Emergent Unitarity from the Amplituhedron

We present a proof of perturbative unitarity for N = 4 SYM, following from the geometry of the amplituhedron. This proof is valid for amplitudes of arbitrary multiplicity n, loop order L and MHV degree k. ar X iv :1 90 6. 10 70 0v 1 [ he pth ] 2 5 Ju n 20 19


Introduction
Unitarity is at the heart of the traditional, Feynman diagramatic approach to calculating scattering amplitudes. It is built into the framework of quantum field theory. Modern on-shell methods provide an alternative way to calculate scattering amplitudes. While they eschew lagrangians, gauge symmetries, virtual particles and other redundancies associated with the traditional formalism of QFT, Unitarity remains a central principle that needs to be imposed. It allowed the construction of loop amplitudes from tree amplitudes via generalized unitarity methods [1][2][3][4][5] and loop level BCFW recursion relations [6,7]. These on-shell methods were particularly fruitful in N = 4 SYM and led to the development of the on-shell diagram approach in [8] and the discovery of the underlying grassmannian structure. Locality and unitarity seemed to be the only guiding principle behind gluing together the on-shell diagrams. The discovery of the amplituhedron [9], [10] revealed the deeper picture behind the process of gluing of on-shell diagrams -positive geometry. Positivity dictated how the on-shell diagrams were to be glued together. The resulting scattering amplitudes were local and unitary! This discovery of the amplituhedron was inspired by the polytope structure of the NMHV for scattering amplitudes first elucidated in [11] and expanded upon in [12]. These motivated a definition of the amplituhedron analogous to the definition of the interior of a polygon. The tree amplituhedron A n,k,0 is the span of k planes Y I α living in (k + 4) dimensions.
where Z I a are positive external data in (k + 4) dimensions, i.e. Z a 1 . . . Z a k+4 > 0 if a 1 < · · · < a k+4 and C αa ∈ G + (k, n). The extension to loop level is more involved and can be found in [9].
The amplituhedron thus replaced the principles of unitarity and locality by a central tenant of positivity. Tree level locality emerges as a simple consequence of the boundary structure of the amplituhedron, which in turn dictated is by positivity. The emergence of unitarity is more obscure. It is reflected in the factorization of the geometry on approaching certain boundaries. This was proved for A 4,0,L in [10]. The extension of this proof to arbitrary multiplicity is cumbersome using this defintion of the amplituhedron. The topological definition of the amplituhedron introduced in [13] is central to extending the proof of unitarity.
The topological definition can be stated entirely in terms of 4 dimensional data (the familiar momentum twistors). The amplituhedron A n,k,L depends on the n momentum twistors corresponding to the external legs of the amplitude {Z 1 , . . . Z n } and the lines in twistor space corresponding to the loop momenta, (AB) 1 , . . . (AB) L .
It satisfies the following constraints.
Tree level ii + 1jj + 1 > 0 and the sequence (1.2) S tree : ii + 1i + 2i + 3 , . . . ii + 1i + 2i − 1 (−1) k−1 has k sign flips. Loop level (AB) a ii + 1 > 0 and the sequence This topological definition has already been used to investigate the structure of "deep" cuts to all loop orders in [14,15]. It is well known that the branch cut structure of amplitudes is intimately tied to perturbative unitarity. This is encapsulated in the optical theorem which related the discontinuity across a double cut to the product of tree amplitudes. However, the location of branch points in loop amplitudes is governed by the boundary structure of the amplituhedron. The discontinuity across a branch cut is calculated by the residue on an appropriate boundary of the amplituhedron. The optical theorem thus translates into a statement about the factorization of the residue on this boundary. We expect this factorization to emerge as a consequence of the positive geometry. In order to make contact with the amplituhedron, it is helpful to rewrite this using momentum twistors. For now, we will focus on the MHV amplitudes A n,0,L . We are interested in the case where one of the loops, call it AB, cuts lines ii + 1 and jj + 1. Let us label the other loops as (AB) a . We can parametrize AB as The term in the L-loop integrand which contributes to this cut is The residue on the cut ABii + 1 = ABjj + 1 = 0 is Unitarity predicts that the function f (x, y, 0, 0, (AB) a ) is related to lower point amplitudes (see figure 1) and is of the form We will show that this structure follows simply from the geometry of the amplituhedron. We will first present a proof for the four point case. This is just a rewriting of the proof found in [10] in the language of sign flips. This proof will then admits a generalization to amplitudes of higher multiplicity.

Proof for 4 point Amplitudes
At four points (n = 4, k = 0) we focus on the unitarity cut AB12 = AB34 = 0. On this cut, we can parametrize (AB) as with x > 0, y > 0. Denoting the uncut loops as (AB) i , the mutual positivity conditions are This can be "factorized" because each term is individually positive. There two possible solutions are If L 1 loops, (AB) a obey (2.2) and L 2 = L − L 1 − 1 loops, (AB) α obey (2.3), it is easy to see that each loop (AB) α and (AB) a is in an amplituhedron with lower n, L.
(AB) α belong to a 4 point, L 2 loop amplituhedron with legs {A, Z 2 , Z 3 , B} as seen from the conditions below. To complete the proof of the factorization of the residue into the product of two lower loop amplitudes, we must show that the mutual positivity between the loops (AB) a and (AB) α imposes no constraints. To see this, we can expand the loop (AB) a in terms of {Z 1 , A, B, Z 4 } as This gives Note that all the terms except for (AB) α A4 and (AB) α B1 are obviously positive.
These follow from (2.4) and we can conclude that (AB) a (AB) α > 0 imposes no further constraints. The residue factorizes into M L and M R .

Proof for MHV amplitudes of arbitrary multiplicity
We will extend the above results to amplitudes of arbitrary multiplicity. However, the existence of higher k sectors beginning with n = 5 complicates the proof. In this section we will focus on a proof of unitarity for MHV amplitudes. This allows us to sketch the essentials of the proof without additional complications. In the next section, we modify the proof to account for higher k sectors.
We are interested in examining the residue of the MHV amplituhedron A n,0,L , with external data {Z 1 , . . . Z n }, on the cut ABii + 1 = ABjj + 1 = 0 where we can parametrize AB as We need to show that the canonical form for every configuration on this cut can written as a product of canonical forms for lower loop, "left" and "right" MHV amplituhedra A L 1,0,L 1 and A R n 2 ,0,L 2 . The precise definitions and the proof that they exist are in the following section. Their existence is tied to the fact that the external data {Z 1 , . . . Z n } and the uncut loops (AB) a are all in the amplituhedron A n,0,L and satisfy the following conditions.
Here, ij ≡ (AB) a ij where (AB) a are the uncut loops.

The left amplituhedron
The left amplituhedron A n 1 ,0,L 1 is defined by three sets of conditions similar to (3.2). In this case, the external data is the set L = {Z 1 , . . . Z i , A, B, Z j+1 , . . . , Z n }. Letting a, b, c, d denote elements of this set and ij ≡ (AB) a ij , the defining conditions are For consistency, we must also verify that the all sequences and S L have the same number of sign flips. Note that the positivity conditions on the loop data ensures that all the first and last entries of these sequences are positive. Furthermore any two sequences in the above set are of the form { ak } and { a + 1k } which satsify The equality of sign flips now follows immediately from the analysis in Appendix[A]. This shows that the left amplituhedron can be consistently defined.
All the tree level, mutual positivity and the loop level positivity conditions are automatically satisfied because of (3.1) and (3.2). The flip condition is the only one that requires a detailed analysis which is presented in Section[3.2].

The right Amplituhedron
The external data for the right amplituhedron A n 2 ,0,L 2 is R = {A, Z i+1 , . . . , Z j , B} and the defining inequalities are listed below. a, b, c, d ∈ L and ij ≡ (AB) a ij with (AB) a being an uncut loop.
Once again, for consistency we should verify that

Factorization on the unitarity cut
The external data divides into two sets L = {Z 1 , . . . , Z i , A, B, Z j+1 , . . . , Z n } and R = {A, Z i+1 , . . . Z J , B} which are both positive as observed in the previous section. To see the factorization at loop level, consider any uncut loop (AB) a and a corresponding sequence S.
We have divided the sequence in a suggestive way. The left half of S looks very similar to S R . It natural to label the different flip patterns of S as S ablr where a, b = ± are the signs of i + 1j and i + 1j + 1 and l, r are the number of flips in the left and right parts of S. In order to compare S L to S, we introduce the sequence S L and S L are obviously connected by a Plücker relation and following Appendix[A], the relation between k L and k L is determined entirely by the signs of the first and last elements S now looks almost like a juxtaposition of S R and S L . Each flip pattern determines whether the corresponding loop (AB) a belongs to the left or the right ampltuhedron.
If i + 1B < 0, then S R has 2 sign flips and it can be shown that k L = 0 by analysis similar to the cases above. This (AB) a belongs to the right ampltiuhedron.
Once again, it is simple to show that S R has two sign flips and S L has 0 sign flips in this configuration.

Trivialized mutual positivity
It remains to be shown that the mutual positivity between a loop (AB) L in the left amplituhedron and a loop (AB) R in the right amplituhedron is trivially true. It is easiest to see this if we expand each loop (AB) L in the left amplituhedron and (AB) R right amplituhedron in a Kermit expansion by using the following parametrizations.
. . , Z j , B}. (AB) L is in one of the following cells of the left one-loop amplituhedron with l 1 < l 2 ∈ {Z 1 , . . . Z i , A, B, Z j+1 , . . . , Z n }. On expanding (AB) L (AB) R , every term is of the form l 1 l 2 r 1 r 2 . Since the external data are positive, i.e. ijkl > 0 for i < j < k < l.
This completes the proof of factorization on the Unitarity cut for MHV amplituhedra.
In the next section, we will demonstrate that this proof can be extended to higher k sectors.

Proof for higher k sectors
The proof of unitarity for higher k is similar in spirit to that for the MHV sector. However, there are a lot additional details that we must take into account. Firstly, we must modify (1.5) to include products with different k. Suppose the left amplitude has n L negative helicity gluons and the right amplitude has n R negative helicity gluons, then we have n L + n R = n + 2. With the MHV degrees are defined as k L = n L − 2, k R = n R − 2, k = n − 2, this equation reads k L + k R = k. Thus we expect Figure 2. Unitarity cut for an N k MHV amplitude We expect that unitarity emerges from a factorization property of the geometry in a manner similar to the MHV case. In order to make this statement more precise, we will have to define analogues of the left and right MHV amplituhedra. We start with A n,k,L , the N k MHV amplituhedron which is defined by the conditions Tree level ii + 1jj + 1 > 0 and the sequence (4.1) S tree : ii + 1i + 2i + 3 , . . . ii + 1i + 2i − 1 (−1) k−1 has k sign flips. Loop level (AB) a ii + 1 > 0 and the sequence Note that this definition is invariant under a twisted cyclic symmetry Z n+1 → (−1) k−1 Z 1 . On the unitarity cut ( ABii+1 = ABjj +1 = 0), there is a natural division of the external data into "left" and "right" sets, {Z 1 , . . . , Z i , A, B, Z j+1 , . . . , Z n } and {A, Z i+1 , . . . , Z j , B}. However, the data obeys a twisted cyclic symmetry with a fixed k. In order to generate data with k L which has a different even/odd parity, we will have to allow for arbitrary signs on the Zs and define two sets of external data.
where σ(k) = ±1. These will be determined by conditions like (4.1) which define the left and right amplituhedra along with the appropriate twisted cyclic symmetry. We will then show that the canonical form for every configuration in A n,k,L can be mapped into a product of canonical forms on suitably defined left and right amplituhedra A L n 1 ,k L ,L 1 and A R n 2 ,k R ,L 2 . We must demand that the set L satisfies all the conditions in (4.1). In addition, this must also be compatible with the fact that the Z i are the external data for A n,k,L .
AB and the Zs automatically satisfy aa + 1bb + 1 > 0 and ABaa + 1 > 0. Thus we have, σ L (a)σ L (a + 1)σ L (b)σ L (b + 1) > 0 and σ L (A)σ L (B)σ L (a)σ L (a + 1) > 0. Furthermore, we have new constraints on A and B coming from Finally, since the set L is the external data for A L n 1 ,k L ,L 1 , it must satisfy a twisted cyclic symmetry aa + 1n1 σ L (a)σ L (a + 1)σ L (n)σ L (1)(−1) k L −1 > 0 (4.5) Since aa+1n1 (−1) k−1 > 0, consistency requires (−1) k+k L σ L (a)σ L (a+1)σ L (n)σ L (1) > 0. This divides into two cases iABj which has the following solutions Each of these regions is characterized by a particular sign of iAkk + 1 and Bj + 1kk + 1 along with a pattern of sign flips for the sequence Each region allows parametrization of the line (AB) as A = ±Z i ± xZ i+1 and B = ±yZ j ± Z j+1 with x > 0, y > 0. In the table below, we list the different possibilities. A similar analysis of the effects of (4.1) on the set R yields the following constraints on {σ R }.
Once again, each region is characterized by different pattern of sign flips of the sequence where we have ignored an overall factor of σ R (A)σ R (i + 1)σ R (i + 2). We list the various parametrizations and sign patterns of S tree Table 2. Parametrization of (AB) in the four regions

Factorization of the external data
We will show that on the unitarity cut, every allowed flip pattern for the sequence S tree , we can find regions L i , R i such that S tree L and S tree R have the flip patterns necessary for A L n 1 ,k L ,L 1 and A R n 2 ,k R ,L 2 . In order to compare S tree L with S tree , it is useful to introduce another sequence S tree L .
S tree Let k, k L , k L , k R be the number of flips in S tree L , S tree L , S tree R , S tree respectively. k L and k L are related to each other due to the existence of the following Plücker relations which hold in all regions (L i , R i ).
As shown in Appendix[A], we can conclude that the relation between k L and k L depends only on the signs of first and last terms which are encoded in the matrix below.
The relation between k L and k L is shown below We will keep track of the flip patterns of S tree by labeling them as S tree ab where a and b are the signs of ii + 1i + 2j and ii + 1i + 2j + 1 respectively. All the possibilities are listed below.
S tree : ii + 1i + 2i + 3 , . . . ii + 1i + 2j ii + 1i + 2j + 1 , . . . , ii + 1i + 2i − 1 (−1) k−1 S tree Table 4. (k L , k R ) in all regions for the configuration S ++ S tree Table 6. (k L , k R ) in all regions for the configuration S −+ Table 7. (k L , k R ) in all regions for the configuration S −− We see that each flip pattern can be covered by many charts (L i , R i ). However we must also ensure that the sequence S AB shown below has k + 1 sign flips.
ABii + 1 , ABii + 2 , . . . , ABij ABij + 1 , . . . ABi − 1i (−1) k = 0, Bii + 1i + 2 , . . . , Bii + 1j Bii + 1j + 1 , . . . Bi + 1i − 1i + 1 (−1) k The number of flips of this sequence is almost determined by k L and k R . However, there is the possibility of an additional flip at the boundary in regions (L i , R i ) such that Bii + 1j and Bii + 1j + 1 have opposite signs. We see from Table 8 that for every configuration S tree ab , there is a a pair of regions (L i , R i ) that satisfy k L + k R = k and in which S AB has k + 1 sign flips. Thus every configuration in the original amplituhedron can be covered by these regions consistent with the expected factorization. The remaining regions exist because they are related to amplitudes via inverse soft factors and have identical canonical forms. However, these are not necessary to cover all regions of the original amplituhedron.

Factorization of loop level data
At loop level, we need to show that each loop (AB) a belongs either to the left or the right amplituhedron. The relevant sequences are (denoting (AB) a ij as ij ) Similar to before, it will be convenient to introduce the sequence S loop S loop : i + 1i + 2 , i + 1i + 3 , . . . , i + 1j i + 1j + 1 , . . . , i + 1i (−1) k−1 S loop ++ : On the cut, the external data factorizes such that k L +k R = k with k L , k R ∈ {0, . . . k}.
It is trivially true that each loop belongs to the left or the right amplituhedron. We must show that if a loop (AB) a belongs to the left amplituhedron, then it cannot belong to the right amplituhedron. First, note that in each configuration, we will have k l = k 2 + l and k r = k 1 + r with r, l = 1 or 2. Now suppose that (AB) a belongs to both the left and right amplituhedra. Then we must have k l = k L + 2 and k r = k R + 2. Expressing k l and k r in terms of k 1 , l, k 2 and r, and using k L + k R = k, we get Clearly, l + r = 5 is impossible since l, r = 1 or 2. We just need to show that l + r = 4 is impossible. Note that this is possible only if l = r = 2. In this case we will must have In all these cases, we must have σ R (A)σ R (B)σ L (A)σ L (B)(−1) k R < 0. It is easy to verify from Section [4.1] that this is always false. Thus each loop belongs either to the left or the right ampltuhedron.

Mutual positivity
To complete the proof of factorization, we need to show that the mutual positivity between a loop in A L n 1 ,k L ,L 1 and one in A R n 2 ,k R ,L 2 is automatically satisfied. This is easier to see while working with (k + 2) dimensional data. A loop in the left amplituhedron can be parametrized as a k L + 2 plane Y L 1 . . . Y L k L A a B a .
This reduces the mutual positivity condition Y L (AB) a Y R (AB) b > 0 to a condition involving k + 4 brackets of the form ijklm . It is easy to see that with positive k + 4 dimensional data ( i 1 . . . i k+4 when i 1 < i 2 < . . . i k+4 ), mutual positivity is guaranteed. The signs σ L (k) and σ R (k) are crucial in making this work.

Conclusions
We have shown that Unitarity can be an emergent feature. The positivity of the geometry inevitably leads to amplitudes identical to those derived from a unitary quantum field theory. This lends further support for the conjecture that the amplituhedron computes all the amplitudes of N = 4 SYM. It also suggests that the notion of positivity is more fundamental than those of unitarity and locality which are the cornerstones of the traditional framework of quantum field theory.