D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes

TL;DR

This work develops a D-independent Mellin representation for conformal field theories, expressing Euclidean and Minkowski Green functions through Mellin amplitudes $M^c$ with linear constraints $\sum_i \delta_{ij}=d_j$, thereby encoding the full operator content via pole structures. By linking OPE data to exact pole-residue factorization in Mellin space, the approach casts scalar and spinning exchanges as dual-resonance-like contributions, with dynamical poles at $\delta^{(43)(21)}=\delta^n_l(d_k)$ and satellites controlled by conformal descendants. The formalism integrates Osterwalder–Schrader positivity, dimensional reduction arguments, and the Symanzik $n$-star machinery to provide a consistent, dimension-agnostic framework that reproduces known OPE limits, Born terms, and the factorization of 4-point functions. The results offer a robust bridge between conformal bootstrap data, resonance-model intuition, and holographic viewpoints, enabling controlled analyses of short-distance behavior and spectral constraints across dimensions. The paper also provides detailed appendices on partial waves, pole structures, and 3-point functions, equipping the reader with concrete computational tools for Mellin-space CFT analysis.

Abstract

The Euklidean correlation functions and vacuum expectation values of products of field operators of some Lorentz spin and dimension are expressed through Mellin amplitudes which depend on complex dimensions subject to linear constraints. The constraints can be solved in terms of conserved momenta whose squares are given by the field dimensions, and related Mandelstam variables s. The Mellin amplitudes furnish a universal representation of conformal field theories without explicit reference to D. The costumary principles of quantum field theory plus conformal invariance and operator product expansions (OPE) say that the Mellin amplitudes are amplitudes of dual resonance models with exact duality and a form of factorization which follows from OPE. Fields in the OPE with spin l and dimension d produce simple poles in the scalar 4-point Mellin amplitude at s=d-l+2n, n=0,1,2,3... with polynomial residues. The leading pole determines the satellites n=1,2,3...

Figures (2)

• Figure 1: Path of the $c$-integration in the complex $c$-plane in the derivation of OPE, for fixed $l$ in the presence of a fundamental field with Lorentz spin $l$ and dimension $d_f=h+c_f$, $c_f<0$: a) before and b) after the shift of the path of integration. To each field $\phi^k$ in the OPE of Lorentz spin $l$ and dimension $d_k=h+c_k$ there corresponds a pole $\bullet$ at $c=c_k$ in $g_{\dots}^{(43)(21)}([l,c])$, and a shadow pole $\circ$ at $c=-c_k$. The contributions in a) from the the circles around the poles at $c=\pm c_f$ represent the Born term.
• Figure 2: Deformation of the path of integration in the complex energy plane. a) Integration over Euklidean space involves integration over imaginary $p^0$. b) The deformed path