Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces
We present a general construction of two types of differential forms, based on any -dimensional subspace in the kinematic space of massless particles. The first type is the so-called projective, scattering forms in kinematic space, while the second is defined in the moduli space of -punctured Riemann spheres which we call worldsheet forms. We show that the pushforward of worldsheet forms, by summing over solutions of scattering equations, gives the corresponding scattering forms, which generalizes the results of Arkani-Hamed:2017mur (). The pullback of scattering forms to subspaces can have natural interpretations as amplitudes in terms of Bern-Carrasco-Johansson double-copy construction or Cachazo-He-Yuan formula. As an application of our formalism, we construct in this way a large class of scattering forms and worldsheet forms, which are in one-to-one correspondence with non-planar MHV leading singularities in super-Yang-Mills. For every leading singularity function, we present a new determinant formula in moduli space, as well as a (combinatoric) polytope and associated scattering form in kinematic space. These include the so-called Cayley cases, where in each case the scattering form is the canonical forms of a convex polytope in the subspace, and scattering equations admit elegant rewritings as a map from the moduli space to the subspace.
A new framework has been proposed in Arkani-Hamed:2017mur (), which naturally merges three lines of development in the study of new structures of scattering amplitudes: remarkable geometric constructions known as the amplituhedron in planar SYM Arkani-Hamed:2013jha (); Arkani-Hamed:2013kca (); Arkani-Hamed:2017vfh (), the Cachazo-He-Yuan (CHY) formulation Cachazo:2013hca (); Cachazo:2013gna (); Cachazo:2013iea () and (ambitwistor) string models Berkovits:2013xba (); Mason:2013sva () for the scattering of massless particles in general dimensions, as well as the color/kinematics duality and Bern-Carrasco-Johansson (BCJ) double copy Bern:2008qj (); Bern:2010ue (). The key ingredient is to consider differential forms in the kinematic space of massless particles, where we replace color factors of cubic tree graphs in color-dressed amplitudes by wedge-products of ’s, due to a basic observation that they satisfy the same algebra Arkani-Hamed:2017mur (). The BCJ color/kinematics duality means that kinematic numerators satisfy Jacobi identities just as the color factors, the form is required to be projective, which provides a geometric origin for the duality.
Any such projective, scattering forms, can be obtained as pushforward of some differential forms in the moduli space , which we call worldsheet forms. This is realized by summing over all solutions of scattering equations, and it is equivalent to CHY formula for the corresponding color-dressed amplitudes. Important examples include the forms for Yang-Mills theory and non-linear sigma model, which are uniquely fixed by gauge invariance and Adler’s zero, respectively. Partial amplitudes of the theory from color decomposition can be obtained from the pullback of the form to certain subspaces encoding the ordering Arkani-Hamed:2017mur (). These forms, however, are not canonical forms Arkani-Hamed:2017tmz () since they are not ’s with unit residues, but rather with residues depending on polarizations or other data.
The primary example for canonical forms is the planar scattering form, , which is a scattering form from summing over planar cubic trees respecting the ordering; it represents a bi-adjoint amplitude with color stripped for one of the two groups. Geometrically, the pullback of to the subspace for this ordering is the canonical form of an associahedron that is beautifully defined in the kinematic space Arkani-Hamed:2017mur (). Moreover, as a geometric reformulation of the CHY formula, the form can be obtained as the pushforward of the Parke-Taylor wordsheet form; the latter is the canonical form of the positive part of moduli space, , and scattering equations naturally provide a one-to-one map from (also an associahedron) to the kinematic associahedron.
Now it is very natural to ask, without the input of physical amplitudes or CHY formulas, what can be said about these scattering forms and corresponding worldsheet forms? More specifically, it is highly desirable to have a mechanism that can generate both forms in a straightforward way. In this note we will show that, starting from a -dimensional subspace of the kinematic space, one obtains general scattering forms and worldsheet forms without any other input. This has been implicitly stated in Arkani-Hamed:2017mur (): both the planar scattering form and the Parke-Taylor form, can be derived only from the subspace given by conditions for non-adjacent ’s with e.g. . In sec. 2 we generalize the associahedron story in a systematic way: for any -dimensional subspace, one constructs a scattering form by dressing each cubic tree with the pullback of its wedge product to the subspace, and a worldsheet form from the pullback of scattering equations to it. We will prove that the two forms, though constructed independently, have the remarkable properties that the latter pushes forward to the former.
The idea that forms can be constructed from subspaces is useful for connecting them to amplitudes, as we show in sec. 3. Just as color-ordered amplitudes are given by pullback to the subspace for an ordering, it is natural to interpret amplitudes as pullback, which is defined by a pair of subspaces. Such amplitudes are exactly those obtained from BCJ double copy, and they are naturally given by CHY formula associated with the two subspaces. This way of thinking allows one to view scattering equations in a novel way. Given any -dimensional subspace, the scattering equations can be rewritten as a manifestly SL(2)-invariant map from to this subspace; this is obtained by exploting the GL redundancy of the equations, with the Jacobian given by the worldsheet form. For the case of , the map was obtained in Arkani-Hamed:2017mur () and we will see that now it naturally generalizes to any subspace. In particular, for an infinite family of subspaces with combinatoric interpretation as spanning trees (called Cayley cases) Gao:2017dek (), the map has an elegant form which follows from simple graphic rules. As presented in Arkani-Hamed:2017mur (), the forms in these cases are canonical forms of so-called Cayley polytopes in kinematic space (generalizations of the associahedron). We will show in sec. 4 that they are naturally derived from the subspaces, and present the explicit construction for these Cayley polytopes.
To illustrate the power of our construction, in sec. 5 we go beyond Cayley cases and discover a much larger class of subspaces that also give forms. The most natural generalization is the so-called “inverse-soft” construction, which gives a class of -pt subspaces for forms from any -pt subspace. Concerning the worldsheet forms, our construction corresponds to the well-known “inverse-soft factor": as we will review, this can be used to recursively build MHV (non-planar) leading singularities in super-Yang-Mills (SYM). The most general MHV leading singularities were classified in Arkani-Hamed:2014bca (), which include but are not restricted to those constructed using inverse-soft factors. These leading singularities correspond to functions/forms in the moduli space that have nice properties such as factorizations Cachazo:2017vkf (). Explicitly, we find very simple subspaces whose worldsheet forms give any leading singularity functions, with a new formula that is very different from the one in Arkani-Hamed:2014bca (). It is intriguing that every MHV leading singularity, viewed as a worldsheet form, now has a scattering form and a (combinatoric) polytope in kinematic space associated with it. All these directly follow from the subspaces we constructed, and we conjecture that they are all simple polytopes.
2 Scattering forms and worldsheet forms from subspaces
In large enough spacetime dimensions, the kinematic space of massless particles, , can be spanned by all independent ’s, thus it has dimension . As shown in Arkani-Hamed:2017mur (), certain -dimensional subspaces of play an important role in the study of scattering forms and in particular canonical forms and positive geometries in kinematic space. In this paper, we initiate a systematic construction, which generalizes the results of Arkani-Hamed:2017mur (), for any generic -dimensional subspace . For any point , we construct two types of closely-related differential forms of dimension : one in the kinematic space and the other in the moduli space of -punctured Riemann spheres, .
The differential form in is a scattering form: a linear combination of ’s of propagators of cubic tree Feynman diagrams with external legs. Let’s denote the collection of all diagrams as ; each is specified by Mandelstam variables that are mutually compatible, which are denoted as . We define the wedge product of ’s for (the overall sign depends on ordering of ’s):
It is natural to consider the pullback of such wedge products to :
where is the Jacobian of with respect to ’s, which depends on the tangent space of at . It is possible that all ’s vanish at , then we say is degenerate at ; we say the subspace is non-degenerate if it is non-degenerate everywhere. It is natural to define a scattering form for at , which is non-vanishing for non-degenerate case
We emphasize that is completely determined by the tangent space of at .
A basic observation in Arkani-Hamed:2017mur () is that all linear relations among these wedge products are given by Jacobi identities which are equivalent to those of color factors. There is one such identity for any triplet of graphs, , that differ only by one propagator, see Figure 1. The wedge products of these three graphs satisfy the Jacobi identity
where the distinct Mandelstam variables are , respectively, and denote the wedge-products of the remaining propagators shared by the three graphs. Denoting the four propagators connecting to the four subgraphs as , the second equality follows from the basic identity implied by momentum conservation, . (4) implies that (3) is a projective form Arkani-Hamed:2017mur (), i.e. it is invariant under a GL transformation for all subsets (with depending on ). As shown in Arkani-Hamed:2017mur (), the projectivity of the form is guaranteed if for any three graphs as in Figure 1, we have
Similarly, we can study the pullback of scattering equations, (for ), to , and define its Jacobian with respect to :
where we delete three rows of the derivative matrix, and compensate with the factor . It is easy to check that by combining with the top-form of , we have a SL-invariant -form we call worldsheet form on :
This is a natural form in associated with the tangent space of at .
The main claim we make here is
The pushforward of gives :
where the sum is over the solutions of scattering equations.
One can show that vanishes if and only if is degenerate at , in which case (8) holds trivially. We will prove (8) in Appendix A, and let’s look at the case for now. is one-dimensional and its tangent space can be written as where guarantees . The projective scattering form in is given by
and the form obtained from the pullback of scattering equations in is
It is straightforward to see that (8) holds by plugging in the solution of .
In general, is a hypersurface and both forms are defined locally on it. In the following, we will consider the special case when is a hyperplane that can be defined by linear constraints on the Mandelstam variables. In this case one uses global coordinates for and any Mandelstam can be written as a linear combination of ’s when pulled back to . For a hyperplane , ’s become constants (independent of ) and the Jacobian becomes a rational function of ’s only. Note that different hyperplanes can give identical forms in and , and we will call them equivalent hyperplanes. In the following we focus on equivalence classes of hyperplanes.
The first and most important example of equivalence classes of was found in Arkani-Hamed:2017mur (). Let denote the class defined by the following constraints: for all non-adjacent of a set of labels e.g. . It has been shown in Arkani-Hamed:2017mur () that excluding any one label from gives equivalent hyperplanes, and in view of our construction the results can be summarized as
for all planar cubic trees with canonical ordering (with a sign flip for any two trees differ by one propagator), and for non-planar cubic trees.
, thus gives the canonical form of .
In this special case, (8) pushes the Parke-Taylor form, to the planar scattering form .
Our construction here can be viewed as a generalization to general (not necessarily ) forms, both in and . It is remarkable that they are completely determined by the choice of the hyperplane , without any other inputs. Generally, the meaning of such generalized scattering forms was discussed in Arkani-Hamed:2017mur (): they are the dual of color-dressed amplitudes in certain theories, where ’s are dual to color factors (the dual of Jacobi identities are given by (4)), and ’s the so-called Bern-Carassco-Johansson (BCJ) numerators that also satisfy Jacobi identities. It is an important open question how to find hyperplanes (or hypersurfaces) such that the ’s become BCJ numerators of a given theory, such as Yang-Mills theory (YM) or non-linear sigma model (NLSM) Du:2016tbc (); Carrasco:2016ldy (); Du:2017kpo (); Du:2018khm (); equivalently, one can try to find such that, on the support of scattering equations, the Jacobian equals the reduced Pfaffian, for YM, or for NLSM Cachazo:2014xea ().
3 Amplitudes from pullback: BCJ and CHY formulas
In the case of planar scattering forms, represents color-dressed amplitudes for one of the color groups in bi-adjoint scalar theory, and it is decomposed to the canonical ordering for the other group. Furthermore, the pullback of to gives the double-partial amplitude , where are the orderings for the two groups. In general, one can study pullback of to any hyperplane . Note that , the pullback reads
and we will call the expression inside the bracket the “amplitude” . This is reminiscent of the BCJ double-copy construction: given two color-dressed amplitudes/forms defined by and , the numerators and satisfy Jacobi identities, and the amplitude for the double-copy is exactly given by ! Note that for any hyperplane, defined from is equal to from , thus the amplitude is symmetric in and . For the special case of for some ordering , the pullback is equivalent to trace-decomposition of e.g. group, as studied in Arkani-Hamed:2017mur (). It gives the partial amplitude , for the form/color-dressed amplitude defined by the subspace . In particular, for , we recover .
Moreover, for both and , one can define the Jacobian of scattering equations, and . An interesting corollary of (8) is that the double-copy amplitude is given by the CHY formula with and :
Now on the LHS we also need to factor out , which means we want to rewrite the pullback as an integral with delta functions imposing scattering equations:
where the scattering equations inside the delta functions are written as a map from ’s to ’s, . Note that, according to (6), the Jacobian of the transformation from these equations to the standard scattering equations , is exactly ,
from which (12) follows directly! We have seen that arises from the pullback of scattering equations to any hyperplane . Equivalently, it is the Jacobian of fixing a GL symmetry of the equations, and the latter can be viewed as a map from to .
Obviously the equations are invariant under any transformation: where is a -dependent matrix. We can exploit the symmetry when considering the pullback to ; denote the constraints defining as for , and it is obvious that the ’s and the ’s form a basis of . In this basis, the scattering equations can be written as a matrix acting on this basis (here after deleting three equations),
It is clear that if is non-degenerate, the matrix formed by the first columns of , denoted as , must be invertible. We can choose its inverse, , as a transformation, and makes the first columns the identity matrix by acting on . After this transformation, we have where denotes the remaining part for and . We arrive at the scattering-equation map from to (recall that )
for . Since ’s are constants, each is expressed as a function of ’s, , by (17). Note that the Jacobian of the transformation depends on those three equations that are deleted, e.g. . To obtain a permutation-invariant Jacobian, we can define the matrix before deletion, , and the reduced determinant is exactly that given by (6): . An important point is that the rewriting, (17), makes the SL(2)-invariance of scattering equations manifest: each must be individually invariant under the SL(2) transformation of ’s since ’s and ’s are independent of ’s, thus it can only depends on cross-ratios of ’s. For a general , these ’s can be rather complicated. In the next section, however, we will encounter a class of hyperplanes where (17) takes an elegant form with ’s given explicitly.
4 Cayley cases: the rewriting, forms and polytopes
As proposed in Gao:2017dek () and studied in Arkani-Hamed:2017mur (), there is a very special class of hyperplanes on which the form , if not zero, can be interpreted as the canonical form of a convex polytope in , just like the associahedron for the planar case. These are the so-called Cayley cases, as each of them can be represented by a Cayley tree, or spanning tree of labelled vertices. We will see how the Cayley cases naturally arise from the simplest way of rewriting scattering equations as a map, how both forms can be naturally extracted from the tree for which (8) can be easily verified, and how to construct polytopes for these cases.
Recall that the kinematic information of the original scattering equations is encoded by ’s. By using momentum conservation, one can eliminate e.g. all ’s and write the equations in terms of ’s with . It is easy to see that the only remaining constraint is that the sum of them vanishes, and a basis of can be chosen as any of them (by eliminating any one of them). To rewrite the equations as a map, the most natural and the simplest way is to choose such that the ’s and ’s exactly form such a basis. By choosing of ’s to be ’s (the constants), one can associate the hyperplane with a graph, with edges corresponding to the ’s as the complement of ’s (the variables). Let’s denote the graph as and the hyperplane under consideration as , and it is clear that any of the complement can serve as ’s, that is the coordinates of . The first claim we make here is that is non-degenerate if and only if is a connected graph! Note that has vertices and edges, thus must be a tree graph as long as it is connected.
If is disconnected, we will see that both the matrix and the hyperplane are degenerate. Recall that now must have a connected component which is a tree with no more than vertices (see Figure 2 (a) for example), then contain only non-constant ’s after using momentum conservation to eliminate . This means that there are less than independent equations with respect to the ’s among the scattering equations, thus the Jacobian , or equivalently the matrix is not invertible. It is also straightforward to observe that vanishes on for every . Since is disconnected, let’s denote the two sets of vertices with no edge between and . The key is, for every cubic graph , there is at least one vertex that is attached to three edges corresponding to , where and (see Figure 2 (b) for example). By pullback to we have
thus . This can also be derived indirectly by using (8) and hence .
When is connected, it must be a spanning tree of vertices, . is defined by constant conditions where is not an edge of ; one can choose any of the ’s where is an edge as the coordinates of . For example, in Gao:2017dek (); Arkani-Hamed:2017mur () two extreme Cayley cases are the linear (or Hamiltonian) tree and the symmetric, star-shaped tree , which are illustrated in Figure 3.
Scattering-equation map and Cayley functions
Now we are ready to derive an elegant rewriting of scattering equations as a map, (17), as well as the Jacobian, . As already studied in Gao:2017dek (), for the Cayley case without label , it is convenient to work in the SL(2) fixing . We will rewrite scattering equations, one for each of the edges of ; each edge, , divides into two parts and (we will omit the subscript and our convention is that , ), see Figure 4. Let’s take the sum of scattering equations with :
It is interesting to see that all terms with both cancel in this sum, and the remaining ones include and those ’s with , :
By multiplying and plugging in we have the scattering-equation map:
for all edges of . One can easily recover the SL(2) invariance and the coefficients of ’s are nice cross-ratios of ’s:
where ’s serve as the coordinates of , thus (21) provides a map from to .
Similar to the case of scattering equations, only of these equations are independent, which can be obtained by deleting any edge, say , of (see Figure 4). By arranging the equations, , in an appropriate order, the transformation matrix from to is a unit triangular matrix with unit determinant, thus the computation of Jacobian simplifies to the product of factors ’s:
where the product is over edges . By ignoring the infinity pre-factor in , we arrive at the SL(2)-fixed Cayley function of Gao:2017dek ():
and the SL(2)-invariance can be recovered by dressing with the prefactor Gao:2017dek (). It follows that the worldsheet form can be nicely written as
and the Cayley function is the Parke-Taylor factor. For with e.g. label at the center, the rewriting and the (fixed and invariant form of) Cayley function read:
where in the first equation we have used and the cross-ratios simplify.
Scattering forms and Cayley polytopes
Now we proceed to scattering forms and the pushforward for Cayley cases, as already studied in Arkani-Hamed:2017mur (); Gao:2017dek (). It is straightforward to show that the projective, scattering forms for any Cayley graph agree with previous results. If a cubic graph has any pole with subset that is not a connected subgraph of , one can show that the pullback to gives zero, by a argument similar to (18). Therefore, any tree that has non-zero pullback consists of poles corresponding to compatible, connected subsets of (except for trivial cases or ) Gao:2017dek (), and let’s denote the collection of such cubic trees as . Furthermore, by choosing any that span as above, the pullback of such always give , and we have a projective form (see Figure 5 for example):
On the other hand, as shown in Gao:2017dek (); Cachazo:2017vkf (), the Jacobian has unit leading singularity at each 0-dimension boundary of where every set of pinching punctures belong to a connected subset of . This discussion implies that the pushforward of coincides with . By construction, the form is projective thus we do not even need to further check the sign of each term in the verification of (8) for any hyperplane .
We have seen that the Cayley case has the advantage that the both the rewriting, (17) and the proof of (8) are simple and clear. Furthermore, there is a nice geometric construction based on , which was first discussed in Arkani-Hamed:2017mur (). When is a Cayley graph, let’s define the top-dimensional, positive region with all poles being positive:
Now by requiring all the constants in defining , being positive, one can check that have non-empty intersection with , which turns out to be a convex polytope that we call Cayley polytope:
For example, for the star-shaped tree, , we have the region and subspace
and the intersection gives as the permutohedron polytope Arkani-Hamed:2017mur (). We will prove the claim for general Cayley polytope in B by studying the geometric factorization: any codimension- boundary of corresponds to a set of compatible poles where each is a connected subset in . Moreover, the canonical form of coincides with the pullback of on , which naturally follows from our construction.
Regions in and relations to graph associahedron
We have seen that a convex polytope can be constructed beautifully in the subspace of for each Cayley case, now we show how to get the same combinatoric polytope as a region in . The Cayley worldsheet form, is the canonical form of the region, which can be pushed forward to yield the canonical form of Cayley polytope.
The region can be understood as the union of with different orderings in a natural way, following the results of Gao:2017dek (). To do this, we need to regard the spanning tree as a directed graph, which also fix the sign convention for . We pick e.g. as the root and define as a directed graph with all arrows pointing towards . Now the sign convention in (which we have not been careful about) is that we have for every edge from to . Interestingly, we have a nice region that goes with this directed graph:
It is the union of associahedra with orderings such that precedes in for each directed edge from to . For instance, is just the positive part since the directed edges in Hamilton graph are those form to , to , and so on. Another example is , all contribute in this case since the only directed edges are those from to for any , thus is the union of associahedra. It is the following non-trivial identity derived in Gao:2017dek () that guarantees that the canonical form of is given by the worldsheet form :
Of course we can choose another label as root which will result in a different region, but they all have the same canonical form (up to a possible sign). In general these regions do not look like a convex polytope in , but has exactly the same boundary structure as the corresponding Cayley polytope, . For example, the boundaries of is exactly those of the permutohedron . One can show this by noting that any co-dimension 1 boundary of corresponds to a subset of for pinching together where induces a connected subgraph in , and so on.
Furthermore, one can show that the scattering-equation map (21) maps all boundaries of to corresponding boundaries of . In particular, it is obvious from (21) that the boundaries of with are mapped to those of with . However, unlike the associahedron case, for any that consists of more than one associahedron, its interior is not mapped to the interior of (let alone any one-to-one map). We expect that instead the image of the is the exterior, or the complement of in the subspace . This of course explains why the form obtained from pushforward of gives , which is the canonical form for the “exterior” as well!
Last but not least, the combinatoric polytopes for are special cases of the so-called graph associahedra MichaelCoxeter (), which are natural generalizations of associahedron and play an important role in Coxeter complexes etc. To see this, consider a graph with vertices one for each edge of , and two vertices are connected iff they are adjacent in (i.e. they share a vertex). For example, is a Hamilton graph and a complete graph, with vertices. Our and are combinatorially the same polytope as the graph associahedron obtained from . On the other hand, there are of course graphs that cannot be obtained from a spanning trees in this way. For example, we have seen that in rewriting the scattering equations, we encounter disconnected graphs that correspond to degenerate . They still give perfectly well-defined and graph associahedra (for example, the cylcohedron for belong to this case), but there is no Cayley polytope for such cases. Thus our construction singles out a special class of graph associahedra that have a nice realization in kinematic space and scattering-equation maps.
5 Beyond Cayley: subspaces and leading singularities
We have studied Cayley cases in detail, where the entire construction is dictated by a spanning tree and the resulting scattering and worldsheet forms are both forms. As already mentioned in Arkani-Hamed:2017mur (), such forms are the most direct generalizations of the planar scattering form. We believe the most general scattering and worldsheet forms can be constructed from the so-called subspace (or hyperplane), as we define now.
A non-degenerate is a hyperplane if for all , non-zero ’s are all equal to each other up to a sign. According to (5), the necessary and sufficient condition for a non-degenerate to be is that for all triplets of graphs as in Figure 1 where are not all zero, exactly one of the three vanishes (thus the remaining two add up to zero); we further restrict to connected case, thus all non-vanishing ones should be related via such triplets. Given a hyperplane, it is natural to choose its coordinates ’s to be ( of a tree with non-vanishing , then and any non-vanishing by definition. Denote the set of trees with non-vanishing ’s as , then the scattering form for reads
with . Similarly the worldsheet form is also a form, i.e. it has unit leading singularities on . Instead of fully classifying these hyperplanes, here we focus on a class of that has a particularly nice interpretation; namely the Jacobians