Racah coefficients for the group $\mathrm{SL}(2,\mathbb{R})$

The paper is devoted to the derivation of a universal integral representation for $6j$-symbols, or Racah coefficients, for the tensor product of three unitary representations of the main series of the group $\mathrm{SL}(2,\mathbb{R})$. The problem of calculating $6j$-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 main series, a closed expression in the form of the Mellin–Barnes integral is obtained for the basic diagram.