%Title: Next-to-leading-log renormalization-group running in heavy-quarkonium  creation and annihilation 
%Author: Antonio Pineda 
%Published: <EM> Phys. Rev. </EM> <B> D66 </B> (2002) 054022. 
%hep-ph/0110216
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\begin{document}
\begin{frontmatter}
\begin{flushright}
\tt{TTP01-27}
\end{flushright}
\vskip 1truecm
\title{Next-to-leading-log renormalization-group running in heavy-quarkonium creation and annihilation}
\author {Antonio Pineda}
\address{Institut f\"ur Theoretische Teilchenphysik\\ 
        Universit\"at Karlsruhe, 
        D-76128 Karlsruhe, Germany}

\begin{abstract}
In the framework of potential NRQCD, we obtain the next-to-leading-log
 renormalization-group running of the matching coefficients for the heavy
 quarkonium production currents near threshold. This allows to obtain
 S-wave heavy-quarkonium production/annihilation observables with
 next-to-leading-log accuracy within perturbative QCD. In particular, we
 give expressions for the decays of heavy quarkonium to $e^+e^-$ and to
 two photons.
\end{abstract}

\vspace{1cm}
{\small PACS numbers: 12.38.Cy, 12.38.Bx, 11.10.St, 12.39.Hg}
\end{frontmatter}

\newpage        

\pagenumbering{arabic}
Heavy quark-antiquark systems near threshold are characterized by the
small relative velocity $v$ of the heavy quarks in their center of
mass frame. This small parameter produces a hierarchy of widely
separated scales: $m$ (hard), $mv$ (soft), $mv^2$ (ultrasoft), ... .
The factorization between them is efficiently achieved by using
effective field theories, where one can organize the calculation as
various perturbative expansions on the ratio of the different scales
effectively producing an expansion in $v$. The terms in these series
get multiplied by parametrically large logs: $\ln v$, which can also
be understood as the ratio of the different scales appearing in the
physical system. Again effective field theories are very efficient in
the resummation of these large logs once a renormalization group (RG)
analysis of them has been performed. This will be the aim of this
paper for annihilation and production processes near threshold.

We will restrict ourselves, in this paper, to the situation where $\lQ
\ll m\als^2$ (to be implicit in what follows), which is likely to be
relevant, at least, for $t$-$\bar t$ production near threshold.

NRQCD \cite{NRQCD} has an ultraviolet (UV) cut-off $\nu_{NR}=\{\nu_p,\nu_s\}$
satisfying $mv \ll \nu_{NR} \ll m$. At this stage $\nu_p \sim
\nu_s$. $\nu_p$ is the UV cut-off of the relative three-momentum of
the heavy quark and antiquark, ${\bf p}$. $\nu_s$ is the UV cut-off of the
three-momentum of the gluons and light quarks.

Potential NRQCD (pNRQCD) \cite{pNRQCD} (see \cite{pNRQED,long} for
details) is defined by its particle content and cut-off
$\nu_{pNR}=\{\nu_p,\nu_{us}\}$, where $\nu_p$ is the cut-off of the
relative three-momentum of the heavy quarks and is such that $|{\bf
p}| \ll \nu_p \ll m$, and $\nu_{us}$ is the cut-off of the
three-momentum of the gluons and light quarks with ${\bf p}^2/m \ll
\nu_{us} \ll |{\bf p}|$. Note that no gluons or light quarks with
momentum of $O(|{\bf p}|)$ are kept dynamical, since they do not appear
as physical (on-shell) states near threshold. Nevertheless, these
degrees of freedom can appear off-shell and, since their momentum is
of the order of the relative three-momentum of the heavy quarks, their
integration produces non-local terms (potentials) in three-momentum
space. In particular, the integration of these degrees of freedom
takes into account the non-analytical behavior in the transfer
momentum of the heavy quark, ${\bf k}={\bf p}-{\bf p}'$, of the order
of the relative three-momentum of the heavy quarks.

The matching process, which basically means the computation of the
potentials, is carried out for a given external incoming (outcoming)
momentum ${\bf p}$ (${\bf p}'$) and one has to sum over all them in
the pNRQCD Lagrangian, since they are still physical degrees of
freedom as far as their momentum is below $\nu_p$. In position space
this means that an integral over ${\bf x}$, the relative distance
between the heavy quarks, appears in the Lagrangian when written in
terms of the heavy quark-antiquark bilinear.

Within pNRQCD, integrals over ${\bf p}$ (or ${\bf x}$) appear when
solving the Schr\"odinger equation that dictates the dynamics of the
heavy quarkonium near threshold. At lower orders these integrals are
finite effectively replacing ${\bf p}$ by $\sim m\als$. Nevertheless,
at higher orders in quantum mechanics perturbation theory and/or if
some singular enough operators are introduced (as it will be the case
of the heavy quarkonium production currents) singularities
proportional to $\nu_p$ appear that must be absorbed by the potentials
or by the matching coefficients of the currents. We will describe how
to resum the logarithms associated to this cutoff within pNRQCD.

Let us now describe the matching between QCD and pNRQCD within an RG
framework. For the case where no divergences proportional to $\nu_p$
appear, the procedure reduces to the results of Ref. \cite{RGmass}, of
which we will closely follow the notation in the following.

We first address the procedure that gives the running of the potentials.
One first does the matching from QCD to NRQCD. The latter depends on
some matching coefficients: $c(\nu_s)$ and $d(\nu_p,\nu_s)$, which can
be obtained order by order in $\als$ (with $\nu_p=\nu_s$) following the
procedure described in Ref. \cite{Manohar}. The $c(\nu_s)$ stand for the
coefficients of the operators that already exist in the theory with only
one heavy quark (ie. HQET) and the $d(\nu_p,\nu_s)$ stand for the
coefficients of the four heavy fermion operators. The starting point of
the renormalization group equation can be obtained from these
calculations by setting $\nu_p=\nu_s=m$ (up to a constant of order
one). In principle, we should now compute the running of $\nu_p$ and
$\nu_s$. The running of the $c(\nu_s)$ can be obtained using HQET
techniques \cite{HQET}. The running of the $d(\nu_p,\nu_s)$ is more
complicated. At one-loop, $\nu_p$ does not appear and we effectively
have $d(\nu_p,\nu_s)\simeq d(\nu_s)$, whose running can also be obtained
using HQET-like techniques \cite{RGmass}. At higher orders the
dependence on $\nu_p$ appears and the running of the $d(\nu_p,\nu_s)$
becomes more complicated. Fortunately, we need not to compute the
running of $d$ in this more general case because, as we will see, the
relevant running of the $d$ for near threshold observables can be
obtained within pNRQCD.

The next step is the matching from NRQCD to pNRQCD. The latter depends
on some matching coefficients (potentials). They typically have the
following structure: ${\tilde V}(c(\nu_s), d(\nu_p,\nu_s),\nu_s,\\
\nu_{us},r)$. After matching, any dependence on $\nu_s$ disappears since
the potentials have to be independent of $\nu_s$. Therefore, they could
be formally written as ${\tilde V}(c(1/r), d(\nu_p,1/r),
1/r,\nu_{us},r)$. These potentials can be obtained order by order in
$\als$ following the procedure of \cite{pNRQCD,pNRQED}. The integrals in
the matching calculation would depend on a factorization scale $\mu$,
which could be either $\nu_s$ or $\nu_{us}$. In the explicit calculation
they could be distinguished by knowing the UV and infrared (IR) behavior
of the diagrams: UV divergences are proportional to $\nu_s$, which
should be such as to cancel the $\nu_s$ scale dependence inherited from
the NRQCD matching coefficients, and IR divergences to $\nu_{us}$. In
practice, however, as far as we only want to perform a matching
calculation at some given scale $\mu=\nu_s=\nu_{us}$, it is not
necessary to distinguish between $\nu_s$ and $\nu_{us}$ (or if working
order by order in $\als$ without attempting any log resummation).

Before going into the rigorous procedure to obtain the RG equations of
the potentials, let us first discuss their structure on physical
grounds. As we have mentioned, the potential is independent of
$\nu_s$. The independence of the potential with respect $\nu_s$ allows
us to fix the latter to $1/r$ that, in a way, could be understood as the
matching scale for $\nu_s$\footnote{In practice, the potential is often
first obtained in momentum space so that one could then set
$\mu=k$. Note, however, that this is not equivalent to $\mu=1/r$ since
finite pieces will appear after performing the Fourier
transform.}. Therefore $1/r$, the point where the multipole expansion
starts, would also provide with the starting point of the
renormalization group evolution of $\nu_{us}$ (up to a constant of order
one).  The running of $\nu_{us}$ can then be obtained following the
procedure described in Ref. \cite{RG,RGmass}.  At the end of the day, we
would have ${\tilde V}(c(1/r),d(\nu_p,1/r),1/r,\nu_{us},r)$, where the
running on $\nu_{us}$ is known and also the running in $1/r$ if the $d$
is $\nu_p$-independent. So far the only explicit dependence of the
potential on $\nu_p$ appears in the $d$. Nevertheless, the potential is
also implicitly dependent on the three-momentum of the heavy quarks
through the requirement $1/r \sim {\bf p} \ll \nu_p$ and also through
$\nu_{us}$ since the latter needs to fulfill ${\bf p}^2/m \ll \nu_{us}
\ll |{\bf p}|$. This latter requirement is fulfilled if we fix
$\nu_{us}=\nu_p^2/m$ (this constraint tells you how much you can run
down $\nu_{us}$ in the potential before finding the cutoff $\nu_p^2/m$
caused by the cutoff of ${\bf p}$).

Within pNRQCD, the potentials should be introduced in the
Schr\"odinger equation. This means that integrals over the relative
three-momentum of the heavy quarks take place. When these integrals
are finite one has ${\bf p} \sim 1/r \sim m\als$ and ${\bf p}^2/m \sim
m\als^2$. Therefore, one can lower $\nu_{us}$ up to $\sim m\als^2$
reproducing the results obtained in Ref. \cite{RGmass}. In some cases,
in particular in heavy quarkonium creation, the integrals over ${\bf
p}$ are divergent and the log structure is dictated by the ultraviolet
behavior of ${\bf p}$ and $1/r$. This means that we can not replace
$1/r$ and $\nu_{us}$ by its physical expectation values but rather by
their cutoffs within the integral over ${\bf p}$. Therefore, for the
RG equation of $\nu_p$, the anomalous dimensions will depend (at
leading order) on ${\tilde
V}(c(\nu_p),d(\nu_p,\nu_p),\nu_p,\nu_p^2/m,\nu_p)$\footnote{Roughly speaking, this result can be
thought as expanding $\ln r$ around $\ln \nu_p$ in the potential ie.
\bea
\nn 
{\tilde V}(c(1/r),d(\nu_p,1/r),1/r,\nu_p^2/m,r) &\simeq& {\tilde
V}(c(\nu_p),d(\nu_p,\nu_p),\nu_p,\nu_p^2/m,\nu_p)
\\
&&
+\ln(\nu_pr)r{d \over d r} {\tilde V}\bigg|_{1/r=\nu_p} + \cdots 
\eea 
The $\ln(\nu_pr)$ terms may give subleading contributions to the
anomalous dimension when introduced in divergent integrals over ${\bf
p}$. The discussion at this stage is not very rigorous and a more
precise discussion would require a full detailed study within
dimensional regularization. Nevertheless, we do not expect it to
change the underlying idea although it deserves
further investigations.} and the running will go from $\nu_p \sim m$
down to $\nu_p \sim m\als$. Note that at this stage a single cutoff,
$\nu_p$, exists and the correlation of cutoffs can be seen. The
importance of the idea that the cutoffs of the non-relativistic
effective theory should be correlated was first realized by Luke,
Manohar and Rothstein in Ref. \cite{LMR} (for an application to QED see \cite{QED}). Note also that at the matching scale $\nu_p \sim m$
what it would be the ultrasoft cutoff is also of order $m$. In this
sense it should be understood the statement in Ref. \cite{LMR} that
ultrasoft gluons appear at the scale $m$, a point that becomes relevant
within a RG approach.

With the above discussion in mind the matching between NRQCD and
pNRQCD could be thought as follows. One does the matching by computing
the potentials order by order in $\als$ at the matching scale
$\nu_p=\nu_s=\nu_{us}$ following the procedure of \cite{pNRQCD,pNRQED} (by
doing the matching at a generic $\nu_p$ some of the running is
trivially obtained).  The structure of the potential at this stage
then reads ${\tilde V}(c(\nu_p),d(\nu_p,\nu_p),\nu_p,\nu_p,\nu_p)$ (and
similarly for the derivatives with respect $\ln r$ of the
potential). This provides the starting point of the renormalization
group evolution of $\nu_{us}$ (up to a constant of order one).  The
running of $\nu_{us}$ can then be obtained following the procedure
described in Ref. \cite{RG,RGmass}. For the final point of the evolution
of $\nu_{us}$ we choose $\nu_{us}=\nu_p^2/m$. At the end of the day, we
obtain ${\tilde V}(c(\nu_p), d(\nu_p,\nu_p),\nu_p,\nu_p^2/m,\nu_p)
\equiv {\tilde V}(\nu_p)$.

The running of $\nu_p$ goes from $\nu_p=m$ (this was fixed when the
matching between QCD and NRQCD was done) up to the physical scale of
the problem $\nu_p \sim m\als$. As far as the running of the NRQCD
matching coefficients is known, the above result gives the complete running of
the potentials. The procedure to get the running of the $c$ is known
at any finite order. For the $d$ it is only known at one-loop order, 
since, at this order, it is just $\nu_s$-dependent. Nevertheless, at
higher orders, dependence on $\nu_p$ appears. Therefore, the above
method is not complete unless an equation for the running of $\nu_p$ is
provided. This is naturally given within pNRQCD. It appears through the
iteration of potentials. One typical example would be the diagram in
Fig. 1, which would contribute to the running of
$D_{d,s}^{(2)}$, the matching coefficient of the delta potential, at
next-to-leading-log (NLL) order as follows (see Ref. \cite{RGmass} for
notation) 
\be
\label{eqDd}
\nu_p {d \over d\nu_p}D_{d,s}^{(2)}(\nu_p) \sim \alpha_{V_s}(\nu_p)D_{d,s}^{(2)2}(\nu_p)+\cdots
\ee
so that even without knowing the running of the $d$ 
(which need to be known at NLL order in this case) we
can obtain the running of the potential (one can also think of trading
Eq. (\ref{eqDd}) into an equation for $d$, which is the only unknown 
parameter within the potential).

%%%%Figure Ddnup%%%%
\medskip
\begin{figure}[h]
\hspace{-0.1in}
\epsfxsize=2.8in
\centerline{\epsffile{Ddnup.eps}}
\caption {{\it Contribution to the running of $D_{d,s}^{(2)}$ at NLL.}}
\label{obs12}
\end{figure}
%%%%End Figure Ddnup%%%%

The matching scale between QCD and NRQCD is $\nu_p \sim \nu_s \sim
m$. On the other hand, the matching scale between NRQCD and pNRQCD is
also the hard scale: $\nu_p\sim \nu^2_p/m \sim m$. Therefore, one could
wonder about the necessity of using the intermediate theory NRQCD. This
is indeed the attitude in Ref. \cite{LMR,QED,vNRQCD2,V1MS}, where they
directly perform the matching between QCD to an effective field theory:
vNRQCD that, once the RG evolution has been performed and the soft
degrees of freedom have been integrated out, should be physically
equivalent to pNRQCD with $\nu_p \sim m\als$. One motivation for going
through NRQCD is that it allows to perform the factorization of the hard
scale within an effective field theory framework. In fact, a full
factorization of the different regions of momentum that ought to be
integrated out is achieved within pNRQCD. This extremelly simplifies the
matching process since one deals with only one scale (region of
momentum) in the loops at each step. In the matching between QCD and
NRQCD only hard loops need to be considered whereas in the matching
between NRQCD and pNRQCD only soft loops need to be
considered. Moreover, the structure of the UV cutoffs of the theory is
better understood in this way. For instance, one can see that all the
{\it explicit} dependence of the potentials on $\nu_p$ is inherited from
the $d$ matching coefficients. Within a diagrammatic approach the
factorization of the different regions of momentum have been achieved
using the threshold expansion \cite{BS}.

Let us now consider the case of the electromagnetic current, which will
provide an example where to apply the above discussion. 
The
procedure is analogous to the potentials. We first do the matching
from QCD to NRQCD:
\be
{\bar Q}\gamma^{\mu}Q(0) \bigg|_{\rm QCD} \dot= c_{1,\rm NR}\psi^\dagger\sigma^i\chi(0)+O(1/m)\bigg|_{\rm NRQCD}\,.
\ee
We will just concentrate in the coefficient $c_{1,\rm NR}$. Within NRQCD, it should
be understood as a function of $\nu_p$ and $\nu_s$,
ie. $c_{1,\rm NR}(\nu_p,\nu_s)$. One should first obtain the matching conditions
at the hard scale. This has been
computed up to two loops \cite{BSSCM} but we will only need the one-loop
expression \cite{Barbieri}:
\be
c_{\rm 1, NR}(m,m)=1-2C_f{\als(m) \over \pi}
,
\ee
since we only aim to a NLL resummation in this paper. If we compare with the previous discussion of the potentials, the
matching coefficients $d$ play the role of $c_1$. Therefore, within
pNRQCD, we will need $c_{\rm 1, NR}(\nu_p,\nu_p)\equiv c_{\rm 1, NR}(\nu_p)$. The matching from NRQCD to pNRQCD creates the
potentials but let $c_1$ unchanged since soft loops or HQET-like
calculations give zero correction to $c_1$. Formally, 
\be
c_{\rm 1, NR}\psi^\dagger\sigma^i\chi(0)\bigg|_{\rm NRQCD}=c_{\rm 1, pNR}\psi^\dagger\sigma^i\chi(0)\bigg|_{\rm pNRQCD}\,,
\ee
or in other words the matching condition reads 
$c_{\rm 1, pNR}(c_{\rm 1, NR}(\nu_p),\nu_{us}=\nu_p)=c_{\rm 1, NR}(\nu_p)$. The running
of $\nu_{us}$ is also trivial as there is none (this has to do with the
fact that we are dealing with an annihilation process). Therefore, we
finally have $c_1(\nu_p) \equiv
c_{\rm 1, pNR}(c_{\rm 1, NR}(\nu_p),\nu_p^2/m)=c_{\rm 1, NR}(\nu_p)$.  We can see that we are in the analogous
situation to the running of $D_{d,s}^{(2)}(\nu_p)$ versus the running of
$d(\nu_p)$. We now need the RG
equation for $c_1(\nu_p)$. This demands to obtain the
ultraviolet corrections to the current within pNRQCD keeping track of
the contributions due to the different potentials. Fortunately, this
calculation has already been done and we can extract the relevant information from
\cite{KP2}. The RG equation reads 
\be 
\label{cseq}
\nu_p {d  \over d\nu_p} c_s =c_s
\left[
-{C_AC_f \over 2}D_s^{(1)}
-{C_f^2 \over 4}\al_{V_s}
	\left\{\al_{V_s}-{4\over 3}s(s+1)D_{S^2,s}^{(2)}
		-D_{d,s}^{(2)}+4D_{1,s}^{(2)}
	\right\} 
\right]
\,, 
\ee
where $C_f=(N_c^2-1)/(2N_c)$, $C_A=N_c$, and the RG-improved potential matching coefficients can be read from 
Ref. \cite{RG,RGmass} with the assignment $1/r \rightarrow \nu_p$ and $\nu_{us}
\rightarrow \nu^2_p/m$ (see also \cite{pot,long} for calculations of the potential
at finite orders in $\als$). We have kept the spin explicit so our
results will also be valid for the pseudoscalar current:
\be
{\bar Q}\gamma^{0}\gamma_5Q(0) \bigg|_{\rm QCD}\dot= c_{0,\rm
NR}\psi^\dagger\chi(0)+O(1/m)\bigg|_{\rm NRQCD}\,, 
\ee
with the matching condition \cite{Barbieri2}:
\be
c_0 (m)=1+\left({\pi^2 \over 4}-5\right){C_f \over 2}{\als(m) \over \pi}
.
\ee

Eq. (\ref{cseq}) gives subleading effects
within an strict expansion in $\als$. Therefore, it can be approximated to 
\be 
\label{cseqap}
\nu_p {d  \over d\nu_p} c_s =
-{C_AC_f \over 2}D_s^{(1)}
-{C_f^2 \over 4}\als
	\left\{\als-{4\over 3}s(s+1)D_{S^2,s}^{(2)}
		-D_{d,s}^{(2)}+4D_{1,s}^{(2)}
	\right\} 
\,,
\ee
and the solution reads
\bea 
c_s(\nu_p)&=&c_s(m)+ 
	A_1 {\als(m) \over w^{\beta_0}} \ln(w^{\beta_0})+ 
	A_2 \als(m) \bigg[z^{\beta_0}-1\bigg] + 
	A_3 \als(m) \bigg[z^{\beta_0-2 C_A}-1  \bigg]
\nn
\\
&&+
	A_4 \als(m) \bigg[z^{\beta_0-13C_A/6}-1 \bigg]+
	A_5 \als(m) \ln(z^{\beta_0})
\,,
\label{cssln}
\eea
where $\beta_0={11 \over 3}C_A -{4 \over 3}T_Fn_f$, $z=\left[{\als(\nu_p) \over \als(m)}\right]^{1 \over
\beta_0}$ and $w=\left[{\als(\nu_p^2/m) \over \als(\nu_p)}\right]^{1 \over
\beta_0}$. The coefficients $A_i$ in Eq.~(\ref{cssln}) read
\begin{eqnarray} 
\label{Acoeffs}
  A_1 &=& {8\pi C_f \over 3\beta_0^2 }\left(C_A^2 +2C_f^2+3 C_fC_A
  \right) \,,\nn\\ 
  A_2 &=& { \pi C_f [3 \beta_0(26C_A^2+19C_A C_f-32C_f^2)-
     C_A(208 C_A^2+651 C_A C_f+116 C_f^2)]\over 78\,\beta_0^2\, C_A } \,, \nn\\
  A_3 &=& -{\pi C_f^2 \Big[  \beta_0 (4s(s+1)-3)+ C_A (15-14s(s+1)) \Big]
     \over 6 (\beta_0-2 C_A)^2 }\,,\nn\\
  A_4 &=& {24 \pi C_f^2 (3\beta_0-11 C_A)(5 C_A+8 C_f) \over 13\, C_A
     (6\beta_0-13 C_A)^2}\,,\nn\\
  A_5 &=& {-\pi C_f^2 \over \beta_0^2\, (6\beta_0-13C_A) (\beta_0-2C_A)}
      \, \Bigg\{ C_A^2(-9C_A+100C_f)
	\nn\\ 
	&&+\beta_0 C_A(-74C_f+C_A(42-13s(s+1)))
	+6\beta_0^2(2C_f+C_A(-3+s(s+1)))\Bigg\}
     \,.
\end{eqnarray}
Our evaluation can be compared with the result obtained using the
vNRQCD formalism \cite{V1MS}. We agree for the spin-dependent terms
but differ for the spin-independent ones. The disagreement still holds
if we consider QED with light fermions ($C_f \rightarrow 1$, $C_A
\rightarrow 0$, $T_F \rightarrow 1$). Agreement is found if we
consider QED without light fermions ($C_f \rightarrow 1$, $C_A
\rightarrow 0$, $n_f \rightarrow 0$, $T_F \rightarrow 1$). If we
expand our results in $\als$ we can compare with earlier results in
the literature. By following the discussion in Ref. \cite{V1MS}, we
can relate our results with the correction to the wave-function at the
origin as defined in Ref. \cite{KP2}. We obtain
\begin{eqnarray} \label{D1}
\Delta \psi^2(0) &=& \left| \frac{c_s(\nu_p)}{c_s(1)} \right|^2-1
  \,
=
-C_f \als^2 \ln(\als) \bigg\{ \Big[2-\frac23 s(s+1) \Big] C_f
+ C_A \bigg\} 
\\
\nn
&&- \frac{C_f}{\pi} \als^3 \ln^2(\als) \Bigg\{ \frac32 C_f^2 +
  \Big[ \frac{41}{12}-\frac{7}{12} s(s+1) \Big] C_f C_A +\frac23 C_A^2
\\
\nn
&&
\qquad\qquad\qquad\qquad
+\frac{\beta_0}{2} \bigg[ \Big( 2-\frac23 s(s+1) \Big) C_f+C_A \bigg] 
  \Bigg\} + \ldots \,,
\end{eqnarray}
where we have expanded up to second order in
$\ln(\nu_p)=\ln(m\als)$ with $\als\equiv\als(\nu_p)$.  The first
term reproduces the leading log term \cite{BSSCM,leadinglog} (see also \cite{HT}), the $\beta_0$-independent
$O(\als^3\ln^2\als)$ terms reproduce Kniehl and Penin results
\cite{KP2} and we agree with the complete $O(\als^3\ln^2\als)$ term
computed by Manohar and Stewart \cite{V1MS} (the sign of difference
for the $\beta_0$-dependent terms displayed in Ref. \cite{V1MS} is due
to the fact that in Ref. \cite{V1MS} the expansion was made with
$\als(m)$ whereas here we have chosen $\als(m\als)$). Nevertheless, 
disagreement with this last evaluation appears at higher orders in the
expansion in $\als$ (we have explicitly checked this for the
$O(\als^4\ln^3\als)$ terms). As far as we can see the disagreement seems
to be due to the fact that they have different expressions for the RG
improved potentials \cite{vNRQCD2,V1MS}.

By setting $\nu_p \sim m\als$, $c_s(\nu_p)$ includes all the large logs at NLL order in any S-wave
heavy-quarkonium production observable we can think of. For instance, the decays to $e^+e^-$ and to two
photons at NLL order read
\bea
\Gamma(V_Q (nS) \rightarrow e^+e^-) &=&
2 \left[ { \alpha_{em} Q \over M_{V_Q(nS)}} \right]^2
\left({m_QC_f\als \over n}\right)^3
\left\{
c_1(\nu_p)(1+\delta \phi_n)
\right\}^2
\\
\nn
&\simeq&
2 \left[ { \alpha_{em} Q \over M_{V_Q(nS)}} \right]^2
\left({m_QC_f\als \over n}\right)^3
\left\{1+2(c_1(\nu_p)-1)+2\delta \phi_n
\right\}
\,,
\\
\Gamma(P_Q (nS) \rightarrow \gamma\gamma) &=&
6 \left[ { \alpha_{em} Q^2 \over M_{P_Q(nS)}} \right]^2
\left({m_QC_f\als \over n}\right)^3
\left\{
c_0(\nu_p)(1+\delta \phi_n)
\right\}^2
\\
\nn
&\simeq&
6 \left[ { \alpha_{em} Q^2 \over M_{P_Q(nS)}} \right]^2
\left({m_QC_f\als \over n}\right)^3
\left\{1+2(c_0(\nu_p)-1)+2\delta \phi_n
\right\}
\,,
\eea
where $V$ and $P$ stand for the vector and pseudoscalar heavy
quarkonium, we have fixed $\nu_p=m_QC_f\als/n$, $\als=\als(\nu_p)$, and
($\Psi_n(z) ={d^n \ln \Gamma (z) \over dz^n}$ and $\Gamma (z)$ is the
Euler $\Gamma$-function)
\be
\delta \phi_n ={\als \over \pi}
\left[-C_A+
{\beta_0\over 4}
\left(\Psi_1(n+1)-2n\Psi_2(n)+{3 \over 2}+\gamma_E+{2\over n}
\right)
\right]
\,,
\ee
which has been read from Ref. \cite{leadinglog} (see also \cite{KPPY}). Working along similar
lines one could easily obtain NLL expressions for other heavy quarkonium
observables in the study of $t$-$\bar t$ production
near threshold or in sum rules of bottomonium.   

In conclusion, we have shown that pNRQCD, if necessary, can
comprehensively incorporate a RG framework by using the method
described in Ref. \cite{RGmass} plus incorporating the idea \cite{LMR}
of correlating the cutoffs of the effective theory. We have used this 
formalism to compute the running of the matching coefficients of the
vector and pseudoscalar currents and disagreement with the results
obtained using the vNRQCD framework have been found \cite{V1MS}. Our
results allow to obtain S-wave heavy quarkonium production observables
with NLL accuracy. We have explicitely illustrated this point for
heavy-quarkonium decays to $e^+e^-$ and to two photons.

{\bf Acknowledgments}\\
We thank A. Hoang, A.V. Manohar and specially J. Soto for useful
discussions. We also thank J. Soto for comments on the
manuscript. 

%%%%%%%%%%%%%%%%%%%%% BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{thebibliography}{99}

\bibitem{NRQCD} W.E. Caswell and G.P. Lepage, Phys. Lett. {\bf B167}, 437 (1986); 
 G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. {\bf D51}, 1125
(1995); (E) ibid. {\bf D55}, 5853 (1997).

\bibitem{pNRQCD} A. Pineda and J. Soto, Nucl. Phys. {\bf B}
(Proc. Suppl.) {\bf 64}, 428 (1998)

\bibitem{pNRQED} A. Pineda and J. Soto, Phys. Lett. {\bf B420}, 391
(1998); Phys. Rev. {\bf D59}, 016005 (1999).

\bibitem{long} N. Brambilla, A. Pineda, J. Soto and A. Vairo, Phys. Rev. {\bf D60} (1999) 091502; 
  Nucl. Phys. {\bf B566}, 275 (2000); Phys. Lett. {\bf B470} (1999) 215.

\bibitem{RGmass} A. Pineda, hep-ph/0109117.

\bibitem{Manohar} A.V. Manohar, Phys. Rev. {\bf D56}, 230 (1997);
A. Pineda and J. Soto, Phys. Rev. {\bf D58}, 114011 (1998).

\bibitem{HQET} E. Eichten and B. Hill, Phys. Lett. {\bf B243}, 427
(1990); A.F. Falk, B. Grinstein and M.E. Luke, Nucl. Phys. {\bf B357},
185 (1991); B. Blok, J.G. Korner, D. Pirjol and J.C. Rojas,
Nucl. Phys. {\bf B496}, 358 (1997). C. Bauer and A.V. Manohar,
Phys. Rev. {\bf D57}, 337 (1998).

\bibitem{RG} A. Pineda and J. Soto, Phys. Lett. {\bf B495}, 323 (2000).

\bibitem{LMR} M.E. Luke, A.V. Manohar, I.Z. Rothstein,
Phys. Rev. {\bf D61}, 074025 (2000).  

\bibitem{QED} A.V. Manohar and I.W. Stewart, Phys. Rev. Lett. {\bf 85},
2248 (2000). 

\bibitem{vNRQCD2} A.V. Manohar and I.W. Stewart, Phys. Rev. {\bf D62},
014033 (2000); Phys. Rev. {\bf D62}, 074015 (2000).  

\bibitem{V1MS} A.V. Manohar and I.W. Stewart, Phys. Rev. {\bf
D63}, 054004 (2001).  

\bibitem{BS} M. Beneke and V.A. Smirnov, Nucl. Phys. {\bf
B522}, 321 (1998). 

\bibitem{BSSCM} M. Beneke, A. Signer and V. Smirnov, Phys. Rev. Lett. {\bf 80}
2535 (1998); A. Czarnecki and K. Melnikov, {\it ibid} 2531.

\bibitem{Barbieri} R. Barbieri, R. Gatto, R. Kogerler and Z. Kunszt,
Phys. Lett. {\bf B57}, 455 (1975). 

\bibitem{KP2} B.A. Kniehl and A.A. Penin, Nucl. Phys. {\bf B577}, 197
(2000); Nucl. Phys. {\bf B563}, 200 (1999).

\bibitem{pot} W. Fischler, Nucl. Phys. {\bf B129}, 157 (1977);
S.N. Gupta, S.F. Radford and W.W. Repko, Phys. Rev. {\bf D26} 3305 (1982);
S. Titard and F.J. Yndur{\'a}in, Phys. Rev. {\bf D49}, 6007 (1994); Y. Schr\"oder, Phys. Lett. {\bf B447}, 321 (1999);
B.A. Kniehl, A.A. Penin, V.A. Smirnov and M. Steinhauser,
hep-ph/0106135; M. Peter, Phys. Rev. Lett. {\bf 78}, 602  (1997).

\bibitem{Barbieri2} I. Harris and L.M. Brown, Phys. Rev. {\bf 105}, 1656
(1957); R. Barbieri, E. d'Emilio, G. Curci and E. Remiddi,
Nucl. Phys. {\bf B154}, 535 (1979).


\bibitem{leadinglog} K. Melnikov and A. Yelkhovsky, Phys. Rev. {\bf D59},
114009 (1999); A.A. Penin and A.A. Pivovarov, Nucl. Phys. {\bf B549},
217 (1999).   

\bibitem{HT} A.H. Hoang and T. Teubner, Phys. Rev. {\bf D58}, 114023
(1998). 

\bibitem{KPPY} J.H. K\"uhn, A.A. Penin and A.A. Pivovarov,
Nucl. Phys. {\bf B534}, 356 (1998);  F.J. Yndur\'ain, hep-ph/0007333. 
 
\end{thebibliography}



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