Celestial Leaf Amplitudes

Celestial amplitudes may be decomposed as weighted integrals of AdS$_3$-Witten diagrams associated to each leaf of a hyperbolic foliation of spacetime. We show, for the Kleinian three-point MHV amplitude, that each leaf subamplitude is smooth except for the expected light-cone singularities. Moreover, we find that the full translationally-invariant celestial amplitude is simply the residue of the pole in the leaf amplitude at the point where the total conformal weights of the gluons equals three. This full celestial amplitude vanishes up to light-cone contact terms, as required by spacetime translation invariance, and reduces to the expression previously derived by Mellin transformation of the Parke-Taylor formula.


Introduction
The subleading soft graviton theorem [1] implies [2] that observables of any consistent 4D quantum theory of gravity in asymptotically flat space are invariant under the action of the local 2D conformal group on the celestial sphere.This brings the powerful tools of 2D conformal field theory (CFT) to bear on the difficult problem of 4D quantum gravity.In particular the 4D gravitational scattering amplitudes obey the same constraints as those of 2D CFT correlators.
Application of the powerful CFT toolkit to constrain quantum gravity scattering amplitudes has been hindered by the fact that the latter, although fully conformally invariant, sometimes take a distributional form which is unfamiliar in studies of 2D CFT.Spacetime translation invariance imposes an infinite number of relations among already-highly-constrained conformally invariant correlators, and at low points these constraints can be satisfied only by distributional expressions.A number of successful efforts have constructed smooth conformally invariant celestial amplitudes by expanding around translation noninvariant backgrounds [3][4][5][6][7][8][9][10][11][12][13][14].
In this paper we show, focusing on the three-gluon MHV amplitude in Klein space 1 , that the translationally-invariant distributional expressions have a simple geometric interpretation as sums of generically smooth amplitudes given by AdS 3 /Z-Witten diagrams.The derivation given herein is intricate but the basic construction is simple.Klein space can be foliated by two sets of hyperbolic slices in the the timelike and spacelike wedges with X2 < 0 and X 2 > 0. The leaves of the slices are AdS 3 /Z spacetimes whose boundaries are copies of the celestial torus CT 2 .A conformally invariant leaf amplitude on CT 2 is then given by an AdS 3 /Z-Witten diagram for each leaf.These are not constrained by translation invariance as an individual leaf preserves only Lorentz/conformal symmetry.They take the familiar 2D CFT form 2 and are smooth up to light-cone singularities.To get the full celestial amplitude, we must integrate over all the leaves.We verify, as required by translation invariance, that the full amplitude vanishes for generic scattering angles due to cancellations between the two wedges.
Care must be exercised when two points lie on a common light cone in CT 2 .In this case we show that the cancellation between the two wedges is incomplete.A contact interaction arises from three-point generalizations of the basic identity where ǫ is a regulator for a light cone singularity in a null coordinate σ.Moreover we find the strikingly simple result that the full amplitude is proportional to the pole appearing in the leaf amplitudes when the net sum of the conformal weights of the gluons equals three.This expression agrees in full detail with the PSS expression [16] for three-gluon MHV scattering obtained via Mellin transforms of the Parke-Taylor formula.Our main result for the iǫ-regulated 3-gluon celestial amplitude is equation (5.3).
Leaf amplitudes are simpler than the full celestial amplitudes which they are used to construct.Moreover, they have a natural holographic interpretation provided by the AdS 3 /CFT 2 correspondence.This suggests that leaf amplitudes may provide useful building blocks for holography in flat space.Also of interest in this regard are the half amplitudes obtained by integrating the leaf amplitudes over one of the two wedges. 3n order to be specific this paper considers only the MHV 3-point amplitude, but we anticipate general constructions of celestial from leaf amplitudes at all orders for any number of particles.We have not considered this in detail.
The rest of this paper is organized as follows.Section 2 briefly reviews Klein space and celestial amplitudes, collects useful formulae and sets our conventions.Section 3 defines leaf amplitudes, relates them to AdS 3 -Witten diagrams and expresses the full celestial amplitudes as weighted integrals over their leaves.The gluon leaf amplitude turns out to be a kinematic factor times a scalar contact Witten diagram.This scalar contact diagram is evaluated in section 4, with care to keep track of the necessary iǫ prescriptions in Lorentzian signature.In section 5 we show that only the simple pole in the leaf amplitude at the point where the gluon conformal weights sum to 3 contributes to the full celestial amplitude.Moreover we find that the coefficient of this pole contains delta functions in the light cone coordinates and matches perfectly the known PSS Mellin transform expression, including the indicator functions separating causal regions.

Klein space kinematics
Let us start by reviewing the geometry of Klein space K = R 2,2 and its foliation by Lorentzian AdS 3 /Z slices or 'leaves'.Its conformal boundary, where the dual CCFT resides, is foliated by celestial tori.Points on the celestial torus parametrize Kleinian null momenta.For further discussion, see [18,19].
Hyperbolic foliation of Klein space.The metric of Klein space with coordinates X µ reads (2.1) Upon excising the light cone of the origin, Klein space splits into a timelike wedge W T on which X 2 < 0, and a spacelike wedge W S on which X 2 > 0. Depending on the region, we write where τ ∈ (0, ∞) is the magnitude of X µ .In either region, slices of constant τ are diffeomorphic to AdS 3 /Z, i.e., Lorentzian anti-de Sitter space with periodically identified time.Taken together, they constitute a hyperbolic foliation of Klein space, mirroring the hyperbolic foliation of Minkowski space used by de Boer and Solodukhin [20].The timelike leaves x2 + = −1 which foliate the timelike wedge W T can be parametrized in terms of standard global coordinates: Here, ρ ∈ (0, ∞) is the radial coordinate on AdS 3 /Z, whereas ψ ∼ ψ + 2π and ϕ ∼ ϕ + 2π are periodic coordinates along timelike and spacelike cycles respectively.The metric is where the bracketed part is the unit metric on AdS 3 /Z.The spacelike wedge W S (x 2 − = +1) can be parametrized by analogous coordinates, but with the roles of spacelike and timelike cycles exchanged: In this region, the metric becomes The ρ → ∞ conformal boundary of each AdS 3 /Z slice is a Lorentzian torus CT 2 = S 1 × S 1 referred to as the celestial torus.In the coordinates ψ, ϕ on CT 2 , it is equipped with the induced conformal metric ds 2 = − dψ 2 + dϕ 2 .It also admits null coordinates in terms of which the conformal metric becomes ds 2 = −4 dσ dσ.They are subject to the coupled periodic identifications In the study of Lorentzian (celestial) CFT, one conventionally works with planar coordinates z, z ∈ R defined by z = tan σ , z = tan σ (2.9) on which the bulk isometry group SL(2, R) × SL(2, R) acts by real and independent Möbius transformations.But these only cover half the celestial torus as they do not distinguish between antipodal points (ψ, ϕ) and (ψ + π, ϕ + π).Consequently, we will find it useful to work directly in terms of the global coordinates σ, σ.Geometrically, z, z are local coordinates on a Minkowski diamond R 1,1 , as the 2D metric reduces to ds 2 = −dz dz up to conformal rescalings.The open set CT 2 − {zz = 0} of the celestial torus is covered by two such diamonds.On this set, if σ is brought to its fundamental domain 0 < σ < π, then the two diamonds are cut out by the coordinate regions 0 < σ < π and π < σ < 2π.Since the celestial torus has only one connected component, the distinction between the two diamonds is purely a choice of coordinate patches and evaporates in global coordinates.
Massless kinematics.Momenta of massless particles in (2, 2) signature are parametrized in planar coordinates as Here i is a particle label, ω i ∈ (0, ∞) denotes the magnitude of the frequency, ε i ∈ {±1} denotes the sign of the frequency, and q µ i are null vectors that are normalized to satisfy q 0 i + q 2 i = 2.These are null with respect to the (− − + +) signature flat metric (2.1).(z i , zi ) describe points on the celestial torus where the massless particles exit.The choice of coordinates z, z divides the torus into two diamonds, and the signs ε i denote the choice of the diamond.This is equivalent to a choice of in vs. out in Minkowski space, and we occasionally continue to use this language for sake of simplicity.
In this work, instead of planar we primarily use global coordinates on the celestial torus.Substituting we find the decomposition where pµ i are null vectors parametrized by points (ψ i , ϕ i ) ∈ CT 2 : In particular, we identify ε i = (p 0 i + p 2 i )/2ω i = sgn(cos σ i cos σi ).The inner products of such a null vector pµ i with the unit timelike or spacelike vectors xµ ± are found to be pi Similarly, the inner product of two null vectors pµ i , pµ j is given by pi where ϕ ij = ϕ i − ϕ j , σ ij = σ i − σ j , etc.We will also find it useful to introduce the following abbreviations for the torus separations: These become the natural variables for writing CFT correlation functions on Lorentzian tori [15].

Celestial amplitudes in global coordinates
Here we give our conventions for celestial amplitudes.Let A(p i , J i ) denote a momentum space scattering amplitude for massless particles carrying momenta p i and helicities J i , where i = 1, . . ., n.In particular, this contains the momentum conserving delta function The corresponding celestial amplitude is given by the Mellin transform, where ǫ > 0 is a small regulator.It depends on the celestial torus positions z i , zi , on the signs ε i , as well as on a choice of conformal weights ∆ i ∈ C.Under 2D conformal transformations, it transforms as a 2D CFT correlator [21] where are the left/right conformal weights.
In Lorentzian signature, a choice of complete basis for the space of external states is provided by conformal primary states with weights ranging over the principal series [22]: ∆ i ∈ 1 + i R. Whenever needed, our expressions will be understood as being defined for the same choice of weights in (2, 2) signature as well.
To rewrite celestial amplitudes in global coordinates, we substitute the global parametrization (2.13) into the definition (2.19) and rescale ω i → ω i | cos σ i cos σi | for each i.This produces where we have defined the torus uplifts of massless celestial amplitudes: Equivalently, this is just the celestial amplitude for scattering particles carrying null momenta If one has already computed the celestial amplitude A(ε i , ∆ i , J i , z i , zi ), one can also derive its torus uplift A(∆ i , J i , σ i , σi ) by simply plugging in and dividing out the Jacobian factor j | cos σ j | 2h j | cos σj | 2 hj .Crucially, as we are working in global coordinates, the result no longer depends on any ingoing or outgoing labels ε i because the parametrization in (2.14) covers the entire celestial torus.
3-point PSS amplitude.Our primary object of interest is the celestial amplitude for three gluons in the maximally helicity violating (MHV) sector.We take this to be the amplitude where gluons 1, 2 have helicity J 1 = J 2 = −1 and gluon 3 has helicity J 3 = +1.To compute this, one starts with the Parke Taylor formula for the associated momentum space amplitude: where we are only considering the color-stripped amplitude.Here are the so-called spinor-helicity brackets that are commonly employed in computations of massless scattering [23].Also, A(1 , and we will use a similar abbreviation when writing celestial amplitudes.On plugging in the parametrization (2.10) and performing the Mellin transforms, one finds the 3-point PSS amplitude [16,24] A(1 where Θ(x) is the Heaviside step function, and |z ij | are real absolute values of the chiral separations z ij . 4Here, and we have introduced the convenient notation along with their antichiral counterparts (2.31) 4 They are not to be confused with (z ij zij ) Because we are using global coordinates, the ingoing/outgoing labels ε i have been eliminated using the relations ε i = sgn(cos σ i cos σi ).We have also used sgn(cos σi cos σj ) = sgn(cos σij ) which is valid on the support of δ(sin σ13 ) δ(sin σ23 ).The step functions ensure that there is no way to divide the celestial torus into an ingoing and outgoing diamond such that all three particles are ingoing or outgoing when continued to Lorentzian signature.
It is interesting and nontrivial to show that (2.31) is single-valued and conformally covariant on CT 2 [15,16], as will be reconfirmed in the derivation herein.In fact these considerations largely fix (2.31).

Leaf amplitudes
Celestial amplitudes are often constructed as Mellin transforms of momentum space amplitudes.Alternately, evaluating celestial amplitudes directly in position space usefully presents them as weighted integrals of Witten diagrams on the leaves of a hyperbolic foliation of spacetime [5,21,25], or 'leaf amplitudes'.In this section we define the leaf amplitude arising in Kleinian MHV gluon scattering, but we expect the construction to generalize.
The color-stripped momentum space Parke-Taylor amplitude is where P µ denotes the total momentum We wish to express the associated celestial amplitude directly as an integral over conformal primary wavefunctions without reference to momentum space.We start by invoking the Fourier representation of the momentum-conserving delta function, We then perform the Mellin transform (2.19) before the spacetime integration.On using z ij = sin σ ij / cos σ i cos σ j , the expression for angle brackets in global coordinates is found by stripping off the cos σ i cos σ j factors Using the notation (2.18), the MHV celestial amplitude (2.19) then takes the desired form where the scalar conformal primary wavefunctions are The Mellin integral here is defined with a small, positive regulator ǫ.In hyperbolic coordinates, these wavefunctions turn into embedding space expressions for bulk-to-boundary propagators on Lorentzian AdS 3 /Z.The expression (3.5) transforms as a correlation function of n conformal primaries of weights We then have hj . (3.8) The next step is to break the integral over Klein space into two integrals, one over the timelike wedge W T (X 2 < 0) and another over the spacelike wedge W S (X 2 > 0).In either region, the integral in (3.5) can be factorized into an integral over τ = |X 2 | ∈ (0, ∞), and an integral over a unit AdS 3 /Z slice.This yields where d 3 x± are the volume forms on AdS 3 /Z (see (4.3) below).Additionally, a simple change of variables shows that the integral over the spacelike region of Klein space can be obtained from the integral over the timelike region by sending σi → −σ i .
Since τ > 0, in the limit ǫ → 0 we can use Φ ∆ (τ x± , pi ) = τ −∆ Φ ∆ (x ± , pi ) to rewrite this as L here is the MHV n-point leaf amplitude, which is proportional to the leaf amplitude C for n massless scalars.The τ integral in (3.10) has been performed using the identity Strictly speaking, this holds when Re β = 0, and δ(β) is to be interpreted as δ(Im β).But it can be analytically continued to other β as explained in [26].As a result, the integral over τ reproduces the δ(β) expected from bulk scale invariance, while the integrals over the unit AdS 3 /Z slices x2 ± = ∓1 generate leaf amplitudes L(σ i , ±σ i ).We see here that the conformal weights for leaf amplitudes are unconstrained: the celestial amplitude constraint on their total sum (for massless particles) comes from the τ integral over leaves.Also of interest are the nontranslationally invariant half amplitudes H, obtained by integrating over the leaves in only the timelike wedge W T .These include the constraint on the weights and are simply related to the full amplitudes by (3.14) We will now evaluate these leaf amplitudes for 3-gluon MHV scattering and show that (3.10) reproduces the standard gluon 3-point celestial amplitude (2.27), even though the leaf amplitudes themselves are non-distributional.

3-point scalar leaf amplitude
The leaf amplitude (3.12) is a contact Witten diagram for massless scalars propagating on Lorentzian AdS 3 /Z.At three points we have

.1)
Here x2 + = −1 as before, and ǫ is a small, positive regulator that picks a choice of branch for each bulk-to-boundary propagator.To be precise, for ∆ ∈ C and x ∈ R − 0, quantities like (x ± iǫ) ∆ are defined by6 understood in the limit ǫ → 0 + .The measure of integration is We will use ψ, ϕ as integration variables instead of σ, σ as this will aid us in separation of integrals.
The constraint that β = 0 arises from τ integration and does not restrict the leaf amplitudes of this section.
Hyperbolic integrals.To evaluate the AdS 3 /Z integrals, we reinstate the Mellin representations of the bulk-to-boundary propagators: Using (2.14), this becomes We now substitute cos(ϕ − ϕ i ) = cos ϕ cos ϕ i + sin ϕ sin ϕ i and similarly expand out cos(ψ − ψ i ).
Using the Bessel function representation 2π 0 dϕ e ix cos ϕ+iy sin ϕ = 2π J 0 ( we can perform the angular ψ, ϕ integrals: Here we have substituted y = sinh ρ and abbreviated (4.8) These have been expressed in terms of ϕ jk = ϕ j − ϕ k and ψ jk = ψ j − ψ k .The integral over y can be performed by using the Schläfli integral representation of the Bessel function,7 valid when Re ν > −1 and δ > 0. We will apply this in the limit δ → 0 + .Using this representation to substitute for J 0 (Ψ 1 + y 2 ) in (4.7), then rescaling u → uΨ 2 /4, we get The y integral is convergent when the real part of u is positive, i.e., δ > 0. It yields The factor of Ψ 2 − Φ 2 in the exponential may be brought to a physically interesting form: where s jk and sjk are as defined in (2.18), and P µ ≡ ω 1 pµ 1 + ω 2 pµ 2 + ω 3 pµ 3 denotes the total momentum in global coordinates.
Mellin integrals.A standard change of variables [29] computes the Mellin integrals involved in 3-point contact Witten diagrams: with t i ∈ (0, ∞) for all i = 1, 2, 3.This yields To simplify the computation, we rescale the iǫ factors by positive factors of ω i to send e −ǫ j ω j → e −ǫ j t j ; this damps the integrals at both large ω i and large t i and will not change the ǫ → 0 limit.Doing this, we find having collected together the exponentials.The exponents ā, b, c were defined in (2.30).
To proceed with the t i integrals, let us analyze the factor in the exponential.Notice that Since u ∈ δ + i R, the t i integrals will converge if δs ij sij + ǫ > 0 for every i, j.This is ensured by sending δ → 0 before we take the ǫ → 0 limit, so this is the order of limits that we will employ.Thereafter, the Mellin integrals lead to where the δ → 0 + limit is implicitly understood.Let u = δ + iv, with v ∈ R. Then we can write That is, we have replaced factors like (ivs 12 s12 + δs 12 s12 + ǫ) ā by (ivs 12 s12 + ǫ) ā using the fact that they describe the same branch prescription for (ivs 12 s12 ) ā in the limit that δ → 0 + faster than ǫ → 0 + (because δs 12 s12 + ǫ stays positive).
The presence of δ in e i/(v−iδ) regulates the singularity of e i/v at v = 0. We can instead set δ = 0 and perform the integral by a principal value prescription.Breaking the v integral into the two domains v > 0 and v < 0 and pulling out factors of i|v| from the denominators yields where we have simplified the phases by recalling that ā+ b+c = 2+β/2.We have also substituted v → −v for v ∈ (−∞, 0) to obtain the second line.
The integrals over v become standard Euler integrals upon substituting v → 1/v.They lead to the following final result for the leaf amplitude c , (4.20) with an overall normalization given by We see that the ±iǫ regulators which distinguish bulk positive and negative frequency conformal primary wave functions in (3.6) also ultimately regulate the light cone singularities of the correlation functions on the celestial torus CT 2 , even though bulk time is not the the same thing as boundary time.
The relative minus sign in (4.20) will be crucial for what follows.Notice that due to phase differences the two terms cancel only in the region where the s jk sjk are all positive.
Similarly, sending σi → −σ i for all i and applying the identity we obtain the leaf amplitude associated to the spacelike wedge W S of Klein space, The key observation is that while the gluon 3-point function in (5.1) may naively appear to vanish because of the δ(β) sin πβ/4 factor, it contains a pole in β that cancels the zero of sin πβ/4.The residue at this pole gives the familiar gluon 3-point amplitude.
The first step in proving this requires reexpressing (5.1) in planar coordinates z i , zi .This is accomplished using the conformal transformation (2.22).Substituting sin σ ij = z ij cos σ i cos σ j , sin σij = zij cos σi cos σj (5.The building blocks B ± entering this decomposition are given by While obtaining these, we have reinstated ε i = sgn(cos σ i cos σi ) by noting that sin σ ij sin σij | cos σ i cos σi cos σ j cos σj | = ε i ε j z ij zij . (5.5) Each building block B ± transforms as a 3-point Lorentzian CFT 2 correlator with weights (h i , hi ).This recasts our hyperbolic decomposition of the 3-gluon amplitude in standard notation.
Recovering the distributions.Next, we show that each of the building blocks B ± behaves like a contact term δ(z 13 )δ(z 23 ) near its singularity z1 = z2 = z3 , with a coefficient containing a simple pole at β = 0. Again, it will suffice to prove this for B + , the proof for B − being completely analogous. 8o start the proof, set We can pull out the factors of z ij from the denominator of B + to factorize it into For separated insertions, B + manifestly contains no poles in β and vanishes when multiplied by δ(β) sin(πβ/4).To show that it contains a distributional term with a pole at β = 0 when all zi are coincident, we need to evaluate the integral where we have substituted z1 = z3 + ǫx, z2 = z3 + ǫy to obtain the second line.Clearly we will be able to take the ǫ → 0 limit trivially once we have taken the β → 0 limit.We can perform the x, y integrals by contour rotation.There are two branch points in each of the x and y complex planes.When all the ζ ij are identical, we place our branch cuts one above the real axis and one below.This is displayed in figure 2  Note that if the ζ ij are not all identical, the branch points will lie on the same side of the real axis in either the x or the y complex plane.So we can close the contour of either the x or the y integral and shrink it to a point.Ergo, we find (5.17) The β → 0 limits of βB ± extract the residues of B ± at β = 0.The latter are independent of ǫ, so the ǫ → 0 limit is taken trivially.Recalling the value of N from (4.21), the factor of 8π 2 /Γ(ā)Γ( b)Γ(c) in (5.15) can be cancelled against N /8π 2 when β = 0. Therefore, we obtain

Figure 1 :
Figure 1: A toric Penrose diagram depicting the hyperbolic foliation of Klein space in the timelike (W T ) and spacelike (W S ) regions.

Figure 2 :
Figure2: The contour deformation used to evaluate the integral(5.8).If the ζ ij are not all identical, both branch cuts would lie on the same side of the origin in either the x plane or the y plane, implying that the integral would vanish.