Dynamical derivation of vacuum operator-product expansion in Euclidean conformal quantum field theory

Abstract

An expansion of the type (@(&ij &(&n~&() = (fr(' (& pl&(&2l&0(@(&p~' @(&n~&p (x, ) Xi is derived, where yi -fl, ci] are labels for infinite-dimensional symmetric tensor representations of the Euclidean conformal group 0 (2@+1, 1), X, =[ l, , -ci], the constants C (y i) are real, and Q)( and u)x have the properties of vacuum expectation values of field products. The starting point is an infinite set of coupled nonlinear integral equations for Euclidean Green's functions in 2h space-time dimensions of the type written some 15 years ago by Fradkin and Symanzik. The Green's functions of the corresponding Gell-Mann-Low limit theory are expanded in conformal partial waves. The dynamical equations imply the existence of poles and factorization of residues in the partial waves as functions of the representation parameters. In proving the validity of the expansion we use some differential relations between partially equivalent exceptional representations of 0& (2 h + 1, 1), established in an earlier paper. This work completes the group-theoretical derivation of the vacuum operator-product expansion undertaken by Mack in 1973.

Arbitrary spin representations in de Sitter from dS/CFT with applications to dS supergravity

“…, t+1) and gives µ(n, s)−µ(n, s−t) degrees of freedom. 10 These representations have been studied in a conformal field theory context in [19,20,21,22].…”

Conformal four point functions and the operator product expansion

“…representations of O(d), corresponding to fields O (ℓ) µ 1 ...µ ℓ which are totally symmetric traceless rank ℓ tensors. Although conformal partial wave expansions were obtained long ago [1,2,3] they have not been in a form which is easy to apply to disentangling the contributions of different operators to the four point functions found through the AdS/CFT correspondence. This has necessitated approximate calculations of just the leading terms in the power expansion for the contribution of operators with non zero spin in the operator product expansion [4].…”

On twist-two operators in SYM

“…7 In [29] the homogeneous polynomial P is rewritten in terms of a single variable (∂a − ∂ b )/(∂a + ∂ b ) and the corresponding differential equation is identified with that for a Gegenbauer polynomial. This is the standard form for twist-two operators in the QCD literature.…”