An algebraic approach to BCJ numerators

Abstract

One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.

Published for SISSA by Springer

Received: January 26, 2013

Accepted: February 13, 2013

Published: March 8, 2013

Chih-Hao Fu,a,b Yi-Jian Duc,d and Bo Fenga,d

aCenter of Mathematical Science, Zhejiang University,

38 Zheda Road Hangzhou, 310027 P.R China

bDepartment of Electrophysics, National Chiao Tung University, 1001 University Street, Hsinchu, Taiwan, R.O.C.

cDepartment of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, P.R China

dZhejiang Institute of Modern Physics, Zhejiang University,

38 Zheda Road Hangzhou, 310027 P.R China

E-mail: [email protected], mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satised. The corresponding color ordered amplitudes satisfy o -shell KK-relation and o -shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.

Keywords: Scattering Amplitudes, Gauge Symmetry

ArXiv ePrint: 1212.6168

c

An algebraic approach to BCJ numerators

JHEP03(2013)050

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP03(2013)050

Web End =10.1007/JHEP03(2013)050

Contents

1 Introduction 1

2 Generators and kinematic structure constant 2

3 Construction of kinematic numerators 43.1 kinematic numerators for 4-point amplitudes 43.1.1 Eliminating contact terms 73.1.2 KK vs. BCJ-independent basis 83.2 5-point numerators 93.2.1 Eliminating contact terms 113.3 n-point numerators 12

4 Fundamental BCJ relations 134.1 The color-order reversed relation 134.2 The o -shell and on-shell BCJ relation 144.3 The KK-relation 17

5 Kinematic ordering of gauge theory amplitude 18

6 Various forms of amplitudes 20

7 Conclusion 23

A O -shell KK relation from Berends-Giele recursion 23

1 Introduction

Recent studies have revealed that there are many new structures for scattering amplitudes unforeseen from lagrangian perspective. One of such examples is the color-kinematic duality discovered by Bern, Carrasco and Johansson [1] (BCJ). In the work it was conjectured that color-ordered amplitudes of gauge theories can be rearranged into a form where kinematic numerators satisfy the same Jacobi identities as the color part does (i.e, the part given by multiplication of structure constants of gauge group according to corresponding cubic Feynman diagrams). These forms (we will call BCJ-form) lead to very nontrivial linear relations among color ordered amplitudes,1 thus we can reduce the number of independent amplitudes to (n 3)! . A further conjecture of color-kinematic dual form

(BCJ-form) is that if we replace the color part by kinematic part in the BCJ-form, we will

1BCJ relations between color-ordered amplitudes has been proved in string theory in [25] and in eld theory in [68] using on-shell recursion relations

1

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get corresponding gravity amplitudes. The double-copy formulation of tree-level gravity amplitudes is equivalent2 to the Kawai-Lewellen-Tye (KLT) relations [9, 10]. However, unitarity suggests that double-copy formulation may be generalized beyond tree-level and therefore provides an extremely useful aspect to understand or calculate gravity amplitudes at loop-levels. Recent discussions on loop-level can be found in [1224]. Because these important applications to gravity amplitudes, the simple and explicit construction of kinematic numerators is very important. In this paper we show that assuming gauge symmetry provides enough degrees of freedom, it is possible to construct kinematic numerators as linear combinations of contributions coming from cubic graphs, with vertices given by generalization of the algebraic structure constant given in [25, 26]. This construction makes many algebraic relations between numerators, such as Jacobi identity, KK-relation and BCJ relations, manifest.

Another interesting consequence of color-kinematic duality is that gauge theory amplitudes may have di erent forms. Two such examples are the color-ordered decomposition Atot =

PSn1 T r(T 1T 2 . . . T n) A() (which we will call Trace form) and the form discovered by Del Duca, Dixon and Maltoni (DDM) [27] Atot =

PSn2 f12x1fx13x2 . . . fxn3n1n A(1, , n) (which we will call the DDM form). The equivalence of two forms gives another proof of Kleiss-Kuijf (KK) [28] relations of the color-ordered amplitudes.3 Within the color-kinematic duality, it is natural to have the Dual Trace form and Dual DDM form as discussed in [11, 31]. However, unlike the Trace form and DDM form, the dual form does not have very simple construction for the dual color part. In this paper, we will give a partial construction of the dual color part.

This paper is organized as follows. In section 2 we introduce the Lie algebra of general di eomorphism in Fourier basis. Upon the sum of cyclic permutations of the structure constant, we get Yang-Mills 3-point vertex. Section 3 is our main part where the construction of kinematic numerators is given. We start with two examples, the 4-point numerator and 5-point numerator, where explicit calculations are given. Then we give a general frame for our construction. In section 4 we discuss relations, such as KK and o -shell BCJ relations, among quantities dened in section 3. In section 5 we derive the dual DDM form using relations from previous section. A few comments on relations between di erent formulations of Yang-Mills amplitudes are given in section 6. After a short conclusion, a proof of KK relation using o -shell recursion relation is included in the appendix.

2 Generators and kinematic structure constant

Our starting point is a generalization of the di eomorphism Lie algebra introduced by Bjerrum-Bohr, Damgaard, Monteiro and OConnell [25, 26]. The generator is dened as

T k,a eikxa, (2.1)

2A proof can be found in [11].

3In [27], the DDM form was derived using the properties of Lie algebra. However, it can also be derived [29] using KLT formulation of Yang-Mills amplitude [30].

2

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1

2

3

2

1 1

3

2

3

Figure 1. From left to right, three diagrams represent f123, f312, f231 in eq. (2.5), where we used

arrows to distinguish upper from lower indices.

with label (k, a), where k is a D-dimensional vector and a, a Lorentz index. The kinematic structure constant can be read out from commutator

[T k1,a, T k2,b] = (i)(ack1b bck2a) ei(k1+k2)xc (2.2)

= f(k1,a),(k2,b)(k1+k2,c) T (k1+k2,c).

In the following we shall use a shorthand notation by writing f(k1,a),(k2,b)(k1+k2,c) as f1a,2b(1+2)c . The upper and lower scripts of Lorentz indices a, b and c are introduced to distinguish whether the corresponding generators are contravariant or covariant under Lorentz symmetry. Jacobi identity coming from cyclic sum of the commutator [[T k1,a, T k2,b], T k3,c]

is given by

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f1a,2b(1+2)e f(1+2)e,3c(1+2+3)d +f2b,3c(2+3)e f(2+3)e,1a(1+2+3)d +f3c,1a(1+3)e f(1+3)e,2b(1+2+3)d = 0.(2.3)

To relate structure constants to Feynman rules, we need to lower or raise Lorentz indices by contracting with Minkowski metric. For example

f1a,2b(1+2)c f1a,2b(1+2)

e ec = (i)(ack1b bck2a), (2.4)

The index-lowered structure constant (2.4) does not enjoy cyclic symmetry. However summing over cyclic permutations of k1, k2 and k3 = k1 k2 produces the familiar color

ordered 3-point Yang-Mills vertex

1p2 f1a,2b

3c + f3c,1a2b + f2b,3c1a

= i

p2 [ab(k1 k2)c + bc(k2 k3)a + ca(k3 k1)b] .

(2.5)

Three terms at the left handed side of (2.5) can be represented by the three arrowed graphs in gure 1. In this representation, two upper indices a, b of fab c are denoted by arrows pointing towards the vertex while lower index c is denoted by an arrow leaving the vertex.

These three terms are related to each other by counter-clockwise cyclic rotation, thus from the left to right, they represent f123, f312, f231.

Note that when expressed in terms of index-lowered structure constants, Jacobi identity becomes

f1a,2b(1+2)ee[tildewide]ef(1+2)[tildewide]e,3c(1+2+3)d + f2b,3c(2+3)ee[tildewide]ef(2+3)[tildewide]e,1a(1+2+3)d

+f3c,1a(1+3)ee[tildewide]ef(1+3)e,2b(1+2+3)d = 0.

(2.6)

3

When we interpret relations between numerators as Jacobi identities, the Minkowski metric e[tildewide]e comes from gluon propagator and connects two structure constants. In discussions below we neglect Lorentz indices of structure constants, which can be easily recovered from the context. Contraction of a structure constant f1a,2b3c with other structure constants should be understood as the same as contracting a tensor f1,23 labelled by legs 1, 2, 3.

3 Construction of kinematic numerators

In this section we present an algorithm to construct the kinematic numerators that satisfy Jacobi identity as proposed by Bern, Carrasco and Johansson [1]. We demonstrate our method through 4-point and 5-point amplitudes, and then present the general picture for arbitrary n-point amplitudes.

3.1 kinematic numerators for 4-point amplitudes

For 4-point amplitudes, we consider two color-ordered ones A(1234) and A(1324), since rest of amplitudes can be obtained from these two with the Kleiss-Kuijf (KK) [28] relations. From the prescription of Bern, Carrasco and Johansson, 4-point color-ordered amplitudes can be divided into contributions of s, t and u-channels [1],4

A(1234) = ns

s

nuu , A(1324) =

ntt +

nuu . (3.1)

Our goal is to construct kinematic (BCJ) numerators ns, nt, nu that satisfy Jacobi identity ns + nt + nu = 0. Let us rst focus on amplitude A(1234). From color-ordered Feynman rules, amplitude A(1234) contains a s-channel and a u-channel graphs with only cubic vertices. Thus it is natural to attribute expressions coming from Feynman rules to numerators ns and nu respectively. In addition we have a contribution from color-ordered 4-point vertex

iacbd

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i2(abcd + adbc) =

i2(acbd adbc) +

i2(acbd abcd), (3.2)

(where the Lorentz indices of particles 1, 2, 3, 4 are a, b, c, d,). We attribute the rst and the second terms of (3.2) at the right handed side to ns and nu.5 Using propagator i p2 , the s-channel numerator ns can be read out

ns =

i 2

f1,2e + (fe,12 + f2,e1)

f3,4e + (fe,34 + f4,e3) (3.3)

2s (acbd adbc),

where we used equation (2.5) to express 3-point Yang-Mills vertex in terms of kinematic structure constants. A star sign of ns was introduced to denote quantities that have

4We follow the sign convention such that ns, nt, nu correspond to cyclic permutations of the three external particles with the fourth one xed, as they appear in the Jacobi identity.

5The reason for assigning the rst term with adbc to s-channel can be understood as collecting contributions that carry the same color dependence as s-channel graph from the complete 4-point Yang-Mills vertex.

+ i

4

1

2

1

2

4

1

2

4

1

2

4

1

2

4

e

e

e

e

e

e

e

e

e

3

3

3

3

1

2

1

2

4

1

2

4

1

2

4

1

2

4

1

2

4

e

e

e

e

e

e

e

e

e

e

e

3

3

3

3

3

Figure 2. Graphical representation of contributions from eq. (3.4) where the rst four graphs correspond to contributions from the rst line of the equation, and the remaining ve graphs, from the second line of the equation.

not been contracted with polarization vectors. Expanding product of kinematic structure constants in the rst line yields the following nine terms

f1,2e(fe,34 + f4,e3) + (fe,12 + f2,e1)f3,4e (3.4)

+f1,2ef3,4e + (fe,12 + f2,e1)(fe,34 + f4,e3),

which can be represented by the graphs in gure 2. From these graphs, several information can be read out.

First we note that contraction of the repeated index e leads to consistent arrow directions for internal lines in rst four graphs but inconsistent arrow directions for internal line in the remaining ve graphs. As we will see in discussions below, contributions from consistent contractions satisfy the Jacobi identity of kinematic structure constants fabc while

inconsistent contractions do not. Because of this reason we shall split contributions from Feynman diagrams consisting of only cubic vertices into two groups: the good ones with consistent contractions and the bad ones with at least one inconsistent contractions.

Secondly, we note that a good graph can only have one outgoing arrow among all external particles. The unique external particle line carrying outgoing arrow plays an important role when we consider identities among graphs. In particular, we shall see that Jacobi identity is separately satised among graphs that have same outgoing leg.

Based on above observations, we can write numerator ns = G + X where G is contributions from good graphs and X is contributions from bad graphs and four-point vertex

i2 s (acbd adbc). As we will explain in section 3.1.1, remainder X can be eliminated

through averaging procedure.

However, for the simple 4-point amplitude, we can do better by digging out some good part from the bad contribution. We note that we are allowed to freely translate between upper and lower script structure constants through the identities

(fe,12 + f2,e1) f1,2e = iab(k1 + k2)e + O(k1a, k2b), (3.5) (fe,34 + f4,e3) f3,4e = icd(k3 + k4)e + O(k3c, k4d), (3.6)

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where Oe(k1a, k2b) denotes longitudinal term Oe(k1a, k2b) = i(eak2b ebk1a), and simi

larly does Oe(k3c, k4d). Both longitudinal terms do not contribute when they are contracted

with physical polarization vectors of external legs. Multiplying (3.5) with (3.6) we obtain the identity

f1,2ef3,4e + (fe,12 + f2,e1)(fe,34 + f4,e3) (3.7)

= f1,2e(fe,34 + f4,e3) + (fe,12 + f2,e1)f3,4e (t u)abcd+O(k1a, k2b, k3c, k4d),

where O(k1a, k2b, k3c, k4d) = Oe(k3c, k4d) Oe(k1a, k2b) iabOe(k3c, k4d) (k1 + k2)e icdOe(k1a, k2b) (k3 + k4)e. Thus ns is given by

ns = i

f1,2e(fe,34 + f4,e3) + (fe,12 + f2,e1)f3,4e

2 [s (acbd adbc) + (t u)abcd] + O(k1a, k2b, k3c, k4d),

and nu, nt can be derived from it by permutations of indices (123) ! (312) and

(123) ! (231) respectively. To obtain the numerators ns, nu, nt in equation (3.1), we

just need to contract ns, nu, nt with physical polarization vectors, thus the longitudinal terms O(k1a, k2b, k3c, k4d) drop out.

Having obtained expressions (3.8) we want to check the Jacabi idenity ns +

nt + nu = 0. First we notice that after contraction, contributions from

i2 [s (acbd adbc) + (t u)abcd] will be trivially zero under cyclic sum. To see con

tributions from the rst line of equation (3.8) give zero, let us expand the rst line of ns into

and write down corresponding terms of nu by permutation (123) ! (231)f2,3efe,14 + f2,3ef4,e1 + fe,23f2,4e + f3,e2f2,4e, (3.10)

and similarly terms of nt by permutation (123) ! (312)f3,1efe,24 + f3,1ef4,e2 + fe,31f2,4e + f1,e3f2,4e. (3.11)

When summing these three contributions (3.9), (3.10) and (3.11) together, the rst terms from each contribution add up to zero because of the Jacobi identity derived from cyclic permutations of legs (123),

f1,2efe,34 + f2,3efe,14 + f3,1efe,24 = 0. (3.12)

Adding up the rest three terms from each contribution again gives zero by following three Jacobi identities (3.13), (3.14) and (3.15),

f1,2ef4,e3 + f2,4ef1,e3 + f4,1ef2,e3 = 0, (3.13)

f3,1ef4,e2 + f1,4ef3,e2 + f4,3ef1,e2 = 0, (3.14)

f2,3ef4,e1 + f3,4ef2,e1 + f4,2ef3,e1 = 0. (3.15)

which correspond to xed leg 3, 2, 1.

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(3.8)

+ i

f1,2efe,34 + f1,2ef4,e3 + fe,12f3,4e + f2,e1f3,4e. (3.9)

It is easy to see that when expressed graphically, terms in equation (3.12) shall all have the outgoing arrow on leg 4, and similarly terms in equations (3.13), (3.14) and (3.15) shall all have the outgoing arrows on legs 3, 2 and leg 1 respectively. Thus Jacobi identity can be translated as the sum of three graphs related to each other by cyclic permutations with a xed leg having outgoing arrow.

3.1.1 Eliminating contact terms

In the discussion above we demonstrated explicitly that contributions from cubic and quartic diagrams together give rise to numerators that satisfy the Jacobi identity. While good parts of these contributions satisfy the identity manifestly, the bad parts, do not. For 4-point amplitudes since structure of amplitudes is simple, we were able to rewrite these bad parts into nicer forms. However this rewriting becomes rather di cult for higher point amplitudes, therefore we resort to an alternative way to solve the problem. The idea is the following. Since the numerator such as ns is calculated from contracting ns with polarization vectors, in a gauge theory we have the freedom to choose di erent gauges(i.e., di erent polarization vectors). Using this freedom we can eliminate bad contributions and keep only good contributions, thus the nal result will satisfy Jacobi identity manifestly.

Now we demonstrate the idea using 4-point amplitudes. For simplicity let us abuse the notation a bit by writing

ns(q) = a11(q1) . . . ann(qn)ns (q) ns , (3.16) where q represents the set of reference momenta {q1, q2, . . . , qn} collectively. Using the

notation that the good contribution given by equation (3.9), (3.10) and (3.11) ass,u andt, we have

ns(q) = i(q) s +

(3.17)

itt + iuu +i2X2 , X2 = Xt + Xu . (3.18)

In above expressions, good contributions intoi, which satisfy Jacobi identity, have been separated from the bad contributions Xi manifestly. Having done the reorganization, next step is to eliminate the Xi parts. To realize it, we consider the average of above two color-ordered amplitudes over three di erent choices of gauges. Since A(1234) is invariant under gauge choices, we can get rid of all Xi parts simultaneously if we impose following three conditions

T3 :

7

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i 2sXs(q),

Xs(q) (q) Xs = (q)

(acbd adbc) +

and similarly for nu(q), nt(q), Xu(q), and Xt(q). Thus the amplitudes are given by

A(1234) = (q)

iss iuu +i2X1 , X1 = Xs Xu

(t u)

s abcd

A(1324) = (q)

1 = c1 + c2 + c3

0 =

P3i=1 ci(qi) X1 0 =

P3i=1 ci(qi) X2

(3.19)

By gauge invariance, the rst condition guarantee that

A(1, 2, 3, 4) = ns

s

nuu , A(1, 3, 2, 4) =

ntt +

nuu (3.20)

where ns =

P3 i=1

ici(qi) s and similarly for nu, nt. Since each (qi) s,u,t satises

Jacobi identity, so do ns, nu, nt. To see that there is indeed a solution for ci, we simply need to show that the following matrix has nonzero determinant

1 1 1 (q1) X1 (q2) X1 (q3) X1 (q1) X2 (q2) X2 (q3) X2

. (3.21)

This can be checked by explicit calculations.

3.1.2 KK vs. BCJ-independent basis

In previous section we have showed how to derive kinematic numerators ns, nt and nu satisfying Jacobi identity by eliminating bad contributions. In the derivation we considered the analytic structures of two color ordered amplitudes A(1234) and A(1324), which serve as a basis when KK-relations [28] are taken into account. Since there were two remainders(i.e., the bad contribution part X) we need to introduce three ci to achieve our goal. But could we do better by introducing fewer variables ci?

Let us consider only A(1234). To eliminate its remainder term, we only need to average over two di erent gauge choices. The constraint conditions for ci are

T2 :

(1 =

ec1 + ec2 0 =

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(3.22)

P2 i=1

eci(qi) X1

which have the solution

ec1 =

(q2)X 1 (q1)X 1(q2)X 1 and

ec2 =

(q1)X 1(q1)X 1(q2)X 1 . Substituting them

back, we have

A(1234) = ns

s

nuu , ns = i(

ec1(q1) +

ec2(q2)) s, nu = i(

ec1(q1) +

ec2(q2)) u(3.23)

Having obtained these two numerators, we can dene an amplitude using

(ns + nu)

nuu (3.24)

It is easy to check that the amplitude just dened satises fundamental BCJ relation [1] by construction,

s21A(1234) + (s21 + s23)

eA(1324) =

t +

eA(1324) = 0, (3.25)

Since the same relation is satised between physical amplitudes, s21A(1234) + (s21 + s23)A(1324) = 0, we conclude that

eA(1324) = A(1324) and in particular, nt = (ns + nu),i.e., the kinematic-dual Jacobi identity we would like to have.

8

Above discussions show that, because of the BCJ relation for color-ordered amplitudes, we can use fewer ci to eliminate remainders. After doing so, Xi in rest of the color-ordered amplitudes automatically disappear, i.e.,

ec1(q1) X2 + ec2(q2) X2 = 0 . (3.26)

Now we have developed two methods to eliminate remainders through averaging over

KK or BCJ basis of amplitudes. We need to clarify the relation between these two methods. To do so, let us assume that we have solution (c1, c2, c3) with gauge choice (q3) = (q1)+ (q2). This gauge choice can be achieved if reference spinors of polarization vectors of three particles, for example, 2, 3, 4 are same for gauge choices (q1), (q2), (q3), but reference

spinors of polarization vector of particle 1 satisfy the relation (q3) = (q1) + (q2). Putting it back to the second equation of T3 given in (3.19) and comparing with the second equation of T2 given in (3.22), we can write down the following solution for T2,

ec1 = c1 + c3 + y(q2) X1, ec2 = c2 + c3 y(q1) X1 (3.27)

where y is determined by

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ec1 +

ec2 = 1 to be y = (

+1)c3((q1)(q2))X 1 . It is easy to check that the above indeed constitutes a solution if we assume + 1 = 0. In this case (3.26) is

automatically satised because of the third equation in (3.19). To see that indeed + = 1, notice that the reference spinors of particle 1 have relation

e3 = a e1 + b e2, so

(

e3) = 1

e3 [1|

e3] =

e1] a [1|

e1] + b [1|

e2]

a [1|

1

e1 [1|

e1] +

e2] a [1|

e1] + b [1|

e2]

b [1|

1

e2 [1|

e2] =) + = 1

This explanation shows that solutions (

ec1,

ec2) can be taken as a special case of solutions

(c1, c2, c3).

3.2 5-point numerators

For 5-point amplitudes KK relations reduce the number of independent color-ordered amplitudes to six. It was shown by Bern, Carrasco and Johansson [1] that these six amplitudes can be written into following forms with fteen numerators suggested by possible cubic graphs:

A(12345) = n1

s12s45 +

n2 s23s51 +

n3 s34s12 +

n4 s45s23 +

n5s51s34 , (3.28)

A(14325) = n6

s14s25 +

n5 s43s51 +

n7 s32s14 +

n8 s25s43 +

n2 s51s32 ,

A(13425) = n9

s13s45 +

n5 s34s51 +

n10

s42s13

n8 s25s34 +

n11

s51s42 ,

A(12435) = n12 s12s35 +

n11

s24s51

n3 s43s12 +

n13

s35s24

n5 s51s43 ,

A(14235) = n14 s14s35

n11

s42s51

n7 s23s14

n13

s35s42

n2 s51s23 ,

A(13245) = n15 s13s45

n2 s32s51

n10

s24s13

n4 s45s32

n11

s51s24 ,

9

Figure 3. A Feynman diagram contributing to n 1

To nd expressions for these ni, as in the 4-point amplitudes, we divide contributions from Feynman rules to good contributions plus a remainder (bad contributions),

A(12345) = n1

s12s45 +

(3.30)

Expanding (3.30) produces 27 terms, ve terms among them have consistent arrows in the internal lines (see gure 4) (so they are good contributions). We assign these ve terms to n1 and the rest to X (these bad contributions), thus we have

n1 = f1,2gfg,3h(fh,45 + f5,h4) + f1,2gfh,g3f4,5h + (fg,12 + f2,g1)f3,hgf4,5h.(3.31)

It is worth noticing that ve terms in n1 correspond to ve possible assignments of single outgoing arrow to external legs in graphical representations. If we use n1,k to denote the consistent graph having leg k with outgoing arrow, for example n1,3 =

f1,2gfh,g3f4,5h, the numerator can be written as n1 =

P5k=1 n1,k. It is straightforward to see that numerators from rest of channels can be written into similar structures.

In particular, (n15) is found to be the same as permutation (123) ! (312) of n1

n15 = f3,1gfg,2h(fh,45 + f5,h4) + f3,1gfh,g2f4,5h + (fg,31 + f1,g3)f2,hgf4,5h,(3.32)

and (n4), the same as permutation (123) ! (231) of n1,

n4 = f2,3gfg,1h(fh,45 + f5,h4) + f2,3gfh,g1f4,5h + (fg,23 + f3,g2)f1,hgf4,5h.(3.33)

6For simplicity we neglect the overall factor (i)

(p2)3 , where 1/p2 comes from (2.5) and (i)2 come from

10

n2 s23s51 +

n3 s34s12 +

n4 s45s23 +

n5s51s34 + X(12345) . (3.29)

As before we use to denote quantities that have not been contracted with polarization

vectors. The denition of ni and X is the following. First we include all contributions that contain at least one 4-point vertex in Feynman diagrams to X. For remaining Feynman diagrams having only cubic vertices like gure 3 for example, we use (2.5) to translate 3-point vertices to kinematic structure constants, thus obtain 6

(fg,12 + f2,g1) + f1,2g (fg,3h +fh,g3 +f3,hg)h(fh,45 + f5,h4) + f4,5h

i

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.

2

two propagators.

1

2 3 4

5

1

2 3 4

5

1

2 3 4

5

1

2 3 4

5

1

2 3 4

5

g

g

h

h

g

g

h

h

g

g

h

h

g

g

h

h

g

g

h

h

Figure 4. Five terms with consistent arrow directions contributing to n 1

When adding up (3.31), (3.32) and (3.33), terms with same outgoing leg will add to zero by Jacobi identity. Thus by our construction, we have Jacobi identity n1 n15 n4 = 0.

Similar argument shows when we permute (345) ! (534) we will produce (n3) and when

we permute (345) ! (453) we will produce (n12). Thus n1 n3 n12 = 0 is guaranteed

by Jacobi identity of kinematic structure constants fabc.

3.2.1 Eliminating contact terms

Having established the form (3.29) as well as similar expressions for other ve amplitudes given in (3.28), we construct the ni given in (3.28) by averaging over di erent choices of gauges. Just like for the 4-point amplitudes, we consider seven gauge choices denoted by (qi) with i = 1, . . . , 7 for polarization vectors under gauge choice qi = {q1,i, q2,i, q3,i, q4,i, q5,i} and impose following seven equations for coe cients ci,

i = 1, . . . , 7:

7

Xi=1ci = 1,

7

JHEP03(2013)050

Xi=1ci(qi) Xj = 0, j = 1, . . . , 6 (3.34)

where six remainders Xj are those given in (3.29). After solving ci from above equations, we can get ni dened in (3.28) as following

ni =

7

Xj=1cj(qj) ni . (3.35)

Since by our construction, ni satisfy Jacobi identity even before contracting with polarization vectors and cj are same for all fteen ni, ni will too satisfy Jacobi identity.

In the prescription above, we use the KK-basis (i.e., the basis under KK-relation) and BCJ relations between amplitudes follow as a consequence of the Jacobi identities among ni. However, if our focus is the construction of these ni numerators, we can take another logic starting point using only BCJ-basis (i.e., the basis under BCJ relation). For 5-point amplitudes, we can take A(12345) and A(13245) as BCJ-basis and consider averaging over three di erent gauge choices

c1 + c2 + c3 = 1,

3

Xi=1ci(qi) X(12345) = 0,

3

Xi=1ci(qi) X(13245) = 0 (3.36)

By imposing these conditions we obtain

ni =

3

Xi=1cii ni, i = 1, 2, 3, 4, 5, 10, 11, 15 . (3.37)

11

Then we construct the remaining seven coe cients using Jacobi identities

n6 = n10 + n1 n3 n4 + n5, n7 = n2 n4, n8 = n3 + n5,n9 = n3 n5 + n6, n12 = n1 n3, n13 = n1 + n2 n3 n4 n6, n14 = n2 + n4 + n6 (3.38)

The relation between these two eliminating methods can be understood similarly to the 4-point example.

3.3 n-point numerators

Having above two examples, it is straightforward to see the structure of kinematic numerators for n-point amplitudes. Generically a color-ordered amplitude can be written as

A =

XiniDi + X (3.39)

where the sum is taken over all cubic graphs. In this expression, ni contain only contributions from cubic graphs that have consistent arrow directions. All other contributions from cubic graphs with inconsistent arrow directions as well as graphs with at least one 4-point vertex are assigned to X part. Furthermore, according to which external particle has been assigned with the outgoing arrow in graphical representation, we can divide kinematic numerator into

ni =

n

JHEP03(2013)050

Xk=1ni,k, (3.40)

so that each ni,k is represented by a single graph. All these ni,k will have Jacobi identities

among themselves with di erent i but same xed k.

Having expressions as in (3.39), we average over amplitudes to eliminate the remainder terms X. This can be done through averaging over either KK-basis or BCJ basis. The average coe cients ci are determined by N1 = (n 2)! + 1 equations

TKK :

(1 =

PN1 i=1 ci 0 =

PN1i=1 ci(qi) Xj, j = 1, . . . , (n 2)!

(3.41)

for KK-basis or N2 = (n 3)! + 1 equations

TBCJ :

(1 =

eci 0 =

PN2 i=1

PN2 i=1

eci(qi) Xj, j = 1, . . . , (n 3)!

(3.42)

PNi=1 ci(qi) nj. Other nis which do not show up in the KK-basis or BCJ-basis can be constructed from various relations including

Jacobi identities. From either method we can construct the numerators proposed by Bern, Carrasco and Johansson in [1].

A technical issue concerning the above averaging procedure is the existence of solution for equation (3.41) and (3.42). The existence for lower point amplitudes can be checked by explicit calculations, but we have not nd a proof for general n. In this paper, we will assume their existence.

12

for BCJ-basis. After the averaging we have nj =

4 Fundamental BCJ relations

In previous section, we have shown how to construct the kinematic numerator satisfying the Jacobi identity by averaging over di erent gauge choices. An important step is to separate contributions from Feynman diagrams to two parts

A =

Xi

Pnk=1 ni,kDi + X, (4.1)

where each ni,k can be represented by a single consistent arrow graph with only cubic vertices. E ectively, we can treat these graphs as if they were built from the Feynman rules with only cubic vertices, where the coupling is given by kinematic structure constant fabc. From this point of view we can dene an n-point color-ordered amplitude for given k as

An;k =

Xini,kDi . (4.2)

The physical amplitude is given by linear combination of these xed-k amplitudes

A =

N

Xi=1ci(qi)

n

Xk=1An;k. (4.3)

An important feature of formula (4.3) is that the part

JHEP03(2013)050

PNi=1 ci(qi) coming from averaging procedure does not depend on the color ordering of external particles.

The amplitudes dened in (4.2) contain similar algebraic structure as these amplitudes A(color)n of color-dressed scalar theory considered in [29]. In that paper we have shown that amplitudes A(color)n satisfy color-order reversed relations, U(1) decoupling relations, KK-relations and both on-shell and o -shell BCJ relations. Because of the similarity between amplitudes A(color)n and An;k, it is natural to ask if the An;k dened by (4.2) obey these same identities. We can not make the naive conclusion since there are di erences between these two theories. First the kinematic coupling constant fabc here is only antisymmetric between a, b while group structure constant fabc of U(N) is totally antisymmetric. In addition, fabc depends on kinematics while fabc is independent of momenta. Bearing these in mind, we discuss properties of new amplitudes in this section.

4.1 The color-order reversed relation

Since each ni,k is given by single graph, it is easy to analyze it directly. Under the color-order reversing, each cubic vertex will gain a minus sign coming from fabc = fbac (See gure 5a for example). For n-points amplitudes, there are (n2) cubic vertices and (n3)

propagators, thus we will get a sign ()n2, i.e., we do have

An;k(123 . . . n) = ()nAn;k(n . . . 321) . (4.4)

To see U(1)-decoupling relation

XcyclicAn;k(C(1, 2, . . . , n 1), n) = 0 (4.5)

13

1

2 3 4

5

2 3 4

5

(a) reversing the color ordering in a ve point graph

1

n

n-1

m+1

n-1

m+1

1

2

.

.

1

2

.

.

.. . .

.. . .

.

.

m

m

n

(b) two typical terms in U(1)-decoupling relation

Figure 5. Demonstration of color-order reversed relation (part (a)) and the U(1)-decoupling relation (part (b))

is satised, we draw two typical terms in the cyclic sum in gure 5b). These two terms have same denominator and same numerator up to a sign since the only di erence between them is the reversing of vertex connecting n, thus contributing () sign. However, the left

term belongs to color ordering (123, . . . , n) while the right term belongs to color ordering (m + 1, . . . , n 1, 1, 2, . . . , m, n), thus we can see the general pair-by-pair cancellation in

U(1) identity given in (4.5).

4.2 The o -shell and on-shell BCJ relation

Just like the color-dressed scalar eld theory, the An;k satises a similar o -shell BCJ relation, which can be represented graphically by

(4.6)

with the momentum of particle n taken o -shell. Depending on the arrow directions of 2, n we have another two similar relations

(4.7)

JHEP03(2013)050

order(12...n) order(m+1...n-1,1,2..m,n)

14

and

(4.8)

There three relations show that the o -shell BCJ relations are, as in the case of color-order reversed and U(1)-decoupling relations, independent of the choice of arrow directions. The proof of relations (4.6) is similar to the proof given in [29] for color-dressed scalar theory.

The case of n = 3 is trivially true from momentum conservation. For n = 4, the left handed side of relation (4.6) consists of following sum of graphs,

We note that graphs (2) and (5) cancel due to antisymmetry of the structure constant. Using Jacobi identity, graphs (1) and (6) combine to produce

which, when added to the rest two graphs (3) and (4), produces the result as claimed using the on-shell conditions of particles 1, 2, 3.

Similar manipulations can be done for (4.7) and (4.8).

Having proven the n = 4 example, let us consider, for example, relation (4.8) for general n. We divide contributions to any amplitude An;k into the two sub-amplitudes that share same cubic vertex with leg n. (See part gure 5b as an illustration.) i.e.,

An;k =

#(nL)=n2

X#(nL)=1

AL({nL}; PL)V3(n, PL, PR)AR(PR; {nR}) (4.9)

where the number of legs in set nL can be 1, 2, . . . , (n 2), and we used V3(n, PL, PR)

to denote the cubic vertex that connects leg n to the two sub-amplitudes. Using this

15

JHEP03(2013)050

decomposition, the left handed side of (4.8) can be expressed by following graphs

n

n

n

1...j 2 j+1 ... n-1

n

( )

n-1

j-1

n-2

+

+

(s + s + ... + s )

21 23 2j

j=1

k=1

k=j+1

1 ... k k+1 n-1

2

1 ... j 2 ...n-1

j+1

+

1 k k+1... n-1

2

(4.10)

We can categorize terms in (4.10) according to whether leg 2 belongs the left or right sub-amplitude. When the 2 belongs to the left, the summation is given by

n

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n

n

p2

L

s21 (s + s )

21 23

n

=

2

1 k k+1... n-1

2 3

+ (s + s + ... + s )

21 23 2k

1 k k+1... n-1

3 2

+ ... +

1 2 k+1... n-1

... k

1 ... k k+1... n-1

(4.11)

where we used the o -shell BCJ relation for left part with fewer points. The value of k in sum (4.11) can be 1, 3, 4, . . . , n2. Similarly, when leg 2 belongs to the right sub-amplitude,

the summation is given by

n

n

n

n

[ ] [ ]

(s + s + ... + s )

(s + s + ... + s )

(s + s + ... + s )

p

+ ... +

+

2

1 ... k 2 n-1

k+1

+ + (s + ... + s )

s

1 k k+1 n-1

2

1 ... k k+1 2

n-1

=

1 ... k k+1... n-1

(4.12)

The above sum can be split into two parts. First there are terms carrying the common factor

Pki=1 s2i, and their sum (

Pki=1 s2i)(AR(2, k + 1, . . . , n 1, PR) + A(k + 1, 2, . . . , n 1, PR)+. . .+A(k+1, . . . , n1, 2, PR)) = 0 by U(1)-decoupling identity. The remaining part

can be simplied by o -shell BCJ relation for fewer points. The value of k for sum (4.12) can be 1, 3, 4, . . . , n 2. The sum given in (4.11) and (4.12) can be further combined to

n

1 ... k k+1... n-1

n

p2

L

p2R

(k + k )2

n 2

+ =

n

2

2

1 ... k k+1... n-1

1 ... k k+1...n-1

2

(4.13)

by using the Jacobi identity derives from permuting the internal 4-point tree

{n, 2, {1, . . . , k}, {k + 1, . . . , n 1}}. When we sum over k, we get (kn + k2)2V3(PL, 2, n)An1(1, 3, . . . , n 1, PL). Finally, result given in (4.13) is combined with

16

Pn1j=1 s2j)V3(PL, 2, n)An1(1, 3, . . . , n 1, PL) coming from the decomposition of An(1, 3, 4, . . . , n 1, 2, n) according to (4.9) ( which is the boundary term that has been

neglected in the sum (4.11) and (4.12)). Putting together, we have the same graph multiplied by (kn + k2)2 2k2 kn = k2n, which is exactly the right handed side of (4.8). In

other words, we have proved the o -shell BCJ relation for An;k amplitudes dened in (4.2). Taking the on-shell limit k2n ! 0, we get the familiar on-shell BCJ relation.

4.3 The KK-relation

The KK-relation found originally in [28] for gauge theory is given by

An(1, . . . , r, 1, 1, . . . , s, n) = (1)rX{}P (O{}O{}T )

An(1, {}, n), (4.14)

where the sum is over all permutations keeping relative ordering inside the set and the set T (where the T means the set with its order reversed), but at the same time allowing all relative orderings between sets and . We show that relation (4.14) still holds if we replace An by the xed k amplitude An;k dened in (4.2).

When {} is empty set (4.14) reduces to the color-order reversed relation (4.4), while

when there is only one leg in the set {} or the set {}, (4.14) reduces to the U(1)-

decoupling identity (4.5). Thus the color-order reversed relation and the U(1)-decoupling identity are just two special cases of KK relation. Since KK-relations coincide with these two relations for n 5, the starting point of our induction proof is checked.

Now we give the proof. Using the graphical representation, when the set {} is not

empty, there are two types of graphs depending on if 1 is at the left or right handed side of n. When leg 1 is at the right sub-amplitude as described by the left graph of (4.15)

n

term (

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1

T(-)# 1

{

1

n

1

2

{

T

n

1

{

{

1

2

U T 1(-)#

2

U T

2

(-)#

(4.15)

2

we can use KK-relation of the right sub-amplitude to get the middle graph of (4.15). After that we reverse the ordering of sub-amplitude 1 and ip it to the right hand side of n.

The nal result is the last graph of (4.15). When leg 1 is at the left handed side of n as given by the left graph of (4.16),

n

1

2 1

n

{

{

{

2

{

{

1

U T

1

(-)#

(4.16)

17

we can use KK-relation for the left sub-amplitude to get the right graph of (4.16). When we combine results from (4.16) and (4.15), we nd they are nothing but the graphical representation of right handed side of equation (4.14) except contributions from following graphs

n

1

U T

= 0 . (4.18)

Sum inside the bracket of (4.18) to be zero can be proved by exactly same method as that given in [29] after using two times of KK-relation for (n 1)-point amplitudes.

5 Kinematic ordering of gauge theory amplitude

As proposed in [30] and proved in [29], the full gauge theory amplitude can be represented by the manifestly (n 2)! symmetric KLT formula (which was found in [34])

An

= ()n

X,

k

Xq>t(it, iq)sitiq) , (5.2)

where (it, iq) is zero when pair (it, iq) has the same ordering in both set I, J and otherwise

it is one. In the KLT formulation above, one copy of the amplitudes

eA is calculated from the color-dressed scalar theory discussed in [29] and other copy A is the familiar color-ordered gauge theory amplitude.

To calculate the sum in numerator of (5.1), let us consider the following sum for given xed ordering of , for example, (2, . . . , n 1) = (2, 3, . . . , n 1),

S[2, 3, . . . n 1|i2, i3, . . . in1]k1 An(1, i2, i3, . . . in1; n) (5.3)

7Generalized U(1)-decoupling equation (4.18) has been written down in [35].

18

{

(4.17)

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These contributions are nothing, but7

V3(n1P )

X

{}P (O{}O{T })

eA(n, (2, . . . , n1), 1)S[(2, . . . , n1)|(2, . . . , n1)]p1A(1, (2, . . . , n1), n) s123...(n1)

(5.1)

where leg kn has to be taken o -shell prior to the summation and the full amplitude is given by the limit k2n ! 0. The momentum kernel S is dened as

S[i1, . . . , ik|j1, j2, . . . , jk]p1 =

k

Yt=1(sit1 +

An1(P1,n, {})

X{ i}Sn2

where the semicolon is used to emphasize that leg n is taken o -shell. For amplitudes given by

An =

N

Xi=1ci(qi)

n

Xk=1 An;k

!

(5.4)

since the part

PNi=1 ci(qi) is same for all color orderings, using the denition of function

S we see that the sum in (5.3) can be written as8

X{ j}Sn3

S[3, . . . n 1|j3, . . . jn1]

hs21An(1, 2, j3 . . . jn1; n) + (s21 + s2j3)n(1, j3, 2, . . . jn1; n) + . . .

i

= p2n p2n2

V3(2nc)

X{ j}Sn3

S[3, . . . n 1|j3, . . . jn1]

n1(1, j3, . . . ; c). (5.5)

where {j} is the set dened by deleting leg 2 from the set {i}. In the last line we have

used o -shell BCJ relation (4.8) as well as the form (5.4). The sum over the new S can be

done similarly and we obtain

p2n p2n2

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V3(2nc) p22n

p2n23

V3(3cc1)

X{ j}Sn4

S[4, . . . n 1|j3, . . . jn1]

n1(1, j4, . . . ; c1) (5.6)

Repeatedly reducing the number of legs contained in the amplitude one by one for An;k we arrive at the graphical representation

n

k

2

n

n

S[3,...,n-1| {i} ]

3,n-1

n-2

S[2,...,n-1| {i} ]

2,n-1

Sn-3

2

1 { i }

2,n-1

1 { i }

3,n-1

= ... =

2 3 4 n - 2 n-1

1

n

k

2

n (5.7)

Putting this result back to amplitudes given by (5.4), the KLT formula (5.1) produces

...

8In this form, we have used V3, which is not exactly right since we have not included the factor PNi=1 ci(qi).

19

naturally the following expression

An =

1

2 3 4 n-2 n-1

n

...

+

1

2

3 4 n-2 n-1

n

...

(5.8)

X (23...(n1))Sn2 (1n)

N

Xj=1cj(qj)

+

1

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2 3 4 n-2 n-1

n

...

2 3 4 n-2 n-1

n

. . .

+

1

...

The graph at the right handed side of (5.8) is very similar to the chain of U(N) group structure constant given in [27]. The manipulation demonstrated above can obviously be applied to KLT relation of gravity theory, so graviton amplitudes can be ordered by the same kinematic structure constants.

6 Various forms of amplitudes

From recent progresses we saw that amplitudes of gauge theory can be expressed in following three formulations [1, 27]:

double copy form : Atot =

XiciniDi (6.1)

Trace form : Atot =

XSn1Tr(T 1 . . . T n)A() (6.2)

DDM form : Atot =

XSn2c1|(2,...,n1)|nA(1, , n) (6.3)

where A are color ordered amplitudes, T a is the matrix of fundamental representation of U(N) group and ci, c1|(2,...,n1)|n are constructed using the structure constants fabc. For example, we have

c1|(2,...,n1)|n = f12x1fx13x2 . . . fxn3n1n (6.4)

The transformation from double-copy formulation to DDM was shown in [29] using the KLT relation, while the transformation from DDM to Trace was given in [27] where the following two properties of Lie algebra of U(N) gauge group were essential

Property One : (fa)ij = faij = Tr(T a[T i, T j]), (6.5) Property Two :

XaTr(XT a)Tr(T aY ) = Tr(XY ) (6.6)

20

A special feature of double-copy formulation is that both ci and ni satisfy the Jacobi identity in corresponding Feynman diagrams with only cubic vertices. Because of this duality, it is natural to exchange the role between ci and ni and consider the following two dual formulations

Dual Trace form : Atot =XSn1 1...n

eA() (6.7)

Dual DDM form : Atot =

XSn2 n1|(2,...,n1)|n

eA(1, , n) (6.8)

eA is color ordered scalar theory with fabc as cubic coupling constants (see the references [29, 30]) and is required to be cyclic invariant. Indeed, the Dual-DDM was given in [11] while the Dual-Trace-form was conjectured in [31] with explicit constructions given for the rst few lower-point amplitudes and a general construction was given in [26]. Although the existence of above two dual formulations were established, a systematic Feynman rule-like prescription to the coe cients and n is not known at this moment. Our result (5.8) n1n for dual-DDM-form serves as a small step towards this goal.

Having above explanation, let us consider following situation where both ci, ni satisfying Jacobi-identity can be constructed by Feynman rule, i.e., the theory can be constructed using cubic vertex with coupling constant Fabc

eFabc. We want to know under this assumption, which dual form comes out naturally. The conclusion we found is that the dual

DDM (6.8) is more compatible with double-copy formulation.To see that, let us note that the total amplitude can be constructed recursively as

A(1, 2, . . . , n) =

n1

Xi=1

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where

XSplit F1e1e2

eF112 A

(e1, u1, . . . , ui)

P 2u1,...,ui A

(e2, v1, . . . , vn2i, n)

P 2v1,...,vn2i,n

, (6.9)

where the second sum is over all possible separations of (n 1) particles into two subsets {u}, {v} with nu = i. Assuming the color-decomposition holds for lower-point amplitude A, we can substitute the lower-point DDM-form into above equation and obtain

A(1, 2, . . . , n)

=

n1

Xi=1

XSplit F1e1ei

eF11i

X

Fe1 1e2Fe2 2e3 . . . Fei1 i1ui

eA(e1, 1, . . . , i1, ui) P2 u1,...,ui

2perm{u1,...,ui1}

X

eA(ei, 1, . . . , ni2, n) P2 v1,...,vni2,n

.

2perm{v1,...,vn2i}

Fei 1ei+1Fei+1 2ei+2 . . . Fen3 ni2n

=

n1

Xi=1

XSplitF1e1eiFe1 1e2 . . . Fei1 i1uiFei 1ei+1 . . . Fen3 ni2n

"

# (6.10)

where for given permutations 1,. . . ,i of us and 1,. . . ,ni2 of vs, the contraction of

Fs has the structure at the left handed side of gure 6. After applying Jacobi identity,

21

eF11i

eA(e1, 1, . . . , i1, ui) P2 u1,...,ui

eA(ei, 1, . . . , ni2, n) P2 v1,...,vni2,n

!

!

!

!

!

!

.

.

.

.

= .

- .

!

" "

!

" "

!

!

" "

!

!

.

. .

.

. .

.

. .

1 n

1 n

1

n

Figure 6. We can use Jacobi identity to reduce the contraction of Fs.

2

n -1

JHEP03(2013)050

.

. .

.

. .

1

n

Figure 7. A DDM chain with contractions of structure constants F12e1Fe13e2 . . . Fen3,n1,n

F1e1eiFe11e2 becomes

F1e1eiFe11e2 = F11e1Fe1e2ei F1e2e1Fe11ei, (6.11)

i.e., the right handed side of gure 6. Iterating this procedure like the one did in [27], we get a sum of 2i1 DDM chains (e.g., gure 7) where the ordered set O{1, . . . , i1} is split into two ordered sets O{} and O{} and the form is given by

(1)sF11e . . . FeteFeuieFese . . . Fe1eFe1e . . . Fen3,n2i,n. All these forms are multiplied by

eF11i

eA(e1, 1, . . . , i1, ui) eA(ei, 1, . . . , ni2, n). Doing same things to other permutations of u1, . . . , ui1s and collecting all terms having same DDM chain structure, we get

F11e1 . . . FeteFeuieFese . . . Fe1eFe1e . . . Fen3,n2i,n

eF11i

1P 2u1,...,ui

X

OP ({} [uniontext]{})

eA(e1, 1, . . . , i1, ui)

1P 2v1,...,vn2i,n

eA(ei, 1, . . . , ni2, n)= F11e1 . . . FeteFeuieFese . . . Fe1eFe1e . . . Fen3,n2i,n

eF11i

1P 2u1,...,ui

eA(e1, 1, . . . , t, ui, s, . . . , 1) 1P 2v1,...,vn2i,n

eA(ei, 1, . . . , ni2, n).(6.12)

where the KK-relation has been used for the sum in square bracket.

22

Putting this result back to recursion relation we reach our nal claim

A(1, 2, . . . , n)

=

n1

Xi=1

XSplit

XS{u}

XS{v1}

hF11e1 . . . FeteFeuieFese . . . Fe1eFe1e . . . Fen3,n2i,n

eF11i

1P 2u1,...,ui

eA(e1, 1, . . . , t, ui, s, . . . , 1) 1P 2v1,...,vn2i,n

eA(ei, 1, . . . , ni2, n)

i

= XSn2 Fa1a2 e1 . . . Fen3an1 an

JHEP03(2013)050

n1

Xi=1

eF11i

eA(e1, 1, . . . , t, ui, s, . . . , 1) P 2u1,...,ui

eA(ei, 1, . . . , ni2, n) P 2v1,...,vn2i,n

eA(1(2 . . . n 1)n) (6.13)

where at the last step we have used the recursion relation for color ordered amplitudes.

7 Conclusion

In this paper we have presented an algorithm which allows systematic construction of the BCJ numerators as well as the kinematic-dual to the DDM formulation. We have shown that assuming gauge symmetry provides enough degrees of freedom, we can express tree-level amplitudes as linear combinations of cubic graph contributions, where Jacobi-like relations between kinematic numerators can be made manifest.

Although our construction is systematically, it is a little bit hard to use practically. In other words, our results is just a small step toward the simple construction of BCJ numerators, which can have important applications for loop calculations of gravity amplitudes.

Acknowledgments

Y.J.Du would like to thank Profs. Yong-Shi Wu and Yi-Xin Chen for helpful suggestions. He would also like to thank Qian Ma, Gang Chen, Hui Luo, Congkao Wen and Yin Jia for helpful discussions. Y. J. Du is supported in part by the NSF of China Grant No. 11105118. CF is grateful for Gang Chen, Konstantin Savvidy and Yihong Wang for helpful discussions. Part of this work was done in Zhejiang University and Nanjing University. CF would also like to acknowledge the supported from National Science Council, 50 billions project of Ministry of Education and National Center for Theoretical Science, Taiwan, Republic of China as well as the support from S.T. Yau center of National Chiao Tung University.B.F is supported, in part, by fund from Qiu-Shi and Chinese NSF funding under contract No. 11031005, No. 11135006, No. 11125523.

A O -shell KK relation from Berends-Giele recursion

The KK-relation was rst written down in [28] without proof. With our knowledge, a proof can be found in [27]. Since the o -shell tensors can be constructed by Berends-Giele

23

= XSn2 Fa1a2 e1 . . . Fen3an1 an

recursion relation [32], it is natural to prove the o -shell KK relation by this recursion relation and in this appendix we provide a proof for readers convenience.

The o -shell KK relation is given as

J(1, {}, n, {}) = (1)n XOP ({} [uniontext]{T })

J(1, , n), (A.1)

where J(1, 2, . . . , n) is an o -shell tensor. After contracting J with on-shell polarization vectors of external legs, it becomes a color-ordered amplitude A(1, 2, . . . , n), and thus the o -shell KK relation becomes the on-shell KK relation.

According to Berends-Giele recursion relation, for a given tensor, we can pick out a leg, for example, the leg 1, to construct whole tensor recursively. In the formula, the leg 1 can be connected to either a three-point vertex or a four point vertex, i.e., we can separate the tensor into J(1, 2, . . . , n) = J(3)(1, 2, . . . , n) + J(4)(1, 2, . . . , n). We will do the same

separation at both sides of (A.1) and show the matching for each part.

Connecting to 3-point vertex: in this case, the R.H.S. of KK relation (A.1) can be expressed by

()n X ! A, B ! A, B

XA 2 OP ({A}

S{B}T ) B 2 OP ({B}

S{A}T )

V 1e1e2(3)

1P 2A,B

J(e1, A) 1

J(e2, B, n),

P 2B,A

JHEP03(2013)050

= ()n

XA,B;A,BV 1e1e2(3)1P 2A,B

X

AOP ({A} [uniontext]{B}T )

J(e1, A)

1P 2B,A

X

(A.2)

where the rst sum is over all possible splitting of set , into two subsets (including the case, for example, A = ;) and the second sum is over all possible relative ordering between

subsets i, j. Now we consider the sum in (A.2) for di erent splitting:

(i) If both A and B sets are nonempty, we can use lower-point generalized U(1)-decoupling identity (4.18)

XAOP ({A} [uniontext]{B}T )

J(e1, A) = 0. (A.3)

Thus this case does not have nonzero contribution.

(ii) If B set is empty, we have

BOP ({B} [uniontext]{A}T )

J(e2, B, n)

XA,BV 1e1e2 1P 2AJ(e1, A) 1P 2B,

J(e2, B, n, ), (A.4)

where we have used lower-point KK relation to sum up the last line in (A.2).

24

(iii) If A is empty, we have

XA,BV 1e2e1 1P 2,AJ(e2, , n, A) 1P 2BJ(e1, B), (A.5)

where we have used lower-point KK relations for the second bracket, the color-order reversed relation for the rst brackets as well as the antisymmetry of three-point vertex V 1,2,3 = (1)V 1,3,2 (so the overall factor ()n disappears).

The sum of contributions from (ii) and (iii) is just the recursive expansion of J(3)(1, , n, ).

Connecting to 4-point vertex: in this case, the R.H.S. of KK relation (A.1) is given as

()n XA,B,C;A,B,CV 1e1e2e3(4)1P 2A,A

JHEP03(2013)050

X

AOP ({A} [uniontext]{TA})

J(e1, A)

1P 2B,B

X

J(e2, B)

1P 2C,C

X

J(e3, C, n)

.

BOP ({B} [uniontext]{TB})

COP ({C} [uniontext]{TC })

(A.6)

where the sum is over all possible splitting of sets , into three subsets (with possible empty subset). For given splittings ! A, B, C, ! A, B, C, there are several cases:

(i) If both {A} and {A} are nonempty or both {B} and {B} are nonempty, we can

use lower-point generalized U(1)-decoupling identity (A.3) and the sum is zero for the rst or the second brackets in (A.6).

(ii) If A = OP ({A}), B = OP ({B}), C 2 OP ({C}

S{T }) , we have nonzero

contribution

XA,B,CV 1e1e2e3(4)1P 2AJ(e1, A) 1P 2BJ(e2, B) 1P 2C,

J(e3, C, n, ), (A.7)

where we have used lower-point KK relation to sum up the last bracket.

(iii) If A = OP ({TC}),B = OP ({TB}), C 2 OP ({}

S{TA}), we have nonzero contri-

bution

XA,B,CV 1e1e2e3(4)1P 2,AJ(e1, , n, A) 1P 2BJ(e2, B) 1P 2CJ(e3, C), (A.8)

where we have used lower-point KK relation for the third bracket and the color-order reversed relation for the rst and second brackets as well as the symmetry of four-vertex V 1234(4) = V 1432(4).

25

(iv) If A = OP ({A}), B = OP ({TB}), C 2 OP ({B}

S{TA}), the nonzero contri-

bution is given as

()n

XA,B;A,BV 1e1e2e3(4)1P 2AJ(e1, A) 1P 2BJ(e2, TB)

. (A.9)

Similarly, If A = OP ({TB}), B = OP ({A}), C 2 OP ({B}S{TA}) , we have

()n XA,B;A,BV 1e2e1e3(4)1P 2BJ(e2, TB) 1P 2AJ(e1, A)

1P 2B,A

X

OP ({B} [uniontext]{TA})

J(e3, , n)

JHEP03(2013)050

1P 2B,A

X

. (A.10)

where it is worth to notice that the 4-point vertex is written as V 1e2e1e3(4). The reason doing so is because the 4-point vertex is

V 1234 = i1324

OP ({B} [uniontext]{TA})

J(e3, , n)

i2(1234 + 1423). (A.11)

so we have following identity

V 1234 + V 1324 = V 1243. (A.12)

Using this identity and lower-point KK relation for the third brackets and the color order reversed relation for the rst or the second brackets (thus the factor ()n

disappears), the sum of above two contributions becomes

XA,B;A,BV 1e1e3e2(4)1P 2AJ(e1, A) 1P 2B,AJ(e3, B, n, A) 1P 2BJ(e2, B).

(A.13)

The sum of (ii), (iii), (iv) is just J(4)(1, , n, ).

Having shown both 3-point vertex part and 4-point vertex part have KK-relation, we have shown the whole o -shell tensor J(1, , n, ) has the KK-relation. In the proof, we have used the antisymmetry of three-point vertex under exchanging a pair of indices as well as the identity between 4-point vertex. This proof shows that if a tensor is constructed only by three-point vertices, it obeys KK relation when the three-point vertex is antisymmetry under exchanging a pair of indices.

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