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\newcommand{\DVslash}{\line(4,1){22} \!\!\!\!\!\!\!\!\!\!\Delta V}
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\title{Renormalization in Reparameterization Invariance}

\author{Yu-Qi Chen}
\address{Institute of Theoretical Physics, Academia Sinica,
Beijing 100080, P.R. of China }



\begin{abstract}
The renormalization issue of the reparameterization invariance in
heavy quark effective theory and NRQCD effective theory is
investigated. I argue that the renormalization of the
transformation of the heavy quark field under the variation of the
velocity parameter $V$ is attributed to the renormalization of the
small component field in the proposed transformation. I show that
the matching condition for determining the renormalized small
component field can be obtained by imposing an infinitesimal
transformation of $V$ on the relations between the Green's
functions in QCD full theory and that in the effective theory. As
an example, I derive the renormalized transformation to order
$1/m^2$.
\end{abstract}

\pacs{11.10.Ef, 11.10.Gh, 11.30.Ly}
 \vfill \eject
\maketitle
 \narrowtext

\section{introduction}
Heavy quark effective theory (HQET)\cite{HQET} and nonrelativistic
QCD (NRQCD)\cite{NRQCD} are powerful tools in dealing with
dynamics of heavy-light and heavy-heavy systems, respectively. In
those systems, the off-shell momentum of the heavy quark is much
smaller than its mass. The effective theories are designed to
reproduce the results of the QCD full theory at the low energy
scale in a simpler way by integrating out the effects at the
energy scale of the heavy quark mass.  In the past decade both
effective theories and their applications have been intensively
studied.

One interesting theoretical problem in those effective theories is
the reparameterization invariance (RPI). The invariance arises
from the introducing of a velocity parameter $V$ in the effective
theory. In constructing the effective lagrangian, one needs to
divide the heavy quark momentum $P$ into a large part and a small
one as $P=mV +k$, where $k$ is a small residual momentum, and to
decompose the Dirac 4-fermion field as large and small two
component fields in respect of $V$ and uses the large two
component field to describe the heavy quark or antiquark. These
procedures lead to an explicit $V$-dependent in the effective
lagrangian. The choice of $V$ which satisfies $V^2=1$ is not
unique. But the physical prediction should be unchanged against
the variation of the velocity parameter $V$. This is the RPI. It
is required by the consistence of the effective theory and also
conducts interesting applications. It is first proposed in HQET.
However, the same invariance also valid in NRQCD effective theory.

To implement RPI, it is essential to find out a proper
transformation of the heavy quark field under the variation of
$V$. It is found to be quite nontrivial. It was first studied by
Manohar and Luke\cite{Luke:1992cs} in HQET. They used the Lorentz
boost of spinor field as the transformation of the heavy quark
field from finite velocity $V\to V'$. Their transformation suffers
from operator ordering ambiguities when it is expanded to a higher
order of $1/m$. Chen\cite{Chen:sx} proposed an  infinitesimal
transformation of the heavy quark field under the velocity
variation  from $V \to V + \Delta V$. Chen's transformation keeps
tree level effective Lagrangian invariant to all orders of $1/m$.
Finkemeier, Georgi, and McIrvin\cite{Finkemeier:1997re} shown that
to order $1/m^2$ the effective lagrangian constrainted  by Manohar
and luke's transformation and Chen's transformation may be related
by a field redefinition. Kilian and Ohl\cite{Kilian:1994mg}
proposed a renormalized transformation with the same form as
Chen's one but substituting the covariant derivative $D^\mu$ by a
general one. Sundrum\cite{Sundrum:1997ut} discussed this issue
using the auxiliary field method and obtained a result similar
with Kilian and Ohl's. However, it is unclear how the generalized
form to reproduce the QCD full theory and how to determine the
coefficient of the new operators by matching conditions from those
papers. Balzereit\cite{Balzereit} carried out an explicit
calculation of the effective lagrangian to order $1/m^3$. By
requiring the effective lagrangian invariant, he tried to
determine the transformation to order $1/m^2$ and found that the
transformation receives radiative corrections at this order. In
all these studies, a general procedure to determine the matching
condition of the renormalized transform has not yet been given.

In this paper, I argue that the renormalization of the
transformation of the heavy quark field under the variation of the
velocity parameter $V$ is attributed to the renormalization of the
small component field in the proposed transformation. I show that
the matching condition for determining the renormalized small
component field can be obtained by imposing an infinitesimal
transformation of $V$ on the relations between the Green's
functions in QCD full theory and that in the effective theory. As
an example, I derive the renormalized transformation to order
$1/m^2$.

The remainder of the paper is organized as follows. In Section II,
I show that the renormalization the transformation of the heavy
quark field is attributed to the renormalization of the small
component field. I then show that the matching condition for
determining it can be obtained by imposing an infinitesimal
transformation on the general relations between the Green's
functions in the QCD full theory, and that in the expanded local
effective theory. As an example, I derive the renormalized
transformation to order $1/m^2$. In section III, an alternative
procedure to determine the effective lagrangian is presented and
the expression of the renormalized small component is given. As an
example, I determine the effective lagrangian to order $1/m$ and
the renormalized small component field to order $1/m^2$ by this
method. In section IV, I show that the renormalized small
component field determined by those two methods are identical and
I prove that the renormalized heavy quark effective lagrangian is
invariant under this renormalized transformation. Section V is a
summary and discussion of this work. Finally, in appendix A, I
derive the general relation between the Green's functions using
generating functional method.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{renormalized transformation of the heavy quark field}

 In heavy quark effective theory, the heavy quark is described
by a two component field. Thus, one needs to decompose the Dirac
four-component field into two-component field. A simple way to
realize this decomposition is given by:
%
\begin{equation}
 h_{V\pm}(x)\;\equiv\;\exp\left(imV\cdot x\right)\;P_{V\pm}\Psi(x)\;,
 \label{hvp}
\end{equation}
%
where
%
\begin{equation}
P_{V\pm} \;\equiv \; \frac{1\pm \not\!{V}}{2}\;,
 \label{ppm}
\end{equation}
%
are the projection operators. The phase factor just removes the
large part of the heavy quark momentum when it is written as $P=mV
+k$. This definition of the field is used by most people. However,
there are some arbitrariness for the definition which leads to
different forms of the effective lagrangian. However, they can be
related to each other by field redefinition. Consequently, they
produce the same physical predictions\cite{Finkemeier:1997re}.

With this definition, the effective lagrangian reads:
%(\ref{hm-tree}):
%
\begin{eqnarray}
{\cal L}_{\rm eff}^0 &=&
 \bar h_{V+} (x)i D \cdot V h_{V+}(x)\,
 - \bar h_{V+}(x) \not\! {D}
 \frac{1}{2m+iD \cdot V}\, P_{V-} \not\!{D} h_{V+}(x)
 \;.
  \label{L-tree}
\end{eqnarray}
%
This is the nonlocal form of the effective theory. Without
expansion, it is equivalent to that of the full theory in the
sense that they produce the same $S-$matrix element.

Obviously, this effective lagrangian depends on the velocity
parameter $V$. The choice of this parameter is not unique. The RPI
implies that the physical prediction by effective theory is
independent of the choice of $V$. In Ref.\cite{Chen:sx},  it was
shown that this effective lagrangian (\ref{L-tree}) is invariant
under an infinitesimal transformation $V\rightarrow V+ \Delta V$
%
\begin{eqnarray}
\Delta h_{V+}(x) &=& \frac{\DVslash}{2} \,
  \Big(   h_{V+}(x) +  h_{V-}(x) \Big)
 \;.
  \label{D-V}
\end{eqnarray}
%
with the $h_{V-}(x)$ being the small component field and given by:
%
\begin{eqnarray}
 h_{V-}(x) &=& \frac{1}{2m+iD\cdot V}\, P_{V-} i \not\! D
 h_{V+}(x)\;.
 \label{hm-tree}
\end{eqnarray}
%
$\Delta V$ is constrained by $\Delta V\cdot V =0$ due to $V^2=1$.
The above equation can also be expanded as inverse power series of
$m$. The RPI is then valid order by order in $1/m$. The
cancellation of the shift of the lagrangian $\Delta L$ at each
order is quite nontrivial.

With expansion in terms of revise power of $m$, the ultraviolet
behavior is changed. The effective lagrangian receives
renormalization. It is not evident if the renormalized effective
lagrangian is still reparameterization invariant. Another problem
here is that even if it is reparameterization invariant, it is
excepted that the transformation (\ref{D-V}) also receives
renormalization. Then one may ask what's the matching condition to
determine its renormalization.


To investigate these problems, let's look at the relations between
the Green's functions in the full theory and that in the effective
theory.

Denote the Green's function in full theory by $G(x,y;B)$ and in
effective theory by $G_V(x,y;B)$, respectively, where $B$ is an
arbitrary background field.  They are defined by
%
\begin{eqnarray}
 G(x,y;B) \equiv \langle 0 | T\Psi(x)\bar{\Psi}(y)| 0 \rangle^B\;,
 \label{G-f}
\end{eqnarray}
%
and
%
\begin{eqnarray}
 G_V(x,y;B) \equiv \langle 0 |T h_{V+}(x)\bar{h}_{V+}(y)| 0 \rangle^B\;,
 \label{G-e}
\end{eqnarray}
%
Any interaction vertex with gluon can then be obtained by
functional differentiating over $B(x)$. When the quark field
$h_{V+}(x)$ is related to field in the full theory by
Eq.(\ref{hvp}), the Green's functions satisfy the following
relation (A derivation of this relation using generating
functional method is given in Appendix A):
%
\begin{eqnarray}
 G_V(x,y;B) & \doteq & P_{V+} \, G(x,y;B) \, P_{V+}\;,
 \label{G-Gv}
\end{eqnarray}
%
here $\doteq$ means that we omit the phase factor $\exp{(i m
V\cdot (x-y)}$ and the renormalization constant $Z(m,\alpha_s(m))$
which arises from the renormalization of the heavy quark field in
the full theory and that in the effective theory. Both sides are
valid to all orders of $1/m$ and $\alpha_s$. This relation ensures
that the $S-$matrix elements in effective theory are identical to
that in the full theory. The local effective theory is gained by
expanding the nonlocal effective lagrangian density
(\ref{L-tree}). This expansion changes the ultraviolet behavior of
the nonlocal effective theory. To reproduce the the result of the
nonlocal effective theory, one needs to add local operators to the
the effective lagrangian. The relation (\ref{G-Gv}) is required to
be satisfied as matching conditions to determining the
coefficients of those local operators.

Obviously, this relation is valid for any value of $V$. This
allows us  impose an infinitesimal transformation $V\rightarrow V+
\Delta V$ on both sides. It follows that:
%
\begin{eqnarray}
 \Delta G_V(x,y;B) & \doteq & \frac{\DVslash}{2} \, G(x,y;B) \, P_{V+} +
  P_{V+} \, G(x,y;B) \, \frac{ \DVslash}{2}\;.
 \label{DG-DGv}
\end{eqnarray}
%
Again the symbol $\doteq$ means we omit the  phase factor and a
term arises from its infinitesimal shift which is trivial under
the transformation.

Given the definitions of the Green's functions in
Eqs.(\ref{G-f}),(\ref{G-e}), we have the following  unique
solution of (\ref{DG-DGv}):
%
\begin{eqnarray}
 \langle 0 |T\big(\Delta h_{V+}(x)\,\bar{h}_{V+}(y)\big)| 0
 \rangle^B
  &\doteq &
 \frac{\DVslash}{2} \,\langle 0 | T\Psi(x)\,\bar{\Psi}(y)| 0
 \rangle^B
 P_{V+}\;.
 \label{D-h}
\end{eqnarray}
%

From Eq.~(\ref{D-h}) it is easy to see that $\Delta h_{V+}(x)$
must be proportional to $\DVslash$. Thus we can generally write it
as a form
%
\begin{eqnarray}
 \Delta h_{V+}(x) &=& \frac{\DVslash}{2} \,
 \Big( P_{V+} h'_{V+}(x) + P_{V-} h'_{V-}(x) \Big) \;.
 \label{D-h-pm}
\end{eqnarray}
%
With this, Eq.~(\ref{D-h}) then can be decomposed into two
equations:
%
\begin{eqnarray}
 \langle 0 |T\big( h'_{V+}(x)\,\bar{h}_{V+}(y)\big)| 0 \rangle^B
  &\doteq &
 P_{V+} \,\langle 0 | T\Psi(x)\,\bar{\Psi}(y)| 0 \rangle^B P_{V+}
 \label{hpp} \,, \\
 \langle 0 |T\big( h'_{V-}(x)\,\bar{h}_{V+}(y)\big)| 0 \rangle^B
  &\doteq &
 P_{V-} \,\langle 0 | T\Psi(x)\,\bar{\Psi}(y)| 0 \rangle^B P_{V+}
 \label{hmp} \;.
\end{eqnarray}
%
Eq.~(\ref{hpp}) is nothing but Eq.~(\ref{G-Gv}) if $h'_{V+}(x)$ is
identical to $h_{V+}(x)$. Eq.~(\ref{hmp}) is a new one which can
be regarded as the definition of the $h'_{V-}(x)$. Some comments
can be given about this equation. The right hand side is still the
Green's functions in the full theory while the left hand is the
Green's functions in the effective theory with insertion of local
operators at point $x$. A sort of tadpole diagrams are involved.
Again both sides are valid at any desired order of $1/m$ and any
order of $\alpha_s$ with all possible interaction with gluons.

At tree level effective lagrangian, this equation is satisfied if
$\Delta h_{V+}(x)$ is given by transformation (\ref{D-V}) with
$h_{V-}$ is given by the expanded expression (\ref{hm-tree}). At
loop level,  it is also valid  at any specific loop momentum which
is smaller than the quark mass. However, in calculating the whole
loop momentum integration, it makes difference.


Let's illustrate it in a hard cut-off regularization. The
situation in dimensional regularization is similar.

Suppose one takes different hard cut-off regularization energy
scales $\Lambda_e$ and $\Lambda_f$ in effective theory and in full
theory, respectively. The $\Lambda_f$ should be much larger than
the heavy quark mass $m$ for including both quark and antiquark
contributions. The $\Lambda_e$ should be much smaller than $m$ for
the validation of the $1/m$ expansion. Thus they satisfy a
hierarchy relation $\Lambda_e \ll m \ll \Lambda_f$. Thus in
calculating loop diagrams, the integration bounds are different on
both sides ( the left hand side is integrated out from 0 to
$\Lambda_e$ while the right hand side is integrated from 0 to
$\Lambda_f$ ). It leads that the tree level expression of
$h_{V-}(x)$ given by the expansion of (\ref{hm-tree}) makes
relation (\ref{hmp}) no longer valid. To make it valid, one has to
add the contributions of the loop momentum integrals from
$\Lambda_e$ to $\Lambda_f$ to the left hand side.  Those
contributions can be expressed as local operators shrunk to point
$x$ ( the loop diagrams not started from point $x$ are excluded
since they are accounted for in the effective lagrangian ). Thus,
to make equation (\ref{D-h}) valid, one has to add some local
terms to $ h_{V-}(x)$ to compensate the difference between the
bounds of the loop momentum integrals in the full theory and that
in the effective theory. Therefore, Eq.~(\ref{hmp}) is just the
matching condition for determining the renormalized small
component field $h_{V-}(x)$. We see that the transformation
(\ref{D-h}) keeps the same form as (\ref{D-V}). This implies that
in order to renormalize the transformation of the heavy quark
field, we only need to renormalize the small component field
$h_{V-}(x)$ in the Chen's proposed transformation\cite{Chen:sx}.
Eq.~(\ref{hmp}) allows one to determine the renormalization of the
$h_{V-}(x)$ to any order of $1/m$ and $\alpha_s$. As a specific
calculation I determine it up to order $1/m^2$ at one loop level
using dimensional regularization in Feynman gauge using this
matching condition. It reads:
%
\begin{eqnarray}
 h_{V-}(x) &=& P_{V-} \Big( {i\not \!D \over 2m} + {d_1(\mu)\over 4m^2 }
   D\cdot V \not \!D  - {d_2(\mu) \over 4m^2} \not \!D  D\cdot V \Big)
    h_{V+}(x)\;,
 \label{hm-1}
\end{eqnarray}
%
where
%
\begin{eqnarray}
%
 d_1(\mu) &=& \; 1+ {\alpha_s(\mu) \over 4\pi } \;
  \Big[ \,\Big (3C_A -4 C_F \big) \,\ln {\mu^2 \over m^2}
  -2C_F+6C_A \Big]\,,
 \label{d1}\\
 d_2(\mu) &=& \Big(2C_F+{3}C_A\Big){\alpha_s(\mu)\over 4\pi}
 \Big( \ln{\mu^2\over m^2}+2\Big )\,,
\label{d2}
\end{eqnarray}
%
with $C_F=4/3$ and $C_A=3$. Comparing to the result in
\cite{Balzereit}, we see that the renormalized transformation
determined here is somewhat different from that determined in
\cite{Balzereit}.

\section{Renormalized  effective lagrangian }

In heavy hadrons, the heavy quark is almost on mass-shell. The
off-shell momentum $k$ is much smaller than the mass of the heavy
quark. It allows us carry out a low energy expansion. In
heavy-light system the expansion parameter is $\Lambda_{\rm
QCD}/m$ while in heavy-heavy system it is the relative velocity
$v$ between both heavy quarks. Effective theory method provides a
systemic expansion by expanding the effective lagrangian  in terms
of the expansion parameter. When an expanded tree level lagrangian
is used to calculate the tree diagrams, it reproduce the same
results as the full theory to any desired order of the expansion
parameter. However, Some differences arise when it is used to
calculate the loop diagrams in which the loop momentum may be
larger than the heavy quark mass, which makes the expansion
failure in the integrand. Fortunately, this difference can be
expressed as contributions from local operators since particles
are highly virtue in this region and can only  propagate in a
short distance in space-time. In order to reproduce the results of
the full theory, it is necessary to add those local operators with
certain coefficients to the tree level expanded lagrangian to
compensate the difference between the full theory and the expanded
theory. The coefficient of each operator which is called as Wilson
short distance coefficient can be determined by matching the
effective theory and the full theory.

In calculating higher order loop diagrams, ultraviolet divergences
may arise. They need to be regularized and then renormalized. With
proper regularization, the renormalization condition for
determining the short-distance coefficients is just given by the
matching conditions. The most frequently used regularization
methods are dimensional regularization and hard cut-off
regularization. They are equivalent in a sense that they produce
the same physical predictions. The first one is convenient in
practical calculations while the last one is intuitive and
relatively easy to be interpreted. In this paper, we take the hard
cut-off regularization to illustrate renormalization of the
effective lagrangian and the transformation of the heavy quark
field. But we use dimensional regularization in real calculations
of the short distance coefficients.

In the conventional method, a renormalized effective lagrangian is
constructed by following steps. First a proper field to describe
the low energy particles is chosen.  In HQET and NRQCD, this
effective field to describe the the heavy quark is just the
two-component field. Then the effective lagrangian in this field
is expanded as sum of local operators in terms of appropriate
counting rule. Then the renormalized short distance coefficients
of these local operators are determined by matching the full
theory and the effective theory. We refer this method as matching
after expansion.

Here we introduce an alternative way to determine the renormalized
effective lagrangian. In this method, renormalized local operators
in the field of the full theory are added in the lagrangian of the
full theory by matching condition. Then it is expanded in terms of
the two component field. We refer this method as matching before
expansion.



Let's illustrate how this works in a hard cut-off regularization.

As discussed in the last section, suppose one takes different hard
cut-off regularization energy scales $\Lambda_e$ and $\Lambda_f$
in effective theory and full theory, respectively. They satisfy a
hierarchy relation $\Lambda_e \ll m \ll \Lambda_f$. In calculating
the one-loop 1PI diagrams in full QCD theory, we need to calculate
the loop momentum integrals from zero to $\Lambda_f$. They can be
separated into integrals from 0 to $\Lambda_e$ and integrals from
$\Lambda_e$ to $\Lambda_f$. The first part is just the same with
that in the effective theory while the second part gives extra
contributions. As argued above, the contributions from this region
can be written as local terms of external momentum and can be
expressed as contributions from local operators. Therefore, once
those local operators are added to the lagrangian, the effective
theory with hard-cutoff $\Lambda_e$ can produce the same result of
the full theory with cut-off $\Lambda_f$. This argument can easily
be generalized to the case of multi-loop.

At this stage, those local operators are written in terms of Dirac
four-component field. A general form of the renormalized effective
lagrangian density with hard cutoff $\Lambda_e$ for heavy quark
field can formally be expressed as:
%
\begin{eqnarray}
{\cal L}_{\rm eff} &=& \bar{\Psi}(x) \, (i{\not D}-m) \, \Psi(x) +
\bar{\Psi}(x) \, O_1(x) \, \Psi(x)\;, \label{L-O1}
\end{eqnarray}
%
where $D^\mu=\partial^\mu - i g A^a_\mu T^a $ is the covariant
derivative. It  may simply be denoted as:
%
\begin{eqnarray}
{\cal L}_{\rm eff} &=& \bar{\Psi}(x) \, O(x) \, \Psi(x)\;,
\label{L-O}
\end{eqnarray}
%
by defining $O(x)\equiv i\not \! D-m + O_1(x)$.

The first term in (\ref{L-O1}) is just the tree-level lagrangian
while the second term arises from the renormalization with a
cut-off $\Lambda_e \ll m$. The operators in this term are
generally the function of the covariant derivative and the heavy
quark mass. It may contains terms such as $D^2+ m^2$, and
$g_sG^{\mu\nu} = i[D^\mu,D^\nu]$, which are suppressed by the
off-shell momentum of the heavy quark or the momenta of the
external gluons. They can be organized via an appropriate power
counting rule. In perturbative calculations, the loop momentum
integral is from zero to $\Lambda_e$. In this region, both the
external and the loop momenta are smaller than $m$, hence the
$1/m$ expansion is allowed and the quark mass dependence is
extracted explicitly. Thus the energy scale $m$ is no longer
involved in the effective theory. I emphasis here that the
effective lagrangian density in this form is independent of the
velocity parameter $V$. Thus it automatically satisfies the RPI.

Higher dimensional operators appear in $O_1(x)$. It implies that
power divergences arise in the loop momentum integrals. In the
full theory the power divergences cancel when both the
contributions from quark and antiquark are included. However, in
the effective theory when we impose a hard cutoff $\Lambda_e\ll m$
on the loop momentum integrals, the contributions from antiquark
are excluded so that the power divergences do not cancel.
Nevertheless, those power divergences are artificial since the
power divergences from the diagram calculations just cancel that
from the short distance coefficients.

At one loop and leading order of $1/m$, the most general form of
the four-component effective lagrangian is
%
\begin{eqnarray}
{\cal L}_{\rm eff} &=& (1+c_0) \bar{\Psi}(x) \, (i{\not D}-m)
\,\Psi(x)
 - {c_1 \over 2m} \bar{\Psi}(x) \, (D^2+m^2) \, \Psi(x)\nonumber\\
 &&
 - {i c_2 \over 4m^2 } \bar{\Psi}(x)\,\Big[ \not\!D
 (D^2+m^2)+(D^2+m^2)\not\!D\Big]\Psi(x)
+ {c_3 \over 4m} \bar{\Psi}(x) \, \sigma^{\mu\nu} G_{\mu\nu} \,
\Psi(x)
  \;. \label{L-1}
\end{eqnarray}
%
The coefficients $c_0-c_3$ can be calculated  by matching the full
theory and the effective theory using the dimensional
regularization. In the matching procedure, the off-shell momentum
of the heavy quark and the momentum of the gluon can be treated as
much smaller than the quark mass hence can be expanded. It leads
to that the remainder part of the loop momentum integral no longer
depends on the heavy quark mass. We are free to chose the infrared
regulator since the infrared divergences cancel on both sides. If
we use a limitation order in which the external off-shell momentum
of heavy quark and the momenta of the gluons go to zero first
followed by  $\epsilon=2-D/2$ going to zero second, as used by
Eichten and Hill in Ref.\cite{HQET}, then all the loop integral on
the effective theory side vanishes. This simplifies the matching
calculations significantly. On the full theory side, at one-loop
level, one needs to calculate the heavy quark self-energy diagram
and two  1PI vertex diagrams. In $\rm \overline{MS}$ scheme and at
one-loop level, those coefficients are determined to be
%
\begin{eqnarray}
&& c_0(\mu)= {C_F\over 4\pi } \alpha_s(\mu)\Big( \ln{\mu^2\over
m^2}+2\Big )\,,
 ~~~~~~~~
 c_1(\mu)= {C_F\over 2 \pi} \alpha_s(\mu)\Big( 2\ln{\mu^2\over m^2}+3\Big )\,,
 \nonumber \\ &&
 c_2(\mu)=-{C_F \over 2\pi} \alpha_s(\mu) \Big( \ln{\mu^2\over m^2}+2\Big )\,,
 ~~~~~~
 c_3(\mu)= \Big(2C_F+{3}C_A\Big){\alpha_s(\mu)\over 4\pi} \Big( \ln{\mu^2\over m^2}+2\Big
 )\,.
\label{c0-3}
\end{eqnarray}
%


Now let's derive the two-component effective lagrangian from it.
The equation of motion now reads:
%
\begin{eqnarray}
P_{V-} \;\overline{O}(x) \;
       \left( h_{V+}(x) + h_{V-}(x)\right) &=& 0\;,
  \label{hpm}
\end{eqnarray}
%
where $\overline{O}(x)$ is the $O(x)$ whose covariant derivative
$iD$ is replaced  by $iD+mV$ due to the phase factor in the field
redefinition. It can be regarded as the renormalized equation of
motion.

From Eq.(\ref{hpm}), we can express $h_{V-}(x)$ as a function of
$h_{V+}(x)$ formally as:
%
\begin{eqnarray}
 h_{V-}(x) &=& \frac{1}{2m+iD(x)\cdot V -P_{V-}\overline{O}_1(x)P_{V-}}\,
  P_{V-} \left(i\not{D} +\overline{O}_1(x)\right) h_{V+}(x)\;,
 \label{hm}
\end{eqnarray}
%
where $\overline{O}_1(x)$ is the $O_1(x)$ whose covariant
derivative $iD$ is substituted by $iD+mV$. This modifies the tree
level expression (\ref{hm-tree}).
 Once the form of $O_1(x)$ is
given, the right hand side of Eq.(\ref{hm}) can be expanded as
power series of $1/m$. With $O_1(x)$ given in (\ref{L-1}), up to
order $\alpha_s$ and $1/m^2$, $h_{V-}(x)$ reads:
%
\begin{eqnarray}
 h_{V-}(x) &=& P_{V-} \Big[ \;{i\not \!D \over 2m} + {(1-c_1+c_2+c_3) \over 4m^2 }
   D\cdot V \not \!D  - {c_3 \over 4m^2} \not \!D  D\cdot V \;\Big]
    h_{V+}(x)\;.
 \label{hm-2}
\end{eqnarray}
%
Comparing this with (\ref{hm-1}), they are in agreement with each
other.

Finally, with equation of motion (\ref{hpm}), the effective
lagrangian (\ref{L-O}) is reduced to:
%
\begin{eqnarray}
{\cal L}_{\rm eff} &=&
 \bar h_{V+}(x) \,
  \overline{O}(x)\Big( h_{V+}(x) + h_{V-}(x)\Big)\nonumber \\
  &=&
 \Big(\bar h_{V+}(x) +\bar h_{V-}(x)\Big)\,
  \overline{O}(x)h_{V+}(x) \;.
  \label{L-O-h}
\end{eqnarray}
%
This is just the two-component effective lagrangian. It can be
expanded as power series of $1/m$. In this way, the four component
effective lagrangian is reduced to the two component effective
lagrangian. Up to order $\alpha_s$ and $1/m$ correction, it reads
%
\begin{eqnarray}
L_{V}^1(x) &=& Z\bar h_{V+} (x)i D \cdot V h_{V+}(x)\, -
 {Z \over 2m} \bar h_{V+} (x) D^2 h_{V+} (x)
  +{ZZ_m \over 4m} h_{V+} (x) \sigma^{\mu\nu} G_{\mu\nu} h_{V+} (x)
  \;,
\label{LV-1}
\end{eqnarray}
%
where
%
\begin{eqnarray}
Z &=& 1+c_0+c_1+c_2 \;=\;{C_F\over 4 \pi} \alpha_s(\mu)\Big(
3\ln{\mu^2\over m^2}+4\Big )\,,
 \nonumber \\
Z_m &=& 1-c_1-c_2+c_3 \;=\; {1\over 4 \pi} \alpha_s(\mu)\Big[ 2C_F
+ C_A\Big( 3\ln{\mu^2\over m^2}+2\Big ) \Big]\,.
\end{eqnarray}
%
This effective lagrangian is in agreement with that obtained by
Eichten and Hill in Ref.\cite{HQET}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{RPI of the renormalized effective lagrangian}
In Sec. II, the matching condition for determining the
renormalized $h'_{V-}(x)$ field is given by Eq.~(\ref{hmp}). In
Sec. III, the $h_{V-}(x)$ field has been obtained by equation of
motion. Its expression is given by Eq.~(\ref{hm}).  In this
section, I will show that the  small component fields obtained by
these two different methods are identical. They uniquely determine
the renormalized transformation of the heavy quark field against
the infinitesimal variation of the velocity parameter $V$.
Finally, I will show that the renormalized heavy quark effective
lagrangian is invariant under this transformation.

Adding both sides of (\ref{hpp}) and (\ref{hmp}) together, we have
%
\begin{eqnarray}
 \langle 0 |T\Big( h_{V+}(x)+h'_{V-}(x)\Big)\,\bar{h}_{V+}(y) | 0
 \rangle^B
  &\doteq &
  \langle 0 | T\Psi(x)\,\bar{\Psi}(y)| 0 \rangle^B P_{V+}
 \label{hpmp} \,.
\end{eqnarray}
%
where $\langle 0 | T\Psi(x)\,\bar{\Psi}(y)| 0 \rangle^B$ is a full
propagator under arbitrary external field $B^\mu(x)$. It is
satisfied order by order in $\alpha_s$. Suppose we calculate the
left hand side at tree level with the renormalized effective
lagrangian. To validate this equation, the right hand side then
should also be calculated to the tree level with containing
contributions of the loop momentum integrals from $\Lambda_e$ to
$\Lambda_f$. This can be calculated by the renormalized
four-component effective lagrangian (\ref{L-O1}) to tree level.
Thus it satisfies the following equation
%
\begin{eqnarray}
  O(x)G(x,y;B)&=&\delta^4 (x-y)
 \label{O-G}  \;.
\end{eqnarray}
%
 Acting an operator $P_{V-}\overline{O}(x)$ on the left hand side
and $P_{V-}{O}(x)$ on the right hand side of (\ref{hpmp}), the
right hand side vanishes immediately since $P_{V-}\cdot P_{V+}=0$.
Since we only calculate them at tree level, the operator
$\overline{O}(x)$ can be moved within the bracket:
%
\begin{eqnarray}
 P_{V-}\langle 0 |T\big[O(x)\big( h_{V+}(x)+h'_{V-}(x)\big)\,
  \bar{h}_{V+}(y)\big]| 0 \rangle^B
  & = & 0
 \label{O-hpm} \,.
\end{eqnarray}
%
Since the argument $y$ in $\bar{h}_{V+}(y)$ is arbitrary and this
correlation function contains interaction with arbitrary
background gluon field, the unique solution of this equation is
%
\begin{eqnarray}
 P_{V-} \overline{O}(x)\Big( h_{V+}(x)+h'_{V-}(x)\Big) & = & 0
 \label{O-hpm1} \,.
\end{eqnarray}
%
This is just identical to (\ref{hpm}) if $h'_{V-}(x)$ is the same
with $h_{V-}(x)$. This implies that the renormalized $h'_{V-}(x)$
obtained from the matching condition (\ref{hmp}) is identical to
that from the equation of motion (\ref{hpm}).

Now let's prove that the renormalized effective lagrangian
(\ref{L-O-h}) is invariant under the transformation (\ref{D-V}) or
(\ref{D-h-pm}) with renormalized small component field.

It follows that from an infinitesimal transformation of the
effective lagrangian (\ref{L-O-h})
%
\begin{eqnarray}
{\Delta \cal L}_{\rm eff} & \doteq &
 \Delta \bar h_{V+}(x) \,
  \overline{O}(x) \,h_{V}(x)
 +\bar h_{V+}(x) \,
  \overline{O}(x)\, \Delta h_{V}(x) \nonumber\\
& = &
  \bar h_{V}(x) \frac{\DVslash}{2}\,
  \overline{O}(x) \,h_{V}(x)
 +\bar h_{V+}(x) \,
 \overline{O}(x) \, \Delta  h_{V}(x)
  \;.
  \label{D-L-1}
\end{eqnarray}
%
We have use a shorthand notation $h_V(x)= h_{V+}(x) + h_{V-}(x)$.
It is emphasized here that the operator $O(x)$ which is from the
four component effective field theory is invariant against the
variation of the velocity $V$. Any change arising from the phase
factor in the definition of the effective field has been omitted
simply because it is trivial under the transformation.


Imposing an infinitesimal transformation on the equation of motion
(\ref{hpm}), we obtain that
%
\begin{eqnarray}
 - \frac{\DVslash}{2} \, \overline{O}(x) \, h_{V}(x) + P_{V-}
 \,\overline{O}(x)\, \Delta h_{V}(x)
  & \doteq& 0 \;.
  \label{D-hpm}
\end{eqnarray}
With it, (\ref{D-L-1}) can be rewritten as
\begin{eqnarray}
{\Delta \cal L}_{\rm eff} & \doteq &
  \bar h_{V}(x)\,\overline{O}(x) \, \Delta h_{V}(x)
   \;.
  \label{D-L-3}
\end{eqnarray}
%
Notice that $P_{V+}\, h_{V-}(x)=0$. Imposing an infinitesimal on
it, we immediately have
%
\begin{eqnarray}
P_{V+} \Delta h_{V-}(x) &=& -\frac{\DVslash}{2}\, h_{V-}(x) \;.
\end{eqnarray}
%
Adding it together with $P_{V+}\Delta h_{V+}(x) = \DVslash /2 \;
h_{V-}(x) $, we have
%
\begin{eqnarray}
P_{V+}\Delta h_{V}(x) = 0. \;,
\end{eqnarray}
%
With it, (\ref{D-L-1}) is  reduced to
\begin{eqnarray}
{\Delta \cal L}_{\rm eff} & \doteq &
  \bar h_{V}(x)\,\overline{O}(x) P_{V-}\, \Delta h_{V}(x)
   \;.
  \label{D-L-4}
\end{eqnarray}
%
It follows that ${\Delta \cal L}_{\rm eff} =0$ from the equation
of motion $ \bar h_{V}(x)\,O(x) \,P_{V-} =0 $. Thus we have shown
that the renormalized effective lagrangian (\ref{L-O-h}) is
invariant against the variation of the velocity parameter $V$
under the infinitesimal transformation (\ref{D-V}) with the
renormalized small component field.

\section{conclusion}
The RPI is an important theoretical problem in the heavy quark
effective theory and the NRQCD effective theory. It is required by
the consistence of the effective theory. It also conducts
interesting
applications\cite{Neubert:1993iv}\cite{Chen:1993us}\cite{Chen:1994dg}.
The transformation of heavy quark field under the variation of the
velocity parameter $V$ proposed by Chen\cite{Chen:sx} with tree
level expression of the small-component field keeps the tree level
effective theory invariant. However, at loop level, the
transformation needs to be renormalized in the renormalized
effective theory. In this paper, I show that the the renormalized
transformation of the heavy quark keeps that same form as Chen's
transform where the small component field needs to be
renormalized. I propose two different approaches to determine it.
In the first one, I propose a new procedure to determine the
renormalized effective lagrangian in which a renormalized
effective lagrangian in Dirac four-component field is constructed
first, followed by its reduction  to the two-component effective
lagrangian. The renormalized small component field is then
obtained by the equation of motion. The four-component effective
lagrangian automatically satisfies RPI. Thus RPI  cannot give any
constraints on any operators in it. When it is reduced to the two
component effective theory, the same operator with certain
coefficient may appear in different terms, the RPI can be used to
connect those terms. In the second approach, I argue that the
general relations between the Green's functions in QCD full theory
and that in effective theory are valid to all orders of $1/m$. I
then show that the matching condition for determining the
renormalized small component field can be obtained by imposing an
infinitesimal transformation of $V$ on these relations.  The
obtained renormalized small component fields by these two methods
turn out to be  identical and remormalized two-component effective
lagrangian is invariant under the renormalized transformation to
all orders of $1/m$.  I use the first approach to determine the
renormalized transformation to order $1/m^2$ and the effective
lagrangian as an example to illustrate the method. %In this way,
%the renormalization issue of the RPI is completely resolved.
I have used  hard cut-off regularization to illustrate all these
methods.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\appendix
\section{generating functional of Green's functions} In this
appendix, we derive the relations between the Green's functions in
QCD full theory and that in the effective theory using generating
functional method. It is similar with that used in
\cite{Mannel:1991mc} and \cite{Chen:sx}. We use the background
field method\cite{background-field,abbott} for gluon field, which
preserves explicitly the gauge covariant.

In QCD full theory, the generating functional reads
%
\begin{eqnarray}
 Z[\eta,\bar{\eta},J,B]\,&=&\,
 \int \delta[\eta,\bar{\eta},A]\exp i\int d^4x ( I_Q(x) +I_g(x))\;,
 \label{Z-qcd}
\end{eqnarray}
%
where  $\eta$, $\bar{\eta}$,$J$ are the external sources for heavy
quark, antiquark and gluon field, $B$ is the background gluon
field,  $I_g$ is given by
%
\begin{eqnarray}
I_g(x) &=& - {1\over 4} F^a_{\mu\nu}F^{a\mu\nu}
      - {1\over 2\xi}(G^a)^2
      + \ln \det\Big[{\delta G^a \over \delta \omega^b}\Big]
      +J_\mu^a A^{a\mu }  \;,
\end{eqnarray}
%
with
\begin{eqnarray}
F^a_{\mu\nu} &=& \partial_\mu (A + B)_\nu^a
                -\partial_\nu (A + B)_\mu^a
                +gf^{abc}(A + B)_\mu^b (A + B)_\nu^c \;,
\end{eqnarray}
%
\begin{eqnarray}
 G^a &=& \partial_\mu A_\mu^a
                +gf^{abc} B_\mu^b A^{c\mu}
                \;,
\end{eqnarray}
being the gauge-fixing term. If $J_\mu$ satisfies the following
relation
%
\begin{eqnarray}
 {\delta W \over \delta B^a_\mu}
 + \int d^4y \Big[
 {\delta {W} \over \delta J^b_\nu}
 {\delta J^b_\nu(y) \over\delta B^a_\mu} \Big]
 &=& -J_\mu^a\;,
\end{eqnarray}
with $W[\eta,\bar{\eta},J,B] = - i \ln Z[\eta,\bar{\eta},J,B]$,
$W[\eta,\bar{\eta},J,B]$ is just the effective action regarding to
the gluon field $B$ with gauge-fixing term
%
\begin{eqnarray}
  G^a &=& \partial_\nu (A-B)_\nu^a
  +gf^{abc} B_\mu^b A_\nu^c   \;,
\end{eqnarray}
%
and $I_Q$ reads
%
\begin{eqnarray}
I_Q(x) &=& \overline{\Psi}(x) (i\not \!{D} -m) \Psi(x)
+\overline{\eta}(x) \Psi(x) +\overline{\Psi}(x) \eta (x) \;,
\end{eqnarray}
%
The quark field can be integrated out formally and then we have
%
\begin{eqnarray}
 Z[\eta,\bar{\eta},J,B]\,&=&\,
 \int \delta[A] \det[i\not\!D-m] \exp i\int d^4x ( I'_Q +I_g)\;,
 \label{Z-A}
\end{eqnarray}
%
where $I'_Q$ remains the same form as $I_Q$. But the quark field
now is related to the external source $\eta(x)$ by the following
equation of motions:
%
\begin{eqnarray}
(i\not\!D -m )\Psi(x) = -\eta(x)\;.
\end{eqnarray}
%
The generating functional of the effective theory is similar with
that of the full theory except the heavy quark action. The
effective lagrangian is substituted by (\ref{L-O}). In the
external source term of the heavy quark  only the large component
effective field defined in (\ref{hvp}) couples to the external
source. The action of the heavy quark is given by:
%
\begin{eqnarray}
 I^{V+}_Q(x) &=& \overline{\Psi}(x) O(x) \Psi(x)
 +\bar{\eta}(x)P_{V+} h_{V+}(x)
 +\bar{h}_{V+}(x)P_{V+}\eta (x) \;.
\end{eqnarray}
%
Similarly, integrating out the heavy quark field, the generating
functional takes the same form as (\ref{Z-A}) with the effective
action of the heavy section is substituted by %
\begin{eqnarray}
 I'^{V+}_Q(x) &=&
 \overline{h}_{V}(x) \overline{O}(x) h_{V}(x)
 +\bar{\eta}(x)P_{V+} h_{V+}(x)
 +\bar{h}_{V+}(x)P_{V+}\eta (x) \;,
 \label{I'V}
\end{eqnarray}
%
with $h_{V}(x)= h_{V+}(x) + h_{V-}(x)$.

 The quark field now is
related to the external source $\eta(x)$ by the following equation
of motions:
%
\begin{eqnarray}
 \overline{O}(x) h_{V+}(x)
 &=& - P_{V+}\eta (x) \;.
\end{eqnarray}
%
Multiplying $P_{V-}$ on both sides, the right hand vanishes and we
obtain the renormalized equation of motion:
%
\begin{eqnarray}
P_{V-} \;\overline{O}(x) \;
       \Big( h_{V+}(x) + h_{V-}(x)\Big) &=& 0\;.
\label{E-Q}
\end{eqnarray}
%
This is just the equation (\ref{hpm}).  The renormalized
$h_{V-}(x)$ can be related to $h_{V+}(x)$ by (\ref{hm}).  With the
equation of motion (\ref{E-Q}), (\ref{I'V}) can be simplified as
%
\begin{eqnarray}
I'^V_Q &=&
 \bar h_{V+} (x)\overline{O}(x) h_{V}(x)\,
 + \bar{\eta}(x)P_{V+} h_{V+}(x)
 +\bar{h}_{V+}(x)P_{V+}\eta (x) \;.
  \label{I'-V-1}
\end{eqnarray}
%
This gives the effective lagrangian density (\ref{L-O-h}).

 The quark determinant in (\ref{Z-A}) is responsible for the
contributions of the heavy quark loop. It is the same in the full
theory and in the effective theory and is suppressed by al least
$1/m^2$. Thus we may ignore it.

The full quark propagator with background field $B^\mu(x)$ is
gained by differentiating over the external sources.
%
\begin{eqnarray}
 G(x,y;B) &=&
 {\delta^2 \over \delta\eta(x) \bar\delta \eta(y) }
 W(\eta,\bar\eta,J,B)\;.
\end{eqnarray}
%
If the hard cutoff energy scales are set to $\Lambda_f$, the same
with that in the QCD full theory, the $O(x)$ is then to be
$i\not\!D-m$, just the same with that in the full theory. The
effective lagrangian is just the nonlocal form (\ref{L-tree}). In
this case the only difference of  the effective theory and the
full theory is the external source term.  One immediately gains
the relation between the Green's function of the full theory and
that in the effective theory (\ref{G-Gv}). This relation ensures
that the nonlocal effective theory is equivalent to the QCD full
theory. The local effective theory with the hard cutoff
regularization scale $\Lambda_e$ is equivalent to that the
nonlocal effective theory with hard cutoff $\Lambda_f$. This
ensures the validation of the relation (\ref{G-Gv}).

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\end{references}


\end{document}

