 in

a free theory involving Ndf Dirac fermions and Ns scalars transforming under the global symmetry. These numbers are counted using the index of the representations [73],

Tr(tastbs) = Nsab, Tr(tadftbdf) = Ndfab. (E.2)

Our method is an extension of the appendix C of [70] to the case of spin-1 currents.In free theories,2 has the following general form:

2 = c1Ns + c2NdfCJ . (E.3)

This follows from the denition of the charge correlator, since the three point function

hJJJi in a free theory is linear in Ns and Ndf. The CJ in the denominator comes from

normalizing the charge correlator with the two point function hJJi [73]:

CJ = 1 S2d

Ns d 2

+ Ndf2d2

(E.4)

d2 in d dimensions. We argue that in a theory of free bosons it is impossible to create two particles propagating back to back perpendicular to the direction of the current, so that the charge correlator has to vanish at n = 0.

We create a state with J+1 and consider the matrix element hp, p| J+|0i, where hp, p|

denotes a two particle state with p1 = 0. Under a reection of the rst axis, J+1 J+1

and p p. Therefore the matrix element vanishes because the operator is antisymmetric

but the state is symmetric under this reection. A similar argument indicates that the fermion charge 1-point function should vanish when and n are parallel.

This xes

c1 = 1 S2d

d 1

d 2

Note that the matrixes are

d 2

, c2 =

2d2

S2d

d 1

d 2

, (E.5)

so consequently we nd

2 = (d 1)

Ns Ndf2d2

Ns + (d 2)Ndf2d2

. (E.6)

43

36Ndf

Setting d = 4, we obtain2 = 3

8Ndf +Ns . This matches with the result in 4 dimensions given in [51]. We also see in general dimensions that if we have an equal number of on-shell bosonic and fermionic degrees of freedom in a representation,2 vanishes and the result is a uniform distribution.

Setting d = 3, we see that the charge is always positive if2 lies between the free eld theory values in 3 dimensions. In a theory with only scalars charged under the relevant global symmetry, there is a zero at = /2 and if there are only charged fermions there is a zero at = 0, .

We can rewrite this in terms of the coe cients CJ and JJJ that parameterize hJJJi.

In a free theory, we have:

JJJ = Ns2(d 2)S3d

+ Ndf2d/2S3d

JHEP02(2016)143

, (E.7)

CJ = Ns

(d 2)S2d

+ Ndf2d/2S2d

. (E.8)

We then obtain

2 = d 1 d 2

2d 3 JJJ2SdCJ (d 1) . (E.9)

If we require hQ()i to be non-negative for all , thenCJ2Sd JJJ

CJ

Sd . (E.10)

For d = 3, this agrees with (4.23).

F Other technical details

Here we will collect some other formulas referenced in the body of the text.

F.1 Change of basis for hT T T iIn general dimensions the matrix (a1)ij is given by

1

2h43h3+h

1h 2h43h3+h

1 2h4h3+h2

1

h(2h2+h1)

2 2h43h3+h

1

2h2+2h

1 4h46h3+2h2

1 h(2h2+h1)

h+3 2h43h3+h

h(h+2)+5 h(h+1)2(2h1)

2(h+2) h2(2h2+h1)

2 2h3h2+h

2(h+3) 2h43h3+h

1 h2+h

h+4 4h46h3+2h2

h+3 2h3h2+h

0 1

2h4h2 0

1

2h4h2 0

1

2h 0

1

2h4h2

0 1

2h1

1

h 0 1

h2h2 0

1

h 0

h+1 h2h2

h(h+2)+3 h(h+1)2(2h1)

h3+5h2+9h+11

4h46h3+2h

h(h+3)+4 h2(2h2+h1)

(h1)(h+3) 2h(2h2+h1)

h(h+2)+5 h(h+1)2(2h1)

,

1

2h

1

h+1

h(h+2)+5 2h(2h2+h1)

0 0 0 0 0 1 0 0 2(h+3)

h(2h2+h1)

4(h+3) h(2h2+h1)

h+2 2h43h3+h2

2h+8h22h3 0

8

h(2h2+h1) 0

h2+3h+4 2h43h3+h2

4 h2h2

0 (

h1)2h2h2 0

1h2h2h2 0

h1h 0

h2+1 h2h2

(F.1)

where h = d2 and the matrix maps the di erential basis Wi (eqs. (D.35)(D.42)) to the standard basis Qi (eqs. (D.15)(D.22)).

44

F.2 SU(N) adjoint crossing matrix

The matrix M used in the crossing symmetry equation for four SU(N) adjoints is given by

Mr

r = (F.2)

1 (N1)(N+1)

2N (N1)(N+1)

2(N2)(N+2) (N1)N(N+1)

(N2)(N+2) (N1)(N+1)

(N3)N2 (N1)2(N+1)

N2(N+3)

(N1)(N+1)2

1

2N

1

2

(N2)(N+2)2N2 0

N3 2(N1)

N+3 2(N+1)

N 2(N2)(N+2)

N2 2(N2)(N+2)

N212 2(N2)(N+2)

N (N2)(N+2)

(N3)N3 2(N2)2(N1)(N+2)

N3(N+3)

2(N2)(N+1)(N+2)2

,

1

2 0 2N

1

2

(N3)N 2(N2)(N1)

N(N+3) 2(N+1)(N+2)

JHEP02(2016)143

1 4

1

2

N+22N N+24N

N2N+2 4(N2)(N1)

N+3 4(N+1)

1 4

12

N2 2N

N24N

N3 4(N1)

N2+N+2 4(N+1)(N+2)

in the basis r = I , Adja , Adjs , (S,)a (A,

S)a , (A,)s , (S, S)s

.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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