Nonplanar onshell diagrams and leading singularities of scattering amplitudes
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Abstract
Bipartite onshell diagrams are the latest tool in constructing scattering amplitudes. In this paper we prove that a Britto–Cachazo–Feng–Witten (BCFW) decomposable onshell diagram process a rational top form if and only if the algebraic ideal comprised the geometrical constraints are shifted linearly during successive BCFW integrations. With a proper geometric interpretation of the constraints in the Grassmannian manifold, the rational top form integration contours can thus be obtained, and understood, in a straightforward way. All rational top form integrands of arbitrary higher loops leading singularities can therefore be derived recursively, as long as the corresponding onshell diagram is BCFW decomposable.
Keywords
Geometric Constraint External Line Grassmannian Manifold Planar Diagram Yangian Symmetry1 Introduction
Scattering amplitudes are of profound importance in high energy physics describing the interactions of fundamental forces and elementary particles. The scattering amplitudes are widely studied for \(\mathcal {N}=4\) super Yang–Mills theory and QCD. At tree level, BCFW recursion relations [1, 2, 3, 4] can be used to calculate npoint amplitudes efficiently. Unitarity cuts [5, 6, 7] and generalized unitarity cuts [8, 9, 10, 11, 12, 13, 14, 15] combined with BCFW for the rational terms work well at loop level [16, 17, 18, 19, 20, 21].
Leading singularities [22] are closely related to the unitarity cuts of looplevel amplitudes. For planar diagrams in \(\mathcal {N} = 4\) super Yang–Mills [23, 24], the leading singularities are invariant under Yangian symmetry [25, 26, 27, 28], which is a symmetry combining conformal symmetry and dual conformal symmetry [29, 30, 31, 32, 33]. The leading singularity can also be used in constructing oneloop amplitudes by taking this as the rational coefficients of the scalar box integrals. Extending this idea to higherloop amplitudes are reported in [10, 34, 35, 36].
A leading singularity can be viewed as a contour integral over a Grassmannian manifold [37, 38, 39, 40, 41, 42]. This expression of the leading singularity keeps many symmetries, in particular, the Yangian symmetry, cyclic and parity symmetries, manifest. On the one hand this new form makes the expression of amplitudes simple and hence easy to calculate. On the other hand it is related to the central ideas in algebraic geometry: Grassmmannian, stratification, algebraic varieties, toric geometry, and intersection theory etc. For leading singularities of the planar amplitudes in \(\mathcal {N}=4\) super Yang–Mills (SYM), ArkaniHamed et al. [43] proposed using positive Grassmannian to study them along with the constructions of the bipartite onshell – all internal legs are put on shell – diagrams [44].
Top forms and the d“log” forms of the Grassmannian integrals are systematically studied for planar diagrams. Each onshell diagram corresponds to a Yangian invariant, as shown in [31] at tree level and [32, 33, 47] at loop level. (See also [48, 49] for earlier work and [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64] for a sample of interesting developments thereafter, and [65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77] for a sample of reviews; also a new book has appeared [78].)
We report, in this paper, our detailed and systematic studies of the nonplanar onshell diagrams which can be decomposed by removing BCFW bridges and applying a U(1) decoupling relation of the four and threepoint amplitudes (or just decomposable diagrams for convenience). For wide classes of leading singularities, the corresponding onshell diagrams are decomposable diagrams. We first construct the chain of BCFW decompositions for the onshell diagrams. During this process we obtain the unglued diagram by cutting an internal line. We prove any unglued diagrams can be categorized into three distinct classes which can be subsequently turned into identity utilizing crucially the permutation relation of generalized Yangian invariants [79]. This construction is presented in Sect. 2.
We then proceed to the study of the geometry of the leading singularities. We are interested in the constraints encoded in the Grassmannian manifolds and how these constraints determine the integration contours in the top forms. As the cyclic order is destroyed by nonplanarity the integrand of Grassmannian integral also needs to be constructed from scratch. To achieve the above goals we attach nonadjacent BCFW bridges to the planar diagrams and observe how the integrands and the Cmatrices transform. Further we can construct the (rational) top form including both integrand and integration contour of any nonplanar leading singularity by attaching (linear) BCFW bridges to the identity diagram in the reverse order of the BCFW decomposition chain from the previous section. This construction is presented in Sect. 3
2 Scattering amplitudes: BCFW decomposition
In an onshell diagram representing an Lloop leading singularity, we are free to pull out a planar subdiagram (unglued diagram) between two internal loop lines – both are also onshell as shown in Fig. 1.^{1} Locally the subdiagram is planar except that we cannot perform BCFW integrations on these two loop lines. We proved that every such subdiagram, upon the removal of all BCFW bridges in the permutations, can be cast into one of the three distinct types of skeleton graphs. A U(1)decoupling relation can be further performed on the latter two types. And the Lth loop is unfolded. Unfolding the loops recursively, we obtain the BCFW decomposition chain for the leading singularities of any Lloop nonplanar amplitudes. In other words the BCFW chain captures all the information of the leading singularities of the Lloop nonplanar graphs. We are thus able to reconstruct the onshell diagrams by attaching BCFW bridges from the identity.
In this section, we will introduce a systematic way of finding the BCFW bridge decomposition chain from the marked permutations^{2} of the unglued diagrams.
2.1 From permutations to BCFW decompositions
 (1)
\( \sigma _1(A) \ne \bar{A} ~ \& \& ~ \sigma _1(\bar{A}) \ne A \);
 (2)
\( \sigma _2^a(A) = \bar{A} ~ \& \& ~\sigma _2^a(\bar{A}) \ne A \), or \( \sigma _2^b(\bar{A}) = A ~ \& \& ~\sigma _2^b(A) \ne \bar{A}\);
 (3)
\( \sigma _3(A) = \bar{A} ~ \& \& ~\sigma _3(\bar{A}) = A\).

External line pair: Most external lines are paired. The external lines next to the internal cut line may also attach to the black/white vertices or be paired with the internal cut line as shown in Fig. 3a. For this type of onshell diagrams, gluing back the internal lines and removing all pairs will lead to the identity.

Black–white chain: In this case, white and black vertices are connected together recursively, as shown in Fig. 3b. To further decompose, we use the amplitudes relation \(\mathcal {A}(a_1, a_2, a_3)=\mathcal {A}(a_1,a_3,a_2)\) (see Fig. 4) to twist one downleg to upleg. Then an adjacent bridge will appear. By removing the new appeared BCFW bridge, the diagram is unfolded into a planar diagram. The diagram can then be decomposed to identity according to its permutation [43].

Box chain: In this case, the diagram is composed of boxes linked to a chain, as shown in Fig. 3c. Using the U(1) decoupling relation [80] of the four point amplitudes (see Fig. 5), this diagram turns into the sum of two diagrams with adjacent BCFW bridges. The nonadjacent legs of the box will become adjacent under this operation. Performing adjacent BCFW decompositions on both diagrams will unfold the loop and arrive at two planar diagrams, which can be decomposed to identity.
3 Scattering amplitudes: the top form
Through the BCFW bridge decompositions we obtain the dlog form characterized by the bridge parameters. The dlog form can be viewed as an explicit parameterization of a more general integration over the Grassmannian manifold, which is invariant under the GL(k) transformations. The invariant form, known as the “top form,” for planar diagrams has been constructed in [43]. In this section, we construct the top form for the nonplanar leading singularities. Recent progress on nonplanar onshell diagrams can be found in [45, 46].
To construct the top form for nonplanar leading singularities, we need to determine the integration contour \(\Gamma \) and the integrand f(C). Since the integration contour is constrained by a set of geometrical relations linear in the \(\alpha \), we make use of the BCFW chain we obtained in Sect. 2 to look for all geometric constraints, fixing \(\Gamma \) in the process. Next we will see, with the BCFW approach extended to loop level, the integrand of the top form can be calculated by attaching BCFW bridges.
3.1 Geometry and the BCFW bridge decomposition
In this subsection we shall introduce the method of searching for geometric constraints in the Grassmannian matrix. Geometric constraints are linear relations among columns of C matrix. In fact, the total space is taken as \((k1)\)dimensional projective space. Each column denoted by the index of the external line can be map to a point in the projective space. Each time we attach a bridge a constraint will be fixed and the geometry constraints change accordingly. In the Grassmannian matrix, adding a white–black bridge on external lines \(\mathbf {a}\), \(\mathbf {b}\) yields a linear transformations of the two columns, \(\mathbf {a}\) and \(\mathbf {b}\), \(\mathbf {a} \longrightarrow \widehat{\mathbf {a}} =\alpha \mathbf {a}+\mathbf {b}\); whereas adding a black–white bridge means \(\mathbf {b}\longrightarrow \widehat{\mathbf {b}}=\alpha \mathbf {b}+\mathbf {a}\).
For convenience, we divide the geometric constraint into two types: simple coplanar constraint and tangled coplanar constraint. Simple coplanarity is just the coplanarity among the points corresponding to the external line. For the tangled coplanar constraint, at least one point is formed by the intersection of superplanes characterized by the point of the external line. We first present an example for each case.
In the Grassmannian matrix, all elements in the three columns are zero. We then reconstruct the diagram through attaching BCFW bridges.
There are eight bridges needed to construct the nonplanar diagram. Each step will diminish one coplanar relation. For instance, the first step is adding a white–black bridge on external line 2 and 6, leaving \(column\ 6\) to become \(c_6+\alpha c_2\). The relation \((6)^{0}\) then becomes \((2,6)^{1}\). Similarly upon attaching bridges (3, 5), (1, 2), (2, 3), the coplanar constraints are \((4)^{0}\), \((2,3,5)^2\), \((1, 2, 6)^{2}\). Upon attaching bridge (3, 4), constraint \((4)^{0}\) becomes \((3,4)^{1}\). This means that points 3 and 4 merge to a single point. Then the constraint \((2,3,5)^2\) can be written as \((2,(3,4)^{1},5)^2\). Attaching the bridges consecutively as shown in Table 1, we can finally get the coplanar constraint \((1,2,6)^{2}\) for the onshell diagram.
Example for tangled coplanarity We consider a nonplanar twoloop diagram, \(\mathcal {A}_{6}^{3}\), as shown in Fig. 7.
The evolution of the geometry constraints under adding BCFW bridges. The first row is the linear relation in the identity diagram and the column on the left represents the bridge decomposition chain
Bridge  Coplanar constraints 

Begin  \((4)^{0}\) \((5)^{0}\) \((6)^{0}\) 
(2, 6)  \((4)^{0}\) \((5)^{0}\) \((2,6)^{1}\) 
(3, 5)  \((4)^{0}\) \((3,5)^1\) \((2,6)^{1}\) 
(1, 2)  \((4)^{0}\) \((3,5)^1\) \((1,2,6)^{2}\) 
(2, 3)  \((4)^{0}\) \((2,3,5)^2\) \((1,2,6)^{2}\) 
(3, 4)  \((3,4)^{1}\) \((2,(3,4)^1,5)^2\) \((1,2,6)^{2}\) 
(2, 3)  \((2,3,4,5)^2\) \((1,2,6)^{2}\) 
(1, 2)  \((3,4,5)^2\) \((1,2,6)^{2}\) 
(1, 4)  \((1,2,6)^{2}\) 
The evolution of the geometry constraints under adding BCFW bridges
Bridge  Coplanar constraints 

Begin  \((4)^{0}\) \((5)^{0}\) \((6)^{0}\) 
(3, 6)  \((4)^{0}\) \((5)^{0}\) \((3,6)^{1}\) 
(3, 5)  \((4)^{0}\) \((3,5)^1\) \((5,6)^1\) 
(2, 3)  \((4)^{0}\) \((2,3,5)^2\) \((5,6)^{1}\) 
(3, 4)  \((34)^{1}\) \((2,3,5)^2\) \((5,6)^{1}\) 
(1, 2)  \((3,4)^{1}\) \((5,6)^{1}\) 
(2, 3)  \((2,3,4)^2\) \((5,6)^{1}\) 
(3, 5)  \((2,3,4)^2\) \((3,5,6)^{2}\) 
(1, 3)  \({(234)\over (214)}{(356)\over (156)}=0\) 
For now we have obtained geometry constraints according to BCFW bridge decomposition chain. We would like to stress that our approach can be applied to seeking all loop leading singularity’s geometry constraints. During the process, we introduced a method that the constraints are independently and completely represented. The constraints of the graph constructed by any “top form bridge” are immediately obtained using our method. Thus the top form integrations’ contour \(\Gamma \) is determined.
3.2 Rational top forms and linear BCFW bridges
Attaching BCFW bridges and using the 3 or 4point amplitude relations reductively, all nonplanar diagrams can be constructed and their dlog forms be found. We should stress, however, that not all nonplanar onshell diagrams have rational top forms; and it is worth to remark on which kind of nonplanar onshell diagrams can have rational top forms. We address this question by building up an equivalent relation between rational top form and linear BCFW bridges. If a BCFW bridge results in the shifted constraint function to be a linear function of \(\alpha \), we call this BCFW bridge a linear BCFW bridge.
A constraint function \(F_i\) is a rational function of the minors of the Grassmannian matrix, C. Altogether they span an algebraic ideal \(\mathcal {I}[\{F_i\}]\). Under a BCFW shift \( X \rightarrow \widehat{X} =X + \alpha Y\), a constraint is eliminated, with C being transformed to \(\widehat{C}\). The transformed \(f^{\prime }(\widehat{C})\) is also rational iff \(\alpha \) is also a rational function of \(\widehat{C}\).
Finally we conclude that upon adding a BCFW bridge the onshell diagram resulted has a rational top form if and only if the shift on the algebra ideal \(\mathcal {I}\) is linear. For a generic onshell diagram, BCFW bridges can be added in an arbitrary manner and the transformations on the constraints are complicated. Top forms can be obtained if and only if when the BCFW parameters shift the constraints linearly. This type of bridges is thus called linear BCFW bridges. In the construction of top forms one should avoid using BCFW bridges that shift the constraints in a nonlinear manner.
3.3 From BCFW decompositions to top forms
To obtain the top form of scattering amplitudes, besides the geometric constraints, we also need to get the integrand, f(C). It must then contain those poles equivalent to the constraints in \(\Gamma \) to keep the nonvanishing of the circleintegration in Eq. 1. Each BCFW bridge removes one pole in f(C) by shifting a zero minor to be nonzero: in tangled cases the poles in the integrand must change their forms accordingly.
To see this we parameterize the constraint matrix, C, using the BCFW parameter, \(\alpha \). In the last BCFW shift \( X\rightarrow \widehat{X}=X+\alpha Y, \) several minors in f(C) become functions of \(\alpha \). There exists at least one minor \( M_0(\widehat{X})=M_0(X)+\alpha R(Y) \) having a pole at \(\alpha =0\). After this shift, \( M_0(X)\rightarrow M_0(\widehat{X}), \) the constraint \(M_0(X)=0\) is removed. And \( \alpha =M_0(\widehat{X})/R(Y) \) is then a rational function of \(\widehat{C}\) and can be subtracted from the other shifted minors to obtain the shiftinvariant minors of \(\widehat{C}\), \( M_i(X)=M_i(\widehat{X}\alpha Y). \) This is demonstrated in Sect. 3.2
3.4 Several examples
The evolution of the geometric constraints with adding BCFW bridges for the diagram in Fig. 12
\((3)^{0}\)  \((4)^{0}\)  \((7)^{0}\)  \((8)^{0}\)  \((9)^{0}\)  \((10)^{0}\)  

(6, 10)  \((3)^{0}\)  \((4)^{0}\)  \((7)^{0}\)  \((8)^{0}\)  \((9)^{0}\)  \((6,10)^{1}\) 
(2, 6)  \((3)^{0}\)  \((4)^{0}\)  \((7)^{0}\)  \((8)^{0}\)  \((9)^{0}\)  \((2,6,10)^{2}\) 
(6, 9)  \((3)^{0}\)  \((4)^{0}\)  \((7)^{0}\)  \((8)^{0}\)  \((6,9)^{1}\)  \((2,6,10)^{2}\) 
(5, 6)  \((3)^{0}\)  \((4)^{0}\)  \((7)^{0}\)  \((8)^{0}\)  \((5,6,9)^{2}\)  \((2,6,10)^{2}\) 
(6, 9)  \((3)^{0}\)  \((4)^{0}\)  \((7)^{0}\)  \((8)^{0}\)  \((5,6,9)^{2}\)  \((2,6,9,10)^{3}\) 
(6, 8)  \((3)^{0}\)  \((4)^{0}\)  \((7)^{0}\)  \((6,8)^{1}\)  \((5,6,9)^{2}\)  \((2,6,9,10)^{3}\) 
(6, 7)  \((3)^{0}\)  \((4)^{0}\)  \((6,7)^{1}\)  \((6,8)^{1}\)  \((5,6,9)^{2}\)  \((2,6,9,10)^{3}\) 
(5, 6)  \((3)^{0}\)  \((4)^{0}\)  \((7,8)^{1}\)  \((5,6,8)^{2}\)  \((5,8,9)^{2}\)  \((2,8,9,10)^{3}\) 
(2, 5)  \((3)^{0}\)  \((4)^{0}\)  \((7,8)^{1}\)  \((6,8,9)^{2}\)  \((2,5,8,9)^{3}\)  \((2,8,9,10)^{3}\) 
(2, 4)  \((3)^{0}\)  \((2,4)^{1}\)  \((7,8)^{1}\)  \((6,8,9)^{2}\)  \((2,5,8,9)^{3}\)  \((2,8,9,10)^{3}\) 
(1, 2)  \((3)^{0}\)  \((1,2,4)^{2}\)  \((7,8)^{1}\)  \((6,8,9)^{2}\)  \((4,5,8,9)^{3}\)  \((4,8,9,10)^{3}\) 
(2, 3)  \((2,3)^{1}\)  \((1,2,4)^{2}\)  \((7,8)^{1}\)  \((6,8,9)^{2}\)  \((4,5,8,9)^{3}\)  \((4,8,9,10)^{3}\) 
(1, 2)  \((1,2,3)^{2}\)  \((1,3,4)^{2}\)  \((7,8)^{1}\)  \((6,8,9)^{2}\)  \((4,5,8,9)^{3}\)  \((4,8,9,10)^{3}\) 
(8, 2)  \((1,2,3,8)^{3}\)  \((1,3,4)^{2}\)  \((7,8)^{1}\)  \((6,8,9)^{2}\)  \((4,5,8,9)^{3}\)  \((4,8,9,10)^{3}\) 
(2, 8)  \((1,2,3,8)^{3}\)  \((1,3,4)^{2}\)  \((2,7,8)^{2}\)  \((6,7,9)^{2}\)  \((4,5,7,9)^{3}\)  \((4,7,9,10)^{3}\) 
(6, 1)  \((2,3,4,8)^{3}\)  \((1,3,4,6)^{3}\)  \((2,7,8)^{2}\)  \((6,7,9)^{2}\)  \((4,5,7,9)^{3}\)  \((4,7,9,10)^{3}\) 
(3, 4)  \((2,3,4,8)^{3}\)  \((1,3,4,6)^{3}\)  \((2,7,8)^{2}\)  \((6,7,9)^{2}\)  \((5,7,9,10)^{3}\)  
(5, 3)  \(((5,3)\cap (2,4,8),1,4,6)^{3}\)  \((2,7,8)^{2}\)  \((6,7,9)^{2}\)  \((5,7,9,10)^{3}\)  
(7, 4)  \((2,7,8)^{2}\)  \((6,7,9)^{2}\)  \((5,7,9,10)^{3}\) 
A tangled twoloop example At multiloop level the geometric constraints for a nonplanar leading singularity can be highly tangled, as the diagrams cannot, in general, be reduced to the planar ones by KK relation [80].

If \(m=1\), the numerator is then \((b2,b1)\) and the integrand can be simplified to a term with its numerator equaling one and f(C) of cyclic orders, i.e. a planar one.

If \(m>1\), we can multiply the numerator and denominator by \((a+1,\widehat{b})\):

If \(m=2\), the second term is planar.

If \(m>2\), we multiply the integrand by \((a+2,b),(a+3,b),\ldots ,(a+m1,b)\) one by one. For each step of multiplication, we utilize the Pluck relation to transform the nonplanar term into a summation of planar terms and a remaining term. The final term left after series of multiplication is
Following these steps, we can finally simplify the top forms of all nonplanar MHV amplitudes into the sum of planar ones. One can easily verify that the simplification process from nonplanar one to planar term’s summation is equivalent to applying the KK relation to the MHV amplitudes.
In this section, we construct the top forms of the nonplanar onshell graphs. The key step is attaching a nonadjacent BCFW bridge to a planar diagram. The cyclic order of f(C) is then broken and we obtain a different integrand from the planar ones. Keep attaching bridges on the identity and we can arrive at the top form of our target – the nonplanar leading singularity. We then break down the top forms of the nonplanar MHV amplitudes into a summation of the planar top forms. For the leading singularities of the oneloop amplitudes, this simplification is similar to the KK relation. For leading singularities of the general amplitudes, the relation between the top form’s simplification and the KK relation will be discussed in our future work (Table 3).
4 Conclusion
We have classified nonplanar onshell diagrams according to whether they possess rational top forms, and we proved its equivalence to linear BCFW bridges. We conclude that when attaching linear bridges, geometric constraints of the nonplanar diagrams – tangled or untangled – can all be constructed systematically. With this chain of BCFW bridges rational top forms of the nonplanar onshell diagrams can then be derived in a straightforward way. This method applies to leading singularities of nonplanar multiloop amplitudes beyond MHV.
Footnotes
 1.
In this paper we assume there are more than two external lines for this planar subdiagram. If there is only one external line in the subdiagram, we need to use another method, which is presented in our following paper.
 2.
Marked permutations refer to permutations with two end points treated specially – the two points are allowed to be marked to themselves or to each other.
 3.
The bridge at least shifts one of the constraints linearly. We will show in next section that this condition is equivalent to that the top form is rational.
Notes
Acknowledgements
GC thanks Nima ArkaniHamed for helpful discussion and useful comments. We thank Peizhi Du, Shuyi Li and Hanqing Liu for constructive discussion. Yuan Xin thanks Bo Feng for introducing the background on the recent developments of scattering amplitude. GC, RX, and HZ have been supported by the Fundamental Research Funds for the Central Universities under contract 020414340080, NSF of China Grant under contract 11405084, the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (No. Y5KF171CJ1). We also thank Y. Gao, T. Han for hospitality and the Key Laboratory of Theoretical Physics for hosting.
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