New Factorization Relations for Yang Mills Amplitudes

A double-cover extension of the scattering equation formalism of Cachazo, He and Yuan (CHY) leads us to conjecture covariant factorization formulas of n-particle scattering amplitudes in Yang-Mills theories. Evidence is given that these factorization relations are related to Berends-Giele recursions through repeated use of partial fraction identities involving linearized propagators.


INTRODUCTION
The CHY-formalism of scattering equations of Cachazo, He and Yuan provide an intriguing novel way of computing gauge and gravity S-matrix elements [1][2][3]. The n-point scattering amplitudes are here expressed in terms of integrals over auxiliary variables z a on the Riemann sphere that become localized on the set of solutions to the scattering equations, Here s ab = 2k a · k b are generalised Mandelstam variables and the index a labels the (ordered) external particles of momenta k a . One remarkable feature of the CHYformalism, and one which shows its fundamental nature, is that it is dimension-agnostic. The defining integral over the variables z a is invariant under an SL(2,C) transformation z a → Az a + B Cz a + D , which needs to be fixed. Fixing three of the variables in the standard manner, only (n − 3) variables z a are left. This matches precisely the (n − 3) independent scattering equations after imposing overall momentum conservation. The number of independent solutions (n − 3)! is nevertheless huge and finding all these solutions is computationally difficult even for moderate values of n. Summing over these independent solutions can fortunately be done more directly, through general integration rules developed in refs. [4,5]. A proof of the CHY formalism has been provided by Dolan and Goddard in ref. [6].
Recently, one of us [8] (see also ref. [21]) showed how the CHY formalism can be given a new formulation in which the basic variables z a live not on CP 1 but on the complex projective plane CP 2 . Dubbed the 'Λ-formalism' in [8], we shall here refer to it as CHY on a double-cover. At first sight it may seem to be a complication to extend the CHY-formalism in this manner. However, as we shall demonstrate in this 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: The double-cover formalism naturally expresses the scattering amplitude so that it is factorized into different channels. The propagator that forms the bridge between two factorized pieces arises as the link between two separate CP 1 pieces, thus intuitively explaining why the double-cover naturally expresses amplitudes in a factorized manner.
In many cases, the factorizations obtained in this way correspond directly to all 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.
We start with a brief review of the CHY-formalism, and then give the corresponding expressions in the double-cover formulation of ref. [8]. Next, we describe how the evaluation of amplitudes on a double-cover produces factorizations into different channels. Finally we write down an explicit factorization expression valid for n gluons in any dimension and relate it to known techniques such as on-shell and Berends-Giele recursions.

THE CHY CONSTRUCTION AND A DOUBLE-COVER
Consider the scattering of n massless particles. The scattering data will then be presented in terms of a set of n momentum vectors {k µ 1 , k µ 2 , . . . , k µ n } and n "wave functions" that encode the spin degrees of freedom. For Yang-Mills amplitudes the latter will correspond to the polarization vectors { µ 1 , µ 2 , . . . , µ n }. Graviton scattering will similarly be characterized by a set of polarization tensors, or, simpler, as outer products of polarization vectors.
It is straightforward to verify that three linear combi- There are thus only (n − 3) independent scattering equations. These equations have (n − 3)! solutions.
Different choices of punctures on CP 1 are to be identified if they differ by a P SL(2, C) transformation. As is well known, the location of three punctures must be fixed. Let us introduce the compact notation of |ijk| z indicating the Vandermonde determinant of variables z i , z j , z k : It is possible to show that for any rational function H(z) which transforms as when the contour integral [2] n a=1,a ={i,j,k} dz a |ijk| z |pqr| z n c=1,c ={p,q,r} S c (z) is independent of the choice of fixed punctures {z i , z j , z k } and of equations eliminated {S p , S q , S r }. One way to see that this is the case is to realize that an explicit (infinitedimensional) realization of the generators of P SL(2, C) is (8) Treating the P SL(2, C) as a redundancy of the integral and using the standard gauge-fixing procedure one can check that the Faddeev-Popov determinant is indeed as claimed above.
The precise form of the integrand H(z) defines different (color-ordered) theories. The simplest case is φ 3 -theory.
(10) Color-ordered φ 3 -amplitudes correspond to integrands with such factors squared: As shown in refs. [10,12] (see also [11]), the basic building blocks of other theories are products of one Parke-Taylor factor with a shuffled Parke-Taylor factor (α indicating a permutation): (12) Such a product of Parke-Taylor factors in the integrand thus forms a basic skeleton for all other theories.
For Yang-Mills theory we have where The 2n × 2n matrix, Ψ n , is defined as with, and Notice the unusual normalization in the A and C matrices. If we put α = 1 we recover the CHY-prescription as originally defined. If instead we choose α = √ 2 the normalization matches with the color-ordered Feynman rules given by Dixon in [13]. In what follows, α can take any value (it only changes the overall normalization of the color-ordered amplitudes, a convention), but we keep it arbitrary at this point to facilitate a comparison with Feynman diagrams based on color-ordered Feynman rules later in this paper. The matrix (Ψ n ) ij ij denotes the reduced matrix obtained by removing the rows and columns i, j from Ψ n , where 1 ≤ i < j ≤ n. For how to use the integration rules [5,7] in the context of Yang-Mills theory, see [10][11][12].

The Double-Cover
A double-cover version of the CHY construction was recently developed by one of us in [8]. In this approach the amplitudes are given as contour integrals on npunctured double-covered Riemann spheres. Restricted to the curves 0 = C a ≡ y 2 a − σ 2 a + Λ 2 for a = 1, . . . , n, the pairs (σ 1 , y 1 ), (σ 2 , y 2 ), . . . , (σ n , y n ) provide the new set of doubled variables. A translation table has been worked out in detail in ref. [8]. Specifically, one defines and ∆(pqr) ≡ y p y q y r and simultaneously imposes scattering equations in the form (momentum conservation k a = 0 is implicitly used throughout) where a = 1, . . . , n. Amplitudes are then derived from the following expression: where the measure dµ Λ n is defined as with the Γ contour being defined by the equations (23) This 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. Note that the integrand now includes a scale Λ. In order to fix this larger GL(2, C) symmetry, we gauge-fix four σ a 's. Then the measure must be multiplied by the Faddeev-Popov determinant ∆(pqr|m) = y p y p (y p +σ p ) y p (y p +σ p ) −1 σ p y q y q (y q +σ q ) y q (y q +σ q ) −1 σ q y r y r (y r +σ r ) y r (y r +σ r ) −1 σ r y m y m (y m +σ m ) y m (y m +σ m ) −1 σ m . (24) Therefore, dµ Λ n becomes [19] dµ Λ n = As in the original CHY approach, the precise form of the integrand I n (σ, y) defines the theory. For example, color-ordered φ 3 -theory corresponds to the integrand where Note the τ 's are neither antisymmetric nor symmetric; the precise definition as given above is correct. 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.

THE YANG-MILLS THEORY IN THE DOUBLE-COVER PRESCRIPTION
Since τ (a,b) = −τ (b,a) , it is not immediately obvious how to define the double-cover analog of the reduced Pfaffian for pure Yang-Mills theory. In order to obtain the double-cover version of the Ψ n matrix, we write (we define (yσ) a ≡ y a + σ a ) on the support, C a = C b = 0, where clearly, T ab = −T ba . Since T ab is anti-symmetric, we establish the single and double-cover identification, 1 σ ab ↔ T ab , so, the double- tice that it is straightforward to rewrite the φ 3 -integrand in terms of T ab , namely with, Following the CHY program developed in [3], the double-cover representation of the ordered Yang-Mills amplitude is obtained by the replacing, with where the (Ψ Λ n ) ij ij matrix is given by removing the rows and columns i, j from Ψ Λ n , with 1 ≤ i < j ≤ n. Therefore, the pure Yang-Mills amplitude at tree-level in the doublecover language is given by the expression

A NEW RELATION FOR YANG-MILLS AMPLITUDES
We now wish to illustrate that the double-cover formalism leads to a new insight into amplitude factorizations. We focus here on the perhaps most interesting case of Yang-Mills theory. As will be shown in great detail elsewhere [19], by integrating the double-cover representation of an ordered Yang-Mills amplitude one is led to the following general formula which factorizes arbitrary n-point Yang-Mills aplitudes into a product of (singlecover) CHY representations of lower-point amplitudes:  Let us be clear: this factorized form of Yang-Mills amplitudes is a conjecture. What the double-cover formalism produces directly are the two first terms plus contributions that come from linking amplitudes together with scalar degrees of freedom. Miraculously, it appears that these scalar contributions can be exactly represented by gluing two Yang-Mills amplitudes together with longitudinal polarizations only. The technical details of how these manipulations arise will be presented elsewhere [19]. Needless to say, in the factorized form on the right hand side the two amplitudes each have one external leg off-shell (although still dressed with the corresponding unphysical polarization vector). Gluing these two amplitudes together proceeds through the polarization sums as described below. It should also be stressed that the above expression comes from the double-cover formalism with Mobius and scale-invariance gauge choices (pqr|m) = (123|4). The removed legs in the Pfaffians are the off-shell ones and the ones by denoted superscripts on the amplitudes. This is important to remark since the above factorization is a gaugefixing dependent expression. Of course, the final result, the left hand side, is the correct full n-point amplitude, but the precise factorized form on the right hand side depends on that generalized gauge fixing. The three punctures which must be fixed in the smaller off-shell Yang-Mills amplitudes are given by the set, ({All punctures in the amplitude} ∩ {1, 2, 3, 4}) ∪ {off − shell puncture} = {fixed punctures}. We denote sums of cyclically-consecutive external momenta (modulo the total number of external momenta) by P i:j ≡ k i + k i+1 + . . . + k j−1 + k j , and such sums with addition of one arbitrary leg by P i:j,l ≡ P i:j + k L . For expressions with only two momenta involved (not necessarily consecutive) we are using the shorthand notation P ij ≡ k i +k j . The above double-cover representation of the amplitude provides an expression that is factorized into different CP 1 -pieces provided we apply the following gluing identities: We have denoted the polarisation degrees of freedom by M and longitudinal by L . Using the simple identity  ( Notice that the poles related to the longitudinal polarization contributions 1/(P i:j,l ) 2 are not physical and indeed these unphysical poles are cancelled by corresponding numerator factors. This is the way local 4-point Yang-Mills interactions appear in this formalism.

Feynman diagrams and Bern-Carrasco-Johansson (BCJ) numerators
We will first consider how the double-cover representation relates to BCJ numerator identities [14]. From the formula (35), we arrive at It is simple to check that in the normalization convention α = √ 2 (corresponding to [13]), the first and second line are just the conventionally normalized Feynman diagrams, Using the above it is simple to check that we have, n s − n T = n u , where n u can be obtained from n s under the permutation, (1, 2, 3, 4) → (1, 3, 2, 4). Extending such ideas to a higher number of points should be a possible avenue and would be very interesting.
Obviously the pole P 2 23 does not depend on z so that this physical factorization channel only contributes at infinity. The most interesting observation is that the spurious poles, P 2 i+1:1,3 (z) cancel out because the longitudinal contributions n i=4, L A (1) n−i+3 P L 4:i,2 , i+1, . . . , 3 × A (4) i−1 P L i+1:1,3 , 2, ..., i are proportional to them. Therefore, the boundary contributions at z = ∞ are related to the unphysical poles that appear in the double-cover, eq. (33). This gives these poles a special significance in the context of BCFW recursion and potentially a new recursive path for dealing with such contributions.

Berends-Giele recursion and the double cover
Another natural question concerns the similarity of the factorized forms from the double-cover method and Berends-Giele recursion [17]. In order to shed light on this we will will focus in the bi-adjoint φ 3 theory in the double-cover formalism. Because of the trivial numerator factors of this case it is far simpler to analyze.
The connection is well illustrated by considering the five-point amplitude. The factorizations from the doublecover method lead to On the support, k 1 + k 2 + k 3 + k 4 + k 5 = 0, and under the on-shell condition, k 2 i = 0, it is trivial to check that the expressions obtained in (40) and (41) are identical. However, the appearance of the unphysical poles in the double-cover framework, (P 2 5:2 − P 2 12 ) −1 = (P 2 34 − P 2 3:5 ) −1 and (P 2 3:5 −P 2 34 ) −1 makes it clear that the two representations are not directly equal. Interestingly, these poles unphysical are related to the physical channel, .
As it happens with the linear propagators at loop-level [20][21][22][23], the CHY-formalism is naturally built of linear propagators that can relate to the usual Feynman propagators by means of partial fractioning.

CONCLUSIONS
We have presented a new set of factorization identities for Yang-Mills theory that naturally arise from a doublecover version of the CHY-formalism. These factorizations glue amplitudes together in what can interpreted as the covariant Feynman gauge, with the additional 4point contact interactions coming from an explicit sum over longitudinal polarizations. The factorizations are at the conjectured level, but there are many hints that they may also derivable from Berends-Giele recursions. Although spurious poles appear, simple checks show that they cancel through repeated use of partial fraction identities. It would be an interesting extension of this work to derive these relations directly from off-shell recursion relations.
Factorizations of amplitudes grow out of the doublecover formalism precisely because it is "double": there are, figuratively speaking, two CHY-integrals involved. The bridge between these two CHY-integrals is an offshell leg, a propagator. In the double-cover formalism this off-shell leg stems from one scattering equation that is not imposed as a delta-function constraint.
These factorizations of Yang-Mills amplitudes are just a small part of more general relations that follow when the double-cover formalism of CHY is analyzed for the known set of theories that can represented in this form. Details of that will be provided by one of us in a subsequent paper [19].