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Conformal Partial Waves: Further Mathematical Results

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.