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Published for SISSA by Springer

Received: July 30, 2014

Accepted: September 3, 2014

Published: September 29, 2014

Three-point functions and su(1|1) spin chains

João Caetanoa,b,c and Thiago Fleuryd

aPerimeter Institute for Theoretical Physics,

Waterloo, Ontario N2L 2Y5, Canada
bDepartment of Physics and Astronomy & Guelph-Waterloo Physics Institute,

University of Waterloo,

Waterloo, Ontario N2L 3G1, Canada
cCentro de F́ısica do Porto e Departamento de F́ısica e Astronomia,

Faculdade de Ciências da Universidade do Porto,

Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
dInstituto de F́ısica Teórica, UNESP — Univ. Estadual Paulista,

ICTP South American Institute for Fundamental Research,

Rua Dr. Bento Teobaldo Ferraz 271, 01140-070, São Paulo, SP, Brasil

E-mail: jd.caetano.s@gmail.com, tfleury@ift.unesp.br

Abstract: We compute three-point functions of general operators in the su(1|1) sector

of planar N = 4 SYM in the weak coupling regime, both at tree-level and one-loop.

Each operator is represented by a closed spin chain Bethe state characterized by a set of

momenta parameterizing the fermionic excitations. At one-loop, we calculate both the two-

loop Bethe eigenstates and the relevant Feynman diagrams for the three-point functions

within our setup. The final expression for the structure constants is surprisingly simple

and hints at a possible form factor based approach yet to be unveiled.

Keywords: Supersymmetric gauge theory, 1/N Expansion, Bethe Ansatz, Integrable

Field Theories

ArXiv ePrint: 1404.4128

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP09(2014)173

mailto:jd.caetano.s@gmail.com
mailto:tfleury@ift.unesp.br
http://arxiv.org/abs/1404.4128
http://dx.doi.org/10.1007/JHEP09(2014)173


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Contents

1 Introduction 1

2 Three-point functions at leading order 3

2.1 The one-loop Bethe eigenstates and structure constants 6

3 One-loop three-point functions 9

3.1 Two-loop coordinate Bethe eigenstates and norms 9

3.2 One-loop perturbative calculation 11

3.3 Final result 14

4 Discussion and open problems 15

A Notation and conventions 17

B One-loop perturbative computation details 18

C Some examples of three-point functions 24

C.1 Three half-BPS operators 24

C.2 Two non-BPS and one half-BPS operators 26

D Wilson line contribution 29

D.1 Wilson line connecting two scalars 30

D.2 Wilson line connecting either a scalar and a fermion or two fermions 30

E A note on the su(1|1) invariance of the final result 30

1 Introduction

Integrability has proven to be a powerful tool for studying the planar N = 4 SYM theory.

In particular, it was successfully used to compute all the two-point functions of the gauge-

invariant single-trace operators for any value of the ’t Hooft parameter λ, see for instance [1–

3]. The predictions from integrability have been extensively tested and they correctly

reproduce the known results obtained in perturbation theory at weak coupling and the

ones obtained by the AdS/CFT conjecture in the strong coupling limit.

The natural next step is computing the three-point functions. Together with the two-

point functions these are the building blocks for all the higher point correlators. With

the help of table 1, let us briefly recall the state of the art concerning the computation of

the three-point functions at weak coupling and explain where our findings fit within this

picture.

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Sector
Tree-level and

Integrability

One-loop

prescription

One-loop and

Integrability

Higher

loops

su(2) [4, 5] [6, 7] [8, 9] unknown

sl(2) [10, 11] [7] (some cases) [10] (some cases) [12, 13]

su(1|1) here here here unknown

so(6) [14] (some cases) [6, 7] unknown unknown

psu(2, 2|4) unknown unknown unknown unknown

Table 1. The current status of the computation of three-point functions.

A single-trace operator of N = 4 SYM is thought of as a closed spin chain state. To

leading order in the ’t Hooft coupling these spin chain states are very well understood and

given by the so-called Bethe ansatz. The problem at tree-level is purely combinatorial and

amounts to cutting and sewing such spin chains. At the end of the day, this boils down

to a computation of some scalar products of Bethe states. Nevertheless, this is a very rich

and non-trivial problem. For instance, scalar products between Bethe states in higher rank

algebras are not known. It is therefore so far unclear how to perform the computation of

the most general psu(2, 2|4) correlators as indicated by the last row of table 1.

This motivates one to start studying the rank one sectors in a systematic way. They

consist of the su(2), su(1|1) and sl(2) sectors and they played a very important role in the

spectrum problem, see for instance [15]. The first three rows of the table 1 summarize

the current knowledge on these sectors. In the su(2) case, the final result for the structure

constants turns out to be given in terms of determinants depending on three sets of numbers

called Bethe rapidities while in the sl(2) sector, it was found a formula given in terms of a

sum over partitions of these Bethe rapidities. In this paper, we will study the remaining

rank one sector.

Whichever sector we consider, there are, at one-loop, two effects that need to be taken

into account.

• Firstly, there is the two-loop correction to the Bethe state, which is of order λ and

thus contributes to the one-loop structure constant. This amounts to correct not only

the S-matrix but also modifying the Bethe ansatz itself by introducing the so-called

contact terms. These are required due to the long-range nature of the dilatation

operator which couples non-trivially neighboring magnons on the spin chain. In this

regard, some surprises were found recently. The contact terms were found to be

ultra-local in sl(2) and much simpler than in the su(2) case. In this paper we find a

remarkably simple form for the contact terms in su(1|1) allowing us to fully construct

the two-loop Bethe state for an arbitrary number of magnons.

• Secondly, there is the perturbative correction from the Feynman diagrams. This can

be effectively described by an insertion of an operator at the splitting points of the

spin chain and this is what we call the prescription for the one-loop computation. So

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far, the prescription was only fully computed for the so(6) sector. For the sl(2) sector,

partial results were obtained in [7] but the complete computation remains to be done.

In this paper, we provide the complete one-loop prescription for the su(1|1) sector.

In the end, combining both loop contributions for su(1|1), we found a strikingly sim-

ple formula for the one-loop structure constant C123. Given three operators Oi with Ni

excitations with momenta {p(i)
j }

Ni
j=1, and length Li (the details of the exact setup will be

given below), we have

C123 = C

3∏
i=1

Ni∏
j<k

f(y
(i)
j , y

(i)
k )

N1∏
i=1

N2∏
j=1

f(y
(1)
i , y

(2)
j )

N1∏
k=1

[
1− (y

(1)
k )L2

N2∏
i=1

(
−S(y

(2)
i , y

(1)
k )
)]

, (1.1)

where y
(i)
j ≡ eip

(i)
j , C is a simple normalization factor given in (3.14), S is the su(1|1)S-

matrix. The most essential ingredient and main result of this work is the function f which

is simply given by

f(s, t) = (s− t)
[
1− g2

2

(
s

t
+
t

s
− 1

s
− s− 1

t
− t+ 2

)
+O(g4)

]
, (1.2)

with g2 = λ
16π2 .

The paper is organized as follows. In section 2, we explain the three-point function

setup that will be used in the remaining of the paper and compute the leading contribution

to the structure constants in terms of a simple expression which is function of the momenta

of the excitations. Section 3 is devoted to the calculation of the one-loop corrected struc-

ture constants. The section begins with the construction of the two-loop eigenstates by

computing the contact terms, then we evaluate the relevant Feynman diagrams needed for

determining the prescription for computing the one-loop corrections. In the end, we put

the different contributions together and we arrive at the formula (1.1). Finally, the section

4 contains our conclusions and perspectives. Several appendices have additional details

omitted during the presentation.

2 Three-point functions at leading order

In this section, we perform the computation of the structure constants at leading order.

The setup that will be used for the calculation involves composite operators made out of

both fermionic and scalar fields. Each of these operators is thought of as a state of a closed

spin chain with the fermionic fields being excitations over a ferromagnetic vacuum. The ad-

vantage of this approach is that the connection with the integrability tools of quantum spin

chains becomes manifest (see for instance [2]) and facilitates the combinatorial problem.

The smallest (closed) sector of N = 4 SYM containing both fermionic and bosonic

fields is the su(1|1). The field content of this sector consists of one complex scalar that we

will denote as Z = Φ34 and a complex chiral fermion that shares with the scalar one R-

charge index, for instance Ψ = ψ4
α=1. The setup for the calculation of the planar three-point

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{ {,

{ {,

{ {,

Figure 1. The leading order contribution to the three-point functions. The solid lines represent

bosonic propagators and the dashed lines represent fermionic propagators. We also indicate our

conventions for labeling the positions of the excitations. Notice that in our setup the first N3

excitations of the operator O2 have always their position fixed.

functions that we will be considering involves an operator O1 given by a linear combination

of single traces made out of products of these fields. More precisely,

O1 =
∑

1≤n1<n2<...<nN1
≤L1

ψ(1)(n1, n2, . . . , nN1) Tr

(
Z . . . Ψ

n1

. . . Ψ
n2

. . . Z

)
, (2.1)

where L1 is the length of the operator, N1 is the number of its fermionic fields and n’s are

the positions of the excitations along the chain of Z’s. We designate the coefficients ψ(1)

in this linear combination by wave-function. It is natural to consider the second operator

O2 made out of the complex conjugate fields, namely

O2 =
∑

1≤n1<n2<...<nN2
≤L2

ψ(2)(n1, n2, . . . , nN2) Tr

(
Z̄ . . . Ψ̄

n1

. . . Ψ̄
n2

. . . Z̄

)
. (2.2)

In our conventions, the complex conjugate fields are given by

Z̄ = (Z)∗ = Φ34 = Φ12 , (2.3)

Ψ̄ = (Ψ)† = ψ̄4, α̇=1̇ .

From now on, we will omit the Lorentz spinorial indices at several places keeping in mind

that they are always kept fixed.

As a consequence of the R-charge conservation, it is clear that we cannot take the third

operator to be also in the same su(1|1) sector to which O1 and O2 belong, if we want to

have a non-vanishing result and avoid extremal correlation functions.1 Instead, we consider

1The extremal case presents additional subtleties related to the mixing with double-trace operators,

see [4]. We will not investigate such issues in this paper and therefore only non-extremal three-point

functions will be considered.

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a “rotated” operator constructed by applying su(4) generators several times to a su(1|1)

operator of the type O1. The idea is to get a composite operator having a term with only Ψ

and Z̄ fields in order to allow non-vanishing Wick contractions between all pairs of operators

(see figure 1 for an example of a non-extremal three-point function). More precisely, let us

suppose that we start with a state made out of Ψ and Z fields. In order to convert a single

Z into a Z̄ we must apply a pair of su(4) generators that rotate its two R-charge indices.

In sum, we can generate a term with Ψ ’s and Z̄’s by considering the following operation

O3 =
1

(L3 −N3)!2
(R2

4R
1
3)L3−N3

∑
1≤n1<...<nN3

≤L3

ψ(3)(n1, . . . , nN3) Tr (Z . . . Ψ . . . Ψ . . . Z) ,

(2.4)

where Ra
b are su(4) generators and they act on the fields inside the trace. Now, the su(4)

generators may also act on the field Ψ which carries one R-charge index. Therefore, this

operation will generate several terms coming from the different ways of acting with the

generators,

O3 =
∑

1≤n1<...<nN3
≤L3

ψ(3)(n1, . . . , nN3)
[
Tr
(
Z̄ . . . Ψ . . . Ψ . . . Z̄

)
+

+ Tr
(
Z̄ . . . ψ2 . . . Ψ . . . Φ14

)
+ . . .

]
, (2.5)

where in the first line we have the term where all the su(4) generators act on the scalar

fields Z. In the second line, we represent the terms where some of the generators also act on

the fermionic fields Ψ . As an example of how the formula given above is evaluated consider,

(R2
4R

1
3) · Tr (ΨZ) = (R2

4R
1

3) · Tr (ψ4 Φ34) = Tr (ψ2 Φ14) + Tr (ψ4 Φ12) .

At tree-level, the terms in the second line of (2.5) do not give any contribution due

to the R-charge conservation. In other words, one always has a zero Wick-contraction.

Therefore, at leading order, only the first line contributes and we get a tree-level diagram

of the type represented in figure 1. At one-loop, the terms in the second line will also need

to be taken into account. We emphasize that the operators O1 in (2.1) and O3 in (2.4)

are spinorial operators with N1 and N3 indices α = 1 respectively. This follows from the

definition of the field Ψ given previously. The operator O2 in (2.2) has N2 Lorentz indices

α̇ = 1̇ associated to each of the fermions Ψ̄ .

In a conformal field theory, the two-point functions are completely fixed by the symme-

tries up to a normalization constant. For two operators having spinorial indices as shown

below, we have

〈Oi ; 11...1Ni
(x1) Ōi ; 1̇1...1̇Ni

(x2)〉 = Ni
(J12,11̇)Ni

|x12|2∆i
, (2.6)

where Ni is a constant associated to the normalization of the operator, ∆i is its conformal

dimension and the tensorial structure is2

Jij,11̇ =
xµij
(
σEµ
)

1 1̇

(2π)2 |xij |
, with xµij = xµi − x

µ
j . (2.7)

2See appendix A for our conventions.

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In the case of three-point functions of generic operators having spinorial indices, one

has many inequivalent tensor structures consistent with the conformal symmetry, and the

result of the correlation function is a linear combination of these structures. The constraints

following from conformal symmetry on the higher point functions were studied for instance

in [16–18]. However, for the setup considered in this work there is only one possible tensor

and the three-point functions are of the form

〈O1;11...1N1
(x1)O2;1̇1...1̇N2

(x2)O3;11...1N3
(x3)〉 = (2.8)

=
(J12,11̇)N1(J23,11̇)N3

√
N1N2N3C123(g2)

|x12|∆1+∆2−∆3 |x13|∆1+∆3−∆2 |x23|∆2+∆3−∆1
,

where we are considering N2 = N1 +N3 and g2 = λ
16π2 with λ the ’t Hooft parameter.

The structure constant C123(g2) has a perturbative expansion when g2 is small, and

its leading order will be designated by C
(0)
123. Using the figure 1, we observe that the only

non-trivial Wick contractions occur between operators O1 and O2. The structure constant

C
(0)
123 is then given by the product of the three wave-functions with a sum over the positions

of the excitations between these two operators,∣∣∣C(0)
123

∣∣∣ = α

∣∣∣∣∣∣ψ(3)
1,...,N3

∑
N3<n1<...<nN1

≤L2

ψ
(1)
L2+1−nN1

,...,L2+1−n1
ψ

(2)
1,...,N3,n1,...,nN1

∣∣∣∣∣∣ . (2.9)

α is a normalization factor that comes from the fact that we are normalizing the operators

such that their two-point functions have the canonical form (2.6) with Ni = 1. It is given by

α =

√
L1L2L3

N (1)N (2)N (3)
, with N (j) =

∑
1≤n1<...<nNj

≤Lj

(ψ(j)
n1,...,nNj

)∗(ψ(j)
n1,...,nNj

) . (2.10)

The main goal of this section is to find a closed formula for C
(0)
123.

2.1 The one-loop Bethe eigenstates and structure constants

To compute C
(0)
123 we must consider states with definite one-loop anomalous dimension [4].

The one-loop su(1|1) integrable Hamiltonian and S-matrix can be found in [15, 19]. The

Hamiltonian is simply the fermionic version of the Heisenberg Hamiltonian and it is written

in terms of the Pauli matrices as

H1 = 2g2
L∑
n=1

(
(1− σ3

n)− 1

2
(σ1
nσ

1
n+1 + σ2

nσ
2
n+1)

)
, (2.11)

where L is the length of the spin chain. At leading order the two-excitation S-matrix is

independent of their momenta and simply given by

S(p1, p2) = −1 . (2.12)

In order to find the eigenstates of the Hamiltonian given above, we use the usual

coordinate Bethe ansatz. A N -magnon state of a spin-chain of length L is of the form

|ψN 〉 =
∑

1≤n1<n2<...<nN≤L
ψN (n1, n2, . . . , nN )|n1, . . . , nN 〉 , (2.13)

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where the ni’s in |n1, . . . nN 〉 indicate the position of the fermionic excitations Ψ on the

chain (for details about the coordinate Bethe ansatz see [2, 4]). Notice that the ket

|n1, . . . nN 〉 represents the trace in (2.1). The wave-function ψN (n1, . . . , nN ) is a com-

bination of plane waves with as many terms as the number of possible permutations of the

momenta with the relative coefficients being the S-matrices. Since the leading order su(1|1)

S-matrix is just −1, the several terms in the wave-function will appear with alternating

signs which we write as

ψN (n1, n2, . . . , nN ) =
∑
P

signP exp(ipσP (1)n1 + ipσP (2)n2 + . . .+ ipσP (N)nN ) (2.14)

where P indicates sum over all possible permutations σP of the elements {1, . . . , N}, and

signP is the sign of the permutation. Moreover, we should impose the periodicity condition

by requiring the momenta pi to satisfy the Bethe equations

eipiL = 1 . (2.15)

The cyclic property of the trace is implemented by imposing the zero momentum condition

of the state,
N∑
i=1

pi = 2π × integer . (2.16)

Having determined the eigenstates of the one-loop su(1|1) Hamiltonian, we can proceed

to compute the leading order structure constant C
(0)
123 given in (2.9) by following some simple

steps. Firstly, we notice that since the positions of the excitations of the third operator

are fixed, we can use (2.14) to write ψ(3) explicitly. It is simple to see that we obtain a

Vandermonde determinant which can be also presented as a simple product,

∣∣∣C(0)
123

∣∣∣ = α

∣∣∣∣∣∣
N3∏
j<k

[
eip

(3)
j − eip

(3)
k

] ∑
N3<n1<...<nN1

≤L2

(ψ(1)
n1,...,nN1

)∗ ψ
(2)
1,...,N3,n1,...,nN1

∣∣∣∣∣∣ . (2.17)

Moreover we have replaced ψ
(1)
L2+1−nN1

,...,L2+1−n1
by (ψ

(1)
n1,...,nN1

)∗ since they differ by at

most a sign.

Notice that the first N3 excitations of the wave-function ψ(2) have their positions fixed

or frozen. In order to make the computation of this sum simpler, we consider an auxiliary

problem where we add N3 extra excitations to the wave-function ψ(1) and liberate the fixed

N3 excitations of ψ(2) with their positions being summed over too,

Saux ≡
∑

1≤n1<...<nN3+N1
≤L2

(ψ(1)
n1,...,nN3+N1

)∗ ψ(2)
n1,...,nN3+N1

. (2.18)

The advantage of considering this auxiliary problem is that the sum (2.18) can be easily

computed due to the form of the wave-functions. Moreover, we can relate it with the original

sum appearing in (2.17) as we now explain. Indeed, let us consider that N3 momenta, say

{p(1)
1 , . . . , p

(1)
N3
}, are complex. We can then dynamically localize the wave-function around

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the original N3 positions by taking the limit of these momenta going to minus infinity.

More precisely, we send {e−ip
(1)
1 , . . . , e

−ip(1)N3} to zero in such a way that

e−ip
(1)
1 � · · · � e

−ip(1)N3 . (2.19)

Thus, given the explicit form of the wave-function (2.14), we observe that in this limit the

sum over the positions of the extra roots in (2.18) is dominated by the term for which

n1 = 1, . . . , nN3 = N3. This procedure of sending roots to a particular limit in order to

freeze their positions is the coordinate Bethe ansatz counterpart of the freezing trick used

in [5] at the level of the six-vertex model. Neglecting all the subleading terms, we get that

in this limit, (2.18) is reduced to

Saux →

(
N3∏
k=1

e−ip
(1)
k k

) ∑
N3<n1<...<nN1

≤L2

(ψ(1)
n1,...,nN1

)∗ ψ
(2)
1,...,N3,n1,...,nN1

, (2.20)

where we recognize precisely the original sum of (2.17).

Returning to our auxiliary problem, we use again that the wave-function is completely

antisymmetric in its arguments to extend the limits of the sum (2.18). In compensation,

we merely have to introduce a trivial overall combinatorial factor. Using the explicit form

of the wave-function we write the sum (2.18) as

Saux =
1

N2!

∑
{ni}

∑
P,Q

signP signQ

N1+N3∏
a=1

e

(
ip

(2)
P (a)
−ip(1)

Q(a)

)
na . (2.21)

We emphasize again that we now sum without restrictions, 1 ≤ ni ≤ L2, for all ni.

These sums over ni can be explicitly computed as they are geometric series. Using the

Bethe equations and the total momentum condition for the operator O2, we can then

simplify (2.21) to

Saux =

[
N1+N3∏
a=1

(
1− e−ip

(1)
a L2

)] 1

N2!

∑
P,Q

signP signQ

N1+N3∏
a=1

1

e
ip

(1)
Q(a) − eip

(2)
P (a)

. (2.22)

The remaining sum in the previous expression is manifestly the definition of a Cauchy

determinant and, therefore, it can be written explicitly as a simple product as follows

Saux =

[
N1+N3∏
a=1

(
1− e−ip

(1)
a L2

)] ∏
j<k

(eip
(1)
j − eip

(1)
k )(eip

(2)
k − eip

(2)
j )

∏
j,k

(eip
(1)
j − eip

(2)
k )

. (2.23)

Notice that this expression contains as a limit the norm of an operator.3 It is given by

N (j) = L
Nj

j . (2.24)

3If we consider p
(1)
j → p

(2)
j we get the expression for N (2) after using the Bethe equations (2.15).

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Finally, we take the limit of (2.23) when {e−ip
(1)
1 , . . . , e

−ip(1)N3} vanish as in (2.19). Plug-

ging the resulting limit and taking into account the overall product multiplying the sum

in (2.20), we obtain our final result

∣∣∣C(0)
123

∣∣∣ =

[
3∏
i=1

L
1−Ni

2
i

] ∣∣∣∣∣∣∣∣∣
N1∏
j=1

(
1− eip

(1)
j L2

)
3∏

a=1

Na∏
j<k

(eip
(a)
j − eip

(a)
k )

N1∏
j=1

N2∏
k=1

(eip
(1)
j − eip

(2)
k )

∣∣∣∣∣∣∣∣∣ . (2.25)

It is now straightforward to confirm that our formula (1.1) given in the introduction,

reduces to this one when g is set to zero.

This result fills the first column for the su(1|1) row of the table 1 in the introduction.

Let us remark that this expression is considerably simpler than the ones found for the su(2)

and sl(2) sectors. This is perhaps not surprising given that at leading order we are dealing

with a theory of free fermions so that the form of the su(1|1) wave-function becomes quite

simple. However, we will see that the one-loop result persists to be simpler than in the

other sectors.

3 One-loop three-point functions

In this section, we compute the structure constants at first order in the ’t Hooft coupling

λ for our setup. There are two main ingredients in this computation. Firstly, one has

to consider Bethe eigenstates that diagonalize the two-loop dilatation operator as these

states are of order λ. Secondly, one has to compute the relevant Feynman diagrams at this

order in perturbation theory. This second contribution can be compactly taken into account

through the insertion of an operator at specific points of the spin chains as will be reviewed.

3.1 Two-loop coordinate Bethe eigenstates and norms

The two-loop Bethe eigenstates are determined by diagonalizing the long-range Hamilto-

nian H [15]

H = H1 +H2 , (3.1)

where H1 is given in (2.11) and

H2 = 4g2
L∑
n=1

(
2(σ3

n − 1)− 1

4
(σ3
nσ

3
n+1 − 1) + (σ1

nσ
1
n+1 + σ2

nσ
2
n+1)

(
9

8
− 1

16
σ3
n+2

)
(3.2)

− 1

16
σ3
n(σ1

n+1σ
1
n+2 + σ2

n+1σ
2
n+2)− 1

8
σ1
n(1 + σ3

n+1)σ1
n+2 −

1

8
σ2
n(1 + σ3

n+1)σ2
n+2

)
,

where σi are the Pauli matrices. In order to diagonalize it, we start with the usual coor-

dinate Bethe ansatz which works when the excitations are at a distance bigger than the

range of the interaction, i.e. when |ni − nj | > 2. In this region all we need is the two-loop

S-matrix which reads

S(p1, p2) = −1− 8ig2 sin
(p1

2

)
sin

(
p1 − p2

2

)
sin
(p2

2

)
. (3.3)

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Given the long-range nature of the Hamiltonian (3.1), we expect the form of the wave-

function to be modified with respect to the usual Bethe ansatz (2.14). In fact, when

magnons are placed at neighboring positions on the spin chain they interact in a non-

trivial way. Therefore, the wave-function must be refined by the inclusion of the so-called

contact terms. For instance, in the case of three magnons we write it as

ψ(n1, n2, n3) = φ123 + φ213S21 + φ132S32 + φ312S31S32 + φ231S31S21 + φ321S32S31S21 ,

where we have used the notation Sab = S(pa, pb) and

φabc = eipan1+ipbn2+ipcn3

(
1 + g2 C(pa, pb) δn2,n1+1δn3>n2+1 + g2C(pb, pc) δn2>n1+1δn3,n2+1

+g2 C(pa, pb, pc) δn2,n1+1δn3,n2+1

)
. (3.4)

The functions C are the contact terms which are fixed by solving the energy eigenvalue

problem. In the case of N -magnons, the wave-function has a similar structure. It consists

of N ! terms coming from the permutations of {p1, . . . , pN} and N − 1 types of contact

terms namely C(pi, pj), . . . ,C(p1, . . . , pN ).

Unexpectedly, we have found that up to seven magnons the contact terms are simply

given by4

C(p1, . . . , pn) =
n− 1

2
. (3.5)

Even though we have not proved the validity of this formula for an arbitrarily high number

of magnons, the pattern emerging up to seven magnons is quite suggestive. Given the

form of the contact terms in the su(2) and sl(2) sectors, the simplicity of the su(1|1) result

is quite surprising. In particular, notice that they are independent of the momenta of

the colliding magnons. This might be pointing towards the existence of a new algebraic

description of these states yet to be unveiled.

As already explained, in order to correctly compute the three-point functions we need

to know the norm of the Bethe eigenstates as we are normalizing the result by the two-point

functions. Remarkably, we have checked numerically up to six-magnons that the two-loop

(coordinate) norm is given by

N = det
j,k≤N

∂

∂pj

Lpk +
1

i

N∑
m6=k

logS(pm, pk)

 . (3.6)

Interestingly, this formula is precisely the well-known Gaudin norm for the one-loop su(2)

Bethe states. Still within the su(2) sector, it was recently shown in [8] that this expression

remains valid at higher loops leading to an all-loop conjecture for the norm. Moreover,

the two-loop norm for sl(2) Bethe states was found to be precisely of the type (3.6) as

described in [10]. In all these cases, the contact terms recombine exactly to preserve the

determinant form. This is very suggestive of an underlying hidden structure that is worth

investigating.

4We thank Tianheng Wang for collaboration on this point.

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3.2 One-loop perturbative calculation

Loop computations will give rise to divergences which require the introduction of a reg-

ularization scheme. A very convenient one and the one that will be used in this work is

the point splitting regularization. At one-loop, only neighboring fields inside any of the

single-trace operators interact and the divergences arise because the two fields are at the

same spacetime point. The idea behind the point splitting regularization is to separate

these two fields by a distance ε which will act as a regulator.5

Consider a su(1|1) bare operator which is an eigenstate of the one-loop dilatation

operator. Its non-vanishing two-point function is of the form

〈Oi ; 11...1Ni
(x1) Ōi ; 1̇1...1̇Ni

(x2)〉 = Ni
(J12,11̇)Ni

|x12|2∆0,i

(
1 + 2g2 ai − γi log

(
x2

12

ε2

))
, (3.7)

where the tensor on the right-hand side was defined in (2.7). In the expression above, ∆0,i

and γi are the free scaling dimension and the one-loop anomalous dimension of the operator

Oi respectively, Ni is a normalization constant and ai is a scheme dependent constant. In

addition, the three-point function of three su(1|1) bare operators that diagonalize the one-

loop dilatation operator is, in our setup, fixed by conformal symmetry and takes the form

(see [6] for details)

〈O1 ; 11...1N1
(x1)O2 ; 1̇1...1̇N2

(x2)O3 ; 11...1N3
(x3)〉 = (3.8)

=
(J12,11̇)N1(J23,11̇)N3

√
N1N2N3

|x12|∆0,1+∆0,2−∆0,3 |x13|∆0,1+∆0,3−∆0,2 |x23|∆0,2+∆0,3−∆0,1
C

(0)
123 ×

×
(

1+g2(C
(1)
123+a1+a2+a3)− γ1

2
log

(
x2

12x
2
13

x2
23ε

2

)
− γ2

2
log

(
x2

12x
2
23

x2
13ε

2

)
− γ3

2
log

(
x2

23x
2
13

x2
12ε

2

))
where we have factored out the tree-level constant C

(0)
123.

To extract the regularization scheme independent structure constant C
(1)
123 from the

expression above, we have to divide the three-point function by the square root of the

two-point functions of all the operators to get rid of the constants ai’s. After performing

this division, one can then read the meaningful structure constant.

From the Feynman diagrams computation point of view, it is actually simpler to cal-

culate C
(1)
123 instead of the combination (C

(1)
123 + a1 + a2 + a3). In fact, because we have to

divide by the square root of the two-point functions, all one-loop diagrams in the three-

point function involving only two operators are canceled. The figure 2 has an example of

a such cancellation.

The conclusion is that one is left with the computation of only genuine three-point

diagrams, i.e., the diagrams involving fields from the three operators.6 The allowed posi-

tions of the spin chains where it is possible to have those genuine diagrams are commonly

5In order to preserve the gauge invariance, one can introduce a Wilson line between the two shifted

fields. This will in principle introduce extra diagrams at one-loop, coming from the gluon emission from the

Wilson line. However, we will show in the appendix D that this additional contribution actually vanishes

at this order in perturbation theory.
6This fact was dubbed the slicing argument in [7].

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O2

O1

O3

O1 − 1
2 O2 = 0− 1

2 Ō1 Ō2

Figure 2. The wavy-line in the figure is just a representation of a one-loop diagram (for example, a

gluon exchange). When the contribution of the square root of the two-point functions is subtracted

(this is the reason for the factor 1
2 ), all the diagrams involving just two operators are canceled.

O2

O1

O3

− 1
2

Ō2

O2

Figure 3. A genuine three-point diagram to which we subtract half of the same diagram but

seen as a two-point process is shown. The constant coming from this combination of diagrams is

regularization scheme and normalization independent.

called the splitting points. We are then seeking the constants coming from the genuine

three-point diagrams subtracted by the constants coming from the same diagrams but now

seen as two-point processes. This is exemplified in the figure 3.

The details of the Feynman diagram computation are given in the appendix B and

here we just provide the results. In the figure 4, we list all diagrams giving a non-zero

contribution to the three-point functions as well as the result of the respective scheme

independent constants. A relevant aspect of this computation is that some terms in the

second line of (2.5) are now important at one-loop level. Indeed, from figure 4 we realize

that the second graph of the second row mixes up the R-charge indices of the scalar and

the fermion. In particular, the scalar Φ14 and the fermion ψ2 in the second line of (2.5)

can be converted into a Ψ and a Z̄ through this diagram. The resulting state can then be

contracted with the remaining external operators and give a non-vanishing contribution.

From the results of figure 4, we can directly read off an operator acting on the two

fields at the splitting points of an external state and that gives those same constants after

contraction with the remaining states. We denote this operator by F and define it by the

following matrix elements

〈ψa ψb | F |ψc ψd 〉 = − δacδbd , (3.9)

〈Φef Φgh | F |Φab Φcd 〉 = 2 δ̄gh,ab δ̄ef,cd − 2 δ̄ef,ab δ̄gh,cd − εabcd εefgh ,

〈Φde ψf | F |Φab ψc 〉 = − δfcδ̄ab,de , 〈ψf Φde | F |ψc Φab 〉 = − δfcδ̄ab,de ,

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Figure 4. These are the relevant one-loop diagrams for the three-point functions. All other graphs

give a zero contribution. The solid, wiggly and dashed lines represent the scalars, gluons and

fermions, respectively. The constants are obtained by combining the three-point and two-point

graphs as illustrated in figure 3. We have used the point splitting regularization and the Feynman

gauge. For three-point diagrams we take the limit where a pair of dots (either top or bottom) are

brought to the same spacetime points. For the two-point function, both pairs of dots (top and

bottom) are brought to the same spacetime points. We are using the definition δ̄abcd ≡ δac δbd − δbcδad .

〈Φde ψf | F |ψc Φab 〉 = δceδ̄ab,df , 〈ψf Φde | F |Φab ψc 〉 = δceδ̄ab,df ,

where δ̄ab,cd ≡ δacδbd− δadδbc and in the second line we recognize the so(6) Hamiltonian [6,

7, 20]. It is simple to check that the operator g2

2 F reproduces the constants of figure 4.

For the specific setup that we are considering only the diagrams of figure 4 are relevant,

since additional diagrams either cancel among them or vanish, see appendix B for details.

In the case of a more general setup, the operator F defined receives corrections from new

diagrams.

In what follows, the operator F will appear with additional indices as Fij , which

indicate the sites in the spin chain where the operator acts. As an example, we have that

〈 . . .
i
Ψ
j

Z . . . | g
2

2
Fij | . . .

i
Ψ
j

Z . . . 〉 = −g
2

2
,

which reproduces the result of the first diagram of the second row of figure 4. It is important

to note that when the operator Fij acts on non-neighboring sites, it can pick up additional

minus signs due to statistics, for example,

〈Ψ . . . Ψ . . . Ψ︸ ︷︷ ︸
n fermions

. . . Z | g
2

2
F1L |Z . . . Ψ . . . Ψ︸ ︷︷ ︸

n fermions

. . . Ψ 〉 = (−1)n
g2

2
,

where n denotes the number of fermionic excitations between the first and last sites and

we have used the last rule of (3.9).

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3.3 Final result

We now give the complete expression for the structure constants up to one-loop in the

setup considered in this work. It reads

C123 = α×
(
〈1f |1 +

g2

2
FL3−N3,L3−N3+1 +

g2

2
FL1,1| Z̄ . . . Z̄︸ ︷︷ ︸

L3−N3

i1 . . . iL2−N3〉 (3.10)

〈Ψ̄ . . . Ψ̄︸ ︷︷ ︸
N3

i1 . . . iL2−N3 |1 +
g2

2
FN3,N3+1 +

g2

2
FL2,1|2〉

)
×

×〈Ψ . . . Ψ︸ ︷︷ ︸
N3

Z̄ . . . Z̄︸ ︷︷ ︸
L3−N3

|1 +
g2

2
FN3,N3+1 +

g2

2
FL3,1|3〉 ,

where we have that

α =

√
L1L2L3

N (1)N (2)N (3)
, (3.11)

with N (i) being the respective norms and we are using the conventions

〈σi1σi2 · · ·σiL |σj1σj2 · · ·σjL〉 = δi1j1δi2j2 · · · δiLjL ,

where σ is any field.

In the formula (3.10), ia can be either Z̄ or Ψ̄ and a sum over all these intermediate

states is implied. Moreover, we have included a superscript f in the bra associated to the

operator O1 to emphasize that the state was flipped,7 see [4] for details. The external states

are the two-loop corrected Bethe eigenstates as described in section 3.1, for instance

|1〉 = |1〉(0) + g2|1〉(1) +O(g4) . (3.12)

We have checked that for the simple case of three half-BPS operators, the one-loop

correction to the structure constant vanishes as expected from the non-renormalization

theorem of [21], see appendix C for details. Additionally, in the appendix E we check that

this result satisfies some constraints from symmetry considerations.

The expression (3.10) can now be evaluated as an explicit function of the Bethe roots

by using the known form of the two-loop Bethe states. As the number of excitations of the

external states increases, such task becomes tedious and the result gets lengthy obscuring

possible simplifications. Nevertheless, we can easily deal with states of arbitrary length but

only a few magnons. It turns out that the manipulation of the resulting expressions for

these simple cases reveals a strikingly compact structure that can be easily generalizable

for arbitrary complicated states. We then resort to the numerical approach in order to

confirm that such generalization actually holds. In the end, we find a formula given by a

7In short, the flipping operation F l introduced in [4] is defined as F l : ψ(n1, . . . , nN )|n1, . . . , nN 〉 7→
ψ(n1, . . . , nN )〈L− nN + 1, . . . , L− n1 + 1| Ĉ, where Ĉ means charge conjugation which exchanges Z ↔ Z̄

and Ψ ↔ Ψ̄ .

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very simple and natural deformation of the tree-level result (2.25), as follows

C123 = C

3∏
k=1

Nk∏
i<j

f(y
(k)
i , y

(k)
j )

N1∏
i=1

N2∏
j=1

f(y
(1)
i , y

(2)
j )

N1∏
k=1

[
1− (y

(1)
k )L2

N2∏
i=1

(
−S(y

(2)
i , y

(1)
k )
)]

, (3.13)

where we are using the notation y
(i)
k = eip

(i)
k and the normalization factor C is given by

C =

√
L1L2L3

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

[
1 + g2

(
N2

3 − 1
)
− 1

4

3∑
i=1

γi

]
, (3.14)

with γi being the anomalous dimension of the operator Oi. As described in the subsec-

tion 3.1, the norms N (i) are given by the formula

N (i) = det
j,k≤Ni

∂

∂p
(i)
j

Lp(i)
k +

1

i

Ni∑
m 6=k

logS(p(i)
m , p

(i)
k )

 . (3.15)

The most important and non-trivial part of the final result is the function f which reads

f(s, t) = (s− t)
[
1− g2

2

(
s

t
+
t

s
− 1

s
− s− 1

t
− t+ 2

)]
. (3.16)

The momenta p
(j)
k of the fermionic excitations must satisfy the Bethe equations which take

the form

eip
(j)
k Lj =

N∏
i 6=k

(
−S(p

(j)
k , p

(j)
i )
)
, (3.17)

and the total momentum condition (2.16). This constitutes the most important result of

this paper and it will be discussed in the next section.

4 Discussion and open problems

In this work, we have computed both the leading order contribution and the one-loop per-

turbative correction at weak coupling to the three-point functions of single-trace operators

of N = 4 SYM in the su(1|1) sector. The su(1|1) sector is closed to all orders in perturba-

tion theory [19] and it is the simplest sector having both fermions and bosons. Representing

each operator by a Bethe eigenstate, we were able to derive a simple expression (2.25) for

the leading order result in terms of the momenta characterizing the states.

In addition, we have also computed the one-loop correction by evaluating the relevant

Feynman diagrams and also determining the two-loop Bethe eigenstates and their norm.

The prescription for computing the scheme independent three-point structure constant

turns out to be given in terms of the insertion of the operator F , defined in (3.9), at the

splitting points of the spin chains.

Regarding the external states, due to the long-range property of the two-loop dilatation

operator in the su(1|1) sector, its diagonalization involves the usual Bethe ansatz corrected

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by the contact terms. These in turn are independent of the momenta of the excitations and

have a very simple expression for an arbitrary number of magnons, see (3.5). The norm of

these states is compactly given by a simple determinant, analogous to the well known case

of the su(2) sector.

The one-loop structure constant in our setup turns out to be given by the simple

formula (3.13) in terms of the Bethe roots. It is tempting to investigate the thermodynamic

limit of our result, namely when we consider one or more long spin chains Li � 1, with a

large number of excitations Ni = O(Li). This might be useful for future comparison with

string theory calculations in a specific limit. An obvious open problem is the computation

of the three-point functions in higher rank sectors at least at tree-level. Our result and the

results of [5, 10] are encouraging in order to find a simple expression for the full psu(2, 2|4).

The main obstacle is the knowledge of the scalar products of Bethe states for generic (super)

algebras, although some progress has been made in the su(3) case [22–25].

The final expression (3.13) is very suggestive and deserves further comments. Apart

from the simple normalization factor C given by the expression (3.14), the structure con-

stant has two distinct contributions. Firstly, the two-loop correction to the S-matrix ap-

pears in a very natural way when we look at the tree-level result (2.25). Secondly, the most

non-trivial part comes from the function f. The one-loop result is achieved by deforming

this function, which bears some similarities to the su(2) and sl(2) cases [8, 10]. As already

pointed out in [10], it would be interesting to deepen the connection of the three-point

function with form factors as started in [26, 27]. In particular, that could shed light on a

(non-perturbative) definition of the function f from the form factors axioms. In fact, such

axiomatic approach was recently explored in the context of the scattering amplitudes [28–

30]. There, the central object called pentagon transition P (u|v) was required to satisfy

some natural constraints from the integrability point of view. These conditions were then

used to bootstrap the function exactly. In this regard, we notice some striking similarities

of the dependence of our final result on this function f with the expression (9) of [28] which

corresponds to a multi-particle transition. We hope that such ideas can be applied for the

calculation of three-point functions at any value of the coupling constant.

Acknowledgments

We would like to thank P. Vieira for many invaluable discussions and suggestions. We

also thank B. Basso, N. Berkovits, N. Gromov, Y. Jiang, H. Nastase, J. Penedones, D.

Serban, A. Sever, C. Sieg, E. Sobko, J. Toledo for comments and discussions and especially

T. Wang for collaboration in a stage of this project. We would like to thank the warm

hospitality of the ICTP-SAIFR, FAPESP grant 2011/11973-4, where part of this work was

done. TF would like to thank the warm hospitality of the Perimeter Institute where this

work was initiated.

JC is funded by the FCT fellowship SFRH/BD/69084/2010. This work has been

supported in part by the Province of Ontario through ERA grant ER 06-02-293. Research

at the Perimeter Institute is supported in part by the Government of Canada through

NSERC and by the Province of Ontario through MRI. This work was partially funded

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by the research grants PTDC/FIS/099293/2008 and CERN/FP/116358/2010. Centro de

F́ısica do Porto is partially funded by FCT under grant PEst-OE/FIS/UI0044/2011. TF

would like to thank FAPESP grant 2013/12416-7 and 2009/50775-3 for financial support.

A Notation and conventions

In this appendix, we fix our conventions for the perturbative computations. The N = 4

SYM with SU(N) gauge group has the following Lagrangian [31, 32]

L = Tr

(
−1

2
FµνF

µν + 2DµΦabDµΦab + 2iψαaσµαα̇(Dµψ̄a)α̇ (A.1)

+2g2
YM[Φab, Φcd][Φab, Φcd]− 2

√
2gYM

(
[ψαa, Φab]ψ

b
α − [ψ̄α̇a, Φ

ab]ψ̄α̇b

))
,

with all the fields in the adjoint representation of the gauge group and the covariant deriva-

tive is Dµ · = ∂µ − igYM [Aµ, · ]. The propagators extracted from this Lagrangian are (we

are suppressing the gauge indices and taking the leading order in N)

〈Φab(x)Φcd(0)〉 =
δ̄abcd
8

1

(2π)2(−x2 + iε)
,

〈ψaα(x)ψ̄β̇ b(0)〉 =
i δab
2
σµ
αβ̇
∂µ

1

(2π)2(−x2 + iε)
,

〈Aµ(x)Aν(0)〉 = −ηµν
2

1

(2π)2(−x2 + iε)
,

where δ̄abcd ≡ δac δbd− δbcδad . We are using the Minkowski metric (+− − −) and the Feynman

gauge. The action of the (classical) supersymmetry generators are given by [33]

[Qα
a, Φ

bc] =
i
√

2

2

(
δbaψ

αc − δcaψαb
)
,

[Qα
a, ψ

b
β] = δbaF

α
β ,

[Qα
a, ψ̄

β̇
b ] = 2

√
2Dβ̇αΦab ,

and the conjugate expressions for the action of Q̄a
α̇ . The action of the R-symmetry

generators is given by

[Ra
b, Φ

cd] = δcb Φ
ad + δdb Φ

ca − 1

2
δab Φ

cd ,

[Ra
b, ψ

c] = δcb ψ
a − 1

4
δab ψ

c .

In the computations of the Feynman diagrams, in particular for the evaluation of the

integrals, we analytical continued to Euclidean space by using

x0 = ix4 , σ0
M = −σ0

E , σiM = iσiE ,

where the subscripts M means Minkowski space and E means Euclidean space,

σ0
M = Id2×2 ,

and, finally, σiM are the usual Pauli matrices.

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Figure 5. The results of the Feynman diagrams computation omitting both terms that must vanish

or cancel when summing all the diagrams (see text) and factors of N . The solid, wiggly and dashed

lines represent the scalars, gluons and fermions, respectively. The δ̄ was defined in appendix A.

B One-loop perturbative computation details

In this appendix, we present the details of the perturbative computation of the three-point

functions at one-loop using the point splitting regularization. As reviewed in the main part

of this paper, in order to obtain scheme and normalization independent structure constants

we also need to know the results of the two-point functions. For completeness, we explicitly

compute the one-loop dilatation operator of the su(1|1) sector as well.

Typically three kinds of integrals will appear in the computations

Y123 =

∫
d4u Ix1uIx2uIx3u ,

X1234 =

∫
d4u Ix1uIx2uIx3uIx4u ,

H12,34 =

∫
d4u d4v Ix1uIx2uIuvIx3vIx4v ,

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where Ixaxb is the (euclidean) scalar propagator defined as

Ixaxb ≡
1

(2π)2(xa − xb)2
.

The Y and X integrals are well-known and explicit expressions for them can be found for

instance in [7, 34]. The integral H is not known analytic, however, only its derivatives will

be needed. In particular, the following combination [34] turns out to be useful

F12,34 ≡
(∂1 − ∂2) · (∂3 − ∂4)H12,34

Ix1x2Ix3x4
(B.1)

=
X1234

Ix1x3Ix2x4
− X1234

Ix1x4Ix2x3
+G1,34 −G2,34 +G3,12 −G4,12 ,

where

Ga,bc =
Yabc
Ixaxc

− Yabc
Ixaxb

.

We will need several limits of the expressions for Y and X, namely when pairs of

distinct points collapse into each other

Y113 ≡ lim
x2→x1

Y123 =

(
2− log

(
ε2

x2
13

))
Ix1x3
16π2

,

X1134 ≡ lim
x2→x1

X1234 =

(
2− log

(
ε2x2

34

x2
13x

2
14

))
Ix1x3Ix1x4

16π2
,

where we are considering xµ2 = xµ1 + εµ with εµ → 0. We can also take a further limit of

the last expression above when x4 → x3 giving

X1133 =

(
1− log

(
ε2

x2
13

))
I2
x1x3

8π2
.

Moreover, we also need limits of the first and second derivatives of both the Y and the X

integrals. We include the results of them below for completeness. The first derivatives are

given by

lim
x2→x1

∂1,µY123 =
εµ
ε2
Ix1x3
8π2

−
(

1− log

(
ε2

x2
13

))
x13,µI

2
x1x3

4
−
x13,νε

νεµI
2
x1x3

2ε2
, (B.2)

lim
x2→x1

∂3,µY123 =

(
1− log

(
ε2

x2
13

))
x13,µI

2
x1x3

2
, (B.3)

lim
x2→x1

∂1,µX1234 =
εµ
ε2
Ix1x3Ix1x4

8π2
−
(

1− log

(
ε2x2

34

x2
13x

2
14

))
x13,µI

2
x1x3Ix1x4 + x14,µIx1x3I

2
x1x4

4

−
x14,ν ε

νεµIx1x3I
2
x1x4

2ε2
−
x13,ν ε

νεµI
2
x1x3Ix1x4

2ε2
,

lim
x2→x1

∂3,µX1234 = −x34,µIx1x3Ix1x4Ix3x4
2

+

(
1− log

(
ε2x2

34

x2
13x

2
14

))
x13,µI

2
x1x3Ix1x4
2

,

As before, one can take further limits of these expressions when needed. The second

derivatives read

lim
x2→x1

∂1,µ∂2,νY123 = −εµεν
ε4

Ix1x3
4π2

+
εµενI

2
x1x3

ε2

(
1

6
+
x13,ρ ε

ρ

ε2
− 16π2x13,ρ ε

ρx13,σ ε
σIx1x3

3ε2

)

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+
εν
ε2
x13,µI

2
x1x3

2
− εµ
ε2
x13,νI

2
x1x3

2
+

8π2x13,ρ ε
ρI3
x1x3

3ε2
(2x13,νεµ−x13,µεν)

+
1

ε2
δµνIx1x3

8π2
− δµνI2

x1x3

(
11

36
+
x13,ρε

ρ

2ε2
− 8π2x13,ρ ε

ρx13,σ ε
σIx1x3

3ε2

)
+

1

12
log

(
ε2

x2
13

)
I2
x1x3δµν +

2π2

9

(
1− 6 log

(
ε2

x2
13

))
x13,µx13,νI

3
x1x3 ,

lim
x2→x1

∂1,µ∂3,νY123 =
εµεν
ε2

I2
x1x3

2
+
εµ
ε2
x13,νI

2
x1x3 − 2π2

(
1− 2 log

(
ε2

x2
13

))
x13,µx13,νI

3
x1x3

+
1

4

(
1− log

(
ε2

x2
13

))
I2
x1x3δµν −

8π2x13,ν εµ x13,ρ ε
ρI3
x1x3

ε2
, (B.4)

lim
x2→x1

∂1,µ∂2,νX1234 = −εµ εν
ε4

Ix1x3Ix1x4
4π2

+
εµ ενI

2
x1x3I

2
x1x4

6ε2

(
1

Ix3x4
− 32π2x13,ρ ε

ρ x14,σ ε
σ

ε2

)
+
εµ εν ε

ρIx1x3Ix1x4
ε4

(x14,ρIx1x4 + x13,ρIx1x3)

−16π2 εµ εν ε
ρ εσIx1x3Ix1x4
3ε4

(x14,ρx14,σI
2
x1x4 + x13,ρx13,σI

2
x1x3)

−εµIx1x3Ix1x4
2ε2

(x14,νIx1x4 + x13,νIx1x3)

+
ενIx1x3Ix1x4

2ε2
(x14,µIx1x4 + x13,µIx1x3)

+
8π2εµ ε

ρIx1x3Ix1x4
3ε2

( 2x14,ν x14,ρ I
2
x1x4 + x13,ρ x14,ν Ix1x3Ix1x4)

+
8π2εµ ε

ρIx1x3Ix1x4
3ε2

( 2x13,ν x13,ρ I
2
x1x3 + x14,ρ x13,ν Ix1x3Ix1x4)

−4π2εν ε
ρIx1x3Ix1x4
3ε2

( 2x14,µ x14,ρI
2
x1x4 + x13,ρ x14,µIx1x3Ix1x4)

−4π2εν ε
ρIx1x3Ix1x4
3ε2

( 2x13,µ x13,ρI
2
x1x3 + x14,ρ x13,µIx1x3Ix1x4)

+
1

ε2
Ix1x3Ix1x4δµν

8π2
− Ix1x3Ix1x4δµν

4
( Ix1x4 + Ix1x3)

−Ix1x3Ix1x4δµν ε
ρ

2ε2
(x14,ρ Ix1x4 + x13,ρ Ix1x3)

+
8π2Ix1x3Ix1x4 δµν ε

ρ εσ

3ε2
(x14,ρ x14,σ I

2
x1x4 + x13,ρ x13,σ I

2
x1x3)

+
8π2Ix1x3Ix1x4 δµν ε

ρ εσ

3ε2
x13,ρ x14,σ Ix1x3Ix1x4

+
1

36

(
−2 + 3 log

(
ε2x2

34

x2
13x

2
14

))
I2
x1x3I

2
x1x4δµν

Ix3x4

+
2π2

9

(
1− 6 log

(
ε2x2

34

x2
13x

2
14

))
x13,µx13,νI

3
x1x3Ix1x4

−2π2

9

(
1 + 3 log

(
ε2x2

34

x2
13x

2
14

))
(x13,νx14,µ + x13,µx14,ν) I2

x1x3I
2
x1x4

+
2π2

9

(
1− 6 log

(
ε2x2

34

x2
13x

2
14

))
x14,µx14,νIx1x3I

3
x1x4 ,

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Figure 6. The results of the Feynman diagrams for the two-point functions. They are obtained by

taking the limits x3 → x2 and x4 → x1 of the expressions in figure 5.

lim
x2→x1

∂1,µ∂3,νX1234 =
εµεν
ε2

I2
x1x3Ix1x4

2
+ 2π2 log

(
ε2x2

34

x2
13x

2
14

)
x13,νx14,µI

2
x1x3I

2
x1x4

+
x13,ν εµI

2
x1x3Ix1x4
ε2

( 1− 4π2x14,ρ ε
ρIx1x4 − 8π2x13,ρ ε

ρIx1x3)

+2π2

(
−1 + 2 log

(
ε2x2

34

x2
13x

2
14

))
x13,µx13,νI

3
x1x3Ix1x4

−1

4

(
−1 + log

(
ε2x2

34

x2
13x

2
14

))
I2
x1x3Ix1x4δµ,ν

+2π2x13,µx34,νI
2
x1x3Ix1x4Ix3x4 + 2π2x14,µx34,νIx1x3I

2
x1x4Ix3x4 ,

lim
x2→x1

∂3,µ∂4,νX1234 =
δµν
2
Ix1x3Ix1x4Ix3x4 − 4π2x34,µx34,νIx1x3Ix1x4I

2
x3x4

−4π2x14,ν x34,µ Ix1x3I
2
x1x4Ix3x4 + 4π2x13,µ x34,ν I

2
x1x3Ix1x4Ix3x4

−4π2x13,µ x14,ν log

(
ε2x2

34

x2
13x

2
14

)
I2
x1x3I

2
x1x4 .

Using the above results, we can proceed to the computation of the two- and three-

point functions. The result of all the non-zero Feynman diagrams relevant for us is given

in figure 5, where we have omitted terms involving εµνρλ that must either vanish when a

pair of point collide or cancel when all the diagrams are summed. This is the case in order

to preserve conformal invariance and parity.

The results of figure 5 only contain derivatives of the function H12,34 and it is possible

to evaluate them explicitly [35]. Consider the case when the derivatives act on either the

first or the second pair of points of H, namely ∂1 · ∂2H12,34, and also the case when they

act on a point belonging to the first pair and a point belonging to the second pair, for

instance ∂1 · ∂4H12,34. The first case is straightforward to compute by using integration by

parts and the property of the euclidean propagator �xIxy = −δ(4)(x− y). The result is

∂1 · ∂2H12,34 =
1

2
(Y134Ix1x2 + Y234Ix1x2 −X1234) . (B.5)

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Figure 7. Using the results of figure 6, the sum of the graphs appearing in this figure gives precisely

the su(1|1) Hamiltonian of (B.11).

For computing the second case, we need the function F12,34 defined in (B.1) and some

identities of H12,34. Firstly, note that H satisfies the equation

(∂1,µ + ∂2,µ + ∂3,µ + ∂4,µ)H12,34 = 0 , (B.6)

which can be proved by integration by parts. Similarly, it is possible to show that the

following identity holds

∂i · ∂jH12,34 =
1

2
(�k + �l −�i −�j)H12,34 + ∂k · ∂lH12,34 (B.7)

for i 6= j 6= k 6= l. In order to get ∂1 · ∂4H12,34, it is convenient to write it as

∂1 · ∂4H12,34 =
1

2
(∂1 · ∂3 + ∂1 · ∂4)H12,34 −

1

2
(∂1 · ∂3 − ∂1 · ∂4)H12,34 . (B.8)

Now using (B.7), one can show that the first term on the right-hand side of (B.8) can be

written as
1

2
(∂1 · ∂3 + ∂1 · ∂4)H12,34 = −1

2
(�1H12,34 + ∂1 · ∂2H12,34) (B.9)

where �iH12,34 can be computed using the equation defining the euclidean propagator and

∂1 · ∂2H12,34 is known from (B.5). Using (B.6), the second term on the right-hand side

of (B.8) can be written as

1

2
(∂1 · ∂3 − ∂1 · ∂4)H12,34 =

1

4
(F12,34Ix1x2Ix3x4 + (�4 −�3)H12,34) . (B.10)

Finally, substituting (B.9) and (B.10) in (B.8), one gets an expression for ∂1 · ∂4H12,34.

The expressions for the remaining cases where the derivatives act on other points can be

deduced analogously.

In order to get the two-point functions, one takes the limit where two pairs of points

collapse into each other that is x4 → x1 and x3 → x2. The results of these limits are given

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Figure 8. The additional Feynman diagrams that do not contribute in the setup considered in this

work. The solid and dashed lines represent the scalars and fermions, respectively.

in figure 6. Summing all the diagrams as illustrated in figure 7, one obtains the one-loop

Hamiltonian operator

H = 2g2(I − SP ) , (B.11)

where SP is the superpermutator which exchanges the fields and picks up a minus sign

when both fields are fermionic. This Hamiltonian is the well known result of [15, 19].

We now proceed to the three-point functions. In order to obtain the constant coming

from each diagram, one takes the limit of the expressions given in figure 5 where a single

pair of points collapses into each other. After taking that limit, the result will have constant

terms, divergent logarithmic terms and eventually Y functions and their derivatives. The

derivatives of the Y functions can be expressed in terms of the Y function itself by using

some of its properties. This will be explained in detail in the appendix C. After this

procedure, the logarithmic terms will contribute to the standard regulator dependence

in (3.8) and the remaining Y functions will cancel with similar contributions from other

diagrams in a way that the conformal invariance is restored. One can then read the constant

part of the diagram. The final step is to subtract one half the constant coming from the

same diagram but when the two pairs of points collapse into each other as described in

figure 3. The results are given in figure 4.

Let us comment now on a detail of this computation. Our final results presented in the

figures 4 and 5 do not contain the Feynman diagrams of figure 8. This is the case because

the first two graphs of this figure turn out to cancel among them. They can give a non-zero

contribution in our setup only when either b = c = 4 and a = 3 or a = c = 2 and b = 3.

However, as these two graphs always appear with the same weight and opposite signs, they

end up canceling. The last graph of the figure 8 must vanish when |x12| → 0 because it is

∝ εγδεγ̇δ̇σµ
Eγ1̇

σν
Eδ1̇

σρE1γ̇σ
λ
E1δ̇

∫
d4v d4u (∂vµIx1v)(∂

v
νIx2v)Ivu(∂uρ Ix3u)(∂uλIx4u) → 0 .

If it was non-vanishing it would produce a term with a different tensor structure of (3.8)

which would violate conformal invariance.

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Figure 9. The tree-level diagrams for the three-point functions of the three half-BPS operators

considered in (C.1)–(C.3). Note that only the first term of O3 in (C.3) gives a non-zero contribution

at this order in perturbation theory as the second term clearly gives a vanishing contribution due

to R-charge conservation.

C Some examples of three-point functions

In this appendix, we give two examples of three-point functions. The first one is the case

of three half-BPS operators. It is well-known that this correlator is protected and therefore

it constitutes a check for our computations. Then we compute a non-protected three-point

function both by brute force and by using our prescription of inserting the operator F at

the splitting points.

C.1 Three half-BPS operators

Consider the following three half-BPS operators

O1 = Tr (ZZ) , (C.1)

O2 = Tr
(
Ψ̄ Z̄
)
, (C.2)

O3 = (R2
4R

1
3) · Tr (ΨZ) = Tr

(
ΨZ̄
)

+ Tr
(
ψ2 Φ14

)
. (C.3)

At tree-level the result is simply given by the sum of the two diagrams of figure 9 and reads

〈O1(x1)O2(x2)O3(x3)〉 =
2

(2π)682x2
12x

2
13

(σµE)11̇∂3,µ
1

2x2
23

. (C.4)

At one-loop, one has to sum the diagrams of figure 10 and use the results given in figure 5

taking the appropriate limits.

Some diagrams will still contain the function Y and its first derivatives. The Y function

depends on the external points in a way that does not respect the spacetime dependence

fixed by conformal symmetry, see equation (3.8). However, when one sums the different

diagrams this non-conformal spacetime dependence turns out to cancel identically.

At the end of the day, summing all the diagrams gives a vanishing one-loop contribution

to the structure constant in agreement with the non-renormalization theorem for the three-

points functions of half-BPS operators introduced in [21]. Equivalently, one can also use our

prescription to reproduce this one-loop result. One simply has to sum over the insertions

represented in figure 11 obtaining zero as expected.

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3Figure 10. The one-loop diagrams contributing to the three-point function of the three half-BPS

operators considered in (C.1)–(C.3). In the last four diagrams of the second row, the second term

of O3 (see expression (C.3)) gives a non-zero result. In all other diagrams only the first term of

O3 contributes.

Figure 11. Inserting the F operator at the splitting points, one reproduces the vanishing result

expected for a three-point function of half-BPS operators. Apart from the graphics in the figure,

there are similar graphics with the operator F acting in the remaining splitting points.

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Figure 12. The tree-level diagrams contributing to the three-point function of the opera-

tors (C.5)–(C.7). Once again, we only need to consider the first term of (C.7) at this order in

perturbation theory.

C.2 Two non-BPS and one half-BPS operators

We consider now a non-protected three-point function. This example serves as an illus-

tration of some of the technical details of the brute force computation. Moreover, we also

use it to check our prescription of the F operator insertion at the splitting points. The

operators at one-loop level that we will consider are

O1 = Tr (ZΨΨZ) , (C.5)

O2 = Tr
(
Z̄Ψ̄ Ψ̄ Ψ̄

)
, (C.6)

O3 = (R2
4R

1
3) · Tr (ΨZ) = Tr

(
ΨZ̄
)

+ Tr
(
ψ2 Φ14

)
. (C.7)

Note that the O1 and O2 are not half-BPS and therefore they will receive corrections as

explained in subsection 3.1. However, to compute the Feynman diagrams contribution we

do not need to take them into account. At tree-level the result is simply the sum of the

two diagrams of figure 12 which gives

〈O1(x1)O2(x2)O3(x3)〉0 = − 2

(2π)1082x2
13x

2
12

×
(

(σµE)11̇∂1,µ
1

2x2
12

)(
(σνE)11̇∂1,ν

1

2x2
12

)(
(σρE)11̇∂3,ρ

1

2x2
23

)
.

The diagrams contributing at one-loop are represented in figure 13.

As in the previous example, the dependence of each diagram on the Y function and

its derivatives will cancel when we sum over all the diagrams. This ensures that we obtain

a conformal invariant result. However, this cancellation is not immediate and it relies on

several properties of the Y function. The first observation is that the function Y is given by

Y123 =
π2φ(r, s)

(2π)4
Ix1,x3 , (C.8)

where r =
x212
x213

and s =
x223
x213

and an explicit expression for φ(r, s) can be found in [34].

The important information for us is that the function φ satisfies the following differential

equations [7]

φ(r, s) + (s+ r − 1)∂sφ(r, s) + 2r∂rφ(r, s) = − log r

s
, (C.9)

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Figure 13. The relevant one-loop diagrams for the three-point function of the operators (C.5)–

(C.7). The second term of (C.7) gives a non-zero contribution namely the first four diagrams of

the third row.

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Figure 14. Inserting the operator F at the splitting points reproduce the result of the one-loop

Feynman diagrams.

φ(r, s) + (s+ r − 1)∂rφ(r, s) + 2s∂sφ(r, s) = − log s

r
,

which can be used to relate the first derivatives of Y with Y itself. In addition, one can

take derivatives with respect to r and s of both the equations above to arrive at a system

of equations that relates second derivatives of φ with first derivatives and the function

φ itself. Using then (C.8), it is trivial to get rid of the second derivatives of Y . These

properties of the function φ ensure that the non-conformal dependence of the three-point

function indeed cancel when all diagrams are summed over.

The final result is given by

〈O1(x1)O2(x2)O3(x3)〉 = 〈O1O2O3〉0
(

1 + 4g2

(
−1 + 2 log

(
ε2

x2
12

))
+O(g4)

)
,

which comparing with (3.8) gives the correct anomalous dimensions of the operators. This

is a non-trivial consistency check of our computation.

The structure constant can now be obtained by also computing the constants from the

two-point functions. We have all the tools at hand to perform such calculation and read

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Figure 15. The one-loop additional graphs coming from the Wilson line.

Figure 16. The Wilson line contributions to the two-point functions. In the combination of the

two- and the three-point diagrams that provide the scheme independent structure constant C
(1)
123

of (3.8), all the extra diagrams coming from the Wilson lines cancel each other at this order in

perturbation theory. In the figure, the diagram corresponding to the emission of a gluon between

the two Wilson lines is not depicted, since it is proportional to ε2 and vanishes in the limit ε→ 0.

the one-loop constant. We obtain the following contribution from the Feynman diagrams

to the structure constant

C
(1)
123

∣∣∣∣
Feynman diagrams contribution

= −3

2
. (C.10)

Recall that this is not the final result, one also has to add the extra contribution from the

corrected two-loop Bethe states.

Finally, it is possible to test our prescription of inserting the F operator at the split-

ting points, see figure 14. Summing over all these insertions gives precisely the contribu-

tion (C.10) to the structure constant.

D Wilson line contribution

As mentioned before, the point splitting regularization breaks explicitly the gauge invari-

ance due to the fact that some fields inside the trace are now at a slightly different space-

time points. The introduction of a Wilson line connecting these fields restore the gauge

invariance at the price of introducing extra Feynman diagrams. In this appendix, we show

that these extra diagrams do not contribute to the scheme and normalization independent

structure constant C
(1)
123 defined in (3.8).

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D.1 Wilson line connecting two scalars

In our conventions the Wilson line operator is defined by

Wl = P exp

[
i gYM

∫
Aµ d~x

µ

]
.

When inserting a Wilson line connecting two scalars, it is necessary to consider the one-loop

graphs corresponding to the gluon emission depicted in figure 15(a). Let us define εµ =

xµ4 −x
µ
3 and at the end of the day we will take the limit εµ → 0. Then we can conveniently

parametrize the Wilson line by xµ(z) = xµ3 + zεµ. The result of the sum of the diagrams is

figure 15(a) =
λ

128

∫ 1

0
dz εµ (Ix2x3 ∂

µ
1 Y1x4 − Ix1x4 ∂

µ
2 Y2x3 + Ix1x4 ∂

µ
3 Y2x3 − Ix2x3 ∂

µ
4 Y1x4) ,

(D.1)

where we have suppressed both the R-charge and the gauge indices which are the same as

in the tree-level case. From the formula (B.3), it follows that the first and second terms of

the above result are of order ε and therefore vanish in the limit ε→ 0. However, from (B.2)

we see that the third and last term give a finite contribution.

In order to compute the scheme and normalization structure constant C
(1)
123 of (3.8),

we have to subtract from the previous result one half of the one-loop diagrams from the

two-point functions as shown in figure 16 (we take both the limits x4 → x3 and x2 → x1).

It is simple to show that the contribution of these diagrams cancels exactly the constant

coming from the expression (D.1). So, at this order in perturbation theory we do not get

any further contribution to C
(1)
123 and therefore we can safely ignore the Wilson lines.

D.2 Wilson line connecting either a scalar and a fermion or two fermions

In the case of a scalar and a fermion connected by a Wilson line, the contribution of the

diagrams depicted in 15(b) is given by

figure 15(b) =
λ

32

∫ 1

0
dz εµ

[
Ix2x3(σEµ11̇∂

4 · ∂1Y1x4 − σνE11̇
∂4
ν∂

1
µY1x4 − σνE11̇

∂4
µ∂

1
νY1x4

−ερµλν σνE11̇
∂4,λ∂1,ρY1x4) + σν

E11̇
∂1
νIx1x4(∂3

µY2x3 − ∂2
µY2x3)

]
. (D.2)

Using the expressions (B.2)–(B.4), one can easily see that this gives a finite contribution

in the limit when ε goes to zero (in particular, the term with ερµλν vanishes). To this

result, we have again to subtract one half of the one-loop diagrams from the two-point

functions as was done in the previous subsection for the case of two scalars. Once again,

the contribution of these diagrams cancels exactly the expression (D.2).

In the case when we have a Wilson line connecting two fermions, the same argument

holds. Hence at one-loop level, we can ignore the Wilson lines contributions in all cases.

E A note on the su(1|1) invariance of the final result

In this appendix, we address the question of the su(1|1) invariance of our formula for

the structure constant. In particular, this serves as a consistency check for the one-loop

prescription we have computed.

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Let us start by checking the tree-level structure constant. Its expression is given

in (3.10) when g → 0. One possible way of implementing a symmetry transformation on

a state at any value of the coupling is to add a Bethe root with zero-momentum. It is

clear from the Bethe equations that we obtain a state with the same energy and therefore

belonging to the same multiplet as the original one. Consider then the states |1〉 and |2〉
with one of their momenta p

(1)
j and p

(2)
i being equal to zero. In this particular case, we can

write apart from possible signs

|2〉 = Q̄ |2, {p̂(2)
i }〉 , 〈1f | = 〈1f , {p̂(1)

j }| S̄ ,

where the hat over a p means that this momentum is absent. The operator Q̄ (S̄) cre-

ates (annihilates) a zero-momentum magnon on a ket and annihilates (creates) a zero-

momentum magnon on the bra (we are omitting the R-charge and Lorentz indices for

simplicity).

For this particular choice of momenta the expression (3.10) for g = 0 becomes

〈1f , {p̂(1)
j }| S̄ Õ3 Q̄ |2, {p̂(2)

i }〉 = 〈1f , {p̂(1)
j }| Õ3 {S̄, Q̄} |2, {p̂(2)

i }〉 , (E.1)

where we denote the operator |Z̄ . . . Z̄i1 . . . iL2−N3〉〈Ψ̄ . . . Ψ̄ i1 . . . iL2−N3 | by Õ3. Moreover

in this equality, we have used that S̄ and Õ3 commute which can be proved by applying

the commutator to a generic su(1|1) state. In addition, we have also used that

S̄ |2, {p̂(2)
i }〉 = 0 , (E.2)

as the state is primary.

Now, the anticommutator {S̄, Q̄} is given by (see for instance the appendix D of [36])

{S̄, Q̄} = L+
1

2
H(g) , (E.3)

where L is the length operator and H(g) is the Hamiltonian operator. When acting on the

state |2, {p̂(2)
i }〉 it gives at leading order the length of the state L2. In conclusion, we have

derived the following equality

| 〈1f | Õ3 |2〉 | = L2 | 〈1f , {p̂(1)
j }| Õ3 |2, {p̂(2)

i }〉 | . (E.4)

The relation above shows how the structure constant changes under su(1|1) transfor-

mations of the states |2〉 and 〈1f | at leading order. It is now simple to check that our

expression for this scalar product given in the main text indeed satisfies this relation.

At one-loop, the final expression for the structure constants given in (3.10) has the

following term

〈1f | Õ′3 |2〉 , (E.5)

where

Õ′3 =

(
1 +

g2

2
FL3−N3,L3−N3+1 +

g2

2
FL1,1

)
Õ3

(
1 +

g2

2
FN3,N3+1 +

g2

2
FL2,1

)
. (E.6)

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If we require that Õ′3 commutes with the generator S̄ then we find that at one-loop

the relation (E.4) becomes

| 〈1f | Õ′3 |2〉 | =
(
L2 +

1

2
γ2

)
| 〈1f , {p̂(1)

j }| Õ
′
3 |2, {p̂

(2)
i }〉 | , (E.7)

where γ2 is the one-loop anomalous dimension of the operator O2. We have verified that

this relation is indeed obeyed, which shows that our prescription respects these symmetry

constraints.

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
	Three-point functions at leading order
	The one-loop Bethe eigenstates and structure constants

	One-loop three-point functions
	Two-loop coordinate Bethe eigenstates and norms
	One-loop perturbative calculation
	Final result

	Discussion and open problems
	Notation and conventions
	One-loop perturbative computation details
	Some examples of three-point functions
	Three half-BPS operators
	Two non-BPS and one half-BPS operators

	Wilson line contribution
	Wilson line connecting two scalars
	Wilson line connecting either a scalar and a fermion or two fermions

	A note on the mathfraksu(1|1) invariance of the final result