RNS derivation of N-point disk amplitudes from the revisited S-matrix approach

In the past year, in arXiv:1208.6066 we proposed a revisited S-matrix approach to efficiently find the bosonic terms of the open superstring low energy effective lagrangian (OSLEEL). This approach allows to compute the ${\alpha'}^N$ terms of the OSLEEL using open superstring $n$-point amplitudes in which $n$ is very much lower than $(N+2)$ (which is the order of the required amplitude to obtain those ${\alpha'}^N$ terms by means of the conventional S-matrix approach). In this work we use our revisited S-matrix approach to examine the structure of the scattering amplitudes, arriving at a closed form for them. This is a RNS derivation of the formula first found by Mafra, Schlotterer and Stieberger in arXiv:1106.2645, using the Pure Spinor formalism. We have succeeded doing this for the 5, 6 and 7-point amplitudes. In order to achieve these results we have done a careful analysis of the kinematical structure of the amplitudes, finding as a by-product a purely kinematical derivation of the BCJ relations (for N=4, 5, 6 and 7). Also, following the spirit of the revisited S-matrix approach, we have found the $\alpha'$ expansions for these amplitudes up to ${\alpha'}^6$ order in some cases, by only using the well known open superstring 4-point amplitude, cyclic symmetry and tree level unitarity: we have not needed to compute any numerical series or any integral involving polylogarithms, at any moment.


Introduction
In the recent years there has been a considerable progress in perturbative String Theory (considering D-brane systems as well). Acheivements have been going on in S-matrix calculations involving Ramond-Ramond massless states and open superstrings [1, 2, 3, 4, 5, 6] 1 , the closed and the open sector of the superstring low energy effective lagrangian [8,9,10], higher loop calculations of closed and open superstring scattering [11,12,13], Mellin correspondence between supergravity and superstring amplitudes [14,15] and a deeper understanding of the α expansion of tree level open and closed superstrings [16,17,18,19], among other things. One of the very important results that has been acheived is Mafra-Schlotterer-Stieberger (MSS) formula for the (tree level) N -point scattering of nonabelian massless open superstrings [20]: Besides the fact of improving the α order of expansions of the past [27,28,23] in various orders [29,16,17], a conjecture has been established for the general form of the α expansion of the F {σ N } (α )'s (for arbitrary N and arbitrarily high order in α [16]) 4 ' 5 and, also, a recursive formula (in N ) for these α expansions has been proved in [18] (by means of a generalization of the work in ref. [30]) 6 . All these results have as a common starting point the MSS formula, given in eq.(1.1). This formula was derived using the Pure Spinor formalism [31], which is manifestly supersymmetric right from the beginning. Formula (1.1) is the final (and simple) result of an ellaborated study involving pure spinor superspace and its cohomology structure [32], first applied in the calculation of Super Yang-Mills amplitudes and afterwards extended to the corresponding calculations in Open Superstring Theory [20]. The purpose of the present work is two-fold. On one side, we show that it is possible to arrive to MSS's formula in (1.1) working only in the Ramond-Neveu-Schwarz (RNS) formalism [33] 7 . For the moment, we only have a proof for 3 ≤ N ≤ 7 within this approach, but we think that a deeper understanding of our procedure can, eventually, lead to the proof for arbitrary N . On the other side, we shed light in how the α expansion of the F {σ N } (α ) momentum factors in (1.1) can be obtained, order by order in α , not by doing the explicit computations of the coefficients of its expansion (which are given in terms of multiple zeta values (MZV's) [35,36,29,16]), but by demanding tree level unitarity of the amplitudes and (presumably) using only the α expansion of the 5-point amplitude 8 .
The basic tool in which our findings are supported is the 'revisited S-matrix approach', found by us in the past year [10]. This method was initially proposed as an efficient tool to determine, order by order in α , the bosonic part of the open superstring low energy effective lagrangian (OSLEEL) but, as we will see in the present work, it has a direct counterpart in the determination of the scattering amplitudes of the theory, allowing us to arrive to (1.1) and also to the α expansion version of it. This method is intrinsically kinematic and supersymmetric, although it is not manifestly supersymmetric. It deals, first, with the pure external gauge boson interactions and only at the end, it incorporates the interactions between external gauge bosons and its fermionic superpartners. The kinematics is present right from the beginning since the main statement of the method has to do with the kinematical structure of the N -point amplitudes of gauge bosons: in Open Superstring Theory they should not contain (ζ ·k) N terms, at any α order [10]. With respect to supersymmetry, it is also present right from the beginning since (we believe that) it is the reason for the absence of those kinematical terms in the amplitudes 9 . The structure of this work is as follows. In section 2 we give a brief review of the revisited S-matrix method. We explain in it why we claim that demanding the absence of the (ζ · k) N terms in the N -point amplitude of gauge bosons and, based on the conjecture of ref. [16] and the main result of ref. [18], using only the α expansion of the 5-point amplitude, that is enough information to find the complete (bosonic part of the) OSLEEL. We claim there, then, that using similar arguments there should be a direct analog of this situation from the perspective of the scattering amplitudes, that is, at a given order in α , knowing the 5-point amplitude is enough information to know any higher N -point amplitude 10 .
We begin the ellaboration of these ideas in section 3 by examining the space of N -point kinematic expressions which are on-shell gauge invariant and which do not contain (ζ · k) N terms. We find that this space is (N − 3)! dimensional (at least for 3 ≤ N ≤ 7). We then check that a BCJ basis for Yang-Mills subamplitudes (see eq.(3.1)) [38] can indeed be chosen as a basis for this space. In light of this important kinematical result, the determination of the explicit expressions of the open superstring subamplitudes (for gauge bosons) and of the BCJ relations themselves become simply a linear algebra problem: we know a vector of the space (that is, a given subamplitude for which we want an expression) and what is left to do is to find the components of this vector with respect to the basis of the vector space. We do this calculation in section 4 for the open superstring subamplitudes (for gauge bosons), arriving precisely to the bosonic part of (1.1), and in Appendix E, for the BCJ relations themselves (arriving to the same result of refs. [39,40]). Then, in the last part of section 4, we briefly explain why once we have found the gauge boson amplitudes in this manifestly gauge invariant way, the corresponding amplitudes involving fermions are immediate, and thus leading to MSS result in eq. (1.1) (this time considering there all possible scattering processes involving external gauge bosons and their supersymmetric partners). In section 5 we apply our revisited S-matrix method to find the α expansion of the F {σ N } (α ) momentum factors of (1.1), in the case of N = 5, 6, 7 11 . In order to acheive this, besides the we would only use the α information from the 4-point amplitude.
The clarifying statement is that, for the calculation purposes of this work, in which at most we have obtained α 6 order results, it will be enough to use the 4-point amplitude information. As explained in section 2.2, the 5-point amplitude information will become important from α 8 order onwards and, as argued in that section, we claim that the pure 5-point amplitude is enough to find the whole α terms of the F {σ N } (α ) momentum factors. 9 It is well known that the (ζ · k) N terms are indeed present in the case of 3 and 4-point amplitudes of massless states in Open Bosonic String Theory [34,37] and, from the general integral formula for the N -point amplitude [34], it is also believed that they are present in this general case. Therefore, it is quite reasonable to conjecture that the absence of the (ζ · k) N terms, in the case of supersymmetric open strings, is a consequence of Spacetime Supersymmetry. 10 See footnote 8 in the previous page. 11 Because of computer limitations, we have done the explicit calculations only up to α 6 order in the first two requirements of the revisited S-matrix method, we demand cyclicity and tree level unitarity to be obeyed by the subamplitudes. We end in section 6 by summarizing our results and conclusions. Throughout this work all scattering amplitudes are tree level ones. Since, at some points, we have needed to deal with huge calculations and formulas, we have considered only the simplest ones in the main body of this work and we have left the more extensive ones for the appendices 12 . These last ones usually do not offer any new conceptual insight, but they have played an important role checking our main statements.

Finding terms of the OSLEEL in an efficient way
Let L eff be the general low energy effective lagrangian (LEEL) for nonabelian gauge bosons in (either bosonic or supersymmetric) Open String Theory . It has the following form: Each of the F n and the D 2p F q terms in (2.1) is an abbreviation of the sum of different contractions of Lorentz indexes for those sort of terms. For example,"F 4 " really denotes an abbrevia- [41], where {b 1 , b 2 , b 3 , b 4 } are the coefficients to be determined. In the second column of the table in (2.2) 13 we have written the number of coefficients that the general LEEL contains at the first orders in α 14 . These coefficients are the ones that the conventional S-matrix approach usually finds by computing the open string N -point amplitudes (from N = 4 up to N = p + 2, at least) at α p order. In the third column of table (2.2) we have written the number of coefficients (which is extremely small!) that the revisited S-matrix approach really needs to find in order to determine the OSLEEL at a given α order. The reason for the smallness of these numbers (in relation to the corresponding ones in the second column) is that in the revisited S-matrix method (only applicable to the case of the supersymmetric string) the N -point amplitudes satisfy the constraint [10] absence of (ζ · k) N terms . (2.3) cases and up to α 4 order, in the last case. But it is clear that the method can be used to obtain higher order α contributions. See more details in section 5. 12 Moreover, there are some extremely long expressions that we have preferred not to include them in the text of this work and only to attach them as 'txt' files, in the version that we have submitted to the hep-th arXiv. 13 This table has been taken from eq.(3.4) of ref. [10]. 14 The terms which are being taken into account in the LEEL are only the ones which remain invariant under field redefinitions.
This constraint implies further linear restrictions that the b j coefficients of the LEEL in (2.1) should satisfy (see section 4 of [10] for more detalis about these restrictions). These restrictions are so strong that only a small number of coefficients remain free: this number is precisely the 'dimension of the constrained basis' appearing in the third column of table (2.2). These same type of contraints had correctly been found about ten years before, by Koerber and Sevrin, using the method of BPS configurations (which is not directly a String Theory one) [42]. In [10] we pointed out that the (probable) reason for the constraint in (2.3) is Spacetime Supersymmetry. Due to the highly constrained form that the OSLEEL lagrangian adopts after demanding the requirement in eq. (2.3), in order to determine the α p order terms of it, a (p + 2)-point amplitude calculation, in Open Superstring Theory, is no longer needed (as in the conventional S-matrix method): very much lower N -point amplitudes (expanded at α p order) are expected to be enough for this purpose. In fact, in [10] we saw that the α expansion of the 4-point momentum factor (given in eq.(F.3)) is enough to determine explicitly the OSLEEL, at least up to O(α 4 ) terms.
2.2 Using the 4 and the 5-point α information to obtain N -point information So, a main idea that arises naturally from the revisited S-matrix method is the fact that the α expansion of the 4-point momentum factor (whose coefficients are completely known in terms of integer zeta values, at any α order, see eq. (F.2)) is enough information to obtain completely the α expansion of higher N -point amplitudes, at least up to a certain order in α . For example, with the calculations that we did in [10] it is clear that the 5 and the 6-point amplitudes (and any higher N -point amplitude) can be completely determined at least up α 4 order, because we found the OSLEEL explicitly up to that order, bypassing 5 and 6-point worldsheet integral calculations.
In fact, in [10] we raised the possibility that the OSLEEL could be determined to any α order by means of the revisited S-matrix method plus the known α expansion of the 4-point momentum factor (see eqs. (F.2) and (F.3)), but this can hardly happen since there are higher order coefficients given in terms of multiple zeta values (MZV's) 15 , like ζ(3, 5), ζ(3, 7), ζ(3, 3, 5), etc., which already show up in the α expansion of the 5-point amplitude (at α 8 , α 10 and α 11 order, respectively [16,29]) 16 . These coefficients are not expected to be given by linear combinations of products of ζ(n)'s (in which the coefficients of these linear combinations are rational numbers) 17 [43], so they are not present in the α expansion of the 4-point amplitude.
Since α 8 is the first order at which these non trivial MZV's arise, we do not have a proof, but we believe that up to α 7 order any N -point amplitude can be found by means of the revisited S-matrix method plus only the known α expansion of the 4-point momentum factor, F {2} , (see eqs. (4.4), (F.2) and (F.3)). From α 8 order onwards we expect the 4-point amplitude to only give a partial (but still important) information for the determination of the OSLEEL terms 18 . For example, the α p D 2p−4 F 4 terms of the OSLEEL (for p = 2, 3, 4, . . .) are still going to be completely determined by the 4-point amplitude [45,46]. One might ask if new MZV's, besides the ones that already appear in the 5-point amplitude, will eventually appear at a given (highly enough) α order. This would be a signal, from this α order onwards, that α information from a 6-point (or eventually higher N -point) amplitude would be required, in order to determine those α terms of the OSLEEL. But this does not seem to be the case. A remarkable observation was done in ref. [16], claiming that the coefficients of the α expansion of the F {σ N } (α ) momentum factors, for N > 5, are always the same as the ones that appear for N = 5: what changes (from one N to another) is only the kinematic polynomial which is being multiplied by this coefficient (see Appendix F.2 for more details) 19 . This conjecture is also consistent with the recent discovery that the F {σ N } (α )'s can be iteratively obtained (from the N -point point of view), to any order in α , from a unique and same Drinfeld associator (which is a generating series for the MZV's) [18]. So, the MZV's that already appear in the α expansion of the 5-point amplitude, will be the same ones that will appear for higher N -point amplitudes 20 . So we are left with the conjecture that our revisited S-matrix method plus the α expansion of the 5-point amplitude, are enough informations to find all the α corrections coming from Open Superstring Theory to the Yang-Mills lagrangian 21 .
In spite of this conjecture, for the calculations that we do in this work, which at most go to α 6 order (see section 5), we will still use only the known 4-point α expansion (see Appendix F.1).

Basis for open superstring and Yang-Mills subamplitudes
In this section we prove that, from a kinematical structure perspective, open superstring and Yang-Mills (tree level) subamplitudes belong to a same space of kinematical N -point expressions (where N ≥ 3) and that a possible basis for this space is given by the (N − 3)! Yang-Mills subamplitudes 22 In order to acheive this, in subsection 3.1 we review some important facts about the structure of gauge boson scattering subamplitudes. Then, based on this previous background, in the following subsections we argue, case by case (from N = 3 up to N = 7), that the set of subamplitudes in (3.1) is indeed a basis for the corresponding space 24 .

Some general facts about the structure of scattering amplitudes of gauge bosons
If we compare open superstring and Yang-Mills (tree level) N -point subamplitudes for gauge bosons, both theories being treated in the Lorentz gauge, from the point of view of the structure of its kinematical terms they have in common the constraint in eq. (2.3), namely, the absence of (ζ · k) N terms. In the first case this has been recently emphasized in [10], together with the strong implications that it has for determining the bosonic terms of the low energy effective lagrangian of the theory in a very simplified way and, in the second case, the claim in (2.3) can easily be confirmed by considering Feynman rules in the construction of tree level scattering amplitudes. So, let us consider the space of all scalar N -point kinematical expressions constructed with the polarizations ζ i and the momenta k i of N external gauge bosons in a nonabelian theory (like Open Superstring Theory or Yang-Mills theory, for example). The momenta and polarizations should satisfy: Mass-shell condition: Transversality (Lorentz gauge) condition: Let us denote this space by V N . We further restrict V N such that its elements T (1, 2, . . . , N ) obey the following conditions: 1. They are multilinear in the polarizations ζ i (i = 1, 2, . . . , N ). 2. They do not contain (ζ · k) N terms.
3. (On-shell) Gauge invariance: whenever any ζ i → k i (i = 1, 2, . . . , N ) then T (1, 2, . . . , N ) becomes zero. (3.5) These four requirements are simply properties of tree level gauge boson subamplitudes in Open Superstring and Yang-Mills theories and we want the elements of V N to also satisfy them. Since these elements, the T (1, 2, . . . , N )'s, are Lorentz scalars, they can only be constructed from linear combinations of 25 where [p] denotes the integer part of p and N ≥ 3. Besides the terms excluded in (2.3) and the ones that we have mentioned in (3.6), there are no more possibilities of kinematical terms that can be constructed from the polarizations ζ i and the momenta k i of the external gauge bosons. The only place where some extra dependence in the momenta could be considered is in the scalar coefficients which in T (1, 2, . . . , N ) go multiplying the kinematical terms in (3.6): these coefficients are allowed to be given in the terms of the k i · k j factors (or equivalently, in terms of the Mandelstam variables) and, eventually, in terms of a length scale (for example √ α , in the case of String Theory) . For example, in the case of N = 3, the Yang-Mills subamplitude contains only (ζ · ζ) 1 (ζ · k) 1 terms. It is given by [34] It is easy to see that it is an element of V 3 . In subsection 3.2.1 we will prove that the only elements of V 3 are multiples of A Y M (1, 2, 3) (see eq.(3.18)), so the fact that A Y M (1, 2, 3) ε V 3 will turn out to be an immediate thing. In order to do a counting of all the possible independent kinematical terms that can be taken from the list in (3.6) to construct the expression for an element T (1, 2, . . . , N ) ε V N , respecting the kinematic conditions in (3.2), (3.3) and (3.4) and also the requirement in (2.3), we need first to analize the (ζ · k) terms. In principle, for each i there are N possible (ζ i · k j ) terms (because j runs from 1 to N ). But taking into account the the transversality condition (3.4) and also momentum conservation (3.2), this implies that for each i (= 1, 2, . . . , N ) we have: So, at the end, for each i there are only (N − 2) independent (ζ i · k j ) terms.
With this information we are in conditions to do a counting of the different independent terms, specified by structure in eq. (3.6), that in principle are allowed to appear in T (1, 2, . . . , N ). This leads us to the following table, for the number of independent allowed kinematical terms of T (1, 2, . . . , N ): It is not difficult to prove that the number of independent (ζ · ζ) j (ζ · k) N −2j terms (where j = 1, 2, . . . , [N/2]), for an arbitrary N , is given by This is the formula that has been used to compute the corresponding number of kinematical terms in table (3.9) 26 . We have only worried to specify in this table the cases N = 3, 4, 5, 6, 7, because these are the ones that we will consider in the present work.

Finding a basis for
In this subsection we will explicitly present the derivation of a basis for V N only in the case N = 3 and N = 4, which involve simple calculations and illustrate the procedure to arrive at that basis. For N = 5, 6, 7 we will just mention the final result and we will leave the details of the main calculations to Appendix D.

Case of N = 3
It is well known that in the case of 3-point amplitudes of massless states, momentum conservation and the mass-shell condition (3.3) imply that the momenta obey the relations According to the table in (3.9), T (1, 2, 3) has only three independent (ζ · ζ) 1 (ζ · k) 1 terms, which, following our prescription in (3.11), leads to: Demanding on-shell gauge invariance for gauge boson 1, A(1, 2, 3) should become 0 when ζ 1 → k 1 .

Case of N = 4
In this case, according to the table in (3.9) the open superstring 4-point subamplitude has a kinematical expression which consists of twenty four (ζ · ζ) 1 (ζ · k) 2 and three (ζ · ζ) 2 terms, all of them being independent, so, following the prescrition in (3.11), leads to: In Appendix C we find that demanding on-shell gauge invariance of the subamplitude when ζ 1 → k 1 we can arrive to the following thirteen linearly independent relations: where s = (k 1 + k 2 ) 2 = 2k 1 · k 2 and t = (k 1 + k 4 ) 2 = 2k 1 · k 4 (3.21) are two of the three Mandelstam variables that appear in the 4-point scattering 28 ' 29 . The set of equations in (3.20) is the 4-point analog to eq.(3.14), found for the 3-point subamplitude when ζ 1 → k 1 .
In the same way, demanding on-shell gauge invariance of the T (1, 2, 3, 4), when ζ 2 → k 2 , ζ 3 → k 3 and ζ 4 → k 4 , we can arrive to a set of thirteen linearly independent equations in each case. The explicit expression of these additional equations and the details of its solution can be found in Appendix C.
The important thing is that in the whole set of fifty two equations that come from demanding onshell gauge invariance (eqs. (3.20), (C.3), (C.6) and (C.9)) only half of them are linearly independent as a whole. The solution of this system is given in Appendix C, in eq. (C. , (3.22) or equivalently, The procedure for N > 4 is exactly the same one presented in subsections 3.2.1 and 3.2.2, but for the corresponding N -point subamplitudes whose general kinematical structure we have mentioned in table (3.9). Since for these cases the calculations are much more involved than the ones that we have already done for N = 4, we leave the details of them for Appendix D. In all these cases it is possible to arrive to an (N − 3)! dimensional basis for V N (dim(V N ) = (N − 3)!) and it is always possible to choose this basis in terms of only Yang-Mills subamplitudes, in particular the one indicated in at the beginning of this section, in eq. (3.1). That set of subamplitudes precisely contains (N − 3)! elements. That set constitutes one of the possible basis that have appeared in the literature as basis for the space of Yang-Mills subamplitudes [38,39,14] and also for the space of open superstring subamplitudes [20], which is the purpose of this whole section to prove. In Appendix D we see that in the cases for N > 4 our calculations have not lead directly to the basis in (3.1), but only after having verified that the dimension of V N is (N − 3)! and that the set , leads to an expression which is only on-shell equivalent to the one in eq. (3.22): it is necessary to use eqs. (C.12) and (C.13) in order to check the equivalence between both formulas. 31 This explicit expression for A Y M (1, 2, 3, 4) can be found in many places in the literature, for example, in section 3 of [22]. But there are some sign differences due to the different convention that we have used for the 4-point Mandelstam variables in eq.(3.21). 32 We have used the curly brackets, '{}', in the index 2, just as a remainder of the rule, mentioned in eq.(3.1), to choose the Yang-Mills subamplitudes of B N . of amplitudes mentioned in (3.1) is linearly independent 33 . So, summarizing, with the calculations and proofs that we have done in Appendix D we can write an arbitrary element of V 5 , V 6 and V 7 , respectively as 34 Once we have accepted that the set B N is indeed a basis for V N , at least for 3 ≤ N ≤ 7, the next step consists in finding the components of an element of V N with respect to B N . We will present here a procedure which we expect to be valid for any N ≥ 3 (even for N > 7) 35 . Let T (1, . . . , N ) ε V N . To find the components of T (1, . . . , N ) with respect to the basis B N we have to find the momentum factors, λ {σ N } 's, such that 27) where σ N = {2 σ , 3 σ , . . . , (N − 2) σ } denotes the same permutation of indexes {2, 3, . . . , (N − 2)} that we referred to in eq.(3.1). The natural (but tedious!) way to find the λ {σ N } 's is by writing down the expression of T (1, . . . , N ), and of each Yang-Mills submplitude in B N , in terms of the kinematical terms listed in eq. (3.6), with the convention (3.11) for the (ζ · k) terms, for example. Then, a linear system for the λ {σ N } 's 33 Our proposal of basis in eq. (3.1) is based in the previously known results of the BCJ relations for N = 5, 6, 7 [38,39,14]. Except for the case of N = 4, without these previously known results it would have been quite difficult to guess this basis and, as seen in Appendix D, we would have had only (N − 3)! known (and very long) N -point kinematical expressions for which we would have no interpretation at all. 34 In eqs. (3.24), (3.25) and (3.26) we are following the same sort of notation that the authors of [20] used to write the linear combinations of the subamplitudes, that is, the momentum factors are labelled by a superscript denoting the corresponding σ N permuation. 35 At least in this subsection we will work with the hypothesis that B N , as specified in (3.1), is a basis of V N for any N ≥ 3.
arises from demanding that the coefficient of each (ζ · ζ) j (ζ · k) N −2j term (where j = 1, . . . , [N/2]), in both sides of (3.27), is the same. Since this linear system is overdetermined, it is not necessary to consider all the equations of it in order to find the (N − 3)! λ {σ N } 's. For small values of N it is easy to see that considering in (3.27) the kinematical terms with the smallest number of (ζ · k) terms (that is, the ones for j = [N/2]), that is enough information to find the λ {σ N } 's. Consider, for example, the cases of N = 3, 4, 5. From the table in eq.(3.9) we see that the 3-point amplitude contains 3 (ζ · ζ) 1 (ζ · k) 1 terms, the 4-point amplitude contains 3 (ζ · ζ) 2 terms and the 5-point amplitude contains 45 (ζ · ζ) 2 (ζ · k) 1 terms. It is clear, then, that considering these particular kinematical terms in (3.27), that would be enough information to find the λ {σ N } 's, since for N = 3 and N = 4 there is only one component and for N = 5 there are two components to determine. But, for sufficiently large N 36 , the number of kinematical terms of the type mentioned above (the ones for j = [N/2]) is less than (N − 3)! and, then, considering only those type of terms will not be enough information to find all the λ {σ N } 's. Since, at this point, we are worried about a strategy to find the components of T (1, . . . , N ), for an arbitrary N , our proposal will be to consider the kinematical terms for j = 1, namely, the (ζ · ζ) 1 (ζ · k) N −2 terms. Those are the ones, among all the possible types of kinematical terms in (3.6), which show up in more amount in the most general expression for T (1, . . . , N ) 37 . So, in the next two sections, when we will look for the momentum factors that are present in the BCJ relations and in the open superstring formula, we will consider the (ζ · ζ) 1 (ζ · k) N −2 terms in our kinematical analysis. . We will first derive MSS formula in the case of pure gauge boson scattering in subsection 4.1 and then, in subsection 4.2, we will quickly see that there is no need to deal with fermion vertex operators (at least for the tree level amplitudes) in order to find the scattering amplitudes involving fermions (once the closed formula has been found for gauge bosons).

For gauge bosons only
Using the RNS formalism, it is known that the N -point subamplitude for gauge bosons in Open 36 Consider N ≥ 12, for example. 37 It is not difficult to see, by considering formula (3.10), that the number of independent (ζ · ζ) 1 (ζ · k) N −2 terms is bigger than (N − 3)!. 38 In the case of Yang-Mills subamplitudes, it was already mentioned in the beginning of subsection 3.1 that they do not contain (ζ · k) N terms. Superstring Theory is given by the following integral formula [48] 39 :  . This interchange of indices should be done in both, the k j 's momenta and the x j 's variables inside the curly brackets. This will become more clear in the case-by-case study that we will consider in the following subsections. Before going into the derivation of the momentum factors we have two remarks: 1. As they stand, formulas (4.1) and (4.2) are not applicable to the case of N = 3. Since it is very well known that in this case A b (1, 2, 3) = A Y M (1, 2, 3), the 3-point momentum factor is simply defined as being 1.
2. There is an equivalent expression that Mafra, Schlotterer and Stieberger give for the momentum factor in eq.(4.2), for N ≥ 5, and it comes by integrating by parts on it [36]: In the following subsections we will see that we reproduce formula (4.2) in the case of N = 4 and formula (4.3) in the cases of N = 5, 6, 7. In the case of N = 4 we will present the derivation with enough details to see the consistency of the relations that arise when we consider the (ζ · ζ) 1 (ζ · k) N −2 terms of the subamplitudes. For N = 5, 6, 7 the procedure will be the same one, but we will not present so many details because the open superstring and the Yang-Mills subamplitudes become too large. For N = 5 we will explain how to arrive at the known expression of the two momentum factors (eq.(4.3)), together with some evidence of the self consitency of the calculations, and for N = 6, 7 we will just explain how to arrive to the corresponding momentum factors.
Notice that comparing other (ζ · ζ) 1 (ζ · k) 2 terms in eq.(4.5) we may arrive to different expressions for F {2} (α ). For example, if we compare the coefficient of (ζ 1 · ζ 3 )(ζ 2 · k 3 )(ζ 4 · k 1 ) and the coefficient of (ζ 2 · ζ 3 )(ζ 1 · k 2 )(ζ 4 · k 1 ), in both sides of (4.5), we find that, respectively, 3 . (4.10) Integrating by parts in the definition of I [4] 2 (and considering the analytic continuation in α s and α t of the three integrals in eq.(4.6)), we have that (4.12) Using relations (4.11) and (4.12) it is easy to see that the two alternative expressions for F {2} (α ), given in eq.(4.10), are equivalent to the one in (4.7) (and, therefore, equivalent to the one in (4.9)). One might even ask what would have happended if we had included the three (ζ · ζ) 2 terms in both sides of (4.5) and compared their corresponding coefficients. The answer is that we would have found two of the three integral representations that we already have for F {2} (α ) plus a new one: It can be easily verified that this expression can be obtained using integration by parts in I [4] 3 , in the second equation in (4.10).

Case of N = 5
In this case the relation (3.24) guarantees that choosing T (1, 2, 3, 4, 5) = A b (1, 2, 3, 4, 5) it is possible to write down (4.14) Following the procedure proposed in subsection 3.3, in eq.(4.14) we consider only the (ζ · ζ) 1 (ζ · k) 3 terms, where we are supposed to write all (ζ · k) terms in the basis given in (3.11) for N = 5. On one side, in section 5 of ref. [22] it has been explained in detail how the x 5 → +∞ limit is taken and how the Grassmann integrations are done in eq.(4.1), in the case of N = 5 43 , so all the terms of the left hand-side of eq.(4.15) are known.
If we examine the coefficient of ( This equation leads directly to an expression for F {23} (α ), which might be written as 45 where x 1 = 0 and x 4 = 1.
The fact that in eq.(4.16 from which we can arrive at Formulas (4.17) and (4.19) correspond to the expressions of the two momentum factors that appear in the case of N = 5. These two formulas agree completely with the expected expression for F {23} (α ) and F {32} (α ), according to formula (4.3).
For the sake of completeness we will examine in eq.(4.15) two other kinematical structures, namely, . This leads, respectively, to the following two relations that F {23} (α ) and F {32} (α ) should satisfy: 2 , (4.20) where Solving the linear system in (4.20) leads to integral expressions for F {23} (α ) and F {32} (α ) which apparently differ from the ones found in (4.16) and (4.18), respectively. But this is exactly the same thing that we analized in eqs.(4.10) and (4.13), for the case of N = 4: there are integration by parts and partial fraction relations that allow us to write a same momentum factor in apparently different integral representations. Since the expressions found for the momentum factors this time are double integrals, there is a much rich variety of relations that can be found for them and it is not always an immediate thing to prove the equivalence between this different integral representations. In refs. [50] and [28] it was explained how using these relations, A b (1, 2, 3, 4, 5) for the first time could be written in a simple form, as a sum of two contributions (in direct analogy to eq.(4.14)).

Case of N = 6 and N = 7
In the case of N = 6 and N = 7 we will not refer any longer to the alternative integral expressions that show up for the momentum factors (and which can always be proved by combining integration by parts and partial fraction techniques). These expressions will simply have to be equivalent to the ones we are looking for, because of the consistency of our method. In this case relations (3.25) and (3.26) guarantee that choosing T (1, 2, 3, 4, 5, 6) = A b (1, 2, 3, 4, 5, 6) and T (1, 2, 3, 4, 5, 6, 7) = A b (1, 2, 3, 4, 5, 6, 7), respectively, it is possible to write down similar relations to the ones in those equations, but relabelling the momentum factors as where σ 6 and σ 7 are all possible S N −3 permutations (for N = 6 and N = 7) of indices {2, 3, 4} and {2, 3, 4, 5}, respectively. The final result is that we succeed in arriving at the six and the twenty four momentum factors of formula (4.3) for N = 6 and N = 7, respectively, that is: and In eq.(4.24) it is understood that As we mentioned after eq.(4.2), the MSS prescription for finding the remaining momentum factors consists in simply doing a permutation of indices inside the curly brackets of (4.24) and (4.25), according to the σ N permutation that is being considered (where N = 6 and N = 7, respectively). For example, doing 2 ↔ 3 in (4.24) and (4.25) we arrive, respectively, at and In Appendix G we specify the kinematical terms that allow us to write a linear system of equations which has a unique solution for the momentum factors and which is precisely given by the ones in eqs.(4.24) and (4.25) (and for the remaining momentum factors, by means of a σ N permutation of the corresponding indices). We also specify, in that appendix, the corresponding linear system of equations for the F {σ6} (α )'s and for the F {σ7} (α )'s.

For gauge bosons and massless fermions
In this section we briefly explain how to find the amplitudes involving fermions directly from the corresponding amplitude involving only gauge bosons. From the point of view of vertex operators, in the RNS formalism, this is a non trivial thing to do since it is known that there arise difficulties when calculating amplitudes of n gauge bosons and 2m massless fermions in the case of m ≥ 3 46 (where n = 1, 2, . . .) [33]. 46 The difficulties have to do with the fact that the total super ghost charge of the involved vertex operators must be −2 and fermion vertex operators are generally known in the +1/2 and −1/2 picture (therefore, four fermions gives 4 × (−1/2) = −2 and this case is fine). We thank E. Hatefi for clarifying this issue to us. In spite of this difficulty in ref. [51] the pure fermion 6-point amplitude was obtained using the RNS formalism.
The shortcut to find the amplitudes involving (any number of even) fermions is that we do not need to compute them from the beginning, by considering correlation functions of vertex operators: the closed formula for the N -point amplitude of gauge bosons and D=10 Supersymmetry are enough ingredients to find the amplitudes involving fermions. We already saw this in section 2.3 of ref. [52], in the case of the open superstring 5-point amplitude, where the ansatz that we proposed for the fermion amplitudes consisted in simply changing the original gauge boson subamplitudes of the 2-dimensional basis (A Y M (1, 2, 3, 4, 5) and A F 4 (1, 2, 3, 4, 5), in the case of ref. [52]) by the corresponding expression of their superpartner subamplitudes. In the N -point case we do exactly the same thing: we change the gauge boson subamplitudes of the basis (in this case given by the Yang-Mills subamplitudes in (3.1)) by their superpartner subamplitudes. The resulting formula is precisely the one obtained by Mafra, Schlotterer and Stieberger [20] (see eq. (1.1)). Besides the expression in (1.1) there is no other possibility for the scattering amplitudes involving fermions since, as analyzed in [48], the global supersymmetry requirement is sufficient to determine the structure of the boson/fermion interactions uniquely 47 . In formula (1.1) this means that the summed variation of all possible boson/fermion subamplitudes, under the supersymmetry transformations , is zero, after using the on-shell and the physical state conditions, together with momentum conservation 48 .

Finding the α expansion of the momentum factors
It was mentioned in ref. [36] that the factorization properties of open superstring subamplitudes could be used as a complementary approach to determine the α expansion of the momentum factors F {σ N } (α ) 49 . We agree completely with this observation and, indeed, we will use these sort of factorization properties in this section as part of the tools to find the α expansion of the momentum factors (see eqs.(5.4) and (5.9)). Our remark at this point is that there is a more general form for the factorization (or tree-level unitarity) relations than the ones considered in ref. [36], which includes as a particular case the collinear limit considered there (see eq.(5.9)). This form, together with the cyclic property of the subamplitude, allows us to obtain all the α terms of the momentum factors, up to quite high α order, by just using the very well known 4-point (Veneziano-type) momentum factor. This is a very non trivial result that will allow us to bypass α expansions of worldsheet integrals for N = 5, N = 6 and N = 7, at least up to α 6 order in the first two cases and α 4 order in the last one 50 , in the same spirit that the revisited S-matrix method succeeds in finding the corresponding α terms of the OSLEEL which would commonly be determined by open superstring N -point amplitudes (where N > 4) [10]. 47 This analysis can be found, for example, in sections 5.3.1 and 7.4.1 of ref. [48]. We thank N. Berkovits for suggesting part of the specific sections of this book to us. 48 The formulas in (4.28) are nothing else than the momentum space version of the supersymmetric transformations of the fields A a µ and ψ a of D=10 Super Yang-Mills theory. 49 See, for example, Appendix C of this reference. 50 Our limitation to go to higher orders in α , by just considering the expansion of the 4-point Gamma factor, is just a computational one since the expressions become extremely huge as the α order grows. As we explained in subsection 2.2, we believe that using using the 4-point amplitude α expansion, we are able to obtain completely the α expansion of any N -point momentum factor up to α 7 order.

Tree level unitarity of the amplitudes
In the case of gauge bosons, the tree level unitarity relations state that the N -point subamplitude obeys 51 [49] where A µ and A µ are, respectively, m and (N + 2 − m)-point subamplitudes and where the pole that A(ζ 1 , k 1 ; . . . ; ζ N , k N ) has in the Mandelstam variable (k 1 + k 2 + · · · + k m−1 ) 2 is being taken to zero: All A subamplitudes in (5.1) include the 'i δ' factor that takes into account momentum conservation (see eq.(1.2)) and D (= 10) is the spacetime dimension.
The remaining unitarity relations arise when the other Mandelstam variables, are individually taken to zero. There are N (N − 3)/2 independent Mandelstam variables and that is the total number of independent unitarity relations. Substituing the explicit dependence of each A subamplitude in the 'i δ' factor in (5.1) and calculating the D-dimensional integral leads to where, now, the A, A µ and A µ subamplitudes no longer include the δ dependence and the momentum conservation on each of them must be implicitly assumed. These subamplitudes are the ones that we have been dealing with throughout this work. A µ and A µ are subamplitudes in which the corresponding polarization vector has been taken away: and where the assymptotic mass-shell condition (5.2) is being taken into account. Formula (5.4) states that the residue that the N -point subamplitude has in (k 1 +k 2 +· · ·+k m−1 ) 2 = 0 is given by the product of two lower-point subamplitudes (an m-point and a (N + 2 − m)-point one). It is valid at any α order. In particular, if α → 0 it means that Yang-Mills subamplitudes also satisfy it. When eq.(5.4) is considered for the particular case of m = 3, it becomes is the (contracted) Yang-Mills vertex. For m = 3, condition (5.2) implies that (k 1 + k 2 ) 2 = 2k 1 · k 2 goes to zero: If k 1 and k 2 are non zero light-like vectors, then condition (5.11) implies that these momenta are parallel (k 1 ||k 2 ) and, therefore, the unitarity relation (5.9) is nothing else than the collinear version of the factorization property of gauge boson subamplitudes [53].
In the case of arbitrary m (= 3, . . . , quently, all Mandelstam variables should tend to zero (because if the momenta tend to be parallel then k i · k j → 0). This is what happens for the Mandelstam variables in the physical region. So, the word of caution when considering eq.(5.4) subject to condition (5.2), is that we are considering there the subamplitude expressions where the Mandelstam variables have been analytically continued in the complex plane. Then, the Mandelstam variables are generally out of the physical region and, therefore, condition (5.2) does not have any implication for any of the other Mandelstam variables: they remain being independent in spite of one of them (namely, (k 1 + k 2 + · · · + k m−1 ) 2 ) tending to zero. This fact will be implicit in the calculations that we will do in subsections 5.3, 5.4 and 5.5. In these subsections we will use formula (5.9) and its generalization, eq.(5.4), to find α terms of the N -point amplitude for N = 5, 6, 7, by just using the well known open superstring 4-point α expansion (see eqs.(F.2) and (F.3)).

Analyticity of the momentum factors
Before going to the details of the α calculations in the N -point amplitudes (where N = 5, 6, 7) an important remark proceeds. From the analysis that we did in section 4 we have that the N -point subamplitude of gauge bosons in Open Superstring Theory is given by where the F {σ N } (α )'s are given by the integral expression (4.2) (or equivalently, if N ≥ 5, by eq.(4.3)).
In any of the two expressions of the momentum factors we see that the α dependence of them is completely enclosed in the dimensionless Mandelstam variables, α s i and α t j . So, on one side, if these variables are analytically continued in the complex plane, then the F {σ N } (α )'s will have a well defined Laurent expansion with respect to α s i = α t j = 0. On the other side, it is well known that the low energy limit of the gauge boson (tree level) subamplitudes in Open Superstring Theory gives the corresponding Yang-Mills subamplitudes: In eq.(5.12) this last condition implies that The analytic behaviour of the F {σ N } (α )'s, when α → 0, will be an important fact which is behind the α expansions which we will obtain in the next three subsections.

Case of the 5-point momentum factors
In the N = 5 case, we will give the details of how using tree level unitarity of the subamplitudes we can arrive at the α 2 order terms of F {23} (α ) and F {32} (α ). The higher order terms of these two momentum factors will be afterwards written as a consequence of repeating the procedure for the corresponding α order. It will turn out to be convenient to introduce here the notation where k i and k j are, respectively, the i-th and the j-th external leg momentum (and i < j). This notation will also be useful in the case of N = 6 (subsection 5.4) and the case of N = 7 (subsection 5.5). Now, in order to find the α expansion of F {23} (α ) and F {32} (α ) within our approach, it will be convenient to write, once again, eq.(4.14) in the present subsection: We begin by looking for the poles of all three subamplitudes involved in eq.(5.16). In figure 1 we have drawn the two type of Feynman diagrams that contribute to the Yang-Mills 5-point amplitude. Examining all possible diagrams of this type we can infer that A Y M (1, 2, 3, 4, 5) and A Y M (1, 3, 2, 4, 5) have first and second order poles in the α ij variables (one and two propagators, respectively) that we list in table (5.17). first order second order On the other hand, in the left side of equation (5.16), the 5-point superstring subamplitude, A b (1, 2, 3, 4, 5), presents only first order poles in the same variables that A Y M (1, 2, 3, 4, 5) does, namely, {α 12 , α 23 , α 34 , α 45 , α 15 } 54 . We will refer to these poles as the physical poles. For reasons that will become clear in the next lines, it will be convenient to write F {23} (α ) and F {32} (α ) in terms of the simple poles that A Y M (1, 2, 3, 4, 5) and A Y M (1, 3, 2, 4, 5) have, respectively: Now, since the momentum factors have a well defined α power series (as it was seen in the previous subsection), we may write that 55 We will find the value of all the a i 's, b j 's, c k 's and d l 's that participate in these two expansions. We will do this by demanding three requirements: 1. Absence of unphysical poles. 54 Since A b (1, 2, 3, 4, 5) agrees with A Y M (1, 2, 3, 4, 5) when α → 0, and A Y M (1, 2, 3, 4, 5) does have second order poles, the statement that A b (1, 2, 3, 4, 5) has only first order poles is only valid perturbatively, for any non zero order in α . 55 In eqs.(5.20) and (5.21) we are already using the fact the F {σ 5 } (α )'s satisfy low energy requirement in eq.(5.14).
Step 1: Absence of unphysical poles in A b (1, 2, 3, 4, 5) According to eq.(5.16), the only unphysical poles that could arise in A b (1, 2, 3, 4, 5) are the ones that come from A Y M (1, 3, 2, 4, 5), namely, α 13 and α 24 . These poles appear always as simple poles in A Y M (1, 3, 2, 4, 5), in spite of they participating, also, in the second order poles of this subamplitude (see table (5.17)). The only possible way to cancel, in the right hand-side of eq.(5.16), the independent simple poles that A Y M (1, 3, 2, 4, 5) has at α 13 = 0 and α 24 = 0, is demanding that F {32} (α ) should be possible to be written with a common α 13 α 24 factor: Step 2: Unitarity relation for A b (1, 2, 3, 4, 5) In Appendix F.3.1 we prove that demanding unitarity of A b (1, 2, 3, 4, 5) with respect to its α 12 pole implies that the 5-point momentum factor, F {23} (α ), is related to the 4-point momentum factor, F {2} (α ), by the following relation: where we have introduced the notation Notice that the two Mandelstam variables in which F {2} is being evaluated in (5.25) are not the same two ones that are implicit in the notation (5.26) 56 . Relation (5.25) is an extremely strong constraint since in the left hand-side F {23} is being evaluated at arguments which do not appear in the right hand-side of this relation (namely, α 23 and α 15 ).
The corresponding full expressions up to α 6 terms are given in the text files that we have submitted, attached to this work, to the hep-th arXiv preprint basis. We have confirmed that our results are in perfect agreement with the ones found previously in [28,29,54].

Case of the 6-point momentum factors
Besides the notation introduced in eq.(5.15), here it will be convenient to introduce the notation where i < j < k.
The notation in eqs.(5.15) and (5.37) will also be valid for the N = 7 case (subsection 5.5). So, we will repeat the procedure that we did in subsection 5.3 for N = 5. For a general N the procedure consists in the following four stages: i) Identify the poles of each of the (N − 3)! Yang-Mills subamplitudes, up to the desired α order, in terms of unknown coefficients. iv) Determine the coefficients of the previous α expansions by following the three steps specified in subsection 5.3, namely, demand absence of unphysical poles, tree level unitarity and cyclic invariance.
Let us do this procedure for the case of N = 6.

Stage i):
In this case there are first, second and third order poles, as we can see in figure 2. Stage ii): So, we will work with the six momentum factors as functions of nine independent Mandelstam variables, according to  Stage iii): The general form of the α power series for the momentum factors is the following: n1,n2,n3,n4, n5,n6,n7,n8,n9=0 In all expansions, from (5.45) to (5.50), the prime on the internal sum means that the set of indices {n 1 , n 2 , n 3 , n 4 , n 5 , n 6 , n 7 , n 8 , n 9 } obeys the constraint n 1 + n 2 + n 3 + n 4 + n 5 + n 6 + n 7 + n 8 + n 9 = N . (5.51) In these expansions we have already used the condition of low energy behaviour, eq.(5.14), for the momentum factors: that is why the α series of F {234} (α ) begins with '1' and the five others begin with O(α 1 ) order contributions 59 .
Stage iv): Here we apply the three steps that will allow us to obtain the coefficients of the α expansions in eqs.(5.45)-(5.50). We will do our calculations up to α 6 order terms.
Demanding absence of unphysical poles in A b (1, 2, 3, 4, 5, 6) implies that there should be cancellations between the unphysical simple poles which come in the five Yang-Mills subamplitudes which are different of A Y M (1, 2, 3, 4, 5, 6) (which has only physical poles). After this procedure has been done, the number of independent coefficients reduces considerably according to table (5.55).
There is also the subtlety that F {234} (α ) and the remaning five momentum factors should be such that they cancel the third order poles that come in A Y M (1, 2, 3, 4, 5, 6). These poles are not present in A b (1, 2, 3, 4, 5, 6) (at non zero α order).
In the case of the α 1 order terms, the requirement of absence of unphysical poles is strong enough to forbid them: the coefficients of those terms are all zero and that is the reason of why the first line in table (5.55) contains only 0's (zeroes). We will not write down here the partial result that we have obtained for the expansions in eqs.(5.45)-(5.50), as we did at the end of Step 1 in subsection 5.3. Instead, for each of the six F {σ6} (α )'s we have presented two data at a given order in α , in table (5.55). The first data is the number of undetermined coefficients after the actual Step (Step 1) and the second data is the number of undetermined coefficients after Step 2. We see there, that before demanding cyclic symmetry (in Step 3) there is still a big number of undetermined coefficients.
Step 2: Unitarity relations for A b (1, 2, 3, 4, 5, 6) Since in the N = 6 case there are two type of simple poles (α ij and β ijk , see table (5.38)), demanding unitarity of A b (1, 2, 3, 4, 5, 6) with respect to each of them will lead to independent unitarity relations. The remaining unitarity relations will be a consequence of the previous ones once cyclic 59 Once the unitarity and the cyclicity relations have been demanded it will naturally happen that the order '1' coefficients of all expansions will be zero, but as a matter of principles here we are just proposing their general α expansions in such a way that they respect the low energy requirement in (5.14) and the analyticity of the momentum factors. 60 In this case, the geral formula for the number of coefficients at α N order is (N + 8)!/(N ! 8!).  For the remaining momentum factors, the number of undetermined coefficients has not changed from Step 1 to Step 2. After demanding cyclic invariance the coefficients of all six momentum factors will be related and it will be possible to find them all, at least up to α 6 order.
Order Step 3: Cyclic invariance of A b (1, 2, 3, 4, 5, 6) In Appendix F.4.2 we prove that demanding cyclic invariance for A b (1, 2, 3, 4, 5, 6), in the closed formula that we have for it, implies the following six relations for the momentum factors: The corresponding full expressions up to α 6 terms are given in the text files that we have submitted, attached to this work, to the hep-th arXiv preprint basis.
We have confirmed that our results are in perfect agreement with the ones found previously in [36,54].

Case of the 7-point momentum factors
Doing the procedure for the N = 7 case is straight forward (although very tedious) once it has been done for N = 5 and N = 6. We will only mention here a few details.  A computational detail that coould be of interest to the reader is that finding the coefficients of the α expansions, along the same lines presented in subsection 5.4 (in four stages, in particular, demanding cyclic symmetry after unitarity has been demanded in one physical pole), becomes a heavy task if compared with solving the coefficients from purely demanding unitarity in all poles. These two methods happen to agree at low orders in α (but not necessarily at high orders 63 ). The advantage of the second method lies in the fact that each of its equations contain a few momentum factors (containing, therefore, a low number of unknowns) while in the first method each of the equations contains many more momentum factors. In Appendix F.3.3 we have written the momentum factor relations that arise when unitarity of the amplitudes is demanded with respect to their α 12 and their β 123 poles. Our complete result is that, when any α ij → 0 there arise six relations and when any β ijk → 0 there arise only two relations.  63 We have verified this computationally. 64 As mentioned before, the reason for not going to higher orders is a purely computational one: we have had memory difficulties to deal with N = 7 α 5 calculations on the computers . In spite of this complication, as we mentioned in subsection 2.2, we believe that only using the 4-point amplitude information our method is, in principle, able to acheive α 7 order calculations, for any number of legs. 65 There are specific cases of momentum factors which, as remarked in [36], have an α expansion which begins at α 4 order. For these cases we have written the α 4 terms.
The corresponding expressions of the α 4 terms for the remaining terms are included in the text files that we have submitted, attached to this work, to the hep-th arXiv preprint basis. We have confirmed that our results are in perfect agreement with the ones found previously in [54].

Summary and conclusions
We have successfully derived a closed formula for the tree level N -point amplitude of open massless superstrings (for 3 ≤ N ≤ 7). This is the same formula first found by Mafra, Schlotterer and Stieberger in [20], using the Pure Spinor formalism [31]. Our approach has consisted in working only within the RNS formalism, so spacetime supersymmetry has been present throughout our approach, but in a non manifestly manner. First, it has been present implicitly in the computation of the N -point gauge boson amplitude in a closed form (see eq.(5.12)) because, in order to arrive to it, the condition of absence of (ζ · k) N terms 66 has been used. And second, it has been used to find uniquely the amplitudes involving fermions, once the N -point formula for gauge bosons, eq.(5.12), has been found. We believe that a deeper understanding of our procedure can eventually arrive to the MSS formula in eq.(1.1), for arbitrary N . The kinematic analysis that we have required to arrive to MSS formula, naturally leads us to a space of N -point gauge boson subamplitudes which is (N − 3)!-dimensional (at least for 3 ≤ N ≤ 7) 67 . At this point is where the basis of Yang-Mills subamplitudes first proposed by Bern, Carrasco and Johansson [38], plays an important role in our procedure. Once this basis has been identified as a basis for V N , then the MSS formula and the explicit BCJ relations themselves become linear algebra problems in which the components of a certain vector, with respect to a given basis, are desired to be found. We have done these calculations in section 4 and Appendix E, respectively. Following the same spirit of the revisited S-matrix approach [10], we have found α correction terms to the open superstring N -point amplitudes (where N = 5, 6, 7) by only using the well known 4-point amplitude α expansion (see eqs.(F.2) and (F.3)) and demanding cyclic symmetry and tree level unitarity for the scattering amplitudes. We have done these calculations to at most α 6 order. It is quite remarkable that we have not needed to compute, for the calculations that we have proposed to do in this work, any coefficient as a numerical series or any integral involving polylogarithms.
66 This is the main observation of our revisited S-matrix method [10]. 67 In the main body of this work we have referred to this space by V N . See section 3 for more details.
We expect that, within our approach, only for α 8 order onwards 68 the α expansion of the 5-point amplitude will be required to obtain some of the coefficients of the series 69 . In the following table we have summarized the existing parallel of the revisited S-matrix method in the determination of the OSLEEL and the corresponding scattering amplitude α calculations.
Step of the revisited Open superstring Open superstring S-matrix method low energy effective lagrangian scattering amplitude at α p order at α p order at α p order Step I : Requirement of absence Finds that the N -point of (ζ · k) N terms in the Reduces the general basis amplitude can be written N -point amplitude to a constrained basis of terms in terms of the B N basis (N = 4, . . . , p + 2) (see eq.(3.1)) Step II : Use of n-point amplitudes information (n p + 2) Determines all the coefficients Determines all the (presumably, only n = 4 of the constrained basis momentum factors and n = 5) (6.1) In a forthcoming work we will use the revisited S-matrix method to compute the α 5 order terms of the OSLEEL (in analogy to the calculations that we did in [10]) and also to compute those terms in the N = 7 scattering amplitude [55].

Acknowledgements
We would like to thank N. Berkovits, R. Boels, E. Hatefi and S. Stieberger for useful e-mail correspondence at different stages in the development of this work. This work has been partially supported by the brazilian agency FAPEMIG (research projects PPM-00617-11 and APQ-00140-12).

A. Conventions
We use the following convention for the Minkowski metric: The field strength is defined by With the remarkable exception of the α 9 order terms, as mentioned in subsection 2.2. 69 These coefficients would be the ones which have dependence in the non trivial MZV's that we referred to in subsection 2.2

B. Known relations for Yang-Mills subamplitudes before BCJ
Before the BCJ relations were found, based on the conjucture between color and kinematics [38], other relations were very well known for the Yang-Mills (tree level) subamplitudes [53]: 1. On-shell gauge invariance (item 3 in eq. (3.5)).

Reflection:
A

C. Calculations that lead to one 4-point kinematical expression
In this appendix we give the details of the results mentioned in subsection 3.2.2, namely, we give the details of how starting with the general expression in (3.19) for T (1, 2, 3, 4) we arrive to (3.23), after demanding on-shell gauge invariance. On-shell gauge invariance of T (1, 2, 3, 4) means that it becomes zero (after demanding momentum conservation, the physical state condition (3.4) and the mass-shell condition (3.3)) whenever any of the polarizations vectors ζ i becomes k i (i = 1, 2, 3, 4). Let us see this: In this case, using that k 1 · k 2 = s/2 and k 1 · k 3 = −(s + t)/2, where s and t are the Mandelstam variables defined in eq.(3.21), we can arrive to (C.1) In (C.1) the fisrt square bracket contains seven (ζ · k) 3 terms while the second square bracket contains six (ζ · ζ) 1 (ζ · k) 1 ones. Since the complete expression in (C.1) should be zero and all its kinematic terms are linearly independent then the coefficient of each of them should be zero. This leads precisely to the linear system of equations of eq. (3.20) in the main text.

2.
Case of ζ 2 → k 2 : In this case, using that k 1 · k 2 = s/2 and k 2 · k 3 = t/2, we can arrive to In a similar way to what was argued in the previous case, the expression in (C.2) should be zero and this leads to the following set of thirteen equations (which happens to be linearly independent): 3. Case of ζ 3 → k 3 : In this case, besides using the expressions for products of momenta in terms of Mandelstam variables (k 1 · k 3 = −(s + t)/2 and k 2 · k 3 = t/2), we need to use the condition which comes from the gauge (or the physical state) condition in (3.4) and momentum conservation. Doing these substitutions, after ζ 3 → k 3 , we can arrive to This time demanding the previous expression to vanish leads to fourteen equations, but clearly, two of them are identical (−λ 21 = 0), so we just have a linear system of thirteen equations (which, once more, happens to be linearly independent) and which is equivalent to the following one: Similarly to the previous case, besides using the expressions for products of momenta in terms of Mandelstam variables (k 1 · k 4 = t/2 and k 2 · k 4 = −(s + t)/2), we need to use the conditions which come from the gauge (or the physical state) condition in (3.4) and momentum conservation.
Doing these substitutions, after ζ 4 → k 4 , we can arrive to As it happended in the case of ζ 3 → k 3 , demanding the previous expression to vanish leads to fourteen equations, where only thirteen of them are linearly independent. The set of equations is equivalent to the following thirteen ones: where λ 24 is completely arbitrary. In this way, substituing the final solution found in eq. (C.10), in the original expression for T (1, 2, 3, 4), given in eq. (3.19), this subamplitude becomes a kinematical expression which contains as a global factor λ 24 . Due to the arbitrariness of λ 24 , for convinience we may rewrite it as Afterwards, using appropriately the relations in (C.7) plus the relations we can finally arrive to the symmetric formula in (3.22). In (3.22) the expression inside the curly brackets is precisely the well known 4-point kinematic factor [34].

D. Finding an N -point basis for the scattering amplitude (N = 5, 6, 7)
In this appendix we give the details of the calculations that support the main claim of section 3, namely, that the B N set of subamplitudes, given in eq.
In eqs.(D.1), (D.2) and (D.3) it is implicit the identification of indexes (6 ↔ 1), (7 ↔ 1) and (8 ↔ 1, 9 ↔ 2), respectively. For example, according to this identification of indexes, in (D.3) we have that t 7 = (k 7 + k 8 + k 9 ) 2 = (k 7 + k 1 + k 2 ) 2 = 2 (k 7 · k 1 + k 7 · k 2 + k 1 · k 2 ) . The remaining scalar products of the momenta that appear in the corresponding N -point scattering process of massless particles can all be written in terms of the corresponding Mandelstam variables, in the following way 74 : As mentioned at the end of subsection 3.3, our proposal is that it will be enough to consider the (ζ · ζ) 1 (ζ · k) N −2 terms of the Yang-Mills N -point subamplitude, in order to find the momentum factors that appear in the BCJ relations (and also the momentum factors that appear in the closed expression of the open superstring scattering subamplitudes). So, in the next subsection of this appendix we will give the explicit expression of those terms in the case of N = 5. We have obtained it by calculating the 5-point subamplitude in Yang-Mills theory (in the Lorentz gauge) by using Feynman rules. We have also written the (ζ · k) terms of them in the basis mentioned in eq. (3.11).
For the Mandelstam variables we are using the conventions of eq. (D.1). In the case of N = 6 and N = 7 we have written in subsection D.2.2 only the general structure of the (ζ · ζ) 1 (ζ · k) 4 and the (ζ · ζ) 1 (ζ · k) 5  74 This comes from demanding momentum conservation and the mass-shell condition for the external massless states. 75 We have specified on these expressions the number of terms that they have. See equations (D.8) and (D.9).

D.2.1 Case of N = 5
D.2.2 Case of N = 6 and N = 7 The (ζ · ζ) 1 (ζ · k) 4 and the (ζ · ζ) 1 (ζ · k) 5 terms of the 6 and the 7-point Yang-Mills subampltiudes have the following structure 76 : In the 6-point case, in eq. (D.8), there is no mistake in the fact that the (ζ 1 · ζ 4 ) term does not carry a factor '2', as every other terms does: this factor will arise when summing the cyclic permutations. and A Y M (1, 2, 3, 4, 5, 6, 7) The color/kinematic duality in Yang-Mills theory [38] and the monodromy relations found for the open string subamplitudes [39,40] succeed in finding all N -point subamplitudes of Yang-Mills theory in terms of a linear combinations of a set of (N − 3)! subamplitudes (which can be chosen to be the ones in the B N set of eq.(3.1)) 77 . The procedure of each of these approaches is self consistent in the sense that it is found a linear system for the subamplitudes which is overdetermined; then, it is found that this linear system has a unique solution (which consists, precisely, in the BCJ relations for the N -point subamplitudes). But this result still does not guarantee that this set B N is indeed a basis of V N : nothing forbids that there could still be additional restrictions that relate the Yang-Mills subamplitudes in B N . In fact, it is known that in D=4, the N = 5 BCJ relations can all be found in terms of only one Yang-Mills subamplitude, instead of two (see section 5.2 of ref. [47]). This does not invalidate the BCJ relations that have been found for N = 5: it simply states that, at least in D=4, the set B 5 is not linearly independent and the BCJ relations can have an even simpler form.
In the way that the BCJ relations have been found [38,39,40], their validity is for any spacetime dimension D. So, what may happen (and the mentioned result of ref. [47] is an explicit evidence for it) is that if N is sufficiently high (as compared to D), there may still be further relations among the Yang-Mills subamplitudes of the B N set in eq.(3.1). The same as in [38] and [39,40], our approach throughout this work has only considered the general case, that is, the one in which the BCJ relations are written being valid for any spacetime dimension D (or equivalently, we will assume that N is sufficiently small, as compared to D, such that no new extra relations between the subamplitudes arise).
Having stated this subtlety about the dependence of the basis of V N in the spacetime dimension D, we will proceed with our analysis only using arguments that are valid for any spactime dimension. In this section of Appendix D we will see that in fact, as mentioned at the end of subsection 3.3, considering only the (ζ · ζ) 1 (ζ · k) N −2 terms of the Yang-Mills N -point subamplitudes, there is enough information (which is even redundant when we consider all those terms) to prove the linear independence of the set of subamplitudes B N , given in eq. (3.1).

D.3.1 Case of N = 5
In order to prove that B 5 = {A Y M (1, 2, 3, 4, 5), A Y M (1, 3, 2, 4, 5)} is a linearly independent set in 77 In the case of the color/kinematic duality approach, the authors of [38] have conjectured the explicit form of BCJ relations for arbitrary N , and they have found evidence for their conjecture up to N = 8, but the proof for arbitrary N has not been given yet. Considering only the (ζ · ζ) 1 (ζ · k) 3 terms in (D.10), we have that and using the explicit expression for the (ζ · ζ) 1 (ζ · k) 3 terms of the 5-point subamplitude, given in (D.7), we have Since the (ζ · ζ) 1 (ζ · k) 3 terms in (D.12) are linearly independent, then this equation is valid if and only if the coefficient of each of these terms is zero: and λ 2 . From the first two equations, explicitly written in (D.13), it is easy to see that the unique solution of this system is the trivial one: (D.14) D.3.2 Case of N = 6 and N = 7 Doing exactly the same procedure of the previous subsection, but for N = 6 and N = 7, using the corresponding expressions of the (ζ · ζ) 1 (ζ · k) 4 and the (ζ · ζ) 1 (ζ · k) 5 terms, mentioned in eqs.(D.8) and (D.9), respectively, we arrive to the same conclusion, namely, that the six λ {σ6} 's and the twentyfour λ {σ7} 's are all zero, implying that the B 6 and B 7 sets are linearly independent in their corresponding spaces, V 6 and V 7 . An interesting remark is that we have not needed to examine all the (ζ · ζ) 1 (ζ · k) N −2 terms of the Yang-Mills N -point subamplitudes (for N = 5, 6, 7), in order to arrive at the conclusion that the λ {σ N } 's are all zero : it has been enough to just analyze the (ζ 1 · ζ 2 )(ζ · k) N −2 kinematical structures (which are just one of the many possible structures of the (ζ · ζ) 1 (ζ · k) N −2 terms).

D.4 Dimension of V N and a basis for it
In the case of N = 3 and N = 4, in subsections 3.2.1 and 3.2.2, respectively, it was checked that the dimension of V N is, in fact, and that the set of subamplitudes B N , given in eq.(3.1), is indeed a basis of this space. In this subsection we will argue that the validity of the formula (D.15) and of the set B N proposed as a basis for V N , given in eq.(3.1), indeed holds for N = 5, 6, 7. We will argue this in subsection D.4.1 for the N = 5 case, with some detail, and in subsection D.4.2 for the N = 6 and the N = 7 case. In these last two cases we will just mention the final results, because the intermediate formulas are simply too big and they do not give any additional idea. Before going into the details of sections D.4.1 and D.4.2 we warn the reader that our calculations have not lead directly to the set B N as a basis for V N (as explicited in eq. (3.1)): what we have found for each N (=5,6,7) is another set of (N − 3)! independent kinematical expressions 78 which become a basis for the corresponding V N space. Within our approach, it would simple be too tricky to arrive, for any N ≥ 5, directly to the set B N . So, at this point we must admit that, without knowing the BCJ result for the basis of Yang-Mills subamplitudes, given in eq.(3.1), demanding on-shell gauge invariance would have lead us to the validity of eq.(D.15), but we would have hardly arrived to a basis in which all the elements can be directly associated to Yang-Mills theory.
have the following form 79 : In ( Since in subsection D.3.1 we proved that the two-element set B 5 is linearly independent in V 5 , it is clear that a change of basis can be done, namely, Let T (1, 2, 3, 4, 5, 6) ε V 6 and T (1, 2, 3, 4, 5, 6, 7) ε V 7 . Although the computational effort becomes greater as N grows, we have succeeded in finding that, repeating exactly the same procedure that we have described in subsection 3.2.2 and in Appendix C for the N = 4 case, and in subsection D.4.1 for the N = 5 case, we arrive that these two elements can be respectively written as T (1, 2, 3, 4, 5, 6) = ρ 1 K 4 (ζ, k) + ρ 1 K 1 (ζ, k) + ρ 2 K 2 (ζ, k) + ρ 3 K 3 (ζ, k) + ρ 4 K 4 (ζ, k) + ρ 6 (ζ, k) + ρ (7) 9 K (7) 9 (ζ, k) + ρ 10 K 10 (ζ, k) + ρ 23 (ζ, k) + ρ   where the K (N ) i (ζ, k)'s are known kinematic expressions (without poles). In the case of N = 6 we have found our result in (D.22) analytically, in the same way as we did in (D.17). In the case of N = 7 we have obtained our result numerically: we have found the explicit form of the K in agreement with formula (D.15). Since in subsection D.3.2 we proved that the six-element set B 6 and the twenty four-element set B 7 are linearly independent in V 6 and V 7 , respectively, due to the matching with the dimensions found in (D.24), these sets can be chosen as basis of the corresponding spaces.

E. Kinematical derivation of BCJ relations
BCJ relations were discovered by Bern, Carrasco and Johansson in 2008 by means of a conjectured duality between color and kinematics in Yang-Mills theories [38]. These relations state that the (N − 1)! Yang-Mills subamplitudes that appear in the N -point formula (see eq.(1.2) 80 ) can be written as linear combinations of a subset which contains only (N − 3)! of them. Although not exactly the same, from the work in [38] it can be easily seen that the set of subamplitudes that we propose in eq. (3.1) can be chosen as a basis for the Yang-Mills subamplitudes in the BCJ relations 81 . A rigorous proof of the BCJ relations was found first using String Theory methods, independently, by the authors of [39] and [40]. Later, a pure field theory proof of these relations was also found in [56] (at least for N = 4, 5, 6). In this appendix we present an alternative derivation of BCJ relations for N = 4, 5, 6, 7. We will not deal with color/kinematic duality of Yang-Mills subamplitudes at any moment. We will not need to consider Kleiss-Kuijf relations [57] either. From our perspective the result is that BCJ relations arise as a natural consequence of the basis found for V N , considered in section 3 of this work. The only unknowns in the BCJ relations will then be the momentum factors that participate in them as the components of a given Yang-Mills subamplitude with respect to the B N basis. As mentioned at the end of subsection 3.3, our proposal is that it is enough to consider the explicit expression of the (ζ ·ζ) 1 (ζ ·k) N −2 terms of the N -point Yang-Mills subamplitudes, in order to obtain the momentum factors that appear in the BCJ relations. These subamplitudes can be obtained using Feynman rules from the Yang-Mills lagrangian (in the Lorentz gauge) or by selecting them in the low energy limit of the corresponding subamplitude of gauge bosons in String Theory. We will find the BCJ relations for a set of (N Here, we will give the details of our derivation of the BCJ relations for the case of N = 4 and N = 5. The procedure for N = 6 and N = 7 is exactly the same one, but the intermediate formulas become too big, so we will just present some examples of these relations. 80 Formula (1.2) was written in the Introduction as valid for gauge bosons in tree level Open Superstring Theory, but it is well knwon that it is also valid for tree level Yang-Mills N -point amplitudes [53]. 81 The reason of our choice of basis is, of course, that we want to arrive to the MSS formula in eq.(1.1), which is written in this basis.

(E.2)
Next, we write all (ζ · k) terms in both sides of (E.2) in the basis mentioned in (3.11), by using the relations (3.8) for this case, namely, Once both sides of (E.2) have been written in terms of the same basis for the (ζ · ζ) 1 (ζ · k) 2 terms, it is possible to compare their coefficients, arriving at a set of eight linearly dependent equations for λ {2} . After using that u = −s − t it can be seen that there is really only one linearly independent equation, namely, λ {2} = t/u, from which it comes that (1, 2, 3, 4) .
Doing exactly the same procedure to find A Y M (1, 4, 2, 3) in terms of A Y M (1, 2, 3, 4) we find that (1, 2, 3, 4) . these (ζ · ζ) 1 (ζ · k) 3 terms in (E.9) we arrive to an equation which has the following form 84 : ( cyclic permutations of indexes (1,2,3,4,5) ) + ( cyclic permutations of indexes (1,3,2,4,5) ) . (E.10) In this equation we have preferred to present explicitly only some of the (ζ · ζ) 1 (ζ · k) 3 terms in order to save some space on it 85 . The peculiar factor (s 4 − s 1 − s 2 ) in some of the denominators of eq. (E.10) simply corresponds to k 1 · k 3 , as given in eq. (D.1). Writing all (ζ · k) terms in both sides of (E.10) in the basis mentioned in (3.11), once again, by appropriately using the relations ( Doing this same procedure we arrive to the following additional relations: (1,3,2,4,5) , (E.12) In this case there are (N −1)!/2 = 60 subamplitudes which are not related by cyclic neither reflection symmetry. The outcome of our procedure is that we have succeeded in finding an expression for 54 of them in terms of the six ones of B 6 , as predicted in eq.(3.25).
We have found that these 54 relations can be classified into two types. In the first type, 42 subamplitudes are in fact given as a linear combination of the six ones of B 6 , like for example 89 : A Y M (1, 3, 2, 4, 5, 6) + (1,4,3,2,5,6) .

(E.24)
Our confidence in the correctness of these (and the remaining 45) relations is the fact that the linear system that we have found for the momentum factors in each of them, following the procedure of subsection 3.3, is overdetermined, and we have found a unique an consistent solution for it.

F. Momentum factors in the open superstring formula
In this appendix we will consider the α expansion of the momentum factors that appear in the N -point open superstring formula. These are the F {σ N } (α )'s that appear in eq.(1.1).
In section F.1 we will consider the momentum factor that appears in the case of N = 4 and in section F.2 we will consider the case of N ≥ 5.

F.1 Gamma factor
As mentioned in [59], using the Taylor expansion for ln Γ(1 + z) 93 , it may be proved that the explicit α expansion for the Gamma factor is given by So, up to O(α 6 ) terms we have that . are called, respectively, the 'depth' and the 'weight' (or the transcendentality) of ζ(n 1 , . . . , n r ).
In the case of r = 1, (F.4) becomes a single zeta value, that is, Riemann's zeta function evaluated at an integer n ≥ 2, ζ(n). In this case the weight is w = n.
The following are a few examples of (F.4) for r = 2 and weights 3, 4 and 5, respectively: In order to compute these double zeta values and arrive to the expressions that we have written in these equations, in terms of single zeta ones, Harmonic sums and its algebraic property [25] can be used, for example. An extremely important data mine of proven results for MZV's is ref. [44]. Higher order coefficients of momentum factor α expansions in String Theory, for N ≥ 5, are based on the MZV's basis of this reference (see [16] and [29], for example). For w ≤ 7 it is always possible to write any MZV as a rational linear combination of products of ζ(n)'s, where n ≤ w (as in (F.6), (F.7) and (F.8), for example). But already at w = 8 there is a first MZV (which is present in the MZV basis of ref. [44]), namely ζ (3,5), which is believed to not admit such sort of expression [43]. Other MZV's, for higher weights, which are also believed that cannot be written as a rational linear combinations of products of ζ(n)'s, and which are present in the MZV basis of ref. [44], are ζ(3, 7), ζ(3, 3, 5), ζ (3,9) and ζ (1,1,4,6). Section 2 of ref. [16] contains tables in which these type of MZV's can be found up to w = 16. We recall that in section 2 of the main body of the present work we referred to these peculiar MZV's as "non trivial MZV's". Following [16], we wil first consider the case N = 5. From eq. (4.14) we can write that where eq.(F.9) is literally the same one written in (4.14) and eq.(F.10) has been obtained from the first one by interchanging indexes 2 and 3, thereforẽ (F.11) In matrix notation, eqs.(F.9) and (F.10) can be written as where A and A Y M are the 2-component vectors given by and where F is a 2 × 2 matrix given by In ref. [16] the authors propose that the α expansion of F can be decomposed in the following way 95 : 95 We will not explain here what is the prescription for the ordering colons,': ... :', in eq.(F.15). The interested reader can find this detail in eq.(3.18) of ref [16].
where the P 2n 's, Q and the M 2n+1 's are also 2 × 2 matrices 96 , given by  Here we give the details of the proof of the relation between the 5 and the 4-point momentum factors, F {23} (α ) and F {2} (α ), respectively (see eq.(5.25)). In the N = 5 case there are five independent Mandelstam variables (which are identified with the poles of the amplitude) and, therefore, there are five unitarity relations like the one in eq.(5.9). We will give the details of the unitarity relation of A b (1, 2, 3, 4, 5) with respect to its pole α 12 . According to eq.(5.9), when α 12 → 0, in this case we have that where the expression for V µ (12) is given in eq.(5.10). We have explicited the case of Yang-Mills subamplitudes in (F.21) because we will soon need this relation. It is understood that the limit α 12 → 0 also is being taken in the 4-point subamplitudes on the right hand-side of eqs.(F.20) and (F.21). The proof of (5.25) goes as follows. Using eq.(4.4), we may write the 4-point amplitude of the right hand-side of (F.20) as where, after using eqs.(D.1) and (D.4), we may prove that 2(k 1 + k 2 ) · k 3 = 2(k 4 · k 5 − k 1 · k 2 ), and since we are considering the limit α 12 → 0, we have that and in (F.22) we are left with (F.24) Substituing (F.24) in (F.20) we have that and identifying the term in the curly brackets (in the α 12 → 0 limit, which is being taken) as the Yang-Mills 5-point subamplitude (see eq.(F.21)) we finally arrive at A(ζ 1 , k 1 ; ζ 2 , k 2 ; . . . ; ζ 5 , k 5 ) ∼ F {2} [ α 45 , α 34 ; α ] A Y M (ζ 1 , k 1 ; ζ 2 , k 2 ; . . . ; ζ 5 , k 5 ) , (F. 26) or even using the simpler notation, In the N = 6 case there are nine independent Mandelstam variables (which are identified with the poles of the amplitude). Six of these variables (the α ij 's) have to do with the poles coming from two adjacent legs and the other three Mandelstam variables (the β ijk 's) have to do with the poles that come from three adjacent legs. Therefore, the first six unitarity relations are like the one in eq.(5.9) and the remaining three ones are like the one in eq.(5.4), with m = 4. Here we will give the details of the unitarity relations of A b (1, 2, 3, 4, 5, 6) with respect to its α 12 pole and of the one with respect to its β 123 pole.