Mellin-Barnes Representation of Correlators

Updated 12 December 2025

The Mellin–Barnes representation is a contour-integral formalism in QFT and CFT that systematically decouples kinematic structures from dynamical data.
It employs multidimensional complex integrals with Gamma functions to expose analytic properties, including pole locations and branch cuts.
Applications range from multi-loop computations and conformal bootstrap methods to holographic and cosmological correlators, offering practical analytic insights.

The Mellin-Barnes (MB) representation of correlators is a contour-integral formalism that provides meromorphic, multidimensional integral representations for correlation functions in quantum field theory and conformal field theory. This approach systematically decouples kinematic structures from dynamical data, exposes analytic properties such as pole locations and branch cuts, and is critical in multi-loop computations, studies of conformal field theory, AdS/CFT, and cosmological correlators.

1. Mellin–Barnes Representation: General Structure

The Mellin–Barnes representation expresses $n$ -point correlation functions as multidimensional contour integrals over several complex variables, called Mellin variables, which are subject to linear “conformal” or “momentum-conservation” constraints. The most general form for an $n$ -point scalar correlator in a conformal field theory (CFT) is

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

with

$\sum_{j \ne i} \delta_{ij} = \Delta_i$

where $M(\{\delta_{ij}\})$ is the Mellin amplitude containing dynamical information, while the Gamma functions $\Gamma(\delta_{ij})$ encode kinematic “double-trace” singularities, analogous to momentum-space phase-space factors in flat space. The integration contours are parallel to the imaginary axis and pass between the left- and right-going series of Gamma-function poles (Paulos et al., 2012, Nizami et al., 2016).

This formalism is universal: it admits direct generalization to operators with spin via the incorporation of tensor structures and discrete labelings, to nonconformal correlators by relaxing constraints and replacing Beta functions with generalized hypergeometric functions, and to integrals in curved backgrounds and more general spaces (Chen et al., 2017, Barvinsky et al., 3 Dec 2025).

2. Basic Examples: Contact and Exchange Diagrams

Contact Interactions

For a non-derivative $n$ -point vertex in a CFT or holographic context, the MB representation reduces to

$M_\text{contact} = 1$

That is, the dynamical Mellin amplitude is trivial, and all kinematic singularities arise from the $\prod \Gamma(\delta_{ij})$ factors. Derivative interactions shift Mellin variables and promote amplitudes to polynomials (Nizami et al., 2016, Pacifico et al., 16 Dec 2024).

Exchange Diagrams

For a four-point function with a tree-level $s$ -channel exchange of a conformal block of dimension $\delta$ , the Mellin amplitude acquires a Beta function propagator structure: $M_\text{exchange}(s_{ij}) = \frac{1}{2\,\Gamma(\delta)} B(\delta - s, D - \delta)$ with $s$ a specific Mandelstam-like combination of Mellin variables. The Beta function develops simple poles at the positions of the exchanged primary and all its descendants. In the Mellin–Barnes integral language, these singularities factorize residues according to OPE data (Nizami et al., 2016, Paulos et al., 2012, Pacifico et al., 16 Dec 2024).

For more general cases, e.g., higher spins, spinning primaries, or loop diagrams, the Mellin amplitude becomes a rational function (or a sum of polynomials and simple poles) of several variables, possibly with additional contour integrations introduced by MB representations of hypergeometric functions arising in multi-loop or multi-scale problems (Chen et al., 2017, Derkachev et al., 2023, Prausa, 2017, Valtancoli, 2011).

3. Construction Methods and Algorithms

Algorithmic Generation

The standard method begins from the Schwinger or Feynman parameter representation, followed by MB transforms to handle sums and exponentials:

Exponentials $\exp[-A(x)]$ are rewritten as MB integrals over Gamma functions.
Multinomials $(A_1+\cdots+A_J)^\alpha$ are split into multiple MB integrals.
Integration over Schwinger parameters introduces bracket notation for constraints.
Bracket elimination selects $K$ Mellin variables for elimination, reducing the dimensionality of the integral. Optimization procedures reduce the number of MB variables by factoring common sub-expressions in Symanzik polynomials (Prausa, 2017, Derkachev et al., 2023).

For instance, the scalar box integral in $d=4+2\epsilon$ is transformed to a one-fold MB integral,

$I(s,t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} dw\, (-s)^{2-\varepsilon+w}(-t)^w \frac{\Gamma(w+1)^2\Gamma(2-\epsilon+w)\Gamma(-w)\Gamma(\epsilon-1-w)}{\Gamma(2\epsilon)}$

where $w$ is the MB parameter (Valtancoli, 2011). This representation permits analytic continuation, Laurent expansion, and resummation of hypergeometric series.

Special Functions and Factorization

The residues of MB integrals are often recognized as hypergeometric ( ${}_2F_1$ , ${}_3F_2$ ) or more general special functions, which can simplify multi-variable integrals into bilinear or tensor-product expressions (Derkachev et al., 2023).

For multi-loop diagrams, each loop introduces new MB variables, but the bracket method can reduce apparent complexity substantially, leading to lower-dimensional final MB integrals.

4. Analytic Structure and Physical Interpretation

The Mellin–Barnes approach foregrounds the analytic structure of the correlator:

Double-trace poles from Gamma functions $\Gamma(\delta_{ij})$ : universal, kinematic, encode power-law OPE data of generalized free fields.
Single-trace poles from Mellin amplitude $M(\delta_{ij})$ : dynamical, encode operator exchanges, OPE coefficients, and dimensions.
Loop contributions increase the dimensionality of the MB integral (additional integrals over internal masses or Schwinger parameters), but do not introduce new external pole series; the pole structure remains that of tree-level exchange (Pacifico et al., 16 Dec 2024, Prausa, 2017).
Branch cuts and discontinuities arise from non-perturbative effects and are tied to the spectral density of the theory via Källén–Lehmann integrals—overall MB factors, such as $\mathcal{M}[\rho](s)$ , represent the Mellin transform of the spectral density and encode nonperturbative analytic properties and unitarity cuts (Barvinsky et al., 3 Dec 2025, Pacifico et al., 16 Dec 2024).

5. Key Applications: Conformal, Holographic, and Celestial Correlators

Conformal Field Theories

The MB representation underlies the modern theory of the conformal bootstrap, Mellin-space Feynman rules, and explicit computation of correlators in weakly and strongly coupled CFTs, including spinning primaries and fermionic correlators (Paulos et al., 2012, Nizami et al., 2016, Faller et al., 2017, Chen et al., 2017). In CFT applications, the MB amplitude is symmetric under crossing, and the structure of poles is prescribed by conformal block/OPE fusion rules.

AdS/CFT and Holography

For Witten diagrams, MB representations make explicit the connection between position- or momentum-space diagrams and the underlying CFT data. In the supergravity regime, correlators reduce to low-dimensional MB integrals with rational or polynomial Mellin amplitudes (“contact” diagrams) or propagate poles corresponding to single- and double-trace exchanges. Advanced constructions, such as the differential representation for higher-point correlators, directly relate position-space polylogarithms to rational Mellin amplitudes acting on a minimal set of “seed” functions (Huang et al., 15 Mar 2024, Aprile et al., 19 Sep 2024).

Cosmological and Celestial Correlators

In cosmological correlators (inflation, dS/CFT), MB representations are used in both position and momentum space to expose late-time analytic structure, manifest the separation between local backgrounds, oscillatory nonlocal “signals,” and allow efficient OPE/EFT expansions. Both tree-level inflaton correlators and “helical inflation” signatures of spinning particles can be constructed using partial MB and boostless bootstrap techniques (Qin et al., 2022, Sleight, 2019, Sleight et al., 2019).

On the celestial sphere, MB amplitudes reorganize Minkowski correlators as meromorphic functions of conformal “Mandelstam” variables, closely paralleling AdS/CFT constructions but adapted to Lorentzian signature and the physics of the celestial sphere. Non-perturbative structure is encoded through the Mellin transform of spectral densities, making the Källén–Lehmann data explicit and capturing unitarity cuts (Pacifico et al., 16 Dec 2024).

6. Advanced Formulations and Operator Functions

For operator functions $f(\hat F)$ of minimal second-order operators on curved backgrounds, the MB representation provides a systematic methodology for constructing the kernel of $f(\hat F)$ in the off-diagonal Schwinger–DeWitt expansion: $\mathbb{B}_\alpha[f \mid \sigma] = \int_{\mathcal{C}} \frac{ds}{2\pi i} \Gamma(-s)\Gamma(\alpha+s)\sigma^s F(s)$ where $F(s)$ encodes the analytic structure of $f$ , and the full expansion separates geometry (through heat-kernel coefficients $\hat a_k$ ) from analytic data (Barvinsky et al., 3 Dec 2025). The extension to kernels with mass terms introduces further MB integrals and nontrivial infrared regularization.

7. Contour Prescriptions, Convergence, and Analytic Continuation

MB integrals require careful choice of contours: lines are chosen so that left- and right-going sequences of Gamma-function poles are separated, ensuring convergence. The deformation of contours, closing them left or right, selects pole series for OPE expansions, EFT expansions, or other asymptotic behaviors. Analytic continuation is implemented by moving contours and is often necessary to reach physical regimes of parameters or to resum divergent series (Valtancoli, 2011, Derkachev et al., 2023, Prausa, 2017, Barvinsky et al., 3 Dec 2025).

The Mellin–Barnes representation is a unifying toolbox for correlators across QFT, CFT, AdS/CFT, cosmology, and perturbative and nonperturbative quantum gravity. Its key strengths are the explicit analytic control it provides, its compactness for multi-loop and multi-scale problems, and the clarity with which it encodes both kinematic (double-trace) and dynamical (single-trace, nonperturbative) spectral data (Nizami et al., 2016, Pacifico et al., 16 Dec 2024, Barvinsky et al., 3 Dec 2025).