text:
\section{Introduction}

The singlet potential between two color sources in the static limit, i.e. with infinite masses,
is one of the most accurately studied objects in QCD. It plays an important role in understanding the
dynamics of QCD. On the one hand it is an essential ingredient in a
Schr\"odinger-like description of heavy quarkonia. On the other hand, a linear growing potential at long distances (in the large
$N_c$ limit) indicates confinement of the strong interactions. This has motivated
a lot of effort to steadily increase the accuracy of the perturbative prediction for
that object~\cite{FSP,Schroder:1999sg,short,KP1,RG,Hoang:2002yy,Brambilla:2006wp,Brambilla:2009bi,Anzai:2009tm,Smirnov:2009fh},
which should be reliable at short distances. A crucial point is obviously to distinguish which region can be described within perturbation
theory and to understand the crossover to the non-perturbative regime~\cite{Recksiegel:2001xq,Pineda:2002se,Lee:2002sn},~\cite{Brambilla:2009bi}.

Besides the singlet also the color octet static potential has attracted some interest in the last decade.
It has been computed with two-loop precision at weak coupling in Ref.~\cite{Kniehl:2004rk}. Together with
the so-called gluelumps, it describes the short-distance behavior of
the hybrid potentials~\cite{Brambilla:1999xf,Bali:2003jq}, which are potentially important for the theoretical prediction
of physical hybrids made of heavy quarks. At long distances the hybrid potentials could also help
to shed light on the question whether the dynamics responsible for confinement is of string type and if so,
to determine its structure exactly.

In this paper, we will focus on the singlet static potential, $V_s(r)$,
and on the singlet static energy, $E_s(r)$, at short distances. $V_s(r)$ is the leading order
potential introduced in a Schr\"odinger-like description of heavy quarkonia.
$E_s(r)$ is the energy of a static color singlet quark-antiquark pair with (fixed) distance $r$,
being the object actually computed on  the lattice.
Its physics at short distances is governed by at least two physical scales.
One is the soft scale $\sim 1/r$, and the other one is the ultrasoft scale $\sim V_s$.
The most convenient framework to perform the corresponding calculations is the effective field theory (EFT)
``potential NRQCD'' (pNRQCD) \cite{Pineda:1997bj} (for a review see \cite{Brambilla:2004jw}).
pNRQCD also has been instrumental in many of the latest perturbative results
quoted above, especially in the determination of the ultrasoft contributions.
In its static limit the theory exploits the scale separation $1/r \gg V_s$. $V_s$
accounts only for the soft scale contributions to the static energy and can be computed in
perturbative QCD (pQCD). $E_s$ also includes effects at the ultrasoft scale, which in the
static limit is generated by a resummation of certain classes of loop diagrams to all
orders and serves as an infrared (IR) regulator~\cite{Appelquist:1977es}. These contributions
can be calculated in pNRQCD from a finite number of diagrams.

In order to gain a deeper understanding of the static potential one can consider how it
is qualitatively affected by changing the number of dimensions from four (4D) to three (3D) or two (2D).
The three-dimensional result is also important on its own. Four dimensional thermal QCD
undergoes effectively a dimensional reduction for large temperatures.
Therefore, determining the renormalization group (RG) structure for the static potential in
three dimensions may open the way to a resummation of logarithms at finite temperature.
Three dimensional space-time is moreover a good testing ground for renormalon issues,
since the linear power divergences associated with renormalons in four dimensions become logarithmic divergences
in three dimensions and can be traced back using dimensional regularization.
The computations in three and two dimensions finally represent consistency
checks of the theoretical approach used to describe the 4D potential.

Partially driven by these motivations the computation of the static potential has been
carried out up to two loops for arbitrary dimension $D$ in Ref.~\cite{Schroder:1999sg}. Whereas in four
dimensions the result is finite, this is not the case in three dimensions.
Beyond one-loop it suffers from IR singularities, whose origin was obscure at the time they were encountered.
In particular, it was not clear whether the space-like end-strings of the rectangular Wilson loop, which
defines the static potential non-perturbatively, could contribute, raising doubts about the
independence of the result on the precise form of the strings.

In 4D such IR singularities first appear at three-loop level and were shown to cancel with the ultrasoft ultraviolet (UV)
singularities in Ref.~\cite{short}. Up to now an analogous ultrasoft computation has been missing in three dimensions. Therefore,
the precision of the 3D static potential has been limited to ${\cal O}(\alpha^2 r)$ from the finite tree-level and one-loop results.
With this work we will show for the color singlet state that the UV divergences
of the ultrasoft pNRQCD calculation in fact cancel the soft pQCD IR singularities exactly.
This allows us, for the first time, to go beyond the one-loop level and to obtain
the logarithmic corrections at two loops as well as the full analytical structure of
the ${\cal O}(r^2)$ correction. We will point out the peculiarities of the 3D computation in detail.

The outline of the paper is as follows.
In Sec.~\ref{pNRQCD} we introduce the theoretical setup for our calculations,
we define the power counting of the EFT in $D<4$ dimensions and review existing
results for the relevant ultrasoft loop diagrams. Sec.~\ref{sec3D} contains the
complete RG improved ultrasoft calculation up to NNLL and partly up to $\rm N^4$LL
order in three dimensions as well as a comparison of the results to lattice data.
In Sec.~\ref{secD2} we discuss the situation in two dimensions and in Sec.~\ref{conclusion} we present our conclusions.

\section{pNRQCD}
\label{pNRQCD}

Up to the next-to-leading order (NLO) in the multipole expansion (and irrespectively of the space-time dimension)
the effective Lagrangian density of pNRQCD in the static limit
 takes the form~\cite{Pineda:1997bj},~\cite{Brambilla:1999xf}:
\begin{eqnarray}
& & {\cal L}_{\rm us} =
{\rm Tr} \Biggl\{ {\rm S}^\dagger \left( i\partial_0  - V_s(r)   \right) {\rm S}
 + {\rm O}^\dagger \left( iD_0 - V_o(r)  \right) {\rm O} \Biggr\}
\nonumber\\
& &\qquad
+ g V_A ( r) {\rm Tr} \left\{  {\rm O}^\dagger {\bf r} \cdot {\bf E} \,{\rm S}
+ {\rm S}^\dagger {\bf r} \cdot {\bf E} \,{\rm O} \right\}
  + g {V_B (r) \over 2} {\rm Tr} \left\{  {\rm O}^\dagger \left\{{\bf r} \cdot {\bf E} , {\rm O}\right\}\right\} +  {\cal O}(r^2)\,.
\label{pnrqcd0}
\end{eqnarray}
We define color singlet and octet fields for the quark-antiquark system by $S = S({\bf r},{\bf R},t)$ and
$O^a =  O^a({\bf r},{\bf R},t)$, respectively. ${\bf R} \equiv ({\bf x}_1+{\bf x}_2)/2$ is the center position of the system.
In order for $S$ and $O^a$ to have the proper free-field normalization in color space they are related to the fields in Eq.~(\ref{pnrqcd0}) as follows:
\begin{equation}
{\rm S} \equiv { 1\!\!{\rm l}_c \over \sqrt{N_c}} S\,, \qquad {\rm O} \equiv  { T^a \over \sqrt{T_F}}O^a,
\label{norm}
\end{equation}
where $T_F=1/2$ for the fundamental representation of SU($N_c$).
All gluon and scalar fields in Eq. (\ref {pnrqcd0}) are evaluated
in ${\bf R}$ and the time $t$, in particular the chromoelectric field ${\bf E} \equiv {\bf E}({\bf R},t)$ and the ultrasoft covariant derivative
$iD_0 {\rm O} \equiv i \partial_0 {\rm O} - g [A_0({\bf R},t),{\rm O}]$.
$V_{\{s,o,A,B\}}(r)$ are the Wilson coefficients of the effective Lagrangian. They are fixed
 at a scale $\nu$ smaller than (or similar to) $1/r$ and larger than the ultrasoft and any other scale in the problem by
 matching the effective and the underlying theory, which in this case is QCD in the static limit.

\subsection{Power counting}

Because the mass dimension of the coupling is $[g^2]=M^{4-D}$, we have (at least) the following physical scales involved in the problem:
\begin{equation}
k \sim 1/r \;({\rm soft})\,,\qquad k \sim V \;({\rm ultrasoft})\,, \qquad k \sim g^{\frac{2}{4-D}} \;(\text{non-perturbative})\,.
\end{equation}
In order for perturbation to make sense at the (soft) matching scale we demand
\begin{equation}
\label{MP}
gr^{\frac{4-D}{2}} \ll 1
\,,\end{equation}
i.e. weak coupling.
Therefore $g^2r^{4-D}$ plays the role of a dimensionless expansion parameter.
It is a good expansion parameter at short distances for $D<4$
(for $D=4$ Eq.~(\ref{MP}) implies $g \ll 1$).

We are left with two scales: $V \sim 1/r \times g^2r^{4-D}$ and $g^{\frac{2}{4-D}}$ much smaller than $1/r$
as we go to short distances.
In principle, this guarantees that the multipole expansion makes sense, but
it says nothing about the relative size of $V$ and $g^{\frac{2}{4-D}}$, i.e. whether one can use
perturbation theory at the ultrasoft scale $V$. On the other hand note that
\begin{equation}
V/g^{\frac{2}{4-D}} \sim r^{3-D}g^{2(\frac{3-D}{4-D})} \sim (rg^{\frac{2}{4-D}})^{3-D}
\,.
\end{equation}
Therefore, $D=3$ is the turning point for the use of perturbation theory at the ultrasoft scale.

For $4>D>3$ we have a perturbative expansion in $g$:
\begin{equation}
1/r \gg V \gg g^{\frac{2}{4-D}}
\,.
\end{equation}
In this case there is a double expansion with the two parameters
\begin{equation}
g^2r^{4-D} \ll 1\,, \qquad \frac{g^{\frac{2}{4-D}}}{V} \ll 1\,.
\end{equation}

For $D<3$ the right expansion parameter would be $V/g^{\frac{2}{4-D}}$.
In principle, this is not a real problem. We just have to expand the exponent in the
ultrasoft correlators, as we will see later, while the multipole expansion is still valid.
This is potentially quite interesting. In two dimensions the exact result is
known and equivalent to the tree-level result. Therefore, strong cancellations are supposed to occur
among the contributions from the different scales: soft, ultrasoft and $g$. We will further discuss this issue in
Sec. \ref{secD2}.

The case $D=3$, the main focus of this paper, deserves a special discussion.
At short distances we find $V \sim g^2\ln(r\nu)$. This implies that
\begin{equation}
g^2/V_s \sim 1/\ln(r\nu) \ll 1\,,
\end{equation}
if $\nu \sim  V$
and we conclude that we can use perturbation theory at the ultrasoft scale $V$. We therefore formally distinguish between
the scale $V$ and $g^2$. Logarithms from the ultrasoft perturbative computation will have the form $\ln ( V/\nu)$ and are rendered small, if we set $\nu \sim  V$.
Thus it is legitimate to consider the ultrasoft regime as perturbative, i.e. the pNRQCD loop expansion makes sense (for sufficiently small $r$).
The inclusion of non-perturbative effects will be discussed in subsection~\ref{npsubsec}.

\subsection{Bare data}

Here we summarize some bare results that will be relevant in the next sections.
In the following we will use the index ``$B$'' to explicitly denote bare quantities. Parameters without this index are understood to be renormalized.

The general expression for the bare singlet potential in momentum space, ${\tilde V}_{s,B}$, in $D$ dimensions can be written as
\begin{equation}
{\tilde V}_{s,B}=-C_Fg_B^2\frac{1}{{\bf k}^2}
\left\{
1
+
\sum_{n=1}^{\infty}g_B^{2n}{\bf k}^{2n\epsilon}\frac{\tilde c_n(D)}{(4\pi)^{nD/2}}
\right\}
\,.
\label{momspacepot}
\end{equation}
The coefficients $\tilde c_1(D)$ and $\tilde c_2(D)$ can be found
in Ref.~\cite{Schroder:1999sg}. Throughout this paper we will use the notation
$D\equiv 4+2\epsilon \equiv n+2\epsilon_n$, where $\epsilon_n=\frac{D-n}{2}$ parametrizes the (typically infinitesimal) difference to the closest integer dimension $n=4$, 3, 2.

After the Fourier transformation to position space Eq.~(\ref{momspacepot}) becomes
(see for instance Ref.~\cite{Pascual:1984zb})
\begin{eqnarray}
\label{VsBD}
V_{s,B}&=&-C_Fg_B^2
\sum_{n=0}^{\infty}\frac{g_B^{2n}r^{-2(n+1)\epsilon}}{r}
\frac{\tilde c_{n}(D)}{(4\pi)^{nD/2}}
\frac{\Gamma[1/2+(n+1)\epsilon]}{2^{2-2n\epsilon}\pi^{3/2+\epsilon}\Gamma[1-n\epsilon]}
\\
\nonumber
&\equiv&
-C_Fg_B^2\sum_{n=0}^{\infty}\frac{g_B^{2n}c_n(D)r^{-2(n+1)\epsilon}}{r}\,.
\end{eqnarray}

The singlet static energy can be considered to be an observable for our purposes.
It consists of the potential, which is a Wilson coefficient, and an ultrasoft
contribution\footnote{If one has enough precision non-perturbative effects at the
hadronization scale $\Lambda_{\rm QCD} \sim g^{\frac{2}{4-D}}$ should also be included. We will discuss
them in Sec.~\ref{sec3D}.}, either bare or renormalized:
\begin{equation}
E_s(r)=
V_{s,B}+\delta E^{us}_{s,B}
=
V_{s,\overline{\rm MS}}+\delta E^{us}_{s,\overline{\rm MS}}
\,.
\label{Estot}
\end{equation}
The ultrasoft contribution can be expressed in a compact form at NLO in the multipole expansion (but exact to any order in $\alpha$) through the
chromoelectric correlator. It reads (in the Euclidean)
\begin{equation}
\delta E^{us}_{s,B}
  = V_A^2{T_F \over (D-1) N_c} {\bf r}^2 \int_0^\infty \!\! dt  e^{-t(V_{o,B}-V_{s,B})}
  \langle vac|
g{\bf E}_E^a(t)
\phi_{\rm adj}^{ab}(t,0) g{\bf E}_E^b(0) |vac \rangle
\label{deltaVUS}
\,.
\end{equation}

Concrete results for the ultrasoft corrections in $D$ dimensions are
known at one loop since Ref.~\cite{Pineda:1997ie} (see also~\cite{short,KP1}).
The pNRQCD one-loop computation yields
\begin{equation}
\delta E^{us}_{s,B}({\rm 1-loop})
=
-g^2C_FV_A^2(1+\epsilon)\frac{\Gamma[2+\epsilon]\Gamma[-3-2\epsilon]}{\pi^{2+\epsilon}}
{\bf r}\,\Delta V^{3+2\epsilon}_B{\bf r}\,,
\label{USbare1loop}
\end{equation}
where we have defined $\Delta V \equiv V_o-V_s$.
The two-loop bare expression can be deduced from
the results obtained in Refs.~\cite{Eidemuller:1997bb,Brambilla:2006wp} and reads
\begin{equation}
\label{USbare2loop}
\delta E^{us}_{s,B}({\rm 2-loop})
=
g^4C_FC_AV_A^2\Gamma[-3-4\epsilon]
\left[
{\cal D}^{(1)}(\epsilon)-(1+2\epsilon){\cal D}_1^{(1)}(\epsilon)
\right]
{\bf r}\,\Delta V_B^{3+4\epsilon}{\bf r}\,,
\end{equation}
where
\begin{equation}
{\cal D}^{(1)}(\epsilon)
=
\frac{1}{(2\pi)^2}\frac{1}{4\pi^{2+2\epsilon}}\Gamma^2[1+\epsilon]g(\epsilon)\,,
\end{equation}
\begin{equation}
{\cal D}_1^{(1)}(\epsilon)
=
\frac{1}{(2\pi)^2}\frac{1}{4\pi^{2+2\epsilon}}\Gamma^2[1+\epsilon]g_1(\epsilon)\,,
\end{equation}
and
\begin{equation}
g(\epsilon)=\frac{2 \epsilon^3+6 \epsilon^2+8 \epsilon+3}{\epsilon \left(2 \epsilon^2+5 \epsilon+3\right)}
-\frac{2 \epsilon \Gamma
   (-2 \epsilon-2) \Gamma (-2 \epsilon-1)}{(2 \epsilon+3) \Gamma (-4 \epsilon-3)}
\,,
\end{equation}
\begin{equation}
g_1(\epsilon)=\frac{6 \epsilon^3+17 \epsilon^2+18 \epsilon+6}{\epsilon^2
\left(2 \epsilon^2+5 \epsilon+3\right)}+\frac{4 (\epsilon+1)
   n_f T_f}{\epsilon (2 \epsilon+3) N_c}+\frac{2
   \left(\epsilon^2+\epsilon+1\right) \Gamma (-2 \epsilon-2) \Gamma (-2 \epsilon-1)}{\epsilon (2 \epsilon+3)
   \Gamma (-4 \epsilon-3)}
\,.
\end{equation}

\subsection{Renormalization, generalities}
\label{RenAndGen}

The bare parameters of the theory are $\alpha_B$ ($g_B$) and the potentials $V_B=V_{\{s,o,A,B\},B}$. In our convention
$\alpha_B$ has integer mass dimension and is related to $g_B$ by
\begin{equation}
\alpha_B=\frac{g_B^2\nu^{2\epsilon_n}}{4\pi}\,,
\end{equation}
where $\nu$ is the renormalization scale.
It has a special status since
it does not receive corrections from other Wilson coefficients of the
effective theory. Therefore,
it can be renormalized multiplicatively:
\begin{equation}
\alpha_B=Z_{\alpha}\alpha
\,,
\end{equation}
where
\begin{equation}
Z_{\alpha}
=1+
\sum_{s=1}^{\infty}Z^{(s)}_{\alpha}\frac{1}{\epsilon_n^s}\,.
\end{equation}
The renormalization group equation (RGE) of $\alpha$ is
\begin{equation}
\nu\frac{d}{d\nu}\alpha\equiv \alpha\beta(\alpha;\epsilon_n)=2\epsilon_n\alpha+\alpha\beta(\alpha;0)\,.
\end{equation}
In the limit $\epsilon_n \rightarrow 0$
\begin{equation}
\nu\frac{d}{d\nu}\alpha\equiv \alpha\beta(\alpha;0)
\equiv \alpha\beta(\alpha)=-2\alpha \frac{d}{d\alpha}Z^{(1)}_{\alpha}\,.
\end{equation}

The bare potentials $V_B$ in position space have integer mass dimensions (note that this is not true in
momentum space) and, due to the structure of the theory, we do not renormalize them multiplicatively,
see the discussion in Ref.~\cite{RG}. We define
\begin{equation}
\label{VBsplitting}
V_B=V+\delta V\,.
\end{equation}
$\delta V$ will generally depend on the (matching) coefficients of the effective theory, i.e. on $\alpha$ and $V$
and on the number of space-time dimensions.
In $D(n)$ dimensions, using the MS renormalization scheme, we define
\begin{equation}
\delta V
=
\sum_{s=1}^{\infty}Z^{(s)}_{V}\frac{1}{\epsilon_n^s}\,.
\end{equation}

From the scale independence of the bare potentials
\begin{equation}
\nu \frac{d}{d\nu}V_B=0
\,,
\end{equation}
one obtains the RGE's of the different
renormalized potentials. They can schematically be written as one (vector-like) equation including all potentials:
\begin{equation}
\nu \frac{d}{d\nu}V=B(V)\,, \label{VRGE}
\end{equation}
\begin{equation}
B(V)\equiv -\left(\nu \frac{d}{d\nu}\delta V\right)\,.
\end{equation}
Note that Eq.~(\ref{VRGE}) implies that all the $1/\epsilon_n$ poles disappear once the derivative
with respect to the renormalization scale is performed.
 This imposes some constraints on $\delta V$:
\begin{eqnarray}
\label{Z1}
{\cal O}(1/\epsilon_n): \qquad
&&B(V)=-2\alpha \frac{\partial}{\partial\alpha}Z^{(1)}_{V}\,,\; \label{1loopBV}
\\
{\cal O}(1/\epsilon^2_n): \qquad
&&
\label{Z2}
B(V)\frac{\partial}{\partial V}Z^{(1)}_{V}
+
\alpha \beta (\alpha)\frac{\partial}{\partial\alpha}Z^{(1)}_{V}
+
2\alpha \frac{\partial}{\partial\alpha}Z^{(2)}_{V}=0\,,
\end{eqnarray}
and so on.

\section{pNRQCD (D=3)}
\label{sec3D}

In three dimensions the purely gluonic sector is superrenormalizable. As a consequence the coupling constant is not renormalized:
\begin{equation}
\alpha_B=\alpha
\,.
\label{alphabare}
\end{equation}

At leading order in the multipole expansion the singlet field of the quark-antiquark system is free, i.e.
it does not interact with gluons, and renormalization scale independent.
In other words its renormalization constant (${\rm S}_B= Z_s^{1/2}{\rm S}$) is
\begin{equation}
\label{ZsLO}
Z_s=1+{\cal O}(r^2)
\,.
\end{equation}
Similarly the singlet potential is not renormalized at ${\cal O}(r^0)$:
\begin{equation}
\delta V_s={\cal O}(r^2)
\,.
\end{equation}
\begin{figure}
\includegraphics[width=0.25 \textwidth]{usoctet3D.eps}
\caption{\it One-loop contribution to the octet propagator. The dotted line
represents the $A^0$ field.}
\label{usoctet3D}
\end{figure}

For the octet field the situation is different. Even at leading order in the multipole expansion it has
a residual interaction with ultrasoft gluons.
The octet potential receives an ultraviolet (UV) divergent correction from
the one-loop self-energy diagram shown in Fig. \ref{usoctet3D}.
It gives
\begin{equation}
Z^{(1)}_{V_o}
=
\frac{C_A}{2}\, \alpha + {\cal O}(r^2)
\,.
\label{ZVo}
\end{equation}
Higher loop diagrams cannot contribute at ${\cal O}(r^0)$.
This is because the coupling and/or the potentials must appear perturbatively (with positive powers)
in the Z's. Since $\alpha$ has positive mass dimension, the potentials would appear with negative powers
in higher loop corrections to Eq.~(\ref{ZVo}), which
is not allowed by renormalizability. By the very same reason the octet field does not require renormalization at ${\cal O}(r^0)$,
i.e.
\begin{equation}
Z_o=1 + {\cal O}(r^2).
\end{equation}
This will be all we need to know about the renormalization of the octet sector.

The operators that appear at ${\cal O}(r)$ in the Lagrangian, $V_{A/B}$,
are not renormalized at ${\cal O}(r^0)$ either, i.e.
\begin{equation}
\label{ZVAB}
Z_{A/B}=1+ {\cal O}(r)
\,.
\end{equation}
The reason is, as mentioned before, that the coupling and the potentials have to appear perturbatively in the counterterms. Otherwise the renormalizability of the theory at LO of the multipole expansion would be spoiled.
Moreover, we can take $V_A=V_B=1$, as ${\cal O}(\alpha)$ soft corrections would be multiplied by factors of $r$ and would move us away from
the precision of $V_s$ aimed for at this paper. Actually, from inspection of the possible diagrams that will contribute
at the soft scale, we know that $V_{A/B}=1+{\cal O}(\alpha^2)$~\cite{Brambilla:2006wp}.

We now focus on the singlet and the
renormalization of $V_s$ beyond ${\cal O}(r^0)$.
The singlet potential is IR safe up to soft one-loop order. At two soft loops in dimensional regularization IR poles up to ${\cal O}(1/\epsilon_3^3)$ appear~\cite{Schroder:1999sg}.
The ultrasoft computation in pNRQCD yields the following results for the counterterms
\begin{eqnarray}
\label{Z1Vs}
Z^{(1)}_{V_s,X}
&=&
-r^2 \Delta V_X^2\alpha C_F V_A^2\frac{1}{4}
-r^2 \Delta V_X \, C_A\alpha^2 C_F V_A^2\frac{1}{4}\
\\
\nonumber
&&
-r^2\alpha^3 C_FV_A^2
\frac{\left(13 \pi ^2-2208\right) C_A^2+8 \left(19\pi^2+144\right)
C_A T_Fn_f-48 T_Fn_f \left(4 \left(\pi ^2-10\right) C_F+\pi ^2T_Fn_f\right)}{2304}\,,
\end{eqnarray}
\begin{equation}
\label{Z2Vs}
Z^{(2)}_{V_s,X}
=
-r^2 \Delta V_X \alpha^2 C_FC_AV_A^2\frac{1}{8}
-r^2\alpha^3 C_FC_A^2V_A^2\frac{1}{24}\,,
\end{equation}
\begin{equation}
\label{Z3Vs}
Z^{(3)}_{V_s}
=
-r^2\alpha^3 C_FC_A^2V_A^2\frac{1}{48}\,,
\end{equation}
\begin{equation}
\label{ZnVs}
Z^{(n)}_{V_s}=0 \quad \forall \quad n>3\,.
\end{equation}

These are the complete  ${\cal O}(r^2)$ results. We used the MS renormalization scheme to derive them
and we checked that their form is invariant under scheme transformations that amount to a change
of the renormalization scale. In other words the form of $Z_{V_o}$ and $Z_{V_s}$ remains unchanged, if we replace $\alpha \rightarrow \alpha\, c_X^{2\epsilon_3}$, where $c_X$ is a $\epsilon_3$-independent constant ($c_{\rm MS}=1$). We refer to this class of renormalization schemes as ``global'' and indicate globally renormalized quantities by an index ``$X$'' representing the scheme (X=MS, ${\overline{\rm MS}}$, etc.) as e.g. $\Delta V_X$ in the above equations.

The fact that one can renormalize the
potential with a finite number of terms at a given order in the multipole expansion (at ${\cal O}(r^2)$: $Z^{(n)}_{V_s}=0$ for $n>3$)
reflects the super-renormalizability of the theory.
Let us now explain how Eqs.~(\ref{Z1Vs}-\ref{Z3Vs}) are obtained.

1) The first term in Eq.~(\ref{Z1Vs}) comes from the $1/\epsilon_3$ divergence of the ultrasoft one-loop correction in Eq.~(\ref{USbare1loop}).
It has a scheme independent form and fixes, together with Eqs.~(\ref{Z1}) and~(\ref{Z2}), the first term of Eq.~(\ref{Z2Vs}) and
Eq. (\ref{Z3Vs}). Of course it is also possible to compute these $1/\epsilon_3^2$ and $1/\epsilon_3^3$
divergences directly. They are generated by the entanglement of the ultrasoft one-loop diagram
with the octet self energy in Fig.~\ref{usoctet3D}.\footnote{
In fact one can even perform an exact resummation of these diagrams (at least of the divergent pieces)
to handle all those divergences at once.} Other types of diagrams with the same number of loops are less divergent.

2) The second term in Eq.~(\ref{Z1Vs}) follows from the remaining $1/\epsilon_3$ divergence in the ultrasoft two-loop computation, Eq.~(\ref{USbare2loop}), once all subdivergences (associated with the octet potential) have been subtracted.
We have checked its (global) scheme independence explicitly.
This result combined with Eq.~(\ref{Z2}) then fixes the
second term of Eq.~(\ref{Z2Vs}).

3) We now compare the results of 1) and 2) (with $V_A=1$) to the two-loop calculation of the soft contribution,
which equals the bare static potential of Ref. \cite{Schroder:1999sg}.
Keep in mind that the comparison is carried out in $D=3+2\epsilon_3$ dimensions and $\Delta V$ is a polynomial in $\epsilon_3$.
We find perfect agreement of the soft IR $1/\epsilon_3^3$ and $1/\epsilon_3^2$ singularities and the corresponding ultrasoft counterterms in Eq.~(\ref{Z2Vs}) and~(\ref{Z3Vs}).\footnote{Actually,
if we compare with Eq. (3.51) of Ref.~\cite{Schroder:1999sg}, we find a discrepancy, but if we
explicitly sum up the expressions for the contributing diagrams given in the same reference the results agree.
Most likely there is a typo in Eq. (3.51) of Ref.~\cite{Schroder:1999sg}.}
That means that the respective divergent parts of the bare quantities in Eq.~(\ref{Estot}) cancel.
Note that this can be understood as a non-trivial check of two independent determinations of these terms.

4) Our findings so far open the way to obtain the missing subleading UV divergence in Eq.~(\ref{Z1Vs}).
The key point is that the counterterms of the potential cannot contain negative powers of $\Delta V$. They would give rise to
negative powers of $\ln (r \nu)$, which cannot be absorbed by the potential (cf. Eq.~(\ref{VsBD})).
Hence $Z^{(1)}_{V_s}$ is a polynomial of $\Delta V$.
From the previous computations we know all terms of the polynomial except the constant at ${\cal O}(\Delta V^0)$. This term can be inferred
from the (soft) result in Ref.~\cite{Schroder:1999sg}.
We simply subtract all the $1/\epsilon_3^3$ and $1/\epsilon_3^2$ counterterms as well as the first two $1/\epsilon_3$ terms according to Eq. (\ref{Z1Vs}). The remaining (IR) $1/\epsilon_3$ divergence must then match the last term in Eq. (\ref{Z1Vs}) in order to render Eq.~(\ref{Estot}) finite.
In this way we fix the $\Delta V^0$ piece of $Z^{(1)}_{V_s}$ indirectly.
A consistency check of this computation is that the result is really independent of $\Delta V$ (i.e. free of $\ln(r)$) and invariant within the class of global renormalization schemes defined above.
Thus we have obtained the complete renormalization structure of $V_s$ at ${\cal O}(r^2)$.

\subsection{pNRQCD RG}

We will now derive the RG evolution of the matching coefficients of the effective theory.
The running of the coupling constant is trivial, since it is not renormalized in three dimensions,
as follows from Eq. (\ref{alphabare}):
\begin{equation}
\nu\frac{d}{d\nu}\alpha=0 + {\cal O}(\epsilon_3)
\,.
\end{equation}
As a consequence of Eq.~(\ref{ZVAB}) $V_{A/B}$ does not run either at ${\cal O}(r^0)$.
Similarly there is no running of the singlet static potential at LO in the multipole expansion, whereas
the running of the octet potential according to Eqs.~(\ref{1loopBV}) and~(\ref{ZVo}) reads
\begin{equation}
\nu\frac{d}{d\nu}V_o=-C_A\alpha
\,.
\end{equation}
We are interested in the running of the octet potential only insofar as it enters the running
of $\Delta V$, which including the tree-level matching condition is then given by
\begin{equation}
\label{VoRG}
\Delta V_X(r;\nu)= V_{o,X}(r;\nu)-V_{s,X}(r;\nu)  =-\alpha C_A\ln(r\nu d_X)+{\cal O}(r^2)+{\cal O}(\epsilon_3)
\,.
\end{equation}
The index $X$ stands for the scheme (e.g. MS or $\overline{\rm MS}$):
\begin{equation}
d_{\rm MS}=e^{\gamma_E/2}\sqrt{\pi} \simeq 2.36546,
\qquad
d_{\rm \overline{\rm MS}}=d_{\rm MS}\, c_{\overline{\rm MS}}^{-1}=e^{\gamma_E}/2 \simeq 0.890536,
\end{equation}
with $c_{\overline{\rm MS}}= e^{1/2(\ln (4\pi) - \gamma_E )}$.
This result will be sufficient for our purpose, which is the computation of the singlet static potential and energy.

From the counterterms determined in the previous subsection, we can derive
the complete running of the singlet static potential at ${\cal O}(r^2)$:
\begin{eqnarray}
\label{VsRGeq}
\nu\frac{d}{d\nu}V_{s,X} &=& \frac{C_F}{2}V_A^2r^2(\Delta V_X)^2\alpha+C_FV_A^2r^2 \Delta V_X C_A\alpha^2
\\
&&
+\,
r^2\alpha^3C_FV_A^2
\frac{(13 \pi ^2-2208) C_A^2+8 (19\pi^2+144)
C_A n_f T_F-48 n_fT_F (4 (\pi ^2-10) C_F+\pi ^2n_f T_F)}{384}\,.
\nonumber
\end{eqnarray}
Its form is globally scheme independent.

By solving the RG equations we obtain for the ${\cal O}(r^2)$ contribution
\begin{equation}
\label{VsRG}
V_s^{\rm MS}(\nu)=V_s^{\rm MS}(r;\nu \!=\! \frac1{r})+V_s^{\rm RG;MS}(r;\nu)\,,
\end{equation}
where
\begin{eqnarray}
 \lefteqn{V_s^{\rm RG;MS}(r;\nu)=C_F r^2 \alpha ^3 }  \nonumber\\
&\times&
 \Bigg\{\frac{1}{6} C_A^2 \ln^3(r \nu ) \nonumber\\
&&
+\frac{1}{4} C_A^2 \big[-2+\gamma_E +\ln \pi \big] \ln^2(r \nu ) \nonumber\\
&&
+\bigg[ C_A^2 \Big(\frac{13\pi ^2}{384}+\frac{1}{8} \big(\gamma_E ^2 -4 \gamma_E - 46 + (2 \gamma_E - 4)\ln \pi+\ln^2 \pi   \big) \Big) \nonumber\\
&&
+n_f T_F \Big(C_A \Big(3+\frac{19 \pi ^2}{48}\Big) +C_F \Big(5-\frac{\pi ^2}{2}\Big)\Big) - (n_f T_F)^2 \frac{ \pi ^2}{8}\bigg] \ln (r \nu)
\Bigg\}
\label{Vsrunning}
\end{eqnarray}
is the running and
\begin{eqnarray}
  \lefteqn{ V_s^{\rm MS}(r;\nu \!=\! \frac1{r}) =
  C_F \alpha \ln (r^2 \nu^2_s\pi e^{\gamma_E} )  +  \frac{\pi}{4} C_F (7 C_A-4 n_f T_F) \alpha^2 r
   } \nonumber\\
&+C_F r^2 \alpha ^3&
 \Bigg\{C_A^2 \bigg[\frac{\pi ^2}{2304} (39 \gamma_E-715 +564 \ln 2+39 \ln \pi ) + \frac{209}{24}-\frac{43}{24} \zeta (3) \nonumber\\
&&
 \qquad+\frac{\gamma_E^3 - 6 \gamma_E^2 - 138\gamma_E +
 3(\gamma_E^2 - 4 \gamma_E -46) \ln \pi +3 (\gamma_E-2) \ln^2 \pi +\ln^3 \pi}{48}\bigg] \nonumber\\
&&
 + n_fT_F C_F \Big[\frac{\pi^2}{24}  (15-6 \gamma_E -8 \ln 2-6 \ln \pi )+\frac{1}{2} (5 \gamma_E-8 +5 \ln \pi )\Big] \nonumber\\
&&
+ n_fT_F C_A \Big[\frac{\pi^2}{288} (57 \gamma_E -97 +12 \ln 2+57 \ln \pi )+\frac{1}{6} (9 \gamma_E -22 +9 \ln \pi +10 \zeta (3))\Big] \nonumber\\
&&
+(n_f T_F)^2 \Big[\frac{\pi^2}{48}(7-3 \gamma_E -\ln (16 \pi ^3) ) \Big] \Bigg\}
\label{Vs1r}
\end{eqnarray}
is the initial matching condition. Note that the tree-level and one-loop matching conditions have been included.
For the two-loop initial condition we used the bare data of Ref.~\cite{Schroder:1999sg}.

At tree-level there is a dependence on another factorization scale $\nu_s \gg 1/r$. The logarithmic dependence of $V_s$ on $\nu_s$ would cancel the (hard) IR scale dependence of a large but finite mass $m$ ($m \gg \nu_s$). Once the mass is included, the static energy reads
\begin{equation}
E_s(r)=2m(\nu_s)+V_s(r;\nu_s;\nu)+\delta E_s(\nu)
\,.
\end{equation}
The $\nu_s$ dependence of $m$ is the three dimensional relic of the four dimensional pole mass renormalon.
In this work we only care about the ultrasoft ($\nu$) scale dependence. Therefore, we do not need to explicitly consider $\nu_s$, which only appears in the initial matching condition, or, if combined with the mass, is replaced by $m$ (up to a constant).
On the other hand, note that for $\Delta V=V_o(\nu)-V_s(\nu)$ the dependence on $\nu_s$ disappears.
This is analogous to the ultraviolet renormalon cancellation that takes place in four dimensions, but does not mean that
$\Delta V$ is renormalon free (in 4D). There is a leftover infrared renormalon, which is reflected by the remaining $\nu$ dependence in three dimensions.

Note that we have now obtained the exact $O(r^2)$ contribution to the static potential (i.e. the soft contribution to the
static energy). There is nothing left. Moreover, by setting $\nu \sim \Delta V$ potentially large logarithms are resummed.

The above results have been presented in the MS scheme. It is possible to change to a different renormalization scheme. If we rewrite Eq.~(\ref{VsRG}) in terms of $\Delta V$ all the global scheme dependence gets encapsulated in the $\Delta V$'s and Eq. (\ref{Vsrunning}) becomes
\begin{eqnarray}
\nonumber
&&
V_s^{\rm RG;MS}(r;\nu)
=
-\frac{C_F}{6C_A}V_A^2r^2\left(\Delta V^3_X(r;\nu)-\Delta V^3_X(r;1/r)\right)
-\frac{C_F}{2}V_A^2r^2\alpha\left(\Delta V^2_X(r;\nu)-\Delta V^2_X(r;1/r)\right)
\\
&&
\nonumber
-
r^2\alpha^2\frac{C_F}{C_A}V_A^2
\frac{\left(13 \pi ^2-2208\right) C_A^2+8 \left(19\pi^2+144\right)
C_A n_fT_F-48 n_fT_F \left(4 \left(\pi ^2-10\right) C_F+\pi ^2 n_fT_F \right)}{384}
\\
   &&
   \qquad
   \times \left(\Delta V_X(r;\nu)-\Delta V_X(r;1/r)\right)
\,.
\label{VsRGMSinDeltaV}
\end{eqnarray}
In a general global renormalization scheme the $[\Delta V_X(r;1/r)]^n$ terms in Eq.~(\ref{VsRGMSinDeltaV}) vanish, when the respective matching condition is added, leaving a (globally) scheme independent constant.

\subsection{$\delta E^{us}_s$: ultrasoft effects up to NNLL order}

We proceed with computing the static singlet energy. We have already derived the RG improved
expression for the potential in the previous subsection. Therefore, the computation that is left (if we neglect non-perturbative effects)
addresses the ultrasoft contribution. The compact expression Eq.~(\ref{deltaVUS}) includes all ultrasoft effects at
${\cal O}(r^2)$ and at any
order in the $g^2/\Delta V$ expansion. At present concrete results are available at one, ${\cal O} (g^2)$,
and two loops, ${\cal O}(g^4)$, given in Eqs.~(\ref{USbare1loop}, \ref{USbare2loop}). After minimal subtraction they read
\begin{equation}
\delta E^{us}_{s,\rm MS}(\text{1-loop})
=
\frac1{4} \alpha\, C_FV_A^2 r^2\, \Delta V_{\rm MS}^{2}
\left(
1+\gamma_E-\ln(4\pi)+2\ln \Big[ \frac{\Delta V_{\rm MS}}{\nu} \Big]
\right)
\end{equation}
\begin{eqnarray}
\nonumber
\delta E^{us}_{s,\rm MS}(\text{2-loops})
&=&
-\frac{1}{2} \alpha^2\, C_F \Delta V_{\rm MS} V_A^2 r^2
\left(
C_A\ln^2\Big[ \frac{\Delta V_{\rm MS}}{\nu } \Big]
+
C_A\ln \Big[ \frac{\Delta V_{\rm MS}}{\nu } \Big]
\left(
\gamma_E-\ln(4\pi)
\right)
   \right.
   \\
   &&
   \left.
   +C_A
   \big(
   6+\frac1{4}\left(\gamma_E-\ln(4\pi)\right)^2
   \big)
   -2n_fT_F
   \right)
   \label{deltaUS2loop}
\end{eqnarray}
Setting $\nu \sim \Delta V_X(\nu)$ resums large logarithms in the potential and minimizes them in the ultrasoft contributions.
Expressed as a double expansion in $\alpha r$ and $1/\ln(r\Delta V)$
the result for the static energy reads with ${\cal O}(\alpha^2r^2)$ and NNLL accuracy
\begin{eqnarray}
E_s(r)&=&
C_F \alpha \ln (r^2 \nu^2_s  \pi e^{\gamma_E}
 )  +  \frac{\pi}{4} C_F (7 C_A-4 n_f T_F) \alpha^2 r+
V_s^{\rm RG;MS}(r;\nu\!=\!\Delta V)+\delta E^{us}_{s,\rm MS}(\nu\!=\!\Delta V) \nonumber\\
&&+{\cal O}(\alpha^3 r^2 \ln^0)  \nonumber
\\
&=&
C_F \alpha \ln (r^2 \nu^2_s \pi e^{\gamma_E}
)   +  \frac{\pi}{4} C_F (7 C_A-4 n_f T_F) \alpha^2 r  \nonumber
\\
&&+C_F \alpha ^3\, r^2
\Bigg\{
\frac{1}{6} C_A^2 \ln ^3(r \Delta V_{\rm MS})
+\frac{1}{4} C_A^2 (2 \gamma_E-1 -\ln 4 ) \ln ^2(r \Delta V_{\rm MS}) \nonumber\\
&&+\bigg[C_A^2 \Big(\frac{13 \pi ^2}{384}+\frac{1}{8} \left(4 \gamma_E ^2+4 \ln ^2 2-\gamma_E  (2+\ln 256)-2 (11+\ln \pi )\right)\Big)\nonumber\\
&&
\hspace{2.5 ex} + n_f T_F \Big(C_A \big(2+\frac{19 \pi ^2}{48}\big) +C_F \big(5-\frac{\pi ^2}{2}\big)\Big)- (n_f T_F)^2 \frac{ \pi
   ^2}{8}\bigg] \ln (r \Delta V_{\rm MS}) \Bigg\} +{\cal O}(\alpha^3 r^2 \ln^0)
\,,
\label{NNLL}
\end{eqnarray}
where the omitted ${\cal O}(\alpha^3 r^2)$ terms do not contain logarithms of $r \Delta V$.
Eq.~(\ref{NNLL}) is renormalization scale independent up to ${\cal O}(\frac{\alpha^4 r^2}{\Delta V})$.
Furthermore the explicit scheme dependence of $V_s^{\rm RG}$ and $\delta E^{us}_{s}$ and the
implicit scheme dependence of Eq.~(\ref{NNLL}) through the logarithms of $\Delta V$ cancel up to ${\cal O}(\alpha^3 r^2 \ln^0)$.

Finally we would like to note that the condition
$\nu_{us} \equiv \Delta V_X(\nu_{us})$, produces a factorization-scale independent scale
that is non-perturbative in $\alpha$,
\begin{equation}
\nu_{us} \equiv \Delta V_X(\nu_{us})=C_A\alpha\, W(1/(C_A\alpha\, d_X r))\,.
\end{equation}
$W(z)$ is the Lambert function and has the following expansion for
large $|z|$
\begin{equation}
\label{Lambert}
W(z)=\ln (z)-\ln (\ln (z))-\sum _{k=0}^{\infty }
   (-1)^k \ln ^{-k}(z) \sum _{j=1}^k \frac{\ln
   ^j(\ln (z)) S_k^{(-j+k+1)}}{j!}
\,,
\end{equation}
where $S_k^{(-j+k+1)}$ is the Stirling number of the first kind\footnote{see e.g.: http://functions.wolfram.com/IntegerFunctions/StirlingS1}.
Therefore, $\nu_{us}$ resums a certain class of logarithms\footnote{
Something similar happens in four dimension when solving the Schr\"odinger equation
with the lowest order Coulomb potential. In that case however one sets $\nu_s=m\,C_F\alpha(\nu_s)$.}.

\subsection{Subleading ultrasoft effects: ${\cal O}(\alpha^3r^2)$}

Let us now consider higher order corrections to the result obtained in Eq. (\ref{NNLL}).
We have already stated that, from Eq. (\ref{VsRG}), the soft contribution to the static energy, i.e. the potential, is known completely
at ${\cal O}(\alpha^3r^2)$. This is not the case for the ultrasoft contribution at that order.
The general structure of $\delta E^{us}$ at ${\cal O}(r^2)$ is
\begin{equation}
\delta E_s^{us}
=
C_F\alpha V_A^2r^2(\Delta V)^2\sum_{n=0}^{\infty}\left(\frac{C_A\alpha}{\Delta V}\right)^{\!\! n}\sum_{s=0}^{n+1}c_{n,s}\ln^s\Big[\frac{\Delta V}{\nu}\Big]
\,.
\label{USstruc}
\end{equation}
Interestingly the RG contains important information on the higher order ultrasoft contributions, namely on the logarithmic terms. Since the singlet static energy $E_s$ is a physical observable and therefore
factorization/renormalization scale independent we find at ${\cal O}(r^2)$ (neglecting non-perturbative effects for the moment)
\begin{equation}
\nu\frac{d}{d\nu}  \delta E_s^{us} =-B(V_s)=-C_F\alpha V_A^2r^2\sum_{n=0}^2 B_n(C_A\alpha)^n(\Delta V)^{2-n}
\,,
\end{equation}
where the $B_n$ coefficients can be read off Eq. (\ref{VsRGeq}).
We obtain the following relation between the coefficients at different orders in $\Delta V$ (note that $\Delta V$ is scale dependent
itself):
\begin{equation}
\label{cnrelation}
B_n\delta_{s,0}+(n-3)c_{n-1,s}-(s+1)(c_{n-1,s+1}+c_{n,s+1})=0
\end{equation}
with $c_{n,s}=0$ if $s > n+1$ or $n<0$ or $s<0$ and $B_n=0$ if $n>2$ or $n<0$.

In particular, Eq. (\ref{cnrelation}) allows to fix all logarithmic terms of the three-loop
ultrasoft computation:
\begin{eqnarray}
 \lefteqn{\delta E_{s,\rm us}^{\rm 3loop;MS}(r;\nu)=C_F r^2 \alpha ^3 }  \nonumber\\
&\times&
 \Bigg\{\frac{1}{6} C_A^2 \ln ^3\Big[\frac{\Delta V_{\rm MS}}{\nu }\Big] \nonumber\\
&&
 + \frac{1}{4} C_A^2  \left(2 + \gamma_E  - \ln (4 \pi ) \right) \ln ^2\Big[\frac{\Delta V_{\rm MS}}{\nu }\Big] \nonumber\\
&&
+ \bigg[
C_A^2  \Big(\frac{13 \pi ^2}{384}-\frac{11}{4}
+\frac{1}{2}\left(\gamma_E-\ln(4\pi)\right)+\frac{1}{8}\left(\gamma_E-\ln(4\pi)\right)^2
\Big)\nonumber\\
&&
+n_f T_F \Big(\big(2+\frac{19 \pi ^2}{48}\big) C_A+ C_F \big(5-\frac{\pi ^2}{2}\big)\Big)
-(n_f T_F)^2\frac{ \pi ^2}{8}\bigg] \ln\Big[ \frac{\Delta V_{\rm MS}}{\nu } \Big]
 +C_A^2c_{2,0}\Bigg\}
\,.
\label{US3loop}
\end{eqnarray}

Note that, though we cannot fix the $\nu$ independent constant $c_{2,0}$ from RG arguments, it can be computed from perturbation theory,
but requires a three-loop pNRQCD computation which has not been performed yet.
The constant term of the static singlet energy at ${\cal O}(\alpha^3r^2)$, which is the only missing
term in Eq.~(\ref{NNLL}) to reach N$^3$LL precision,
is then given by
\begin{equation}
\label{NNNLLmatching}
V_s^{\rm MS}(r;\nu \!=\! \frac1{r})\Big|_{{\cal O}(\alpha^3)} + C_F C_A^2r^2 \alpha^3 c_{2,0}^{\rm MS}
\,.
\end{equation}

\subsection{Non-perturbative effects: ${\cal O}(\alpha^4r^2/\Delta V)$}
\label{npsubsec}

At even higher orders in the $\alpha/\Delta V$ expansion, non-perturbative effects start to contribute.
In order to study these effects related to loop momenta $k \sim \alpha$, we integrate out the $\Delta V$ scale.
This means integrating out the octet field and ultrasoft gluons. The degrees of freedom left are the
singlet field and non-perturbative gluons with energy and momentum of order $\alpha$.
The resulting Lagrangian, including the leading order non-perturbative effects at ${\cal O}(r^2)$, reads
\begin{eqnarray}
 {\cal L}_{\rm np} =
{\rm Tr} \Biggl\{ {\rm S}^\dagger \left( i\partial_0  - V_s(r) -\delta E_s^{us}  \right) {\rm S} \Biggr\}
-  \frac{C_{np}}{\Delta V} {\rm Tr} \left\{  {\rm S}^\dagger {(g\bf E \cdot r})^2 \,{\rm S}\right\}
\label{pnrqcdnp}
\end{eqnarray}
for the case without light fermions ($n_f=0$). If we were to include light fermions there would also be operators of the form
\begin{equation}
\delta {\cal L} \sim \frac{r^2}{\Delta V} {\rm Tr} \left\{  {\rm S}^\dagger \alpha^2\bar q q \,{\rm S}\right\}\,.
\end{equation}
They could generate corrections to the static energy, due to the quark condensate, which are of the same parametric order as the
purely gluonic ones. In the following we restrict ourselves to the purely gluonic case ($n_f=0$).

The coefficient of the non-perturbative operator in Eq.~(\ref{pnrqcdnp}) is $C_{np}=1$ at leading order in the
$\frac{\alpha}{\Delta V}$ expansion. This result is obtained
by matching to a pNRQCD tree-level diagram, where two gluons couple to the singlet field, as sketched in Fig.~\ref{NonPertMatch}.
\begin{figure}
\includegraphics[width=0.6 \textwidth]{NonPertMatch.eps}
\caption{\it Schematic tree-level matching procedure for the non-perturbative interaction operator in Eq.~(\ref{pnrqcdnp}). The pNRQCD octet field in the left diagram has been integrated out on the right hand side.}
\label{NonPertMatch}
\end{figure}

The interaction with non-perturbative gluons produces a shift of the energy which is proportional to the gluon
condensate in three dimensions\footnote{This is the first term of a Taylor series that can be easily obtained by expanding
 Eq. (\ref{deltaVUS}) in $1/\Delta V$, see also Ref. \cite{Pineda:1996uk}:
\begin{equation}
\delta E^{np}=\sum_{i=0}^{\infty}C_iO_i \,,
\label{higherNPcorr}
\end{equation}
where
\begin{equation}
C_i=(-1)^{2i+1}\frac{r^2}{(\Delta V)^{2i+1}}
\end{equation}
and ($H_g$ represents the gluonic Hamiltonian)
\begin{equation}
O_i =
{T_F g^2 \over N_c(D-1)}
\langle vac \vert E^{a}_j
H_g^{2i}
E^{a}_j \vert  vac \rangle\,.
\end{equation}
Using the equation of motion, the gauge fixing and Lorentz
covariance we obtain
\begin{eqnarray}
&& O_i =-
{T_F g^2 \over N_c^2(D-1)}
v^{\beta_0}...v^{\beta_i}v^{\alpha_0}...v^{\alpha_i}
\\
\nonumber
&&
\times
\langle 0 \vert
{\rm Tr} \left(
[D_{\beta_1}(0),[...[D_{\beta_i}(0),G_{\beta_0 \rho}(0)]...]
[D_{\alpha_1}(0),[...[D_{\alpha_i}(0),{G_{\alpha_0}}^{\rho}(0)]...]
\right) \vert  0 \rangle
\,,
\end{eqnarray}
where $v$ is the velocity of the center of mass frame with $v^2=1$
(in the comoving frame $v=(1, {\bf 0})$) and the trace is taken in the
adjoint representation. Generally, however, Eq. (\ref{higherNPcorr}) does not cover all possible non-perturbative corrections at ${\cal O}(r^2)$.
}
\begin{equation}
\delta E^{np}_{s,B}=C^B_0O^B_0=\frac{r^2}{\Delta V_B}\frac{2\pi}{N_c(D-1)D}\langle \alpha\, G_{\mu \nu}^a G^{\mu \nu,a} \rangle_B\,.
\label{deltaEnp}
\end{equation}
The leading ultraviolet divergence of the gluon condensate has been calculated in perturbation theory at
four loops \cite{Schroder:2003uw}. The determination of the finite piece requires lattice simulations \cite{Hietanen:2004ew,Hietanen:2006rc}
and a computation to change from the lattice to dimensional regularization \cite{DiRenzo:2006nh}.
Taking the result (in the Euclidean) from the last reference we have
\begin{eqnarray}
\lefteqn{\langle \alpha\, G_{\mu \nu}^a G^{\mu \nu,a} \rangle_B = -\langle \alpha\, G_{\mu \nu}^a G^{\mu \nu,a} \rangle_{B,\rm Euclidean} =}&& \nonumber\\
&& -6 \frac{\alpha^4}{\pi} C_F C_A^4
 \biggl[
  \biggl( \frac{43}{12} - \frac{157}{768} \pi^2 \biggr)
 \biggl( -\frac{1}{8\epsilon_3}\left(\frac{e^{\gamma_E}}{4\pi}\right)^{4\epsilon_3}+
 \ln\frac{\nu}{2 C_A g^2} -\fr13
 \biggr)
 + {\beta_G}
 + {\cal O}(\epsilon_3)
 \biggr]
 \,,
 \label{eq:condensate}
\end{eqnarray}
\begin{eqnarray}
B_G^{(SU(3))}=-0.2\pm 0.4({\rm MC}) \pm 0.4(\rm NSPT)\,. \nonumber
\end{eqnarray}

Eq.~(\ref{deltaEnp}) is the analog to the Voloshin-Leutwyler correction to
the static potential in four dimensions \cite{Flory:1982qx,Bertlmann:1983pf}, which scales like $\frac{r^2}{\Delta V}\langle \alpha G^2 \rangle
\sim r^3\langle \alpha G^2 \rangle$. In three dimensions we have instead $\frac{r^2}{\Delta V}\langle \alpha G^2 \rangle
\sim \frac{r^2}{\ln r}\langle G^2 \rangle$.

Renormalizing the bare expression in Eq.~(\ref{deltaEnp}) in the $\rm MS$ scheme yields
\begin{eqnarray}
\delta E_{s, \rm MS}^{np}(\nu)
=-\frac{C_A^3 C_F r^2 \alpha^4 }{\Delta V_{\rm MS}} \bigg[
\Big(\frac{43}{6} - \frac{157}{384} \pi^2 \Big) \Big( \ln \Big[\frac{\nu}{C_A \alpha}\Big] - \frac12 (\ln (16 \pi)+ \gamma_E) - \frac{1}{8}\Big) + 2 B_G
\bigg]\,.
\label{NPlog}
\end{eqnarray}

To be consistent with our power counting this result has to be combined with the four-loop ultrasoft computation. Again the logarithm structure can be determined by
demanding scale independence of $E_s(r)$, which results in the following renormalization group equation for the ultrasoft part
\begin{equation}
\nu\frac{d}{d\nu}\delta E^{us}_s=-B(V_s)-\nu\frac{d}{d\nu}\delta E^{np}\,.
\end{equation}
This equation allows us to fix the coefficients of the four-loop ultrasoft contribution:
\begin{equation}
c_{3,4}=0\,,\qquad c_{3,3}=-c_{2,3}=-\frac{1}{6}\,,\qquad
c_{3,2}=-c_{2,2}=-\frac{1}{2}-\frac{1}{4}\left(\gamma_E-\ln(4\pi)\right)\,, \qquad c_{3,1}=c_{3,1}^{UV}+c^{np}_{3,1}
\,.
\end{equation}
$c_{3,1}^{UV}=-c_{2,1}$ represents the coefficient of the ultraviolet logarithm and follows from Eq. (\ref{cnrelation}).
\begin{equation}
c_{3,1}^{np}=-\frac{43}{6} + \frac{157}{384} \pi^2
\end{equation}
is associated with the infrared logarithm in the ultrasoft contribution, that cancels the renormalization scale dependence from the non-perturbative logarithm in Eq.~(\ref{NPlog}).

Thus in order to obtain the complete solution for the singlet static energy up to ${\cal O}(r^2\alpha^4/\Delta V)$, the missing pieces are purely
perturbative: $c_{2,0}$ and $c_{3,0}$, i.e. the non-logarithmic three- and four-loop ultrasoft contributions.

The results given in Eqs.~(\ref{NNLL}),~(\ref{US3loop}) and~(\ref{NPlog})
have been written as an expansion in $1/\ln(r \Delta V)$. Alternatively one can use
$1/\ln(C_A r\alpha)$ as the expansion parameter when setting $\nu_{us}=\Delta V_X$ and
using Eq.~(\ref{Lambert}) to expand $\Delta V_X$ in powers of $(\ln(\ln(r\alpha)))^{n \le s} /\ln^s(r\alpha)$.
This expansion might make sense
as soon as non-perturbative terms like Eq.~(\ref{NPlog}) are included, since they can be organized in the same series.
Up to ${\cal O}(r^2 \alpha^2)$ in the multipole expansion and $\rm N^3$LL order perturbation theory the singlet static energy then reads
\begin{eqnarray}
E_s(r)&=&
C_F \alpha \ln (r^2 \nu^2_s \pi e^{\gamma_E})   +  \frac{\pi}{4} C_F (7 C_A-4 n_f T_F) \alpha^2 r  \label{FinalNNNLL}\\
&+& C_A^2 C_F r^2 \alpha^3 \Bigg\{ \frac{1}{6} \ln^3(C_A r \alpha ) \nonumber\\
&+& \ln^2(C_A r \alpha )
\bigg[
\frac{1}{2}\ln[-\ln(C_A r \alpha )]+\frac{1}{4} (2 \gamma_E -\ln 4-1)
\bigg] \nonumber\\
&+& \ln(C_A r \alpha )
\bigg[
\frac{1}{2}\ln[-\ln(C_A r \alpha )]^2+(\gamma_E -\ln 2) \ln[-\ln(C_A r \alpha )]+ K
\bigg] \nonumber\\
&+&
\frac{1}{6}\ln[-\ln(C_A r \alpha )]^3+\frac{1}{2} (\gamma_E -\ln 2+1) \ln[-\ln(C_A r \alpha )]^2+(K+\gamma_E -\ln 2) \ln[-\ln(C_A r \alpha )] \nonumber\\
&&+\frac{n_f T_F  \big(\pi ^2  (\frac{\ln 2}{8}-\frac{97}{96} )-11+5 \zeta(3) \big) }{3C_A}
+\frac{\frac{\pi ^2}{48}  (7-4 \ln 2) (n_f T_F)^2 - C_F \left(4 - \pi ^2 \left(\frac{5}{8}-\frac{\ln 2}{3}\right)\right)
   n_f T_F}{C_A^2} \nonumber\\
&&-\frac{X^3}{6}-\frac{X^2}{2}  (\gamma_E -\ln 2 + 1) - X (K+\gamma_E - \ln 2)
 +\frac{209 - \pi ^2 (\frac{715}{96} - \frac{47 \ln 2}{8})-43 \zeta (3)}{24} +c_{2,0} \nonumber
\Bigg\}
\end{eqnarray}
for a general renormalization scheme with $c_X=e^{\frac12(\gamma_E + \ln \pi) + X}$.
\begin{eqnarray}
\lefteqn{K = B_2 - c_{1,0}^{\rm MS} + \frac{1}{8} (\gamma_E +\ln \pi ) (3 \gamma_E -\ln (16 \pi )) =}&&  \\
&&=\frac{(2+\frac{19 \pi ^2}{48}) n_f T_F}{C_A} - \frac{\frac{1}{8} \pi ^2 (n_f T_F)^2 + \frac{1}{2} C_F (\pi ^2-10) n_f T_F}{C_A^2}
+ \frac12 (\gamma_E -\ln 2)^2-\frac{11}{4} + \frac{13 \pi ^2}{384} \nonumber
\end{eqnarray}
is a manifestly scheme independent number. Thus the yet unknown ultrasoft constant $c_{2,0}$ defined by Eq.~(\ref{USstruc}) is expected to absorb the remaining scheme ($X$) dependence.

For $n_f=0$ we can even go one step further in the $1/\ln(C_A r\alpha)$ expansion and include the leading non-perturbative (Eq.~(\ref{NPlog}))
as well as the N$^4$LL (ultrasoft four-loop) contribution to the static singlet energy by adding
\begin{eqnarray}
\delta E_s|_{\rm N^4LL}=
\lefteqn{\frac{C_A^2 C_F r^2 \alpha^3 }{\ln(C_A r \alpha )} \bigg[
\frac{1}{6}\ln[-\ln(C_A r \alpha )]^3+\frac{1}{2} (\gamma_E -\ln 2+1) \ln[-\ln(C_A r \alpha )]^2 } && \nonumber\\
&& +\Big(K+\gamma_E -\ln 2 +\frac{43}{6} -\frac{157 \pi ^2}{384}\Big) \ln[-\ln(C_A r \alpha )] - \Big(\frac{43}{6}-\frac{157 \pi ^2}{384}\Big) \big(\gamma_E +\ln (4 \pi )+\frac{1}{8}\big) \nonumber\\
&& -\frac{X^3}{6} - \frac{1}{2} X^2 (\gamma_E -\ln 2 +1) - X \Big(K+\gamma_E -\ln 2 + \frac{43}{6} -\frac{157 \pi^2}{384} \Big) +2 B_G- c_{3,0}
\bigg]
\label{Final4loopNP}
\end{eqnarray}
to Eq.~(\ref{FinalNNNLL}). The $X$ dependent terms must be subtracted by corresponding terms in the ultrasoft four-loop constant $c_{3,0}$.

Another convenient choice for the renormalization scale is $\nu = C_A \alpha$, thus keeping the ultrasoft logarithms, but eliminating the logarithms in the non-perturbative finite parts. Eqs.~(\ref{FinalNNNLL}) and (\ref{Final4loopNP}) as parts of an expansion up to ${\cal O}( \ln^{-1}(C_A r \alpha))$ remain of course unchanged.

\subsection{Comparison with lattice data}

\begin{figure}
\includegraphics[width= \textwidth]{All3DPlots.eps}
\caption{\it Plots of the analytic results for the force $F(r)=\frac{dE_s(r)}{dr}$ in $r_0$ units with $n_f=0$ in comparison to existing lattice data (black dots):\\
a) $F(r)$ for $N_c=2$. The dotted (LO) and dashed (LO+NLO) curves are due to Ref.~\cite{Schroder:1999sg}, the results including the NNLO order ($\alpha^3 r^2$) in the multipole expansion are new. The labels NNLL and $N^3$LL of these curves refer to the order in the $\alpha/\Delta V$ expansion. The most precise result displayed here includes the leading non-perturbative logarithm of Eq.~(\ref{NPlog}) in addition to all corrections up to $N^3$LL. We have plotted our results in the $\rm MS$ and the ${\overline{\rm MS}}$ scheme in order to make the scheme dependence visible and set $\nu=\Delta V_X$.\\
b) $F(r)$ for $N_c=3$. The corresponding curves to a) are shown.\\
To further illustrate the relative size of the different orders in the $\alpha/\Delta V$ expansion, we have displayed the SU(2) results for $r\,F(r)$ starting from the NLO and subsequently adding higher corrections up to $N^3$LL including the logarithm of Eq.~(\ref{NPlog}). Plot c) and d) show the situation in the ${\overline{\rm MS}}$ and the $\rm MS$ scheme respectively. The LL, NLL, etc. curves are understood to be at NNLO precision regarding the multipole expansion.\\
Depending on the scheme we have used the values in the first column and the first/third line of table~\ref{c20} for $c_{2,0}$ in the $N^3$LL plots.
}
\label{Plots}
\end{figure}

The aim of this subsection is, on the one hand, to see how well the short-distance 3D lattice data can be reproduced by our theoretical
expressions for the static singlet energy and, on the other hand, to try to extract numerical values for the ultrasoft
three- and four-loop constants $c_{2,0}$ and $c_{3,0}$ from fits to this data.

The authors of Refs.~\cite{HariDass:2007tx,Luscher:2002qv,Meyer:2006gm} performed quenched ($n_f=0$) three dimensional lattice simulations
of the force $F(r)=\frac{dE_s(r)}{dr}$ between two static color sources in the fundamental representation of SU($N_c$) that form a color
singlet.
In Ref.~\cite{HariDass:2007tx} the number of colors is $N_c=2$ and in Ref.~\cite{Luscher:2002qv} $N_c=3$.\footnote{We don't
consider the lattice data for $N_c=5$ in Ref.~\cite{Meyer:2006gm}, because we focus on short distances.} Unlike the static energy, the
force does not contain the unphysical logarithmic (UV) divergence of the LO static potential, which depends on the regularization scheme.
It is therefore the preferred quantity to study on the lattice.

The value of $\alpha$ to be used in our perturbative results for a comparison with the SU(3) data is given by $\alpha = \frac{0.18}{r_0}$~\cite{Luscher:2002qv},
where $r_0=0.5\,fm$ is the Sommer scale. The corresponding value for $\alpha$ in Ref.~\cite{HariDass:2007tx} was obtained from a fit
using a wrong ansatz to parametrize the corrections higher than NLO. We repeated the fit as described in Ref.~\cite{HariDass:2007tx}
using the perturbative result corresponding to Eq.~(\ref{NNLL}) in the scheme for which $d_X=1$ (i.e. $c_X=e^{\frac12(\gamma_E + \ln \pi)}$) and find $\alpha=\frac{0.29}{r_0}$.
This number is quite similar to
the one of Ref.~\cite{HariDass:2007tx} ($\frac{0.30}{r_0}$) and indicates that the multipole expansion works quite well for
the available SU(2) short distance lattice data.

To illustrate the relative size of the contributions at different orders in the multipole
and the ultrasoft $\alpha/\Delta V$ expansion, we present two plots of the force $F(r)$
and two plots of $r F(r)$ (normalized to a dimensionless number by factors of $r_0$) as a
function of $r/r_0$ in Fig.~\ref{Plots}. We also display the lattice points of
Refs.~\cite{HariDass:2007tx} and~\cite{Luscher:2002qv} (black dots) for comparison.

Figs.~\ref{Plots} a) and b) show the results with increasing precision in the multipole
expansion for SU(2) and SU(3), respectively. At the highest order, i.e. NNLO, there are two curves
corresponding to Eq.~(\ref{NNLL})  in the $\rm MS$ and ${\overline{\rm MS}}$ scheme with NNLL precision.
If we compare the LO, NLO and the NNLO curves with NNLL precision,
we find a convergent pattern, in particular in the short distance limit.
On the other hand the $\alpha/\Delta V$ expansion does not work that well,
even for the shortest distances that were probed on the lattice. Indeed, already at
$r/r_0 \simeq 0.22$ for SU(2) and $r/r_0 \simeq 0.35$
for SU(3) we have
$\big[\frac{C_A \alpha}{\Delta V_{{\overline{\rm MS}}}} \big]_{SU(2)} \simeq 0.60$ and
$\big[\frac{C_A \alpha}{\Delta V_{{\overline{\rm MS}}}} \big]_{SU(3)} \simeq 0.69$, respectively.
A reasonable convergence of the ultrasoft expansion, say
$\frac{C_A \alpha}{\Delta V_{{\overline{\rm MS}}}} \lesssim 0.25$, is however only assured for
$r/r_0 \lesssim 0.01$ with the given values of $\alpha$. In the $\rm MS$ scheme it is even worse, since generally
$\Delta V_{\rm MS}<\Delta V_{{\overline{\rm MS}}}$.

The lack of convergence is highlighted in Figs.~\ref{Plots} c) and d):
The LL and NLL contributions are
very small (maybe anomalously small), much smaller than the NNLL contribution, which, however improves the agreement
with the lattice data points. The pattern is somewhat similar in the $\rm MS$ and ${\overline{\rm MS}}$
scheme despite $\frac{C_A \alpha}{\Delta V_{{\overline{\rm MS}}}}<\frac{C_A \alpha}{\Delta V_{\rm MS}}$.
We have also studied the convergence of Eq.~(\ref{FinalNNNLL}), which does not incorporate
residual subleading (N$^4$LL) scheme dependences (and is therefore, with the precision
of the computation, scheme independent). If we consider the equivalent
NNLL precision, we find a similar pattern as for the $\rm MS$ and ${\overline{\rm MS}}$ scheme, though a bit closer to ${\overline{\rm MS}}$.

We now consider the corrections higher than NNLL in the $\alpha/\Delta V$ expansion.
At N$^3$LL precision we add Eq.~(\ref{NNNLLmatching}) to
Eq.~(\ref{NNLL}) in the $\rm MS$ scheme (and accordingly in the ${\overline{\rm MS}}$ scheme). In the
$1/\ln(C_A r\alpha)$ expansion, the corresponding result is Eq. (\ref{FinalNNNLL}).
At this order the constant $c_{2,0}$ appears in the analytic expressions. It is unknown at present.
To reach N$^4$LL precision we have to add Eq.~(\ref{NPlog}) with $\nu=\Delta V$ to
Eq.~(\ref{NNLL}) plus Eq.~(\ref{NNNLLmatching}). In the
$1/\ln(C_A r\alpha)$ expansion, N$^4$LL precision corresponds to adding Eq.~(\ref{Final4loopNP}) to Eq.~(\ref{FinalNNNLL}).\footnote{In the following we will only take into account the logarithmic terms in Eqs.~(\ref{NPlog}) and~(\ref{Final4loopNP}), since the constant term includes $c_{3,0}$.}

From the previous analysis we have seen that the convergence in the
$\alpha/\Delta V$ expansion is not good. Therefore, we consider any attempt to fit $c_{3,0}$ as unrealistic
and focus on $c_{2,0}$, for which we only expect an order of magnitude estimate from a fit to the available lattice data.

The strategy we follow is to fit our analytic expressions on the lattice data at the shortest available distances
in order to obtain $c_{2,0}$. There is some scheme dependence in this determination,
which we use for a rough error estimate. On the one hand, we use {\bf A)}:
Eq. (\ref{NNLL}) plus Eq. (\ref{NNNLLmatching}) plus the $\ln (\frac{\nu}{C_A \alpha})$ term of Eq.~(\ref{NPlog}) for
$\nu=\Delta V_X$ in the $X=$MS and $X=\overline{\rm MS}$ schemes and, on the other hand {\bf B)}:
Eq. (\ref{FinalNNNLL}) plus all the logarithms of Eq. (\ref{Final4loopNP}).
Ideally most of the scheme dependence should originate from the $X$ dependent terms
in Eq.~(\ref{FinalNNNLL}).
We also vary the number of lattice data points in the fit to see the dependence of the results.
The numerical value should not depend much on the number of data points if we are at short enough distances.
Finally, we also consider determinations for $N_c=2$ and $N_c=3$, as we expect
$c_{2,0}$ to be independent of $N_c$ (for $n_f=0$).

\begin{table}[t]
\begin{tabular}{|l|c|c|c|c|}
\hline
gauge group: & SU(2) & SU(2) & SU(2) & SU(3)\\
\hline
\# of fit points: & 1 & 3 & 5 & 1 \\
\hline
\hline
{\bf (A)}: ${\overline{\rm MS}}$ fit & $-0.04$ & $-0.42$ & $-0.82$ & $+0.41$ \\
\hline
{\bf (A)}: $\rm MS$ fit & $+0.92$ & $+0.18$ & $-0.67$ & $+0.85$ \\
\hline
{\bf (A)}: $\rm MS$ from ${\overline{\rm MS}}$ fit & $+2.64$ & $+2.26$ & $+1.87$ & $+3.09$ \\
\hline
{\bf (A)}: ${\overline{\rm MS}}$ from $\rm MS$ fit & $-1.76$ & $-2.51$ & $-3.35$ & $+0.18$ \\
\hline
{\bf (B)}: ${\overline{\rm MS}}$ fit & $-0.74$ & $-1.22$ & $-1.75$ & $-0.43$ \\
\hline
{\bf (B)}: $\rm MS$ fit & $+1.95$ & $+1.47$ & $+0.93$ & $+2.25$ \\
\hline
\end{tabular}
\caption{\it Results for $c_{2,0}$ from fitting the force derived from
{\bf (A)}: Eq. (\ref{NNLL}) plus Eq. (\ref{NNNLLmatching}) plus the logarithm of Eq.~(\ref{NPlog}),
and {\bf (B)}: Eq. (\ref{FinalNNNLL}) plus all the logarithms of Eq. (\ref{Final4loopNP}), to lattice
data for $N_c=2$ and $N_c=3$ in the $X=\rm MS$ and $X=\overline{\rm MS}$ scheme.
The (first/first three/first five) lattice points with smallest distances $r$ have been used for the fit.
To obtain the values in the third and the fourth line we have transformed the ${\overline{\rm MS}}$ fit values to
the $\rm MS$ scheme, and vice versa, by adding/subtracting the difference coming from the scheme ($X$) dependent terms in Eq.~(\ref{FinalNNNLL}).
Note that for option {\bf (B)} such a scheme transformation reproduces the same results as from the direct fits. There is therefore no uncertainty from the scheme dependence in this case.
\label{c20}}
\end{table}

Table~\ref{c20} lists the numbers for $c_{2,0}$ resulting from the fits to the data points at
the smallest available distances $r$ in the renormalization schemes $\rm MS$ and ${\overline{\rm MS}}$.
Albeit with large uncertainties, we see that the numbers are in the same ballpark.
We should moreover not equally weight all the determinations. We believe for instance
that the $\rm MS $ fit ${\bf (A)}$ is less reliable, since $\Delta V_{\rm MS} < \Delta V_{{\overline{\rm MS}}}$.
We also expect that increasing the number of included data points (with larger $r$) decreases the fit quality.
In any case, we should not forget about the poor convergence, which makes a precise determination of $c_{2,0}$ impossible.
On the other side we find it reassuring and nontrivial that the SU(2) and SU(3) determinations, just as the ${\bf (A)}$ and ${\bf (B)}$
${\overline{\rm MS}}$ results, are quite close. We have also performed fits not including the non-perturbative logarithms.
Their effect is small and within the uncertainties, which, from the above discussion, we roughly estimate to be as follows:
\begin{equation}
\label{c20fit}
c_{2,0}^{\overline{\rm MS}} \simeq -0.04 \pm 2\,.
\end{equation}
We took the central value from fitting only to the first lattice point at $r/r_0 \simeq 0.22$,
where we expect the smallest theoretical error from the truncation of the perturbation series.

We use this value for the ${\overline{\rm MS}}$ N$^3$LL and N$^3$LL+non-perturbative-logarithm curves shown in Figs.~\ref{Plots}.
In other words we have fitted the N$^3$LL+non-perturbative-logarithm curve, which includes all mentioned corrections in the ${\overline{\rm MS}}$ scheme,  to the first data point for $N_c=2$. That's why this point lies exactly on
the line, whereas in Fig.~\ref{Plots} b), where $N_c=3$ and the same value for $c_{2,0}^{\overline{\rm MS}}$
has been used, it does not. As we have already stated the convergence is not good. This is also seen
in the difference between the NNLL and N$^3$LL curves, although the difference between the N$^3$LL and the
N$^4$LL curves due to the non-perturbative logarithms is small at short distances. The pattern in the
MS scheme is similar even though the convergence is worse. Working with
Eq. (\ref{FinalNNNLL}) plus all the logarithms of Eq. (\ref{Final4loopNP}) produces curves
similar to those obtained in the ${\overline{\rm MS}}$ scheme.

From the above analysis, we conclude that the lattice simulations so far have produced data at short enough
distances to test the multipole ($\alpha r$) expansion (cf. Figs.~\ref{Plots} a) and b)). A Precise test of the $\alpha/\Delta V$ expansion is however not possible with the existing data.

\section{pNRQCD(D=2)}
\label{secD2}

It is also interesting to investigate the static quark-antiquark potential in two space-time dimensions.
In the following we set $n_f =0$.

 For $D=2$ the exact result for the singlet static potential is known. It can be derived from its standard (non-perturbative) definition via a Wilson-loop:
\begin{equation}
E_s(r) := \lim_{T \to \infty} \frac{i}{T} \ln\, \langle {\frac1{N_c} \rm Tr} \,{\cal P} \exp\big(-i g \oint_{\Gamma} A_\mu dx^\mu\big) \rangle\,,
\label{Wilsonloop}
\end{equation}
where the contour $\Gamma$ of the integral in Eq.~(\ref{Wilsonloop}) is a rectangle with spatial extension $r$ and extension $T$ along the time coordinate axis in Minkowski space. Choosing axial ($A_1 \equiv 0$) gauge in exactly two dimensions, the only non-vanishing components of the gluon field-strength tensor are $F_{01}=-F_{10}=-\partial_1 A_0$. Hence, ($A_0$) gluons do neither interact among themselves, nor propagate in time, since no time derivative is left in the Lagrangian.
Therefore only planar ``ladder'' diagrams (with ``potential'' gluons) contribute to the Wilson-loop.
In such a quasi-Abelian situation, one can forget about the path-ordering operator
${\cal P}$ in Eq.~(\ref{Wilsonloop}) and treat the SU($N_c$) generators $T^a$ in the exponential
as if they were proportional to the unit matrix. Finally we arrive at the result for the
corresponding Abelian Wilson-loop multiplied by $\frac1{N_c}{\rm Tr}\, T^aT^a=C_F$. This result
is identical to the perturbative tree-level contribution
\begin{equation}
E_s(r) = 2 \pi C_F \alpha\, r
\label{ExactPot}
\end{equation}
and therefore proving exponentiation.

The computation in pNRQCD is organized differently, because of the factorization of the different scales that contribute
to the potential. For exactly $D=2$ pNRQCD recovers Eq. (\ref{ExactPot}), since the soft contribution is precisely Eq.
(\ref{ExactPot}) and there are neither ultrasoft nor non-perturbative gluons. For $D > 2$ one would expect the result to
converge to Eq. (\ref{ExactPot}), when $D \rightarrow 2$, provided the limit is smooth.
The soft two-loop calculation \cite{Schroder:1999sg} yields however
\begin{equation}
V_s(r) = 2 \pi C_F  \alpha\, r \,-\, 2  \pi C_A C_F \alpha^2 r^3  \,+\, \frac{\pi}{27} (33+2 \pi ^2) C_A^2 C_F \alpha^3 r^5
\label{Vsoft2D}
\end{equation}
for $D \to 2$ and $n_f =0$.
From the effective theory viewpoint, we expect that the ultrasoft contributions to the static energy,
calculated in pNRQCD for $D \to 2$, cancel the $\alpha^2$ and $\alpha^3$ terms in Eq.~(\ref{Vsoft2D}). Though, checking this is a delicate issue,
since as mentioned in subsection~\ref{RenAndGen}
the proper ultrasoft expansion parameter is now $\frac{\Delta V}{g} \sim g r$, i.e. non-perturbative. Thus the intrinsically
non-perturbative ultrasoft ($\Delta V/g$) expansion and the perturbative multipole ($g r$) expansion mix.

The non-perturbative nature of the ultrasoft expansion does not allow a calculation in terms of pNRQCD loop diagrams for $D \not= 2$.
Expressions like Eq.~(\ref{deltaVUS}) should rather be expanded for small $\Delta V$ and the full non-perturbative result
for the chromoelectric correlator should be used. Of course the latter is not known in dimensional
regularization.

Despite that we have observed interesting cancellations. The ultrasoft one-loop contribution at the
lowest non-vanishing order in the multipole expansion, Eq.~(\ref{USbare1loop}), gives
\begin{equation}
\delta E^{us}({\rm 1-loop}) = 2 \pi C_A C_F  \alpha^2 r^3
\end{equation}
in the limit $D\to2$ and exactly cancels the soft $\alpha^2$ contribution in Eq.~(\ref{Vsoft2D}).

At this point we are unfortunately not able to check the (anticipated) cancellations at higher orders in $\alpha$, as
they may require higher order terms in the multipole expansion. We are also not sure about the relevance of the above result, as
it follows from a perturbative ultrasoft computation.
Nevertheless the observed cancellation could be interpreted as another check of the applied effective field theory
approach, which uses perturbative methods to derive meaningful results for the singlet static energy in arbitrary dimensions.

\section{Conclusions}
\label{conclusion}

The main achievement of this paper is a precise determination of the static energy
of a color singlet quark-antiquark system in three space-time dimensions for $\alpha\, r \ll 1 $.
We have complemented the strictly perturbative (soft) two-loop QCD result~\cite{Schroder:1999sg}
by an effective theory calculation in pNRQCD to account for effects at the ultrasoft scale
($\Delta V=V_o\!-\!V_s$), which are not accessible to pQCD.

We observe a perfect cancellation of the IR singularities in the soft pQCD result at ${\cal O}(\alpha^3r^2)$
and the UV divergences in the one- and two-loop pNRQCD corrections,
i.e. at NLL and NNLL order in the ultrasoft $\alpha/\Delta V$ expansion. This represents an
independent and nontrivial check of the soft pQCD and the ultrasoft pNRQCD contributions,
because the singlet static energy is a physical observable and must be IR finite.
We also observe a cancellation of logarithmic divergences that can be associated with the
leading renormalons of the potential and the quark mass in four dimensions.

Demanding complete IR finiteness we have determined indirectly the UV divergences at ultrasoft
three-loop level, without having calculated the actual diagrams. In fact the only missing piece in
our N$^3$LL prediction for the singlet static energy up to ${\cal O}(r^2)$, Eq.~(\ref{FinalNNNLL}),
is the ultrasoft three-loop constant $c_{2,0}$. We also provide expressions for the N$^4$LL correction
and the known leading non-perturbative contribution from the gluon
condensate~\cite{Schroder:2003uw,DiRenzo:2006nh,Hietanen:2004ew,Hietanen:2006rc},
which is parametrically of the same order. They are complete up to the ultrasoft four-loop constant
$c_{3,0}$. Thus the precision is currently limited by missing perturbative three- and four-loop computations.

From a fit to SU(2) lattice data~\cite{HariDass:2007tx} we were able to extract a rough value for
$c_{2,0}$. The uncertainty is however rather large, because data points for distances $r$
sufficiently small to ensure reasonable convergence of the perturbative $\alpha/\Delta V$ series
are currently not available. The $\alpha r$ (multipole) expansion in contrast can be tested with the present
precision of the lattice simulations.

With lattice data at smaller distances and analytic results for $c_{2,0}$ and $c_{3,0}$,
an interesting cross-check of the three-dimensional lattice simulation for the gluon
condensate~\cite{Hietanen:2004ew,Hietanen:2006rc} in Eq.~(\ref{eq:condensate}) would be feasible.

We have also studied the two dimensional case, for which the exact result is
known and equivalent to the tree-level result. Therefore, strong cancellations have to
occur among the contributions from the different scales: soft, ultrasoft and $g$ in the $D \rightarrow 2$ limit.
We have found a cancellation between the one-loop soft and the one-loop ultrasoft results in dimensional regularization for $D \rightarrow 2$.
The relevance of this cancellation is however not clear to us, because it follows from a perturbative ultrasoft computation.
Nevertheless, we think that it could be worthwhile to explore the $D \rightarrow 2$ limit
in more detail, and see whether it could provide nontrivial information on the structure of the
perturbation series for general dimensions. We leave this issue open in this paper.

\bigskip

\acknowledgments{
This work was partially supported by the EU network contract
MRTN-CT-2006-035482 (FLAVIAnet), by the Spanish
grant FPA2007-60275 and by the Catalan grant SGR2009-00894.
}

redpajama_set_name: RedPajamaArXiv


University of Missouri students visit ENRD Contact Point.
A group of US university students from the University of Missouri visited the ENRD Contact Point on 12th January 2012. The group was keen to learn about the future of the CAP post-2013. They made their visit to the ENRD CP as part of a wider study trip to Brussels to learn about EU institutions, policies and issues.


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Rein[Forced] – Pre-Existing Conditions – Music Stream
Remixes from the band's 2012 release, X Amount of Stab Wounds in the Back, and this reworking by Imperative Reaction is even more harrowing than the original.
Pillar Point – Diamond Mine – Music Stream
It's tough to describe the smiling sounds of tracks like "The Reeling" and "Sleepyhead" as anything but celebratory.
SMP – Death of the Format – Music Stream
SMP delivers "Metal Madness" with metal aggression (and a delightfully wanky guitar solo) over a dance/industrial skeleton, along the lines of Rammstein.
Essence of Mind – Indifference – Music Stream
Norway's answer to anthemic synth pop, while there's nothing here to topple Apoptygma Berzerk or Zeromancer, there are many potential singles here.
Siva Six – Superstition – Music Stream
The Greek Gothers nailed it with 2011's The Twin Moons, and this 15-track Superstition EP continues the distorted whispers, frantic beats, and industrial chaos.
Helalyn Flowers – White Me In / Black Me Out – Music Stream
Helalyn Flowers combine a strong, beautiful female singer and a strong electro metal-tinged male rock beast, so it'd be hard to go wrong, yes? Well, no.
Ayria – Plastic Makes Perfect – Music Stream
Plastic Makes Perfect is a great album title, but Jennifer Parkin is no stranger to great looks, writing styles, titles, cover art, and, of course, songs.
Implant – The Productive Citizen – Music Stream
Cold, classic EBM with distorted vocals, news samples, plenty of trance-inducing repetition, and harsh keyboards replacing the crunch of "metal guitars."
Deviant UK – Very Bad Things – Music Stream
13 tracks and 3 remixes exclusive to this WWII release, and the download includes eight additional remixes not even listed. Whew!
Lovelorn Dolls – The House of Wonders – Music Stream
Debut full-length from this Belgian duo, Ladyhell and Corpus Christi. They love Evanescence, Garbage, and Lacuna Coil and it shows.
Aesthetische – Nachtbrenner – Music Stream
Aghast View and Biopsy side project returns with a 15 track release, few new songs, bunch of reworks. EBM, pure and simple.
Mechanical Cabaret – Selective Hearing – Music Stream
Selective Hearing is a greatest hits record, and therefore a great introduction. The words dour and deadpan pop up in numerous reviews, as they should.
Kant Kino – Father Worked In Industry – Music Stream
Combining cold distortion of '90s-style dark industrial with the anthemic synthpop of 2000's faves like Apoptygma Berzerk, including chillingly female vocals.
The Orb – The Orbserver in the Star House – Music Stream
Synopsis: Well-respected UK electro/dance duo team up with wacko Reggae legend. Cool songs like "Hold Me Upsetter" ensue.
Marilyn Manson – Born Villain – Music Stream
Nasty guitars, smart lyrics to whisper while fucking all your little sister's friends, and a snarl that'd make a lion tuck tail.
Suicide Inside – Homicide – Music Stream
An eclectic blend of aggressive electronic music and furious industrial rock, with heavy breakbeats, rhythmic noise, and Natasha's razor-sharp provocative female vocals.
Stray – Letting Go – Music Stream
Second release by Erica Dunham of Unter Null's side project. One can perceive hope and light at the end of this trancey tunnel.
Skinny Puppy – HanDover – Music Stream
Skinny Puppy are as wild and weird as ever on their 14th album. The big beats of "Village" stand on their own, but they're just one of many puzzle pieces.
Schwarzblut – Maschinenwesen – Music Stream
A five-piece band from the Netherlands that blends romantic, melancholic Weimar poetry with danceable dark electronic music.
Freakangel – Let It All End – Music Stream
"It's an album about all of us… how brain-dead we act in this dying world of ours, how we have lost faith in ourselves and how we prefer to believe in television nonsense and pseudo-Gods."
Alien Vampires – Clubbers Die Younger – Music Stream
Louder, faster, harder: Upfront industrial dance full of distorted noise, vibrating bass-drums, hypnotic turbo beats, and provocative, harsh vocals.
The Gothsicles – Industrialites & Magic – Music Stream
Self-described practitioners of the "nerdy shitheads who jump around-dustrial" sound, The Gothsicles are nothing if not immediate on "Drunk Cuts."
Stromkern – Dead Letters – Music Stream
Wisconsin's Stromkern mixes minimalist beats with hip-hop, filling the empty spaces of their Kraftwerk-style framework with fast, catchy wordplay.
Stiff Valentine – America Inbreeding – Music Stream
Stiff Valentine brings a lighthearted vibe to their raucous electronic attack, and "Somebody to Hate" is a thrashing rallying call to the likeminded.
Skrillex – More Monsters And Sprites – Music Stream
The dubstep maestro whose songs grace every movie trailer or commercial wishing to make a loud statement.
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redpajama_set_name: RedPajamaCommonCrawl


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Directory of Abroad Resources
Home > Regional > North America > Mexico > Government > Embassies and Consulates > Abroad
Belize - Belize City
http://embamex.sre.gob.mx/belice/
Lists and explains the consular services of the Mexican Embassy in Belize, and has opening hours and contact information.
Canada - Ottawa
http://embamex.sre.gob.mx/canada/
With economic, political, education, energy and tourism sections.
http://www.mexico.com.hk/
Consulate General of Mexico in Hong Kong, with information about Mexico and its visa requirements.
India, Bangladesh, Nepal and Sri Lanka
http://embamex.sre.gob.mx/india/
Mexican mission located in New Delhi and accredited to India, Bangladesh, Nepal and Sri Lanka, with a brief summary of the bilateral and multilateral relations, and of issues related to economic, cultural and technical cooperation and current treaties and agreements. In English and Spanish.
Japan - Tokyo
http://embamex.sre.gob.mx/japon/
Includes a history of relations between Mexico and Japan, and contact details for the consular affairs section.
http://embamex.sre.gob.mx/malasia/
With contact details for the consular section and the trade office of the Embassy of Mexico in Kuala Lumpur, information on doing business with Mexico and visa application forms.
Philippines - Manila
http://embamex.sre.gob.mx/filipinas/
Embassy of Mexico in Manila, detailing the requirements for the various types of visa.
http://embamex.sre.gob.mx/suecia/
Embassy of Mexico to Sweden, also accredited to the Republic of Latvia and the Republic of Lithuania. Includes contact information for the Honorary Consulates in Gotemburgo, Umeå, Malmö, and Riga.
United States - Washington DC
http://embamex.sre.gob.mx/eua/
Embassy of Mexico in the United States of America. Has information about the embassy's activities, services, staff personal, a press room and information related to tourism.
Home > Regional > North America > Government > Embassies and Consulates > By Country of Origin
Home > Society > Government > Embassies and Consulates > By Country of Origin


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Due to Covid-19 School is now closed until further notice - please check your emails. (04.01.2021)
Year 5&6 Pop Lacrosse Team (District WINNERS)
Thursday 30th April - Pop Lacrosse Regional Finals
After a successfully winning the Gedling pop lacrosse competition in March it was the teams turn to compete in the regional finals at Nottingham Trent University. With a fantastic venue and the best teams in the region competing the team knew they were in for a hard afternoon. The team played 3 league games, winning their first 2 but losing their final game, this secured them second spot in their league and a place in the semi finals. Unfortunately we lost the semi final 6-3 but went away from the event feeling very proud and very pleased with our performance.
Friday 27th March - Gedling Pop Lacrosse Competition (Carlton Academy)
The team travelled to The Carlton Academy with high hopes and with great excitement. After a long warm up all the teams were finally brought together for a run through of the rules. With this explained the tournament got under way.
The team played 4 games against some very good opposition but our teams ability shone through as they claimed 4 victories out of 4. This fantastic effort meant that they won the competition and progressed through to the regional finals, which take place on Thursday 30th April at Nottingham Trent University.
Friday 5th December - Gedling Pop Lacrosse Competition (Carlton Academy)
Our pop lacrosse team braved the cold weather conditions and took part in the 8 team Gedling competition. The team were looking forward to the afternoon but were all feeling slightly nervous as it was their first competitive competition. Upon arrival the 8 teams were split into 2 leagues of 4.
The team began slowly, falling behind to Salterford House (the eventual winners) in the opening game. However team AMPS showed excellent resilience and managed to secure a draw (we were the only team to take points of the afternoon winners). The name game saw the team win against Killisick but unfortunately they lost the third game against a strong Willow Farm Team.
As a team we qualified in second place and went into a third and fourth play off against St John Baptist school. Team AMPS upped their game and produced an outstanding display, beating the opposition comfortably and securing a 6-1 victory to claim 3rd place at the event.
The players were extremely pleased with their performance and they all thoroughly enjoyed the afternoon - once again it was a pleasure watching them play and grow in confidence.
The team is now looking forward to the next competition in 2015.
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Just over a century ago, the Canadian Army built the Royal George Theatre in Niagara-on-the-Lake as part of a vast training base, preparing troops to go the bloodbath of the Western Front.
The war started badly and descended into military mass murder.
The Shaw Festival decided bringing "The Great War" back to the Royal George was the way to go, along with mixing in some of Canada's baggage, both from the war and the issues of getting French Canadians, Blacks and first nations to participate in the war and the aftermath.
The mix of the war and the excesses and abuses of history just clangs.
War often brings out underlying and hidden issues in a nation, as the U.S. learned in both world wars about African-Americans, as the U.S. military tried to have Black soldiers without actually using them in combat.
The needs of war and the casualty lists mixed cannon fodder needs and military segregation and changed some of that.
The Canadian situation was similar in WWI, along with the difficulties of persuading French Canadians to join in.
Director Peter Hinton has this show set in the Royal George way back then, as the show is designed.
It's a music hall, with occasional excursions out into the audience and effective use of the doors, balcony and aisles.
Allan Louis is the chairman, the British music hall equivalent of the American MC.
The reference to Picardy is interesting because that area of the Western Front is where wars have been fought for centuries, including the area of Agincourt where "Henry V" is set and in the Shaw production set in those World War I trenches.
The "Lovely War" show centers on one of the most controversial of the incompetent military leaders of World War I, a group filled by generals from both sides.
That's Field Marshall Haig, the British commander, who kept telling his political masters that one last big push would end the war and he put on several of those big pushes, costing hundreds of thousands of soldiers their lives and not winning the war.
On the Somme, 60,000 Brits died the first day and the attack went on for months.
The show is so savage about Haig that his family tried to have the show ordered off the stage by the Lord Chancellor, then the official censor of entertainment.
Considering he had been a key figure in a book called "The Donkeys" the Haig family lost and the public won.
As with the original production back in the day, the cast of this production isn't filled with great singers although the music doesn't require it.
Probably more Canadians than Americans will recognize the music, songs like "The Maple Leaf Forever" or "Your King and Country" along with those background songs of the era, like "Pack Up Your Troubles in Your Old Kitbag," "Roses of Picardy" and the lament in the trenches of "Keep the Home-Fires Burning" with its heart tugs at home.
There's almost no one left in Britain from World War I, soldier or civilian, but the war lives on in the monuments, the memorials to those who died "For King and Country" and the list of the dead in so many small towns across the world, often names still surrounding them.
In Niagara-on-the-Lake, just walk to the clock tower, which is actually a cenotaph for the dead, names of the local dead from World War I and II, still there a century after troops marched down Queen Street and to war.
A strong cast, Louis and others, does a good job on Teresa Przybylski's wondrous set.
It's just that "Oh What A Lovely War" needs more connection from 21th Century social issues to the bloodbath of the 20th Century which ended the imperial 19th Century.
Sometimes, good intentions need crafts people to make it work.


PreSys is now available to users world-wide. This offering will allow NISA users to take advantage of PreSys' streamlined user interface, CAD interoperability and advanced graphics capabilities, thereby improving the overall NISA user experience.
PreSys will allow NISA users to create even complex finite element models easily. It is a full-featured, core mechanical engineering software solution that can be used on its own or with a variety of available add-on applications.
It also offers a scripting interface to automate processes and interact with 3rd party solutions, as well as a model explorer feature to provide in-depth data navigation.
When the user opens PreSys for the first time, they will immediately notice the refreshed user interface, with drop-down menus and toolbars. They also quickly notice the graphics speed and added graphics capabilities of PreSys. The user interface also allows user to quickly find the tools they need, has fewer levels of menus and is easier to learn.
Developers focused not only on packaging the existing tools into the new user interface, but took the opportunity to streamline as many functions as possible. This means that fewer operation steps will be required, and that the users will have a minimum amount of data to enter. The result is a faster path to your FE analysis.
A significant change in the software is the addition of the Model Explorer. This essentially adds another dimension to the information available to the user at all times. The Model Explorer allows the user to manipulate all of the entities in the model, and also allows users to access non-graphical data through a Keyword tree. The Keyword tree allows access to editable card images for virtually any entity in the model.
As the needs of the CAE engineer expand, the software tools must expand along with them. Inventium's products allow users to open and operate on multiple models, as well as opening finite element analysis (fea) simulation results files along with finite element data. This feature provides a unique way for users to get the most out of their software and increase productivity.
The Inventium architecture allows the user to capture the process, replay and modify it. This process capture is based on a fundamental capability of the software to execute any function through a scripting interface.
To be truly open and accessible for outside software tools and configurable for processes, each function within the software must be accessible through scripting access. Passing variables or data between software products provides the foundation for parametric design optimization, where the FE modeling software reacts to the results or direction dictated from external software, and automatically updates the model. In this case the design may be 'driven' by outside software.
This type of capability, combined with powerful, robust and accurate modeling functions promises to change the way that engineers develop products, and will unlock the potential of optimization software for the use in everyday problems.


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The Power of Good Food
Drop-In Meals
Extreme Weather Centre
Peer Advocacy
Our kitchen staff prides themselves on serving fresh, healthy, and well-balanced meals to our neighbours. Between breakfast & lunch five times a week, and dinner four times a week, we serve over 1200 meals weekly.
Breakfast is served between 9:00 am and 10:00 am, Monday-Friday.
Lunch is served between 11:30 am and 1:15 pm, Monday-Friday
Dinner is served between 4:00 pm and 5:30 pm on Mondays, Tuesdays, Fridays, and Saturdays.
Discover different areas for donation, and learn about new ways to donate to The Gathering Place.
Resources that Feed the Mind and Soul
Stay connected, find new recipes and more in our resources section.
1181 Cassells Street, North Bay, ON P1B 4B4 Phone: 705.474.7687
Please note that we answer the phone only during office hours.
Some of yesterdays garden harvest! #tgp #tgpgarden #communitygarden #northbay #plottoplate #growarow https://t.co/2cESXjl7C4
© Copyright 2023 Gathering Place


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Review: American Assassin (2017)
Directed by: Michael Cuesta
Premise: Based on the novel by Vince Flynn. An American survivor (Dylan O'Brien) of a terrorist attack is recruited by a covert agency that assassinates terrorist cells. Things get complicated when a former member of the team shows up in the middle of a nuclear bomb plot.
What Works: The opening sequence of American Assassin is appropriately shocking. Apparently based on a real life incident in Tunisia, a group of jihadist gunman storm a beachfront resort and murder Western tourists. The opening is well handled and the filmmakers capture the horror of terrorist violence in a way that lends some emotional stakes to the rest of the movie. Dylan O'Brien plays Mitch Rapp, a young man whose fiancé was murdered on the beach, and O'Brien does an excellent job in the dramatic moments. His character is consumed by rage and the grief over his loss simmers underneath O'Brien's performance. Swearing revenge on the people who killed his fiancé, Rapp trains himself in combat techniques and infiltrates the terrorist organization only to be picked up by an American counterterrorism agency and drafted into its service. From there Rapp is put under the mentorship of a grizzled military veteran, played by Michael Keeton. As usual, Keeton brings a lot to the role and his energy and charisma elevates what might otherwise be a mundane character. As an action movie, American Assassin is acceptable. This is the post-9/11 version of the movies that Chuck Norris and Steven Seagal were making thirty years ago. Taken simply as an action picture, American Assassin is satisfactory. The action sequences are mostly credible and competently staged and the film establishes dramatic stakes and follows them through.
What Doesn't: Nothing about American Assassin is particularly distinguished and nothing in it is memorable. The movie's conventionality is more of a liability for American Assassin than it would be for the average action picture because the premise flirts with big ideas and real world implications. To invoke the specter of radical jihadist violence and to begin the movie with a fictionalized version of an actual terrorist attacks signals to the viewer that this movie has something to say and that it wants to be taken seriously. But American Assassin is unable to martial its ideas and its story into anything more than a by-the-numbers elite solider story. Part of the problem is that the moviemakers don't seem to have a firm perspective on violence. American Assassin begins as a story of how grief and anger can become a self-destructive obsession. Then the film shifts to suggest that terrorist guerilla violence must be met with the same and then it shifts again to imply that counterterrorism breeds its own villains. There's the potential for something interesting and even provocative here but the ideas aren't treated with any kind of competence. American Assassin's lackluster theme is exacerbated by its lack of character. Rapp is only defined by his grief and the movie doesn't provide him with a meaningful character arc. Everyone else in the movie is generic. Michael Keaton's character is the typical grizzled military figure and the film squanders the kernel of an interesting subplot while Sanaa Lathan is cast as the no-nonsense pantsuit-wearing professional woman done much better in the Jason Bourne films. For a movie that wants to be taken seriously, not much in American Assassin makes much sense. The very idea that this self-taught recruit with no combat experience would be preferable to a military veteran is stupid and it's never clear why O'Brien's character is somehow better suited for the job.
Bottom Line: American Assassin meets the basic minimum standards for a movie like this. It is competently made and has just enough action to keep the viewer entertained but it's also entirely forgettable and squanders a good cast.