Explicit BCJ numerators from pure spinors

TLDR: In this paper, local kinematic numerators for gauge theory tree amplitudes are derived from the low energy limit of superstring amplitudes computed with the pure spinor formalism.

Abstract: We derive local kinematic numerators for gauge theory tree amplitudes which manifestly satisfy Jacobi identities analogous to color factors They naturally emerge from the low energy limit of superstring amplitudes computed with the pure spinor formalism The manifestation of the color-kinematics duality is a consequence of the superstring computation involving no more than (n − 2)! kinematic factors for the full color dressed n point amplitude The bosonic part of these results describe gluon scattering independent on the number of supersymmetries and captures any N k MHV helicity configuration after dimensional reduction to D = 4 dimensions

Figures

Table 1. The number of BCJ numerators in n-point amplitudes containing 2j basis kinematics Klσ for j = 0, 1, . . ., n− 3.

Fig. 1 The four- and five-point amplitudes in terms of cubic graphs.

Fig. 2 Triplet of subdiagrams where the sum over the associated color factor vanishes due to the Jacobi identity fe[abf c]de = 0.

Frequently asked questions

Q1. What is the kinematic factor ni for gauge theory?

In this paper the authors have developed a method based on string theory to construct kine-matic factors ni for gauge theory amplitudes which manifestly obey Jacobi identities dual to the color algebra ci + cj + ck = 0. 

Q2. What is the role of contact terms in the amplitudes of gauge theory?

It plays a crucial role for taming the non-planar sector of SYM andfor understanding gravity as the double copy of gauge theories [2] 

Q3. What is the grading of kinematic numerators?

The authors suspect that the grading of kinematic numerators according to their Klσ content is connected with the factorization pattern (4.8) of Pρ (l,σ) entries. 

Q4. What is the symmetry property of f bac?

More importantly,it makes the dual structure constant contraction available from which one can infer the symmetry properties fabc = −f bac and the Jacobi identities f b[a1a2fa3]bc = 0. 

Q5. What is the ni of the field theory limits?

Since the field theory limits of the integrals in (3.5) involve no other coefficients than 0 and ±1 for the propagators, these relations must be of the formni1 ± ni2 ∓ ni3 ± . . .∓ nip−1 ± nip = 0, (3.6)with a so far unspecified number p of terms. 

Q6. What are the supersymmetric momentum and Green-Schwarz constraint?

where V i and U i are vertex operators writen in terms of the SYM superfieldsV i = λαAiα(x, θ), U i = ∂θαAiα +Π mAim + dαW α i +1 2 F imnNmn. (2.3)The bosonic ghost field λα(z) is a pure spinor satisfying λαγmαβλ β = 0 andΠm(z) = ∂Xm + 12 (θγm∂θ), dα(z) = ∂α −1 2 (γmθ)α∂Xm − 1 8 (γmθ)α(θγm∂θ) (2.4)are, respectively, the supersymmetric momentum and Green-Schwarz constraint. 

Q7. What is the ambiguity in the contact term?

In [6,7] the contact term ambiguity was shown to arise from thedouble pole in the OPE of two integrated vertices in the field-theory limit of the stringamplitude. 

Q8. What is the BRST variation of the integrated vertex U(z)?

It can be shown that imposing QV = 0 puts all superfields on-shell,which also implies that the BRST variation of the integrated vertex U(z) can be writtenas QU = ∂V [10] (see also [14]). 

Q9. What is the kinematic amplitude of the n point tree?

According to the hypothesis of BCJ, the color dressed n point tree amplitude in gaugetheories can be parametrized asAn = ∑icini ∏αi sαi(3.1)such that the kinematic factors ni satisfy Jacobi-like relations in one-to-one correspondence with the group-theoretic Jacobi identities for the color factors ci,ci ± cj ± ck = 0 ⇒ ni ± nj ± nk = 0. (3.2)The relative signs depend on the choice of signs when defining the color factors. 

Q10. What is the i sum of the Jacobi triplets?

By taking appropriate permutations of (3.7) and decomposing the occurring subam-plitudes in pole channels, one can derive identities between Jacobi triplets (nik , nil , nim) dual to color factors with cik + cil + cim = 0 of the following form [7,29]∑inik + nil + nim ∏n−4αi sαi= 0. (3.8)The i sum runs over n − 1 point channels of total number 2n−3(2n− 7)!!(n− 3)/(n− 2)! 

Q11. What is the ni constructed in this way?

The ni constructed this way can therefore be recycled from planar N = 4 SYM to the non-planar sector and used for N = 8 supergravity by means of the double copy construction, cf. [8] and references therein. 

Q12. What is the field-theoretic limit of the superstring amplitude?

As will now be discussed,the field-theory limit of the superstring amplitude can be used to find supersymmetric andlocal n-point BCJ-satisfying kinematic numerators [1] in a straightforward manner. 

Q13. what is the number of kl forming the individual BCJ numerators?

It is interesting to note that the number of Klσ forming the individual BCJ numerators is always a power of two, i.e. 1, 2, 4 or 8 in this case. 

Q14. What is the amplitude of the gauge theory?

The BCJ organization scheme represents gauge theory amplitudes in terms of diagramswith cubic vertices only (in short: cubic diagrams). 

Q15. What is the amplitude of the ni in the diagram?

An = ∑icini ∏αi sαi(1.1)The ci denote color factors made of n−2 structure constants fabc of the gauge group, and their dual numerators ni are constructed in this work. 

Q16. What is the symmetry property of the BRST building blocks?

In many instances, the symmetry properties (2.11) of the BRST building blocks withinKlσ allow to rewrite sums over several basic kinematics occurring in some ni as a single superfield, e.g.n2 = K1(23) −K1(32) = 〈(T123 − T132)V4V5〉 = 〈T321V4V5〉.However, the right hand side is outside the five point basis of kinematics, so the Jacobirelations between numerators are rather obscured by this building block manipulations.