  At the long-wavelength approximation, $E1$ transitions are forbidden between
isospin-zero states. Hence $E1$ radiative capture is strongly hindered in
reactions involving $N = Z$ nuclei but the $E1$ astrophysical $S$ factor may
remain comparable to, or larger than, the $E2$ one. Theoretical expressions of
the isoscalar and isovector contributions to $E1$ capture are analyzed in
microscopic and three-body approaches in the context of the
$\alpha(d,\gamma)^6$Li reaction. The lowest non-vanishing terms of the
operators are derived and the dominant contributions to matrix elements are
discussed. The astrophysical $S$ factor computed with some of these
contributions in a three-body $\alpha+n+p$ model is in agreement with the
recent low-energy experimental data of the LUNA collaboration. This confirms
that a correct treatment of the isovector $E1$ transitions involving small
isospin-one admixtures in the wave functions should be able to provide an
explanation of the data without adjustable parameter. The exact-masses
prescription which is often used to avoid the disappearance of the $E1$ matrix
element in potential models is not founded at the microscopic level and should
not be used for such reactions. The importance of capture components from an
initial $S$ scattering wave is also discussed.
