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\begin{document}

\title{Matching functions for heavy particles}
\author{S.D. Bass}
\affiliation{High Energy Physics Group,
Institute for Experimental Physics and
Institute for Theoretical Physics, Universit\"at Innsbruck,
Technikerstrasse 25, A 6020 Innsbruck, Austria}
\author{R.J. Crewther}
\affiliation{Department of Physics and Mathematical Physics and \linebreak
Special Research Centre for the Subatomic Structure of Matter (CSSM),
University of Adelaide, SA 5005, Australia}
\affiliation{Centre for Particle Theory, Department of Mathematical
Sciences, University of Durham, South Rd, Durham DH1 3LE, United
Kingdom}
\author{F.M. Steffens}
\affiliation{Instituto de Fisica Teorica,
Rua Pamplona 145, 01405-900 Sao Paulo - SP, Brazil}
\affiliation{Mackenzie University - FCBEE, Rua da Consolacao 930,
01302-907, Sao Paulo - SP, Brazil}
\author{A.W. Thomas}
\affiliation{Department of Physics and Mathematical Physics and \linebreak
Special Research Centre for the Subatomic Structure of Matter (CSSM),
University of Adelaide, SA 5005, Australia}

\begin{abstract}
We introduce \textit{matching functions} as a means of summing 
heavy-quark logarithms to any order.  Our analysis is based on 
Witten's approach, where heavy quarks are decoupled one at a
time in a mass-independent renormalization scheme.  The outcome is a
generalization of the matching conditions of Bernreuther and Wetzel:  
we show how to derive closed formulas for summed logarithms to any order, 
and present explicit expressions for leading order (LO) and 
next-to-leading order (NLO) contributions.
\end{abstract}
\pacs{11.10.Hi, 12.38.Cy, 12.39.Hg}
\preprint{ADP-02-100/T538}
\maketitle

%
\section{Introduction}\label{I}
%

Decoupling a heavy quark when the renormalization scheme is mass 
independent was originally discussed by Witten \cite{witten}.  
He showed that the results can be elegantly expressed in terms of 
a renormalization group (RG) invariant running coupling 
$\run{h}$ associated with the mass $m_h$ of the heavy quark $h$.  
Subsequently, Bernreuther and Wetzel \cite{BW,wetzel,B1,B2} 
proposed a systematic method for dealing with the matching problem, 
i.e.\ the lack of explicit decoupling in mass independent schemes.  
They applied the Appelquist-Carrazone decoupling theorem 
\cite{AC} to the gluon coupling $\alpha_Q^{\rm MO}$ in the momentum
subtraction (MO) scheme, i.e.\ renormalized at space-like momentum $Q$
%%:
%
\begin{equation}
\alpha^{\mbox{\tiny MO}}_Q\bigr|_{\mathrm{with}\ h}
   = \alpha^{\mbox{\tiny MO}}_Q\bigr|_{\mathrm{no}\ h}\,
          +\,  O(m^{-1}_h)
\label{aa1}
\end{equation}
%
and compared calculations of $\alpha^{\mbox{\tiny MO}}_Q$ in the 
$F=f\!+\!1$ and $f$ flavor $\overline{\mbox{\small MS}}$ (modified 
minimal subtraction) theories. So, in principle, the strong coupling 
$\alpha_F$ for the $F$-flavor $\overline{\mbox{\small MS}}$ theory
is calculable as a power series in its $f$-flavor counterpart 
$\alpha_f$ and logarithms of $m_h$, with coefficients 
constrained by the renormalization group. Results for the first
few loops of perturbation theory appear in the literature
\cite{BW,wetzel,B1,B2,larin,CKS1}. Bernreuther has constructed a 
similar matching procedure to deal with the effects of mass renormalization 
\cite{B1}.


In this paper, we introduce {\it matching functions} 
${\cal F}(\run{h})$ and ${\cal G}(\run{h})$ associated with 
coupling constant and mass renormalization, and show how to obtain 
explicit matching conditions valid to a given \emph{logarithmic} 
order: leading order (LO), next-to-leading order (NLO), 
next-to-next-to-leading order (NNLO), and so on.  Both
${\cal F}(\run{h})$ and ${\cal G}(\run{h})$ vanish for LO and NLO, 
but then there are contributions from successive terms in their 
power series in $\run{h}$, starting with NNLO for ${\cal F}(\run{h})$ 
and NNNLO for ${\cal G}(\run{h})$.  The results of the analysis are 
simple closed expressions for coupling constants, running couplings 
and light masses, with all mass logarithms of a given order already 
summed, as required in realistic applications.  All results are for
quantum chromodynamics (QCD) with three colors, but our technique can
be applied to any renormalizable theory.


{}For coupling constant renormalization, the key equations, which 
we introduce formally in Sec.~\ref{mf}, are
\begin{align}
\ln\frac{\bar{m}_h}{\bar{\mu}}
&= \int_{\alpha_F}^{\run{h}}\hsp{-2.5}dx\,\frac{1}{\beta_F(x)} 
\label{c8} \\[1mm]
\ln\frac{\bar{m}_h}{\bar{\mu}}
&= \int_{\alpha_f}^{\run{h}}\hsp{-2.5}dx\,\frac{1}{\beta_f(x)} 
       + {\cal F}_{F\to f}(\run{h}) 
\label{c9} \end{align}
for the case $F=f+1$.  Here ${\bar{m}_h}$ is Witten's renormalization 
group invariant heavy-quark mass \cite{witten}, and $\beta_f(x)$ is 
the $\beta$-function for the $f$-flavor theory.  The matching
function ${\cal F}_{F\to f}$ is a series in $\run{h}$ whose 
coefficients can be determined perturbatively by comparison 
with (\ref{aa1}).  The matching relation between $\alpha_F$ and 
$\alpha_f$ is then just a matter of eliminating 
$\run{h}$ from Eqs.~(\ref{c8}) and (\ref{c9}):
%
\begin{equation}
\alpha_F = \alpha_F\bigl(\alpha_f,\ln(\bar{m}_h/\bar{\mu})\bigr) .
\label{d0}
\end{equation}
%
Included in this paper are derivations of decoupling formulas for 
$\alpha_{f+1}$ and $\run{h}$ quoted previously by us \cite{BCST}.


Sections \ref{wm} and \ref{mcc} are brief summaries of Witten's treatment 
of heavy-quark decoupling in QCD and the matching procedure of Bernreuther 
and Wetzel for coupling constants.  This lays the foundation for the RG 
analysis in Sec.~\ref{mf}, from which we are led to construct the matching 
function ${\cal F}$ for coupling constant renormalization.  Perturbation
theory for ${\cal F}$ is considered in Sec.~\ref{pt}, with the result that
the first non-zero term (NNLO) in ${\cal F}$ is obtained.  In Sec.~\ref{ce}, 
we show that Eqs.~(\ref{c8}) and (\ref{c9}) lead directly to closed 
expressions for heavy-quark logarithms to a given logarithmic order, and
present explicit NLO expressions.  Section~\ref{mmf} is an extension of
our RG analysis to deal with the matching problem for mass renormalization.  
It is here that we introduce the mass-matching function ${\cal G}$.  
In Sec.~\ref{asd}, we consider the limit where more 
than one heavy-quark is decoupled sequentially, for example  
$\ln (m_t / {\bar \mu}) \gg \ln (m_b / {\bar \mu}) \rightarrow \infty$,
and derive the NLO closed formula for coupling constant 
renormalization in this limit.  Possible applications of our technique
are discussed in the concluding Sec.~\ref{c}.


%
\section{Witten's Method}\label{wm}
%
This section summarizes some key points of Witten's procedure
\cite{witten,BCST}. 


Let $\mu$ be the scale used to define dimensional regularization and
renormalization, and let   
\begin{equation}
\bar{\mu} = \mu\sqrt{4\pi}e^{-\gamma/2}\ ,\ \ \gamma = 0.5772 \ldots
\end{equation}
be the corresponding $\overline{\mbox{\small MS}}$ scale. By convention,
the same scale $\bar{\mu}$ is used for the initial $F$-flavor and all
residual $f$-flavor theories.  Whenever heavy quarks (masses $m_h$) are
decoupled,
\[ F\to f\ \mbox{flavors},\,\ m_h \to \infty  \]
all parameters of the \emph{residual} $f$-flavor theory are 
held fixed: the scale $\bar{\mu}$, all momenta $\mathbf{p}$,
the coupling $\alpha_f$, and all light-quark masses $m_{lf}$.  
In any order of perturbation theory, amplitudes
\[ {\cal A}_F =
{\cal A}_F\bigl(\mathbf{p}, \bar{\mu}, \alpha_F, m_{lF}, m_h\bigr)
\]
are power series in $\smash{m_h^{-1}}$ with each power modified by a
polynomial in $\ln(m_h/\bar{\mu})$.  We will consider the leading power 
$\widetilde{\cal A}_F$:
\begin{equation}
{\cal A}_F = \widetilde{\cal A}_F\{1 + O(m_h^{-1})\}.
\end{equation}
The notation $O(m_h^{-1})$ refers to any sub-leading power, including 
its logarithmic modifications.


Logarithms in $\widetilde{\cal A}_F$ for $m_h\sim\infty$ are generated
by 1PI (one-particle irreducible) subgraphs with at least one 
heavy-quark propagator and with degree of divergence at least logarithmic.  
It is as if all contributing 1PI parts were shrunk to a point.  All 
$F$-flavor amplitudes $\widetilde{\cal A}_F$ tend to amplitudes 
${\cal A}_f$ of the residual $f$-flavor theory, apart from 
$m_h$-dependent renormalizations of the coupling constant, light masses, 
and amplitudes \cite{AC}:
\begin{align}
\widetilde{\cal A}_F\bigl(\mathbf{p}&, \bar{\mu},\alpha_F,m_{lF},m_h\bigr)
  \nonumber \\
&=\, \sum_{{\cal A}'}{\cal Z}_{{\cal AA}'}(\alpha_F, m_h/\bar{\mu})
{\cal A}'_f\bigl({\bf p},\bar{\mu},\alpha_f,m_{lf}\bigr)
\label{a} \\
\alpha_f =\ &\alpha_f(\alpha_F, m_h/\bar{\mu}) \hsp{1},\hsp{1}
m_{lf} = m_{lF}D(\alpha_F, m_h/\bar{\mu}).
\label{a1}\end{align}
For practical applications, Eq.~(\ref{a1}) has to be inverted, so that
$\alpha_f$ and $m_{lf}$ become the dependent variables instead of 
$\alpha_F$ and $m_{lF}$. That is because we hold $\alpha_f$ and 
$m_{lf}$ fixed as $m_h \to \infty$.


For any number of flavors $f$ (including $F$), let
\begin{equation}
{\cal D}_f\, =\, \bar{\mu}\frac{\del\ }{\del\bar{\mu}}
            + \beta_f(\alpha_f)\frac{\del\ \ }{\del\alpha_f}
        + \delta_f(\alpha_f)\sum_{k=1}^f {m_k}_{\hsp{-0.2}f}
               \frac{\del\hsp{4.6}}{\del {m_k}_{\hsp{-0.2}f}}
\label{b7}
\end{equation}
be the corresponding Callan-Symanzik operator.  Since ${\cal A}_F$ 
satisfies an $F$-flavor improved Callan-Symanzik equation 
\cite{weinberg1}, so also does its leading power:
\begin{equation}
\bigl\{{\cal D}_F + \gamma_F(\alpha_F)\bigr\}\widetilde{\cal A}_F\, =\, 0 .
\label{d}
\end{equation}
In general, both  $\gamma_F$ and
${\cal Z} = \bigl({\cal Z}_{{\cal AA}'}\bigr)$ are matrices. 


If we substitute (\ref{a}) in (\ref{d}) and change variables,
\begin{equation}
{\cal D}_F\, =\, \bar{\mu}\frac{\del\ }{\del\bar{\mu}}
   + \bigl({\cal D}_F\alpha_f\bigr)\frac{\del\ \ }{\del\alpha_f}
   + \sum_{k=1}^f\bigl({\cal D}_F{m_k}_{\hsp{-0.2}f}\bigr)
              \frac{\del\hsp{4.6}}{\del {m_k}_{\hsp{-0.2}f}}
\end{equation}
the result is an improved Callan-Symanzik equation for each residual
amplitude,
\begin{equation}
\bigl\{{\cal D}_f + \gamma_f(\alpha_f)\bigr\}{\cal A}_f\, =\, 0
\end{equation}
where the functions \cite{witten,BW}
\begin{eqnarray}
\beta_f(\alpha_f) &=& {\cal D}_F\alpha_f
\label{e1} \\
\delta_f(\alpha_f) &=& {\cal D}_F\ln m_{lf}
\label{eq:e2} \\
\gamma_f(\alpha_f) &=&
     {\cal Z}^{-1}\bigl(\gamma_F(\alpha_F) + {\cal D}_F\bigr){\cal Z}
\label{e3}
\end{eqnarray}
depend \emph{solely} on $\alpha_f$.  The absence of $m_l$ dependence
in the renormalization factors in (\ref{a}) and (\ref{a1}) ensures
mass-independent renormalization for the residual theory.

While these equations hold for any $f<F$, their solutions can be readily
formulated in terms of running couplings only when the heavy quarks 
are decoupled one at a time.  Indeed, Witten's running coupling
\begin{equation}
\run{h} = \run{h}\bigl(\alpha_F, \ln(m_h/\bar{\mu})\bigr)
\end{equation}
is defined for the case $F = f+1$ where just one quark $h$ is heavy, with 
$\overline{\mbox{\small MS}}_F$ renormalized mass $m_h$. The definition
of $\run{h}$ is formulated implicitly \cite{witten}:
\begin{equation}
\ln(m_h/\bar{\mu})
= \int^{\run{h}}_{\alpha_F}\! dx\,\bigl(1-\delta_F(x)\bigr)
    \!\bigm/\!\beta_F(x) .
\label{f3}
\end{equation}
It satisfies the constraints
\begin{equation}
\run{h}(\alpha_F,0) = \alpha_F \hsp{3},\hsp{3}
\run{h}(\alpha_F,\infty) = 0
\end{equation}
where the latter follows from the asymptotic freedom of the $F$-flavor 
theory ($F \leqslant 16$).  Eqs.~(\ref{b7}), (\ref{e1}) and (\ref{f3}) 
imply that $\run{h}$ is renormalization group (RG) invariant: 
\begin{equation}
{\cal D}_F \run{h} = 0 .
\end{equation}


\section{Matching Coupling Constants}\label{mcc}

Generally, the solutions of Witten's equations depend on 
renormalized parameters $\alpha_F$ and $m_{lF}$ of the original
$F=f+1$ flavor theory, whereas the limit $m_h \to \infty$ is to be taken
with parameters $\alpha_f$ and $m_{lf}$ of the \emph{residual} theory
held fixed. To complete the analysis, it is necessary to derive 
asymptotic series in $\ln(m_h/\bar{\mu})$ which relate the initial 
and residual parameters, i.e.\ to ``match'' $\alpha_F$ and $m_{lF}$ 
with $\alpha_f$ and $m_{lf}$.  As noted in Sec.~\ref{I}, Bernreuther 
and Wetzel \cite{BW,wetzel,B1,B2} have set up a systematic 
procedure for this.  This section is a brief account of their
scheme for the case of coupling-constant matching.


The decoupling formula (\ref{aa1}) works to any order of perturbation 
theory, so the task is to express the leading power of the
RG-invariant gluon coupling $\alpha^{\mbox{\tiny MO}}_Q$ with and 
without the heavy-quark $h$ as perturbative series in $\alpha_F$ and 
$\alpha_f$ respectively. 
Generally this involves gluon and other self-energy insertions and a 
vertex amplitude such as fermion-gluon \cite{BW, wetzel} or ghost-gluon 
\cite{B1}. 

For one-loop contributions \cite{weinberg2}, vertex and propagator 
corrections cancel ($Z_1=Z_2$), so only the gluon self-energy amplitude
\[ \Pi^{ab}_{f\mu\nu} 
   = i\delta^{ab}(g_{\mu\nu}q^2 - q_\mu q_\nu)\Pi_f(\sqrt{-q^2}) \] 
is needed.  In that case, we can make the replacement
\[ \alpha^{\mbox{\tiny MO}}_Q\bigr|_{f\ \mathrm{flavors}}
   \longrightarrow \alpha_f\!\bigm/\!\bigl(1 - \Pi_f(Q)\bigr) \]
in Eq.~(\ref{aa1}), with the result
%
\begin{align}
\alpha_{f+1}^{\,-1} - \alpha_{f+1}^{\,-1}&\Pi_{f+1}(Q)  \nonumber \\ 
&= \alpha_f^{\,-1} - \alpha_f^{\,-1}\Pi_f(Q)
      +  O(m_h^{-1},\alpha_f^2) . 
\label{a3}\end{align}
%
Comparing the original and residual theories, we have
\begin{align}
\Pi_{f+1}(Q) &= \alpha_{f+1}\bigl\{\Gamma_{h\mathrm{-loop}} 
             + \Gamma_\mathrm{other}\bigr\} + O(\alpha_{f+1}^2)
\nonumber \\
\Pi_f(Q) &= \alpha_f\Gamma_\mathrm{other} + O(\alpha_f^2)
\label{a4}\end{align}
where
\begin{equation}
\Gamma_{h\mathrm{-loop}} = \frac{1}{\pi}\int^1_0\!\!ds\, s(1\!-\!s)
         \ln\biggl(\frac{m_h^2 + s(1\!-\!s)Q^2}{\bar{\mu}^2}\biggr)
\end{equation}
is the contribution of the heavy-quark loop, and $\Gamma_\mathrm{other}$
represents other one-loop terms. 

The leading power contributed by the heavy-quark loop is
\begin{equation}
\Gamma_{h\mathrm{-loop}} = C_\mathrm{LO}\ln\frac{m_h}{\bar{\mu}}
     + C_\mathrm{NLO} + O\bigl(Q^2/m_h^2\bigr)
\label{a5}\end{equation}
with coefficients for leading and non-leading logarithmic orders given by
\begin{equation}
C_\mathrm{LO} = 1/(3\pi) \hsp{3}\mbox{and}\hsp{3} C_\mathrm{NLO} = 0 .
\label{a5a}
\end{equation} 
The vanishing of the NLO constant term is a well-known characteristic 
of the $\overline{\mbox{\small MS}}$ gluon self-energy \cite{weinberg2}.


Eliminating $\Gamma_\mathrm{other}$ from Eq.~(\ref{a4}) and combining the
result with Eqs.~(\ref{a3}) and (\ref{a5}), we recover the standard 
one-loop matching condition
\begin{equation}
\alpha_{f+1}^{\,-1} - \alpha_f^{\,-1}
 = \frac{1}{3\pi}\ln\frac{m_h}{\bar{\mu}} +  O\bigl(\alpha_f,m_h^{-1}\bigr)
\label{a6}
\end{equation}
or equivalently
\begin{equation}
\alpha_{f+1} = \alpha_f - \frac{\alpha_f^2}{3\pi}\ln\frac{m_h}{\bar{\mu}}
                +  O\bigl(\alpha_f^3,m_h^{-1}\bigr)   .
\label{a7}
\end{equation}


The two-loop analysis is much more complicated, so we simply quote the
result \cite{BW,wetzel,B1,B2}, taking account of a 
subsequent correction \cite{BW,larin,CKS1}.  We find it convenient to 
consider the inverse form where $\alpha_{f+1}$ is written as a series 
in $\alpha_f$.  For the special case of three colors, the result is:
\begin{align}
\alpha_{f+1}
= \alpha_{f} 
   &- \frac{\alpha_{f}^2}{6\pi}\ln\frac{m_h^2}{\bar{\mu}^2}
   + \frac{\alpha_{f}^3}{36\pi^2}\ln^2\frac{m_h^2}{\bar{\mu}^2}
\nonumber \\
   &- \frac{11\alpha_{f}^3}{24\pi^2}\ln\frac{m_h^2}{\bar{\mu}^2}
   - \frac{11\alpha_{f}^3}{72\pi^2}\, +\, O(\alpha_{f}^4) .
\label{e3c}
\end{align}
The first three terms of the right-hand side belong to the leading order
LO, i.e.\ they are proportional to $\alpha_f$ times a power of 
$\{\alpha_f\ln(m_h/\bar{\mu})\}$.  Only the fourth term is NLO; there 
is no $O(\alpha_f^2)$ term independent of $m_h$ because the 
NLO constant in Eq.~(\ref{a5a}) vanishes.  The fifth term is 
$O(\alpha_f^3)$ and $m_h$-independent, so it is the first example of a
NNLO term.  The three-loop result, including the NNNLO constant 
term, is now known \cite{CKS1}.


Now we would like to know what the renormalization group implies for 
matching relations of this type.  Some results for coefficients 
to a given order of perturbation theory already appear in 
\cite{BW,wetzel,B1}.  Consider Eq.\ (3) of Ref.~\cite{BW},
\begin{equation}
\frac{\alpha_f}{\pi} 
= \frac{\alpha_{f+1}}{\pi} 
   + \sum_{k=1}^\infty\Bigl(\frac{\alpha_{f+1}}{\pi}\Bigr)^{k+1}
             C_k\Bigl(\ln\frac{m_h^2}{\mu^2}\Bigr) + O(m_h^{-1})
\label{b1}
\end{equation}
where $C_k$ is a polynomial of degree $k$, as noted below Eq.~(8) of
Ref.~\cite{BW}:
\begin{equation}
C_k\, =\, c_{k,k}\bigl(\ln(m_h^2/\mu^2)\bigr)^k +\, \ldots\, + c_{k,0} .
\label{b2}
\end{equation}
The constants $c_{1,0},\,c_{2,0},\,c_{3,0},\,\ldots$ are the remainders
left when all terms depending on $\ln(m_h^2/\mu^2)$ are subtracted from
the leading-power functions\, $C_1,\,C_2,\,C_3,\ldots$.  Then, if all 
coefficients and RG functions are known to $k-1$ loops, the RG determines 
all $k$-loop coefficients $c_{k,j}$ in $C_k$ \emph{except} 
for $c_{k,0}$. The latter \emph{cannot} be deduced from the RG, to any 
number of loops; rather, $c_{k,0}$ must be calculated explicitly via a 
separate $k$-loop matching calculation.  For example, the NNLO coefficient 
$-11/(72\pi^2)$ in (\ref{e3c}) is just $-c_{2,0}$.


Instead of Eq.~(\ref{b1}), we prefer to consider the inverse relation
\begin{equation}
\alpha_{f+1} = \alpha_f + \sum_{k=1}^\infty \alpha_f^{k+1}
               P_k\Bigl(\ln\frac{m_h}{\bar{\mu}}\Bigr) 
               + O(m_h^{-1})
\end{equation}
because that is what is required in order to take $m_h\to\infty$ with 
$\alpha_f$ held fixed.  The analogue of Eq.~(\ref{b2}) is
\begin{align}
P_k = p_{k,k}\bigl(\ln(m_h/\bar{\mu})\bigr)^k 
          &+ p_{k,k-1}\bigl(\ln(m_h/\bar{\mu})\bigr)^{k-1} 
\nonumber \\
          &+ \ldots + p_{k,0} .
\label{b4}
\end{align}
An analysis in the style of Bernreuther and Wetzel produces
identical conclusions for the remainder constants $p_{k,0}$: 
given $p_{1,0}, p_{2,0},\ldots p_{k-1,0}$, one can use the RG 
to deduce $p_{k,k}, \ldots , p_{k,1}$ but not $p_{k,0}$.


Most practical applications require that terms of the same logarithmic 
order be summed.  This is straightforward for LO logarithms, because 
the LO coefficients $c_{k,k}$ in (\ref{b2}) obey a simple relationship 
\cite{BW}
\begin{equation}
c_{k,k} = \bigl(c_{1,1}\bigr)^{k} 
\end{equation}
which makes the series geometric:
\begin{equation}
\alpha_f\, 
\underset{\mbox{\scriptsize LO}}{=}\, \alpha_{f+1}\!\Bigm/\!\Bigl(1 
    - \frac{\alpha_{f+1}}{3\pi}\ln\frac{m_h}{\bar{\mu}}\Bigr) .
\label{a9}
\end{equation}
This expression is leading order (LO) with respect to powers of 
$\alpha_{f+1}=\alpha_F$ and $\ln(m_h/\bar{\mu})$.  Eq.~(\ref{a9}) 
implies that the term $O(\alpha_f, m_h^{-1})$ in (\ref{a6}) is NLO 
or higher order:
\begin{equation}
\alpha_{f+1}^{\,-1} - \alpha_f^{\,-1}\,
 \underset{\mbox{\scriptsize LO}}{=}\, \frac{1}{3\pi}\ln\frac{m_h}{\bar{\mu}} .
\label{a9a}
\end{equation}
This leads directly to the inverse of (\ref{a9}), 
\begin{equation}
\alpha_{f+1}\, \underset{\mbox{\scriptsize LO}}{=}\, \alpha_f\!\Bigm/\!\Bigl(1 
            + \frac{\alpha_f}{3\pi}\ln\frac{m_h}{\bar{\mu}}\Bigr)
\label{a8}\end{equation}
where now LO refers to powers of $\alpha_f$ and $\ln(m_h/\bar{\mu})$.
Note that the LO coefficients $p_{k,k}$ in Eq.~(\ref{b4}) are given by
\begin{equation}
p_{k,k}\, =\, \bigl(-1\!\bigm/\!3\pi\bigr)^k .
\label{b5}
\end{equation}


Beyond LO, formulas for all the relevant coefficients become
complicated, making order-by-order summation too cumbersome to be practical.
The rest of this paper is concerned with a RG analysis which allows us
to consider matching relations to a given logarithmic order without 
having to expand in perturbative order.


\section{Matching function}\label{mf}


Any RG analysis of decoupling involves ar least \emph{two} renormalization
groups:  one for the initial $F$-flavor theory, and one for each 
$f$-flavor theory produced as a heavy particle decouples.  We append
a flavor subscript to make the distinction, \emph{viz.}\ RG$_F$ or RG$_f$.

 
A key observation is that any quantity which is RG$_{F}$ invariant
must also be RG$_{f}$ invariant ($f < F$).  For example, Witten's RG$_F$
invariant running coupling $\run{h}$ must satisfy the condition
\begin{equation}
{\cal D}_f\run{h} 
= \biggl(\bar{\mu}\frac{\del\;}{\del\bar{\mu}}
            + \beta\frac{\del\ }{\del\alpha}\biggr)_{\!f}\run{h}
= 0 .
\label{b9}\end{equation}
Generally, the substitution
\begin{equation}
{\cal D}_F\ \ \longrightarrow\ \ {\cal D}_f  
\end{equation}
works when applied to any quantity which survives the limit
$m_h \to \infty$. An example is the formula
\begin{equation}
{\cal D}_6 \alpha_5\, =\, {\cal D}_5\alpha_5\, =\, \beta_5(\alpha_5)  .
\label{c1}\end{equation}
which agrees with the general result (\ref{e1}).
However, \emph{the converse is not generally true}.  For example, the 
top-quark mass $m_t$ is RG$_{f=5}$ invariant, but it is certainly 
\emph{not} RG$_{f=6}$ invariant.  Therefore a study of the RG for the 
original $F$-flavour theory is both necessary and sufficient for the 
full implications of the RG to be understood.


Our starting point is Witten's definition (\ref{f3}) of the invariant 
running coupling $\run{h}$.  Let us regard this as a formula for 
$\ln(m_h/\bar{\mu})$ in terms of $\run{h}$ and $\alpha_F$.  Specifically,
the right-hand side is an integral from $\alpha_F$ to $\run{h}$ involving 
RG$_F$ functions $\beta_F$ and $\delta_F$.  Can a similar formula be 
constructed from RG$_f$ functions such that this mass logarithm becomes
a function of $\run{h}$ and $\alpha_f$? 


If such a formula exists, it must be consistent with the requirements 
of the RG$_F$ group for the \emph{original} theory.  However, mass
renormalization produces an unwelcome $F$ dependence in equations 
such as
\begin{equation}
{\cal D}_F\ln(m_h/\bar{\mu}) = \delta_F(\alpha_F) - 1 .
\label{c1a}
\end{equation}
So let us amend the proposal: instead of the 
$\overline{\mbox{\small MS}}_F$ mass $m_h$, consider Witten's
invariant mass%
%
\footnote{See Eq.~(16) of \cite{witten}.  Similar effective masses have 
been invented for the cases of large momenta \cite{weinberg1} and light 
quarks \cite{floratos}.  Their RG invariance makes them useful in
phenomenology \cite{witten,floratos} and lattice calculations \cite{CKS2}.}
%
\begin{equation}
\bar{m}_h 
 = m_h\exp\int^{\run{h}}_{\alpha_F}\!\!dx\, \delta_F(x)/\beta_F(x) .
\label{c5}
\end{equation}
Since $\bar{m}_h$ is RG$_F$ invariant,
\begin{equation} 
{\cal D}_F \bar{m}_h = 0
\end{equation}
replacing $m_h$ by $\bar{m}_h$ in (\ref{c1a}) eliminates the unwanted 
dependence on $\delta_F$:
\begin{equation}
{\cal D}_F\ln(\bar{m}_h/\bar{\mu}) = - 1 .
\label{c5a}
\end{equation}
Notice that the formula (\ref{c8}) for $\ln(\bar{m}_h/\bar{\mu})$ is an
immediate consequence of the definitions (\ref{f3}) and (\ref{c5}) of 
$\run{h}$ and $\bar{m}_h$.  As a check, ${\cal D}_F$ can be applied to the
right-hand side of (\ref{c8}) to give the result $-1$, in agreement with
Eq.~(\ref{c5a}).


Now observe that, because of Eq.~(\ref{e1}), the replacement $F \to f$ 
everywhere on the right-hand side of (\ref{c8}) produces a quantity
which transforms in the same way under the RG$_F$ of the \emph{original} 
theory:
\begin{equation}
{\cal D}_F\int_{\alpha_f}^{\run{h}}\hsp{-2.5}dx\,\frac{1}{\beta_f(x)}  
 = - 1 .
\label{c5b}
\end{equation}
Comparing Eqs.~(\ref{c5a}) and (\ref{c5b}), we see that a RG$_{F}$ 
invariant quantity ${\cal F}$ can be defined as follows:
\begin{equation}
\ln\frac{\bar{m}_h}{\bar{\mu}}\,
=\, \int_{\alpha_f}^{\run{h}}\hsp{-2.5}dx\,
      \frac{1}{\beta_f(x)}\, + {\cal F} \ \ \Longrightarrow\ \
{\cal D}_F {\cal F}\, =\, 0 .
\label{c6}\end{equation}
Since ${\cal F}$ is dimensionless, it can depend on $\run{h}$, 
but RG$_{F}$ invariance forbids dependence on other dimensionless 
variables, such as $\alpha_F$, $\alpha_f$, $\bar{m}_h/m_h$, or 
$m_h/{\bar \mu}$. We call it the
\textsl{matching function}: 
\begin{equation}
{\cal F}\, =\, {\cal F}_{F\to f}(\run{h}) .
\label{c7}\end{equation}
This completes the derivation of the key equations (\ref{c8}) and 
(\ref{c9}):
\begin{align*}
\ln\frac{\bar{m}_h}{\bar{\mu}}\,
&=\, \int_{\alpha_F}^{\run{h}}\hsp{-2.5}dx\,
      \frac{1}{\beta_F(x)}   \\[1mm]
\ln\frac{\bar{m}_h}{\bar{\mu}}\,
&=\, \int_{\alpha_f}^{\run{h}}\hsp{-2.5}dx\,
      \frac{1}{\beta_f(x)}\, + {\cal F}_{F\to f}(\run{h}) .
\end{align*}
When $\run{h}$ is eliminated from these equations, the desired matching 
relation (\ref{d0}) is obtained:
\[
\alpha_F\, =\, \alpha_F\bigl(\alpha_f,\ln(\bar{m}_h/\bar{\mu})\bigr),
\ \ F=f+1 . \]


\section{Perturbation theory for the matching function}\label{pt}


Perturbative matching relations can be used to determine successive
coefficients in the Taylor series of the matching function 
${\cal F}_{F\to f}(\run{h})$.  One needs to calculate the 
mass-independent terms $c_{k,0}$ in (\ref{b2}) or equivalently 
$p_{k,0}$ in (\ref{b4}).  As noted in Sec.~\ref{mcc}, 
these constants cannot be deduced from the RG. This corresponds
to the fact that ${\cal F}_{F\to f}$ cannot be deduced from the 
RG because it is RG$_{F}$ invariant.

 
To illustrate the procedure, let us deduce the consequences for 
${\cal F}_{f+1 \to f}$ of the perturbative matching relation (\ref{b1}). 
We need the two-loop $\beta$-function for three colors:
\begin{equation}
\beta_f(x) 
 = - \frac{x^2}{6\pi}(33-2f) - \frac{x^3}{12\pi^2}(153-19f) + O(x^4) .
\label{d1}
\end{equation}
Its reciprocal is
\begin{equation}
\bigl\{\beta_f(x)\bigr\}^{-1}\, 
=\, - \frac{6\pi}{33-2f}
      \Bigl(\frac{1}{x^2} - \frac{b_f}{x} + b_f'\Bigr) + O(x)
\label{d2}
\end{equation}
where $b_f$ stands for the constant
\begin{equation}
b_f\, =\, \frac{1}{2\pi}\frac{153-19f}{33-2f}
\label{d3}
\end{equation}
and $b_f'$ is another constant whose precise value is not of concern here.


The expansion (\ref{d2}) inserted into (\ref{c8}) and (\ref{c9}) 
yields the following equations:
\begin{align}
\frac{33-2(f+1)}{6\pi}\ln\frac{\bar{m}_h}{\bar{\mu}}
&= {\run{h}}^{-1} - \alpha_{f+1}^{-1} 
     + b_{f+1} \ln\frac{\run{h}}{\alpha_{f+1} }  \nonumber \\
&\phantom{=\ } + O(\alpha_f^2)
\label{d4}  \\[1mm]
\frac{33-2f}{6\pi}\ln\frac{\bar{m}_h}{\bar{\mu}}%  \nonumber \\
&= {\run{h}}^{-1} - \alpha_f^{-1} 
     + b_f\ln\frac{\run{h}}{\alpha_f} + O(\alpha_f^2)
\nonumber \\
&\phantom{=\ }
    + \frac{33-2f}{6\pi}{\cal F}_{f+1\to f}(\run{h}).
\label{d5} 
\end{align}
Note that the corrections are $O(\alpha_f^2)$: contributions to 
(\ref{d4}) and (\ref{d5}) from the constant term in (\ref{d2}) are
\[ b_{f+1}'\bigl(\run{h} - \alpha_{f+1}\bigr) = O(\alpha_f^2) 
\ \ \mbox{and}\ \ 
b_f'\bigl(\run{h} - \alpha_f\bigr)  = O(\alpha_f^2)   \]
because $\run{h}$, $\alpha_{f+1}$ and $\alpha_f$ differ by $O(\alpha_f^2)$.


The next step is to eliminate $\run{h}$.  This is partially achieved by
subtracting (\ref{d4}) from (\ref{d5}):
\begin{align}
\frac{1}{3\pi}\ln\frac{\bar{m}_h}{\bar{\mu}}
 =\ &\alpha_{f+1}^{-1} - \alpha_f^{-1} + b_f\ln\frac{\run{h}}{\alpha_f}
     - b_{f+1}\ln\frac{\run{h}}{\alpha_{f+1}}
\nonumber \\ 
    &+ \frac{33-2f}{6\pi}{\cal F}_{f+1\to f}(\run{h}) + O(\alpha_f^2) .
\label{d6}
\end{align}
Since $\ln(\run{h}/\alpha_f)$ and $\ln(\run{h}/\alpha_{f+1})$ 
are both $O(\alpha_f)$, Eqs.\ (\ref{a6}) and (\ref{d6}) imply
\[ {\cal F}_{f+1\to f}(\run{h}) = O(\alpha_f)  \]
and so, from Eqs.~(\ref{d4}) and (\ref{d5}), we conclude
\begin{align}
\alpha_{f+1}/\run{h} 
 &= 1 + \frac{\alpha_f}{6\pi}\Bigl( 33-2(f+1) \Bigr) \ln\frac{m_h}{\bar{\mu}}
      + O(\alpha_f^2)
\nonumber \\
\alpha_f/\run{h} 
 &= 1 + \frac{\alpha_f}{6\pi} \Bigl( 33-2f \Bigr) \ln\frac{m_h}{\bar{\mu}}
      + O(\alpha_f^2) .
\label{d9}
\end{align}
The logarithms of these expressions can then be substituted back into
Eq.~(\ref{d6}), with the result
\begin{align}
\frac{1}{3\pi}\ln\frac{\bar{m}_h}{\bar{\mu}}
  =\ &\alpha_{f+1}^{-1} - \alpha_f^{-1} 
    - \frac{19\alpha_f}{12\pi^2}\ln\frac{m_h}{\bar{\mu}}
\nonumber \\ 
    &+ \frac{33-2f}{6\pi}{\cal F}_{f+1\to f}(\run{h}) + O(\alpha_f^2) .
\label{d9a}
\end{align}



The next step is to relate the logarithms of $\bar{m}_h$ and $m_h$.
First substitute the three-color formula
\begin{equation}
\delta_f(x) = - \frac{2x}{\pi} + O(x^2)
\label{g1}
\end{equation}
into the definition (\ref{c5}) of $\bar{m}_h$,
\begin{equation}
\ln\frac{\bar{m}_h}{m_h} 
 = \frac{12}{33-2(f+1)}\ln\frac{\run{h}}{\alpha_{f+1}} + O(\alpha_f^2)
\end{equation}
and then substitute Eq.~(\ref{d9}) for $\alpha_{f+1}/\run{h}$.  The 
result is:
\begin{equation}
\ln\frac{\bar{m}_h}{\bar{\mu}} 
 = \Bigl(1-\frac{2\alpha_f}{\pi}\Bigr)\ln\frac{m_h}{\bar{\mu}} 
      + O(\alpha_f^2) .
\label{g1a}
\end{equation}


Then the logarithm of $\bar{m}_h$ can be eliminated from 
Eqs.~(\ref{d9a}) and (\ref{g1a}):
\begin{align}
\alpha_{f+1}
= \alpha_{f} 
   &- \frac{\alpha_{f}^2}{3\pi}\ln\frac{m_h}{\bar{\mu}}
   + \frac{\alpha_{f}^3}{9\pi^2}\ln^2\frac{m_h}{\bar{\mu}}
   - \frac{11\alpha_{f}^3}{12\pi^2}\ln\frac{m_h}{\bar{\mu}}
\nonumber \\
   &+ \frac{\alpha_{f}^2}{6\pi}(33-2f){\cal F}_{f+1\to f}(\run{h})
   + O(\alpha_{f}^4) .
\label{g1b}
\end{align}
Comparing this with the two-loop matching condition (\ref{e3c}), 
we see that all mass logarithms are correctly reproduced, and that 
the first non-zero term in the matching function can be deduced from 
the constant NNLO term in (\ref{e3c}):
\begin{equation}
{\cal F}_{f+1\to f}(\run{h}) 
 =  - \frac{11}{12\pi(33-2f)}\hsp{0.2}\run{h} + O(\run{h}^2) .
\label{e3d}
\end{equation}


The $O(\run{h}^2)$ term in ${\cal F}$ can be found by substituting 
(\ref{e3d}) back into Eq.~(\ref{c9}) and repeating the above process 
using the three-loop $\beta_f$ and two-loop $\delta_f$ functions.  
The answer follows by comparison with the known three-loop matching 
condition \cite{CKS1}.  That is the limit of current calculations, 
but in principle, this strategy could be pursued to any order, with 
all mass logarithms correctly reproduced.



\section{Closed expressions for heavy-quark logarithms}\label{ce}


The importance of the matching function ${\cal F}$ is that it allows 
us to work to a given logarithmic order without having to sum mass
logarithms order-by-order in perturbation theory.  Indeed, the role
of ${\cal F}$ is essentially the same as that of the RG functions 
$\beta$, $\gamma$ and $\delta$: each term in the series for ${\cal F}$ 
corresponds to a particular logarithmic order.  For LO and NLO,
${\cal F}$ does not contribute, but NNLO requires that the $O(\run{h})$ 
term in (\ref{e3d}) be included, NNNLO requires the $O(\run{h}^2)$ term,
and so on.


To illustrate, let us derive the closed NLO formula for the
matching relation between $\alpha_{f+1}$ and $\alpha_f$ which we 
announced in \cite{BCST}. 


As in the previous section, we insert the expansion (\ref{d2}) 
into (\ref{c8}) and (\ref{c9}), but this time we omit the 
NNLO term ${\cal F}_{f+1\to f}$:
\begin{align}
\frac{33-2(f+1)}{6\pi}\ln\frac{\bar{m}_h}{\bar{\mu}}
&\underset{\mbox{\scriptsize NLO}}{=} 
 {\run{h}}^{-1} - \alpha_{f+1}^{-1} + b_{f+1}\ln\frac{\run{h}}{\alpha_{f+1}}  
\label{d5a}\\
%
\frac{33-2f}{6\pi}\ln\frac{\bar{m}_h}{\bar{\mu}}
&\underset{\mbox{\scriptsize NLO}}{=} 
 {\run{h}}^{-1} - \alpha_f^{-1} + b_f\ln\frac{\run{h}}{\alpha_f} .
\label{d5b} 
\end{align}
The difference of these two equations is
\begin{align}
\frac{1}{3\pi}\ln\frac{\bar{m}_h}{\bar{\mu}}
\underset{\mbox{\scriptsize NLO}}{=}
\alpha_{f+1}^{-1} - \alpha_f^{-1} 
     &- (b_f-b_{f+1})\ln\frac{\alpha_f}{\run{h}}
\nonumber \\
     &- b_{f+1}\ln\frac{\alpha_f}{\alpha_{f+1}} .
\label{d8}
\end{align}
The logarithms on the right-hand side are already NLO, so we can use
the LO parts of (\ref{d5b}) and (\ref{d8}) to approximate their arguments:
\begin{align}
\alpha_f/\run{h} &\underset{\mbox{\scriptsize LO}}{=}
   1 + \frac{\alpha_f}{6\pi}(33-2f)\ln\frac{m_h}{\bar{\mu}}
\label{d9b} \\
%
\alpha_f/\alpha_{f+1} &\underset{\mbox{\scriptsize LO}}{=}
  1 + \frac{\alpha_f}{3\pi}\ln\frac{m_h}{\bar{\mu}} .
\label{d9c}
\end{align}
The result is a NLO generalisation of (\ref{a8}):
\begin{align}
&\alpha_{f+1} \underset{\mbox{\scriptsize NLO}}{=}
\nonumber \\
&\alpha_f \!\Bigm/\! \Bigl\{  
   1 + \frac{\alpha_f}{3\pi}\ln\frac{\bar{m}_h}{\bar{\mu}}
+ \alpha_f b_{f+1} \ln
 \Bigl(1 + \frac{\alpha_f}{3\pi}\ln\frac{m_h}{\bar{\mu}}\Bigr)
\nonumber \\
&\phantom{\alpha_f}\
+ \alpha_f (b_f - b_{f+1})\ln \Bigl[1 + 
   \frac{\alpha_f}{6\pi}(33-2f)\ln\frac{m_h}{\bar{\mu}}\Bigr]\Bigr\}  .
\label{e0}
\end{align}


 
If desired, $\ln(\bar{m}_h/\bar{\mu})$ can be eliminated in favor of 
$\ln({m}_h/\bar{\mu})$.  The leading NLO effects of mass
renormalization are due to the one-loop term of $\delta_f$ given by 
Eq.~(\ref{g1}).  When this term is substituted into the definition
(\ref{c5}) of $\bar{m}_h$, keeping all logarithms to this order, we
find an expression
\begin{equation}
\ln\frac{\bar{m}_h}{\bar{\mu}} 
 \underset{\mbox{\scriptsize NLO}}{=} \ln\frac{m_h}{\bar{\mu}} 
    + \frac{12}{31-2f}\Bigl(\ln\frac{\alpha_f}{\alpha_{f+1}}
            - \ln\frac{\alpha_f}{\run{h}}\Bigr) 
\label{e2}\end{equation}
to which Eqs.~(\ref{d9b}) and (\ref{d9c}) can be readily applied:
\begin{align}
\ln\frac{\bar{m}_h}{\bar{\mu}}
&\underset{\mbox{\scriptsize NLO}}{=} \ln\frac{m_h}{\bar{\mu}}
  + \frac{12}{31\!-\!2f}\ln\Bigl(1 
       + \frac{\alpha_f}{3\pi}\ln\frac{m_h}{\bar{\mu}}\Bigr) 
\nonumber \\
&\phantom{\underset{\mbox{\scriptsize NLO}}{=}}\
   - \frac{12}{31\!-\!2f}\ln\Bigl(
      1 + \frac{\alpha_f}{6\pi}(33\!-\!2f)\ln\frac{m_h}{\bar{\mu}}\Bigr) .
\label{y1}
\end{align}
So, by combining (\ref{e0}) and (\ref{y1}), we arrive at a complete NLO
formula for the matching condition:
%
\begin{align}
\alpha_{f+1}^{-1} &\underset{\mathrm{NLO}}{=}
\alpha_f^{-1} + \frac{1}{3\pi}\ln\frac{m_h}{\bar{\mu}} +
c_f\ln\Bigl[1 + \frac{\alpha_f}{3\pi}\ln\frac{m_h}{\bar{\mu}}\Bigr]
\nonumber \\[1mm]
&\phantom{\underset{\mathrm{NLO}}{=} \alpha_f^{-1}\ }
+ d_f\ln\Bigl[1 + \frac{\alpha_f}{6\pi}(33-2f)
      \ln\frac{m_h}{\bar{\mu}}\Bigr] 
 , \nonumber \\
c_f &=  \frac{142-19f}{2\pi(31\!-\!2f)} \hsp{2},\hsp{2}
d_f =  \frac{57+16f}{2\pi(33\!-\!2f)(31\!-\!2f)} .
\label{y3}
\end{align}
%

The same equations can also be used to obtain equations for the RG 
invariant $\run{h}$ (also announced in \cite{BCST}). Eqs.~(\ref{d3}),
(\ref{d5b}) and (\ref{d9b}) imply the NLO formula
\begin{align}
\run{h}^{-1} &\underset{\mathrm{NLO}}{=}
  \alpha_f^{-1} + \frac{1}{6\pi}(33-2f)\ln\frac{\bar{m}_h}{\bar{\mu}}
\nonumber \\
&\phantom{\underset{\mathrm{NLO}}{=}}
  + \frac{153-19f}{2\pi(33\!-\!2f)}
  \ln\Bigl[1 + \frac{\alpha_f}{6\pi}(33-2f)\ln\frac{m_h}{\bar{\mu}}\Bigr] .
\label{y2}
\end{align}
Again, Eq.~(\ref{y1}) can be used to write $\ln(\bar{m}_h/\bar{\mu})$ 
in terms of $\ln(m_h/\bar{\mu})$.  This leads to the following 
asymptotic formula for $\run{h}$ as $m_h \to \infty$:
%
\begin{gather}
\run{h}\ \sim\ 6\pi \!\Bigm/\! \Bigl\{
    (33-2f)\ln\frac{m_h}{\bar{\mu}} 
  + K_f\ln\ln\frac{m_h}{\bar{\mu}} + O(1)\Bigr\}  
 , \nonumber \\
K_f = \frac{3(153-19f)}{33-2f} - \frac{12(33-2f)}{31-2f} .
\end{gather}
%
These results show that we have complete control over the matching process.
Once closed expressions such as (\ref{y3}) and (\ref{y2}) have been 
obtained, RG invariance can be maintained for each logarithmic order, 
and so there is no need to truncate to a given order of perturbation 
theory. 




\section{Mass-matching function}\label{mmf}


Most of the analysis above is restricted to the case of just one heavy 
quark $h$, but it can be readily extended to include \emph{sequential} 
decoupling, where heavy quarks are decoupled \emph{one at a time}.  
The new feature which arises is the need to match the mass of the 
second heavy quark.  For example, suppose that, having decoupled the 
$t$ quark in $F=6$ flavor QCD, we would like to decouple the $b$ quark 
as well:
\begin{equation} 
m_t \to \infty\ \ \mbox{first},\ \mbox{then}\ \ m_b \to \infty.
\label{j8}
\end{equation}
Then it will be necessary to match the six-flavor definition 
${m_b}_6 = m_b$ of the bottom quark mass to its five-flavour 
definition ${m_b}_5$.


As for the matching of couplings, the key is to find a RG invariant 
definition of mass to which the Appelquist-Carrazone theorem \cite{AC} 
can be applied.  This problem was solved by Bernreuther \cite{B1}, 
again by recourse to the momentum subtraction (MO) scheme.


Let $A_\ell(p^2)$ and $B_\ell(p^2)$ denote the form factors
for the unrenormalized 1PI light-quark self-energy amplitude\, 
$-i(/\hsp{-2}p A_\ell - {m_0}_\ell B_\ell)$.  This corresponds 
to the unrenormalized quark propagator
\begin{eqnarray}
S(p)
&=& \frac{i}{/\hsp{-2}p(1\!-\!A_\ell) 
                           - {m_0}_\ell(1\!-\!B_\ell)}
\nonumber \\
&=& \bigl(1\!-\!A_\ell\bigr)^{-1}
  \frac{i}{/\hsp{-2}p - {m_0}_\ell(1\!-\!B_\ell)/(1\!-\!A_\ell)}.
\end{eqnarray}
Define MO light-quark masses $M^\mathrm{MO}_\ell(Q)$ at a fixed 
space-like point $p^2 = -Q^2$:
\begin{equation}
M^\mathrm{MO}_\ell(Q) 
= {m_0}_\ell\bigl(1\!-\!B_\ell(-Q^2)\bigr)\!\bigm/\!
                            \bigl(1\!-\!A_\ell(-Q^2)\bigr) .
\end{equation}
This mass is RG invariant because, expressed in terms of 
renormalized quantities, it is finite.  The choice of space-like 
subtraction point $-Q^2$ means that mass renormalization, as well as 
coupling-constant and wave-function renormalization, is performed 
off-shell:%
%
\footnote{A similar MO definition for heavy-quark masses $M_h$ 
\cite{Appelquist} yields a $\beta$-function $\beta(g,M_h/Q)$ 
\cite{RGGP} with smooth threshold behavior at $Q \sim M_h$.}
\begin{equation} iS^{-1}(p)\bigr|_{p^2 = -Q^2}
  = /\hsp{-2}p - M^\mathrm{MO}_\ell(Q) 
  \not= 0 \ \mbox{at}\  p^2 = -Q^2.
\end{equation}
This avoids problems with Bloch-Nordsieck logarithms 
produced by $n$-loop perturbation theory at the on-shell 
point $p^2 \sim m^2$ \cite{AC}:
\[ iS^{-1}(p) \sim (p^2\hsp{-0.2}-\hsp{-0.2}m^2)^{-1}
 \bigl[\ln(p^2\hsp{-0.2}-\hsp{-0.2}m^2)\bigr]^n .  \]



A complication familiar to many authors \cite{gauge-dep} is that, 
unlike an on-shell renormalized mass, $M^\mathrm{MO}_\ell(Q)$\, is 
gauge dependent.  Despite this, the resulting mass-matching relation 
between $m_{\ell f}$ and $m_{\ell(f+1)}$ is gauge invariant \cite{B1} 
because the relations between $\overline{\mbox{\small MS}}$ masses and 
their bare counterparts are gauge invariant.  In two-loop perturbation 
theory, the result \cite{B1} for QCD with three colors and $f$ light 
flavours is
\begin{align}
\frac{m_{\ell f}}{m_{\ell(f+1)}} =
  1 &+ \frac{\alpha_{f+1}^2}{12\pi^2} 
      \biggl(\ln^2 \frac{m_h^2}{\bar{\mu}^2} + 
        \frac{5}{3} \ln\frac{m_h^2}{\bar{\mu}^2} +
        \frac{89}{36} \biggr)    \nonumber \\
    &+ O\bigl(\alpha_{f+1}^3\bigr) . 
\label{g0a}  
\end{align}


We would like to extend this result to include all terms of the same
\emph{logarithmic} order.  This is achieved by introducing our second 
matching function, ${\cal G}$ -- the matching function for mass 
renormalization.  On order to reduce notational complexity, we
consider the special case $f=5$ mentioned at the beginning of this
section. 


Consider the RG equation
\begin{equation}
{\cal D}_6 \ln\frac{{m_b}_6}{{m_b}_5}\;
=\; \delta_6(\alpha_6) - \delta_5(\alpha_5)   
\label{g1d} 
\end{equation}
The leading power in a large-$m_t$ expansion of\, ${m_b}_6/{m_b}_5$\, 
is a function of\, $\alpha_5$ and\, $\ln(m_t/\bar{\mu})$\, but does not
depend on light-quark masses, so the general solution of (\ref{g1d}) is
\begin{eqnarray}
\ln\frac{{m_b}_6}{{m_b}_5}
&=& \int_{\alpha_5}^{\run{t}}\hsp{-2}dx\,
                   \frac{\delta_5(x)}{\beta_5(x)}
   - \int_{\alpha_6}^{\run{t}}\hsp{-2}dx\,
                   \frac{\delta_6(x)}{\beta_6(x)}\,
\nonumber \\[1mm]
&\phantom{=}&
   +\ {\cal G}_{6\to 5}(\run{t})\, +\, O(m_t^{-1}) 
\label{g2} 
\end{eqnarray}
where ${\cal G}_{6\to 5}(\run{t})$ is the \textsl{mass-matching function}. 
Like\, ${\cal F}_{6\to 5}(\widetilde{\alpha}_t)$\, in Eq.~(\ref{c6}), 
${\cal G}_{6\to 5}$ arises as an integration constant of a
RG equation, so it can depend only on the RG invariant\, $\run{t}$.
Also like ${\cal F}$, it cannot be deduced from the RG and must be 
calculated separately. 


At one-loop order, there are no top-quark corrections (Fig.~\ref{fig1}),
so the mass-matching condition is trivial \cite{B1}: 
\begin{eqnarray} 
M_b^\mathrm{MO}
&=& 
m_{b,6} \bigl[1 + \alpha_6\{\mbox{1-loop}\}\bigr]\,
                         + O(\alpha_6^2)\,
\nonumber \\
&=&  
m_{b,5} \bigl[1 + \alpha_5\{\mbox{1-loop}\}\bigr]\,
                         +\, O(\alpha_5^2) .
\end{eqnarray}
Here $\alpha_f\{\mbox{1-loop}\}$ denotes the self-energy amplitude
derived from Fig.~\ref{fig1}.  Eliminating\, $M_b^\mathrm{MO}$, we find
\begin{equation}
\ln\frac{{m_b}_6}{{m_b}_5}\, 
 =\, (\alpha_5 - \alpha_6)\{\mbox{1-loop}\}\, +\, O(\alpha_5^2)\,
 =\,O(\alpha_5^2) . 
\label{g4} 
\end{equation}

\begin{figure}[b] 
\begin{center}
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\begin{picture}(28.6,6.88)(0,-4.3)
\linethickness{0.25mm}
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\put(14.3,-4.3){\makebox[0mm]{\small $b$}}
\end{picture}
%
{\caption[Delta]
{One-loop correction to mass renormalization.} \label{fig1}} 
\end{center} 
\end{figure}


At this point, we need to specify what is LO, NLO, and so on.  If we were
talking only about corrections to mass, we might consider terms\,
$\sim \bigl(\alpha_f\ln m_t/\bar{\mu}\bigr)^n$\, as LO, but in general,
it is more convenient to regard them as NLO.  That is because mass
renormalization does not contribute to physical amplitudes in LO.
With this convention, Eqs.~(\ref{d2}), (\ref{d9}), and (\ref{g1}) imply
%
\begin{align}
\int_{\alpha_f}^{\run{t}}\hsp{-2}dx\,
               \frac{\delta_f(x)}{\beta_f(x)}\;
&\underset{\mbox{\scriptsize NLO}}{=}\; 
      \frac{6}{\frac{33}{2}\!-\!f}
      \ln\frac{\run{t}}{\alpha_f}
\nonumber \\
&\underset{\mbox{\scriptsize NLO}}{=}\; 
      -\,\frac{6}{\frac{33}{2}\!-\!f}
  \ln\biggl[ 1 + \frac{\alpha_f}{3\pi}\biggl(\frac{33}{2}-f\biggr)
         \ln\frac{m_t}{\bar{\mu}}\biggr] .
\label{g5} 
\end{align}
In lowest-order perturbation theory, we have
%
\begin{align}
\int_{\alpha_5}^{\run{t}}\hsp{-2}dx\,
              &\frac{\delta_5(x)}{\beta_5(x)}
 - \int_{\alpha_6}^{\run{t}}\hsp{-2}dx\,
               \frac{\delta_6(x)}{\beta_6(x)}\,
\nonumber \\
&=
\frac{2}{\pi}\bigl(\alpha_6 - \alpha_5\bigr)
       \ln\frac{m_t}{\bar{\mu}} + O\bigl(\alpha_5^2\bigr)
= O\bigl(\alpha_5^2\bigr)
\label{g6} 
\end{align}
%
since the couplings $\alpha_6$ and $\alpha_5$ differ by $O(\alpha_5^2)$.
{}From Eqs.~(\ref{g2}), (\ref{g4}) and (\ref{g6}), we conclude
%
\begin{equation}
{\cal G}_{6\to 5}(\run{t})\, =\, O(\run{t}^2) .
\end{equation}
%

Having established that ${\cal G}$ is irrelevant at NLO, we neglect it
in Eq.~(\ref{g2}) and substitute (\ref{g5}). This yields the complete 
NLO expression:
\begin{align}
\ln\frac{{m_b}_6}{{m_b}_5}\;
&\underset{\mbox{\scriptsize NLO}}{=}\;
 \frac{6}{\frac{33}{2}\!-\!6}
  \ln\biggl[ 1 + \frac{\alpha_6}{3\pi}\biggl(\frac{33}{2}-6\biggr)
         \ln\frac{m_t}{\bar{\mu}}\biggr]
\nonumber \\
&\phantom{\underset{\mbox{\scriptsize NLO}}{=}}\;
  - \frac{6}{\frac{33}{2}\!-\!5}
  \ln\biggl[ 1 + \frac{\alpha_5}{3\pi}\biggl(\frac{33}{2}-5\biggr)
         \ln\frac{m_t}{\bar{\mu}}\biggr] . 
\label{g8} 
\end{align}
If desired, $\alpha_6$ can be eliminated in favour of $\alpha_5$ via
Eq.~(\ref{a8}).  Note that the $O(\alpha_5^2)$ NLO term is a double 
logarithm which arises from the diagram with a $t$-loop inserted in the 
gluon propagator (Fig.~\ref{fig2}):
%
\begin{figure}[b] 
\begin{center}
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\begin{picture}(28.6,10)(0,-4)
\linethickness{0.25mm}
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\put(22.6,0.7){\makebox[0mm]{\small $b$}}
\put(14.3,-4){\makebox[0mm]{\small $b$}}
\put(14.3,4.4){\makebox[0mm]{\small $t$}}
%
\end{picture}
{\caption[Delta]
{Two-loop heavy-quark contribution to mass renormalization of the bottom
quark.} \label{fig2}} 
\end{center} 
\end{figure}
%
%
\begin{align}
\ln\frac{{m_b}_6}{{m_b}_5}\;
&\underset{\mbox{\scriptsize NLO}}{=}\; 
 \frac{2}{\pi}\bigl(\alpha_6 - \alpha_5\bigr)
       \ln\frac{m_t}{\bar{\mu}}
 - \frac{\alpha_6^2}{3\pi^2}\biggl(\frac{33}{2}\!-\!6\biggr)
       \ln^2\frac{m_t}{\bar{\mu}}
\nonumber \\
&\phantom{\underset{\mbox{\scriptsize NLO}}{=}}\;\
 + \frac{\alpha_5^2}{3\pi^2}\biggl(\frac{33}{2}\!-\!5\biggr)
       \ln^2\frac{m_t}{\bar{\mu}} + O\bigl(\alpha_5^3\bigr) 
\nonumber \\
&\underset{\mbox{\scriptsize NLO}}{=}\;
 -\ \frac{1}{3}\biggl(\frac{\alpha_5}{\pi}\biggr)^2
       \ln^2\frac{m_t}{\bar{\mu}} + O\bigl(\alpha_5^3\bigr). 
\label{g9} 
\end{align}
This reproduces the NLO term of Bernreuther's result (\ref{g0a}) for%
%
\footnote{
Note that Bernreuther expands in $\alpha_6$ instead of $\alpha_5$, 
but to this order the coefficient is the same.}
% 
$m_{f-1}/m_f$. 



Eq.~(\ref{g8}) generates the complete set of NLO logarithms.  They 
correspond to diagrams with a string of one-loop $t$-quark bubbles 
inserted in the gluon propagator of the one-loop $b$-quark self-energy 
amplitude. 



The constant term $89\alpha_f^2/(432\pi^2)$ in Eq.~(\ref{g0a})
corresponds to the first non-zero contribution to the matching 
function ${\cal G}$.  We state the result for any value of $f$:
%
\begin{equation}
{\cal G}_{f+1 \to f}(\run{h}) 
 = - \frac{89}{432\pi^2}\run{h}^2 + O(\run{h}^3).
\label{h2} 
\end{equation}
%
This term is required if NNNLO corrections are being calculated.


\section{APPLICATION TO SEQUENTIAL DECOUPLING}\label{asd}


When decoupling the $b$ quark, it is natural to define five-flavor
quantities $\mbox{$\run{b}$}_5$ and $\mbox{$\bar{m}_b$}_5$ 
by analogy with the six-flavor running coupling $\run{t}$ 
and mass $\bar{m}_t$ for the top quark: \vspace{-1mm}
%
\begin{eqnarray}
\ln\frac{{m_b}_5}{\bar{\mu}}
&=& \int_{\alpha_5}^{\mbox{$\scriptstyle \run{b}$}_5}
      \hsp{-1}dx\,\frac{1-\delta_5(x)}{\beta_5(x)}
\label{h3a} \\[0.5mm]
\ln\frac{\mbox{$\bar{m}_b$}_5}{{m_b}_5}
&=& \int_{\alpha_5}^{\mbox{$\scriptstyle \run{b}$}_5}
      \hsp{-1}dx\,\frac{\delta_5(x)}{\beta_5(x)} .
\label{h3}
\end{eqnarray}
%
Clearly, both $\mbox{$\run{b}$}_5$ and $\mbox{$\bar{m}_b$}_5$ 
are RG$_{f=5}$ invariant,
%
\begin{equation}
{\cal D}_5\mbox{$\run{b}$}_5 = 0
\hsp{4},\hsp{4}
{\cal D}_5\mbox{$\bar{m}_b$}_5 = 0
\end{equation}
but in fact, they are also RG$_{f=6}$ invariant as a consequence of
Eqs.~(\ref{e1}) and (\ref{eq:e2}):
\begin{equation}
{\cal D}_6\mbox{$\run{b}$}_5 = 0
\hsp{4},\hsp{4}
{\cal D}_6\mbox{$\bar{m}_b$}_5 = 0 .
\end{equation}



This means that $\mbox{$\widetilde{\alpha}_b$}_5$ and 
$\mbox{$\bar{m}_b$}_5$ can be expressed in terms of invariants
of the original six-flavor theory.%
\footnote{  
This property is essential for any generalization of the analysis 
to simultaneous decoupling, where all couplings and masses, running 
or otherwise, will have to be defined \emph{only} in terms of 
the initial theory ($F=6$) or the residual theory 
($f=4$ if just the $t$ and $b$ are being decoupled), with 
no reference to five-flavor couplings or masses.}
%
To see this, first combine Eqs.~(\ref{h3}) and (\ref{c5}), and 
then (\ref{g2}):
%
\begin{align}
\ln\frac{\bar{m}_t}{m_t}
-\ln\frac{\mbox{$\bar{m}_b$}_5}{{m_b}_5}
&= \int_{\alpha_6}^{\run{t}}
      \hsp{-1}dx\,\frac{\delta_6(x)}{\beta_6(x)}
   -\int_{\alpha_5}^{\mbox{$\scriptstyle \run{b}$}_5}
      \hsp{-1}dx\,\frac{\delta_5(x)}{\beta_5(x)}
\nonumber \\[0.5mm]
&= - \ln\frac{{m_b}_6}{{m_b}_5} 
 + \int_{\mbox{$\scriptstyle \run{b}$}_5}^{\run{t}}
      \hsp{-1}dx\,\frac{\delta_5(x)}{\beta_5(x)}
\nonumber \\[0.5mm]
&\phantom{+}\
 + {\cal G}_{6\to 5}(\run{t}) + O(m_t^{-1}) .
\end{align}
%
This equation simplifies to
\begin{align}
\ln\frac{\bar{m}_t}{\mbox{$\bar{m}_b$}_5}\,
&=\, \ln\biggl(\frac{m_t}{m_b}\biggr)_{\!6} 
  + \int_{\mbox{$\scriptstyle \run{b}$}_5}^{\run{t}}
      \hsp{-1}dx\,\frac{\delta_5(x)}{\beta_5(x)}
\nonumber \\
&\phantom{=}\ + {\cal G}_{6\to 5}(\run{t}) + O(m_t^{-1}) .
\label{h7} 
\end{align}
%
The sum of Eqs.~(\ref{h3a}) and (\ref{h3})
\begin{equation}
\ln\frac{\mbox{$\bar{m}_b$}_5}{\bar{\mu}}
= \int_{\alpha_5}^{\mbox{$\scriptstyle \run{b}$}_5}
     \hsp{-1}dx\,\frac{1}{\beta_5(x)}  .
\end{equation}
can be subtracted from Eq.~(\ref{c9}), with the result
\begin{equation}
\ln\frac{\bar{m}_t}{\mbox{$\bar{m}_b$}_5} = 
\int_{\mbox{$\scriptstyle \run{b}$}_5}^{\run{t}}
\hsp{-1}dx\, \frac{1}{\beta_5(x)} 
+ {\cal F}_{6\to 5}(\run{t})
+ O(m_t^{-1}) .
\label{i0} \end{equation}
If (\ref{i0}) is now combined with (\ref{h7}), we find that 
$\mbox{$\run{b}$}_5$ and hence $\mbox{$\bar{m}_b$}_5$
can be expressed in terms of RG invariants of the original six-flavor
theory, \emph{viz.}~$\run{t}$ and the ratio $(m_t/m_b)_6$:
\begin{align}
\ln\biggl(\frac{m_t}{m_b}\biggr)_{\!6}\, 
&=
\int_{\mbox{$\scriptstyle \run{b}$}_5}^{\run{t}}
\hsp{-1}dx\, \frac{1-\delta_5(x)}{\beta_5(x)} 
\nonumber \\
&\phantom{=}\
+ {\cal F}_{6\to 5}(\widetilde{\alpha}_t) 
- {\cal G}_{6\to 5}(\widetilde{\alpha}_t)\, 
+\, O(m_t^{-1}) .
\label{i1} \end{align}


The sequential decoupling of the $t$ and $b$ quarks refers to the
limiting procedure
\begin{equation}
\ln(m_t/\bar{\mu}) \gg \ln(m_b/\bar{\mu}) \rightarrow \infty 
\end{equation}
where we choose a six-flavor definition for $m_b$ as well as $m_t$. 
Leading-power six-flavor amplitudes are represented by logarithmic 
expansions for $t$-quark decoupling
\begin{equation}
\widetilde{\cal A}_6 \sim \sum_{p \geqslant 0} \widetilde{\cal C}_{p5} 
        \ln^p\bigl(m_t/\bar{\mu}\bigr) 
\end{equation} 
where each five-flavor coefficient $\widetilde{\cal C}_{p5}$ is a 
leading-power asymptotic expansion for $b$-quark decoupling:
\begin{equation}
\widetilde{\cal C}_{p5} \sim \sum_{q \geqslant 0} {\cal C}_{pq4} 
        \ln^q\bigl(m_b/\bar{\mu}\bigr) 
\end{equation} 
The last decoupling (that of the $b$ quark) is carried out with 
$\alpha_4$ held fixed.  Therefore we seek formulas for couplings 
such as $\alpha_6$ and $\run{t}$ in terms of $m_t$, $m_b$ and $\alpha_4$.



As always, the key step in the derivation of NLO formulas is the neglect
of some matching functions.  In this case, we neglect the NNLO functions
${\cal F}_{6 \to 5} = O(\run{t})$ and 
${\cal F}_{5\to 4} = O(\mbox{$\run{b}$}_5)$
for matching $\alpha_6$ to $\alpha_5$ and  $\alpha_5$ to  $\alpha_4$,
and the NNNLO function ${\cal G}_{6 \to 5} = O(\run{t}^2)$ for 
$\mbox{$m_b$}_5$  to be matched to $\mbox{$m_b$}_6 = m_b$.



We start with the NLO formula (\ref{g8}) for the five-flavor 
mass $\mbox{$m_b$}_5$.  To this order, all dependence on $\alpha_5$ 
and $\alpha_6$ can be eliminated via LO formulas derived 
from Eq.~(\ref{a8}),
\begin{align}
\alpha_5\, &\underset{\mbox{\scriptsize LO}}{=}\, \alpha_4\!\Bigm/\!\Bigl(1 
            + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\Bigr)
\label{p0}\\
\alpha_6\, &\underset{\mbox{\scriptsize LO}}{=}\, \alpha_4\!\Bigm/\!\Bigl\{1 
            + \frac{\alpha_4}{3\pi}\Bigl(
   \ln\frac{m_t}{\bar{\mu}} + \ln\frac{m_b}{\bar{\mu}}\Bigr)\Bigr\}
\end{align}
where (again to this order) $\mbox{$m_b$}_5$ may be replaced by 
$m_b$ on the right-hand side. Then Eq.~(\ref{g8}) becomes
\begin{align}
\ln\frac{\mbox{$m_b$}_5}{\bar{\mu}}\; 
 &\underset{\mbox{\scriptsize NLO}}{=}\;
    \ln\frac{m_b}{\bar{\mu}} 
  - \frac{12}{23}\ln\Bigl(1 + 
        \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\Bigr)
\nonumber \\
 &\phantom{\underset{\mbox{\scriptsize NLO}}{=}}
  - \frac{8}{161}\ln\bigg(1 + \frac{23\alpha_4}{6\pi}\ln\frac{m_t}{\bar{\mu}}
       + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\biggr)
\nonumber \\
 &\phantom{\underset{\mbox{\scriptsize NLO}}{=}}
  + \frac{4}{7}\ln\bigg(1 + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}
      + \frac{\alpha_4}{3\pi}\ln\frac{m_t}{\bar{\mu}}\biggr).
\label{mb5}
\end{align}


Similarly, consider the NLO relation (\ref{y3}) between
$\alpha_{f+1}$ and $\alpha_f$.  For the case $f=4$, the
heavy-quark mass $m_h$ in (\ref{y3}) is $\mbox{$m_b$}_5$, but we can use 
Eq.~(\ref{mb5}) to eliminate $\mbox{$m_b$}_5$ in favor of $m_b$,
\begin{align}
\alpha_5^{-1} \underset{\mathrm{NLO}}{=} \alpha_4^{-1} 
 &+ \frac{1}{3\pi}
      \ln\frac{\mbox{$m_b$}_5}{\bar{\mu}}\biggr|_\mathrm{NLO}\! 
  + c_4\ln\Bigl[1 + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\Bigr]
\nonumber \\[1mm]
%&\phantom{\underset{\mathrm{NLO}}{=} \alpha_4^{-1}\ }
 &+ d_4\ln\Bigl[1 + \frac{25\alpha_4}{6\pi}
      \ln\frac{m_b}{\bar{\mu}}\Bigr] 
\label{p1}
\end{align}
where the constants $c_4$ and $d_4$ are given by
\begin{equation}
c_4 = 33/(23\pi)        
\hsp{4},\hsp{4}
d_4 = 121/(1150\pi). 
\end{equation}
For $f=5$, $m_h$ in (\ref{y3}) is the six-flavor mass $m_t$. Any
$\alpha_5$ dependence can be removed via Eqs.~(\ref{p0}) or (\ref{p1}):
\begin{align}
\alpha_6^{-1} \underset{\mathrm{NLO}}{=} \alpha_5^{-1}\Bigr|_\mathrm{NLO} 
 &+ \frac{1}{3\pi}\ln\frac{m_t}{\bar{\mu}}
    + c_5\ln\biggl[1 + \frac{\alpha_5}{3\pi}\Bigr|_\mathrm{LO}
           \ln\frac{m_t}{\bar{\mu}}\biggr]
\nonumber \\[1mm]
 &+ d_5\ln\biggl[1 + \frac{23\alpha_5}{6\pi}\Bigr|_\mathrm{LO}
      \ln\frac{m_t}{\bar{\mu}}\biggr] 
\label{p2}
\end{align}
The coefficients $c_5$ and $d_5$ have numerical values
\begin{equation}
c_5 = 47/(42\pi)        
\hsp{4},\hsp{4}
d_5 = 137/(966\pi). 
\end{equation}
The result of combining these formulas is:
\begin{align}
\alpha_6^{-1} &\underset{\mathrm{NLO}}{=} \alpha_4^{-1}
  + \frac{1}{3\pi}\Bigl(\ln\frac{m_b}{\bar{\mu}}
                        + \ln\frac{m_t}{\bar{\mu}}\Bigr)
\nonumber \\
&\phantom{\underset{\mathrm{NLO}}{=}}\
 + \frac{55}{42\pi}\ln\bigg(1 
      + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}
      + \frac{\alpha_4}{3\pi}\ln\frac{m_t}{\bar{\mu}}\biggr)
\nonumber \\
&\phantom{\underset{\mathrm{NLO}}{=}}\
 + \frac{121}{966\pi}\ln\biggl(1 
      + \frac{23\alpha_4}{6\pi}\ln\frac{m_t}{\bar{\mu}}
      + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\biggr) 
\nonumber \\
&\phantom{\underset{\mathrm{NLO}}{=}}\
 + \frac{121}{1150\pi}\ln\biggl(1 
            + \frac{25\alpha_4}{6\pi}\ln\frac{m_b}{\bar{\mu}}\biggr).
%\label{p3}
\end{align}

The same procedure can be applied to the NLO formula (\ref{y1}) for 
the RG invariant mass $\bar{m}_h$. For $f=4$, the result is 
\begin{align}
\ln\frac{\mbox{$\bar{m}_b$}_5}{\bar{\mu}}
 &\underset{\mbox{\scriptsize NLO}}{=} \ln\frac{m_b}{\bar{\mu}} 
  - \frac{12}{23}\ln\Bigl(1 + 
        \frac{25\alpha_4}{6\pi}\ln\frac{m_b}{\bar{\mu}}\Bigr)
\nonumber \\
 &\phantom{\underset{\mbox{\scriptsize NLO}}{=}}
  - \frac{8}{161}\ln\bigg(1 + \frac{23\alpha_4}{6\pi}\ln\frac{m_t}{\bar{\mu}}
       + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\biggr)
\nonumber \\
 &\phantom{\underset{\mbox{\scriptsize NLO}}{=}}
  + \frac{4}{7}\ln\bigg(1 + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}
      + \frac{\alpha_4}{3\pi}\ln\frac{m_t}{\bar{\mu}}\biggr)
\label{p4}
\end{align}
after substitution of Eq.~(\ref{mb5}) for $\mbox{$m_b$}_5$.
For $f=5$, Eq.\ (\ref{y1}) expresses $\bar{m}_t$ in terms of $\alpha_5$, 
for which the LO formula (\ref{p0}) may be substituted, with the result
\begin{align}
\ln\frac{\bar{m}_t}{\bar{\mu}}
&\underset{\mbox{\scriptsize NLO}}{=} \ln\frac{m_t}{\bar{\mu}}
  + \frac{4}{7}\ln\Bigl(1 
       + \frac{\alpha_4}{3\pi}\ln\frac{m_t}{\bar{\mu}}
       + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\Bigr) 
\nonumber \\
&\phantom{\underset{\mbox{\scriptsize NLO}}{=}}\
   - \frac{4}{7}\ln\Bigl(1
       + \frac{23\alpha_4}{6\pi}\ln\frac{m_t}{\bar{\mu}}
       + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\Bigr) .
\label{p5}
\end{align}
If desired, inverses of (\ref{p4}) and (\ref{p5}) can be constructed
and used to express quantities such as $\alpha_6$ in terms of the 
invariant masses $\mbox{$\bar{m}_b$}_5$ and $\bar{m}_t$ instead of 
$m_b$ and $m_t$.





Finally, we extract NLO formulas for the invariant running couplings
for sequential $t,b$ decoupling from Eq.~(\ref{y2}) for $\run{h}$.
For the case $f=4$, Eqs.~(\ref{y2}) and (\ref{p4}) imply
\begin{eqnarray}
\mbox{$\run{b}$}_5^{-1} 
 &\makebox[1em]{$\underset{\mathrm{NLO}}{=}$}& \alpha_4^{-1}
    + \frac{25}{6\pi}\ln\frac{m_b}{\bar{\mu}}
   - \frac{729}{1150\pi}\ln\biggl(1 
            + \frac{25\alpha_4}{6\pi}\ln\frac{m_b}{\bar{\mu}}\biggr)
\nonumber \\
&\phantom{=}& 
  -\, \frac{100}{483\pi}\ln\biggl(1 
      + \frac{23\alpha_4}{6\pi}\ln\frac{m_t}{\bar{\mu}}
      + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\biggr) 
\nonumber \\
&\phantom{=}&
  +\, \frac{50}{21\pi}\ln\bigg(1 
      + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}
      + \frac{\alpha_4}{3\pi}\ln\frac{m_t}{\bar{\mu}}\biggr).
\end{eqnarray} 
For $f=5$, it is necessary to combine Eq.~(\ref{y2}) with 
Eqs.\ (\ref{mb5}), (\ref{p1}) and (\ref{p5}):
\begin{align}
\run{t}^{-1} &\underset{\mathrm{NLO}}{=} \alpha_4^{-1}
  + \frac{23}{6\pi}\ln\frac{m_t}{\bar{\mu}}
  + \frac{1}{3\pi}\ln\frac{m_b}{\bar{\mu}}     
\nonumber \\  
&\phantom{\underset{\mathrm{NLO}}{=}}\
 - \frac{457}{483\pi}\ln\biggl(1 
      + \frac{23\alpha_4}{6\pi}\ln\frac{m_t}{\bar{\mu}}
      + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}\biggr) 
\nonumber \\
&\phantom{\underset{\mathrm{NLO}}{=}}\
 + \frac{50}{21\pi}\ln\bigg(1 
      + \frac{\alpha_4}{3\pi}\ln\frac{m_b}{\bar{\mu}}
      + \frac{\alpha_4}{3\pi}\ln\frac{m_t}{\bar{\mu}}\biggr)
\nonumber \\
&\phantom{\underset{\mathrm{NLO}}{=}}\
 + \frac{121}{1150\pi}\ln\biggl(1 
            + \frac{25\alpha_4}{6\pi}\ln\frac{m_b}{\bar{\mu}}\biggr).
\end{align} \pagebreak 
A useful check of the formalism is to show that the difference 
$\run{t}^{-1}-\mbox{$\run{b}$}_5^{-1}$ is correctly given by 
Eq.~(\ref{i1}) in NLO.




\section{Conclusions}\label{c} 


The introduction of the matching functions ${\cal F}$ and ${\cal G}$ 
for coupling constants and masses (Eqs.~(\ref{c9}) and (\ref{g0a}))
completes the theoretical structure needed for a systematic application 
of the RG to the decoupling of heavy particles.  The discussion of this 
paper is restricted to decoupling one heavy particle at a time, but, 
as indicated in our work \cite{BCST} on NLO heavy-quark effects in 
axial charges of nucleons, the analysis can be generalized to include 
simultaneous decoupling of several heavy particles.  We will present 
this extension of the theory in a forthcoming publication.


One can also anticipate generalizations to situations where momentum 
and mass logarithms grow large together.  Examples from the literature
occur in collider physics \cite{denner}, Higgs and supersymmetric 
particle production \cite{dawson}, and deep-inelastic scattering 
through thresholds for heavy-particle production \cite{blumlein}.


%
\begin{acknowledgments}
This work was supported by the Australian Research Council and the University 
of Adelaide.
SDB is supported by a Lise Meitner Fellowship, M683, from the Austrian FWF.
FMS is supported by contract number PV-IFT/005.
RJC thanks Professor Wojtek Zakrzewski for his hospitality at Durham.
SDB thanks the CSSM for its hospitality at an important stage of this work.
\end{acknowledgments}
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