The last of the seven-parton tree amplitudes

We compute the four-quark plus three-gluon and six-quark plus one-gluon tree level amplitudes using on-shell recursion relations. They are needed for the calculation of the 5-jet cross-section at the Born level, and constitute an essential ingredient for next-to-leading order 4-jet and next-to-next-to-leading order 3-jet production at hadronic colliders. Very compact expressions for all possible helicity configurations are provided, allowing direct implementation in computer codes. With the results presented in this paper, the full set of seven-parton tree amplitudes becomes available.


Introduction
With the forthcoming of the Large Hadronic Collider (LHC), the possibility to explore a wide range of new phenomena, like Higgs production, dark matter candidates, SUSY and new quark flavors will open. In order to study new physics, accurate calculations of cross-sections involving many partons are needed, for both signal and background. Such computation involves typically of the order of a thousand Feynman diagrams, so many that traditional methods become obsolete. During the 80's, some new ideas were introduced [1 -7] to allow a great simplification of the calculus, making possible the implementation of automatic computer algorithms (like MadEvent [8]) to perform this task. However, that still implies a huge computational effort, and the calculation of cross-sections for processes involving many partons remains as an unsolved problem.
In the last few years, the situation has drastically changed. Witten first noted the duality between tree level amplitudes and twistor string theory [9]. This lead to the introduction of the off-shell recursion relations by Cachazo, Svrček and Witten [10] (CSW), allowing to compute amplitudes for several partons by iterating the simplest MHV (Maximally Helicity Violating) ones [11]. Britto, Cachazo and Feng presented, as an ansatz,
At tree level, six-parton processes were studied long time ago [20,21], and many of the amplitudes have been recomputed using the recursion relations recently [22,23]. Seven-gluon amplitudes were first calculated in [24], while certain eight-gluon results can be found in [12,25]. The two-fermion plus five-gluon amplitudes were presented in our previous paper [26]. At the loop level, the recursion relations become more complicated. However, calculations have reached the level of one-loop six-gluon amplitudes, and also some results for certain helicity configurations with an arbitrary number of partons were presented [27 -38].
Recent work [39,40] also focused on the numerical implementation of the on-shell recursion relations. The result turns out to be that for a large number of partons, the "old" Berends-Giele [6] recursion relation performs better than the twistor inspired methods. Nevertheless, counting with analytical results for non trivial amplitudes would certainly speed up numerical calculations, reducing considerably the CPU-time required by any automatic algorithm. Therefore, obtaining very simple and compact results for the amplitudes is strongly desirable.
In this paper we present all four-quark plus three-gluon and six-quark plus one-gluon tree level helicity amplitudes, searching always for their most compact expressions. Our results have been checked against the factorization properties of the amplitudes, which provides a very strong consistency test. Combined with previous calculations [12,26], the full set of seven-parton tree level amplitudes is now available.
This article is organized as follows: in section 2 we review the color decomposition of tree level processes, and the main ingredients of the BCFW formulation, recalling its limitations when dealing with fermions. In sections 3 and 4 we introduce the main results for the complete set of seven-parton amplitudes with two and three quark-antiquark pairs, respectively, while in section 5 we present our conclusions. The necessary six-parton formulae for our calculation are collected in the appendix.

Color decomposition, helicity and BCFW
As is well known in the QCD literature, the color factor for the m qq pair plus n gluon amplitude is [1] . . , i m are the color indices of the quarks and α 1 , . . . , α m are the ones for the antiquarks. The indices n 1 , . . . , n m−1 (with 1 ≤ n i ≤ n) correspond to an arbitrary partition of an arbitrary permutation of the n gluon indices. Each external quark is connected by a fermionic line to an external antiquark. When we want to indicate that a quark with color JHEP11(2006)080 index i k and an antiquark with color index α k are in fact connected by a fermionic line, we identify the index α k with the indexī k . Then the string {α} = (α 1 , . . . , α m ) is a generic permutation of the stringī = (ī 1 , . . . ,ī m ). The power p is determined by the number of times α k =ī k . If α ≡ī, then p = m − 1. As usual, a product of zero T matrices has to be understood as a Kronecker delta.
The full amplitude typically reads n+m are known as the partial amplitudes. They depend only on the kinematic of the process, namely, the momenta and helicities of the particles.
In the framework of the helicity formalism [3 -5], with the spinors denoted as the partial amplitudes can be written in terms of the spinors inner-products In our convention all particles are considered to be outgoing and, following the QCD literature [1,2], we fix the sign of the inner products such that ij [ji] = s ij 1 .
The BCFW recurrence relation is based on the analytical properties of the amplitude when the spinors of two external legs (denoted by j and l) are shifted as  s−r+2 (r λr , . . . ,ĵ λ j , . . . , s, −K λ rs ) The sum on eq. (2.7) is over all the possible configurations where the j particle belongs to one of the amplitudes and the l particle to the other one. It includes also the 3-particle sub-amplitudes, which are usually vanishing for on-shell particles but become non zero because of the BCFW shift in eq. (2.6).
When tagging the particles, one has to fulfill certain restrictions, i.e.: some configurations of j and l are forbidden. The helicities of the reference particles can not be chosen as (λ j , λ l ) = (+, −).
Quarks and antiquarks of the same flavor can not be selected if they are adjacent. Furthermore, one can not chose two adjacent fermions of different flavor and the same helicity. For adjacent quarks and gluons the helicities should better be opposite. [22].
Under a general tag, some sub-amplitudes may vanish. A clever choice of j and l leads to most compact expressions for the same amplitudes. In this paper, we have chosen the allowed tag that gives the minor number of terms. Nevertheless, in some calculations we have tried some forbidden tags, in the search for results with fewer terms and simpler structure. In that case we explicitly checked that the final expressions agreed with the longer forms obtained using allowed legs in the BCFW recursion relation.
The main ingredient of the recursion relations, that permits to construct the tree level amplitude for n-partons just by iterations, are the ggg and qqg MHV amplitudes [11], which with our phase conventions read The corresponding MHV amplitudes can be obtained from those above by using some simple properties. The "parity" operation inverts the helicities of all the particles, and at the level of the amplitudes implies to perform the exchange ij ↔ [ji] and add an extra factor of (-1) for each fermionic pair involved. Charge conjugation turns quarks into antiquarks and vice-versa. The partial amplitudes are C-invariant: they do not change under the action of this operator. Also one can profit from the reflection and cyclic symmetries of the amplitudes. The reflection property states that the following equality holds while cyclic symmetry implies that the amplitude is invariant under cyclic permutations of 1, 2, . . . , n. The factorization properties of the amplitudes in the collinear and soft channels are an useful tool to check for the consistency of the results. Further details on the method of calculation can be found in [26].

Four-quark plus three-gluon amplitudes
Seeking for the most general presentation, we concentrate on the case of different quark flavors, that we denote as q and p. To obtain the equal flavor two-quark pair amplitudes, it is enough to substract from a given partial amplitude the same term with the antiquarks exchanged (and not the quarks) [20,41]. From eq. (2.1), we can read off the two different color structures contributing to the two-quark pair plus gluon amplitudes. For p = 0, it follows α 1 =ī 2 , α 2 =ī 1 and one obtains the leading contribution For leading color, the partial amplitude corresponds to the following string of quarks and while for the sub-leading color the ordering is In order to maintain a similar notation (the first particle being always q) to the one used in [26] for one-quark pair amplitudes, we will present the results with the following format A(q,q, g σ 1 , . . . , g σ j , p,p, g σ j+1 , . . . , g σ l ) (3.5) and A(q,p, g σ 1 , . . . , g σ j , p,q, g σ j+1 , . . . , g σ l ) , for the leading and sub-leading color structures, respectively. Those corresponding exactly to the ordering in eq. (3.3) and eq. (3.4) can be easily obtained from the ones above by C operation and relabelling of the partons under cyclic symmetry. Several amplitudes are accounted for by the MHV formula for two-quark pairs and n − 4 gluons, which is given by where i and j are the two fermions with negative helicity while all the other particles have positive helicity [1]. We have reobtained eq. (3.7) using the BCFW recursion relation and eq. (2.9). For the sub-leading case exactly the same formula holds. When facing the NHMV case, we denote the leading and sub-leading color contributions as "A" and "B", respectively. There are fourteen independent "A" amplitudes and JHEP11(2006)080 twelve independent "B" amplitudes. For simplicity, we will refer to them as A 1 , A 2 , . . . , A 14 and B 1 , B 2 , . . . , B 12 . The full list is for type A processes and for type B processes. The explicits results for those amplitudes are presented in the next sub-sections.

4q+3g leading amplitudes
We present now the most compact expressions for the 14 independent leading amplitudes. We have taken j = 5, l = 6 for the calculation of A 1 , leading to

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We have computed A 8 by selecting j = 6, l = 7  Choosing j = 2, l = 3, we arrive at the following expression for A 9  The computation of A 10 has been performed by taking j = 2, l = 3, leading to We have selected j = 3, l = 4 for the calculation of A 11 , arriving at   Picking j = 4, l = 5, A 13 is given by  Finally, A 14 has been computed by selecting j = 1, l = 2 (3.23)

4q+3g sub-leading amplitudes
We anticipate that the sub-leading amplitudes have fewer and simpler terms than the leading ones,mainly due to the fact that the ordering of the quarks causes the vanishing of many BCFW sub-amplitudes.
The calculation of B 1 was performed by taking j = 5, l = 6     (3.26) The calculation of B 4 has been performed by selecting j = 3, l = 4  Setting j = 4, l = 5, we have computed B 5 , arriving at the following expression (3.28) We have calculated B 6 by choosing j = 3, l = 4  Picking j = 4, l = 5, B 7 is simply given by

NMHV six-quark plus one-gluon amplitudes
Six-quark processes in QCD were first considered in [21]. We discuss here the three different flavor (p, q and r) case. In order to obtain the amplitudes for two or three equal flavors one has to antisymmetrize the expression by adding and substracting amplitudes with the antiquarks exchanged; for more details see [21]. From eq. (2.1), one can notice that three different color structure arise: the leading, sub-leading and sub-sub-leading contributions, corresponding to p = 0, p = 1, and p = 2, respectively.
To complete the color basis, one has to add also those structures corresponding to the 2 ↔ 3 exchange, but those terms do not lead to independent amplitudes. For p = 1, the three following color structures have to be taken into account: α 1 = i 1 , α 2 =ī 3 , α 3 =ī 2 , α 1 =ī 2 , α 2 =ī 1 , α 3 =ī 3 and α 1 =ī 3 , α 2 =ī 2 , α 3 =ī 1 . It is easy to see that the second structure can be related to the first one by cyclic permutation, and similarly for the third-one. Therefore the only independent sub-leading color form factor corresponds to the string When p = 2 things become simpler than in the previous cases, because the only possible configuration is α ≡ i. Then the sub-sub-leading color structure is given by At the level of the partial amplitudes, the string for leading color is written in a generic form as [q(g σ 1 . . . g σ j )p][p(g σ j+1 . . . g σ l )r][r(g σ l+1 . . . g σ l )q] (4.5) for first leading-color structure and [q(g σ 1 . . . g σ j )r][p(g σ j+1 . . . g σ l )q][r(g σ l+1 . . . g σ l )p] (4.6) for the second one. For the sub-leading color one has and for the sub-sub-leading case the structure is In order to maintain the same notation along this paper we present the results for the amplitudes using the following order of partons (q always first) A(q,q, g 1 , . . . , g j , p,p, g j+1 , . . . , g k , r,r, g k+1 , . . . g l ) (4.9) or A(q,p, g 1 , . . . , g j , r,q, g j+1 , . . . , g k , p,r, g k+1 , . . . , g l ) (4.10)
For the sake of presentation, we denote the leading contributions as "A" amplitudes, the sub-leading as "B" and the sub-sub-leading "C". We have found that the number of independent amplitudes is in principle twenty-four: eight for the leading color (A 1 , . . . , A 8 ), twelve for the sub-leading (B 1 , . . . , B 12 ) and four for the sub-sub-leading (C 1 , . . . , C 4 ).

Leading amplitudes
In order to obtain a leading amplitude the only available gluon should sit in between two fermions of different flavor. It is therefore equivalent, just by flavor relabelling, to insert the gluon in any possible place compatible with the color structure. We have chosen the gluon to be the third particle. Besides, as there exist already three fermions with negative polarization, the gluon should necessarily have positive helicity to build an NMHV amplitude. Those coming from the first color structure are while those from the second color string are all vanishing. Below we present the most compact expressions for the four leading amplitudes. A 1 has been obtained by picking j = 5, l = 6  Picking j = 2, l = 3, A 3 is given by relations. With these results the full set of tree level helicity amplitudes, up to seven partons, becomes available. Our formulae have been tested in all possible collinear and soft limits, which provides a very strict check for the correctness of the amplitudes. 2 .
These amplitudes are a main ingredient for the calculation of multijets cross-sections in hadronic colliders. As expected, we have obtained very compact expressions, allowing for a more convenient implementation in computer codes than those coming from automatic tree level computational methods.