The analytic bootstrap and AdS superhorizon locality

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We take an analytic approach to the CFT bootstrap, studying the 4-pt correlators of d > 2 dimensional CFTs in an Eikonal-type limit, where the conformal cross ratios satisfy |u| |[upsilon]| < 1. We prove that every CFT with a scalar operator must contain infinite sequences of operators ... with twist approaching [tau] [arrow right] 2[Delta]^sub ^ + 2n for each integer n as [arrow right] ∞. We show how the rate of approach is controlled by the twist and OPE coefficient of the leading twist operator in the × OPE, and we discuss SCFTs and the 3d Ising Model as examples. Additionally, we show that the OPE coefficients of other large spin operators appearing in the OPE are bounded as [arrow right] ∞. We interpret these results as a statement about superhorizon locality in AdS for general CFTs.

Published for SISSA by Springer

Received: June 26, 2013

Accepted: November 22, 2013

Published: December 2, 2013

A. Liam Fitzpatrick,a Jared Kaplan,a,b David Polandc and David Simmons-Du nd

aStanford Institute for Theoretical Physics, Stanford University,

Stanford, CA 94305, U.S.A.

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

cDepartment of Physics, Yale University,

New Haven, CT 06520, U.S.A.

dSchool 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] , mailto:[email protected]

Web End [email protected]

Abstract: We take an analytic approach to the CFT bootstrap, studying the 4-pt correlators of d > 2 dimensional CFTs in an Eikonal-type limit, where the conformal cross ratios satisfy |u| |v| < 1. We prove that every CFT with a scalar operator must

contain innite sequences of operators O, with twist approaching 2 + 2n for each

integer n as . We show how the rate of approach is controlled by the twist and OPE

coe cient of the leading twist operator in the OPE, and we discuss SCFTs and the

3d Ising Model as examples. Additionally, we show that the OPE coe cients of other large spin operators appearing in the OPE are bounded as . We interpret these results

as a statement about superhorizon locality in AdS for general CFTs.

Keywords: Conformal and W Symmetry, Nonperturbative E ects, AdS-CFT Correspondence

ArXiv ePrint: 1212.3616

c

The analytic bootstrap and AdS superhorizon locality

JHEP12(2013)004

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP12(2013)004

Web End =10.1007/JHEP12(2013)004

Contents

1 Introduction and review 11.1 Lightning bootstrap review 3

2 The bootstrap and large operators 42.1 An elementary illustration from mean eld theory 42.2 Existence of twist 2 + 2n + (n, ) operators at large 62.2.1 Relation to numerical results and the 3d Ising model 102.3 Properties of isolated towers of operators 112.3.1 Implications for SCFTs in 4d 142.4 Bounding contributions from operators with general twists 152.5 Failure in two dimensions 16

3 AdS interpretation 17

4 Discussion 18

A Properties of conformal blocks 20A.1 Factorization at large and small u 20 A.1.1 Factorization in 2 and 4 dimensions 20 A.1.2 Extension to even dimensions via a recursion relation for F (d)(, v) 21

A.1.3 Extension to odd dimensions 22 A.1.4 Further approximations for the function k2(1 z) 22

A.2 Positivity of coe cients and exponential fallo at large 23

B () and its crossing equation 24B.1 Existence of () 24B.2 Crossing symmetry for () 26B.3 Bounds on OPE coe cient densities 27

C Generalization to distinct operators 1 and 2 29

D Comparison with gravity results 30

1 Introduction and review

The last few years have seen a remarkable resurgence of interest in an old approach to Conformal Field Theory (CFT), the conformal bootstrap [1, 2], with a great deal of progress leading to new results of phenomenological [39] and theoretical [1015] import. Most of these new works use numerical methods to constrain the spectrum and OPE coe cients

1

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of general CFTs. In a parallel series of developments, there has been signicant progress understanding e ective eld theory in AdS and its interpretation in CFT [10, 13, 1619]. This has led to a general bottom-up classication of which CFTs have dual [2022] descriptions as e ective eld theories in AdS, providing an understanding of AdS locality on all length scales greater than the inverse energy cuto in the bulk. In fact, these two developments are closely related, as the seminal paper [10] and subsequent work begin by applying the bootstrap to the 1/N expansion of CFT correlators. This approach has been fruitful, especially when interpreted in Mellin space [19, 2328], but it is an essentially perturbative approach analogous [29] to the use of dispersion relations for the study of perturbative scattering amplitudes.

In light of recent progress, one naturally wonders if an analytic approach to the bootstrap could yield interesting new exact results. In fact, in [14] bounds on operator product expansion (OPE) coe cients for large dimension operators have already been obtained. We will obtain a di erent sort of bound on both OPE coe cients and operator dimensions in the limit of large angular momentum, basically providing a non-perturbative bootstrap proof of some results that Alday and Maldacena [30] have also discussed.1

Specically, we will study a general scalar primary operator of dimension in a CFT in d > 2 dimensions. We will prove that for each non-negative integer n there must exist an innite tower of operators O, with twist 2 + 2n appearing in the OPE

of with itself. This means that at large , and we can dene an anomalous dimension (n, ) which vanishes as . If there exists one such operator at each n and , we will

argue that at large the anomalous dimensions should roughly approach

(n, )

nm , (1.1)

where m is the twist of the minimal twist operator appearing in the OPE of with itself. Related predictions can be made about the OPE coe cients. Finally, we will show that the OPE coe cients of other operators appearing in the OPE of with itself at large must be bounded, so that they fall o even faster as . Similar results also hold for the

OPE of pairs of operators 1 and 2, although for simplicity we will leave the discussion of this generalization to appendix C.

Our arguments fail for CFTs in two dimensions, and in fact we will see that the c = 1/2 minimal model provides an explicit counter-example. Two dimensional CFTs are distinguished because there is no gap between the twist of the identity operator and the twist of other operators, such as conserved currents and the energy-momentum tensor.

Our results can be interpreted as a proof that all CFTs in d > 2 dimensions have correlators that are dual to local AdS physics on superhorizon scales. That is, CFT processes that are dual to bulk interactions will e ectively shut o as the bulk impact parameter is taken to be much greater than the AdS length. This can also be viewed as a strong form of the cluster decomposition principle in the bulk. Since the early days of AdS/CFT it

1The authors of [30] explicitly discuss minimal twist double-trace operators in a large N gauge theory; however their elegant argument can be applied in a more general context, beyond perturbation theory and for general twists. We thank J. Maldacena for discussions of this point.

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has been argued that this notion of coarse locality [10] could be due to a decoupling of modes of very di erent wavelengths, but it has been challenging to make this qualitative holographic RG intuition precise. The bootstrap o ers a precise and general method for addressing coarse locality.

For the remainder of this section we will give a quick review of the CFT bootstrap. Then in section 2 we delve into the argument, rst giving an illustrative example from mean eld theory (a Gaussian CFT, with all correlators xed by 2-pt functions, e.g. a free eld theory in AdS). We give the complete argument in sections 2.2 and 2.4, with some more specic results and examples that follow from further assumptions in section 2.3. We provide more detail on how two dimensional CFTs escape our conclusions in section 2.5. In section 3 we connect our results to superhorizon locality in AdS, and we conclude with a brief discussion in section 4. In appendix A we collect some results on relevant approximations of the conformal blocks in four and general dimensions. In appendix B we give a more formal and rigorous version of the argument in section 2. In appendix C we explain how our results generalize to terms occurring in the OPE of distinct operators 1 and 2.

In appendix D we connect our results with perturbative gravity computations in AdS.

Note added. After this work was completed we learned of the related work of Komargodski and Zhiboedov [31]; they obtain very similar results using somewhat di erent methods.

1.1 Lightning bootstrap review

In CFTs, the bootstrap equation follows from the constraints of conformal invariance and crossing symmetry applied to the operator product expansion, which says that a product of local operators is equivalent to a sum

(x)(0) =

XOcOfO(x, )O(0). (1.2)

Conformal invariance relates the OPE coe cients of all operators in the same irreducible conformal multiplet, and this allows one to reduce the sum above to a sum over di erent irreducible multiplets, or conformal blocks. When this expansion is performed inside of a four-point function, the contribution of each block is just a constant conformal block coe cient PO c2O for the entire multiplet times a function of the xis whose functional

form depends only on the spin O and dimension O of the lowest-weight (i.e. primary)

operator of the multiplet:

h(x1)(x2)(x3)(x4)i =

1 (x212x234)

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XOPOgO,O(u, v), (1.3)

where xij = xi xj, the twist of O is O O O, and

u =

x212x234 x224x213

, (1.4)

are the conformally invariant cross-ratios. The functions gO,O(u, v) are also usually re

ferred to as conformal blocks or conformal partial waves [3235], and they are crucial elementary ingredients in the bootstrap program.

3

, v =

x214x223 x224x213

In the above, we took the OPE of (x1)(x2) and (x3)(x4) inside the four-point function, but one can also take the OPE between di erent pairs of operators, and the result should be the same. For example, swapping 2 3 gives the bootstrap equation

1 (x212x234)

XOPOgO,O(u, v) =1 (x214x223) XO

POgO,O(v, u). (1.5)

(Meanwhile, swapping 1 2 or 3 4 gives the constraint that only even spin operators

can appear in the OPE. Other permutations give no new constraints.) Much of the power of this constraint follows from the fact that by unitarity, the conformal block coe cients PO must all be non-negative in each of these channels, because the PO can be taken to be

the squares of real OPE coe cients.

2 The bootstrap and large operators

Although some of the arguments below are technical, the idea behind them is very simple. By way of analogy, consider the s-channel partial wave decomposition of a tree-level scattering amplitude with poles in both the s and t channels. The center of mass energy is simply s, so the s-channel poles will appear explicitly in the partial wave decomposition. However, the t-channel poles will not be manifest. They will arise from the innite sum over angular momenta, because the large angular momentum region encodes long-distance e ects. Crossing symmetry will impose constraints between the s-wave and t-wave decompositions, relating the large behavior in one channel with the pole structure of the other channel.

We will be studying an analogous phenomenon in the conformal block (sometimes called conformal partial wave) decompositions of CFT correlation functions. The metaphor between scattering amplitudes and CFT correlation functions is very direct when the CFT correlators are expressed in Mellin space, but in what follows we will stick to position space. In position space CFT correlators, the poles of the scattering amplitude are analogous to specic power-laws in conformal cross-ratios, with the smallest power-laws corresponding to the leading poles.

2.1 An elementary illustration from mean eld theory

Let us begin by considering what naively appears to be a paradox. Consider the 4-point correlation function in a CFT with only Gaussian or mean eld theory (MFT) type correlators. These mean eld theories are the dual of free eld theories in AdS. We will study the 4-pt correlator of a dimension scalar operator in such a theory. By denition, in mean eld theory the 4-pt correlator is given as a sum over the 2-pt function contractions:

h(x1)(x2)(x3)(x4)i =

. (2.1)

Since this is the 4-pt correlator of a unitary CFT, it has a conformal block decomposition in every channel with positive conformal block coe cients. The operators appearing in

4

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1 (x212x234)

+ 1

(x213x224)

+ 1

(x214x223)

,

= 1

(x213x224)

u + 1 + v

[parenrightbig]

the conformal block decomposition are just the identity operator 1 and the double-trace operators On, of the schematic form

On, (2)n1 . . . , (2.2)

with known [29] conformal block coe cients P2 +2n, and twists n, = 2 + 2n. Factoring out an overall (x213x224) , the conformal block decomposition in the 14 23

channel reads

u + 1 + v = v + v

Xn,

P2 +2n, g2 +2n,(v, u), (2.3)

where the v on the r.h.s. is the contribution from the identity operator. If we look at the behavior of the conformal blocks g2 +2n,(v, u), we notice a simple problem with this equation: it is known that the conformal blocks g2 +2n,(v, u) in the sum on the r.h.s.

each have at most a log u divergence at small u, but the l.h.s. has a u divergence. Thus the l.h.s. cannot be reproduced by any nite number of terms in the sum. To be a bit more precise, the conformal blocks have a series expansion around u = 0 with only non-negative integer powers of u and at most a single logarithm appearing, so in particular we can write2

v g2 +2n,(v, u) = f0(v) + uf1(v) + u2f2(v) + . . .

+ log(u) [parenleftBig]

. (2.4)

But this means that if the sum on the right-hand side of equation (2.3) converges uniformly, it cannot reproduce the left-hand side, which includes the negative power term u and

does not include any logarithms.

The simple resolution of this paradox is that the sum over conformal blocks does not converge uniformly near u = 0. In fact, the sum does converge on an open set with positive real u, but when Re[u] < 0 the sum diverges. So we must dene the sum over conformal blocks for general u as the analytic continuation of the sum in the convergent region. Crucially, the analytic continuation of the sum contains the power-law u that

is not exhibited by any of the individual terms in the sum.

Let us see how this works in a bit more detail, so that in particular, we can see that the sum over twists = 2 + 2n at xed converges in a neighborhood of u = 0, but the sum over angular momentum diverges for u < 0. For the purpose of understanding convergence, we need only study the conformal blocks when or are very large. In the very large limit with |u|, |v| < 1 the blocks are always suppressed by u

2 or v

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~f0(v) + u ~f1(v) + u2 ~f2(v) + . . .

[parenrightBig]

2 . The

conformal block coe cients are bounded at large [14]. This means that for small |u| and |v|, the sum over will converge. In fact, once we know that the sum converges for some

particular u0, v0 we see that for u < u0 and v < v0, the convergence at large becomes exponentially faster.

2See for example equation (2.32) in [32]. The sum over powers of (1 v) can be performed explicitly to

nd a hypergeometric function, whose singularities at v 1 are known.

5

Now consider the dependence. At large and xed , we establish in appendix A that the crossed-channel blocks in the |u| |v| 1 limit behave as

g,(v, u)

1

2 2+2

u1

2+21v

v

2 K0 2u

[parenrightbig]

2 e2

u

4

u , (2.5)

where K0 is a modied Bessel function. Notice that for Re[u] > 0 there is an exponential suppression at very large , but for Re[u] < 0 there is an exponential growth. Note also that at small v the lowest twist terms (n = 0) will dominate.

Now, the mean eld theory conformal block coe cients in any dimension d are [29]

P2 +2n, =

1 + (1)

[bracketrightbig]( d2 + 1)2n( )2n+!n!( + d2)n(2 + n d + 1)n(2 + 2n+1)(2 + n + d2)n, (2.6)

where the Pochhammer symbol (a)b (a + b)/ (a). In particular, for n = 0 and at large

even we can approximate

P2 , 1

q

22 +2 2

32 , (2.7)

where q is an -independent prefactor.3 Thus the sum in eq. (2.3) at large and |u| |v| 1 takes the form

v

Xn, large

P2 +2n, g2 +2n,(v, u) q

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Xlarge even 2 1K0 2u

[parenrightbig]

. (2.8)

This sum converges at large for positive real u, and so we will dene it by analytic continuation elsewhere in the complex u plane. As can be easily seen by approximating the sum with an integral, the result reproduces the u power-law term on the left-hand side of equation (2.3), as desired. Thus we have seen that general power-laws in u are reproduced by conditionally convergent large sums in the conformal block decomposition, with a power-law dependence on producing a related power of u as u 0.

2.2 Existence of twist 2 + 2n + (n, ) operators at large

In section 2.1 we saw how in MFT the sum over large conformal blocks in the crossed 14 23 channel controls the leading power-law behavior in u in the standard 12 34

channel. Now we will use the bootstrap equation (1.5) to turn this observation into a powerful and general method for learning about the spectrum and the conformal block coe cients P, at large in any CFT.

Separating out the identity operator, the bootstrap equation reads

1 +

X,

P, u2 f,(u, v) =

u v

X,

P, v2 f,(v, u)

, (2.9)

1 +

3Explicitly, q = [parenleftBig]

8

( )2

.

6

where we have written the conformal blocks as g,(u, v) = u

d2

2 ( = 0), d 2 ( 1),

strictly separates the identity operator from all other operators.

The arguments in this section will follow from an elementary point:

In the small-u limit, the sum on the right-hand side of equation (2.9) must correctly

reproduce the identity contribution on the left-hand side.

We will show that this implies the existence of towers of operators with increasing spin whose twists approach 2 + 2n, for each integer n 0. Together with the results in

appendix A and the more rigorous arguments in appendix B, we will provide a rigorous proof of this claim. In subsequent sections, we will consider subleading corrections to the small-u limit of eq. (2.9) coming from operators of minimal non-zero twist.

For the remainder of this section we will use the approximate relation

1

u v

2 f,(u, v) to emphasize their leading behavior at small u and v. We will work in d > 2 so that the unitarity bound on twists

[braceleftBigg]

(2.10)

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X,

P, g,(v, u), (u 0), (2.11)

valid up to strictly sub-leading corrections in the limit u 0.4 As we saw in section 2.1,

no nite collection of spins on the right-hand side of (2.11) can give rise to the left-hand side. This is true even including an innite sum over large . To understand how these terms are reproduced, we must study the large region of the sum on the right-hand side of equation (2.11). For this purpose we need a formula for the conformal blocks, g,(v, u),

at |u| 1 and large .

We show in appendix A that the blocks can be approximated in this limit by

g,(v, u) k2(1 z)v/2F (d)(, v) (|u| 1 and 1), (2.12)

k(x) x/22F1(/2, /2, , x), (2.13)

where z is dened by u = zz, v = (1 z)(1 z), and the function F (d)(, v) is positive and analytic near v = 0.5 The exact expression for F (d) will not be important for our discussion. Note that z 0 at xed z is equivalent to u 0 at xed v.

A key feature of eq. (2.12) is that the , z dependence of g, factorizes from the , v dependence in the limit z 0, . Thus, we expect operators with large spin

4The sum over conformal blocks on the left-hand side of the crossing relation is necessarily a subleading correction at small u. The reason for this is that we take v to a small but xed value when we take the small u limit, so the conformal blocks factorize at large spin as shown in eq. (2.5) (with v and u interchanged). The sum over spins on the left-hand side therefore manifestly cannot produce additional singularities in u. The sum over twists is regulated by the u 2 factor.

5For example, F (2)(, v) is given by eq. (A.8). In appendix A, we give recursion relations which allow one to generate F (d) in any even d. In odd d, one must resort to solving a di erential equation.

7

to be crucial for reproducing the correct z-dependence on the left-hand side of eq. (2.11). However, a particular pattern of twists should also be necessary for reproducing the correct v-dependence in eq. (2.11). What is this pattern?

To study this question, it is useful to introduce a conformal block density in twist space,

Du,v()

u v

X,

P,( )g,(v, u). (2.14)

By integrating Du,v() against various functions f(), we can study the contributions to the crossing equation from operators with di erent twists. One should think of it as a tool for studying the spectrum of operators in twist-space. Since the conformal blocks are positive in the region 0 < z, z < 1, and the coe cients P, are positive, Du,v is positive as well.

The full conformal block expansion comes from integrating Du,v() against the con

stant function 1. Thus, the crossing eq. (2.11) in the small u limit reads6

1 = lim

u0

[integraldisplay]

d2

Du,v(). (2.15)

As we discussed above, the u 0 limit on the r.h.s. is dominated by the sum over

large , so we are free to substitute the asymptotic form of the blocks eq. (2.12) into the denition of Du,v and maintain the same u 0 (equivalently z 0) limit,

lim

u0

Du,v() =

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dDu,v() =

[integraldisplay]

d2

d lim

u0

[parenleftbigg]

lim

z0

z

X,

P,k2(1 z)( )

v 2

(1 v) F (d)(, v),(2.16)

where we have used z = 1 v + O(z). Note that after this substitution, the and v-

dependence factors out into an overall function v

2 (1 v) F (d)(, v), while the u and

dependence is encapsulated in a particular weighted sum of OPE coe cients,

() lim

z0

z

X,

P,k2(1 z)( ). (2.17)

By crossing symmetry eq. (2.15), the density () satises

1 = (1 v) [integraldisplay]

d2

2 F (d)(, v). (2.18)

We claim that the only way to solve this relation with positive () is if is given by its value in mean eld theory, namely a sum of delta functions at even integer-spaced twist:

() =

Xn=0,1,...

P MFT2 +2n( (2 + 2n)). (2.19)

6We justify switching the limit and integration in appendix B. Roughly, it follows from the fact that the integral of Du,v() over regions with large falls exponentially with .

8

d()v

where the mean eld theory coe cients were given as a function of n and in equation (2.6). Expanding these coe cients at large and performing the sum in eq. (2.17) gives7

2nn!(2 + n d + 1)n. (2.20)

Let us give a brief argument for why this is the case. Note that the function F (d)(, v)

is analytic and positive near v = 0, so the small v behavior of the above integral comes from the term v

d2 + 1

2 . Since the right-hand side is independent of v, and the density () is nonnegative, we see that () must be zero for < 2 and also have a contribution proportional to ( 2 ).

The necessity of the other terms ( (2 + 2n)) now follows from the fact that

the l.h.s. is independent of v, while the rst term ( 2 ) contributes a power series

in v about v = 0, namely F (d)(2 , v). Terms with n > 0 are needed to cancel successive powers of v from this rst term. To see the necessity of this result most clearly, it is helpful to subtract the contribution of ( 2 ) from both sides of eq. (2.18):

O(v) = (1 v) [integraldisplay]

z

0 dfn()k2(1 z), (2.23)

7To perform this computation explicitly, we use the fact that the sum is dominated by the region of xed z2, so that one may use the approximation of eq. (A.21).

8We also note that one might reproduce this conclusion directly by doing a projection of the l.h.s. and r.h.s. onto specic high-spin conformal blocks using the method of conglomerating [29].

9We justify interchanging the limit and integration in appendix B.

9

P MFT2 +2n =

1

22 +2n

2 F (d)(, v) (2.21)

where () = () P MFT2 ( 2 ). We are left with an O(v) term on the l.h.s. which

must now be matched by a ((2 +2)) term in . Repeating this algorithm iteratively,

one can x () to be given by its value in MFT.8

The result eq. (2.19) has several consequences. Firstly, it implies the existence of a tower of operators with increasing spin whose twists approach = 2 + 2n, for each

n 0. To see this, let us integrate eq. (2.19) over a bump function h() with some width

around = 2 + 2n. Using the denition in eq. (2.17), we obtain9

lim

z0

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d()v

X,

P,h()k2(1 z) = P MFT2 +2n. (2.22)

The limit vanishes termwise on the l.h.s. , so a nite result can only come from the sum over an innite number of terms. Thus, for any , there are an innite number of operators with twist = 2 + 2n + O().

We can also be more precise about the contribution of these operators to the conformal block expansion at large . The sum above can be written as an integral over the OPE coe cient density in , for operators with twist near 2 + 2n

[integraldisplay]

X

2 +2n,

P,k2(1 z) =

where we dene an OPE coe cient density

fn() X 2 +2n,

P, ( ). (2.24)

For simplicity, we no longer show the bump function h() explicitly, but indicate its presence by writing 2 + 2n.

One intuitively expects that the OPE coe cient density fn() must be constrained at large in order to reproduce the identity operator in the crossed channel. In mean eld theory fn() has a power-law behavior, and so we expect that, in an averaged sense, fn()

must be similar in any CFT. This motivates introducing an integrated density

Fn(L) [integraldisplay]

L (2)

()2 fn(). (2.25)

In appendix B.3 we prove both upper and lower bounds

AUL2 > Fn(L) > AL L2 log(L) (2.26)

for some coe cients AU and AL in the limit of very large L. We expect that the lower bound can be improved to eliminate the logarithm and make a prediction Fn(L) = AnL2 .10 We

calculate in appendix B.3 that such a prediction would necessarily x An to be

lim

L

L2 Fn(L) =

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P MFT2 +2n

( )2 . (2.27)

In summary, we have shown that in any CFT we must have operators accumulating at twists 2 + 2n at large . In simple cases where these accumulation points are populated by a single operator at each , as in all perturbative theories, we can obtain specic relations for sums over the anomalous dimensions (n, ) and the conformal block coe cients P2 +2n,. We will explore these relations in the next subsection. In appendix C we briey explain how these results generalize when we have distinct operators 1 and 2, so that there must exist operators accumulating at twist 1 + 2 + 2n as .

2.2.1 Relation to numerical results and the 3d Ising model

In [8], the authors used numerical boostrap methods to constrain the dimensions of operators appearing in the OPE of a scalar with itself in 3d CFTs. They found numerical evidence that the minimum twist at each spin in the OPE satises

2 . (2.28) (More precisely, they compute a series of numerical bounds on , which appear to approach the presumably optimal bound eq. (2.28), at least for = 2, 4, 6.) Here, we note that

10In the case where the functions k(1 z) are replaced by their exponential approximation eq. (A.19),

the Hardy-Littlewood Tauberian theorem says that the upper and lower bound at large L are the same, and it xes their coe cient. It seems likely to us that an analogous theorem could be proven for the case at hand.

10

eq. (2.28) follows from our results, together with Nachtmanns theorem [36]. We have seen, among other things, that there exist operators in the OPE with twist arbitrarily close

to 2 and arbitrarily high spin . Nachtmanns theorem states that is an increasing function of for > 0, which implies eq. (2.28) for all > 0.

The 3d Ising Model is a particularly interesting example in light of this result. The theory contains a scalar with dimension 0.518. Thus, the minimum twist operators

at each have twist less than 1.036, which is very close to the unitarity bound. One

might interpret these operators for 4 as approximately conserved higher-spin currents.

It would be interesting to understand what the existence of these approximate currents implies for structure of the theory.

2.3 Properties of isolated towers of operators

So far, we have made no assumptions about the precise spectrum of operators that accumulate at the special values 2 + 2n. However, it is interesting to consider the

case where an accumulation point 2 + 2n is approached by a single tower of operators

O2 +2n, with = 0, 2, . . . , which are separated by a twist gap from other operators in the spectrum at su ciently large spin. This occurs in every example we are aware of, including all theories with a perturbative expansion parameter such as 1/N (see appendix D) or the t Hooft coupling in e.g. N = 4 SYM, and we will also discuss some non-perturbative

examples at the end of this subsection. It would be very interesting to identify any CFTs without operators with twists near 2 + 2n for every su ciently large value of .

With this additional assumption, we will be able to characterize subleading corrections to the bootstrap equation, eq. (1.5), in the small u limit. On the left-hand side, these corrections come from the operators Om with minimal nonzero twist. Thus, we have the

approximate relation

1 +

2

JHEP12(2013)004

u f,(v, u), (2.29)

which is valid up to subleading corrections in u in the limit u 0. We have assumed that

m 2 because higher spin operators either have twist greater than that of the energy-

momentum tensor or, as argued in [37, 38], they are part of an innite number of higher-spin currents that couple as if they were formed from free elds.11 Dolan and Osborn [34] have given a formula in general d appropriate for the conformal blocks corresponding to Om

exchange on the left-hand side, where we have expanded at small u. This is

fm,m(0, v) = (1 v)m2F1 [parenleftBig]

m

2 + m,

m

2 + m, m + 2m, 1 v

Xm=0Pmum2 fm,m(0, v)

X,

P, v 2

[parenrightBig]

. (2.30)

It will be important that this hypergeometric function can be expanded in a power series at small v with terms of the form vk(ak + bk log v). The logarithms will be related to the anomalous dimensions that emerge at large .

11Strictly speaking, the arguments in [37, 38] assumed d = 3 and studied correlators of currents, but it is likely that they can be extended to d 3. In any case, one can view m 2 as an assumption.

11

Now we expect there to be a nite separation between the lowest twist m and the other twists in the theory. To prove this, we consider two cases separately, that of m < d2 and that of m d 2. In the former case, Om must be a scalar operator due to the unitarity

bound (2.10). We will assume that there are a nite number of scalar operators with dimension below any given value,12 which immediately implies that the twist of Om must

have a nite separation from the other twists in the theory. To have a non-vanishing 4-pt correlator must be uncharged, so in the absence of lower twist scalars we have m = d2, because the energy-momentum tensor will appear in the (x)(0) OPE. We can then apply the Nachtmann theorem, which says that minimal twists must be non-decreasing functions of , to conclude that m is separated from the other twists in the theory. Note that, crucially, unless is a free scalar eld we have m2 < 0, so the sub-leading powers of

u grow as u 0. Taken together, these comments imply that there is a limit u 0 where

the exchange of the identity operator plus a nite number of Om dominates the left-hand

side of equation (2.9).

Assuming the existence of an operator with twist approaching 2 + 2n for each , we would like to constrain the deviation of their conformal block coe cients P2 +2n, from

MFT and their anomalous dimensions (n, ) On, 2 2n . This is possible because the Om contribute a dominant sub-leading contribution at small u, with a known

v-dependence that can be expanded in a power series with integer powers at small v. The fact that we have only integer powers vn and vn log v multiplying u

m

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2 on the left-hand side of equation (2.29) means that the right-hand side can reproduce these terms only with the conformal blocks we just discovered above, namely those with twists approaching the accumulation points (n, ) = 2 + 2n + (n, ) so that v

(n,)

2 is an integer power in

the limit.

In fact, expanding eq. (2.30) for the Om conformal blocks at small v gives

fm,m(v) = (m + 2m)(1 v)m

2 m2 + m

[parenrightbig]

Xn=0

m

2 + m

n n!

!2

vn

h2

(n + 1)

m2 + m

[parenrightBig][parenrightBig]

log v

[bracketrightBig]

, (2.31)

where (x) = (x)/ (x) is the Digamma function, and (a)b = (a + b)/ (a) is the Pochhammer symbol. The details of this formula are not especially important, except insofar as it makes explicit the connection between the coe cients of vn and vn log v in the series expansion. We will now see that the vn terms must come from P2 +2n, while the vn log v terms are a consequence of (n, ). This means that the anomalous dimension and the correction to the conformal block coe cients must be related at large . These quantities were seen to be related [10] to all orders in perturbation theory [29] in the presence of a 1/N expansion, so our result extends this relation to a non-perturbative context.

The vn log v terms in equation (2.31) can only be reproduced by expanding v

2 in

(n, ) in the large conformal blocks. For simplicity let us consider the situation where

12This assumption would follow, for instance, from the assumption that the CFT has a well-dened partition function at non-zero temperature, or that the four-point function of the energy-momentum tensor is nite.

12

there is only one operator accumulating near 2 + 2n for each , and that the conformal block coe cients approach P MFT2 +2n,. In this case we can write the r.h.s. of the crossing relation as

X

P MFT2 +2n,

z K0(2z)vn(1 v) F (d)(2 + 2n, v), (2.32)

where we have used a Bessel function approximation to k2(1 z) discussed in ap

pendix A.1.4.13 In order for the sum to produce an overall factor of u

m 2

(n, )2 log v

[bracketrightbigg]

1

2 4

z

m

2 , we

must have power law behavior in (n, ) at very large ,

(n, ) = n

m , (2.33)

with a coe cient n related to the OPE coe cient Pm of the leading twist operator Om in

equation (2.29). In large N theories, the coe cients n are suppressed by powers of 1/N as discussed in [30]. However, we stress that all we need in order to expand v

(n,)

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2 in log v

in the large sum is the property that it is power law suppressed as , which is true

even if the coe cients n are O(1) or larger.The integer powers of v in equation (2.31) must then be reproduced by

X

P MFT2 +2n,

vn 12 4 z K0(2z)(1v) F (d)(2 + 2n, v),

(2.34)

where the (n, ) ddn piece comes from expanding the dependence of the conformal block in small (n, ). Again requiring that we correctly produce the overall u

m 2

P2 +2n, + 12(n, )d dn

z

m

2 behavior,

we must have

m (2.35)

to leading order in 1/ in the large limit, with a coe cient cn related to n.

As an example, it is particularly simple to do this matching explicitly for the leading twist tower with n = 0. In this case, matching the log v and v0 terms gives the relations

0 = Pm

2 ( )2 (m + 2m) m2

P2 +2n, = cn

h

m2 + m

[parenrightBig]

+ + log 2 [bracketrightBig]

0, (2.36)

where is the Euler-Mascheroni constant. It is important to note the relative sign between 0 and Pm (which is strictly positive), since this is required in order to satisfy the

Nachtmann theorem asymptotically at large . It is then straightforward to continue this matching to higher orders in v.

13Strictly speaking, the Bessel function approximation breaks down for 1/z, so we are here implicitly

using the fact that the sum including the MFT coe cients is dominated by the region of xed 2z, where the approximation is valid. If the reader is concerned about this, one can instead write these sums using the hypergeometric function k2(1 z). However, we nd the Bessel function formulae to be useful for

explicitly doing computations using the integral approximations to these sums.

2

m2 + m

2 , c0 =

13

2.3.1 Implications for SCFTs in 4d

In [7, 12] the crossing relation was examined for a chiral primary operator of dimension in an N = 1 SCFT, and it was observed that in the OPE of (x) (0) there exists an

innite tower of operators with spin and dimension exactly 2 + . Supersymmetry and unitarity protect the dimensions of these operators, and further require a gap in twist before non-protected operators can appear. Thus, these operators form an isolated tower with vanishing anomalous dimensions. This immediately implies that the correlator in the channel must satisfy

h (x1) (x2) (x3) (x4)i =

1 (x212x234)

1 + u1 (0 log v + c + . . .) + . . .

[parenrightbig]

(2.37)

where the . . . denote higher order terms in u and v, and the power u1 comes from the U(1)R current multiplet (containing the stress tensor), which has 2 = 1. The point is that the term u1 log v must be absent, because it could only arise from the anomalous dimensions of operators with twist 2 + () in the channel, but we know that due to

supersymmetry, () = 0 exactly. The explicit results of [12] show that the superconformal block relevant for the channel is14

G=2,m(u, v) = (1)m [bracketleftbigg]g2,m(u, v)

[parenleftbigg]

m + 1

4m + 6

JHEP12(2013)004

g2,m+1(u, v)

[bracketrightbigg](2.38)

in terms of the usual 4d blocks given in equation (A.10). Taking m = 1 for the U(1)R current, one can easily verify that the u log v term cancels in this linear combination of conformal blocks in the limit of small u and v.15 In the case where there are non-R currents in the OPE, these currents would also appear in multiplets that contain

scalar components with 2 = 1. Consequently, the cancellation also has to occur for m = 0, as one can easily verify in the blocks themselves.16 This provides a non-trivial consistency check of our results and those of [12].

We can proceed to consider the OPE coe cients of the twist 2 tower, which were bounded as a function of in [7]. Our results predict that these should approach the mean eld theory conformal block coe cients at a rate 2, and this rate of convergence could easily be matched to bounds from the numerical bootstrap in the future.

14Our normalization for the blocks removes a factor of 12[parenrightbig] compared to that used in [12].

15Note that if instead one considers the s-channel expansion of the correlator h i then there is no

longer a relative sign between the even and odd spins in the superconformal block [12], so a u1 log v term is present. However, the conformal block expansion in the s-channel cannot be immediately compared to the channel because passing between these two di erent OPE limits requires changing the radial

ordering of operators, which introduces phases (1) from crossing branch cuts.

16More generally, there exist theories with an innite number of higher-spin currents, and the anomalous dimensions of the n = 0 operators should be protected in such cases as well. Additionally, while twists greater than 2 would not be the minimal twist and therefore not obviously constrained by our results, the presence of u 2 v0 log v terms would at the very least impose non-trivial constraints that would have to be satised to be consistent with vanishing anomalous dimensions in the cross-channel. In any case, any fears of a possible contradiction are readily allayed: it is easy to verify that in fact the u

m

2 v0 log v terms in the

Gm,m super-conformal blocks cancel for any m and any m.

14

2.4 Bounding contributions from operators with general twists

Finally, let us show that as , the contribution from accumulation points a other than

2 + 2n is strictly bounded. An analogous generalization for distinct operators follows from observations in appendix C. The idea of the argument is very simple specic power-law behaviors in for the conformal block coe cients P, result in related power-law contributions at small u. Since we explicitly know the leading and sub-leading behavior as u 0, we can obtain a bound on the conformal block coe cients using the crossing

symmetry relation, eq. (2.29). The remainder of this subsection formalizes these claims.

Consider all terms on the right-hand side of (2.29) at large with | (2 + 2n)| > for some > 0 small but xed. This bound separates out the contributions we studied in the previous subsections. Furthermore, let us consider only operators with twists <

for some arbitrary choice of . The reason for imposing this bound on is that we wish to

constrain the CFT spectrum and the conformal block coe cients at large , and by this we mean large with xed . In the analogy with scattering, we are studying the scattering amplitude at large impact parameter and xed center of mass energy.

Let us dene a quantity that is the partial sum of the right-hand side of (2.29) keeping only operators with < , > 1/z, and | (2 + 2n)| > :r.h.s. (, )

X<,>

6=2 +2nO()

P, v

2 u f,(v, u). (2.39)

JHEP12(2013)004

Then we can approximate

r.h.s. (, )

1/z

(1 v) F (d)(, v)[bracketrightBig]. (2.40)

The idea will be to combine together all the various values of for each . Since the conformal block coe cients satisfy P, > 0 by unitarity, a weighted sum of them will also be positive. Furthermore, if we can bound their weighted sum then we can bound each individual term. For all physical the function v

z

X<,>

P,k2(1 z)

hv 2

2 (1 v) F (, d, v) will be

bounded from above by some B(, d, v), so we can write an inequality

r.h.s. (, ) < B(, d, v)z

X<,>

P,k2(1 z). (2.41)

For each value of , there can be only a nite number of operators with < . This means

that we can dene a new quantity that includes the contributions of all these operators at xed :

Q,

X<

P,. (2.42)

Now we have the bound

r.h.s. (, ) < B(, d, v)z

X>

Q,k2(1 z). (2.43)

15

Again, the lower-bound can be taken arbitrarily large since only the innite sum over

produces additional u1 singularities; operators that do not belong to an innite tower of spins are irrelevant.

Now, for the purposes of this argument we can also approximate the > sum by

an integral. In order to avoid producing non-integer powers of v on the l.h.s. of (2.29), we must then have that

lim

z0

z m 2

[integraldisplay]

d Q,k2(1 z)

[bracketrightbigg]

= 0. (2.44)

If we use the K0 approximation of eq. (A.21), then performing a change of variables to y = z immediately shows that since Q, > 0, we expect to have an asymptotic bound

Q, < 42

3 2

JHEP12(2013)004

m in the large limit, at least in an averaged sense when we smear over a large number of . More precisely, we can use the arguments in appendix B.3 to show that

[integraldisplay]

()2 Q, < A L2 m (2.45)

for some positive constant A at very large L. This provides a general smeared bound for every sequence of P, as . Note, however, that our method cannot strictly exclude

examples where large but extremely rare conformal block coe cients occasionally appear at large .

2.5 Failure in two dimensions

In the previous sections, we had to restrict to d > 2 dimensions in order to have a gap between the twist of the identity operator and m. It is illuminating to see how the absence of such a gap in d = 2 theories explicitly leads to violations of our conclusions in specic examples. We will focus here on the simplest of such examples, the c = 1

2 minimal (i.e.

d = 2 Ising) model (see e.g. [39] for a review). This theory contains three Virasoro primary operators, all scalars: 1, , and , of dimensions 0, 18, and 1 respectively, as well as all their Virasoro descendants. Consider the operator ; the OPE in this case can be

summarized succinctly as

[][] = [1] + [], (2.46)

where [O] denotes the full Virasoro conformal block associated with an operator. Now,

since and 1 both have integer dimensions, and the Virasoro operators just raise the dimension by integers, this means that every operator that appears in the conformal block decomposition has integer twist, violating our conclusion in d > 2 that there must be operators with twist = 2 + 2n = 14 + 2n. To see what has gone wrong, examine the bootstrap equation in this theory at |u| |v| 1:u

18 +

L d (2)

X,

P,v2 18 f,(v, u) + subleading in 1/u. (2.47)

In this case there is no gap between the twist of the identity operator and m. Furthermore, our assumption from the analysis of [37, 38] that there is no non-trivial innite tower of

16

X

P0,u18 f0,(u, v) =

conserved higher-spin currents with () = d 2 for > 2 is also violated. Far from having

an isolated dominant contribution from the identity operator at small u followed by a nite number of isolated contributions from twist m operators (followed by everything else), we immediately have an innite tower of contributions all at = 0. Now we see why there are no operators in this theory with twist = 2 : the existence of this low-twist tower means that the identity operator can be (and is) cancelled by contributions on the same side of the crossing relation. In fact, in this case, the = 0 tower contributes not only the same u singularity, but it also contributes a v coe cient, for a total of (uv) .

The resulting singularity in the cross-channel can be seen explicitly in the exact four-point function, which contains a leading singularity at small u and v of the form

G(z, z)

1(uv) , (2.48)

as opposed to the usual u . It is interesting to note that the constraints from the Virasoro algebra that make many d = 2 CFTs solvable also directly cause them to di er quite drastically in their behavior at large spin from essentially all other CFTs.

3 AdS interpretation

To the uninitiated, results concerning the CFT spectrum and conformal block coe cients may appear rather technical. However as recent work has shown [16, 40, 41], both anomalous dimensions and OPE or conformal block coe cients have a very simple interpretation as amplitudes for scattering processes in AdS space. This follows from the fact that in global AdS, time translations are generated by the dilatation operator D of the dual CFT, so anomalous dimensions in the CFT represent energy shifts of bulk states due to interactions. By the Born approximation, these are related to scattering amplitudes in the perturbative regime [16]. A thorough investigation of this connection in the context of gravitational scattering in AdS at large impact parameter was performed in [42, 43], and in appendix D we explicitly compare our results to theirs in the region of overlap.

To understand the connection to AdS, consider any scalar primary operator with dimension , which creates a state |i = |0i when acting on the vacuum of the CFT.

If we were working at large N and was single-trace, then we could interpret |i as a

single-particle state in AdS. Furthermore, we could interpret the operators O, appearing

in the OPE

(x)(0) =

X,c,f,(x, )O,(0) (3.1)

of with itself as 2-particle states whose anomalous dimensions were due to bulk interactions. The operators O, at large correspond to states with large angular momentum in

AdS, so that the two particles are orbiting a common center with a large angular momentum. This obviously implies that at large the pair of particles will become well-separated, although due to the warped AdS geometry, their separation or impact parameter b is

b RAdS log [parenleftbigg]

JHEP12(2013)004

[parenrightbigg]

(3.2)

at large . So we need to study very large to create a large separation in AdS units.

17

In the absence of large N, we certainly cannot interpret the state |i as a bulk particle,

but we can still view it as some de-localized blob in AdS. Without large N we would also expect to lose the interpretation of operators in the (x)(0) OPE as 2-particle states. The results of the previous sections show that on the contrary, at large there are always operators O, in the OPE that we can interpret as creating 2-blob states, where the blobs

are orbiting each other at large separation in AdS. The fact that we must always have innite towers of operators in the OPE with twist = 2 + 2n + (n, ) and (n, ) 0

as shows that at large , the interactions between these orbiting AdS blobs are

shutting o . In particular, let us assume that there is exactly one operator at each n and and that the (n, ) 0 smoothly. In this case we obtain a specic power-law dependence

on that can be written as

(n, ) = n

m n exp mb RAdS

[bracketrightbigg]

, (3.3)

so the interactions between the blobs are shutting o exponentially at large, superhorizon distances in AdS. This is the sense in which our results prove superhorizon locality in the putative AdS dual of any d > 2 CFT.

To emphasize the generality of this result, and the fact that really create blobs, note that we can even apply our results to the scalar primary operators that create large black holes in AdS theories dual to CFTs with large N and large t Hooft coupling. In that case, our results show that if the AdS black holes orbit each other with su ciently large angular momentum, then their interactions become negligible. It should be noted that since these black hole states/operators couple to the energy momentum tensor with OPE coe cients proportional to their dimension, they will not decouple until very large .

4 Discussion

The recent revival of the conformal bootstrap has led to a great deal of progress, but perhaps the best is yet to come. Thus far much of the work on the bootstrap has been numerical and has focused on questions of phenomenological interest, so further studies of superconformal theories [12], AdS/CFT setups [10], and even quantum gravity [41] may yield important results. Our results in this paper followed from a seemingly elementary consideration of how singularities in one channel of the conformal block expansion can be reproduced in the crossed channel, yet they have powerful implications for general CFTs.

We have shown that the OPE of a scalar operator with itself has a universal leading behavior in the limit of large with xed twist. In particular, there always exist operators that we could call []n, at very large which have twist 2 + 2n + (n, ), with (n, ) 0 as . This is directly analogous to the structure of double-trace

operators in large N theories, but it holds in any CFT. Furthermore, we saw that with reasonable assumptions, we could make specic predictions for the fall-o of (n, ) and of the related OPE coe cients. We proved that all other contributions to the OPE must be sub-dominant at large . Our bootstrap methods apply in a simple way only to the OPE of scalar operators, but it seems very likely that equivalent results also hold for the

18

JHEP12(2013)004

OPE of higher-spin operators. Perhaps in the future the results of [44, 45] could be used to prove these statements. Another interesting extension would involve studying further sub-leading corrections to the bootstrap as u 0; analyzing these corrections could lead to

a more general proof of the Nachtmann theorem that does not rely on conformal symmetry

breaking in the IR.

CFTs with dual AdS descriptions that are local at distances much smaller than the bulk curvature scale must have special features [10, 19]. However, we have seen that in a certain technically precise sense, all d > 2 CFTs can be viewed as dual to AdS theories that are local at superhorizon distances. The question of superhorizon AdS locality has often been discussed in the context of the holographic RG [4653], although the general success of this interesting approach has not been manifest. It would be interesting if our results could be related to or shed light on the holographic RG.

Our arguments fail for CFTs in two dimensions. In fact as we discussed in section 2.5, minimal models provide an immediate counter-example, as they have scalar operators of dimension without corresponding operators of twist 2 + 2n at large . The reason

is that in two dimensions, there is no separation between the dimension of the identity operator and the twists of conserved currents and the energy-momentum tensor. One might try to interpret this in AdS as the statement that there is no clear separation between free propagation and interactions, perhaps due to the fact that gravitational interactions produce a decit angle in three dimensions; it would be interesting to explore this issue further.

One inspiration for our approach was the structure of conformal blocks in Mellin space [23, 24, 29, 41, 54], where the blocks imitate the momentum-space partial waves of scattering amplitudes more transparently. The leading behavior at small u in position space translates into the presence of a leading pole in the Mellin amplitude which must be reproduced by an innite sum over angular momenta in the crossed channel. Through further work it should be possible to use our results to shed light on the convergence properties of the CFT bootstrap in Mellin space. Our results seem to suggest that the sum of conformal blocks in Mellin space will only converge away from the region where the Mellin amplitude has poles. A more precise version of this observation could be useful for further work using the CFT bootstrap, both analytically and numerically.

Acknowledgments

We are grateful to Sheer El-Showk, Ami Katz, Juan Maldacena, Miguel Paulos, Joo Penedones, Slava Rychkov, Alessandro Vichi, and Alexander Zhiboedov for discussions. We would also like to thank the participants of the Back to the Bootstrap II workshop for discussions and the Perimeter Institute for hospitality during the early stages of this work. ALF and JK thank the GGI in Florence for hospitality while this work was completed; JK also thanks the University of Porto. This material is based upon work supported in part by the National Science Foundation Grant No. 1066293. ALF was partially supported by ERC grant BSMOXFORD no. 228169. JK acknowledges support from the US DOE under contract no. DE-AC02-76SF00515.

19

JHEP12(2013)004

A Properties of conformal blocks

Our arguments rely on a few key properties of crossed-channel conformal blocks in the small-u (equivalently small-z) limit at asymptotically large values of and . In this appendix we will establish these properties in general space-time dimensions. We also establish some useful results that hold for general kinematics, including the positivity of the coe cients in the series expansion of the conformal blocks.

A.1 Factorization at large and small u

First, we would like to establish that in the large- and small-u limits, the - and -dependence of the crossed-channel blocks factorizes. More precisely, for the blocks in d dimensions, we would like to show that

g(d),(v, u)

1

u1

= k2(1 z)v/2F (d)(, v) (1 + O(1/

JHEP12(2013)004

, z)), (A.1)

where

k(x) x/22F1 (/2, /2, , x) , (A.2)

and F (d)(, v) is an analytic function that is regular and positive at v = 0. In these expressions we are using the identications u = zz and v = (1 z)(1 z). The error

term may depend arbitrarily on and v, but must have the indicated dependence on and z

u 1v .

A.1.1 Factorization in 2 and 4 dimensions

Let us start by establishing this factorization in d = 2. In this case the crossed channel blocks take the form

g(2),(v, u) = k2+(1 z)k(1 z) + k2+(1 z)k(1 z), (A.3)

Since we will be in the regime with (1 z) < 1, the second term is exponentially suppressed

at large and we can ignore it.Now, the hypergeometric function k2+(1 z) has the integral representation

k2+(1 z) =

(2 + )

( + /2)2

[integraldisplay]

1

dt t(1t)

(1 z)t(1 t) 1 t(1 z)

/2

(1 z)t(1 t) 1 t(1 z)

, (A.4)

0

where we have factored the integrand into a -dependent piece and an -dependent piece. When is large, the integrand is sharply peaked near the value t = 1

z1z , with a width

that goes like 1/. Meanwhile, the -dependent part of the integrand varies slowly over the peak, and thus contributes its value at t (up to small corrections)

(1 z)t(1 t) 1 t(1 z)

1 + O

z, 1/

[parenrightBig]

. (A.5)

Plugging this in, and using Stirlings approximation for the -functions, we nd

k2+(1 z) = 2k2(1 z) [parenleftBig]

1 + O

z, 1/

[parenrightBig][parenrightBig]. (A.6)

20

In the small z limit, we have 1 z = v + O(z), so that g(2),(v, u) = k2(1 z)2k(v) [parenleftBig]

1 + O

z, 1/

[parenrightBig][parenrightBig]. (A.7)

This veries eq. (A.1) in 2d with

F (2)(, v) 22F1 [parenleftBig]

2 ,

. (A.8)

It is straightforward to repeat these steps in d = 4. In this case, the crossed-channel blocks are given by

g(4),(v, u) =

(1 z)(1 z)

z z

(k2+(1 z)k2(1 z) k2+(1 z)k2(1 z)). (A.9)

Again, one can neglect the second term in the large limit, and accounting for the pre-factor one straightforwardly nds

g(4),(v, u) = k2(1 z)

2v

1 v

2 , , v

[parenrightBig]

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k2(v)

1 + O

1/, z

[parenrightBig][parenrightBig]

. (A.10)

This veries eq. (A.1) in 4d with

F (4)(, v)

21 v 2

F1

2 1,2 1, 2, v

[parenrightBig]

. (A.11)

A.1.2 Extension to even dimensions via a recursion relation for F (d)(, v)

One can easily extend this to all other even d using a recursion relation relating the conformal blocks in d dimensions to those in (d + 2) dimensions. Concretely, the blocks satisfy the relation [33]

zz zz

2g(d+2) ,(u, v) = g(d) 2,+2(u, v) 4( 2)(d + 1)(d + 2 2)(d + 2)g(d) 2,(u, v) (A.12)

4 (d 1)(d ) (d 2 )(d 2 + 2)

[bracketleftbigg]

( + )216( + 1)( + + 1)

g(d) ,+2(u, v)

(d + 2)(d + 1)(d + )2

4(d + 2 2)(d + 2)(d + 1)(d + + 1)

g(d) ,(u, v)

[bracketrightbigg]

.

In the large but xed = limit, this recursion relation simplies to

z z zz

2g(d+2),(u, v) g(d)4,+2 g(d)2,(u, v) 116g(d)2,+2(u, v) (A.13)

+ (d )2

16(d 1)(d + 1)

g(d),(u, v).

Inserting the factorized form eq. (A.1) and using the property k2(+2)(1z) 24k2(1z), one nds that the function F (d)(, v) should satisfy the recursion relation

(1 v)2F (d+2)(, v) 16F (d)( 4, v) 2vF (d)( 2, v) (A.14)

+ (d )2

16(d 1)(d + 1)

v2F (d)(, v).

This makes it trivial to generate the function F (d)(, v) for any even d. Moreover one can verify that this relation holds between the functions F (2)(, v) and F (4)(, v) ob

tained above.

21

A.1.3 Extension to odd dimensions

Finding the function F (d)(, v) in odd d is less straightforward, but it is easy to see that the factorization property should continue to hold. One approach is to note that the conformal blocks are determined as the solutions to the Casimir equation [3235]

Dg(d),(v, u) =

1

2C(d),g(d),(v, u) (A.15)

where D is the conformal Casimir as a di erential operator

D = (1 v u)uuu + vv(2vv d) (1 + v u)(vv + uu)2 (A.16)

and C(d), = 2( + 1) + ( d). One can immediately see that the d-dependence

enters the Casimir at O(1/2) at large , so the dominant dependence is in a part of the di erential operator which depends only on v. Furthermore, it is also clear that we can dene the blocks by integrating this di erential equation, because from [34] we have that in the small v limit

g(d),(v, u) = v

2 (1 u)2F1 [parenleftBig]

2 + ,

2 + , + 2, 1 u

[parenrightBig]

+ O(v) (A.17)

This gives a boundary condition as v 0 for any value of u, so to obtain the conformal

blocks at any v we need only integrate equation (A.15) a small distance, over which the

O u, 1 [parenrightbig]

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errors cannot accumulate. Since the equation (A.15) is satised by the ansatz in equation (A.1) to leading order, the factorized ansatz su ces in all d.

A.1.4 Further approximations for the function k2(1 z)

The function k2(1z) = (1z)2F1(, , 2, 1z) can be approximated further depending

on the relative size of and z. In the extreme asymptotic regime 1/z one can use

the saddle point approximation

()2 (2)

2F1(, , 2, 1 z) =

[integraldisplay]

1

dt t(1 t)

t(1 t)

1 t(1 z)

=

[integraldisplay]

1

dt t(1 t)

eln[parenleftBig]

t(1t)

1t(1z) [parenrightBig]

0

[radicaltp]

[radicalvertex]

[radicalvertex]

[radicalbt]

2

1 t(1 t)

t(1t)

0 1t(1z) [parenrightBig]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

d2dt2 ln

[parenleftBig]

t(1t)

1t(1z)

eln[parenleftBig]

t= 1z

1z

r14z(1 + z)21 [parenleftbigg]1 + O

[parenleftbigg]

1 z

[parenrightbigg][parenrightbigg]

. (A.18)

The corrections can be easily obtained by expanding to the next order. This approximation is best when we take at a xed value of z. In the extreme limit where z 1 but

z 1, one sees an exponential decay

()2 (2)

2F1(, , 2, 1 z)

r e2z 4 z

1 + O

z, 1z[parenrightbigg][parenrightbigg]. (A.19)

22

Alternatively, we could consider the regime where we take holding y z2

xed. In this case, we can approximate

()2 (2)

F (, , 2, 1 z) =

[integraldisplay]

1

dt t(1 t)

t(1 t)

1 t(1 x)

[integraldisplay][parenleftBigg]23 ). (A.20)

The higher order terms in the exponent can be neglected at large with y [lessorsimilar] O(1) held xed. If we dene a new variable s

t1t , we can rewrite the integral as

()2 (2)

1

0 1 t1

0 1 t

e

ty (1t) +O(

1 (1t)

F

, , 2, 1 y 2

[parenrightBig] [integraldisplay]

0

dss e

sy

1s

= 2K0(2y) + O

1

, (A.21)

where K0 is a modied Bessel function of the second kind. We stress that this approximation breaks down when y = z2 1 (though it does correctly reproduce the exponential

decay in eq. (A.19)), but provides a good description of the regime with y [lessorsimilar] O(1).

A.2 Positivity of coe cients and exponential fallo at large

In this section we will prove that in general d, the coe cients in the power series expansion of the conformal blocks

g,(z, z) =

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Xm,n0amnz/2+mz/2+n (A.22)

satisfy

amn > 0 (A.23)

for all m, n, where we recall that u = zz and v = (1 z)(1 z). One can verify the claim

directly in the case of d = 2, 4, 6 where explicit formulas are known. Similar positivity results are fairly well-known [14] when the conformal blocks are expanded with z = z, but as far as we are aware this more general result has not been discussed previously.

This result will be useful because it implies that for real z and z, if we take z z/

with > 1 then we must have

g,(z, z/) <

2 g,(z, z). (A.24)

This allows us to conclude that contributions from innite sums over large are exponentially small for appropriate choices of kinematics, because for 0 < z, z < 1 we know that the conformal block expansion converges [14].

The proof is simple, and uses the clever choice of coordinates in [14], where we set x3 = ei, x4 = ei, while x1 = rei and x2 = rei with , , r all real and 0 r < 1.

In these coordinates we have rei() given in terms of z by

(z) = z

(1 + 1 z)2

, (A.25)

23

and we can dene a quantity

(z) similarly. The formula can be inverted to give z(). A crucial point is that when we expand (z) in a power series in z, all of the coe cients will be positive. This implies that if g,(,

) has a power series expansion in ,

with positive

coe cients, then the same property holds when it is viewed as a function of z, z. Note that

L +

L2 = r cos(L( )), (A.26)

so we can expand the conformal blocks as

g,(,

) =

Xs,Lcs,L()2 +s+ L2 cos(L( )). (A.27)

If we can prove that cs,L > 0 then we will have proven that amn > 0. However, we can immediately see that

Xs,L

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[integraldisplay]

2

0 d cos(m) [integraldisplay]

2

0 d cos(m)hO(ei)O(ei)|O+2s,LihO+2s,L|O(rei)O(rei)i

(A.28)

is the norm of some denite linear combination of descendants of the primary O, whose

conformal block we are considering. Therefore this norm will be positive. Applying this norm to the series expansion of the block above, we nd

Xscs,L r+2s+L > 0 (A.29)

for every L. One can similarly project onto denite powers of r by smearing O(x1)O(x2) and O(x3)O(x4) in the radial direction. For example, we can promote x3 e34ei and

x4 e34ei, and similarly take x1 = e12ei and x2 = e12ei, so that we have r =

e1234. Then smearing the conformal block against wavefunctions such as cos(1212) and cos(3434) projects out denite values for s, and we have positivity for the case 12 = 34.

A more concise way of obtaining the same conclusion is to consider the cs,L as the

norms of states on the subspaces of denite dimension and angular momentum on the circle. In any case, we nd that all cs,L > 0, and so we conclude that amn > 0 as claimed.

B () and its crossing equation

In this appendix, we present several details that were suppressed in section 2.2 for the sake of readability. Specically, we will give a rigorous denition of the asymptotic density (), and a derivation of its crossing equation.

B.1 Existence of ()

Let us dene a conformal block density in -space on the r.h.s. of the crossing relation,

Dv,u()

u v

X,( )P,g,(v, u). (B.1)

24

One should always think of Dv,u() as being integrated against some function f(). More formally, Dv,u denes a linear functional given by the pairing

Dv,u[f]

u v

X,f()P,g,(v, u). (B.2)

Since the conformal block decomposition is absolutely convergent, Dv,u is well-dened on any bounded function f. Further, since the conformal blocks and coe cients P, are

positive, Dv,u is positive as well. We will use Dv,u as a tool for slicing up the r.h.s. of the crossing relation into di erent contributions at various twists.

Our goal is to characterize the limit limu0 Dv,u in terms of a density in twist-space at

asymptotically large . To disentangle asymptotic behavior in from asymptotic behavior in , its extremely useful to restrict the twist to lie in some nite range, and study the contribution of only the operators in this range. In terms of the functional Dv,u, this means

we should consider its action on continuous functions with compact support, f Cc(

R+).17

Note in particular that when f has compact support, then there are only a nite number of nonzero terms at each on the r.h.s. of eq. (B.2). This will be important below.

Let us suppose f Cc(

R+), and consider the limit

lim

u0

Dv,u[f] = lim

u0

u v

X,f()P,g,(v, u). (B.3)

The limit of each individual term above vanishes, so the limit of the sum is unchanged if we discard any nite number of terms. Since f has compact support, we may restrict L

for any nite L without changing the result,

lim

u0

Dv,u[f] = lim

u0

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u v

X0,Lf()P,g,(v, u) (B.4)

= lim

z0

u v

X0,Lf()P,k2(1 z)v/2F (d)(, v) [parenleftBig] 1+O

1/L, z

[parenrightBig][parenrightBig]

[parenleftbigg]

lim

= z0

u v

X,0f()P,k2(1 z)v/2F (d)(, v)[parenrightbigg] [parenleftBig]1 + O

1/L[parenrightBig][parenrightBig].

In the second line, we have substituted the small-z and large- form of the crossed-channel blocks eq. (A.1). In the third line, we have used the fact that the limit still vanishes termwise after this substitution. The above equation holds for all nite L, so we may drop the error term to obtain

lim

u0

Dv,u[f()] = lim

z0

u v

X,0f()P,k2(1 z)v/2F (d)(, v)

= (1 v) D [bracketleftBig]

, (B.5)

17Of course, the twist is restricted by unitarity to lie above some minimum value depending on the spacetime dimension. This is unimportant to the present analysis; for convenience, we will allow f to be a function on all of R+.

v/2 F (d)(, v)f()

[bracketrightBig]

25

where we have used u z(1 v) and dened the functionalD[f] lim

z0

z

X,

P,f()k2(1 z). (B.6)

D is a positive linear functional on Cc(R). By the Reisz Representation Theorem (RRT), we can write it as the integral of a density in -space. Specically, the theorem states that there exists a unique regular Borel measure, which we denote by ()d, such that

D[f] =

[integraldisplay]

0 f()()d (B.7)

for all f in Cc(R).18

In summary, we may formally dene a density () by

() = lim

z0

z

X,

P,( )k2(1 z). (B.8)

When integrating () against some function f, one might worry about switching the order of the limit and the integration. We can interpret the RRT as saying that switching the limit and integration is justied whenever f has compact support.

B.2 Crossing symmetry for ()

Crossing symmetry states that

1 = lim

u0

Dv,u[1] = (1 v) D [bracketleftBig]

v/2 F (d)(, v)

JHEP12(2013)004

[bracketrightBig]

. (B.9)

We cant immediately write this in terms of the density () because a priori eq. (B.7) only holds for functions f with compact support. This is the order of limits and integration issue mentioned in section 2.2. For example, one might worry that the limit limu0 Dv,u[1]

is dominated by operators with 1/u. Then D[f] would vanish on functions with

compact support (and () would be identically zero), while eq. (B.9) might still hold.

We can show that this does not happen by using the fact that the blocks die exponentially at large . Specically, let us study the contribution of conformal blocks with twist less than some large . Since the function ( ) has compact support, eqs. (B.5)

and (B.7) imply19

(1 v) [integraldisplay]

0 v/2 F (d)(, v)()d = lim

u0

Dv,u[( )],

= lim

u0

u v

X

P,g,(v, u). (B.10)

18The density () can be reconstructed by considering the value of D[f] on functions with support in very small neighborhoods. The non-trivial content of this theorem is that positivity and linearity imply that the density so-constructed is unique, and gives the correct value of the functional on any set with compact support.

19Strictly speaking, we should use a slightly smoothed -function, since itself is not continuous. This subtlety will not be important in the analysis.

26

Our goal is to bound the discrepancy between this quantity and the full conformal block expansion. Let us choose some constant satisfying 1 < < 1

1z. Eq. (A.24)

then implies

lim

u0

u v

u v

X>

P,g,(v, u)|zz ,

lim

u0

X>

P,g,(v, u) lim u0

u v

X,

P,g,(v, u)|zz ,

= , (B.11)

where in the second line weve used positivity of each term in the conformal block expansion to enlarge the sum to all operators, and in the last line weve used crossing symmetry at the point (z, z).

Combining eqs. (B.10) and (B.11), we nd

1 + O() = (1 v) [integraldisplay]

0 v/2 F (d)(, v)()d, (B.12)

so that we can safely extend the region of integration out to innity and conclude

1 = (1 v) [integraldisplay]

0 v/2 F (d)(, v)()d. (B.13)

JHEP12(2013)004

This is eq. (2.18) in the text.

B.3 Bounds on OPE coe cient densities

In this appendix, we will prove a bound on the asymptotic behavior of the integral over f(), which is dened in eq. (2.24). Specically, we will show that given a function L(x)

with the representation

L(x) =

[integraldisplay]

0 df()k(1 x2) (B.14)

that behaves like xa at small x, then there exist numbers AL, AU such that the integrated value

F (B)

[integraldisplay]

B

0 d

f(), f()

()

2(/2)f(), (B.15)

is bounded at large B by

AUBa > F (B) > AL Balog(B). (B.16)

We will begin with the upper bound. First, we dene

h(, x)

2(/2)

() k(1 x2), (B.17)

27

which is a positive, decreasing function of at xed x. Since f()h(, x) is non-negative, we therefore have for any B that

L(x)

[integraldisplay]

B

0 d

f()h(, x) h(B, x)F (B). (B.18)

Fixing to a positive real number (its value is not important) and taking x = /B, we therefore nd the bound

F (B)

Ba ah B, B

[parenrightbig]

(B.19)

at su ciently large B. Furthermore, as shown in appendix A.1.4, in the limit of large B with xed, h(B, B ) approaches a nite value independent of B:

lim

B

h

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B, B

[parenrightbigg]

= 2K0(), (B.20)

where K0() is a modied Bessel function. Thus, we have proven an upper bound

lim

B

BaF (B)

1a2K0(). (B.21)

One can try to improve this bound by maximizing aK0() as a function of , but any value of proves the existence of some AU.

Now, let us turn to proving the lower bound. It will be convenient to dene the function (, u) by

(, x) =

h(, x), (B.22)

which is positive since h(, x) is decreasing as a function of . Then, we can write

L(x) =[integraldisplay]

0 dBF (B)

(B, x). (B.23)

The limit limx0 xaL(x) is una ected by adding a xed lower-bound on the d integration,

so we can dene the function

~L(x) [integraldisplay]

0 dBF (B)

(B, x), (B.24)

which also approaches xa at small x. Furthermore, F (B) is an integral over non-negative terms, so F (B) F (B) for B B. Thus, we have

~L(x) = [integraldisplay]

M

0 dBF (B)

(B, x) +

[integraldisplay]

M dBF (B)

(B, x),

M dBB2a

F (M) (h(0, x) h(M, x)) + AU

[integraldisplay]

(B, x),

M dBB2a

F (M)h(0, x) + AU

[integraldisplay]

(B, x). (B.25)

28

Let us take x = /M such that 1 < < M. The second integral on the right-hand-side above is just an integral over known functions and by using the large B, large xB approximation to(B, x), it can be seen to vanish like A2e at large and M for some

A2. Finally, the small x behavior of h(0, x) is proportional to log(1/x), so we have h(0, x) < A3 log(1/x) for some A3, and thus

~L(/M) F (M)A3 log(M/) + AUA2e. (B.26)

By making large, we can make the last term as small as we like, so we have

lim

M

log(M)MaF (M) > 1

A3a , (B.27)

which proves the lower bound.

Finally, note that if we knew that the bounds could be improved such that the upper and lower bound were the same (F (B) AMBa for some constant AM that depended only

on a), then we could calculate AM using the special case f() a1 to be

F (B)

[integraltext]

B0 da1 limx0 xa

[integraltext]

0 da1h(, x)

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1a Ba

[integraltext]

= 0 da12K0()

=

B 2

a 1a2 2(a2). (B.28)

C Generalization to distinct operators 1 and 2

For simplicity we have presented the argument in the body of the draft for a single operator . However, our results extend to distinct operators 1 and 2, as might be expected from

AdS intuition. Specically, by studying the correlator

h1(x1)1(x2)2(x3)2(x4)i (C.1)

and using the crossing relation in the 12 34 and 13 24 channels, we can prove that

there must exist an innite sequence of operators with twists 1 + 2 + 2n + (n, ) at

large . If we have a unique such operator at each , we can also show that

(n, ) = n

m and Pn, =

cnm (C.2)

as we found for the case 1 = 2 in the body of the paper, where m is the minimal non-vanishing twist appearing in the OPE of 1 with itself. The main complication compared to the case of identical operators is that the conformal blocks now depend on 2 12 = 1 2, although not in any way that qualitatively a ects the results in the u v < 1 limit. Note

also that in the 12 34 channel we cannot assume that the conformal block coe cients

are positive, although this will not be necessary to obtain a proof.

Note that for example in mean eld theory we would have the single term

h1(x1)1(x2)2(x3)2(x4)i =

1

2 (C.3)

and we can factor this dependence out of all conformal blocks. Then the crossing relation takes the form

u

1+ 2

2 + u

(x212)

1 (x234)

X,

P 12,12,g12,12,(v, u) (C.4)

1+ 2 2

X,

P 11,22,g11,22,(u, v) = v 1+ 2 2

29

where we have labeled the conformal blocks and their coe cients according to the conguration of operators. Note that on the left-hand side the blocks g11,22,(u, v) have leading terms at small u proportional to u

2 , and for d > 2 this is isolated from the identity. In fact, g11,22, = g, that we used in the case of identical operators [34], as the di erence 1 6= 2

is irrelevant in this channel. On the right-hand side the blocks g12,12,(v, u) again begin as v

2 at small v [34].

Now we can proceed as in the body of the draft, expanding at small u and keeping only the identity and the rst sub-leading term on the left-hand side. This gives an approximate crossing relation

u

, z)), (C.8)

so we obtain the same leading order behavior at large as in the case of identical operators, and F has some convergent series expansion in integer powers of v. This means that the arguments from the body of the text can now be followed as before with 1 and 2, leading to analogous conclusions.

D Comparison with gravity results

The calculation of anomalous dimensions of double-trace operators starting with a perturbative AdS description is usually simplest for the lowest-spin states [16, 55], and becomes increasingly more involved for higher spins. However, a signicant simplication occurs in the limit of very large spin, which is described by the Eikonal limit of 2-to-2 scattering in AdS space, as was worked out in detail in [42, 43]. There, it was shown that the anomalous dimensions could be calculated by treating one of the two particles in AdS as

30

X,

P 12,12,v 1 22 f12,12,(v, u) (C.5)

where we have separated out the leading u dependence from the blocks to make the behavior clear. This means that the sub-leading term on the left-hand side has a convergent series expansion in integer powers vk and vk log v, which we can use as in the case of identical operators to conclude that on the right-hand side we need contributions from towers of operators with twist approaching 1 + 2 + 2n.

As we considered in appendix A, the conformal blocks g12,12, can be determined as the solutions to the generalized Casimir equation [3235]

Dg(v, u) = 12C(d) ,g(v, u) (C.6)

where D is the conformal Casimir as a di erential operator that for 12 6= 0 is

D = (1 v u)u(uu 12) + vv(2vv d)

(1 + v u)(vv + uu 12)(vv + uu), (C.7) and C(d), = 2( + 1) + ( d) as before. This is su cient to conclude that

g12,12,(v, u) k2(1 z)v/2F (d)(, 12, v) (1 + O(1/

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1+ 2

2 + u

m 1 2

2 P 11,22mfm,m(u, v)

a classical shock wave that sourced the deformation of the geometry in which the other particle traveled. The result of their calculations was that in the limit of large spin with xed and large, the anomalous dimensions due to semi-classical gravitational interactions behave as

(, ) 1 25 mjm1

d

m . (D.1)

The gravity case takes m = d, jm = 2, and the coupling G dened in [42, 43] determines the strength of gravitational interactions, i.e. G GN. This can then be compared to our

formula (2.36)

0 = Pm

2 ( )2 (m + 2m) m2

2 G ( m 1)

m d2 + 1

[parenrightbig]

m+jm2

JHEP12(2013)004

2 (D.2)

combined with the specic gravity prediction for Pm. Taking the specic values of m = d2

and m = 2, this reduces to

0 =

1

Pm d2

2 (d + 2)

2 d2 + 1

. (D.3)

2

m2 + m

In d = 4 the OPE coe cient of the stress tensor is Pm =

2

360c , and therefore for-

mula (2.36) gives

4

(2 , )

1 6c

2 . (D.4)

This is to be compared with (D.1) for the gravity case with = 2 and d = 4, which gives

(2 , )

8G

4

2 . (D.5)

The anomaly coe cient c is proportional to 1/GN in AdS gravity theories, so our results clearly agree parametrically with the perturbative AdS computations, with exact agreement for c =

48G .

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