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

Guo-Li Wang; Qiang Li; Tianhong Wang; Tai-Fu Feng; Xing-Gang Wu; Chao-Hsi Chang

doi:10.1140/epjc/s10052-022-10997-4

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2022) 82: 1027
https://doi.org/10.1140/epjc/s10052-022-10997-4

Regular Article - Theoretical Physics

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

Guo-Li Wang¹^,2^a, Qiang Li³, Tianhong Wang⁴, Tai-Fu Feng¹^,2, Xing-Gang Wu⁵ and Chao-Hsi Chang⁶^,7

¹ Department of Physics, Hebei University, 071002, Baoding, China
² Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province, 071002, Baoding, China
³ School of Physical Science and Technology, Northwestern Polytechnical University, 710072, Xi’an, China
⁴ School of Physics, Harbin Institute of Technology, 150001, Harbin, China
⁵ Department of Physics, Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, 401331, Chongqing, China
⁶ Institute of Theoretical Physics, Chinese Academy of Science, 100190, Beijing, China
⁷ CCAST (World Laboratory), P.O. Box 8730, 100190, Beijing, China

^a wgl@hbu.edu.cn

Received: 4 October 2022
Accepted: 1 November 2022
Published online: 15 November 2022

Abstract

Using an almost complete relativistic method based on the Bethe–Salpeter equation, we study the mixing angle $θ$ $\theta$ , the mass splitting $△ M$ $\bigtriangleup M$ , the strong decay widths $Γ (D_{1}^{(')})$ $\Gamma (D^{({\prime })}_1)$ and the weak production rates $B r (B \to D_{1}^{(')} ℓ ν_{ℓ})$ $Br(B\rightarrow D^{({\prime })}_1\ell \nu _{\ell })$ of the $D_{1} (2420)$ $$D_1(2420)$$ and $D_{1}^{'} (2430)$ $D_1^{\prime }(2430)$ . We find there is the strong cancellation between the $^{1} P_{1}$ $$^1P_1$$ and $^{3} P_{1}$ $$^3P_1$$ partial waves in $D_{1}^{'} (2430)$ $D_1^{\prime }(2430)$ with $θ \sim - 35 . 3^{\circ}$ $\theta \sim -\,35.3^{\circ }$ , 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 $△ M$ $\bigtriangleup M$ is linear varying and not small, the $θ$ $\theta$ , $Γ (D_{1}^{(')})$ $\Gamma (D^{({\prime })}_1)$ and $B r (B \to D_{1}^{(')} ℓ ν_{ℓ})$ $Br(B\rightarrow D^{({\prime })}_1\ell \nu _{\ell })$ remain almost unchanged but conflict with data. While in a special range around the mass inverse point where $△ M = 0$ $\bigtriangleup M=0$ and $θ = \pm 90^{\circ}$ $\theta =\pm \, 90^{\circ }$ , they change rapidly but we find the windows where $△ M$ $\bigtriangleup M$ , $Γ (D_{1}^{(')})$ $\Gamma (D^{({\prime })}_1)$ and $B r (B \to D_{1}^{(')} ℓ ν_{ℓ})$ $Br(B\rightarrow D^{({\prime })}_1\ell \nu _{\ell })$ are all consistent with data. The small $△ M$ $\bigtriangleup M$ confirmed by experiment, is crucial to solve the ‘1/2 vs 3/2’ puzzle.

© The Author(s) 2022

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