\documentstyle[12pt]{article}
\renewcommand{\baselinestretch}{1.3}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\hoffset=-1.5cm
\voffset=-1.5cm
\textheight=22.0cm
\textwidth=16.5cm
\begin {document}

\begin{center}
{\Large \bf Finite W boson width effects in the top quark width}\\
\vspace{1.3cm}

{\large G. Calder\'on and G. L\'opez Castro}\\

{\it Departamento de F\'{\i}sica, Cinvestav del IPN \\
Apartado Postal 14-740, 07000 M\'exico D.F. M\'exico}
\end{center}
\vspace{1.3cm}

\begin{abstract}
 In the standard model, the top quark decay width $\Gamma_t$ is 
computed from the exclusive $t \rightarrow bW$ decay. We argue in favor of
using the three-body decays $t \rightarrow bf_i\bar{f}_j$ to compute
$\Gamma_t$ as a sum over these exclusive modes. As dictated by the 
S-matrix
theory, these three-body decays of the top quark involve only asymptotic
states and incorporate the width of the $W$ boson resonance in a natural
way.  The convolution
formula commonly used to include the $W$ boson finite width effects is
found to be valid in the limit of massless fermions $f_i\bar{f}_j$. The
relation $\Gamma_t=\Gamma(t\rightarrow bW)$ is recovered by taking the
limit of massless fermions followed by the $W$ boson narrow width
approximation. Although both calculations of $\Gamma_t$ are different at
the formal level, their results would differ  only by tiny effects
induced by first (second) order electroweak (QCD) radiative corrections.
\end{abstract}
{\it PACS: 14.45.Ha, 13.38.Be, 11.10.St}\\
{\it Keywords: Top quark width, unstable particles}

\newpage

The precise calculation and measurement of the top quark 
decay width $\Gamma_t$ is important to provide a consistency check of
the standard model between the mass and the width of the top
quark. Moreover, the large mass of the top quark implies a large
width, which makes the top to behave almost like a free particle.
This feature makes attractive the application of perturbative QCD methods
to evaluate the quantum
corrections. In particular, one expects that the top quark decay would
provide the QCD analog of muon decay at the level of radiative
corrections. In this context, a definition of $\Gamma_t$ that satisfies
general properties of the relativistic quantum theory is welcome.

As is known, theoretical predictions have to match an accuracy
comparable to or better than the experimental error bars. On the
experimental side, the mass of the top quark $m_t$ is expected to
be measured with an uncertainty of up to 3 GeV in the Run II at the
Tevatron Collider \cite{willenbrock}, while $\Gamma_t$ can eventually be
measured with a precision of 5-7\% from the
forward-backward asymmetry at planned linear colliders
\cite{fujii97}. On the theoretical side, it is customary 
to compute the top quark width from the electroweak 
$t\rightarrow bW$ process\footnote{The decay modes $t\rightarrow
sW, dW$ would be important only for a calculation aiming an
accuracy below the 0.1\% level.} 
(see p. 385 in \cite{pdg2000} and Refs.
\cite{jk,alpha_s2,alpha}). This is the so-called narrow width
approximation because the W boson is considered as a stable
particle. Different radiative corrections to this process have
been reported within the standard model. The order $\alpha_s$
 and $\alpha_s^2$ QCD corrections to this rate turn out to be at the 10\%
\cite{jk} and 1$\sim$2 \% \cite{alpha_s2} level,
respectively. On the other hand, the one-loop electroweak
corrections are found to affect the decay rate at the 1$\sim$2\%
level \cite{alpha}. Even some tiny effects in $\Gamma_t$ of
$O(10^{-5})$ arising from the renormalization of the $V_{tb}$
matrix element, have been reported recently in the literature
\cite{renorm}. Finally, given the large mass of the top quark,
virtual effects of hypothetical heavy particles might contribute to
$\Gamma(t \rightarrow bW)$ at a few of percent level (see for example
\cite{higgs}).

   Since measurements of the top quark width by different experiments can
be intimately related to a particular definition of $\Gamma_t$, let us  
comment on the case where our discussion can be relevant. The Tevatron
collider may eventually provide an indirect measurement of $\Gamma_t$ from
the observation of single top quark production. In this case, the
definition of $\Gamma_t$ based on the two-body decays can be more
appropriate because the {\it production} mechanisms involve only the $tbW$
vertex. On the other hand, the measurement of $\Gamma_t$ from the
threshold region for $t\bar{t}$ production at linear colliders or from the
pole position of their resonant energy decay distribution, do not involve
directly the $tbW$ vertex and a model-independent definition would be more
suitable. 

As it was indicated above, most of these calculations of $\Gamma_t$ (with
the exception of Refs. \cite{jk}) assume that the $W$ is a stable
particle. The analysis
of Refs.\cite{jk} has found that the finite width of the $W$ boson
can induce an additional correction of 1$\sim$2\% to the
$t\rightarrow bW$ decay rate. In practice, the $W$ boson is not a
real particle that can be reconstructed from their decay products with a
well defined invariant mass $m_W$. Instead, 
the $W$ boson is a resonance and  it can not formally be
considered as an asymptotic state to be used in the evaluation of
S-matrix amplitudes. Furthermore, by cutting the production and decay
mechanisms of a resonance can lead to miss important interference effects
in the real observables which necessarily are related to the
{\it detection} of (quasi)stable particles \cite{kz}.

It is the purpose of the present paper to compute the top quark decay
width by using the three-body decays $t
\rightarrow bf_i\bar{f}_j$, such that the decay width is defined  by the
relation:
 \be
\Gamma_t \equiv \sum \Gamma(t \rightarrow bf_i\bar{f}_j)\
,\label{decay}
\ee
 where the sum is carried over flavors and colors
of the pair of fermions $f_i\bar{f}_j$ that are allowed by kinematics.
Since the lifetimes of fermions in
the final states are much larger than the typical interaction time scales 
(the $W$ and $t$ decay times), they can be considered as asymptotic states
 of this process.

  We show in this paper that, in the limit of massless fermions, the
tree-level expression of Eq. (\ref{decay}) give rise to a convolution
formula
commonly used (see for example Refs. \cite{muta,tbwz}) to include the
finite width effects of the $W$ boson. Later, we let the $W$ boson 
width to vanish and we show that the r.h.s of Eq. (1) reduces to 
$\Gamma(t\rightarrow bW)$. Finally, we briefly comment on the implications
of Eq. (\ref{decay}) for the calculation of $\Gamma_t$ at the level of the
QCD radiative corrections.


Let us start our evaluation of the top decay width by
computing of the $t \rightarrow bf_i\bar{f}_j$ decay rate. The
tree-level amplitude for this process is given by: \be {\cal
M}=\frac{g}{2\sqrt{2}}V_{tb}\bar{u}(p_b)\gamma^{\mu}(1-\gamma_5)u(p_t)
\cdot (-iD_{\mu\nu}(Q))
\frac{g}{2\sqrt{2}}V_{ij}^*\bar{v}(p_j)\gamma^{\nu}(1-\gamma_5)u(p_i)\
. \label{treelevel}\ee In this expression $V_{kl}$ denote the
corresponding Kobayashi-Maskawa matrix elements associated to the
$kl$ fermionic current ($V_{ij}=1$ for lepton doublets), $g$ is the
strength of the weak charged
current and the momentum-transfer is defined by
$Q=p_t-p_b=p_i+p_j$. In the unitary gauge, the full $W$ boson
resonant propagator $D_{\mu\nu}(Q)$ \cite{wprop1} can be divided
into its spin-1 (transverse) and spin-0 (longitudinal) pieces
according to: \ba D_{\mu\nu}(Q) =
\frac{g_{\mu\nu}-\frac{\textstyle Q_{\mu}Q_{\nu}}{\textstyle
Q^2}}{Q^2-m_W^2+i Im\Pi_T(Q^2)}- \frac{Q_{\mu}Q_{\nu}}{Q^2}
\frac{1}{m_W^2-iIm\Pi_L(Q^2)}\ . \label{propaW}
\ea 
As it was explained in Ref.\cite{wprop1}, we consider $m_W$ to be the
renormalized mass of the $W$ boson and we include only the
(finite) absorptive corrections
to the $W$ propagator as done in the context of the fermion-loop scheme
\cite{wprop1,fls}.
In Eq.(\ref{propaW}) we have defined $Im
\Pi_T$ and $Im\Pi_L$ as the transverse and longitudinal
projections of the $W$ boson self-energy tensor, namely
\cite{wprop1}: \be Im \Pi^{\alpha\beta}(Q)=\left(
g^{\alpha\beta}-\frac{Q^{\alpha}Q^{\beta}}{Q^2} \right)
Im\Pi_T(Q^2) + \frac{Q^{\alpha}Q^{\beta}}{Q^2}Im \Pi_L(Q^2)\ . \ee

Using cutting rules techniques, we can compute these absorptive parts of
the $W$ boson self-energy. The expressions obtained at the lowest
order in $g$ are given by:
\ba Im\Pi_T(Q^2) &=& \sqrt{Q^2} \sum_{k,l}
\Gamma^0(W(Q^2)
\rightarrow f_k\bar{f}_l) \theta(Q^2-(m_k+m_l)^2)\nonumber \\
Im\Pi_L(Q^2) &=& -\sum_{k,l}N_C \frac{g^2}{8}|V_{kl}|^2
\frac{Q^2}{4\pi} f(x_k^2,x_l^2)\lambda^{1/2}(1,x_k^2,x_l^2)
\theta(Q^2-(m_k+m_l)^2)\ , \label{absorW}\ea
where the sum extends
over flavors ($k=u,c,e,\mu,\tau;\ l=s,d,b, \nu_e,\nu_{\mu},
\nu_{\tau}$) in fermion loops that are allowed in the $W$ boson decay
when its mass $\sqrt{Q^2}
> m_k+m_l$. Let us emphasize that we have kept the masses of all fermions
in these corrections in order to remain consistent when the
masses of fermions in the final states of top decay are finite.

In the above expressions we have defined the (tree-level) partial decay
width of off-shell $W$ bosons as follows: 
\be \Gamma^0(W(Q^2) \rightarrow
f_k\bar{f}_l) =N_C\left( \frac{g^2}{8} \right)
\frac{|V_{kl}|^2}{12\pi}\sqrt{Q^2} \left[ 2-f(x_k^2,x_l^2)\right]
\lambda^{1/2}(1,x_k^2,x_l^2)\ , \ee where $N_C$ is the number of
colors, $x_i\equiv m_i/\sqrt{Q^2}$ and we have defined the
functions: $f(r,s)\equiv r+s-(r-s)^2$ and $ \lambda(a,b,c)\equiv
a^2+b^2+c^2-2(ab+ac+bc)$.

The top quark decay width reported in the literature
is calculated from the two-body $t\rightarrow bW$ process. If the $W$
boson is off its mass-shell, with a mass $\sqrt{Q^2}$, the decay width
of the top quark becomes: 
\be \Gamma^0(t \rightarrow
bW(Q^2)) = \frac{G_F^0m_t^3}{8\pi \sqrt{2}}|V_{tb}|^2
\frac{m_W^2}{Q^2} \left[(1-x^2)(1+2x^2)-y(2-x^2-y^2) \right]
\lambda^{1/2}(1,x^2,y^2)\ \label{top},\ee 
where $x\equiv
\sqrt{Q^2}/m_t$ and $y\equiv m_b/m_t$. Note that we have replaced
$g^2 \rightarrow 8G_F^0m_W^2/\sqrt{2}$, where $G_F^0$ is the Fermi
constant at the tree-level.

An straightforward calculation of the top quark decay width using 
the formula (\ref{decay}) leads to the following result:
 \ba
\Gamma_t^0 &=& \sum_{i,j}\Gamma^0(t\rightarrow bf_i\bar{f}_j)\nonumber  \\
&=&\frac{1}{\pi}\sum_{i,j} \int_{(m_i+m_j)^2}^{(m_t-m_b)^2} dQ^2
\ \Gamma^0(t \rightarrow bW(Q^2)) \frac{\sqrt{Q^2}\
\Gamma^0(W(Q^2) \rightarrow
f_i\bar{f}_j)}{(Q^2-m_W)^2+(Im\Pi_T(Q^2))^2}
\nonumber \\
&& +\ \frac{G_F^2m_W^4m_t^3}{64\pi^3}
\sum_{i,j}|V_{tb}V_{ij}^*|^2\int_{(m_i+m_j)^2}^{(m_t-m_b)^2} dQ^2
\lambda^{1/2}(1,x^2,y^2)\lambda^{1/2}(1,x_i^2,x_j^2) \nonumber \\
&& \ \ \ \ \ \ \ \ \times \left[(1-y^2)^2-x^2(1+y^2)\right]
\frac{f(x_i^2,x_j^2)}{m_W^4+(Im\Pi_L(Q^2))^2} , \label{prime}\ea

The first term in Eq.(\ref{prime}) arises from the spin-1 degrees
of freedom in the $W$ boson propagator and the second one comes
from its spin-0 component. The  interference term in
Eq.(\ref{prime}) vanishes due to the orthogonality of the
amplitudes arising from the decomposition in
Eqs.(\ref{treelevel},\ref{propaW}). Observe that the second term
in the r.h.s of Eq.(\ref{prime}) vanishes in the limit of massless
fermions ($x_i,x_j=0$); if we keep the finite masses of these fermions,
the second term in Eq.(\ref{prime}) will give a contribution of order
$10^{-5} \sim 10^{-6}$ GeV to $\Gamma_t$.

  Let us now consider the massless limit for the fermions that participate
in the self-energy correction and the decay of the $W$ boson.  In this
case we have, from Eqs.(\ref{absorW}), $Im\Pi_L(Q^2)=0$ and: 
\ba
Im\Pi_T(Q^2)&=& \sum_{k,l}\sqrt{Q^2}\Gamma(W(Q^2)\rightarrow
f_k\bar{f}_l)
\nonumber \\
&=& \frac{Q^2}{m_W}\Gamma^0_W\ , \ea where
$\Gamma^0_W=\sum_{k_l}N_Cg^2m_W/48\pi$ is the on-shell decay width
of the $W$ boson in the limit of massless fermions. Thus, in the
limit of massless (light) fermions, Eq.(\ref{prime} is: 
\be \Gamma^0_t
=\int_0^{(m_t-m_b)^2} dQ^2\ \Gamma^0(t \rightarrow
bW(Q^2)) \rho_W(Q^2)\ ,
\ee
where, 
\be
\rho_W(Q^2)=\frac{1}{\pi} \cdot \frac{\frac{\textstyle Q^2}{\textstyle
m_W}\Gamma_W^0}{(Q^2-m_W^2)^2+\left (\frac{\textstyle
Q^2}{\textstyle m_W}\Gamma_W^0 \right)^2}\ \label{convo}. 
\ee

A few comments about Eqs. (10)-(11) can make more clear our point. The
convolution kernel $\rho_W(Q^2)$ in Eq. Eq.(\ref{convo}) coincides
with the one  used in Refs.\cite{muta,tbwz} to include the finite width
effects of gauge bosons in final states. The factor $Q^2$ in the numerator
of the
convolution kernel serves to cancel the $Q^2$ factor appearing in
the denominator of the $t\rightarrow bW(Q^2)$ decay rate --see
Eq.(\ref{top})-- and avoids that the integrand in Eq.(\ref{convo})
diverges in the limit $Q^2 \rightarrow 0$. This result is
consistent with the fact that in the limit $Q^2 \rightarrow 0$,
the $W$ boson produced in the $t\rightarrow bW(Q^2)$ decay has
only two degrees of freedom and some care must be taking when
using Eq.(\ref{top}) in that limit. On the other hand, Eq.(\ref{convo})
(and, as a matter of fact, the results of Refs. \cite{muta,tbwz}) can be
viewed as a factorization of the production and decay subprocesses
of the $W$ gauge-boson. As already explained, this approximation (which
can be justified on probabilistic grounds when the production and decay
mechanisms of the $W$ boson are independent)
be introduced on statistical grounds) can be valid only in the limit of
massless fermions.

Next we focus in the narrow $W$ boson width approximation. In the limit
$\Gamma_W^0 \rightarrow 0$, Eq.(\ref{convo}) reduces to: 
\be
\Gamma_t^0=\Gamma^0(t \rightarrow bW) \label{usual}\ee 
(with the
$W$ boson on its mass-shell)  by virtue of the representation of
the Dirac delta function given by $\lim_{\epsilon \rightarrow 0}
\epsilon/(x^2+\epsilon^2) =\pi\delta(x^2)$. Thus, we consistently
recover the tree-level decay rate of the two-body decay, 
Eq.(\ref{top}), which is used as the starting point to implement
the radiative corrections to  $\Gamma_t$ in other
calculations
\cite{jk}-\cite{higgs}. This result reinforces our arguments that the top
quark decay width {\it and} its radiative
corrections must be computed using Eq.(\ref{decay}) as the
starting point instead of Eq.(\ref{usual}).

Let us now briefly comment on the effects of radiative corrections. QCD
corrections introduce two important differences between the calculation 
of $\Gamma_t$ based on Eqs. (\ref{decay}) and (\ref{usual}): the energy
scale to evaluate
$\alpha_s$ and the role of box diagrams. The $O(\alpha_s)$ QCD radiative
corrections to the $t\rightarrow
bW$ decay are reported in Ref. \cite{jk} and turns out to be of the order 
of 10\%. These corrections include quark self-energies and gluon
exchanges between quark lines of the  $tbW$ and $Wq_i\bar{q}_j$ vertices
\cite{jk}.
Since $t\rightarrow bW$ is an exclusive process,
the QCD energy scale used to evaluate the one-loop corrected width is
the mass of the top quark, namely  $\alpha_s(m_t)$ \cite{jk}. According to
Eq. (1),
$\Gamma_t$ is an inclusive quantity which is obtained from a sum over
exclusive three-body decays. Thus, for an invariant mass $\mu$ of the
fermion pair $f_i\bar{f}_j$, there are indeed two mass scales involved in
the problem: $m_t$ and $\mu$.
However, when $\mu$ is peaked around the $W$ boson mass (as it is the case
of the dominant contributions in three-body top quark decays) no large
logarithms will be induced
\cite{fjy}. Thus,
in order to have an estimate of the higher order contributions one should
compare the QCD corrections to $\Gamma_t$ at the
scales $\mu=m_W$ and $\mu=m_t$.
For the case of the $O(\alpha_s)$ corrections, the top decay rate
decreases by 1.03\% when the scale $\mu$ changes from $m_t$ to $m_W$.

  On the other hand, the one-loop electroweak and QCD box diagrams will
appear in 
three-body decays of the top quark, but will be absent in $t \rightarrow
bW$. Fortunately, these corrections do not contribute to the $O(\alpha_s)$
in $t \rightarrow bf_i\bar{f}_{j}$. This happens because the interference
of the
box diagram and tree-level amplitudes involves the trace over the product
of a color-singlet and a color-octet fermionic currents, which vanishes
identically \cite{smith}.
However, the four (one-loop) box diagrams for the $t \rightarrow
bf_if_{j}$ decay will contribute to the order $\alpha_s^2$ corrections in 
$\Gamma_t$. On the other hand, the interference of the one-loop
electroweak box diagram and the tree-level amplitudes will not vanish 
and they will contribute to the order $\alpha$ corrections. Their
calculation are, however, beyond the scope of the present paper.

 In summary, the purpose of the present paper is to call the attention on
the fact
the top quark decay width must be evaluated from its three-body decays ($t
\rightarrow bf_i\bar{f}_j$)  instead of using the decay $t \rightarrow
bW$. The former modes involve
only final particles that can be considered as asymptotic states required
to correctly define S-matrix amplitudes. On the other hand, their
amplitudes incorporate the finite-width effects of the $W$ boson in a
natural way. Although, we do not find significant numerical differences
for the top decay width at the tree-level, some differences can appear at
the level of the $O(\alpha_s^2)$ and $O(\alpha)$ radiative corrections.

   Before closing this paper, let us comment that the convolution integral
used to include
the finite width effects of massive gauge bosons in final states (see for
example \cite{muta,tbwz}) turns out to be an approximation valid in the
limit of massless fermions produced in the decays of these gauge bosons.
On another hand, the validity 
of the  convolution formula is based on the independence of the production
and decay probabilities of the gauge bosons. In the present case, this
independence can be justified, at the tree-level, on the fact that the
spin-0 component of the $W$ propagator decouples in the limit of massless
fermions.

\

  {\bf Acknowledgements}. We are grateful to J. Pestieau, F. J. Yndur\'ain
and C. P. Yuan for useful correspondence. This work has been partially
supported by Conacyt (M\'exico) under contracts E-32429 and W35792-E.

\

\begin{thebibliography}{40}
\bibitem{willenbrock}
S. Willenbrock, Rev. Mod. Phys. {\bf 72}, 1141 (2000); P. C.
Bhat, H. B. Prosper and S. S. Snyder, Int. J. Mod. Phys. {\bf A13},
5113 (1998); M. Beneke et al., Proc. of the Workshop on the Standard Model
Physics at the LHC, eprint .
\bibitem{fujii97}
K. Fujii, Nucl. Phys. (PS){\bf 59}, 331 (1997).
\bibitem{pdg2000}
D. E. Groom et al, Particle Data Group, Eur. Phys. Jour. {\bf
C15}, 1 (2000).
\bibitem{jk}
M. Jezabek and J. H. K\"uhn, Nucl. Phys. {\bf B314}, 1 (1989); M.
Jezabek and J. H. K\"uhn, Phys. Rev. {\bf D48}, R1910 (1993).
\bibitem{alpha_s2}
A. Czarnecki and K. Melnikov, Nucl. Phys. {\bf B544}, 520 (1999);
K. G. Chetyrkin, R. Harlander, T. Seidensticker, and M.
Steinhauser, Phys. Rev. {\bf D60}, 114015 (1999).
\bibitem{alpha}
A. Denner and T. Sack, Nucl. Phys. {\bf B358}, 46 (1991); G.
Eilam, R. R. Mendel, R. Migneron, and A. Soni, Phys. Rev. Lett.
{\bf 66}, 3105 (1991).
\bibitem{renorm}
S. M. Oliveira, L. Bruecher, R. Santos, and A. Barroso, Phys. Rev. {\bf
D64}, 017301 (2001).
\bibitem{higgs}
B. Grzadkowski and W. Hollik, Nucl. Phys. {\bf B384}, 101 (1992);
A. Denner and A. Hoang, Nucl Phys. {\bf B397}, 483 (1993).
\bibitem{kz} See for example: N. Kauer and D. Zeppenfeld, eprint
.
\bibitem{muta}  
T. Muta, R. Najima and S. Wakaizumi, Mod. Phys. Lett. {\bf A1},
203 (1986)
\bibitem{tbwz}
G. Altarelli, L. Conti and V. Lubicz, Phys. Lett. {\bf B502}, 125 (2001).
\bibitem{wprop1}
M. Beuthe, R. Gonz\'alez Felipe, G. L\'opez Castro and J.
Pestieau, Nucl. Phys. {\bf B498}, 55 (1997); D. Atwood, G. Eilam,
R. Mendel, R. Migneron and A. Soni, Phys. Rev. {\bf D49}, 289
(1994).
\bibitem{fls}
E. N. Argyres et al, Phys. Lett. {\bf 358}, 339 (1995); U. Baur
and D. Zeppenfeld, Phys. Rev. Lett. {\bf 75}, 1002 (1995).
\bibitem{fjy}
Private communication with F. J. Yndur\'ain.
\bibitem{smith}
M. C. Smith and S. Willenbrock, Phys. Rev. {\bf D54}, 6696 (1996)
\end{thebibliography}

\end {document}

