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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 theCOMPASS collaboration seems unlikely to be an ordinary $\bar qq$ mesonic state.Available theoretical explanations include tetraquark or rescattering effectsdue 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 $a_1(1260)$. Inthis work, we propose to explore the ratios of branching fractions of heavymeson weak decays into the $a_1(1420)$ and $a_1(1260)$, and testing theuniversality of these ratios would be a straightforward way tovalidate/invalidate the rescattering explanation. The decay modes include inthe charm sector the $D^0\to a_1^-\ell^+\nu$ and $D^0\to \pi^\pm a_1^\mp$, andin 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 $\Lambda_b\to \Lambda_c a_1$. Wecalculate the branching ratios for various decay modes into the $a_1(1260)$.The numerical results indicate that there is a promising prospect to studythese decays on experiments including BES-III, LHCb, Babar, Belle and CLEO-c,the forthcoming Super-KEKB factory and the under-design CircularElectron-Positron Collider. Experimental analyses in future will lead to adeeper understanding of the nature of the $a_1(1420)$.

$a_1(1260), a_1(1420)$ and the production in heavy meson decays