Note on identities inspired by new soft theorems

The new soft theorems, for both gravity and gauge amplitudes, have inspired a number of works, including the discovery of new identities related to amplitudes. In this note, we present the proof and discussion for two sets of identities. The first set includes an identity involving the half-soft function which had been used in the soft theorem for one-loop rational gravity amplitudes, and another simpler identity as its byproduct. The second set includes two identities involving the KLT momentum kernel, as the consistency conditions of the KLT relation plus soft theorems for both gravity and gauge amplitudes. We use the CHY formulation to prove the first identity, and transform the second one into a convenient form for future discussion.


Introduction
Scattering amplitudes often have an universal soft behavior when the momentum of one external leg tends to zero. This soft limit can be traced back to the works [1][2][3][4][5][6]. Recently, a new soft theorem for tree level gravity amplitudes was studied in [7]. By using the on-shell recursion relation [8,9] and imposing the holomorphic soft limit, Cachazo  (1.1) here for M n and M n−1 , the unmentioned external kinematic data are un-deformed and we prefer to suppress them for conciseness. Taylor expansion in ε exhibits three singular terms in orders ε −3 , ε −2 and ε −1 , while higher order terms in ε will be mixed with the less interesting O(ε 0 ) parts. A similar relation for tree level Yang-Mills amplitudes using the on-shell recursion relation, proved by Casali [10], takes the form A n (λ n → ελ n ) = 1 ε 2 n − 1, 1 n − 1, n n1 A n−1 λ n−1 →λ n−1 +ε 1n 1,n−1 λ n ,λ 1 →λ 1 +ε n−1,n n−1,1 2) where two singular terms in orders ε −2 and ε −1 appear after Taylor expansion. The mixing between higher order terms from the deformed A n−1 and O(ε 0 ) parts also persists to this case.
Among these studies, we have met two sets of identities which have not been proved so far. We will present the proof in this note.
One identity of the first set was mentioned in [11], which explored loop correction to the soft theorem. It involves the so-called half-soft function h (first defined in [12] and reinterpreted in [13]), which appears naturally for all-plus one-loop gravity amplitude. Its general proof was not given in [11], but explicit checks up to 12 points had been done. The identity reads where M, N are two nonempty partition sets of the (n − 2) particles other than b and n, and K M and K N are the corresponding total momenta. During the proof, we had also discovered another simpler identity, which can serve as its logical preliminary. It reads where the ψ matrix is related to h, and other symbols above will be explained shortly. The second set of identities was conjectured in [14], which is a consequence of consistency conditions between the soft theorems for gravity and gauge amplitudes, under the well-known KLT relation [15]. It involves the KLT momentum kernel [12,[16][17][18], and the transformation matrices (D and C below) between BCJ basis of gauge amplitudes [19]. These two identities are where S[α t |β t ] p n−1 is the KLT momentum kernel of pivot p n−1 , and J t ≡ J t ,αβ is the anti-holomorphic angular momentum operator. We will use the CHY formulation [20][21][22] to prove the first identity and discuss the second one. This note is organized as follows. In section 2, we prove identity (1.3) of the half-soft function, and also the byproduct identity (1.4). In section 3, we prove identity (1.5) of the KLT momentum kernel by using the CHY formulation, while we transform identity (1.6) into a convenient form for possible future attempts and end with some discussion.

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2 Two identities of the half-soft function In this section we will prove (1.3) and (1.4), first let's set up a bit convenient facilitation. For reader's reference, we write (1. where M, N are two non-overlapping nonempty sets satisfying M ∪ N = {1, . . . , n − 1} \ b, and momentum conservation enforces k b + k n + K M + K N = 0. The half-soft function h above is defined as [13] h(b, n, N ) = 1 where |Ψ| r r denotes the determinant of matrix Ψ after deleting its r-th row and r-th column, and ||Ψ|| indicates this quantity is independent of the choice r ∈ N . If there is only one row and one column, the determinant is 1 after deletion. The matrix Ψ is defined as where b and n serve as auxiliary spinors. The sum of each row is zero, so Ψ is degenerate.
Observe that the summand in (1.3) has even power of K M and K N , by momentum conservation this sum is symmetric between M and N , then we can replace K M by −K N and rewrite (1. for brevity N stands for K N in spinorial products (and later N also represents the number of elements in the set N , depending on the context).
To simplify the proof, we define the matrix ψ as where the common factor ib in of the i-th row in Ψ has been stripped off. One can easily verify that where N has been added to ψ to label the corresponding set, note that |ψ N | x x / xb xn is independent of the choice x ∈ N . Then we have

A simpler byproduct identity
In the proof of (2.8), we happened to discover (1.4). For reader's reference, it is given below where N, M are two non-overlapping nonempty sets satisfying N ∪ M = {2, . . . , n − 1}, and the auxiliary spinors are 1 and n. Also note that w ∈ N ∪ M , x ∈ N , y ∈ M and it is free to switch the choices w, x, y within each set. Since this is mandatory for (2.8) to hold, we will prove it first as the tricks used here are analogous to those for (2.8). Now we will adopt the BCFW deformation and reduce it into an identity of the same form, but with one particle removed, in other words, we will perform an inductive proof. Before induction, the identity is confirmed analytically at lower points for n = 4, 5, 6. For later convenience, we multiply it by a non-zero factor, yields which is of course equivalent to (1.4). But now there are two advantages: the large z behavior of its l.h.s. is improved, and it has the desired simple pole for residue evaluation, as we will soon see. For generic n, consider BCFW deformation 1|n] and a particular pole 21 . Note that particles 1 and n are special while the rest (n − 2) ones are symmetric, so it is sufficient to consider the residue of 21 only, as all i1 's with i ∈ {2, . . . , n − 1} behave similarly. At 21 = 0, we have by which we mean to combine the momenta of particle1 and 2 into that of particle2, or more physically, particles1 and 2 merge into particle2. Including the deformed particlen, the set {1, 2, . . . , n} now shrinks into {2, . . . ,n} while momentum conservation still holds, as what induction requires.
To locate pole 21 in (2.10), we immediately find one in the overall factor. Naively, there might be another one under |ψ N | x x if we take x = 2, for example. However, the expansion of |ψ N | x x in terms of 21 will cancel this pole. In other words, |ψ N | x x / x1 xn is a polynomial of 21 (one may also choose x = 2 to invalidate this pole), that's why the overall factor is mandatory.

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The next step is to analyze the large z behavior of the l.h.s. in (2.10) before evaluating its residues at finite locations. To clarify the analysis, we further separate the second term in the parenthesis, and from now on we redefine N, M to exclude particle 2 from them while N , M denote the original sets. Depending on whether N or M contains particle 2, the set {2, . . . , n − 1} has three types of splitting: [12] |ψ N ∪M | w w w1 wn . (2.13) Also, the first term in (2.10) can be written as (2.14) Since the three ψ's in (2.14) and the first and second terms of (2.13) contain particle 2, we can choose to delete its corresponding row and column. Large z power counting shows that all four terms in (2.13) and (2.14) behave as z N +M −1 = z n−4 under 1|n], but the overall factor in the front of (2.10) behaves as z −(n−2) , which renders the entire expression as z −2 , so there is no boundary contribution. Therefore, via contour integration, the l.h.s. of (2.10) (denoted I below) can be expressed as if the residue at 21 = 0 vanishes, by the symmetry among particles {2, . . . , n − 1} the entire un-deformed expression must also vanish. Note the contribution from the overall factor in (2.10) is universal, so it can be dropped. At 21 = 0, after some algebra, the residue evaluation gives after assuming the identity of (n − 1) particles holds. This finishes the inductive proof of (1.4).

Proof of the first identity
Now we move to prove ( . . , n − 1} \ b, but the last two will not contribute to the residue of 21 and hence the corresponding terms are neglected, which will be explained shortly. According to the splittings above, we can write  After the separation, we now analyze the large z behavior. Under 1|n], large z power counting shows that I 1 ∼ z −2 , I 2 ∼ z −1 and I b =1,2,n ∼ z −1 , so there is no boundary contribution. Then we can repeat the contour integration (2.15). Again, thanks to the symmetry among particles {2, . . . , n − 1}, it is sufficient to consider the residue of 21 only. To settle this leftover, we look back to I b =1,2,n in (2.23) and find where again we have used the independence of choice x to switch the deleted row and column. Now 21 1n which returns to the form of (2.8) for the set {2, . . . ,n}! It vanishes after assuming the identity of (n − 1) particles (without particle 1) holds. Similar to N , M , here N b , M b denote the sets including2 but not b. This finishes the inductive proof of (1.3).

Two identities of the KLT momentum kernel
In this section we will prove (1.5) and (1.6) as conjectured in [14]. To understand these relations, we must first define the transformation matrices D and C between BCJ basis of gauge amplitudes via where α t and β t denote the permutations of (n − 3) particles other than t , (n − 1) and n. In a tensorial sense, D and C are the transformation matrices with respect to the summation of all (n − 3)! permutations, which is defined as the inner product. For reader's reference, we write (1.5) and (1.6) again below where S[α t |β t ] p n−1 is the KLT momentum kernel of pivot p n−1 , and J t ≡ J t ,αβ is the anti-holomorphic angular momentum operator. Here we follow the convention of S in [16][17][18], namely

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where s ij is each Mandelstam variable, and θ(α i , α j ) is zero when the pair (α i , α j ) has the same ordering at both sets {α 1 , . . . , α k } and {β 1 , . . . , β k }, and unity otherwise. For the first identity, its physical interpretation is straightforward: if we regard the KLT momentum kernel S as the metric, it is simply the tensorial transformation rule for metric. In fact, such a tensorial formulation had been established in [20,22] (known as the KLT orthogonality or the CHY formulation) and we will use it to formally prove the first identity shortly. The second identity is however more intricate, as it roughly represents angular momentum conservation in an entangled way. The CHY formulation can help transform it into a relation that may reveal very nontrivial properties of scattering process, while to prove it directly is yet beyond our understanding.

Proof of the first identity
Before the proof, we must first rewrite gauge amplitudes in the CHY formulation [22] which is based on the scattering equations [21]. It tells that where σ (i) denotes the i-th solution to the scattering equations b =a s ab σ ab = 0, (3.8) with σ ab = σ a − σ b , and there are (n − 3)! solutions in total. The definitions of det (Φ) and Pf Ψ, namely the reduced determinant of Jacobian Φ and the reduced Pffafian of antisymmetric matrix Ψ, can be found in [22]. The object mainly concerns us is .

(3.9)
On the other hand, the KLT relation gives  which is the KLT orthogonality, if we define matrices . (3.13) From this matrix relation we immediately get (3.14) Back to (3.6) and (3.7), if we further define the row vector Plugging them back into (3.1) and (3.2), and assuming their independence of basis Θ i , we get 17) or equivalently, Finally we plug them back into the l.h.s. of (1.5) and interchange t and t , together with (3.14) we get

Discussion of the second identity
Now we move to prove (1.6). Equipped with the matrices defined in the previous subsection, the l.h.s. of (1.6) can be simplified as 20) assuming the two matrices in the front and end of the last line are non-degenerate, we should prove For the first term above, the summation over t is trivial since the matrix product involves t only, so it is in fact due to angular momentum conservation, as the absence of J n−1 and J n does not matter sinceλ n−1 andλ n have been solved by momentum conservation (see [14] for more details). Therefore we are left with where the dummy variable t has been replaced by t. We can continue to transform it into a convenient form for further attempts to prove, by isolating its real matrix content. Let's define where in the third line, the second term vanishes again due to angular momentum conservation. Finally, we are left with which can no longer be further simplified.
To get some sense of this very nontrivial identity, it is helpful to see the first nontrivial case n = 4, which corresponds to the first nonempty β t . Recall that α(1),α (2) . . . σ (3.28) which trivially holds by the antisymmetry of σ ab ! But as n increases, even for n = 5 this identity will be much more entangled and simple antisymmetry is insufficient for its proof. The potential toolkit for this purpose includes: (1) relations of spinor derivatives on scattering equations; (2) KK and BCJ relations of Σ jβt ; (3) induction, which may involve contour integration. We will come back to this point in the future after better understanding the scattering equations and their solutions.
A last comment is that in (1.6), the anti-holomorphic angular momentum operator J t ,αβ should be generalized to J t ,µν in arbitrary dimensions. Since in 4-dimension J µν ∼ ε αβ Jαβ + εαβJ αβ , and the soft theorem must hold for both holomorphic and antiholomorphic soft limits, it is more natural to use J µν as all other quantities are already defined for arbitrary dimensions.