  The $a_1(1420)$ with $I^G(J^{PC})= 1^-(1^{++})$ observed in the $\pi^+
f_0(980)$ final state in the $\pi^-p\to \pi^+\pi^-\pi^- p$ process by the
COMPASS collaboration seems unlikely to be an ordinary $\bar qq$ mesonic state.
Available theoretical explanations include tetraquark or rescattering effects
due to $a_1(1260)$ decays. If the $a_1(1420)$ were induced by the rescattering,
its production rates are completely determined by those of the $a_1(1260)$. In
this work, we propose to explore the ratios of branching fractions of heavy
meson weak decays into the $a_1(1420)$ and $a_1(1260)$, and testing the
universality of these ratios would be a straightforward way to
validate/invalidate the rescattering explanation. The decay modes include in
the charm sector the $D^0\to a_1^-\ell^+\nu$ and $D^0\to \pi^\pm a_1^\mp$, and
in the bottom sector $\overline B^0\to a_1^+ \ell^- \bar\nu$, $B\to D a_1,
\pi^\pm a_1^\mp$, $B_c\to J/\psi a_1$ and $\Lambda_b\to \Lambda_c a_1$. We
calculate the branching ratios for various decay modes into the $a_1(1260)$.
The numerical results indicate that there is a promising prospect to study
these decays on experiments including BES-III, LHCb, Babar, Belle and CLEO-c,
the forthcoming Super-KEKB factory and the under-design Circular
Electron-Positron Collider. Experimental analyses in future will lead to a
deeper understanding of the nature of the $a_1(1420)$.
