Transmuting off-shell CHY integrals in the double-cover framework

In this paper, by defining off-shell amplitudes as off-shell CHY integrals, and redefining the longitudinal operator, we demonstrate that the differential operators which link on-shell amplitudes for a variety of theories together, also link off-shell amplitudes in the similar manner. Based on the algebraic property of the differential operator, we also generalize three relations among color-ordered on-shell amplitudes, including the color-ordered reversed relation, the photon decoupling relation, the Kleiss-Kuijf relation, to off-shell ones. The off-shell CHY integrals are chosen to be in the double-cover framework, thus, as a by product, our result also provides a verification for the double-cover construction.


I. INTRODUCTION
The Cachazo-He-Yuan (CHY) formalism reveal the marvelous unity among on-shell treelevel amplitudes of various theories [1][2][3][4][5]. In the CHY formulae, different theories correspond to different CHY integrands. Through the so-called dimensional reduction, squeezing, as well as the generalized dimensional reduction procedures, on-shell CHY integrands for a wide range of theories can be generated from the on-shell CHY integrand for the gravity (GR) theory 1 [5].
Similar unified web for on-shell tree-level amplitudes of different theories can be achieved via the differential operators proposed by Cheung, Shen and Wen. These differential operators, which act on Lorentz invariant kinematic variables, transmute the on-shell amplitude of one theory to that of another theory [6]. By applying these operators, amplitudes for a variety of theories, can be generated from the GR amplitude. The similarity between two unified webs implies the underlying relationship between two methods. This relationship has been spelled out in [7,8], by acting differential operators on CHY integrals for different theories.
Based on the relations among on-shell amplitudes expressed by differential operators, several other relations for on-shell amplitudes can be derived. First, using the algebraic property of the differential operator, one can derive the generalized color-ordered reversed relation, the generalized photon decoupling relation, as well as the generalized Kleiss-Kuijf (KK) relation, which are valid for all color-ordered on-shell amplitudes [9]. Secondly, a dual version of the unified web, which reflects the relations among amplitudes by expanding the amplitude of one theory to amplitudes of another theory, can be established via the differential operators [9][10][11]. Notice that the expansions of amplitudes have been studied from various angles in the literature [12][13][14][15][16][17][18][19][20][21]. Among these angles, the approach used in [9][10][11] manifests the duality between the differential operators and the coefficients of basis in the expansions.
In this paper, we demonstrate that the differential operators transmute off-shell amplitudes in the manner similar as that for on-shell amplitudes. The off-shell tree-level amplitudes in this paper are defined by off-shell CHY integrals. There are three motivations for considering off-shell amplitudes in the CHY framework. First, the massive external momentum with k 2 i = 0 can be treated as a special case of the off-shell massless momentum, thus, considering off-shell CHY integrals is an effective way to generalize the CHY formula to massive external states. Secondly, in the double-cover prescription proposed by Gomez, the evaluation of tree-level amplitudes in the CHY framework can be reduced to lower-point off-shell amplitudes, due to the factorizations realized by the double-cover method [22][23][24][25][26][27]. This is another important application of off-shell CHY integrals. Thirdly, from the theoretical point of view, it is interesting to broaden the CHY formalism to reproduce off-shell tree-level amplitudes, and for loops to be computed, without a Lagrangian.
The main method used in this paper is similar as that used in [7,8], and can be summarized as follows. The off-shell tree-level amplitude in the CHY formula arise from the integral over auxiliary coordinates as A n = dµ n I CHY . (1) The measure part dµ n is universal, while the CHY integrand I CHY depends on the theory under consideration. All the operators which will be considered in this paper act on Lorentz invariants i · j or i · k j , where k i and i are the momentum and the polarization vector of the i th leg, respectively. Thus, these operators will not affect the measure part which is independent of polarization vectors. In other words, differential operators are commutable with the integral over auxiliary variables. Thus, transmuting an amplitude is equivalent to transmuting the corresponding CHY integrand. More explicitly, suppose O is a differential operator, it satisfies Thus, if two amplitudes A α and A β are related by an operator O as A α = OA β , analogous relation I CHY for two integrands must hold, and vice versa. Consequently, one can derive the unifying relations systematically by applying differential operators to CHY integrands.
Using the above method, we will show that the unifying relations provided in [6], which link on-shell amplitudes of different theories together, also hold for off-shell amplitudes defined by off-shell CHY integrals. There are three types of basic operators, the trace operator T [i, j], the insertion operator I ikj , as well as the longitudinal operators L i and L ij , will be considered in this paper. The definitions of the operators T [i, j] and I ikj are the same as in [6][7][8], while the definitions of the longitudinal operators L i and L ij will be modified in the off-shell case. By applying the combinatory operators constructed by these three basic operators to the GR CHY integral, one can get the CHY integrals for theories including: Einstein-Yang-Mills (EYM) theory, Yang-Mills (YM) theory, Einstein-Maxwell (EM) theory, Einstein-Maxwell theory with photons carry flavors (EMf), Born-Infeld (BI) theory, Yang-Mills-scalar (YMS) theory, special Yang-Mills-scalar (sYMS) theory, bi-adjoint scalar (BAS) theory, non-linear sigma model (NLSM), φ 4 theory, Dirac-Born-Infeld (DBI) theory, extended Dirac-Born-Infeld (exDBI) theory, special Galileon (SG) theory. Since the unifying relations can be extended to off-shell amplitudes, the relevant relations among colorordered amplitudes, including the generalized color-ordered reversed relation, the generalized photon decoupling relation, and the generalized KK relation, can also be generalized to the off-shell case.
In [7,8], CHY integrals are chosen to be in the original single-cover version. In this paper, we choose CHY integrals in the double-cover formulae [22][23][24][25][26][27]. Since on-shell amplitudes can be regarded as the special case of off-shell ones, if the unifying relations hold for offshell amplitudes, they also hold for on-shell ones. Thus, our result in this paper provides a verification of the double-cover construction, because we have shown that the unifying relations for on-shell amplitudes in the single-cover formulae, which are already proved in [7,8], are also correct for off-shell amplitudes in the double-cover framework.
The remainder of this paper is organized as follows. In §II, we give a brief introduction of the off-shell CHY formalism, and the double-cover prescription, which are necessary for subsequent discussions. In §III, we study the effects of three basic operators when applying them to building blocks of CHY integrands. In §IV we consider the effects of combinatory operators constructed by these basic operators. The relations among amplitudes, based on the previous preparations, are presented in §V. Finally, we end with a brief discussion in §VI.

II. BACKGROUND
For reader's convenience, in this section we rapidly review the off-shell CHY formalism, and the double-cover prescription.

A. Off-shell CHY formalism
The off-shell CHY formalism bears strong similarity with the on-shell one, except the correction elements ∆ ij and η ij in scattering equations and matrix elements. In this subsection, we only introduce the off-shell CHY formalism. The on-shell one can be reproduced by taking ∆ ij → 0, η ij → 0.
In the off-shell CHY formula, the tree-level amplitude for n massless particles arises from a multi-dimensional contour integral over the moduli space of genus zero Riemann surface with n punctures, M 0,n . It can be expressed as which possesses the Möbius SL(2, C) invariance. Here k i , i (or i ) and z i are the momentum, polarization vector, and puncture location for the i th external leg, respectively. The measure is defined as The factor |pqr| is given by |pqr| ≡ z pq z qr z rp , where z ij ≡ z i − z j . The off-shell scattering equations are given as 2 [28][29][30] E i (z) ≡ j∈{1,2,...,n}\{i} where These scattering equations yield correct propagators in the Feynman gauge, and satisfy the condition which protects the SL(2, C) invariance. The (n − 3) independent scattering equations define the map from the space of kinematic variables to M 0,n , and fully localize the integral on their solutions. After fixing the SL(2, C) gauge, the measure part is turned to The integrand in (3) depends on the theory under consideration. For any theory known to have a CHY representation, the corresponding integrand can be factorized into two parts I L and I R , as can be seen in (3). Either of them are weight-2 for each puncture coordinate z i under the SL(2, C) transformation. In Table I, we list integrands for theories which will be considered in this paper 3 [5].
We now explain each building block in turn. There are five kinds of n × n matrices, which are defined through 2 In this paper, we choose 2k i · k j rather than s ij to define the scattering equations. Two choices are un-equivalent for off-shell momenta. 3 For theories contain gauge or flavor groups, we only show the integrands for color-ordered partial amplitudes instead of full ones. Theory Elements η ij are given as [29] η j±1,j = 1 They make C ii to be weight-2 under the SL(2, C) transformation, therefore keep the SL(2, C) invariance of the whole theory. δ I i ,I j is the Kronecker symbol, which forbids the interaction between particles with different flavors I i and I j . When the dimension of a matrix need to be clarified, we often denote the n × n matrix S as [S] n . The 2n × 2n antisymmetric matrix Ψ can be constructed from the matrices A, B and C, in the following form The reduced Pfaffian of Ψ is defined as Pf Ψ = (−) i+j z ij Pf Ψ ij ij , where the notation Ψ ij ij means the i th and j th rows and columns in the matrix Ψ have been removed (with 1 ≤ i < j ≤ n). Analogous notation holds for Pf A.
It is worth to emphasize the definition of Pfaffian, since it is crucial for the work in this paper. For a 2n × 2n antisymmetric matrix S, Pfaffian is defined as where S 2n denotes permutations of 2n elements and sgn(σ) is the signature of σ. More explicitly, let Π be the set of all partitions of {1, 2, · · · , 2n} into pairs without regarding to the order. An element α in Π can be written as with i k < j k and i 1 < i 2 < · · · < i n . Now let π α =   1 2 3 4 · · · 2n − 1 2n be the corresponding permutation of the partition α. If we define then the Pfaffian of the matrix S is given as From the definition (18), one can observe that in every term S α of the Pfaffian, each number in the set {1, 2, · · · , 2n}, which serves as the subscript of the matrix element, will appear once and only once. This simple observation indicates that: each polarization vector i appears once and only once in each term of the reduced Pfaffian Pf Ψ. This conclusion is important for latter discussions.
The definition of Ψ can be generalized to the (2a The definitions of elements in A, B and C are the same as in (9). The reduced Pfaffian Pf [Ψ] a,b:a can be defined in the same manner.
In the second line of (20), the reduced Pfaffian is calculated by removing rows and columns i m and j m . Based on the definition of P {i,j} (n, l, m) in the second line of (20), the polynomial where the sum is over all possible choices of pairs in each trace-subset Tr k . In the on-shell case, the polynomial {i,j} P {i,j} (n, l, m) defined above equals to the reduced Pfaffian of the matrix Π, which is constructed from Ψ via the squeezing procedure [5]. In this paper, we will not use the matrix Π. The advantage of this choice is, the polynomial {i,j} P {i,j} (n, l, m) defined in (22) is manifestly weight-2 under the SL(2, C) transformation, while the reduced Pfaffian of Π does not have manifest weight. Finally, the Parke-Taylor factor for the ordering σ is given as It indicates the color-ordering (σ 1 , σ 2 , · · · , σ n−1 , σ n ) among n external legs. With ingredients introduced above, off-shell CHY integrands for various theories can be defined as in Table I. Some remarks are in order. First, the SL(2, C) symmetry plays the central role in the construction of off-shell CHY integrals. From on-shell integrals to off-shell ones, all corrections ∆ ij and η ij are required by the SL(2, C) invariance. It is straightforward to see that both I L and I R defined in Table I are weight-2 under the SL(2, C) transformation, guarantee the SL(2, C) invariance of the whole integral. Secondly, all the reduced Pfaffians appear in off-shell CHY integrands in Table I are independent of the removed rows and columns, similar as those in on-shell integrands. One can follow the method used in [2], and use as well as scattering equations, to prove this. Thirdly, the gauge invariance no longer exist in general. For example, under the replacement i → k i , we do not have Pf Ψ = 0 anymore, unless we take all ∆ ij and η ij to be 0. But it is quite natural that the off-shell or massive external states violate the gauge invariance.

B. Double-cover prescription
In next sections, all CHY integrals under consideration are in the double-cover formalism. In this subsection, we introduce how to reformulate the single-cover CHY integral defined in (3) into the double-cover one.
The double-cover prescription of CHY construction is given as a contour integral on n-punctured double-covered Rieman spheres [22][23][24][25][26][27]. Restricted to the curves 0 = C i ≡ y 2 i −σ 2 i +Λ 2 for i ∈ {1, 2, · · · , n}, the pairs (y 1 , σ 1 ), (y 2 , σ 2 ), · · · , (y n , σ n ) serve as coordinates. Then, all 1 z ij in the single-covered formula (3) are replaced by Especially, the scattering equations are turned to Amplitudes in such framework are expressed as the contour integral where the measure dµ Λ n is defined through with ∆ pqr ≡ (τ pq τ qr τ rp ) −1 . Correspondingly, the contour is determined by poles Λ = 0, C i = 0, as well as E τ j = 0 for j = p, q, r, m. Eliminating the GL(2, C) gauge redundancy turns the measure to be where Then we consider Table I via the replacement 1 z ij → τ ij . To obtain the double-cover form of the integrand which is more convenient for our consideration, we rewrite τ ij as on the support C i = C j = 0. The advantage of this reformulation is that T ij carries algebraic properties similar to 1 z ij , such as antisymmetry, and This similarity allows us to use a lot of technics for single-cover CHY integrals. Under the replacement (31), we have 5 For example, the Parke-Taylor factor becomes Thus, start from an integrand in the single-cover version, one can first replace all 1 z ij by T ij , then times the resulting formula by the factor Let us take Pf Ψ as the example, the reduced Pfaffian with new coordinates y i and σ i is given by where the matrix Ψ Λ is obtained from Ψ via the replacement 1 z ij → T ij . 5 Maybe the notations I T L (T ij , k i , i ) and I T R (T ij , k i , i ) are more suitable. However, when encounter matrices, notations such as Ψ T , A T will cause some ambiguity, since we always use T to denote the transpose of the matrix. Thus we choose the superscript to be Λ rather than T .

III. EFFECTS OF BASIC OPERATORS
As discussed in §I, we are interested in acting differential operators on off-shell CHY integrals in the double-cover version. To achieve the goal, it is sufficient to apply differential operators to I Λ L (T ij , k i , i ) and I Λ R (T ij , k i , i ), since the operators under consideration will not affect both the measure and the factor (yσ) i y i . In this section, we will consider the effects of applying three basic differential operators to the elementary building-blocks of :a , as well as {i,j} P Λ {i,j} (n, l, m). Among the operators which will be discussed in this section, the trace and insertion operators are the same as those defined in [6], while the definition of longitudinal operators will be modified.

A. Trace operator
The trace operator T [i, j] is defined as [6] T If one applies T [i, j] to the reduced Pfaffian Pf Ψ Λ , only terms containing factor ( i · j ) (i.e.,element Ψ Λ i+n,j+n ) provide non-vanishing contributions. Since i and j appear once and only once in each term of the reduced Pfaffian, performing the operator T [i, j] is equivalent to the replacement where V denotes vectors k l or l =i,j . This manipulation is equivalent to the dimensional reduction procedure in [5]. Then, we arrive at a new matrix Ψ Λ , satisfies Without lose of generality, we assume {i, j} = {n − 1, n} 6 , then the new matrix Ψ Λ is given by Thus, we find Applying the similar procedure to the matrix [Ψ] Λ n−2,2:n−2 gives By repeating the manipulation, we find the multiple action of trace operators gives the following recursive pattern where the polynomial P Λ {i,j} (n, l, m) is obtained from P {i,j} (n, l, m) in (20) via the replacement 1 z ij → T ij , and we have arranged elements as We want to point out that the factor (T ij T ji ) appear in (43) is the simplest Parke-Taylor factor C Λ 2 defined in (34), indicates the simplest color-ordering (i, j). More general colororderings can be generated from it by inserting other elements.

B. Insertion operator
The insertion operator is defined by [6] In this subsection, we discuss the effect of acting this operator on the polynomial {i,j} P Λ {i,j} (n, l, m). As will be seen immediately, the most important effect is replacing T ij in the Parke-Taylor factor by T ik T kj . In other words, this operator transmutes the colorordering (· · · , i, j, · · · ) to (· · · , i, k, j, · · · ). To show this, one need to assume i th and j th legs belong to the same trace subset, i.e., i, j ∈ Tr k . For simplicity, we assume i, j ∈ Tr m , and take the expansion (20) where rows and columns i m , j m ∈ Tr m have been removed (with replacing 1 z ij by T ij ). Since in the off-shell case the reduced Pfaffian is also independent of the choice of removed rows and columns, as pointed out in the previous section, assuming i and j belong to any other Tr k will not change the conclusion, although the calculation will be more complicate..

C. Longitudinal operator
For on-shell amplitudes, the longitudinal operators are defined via [6] and For the off-shell case, they should be modified to and .
The reason for the above modifications will be seen in the next section. We now discuss the effects of acting them on the reduced Pfaffian Pf [Ψ] Λ a,b:a . We first consider the operator L ij . It turns ( i · j ) to (k i · k j + ∆ ij ), and annihilates all other ( i · V ), ( j · V ). Using the observation that i and j appear once and only once respectively, one can conclude that L ij transmutes the reduced Pfaffian of the matrix [Ψ] Λ a,b:a as follows Then, we consider the operator L i . It is straightforward to see thus the operator L i transmutes every (k j · i +η ji ) to (k j ·k i +∆ ij ). Under such replacement, the diagonal elements of the matrix C Λ are transmuted to and vanishes due to scattering equations Thus, the operator L i has the following effect The results (55) and (59) are crucial for generating the ingredient (Pf A) 2 .

IV. EFFECTS OF COMBINATORY OPERATORS
The combinatory operators are constructed by three types of basic operators. In this section, by using the results obtained in the previous section, we will consider three kinds of combinatory operators, especially their action on the reduced Pfaffian Pf Ψ Λ , which is the fundamental building-block for the GR integrand.
To verify this expectation, let us compute the result of applying the operator T [α] to Pf Ψ Λ . At the first step, performing T [α 1 , α m ] gives where the result (41) have been used. Then one can act I α 1 α 2 αm on the resulting object, and use (49) to get 7 We adopt the convention in [6] that the product of two operators O 1 · O 2 acts on an amplitude as

Similar manipulation gives
and the recursive pattern can be observed. Repeating the above procedure, one will arrive at Especially, when m = n, we have The procedure of generating the color-ordering (α 1 , · · · , α m ) can be understood as follows. At the first step, two reference points α 1 and α m are created by the operator T [α 1 , α m ]. Then, other legs in the set α are inserted between α 1 and α m , via insertion operators. This interpretation allows the trace operator T [α] to have a variety of equivalent expressions, since different legs can be inserted by a variety of equivalent ways. As an example, let us consider the color-ordering (1,2,3,4,5). To generate it, the firs step can be creating two reference points 1 and 5 via the trace operator T [1,5]. Then, one can insert other legs between 1 and 5 by the following order: • inserts 2 between 1 and 5, • inserts 3 between 2 and 5, • inserts 4 between 3 and 5.
This order yields the operator coincide with the initial definition (60). However, one can also choose other equivalent orders, for instance: • inserts 3 between 1 and 5, • inserts 2 between 1 and 3, • inserts 4 between 3 and 5, and arrive at the operator Furthermore, based on the cyclic symmetry of the color-ordering, one can chose arbitrary two points α a and α b as reference points, then insert α a+1 , · · · , α b−1 between α a and α b , and insert α b+1 , · · · , α a−1 between α b and α a . For example, one can also choose the trace operator to be The previous result in the current subsection can be generalized to multi-trace cases T [α 1 ] · T [α 2 ] · · · , with the constraint [α i ] ∩ [α j ] = ∅ for arbitrary i and j. Let us consider, for example, From the first line to the second line, the commutativity of the operators has been used. First, we apply the operators T [α 1 , α m ] and T [β 1 , β l ] to Pf Ψ Λ . Using (43), we obtain where {i,j} Thirdly, we use (49) again to obtain Combining (70), (72) and (73) together, we get The most general formula is given by where |α i | denotes the length of the set α i . It can be obtained recursively, by applying the extremely similar technic.

B. Operators T [a, b] · L and T [a, b] · L
The operator L is defined through longitudinal operators as [6] L ≡ Here the set of pairs {(i 1 , j 1 ), (i 2 , j 2 ), · · · , (i m , j m )} is a partition of I with conditions i 1 < i 2 < ... < i m and i t < j t , ∀t. At the algebraic level, two operators L and L are not equivalent to each other. However, for on-shell integrands, if we apply the combinatory operators T [a, b] · L and T [a, b] · L to Pf Ψ Λ with even number of external legs, and let subscripts of L i and L ij run through all nodes in {1, 2, · · · , n} \ {a, b}, two operators have the same effect [6][7][8], providing up to an overall sign. We now show that the relation (77) also holds for off-shell integrands. We first consider the effect of the manipulation T [a, b] · L Pf Ψ Λ . Acting T [a, b] on Pf Ψ Λ gives (41), which is the reduced Pfaffian of the matrix (39). Using the previous result (55), it is straightforward to see L transmutes the matrix (39) to The Pfaffian of the matrix Then, we consider the effect of acting L on T [a, b] Pf Ψ Λ . Using (59) we know the operator L transmutes the matrix (39) to thus the reduced Pfaffian Pf Ψ Λ is turned to where For simplicity, we choose a th and b th rows and columns to be removed when evaluating the reduced Pfaffian of A Λ . Then we need to compute Using the definition of Pfaffian (18), one can find that the non-vanishing contributions for (84) come from rows i ∈ {1, · · · , n − 2} and columns j ∈ {n − 1, · · · , 2n − 4}, which give rise to the determinate of the matrix (A Λ ) ab ab . Thus the reduced Pfaffian of A Λ can be obtained as where we have used (−) (n−2)(n−3) 2 = (−) n−2 2 , due to the fact n is even. Putting it back to (82), we obtain Above calculations show that It is worth to notice that this result is independent of the choice of a and b.
Although the relation (77) for on-shell amplitudes also holds for off-shell ones, we must emphasize that the definitions of operators L and L are different for the on-shell and off-shell cases, since for the off-shell case, corrections ∆ ij are introduced when defining longitudinal operators L i and L ij . We have shown that, with the redefined L i and L ij , one can transmute Pf Ψ Λ to (Pf A Λ ) 2 , which serves as a building block for BI, DBI, exDBI, NLSM and SG integrands, therefore have meaningful interpretation. If we insist the original definition, the resulting final object can not be interpreted physically.

C. Operators T X 2m and T X 2m
In this subsection, we consider the combinatory operators T X 2m and T X 2m , which generate Pf [X] Λ 2m and Pf [X ] Λ 2m from Pf Ψ Λ , respectively. For a given length-2m set I, the operator T X 2m is defined as The notation ρ∈pair i k ,j k ∈ρ is explained after (76). Using the result in (43), and the definition of Pfaffian (18), one can find that the operator T X 2m transmutes Ψ Λ to a new matrix so that therefore provides the building block Pf [X] Λ 2m . The operator T X 2m is defined in a similar form as where δ I i k ,I j k forbids the interaction between particles with different flavors. These δ I i k ,I j k turn the matrix [X] Λ 2m to [X ] Λ 2m . Thus, we have which gives the building block Pf [X ] Λ 2m .

V. RELATIONS AMONG AMPLITUDES
With preparations in previous sections, we are ready to exhibit relations among different off-shell amplitudes. We will re-establish the unified web in [6] for off-shell amplitudes. Then, we will claim that three important relations among color-ordered on-shell amplitudes, including the color-ordered reversed relation, the photon decoupling relation, as well as the KK relation, can be generalized to the off-shell case.

A. Transmuting amplitudes by differential operators
In this subsection, we will show the combinatory differential operators discussed in §IV transmute the GR amplitude to amplitudes of a variety of other theories. It is quite natural to take the GR amplitude whose external states carry highest spins as the starting point, since all operators decrease the spins of external legs. The GR integrand in the single-cover formula is shown in the first line of Table I. To achieve the double-cover expression, we can get I Λ L and I Λ R from I L and I R , via the replacement 1 z ij → T ij , then time both of them by the factor . Thus, we have We will consider the effect of applying differential operators to the above GR CHY integral. As introduced in §I, the key point is, differential operators are commutable with the integral over auxiliary coordinates, thus transmuting an amplitude is equivalent to transmuting the corresponding CHY integrand. Further more, since differential operators do not affect the factor (yσ) i y i , transmuting I τ L or I τ R is equivalent to transmuting I Λ L or I Λ R . In summary, we have In the GR integrand, two parts I Λ L and I Λ R depend on two independent sets of polarization vectors { i } and { i }, respectively. It means we can define two independent sets of differential operators, through two sets of polarization vectors. The operators defined via { i } only act on I Λ L , while the operators defined via { i } only act on I Λ R . This property protects the manifest double copy structure of CHY integrands. Without lose of generality, we can restrict the effect of operators on the I Λ L part, by defining them via { i }.
Performing operators on the I Λ L part and using (43), (65), (75), (87), (90), as well as (92), after comparing with the middle column of Table I, we get following relations: up to an overall sign. Similarly, applying operators to the pure YM integrand, we obtain relations: up to an overall sign. Notice that the amplitude of φ 4 theory is generated via a special T X 2m with 2m = n. Applying operators to the BI integrand, we get relations: up to an overall sign. Relations presented in (95), (96) and (97) can be organized as we get Differential operators connect not only amplitudes of different theories, but also amplitudes of the same theory. For example, using the relation (49) one can get The notation ({i h 1 , · · · , i h m }||j g 1 , · · · , j g n ) means there is no color-ordering among elements in the set {i h 1 , · · · , i h m }, while elements (j g 1 , · · · , j g n ) are color-ordered. The insertion operators turn gravitons i h 1 and i h 2 to gluons, and insert them between gluons j g 1 and j g 2 in the color ordering (j g 1 , · · · , j g n ). Consequently, insertion operators T i 1 i 2 j 2 , T j 1 i 1 j 2 transmutes the EYM amplitude A , EYM ({i h 1 , · · · , i h m }||j g 1 , · · · , j g n ) to the EYM amplitude A , EYM ({i h 3 , · · · , i h m }||j g 1 , i g 1 , i g 2 , j g 2 , · · · , j g n ).

B. Three generalized relations for color-ordered amplitudes
In this subsection, we demonstrate that three generalized relations for on-shell colorordered amplitudes, also hold for off-shell amplitudes. From Table II, one can see that each color-ordered amplitude can be generated by applying the general trace operator T [α], formally expressed as Thus, we will derived three relations by using the algebraic property of the general trace operator.
This algebraic relation indicates that The shuffle of two ordered sets α ¡ β is the permutation of the set α ∪ β while preserving the ordering of α and β.
where α and β are two ordered sets, and β T is obtained from β by reversing the ordering of elements. To derive it, let us reformulate the trace operator T [a 1 , · · · , a n ] as T [a 1 , · · · , a n ] = T [a 1 , a n ] · The operator T [a 1 , a n ] · k i=1 I a i−1 a i an generates the color-ordering (a 1 , a 2 , · · · , a k , a n ), which is equivalent to (a n , a 1 , · · · , a k ), due to the cyclic symmetry. The operator (−) n−k−1 n−1 j=k+1 I ana j a j−1 can be interpreted as inserting {a n−1 , a n−2 , · · · , a k+1 } between a n and a k . More explicitly, using the definition of the insertion operator we know that Repeating this decomposition, we find that the operator I ana k+1 a k gives rise to ¡ A(a n , {a 1 , · · · , a k−1 } ¡ a k+1 , a k ) .

VI. DISCUSSION
With modifying the definition of the longitudinal operators, we have generalized various relations for on-shell amplitudes to off-shell ones. The CHY formulism serve as a powerful tool for this work. As explained in §I, our result also provides a verification for the doublecover construction.
At the end of the paper, we want to point out three things. First, the expansions of on-shell amplitudes, which can be derived via relations in Table  II, can not be generalized to the off-shell case, since in the derivation proposed in [9][10][11], the gauge invariance is a necessary tool. As mentioned in §II, for the off-shell case, the gauge invariance is lost. For some special case, for example the GR amplitude including two massive external legs with the same mass (a special case of the off-shell external massless states), one can avoid the appearing of corrections ∆ ij and η ij by choosing the removed rows and columns in the reduced matrix to be two massive ones, and obtain the reduced matrix totally the same as that for the massless on-shell amplitude. Then one can claim that the expansion of on-shell massless GR amplitude to BAS amplitudes also hold for this massive (or off-shell) GR amplitude, since from the CHY point of view, this expansion is just expanding Pf Ψ Λ to Parke-Taylor factors C Λ n (σ). But this manipulation can not be generalized to general off-shell amplitudes.
Secondly, the differential operators discussed in this paper will not affect any k 2 i . This fact has its physical meaning when treating amplitudes with massive external states as off-shell CHY integrals. Thus, for the massive amplitudes of different theories related by differential operators, the mass for a particular external state is unique, due to the invariance of k 2 i . For example, suppose the leg 1 is a vector particle in the amplitude A 1 , and is a scalar particle in the amplitude A 2 . If A 1 and A 2 are related by a differential operator, then the vector particle and the scalar particle denoted by 1 have the same mass.
Thirdly, in the literature [6], the gauge invariance is one of conditions for constructing differential operators. In the off-shell case, the gauge invariance is lost. However, differential operators transmute off-shell amplitudes in the same manner as transmuting on-shell ones. It implies that maybe it is not suitable to regard the gauge invariance as the fundamental principle for constructing differential operators. The underlying principle which determines differential operators discussed in this paper is an interesting question.