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

Abstract

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