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 θ, the mass splitting △ M, the strong decay widths Γ(D1(′)) and the weak production rates Br(B→D1(′)ℓνℓ) of the D₁(2420) and D1′(2430). We find there is the strong cancellation between the ¹P₁ and ³P₁ partial waves in D1′(2430) with 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 △ M is linear varying and not small, the θ, Γ(D1(′)) and Br(B→D1(′)ℓνℓ) remain almost unchanged but conflict with data. While in a special range around the mass inverse point where △ M= 0 and θ=±90∘, they change rapidly but we find the windows where △ M, Γ(D1(′)) and Br(B→D1(′)ℓνℓ) are all consistent with data. The small △ M confirmed by experiment, is crucial to solve the ‘1/2 vs 3/2’ puzzle.