Mellin Space Method for Boundary Correlators

• Mellin space method is a unified algebraic approach that encodes boundary correlation functions with meromorphic amplitude representations.
• Its pole structure and factorization properties echo momentum-space scattering, offering clarity in operator exchanges and conformal block expansions.
• The framework generalizes to spinning fields, boundary defects, and cosmological/ celestial contexts, bridging holographic CFTs with flat-space S-matrix limits.

The Mellin space method for boundary correlation functions provides an algebraic framework to encode, analyze, and compute correlation functions in conformal field theories (CFTs), their holographic AdS/CFT duals, and celestial and cosmological adaptations. Mellin amplitudes are meromorphic functions of variables conjugate to boundary cross-ratios; their pole structure and residues capture operator exchanges, factorization, and conformal block expansion in direct analogy to momentum-space scattering amplitudes. This formulation unifies tree-level Witten diagrams, Feynman-like diagrammatics, and conformal bootstrap constraints, and allows direct generalizations to correlators with spin, boundaries/defects, cosmological observables, and celestial holographic amplitudes.

1. Mellin Representation of Boundary Correlators

Any $n$-point boundary correlator for scalar primaries of dimension $\Delta_i$ admits a Mellin representation (Fitzpatrick et al., 2011, Penedones, 2010, Nizami et al., 2016):

$$G_n(x_1,\dots,x_n) = \int[d\delta_{ij}]\, M_n(\{\delta_{ij}\}) \prod_{i<j} \Gamma(\delta_{ij}) (x_{ij}^2)^{-\delta_{ij}}$$

where $\delta_{ij}=\delta_{ji}$ are Mellin variables obeying $n$ linear “on-shell” constraints for conformal covariance:

$$\forall i:\quad \sum_{j \neq i} \delta_{ij} = \Delta_i$$

yielding $n(n-3)/2$ independent variables related to the conformal cross-ratios. The Gamma-factor prefactor ensures correct scaling under $x_i\to\lambda_i x_i$. The Mellin amplitude $M_n(\delta_{ij})$ encodes all dynamical information.

In momentum-space, boundary correlators in (A)dS can be expressed via Mellin-Barnes integrals over momenta $k_i$ and Mellin variables $s_i$ conjugate to $k_i$:

$$G(\{k_i\}) = \int \prod_i \left( \frac{ds_i}{2\pi i} \right) M(s_i) \prod_i \left(\frac{k_i}{2}\right)^{-2s_i + i\nu_i}$$

subject to $(\sum_i s_i) = d(n-2)/4$ (Sleight, 2019, Sleight et al., 2019).

2. Analytic Structure and Factorization

The essential structure of Mellin amplitudes is their pole decomposition, paralleling the OPE and scattering factorization (Fitzpatrick et al., 2011, Gonçalves et al., 2014). The exchange of a primary of dimension $\Delta$, spin $J$ in an OPE channel leads to a family of poles in a “Mandelstam-like” invariant $\delta_{LR}$:

$\delta_{LR} = \Delta - J + 2m, \quad m=0,1,\dots$

The residue at each pole factorizes:

$$\operatorname{Res}_{\delta_{LR} = \Delta - J + 2m} M_n = \mathcal Q_m = \text{(kinematic factor)} \cdot L_m \cdot R_m$$

with $L_m$ and $R_m$ the subdiagram Mellin amplitudes. For scalars ($J=0$), the kinematic factor is simple; for spinning exchange ($J>0$), polynomial structures capture tensor index contractions.

This factorization is direct at tree-level in cubic bulk theories, where diagrammatic rules assign propagator poles and vertex factors to each internal line and sum over non-negative indices labeling descendants (Fitzpatrick et al., 2011, Nandan et al., 2011).

3. Diagrammatic Rules and Spinning Generalization

Tree-level Mellin-space Feynman rules are algebraic (Fitzpatrick et al., 2011, Nandan et al., 2011, Chen et al., 2017). For a cubic scalar theory, each internal propagator with channel variable $\delta$ and dimension $\Delta$ provides poles at $\delta=\Delta+2m$ and vertex factors built from Pochhammer symbols and hypergeometric functions (${_3F_2}$). The amplitude is a sum over internal indices $m_i$:

$$M = \sum_{\{m_i\}} \prod_{\text{props}} \text{Pole Factors} \times \prod_{\text{vertices}} \text{Vertex Factors}$$

Spinning correlators require additional discrete Mellin variables $a_{ij},b_{ij}$ cataloging tensor structures that encode polarization contractions (Chen et al., 2017). For a four-point function with external spins $\ell_i$, the Mellin representation becomes:

$$\langle O_{\Delta_1,\ell_1} \cdots O_{\Delta_4,\ell_4} \rangle = \sum_{\text{structures}} \int ds\, dt\, d\nu\, \text{Spectral Prefactors} \times \text{Kinematical Mack Polynomials} \times \prod_{i<j} \Gamma(\delta_{ij}) P_{ij}^{-\delta_{ij}}$$

The kinematical polynomials generalize the Mack polynomial, forming a basis for solutions to conformal Casimir equations in Mellin space.

4. Boundary, Defect, and Interface Extensions

The formalism generalizes to BCFTs and interface CFTs by introducing Mellin variables for bulk-bulk, bulk-boundary, and boundary-boundary invariants ($\delta_{ij}, \gamma_{iI}, \beta_{IJ}, \alpha_i$) (Rastelli et al., 2017). The correlator reads:

$$\mathcal C_{n,m} = \int [d\delta][d\gamma][d\beta][d\alpha]\, \frac{M}{\prod \Gamma(\delta)\Gamma(\gamma)\Gamma(\beta)\Gamma(\alpha)} \frac{\Gamma(-{\cal P}^2)}{\Gamma(-{\cal P}^2/2)} \prod (-2\,\cdots)^{-\cdots}$$

with linear constraints reflecting boundary/defect quantum numbers. Each OPE channel manifests as a family of poles and factorized residues, with new towers arising from normal derivatives of bulk fields restricted to the boundary.

5. Celestial Correlators and Cosmological Mellin Techniques

Celestial holography and cosmological correlators adapt the Mellin representation to new geometries and spectral decompositions (Pacifico et al., 16 Dec 2024, Jiang, 2022, Sleight, 2019, Sleight et al., 2019). Celestial Mellin amplitudes employ radial Mellin transforms of Minkowski correlators onto the celestial sphere, with Gamma-factor prefactors and constraints as in standard boundary Mellin:

$$\langle O_1 \cdots O_n \rangle = \int_{\!-i\infty}^{+i\infty} \prod_{i<j} \frac{d\delta_{ij}}{2\pi i} \Gamma(\delta_{ij}) M(\{\delta_{ij}\}) \prod_{i<j} (-2Q_i \cdot Q_j)^{-\delta_{ij}}$$

Meromorphicity remains, with contact diagrams polynomial and exchanges yielding pole families encoding operator dimensions. Celestial Mellin block expansions, inversion formulae, and direct energy/cross-ratio Mellin transforms recover OPE data.

In cosmological contexts, the late-time correlators in $(d+1)$-dimensional de Sitter employ Mellin-Barnes representations in momentum space, with Gamma-pole expansions revealing both OPE-like and EFT expansions (Sleight, 2019, Sleight et al., 2019).

6. Flat-Space Limit and S-Matrix Correspondence

The Mellin amplitude becomes a holographic pre-image of the flat-space S-matrix under large-dimension and large Mellin-variable limits (Fitzpatrick et al., 2011, Penedones, 2010, Gonçalves et al., 2014). Explicitly, for $n$-point functions:

$$M_n(\delta_{ij}) \sim \int_0^\infty d\beta\, \beta^{\frac12 \sum \Delta_i - h - 1} e^{-\beta} T(p_i \cdot p_j = 2\beta \delta_{ij})$$

As $\delta_{ij} \to \infty$ and $\Delta_i \to \infty$, AdS factorization poles coalesce to propagator poles of the flat-space S-matrix and the residue factorizes into products of lower-point S-matrices, establishing a direct Laplace-analytic bridge between boundary CFT data and bulk scattering amplitudes.

7. Applications: Bootstrap, Loop Diagrams, and Superconformal Theories

The Mellin space method streamlines analytic bootstrap approaches (especially at large $N$) (Penedones, 2010, Rastelli et al., 2016), as crossing symmetry reduces to rational functional constraints on $M_4(s,t)$ rather than infinite conformal block sums. Loop corrections reorganize into Mellin integrals with meromorphic structure—single-trace singularities are manifest (Chen et al., 2017, Pacifico et al., 16 Dec 2024). In maximally supersymmetric settings, such as $AdS_5 \times S^5$, compact rational Mellin formulas for half-BPS correlators emerge by imposing crossing, Ward identities, and large-$N$ analytic structure (Rastelli et al., 2016).

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The Mellin space technology for boundary correlation functions unifies conformal, holographic, and scattering amplitude analyses. It grants factorization, crossing symmetry, and analytic control over operator exchanges, and admits systematic diagrammatic and computational methodologies. The approach further readily generalizes to spinning fields, boundary/interface configurations, celestial and cosmological observables, and provides the correct holographic correspondence to the flat-space S-matrix in the bulk limit.