Simple tests for CP or P violation by sequential decays: $V_1$$V_2$ modes with decays into l${¯}_A$$l_B$ and/or q${¯}_A$$q_B$

In a φJ, JJ, or JΥ decay channel sin 2φ and/or sin φ terms in the azimuthal distribution would imply that P and CP separately are violated. Similarly in a φ$Z^0$, ${\mathrm{JZ}}^0$, $Z^0$ $Z^0$, or $W^{+}$ $W^{\mathrm{-}}$ decay channel sin 2φ terms would imply that CP is violated. For these latter modes there are several distinct α and γ signatures for CP violation. Analogous signatures for expected P violation occur in a charged φ$W^{\pm }$, ${\mathrm{JW}}^{\pm }$, or $Z^0$ $W^{\pm }$ decay channel. Stronger tests for violations occur when the two vector bosons are identical and the X has odd spin because for J=1 (>1) there are only two (three) independent amplitudes. For X of spin 0 many signatures and consistency checks are tabulated for (i) the violation of both P invariance and CP invariance in a φ${\rho }^0$, φφ, $K^{\mathrm{*}+}$ $K^{\mathrm{*}\mathrm{-}}$, φJ, JJ, or JΥ decay channel, for (ii) the violation of CP invariance in a φ$Z^0$, ${\mathrm{JZ}}^0$, $Z^0$ $Z^0$, or $W^{+}$ $W^{\mathrm{-}}$ decay channel, and for (iii) the expected violation of P invariance in a charged φ$W^{\pm }$, ${\mathrm{JW}}^{\pm }$, or $Z^0$ $W^{\pm }$ decay channel.