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%% section 4 Applications [slacpub7157004 in slacpub7157004: slacpub7157005]
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% author definitions added by nc_fix
\newcommand{\ie}{{\it i.e.}}
\newcommand{\ms}{$\overline{\mbox{MS}}$}
\newcommand{\ams}{\mbox{$\alpha_{\overline{\mbox{\tiny MS}}}$}}
\newcommand{\amst}{\mbox{${\widetilde{\alpha}_{\overline{\mbox{\tiny MS}}}}$}}%$
\newcommand{\av}{\mbox{$\alpha_{V}$}}
\newcommand{\ar}{\mbox{$\alpha_{R}$}}
\newcommand{\nfz}{\mbox{$N_{F,V}^{ (0) }$}}
\newcommand{\nfo}{\mbox{$N_{F,V}^{ (1) }$}}
\newcommand{\nfdt}{\mbox{$N_{F,DT}$}}
\newcommand{\nfct}{\mbox{$\widetilde{N}_{F,\overline{\mbox{\tinyMS}}}^{(0)}(Q)$}}
\newcommand{\nf}{\mbox{$N_F$}}
\newcommand{\mz}{\mbox{$M_Z$}}
\newcommand{\mb}{\mbox{$m_b$}}
\newcommand{\mc}{\mbox{$m_c$}}
\newcommand{\betz}{\mbox{$\beta^(0)$}}
\newcommand{\adt}{\mbox{$\alpha_{DT}$}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% end of definitions added by nc_fix
\section{\usemenu{slacpub7157::context::slacpub7157004}{Applications}}\label{section::slacpub7157004}
In this section we will show how to compute an observable using the
analytic extension of the \ms\ scheme and compare with the standard
treatment of quark mass threshold effects in the \ms\ scheme.
The essential difference between the
perturbative expansions in the \ams\ and \amst\ couplings are terms that
contain quark masses.
In the analytic scheme the quark mass
effects are automatically included whereas in the \ms\ scheme they have
to be included by hand for each observable.
For some observables, such as the hadronic width of the Zboson and
the $\tau$ lepton semihadronic decay rate, corrections due to nonzero
quark masses have been calculated within the \ms\ scheme
\cite{18,19,20,9}. To be specific we
are interested in the so called double bubble diagrams where the outer
quark loop which couples to the weak current is considered massless
and the inner quark loop is massive. Other types of mass corrections,
such as the double triangle graphs where the external current is
electroweak, are not taken into account by the analytic extension of
the \ms\ scheme. (For a recent review of higher order corrections to
the Zboson width see \cite{44}.)
To illustrate how to compute an
observable using the analytic extension of the \ms\ scheme and compare
with the standard treatment in
the \ms\ scheme we consider the
QCD corrections to the quark part of the nonsinglet hadronic width of
the Zboson, $\Gamma_{had,q}^{NS}$. Writing the QCD corrections in terms
of an effective charge we have
\begin{equation}
\Gamma_{had,q}^{NS}=\frac{G_FM_Z^3}{2\pi\sqrt{2}}
\sum_{q}\{(g_V^{q})^2+(g_A^{q})^2\}
\left[1+\frac{3}{4}C_F\frac{\alpha_{\Gamma,q}^{NS}(s)}{\pi}\right]
\end{equation}
where the effective charge $\alpha_{\Gamma,q}^{NS}(s)$ contains all
QCD corrections,
\begin{eqnarray}
\frac{\alpha_{\Gamma,q}^{NS}(s)}{\pi} & = &
\frac{\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}(\mu)}{\pi}
\left\{1+\frac{\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}(\mu)}{\pi}
\left[\sum_{q=1}^{N_L}\left(\frac{11}{12}+\frac{2}{3}\zeta_3
+ F\left(\frac{m_q^2}{s}\right)
\frac{1}{3}\ln\left(\frac{\mu}{\sqrt{s}}\right)\right)
\right. \right. \nonumber \\ && \left. \left.
+\sum_{Q=N_L+1}^{6}G\left(\frac{m_Q^2}{s}\right)\right] + \ldots \right\}
\end{eqnarray}
The functions $F$ and $G$ are the effects of nonzero quark masses
for light and heavy quarks, respectively.
In the following we will not restrict ourselves to the case $\sqrt{s}=M_Z$
since we want to compare the two treatments of masses for arbitrary $s$.
Thereby the number of light flavors $N_L$ will also vary with $s$.
We will also assume that the matching of $N_F$ is done
at the quark masses. Thus a quark with mass $m<\mu$ is considered
as light whereas a quark with mass $m>\mu$ is considered as heavy.
To calculate $\alpha_{\Gamma,q}^{NS}(s)$ in the
analytic extension of the \ms\ scheme one first has
to apply the BLM scalesetting procedure which absorbs all the massless
effects of nonzero $N_F$ into the running of the coupling.
This gives,
\begin{eqnarray}
\label{eq:agms}
\frac{\alpha_{\Gamma,q}^{NS}(s)}{\pi} & = &
\frac{\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}(Q^*)}{\pi}
\left\{1+\frac{\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}(Q^*)}{\pi}
\left[\sum_{q=1}^{N_L}F\left(\frac{m_q^2}{s}\right)
%\right. \right. \nonumber \\ && \left. \left.
+\sum_{Q=N_L+1}^{6}G\left(\frac{m_Q^2}{s}\right)\right] + \ldots \right\}
\end{eqnarray}
where
\begin{equation}
Q^*=\exp\left[3\left(\frac{11}{12}+\frac{2}{3}\zeta_3\right)\right]\sqrt{s}
=0.7076\sqrt{s}.
\end{equation}
Operationally, one next simply drops
all the mass dependent terms in the above expression and replaces the
fixed $N_F$ coupling $\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}$
with the analytic \amst. (For an observable calculated with massless quarks
this step reduces to replacing the coupling.)
In this way both the massless $N_F$ contribution
as well as the mass dependent contributions from double bubble diagrams
are absorbed into the coupling
and we are left with a very simple expression,
\begin{eqnarray}
\label{eq:aganalytic}
\frac{\alpha_{\Gamma,q}^{NS}(s)}{\pi} & = &
\frac{\amst(Q^*)}{\pi}.
\end{eqnarray}
This simple expression reflects the fact that the effects of quarks in the
perturbative coefficients, both massless and massive, should be absorbed
into the running of the coupling.
To compare with the ordinary \ms\ treatment we need the functions $F$ and $G$
in Eq.~(\docLink{slacpub7157004.tcx}[eq:agms]{23}).
Expansions in terms of $m^2/s$ and $s/m^2$ can be found in
\cite{18,19,9} whereas they have been calculated numerically
in \cite{20}. In addition the
$\alpha_{\mbox{\scriptsize{s}}}^3$ correction due to
heavy quarks has been calculated as an expansion in $s/m^2$ in \cite{9}.
It should also be noted that the function $G$ was first calculated for QED
\cite{45}.
Here we will use the following expansions,
\begin{eqnarray}
F\left(\frac{m^2}{s}\right) & = &
\left(\frac{m^2}{s}\right)^2\left[\frac{13}{3}
4\zeta_3\ln\left(\frac{m^2}{s}\right)\right]
\nonumber \\ &&
+ \left(\frac{m^2}{s}\right)^3\left[\frac{136}{243}+\frac{16}{27}\zeta_2
+\frac{56}{81}\ln\left(\frac{m^2}{s}\right)
\frac{8}{27}\ln^2\left(\frac{m^2}{s}\right)\right]
\\
G\left(\frac{m^2}{s}\right) & = &
\frac{s}{m^2}\left[ \frac{44}{675}
+\frac{2}{135}\ln\left(\frac{s}{m^2}\right)\right]
%\nonumber \\ &&
+\left(\frac{s}{m^2}\right)^2\left[\frac{1303}{1058400}
\frac{1}{2520}\ln\left(\frac{s}{m^2}\right)\right]
\end{eqnarray}
which are good to within a few percent for $m^2/s<0.25$
and $s/m^2<4$ respectively.
We will also use the relation \cite{20},
\begin{equation}
F\left(\frac{m^2}{s}\right) =
G\left(\frac{m^2}{s}\right) + \frac{1}{6}\ln\left(\frac{m^2}{s}\right)
\left(\frac{11}{12}+\frac{2}{3}\zeta_3 \right)
\end{equation}
to get $F$ in the interval $0.25 < m^2/s < 1$ since the expansion of
$F$ in terms of $m^2/s$ breaks down for $m^2/s > 0.25$.
Before carrying out the comparison of the analytic extension of the \ms\
scheme with the standard treatment it is instructive to look at the effective
contribution to $\alpha_{\Gamma,q}^{NS}(s)$ from one flavor with mass $m$ as
a function of $s$. To make the arguments more transparent we will use
the renormalization scale $\mu=\sqrt{s}$ when doing this.
For small $s$, when the quark is considered heavy the
contribution is given by $G(m^2/s)$ whereas for larger $s$ the quark is
considered as light and contributes with $F(m^2/s)
\frac{11}{12}+\frac{2}{3}\zeta_3$.
Normalizing to the massless contribution $\frac{11}{12}+\frac{2}{3}\zeta_3 $
gives the contribution to the effective $N_F$ in the
$\alpha_{\mbox{\scriptsize{s}}}^2$coefficient,
\begin{equation}
N_{F,\overline{\mbox{\tiny MS}}}^{\mbox{\scriptsize{eff}}}
\left( \frac{s}{m^2} \right) =
\left\{
\begin{array}{lc}
{\displaystyle \frac{G\left({ \frac{m^2}{s}}\right)}
{ { \frac{11}{12}+\frac{2}{3}\zeta_3}} }
& \quad \text{for $\sqrt{s}m$ }
\end{array}
\right. \quad ,
\end{equation}
which is shown in Fig.~\docLink{slacpub7157004.tcx}[fig:n_eff_MS]{6} as a function of $s/m^2$.
\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=neffms.eps,width=10cm}}
\end{center}
\caption[*]{The effective contribution to $N_F$ in the
$\alpha_{\mbox{\scriptsize{s}}}^2$coefficient in the standard \ms\ scheme
from a quark with mass $m$ as a function of $s/m^2$ (using $\mu=\sqrt{s}$).
The discontinuity
between the two expansions in $s/m^2$ and $m^2/s$ can be seen
at the nonanalytic point $s/m^2=4$.}
\label{fig:n_eff_MS}
\end{figure}
At first it might seem unnatural that the effective contribution to $N_F$
in the $\alpha_{\mbox{\scriptsize{s}}}^2$coefficient is
negative for heavy quarks. However, this is a characteristic feature
of the standard \ms\ scheme which arises from the fact that the number of
flavors in the running of the coupling is kept constant.
Starting from a scale well below the threshold the number of flavors in the
running as well as in the $\alpha_{\mbox{\scriptsize{s}}}^2$
coefficient is not affected by the heavy quark.
As the threshold is approached from below the number of flavors in the running
should increase which would make the running of the coupling slower (since
$\psi^{(0)}$ would be smaller) which in turn should lead to a larger \ams.
But, since the number of flavors is kept constant in the running this
effect has to be taken into account by adding a positive contribution
to the $\alpha_{\mbox{\scriptsize{s}}}^2$coefficient,
i.e. the function $G$. Since the massless contribution
is negative this means that the
contribution to $N_F$ becomes negative for a heavy
quark. Once the threshold has been
crossed the number of flavors in the running changes and the need to compensate
for a too small \ams\ vanishes rapidly as the scale is increased above the
threshold. For scales well above the threshold the masseffects are negligible
and the massless result is regained as $F$ goes to zero. This should be
compared with the analytic \ms\ scheme where $N_F$ is increased continuously
in the running.
To compare the analytic extension of the \ms\ scheme
with the standard \ms\ result for
$\alpha_{\Gamma,q}^{NS}(s)$ we will apply
the BLM scalesetting procedure also for the standard \ms\ scheme.
This is to ensure that any differences are due to the different
ways of treating quark masses and not due to the scale choice.
In other words we want to compare Eqs.~(\docLink{slacpub7157004.tcx}[eq:agms]{23})
and (\docLink{slacpub7157004.tcx}[eq:aganalytic]{25}). As the normalization point we use
$\alpha_{\overline{\mbox{\tiny MS}}}^{(5)}(M_Z)=0.118$
which we evolve down to $Q^*=0.7076M_Z$ using leading order
massless evolution with $N_F=5$. This value is then used to calculate
$\alpha_{\Gamma,q}^{NS}(M_Z)=0.1243$ in the \ms\ scheme using
Eq.~(\docLink{slacpub7157004.tcx}[eq:agms]{23}). Finally, Eq.~(\docLink{slacpub7157004.tcx}[eq:aganalytic]{25}) gives the
normalization point for $\amst(Q^*)$.
\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=gdiff.eps,width=10cm}}
\end{center}
\caption[*]{The relative difference between the calculation of
$\alpha_{\Gamma,q}^{NS}(s)$ in the analytic extension of the \ms\ scheme
and the standard treatment of masses in the \ms\ scheme. The discontinuities
are due to the mismatch between the $s/m^2$ and $m^2/s$ expansions of the
functions $F$ and $G$.}
\label{fig:gdiff}
\end{figure}
Fig.~\docLink{slacpub7157004.tcx}[fig:gdiff]{7} shows the relative difference between the two
expressions for $\alpha_{\Gamma,q}^{NS}(s)$ given by Eqs.~(\docLink{slacpub7157004.tcx}[eq:agms]{23})
and (\docLink{slacpub7157004.tcx}[eq:aganalytic]{25}) respectively. As can be seen from the figure the
relative difference is smaller than 0.2\% for scales above 1 GeV. Thus
the analytic extension of the \ms\ scheme takes the mass corrections
into account in a very simple way without having to include an infinite
series of higher dimension operators or doing complicated multiloop
diagrams with explicit masses.
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