Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame

Using the Cachazo-He-Yuan (CHY) formalism, we prove a recursive expansion of tree level single trace Einstein-Yang-Mills (EYM) amplitudes with arbitrary number of gluons and gravitons, which is valid for general spacetime dimensions and any helicity configurations. The recursion is written in terms of fewer-graviton EYM amplitudes and pure Yang-Mills (YM) amplitudes, which can be further carried out until we reach an expansion in terms of pure YM amplitudes in Kleiss-Kuijf (KK) basis. Our expansion then generates naturally a spanning tree structure rooted on gluons whose vertices are gravitons. We further propose a set of graph theoretical rules based on spanning trees that evaluate directly the pure YM expansion coefficients.


Introduction
Recently, non-trivial relations between Einstein-Yang-Mills (EYM) and pure Yang-Mills (YM) amplitudes 1 have been studied in several papers [1][2][3][4][5][6][7][8]. These works have more or less the same goal: to find the relations that expand the EYM amplitude with n gluons and m gravitons in terms of a linear sum of (n + m)-point pure YM amplitudes, although it is studied from different points of view. The starting point of our paper is the Cachazo-He-Yuan (CHY) formulation [9][10][11][12][13][14], which is the same as most of the above mentioned discussions. The key idea in this pursuit is to use various identities to reformulate the CHY integrand for EYM into an appropriate form. In particular, the cross-ratio identity and other off-shell identities of integrands [15,16] play a very crucial rule 2 . With the help of these techniques, the explicit expansions of EYM amplitudes with up to three gravitons have been provided in [2]. Based on these explicit results, an recursive expansion for EYM amplitudes with an arbitrary number of gravitons has been conjectured in [8], using the gauge invariance principle advocated in [18][19][20][21].
One of the main subjects in this paper is to investigate directly in the CHY frame the aforementioned recursive structure proposed in [8]. Especially, we want to see how this structure appears at the level of CHY integrands. Intuitively, we believe that the CHY formalism sets one of the best frameworks to understand such a recursive structure. The reason is that the information related to graviton polarizations is packed into a Pfaffian, which can be recursively evaluated by the Laplace expansion: for a 2n × 2n antisymmetric matrix M = (m ij ), its Pfaffian can be calculated by where θ(i − j) is a step function and M ij ij denotes the matrix obtained from M by deleting the i-th and j-th row and column. As we will see later in Section 3 and 4, such a Laplace expansion of the m-graviton EYM integrand always contains an (m − 1)-graviton integrand, plus other algebraic minors. Then if we can prove that those additional algebraic minors reduce to a linear combination of pure YM and EYM integrands with m − 2 gravitons at most, we essentially derive a recursive structure! The proof requires an elegant arrangement of the Laplace expansion for Pfaffian, and a proper use of various on-shell and offshell identities, to be spelled out later. This technique can possibly be generalized to a variety of theories constructed in, for example, [13,14,22].
Knowing the recursive relation, we can perform the expansion of generic EYM amplitudes level by level in a well-controlled way. Eventually, this process leads to an expansion in terms of pure YM Kleiss-Kuijf (KK) basis [23]. Although the final result involves a huge number of terms, it is actually very well-organized and enjoys a very elegant spanning tree structure. We find that our recursive relation derives an algorithm that evaluates directly the spanning trees, yielding the expansion coefficients we want. Spanning tree structures have appeared in the literature before for EYM and gravity in four dimensional spacetime with the MHV helicity configuration [5,24,25]. In contrary, the algorithm we propose here works in arbitrary spacetime dimensions for EYM amplitudes with general helicity configurations. Since EYM amplitudes can be realized as a double copy of YM and YM-scalar, even at multitrace levels [14,26,27], a very promising application of this technique will be the direct evaluation of Bern-Carrasco-Johansson (BCJ) numerators [28,29] for YM and YM-scalar.
The structure of this paper is the following. In Section 2, we give the essential background knowledge of the CHY formalism. The emphasis is put on the definition of the tree level YM and single trace EYM integrand. Then in Section 3 we give several explicit examples, up to four gravitons, on how to expand the EYM integrands, such that one can clearly observe the pattern and summarize it into a recursive relation for generic cases, which is then proved by induction in Section 4. We propose in Section 5 a set of graphic rules based on spanning trees to evaluate directly the expansion coefficients of EYM amplitudes in terms of pure YM ones in the KK basis [23]. These rules of course come from the recursive relation we proved in Section 4, which is demonstrated in Appendix A. Finally, we give the conclusion and discussion in Section 6.

CHY integrand for Yang-Mills, Einstein-Yang-Mills and gravity
The central object we are going to study is the tree-level single trace color ordered EYM amplitude with n gluons and m gravitons, denoted by For convenience, we define the gluon and graviton set as The CHY integrand for this amplitude is [13] I where the Parke-Taylor factor: encodes the color ordering of the gluons. The 2n × 2n matrix Ψ is defined as: The three n × n matrices A, B and C have the form: a · k c σ ac a = b , (2.5) where the indices a, b and c takes value within G ∪ H. The reduced Pfaffian of Ψ is defined as: The matrix Ψ ij ij is reduced from Ψ by deleting the i-th and j-th row and column. Finally, the 2m × 2m matrix Ψ H is defined as: where A H , B H and C H are respectively the m × m submatrices of A, B and C whose rows and columns take value only within H. In order to obtain the amplitude A EYM n,m , we integrate over the measure Ω CHY that localizes all the σ's to the solutions of the scattering equation for n + m massless particles: The expression of Ω CHY is not needed in this work, and we refer the interested readers to the original works of CHY [9][10][11][12][13][14]. Similarly, we have the following integrands for tree-level YM and Einstein gravity: from which the amplitudes are obtained through a similar integration as in Eq. (2.8). Then Eq. (2.2) and (2.9) suggest that there should exist an expansion of the tree-level EYM integrands in terms of the YM ones, namely: where the equal sign holds when the σ i 's are solutions to the scattering equations.
The general expansion 3 at the amplitude level has been proposed in a recent paper [8] based on the gauge invariance. Let us first recall some examples: • With only one graviton, the expansion is given by: where ¡ is over the shuffle product ρ ρ ρ ¡ω ω ω of two ordered sets ρ ρ ρ and ω ω ω (i.e., all the permutations of ρ ρ ρ ∪ ω ω ω that preserving the ordering of ρ ρ ρ and ω ω ω respectively), and Y p denotes the sum of the momenta of all the gluons ahead of the leg p in the color-ordered YM amplitude. Thus Y p depends on the orderings contained in the shuffle product. We keep this dependence implicit throughout this work in order not to make our notation too cluttered.
• With two gravitons, the expansion is given by: (2.12) where we have used the field strength tensor: • With three gravitons, the result is: (2.14) All the above relations are written in terms of amplitudes. However, using Eq. (2.8), we can rewrite them in terms of CHY integrands, making the pattern even more transparent. With one graviton, we get: which can be interpretted as turning the graviton into a gluon. With two gravitons, we get: (2.16) With three gravitons, we get: (2.17) Now it is clear that the general patterns observed in [8] indicates a pattern for CHY integrand expansion. In Section 3 and 4, we will derive explicitly how this pattern appears. We note that the sign difference between the amplitude/integrand expansion originates from the phase factor in Eq. (2.8).

Some examples
In this section, we will use various on-shell and off-shell identities to show the pattern of CHY integrand expansion with some examples, from which one can easily find the generalization. To make our discussion clear, we will carefully distinguish the identities we derive. The off-shell identities are those that hold at the pure algebraic level. The on-shell identities are true only when certain on-shell conditions have been imposed. These on-shell conditions can be further divided into two types: the first type is the momentum conservation and the transverse condition of polarization vectors, and the second type is that the σ's in the integrands are solutions to the scattering equations. To manifest the distinction, we will use = for off-shell identities, .
= for on-shell identities of the first type and ∼ = for on-shell identities of the second type.

One graviton
The relevant part of the EYM integrand I EYM (12 . . . n | h 1 ) is where Pf (Ψ h 1 ) can be easily calculated as: To move on, we need the following very useful identity: The proof is very simple: where we have used σ h 1 ,i+1 = σ h 1 i + σ i,i+1 to go from the first line to second. We want to remark that many papers use instead C ii = j =t i · k j σ jt σ ji σ it . Although this equivalent form has the right SL(2, C) weight for the i-th node, we find that our form of C ii is more convenient for our purpose, for example, the recursive manipulation in Eq. (3.4). Furthermore, we emphasize that Eq. (3.3) holds at the algebraic level for arbitrary set of σ's, not necessarily those of the solutions to the scattering equations. In this sense, it is an off-shell identity. Now with the help of Eq. (3.3), we find that where we have used the on-shell condition h 1 · ( n i=1 k i ) = − h 1 · k h 1 = 0 at the second line. Therefore, Eq. (3.5) holds under the on-shell condition of the first type, as indicated by the . = sign in the last line. With the help of Eq. (3.5), we arrive at 6) or, in terms of the amplitudes, This is nothing but (2.15). The sign change is accounted by the phase factor of Eq. (2.8).
For future use, we define the following level-zero-current CHY integrand T µ Like the usual CHY integrands, T µ has weight two for all legs, so it is well defined even without imposing the scattering equations. However, T µ has some special properties. First, its parameters are divided to two parts. The first part contains two ordered sets {2 . . . n − 1} and {h 1 . . . h i }. The second part contains an unordered set. For Eq. (3.8) it is the empty set ∅, and this is the reason why Eq. (3.8) is called the level-zero-current. Secondly, Eq. (3.8) contains the Lorentz indices through Y µ h 1 . It is worth to notice that h 1 is the first element of the ordered list {h 1 , . . . , h i }. With this new notation, Eq. (3.6) can be written as: (3.9)

Two gravitons
The relevant part of the EYM integrand I EYM (12 . . . n | h 2 h 1 ) is given by To continue, we expand Pf (Ψ h 2 h 1 ) as where the superscript n + 2 in Ψ n+2 is to remind us that we are now working with the (n + 2)-particle kinematics, although the form of Ψ n+2 h 1 resembles the Ψ h 1 in Section 3.1. Similar to Eq. (3.5), when C h 2 h 2 hits PT (12 . . . n), we get: Now we give some remarks to this equation. First, when using Eq. (3.3) to insert h 2 into gluons, only the gluon momenta contribute, so we get h 2 · Y h 2 . Secondly, going from the second equality to the third, we have used the momentum conservation and h 2 · k h 2 = 0, as indicated by the . = symbol. Thirdly, the above manipulation is, in fact, the action of changing one graviton to gluon, which results in the first term of Eq. (3.12). However, in the presence of other gravitons, we need some corrections [such as the second term in Eq. (3.12)]. As we will see shortly, the recursive structure is hidden in these corrections.
The relation (3.12) can be easily generalized to the cases with more gravitons as for any h i ∈ H = {h m h m−1 . . . h 1 }. We will use (3.13) in later computations again and again. A further observation is that, when we make the replacement h i → k h i and impose the scattering equations, we will make C h i h i → 0 at the left hand side, such that Eq. (3.13) leads to: (3.14) We have used the ∼ = symbol to emphasize that this equation holds only under the on-shell condition of the second type, namely, localizing σ's to be the solutions to the scattering equations. Now plugging Eq. (3.12) back into Eq. (3.10), we can rewrite Eq. (3.10) into: The first line of Eq. (3.15) is now in its final form, which gives: after the CHY integration. Now we consider the second line of (3.15), which contains two terms. The first term contains the factor Pf (Ψ n+2 h 1 ) PT (12 . . . n) .
It would just be the one-graviton case solved in Section 3.1, had h 2 joined in the Parke-Taylor factor. This observation guides us to turn PT (12 . . . n) into PT (12 . . . nh 2 ) by inserting the factor σ n1 /(σ nh 2 σ h 2 1 ), such that the previous result can be used directly: (3.16) To go from the second line to third line, we have separated the orderings in the shuffle {2 . . . n} ¡{h 1 } into two groups, in which: (1) n being the last element; (2) h 1 being the last element. The first group gives the first term in the third line of (3.16). For the second group, we get Putting (3.16) back into the second row of Eq. (3.15), we get: which is its final form. To derive the last line of (3.17), we have to use an insertion relation, which can be derived as follows: We want to emphasize that Eq. (3.18) is an off-shell identity. This manipulation can be generalized to With this equation, the last row of Eq. (3.15) becomes (3.21) In fact, the relation between the second line and the third line of Eq. (3.15) can be more easily seen by noticing that Putting together Eq. (3.17) and (3.21), we can write down the final result for Eq. (3.15): where we have used the definition (2.13) to combine the second and third line of Eq. (3.15). Namely, we can expand the two-graviton EYM integrand in terms of one-graviton integrands and pure gluon ones.
Eq. (3.23) leads to the following amplitude relation: The sign difference between Eq. (3.24) and Eq. (3.23) is correctly adjusted according to the phase factor in the CHY integration (2.8) for EYM amplitudes. Similar to what we have done in Eq. (3.8), we define the following level-one-current CHY integrand It is easy to see that after using Eq. (3.8) at the second line, we can re-express Eq. (3.23) as: which holds for an arbitrary number of gluons.

Three gravitons
As the first nontrivial example, we come to expand the three-graviton EYM integrand: In this equation, the matrix ψ(32|1) is given by: Namely, ψ(32|1) is obtained from Ψ h 3 h 2 h 1 by deleting the row and column intersected at ±C h 3 h 2 . Then ψ(31|2) is obtained from ψ(32|1) by the exchange h 2 ↔ h 1 . On the other hand, the matrix ψ(32|1) is obtained from ψ(32|1) by the replacement h 2 → k h 2 , and similarly for ψ(31|2). Now using Eq. (3.13) for the first term of (3.27) to turn the graviton h 3 into a gluon, we arrive at: At this moment, after comparing Eq. (3.29) with (3.15), one can see a pattern starting to emerge. Another thing we want to point out is the relative sign difference between the first line of (3.29) and (3.15). In Eq. (3.29), the first row is already in our desired form. We then focus on the second row, while the third row can be obtained by simply exchanging h 1 and h 2 . We would like to reuse Eq.
To do so, we again use the trick of inserting h 3 into the Parke-Taylor factor. Then using Eq. (3.23), we obtain (3.30) In this equation, we treat h 3 as a gluon, playing the same role as n in Eq. (3.23). In the first line of Eq. (3.30), the shuffle product can be divided into two groups: one with n near h 3 and the other with h 2 near h 3 : After this separation, the first line of Eq. (3.30) becomes: has been used to derive the expression after the first equal sign. The final expression is then obtained by acting Pf (Ψ n+3 h 1 ) onto PT (12 . . . nh 2 h 3 ) using Eq. (3.6). In the last line of Eq. (3.31), we have introduced a new symbol X h 1 . This is because when we apply (3.6) here, h 1 will view h 2 as a gluon, so we use the X h 1 instead of Y h 1 to emphasize this difference. From now on, we will use X i to stand for the sum of momenta of all the legs at the left hand side of leg i in the color ordering. With this explanation, we can expand the last line of (3.31) as: Now we move to the second row of Eq. (3.30). Again we separate the shuffle product into three groups This separation leads to: Summing up Eq. (3.33) and (3.34), we find that after a few algebras, Eq. (3.30) becomes: The first two lines of this equation are in our desired form, while the rest of them look very messy at the first glance. Very remarkably, they can actually be organized into Pf [ψ(32|1)]. To show this, we first use the identity: to group Eq. (3.35a), (3.35b) and (3.35c) into: Next, by extracting h 2 and h 3 out of the Parke-Taylor factor, we can write Eq. (3.35d) as: As the last step, we replace the h 1 by k h 1 in Eq. (3.36), which leads to the on-shell identity: With the help of this result, Eq. (3.35e) and (3.35f) combine into: Namely, we have proved the following identity: With all these preparations, we can now return to tackle the second row of Eq. (3.29). Using Eq. (3.42), we find that: Finally, using the insertion identity (3.18), we get: (3.45) Thus combining (3.44) and (3.45) together, we arrive at: Finally, putting everything altogether, we reach the following result: It is wroth noting that after considering the relative signs encoded in Eq. (2.8), this result indeed gives Eq. (2.14) after the CHY integration. Eq. (3.47) demonstrates a very clear recursive structure, while we can make it more compact by defining the level-(m − 1)-current CHY integrand [like the one given in Eq. (3.25)]: This is a recursive definition, and one can move down by reducing the number of gravitons in the right list until it becomes the empty set. Then Eq. (3.8) terminates the whole recursive construction. We can use T µ to rewrite Eq. (3.47) as: This three-graviton example makes the recursive pattern very clear. Our next job is of course to prove that it works for arbitrary number of gravitons. To get more confidence and insight, we do one more example with four gravitons. The techniques used in this calculation can be very easily generalized to prove the generic case, presented in Section 4.

Four gravitons
As the last example, we expand the Pfaffian part of the four-graviton integrand Pf (Ψ h 4 h 3 h 2 h 1 ) PT (12 . . . n) as follows: where Ψ n+4 h 3 h 2 h 1 has a very similar form to that of Eq. (3.27). However, now the σ's satisfy instead the n + 4 point scattering equations, while in the diagonal C elements, the summation is over all the n + 4 particles. The matrix ψ(43|21) is obtained from Ψ h 4 h 3 h 2 h 1 by deleting the column and row intersected at ±C h 4 h 3 : Putting it back to (3.50) and reorganizing, we get: such that it can be separated in the following way, according to the definition (3.48): (3.55) Now using the following identities proved in the three-graviton example: 5

57)
After we replace the T 's in the last two lines of Eq. (3.55) by the above eqaution, and dot the entire Eq. (3.55) with h 3 , it becomes: If we put them back into Eq. (3.54) and then Eq. (3.53), we get: (3.59) If we make the replacement h 3 → k h 3 above, we then get:  Finally, using the definition (3.48), one can express the four-graviton EYM integrand as: The recursive nature of this construction is now clear, and we are going to prove the generic recursive relation for expanding EYM integrands with an arbitrary number of gravitons.

General recursive relation
The calculations in Section 3, especially the cases of three and four gravitons, suggest us to construct the recursive definition (3.48), which we recall here as:    [28].
The m = 1, 2, 3 and 4 cases have been explicitly checked in the previous section. Now suppose that Eq. (4.2) holds for p gravitons with p m − 1, as our induction assumption, then we move to the case of m gravitons. The 2m × 2m matrix Ψ H has the following form: To calculate the Pfaffian, we expand along the row starting with C hmhm [which is the (m + 1)-th row]: with the help of Eq. (3.13). Then from Eq. (4.4), we get:  Our proof completes if we can prove the following two identities: Now our inductive assumption says: 6 As a result, we get: which proves Eq. (4.7). Then replacing h m−1 by k h m−1 , we also proves Eq. (4.8): With the help of these two equations, we can immediately see that Eq. (4.6) gives exactly our statement: We have now completed the proof of our general recursive relation Eq. (4.2).

Expansion in terms of pure YM amplitudes
If we keep using the recursive relation (4.2), we can finally expand generic EYM tree amplitudes in terms of pure YM amplitudes. In this representation, the final expression does not enjoy the explicit permutation invariance of the gravitons. The reason is that in Eq. (4.2), we have defined a standard graviton order {h m h m−1 . . . h 1 }. At each level of recursion, we always convert the first graviton remained in this standard order into a gluon according to the definition of T , which breaks the explicit permutation invariance. 7 In this section, we give these expansions a graph theory picture, derived from our recursive expansion (4.2). More details on this derivation is presented in Appendix A

Some expansion examples
In this part, we will give the expansions of the EYM amplitudes with two and three gravitons. It serves two purposes: (1) establishing our convention; (2) providing explicit expressions to compare with known results as well as a better understanding of the graphic rules.
Let us start with the example of two gravitons, whose recursive construction is given in Eq. (3.24). Since the last term of Eq. (3.24) contains only YM amplitudes. We only need to further expand the first term using Eq. (3.7): where Y h i is defined as the sum of all original gluon momenta before the graviton h i in each ordering of the shuffle product. The important difference between Y h i and X h i is that X h i is the sum of all momenta before h i , including those gluons converted from gravitons. Now using the associativity of the shuffle product: we can write the second line of Eq. (5.1) as Therefore, if we further expand Eq. (3.24) down to the pure YM level, we get: We want to emphasize that in the above expansion, the coefficients contain only the kinematic variables Similar manipulations can be performed for EYM amplitudes with three gravitons. By recursively using Eq. (2.12) and (2.11) to Eq. (2.14), one can write the final result as: 8 For each graviton permutation σ σ σ, the coefficient C [321] (σ σ σ) has the following expression: The subscript [321] stands for the standard reference order of converting the gravitons into gluons, whose role will be discussed later. Note that these coefficients under the standard order [321] do not enjoy the explicit permutation invariance, namely, for example. This is because the expansion basis is KK, which has some redundancy, such that a certain gauge choice is allowed. It is worth to compare our C [321] (σ σ σ) with the results given in a previous paper [8], where the kinematic variable Z has been used and the coefficients contain different number of terms. Here, although C [321] (σ σ σ) are not symmetric under relabeling, the total number of terms contained in each C [321] (σ σ σ) is the same, namely, 6 = 3! terms for this three-graviton example. In fact, with some further observations, we find the following features: • In terms of only the , F , and Y variables, each coefficient is given by a sum of six terms. Then each of these six terms is a product of a few scalar factors, enclosed in parentheses.
• Now we count the number of elements in each parenthesis. For examples, Then we find that for each term that appears in any C [321] (σ σ σ), we have: If we represent each graviton by a vertex, and supplement them with a common root, we can then associate each factor with a path starting at a graviton vertex labeled by and ending at either a graviton vertex (labeled by k) or the root (labeled by Y ), while n i is just the number of vertices along the path. One can immediately observe that each term of C [321] (σ σ σ) provides a tree structure. The sum all factors (n i − 1) gives nothing but the total number of gravitons. This observation motivates us to construct the following graph theory rules based on spanning trees.

Graphic rules for coefficients of pure YM expansion
In this part, we propose a set of graphic rules for a direct reading of the pure YM expansion coefficients, based on spanning trees. The recursive relation (4.2) indicates that for a generic m-graviton EYM amplitude, we can always expand it in terms of pure YM amplitudes in the KK basis, with the following form: 9 where σ σ σ is a permutation of the m gravitons {h 1 . . . h m } and ρ ρ ρ is a reference order. The coefficients C ρ ρ ρ (σ σ σ) can be evaluated from the spanning trees of m + 1 vertices. Among these vertices, m of them correspond to the gravitons {h 1 , h 2 . . . h m }, and a single vertex g as the root, representing the gluon set {2 . . . n − 1}.
The coefficients C ρ ρ ρ (σ σ σ) can be read out by the following algorithm: Step 1: constructing the trees contributing to the order σ σ σ. We construct the increasing trees with 9 As kept implicit here, Cρ ρ ρ(σ σ σ) also depends on the shuffle product, since it contains Y h i , the sum of gluon momenta before respect to the order σ σ σ, i.e., g ≺ h σ(1) ≺ h σ(2) ≺ . . . ≺ h σ(m) . Such trees can be obtained by arranging vertices h σ(1) , h σ(2) , . . . , h σ(m) from the left to right and then drawing all possible tree diagrams after avoiding the following situations: if two vertices A ≺ B are connected by a path originated from the root, then B can not be the one closer to the root. We emphasize that the construction of the trees only depends on σ σ σ, not the reference order ρ ρ ρ.
For example, when there are three gravitons, we have the following trees contributing to σ σ σ = {123}: trees that contribute to tree 6 (5.14) In these trees, all vertices in a path starting from the root and ending at a leaf respect the order σ σ σ. There are exactly m! such trees. This can be easily seen by a recursive construction. For example, from each tree shown in Eq. (5.14), we can construct exactly four increasing trees for {1234} by either connecting h 4 to any h i , with i = 1, 2, 3, or connecting h 4 directly to the root. More explicitly, the first tree gives: Step2: Reading out the expression for a tree. Having constructed the increasing trees, we assign an expression to each tree. At this moment, the choice of ordering ρ ρ ρ will play an important role. We will accomplish it by following steps: • First, we draw ordered paths. We will start from ρ(1), the first graviton in list ρ ρ ρ, and then draw a path from the root g to ρ(1). If there are vertices along this path, we will set φ 1 = ρ(1), and the subsequent vertices φ 2 , φ 3 , etc, until the root φ = g. We will denote such a path by P[1] = {ρ(1), φ 2 , ..., φ −1 , g}.
An illustration of such path P [1] is given as the following: It is worth to emphasize that ρ(1) may or may not be a leaf.
• As the first path is done, we construct the second path from the remaining vertices: Now we construct P [2] by considering the path from the root g to the first element of ρ ρ ρ 1 , denoted as ρ 1 (1). Let us denote this tentative path as T 2 . Starting from the root, the path T 2 will in general coincide with other previous constructed paths until the vertex V 2 , the last vertex (counted from the root) along T 2 that belongs to previous constructed paths. The crossroad vertex V 2 can either be a graviton or just the root. Now the part of T 2 from V 2 and beyond, until ρ 1 (1), is our P [2]. It has the form P This process is illustrated as: (5.18) • Repeat this procedure until we have exhausted all the graviton vertices.
• Now we assign each path a factor. The first element in the path is replaced by , the middle element by F . For the last element, if it is the root, then we replace it by by Y t if it is the root g (the t is the next-to-last element); if it is a graviton vertex a, we replace it by k a . Finally, multiplying all these factors together, we get the term contributed by the tree under our consideration.
• We emphasize that given a tree, the evaluation only depends on the reference order ρ ρ ρ. Actually, a tree can contribute to more than one σ σ σ in general, which will be further clarified later in this section.
To demonstrate how these rules work, we again take ρ ρ ρ = {321} and consider the six trees given in (5.14) as an example. These trees all contribute to σ σ σ = {123}. It is easy to see that we have One can check that they exactly reproduce the C [321] (123) given in Eq. (5.6). With the reference order ρ ρ ρ = {321}, another example is This tree contributes to σ σ σ = {321} only. As some less trivial examples, we consider several typical trees that appear in the evaluation of the coefficients for four-graviton EYM amplitudes. With the reference order ρ ρ ρ = {4321}, we have: Figure 1. The rooted tree with three levels.
In the above discussion, we have given the graphic rule to read out the corresponding coefficients of a given KK basis. In Eq. (5.13), there are m! coefficients C ρ ρ ρ (σ σ σ), each of which contains m! terms given by m! increasing trees. Thus naively we would expect that there are in all (m!) 2 trees involved. However, there are only (m + 1) m−1 distinct spanning trees for m + 1 vertices (m graviton vertices and a gluon root). Since (m!) 2 > (m + 1) m−1 for m 2, some trees must contribute to multiple C ρ ρ ρ (σ σ σ) coefficients. The next problem we want to address is that given a spanning tree, how to find out all the C ρ ρ ρ (σ σ σ) coefficients it can contribute to.
The answer to this question is simple: suppose we define the set of all the increasing trees respecting the order σ σ σ as IT (σ σ σ), then given a spanning tree T, it will contribute to all those σ σ σ's whose IT (σ σ σ) contains T. These σ σ σ's can be read out in the following way. First let us define the level structure of a spanning tree, demonstrated by Figure 1: • From the root, there are three outgoing arrows, so we have three subsets in the first level.
• The subsets of a and b are simple, which are {a 1 , a 2 , a 3 } and {b 1 , b 2 , b 3 } respectively.
• For the middle subset, it is much more complicated. Along the arrow, first we have {c 1 , c 2 , c 3 }.
However, there are two outgoing branches starting from c 3 . Thus we have a second level with two branches: one is {d 1 , d 2 } and the other is {e 1 , e 2 }. Again, starting from e 2 there are two branches: {f 1 , f 2 } and {g 1 , g 2 }, which give us the third level.
• Putting everything together, we have the following level structure: After reading out the level structure, we can write down the orderings each level gives by shuffle actions. First, there are two branches at level 3, such that this level gives: Then at the second level, there are also two branches. The first is simply {d 1 , d 2 }, while the second one is {e 1 , e 2 } concatenating level3. The orderings given by this level are: Finally, at the first level, we have: After the shuffle products are fully expanded, level1 gives the set of orderings this tree contributes to. Next, we check this rule with our three-graviton expressions. In Eq. (5.14), the first tree only has one level, such that contributes to Indeed, this term is contained in all six C ρ ρ ρ (σ σ σ), given in Eq. (5.6) to (5.11). For tree 5 in Eq. (5.14), there is only one branch at level one but two branches at level two, such that  contributes to contributes to Before closing this section, we note that the above rules can be recursively defined for generic trees. First, the level splitting happens only when a branch in the tree splits. Then for a subtree with the structure of Figure 2, it will contribute to the following set of orderings: where, for example, Branch1 is the set of orderings contributed by the first branch, etc, which can be recursively constructed using the above relation, until the highest level.

Conclusion and discussion
In this paper, we have established a recursive expansion (4.2) for tree level single trace EYM amplitudes with m gravitons and n gluons, facilitated by expanding the CHY integrand Pf (Ψ H ) along one row. It works for arbitrary number of gravitons and gluons, as we have proved in Section 4. This recursive construction agrees with the one proposed in a recent paper [8], which is observed simply by imposing gauge invariance. Our paper gives a firm proof of the observation. We note that our recursive expansion holds only on-shell, namely, the momentum conservation, transversal condition and scattering equations should all be satisfied.
Very remarkably, once we fully expand the m-graviton and n-gluon EYM amplitudes in terms of pure YM amplitudes with n + m gluons in the KK basis, all the expansion coefficients can be written down through a set of elegant graph theory rules. To be specific, the coefficient of each KK basis amplitude contains exactly m! terms, each of which corresponds to an increasing tree with respect to the order of the gravitons in a certain KK basis amplitude. Then the evaluation of each tree relies on a reference order. On one hand, the origin of this reference order is how we expand Pf (Ψ H ) step by step. On the other hand, it serves as a "gauge choice" in the over-complete KK basis YM amplitudes.
From these results, one very promising direction is of course the construction BCJ numerators [28,29]. It is pointed out in [8] that the YM integrand Pf (Ψ) can be readily expanded in terms of the EYM integrands with polynomial coefficients. Then our results essentially provides a working algorithm to write down the polynomial BCJ numerators for tree level YM, especially the spanning tree structure which has not been explicitly spelled out in this paper. This study may shed lights on new understandings of the numerator algebra, and also on possible new formulations of YM that manifest the numerator algebra. Such a formulation already exists for NLSM [30], and it will be fascinating if it also exists for YM.
Our work shows that a controlled expansion of CHY integrands can produce very fruitful results. It will be interesting to see, for example, how the expansion of Pf (A), the CHY integrand for NLSM 10 , interplays with [30]. One can also try the CHY integrand at the one-loop level for YM theory [34][35][36]. Another topic that worths more attention is the study of multitrace EYM integrands [2,4,14], and its relation with the known results constructed from double copy [26,27].
Note added During the final completion of this work, we came aware of [37], which partially overlaps with our results.
· · · · · · · · · In the first term, the factor ( hm · Y hm ) can be viewed as connecting the graviton vertex h m to the root, while in the second term, ( hm · F hs ) µ can be interpreted as connecting h m to another graviton h s , since in the ordering {h s h m }, h s is at the left hand side of the h m . For the second term, further expansion will again lead to two term, one with F hs connected to the root, giving ( hm · F hs · Y hs ) and an (m − 2)-graviton EYM integrand: We can do it recursively and this procedure gives, in fact, a construction of those increasing trees presented in Step 1 in the Section 5. Furthermore, the starting vertex h m plays the role of ρ(1) in the reference order ρ ρ ρ. For convenience, we show this expansion process schematically in Figure 3.
Now we work out more details for the above construction by an example. We first keep following the red arrows in Figure 3 to connect more gravitons until the vertex φ , which is then connected to the gluon root. In this way, we get a chain evaluated as where we have defined the first vertex φ 1 ≡ h m . In the scheme of Figure 3, terminating a graviton chain at the gluon root essentially means that we switch to a blue arrow and arrive at the block . . h m }. Now we come across a Pfaffian again, and we have to choose another graviton along which we expand this Pfaffian. Thus here comes the origin of the reference order ρ ρ ρ: it is a list of priority we give to each graviton, such that once we need to expand a Pfaffian, we always choose to expand along the one with the highest priority according to ρ ρ ρ. In this case, we simply delete {φ 1 . . . φ } from ρ ρ ρ and pick up the first remaining graviton, say θ 1 . We then stay with the red arrows until we deplete all the gravitons, which gives us another chain. The final block we reach gives: Since we have treated {φ . . . φ 1 } as gluons when expanding the θ-chain, we need to include k φ i into X θt if there are some φ i before θ t : After fully expanding the shuffle {φ . . . φ 1 }¡{θ t . . . θ 1 }, one can notice that the factor ( θ 1 · F θ 2 · · · F θt · Y θt ) is contained in the coefficients of all the corresponding KK basis: In addition, the factor ( θ 1 · F θ 2 · · · F θt · k φ ) appears when φ is ahead of θ t : Generally, we have: Now one can understand how the structure of increasing tree emerges. We can graphically represent Eq. (A.6) to (A.8) as follows: It is interesting to observe that the tree in (A.9) is an increasing tree respecting any σ σ σ ∈ {φ . . . φ 1 } ¡ {θ t . . . θ 1 }. Similarly, the tree in (A.10) is an increasing tree respecting any σ σ σ ∈ {φ . . . φ i , {φ i−1 . . . φ 1 } ¡ {θ t . . . θ 1 }}. Our derivation also shows that the terms evaluated from Eq. (A.9) and (A.10) do contribute to all such σ σ σ's represented by the two trees. Finally, this example shows how we should close a chain that ends on a graviton vertex, as given in Step 2 in Section 5 The above two-chain calculation demonstrates how the graph theory rules presented in Section 5, in their primitive forms, originates from our recursive relation. Of course, we have already refined these rules into a set of algorithms in Section 5 such that they can be applied to the most generic cases.