General spin analysis from angular correlations in two-body decays

Determining the spin of any new particle and measuring its couplings to other particles and/or itself are crucial in reconstructing the structure of any quantum field theory containing the particle. A general helicity formalism is employed to describe the polarization of the particle Y in a two-body decay X₂→ YX₁ with polarized X₂ for the purpose of diagnosing the dynamical properties of three involved particles and for determining their spins altogether. We perform a general and comprehensive analytic analysis with our special focus on grasping fully how to connect the decay helicity amplitudes and decay distributions in the X₂ rest frame and those in a laboratory frame with X₂ moving with a non-zero velocity through Wick helicity rotation on helicity states and amplitudes. This theoretical framework is demonstrated in a detailed illustrative manner with the Standard Model (SM) processes, the sequential process e^-e⁺→ Z→ τ^-τ⁺ followed by τ^-→ ρ^-ν_τ→ (π^-π) ν_τ and the sequential process e^-e⁺→ tt¯ followed by t→ W⁺b→ (ℓ⁺ν_ℓ) b, and one non-standard decay process of a new vector-like heavy top quark, T→ Zt, followed by Z→ ℓ^-ℓ⁺. All the useful formulas directly applicable to any combinations of spins and any types of couplings in the two-body decay X₂→ YX₁ followed by suitable Y two-body decays processes are collected and described in detail.