Celestial Operator Product Expansion

• Celestial OPE is the short-distance expansion of conformal primaries in celestial CFT, converting 4D scattering amplitudes into 2D correlators with Beta-function driven coefficients.
• It encodes collinear factorization and asymptotic symmetries, with tree-level interactions revealing self-duality and the emergence of soft algebras.
• Extensions to infinite descendant towers, multiparticle channels, and shadow sectors highlight its role in effective-field theories and test consistency through associativity and Jacobi identities.

Searching arXiv for papers on celestial OPE and related developments. {"query":"celestial operator product expansion associativity self-duality higher spins arXiv (Serrani, 22 Aug 2025)", "max_results": 5} {"query":"Celestial Operator Products of Gluons and Gravitons (Pate et al., 2019)", "max_results": 5} The celestial operator product expansion (OPE) is the short-distance expansion of conformal primary operators in celestial conformal field theory (CCFT), the two-dimensional framework in which four-dimensional massless scattering amplitudes are rewritten as correlators on the celestial sphere. In this setting, Mellin-transformed asymptotic states become operators $\mathcal O^\pm_{\Delta,s}(z,\bar z)$ with conformal weights fixed by their boost weight $\Delta$ and helicity $s$, and the coefficients of the OPE are in one-to-one correspondence with bulk three-point couplings. The celestial OPE therefore furnishes a direct bridge between collinear factorization of the $S$-matrix, asymptotic symmetries, conformal block decompositions, and—within chiral and self-dual sectors—associativity, Jacobi identities, vanishing tree amplitudes, and light-cone consistency conditions (Serrani, 22 Aug 2025).

1. Definition in celestial conformal field theory

In CCFT, an $n$-point momentum-space amplitude $\mathbb M_n(p_i)$, with $p_i^\mu=\epsilon_i\omega_i q_i^\mu(z_i,\bar z_i)$, is Mellin transformed in the energies $\omega_i$ to a celestial amplitude that transforms as a two-dimensional conformal correlator. The corresponding conformal primary wavefunction is labeled by $(\Delta,J;z,\bar z)$, and its left- and right-moving weights are

$$h=\tfrac12(\Delta+s),\qquad \bar h=\tfrac12(\Delta-s),$$

with $\Delta$0 for the conventions used in the celestial literature (Bhattacharyya et al., 5 May 2025).

The local OPE of two celestial primaries takes the schematic form

$\Delta$1

where $\Delta$2 are the celestial OPE coefficients. For massless tree-level amplitudes, these coefficients arise by combining the bulk collinear limit with the Mellin transform; the resulting expressions contain Euler Beta functions that encode the descendant tower, while the overall normalization is set by the flat-space three-point coupling (Serrani, 22 Aug 2025). In a compact general-spin form, the leading holomorphic singular terms are fixed by coefficients

$\Delta$3

showing that the Beta-function dependence is not incidental but built into the Poincaré-covariant structure of the celestial expansion (Himwich et al., 2021).

This definition makes the celestial OPE the boundary avatar of bulk collinear factorization. In the simplest tree-level sectors, it functions exactly as an OPE in an ordinary two-dimensional CFT; in more elaborate settings, including multi-particle channels and shadow sectors, the same basic structure persists but requires extensions of the exchanged operator spectrum.

2. Tree-level coefficients, collinear limits, and symmetry determination

The leading celestial OPEs of gluons and gravitons were first extracted directly from tree-level Einstein–Yang–Mills amplitudes. For identical-helicity gluons, the holomorphic singularity is

$\Delta$4

with analogous formulas for mixed-helicity gluons, gluon–graviton OPEs, and graviton–graviton OPEs. In tree-level Einstein–Yang–Mills theory, the leading coefficients are completely fixed by the combination of $\Delta$5 covariance, the translation Ward identity, subleading soft-gluon and subsubleading soft-graviton Ward identities, and analyticity in $\Delta$6; the unique solution is an Euler Beta function (Pate et al., 2019).

The poles of these Beta functions are physically significant. The simple poles at integer values of $\Delta$7 generate conformally soft currents, such as the leading soft gluon current $\Delta$8, together with its subleading analogues. In this way, large gauge transformations and large diffeomorphisms appear as residues of celestial OPE coefficients rather than as external add-ons to the operator algebra (Pate et al., 2019).

A broader statement holds for arbitrary massless spin. Poincaré symmetry implies recursion relations on the leading holomorphic OPE coefficients, and these recursions are again solved by Beta functions whose arguments depend simply on the right-moving weights. The same coefficients match Mellin transforms of tree-level collinear splitting functions obtained from BCFW analysis, and they are compatible with an infinite family of $\Delta$9-type symmetry transformations acting on celestial primaries of all spins, including fermions and theories with higher-derivative non-minimal couplings (Himwich et al., 2021).

The overall picture is that the leading celestial OPE is not merely extracted from amplitudes; in many sectors it is overdetermined by symmetry and analyticity, with the collinear $s$0-matrix serving as the consistency check that fixes the normalization.

3. Descendants, soft algebras, and all-order expansions

Beyond the leading singular term, the celestial OPE contains an infinite tower of descendants. For positive-helicity gravitons in Einstein gravity, the first subleading correction can be organized into BMS descendants. The leading pole creates the supertranslation descendant $s$1, while the first subleading terms involve $s$2, $s$3, and the “holomorphic singular supertranslation” $s$4. Their coefficients are fixed algebraically by BMS commutators once the leading Beta-function coefficient is known (Banerjee et al., 2020).

A complementary derivation comes from four-dimensional ambitwistor strings. There, the worldsheet OPE of vertex operators localizes precisely on the celestial short-distance limit, and the worldsheet CFT dynamically produces the known leading celestial OPE coefficients together with infinite towers of $s$5 descendants for gluons and gravitons of all helicities and incoming/outgoing configurations. The same construction realizes the soft-current algebras explicitly: for gluons, soft modes generate a Kac–Moody algebra, while for gravitons they generate a loop algebra of $s$6 (Adamo et al., 2021).

In the maximally helicity violating sector, all-order holomorphic OPEs can be derived by on-shell recursion. Inverse-soft recursion for MHV amplitudes becomes dual to OPE recursion on the celestial sphere: adding a soft leg in the bulk is equivalent to inserting a soft-current descendant in the boundary correlator. The resulting tree-level OPEs close on towers of soft-current descendants and antiholomorphic translations, without requiring independent holomorphic Virasoro descendants in that sector (Ren et al., 2023). Twistor-string methods sharpen this statement further by providing master formulas that include both singular and regular terms in the simultaneous holomorphic and antiholomorphic expansions of MHV gluon OPEs (Adamo et al., 2022).

These results collectively show that the celestial OPE is not exhausted by its leading collinear pole. In sectors with sufficient control—BMS, worldsheet, or inverse-soft recursion—the entire descendant structure can be computed to all orders and organized in terms of soft symmetry generators.

4. Associativity, Jacobi identities, and self-dual sectors

Associativity is the central consistency condition of the celestial OPE. In CCFT, the requirement that a four-point correlator admit consistent OPE expansions in different pairings can be expressed as a vanishing double-residue condition, or equivalently as the Jacobi-like identity

$s$7

For the most general class of chiral cubic vertices in flat space, this associativity constraint is algebraically identical to three other conditions: the vanishing of the corresponding four-point tree-level amplitude, the Jacobi identity of an emergent “gauge” algebra extracted from cubic couplings, and the light-cone holomorphic quartic constraint. In this precise sense, celestial OPE associativity becomes equivalent to self-duality in a large class of chiral higher-spin theories (Serrani, 22 Aug 2025).

The equivalence is explicit in familiar cases. For self-dual Yang–Mills, the associativity condition reduces to the standard color Jacobi identity. For self-dual gravity, the Schouten identity underlies the Jacobi identity of the kinematic algebra and the celestial OPE constraint. For the infinite chiral higher-spin tower, one recovers the Metsaev gamma-function solution to the holomorphic light-cone constraint and verifies celestial associativity to all spins. These theories and their self-dual gauge/gravity truncations are singled out as the unique consistent perturbatively local flat-space theories with only cubic vertices, modulo a specific abelian subtlety (Serrani, 22 Aug 2025).

That subtlety concerns zero-exchange cubic terms of the form $s$8. Both the light-cone holomorphic constraint and celestial associativity leave these unconstrained; they correspond to Born–Infeld-type vertices that can give non-zero amplitudes without spoiling self-duality of the cubic chiral sector (Serrani, 22 Aug 2025).

Self-dual gravity also provides an instructive refinement of descendant structure. In the celestial OPE of two outgoing positive-helicity gravitons, only a finite number of $s$9 descendants contribute at any fixed order, despite the unbounded celestial spectrum. This truncation is enforced by explicit null-state relations such as $S$0 and $S$1, and is consistent with representation-theoretic analyses of the $S$2 wedge algebra (Banerjee et al., 2023). A plausible implication is that self-dual sectors organize the celestial operator algebra more rigidly than generic flat-space theories.

5. Multiparticle channels, shadow sectors, and block formulations

Single-particle exchanges do not exhaust the celestial OPE. A first systematic treatment of multiparticle contributions introduced two-particle celestial primaries defined by regularized coincident limits of single-particle operators and showed that tree-level factorization channels determine single-particle exchange coefficients, whereas genuine two-particle exchanges arise from non-factorizing $S$3 sectors. This establishes that the full celestial OPE must contain multiparticle operators already at tree level (Guevara et al., 2024).

A later boundary derivation sharpened this point. Multi-particle operators can be defined as ordinary normal-ordered products, and the singular terms in their OPEs are determined entirely by the singular terms in the OPEs of their single-particle constituents. Three independent derivations—generalized Wick expansion on the boundary, bulk multicollinear limits, and imposition of four-dimensional translational invariance as a global symmetry of the celestial dual—lead to identical triple-Beta-function coefficients for multi-particle OPEs (Calkins et al., 7 Jan 2026). This makes the multi-particle extension structurally close to standard two-dimensional CFT, even though its bulk interpretation involves subtle multi-collinear kinematics.

Shadow sectors provide a second extension. In the shadow conformal primary basis, celestial amplitudes become manifestly local on the celestial sphere and exhibit standard OPE behavior in short-distance limits. For tree-level scalar amplitudes, the four-point shadow celestial amplitude admits a conformal block expansion whose coefficients factorize into products of three-point OPE coefficients, exactly as in an ordinary CFT (Chang et al., 2022). OPE blocks, defined as conformally covariant packages of a primary together with its global descendants, are the natural language for this extension and also encode shadows and light-transformed partners after analytic continuation to Lorentzian kinematics (Guevara, 2021).

Recent work has made the shadow dependence more explicit. Universal transformation laws determine how OPE coefficients change when one shadows an external leg or the exchanged operator; this gives the coefficients of all three-point functions involving arbitrary mixtures of celestial and shadow primaries, and yields concrete sign flips in $S$4-current and stress-tensor OPEs (Himwich et al., 14 May 2025). A complementary charge-bracket construction derives shadow OPEs algorithmically by demanding that the correspondence between soft-theorem charge brackets and celestial OPEs persist under shadow transformation; in this form it also resolves the mixed-helicity double-soft ambiguity by a specific “first entry soft first” prescription (Pranzetti et al., 30 Oct 2025).

The strongest recent claim is that the celestial OPE does not close on Mellin-basis exchanges alone. A shadow-completed OPE includes, for every exchanged bulk particle, both the Mellin-basis primary and its shadow-basis representative, with the shadow coefficient fixed by the ordinary collinear coefficient times a universal shadow factor. This proposal is supported by tree-level regular celestial amplitudes, including scalar $S$5 kinematics and explicit five-point examples (Liu et al., 18 Jun 2026). If maintained, this would mean that closure of the local boundary operator algebra requires both representatives of the same bulk degree of freedom.

6. Effective-field-theory realizations, deformations, and unresolved issues

The celestial OPE has also become a computational tool in effective field theory. In quadratic EFTs of gravity, Mellin-transformed amplitudes lead to celestial correlators whose conformal block decomposition can be analyzed in closed form. The Burchnall–Chaundy expansion extracts scalar-exchange OPE coefficients, while the Euclidean OPE inversion formula yields arbitrary-spin coefficients. The resulting coefficients are meromorphic in $S$6, reduce consistently between the two methods at $S$7, and decay factorially at large $S$8, reflecting the improved ultraviolet behavior of the quadratic EFT and ensuring convergence of the block expansion (Bhattacharyya et al., 5 May 2025).

A different deformation arises from Liouville-theoretic constructions of MHV celestial amplitudes. Partial differential equations governing the $S$9-dependent amplitudes generate $n$0 corrections to the celestial OPE. For gluons, the resulting deformation is isomorphic to the one-loop correction of the pure Yang–Mills celestial OPE; the correction contains derivatives with respect to conformal dimensions together with logarithmic $n$1 terms. For gravitons, an analogous logarithmic deformation is obtained, but its physical interpretation remains open because of subtleties in loop-corrected holomorphic collinear limits and gravitational anomalies in the self-dual sector (Mol, 2024).

The main controversy concerns associativity beyond the single-particle sector. In massless $n$2 theory, the celestial three-OPE contains a term with a branch cut inherited from the $n$3 pole of the bulk three-collinear splitting function. This branch cut obstructs the double-residue condition that underlies celestial Jacobi identities if one restricts the OPE to single-particle exchanges. The proposed resolution is not non-associativity of the OPE itself, but enlargement of the operator algebra to include a two-particle operator channel whose contribution cancels the obstruction (Ball et al., 2023). This suggests that branch-cut singularities and continuum multiparticle labels are intrinsic, not pathological, features of a non-rational celestial CFT.

Taken together, these developments show that the celestial OPE is simultaneously a reformulation of flat-space collinear physics, a laboratory for asymptotic symmetry algebras, and a stringent consistency condition on flat-space dynamics. In its most constrained manifestations, notably self-dual and chiral higher-spin sectors, associativity reduces to amplitude vanishing and algebraic Jacobi identities. In its more general form, it appears to require multi-particle and shadow completion, with effective-field-theory and loop-level deformations providing increasingly precise tests of how a full celestial operator algebra should be defined.