Total Born cross section of $e^{+}{e}^{-}$-pair production by an electron in the Coulomb field of a nucleus

We calculate the total Born cross section of the $e^{+}{e}^{-}$-pair production by an electron in the field of a nucleus (trident process) using the modern multiloop methods. For general energies we obtain the cross section in terms of converging power series. The threshold asymptotics and the high-energy asymptotics are obtained analytically. In particular, we obtain additional contribution to the Racah formula due to the identity of the final electrons. Besides, our result for the leading term of the high-energy asymptotics reveals a typo in an old Racah paper [Racah1937].