Simplifying the Tree-level Superstring Massless Five-point Amplitude

We use the pure spinor formalism to obtain the supersymmetric massless five-point amplitude at tree-level in a streamlined fashion. We also prove the equivalence of an OPE identity in string theory with a subset of the Bern-Carrasco-Johansson five-point kinematic relations, and demonstrate how the remaining BCJ identities follow from the different integration regions over the open string world-sheet, therefore providing a first principles derivation of the (supersymmetric) BCJ identities.


Introduction
Since the discovery of the pure spinor formalism [1] a new method to efficiently compute supersymmetric scattering amplitudes is available. Although its simplifying features manifest themselves more vividly in explicit one-and two-loop computations [2,3,4,5,6] and provide hope 2 for higher-loop extensions [10,11,12], tree-level amplitudes [13] also benefit from the streamlined nature of the formalism 3 . In particular, having results written in terms of pure spinor superspace expressions [15,6] sheds new light into finding supersymmetric completions [16]. This paper simplifies the long (bosonic) RNS five-point computations of [17,18] while naturally extending them to the full supersymmetric multiplet using the pure spinor superspace. In doing that we uncover the superstring origin of the Bern-Carrasco-Johansson (BCJ) kinematic identities of [19], proving that some of them come from an OPE identity and that they are supersymmetric. And in view of the string theory proof for the fourpoint BCJ identity we will demonstrate that the remaining BCJ relations follow from the different integration regions of the open string world-sheet. These integrations over the various domain of integrations are related to the monodromy identities [20,21] between the string theory amplitudes which have been used in [20] to prove that the number of partial amplitudes is (N − 3)!, and is ultimately related to the BCJ identities of [19].
2 Useful knowledge can be obtained even without fully explicit higher-loop computations [7,8,9]. 3 For a review of scattering amplitudes in the pure spinor formalism, see [14].
The kinematic factorsL ijkl are given by simple pure spinor superspace expressions which satisfy the supersymmetric BCJ relations, The paper is organized as follows. In section 2 we compute the five-point amplitude at tree-level and express it in terms of simple pure spinor superspace expressions. In section 3 we prove an OPE identity from which the supersymmetric generalizations for some of the BCJ relations can be obtained. Furthermore, using the analogy with the four-point amplitude derivation of the BCJ identity we show how to obtain the remaining ones.
The pure spinor superspace computations are presented in Appendix A, together with the explicit proof of (1.2) directly in superspace. Of particular importance is the simplified expression for the OPE of two integrated vertices presented in (A.2). The Appendix B is devoted to writing down an ansatz for a simplified expression of A F 4 (θ), whose bosonic component expansion agrees with the expression obtained in section 2. The Appendix C is a formal rewriting of the ten-dimensional results using the four-dimensional spinor helicity formalism, and we show agreement with the expressions of [22,23,24,25]. Finally, in Appendix D we derive the relations obeyed by the integrals K j which were used in section 2.

The five point amplitude in pure spinor superspace
Following the tree-level prescription of [1] the open superstring 5-point amplitude is where the partial amplitude A 5 (1, 2, 3, 4, 5) is given by The SL(2, R) symmetry of the disc requires the fixing of three positions, chosen as (z 1 , z 4 , z 5 ) = (0, 1, ∞). Therefore the integrals are over the region 0 ≤ z 2 ≤ z 3 ≤ 1.
Using the OPEs of the pure spinor formalism to integrate out the conformal weight-one variables, (2.1) assumes the following form where the kinematic factors L ijkl are given by the following pure spinor superspace expressions (from now on we set 2α ′ = 1), while the other L ijkl are obtained by exchanging labels appropriately. All the terms containing factors of (k i · k j )(A k W l ) are "total derivative" terms and will be shown to cancel in the final result. Furthermore, the double pole in the OPE of U 2 (z 2 )U 3 (z 3 ) gives rise to the following expression for L 2314 As will become clear later, the factor of (1 + α 23 ) appearing in (2.4) is essential to obtain a simple answer for the amplitude. That this is possible can be traced back to the fact that the pure spinor Lorentz currents have level −3 (see the computations of Appendix A).
With the notation of [17] for the integrals appearing 4 in (2.2), the amplitude can be written [18] and where (2.6) will be proved as an OPE identity in the next section. Plugging in the expressions for K j in terms of T and K 3 derived in the Appendix D, the amplitude (2.5) becomes where A Y M (θ) and A F 4 (θ) are superfields,  where we used α 24 = α 51 − α 23 − α 34 and α 13 = α 45 − α 23 − α 12 and the redefinedL ijkl are given bỹ The identities (2.6) continue to hold with these redefinitions, that is 14) The use of (2.13) also removes the contact terms appearing in (2.9), simplifying it. In fact, using (2.13) the supersymmetric string theory partial amplitude (2.1) becomes, The component expansions of (2.9) and (2.8) can be computed 5 using the methods of [28,3,29]. When all external states are bosonic the RNS results of [17,18] are recovered, The higher α ′ expansion in (2.7) is determined solely by the expansions of T and K 3 , and all the (supersymmetric) information about the external states is encoded in the superfield expressions A YM (θ) and A F 4 (θ), in accord with the observations of [30]. This is in fact a generic feature of the amplitudes computed in the pure spinor formalism. The kinematic factors of bosonic and fermionic states are always multiplied by the same "form factors", which come from the integrals over the world-sheet.

Derivation of the BCJ kinematic identities
In reference [19], the massless four-point partial amplitudes at tree-level were represented in terms of its poles as and the identity n u = n s − n t was explicitly shown to be true. Furthermore, the five-point amplitudes were written as and by analogy with the Jacobi-like four-point kinematic relation, the numerators were required to satisfy n 3 − n 5 + n 8 = 0, n 3 − n 1 + n 12 = 0, n 4 − n 1 + n 15 = 0, n 4 − n 2 + n 7 = 0, n 5 − n 2 + n 11 = 0, n 7 − n 6 + n 14 = 0, n 8 − n 6 + n 9 = 0, n 10 − n 9 + n 15 = 0, which was explicitly verified to be true. Extending the same reasoning to higher points, it was argued that those kind of relations impose additional constraints which reduce the number of independent N -point color-ordered amplitudes to (N − 3)!. This conclusion was later demonstrated in [20] using the field theory limit of string theory. We will now prove the identity (2.6) and discuss its relation 6 with the 5-point BCJ identities of [19]. 6 I thank Pierre Vanhove for several discussions about this.
To prove (2.6) it suffices to note that in the computation of a kinematic identity can be obtained by considering the different orders in which the OPE's are evaluated. By computing first the OPE's of U 2 (z 2 ) followed by U 3 (z 3 ) one gets, while in reverse order, As the integrated vertex U I is bosonic, (3.4) and (3.5) must be equal. Therefore we get where we used L 3221 = −L 2331 and L 2314 = L 3214 . To see this one notes that After absorbing the contact terms as in (2.13), the field theory limit of the string partial amplitudes A(1, 2, 3, 4, 5) and A(1, 3, 2, 4, 5) are given by (2.11) and (2.12), respectively.
Note that there is an ambiguity (or freedom) on how to absorb the contact terms, as there is no unique way in doing so. We chose to absorb them while preserving the kinematic identities 7 (3.7). This is in agreement with the discussions of [19], where it is emphasized that the BCJ identities would not be satisfied by any choice of absorbing contact terms.
Using s+t+u = 0 and taking the field theory limit one can easily derive the supersymmetric generalization of the four-point BCJ relation n u = n s − n t by comparing (3.11) with (3.1).
That is, n s = −K 21 , n t = K 23 and n u = −K 21 − K 23 .
Therefore, computing the integrals appearing in the five-point scattering amplitude for each of the twelve regions of integration should provide the remaining five-point BCJ identities in a supersymmetric fashion. For example, the partial amplitude A YM (1, 4, 2, 3, 5) is obtained by integrating (2.2) over 1 ≤ z 2 ≤ z 3 ≤ ∞, and in this case the kinematic factors for the different poles appearing in the last equation of (3.2) will be given by combinations of the factors already present in (2.11), so that new identities will have to arise. In fact, using the transformations y 3 = (z 3 − 1)/z 3 and y 2 = (z 2 − 1)/z 2 the integrals which allow them to be written in terms of K j and L j of [17], provided that α 13 → α 34 , α 34 → α 35 , α 12 → α 24 , α 24 → α 25 , α 51 → α 14 , α 23 → α 23 , α 45 → α 51 (3.12) The only "new" integral which is not already computed in [17] is the one associated to . (3.13) However, (3.13) is easily seen to be equal to K ′ 1 + K 3 + L 3 ≡ L 8 . Finally, the amplitude (2.2) integrated over 1 ≤ z 2 ≤ z 3 ≤ ∞ is given by where the tildes mean that the substitution (3.12) must be performed. Using the explicit results of [17] for the integrals, the field theory limit of (3.14) is given by and together with n 8 − n 6 + n 9 = 0 and n 14 + n 13 − n 12 = 0, which follow as OPE identities using U 2 and U 4 or U 3 and U 4 as integrated vertices, we get the same solution as (4.12) of [19]. Therefore the BCJ identities of [19] were obtained from first principles. And by using the pure spinor formalism and its pure spinor superspace, we have shown that the BCJ relations are in fact supersymmetric.
Acknowledgements: I deeply thank Pierre Vanhove for reading an early draft and for suggesting the connection between the identities (2.6) and the BCJ relations, and also for several discussions.

Appendix A. Computation of the kinematic factors
In this section we compute the OPE's appearing in the amplitude (2.1) to obtain the explicit expression for the kinematic factors L ijkl in pure spinor superspace.
Using the OPE's and the equations of motion a long computation leads to the OPE between two integrated vertices, where we dropped the total derivative terms with respect to z 2 which appear when Taylor expanding the superfields in the double pole. The super-Yang-Mills equations of motion (A.1) have been used judiciously to arrive at the simple answer (A.2). For example, the terms which contribute to the double pole are given by, Using the OPE's one obtains (omitting (z − w) −2 ) where the last term comes from the level −3 double pole of the pure spinor Lorentz currents.
One can now use D α A m = (γ m W ) α + k m A α and the fact that k m (γ m W ) α = 0 to simplify (A.3) to, where we used tr(γ mn γ pq ) = −32δ mn pq and − 1 . Using the same kind of manipulations as [6] one can also prove the following OPE identity as z 2 → z 1 where M(x, θ) is any superfield. Furthermore, if QM = 0 then the following holds true and One can also show by using gamma matrix identities, the pure spinor constraint and the SYM equations of motion (A.1) that from which the following expressions can be read for L 2331 and L 2314 , and where we used that which can be checked by writing k 1 m (λA 1 ) = QA 1 m − (λγ m W 1 ) in the last term of the LHS and integrating the BRST charge by parts.
The expression for L 2131 can be deduced from the OPE as z 2 → z 1 followed by z 3 → z 1 .
Using (A.5) we obtain the singularity as z 2 → z 1 whose OPE computation for z 3 → z 1 implies, after some manipulations in superspace, that while L 2434 and L 3121 are obtained by exchanging 1 ↔ 4 and 2 ↔ 3, respectively.
The kinematic factor L 2134 is given by the coefficient of the OPE as z 2 → z 1 followed by z 3 → z 4 . Using (A.5) the first limit becomes and using (A.5) again to evaluate as z 3 → z 4 we obtain . From (A.13) we get the expression for L 2134 , To see this first note that all terms containing (k i · k j ) trivially match on both sides of (A.14). Using that (λγ we get, after some trivial cancellations, which after using F 2 mn = k 2 m A 2 n − k 2 n A 2 m is equal to zero, as we wanted to show. are proportional, therefore the result of this factorization should also be captured by the tree-level massless five-point amplitude at the correct α ′ order. This is given by the A F 4 superfield above. As discussed in [31], the factorization in the (12)-channel ((23)-channel) is given by L 12 /α 12 (K 23 /α 23 ), where   where N (ijklm) = ij jk kl lm mi . Using the results above it is straightforward to obtain, in the NS sector, agreeing with (5.45) of [25] and (37) of [23], apart from the overall coefficient.