Pfaffian Diagrams for Gluon Tree Amplitudes

Pfaffian diagrams are formulated to represent gluon amplitudes computed from the Cachazo-He-Yuan (CHY) formula. They may be regarded as a systematic regrouping of Feynman diagrams after internal momenta are expanded and products of vertex factors are evaluated. This reprocessing enables gluon amplitudes expressed in Pfaffian diagrams to contain less terms. For example, there are 19 terms for the four-point amplitude in Pfaffian diagrams, and 35 terms in Feynman diagrams. Gauge invariance is simpler and more explicit in Pfaffian diagrams, in that subset of diagrams with the same root configuration are already gauge invariant in all lines but two. In getting to these results, several technical difficulties must be overcome. Double poles must be converted to simple poles, integrations must be carried out directly and formulated into simple rules, and the three \M constant lines must be suitably chosen to minimize the number of terms present.


I. INTRODUCTION
It is well known that the n-gluon scattering amplitude contains many terms, even in the tree approximation. For n = 4, there are four Feynman diagrams, the s, t, u-channel as well as the four-gluon diagrams. Each of the former three consists of 3 2 = 9 terms, and the four-gluon diagram contains 3 terms, making a total of 3×9+3 = 30 terms. To express the amplitude in measurable quantities, the polarization vectors i and the outgoing momenta k i , internal momenta must be expanded into sums of k i , thereby further increasing the number of terms. For larger n, the number of diagrams grows rapidly, reaching over ten million for n = 10. The number of terms in each diagram also increases exponentially, being already 3 n−2 if the diagram consists of triple-gluon vertices alone, and much more after the internal momenta are expanded into sums of external momenta.
Simplification occurs after the color factor, say Tr(λ 1 λ 2 · · · λ n ), is factored out. The resulting color-stripped amplitude has less terms, and is cyclically invariant in the order of the color trace. Only such color-stripped amplitudes with the natural order (123 · · · n) will be considered in this article. Even so, the number of terms is still huge, so any method that can further reduce it would be welcome. This note is an attempt to do that by formulating the diagrams in a completely different way.
In this connection, one might think of using the Cachazo-He-Yuan (CHY) [1][2][3][4][5] formula, since the tree amplitudes are given there by a single integration over n − 3 variables σ i . This formula, valid for any number of dimensions, is very efficient in understanding general properties such as gauge invariance in gluon scattering, but to obtain explicit expression in terms of external momenta and polarizations, the n − 3 integrations must still be carried out. Unfortunately, in the case of bi-adjoint φ 3 amplitudes, integration simply reverts the amplitude back to a sum of Feynman diagrams.
Nevertheless, there is still hope for gluon amplitudes because the CHY formula contains no internal momenta so at least extra terms coming from their expansion are avoided. Its dependence in the numerator on i and k i comes directly from the 'reduced Pfaffian' in the integrand. To get the gluon amplitude from the CHY formula, this reduced Pfaffian must be expanded, and the n−3 integrations carried out. There are a number of technical difficulties to be overcome, but at the end a set of 'Pfaffian diagrams' and 'Pfaffian rules' can be devised to express the final result. These new diagrams can be regarded as a regrouping of the familiar Feynman diagrams and Feynman rules, although this regrouping cannot be easily derived without the CHY formalism. When the amplitude is expressed in Feynman diagrams, a single diagram contains terms with a common denominator, the product of the propagators in the diagram. The numerator is computed from the product of vertex factors after expanding the internal momenta into sums of external momenta. When the amplitude is expressed in Pfaffian diagrams, terms with similar products of i · j , i · k j , k i · k j in the numerator are grouped together, with a coefficient given by sums of products of propagators.
This reprocessing generally reduces the number of terms in a Pfaffian amplitude compared to a Feynman amplitude, though of course the two amplitudes must be the same after summation. For example, for n = 4, the number of terms in the Pfaffian amplitude is 19, far less than the 35 terms appearing in the (color-stripped) Feynman amplitude.
The other advantage of Pfaffian diagrams is that gauge invariance is simpler and more explicit. A single Feynman diagram is never gauge invariant; a sum of several diagrams is needed for the gauge invariance of a single external line i. To make it gauge invariant for two i's, a sum of more diagrams is required. The larger the set of external lines, the more diagrams must be summed to gain gauge invariance for members of the set. Finally, for the set of all external lines, every Feynman diagram must be included to attain gauge invariance. For large n, this is an astronomical number.
In contrast, there are many small subsets of Pfaffian diagrams which are gauge invariant for every i, except two special ones λ, ν picked out from the very start in constructing the Pfaffian diagrams. This property can be used in practice to check and to simplify calculations. To be gauge invariant for λ or ν, again a sum of all diagrams is required.
Reduced Pfaffians can be expanded into permutation cycles [6], but that causes problems in integration. In the case of a bi-adjoint φ 3 amplitude, its integrand consists only of simple poles in the integration variables σ i , so residue calculus can be used to compute the integral easily. The terms in the expanded Pfaffian however contain double poles, making it much harder to apply residue calculus. It was realized in [6] in low-n examples that scattering equations can be used to convert double poles into simple poles, after which integrations can be easily carried out. In this note we find a systematic way to apply this idea to all n to get rid of the double poles, thereby allowing one to formulate a set of rules to compute gluon amplitudes in terms of the Pfaffian diagrams.
There is some overlap with previous work [7][8][9][10][11], but the present approach is more eco-nomical and results in less number of terms. This is so partly because we need not expand our expressions into KK, BCJ, or Cayley basis, by adding and subtracting additional terms.
Integrations are carried out directly in this article. Further simplification is achieved by a proper choice of the three special Möbius lines r, s, t, as well as the two special rows and columns λ, ν in the reduced Pfaffian. Note that only lines r, s, t, λ, ν are chosen, but the specific values of σ r , σ s , σ t are left arbitrary to preserve Möbius invariance of the amplitude.
The expansions of Pfaffians and reduced Pfaffians are reviewed in Section II, both analytically and graphically. The method to convert bi-adjoint φ 3 integrations into Feynman diagrams is reviewed in Section III, then generalized to enable it to integrate gluon amplitudes after double poles are converted into simple poles. How double poles are converted is discussed in Section IV, analytically and graphically. These prerequisites allow Pfaffian diagrams and Pfaffian rules to be formulated in Section V. That Section also contains a comparison between Pfaffian diagrams and Pfaffian rules with Feynman diagrams and Feynman rules. Four point amplitudes are computed in Section VI using both Pfaffian diagrams and Feynman diagrams, to illustrate how Pfaffian rules are used, and how the two ways to compute differ. Finally, Section VII contains a summary, and Appendix A consists of detailed explanations of the material in Section III.
The scattering functions f i are defined by with k i being the outgoing momentum of the ith gluon. The three special lines r, s, t for the Möbius constants σ r , σ s , σ t will be referred to as constant lines, the rest variable lines. The reduced Pfaffian Pf (Ψ) is related to the Pfaffian of a matrix Ψ λν λν by where Ψ λν λν is obtained from the matrix Ψ with its λth and νth columns and rows removed. The antisymmetric matrix Ψ is made up of three n × n matrices A, B, C, The non-diagonal elements of these three sub-matrices are where i is the polarization of the ith gluon, satisfying i ·k i = 0. The diagonal elements of A and B are zero, and that of C is defined by so that the column and row sums of C is zero. A similar property is true for A if the scattering equations f i = 0 are obeyed. This is the case because the integration contour Γ encloses these zeros anticlockwise.
For massless particles satisfying momentum conservation, the amplitude M α is Möbius invariant, and is independent of the choice of r, s, t, λ, ν, as well as the values of σ r , σ s , and σ t . It is also gauge invariant, in the sense that when any i is replaced by k i , then the amplitude is zero.
It is important to note that each cycle factor Ψ I , · · · , Ψ L is gauge invariant in all its external lines. If a is replaced by k a , every cycle factor vanishes because U a is then zero.
When expanded, the trace gives rise to 2 u terms. Each is a product of u factors of d ii , There are several general features worth noting about this expansion. First of all, the structure of U a in Eq.(10) dictates that there is one a and one k a at each juncture a. Thus a x·k a for any x must be following by a a ·y for some y, never a k a ·y. In other words, the four letters a, b, c, c t must be assembled according to the orders given in Fig. 1, where the circles around c and c t indicate that c can be followed by another c, and c t can be followed by another c t . Fig. 1. Allowed orders of a, b, c, c t . For example, b may follow c but never followed by c, though it can follow or be followed by a. c can be followed by b or another c, but never a.
Secondly, cyclical invariance of the trace is preserved and is reflected in terms 2,3,4 of Eq.(11), as well as in terms 5,6,7. Thirdly, Tr( shows that to each term present there must be another term read backward, with a sign given by (−1) u . This is so for the pair of terms (1,8), (2,5), (3,6), and (4,7). Note also that the number of b's and a's must be the same to keep the total number of 's and k's equal.
With these observations, it is easy to write down the 2 u terms in any Ψ I . A term with the product of u c's is always present, with a + sign. Starting from this term, one can obtain other terms by making the replacement cc → −ba. In addition, we may have to add in terms to make it cyclically invariant, and terms read backward with a (−1) u sign.
The terms constructed this way never contain c and c t at the same time, but such terms are allowed by Fig. 1 so must be added in. Such terms are of the form ac m b(c t ) n , or b(c t ) n ac m , for any m and n. They involve at least four factors, hence absent in Eq.(12), but they are generally present.
Using these rules, the 16 terms of a 4-cycle Tr(U I ) can easily be written down to be (cccc)−(bacc+accb+ccba+cbac)+(baba+abab)+(acbc t +cbc t a+bc t ac+c t acb) where cyclic partners are grouped together in round parentheses, and the second line contains additional terms from the first line read backward.
The cycle factor Ψ I = Tr(U I )/σ I is obtained by dividing every d ii in Tr(U I ) by σ ii .
According to Eq.(6), this simply turns the lower-case symbols into capital symbols: Thus from Eq.(11) we get and for any 4-cycle I we have Note that because of the minus sign in the last expression of Eq.(13), now all terms read backward have the same sign as terms read forward in both cases.
It would be useful to devise a diagrammatic representation for the cycle factors Ψ I . We shall use a heavy dot (•) at node i to represent the presence of i , and an arrow (→) to represent the presence of k i . A line linking two neighboring nodes i and i contains the scalar product between these two factors divided by σ ii . In this way we arrive at the graphic representation for A ii , B ii , C ii , C t ii shown in Fig. 2(a). Furthermore, a vertical bar inserted at node i represents the factor U i . It contains two terms, shown in Fig. 2(b). With this notation, Eq.(14) can be displayed graphically as shown in Fig. 3.

C. Expansion of the reduced Pfaffian
What is needed in the CHY formula is the reduced Pfaffian defined in Eq.(4), which can be obtained from the Pfaffian by the formula Since a λν is present in a cycle only when that cycle contains λ and ν in an adjacent position, differentiation with respect to a λν simply removes that factor and opens up the cycle into a line bounded on the left by λ and on the right by ν .
In this way one gets where the sum is now taken over all permutations of n − 2 numbers, consisting of 1 to n except λ and ν. The open cycle factor is Since U a is gauge invariant, every factor of every term in Eq.   The bi-adjoint φ 3 amplitude is also given by Eq.(1) but with for some β ∈ S n [2]. Unlike the gluon amplitude, it has only simple poles in σ i , so integrations can be carried out easily using residue calculus, resulting in a sum of Feynman diagrams determined by the choice of β. Details of how to do that will be reviewed later.
For gluon amplitudes, double poles are present in Pf (Ψ). For example, Ψ (ab) = Tr(U a U b )/2σ ab σ ba contains a double pole in σ a . Similarly, the product of σ ij in any cycle loops back so it also causes a double pole to occur. Certainly residue calculus can still be used to carry out integrations in the presence of double poles, but then derivatives of the rest of the integrand must be computed, making it very difficult to obtain general rules.
This problem is solved in the next section by using the scattering equations to convert double poles into simple poles, thus allowing a systematic computation whose results can be formulated into Pfaffian diagrams and Pfaffian rules. As a preparation, we shall spend the rest of this Section to review how integrations involving only simple poles can be carried out to yield Feynman diagrams.
There are many ways to do integrations [3,[12][13][14][15] but we shall follow the method discussed in [16]. This method differs from the others in that an explicit choice of the three constant lines r, s, t in Eq.(2) is required, though the values of their respective Möbius constants σ r , σ s , σ t remain unspecified. The latter allows Möbius invariance of the amplitude to be explicitly verified, by checking the final expression of Eq.(1) to be independent of σ r , σ s , and σ t . More importantly, in the case of gluon amplitude, doing the integral with specific choices of r, s, t avoids the necessity of expanding the integrand in some universal basis by adding and subtracting terms, thereby reducing the final number of terms in the amplitude. Also, as we shall see, a judicious choice of r, s, t can further reduce the total number of terms.

A. Feynman diagrams and their symbolic representations
In order to specify which Feynman diagrams emerge from the integrations, it is convenient to have a symbolic way to describe the diagrams without resorting to pictures. The diagrams in question are planar, massless with cubic vertices, and have their external momenta k i arranged in cyclic order (123 · · · n). For such diagrams, an internal momentum is always equal to a sum of some consecutive external momenta k i + k i+1 + · · · + k i+m , thus allowing an internal line or a propagator to be denoted by (i, i+1, · · · , i+m). Taking m = 0, the parenthesis symbol can also be used to denote an external line i = (i).
This description can be refined to review the structure of the Feynman diagram. If we denote a line a obtained by joining (external or internal) line b and line c as (bc) (Fig. 6), then its momentum is simply the momentum of those in b and c. If b = (i, i + 1, · · · , j) and c = (j + 1, · · · , i + m), then the structure a = (bc) can be revealed by putting a pair of inner parentheses at the appropriate place in a = (i, i+1, · · · , i+m), to split it up into The symbol (bc) can also be understood to be a vertex where line b merges with line c to form line a = (bc), though this notation is not symmetrical in the three lines forming the cubic vertex. A symmetrical notation to denote the same vertex is (a)(b)(c). A whole Feynman diagram can be described using these parenthesis symbols. Start from any vertex v = (R)(S)(T ) (Fig. 7), then proceed to split the internal lines repeatedly to expose the inner structure until they cannot be split up anymore. In this way the whole structure of the diagram is revealed and this can be used as a symbolic representation of the whole diagram. For example, taking v = u, Fig. 8(a) can be represented as  67)8)(91)(2(3(45))), (21) which amounts to a regrouping of the parentheses in the u-representation.
The following requirement can be and will be imposed to reduce the number of allowed representations. Take any three external lines r, s, t and require r ∈ R, s ∈ S, and t ∈ T .
If we apply that to Fig. 8(a) with (rst) = (123), then the u-representation is the only one allowed. This requirement may seem artificial, but when it comes to the CHY formula in Eq. (1), there are actually three such special lines, and the integral must depend on them.
These three lines must somehow makes their appearance in the resulting Feynman diagram, and this is how they appear.
A φ 3 Feynman diagram can be drawn in many different ways by flipping lines at vertices. Such swapping changes the order of number within parentheses, but one thing never changes: the momentum sum of all the external lines within each parenthesis remain the same consecutive sum, though the order of the terms k i +k i+1 +· · ·+k i+m may be interchanged.
We shall refer to an ordered set of numbers as a consecutive set if it can be obtained by permuting a set of consecutive numbers. For example, (86719) is a consecutive set because if can be obtained from (67891) by a permutation, but (86729) is not. Thus, no matter how the external lines are flipped, the numbers within each parenthesis always form a consecutive set. When we refer to a parenthesis from now on, we always assume the numbers within the parenthesis form a consecutive set.
In conclusion, a Feynman diagram can always be represented by a set of parentheses within parentheses, such that each parenthesis contains exactly two members, representing the vertex in Fig. 6. This representation will be referred to as a triple binary split, indicating how one starts at a vertex v that divides the diagram into three parts, and how the internal lines in every part keeps on making binary splits at each subsequent vertices. It will also be simply called pairing, to conjure up the reverse procedure of constructing a diagram by repeatedly merging a pair of lines at a vertex.

B. Feynman diagrams of a φ 3 amplitude
Details of how to integrate a bi-adjoint φ 3 amplitude is discussed in [16] and summarized in Appendix A. The resulting Feynman diagrams for a given σ (β) can be obtained by reversing the discussion of the last subsection.
Starting from a list (β) of numbers, the first task is to divide the numbers into three consecutive sets so that r ∈ R, s ∈ S, and t ∈ T , where r, s, t are the constant lines in Eq.(1). This amounts to picking a vertex v to unravel the diagram.
Next, take each of these three sets and divide it into two consecutive subsets, thereby exposing a vertex at which this line is split into two lines defined by these two consecutive subsets. Continue this way with every subset until no more split is possible. This then creates a representation of a Feynman diagram. As before, we shall refer this procedure as triple binary splitting, or simply as pairing.
The point is, the integral Eq.(1) with I = 1/σ (β) is equal to the sum of the Feynman amplitudes for the Feynman diagrams created this way by triple binary splits. If such splits cannot be completed, which would happen if at some point a set can no longer be divided into two consecutive sets, then the integral is zero. For example, (7546) → (7(546)) is a consecutive set that can be so split, but (5746) cannot be split into two consecutive sets. If several inequivalent splits exist, then the integral is equal to the sum of them. This is so because every such split defines a dominant integration region in which the integral can be computed and expressed as a Feynman diagram.

C. Simple-poles in two dimensional patterns
In a bi-adjoint φ 3 amplitude, the product of σ ij in Eq.(19) is sequential along a 1dimensional list (β).
For gluon scattering, a more erratic pattern of product emerges after double poles are converted into simple poles. Instead of a 1-dimensional pattern like (β), the result can usually be displayed only in a 2-dimensional connected tree, as illustrated in Fig. 9(a). The bottom horizontal line of such a pattern will be called a root, and all the other lines with an arrow at the end branches. Branches could be horizontal, vertical, or oblique. The factors σ ij in the pattern are read circularly from left to right on the root, and read along the direction of the arrow on a branch, ending at the arrow.
Integrations can be carried out similar to 1-dimensional patterns, and the result can also be obtained from triple binary splits. In this case, when a set is divided into two consecutive sets, each of the daughter sets must again be a connected tree.
The result of the integral Eq.(1) with this I(σ) is shown in the rest of Fig. 9, both in terms of Feynman diagrams and the triple binary split representations of them.   1), which means that the integration of I/σ (12···n) must produce a factor proportional to 1/σ 2 (rst) . What makes it interesting and complicated is that this may not happen in every term in the expansion of Pf (Ψ).
As shown in Appendix A, the Möbius factors left behind after integration can be obtained by merging every variable σ's into the Möbius constant contained in that part of the triple split. If we choose r < s < t, the Park-Taylor factor 1/σ (12···n) in the integrand always yields a factor 1/σ (rst) , making it necessary for I in Eq.(1) to yield another factor proportional to 1/σ (rst) . This is clearly so when I = 1/σ (β) , but for 2-dimensional patterns, this is not always the case term by term.
For example, let (rst) = (123) in Fig. 9(a). 4 and 5 merge into 3 so the factor σ 25 becomes σ 23 . Thus the leftover factor of this tree is 1/σ (13) σ 23 , and not 1/σ (123) . To render the gluon amplitude Möbius invariant, another tree(s) with the leftover factor 1/σ (13) σ 21 is needed, with the same coefficient and an opposite sign, so that the combination becomes 1/σ (123) , making the gluon amplitude Möbius invariant. How this can happen and its implication will be discussed in Sec. VI.

E. decomposition of 2-dimensional into 1-dimensional patterns
Integration of a 2-dimensional pattern can always be converted into a sum of integrations of 1-dimensional patterns, by artificially adding and subtracting terms. For example, the upper left corner of Fig. 10 shows a 2-dimensional pattern, whose split into three groups with (rst) = (123) is shown to its right. This pattern is equivalent to the sum of the three 1-dimensional patterns below the horizontal line, obtained by inserting 5 in front of every number between 1 and 7. Such 1-dimensional patterns either generate no allowed split and hence gives a zero integral, or they generate additional splits that cancel one another.
This simple example can be generalized [11] to any 2-dimensional tree. The advantage of such a decomposition is that it can be carried out using a set of rules without thinking. The disadvantage is that it adds many additional terms that finally cancel one another. Since we are interested in getting as few terms as possible for the gluon amplitude, we shall not employ such decompositions. Double poles are also present in the cycle factors Ψ J , · · · , Ψ L of Eq.(17). To get rid of them, the following trick can be used to remove one σ factor that causes loop back and double poles to occur.
It is convenient to designate any Pfaffian with m lines as Pf(Ψ(m)). With this notation, the original Pfaffian is Pf(Ψ(n)). Let u be the number of nodes in root I. The crucial observation is that terms in Pf (Ψ) with the same u add up to a Pf(Ψ(m)) with m = n−u.
Once so grouped, Eq. (17) can be rewritten as provided Pf(Ψ(0)) ≡ 1. ξ I is a sign factor that turns out to be irrelevant and can be ignored.
Concentrate on the first m columns and m rows of the 2m × 2m matrix Ψ(m), and label them as 1, 2, · · · , m. The ath column (1 ≤ a ≤ m) contains matrix elements C ba and A ba , The trick mentioned above consists of adding the remaining (m − 1) columns to the ath column, and the remaining (m−1) rows to the ath row. This does not alter Pf(Ψ(m)), but it changes the ath column to where X is the set of all numbers not in Ψ(m). This is a consequence of the scattering equation and the definition of C aa which state that A similar change occurs in the ath row.
This manipulation can be expressed diagrammatically as opening the cycle into a line.
Recall that each node a consists of two terms, with a heavy dot • on one side and an arrow (→) on the other side. See Figs. 2 and 3. On the dotted side C ab or B ab appears, and on the arrowed side it is A ba or C ba , with the arrow pointing into node a. What Eq.(24) does is to move the arrow from a to x, a point beyond all the cycles in Pf(Ψ(m)). See Fig. 11.
In this way one breaks the cycle and removes the double pole. The opened line begins with a dot at node a and ends with an arrow at point x. It shall be referred to as a branch, or the a-branch.
Before we move on there are several details to be settled first. To start with, there is the question of numerical factors. A factor 1 2 is present in Eq.(10) for every Ψ I , but there are also two terms at every node a, corresponding to reading the scalar products clockwise and anticlockwise. For x > 2, to every labelling of a cycle there is also another labelling in the opposite way. The end result is that there are no factors of 2, and the branch consists of both orderings, as illustrated in Fig. 11 for u = 3 and a = 2. There is a minus sign in Eq.(24) when a cycle is opened up, so there is a total sign of (−1) γ in a term with γ cycles. It cancels a similar sign in Eq.(17) and turns that equation whereΨ J is the cycle factor Ψ J opened up, namely, with node a at the arrow end replaced by x ∈ X:Ψ The replacement in Eq.  Lastly, there is the question of what the nodes a, a , · · · are. In principle they are completely arbitrary and their choice should not affect the result. To be consistent and to make it easy to remember, we shall choose them in numerical order, namely a < a < · · · . In practice this means the following. We did decide to put two of them at the two ends of the root, but we have not yet specified what they are. To simplify the eventual outcome, we shall choose these three constant lines to be consecutive, and to be 1, 2, 3 without loss of generality. Line 1 is placed at the left end of the root, and line 3 at the right end. Line 2 can either be on the root or on one of the branches.
The purpose of this choice is to simplify integration. After insertions, the branches and the root together form a two dimensional tree. For example, Fig. 13 is an n = 9 tree possessing four branches: a 2-branch, a 4-branch, a 6-branch, and an 8-branch. Its two dimensional pattern is essentially that of Fig. 9(a). The only difference is that the paraphernalia of Pf (Ψ) expansion, indicating and k dependences (•, | ), do not appear in Fig. 9(a), but that does not affect how integrations are carried out. These remarks can be summarized into a set of branch rules: 1. the top (the • end) of a branch always consists of the smallest number in that branch; 2. a larger branch should be inserted into the root and smaller branches in all possible ways, but a smaller branch should never be inserted into a larger branch; 3. no branch is allowed to be inserted on top of the 2-branch. Although Pfaffian diagrams are designed to describe the reduced Pfaffian, they also represent the whole gluon amplitude. With the interpretation of Fig. 2  Remember from the last Section that there is a demarkation point on the root separating the tree into a part containing 1 and a part containing 3. If 2 appear on the root, then it must be the demarkation point, for otherwise one part would contain two Möbius constant lines. If 2 appears on a branch, then situations can occur where roughly 'half' the diagrams can be dropped, further reducing the final number of diagrams and terms. This 'half-2 rule' will be discussed in the next Subsection. (1) to render the amplitude Möbius invariant. If it ends up above 3, as is the case in Fig. 9 and Fig. 13, then the constant Möbius factors left over is 1/σ (13) σ 23 .

and with
If it ends up above 1, then the factor is 1/σ (13) σ 21 . Since Möbius invariance of the amplitude requires 1/σ (123) to be left over, these two cases must combine according to Eq.(22).
To be able to combine, the two terms must have equal and opposite coefficients. This requirement provides a tool to check for errors in computation.
This requirement also allows all the diagrams proportional to 1/σ (13) σ 21 to be dropped, or all the diagrams proportional to 1/σ (13) σ 23 to be dropped, because the other half gives only redundant information. In this way, we can get rid of half the diagrams, hence the name 'half-2 rule'. In practice, the side possessing more diagrams would be dropped, so the half-2 rule actually ends up with less than half of the diagrams.

C. C-rule
A special case of the half-2 rule can be implemented easily, leading to a 'C-rule' which proves to be very useful in actual calculations.
A branchΨ aj 2 ···jvx with v nodes has 2 v terms, one of them being C aj 2 C j 2 j 3 · · · C jvx ≡ Ψ C aj 2 ···jvx . Diagrammatically it is represented by a branch of arrows. We shall refer to such a term as a C branch, or a aC branch to specify it starts with node a.
Consider any diagram D made up of a root and a number of C branches, but without line 2 present, such as Fig. 14(a). Let D2 denote the sum of all diagrams with 'possible' and 'impossible' 2-insertions into D, as in Fig. 14(b). Then the C-rule states that the amplitude of the sum is related to the amplitude of D by the formula where k right is the sum of external momenta to the right of the demarkation point (thick vertical line on the root), and k lef t is the sum to the left. This rule, which is a special case of the half-2 rule, will be proven later and can be used to simplify actual calculations.  Proof of the C-rule is simple because C 2x = 2 ·k x /σ 2x . After integration, node x either merges into node 1, or node 3, depending on which side of the demarkation mark it belongs to. The sum of all the former cases results in an extra factor 2 ·k lef t /σ 12 σ (13) , and the sum of all the latter cases results in an extra 2 ·k right /σ 23 σ (13) . Eq.(28) then follows from momentum conservation and Eq.(22).

D. Comparison between Pfaffian and Feynman diagrams
The It is only when all the branches associated with the same Pf(m) added together that becomes gauge invariant. In other words, only the subsets with a given root configuration.
An important attribute of the Pfaffian amplitude is that it usually has less terms than the Feynman amplitude. This is so because there is no need for internal momenta expansion, and because many diagrams can be discarded using the half-2 rule, the C-rule, and the integration rule. See the next Section for a concrete example.
The rest of this Subsection is devoted to a diagrammatic comparison between Feynman diagrams and Pfaffian diagrams. Let us start with n = 3.
Feynman amplitude from the triple gluon vertex has three terms that can be represented in three separate diagrams in Fig. 16. As before, • represents , and the box represents the difference of two k's. Using momentum conservation and setting g = 1 2 , these three terms can be transformed into Fig. 17, where as usual an arrow at node i means k i . Fig. 17. These three diagrams are equivalent to the three diagrams in Fig. 16.
In this form they can be seen to be identical to the Pfaffian diagrams shown in the second line of Fig. 18.   Fig. 18. Pfaffian diagrams for n = 3.
Thus at least for n = 3, Pfaffian diagrams are essentially the same as Feynman diagrams after momentum conservation is used. For a larger n, it is almost impossible to do the expansion of internal momenta and vertex products by hand, and then to arrange them into Pfaffian diagrams in a systematic way, though one can still see a a connection between the two. For example, take the Pfaffian diagram in Fig. 13, which has the 2-dimensional pattern of Fig. 9(a) and contains the Feynman diagram Fig. 8(b). This Feynman diagram contains 7 triple-gluon vertices and therefore 3 7 terms, even before internal momenta are expanded into sums of external momenta. One of these terms is shown in Fig. 19(a), where a box represents the difference of two momenta as before. It can be seen that it does contain the Pfaffian diagram term shown in Fig. 19(b) after its external momenta are suitably replaced by its external momenta by using momentum conservation.
Next consider IIA, which consists of 2-insertions into the root. The two allowed pairings of the root are (14) accounting for part of the four-gluon vertex contributions in Feynman diagrams. The total contribution from column III is therefore Since each U i has two terms, there are 8 terms in Eq.(29), 4 terms each in Eq.(30) and Eq.(31), and 3 terms in Eq.(34), making a total of 19 terms.
For lines 2 and 4, gauge invariance shows up in each IA, IB, IIA, IIB, and IIIA+IIIB, because each of them has a different root configuration.
In order to compare and to verify, let us compute the Feynman amplitude from Fig. 21. The s 12 -channel amplitude consists of 16 terms, the s 41 -channel amplitude contains another 16 terms, and the four-gluon contact amplitude contains another 3 terms, making a total of 35 terms, compared to the 19 terms in the Pfaffian amplitude. With g = 1 2 , I have explicitly verified that the Pfaffian amplitude is identical to the Feynman amplitude.

VII. CONCLUSION
The CHY formula for gluon amplitude in any dimension can be expressed as a sum of Pfaffian diagrams, whose ingredients are roots and branches. Each diagram is composed of several branches grown on one root, from which the amplitude can be read out. Pfaffian diagrams may be considered as a symmetric rearrangement of the terms in Feynman diagrams, after internal momenta are expanded into sums of external momenta and products of vertex factors are computed. This processing generally renders Pfaffian diagrams to carry less terms than Feynman diagrams. Pfaffian diagrams have the further advantage that gauge invariance is explicit in any subset of diagrams with a fixed root configuration, for all lines but two. This is a property not share by the Feynman diagrams.
There are three critical technical bottlenecks to get through before these Pfaffian diagrams can be obtained. Double poles in the expansion of the reduced Pfaffian must be systematically converted to simple poles. Integration in the resulting simple poles must be efficiently carried out and formulated into simple rules. Finally, the three Möbius constant lines must be suitably chosen to simplify the Pfaffian rules and to minimize the number of terms present.
Considerable simplification in the computation occurs in four dimensions [17]. The technique to get rid of double poles can also be used on other CHY theories where reduced Pfaffians are present [3][4][5], though the method of integration must be adjusted because of the modification or the absence of the Parke-Taylor factor in these other theories.
I am grateful to Song He for interesting discussions.
There is no way to evaluate the integral directly from the contour Γ because solutions of f i (σ) = 0 are unknown for large n. A way out is to distort Γ to surround and to evaluate at the simpler and explicit singularities of I(σ)/σ (12...n) . In order to avoid dealing with multidimensional topology needed for a multi-variable complex integration, the integrations will be carried out one at a time, each time distorting the contour away from one f i = 0. There are two kinds of parentheses in I(σ), seeded and unseeded. A seeded parenthesis is one that carries a number c (the 'seed') whose σ c is not an integration variable. An unseeded parenthesis is one in which all σ i are integration variables. The parentheses for R, S, T are seeded with seeds r, s, t, but many of their sub-parentheses are unseeded. We shall see that a new seed is produced each time an integration is carried out, thereby turning some unseeded parentheses into seeded ones. This makes it possible for the the following method which works only for seeded parentheses to be used repeatedly to carry out all the integrations in Eq.(A1).

Consider the integral
for a seeded parenthesis D in I(σ). It contains d + 1 numbers including the seed c. The contour Γ D encircles f i = 0 ∀i ∈ D\c counter-clockwise. I D (σ) is the portion of I(σ) of the form i,j∈D (1/σ ij ), and σ D is the portion of the Parke-Taylor factor of the form σ D = i∈D,i+1∈D σ i,i+1 . Both contain d factors.
Pick any p ∈ D which is different from c. Distort the contour Γ D away from f p = 0 to surround the singularities of I D (σ)/σ D . Make the change of variables σ ij = σ ij for i, j ∈ D, with = σ pc so that σ pc = 1. The integration measure is then converted to i∈D,i =c Since the explicit singularities lie in the small σ ij region, to evaluate the integral we need to consider what happens to the integrand in the → 0 limit.
In that limit, f i (σ) → f D i (σ )/ , where If the d+1 numbers in D did not form a consecutive set, then I D (σ) would yield a smaller power of than d , causing the simple pole to disappear and the integration to be zero. This is why the dominant integration region at small σ ij 's must come from a consecutive set.
At = 0, f p becomes f D p = k 2 D . If the parenthesis D splits into two parentheses D 1 and D 2 , with d 1 + 1 and d 2 + 1 members respectively, then one of them, say D 2 , must contain c. We will choose p to be situated in D 1 so that both D 1 and D 2 are seeded parentheses, with seeds p and c respectively. Moreover, I D (σ ) = I D 1 (σ )σ xy I D 2 (σ ), and σ D = σ D 1 σ ab σ D 2 , where x ∈ D 1 is the point next to y ∈ D 2 on the tree, and a ∈ D 1 is the number next to the number b ∈ D 2 . See Fig. 22.   Fig. 22. A seeded parenthesis D with seed c is split into two seeded parentheses D 1 and D 2 with seeds c and d respectively.
If d i = 0, then K D i = ±1, with the sign determined by the method in Section IIId. If d i > 0, then the treatment for Eq.(A3) can be repeated on K D i to factorize it into two K-integrals for its sub-parentheses. This procedure can be repeated over and over again until all the integrations of Eq.(A3) are carried out. According to Eq.(A8), each integration produces a propagator 1/k 2 D appropriate to that parenthesis, so the result of all integration would produce a product of d propagators for the d parentheses in D, up to a sign.
In getting to Eq.(A8), the factors σ xy in I D (σ ) and the factor σ ab in σ D have been set equal to σ pc = 1, in anticipation of subsequent integrations of K D 1 and K D 2 which would put all the σ i for i ∈ D 1 to be equal to σ p , and all the σ j for j ∈ D 2 to be equal to σ c .
Let us now apply this method to Eq.(A1). As a result of triple splitting, I(σ) = I R (σ)I S (σ)I T (σ)ρ(σ) and σ (12···n) = σ R σ S σ T ρ (σ). What ρ(σ) and ρ (σ) are depend on r, s, t and where they are situated in the tree. Consequently, so up to a sign and the ρ T = σ 2 (rst) /ρ(σ)ρ (σ) factor, the value of M for any triple binary split is given by the product of the n−3 propagators determined by the parentheses of that split. At the end of all integrations, σ i = σ r ∀i ∈ R, σ j = σ s ∀j ∈ S, σ k = σ t ∀k ∈ T , so ρ T can be evaluated with these substitutions.
Let us see what the left-over factor ρ T is for the Pfaffian diagrams, where (rst) = (123), with 1 at the left end of the root and 3 at the right hand of the root. In that case, σ (rst) = σ (123) , ρ (σ) = σ (123) . The value of ρ(σ) is equal to σ (123) if 2 is on the root, and is equal to σ (13) σ 23 if the 2 appears on a branch, and if the 2-branch is to the right of the demarkation point. It is equal to σ (13) σ 21 if the 2-branch is to the left of the demarkation point.