Standard Model and Beyond · Problem 2.26

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · lambda-hyperon-decay

Problem

The Λ\Lambda is a heavy spin-12\frac{1}{2} hyperon that decays into pπ−p\pi^- via the non-leptonic weak interactions. The decay interaction can be modeled by LI=ψˉp(gS−gPγ5)ψΛϕπ++h.c.,\mathcal{L}_I = \bar{\psi}_p(g_S - g_P\gamma^5)\psi_\Lambda\phi_{\pi^+} + h.c., where gSg_S and gPg_P are complex constants that lead, respectively, to SS and PP-wave final states. (a) Calculate the width Γ\Gamma and the differential width dΓ/dcos⁡θd\Gamma/d\cos\theta in the Λ\Lambda rest frame for a polarized Λ\Lambda, where θ\theta is the angle between s^Λ\hat{s}_\Lambda and the proton momentum p⃗p\vec{p}_p. Use trace techniques. (b) Show that dΓ/dcos⁡θd\Gamma/d\cos\theta is not reflection invariant for Re(gPgS∗)≠0\mathfrak{Re}(g_P g_S^*) \neq 0, i.e., that it is not invariant under p⃗p→−p⃗p\vec{p}_p \to -\vec{p}_p, s^Λ→s^Λ\hat{s}_\Lambda \to \hat{s}_\Lambda. (c) Repeat (a), but use explicit expressions for the Λ\Lambda and pp spinors in the Pauli-Dirac representation. Justify the claim that gSg_S and gPg_P generate SS and PP-wave amplitudes.

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