The solution to the `$1/2$ vs $3/2$' puzzle

Using an almost complete relativistic method based on the Bethe-Salpeter equation, we study the mixing angle $\theta$, the mass splitting $\bigtriangleup M$, the strong decay widths $\Gamma(D^{({\prime})}_1)$ and the weak production rates $Br(B\to D^{({\prime})}_1\ell\nu_{\ell})$ of the $D_1(2420)$ and $D_1^{\prime}(2430)$. We find there is the strong cancellation between the $^1P_1$ and $^3P_1$ partial waves in $D_1^{\prime}(2430)$ with $\theta \sim-35.3^{\circ}$, which leads to the `$1/2$ vs $3/2$' puzzle. The puzzle can not be overcome by adding only relativistic corrections since in a large parameter range where $\bigtriangleup M$ is linear varying and not small, the $\theta$, $\Gamma(D^{({\prime})}_1)$ and $Br(B\to D^{({\prime})}_1\ell\nu_{\ell})$ remain almost unchanged but conflict with data. While in a special range around the mass inverse point where $\bigtriangleup M=0$ and $\theta =\pm 90^{\circ}$, they change rapidly but we find the windows where $\bigtriangleup M$, $\Gamma(D^{({\prime})}_1)$ and $Br(B\to D^{({\prime})}_1\ell\nu_{\ell})$ are all consistent with data. The small $\bigtriangleup M$ confirmed by experiment, is crucial to solve the `$1/2$ vs $3/2$' puzzle.