Generalized Additivity in Unitary Conformal Field Theories

faculteit Wiskunde en Natuurwetenschappen

Generalized Additivity in Unitary Conformal Field Theories

Master research Theoretical physics

August 2014 Student: G. Vos

First supervisor: K. Papadodimas Second supervisor: E. Pallante

Generalized Additivity in Unitary Conformal Field Theories

Author: Gideon Vos,

Supervisor: prof. dr. Kyriakos Papadodimas

1CTN, University of Groningen,

2CTN, University of Groningen, Theory department, CERN

∗To whom correspondence should be addressed; E-mail: g.vos.5@student.rug.nl.

Abstract:

In the past it was demonstrated in [1],[2] that in d = 4 unitary CFT’s if scalar operators with conformal dimension ∆1 and ∆2exist in the operator spectrum, then the conformal bootstrap demands that large spin primary operators have to exist in the operator spectrum with a conformal twist close to ∆₁+ ∆₂. In this paper the methods in [1] have been generalized to demonstrate that similar limit points in twist space exist at ∆1+ ∆2+ 2N for every integer N . The anomalous dimension of these operators have been derived aswell. In AdS these operators can be interpreted as the excited states of the product states of objects that were found in other works.

Contents

1 Introduction 2

2 A review on conformal field theory 4

2.1 The conformal group . . . 4 2.2 The operator product expansion . . . 7

2.3 Conformal partial wave decomposition . . . 8 2.4 The conformal bootstrap . . . 10

3 A short review of [1]. 11

3.1 Evaluating the s-channel CPW decomposition . . . 14 3.2 The anomalous dimension coefficient . . . 17

4 Generalizing to arbitrary N 18

4.1 proof of N-independance . . . 23

5 A more general four-point function 24

6 Implications to AdS/CFT 27

7 Conclusion 28

8 Acknowledgements 30

Appendix A: Solving the sum over all spins 30

Appendix B: Recreating the identity operator contribution 33

Appendix C: The geodesic separation in AdS 37

1 Introduction

Over the last 40 years conformal field theories (CFT’s) have been carefully studied. They have both nice properties from a mathematical point of view such as the fact that up to normalization the two- and three-point functions are fully constricted by symmetry. But they also have practical applications such as in the effective field theory description of condensed matter systems near critical points and in the fixed points of RG-flow. For the purpose of

this paper the main point of interest as to why these theories are worth studying is their application to AdS/CFT. In the late ninetees it was conjectured that there is a duality between type IIb string theory in the bulk of AdS and N = 4 SYM on the boundary [3].

Since conformal transformations can be applied to map any set of four points onto a plane it has always been natural for CFT’s (that live in a space that contains at least two spatial dimension) to map four points onto a plane spanned by two spatial basis vectors to create an effective Euclidian theory. A while ago though it was demonstrated in [1][2] that by considering CFT’s on a Minkowski background an interesting property reveals itself.

By assuming only that a CFT satisfies unitarity (ie. All OPE coefficients are positive definite and the conformal dimension of local operators satisfy a unitarity bound.) and a crossing symmetry, they have demonstrated that any CFT contains a sector of operators with large spin that behaves as a generalized free theory (GFT). Furthermore they have explicitly calculated the anomalous dimension associated with these large spin operators for the collinear blocks that contribute to leading order.

In this paper their analysis has been expanded to the subleading terms in the conformal partial wave expansion of the s-channel four-point function. The result is that as was predicted in [1] [2] that not only do operators have to exist with a conformal twist that lies in a region around 2∆ but there actually has to exist a tower of limit points in twist space at 2∆ + 2N for any integer N . This supports the conclusion that every CFT has a region which behaves as a GFT. From this the conclusion can be drawn that applying the AdS/CFT dictionary to any CFT results in a theory which has a high spin region that can be interpreted as a Fock space, which implies that that region can be interpreted as a gravitational particle theory in the weak coupling limit.

Beyond that the main focus of this paper is the anomalous dimensions associated with these higher order conformal blocks. These have been calculated and have been shown to

just as for the N = 0 case to be inversely proportional to spin. The interpretation that the large spin limit in AdS behaves as a free field theory stays intact.

This thesis has been divided in the following way. In section 2 a short review will be given on conformal field theory and some useful techniques. In section 3 a summary will be given of the methods and results obtained in [1]. Section 4 will contain an argument to the existence of operators with conformal dimension close to 2∆ + 2N and contains the deriviation of an expression for the anomalous dimensions of these double trace operators.

Section 5 contains a similar analysis as section 4 but with a slightly more general four-point function which leads to a significantly more powerful result. Section 6 will contain some arguments from AdS that demonstrates that these anomalous dimensions reproduce the right leading order behaviour if the conformal dimension ∆ of the operators that appear in the four-point function is large.

2 A review on conformal field theory

To keep this thesis self-contained a review will be given on the conformal group and the CFT techniques that will be employed throughout the rest of the paper. For other similar reviews see [3],[4],[11]

2.1 The conformal group

The conformal group is the set of all coordinate transformations such that any two arbitrary intersecting curves get mapped to curves that intersect with the same angle. On a Minkowski background the Poincar´e is forms a subgroup of this group, this follows trivially if we write the angle of two 4-vectors x^µ and y^µ tangent to the curves in terms of their scalar products

cos(θ) = xµy^µ

√ x²p

y² (1)

Since all scalar factors on the right hand side are invariant under poincar´e transformations so is the left hand side. Another intuitive element of the conformal group is scale invariance.

Expanding or contracting all space time coordinates by a constant factor only changes the magnitude of vectors but keeps their relative angles intact (i.e. if you transform x^µ→ λx^µ, y^µ → λy^µ then all factors λ will cancel from equation (1)).

The final and least intuitive conformal transformation is the special conformal transfor- mation. It consists of an inversion (x^µ → ^x_x^µ2) followed with a translation by an arbitrary global vector b and finished with another inversion. The combined result is

x^µ→ x^ν− b^νx²

1 − 2bµx^µ+ b²x². (2)

One of the special things about the conformal group and definitly one of the more interesting features of CFTs is that these symmetries up to normalization fix both the two-point function

hO1(0)O2(x)i = c12

|x|^2∆, (3)

and the three-point function

hO₁(x)O₂(y)O₃(z)i = c123

|x − y|^{∆1+∆2−∆3}|y − z|^{∆2+∆3−∆1}|z − x|^{∆3+∆1−∆2}. (4) For the generators we use Mµν to designate the Lorentz generators, Pµ for the translation generators, D for the scale invariance generator and finally Kµ for the special conformal transformation generators. The coordinate representation of these generators is given by

M_µν = i(x_µ∂_ν− xν∂_µ) P_µ= −i∂_µ

D = −ix_µ∂^µ

Kµ= i(x²∂µ− xµxν∂^ν)

(5)

The Lie-algebra associated with these generators is given by [Kµ, Mνρ] = i(ηµνKρ− ηµρKν)

[Kµ, Pν] = 2i(ηµνD − Mµν) [D, Pµ] = iPµ

[D, Kµ] = −iKµ

(6)

Where the Lie-subalgebra due to the Lorentz algebra has been omitted.

We will distinguish between conformal primary operators and descendants. The con- formal primaries are operators wich when inserted at the origin create eigenstates of the scale invariance generator, this eigenvalue is −i∆ the factor ∆ is the scaling dimension or conformal dimension of the operator. The effect of the generators on the primaries in the spectrum of the CFT is given by [3]

[Pµ, O(x)] = i∂µO(x)

[D, O(x)] = i(−∆ + x^µ∂µ)O(x)

[Kµ, O(x)] = (ix²∂µ− 2ixµx^ν∂ν+ 2ixµ∆ − 2x^νΣµν)O(x)

(7)

Where ∆ is the scaling dimension associated with the operator O(x) and Σµν form a finite dimensional represenation of the Lorentz group. The two-point function (3) shows that the operators in the spectrum have a physical dimension x^−∆. Therefore dimensional analysis show that acting on an operator with the translation generators has the effect of raising the conformal dimension, whereas acting with the generators of the special conformal transfor- mation has the effect of lowering the conformal dimension.

For this reason it is that we can distinguish between two classes of operators in the spectrum of a CFT. We define the set of all conformal primaries as the set of all operators that when inserted at the origin are annihilated by all special conformal transformation generators. All other operators in the spectrum we can classify as conformal descendants, they can be generated by succesively acting on conformal primaries with Pµ.

A note on normalization, after we have classified the conformal primaries we can make linear combinations in such a way that the two-point function (3) is diagonalized with respect to families of operators. Meaning that the coefficient c₁₂ = 0 if O₁ and O₂ descend from different primaries and c126= 0 if they descend from the same primary.

2.2 The operator product expansion

The operator product expansion is the most important conformal field theory technique applied in this thesis, as all other techniques are derived from this one. It is the statement

that any bilocal product of operators can be decomposed into a sum over all local operators in the spectrum, i.e.

O1(0)O2(x) =X

O^(k)

c_12k

|x|^{∆1+∆2−∆k}

x^µ1...x^µ2

|x|^s O^(k)_µ1...µs(x) (8)

Note that it is implied that this decomposition only makes sense when both the left and right hand side are located within an n-point function. The reason as to why this can be done is quite special, to demonstrate this we will need to do a coordinate transformation to perform radial quantization. To this end first compactify all the spatial dimension by adding a point at infinity, similarly as to how one would compactify the complex plane to the Riemann sphere. The result is a generalized cylinder R × S^d−1. From this point on we can abuse scale invariance by application of an exponential mapping. Take any coordinate system for a sphere of radius R and perform the coordinate transformation R → e^t. The point at the infinite past now maps back to the origin. This is displayed in figure 1.

Figure 1: This figure gives a pictorial represenation of the exponential map. Note that the entire infinite past is mapped back to a single point.

This brings to light a very important feature. A quantum state is an inherently non-local object defined over the entirety of space, but for a unitary theory the time development- operator will map it back to the infinite past in a well-defined way, at the origin we can consider the state to be the effect of a local operator inserted at the origin acting on the vacuum The result is that there is a one-to-one correspondence between a non-local object (the state) and a local object (the operator inserted at the origin), this is typically called

the state-operator map.

The OPE that was claimed seems to make more sense now. The bilocal operator will create a state when acting on the vacuum. A state is a well-defined object living in a Hilbert space, we can consider any basis decomposition of this state we like. Mapping the basis vectors back to local operators using the state operator map demonstrates that the form of the operator product expansion makes sense. Note that the OPE is therefore a true basis decomposition and not a power series approximation, this is stressed in the literature (for example [4]) hence it is repeated here aswell. Another note the spatial dependance of all the terms on the right hand side of the OPE looks unusual but it follows directly from dimensional analysis. One final remark, consistency with the three-point function (4) demands that the coefficient that appears in the OPE is the same as the one that appears in the three-point function.

Since conformal invariance fixes the two and three-point functions and since we can apply the OPE to reduce any n-point function to a (n−1)-point function we know everything there is to know about a CFT if we know all of the operators in the spectrum and all of their respective OPE coefficients.

2.3 Conformal partial wave decomposition

Unlike the two- and three-point function the four-point function is not fixed by symmetry.

The reason for this is that for four points in spacetime we can construct two conformal invariants

u = ^(x_(x^{1−x2)2(x3−x4)2}

1−x3)²(x2−x4)², v = ^(x_(x^{1−x4)2(x2−x3)2}

1−x3)²(x2−x4)², (9) typically called the conformal cross ratios or the anharmonic ratios. As a result any analytic function of these ratios f (u, v) satisfies the required symmetry properties and just on sym- metry alone we can not find a unique expression for the four-point function. What can be

done is to apply the OPE to any two pairs of operators within the four-point function, the result is a double sum over two-point functions. Due to the diagonalization of the two-point function each term in one sum will only connect to terms in the other sum that are from the same conformal family (i.e. descend from the same primary).

hO1(x1)O2(x2)O3(x3)O4(x4)i = ¹

(x₁−x₂)¹2(∆1+∆2)(x₃−x₄)¹2(∆3+∆4)

x₂−x₄ x₁−x4

¹₂(∆1−∆2)

x1−x4

x₁−x₃

¹₂^(∆3−∆4)

P

kc12kc34kG(¹₂(∆k− ∆1+ ∆2− s),¹₂(∆k− ∆3+ ∆4− s), ∆k; u, v), (10) where the sum over k represents a sum over all primaries. For all purposes in this paper all operators that appear in the four-point functions are chosen to be scalar operators.

Since the raising operator of the conformal algebra acts as a derivative on the operators, it makes sense to group all contributions from a single conformal family into a single function g(u, v), these functions are typically called either conformal partial waves or conformal blocks. Since the earlier days of conformal field theory series representations for these conformal blocks have been known, but at the start of the 21st century F.A. Dolan and H.

Osborn have famously been able to find closed form expressions for these functions under certain conditions [12],[13],[14].

By use of the coordinate transformations

u = z ¯z , v = (1 − z)(1 − ¯z), (11)

they were able to solve the eigenfunctions of the Casimir operator of the conformal algebra.

These lightcone coordinates have a simple interpretation if you use conformal transforma- tions to map three out of the four points onto a straight line, this is demonstrated in figure 2. On an euclidian plane these coordinates play the role of the usual complex coordinates z = x − iy and ¯z = x + iy. On a minkowski plane they can be interpreted as lightcone coor- dinates, in this paper the points are chosen be on a minkowski plane. These eigenfunctions

are of the form

Figure 2: In this picture the coordinate system is displayed in which z and ¯z take on a more clear meaning as lightcone coordinates. The point x1 is located at (0,0), x3 at (1,1) and x4

at (L,L). The point x2is free and is considered at z → 0, ¯z → 1.

G_∆k,s(z, ¯z) = ^{(z ¯z)}

1 2(∆k−s)

z−¯z [(−¹₂z)^sz₂F₁(¹₂(∆_k− ∆₁+ ∆₂− s − 2),¹₂(∆_k− ∆₃+ ∆₄− s − 2), ∆_k− s − 2, ¯z)

×₂F₁(¹₂(∆_k− ∆₁+ ∆₂+ s),¹₂(∆_k− ∆₃+ ∆₄+ s), ∆ + s, z) − z ↔ ¯z]

(12) These functions because they are the eigenfunctions of the conformal Casimir are by con- struction conformal invariants and eigenfunctions of the scale invariance generator. As a result they can be interpreted as a function representation of the primaries that appear in the OPE. Also as such the conformal partial wave decomposition given above is analogous to decomposing functions into spherical harmonics.

2.4 The conformal bootstrap

In the previous section a certain choice has been made in the way the operators in the four- point function have been contracted. But the way pairs of operators have been selected in the four-point function should be in some sense arbitrary. The predicted outcomes should be related to each other by some sort of conformal transformation such that both insertion locations of the operators within an OPE pair are such that the OPEs converge. This is represented in a graphical way in figure 3, here we distinguish between an s-channel CPW decomposition and a t-channel decomposition. The relation between these two will play a

critical role in the upcoming sections.

Figure 3: The difference between the t-channel (left) and s-channel (right) is in the way the pairs of operators in the four-point function have been contracted by taking the OPE’s

This consistency constraint on the CFT is in the literature typically called the conformal bootstrap or the crossing symmetry. It places very heavy demands on a consistent CFT, for a very long time some people have suggested that it might be strong enough to find a general solution for a CFT [11], since as mentioned in a previous section all of the dynamic information of a CFT is contained in its operator spectrum and it’s OPE coefficients.

3 A short review of [1].

This section will contain a very short review on the argument developed in the paper [1]

where they derive the existence of operators with dimensions 2∆ + s and their anomalous dimension¹, assuming the existence of a conformal primary with dimension ∆. Or to put it more explicitly, their result is that if a unitary CFT contains a scalar operator of dimension

∆ then at large s a tower of double trace operators exists with conformal dimension given by

τ = 2∆ −2cτ_m

s^τm , (13)

1To be fair; that paper gives a more general result than that and demonstrates that if operators with dimension ∆1 and ∆2 exist there should exist large spin operators with dimension arbitrarily close to

∆1+ ∆2. The special case ∆1= ∆2will be reviewed here for its lighter notation

where τ is the conformal twist defined as τ = ∆k − s. The coefficient of the anomalous dimension is given by

c_τm = f² Γ(τ_m+ 2s_m) (−2)^smΓ(^τm+2s₂ ^m)²

Γ(∆)²

Γ(∆ −^τ₂^m)², (14)

where the subscript m designates that it is a property of the local operator in the spectrum with lowest twist (excluding the identity operator). The constant f designates the coeffi- cient of the minimal twist operator that appears in the OPE of O(x) with itself. The first observation that has to be made is that the operator product expansion on a Minkowski background due to dimensional analysis has the form of equation (8)

O₁(0)O₂(x) =X

O^(k)

c_12k

|x|^{∆1+∆2−∆k}

x^µ1...x^µ2

|x|^s O^(k)_µ

1...µ_s(x), (15)

where the first sum is over the conformal primaries and the second is over all spins.

The argument now depends on the evaluation of the four-point function

F (x1, x2, x3, x4) = hO(x1)O^†(x2)O(x3)O^†(x4)i, (16)

where conformal transformations have been applied to map all the points onto a plane. The points are chosen to be at the lightcone coordinates x1 = (0, 0), x2 = (z, ¯z), x3 = (1, 1) and x4 = (L, L) where L is a large constant in the sense that to any arbitrary precision

|x1− x4| ≈ |x2− x4| ≈ |x3− x4|, see figure 2. As was mentioned in the previous section two different conformal partial wave decompositions can be found; firstly the s-channel where one first takes the OPE between O(x2) and O(x1). which results in

F (z, ¯z) = 1 z ¯z

^∆ X

∆_k

∞

X

s=0

c_s,∆kG_s,∆k(z, ¯z). (17)

where Gs,∆(z, ¯z) are the conformal blocks associated to the double-trace operators and their descendants that occur in the OPE and are found by applying equation (12)² to our

2Note that the factor ₂¹s in equation (12) has been suppressed in equations (18) and (19), if you wish to keep it just remember to multiply the CPW coefficients with a factor 2^s

situation:

G∆,s= (−1)^s z ¯z z − ¯z



z^{τ +2s2} z¯^{τ −22} 2F1(τ + 2s

2 ,τ + 2s

2 , τ + 2s, z)2F1(τ − 2 2 ,τ − 2

2 , τ − 2, ¯z) + z ↔ ¯z

 . (18) By taking the limit z → 0 the last expression is to leading order given by the collinear blocks

g∆,s≈ z^τ2z¯^{τ +2s2} 2F1(τ + 2s

2 ,τ + 2s

2 , τ + 2s, ¯z) (19)

Similarly one could have chosen to first take the OPE between O(x2) and O(x3) which would have resulted in a decomposition which will be called the t-channel:

F (z, ¯z) =

 1

(1 − z)(1 − ¯z)

∆

X

∆_k

∞

X

s=0

c_s,∆kg_s,∆k(1 − z, 1 − ¯z). (20)

The distinction between these channels was displayed graphically in figure (3). It is important now to note that in the limit z → 0 and ¯z → 1 the t-channel conformal blocks can be approximated as

g∆,s(1 − z, 1 − ¯z) → − Γ(τ + 2s)

(−2)^sΓ(^{τ +2s}₂ )²(1 − ¯z)^τ2(log(z) + O(1)) (21) At this point the constraint of unitarity comes in, a unitary CFT places a contraint on the dimensions of the operator spectrum of the CFT, specifically τ ≥ d − 2 for operators with spin greater then 1 and ∆ ≥ ^d−2₂ for scalar operators. There is one exception in the form of the identity operator which has scaling dimension equal to 0. The result is that in a CFT that lives in d > 2 there exists a so-called twist gap i.e. there exists a certain operator with minimal twist τm. Hence by taking the limit ¯z → 1 we can keep the conformal blocks related to the identity and the minimal twist operator and neglect the rest of the terms in the CPW decomposition since they are supressed by higher powers of (1 − ¯z)³. The result is the following approximation for the t-channel CPW decomposition:

F (z, ¯z) =

 1

(1 − z)(1 − ¯z)

^∆

1 − f² Γ(τm+ 2sm)

(−2)^smΓ(^τm+2s₂ ^m)²(1 − ¯z)^τm2 log(z) + ...

!

, (22)

3In other words in the regime we are interested in we can think of the CFT as one where the only operator that couples to the operators O is the minimal twist operator.

where f is the OPE-coefficient related to the minimal twist operator in the OPE between O(x2) and O(x3). Note that the minimal twist operator does not have to be unique. If the theory under consideration contains multiple primaries with conformal dimension τm then we simply add up their respective conformal blocks.

A key point is that both channels can be made to converge simultaneously by taking x₂ to ¯z = 1 and z = 0 since at this point the lightcones of both x₁ and x₃ intersect therefore x2 can be made arbitrarily close to both points simultaneously. This leads to the following crossing equation

X

∆_k

∞

X

s=0

c_s,∆kg_s,∆k(z, ¯z) =

 z ¯z (1 − z)(1 − ¯z)

∆

1 − f² Γ(τ_m+ 2s_m)

(−2)^smΓ(^τm+2s₂ ^m)²(1 − ¯z)^τm2 log(z) + ...

! . (23) At this point it can be argued that the operator spectrum of the CFT needs to contain double-trace operators of conformal dimension close to 2∆ + s, since it is the only set of collinear blocks (19) that can reproduce the leading order dependance on z of the righthand side.

3.1 Evaluating the s-channel CPW decomposition

To demonstrate that the anomalous dimension can be argued to be of the form given in (13) start with the decomposition into collinear blocks of the free field solution of the four point function

 1 z ¯z

^{∆ ∞} X

s=0

c_s,∆g_s,∆(z, ¯z) = (z ¯z)^−∆+ 1 + (1 − z)(1 − ¯z))^−∆.(24)

In the region we are interested in this can reasonably be approximated by

∞

X

s=0

c_s,∆g_s,∆(z, ¯z) =

 z ¯z (1 − z)(1 − ¯z)

∆

, (25)

which is exactly the contribution of the identity operator to the crossing equation (23). The coefficients of the CPW decomposition are known and have been calculated in for instance

[6] by:

c_N,s= (1 + (−1)^s)(∆ −^d₂+ 1)²_N(∆)²_{N +s}

Γ(s + 1)Γ(N + 1)(s +^d₂)N(2∆ + N − d + 1)N(2∆ + 2N + s)s(2∆ + N + s −^d₂)N

, (26) where d is the number of spacetime dimensions and where (a)b denotes the Pochhammer symbol, which is defined as (a)b= Γ(a + b)/Γ(a). At the moment we are interested only in the leading N = 0 terms, in which case the coefficients reduce to:

cs= (1 + (−1)^s) Γ(∆ + s)²Γ(2∆ + s − 1)

Γ(s + 1)Γ(∆)²Γ(2∆ + 2s − 1). (27) Plugging the collinear blocks from equation (19) into the decomposition (25) and replacing the sum by an integral over s leads to the expression

Z ∞ 0

ds cs,∆z^∆z¯^∆+s2F1(∆ + s, ∆ + s, 2∆ + 2s, ¯z), (28)

note that this means that we implicitly expect the sum over large spins to be most relevant.

First apply an integral representation of the confluent hypergeometric function to obtain the expression

Z ∞ 0

ds Z 1

0

dt 2 Γ(∆ + s)²Γ(2∆ + s − 1) Γ(s + 1)Γ(∆)²Γ(2∆ + 2s − 1)

Γ(2∆ + 2s) Γ(s + ∆)²

_zt(1−t)¯

1−¯zt

s+∆

t(1 − t) . (29) Next perform the coordinate transformations:

z = e^−2β,

¯

z = 1 − e^−2σ, ˆ

s = s + ∆.

(30)

This results in the expression Z 1

0

dt Z ∞

∆

dˆs 2 Γ(∆ + ˆs − 1)Γ(2ˆs) Γ(ˆs − ∆ + 1)Γ(∆)²Γ(2ˆs − 1)

_(1−e−2σ)t(1−t) 1−(1−e^−2σ)t

ˆs

t(1 − t) . (31)

The quotient of gamma-functions can be significantly simplified by applying in turn the recurrence relation Γ(x + 1) = xΓ(x), the Stirling approximation Γ(x) =q

2π

xe^{x log(x)−x}(1 + O(¹_x)) and only keeping the leading order term in an expansion in ¹_ˆs. This all results in

4 Γ(∆)²

Z 1 0

dt Z ∞

∆

dˆs ˆ

s^(2∆−1)_(1−e−2σ)t(1−t) 1−(1−e^−2σ)t

^sˆ

t(1 − t) (32)

The next point is to determine which spins dominate the above integral. To do so perform a saddle point analysis. Rewriting numerator in the integral as an exponent leads to the dominant spin

ˆ

s₀= − 2∆ − 1 log_{(1−e−2σ)t(1−t)}

1−(1−e^−2σ)t

 (33)

Resulting expression is dependant on t. To eliminate the t-dependance first use the Laplace method to eliminate the integral over s and then find the saddle point for the integral over t. The first step leads to

4e^1−2∆√

π(2∆ − 1)^2∆−12 Γ(∆)²

Z 1 0

dt

− log_(e2σ−1)(1−t)t e^2σ(1−t)+t

−2∆

t(1 − t) (34)

Doing a series approximation around (1 − t) << 1 keeping the resulting three singular terms and determining the dominant contribution to t leads to the approximate form

t0= 1 −r 2∆ − 1

2∆ + 1e^−σ+ O(e^−2σ). (35)

Now that we got this, the value for s that dominates this sum can be expressed as

log(s0) = σ + log (2∆ − 1)³²(2∆ + 1)¹² 4∆

!

+ O(e^−σ). (36)

This demonstrates that an anomalous dimension of the form shown in (13) will accurately reproduce the ¯z dependance of the subleading term on the righthand side of the crossing equation (23).

3.2 The anomalous dimension coefficient

In this section the deriviation of the coefficient (14) will be reviewed. First look at the crossing equation (23), the important aspect is that the only place where τ occurs and the anomalous dimension would not be negligable compared to s is in the power of z.

Performing the series expansion z^∆−= z^∆(1 − log(z) + O(²)) demonstrates that the this will reproduce the factor log(z) in the subleading term on the righthand side. This gives

a good indication that this approach is on the right track. Doing this expansion in the s-channel CPW decomposition and equating the resulting subleading terms in (23) results in

P∞ s −cs

c_τm

s^τmz^∆log(z)¯z^∆+s2F1(∆ + s, ∆ + s, 2∆ + 2s)

=

z ¯z (1−z)(1−¯z)

^∆

f^{2 Γ(τm+2sm)}

(−2)^smΓ(^τm+2sm₂ )²(1 − ¯z)^τm2 log(z) (37) Taking on the righthand side only the leading z^∆part in the region z << 1 and ignoring all powers of ¯z results in the following defining equation for c_τm.

cτ_m = f² Γ(τm+ 2sm) (−2)^smΓ(^τm+2s₂ ^m)²

(1 − ¯z)^τ2−∆

P∞

s c_ss^−τm₂F₁(∆ + s, ∆ + s, 2∆ + 2s, ¯z) (38) Therefore the only thing left to do is to evaluate the sum in the denominator

∞

X

s

css^−τm2F1(∆ + s, ∆ + s, 2∆ + 2s, ¯z). (39)

To do that first perform the transformation ∆ + s = ^√A_. Replace the sum over s by an integral over A

Z ∞ 0

dA√

2

Γ(^√A_)²Γ(^√A_ + ∆ − 1)

Γ(^√A_− ∆ + 1)Γ(∆)²Γ(^2A√_− 1)(A

√− ∆)^−τm2F1( A

√, A

√,2A

√, 1 − ) (40)

where the final argument of the hypergeometric function is due to the just established approximate relation s ≈ ^√_1−¯¹ _z. Next take the limit  → 0, such as to exploit the previous assertion that it is the large spin contribution that dominates the sum. In this limit the hypergeometric function behaves as

2F₁( A

√, A

√,2A

√, 1 − ) → A

√ 4^√A

¹⁴ K₀(2A), (41)

where K0(x) is the modified Bessel function of the second kind. Using this and applying the Stirling approximation for the gamma-functions in the coefficient results in the following form for (39)

4 Γ(∆)²^∆−τ2

Z ∞ 0

dA A^{2∆−τm−1}K0(2A). (42)

Applying the following integral identity of the Bessel functionR∞

0 dx x^2c−1K0(2x) = ¹₄Γ(c)² the previous integral can be solved to give

4 Γ(∆)²^∆−τ2

Z ∞ 0

dA A^{2∆−τm−1}K0(2A) = 1

^∆−τ2

Γ(∆ −¹₂τm)²

Γ(∆)² . (43)

Finally replacing the sum in (38) by this expression results in equation (14) that was quoted at the start of this section:

cτ_m = f² Γ(τm+ 2sm) 2^smΓ(^τm+2s₂ ^m)²

Γ(∆)²

Γ(∆ −^τ₂^m)². (44)

4 Generalizing to arbitrary N

In this section we will build on from the previously stated results by applying the techniques developed in the paper [1] to show that operators have to exist in the spectrum with twist close to 2∆ + 2N where N is any positive integer. The result is an expression for the general anomalous dimension of these large spin operators, which turns out to be identical to the N = 0 anomalous dimension for all integers N . To do this make the stronger assumption that the crossing equation from before

X

∆_k

∞

X

s=0

cs,∆_kgs,∆_k(z, ¯z) =

 z ¯z (1 − z)(1 − ¯z)

^∆

1 − f² Γ(τm+ 2sm)

(−2)^smΓ(^τm+2s₂ ^m)²(1 − ¯z)^τm2 log(z) + ...

!

(45) not only holds true in the limit z → 0 but also at finite but small separation from 0. To do that take the full dependance of z into account of the crossing equation (23). By considering the expression _(1−z)¹ ∆ on the righthand side and expressing it in series

P

∆k

P∞

s=0cs,∆_kGs,∆_k(z, ¯z) =

¯ z (1−z)

∆ P∞

q=0

Γ(∆+q)

Γ(∆)Γ(q+1)z^∆+q

×

1 − f^{2 Γ(τm+2sm)}

(−2)^smΓ(^τm+2sm₂ )²(1 − ¯z)^τm2 log(z) + ... .

(46)

Note that instead of just z^∆we now have an infinite series of terms that scale with z^∆+q to recreate on the left-hand side. In the limit ¯z → 1 it seems natural that these terms would be recreated on the left hand side by assuming that conformal blocks with twist such that

τ

2 = ∆ + q have to enter the s-channel OPE, since it is known that they are required to build four-point function of the generalized free theory.

Unfortunately the collinear blocks (19) that have been used so far are only the leading order term that one obtains when expanding the full conformal blocks (18) in powers of z therefore if we want to take the right-hand side at finite separation we have to be fair and also take the full z dependance of the s-channel conformal blocks into account. Therefore to find the anomalous dimension for every value of N the left-hand side needs to be fully expanded in powers of z and both sides of the equation have to matched term by term.

First focus on the full left hand side of the above equation P

∆_k

P∞

s=0cs,∆_kGs,∆_k(z, ¯z) = P

∆_k

P∞

s=0c_s,∆k((−1)^{s z ¯}_z−¯^z_zz^{∆+N +s}z¯^{∆+N −1}₂F₁(∆ + N + s, ∆ + N + s, 2(∆ + N + s), z)

×₂F₁(∆ + N − 1, ∆ + N − 1, 2(∆ + N − 1), ¯z) + z ↔ ¯z),

(47) where the conformal dimension of the double trace operators has already been applied. The defining sum representation of the hypergeometric function is given by

2F1(a, b, c, x) =

∞

X

n=0

(a)n(b)n

(c)_n zⁿ

n! (48)

Therefore by expanding the prefactor _z−¯^{z ¯z}_z in (47) in series of z and multiplying this with a series expansion of the hypergeometric series that have z as an argument a full power series of in z can be found:

P

N

P

s

P

k−cs,n(−1)^{s zk+1}_¯zk (¯z^{∆+N −1}₂F₁(∆ + N − 1, ∆ + N − 1, 2∆ + 2N − 2, ¯z) P

l

Γ(∆+N +s+l)²Γ(2∆+2N +2s)

Γ(∆+N +s)²Γ(l+1)Γ(2∆+2n+2s+l)z^{∆+N +s+l}− ¯z^{∆+N +s}₂F₁(∆ + N + s, ∆ + N + s, 2∆ + 2N + 2s, ¯z) P∞

l⁰=0

Γ(∆+N +l⁰−1)²Γ(2∆+2N −2)

Γ(∆+N −1)²Γ(l⁰+1)Γ(2∆+2N +l⁰−2)z^{∆+N −1+l0})

(49) These two quadruple sums can be reordered into a single quadruple sum in the following way

P

N

P

s

P

kc_s,n(−1)^s

−P∞ k=0

z^k+1

¯ z^k

P∞

l=0(¯z^{∆+N −1}θ_s+1(l)₂F₁(∆ + N − 1, ∆ + N − 1, 2∆ + 2N − 2, ¯z)

Γ(∆+N +l−1)²Γ(2∆+2N +2s)

Γ(∆+N +s)²Γ(l−s)Γ(2∆+2N +s+l−1)− ¯z^{∆+N +s}₂F₁(∆ + N + s, ∆ + N + s, 2∆ + 2N + 2s, ¯z)

Γ(∆+N +l−1)²Γ(2∆+2N −2)

Γ(∆+N −1)²Γ(l+1)Γ(2∆+2N +l−2))z^{∆+N −1+l}

(50) Where the step function θk(l) is defined as:

θk(l) =

 0 if l < k 1 if l ≥ k (51)

The mayor simplifying argument is that if we keep the power of z fixed then l is also fixed, but as was shown in [1] it is known that the sum over s in the CPW decomposition will be dominated by the high spin terms. The left term in the sum over l is exactly zero though if s + 1 > l therefore if doing a saddle point analysis over the sum over s is justified then it should be safe to ignore the entire left most term. This is nice because the only difference between the rightmost term for every index pair (l, k) and the collinear block is a factor that although large is independant of s. Therefore if it was not the case already that the terms ignored are negligable then we could just take ¯z closer to 1 such that the saddle-point of the important points gets shifted to larger values.

This is a very heuristic argument, but doing the actual sum over all spins over the terms that are not neglected (as is done in appendix B) demonstrates that this sum will scale as _1−¯¹_z^∆, and therefore reproduces the scaling of the identity operator contribution to the right-hand side. The finite number of terms that have been neglected have their scaling with variation in ¯z determined by the hypergeometric functions ₂F₁(∆ + N − 1, ∆ + N − 1, 2∆ + 2N − 2, ¯z). Which will to leading order diverge logarithmically in the limit ¯z → 1. Therefore these terms only form logarithmic corrections and can safely be ignored. Furthermore this means that this can be made arbitrarily precise by considering values of ¯z as close to 1 as one likes.

Also doing a parameter transformation q = N + K + l and eliminating l in favor of q leads to the expression

P∞ q=0

Pq N =0

Pq−N k=0

P∞

s=0(−1)^scs,N Γ(∆+q−k−1)²Γ(2∆+2N −2) Γ(∆+N −1)²Γ(q−N −k+1)Γ(2∆+N +q−k−2)

×2F1(∆ + N + s, ∆ + N + s, 2∆ + 2N + 2s, ¯z)z^∆+q

(52)

The nice intermediate result is that you have an explicit sum over the powers of z so now you can equate the conformal bootstrap equation term-by-term. Also notice that the problem:

how many terms have the same power of z. Is equivalent to the problem in how many ways can I distribute q identical objects over 3 distinct containers. Therefore the number of terms with the same power is equal to2 + q

q



= ¹₂(q + 1)(q + 2).

As before introduce an anomalous dimension in the exponent of z of the form γ(N, s) =

−^2cτm_sτm^{(N )} note that it is anticipated that the anomalous dimension coefficient will depend on N . This anomalous dimension will recreate the logarithmic dependance on the right hand side via the expansion z^∆+= z^∆(1 + log(z) + O(²)). This results in a crossing equation of the form

P∞ q=0

Pq N =0

Pq−N k=0

P∞

s=0(−1)^scs,N Γ(∆+q−k−1)²Γ(2∆+2N −2) Γ(∆+N −1)²Γ(q−N −k+1)Γ(2∆+N +q−k−2)

×2F1(∆ + N + s, ∆ + N + s, 2∆ + 2N + 2s, ¯z)z^∆+q

1 − 2 log(z)^cτm_sτm^{(N )}

=

¯ z (1−z)

∆ P∞

q=0

Γ(∆+q)

Γ(∆)Γ(q+1)z^∆+q 

1 − f^{2 Γ(τm+2sm)}

(−2)^smΓ(^τm+2sm₂ )²(1 − ¯z)^τm2 log(z) + ... . (53) By arguing that the above crosssing equation should also hold at finite separation, both sides can be equated term by term in powers of z. Now equate the two first subleading terms that scale with log(z), this results in the following more complete crossing equation

Pq N =0

Pq−N k=0

P∞

s=0(−1)^s+1cs,N Γ(∆+q−k−1)²Γ(2∆+2N −2) Γ(∆+N −1)²Γ(q−N −k+1)Γ(2∆+N +q−k−2)

×2F₁(∆ + N + s, ∆ + N + s, 2∆ + 2N + 2s, ¯z)z^∆+q2 log(z)^{cτm(N )}_sτm

= −

¯ z (1−z)

^∆ _Γ(∆+q)

Γ(∆)Γ(q+1)z^∆+qf^{2 Γ(τm+2sm)}

(−2)^smΓ(^τm+2sm₂ )²(1 − ¯z)^τm2 log(z).

(54)

The sum over s in equation (54) can be resolved with the assumption that it is the terms

at large s that dominate the sum. The calculation is done in appendix A. The result is:

Pq N =0

Pq−N

k=0 cτ_m(N ) ^{Γ(∆+q−k−1)2}Γ(2∆+2N −2) Γ(∆+N −1)²Γ(q−N −k+1)Γ(2∆+N +q−k−2)

×¹₂α(N )(1 − ¯z)^12τm−∆Γ(∆ −¹₂τm)z^q,

(55)

where the factor α(N ) is defined as

α(N ) ≡ 2Γ(∆ + N − 1)²Γ(2∆ + N − 3)

(∆ − 1)²Γ(∆ − 1)⁴Γ(2∆ + 2N − 3). (56) The result is the following crossing equation for the first subleading term

Pq N =0

Pq−N

k=0 2c_τm(N ) ^{Γ(∆+q−k−1)2}Γ(2∆+2N −2) Γ(∆+N −1)²Γ(q−N −k+1)Γ(2∆+N +q−k−2)

×¹₂α(N )(1 − ¯z)^12τm−∆Γ(∆ −¹₂τm)z^∆+q = 

¯ z (1−z)

^∆ _Γ(∆+q)

Γ(∆)Γ(q+1)z^∆+q

×f^{2 Γ(τm+2sm)}

(−2)^smΓ(^τm+2sm₂ )²(1 − ¯z)^τm2 log(z).

(57)

From this point onwards all the anomalous dimension coefficients cτ_m(N ) can be found recursively. To do this take the term related to the greatest value of N (i.e. cτ_m(q)) and pull it out of the sum. The result is

cτ_m(q) = _{α(q)Γ(∆−}² 1 2τ )²

h _Γ(∆+q)

Γ(∆)Γ(q+1)f^{2 Γ(τm+2sm)}

(−2)^smΓ(^τm+2sm₂ )²

−Pq−1 N =0

Pq−N

k=0 c_τm(N )^{Γ(∆+q−k−1)2}Γ(2∆+2N −2)α(N )Γ(∆−¹₂τ_m)² 2Γ(∆+q−1)²Γ(q−N −k+1)Γ(2∆+q+N −k−2)].

(58)

Reinstating the cumbersome factors α(N ) leads to cτ_m(q) = Γ(2∆+2q−3)Γ(∆−1)²Γ(∆)Γ(q+1)

Γ(∆+q−1)²Γ(2∆+q−3)Γ(∆−¹₂τ )²

h_Γ(∆+q)

Γ(q+1)f^{2 Γ(τm+2sm)}

(−2)^smΓ(^τm+2sm₂ )²

−Pq−1 N =0

Pq−N

k=0 c_τm(N ) ^{Γ(∆+q−k−1)2}(2∆+2N −3)Γ(2∆+N −3)Γ(∆−¹₂τ_m)² Γ(2∆+N +q−k−2)Γ(q−N −k+1)Γ(∆−1)²Γ(∆)Γ(N +1)].

(59)

Which trivially reduces to the result of [1],[2] in the case q = 0. To keep the interpretation clear. The one positive term in square brackets is the contribution from the anomalous dimension of the conformal block G_∆,qthe negative terms are corrections due to resumming over the subleading terms of the conformal blocks G_∆,N where N < q.

From this point it is clear though that the anomalous dimension coefficients are unique in the sense that they are fixed by all preceding anomalous dimension coefficients, including the q = 0 case which is known.

4.1 proof of N-independance

The defining equation for the anomalous dimension (59) derived so far can be shown to be constant for all q. The proof for that statement follows from complete induction. First assume that cτ(i) = f^{2 Γ(τm+2sm)}

(−2)^smΓ(^τm+2sm₂ )²

Γ(∆)²

Γ(∆−¹₂τ_m)² = Λ for all i < q. In this case the coefficients can be pulled out of the double sum in (59).

The most important tool for this proof is the consistency relation noted in appendix B:

q

X

N =0 q−N

X

k=0

Γ(∆ + q − k − 1)²(2∆ + 2N − 3)Γ(2∆ + N − 3)

Γ(q − N − k + 1)Γ(2∆ + q + N − k − 2)Γ(∆ − 1)²Γ(N + 1) = Γ(∆ + q) Γ(∆)Γ(q + 1).

(60) Adding and subtracting a term for N = q in equation (59) and applying the equation above leads to

c_τm(q) = Γ(2∆+2q−3)Γ(∆−1)²Γ(∆)Γ(q+1) Γ(∆+q−1)²Γ(2∆+q−3)Γ(∆−¹₂τ )²

h_Γ(∆+q)

Γ(q+1)Λ^Γ(∆−_Γ(∆)^12τ₂^m)2

−Λ^{Γ(∆+q)Γ(∆−}_Γ(∆2)Γ(q+1)^12τm)2 + Λ^{Γ(∆+q−1)}

2Γ(2∆+q−3)Γ(∆−¹₂τ )² Γ(2∆+2q−3)Γ(∆−1)²Γ(∆)Γ(q+1)] = Λ.

(61)

Take as a trivial induction step q = 1 to complete the proof. The conclusion that to the precision in which we have considered the t-channel four-point function, the anomalous dimension derived in [1],[2], i.e.

cτ(N ) = f² Γ(τm+ 2sm) (−2)^smΓ(^τm+2s₂ ^m)²

Γ(∆)²

Γ(∆ −¹₂τm)², (62) is valid for all double-trace operators of the form O^N∂^sO that appear in the OPE of O with itself.

5 A more general four-point function

Too keep up the parallel with [1] consider a more general four-point function. A much stronger result can be obtained by considering crossing symmetry of a four point function of the form

F (z, ¯z) = hO1(0, 0)O2(z, ¯z)O2(1, 1)O1(L, L)i, (63)