  Indirect methods have become the predominant approach in experimental nuclear
astrophysics for studying several low-energy nuclear reactions occurring in
stars, as direct measurements of many of these relevant reactions are rendered
infeasible due to their low reaction probability. Such indirect methods,
however, require theoretical input that in turn can have significant
poorly-quantified uncertainties, which can then be propagated to the reaction
rates and have a large effect on our quantitative understanding of stellar
evolution and nucleosynthesis processes. We present two such examples involving
$\alpha$-induced reactions, $^{13}$C($\alpha,n)^{16}$O and
$^{12}$C$(\alpha,\gamma)^{16}$O, for which the low-energy cross sections have
been constrained with $(^6$Li$,d)$ transfer data. In this Letter, we discuss
how a first-principle calculation of $^6$Li leads to a 21% reduction of the
$^{12}$C$(\alpha,\gamma)^{16}$O cross sections with respect to a previous
estimation. This calculation further resolves the discrepancy between recent
measurements of the $^{13}$C$(\alpha,n)^{16}$O reaction and points to the need
for improved theoretical formulations of nuclear reactions.
