Boundary Operator Product Expansions

• Boundary Operator Product Expansions are composite operator techniques that use regulated coincident products and Mellin transforms to formalize multi-particle singularities.
• They employ recursive normal ordering and a generalized Wick method to ensure associativity and factorize bulk collinear limits into boundary conformal data.
• Applications span flat-space holography to boundary CFTs, providing analytic bootstrap frameworks that bridge operator algebra with scattering amplitude dynamics.

A multi-particle celestial operator is a composite conformal primary operator on the celestial sphere associated to the Mellin transform of regularized, coincident multiparticle states in four-dimensional asymptotically flat spacetimes. Such operators, together with their operator product expansions (OPEs), provide a powerful bridge between the kinematic structure of scattering amplitudes in the bulk and CFT techniques on the boundary. The study of their OPEs, structure constants, and factorization properties lies at the intersection of celestial holography, boundary CFT, and the mathematical formulations of operator algebraic expansions. Multi-particle celestial OPEs unify singularity structures from bulk collinear limits, boundary conformal methods, and symmetry constraints, comprising an essential organizing principle in flat-space holography and related contexts.

1. Construction of Multi-Particle Celestial Operators

The celestial map associates a massless bulk momentum eigenstate $\mathfrak{a}(\epsilon\omega, \hat{p}(z,\bar{z}))$ to a celestial operator via a Mellin transform in energy: $$\mathcal{O}_{\Delta,J}(z,\bar{z}) = \int_0^\infty d\omega\, \omega^{\Delta-1}\, \mathfrak{a}(\epsilon\omega, \hat{p}(z,\bar{z}))$$ where $$(h, \bar{h}) = \left(\tfrac{\Delta+J}{2},\,\tfrac{\Delta-J}{2}\right)$$ and $(z,\bar{z})$ parametrizes the celestial sphere. Multi-particle celestial operators are constructed as regulated coincident products of single-particle celestial primaries: $$:\!\mathcal{O}_1 \mathcal{O}_2 \!\!:(z,\bar{z}) = \lim_{w \to z} \left[\mathcal{O}_1(w,\bar{w})\mathcal{O}_2(z,\bar{z}) - \text{singular terms}\right]$$ This normal ordering projects out all singular (non-integrable) terms in the OPE as $w \to z$, defining a composite conformal primary at coincident coordinates. The definition extends recursively to multi-particle operators of higher rank as nested normal-ordered products. Their transformation properties under the Lorentz group $(SU(2)\times SU(2))$ follow from the tensor product decompositions of single-particle weights, while their explicit OPEs manifest the rich collinear singularity algebra inherited from the parent amplitudes (Calkins et al., 7 Jan 2026).

2. Operator Product Expansions: Theory and Structure

The OPE of two celestial primary operators is determined by conformal covariance and factorizes as

$$\mathcal{O}_1(z_1,\bar{z}_1)\,\mathcal{O}_2(z_2,\bar{z}_2) \sim \sum_{k} \frac{C^{(k)}_{12}}{z_{12}^{h_1+h_2-h_k}\,\bar{z}_{12}^{\bar{h}_1+\bar{h}_2-\bar{h}_k}}\, \mathcal{O}_k(z_2,\bar{z}_2) + \cdots$$

In celestial holography, these singularities are the reflections of bulk collinear (soft and collinear) limits. For multi-particle composites, the OPE is constructed using a generalized Wick theorem, recursively applying the single-particle OPE and evaluating all singular pairings: $$\mathcal{O}_1(z_1)\,:\!\mathcal{O}_2\,\mathcal{O}_3\!: (z_3) \sim \sum_{I,J} \gamma_{s_I}^{s_1 s_2}\gamma_{s_J}^{s_I s_3} \frac{\bar{z}_{13}^{p_{12I}+p_{I3J}}}{z_{13}^2} B\big(2\bar{h}_1+p_{12I}+p_{I3J}, 2\bar{h}_2+p_{12I}, 2\bar{h}_3+p_{I3J}\big) \mathcal{O}_J(z_3) + \cdots$$ where the coefficients are products of single-particle OPE tensors $\gamma$ and generalized Euler Beta functions, and $p_{ijK}$ and $s_K$ encode fusion and spin selection rules. The associativity and locality of the OPE ensure that further nested compositions yield coefficients fully determined by repeated convolution of the single-particle data and the normalization conventions (Calkins et al., 7 Jan 2026).

3. Connection to Bulk Collinear Limits and Symmetry Constraints

A key structural result is the precise correspondence of multi-particle celestial OPE singularities with the multi-collinear limits of bulk scattering amplitudes in four dimensions. For instance, the $n\to 1$ collinear splitting functions in Yang-Mills and (Einstein) gravity factorize as sums of channels: $$\mathcal{A}_n(1,2,\ldots) \sim \sum_J \text{Split}[1 2 3 \to J]\,\mathcal{A}_{n-2}(J,\ldots)$$ Collinear singularities in the bulk translate, under Mellin transformation and $(z,\bar{z})$ expansion, to the celestial OPE coefficients computed with the boundary Wick method. Symmetry constraints—especially the enforcement of four-dimensional translation invariance, realized as a specific $(h,\bar{h})=(\tfrac{3}{2},-\tfrac{1}{2})$ current on the celestial sphere—further fix the structure constants via recursion and translation Ward identities, in direct correspondence with the leading OPE and composite terms (Calkins et al., 7 Jan 2026). Conformal covariance (SL(2,$\mathbb{C}$) invariance) is imposed at each step, enforcing the universality and kinematic factorization of the coefficients.

4. Mathematical Foundations: OPE Algebra and Regularity

The mathematical regularity and analytic structure of multi-particle celestial operators and their OPEs are rooted in the underlying framework of boundary CFT, and, in two dimensions, vertex operator algebras (VOAs). Within this algebraic context, the OPE becomes a convergent Laurent expansion on configuration spaces governed by operadic structures (e.g., the Swiss-cheese operad for boundaries) (Moriwaki, 2024). The associativity, independence of insertion ordering, and convergence of the iterated OPE are enforced by absolute convergence theorems and the $C_1$-cofiniteness properties of the underling algebra modules. In higher dimensions and for nonlocal constructs (such as in celestial CFTs), factorization and convergence generally persist on open regions, so long as the regularization of coincident limits and subtraction of singularities are under analytic control (Calkins et al., 7 Jan 2026).

5. Applications Across AdS, dS, and Mixed Boundary Contexts

Composite OPEs and their multi-particle operator algebra have broad applicability:

• Celestial holography: The multi-particle OPE organizes the structure of the celestial CFT dual to flat space. Its coefficients encode the complete set of kinematic singularities from bulk S-matrix elements and provide a bootstrap framework for analytic continuation to nonperturbative regimes (Calkins et al., 7 Jan 2026).
• Worldsheet/Boundary matching in AdS/CFT: Multi-particle OPEs and their recursion relations explicitly reproduce the recursion relations and fusion rings of extremal ($p$-point) correlators in the context of AdS$_3$/CFT$_2$, matching both worldsheet CFT and boundary symmetric orbifold constructions (Kirsch et al., 2011).
• Boundary CFT and BCFT: The structure and universal coefficients of multi-particle (bulk and boundary) operator expansions are essential in classifying the boundary spectra and determining universal quantities such as the critical Casimir force, boundary susceptibility, and surface critical exponents, with numerically precise predictions in e.g. the Ising and $O(N)$ universality classes (Przetakiewicz et al., 20 Feb 2025, Dey et al., 2020, Burkhardt et al., 2020).

6. Advanced Properties and Open Directions

Every coefficient and singular term in the multi-particle OPE is rigorously controlled by the single-particle OPE algebra. For multi-particle operators formed by nested normal-orderings, the OPE coefficients factorize as products of lower-level coefficients and generalized Beta functions, directly reflecting the associativity of consecutive collinear limits. Translational, Lorentz, and conformal symmetry constraints fully determine the allowed operator content and selection rules, with soft current algebras (e.g., $w_{1+\infty}$, S-algebra) expected to govern higher-level recursion relations.

Several nontrivial extensions are currently under active investigation:

• Extension to higher loops and branch cuts: While the generalized Wick method is purely kinematical at tree level, the analytic continuation through branch cuts (corresponding to multi-valuedness and unitarity) and loop corrections in the operator algebra remain an open avenue (Calkins et al., 7 Jan 2026).
• Algebraic characterization via operads: Extension to $n$-ary operadic frameworks and coherence, especially with boundary condensates or defects, is a focal point of current mathematical investigation, unifying the local algebraic and global geometric (operadic) properties (Moriwaki, 2024).
• Essential singularities and non-highest weight representations: In unconventional contexts such as dS/CFT, operator expansions admit essential singularities due to the presence of principal/complementary series representations, leading to OPEs with infinitely many undetermined coefficients at each order and a drastic failure of locality—highlighting foundational distinctions between highest-weight and general series (Chatterjee et al., 2016).

7. Representative Coefficient Formulas and Examples

For Yang-Mills (spin $s_i = \pm 1$), gravity ($s_i = \pm 2$), and general composite operator structures, the holomorphic part of the multi-particle celestial OPE coefficient is given by: $$C_p^{(m)} = \frac{\gamma_{s_I}^{s_1,s_2}}{m!}\;B\big(2\bar{h}_1 + p + m,\,2\bar{h}_2 + p\big)$$ where $B$ denotes the Beta function, $\gamma_{s_I}^{s_1,s_2}$ the relevant coupling, and $p$ encodes fusion channel shifts.

The full OPE for a three-particle composite (holomorphic contribution) reads: $$\mathcal{O}_1 :\!\mathcal{O}_2 \mathcal{O}_3\!:(z) \sim \sum_{I,J} \gamma_{s_I}^{s_1,s_2} \gamma_{s_J}^{s_I,s_3} \frac{\bar{z}^{p_{12I} + p_{I3J}}}{z^2}\, B\big(2\bar{h}_1 {+} p_{12I} {+} p_{I3J},\, 2\bar{h}_2{+}p_{12I},\,2\bar{h}_3{+}p_{I3J}\big)\,\mathcal{O}_J(z) + \text{composite subtractions}$$ This structure upholds the expected associativity and recursive generation via nested collinear singularities (Calkins et al., 7 Jan 2026).

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In summary, multi-particle celestial operators encapsulate the compositional and algebraic complexity of the celestial CFT and its holographic connection to the infrared structure of gravitational and gauge amplitudes. Their OPEs, symmetry properties, and analytic foundations serve as universal building blocks central to celestial holography and boundary operator algebra, with structural links to bulk factorization, BCFT, and the broader operadic and representation-theoretic machinery of quantum field theory.