Celestial Amplitudes in Conformal Scattering

• Celestial amplitudes are scattering amplitudes recast via Mellin transforms into 2D conformal correlators, enabling a conformal description of 4D S-matrix elements.
• They reorganize infrared data, asymptotic symmetries, and collinear singularities into conformal primary operators with clear SL(2,C) covariance.
• The framework extends to include massive states and loop corrections, offering novel conformal block structures and ties to flat-space holography.

Celestial amplitudes are scattering amplitudes written in a conformal basis on the celestial sphere, obtained by Mellin-transforming external on-shell energies. In four-dimensional massless kinematics, this recasts the $S$-matrix into correlation functions of conformal primary operators inserted at points $(z,\bar z)$ on the two-dimensional celestial sphere, with boost weights $\Delta_i$ replacing energies and with the Lorentz group acting as the global conformal group $SL(2,\mathbb C)$ (Pasterski, 2021). Within this reformulation, infrared data, asymptotic symmetries, collinear singularities, conformal block expansions, massive states, and loop effects are reorganized in terms that are natural for a two-dimensional conformal description (Nandan et al., 2019, Liu et al., 2024, Donnay et al., 2023).

1. Kinematics, Mellin transform, and conformal covariance

The basic massless kinematic parametrization writes each null momentum as

$$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$

with $\omega_i>0$ and $(z_i,\bar z_i)$ stereographic coordinates on the celestial sphere (Pasterski, 2021). The proper orthochronous Lorentz group $SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$ acts on $(z,\bar z)$ by Möbius transformations,

$$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$

so the celestial sphere carries the standard global conformal action of $(z,\bar z)$0 (Pasterski, 2021).

The celestial transform Mellin-transforms each external energy,

$(z,\bar z)$1

where $(z,\bar z)$2 is the boost weight and $(z,\bar z)$3 is the four-dimensional helicity (Pasterski, 2021). In the radiative sector one uses the principal series $(z,\bar z)$4, with analytic continuation allowed. The associated two-dimensional weights are

$(z,\bar z)$5

The conformal basis therefore diagonalizes boosts rather than energies, and global Lorentz generators act as the standard $(z,\bar z)$6 generators on celestial correlators (Pasterski, 2021).

Poincaré symmetry is more subtle in this basis. Lorentz transformations are manifest, but translations act by dimension-shifting operators. In particular, momentum operators raise $(z,\bar z)$7 by one unit on the leg they act upon, so momentum conservation becomes a set of functional constraints rather than a purely algebraic $(z,\bar z)$8 statement (Stieberger et al., 2018). For low multiplicity, these constraints force singular support. In four-point massless kinematics, momentum conservation enforces reality of the cross-ratio, so celestial correlators are supported on $(z,\bar z)$9 together with channel-dependent ranges such as $\Delta_i$0 in the $\Delta_i$1 channel (Nandan et al., 2019).

2. Conformal primary wavefunctions and the celestial dictionary

The external conformal basis is built from conformal primary wavefunctions $\Delta_i$2 that transform simultaneously as four-dimensional spin-$\Delta_i$3 fields and two-dimensional conformal primaries. Their defining covariance is

$\Delta_i$4

with $\Delta_i$5 the spin-$\Delta_i$6 Lorentz representation (Pasterski, 2021).

A convenient construction uses a null tetrad built from the null direction $\Delta_i$7, the spacetime point $\Delta_i$8, and polarization vectors $\Delta_i$9. The basic massless scalar primary is

$SL(2,\mathbb C)$0

while gauge and graviton primaries are obtained by dressing $SL(2,\mathbb C)$1 with tetrad vectors: $SL(2,\mathbb C)$2

$SL(2,\mathbb C)$3

Incoming and outgoing solutions are distinguished by the $SL(2,\mathbb C)$4 prescription $SL(2,\mathbb C)$5 (Pasterski, 2021).

The shadow transform is intrinsic to the celestial basis. For a primary $SL(2,\mathbb C)$6, its shadow $SL(2,\mathbb C)$7 is obtained by an integral transform on the sphere with kernel

$SL(2,\mathbb C)$8

with $SL(2,\mathbb C)$9 chosen so that the transform squares to $$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$0 (Pasterski, 2021). This shadow structure enters both the operator dictionary and the treatment of low-point amplitudes.

Celestial operators are defined by pairing bulk fields with conformal primary wavefunctions through an inner product,

$$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$1

so incoming and outgoing states become operator insertions labeled by $$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$2 (Pasterski, 2021). This realizes the basic holographic dictionary: four-dimensional external states map to two-dimensional operator insertions; helicity maps to celestial spin via $$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$3; and energies Mellin-transform to boost weights $$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$4 (Pasterski, 2021).

Massive external states require a different conformal basis. Their momenta are parametrized on the unit hyperboloid by

$$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$5

and the corresponding conformal basis is constructed by integrating over $$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$6 rather than Mellin-transforming a single energy variable (Pasterski, 2021). Later work computed scalar three-point celestial amplitudes with two massive scalars in terms of a hypergeometric function and with three massive scalars as a triple Mellin-Barnes integral, extending the dictionary beyond the massless sector (Liu et al., 2024).

3. Soft theorems, asymptotic symmetries, and celestial currents

A defining structural feature of celestial amplitudes is that infrared theorems become operator relations. In Bondi coordinates $$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$7, asymptotically flat gravity has free data $$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$8 obeying the standard Bondi constraints, and residual diffeomorphisms preserving the falloffs are supertranslations $$p_i^\mu=\omega_i\,q^\mu(z_i,\bar z_i),\qquad q^\mu(z,\bar z)=(1+z\bar z,\;z+\bar z,\;-i(z-\bar z),\;1-z\bar z),$$9 and superrotations $\omega_i>0$0 (Pasterski, 2021). The associated charges satisfy a diagonal Ward identity across $\omega_i>0$1,

$\omega_i>0$2

with the charges splitting into soft and hard parts $\omega_i>0$3 (Pasterski, 2021).

In momentum space, the leading soft photon and graviton theorems read

$\omega_i>0$4

$\omega_i>0$5

and, after Mellin transformation, these soft insertions become two-dimensional currents (Pasterski, 2021). Leading soft photons generate a Kac–Moody current $\omega_i>0$6, while leading soft gravitons generate a stress tensor $\omega_i>0$7. In gravity the stress tensor can be written as

$\omega_i>0$8

and its Ward identity has the standard conformal form

$\omega_i>0$9

(Pasterski, 2021).

For Yang–Mills, the conformally soft limit $(z_i,\bar z_i)$0 of a positive-helicity gluon produces a holomorphic current $(z_i,\bar z_i)$1, and field-theoretic analysis shows that this limit is dominated by the soft-energy region $(z_i,\bar z_i)$2 (Fan et al., 2019). The corresponding current Ward identity is

$(z_i,\bar z_i)$3

and the current–primary OPE is

$(z_i,\bar z_i)$4

(Fan et al., 2019).

The same logic extends to conformally soft gravitons. In Lorentzian three-point celestial amplitudes, a soft graviton at $(z_i,\bar z_i)$5 gives a primary of weights $(z_i,\bar z_i)$6, and its level-one descendant is identified with the supertranslation current $(z_i,\bar z_i)$7 (Chang et al., 2022). This sharpens the link between soft graviton theorems, BMS supertranslations, and celestial current algebra (Chang et al., 2022).

4. Collinear limits, OPEs, and conformal block structure

A central operational statement is that collinear limits of four-dimensional amplitudes become operator product expansions on the celestial sphere. For gravitons, taking the holomorphic collinear limit $(z_i,\bar z_i)$8 with $(z_i,\bar z_i)$9 fixed and Mellin-transforming the two energies produces Euler Beta functions. The resulting celestial OPEs include

$SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$0

and

$SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$1

(Pasterski, 2021). In Yang–Mills, analogous OPE coefficients arise from Beta functions multiplied by Mellin normalization ratios. For example, the same-helicity gluon OPE coefficient is

$SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$2

with corresponding formulas for $SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$3 and mixed-helicity channels (Fan et al., 2019).

Low-point celestial amplitudes exhibit both the promise and the singularity structure of the formalism. Four-point massless scalar and gluon celestial amplitudes admit conformal partial wave decompositions in a basis of global $SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$4 conformal partial waves $SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$5, and the four-point correlator can be expanded as

$SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$6

(Nandan et al., 2019). In the scalar exchange example, only $SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$7 partial waves appear, whereas the four-gluon MHV amplitude contains all integer spins $SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$8 in its partial-wave decomposition (Nandan et al., 2019). Crossing is implemented by Möbius transformations $SO^+(1,3)\simeq SL(2,\mathbb C)/\mathbb Z_2$9, $(z,\bar z)$0, and $(z,\bar z)$1, with the appropriate conformal weight factors (Nandan et al., 2019).

Three-point amplitudes are more delicate. In plane-wave Minkowski kinematics, real massless three-point amplitudes vanish for finite momenta because momentum conservation cannot be satisfied away from soft and collinear regions. Nevertheless, celestial three-point amplitudes are nonzero because the momentum-conservation delta function has support on soft and collinear configurations, and Mellin or shadow integrals detect that support (Chang et al., 2022). In a conformal-in / shadow-out basis, massless scalar, photon, gluon, and graviton three-point celestial amplitudes take the standard two-dimensional CFT form, and the three-point gluon amplitudes realize a spin-one current with two spin-one primaries (Chang et al., 2022).

Beyond four points, conformal block expansions reveal multi-particle operator content. In the comb channel, five-point scalar celestial amplitudes exhibit two-particle operators, while six-point scalar amplitudes exhibit new three-particle operators (Liu et al., 2024). A plausible implication, explicitly conjectured in that work, is that $(z,\bar z)$2-particle exchanges appear in the comb-channel conformal block expansion of $(z,\bar z)$3-point massless celestial amplitudes (Liu et al., 2024).

5. Massive states, loop amplitudes, and lower-dimensional variants

Massive celestial amplitudes retain conformal covariance but replace the one-dimensional Mellin integral over energy by hyperbolic integrals over on-shell massive momenta. This enables genuinely new three-point structures and higher-point conformal block expansions. In particular, the three-point coefficient with two massive scalars contains a hypergeometric function, and the coefficient with three massive scalars admits a triple Mellin-Barnes representation (Liu et al., 2024). Turning on the mass of one incoming particle in five-point scalar amplitudes produces a new operator series whose leading member can be interpreted as a two-particle exchange in the OPE of one massive and one massless scalar (Liu et al., 2024).

Loop amplitudes sharpen the ultraviolet and infrared sensitivity of the celestial basis because Mellin integrals probe all energies. The 2021 lecture notes emphasize that tree-level four-graviton celestial amplitudes are formally divergent on the principal series and that stringy form factors regulate the ultraviolet behavior (Pasterski, 2021). A concrete string-theoretic analysis of genus-one open-string four-gluon scattering shows that the Mellin transform commutes with the appropriate field-theory degeneration limit in worldsheet moduli space, and that the one-loop celestial form factor carries an overall $(z,\bar z)$4 instead of the tree-level $(z,\bar z)$5, making $(z,\bar z)$6 the natural dimensionless loop-expansion parameter in the celestial basis (Donnay et al., 2023).

On the field-theory side, one-loop rational all-plus and single-minus gluon and graviton amplitudes provide explicit finite celestial correlators beyond tree level (Albayrak et al., 2020). Their Mellin transforms yield closed-form four- and five-point celestial amplitudes with the expected $(z,\bar z)$7 covariance and with the familiar four-point support on real cross-ratios (Albayrak et al., 2020). A different development recasts MHV celestial amplitudes in terms of Liouville-theoretic partial differential equations; the resulting $(z,\bar z)$8 corrections are logarithmic for both gluons and gravitons, and in the gluon case the deformed celestial OPE is isomorphic to the one-loop correction of the celestial OPE in pure Yang–Mills theory (Mol, 2024).

Lower-dimensional analogues modify the transform itself. In two-dimensional massive theories, Lorentz boosts act additively on the rapidity $(z,\bar z)$9, so the celestial transform becomes a Fourier transform in rapidity rather than a Mellin transform in energies,

$$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$0

or, with an $$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$1 prescription, retarded and advanced transforms $$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$2 (Duary, 2022). In the $$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$3 nonlinear sigma model, exact two-dimensional celestial amplitudes are Meijer $$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$4-functions, and crossing becomes

$$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$5

(Stolbova, 2023). These two-dimensional constructions isolate analyticity, crossing, and unitarity in a particularly transparent form (Duary, 2022, Stolbova, 2023).

6. Reformulations, conceptual tensions, and current directions

Several later developments concern the status of distributional support and the correct notion of a celestial correlator. In $$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$6, an ambidextrous basis built by composing the Mellin map with improved light transforms yields low-point massless celestial amplitudes that are non-distributional in the celestial coordinates and transform covariantly in all scattering channels (Jorge-Diaz et al., 2022). For the four-gluon tree amplitude, this basis gives a streamlined Grassmannian representation and an alpha-space decomposition that exposes towers of exchanged operators and selection rules between symmetric and antisymmetric $$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$7 sectors (Jorge-Diaz et al., 2022).

A different reorganization decomposes celestial amplitudes into integrals of $$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$8-Witten diagrams associated with leaves of a hyperbolic foliation of spacetime. For the Kleinian three-point MHV amplitude, each leaf subamplitude is smooth except for the expected light-cone singularities, and the full translationally invariant celestial amplitude is the residue of the pole at $$z\to \frac{az+b}{cz+d},\qquad \bar z\to \frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$9 (Melton et al., 2023). This suggests a precise relation between celestial amplitudes and $(z,\bar z)$00 contact diagrams, with translation invariance emerging only after summing over leaves (Melton et al., 2023).

The most explicit recent critique of the conventional Mellin definition is the claim that conventional massless celestial amplitudes are distributional and fail to realize the celestial OPE, most sharply in the non-MHV paradox: OPE arguments predict nonzero tree-level celestial amplitudes with helicities $(z,\bar z)$01, whereas the conventional Mellin transform of the momentum-space amplitude vanishes (Liu et al., 5 Dec 2025). The proposed resolution is a class of regular celestial amplitudes obtained by smearing slightly off shell and combining massive and tachyonic conformal bases. At tree level these amplitudes are non-distributional and consistent with celestial OPEs, leading to the proposed revised dictionary that CCFT correlators are the regular, not conventional, celestial amplitudes (Liu et al., 5 Dec 2025). This is a substantive conceptual shift rather than a settled consensus.

A complementary geometric reformulation appears in positive-geometry constructions. For arbitrary multiplicity $(z,\bar z)$02 of massless states in sufficiently high spacetime dimension $(z,\bar z)$03, all Mellin integrations of tree-level $(z,\bar z)$04 amplitudes can be carried out and the result identified with the canonical form of a celestial associahedron (Dong et al., 17 Dec 2025). In that construction, distributional support on the celestial sphere is not imposed separately but arises geometrically from the positivity domain, and the same Mellin machinery extends through scalar scaffolding to gluons and gravitons (Dong et al., 17 Dec 2025). This suggests a geometric interpretation of celestial amplitudes in terms of boundary kinematics, factorization facets, and scattering-equation variables (Dong et al., 17 Dec 2025).

Other structural extensions include a celestial double copy, in which momentum-space kinematic numerators are promoted to generalized differential operators acting on conformal primary wavefunctions and then squared, reproducing three- and four-point gravity celestial amplitudes from Yang–Mills data (Casali et al., 2020). The 2021 lecture notes also emphasize unresolved issues: constructing complete celestial CFTs, determining central charges and Kac–Moody levels, understanding loops and IR/UV regularization, incorporating massive spectra systematically, and identifying a nonperturbative flat-space holographic dual (Pasterski, 2021). Taken together, these developments indicate that celestial amplitudes are not merely Mellin transforms of known $(z,\bar z)$05-matrix elements; they are a rapidly evolving framework in which the correct basis, operator algebra, and even the definition of the correlator remain active subjects of research.