Scattering Equations and a new Factorization for Amplitudes I: Gauge Theories

In this work we show how a double-cover (DC) extension of the Cachazo, He and Yuan formalism (CHY) can be used to provide a new realization for the factorization of the amplitudes involving gluons and scalar fields. First, we propose a graphic representation for a color-ordered Yang-Mills (YM) and special Yang-Mills-Scalar (YMS) amplitudes within the scattering equation formalism. Using the DC prescription, we are able to obtain an algorithm (integration-rules) which decomposes amplitudes in terms of three-point building-blocks. It is important to remark that the pole structure of this method is totally different to ordinary factorization (which is a consequence of the scattering equations). Finally, as a byproduct, we show that the soft limit in the CHY approach, at leading order, becomes trivial by using the technology described in this paper.

In the usual CHY formalism 1 , amplitudes are contour integrals over the moduli space of n-punctured Riemann spheres (M 0,n ). This contour integral is localized on the solutions of the so-called scattering equations, S a ≡ b =a ka·k b σ ab = 0, σ ab ≡ σ a − σ b , where (σ 1 , . . . , σ n ) denote local coordinates over M 0,n and the index "a" labels the external particles of momentum k a (and polarization vector a ) at the puncture "σ a ". The prescription for the tree-level S-matrix of any quantum field theory may be given by the expression where, ∆(pqr) ≡ σ pq σ qr σ rp (Faddeev-Popov determinant), dµ CHY n ≡ n a =p,q,r dσa Sa , and the Γ contour is defined by the n − 3 independent equations 2 , S a = 0, a = p, q, r. A different integrand describes a different theory, in particular we focus our attention in pure Yang-Mills and special Yang-Mills-Scalar theories. For example, for a colorordered pure Yang-Mills amplitude, A YM n (1, 2, ..., n), the CHY integrand is given by, with the Parke-Taylor factor (PT) and the reduced Pfaffian (Pf Ψ n ) define as PT (1,2,...,n) = 1 σ 12 σ 23 · · · σ n1 , where the 2n × 2n matrix, Ψ n , is given by the blocks 3 (1.4) 1 We call this formalism the single-cover approach. 2 It is simply to verify that, n a=1 S a = n a=1 σ a S a = n a=1 σ 2 a S a = 0, on the support of momentum conservation and on-shell conditions, i.e. k 1 + · · · + k n = 0 and k 2 a = 0. 3 Let us recall ourselves that the on-sell polarization vectors, µ a , obey the transverse condition, a · k a = 0. The gauge symmetry of the theory is given by the shifting, µ a → µ a + k µ a .
These blocks are given by the expressions c =a a · k c σ ac .
(1.5) The reduced matrix, (Ψ n ) ij ij , is built by removing the rows and columns (i, j) from Ψ n , with 1 ≤ i < j ≤ n.
Recently, in [7] (see also ref. [8]), we showed how the CHY approach can be written in a new formulation in which the basic variables σ a live not on CP 1 but on a quadratic algebraic curve embedded in the complex projective plane CP 2 . Dubbed the "Λ-formalism" in [7], we shall here refer to it as CHY on a double-cover (DC). At first sight it may seem to be a complication to extend the CHY formalism in this manner. For example, the reduced Pfaffian in the double cover representation is given by (see sections 2 and 3) However, as we shall demonstrate in the present paper the double-cover formalism adds a new ingredient to the standard CHY formalism that is much more difficult to extract in the single cover formulation. Briefly stated, it is this: after integrating the new auxiliary variables, (y 1 , ..., y n , Λ), the double-cover formalism naturally expresses the scattering amplitudes so that they appear factorized into channels. Nevertheless, this procedure must be performed with care, broadly speaking, the integral over the auxiliary variables is based on two main points: 1. The number of fixed punctures (colored vertices) over each Riemann sheet.
2. The number of arrows cut by the branch-cut when it is getting closed.
In Fig. 2 and section 4 we will give detail about this subject. The propagator that forms the bridge between two factorized pieces arises as the link between two separate CP 1 in the single-cover approach, thus intuitively explaining why the double-cover naturally expresses amplitudes in a factorized manner. Since in most cases this process can be iterated, then, we will not need to solve any scattering equation, which is one of the virtues of this approach. In Fig. 1 we give a few characteristics of the single and double cover approach, in order to reference some general differences among these two prescriptions. It is interesting to remark that, in many cases, the factorizations obtained in this way corresponds directly to the physical channels. Interestingly, there are instances where, unavoidably, the factorizations proceed in a slightly different manner: some physical channels appear immediately, but others only resurface after pole-cancelling terms have rearranged the expressions. This seems to lead to an intriguing connection to on-shell BCFW recursion [9]. It turns out that momentum shifts that lead to poles at infinity become evaluated in a quite straightforward way by means of the double-cover formalism.
On the other hand, several methods have been developed to compute the CHY contour integral given in (1.1), most of them are applied to φ 3 or focused on solving the scattering equations [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In this work, from the double-cover representation, we have been able to achieve a graphic off-shell algorithm to carry out any color-ordered scattering of n-gluons and interactions with scalar fields, resulting in an expansion in terms of three-point amplitudes 4 .
In view that the algorithm obtained in this work is an off-shell method 5 , then a natural question arises, Is there any connection with the method proposed by Berends-Giele in [26]? In order to illustrate the answer, we focus on the bi-adjoint φ 3 theory (since its integration rules are simpler [7]). Usually, the double-color partial amplitude for the bi-adjoint φ 3 theory is denoted as m (0) (α|β), where α and β are two partial orderings [4]. When α = β, we denote 4 Notice that, although the methods presented in [14,25] look somewhat similar to the one developed in this paper, the process and the form of results obtained by us are different. 5 Since all intermediate particles are off-shell, including their polarization vectors, i.e.
[i] · k [i] = 0, and as it is an iterative program, then, we can consider this algorithm as an off-shell method. m (0) (α|α) as A φ 3 (α) ≡ m (0) (α|α). In particular, let us analyse the five point example, A φ 3 (1,2,3,4,5), where the red label means an off-shell particle, k 2 5 = 0. Following the recurrence relation obtained by us et al. from the double cover approach (see equation (C.7) in [13]), one has A φ 3 (1, 2, 3, 4, 5) = A φ 3 (3, 4, [ with, A φ 3 (a, b, c, d) ≡ 1 s dc + 1 s da , where {k a , k b , k c } can be off-shell. We have also introduced the notation, k [a 1 ,...,ap] ≡ k a 1 + · · · + k ap ,s a 1 a 2 ...ap ≡ i<j k a i · k a j and s a 1 a 2 ...ap ≡ 1 2 k 2 [a 1 ,...,ap] . Pictorially, (1.7) looks like [7], . (1.8) Although the color in each vertex has an important meaning, as we will explain later, roughly speaking, the colored vertices symbolize the punctures have been fixed by the global isometry generators (red means off-shell), while the black one means is unfixed puncture over the double cover sphere 6 . Otherwise, Mafra was able to obtain the Berends-Giele-like currents for the biadjoint φ 3 theory [27], where he used the Perturbiner method 7 . For instance, the n-point amplitude, A φ 3 (1, . . . , n), is given by the recurrence relation The notation XY =P means a sum over all possible ways to deconcatenate the word P in two non-empty words X and Y . For example, φ 1234 = 1 . Therefore, from (1.9), it is straightforward to see Graphically, (1.10) is represented by the diagram, , (1.11) 6 Notice that the first diagram on the third column in (1.8) looks like the same to the fourth one. However, they are different because during the Λ-algorithm process appeared spurious poles, as one can see in (1.7). For instance, the first diagram is given by the expression, 1 s234s 2 [5,1] , and the fourth one by, 1 s34s 2 [3,4] . 7 Currently, the Perturbiner method has been successfully used for several theories, [27][28][29][30].
where the dashed line is an off-shell particle (k 2 5 = 0). Clearly, the results obtained in (1.7) and (1.10) are related by the partial fraction decomposition [3,4] , (1.12) which is the same phenomenon found at loop-level [8,[31][32][33][34][35][36][37][38]. Nevertheless, although in Yang-Mills theory the double-cover approach also produces spurious poles, such as those on the right-hand side in (1.12) (it can be checked in section 8 in the five-point example), we have no idea how to relate our method with the one developed by Berends-Giele in [26]. At four-point, the relation among these two methods is straightforward, but at five-point, we do not know how to get it. This could be a possible future direction in order to understand better the double-cover representation.
It is important to mention that in most of the case we will consider on-shell external particles. Nevertheless, our method is able to support up to two off-shell gluons, such as the example at four-point shows in section 8.
On the other hand, on the support of the scattering equations, the reduced Pfaffian can write as a linear combination of Parke-Taylor factors [4,[39][40][41] Pf Ψ n = ρ∈S n−2 PT (1,ρ(2,...,n−1),n) N tree 1|ρ(2,...,n)|n , (1.13) where the N tree 1|ρ(2,...,n)|n terms are the tree-level Bern-Carrasco-Johansson (BCJ) numerators [42] and S n−2 is the group of all possible permutations of the labels (2, ..., n − 1). Clearly, from the expansion in (1.13), the n-point ordered YM amplitude, A YM n (1, ..., n), is written in terms of the (n − 2)! φ 3 -amplitudes, m (0) (1, ..., n|1, ρ(2, ..., n − 1), n), times its corresponding BCJ factor, N tree 1|ρ(2,...,n)|n . Although this formula looks simple, notice that the expansion grows quickly with the number of points, additionally, when the number of points is large, the BCJ numerators are not straightforward to carry out. Conversely, in the proposal given by us here, the n-point color-ordered YM amplitude is represented just by one graph, and, after using iteratively the integration rules from Fig. 2, the final answer is expressed as a product of the three-point amplitude, A YM 3 . To end, it is interesting to remember that the usual CHY approach of the n-point color-ordered YM amplitude can be seen as the d-dimensional version of the Roiban, Spradlin and Volovich formula (RSV) [43] (which is called the connected prescription). Otherwise, Cachazo, Svrcek and Witten (CSW) proposed an alternative 8 formulation that describes the same physical object given by the RSV formula (disconnected prescription) [44]. The relation between the RSV and CSW was shown in Refs. [45][46][47]. Since the double-cover representation is able to express the n-point amplitude as a product of three-point building-blocks, then, we think this is the disconnected version of the usual CHY approach. It would be very interesting to obtain a relation between the CSW prescription and the CHY double-cover representation. Outline: The present work is organized as follows: In section 2, we give a simple review of the double-cover prescription for the double color-ordered φ 3 theory, for detail see [7].
In section 3, we introduce a new deferential form given by, T ab dσ a ≡ dσa (ya+σa)−(y b +σ b ) , which is the key to define the CHY matrices in the double-cover approach. In the original paper where the double-cover prescription was introduced [7], the fundamental object is given by the expression, τ (a,b) dσ a ≡ 1 2ya ya+y b σ ab + 1 dσ a , and although the relationship between T ab and τ (a,b) looks pretty simple, 14) on the support of y 2 a = σ 2 a − Λ 2 and y 2 b = σ 2 b − Λ 2 , this is non-trivial. Now, since T ab is antisymmetric, we define the mapping, 1 σ ab → T ab , to translate the most familiar CHY matrices in the double-cover language. However, the Π α 1 ,...,αp matrix [48] must be treated with care [49].
To end the section, we propose the DC integrands to compute the scattering amplitudes of n-gluons and interactions with scalar fields. Section 3.1 is conceptually important, since here we introduce a graphical representation for CHY integrals, as much as in the double-as in the single-cover. These graphics are essentials in this work, because all our computations are performed over them. Additionally, we clarify the color notation of the vertices.
In section 4, we formulate the integration rules, perhaps the central part of the paper. We perform an extensive and careful analysis of the double cover representation and the integration over the auxiliary variables, (y 1 , ..., y n , Λ).
First, we study a general situation, i.e. for any CHY integrand. In that case, we obtain the rule-I, which basically claims that if any given cut does not encircle two colored-vertices, then, that cut vanishes trivially. Next, we go to a particular case, the Yang-Mills integrand. Thus, studying the Pfaffian, we obtain three more integration rules. Finally, in Fig. 2, we have summarized all rules in a flowchart.
In sections 5 and 6, we compute the fundamental three-point building blocks. This method is able to break a big graph as a product of three-point amplitudes, similar as it was done in Fig. (1.8). Subsequent, we give the simplest example, the four-point Yang-Mills amplitude. Applying the algorithm schematized in Fig. 2 over this fourpoint graph, we obtain two standard factorization cuts and an additional (apparently) non-physical contribution (which we call a strange-cut).
Next, in section 7, following the Pfaffian identities given in appendix B, we interpret the strange-cuts as longitudinal contributions from some off-shell Yang-Mills amplitude.
To be more precise, we obtain the equality (up to overall sign), , (1.15) where the integrand on the first line is a generic strange-cut, and the fixed puncture, σ [1,2] , is an off-shell vertex, (i.e. k µ [1,2] . We also analyze the standard factorization cuts (standard-cuts), and the different ways to glue their resulting-graphs. It is crucial in order to obtain a recursive method.
In section 7.2, we basically generalize the results found previously in section 7 to more than one off-shell particle. Additionally, we use reverse engineering to compute (applying the algorithm in Fig. 2) some non-trivial standard factorization contributions.
In section 8 we apply all technology developed up to this point. Here, we carry out analytically (and in a simple way) the five-point amplitude, A YM 5 (1, 2, 3, 4, 5). In sections 9 and 9.1, we see that the same ideas developed for Yang-Mills theory are naturally extended to the special Yang-Mills-Scalar theory. We give some simple The graph is broken Figure 2. Integration rules as a Flowchart. Given a CHY integral, we build its graph representation, section 3.1. Each vertex of the graph represents a puncture on a Riemann sphere, including its physical parameters (momentum and polarization vector). On the other hand, the arrows of the graph represent the factors " 1 σ ab " while the reduced Pfaffian is always implicit in it. The integration rules give a program to break any graph in terms of two smaller graphs (resulting-graphs). The first step is to choose four vertices (colored vertices), we call to this process a gauge fixing. The condition over the vertices, to obtain a good splitting of the graph (non-vanishing contribution), is that each split part must contain two colored vertices. After getting a good splitting, we can focus on the arrows. If the graph is split by cutting less than three arrows, then, this is a singular-cut (section 4) and the method can not be applied, therefore, one must come back to choose a new gauge fixing. Solving the singular-cut issue by choosing an appropriate gauge fixing, we have three options: 1. The graph is split by cutting three arrows: in this case the two resulting graphs are glued by an off-shell vector field. 2. The graph is split by cutting four arrows: in this case the two resulting graphs are glued by a longitudinal vector field or scalar field. 3. The graph is split by cutting more than four arrows: then, this contribution vanishes trivially. These rules are going to be clearer after section 4. examples, for instance, two gluons interacting with two scalars, and four and six scalars. Additionally, in this section we examine as the A matrix (the kinematic matrix) is factored into a product of two matrices that involve scalars and gluons (usually denoted by Ψ g,s:g ).
In section 10, we obtain a connection between the Yang-Mills-Scalar amplitudes and the strange-cuts. In other words, a strange-cut factorizes the reduced Pfaffian, Pf Ψ n , into a product of two Pfaffians that involve the matrix Ψ g,s:g , which contains both gluons and scalars.
Finally, in section 11, we see as the soft limit, at leading order, becomes simple to carry out after using the ideas developed in this paper.
Some conclusions are presented in section 12, and in appendix A and B, we give a small glossary and some properties of the off-shell Pfaffians, ijk . In this work we describe carefully the results found by the author et al. in the recent paper [50].

A Brief Review of the Double Cover Representation
The double cover representation of the CHY construction is given as a contour integral on n-punctured double-covered Riemann sphere. Restricted to the curves, C a ≡ y 2 a − σ 2 a + Λ 2 = 0 for a = 1, . . . , n, the pairs, (σ 1 , y 1 ), (σ 2 , y 2 ), . . . , (σ n , y n ), provide a set of doubled variables (a translation table has been worked out in detail in ref. [7]).
As a fast overview, in the DC approach, a CHY-like integrand is built using the third kind form, τ (a,b) dσ a ≡ 1 2 y b ya + 1 1 σ ab + 1 2 ya dσ a , on the support C a = C b = 0, and the integration measure is given by where the Faddeev-Popov determinants, ∆ (pqr) ∆ (pqr|m) , and the scattering equations, S τ a 's, are defined as (we use the notation (yσ) a ≡ y a + σ a ) Here is important to remind ourselves where the factors, ∆ (pqr) and ∆ (pqr|m) , come from. Such as in the single cover approach, the number of independent scattering equations, S τ a = 0, is "n − 3", which is straightforward to see of the identities (on the support C a = 0), n a=1 y a S τ a = n a=1 y a (y a + σ a )S τ a = n a=1 y a y a + σ a S τ a = 0. (2.5) From these identities, we can read the SL(2, C) generators given by global vectors 6) The Faddeev-Popov determinant to fix the (2.5) redundancy is the factor defined in (2.2) that we denoted by ∆ (pqr) .
The amplitudes are derived from the integral where the Γ contour is defined by the equations 9 , Λ = 0, S τ d = 0, for d = {p, q, r, m}, C a = 0, ∀ a .
Similarly, other theories correspond to products of such modified Parke-Taylor factors with additional expressions, much like in the original CHY formalism. Again, the integrands for these other theories can be broken down to products of shuffled Parke-Taylor expressions.

Matrices in The DC prescription
Since τ (a,b) = −τ (b,a) , it is not immediately obvious how to define the CHY antisymmetric matrices to describe the double-cover analog of the pure Yang-Mills, Gravity, NLSM theory, among others. In order to obtain a double-cover version for the CHY matrices, we rewrite τ (a,b) as Clearly, T ab is an anti-symmetric form, T ab = −T ba , thus, we can establish the simple map, 1 σ ab → T ab , in order to define the matrices in the double-cover representation.
For example, the A Λ matrix is obtained from the A matrix by the replacement, A Λ ≡ 9 The rewriting of the amplitude in terms of this contour "Γ", which does not encircle the scattering equation S τ m , follows from the global residue theorem. 10 At loop level see [37,38,51]. (see [48]), but, there is one matrix that must be handled with care, the Π α 1 ,...,αm matrix. This matrix has elements such as, i ∈α p ,j∈αq , and we in [49] will explain how to deal with that type of terms.
Before analyzing the pure and scalar Yang-Mills theories in the double-cover formalism, it is useful to see how φ 3 -integrands may be rewritten in terms of T ab , where PT T (a 1 ,a 2 ,...an) ≡ T a 1 a 2 T a 2 a 3 · · · T ana 1 . Following the CHY program developed in [4,48], the DC prescription for the colorordered scattering amplitudes of the pure Yang-Mills theory can be obtained from (3.2) by where (Ψ Λ n ) ij ij is built by removing the rows/columns i, j from Ψ Λ n , with 1 ≤ i < j ≤ n. In a similar way, the DC integrand for the special Yang-Mills-Scalar theory is given by Here, the set of gluons is denoted as "g" (g = {g 1 , ..., g p }) while that the set of scalars 12 as "s" (s = {s 1 , , ..., s 2m }). The symbol "p.m." means perfect matching and, Since the (3.4) expansion comes from the pfaffian, Pf(X Λ s ) [48], then sgn ({a,b}) means the corresponding signature (more details about this theory in section 9).
Finally, the pure Yang-Mills and the special Yang-Mills-Scalar amplitudes at tree-level in the DC language are given by the integrals where (α) ≡ (α 1 , α 2 , . . . , α n ) is a partial (generic) ordering.
The following sections will be dedicated to the pure Yang-Mills theory. The generalization to the special Yang-Mills-Scalar theory is straightforward, and we will come back to this model in section 9.
It is important to remark that due to the normalization of the kinematic parameters and the polarization vectors chose in these notes, we are going to obtain an extra overall factor, (2) , compared with color-ordered Feynman rules given in 13 [52].

Graphical Representations of DC Integrands
Since the method that we are going to describe in this work is based on so called the Λ−algorithm, which is given by graphic rules [7], we introduce a simple graph representation for the amplitude A YM n (α) in (3.5). Like in φ 3 , a Parke-Taylor factor is drawn by lines join the vertices in a sequence way.
In order to specify the ordering we replace the lines by arrows, for example On the other hand, due to the factor, Pf (Ψ Λ n ) ij ij , is a non-ordered object, we do not know how to build a graph representation for it. Nevertheless, there is a term in Pf Ψ Λ n that one can draw, specifically T ij , which we sketch by red arrow, namely, T ij ≡ i → j. Finally, so as in [7], the factor, , is symbolized by yellow vertices for (σ p , σ q , σ r ) and a green one for the σ m -puncture. Therefore, the complete graphs for the YM integrand in the 13 In order to obtain the normalization given by Dixon in [52] for color ordered amplitudes, we just need to perform the replacement, k µ a → DC representation is 14 where, without loss of generality, we have chosen (pqr|m) = (123|4) and (i, j) = (2, p). The amplitude, A YM n (1, 2, ..., n), is given by where the superscript denotes the (i, j) choosing, always with i < j in order to give the direction to the red arrow, this is import in order to obtain the cyclic property. We call to (3.9) a YM-graph.
Obviously, when all polarization vectors are transverse and all particles are on-shell (i.e. k a · a = k 2 a = 0), the above expression is independent of the red arrows and colored vertices [3,48] (physical amplitudes). However, when there is one off-shell particle, this notation becomes important and that expression depends on the colored vertices and red arrows, such as it will be shown later.
The generalization of the graph representation from the DC prescription to the singlecover approach is simple, it is just to replace the green vertex by a black one, since the single-cover is not scale invariant, namely dµ Λ On the right hand side, the colored vertices mean the punctures (σ 1 , σ 2 , σ 3 ) are fixed, i.e. the Faddeev-Popov determinant, ∆(123) 2 = (σ pq σ qr σ rp ) 2 . The map from the double-cover to the single one can be seen in [7] At last, it is useful to observe the following two properties 14 From this graph representation we can conclude that, a vertex with two black arrow is a gluon and, a vertex with two black arrows and a red one represents a gluon such that its row/column (among 1 and n) must be removed from the matrix.
We would like to clarify that, although right now the notation, A (i,j) n (1, 2, . . . , n), looks useless, it will become important since the method developed here depends of the gauge fixing, i.e. of the choosing of (pqr|m) and (i, j).

The double-cover integration rules
In this section, we schematize how the double cover formalism works. Following, we obtain some integration rules, which are applied to the YM-graphs, in order to solve the CHY integral.
Since the amplitudes are computed in a manner that differs in detail substantially from the original CHY prescription, we will provide a few explicit examples. Let us start the discussion with the integration measure of the double cover prescription, namely where we have fixed (pqr|m) = (123|4). Like it was shown in [7], after integrating the y a coordinates around the solutions, . 15 Since the Z 2 symmetry, y a → −y a , a = 1, 2, . . . , n, there are 2 n−1 nonequivalents configurations. In other words, the upper and lower spheres are indistinguishable. 16 The factor (1/2 2 ) fixes the discrete symmetry, see [7].
where upper/lower spheres are represent by the local coordinates, (y a = σ 2 a − Λ 2 , σ a ) and (y a = − σ 2 a − Λ 2 , σ a ), respectively. The neck that joins the spheres represents the branch-cut and the branch-points, −Λ and Λ, give the width of the neck.
Integrating Λ (keeping Λ ∼ O(0)), the double cover formulation factorized into two single covers attached by an off-shell propagator (the free scattering equation in the DC formalism reduce to the propagator when one performs the integration over Λ). For instance, let us consider the punctures {σ p + 1, . . . , σ n , σ 1 , σ 2 } on the upper sheet and {σ 3 , σ 4 , . . . , σ p }, on the lower one. The measure and the Faddeev-Popov determinant turn into 17 (expanding around This analysis gives us the first integration rule [7] 17 Let us remember that in this work we are using the momentum notation, k [a1,a2,...,ap] ≡ k a1 + · · · + k ap ,s a1a2...ap ≡ p i<j k ai · k aj and s a1a2...ap ≡ k 2 [a1,a2,...,ap] . • Rule-I. All configurations (or cuts) where there are less (or more) than two colored vertices (yellow or green) on each branch, vanish trivially .
For simplicity, we represent a cut (or configuration) over a YM-graph by a dashed red line, which separates (encircles) the punctures localized on the upper (or lower) sheet. For example, where cut-1 vanished trivially by rule-I, but cut-2 could give a non-zero contribution.

Specific rules
The rule-I is a general rule which works for any integrand I Λ (σ, y). In this section, we are going to formulate some integration rules which depend on the integrand similar to those discussed in [7] for φ 3 .
Following the rule-I, it is enough to focus on all possible non-vanishing configurations.
Expanding the YM integrand of each possible non-zero cut around Λ = 0, we notice that The dashed red line cuts more than four arrows.
The dashed red line cuts three or four arrows.
The dashed red line cuts two arrows (singular cut). Therefore, considering the expansion obtained in (4.2), the next rule is obvious • Rule-II. If the dashed red line cuts less than three arrows over the YM-graph, the integrand must be expanded next to leading order. If the dashed red line cuts three or four arrows, the leading order expansion is sufficient. Otherwise, the cut is zero.
This rule is equivalent to the Λ−theorem given in [7].
Finally, the last rule we are going to formulate is only applied when the leading order expansion is sufficient, i.e. when the dashed red line cuts three or four arrows.
First, let us consider the case when the dashed red line cuts three arrows over a YM-graph (see cut-2 in (4.4)). The only way to break the Pfaffian into two pieces is by introducing new polarization vectors, one for each new puncture (one for σ [up] and one for σ [down] ) [3,14].
These two resulting graphs must be glued by the identity, r On the other hand, when the dashed red line cuts four arrows, the Pfaffian (Pf (Ψ Λ n ) ij ij ) breaks spontaneously into two pieces. For instance 20 , let us consider again the matrix, (Ψ Λ 4 ) 13 13 , and let us expand its Pfaffian around Λ = 0, but now, when the punctures, (σ 1 , σ 3 ), are on the upper sphere (and (σ 2 , σ 4 ), are on the lower one). It is simple to show that the leading order contribution is given by 18 Here, we are assuming the dashed red line is cutting two arrows of the Parke-Taylor factor, i.e.
19 It is useful to recall that, 2 · k [3,4] 20 Here, we are assuming the dashed red line is cutting four arrows of the Parke-Taylor factor, i.e. [1,3] and σ [2,4] = σ [1,3] = 0. The same behavior can be checked at higher number of points. Finally, notice that the matrices in (4.7) do not have any rows/columns associated with the new punctures, σ [2,4] and σ [1,3] . So, we have the last integration rule • Rule-IIIb. When the dashed red line cut four arrows, the YM-graph breaks spontaneously into two resulting graphs, which are written in the single-cover language (times a propagator given by (4.2)). All rows/columns related to the new resulting vertices (punctures with four arrows) must be removed from the resulting matrices.
We call to this type of cut a strange-cut, since it produces spurious poles, such as we will show in the next section.
In general, a puncture with four arrows represents a scalar particle [7] (we will come back to this point later). Nevertheless, in pure Yang-Mills a puncture with four arrows can be interpreted as a longitudinal gluon, it will be explained in detail later. Therefore, this means that two new punctures, σ [up] and σ [down] , are longitudinal off-shell gluons when the dashed red line cuts four arrows.
We would like to draw attention to the importance of the green vertex. This is differentiated since its scattering equation is responsible for generating the propagator of a given cut, via equation (4.2).
As a final observation, in order to formulate a well-defined method, we remark that the integration rules obtained in this section are independent of the embedding. The only thing that one must keep in mind is the following additional rule This means that when the dashed red-line cuts an even number of times an arrow, it is always possible to find an embedding such that the dashed red-line does not cut with that arrow. In a similar way, when the dashed red-line cuts an odd number of times an arrow, it is always possible to find an embedding such that the dashed red-line cuts just one time with that arrow.

Three-point building-block
Before giving simple examples, it is going to be useful to introduce the three-point functions that will work as building blocks., Additionally, in appendix A we give a small glossary in order to remember the notation.
The first three-point function, which is important to remark its normalization, is the where the punctures are off-shell, k 2 It is obvious to note that any three-point computation is just algebraic, i.e. there is no an integral (its integration measure is trivial, dµ CHY 3 = 1). Therefore, for the rest of the paper, we will always omit the symbol dµ and [c] · k [c] = 0. Notice that the transverse constraint,     (6.2)

Four-point
Using the rules-II, III and the expansion in (4.2), the cuts become where let us remind the red vertices mean they are fixed and off-shell punctures.
Notice the upper index "r" over the red vertices, for instance [a 1 , . . . , a i ] r , means the off-shell punctures, σ [a 1 ,...,a i ] , have as associated polarization vector, r,µ [a 1 ,...,a i ] . As it was said above, the four black-arrows on the off-shell punctures over the resulting graphs in cut-3 mean all rows/columns related with them must be removed from Ψ matrix (rule-IIIb). In other words, these off-shell vertices have an associated polarization vector proportional to their momentum, i.e. longitudinal gluons. The explicit computation will be performed in (6.4), and in the next section we will give more details about this issue.
Using the three-point off-shell building-block, A Analogously, , with, r r,µ [2,3] r,ν [4,1] = η µν . Finally, to compute cut-3, we just read the resulting graphs, namely [1,3] and C 44 = − 4 ·k 2 σ 42 + 4 ·k [1,3] σ 4 [1,3] . Therefore It is straightforward to check that, in fact, A YM 4 (1, 2, 3, 4) = cut-1+cut-2+cut-3. As a final remark, it is interesting to see that the strange-cut 3 is related with the quartic vertex, Tr [A µ , A ν ] 2 . First, notice that the cut-3 can be rewritten as On other hand, the color-ordered contact vertex is given by 21 Clearly, the first term matched perfectly, but, the others two are not present in cut-3. This fact confirms that the integration rules proposed in this paper are not the Feynman rules, as a consequence, we obtain spurious poles. 21 See Dixon normalization in [52].
Thus, applying the integration rules, one has the cutting expansion This expansion has been verified numerically, up to an overall sign, and we will compute it explicitly in the last section.
Since one of our objectives is to describe a Yang-Mills CHY-algorithm, we need to understand how to apply the integration rules over the resulting graphs. For example, in the φ 3 algorithm obtained in [7], the resulting gauge fixing can not be modified on the go to compute the resulting graphs. As an illustration, let us consider cut-2 in fig. ( and r,ν [1,2] , are not transverse ( r [3,4,5] · k [3,4,5] = 0, r [1,2] · k [1,2] = 0). In the next section, we are going to solve this drawback.

Longitudinal and Transverse Gluons
Additionally to the standard factorization cuts, where a YM-graph is splitting in two smaller ones with an off-shell gluon propagating among them (standard-cuts), we have also obtained some strange contributions (strange-cut), with no obvious physical interpretation. In this section, we study these strange-cuts.
Under the gauge fixing, (pqr|m) = (123|4) and with the red arrow over, (i, j) = (1, 3), a generic strange-cut encircles the vertices, (1, 3, p + 1, ..., n), produces the following two types of resulting graphs dµ CHY (n−p+3) where we used the property-I (appendix B) to obtain the equality. Notice that from the rule-IIIa, the associated matrices of these two resulting graphs are given by, (Ψ g,s:g ) .
It is clear that this equality give us the following physical interpretation for the strange cuts • All strange-cuts can be rewritten as a product of two YM-graphs, which must be glued by a longitudinal gluon. 22 This identity can be easily extended to another setup.

Transverse Gluons
On the other hand, despite to the result obtained in (7.2), we still do not know how to deal with some resulting graphs, for example, the four-point in (6.9).
Nevertheless, from the properties-III, IV in appendix B, it is straightforward to obtain the following identities for a general standard cut dµ CHY (p−1)      k [3,4,5] ·k [1,2] . This is very important to observe that the integration rules can not be applied on the left-hand side, while that on the right-hand side they work perfectly.
Up to this point, under the identities set down in this section, we are able to write an ordered on-shell YM-amplitude as a sum of the product of two smaller partial off-shell YMamplitudes, which must be glued by off-shell gluons. However, it is important to remark that those graphical identities proposed here involve only one red-vertex over each resulting graph (off-shell gluon). In the next section, we will refine the gluing process to go beyond more than one off-shell puncture.

More off-shell gluons
Since the above identities can only be applied to graphs with one off-shell particle, we need to discuss what happens when there is more than one red puncture; this will help us to develop a graph-algorithm for more general cases. In other words, we would like to generalize the properties given in appendix B.
Let us start by considering the following simple example, the cut-2 given in (6.9). Using r,ν [4,5] = η µν . The computation of these terms is straightforward and the final result has been checked numerically. Note the emergence of the spurious pole,s 5 [1,2] = k 5 · (k 1 + k 2 ), which does not appear in any known method before 23 . 23 These kind of poles are a direct consequence of the scattering equations, similar to the linear This example showed us that the integration rules work perfectly over a YM-graph with one off-shell particle (k 2 [1,2] = 0 and A [1,2] · k [1,2] = 0). In fact, we have successfully tested them over bigger graphs and with more than one off-shell particles 24 . Therefore, we claim that our graph-method is recursive over YM-graphs. Now, we want to know what happens with the strange-cuts, i.e. Can the conjecture in (7.2) be generalized to more than one off-shell puncture?
On the other hand, the generalization of the second strange-graph in (7.6) is given by where, unlike to the graph in (7.9), the polarization vector P,µ [p] must be transverse ( P [p] · k [p] = 0), this in order for the computation to not depend on σ s (PSL(2, C) symmetry). Note that we have only considered two off-shell punctures, k 2 [C] = 0 and k 2 [p] = 0, which is enough since after gluing two strange graphs, such as those given in (7.10) and (7.11), one must obtain a cut from a YM-graph, who can just have up to three off-shell particles.
Following the properties-I,II in appendix, we should focus on the graph (let us remind [ p ] q P . (7.12) Visibly, we obtained a matching with (7.11).
The same behavior is observed at four and five points, therefore, the generalization of the identity obtained in (7.2) is direct , and with P [p] · k [p] = 0. It is useful to remember that each graph satisfies the momentum conservation condition, , must be imposed in order to glue the graphs.

Standard-cuts
Naively, one can think to achieve a recursive method the identities in (7.3) should be generalized. However, when there is more than one off-shell particles with non-transverse polarization vectors in a YM-graph, this generalization is not possible. Furthermore, notice that the (7.3) relationships are not enough to carry out the four-point resulting graph obtained from the cut-3 in the five-point amplitude. For instance, applying the third identity given in (7.3)  k [4,5,1] ·k [2,3] . Clearly, the integration rules do not work over the above four-point graphs.
Fortunately, we have found two ways to face this issue, the first one is simple and intuitive, and the second one is more systematic.
By the property III in appendix B, over the transverse sector, we can move the red arrow in the four-point graph from (i, j) = (1, [2,3] T ) → (i, j) = (1, 4), and now the integration rules can be applied.
On the other hand, although over the longitudinal sector the same trick doesn't work, we can make use reverse engineering with the help of (7.2) identity. To be more precise, from the properties-I,II of the appendix B, it is straightforward to see the equality, k [2,4] · k [1,4,5] 2 × dµ CHY 4 4 1 5 [2,3 ] .
(7.15) Clearly, the right-hand side graph is trivial to be computed via the integration rules, however, it has a strange shape. Furthermore, observe that the matrix associated with this graph is, (Ψ g,s:g ) This method can be extended to a higher number of particles or more off-shell vertices.
From simple examples, we have presented two more ways to deal with that kind of graphs.
A generalization of the graph in (7.17) is given by

Examples
As a final illustration, in this section we would like to apply the previous ideas to compute, explicitly, the five-point amplitude A YM 5 (1, 2, 3, 4, 5). The plan is to write its five cuts in terms of the three-point building-block, A YM 3 . Before computing the cuts obtained in (6.8) from A (1, 2, 3, 4, 5), it is useful to carry out the off-shell four-point amplitude, 25 Let us recall that the dashed arrow (anti-line) on this graph means the factor "σ 46 " is in the numerator. In addition, so as in [7], when the integration rules are applied on this type of graphs, the anti-line subtracts in one the total number of arrows cut by a given configuration (dashed red line).
, and the particles, This four-point result has been checked numerically.
Using the off-shell amplitude, A . Finally, it is not hard to verify that, .
As a last point, notice the non conventional structure of the poles, for example, in the cut-1 and cut-2 one has,s 5 [3,4] ands 5 [1,2] = −s 5 [3,4] , respectively. This fact is a consequence from the scattering equations and the (4.2) expansion.

Special Yang-Mills-Scalar Theory
After giving an extended analysis and obtaining an alternative algorithm of the pure Yang-Mills theory in the CHY framework, the generalization to the special Yang-Mills-Scalar theory is simple.
The Lagrangian for this theory is given by the expression where the gauge group is U (N ) and the scalars have a flavor index from a global symmetry group, SO(M ).
In [48], it was conjectured that the tree-level color-ordered amplitude for a set "g" of p gluons (i.e. g = {g 1 , . . . , g p }) and a set "s" of 2m scalars 27 (i.e. s = {s 1 , . . . , s 2m }) is given by the CHY integral (note that, p + 2m = n, where n is the total number of particles) where the Ψ g,s:g matrix is given by the blocks It is useful to remind that "p.m." means perfect matchings, and note, where, C 11 = − 1 ·k 2 σ 12 + 1 ·k 3 σ 13 . As it was found in (3.4), the double-cover version of the integrand, I YMS g:s , is given by the expression where, Pf Ψ Λ g,s:g ≡ (−1) i+j T ij n a=1

Special Yang-Mills-Scalar Examples
The ideas presented up to this point can be easily extrapolated to this special Yang-Mills-Scalar theory. Here, we show some simple examples to illustrate how the integration rules work over the amplitude, A YMS g:s . Such as in Yang-Mills, we will introduce a superscript in order to indicate the red arrow, (i, j).
Let us consider the four-point example, g = {1, 2} and s = {3, 4}. The color-ordered amplitude is given by the integral, A YMS g:s (g 1 , g 2 , s 3 , s 4 ) (3,4) 34 Pf Ψ Λ g,s:g , where we have chosen, (pqr|m) = (123|4). To avoid singular configurations (see (4.5)), we pick out the red arrow to join the vertices, (i, j) = (1, 3). Therefore, following the graphical construction in section 3.1, the partial amplitude, A YMS g:s (g 1 , g 2 , s 3 , s 4 ) (3,4) , is represented by the graph (which we call as YMS-graph) where we have applied the rules-I,II. It is very important to note that, the vertices with two black arrows are gluons, while the vertices with three black arrows are scalars. Additionally, from the Pf Ψ Λ g,s:g definition, the scalar vertex with four arrows (three black and one red) does not appear in the Ψ Λ g,s:g matrix, i.e. all its associated rows/columns must be removed of Ψ Λ g,s:g (such as it was said at the end of section 4.1). Applying the rule-III over the above cuts one has where we have used the gluing identity (by the rule-III), r r,µ [3,4] r,ν [1,2] = η µν , and the threepoint building-block in (5.2) . Finally, it is not hard to show that, A YMS g:s (g 1 , g 2 , s 3 , s 4 ) (3,4) =cut-1 + cut-2, up to overall sign.
It is straightforward to note that the three-point function, A 2] ·k [1,2] ) 1 2 , is the Feynman vertex, Tr A µ φ I , ∂ µ φ I (Lagrangian (9.1)), plus an extra term that measures the transversality of the polarization vector r,µ [1,2] (like in pure Yang-Mills, expression (5.2)). where the rules-I,II have been applied. By the rule-IIIb, it is simple to carry out cut-1 where we have taken into account that a scalar vertex with four arrows means one must remove its row/column of the matrix. This simple result matched with the one given in [48] (equation (4.5)), in addition, an interesting question arises, what is the physical meaning of the three-point resulting graphs obtained in (9.11)? (there is no cubic scalar vertex 28 in (9.1)).

Scalar Amplitudes, from
Such as we learned in pure Yang-Mills (section 7), the spurious pole contribution in (9.11) can be seen as a longitudinal gluon contribution. k [2,3] ·k [1,4] . This expression means that the φ 4 vertex can be factorized as a product of two Yang-Mills-Scalar three-point amplitudes glue by a longitudinal off-shell gluon 29 . We will give a non-trivial example later.
Let us consider the next example, , A YMS g:s (s 1 , s 2 , s 3 , s 4 ) (1,2:3,4) . Its YMS-graph is given by where we have used the rules-I,II. While the cut-1 was computed previously in (9.11), cut-1 = 1, the cut-2 is not obvious to carry out. If one applies the rule-IIIa, the cut-2 should be 28 As one can note from the previous examples (YM and YMS), the number of arrows cut by a given configuration (red dashed line) is related with the Feynman vertex. In the above example, (9.10), the YMS-graph has a configuration that cuts four arrows, therefore, this must be related with the Feynman vertex, Tr φ I , φ J 2 . Since this cut is the only one contribution, we could assume in advance that the final answer was 1.
29 This is the equivalent to the identity found in ( Now the question is, How to read these resulting graphics? Notice that these resulting graphs have an identical structure as the second one obtained in (9.9), and so, their meaning is the same. In other words, each graphs in (9.14) have a off-shell gluon vertex with three arrows (i.e. its row/column must be removed of the Ψ g,s:g matrix) and two on-shell scalar vertices, one of them with four-arrows (its row/column must be removed of the Ψ g,s:g matrix). Thus, the first graph becomes g:s (g [3,4] r , s 1 , s 2 ) (1,2) = (σ 12 σ 2 [3,4] σ [3,4] 3,4] ). (9.15) In a similar way we compute the second resulting graph, therefore the cut-2 turns into, cut-2 which is the same aswer found in [48] (up to overall sign).
Roughly speaking, one of the interesting things here is that the Ψ g,s:g matrix can emerge exponentially after factoring the A matrix, to be more precise, Pf (9.14). This implies there are virtual gluons in a process that involves just scalar particles, which in the context of the special Yang-Mills-Scalar Lagrangian in (9.1) is a natural fact.
However, it can also be seen in pure 30 φ 4 , and additionally, a similar phenomenon appears in the effective field theories framework, which will be deeply studied in [49].
Finally, we consider the six-point amplitude, A where we have chosen, (pqr|m) = (124|6) and (i, j) = (6, 1). It is simple to see that the cut-1 is factorized by a vector field, while the cut-2 by a scalar one, namely, k [2,3,4,5] ·k [1,6] . This identity can be extended a higer numeber of points, such as it was done for pure Yang-Mills in section 7.

Strange-cuts and special Yang-Mills-Scalar Amplitudes
From the previous section, we learned that a vertex with four arrows may be interpreted as a scalar particle. Additionally, we saw as the Ψ g,s:g matrix can emerge after factorizing the A matrix.
In another way, the resulting graphs obtained from a strange-cut in pure Yang-Mills have an off-shell vertex with four arrows, thus, in this section we come back to this point in order to understand its relationship with the scalar particles in a special Yang-Mills-Scalar theory.
The first observation we would like to do is that there are two types of resulting graphs obtained from a strange-cut, which are related by (7.1) (property-I of the appendix B).
Therefore, it is enough just to work with one of them, to be more precise . (10.1) It is simple to note that the integration rules can be applied over above graph.
Such as it was remarked in section 7, the associated matrix to the above graph is given  [2,3,...,p] " looks like an off-shell scalar particle, nevertheless, the CHY integrand of this graph does not fit over any theory known. It is important to note that the Ψ g:s,g matrix has appeared after a factorization process, instead to squeeze or compactify the theory. For instance, in (6.4) we wrote the matrices, (Ψ g:s,g ) 13 [2,4] 13 [2,4] and (Ψ g:s,g ) [1,3]  In order to go beyond to the longitudinal gluon interpretation given in section 7, we are going to apply the integration rules over the graph in (10.1). Before carrying out the general case, we would like give a simple example. Let us consider the four-point computation, where we have used the integration rules and, r r,µ [1,5] r,ν [2,3,4] = η µν . Notice that second resulting graph obtained in the first bracket was already found in the previous section, after factorizing a scattering among two gluons with two scalar particles, equation (9.9 Clearly, on the first line, the puncture "σ [2,4] " behaves as an off-shell scalar particle, which interacts with one more off-shell scalar and an on-shell gluon 31 . Therefore, from the factorization method developed in this paper, a scattering of pure gluons is able to generate an off-shell scalar-like particle 32 . The above example is straightforward to generalize to higher number of points. Applying the integration rules over the graph in (10.1), we obtained two types of resulting graphs,   and cut-2 = . Observe the non-conventional pole,s 5 [2,4] . 32 Additionally, the replacement, L,µ [2,4,5] → −k µ [2,4,5] k [2,4,5] ·k [1,3] , also appears in [29], in the context of the boundary conditions for the Berends-Giele currents obtained by using the new NLSM action proposed in [56,57] .
(10.6) Clearly, we have obtained an off-shell Yang-Mills-Scalar amplitude multiplied by a graph with the same structure as the original one, therefore, we can applied (10.4) again.
One of the important things to remark here is to see as some off-shell scalar particles (in a special Yang-Mills-Scalar theory) can emerge from the strange cuts. This fact confirms the interpretation of seeing the off-shell puncture, "σ [2,...,p] ", as a scalar-like particle.

Soft Limit
As a final section, we would show how the soft limit in pure Yang-Mills theory [58], at leading order, can be computed in a simple way by using the factorization method developed in this paper.
Before going to the general case, let us consider the four-point example, A YM 4 (1, 2, 3, 4). Such as it was shown in section 6.1, this amplitude can be written in the following way r,ν [4,1] = η µν . This important to recall that the setup used was, (pqr|m) = (123|4), and the red arrow by joining the punctures (i, j) = (1, 3). Now, we are taking the particle at the puncture "σ 4 " as the soft particle, i.e. k µ 4 = z q µ 4 , with z → 0. Since that the second resulting graph on the second line in (11.1) is proportional tos 42 , the only leading order contribution comes from the first and second terms (with poles,s 43 = z(q 4 · k 3 ) ands 41 = z(q 4 · k 1 ), respectively). Notice that when, z → 0, the off-shell momenta of the particles at σ [3,4] and σ [4,1] become the on-shell momenta, k µ [3,4]  [j] = η µν . By the property in (7.1), the resulting graphs obtained from a strange-cut always cancel out the pole (spurious pole), therefore, those configurations do not contribute to the soft limit at leading order. On the other hand, the leading order contribution from the second line in (11.2) is given when O * i is an empty set, i.e. when i = n − 4 (O * n−4 = ∅). Thus, when z → 0 the leading order contribution of (11.2) is given by Now, performing the same procedure as in the previous four-point example, it is trivial to see that A YM n (1, . . . , z q n ) = 1 z n · k 1 q n · k 1 − n · k n−1 q n · k n−1 A YM n−1 (1, 2, . . . , n − 1) + O(z 0 ) , (11.4) which is the right answer [58].

Conclusions
In this work, we have obtained a new graphical method to compute the scattering of ngluons and interactions with scalar particles, which was developed under the double-cover formulation of the CHY-approach. The main idea was to define a graph representation for an ordered amplitude from the double-cover (and single-cover) prescription. By integrating the extra parameter, "Λ", the double-cover breaks into two single-cover and the integration rules arose naturally. This process generates smaller YM/YMS-graphs and, as a byproduct, some strange-graphs. The strange-graphs did not seem to have a simple physical interpretation, however, we were able to conjecture that these are just longitudinal contributions from YM/YMS amplitudes. To be more precise, under the gauge fixing, (pqr|m) = (123|4), and by choosing the red arrow from the vertices, (i, j) = (1, 3), the integration rules over the Yang-Mills amplitude, A YM n (1, ..., n), produce the off-shell general formula 35 : • When n = 2m + 1 35 When the number of particles is even, there is an overall sign, (−1) i , missing in the equations (43) and (44) in [50].  (1, [2,3]  One of the more amazing things we would like to remark is the computational information that there is behind of the identities obtained in section 7 (properties-I, II in appendix). For example, in the identity obtained in (7.1), we related a graph with a matrix 2(p − 2) × 2(p − 2) to another graph with a matrix 2(p − 2) − 2 × 2(p − 2) − 2, and the relationship is given just for an overall factor. Computationally, this is very important, let us suppose that p = 5, we are simplifying the problem from a matrix 6 × 6 to a new one 4 × 4.
On the other hand, although the integration rules emerged from the pure Yang-Mills amplitudes, these can be applied naturally over a large spectrum of graphs, such as special Yang-Mills-Scalar theory, effective field theories [49], and an others strange graphs. In addition to this fact, it was very interesting to see as the A and Ψ n matrices can be factorized in terms of the off-shell Ψ g,s:g matrix 36
As a final point, we would like to understand the connection among our graphic method, the ambitwistor string [5] and the gluing operator in [25]. III.