Feynman Rules of Higher-order Poles in CHY Construction

In this paper, we generalize the integration rules for scattering equations to situations where higher-order poles are present. We describe the strategy to deduce the Feynman rules of higher-order poles from known analytic results of simple CHY-integrands, and propose the Feynman rules for single double pole and triple pole as well as duplex-double pole and triplex-double pole structures. We demonstrate the validation and strength of these rules by ample non-trivial examples.


Introduction
In the last few years, a new formulation for tree-level amplitudes of massless theories has been presented by Cachazo, He and Yuan [CHY] in a series of papers [1][2][3][4][5]. The formula is given by where z i are puncture locations of n external particles in CP 1 and dω = dzrdzsdzt zrszstztr comes after we use the Möbius SL(2, C) symmetry to fix the location of three of the variables z r , z s , z t by the Faddeev-Popov method. The Ω is defined by where E a 's are the scattering equations defined as As in the formula (1.1), the CHY construction involves two parts: solving the scattering equations E a which are universal for all theories, and formulating the CHY-integrand I for a given theory.
Although conceptually, the CHY approach is remarkable and very useful for many theoretical studies on the properties of scattering amplitudes, when applying to practical evaluation, one confronts the problem of solving scattering equations. The scattering equation problem can be related to solving a polynomial equation system of degree (n − 3)! [6]. It is well-known that when n ≥ 6, in general there is no analytic way to solve it 1 . To deal with this problem, many methods have been proposed. In [9], using classical formulas of Vieta, which relates the sums of roots of polynomials to the coefficients of these polynomials, analytic expression can be obtained without solving roots explicitly. The view of algebraic geometry has been further developed in several works. In [10] the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations has been proposed. In [11], the Bezoutian matrix method has been used to facilitate the calculation of amplitudes. In [12][13][14], the elimination theory has been exploited to great details for the group of scattering equations with (n − 3) variables. In [15,16] an efficient method has been developed to evaluate the Yang-Mills theory.
Different from the algebraic geometry methods, two more generic algorithms have been proposed. In one approach [17], using known results for scalar φ 3 theory, one can iteratively decompose the 4-regular graph determined by the corresponding CHY-integrand to building blocks related to φ 3 theory, Which ends the evaluation. In another approach [18,19], by careful analysis of pole structures, the authors wrote down an integration rule, so that from the related CHY-integrand, one can read out contributions from the corresponding Feynman diagrams directly if there are only simple poles appear. The later is really an efficient tool, however when higher-order poles are presented in a CHY-integrand, it can not proceed furthermore because of lacking Feynman rules for them.
In this paper, we would like to improve the efficiency of solving scattering equations by generalizing integration rule from only simple poles to higher-order poles, in the sprint of [18,19]. The key-point of our generalization is to realize that, even for higher-order poles, one can find properly-defined Feynman rules. Although in current stage, we are not able to derive these generalized Feynman rules for higher-order poles, we have provided substantial evidences to support our claim.

Review of integration algorithm
Before proposing the Feynman rules for higher-order poles, let us for reader's convenience review the integration algorithm presented in [18,19,21] 2 . The whole algorithm can be roughly described as two parts: (1) given a CHY-integrand, find out the corresponding Feynman diagrams; (2) work out corresponding Feynman rules (or generalized Feynman rule while considering higher-order poles), with which write down the result directly for given Feynman diagrams.
For a n-point amplitude, the CHY-system is described by n complex variables {z 1 , z 2 , . . . , z n } before gauge fixing. The CHY-integrand, as a rational function of z ij ≡ z i −z j , can be represented by a graph of n nodes with many lines connecting them. A factor z ij in denominator(or numerator) can be represented by a solid line(or dashed line) connecting nodes i and j with an arrow pointing from i to j. A CHY-integrand which is invariant under Möbius transformation is then represented by a 4-regular graph, i.e., a graph such that for each node, the number of attached solid lines minus the number of attached dashed lines is four. This is illustrated by an example with only simple poles as shown in Figure 1. The corresponding CHY-integrand for this 4-regular graph 3 is 1 (1,2,3,4,5,6)(1, 2, 3, 4, 6, 5) = 1 z 2 12 z 2 23 z 2 34 z 2 56 z 45 z 61 z 46 z 15 . (2.1) The purpose of integration algorithm is to read out the result of a given CHY-integrand directly from the 4-regular graph without solving any scattering equations. As described in [18,19,21], this is readily programmed for situations where only simple poles are present. Now we present a review on integration algorithm. For a n-point system, let A = {a 1 , a 2 , . . . , a m } ⊂ {1, 2, . . . , n}, which is a subset of nodes. For each subset, let us define following pole index

2)
2 In these literatures, it is called integration rule. To not get confused with Feynman rules, we will call it integration algorithm. 3 Throughout this paper we have omitted the arrows of lines in the graph for simplicity. Also when we read out result using integration algorithm, there is an overall sign one need to pay attention to.
where L[A] is the number 4 of lines connecting these nodes inside A, and |A| is the number of nodes. The condition will be called the pole condition for a given subset. Explicitly, each subset gives a non-zero pole contribution if and only if χ(A) ≥ 0. The pole will be in the form 1 s , where s A = (p a 1 + p a 2 + · · · + p am ) 2 for massless momentum p i . Also, because of the momentum conservation, we will treat the subset A to be equivalent to its complement A = {1, 2, . . . , n}−A, thus we can impose the length condition 2 ≤ |A| ≤ n 2 . Once the pole structures of subsets are clear, we define the compatible condition for two subsets A 1 , A 2 : two subsets are compatible to each other if one subset is completely contained inside another subset or the intersection of two subsets is empty. Now we state the integration algorithm: • (1) Given a CHY-integrand, draw the corresponding 4-regular graph. For example, for the CHYintegrand (2.1) we draw the graph in Figure 1.
For example (2.1) (see Figure 1), since n = 6, we have 2 ≤ |A| ≤ 3. From above integration algorithm, one can see that all the five steps are in fact computer task except one, i.e., the determination of Feynman rules. For simple pole, the Feynman rule is nothing but the usual Feynman propagator 1 s A . For higher-order poles, the story becomes complicated, and they are the main context we would discuss in the following sections.
Before going on, there are some general remarks. The first is, why we believe there are Feynman rules for higher-order poles? The first hint comes from the pinching picture given in [22]. When discussing contributions with a given pole structure, it is found that we can group some nodes together to reduce a CHY-graph to two CHY-graphes with fewer nodes. Thus the contribution can be roughly written as where T L , T R have natural tree amplitude structure. Iterating the pinching procedure we can reduce a big CHY-graph to some building blocks. This picture is nothing but the familiar one when cutting a propagator in Feynman diagram to reduce a big diagram to two sub-diagrams, while the building block is nothing but the Feynman rules.
Keeping above picture in mind, now we present a more detailed discussion on the pinching operation. Let us divide n nodes to the subset A ≡ {z a 1 , z a 2 , . . . , z am } and its complement B ≡ {z a m+1 , z a 2 , . . . , z an }. The pinching corresponds to representing the whole subset by a new node and then adding this new node to the graph, which means we will get two new sets A ≡ {z a 1 , z a 2 , . . . , z am , z B } and B ≡ {z A , z a m+1 , z a 2 , . . . , z an }. Furthermore, all lines z ab with z a ∈ A and z b ∈ B will be modified as follows: z ab → z aB = z a − z B in the new set A and z ab → z Ab = z This condition leads to χ(A) = 0, i.e., the subset contributes to simple pole 1 s A . It means we can only pinch nodes with simple poles.
Above pinching condition has an important implication. Using pinching, we can reduce a bigger CHY-graph to a smaller one as shown in following figure: Iterating the procedure, we can reduce any graphs to a much simpler primary graph. No matter what the reduced graph looks like, its higher pole structure is invariant, i.e., the number and the degree of higher-order poles are not changing. The pinching picture has also inspired us to find the Feynman rule for a given higher-order pole with the strategy: solving the simplest CHY-integrand containing such pole structure and then properly generalizing it! This will be the approach we use in the paper. Our result of Feynman rules for higher-order poles has also revealed that, while the propagator of simple pole structure should be considered to be complete local (i.e., its Feynman rule 1 s A depends only on the momentum flowing in between), the propagator of higher-order poles should be properly considered as quasi-local, i.e., its rule will depend also on the momentum as well as the types of poles that connecting to the four corners of propagators.
3 The Feynman rules for higher-order poles As reviewed in §2, the integration algorithm can be readily generalized to CHY-integrand with higher-order poles if we have corresponding Feynman rules for them. Once the Feynman rules of higher-order poles are in order, we can produce the results of any CHY-integrands instantly, following the standard five steps of integration algorithm.
In this section, we will provide Feynman rules for some higher-order poles. The general strategy of deducing these Feynman rules are described as follows. Firstly, we find out the simplest CHY-integrand which would contain the required higher-order pole, and obtain the analytic result by any reliable method. In many cases, one would find more than one Feynman diagram for a given CHY-integrand, and from them we need to isolate the contribution that only generated by Feynman rule of that higher-order pole. Then we generalize the result to generic situations by considering the symmetry structures of poles, the kinematics as well as when the external legs are massive. This involves trial-and-error by some other CHY-integrands to fix ambiguities. Once the result is formulated as a rule, then it can be applied in the computation of any CHY-integrands that containing corresponding higher-order poles. R I  ule for a single double pole  The Feynman rule For external legs as labeled in the right-most diagram of Figure 2, the Feynman rule for a single double pole is formulated as

The Feynman rule
where P with capital letters as subscripts denotes a sum of several massless legs, e.g., P A = m i=1 p a i . Terms such as 2P A P B is understood to be the Minkowski products 2P A · P B = s AB − P 2 A − P 2 B . We could notice the difference between 2P A P B and s AB when external legs are massive.
Two remarks before proceed. Firstly, the Feynman diagram in Figure 2 is blind with ordering. But for a given 4-regular graph of CHY-integrand as shown in Figure 2, especially paying attention to the This point is very important when applying the rules of higher-order poles. One should be very careful when drawing the Feynman diagrams for CHY-integrands, especially for those legs connecting to the propagators of double poles. Secondly, different from the simple pole which is completely local, rule for higher-order pole depends not only on the total momentum P A + P B flowing through the propagator, but also momenta P A , P B , P C , P D at four corners. Hence we call this propagator to be quasi-local. As we shall see very soon, when two propagators of double poles are connected by a vertex, the Feynman rule for this kind of pole structure will be different, i.e., we can not apply rule (3.1) to those two propagators, but consider them as a single object. This quasi-local property is the reason that deducing Feynman rules for higher-order poles is quite difficult.

Deducing the Feynman rule
As mentioned, the strategy is to find out the simplest CHY-integrand that containing the required higherorder pole structure and obtain the analytic result by any means. Thus we start from the simple four-point CHY-integrand, It can be trivially computed by directly solving scattering equations, with the result Now we try to reproduce above result by integration algorithm reviewed in previous section. We draw the 4-regular graph for this CHY-integrand, as shown in Figure 2. It is easy to see that there are only two independent subsets of nodes, i.e., {1, 2} and {2, 3}. While χ[{2, 3}] = 1 − 2(2 − 1) = −1, and χ[{1, 2}] = 3−2(2−1) = 1, we conclude that the only compatible combination is {1, 2}, where an underline is to emphasize that it corresponds to a double pole. There is only one Feynman diagram corresponding to this four-point CHY-integrand, as shown in Figure 2 besides the 4-regular graph, where we use a double line to denote the double pole.
In order to deduce the Feynman rule for this double pole, let us try to get some hints from the result (3.3). Note that the 4-regular graph (originates from the CHY-integrand) apparently possesses symmetries under exchanging of nodes (3.6) However, expression (3.6) is not the final answer, since s 12 is generally not equal to 2p 1 · p 2 if p 1 , p 2 are massive. Since the numerator is not constrained by any considerations, both s ij and 2p i · p j could be possible candidates. This motives us to consider the case when external momenta are massive as shown in the right-most diagram of Figure 2, which will be a sum over several massless momenta for generic CHY-integrands. It forces us to determine a proper formulation for (3.6) that can be applied to the computation of more generic CHY-integrands. The denominator, since contains only physical poles, should be s AB but not 2P A P B . In this case, s AB = s CD because of momentum conservation and we can take s 2 AB as denominator. While for the numerator, a five-point example of CHY-integrand would suffice to tell us which one to choose. With trial-and-error method, one find that for massive case, we should use 2P A P B instead of s AB in the numerator. Although it has no difference in the massless limit(i.e., the four-point CHY-integrand), it indeed introduces important corrections to guarantee the correctness for generic CHY-integrands. This leads us to formulate the Feynman rule of a single double pole as in (3.1).

An illustrative example
Let us illustrate the Feynman rule R I ule by a five-point CHY-integrand  From them we can draw two Feynman diagrams which have the required propagators as shown in Figure  3. Note that for simple poles, the Feynman rule is not affected by the ordering of external legs attached to the propagator. For example, switching leg 3 and leg 4 of the first Feynman diagram in Figure 3, we still get 1 s 34 for the propagator. However, the Feynman rules of higher-order poles do depend on the ordering of external legs, as can be seen in the definition (3.1). The ordering of legs can be traced back to the four-point 4-regular graph. For example, legs 3 and 4 in the first Feynman diagram are combined together and become a massive leg of the double pole, and the ordering of legs are determined by the ordering of nodes in the four-point 4-regular graph besides the Feynman diagram. When applying the Feynman rule R I ule for the double pole, we should refer to the ordering marked by the 4-regular graph. In order to use the Feynman rule of double pole directly in the Feynman diagram, we should draw the external legs in a definite ordering by using the four-point 4-regular graph as assist. Hereafter, we will always draw Feynman diagrams with definite ordering of legs attached to the higher-order poles so that we can read out the rules directly from them, but keep in mind that this ordering of legs is determined by the 4-regular graph. in the result is in fact a correction term introduced by using 2P A P B for massive legs instead of s AB in the Feynman rule. This result agrees with that in [17].  For external legs as labeled in the right-most diagram of Figure 4, the Feynman rule for a single triple pole is formulated as Note that terms in the second line are zero when all external legs are massless, so they can not be deduced from the result of four-point CHY-integrand. Nevertheless, it is very important in order to produce correct answer for general situations.

Deducing the Feynman rule
Again let us start from a simple four-point CHY-integrand, which contains this pole structure and is given by The result can be trivially obtained by solving scattering equations as (3.11) Now we follow the integration algorithm for this example. In the 4-regular graph as shown in Figure  4, it is easy to see that which means there is only one subset {1, 2} corresponding to a triple pole (a two-fold underline is introduced to emphasize it).
So we can draw the Feynman diagram for this CHY-integrand, as shown in Figure 4 besides the 4-regular graph, where a triple line is introduced to denote the triple pole.
To deduce (but not derive!) the Feynman rule for a single triple pole from result (3.11), we follow the similar considerations as in previous subsection. Apparently, the 4-regular graph is invariant under exchanging of nodes 1, 2 as well as exchanging of nodes 3, 4. It is also invariant under exchanging i → Mod[i + 2, 4]. From these considerations, intuitively we can propose a symmetrization for (3.11) as s 14 s 13 + s 24 s 23 + s 41 s 42 + s 31 s 32 4s 3 using identities s 13 = s 24 , s 14 = s 23 , and also the rule shown above. However, when computing the fivepoint CHY-integrand as shown in Figure 5, the above rule fails to produce correct answer. But another rule although not preserving the required symmetries, can produce correct answer. Anyway, we need a rule that preserving the symmetries of 4-regular graph, and this motives us to inspect the difference between above two expressions (3.12), (3.13). Notice that for the five-point CHY-integrand in Figure 5, we have applied the rule under the condition that P A = p 1 , P B = p 2 are massless, while P C , P D could be massive. After taking the difference of (3.13) and (3.12), we get (3.14) This indicates that R plus the extra term would produce correct result for CHY-integrand in Figure 5.
Considering symmetries of 4-regular graph, it is reasonable to refine the rule as 6 (3.15) However, this is still not the complete rule. For CHY-integrand in Figure 5, we do not confront situations that two or more external legs are massive. We found that, when both P A , P B or both P C , P D are on-shell, the rule (3.15) still works, otherwise it fails. This motive us to add another correction term, with the property that it becomes zero when P 2 A = P 2 B = 0 or P 2 C = P 2 D = 0, but non-zero when P 2 A = P 2 C = 0 or 6 We comment that, if only relying on the symmetry consideration, there would be many possible terms to be added in the rule, for example So if there is no trial-and-error process with other CHY-integrands, it is in fact difficult to formulate a valid rule from these possible terms due to too many ambiguities.
, and after trial-and-error, we find that a correction term 2 9 would suffice to produce correct results for applying the rule of triple pole to all possible situations of massive legs. Then we formulated the Feynman rule R II ule for triple pole as shown in (3.9).

An illustrative example
Let us illustrate the Feynman rule R II ule by a five-point CHY-integrand The 4-regular graph for this CHY-integrand is drawn in Figure 5. By counting the number of lines connecting subsets of nodes, we find that {1, 2} is associated with a triple pole, while {3, 4}, {4, 5}, {5, 3} are associated with simple poles. We need to select 5−3 = 2 subsets to construct the compatible combinations, and there are three, From them we can draw three Feynman diagrams, as shown in Figure 5 besides 4-regular graph. Note that in the definition of R II ule , the exchanging P A ↔ P B or P C ↔ P D will not affect the rule( which is not true for R I ule ). So there is no definite ordering for external legs attached to the same end of triple pole propagator. Then it is straightforward to write down the result by applying Feynman rules of triple pole

The Feynman rule
For external legs as labeled in the right-most diagram of Figure 6, the Feynman rule for duplex-double pole is formulated as Notice that in the five-point CHY-integrand, we have P 2 E = p 2 3 = 0, which means that the second term in fact can not be deduced from five-point result. But it is non-zero for generic CHY-integrands, and is crucial in order to produce the correct answer.

Deducing the Feynman rule
Although the Feynman diagram in Figure 6 contains only double pole, these two propagators of double poles are connected at a vertex. Practically, we find it impossible to use the Feynman rule R I ule separately for each double pole, and should treat them as a single object which we call duplex-double pole. In order to deduce the Feynman rule for this duplex-double pole, let us start from a simple five-point CHY-integrand, Although it is not as trivial as the four-point CHY-integrands, we can still compute it by solving scattering equations directly. After some simplification, we can write the result as By inspecting the kinematic variables, it is easy to see that the result (3.20) already possesses above antisymmetries as well as symmetries, so it is not necessary to further symmetrize it. With experiences of previous Feynman rules of higher-order poles, it is straightforwardly to propose a Feynman rule for massive legs as shown in the right-most diagram of Figure 6. However, it is not yet the complete rule.
After trial-and-error, we find that the second term in R III ule (3.18) should be included. In the five-point CHY-integrand, since P 2 E = p 2 3 = 0, it vanishes and we can never deduce it from (3.20). But it is crucial in order to produce the correct result for other generic CHY-integrands, and we will show this in §4. 4 with examples containing mixed types of higher order poles.

An illustrative example
We will illustrate the Feynman rule R III ule by a six-point CHY-integrand The 4-regular graph of this CHY-integrand is shown in Figure 7. Following the integration algorithm, firstly we list all the subsets of nodes that contribute to either simple pole or higher-order poles, as Secondly, we need to select 6−3 = 3 subsets out of above four subsets to construct compatible combinations. They are given by Immediately we draw two Feynman diagrams as shown in Figure 7, which have the required propagators according to compatible combinations respectively. We have intentionally labeled the external legs in certain ordering such that we can directly read out the Feynman rules from the diagram. Then according to the rules, we get the result . (3.23) The last term is in fact the correction term introduced by using 2P A P B instead of s AB for massive legs in the Feynman rules. This result is confirmed numerically.

The Feynman rule R IX ule for triplex-double pole
We have presented the Feynman rules R I ule , R II ule , R III ule for double pole, triple pole and duplex-double pole. They are reasonably simple. But when going to the higher-point CHY-integrands, new structures of poles would appear. Deducing rules for them could be very difficult, and the rules themselves could be complicated. To illustrate, let us present another Feynman rule R IX ule for a pole structure that first appear in six-point CHY-integrand. 1 2

The Feynman rule
For external legs as labeled in the right-most diagram of Figure 8, the Feynman rule for triplex-double pole is formulated as (for a better presentation, we define the stripped Mandelstam variables s AB = 2P A P B as well as where and Note again that the last term in (3.24) vanishes for six-point CHY-integrand, and it can not be deduced from there. However it indeed contributes when any of external legs is sum of more than one massless momenta. We need this correction term to produce correct answer.

Deducing the Feynman rule
When applying integration algorithm to a six-point CHY-integrand that will contribute to poles, of which three are associated to double poles. From them we can construct four compatible combinations, and the corresponding four Feynman diagrams are shown in Figure 8. The first three Feynman diagrams can be computed by rule for simple poles and R I ule . However, for the last Feynman diagram, three double poles are connected to one vertex, and we can neither use rule for double pole nor rule for duplex-double pole to compute it. Therefore, we need to create a rule R IX ule for this pole structure, which we call triplex-double pole. It has 1/(s 2 12 s 2 34 s 2 56 ) pole structure. Recall that when deducing Feynman rules R I ule , R II ule , R III ule for double, triple and duplex-double poles, we always start from known results of the simplest CHY-integrands which exactly contain one Feynman diagram of that pole structure. However, the six-point CHY-integrand (3.35) is the first one that appears the triplex-double pole, and it has four Feynman diagrams. It is impossible to find another CHY-integrand that contains only one Feynman diagram, of which is exactly the triplex-double pole. So in order to deduce the Feynman rule R IX ule , we have no choice but play with the result of CHY-integrand (3.35). In order to get the result of (3.35), one can use the Pfaffian identity, as is already shown in [18,19]. Starting from a template of 3-regular graph, The first diagram in the first line and second line is the one we want to compute, and now we can instead compute the remaining four in the identity, which could be CHY-integrands with only simple poles or higher-order poles. In paper [18,19], one need more Pfaffian identities if some of the remaining CHYintegrands still possess higher-order poles. However, since now we have the Feynman rule for double pole, the remaining four CHY-integrands can be instantly computed, without reducing to those only with simple poles. Anyway, we can phrase the result as On the other hand, summing over all four Feynman diagrams in Figure 8, we should also get the result From results F 1 , F 2 , F 3 it is immediately to see that, the terms T 1 , i.e., with simple poles s 123 , s 234 , s 345 , is really computed by F 1 , F 2 , F 3 . This is consistent with the fact that the triplex-double pole can not produce simple poles s 123 , s 234 , s 345 . The equality and similarly other exchanging of legs. We should reformulate F 4 in such a way that these symmetries are still kept. However, there are too many possibilities to reformulate F 4 due to the momentum conservation and massless conditions, and we have no strategy to choose the one that can be generalized as rule. There is another very important difference here compared to the previous rules. For R I ule , R II ule , R III ule , it is possible to reformulate the numerator of the known results such that the s variable of higher-order pole do not present in the numerator. So we do not need to consider the possibility that whether s in the numerator would cancel a factor of higher-order pole or not. Although not explicitly mentioned, this is a guide line to deduce the Feynman rules R I ule , R II ule , R III ule . However, here it seems impossible to completely eliminate s 12 , s 34 , s 56 dependence in the numerator of F 4 by momentum conservation, thus it is also a possibility that they would cancel a factor of double poles. All these ambiguities makes things difficult. By extensive trial-and-error method with hints from a seven-point CHY-integrand, we finally formulate a valid Feynman rule R IX ule for triplex-double pole as presented in (3.24), and at least it works for the examples we checked up to nine points.

Supporting examples
The Feynman rules R I ule , R II ule , R III ule , R IX ule are deduced but not derived, and their validation is supported by ample examples. The integration algorithm, as reviewed in §2, involves nothing but selecting subsets of nodes according to pole condition, constructing compatible combinations and performing the pattern matching to apply the rules. All these steps can be implemented by a few lines of codes in Mathematica. One is able to produce the analytic result within seconds, while if trying to obtain a result from solving  scattering equations, even numerically it would take for example two or more hours for nine-point CHYintegrands in a laptop(this is the reason why we do not check the rules by ten or even more point examples).
For those who are not familiar with integration algorithm, we present here the results of various CHYintegrands by rules of higher-order poles, and all have been checked numerically. For reader's convenience, we collect all the 4-regular graphs of examples in Figure 9, Figure 10, Figure 11 and Figure 12. In case that some readers are specially interested in certain CHY-integrand, they can find the 4-regular graph in these figures and jump to corresponding subsection for details.

Examples with only double poles and simple poles
In this subsection, we will examine the Feynman rule R I ule of double pole by four examples in Figure 9.   From these combinations we can draw two corresponding Feynman diagrams as shown in Figure 13, and using R I ule we can read out     From them we can draw eight Feynman diagrams as shown in Figure 15, According to the rule R I ule , the

Examples with only triple poles and simple poles
In this subsection, we will examine the Feynman rule R II ule of triple pole by eight examples in Figure 10.
Example Figure 10.a: This is a six-point example with CHY-integrand The five Feynman diagrams then follows in Figure 17.   Figure 18.   Figure 19. Then the answer is trivially given by  Example Figure 10.d: This is a seven-point example with CHY-integrand With them we can draw 15 Feynman diagrams as presented in Figure 20. According to rule R II ule , the 7 6 5 4 6 7 5 4 7 6 5 4 7 6 5 4 7 6 4 5  Example Figure 10.e: This is a seven-point example with CHY-integrand The 30 Feynman diagrams corresponding to this CHY-integrand is shown in Figure 21. Then the answer is given by   The Feynman diagrams for them are presented in Figure 22. Then the answer is given by  where in order to simplify the presentation we use product to denote The corresponding 42 Feynman diagrams are presented in Figure 23. By R II ule we immediately arrive at

Examples with only duplex-double poles and simple poles
In this subsection, we examine Feynman rule R III ule of duplex-double pole by four examples in Figure 11.
Example Figure 11.a: This is a seven-point example with CHY-integrand    The 10 Feynman diagrams can be accordingly drawn as in Figure 27. So according to the rule R III ule , we can write down the answer as

Examples with mixed types of higher-order poles
In this subsection, we will examine Feynman rules for higher-order poles by examples in Figure 12. They all contain Feynman diagrams with mixed types of higher-order poles, thus really serve as extremely non-trivial supporting for the validation of rules.  . (4.31) Example Figure 12.b: This is a seven-point example with CHY-integrand The corresponding eight Feynman diagrams are shown in Figure 30. So applying the rules we get the result  Example Figure 12.c: This is an eight-point example with CHY-integrand From them we can draw nine Feynman diagrams as shown in Figure 31. So according to the rules, the    The corresponding nine Feynman diagrams are shown in Figure 33. Applying the rules, we simply get the result

Discussion and conclusion
The integration rule presented in [18,19] is quite efficient and elegant. As the word rule suggests, there is actually no practical computation but just pattern matching. But the integration algorithm is limited to CHY-integrands with simple poles therein. In order to compute CHY-integrands with higher-order poles, one need to use for example the Pfaffian identities. Although the Pfaffian identities can relate a CHYintegrand of higher-order poles with those of simple poles, the number of terms in the identities suffers from factorial increasing. Also generally one Pfaffian identity is not enough to completely decompose a CHY-integrand of higher-order poles into terms with only simple poles, which makes computation involved. What is worse, some CHY-integrands, for example the one in Figure 10.c, as already stated in [18], has no corresponding 3-regular graph. So it can not be related to other CHY-integrand by Pfaffian identities at all. In this paper, we sharpen the integration algorithm by providing Feynman rules for higher-order poles. With these rules, we can deal with CHY-integrands with corresponding higher-order poles exactly in the same sprint as in [18,19]. For example again the CHY-integrand Figure 10.c, previously not solved, can be instantly obtained by summing over nine Feynman diagrams of our Feynman rule R II ule . Ample examples have been checked, not limited to those provided in §4 of this paper, to support the validation of these rules. Again as the name rule suggests, if we forget the hard work spent in working out these rules, then the computation in fact involves no computation but just pattern matching, and all results come out instantly in Mathematica.
In a recent paper [20], Gomez proposed a so-called Λ-algorithm, to compute CHY-integrands with higher-order poles, in the framework of Λ scattering equations. The Λ-algorithm is quite elegant and pole to another, there might be other better approaches, and one need to figure out the details. Finally, with the Feynman rules of higher-order poles, one is ready to apply it in explicit theories and exploit details therein.