CHY formulae in 4d

In this paper, we develop a rather general way to reduce integrands with polarization involved in the Cachazo-He-Yuan formulae, such as the reduced Pfaffian, its compactification and its squeezing, as well as the new object for F3 amplitude. We prove that the reduced Pfaffian vanishes unless evaluated on a certain set of solutions. It leads us to build up the 4d CHY formulae using spinors, which strains off many useless solutions. The supersymmetrization is straightforward and may provide a hint to understand ambitwistor string in 4d.


Introduction
A new formulation for S-matrix of massless particles in arbitrary dimensions, dubbed as Cachazo-He-Yuan (CHY) formulation, has been developed for a large variety of theories [1][2][3][4]. It expresses tree-level S-matrix as an integral over the moduli space of Riemann spheres, which are localized by a set of constraints, known as scattering equations [1,5,6]  where s a b = (k a +k b ) 2 = 2k a ·k b , σ a denotes the position of the a th puncture and we denote σ a b := σ a − σ b .It has been argued that what underpins the formulation is the ambitwistor string theory [7][8][9]. The formulation has been inspired by Witten's revolutionary twistor string theory for N = 4 super-Yang-Mills theory (SYM) in four dimensions [10], and in particular the Roiban-Spradlin-Volovich-Witten (RSVW) formulae for all tree amplitudes in the theory [11]. Originally CHY discovered scattering equations in attempts to rewrite the equations in the delta functions of RSVW formulae without using 4d spinor helicity variables [5], thus by construction they reduce to RSVW equations in four dimensions. More precisely, we have n−3 different sets of 4d equations, which are polynomial equations of degree JHEP07(2017)069 d = 1, 2, . . . , n−3. The n−3 sectors are labeled by k = d+1 = 2, . . . , n−2, which coincide with helicity sectors. A set of equations, which are completely equivalent to RSVW equations, have been proposed in [12] based on ambitwistor string theory in four dimensions. It turns out that they are more convenient for our purposes, and in particular for helicity amplitudes. To write the equations in sector k , we divide n particles into two sets of k and n−k particles denoted as − and + respectively: 2) Here the variables are σ's and t's, which can be combined into n variables in C 2 , σ α a = 1 ta (σ a , 1). The σ b p is the abbreviation of σ b −σ p . The − and + are arbitrary two sets of the n external particles, with their length equal to k and n − k respectively. Different choices just correspond to different link representation [13,14], which share the same solution of σ's. In this paper, we reserve − and + as the negative and positive helicity sets of external particles and k the length of −, i.e. the number of external particles of negative helicity. A priori there is no relation between solution sector and helicity sector. We refer the readers to [15] for the direct derivation of (1.2) from (1.1); in the same paper, it has been shown that (1.2) is equivalent to RSVW equations, and one can freely translate between the two forms. Each solution of (1.1) corresponds to a unique solution {σ a , t a } of (1.2) for some k , with identical cross-ratios of the σ's. For each k , (1.2) have an Eulerian number of solutions, E n−3,k −2 , and the union of them for all sectors give (n−3)! solutions of (1.1), with (n−3)! = n−2 k =2 E n−3,k −2 [5,16]. It is highly non-trivial to reduce the localized integral measure of CHY formula, with delta functions of (1.1), to that of 4d formula, with (1.2), for some k sector. The reduction requires a sum over all sectors, and for each of them it results in a complicated conversion factor that depends on k . In addition, after we plug in spinor-helicity variables for e.g. Yang-Mills amplitudes, the CHY integrand behaves very differently in different helicity and solution sectors. As we will see, the Pfaffian plays the role of "solution-filter": it is non-vanishing only on the solution sector that coincides with the helicity sector, which is why we have a 4d formula for each helicity sector. What is even more interesting is that in the right sector, the polarization part of the CHY integrand exactly cancels the k -dependent conversion factor from the measure! Thus two complications cancel out, and for YM we are left with a trivial Parke-Taylor factor in 4d.
Let's make the statement more precisely. For gauge theory and gravity, the most important ingredient is a 2n × 2n skew matrix Ψ n and we define its reduced Pfaffian Pf Ψ n := (−) a+b σ a b Pf|Ψ n | a b a b with 1≤a<b≤n. We try to factorize the Pf Ψ n into two parts depending on particles of negative and positive helicity respectively. Then we show in the right sector that is consistent to the JHEP07(2017)069 helicity sector, each of the parts combines to a reduced determinant while in other sector one of the part must vanish. That is, Pf Ψ n k = δ kk det h k det h n−k , (1.4) Here the two matrices, the k × k matrix h k and (n−k) × (n−k) oneh n−k essentially introduced in [12] (see also [17,18]) are given by and we define det h k = det |h k | a b /(t a t b ) (similarly for det h n−k ) where we use |h k | a b to denote the minor with any row a and column b deleted.
We rearrange the Pf Ψ n using some fundamental gauge invariant or almost gauge invariant objects. It is either a (modified) trace of linearised field strength ornamented with some σ's or C aa . We view the 4d scattering equations (1.2) as a change of variables: we refer to λ b∈− ,λ p∈+ and t a , σ a as "data" and the 4d scattering equations (1.2) as writing λ b∈− and λ p∈+ in terms of the data. After plugging in this change of variables, the C aa in Ψ n directly reduces to object made up of spinors. What left to do is to deal with all kind of trace. After all, somehow, we find the reduced Pfaffian reduces to the two reduced determinants. This way of reduction is rather general: not only the reduced Pfaffian, but also many other integrands, such as the reduced compactified Pfaffian used in EM, YMS, DBI amplitudes or the new object P n used in F 3 , R 2 , R 3 amplitudes are also related to these two (extended) matrices. It may even be applied at loop level [19].
The paper is organized as follows. In section 2, we introduce the CHY formulae in 4d. In section 3, we study the reduction of Pfaffians to 4d for k =k. First we see how PfΨ n factorizes in 4d in a manifeslty gauge-invairant way, which naturally leads to the 4d matrices h k andh n−k . Then we present the beautiful reduction of Pf Ψ n , in a similar but more non-trivial way. In section 4, we move to general case with arbitrary k , which requires generalized version of h k k andh k n−k matrices. We show that both Pf Ψ n and P n reduce nicely into the generalized h k k andh k n−k ; while Pf Ψ n directly vanishes when k = k, P n does not and gives interesting formulae in 4d. The reduction of the reduced compactified Pfaffian and squeezed Pfaffian is put in appendix B, C.

4d CHY formulae
We start with CHY formula for tree-level S-matrix of n massless particles: where the precise definition of the integral measure including delta functions can be found in [1], and I n is the CHY integrand defines the theory. In the second equality one sums over JHEP07(2017)069 (n−3)! solutions of (1.1), evaluated on the integrand and the Jacobian, which is defined as a reduced determinant: 2) where the n × n matrix (Φ n ), with entries {Φ a b } := −∂{E a }/∂{σ b }, is the derivative matrix; the rows p, q, r and columns a, b, c are deleted (corresponding to deleted equations and variables, respectively), and we have two Fadeev-Popov factors, defined as |a b c| := σ a b σ b c σ c a .
For gauge theory and gravity, the most important ingredients is the reduced Pfaffian Pf Ψ n given in (1.3). Many other integrands can be abtained by doing some operation on it. The CHY integrand for n-point Yang-Mills tree amplitudes reads where C n is the color-dressed Parke-Taylor factor, with the sum over (n−1)! inequivalent permutations.
The general 4d formula in solution sector k for n-point amplitudes reads: where d 2 σ a := dσ a dta ta , and in addition to 4 deleted variables by GL(2), 4 redundant equations in (1.2) are deleted which give overall delta functions for momentum conservation. In the second equality, one sums over the Eulerian number, E n−3,k −2 , solutions in sector k . The J n,k is the Jacobian of the localized 2n − 4 integrals where we have chosen to eliminate t m , σ u , σ v , σ w and Eα b=c,d , with the FP factor c d 2 (for E α p =q,r the FP factor is [q r] 2 ). The relation between the two jacobians is simple. Viewing (1.2) as a change of variables and plugging in it, we find Here we don't need to plug in any solutions, but simply make a change of variables, so this is really an equality between rational functions of the data, i.e. λ b 's,λ p 's, and σ a , t a 's. The two reduced determinants det h k and det h n−k can be thought as two resultants and are divided by det Φ n as discussed in [20]. We find that the quotient is just J n,k . A conjecture about the closed form of J n,k is put in appendix A.

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Thanks to (1.4), then for gluon amplitudes, the integrand is nothing but the (colordressed) Parke-Taylor factor I YM n = C n . Different from (2.1), any σ bc t b tc σpq tptq or σ bp t b tp with b, c ∈ − and p, q ∈ + is GL(2, C) invariant and any known 4d integrand added with these objects could be a new 4d integrand, for example we add some σ bp t b tp to the I YM n and we get those for QCD in [21].
In this paper, we explicitly demonstrate the first identity (1.4). Compared to this identity, the second one (2.6) is a more boring one, as there is no polarization involved and just kinematics reducing to 4 dimensions. One can check as many points as we want, without any difficulties (we have checked up to 50 points with all solution sectors numerically). A proof based on direct inspection should be straightforward.
3 Reduced Pfaffian in 4d for the k = k sector In this section, we show in a constructive way how the reduced Pfaffian factorizes in four dimensions for the solution sector that coincides with its helicity sector, k = k. We will proceed in two steps: as a warm up, we show how it works for the vanishing Pfaffian Pf Ψ n , which factorizes into two vanishing determinants in 4d; then we apply it to the more nontrivial case of the reduced Pfaffian and show Pf Ψ n = det h det h . The reason for doing so is that both PfΨ n and Pf Ψ n have similar expansions, as first studied in [22], and we review them here.
From the definition of Pfaffian and thanks to the special structure of 2n × 2n matrix Ψ n , we can expand PfΨ n as a sum over n! permutations of labels 1, 2, . . . , n, denoted as where sgn(p) denotes the signature of the permutation p and in the second equality, we use the unique decomposition of any permutation p into disjoint cycles I, J, · · · , K given by I = (a 1 a 2 · · · a i ), J = (b 1 b 2 · · · b j ) , · · · , K = (c 1 c 2 · · · c k ) ; (3.2) each Ψ p is the product of its "cycle factors" Ψ I Ψ J · · · Ψ K , which we define now. When the length of a cycle equals one, its cycle factor Ψ (a) is given by the diagonal of C-matrix: and when the length exceeds one e.g. i > 1, the cycle factor is given by Here σ (a 1 a 2 a 3 ···a i ) = σ a 1 a 2 σ a 2 a 3 · · · σ a i a 1 . The trace is over Lorentz indices and f µν is the linearized field strengths of gluons. Note that the decomposition is manifestly gauge invariant: for cycle factors with length more than 1 (3.4), the trace of f µν is gauge invariant, while for 1-cycles, (3.3), the factor is gauge invariant on the support of scattering equations (1.1).

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The reduced Pfaffian Pf Ψ n , as discussed in [22], is different from PfΨ n . Because the 1 th , n th columns and rows have been deleted, the numerator of the cycle containing 1 and n becomes 1 2 1 · f a 2 f a 3 · · · f a i−1 · n instead of a trace. Then with Here I, J, · · · K are the cycles of permutation p. The prime on the summation sign indicates that the sum is taken over all p ∈ S n such that 1 is changed into n. There are (n − 1)! such permutations in S n so the sum consists of (n − 1)! terms.
The key observation in [22] allows us to expand the reduced Pfaffian in terms of building blocks, each of which is either the product of various closed cycles or an open cycle involving the two deleted labels. Closed cycles have a very good property that they will vanish unless all of their elements belong to same helicity. While the open cycle is much tougher, as it's not gauge invariant individually (dependent on the gauge of the two deleted particles) and wont't vanish when their elements come from different helicity sets. As a warm up, we show in the first subsection the Pfaffian, which is the product of only closed cycles [22], factorizes. Though the Pfaffian equals zero, it very non-trivially factorizes into determinants of two matrices. Also it is the naturel way to introduce the two matrices h k andh n−k (1.5). In the next subsection we carefully deal with the open cycle and finally factorize the reduced Pfaffian to two reduced determinants.

The Pfaffian in 4d
Let's start with the Pfaffian, PfΨ n . In 4 dimension, f µν reduces to a self-dual part and an anti-self-dual part: f µν → αβ fαβ + αβ f αβ . We denote these two parts as f − and f + respectively. An important property is that any two adjoint linearised strength fields f − b f + p in the trace can exchange their place if the helicity of b, p are different, i.e.
So we can always reduce those traces where particles of negative or positive helicity are mixed each other to split ones which have a simple reduction in 4d. Then Here b 1 , b 2 , · · · , b x are all the particles of negative helicity from a 1 , a 2 , · · · , a i with their ordering unchanged and similarly p 1 , p 2 , · · · , p y are all the particles of positive helicity from a 1 , a 2 , · · · , a i with their ordering unchanged. Note that tr (f a 1 f a 2 · · · f a i ) directly vanishes if there is only one particle of negative helicity or only one particle of positive JHEP07(2017)069 helicity in a 1 , a 2 , · · · , a i . However we see that the remaining case still effectively vanish as we always add up all permutations (see (3.1)) while Here the sum is over ordered permutations "OP", namely permutations of the labels in the joined set {b 1 , b 2 , · · · , b x }, {p 1 , p 2 , · · · , p y } such that the ordering within {b 1 , b 2 , · · · , b x } and {p 1 , p 2 , · · · , p y } is preserved. Therefore, in the sum of (3.1), we can effectively write tr (f a 1 f a 2 · · · f a i ) in 4d in a remarkably simple way: Motivated by (3.10), we recall the off-diagonal elements of the k × k matrix h k and (n−k) × (n−k) oneh n−k essentially introduced in [12] (see also [17,18]): It is clear that when we have any cycle factor with length at least 2, it must be given by the chain product of such off-diagonal elements To this point we have not used scattering equations and solution sectors in four dimensions. The non-trivial part of the reduction concerns 1-cycle, or the diagonal entries of C-matrix.
Note that Ψ (a) = C aa is only gauge invariant on the support of scattering equations, so it is not surprising that to reduce it nicely one needs to use scattering equations in four dimensions. Now we derive the explicit expression of C aa . When a ∈ − and a ∈ − , we have Note that C aa depends on σ and because of the 4d scattering equations (1.2), we can make the change of variables 14)

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Such that C − aa reduces to: In the last equality, we have collected the denominators together such that σ bp is canceled. Now C − aa factorizes into two factors All gauge dependence is in the latter factor and it can be eliminated by scattering equations as t a p∈+ tpλα p σa p =λα a (3.14). Similarly we can work out the case of a ∈ + and a ∈ + We first discuss the k = k case and without loss of generality let's consider − =−, which makes our discussion simpler. Then the above two cases are already enough here, postponing other two cases in the following sections. Miraculously, C aa reduces to diagonal entries of h k orh n−k [12] depending on the helicity: The important thing is that the diagonal entry is a linear combination of off-diagonal entries in that row/column. With these diagonal entries of h k orh n−k , the reduction for Ψ (a 1 a 2 ···a i ) with i > 1 or i = 1 (for k = k) are both spelled out in one nice formula, (3.12). We find h a 1 a 2 h a 2 a 3 · · · h a i a 1 in (3.12) is just the ingredient of det h k , where the sum is over all permutations of particles of negative helicity, i.e. q ∈ S k and I 1 , I 2 , · · · , I s are the cycles of the permutation q. Similarly works forh a 1 a 2h a 2 a 3 · · ·h a i a 1 . Then, we see that PfΨ n factorizes to two parts depending on particles of negative or positive helicity respectively, with most of the terms vanishing and the surviving terms combining to det h k or deth n−k , (3.20) Obviously both det h k and deth n−k vanish since they both have a null vector; this is consistent with the fact that PfΨ n vanishes due to the two null vectors.

Reduced Pfaffian in 4 dimensions
Now we turn to Pf Ψ n . Now we need to deal with the open cycle. Similarly, we can always reduce these mixed open brackets into split one as any two adjoint linearised strength fields Note that this equality is true no matter what the helicity of 1 and n are. In the following demonstration we need to delete two columns and rows from negative and positive helicity set respectively, so we assign 1 − and n + . Using this property, we can always rearrange the kinematic numerator in a split form with the ordering of particles of negative helicity and the ordering of particles of positive helicity unchanged respectively. For example, with n>6, Further on, all such kinematic numerator can reduce to a product of some simple angle brackets and square brackets as shown in the last equality. Here |µ], |µ are the reference of 1,n respectively.
For the general case with x particles of negative and y particles of positive helicity between 1 and n, there are x x+y cycles whose kinematic numerators are equal to those of a certain split open cycles and they all reduce to a product of some simple angle brackets and square brackets, (3.23) Here |µ], |µ are the reference of 1,n respectively, i.e. − 1 = |1 [µ| [1µ] , + n = |n] µ| nµ and we have used the reversed ordering p y , p y−1 , · · · , p 1 for later convenience.

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In the first equality, we have plugged in (3.23). In the second equality, we have defined [1µ]σ 1py . Here we can treat 1 as b 0 and if there is no particles of negative helicity between 1 and n, bxµ nµ σ bxn reduces to 1µ nµ σ 1n ; similarly we can treat n as p 0 and if there is no particles of positive helicity between 1 and n, [µpy] [1µ]σ 1py reduces to [µn] [1µ]σ 1n . Note that these prefactors bxµ nµ σ bxn , 1µ nµ σ 1n only depend on b x or c y respectively. Though Ψ [1a 2 ···a i−1 n] has particles with mixed helicity, h [1b 1 b 2 ···bx] andh [np 1 p 2 ···py] do have only particles of negative or positive helicity respectively. Adding that closed cycles vanish unless all of their elements have same helicity, Pf Ψ n decouples to two parts which are dependent on particles of negative and positive helicity respectively, (3.26) Here we have explicitly written out the open cycles to emphasis them. β, I, · · · , J are the cycles of permutations r of negative helicity particles except 1 and ρ, K, · · · , L are the cycles of permutationsr of positive helicity except n.

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Since (3.30) we write the first part in the r.h.s. of (3.26) as a sum over all possible b x , i.e. b x = 1, 2, · · · k. This equality can also be seen by collecting terms with the same prefactor bxµ nµ σ bxn , J are the cycles of permutations of particles of negative helicity except 1 and b x . Then each term of the summation in r.h.s. of the above equation equals det |h k | bx 1 up to a prefactor. Summing over all possible b x , i.e. b x = 1, 2, · · · k, gives the left parenthesis of r.h.s. in (3.26). Similar derivations leads to the right parenthesis. Then We insert with p y = k + 1, k + 2, · · · , n reduce to det h n−k . Then All gauge dependence of particle 1 and n combine to one factor respectively and on the support of 4d scattering equation (1.2),

Extension to all solution sectors
We have arrived at (3.32) without using the explicitly form of 1-length cycle, i.e. C aa . When extended to all solution sectors, those cycles whose length are longer than 1 don't change, while the 1-length cycles change to C aa with the solutions of k sectors plugged in. That is, we need to enhance the origin two matrices to h k k andh k n−k with their diagonal entries depending on the solution sector k while the off-diagonal entries unchanged. The expression of C aa with a ∈ − and a ∈ − has been given in (3.16). Note that this expression is true even when k = k, a ∈ − and a ∈ − . (4.1) Now we derive the expression of C aa with a not consistent in helicity sector and solution sector. When a ∈ − but a / ∈ − , we have After we plug in the changes of variables (3.14), unlike (3.15), terms with a ∈ + and p = a both contribute. 3) The first term on the r.h.s. also factorizes into two parts following the trick used in (3.15), (3.16), then we see that C aa only has contribution from the term of p = a, and we obtain Consequently, we have By a parity transformation, we can directly obtainh aã When k = k, these extended matrices come back to their original ones. When k = k, one of det h k k , deth k n−k must vanish. Further more, when k < k, after deleting appropriate row and column of h k k , the determinant of the remaining matrix still vanishes, so does theh k n−k when k > k, which results in the vanishing of Pf Ψ n in k = k sectors. We will discuss this in section 4.1. Some integrands receive the contribution from the k = k sectors, such as P n , which will be discussed in section 4.2.

The vanishing of reduced Pfaffian in other sectors
We start from the equation (3.32). Note that we have got this by deleting the 1 th and n th rows and columns of Ψ n . We can also delete other rows and columns to get a similar expression. What's more, along the demonstration of (3.5) to (3.32), we don't use the scattering equations (1.2), in other words (3.32) is true for any solutions. After (3.32), the scattering equation is used and we demonstrate (3.36). Now we move to other solution sectors. Without loss of generality, let's consider − = {1, 2, . . . , k} while − = {1, 2, . . . , k }, which makes our discussion simpler. Then when k < k, further on, we can also and always delete (k + 1) th and n th column and row instead of the of 1 th and n th ones, then the reduced Pfaffian becomes

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Notice that we still have to calculate the determinants of series of matrices. Instead of both summations in r.h.s. of (4.9) being combined to simple factors as shown in (3.34), we show that the determinants of matrices |h k k | bx k +1 with b x = 1, 2, · · · , k in the first summation vanish identically.
These matrices all come from the original matrix h k k with the (k +1) th column deleted and the 1 th , 2 th , · · · , k th row deleted respectively. An important observation is that the first k columns of these matrices are linearly dependent as t 1 η 1 + t 2 η 2 + · · · t k η k = 0 (4.10) here η 1 , η 2 , · · · , η k are the 1 th , 2 th , · · · , k th columns of anyone of matrices |h k k | bx k +1 with b x = 1, 2, · · · , k. This is equivalent to say that k b=1 t b h ab = 0, for a = 1, 2, · · · , k (4.11) These equations come from two totally different origins as a ≤ k or a > k . For the case of a ∈ − i.e. a = 1, 2, · · · , k , the establishment of (4.11) come from the fact that the diagonal elements h aa are a linear combination of some off-diagonal entries as shown in (4.1) with some appropriate coefficients.
While for the cases of a > k , note that a now belongs to the set + , and the validity of (4.11) come from the change of variables (3.14). What we need here is the cases of a = k + 1, k + 2, · · · , k, for a = k + 1, k + 2, · · · , k (4.12) Obviously after we act λ a on both sides of above equation, both sides vanish, that is t b ab σ ab = 0 for a = k + 1, k + 2, · · · , k (4.13) After understanding (4.10), now it is easy to understand the vanishing of all matrices |h k k | bx k +1 with b x = 1, 2, · · · , k. We take a multiple of the a th row of these matrice by t a for a = 1, 2, · · · , k , and then add 2 th , 3 th , · · · , k th column to the 1 th column; in this way we obtain a new 1 th column whose entries all equal to zero because of (4.10). Since we just do some fundemantal operation on these matrices and we obtain a column with all entries equal to zero, all determinants of these matrices vanish.
When k < k, the determinants of matrices in the first summation in (4.9) vanish; when k > k, the determinants of matrices in the second summation in (4.9) vanish. Pf Ψ n only receives the contribution of k = k sector, then we proved the identity (1.4) given in the introduction.
In the reduction of Pf Ψ n , we reorganize the Pfaffian using some fundamental (almost) gauge invariant objects and then deal with these objects, finally we reconstruct the reduced Pfaffian using det h k and deth n−k . This procedure is quite general. The reduced

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Pfaffian can be thought as putting two kinematic of the deleted particles in higher dimensions and the Lorentz contraction of them and anything else vanish unless contraction of them each other equal to 1. 1 Similarly, we can put polarizations of m pairs of particles in higher dimension and then we get the reduced compactified Pfaffian, which is the integrand of Einstein-Maxwell, Einstein-Maxwell-scalar, Yang-Mills-scalar, Born-Infeld amplitudes. This can be viewed as the "fancy reduced Pfaffian" as we may meet several open brackets in each term of the expansion. Besides, the valid solution sector for the reduced compacted Pfaffian is shifted to k = k + m sector with 2m the number of the particles whose polarization are set in higher dimension. These are presented in the Appdendix B. Further more, the reduced squeezed Pfaffian, which is the integrand of Einstein-Yang-Mills, can be obtained by some combination of the reduced compactified Pfaffian. So all these methods can be applied in the reduced squeezed Pfaffian, seen in Appdendix C.
There are some integrands that can't be organized as a matrix, let alone its Pfaffian, such as the P n used in F 3 , R 2 , R 3 amplitudes. We can still reduce it into objects related to det h k and deth n−k and make some properties manifest. Besides, it receives the contribution from several sectors and one need to add up all of these to get the corresponding amplitudes, as discussed in the following subsection.

The new object P n for higher dimension operator
As shown in [21], as a generalization of the reduced Pfaffian in Yang-Mills theory, P n is a new, gauge-invariant object that leads to gluon amplitudes with a single insertion of F 3 , and gravity amplitudes by Kawai-Lewellen-Tye relations. When reduced to four dimensions for given helicities, this new object vanishes for any solution of scattering equations on which the reduced Pfaffian is non-vanishing. This intriguing behavior in four dimensions explains the vanishing of graviton helicity amplitudes produced by the Gauss-Bonnet R 2 term, and provides a scattering-equation origin of the decomposition into self-dual and anti-self-dual parts for F 3 and R 3 amplitudes.
No matter what P n is, it must be gauge invariant. It's most natural to start from the expansion of Pfaffian in a manifest gauge invariant way (3.1). Reorganize these gauge invariant objects according to their length and we define the minimal gauge invariant and guage invariant objects P as P i 1 i 2 ··· ir := |I 1 |=i 1 ,|I 2 |=i 2 ,··· ,|Ir |=ir with i 1 + i 2 + · · · + i r = n and the convention i 1 ≤ i 2 · · · ≤ i r . Then Pfaffian can be written as 1 Here is some subtlety. We have to complex k1, kn set in higher dimensions such that they dotting anything else equal to 0, while they dotting each other equal to 1. It is just a mathematics trick after all nothing about k1, kn changes beyond the reduced Pfaffian.

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Decorated with some appropriate coefficient, P can be used to build up some unknown CHY integrands, such as the new integrand is defined as (4.16) where N i>1 denotes the number of indices in i 1 , i 2 , · · · , i m which are larger than 1, or the number of cycles with length at least 2; c is just any constant because we can add any multiplet of (4.15) without changing the answer. In 4d, PfΨ n reduce to the determinants of h k k andh k n−k . Similarly we define with i 1 + i 2 + · · · + i = k and the convention i 1 ≤ i 2 · · · ≤ i . Then det h k k can be rewritten as a sum of H and similarly works deth k n−k , where we have introduced shorthand notation for the summation range, {i} k means i 1 + i 2 + . . . i = k and i 1 ≤ i 2 ≤ · · · ≤ i and similarly for {ĩ}˜ n−k .
Further, similar to the definition of (4.16), we define two auxiliary objects H k k andH k n−k , Thanks to (3.12), P reduces to several products H andH in 4d, Here the sum is over all distinct partition of i 1 i 2 · · · i m into two parts j 1 j 2 · · · j and j 1j2 · · ·j˜ , with j 1 + j 2 + · · · + j = k andj 1 +j 2 + · · · +j˜ = n − k. Dividing N i>1 in (4.16) into two parts N j>1 and Nj >1 (set c = 0) which depend on − and + sets respectively, P n reduces to: where each mix summation over i andĩ decouples to two independent summation over i and overĩ respectively. Then

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When k = k, both terms in the r.h.s. of (4.22) vanish, which is orthogonal to Pf Ψ n and answers the vanishing of R 2 theory which is a Gauss-Bonnet term in 4 dimensions. When k < k, the second term vanishes and P n reduces to H k deth k n−k , which gives the self-dual amplitude of F 3 , R 3 theory etc. When k > k, the first term vanishes and P n reduces to det h k kH n−k , which gives the anti-self-dual amplitude of those theory. For example, the integrand for F 3 theory reads I F 3 n = C n P n . In 4 dimension, when k < k, I F 3 n reads Here det h k det h n−k comes from the transition of two forms of scattering equations as shown in (2.6).

Discussion
In CHY representation, the fundamental gauge invariant objects are quite common, either C aa or the trace of some linearised field strength together with some σ's. In this paper, we find a rather general way to reduce this gauge invariant objects into that made up of spinors using 4d scattering equations. Particularly, we show how the reduced Pfaffian reduces to some determinants and why it vanishes on the support of most solutions. This explains why only some particular solutions contribute to the YM or GR amplitudes according to their helicity structure in 4d and provides a basis for dividing the solutions of scattering equations into MHV,NMHV,· · · ,MHV sectors which contributes to corresponding YM or GR amplitudes respectively, also seen in [23]. We extend this methods to reduced compactified Pfaffian where some polarizations are set in higher dimensions, which is building block for EM, EMS, YMS or DBI amplitudes. We give the explicit reduction results of the compactified Pfaffian up to 3 pairs of particles whose polarizations are set in higher dimension and provides the general way to get the reduction with arbitrary pairs. Another interesting integrand with polarization involved is the reduced squeezed Pfaffian, which is the building block of EYM theory [3,4,24,25]. As it can be expressed by some combination of the reduced compactified Pfaffian, its reduction in 4d is directly obtained from that of the reduced compactified Pfaffian. The results of one and two gluon color traces are explicitly presented in the appendix C.
Even some integrands which can't be organized as a matrix, let alone its Pfaffian, such as the new integrand P n used in F 3 , R 2 , R 3 theory, also can be enclosed in this procedure. We decompose P n to some fundamental gauge invariant objects,reduce these fundamental gauge invariant objects first and then organize them into a compacted form, which apparently shows most information of the P n , and explains the orthogonality of F 3 and YM amplitudes, the vanishing of Gauss-Bonnet term R 2 and the self-dual and antiself-dual amplitudes of F 3 or R 3 amplitudes in 4d. In fact, we use these properties to guess what the compacted form in 4d of P n should be, then fix the coefficient of the fundamental gauge invariant objects and finally get the P n . This is quite general to find the CHY integrand of an unknown theory. Even when the scattering equations or σ dependence of JHEP07(2017)069 the entries in the matrix Ψ n has been changed, their CHY integrand are very likely to be decomposed into some C aa or trace like fundamental gauge invariant objects. And we can reduce these objects first, organize them into a form manifest showing some properties the theory requires and finally confirm their CHY integrand. This can even be applied at loop level, as shown in [19,26].
Instead of reducing all kinds of integrands in 4d, we now turn to the general 4d CHY formulae. After overcoming the obstacles to reduce integrands with polarization involved, the calculation of CHY formulae becomes much simpler. We develop the 4d CHY formulae to directly calculate the amplitude of the some theory. The reduced Pfaffian behaves like a solution filter, making the building of 4d CHY formulae natural. As if the general CHY formulae has been reduced to 4d CHY formulae and the number of solutions decrease from (n − 3)! to E n−3,k −2 .
We have discussed the reduction of the reduced compactified Pfaffian and squeezed Pfaffian in appendix B, C and discussed how the valid solution sector shift from the helicity sector. The more polarizations there are, the more efficient our procedure is. Even when there is no polarization involved, and the reduced compactified Pfaffian totally reduce to Pf A n times something, our reduction procedure still holds and it tells us only the k = n/2 solution sector contributes. This means CHY formulae of some effective field theory such as Born-Infeld, Dirac-Born-Infeld, Non-Linear Sigma Model, Special Galileon theories with Pf A n acting as CHY integrand also reduce to a set of 4d CHY formulae. Even for some theories that receive the contribution of several solution sectors such as those with P n acting as CHY integrand, the physical meaning of 4d CHY formulae is also apparent: the contribution from the k < k sectors gives self-dual amplitude and that of k > k gives anti-self-dual amplitudes.
Many good properties shared by CHY formulae are still inherited by the 4d CHY formulae. The soft limits has been discussed in [15,[27][28][29][30]. Factorisation should also be easy to study. Not only the CHY formulae have a simple representation in 4d, 4d CHY formulae can also help us understand the CHY formulae in general dimension. Besides, the supersymmetrization of the 4d CHY formulae is directly and we just need to replace theλ α a with {λ α a |η A a } in the scattering equation in (2.4) as shown in [15]. This way we can involve fermions in CHY formulae, for example we use SYM amplitudes to build up QCD amplitudes as shown [31]. In the same paper, we use two set of spinors to describe the massive Higgs, which has been generalized to calculate form factors [32,33].
We tentatively study whether the solutions divide beyond in 4d. Especially we hope something interesting come out in 6 dimensions where we also have a good spinor representation [34][35][36] and some nice result of CHY formulae in 6d comes out. We can treat a massive particle in 4d as a massless particles in 6d, especial the massive loop particles in 4d. Up to now, our result is negative and we didn't find the solutions of scattering equation divide again in other dimension.
CHY formulae has been extended to 1-loop level, as discussed in [37,38]. It has been known that what underpins the CHY formulae is ambitwistor string. And ambitwistor string theory has been extended to 1-loop level, as shown in [7,39,40]. However we find the solutions at 1-loop level don't divide into several sectors again in SYM or SUGRA JHEP07(2017)069 theory. CHY formulae has singular solutions at 1-loop, how about 4d CHY formulae and how does it contribute to the divergence bubble or tadpole? Also it is interesting to check whether the integrand still factorizes to two objects that depend on particles of negative helicity or positive helicity respectively. Also it will help us to build the general 1-loop CHY integrand.
As discussed in [41], also it is very useful to study the positivity of the jacobian or integrand in 4d CHY formulae. (2.6) is a useful identity as it link several objects. As shown in [41], det Φ n ({s ab , σ a }) is positive at the positive region. We know that det h k det h n−k is exactly the result of Pf Ψ n with k external particles of negative helicity. If the 4d jacobian J n,k is also positive at the positive region, it strongly supports that the YM amplitude is also positive in some regions.
(A.4) Here d 1 , d 2 is a particle label as the breviate of d 11 , d 21 and {b 1 , c 1 , d 11 } = − ,{b 2 , c 2 , d 21 } = − . r 1 , r 2 is a particle label as the breviate of r 11 , r 12 and {p, q, r 11 , r 12 } = + . Note that this restrain doesn't fix r 11 , r 12 totally as r 11 , r 12 can exchange their value. However it doesn't affect the value of J 7,3 as we always sum over all b 1 < c 1 , b 2 < c 2 .
Here d x is a particle label as the breviate of d x1 and {b x , c x , d x } = − . r x is a particle label as the breviate of r 1x and {r 11 , r 12 , · · · r 1,n−5 } = + \{p, q}.
For NNMHV solution sector,

B The reduced compactified Pfaffian in 4d
In the main text, we have discussed the reduced Pfaffian which we delete 1 th and n th rows and columns of the matrix Ψ n . Then we introduce the open cycle to reduce it into two reduced determinants. Also we can effectively think that we set the momenta of the particles 1, n in higher dimension and they dotting everything equal to zero unless they dotting themselves equal to 1 to make the complement 1 σ 1n . Then we can still decompose the reduced Pfaffian into some (modified) closed cycles. One closed cycle must contain 1, n both or vanish if it just contains one of them, and then it reduces to open cycle as k 1 , k n are set in higher dimension. This can be extended to other cases as now we set the JHEP07(2017)069 polarisation of some pairs of particles in higher dimension. We call it reduced compactified Pfaffian which is the building block for EM, EMS, YMS amplitudes, as discussed in [4]. Then we can use the similar trick to reduce the reduced compactified Pfaffian in 4d.
We donate the set of particles whose polarisation are set in higher dimension as h. Obviously, the length of set h must be even. Further on, we let the polarisation of particles in h be anyone of an orthogonal bases and they dotting each other equal to 1 or 0, donated as δ IaI b . Then the compactified Pfaffian can be think of as being deleted the rows and columns of h in the last n rows and columns of the matrix Ψ n donated as Pf |Ψ n | {h} {h} and complement it with a Pfaffian. That is, Here 2m is length of the set h. First we considering the case with m = 1, that is only one pair of particles donated as e 1 , e 2 that needs dimension reduction. For simplicity, we also delete the rows and columns of e 1 , e 2 in the first n rows and columns to satisfy the mass dimension, i.e. we effectively set the momenta of e 1 , e 2 in higher dimension. Then in the expansion of the reduced compactified Pfaffian, all cylces that contain e 1 , e 2 vanish unless they contain and only contain both e 1 , e 2 . Then this cycle, which equals to δ Ie 1 Ie 2 σe 1 e 2 , factor out, leaving all other cycles normal as if e 1 , e 2 not existed. They factor into two determinants of two matrices in 4d, just like the factorisation of Pfaffian in (3.20), with the diagonal elements equal to C aa plugging a certain solution sector k , as expressed in (4.1), (4.7), (4.8). One of two determinants of these two matrices will vanish trivially unless k = k + 1. That is, we need to assign e 1 , e 2 to into two sets, for example, we let − = − ∪ {e 1 } and + = + ∪ {e 2 }. Then the reduced compactified Pfaffian with only one pair of particles needing dimension reduction reduces to Pf Ψ n;1 k = δ k+1,k 1 σ 2 e 1 e 2 det |h k | e 1 e 1 det |h n−k | e 2 e 2 δ Ie 1 Ie 2 . (B. 3) The expression of h ab with a, b ∈ − andh ab with a, b ∈ + are given in (3.11), (4.1), (4.8).
Here we can extend the definition domain from − to − ∪ {e 1 } and from + to + ∪ {e 2 }, to enclose h e 1 b orh e 2 b , though it is not important as such entries will always been deleted from the matrices h k andh n−k in the above equation. In this case, the exchange of e 1 ↔ e 2 will affect the expression of the diagonal elements of h k andh n−k but it won't affect the final result. For later convenience, we write the above equation in a slightly different way, Now we consider the case with m = 2, i.e. 4 particles donated as e 1 , e 2 , e 3 , e 4 need dimension reduction. There are be 3 perfect matching to make pairs in the expansion of Pf[X ] h , as shown in (B.2). We can take these perfect individually and at last add them JHEP07(2017)069 up. For example, we consider a perfect matching that e 1 , e 2 a pair and e 3 , e 4 a pair. Still we effectively set the momenta of e 1 , e 2 in higher dimension and then δ Ie 1 Ie 2 σe 1 e 2 factors out. The left pair e 3 , e 4 must be adjoint and enclosed in one cycle and it reduces to an open cycle similar to (3.6) with the polarisations on the ends replaced by kinematics as Here f h e just means the polarisation of particle e is set in higher dimension and f ± a means when reduced to 4 dimension, the helicity of particle a can be negative or positive. Also any two adjoint linearised strength fields f − b f + p in the trace can exchange their place if the helicity of b, p are different. So we use this property to split the kinematic numerator of this open cycle in respect of negative and positive helicity first, then use the partial fraction identity to spilt the denominator, and finally cut the open cycle into two closed cycles,

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Now we consider the case with m = 3, i.e. 6 particles donated as e 1 , e 2 , e 3 , e 4 , e 5 , e 6 need dimension reduction. Still we consider the perfect matching with {e 1 , e 2 } a pair,{e 3 , e 4 } a pair and {e 5 , e 6 } a pair first. Still we effectively set the kinematics of e 1 , e 2 in higher dimension and then δ Ie 1 Ie 2 σe 1 e 2 factors out. The left two pairs {e 3 , e 4 } and {e 5 , e 6 } must be adjoint in the trace respectively, or vanish. However these two pairs can be enclose in two different cycles or in a common cycle. The former case is simpler. We split the open cycles, cut it into modified open cycles and rearrange them together with other closed cycles into determinants. That is, those of the expansion of Pf Ψ n;2 k that contain such cycles tr(f h e 3 ···f h e 4 ) σ (e 3 ···e 4 ) , tr(f h e 5 ···f h e 6 ) σ (e 5 ···e 6 ) reduce to The factor δ k+3,k come from the fact that we must assign each pair of particles in this perfect matching into different sets, for example − = − ∪ {e 1 , e 3 , e 5 } and + = + ∪ {e 2 , e 4 , e 6 }. In this case, the exchange of particles in each pair doesn't affect the final result. The exchange of different pairs also does't affect the final result. However the case where {e 3 , e 4 } and {e 5 , e 6 } are enclosed in a common cycle also contribute and we need to deal with it more carefully. We find that the equation (B.5) can be extended to tr · · · f h eτ 2 f ± a 3 f ± a 4 · · · f ± a i f h eτ 3 · · · · · · σ eτ 2 a 3 σ a 3 a 4 ··σ a i−1 a i σ a i eτ 3 σ eτ 3 eτ 4 · · · = · · · h eτ 2 b 1 h b 1 b 2 ··h b x−1 bx h bxeτ 3h eτ 2 pyhpyp y−1 ··h p 2 p 1h p 1 eτ 3 σ eτ 3 eτ 2 σ eτ 3 eτ 4 · · · (B.11)

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Here we have given the general form with arbitrary pairs of particles in h enclosed in a common trace. . We can think there is also a factor , e 3 groups with e 6 and e 4 groups with e 5 . In the former case, the group pairs are consistent with the perfect matching pairs and the exchanging of particles in the same perfect matching pair is identical, while in the latter cases, the group pairs are not consistent with the perfect matching pairs and the exchanging of particles in the same perfect matching pair is two different contributions. However it is still not tough. The reduction of those that contain such cycles The minus before the exchanging of e 5 ↔ e 6 comes from σ e 6 e 5 = −σ e 5 e 6 and σ e 5 e 6 is absorbed in Pf[X ] h . When it comes to the cases with m ≥ 3, there are no more new objects coming out and just some more calculation and we can always reduce the reduced compactified Pfaffian into some determinants. Tr T Ia 2 T I b 2 σ a 2 ,b 2 σ b 2 ,a 2 · · · Tr T Ia m T I bm σ am,bm σ bm,am ×sgn({a, b})σ a 1 ,b 1 σ a 2 ,b 2 · · · σ am,bm Pf|Ψ| H,a 1 ,b 1 ,a 2 ,b 2 ,···a m−1 ,b m−1 ;H (C.1)
We denote the set of gluons as g and the subsets sharing in the same color trace as Tr 1 , Tr 2 , · · · , Tr m . Then the half integrand for EYM are given by C Tr 1 · · · C Trm a 1 <b 1 ∈Tr 1 ··· a m−1 <b m−1 ∈Tr m−1 sgn({a, b})σ a 1 ,b 1 · · · σ a m−1 ,b m−1 Pf|Ψ| H,a 1 ,b 1 ,a 2  Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.