Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes

We find a permutation relation among Yangian Invariants -- two Yangian Invariants with adjacent external lines exchanged are related by a simple kinematic factor. This relation is shown to be equivalent to U(1) decoupling and Bern-Carrasco-Johansson (BCJ) relation at the level of maximal helicity violating (MHV) amplitudes. Together with the newly found permutation relation and using unitarity cuts to remove ambiguity in the definition of loop momenta of cut amplitudes due to the nonplanar legs, we propose a systematic way of reconstructing the the integrands of nonplanar MHV amplitudes up to a rational function which vanishes under all possible unitarity cuts. This method when applied to planar diagrams reproduces results from the single cut method. As explicit examples the construction of one-loop double-trace MHV amplitudes of 4- and 5-point interactions are presented using on-shell diagrams. The kinematic factors and the resultant planar diagrams are carefully dealt with using the unitarity cut condition. The first next-to-MHV amplitudes are addressed using generalized unitarity cuts. Their leading singularities can be identified as residues of the Grassmanian integral. This example also serve to demonstrate the power of the newly found relation of Yangian Invariants.

Besides the progress on calculations interesting and useful relations among colorordered partial amplitudes have been uncovered. A relation of such kind was proposed by Bern, Carrasco and Johansson [27], the BCJ relation. Together with the KK relation proposed earlier by Kleiss and Kuijf [28], these two relations have since then been widely used to simplify calculations at tree level [29][30][31][32][33][34].
Lately Arkani-Hamed et al [35] proposed using positive Grassmannian to study N = 4 super Yang-Mills along with the constructions of the bipartite ribbon on-shell diagrams [36]-in which all internal legs are on shell-for planar Yang-Mills interactions. In such a construction each on-shell bipartite diagram is automatically gauge invariant; and a direct relationship between planar amplitudes in N = 4 super Yang-Mills (SYM) and the positive Grassmannian structures is presented. Furthermore they also prescribe a permutation rule for characterizing on-shell diagrams of tree level amplitudes as well as the leading singularities [37,38] in planar loop-level amplitudes. Specifically each onshell diagram corresponds to a Yangian invariant [39,40]; and the scattering amplitudes can be obtained by summing over the underlying Yangian Invariants. All can be done in either the momentum space or the momentum twistor space [41][42][43].
On-shell bipartite diagrams fall into equivalence classes under square moves and mergers. Such equivalence operations leave the corresponding Yangian Invariants unchanged. However if we only require the corresponding Grassmannian geometry unchanged, by intuition, there should be new generators of equivalence operations. In this paper, we will discuss a class of such operations generated by a black & white (B&W) square-which for the rest of the paper will be called a "basic box"-leading to permutation relations in the on-shell diagrams. These, in turn, induce relations among Yangian invariants.
Another motivation for this work is to present a systematic method to construct the local integrands [22,44] of Yang-Mills scattering amplitudes from unitarity cuts for planar diagrams. Under each unitarity cut, the integrand of the amplitude is welldefined and can be obtained by gluing tree-level amplitudes. According to the onshell diagrams of the tree-level amplitudes, we can directly remove the unitarity cut constraints in the frame of on-shell diagram. Then for each unitarity cut we obtain a simple form of the integrand up to a rational function which will vanish on the unitary cut. After introducing a proper operation to combine the integrands for all kinds of the unitarity cuts, we can get an integrand for an general amplitude up to a rational function which will vanish under all the unitarity cuts. According to the unitarity constructible condition for the amplitudes in super Yang-Mills theory [5,6], the final ambiguity of the integrand can be fixed by setting the rational function to zero. Then we obtain the final form of the integrand for the amplitude. This method enjoys a direct extension to nonplanar diagrams when combined with the newly found on-shell permutation relations. In nonplanar diagrams 1 it is still possible to define an integrand up to rational functions which will vanish under all the unitarity cuts [45][46][47][48][49].
Unitarity cuts are deployed, in nonplanar diagrams, to remove the well-known ambiguity in loop momentum definition due to the nonplanar leg(s), as opposed to single cuts used by Arkani-Hamed et al [35] in the construction of planar amplitudes. A good definition of loop momentum under a unitarity cut in the nonplanar loop diagrams will be presented. The resulted diagrams after a unitarity cut of a given one loop nonplanar diagram (by which we mean one-loop double-trace amplitudes) can be transformed into the on-shell diagrams. Using our newly found on-shell permutation relation a nonplanar on-shell diagram could be subsequently converted to (a linear combinations of) planar diagrams with kinematic functions as coefficients. One then needs to sum up the resultant planar diagrams, from all possible unitarity cuts of a given nonplanar diagram, to obtain the total nonplanar amplitudes after unitarity cuts.
The final step of our construction is to use appropriate BCFW bridges to reconstruct the total on-shell diagrams for a given nonplanar diagram. All possible but inequivalent connections by BCFW bridges need to be taken into account. The most crucial step is the discovery of the permutation relation for bipartite on-shell diagrams that enable us to convert nonplanar on-shell sub-diagrams into planar ones, which, in turn, enable the straight forward application of the existing techniques developed for on-shell planar diagrams. This method works is well-adapted for higher loops; and we believe that it can be generalized to higher-loop nonplanar diagrams (work in progress).
We shall show by explicit computations in Section 4 that the total on-shell diagrams constructed by unitarity cuts for MHV nonplanar one-loop amplitudes in N = 4 super Yang-Mills give the correct local integrands. The total on-shell diagrams constructed for NMHV nonplanar one-loop amplitudes by generalized unitarity cuts reproduce the correct integrals [50], as presented in Section 5.

A permutation relation among Yangian Invariants in onshell diagrams
In dealing with nonplanar amplitudes it is crucial that there be a relation to enable the transformation of nonplanar elements into planar ones. To our pleasant surprise there exists such a simple relation, represented pictorially in Fig tion relation relating different Yangian invariants is completely analogous to the BCJ relation [27] for 4-point amplitude. We shall henceforth call such transformation the permutation relation among Yangian invariants.
In N = 4 SYM the amplitudes can be constructed by unitary cuts or generalized unitary cuts. After cuts the loop amplitude is a combination of tree level amplitudes. As we want to transform the nonplanar amplitudes, after unitarity cut, into planar ones we need to know how the constituent tree amplitudes change under the permutations of legs. Since a permutation of legs is generated by a pairwise exchange of two consecutive legs we only need to know the transformations of the tree amplitudes under an exchange of two consecutive legs.

Permutation relation of box in bipartite diagrams
The set of rules governing the permutations of external legs for the bipartite on-shell diagrams have been introduced in [35]. Let us take, again, the box as an example: the 4-point tree amplitude has only one Yangian invariant. The corresponding permutation is 1 2 3 4 3 4 5 6 , In a on-shell diagram of a tree-level amplitude a permutation is in one to one correspondence to a Yangian invariant, we can therefore use permutations to characterize Yangian invariants. Without loss of generality, we take the permuted external legs to be 3 and 4. It is then easy to see This is exactly what has been depicted in Fig. 1 above.
This new permutation relation holds, furthermore, for an analogous class of Yangian invariants in non-MHV amplitudes. To aid in the discovery we first establish a criterion suitable for this class of amplitudes. Firstly, we should define a modified BCFWdecomposition [35]. The on-shell diagram can be decomposed as taking apart a BCFW bridge from the diagram with a sub-diagram left. Then the permutation of the diagram σ can be decomposed as (ij) • σ ′ , where (ij) is the permutation of the BCFW bridge on i, j and σ ′ is the permutation of the left sub-diagram.
A BCFW-Bridge decomposition to a Box: Starting with a given permutation σ and picking two consecutive legs i and i+1, if σ(i) = i mod n and σ(i+1) = i+1 mod n and σ for the other legs is not identical to the identity modulus n (a "dressed" identity 2 ), one can decompose σ as (j 1 j 2 ) • σ ′ , where 1 j 1 < j 2 n and σ(j 1 ) < σ(j 2 ), with j 1 = {i, i + 1}, and j 2 = {i, i + 1}. The legs j 1 and j 2 are being separated only by the unpermuted legs or leg i, or i + 1, keeping the order of σ −1 (i), σ −1 (i + 1) invariant.
One repeats the process until σ is an identity for all the legs except the legs i, i + 1 and σ(i) mod n, σ(i + 1) mod n. We denote the final permutation asσ. If, on the other hand,σ (i) <σ(i + 1) it is easy to see that theσ corresponds to a four point amplitude for legs (i, i + 1, σ(i) mod n, σ(i + 1) mod n). Furthermore σ is obtained by putting BCFW bridges onσ. Hence we conclude that the on-shell diagram corresponding to σ can have "a box" connecting directly to these two marked legs.
Moreover, in a BCFW-Bridge decomposition, we always keep the order of σ −1 (i) and σ −1 (i + 1). Then the condition (2.7) is equivalent to which is a convenient criterion on permutations to check whether a "box" can enjoy direct connection to a pair of adjacent legs in a on-shell bipartite diagram.

Conclusion:
For planar bipartite on-shell diagrams, we can use the criterion (2.8) to justify if any two adjacent legs fall into a box.

Kinematic factors
Let us now turn our attention to the kinematic factors-the remaining obstacle in the construction of total integrands for nonplanar amplitudes. How to deal with the resultant planar diagrams as well as the concrete steps of reconstruction will be presented in Section 4 and 5. In this subsection, we also study the behavior of Yangian invariants 1 1 Figure 3. A Yangian invariant relation of a 4-point tree amplitude, with the dark circle denoting a general on-shell diagram.
Y k n and the corresponding Grassmannian cells, (k × n)-matrix C, under a permutation of two external legs. Each Grassmannian cell C is a point in the Grassmannian G(k, n) which characterize the k-plane in the n-dimensional space.
To compute the kinematic factor we only need to determine the momenta of the two internal lines connecting to Y (2) 4 , which can be done recursively by BCFW method. We set a variable α to exhibit the momentum shift between i + 1 and i, α λ i+1λi . Then the shifted momenta are The internal momentum connecting these two legs, λ IλÎ , is determined by momentum conservation from the left (or right), where the index k in the sum runs through all the external momenta on the left. The variable α can thus be solved by the condition λ IλI being on-shell The momenta λ IλI and λĵλĵ are fully determined by the spinors of the external momenta. And the Yangian invariant on the right is, in turn, determined.
We repeat the above operation until only a Y According to the arguments in [35] each on-shell diagram or Yangian invariant is associated with a differential form dΩδ(C · η)δ(C · λ )δ( λ · C ⊥ ), (2.13) where dΩ is the Grassmannian integration measure and C ⊥ is orthogonal to C. The C can be taken as the matrix associated with the linear constraints δ(C · η), δ(C · λ ), δ( λ · C) for the external spinors λ, λ , η. The Grassmannian cell for a MHV amplitude is always A permutation of two external lines does not change any of the linear constraints in (2.13). The Grassmannian cell is thus not affected by permutations when we fix the vector of external spinors. In fact this rule can be generalized to any on-shell diagrams in tree-level amplitudes. Any permutation of two external legs attached to a box does not affect the Grassmannian cell for a given vector of external spinors. This is easy to prove. Without lose of generality we take the permuted legs n − 1 and n. According to a BCFW decomposition to a box, the C matrix of an on-shell diagrams can be generalized by performing BCFW bridge on the C 0 matrix corresponding toσ. The rows of C 0 are denoted by the i w 's which satisfyσ(i w ) = i w + n,σ(n − 1) − n and σ(n) − n. For the tree level on-shell diagrams the δ-functions are just enough to fix the parameters α I 's. Hence the total number of BCFW bridge acting on a box is 2n − 8; we obtain C = C 0 B(i 5 , j 5 ; α 5 ) · B(i 6 , j 6 ; α 6 ) . . . B(i I , j I ; α I ) . . . B(i 2n−4 , j 2n−4 ; α 2n−4 ), (2.14) where j I 1 0 · · · 0 · · · 0 · · · 0 0 1 · · · 0 · · · 0 · · · 0 . . . . . . . . . 0 · · · 0 · · · 0 i I 0 0 0 1 · · · α I · · · 0 . . .
are the element in i w row and i w columns of C 0 . Other elements in C 0 are zero. It is obvious that the permutation on the box will not affect C 0 and all the delta functions associate with C 0 . Furthermore all the parameters α I are fixed by the delta functions in the box. And none of vertices and internal lines outside the box in the on-shell diagram can be affected by permutations of the external legs. Hence all BCFW bridges are invariant under the external leg permutations, which in turn implies that the Grassmannian cell C is invariant under the permutations. We can make a stronger generalization: any permutations of two legs attached to a box in bipartite diagram will not affect the Grassmannian cell. The proof is completely analogous; an example will be presented in Section 5.

Permutation relations for NMHV amplitudes-a twin-box connected by a BCFW bridge
In general, however, not any two consecutive legs can enjoy direct connection to a box. From NMHV amplitude to more general amplitudes, the next basic object arisen in a permutation of two adjacent legs is a twin-box with the two permuting legs connected by a BCFW bridge in Fig 5, showing the first NMHV example of a six point Yangian invariant corresponding to the permutation σ 1 = {4, 5, 6, 8, 7, 9} with 5 and 6 being the permuting legs. According to σ 1 (o1) 3 , the permutation relation is easy to obtain, In fact for a Yangian Invariant in NMHV amplitudes, a bipartite diagram is composed of BCFW-bridged box glued with k-preserving inverse soft factor Y . This is because we can choose a BCFW bridge for at least one pair of consecutive legs such that Y k L n L and Y k R n R with n L > 3, n R > 3 and k L = k R = 2. And according to the analysis in Section 2.1 the general structure and the permutation relation of an on-shell diagram in NMHV is as shown in Fig. 6.
For general amplitudes beyond NMHV this permutation relation for the BCFWbridged twin-box can easily be shown to exist for a class of Yangian invariants. We shall establish a convenient criterion for them. To this end, we define a revised BCFW-Bridge decomposition [35].
BCFW-Bridge decomposition of a bridged twin-box: Staring with a given permutation σ, we mark two consecutive legs i and i+1 and other two legs σ(i) mod n, σ(i+ 1) mod n. Now one chooses another n − 6 legs, other than the four chosen ones. The left 6 legs are left fixed if σ(i) mod n = i and σ(i + 1) mod n = i + 1 and the the box contact condition (2.8) does not hold. If σ for other legs is not a "dressed" identity 4 we decompose σ as (j 1 j 2 ) • σ ′ , where 1 j 1 < j 2 n, σ(j 1 ) < σ(j 2 ), j 1 = i, i + 1, j 2 = i, i + 1 and j 1 , j 2 are separated only by marked legs or legs i, i + 1, keeping the order of the 6 fixed legs. This procedure is repeated until σ becomes the identity for all the mobile legs and the resultant permutation is denoted byσ.
Ifσ is a permutation of Y 3 6 then the bipartite diagram is of the form shown in Fig. 7. When we permute the legs "i" and "i + 1", the effect of the permutation on the Grassmannian matrix C will be partially blocked by the box. In fact the total number of BCFW bridges acting on a bridged twin-box is 2n − 12; and we obtain Similar to a permutation on the box all the parameters α I are fixed by delta 4 For example, for n = 6, k = 3, a "dressed" identity is {7, 8, 3, 10, 5, 6}.
functions of momentum conservation in the bridged twin-box. None of the vertices and internal lines outside of the bridged twin-box are affected by the permutations of the external legs. Hence all BCFW bridges are invariant under these permutations. Nevertheless one row in C 0 does change while the other rows of are invariant under a leg permutation on the twin-box. Such transformation relations on C is therefore useful for classifying Yangian invariants related by a given permutation of legs.

Unitarity cuts and generalized unitarity cuts
In N = 4 supersymmetry Yang-Mills theory "single cut" is an efficient way of constructing all loop integrands for planar loop amplitudes. However in the nonplanar case the resultant diagram after a single cut is often not a well-defined Feynman diagram. One can also view this problem as a difficulty to endow the loop momentum with a canonical definition because the nonplanar leg(s) can fall between any two planar legs inside a loop. In fact all such possibilities should be taken into account.
The general loop amplitudes after the unitarity cuts and generalized unitarity cuts can be regarded as tree level amplitudes being glued together. Hence the unitarity and generalized unitarity cut loop amplitudes are well-defined and can be taken as the foundation to construct the integral of the amplitudes. From this point of view, the major difference for planar and nonplanar diagrams under unitarity cuts is that all the gluing lines in each tree-level amplitude are adjacent for planar diagrams while in nonplanar diagrams at least a pair of gluing lines is nonadjacent. Furthermore, for the unitarity cuts, together with the on-shell diagrams for the tree-level amplitudes, it is also possible to construct the integrand of the general loop amplitudes systemically, which we will discuss in Section 4.

Unitarity cut
Given a nonplanar diagram one should consider all possible diagrams resulted from the nonplanar leg(s) taking all probable positions when traversing around the loop. The simplest example is the four-point one-loop with one nonplanar leg-which we shall call the "(3+1)" case for short in the rest of the article-as shown in Fig. 8. The nonplanar The ambiguity in defining the loop momentum is resolved as follows: if we start with the external line marked "4" we can call the momentum on the first cut loop line l, andl the loop momentum on the other cut line. In the clockwise order for the color-ordered amplitudes one easily checks that in each resultant tree diagram the momentum is well defined for each of the external legs.
We can keep track of the order of unitarity cuts as well: under each cut the diagram will be divided into a diagram with one fewer loops in addition to a tree-level diagram. A typical higher loop case is shown in Fig. 9.
The ambiguity in the definitions of the loop momenta is resolved by a series of unitarity cuts. In fact we can define each loop momentum clockwise from a reference external line. For the typical example in (a) of Fig. 9, if we start with the external line marked "1"-in the clockwise order for the color-ordered amplitudes-we can call the momentum on the first cut loop-line l 1 andl 1 the loop momentum on the other cut line. After setting l 1 as the reference line in the first loop the momentum on the other cut line becomesl 1 = l 1 − P R where P R being the sum of all external momenta between these two cut lines. The momenta on the second loop can thus be fixed to be l 2 andl 2 = l 2 − P ′ R with P ′ R denoting the sum of all the external momenta to the right of l 2 , and so forth.
Obvious in this construction, topological information of nonplanarity is preserved: one unitarity cut can only fix two components of the 4-momentum integrals leaving the other two integrations unconstrained. Furthermore, the integrand under the unitarity cut is also well-defined. This means that the integrand contains enough information for the characterization of the loop topology of nonplanar diagrams. And the loop topology and geometric properties of the underlying Grassmannian arisen in nonplanar amplitudes will be manifest once we construct nonplanar amplitudes in on-shell bipartite diagrams [35].
The permutation relations of bipartite on-shell diagrams-each of them corresponding to a Yangian with its gauge invariance-are instrumental in constructing the whole amplitude from unitarity cut or generalized unitarity cut diagrams. To construct the bipartite on-shell diagrams after a unitarity cut we need to convert each resultant tree amplitudes in Fig. 9 into the corresponding bipartite diagrams. The loop lines connecting the tree amplitudes now denote the same integration as the internal lines in tree-level bipartite on-shell diagrams. Since the construction of bipartite on-shell diagram for each tree level diagram is well established in [35], we only need to verify the gluing lines in bipartite diagram are equivalent to a unitarity cut of loop amplitudes.
In the language of bipartite on-shell diagram, the gluing line represents an extra integral: . This can be further simplified, which is exactly the expression of a loop level amplitude after a unitarity cut.
Let us turn, again, to our lovely "3+1" example ( Fig. 8), the bipartite on-shell diagrams correspond to the tree amplitudes resulted from a unitary cut are shown in Fig. 10. After the cuts, only two four-point tree amplitudes are left. Each tree (c) Figure 10. The bipartite on-shell diagrams of an s-channel cut (a), and a t-channel cut (b), and a u-channel cut (c). Note: In our convention a horizontal square denotes the planar tree amplitude while a rhombus (at 45 degrees) denotes a nonplanar amplitude with the plane of rhombus being perpendicular to the plane of the paper where the points marked "3" and "4" are at equal distance from the vertical edges of the (planar) square.
amplitude is a box. Now we can add two lines to connect these two boxes to represent the cut amplitude. There are three different cuts-the s-channel cut, the t-channel cut, and the u-channel cut. Each of them can be represented in on-shell diagrams in the ways shown in Fig. 10.
We now present our strategy for constructing the full scattering amplitudes in the bipartite on-shell language for the corresponding the nonplanar Feynman diagrams. In this work we only focus on the one-loop diagrams. We would like to stress that our strategy can be extended to the higher loops cases, with generalized unitarity cuts, in a straightforward way. Detailed descriptions, together with carefully worked out examples, of the one-loop amplitudes will be presented in Section 4.
• Perform all possible unitarity cuts on a given nonplanar Feynman diagram.
We convert the resultant diagrams from each possible (series of) unitarity cuts into on-shell bipartite diagrams. Each bipartite diagram corresponds to a Yangian invariant.
• Remove all unphysical poles in loops, the structures of which depend on the loop momenta.
They occur because Yangian invariants in general contain unphysical poles. However the unphysical poles will cancel each other upon summing over all Yangian invariants of a given amplitude-only physical poles remain. Furthermore the unphysical poles in loops are not allowed in the total amplitudes. We need to ensure that no unphysical poles of the loop momenta appear in the final expressions.
• Sum over all the inequivalent terms from each series of unitarity cuts. After removing the unitarity cut conditions, we will get an integral with respect to all the loop momenta. If there appears the same integral when reconstructing from a different unitarity cut then it suffices to count it once.

Generalized unitarity cuts
Generalized unitarity cut can also be used to construct the full loop level amplitudes. For N = 4 SYM after a quadruple cuts on each loop only the leading singularity of the loop level amplitudes remains. All loop momenta are fixed by the cut constraints. Absent is the possibility of having a rational function in loop momenta. However, from the bipartite on-shell diagrams, lots of geometric information of the Grassmannian can be read off from the leading singularity, as shown in Fig. 11.
In a one loop nonplanar diagram, Fig. 12, the leading singularity is of form (3.3) Similar to the case of planar diagrams the leading singularities of nonplanar diagrams can also be identified as residues of the Grassmannian integral, a specific example of which will be given in Section 5. In the case of N = 4 SYM double unitarity cuts (quadruple cuts) can fully determine the full amplitudes. Compared to a unitarity cut double unitarity cuts are more convenient to the full amplitudes' reconstruction because all the poles in loop momenta are automatically physical upon such cuts. Furthermore different double unitarity cuts lead to different scalar integrals: we do not need to consider equivalent integrals as we do with unitarity cuts. The general procedures of reconstructing the full amplitudes by double cuts are as follows.
• Perform all possible double unitarity cuts on the nonplanar Feynman diagrams with loops. For each possible series of quadruple cuts we convert the resultant tree-level amplitudes to the bipartite diagrams. We then glue the cut loop lines according to the Feynman diagram. Similar to the planar case these reproduce the leading singularities of the nonplanar amplitudes.
• Transform the nonplanar leading singularities into planar ones by the permutation relations of the Yangian invariants.
• Multiply the leading singularities by a standard integral.
• Sum over all contributions from each series of quadruple cuts.
• To elucidate the geometric properties of the Grassmannian we group terms according to their underlying Grassmannian geometry.

MHV Loop Amplitudes
In U(N) Yang-Mills theory, the one loop amplitudes can be decomposed as [52] A 1-loop where A n;c are the partial amplitudes, Z n and S n;c are the subsets of S n that leave the corresponding single and double trace structures invariant, and ⌊x⌋ is the greatest integer less than or equal to x. In this paper the diagrams corresponding to single trace and double trace partial amplitudes are regarded as planar and non-planar diagrams respectively. We will focus on the partial amplitudes of planar diagram, A P ≡ A n;1 , as well as nonplanar diagrams, A N P ≡ A n;2 , of 4-point, 5-point interactions.

MHV planar amplitudes and unitarity cuts
Using single cuts techniques, Arkani-Hamed et al has thoroughly studied planar amplitudes of all loops in momentum twistor space [21]. On the other hand Bern et al introduced unitarity cuts as a way to reconstruct planar MHV amplitudes in momentum space [5]. At this point, however, no systematic method of MHV amplitudes reconstruction from unitarity cuts in momentum twistor space exists.
In this section, we present a detailed method of constructing MHV one-loop amplitudes from unitarity cuts. We build relations of bipartite on-shell diagrams and express them in momentum twistor space. This method leads us naturally to simple results without unphysical poles, in addition to the final integrands being the same as those from single cuts. This is to be contrasted with the way proposed by Bern et al [5] in dealing with the box integrals. Given these advantages it is thus a meaningful exercise to study MHV amplitudes in momentum twistor space together with unitarity cuts. Example: Integrands of five-point one loop amplitudes Now we give an example to explicit the above procedure. First non-trivial example is five-point planar amplitude.
Step I: In five-point situation, there are five different unitarity cuts A c (i, i + 1|i + 2, i + 3, i + 4), i = 1, 2, 3, 4, 5. A c (12|345), for instance, can be constructed as gluing two tree level amplitudes A L (12ll) and A R (ll345) as shown in Fig. 9 for the general case. The corresponding on-shell diagrams is shown in (a) of Fig. 13.
Step II: Add BCFW bridges to (23) and (15) to remove the cut constraints (shown in Fig. 13). We can simply write the integrand in momentum twistor space, based on the four-point one-loop situation, as where A and B denote the points of the cut lines in momentum twistor space (Appendix A).
Step III: There are unphysical poles in the denominator of the previous equation, such as AB35 . These poles can be converted to physical ones using unitarity condition, AB23 = 0, AB51 = 0. We could build a relation between Step IV: Repeat the above three steps on another unitarity cut, we obtain the result of another cut A 1 (2, 3|4, 5, 1) as Now we need to combine these two results from different cuts. Z 5 could be expanded based on Z 1 , Z 2 , Z 3 , Z 4 , to get AB25 4123 = AB23 4125 + AB24 5123 where AB12 vanishes due to unitarity cut condition. Obviously the term AB24 is the same in these two terms, so we only need to count its contribution once. Other terms in these two equations could be directly added together not affecting the results under both unitarity cuts. Combining other three cuts with the same method we get the final integrand of planar one loop five-point MHV amplitude as which is the same result as obtained from single cuts in [21].
This simple example serves to illustrate our strategy to compute Yang-Mills integrands using unitarity cuts. This is to be contrasted with the single cuts method proposed by Arkani-Hamed et al [21,22] as well as constructing the scattering amplitudes integral after unitarity-cutting the Feynman diagrams as done by Bern et al at a much earlier attempt [5,6]. We shall proceed with a general discussion of the higher-point results in the rest of the section.

Property of MHV planar amplitudes under unitarity cuts
To study MHV loop amplitudes under unitarity cuts we first tackle MHV tree amplitudes, of which Yangian Invariant is Y  We can therefore obtain the MHV amplitude after unitarity cuts as where A c denotes the amplitude under each of the possible unitarity cuts. '|' denotes the cut line between these two legs and the other cut line between the first and last legs in the bracket. This relation shows that an n-point MHV amplitude is the same as a four-point amplitude under unitarity cuts in momentum twistor space.

Removing the unitarity cut constraints
The standard way of reconstructing amplitudes from single cuts is by adding BCFW bridges across cut lines on a pair of external legs. For amplitudes under unitarity cuts we do not necessarily have to add bridges on external legs. In fact the unitarity-cut amplitudes in MHV case can be reduced to the basic "twin-box" in momentum twistor space (Fig. 14). They contain all the essential information of the whole amplitudes after unitarity cuts. This is also apparent in momentum space. After four internal integrals, all δ-functions from the three blocks vanish, reducing the number of external legs by one. Recursively, the whole amplitude can be reduced to a basic "twin-box" with four new external on-shell momenta. This means that adding bridges to the "twin-box" recover the same result from a unitarity cut. This new way of adding BCFW bridges will greatly simplify our subsequent computations. In on-shell diagram it amounts to Fig. 15.
The final result of this planar part is, thus, nothing but a one-loop four-point planar amplitude. We denote the cut loop momenta, l andl, by the variables A and B in momentum twistor space. 6 A 0 (1, . . . , i|i + 1, . . . , n) = 1ii + 1n ii + 1n1 where A 0 denotes the amplitude after adding two BCFW bridge to the "twin-box".
Conversion from unphysical poles to physical ones: Since the form of planar amplitudes after reconstruction is actually from a fourpoint (Z i , Z i+1 , Z 1 , Z n ) (with the subscripts denoting the momenta of the four legs connected to the "twin-box" as in Fig. 14) amplitude, some propagators, say A B i + 1 n and A B i n , becomes unphysical poles inside an n-point amplitude (A and B as before denote the loop momenta, l andl, in the momentum twistor space.).
We at present present a way to convert unphysical poles to physical ones using the unitarity cut condition. We show by an example of an amplitude with color ordering (1, 2, . . . , n) and unitarity cut ABii + 1 ABjj + 1 as shown in Fig. 16. In a particular order, this is just a planar diagram, we can discuss the integrand. The unitarity cut condition is ABii + 1 = 0, ABjj + 1 = 0. And the poles ABi + 1j and ABj + 1i become unphysical.
Imposing the unitarity cut condition, where all of the poles in the denominator are physical. Here we convert it to another form related to the intersection of two planes (ii+1i+2) and (j-1jj+1), which is useful 6 We omit the terms related to integral variables ABd 2 z A ABd 2 z B in this paper. Since we deal with the integrand of amplitudes, we leave this as a common factor of in the integrand. in the latter discussion. ii We use the same method to deal with pole ABj + 1i . According to (4.10) and combining these two parts, In this paper c ij with 1 i < j n − 1 denotes the cuts that divide the external lines into two groups i · · · j and j + 1 · · · i − 1. For convenience, we also use C ij to denote a cut set containing all the cuts c 12 · · · c ij in the union order. The corresponding A C ij denote the union of all the integrands from the cuts in C ij . And A c ij , A C ij may have the ambiguity of a rational function which will vanish under all the cuts in C ij . We denote the corresponding arbitrary rational functions as R C ij or R c ij .

Combination of different unitarity cuts
In the above paragraphs, we obtain an integrand A c from each cut amplitudes. Such integrand contains only physical poles and is well-defined up to rational functions which will vanish under unitarity cuts. In this paragraph, we need to find a way to get the rational function A C 1n of the integrand such that it is equal to A c ij at the corresponding unitarity cuts. To this end, we define a operation ∪ to the constructed integrand, which means an union of two integrand from different unitarity cuts. After the union of all the possible unitarity cuts, we get the integrand A C 1n automatically.
According to the analysis in [5] on the unitary cut constructible for super-Yang-Mills theory, we can get A P = A C 1n .
Now we define the operation ∪ on two rational functions in the function set group A c ij and A C ij as (4.12) The operation can be divided into two steps: First, choose a proper R C ij such that all the terms T (ij),(i ′ j ′ ) , which have one cut in C ij and another cut in C i ′ j ′ , are the same in both A C ij and A C i ′ j ′ , while other terms T (ij) or T (i ′ j ′ ) in A C ij and A C i ′ j ′ can only have one cut either in C ij or C i ′ j ′ respectively 7 ; Second, add all the terms of same formula once and all other terms. Hence we can get Such definition is similar for the union with A c ij .
We unite the integrand from all the unitarity cuts in the union order one by one. We begin from the integrand A C 12 from the unitarity cut c 12 , and then unite integrand A C 34 and A C 12 and then others in the order. Finally we can obtain A C 1n . To this end, we first introduce some lemmas.

Lemma 1 For a pentagon integrand defined by five lines
and an arbitrary plane P under the unitarity cuts of the underlined propagators, where Y i is proportional to a line which will keep the scalar invariance of the integrand.
Proof: We transform the integrand with numerator ABP x ∩ P j as following.
ABP x ∩ P j ABi − 1i ABii + 1 ABj − 1j ABjj + 1 ABj + 1j + 2 (4.14) Then we expand the intersection of two plane as where the coefficients a i are constant which is related to the plane P x . It is obvious that only the term with line L j−1j+1 are not obviously of the form in (4.14). The numerator of this term is Since any point in twistor space of CP 3 can be expand as four independent point. Here we choose four base points {Z j , Z i−1 , Z i , Z i+1 } to expand points {Z m ′ Z m ′ +1 }. Then we can expand bi-twistor (m ′ m ′ + 1) based on (ji − 1), (ji), (ji + 1), (i − 1i), (ii + 1), (i − 1i + 1). Then According to the Schouten identity (A.7), it is easy to see that only the terms with line L i−1i+1 is not obvious to of the form (4.14) ABi − 1i + 1 ABj − 1j + 1 . Then expanding point Z j−1 as {Z j , Z j+11 , Z j+2 , Z i }, (4.17) can be transformed as Finally all the terms are obviously of form (4.14) according to (A.7). The integrand with numerator ABP i ∩ P x can be also proved similarly.

Lemma 2 For any unitarity cuts c i+1j , we can choose a line
where The geometry of the terms are shown in Fig. 17.
Proof: According to Lemma 1, it is easy to see that the integrand (4.11) can be transformed to pentagons and boxes with one propagator ABm ′ m ′ + 1 . The explicit form can be calculated directly. First, we can expand bi-twistor (Z m ′ Z m ′ +1 ) based on six bi-twistors in the set where (Z k Z l ) is the line in set L which do not have the common point with (Z m Z n ). Then we add (Z A Z B ) to get a ABm ′ m ′ + 1 as where the unitarity cut condition ABii + 1 , ABjj + 1 = 0 has been applied.
Then we add term ABm ′ m ′ + 1 to both denominator and numerator of A c i+1j , which will not affect the final answer. We first deal with the term A c i+1j ABm ′ m ′ +1 . According to (4.10) and (4.11), we can get Put them together, the term ABij + 1 vanishes, resulting (4.21) and this is exactly one term of the final answer. Similarly, we can get In order to get the other two terms, we need to transform as based on Schouten identity and unitarity cut condition. Also, use the relation between dash line and wavy line, we can get if we use this equation to replace ABij , while the remaining ABj + 1i + 1 will be canceled by terms in ABm ′ m ′ + 1 , and will become Lemma 3 Under the unitarity cut c i+1n−1 we have where Proof: First we expand ABP j ∩P i+1 and add a term ABi−1i both in the numerator and denominator. Since two parts have the same denominator, we only deal with the numerators. We first deal with ABP j ∩ P i+1 ABi − 1i = ABj − 1j j + 1ii + 1i + 2 + ABjj + 1 j − 1ii + 1i + 2 + ABj − 1j + 1 ii + 1i + 2j .

Lemma 4
The integrand A c i+1n−1 is equal to the integrand A C 1n−2 with reference line L n−1 under unitarity cut c i+1n−1 .
Proof: First we find out the terms in A C 1n−2 which contain both ABii + 1 and ABn − 1n . We need to prove After sum over all possible terms of c i+1n−1 , only four terms left For every term in previous equation, This result is the same as

Theorem 1 The integrand of MHV one loop amplitudes can be constructed by the union over all the integrand from each unitarity cut. If we choose the reference line to be L n−1 , the integrand of MHV one loop amplitudes is
Proof: As discuss above, for convenience, we unite the A c ij in a union order. In general, when uniting integrands A c i−1j and A C ij , the only cut-related terms in A C ij which may influence the unitarity cuts of c i−1j are {c i−1j−1 , c ij , c ij−1 }. On the other hand, A c i−1j may affect the cuts {c i−1j−1 , c ij , c ij−1 } in C ij . Hence we only need verify that the cut-related terms in A C ij are either same with the terms in A c i−1j or cut un-related with each other. This is easy to prove according to Lemma 2. Hence we can unite over all the terms except the last column in 1. According the Lemma 4, we do not need to unite over the integrand from the last column and the union of C 1n−2 is just the final integrand from the unitarity cuts up to a rational function on all the unitarity cuts. We find such construction of integrand is equivalent to the integrand from single cuts in [21,22].

MHV nonplanar amplitudes and unitarity cuts
In this section, we will present a general recipe for dealing with MHV nonplanar amplitudes. One-loop four-point and five-point amplitudes are carefully worked out as examples.

Properties of MHV nonplanar amplitudes under unitarity cuts
We consider first the situation with only one nonplanar leg. Based on the permutation relations of Yangian Invariants (2.1) this amplitude under a unitarity cut can be converted to a planar one at a price of a simple factor f kin in Fig. 18  This step removes one nonplanar leg. For more than one nonplanar legs, say m, we can repeat this operation m times to arrive at a planar diagram. Planar MHV amplitudes under unitarity cuts have been discussed in section 4.1 we can therefore apply the planar results to obtain the nonplanar MHV amplitude after a unitarity cut by A c (1, 2, . . . , j|j + 1, . . . , n − 1,ṅ) = f kin A c (1, j|j + 1, n), (4.36) where 'i' labels the nonplanar leg and A is planar if there is no dotted leg in the arguments. This equation clearly shows that the difference between the planar result and the nonplanar one lies in the kinematic factors.

Reconstructing the kinematic factors:
Amplitudes under a unitarity cut only contains two variables related to the loop momentum. Unitarity cut condition sets two cut propagators to zero, while fixing the other two variables. With this in mind we need to reconstruct the amplitudes to a function of four variables. A unitarity cut on nonplanar diagrams contains two pieces of information, the kinematic factor and the corresponding planar amplitudes. We need to discuss them separately. As reconstruction of planar part is the same as above we deal with the kinematic factors here.
For one nonplanar leg (labelled n), we can read off from Fig. 18 the kinematic factor as f kin = n − 1n l l n − 1l nl (4.37) Using Schouten identity, and expanding spinor λ n−1 based on λ j+1 and λ n as λ n−1 = n − 1, n j + 1, n λ j+1 + j + 1, n − 1 j + 1, n λ n , f kin can be expanded as f kin = −1 + ln l j + 1 ln − 1 l n n − 1, n j + 1, n + l n l, n − 1 Under a unitarity cut, variables l andl in momentum space are denoted by variables A and B in momentum twistor space. So ln l j + 1 can be rewritten as A n B j + 1 . Using the equation we express f kin as The next step is to combine kinematic factors with the planar amplitudes. Recall that the planar amplitudes A 0 (1, . . . , j|j + 1, . . . , n) (Here we add the MHV tree amplitudes as a coefficient because, in nonplanar situation, this coefficient is important and can not be omitted). It is then clear that the numerator A n B j + 1 o1 will cancel the same term in the denominator of A 0 (1, 2, . . . , j|j + 1, . . . , n) by adding a new term related to o2 or o3, and consequently changing the color ordering. Hence This is nothing but A 0 with unphysical poles, which needs to be converted to A 1 , which only contains physical poles, to arrive at the final results. This can be achieved in a way completely analogous to the procedures presented in Section 4.1.
Comparison with U(1) decoupling relation As alluded above a relation of reconstructed amplitudes from permutation relation of Yangian Invariants has only these three terms (or two terms in the case of four-point). (1, 2, . . . , j|j + 1, . . . , n − 1,ṅ) = f kin A 1 (1, 2, . . . , j|j + 1, . . . , n) = − A 1 (1, 2, . . . , j|j + 1, . . . , n) − A 1 (1, 2, . . . , j|n, . . . , n − 1) − n n − 2 n − 1 j + 1 n − 2 n − 1 n j + 1 · A 1 (1, 2, . . . , j|j + 1, . . . , n, n − 1) When considering the U(1) decoupling relation more terms will appear. However, due to the special property of MHV amplitudes, the exchange of external legs which are not the four legs connecting the "basic twin-box" will not affect the final result of this cut. That is to say, all of the possible cases in this set have the same cut amplitudes. The only difference is the pre-factor, MHV tree amplitude. If we expand the factor n n − 2 n − 1 j + 1 n − 2 n − 1 n j + 1 we can find cuts with all possible color ordering, which are the same as obtained from U(1) decoupling. This form is obviously more compact than U(1) decoupling, since it combines, in one integral, terms related to planar amplitudes of different orders but having same result under a given unitarity cut.

Final results of MHV nonplanar amplitudes:
Final results of MHV nonplanar amplitudes are the union of all possible f kin A 1 where A N P stands for the final results of nonplanar amplitudes and A P is the final results of planar counterparts.
This formula just give a procedure, which is the same as the planar situation, to get the final result. However, to get the general formula of integrand, a lemma needs to be proved. When two terms in two different cut results are cut related (one term has propagators of the other unitarity cuts), one of them can convert to terms which are the same as the other one and others that are cut un-related. We will prove it in future work. In the following sections, we will show this method is valid in some particular examples of four-point and five-point situation.

One-loop four-point MHV nonplanar amplitudes
In this section we describe a nonplanar four point amplitude with one nonplanar leg (we call "(3+1)" case) as an explicit example of our construction.
We start with the cut result of this case, which is shown in Fig. 10. First, we convert this cut amplitude to planar one with a kinematic factor. in Fig. 19.  Using "square moves" and "merges" to represent the diagram as shown in [35].
Using the unitarity cut condition AB23 = 0, AB41 = 0,we can rewrite the kinematic factor as Here we define two color orders, o 1 means color order (1,2,3,4), and o 2 means (1,2,4,3). Combine the factor and planar result, we can get Other possible unitarity cuts should be taken into consideration. The steps to deal with all possible unitarity cuts are as described above. Since each term is actually the planar four-point one-loop amplitude with a certain color ordering, we can simplify the expression as Now we need to unite A 1 (2, 3|1,4) and A 1 (1, 2|3,4). Obviously, the terms with o1 are the same in these two. While it is not obvious to judge whether other terms are cut related or not. However, we can set one of the cut propagator in all terms as l 2 in momentum space and write down the denominators of these three terms.

One-loop five-point MHV nonplanar amplitudes
We first obtain all the unitarity cuts of five point nonplanar amplitudes A c (1, 2, 3, 4,5) as following Cut Type I For the Cut Type I, the tree level part of nonplanar leg is four point amplitude. While for the Cut Type II, the tree level part of nonplanar leg is five point amplitude. In both cases, all the tree part under a unitarity cut are MHV amplitudes.
Since case I are actually the same situation as the four-point ones, we just simply write down the result.  The planar part can be done with the same strategy above, and covert to box integrand. Here we first deal with the kinematic factor f kin = 45 ll l4 l 5 Considering different color order, we can get Uniting A 1 (1, 2|3, 4,5) and A 1 (2, 3|4, 1,5) based on the same method in four-point case, we can find that every term is only cut-related with the results of its own color order, while not affect those of other orders under unitary cuts. The final non-planar integrand is the union of all five cuts, This result contains all of the possible results of unitary cuts in all possible color orders (i, i + 1, i + 2, i + 3, 5), i = 1, 2, 3, 4 of planar amplitudes. For instance, we can find all unitary cuts A 1 (i, i + 1|i + 2, i + 3, i + 4), i = 1, 2, 3, 4, 5 of order (1,2,3,4,5). According to the discussion in planar MHV amplitudes, the union of these cuts can get A P (1,2,3,4,5). In the same way, we can get A P (1, 2, 3, 5, 4), A P (1, 2, 5, 3, 4), A P (1, 5, 2, 3, 4). Since results from different orders do not affect each other under unitary cuts, the union of all non-planar results equals to the sum of the unions of every order A N P (12345) = A P (12345) + A P (12354) + A P (12534) + A P (15234). (4.54)

NMHV nonplanar amplitude from double unitarity cuts
In this section we will present results of a 6-point one loop NMHV nonplanar amplitude, by double unitarity cuts, in the invariant top form. It is convenient to see the geometric structures of the amplitudes from this invariant top form [35]. In order to write the total amplitude in the top form the newly found permutation relation of the Yangian Invariants again comes in handy.
Before investigating specific examples we propose the general procedures of constructing total amplitudes for one loop nonplanar Feynman diagrams. At one loop level the planar diagrams corresponds to the single-trace partial amplitudes in color-order decomposition. The planar on-shell diagrams are associated with the (kxn) Grassmannian Matrices [35], C, It is of convenience to view the Grassmannian cell C as a collection of k-dimensional columns { c 1 , c 2 · · · c n }. There are k×(n−k) parameter for a generic Grassmanian matrix C, of which 2n − 4 parameter are determined by the δ-functions δ(C · λ ) and δ( λ · C ⊥ ) in (2.13) and others are determined by the linear-structures of the on-shell reduced diagrams. As discussed in [35], for planar diagrams, such linear-structures of lineardependencies among consecutive chains of columns is known as positroid stratification [53,54]. The top form of correct singularities should be Ω = d k×n C vol(GL(k)) 1 (1 · · · k) · · · (n · · · k − 1) , where (1 · · · k) is the minor of matrix { c 1 · · · c k }. Hence the top forms of planar diagrams are characterized by consecutive minors of the Grassmannian matrix.
The linear-dependencies of an on-shell diagram correspond to k × (n − k) − (2n − 4) minors m i in M vanishing. The set of these k × (n − k) − (2n − 4) minors is denoted as M I . As conclusion, the on-shell bipartite diagram of the leading singularities of the loop amplitudes correspond to the Grassmannian cell C k×n whose values can be completely fixed by the δ-function in (2.13) and the constraints m i = 0 for arbitrary m i ∈ M I . In this way terms with the same Grassmannian geometry are collected together. We shall be using these properties in a crucial way to study the singularity structures of nonplanar NMHV amplitudes in this section. We shall as before use a specific example, in this case a 6-point NMHV nonplanar amplitude, to assist a general discussion whenever appropriate.
For the on-shell diagrams with same linear-structures characterized by M I , we define a function picks up the residue on one minor m i = 0 upon an integration along the contour, C inḠ m i =0 . In order to form an invariant top form and to include all existent poles, P g ≡ n i=1 m g i and P g should scale uniformly as P g (tC) = t k×n P g (C) and contain all the factors m i ∈ M I . We therefore propose a general formula for nonplanar one-loop diagram in the invariant top form A k n (1 · · · j,j + 1 · · ·ṅ) = M I ⊂M F M I (1 · · · j,j + 1 · · ·ṅ), (5.5) where F M I (1 · · · j,j + 1 · · ·ṅ) ≡ g N g F g M I . The sum runs over all the top forms with poles on the hypersurfaces, defined by m i = 0, in G(k, n). The coefficients, N g , do not depend on C.
We consider a 6-point one-loop amplitude A(1, 2, 3, 4, 5,6) with "6" being the nonplanar leg. More general one-loop amplitudes and higher-loop amplitudes will be left to a future publication. The set of minors is   We need to verify that the amplitude is in the form And we also need to determine the N j . To this end we classify all the leading singularities into three type: Type I: The nonplanar leg belongs to a 3-point amplitude after a double unitarity cut, as shown in Fig. 24, which are the same as a planar diagram up to a minus sign.
Type II: The nonplanar leg belongs to a 4-point amplitude after a double unitarity cut, as shown in Fig. 25, which can be transformed into a planar diagram up to a kinematic factor.
Type III: The nonplanar leg lies in a 5-point amplitude after a double unitarity cut, as shown in Fig. 26, which can also be transformed into a planar diagram up to an overall coefficient.
These three types of singularities are presented in Figs. 24, Fig. 25, and Fig. 26, respectively.
To arrive at the top form (5.9) we need to group terms with the same vanishing minor together. For example, if we consider (123) = 0, then according to Fig. 24,  Fig. 25  and it works similarly for others. For Type II and Type III, the coefficients in (5.10) are obtained by the permutation relation of a "box." As it is explained in detail in Section 2.1, the permutation relation do not change the geometry of the Grassmannian cell.

Conclusion and Outlook
In this paper we present a new and useful permutation relation of Yangian Invariants. Different from KK and BCJ relations working at the level of amplitudes, it unveils a relation between two Yangian Invariants with two consecutive legs exchanged. Interesting properties governing the permutations of Yangian Invariants can be uncovered in the bipartite on-shell diagram. For instance all Yangian Invariants have at least one "box" connecting to two external legs. When these two legs are exchanged the Grassmannian matrix does not change but maintain the same geometric property. The two Yangians are related by a simple kinematic factor which can be calculated recursively by BCFW method.
However, it is not always obvious to find the "box" due to the equivalence of bipartite on-shell diagrams. To this end we give a simple criterion from the associated permutation to check whether a given pair of consecutive legs are connected to a "box". Because all consecutive legs in MHV amplitudes 8 lie in a "box," we can exchange any two legs at the expense of the kinematic factor. Most importantly, for a general diagram, if we exchange two lines-either internal or external-connecting to a "box" the geometry of the underlying Grassmannian will not be affected. This property can be interpreted as a new generator of equivalence in bipartite on-shell diagram-apart from square moves and mergers already observed in [35].
In the case of NMHV amplitudes there will be a special case-but only one case-that cannot be molded into a "box." There arises a second basic building block in bipartite diagram, a "bridged twin-box" (Fig. 6), the permutation relation of which is discussed in Section 2.3. With these two permutation relations we can resolve all permutations in NMHV amplitudes in the process of constructing their total on-shell integrals.
In this paper we also present a systematic way to deal with the integrands of scattering amplitudes using unitarity cuts. Momentum twistor space is a natural language to reconstruct integrand without unphysical propagators. We discover a new way to add BCFW bridges and a new operation called "union" is introduced to combine results from different cuts to arrive at the total integrands. For one-loop planar MHV amplitudes our results coincide with those obtained from single cuts. The advantage of our proposal is its easy extension to NMHV and higher loops.
For nonplanar loop amplitudes we apply unitarity cuts to fix the loop momenta endowing them with a reasonable definition in the loop integrand. A crucial relation between planar and nonplanar elements has been discovered which, in turn, enable us to turn nonplanar components into planar ones at the expense of a simple kinematic factor. With on-shell diagrams we present detailed and systematic constructions of the total integrands for four-and five-point one-loop nonplanar MHV amplitudes. The kinematic factors as well as the corresponding planar amplitudes are separately dealt with using unitarity conditions. Final results are the "union" of all results reconstructed from all possible unitarity cuts.
Generalized unitarity cuts are used to address NMHV amplitudes. With six-point one-loop nonplanar as an explicit example, the amplitude after double unitarity cutswith all loop momenta being fixed by the cut constraints-is a leading singularity without any variables. Interesting geometric properties, nevertheless, can be found in the nonplanar leading singularities: it is the result of top-forms integrating around different poles.
There is an abundance of interesting open questions generated from these ideas. In the next paper we will present findings on the leading singularities in bipartite on-shell diagrams as well as a systematic way of building these diagrams in the twistor space. This way of dealing with leading singularities lends itself straightforward applications to higher loops. Furthermore, according to the geometric properties of the Yangian Invariants, say, collinearity or coplanarity of several points, we can further classify the permutation relations; and we will probably find permutation relations of nonadjacent legs. Moreover, interesting geometric shapes, such as knots, will appear in two loops. Ideas and methods in topology are called for to deal with higher-loop nonplanar amplitudes. Last but not the least we will apply our methodology to N < 4 SYM or gauge theories in other dimensions.
In momentum twistor space, any arbitrary set of five twistors {Z a , Z b , Z c , Z d , Z e } will satisfy the following identity, Z a bcde + Z b cdea + Z c deab + Z d eabc + Z e abcd = 0.
(A. 5) According to this, we could obtain the 5-term identity also called a Schouten identity: f gha bcde + f ghb cdea + f ghc deab + f ghd eabc + f ghe abcd = 0. (A.6) We will show another frequently used identity related to A and B, which is very analogous to (A.4),