Covariant canonical-spinor amplitudes for partial wave analysis

Abstract

We propose a covariant orbital-spin ($LS$) decomposed amplitude for the partial wave analysis using the massive spinor-helicity formalism. First we review the traditional-$LS$ method in the little group space and the Zemach tensor method in the double cover of the $\mathrm{SO}(3)$ space. To recover the $\mathrm{SO}(3,1)$ Lorentz covariance, several Lorentz covariant $LS$ tensors have been constructed in several different methods: covariant tensor, covariant projection tensor in pure-spin and general-spin schemes, but performing a intrinsic separation between $LS$ coupling while maintaining covariance is not obvious. We utilize the massive canonical-spinor variables to determine general three-point amplitudes, in which the spin-orbital decomposition is realized in single little group space by projecting little group indices of each particles into one, while the Lorentz covariance is ensured by the spinor form naturally. This covariant spinor method allows direct evaluation in any frame and a streamlined treatment of cascade decays within a single frame without additional alignment rotations in non-covariant treatment. As a benchmark, we implement the method in TF-PWA and analyze $Λ_c^+\toΛπ^+π^0$, finding consistent fit results across the helicity, traditional-$LS$, and canonical-spinor amplitudes. This validates the canonical-spinor amplitude as a practical tool for modern partial wave analyses of complex decay chains.

Figures (9)

• Figure 1: Canonical-standard boost for a spin-$j$ particle. The blue arrow indicates the spin polarization direction and the red arrow indicates the momentum direction.
• Figure 2: Helicity-standard boost for a spin-$j$ particle. The blue arrow indicates the spin polarization direction and the red arrow indicates the momentum direction.
• Figure 3: Rotation acting on a canonical single-particle state. The brown line indicates the spin quantization axis, the red line represents the momentum direction of the particle, and the blue line denotes the spin polarization direction of the particle.
• Figure 4: Rotation acting on a helicity single-particle state. The red line indicates the momentum direction of the particle, which is also the spin quantization axis, and the blue line denotes the direction of the particle's spin polarization.
• Figure 5: Two boost paths from the lab frame to the rest frame of particle-$i$: a direct boost $L_{i,c}^{-1}(\textbf{p}_i)$ and the two-step boost $L_{i,c}^{-1}(\mathbf{p}_i^\ast)L_{3,c}^{-1}(\mathbf{p}_3)$. Their difference is the Wigner rotation acting on the little group indices of particle-$i$.