Derivation of Feynman rules for higher order poles using cross-ratio identities in CHY construction

In order to generalize the integration rules to general CHY integrands which include higher order poles, algorithms are proposed in two directions. One is to conjecture new rules, and the other is to use the cross-ratio identity method. In this paper, we use the cross-ratio identity approach to re-derive the conjectured integration rules involving higher order poles for several special cases: the single double pole, single triple pole and duplexdouble pole. The equivalence between the present formulas and the previously conjectured ones is discussed for the first two situations.


Introduction
The well-known Cachazo-He-Yuan (CHY) formulation [1][2][3][4][5] is an elegant new representation of tree-level amplitudes for massless particles in arbitrary dimensions, as given by where s ab ≡ 2k a · k b is the Mandelstam variable and z ab is defined as z ab ≡ z a − z b . The scattering equations and the integration measure are universal, while the integrand I CHY n obeys some general constraints (such as of weight-4 under Möbius transformations), and it also depends on the specific field theory. This formulation indicates that a tree amplitude can be calculated by solving scattering equations and summing over different solutions. However, it is hardly possible to get direct solutions beyond five points due to the Abel-Ruffini theorem for algebraic systems. Thus, to search for a new computation method to avoid the explicit solutions of algebraic equations JHEP06(2017)091 becomes a crucial challenge. Among investigations from various directions [6][7][8][9][10][11][12][13][14][15][16][17][18], the socalled integration rule method inspired by string theory is one of the most efficient and systematic approaches [17][18][19]. This approach replies only on the CHY integrands, without mentioning the solutions of scattering equations and the integration measure. Applying this method, one can extract all the correct pole structures from a given integrand, and directly obtain the result via the corresponding Feynmann diagrams, rather than solving the scattering equations. However, one shortcomings of the original integration rules [17][18][19] is that it requires the CHY integrand under consideration containing simple poles only, therefore cannot be applied to arbitrary physically acceptable integrands in general.
To handle this disadvantage, there are two alternative approaches. One is to derive integration rules for higher order poles [20], and the other is to reduce terms containing higher order poles into those with simple poles only [17,21]. In the first direction, integration rules for several special configurations of CHY integrands with higher order poles are conjectured in [20]. Although its analytic proof is absent, these rules are numerically verified. A more hopeful approach comes from the second direction, thanks to the discovery of the cross-ratio identities, which reveals relations between different rational functions of z ij 's [22]. By applying these identities iteratively, one can expand a term involving higher order poles as terms with simple poles only. After this decomposition, one can obtain the integrated result through the original integration rules. The fesasibility of this algorithm is verified in [26]. Further applications of the cross-ratio identities can be seen in [23][24][25].
It is natural to ask: can we prove the conjectured integration rules of higher order poles in [20] via the cross-ratio identity method? In this paper, we will derive these rules analytically by applying those identities. Our derivation depends on the choices of the cross-ratio identities, thus different choices yield different expressions for the same pole configuration. The expressions of rules in this paper will be different from those conjectured in [20], and we will prove their equivalence for two cases.
This paper is organized as follows. In section 2, we summarize some useful notations, and some general properties of the CHY integrands and cross-ratio identities. In section 3, we derive the integration rule for CHY integrands containing a single double pole. In section 4, we derive the integration rule for that containing a single triple pole. For this case, the equivalence between our formula and the conjectured one is rather non-trivial, and the proof of their equivalence indicates a new kind of integration rule, as will be discussed explicitly. In section 5, we derive the integration rule for those containing duplex-double poles, regarding a simplest special case. A brief conclusion is given in section 6.

Preparation
Before going to the details, for clarity we give a summary of notations. Each CHY integrand corresponds to a weight-4 graph, in which n nodes are connected by a number of lines. Each line corresponds to a factor z ij . 1 Furthermore, since factor z ij can appear in both the numerator and the denominator, to distinguish them, we use a solid line to represent z ij in the denominator and a dashed line to represent that in the numerator. With this JHEP06(2017)091 assignment, the weight-4 condition becomes that there are four lines connecting to each node, where a solid line is counted as +1 and a dashed line −1.
For a set Λ containing |Λ| points of z i , we call a line connecting two points in Λ the internal line of Λ, and a line connecting at least one point inside Λ and the other outside Λ the external line of Λ. The number of internal lines of Λ is denoted by L[Λ], and that of external lines by E[Λ]. 2 Furthermore, we denote the number of lines connecting two sets Λ 1 and Λ 2 as The order of poles corresponding to the set Λ is defined as which will be useful later. Then, we have the following corollaries: • A simple set has 4 external lines.
• A double set has 2 external lines.
• A triple set has 0 external lines.

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Each CHY integrand may give a number of terms and each term can be represented by a Feynman diagram with only cubic vertices. Except the special case of n = 3 (i.e., there are only three external nodes), each cubic vertex contains at least one internal propagator. At each endpoint of this internal propagator (there are two), two branches are produced. They can be two external nodes, or one external node and one internal propagator, or two internal propagators. No matter in which situation, this internal propagator will be associated with a subset Λ and its complement Λ, thus we have where Λ 1 and Λ 2 are two branches (two subsets with Λ 1 Λ 2 = Λ) associated with the endpoints of subset Λ. The summation is over all correct divisions of Λ 1 , Λ 2 , and a special division is denoted as Λ 1 Λ 2 . The major machinery we use in this paper is the cross ratio identity given in [22]. Let us give some explanations of (2.6): (1) (p, q) is the gauge choice. Although different gauge choices give equivalent expressions, some choices will simplify the calculation.
(2) We have double sums over all subsets Λ and Λ.
(3) For each term in the sum, we have a kinematic factor s ij . The two denominators z ij and z pq (fixed for all terms in the sum) between subsets Λ and Λ increase E As when applying the cross-ratio identities, one needs to choose a gauge which includes two points and a set corresponding to a pole, for simplicity we use [p, q, Λ] to denote the gauge choice, as well as the corresponding pole.

Rule I: single double pole
In this section, we will derive the Feynman rule I for a single double pole. The corresponding conjectured formula in [20] is given as (3.1)

CHY configuration
First we try to understand CHY configurations with only one double pole s Λ . This means that all subsets A have χ[A] ≤ 0, except one subset Λ with χ[Λ] = 1. Furthermore, for simplicity we will assume the numerator of CHY integrand is just 1. 4 With the assumption above, we can give some statements of CHY configurations. First, we have which means that there are two and only two lines connecting subset Λ and its complement Λ. Now we will show that these two lines cannot meet at the same node. Let us assume the two lines are given by a point a ∈ Λ connecting to Λ, then where we have used the fact that point a has four connecting lines, so there are two lines connecting point a and the subset Λ/{a}. Since we have assumed that all χ[A] ≤ 0 with A = Λ, (3.3) contradicts with this assumption. This means that there are two points a 1 , a 2 ∈ Λ and two points b 1 , b 2 ∈ Λ, such that there are one line connecting a 1 , b 1 and the other connecting a 2 , b 2 . As will be discussed cautiously, when we apply the cross-ratio identities to find the Feynman rule, the good gauge choice is either [a 1 , b 2 , Λ] or [a 2 , b 1 , Λ]. 5 Second, we will show that there is no subset A ⊂ Λ containing both a 1 , a 2 satisfying χ[A] = 0. If such a subset exists, we can consider its complement A = Λ/A. Since Third, we will show that there is no subset Σ = α β, such that α ⊂ Λ, β ⊂ Λ satisfying χ[Σ] = 0. If such a subset exists, we have which contradicts with the second observation in the previous paragraph. 4 Although we will not give a rigorous proof for CHY integrands with nontrivial numerators in this paper, we believe Feynman rule I will be applicable to this more general situation. In fact, when we derive the rule III, we will meet the situation where although the numerator is not one, the same rule I has been applied to get the correct results. 5 As we have remarked, the claim above, i.e., there are only two lines z a 1 b 1 and z a 2 b 2 , has neglected the possibility that there are nontrivial numerators in CHY integrands, which will bring more solid lines (i.e., factors in the denominator) connecting Λ, Λ. To deal with this case, one can use, for example, z ab z dc zacz bc = z ad zac − z bd z bc , (3.4) to get rid of the numerator. In this paper, for simplicity we will not discuss such more general configurations.  With these observations for single pole structures, we move to the maximal compatible combinations of poles, i.e., the possible Feynman diagrams. The claim is: all maximal compatible combinations must contain the double subset Λ. The reason is: to get nonzero contributions, the number of compatible subsets must be n − 3. Since we have argued that there is no single pole Σ = α β with α ⊂ Λ, β ⊂ Λ, all single poles are either in Λ or in Λ. But Λ can contain at most (|Λ| − 2) compatible poles, while Λ can contain at most (|Λ| − 2) ones. Thus their combinations contribute only (n − 4) compatible single poles, and we need to include the double pole to get nonzero contributions.

Some examples
To demonstrate how to derive the Feynman rule using cross ratio identities, we present several examples in this subsection. The same picture will be used when derive other Feynman rules later although explicit examples will not be given.
Let us start with the simplest example with only four points with the CHY integrand This example contains only one double pole s Λ with Λ = {1, 2}. There are two lines [14], [23]

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We see that for the first gauge choice, pole s 23 is introduced since in addition to original lines z 12 , z 34 , new lines z 23 , z 14 have been added by the cross ratio identity. For the second gauge choice, no new pole is introduced, so its calculation is simpler.
This phenomenon in fact suggests the general pattern. From the discussion in the previous subsection, for a given Feynman diagram, Λ has been split to two subsets A 1 , A 2 and Λ has been split to two subsets , it is possible to find two subsets α ⊂ A 2 , β ⊂ B 2 satisfying the following conditions (3.10) Thus after multiplying the CHY integrand by s ij where the extra +1 comes from the denominator z ij for any i ∈ α, j ∈ β. In other words, we will get a new pole s α β , which does not exist in the original integrand. If we adopt the gauge p ∈ A 2 , q ∈ B 1 , such a phenomenon will not happen since L[A 2 , B 1 ] = 0. Thus to avoid this problem, from now, we will always adopt the gauge p ∈ A 2 , q ∈ B 1 or p ∈ A 1 , q ∈ B 2 . Furthermore, since the splitting of Λ into A 1 , A 2 is, in general, arbitrary, it is hard to guarantee that a node a always stays in A 1 . The only exception is that nodes a 1 , a 2 (which connect to b 1 , b 2 ∈ Λ) from the second observation in the previous subsection, i.e., a 1 and a 2 are always in different subsets of the split Λ. Thus there is a universal gauge choice for all Feynman diagrams, which can be either [ The next example is the 5-point CHY integrand which matches the Feynman rule (3.13). Although simple, this example reveals that: (1) It seems that we can define different "Feynman rules"; (2) Different Feynman rules come from different gauge choices.
One can use more examples to better understand these two observations, as they persist to all configurations of the current category. In the next subsection, we will give an analytic proof.

Analytic proof
Having understood those examples, now we can give a general analytic proof. First, from the assumption of CHY integrands, i.e., there is one and only one subset satisfying χ[Λ] = 1 and for all others χ[A] ≤ 0, we have the following statement: • All maximal compatible combinations contain the subset Λ, i.e., all nonzero Feynman diagrams contain the double pole 1 • There are four special nodes a 0 , b 0 ∈ Λ and c 0 , d 0 ∈ Λ, such that there is one line connecting nodes a 0 , d 0 and one connecting nodes b 0 , c 0 . 8

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Now we consider the Feynman rule with different gauge choices. For [a 0 , c 0 , Λ], the corresponding cross ratio identity is as shown in figure 1b. Now we consider new CHY integrands I org for each (i, j) pair. First, all integrands contain only simple poles in our construction. Second, as we have argued, under this gauge choice, all possible poles are those already appeared in the original I org and no new pole will appear. Based on these two facts, now we focus on the contributions to a particular Feynman diagram with the pole structure 9 from these CHY integrands. For this pole structure, summation after inserting the cross ratio identity can be divided into the following four parts: will not contain the pole 1 s A . Similarly, the numerator z jc 0 will lead to the fact that there is no pole 1 s C . Altogether, we find the G(a 0 , c 0 ) I part will not contribute to the pole structure 1 s A s B s C s D . The same argument tells that, we can exclude the contribution from G(a 0 , c 0 ) II and G(a 0 , c 0 ) III . For the G(a 0 , c 0 ) IV part, each term gives the same contribution 10 to the particular pole structure, and we find the total coefficient is given by Summing over all possible splittings A, B, C, D, we get (3.20) 9 As we have proven, four special nodes must be in four different corners. 10 It is easy to see that each term contributes the same pole structure, but it is hard to see that each term gives with same sign. The sign can, in principle, be determined by using either the method in [3] or that in [22]. The discussion of sign is too complicated for us to give a general, simple argument.

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which is one possible Feynman rule. From the same argument, one can see that for the gauge choice [b 0 , d 0 , Λ], among four parts, only one gives nonzero contribution with the coefficient thus the other possible Feynman rule is Averaging over these two contributions, we get the Feynman rule

Rule II: single triple pole
In this section, we consider CHY configurations with only one triple pole s Λ . The equivalence between our formula and the conjectured one proposed in [20] is rather subtle, and it indicates a new kind of integration rule involving quartic vertices, as will be discussed cautiously.

CHY configuration
The assumption of only single triple pole requires that all subsets A have χ[A] ≤ 0, except one subset Λ with χ[Λ] = 2. Furthermore, for simplicity, we will assume the numerator of CHY integrand is just 1. 11 With above assumption, we can make some statements of CHY configurations. First, the triple set Λ satisfies E[Λ] = 0, thus there is no line between Λ and its complement subset Λ. In other words, subset Λ and Λ give weight-4 graphes by themselves (or a legitimate CHY integrands for smaller nodes). As a consequence, if a subset A ⊂ Λ gives a simple pole, so is Similarly, if subset C ⊂ Λ gives a simple pole, so is D = Λ/C. To make things simpler, we will assume node 1 ∈ Λ and node n ∈ Λ. Furthermore, when split at the two ending points of internal propagator s Λ , we will assume that 1 ∈ A, n ∈ D and denote the splitting as ABCD with B = Λ/A and C = Λ/D. The corresponding configuration is shown in figure 2a. 11 Again, although we do not have the proof, we think the Feynman rule could be applied to the case with nontrivial numerators.   Secondly, using (3.5), one can observe that there is no pole corresponds to the set Σ = α β such that α ⊂ Λ and β ⊂ Λ, since χ α ≤ 0, χ β ≤ 0 and L[α, β] = 0. Finally, for any maximal compatible combination, by the same argument as in the previous section, one can see that it must contain the subset Λ. Thus, the CHY integrand will give the contribution like i∈A⊂Λ n∈D⊂Λ

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where the factor X is what we need to derive for the Feynman rule.

Derivation of new Feynman rules
Having understood the configurations, now we drive the rule. Since it is the triple pole, we need to use the cross-ratio identities twice to reach simple poles. At the first step, we chose the gauge [1, n, Λ] to get as can be seen in figure 2b. For given (i, j), the CHY integrand I org z jn z 1i z ij z 1n is nothing but the configuration we have considered in previous section, i.e., the one with only one double pole. It is easy to see the point: the denominators z ij and z 1n have created two connecting lines between Λ and Λ. One may worry about the numerator z jn in Λ and z 1i in Λ. But as we have remarked, although our proof is given for the case with numerator 1, we believe the rule holds for nontrivial numerator as current situation. Furthermore, at least for the current CHY graph, there is always a closed Hamiltonian circle containing 1 and i in the subset Λ (or n and j in the subset Λ). Thus, one can use the open-relation in [23] (eq. (3.4)) or [25] (eq. (A12)) to eliminate the numerator (dashed line) to the case with numerator 1.

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Since the problem has been reduced to the case with only one double pole, we can use the previous result to write down

C[A]C[B]C[C]C[D]
, for the gauge choice [1, j, Λ] for the second step. Or

C[A]C[B]C[C]C[D]
,  for the gauge choice [1, j, Λ]. Or for the gauge choice[i, n, Λ]. Results (4.7) and (4.8) are, in fact, two possible Feynman rules for the triple pole. 12 It is worth to emphasize that when exchanging the ordering of the sum, the i∈Λ\{1} j∈Λ\{n} s ij produces (2p B · p C ). Also, changing of summation ordering is allowed because we have fixed the first gauge choice 1 ∈ Λ, n ∈ Λ for all splittings of ABCD . The first gauge choice is crucial for the previous Feynman rule, while the second gauge has some natural choice. Since we will use different gauge choices, we will use [1, n; 1; Λ] for the gauge choice leading to the rule (4.7) and [1, n; n; Λ] for the gauge choice leading to the rule (4.8).

Comparison with conjectured formula
The Feynman rules found in the previous subsection is not the one conjectured in [20], which is given by Comparing these different Feynman rules, we see that the major difference is that the rule (4.9) is gauge independent, while rules (4.7) and (4.8) depend on both the first gauge choice and the second gauge choice. Because of this, the later two Feynman rules are simpler than the first one. In this subsection, we discuss how to arrive at (4.9) from (4.7) and (4.8).
For simplicity, let us assume that Λ = {1, 2, . . . , m}, Λ = {m + 1, m + 2, . . . , n}. The final result of a given CHY integrand in our case will be the sum of different splittings ABCD where the rule (4.7) has been used. For the second gauge choice [1, n; n; Λ], we get where the rule (4.8) has been used. For the third gauge choice [m; n; m; Λ], the thing is a little bit complicated: since we have fixed the subset A to be the one containing node 1, we need to consider two different cases. When m is also in the subset A, we get When m is not in the subset A, we get (4.14)

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Adding these two parts (4.13) and (4.14) together, we get the expression for the third gauge choice
Summing over (4.16) and (4.17), the expression of the fourth gauge choice is

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Thus to reproduce the rule (4.9), we need to show that To make the comparison explicitly, we rewrite (2p and similarly where We will prove that 1,m∈A⊂Λ  Figure 3. Creation of the quartic vertex from a cubic vertex. and 1,m∈A⊂Λ n∈D⊂Λ This procedure removes the propagator 1/s A from the cubic vertex, and creates a new quartic vertex by pushing A 1 and A 2 to the original cubic vertex, as shown in figure 3. Similarly, if B also contains at least two points, In other words, we will have (4.29) where to fix the ambiguity we will assume that 1 ∈ A 1 . The summation in the first term is over all correct divisions of A 1 and A 2 , and the summation in the second term is over all correct divisions of B 1 and B 2 . We emphasize that a correct division must ensure that A 1 , A 2 , B 1 and B 2 are simple sets since they corresponding to Feynman lines in Feynman diagrams.
If B contains only one point, we have

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We fix the configuration as 1 ∈ Λ 1 . Thus we can consider the coefficient of under a special division Λ 1 Λ 2 Λ 3 . There are several cases: (1) If m ∈ Λ 1 , there are three correct configurations which are Notice that for the third configuration above, {A = Λ 1 , B 1 = Λ 3 , B 2 = Λ 2 } gives the same situation B = Λ 2 Λ 3 as {A = Λ 1 , B 1 = Λ 2 , B 2 = Λ 3 }, so we chose only one of them. An alternative way is to add them together and divide by the symmetry factor 2.
(2) If m ∈ Λ 2 (m ∈ Λ 3 gives the same situation), there is only one correct configuration One may observe that (4.30) can be ignored when counting the coefficient of The reason is, if we consider the configuration {A = Λ 1 , B 1 = Λ 2 , B 2 = Λ 3 }, we have assumed that B contains at least two points. If we consider other configurations, the structure of B is not important. The set Λ can always be divided into three simple subsets if it contains at least three points. We can take the perspective that A and B are constructed by three subsets, then the irrelevant cases (such as B can not be divided into two simple subsets) will be neglected automatically.
Having considered the left handed side of relation (4.26), we move to the right handed side, i.e., the (s A − s B ) 2 C[A]C[B] part. By similar argument, we can write it as (4.32) If one of A and B is a single point, one term in the bracket vanishes.
which is over all divisions AB , without the constraint m ∈ A. There are three correct configurations, Notice that if some of Λ i 's are single points, i.e., s Λ i = 0, the above relation still holds, as can be verified directly. Thus, after summing over contributions from different divisions AB , We get
Then we turn to the most difficult relation (4.28). We divide into following three parts: (4.36) (4.37) and will treat them one by one. First, using (4.31) we get (4.39) We assume that n ∈ D 1 . We can consider the coefficient of with n ∈ Λ 1 . There are three correct configurations which are Secondly, considering Y 2 gives Substituting it into (4.41) we get where (4.31) has been used again. In the last line, Λ i1 and Λ i2 are two simple subsets ] with n ∈ λ 1 and λ 1 λ 2 λ 3 λ 4 = Λ, its coefficient under the summation over divisions Λ 1 Λ 2 Λ 3 can be counted by following six configurations: where C 1 and C 2 are two simple subsets of C, D 1 and D 2 are two simple subsets of D with n ∈ D 1 . Considering the coefficient of under the summation over divisions CD , one can find three correct configurations:

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Adding Y 1 , Y 2 and Y 3 together, we can see that the quintic vertexes cancels each other, and the result is given as (4.47) Similar calculation gives 1∈A⊂Λ n,m+1∈D⊂Λ To finish our proof, let us we consider the last term of relation (4.28) 1∈A⊂Λ n∈D⊂Λ (4.50) Thus, we have 1,m∈A⊂Λ n∈D⊂Λ This ends the proof of relation (4.28).

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Before ending this section, we would like to point out a by-product in this section. We have shown that the final result of the integration can be expressed as where X is defined in (4.20). This formulation can be considered as a new kind of integration rule. Now instead of just cubic vertexes like Λ → AB and Λ → CD , there are new quartic vertexes like Λ → Λ 1 Λ 2 Λ 3 and Λ → Λ 1 Λ 2 Λ 3 .

Rule for duplex-double pole
In this section, we will derive the Feynman rule for duplex-double pole. In [20], the Feynman rule is conjectured to be In this section, we will derive a different rule for the special case p 2 E = 0 for simplicity. The new rule will be different from the one given in (5.1). Since we have restricted ourselves only on the special case, we will not present the equivalent proof of these two rules in this paper.

CHY configuration
Like other two cases, first let us specify the CHY configurations in this section, i.e., all poles are simple poles except two double poles s Λ 1 and s Λ 2 (Λ 1 Λ 2 = ∅). Furthermore, to simplify the problem, we have assumed the numerator is one for CHY integrands and the E = Λ 1 Λ 2 is just a single node e. With above specifications, we can derive the following statements: (1) By the same argument for the case with only one double pole, for the subset Λ 1 , there are two points a, b having a line connecting to subset Λ 1 . Similarly, for the subset Λ 2 , there are two points c, d having a line connecting to subset Λ 2 .
(2  Putting it back, we get In the second case, A = A and B = B E. Unlike the first case, we know for sure that Here node e can be attached to some propagators in the sub-Feynamn diagrams of subset B. We will show that the final result for the second case is which implies that the X in (5.3) is given by The Feynman rule (5.3) with X given by (5.11) has been checked numerically with several examples. Now we explain the result (5.9) for the second splitting of Λ 1 = A B with A = A and B = B E. For a given subset B, there are several compatible combinations, i.e., several sub-Feynman diagrams. Let us focus on a particular sub-Feynman diagram Γ, which is shown in figure 5a. For this Γ, there is a sequence of single subsets such that B m ⊂ B m−1 ⊂ · · · ⊂ B 1 ⊂ B with B m = {b}. The reason we consider this sequence is that by our previous argument, it is exactly these subsets B t , which can combine with the subset E = {e} to create a new single pole when the sum i ∈ B t in the second part of (5.4). In other words, for the second splitting of Λ 1 , the node e will attach exactly to these propagators B t in the Feynman diagram Γ, as can be seen in figure 5b. Now we can write down the expression when node e is attached to propagator B t as Let us explain the meaning of (5.12). First, to be able to attach e to the propagator B t , the sum i in the second part of (5.4) can only be those i ∈ B t , so we get the numerator 2p Bt · p E . Secondly, along the sequence of propagators, when the node e has been attached, the later propagators will carry corresponding momentum, so we have the propagators s BtE , s B t−1 E , s B t−2 E until s BE . Thirdly, γ = t γ Bt is the other part of propagators, which are not affect by node e, as shown in figure 5. It will be the same for all T Γ;Bt .
14 The relative minus sign is because for the second part we have inserted another cross ratio identity.  Having obtained the expression, we can carry out the sum. It is easy to see that where at the second line we have used the fact p 2 E = 0 since E = {e} is just a single node. Adding T Γ;B m−2 , we get (5.14) Now we can see the recursive pattern, which leads to This is just the expression of Γ itself. This summation can be understood diagrammatically as in figure 6. This calculation has also shown how the new created poles have been canceled out when summing over all terms. Having above preparations, we are ready to write down the contribution from the second splitting of Λ 1 as

Conclusion
In this paper, we use the cross-ratio identity to derive the Feynman rules of higher-order poles in the CHY construction conjectured in [20]. The first rule is valid for CHY integrands containing a double pole, the second rule is valid for integrands containing a triple pole, while the third rule is valid for integrands containing a duplex-double pole. The new expression of rules obtained in this paper depends on the gauge choices of cross-ratio identities, however the final results of integrations after summing over all correct compatible combinations are gauge invariant. For the first and second rule, we make the comparison of them with those conjectured in [20]. For the second rule, the equivalence between our formula and the conjectured one is non-trivial. We have found that the result of integration can be arranged into a new version of rule which contains quartic vertexes. This is an interesting phenomenon.
For the first two rules, we have performed special gauge choices to ensure that new poles which do not satisfy the desired compatible combination will not emerge. For the third rule however, this idea can not be realized, i.e., one can not avoid the appearing of new poles. Thus we have provided a treatment of new poles for this rule.
In general, searching for integration rules for higher-order poles is not efficient enough for practical computation, since one can not reduce the most general CHY integrands which contain higher order poles into a few configurations. Rules in this paper or rules in [20] only cover some special cases. When encountering new cases, we need new rules. However, since we have proven the conjectured rules through the cross-ratio identities, we can conclude that the cross-ratio identity method is a powerful and universal tool for analytic calculation.