Stage-two spin-correlation functions: Tests for non-CKM-type leptonicCPviolation in τ→ρν decay

There are two tests for leptonic CP violation in τ→ρν decay by inclusion of ρ polarimetry observables in the energy correlation function for ${\mathit{Z}}^0$ or ${\gamma }^{\mathrm{*}}$→${\mathrm{\tau }}^{\mathrm{-}}$ ${\mathrm{\tau }}^{+}$→(${\mathrm{\rho }}^{\mathrm{-}}$ν)(${\mathrm{\rho }}^{+}$ν¯). By CP invariance the moduli ratio of, and phase difference between, the two helicity amplitudes for ${\mathrm{\tau }}^{\mathrm{-}}$→${\mathrm{\rho }}^{\mathrm{-}}$ ${\nu }_{\mathrm{\tau }}$ decay should equal those for ${\mathrm{\tau }}^{+}$→${\mathrm{\rho }}^{+}$ν${\mathrm{¯}}_{\mathrm{\tau }}$ decay. The full angular distribution for the above process, including the ${\mathrm{\pi }}^{\mp }$ momentum direction versus that of the ${\mathrm{\rho }}^{\mp }$ momentum, can be used to test for such a non-CKM-type leptonic CP violation in τ→ρν 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 τ→ρν decay are calculated for possible application at the ${\mathit{Z}}^0$, at a B factory, or at the τ-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.