Note on differential operators, CHY integrands, and unifying relations for amplitudes

An elegant unified web for amplitudes of various theories was given by Cachazo, He and Yuan in the CHY framework a few years ago. Recently, similar web has also been constructed by Cheung, Shen and Wen, which relies on a set of differential operators. In this note, by acting these differential operators on CHY-integrands systematically, we have established the relation between these two approaches. Thus, amplitudes for all theories which have CHY representations, include gravity theory, Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory and its extension, bi-adjoint scalar theory, $\phi^4$ theory, non-linear sigma model, as well as special Galileon theory, have been included in the unified web rooted from gravity theory.


Introduction
The unification of different theories is always one of interesting problems in theoretical physics. The modern researches on S-matrix have exhibited amazing structures within amplitudes of gauge and gravity theories, such as the Kawai-Lewellen-Tye (KLT) relations [1], Bern-Carrasco-Johansson (BCJ) color-kinematics duality [2][3][4], which are invisible in the traditional Lagrangian formulism of quantum field theory. These discoveries hint the existence of some long hidden unifying relations for on-shell amplitudes. A strong evidence for the marvelous unity among amplitudes of different theories has been spelled out in [9] by using the CHY formulae [5][6][7][8][9]. More explicitly, different theories are defined by different CHY-integrands, while they found that CHY-integrands for a wide range of theories can be generated from the CHYintegrand for gravity theory 1 , through the so called compactifying, squeezing as well as the generalized dimensional reduction procedures [9].
Recently, Cheung, Shen and Wen discovered similar unifying relations for on-shell tree-level amplitudes of a variety of theories from a different angle: by acting some Lorentz and gauge invariant differential operators, one can transmute the physical amplitude of a theory into the one of another theory [10]. In their unified web, amplitudes of various theories include Einstein-Yang-Mills theory, Einstein-Maxwell theory, Born-Infeld theory, Dirac-Born-Infeld theory, special Galileon theory, non-linear sigma model, as well as bi-adjoint scalar theory, can be generated by transmuting the amplitudes of gravity theory. The role of these differential operators has been understood and checked from various angels, such as several explicit examples, factorization property, double copy structure, soft behavior, etc.
Since the similar unified webs for amplitudes of various theories have been given both in [9] and [10], it is very natural to investigate the relation among these two different approaches. In this note, we will establish the exact relation through the CHY formulae [5][6][7][8][9]. Tree-level amplitudes in the CHY formulae are represented as integrals over auxiliary variables as A n = dµ n I CHY , (1.1) the effects of operators built by these basic operators. The unified web and other relations for amplitudes will be presented in §5. Finally, we end with a summary and discussions in §6.

Review of Pfaffian and CHY formulae
For reader's convenience, we will briefly discuss the definition of Pfaffian, and rapidly review the CHY formulae.

Definition of Pfaffian
The definition of Pfaffian is essential for the work in this note. For a 2n × 2n skew symmetric matrix S, Pfaffian is defined as where S 2n is the permutation group 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 regard 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 be the corresponding permutation of the partition α. If we define then the Pfaffian of the matrix A is given as Both representations (2.1) and (2.5) will be used later. From the (2.5) one can observe that in every term S α of the Pfaffian, each number of {1, 2, · · · , 2n}, as the subscript of the matrix element, will appear once and only once. This observation is simple but useful for latter discussions.

CHY formulae
With the definition of Pfaffian described above, now we can introduce the CHY formulae [5][6][7][8][9]. In the CHY formulae, tree level amplitudes for n massless particles arise from a multi-dimensional contour integral over the moduli space of genus zero Riemann surfaces with n punctures, M 0,n . It can be expressed as which possesses the Möbius SL(2, C) invariance. Here k i , i and z i are the momentum, polarization vector, and puncture location for i th particle, respectively. The measure is defined as (2.7) The δ-functions impose the scattering equations where s ij ≡ (k i + k j ) 2 is the Mandelstam variable, and z ij ≡ z i − z j . The scattering equations define the map from the space of kinematic variables to M 0,n , and fully localize the integral on their solutions.
The integrand in (2.6) depends on the theory under consideration, and carries all kinematical information of external particles. For any theory known to have a CHY representation, the corresponding integrand can be split into two parts I L and I R , as can be seen in (2.6). Either of them are weight-2 for each variable z i under the Möbius transformation. We list integrands for various theories as in Table 1[9] 3 . Theory and (2.10) where δ I i ,I j forbids the interaction between particles with different flavors. To clarify the dimension, we denote the n × n matrixes X and X as [X] n , [X ] n . The 2n × 2n antisymmetric matrix Ψ is given by The reduced Pfaffian of Ψ is defined as , where the notation Ψ [i,j] means the rows and columns i, j of the matrix Ψ have been deleted (with 1 ≤ i, j ≤ n). It can be proved that this definition is independent of the choice of i and j. Analogous notation holds for Pf A.
The definition of Ψ can be generalized to the (2a here A is a (a + b) × (a + b) matrix, C is a (a + b) × a matrix, and B is a a × a matrix. The definitions of elements of A, B and C are the same as before. The reduced Pfaffian Pf [Ψ] a,b:a is defined in the same manner. With the definition of the reduced Pfaffian, one can observe that: each polarization vector i appears once and only once in each term of the reduced Pfaffian. Furthermore, starting from the 2n × 2n matrix Ψ, the polynomial P {i,j} (n, l, m) is defined by where i k < j k ∈ Tr k and Tr k 's are m sets satisfy 4 Tr 1 ∪ Tr 2 ∪ · · · ∪ Tr m = {n − l + 1, n − l + 2, · · · , n} . (2.14) In the notation [Ψ] n−l,i 1 ,j 1 ,...,im,jm:n−l , we explicitly write {i 1 , j 1 , . . . , i m , j m } instead of 2m to emphasize the locations of 2m rows and 2m columns in the original matrix Ψ. Two signatures sgn({i, j}) and sgn({i, j} ) correspond to partitions {(i 1 , j 1 ), · · · , (i m , j m )} and {(i 1 , j 1 ), · · · , (i m−1 , j m−1 )} respectively, and one can verify sgn({i, j}) = sgn({i, j} ). In the second line of (2.13), the reduced Pfaffian is calculated by removing rows and columns i m and j m , and (−) (n−l+2m−1)+(n−l+2m) = (−) have been used. Under the definition of P {i,j} (n, l, m) in the second line, the summation {i,j} P {i,j} (n, l, m) means where the sum is over all possible choices of pairs in each trace subset. Notice that one can choose to delete rows and columns belong to any Tr k when computing the reduced Pfaffian, and {i,j} P {i,j} (n, l, m) is independent of the choice since it is equal to the reduced Pfaffian of Π, which is constructed using the squeezing procedure [9]. Finally, the Parke-Taylor factor for ordering σ is given as it implies the color order {σ 1 σ 2 · · · σ n−1 σ n } for the partial amplitude.

Basic operators
In this section, we will consider the effects of acting three basic differential operators given in [10] on the elementary building-blocks of CHY-integrands such as Pf Ψ, Pf [Ψ] a,b:a , as well as {i,j} P {i,j} (n, l, m).

Trace operator
The trace operator T ij is defined as [10] T ij ≡ ∂ i j . (3.1) Here i j means i · j and the differential operator is to take derivative regarding to the combination i · j . Similar understanding holds for all operators in this note. If one apply T ij on the reduced Pfaffian Pf Ψ, only terms containing factor i j (i.e.,element Ψ i+n,j+n ) provide non-vanishing contributions. Thus performing the operator T ij is equivalent to the replacement where V denotes vectors k l 's or l =i,j 's, since i and j appear once and only once in each term of the reduced Pfaffian respectively. As noted in [10], the effect is nothing but the dimensional reduction (or the "compactifying" procedure in [9]). Thus we arrive at a new matrix Ψ satisfies Without lose of generality, one can assume {i, j} = {n − 1, n} 5 , then the new matrix Ψ is given by The reduced Pfaffian of the matrix Ψ can be calculated straightforwardly as Thus, we find Repeating the manipulations, multiple action of trace operators give following generalization of (3.6) as where P {i,j} (n, l, m) is defined in (2.13), and we have arranged elements as We want to emphasize that the multiple action of T ij on Pf Ψ produce the structure P {i,j} (n, l, m), which is crucial for many theories. This is why it is called the trace operator.

Insertion operator
The insertion operator is defined by [10] As pointed out in [10], T ikj itself is not a gauge invariant operator, but when it acts on objects obtained after acting one trace operators, it is effectively gauge invariant. Thus we consider the effect of acting this operator on the polynomial {i,j} P {i,j} (n, l, m) only. According to discussions in [10], one should assume that k ∈ {1, 2, · · · , n − l} and i, j ∈ Tr i to protect the gauge invariance. For simplicity, we assume i, j ∈ Tr m and taking the expansion (2.13) where Tr m has been deleted. This gauge choice will greatly simplify our discussion, since with this choice k i k can appear in (2.13) only through C kk .
where the definition in (2.5) has been used. The element [Ψ] a i b i is at the a th i row and b th i column of the matrix [Ψ] n−l,i 1 ,j 1 ,...,i m−1 ,j m−1 :n−l , and we have defined n = n − l, m = m − 1. Since k appears only in C kk , when acting ∂ k i k on (3.11), only terms containing element [Ψ] k,n +2m +k (see the expression (2.12)) can survive. Consider such a term, the remaining part after the action corresponds to a partition of the the set {1, 2, · · · , 2(n + m )} \ {k, n + 2m + k}, which has the length 2(n + m − 1). Such a term appears in the Pf ..,i m−1 ,j m−1 :n−l by deleting k th and (n +2m +k) th rows and columns, and sgn(π α ) corresponds to the partition of the length-2(n +m −1) set. By comparing these two special partitions, where one belongs to the original matrix and one belongs to the new one, one can get sgn(π α ) = sgn(π α ) since α is obtained from α by deleting the pair (k, n + 2m + k). Using above observation, when we sum all contributions together, we will have Applying this result to (2.13), we get immediately Let us give a little bit explanation of the result (3.14). There are two parts. The part {i,j} P {i,j} (n, l + 1, m) means we have added a new element k into the set Tr m . Here is no ordering of the set and every element is at the same footing. The ordering information comes from the part −z ij z ik z kj , especially the denominator factor z ik z kj gives a line connecting i to k and then k to j, i.e., one has inserted the element k between i, j.
To really achieve the goal, from Table 1, one can see that {i,j} P {i,j} (n, l, m) always appears together with a series of Parke-Taylor factors C Tr 1 · · · C Trm . If the original C Trm contains 1/z ij , multiplying the factor z ij /(z ik z jk ) replaces it with 1/z ik z kj , therefore implies the new color order {...ikj...}, i.e., the insertion of the element k between i, j. This explanation tells us how to systematically insert elements into a trace one by one with a well defined sequence of insertion operators.
Since the polynomial {i,j} P {i,j} (n, l, m) is independent of the choice of the deleted rows and columns, assuming i and j belong to any other Tr k will lead to the same conclusion, although the calculation will be more complicate.

Longitudinal operator
The longitudinal operators are defined via [10] and Among these two, the L ij is intrinsically gauge invariant, but L i is not 6 . We now discuss the effects of acting them on the reduced Pfaffian Pf [Ψ] a,b:a . We first consider the operator L ij . It turns i j into k i k j , and annihilates all other i V 's, j V 's. Using the observation that i and j can appear once and only once respectively, one can conclude that L ij changes the reduced Pfaffian of the matrix [Ψ] a,b:a as Next, we turn to the operator L i , which replaces every k j i with k j k i . Under such replacement, the diagonal elements of the matrix C become which will vanish due to the scattering equation. Thus, the effect of L i is given by At this moment, the meaning of (3.17) and (3.19) is not clear. Actually, the longitudinal operators can not be performed individually to generate any object belongs to physical integrands. Instead, they should be used in a special manner, which will be discussed in the next section.

Products of basic operators
Using the products of basic operators, more operators will be constructed. In this section, we will discuss these composed operators, especially their action on the reduced Pfaffian Pf Ψ, which is the fundamental building-block for the integrand of gravity theory.

Operator L · T ab
The operator L is defined through longitudinal operators as [10] L ≡ The expression (4.16) means that at the algebraic level, the effect of i L i is different from that of ρ∈pair i,j∈ρ L ij . However, if one consider the combination L · T ab Pf Ψ, and let subscripts of L i 's and L ij 's run through all nodes in {1, 2, · · · , n} \ {a, b}, the effects of i L i and ρ∈pair i,j∈ρ L ij are same, give a result which has a meaningful explanation.
Let us first study the effect of the operation L · T ab Pf Ψ. Since L and T ab are commutable, i.e., L · T ab = T ab · L, we will apply the operator T ab on Pf Ψ firstly to get (3.6), then act L on it. It is straightforward to see ρ∈pair i,j∈ρ L ij changes the matrix Next we consider the effect of acting i L i on T ab Pf Ψ. Using (3.19) we know the operator i L i turns the matrix (3.4) into thus the reduced Pfaffian is To compute the reduced Pfaffian of A, we choose a th and b th rows and columns of A n×n to be removed. Furthermore one can use the relation that for the matrix S with the block structure when M is invertible, the Pfaffian of S satisfies Using this, the reduced Pfaffian of A can be calculated as Putting it back we obtain (4.26) Above calculations show that It is worth to notice that this result is independent of the choice of a and b.

New operators T X 2m and T X 2m
As can be seen in Table 1, the CHY-integrands for several theories require the ingredients Pf [X] 2m and Pf [X ] 2m . These objects can also be created from the original matrix Ψ via appropriate operators. Now we give the definition of these new operators.
For a given length-2m set I, we define a new operator as 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. Using the result in (3.8) as well as the (2.5), one can conclude that the operator T X 2m generates a new matrix such that acting T X 2m on the reduced Pfaffian Pf Ψ gives which provides the desired building block Pf [X] 2m . By similar argument, we can also define the operator T X 2m as which is the generalization of T X 2m . The δ I i k ,I j k 's turn the matrix [X] 2m into [X ] 2m , therefore we get which gives the required building block Pf [X ] 2m . Before ending this part, we want to emphasize one important point: since T ij is intrinsically gauge invariant, so are T X 2m and T X 2m .

Unifying relations for amplitudes
With preparations in previous sections, we are ready to exhibit relations between amplitudes. As discussed in §1, the idea is, differential operators are commutable with the integration over complex variables z i 's, thus the effects of acting them on amplitudes can be realized as acting on corresponding CHY-integrands, and vice versa. Our previous calculations have explicitly established the relation between two approaches in [9] and [10]. In this section, we will apply our results in sections §3 and §4 to write down relations between different scattering amplitudes, as did in [9] and [10].

The unified web
Now we act the operators on CHY integrands for various theories to get the unifying relations for amplitudes. The starting point is the formulation for the gravity theory. The reason is, all operators decrease the spins of external particles, thus the unified web must start from the amplitudes for gravitons which carry highest spins. The integrand of gravity theory is shown in the first line of Table 1, two parts I L and I R depend on two independent sets of polarization vectors { } and { }, respectively. Since all operators are defined through partial differentials of some Lorentz invariants contain polarization vectors, it is natural to restrict the effect of them on the I L part (or equivalently the I R part), by defining operators via (or ). Performing operators on the I L part and using (3.8), (4.8), (4.15), (4.27), (4.30) and (4.32), after comparing with the middle column of Table 1, we get following relations: up to an overall sign. Here A G , A EYM , A YM , A EM , A EM flavor , A BI denote amplitudes of gravity theory, Einstein-Yang-Mills theory, pure Yang-Mills theory, Einstein-Maxwell theory, Einstein-Maxwell theory that photons carry flavors, Born-Infeld theory, respectively.
For the pure Yang-Mills integrand, there is only one copy pf Ψ depends on polarization vectors, thus operators can be performed directly. Starting from the pure Yang-Mills integrand, we obtain relations: up to an overall sign, where A YMS , A YMS special , A BS , A NLSM , A φ 4 denote amplitudes of Yang-Mills-scalar theory, special Yang-Mills-scalar theory, bi-adjoint scalar theory, non-linear sigma model, as well as φ 4 theory, respectively. Notice that the amplitude of φ 4 theory is generated via a special T X 2m that 2m = n.
Applying operators on the Born-Infeld integrand, we get relations: up to an overall sign, where A DBI ex , A DBI , A NLSM , A SG denote amplitudes of extended Dirac-Born-Infeld theory, Dirac-Born-Infeld theory, non-linear sigma model, special Galileon theory, respectively.
Our results (5.1), (5.2) and (5.3), gives not only unified relations presented in [10], but also other relations among theories having CHY representations in [9]. We want to remark that a result in this paper is different from the one in [10], i.e., the Einstein-Maxwell theory: their differential operator is just one term of the operator T X 2m defined in (4.28).
Relations presented above can be organized into   Table 2 can be summarized as where O and O denote operators which are defined through { } and { } respectively. Since the manifest double copy structure of the CHY integrands, O and O are applied on two copies independently at the integrand-level.

Other relations
Differential operators connect not only amplitudes from different theories, but also amplitudes of same type of theory. For example, let us consider the Einstein-Yang-Mills theory. Let us start from a (m + n)point color-ordered amplitude A EYM (i h 1 , · · · , i h m ; j g 1 , · · · , j g n ), where h and g denote gravitons and gluons respectively with the color order of gluons as {j 1 , j 2 , · · · , j n }. Using the relation (3.14), one can act insertion operators to turn gravitons into gluons at any desired positions, such as following: In above expressions, we have turned two gravitons into gluons, with different orderings: the first one with ordering {j 1 , i 1 , i 2 , j 2 , · · · , j n } and the second one, {j 1 , i 1 , j 2 , i 2 , j 3 , · · · , j n }, respectively. Situations for other theories can be analyzed similarly. One can also seek amplitudes for other theories beyond these given in Table 2, by acting on the amplitude of gravity theory via other combinations of differential operators. The operator O in (5.4) has 6 choices which are I, T [Tr 1 ] · · · T [Tr m ], T [i 1 · · · i n ], T X 2m , T X 2m , L·T ab , and so does O . Thus, starting from the CHY-integrand of gravity theory, there are 21 kinds of CHY-integrands can be obtained by performing operators. We now list the remaining 8 cases as following: Using results in §4, one can get the corresponding integrands generated by them. If some of these integrands correspond to physical amplitudes, then new unifying relations occurs. The complete analyis of various combinations in (5.6) is beyond the scope of this note and we will leave it to future work. Here we just give some brief discussions. For the first case T X 2m · T X 2m , when 2m = 2m = n, it yields the integrand where I G ( , , k, z) denotes the integrand for gravity theory. This is the integrand for Einstein-Maxwellscalar theory, with all external particles are scalars [11]. This result is just a special case with n = 2m = 2m . As we have emphasized, since T Xn is intrinsically gauge invariant, we can take any length for this operator. Furthermore, the role of T ij is just to do the dimension reduction. With this understanding, one can see that for general m and m the T X 2m · T X 2m A G ( , , k) will give the theory obtained from gravity theory by dimension reduction, i.e., the general Einstein-Maxwell-scalar amplitudes, whose external particles can be either gravitons, photons, as well as scalars, i.e., For the second case T [i 1 · · · i n ] · T X 2m , when 2m = n, we get which is the φ 4 theory. Again, the operator T X 2m can be any length. When 2m < n, we get the theory obtained by doing dimension reduction from Yang-Mills theory, which is the special Yang-Mills-Scalar theory T [i 1 · · · i n ] · T X 2m I G ( , , k, z) = C n Pf [Ψ] n−2m,2m:n−2m Pf [X] 2m . (5.10) Here if we replace T X 2m by T X 2m , we will get the special Yang-Mills-Scalar theory with multiple kinds of scalars, as can be seen in Table 2.
Other cases in (5.6) can be discussed similarly. One can obtain more possible integrands via products O = O 1 · · · O a and O = O 1 · · · O b . In general, for any O · O I G ( , , k, z), information of external particles such as spins and gauge structures can be read out directly from the obtained integrand, but pin down the form of interaction is a hard work.

Summary and discussion
To summarize, we have provided manifest connection between two approaches, i.e., the differential operator in [10] and various manipulations (such as compactification and squeezing procedures) in [9]. Using this connection, by acting differential operators on the CHY integrand of gravity theory, one can systematically derive unifying relations for amplitudes of various theories, include Einstein gravity, Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory and its extension, bi-adjoint scalar theory, φ 4 theory, non-linear sigma model, as well as special Galileon theory. Along the line, all unifying relations in [10] have been reproduced, and all theories which have CHY representations in [9] have been included in the unified web. We have also discussed other new relations for amplitudes, which are indicated by our method.
The manifest double copy structure of the CHY integrand permits two sets of operators O and O to be applied independently. This advantage simplifies the derivation: it is sufficient to consider the effects of acting operators on the reduced Pfaffian Pf Ψ.
A natural question will be, why these operators? From discussions in [10], one critical condition is the gauge symmetry. The trace operators protect the gauge invariance while others do not. This is why the insertion and longitudinal operators should be performed after the trace operator. There are other operators, such as T ijkl , have not been used in the construction. Thus it will be interesting to consider broader form of differential operators. Furthermore, how to understand these physical conditions from the point of view of CHY formulae is also important.
Our result can also be used to other studies. For example, recent studies [12][13][14] have shown how to expand the Einstein-Yang-Mills amplitudes by the Yang-Mills ones. If one act the differential operator at both sides of the expansion, a differential equation connecting amplitudes of two different theories will be obtained. Solving this differential equation (or doing the integration), we should find amplitudes for particles with higher spins from other ones with lower spins. This is opposite to current construction of united web by starting from highest spin state, i.e., gravitons.