[Paper Review] Conformal Partial Waves: Further Mathematical Results

F.A. Dolan, H. Osborn|arXiv (Cornell University)|Aug 31, 2011

Black Holes and Theoretical PhysicsPhysics and Astronomy27 references175 citations

TL;DR

This paper derives further mathematical results for conformal partial waves in CFTs, focusing on their Mellin transform representations and differential operators that shift parameters a, b, and dimension d. It provides explicit polynomial expressions for the Mellin transform of conformal partial waves and identifies shift operators that modify scale dimension Δ and spin ℓ, recovering known results for d = 2, 4, 6 and extending them to d = 1, 3, and general even dimensions.

ABSTRACT

Further results for conformal partial waves for four point functions for conformal primary scalar fields in conformally invariant theories are obtained. They are defined as eigenfunctions of the differential Casimir operators for the conformal group acting on two variable functions subject to appropriate boundary conditions. As well as the scale dimension $Δ$ and spin $\ell$ the conformal partial waves depend on two parameters $a,b$ related to the dimensions of the operators in the four point function. Expressions for the Mellin transform of conformal partial waves are obtained in terms of polynomials of the Mellin transform variables given in terms of finite sums. Differential operators which change $a,b$ by $\pm 1$, shift the dimension $d$ by $\pm 2$ and also change $Δ,\ell$ are found. Previous results for $d=2,4,6$ are recovered. The trivial case of $d=1$ and also $d=3$ are also discussed. For $d=3$ formulae for the conformal partial waves in some restricted cases as a single variable integral representation based on the Bateman transform are found.

Motivation & Objective

• To derive further mathematical properties of conformal partial waves in conformal field theories, particularly their Mellin transform representations.
• To identify differential operators that shift the parameters a, b (related to operator dimensions) and the spacetime dimension d by ±1 and ±2, respectively.
• To recover and generalize previous results for d = 2, 4, 6, and extend the formalism to d = 1 and d = 3, including integral representations for d = 3.
• To establish connections between conformal partial waves and symmetric polynomials, particularly Jack polynomials and Gegenbauer polynomials, via the Bateman transform and hypergeometric functions.
• To provide a systematic framework for constructing conformal partial waves using recurrence relations and orthogonal polynomial structures in the x, x̄ variables.

Proposed method

• Define conformal partial waves as eigenfunctions of the quadratic Casimir operator of the conformal group, with eigenvalues determined by scale dimension Δ and spin ℓ.
• Introduce the variables x, x̄ (related to conformal invariants u, v) to simplify the eigenvalue equation, reducing it to a form involving products of hypergeometric functions.
• Derive the Mellin transform of conformal partial waves as finite sums of polynomials in Mellin variables, enabling explicit algebraic computation.
• Construct differential operators that shift a → a±1, b → b±1, d → d±2, and simultaneously alter Δ and ℓ, using recurrence relations in the Mellin space.
• Use the Bateman transform to derive single-variable integral representations for conformal partial waves in d = 3, particularly in restricted cases.
• Express solutions in terms of Jack polynomials and relate them to Gegenbauer polynomials, especially for ε = 1/2 (corresponding to d = 2, 4), where they reduce to Legendre polynomials.

Experimental results

Research questions

• RQ1 How can the Mellin transform of conformal partial waves be expressed as a finite polynomial in Mellin variables?
• RQ2 What differential operators exist that simultaneously shift the parameters a, b, and the spacetime dimension d, along with Δ and ℓ?
• RQ3 How do the conformal partial wave solutions behave in the limiting cases d = 1 and d = 3, and can they be represented via integral transforms?
• RQ4 What is the connection between conformal partial waves and symmetric polynomials such as Jack polynomials and Gegenbauer polynomials?
• RQ5 Can the leading twist case (Δ = ℓ + d − 2) be systematically solved using additional second-order differential operators?

Key findings

• The Mellin transform of conformal partial waves is expressed as a finite sum of polynomials in the Mellin variables, providing an explicit algebraic representation.
• Differential operators are constructed that shift a → a±1, b → b±1, d → d±2, and simultaneously change Δ and ℓ, generalizing known shift relations.
• For d = 3, conformal partial waves are expressed as single-variable integral representations using the Bateman transform, valid in restricted cases.
• The results recover and generalize previous expressions for d = 2, 4, 6, with the d = 4 case corresponding to ε = 1 and d = 2 to ε = 1/2.
• For ε = 1/2, the conformal partial waves are shown to be equivalent to Legendre polynomials via Jack polynomial expansions, confirming consistency with known results.
• The formalism using Jack polynomials provides a systematic framework for constructing solutions, with truncation observed when λ₂ = ε − b − N or λ₂ = ε − a − N, leading to finite series.