Form Factor and Boundary Contribution of Amplitude

The boundary contribution of an amplitude in the BCFW recursion relation can be considered as a form factor involving boundary operator and unshifted particles. At the tree-level, we show that by suitable construction of Lagrangian, one can relate the leading order term of boundary operators to some composite operators of N=4 super-Yang-Mills theory, then the computation of form factors is translated to the computation of amplitudes. We compute the form factors of these composite operators through the computation of corresponding double trace amplitudes.


Introduction
The ongoing research on the on-shell techniques has gone beyond its primal scattering amplitude domain, to the computation of form factor in recent years. The form factor, sometimes stated as a bridge linking on-shell amplitude and off-shell correlation function, is a quantity containing both on-shell states(ingredients for amplitudes) and gauge invariant operators(ingredients for correlation functions). Its computation can be traced back to the pioneering paper [1] nearly 30 years ago, where the Sudakov form factor of the bilinear scalar operator Tr(φ 2 ) is investigated up to two loops. At present, many revolutionary insights originally designed for the computation of amplitudes 1 , such as MHV vertex expansion [5], BCFW recursion relation [6,7], color-kinematic duality [8,9], unitarity cut [10,11] method(and its generalization to D-dimension [12,13]), generalized unitarity [14,15], etc., have played their new roles in evaluating form factors.
These progresses are achieved in various papers. In paper [16], the BCFW recursion relation appears for the first time in the recursive computation of tree-level form factor, mainly for the bilinear scalar operator. As a consequence, the solution of recursion relation for split helicity form factor is conquered [17]. Intensive discussion on the recursion relation of form factor is provided later in [18]. A generalization to the form factor of full stress tensor multiplet is discussed in [17] and [19], where in the former one, supersymmetric version of BCFW recursion relation is pointed out to be applicable to super form factor. Shortly after, the color-kinematic duality is implemented in the context of form factor [20], both at tree and loop-level, to generate the integrand of form factor. Most recently, the elegant formulation of amplitudes based on Grassmannian prescription [21] is also extended to tree-level form factors [22]. At loop-level, the form factor is generally computed by unitarity cut method. The generic Maximal-Helicity-Violating(MHV) super form factor as well as some Next-MHV(NMHV) form factor at one-loop are computed in [17,[23][24][25] with compact results. The Sudakov form factor is computed to three loops in [26][27][28]. The three-point two-loop form factor of half-BPS operator is achieved in [29], and the general n-point form factor as well as the remainder functions in [30]. The scalar operator with arbitrary number of scalars is discussed in [19,30,31]. Beyond the half-BPS operators, form factors of non-protected operators, such as dilatation operator [32], Konishi operator [33], operators in the SU (2) sectors [34], are also under investigation. Furthermore, the soft theorems for the form factor of half-BPS and Konishi operators are studied at tree and one-loop level [35], showing similarity to amplitude case. Carrying on the integrand result of [20], the master integrals for four-loop Sudakov form factor is determined in [36]. An alternative discussion on the master integrals of form factor in massless QCD can be found in [37]. Similar unitarity based studies on Sudakov form factor of three-dimensional ABJM theories are also explored [38][39][40].
The above mentioned achievements encode the belief that the state-of-art on-shell techniques of amplitude would also be applicable to form factor. Recently, the advances in the computation of boundary contribution have revealed another connection between form factor and amplitude. When talking about the BCFW recursion relation of amplitude, the boundary contribution is generally assumed to be absent. However this assumption is not always true, for example, it fails in the theories involving only scalars and fermions or under the "bad" momentum deformation. Many solutions have been proposed(by auxiliary fields [41,42], analyzing Feynman diagrams [43][44][45], studying the zeros [46][47][48], the factorization limits [49], or using other deformation [50][51][52]) to deal with the boundary contribution in various situations. Most recently, a new multi-step BCFW recursion relation algorithm [53][54][55] is proposed to detect the boundary contribution through certain poles step by step. Especially in paper [54], it is pointed out that the boundary contribution possesses similar BCFW recursion relation as amplitudes, and it can be computed recursively from the lower-point boundary contribution. Based on this idea, later in paper [56], the boundary contribution is further interpreted as form factor of certain composite operator named boundary operator, while the boundary operator can be extracted from the operator product expansion(OPE) of deformed fields.
The idea of boundary operator motives us to connect the computation of form factor to the boundary contribution of amplitudes. Since a given boundary contribution of amplitude can be identified as a form factor of certain boundary operator, we can also interpret a given form factor as the boundary contribution of certain amplitude. In paper [56], the authors showed how to construct the boundary operator starting from a known Lagrangian. We can reverse the logic and ask the question: for a given operator, how can we construct a Lagrangian whose boundary operator under certain momentum deformation is exactly the operator of request? In this paper, we try to answer this question by constructing the Lagrangian for a class of so called composite operators. Once the Lagrangian is ready, we can compute the corresponding amplitude, take appropriate momentum shifting and extract the boundary contribution, which is identical(or proportional) to the form factor of that operator. By this way, the computation of form factor can be considered as a problem of computing the amplitude of certain theory.
This paper is structured as follows. In §2, we briefly review the BCFW recursion relation and boundary operator. We also list the composite operators of interest, and illustrate how to construct the Lagrangian that generates the boundary operators of request. In §3, using Sudakov form factor as example, we explain how to compute the form factor through computing the boundary contribution of amplitude, and demonstrate the computation by recursion relation of form factor, amplitude and boundary contribution. We show that these three ways of understanding lead to the same result. In §4, we compute the form factors of composite operators by constructing corresponding Lagrangian and working out the amplitude of double trace structure. Conclusion and discussion can be found in §5, while in the appendix, the construction of boundary operator starting from Lagrangian is briefly reviewed for reader's convenience, and the discussion on large z behavior is presented.

From boundary contribution to form factor
The BCFW recursion relation [6,7] provides a new way of studying scattering amplitude in S-matrix framework. Using suitable momentum shifting, for example, one can treat the amplitude as an analytic function A(z) of single complex variable, with poles in finite locations and possible non-vanishing terms in boundary, while the physical amplitude sits at z = 0 point. Assuming that under certain momentum shifting, A(z) has no boundary contribution in the contour integration 1 2πi dz z A(z), i.e., A(z) → 0 when z → ∞, then the physical amplitude A(z = 0) can be purely determined by the residues of A(z) at finite poles. However, if A(z) does not vanish around the infinity, for example when taking a "bad" momentum shifting or in theories such as λφ 4 , the boundary contribution would also appear as a part of physical amplitude. Most people would try to avoid dealing with such theories as well as the "bad" momentum shifting, since the evaluation of boundary contribution is much more complicated than taking the residues of A(z).
Although it is usually unfavored during the direct computation of amplitude, authors in paper [56] found that the boundary contribution is in fact a form factor involving boundary operator and unshifted particles, where Φ(p i ) denotes arbitrary on-shell fields, and momenta of Φ(p 1 ), Φ(p 2 ) have been shifted according to eqn.(2.1). The momentum q carried by the boundary operator is q = −p 1 − p 2 = n i=3 p i . Eqn. (2.2) is identical to a (n − 2)-point form factor generated by operator O 1|2] with off-shell momentum q 2 = 0. The observation (2.2) provides a new way of computing form factor, 1. Construct the Lagrangian, and compute the corresponding amplitude, 2. Take the appropriate momentum shifting, and pick up the boundary contribution, 3. Read out the form factor from boundary contribution after considering LSZ reduction.
In paper [56], the authors illustrated how to work out the boundary operator O Φ i |Φ j ] from Lagrangian of a given theory under momentum shifting of two selected external fields. Starting from a Lagrangian, one can eventually obtain a boundary operator. For example, a real massless scalar theory with φ m interaction under momentum shifting of two scalars(say φ 1 and φ 2 ) will produce a boundary operator Hence the boundary contribution of a n-point amplitude A n (φ 1 , . . . , φ n ) in this κφ m theory under φ 1 |φ 2 ]-shifting is identical to the (n − 2)-point form factor However, this form factor is not quite interesting. We are interested in certain kind of operators, such as bilinear half-BPS scalar operator Tr(φ AB φ AB ) or chiral stress-tensor operator Tr(W ++ W ++ ) in N = 4 super-Yang-Mills(SYM) theory, where W ++ is a particular projection of the chiral vector multiplet superfield W AB (x, θ) in SYM. What we want to do is to compute the form factor for a given operator, but not the operators generated from arbitrary Lagrangian. More explicitly, if we want to compute the form factor of operator O, we should first construct a Lagrangian whose boundary operator is identical(or proportional) to O. With such Lagrangian in hand, we can then compute the corresponding amplitude, take the momentum shifting and pick up the boundary contribution. So the problem is how to construct the corresponding Lagrangian.

The operators of interest
It is obvious that the construction of Lagrangian depends on the operators we want to produce. In this paper, we will study the so called gauge-invariant local composite operators, which are built as traces of product of gauge-covariant fields at a common spacetime point. These fields are taken to be the component fields of N = 4 superfield Φ N =4 [57], given by six real scalars φ I , I = 1, . . . , 6(or 3 complex scalars φ AB ), four fermions ψ A α = ABCD ψ BCDα , four anti-fermionsψ Aα and the field strength F µν , where α, β,α,β = 1, 2 are spinor indices, A, B, C, D = 1, 2, 3, 4 are SU (4) R-symmetric indices, and µ, ν = 0, 1, 2, 3 are spacetime indices. The field strength can be further split into self-dual and anti-self-dual parts F αβ ,Fαβ: corresponding to positive gluon and negative gluon respectively. The number of fields inside the trace is called the length of operator, and the simplest non-trivial ones are the length two operators. There is no limit on the length of operator, for example, the bilinear half-BPS scalar operator Tr(φ I φ J ) is length two, while we could also have length L scalar operator Tr(φ I 1 · · · φ I L ). The operators can also carry spinor indices, such as O αβαβ = Tr(ψ Aα ψ BβFαβ ) in the (1, 1) representation under Lorentz group SU (2) × SU (2).
We will mainly focus on the length two operators. These operators can be classified by their spins and labeled by their representations under SU (2) × SU (2) group. For spin-0 operators in (0, 0)-representation, we have For operators of the same class, we can apply similar procedure to construct the Lagrangian. The operators with length larger than two can be similarly written down, and classified according to their spins and representations. For those whose spins are no larger than 2, we can apply the same procedure as is done for length two operators. while if their spins are larger than 2, we need multiple shifts. Some of above operators are in fact a part of the chiral stress-tensor multiplet operator in N = 4 SYM [58,59], and their form factors are components of N = 4 super form factor. However, we have assumed that, all indices of these gauge-covariant fields are general, so above operators are not limited to the chiral part, they are quite general.

Constructing the Lagrangian
One important property shared by above operators is that they are all traces of fields. Tree-level amplitudes of ordinary gauge theory only possess single trace structure. From the shifting of two external fields, one can not generate boundary operators with trace structures, which can be seen in [56]. The solution is to intentionally add a double trace term in the standard Lagrangian. The added term should be gauge-invariant, and generate the corresponding operator under selected momentum shifting.
For a given operator O of interest, let us add a double trace term ∆L to the N = 4 Lagrangian L SYM , where SU (N ) group is assumed, κ,κ are coupling constants for the double trace interactions(which can be re-scaled to fit the overall factor of final result) and Φ α , Φ † α denotes 2 any type of fields among φ I , ψ Aα ,ψα A , F αβ ,Fαβ. The spinor indices are not explicitly written down for Φ, Φ † , however we note that they should be contracted with the spinor indices of the operator, so that the added Lagrangian terms are Lorentz invariant. We will show that at the large N limit, momentum shifting of two fields in ∆L indeed generates the boundary operator O.
The tree-level amplitudes defined by Lagrangian L O can have single trace pieces or multiple trace pieces. A full (n + 2)-point amplitude thus can be decomposed into color-ordered partial amplitudes A as A full n+2 = A n+2 (1, 2, . . . , n + 2) Tr(t a 1 · · · t an t a t b ) + · · · (2.15) + 1 N A k;n+2−k (1, . . . , k; k + 1, . . . , n + 2) Tr(t a 1 · · · t a k ) Tr(t a k+1 · · · t a t b ) + · · · where A n denotes n-point single trace amplitude, A k;n−k denotes n-point double trace amplitude. We use i to abbreviate Φ i , and · · · stands for all possible permutation terms and other higher order multiple trace pieces. Since the operator O we want to generate is single trace, the terms with higher multiple trace in · · · is then irrelevant for our discussion, and also they can be ignored at large N . Now let us contract the color indices a, b, which gives 3 (1, 2, . . . , n + 2) Tr(t a 1 · · · t an ) + · · · (2.16) . . , k; k + 1, . . . , n + 2) Tr(t a 1 · · · t a k ) Tr(t a k+1 · · · t an ) + · · · In this case, the O(N ) order terms in (2.16) come from two places, one is the single trace part in (2.15) when t a and t b are adjacent, the other is the double trace part in (2.15) whose color factor has the form Tr(· · · ) Tr(t a t b ). So when color indices a, b are contracted, the leading contribution of the full (n + 2)-point amplitude is A full n+2 = N Tr(t a 1 · · · t an )K(1, 2, . . . , n) + possible permutation{1, 2, . . . , n} , (2.17) where K(1, . . . , n) ≡ A n+2 (1, . . . , n, n + 1, n + 2) + A n+2 (1, . . . , n, n + 2, n + 1) +A n;2 (1, . . . , n; n + 1, n + 2) . (2.18) The first two terms in K are the same as the corresponding color-ordered single trace amplitudes, since the other double trace terms in the Lagrangian will not contribute to the O(N ) order at tree-level. The third term in K is double trace amplitude of the trace form Tr(· · · ) Tr(t a t b ), and the Feynman diagrams contributing to this amplitude are those whose Φ n+1 and Φ n+2 are attached to the same double trace vertex, while the color indices of Φ n+1 , Φ n+2 are separated from others. Now let us examine the large z behavior of the amplitude under momentum shifting Φ α n+1 n+1 |Φ α n+2 n+2 . Since the color indices of two shifted legs are contracted, it is equivalent to consider the large z behavior of K(1, 2, . . . , n) under such shifting. Following [56], we find that at the large N limit, the leading interaction part V is given by where T αβ is defined through Φ α = T αβ Φ † β , and α 1 = α n+1 , α 2 = α n+2 , indicating that the shifted fields Φ n+1 , Φ n+2 are the two fields of Tr(Φ α 1 Φ α 2 ) in (2.14) with specific field type. In general, the OPE of shifted fields has the form [56] where n+1 α , n+2 β are external wave functions of Φ n+1 , Φ n+2 . The terms with (D −1 0 ) k correspond to Feynman diagrams with k hard propagators. The Z(z) for L O contains two parts, one from the single trace and the other from double trace. The single trace amplitudes in K originate from Feynman diagrams with vertices of N = 4 Lagrangian, thus their Z(z) can be directly obtained by replacing V αβ with V αβ SYM . The double trace amplitudes in K originate from Feynman diagrams with double trace vertices. Because the two shifted fields Φ n+1 , Φ n+2 should be attached to the same double trace vertex, in this case the hard propagator will not appear in the corresponding Feynman diagrams. Thus for this part, we only need to keep the first term in (2.20)(more explicitly, the terms with single O orŌ in (2.19)). Combined together, we have The summation of α, β runs over all types of fields. For a given momentum shifting α 1 = α n+1 , α 2 = α n+2 , we can choose the wave function such that n+1 α n+1 n+2 α n+2 = 0 but all other types of contractions vanish. In this case, the second line of (2.21) contains a factor , it is clear that this factor is zero when the two shifted fields are not complex conjugate to each other. So we have, However, if the two shifted fields are complex conjugate to each other, then in the definition of Lagrangian (2.14),Ō is in fact identical to O. This means that there is only one term in ∆L but not two, and consequently there is only the first line in (2.21). After the choice of wave functions, we again get (2.22). From eqn.(2.22), we know that the large z behavior of L O under Φ|Φ]-shifting depends on the large z behavior of N = 4 SYM theory as well as the double trace term ∆L. In fact(please refer to Appendix B for detailed discussion), for all the shifts we use in this paper 4 , Z SYM (z) has lower power in z than the second term in (2.22) at large z. This means that the boundary operator(or the operator defined by the leading z order) is always determined by the second term in (2.22), So it produces the desired operatorŌ, up to certain possible pre-factor from the external wave functions.

Sudakov form factor and more
In this section, we will take the bilinear half-BPS scalar operator as an example to illustrate the idea of computing form factor from boundary contributions. The form factor is defined as Here |s is a n-particle on-shell states, and each state in |s is on-shell, with a momentum p 2 i = 0, while the operator, carrying momentum q = n i=1 p i , is off-shell. The simplest example is given by taking |s = |φ I (p 1 )φ J (p 2 ) , i.e., the Sudakov form factor, and it is simply 5 A more complicated one is given by taking the on-shell states as two scalars and (n − 2) gluons. Depending on the helicities of gluons, it defines the MHV form factor, NMHV form factor and so on.
In order to compute the form factor (3.1) as boundary contribution of certain amplitude under BCFW shifting, we need to relate the operator O 2 with certain boundary operator. This can be done by constructing a new Lagrangian L O 2 by adding an extra double trace term ∆L in the N = 4 Lagrangian as where κ is the coupling constant. Since we are dealing with real scalars, there is no need to add the corresponding complex conjugate term. This new term provides a four-scalar vertex, and it equals to iκ, as shown in Figure (1). If we split two scalars into ordinary part and hard part φ Ia → φ Ia + φ ΛIa and φ Jb → φ Jb + φ ΛJb (the hard part φ Λ corresponds to the large z part), then the quadratic term φ ΛIa φ ΛJb of L SYM part can be read out from the result in Appendix B of [56] by setting A = (A µ , φ I ), which is given by The quadratic term φ ΛIa φ ΛJb of ∆L part is simply(at the leading N order) Thus the boundary operator under two-scalar shifting is Notice that the traceless part (while I = J) of boundary operator (3.5) is proportional to the operator O 2 . This means that if the two shifted scalars are not the same type of scalar, i.e., I = J, the corresponding boundary contribution B φ Ia |φ Jb ] of amplitude defined by Lagrangian L O 2 is identical to the form factor of O 2 = Tr(φ K φ L ), up to some over-all factor which can be fixed by hand. More explicitly, let us consider the color-ordered form factor 1, 2, . . . , n|O 2 |0 , where i denotes an arbitrary field. It is dressed with a single trace structure Tr(t 1 t 2 · · · t n )O 2 . In the amplitude side, O 2 is generated from the double trace term ∆L, and the corresponding trace structure of color-ordered amplitude is Tr(t 1 t 2 · · · t n ) Tr(t n+1 t n+2 ). We denote the amplitude of double trace structure as A n;2 (1, 2, . . . , n; φ n+1 , φ n+2 ). It only gets contributions from the Feynman diagrams where φ n+1 , φ n+2 are attached to the sole four-scalar vertex of ∆L. Then the form factor 1, 2, . . . , n|O 2 |0 is just the boundary contribution of A n;2 (1, 2, . . . , n; φ n+1 , φ n+2 ) under BCFW shifting of two scalars φ n+1 , φ n+2 ! As a simple illustration, let us consider four-point scalar amplitude . In this case, the only possible contributing diagram is a four-scalar vertex defined by ∆L, and we can directly work out as A 2;2 (φ 1 , φ 2 ; φ 3 , φ 4 ) = iκ. After appropriate normalization, it can be set as 1. Since it has no dependence on any external momenta, after momentum shifting the amplitude still remains the same, while the boundary operator is Thus we confirm the tree-level Sudakov form factor Now we have three different ways of studying form factor. The first, as stated in [16], form factor obeys a similar BCFW recursion relation as amplitude. This enables us to compute a form factor recursively from lower-point ones. The second, we can compute the corresponding amplitude. Once it is obtained, we can take the BCFW shifting φ n+1 |φ n+2 ] and extract the boundary contribution B φ n+1 |φ n+2 ] , which equals to the corresponding form factor after identification. The third, as stated in [54], the boundary contribution also obeys a similar BCFW recursion relation as amplitude. We can compute boundary contribution recursively from lower-point boundary contributions, and once it is obtained, we can work out the form factor after identification.
In the following subsection, we will take MHV form factor of operator O 2 as an example, to illustrate these three ways of understanding.

MHV case
The n-point color-ordered MHV form factor of operator O 2 is given by

BCFW recursion relation of form factor
The result (3.8) has been proven in paper [16] 6 by BCFW recursion relation of form factor. As stated therein, after taking BCFW shifting of two momenta p i 1 , p i 2 , the form factor can be computed as summation of products of lower-point form factor and lower-point amplitude, as long as the large z behavior F(z)| z→∞ → 0 is satisfied under such deformation. The n external legs will be split into two parts, with p i 1 , p i 2 in each part separately. The operator, since it is color-singlet, can be inserted into either part. So it is possible to build up a n-point form factor recursively from three-point amplitudes and three-point form factors. Since this method has already been described in [16], we will not repeat it here.

BCFW recursion relation of amplitude
Instead of computing form factor directly, we can first compute the corresponding (n + 2)point amplitude This amplitude can be computed via BCFW recursion relation. If we choose one shifted momentum to be gluon, A n;2 (z) will be vanishing when z → ∞, i.e., there is no boundary contribution. So we can take g + |φ]-shifting in the computation. The four-point amplitude is trivially , we can take g + 3 |φ 1 -shifting. There is only one contributing term as shown in Figure (2.a), which is given by Similarly, for general amplitude A n;2 , we can take g + i+1 |φ i ]shifting 7 . If j = (i + 2), we need to consider two contributing terms as shown in Figure  (2.b) and (2.c), while if j = (i + 2), we need to consider two contributing terms as shown in Figure (2 . 6 Note that we have introduced an over-all minus sign in the expression (3.8), so that the Sudakov form factor is defined to be FO 2 ,2(φ1, φ2; q) = 1. 7 Because of cyclic invariance, we can always do this.
are the contributing diagrams for general A n;2 when j = i + 2 while (b)(d) are the contributing diagrams for A n;2 when j = i + 2.

Assuming that
is true for A n−1;2 , then (3.14) Similar computation shows that for j = i + 2 case, (3.12) is also true for all n. Thus we have proven the result (3.12) by BCFW recursion relation of amplitude.
As discussed, φ n+1 |φ n+2 ]-shifting generates the boundary operator O 2 , and the corresponding boundary contribution is identical to the form factor of operator O 2 . Here, A n;2 does not depend on momenta p n+1 , p n+2 , thus and correspondingly which agrees with the result given by BCFW recursion relation of form factor.

Recursion relation of boundary contribution
We can also compute the boundary contribution directly by BCFW recursion relation without knowing the explicit expression of amplitude, as shown in paper [54]. The boundary contribution of four and five-point amplitudes can be computed directly by Feynman diagrams. For four-point case, there is only one diagram, i.e., four-scalar vertex, as shown in Figure (3.a), and B φ 3 |φ 4 ] 2;2 (φ 1 , φ 2 ; φ 3 , φ 4 ) = 1. For five-point case, under φ 4 |φ 5 ]-shifting, only those Feynman diagrams whose p 4 , p 5 are attached to the same four-scalar vertex contribute to the boundary contribution. There are in total two diagrams as shown in Figure  (3 where the polarization vector ± µ (p) is defined to be with r an arbitrary reference spinor. From these lower-point results, it is not hard to guess that B φ n+1 |φ n+2 ] n;2 can be fully determined by the pole terms under i 1 |φ n+2 ]-shifting. Otherwise we should take a third momentum shifting and so on, until we have detected the complete boundary contribution.
Fortunately, if p i 1 is the momentum of gluon, a second shifting g + i 1 |φ n+2 ] is sufficient to detect all the contributions [54]. For a general boundary contribution B φ n+1 |φ n+2 ] n;2 , we can take g + 1 |φ n+2 ]-shifting. It splits the boundary contribution into a sub-amplitude times a lower-point boundary contribution, and only those terms with three-point amplitudes are non-vanishing. Depending on the location of φ i , φ j , the contributing terms are different. Assuming that (3.19) is true for B n−1;2 , if i, j = 2, n, we have , ) .
All of them lead to the result (3.19), which ends the proof. Again, with the result of boundary contribution, we can work out the corresponding form factor directly. We have shown that the BCFW recursion relation of form factor, amplitude and boundary contribution lead to the same conclusion. This is not limited to MHV case, since the connection between form factor and boundary contribution of amplitude is universal and does not depend on the external states. In fact, for any form factor with n-particle onshell states |s , we can instead compute the corresponding amplitude A n;2 (s; φ n+1 , φ n+2 ) defined by Lagrangian L O 2 , and extract the boundary contribution under φ n+1 |φ n+2 ]shifting. There is no difference between this boundary contribution and form factor of O 2 . For example, in [17], the authors showed that the split-helicity form factor shares a similar "zigzag diagram" construction as the split-helicity amplitude given in [60]. It is now easy to understand this, since the form factor is equivalent to the boundary contribution of the amplitude, and it naturally inherits the "zigzag" construction with minor modification.
The tree amplitude A n;2 (1, . . . , n; n+1, n+2) associated with the double trace structure is cyclically invariant inside legs {1, 2, . . . , n} and {n + 1, n + 2}, so no surprisingly, the color-ordered form factor is also cyclically invariant on its n legs. Since the trace structure Tr(t n+1 t n+2 ) is completely isolated from the other color structure, while the later one is constructed only from structure constant f abc . Thus for amplitudes A n;2 , we also have Kleiss-Kuijf(KK) relation [61] among permutation of legs {1, 2, . . . , n} as where n β is the length of set β, β T is the reverse of set β, and OP is the ordered permutation, containing all the possible permutations between two sets while keeping each set ordered. This relation can be similarly extended to form factors. Especially for operator O 2 , we can relate all form factors to those with two adjacent scalars, (3.24)

Form factor of operator
Let us further consider a more general operator O k ≡ Tr(φ M 1 φ M 2 · · · φ M k ) and the form factor F O k ,n (s; q) = s|O k (0)|0 . In order to generate the operator O k under certain BCFW shifting, we need to add an additional Lagrangian term to construct a new Lagrangian L O k = L SYM + ∆L. Then the boundary contribution of corresponding amplitude A n;2 (s; φ n+1 , φ n+2 ) under φ n+1 |φ n+2 ]-shifting is identical to the form factor F O k ,n (s; q). To see that the boundary operator O φ Ia |φ J b ] is indeed the operator O k , we can firstly compute the variation of Lagrangian L O k from left with respect to φ Ia , and then the variation of δL O k δφ Ia from right with respect to φ J b , which we shall denote as ← − δ δφ J b to avoid ambiguities. The variation of L SYM part is given in (3.3), while for ∆L part, we have and The first term contains O(N ) order result, with a single trace proportional to Tr(φ k ), while the second term is O( 1 N ) order, and the third term is also O( 1 N ) order with even triple trace structure. Thus at the leading N order, the boundary operator of L O k is Similar to the O 2 case, the traceless part of (3.27) is proportional to the operator O k . The ∆L term introduces a (k + 2)-scalar vertex, besides this it has no difference to O 2 case. We can compute the amplitude A n;2 (s; φ n+1 , φ n+2 ), take φ n+1 |φ n+2 ]-shifting and extract the boundary contribution. Then transforming it to form factor is almost trivial. For instance, A k;2 (φ 1 , . . . , φ k ; φ k+1 , φ k+2 ) = 1, thus F O k ,k (φ 1 , . . . , φ k ; q) = 1. It is also easy to conclude that, since the Feynman diagrams of amplitude A n;2 (φ 1 , · · · , φ k , g + k+1 , . . . , g + n ; φ n+1 , φ n+2 ) defined by L O k have one-to-one mapping to the Feynman diagrams of amplitude A n−(k−2);2 (φ 1 , φ k , g + k+1 , . . . , g + n ; φ n+1 , φ n+2 ) defined by L O 2 by just replacing the (k + 2)-scalar vertex with four-scalar vertex, we have Thus we get F O k ;n (φ 1 , . . . , φ k , g + k+1 , . . . , g + n ; q) = − 1 k k, k + 1 k + 1, k + 2 · · · n 1 . (3.29)

Form factor of composite operators
Now we move to the computation of form factors for the composite operators introduced in §2.1. For convenience we will use complex scalars φ AB ,φ AB instead of real scalars φ I in this section. We will explain the construction of Lagrangian which generates the corresponding operators, and compute the MHV form factors through amplitudes of double trace structure.

The spin-0 operators
There are three operators with their complex conjugate partnersŌ III . For these operators, in order to construct Lorentz invariant double trace Lagrangian terms ∆L, we need to product them with another spin-0 trace term. Since shifting a gluon is always more complicated than shifting a fermion, and shifting a fermion is more complicated than shifting a scalar, we would like to choose the spin-0 trace term as trace of two scalars, as already shown in operator O 2 case.

(4.4)
It is easy to verify above result by BCFW recursion relation of amplitude, for example, by taking g + 1 |ψ i ]-shifting. Similar to the O 2 case, only those terms with three-point subamplitudes can have non-vanishing contributions, and after substituting the explicit results for A 3 and A n−1;2 , we arrive at the result (4.4). The boundary contribution of amplitude (4.4) under φ n+1 |φ n+2 ]-shifting keeps the same as A n;2 itself, thus consequently we get the form factor It is also interesting to consider another special n-point external states, i.e., two fermions with (n − 2) gluons of negative helicities. For five-point amplitude A 3;2 (ψ 1 ,ψ 2 , g − 3 ;φ 4 ,φ 5 ), the contributing Feynman diagrams can be obtained by replacing g + 3 as g − 3 in amplitude A 3;2 (ψ 1 ,ψ 2 , g + 3 ;φ 4 ,φ 5 ), so we have More generally, we have This result can be proven recursively by BCFW recursion relation. Assuming eqn. (4.7) is valid for A n−1;2 , then taking φ n+2 |g n ]-shifting, we get two contributing terms 8 for A n;2 . The first term is while the second term is Summing above two contributions, we get the desired eqn. (4.7). Note that q = −p n+1 − p n+2 shows up in result (4.7), which is the momentum carried by the operator in form factor. The φ n+1 |φ n+2 ]-shifting assures that p n+1 + p n+2 = p n+1 + p n+2 , thus we get the form factor . (4.10) III , we can also construct the Lagrangian as and rewrite the Lagrangian as The off-shell Feynman rules for the four-point vertices defined by the corresponding terms inside Tr(φφ) Tr(F + F + ) or Tr(φφ) Tr(F − F − ) of ∆L are given by where p i 1 , p i 2 are the momenta of two gluons. In fact, M + µν can only attach gluons with positive helicities while M − µν can only attach gluons with negative helicities, since And the four-point amplitudes defined by these vertices are given by In order to compute the five-point amplitude A 3;2 (g − 1 , g − 2 , g + 3 ;φ 4 ,φ 5 ), we also need the Feynman rule for five-point vertex defined by the corresponding terms inside Tr(φφ) Tr(F + F + ) or Tr(φφ) Tr(F − F − ), which is given by V abc µνρ (4.14) There are in total three contributing Feynman diagrams, as shown in Figure (4). We need to sum up all of three results. The first diagram gives III . All external particles are out-going.
The third diagram (4.c) is defined by the five-point vertex (4.14), while the result of first three terms in the bracket of (4.14) is (4.17) Using the last term in the bracket of (4.14) can be computed as (4.18) Summing above contributions, we get More generally, we have which can be trivially proven by BCFW recursion relation. This expression is exactly the same as the pure-gluon n-point MHV amplitude of Yang-Mills theory. By taking φ n+1 |φ n+2 ]-shifting, we can get the form factor as Again, let us consider another configuration of external states, i.e., n gluons with negative helicities and two scalars. Computation of A 3;2 (g − 1 , g − 2 , g − 3 ;φ 4 ,φ 5 ) is almost the same as A 3;2 (g − 1 , g − 2 , g + 3 ;φ 4 ,φ 5 ), and we only need to replace + 3 by − 3 . Direct computation shows that, contributions of all three diagrams lead to This result can be generalized to A n;2 as and can be proven recursively by BCFW recursion relation. In fact, assuming eqn. (4.23) is true for A n−1;2 and taking g − n |g − 1 ]-shifting, there is only one non-vanishing term in BCFW expansion, which gives So the corresponding form factor is In order to generate operator O , we should shiftφ n+1 ,ψ n+2 . However, there are two ways of shifting, and their large z behaviors are different. If we consider φ n+1 |ψ n+2 ]shifting, the leading term in z is O(z 0 ), and the boundary operator after considering the LSZ reduction is hence it has a λ n+2,α factor difference with O  . However, it is better to take the shifting where the leading z term has lower rank, preferably O(z 0 ) order, since the computation would be simpler.

II
. All external particles are out-going.
II . All external particles are out-going.
For the operator O  It introduces new four-point verticesψ-g + -ψ-g + and ψ-g − -ψ-g − , as well as five, six-point vertices.

Summary and discussion
The boundary operator is initially introduced as a formal technique to study the boundary contribution of amplitude when doing BCFW recursion relation in paper [56]. It defines a form factor, and practically this off-shell quantity is difficult to compute. In this paper, we take the reversed way to study the form factor from boundary contribution of amplitude of certain theory. We show that by suitable construction of Lagrangian, it is possible to generate boundary operators which are identical(or proportional) to the given operators of interest. This means that the form factor of given operator can be extracted from the boundary contribution of corresponding amplitude defined by that Lagrangian. We demonstrate this procedure for a class of composite operators by computing amplitudes of double trace structure and reading out the form factors from corresponding boundary contribution. Thus the computation of form factor becomes a problem of computing the scattering amplitude.
We have considered a class of composite operators, which are traces of product of two component fields from N = 4 SYM, and the sum of spins of those two fields is no larger than two. In fact, the construction of Lagrangian has no difference for other operators with length(the number of fields inside the trace) larger than two, provided the sum of their spins is no larger than two. This is because we can always product them with a length-two trace term to make a Lorentz-invariant Lagrangian term, and deform the two fields in the extra trace term to produce the required boundary operators. However, if the operator has spin larger than two, in order to make a Lorentz-invariant Lagrangian term, the length of extra trace term should be larger than two. Then deformation of two fields in the extra trace term is not sufficient to produce the desired boundary operators, and we need multi-step deformation. It would be interesting to investigate how this multi-step deformation works out. It would also be interesting to find out how to apply this story to other kind of operators such as stress-tensor multiplet or amplitude with off-shell currents.
Note that all the discussions considered in this paper are at tree-level. While it is argued [56] that the boundary operator is generalizable to loop-level since the OPE can be defined therein, it is interesting to see if similar connection between form factor and amplitude also exists at loop-level or not. For this purpose, it would be better to study the loop corrections to the boundary operators, which is under investigation.
With these notations, the quadratic term in the Lagrangian is Following the standard procedure of computing generating functions, one can get is a function of Φ, and in general can be decomposed into a free part D 0 and an interaction part V as D α β (Φ) = (D 0 ) α β + V α β (Φ). The Z Λ (Φ) can be dropped at treelevel. After some evaluation including LSZ reduction for fields H(p 1 + zq), H(p n − zq), the remaining part yields Then we can read out the z-dependence from above result.

B Discussion on the large z behavior
Let us start from the Lagrangian of N = 4 SYM in component fields, where T IAB is the transformation matrix between SO(6) and SU (4) representations of scalar fields φ AB = 1 √ 2 φ I T IAB . The gauge fixing term is In order to get the quadratic terms of shifted hard fields, we need to compute the second order variation of L. Since where we can write D −1 we have (B.9) It is very crucial to have d 0 X = Xd 0 = 0, then the expansion Now let us first consider g − 1 |g + n ]-shifting, and determine the leading order of Z g − 1 |g + n ] (z). The helicity vectors of g − 1 , g + n both introduce a factor of z, while zX(1 + d 1 X) −1 introduce another factor of z. Notice that both d 1 and X are block-diagonal, which means fermion operators will not appear. It implies that, in this order, the Z g − 1 |g + n ] (z) of N = 4 SYM is the same as its bosonic sub-theory, which is a 4-dimensional reduction of 10-dimensional Yang-Mills theory. According to [56], Z g −a 1 |g +b n ] (z) = −2iz 3 gf abc (p 1 · p n )q µ A µc + O(z 2 ) . (B.11) The leading order is z 3 . If the color indices of two shifted fields are contracted, then the first term vanishes due to f abc = 0 when a = b, and the leading order becomes z 2 , while we know in §4.5 that the leading order of double trace term under such shifting is z 4 . In paper [62], it was proved that the large z behavior of Φ U 1 a |Φ U 2 b -shifting is We would like to refine their result as where L c is an arbitrary operator, and there is always a f abc associated with the leading order term. We already proved (B.13) for g −a |g +b ]-shifting. Since all states in N = 4 SYM are related by SUSY, (B.13) also holds for any shifting, and the proof will be complete parallel to §7.1 of [62]. This means that after contracting the indices a, b, the first term vanishes, and the large z behaves even better than expected.