  We study the SL(2,C) Clebsch-Gordan coefficients appearing in the lorentzian
EPRL spin foam amplitudes for loop quantum gravity. We show how the amplitudes
decompose into SU(2) nj-symbols at the vertices and integrals over boosts at
the edges. The integrals define edge amplitudes that can be evaluated
analytically using and adapting results in the literature, leading to a pure
state sum model formulation. This procedure introduces virtual representations
which, in a manner reminiscent to virtual momenta in Feynman amplitudes, are
off-shell of the simplicity constraints present in the theory, but with the
integrands that peak at the on-shell values. We point out some properties of
the edge amplitudes which are helpful for numerical and analytical evaluations
of spin foam amplitudes, and suggest among other things a simpler model useful
for calculations of certain lowest order amplitudes. As an application, we
estimate the large spin scaling behaviour of the simpler model, on a closed
foam with all 4-valent edges and Euler characteristic X, to be N^(X - 5E +
V/2). The paper contains a review and an extension of results on SL(2,C)
Clebsch-Gordan coefficients among unitary representations of the principal
series that can be useful beyond their application to quantum gravity
considered here.
