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% *** Marcus Mauser and Juergen Koerner                                    *** %
% *** O(alpha_s) radiative corrections to                                  *** %
% *** polarized top decay into a charged Higgs                             *** %
% *** Johannes Gutenberg University, 2002                                  *** %
% *** version: 3         date: 01.11.2002                                  *** %
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\begin{document}
\thispagestyle{empty}
\begin{flushright}
        MZ-TH/02-14\\
        \\
        November 2002\\
\end{flushright}
\vspace{0.5cm}
\begin{center}
 {\Large\pf $ O(\alpha_s) $}
 {\Large\bf radiative corrections to polarized top decay into a charged Higgs }
 {\Large\pf $ t(\uparrow) \rightarrow H^+ + b $}\\[7mm]
 {\large J.G.~K\"orner and M.C.~Mauser}\\[13mm]
         Institut f\"ur Physik, Johannes Gutenberg-Universit\"at\\[2mm]
         Staudinger Weg 7, D-55099 Mainz, Germany\\[25mm]
\end{center}

% **************************************************************************** %
% *** abstract                                                             *** %
% **************************************************************************** %

\begin{abstract}
 \noindent We calculate the $ O(\alpha_s) $ radiative corrections to
 polarized top quark decay into a charged Higgs and a bottom quark in
 Two-Higgs-Doublet-Models for nonzero bottom quark masses $ m_b \neq 0 $.
 The radiative corrections to the polarization asymmetry parameter
 of the decay can become as large as $ 25 \, \% $.
 We provide analytical formulae for the unpolarized and
 polarized rates for $ m_b \neq 0 $ and for $ m_b = 0 $.
 For $ m_b = 0 $ our closed-form expressions for the
 unpolarized and polarized rates become rather compact.
\end{abstract}

\newpage

% **************************************************************************** %
% *** introduction                                                         *** %
% **************************************************************************** %

\section{\bf Introduction}

 The purpose of this paper is the evaluation of the first order QCD corrections
 to the decay of a polarized top quark into a charged Higgs boson and a bottom
 quark. Highly polarized top quarks will become available at hadron colliders
 through single top production which occurs at the $ 33 \, \% $ level of the
 top quark pair production rate~\cite{MaPa97,EsMa02}.
 Future $ e^{+} e^{-} $ -- colliders will also be copious sources of polarized
 top quark pairs. The polarization of these can be easily tuned through the
 availability of polarized beams~\cite{PaSh96,KoMePr02}.
 Measurements of the decay rate and the asymmetry parameter in the decay
 $ t(\uparrow) \rightarrow H^{+} + b $ will be important for future tests
 of the Higgs coupling in the minimal supersymmetric standard model (MSSM).

 The $ O(\alpha_s) $ corrections to the decay rate $ t \rightarrow  H^{+} + b $
 have been calculated previously in Refs.~\cite{Czar93(1),Czar93(2),Lial92} and
 in Refs.~\cite{LiYa92,RTLS91,LiYu90,LiYa90}, and have been found to be
 important.
 The present paper provides the first calculation of the $ O(\alpha_s) $
 radiative corrections to the asymmetry parameter in polarized top decay
 $ t(\uparrow) \rightarrow H^{+} + b $.
 Depending on the value of $ \tan \beta $ the radiative corrections to the
 asymmetry parameter can become quite large (up to $ 25 \, \% $) and must
 therefore be included in a decay analysis.

 The decay $ t \rightarrow H^{+} + b $ is analyzed in
 the rest frame of the top quark (see Fig.~1).
 The $ 3 $--momentum $ \vec{q} $ of the $ H^+ $ boson
 points into the positive $ z $--axis.
 The polar angle $ \theta_P $ is defined as the angle
 between the polarization vector $ \vec{P} $ of the
 $ t $ quark and the $ z $--axis.

 In terms of the unpolarized rate $ \Gamma $ and the polarized
 rate $ \Gamma^P $ the differential decay rate is given by

 \begin{equation} % *** differential decay rate *** %
  \label{difrate} % *** Eq.(1) *** %
  \frac{d \Gamma}{d \! \cos \theta_P} =
  \frac{1}{2} \, \Big(\Gamma + \mbox{P} \, \Gamma^P \cos \theta_P \Big) =
  \frac{1}{2} \, \Gamma \Big(1 + \mbox{P} \alpha_H \cos \theta_P \Big),
 \end{equation}

 \noindent where the asymmetry parameter $ \alpha_H $ is
 defined by $ \alpha_H = \Gamma^P / \Gamma $.
 Alternatively one can define a forward-backward asymmetry
 $ A_{F \! B} $ by writing

 \begin{equation} % *** forward-backward asymmetry *** %
  \label{asymm}   % *** Eq.(2) *** %
   A_{F \! B} = \frac{\Gamma_F - \Gamma_B}{\Gamma_F + \Gamma_B},
 \end{equation}

 \noindent where $ \Gamma_F $ and $ \Gamma_B $ are the
 rates into the forward and backward hemispheres, respectively.
 The two measures are related by
 $ A_{F \! B} = \frac{1}{2} \mbox{P} \alpha_H $.
 In our numerical results we shall always
 set $ \mbox{P} = 1 $ for simplicity.

 Technical details of our calculation can be found in \cite{FGKM02}.
 As in \cite{FGKM02} we use dimensional regularization
 ($ D = 4 \!-\! 2 \, \varepsilon $ with $ \varepsilon \ll 1 $) to
 regularize the ultraviolet divergences of the virtual corrections.
 We regularize the infrared divergences in the virtual one--loop
 corrections by introducing a finite (small) gluon mass $ m_g \neq 0 $
 in the gluon propagator.
 In the tree graph corrections the phase space boundary becomes deformed
 away from the IR singular point through the introduction of a (small)
 gluon mass.
 The logarithmic gluon mass dependence resulting from the regularization
 procedure cancels out when adding the virtual and tree--graph
 contributions.
 We have checked consistency with the Goldstone boson equivalence
 theorem which, in the limit $ m_{W^+} / m_t \rightarrow 0 $ and
 $ m_{H^+} / m_t \rightarrow 0 $ relates $ \Gamma $ and $ \Gamma^P $
 for $ t \rightarrow H^{+} +b $ to the unpolarized and polarized
 longitudinal rates $ \Gamma_L $ and $ \Gamma_L^P $ in the decay
 $ t \rightarrow W^{+} + b $ calculated in \cite{FGKM02}.

% **************************************************************************** %
% *** Born term results                                                    *** %
% **************************************************************************** %

\section{Born term results}

 The coupling of the charged Higgs boson to the top and bottom quark in the
 MSSM can either be expressed as a superposition of scalar and pseudoscalar
 coupling factors or as a superposition of right-- and left--chiral coupling
 factors. The Born term amplitude is thus given by

 \begin{equation} % *** born amplitude, Eq.(3) *** %
  {\cal M}_0 = \bar{u}_b (a \1 + b \gamma_5) u_t =
  \bar{u}_b \left\{
  g_t \frac{\1 + \gamma_5}{2} +
  g_b \frac{\1 - \gamma_5}{2} \right\} u_t,
 \end{equation}

 \noindent where $ a \!=\! \frac{1}{2} (g_t + g_b) $ and
 $ b \!=\! \frac{1}{2} (g_t - g_b) $.
 The inverse relation reads $ g_t = a + b $ and $ g_b = a - b $.

 In the MSSM with its two Higgs doublets one of the charged Higgs
 bosons is real and the top quark can decay to $ (H^{+} + b) $
 provided $ m_t > m_{H^+} + m_b $.
 In order to avoid flavor changing neutral currents (FCNC) the
 generic Higgs coupling to all quarks has to be restricted.
 In the notation of \cite{GuHaKaDa90} in model~1 the doublet
 $ H_1 $ couples to all bosons and the doublet $ H_2 $
 couples to all quarks. This leads to the coupling factors

 \alpheqn
 \begin{Eqnarray} % *** model 1, Eqs.(4a,b) *** %
  \fbox{model 1:} & &
  a = \frac{g_w}{2 \sqrt{2} m_W}
  V_{tb}(m_t - m_b) \cot \beta, \\[3mm] & &
  b = \frac{g_w}{2 \sqrt{2} m_W}
  V_{tb}(m_t + m_b) \cot \beta,
 \end{Eqnarray}
 \reseteqn

 \noindent where $ V_{tb} $ is the CKM-matrix element and
 $ \tan \beta = v_2 / v_1 $ is the ratio of the vacuum expectation values
 of the two electrically neutral components of the two Higgs doublets.
 The weak coupling factor $ g_w $ is related to the usual Fermi coupling
 constant $ G_F $ by $ g_w^2 = 4 \sqrt{2} m _W^2 G_F $.

 In model~2, the doublet $ H_1 $ couples to the right--chiral
 down--type quarks and the doublet $ H_2 $ couples to the
 right--chiral up--type quarks. Model~2 leads to the coupling factors

 \alpheqn
 \begin{Eqnarray} % *** model 2, Eqs.(5a,b) *** %
  \fbox{model 2:} & &
  a = \frac{g_w}{2 \sqrt{2} m_W}
  V_{tb}(m_t \cot \beta + m_b \tan \beta), \\[3mm] & &
  b = \frac{g_w}{2 \sqrt{2} m_W}
  V_{tb}(m_t \cot \beta - m_b \tan \beta).
 \end{Eqnarray}
 \reseteqn

 Since $ m_b \ll m_t $ the $ b $ mass can always be safely neglected in model~1.
 In model~2 the left--chiral coupling term proportional to $ m_b \tan \beta $
 can become comparable to the right--chiral coupling term $ m_t \cot \beta $
 when $ \tan \beta $ becomes large.
 One cannot therefore naively set $ m_b = 0 $ in all expressions.
 One has to distinguish between the cases when the scale of the
 $ b $ mass is set by $ m_t \cot \beta $ and when the scale of
 the $ b $ mass is set by $ m_t $.
 In the former case one has to keep the $ b $ mass finite.
 In the latter case the $ b $ mass can safely be set to zero
 except for logarithmic terms proportional to $ \ln (m_b / m_t) $
 that appear in the NLO calculation.
 Keeping this distinction results in very compact closed form
 expressions for the radiatively corrected unpolarized and polarized
 rates which we shall list in the main text.
 Our numerical results are calculated from these closed form expressions
 where $ m_b $ has been set to zero with the above two provisions.
 We shall, however, also present general $ m_b \ne 0 $
 results along with the $ m_b = 0 $ results.
 It is quite evident that one has to use the  $ m_b \ne 0 $ expressions
 for charged Higgs masses close to the $ W $--boson mass.

 For the unpolarized and polarized Born term rates one obtains

 \alpheqn
 \begin{Eqnarray} % *** decay rates at born level *** %
  \label{born}    % *** Eqs.(6a,b) *** %
  \Gamma_{Born} & = &
  \frac{\sqrt{\lambda}}{16 \pi m_t^3} \Big\{
  (a^2 + b^2) \eta + 2 (a^2 - b^2) m_b m_t \Big\}, \\[3mm]
  \Gamma_{Born}^P & = &
  \frac{\sqrt{\lambda}}{16 \pi m_t^3} \Big\{
  2 a b \sqrt{\lambda} \Big\},
 \end{Eqnarray}
 \reseteqn

 \noindent where we have used the abbreviations
 $ \lambda = \lambda(m_t^2, m_{H^+}^2, m_b^2) $
 for the K\"all\'en function
 $ \lambda(a,b,c) = a^2 + b^2 + c^2 - 2 (a \, b + b \, c + c \, a) $
 and $ \eta = m_t^2 \!-\! m_{H^+}^2 \!+\! m_b^2 $.
 The factor $ \mbox{PS}_2 = (\sqrt{\lambda})/(16 \pi m_t^3) $
 is a two-body phase space factor.
 In the following we shall use the abbreviations
 $ x = m_{H^+} / m_t $ and $ y = m_b / m_t $.

 In the $ m_b \rightarrow 0 $ case the unpolarized
 and polarized Born term rates simplify to

 \alpheqn
 \begin{Eqnarray} % *** decay rates at born level *** %
  \label{bornzerobmass} % *** Eqs.(7a,b) *** %
  \Gamma_{Born}(m_b \rightarrow 0) & = &
  (a^2 + b^2) \big( 1 + \frac{a^2 - b^2}{a^2 + b^2}
  \frac{2 y}{1 - x^2} \big) \, \hat{\Gamma}, \\[2mm]
 \Gamma_{Born}^P(m_b \rightarrow 0) & = & 2  a  b \, \hat{\Gamma},
 \end{Eqnarray}
 \reseteqn

 \noindent where $ \hat{\Gamma} = m_t(1 - x^2)^2/(16 \pi) $.
 The asymmetry parameter is then simply given by

 \begin{equation} % *** asymmetry parameter *** %
 \label{asym}     % *** Eq.(8) *** %
  \alpha_H(m_b \rightarrow 0) =
  \frac{\Gamma^P_{Born}(m_b=0)}{\Gamma_{Born}(m_b=0)} =
  \frac{2 a b}{a^2 + b^2}
  \Bigg( 1 + \frac{a^2 - b^2}{a^2 + b^2} \frac{2 y}{1 - x^2} \Bigg)^{-1}.
 \end{equation}
 
 \noindent As was mentioned before, the $ y $--dependent terms in 
 Eqs.~(\ref{bornzerobmass}) and (\ref{asym}) can be safely dropped in
 model~1 but not in model~2.

 In model~1 the asymmetry parameter $ \alpha_H $ is
 independent of $ \tan \beta $ and the value of the Higgs mass for $ m_b = 0 $.
 It attains its maximal value $ \alpha_H = 1 $ for $ m_b \rightarrow 0 $
 and, as Fig.~2 shows, stays very close to its maximal value for nonzero bottom
 masses except for the endpoint region when $ m_{H^+} $ approaches $ m_t $.
 Contrary to this, in model~2 the asymmetry parameter $ \alpha_H $ shows a
 strong dependence on the value of $ \tan \beta $ as shown in Fig.~3b.
 The main qualitative features of the behavious of $ \alpha_H $
 in model~2 can again be extracted from the simple $ m_b \rightarrow 0 $
 formula Eq.~(\ref{asym}).
 The asymmetry parameter $ \alpha_H $ is positive/negative
 for small/large values of $ \tan \beta $ and goes through
 zero for $ \tan \beta = \sqrt{m_t / m_b} $.
 The exact numerical analysis of the unpolarized and polarized Born term
 rates and the asymmetry parameter will be deferred to Sec.~5 where they
 are discussed together with their radiatively corrected counterparts.

% **************************************************************************** %
% *** virtual corrections                                                  *** %
% **************************************************************************** %

\section{Virtual corrections}

 The virtual one--loop corrections to the
 $ (t \, H^+ \, b) $--vertex due to one--gluon
 exchange between the quark legs lead to
 IR-- and UV--singularities.
 As mentioned before, the UV--singularities are
 regularized in $ D = 4 \!-\! 2 \, \epsilon $ dimensions.
 The IR--singularities are regularized by a small
 gluon mass $ m_g $ in the gluon propagator.
 The renormalization of the UV--singularities
 is done in the ``on--shell'' scheme.

 We begin our discussion by describing the renormalization
 of the virtual one--loop vertex for the $ m_b \ne 0 $ case.
 We then take the $ m_b \rightarrow 0 $ limit of the
 renormalized $ m_b \ne 0 $ vertex which results in simple
 compact expressions which are listed at the end of this section.

 The renormalized amplitude of the virtual corrections in
 the right-- and left--chiral representation can be written as

 \begin{equation} % *** loop amplitude, Eq.(9) *** %
  {\cal M}_{loop} = \bar{u}_b \Big\{
  \Big( a \1 + b \gamma_5  \Big) \Lambda_1 +
  a \Lambda_2 + \delta \Lambda \Big\} u_t,
 \end{equation}

 \noindent where the functions $ \Lambda_1 $ and $ \Lambda_2 $ are given by

 \alpheqn
 \begin{Eqnarray} % *** structure functions, Eqs.(10a,b) *** %
  \Lambda_1 & = & \frac{\alpha_s}{2 \pi} C_F \Big\{ \eta \, C_0 +
  (\eta + m_t (m_t + m_b)) \, C_1 +
  (\eta + m_b (m_t + m_b)) \, C_2 + \\[3mm] & + &
  2 m_t^2 C_{11} + 2 m_b^2 C_{22} + \nonumber
  2 \eta \, C_{12} + 8 \, C_{00} - 2 \Big\}, \\[3mm]
  \Lambda_2 & = & \frac{\alpha_s}{\pi} C_F \, \Big\{ \!\! -
  m_b m_t (C_1 + C_2) \Big\}.
 \end{Eqnarray}
 \reseteqn

 \noindent The standard one--loop integrals $ C_0 $, $ C_i $
 and $ C_{ij} $ are defined e.g. in Sec.~4 of Ref.~\cite{Denn93}.
 After a linear transformation these loop functions can be seen
 to agree with Eqs.~(8) and (9) in Ref.~\cite{Lial92}.
 Following Refs.~\cite{Czar93(2),Lial92,LiYu90,BrLe80}
 the counter term of the vertex is given by

 \begin{Eqnarray} % *** counter term, Eq.(11) *** %
  \delta \Lambda & = &
  ( a + b ) \frac{\1 + \gamma_5}{2} \Bigg(
  \frac{1}{2} (Z_2^t - 1) + \frac{1}{2} (Z_2^b - 1) -
  \frac{\delta m_t}{m_t} \Bigg) +
  \\[2mm] & + & \nonumber
  ( a - b ) \frac{\1 - \gamma_5}{2} \Bigg(
  \frac{1}{2} (Z_2^t - 1) + \frac{1}{2} (Z_2^b - 1) -
  \frac{\delta m_b}{m_b} \Bigg).
 \end{Eqnarray}

 In the ``on--shell'' scheme the wavefunction renormalization constant
 $ Z_2^q $ and the mass renormalization constant $ \delta m_q $ can be
 calculated from the renormalized QCD self-energy $ \Sigma_q (p) $ of
 the quarks $ q = t,b $.
 The evaluation of the two conditions $ \Sigma_q (p) |_{p \!\!\!/ = m_q} = 0 $
 and $ \partial \Sigma_q (p) / \partial \slp |_{p \!\!\!/ = m_q} = 0 $ lead to
 the following renormalization constants

 \alpheqn
 \begin{Eqnarray} % *** renormalization constants, Eqs.(12a,b) *** %
  Z_2^q & = &
  1 - \frac{\alpha_s}{4 \pi} C_F \Bigg\{
  \frac{1}{\varepsilon} - \gamma_E +
  \ln \frac{4 \pi \mu^2}{m_q^2} +
  2 \ln \frac{m_g^2}{m_q^2} + 4 \Bigg\}, \\[2mm]
  \delta m_q & = &
  \frac{\alpha_s}{4 \pi} C_F \, m_q \, \Bigg\{
  \frac{3}{\varepsilon} - 3 \gamma_E +
  3 \ln \frac{4 \pi \mu^2}{m_q^2} + 4 \Bigg\},
 \end{Eqnarray}
 \reseteqn

 \noindent where $ p $ is the 4--momentum and $ m_q $
 is the mass of the relevant quark.

 Putting everything together the virtual one--loop
 contributions to the unpolarized and polarized rates read

 \alpheqn
 \begin{Eqnarray} % *** loop contribution *** %
  \label{GloopA}  % *** Eqs.(13a,b) *** %
  \Gamma_{loop} & = &
  \Gamma_{Born} \Bigg( Z_2^t - 1 + Z_2^b - 1 -
  \frac{\delta m_t}{m_t} - \frac{\delta m_b}{m_b} +
  2 \Lambda_1 \Bigg) + \\[2mm] & + &
  \frac{\sqrt{\lambda}}{16 \pi m_t^3} \nonumber \Bigg(
  2 \, a^2 \, \Lambda_2 (\eta + 2 m_b m_t) - 2ab \, \eta
  \Bigg( \frac{\delta m_t}{m_t} -
  \frac{\delta m_b}{m_b} \Bigg) \Bigg), \\[2mm]
  \label{GloopB}
  \Gamma_{loop}^{P} & = &
  \Gamma_{Born}^P \Bigg( Z_2^t - 1 + Z_2^b - 1 -
  \frac{\delta m_t}{m_t} - \frac{\delta m_b}{m_b} +
  2 \Lambda_1 \Bigg) + \\[2mm] & + &
  \frac{\sqrt{\lambda}}{16 \pi m_t^3} \nonumber \Bigg(
  2ab \, \Lambda_2 \sqrt{\lambda} - (a^2 + b^2) \, \sqrt{\lambda}
  \Bigg( \frac{\delta m_t}{m_t} -
  \frac{\delta m_b}{m_b} \Bigg) \Bigg).
 \end{Eqnarray}
 \reseteqn

 The renormalized virtual one--loop correction to the unpolarized rate
 (\ref{GloopA}) is in agreement with Ref.~\cite{Lial92}.
 The result for the virtual one--loop correction to the polarized rate is new.
 The infrared divergent terms residing in the renormalization factor $ Z_2^q $
 and in the integral term $ C_0 $ in $ \Lambda_1 $ are proportional to the
 Born term rates $ \Gamma_{Born} $ and $ \Gamma_{Born}^P $, respectively.

 We now proceed to take the limit $ m_b \rightarrow 0 $ of
 (\ref{GloopA}) and (\ref{GloopB}). One obtains

 \alpheqn
 \begin{Eqnarray} % *** loop contribution for mb = 0 *** %
  \label{GloopA0} % *** Eqs.(14a,b) *** %
  \Gamma_{loop}(m_b=0) & = &
  (a^2 + b^2) \, 
  \Bigg( 1 + \frac{a^2 - b^2}{a^2 + b^2} \frac{2 y}{1 - x^2} \Bigg)
  \hat{\Gamma} \Delta_0 -
  2 a b \, \frac{\alpha_s}{\pi} C_F \, \frac{3}{2}
  \hat{\Gamma} \ln \frac{m_b}{m_t}, \\[2mm]
  \label{GloopB0}
  \Gamma_{loop}^P(m_b=0) & = &
  2 a b \, \hat{\Gamma} \Delta_0 -
  (a^2 + b^2) \, \frac{\alpha_s}{\pi} C_F \, \frac{3}{2}
  \hat{\Gamma} \ln \frac{m_b}{m_t},
 \end{Eqnarray}
 \reseteqn

 \noindent where the common IR and mass divergent factor is given by

 \begin{Eqnarray} % *** DELTA(IR) for mb = 0 *** %
  \label{deir}    % *** Eq.(15) *** %
  \Delta_0 & = &
  \frac{\alpha_s}{\pi} C_F \Bigg\{
  \Bigg[ \ln \Bigg( \frac{m_t^2 - m_{H^+}^2}{m_b \, m_t} \Bigg) - 1 \Bigg]
  \ln \frac{m_g^2}{m_t^2} + \ln \frac{m_b}{m_t} +
  \ln^2 \frac{m_b}{m_t} \nonumber \\[2mm] & & - 1 +
  \frac{m_t^2}{m_H^2} \ln \Bigg( \frac{m_t^2 - m_{H^+}^2}{m_t^2} \Bigg) -
  \ln^2 \Bigg( \frac{m_t^2 - m_{H^+}^2}{m_t^2} \Bigg) -
  \Li \Bigg( \frac{m_{H^+}^2}{m_t^2} \Bigg) \Bigg\}.
 \end{Eqnarray}

% **************************************************************************** %
% *** tree graph contributions                                             *** %
% **************************************************************************** %

\section{Tree graph contributions}

 The one-gluon emission amplitude reads

 \begin{equation} % *** tree amplitude, Eq.(16) *** %
  {\cal M}_{tree} = g_s \frac{\lambda^a}{2} \bar{u}_b \Bigg\{
  \frac{2 p_t^{\sigma} - \slk \gamma^{\sigma}}{2 k \!\cdot\! p_t} -
  \frac{2 p_b^{\sigma} + \gamma^{\sigma} \slk}{2 k \!\cdot\! p_b} \Bigg\}
  \Bigg\{ a \1 + b \gamma_5 \Bigg\} u_t \varepsilon_{\sigma}^{\ast}(k),
 \end{equation}

 \noindent where the first and second term in the first curly bracket refer to
 gluon emission from the $ t $ quark and the $ b $ quark, respectively.

 After squaring the tree graph amplitude it is convenient to isolate the
 IR--divergent part resulting from soft gluon emission in terms of the
 unintegrated soft--gluon (or eikonal) factor $ |{\cal M}|^2_{SGF} $ given by

 \begin{equation} % *** soft gluon factor, Eq.(17) *** %
  |{\cal M}|^2_{SGF} = -
  \frac{\alpha_s}{4 \pi} C_F \Bigg\{
  \frac{m_t^2}{(k \!\cdot\! p_t)^2} +
  \frac{m_b^2}{(k \!\cdot\! p_b)^2} - 2
  \frac{p_b \!\cdot\! p_t}
  {(k \!\cdot\! p_b)(k \!\cdot\! p_t)} \Bigg\}.
 \end{equation}

 \noindent The universal soft gluon factor $ |{\cal M}|^2_{SGF} $ multiplies
 the leading order Born term amplitude squared $ |{\cal M}_{0}|^2 $.
 The same universal soft gluon factor appears in the calculation of the
 radiative corrections to $ t \rightarrow W^{+} + b $.
 We can therefore take the result of its phase space integration
 (with $ m_g \ne 0 $!) from Eq.~(63) in Ref.~\cite{FGKM02}.

 To isolate the IR--divergent part we write

 \begin{equation} % *** tree graph contribution *** %
 \label{IR-isolation} % *** Eq.(18) *** %
  |{\cal M}_{tree}|^2 =
  \Big\{ |{\cal M}_{tree}|^2 - |{\cal M}_{0}|^2 |{\cal M}|^2_{SGF} \Big\} +
  \Big\{ |{\cal M}_{0}|^2 |{\cal M}|^2_{SGF} \Big\}.
 \end{equation}

 \noindent The first term in (\ref{IR-isolation}) is now IR--convergent
 and can be integrated without a gluon mass regulator.
 It is technically advantageous to further add and subtract the term
 $ |\widetilde{\cal M}_{0}|^2 |{\cal M}|^2_{SGF} $
 to the convergent piece in (\ref{IR-isolation}), where
 $ |\widetilde{\cal M}_{0}|^2 $ is the Born term amplitude
 squared but evaluated for $ p_t = p_b + q + k $.
 Defining $|{\cal M}_{rest}|^2 = |{\cal M}_{tree}|^2 - $
 $ |\widetilde{\cal M}_{0}|^2 |{\cal M}|^2_{SGF} $
 one finally has

 \begin{equation} % *** tree graph contribution, Eq.(19) *** %
  |{\cal M}_{tree}|^2 =
  \Big\{ |{\cal M}_{rest}|^2 +
  \Big( |\widetilde{\cal M}_{0}|^2 - |{\cal M}_{0}|^2 \Big)
  |{\cal M}|^2_{SGF} \Big\} +
  \Big\{ |{\cal M}_{0}|^2 |{\cal M}|^2_{SGF} \Big\}.
 \end{equation}

 The IR--convergent part of $ |{\cal M}_{tree}|^2 $ is thus given by

 \begin{Eqnarray} % *** convergent part, Eq.(20) *** %
  |{\cal M}_{rest}|^2 + \Big( |\widetilde{\cal M}_{0}|^2 -
  |{\cal M}_{0}|^2 \Big) |{\cal M}|^2_{SGF} & = & \nonumber
  \frac{\alpha_s}{2 \pi} C_F \frac{k \!\cdot\! \tilde{q}}
  {(k \!\cdot\! p_t)(k \!\cdot\! p_b)} \Bigg\{
  (a^2 + b^2) (k \!\cdot \tilde{q}) - 2 a  b \, m_t
  \times \nonumber \\[4mm] & & \hspace{-6cm} \times \Bigg[
  \Bigg( \frac{m_t^2 + m_{H^+}^2 - m_b^2}{2 k \!\cdot\! p_t} - 1 -
  \frac{m_{H^+}^2}{k \!\cdot\! \tilde{q}} \Bigg) (k \!\cdot\! s_t) +
  \Bigg( \frac{m_t^2}{2 k \!\cdot\! p_t} -
  \frac{m_b^2}{2 k \!\cdot\! p_b} - \frac{1}{2}  -
  \frac{m_{H^+}^2}{2 k \!\cdot\! \tilde{q}} \Bigg)
  (\tilde{q} \!\cdot\! s_t) \nonumber \\[4mm] & & \hspace{-6cm} + 2
  \Bigg( \frac{m_t^2}{2 k \!\cdot\! p_t} -
  \frac{m_b^2}{2 k \!\cdot\! p_b} - 1 -
  \frac{m_{H^+}^2}{2 k \!\cdot\! \tilde{q}} \Bigg) (q \!\cdot\! s_t)
  \Bigg] \Bigg\},
 \end{Eqnarray}

 \noindent where $ k $ is the $ 4 $--momentum of the emitted gluon,
 $ \tilde{q} = p_t - p_b - k $ is the $ 4 $--momentum of the $ H^{+} $ boson
 for the three body final state and $ q = p_t - p_b $ is the $ 4 $--momentum
 of the $ H^{+} $ boson for the two body final state.
 $ s_t $ is the $ 4 $--polarization vector of the top quark.
 It is quite remarkable that the unpolarized and polarized pieces of the
 convergent tree-level contribution are proportional to $ (a^2 + b^2) $
 and $ 2 a b $, respectively.

 The phase space integration is done w.r.t.~the gluon energy $ k_0 $
 and the $ H^{+} $ boson energy $ \tilde{q}_0 $, where the $ k_0 $
 integration is done first.
 Details of the phase space integration can be found in Ref.~\cite{FGKM02}.
 After the substitution $ E_{H^+} = (m_t^2 + m_{H^+}^2 - m_t^2 z)/(2 m_t) $,
 which introduces a dimensionless integration variable $ z $, and the
 definition of the dimensionless quantities

 \begin{equation} % definition of x,y,Lambda, Eq.(21) *** %
  x := \frac{m_{H^+}}{m_t}, \qquad
  y := \frac{m_b}{m_t}, \qquad
  \Lambda := \frac{m_g}{m_t}, \qquad
  \lambda^{\prime} := \lambda(z,x^2,y^2)
 \end{equation}

 \noindent the remaining phase space integration can
 be reduced to the following classes of integrals

 \alpheqn
 \begin{Eqnarray} % *** integrals *** %
  \label{master}  % *** Eqs.(22a,b,c,d) *** %
  R(n) & := & \int\limits_{z_{min}}^{z_{max}}
    \frac{dz}{(z - y^2) \sqrt{\lambda^{\prime n}}}, \\[2mm]
  R(m,n) & := & \int\limits_{z_{min}}^{z_{max}}
    \frac{z^m \, dz}{\sqrt{\lambda^{\prime n}}}, \\[2mm]
  S(n) & := & \int\limits_{z_{min}}^{z_{max}}
    \frac{1}{(z - y^2) \sqrt{\lambda^{\prime n}}} \ln \Bigg(
    \frac{1 - x^2 + z + \sqrt{\lambda^{\prime}}}
    {1 - x^2 + z - \sqrt{\lambda^{\prime}}} \Bigg) \, dz, \\[2mm]
  S(m,n) & := & \int\limits_{z_{min}}^{z_{max}}
    \frac{z^m}{\sqrt{\lambda^{\prime n}}} \ln \Bigg(
    \frac{1 - x^2 + z + \sqrt{\lambda^{\prime}}}
    {1 - x^2 + z - \sqrt{\lambda^{\prime}}} \Bigg) \, dz.
 \end{Eqnarray}
 \reseteqn

 The coefficient functions $ \rho_{(n)} $, $ \rho_{(m,n)} $,
 $ \sigma_{(n)} $ and $ \sigma_{(m,n)} $ that are needed in
 the present application are listed in the Appendix.
 The associated integrals $ S(n) $, $ S(m,n) $, $ R(n) $ and
 $ R(m,n) $ that multiply the coefficient functions are too
 lengthy to be listed here but can be taken from Appendix~A
 of Ref.~\cite{FGKM02}.
 Finally, the tree graph contributions are given by

 \alpheqn
 \begin{Eqnarray} % *** tree graph contribution, Eq.(23a) *** %
  \Gamma_{tree} & = & - \frac{1}{4 \pi m_t} \Bigg[
  \frac{\alpha_s}{4 \pi} C_F \, m_t^2 (a^2 + b^2) \Bigg\{
  \sum\limits_{m,n} \sigma_{(m,n)} S(m,n) +
  \sum\limits_{m,n} \rho_{(m,n)} R(m,n) \Bigg\}
  \nonumber \\ & & \hspace{3cm} +
  \mbox{PS}_2^{-1} \, \Gamma_{Born} \, \Delta_{SGF} \Bigg],
 \end{Eqnarray}

 \noindent and

 \begin{Eqnarray} % *** polarized tree graph contribution, Eq.(23b) *** %
  \Gamma_{tree}^{P} & = & - \frac{1}{4 \pi m_t} \Bigg[
  \frac{\alpha_s}{4 \pi} C_F \, m_t^2 \, 2 a b \, \Bigg\{
  \sum\limits_{m,n} \sigma_{(m,n)} S(m,n) +
  \sum\limits_{m,n} \rho_{(m,n)} R(m,n)
  \hspace{1.4cm} \nonumber \\[2mm] & & \hspace{+3cm} +
  \sum\limits_{n} \sigma_{(n)} S(n) +
  \sum\limits_{n} \rho_{(n)} R(n) \Bigg\} +
  \mbox{PS}_2^{-1} \, \Gamma_{Born}^P \, \Delta_{SGF} \Bigg],
 \end{Eqnarray}
 \reseteqn

 \noindent where, as before,
 $ \mbox{PS}_2 = (\sqrt{\lambda})/(16 \pi m_t^3) $
 is a two--body phase--space factor.

 Using the coefficient functions listed in the Appendix and the tree graph
 integrals given in Ref.~\cite{FGKM02} one can take the $ m_b \rightarrow 0 $
 limit and obtains

 \begin{Eqnarray} % *** Delta(SGF,mb=0), Eq.(24) *** %
  \Delta_{SGF}(m_b=0) & = &
  \frac{\alpha_s}{4 \pi} C_F \Bigg\{
  (1 - x^2) \Bigg[
  \Bigg[ \ln \Bigg( \frac{1 - x^2}{y} \Bigg) - 1 \Bigg]
  \ln \Lambda^2 - \ln y + \ln^2 y
  \nonumber \\[3mm] & & \hspace{-26mm} -
  4 + \frac{\pi^2}{3} +
  \Bigg[ 3 - \ln \Bigg( \frac{1 - x^2}{x^2} \Bigg) \Bigg]
  \ln \Big(1 - x^2 \Big) +
  \Li \Big(x^2 \Big) \Bigg] +
  x^2 \ln x^2 \Bigg\},
 \end{Eqnarray}

 \alpheqn
 \begin{Eqnarray} % *** Gamma(tree,mb=0), Eq.(25a) *** %
  \Gamma_{tree}(m_b = 0) & = &
  \frac{\alpha_s}{\pi} C_F
  \hat{\Gamma} (a^2 \!+ b^2) \Bigg\{ \!\!-\!\!
  \frac{a^2 - b^2}{a^2 + b^2} \frac{2 y}{(1 - x^2)^2}
  \frac{4 \pi}{\alpha_s C_F} \Delta_{SGF}(m_b = 0)
  \nonumber \\[3mm] & & \hspace{-26mm} -
  \Bigg[ \ln \Bigg( \frac{1 - x^2}{y} \Bigg) \!-\! 1 \Bigg]
  \ln \Lambda^2 + \frac{1}{2} \ln y - \ln^2 y +
  \frac{13}{4} - \frac{\pi^2}{3}
  \nonumber \\[3mm] & & \hspace{-26mm} -
  \Bigg[ \frac{5}{2} - \ln \Bigg( \frac{1 - x^2}{x^2} \Bigg) \Bigg]
  \ln \Big( 1 - x^2 \Big) -
  \frac{x^2}{1 - x^2} \ln x^2 -
  \Li \Big( x^2 \Big) \Bigg\},
 \end{Eqnarray}

 \begin{Eqnarray} % *** Gamma^P(tree,mb=0), Eq.(25b) *** %
  \Gamma_{tree}^P(m_b=0) & = &
  \frac{\alpha_s}{\pi} C_F
  \hat{\Gamma} \, 2ab \, \Bigg\{ \!\! - \!\!
  \Bigg[ \ln \Bigg( \frac{1 - x^2}{y} \Bigg) - 1 \Bigg]
  \ln \Lambda^2 + \frac{1}{2} \ln y - \ln^2 y
  \nonumber \\[3mm] & & \hspace{-26mm} -
  \frac{7 - 14 x - 11 x^2}{4 (1 + x)^2} +
  \frac{1 + 2 x^2}{(1 - x^2)^2} \frac{\pi^2}{6} +
  \ln^2 \Big( 1 - x^2 \Big) -
  \frac{7 - x^2}{2 (1 - x^2)} \ln \Big( 1 + x \Big)
  \nonumber \\[3mm] & & \hspace{-26mm} -
  \frac{5}{2} \ln \Big( 1 - x \Big) +
  \frac{6 - 4 x^2 + 4 x^4}{(1 - x^2)^2}
  \Li \Big( - x \Big) \Bigg\}.
 \end{Eqnarray}
 \reseteqn

% **************************************************************************** %
% *** results                                                              *** %
% **************************************************************************** %

\section{Results}

 The complete $ O(\alpha_s) $ results are obtained by summing the Born term,
 the virtual one--loop and the tree graph contributions.
 As mentioned before the IR and mass singularities cancel in the sum of the
 one--loop and the tree graph contributions.
 We shall not explicitly write out the complete $ m_b \ne 0 $ expressions
 since they can easily be put together using the results listed in Secs.~2,
 3 and 4.
 Instead we discuss various limiting cases for the unpolarized and polarized
 rates which, among others, serve to check on the reliability of our results.
 We have checked that our $ m_b \ne 0 $ results for the unpolarized rate
 agree with those given in \cite{Czar93(2)}.
 We do not, however, agree with the $ m_b \ne 0 $ results of \cite{Lial92}.

 The limit $ m_{H^+} \rightarrow 0 $ is of interest since, according to the
 Goldstone equivalence theorem, the unpolarized and polarized rates for
 $ t \rightarrow H^{+} + b $ become related to the unpolarized and polarized
 longitudinal rates for $ t \rightarrow W^{+} + b $.
 For $ m_{H^+} \rightarrow 0 $ one has

 \alpheqn
 \begin{Eqnarray} % *** correction for mH=0, unpolarized *** %
  \label{mHzerounpol} % *** Eq.(26a) *** %
  \lim\limits_{m_{H^+} \rightarrow 0} \Gamma^{\phantom{P}} & = &
  \frac{m_t^3}{8 \pi} \frac{G_F}{\sqrt{2}}
  |V_{tb}|^2 (1 - y^2)^3 \Bigg( 1 + \frac{\alpha_s}{\pi} C_F
  \frac{1 + y^2}{1 - y^2} \Bigg(
  \frac{5 - 22 y^2 + 5 y^4}{4 (1 - y^4)} -
  2 \ln (y) \ln (1 - y^2)
  \nonumber \\[3mm] & & \hspace{-2cm} -
  2 \frac{1 - y^2}{1 + y^2} \ln \Big( \frac{1 - y^2}{y^2} \Big) -
  \frac{4 - 5 y^2 + 7 y^4}{(1 - y^2) (1 - y^4)} \ln (y) -
  2 \Li (1 - y^2) \Bigg) \Bigg),
 \end{Eqnarray}

 \begin{Eqnarray} % *** correction for mH=0, polarized *** %
  \label{mHzeropol} % *** Eq.(26b) *** %
  \lim\limits_{m_{H^+} \rightarrow 0} \Gamma^{P} & = &
  \frac{m_t^3}{8 \pi} \frac{G_F}{\sqrt{2}}
  |V_{tb}|^2 (1 - y^2)^3 \Bigg( 1 + \frac{\alpha_s}{\pi} C_F
  \frac{1}{1 - y^2} \Bigg( - \frac{3}{4} (5 + y^2) -
  2 (1 + y^2) \ln (y) \times \quad
  \nonumber \\[3mm] & & \hspace{-2cm} \times
  \ln (1 - y^2) - 2 (1 - y^2) \ln \Big( \frac{1 - y^2}{y^2} \Big) -
  \frac{4 + 5 y^2}{1 - y^2} \ln (y) +
  (1 - 2 y^2) \Li (1 - y^2) \Bigg) \Bigg).
 \end{Eqnarray}
 \reseteqn

 \noindent We have checked that for model~1 and $ \tan \beta = 1 $ the limiting
 expressions Eqs.~(\ref{mHzerounpol}) and (\ref{mHzeropol}) agree exactly with
 the $ m_{W^+} \rightarrow 0 $ limit of the corresponding longitudinal and
 polarized longitudinal rates in the process $ t \rightarrow W^+ + b $ listed
 in \cite{FGKM02}.
 This is nothing but the statement of the Goldstone equivalence theorem.
 Except for a minor typo in \cite{Czar93(2)} our unpolarized result
 (\ref{mHzerounpol}) agrees with the corresponding result in \cite{Czar93(2)}.

 Taking both $ m_{H^+} \rightarrow 0 $ and $ m_b \rightarrow 0 $ one obtains

 \alpheqn
 \begin{Eqnarray} % *** equivalence theorem *** %
  \label{gbet}    % *** Eq.(27a,b) *** %
  \lim\limits_{m_{H^+} \rightarrow 0} \Gamma^{\phantom{P}} & = &
  \frac{m_t^3}{8 \pi} \frac{G_F}{\sqrt{2}} |V_{tb}|^2 \cot^2 \beta
  \Bigg( 1 + \frac{\alpha_s}{2\pi} C_F
  \Bigg( \frac{5}{2} - \frac{2 \pi^2}{3} \Bigg) \Bigg), \\[3mm]
  \lim\limits_{m_{H^+} \rightarrow 0} \Gamma^P & = &
  \frac{m_t^3}{8 \pi} \frac{G_F}{\sqrt{2}} |V_{tb}|^2 \cot^2 \beta
  \Bigg( 1 - \frac{\alpha_s}{2\pi} C_F
  \Bigg( \frac{15}{2} - \frac{\pi^2}{3} \Bigg) \Bigg)
 \end{Eqnarray}
 \reseteqn

 \noindent which, when setting $ \cot \beta = 1 $, agree
 exactly with Eqs.~(48) and (49) of \cite{FGKM02}.
 The unpolarized rate in this limit agrees with the
 corresponding results in \cite{Czar93(1),Czar93(2),LiYa92}.

 When $ m_{H^+} $ approaches $ m_t $ for $ m_b \rightarrow 0 $ one has

 \alpheqn
 \begin{Eqnarray} % *** limit mt->mH+ *** %
  \label{limit}   % *** Eq.(28a,b) *** %
  \lim\limits_{m_{H^+} \rightarrow m_t}
  \frac{\Gamma}{\Gamma_{Born}} & = &
  1 + \frac{\alpha_s}{2 \pi} C_F \Bigg(\frac{13}{2}  - \frac{4 \pi^2}{3} -
  3 \ln (1 - x^2) \Bigg), \\[3mm]
  \lim\limits_{m_{H^+} \rightarrow m_t}
  \frac{\Gamma^P}{\Gamma_{Born}^P} & = &
  1 + \frac{\alpha_s}{2 \pi} C_F \Bigg( 1 - \pi^2 -
  3 \ln (1 - x^2) \Bigg).
 \end{Eqnarray}
 \reseteqn

 \noindent For the unpolarized rate the limiting expression
 agrees with the corresponding limit given in \cite{Czar93(2)}.

 Finally, we consider the limit $ m_b \rightarrow 0 $
 keeping the charged Higgs mass finite.
 This results in very compact expressions for the
 unpolarized and polarized rates.
 Due to the smallness of the bottom quark mass and
 the fact that the bottom quark mass corrections are of
 $ O(m_b^2 / m_t^2) $ the $ m_b \rightarrow 0 $ formulae
 give quite good approximations to the exact formulae
 for Higgs masses not close to the top quark mass.
 
 One obtains
 
 \alpheqn
 \begin{Eqnarray} % *** correction for mb=0, unpolarized *** %
 \label{mbzerounpol} % *** Eq.(29a,b) *** %
   \Gamma(m_b \rightarrow 0) & = &
   \frac{m_t}{16 \pi}(1 - x^2)^2 (a^2 + b^2) \Bigg\{
   1 + \frac{a^2 - b^2}{a^2 + b^2}
   \frac{2 y}{1 - x^2} + \frac{\alpha_s}{2\pi} C_F
   \Bigg( \frac{9}{2} - \frac{2 \pi^2}{3} -
   \frac{4 x^2}{1 - x^2} \ln x
   \nonumber \\[3mm] & & \hspace{-20mm} +
   \Big( \frac{2 - 5 x^2}{x^2} -
   4 \ln x \Big) \ln (1 - x^2) -
   4 \Li (x^2) + \frac{(a - b)^2}{a^2 + b^2} 3 \ln y \Bigg) \Bigg\}
 \end{Eqnarray}

 \noindent and

 \begin{Eqnarray} % *** correction for mb=0, polarized *** %
   \label{mbzeropol}
   \Gamma^P(m_b \rightarrow 0) & = &
   \frac{m_t}{16 \pi}(1 - x^2)^2 2 a b \Bigg\{
   1 + \frac{\alpha_s}{2\pi} C_F \Bigg( -
   \frac{11 - 6 x - 7 x^2}{2 (1 + x)^2} +
   \frac{1 + 2 x^2}{(1 - x^2)^2} \frac{\pi^2}{3}
   \nonumber \hspace{20mm} \\[3mm] & & \hspace{-20mm} +
   \frac{2 - 9 x^2 + x^4}{(1 - x^2) x^2} \ln (1 + x) +
   \frac{2 - 5 x^2}{x^2} \ln (1 - x) -
   4 \Li (x) + \frac{8 + 4 x^4}{(1 - x^2)^2} \Li (-x)
   \nonumber \\[3mm] & & \hspace{-2.0cm} -
   \frac{(a - b)^2}{2 a b} 3 \ln y \Bigg) \Bigg\}.
 \end{Eqnarray}
 \reseteqn

 \noindent Note that the seemingly mass singular terms proportional to
 $ \ln y $ in (\ref{mbzerounpol}) and (\ref{mbzeropol}) are not in fact
 mass singular since they are multiplied by the factor $ ( a - b )^2 $ which
 is proportional to $ m_b^2 $ in both models~1 and 2.
 Although the contributions proportional to $ y^2 \ln y $ formally
 vanish for $ m_b \rightarrow 0 $ they can become numerically quite
 large for $ m_b = 4.8 $ GeV in model~2 depending, of course, on the
 value of $ \tan \beta $.
 This can be seen by calculating the ratios of the
 coupling factor expressions in model~2, i.e.
 $ (a-b)^2 / (a^2+b^2) = 2 y^2 \tan ^4 \beta /(1 + y^2 \tan ^4 \beta) $ and
 $ (a-b)^2 / (2ab) = 2 y^2 \tan ^4 \beta /(1 - y^2 \tan ^4 \beta) $.
 For example, for $ \tan \beta = 10 $ one finds
 $ (a - b)^2 / (a^2 + b^2) = 1.77 $ and
 $ (a - b)^2 / (2ab)= - 2.31 $.
 With a little bit of algebra one finds that the NLO corrections in
 model~2 are in fact dominated by the $ y^2 \ln y $ contributions for
 larger values of $ \tan \beta $ as also noted in \cite{Czar93(2)}.
 This is evident in Fig.~4 where the radiative corrections to the
 Born term result can be seen to be as large as $ - 50 \% $ compared
 to the $ \approx - 10 \% $ expected from the corresponding corrections
 in the decay $ t \rightarrow W^{+} + b $ \cite{FGKM02}.

 We now turn to our numerical results.
 In Fig.~2 we show the radiative corrections to the polarization asymmetry
 $ \alpha_H $ in model~1 as a function of $ x = m_{H^+} / m_t $.
 The radiative corrections are quite small and lower
 $ \alpha_H $ by only $ \approx 3 \permille $.
 We do not present a curve for the dependence of $ \alpha_H $
 on $ \tan \beta $ because the radiative corrections are
 again quite small $ \approx 3 \permille $ over a large
 range of $ \tan \beta $--values and thus do not change
 the flat Born term behaviour.
 Figs.~4a and 4b show the LO and NLO model~2 results for the
 unpolarized and polarized rates as a function of the mass ratio
 $ x = m_{H^+} / m_t $ for $ \tan \beta = 10 $.
 As input values for our numerical evaluation we use
 $ m_b = 4.8 \mbox{ GeV} $ and $ m_t = 175 \mbox{ GeV} $.
 The strong coupling constant is evolved from
 $ \alpha_s(M_Z) = 0.1175 $ to
 $ \alpha_s(m_t) = 0.1070 $ using two-loop running.
 The unpolarized and polarized rates are largest for $ m_{H^+} = 0 $
 and become smaller towards the phase-space boundary $ x = 1 - y $.
 The radiative corrections are substantial due to the
 $ y^2 \ln y $ contribution discussed above.
 The dotted curves in Figs.~4a and 4b are drawn using the ``kinematical''
 $ m_b \rightarrow 0 $ approximations Eqs.~(\ref{bornzerobmass}) (LO) and
 Eqs.~(\ref{mbzerounpol}) and (\ref{mbzeropol}) (NLO) where the bottom
 quark mass has been set to zero whenever the scale of $ m_b $ is set by
 $ m_t $ as in the kinematical factors and not by $ m_t \cot \beta $
 as in the coupling factors.
 As Fig.~4 shows the ``kinematical'' $ m_b \rightarrow 0 $
 approximation is an excellent approximation for both the unpolarized and
 polarized rate.

 In Fig.~3 we show the model~2 LO and NLO results
 for the asymmetry parameter $ \alpha_H $.
 In Fig.~3a we have again fixed $ \tan \beta $ at
 $ \tan \beta = 10 $ and show the dependence of $ \alpha_H $
 on the mass ratio $ x = m_{H^+} / m_t $.
 The asymmetry parameter is large and negative with only
 little dependence on the Higgs mass except for the region
 close to the phase-space boundary where $ \alpha_H $
 approaches zero.
 The radiative corrections to the LO Born term result are
 substantial and reduce the size of the asymmetry parameter
 by $ \approx 25 \% $ over much of the range of the Higgs mass.
 As anticipated from the results for the unpolarized and polarized rate
 the $ m_b \rightarrow 0 $ approximation is excellent. At the scale of
 Fig.~3 the $ m_b \ne 0 $ corrections are barely visible.
 In Fig.~3b we fix the mass of the charged Higgs boson at
 $ m_{H^+} = 120 \mbox{ GeV}$ and vary $ \tan \beta $
 between $ 0 $ and $ 40 $.
 For small values of $ \tan \beta $ the asymmetry parameter
 $ \alpha_H $ is negative and rapidly approaches zero around
 $ \tan \beta = 7 $.
 The LO zero position $ \tan \beta = \sqrt{m_t/m_b} $ is shifted
 upward by approximately one unit by the radiative corrections.
 Beyond the zero position the asymmetry parameter rapidly approaches
 values close to $ \alpha_H = - 1 $.
 The radiative corrections are
 largest around the zero position of $ \tan \beta $ at
 $ \tan \beta  \approx 7 $.

% **************************************************************************** %
% *** Concluding remarks                                                   *** %
% **************************************************************************** %

\section{Concluding remarks}

 We have calculated the $ O(\alpha_s) $ radiative corrections to polarized
 top quark decay into a charged Higgs and a bottom quark in two variants
 of the Two-Higgs-Doublet model.
 We have checked our unpolarized results against
 known results and find agreement.
 Using the same techniques we have calculated the polarized rate.
 Further, we have compared our polarized results with
 the corresponding polarized results in the decay $ t \rightarrow W^{+} + b $
 appealing to the Goldstone equivalence theorem.
 Because of our numerous cross-checks we are quite confident that our
 new results on the polarized rates are correct.
 We have found very compact $ m_b = 0 $, $ O(\alpha_s) $ expressions for
 the unpolarized and polarized rates which can be usefully employed to scan
 the predictions of the 2HDM $ (m_{H^+},\tan \beta) $ parameter space.
 We have found that the radiative corrections to the unpolarized and
 polarized rates, and the asymmetry parameter of the decay can become
 quite large.

% **************************************************************************** %
% *** acknowledgements                                                     *** %
% **************************************************************************** %

\vspace{1truecm} \noindent {\bf Acknowledgements:}
 M.~Mauser was partly supported by the DFG (Germany) through the
 Graduiertenkolleg ``Eichtheorien'' at the University of Mainz and
 by the BMBF (Germany) under contract 05HT9UMB/4.
 The author would like to thank K.~Schilcher for his support.

% **************************************************************************** %
% *** appendix                                                             *** %
% **************************************************************************** %

\section{Appendix}

 In this appendix we list the coefficient functions $ \sigma_{(n)} $,
 $ \sigma_{(m,n)} $, $ \rho_{(n)} $ and $ \rho_{(m,n)} $ which multiply
 the basic set of integrals $ S(n) $, $ S(m,n) $, $ R(n) $ and $ R(m,n) $
 in Eqs.~(\ref{master}-d).
 We define $ \lambda_0 = \lambda(1,x^2,y^2) $ with $ x = m_{H^+} / m_t $
 and $ y = m_b / m_t $, as before.
 The coefficient functions for the unpolarized and polarized rates
 $ \Gamma_{tree} $ and $ \Gamma_{tree}^{P} $ can be obtained from
 the unpolarized and polarized scalar rates ($ i = S $) in Appendix
 B.7 and ($ i = S^P $) in B.8 of Ref.~\cite{FGKM02} by multiplication
 with the factors $ - x^2/(4 + 4 y^2) $ and $ - x^2/(4 - 4 y^2) $,
 respectively.

\subsection{Coefficient functions for \pf $ \Gamma_{tree} $}

 \alpheqn
 \begin{Eqnarray} % *** unpolarized coefficients *** %
  \sigma_{(0,0) \pp \pp} & = & \pp \frac{1}{2} y^2, \hspace{98truemm} \\
  \sigma_{(1,0) \pp \pp} & = & - \frac{1}{2}, \\
  \rho_{(-2,-1)}     & = & \pp \frac{1}{4} (1 - x^2) y^2, \\
  \rho_{(-1,-1)}     & = & - \frac{1}{4} (1 - x^2 + 3 y^2), \\
  \rho_{( 0,-1) \pp} & = & \pp \frac{3}{4}. \hspace{90truemm}
 \end{Eqnarray}
 \vspace*{-5truemm}
 \reseteqn

\subsection{Coefficient functions for \pf $ \Gamma_{tree}^{P} $}

 \alpheqn
 \begin{Eqnarray} % *** polarized coefficients *** %
  \sigma_{(0) \pp \pp} & = & \pp
  (1 - x^2 + y^2) \lambda_0^{1/2}, \\[2mm]
  \sigma_{(1) \pp \pp} & = & -
  (1 - x^2 + y^2) \lambda_0, \\[2mm]
  \sigma_{(0,0) \pp} & = & \pp  \lambda_0^{1/2}, \\[2mm]
  \sigma_{(0,1) \pp} & = & \pp \frac{1}{2}
  (4 x^2 (1 - x^2) + (7 + 5 x^2) y^2 - 2 y^4), \\[2mm]
  \sigma_{(1,1) \pp} & = & - \frac{1}{2} (3 - 3 x^2 + y^2), \\[2mm]
  \sigma_{(2,1) \pp} & = & - \frac{1}{2}, \\[2mm]
  \rho_{(-1)\pp}     & = & - 2 \lambda_0^{1/2}, \\[2mm]
  \rho_{(0) \pp \pp} & = & \pp 2 \lambda_0, \\
  \rho_{(-2,0)}  & = & \pp \frac{1}{4} (1 - x^2)^2 y^2, \\
  \rho_{(-1,0)}  & = & - \frac{1}{4}
  ((1 - x^2)^2 + 2 (3 - x^2) y^2), \\
  \rho_{(0,0) \pp} & = & - \frac{1}{4} (2 + 10 x^2 - y^2), \\
  \rho_{(1,0) \pp} & = & \pp \frac{7}{4}. \hspace{105truemm}
 \end{Eqnarray}
 \vspace*{-5truemm}
 \reseteqn

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% *** bibliography                                                         *** %
% **************************************************************************** %

 \newpage

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% **************************************************************************** %
% *** pictures                                                             *** %
% **************************************************************************** %

 \newpage \thispagestyle{empty} \vspace*{4truecm}

 \begin{center} % *** definition of theta_P *** %
  \includegraphics[width=150mm]{feyn/theta.ps}
 \end{center}

 \centerline{\Large \bf Figure 1} \vspace*{5truemm}

 \centerline{Definition of the polar angle $ \theta_P $.
  $ \vec{P} $ is the polarization vector of the top quark.}

% **************************************************************************** %
% ***                                                                      *** %
% **************************************************************************** %

 \newpage \thispagestyle{empty}

 \vspace*{4truecm}

 \begin{center} % *** asym. parameter mh/mt *** %
 \begin{picture}(120,075)
  \includegraphics[width=120mm]{plot/asym_ah3.eps}
  \put(-055,-04){$ m_{H^+}/m_t $}
  \put(-125,+37){\rotatebox{90}{$ \alpha_H $}}
  \put(-090,+50){$ O(\alpha_s) $}
  \put(-015,+55){Born}
 \end{picture}
 \end{center}

 \vspace*{5truemm}

 \centerline{\Large \bf Figure 2} \vspace*{5truemm}

 \noindent Asymmetry parameter $ \alpha_H $ for model~1 with
  $ m_b = 4.8 \mbox{ GeV} $ and $ m_t = 175 \mbox{ GeV} $
  (Born term: full line; $ O(\alpha_s) $: dashed line)
  as function of $ m_{H^+}/m_t $ for $ \tan \beta = 10 $.

% **************************************************************************** %
% ***                                                                      *** %
% **************************************************************************** %

 \newpage \thispagestyle{empty}

 \begin{center} % *** asym. parameter mh/mt *** %
 \begin{picture}(120,070)
  \includegraphics[width=120mm]{plot/asym_ah1.eps}
  \put(-055,-04){$ m_{H^+}/m_t $}
  \put(-125,+37){\rotatebox{90}{$ \alpha_H $}}
  \put(-090,+28){$ O(\alpha_s) $}
  \put(-020,+10){Born}
  \put(-015,+65){\bf 3a}
 \end{picture}
 \end{center}

 \begin{center} % *** asym. parameter tan beta*** %
 \begin{picture}(120,080)
  \includegraphics[width=120mm]{plot/asym_ah2.eps}
  \put(-055,-04){$ \tan \beta $}
  \put(-125,+37){\rotatebox{90}{$ \alpha_H $}}
  \put(-078,+37){$ O(\alpha_s) $}
  \put(-100,+18){Born}
  \put(-010,+65){\bf 3b}
 \end{picture}
 \end{center}

 \vspace*{5truemm}

 \centerline{\Large \bf Figure 3a, b} \vspace*{5truemm}

 \noindent Asymmetry parameter $ \alpha_H $ for model~2 with
  $ m_b = 4.8 \mbox{ GeV} $ and $ m_t = 175 \mbox{ GeV} $
  (Born term: full line; $ O(\alpha_s) $: dashed line) as function of
  $ m_{H^+}/m_t $ (Fig.~2a, $ \tan \beta = 10 $) and as function of
  $ \tan \beta $ (Fig.~2b, $ m_{H^+} = 120 \mbox{ GeV} $).
  The barely visible dotted lines show the corresponding
  $ m_b \rightarrow 0 $ curves.

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% ***                                                                      *** %
% **************************************************************************** %

 \newpage \thispagestyle{empty}

 \begin{center} % *** total decay rate *** %
 \begin{picture}(120,75)
  \includegraphics[width=120mm]{plot/corr.eps}
  \put(-050,+50){Born}
  \put(-096,+43){$ O(\alpha_s) $}
  \put(-010,+65){\bf 4a}
  \put(-055,-05){$ m_{H^+}/m_t $}
  \put(-125,+35){\rotatebox{90}{$ \Gamma \mbox{  [GeV]} $}}
 \end{picture}
 \end{center}

 \begin{center} % *** polarized decay rate *** %
 \begin{picture}(120,80)
  \includegraphics[width=120mm]{plot/corrp.eps}
  \put(-050,+25){Born}
  \put(-096,+38){$ O(\alpha_s) $}
  \put(-010,+60){\bf 4b}
  \put(-055,-05){$ m_{H^+}/m_t $}
  \put(-125,+35){\rotatebox{90}{$ \Gamma^P \mbox{  [GeV]} $}}
 \end{picture}
 \end{center}

 \vspace{5truemm}

 \centerline{\Large\bf Figure 4a, b} \vspace*{5truemm}

 \noindent Unpolarized (Fig.~4a) and polarized (Fig.~4b) decay rate
  for model~2 with $ m_b = 4.8 \mbox{ GeV} $, $ m_t = 175 \mbox{ GeV} $
  and $ \tan \beta = 10 $ (Born term: full line; $ O(\alpha_s) $:
  dashed line). The barely visible dotted lines show the corresponding
  $ m_b \rightarrow 0 $ curves.

\end{document}

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