ABJM Amplitudes in U-gauge and a Soft Theorem

We report progress in computing and analyzing all tree amplitudes in ABJM theory. Inspired by the isomorphism between the orthogonal Grassmannian and the pure spinor geometries, we adopt a new gauge, called u-gauge, for evaluating the orthogonal Grassmannian integral for ABJM amplitudes. We carry out the integral explicitly for the 8-point amplitude and obtain the complete supersymmetric amplitude. The physical and spurious poles arise from the integral as expected from on-shell diagrams. We also derive a double scalar soft theorem of ABJM amplitudes and verify it for known amplitudes.


Introduction and discussions
Recent years have witnessed major breakthroughs in computing and understanding scattering amplitudes of gauge theory and gravity (see, e.g., [1] for reviews). At many stages of the development, the N = 4 supersymmetric Yang-Mills theory (SYM) proved to be an extremely fruitful testing ground. Many novel ideas, such as twistor string theory [2], dual superconformal symmetry [3], Grassmannian formulation [4], on-shell diagram representation [5] and amplituhedron [6,7], are realized in their simplest forms in N = 4 SYM and then generalized to less symmetric theories.
The three dimensional N = 6 supersymmetric Chern-Simons-matter theory [8][9][10][11][12][13][14], often referred to as the ABJM theory, is a close sibling of the N = 4 SYM in many respects. For instance, the scattering amplitudes of the ABJM theory exhibit dual superconformal symmetry [15][16][17][18] and admits Grassmannian [19], twistor string [20][21][22], and on-shell diagram formulations [23][24][25]. Despite these parallel successes, the study of ABJM amplitudes fall short of those of N = 4 SYM in many respects. One of the most pressing problem is the lack of a "momentum twistor" formulation [26][27][28] in which the dual superconformal symmetry would become manifest (see [29] for a recent attempt in this direction). Closely related to momentum twistors are the dual superconformal R-invariants [3], which serves as building blocks for an explicit formula for all tree amplitudes [30] and a starting point for the construction of amplituhedron.
In principle, all ingredients to compute the ABJM tree amplitudes are available in the literature. The Grassmannian integral [19], supplemented by the contour prescription from on-shell diagrams [23][24][25], will produce the amplitudes. A mundane, yet seemingly unavoidable, problem is that each BCFW bridge in the on-shell diagram introduces a quadratic equation in the integration variables. The solutions to quadratic equations generically contain square-roots, which must cancel out when summed over all solutions and produce a rational function of kinematic variables. Mainly for this technical reason, explicit results for ABJM tree amplitudes to date are limited to 4-and 6-point amplitudes [15,18,19] which are free from square-roots due to limited kinematics, and a partial result for 8-point amplitude [18] without manifest supersymmetry.
The goal of this paper is to take a few steps toward the computation of all ABJM tree amplitudes. Our two main results are a complete evaluation of the supersymmetric 8-point amplitude and a derivation of a double soft theorem valid for all tree amplitudes.
In evaluating the Grassmannian integral for 8-point or higher amplitudes, we find it convenient to use a new gauge, which we call "u-gauge". The u-gauge is inspired by the isomorphism between the orthogonal Grassmannian and the pure spinor geometries; both of them admit the SO(2k)/U (k) coset description. A particular set of coordinates of the coset space introduced in [31] trivially solves the orthogonality constraint and can be easily generalized to arbitrary k. This fact makes the u-gauge, at least in some contexts, more convenient than conventional gauges involving Euler angle coordinates.
Although the u-gauge do not circumvent the square-root problem mentioned above, the quadratic equations in the u-gauge tend to be simpler, which allow us to combine all residues in the contour integral. For the 8-point amplitude, the integral is effectively onedimensional. We can express the denominators of the amplitude in terms of the standard cross-ratios among solutions to quadratic equations. It is easy to see that the cross-ratios can in turn written in terms of the coefficients of the quadratic equations, thereby avoiding the need to solve the equations explicitly.
The final result for the 8-point amplitude takes the form, (1.1) The (1 + π) factor accounts for the sum over two disjoint branches of the orthogonal Grassmannian. The two rational functions in the big parenthesis corresponds to the two on-shell diagrams contributing to the 8-point amplitude. The numerators F (1), F (3) as well as the ∆ ij factors in the denominators are polynomials in kinematic variables. The on-shell diagrams suggest that ∆ 12 , ∆ 14 , ∆ 32 , ∆ 34 should be proportional to physical poles of the amplitude whereas ∆ 13 = ∆ 31 should be spurious. We confirm the expectation by explicitly proving that ∆ ij for the physical poles are proportional to p 2 klm factors for adjacent particles.
In the second half of this paper, we consider the double-soft limit of ABJM amplitudes. Soft limits of scattering amplitudes in gauge and gravity are well known to exhibit universal behavior and have bearing on gauge symmetries and spontaneously broken global symmetries. Our motivation to study the soft limit is more modest. As we make progress in computing higher point amplitudes, we wish to use the soft theorem to test the consistency of the methods we use. Our derivation of the soft theorem will closely follow that of ref. [32], where a similar double-soft theorem was derived for three dimensional supergravity theories. We show that the (2k + 2)-point amplitude A 2k+2 reduces to the 2k-point amplitudes A 2k with universal leading and sub-leading soft factors in the double soft limit, As in [32], the proof of the soft theorem is based on the BCFW [33,34] recursion relation of the ABJM theory [18]. We confirm that the universal soft factors respect all the symmetries of the ABJM amplitudes for all k. For 6-point amplitude to the sub-leading order, and for 8-point amplitude to the leading order, we explicitly take the soft limit of the known amplitude and verify that the soft theorem holds.
Although we still have explicit form of tree amplitudes only up to 8-point, we expect that the findings in the present paper, such as the u-gauge, cross-ratios among different poles in the contour integral, and the double soft theorem, will lay the groundwork for a complete construction of all tree amplitudes of ABJM theory in terms of momentum twistors and/or dual superconformal R-invariants.
This paper is organized as follows. In section 2, we give a short review of the general structure of the ABJM tree amplitudes and the Grassmannian integral. Then we introduce the u-gauge and compare it with other well-known gauges. In section 3, we use the u-gauge to compute some tree amplitudes. After reproducing the 4-point and 6-point amplitudes, we present the details of how to evaluate the 8-point amplitude. In section 4, we propose the double soft theorem of the ABJM amplitudes and prove it using the BCFW recursion relation. We take the double soft limit of the 6-and 8-point amplitudes, and verify explicitly that the theorem holds.
2 Grasssmannian integral in the U-gauge

Preliminaries
Here we briefly review the 3-dimensional spinor helicity formalism [15] and introduce the Grassmannian integral formula for tree level amplitudes of planar ABJM theory [19].
Each spinor in three dimensions transforms under SL(2, R), and a null momentum can be written in the bi-spinor form (2.1) Our convention for spinors and gamma matrices are such that p αβ is real for real p µ , and λ α is real (purely imaginary) for outgoing (incoming) particles. The spinors are contracted as ij ≡ λ α i λ jα . We normalize the norm of vectors such that p 2 ij = (p i + p j ) 2 = ij 2 when both λ i and λ j are real.
The on-shell superfield notation for ABJM amplitudes is built on three fermionic coordinates η I , in addition to λ α , which transform as 3 under the U(3) subgroup of the SO(6) R-symmetry group. The particle/anti-particle superfields take the form A collective notation Λ = (λ; η) will be used when appropriate. The fact that (2.1) is invariant under λ → −λ, while the wave-functions of fermions pick up a minus sign, implies the so-called "λ-parity" of the super-amplitudes.
The super-conformal generators of the superconformal symmetry come in three types: For the second type, we will use the notation The super-momentum conservation is denoted as delta functions by The Grassmannian integral formula for the tree level amplitudes of planar ABJM theory, first proposed in [19], is It was shown in [19] that this formula satisfies the same cyclic symmetry and superconformal symmetry as the tree-level (2k)-point amplitude. Yangian invariance of the formula was first argued in [19] and explicitly proved later in [35].
The integral (2.8) should be considered as a contour integral on the moduli space of rank k, (k ×2k) matrices C with the constraint C ·C T = 0 and the equivalence relation C ∼ gC (g ∈ GL(k)). This moduli space is known as the orthogonal Grassmannian OG(k, 2k). The dimension of OG(k, 2k) is determined by the aforementioned two conditions: (2.10) Integrating out the bosonic delta function δ 2k (C · λ) leaves the momentum conserving delta function and a contour integral over (k − 2)(k − 3)/2 variables. The geometry and combinatorics behind the Grassmannian integral for all tree amplitudes, as well as some loop amplitudes, have been elucidated in [23][24][25]. On the other hand, explicit computation of amplitudes has never proceeded beyond 8-point [18].

U-gauge
We take a real slice of the complex orthogonal Grassmannian with the split signature, where the "metric" in the particle basis is g = diag(−, +, −, +, · · · ) . (2.11) In this basis, the momenta and their spinor variables are related by We find it convenient to switch between the particle basis and the light-cone basis: In the light-cone basis, a GL(k) R subgroup of the O(k, k) symmetry group remains manifest. We will use a notation with covariance under SL(k) ⊂ GL(k) R and adopt the summation convention. The invariant tensor of SL(k, R) will be denoted by m 1 ···m k .
The spinor-helicity variable for particles are denoted by λ α i (i = 1, . . . , 2k, α = 1, 2). We use the same letters w, v for the light-cone combinations of the spinor variables: (2.14) The scalar product of two spinors are defined in a usual manner.
To avoid confusion, we reserve the shorthand notation 12 = λ 1 λ 2 exclusively for the particle basis. In the light-cone basis, we will use w m w n , w m v n and so on. The overall momentum conservation is written as The light-cone components of the fermionic coordinates η I i are denoted bȳ The supermomentum components are rewritten as In summary, the metric and the kinetic variables in the light-cone basis take the form The light-cone form of the C-matrix before a gauge fixing is C = t a n | s an . (2.20) A priori, the GL(k) L index a is not correlated with the light-cone index n. We choose to fix the gauge by locking GL(k) L and GL(k) R : The orthogonality condition implies that u mn is anti-symmetric: Since the decomposition of u mn into the symmetric and anti-symmetric parts is a linear operation, the delta-function does not produce any u-dependent Jacobian factor.
We will call this gauge fixing the "u-gauge". This gauge was inspired by the fact that the orthogonal Grassmannian and the pure spinor admit the same SO(2k)/U (k) coset description and that the u mn coordinates were used in ref. [31] to solve the non-linear constraints of the pure spinors in order to construct higher dimensional twistor transforms.
It is well known that C · λ = 0 and C · g · C T = 0 implies the overall momentum conservation. In the light-cone gauge, C · λ = 0 is written as This equation admits a particular form of SL(k, R)-invariant solution for all k: To verify that (2.24) is indeed a solution to (2.23), it suffices to use the Schouten identity 25) and the momentum conservation (2.16).
The light-cone basis before the gauge fixing respects the symmetry exchanging w n and v n . Thus it is natural to consider the "dual u-gauge" in which the roles of w n and v n are reversed: In the dual u-gauge, the C · λ = 0 condition reads v m +ū mn w n = 0 , which admits a particular solution,ū The dual u-gauge will be useful in a later discussion on the λ-parity for odd k. Using the energy momentum conservation and Schouten identity, one can show that For k = 2 and k = 3, (2.24) is the unique solution to (2.23). For higher k, there is a (k − 2)(k − 3)/2-dimensional solution space containing (2.24). For instance, for k = 4, the general solution can be parametrized bŷ The general solution for k = 5 iŝ The "vector" z p appears to have five components, but only three of them are independent due to the equivalence relation, which follows from the fact thatū rs * ∝ v m v n and the Schouten identity. Along the same line of reasoning, we can write the general solution for k ≥ 4 aŝ In the (k −2)-dimensional space surviving the quotient z p ∼ z p +c α v pα , the tensor z p 1 ···p k−4 spans a (k − 4)-plane. The effective number of components for z p 1 ···p k−4 is, as expected, (2.34)

Broader class of u-gauges
Most features of the u-gauge survives permutations of columns. For instance, instead of the alternating signature metric (2.11), we may take and define light-cone coordinates by One of the useful features of all u-type gauges is that the Jacobian factors arising from the computation of amplitudes are always powers of the R factor defined in (2.24), although the numerical value of R does depend on the particular gauge.
There are (2k)!/(k!) 2 different ways to distribute (−1) and (+1) in the diagonal entries of the metric. An overall flip of the signs is irrelevant, so there are (2k − 1)!/(k!(k − 1)!) inequivalent metrics. Given a fixed metric, there are k! inequivalent ways to pair the coordinates to define light-cone coordinates. To sum up, the number of different u-type gauges is ( Among all possibilities, we will mostly focus on the two choices we mentioned explicitly above. Both of them generalizes to arbitrary k straightforwardly. The alternating signature gauge defined (2.11), (2.13) is the only choice which respects the cyclic symmetry. For this reason we will call this gauge "u-cyclic gauge". As we will see later, the other gauge defined by (2.35), (2.36) is convenient when we examine the factorization of A 2k into two copies of A k+1 when k is odd. We will call this choice "u-factorization gauge".

Lambda-parity in the u-gauge
Let us examine how the lambda parity is reflected in the u-gauge. We will show that, for odd k, the lambda parity induces the exchange, (2.37) For notational convenience, we will work in the u-factorization gauge, but the same arguments hold in all u-type gauges.
With a usual gauge fixing in the particle basis [18,19], the C-matrix is given by In the light-cone basis, the C-matrix translates tô Note that the following identities hold for odd-dimensional orthogonal matrices: When det(O) = −1, a GL(k) gauge transformation gives rise tô This establishes the relation between the u-gauge and the usual gauge in the particle basis.
The sign flip has the same effect as flipping the signs of all λ α m for m = k + 1, · · · , 2k. Up to an overall SO(2k) rotation, this is the same as the exchange (2.37). Thus we have proved that the lambda parity induces the exchange of w m and v m .

4-point
The momentum conservation in the particle basis reads, where we suppressed the spinor indices. In terms of the Lorentz scalars, ij , we obtain The sign factor σ in (3.2) specifies a branch of OG 4 . Without loss of generality, we will work in the σ = +1 branch for the rest of this subsection.
In the light-cone basis, the gauge-fixed C-matrix and the metric are (u = u 12 ) In the particle basis, To avoid confusion, we put hats on the objects in the light-cone basis.
In the evaluation of the Grassmannian integral, the kinematic delta-function gives The value of u * is determined by (2.24): The equality of three expressions follow from (3.2) with σ = +1 and (3.3). The Jacobian factor in (3.6) is The fermionic delta function gives The denominator at u = u * is Collecting all ingredients, we reproduce the standard form of the 4-point amplitude,

6-point
It is well known that the Grassmannian integral for the 6-point amplitude is fully localized by the delta functions and leaves no contour integral. In the particle basis, the gauge-fixed C matrix in the (+)-branch is The kinematic delta-function can be transformed into The value of u * mn is determined by (2.24) and the Jacobian factor is J B 6 = 1/2. The fermionic delta function gives (3.14) The Jacobian factor from the fermionic delta function is .

(3.16)
Collecting all ingredients, we obtain the result of the Grassmannian integral in the u-gauge: .
The full amplitude is the sum of the contributions from the two branches related to each other by λ-parity. As explained in the previous section, for odd k, λ-parity exchanges the two types of light-cone coordinates. In terms of the super-space variables, the exchange means The explicit form of the C-matrix in the conjugate branch is Summing up the two terms, we obtain the full 6-point amplitude: The planar 6-point amplitude can be factorized by two 4-point amplitudes in three different channels. At first sight, it is not clear how the result (3.20) can exhibit the factorization properties. Remarkably, it is possible to show that the consecutive minors from the two branches combine to produce the desired physical poles. In the u-gauge, we have where p 2 ijk··· = (p i + p j + p k + · · · ) 2 . A proof of this relation and discussion on its gauge (in)dependence is presented in appendix A.1.

Contour integral
As discussed in section 2.2, the general solution to C · λ = 0 in a u-gauge iŝ The Grassmannian integral reduces to a contour integral in z through the relation with J B 8 = 1/(2R). Up to an overall sign, the full 8-point amplitude is obtained when the contour separates the poles of M 1 and M 3 from those of M 2 and M 4 [18].
The minors of C-matrix can be at most quartic inû mn (z). But, explicit computations show that all quartic terms can be absorbed into the square of the quadratic polynomial, Similarly, all cubic terms can be rewritten as the same polynomial (3.24) times a linear combination ofû mn (z). These two statements imply that all minors of C, including the consecutive ones, are quadratic in z: The fermionic delta function produces where the fermion bilinears (A I , B I , C I ) are defined as follows: We wish to evaluate the contour integral . .
These integrals share two crucial features. One is that they are homogeneous functions of the variables (a i , b i , c i ) with degree (−1) for i = 1, . . . , n + 1 and (+1) for i = n + 2, . . . , 2n + 1. The other is that they are invariant under the SL(2, C) transformation, It is instructive to consider the generators of SL(2, C) one by one:

31)
Inversion: z → −1/z , (3.32) The change in z can be reproduced exactly by the change in the coefficients: Inversion: Translation: The integral (3.29) should be invariant under the SL(2, C) action on z, provided that the contour transforms accordingly. It follows that the result of the integral should be invariant under the change of coefficients listed above.
To be specific, let us focus on the contribution of the contour C 1 enclosing the two poles z ± 1 only. The residue theorem gives where we defined The product D n (z + 1 )D n (z − 1 ) is easy to evaluate. Using the relations we find where we defined short-hand notations The new symbols (α ij , β ij , γ ij ) obey simple SL(2, C) transformation rules, Translation: To summarize what we have done so far, The remaining z ± 1 -dependent part may look complicated as both N n (z) and D n (z) are degree 2n polynomials in z. However, since we only need their values at the two solutions of M 1 (z) = 0, we can take the polynomial quotients. If we denote the quotient and the remainder by the integral gives The denominator n+1 i=2 ∆ 1i has degree (2n; 2, · · · , 2; 0, · · · , 0). It remains to express the numerator F n ≡ (a 1 ) 2n−1 (R nSn − S nRn ), which has degree (2n − 1; 1 · · · , 1; 1, · · · , 1), in an SL(2, C) invariant way. To do so, we introduce a few additional SL(2, C)-invariants: (3.48) For n = 0, the integral vanishes trivially as the contour can be pushed to infinity without encountering any poles. For n = 1, the numerator F n should be of degree (1; 1; 1) in three groups of variables and anti-symmetric with respect to the last two. It appears that J 123 is the only SL(2, C)-invariant with required properties. An explicit computation indeed shows that (3.49) For n = 2, we look for a polynomial of degree (3; 1, 1; 1, 1) with total symmetry under permutations in the same group and anti-symmetry between the last two groups. The answer indeed respects all the desired properties: Finally, we turn to n = 3, our original problem. There are a number of ways to combine J ijk , K ij , L ijkl to construct SL(2, C)-invariants with correct symmetry properties.
Remarkably, the answer can be organized using only two such combinations: A remark is in order. The integrals I n (C) are defined in such a way that if we set, say, M 3 (z) = M 5 (z), I 2 (C 1 ) should reduce to I 1 (C 1 ). In terms of F n , we should have F 2 (12345)| "3=5" = ∆ 13 F 1 (124) . (3.52) The reduction does not look obvious from the expression (3.50). Similarly, it is not obvious how the reduction from I 3 to I 2 occurs: It is conceivable that the decompositions (3.50) and (3.51) are not unique, and some alternative decomposition will make the reduction more obvious.

8-point amplitude: the result
In summary, the 8-point amplitude can be written as with the Jacobian factors (3.55) With the λ-parity operator π, the (1 + π) factor denotes the sum over two branches of the orthogonal Grassmannian. In (3.54), we removed the subscript from F 3 and and added reference to the contour by F (i). Note that this form of the 8-point amplitude is valid in any u-type gauges.

Physical and spurious poles
The 8-point amplitude (3.54) is the sum of two contour integrals, I(C 1 ) encircling the poles from M 1 and I(C 3 ) encircling the poles from M 3 . Each term carries physical and spurious poles. The most convenient tool to analyze the pole structure is the on-shell diagram pioneered by [5] and elaborated for ABJM amplitudes in [23][24][25].  The poles of the amplitude corresponds to boundaries of the on-shell diagrams. Each on-shell diagram has five vertices. Barring disconnected diagrams, each vertex yields exactly one boundary term. Figure 2 shows the five boundary terms from the on-shell diagram for I(C 1 ). Using the canonical coordinates for on-shell diagrams introduced in [23-25], we can easily see which consecutive minors vanish as we approach each of the five boundary components. To be specific, we adopt the coordinates of [24] associated with the OG tableaux. The tableau for I(C 1 ) is depicted in Figure 3. It can be translated to the C-matrix according to the rules explained in [24]. Let C i be the i-th column of the C-matrix. We begin by setting the 'source' columns (C 1 , C 2 , C 3 , C 5 ) to form an identity matrix. We assign a coordinate t v to each vertex. To fill in the 'sink' columns (C 4 , C 6 , C 7 , C 8 ), we consider all paths from a source to a sink which may move upward and to the right but not downward or to the left. The path picks up ± sinh(t v ) if it passes through the vertex, or ± cosh(t v ) if it makes a turn at the vertex. The final matrix element is given by a polynomial of the form, schematically, (3.56) We refer the readers to [24] for details. All we need here is the remarkable fact that the consecutive minors are given by monomials of the sinh(t) factors.: In these coordinates, the boundary operation amounts to taking one of the coordinate variables to zero or infinity. The orientation of the untied diagram in the OG tableaux is shown in Figure 4. The rescaled minor M 4 vanishes in the limit s 2 → ∞ or s 5 → 0. Through the prescriptions in Figure 3 and 4, the two limits give the two boundary diagrams on top of Figure 2, which in turn corresponds to the factorization channels for p 2 123 and p 2 567 , respectively. In the contour integral obtained earlier, the simultaneous vanishing of M 1 and M 4 , or equivalently the 'collision' of poles from M 1 and M 4 , would result in the vanishing of ∆ 14 . It is then natural to expect that ∆ 14 , a polynomial of kinematic variables, is proportional to p 2 123 p 2 567 . In the u-gauge, we can explicitly verify the proportionality between ∆ ij and physical poles. By symmetry, we expect that all of the eight physical poles are indeed associated with "collision" of roots of the minors: The powers of R are fixed on dimensional ground. We leave the details of the verification, including the numerical coefficients, to appendix A.2.
We can identify the poles for ∆ 13 = ∆ 31 in (3.54) as spurious poles. A standard argument in the Grassmannian integral uses the fact that Since ∆ 13 = ∆ 31 arises from I(C 1 ) and I(C 3 ) but not from I(C 2 ) or I(C 4 ), it must be spurious. The physical poles (3.59), in contrast, appear in both contour prescriptions. A related observation is that the on-shell diagram for ∆ 13 = 0 in Figure 2 can cancel against the same diagram from the boundary of I(C 3 ) if sign factors are properly assigned.
We conclude this section with a few remarks on the generalization of the methods we used. The u-gauge has some advantages over more familiar gauges based on Euler angles. One of them is the decomposition of the fermionic delta-function, withû pq linear in the z coordinates in (2.33). Another advantage is that, as explained in appendix A.2, the minors take a relatively simple form in the u-gauge.
Finally, in anticipation of the generalization to 10-point or higher amplitudes, we note that the SL(2, C) invariants are related to cross-ratios. For instance, Higher point amplitudes would inevitably give rise to more complicated "collision of poles" and it would be crucial to introduce higher dimensional analogs of ∆, J, K, L invariants to work without explicitly solving quadratic equations for the z coordinates.

Soft theorem for ABJM amplitudes
Soft theorems in gauge (gravity) theories explore the limit in which one or more gluon (graviton) approaches vanishing momenta. It is well-known that the soft limit of a nonvanishing tree amplitude is divergent and that the leading divergent term takes a universal form. More recently the sub-leading terms in the soft limit were calculated by using on-shell techniques [36] and spurred renewed interest in soft theorems and their applications. In this section, we derive a soft theorem for ABJM tree amplitudes, following a similar analysis for three-dimensional supergravity theories [32] (see also [23] for an early consideration of the double soft limit of ABJM theory).
Since the ABJM amplitudes are well-defined only for even number of external particles, it is natural to define the double soft limit of the (2k + 2)-point amplitude A 2k+2 by scaling the momenta of the last two particles, and taking the → 0 limit. In spinor variables, the scaling rule is In view of the soft theorems in gauge theories in various dimensions, we anticipate that A 2k+2 in the soft limit reduces to the A 2k up to a universal soft factor S( ), We will find that the soft factor consists of a leading and a sub-leading term:

Recursion relation for soft limit
Following the approach of ref. [32], we will use the BCFW recursion relation for ABJM amplitudes to analyze the double soft theorem. It is convenient to choose the two reference particle in the BCFW recursion to be (2k) and (2k + 1), namely, neighboring soft and hard particles. The BCFW-shifted kinematic variables are given bŷ where c = cosh t and s = sinh t with c 2 − s 2 = 1 and z ≡ c + s = e t .
As explained in [32,36] for soft graviton theorems, only one of the terms in the BCFW recursion formula contributes to the divergent soft factors. In our notation, the term is depicted in Figure 5. Let us briefly review why this is the case. The recursion formula schematically takes the form: is the BCFW kernel introduced in ref. [18].
Note that, in (4.5), the first correction terms carry 2 weight relative to the leading terms for bosonic variables, whereas the relative weight is 1 for fermionic variables. In order to compute the leading and sub-leading terms of the soft limit, we need only the leading correction terms for bosonic variables, but we should keep track of leading and next to leading corrections for fermions.
Collecting all ingredients, we find the z + contribution to the soft limit of A 2k+2 : where Expanding explicitly in powers of , we obtain where we introduced … … Figure 6. Soft limit from the on-shell diagram perspective.
Note that R i,j are R-symmetry generators of the ABJM theory. Note also that we could have obtained exactly the same result if we had chosen external particles (2k + 2) and (1) as the reference legs for the BCFW recursion. In this sense, the symmetry between (2k, 2k + 1) and (2k + 2, 1) has been restored. This is natural from the on-shell diagram perspective as illustrated in Figure 6.
Finally, we add the two contributions to obtain the leading and the sub-leading soft factor of the double soft limit where the leading and sub-leading soft factors are ) . (4.27)

Soft limit of the 6-point amplitude
For simplicity, we will use λ-parity operator π which is given by π : Λ 6 → −Λ 6 . (4.28) If we use this operator, we can consider only one part of 6-point amplitude.
The soft limit with particle 5 and 6 soft is realized in the light-cone basis as As we observed earlier, the bosonic kinematic invariants receive leading corrections at the 2 order. So we can freely use the 4-point kinematic relations. For example, (4.31) In the soft limit, up to O( 2 ) terms, the minors become Recall that the 6-point amplitude (3.20) contains two fermionic parts δ 3 (ζ + ) , δ 6 (Q 6 ) . Neglecting O( 2 ) terms, we observe that The second identity follows from where we used (super)-momentum conservation and Schouten identities.
We now move on to the δ 6 (Q 6 ) factor. To check our result, it is better to start with our conjecture. From our recursion relation result, A 4 part gives the super-momentum conservation like The last equality holds on the support of (4.36). So we can conclude that the six-point supermomentum conservation becomes the four-point supermomentum conservation with next-leading soft correction. Finally, our 6-point amplitude becomes if we expand the second line of above equations in terms of up to leading and sub-leading orders.

Soft limit of the 8-point amplitude
In this last subsection, we examine the soft limit of the 8-point amplitude we computed in section 3.3. In view of the computational complexity, we content ourselves with checking the leading order soft factor S (0) .

8-point amplitude with u-cyclic gauge
To take the double soft limit of the 8-point amplitude, we revisit the computation of section 3.3 with two slight changes. The first is that, to be specific, we work in the u-cyclic gauge. The C-matrix is given by In this gauge, the fermionic delta function reduces to (4.41) The fermionic bilinear coefficients are The second, more important change compared to section 3.3 is that, in order to expose the soft limit more clearly (more on this below), we use the contours C 2 and C 4 instead of C 1 and C 3 . Of course the two choices are equal up to an overall sign. In the notations of section 3.3, the result is The λ-parity operator π acts on A 8 as The numerators F (2) and F (4) are given by The derivation of these relations is essentially the same as the one given in appendix A.2. The factor ∆ 24 corresponds to spurious poles.
When we consider the limit in which particles 7 and 8 become soft, divergent terms come from ∆ 21 and ∆ 23 . If we use the contours C 1 and C 3 as in section 3.3, the two contributions are divided into two different on-shell diagrams. But, if we use the contours C 2 and C 4 , both contributions come from the residues of M 2 (z) and we can ignore the residues of M 4 (z).

Soft limit of 8-point amplitude
In the lightcone coordinates, the the double soft limit of the 7 and 8 is realized by (4.48) In the → 0 limit, u * m4 andū n4 * are of order . As we discussed earlier, kinematic invariants receive 2 corrections, so we can freely use the kinematic relations of the 6-point amplitude. For example, the identity (2.29) in the soft limit implies that 1 + u *  If we focus on the leading order only, the supermomentum-conserving delta function of A 8 trivially reduces to that of A 6 : (4.51) The only non-vanshing contribution from the fermionic part in the numerator F (2) is One can easily check that fermionic bilinears a 4+I = A I and c 4+I = C I become We observe that the following useful identities hold in the soft limit: 57) 58) where M ± i here denote consecutive minors of C ± contributing to A 6 . Collecting all ingredients, we obtain the soft limit of the 8-point amplitude in the leading order a 5 a 6 a 7 + 1 c 4 α 12 α 23 c 5 c 6 c 7 In the final step, we used the following non-trivial identity In the second step, we used momentum conservation. Note that the alternating signature metric is reflected in the square of a partial sum of momenta as p 2 ijk = (−1) i+j ij 2 + (−1) j+k jk 2 + (−1) k+i ki 2 .
In the third and fifth line, we used R 2 = v 12 w 12 + v 23 w 23 + v 31 w 31 , which is the same as (2.29). To conclude, we have verified (A.1) that in the u-cyclic gauge,