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Published for SISSA by Springer

Received: August 16, 2016

Revised: January 29, 2017

Accepted: March 7, 2017

Published: March 15, 2017

Extracting OPE coefficient of Konishi at four loops

Vasco Goncalves

ICTP South American Institute for Fundamental Research Instituto de Fisica Teorica,

UNESP, Universidad Estadual Paulista,

Rua Dr. Bento T. Ferraz 271, 01140-070, Sao Paulo, SP, Brasil

E-mail: vasco.dfg@gmail.com

Abstract: In this short note we compute the OPE coefficient of two 20′ operators and the

Konishi operator. To this end, we use the OPE decomposition of a four point function of

four 20′ operators and the method of asymptotic expansions to compute the leading term,

in the OPE limit, of all integrals contributing to the four point function.

Keywords: AdS-CFT Correspondence, Conformal Field Theory

ArXiv ePrint: 1607.02195

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP03(2017)079

mailto:vasco.dfg@gmail.com
https://arxiv.org/abs/1607.02195
http://dx.doi.org/10.1007/JHEP03(2017)079


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Contents

1 Introduction 1

2 Four point function and OPE limit 2

2.1 Asymptotic expansions 3

2.2 Konishi from OPE limit 5

3 Conclusions 6

A Master integrals 7

1 Introduction

Correlation functions of local operators in a CFT are completely determined by dimensions

of all operators and their OPE coefficients. Over the last years there has been a significant

progress in computing the dimensions and OPE coefficients of local operators in N = 4

SYM [1]. Integrability of the planar sector of this particular CFT has allowed the deter-

mination of the spectrum of single trace operators at any value of the coupling. Recently,

it was proposed a method (hereafter called the hexagon approach) to compute OPE coef-

ficients of single trace operators at any value of the coupling [2]. This new approach has

passed several non-trivial checks [2–4]. At weak coupling, there are new features appearing

at each loop order and in the past it was useful to have these OPE coefficients computed

by other means in order to check the correctness of the integrability result. The interest

in the four loop stems from the appearence of a new effect in the hexagon approach due

to wrapping effects [2, 3]. Thus, reproducing the result of this note will be an important

non-trivial check of the integrability computation.

We compute the OPE coefficient of two 20′ operators and the Konishi operator in the

four loop level by doing the OPE decomposition of a four point function 20′ operators. This

four point function is known only at the integrand level, so to extract the OPE coefficient

we will use the method of asymptotic expansions that allows to obtain a series expansion

of all integrals in OPE limit. This method has already been implemented in the past to

determine the OPE coefficient at three loops [5].

In the next section we will define the four point function that we will be working with.

Then we briefly review the method of asymptotic expansions and finally we extract the

OPE coefficient by considering a limit of the four point function.

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2 Four point function and OPE limit

In N = 4 SYM there are special operators (often called protected) that do not receive

quantum corrections to their dimension and OPE coefficients or in other words, their two

and three point function are the same at any coupling. However, a four point function of

these operators does get corrected. One way to understand this is by writing the four point

function as a sum of two three point functions, i.e. by doing the OPE decomposition

〈O(x1)O(x2)O(x3)O(x4)〉 =
∑
k

c2
OOOk

(x2
12x

2
34)∆O

G∆k,Jk(u, v), u =
x2

12x
2
34

x2
13x

2
24

, v =
x2

14x
2
23

x2
13x

2
24

(2.1)

where cOOOk is an OPE coefficient, G∆,J(u, v) is a conformal block (that resums the con-

tribution of a conformal family to a four point function) and u and v are cross ratios.

In general the OPE coefficient cOOOk and dimension ∆k depend on the coupling. Conse-

quently, the four point function will inherit this dependence.

Our main goal is to extract an OPE coefficient that has wrapping. This effect starts to

be present at four loops for small operators, like the Konishi. The correlation function of 20′

operators is the only one that has been computed at the four loop level. For completeness

let us define the 20′ operators

O(x, y) = YIYJOIJ20′(x) = YIYJtr
(
ΦI(x)ΦJ(x)

)
, Y 2 = YIYI = 0. (2.2)

where the null variables Y insure that the operator is symmetric and traceless in the R-

charge indices. The four point function depends on the polarization vectors YI . Naively,

one would expect a nontrivial dependence on these variables but it turns out that this

dependence factorizes and consequently the four point function can be written as [16]

G4 = 〈O(x1, y1) . . .O(x4, y4)〉 =

∞∑
l=0

alG
(l)
4 (1, 2, 3, 4), (2.3)

with the tree level result given by

G(0)(1, 2, 3, 4) =
(N2 − 1)2

4(4π2)4

(
y4

12y
4
34

x4
12x

4
34

+
y4

13y
4
24

x4
13x

4
24

+
y4

14y
4
23

x4
14x

4
23

)
+
N2 − 1

(4π2)4

(
y2

12y
2
23y

2
34y

2
41

x2
12x

2
23x

2
34x

2
41

+
y2

12y
2
24y

2
43y

2
31

x2
12x

2
24x

2
43x

2
31

+
y2

13y
2
32y

2
24y

2
41

x2
13x

2
32x

2
24x

2
41

)
, yij = Yi · Yj (2.4)

and the loop level by

G
(l)
4 =

2(N2
c − 1)

(4π2)4
R
x2

12x
2
13x

2
14x

2
23x

2
24x

2
34

l!(−4π2)l

∫
d4x5 . . . d

4x4+lf
(l)(x1, . . . , x4+l), (for l ≥ 1)

where a is the t’Hooft coupling a = g2Nc/(4π
2) and R contains all the dependence on the

polarization vectors Y

R =
y2

12y
2
23y

2
34y

2
41

x2
12x

2
23x

2
34x

2
41

(
x2

13x
2
24 − x2

12x
2
34 − x2

14x
2
23

)
+
y2

12y
2
24y

2
43y

2
31

x2
12x

2
24x

2
43x

2
31

(
x2

14x
2
23 − x2

12x
2
34 − x2

13x
2
24

)
+
y2

13y
2
32y

2
24y

2
41

x2
13x

2
32x

2
24x

2
41

(
x2

12x
2
34 − x2

13x
2
24 − x2

14x
2
23

)
+
y4

12y
4
34

x2
12x

2
34

+
y4

13y
4
24

x2
13x

2
24

+
y4

14y
4
23

x2
14x

2
23

. (2.5)

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The function f (l)(x1, . . . , x4+l) possesses a hidden permutation symmetry S4+l and this,

together with imposing the correct OPE behavior, has led to a complete description of its

form up to a high loop order [6]. An useful representation for f (l)(x1, . . . , x4+l) is [6, 7]

f (l)(x1, . . . , x4+l) =
P (l)(x1, . . . , x4+l)

Π1≤i<j≤4+lx
2
ij

(2.6)

where P (l)(x1, . . . , x4+l) is a symmetric polynomial depending only on distances x2
ij and is

homogeneous of degree (l − 1)(l + 4)/2 in each point. The planar part of this polynomial

is given up to four loops by [7]

P (1) = 1, P (2) =
1

48
x2

12x
2
34x

2
56 + S6 perm, P (3) =

1

20
(x2

12)2x2
34x

2
45x

2
56x

2
67x73 + S7 perm

P (4) =
1

24
x2

12x
2
13x

2
16x

2
23x

2
25x

2
34x

2
45x

2
46x

2
56x

6
78 +

1

8
x2

12x
2
13x

2
16x

2
24x

2
27x

2
34x

2
38x

2
45x

4
56x

4
78

− 1

16
x2

12x
2
15x

2
18x

2
23x

2
26x

2
34x

2
37x

2
45x

2
48x

2
56x

2
67x

2
78 + S8 permutations. (2.7)

The integrals appearing up to three loops can be expressed in terms ladder integrals and two

functions called Easy and Hard integrals [7, 8]. We have not tried to count the minimum

number of independent integrals that appear at four loop level because the computer time

saved is not considerable since we are only interested in the contribution of the Konishi

operator.

2.1 Asymptotic expansions

Each integral appearing in the planar part of the four point function described above is

both UV and IR finite. They are also conformal, consequently they depend on two cross

ratios u and v. Moreover, it is possible to send one of the points to infinity since all the

integrals have conformal symmetry. Most of the integrals at four loops are not known

explicitly as a function of these cross ratios, however the method of asymptotic expansions

can be used to reduce the computation of these four point integrals to the evaluation of

simpler integrals involving just two points. The expansion of the integrals in terms of the

cross ratios can then be used to extract the dimension and OPE coefficients of the operators

that can couple to the external ones. This method has been used in the past to compute

the OPE coefficient of twist two operators at three loops [5].

We will review briefly how the method works on the four-loop integral that appears in

this four point function1

I =

∫
x2

15x
2
67 d

4x5d
4x6d

4x7d
4x8

x2
16x

2
17x

2
18x

2
25x

2
26x

2
27x

2
28x

2
35x

2
36x

2
45x

2
47x

2
56x

2
57x

2
68x

2
78

=

∫
x2

5x
2
67 d

4x5d
4x6d

4x7d
4x8

x2
6x

2
7x

2
8x

2
25x

2
26x

2
27x

2
28x

2
35x

2
36x

2
56x

2
57x

2
68x

2
78

(2.8)

where the point x4 was sent to infinity and x1 to 0 using conformal invariance of the

integral. The cross ratios, in these coordinates, are given by u = x2
2/x

2
3 and v = x2

23/x
2
3.

1More details on the method can be found in [5, 15].

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The method of asymptotic expansions can be used obtain a series expansion in small u and

(1− v) to any desired order. Powers of u in a four point function control the twist (recall

that twist is defined by ∆− J) and powers of (1− v) control the spin, J , of the operators.

We are only interested in extracting the OPE coefficient of the Konishi operator, so we can

focus only on the leading term of the expansion.

The main idea of the method is to divide the range of the integration of each integration

variable in two regions, one where it is of the order of x2 � 1, that is assumed to be

small, and other where it is of the order of x3. There are four integration variables and

consequently 24 = 16 integration regions. The goal of dividing into these regions is that

it allows to simplify the integrand. For example in the region where all the integration

variables are of the order x3 we can use

1

x2
2j

=
∞∑
n=0

(2x2 · xj − x2
2)n

(x2
j )

1+n
, for j = 5, . . . , 8. (2.9)

Obviously, this equation is only valid when the region of integration satisfies x2
2 ≤ x2

j ,

however we can extend this region of integration to all space at the expense of introducing

a regulator d = 4− 2ε. The integrals will have poles in ε as a consequence of extending the

integration region. However, the sum of all regions needs to give a finite result in the limit

of ε → 0 since we are dealing with finite integrals. We have verified that this happens for

all integrals that we have analyzed.

At the end of the day each integration region is expressed in terms of integrals of the

propagator type∫
ddx5d

dx6d
dx7d

dx8

(x2
5)a1(x2

25)a2(x2
56)a3(x2

57)a4(x2
58)a5(x2

6)a6(x2
26)a7(x2

67)a8(x2
68)a9

× (2.10)

× 1

(x2
7)a10(x2

27)a11(x2
78)a12(x2

8)a13(x2
28)a14

, ai ∈ Z

and fortunately all integrals that are needed have been computed before [7, 13].

Let us go back to the example where all the integration variables are of the order x3

in the integral I

∞∑
ni=0

∫ ∏8
i=5(2xi ·x2 − x2

2)nix2
67 d

dx5 . . . d
dx8

(x2
5)n5(x2

6)2+n6(x2
7)2+n7(x2

8)2+n8x2
35x

2
36x

2
56x

2
57x

2
68x

2
78

. (2.11)

Higher powers of x2
2 encode the contribution of higher twist operators, since we are only

interested in twist two, we can neglect the factor x2
2 in (2xi·x2−x2

2). A simple dimensional

analysis tells that the result after integration will be a combination of terms∑
k

ck
(x2

2x
2
3)ni−k(x2 · x3)k

(x2
3)10+

∑8
i=5 ni−2d

. (2.12)

for certain coefficients ck. In particular, this shows that non-zero values of ni will give either,

higher powers of x2
2 or x2 · x3 compared with the case ni = 0. So, one can safely restrict

to ni = 0 since we are only interested in the leading contribution for small u and 1− v.

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There are also other regions that contribute to this integral, one of them is characterized

by x5, x6 ∼ x2 and x7, x8 ∼ x3. The integrand also simplifies in this case after doing the

following changes

1

x2
2j

=
∞∑
n=0

(2x2 · xj − x2
2)n

(x2
j )

1+n
,

1

x2
3i

=
∞∑
n=0

(2x3 · xi − x2
i )
n

(x2
3)1+n

,
1

x2
ij

=
∞∑
n=0

(2xi · xj − x2
i )
n

(x2
j )

1+n

(2.13)

where i = 5, 6 and j = 7, 8. The contribution coming from this region is given by

∞∑
ni,nij=0

x2
5

∏8
j=7(2x2 ·xj − x2

2)nj
∏6
j=5(2x3 ·xi − x2

i )
ni(2x5 ·x7 − x2

5)n57(2x6 ·x8 − x2
8)n68x2

67

x2
6x

2
56x

2
25x

2
26x

2
78(x2

3x
2
3)2+n5+n6(x2

7)3+n7+n57(x2
8)3+n8+n68

.

(2.14)

Notice that the integrals in this region can be viewed as the product of two loop propagators

with numerators. In fact this is a feature of the method, an l-loop four point conformal

integral can be written in terms of (l−k)-loop propagator type integral with k = 0, . . . , l−1.

For the same reason all terms with non-zero ni are subleading compared to the case

with ni = 0. It is also simple to estimate the dependence on the position of a given region,

it just amounts to doing dimensional analysis. It turns out to be quite useful to do this for

all regions since some of them are subleading in the OPE limit.

It is often the case where one has to deal with integral with open indices or in other

words, an integral where the numerator is contracted with an external vector. An example

of this is ∫
x2

5(x2
6 − 2x6 ·x7 + x2

7)ddx5d
dx6

x2
6x

2
56x

2
25x

2
26

. (2.15)

These can be expressed in terms of integrals of the form (2.11). The procedure is simple

and it is explained in section 3 of [5].

After all the integrals appearing in the asymptotic expansions are expressed in terms

of integrals of the form (2.11) one just uses a program such as LiteRed [10] or FIRE [9] to

reduce them to master integrals. The number of master integrals of the propagator type

depends on the loop order, at one, two, three and four loops the number of master integrals

is 1, 5, 9 and 24 respectively. At this point any conformal integral is given by a combination

of master integrals, whose values have been determined in an ε expansion [7, 13]2

jk =
∞∑

i=−4

εici,k jk,i, k = 1, . . . , 24. (2.16)

2.2 Konishi from OPE limit

The OPE decomposition of a four point function can be done in every conformal field

theory. In the present case we are interested on the contribution of the Konishi operator

2For planar integrals there is no difference between momenta and position space since they are related

by a simple change of variables [15]. There is no relation of this type for non-planar integrals. Fortunately

all the non-planar integrals that we need have been computed in [7].

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K(x) in the OPE of two O(x, y) operators

O(x1, y1)O(x2, y2) = cI
y2

12

(x2
12)2
I + cK

y2
12

(x2
12)1−γK/2

K(x2) + cO
y12

x2
12

Y I
1 Y

J
2 OIJ20 (x2) + . . . .

(2.17)

The operators that flow in the four point function depend on the polarizations vectors

of the external operators. To understand what operators are exchanged remember that

the tensor product of two 20’s decomposes in six irreducible representations. It is more

convenient to do the OPE in the channel 20 since there is just one operator flowing with

twist two per spin [12]. Since we are just interested in the of the Konishi we can focus on

the leading term for small u and 1− v∑
l≥1

al
x2

12x
2
13x

2
14x

2
23x

2
24x

2
34

l!(−4π2)l

∫
d4x5 . . . d

4x4+lf
(l)(x1, . . . , x4+l) →︸︷︷︸

x1→x2, x3→x4

(2.18)

→ 1

6x4
13

(c2
K(a)u

γK
2 − 1)(1 +O(u) +O(1− v)). (2.19)

The method of asymptotic expansions gives the following leading term for the sum of all

planar integrals

x4
13

∫
d4xif

(4)(xi) = 4
(
148ζ2

3 + (1312− 60ζ4) ζ3 + 5020ζ5 − 1250ζ6 + 8305ζ7 + 9952
)

− 64 (78ζ3+55 (3ζ5+8)) lnu+576 (3ζ3+14) ln2 u−1152 ln3 u+72 ln4 u+O(u)+O(1−v).

Now we can use the lower loop data for the anomalous dimension and OPE coefficient (that

can also be extracted from the lower loop four point function)

γK = 12a− 48a2 + 336a3, cK =
4

3
− 16a+ a2(224 + 96ζ3)− a3(3072 + 512ζ3 + 1600ζ5)

(2.20)

to extract the OPE coefficient and anomalous dimension in the four loop level

c
(4)
K = a4 32

(
36ζ2

3 + 164ζ3 + 490ζ5 + 735ζ7 + 1244
)

(2.21)

γ
(4)
K = 256a4

(
9ζ3

4
− 45ζ5

8
− 39

4

)
. (2.22)

The value of the anomalous dimension was known before [17] and served as a further check

of the method.

3 Conclusions

We have computed the OPE coefficient between two O20′ and the Konishi operator. One of

the main motivations to obtain this result was that it will allow to check the integrability

computation. The result at four loops is particularly important since in the hexagon

approach there is a new effect that will only kick in at this order [3]. Recall that it

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is also at this loop order that the wrapping effects in the spectrum start to contribute.

Reproducing the results of this note with the hexagon approach is an important non-trivial

check. As a curiosity, notice that the OPE coefficient at four loops continues to be given

entirely by odd zeta values.

We have focused on the OPE coefficient of the Konishi operator but it is also possible

to obtain the OPE coefficients of twist two operators with higher spin. The main hurdle

is to decompose a given integral in terms of master integrals. The packages LiteRed and

FIRE can do this decomposition but it will demand more computer time.

There are two more interesting directions, one is the evaluation of the non-planar

corrections at four loops and the other is to repeat this procedure for the OPE coefficient

but at five loops.

Acknowledgments

We are grateful to Pedro Vieira and Benjamin Basso for encouraging to do this project and

for discussions. We would like to thank Gregory Korchemsky for discussions. V.G. would

also like to thank FAPESP grant 2015/14796-7, CERN/FIS-NUC/0045/2015 and CGI in

Florence where this work has been completed.

A Master integrals

Every conformal integral can be expressed as a linear combination of master integrals of

the propagator type for each term in the series expansion in the cross ratios. These were

computed in the literature in momenta space at four loop level [13]. A duality between

planar integrals in momenta space and position space can be used to determine the coeffi-

cients ci,k of (2.16) in the planar sector. There are 2 more master integrals that contribute

to the leading term of this four point function

j21 =
n4

0(ε)

π2d

∫
ddx5d

dx6d
dx7d

dx8

x2
25x

2
56x

2
57x

2
6x

2
68x

2
7x

2
78x

2
28

, j22 =
n4

0(ε)

π2d

∫
ddx5d

dx6d
dx7d

dx8

x2
25x

2
56x

2
57x

2
6x

2
68x

2
7x

2
78(x2

28)2

(A.1)

where n0(ε) = e−γeεΓ(2−2ε)/(Γ(1+ε)Γ2(1−ε)) converts the integrals to the G-scheme [13].

These two integrals were computed in [14] up to order ε.

The method to compute the first two integrals in (A.1) was presented in the appendix

of [14]. The method is nice and we will review the main idea here. We are interested in

evaluating the integrals j21 and j22 in a Laurent expansion in 2ε = 4− d

j21 =
j21,−1

ε
+ j21,0 + j21,1ε, j22 =

j21,−1

ε
+ j21,0 + j21,1ε. (A.2)

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Imposing that the each conformal integral that appears in the planar part of the four point

function (2.7) are finite in the limit ε→ 0 and conformal fixes their values3

j21,−1 = 5ζ5, j21,0 =
5π6

378
− 13ζ2

3 − 5ζ5, j22,−1 = −20ζ5, j22,0 = −8ζ2
3 + 120ζ5 −

10π6

189
.

(A.3)

To obtain higher order terms in ε one can consider the following finite integrals

I3(κ) =
n4

0(ε)

π2d

∫
ddx5d

dx6d
dx7d

dx8

(x2
25x

2
28x

2
56x

2
57x

2
6x

2
7x

2
67x

2
68x

2
78)1−εκ ,

I4(κ) =
n4

0(ε)

π2d

∫
ddx5d

dx6d
dx7d

dx8

(x2
25x

2
26x

2
28x

2
56x

2
57x

2
6x

2
7x

2
68x

2
78)1−εκ .

Both I3 and I4 admit a power series in ε

Ii(κ) = bi + ε(ci + κdi) +O(ε2). (A.4)

It turns out that it is easier to evaluate these integrals than j21 and j22 for particular

values of κ. Then we use the fact that the power series expansion in ε is linear in κ at first

order in ε to obtain bi and ci. These constants are related to j21,0, j21,1, j22,0 and j22,1 by

integration by parts4

b3 + c3 ε =

(
830ζ5

3
− 2j21,0

3
− 7j22,0

3
+

26ζ2
3

3
− 65π6

567

)
+ (A.5)

ε

(
14j21,0

3
+

14j22,0

3
− 2j21,1

3
− 7j22,1

3
− 208ζ2

3

3
+

13π4ζ3

45
− 3220ζ5

3
− 4667ζ7

6
+

520π6

567

)
+

b4 + c4 ε =

(
235ζ5 − j21,0 − 2j22,0 + 7ζ2

3 −
5π6

54

)
(A.6)

+ ε

(
2j21,0 − j21,1 − 6j22,0 − 2j22,1 − 21ζ2

3 +
7π4ζ3

30
+ 285ζ5 −

4193ζ7

4
+

5π6

18

)
.

We will use the values κ = 1 and κ = 1
2 . These have been computed in [14]

I3

(
1

2

)
= I4

(
1

2

)
= n4

0(ε)
[ (

144ζ2
3 + 108ζ4ζ3

)
ε+ 36ζ2

3

]
+O(ε2) (A.7)

I3(1) = n4
0(ε)

[ (
288ζ2

3 + 108ζ4ζ3 − 378ζ7

)
ε+ 36ζ2

3

]
+O(ε2), (A.8)

I4(1) = n4
0(ε)

[(
108ζ2

3 + 108ζ4ζ3 +
189ζ7

2

)
ε+ 36ζ2

3

]
+O(ε2) (A.9)

3In our computations we have used G-scheme and we have used the results of [13] for the planar master

integrals.
4We have used LiteRed [10] package to do this reduction. There are terms in the following equation

that are different from [14]. This is not strange since [14] used a different scheme for the master integrals.

However, we were able to verify the four loop anomalous dimension which also gives us some confidence of

the correctness our result.

– 8 –



J
H
E
P
0
3
(
2
0
1
7
)
0
7
9

The constants bi and ci are obtained using

2Ii

(
1

2

)
− Ii (1) = bi + εci (A.10)

b3 = 36ζ2
3 , c3 =

6

5

(
315ζ7 − 240ζ2

3 + π4ζ3

)
(A.11)

b4 = 36ζ2
3 , d4 =

3

10

(
4π4ζ3 − 360ζ2

3 − 315ζ7

)
. (A.12)

Plugging these values in (A.5) we obtain again (A.3) and also the values for j22,1 and j22,1

j21,1 = 13ζ2
3 −

13π4ζ3

30
+ 35ζ5 +

345ζ7

4
− 5π6

378
(A.13)

j22,1 = 48ζ2
3 −

4π4ζ3

15
− 240ζ5 − 520ζ7 +

20π6

63
. (A.14)

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

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	Introduction
	Four point function and OPE limit
	Asymptotic expansions
	Konishi from OPE limit

	Conclusions
	Master integrals