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

G. Mack|ArXiv.org|Jul 14, 2009

Black Holes and Theoretical Physics48 references139 citations

TL;DR

This paper introduces a D-independent formulation of conformal field theories (CFTs) in D dimensions by mapping Euclidean correlation functions to Mellin amplitudes that depend on complex kinematic variables δ_ij, which are related to conserved momenta and Mandelstam invariants. The key contribution is a universal representation of CFTs via dual resonance models with exact duality and OPE-induced factorization, enabling D-independent analysis of scalar amplitudes and revealing spin-dependent poles at s_ij = d - l + 2n with polynomial residues in l.

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...

Motivation & Objective

• To develop a D-independent formulation of conformal field theories that abstracts away explicit dependence on spacetime dimension D.
• To express n-point correlation functions of scalar and spinning fields through Mellin amplitudes parameterized by complex dimensions δ_ij subject to linear constraints.
• To establish that Mellin amplitudes of CFTs naturally realize the structure of dual resonance models, including exact duality and factorization from OPE.
• To show that the OPE structure of CFTs manifests as simple poles in Mellin amplitudes at s_ij = d - l + 2n with polynomial residues in spin l.
• To provide a universal framework for analyzing CFTs across different dimensions using auxiliary dual resonance models, enabling dimensional induction and holographic insights.

Proposed method

• Represent n-point Euclidean correlation functions via Mellin amplitudes M_{k_n...k_1}(δ_ij), where δ_ij are complex variables constrained by ∑_j δ_ij = d_i.
• Map the δ_ij variables to conserved momenta p_i with p_i² = d_i, and define Mandelstam invariants s_ij = (p_i + p_j)², so that δ_ij = -p_i p_j.
• Use the operator product expansion (OPE) to derive factorization properties of Mellin amplitudes, ensuring duality and consistency with quantum field theory principles.
• Construct Mellin amplitudes as generating functions for CFT data, with poles at s_ij = d - l + 2n for fields of spin l and dimension d.
• Employ differential operators D_l of order l to express conformal three-point functions, enabling the derivation of un-amputated OPE coefficients in momentum space.
• Derive the un-amputated OPE coefficient Q^u as an entire function of momentum p, involving Bessel functions and hypergeometric-type differential operators.

Experimental results

Research questions

• RQ1 How can conformal field theories in D dimensions be represented without explicit dependence on D?
• RQ2 What kinematic structure underlies Mellin amplitudes of CFTs, and how do they relate to dual resonance models?
• RQ3 How do operator product expansions (OPE) in CFTs manifest in the Mellin amplitude formalism?
• RQ4 What determines the location and residue structure of poles in Mellin amplitudes for spinning operators?
• RQ5 Can the Mellin amplitude framework unify CFTs across different dimensions through a universal dual resonance model structure?

Key findings

• The Mellin amplitude M_{k_4...k_1} of a 4-point function exhibits simple poles at s_ij = d - l + 2n for fields of spin l and dimension d, with residues that are polynomials in l of degree n.
• The residues of the leading pole (n=0) determine the entire tower of satellite poles (n=1,2,3,...), encoding the full OPE contribution of a field.
• The Mellin amplitude framework realizes exact duality and factorization properties derived from the OPE, ensuring consistency with quantum field theory and conformal invariance.
• The un-amputated OPE coefficient Q^u is an entire function of momentum p, expressed through a differential operator D_l and an integral over u involving Bessel functions.
• The Mellin amplitude provides a universal representation of CFTs independent of D, with all D-dependence encoded in the coupling constants and normalization factors.
• The formalism allows analytic continuation between Euclidean and Minkowski space correlation functions via iε prescriptions, consistent with the spectrum condition.