Conformal Partial Waves: Further Mathematical Results (1108.6194v2)

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 $\Delta$ 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 $\Delta,\ell$ 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.

• The paper introduces novel Mellin transform formulations and eigenvalue equations for conformal partial waves, enhancing the analysis of four-point functions in CFTs.
• It provides integral representations and recursion relations that simplify complex computations across dimensions, notably refining methods in d=3, 4, and 6.
• The study leverages classical harmonic and Jacobi polynomial techniques to extend analytic tools in CFT, paving the way for future research in conformal bootstrap dynamics.

Overview of "Conformal Partial Waves: Further Mathematical Results"

The paper elaborates on the mathematical description and properties of conformal partial waves in conformal field theories (CFTs), as investigated by F.A. Dolan and H. Osborn. The work builds on the understanding of four-point functions in CFTs, formulating these functions in terms of eigenfunctions of the differential Casimir operators for the conformal group. The central focus concerns the conformal partial waves, which depend on the scale dimension $\Delta$ and spin $\ell$, expressed through the Mellin transform in various dimensions.

A key achievement here is the formulation of the Mellin transform expressions applicable to conformal partial waves and their representation using Bateman transforms. The framework recovers prior results for $d = 2, 4, 6$ and trivial cases like $d = 1$, while presenting new analytical tools for $d = 3$. Utilizing these transforms, the paper discusses the symmetric functions of variables $x$ and $\bar{x}$ representing the conformal invariants $u$ and $v$.

Mathematical and Analytical Results

1. Eigenvalue Equations and Symmetry Relations: The eigenvalue equations for conformal partial waves are systematically derived using differential operators acting on conformal invariants. Significant emphasis is placed on symmetry properties under variable transformations, yielding remarkably simple expressions in specific even dimensions. These mathematical deductions hold broad applicability in simplifying the bootstrap equations within CFT analyses.
2. Integral Representations: Integral representations for conformal partial waves are presented, highlighting a comprehensive treatment for both the primary and shadow operators. The paper investigates various recursion relations leveraging Mellin-Barnes integrals, elucidating the complexity encountered with increased spin $\ell$, yet offering simplifications in special cases involving leading twist operators.
3. Recurrence Relations and Shift Operators: A detailed exploration into recursion relations is exhibited, targeting both general dimension models and special cases adhering to specified symmetries. The manuscript effectively constructs higher-order differential operators to generate finite sequences of conformal partial waves, showcasing how dimension-raising operators can facilitate deriving results for specific cases like $d = 4, 6$.
4. Harmonic Polynomials and Jacobi Extensions: The work draws parallels between conformal partial waves and harmonic polynomials, employing classical Jacobi polynomials and extending these to multiple variables. This analogy allows for insightful derivations of recurrence relations and expressions that align with compact harmonics literature.
5. Special Cases and Dimensional Analysis: An emphasis is placed on the role of conformal partial waves in lower dimensions (e.g., $d=1$ and $d=3$) where such waves simplify or reduce to known forms. The application in three dimensions is particularized through integral representations employing the Bateman transform, crucial for advancing the utility in analyzing odd-dimensional CFTs.

Implications and Future Directions

The implications of this research are multifold. The paper provides analytical tools that are pivotal for CFT practitioners attempting to leverage the conformal bootstrap approach, constraining operator dimensions and refining theoretical models. The recurrence relations and integral formulations represent a foundational advance towards handling complex computations in a theoretically grounded manner.

The adaptable formulae and methods developed here also suggest a pathway for implementing similar ideas in superconformal field theory analyses across diverse dimensions, elucidating relationships between short and long representations. The methodologies adopted promise enhanced computational precision and utility in other physical models where conformal symmetry plays a fundamental role.

Future developments could extend these results to more intricate CFTs and possibly exploring numerical implementations for higher-spin contributions, further enriching the domain of theoretical physics by providing insight into unresolved problems through the bootstrap paradigm or even guiding experimental physicists in identifying novel phenomena linked with critical exponents and scaling behaviors.

Overall, this paper stands as a substantial academic contribution, deepening both the theoretical and practical understanding of conformal field theory's mathematical backbone, enabling further explorations into the rich domain of higher-dimensional physics.