The $6j$-symbols for the $SL(2,\mathbb C)$ group

We study $6j$-symbols or Racah coefficients for the tensor products of infinite-dimensional unitary principal series representations of the group $SL(2,\mathbb C)$. Using the Feynman diagram technique, we reproduce the results of Ismagilov in constructing these symbols (up to a slight difference associated with equivalent representations). The resulting $6j$-symbols are expressed either as a triple integral over complex plane or as an infinite bilateral sum of integrals of the Mellin–Barnes type.