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

Abstract

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

Comments: 18 pages, 5 figures