Racah Coefficients for the Group SL(2,ℝ)

The paper is devoted to the derivation of a universal integral representation for 6 j -symbols, or Racah coefficients, for the tensor product of three unitary representations of the principle series of the group SL(2, ℝ). The problem of calculating 6 j -symbols admits a natural reformulation in the language of Feynman diagrams. The original diagram can be significantly simplified and reduced to a basic diagram using the Gorishnii–Isaev method. In the case of representations of the principle series, we obtained a closed expression in the form of the Mellin–Barnes integral for the basic diagram.