E2 ¯ − Parametrization of SL(2, C)

We consider the $\overline{\text{E2}}-\mathrm{parametrization}$ of unimodular 2 × 2 matrices A ∈ SL(2, C), which is of the form $${\mathrm{A=E}}_1{e}^{-\frac{1}{2}{\mathrm{a\sigma }}_3}{\mathrm{VE}}_2\mathrm{,\hspace{0.17em}\hspace{0.17em}}\mathrm{with}{\mathrm{\hspace{0.17em}\hspace{0.17em}V=e}}^{\frac{1}{2}{\mathrm{i\pi \sigma }}_2}\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{and}\mathrm{\hspace{0.17em}\hspace{0.17em}E=}\left(\begin{array}{cc}{e}^{-\frac{1}{2}\mathrm{i\phi }}& 0\\ z& e^{\frac{1}{2}\mathrm{i\phi }}\end{array}\right)\in \overline{\text{E2}}$$ . $\overline{\text{E2}}$ is a covering of the group of Euclidean motions in the plane. We compute the correspondingly factorized matrix elements of the unitary representations of SL(2, C) in an $\overline{\text{E2}}-\mathrm{basis}$ the result is given in Eq. (6). As a fringe benefit we obtain an integral transform which amounts to expansion in terms of Meijer G‐functions and which generalizes the familiar Hankel transform. The results of this paper are useful, e.g., for computing vertex functions in the theory of massless particles with continuous spin.