Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes
Baoyi Chen, Gang Chen, Yeuk-Kwan E. Cheung, Yunxuan Li, Ruofei Xie, Yuan Xin
aa r X i v : . [ h e p - t h ] F e b Noname manuscript No. (will be inserted by the editor)
Nonplanar On-shell Diagrams and LeadingSingularities of Scattering Amplitudes
Baoyi Chen, Gang Chen, Yeuk-Kwan E.Cheung, Yunxuan Li, Ruofei Xie, YuanXin
Received: date / Accepted: date
Abstract
Bipartite on-shell diagrams are the latest tool in constructing scat-tering amplitudes. In this paper we prove that a Britto-Cachazo-Feng-Witten(BCFW)-decomposable on-shell diagram process a rational top-form if andonly if the algebraic ideal comprised of the geometrical constraints is shiftedlinearly during successive BCFW integrations. With a proper geometric inter-pretation of the constraints in the Grassmannian manifold, the rational top-form integration contours can thus be obtained, and understood, in a straight-forward way. All rational top-form integrands of arbitrary higher loops leadingsingularities can therefore be derived recursively, as long as the correspondingon-shell diagram is BCFW-decomposable.
Keywords
Nonplanar Amplitudes, Non-positive Grassmannians, N=4 SuperYang-Mills, Unitarity Cuts, BCFW
Scattering amplitudes are of profound importance in high energy physics de-scribing the interactions of fundamental forces and elementary particles. Thescattering amplitudes are widely studied for N = 4 super Yang-Mills theoryand QCD. At tree level, BCFW recursion relations [1,2,3,4] can be used tocalculate n-point amplitudes efficiently. Unitarity cuts [5,6,7] and generalized G. ChenDepartment of Physics, Zhejiang Normal University, Jinhua, Zhejiang Province, ChinaDepartment of Physics, Nanjing University, 22 Hankou Road, Nanjing 210093, P. R. ChinaB.Y. Chen, Y.E. Cheung, Y.X.Li, R.F. Xie, Y. XinDepartment of Physics, Nanjing University, 22 Hankou Road, Nanjing 210093, P. R. ChinaPlease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle unitarity cuts [8,9,10,11,12,13,14,15] combined with BCFW for the rationalterms work well at loop level [16,17,18,19,20,21].Leading singularities [22] are closely related to the unitarity cuts of loop-level amplitudes. For planar diagrams in 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 sym-metry [29,30,31,32,33]. The leading singularity can also be used in construct-ing one-loop amplitudes by taking this as the rational coefficients of the scalarbox integrals. Extending this idea to higher loop amplitudes are reported in [10,34,35,36].A leading singularity can be viewed as a contour integral over a Grass-mannian manifold [37,38,39,40,41,42]. This expression of the leading singu-larity keeps many symmetries, in particular, the Yangian symmetry, cyclicand parity symmetries, manifest. On the one hand this new form makes theexpression of amplitudes simple and hence easy to calculate. On the otherhand 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 N = 4 super Yang-Mills(SYM) , Arkani-Hamed et al [43] proposed using positive Grassmannian tostudy them along with the constructions of the bipartite on-shell–all internallegs are put on shell–diagrams [44].Top-forms and the d“ log ” forms of the Grassmannian integrals are system-atically studied for planar diagrams. Each on-shell diagram corresponds to aYangian invariant, as shown in [31] at tree level and [32,33,47] at loop level.(See also [48,49] for earlier works 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 and a new book [78].)We report, in this paper, our detailed and systematic studies of the non-planar on-shell diagrams which can be decomposed in by removing BCFW-bridges and applying U(1) decoupling relation of the four- and three-pointamplitudes (or just decomposable diagrams for convenience). For wide classesof leading singularities, the corresponding on-shell diagrams are decomposablediagrams. We first construct the chain of BCFW-decompositions for the on-shell diagrams. During this process we obtain the unglued diagram by cuttingan internal line. We prove any unglued diagrams can be categorized into threedistinct classes which can be subsequently turned into identity utilizing cru-cially the permutation relation of generalized Yangian Invariants [79]. Thisconstruction is presented in Section 2.We then proceed to study the geometry of the leading singularities. Weare interested in the constraints encoded in the Grassmannian manifolds andhow these constraints determine the integration contours in the top-forms. Asthe cyclic order is destroyed by non-planarity the integrand of Grassmannian onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 3 integral also needs to be constructed from scratch. To achieve the above goalswe attach non-adjacent BCFW-bridges to the planar diagrams and observehow the integrands and the C-matrices transform. Further we can constructthe (rational) top-form including both integrand and integration contour ofany nonplanar leading singularity by attaching (linear) BCFW-bridges to theidentity diagram in the reverse order of the BCFW decomposition chain fromthe previous section. This construction is presented in Section 3 In an on-shell diagram representing an L -loop leading singularity, we are freeto pull out a planar sub-diagram (unglued diagram) between two internal looplines–both are also on-shell as shown in Fig. 1 . Locally the sub-diagram is Fig. 1
The L-loop’s d log form can be obtained by reducing to (L-1)-loop problem. planar except that we cannot perform BCFW integrations on these two looplines. We proved that every such sub-diagram, upon the removal of all BCFWbridges in the permutations, can be casted into one of the three distinct typesof skeleton graphs. U (1)-decoupling relation can be further performed on thelatter two types. And the L -th loop is unfolded. Unfolding the loops recursively,we obtain the BCFW decomposition chain for the leading singularities of any L -loop nonplanar amplitudes. In other words the BCFW chain captures allthe information of the leading singularities of the L-loop nonplanar graphs.We are thus able to reconstruct the on-shell diagrams by attaching BCFW-bridges from the identity.In this section, we will introduce a systematic way of finding the BCFWbridge decomposition chain from the marked permutations of the unglueddiagrams In this paper we assume there are more than two external lines for this planar sub-diagram. If there is only one external line in the sub-diagram, we need to use other methodwhich is presented in our following paper. Marked permutations refer to permutations with two end points treated specially–thetwo points are allowed to marked to themselves or to each other.Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle
From unglued diagram to skeleton diagram
All unglued diagrams can be cate-gorized into three groups depending on the permutations of the two cut lines,denoted as A and ¯ A , (1) σ ( A ) = ¯ A && σ ( ¯ A ) = A ; (2) σ a ( A ) = ¯ A && σ a ( ¯ A ) = A , or σ b ( ¯ A ) = A && σ b ( A ) = ¯ A ; (3) σ ( A ) = ¯ A && σ ( ¯ A ) = A .To decompose an unglued diagram, the first step is the full removal of twotypes of adjacent bridges on the target diagram: the white-black bridge andthe black-white bridge as shown in Fig.2. The changes to the permutation af-ter removing either of them are, respectively, σ → σ ′ = Z ( k, k + 1) · σ and σ → σ ′ = Z ( σ − ( k ) , σ − ( k + 1)) · σ , where Z ( k, k + 1) is a Z permutationbetween line k and k + 1[43]. For an unglued diagram arisen from a nonpla- Fig. 2 white-black bridges and black-white bridges nar leading singularity, the cut line should not be involved in BCFW bridgedecompositions as the pair of marked lines are to be glued back eventually.Thus we should restrict the set of allowed BCFW bridge decompositions tothose preserving the two marked legs. By following this restriction, the groupour target unglued diagram originally belongs to will not alter during bridgedecompositions.Due to the existence of nonadjacent bridges, an unglued diagram cannotbe fully decomposed and will pause at a certain diagram. It is easy to seethat after removing all BW- and WB- bridges the three groups of unglueddiagrams will fall into
External line pair , Black-White Chain and
Box Chain respectively. The three categories are named after their general patterns as onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 5 shown in Fig. 3. We refer to any diagram belonging to the above three cate-gories as the “skeleton diagram” , naming after its skinny looks. As long as wecan fully decompose all three skeleton diagrams, it is then direct to obtain thecomplete decomposition chain of any unglued diagram. (a) External line Pair (b) Black-White Chain (c) Box Chain
Fig. 3
Skeleton Diagrams
Fig. 4 U (1) decoupling relation of 3-point amplitudes. Fig. 5 U (1) decoupling relation of 4-point amplitudes. From skeleton diagram to identity – External Line Pair:
Most external lines are paired. The external linesnext to the internal cut line may also attach to the black/white verticesor be paired with the internal cut line as shown in Fig 3(a). For this typeof on-shell diagrams, gluing back the internal lines and removing all pairswill lead to the identity. – Black-White Chain:
In this case, white and black vertices are connectedtogether recursively, as shown in Fig 3(b). To further decompose, we usethe amplitudes relation A ( a , a , a ) = −A ( a , a , a ) (see Fig 4) to twist Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle one down-leg to up-leg. Then an adjacent bridge will appear. By removingthe new appeared BCFW bridge, the diagram is unfolded into a planardiagram. The diagram can then be decomposed to identity according to itspermutation [43]. – Box Chain:
In this case, the diagram is composed by boxes linked to achain, as shown in Fig 3(c). Using the U (1) decoupling relation [81] of thefour point amplitudes (see Fig 5), this diagram turns into the sum of twodiagrams with adjacent BCFW bridges. The non-adjacent legs of the boxwill become adjacent under this operation. Performing adjacent BCFWdecompositions on both diagrams will unfold the loop and arrive at twoplanar diagrams, which can be decomposed to identity. Top–form
Through the BCFW bridge decompositions we obtain the d log form charac-terized by the bridge parameters. The d log form can be viewed as an explicitparameterization of a more general integration over the Grassmannian mani-fold, which is invariant under the GL ( k ) transformations. The invariant form,known as the “top-form,” for planar diagrams has been constructed in [43]. Inthis section, we construct the top-form for the nonplanar leading singularities.Recent progress on nonplanar on-shell diagram can be find in [45,46]For planar diagrams, the top-form manifests the Yangian symmetry: theleading singularities can be written as multidimensional residues in the Grass-mannian manifold G ( k, n ), T kn = I C ⊂ Γ d k × n Cvol ( GL ( k )) δ k × ( C · e η ) f ( C ) δ k × ( C · e λ ) δ × ( n − k ) ( λ · C ⊤⊥ ) , (1)where Γ is a sub-manifold of G ( k, n ). Γ is constrained by a set of linear relationsamong the columns of C –certain minors of C be zero. As any function of theminors of C , f ( C ), has the scaling property f ( tC ) = t k × n f ( C ).To construct the top-form for nonplanar leading singularities, we need todetermine the integration contour Γ and the integrand f ( C ). Since the in-tegration contour is constrained by a set of geometrical relations linear in α ’s, we make use of the BCFW chain we obtained in Section 2 to look forall geometric constraints, fixing Γ in the process. Next we will see, with theBCFW approach extended to loop-level, the integrand of the top-form can becalculated by attaching BCFW bridges. onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 7 C matrix. In fact, the total space is taken as ( k − a , b yields a linear transformations of the two columns, a and b , a −→ b a = α a + b ;whereas adding a black-white bridge means b −→ b b = α b + a .For convenience, we divide the geometric constraint into two types: Simplecoplanar constraint and tangled coplanar constraint. Simple coplanarity is justthe coplanarity among the points corresponding to the external line. For thetangled coplanar constraint, at least one point is formed by the intersection ofsuper-planes characterized by the point of the external line. We first presentan example for each case. Example for simple coplanarity
As an explicit example, we work out A ’sgeometry shown in Fig. 6. This diagram becomes planar upon removal of awhite-black bridge (1,4). The remaining BCFW bridge decomposition is:(1 , → (2 , → (3 , → (2 , → (1 , → (3 , → (2 , , (5) , (6) In the Grassmannian matrix, all elements in the three columns are zero. We
Fig. 6
The BCFW bridge to open the loop of the diagram.Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle
Table 1
The evolution of the geometry constraints under adding BCFW bridges. The firstrow is the linear relation in the identity diagram and the column on the left represents thebridge decomposition chain.Bridge coplanar constraints begin (4) (5) (6) (2,6) (4) (5) (2 , (3,5) (4) (3 , (2 , (1,2) (4) (3 , (1 , , (2,3) (4) (2 , , (1 , , (3,4) (3 , (2 , (3 , , (1 , , (2,3) (2 , , , (1 , , (1,2) (3 , , (1 , , (1,4) (1 , , then reconstruct the diagram through attaching BCFW bridges. There areeight bridges needed to construct the nonplanar diagram. Each step will di-minish one coplanar relation. For instance, the first step is adding a white-black bridge on external line 2 and 6, leaving column c + αc . The relation (6) then becomes (2 , . Similarly upon attaching bridges(3 , , (1 , , (2 , , (2 , , , (1 , , . Uponattaching bridge (3 , becomes (3 , . This means that point3 and 4 merge to a one point. Then the constraint (2 , , can be written as(2 , (3 , , . Attaching the bridges consecutively as shown in Tab. 3.1, wecan finally get the coplanar constraint (1 , , for the on-shell diagram. Example for tangled coplanarity
We consider a nonplanar 2-loop diagram, A as shown in Fig. 7. The BCFW decomposition chain is Fig. 7
An example of nonplanar two loop diagram A . (1 , → (3 , → (2 , → (1 , → (3 , → (2 , → (3 , → (3 , . (2)Then, the top-form of the diagram can be reconstructed by attaching thesebridges one by one. There are eight bridges and each one diminish a coplanarconstraint as shown in Tab. 3.1The first seven bridges attached yield simple coplanar constraints. Then thegeometry constraints is (2 , , , (3 , , , which indicates that points 2 , , onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 9 Table 2
The evolution of the geometry constraints under adding BCFW bridges.Bridge coplanar constraints begin (4) (5) (6) (3,6) (4) (5) (3 , (3,5) (4) (3 , (5 , (2,3) (4) (2 , , (5 , (3,4) (34) (2 , , (5 , (1,2) (3 , (5 , (2,3) (2 , , (5 , (3,5) (2 , , (3 , , (1,3) (234)(214) − (356)(156) = 0 and 3 , , , ˆ3 , , , b ∩ (2 4) lie in the line of (5 6). Forconvenience, we denote this tangled coplanar relation as (cid:0) (1 b ∩ (2 4) 5 6 (cid:1) .Thegeometry evolution under the last bridge (1 ,
3) is shown in Fig. 8.
Fig. 8
The constraint of the diagram is (cid:0) (1 b ∩ (2 4) 5 6 (cid:1) . In projective space, it means line(1 b General simple coplanar constraints
We first discuss the cases without tangledcoplanar constraints. We can classify the coplanar constraints into four sets \authorrunning and \titlerunning prior to \maketitle according to the elements: 1 . ( a , b , , ...m ) k . ( a , , ...m ) k . ( b , , ...m ) k . (1 , , ...m ) k where k to k are the ranks of the minors. Without loss of generality, we shallmake k < m + 2, k < m + 1, k < m + 1 and k < m . We call the minor complete if only if adding any other column to the matrix will make the rankincrease by one. From now on we shall assume that all the minors in the abovesets are complete in the following discussion. In fact, incomplete minors canalways be transformed into the complete ones by adding to the bracket all thenecessary elements while keeping the rank unaltered.Attaching a white-black bridge does not change the rank of minors in set a and b are both in this set. The minors in set set b is excluded from these two sets. ( α a + b ) / ∈ span { b , , ... m } ,thus after adding a white-black bridge the only set with its rank altered is set set a, b, , ...m ) k +1 and (1 , ...m ) k . Similarly, upon attaching a black-white bridge, the minors in set a, b, , ...m ) k +1 and (1 , ...m ) k . We have completed thediscussion of how constraints alter during each step of bridge decompositions.Next we turn to attaching bridges starting from the identity with the iden-tity diagram being a matrix with n − k columns of zero vectors. Each timewe attach a BCFW bridge, the number of independent geometric constraintswill decrease by one. This can be proved through the following procedure.Attaching the bridge (a, b) affects the linear relation involving b . The onlyexceptions are the relations containing both a and b , which will not be affectedby the bridge (a, b).( b, , ...m ) k −→ (cid:26) ( a , b , (1 , ... m ) k ) k +1 (1 , ...m ) k If k = m , the linear relation (1 , ...m ) k does not give rise to any constraint, thus( a, b, , ...m ) k +1 has one higher rank than ( b, , ...m ) k . The constraints’ num-ber is then diminished by one upon attaching the bridge. If k < m , the coplanarconstraints ( b, , ...m ) k can be decomposed to ( b, (1 , ...m ) k ) k and (1 , ...m ) k .The independent constraints after attaching the bridge are ( a, b, (1 , ...m ) k ) k +1 and (1 , ...m ) k . Comparing the constraints between ( b, (1 , ...m ) k ) k and ( a, b, (1 , ...m ) k ) k +1 ,the number of constraints is reduced by one upon adding the BCFW bridge. General Tangled Coplanar Constraints
Now we discuss the tangled coplanarconstraints. When we attach a BCFW bridge, X → b X = X + αY . There are The bridge at least shifts one of the constraints linearly. We will show in next sectionthat this condition is equivalent to that the top-form is rational.onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 11 two constraints ( XA A . . . A i ) i and ( XB B . . . B i ) i both containing theshifting leg. Without losing of generality, we assume that there is no constraintsamong A , A . . . A i and B , B . . . B i . Upon attaching a BCFW bridge atangled constraint could be obtained: α = V i +1 ( b XA . . . A i ) V i +1 ( Y A . . . A i ) = V i +1 ( b XB . . . B i ) V i +1 ( Y B . . . B i ) , (3)where V stands for the volume of the hyperpolyhedron. It is indeed a geometric Fig. 9
The diagrammatic notation of Eq. 4. The plane or hyperplane A = Span ( XA A . . . A i ) and B = Span ( XB B . . . B i ). α can be regarded as the ratio oftwo hyperpolyhedron’s volumes. constraint on the Grassmannian manifold–the point of intersection of one lineand one hyperplane lying on another hyperplane: (cid:16) ( b XY ) ∩ ( B B . . . B i ) A A . . . A i (cid:17) . (4)We denote the intersection point of ( b XY ) ∩ ( B B . . . B i ) as Q = C b X + C Y .Since Q also lies in the plane ( B B . . . B i ), C V i +1 ( b XB B . . . B i ) + C V i +1 ( Y B B . . . B i ) = 0 . Thus the initial constraint ( QA A . . . A i ) i directly yields Eq. 3. One mayhave noticed that the point Q is precisely the point X before shifting. If wego on attaching another bridge that involves this tangled constraint, the setof constraints can again be written as minors of the Grassmannian matrix.Therefore we conclude that general constraints can always be labelled us-ing nested spans and intersections. Consider attaching a linear BCFW bridge( Y, X ) in an arbitrary amplitude, a constraint to be shifted is M ( X ) ≡ (cid:16) · · · ( XA (0)1 A (0)2 · · · A (0) a ) ∩ ( B (1)1 B (1)2 · · · B (1) b ) A (1)1 A (1)2 · · · A (1) a ) ∩ ( B (2)1 B (2)2 · · · B (2) b ) A (2)1 A (2)1 · · · A (2) a ) · · · ∩ ( B ( m )1 B ( m )2 · · · B ( m ) b m ) A ( m )1 A ( m )2 · · · A ( m ) a m (cid:17) , \authorrunning and \titlerunning prior to \maketitle with X being the external line to be shifted, A ( · ) and B ( · ) denoting twosets of external lines. If column X ∈ Set [ A ( · ) ] or X ∈ Set [ B ( · ) ], they canbe freely replaced by ˆ X after attaching the bridge involving X . Otherwise theconstraint will be a nonlinear function of α , resulting in an irrational top-form.We present a counter example in Appendix to illustrate this point. We canthen simplify M ( X ) as follows (cid:16) L ( m ) ( X ) ∩ ( B ( m )1 B ( m )2 · · · B ( m ) b m ) A ( m )1 ¯ A ( m )2 · · · ¯ A ( m )¯ a m (cid:17) , where ¯ A ( · ) are some points or hyperplanes composed by A ( · ) and B ( · ) and areeasily obtained through a simple relation,( A A A ) ∩ P = ((( A A ) ∩ P ) (( A A ) ∩ P )) = (cid:0) (( A A ) ∩ P ) ¯ A (cid:1) . After the shift the constraint M ( X ) is removed. In order to obtain the otherconstraints, we look for the b C representation of X . This is achieved by unfold-ing the nested intersections level by level.To write a constraint in a compact form and make the geometric relationsencoded manifest, We define a line L ( X ), for i ∈ [2 , m ] L ( i ) ( X ) ≡ (cid:16) L ( i − ( X ) ∩ ( B ( i − · · · B ( i − b i − ) A ( i − (cid:17) . We further define a point R ( i ) ( X ) ≡ L ( i ) ( X ) ∩ ( B ( i )1 · · · B ( i ) b i ), for i ∈ [2 , m ].Given a minor M ( X ), we could obtain the point R ( m ) ( X ) as R ( m ) ( X ) = L ( m ) ( b XY ) ∩ ( B ( m )1 · · · B ( m ) b m ) ∩ ( A ( m )1 · · · ¯ A ( m )¯ a m ) . All levels of R ( i ) ( X ) can be recursively obtained according to R ( i − = ( R ( i ) A ( i − ) ∩ (cid:16) L ( i − ( b XY ) ∩ ( B ( i − · · · B ( i − b i − ) (cid:17) . The geometrical relations is shown in Fig. 10. Finally we are able to denote thecolumn X using the columns in the shifted Grassmannian, X = ( R (1) A (0)1 ) ∩ ( b XY ) . After removing the constraint M ( X ) = 0, the remaining constraintsare invariant under the b C representation of X , making them independent ofthe shift ( Y, X ).For now we have obtained geometry constraints according to BCFW bridgedecomposition chain. We would like to stress that our approach can be ap-plied to seeking all loop leading singularity’s geometry constraints. During theprocess, we introduced a method that the constraints are independently andcompletely represented. The constraints of the graph constructed by any “top-form bridge” are immediately obtained using our method. Thus the top-formintegrations’ contour Γ is determined. onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 13 →→ Fig. 10
The geometric relation indicated by the ground level (the upper figures) and the i -th level (the lower figures) of the nested minor. The left figures are the relations before theBCFW shift. The right figures are the relations after the BCFW shift. d log forms befound. We should stress, however, that not all nonplanar on-shell diagramshave rational top-forms; and it is worth to remark on which kind of non-planar on-shell diagrams can have rational top-forms. We address this ques-tion by building up an equivalent relation between rational top-form and linearBCFW bridges . If a BCFW bridge results in the shifted constraint function tobe a linear function of α , we call this BCFW bridge a linear BCFW bridge .A constraint function F i is a rational function of the minors of Grass-mannian matrix, C . Altogether they span an algebraic ideal I [ { F i } ]. Undera BCFW shift X → b X = X + αY , a constraint is eliminated, with C being \authorrunning and \titlerunning prior to \maketitle transformed to b C . The transformed f ′ ( b C ) is also rational iff α is also a rationalfunction of b C .Next we need to show that rationality of α is guaranteed by the linearBCFW shifts. To prove their equivalence, assuming α = P ( b C ) /Q ( b C ) , where P, Q are polynomials of minors of b C . Expanding P, Q as polynomials of α aswell as the minors of C , α = P + P α + P α + · · · + P N α N Q + Q α + · · · + Q N α N − . The coefficient in each power of α , such as P , P − Q , P − Q , · · · , P N − Q N ,is supposed to vanish. If any of them appears nonzero, it must fall into theideal I [ { F i } ]. This means that all the coefficients are constraints of C . Underthe shift, P i and Q i become b P i = P i + N − i X j =1 (cid:18) ji + j (cid:19) P i + j α j b Q i = Q i + N − − i X j =1 (cid:18) ji + j (cid:19) P i + j α j . (5)The constraint P N − Q N remains the same after the shift b P N − b Q N = P N − Q N .The constraint P N − − Q N − = 0 appears linearly dependent on α upon b P N − − b Q N − = [ N P N − ( N − Q N ] α. If [
N P N − ( N − Q N ] does not vanish then the constraint P N − − Q N − = 0is the removed. Otherwise P N = Q N = 0 and the constraint P N − − Q N − = 0is shifted linearly, b P N − − b Q N − = [( N − P N − − ( N − Q N − ] α. Since P ( b C ) cannot be totally independent of α , we can trace the constraintsfrom N to 0 until we find one constraint ( P i − Q i ) which is a linear functionof α after a shift. This constraint is then the constraint being removed.The proof of the reverse is also straightforward: if one constraint becomesa linear function of α under a shift, for instance F i ( C ) → F i ( b C ) = F i ( C ) + F ′ i ( C ) α = F ′ i ( b C ) α, where F i ( C ) vanishes and F ′ i ( C ) is invariant under the shift, we have α = F i ( b C ) /F ′ i ( b C ) . Note that the remaining s − F i ( · · · b X − αY · · · ) = 0, for i ∈ [2 , s ], which are invariant under the shift.Finally we conclude that upon adding a BCFW bridge the on-shell diagramresulted has a rational top-form if and only if the shift on the algebra ideal I is linear. For a generic on-shell diagram, BCFW-bridges can be added in anarbitrary manner and the transformations on the constraints are complicated.Top-forms can be obtained if and only if when the BCFW parameters shift theconstraints linearly. This type of bridges is thus called linear BCFW bridges .In the construction of top-forms one should avoid using BCFW bridges thatshift the constraints in a nonlinear manner. onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 15 f ( C ). It must then contain thosepoles equivalent to the constraints in Γ to keep the non-vanishing of the circle-integration in Eq. 1. Each BCFW bridge removes one pole in f ( C ) by shiftinga zero minor to be nonzero: in tangled cases the poles in the integrand mustchange their forms accordingly.To see this we parameterise the constraint matrix, C , using the BCFWparameter, α . In the last BCFW shift X → b X = X + αY, several minorsin f ( C ) become functions of α . There exists at least one minor M ( b X ) = M ( X )+ αR ( Y ) having a pole at α = 0. After this shift, M ( X ) → M ( b X ) , theconstraint M ( X ) = 0 is removed. And α = M ( b X ) /R ( Y ) is then a rationalfunction of b C and can be subtracted from other shifted minors to obtain theshift-invariant minors of b C , M i ( X ) = M i ( b X − αY ) . This is demonstrated inSec. 3.2We can further attach a BCFW bridge to the integrand, f ( b C ) = M ( b X ) Y i M i ( b X − αY ) × (cid:18) minorswithout α (cid:19) . (6)In this way top-forms of leading singularities of scattering amplitudes can thusbe obtained–be it planar or nonplanar–and from tree level to all loops. We il-lustrate our method below with several examples: searching for the constraintsand calculating the top-form integrand.3.4 Several Examples A one loop example of attaching a nonadjacent bridge
As an application,we take the nonplanar diagram in Fig. 11 as an example. According to the
Fig. 11
A nonplanar on-shell diagram of A \authorrunning and \titlerunning prior to \maketitle permutation of the planar diagrams before attaching the bridge (1 , , , , and (8 , , , . And the top-form is T = I Γ d C (1234)(2345)(3456)(4567)(5678)(6781)(7812)(8123) , (7)where we have omitted the delta functions for clarity. Attaching bridge (1 , , , , is unaffected while (4 , , , → (4 , , , ((4 , , ,where ˆ6 = 6 + α
1. We choose the shifted pole as (3456). The shift parameter α can be written as α = (345ˆ6)(3451) . According to Eq. 6, we get T = I Γ d C (1234)(2345)(3456)(45(ˆ6 − α − α , (8)where (45(ˆ6 − α − (345ˆ6)(3451) (4517) = − (3457)(1456)(3451)and (5(ˆ6 − α − (345ˆ6)(3451)(5178) = − (3458)(1567)(3451) . Finally we obtain top-form of A : c T = I b Γ d × b Cvol ( GL (4)) (1345) (1234)(2345)(3456)(3457)(1456)(3458)(1567)(6781)(7812)(8123) . Therefore, we can always construct the top-from of nonplanar diagrams byattaching adjacent and nonadjacent bridges on identity diagram step by step.
A tangled two-loop example
At multi-loop level the geometric constraints fora nonplanar leading singularity can be highly tangled, as the diagrams cannot,in general, be reduced to the planar ones by KK-relation [81].The diagram is a planar one before attaching the bridge (3,5) in the seventhstep and the top-form is T = I Γ d C (123)(234)(345)(456)(561)(612) , where we have omitted the delta functions for clarity. After attaching BCFWbridge (3 , T = I Γ d C (361)(123)(234)(345)(356)(146)(561)(612) , (9) onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 17 here the constraints Γ is determined by the linear relation (2 , , and (3 , , as shown in Table 3.1. We do not distinguish the labels Γ for each step inadding the BCFW bridges. According the discussion above, upon attachingthe last BCFW bridge (1 , → (234) = 0 , (234)(214) − (356)(156) = 0 . Adding the bridge (1 ,
3) and the elimination of(234) = 0 leaves (234)(214) − (356)(156) = 0 invariant. Applying the same linear trans-formation to the denominator:1(234)(356) → − b (cid:16) − (156)(2 b + ( b (cid:17) , we extract the top-form c T = I Γ d b C (214) ( b b b (cid:16) (1 b ∩ (24)56 (cid:17) (cid:16) (1 b ∩ (24)45 (cid:17) (146)(561)(612) . MHV top-form and its simplification
The MHV top-form can be further sim-plified. We can always transform any nonplanar MHV top-form into a summa-tion of several top-forms. These generated top-forms share features that theirnumerators of the integrands equal 1 and the minors in f ( C ) are of cyclicorders, which are exactly those of planar MHV top-forms. This yields a strongproof that any nonplanar MHV amplitude is a summation of several planaramplitudes.To see this, let us consider the top-form of A n . Attaching a nonadjacentbridge ( a, b ) to a planar diagram yields c T n = Z dαα I Γ d × n Cvol ( GL (2)) δ ( C )(12) · · · ( b − , b )( b, b + 1) · · · ( n b − , b ) = 0. Following thesame procedure illustrated in Eq. 1, we obtain1 f ( b C ) = ( a, b − · · · ( b − , b − b − , b b )( a, b b )( b + 1 , b − b + 1 , b + 2) · · · ( n a < b −
1, we define a + m = b − m ∈ Z + ). – If m = 1, the numerator is then ( b − , b −
1) and the integrand can besimplified to a term with its numerator equaling one and f ( C ) of cyclicorders, i.e. a planar one. \authorrunning and \titlerunning prior to \maketitle – If m > a + 1 , b b ): f ( C ) = ( a,b − a +1 , b b )(12) ··· ( b − ,b − b − , b b )( a, b b )( b +1 ,b − b +1 ,b +2) ··· ( n a +1 , b b ) = ( a,a +1)( b b,b − a, b b )( a +1 ,b − ··· ( b − ,b − b − , b b )( a, b b )( b +1 ,b − b +1 ,b +2) ··· ( n a +1 , b b ) = ··· ( a − ,a )( a +1 ,a +2) ··· ( b − ,b − a, b b )( b − ,b +1)( b +1)( b +2) ··· ( n a +1 , b b ) + ( a +1 ,b − ··· ( b − ,b − b − , b b )( a +1 , b b )( b +1 ,b − b +1 ,b +2) ··· ( n . The first term is already planar, while the second is not obvious. – If m = 2, the second term is planar. – If m >
2, we multiply the integrand by ( a + 2 , b ) , ( a + 3 , b ) , · · · , ( a + m − , b ) one by one. For each step of multiplication, we utilize thePluck relation to transform the nonplanar term into a summation ofplanar terms and a remaining term. The final term left after series ofmultiplication is ( a + m − , b − · · · ( b − , b − b − , b b )( a + m − , b b )( b + 1 , b − b + 1 , b + 2) · · · ( n . Since a + m = b −
1, this term is also planar.Following these steps, we can finally simplify the top-forms of all nonplanarMHV amplitudes into the sum of planar ones. One can easily verify thatthe simplification process from nonplanar one to planar term’s summation isequivalent to applying KK relation to MHV amplitudes.In this section, we construct the top-forms of the nonplanar on-shell 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 integrandfrom the planar ones. Keep attaching bridges on the identity and we can arriveat the top-form of our target–the nonplanar leading singularity. We then breakdown the top-forms of the nonplanar MHV amplitudes into a summation ofthe planar top-forms. For the leading singularities of the one-loop amplitudes,this simplification is similar to the KK relation. For leading singularities of thegeneral amplitudes, the relation between the top-form’s simplification and theKK relation will be discussed in our future work. We have classified nonplanar on-shell diagrams according to whether theyposses rational top-forms, and proved its equivalence to linear BCFW bridges.We conclude that when attaching linear bridges, geometric constraints of thenonplanar diagrams–tangled or untangled–can all be constructed systemati-cally. With this chain of BCFW bridges rational top-forms of the nonplanaron-shell diagrams can then be derived in a straightforward way. This method onplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes 19 applies to leading singularities of nonplanar multi-loop amplitudes beyondMHV.
Acknowledgements
GC thanks Nima Arkani-Hamed for helpful discussion and usefulcomments. We thank Peizhi Du, Shuyi Li and Hanqing Liu for constructive discussion. YuanXin thanks Bo Feng for introducing the background on the recent developments of scatteringamplitude. GC, RX and HZ have been supported by the Fundamental Research Funds forthe Central Universities under contract 020414340080, NSF of China Grant under contract11405084, the Open Project Program of State Key Laboratory of Theoretical Physics, In-stitute of Theoretical Physics, Chinese Academy of Sciences, China (No.Y5KF171CJ1). Wealso thank Y. Gao, T. Han for hospitality and Key Laboratory of Theoretical Physics forhosting.
All rational top-forms can be constructed by our method described above. Inour discussion we have assumed that the BCFW bridges are the linear BCFWbridges –each successive α -shift transforms the constraints linearly and can thusbe represented by a rational function of minors of the underlying constraintC-matrix. However not all on-shell diagrams are made up completely of suchbridges: the constraints can be nonlinear in alpha and cannot be written asrational functions under some shift. Such an on-shell diagram will not have arational top-form.We present a counter example, A . Upon attaching the bridge (3 , , , , and (1 , , , emerge (the superscripts denote thenumber of independent column vectors), with two columns, 3 and 4, beingthe same. Attaching the bridge (5 , , ∩ (2 , , , , , . If we go on attaching thebridge (7 , α -shift cannot be represented linearly bysome minor being zero, violating our linearity requirements in the constructionof rational top-forms. References
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