Soft and Collinear Limits in $\mathcal{N}=8$ Supergravity using Double Copy Formalism

It is known that $\mathcal{N}=8$ supergravity is dual to $\mathcal{N}=4$ super Yang-Mills (SYM) via the double copy relation. Using the explicit relation between scattering amplitudes in the two theories, we calculate the soft and collinear limits in $\mathcal{N}=8$ supergravity from know results in $\mathcal{N}=4$ SYM. In our application of double copy, a particular self-duality condition is chosen for scalars that allows us to constrain and determine the R-symmetry indices of the supergravity states in the collinear limit.


Introduction
Symmetries in a quantum field theory are their most important features. The difficulty of solving a theory, which means to compute the scattering amplitudes in terms of the correlation functions, is often dictated by the amount of symmetries the theory possesses. This is because symmetries in the theory are reflected in the amplitudes via the Ward identities and one can use these identities to constrain the correlation functions. This is called bootstrapping and has been a very effective tool in conformal field theory [1][2][3]. This also goes in the reverse direction, that is knowledge about the nature of amplitudes can help us discover non-trivial symmetries of the theory. Important examples include soft and collinear limits of amplitudes. Soft limit of an amplitude is defined by taking the momenta of an external particle to be zero. 1 In soft limit, the amplitude factorises into a universal soft factor which contains the divergence of the amplitude times the amplitude without the soft particle. The soft factor is universal in the sense that it does not depend on the intricate detailes of the theory, but only the helicity of the external particles. This universal factorisation can be extended to subleading order in electromagnetism and to subsubleading order in gravity [4][5][6][7][8][9][10][11][12][13][14][15][16]. Soft limits of amplitudes at tree level provided important new insights about the symmetries of certain theories when they were interpreted as Ward identities of certain non-trivial symmetries [17][18][19][20][21]. For example, soft gluon theorem in Yang-Mills theory is related to large gauge transformations [14,16,[22][23][24] and soft graviton theorem in Einstein's gravity is related to the so called Bondi-Metzner-Sachs (BMS) symmetries [24][25][26][27]. Thus studying soft limits of amplitudes even at tree level teach us more about the symmetries of the theory. Another important limit in which one can study the amplitudes is the collinear limit, in which the momenta of two external particles is taken to be collinear. Again the amplitude factorises into a collinear factor containing the divergence times the amplitude with the collinear particles replaced by another particle (see Section 3 and Section 5 for precise details). Collinear limits have played important role in flat space holography [28,29]. The collinear limit of amplitude turns into an operator product expansion (OPE) of conformal operators of the celestial conformal field theory (CCFT) on the celestial sphere 2 on the boundary [29][30][31][32][33][34]. These OPEs can be used to calculate the non-trivial asymptotic symmetries of the theory. The usual method of calculating asymptotic symmetries is by finding conformal Killing vectors and spinors becomes intractable in the presence of other fields in the theory. That is where CCFT becomes important. A recent proposal of Taylor et. al. asserts that one can calculate the asymptotic symmetries of gravity theories using soft and collinear limit of amplitude in the framework of CCFT. This has been confirmed to give consistent results in the few cases it has been implemented [35][36][37]. Hence the study of soft and collinear limits in gravity theories becomes important from this perspective. N = 4 supersymmetric Yang-Mills and N = 8 supergravity are maximally supersymmetric theories and are rich in symmetries. Due to enormous symmetries, one can compute higher and higher loop amplitudes and show that they are finite 1 Of course the particle has to be massless for such a limit to make sense. 2 In CCFT, one describes the four dimensional physics in terms of the conformal correlators of two dimensional CFT on the celestial sphere living at the null infinities of the Minkowski flat spacetime. The map from amplitudes in the bulk to conformal correlators on the boundary is the Mellin transform. [38]. In fact people argue that these are one of the simplest quantum field theories [39]. One can then study the soft and collinear limits of amplitudes in these theories to learn more about their symmetries. The study of soft and collinear limits in N = 4 SYM has already been done [40][41][42] and the corresponding CCFT was studied in [43]. On the other hand, recent investigations into gravity and gauge theory amplitudes have resulted in non-trivial relationships between the two [44]. Gravity tree level amplitudes can be expressed in terms of sums of products of gauge theory tree level amplitudes. This can be described by different double copy formalisms [45][46][47][48]. One can then naturally ask if it is possible to relate the soft and collinear limits in N = 4 SYM to soft and collinear limits in N = 8 supergravity. Indeed this can be done [49][50][51][52]. The relevant double copy formalism are reviewed in [53] which was originally formulated as a relation between open and closed string amplitudes [45]. The corresponding relation in the low energy effective theory gives a relation between gauge theory and gravity amplitudes. Thus one can explicitly calculate soft and collinear limits of amplitudes in gravity using the corresponding results in gauge theory.
In this paper we explicitly calculate the soft and collinear limits of all possible helicity combination in N = 8 supergravity using the known double copy relation to N = 4 SYM. General formulas for the double copy of amplitudes exists in literature [51,52] but to our knowledge, they have not been worked out explicitly. These relations are explicitly derived and stated in this paper.
The paper is organised as follows. In Section 2, we set up the notations that we follow throughout the paper. In Section 3, we briefly review soft and collinear limits in N = 4 SYM which we use later in the paper. Double copy formalism and the relevant formula relating the amplitudes are reviewed in Section 4. In Section 5 we recall some basic facts about N = 8 supergravity and state our conventions for its factorisation into a pair of N = 4 SYM theories. Finally in Sections 6 and 7 we record the explicit soft and collinear limits of supergravity amplitudes. In the main body of the paper we have tabulated the collinear and soft limits of the amplitudes with the appropriate R-symmetry indices and the detailed calculations have been postponed to the appendices for reference. The appendices also include spinor-helicity formalism and a list of computational results.

Notations
The Minkowski space can be parameterized using the Bondi coordinates (u, r, z,z) where (z,z) parameterises the celestial sphere CS 2 at null infinity. The Lorentz group SL(2, C) acts on CS 2 as follows: A general null momentum vector p µ can be parametrized as where q µ is a null vector, ω is identified with the light cone energy and all the particles momenta are taken to be outgoing. Under the Lorentz group the four momentum transforms as a Lorentz vector p µ → Λ µ ν p ν . This induces the following transformation of ω and q µ as It is useful to introduce the bispinor notation at this stage. We can write the basic null momentum vector q µ as Here σ αα µ = (1, σ x , σ y , σ z ).We can thus introduce the familiar angle and square bracket spinor notation (see Appendix A for a brief review of spinor-helicity formalism) for the left and right-handed momentum spinors: where we write To shorten the notation, we denote the spinors for momenta p i by i| α and |i]α respectively. The inner product of momentas p i and p j can then be written in terms of the angle and square brackets of the corresponding spinors which are now given by

Soft and Collinear Limits in N = 4 SYM
As detailed in the introduction, in this paper we shall be studying the interesting limits of supergravity amplitudes using double copy relations. For this purpose we use N = 4 SYM as a machinary to find our desired results for gravity. Let us briefly recall some of the prime properties of N = 4 SYM. There are 16 different fields in N = 4 SYM, all of which can be packaged in a single superfield. Let {η a } 4 a=1 be the Grassmann odd coordinates on the superspace. Then the superfield for N = 4 SYM can be written as where G ± (p) denote positive and negative helicity gluons, Γ a + , Γ − a denote positive and negative helicity gluinos respectively and Φ ab denotes the scalars. The superamplitude of n such superfields is then given by the n-point correlation function We sometimes suppress the momenta p i and superspace Grassmann coordinates η i and simply write A n (1, 2, . . . , n). Expanding both sides in η and comparing, one gets the scattering amplitude of all the component fields. Next we find the soft and collinear limits of the superamplitude. We begin with the soft theorem following [54]: where p s is the momenta of the soft superfield and a, b are the adjacent superfields. The soft factor Soft SYM (a, s, b) is given by [49] Soft SYM hol (a, s, b) = where (0) and (1) indicate the leading and subleading terms. Let us explain the above notation. We associate a pair of spinors h s ,h s with every soft momenta p s .
The limit h s ,h s , η s → εh s ,h s , η s with ε → 0 and h s some fixed spinor (namely h s → 0) is known as the holomorphic soft limit. The holomorphic soft factor is then given by [49] Soft(k) SYM hol (a, s, b) = 1 k! ab as sb sa ba hα Similarly the limit h s ,h s , η s → h s , εh s , η s withh s a fixed spinor (namelyh s → 0) is known as the anti-holomorphic soft limit. The anti-holomorphic soft factor is given by The physical soft limit p s → 0 is equivalent to considering both h s ,h s → 0 simultaneously. Thus in the physical soft lomit, the soft factor splits as the sum of holomorphic as well as the anti-holomorphic soft factors. We use these results in Section 7 to compute soft limits in supergravity. Next we discuss the collinear limits. In collinear limit, we take the momenta of two adjacent superfields p 1 and p 2 to be collinear. Under this limit, the two supefields can fuse to give another supefield with momentum p 12 = p 1 + p 2 . We parametrize the momenta of the collinear superfields as where z corresponds to the combined momentum p 12 on the celestial sphere CS 2 .
Since p 1 + p 2 = p 12 , we see that, for massless fields, the collinear limit p 1 ||p 2 implies p 1 · p 2 ∝ p 2 1 = 0 which is equivalent to the condition p 2 12 → 0. Now the collinear limit in N = 4 SYM is given by [43,55] A n (1, 2, 3, · · · , n) The l = 1, 2 terms in the collinear limits are called the helicity-preserving and helicitydecreasing processes. The collinear singularity is contained in the split factors. The split factor of helicity preserving process is given by [55] Split 0 z; η 1 , η 2 , η p 12 = 1 z(1 − z) Whereas for helicity-decreasing process, the split factor is given by [55] Split −1 z; η 1 , η 2 , η p 12 = 1 z(1 − z) (3.10) The integral over η p 12 can be performed using general results of Grassmann integration. 3 Here 3 As an example for any function f (η) we have [43] where, δ (4) Using these, we express the collinear limit (3.8) as A n (1, 2, 3, · · · , n) (3.12) Expanding both sides in η 1 and η 2 , we can get collinear limits of the component fields. For collinear limit of component fields, we use the following notation: where A n is the amplitude of n different fields in the theory and the sum is over all helicities in the theory. Note that the split factor is trivial for helicities h which does not have corresponding interaction with h 1 and h 2 . The split also satisfies the conjugation relation [55] The split factor of component fields in SYM has two parts, the kinematic part and the index structure part. Kinematic part only depends on the momenta of the collinear particles while the index structure consists of the SU(4) R-symmetry indices of the component fields. As indicated earlier, one can compute the kinematic part of the split factors for various combinations of helicities by expanding both sides of (3.12) in η 1 , η 2 and then comparing the coefficients. This has been done using mathematica. The non-trivial split factor for collinear gluons are: The split factor for collinear gluinos and scalars are: Finally the split factor for mixed helicities are All the other split factors can easily be obtained from Eq. (3.14). We now list the index structure part of the split factors for various component fields. We obtain it by expanding both sides of (3.12) in η 1 and η 2 and comparing the coefficients. Some of the index structures have been worked out in [43]. We complete the list here. Note the index structure in the collinear limit of a gluon with any other component field is trivially determined, hence we omit them from Table 1.

Double Copy : a brief review
Let us briefly review double copy (DC) technique that plays a crucial role in our analysis. It is a multiplicative bilinear operation to compute the amplitudes in one theory using amplitudes from other simpler theories. This is a method to express gravity tree level amplitudes in terms of sums of products of gauge theory tree level amplitudes. There are three different double copy formalisms for tree level amplitudes: KLT (named after Kawai, Lewellen, and Tye) [45], BCJ (named after Bern, Carrasco, and Johansson) [46] and CHY (named after Cachazo, He, and Yuan) [47,48] formalism. We refer to [53] for detailed review of these formalisms. Here we restrict our discussion to the application of double copy to soft and collinear limit of gravity amplitudes in terms of soft and collinear limits of gauge theory amplitudes.

Double copy and collinear limit
The KLT double copy was originally discovered in string theory as a relation between open and closed string amplitudes. Once the large string tension limit (also called the field theory limit) is taken, the KLT relation turns into a relation between gravity and gauge theory tree level amplitudes [51]. The general KLT relation for a general gravity tree level amplitude M tree n (1, 2, . . . , n) with n external legs (we have assumed n to be even below but the odd case can also be written in a similar way with appropriate modifications) with color-ordered gauge theory tree level amplitude A tree n (1, 2, . . . , n) is given by [51].
The functions f andf are defined as Thus every gravity state j on the LHS can be interpreted as the tensor product of the two gauge theory state on the RHS. Note that the doubling of supersymmetry in this double copy relation can be understood by counting the degrees of freedom on the two sides. Indeed N = 8 supergravity has 264 states which is twice the 128 states in N = 4 SYM. One can take collinear limit on both sides of the KLT relation (4.1) to obtain a relation between the split factor for collinear states in gravity to the split factors in gauge theory. We describe this relation below. The collinear limit in gravity is written as [52] M tree Using the KLT relation, the gravity split factor can be related to the "square" of gauge split factors as [51], Here a state h +h in gravity theory is written as product of states h,h in the two gauge theories and s 12 = 12 [21]. We will explain the explicit factorisation of states for the case of N = 8 supergravity into N = 4 super Yang-Mills states in Section 5.

Double copy and soft limit
Similarly, one can take the soft limit of the double copy relation to relate the soft factors in gravity and gauge theories. Let us start with the universal soft behaviour of the tree level n-gluon amplitude. The soft factor when the i-th particle is taken to be soft, for either helicities, is given by, Here the soft limit is parameterized by a factor ε → 0, as described in the last section. The factors S Gauge and S Gauge contains the soft divergences to leading and subleading order in the gauge theory. Similarly, the gravity amplitude also has this universal soft behaviour with i-th particle going soft and is given by, Gravity (i, a, b) Gravity and S Gravity are leading, subleading and subsubleading soft factors in the gravity theory. Double copy relates these soft factors as follows [46,54] 1 Gravity (s, n, 1) . This completes a brief review of double copy relations that we shall be using in the present work.

N =8 Supergravity
In this section, we briefly review the field contents and basic properties of the theory and also establish notations that we follow in the remainder of the paper.
Let {η A } 8 A=1 be the Grassmann coordinates on the N = 8 superspace. The degrees of N = 8 supergravity for an on-shell superfield is defined as where we have introduced the notation The fields H ± represent graviton, G AB + and G − AB represent gluons, ψ A + and ψ − A represent gravitinos, χ ABC + and χ − ABC represent gluinos and finally Φ ABCD represent the real scalars. The (sub)super scripts ± denote positive and negative helicity of various fields. The superamplitude is then defined by We now explain the factorisation of states in supergravity into tensor product of states in super Yang-Mills. We begin by counting the degrees of freedom in the two theories. It is summarised in Table 2 below. The precise factorisation of fields and operators are given in [50]. We summarise the factorisation in Table 3 below. The second factor of N = 4 SYM is written with a tilde to emphasize that the factors of the two gauge theories are not identical. The following notation is used in the table below and in the rest of the paper: uppercase indices A, B, C, D, ... ∈ {1, . . . , 8} will denote indices in N = 8 supergravity, lower case indices a, b, c, d ∈ {1, 2, 3, 4} correspond to first SYM factor and r, s, t, u ∈ {5, 6, 7, 8} correspond to second SYM factor. In particular, in equations where both upper and lower case indices have been used, we will assume A = a and A = r and so on when 1 ≤ A ≤ 4 and 5 ≤ A ≤ 8 respectively.  In this section, we compute the collinear limits using the component field formalism.
The double copy relation of collinear limits in component formalism is given by where the split factor Split SG −h (z, 1 h 1 , 2 h 2 ) is given in terms of the split factors in N = 4 super Yang-Mills theory as follows: where (h +h) is the factorisation of N = 8 supergravity state with total spin h +h in terms of two copies of N = 4 super Yang-Mills states with spins h,h respectively according to Table 3 and s 12 = 12 [21]. The sum over all N = 8 supergravity states is interpreted as a double sum over a tensor product of N = 4 SYM states [51]. The calculation of collinear limit then involves two steps: 1. Calculate the split factors for all possible factorisation channels, that is, for all possible values of spin and helicity states h in N = 8 supergravity. This can be done in such that the factorisation h = h 1 + h 2 into different spin and helicity states in N = 4 SYM from Table 3 gives nontrivial split factors. In general one only needs to calculate half of all possible combinations of helicities. The remaining split factors can be calculated using Write the collinear limit of amplitudes by consistently matching the R-symmetry factors using Table 1 which is non-trivial in case of N > 1 theories.

Collinear limits of like spins
Here we compute the collinear amplitudes from the splits for states of same spin. We will show the computation for some cases and summarise the results for the rests in tabular form and refer the reader to Appendix C.1 for all the details of the computations. Moreover we only summarise the collinear limits for independent cases not related by Eq.(6.3).

Gravitons:
When both collinear gravitons are of same helicity (positive or negative), then from Table 3, we see that A similar factorisation is true for opposite helicities. Thus split factors in N = 8 supergravity for two collinear gravitons is Writing the momenta of the collinear particles as p i = ω i q i , i = 1, 2, the momenta along the collinear channel is p = p 1 + p 2 = ω p q p with ω p = ω 1 + ω 2 and we can write Note that q p = q 1 = q 2 and hence With this parametrization, the collinear limits can be tabulated as, . . , n) Table 4: Amplitude corresponding to two collinear gravitons Here in LHS 1, 2, . . . , n refers to external particles with momenta p 1 , p 2 , . . . , p n and p 1 is taken collinear to p 2 according to the parametrization in Eq. (6.5). We will carry this notation throughout the paper.
Note that the collinear limit of two negative helicity gravitons from the collinear limit of two positive helicity gravitons by flipping the helicities throughout and z 12 ↔z 12 . This is reminiscent of Eq.(6.3).

Gravitinos:
The non-trivial split factors in N = 8 Supergravity for two collinear gravitinos are given by 12 [12] , Split SG +2 z, 1   Table 6: Amplitude corresponding to two collinear graviphotons In writing the collinear limit of opposite helicity graviphotons, we made a choice of self-duality factors α 4 =α 4 = −1, α 8 = 1. This choice is unique and motivated by our aim to make the R-symmetry indices consistent in both sides of the amplitude calculations. See Appendix C for details.

Graviphotinos:
Following the factorisation in Eq. (6.2), the non-trivial split factors for this channel in N = 8 supergravity are given in Appendix B.2.

Collinear limits of Mixed Spins
In this section, we list the collinear limit of states with different spins. The nontrivial split factors are listed in Appendix B and the detailed calculation is done in Appendix C.2.

Graviton soft limit
Recall that in the physical soft limit p s → 0 or equivalently h s ,h s → 0, the leading soft factor in SYM is given by the sum of leading soft factors in holomorphic and anti-holomorphic soft limit: Comparing the coefficients of ε powers, we get Thus the double copy relation gives the sum of leading, subleading and subsubleading soft factors in supergravity in terms of the leading and subleading soft factors in SYM. It is clear that the leading and subleading soft factors in supergravity are given by We now substitute (3.5) and (3.6) into (7.5) and (7.6) to get the leading and subleading soft factors in supergravity. Note that the nonholomorphic soft factor in SYM includes the Grassmann delta function δ 4 (η). So while squaring the nonholomorphic soft factor of SYM, the square of this delta function in double copy is interpreted as the Grassmann delta function on N = 8 superspace: where the indices have the usual meanings with a running from 1 to 4 and A running from 1 to 8. The leading soft factor is then given by [si] si [ai] 2 [as] 2 δ 8 (η A ) (7.8) We now evaluate the subleading soft limit. From (3.6) and (7.6) we have, where we used the momentum conservation Note that in the soft superfield, η s → 0 gives the positive helicity soft graviton and δ 8 (η s ) gives the negative helicity soft graviton. Thus we only retain these terms in the soft factor. Thus we get which is the sum of soft factor for positive and negative helicity soft graviton in pure gravity [54,Eq. 2.9]. Note that in the above formula, the momenta p a acts as reference vector and hence can be taken to be any null vector r. This is an indication of the diffeomorphism symmetry of gravity amplitudes. We can thus rewrite the soft factor as [is] ir si sr hα

Leading soft gravitino limit
To calculate the soft limit of gravitinos, we use the results of [49]. Under the holomorphic soft limit of the superfield, we have (7.10) The leading soft factor 4 is same with the one in pure gravity: [si] ri 2 si rs 2 = S (0) . (7.11) The sub-leading soft operator is given by [si] ri si rs ∂ ∂η iA Here the leading soft gravitino operator involves the first order derivatives with respect to the Grassmannian variables η i 's. These term will preserves the total helicity as well as SU(8) R-symmetry. The sub-sub-leading soft factor is given by [si] si ∂ 2 ∂η aB ∂η aA . (7.14) We now expand the generic superamplitude on the left hand side of (7.10) in the grassmann odd variable η s of the soft superfield: . . , Ψ n + · · · (7.15) and compare with the right hand side of (7.10) to get the following soft limits:

Conclusion
In this work we have computed the soft and collinear limits of the maximally supersymmetric N = 8 supergravity theory in four spacetime dimensions using the double copy relations in both soft and collinear sectors of N = 4 Super Yang-Mills. The computations are done in the celestial basis appropriate for applications to celestial holography. An important point in our application of double copy is a different choice of self-duality condition for scalars. The constraints imposed here differs in signs: α 4 =α 4 = −1 and α 8 = 1. This choice is motivated by our desire to combine the collinear limits for different factorisations of N = 4 SYM to N = 8 supergravity. Based on the factorisation of states in the gravity theory in terms of states in the gauge theory, we are able to constrain and determine the R-symmetry indices in the collinear limit. This is also the novelty of this work.
The goal of this work is twofold: first we would like to construct the dual celestial CFT corresponding to the bulk N = 8 supergravity in four dimensions. This requires the collinear limits of bulk amplitudes as they imply the OPEs of (super)conformal operators in the CCFT. Second, we would like to determine the asymptotic symmetries of N = 8 supergravity using celestial holography. As discussed in [56], our final goal is to determine the contribution of (super)BMS hairs to black hole entropy 5 . The first step to such an analysis would be to understand the extension of the BMS group to super BMS group in N = 8 supergravity. The corresponding N = 1 supergravity case has already been worked out in [35] and a primary construction of the same for higher supersymmetric cases, purely from algebraic perspective, has been addressed in [58]. However a thorough gravity analysis with higher supersymmetry is still missing. This issue has been addressed in a companion paper [56], where the asymptotic symmetry algebra of the N = 8 supergravity has been derived.

A A brief review of spinor-helicity formalism
Recall that helicity spinors are left and right handed representations of the Lorentz group SO(1, 3) ∼ SL(2, C). We denote the left and right handed helicity spinors by h α andhα respectively. Lorentz invariant contractions of spinors is defined using the completely antisymmetric rank 2 tensor ǫ αβ defined as The contractions are then defined as Wherever we have angular brackets, we understand that it is the contraction of left handed spinor whereas the square bracket is the contraction of the right handed spinor. We thus suggestively denote left handed spinor by |λ α and right handed spinor by [λ|α. A given null momentum p µ can be written as a bispinor for p i = |i [i|. i be any one out of the n external momenta. One can also express polarisations in terms of spinors but we will not need it explicitly in our discussions.

B Split Factors
Here we list all of the split factors corresponding to both like and unlike spins in our supergravity theory. Gravi-photons splits: Gravi-photinos splits: Split SG +2 z, 1 12 .

C Explicit computations of Amplitudes
In this appendix, we explicitly calculate the collinear limits of states various spin combinations.

C.1 Like spins
The collinear limits of gravitons is calculated in Section 6.1 in detail. So we start with collinear limit of gravitinos. × M n−1 p +2 , · · · , n + Split SG +2 z, 1 Note that the above expression contains 16 terms but only four terms are nonzero since δ a r = 0.