Permutation in the CHY-Formulation

The CHY-integrand of bi-adjoint cubic scalar theory is a product of two PT-factors. This pair of PT-factors can be interpreted as defining a permutation. We introduce the cycle representation of permutation in this paper for the understanding of cubic scalar amplitude. We show that, given a permutation related to the pair of PT-factors, the pole and vertex information of Feynman diagrams of corresponding CHY-integrand is completely characterized by the cycle representation of permutation. Inversely, we also show that, given a set of Feynman diagrams, the cycle representation of corresponding PT-factor can be recursively constructed. In this sense, there exists a deep connection between cycles of a permutation and amplitude. Based on these results, we have investigated the relations among different independent pairs of PT-factors in the context of cycle representation as well as the multiplication of cross-ratio factors.


Introduction
It was first proposed by Cachazo, He and Yuan (CHY) [1][2][3][4] that the tree-level scattering amplitudes of many massless theories are supported by the solutions to the scattering equations where s ij = (p i + p j ) 2 are Mandelstam variables, and σ ij = σ i − σ j are moduli space variables. The CHYformulation consists of an integrand I n that specifies the theory, and a measure on the moduli space that fully localizes the integration to the solutions of the scattering equations (1.1): dµ CHY = (σ rs σ st σ tr ) 2 i =r,s,t dσ i δ(f i ) . (1. 2) The moduli space integration indicates that the CHY-formulation should appear as a certain limit of the string amplitudes, which effectively reduces the path ordered string measure into the color ordered CHY measure. It has been shown that the CHY-formulation naturally emerges as the infinite tension limit of ambitwistor strings [5][6][7], chiral strings [8,9] and pure spinor formalism of superstrings [10,11]. In the context of conventional string theory, the CHY-formulation can also appear as the zero tension limit of an alternative dual model [12]. The scattering equations (1.1) have (n − 3)! solutions, and to obtain the amplitudes, naively one needs to get all the solutions and sum up their contributions. However, solving the equations becomes computationally unavailable as the number of particles grows. This difficulty can be circumvented by using integration rules [13][14][15][16]. The idea behind this approach is that one can obtain the sum over algebraic combinations of the solutions in terms of the coefficients of the original polynomial equations without knowing each individual solution. Using the original integration rules, we can extract the correct amplitudes of those theories without the appearance of higher order poles, for example, the bi-adjoint scalar theory whose integrand consists of two Parke-Taylor (PT) factors, I n = PT(α α α) × PT(β β β) = α 1 α 2 · · · α n × β 1 β 2 · · · β n = 1 σ α 1 α 2 σ α 2 α 3 · · · σ αnα 1 × 1 σ β 1 β 2 σ β 2 β 3 · · · σ βnβ 1 , where α α α = {α 1 , . . . , α n } and β β β = {β 1 , . . . , β n } are two permutations of external particles. On the other hand, theories with more complicated integrands usually involve (spurious) higher order poles. For example, by expanding the Yang-Mills integrand following the way of Lam and Yao [17], one has to develop various techniques to evaluate the higher order poles and show that they indeed cancel towards the end [18][19][20][21][22]. Alternatively, one can expand the integrand in terms of linear combinations of bi-adjoint scalar ones with local coefficients, such that the calculation of higher order poles can be avoided. This approach has succeeded in Yang-Mills, Yang-Mills-scalar and nonlinear sigma model [23][24][25][26][27][28]. The latter approach has an extra benefit that the expansion coefficients are automatically the Bern-Carrasco-Johansson numerators [4,29,30]. It is not surprising that the amplitudes of various theories land on the bi-adjoint scalar ones after a recursive expansion. The reason is that these bi-adjoint scalar amplitudes capture exactly the physical poles associated with various diagrams that are planar under certain color ordering, while different theories just dress these diagrams with different kinematic numerators. The above discussion shows the fundamental role of the bi-adjoint cubic scalar amplitudes in the understanding of CHY-formulation, so it is not surprising that many different approaches have been proposed towards the evaluation and better understanding of CHY-integrand with product of two PT-factors. While many of them focus on the rational function of complex variables σ i 's, in paper [31] the authors have related the PT-factors to the partial triangulations of a polygon with n edges (n-gon), and the PT-factors are deeply connected to the associahedron. It also brings permutation group S n into the story, since each PTfactor is accompanied by a color trace with a definite ordering. We should find one-to-one correspondence between action of S n onto the PT-factors and partial triangulations of n-gon. It would be a very natural idea to understand the CHY-integrand from the knowledge of permutations. Certain progress along this direction has been made in [32] by investigating pairing of external legs, whose results are presented in terms of illustrative objects like crystal and defect. In fact, for the two PT-factors in the CHY-integrand of bi-adjoint cubic scalar theory, if we set one PT-factor as the natural ordering 12 · · · n , corresponding to identity element in the permutation group, then the other PT-factor can be interpreted as a permutation acting on the identity. The physical information, i.e., the poles and vertices of the Feynman diagrams that this CHY-integrand evaluates to, should find its clue in the structure of permutations. In this paper, we try to understand the cubic scalar amplitude by inspecting the structure of permutations. We demonstrate how the physical information is encoded in the permutations, and explore the relations of different PT-factors from the means of permutations as well as other algebraic methods stemmed from the integration rules. This paper is organized as follows. In §2, we set our convention and provide some necessary backgrounds. In §3, we show how the structure of those Feynman diagrams produced by a CHY-integrand can be extracted from the cycle representations of the corresponding PT-factor viewed as a permutation. In §4, we study the inverse problem on how to write out the PT-factor for an arbitrary given Feynman diagram. In the form of cycle representation, we propose a recursive method to construct an n-point PT-factor recursively from lower point PT-factors. In §5, we investigate the relations among different PT-factors via the merging and splitting of cycle representations, as well as via multiplying cross-ratio factors. Conclusion is presented in §6, and in Appendix A, we comment on an interesting interplay between the associahedron and cycle representations of permutation.

The setup
In this section, we give the definitions of some important objects to be used in later.

The canonical PT-factor
Since the 2n PT-factors obtained by acting cyclic rotations and reflections evaluate to the same amplitude, despite an overall sign (−1) n for the latter case, all these 2n PT-factors form an equivalent class. Thus the number of independent PT-factors is n!/(2n). We can represent each independent PT-factors by a canonical ordering α 1 α 2 · · · α n that satisfy two conditions: (1) the first element α 1 is fixed to be 1 to eliminate the cyclic ambiguity, (2) the second element α 2 should be smaller than the last element α n to eliminate the reversing ambiguity. The complete equivalent class can be generated from these independent PT-factors by acting the cyclic rotation and reversing. For example, up to n = 5, we can choose the independent PT-factors as follows, The CHY-integrand for bi-adjoint cubic scalar amplitudes is given by PT(α α α)×PT(β β β), as shown in Eq. (1.3).
A simultaneous permutation acting on α α α and β β β merely leads to the same result up to a relabeling of external legs. Hence we can fix one of the PT-factors to be the natural ordering PT(α α α) = 12 · · · (n − 1)n , and consider the other PT-factor PT(β β β) as a permutation acting on α α α. Thus all the dynamical information is encoded in PT(β β β), from which one can read out the amplitude.

Permutation, cycle representation and PT-factor
Permutations, as group elements of the n-point symmetric group S n , can be defined by their action onto the space spanned by the elements of S n themselves, for example, β β β|e e e = |β β β , γβ γβ γβ|e e e = γ γ γ|β β β = |γβ γβ γβ , (2.1) where β β β and γ γ γ are two generic elements of S n , and e e e is the identity element defined as the natural ordering {1, 2, . . . , n}. Each permutation can be represented by a product of disjoint cycles (i 1 i 2 · · · i s ), which stands for the map i 1 → i 2 , i 2 → i 3 , . . ., i s → i 1 . For example, (123)(4)(5) · · · (n)|1234 · · · n = |2314 · · · n (2.2) stands for the permutation in Cauchy's two-line notation 1 2 3 4 · · · n 2 3 1 4 · · · n . (2.3) Cycles are defined up to a cyclic ordering, for example, (123) = (231) = (312) gives the same permutation. It is also obvious that two disjoint cycles commute, i.e., (123)(45) = (45)(123). Each permutation has a unique decomposition in terms of disjoint cycles, modulo the cyclicity of each cycle and the ordering of disjoint cycles. 1 We call this unique decomposition the cycle representation of a permutation in the rest of this paper. The number of disjoint cycles in a cycle representation is called the length of this cycle representation, and the number of elements in a cycle is called the length of cycle.

From permutations to Feynman diagrams
One of our motivations is to explore the information encoded in the PT-factors, described in the form of permutations. As mentioned in the previous section, in our setup the amplitude result is determined by the second PT-factor PT(β β β), considered to be a permutation acting on the identity element. It determines an equivalent class containing 2n elements evaluating to the same amplitude. Thus we need to consider all the permutations in the equivalent class. For example, when working with the CHY-integrand PT(α α α)×PT(β β β) = 1234 × 1243 , we need the equivalent class of 1243 , containing eight elements, 1243 , 2431 , 4312 , 3124 , 3421 , 4213 , 2134 , 1342 , (3.1) or in the form of cycle representations, Our purpose is to relate these cycle representations to the Feynman diagrams contributing to the amplitude. All the above eight cycle representations in (3.2) can be used to reconstruct the PT-factor PT(β β β) by acting them on the natural ordering 1234 , so each one encodes the complete information for evaluation. However, they have different structures. In Eq. (3.2), two cycle representations are length-1, four are length-2, while the remaining two are length-3. How can we read useful information out of these different cycle structures?
To answer this question, let us recall the integral result of the CHY-integrand 1234 × 1243 , which is − 1 s 12 . It corresponds to a Feynman diagram with two cubic vertices, one connecting the legs 1, 2 and the internal propagator P 12 := p 1 + p 2 , while the other connecting the legs 3, 4, and the internal propagator −P 12 . It is very plausible to conjecture that, the two cycle representations (1)(2) (34) and (12)(3)(4) in fact describe respectively the two cubic vertices. In (1)(2)(34), the two cycles (1) and (2) describe respectively the external legs 1 and 2 attached to a vertex, while the cycle (34) describes the corresponding internal propagator of that vertex. It also indicates that 1 s 34 = 1 s 12 is an internal pole. Similar analysis can be carried out for the cycle representation (12)(3)(4).
Above discussion tells us that, although each cycle representation contains the complete information of amplitude, the pole structure is manifest in some of them but not all. The complete picture of Feynman diagrams is determined collectively by all cycle representations whose pole and/or vertex structures are manifest. With this understanding, we only need to consider those good cycle representations, i.e., the pole and/or vertex structure are manifest. For bi-scalar theory, the physical pole would appear only for the external legs with consecutive ordering. So let us define the good cycle representations as those satisfying the following criteria: • the cycles in the considered cycle representation can be separated into at least two parts, while the union of cycles in each part is consecutive (later called planar separation).
• in case that the cycle representation can only be separated into two parts, then each part should contain at least two elements.
Moreover, if the planar separation of a good cycle representation contains at least three parts, we call it a vertex type (V-type) cycle representation. Otherwise, we call it a pole type (P-type) cycle representation. Let us give a few more examples. At six point, both (12)(34)(56) and (12)(35)(46) are good cycle representations. The former is a V-type one since it can be separated into three parts (12), (34) and (56), while the latter is a P-type one since it can only be separated into two parts (12) and (35) Combining these two, we do produce all the vertices in the effective Feynman diagram.
The above example shows that by collectively combining the information from all good cycle representations, we can read out the complete Feynman diagram result. This provides one method of analysis. On the other hand, we can arrive at the complete final result by relying on only one good cycle representation, since each one should contain the complete information of PT-factor PT(β β β). Hence we should have another method of analysis. Observations from practical computation show that, (A) All V-type cycle representations together manifest the vertex structure of the corresponding effective Feynman diagram.
(B) It is also possible to reproduce the effective Feynman diagram from one V-type cycle representation if we recursively use the lower multiplicity results.
(C) One P-type cycle representation is not sufficient to reproduce the complete result, and in order to get the correct answer we should consider all P-type cycle representations.
We use again the example (3.4) to demonstrate our observations. We start with the V-type cycle representation (1)(2)(38)(4)(56) (7). In the four-part separation (3.5a), we first coarse grain the part {8123} and {56} by replacing them by a single propagator. This leads to an effective quartic vertex, , which gives the contribution   where the {P 12 , 1, 2} part is another cubic vertex, following the analysis of (3.7). Thus we get the contribution 1 s 812 + 1 s 123 . When combining with (3.6), we do get the complete result (3.3). The above calculation shows that by recursively looking into the V-type cycle representations of each substructure, we can reproduce the full Feynman diagram result. Now we move to the P-type cycle representation, for example, the planar separation (132)| | |(4875) (6). We need to analyze the two substructures given by cycle representations (P 123 )(132) and (P 123 )(4875) (6). Using the algorithm given in (2.6), the substructure (P 123 )(132) gives the following eight equivalent cycle representations V-type :   With the above example in mind, let us move on to the systematic investigation of four, five and six point PT-factors. For presentation purpose, we shall organize the independent PT-factors into categories according to the topology of corresponding Feynman diagrams. In the same category, different PT-factors are related by group actions, and can be analyzed in the same manner. Concretely, we can define the group action as follows. In the space of n! 2n equivalent classes b[β β β], we define the permutation action where C and R also generate a dihedral group D n . 3 The action of D n further separates the space of b[β β β] into different orbits. The number of elements inside each orbit depends on the symmetric property of such orbit.
For example, the identity permutation PT(β β β) = 12 · · · n is invariant under D n action such that it forms a one dimensional orbit by itself. A nontrivial example is that by acting D n onto PT(β β β) = 12846573 , we get an orbit with four elements: The above discussion is useful because, all PT-factors in the same orbit of D n share the same structure of the cycle representations. Most importantly, starting from the V-type and P-type cycle representations of one PT-factor in the orbit, we can get the V-type and P-type cycle representations of all the other PT-factors in this orbit simply by the mapping i → i + 1 or i → n + 1 − i. Later on we will only study one PT-factor for each orbit.
After above general discussion, we present more example to further elaborate our algorithm. In appendix A, we will give some further discussions on the cycle structure of PT-factors and Feynman diagrams. For the PT-factor 1234 , we can read out the complete vertex information from the sole V-type cycle representation (1)(2)(3)(4). This PT-factor gives all trivalent four point Feynman diagrams whose four external legs are connected at cubic vertices respecting the color ordering, and the result is simply 1 s 12 + 1 s 23 . In the language of planar separation, the V-type cycle representation can be separated into four parts (1)| | |(2)| | |(3)| | |(4), which can be explained as defining an effective quartic vertex with exactly the same meaning as mentioned above. This structure will be one of the building blocks for the analysis of higher point Feynman diagrams.

The three and four point cases
However, the two P-type cycle representations alone only provide partial result. For example, the planar separation of cycle representation (41)| | | (23) indicates that, legs 2 and 3 are connected to the same cubic vertex while legs 4 and 1 are connected to another, resulting in a contribution of 1 s 23 . This is half of the complete answer, and the remaining part is given by the other P-type cycle representation (12)| | |(34), leading to 1 s 12 .
(4) The cycle representations for the last PT-factor 13524 are There is no good cycle representation at all, so the contribution is zero, which is indeed the case.

The six point case
There are in total 6! 12 = 60 independent PT-factors for the six point case. According to the number of trivalent Feynman diagrams they evaluate to, we can distribute them into different groups, with the number of PT-factors in each group as # of trivalent Feyn. diagrams We will study them group by group in the following paragraphs.
With 14 Feynman diagrams: There is only one PT-factor PT(β β β) = 123456 that evaluates to 14 Feynman diagrams. In the equivalent class, the good cycle representations are The sole V-type one indicates that the six external legs form all possible cubic diagrams respecting the color ordering, contributing to 14 terms, Again, the planar separation (1)| | |(2)| | |(3)| | |(4)| | |(5)| | |(6) tells us that the above 14 terms in (3.18) can be effectively represented by a six point vertex, which becomes a building block for higher point analysis.
Finally, each of the three cycle representations in the second row of (3.17b) gives four terms in Eq. (3.18). For example, the planar separation (1)(26)| | |(35)(4) gives a propagator s 612 , together with two four point substructures (P )(1)(26) and (P )(35)(4). In their equivalent classes, the former has a V-type cycle representation (P )(6)(1)(2) while the latter has (P )(3)(4)(5), both of which are effective quartic vertices, so that their contribution is Summing over the contributions of these three P-type cycle representations, we reproduce the twelve terms in Eq. (3.18) that contain a three-particle pole s ijk .
With 5 Feynman diagrams: There are six PT-factors in this category, The last five PT-factors can be generated from the first one by cyclic permutation i → i + 1. They actually form an orbit under D 6 action. In the equivalent class of 123465 , the good cycle representations are As an alternative approach, we repeat the result by using only one V-type cycle representation and recursively those of substructures. For example, besides the cubic vertex, the separation (14)(23)| | |(5)| | |(6) also indicates a substructure (P )(14) (23), which has the V-type cycle representation (P )(1)(2)(3)(4) in its equivalent class generated by (2.6). Thus we again recover the effective five point vertex.
For the P-type cycle representations, each one contributes two terms in (3.26). For example, the planar separation (12)| | |(3456) gives a pole s 12 and a substructure (P 12 )(3456), and the substructure has an equivalent V-type cycle representation (P 12 )| | |(3)| | |(5)(46) in the equivalent class. We can then read out the pole s 456 and another substructure (P 456 )(5)(46). This substructure further gives a V-type cycle representation (P 456 )(4)(5)(6), indicating a quartic effective vertex. Putting them together, we get Similar analysis can be done for other three P-type cycle representations, and the results are  The planar separations in the V-type cycle representations indicate the following vertex structures, Combining them together, we get an effective Feynman diagram with two cubic vertices and one quartic vertex, and the result is We can also reproduce above result by considering only one V-type cycle representation and its substructures. For example, the planar separation (1)| | |(2)| | |(3546) gives the substructure (P )(3546), which has a V-type cycle representation (P )(3)(4)(56) generated by (2.6). Thus the substructure contains a quartic vertex with legs P 12 , 3, 4, P 56 , and a cubic vertex with legs 5, 6, P 56 .
Both of them recover the result (3.33) respectively. For the P-type cycle representations, we note that (12) There is no P-type cycle representation. This can be understood as follows. Since the final result only contains one term, so that we do not have partial result. From the V-type cycle representations, we see that each planar separation of (1 (5) and (2) also indicates a substructure (P )(465), which contains a V-type cycle representation (P )(45) (6), giving exactly the other two cubic vertices in Eq. (3.36). Analysis for the other V-type cycle representations is the same. For the category S 2 of (3.34), we analyze 125463 as example. Its good cycle representations are  Both of them manifest a pole P 12 , together with a substructure (P )(3465) and (P )(3564) respectively. However, both substructures are members of (3.15), which give zero contribution. This can be seen clearly if we replace P by 2, and then replace i → i − 1 for the rest. Thus we see that the category S 1 evaluates to zero because it contains a substructure with zero contribution. Actually, the category S 2 of (3.41) evaluates to zero for the same reason. For example, 125364 has good cycle representations V-type : (1)(2)(3564) , P-type : (12)(3465) ,   .
Collectively considering all these four V-type cycle representations, and gluing them via propagators, we get the Feynman diagram as By connecting the substructures together, we obtain the effective Feynman diagram as in (3.45).
There are also four P-type cycle representations. As mentioned previously, they should be considered collectively in order to produce the complete result, while each one only contributes a partial result. From the planar separations of these cycle representations and their substructures, we can work out the contribution of each P-type cycle representation. We will not repeat the detailed analysis here, but only give the result as, The first planar separation indicates a five point vertex, which corresponds to five possible terms, while the second planar separation indicates a quartic vertex which corresponds to two possible terms. So gluing them together we get the nine point effective Feynman diagram as Alternatively, we derive the above result from one V-type cycle representation and its substructures. For instance, the planar separation (1)| | |(2)| | |(3)| | |(48)(5)(67)| | |(9) indicates a five point vertex, while the substructure (P 1239 )(48)(5)(67) has further structure. By working out the equivalent class of (P 1239 )(48)(5)(67), we find a V-type cycle representation of this substructure that allows the planar separation (4)| | |(567)| | |(8)| | |(P 1239 ), indicating a quartic vertex and a four point substructure (567)(P 567 ). Then the equivalent V-type cycle representation (5)| | |(67)| | |(P 567 ) of this four point substructure indicates two cubic vertices with legs 5, P 67 , P 567 and 6, 7, P 67 respectively, according to our four point discussion in §3.1. The above recursive process can be graphically represented by Hence from one V-type cycle representation it is sufficient to obtain the complete result.
On the contrary, if taking only one P-type cycle representation, we will end up with a partial result. For instance, if we take the P-type cycle representation (19432)(586)(7), it has only one planar separation (19432)| | |(586)(7) that splits the external legs into two parts. For both the substructures (19432)(P 5678 ) and (P 5678 )(586) (7), we can work out their contributions by recursively going into their substructures. We will not repeat the details but show the result as follows, The corresponding effective Feynman diagram obtained by gluing these two subdiagrams contributes only half of the full result, since the leg 4 and 8 is connected in just one way of the two allowed by the quartic vertex {P 9123 , 4, P 567 , 8} in the original Feynman diagram. We need to include all the P-type cycle representations to reproduce the complete result.

From Feynman diagrams to permutations
In §3, we have addressed the problem that given a PT-factor as a permutation acting on the identity element, how we can determine the Feynman diagrams the CHY-integrand evaluated. In this section, we will consider the inverse problem, namely, given an effective Feynman diagram, how to obtain directly the corresponding good cycle representations. We will show that there is a recursive construction to produce the good cycle representations of a given Feynman diagram from the relation between subdiagrams and planar separations. Later, we will use the eight point example given in Fig. 1 to illustrate general discussions. We remind the readers that an m-point vertex in the Feynman diagram stands for the sum of all the m-point trivalent diagrams. For example, a quartic vertex gives the sum of s and t channel trivalent diagrams. The Feynman diagram in Fig. 1a corresponds to the CHY-integrand 12345678 × 12783654 , which evaluates As we have mentioned in previous section, the PT-factor PT(β β β) determines a permutation acting on identity element and it encodes the pole structure of the Feynman diagram. Conversely, the pole structure in the Feynman diagram also encodes the information of permutation. To describe the way of reading out the PT-factor and cycle representations, we find it is more convenient to introduce the polygon with n edges (n-gon) that is dual to the n-point Feynman diagram under inspection. An n-point effective Feynman diagram can be described as partial triangulation of n-gon diagram [31,33,34], and an example of our considered eight point Feynman diagram is presented in Fig. 1. Each external leg is dual to an edge of the n-gon, and each vertex is dual to a subpolygon in the interior. A triangulation line inside the n-gon, which cut it into two subpolygons, is dual to a propagator. If the Feynman diagram considered is trivalent diagram with only cubic vertices, the corresponding n-gon is completely triangulated, while if it is an effective Feynman diagram with also higher point vertices, the n-gon diagram is partially triangulated. If we use E i to denote the number of edges of a subpolygon inside the original n-gon, then the number of terms in the final result is given by i∈all polygons where C(n) is also the number of all possible n-point color ordered trivalent Feynman diagrams. The blue line in Fig. 1b represents the triangulation of our considered example, and the dashed gray line gives the Feynman diagram dual to the partial triangulation of n-gon. Discussion on the Feynman diagram can as well be applied to the n-gon diagram, and the latter is naturally enrolled in the associahedron [31,34].

The zig-zag path and cycle-representation of permutation
Now let us return to the problem of reading out the PT-factor from Feynman diagram. A solution for this problem has been provided in paper [15], where a pictorial method has been proposed to write the PT-factor for a given Feynman diagram. Based on their discussion, we will rephrase it by the language of zig-zag path. 5 The basic idea comes as follows, • Any tree-level Feynman diagram can be placed as a planar diagram, while the external legs lying in the plane apparently define an ordering, identified as PT-factor PT(α α α).
• Starting from any external leg, we can draw a zig-zag path along the boundary of diagram, which crosses each internal line it meets and closes at the starting point. The ordering of legs along the direction of zig-zag path is identified as PT-factor PT(β β β).
The zig-zag path for our considered example is shown in Fig. 2, both in the Feynman diagram and n-gon diagram. In the Feynman diagram, the zig-zag path is along the external legs, while in the n-gon diagram, it is along the interior of edges. In both diagrams, the path crosses the lines whenever they are propagators. It is easy to tell that, for the Feynman diagram shown in Fig. 2, we have PT(α α α) = 12345678 , while along the arrows of zig-zag path, we can read out PT(β β β) = 12783654 , as it should be. However, there are two subtleties we should pay attention to. The PT-factor PT(β β β) = 12783654 is obtained under the condition that we read out the zig-zag path from leg 1 towards a specific direction. If starting from the same leg 1 but with opposite direction, we will get PT(β β β) = 14563872 . If we start from another leg and a chosen direction, we will get another PT-factor, though they are in the same equivalent class. Special attention should be paid to the orientation of zig-zag path. It is a local but not global property, and it only make sense with respect to a vertex. For example, for the cubic vertex where legs {1, 2} attached to, the zig-zag path around it is clockwise, while for the quartic vertex where legs {4, 5, 6} attached to, the zig-zag path around it is anti-clockwise. The orientation of zig-zag path is more obvious in the n-gon diagram, as shown in Fig. 2b. For each polygon inside the n-gon, the zig-zag path can be considered as a closed loop with definite orientation. If the zig-zag path in the triangle with edge 1, 2 is clockwise, then the zig-zag paths in the triangles with legs 7, 8 and with legs 3 are also clockwise, while zig-zag paths in the triangle in middle and the quadrangle are anti-clockwise. To the whole Feynman diagram or n-gon diagram, we do not need to worry about the ambiguity of the zig-zag path orientation, since our canonical definition of PT-factors in §2 fixes which leg to start and which direction it should ahead. However, as we would discuss soon, the orientation of zig-zag path is important in the recursive construction of PT-factors. Now we consider splitting the Feynman diagram into two subdiagrams (labeled as L and R) at the propagator P 7812 that connects the vertex V 1 and V 2 , as shown in Fig. 3. In each subdiagram, the zig-zag path form a closed loop, from which we can read out the PT-factors. In order to define the canonical ordering of PT-factors, we read out the α α α-orderings from both subdiagrams in the clockwise direction as PT(α α α L ) = 7812P , PT(α α α R ) = P 3456 . Next, we read out PT(β β β L,R ) from the zig-zag paths of both subdiagrams, starting from the leg P . We emphasize that the zig-zag path for subdiagram L is anti-clockwise with respect to the vertex V 1 , while the zig-zag path for subdiagram R is clockwise with respect to V 2 . The orientations of two zig-zag paths are always opposite with respect to the two vertices connect by the split propagator. This is a generic and important feature since, as we mentioned before, the orientations of zig-zag paths of two adjacent polygons in the n-gon diagram are always opposite. This feature will play a consequential role in determining the cycle representations for subdiagrams. Now we write down the PT-factors for the subdiagrams according to the arrows in the zig-zag paths of Fig. 3a as PT(β β β L ) = P 1278 , PT(β β β R ) = P 3654 .   17)(28)(3)(46)(5). 6 To obtain the above result, we have to select two specific cycle representations for the subdiagrams out of the 10 equivalent ones of β β β L,R , β β β L : (P ) (17) First of all, we choose those that leave P in a single cycle as (P ) in the equivalent class. This is reasonable since when gluing subdiagrams, P should get eliminated without affecting other cycles, which is possible only if P is in a single cycle. This limits us to the two in the first column of (4.6). Next, we need to answer which one to choose among the two. Looking back to (4.5), we find that β β β R is a V-type cycle representation that manifests the vertex V 2 , while β β β L is not a good cycle representation. If we color the legs according to how they are separated by V 1 and V 2 , then for PT(β β β R ), we have (P )(3)(46)(5), namely, each cycle contains elements with the same color, i.e., elements from the same part of the V 2 splitting. We say that this cycle representation satisfies planar splitting for short. In contrary, for PT(β β β L ), some cycles contain elements with different colors, i.e., the elements inside one cycle are from different parts of the V 1 splitting. We call it cycle representation non-planar splitting for short. One notices that among the two subdiagrams of Fig. 3a, one cycle representation is planar splitting while the other is non-planar splitting. For instance, if the arrows in Fig. 3a is reversed globally, then for PT(β β β L ) the consequent cycle representation is (P )(78)(12) while for PT(β β β R ) it is (P )(3456). Again, one cycle representation is planar splitting and the other is non-planar 6 Now the operation ⊕ can be properly defined as follows: if β β β1 and β β β2 are two permutations that have only one overlap element P , and in both of them P sits in a single cycle, namely, β β β1,2 = (P )β β β 1,2 , we have β β β1 ⊕ β β β2 = β β β 1 β β β 2 .
splitting. By gluing them together, we get (78)(12)(3456), which is another V-type cycle representation of the complete Feynman diagram. This feature results from the fact that the orientations of the subdiagram zig-zag paths are opposite with respect to the two vertices connected by the split propagator. It is generally true no matter how we cut the complete Feynman diagram. The above discussion indicates clearly that the permutation representation of PT-factor can be recursively constructed by breaking a complete Feynman diagram into subdiagrams along internal propagators. We will provide a systematic and abstract construction in next subsection.

The recursive construction of PT-factor via cycle representation
We already have the experience that, (1) in the gluing of two subdiagrams to one complete diagram, cycle representation of one subdiagram should be planar splitting, and that of the other should be non-planar splitting, (2) the planar splitting cycle representation corresponds to a zig-zag path in clockwise direction, while the non-planar splitting cycle representation corresponds to a zig-zag path in anti-clockwise direction. The orientation of zig-zag path is related to the planar or non-planar splitting of cycle representations because we use the convention that PT(α α α) is obtained by traversing the external legs in clockwise direction.
Let us head to a more systematic and abstract discussion on the recursive construction of cycle representation for a Feynman diagram. where the total number of vertices v m falls between 1 and n − 2. An illustration of the n-point Feynman diagram as well as the dual n-gon diagram with also the zig-zag path is shown in Fig. 4.
Let us focus on a m-point vertex, marked as a black dot in the middle of the blue octagon 7 in Fig. 4. This vertex connects m subdiagrams via m propagators P k with k = 1, 2 . . . m. Our goal is to write the cycle representation of n-point PT-factor into a form that manifests the factorization into those cycle representations of the m subdiagrams connected to the m-point vertex. Since according to our convention, the PT(α α α m ) is read in clockwise direction, we have PT(α α α m ) = P 1 P 2 · · · P m (4.8) for the subdiagram inside the blue octagon. We intentionally choose the direction of zig-zag path as clockwise with respect to the considered m-point vertex, so that for this subdiagram, we have PT(β β β m ) = P 1 P 2 · · · P m ∼ (P 1 )(P 2 ) · · · (P m ) . (4.9) Note that among the 2m equivalent cycle representations of PT(β β β m ), this is the only one that allows every P k appear as a single element in a cycle. Now consider the m subdiagrams in the other side of P k , denoted as A 1 , A 2 , . . ., A m . Since P k is an external leg of A k , we know a priori that there must be a cycle representation is to be determined. So when gluing all the m subdiagrams to the one inside the octagon by propagator P k 's, we obtain the follow factorization form β β β cyc-rep = (P 1 )(P 2 ) · · · (P m ) ⊕ (P 1 )β β β cyc-rep (4.10) namely, it allows a planar separation into m parts. Once the cycle representations of PT-factors for subdiagrams are known, the complete cycle representation is simply a combination of them. Note that the factorization (4.10) is based upon a given vertex. Now we can go into each subdiagram A i and perform the same construction, until we reach a subdiagram with only one vertex. A remaining problem is that suppose all the cycle representations of a subdiagram A i is known, how do we choose the β β β cyc-rep A i that is used as the building block of (4.10). According to Fig. 4, the subset A i and propagator P i form a (|A i | + 1)-point subdiagram, connected to the remaining parts via the propagator P i . In order to connect it with the subdiagram inside octagon, we already constrain the cycle representation of this subdiagram as (P i )β β β cyc-rep From the definition of equivalent class (2.6), we know that there are two cycle representations satisfying this condition. One of them can be constructed according to (4.10): suppose the propagator P i is connected to an (s + 1)-point vertex in subdiagram A i , marked as black dot in the interior of blue rectangular in Fig. 4. This (s + 1)-point vertex splits the subdiagram A i into s disjoint sub-subsets a i 1 to a is via s propagator P i with = 1, . . . , s. Then following (4.10), we should choose the zig-zag path inside the rectangle in clockwise direction, from which we can obtain a cycle representation satisfying the planar splitting, However, since we have already set the zig-zag path around the octagon to be clockwise, the zig-zag path around the rectangle must be anti-clockwise, and what we should use is the cycle representation other than (4.11) with P i in a single cycle. Thus we can obtain (P i )β β β cyc-rep A i by acting reversing and cyclic rotation onto (4.11). Moreover, the β β β cyc-rep A i obtained this way must be non-planar splitting, namely, at least one cycle of β β β cyc-rep A i contains elements from different subsets a i . In other words, there must exist at least one cycle that can not be a part of any β β β cyc-rep a i . To prove this point, it is suffices to study the following problem. Given an identity element in the permutation group, which splits into three planar parts {P i }, {a 1 , . . . , a i } and {b 1 , . . . , b j }. Let us consider the planar splitting cycle representation (P i )a a a cyc-rep b b b cyc-rep that maps the identity element (4.12) into another permutation as (4.14) Since b j ∈ {b 1 , . . . , b j } and a 1 ∈ {a 1 , . . . , a i }, it is clear in (4.14) that the legs in two subsets {a 1 , . . . , a i }, {b 1 , . . . , b j } must appear together in at least one cycle in the cycle representation.
To recap, the cycle representation of a Feynman diagram can be written as a simple combination of cycle representations of subdiagrams as presented in (4.10). This leads to a recursive construction of cycle representation from those of lower point Feynman subdiagrams. Crucially, the cycle representation of subdiagram A i that used in the complete cycle representation (4.10) should be the one with P i as a single cycle (P i ) and be non-planar splitting with respect to its vertex connected to P i . Then we can work out the permutation from cycle representation and eventually the PT-factor PT(β β β). Note that, it is possible to start the recursive construction from any vertex of a Feynman diagram, and different choice leads to different cycle representation but they are all in the same equivalent class. We will show in Appendix A that different planar splittings characterize the shapes of the associahedron boundaries.

Examples
Let us now present some nontrivial examples to illustrate the recursive construction of cycle representation. First we consider the three point diagram. The cubic vertex splits diagram into three subdiagrams, each one is trivially a single external leg. This is also true for the diagrams with only a single vertex. So following (4.10), we get  In this case, we should use the non-planar splitting cycle representation (a 1 a 2 )(P ) instead of (a 1 )(a 2 )(P ). To see this, let us proceed to a four point Feynman diagram as shown below. If we start from the vertex marked by red dot, the diagram splits to three subdiagrams, two of which are single external legs and one is three point subdiagram. The non-planar splitting for three point subdiagram appears as (12)(P 12 ), so according to (4.10), the recursive procedure is described as follows, (4.17) From above results, we can recursively compute the cycle representation of PT(β β β) for five point CHYintegrand. Here we present an example as follows, and construct the cycle representation starting from two different vertices respectively. If we start from vertex V 1 , then the diagram is split to three parts: the external leg 4, 5 and a four point subdiagram. As mentioned above, (12)(3)(P 123 ) is a planar splitting cycle representation with respect to V 2 , and we should take the non-planar splitting one in the equivalent class, namely, (132)(P 123 ), to form the complete cycle representation, which is obtained from (12) Alternatively, we can start from vertex V 2 . Then the diagram is split to another three parts, two of which are three point subdiagrams with non-planar splitting cycle representation (12)(P 12 ) and (45)(P 45 ), while the other is the single external leg 3. Connecting them via the vertex (P 12 )(P 3 )(P 45 ), we obtain (12)(3)(45). We see that different splitting of diagram leads to different cycle representations. However, all of them are in the same equivalent class. In fact, both (132)(4)(5) and (12)(3)(45) lead to the PT-factor PT(β β β) = 12453 .
Next, we give a seven point example. The Feynman diagram is shown below, together with the resultant cycle representations when we carry out the recursive construction at different vertices, Some brief explanation is in order. For the vertex V 4 , the planar splitting cycle representation is (P )(56) (7), while its non-planar splitting one is [(7)(56)] · [(75)(6)] = (576). The former can be used in the recursive construction starting from vertex V 4 , while the latter can be used in the recursive construction starting from vertex V 3 . Similarly, for the vertex V 3 , the planar splitting cycle representation is (P )(4)(576), while the non-planar splitting one is [(4)(576)] · [(47)(56)] = (467) (5). The latter can be used in the recursive construction starting from vertex V 2 . This recursive construction can be easily taken to higher points, and we have run extensive checking up to eight point diagrams.

Relations between different PT-factors
After clarifying the relations between permutations of PT-factors and the Feynman diagrams, we move on to the relations between different PT-factors in the language of permutation and cycle representation. This topic has been discussed from the associahedron point of view [31] and before proceeding let us briefly review their result. The major conclusion is that, the canonical form of an (n − 3)-dimensional associahedron is the n-particle tree-level amplitude of bi-adjoint scalar theory with identical ordering. A consequence is that, the codimension d faces of an associahedron are in one-to-one correspondence with the partial triangulations with d diagonals, while the partial triangulations are dual to cuts on planar cubic (1)(23)(4) (1)(2)(3) (4) . Figure 5: The associahedron for four point amplitudes and PT-factors. diagrams with each diagonal corresponding to a cut. Hence the faces of the associahedron are dual to the singularities of cubic scalar amplitude. In this sense, PT-factors can also be related to corresponding faces of associahedron. For instance, a three point amplitude is dual to a triangle, allowing only one trivial triangulation. Thus the corresponding associahedron is just a zero dimensional point, on which sits the only independent PT-factor PT(β β β) = 123 . A four point amplitude is dual to a box, while the associahedron formed by its (partial) triangulations is a one dimensional line as shown in Fig. 5. The vertices correspond to all complete triangulations of the box, while the edge corresponds to partial triangulations. The edge is related to PT(β β β) = 1234 with amplitude 1 s 12 + 1 s 23 , while the two ending vertices are related to PT(β β β) = 1243 and 1324 respectively with amplitude 1 s 12 and 1 s 23 . The relations between different PT-factors are manifest in this geometric picture.
A five point amplitude is dual to a pentagon, and the associahedron constructed from all its (partial) triangulations is also a pentagon, as shown in Fig. 6, where the thick black lines form the associahedron and the blue line is the triangulations of pentagon. The face corresponds to 12345 , while the edges correspond to 12543 , 12354 , 13245 , 14325 , 12435 , and the vertices correspond to 12453 , 13254 , 14235 , 13425 , 12534 . The PT-factor of each vertex evaluates to a single Feynman diagram. An edge connects two vertices, which means that the PT-factor of the edge evaluates to two Feynman diagrams. Two edges share a common point, which means that there is common Feynman diagram shared by them. The face contains five vertices, such that its PT-factor evaluates to five Feynman diagrams. Therefore the relation among different PT-factors is obvious in the diagram. For higher point amplitude, the associahedron would be much more complicated geometric objects, however the correspondence is similar.

Relation analysis via cycle representation
From the n-gon picture, we see that by adding one more triangulation line (i.e., fix one more propagator), we get the immediate child amplitude. In contrary, by removing one triangulation line (i.e., unfix a propagator),  Figure 6: The associahedron for five point amplitudes and the PT-factors.
we recover the mother amplitude. In terms of zig-zag path, we have the following picture, The extra triangulation separates the n-gon into two subpolygons. Notably, in one of them we need to reverse the ordering. In this section, we study how the above picture is realized by good cycle representations, namely, how to merge or split certain parts in good cycle representations to fix or unfix a propagator. We start with the general discussion. The left hand side of Eq. where β β β r simply flips the ordering β β β r |i, i + 1 . . . j = |j . . . i + 1, i . Similar to Eq. (2.5), β β β r has the cycle representation Therefore, the process of (5.1) can be realized in an algebraic way as 8 β β β lower β β β upper ⇐⇒ β β β lower β β β upper β β β r .
If the propagator manifested in (5.2) is an overall one, then the above process gives the immediate mother amplitude that has the original amplitude as a part. Otherwise, the above process gives the immediate child amplitude that contains this propagator as an overall factor, and is a part of the original amplitude. Next, we use the example given in After showing how to get the child amplitudes, we now discuss the case of mother amplitudes. This happens when we perform the prescription (5.4) to a separation that manifests an overall propagator. For the case (5.7), there are three common poles s 12 , s 56 and s 8123 . Relaxing any one of them, we can get a mother amplitude. The procedure is similar to the above discussion, and we again do it one by one.

Relation analysis via cross-ratio factor
We can also study the relations of PT-factor from another approach. As we have already known, the relations between different PT-factors can be seen as selecting terms corresponding to specific pole structures in the evaluated results. For instance, 1234 1234 evaluates to 1 s 12 + 1 s 23 while 1234 1243 evaluates to 1 s 12 . It means that, by selecting terms with pole 1 s 12 in 1234 1234 , we can reproduce the result of 1234 1243 . To achieve this goal at the CHY-integrand level, we can use the cross-ratio factor given in paper [38], which we will call it the selecting factor To pick up the Feynman diagrams with a pole 1 s A from a given CHY-integral result, where A follows a certain color ordering, we propose to multiply the CHY-integrand with a selecting factor f select [A −1 , A 1 , A −1 , A 1 ], where A 1 , A −1 are the first and last elements of the subset A respectively, and A is the complement subset of A, i.e., in order to pick up terms with pole 1 s A , the arguments b, c in the selecting factor should be the two ending legs in the set A, while the arguments a, d are the nearby two legs of b, c outside the set A respectively. As an illustration, let us consider the above mentioned PT-factors 1234 and 1243 . We want to select terms with 1 s 12 pole in the evaluated result of 1234 1234 , which means that we need to take the selecting factor f select [4,1,2,3], There is a subtlety in the choice of the selecting factor. The bi-adjoint scalar theory has two color orderings, given by PT(α α α) and PT(β β β). We should choose A to follow one of the orderings. If α α α = β β β, there is no ambiguity in defining the selecting factor, which is the situation discussed in [38]. However if α α α = β β β, the multiplication has two choices for a given set A. The arguments of f select depend on the color ordering of legs, and we have two color orderings to rely on. It can be shown that although we would get two different selecting factors, the resulting CHY-integrands are equivalent in the sense that the difference of two CHY-integrands evaluates to zero. We note that this happens only when 1 s A is indeed a physical pole of the integrated result of PT(α α α) × PT(β β β), but not an overall one. Now let us present a brief explanation about why the selecting factor f select is able to pick up terms with specific poles. It is known that from [14][15][16], the order of pole 1 s A in the evaluated result is characterized by the pole index The linking number L[A] can be read out from the so-called 4-regular diagrams, which is discussed in details in [14][15][16].
a further look on the selecting factor f select , assuming that 1 s A is not an overall pole. The combinations in both the numerator and denominator of f select represent lines connecting elements in subset A and its complement A, so that f select will not change the linking number of A itself: after multiplying f select , we still have χ[A] = 0 and the pole 1 s A remains unchanged. Now suppose there is another pole 1 s B , where B has nonempty overlap with both A and A, then it will be removed by f select . The reason is that in the denominator of PT-factor there are which is a completely irrelevant answer.
In the second situation, the pole 1 s A we pick is overall to all the terms. By multiplying the selecting factor, we produce the mother amplitude, obtained by pinching the propagator 1 s A in the Feynman diagram. For example, there are two overall poles 1 s 12 and 1 s 123 in (5.25). If we follow the color ordering of the first PT-factor, and multiply f select [ indicating that (5.25) can be part of the mother amplitudes with different color orderings.
Before closing, we give a criterion on whether a pole s A is overall with fixed PT(α α α). First, s A is a physical pole iff A is consecutive with both PT(α α α) and PT(β β β). Next, we define {A −1 , A 1 , A −1 , A 1 } according to PT(β β β), and put PT(α α α) into the unique form PT(α α α) = A 1 . . . A −1 . . . . Then s A is an overall pole iff in PT(α α α) we have A −1 ≺ A −1 ≺ A 1 , namely, A −1 precedes A 1 . Otherwise, s A is not an overall pole. This can be easily understood by the zig-zag path in n-gon. Since an overall pole corresponds to a partial triangulation line in the n-gon, a reverse of ordering must happen when the zig-zag path cross this triangulation line.

Conclusion
The CHY-integrand of bi-adjoint cubic scalar theory consists of two PT-factors as PT(α α α) × PT(β β β). Once we fix the color ordering of first PT-factor PT(α α α) as the natural ordering e e e = 12 · · · n , the second PTfactor PT(β β β) then can be interpreted as a permutation acting on the identity element. It is shown in this paper that, the pole structure and vertex information of Feynman diagrams evaluated by a CHYintegrand is completely encoded in the permutations of corresponding PT-factors. The cycle representation of permutation, which neatly organizes the external legs into disjoint cycles, manifests the pole and vertex information. More concretely, since a PT-factor is invariant under cyclic rotations and gains at most a sign (−) n under reversing of color ordering, we are actually considering 2n equivalent permutations of a PT-factor. We then write all the equivalent permutations of PT(β β β) into the cycle representation, and pick out the good ones. Those that can be separated to at least three consecutive parts with respect of PT(α α α) are called V-type cycle representations. Those that can only be separated into two parts, while each part contains more than two elements, are called P-type cycle representations. We show that the CHY-integrand PT(e e e) × PT(β β β) gives nonzero contributions if and only if the ways of planar separations allowed by all the V-type representations satisfy the constraint (3.44). The Feynman diagram of a CHY-integrand can be completely determined by one V-type cycle representation by going into its substructures, or collectively determined by all P-type cycle representations of a PT-factor. The vertex structure can be obviously seen from the planar separation of V-type or P-type cycle representations. We presented the algorithm to read out the physical poles and vertices from them.
On the other hand, given an effective Feynman diagram, with possible effective higher point vertices, we have proposed a recursive algorithm to obtain directly the correct cycle representation of corresponding PT-factor PT(β β β). We show that cycle representations of any Feynman diagram allow a factorization as Eq. (4.10) with respect to an arbitrary m-point vertex, called a planar splitting. We have figured out that, the cycle representations of subdiagrams that used in the factorization (4.10) are the non-planar splitting ones in the equivalent class of PT-factors of subdiagrams. The same algorithm applies to the subdiagrams as well, so we can reconstruct the cycle representation of any n-point PT-factor basically from three point PT-factor. We show that all the discussions are parallel in the Feynman diagram and n-gon diagram, while the latter also takes its role in the associahedron discussion.
It is shown in [31] that different PT-factors are neatly connected in the associahedron picture. In this paper, we also investigate the relations among different PT-factors via the reversing permutation on cycles, which corresponds to adding or removing triangulation line in the n-gon diagrams. The merging and splitting of cycles in a cycle representation mainly select terms with the same poles in a result. From the same thought, we further study the relations among PT-factors via the multiplication of certain cross-ratio factor which we call selecting factor. They all give similar topology about how the PT-factors are connected.
Finally, since the planar diagram possess a natural interpretation as the vertices and boundaries of an associahedron, the structure of good cycle representation introduced in this paper can be used to characterize certain boundaries. We have shown how this can be achieved by merging of cycles, in the equivalent class of a PT(β β β), the number of different factorizations into disjoint permutations describes the shape of boundaries of the associahedron.
where v i satisfies the constraint of (4.7). For instance, at n = 6, the three-dimensional associahedron K 5 has two kinds of faces (two-dimensional boundaries): pentagon (K 4 ) and rectangle (K 3 × K 3 ). They correspond to the diagrams with v v v = (1, 0, 1, 0) and (0, 2, 0, 0) respectively. All the boundaries of K n−1 can be obtained by direct products of lower dimensional associahedrons. We use N n (v 3 , v 4 , . . . , v n ) to denote the number of boundaries K n−1 that are of the form In particular, N n (n − 2, 0, . . . , 0) gives the number of vertices of K n−1 , which is the Catalan number C n = (2n−4)! (n−1)!(n−2)! , and N n (0, 0, . . . , 0, 1) = 1 just stands for the K n−1 itself. Knowing the relation between cycle representations and Feynman diagrams, we can give the number N n (v 3 , v 4 , . . . , v n ) another interpretation. From §4.2, we know that a PT-factor PT(β β β) corresponds to a Now comparing with Eq. (A.5), we see that the two four-part planar splitting is due to the fact that the corresponding Feynman diagram consist of two quartic vertices. Since each subdiagram can also be factorized to sub-subdiagrams, this allows a recursive computation for N n (v v v), down to the pieces where only cubic vertices exist and only length-1 or length-2 cycles appear in the cycle representation. Thus 123465 contributes to the counting N 6 (0, 2, 0, 0). Next, we consider an eight point example with v v v = (2, 2, 0, 0, 0, 0), In the equivalent class of 12846573 , there are again two cycle representations that have planar splittings, but this time there are two ways to split them into three parts (blue partitions), and two ways to split them into four parts (red partitions), Again, the cycle splitting pattern agrees exactly with the vertex number vector (2, 2, 0, 0, 0, 0). Thus the PT-factor 12846573 contributes to the counting N 8 (2, 2, 0, 0, 0, 0). The above machinery can also be used to pick out those PT-factors that do not give any Feynman diagram. We again illustrate this point by a few examples first. At n = 6, we consider the PT-factor 124635 . In its equivalent class, there is only one three-part cycle partition,  .7), 124635 does not correspond to any Feynman diagram compatible with the planar order 123456 . Similarly, the PT-factor 135264 does not have any valid cycle partition (namely, more than three parts) in its equivalent class, it must also give zero Feynman diagram. In general, by inspecting how the cycle representations of a PT-factor split, we can get a vertex number vector v v v. If this v v v fails to satisfy the constraint (4.7), the PT-factor under consideration must give zero Feynman diagram.