Multi-Particle Celestial Operators in Holography

• Multi-particle celestial operators are composite entities on the celestial sphere built from normal-ordered single-particle primaries, encoding bulk scattering and boundary OPE data.
• They employ recursion relations derived from conformal symmetry, generalized Wick theorems, and multi-collinear limits to obtain universal operator product expansions.
• The analysis confirms holographic dualities by establishing precise factorization and associativity between bulk quantum field theories and boundary conformal data.

A multi-particle celestial operator is a composite local operator on the celestial sphere, constructed from coincident or normal-ordered products of single-particle celestial primary operators. In the context of @@@@1@@@@, these multi-particle operators encode information about multi-particle bulk scattering states and their operator product expansions (OPEs) are governed by the structure of underlying conformal field theory (CFT). The study of multi-particle celestial operators brings together techniques from 2D CFT, boundary operator expansions, and the collinear kinematics of scattering amplitudes, establishing a direct link between bulk quantum field theory and boundary conformal data.

1. Celestial Operators and the Framework of Celestial Holography

In four-dimensional asymptotically flat spacetimes, celestial holography posits a duality between bulk scattering amplitudes and correlators of conformal primary operators on the celestial sphere. A massless momentum eigenstate with momentum

$$p^\mu = \epsilon\, \omega\, (1+z\bar z,\, z+\bar z,\, -i(z-\bar z),\, 1-z\bar z), \quad \epsilon = \pm 1$$

is mapped to a celestial operator via a Mellin transform over energy,

$$\mathcal{O}_{\Delta,J}(z,\bar z) = \int_0^{\infty} d\omega\,\omega^{\Delta-1}\,a(\epsilon\,\omega,\hat p(z,\bar z)),$$

where $(\Delta,J)$ label conformal dimension and spin on the 2D celestial sphere. Lorentz $SU(2)\times SU(2)$ acts as global $SL(2,\mathbb{C})$ on $(z, \bar z)$. The operator product structure of such conformal primaries, especially when taken at coincident points or close separation, encapsulates the OPE data relevant for the correspondence between bulk factorization limits and boundary singularities (Calkins et al., 7 Jan 2026).

2. Single-Particle and Multi-Particle Celestial Operator Product Expansions

The OPE for two single-particle primaries of dimensions $(h_1, \bar h_1)$ and $(h_2, \bar h_2)$ in a holomorphic channel is given by

$$\mathcal{O}_1(z_1, \bar z_1) \, \mathcal{O}_2(z_2, \bar z_2) \sim \frac{1}{z_{12}}\sum_{I}\sum_{n=0}^\infty \frac{\gamma_{s_I}^{s_1,s_2}}{n!}\,B\bigl(2\bar h_1+p_{12I}+n,\,2\bar h_2+p_{12I}\bigr)\,\bar z_{12}^{\,p_{12I}+n}\, \bar\partial_2^n \mathcal{O}_I(z_2,\bar z_2) + \text{reg.}$$

with $p_{12I} = s_1 + s_2 - s_I - 1 \ge 0$ and $\gamma_{s_I}^{s_1,s_2}$ proportional to the bulk three-point coupling (Calkins et al., 7 Jan 2026). This singular structure is the boundary manifestation of collinear factorization in the bulk.

A multi-particle celestial operator, formally denoted $$:\!\mathcal{O}_1 \mathcal{O}_2 \ldots \mathcal{O}_n\!:(z, \bar z)$$, is defined by a regularized coincident limit of single-particle primaries implemented by normal ordering or specific contour integrations. Its OPE with a further primary or composite operator can be constructed using recursive CFT techniques and generalized Wick contractions, leading to singular terms completely determined by the single-particle OPE via associativity (Calkins et al., 7 Jan 2026). For three operators, truncating to the leading holomorphic poles,

$$\begin{aligned} &\mathcal{O}_1(z_1):\mathcal{O}_2\mathcal{O}_3:(z_3) \sim \sum_{I,J} \gamma_{s_I}^{s_1,s_2} \gamma_{s_J}^{s_I,s_3}\, \frac{\bar z_{13}^{p_{12I}+p_{I3J}}}{z_{13}^2}\, B\Big(2\bar h_1 + p_{12I} + p_{I3J}, 2\bar h_2 + p_{12I}, 2\bar h_3 + p_{I3J}\Big)\, \mathcal{O}_J(z_3) \ &\quad - \text{composite terms} \end{aligned}$$

where all terms factorize into products of single-particle OPE coefficients and generalized Euler Beta functions (Calkins et al., 7 Jan 2026).

3. Methods of Derivation and Consistency Checks

Three independent approaches yield consistent multi-particle OPE coefficients:

a) Conformal Symmetry and Recursion: Imposing $SL(2,\mathbb{C})$ covariance and translation-invariance leads to a unique recursion for OPE coefficients. The results are verified by enforcing covariance under the translation current on the sphere, a $(h,\bar h) = (3/2, -1/2)$ primary,

$$\mathcal{P}(w,\bar w) = \frac{1}{\kappa} \lim_{\Delta \to 1} (\Delta-1) G^+_\Delta(w,\bar w),$$

with action on $\mathcal{O}_{h,\bar h}$ determined by the corresponding OPE (Calkins et al., 7 Jan 2026).

b) Generalized Wick Theorem: The OPE of a multi-particle operator is constructed by systematically applying the Wick theorem to propagate the singularities of the single-particle OPEs (Calkins et al., 7 Jan 2026).

c) Bulk Multi-Collinear Limit: Direct analysis of the bulk $n$-point amplitude in the multi-collinear limit, in either gauge theory or gravity, followed by Mellin transformation and expansion in the relevant coordinate limits, demonstrably reproduces the boundary OPE coefficients, including all dependence on the three-point couplings and kinematic prefactors (Calkins et al., 7 Jan 2026).

These approaches establish that the multi-particle celestial OPE coefficients are fixed entirely by the single-particle data and adhere to complete associativity inherited from the conformal algebra.

4. Multi-Particle Operators and Boundary Operator Product Expansions in Diverse Contexts

Boundary OPEs play a central role in multiple settings beyond celestial holography.

• BCFT and Higher Dimensions: In BCFT, the boundary operator product expansion arises in expressing a bulk local operator near a boundary as a sum over local boundary operators with universal coefficients fixed by conformal symmetry and boundary universality class, as precisely determined for the three-dimensional Ising universality class (Przetakiewicz et al., 20 Feb 2025). The expansion coefficients can be accurately computed by high-precision Monte Carlo simulations and controlled finite-size scaling, confirming the role of surface operators and their scaling dimensions in critical boundary phenomena.
• Chiral Algebras and Vertex Operator Algebras: In 2D BCFTs, the structure of boundary OPEs is further formalized using the language of operads and module categories, such as the Swiss-cheese operad. The iterated OPEs converge absolutely and define real analytic correlation functions that are independent of order and parenthesization, provided the module category is $C_1$-cofinite (Moriwaki, 2024). The general formulation guarantees consistency and associativity of multi-particle boundary operator expansions.
• AdS$_3$/CFT$_2$ and Worldsheet CFT: The worldsheet OPEs for chiral primaries in AdS$_3 \times$ S$^3 \times$ T$^4$ string theory give rise to recursion relations for extremal $p$-point correlators of symmetric orbifold CFTs, demonstrating how multi-cycle and multi-particle OPEs in the boundary are built up from fundamental three-point data (Kirsch et al., 2011).
• Defect and Mixed Boundary Conditions: In 2D CFTs with mixed boundary conditions, the boundary-operator expansion formalism separates short-distance expansions away from or at switching points of boundary conditions, incorporating boundary-condition changing fields and fusion rule constraints (Burkhardt et al., 2020). The asymptotic expansions express multi-point and multi-particle boundary effects in terms of local operator content, including the Casimir interaction and corrections induced by distant boundaries.

5. Universal Properties, Factorization, and Physical Significance

The essential features of multi-particle celestial operators and their OPEs are:

• Universality and Factorization: Multi-particle OPE coefficients factorize into products of single-particle OPE (or three-point) coefficients and generalized Beta functions. This mirrors the universal associativity of the CFT OPE and the physical factorization of amplitudes in collinear bulk limits (Calkins et al., 7 Jan 2026).
• Associativity and Recursion: The recurrence and consistency of OPE coefficients are determined by translation symmetry, conformal covariance, and the fusion rules of the underlying symmetry algebra. This property is vital for maintaining crossing symmetry and analytic structure in correlators.
• Probes of Physical and Holographic Dualities: The agreement between boundary (CFT) and bulk (QFT) computations of multi-particle coefficients is a nontrivial check of celestial holography, confirming that the full multi-particle structure of the celestial sphere encodes the detailed IR and collinear physics of the bulk S-matrix.

Table: Summary of Multi-Particle Celestial OPE Approaches

Approach	Description	Output Coefficients
Boundary CFT	Recursive symmetry, Wick theorem	Products of $\gamma$, Beta
Bulk QFT	Multi-collinear limit, amplitude factorization	Splitting functions, match
Symmetry	Translational invariance constraints	Unique solution fixed

6. Extensions, Loops, and Open Problems

While the structure of multi-particle celestial OPEs is primarily kinematic and governed by CFT axioms, several open directions remain:

• Loop Corrections and Branch Cuts: The OPE analysis at tree-level can be extended, in principle, to include loop corrections, although analytic properties such as branch cuts and multi-valuedness may arise (Calkins et al., 7 Jan 2026).
• Jacobi and Higher Associativity: Checking associativity and Jacobi-like identities in the presence of multiple fusion channels and higher-particle composites is technically involved and remains an active area of investigation.
• Relations to Higher Symmetry Algebras: Symmetry derivations of OPE coefficients suggest relations to universal enveloping algebras such as $w_{1+\infty}$ and the celestial $S$-algebra, hinting at deeper algebraic structures underlying multi-particle celestial operator expansions (Calkins et al., 7 Jan 2026).
• Applicability in BCFT and Defect CFT: The BCFT results on boundary OPE coefficients and the associated finite-size scaling amplitudes provide benchmarks and quantitative tests for analytic and numeric approaches in statistical and condensed matter systems (Dey et al., 2020, Przetakiewicz et al., 20 Feb 2025).

7. Broader Connections and Contrasts

In other contexts, such as dS/CFT, the operator algebra admits even richer—though more problematic—structures, with principal and complementary series representations leading to essential singularities and non-local features in the OPE, in contrast to the well-structured highest-weight CFTs underlying multi-particle celestial OPEs (Chatterjee et al., 2016).

The combination of boundary, bulk, and symmetry-based methods demonstrates the unifying role of multi-particle operator expansions in bridging the collinear limit physics of four-dimensional QFTs, universal behaviors in statistical systems, and boundary-induced algebraic structures in diverse quantum field theories.