The solution to the ‘1/2 vs 3/2’ puzzle

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