text:
\subsection{Introduction}
The similarities between Quantum Electrodynamics in three space-time
dimensions (QED3) and Quantum Chromodynamics in four space-time
dimensions (QCD4) and the simplicity of the
theory make QED3 attractive for the study of non-perturbative methods.
QED3 is an abelian theory and provides a logarithmic
confining $e^-$-$e^+$ potential~\cite{BPR92}.

Our approach to positronium states in QED3 is via a solution to the
homogenous Bethe-Salpeter equation with fermion propagator
input from the Schwinger-Dyson equation.
The full Schwinger-Dyson and Bethe-Salpeter equations are
intractable. Here we consider a solvable system of integral
equations within the quenched, ladder approximation.  This crude
truncation of the full equations does break gauge covariance but
has very attractive features and has been employed extensively
in QCD4 spectrum calculations~\cite{BSpapers,SC94}.

This study continues on from a previous study~\cite{Bu92} which uses
a four-component fermion version of QED3.  In this version, the massless case
exhibits a chiral-like $U(2)$ symmetry broken into a
$U(1) \times U(1)$
symmetry by the generation of a dynamical fermion mass, resulting in
a doublet of Goldstone bosons.  This pion-like solution is important
for drawing similarities between QED3 and QCD4.  The
four component
version of QED3 is also preferred to the two component
version because
the Dirac action in the two component version is not parity
invariant for massive fermions.  QCD4 is parity invariant
and we aim to have as much in common with
that theory as possible.

The previous work was restricted to zero bare fermion
mass, while in this study the bare mass is increased from
zero to large values in order to compare with results in the
non-relativistic limit.  This study also takes a closer
look at the choice of fermion propagator input.  Knowledge
of the analytic properties of the fermion propagator is
important for determining the approximation's ability
to provide confinement and whether or not any singularities
will interfere with a Bethe-Salpeter
solution.    Based on the work of Maris~\cite{Ma93,Ma95} the
occurrence of mass-like complex singularities
is expected which have the potential to influence our calculations.

In section 2 we look at the Bethe-Salpeter and Schwinger-Dyson
approximations used in this work and the method used to find the
bound state masses.  A brief review of transformation properties in
QED3 is given in the appendix.
These transformation properties are of vital importance for
an understanding of the structure of the $e^-$-$e^+$ vertex
function and the classification of the bound states.
Section 3 describes the non-relativistic limit and the connection
between the Bethe-Salpeter and Schr\"{o}dinger equations for QED3.

In section 4 the approximation to the fermion propagator is
detailed. The structure of the propagators will be analysed in
the complex plane where we attempt to locate the expected
mass-like singularities.
In section 5 the Bethe-Salpeter
solutions are reported and comparisons are made with
non-relativistic limit calculations.  The results are discussed
and conclusions given in section 6.

\setcounter{equation}{0}
\subsection{Solving the Bethe-Salpeter Equation}
The Bethe-Salpeter (BS) kernel for this work is a simple one-photon
exchange (ladder approximation) which is a commonly used starting
point.  For convenience we use the quenched approximation,
work in Feynman gauge and work only with the Euclidean metric.
Fig.~1 shows the Bethe-Salpeter equation in the quenched ladder
approximation.  The corresponding integral equation is
\begin{equation}
\Gamma(p,P) = -e^2 \int \mbox{$ \, \frac{d^3q}{(2\pi)^3} \,$}
D(p-q) \gamma_\mu S(\mbox{$ \, \frac{1}{2} \,$} P+q)
       \Gamma(q,P)S(-\mbox{$ \, \frac{1}{2} \,$} P+q)
       \gamma_\mu, \label{eq:BS}
\end{equation}
where $\Gamma(p,P)$ is the one fermion irreducible
positronium-fermion-antifermion vertex with external legs
amputated.  The photon
propagator $D(p-q)$ in Feynman gauge is $1/(p-q)^2$.  The
fermion propagator $S$ is the solution to a truncated
Schwinger-Dyson (SD) equation.  A fermion propagator has been chosen
which supports spontaneous mass generation necessary
for the formation of the Goldstone bosons.
The truncated SD equation for a fermion of bare mass $m$ is
\begin{equation}
\Sigma(p)= S(p)^{-1} - (i\!\not \! p + m) =
 e^2 \int \mbox{$ \, \frac{d^3q}{(2\pi)^3} \,$} D(p-q) \gamma_\mu
 S_F(q) \gamma_\mu.  \label{eq:SD}
\end{equation}
This approximation is the quenched, rainbow approximation
named so because the photon propagator has been replaced by the
bare photon propagator and the vertex function $\Gamma$ has been
replaced by the bare vertex $\gamma$, resulting in a series
of Feynman diagrams which resemble rainbows.
In the quenched approximation the SD and BS equations can be
recast in terms of a dimensionless momentum $p/e^2$ and bare fermion
mass $m/e^2$.  From here on we work in dimensionless units and set
$e^2 = 1$.

We use either of the two following equivalent
representations of the fermion propagator,
\begin{equation}
S(p)= -i \not\! p \sigma_V(p^2)+\sigma_S(p^2)
\label{eq:SIGDEF}
\end{equation}
or
\begin{equation}
S(p)= \frac{1}{ i \not\! p A(p^2)+B(p^2) }.   \label{eq:ABDEF}
\end{equation}
The generation of a dynamical fermion mass and the breaking of
chiral symmetry is signalled (in the massless limit)
by non-zero $B(p^2)$.  The vector and
scalar parts $\sigma_V$ and $\sigma_S$ of the propagator are
related to the functions $A$ and $B$ simply by dividing
these functions ($A$, $B$)
by a quantity $p^2 A^2(p^2) + B^2(p^2)$.

Note that a substitution of this fermion propagator into the
Ward-Takahashi identity shows that the bare vertex approximation
breaks gauge covariance.  However, this model is simple and does
meet the requirement that the appropriate Goldstone
bosons are formed~\cite{DS79}.
It is not difficult to derive a zero mass
solution to our BSE analytically. A vertex proportional to the
matrix $\gamma_4$ or $\gamma_5$, defined in the appendix,
will reduce the quenched ladder
BSE to the quenched ladder (rainbow) SDE in the case of zero bound
state mass thus forming a doublet of massless states.  According
to the terminology used in the appendix this is an axi-scalar
doublet.  These solutions will be seen in section 5.

Once the photon and fermion propagators are supplied, the BS
equation can be written as a set of numerically tractable
integral equations.  To do this, we write the BS
amplitude $\Gamma$ in its most general form consistent with
the parity and charge conjugation of the required bound
state, and then project out the coefficient functions for
the individual Dirac components.
It is convenient to work
in the rest frame of the bound state by setting
$P_{\mu}=(0,0,iM)$.  Then the scalar and axi-scalar vertices
given in the appendix by eqs.~(\ref{eq:GS}) and (\ref{eq:GAS})
can be written as
\begin{eqnarray}
\lefteqn{\Gamma^S(q,P) =
f(q_3,\mbox{$\left| {\bf q} \right|$};M) - \frac{i q_j \gamma_j}
{\mbox{$\left| {\bf q} \right|$}} U(q_3,\mbox{$\left| {\bf q}
\right|$};M) } \nonumber \\
& & + \frac{i q_j^{\perp} \gamma_j}{\mbox{$\left| {\bf q}
\right|$}}\gamma_{45}
V(q_3,\mbox{$\left| {\bf q} \right|$};M)  + i \gamma_3
W(q_3,\mbox{$\left| {\bf q} \right|$};M), \label{eq:GSNEW}
\end{eqnarray}
\begin{eqnarray}
\Gamma^{AS}(q,P) & = & \left(\begin{array}{c} \gamma_4 \\
 \gamma_5\end{array} \right) f(q_3,\mbox{$\left| {\bf q}
 \right|$};M) - \frac{i q_j \gamma_j}{\mbox{$\left| {\bf q} \right|$}}
\left(\begin{array}{c} \gamma_4 \\ \gamma_5\end{array} \right)
 U(q_3,\mbox{$\left| {\bf q} \right|$};M) \nonumber \\
 &  & + \frac{i q_j^{\perp} \gamma_j}{\mbox{$\left| {\bf q} \right|$}}
\left(\begin{array}{c} \gamma_5 \\ -\gamma_4\end{array} \right)
   V(q_3,\mbox{$\left| {\bf q} \right|$};M) - \gamma_3 \left(
   \begin{array}{c} \gamma_{4} \\
\gamma_{5}\end{array} \right) W(q_3,\mbox{$\left| {\bf q}
\right|$};M), \label{eq:GASNEW}
\end{eqnarray}
where the index $j$ takes on values 1 and 2 only,
$\mbox{$\left| {\bf q} \right|$}=(q_1^2+q_2^2)^{\frac{1}{2}}$ and
$\mbox{$ \, {\bf q} \,$}^{\perp}=(-q_2,q_1)$.
The pseudoscalar and axi-pseudoscalar vertices are obtained
from the scalar and axi-scalar vertices by multiplication by
the matrix $\gamma_{45}$.

It is found that the same coupled integral
equations result when a vertex is multiplied by the matrix
$\gamma_{45}$ and so (scalar, pseudoscalar) and (axi-scalar,
axi-pseudoscalar) form two pairs of degenerate states.

The four equations derived from the BSE, after some manipulation
including an angular integration, are~\cite{Bu92}:
\begin{eqnarray}
f(p) & = & \frac{3}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \,
     \int^{\infty}_0 \mbox{$\left| {\bf q} \right|$} d\mbox{$\left|
     {\bf q} \right|$} \, \frac{1}{(\alpha^2-\beta^2)^{\frac{1}{2}}}
       \times\nonumber \\
 & &  (T_{ff}f(q)+T_{fU}U(q)+T_{fV}V(q)+T_{fW}W(q)) \nonumber \\
U(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \,
  \int^{\infty}_0 \mbox{$\left| {\bf q} \right|$} d\mbox{$\left|
  {\bf q} \right|$} \,
     \frac{(\alpha^2-\beta^2)^{\frac{1}{2}}-\alpha}
             {\beta(\alpha^2-\beta^2)^{\frac{1}{2}}}
       \times \nonumber \\
 & &   (T_{Uf}f(q)+T_{UU}U(q)+T_{UV}V(q)+T_{UW}W(q)) \nonumber \\
V(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \,
  \int^{\infty}_0 \mbox{$\left| {\bf q} \right|$} d\mbox{$\left|
  {\bf q} \right|$} \,
     \frac{(\alpha^2-\beta^2)^{\frac{1}{2}}-\alpha}
             {\beta(\alpha^2-\beta^2)^{\frac{1}{2}}}
       \times\nonumber \\
 & &   (T_{Vf}f(q)+T_{VU}U(q)+T_{VV}V(q)+T_{VW}W(q)) \nonumber \\
W(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \,
   \int^{\infty}_0 \mbox{$\left| {\bf q} \right|$} d\mbox{$\left|
   {\bf q} \right|$} \, \frac{1}{(\alpha^2-\beta^2)^{\frac{1}{2}}}
       \times\nonumber \\
 & &  (T_{Wf}f(q)+T_{WU}U(q)+T_{WV}V(q)+T_{WW}W(q)) ,    \label{eq:IE}
\end{eqnarray}
where
$$
\alpha=(p_3-q_3)^2 + \mbox{$\left| {\bf p} \right|$}^2 + \mbox{
$\left| {\bf q} \right|$}^2, \;\;\;\;\;\;
\beta=-2 \mbox{$\left| {\bf p} \right|$} \mbox{$\left| {\bf q} \right|$}.
$$
Now define the momentum Q by
\begin{equation}
Q^2=q_3^2 + \mbox{$\left| {\bf q} \right|$}^2 -\frac{1}{4}M^2
+ iMq_3 , \label{eq:QDEF}
\end{equation}
and use the abbreviations $\sigma_V=\sigma_V(Q^2)$ and
$\sigma_S=\sigma_S(Q^2)$ for use in the definition of the
functions $T_{ff},T_{fU},\ldots$ which are analytic
functions of $q_3, \mbox{$\left| {\bf q} \right|$}$, and $M$.
The diagonal $T$'s are given by
\begin{eqnarray}
T_{ff}&=&(\frac{1}{4}M^2 + q_3^{\,2} + \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2
  \mp |\sigma_S|^2, \nonumber \\
T_{UU}&=&(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2
  \pm |\sigma_S|^2, \nonumber \\
T_{VV}&=&(\frac{1}{4}M^2 + q_3^{\,2} + \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2
  \pm |\sigma_S|^2, \nonumber \\
T_{WW}&=&-(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2
  \pm |\sigma_S|^2,
\end{eqnarray}
where the upper sign applies to the scalar equations
and the lower sign to the axi-scalar equations.
The off-diagonal $T$'s are, for the scalar positronium states:
\begin{equation}
\begin{array}{ccccl}
T_{fU}&=&T_{Uf}&=&(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)
\mbox{$\left| {\bf q} \right|$}, \\
T_{fV}&=&T_{Vf}&=&-M\mbox{$\left| {\bf q} \right|$}|\sigma_V|^2, \nonumber \\
T_{fW}&=&T_{Wf}&=&-(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)q_3
           +\frac{i}{2}(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)M,  \\
T_{UV}&=&T_{VU}&=&-[\frac{1}{2}(\sigma_V^{\ast}\sigma_S+
                                     \sigma_S^{\ast}\sigma_V)M
                 +iq_3(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)],  \\
T_{UW}&=&T_{WU}&=&2q_3\mbox{$\left| {\bf q} \right|$}|\sigma_V|^2,  \\
T_{VW}&=&T_{WV}&=&-i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)
\mbox{$\left| {\bf q} \right|$},
\end{array}
\end{equation}
and for the axi-scalar states:
\begin{equation}
\begin{array}{ccccl}
T_{fU}&=&T_{Uf}&=&i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)
\mbox{$\left| {\bf q} \right|$},  \\
T_{fV}&=&-T_{Vf}&=&M\mbox{$\left| {\bf q} \right|$}|\sigma_V|^2,  \\
T_{fW}&=&-T_{Wf}&=&-i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)q_3
          -\frac{1}{2}(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)M,  \\
T_{UV}&=&T_{VU}&=&-
       [\frac{i}{2}(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)M
                 -q_3(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)],  \\
T_{UW}&=&-T_{WU}&=&-2q_3\mbox{$\left| {\bf q} \right|$}|\sigma_V|^2, \\
T_{VW}&=&-T_{WV}&=&(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)
\mbox{$\left| {\bf q} \right|$}.
\end{array}
\end{equation}
This is the same set of equations solved in Ref.~\cite{Bu92}
with only the fermion propagator input altered.  The bare fermion
mass $m$ only comes into the calculation through this input.

The solution to the BSE involves iteration of the coupled integral
equations in Eq.~(\ref{eq:IE}).  These equations may be rewritten as
\begin{equation}
{\bf f}(\mbox{$\left| {\bf p} \right|$},p_3;M) = \int dq_3
 \int d\mbox{$\left| {\bf q} \right|$} \, K(\mbox{$\left| {\bf p}
 \right|$},p_3;\mbox{$\left| {\bf q} \right|$},q_3;M) {\bf f}
 (\mbox{$\left| {\bf q} \right|$},q_3;M),
                      \label{eq:BSI}
\end{equation}
where ${\bf f}=(f,U,V,W)^{\rm T}$. For each symmetry case and each
fermion mass this is solved as an eigenvalue problem of the form
\begin{equation}
\int dq\,K(p,q;M) {\bf f}(q) = \Lambda(M) {\bf f}(p), \label{eq:EV}
\end{equation}
for a given test mass $M$.  This is repeated for different test
bound state masses until an eigenvalue $\Lambda(M)=1$ is obtained.

\setcounter{equation}{0}
\subsection{Non-Relativistic Limit}
We consider now the non-relativistic limit $m\rightarrow \infty$
of our BS formalism in order to enable comparisons with existing
numerical calculations \cite{YH91,THY95} of the Schr\"odinger
equation for QED3, and with the large $m$ solution of
Eq.~(\ref{eq:IE})

The Schr\"odinger equation with a confining logarithmic potential
is an interesting problem in its own right.  Initially
one is faced with the problem of setting the scale of the
potential, or equivalently, setting the zero of energy of the
confined bound states.
A solution to this problem was proposed by Sen~\cite{S90} and
Cornwall~\cite{C80} in terms of cancellation of infrared
divergences in perturbation theory.  They introduce a regulating
photon mass $\mu$ in order to set
the potential as the 2-dimensional Fourier transform of the photon
propagator $1/(k^2 + \mu^2)$, leading to a potential proportional to
$\ln (\mu r)$.  They further interpret the sum of the bare fermion mass and the fermion self energy evaluated at the bare fermion
mass shell as a renormalised fermion mass, leading to a mass
renormalisation $\delta m \propto \ln (m/\mu)$.  The logarithmic
divergences in the photon potential and the fermion self energy
then conspire to cancel leaving a finite positronium mass.

The first numerical treatment of the Schr\"odinger
equation for QED3 using this line of argument was carried out by
Yung and Hamer \cite{YH91}. In a subsequent, improved calculation by
Tam, Hamer and Yung \cite{THY95}, the formalism was shown to
be consistent with an analysis of QED3 from the point of view of discrete
light cone quantisation.  Their resulting expression for the bound
state energy, obtained as a solution to the differential equation
\begin{equation}
\left\{ -\frac{1}{m} \nabla^2 + \frac{1}{2\pi} \left(C + \ln(mr)
 \right)\right\}  \phi({\bf r}) = (E - 2m) \phi({\bf r}), \label{eq:THYDE}
\end{equation}
where $C$ is Euler's constant, is
given in terms of the bare fermion mass $m$ as
\begin{equation}
E  =  2m + \frac{1}{4\pi} \ln m  + \frac{1}{2\pi} \left(\lambda -
\frac{1}{2} \ln\frac{2}{\pi} \right).
       \label{eq:THY1}
\end{equation}
The lightest s-wave positronium state and first exited state are given by
$\lambda_0 = 1.7969$ and $\lambda_1 = 2.9316$ respectively.  The first five
states are provided in ref.~\cite{THY95}.

Here we present a treatment of the non-relativistic limit of the
QED3 positronium spectrum in terms of our SD--BS equation formalism.
We begin with the fermion propagator in the limit
$m\rightarrow \infty$.
For large fermion mass we expect the
residual effect of the chiral symmetry breaking contribution
to the fermion self energy to be small compared with contribution from the
perturbative loop expansion.  We shall therefore assume to begin with that
the self energy is reasonably well approximated by the one-loop result.
The validity of this approximation for spacelike momenta will be
demonstrated  numerically in the next section.

The 1-loop fermion self energy, with the functions $A$ and $B$
defined in Eq.~(\ref{eq:ABDEF}), is given by
\begin{equation}
A = 1 + \frac{\Sigma_A(\frac{p^2}{m^2})}{m},
\label{eq:A1LOOP}
\end{equation}
where
\begin{equation}
\Sigma_A(x^2)=\frac{1}{8\pi x^2} \left[ 1 - \frac{1-x^2}{2x}
   \arccos\left( \frac{1-x^2}{1+x^2} \right) \right],
\label{eq:SIGA1}
\end{equation}
and
\begin{equation}
B = m \left( 1 + \frac{\Sigma_B(\frac{p^2}{m^2})}{m} \right),
\label{eq:B1LOOP}
\end{equation}
where
\begin{equation}
\Sigma_B(x^2)=\frac{3}{8\pi x} \arccos\left( \frac{1-x^2}{1+x^2} \right) .
\label{eq:SIGB1}
\end{equation}
This result is valid for (Euclidean) spacelike momenta $p^2 > 0$.  An analytic
continuation of the $\Sigma$ functions valid for $|p|^2 < m^2$, or
$\left| x \right| <1$, is
\begin{equation}
\Sigma_A(x^2)=\frac{1}{8\pi i x} \left[ \frac{x^2-1}{2x^2}
   \ln\left( \frac{1+ix}{1-ix} \right) + \frac{i}{x} \right],
\end{equation}
\begin{equation}
\Sigma_B(x^2)=\frac{3}{8\pi i x} \ln\left( \frac{1+ix}{1-ix} \right) .
\end{equation}
Note that this representation exposes a logarithmic infinity in the
self energy at the bare fermion mass pole $p^2=-m^2$.
This is the
infrared divergence in the renormalised fermion self energy as defined by
Sen~\cite{S90} referred to above.  However, in our formalism, this
singularity does not lead to a pole in the propagator functions
$\sigma_V$ and $\sigma_S$ defined in Eq.~(\ref{eq:SIGDEF}),
which would signal the propagation of a free fermion~\cite{RW94},
but a logarithmic zero.

Using Eqs.~(\ref{eq:A1LOOP}) and (\ref{eq:B1LOOP}) we obtain
\begin{equation}
\sigma_V(p^2) = \frac{1}{m^2}\frac{1 + \frac{\Sigma_A}{m}}
     {\epsilon\left(1 + \frac{\Sigma_A}{m}\right)^2 +
     2\left(1 + \frac{\Sigma_A + \Sigma_B}{2m}\right)
                         \frac{\Sigma_B - \Sigma_A}{m}},
\end{equation}
\begin{equation}
\sigma_S(p^2) = \frac{1}{m}\frac{1 + \frac{\Sigma_B}{m}}
     {\epsilon\left(1 + \frac{\Sigma_A}{m}\right)^2 +
     2\left(1 + \frac{\Sigma_A + \Sigma_B}{2m}\right)
                         \frac{\Sigma_B - \Sigma_A}{m}},
\end{equation}
where we have defined
\begin{equation}
\epsilon = \frac{p^2 + m^2}{m^2}.
\end{equation}
The functions $\sigma_V$ and $\sigma_S$
are plotted in Figs.~2a and 2b for $m=$ 1, 2, 4, 8 and $\infty$,
the final curve being the bare propagator.  From these plots we see that
for large $m$, the deviation from the bare propagator due to the 1-loop
self energy is dominated by the logarithmic contribution near the bare
fermion mass shell, $\epsilon = 0$.  With this in mind, we shall use the
approximation
\begin{equation}
\Sigma_A(x^2) \approx -\frac{1}{8\pi} \ln\frac{\epsilon}{4},
\label{eq:SIGAP1}
\end{equation}
\begin{equation}
\Sigma_B(x^2) \approx -\frac{3}{8\pi} \ln\frac{\epsilon}{4}.
\label{eq:SIGAPX}
\end{equation}
Taking $\epsilon$ to be of order $1/m$ for the purposes of the BS equation
(see Eq.~(\ref{eq:EPSAPX}) below) these approximations give
\begin{equation}
S(p) = \frac{-i \! \not \! p + m}{m} \cdot \frac{1}{m\epsilon
                  - \frac{1}{2\pi} \ln \frac{\epsilon}{4}}
         \left(1 + O\left(\frac{\ln m}{m}\right)\right).  \label{eq:SAPX}
\end{equation}
The vector and scalar parts of this approximate propagator (without
the $O(\ln m /m)$ corrections) are also plotted in Figs.~2a and 2b for
comparison.

Turning now to the BS equation, we set the bound state momentum
in Eq.~(\ref{eq:BS}) equal to
$P_{\mu}=(2m+\delta)iv_{\mu}$, where $v_{\mu}=(0,0,1)$ and
$-\delta$ is a ``binding energy''.  This gives (according to the
momentum distribution in Fig.~1)
\begin{equation}
\Gamma(p) = - \int \frac{d^3q}{(2\pi)^3} D(p-q) \, \gamma_\mu \,
S\left[-(m+\frac{\delta}{2})iv_\mu+q_\mu\right]
       \Gamma(q) \, S\left[(m+\frac{\delta}{2})iv_\mu+q_\mu\right] \gamma_\mu.
\label{eq:NRBSE}
\end{equation}
Setting
\begin{equation}
\epsilon \, = \, \frac{2}{m}\left(-\frac{1}{2}\delta+iq_3+
       \frac{\left|{\bf q}\right|^2}{2m}\right) \, + \,
O\left(\frac{1}{m^2}\right)
\label{eq:EPSAPX}
\end{equation}
in Eq.~(\ref{eq:SAPX}) gives
\begin{equation}
S\left[(m+\frac{\delta}{2})iv_\mu+q_\mu \right]
 = \frac{1+\gamma_3}{2}\, \frac{1}{(\mbox{$ \,
 -\frac{1}{2} \,$} \delta+iq_3+\frac{\left|{\bf q}\right|^2}{2m})-
\frac{1}{4\pi} \ln \frac{\epsilon}{4}} \, + \, O\left(\frac{\ln m}{m}\right).
\label{eq:SPLUS}
\end{equation}
Similarly
\begin{equation}
S\left[-(m+\frac{\delta}{2})iv_\mu+q_\mu \right]
 = \frac{1-\gamma_3}{2}\, \frac{1}{(\mbox{$ \, -\frac{1}{2} \,$}
 \delta-iq_3+\frac{\left|{\bf q}\right|^2}{2m})-
\frac{1}{4\pi} \ln \frac{\epsilon^*}{4}} \, + \, O\left(\frac{\ln m}{m}\right).
\label{eq:SMINUS}
\end{equation}
The $\left|{\bf q}\right|^2/2m$ term has been retained here to ensure
convergence of the $\left|{\bf q}\right|$ integral in the BS equation below.
Since the vertex $\Gamma$ is defined with the fermion legs
truncated, and $S \propto \frac{1}{2}(1 \pm \gamma_3)$, the only
relevant part of $\Gamma$ is the projection
$\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \, \Gamma \,
\mbox{$ \, \frac{1}{2} \,$}(1+\gamma_3)$.
With this in mind,
the general forms in eqs.~(\ref{eq:GSNEW}) and (\ref{eq:GASNEW})
become,
$$
\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \, \Gamma^S \,
\mbox{$ \, \frac{1}{2} \,$}(1+\gamma_3)  =
\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \frac{q_j
\gamma_j}{\mbox{$\left| {\bf q} \right|$}}  g(q_3,\mbox{$\left|
{\bf q} \right|$})
$$
\begin{equation}
\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \, \Gamma^{AS} \,
\mbox{$ \, \frac{1}{2} \,$}(1+\gamma_3)  =
\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \left(\begin{array}{c}
\gamma_4 \\
 \gamma_5\end{array} \right)  g(q_3,\mbox{$\left| {\bf q} \right|$}).
\label{eq:SASEQ}
\end{equation}

Substituting eqs.~(\ref{eq:SPLUS}), (\ref{eq:SMINUS}) and
(\ref{eq:SASEQ}) into Eq.~(\ref{eq:NRBSE}) one obtains for
the scalar states the single integral equation
\begin{equation}
g(p) \approx\int \frac{d^3q}{(2\pi)^3} \,
     \frac{1}{(p - q)^2}
     \frac{\bf{p.q}}{\left|{\bf p}\right|\left|{\bf q}\right|}
     \frac{g(q)}{ \left|-\frac{1}{2}\delta+
     iq_3 + \frac{\left|{\bf q}\right|^2}{2m} -
     \frac{1}{4\pi} \ln \frac{1}{2m}
     (-\frac{1}{2}\delta+iq_3+\frac{\left|{\bf q}\right|^2}{2m})
                        \right|^2 },
\label{eq:NREQ1}
\end{equation}
and for the axi-scalar states the single equation
\begin{equation}
g(p) \approx \int \frac{d^3q}{(2\pi)^3}\,
     \frac{1}{(p - q)^2}
     \frac{g(q)}{ \left|-\frac{1}{2}\delta+
     iq_3 + \frac{\left|{\bf q}\right|^2}{2m} -
     \frac{1}{4\pi} \ln \frac{1}{2m}
     (-\frac{1}{2}\delta+iq_3+\frac{\left|{\bf q}\right|^2}{2m})
                      \right|^2 }.
\label{eq:NREQ1A}
\end{equation}
Note that, without the $\left|{\bf q}\right|^2/2m$ term in the
denominator, translation
invariance of the integrand implies that $g$ is independent of
$\left|{\bf q}\right|$.  In reality, $g$ is a slowly varying function of
$\left|{\bf q}\right|$, and this extra $O(1/m^2)$ term must be retained
in $\epsilon$ to account for the fact that the relevant region of
integration in Eqs.~(\ref{eq:NREQ1}) and (\ref{eq:NREQ1A}) extends out to
$O(\sqrt{m})$ in the $\left|{\bf q}\right|$ direction, but only $O(1)$ in
the $q_3$ direction.  Numerical solutions of Eqs.~(\ref{eq:NREQ1}) and
(\ref{eq:NREQ1A}) will be given in section 5.
The function $g$ is an even or odd function of $q_3$ corresponding
to positronium states which are even or odd respectively under
charge conjugation.

In order to obtain a Schr\"{o}dinger equation, we now rewrite the
axi-scalar equation in the form
\begin{eqnarray}
g(p) & = & \int \frac{d^2{\bf }q}{(2\pi)^2} \int_{-\infty}^\infty
    \frac{dq_3}{2\pi}\,
     \frac{1}{(p_3 - q_3)^2 + \left|{\bf p - q}\right|^2}
     \times \nonumber \\
   & &   \!\!\!\!\! \frac{g(q)}{ \left(-\frac{1}{2}\delta+
     iq_3 + \frac{\left|{\bf q}\right|^2}{2m}
     + \Sigma_+(q_3,\left|{\bf q}\right|)\right)
     \left(-\frac{1}{2}\delta -
     iq_3 + \frac{\left|{\bf q}\right|^2}{2m}
     + \Sigma_-(q_3,\left|{\bf q}\right|)\right)},
\label{eq:NREQ2}
\end{eqnarray}
where
\begin{equation}
\Sigma_\pm(q_3,\left|{\bf q}\right|) =
     - \frac{1}{4\pi} \ln \frac{1}{2m}
     \left(-\frac{1}{2}\delta \pm iq_3
           +\frac{\left|{\bf q}\right|^2}{2m}\right).  \label{eq:SIGPM}
\end{equation}
Assuming the integrand dies off sufficiently rapidly as
$q_3 \rightarrow -i\infty$, we deform the contour of integration around
the pole at
\begin{equation}
q_3^{\rm pole} = -i \left(-\frac{1}{2}\delta +
          \frac{\left|{\bf q}\right|^2}{2m}
              + \Sigma_-(q_3^{\rm pole},\left|{\bf q}\right|) \right),
                   \label{eq:Q3POLE}
\end{equation}
to obtain
\begin{equation}
g(p) = \int \frac{d^2{\bf q}}{(2\pi)^2} \,
     \frac{1}{(p_3 - q_3^{\rm pole})^2 + \left|{\bf p - q}\right|^2}
     .\frac{g(q_3^{\rm pole},\left|{\bf q}\right|)}
      {-\delta + \frac{\left|{\bf q}\right|^2}{m}
     + 2\Re\Sigma_-(q_3^{\rm pole},\left|{\bf q}\right|)}.
\label{eq:NREQ3}
\end{equation}
(We could equally well deform the contour round the pole at
$(q_3^{\rm pole})^*$ if the integrand decays in the opposite direction,
without affecting our final result.)

Defining
\begin{equation}
\Phi(p_3,\left|{\bf p}\right|) =  \frac{g(p_3,\left|{\bf p}\right|)}
      {-\delta + \frac{\left|{\bf p}\right|^2}{m}
     + 2\Re\Sigma_-(p_3,\left|{\bf p}\right|)},
\end{equation}
gives
\begin{eqnarray}
\lefteqn{\left\{-\delta + \frac{\left|{\bf p}\right|^2}{m}
     + 2\Re\Sigma_-(p_3,\left|{\bf p}\right|)\right\}
        \Phi(p_3,\left|{\bf p}\right|) } \hspace{30 mm}\nonumber \\
 & = & \int \frac{d^2{\bf q}}{(2\pi)^2} \,
     \frac{1}{(p_3 - q_3^{\rm pole})^2 + \left|{\bf p - q}\right|^2}
        \Phi(q_3^{\bf pole},\left|{\bf q}\right|). \label{eq:NRPHI1}
\end{eqnarray}
In order to isolate the logarthmic infrared divergence we set
\begin{equation}
p_3 = p_3^{\rm pole} + \mu,   \label{eq:P3MU}
\end{equation}
with $\mu$ small and real, and $p_3^{\rm pole}$ defined by analogy with
Eq.~(\ref{eq:Q3POLE}).  The right hand side of Eq.~(\ref{eq:NRPHI1})
then becomes
\begin{eqnarray}
\mbox{r.h.s.} & = & \int \frac{d^2{\bf q}}{(2\pi)^2} \,
     \frac{1}{\mu^2 + O(\mu(\left|{\bf p}\right| - \left|{\bf q}\right|))
              + \left|{\bf p - q}\right|^2} \phi({\bf q}) \nonumber \\
 & = & \mbox{F.T. of } \frac{-1}{2\pi} \left[C +
        \ln\left(\frac{\mu r}{2}\right)\right] \phi({\bf r})
         \hspace{5 mm} \mbox{as $\mu \rightarrow 0$}, \label{eq:RHS}
\end{eqnarray}
where $\phi({\bf q}) = \Phi(q_3^{\bf pole},\left|{\bf q}\right|)$.

Following the reasoning of refs.~\cite{S90} and \cite{C80}, this
logarithmic divergence should be cancelled by the fermion self energy
contribution $2\Re\Sigma_-(p_3,\left|{\bf p}\right|)$.  However,
from Eq.~(\ref{eq:SIGPM}), we see that the logarithmic divergence in
the self energy occurs at the bare fermion mass pole
$p_3^{\rm bare} = -i (-\delta/2 + \left|{\bf p}\right|^2/2m)$, and not
the dressed pole $p_3^{\bf pole}$.  The problem lies in the use of the
1-loop approximation.  If instead the fermion self energy is calculated
to all orders in rainbow approximation, the self energy feeds back into
the loop integral via the propagator to replace Eq.~(\ref{eq:SIGPM}) by
\begin{equation}
\Sigma_-(p_3,\left|{\bf p}\right|) =
     - \frac{1}{4\pi} \ln \frac{1}{2m}
     \left(-\frac{1}{2}\delta - ip_3
           +\frac{\left|{\bf p}\right|^2}{2m} +
  \Sigma_-(p_3,\left|{\bf p}\right|)\right),  \label{eq:SIGM}
\end{equation}
which provides a rainbow SD equation for $\Sigma_-$
in the non-relativistic limit.
Then using Eqs.~(\ref{eq:NRPHI1}), (\ref{eq:Q3POLE}), (\ref{eq:P3MU}),
(\ref{eq:RHS}) and (\ref{eq:SIGM}) and fourier transforming we finally
obtain
\begin{equation}
\left\{ -\frac{1}{m} \nabla^2 + \frac{1}{2\pi} \left(C + \ln(mr)
 \right)\right\}  \phi({\bf r}) = \delta \phi({\bf r}),
\end{equation}
agreeing with Eq.~(\ref{eq:THYDE}).
Had we started from the scalar equation (\ref{eq:NREQ1}) in place of the
axi-scalar equation, the same result would have been obtained at
Eq.~(\ref{eq:RHS}), leading to an identical Schr\"{o}dinger equation.

The important point to notice in this derivation is the significance
of a non-perturbative solution to the SD equation in cancelling the
infrared divergences.  In the massless fermion limit, it is well known
that chiral symmetry breaking plays a pivotal role in determining the
bound state spectrum.  It appears also that, even in the
non-relativistic limit, the remnant effects of chiral symmetry
breaking, via a non-perturbative solution to the SD equation, have a
role to play.

\setcounter{equation}{0}
\subsection{The Fermion Propagator}
The BSE described in Section 2 requires a fermion
propagator input in the form of Eq.~(\ref{eq:SIGDEF}) or
Eq.~(\ref{eq:ABDEF}) and this
needs to be available over a region in the complex $p^2$ plane
defined by Eq.~(\ref{eq:QDEF}) for $q_3$ and $\mbox{$\left| {\bf q}
\right|$}$ real.  This
is the region~\cite{SC92,SC94}
\begin{equation}
\Omega = \left\{ Q^2 = X + iY \left| X > \frac{Y^2}{M^2}
- \frac{1}{4} M^2 \right. \right\}.
\label{eq:REGION}
\end{equation}
In this section we investigate ways of obtaining a solution to the
SDE over $\Omega$.  The fermion propagator, and thus the functions
$\sigma_V$ and $\sigma_S$, must be well behaved over this region.

The solution to the SDE~(\ref{eq:SD}) is quite simple
along the positive real (spacelike)-$p^2$ axis. Substitution of the
general expression for the fermion propagator~(\ref{eq:ABDEF})
into the SDE
gives an integral equation involving $A$ and $B$ functions
which can be split into two coupled integral equations by
simple projections.  Angular integrals can be performed to
leave one dimensional integrals (over the modulus of the $q$
vector),
$$
A(p^2) - 1 = \frac{1}{4\pi^2p^2} \int^{\infty}_{0}dq
     \frac{q A(q^2)}{q^2 A^2(q^2)+B^2(q^2)}
     \left({\frac{p^2+q^2}{4 p}
     \ln\left({\frac{p+q}{p-q}}\right)^2 - q}\right),
$$

\begin{equation}
B(p^2) - m = \frac{3}{8\pi^2p} \int^{\infty}_{0}dq
     \frac{q B(q^2)}{q^2 A^2(q^2)+B^2(q^2)}
     \ln\left({\frac{p+q}{p-q}}\right)^2
      \label{eq:ABINTS}
\end{equation}
The integrations range from 0 to some UV cutoff along
the positive real axis.  This theory is
super-renormalisable and thus has no ultraviolet divergences and
so this cutoff is merely a numerical limit made large enough so
that it has no bearing on the results.

For a set of points $p$ corresponding
to the set of $q$ points in the integration, the equations
are iterated until convergence to leave the solution along the
positive real axis.  However, the solution is required for complex
$p^2$.  We see three possibilities. The first is to use the
converged functions $A(q^2)$ and $B(q^2)$ in the integrals over the
same contour (positive real $q^2$) and
supply the complex point $p$ desired.  The integrals should
provide the solution at that point $p$.  However the analytic structure
of the integrands in Eq.~(\ref{eq:ABINTS}) will not allow an analytic
continuation by this method, because a pinch singularity in the
integrand forces us to integrate through the point $p$~\cite{Ma95}.

The second possibility is to rotate the contour through an angle
$2\phi$ in the $p^2$ plane so that it passes
through the desired point $p$~\cite{SC92}. In this way a cancellation of the
complex parts within the logarithms occurs.  Fig.~3 shows the
first and second contours ($C_1$ and $C_2$ respectively).
It can be seen from Eq.~(\ref{eq:ABINTS}) that the
logarithms will have real arguments along the radial portion of $C_2$, while
the arc portion contributes nothing to the integral because the integrand
falls off sufficiently quickly in the ultraviolet~\cite{Ma95}.

Based on the Landau gauge calculations of Maris~\cite{Ma93,Ma95} we
expect conjugate singularities to occur
in the second and third quadrants of the $p^2$ plane away from
the negative real (timelike) axis. Thus, as $2\phi$ increases towards $\pi$
from zero (and the negative real $p^2$ axis is approached)
a singularity interferes and we may have
convergence problems.
It will be seen that these singularities can lie a fair way
from $\phi=\frac{\pi}{2}$ and
convergence problems can occur for $\phi$ not much more
than $\frac{\pi}{4}$ (ie: barely reaching into the second
quadrant of $p^2$).
For the case $m=0$ to be considered shortly,
no solution could be found for $\phi$ greater than
0.90 radians (with a reasonable convergence criterion
$\frac{\Delta B}{B}<0.001$).
For this solution to be applied to the BSE we need to know
the value of the fermion
propagator for $\phi$ from $0$ to $\frac{\pi}{2}$ and so this
method is not practical.
However, although a slowing of convergence as $\phi$ increases prevents
a solution being attained in all of $\Omega$, it does provide an
accurate solution in a large portion of $\Omega$.
We therefore
have a test for any $A$ and $B$ functions we wish to use in the
BSE.

The third possibility, and the one employed here and in ref.~\cite{Bu92},
is to find a good analytic fit along the positive real $p^2$
axis and extend the
solution into the complex plane by analytic continuation.
These fits may be for $A$ and $B$ or for the functions
$\sigma_V$ and $\sigma_S$.  The work of Maris~\cite{Ma95} suggests
that it is not necessary for $\sigma_V$ and $\sigma_S$ to be
entire functions for the fermions to be confined, only that there
be no poles on the timelike $p^2$ axis.
Fits to functions $A$ and $B$ used in previous work~\cite{Bu92}
based on the known asymptotic infrared and ultraviolet
behaviour of these functions were
tested by comparing them with the direct solution for
various angles $\phi$.
The fits, adjusted to allow variable fermion mass $m$, are given by
$$
A_{\rm fit}(p^2)= \frac{a_1}{(a_2^{\,2} +p^2)^{\frac{1}{2}}}
             +a_3 e^{-a_4p^2} +1,
$$
\begin{equation}
B_{\rm fit}(p^2)= \frac{b_1}{b_2 +p^2}+b_3 e^{-b_4p^2} + m.  \label{eq:FIT}
\end{equation}
The parameters $a_n$,$b_n$ are functions of fermion mass.
The numerical solution to which these functions were
fitted is an
iterative solution to the SDE using a non-uniform 51 point grid
along the positive real axis up to a momentum cutoff
$p = 1000$ using a $0.1\%$ tolerance in the
integration routine.
Plots of the numerical solutions and function fits for various
$m$ values are given in Fig.~4.

Note that it is the $\sigma$ functions that are important in
Eq.~(\ref{eq:BS}) and
not $A$ and $B$, and thus the effect of the fit on the denominator
$p^2 A^2 + B^2$ relating these must be considered.
Conjugate poles exist where the factor $p^2 A^2 + B^2$
appearing in the denominator of the BSE integrand is zero.
Table 1 lists the conjugate poles arising from the fits for each fermion mass
and the corresponding maximum bound state masses allowed.  The
maximum $M$ allowed is the value for which the boundary of
$\Omega$ in Eq.~(\ref{eq:REGION}) coincides with the
conjugate poles.
No comment about the viability of our model BSE can be made
until solutions are attempted because the integration
region depends on the solution mass $M$.

The location of the conjugate singularities for the $m=0$
case in Table 1 is slightly different to that reported in
Ref.~\cite{Bu92} where it is $-0.00400 \pm i0.00666$.
This is because of the flexibility of the fitting functions.
The fit in this work and that in Ref.~\cite{Bu92} for the zero
fermion mass case had similar accuracy along the positive real
$p^2$ axis but had the freedom to take on slightly different
forms throughout the complex plane.  This is because along the
positive real $p^2$ axis the non-asymptotic form fixing
parameters ($a_4$ and $b_4$) are only
loosely determined.  Despite the difference in the two
results, the BSE calculation for bound state masses should show close
agreement as each fit adequately models the direct solution
throughout the complex plane.

The singularities in the $\sigma$ fits
for fermion masses greater
than or equal to $0.1$ lie on the negative real $p^2$-axis.
This suggests
that free propagation occurs at these masses and the model
is not confining.  An accurate location of the singularities in
the SDE solution would be needed before it can be said
whether this result is due to the fits or the rainbow
approximation used in the SDE solution.
According to Ref.~\cite{Ma95} the rainbow approximation
SDE solution is expected to be confining even for
large fermion mass.  Thus we assume our result is due to the
lack of accuracy in our fits near the negative real
$p^2$ axis
and that it is likely that the singularities move close
to that axis as $m$ increases but never actually lie on that
axis.

Figs.~5a and 5b show plots of $\sigma_V$ and $\sigma_S$ moduli
respectively for zero fermion mass and angles $\phi=0$,
$\phi=\frac{\pi}{8}$ and $\phi=\frac{\pi}{4}$ against the
$p$ modulus (with a range far smaller than the UV cutoff
used in our calculations).
The direct solutions to the SDE and the fits are compared.  It can
be seen that the functions are very good fits along the positive
real axis ($\phi=0$), where both $\sigma_V$ and $\sigma_S$ are real.
The fit is also good for $\phi=\frac{\pi}{8}$.  Real and imaginary
components have not been given separately as they show similar
agreement.  In the case $\phi=\frac{\pi}{4}$ the fitting function
has begun to deviate from the SDE solution.  This is mostly due
to the apparent difference in the location of a spike.
Based on the largest bound state mass for $m=0$ reported
in the next section, the BSE integration region $\Omega$ extends
along the
direction $\phi=\frac{\pi}{4}$ out to a modulus of approximately $0.083$.
In this range the small angle solutions are very accurate but
for larger $\phi$, much of the error due to the difference
in the location
of the spike will be experienced.
As the angle is increased further
convergence problems occur until eventually no solution can be
found at all ($\phi>0.90$).

The spike forming in these plots signals that, as $\phi$
is increased, the contour of integration approaches a singularity.
In fact, the conjugate poles which lie just off (or on as is the
case for larger $m$) the negative real $p^2$ axis
($\phi=\frac{\pi}{2}$) are approached.
It is important that both the direct solution
to the SDE and the fits used in this work have this feature.
This spike was not seen in any other fits which we attempted.
Based on Fig.~5 it seems clear that
the direct solution to the SDE must have singularities
close to those in the fitting functions.

Because the spikes are not in
exactly the same places some error will be introduced
in the contributions from the large $\phi$ part of $\Omega$.
When the bound state
mass becomes large, the large $\phi$ contributions will become
more important and thus we expect the error in the position of
the spikes to result in some noise in the solutions to the
BS equation for large fermion mass.

The $\sigma$ functions were studied for all fermion masses used
in this work in the same fashion.  The results were similar
to the $m=0$ case and need not be shown here.  In each case,
when $\phi$ was increased
far enough, a spike was observed in both the fit and solution,
after which lack of convergence prevented an SDE solution.

However, for very large fermion masses, the accuracy of the fits
decreases as $m$ increases, and with good reason.
As $m$ tends to infinity, the functions
$A$ and $B$ approach constants ($1$ and $m$ respectively).  For
moderately large fermion masses experienced in this work, these
functions become almost constant along the positive real $p^2$
axis while having a singularity near the negative real $p^2$-axis.
It is too much to ask for simple four parameter fits along
the positive real $p^2$-axis to reproduce accurately complex behaviour
deep into the real timelike $p^2$-axis.  The 1-loop propagators,
Fig.~2 described in section 3 illustrate this well.  There
one can see how smooth and level the $\sigma$ functions are
along the positive real $p^2$ axis and also how steep the functions
become back along the negative real $p^2$ axis.

Before moving on to the next section, we return briefly to the
1-loop approximation to the fermion propagator necessary
for the non-relativistic approximations described in section 3.
Fig.~6a compares our rainbow approximation solution $A$
to the 1-loop result given in Eqs.~(\ref{eq:A1LOOP})
and (\ref{eq:SIGA1}).
Fig.~6b compares $B$ from our rainbow approximation
solution and the result in Eqs.~(\ref{eq:B1LOOP}) and
(\ref{eq:SIGB1}).
Both of these comparisons were made at a large fermion mass
($m=5$).  It can be seen
that the curves in each case are in reasonable agreement, at least
for spacelike momenta.

\setcounter{equation}{0}
\subsection{Numerical Solution of the Bethe-Salpeter Equation}
The fits given by Eq.~(\ref{eq:FIT}) to the fermion propagator for a range
of fermion masses were used in the solution of the
Bethe-Salpeter coupled integral equations Eq.~(\ref{eq:IE}).  This
problem was restated in Eq.~(\ref{eq:EV}) as an eigenvalue problem.

A grid of $25 \times 25$ ($\mbox{$\left| {\bf q} \right|$}$,$q_3$)
tiles were used for the
iterative procedure with linear interpolation on each of those
tiles used for the sums ($T_{ij}f_j$) which are
supplied at the corners of the tiles from the previous
iteration.  The tiles were non-uniform in size and an upper limit
to the momentum components ($\mbox{$\left| {\bf q} \right|$}$ and $q_3$) of
between $3.0$ and $9.0$ was
used.  The equations were
iterated to convergence each time
to determine eigenvalues for a given test bound state mass $M$.
The Bound state masses were located by repetitive linear
interpolation or extrapolation to search for the point where the
eigenvalue $\Lambda$ of Eq.~(\ref{eq:EV}) is 1.
This was repeated for each of the fermion masses
ranging from 0 to 5.0.  This procedure was used for each of the
four non-degenerate bound state symmetries described in the appendix.

Table 2 shows the bound state masses for each of the four symmetries
(scalar ${\cal C}=+1$, scalar ${\cal C}=-1$, axi-scalar
${\cal C}=+1$ and
axi-scalar ${\cal C}=-1$) for all fermion masses considered.  Fig.~7a
displays the solutions $M$ for fermion mass 0--0.1.  Fig.~7b
shows $M-2m$ over the greater range of 0--5.
The axi-scalar ${\cal C}=+1$ solution is a degenerate
axi-scalar/axi-pseudoscalar pair of Goldstone bosons for the case
$m=0$, as seen in previous work~\cite{Bu92}.  Minor differences
between Ref.~\cite{Bu92} and the current work at $m=0$
are due to small differences in the propagator fits, as
explained in section 4.

For small $m$ the bound state masses rise rapidly with
with increasing fermion mass.  The mass of the ``Goldstone''
axi-scalar ${\cal C} = +1$ state scales roughly with the square
root of the fermion mass, in agreement with the Gell-Mann--Okubo mass formula~\cite{FS82}.  In fact, for fermion masses 0 to 0.1
a linear regression against $\sqrt{m}$ has correlation coefficient
0.9964 with the mass growing as approximately
$1.27 \times \sqrt{m}$.
(The accuracy of the solution at $m = 0.001$, which comes out with
an anomalously low bound state mass, is severely affected by
numerical inaccuracy arising from the sensitivity the bound state
mass to the eigenvalue $\Lambda$ in Eq.~(\ref{eq:EV}).)  For large
fermion masses, the bound state mass rises predominantly as twice
the fermion mass plus possible logarithmic corrections.  However,
there appears to be a good deal of noise in the large $m$ solutions,
reflecting the difficulty in accurately modelling the fermion
propagator deep into the timelike region from spacelike fits.
No solutions corresponding to states of negative charge parity were
found for $m>1.0$.

Numerical solutions to the integral equations (\ref{eq:NREQ1}),
(\ref{eq:NREQ1A})
arising from our non-relativistic treatment are listed in Table~3
and plotted in Fig.~7b.  Solutions with positive
$\delta$ were found for fermion masses $m \geq 1.0$ in the positive
charge parity sector.  We were unable to locate any solutions to
Eqs.~(\ref{eq:NREQ1}),~(\ref{eq:NREQ1A}) corresponding to
negative charge parity states
over a broad range of $\delta$.  Also given in Table~3 and
Fig.~7b are the two lowest lying s-wave solutions to the
Schr\"{o}dinger equation from the numerical work of
Tam et al.~\cite{THY95}, given by Eq.~(\ref{eq:THY1}).

The lack of exact agreement between the non-relativistic, 1-loop
approximations Eqs.~(\ref{eq:NREQ1}) and (\ref{eq:NREQ1A}), and
the Schr\"{o}dinger equation result Eq.~(\ref{eq:THY1}) is to be
expected.  As pointed out in Section~3, a complete cancellation of
infrared divergences can only occur if the fermion self energy is
calculated non-perturbatively to all orders.  From Table~3, we see that
at very high fermion masses, the accuracy of the 1-loop approximation
is significantly affected as the conjugate poles in the propagator,
measured in momenta scaled by the fermion mass, move closer to the
bare fermion mass pole (see Fig.~2).  At more moderate fermion masses,
$m\approx 5$, the 1-loop approximation is more respectable.

We see no clear agreement between the numerical
results of Eq.~(\ref{eq:IE}), and either non-relativistic approximation
Eqs.~(\ref{eq:NREQ1}),~(\ref{eq:NREQ1A}), or the Schr\"{o}dinger
equation result Eq.~(\ref{eq:THY1}).  Our analysis of the
non-relativistic limit of the BS equation exposes the importance
of the analytic structure of the fermion propagator in the vicinity
of the bare fermion mass pole $p^2 = -m^2$.  The uneven nature of
the lower two curves in Fig.~7b indicates that the determination
of the timelike fermion propagator by an analytic fit to the spacelike
propagator is inadequate for fermion masses $m \geq 1$.
It is clear that a more careful analysis of the
timelike nature of the fermion propagator, possibly involving a
fully non-perturbative treatment of the SD equation to include
remnant chiral symmetry breaking, is necessary for determining the
bound state spectrum for even moderately large fermion masses.

It is important to note that the poles in the
fermion propagator fits listed in Table~1 lie outside the BS
integration region $\Omega$ for all
solutions obtained.  This can be verified by observing that all
masses in Table~2 are lower than the values $M_{max}$ listed in Table~1.
A similar situation arises for the non-relativistic limit calculations.
Listed in Table~3 are maximum allowed $\delta$ values if the integration
region sampled by Eqs.~(\ref{eq:NREQ1}) and (\ref{eq:NREQ1A}) is not to
impinge on the conjugate propagator poles $q_3^{\rm pole}$ and
$(q_3^{\rm pole})^*$ defined in Eq. (\ref{eq:Q3POLE}).  In all cases the
numerical results lie within the permitted region.  This requirement is
equivalent to demanding that $q_3^{\rm pole}$ should not cross the real
$q_3$ axis as as $\left|{\bf q}\right|$ ranges from $0$ to $\infty$.
Interestingly, such a crossing would entail a more careful evaluation
of residues than that carried out in Section~3 leading to the
Schr\"{o}dinger equation.

We note that the Schr\"{o}dinger equation results of
Ref.~\cite{THY95} include the first five s-wave states.
It would certainly be of interest to locate the excited states within the
framework of our BS treatment of QED3.
We have searched for solutions to the eigenvalue equation
(\ref{eq:EV})
corresponding to excited states, and find in general no solutions
within the mass ranges allowed by the values $M_{max}$ in Table 1.
Since there is no reason to assume that the s-wave spectrum should
be bounded above, it seems likely that there will be solutions to
the BS equation for which the region of integration $\Omega$ does
include the conjugate propagator poles discussed in Section~4.  It
follows that the functions $f$, $U$, $V$ and $W$ in the BS
amplitudes of these states
should have compensating zeros, in order that the right hand side of
the BS equation be integrable.  We conjecture that, if the fermion
propagator has an infinite set of poles, there will be a sequence of
excited states, the $n$th excited state having $n$ pairs of zeros in
its BS amplitude.  This conjecture is consistent with the the first
excited state of the Schr\"{o}dinger equation, also listed in Table~3,
for which the wave function has a single zero.

Although we are unable to determine accurately the spectrum in the
large fermion mass limit, our calculations strongly suggest that
there are no scalar or axi-scalar states with negative charge parity
in this limit.
This is consistent with the non-relativistic quark model in four
dimensions in which negative charge parity scalar and pseudoscalar states are forbidden
by the generalised Pauli exclusion principle~\cite{FS82}.  We note,
however, that there is nothing to exclude such states in a fully
relativistic BS treatment~\cite{LS69}, and indeed, negative charge
parity scalar and axi-scalar states are found within the current
model for light fermions.

\setcounter{equation}{0}
\subsection{Conclusions}

In this paper we have solved the combination of rainbow
Schwinger-Dyson and homogeneous Bethe-Salpeter
equations in the quenched ladder approximation for three
dimensional QED with massive fermions.  QED3
was chosen because, like QCD, it is confining but
without the complications of being non-abelian.
A four-component version of this theory is used because, also like
QCD, it provides a parity invariant action with a spontaneously
broken chiral-like symmetry in the massless limit.  The
approximation is amenable to numerical solution, and should help
assess the limitations
of a technique frequently employed in models of QCD~\cite{BSpapers}.

The work in this paper carries on from a previous study of the same
subject~\cite{Bu92}, but with the following extensions.  Firstly,
non-zero fermion masses is considered.  Secondly, an analysis of
the fermion propagator in the complex plane is carried out in order
to assess the appropriateness of the approximations involved.
Thirdly, an analysis of the non-relativistic limit, i.e., large
bare fermion mass, is made in an attempt to compare with existing
Schr\"{o}dinger equation studies of QED3.

The rainbow SD equation was solved in Euclidean space to give
a fermion propagator for spacelike momenta, Euclidean $p^2 > 0$.
The propagator is chirally asymmetric, and in the massless
fermion limit, gives rise to a doublet of massless Goldstone positronium
states analogous to the pion.
Solution of the BS equation for massive positronium states
requires knowledge of the fermion propagator $S(p)$ in the
complex $p^2$-plane extending away from the spacelike axis,
and a finite distance into the timelike axis $p^2 < 0$.  By
rotating the contour of integration we were able to extend the
spacelike solution into part of the complex plane.  However,
the occurrence of complex conjugate poles in
the fermion propagator prevented a numerical solution to the
SD equation throughout the complete region of the complex plane sampled
by the BS equation.  This forced us to apply analytic fits to the
propagator along the positive real $p^2$-axis for use over
the required part of the complex plane.

Our propagator fits were found to have conjugate
poles located close to those of the direct solution for small to
moderate fermion masses.  This,
combined with the accuracy of the fits throughout much the complex
$p^2$ plane, made our choice of propagator very attractive.
The singularities in the fits were found to move
onto the negative real
$p^2$-axis as the fermion mass increased.  This was not
interpreted as a loss of confinement but instead attributed to
a lack of accuracy in the fits deep into the timelike region as
the fermion mass became large.
This reduction in accuracy of the fits for large $m$ was
due to the nature of the functions along the
positive real $p^2$-axis where the fits were made, and the presence
of a singularity near the negative real $p^2$-axis in the vicinity
of the bare fermion mass pole $p^2 = -m^2$, but off the timelike
axis.

BS solutions were found for four pairs of parity degenerate states.
These pairs were the scalar/pseudoscalar ${\cal C}=+1$ and
${\cal C}=-1$ and the
axi-scalar/axi-pseudoscalar ${\cal C}=+1$ and ${\cal C}=-1$ states.
For small to
moderate fermion mass the bound state mass was found to increase
smoothly with $m$.  The axi-scalar {\cal C}=+1 doublet, analogous to
the pion, was the lowest in energy, with a mass rising roughly with
the square root of the bare fermion mass.  For moderately large bare
fermion masses ($m/e^2$ greater than unity) the positronium masses
rise as twice the bare fermion mass, plus a possible logarithmic
correction.  However, an unacceptable level of noise was found to
develop in our results for these larger masses, which we attribute
to inaccuracies in the analytically
continued fermion propagators in the important region near the bare
fermion mass pole.  No negative charge parity
(${\cal C}=-1$) solutions were found for bare fermion masses above
$m/e^2 \approx 1.0$, consistent with the generalised Pauli exclusion
principle of non-relativistic QCD4.

The conjugate poles in the fermion propagators were found
to keep clear of the integration regions required for the BS
solutions
for the lowest state in each of the four space parity/charge parity
sectors considered.  However, it appeared that this would not be so
for any excited states.  We therefore conjecture that the excited
positronium states have zeros in their BS amplitudes positioned so
as to cancel the poles in the propagators encountered within the
integral in the BS equation (\ref{eq:BS}).  This requirement
of compensating zeros was too demanding on our current numerical code,
and as a result, no excited states were found.

In vector calculations under way at present,
where the bound state masses are expected to be
larger, the conjugate poles in the fermion propagator
seen in this work may interfere.  Since the fits used
in this work appear to have their
singularities close to those in the actual Schwinger-Dyson solution,
we may find that the rainbow approximation and the resulting
propagator fits will be inadequate for a
study of vector states in QED3.  This is a very challenging problem
and we hope to report on our results in the near future.

A non-relativistic analysis of the BS equation was also carried out
assuming, in the first instance, a 1-loop approximation to the
fermion propagator.  However it was shown that, in order to cancel
infrared divergences completely between the photon propagator and
fermion self energy, as proposed by Sen~\cite{S90} and Cornwall~\cite{C80},
it is necessary to evaluate the fermion self
energy non-perturbatively.  Only if this is done can the Schr\"{o}dinger
equation be rigorously obtained in the large fermion mass limit.
In spite of this, numerical solutions of the 1-loop equations
give reasonable agreement with the Schr\"{o}dinger equation for
moderately large fermion masses $m/e^2 \approx 5$.

In summary, we were able to carry out an acceptable analysis of the bound
state spectrum of QED3 near the chiral limit $m \rightarrow 0$ by using
analytic fits to the spacelike fermion propagators in the BS
Bethe-Salpeter equation, and in the non-relativistic limit
$m \rightarrow \infty$ by expanding to lowest order in inverse powers
of the fermion mass to obtain a Schr\"{o}dinger equation.  However,
there remains an intermediate mass range $m/e^2 \approx 1$ for which
neither of these techniques is adequate.  It is clear that a more careful
non-perturbative analysis of the fermion propagator in the vicinity
of the bare fermion mass pole is necessary before an accurate determination
of the QED3 positronium spectrum at intermediate fermion masses can be
made.  If a direct analogy with QCD models based on the
Bethe-Salpeter equations is made, we conclude that particular care must
be taken in modelling quark propagators for quarks whose mass is close to
the mass scale of the theory, namely charm quarks.

\renewcommand{\theequation}{\Alph{subsection}.\arabic{equation}}
\setcounter{subsection}{1}
\setcounter{equation}{0}
\subsection*{Appendix - Transformation Properties in QED3}
The four-component QED3 action in Minkowski space~\cite{Pi84}
\begin{equation}
S[A,\overline{\psi},\psi ] =
       \int \mbox{$ \, d^3x \,$} [ -\frac{1}{4} F_{\mu \nu} F^{\mu \nu}
       +\overline{\psi} \gamma_{\mu} (i\partial^{\mu} +eA^{\mu})\psi
       + m \overline{\psi} \psi ],
                               \label{eq:ACT}
\end{equation}
involves $4 \times 4$ matrices $\gamma_{\mu}$ which satisfy
$\{\gamma_{\mu},\gamma_{\nu}\}=2\eta_{\mu \nu}$ where
$\eta_{\mu \nu}={\rm diag}(1,-1,-1)$ with $\mu$ = 0, 1 and 2.
These three matrices belong to a complete set of 16 matrices
$\{\gamma_A\}=\{I,\gamma_{4},\gamma_{5},\gamma_{45},\gamma_{\mu},
         \gamma_{\mu 4},\gamma_{\mu 5},\gamma_{\mu 45} \}$
satisfying $\frac{1}{4} {\rm tr}(\gamma_A \gamma^B) = \delta^B_A$;
$$
      \gamma_0= \left( \begin{array}{cc} \sigma_3 & 0 \\
                                0 & -\sigma_3    \end{array}\right),\;\;\;
      \gamma_{1,2}= -i\left( \begin{array}{cc} \sigma_{1,2} & 0 \\
                                0 & -\sigma_{1,2}    \end{array}\right),
$$
$$
      \gamma_4=\gamma^4= \left( \begin{array}{cc} 0 & I \\
                                      I & 0    \end{array}\right),\;\;\;
      \gamma_5=\gamma^5= \left( \begin{array}{cc} 0 & -iI \\
                                      iI & 0    \end{array}\right),\;\;\;
      \gamma_{45}=\gamma^{45}= -i \gamma_4 \gamma_5,
$$
$$
      \gamma_{\mu 4}=i \gamma_{\mu} \gamma_4,\;\;\;
      \gamma_{\mu 5}=i \gamma_{\mu} \gamma_5,\;\;\;
      \gamma_{\mu 45}=-i \gamma_{\mu} \gamma_4 \gamma_5,\;\;\;
      \gamma^{\mu 4,\mu 5\,{\rm or} \,\mu 45} =
      \eta^{\mu \nu}  \gamma_{\nu 4,\nu 5\,{\rm or} \,\nu 45}
$$

The three $\gamma_{\mu}$, and $\gamma_4$ and $\gamma_5$ are five mutually
anti-commuting matrices. This is unlike the 4-dimensional case
where no analogue of $\gamma_4$ exists.

The action Eq.~(\ref{eq:ACT}) in the massless case $m=0$
exhibits global $U(2)$ symmetry with generators
$\{I,\gamma_{4},\gamma_{5},\gamma_{45}\}$ which is broken
by the generation of a dynamical fermion mass \cite{DK89,Pi84}
to a $U(1)\times U(1)$ symmetry $\{I,\gamma_{45}\}$.
The action is also invariant with respect to discrete parity and
charge conjugation symmetries, which for the fermion fields are
given by
\begin{equation}
\mbox{$\psi(x)$} \rightarrow \psi^\prime(x^\prime) =
\Pi \mbox{$\psi(x)$}, \;\;\;
\mbox{$\overline{\psi}(x)$} \rightarrow \overline{\psi}^\prime
(x^\prime) = \mbox{$\overline{\psi}(x)$} \Pi^{-1},
                  \label{eq:PAR}
\end{equation}
\begin{equation}
\mbox{$\psi(x)$} \rightarrow \psi^\prime(x) =
C \overline{\psi}(x)^{\rm T}, \;\;\;
\mbox{$\overline{\psi}(x)$} \rightarrow \overline{\psi}^\prime(x)
= -\mbox{$\psi(x)$}^{\rm T} C^{\dagger},
                    \label{eq:CH}
\end{equation}
where $x^{\prime}=(x^0,-x^1,x^2)$. The matrices $\Pi$ and $C$ are
each determined only up to an arbitrary phase by the condition that
the action Eq.~(\ref{eq:ACT}) be invariant~\cite{Bu92}:
\begin{equation}
\Pi=\gamma_{14} e^{i\phi_P \gamma_{45}}, \;\;\;
 C=\gamma_{2} e^{i\phi_C \gamma_{45}}, (0\leq \phi_P,\phi_C < 2\pi)
\end{equation}

Scalars, pseudoscalars, axi-scalars and axi-pseudoscalars are
defined by the following transformation properties under
parity transformations

\begin{eqnarray}
\Phi^{S}(x) & \rightarrow &
          \Phi^{S\prime}(x^{\prime}) = \Phi^{S}(x),\nonumber \\
\Phi^{PS}(x) & \rightarrow & \Phi^{PS\prime}(x^{\prime})
                             = -\Phi^{PS}(x), \nonumber \\
\Phi^{AS}(x) & \rightarrow &
          \Phi^{AS\prime}(x^{\prime}) = R_P\Phi^{AS}(x),\nonumber \\
\Phi^{APS}(x) & \rightarrow & \Phi^{APS\prime}(x^{\prime})
                             = -R_P\Phi^{APS}(x), \label{eq:PTY}
\end{eqnarray}
where $\Phi^{AS}$ and $\Phi^{APS}$ are doublet states
$\Phi=(\Phi_4,\Phi_5)^T$, and
\begin{eqnarray}
R_P & = & \left(\begin{array}{cc} -\cos 2\phi_P & -\sin 2\phi_P \\
        -\sin 2\phi_P & \cos 2\phi_P  \end{array}  \right).
\label{eq:RL}
\end{eqnarray}
Similar transformation properties exist for charge conjugation.

The most general forms of the Bethe-Salpeter amplitudes~\cite{LS69} for
bound scalar and pseudoscalar states are
\begin{eqnarray}
\Gamma^S(q,P) & = & If+\not \! qg +\not \! Ph +\epsilon_{\mu \nu \rho}
                        P^{\mu}q^{\nu}\gamma^{\rho 45} k , \label{eq:GS}\\
\Gamma^{PS}(q,P) & = & \gamma_{45} \Gamma^S(q,P), \label{eq:GP}
\end{eqnarray}
where $f,g,h$ and $k$ are functions only of $q^2,P^2$ and
$q\cdot P$.
BS amplitudes corresponding to the components $\Phi_4$ and
$\Phi_5$ of
axi-scalars and axi-pseudoscalars take the general form
\begin{equation}
  \left(\begin{array}{c} \Gamma^{(4)}(q,P) \\ \Gamma^{(5)}(q,P)
        \end{array} \right)^{AS} =
\left(\begin{array}{c} \gamma_4 \\
           \gamma_5\end{array} \right)f +
          \left(\begin{array}{c} \gamma_{\mu 4} \\
     \gamma_{\mu 5}\end{array} \right) (q^{\mu}g+P^{\mu}h)
+\epsilon_{\mu \nu \rho} P^{\mu} q^{\nu}
  \left(\begin{array}{c} \gamma^{\rho 5} \\
              - \gamma^{\rho 4}\end{array} \right)k,  \label{eq:GAS}
\end{equation}
and
\begin{equation}
  \left(\begin{array}{c} \Gamma^{(4)}(q,P) \\ \Gamma^{(5)}(q,P)
        \end{array} \right)^{APS} = \gamma_{45}
\left(\begin{array}{c} \Gamma^{(4)}(q,P) \\ \Gamma^{(5)}(q,P)
        \end{array} \right)^{AS}.
 \label{eq:GAPS}
\end{equation}

Furthermore, the charge parity ${\cal C}=\pm 1$ of the bound states
is determined by
the parity of the functions $f,g,h$ and $k$ under the transformation
$q\cdot P \rightarrow -q\cdot P$.  The quantity
$q\cdot P$ is the only Lorentz invariant which
changes sign under charge conjugation and thus determines the
charge parity of those functions.

Our conventions for Euclidean space quantities are summarised in
Appendix A of Ref.~\cite{Bu92}.  In particular Euclidean momenta and
Dirac matrices are defined by
$$
P_3^{({\rm E})} = -iP_0^{({\rm M})},\hspace{5 mm}
P_{1,2}^{({\rm E})} = P_{1,2}^{({\rm M})},\hspace{5 mm}
\gamma_3^{({\rm E})} = \gamma_0^{({\rm M})}, \hspace{5 mm}
\gamma_{1,2}^{({\rm E})} = i\gamma_{1,2}^{({\rm M})}.
$$
\subsection*{Acknowledgments}
We are grateful to C. J. Hamer, A. Tam and P. Maris for helpful
discussions, and the National Centre for Theoretical Physics at
the Australian National University for hosting the Workshop on
Non-Perturbative Methods in Field Theory where part of this work was
completed.
\pagebreak

redpajama_set_name: RedPajamaArXiv


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text:
Q: Apache archiva not able to connect to central I setup apache archiva using simple way
archiva start

I am using default internal repository to test connection to central by executing following from my browser
http://localhost:8080/repository/internal/junit/junit/3.8.1/junit-3.8.1.jar

I get it in ./logs/archiva.log
2016-04-28 13:11:42,351 [qtp957020437-29] WARN  org.apache.archiva.proxy.DefaultRepositoryProxyConnectors [] - Transfer error from repository central for artifact junit:junit:3.8.1::pom , continuing to next repository. Error message: Download failure on resource [https://repo.maven.apache.org/maven2/junit/junit/3.8.1/junit-3.8.1.pom]:Connection reset (cause: java.net.SocketException: Connection reset)
    2016-04-28 13:11:42,351 [qtp957020437-29] ERROR org.apache.archiva.webdav.ArchivaDavResourceFactory [] - Failures occurred downloading from some remote repositories:
            central: Download failure on resource [https://repo.maven.apache.org/maven2/junit/junit/3.8.1/junit-3.8.1.pom]:Connection reset (cause: java.net.SocketException: Connection reset)
    org.apache.archiva.policies.ProxyDownloadException: Failures occurred downloading from some remote repositories:
            central: Download failure on resource [https://repo.maven.apache.org/maven2/junit/junit/3.8.1/junit-3.8.1.pom]:Connection reset (cause: java.net.SocketException: Connection reset)
            at org.apache.archiva.proxy.DefaultRepositoryProxyConnectors.fetchFromProxies(DefaultRepositoryProxyConnectors.java:367) ~[archiva-proxy-2.2.0.jar:?]
            at org.apache.archiva.metadata.repository.storage.maven2.Maven2RepositoryStorage.applyServerSideRelocation(Maven2RepositoryStorage.java:670) ~[maven2-repository-2.2.0.jar:?]
            at org.apache.archiva.webdav.ArchivaDavResourceFactory.fetchContentFromProxies(ArchivaDavResourceFactory.java:818) [archiva-webdav-2.2.0.jar:?]
            at org.apache.archiva.webdav.ArchivaDavResourceFactory.processRepository(ArchivaDavResourceFactory.java:629) [archiva-webdav-2.2.0.jar:?]
            at org.apache.archiva.webdav.ArchivaDavResourceFactory.createResource(ArchivaDavResourceFactory.java:325) [archiva-webdav-2.2.0.jar:?]
            at org.apache.archiva.webdav.RepositoryServlet.service(RepositoryServlet.java:125) [archiva-webdav-2.2.0.jar:?]
            at javax.servlet.http.HttpServlet.service(HttpServlet.java:848) [javax.servlet-3.0.0.v201112011016.jar:?]
            at org.eclipse.jetty.servlet.ServletHolder.handle(ServletHolder.java:684) [jetty-servlet-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.servlet.ServletHandler.doHandle(ServletHandler.java:501) [jetty-servlet-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.handler.ScopedHandler.handle(ScopedHandler.java:137) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.security.SecurityHandler.handle(SecurityHandler.java:557) [jetty-security-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.session.SessionHandler.doHandle(SessionHandler.java:231) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.handler.ContextHandler.doHandle(ContextHandler.java:1086) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.servlet.ServletHandler.doScope(ServletHandler.java:428) [jetty-servlet-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.session.SessionHandler.doScope(SessionHandler.java:193) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.handler.ContextHandler.doScope(ContextHandler.java:1020) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.handler.ScopedHandler.handle(ScopedHandler.java:135) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.handler.ContextHandlerCollection.handle(ContextHandlerCollection.java:255) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.handler.HandlerCollection.handle(HandlerCollection.java:154) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.handler.HandlerWrapper.handle(HandlerWrapper.java:116) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.Server.handle(Server.java:370) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.AbstractHttpConnection.handleRequest(AbstractHttpConnection.java:494) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.AbstractHttpConnection.headerComplete(AbstractHttpConnection.java:971) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.AbstractHttpConnection$RequestHandler.headerComplete(AbstractHttpConnection.java:1033) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.http.HttpParser.parseNext(HttpParser.java:644) [jetty-http-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.http.HttpParser.parseAvailable(HttpParser.java:235) [jetty-http-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.server.AsyncHttpConnection.handle(AsyncHttpConnection.java:82) [jetty-server-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.io.nio.SelectChannelEndPoint.handle(SelectChannelEndPoint.java:667) [jetty-io-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.io.nio.SelectChannelEndPoint$1.run(SelectChannelEndPoint.java:52) [jetty-io-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.util.thread.QueuedThreadPool.runJob(QueuedThreadPool.java:608) [jetty-util-8.1.14.v20131031.jar:8.1.14.v20131031]
            at org.eclipse.jetty.util.thread.QueuedThreadPool$3.run(QueuedThreadPool.java:543) [jetty-util-8.1.14.v20131031.jar:8.1.14.v20131031]
            at java.lang.Thread.run(Thread.java:745) [?:1.8.0_73]

I setup network proxy in archiva web interface and choose it in "Remove repository management" / "Proxy for Remote Download Index"
Also we are using non standard root cecurity certificate that was added to jdk trusted storage (local maven repository works fine).
Any idea how to make it work?

A: In accoriding to: here proxy is not working in archiva. I've tried to get snapshot-2.2.1 build from source but it was too slow. Switched to Nexus

A: The following solution is for http only.

*

*Modify the default repository entry: switch to Administration > Repositories > Remote Repositories Management

*click "edit" on the default repository entry (Name: Central Repository)

*Change the URL entry from "https://repo.maven.apache.org/maven2" to "http://repo.maven.apache.org/maven2"

*Configure your proxy in Archiva: http://:/#networkproxies

*Switch to Administration > Repositories > Remote Repositories Management

*click "edit" on the default repository entry (Name: Central Repository)

*Locate the entry "Proxy for Remote Download Index" and choose the proxy configured in step 1. My setup has both http and https proxies defined.

*Now click on "Repositories" and click the green button "Download Remote Index" for the "Central Repository"
Again: the solution is for http only. If one wants to use https the appropriate certificates have to be downloaded and manually imported. Nexus 3 offers a nice import functionality, but lacks other features and is not an option for me.

redpajama_set_name: RedPajamaStackExchange


Arnold Bax was an English composer who made many visits to County Donegal, lived in Glencolumbkille for a number of years during the 1920s and was heavily influenced by Irish literature and folklore. He was knighted in 1937 and in 1941 became Master of the King's (later Queen's) Musick. He wrote seven symphonies and several symphonic poems, including The Garden of Fand and Tintagel as well as a violin concerto, a cello concerto, piano solos and music for choral and chamber groups. He wrote several novels under the pseudonym Dermot O'Byrne. In his autobiography, Farewell My Youth, he writes of his dying wish as being 'to gaze from his window at Glen Head in late evening'.


text:
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-Elliott Erwitt
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She's not only incredibly talented, she's extremely easy and fun to work with. Angela takes the time to know the couples she works with and does a great job connecting to their story and vibe. I'm amazed at the way she captures truly intimate and candid moments so beautifully and artistically.
Kristina B.
Her vision and work is truly amazing...each photo tells a story through movement and light. Her passion is evident and she has a true talent for capturing the energy of the moment through photography. We are so excited to look back on these images for years to come.
Named in Top Chicago Wedding Photographers by Brides. Printed in multiple magazines including the Knot, Chicago Style Weddings, Modern Luxury Brides. Featured on Style Me Pretty, Ruffled, Brides, Rustic Wedding Chic, Style Unveiled and more.
@angelareneephotography
Copyrights ©Angela Renée Photography 2023

redpajama_set_name: RedPajamaCommonCrawl


Hi, Tim Blankenship here with divorce661.com. In this episode, we're talking about California child custody agreements and parenting plans. Basically, what you can come up with and what is required by the courts as far as a parenting plan?
People have asked me what's required. Like how much detail does the court need as far as a custom plan? Here's the answer. The courts do not need a detailed parenting plan at all. All they need to know is how you are going to label your custody.
Meaning, are you going to have joint legal and joint physical or is one party going to be the sole physical custodial parent etc.? You don't need an actual detail plan. For people who want to have an open parenting plan and not a specified plan, you can simply say joint legal, joint physical, and reasonable right of visitation to the other party.
That's sufficient enough for the court. They're not going to kick that back and say no we need a parenting plan. If you want an open schedule for people that maybe have schedules that change frequently, law enforcement officers, nurses, firefighters, people that have a changing schedule, you may not want a detailed parenting plan.
It may not be workable because maybe your schedule changes frequently. Now in the alternative, you can have a detailed parenting plan so if you have a good relationship with your spouse and the open parenting plan works that's great. But if you want to have some detail or specificity of your parenting plan, you can have as detailed as a plan as you want.
You can attach the attachments that are part of the court orders. You can have a joint legal custody attachment. You can have a physical custody attachment. You can have a detailed plan where you are laying out every hour and every minute who picks up in what vehicle. I mean you can get as detailed as you want. It's not necessary, but you can have an open plan, a very detailed plan, or anything in the middle.


text:
/*****************************************************************************
   Licensed under the Apache License, Version 2.0 (the "License");
   you may not use this file except in compliance with the License.
   You may obtain a copy of the License at

		http://www.apache.org/licenses/LICENSE-2.0

   Unless required by applicable law or agreed to in writing, software
   distributed under the License is distributed on an "AS IS" BASIS,
   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   See the License for the specific language governing permissions and
   limitations under the License.

   See NOTICE file for details.
 *****************************************************************************/
#include "jpype.h"
#include "pyjp.h"
#include "jp_stringtype.h"
#include <structmember.h>

#ifdef __cplusplus
extern "C"
{
#endif
PyTypeObject *PyJPPackage_Type = NULL;
static PyObject *PyJPPackage_Dict = NULL;

static PyObject *PyJPPackage_new(PyTypeObject *type, PyObject *args, PyObject *kwargs)
{
	JP_PY_TRY("PyJPPackage_new");
	PyObject *name = NULL;
	if (!PyArg_Parse(args, "(U)", &name))
		return 0;

	// Check the cache
	PyObject *obj = PyDict_GetItem(PyJPPackage_Dict, name);
	if (obj != NULL)
	{
		Py_INCREF(obj);
		return obj;
	}

	// Otherwise create a new object
	PyObject *self = PyModule_Type.tp_new(PyJPPackage_Type, args, NULL);
	int rc = PyModule_Type.tp_init(self, args, NULL);
	if (rc != 0)
	{
		// If we fail clean up the mess.
		Py_DECREF(self);
		return 0;
	}

	// Place in cache
	PyDict_SetItem(PyJPPackage_Dict, name, self);
	return self;
	JP_PY_CATCH(NULL); // GCOVR_EXCL_LINE
}

static void dtor(PyObject *self)
{
	JPContext *context = JPContext_global;
	if (context == NULL || !context->isRunning())
		return;
	jobject jo = (jobject) PyCapsule_GetPointer(self, NULL);
	if (jo == NULL)
		return;
	JPJavaFrame frame = JPJavaFrame::outer(context);
	frame.DeleteGlobalRef(jo);
}

static jobject getPackage(JPJavaFrame &frame, PyObject *self)
{
	PyObject *dict = PyModule_GetDict(self); // borrowed
	PyObject *capsule = PyDict_GetItemString(dict, "_jpackage"); // borrowed
	jobject jo;
	if (capsule != NULL)
	{
		jo = (jobject) PyCapsule_GetPointer(capsule, NULL);
		return jo;
	}

	const char *name = PyModule_GetName(self);
	// Attempt to load the object.
	jo =	frame.getPackage(name);

	// Found it, use it.
	if (jo != NULL)
	{
		jo = frame.NewGlobalRef(jo);
		capsule = PyCapsule_New(jo, NULL, dtor);
		PyDict_SetItemString(dict, "_jpackage", capsule); // no steal
		//		Py_DECREF(capsule);
		return jo;
	}

	// Otherwise, this is a bad package.
	PyErr_Format(PyExc_AttributeError, "Java package '%s' is not valid", name);
	return NULL;
}

/**
 * Get an attribute from the package.
 *
 * This will auto load packages and classes when encounter,
 * but first checks the cache.  This acts like an standard Python
 * module otherwise.
 *
 * @param self
 * @param attr
 * @return
 */
static PyObject *PyJPPackage_getattro(PyObject *self, PyObject *attr)
{
	JP_PY_TRY("PyJPPackage_getattro");
	if (!PyUnicode_Check(attr))
	{
		PyErr_Format(PyExc_TypeError, "attribute name must be string, not '%s'", Py_TYPE(attr)->tp_name);
		return NULL;
	}

	PyObject *dict = PyModule_GetDict(self);
	if (dict != NULL)
	{
		// Check the cache
		PyObject *out = PyDict_GetItem(PyModule_GetDict(self), attr);
		if (out != NULL)
		{
			Py_INCREF(out);
			return out;
		}
	}

	string attrName = JPPyString::asStringUTF8(attr).c_str();
	// Check for private attribute
	if (attrName.compare(0, 2, "__") == 0)
		return PyObject_GenericGetAttr((PyObject*) self, attr);

	JPContext* context = JPContext_global;
	if (!context->isRunning())
	{
		PyErr_Format(PyExc_RuntimeError,
				"Unable to import '%s.%U' without JVM",
				PyModule_GetName(self), attr);
		return 0;
	}
	JPJavaFrame frame = JPJavaFrame::outer(context);
	jobject pkg = getPackage(frame, self);
	if (pkg == NULL)
		return NULL;

	JPPyObject out;
	jobject obj;
	try
	{
		obj = frame.getPackageObject(pkg, attrName);
	}		catch (JPypeException& ex)
	{
		JPPyObject h = JPPyObject::accept(PyObject_GetAttrString(self, "_handler"));
		// If something fails, we need to go to a handler
		if (!h.isNull())
		{
			ex.toPython();
			JPPyErrFrame err;
			err.normalize();
			err.clear();
			JPPyObject tuple0 = JPPyObject::call(PyTuple_Pack(3, self, attr, err.m_ExceptionValue.get()));
			PyObject *rc = PyObject_Call(h.get(), tuple0.get(), NULL);
			if (rc == 0)
				return 0;
			Py_DECREF(rc); // GCOVR_EXCL_LINE
		}
		throw; // GCOVR_EXCL_LINE
	}
	if (obj == NULL)
	{
		PyErr_Format(PyExc_AttributeError, "Java package '%s' has no attribute '%U'",
				PyModule_GetName(self), attr);
		return NULL;
	} else if (frame.IsInstanceOf(obj, context->_java_lang_Class->getJavaClass()))
		out = PyJPClass_create(frame, frame.findClass((jclass) obj));
	else if (frame.IsInstanceOf(obj, context->_java_lang_String->getJavaClass()))
	{
		JPPyObject u = JPPyObject::call(PyUnicode_FromFormat("%s.%U",
				PyModule_GetName(self), attr));
		JPPyObject args = JPPyObject::call(PyTuple_Pack(1, u.get()));
		out = JPPyObject::call(PyObject_Call((PyObject*) PyJPPackage_Type, args.get(), NULL));
	} else
	{
		// We should be able to handle Python classes, datafiles, etc,
		// but that will take time to implement.  In principle, things
		// that are not packages or classes should appear as Buffers or
		// some other resource type.
		PyErr_Format(PyExc_AttributeError, "'%U' is unknown object type in Java package", attr);
		return NULL;
	}
	// Cache the item for now
	PyDict_SetItem(dict, attr, out.get()); // no steal
	return out.keep();
	JP_PY_CATCH(NULL);  // GCOVR_EXCL_LINE
}

/**
 *  This next method is required, I have no clue why.  Seems
 * likely that the default PyObject traverse does not agree
 * with modules.
 */
static int PyJPPackage_traverse(PyObject *m, visitproc visit, void *arg)
{
	return PyModule_Type.tp_traverse(m, visit, arg);
}

static int PyJPPackage_clear(PyObject *m)
{
	return PyModule_Type.tp_clear(m);
}

static PyObject *PyJPPackage_str(PyObject *self, PyObject *args, PyObject *kwargs)
{
	JP_PY_TRY("PyJPPackage_str");
	return PyModule_GetNameObject(self);
	JP_PY_CATCH(NULL);
}

static PyObject *PyJPPackage_repr(PyObject *self, PyObject *args, PyObject *kwargs)
{
	JP_PY_TRY("PyJPPackage_repr");
	return PyUnicode_FromFormat("<java package '%s'>", PyModule_GetName(self));
	JP_PY_CATCH(NULL);
}

static PyObject *PyJPPackage_call(PyObject *self, PyObject *args, PyObject *kwargs)
{
	JP_PY_TRY("PyJPPackage_call");
	PyErr_Format(PyExc_TypeError, "Package `%s` is not callable.", PyModule_GetName(self));
	JP_PY_CATCH(NULL);
}

static PyObject *PyJPPackage_package(PyObject *self)
{
	return PyUnicode_FromFormat("java");
}

static PyObject *PyJPPackage_path(PyObject *self)
{
	return PyList_New(0);
}

static PyObject *PyJPPackage_dir(PyObject *self)
{
	JP_PY_TRY("PyJPPackage_dir");
	JPContext* context = PyJPModule_getContext();
	JPJavaFrame frame = JPJavaFrame::outer(context);
	jobject pkg = getPackage(frame, self);
	if (pkg == NULL)
		return NULL;

	jarray o = frame.getPackageContents(pkg);
	Py_ssize_t len = frame.GetArrayLength(o);
	JPPyObject out = JPPyObject::call(PyList_New(len));
	for (Py_ssize_t i = 0;  i < len; ++i)
	{
		string str = frame.toStringUTF8((jstring)
				frame.GetObjectArrayElement((jobjectArray) o, (jsize) i));
		PyList_SetItem(out.get(), i, PyUnicode_FromFormat("%s", str.c_str()));
	}
	return out.keep();
	JP_PY_CATCH(NULL);
}

/**
 * Add redirect for matmul in package modules.
 *
 * This will be used to support "java@obj" which will be used
 * to force cast a Python object into Java.
 *
 * @param self
 * @param other
 * @return
 */
static PyObject *PyJPPackage_cast(PyObject *self, PyObject *other)
{
	JP_PY_TRY("PyJPPackage_cast");
	PyObject *dict = PyModule_GetDict(self);
	PyObject* matmul = PyDict_GetItemString(dict, "__matmul__");
	if (matmul == NULL)
		Py_RETURN_NOTIMPLEMENTED;
	JPPyObject args = JPPyObject::call(PyTuple_Pack(2, self, other));
	return PyObject_Call(matmul, args.get(), NULL);
	JP_PY_CATCH(NULL);
}

static PyObject *PyJPPackage_castEq(PyObject *self, PyObject *other)
{
	PyErr_Format(PyExc_TypeError, "Matmul equals not support for Java packages");
	return NULL;
}

static PyMethodDef packageMethods[] = {
	{"__dir__", (PyCFunction) PyJPPackage_dir, METH_NOARGS},
	{NULL},
};

static PyGetSetDef packageGetSets[] = {
	{"__all__", (getter) PyJPPackage_dir, NULL, ""},
	{"__name__", (getter) PyJPPackage_str, NULL, ""},
	{"__package__", (getter) PyJPPackage_package, NULL, ""},
	{"__path__", (getter) PyJPPackage_path, NULL, ""},
	{0}
};

static PyType_Slot packageSlots[] = {
	{Py_tp_new,      (void*) PyJPPackage_new},
	{Py_tp_traverse, (void*) PyJPPackage_traverse},
	{Py_tp_clear,    (void*) PyJPPackage_clear},
	{Py_tp_getattro, (void*) PyJPPackage_getattro},
	{Py_tp_str,      (void*) PyJPPackage_str},
	{Py_tp_repr,     (void*) PyJPPackage_repr},
	{Py_tp_call,     (void*) PyJPPackage_call},
	{Py_nb_matrix_multiply, (void*) PyJPPackage_cast},
	{Py_nb_inplace_matrix_multiply, (void*) PyJPPackage_castEq},
	{Py_tp_methods,  (void*) packageMethods},
	{Py_tp_getset,   (void*) packageGetSets},
	{0}
};

static PyType_Spec packageSpec = {
	"_jpype._JPackage",
	-1,
	0,
	Py_TPFLAGS_DEFAULT | Py_TPFLAGS_BASETYPE | Py_TPFLAGS_HAVE_GC,
	packageSlots
};

#ifdef __cplusplus
}
#endif

void PyJPPackage_initType(PyObject* module)
{
	// Inherit from module.
	JPPyObject bases = JPPyObject::call(PyTuple_Pack(1, &PyModule_Type));
	packageSpec.basicsize = PyModule_Type.tp_basicsize;
	PyJPPackage_Type = (PyTypeObject*) PyType_FromSpecWithBases(&packageSpec, bases.get());
	JP_PY_CHECK();
	PyModule_AddObject(module, "_JPackage", (PyObject*) PyJPPackage_Type);
	JP_PY_CHECK();

	// Set up a dictionary so we can reuse packages
	PyJPPackage_Dict = PyDict_New();
	PyModule_AddObject(module, "_packages", PyJPPackage_Dict);
}

redpajama_set_name: RedPajamaGithub


Have the banking regulators gone too far?
The effectiveness of regulation is still unknown nearly a decade after the global financial crisis, according to a leading finance sector researcher at the University of Sydney Business School.
Associate finance professor Eliza Wu made the comment ahead of the arrival in Sydney of the world's leading banking experts and banking regulators for the Sydney Banking and Financial Stability conference, which kicks off on Friday, 1 December.
The Future of Banking Regulation will be the focus of the conference's Policy Panel Discussion between Dr Iikka Korhonen from the Bank of Finland, Dr Frank Packer from the Bank for International Settlements and Mr Aidan O'Shaughnessy from the Australian Bankers' Association.
The keynote address, to be delivered by Steven Ongena, professor of banking at the University of Zurich, will focus on the spillovers of macro-prudential policies.


Doctors use the same tests and surgeries to figure out treatments for triple-negative breast cancers as they do for other kinds of breast cancer. Your treatment will be based on tumor size, tumor grade and whether the cancer has traveled to your lymph nodes or other parts of your body. Standard treatment options for triple negative breast cancer include chemotherapy, surgery and radiation.
Your doctor will likely recommend some type of surgery, with the goal of removing the cancer from your breast. In lumpectomy, also called breast-conserving surgery, the surgeon removes the tumor plus a small rim of normal tissue around the tumor, called a margin.
You are likely to receive chemotherapy, medicine that kills cancer cells everywhere in your body. This type of treatment is called systemic, or whole-body, therapy, and it may be given by vein or in some cases by pill. The goal of chemotherapy is to prevent metastasis, when breast cancer comes back and spreads to other parts of the body. A metastatic recurrence occurs when cancer cells travel away from the breast and start growing in other organs such as the bones, liver, lungs or brain.
Chemotherapy may be given before or after surgery. If you have a tumor that is very large or you have a sizable tumor and want a lumpectomy, your doctor may recommend chemotherapy before surgery, also called neoadjuvant therapy. This therapy shrinks the tumor and helps your doctor learn how sensitive the tumor is to chemotherapy.
Chemotherapy is the most effective systemic treatment for triple negative breast cancer. The reason is that chemotherapy works better than other treatments at killing cancer cells that divide quickly, which is very common in triple negative disease. When triple negative breast cancers are found early, response rates to chemotherapy are high. Doctors try to lessen the chance of a metastatic recurrence by treating the whole body, including any areas where very tiny cancer cells may have traveled. Often, the chemotherapy that you receive will be the same type given to people with hormone receptorpositive or HER2-positive breast cancer. Studies show chemotherapy works better against triple negative cancers than hormone receptor positive breast cancers. There are many types of chemotherapy, and you and your doctor will choose the best one for you.
In rare cases, you might not receive chemotherapy; for example, if you have a very low-grade tumor (the cancer cells are not dividing quickly), if the tumor is very small, or if the risks of chemotherapy outweigh the benefits. Because chemotherapy is a common treatment for triple-negative breast cancer, always ask your doctor to explain the reasons why you would not receive it.
Radiation is a local therapy that kills cancer cells left after surgery in the area where the breast cancer was found. It helps protect you from a local recurrence, cancer coming back in the same place. Radiation usually is given from outside your body by an external beam. It can be given inside the body in some circumstances. If you have a lumpectomy, you will need radiation to kill any cancer cells left in the breast and sometimes in the underarm area. Radiation also may be given after mastectomy if your surgeon found cancer close to your chest wall or in your lymph nodes. Ask a member of your healthcare team to explain the reasons why you need the treatment.
TNBC Foundation offers two clinical trial matching services to help connect patients with trials that match their exact diagnosis, stage and treatment history.


The Eagle County Regional Airport (EGE) has conducted advanced terminal planning based on recommendations from the recently completed Master Plan Update (2014 Master Plan). The purpose for conducting this additional analysis was to create and refine landside and terminal facility concepts and assist Eagle County Air Terminal Corporation in preparing to begin terminal construction by 2017. The purpose of the Terminal Area Plan Report is to establish the quantitative basis for defining issues and opportunities, and for the development and evaluation of concepts to address them. This project kicked-off in March 2015 and culminated in November 2015. The Terminal Area Plan Report process and timeline included numerous conceptual and informational meetings with Eagle County Commissioners, Stakeholder Committees, and Airport Staff.


text:
COVID Cases Have Increased By Over 500% Recently in New York
Gov. Cuomo says the highly contagious Delta COVID variant "poses a great risk" to New York.
The CDC confirmed the Delta variant now makes up over 80-percent of new COVID cases. Dr. Gregory Poland of Mayo Clinic believes the Delta variant will "find everybody" who is vaccinated.
"This is a very different variant, it will find you," Poland said.
On Thursday, Gov. Andrew Cuomo updated New Yorkers on the state's progress combating COVID-19 and announced 1.70 percent of tests came back positive in the previous 24 hours.
"COVID, especially the Delta variant, poses a great risk to those who remain unvaccinated. Get your vaccine before it's too late," Cuomo said.
Stay Informed: CDC Website About Coronavirus
Cuomo noted 1,584 tests came back positive out of 93,174. Back on Monday, July 5, 255 tests came back positive.
Earlier this week, Cuomo said the vaccine is the "critical factor" in beating COVID and rebuilding New York State.
"As we near the midpoint of summer, life is picking back up and people are living again after putting everything on hold last year. The best way to take advantage of all that summer in New York has to offer is to get vaccinated - protect yourself and your loved ones," Cuomo added.
The state's 7-day COVID percent positive rate has increased to 1.50 percent. According to Cuomo, 74.1 percent of all New Yorkers 18 and older have received at least one vaccine while 67.6 percent have completed their vaccine series. Nearly 62 percent of all eligible New Yorkers have at least one vaccine dose.
LOOK: Answers to 30 common COVID-19 vaccine questions
While much is still unknown about the coronavirus and the future, what is known is that the currently available vaccines have gone through all three trial phases and are safe and effective. It will be necessary for as many Americans as possible to be vaccinated in order to finally return to some level of pre-pandemic normalcy, and hopefully these 30 answers provided here will help readers get vaccinated as soon they are able.
Hidden Treasure Worth Thousands Found in New York Home
40 Children Have Gone Missing From the Hudson Valley
Gun Violence Emergency Declared in New York, Lawmakers Respond
Gov. Cuomo declared the first-in-the-nation gun violence disaster emergency in New York.
KEEP LOOKING: See what 50 company logos looked like then and now
Gypsy Moth 'Accidentally' Released Causing 'Nightmares' in New York
After 10 to 15 years a gypsy moth that was once "accidentally" released has returned and is causing "nightmares" in New York.
Cops Dismantle 'Local Criminal Enterprise' at Hudson Valley Deli
Six were arrested for allegedly selling drugs and more at a Hudson Valley deli.
Historic Hudson Valley Building Revived as New Business
A historic Hudson Valley building is getting a second life as a new business that will show off the beauty of the region.
Then And Now: How Fast Food Restaurants in The Hudson Valley Have Changed
While many of your favorite Hudson Valley fast-food restaurants have remained in the same spot for decades, their buildings are almost unrecognizable from what they used to look like.
Fast Food Change
Mustang Pinned Under Tractor-Trailer Near I-84 in Hudson Valley
A Mustang crashed into a tractor-trailer near I-84 in the Hudson Valley.
Photos: Truck Drives Into Building in Dutchess County
A truck drove into a building in Dutchess County, causing significant damage.
The 20 Most Surprising Things Found at LEGOLAND New York in Goshen
LEGOLAND New York in Goshen is officially opening this summer. Right now special previews are going on so that guests can get a sneak peek at what's in store when the park opens its gates. While we won't spoil all of the surprises, here are just some of the incredible little details that await you during your first visit to this incredible theme park.
Filed Under: Coronavirus in New York