# Note on Identities Inspired by New Soft Theorems

###### Abstract:

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.

## 1 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]. Recently, a new soft theorem for tree level gravity amplitudes was studied in [4]. By using the on-shell recursion relation [5, 6] and imposing the holomorphic soft limit, Cachazo and Strominger have proved that

(1) | ||||

here for and , the unmentioned external kinematic data are un-deformed and we prefer to suppress them for conciseness. Taylor expansion in exhibits three singular terms in orders , and , while higher order terms in will be mixed with the less interesting parts.

A similar relation for tree level Yang-Mills amplitudes using the on-shell recursion relation, proved by Casali [7], takes the form

(2) | ||||

where two singular terms in orders and appear after Taylor expansion. The mixing between higher order terms from the deformed and parts also persists to this case.

Based on this new discovery, many related studies have been done. In [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], the soft theorem has been generalized to arbitrary dimensions and other theories or categories: string theory, ABJM theory, theories with fermions or massive particles, and form factors. In [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55], the theorem has been understood from various perspectives, especially those of symmetries and invariance. In [8, 25, 56, 57, 58, 59, 60], its generalization to loop level has been discussed. In [61, 62, 63, 64, 65, 66, 67, 68], the relevant double (or multiple) soft theorem has also been discussed.

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 [8], which explored loop correction to the soft theorem. It involves the so-called half-soft function (first defined in [9] and reinterpreted in [10]), which appears naturally for all-plus one-loop gravity amplitude. Its general proof was not given in [8], but explicit checks up to 12 points had been done. The identity reads

(3) |

where are two nonempty partition sets of the particles other than and , and and are the corresponding total momenta. During the proof, we had also discovered another simpler identity, which can serve as its logical preliminary. It reads

(4) |

where the matrix is related to , and other symbols above will be explained shortly.

The second set of identities was conjectured in [11], which is a consequence of consistency conditions between the soft theorems for gravity and gauge amplitudes, under the well-known KLT relation [12]. It involves the KLT momentum kernel [9, 13, 14, 15], and the transformation matrices ( and below) between BCJ basis of gauge amplitudes [16]. These two identities are

(5) |

(6) |

where is the KLT momentum kernel of pivot , and is the anti-holomorphic angular momentum operator. We will use the CHY formulation [17, 18, 19] to prove the first identity and discuss the second one.

This note is organized as follows. In section 2, we prove identity (3) of the half-soft function, and also the byproduct identity (4). In section 3, we prove identity (5) of the KLT momentum kernel by using the CHY formulation, while we transform identity (6) into a convenient form for possible future attempts and end with some discussion.

## 2 Two Identities of the Half-soft Function

In this section we will prove (3) and (4), first let’s set up a bit convenient facilitation. For reader’s reference, we write (3) again below

(7) |

where are two non-overlapping nonempty sets satisfying , and momentum conservation enforces . The half-soft function above is defined as [10]

(8) |

where denotes the determinant of matrix after deleting its -th row and -th column, and indicates this quantity is independent of the choice . If there is only one row and one column, the determinant is 1 after deletion. The matrix is defined as

(9) |

where and serve as auxiliary spinors. The sum of each row is zero, so is degenerate. Observe that the summand in (3) has even power of and , by momentum conservation this sum is symmetric between and , then we can replace by and rewrite (3) as

(10) |

for brevity stands for in spinorial products (and later also represents the number of elements in the set , depending on the context).

To simplify the proof, we define the matrix as

(11) |

where the common factor of the -th row in has been stripped off. One can easily verify that

(12) |

where has been added to to label the corresponding set, note that is independent of the choice . Then we have

(13) |

where is a common factor independent of so it can be dropped, hence (10) becomes

(14) |

### 2.1 A simpler byproduct identity

In the proof of (14), we happened to discover (4). For reader’s reference, it is given below

(15) |

where are two non-overlapping nonempty sets satisfying , and the auxiliary spinors are 1 and . Also note that , , and it is free to switch the choices within each set. Since this is mandatory for (14) to hold, we will prove it first as the tricks used here are analogous to those for (14).

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 . For later convenience, we multiply it by a non-zero factor, yields

(16) |

which is of course equivalent to (4). But now there are two advantages: The large behavior of its LHS is improved, and it has the desired simple pole for residue evaluation, as we will soon see.

For generic , consider BCFW deformation and a particular pole . Note that particles 1 and are special while the rest ones are symmetric, so it is sufficient to consider the residue of only, as all ’s with behave similarly. At , we have

(17) |

and

(18) |

by which we mean to combine the momenta of particle and 2 into that of particle , or more physically, particles and 2 merge into particle . Including the deformed particle , the set now shrinks into while momentum conservation still holds, as what induction requires.

To locate pole in (16), we immediately find one in the overall factor. Naively, there might be another one under if we take , for example. However, the expansion of in terms of will cancel this pole. In other words, is a polynomial of (one may also choose to invalidate this pole), that’s why the overall factor is mandatory.

The next step is to analyze the large behavior of the LHS in (16) 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 to exclude particle 2 from them while denote the original sets. Depending on whether or contains particle 2, the set has three types of splitting: , and , where . So the second term becomes

(19) | ||||

Also, the first term in (16) can be written as

(20) |

Since the three ’s in (20) and the first and second terms of (19) contain particle 2, we can choose to delete its corresponding row and column. Large power counting shows that all four terms in (19) and (20) behave as under , but the overall factor in the front of (16) behaves as , which renders the entire expression as , so there is no boundary contribution. Therefore, via contour integration, the LHS of (16) (denoted below) can be expressed as

(21) |

if the residue at vanishes, by the symmetry among particles the entire un-deformed expression must also vanish. Note the contribution from the overall factor in (16) is universal, so it can be dropped. At , after some algebra, the residue evaluation gives

(22) |

recall that above is not a pole, while the real pole comes from the overall factor. Here is replaced by up to a factor, after recalling (17). By expanding the determinant to the first order of , then using the independence of choice to switch the deleted row and column for each term, we can collect a factor as above. The similar (and simpler) story happens for

(23) |

Plugging them back, up to a factor , the sum of (19) and (20) becomes

(24) | ||||

By momentum conservation, up to a factor , it can be simplified into

(25) | ||||

after assuming the identity of particles holds. This finishes the inductive proof of (4).

### 2.2 Proof of the first identity

Now we move to prove (14) by applying the similar pack of tricks: to consider deformation acting on its LHS, and the pole . First, we separate the expression into three parts corresponding to , and , namely

(26) |

Similarly, we now redefine and to exclude particles 2 and 1, with respect to and . For , the set has three types of splitting: , and , where . For , we have , and . For , there are four types: , , and , where , but the last two will not contribute to the residue of and hence the corresponding terms are neglected, which will be explained shortly.

According to the splittings above, we can write

(27) | ||||

(28) | ||||

(29) | ||||

For one can verify that, only terms for which 1 and 2 are in the same splitting set, have pole and hence contribute to the residue, which explains why we only need the first two terms. Moreover, in can be empty (similarly for ). While for , in cannot be empty, otherwise such a splitting belongs to type (similarly for ).

After the separation, we now analyze the large behavior. Under , large power counting shows that , and , so there is no boundary contribution. Then we can repeat the contour integration (21). Again, thanks to the symmetry among particles , it is sufficient to consider the residue of only.

Recalling (22) and (23), at the residue evaluation gives

(30) | ||||

or after a bit simplification,

(31) | ||||

Similarly for ,

(32) | ||||

Combining and , we find

(33) | ||||

after using the following identity

(34) | ||||

Now note the second and third terms in (33) can be regrouped as

(35) |

which is exactly identity (4) for the set ! Therefore we are left with

(36) |

To settle this leftover, we look back to in (29) and find

(37) |

where again we have used the independence of choice to switch the deleted row and column. Now

(38) | ||||