Celestial superamplitude in $\mathcal N=4$ SYM theory

Celestial amplitude is a new reformulation of momentum space scattering amplitude and offers a promising way for flat holography. In this paper, we study the celestial amplitude in $\mathcal N=4$ Super-Yang-Mills (SYM) theory aiming at understanding the role of superconformal symmetry in celestial holography. We first construct the superconformal generators acting on the celestial superfield which assembles all the on-shell fields in the multiplet together in terms of celestial variables and Grassmann parameters. These generators satisfy the superconformal algebra of $\mathcal N=4$ SYM theory. We also compute the three-point and four-point celestial super-amplitude explicitly. They can be identified as the correlation functions of the celestial superfields living at the celestial sphere. We further study the soft and collinear limits which give rise to the super-Ward identity and super-OPE on the celestial sphere, respectively. Our results initiate a new perspective of understanding the well-studied $\mathcal N=4$ SYM amplitude via 2D celestial conformal field theory.


Introduction
In recent years, a new formalism of scattering amplitude was developed in the study of flat holography [1]. In contrast to the conventional momentum basis manifesting the translational invariance, the new formalism makes full use of the Lorentz symmetry of spacetime and expands the wave functions of particles in terms of boost eigenstates. Mathematically, by performing the Mellin transformation on the momentum space scattering amplitude [1][2][3], one obtains the so-called celestial amplitude which behaves just like a correlation function in 2D CFT under Lorentz transformation [1,4,5]. As such, this reformulation offers a promising approach to establish flat holography by relating the quantum gravity scattering process in the 4D bulk Minkowski spacetime with the observables of the 2D celestial conformal field theory (CCFT) living on the boundary celestial sphere. 1 This goes beyond the paradigm of AdS/CFT [6] which is well-understood now after two decades of efforts. Guided by the general holographic principle [7,8] and considering the great success of AdS/CFT, it is natural to ask how to concretely realize the holographic duality for quantum gravity in the absence of cosmological constant? Recent fruitful results are showing that celestial holography in terms of CCFT opens up an interesting avenue to this question! Besides the connection to flat holography, CCFT also offers a very different perspective to understand scattering amplitude itself. For example, the soft limit and collinear limits feature some universal properties of quantum fields and are vital for the self-consistency of S-matrix. It turns out that they just correspond to the standard Ward identity and OPE in CCFT [9][10][11][12][13]. CCFT also enables us to succinctly characterize the infinite number of non-trivial symmetries in asymptotically flat spacetime [14]. Furthermore, CCFT provides an advantageous language in describing the UV and IR behavior of scattering amplitude [15].
Despite its compelling role, CCFT itself remains poorly understood. On the one hand, the celestial conformal field theory behaves in many aspects like an ordinary two dimensional conformal field theory. Many techniques in the ordinary CFT can be borrowed and applied to CCFT. For example, one can construct the stress tensor [3,16] and it is meaningful to discuss the operator production expansion (OPE) of various operators [9,13]. On the other hand, CCFT features lots of peculiarities. In particular, the spectra of operators and the notion of inner products, conjugation seem to be drastically different from the conventional CFTs [4]. See [17,18] for recent discussions. Nevertheless, the rich symmetries and various self-consistency conditions already impose stringent constraints on CCFT.
In this paper, we add one more ingredient to the CCFT by considering the quantum field theories in 4D with superconformal symmetry. 2 In particular, we will be interested in the N = 4 super-Yang-Mills (SYM) theory. The N = 4 SYM theory enjoys a huge amount of symmetries which make it tractable or even exactly solvable in many situations. Especially, the scattering amplitude in N = 4 SYM has been intensively studied. Furthermore, N = 4 SYM theory provides the first and most successful example of AdS/CFT correspondence. Given its unique role, we are naturally led to study the celestial amplitude of N = 4 SYM theory and its role in flat holography.
We begin our studies with symmetry, which is arguably one of the most important guiding principles in physics. For celestial amplitude, a very natural question is how the various symmetries act on the celestial amplitude and how to constrain the structure of celestial amplitude by imposing symmetries. For Poincaré symmetry and conformal symmetry, the symmetry generators acting on celestial amplitude are found in [19]. While for supersymmetry, it was also discussed in [20,21]. Here we generalize the discussion to superconformal symmetry and especially the maximal superconformal symmetry enjoyed by N = 4 super-Yang-Mills theory. To this aim, we make full use of the on-shell superfield formalism by introducing Grassmann variables. We construct the superconformal generators in celestial superspace and checked that they indeed satisfy the superconformal algebra. These superconformal generators act on the on-shell celestial superfield which we constructed explicitly and fulfills a representation of the superconformal algebra. These superconformal generators can also act on the celestial super-amplitude, which is the Mellin transformation of momentum space superamplitude, and can be regarded as the super-correlator of on-shell celestial superfields. The superconformal symmetry thus imposes non-trivial constraints on the super-correlators: they must be invariant under the action superconformal generators.
Using Mellin transformation, we also explicitly compute the three-point and four-point celestial superamplitude. In particular, we are able to recast the celestial superamplitude of N = 4 SYM theory in a form with not only manifest Lorentz symmetry but also the obvious dilatation symmetry and supersymmetry, in contrast to the manifest translational invariance and supersymmetry of momentum space superamplitude. By expanding the celestial superamplitude in Grassmann variables, one can obtain all the component amplitudes including the gluon amplitude.
We will also study the soft limit and collinear limit of scattering amplitude. In these limits, the amplitudes diverge, featuring some universal behavior of quantum fields. For pure Yang-Mills (YM) theory, these two limits enable us to establish the Ward identity and extract the OPE of celestial operators, respectively [9][10][11][12][13]. We generalize these discussions to our N = 4 SYM theory. In particular, the leading supersoft theorem in momentum space turns into the conformally supersoft theorem on celestial sphere. Taking into account the color factors, this further leads to the Ward identity relating celestial super-correlators with and without soft "super"-current insertion. Physically, this arises from the asymptotic large gauge transformation of gluons, and the soft "super"-current is essentially the soft gluons with both helicities, generating the "super"-Kac-Moody symmetry on the celestial sphere. Again, the "super" here refers to the bulk supersymmetry and differs from the standard super-Kac-Moody in 2D.
While in the collinear limit, the two particles pass through the nearby points on the celestial sphere, thus corresponding to the coincident limit of operators. So the collinear limit enables us to derive the OPE of operators in celestial CFT [9,13]. Indeed, using the super-split factor for the collinear limit of SYM theory, we are able to compute the OPE between two super-operators in N = 4 SYM theory. The super-OPE encodes all the component OPEs which can be obtained easily by expanding in Grassmann variables.
We further consider the super-OPE in the limit ∆ → 1, 1/2, 0, · · · . The resulting super-OPEs manifest themselves with a pole in conformal dimension ∆. The operators with these special values of conformal dimension are thus the soft "super"-current. In particular, when ∆ → 1, the resulting OPE agrees with the one derived directly from the soft gluon theorem, thus providing a consistent check of CCFT. While for ∆ = 1/2, 0, they correspond to soft gluinos and soft scalars, which are related to soft gluons via supersymmetry. This paper is organized as follows. In section 2, we start with symmetry aspect of celestial amplitude by constructing explicitly all the generators of superconformal symmetry as well as the corresponding superfield. In section 3, we compute the three and four-point celestial super-amplitude explicitly and comment on the general structure at higher points. In section 4, we study the soft limit of N = 4 SYM theory which results in the conformally soft theorem and Ward identity for celestial operators. In section 5, we compute the OPE of super-operators from the collinear limit of N = 4 SYM theory. We summarize the results of this paper in section 6 and outlook possible future directions. We also include appendix A, where we review the soft and collinear limit of YM and N = 4 SYM theory.
Noted added: After the completion of this work, we learned that Andreas Brandhuber, Graham R. Brown, Joshua Gowdy, Bill Spence and Gabriele Travaglini, who are in the same group at QMUL, had been working independently on a very similar problem [22], which overlaps with our section 2 and 3. The results in these two papers are consistent although using different approaches. We were not aware of the work of each other until the final stage due to Covid restrictions.

Superconformal symmetry on the celestial sphere
In this section, we will study the symmetry aspect of CCFT. We will construct the superconformal generators (2.78) -(2.89) in terms of the celestial sphere coordinates and Grassmann variables. They act on the celestial on-shell superfield (2.66) which can be obtained as the Mellin transformation of the on-shell superfield in momentum space. These generators are shown to satisfy the superconformal algebra and impose Ward identity constraints on the celestial superamplitude.
In our discussion, we make full use of the on-shell superfield formalism which is especially powerful for N = 4 SYM theory and brings a lot of simplifications. Note that the role of supersymmetry was studied before in [20] where the discussions were mainly based on the component form of each multiplet.

Kinematics on the celestial sphere
Let us start the celestial story with kinematics. A massless particle travels along light-like direction in spacetime and is described by null momentum p µ satisfying p µ p µ = 0. We can parametrize the null momentum in terms of complex coordinates (z,z) on the so-called celestial sphere where ω > 0 and the null vector q µ is The point (z,z) can be regarded as the point on the celestial sphere where the massless particle crosses, while ω is the energy along this null direction. Under SL(2, C), the (z,z) coordinates on the celestial sphere transform as As a consequence, we have where Λ µ ν is the associated Lorentz transformation satisfying Λ T ηΛ = η. This is not surprising as SO(3, 1) SL(2, C). Therefore the Lorentz transformation in 4D Minkowski spacetime induces the conformal transformation on the 2D celestial sphere, suggesting the existence of a conformal field theory living there.
A nice property of null vector q µ is (2.5) A spinning massless particle also carries internal degrees of freedom which can be described in terms of polarization vectors. For gauge boson, the two polarizations can be naturally chosen as We are interested in the particles involved in a scattering process where crucially, we need to distinguish whether the particle is incoming or outgoing. So the precise way to parametrize a null momentum p µ in scattering amplitude is where the extra factor ε = sgn(p 0 ) further takes into account whether the particle is incoming (ε = −1) or outgoing (ε = +1).
For the null vector introduced in (2.2), we have This suggests that we can take where we introduce the following notation for later convenience (2.12) The angle and square brackets of spinors are (2.14)

Celestial amplitude
The scattering amplitude in momentum space is given by where ε i labels the ingoing (−) and outgoing (+) particle, J i is the helicity of massless particle, and δ-function is to enforce the momentum conservation. The scattering amplitude should respect the symmetry of the theory, especially the Poincaré symmetries which consist of translations and Lorentz transformations. In momentum space, this means where the generators of translations and Lorentz transformations act in momentum space as The amplitude in the form of (2.15) obviously satisfies the first equation in (2.16), while the second equation in (2.16) is not manifest anymore. So, the momentum space scattering amplitude makes the translation symmetry manifest but leaves the Lorentz symmetry obscure. By contrast, the celestial amplitude makes the Lorentz symmetry evident, while sacrificing the manifestation of translational invariance. The celestial amplitude is defined as the Mellin transformation of momentum space amplitude where we parametrize the momentum of each particle as (2.8). As such, the celestial amplitude can be regarded as a conformal correlator on the celestial sphere and transforms under SL(2, C) as where a, b, c, d ∈ C, ad − bc = 1. This SL(2, C) symmetry on the celestial sphere is just the Lorentz symmetry of the spacetime. So the Lorentz symmetry is manifest, while the translation symmetry is not obvious anymore. To show that Poincaré symmetries are indeed preserved, we first need find the generators of those symmetries in the celestial basis. Note that the conformal dimension and spin/helicity are related to the holomorphic and antiholomorphic weights as where J in 2d is the spin of the operator, while in 4d it is the helicity of the particle. In order to have a complete and delta-function-normalizable conformal primary wavefunctions in conformal basis, the conformal dimension should reside in the principal continuous series of unitary representations of SL(2, C) [4] ∆ = 1 + iλ , λ ∈ R . (2.22) We will also use the following relations later:

Poincaré symmetry
As in the ordinary 2d CFT case, the SL(2, C) is generated by L k ,L k , k = 0, ±1 whose actions on primary operator O h,h with conformal weights h,h are [23] where the differential operators L k ,L k are given by while L k ,L k should be regarded as charges in the field theory. 3 They satisfy the commutation relations and Note that there is an extra minus sign in (2.28). The difference of the extra minus comes from the fact that transformations compose in the opposite order when acting on the coordinates. 4 We will mostly work with L k ,L k when discussing the conformal symmetry. The generators of Lorentz transformations are the linear combinations of these SL(2, C) conformal generators L k ,L k . Let us now switch to translation symmetry. In momentum space, the action of translation just multiplies the amplitude by the momentum (2.16). In celestial basis, the insertion of momentum P µ j = ω j q µ j in the integrand (2.18) implements a multiplication of q µ j and a shift of conformal dimension ∆ j → ∆ j + 1 (without changing the helicity). As a result, h j → h j + 1 2 ,h j →h j + 1 2 . Therefore the translation generators in the celestial basis are given by [19] Using (2.9), we can rewrite it as 5 whose form further suggests us to write in the product form:

Bulk conformal symmetry
Besides the Poincaré symmetry considered above, we can further consider the 4D conformal symmetry acting on bulk spacetime. Especially, the special conformal generator in the celestial basis has been worked out in [19]. As translation generators, we find special conformal generators can also be compactly rewritten as (2.32) Finally, the dilatation is given by We can then work out the full conformal algebra in this basis. Let us first show two useful relations: (ε 12 = −ε 21 = 1) and its conjugate We can then construct See the appendix of [24] for more detailed discussions on this point. 5 Here we introduce the factor 1/2 for simplicity, otherwise there will be many factors of √ 2 flowing around.
which enables us to write and likewise for Lαβ. It turns out that these are just the L k ,L k in (2.26). More precisely, we have So L αβ , Lαβ are just the generators of SL(2, C). 6 With these generators, one can show that they indeed satisfy the conformal algebra so(4, 2): For any field theory with conformal symmetry, the amplitude should respect these symmetries.
where the subscript j means the action of the generators on the j-th particle.
For comparison, we also write down the conformal generators in the spinor-helicity basis [25]: (2.47)

Bulk superconformal symmetry and N = 4 SYM
Now we turn to supersymmetry. Our main interest in this paper is N = 4 SYM theory which enjoys the maximally superconformal symmetry psu(2, 2|4) in 4D. So to study the celestial amplitude in N = 4 SYM theory, we also would like to find the generators of the superconformal symmetry. We will first review the super-amplitude in spinor-helicity basis and then generalize to the celestial basis. Let us first recall the field content of N = 4 SYM theory which includes the spin-1 gluon, four spin-1/2 gluinos and six real scalars. All of them transform in the adjoint representation of the gauge group. For scattering amplitude, we only need to consider the on-shell degrees of freedom (d.o.f). There are 8 B + 8 F on-shell degrees of freedom and we list them in the following table 2.5: (1) anti-symmetric (6) fundamental (4) anti-fundamental (4) Table 1. On-shell degrees of freedom of N = 4 SYM. Note that the color index for gauge group is suppressed.

Superconformal symmetry in spinor-helicity basis
It is convenient to introduce the anti-commuting Grassmann variables η A with A = 1, · · · , 4. Then the on-shell degrees of freedom of N = 4 SYM theory can be packaged into an on-shell superfield as follows [25]: Note each component field in this superfield has different helicities. We can formally assign η A helicity 1 2 , then each term in the on-shell superfield (2.48) carries the same helicity. Mathematically, we can introduce the following generators (2.49) whose action on the superfield is then ΩΨ = Ψ. The Poincaré and conformal supersymmetry generators acting on the superfield (2.48) are given by [25]: Besides, the R-symmetry generators are given by [25]: Together with those in (2.46) and (2.47), they generate the full superconformal symmetry of N = 4 SYM.
The superamplitude provides a compact way to write down the amplitude of SYM theory: Performing the η expansion and comparing both sides, one can get the scattering amplitude of all the component fields. The superamplitude is severely constrained by the superconformal symmetry of SYM theory. Especially this implies We will not spell out all the details here which can be found in [25]. We can then read off the Q-supersymmetry variation of the component field defined by 7 Expanding both sides in components, we have which gives the supersymmetry variations of various component fields:

Quasi-supersymmetry generators in celestial basis
Now we would like to perform a similar construction in the celestial basis. First we need to find the supersymmetry generators in terms of the celestial coordinates. Motivated by the construction in the spinor-helicity basis in (2.50) and (2.51), it is natural to guess that the Poincaré and conformal supersymmetry generators take the following form: where p,p, k,k are given in (2.31) and (2.32). In particular, it is easy to check that they satisfy the supersymmetry algebra as desired: Besides, we also have the R-symmetry generators which take the same form as (2.52): as well as We can further compute where we used the identities (2.34) and (2.37). Note that when acting on on-shell superfield or super-amplitude, we have Ω = 1.
One can also check all the rest of the commutators. Indeed they generate the psu(2, 2|4) superconformal algebra, which is the symmetry of N = SYM theory. However, it turns out that the generators constructed in this way are a bit subtle as they don't act on the physical on-shell superfield as we expected. For this reason, the generators constructed above will be called quasi-superconformal generators. We will explain this point in detail in the following.

Celestial on-shell superfield and celestial superamplitude
Now we also would like to have the analog of the superamplitude (2.53) in the celestial basis. For this purpose, we need to generalize the notion of superfield in (2.48). The superconformal generators can then act on the celestial on-shell superfield and realize a representation of the superconformal algebra. It turns out that this brings a subtlety which will be resolved at the end of this section.
Celestial on-shell celestial superfield from Mellin transformation A natural way to obtain the superfield in celestial basis is to perform the Mellin transformation of (2.48): (2.65) Since the η-expansion and Mellin integral commute, we can perform Mellin transformation component by component. Each component has the same conformal dimension, but different helicity. The results can thus be compactly written as with the extra condition h −h = 1. Then obviously every component indeed has the same conformal dimension ∆ = h +h and its own correct helicity (note that the Grassmann variable η has ∆ = 0 and J = 1/2). The superamplitude is given by However, this on-shell superfield is not appropriate for the supercharge Q,Q constructed in (2.50). From the supersymmetry variation in (2.57), we obtain However, applying the supercharge (2.58) to the (2.66), the supersymmetry variation rule δ Φ = ) which is not consistent with the above result (2.69). To reproduce the expected supersymmtry variations, one needs to modify the rule as follows: (2.71) These supercharges are just those used in [20,21]. However, at this moment, it is not obvious how to similarly modify the rest of generators in order to fulfill a representation acting on (2.66). This will be achieved at the end of this subsection. Before that, let us stick to the quasi-superconformal generators that we constructed in the previous subsection and try to find a way to fulfill its representation.

Quasi-on-shell superfield
It turns out that we can introduce the following superfield Applying the Poincaré supersymmetry generators (2.58) to the above superfielld, the corresponding supersymmetry transformation rules indeed give the right variation (2.69). However, there is a subtle point related to the superfield (2.72). Different components there can not become on-shell simultaneously due to their different helicities but the same value of h −h. For this reason, we will call (2.72) quasi-on-shell superfield. Nevertheless, this subtlety does not bring any physical pathologies because we always need to pick exactly one component in the multiplet for each external leg to get the amplitude in a physical scattering process. So we can regard (2.72) as a formal sum where the conformal weights h,h are arbitrary a priori. The on-shell condition for h −h is imposed appropriately only after the component field is specified.
The corresponding superamplitude is given then by To obtain the component field amplitude, we first perform the η-expansion, choose a specific component for each external leg, and finally impose the helicity constraint for each h i −h i .
Celestial on-shell superfield, celestial superamplitude and superconformal generators In the above discussions, we have constructed the superconformal generators (2.58), (2.59) whose action on the quasi-on-shell superfield (2.72) fulfills the representation of the superconformal algebra. Although we have argued that the quasi-on-shell superfield (2.72) does not bring physical pathologies, its interpretation is still obscure conceptually. In the following, we will construct a new set of superconformal generators. They act on the celestial on-shell-superfield (2.66) and also realize a representation of the superconformal algebra.
By comparing the component expansion of two superfields (2.66) and (2.72), it is not difficult to find that they can be related as follows: where e − 1 2 η A ∂ A ∂ J implements the shift of h andh according to the number of η's. We have shown that Ψ fulfills the representation of the superconformal algebra with generators {P, L, K, D, Q, S, R}. To find the generators acting on the on-shell superfield Ψ, we can naturally modify generators in the following way where O ∈ {P, L, K, D, Q, S, R} is any superconformal generators we constructed before. Now O acts on the on-shell superfield Ψ in (2.66). By construction, this should also fulfill the superconformal representation and in particular the generators should satisfy the superconformal algebra. In the followings, we will write down the generators { P , L, K, D, Q, S, R} explicitly and verify these statements.
Let us start by observing the following useful relation: which can be proved by inserting 1 = e η A ∂ A ∂x e −η A ∂ A ∂x between ∂ B l and f (x). 9 This immediately implies that and It is also easy to find that The special conformal generators are now: wherek It is worth comparing the special conformal generators here with that in (2.32): the only difference is the shift of h,h by ∓ 1 4 η A ∂ A . Looking at the superfield (2.66), the shift is indeed reasonable because it ensures that the special conformal generators act on each component in a correct way with its right value of h,h.
The Lorentz generators become: The translation generators remain the same: The Q-supersymmetry generators are

86)
where we used the relation (2.77). The Q-supercharges constructed now agree with (2.71). Making full use of (2.77), one also finds the S-supersymmetry generators:

88)
These are all the superconformal generators acting on the on-shell superfield Φ. As we said, by construction they satisfy the superconformal algebra. Now we can compute some commutators of these generators and verify this claim.
First of all, it is easy to see that where we used the identity and noticed that The identities (2.34) and (2.37) are generalized to: They enable us to check All the rest of the commutators can also be checked similarly and indeed they generate the superconformal algebra psu(2, 2|4). The superamplitude should be invariant under the action of these symmetries, so it must satisfy the Ward identities for all the superconformal generators:

96) in addition to the helicity constraint
Especially, the Lorentz invariance is manifest. By generalizing (2.20), the celestial superamplitude transforms under SL(2, C) as The second equality follows because the celestial superamplitude is the sum of different component terms where each term contains specific number of η's; shifting the power in cz j + d,cz j +d according to the number of η j just implements the transformation of η j itself under SL(2, C), which is also consistent with its conformal weight (h η ,h η ) = ( 1 4 , − 1 4 ). One can also check explicitly that all the superconformal generators leave the conformal weights invariant as desired, otherwise different terms in the sum in (2.96) carry different weights due to the action of generators on different particles and thus break the SL(2, C) covariance.
In the following, we will be mainly using the on-shell superfield Ψ as its physical meaning is clearer and has manifest SL(2, C) covariance. To ease the notation, we will ignore the hat on the superfield (2.66), the superamplitude (2.67) and generators (2.78) -(2.89).

Celestial superamplitude in N = 4 SYM theory
In this section, we will compute the (color-ordered) celestial superamplitude in N = 4 SYM theory, focusing especially on the three-point and four-point case. This is done by performing a Mellin transformation on the momentum space superamplitude. As we will see explicitly, the resulting celestial superamplitude indeed can be regarded as the correlators of celestial superoperators and it has also manifest conformal invariance and supersymmetry. The computation technique here closely follows that in pure YM case [5]. We will first discuss the four-point amplitude, and then the threepoint case.

Celestial amplitude in YM theory
Before considering the superamplitude in SYM, let us first review the YM case. Due to the conformal symmetry, the YM amplitude has the following scaling property Then the celestial amplitude is given by the Mellin transform (2.18). For four-particle scattering 1, 2 → 3, 4, actually the scaling property of YM amplitude ensures that the celestial amplitude takes the following form [5]: where Θ equals to 0 (1) for negative (positive) argument and The cross-ratio is defined as The different factors in (3.3) can be explained as follows. The factor δ( i λ i ) accounts for the scaling property (3.1) as well as the following integral Physically this is just the manifestation of the conformal symmetry: it ensures that the Ward identity (2.96) for dilatation (2.33) is identically satisfied. The factor δ(χ −χ) just comes from momentum conservation which ensures that in the four particle scattering process the momenta of for particles should lie on the same plane. The intersection of such a plane with celestial sphere forces the crossratio to be real. For concreteness, let us consider the four-gluon scattering with 1 − , 2 − → 3 + , 4 + , namely ε 1,2 = −1, ε 3,4 = 1 and J 1,2 = −1, J 3,4 = +1. The color-ordered amplitude in momentum space is given by Plugging into (3.3), we get the corresponding celestial amplitude 10 So the four-gluon celestial amplitude indeed takes the form of four-point function in CFT with weights h i ,h i indicated above.

Four-point celestial superamplitude in SYM
Now we move to the amplitude in SYM theory. The color-ordered superamplitude is given by [25] A 4 (ω j , z j ,z j ) = δ 4 (P )δ 8 (Q) 12 23 The delta function δ 4 (P ), δ 8 (Q) guarantees that the amplitude conserves the momentum and is supersymmetric invariant. The celestial amplitude can be again obtained through Mellin transformation.
For the superamplitude in (3.10), we can easily compute where we use the fact that the cross-ratio is real χ =χ. 10 Actually, there is also an overall factor (−1) iλ 2 +iλ 3 which will be suppressed here and below.
Therefore the four-point celestial super-amplitude (2.67) can be written as: The celestial super-amplitude can be regarded as the conformal correlation function of superoperators on the celestial sphere. 11 The shift operation e ∂ ∆ i /2 in the fermionic delta function seems to be unconventional; but actually it can be replaced with the multiplication factor K −1 e ∂ ∆ i /2 K = k =i (z ikzik ) − 1 4 once acting on the square bracket. Performing the η expansion, we get all the component amplitude. For example, consider the 4 A=1 η A 1 η A 2 term in the expansion. Using This agrees with the four-gluon celestial superamplitude in (3.8).
Similarly, we can consider the four gluino amplitude by picking up the term η 1 1 η 1 Φ h 1 ,h 1 (z 1 ,z 1 , η 1 ) · · · Φ hn,hn (z n ,z n , η n ) where the conformal weights are This corresponds to the four-gluino scattering 1 The result can be checked to agree with the one obtained from Mellin transformation directly. All the rest of component amplitudes can be obtained in a similar way.

Structure at higher points
Just like the super-amplitude (3.10) written in a form with manifest translation symmetry and supersymmetry, the celestial superamplitude (3.18) can also be written as where the dilatation and supercharges are (3.29) They follow from our superconformal generators defined in (2.80) and (2.86). The celestial amplitude written in this form has thus manifest dilatation symmetry and supersymmetry. This structure can be generalized to higher points. The n-point superamplitude in momentum space takes the following form [25] A n = δ 4 (P )δ 8 (Q) Each term in the expansion corresponds to MHV, NMHV, · · · , MHV, respectively. For MHV, H Similarly, the corresponding celestial superamplitude is supposed to have the following general structure: where δ(D), δ 8 (Q) are given in (3.28) and (3.29). The function F n can be regarded as an n-point function in CCFT and transforms covariantly under SL(2, C). 12 It also admits an expansion as (3.31) and each term can be written as the product of the kinematic factor (namely the n-point generalization of (3.9)), the function of cross-ratios, and Grassmann variables. The explicit form of F n can be obtained from Mellin transformation as in the YM case. 13 The expression (3.32) reduces to (3.18) when n = 4 as expected. So the general structure in (3.32) has not only the manifest Lorentz invariance, but also the manifest dilatation symmetry and supersymmetry. In principle, we can compute all the higher-point celestial superamplitudes explicitly. For pure YM theory, the higher point was considered in [26]. Instead of doing tedious higher point computation, we will consider the three-point celestial amplitude in the rest of this section.

Three-point celestial superamplitude in SYM
The three-point amplitude vanishes on-shell identically in Minkowski signature due to the kinematic constraints. But we can define the three-point celestial in the (2,2) split signature. As such, the z ij andz ij become real and independent variables. And there are two helicity configurations for three-point amplitude, MHV and MHV. 12 More precisely, both Mn and Fn obey (2.97). Note δ(D), δ 8 (Q) are left invariant under SL(2, C). 13 Considering the rich symmetry of N = 4 SYM, it would be interesting to see whether it is possible to bootstrap Fn without performing Mellin transformation.
For three-point gluon amplitude A 3 (ω j , z j ,z j ) = A 3 (ω j , z j ,z j )δ( i ε i ω i q i ), one can obtain the celestial amplitude by performing the Mellin transformation and using agin its scaling property. The momentum conservation leads to two branches. In particular, on the branch where z ij = 0, the celestial amplitude reads 14 The celestial amplitude on the other branchz ij = 0 is similarly obtained by exchanging z ij ↔z ij . For MHV amplitude 1 − 2 − 3 + in YM theory, we have The corresponding celestial amplitude is thus given by (3.36) The three-point MHV superamplitude in SYM theory is where where we present two equivalent representations. Written in terms of the celestial variables and replacing ω i → e ∂ ∆ i in the Mellin integral, δ 8 (Q) becomes (3.39) Then one can derive the three-point celestial superamplitude which generally takes the form (3.40) 14 Here we used the equation: Applying to MHV in (3.36), we get the celestial super-amplitude where we also present an equivalent expression in the second equality for later discussions. Picking up the term η 4 1 η 4 2 , one gets This agrees with (3.36). One can also consider the MHV superamplitude: , (3.45) whose celestial counterpart can be similarly derived wheres * i is given in (3.34) by replacing z →z. Picking up the term η 4 3 , one recovers the MHV for gluon three-point amplitude . (3.47) The full three-point celestial super-amplitude is the sum of two helicity configurations:

Soft theorem and Ward identity
In this section we study the soft limit of N = 4 SYM theory. The soft limit captures some universal properties of quantum field theory. Indeed it has been understood in recent years that the soft theorems can be regarded as the Ward identity of various asymptotic symmetries. See [27] for a review. For graviton, the soft theorems arise from the BMS symmetries which include super-translation and super-rotation [28]. For gluon, the underlying symmetry for the leading soft theorem is the large gauge transformation [29]. However, the relations between symmetry and soft theorem are obscure in the conventional momentum space where the soft limit corresponds to taking one particle soft with vanishing energy.
Such a connection becomes much more obvious once we go to the celestial sphere and study the celestial amplitude. It turns out in the celestial basis, the energetically soft theorem becomes the conformally soft theorem [9][10][11][12]. Such a conformally soft theorem can be further translated into the Ward identity in CCFT. The Ward identity is just the standard Ward identity in CFT which relates the CFT correlators with and without the insertion of the soft symmetry current. For the leading soft gluon theorem, the corresponding soft symmetry current is just the Kac-Moody current.
We will generalize all these discussions to N = 4 SYM theory. We will show the conformally supersoft theorem where one operator has dimension ∆ → 1. Translating into the language of CCFT, this corresponds to the super-Ward identity.

Conformally soft gluon theorem in YM
Let us review the conformally soft gluon theorem in YM theory first [10]. In the definition of celestial amplitude (2.18), one can choose the j-th particle and take the limit λ j → 0 which leads to It is easy to recgonize that the right hand side just picks the residue of momentum space amplitude A n (ω i , z i ,z i ) at ω j = 0 = p j . The (energetically) soft theorem for color-ordered partial amplitide is given by in (A.1), (A.2). In terms of the celestial coordinates, the leading soft (positive helicity gluon) theorem takes the form: A n−1 (1, · · · , j − 1, j + 1, · · · , n) , (4.2) where the j-th particle is omitted on the right hand side. Plugging this into (4.1), we get the tree-level conformally soft theorem [10] lim M n−1 (1, · · · , j − 1, j + 1, · · · , n) , where the factor can also be written as 1/z j−1,j + 1/z j,j+1 . The conformal correlator is related to the full amplitude which includes the color factors and takes the following form at tree level: 4) where σ acts by permuting the label {2, 3, · · · , n}.
If we define J a (z) = lim which has conformal weights (h,h) = (1, 0), then the conformally soft theorem (4.12) becomes the Ward identity of a current algebra (after choosing normalization properly): where f abc is the structure constant of the Lie algebra.
In terms of OPE, we have We can also consider the soft theorem for negative helicity gluon. Especially we have an antiholomorphic current defined asJ a (z) = lim where K a (z,z) = lim ∆→0 ∆O a ∆,+ . (4.10)

Conformally supersoft theorem in SYM
The (leading) super-soft theorem of SYM is given by (A.3) and (A.6). In terms of celestial coordinates, it reads Using the following identity [10] lim Secondly, one can also show as well as its generalization were studied in [31]. It is straightforward to generalize our celestial superspace to those theories with less supersymmetry. Therefore, our formalism provides a natural language to account for bulk supersymmetry and superconformal symmetry in CCFT.
In particular, it would also be interesting to generalize our celestial superfield techniques to the supergravity theories, especially the N = 8 supergravity. 20 The asymptotic symmetries of those theories are given by the supersymmetric extension of BMS symmetry. The interplay between supertranslation, superrotation, and maximal supersymmetry is supposed to lead to a rich story on the celestial sphere. One could furthermore consider the celestial double copy relation between N = 4 SYM and N = 8 supergravity by generalizing the non-supersymmetric celestial double copy in [32].
One reason for us to focus on N = 4 SYM theory in this paper is because of its maximal superconformal symmetry in 4D. However, a very remarkable feature of N = 4 SYM theory is that the scattering amplitude (in the planar limit) has been shown to enjoy much larger symmetries, the Yangian symmetry, which can be regarded as the combination of superconformal and dual superconformal symmetry [33]. As superconformal symmetry, it is then natural to ask how to realize superconformal and Yangian symmetry on the celestial sphere. Once all various types of symmetries are understood, a bootstrap program for the CCFT of N = 4 SYM theory may become feasible. A closely related question is how to relate the celestial sphere with twistor space where the N = 4 SYM theory was interpreted as B-model topological string theory [34].
Another virtue of N = 4 SYM theory is that the quantum corrections have been much more well-understood compared to pure YM theory. 21 So it would be interesting to understand the fate of loop corrections in celestial amplitude. This is important for further understanding the full-fledged quantum gravity via celestial holography. Even at the tree level, in the discussion of soft and collinear limits, we were mainly discussing the leading term. It would be nice to work out the other sub-leading corrections.
Finally, a useful operation for celestial operator is shadow transformation [4]. Therefore, it would be interesting to see how to perform the shadow transformation for a super-operator. The shadow transformation flips the spin of the operator and in particular, should be compatible with supersymmetry. However, the supersymmetries considered here are 4D supersymmetry in the bulk instead of the ordinary 2D supersymmetry on the celestial sphere. So it seems to be not sensible to combine (z 1 ,z 1 , η A 1 ) and (z 2 ,z 2 , η A 2 ) to form a distance that could be used to construct the supershadow. Nevertheless, it is worth studying whether one can assemble the shadow of each component operator together in a supersymmetric way.

A.1 Soft theorem in YM and SYM
The soft theorem for (color-ordered) YM gluon amplitude is A n (· · · , a, s ± , b, · · · ) ps→0 −−−→ Soft YM (a, s ± , b)A n (· · · , a, b, · · · ) , (A.1) where the soft factor for positive helicity soft gluon at tree level is given by [36] Soft YM (a, s + , b) = ab as sb + 1 sb The first term is the leading soft factor of order O(1/p s ), while the second term in parentheses is the subleading soft factor of order O(1/p 0 s ). For negative helicity soft gluon, the soft factor is given by the conjugate λ i ↔λ i .
In the physical soft limit p s → 0 or equivalently λ s ,λ s → 0, the leading soft factor is given by the sum of leading soft factors in holomorphic and anti-holomorphic soft limit: The two terms obviously correspond to the soft gluons of positive and negative helicity. Other components are absent here because their corresponding soft factors are sub-dominant (for example, the leading soft factor for gluinos scales as 1/ √ p s [20]).

A.2 Collinear limit in YM
When two particles point to the same direction, the amplitude also becomes singular, knowns as the collinear singularity. More specifically, consider two particles with momenta p 1 and p 2 . The two particles can fuse into one particle with momentum P 12 ≡ p 1 + p 2 and one can parametrize the collinear momenta as p 1 = zP 12 , p 2 = (1 − z)P 12 . In the collinear limit, the gluon amplitude in YM theory satisfies A n (1 h 1 , 2 h 2 , · · · ) where the split factors Split −h (1 h 1 , 2 h 2 ) are given by [25,37] Split + (z; 1 + , 2 − ) = (1 − z) 2 z(1 − z) and Split − (z; 1 −h 1 , 2 −h 2 ) = Split + (z; 1 h 1 , 2 h 2 )| 12 ↔ [12] . (A.12) One can actually derive the soft theorem by taking the consecutive collinear limits of the same type [37]. The soft factor derived in this way is in agreement with the leading term in (A.2) as well as its conjugate.
This is in agreement with the leading supersoft factor in (A.6).