Three-Point Functions and su(1|1) Spin Chains

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.


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][2][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 twopoint 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.
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 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) unknown su(1|1) here here here unknown so (6) [12] (some cases) [6], [7] unknown unknown psu(2, 2|4) unknown unknown unknown unknown 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 [13]. 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 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 simple formula for the one-loop structure constant C 123 . Given three operators O i with N i excitations with momenta {p (i) j } N i j=1 , and length L i (the details of the exact setup will be given below), we have f(y (1) i , y (2) j ) −S(y (2) i , y where y (i) j ≡ e ip (i) j , C is a simple normalization factor given in (3.14), S is the su(1|1) Smatrix. The most essential ingredient and main result of this work is the function f which is simply given by with g 2 = λ 16π 2 . 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 structure 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.

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 advantage 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 = Φ 34 and a complex chiral fermion that shares with the scalar one R-charge index, for instance Ψ = ψ 4 α=1 . The setup for the calculation of the planar three-point functions that we will be considering involves an operator O 1 given by a linear combination of single traces made out of products of these fields. More precisely, where L 1 is the length of the operator, N 1 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 O 2 made out of the complex conjugate fields, namely In our conventions, the complex conjugate fields are given bȳ 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 O 1 and O 2 belong, if we want to have a non-vanishing result and avoid extremal correlation functions 1 . Instead, we consider a "rotated" operator constructed by applying su(4) generators several times to a su(1|1) operator of the type O 1 . The idea is to get a composite operator having a term with only Ψ andZ 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 aZ 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 andZ's by considering the following operation 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, (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, 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 O 1 in (2.1) and O 3 in (2.4) are spinorial operators with N 1 and N 3 indices α = 1 respectively. This follows from the definition of the field Ψ given previously. The operator O 2 in (2.2) has N 2 Lorentz indicesα =1 associated to each of the fermionsΨ .
In a conformal field theory, the two-point functions are completely fixed by the symmetries up to a normalization constant. For two operators having spinorial indices as shown below, we have where N i is a constant associated to the normalization of the operator, ∆ i is its conformal dimension and the tensorial structure is 2 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 [14][15][16]. However, for the setup considered in this work there is only one possible tensor and the three-point functions is of the form where we are considering N 2 = N 1 + N 3 and g 2 = λ 16π 2 with λ the 't Hooft parameter. The structure constant C 123 (g 2 ) has a perturbative expansion when g 2 is small, and its leading order will be designated by C 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, α is a normalization factor that comes from the fact that we are normalizing the operators such that their two-point functions has the canonical form (2.6) with N i = 1. It is given by The main goal of this section is to find a closed formula for C

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 [17,13]. The Hamiltonian is simply the fermionic version of the Heisenberg Hamiltonian and it is written in terms of the Pauli matrices as 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 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 |ψ N = 1≤n 1 <n 2 <...<n N ≤L ψ N (n 1 , n 2 , . . . , n N )|n 1 , . . . , n N , (2.13) where the n i 's in |n 1 , . . . n N 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 |n 1 , . . . n N represents the trace in (2.1). The wave-function ψ N (n 1 , . . . , n N ) is a combination 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 ψ N (n 1 , n 2 , . . . , n N ) = P sign P exp(ip σ P (1) n 1 + ip σ P (2) n 2 + . . . + ip σ P (N ) n N ) (2.14) where P indicates sum over all possible permutations σ P of the elements {1, . . . , N }, and sign P is the sign of the permutation. Moreover, we should impose the periodicity condition by requiring the momenta p i to satisfy the Bethe equations The cyclic property of the trace is implemented by imposing the zero momentum condition of the state, 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. First, 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, (2.17) Moreover we have replaced ψ (1) n 1 ,...,n N 1 ) * since they differ by at most a sign.
Notice that the first N 3 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 N 3 extra excitations to the wave-function ψ (1) and liberate the fixed N 3 roots of ψ (2) with their positions being summed over too, (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 N 3 momenta, say {p N 3 }, are complex. We can then dynamically localize the wavefunction around the original N 3 positions by taking the limit of these momenta going to minus infinity. More precisely, we send 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 n 1 = 1, . . . , n N 3 = N 3 . 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 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 We emphasize again that we now sum without restrictions, 1 ≤ n i ≤ L 2 , for all n i . These sums over n i can be explicitly computed as they are geometric series. Using the Bethe equations and the total momentum condition for the operator O 2 , we can then simplify (2.21) to .
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 . (2.23) Notice that this expression contains as a limit the norm of an operator. 3 It is given by Finally, we take the limit of (2.23) when (2.19). Plugging the resulting limit and taking into account the overall product multiplying the sum in (2.20), we obtain our final result .

(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.

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.

Two-loop coordinate Bethe eigenstates and Norms
The two-loop Bethe eigenstates are determined by diagonalizing the long-range Hamiltonian H [13] H where H 1 is given in (2.11) and 3 If we set N 3 = 0 and consider p we get the expression for N (2) after using the Bethe equations (2.15).
where σ i are the Pauli matrices. In order to diagonalize it, we start with the usual coordinate Bethe ansatz which works when the excitations are at a distance bigger than the range of the interaction, i.e. when |n i − n j | > 2. In this region all we need is the two-loop S-matrix which reads Given the long-range nature of the Hamiltonian (3.1), we expect the form of the wavefunction 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 nontrivial 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 where we have used the notation S ab = S(p a , p b ) and 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 {p 1 , . . . , p N } and N − 1 types of contact terms namely C(p i , p j ), . . . , C(p 1 , ..., p N ). Unexpectedly, we have found that up to seven magnons the contact terms are simply given by 4 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 twopoint functions. Remarkably, we have checked numerically up to six-magnons that the two-loop (coordinate) norm is given by 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.

One-loop perturbative calculation
Loop computations will give rise to divergences which require the introduction of a regularization 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 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 where the tensor on the right-hand side was defined in (2.7). In the expression above, ∆ 0,i and γ i are the free scaling dimension and the one-loop anomalous dimension of the operator O i respectively, N i is a normalization constant and a i 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) where we have factored out the tree-level constant C 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 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. 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.
two-point functions of all the operators to get rid of the constants a i '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 calculate C (1) 123 instead of the combination (C (1) 123 + a 1 + a 2 + a 3 ). 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 positions of the spin chains where it is possible to have those genuine diagrams are commonly 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 Φ 14 and the fermion ψ 2 in the second line of (2.5) can be converted into a Ψ and aZ 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 whereδ ab,cd ≡ δ ac δ bd − δ ad δ bc and in the second line we recognize the so (6) Hamiltonian [18,7,6]. It is simple to check that the operator g 2 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 F ij , which indicate the sites in the spin chain where the operator acts. As an example, we have that which reproduces the result of the first diagram of the second row of figure 4. It is important to note that when the operator F ij acts on non-neighboring sites, it can pick up additional minus signs due to statistics, for example, where n denotes the number of fermionic excitations between the first and last sites and we have used the last rule of (3.9).

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 where we have that 11) with N (i) being the respective norms and we are using the conventions where σ is any field. In the formula (3.10), i a can be eitherZ orΨ and a sum over all these intermediate states is implied. Moreover, we have included a superscript f in the bra associated to the operator O 1 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, (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 [19], 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 on 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 very simple and natural deformation of the tree-level result (2.25), as follows where we are using the notation y (i) k = e ip (i) k and the normalization factor C is given by with γ i being the anomalous dimension of the operator O i . As described in the section 3.1, the norms N (i) are given by the formula The most important and non-trivial part of the final result is the function f which reads The momenta p (j) k of the fermionic excitations must satisfy the Bethe equations which take the form 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.

Discussion and open problems
In this work, we have computed both the leading order contribution and the one-loop perturbative 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 perturbation theory [17] 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 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 L i 1, with a large number of excitations N i = O(L i ). 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 [20][21][22][23].
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 constant has two distinct contributions. Firstly, the one-loop correction to the S-matrix appears 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 [24,25]. 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 [26][27][28]. 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 [26] which corresponds to a multi-particle transition. We hope that such ideas can be applied for the calculation of three-

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 [29,30] with all the fields in the adjoint representation of the gauge group and the covariant derivative is The propagators extracted from this Lagrangian are (we are suppressing the gauge indices and taking the leading order in N ) We are using the Minkowski metric (+− − −) and the Feynman gauge. The action of the (classical) supersymmetry generators are given by [31] and the conjugate expressions for the action ofQ aα . The action of the R-symmetry generators is given by In the computations of the Feynman diagrams, in particular for the evaluation of the integrals, we analytical continued to Euclidean space by using where the subscripts M means Minkowski space and E means Euclidean space, and, finally, σ i M are the usual Pauli matrices.

B One-loop perturbative computation details
In this Appendix, we present the details of the perturbative computation of the threepoint 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 where I xax b is the (euclidean) scalar propagator defined as The Y and X integrals are well-known and explicit expressions for them can be found for instance in [32,7]. The integral H is not known analytic, however, only its derivatives will be needed. In particular, the following combination [32] turns out to be useful where We will need several limits of the expressions for Y and X, namely when pairs of distinct points collapse into each other where we are considering x µ 2 = x µ 1 + µ with µ → 0. We can also take a further limit of the last expression above when x 4 → x 3 giving 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 x 13,µ I 2 As before, one can take further limits of these expressions when needed. The second derivatives read lim x 13,ν I 2 x 13,µ x 13,ν I 3 x 13,µ x 13,ν I 3 x 13,µ x 13,ν I 3 x 13,ν x 14,µ I 2 x 1 x 3 I 2 x 13,µ x 13,ν I 3 Using the above results, we can proceed to the computation of the two-and threepoint functions. The result of all the non-zero Feynman diagrams relevant for us is given in figure 5, where we have omitted terms involving µνρλ 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 H 12,34 and it is possible to evaluate them explicitly [33]. Consider the case when the derivatives act on either the first or the second pair of points of H, namely ∂ 1 · ∂ 2 H 12,34 , 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 · ∂ 4 H 12,34 . The first case is straightforward to compute by using integration by parts and the property of the euclidean propagator x I xy = −δ (4) For computing the second case, we need the function F 12,34 defined in (B.1) and some identities of H 12,34 . Firstly, note that H satisfies the equation which can be proved by integration by parts. Similarly, it is possible to show that the following identity holds for i = j = k = l. In order to get ∂ 1 · ∂ 4 H 12,34 , it is convenient to write it as Now using (B.7), one can show that the first term on the right-hand side of (B.8) can be written as ( 1 H 12,34 + ∂ 1 · ∂ 2 H 12,34 ) (B.9)  where i H 12,34 can be computed using the equation defining the euclidean propagator and ∂ 1 · ∂ 2 H 12,34 is known from (B.5). Using (B.6), the second term on the right-hand side of (B.8) can be written as In order to get the two-point functions, one takes the limit where two pairs of points collapse into each other that is x 4 → x 1 and x 3 → x 2 . The results of these limits are given in figure 6. Summing all the diagrams as illustrated in figure 7, one obtains the one-loop Hamiltonian operator 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 [17,13].
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 |x 12 | → 0 because it is If it was non-vanishing it would produce a term with a different tensor structure of (3.8) which would violate conformal invariance.

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  At tree-level the result is simply given by the sum of the two diagrams of figure 9 and reads (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 [19]. 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.   . 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 illustration 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 Note that the O 1 and O 2 are not half-BPS and therefore they will receive corrections as explained in section 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 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  where r = and an explicit expression for φ(r, s) can be found in [32]. The important information for us is that the function φ satisfies the following differential equations [7] φ(r, s) + (s + r − 1)∂ s φ(r, s) + 2r∂ r φ(r, s) = − log r s , (C.9) φ(r, s) + (s + r − 1)∂ r φ(r, s) + 2s∂ s φ(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 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 the one-loop constant. We obtain the following contribution from the Feynman diagrams to the structure constant 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 splitting points, see figure 14. Summing over all these insertions gives precisely the contribution (C.10) to the structure constant.

D Wilson line contribution
As mentioned before, the point splitting regularization breaks explicitly the gauge invariance due to the fact that some fields inside the trace are now at a slightly different spacetime 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).

D.1 Wilson line connecting two scalars
In our conventions the Wilson line operator is defined by When inserting a Wilson line connecting two scalars, it is necessary to consider the oneloop graphs corresponding to the gluon emission depicted in figure 15(a). Let us define µ = x µ 4 − x µ 3 and at the end of the day we will take the limit µ → 0. Then we can conveniently parametrize the Wilson line by x µ (z) = x µ 3 + z µ . The result of the sum of the diagrams is figure 15(a) = λ 128 , (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 and therefore vanish in the limit → 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 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 x 4 → x 3 and x 2 → x 1 ). 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 Using the expressions (B.2-B.4), one can easily see that this gives a finite contribution in the limit when goes to zero (in particular, the term with ρµλν 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.
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 =Q |2, {p (2) i } , j }|S , where the hat over a p means that this momentum is absent. The operatorQ (S) creates (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 j }|SÕ 3Q |2, {p (2) i } = 1 f , {p where we denote the operator |Z . . .Zi 1 . . . i L 2 −N 3 Ψ . . .Ψ i 1 . . . i L 2 −N 3 | byÕ 3 . Moreover in this equality, we have used thatS andÕ 3 commute which can be proved by applying the commutator to a generic su(1|1) state. In addition, we have also used that as the state is primary. Now, the anticommutator {S,Q} is given by (see for instance the Appendix D of [34]) where L is the length operator and H(g) is the dilatation operator. When acting on the state |2, {p (2) i } it gives at leading order the length of the state L 2 . In conclusion, we have derived the following equality j }|Õ 3 |2, {p (2) i } | . (E.4) The relation above shows how the structure constant changes under su(1|1) transformations of the states |2 and 1 f | 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 If we require thatÕ 3 commutes with the generatorS then we find that at one-loop the relation (E.4) becomes where γ 2 is the one-loop anomalous dimension of the operator O 2 . We have verified that this relation is indeed obeyed, which shows that our prescription respects these symmetry constraints.