$T$-odd correlations in polarized top quark decays in the sequential decay $t(uparrow) to X_b+W^+(to ell^+ + u_ell)$ and in the quasi three-body decay $t(uparrow) to X_b+ ell^+ + u_ell$

We identify the $T$-odd structure functions that appear in the description of polarized top quark decays in the sequential decay $t(uparrow) to X_b+W^+(to ell^+ + u_ell)$ (two structure functions) and the quasi-three-body decay $t(uparrow) to X_b+ ell^+ + u_ell$ (one structure function). A convenient measure of the magnitude of the $T$-odd structure functions is the contribution of the imaginary part Im $g_R$ of the right-chiral tensor coupling $g_R$ to the $T$-odd structure functions which we work out. Contrary to the case of QCD the NLO electroweak corrections to polarized top quark decays admit of absorptive one-loop vertex contributions. We analytically calculate the imaginary parts of the relevant four electroweak one-loop triangle vertex diagrams and determine their contributions to the $T$-odd helicity structure functions that appear in the description of polarized top quark decays.