Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program

Using only the Britto-Cachazo-Feng-Witten(BCFW) on-shell recursion relation we prove color-order reversed relation, $U(1)$-decoupling relation, Kleiss-Kuijf(KK) relation and Bern-Carrasco-Johansson(BCJ) relation for color-ordered gauge amplitude in the framework of S-matrix program without relying on Lagrangian description. Our derivation is the first pure field theory proof of the new discovered BCJ identity, which substantially reduces the color ordered basis from $(n-2)!$ to $(n-3)!$. Our proof gives also its physical interpretation as the mysterious bonus relation with ${1\over z^2}$ behavior under suitable on-shell deformation for no adjacent pair.


INTRODUCTION
S-matrix program [1] is a program to understand the scattering amplitude of quantum field theory based only on some general principles, like the Lorentz invariance, Locality, Causality, Gauge symmetry as well as Analytic property. The significance of this approach is its generality: results so obtained do not rely on any detail information of theories, such as the Lagrangian description of theories.
However, exactly because its generality with so little assumptions, there are not much tools available and its study is very challenging. One big step along S-matrix program is the unitarity cut method proposed in [2], where on-shell tree amplitudes have been applied to the calculation of loop amplitudes without drawing many many Feynman diagrams.
Another breakthrough is the Britto-Cachazo-Feng-Witten(BCFW) on-shell recursion relation [3,4]. One way to see it is to pick two momenta to do the BCFW-deformation p i = p i + zq, p j = p j − zq with proper chosen auxiliary nullmomentum q, thus the amplitude A n becomes the analytic function A n (z) of z with only single pole structure. By the familiar complex analysis we can completely determine the function if we know locations of all poles and their residues. Pole happens when a propagator reaches mass-shell and the amplitude is effectively divided into two sub-amplitudes at the left and right (so called factorization property) of the propagator. Summing up all poles we obtain where sum is over all color-order preserved splitting of nparticles into two on-shell amplitudes with condition p i ∈ I, p j ∈ J while z IJ indicates the particular splitting. The brief form (1) is enough for the understanding of our paper. As reader will see, what we do in whole paper is to use (1) to expand, recombine and reshuffle different amplitude components. It is surprising that with such simple algebraic manipulations we can get some very deep results.
The derivation of BCFW relation beautifully demonstrates the idea of S-matrix program and its generality has inspired many works, one of them is the work of Benincasa and Cac-hazo [5]. In the paper, by assuming the applicability of BCFW recursion relation they have easily re-derived many well known (but difficult to prove) fundamental facts in Smatrix, such as the Non-Abelian structure for gauge theory and all matters couple to gravity with same coupling constant.
These four identities have been understood from different perspectives. The properties (1) and (2) can be shown from Lie-algebra structure. Property (3) is inspired from string theory and then shown in field theory [8] using different color decomposition. Property (4) is conjectured through the Jacobiidentity but has only been proved from string theory [9] (see further study [10]).
These four identities, especially the KK and BCJ relations, contain unexpected important properties of gauge theory. Our proof in S-matrix frame unifies the treatment of them all and makes them hold in general ground. Especially our proof is the first pure field theory proof of BCJ relation. Furthermore our method can be applied to the field theory understanding of another very important Kawai-Lewellen-Tye(KLT) relation [11,12], which has only been shown from string theory. The importance of BCJ and KLT relations lies in the mysterious observation: on-shell gravity likes the square of gauge theory while their off-shell Lagrangian descriptions are completely different (one is normalizable and another one, unnormalizable). Understanding these observations in field theory will help us with the searching of consistent quantum gravity theory, which is still one of most fundamental open problems in physics.

THE COLOR-ORDER REVERSED RELATION
One basic observation of [5] is that color-ordered three particle amplitude is completely fixed by Lorentz symmetry and satisfy A(1, 2, 3) = (−)A(3, 2, 1) without using any Liealgebra property. Using the BCFW recursion relation with pair (n, 1), we get where we have expanded amplitude at the second line, then used induction to reshuffle at the third line and finally recombined at the fourth line. These manipulations are exactly patterns we will follow in whole paper. It is worth to notice that we do not need to specify details like the helicity and shifting of (n, 1) as well as explicit expressions of A n as long as BCFW on-shell recursion relation without boundary value is applicable. Thus our conclusion holds for any helicity configuration.

THE U (1)-DECOUPLING RELATION
The n = 4 case is easy to check after using the colorreversed relation in the BCFW expansion. To get more idea of proof, let us present example of n = 5 given in (4). At each line we use (1) to expand left hand side into the right hand side. To make formula compact we have used, for example P 523 to represent amplitude A( 5, 2, 3, − P 523 )/s 523 , thus A(1, 4, P 523 ) really represents A( 5, 2, 3, − P 523 )A(P 523 , 1, 4)/s 523 . By our purposely arrangement, it is easy to see that the sum of each column at the right hand side is zero after we use the U (1)-decoupling equation for n = 3 and n = 4 by induction.
Having the experience of n = 5, the proof for general n by induction is again by BCFW expanding each amplitude first, then regrouping every piece into U (1)-identity for the lower m. For example, with (1, 2)-shift the expansion of a general amplitude where [i, . . . , j] means sum of all divisions between legs i to j and (k, t) means there are t particles in front of 2. It can be checked that with fixed t, the sum of k is indeed the U (1)-decoupling identity with lower m and is zero. Having all possible t we get the identity for n, thus finished the proof.
To count terms, it is easy to see that there are C i i+m−j and C j j+k−i terms for each factor respectively at the right hand side of (6). Thus the total number of terms at the right hand side of (6) is where −2 (m+k)! m!k! counts the two excluded cases. The right hand side of KK-relation (2) will be (k+m)! k!m! (k + m − 1) after we have used the BCFW to expand each amplitude into (k + m − 1) terms. These two numbers match up as it should be.

THE BCJ RELATION
The BCJ relation (3) is more complicated since the appearance of various dynamical factors s ij . In its most general form, the set α, β can be arbitrary. However, we want to show that all other equations are redundant except the one where the set α has only one element, which we call the "fundamental BCJ-relation". More accurately we want to show that if these fundamental BCJ-relations are true, combining with U (1)-decoupling relation and KK-relation we can express any amplitude by (n − 3)! amplitudes of the form A(1, 2, 3, σ(4, .n)). This is exact the same statement given by general BCJ-relation.
Having done for n = 4, we move to the general proof using the induction. To make the step clear, we consider the case n = 6 and arbitrary n is easily dealt with same method. Taking the 2|1]-shifting and using the BCFW recursion relation to expand each amplitude in I 6 , we will get three different splitting for each amplitude. Let us consider the splitting where the splitting parameter [2] means there are two particles at the left hand side. All terms of I [2] 6 can be divided into two