Stage-two spin-correlation functions: Tests for non-CKM-type leptonic<i>CP</i>violation in τ→ρν decay

Abstract

There are two tests for leptonic CP violation in \ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\nu} decay by inclusion of \ensuremath{\rho} polarimetry observables in the energy correlation function for ${\mathit{Z}}^{0}$ or ${\ensuremath{\gamma}}^{\mathrm{*}}$\ensuremath{\rightarrow}${\mathrm{\ensuremath{\tau}}}^{\mathrm{\ensuremath{-}}}$${\mathrm{\ensuremath{\tau}}}^{+}$\ensuremath{\rightarrow}(${\mathrm{\ensuremath{\rho}}}^{\mathrm{\ensuremath{-}}}$\ensuremath{\nu})(${\mathrm{\ensuremath{\rho}}}^{+}$\ensuremath{\nu}\ifmmode\bar\else\textasciimacron\fi{}). By CP invariance the moduli ratio of, and phase difference between, the two helicity amplitudes for ${\mathrm{\ensuremath{\tau}}}^{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}${\mathrm{\ensuremath{\rho}}}^{\mathrm{\ensuremath{-}}}$${\ensuremath{\nu}}_{\mathrm{\ensuremath{\tau}}}$ decay should equal those for ${\mathrm{\ensuremath{\tau}}}^{+}$\ensuremath{\rightarrow}${\mathrm{\ensuremath{\rho}}}^{+}$\ensuremath{\nu}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathrm{\ensuremath{\tau}}}$ decay. The full angular distribution for the above process, including the ${\mathrm{\ensuremath{\pi}}}^{\ensuremath{\mp}}$ momentum direction versus that of the ${\mathrm{\ensuremath{\rho}}}^{\ensuremath{\mp}}$ momentum, can be used to test for such a non-CKM-type leptonic CP violation in \ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\nu} decay. Since this adds on spin-correlation information from the next stage of decays in the decay sequence, we call such an energy-angular distribution a stage-two spin-correlation (S2SC) function. Ideal statistical errors for \ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\nu} decay are calculated for possible application at the ${\mathit{Z}}^{0}$, at a B factory, or at the \ensuremath{\tau}-charm threshold. S2SC functions should be useful for testing for possible non-CKM-type CP violation in top quark and in W boson decay processes.