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\begin{document}
%{\par\raggedleft Preprint JINR-E2-99-220 , RUB--TPII--08/99 \\
%  \par} \vspace{2mm}

\title{QCD vacuum tensor susceptibility and properties
       of transversely polarized mesons}
\author{A. P. Bakulev and S.~V.~Mikhailov}
\address{
Bogoliubov Theoretical Laboratory, Joint Institute for Nuclear
Research, 141980, Dubna, Russia}
\date{\today}
\maketitle
\begin{abstract}
We re-estimate the tensor susceptibility of QCD vacuum, $\chi $,
and to this end, we re-estimate the leptonic decay constants for
transversely polarized $\rho $- ,~$\rho'$- and $b_1$-mesons. The
origin of the susceptibility is analyzed using duality between
$\rho$- and $b_1$- channels in a 2-point correlator of tensor
currents. We confirm the results of \protect\cite{bib-GRVW:87} for
the 2-point correlator of tensor currents and disagree with
\protect\cite{bib-BO:97} on both OPE expansion and the value of
QCD vacuum tensor susceptibility. Using our value for the latter
we determine new estimations of nucleon tensor charges related to
the first moment of the transverse structure function $h_1$ of a
nucleon.
\newline
PACS number(s): 11.15.Tk, 12.38.Lg, 14.40.Cs
\end{abstract}

\section*{Introduction}
  In this paper, we  investigate the low-energy properties
of the lightest transversely polarized mesons with quantum numbers
$J^{PC} = 1^{- -}(\rho),~1^{+ -}(b_1)$ in the framework of QCD sum rules
(SRs) with nonlocal condensates (NLCs) as well as with the standard ones.
  This work was started in ~\cite{bib-BM:98}
where the ``mixed parity" NLC SR for the light-cone distribution
(LCD) amplitudes of both $\rho$- and $b_1$-mesons was constructed.
  It was concluded that to obtain a reliable result
we should reduce model uncertainties due to the nonlocal gluon
contribution into the SR for LCD.
  Different SRs for each P-parity could be preferable
for this purpose.
  As a first step, to obtain twist 2 meson LCD,
we concentrate on the meson static properties using the ``pure
parity" NLC SR for each meson separately:

1) we re-estimate the leptonic decay constants $f^T_{m}$ for
transversely polarized $\rho(770)$, $\rho'(1465)$-mesons
($1^{--}$) and the $b_1 (1235)$-meson ($1^{+-}$) ~\cite{Book98};

2) we correct the previous consideration by Belyaev and Oganesyan
(B\&O)~\cite{bib-BO:97} and provide a new estimation for the
vacuum tensor susceptibility (VTS) introduced in
~\cite{bib-HJ:95,bib-HJ:96}.

  The static characteristics, the decay constants $f^T_m$ and
``continuum thresholds" $s_m$ (parameters of phenomenological
models for spectral densities) of the lightest transversely
polarized mesons in the channels with $J^{PC}=1^{--}$ and $1^{+
-}$, are tightly connected with the value of VTS. Namely, the
difference of the meson properties in these channels fixes the
non-zero value of VTS: in a hypothetical Nature, \eg, where the
properties of these mesons are the same, VTS is identically equal
to zero. For the reason that these meson constants should appear
in VTS in a form of a difference, we have to define them more
precisely and in the framework of a unified approach.

  The method of NLC SRs was successfully applied for
the determination of meson dynamic characteristics (LCD
amplitudes, form factors, see, \eg, \cite{bib-BM:98,bib-MR:92} and
refs therein).
  One of its basic components is a non-zero characteristic scale,
$\lambda_q$, of quark-gluon correlations in the QCD
vacuum~\cite{bib-MR:92}. This parameter fixes the average
virtuality of vacuum quarks which flow through vacuum with the
momentum $k_q$, ~$\va{k_q^2} = \lambda_q^2 \approx
0.4-0.5$~GeV$^2$~\cite{piv91}, this value is of an order of
hadronic scale, $m^2_\rho \approx 0.6~\gev{2}$, and is of
importance in the calculations. Note here that quite recently the
nonlocal character of the quark condensate has been confirmed in
the lattice calculations~\cite{bib-Di:99}, where an attempt to
estimate $\lambda_q$ was made. NLC approach can improve the
stability and accuracy of SRs even for the case of decay constant
determination where the NLC effect is of an order of the
%not smaller than the
radiative correction contribution. Therefore, we revise their
values in pure parity NLC SRs, despite the presence of different
estimations for these quantities in
literature~\cite{bib-BB:96,bib-BO:97,bib-BM:98}, obtained by
different ways. For comparison we also calculate all these
quantities by the standard way, that corresponds to processing our
NLC SR on the limit $ \lambda_q^2 \to 0$.

  The source of the above-mentioned difference of
meson properties is the peculiarity of four-quark condensate
contribution to the ``theoretical part" of SRs.
This contribution is invariant under the duality\footnote{%
We are grateful to O.~V.~Teryaev who involved us in an
investigation of this transformation and suggested this name for
it.} transformation in contrast to all other condensate
contributions which change the sign under the same transformation.
This peculiarity of four-quark condensate contribution will be
considered in detail.

  The plan of presentation is the following:
firstly, we discuss the QCD SR approach to investigation of the
4-rank tensor 2-point correlator for transversely polarized $\rho
$-, $ \rho'$- and $b_1$-mesons. Then, we define the duality
transformation and draw its consequences for the constructed SRs.
Finally, we derive a new estimation for the QCD VTS and nucleon
tensor charges and discuss what is wrong in the consideration
of~\cite{bib-BO:97}.


\section*{Decay constants of transversely polarized
($J^{PC}=1^{--},1^{+-}$)-mesons}

We start with a 2-point correlator of tensor currents $J^{\mu \nu
}(x)=\bar u(x)\sigma^{\mu \nu }d(x)$,
\be
 \Pi^{\mu \nu ;\alpha \beta }(q) =
  \int d^{4}x\ e^{iq\cdot x}\va{0|T[J^{\mu\nu
  +}(x)J^{\alpha\beta}(0)]|0}.
 \label{eq-corr_mnab}
\ee (Note here that due to isospin symmetry this is the same
correlator which was studied in~\cite{bib-BO:97}.) This correlator
can be decomposed in invariant form factors $\Pi_\pm$,
\cite{bib-GRVW:87,bib-BB:96}
\be
 \Pi^{\mu \nu ;\alpha \beta }(q) =
   \Pi_-(q^2)P^{\mu \nu ;\alpha \beta}_1
  +\Pi_+(q^2)P^{\mu \nu ;\alpha \beta}_2 \,
\label{eq-4} \ee where the projectors $P_{1,2}$ are defined by the
expressions \ba
 P_1^{\mu \nu;\alpha \beta} &\equiv &
  \frac1{2q^2}\left[g^{\mu\alpha}q^\nu q^\beta - g^{\nu\alpha}q^\mu q^\beta
                  - g^{\mu\beta}q^\nu q^\alpha + g^{\nu\beta}q^\mu q^\alpha
               \right] \ ;\label{eq-2} \\
 P_2^{\mu \nu ;\alpha \beta} &\equiv &
  \frac12\left[g^{\mu\alpha}g^{\nu\beta} - g^{\mu\beta}g^{\nu\alpha}\right]
 - P_1^{\mu \nu;\alpha \beta} \ ,
\label{eq-3} \ea which obey the projector-type relations
\ba
\label{project}
 \left( P_i\cdot P_j \right)^{\mu \nu ;\alpha \beta}
\equiv
 P_i^{\mu \nu ;\sigma \tau}P_j^{\sigma \tau ;\alpha \beta} =
 \delta_{ij}P_i^{\mu \nu ;\alpha \beta}
 ~\mbox{(no sum over $i$)},~P_i^{\mu \nu ;\mu \nu}=3.
\ea
  Then, for the form factors $\Pi _\pm(q^2)$ it is
possible to use dispersion representations of the form
\be
\Pi_\pm(q^2)
   = \frac1\pi\int^\infty_0\frac{\rho_\pm(s) ds}{s-q^2}
   + \mbox{subtractions} \ , \label{eq-Disp}
\ee
which after the Borel transformation (with Borel parameter $M^2$) become
\be
\Pi_\pm(q^2) \to B\Pi_\pm(M^2)
  = \frac1{\pi M^2}\int^\infty_0\rho_\pm(s)
     e^{-s/M^2}ds \ .
\label{eq-5} \ee
A phenomenological model for the spectral density
$\rho^{\phen}(s)$ is usually taken in the form of ``lowest
resonances + continuum''
\be
 \rho_\pm^{\phen}(s)
  = \pm 2 \pi \left|f^T_{m}\right|^2 s \cdot \delta(s-m^2_m)
  + \rho_{\pm}^{\pert}(s)\theta(s-s_\pm) \ ,
\label{eq-ro_phen} \ee where $f^T_{m}$ and $m_m$ are the decay
constants and masses of the lowest meson resonances, $m=\rho,
\rho', b_1$, contributing to the correlator of interest, and
$\rho_\pm^{\pert}(s)$ are the corresponding spectral densities of
perturbative contributions to the correlators $\Pi_\pm(q^2)$. The
decay constants are defined via the parameterization of the unit
helicity ($|\lambda| = 1$) states of $\rho$-, $\rho'$- and
$b_1$-mesons \ba
 \va{0\left|\bar u(x)\sigma_{\mu\nu}d(x)
       \right|\rho^+(p,\lambda)(\rho'^+)}
 &=& i f^T_{\rho, \rho'} \left(\varepsilon_{\mu}(p,\lambda)p_{\nu}
          - \varepsilon_{\nu}(p,\lambda)p_{\mu}\right)\ ;
 \label{eq-6} \\
 \va{0\left|\bar u(x)\sigma_{\mu\nu}d(x)
       \right|b_1^+(p,\lambda)}
 &=& f^T_{b_1} \epsilon_{\mu\nu\alpha\beta}
        \varepsilon^\alpha(p,\lambda)p_{\beta} \ ,
 \label{eq-7}
\ea here $\varepsilon^\mu(p,\lambda)$ is the polarization vector
of a meson with momentum $p$ and helicity $\lambda$. To construct
SRs, one should calculate OPE of the correlators $\Pi_\pm(M^2)$
\be
 B\Pi_\pm(M^2)
  = \frac1{\pi M^2}\int^\infty_0
     \rho_{\pm}^{\pert}(s)e^{-s/M^2}ds
  + \frac{a_\pm}{M^2}\va{\frac{\as}\pi G^{2}}
  + \frac{b_\pm}{M^4}\pi\va{\sqrt{\as}\bar qq}^2\ .
 \label{eq-OPE}
\ee
   We perform these calculations in the approach of QCD SRs
with NLCs (see~\cite{bib-BM:98}), where the coefficients $a_\pm,
b_\pm$ become functions $a_\pm(M^2), b_\pm(M^2)$ of the Borel
parameter $M^2$ which tend to their standard values for large
$M^2$, $M^2 \gg \lambda_q^2$, \eg, $\Ds b_\pm =
\lim\limits_{(\lambda_q^2/M^2) \to 0}b_\pm(M^2)$. The functions
$a_\pm(M^2), b_\pm(M^2)$ can be considered as accumulating an
infinite subset of the standard condensate
$\left(\lambda_q^2/M^2\right)^j$-contributions~\cite{bib-MR:92} in
OPE. All needed NLC expressions are given in Appendix A, while the
standard coefficients $a_\pm, b_\pm $, corresponding to the limit
$\lambda_q^2/M^2 \to 0$, are explicitly written below. Their
values are in full agreement with the preceding calculations
performed in~\cite{bib-GRVW:87} and \cite{bib-BB:96} \ba
 \frac1{(\pm 2)}\rho^{\pert}_\pm(s)
 &=& \rho^{\pert}_0(s)\equiv
    \frac{s}{8\pi}
     \left[1+\frac{\as(\mu^2)}{\pi}
             \left(\frac79+\frac23\log\frac{s}{\mu^2}
              \right)
      \right] \ ; \label{eq-ro_pert} \\
 \frac1{(\pm 2)}a_\pm &=& \frac1{24} \ ;\label{eq-aGG} \\
  \frac1{(- 2)}b_-  &=& \frac{-16+80+144}{81}\ =\ \frac{208}{81} \;
   \label{eq-bq-}\\
 \frac1{(+ 2)}b_+ &=& \frac{-16+80-144}{81}\ =\ \frac{-80}{81}\ .
\label{eq-bq+} \ea Here $\mu$ is the renormalization scale ($\mu^2
\simeq 1~\gev{2}$) and the coefficients listed in the central
parts of the last two lines correspond to the vector $\va{\bar q
\gamma_m q}$,
 quark-gluon-quark $\va{\bar q G_{\mu\nu} q}$ and the four-quark
$\va{\bar q q \bar q q}$ vacuum condensate contributions (see
details in Appendix A,~\cite{bib-MR:92}). We write down these
coefficients explicitly in order to reveal the discrepancy between
our results and those obtained by B\&O~\cite{bib-BO:97}, who
found, instead, in the last line $$\frac{-16-48-144}{81} =
\frac{-208}{81},$$ a result larger than ours by a factor of $2.6$.
We conclude that in~\cite{bib-BO:97} there is a wrong contribution
due to the quark-gluon-quark vacuum condensate.

Collecting all parts~(\ref{eq-5}), (\ref{eq-ro_phen}),
(\ref{eq-OPE}) together, one obtains the following SRs:
\ba
\label{eq-SRrho}
 \left|f^T_{\rho}\right|^2 m_\rho^2 e^{-m_\rho^2/M^2}
 + (\rho \to \rho')
  = \frac1\pi \int^{s_\rho}_0 \rho_0^{\pert}(s) e^{-s/M^2}ds
  - \frac{a_-}{2}\va{\frac{\alpha_s}\pi G^{2}}
  - \frac{b_{-}(M^2)}{2M^2}\pi \va{\sqrt{\as} \bar qq}^2 \ ; && \\
  \label{eq-SRb1}
 \left|f^T_{b_1}\right|^2 m_{b_1}^2 e^{-m_{b_1}^2/M^2}
  = \frac1\pi \int^{s_{b_1}}_0 \rho_0^{\pert}(s) e^{-s/M^2}ds
  + \frac{a_+}{2}\va{\frac{\alpha_s}\pi G^{2}}
  + \frac{b_{+}(M^2)}{2M^2}\pi \va{\sqrt{\as} \bar qq}^2.&&
\ea The role of NLC, concentrated in $a_\pm,b_\pm(M^2)$, is
important here, \ie, at $M^2=0.6~\gev{2}$ the total condensate
contribution in the SR reduces twice in comparison with the
standard (local) one.
In accordance with QCD SR practice the processing of these NLC SRs
are performed within the validity window
$M_-^2 \leq M^2 \leq M_+^2$ (see details
in~\cite{bib-SVZ:79,bib-BM:98}).
These  windows are determined by two conditions:
the lower bound $M^2_-$ by demanding that the relative value
of $\langle GG\rangle$- and  $\langle\bar qq\rangle$-contributions
to OPE series should not be larger than 30\%,
the upper one $M^2_+$ by requiring that a relative contribution
of higher states in the phenomenological part of SR should not be
larger than 30\%.
The processing with the standard values of vacuum
condensates (see Appendix A) gives the decay constants
\ba
 f_{\rho}^T &=& 0.157 \pm 0.005~\Gev, \ \
  f_{\rho'}^T\ =\ 0.140 \pm 0.005~\Gev,  \ \
  s_{\rho,\rho'}^T\ =\ 2.8~\gev{2}
   \ ; \label{eq-fras} \\
 f_{b_1}^T  &=& 0.184 \pm 0.005~\Gev, \quad
  s_{b_1}^T\ =\ 2.87~\gev{2}
   \ , \label{eq-fbas}
\ea which are presented at normalization point $\mu^2=1$~\gev{2}.
Very stable curves in wide validity windows have been obtained for
all of these quantities.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% FIGURE SR-results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input{psbox}
 \begin{figure}[hb]\noindent
  \hspace*{0.05\textwidth}
   \begin{minipage}{0.9\textwidth}
    $$\psannotate{\psboxto(0.75\textwidth;0cm){dual_1.eps}}%
       {\at(4.3\pscm;-0.5\pscm){\large $M^2~[\gev{2}]$}}%
       {\at(-9.5\pscm;6.1\pscm){\large{\Large $f^T_{\rho}$}$~[\Gev]$}}%
       {\at(-8.9\pscm;4.25\pscm){\large $M^2_{-}$}}%
       {\at(-5.5\pscm;6.1\pscm){\large $M^2_{+}$}}%
       {\at(-2.08\pscm;3.7\pscm){\large $M^2_{+}$}}%
       {\at(-9.9\pscm;0.85\pscm){\large $M^2_{-}$}}%
    $$\vspace*{1pt}
      \caption{The curves of  $f^T_{\rho}$ in $M^2$;
       the solid line corresponds to NLC SR with the
       $\rho'$-meson taken into account,
       the long arrows show its validity window;
       the short-dashed line corresponds to the standard SR without
       $\rho'$-meson,
       the small arrows show reduced validity window for this case;
       the dashed line corresponds to B\&B analyses.}
       \end{minipage}
        \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The processing of the ``local" version
(at $\lambda_q^2 \to 0$) of the SRs
(\ref{eq-SRrho})-(\ref{eq-SRb1})
leads to the values~\footnote{%
To provide a clear comparison with the results of B\&O, who do not
take into account $\rho'$-meson contribution, condensate
nonlocality and $\alpha_s$-corrections in the perturbative
spectral density, we write down the results of processing our SRs
in the same approximation in parentheses.}: \ba
 f_{\rho}^T &=& 0.179 (0.170) \pm 0.007~\Gev, \ \
  f_{\rho'}^T\ \sim 0~\Gev,  \ \
  s_{\rho,\rho'}^T\ =\ 2.1~\gev{2}
   \ ; \label{eq-fras_loc} \\
 f_{b_1}^T  &=& 0.191 (0.178) \pm 0.009~\Gev, \quad
  s_{b_1}^T\ =\ 3.2~\gev{2}
   \ , \label{eq-fbas_loc}
\ea
which accuracy looks worse.
 Really, the curve corresponding to
$ f_{\rho}^T$ in $M^2$ is shown in Fig. 1 (solid line) in
comparison with the result of the standard approach without
$\rho'$-meson (short-dashed line). For the first case, the
validity window expands in all the region $(0.55-1.20)~\gev{2}$
(long arrows) while for the latter case it shrinks twice to the
region denoted in figure by the small arrows
   $M^2_-$ and $M^2_+$. Note, that the standard SR ``pushes out" the
$\rho'$ meson and does not allow to obtain his parameters, while
the NLC SR is sensitive to this meson and even allows to determine
its mass~\cite{bib-BM:98}. We demonstrate on the same figure the
curve  for $f_{\rho}^T$ (dashed line) obtained in~\cite{bib-BB:96}
by Ball and Braun (B\&B) in the framework of the standard
approach, the same small arrows denoting its real ``working
window". Note here that processing B\&B SR just in this thin
working window results in a curve very similar in shape to the
upper short-dashed one with the average value $f_{\rho}^T(1
\gev{2}) = 0.171~\Gev$. Note, that recently performed lattice
estimates~\cite{Beci98} \footnote{We are indepted to D.~Becirevic,
who informed us about these interesting papers, contaning lattice
estimates mass and decay constants of mesons.} give $f^T_{\rho
Latt}(4 \gev{2}) =0.165(11)~\gev{}$, that approximately agree with
both the NLC (\ref{eq-fras}) and the ``standard"
(\ref{eq-fras_loc}) values.
 So, we can conclude that our improved
SRs~(\ref{eq-SRrho})--(\ref{eq-SRb1}) are really justified and
produce reliable and stable results. All the results obtained by
processing ``pure parity" (\ref{eq-SRrho}, \ref{eq-SRb1}) and
``mixed parity" NLC SR~\cite{bib-BM:98} are collected in Table 1,
in comparison with the previous results
in~\cite{bib-BB:96,bib-BO:97}.
 \vspace*{3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%  TABLE 1   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent\hspace*{0.1\textwidth}
\parbox[h]{0.80\textwidth}{\hspace*{0.05\textwidth}
\begin{tabular}{|c||c|c|c||c|c||}\hline
%___________________________________________________________________
 &\multicolumn{3}{|c||}{\strut\vphantom{\vbox to 6mm{}}
  ``pure parity" SR
   $_{\vphantom{\vbox to 4mm{}}}$}
       &\multicolumn{2}{|c||}{\strut\vphantom{\vbox to 6mm{}}
  ``mixed parity" SR
  $_{\vphantom{\vbox to 4mm{}}}$}\\
 &\multicolumn{3}{|c||}{\strut\vphantom{\vbox to 6mm{}}
   based on $\Pi_\mp$ ($-$ for $\rho$, $+$ for $b_1$)
   $_{\vphantom{\vbox to 4mm{}}}$}
       &\multicolumn{2}{|c||}{\strut\vphantom{\vbox to 6mm{}}
   based on $(\Pi_- - \Pi_+)/q^2$
  $_{\vphantom{\vbox to 4mm{}}}$}\\ \hline
\hspace{4mm}Source\hspace{4mm}
 &\hspace{4mm}{\strut\vphantom{\vbox to 6mm{}}
                Here$_{\vphantom{\vbox to 4mm{}}}$}\hspace{4mm}
   &\hspace{2mm}B\&B ~\cite{bib-BB:96}\hspace{2mm}
     &\hspace{2mm}B\&O ~\cite{bib-BO:97}\hspace{2mm}
       &\hspace{4mm}Here\protect\footnotemark{}\hspace{4mm}
         &\hspace{2mm}B\&B ~\cite{bib-BB:96}\hspace*{2mm}\\ \hline \hline
%___________rho_____________________________________________________
 \hspace{3mm}{\strut\vphantom{\vbox to 6mm{}}
  $f^T_\rho \left[\mbox{MeV}\right]%
    _{\vphantom{\vbox to 4mm{}}}$}\hspace{3mm}
 &$157(5)$
   &$160(10)$
     &$-$
       &$166(6)$
          &$163(5)$ \\ \hline
%___________rho'____________________________________________________
 \hspace{3mm}{\strut\vphantom{\vbox to 6mm{}}
  $f^T_{\rho'}\left[\mbox{MeV}\right]%
    _{\vphantom{\vbox to 4mm{}}}$}\hspace{3mm}
 &$140(5)$
   &$-$
     &$-$
       &$-$
         &$-$ \\ \hline
%___________s_rho___________________________________________________
 \hspace{3mm}{\strut\vphantom{\vbox to 6mm{}}
  $s_{\rho}~\left[\gev{2}\right]%
    _{\vphantom{\vbox to 4mm{}}}$}\hspace{3mm}
 &$2.8$
   &$1.5$
     &$-$
       &$1.5$
         &$2.1$ \\ \hline \hline
%___________b_1_____________________________________________________
 \hspace{3mm}{\strut\vphantom{\vbox to 6mm{}}
  $f^T_{b_1}\left[\mbox{MeV}\right]%
    _{\vphantom{\vbox to 4mm{}}}$}\hspace{3mm}
 &$184(5)$
   &$180(10)$
     &$178(10)$
       &$179(7)$
         &$180^{\mbox{\tiny fixed}}$ \\ \hline
%___________s_b_1___________________________________________________
 \hspace{3mm}{\strut\vphantom{\vbox to 6mm{}}
  $s_{b_1}~\left[\gev{2}\right]%
    _{\vphantom{\vbox to 4mm{}}}$}\hspace{3mm}
 &$2.87$
   &$2.7$
     &$3.0$
       &$2.93$
         &$2.1$ \\ \hline \hline
%___________________________________________________________________
\end{tabular}
\\[2mm]
\noindent{\bf Table 1}. Estimates for decay constants
$f^T(1\gev{2})$ of transversely polarized
$\rho(770),~\rho'(1465)$- and $b_1 (1235)$-mesons based on
processing QCD SRs in different approaches. }
\vspace*{2mm}
\footnotetext{%
The estimates presented in this column have been obtained by
processing the ``mixed parity" SR established in~\cite{bib-BM:98}.
We improve the model for phenomenological spectral density using
the features of phenomenological spectral densities of ``pure
parity" SRs.}


 It is interesting to note that in spite of
the discrepancy in the OPE coefficients, the authors
of~\cite{bib-BO:97} obtain for $f_{b_1}^T$ a value of $178 \pm 10$
MeV which is quite close to the value found by B\&B
~\cite{bib-BB:96}: $180 \pm 10$ MeV. This compensation effect
happens due to the fact that both groups of authors used different
sets of the condensate input-parameters in the SR and this
resulted in approximately the same overall contributions of the
quark condensate: B\&B had $(\frac12 b_+)\pi\as\va{\bar qq}^2 = -
4.22\ 10^{-4}~\gev{6}$; and B\&O, $(\frac12 b_+)\pi\as\va{\bar
qq}^2 = - 4.92\ 10^{-4} \gev{6}$, ~see Appendix B.



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

\section*{Duality and its breakdown}
Let us consider now an operator $\hat D$ transforming any rank-4
tensor $T^{\mu \nu;\alpha \beta}$ to another rank-4 tensor
$T_D^{\mu \nu;\alpha\beta} = (\hat D T)^{\mu \nu;\alpha \beta}$
with
\be
 D^{\mu \nu;\alpha \beta}_{\mu' \nu';\alpha' \beta'} =
 \frac{-1}4 \epsilon^{\mu\nu}_{\phantom{\mu\nu}\mu'\nu'}
             \epsilon_{\alpha'\beta'}^{\phantom{\alpha'\beta'}\alpha\beta}\
             ~\mbox{and} ~\hat D^2 = 1 .
\label{eq-13} \ee Our projectors $P_1^{\mu \nu;\alpha \beta}$ and
$P_2^{\mu \nu;\alpha \beta}$ under the action of this operator
transform into each other
\be
 \left(\hat D P_1\right)^{\mu \nu;\alpha \beta}
   = P_2^{\mu \nu;\alpha \beta} \ ;\ \ \ \
 \left(\hat D P_2\right)^{\mu \nu;\alpha \beta}
   = P_1^{\mu \nu;\alpha \beta}, \
 \label{eq-14}
\ee
whereas the correlator $\Pi^{\mu \nu;\alpha \beta} (q)$ transforms into
the correlator of dual tensor currents
$J_5^{\mu\nu}(x) = \bar u(x)\sigma^{\mu\nu}\gamma_5 d(x)$
\be
  (\hat D\Pi)^{\mu\nu;\alpha\beta}(q) =
   \int d^{4}x\  e^{iq\cdot x}
    \va{0|T[J_{5}^{\mu\nu +}(x) J_5^{\alpha\beta}(0)]|0}\ .
 \label{eq-16}
\ee There is a good question: how are
$\Pi^{\mu\nu;\alpha\beta}(q)$ and $(\hat
D\Pi)^{\mu\nu;\alpha\beta}(q)$ connected? \\ In perturbative QCD
with massless fermions, taking into account the standard
anticommutations, one easily arrives at
\be
 (\hat D\Pi)^{\mu\nu;\alpha\beta}_{\pert}(q) =
 -\Pi^{\mu\nu;\alpha\beta}_{\pert}(q),
 \label{eq-18}
\ee from which it follows that
$\Pi^{\mu\nu;\alpha\beta}_{\pert}(q)$ is anti-self-dual. The same
(anti-dual) character is inherent in the phenomenological models,
see Eq.~(\ref{eq-ro_phen}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% FIGURE 4Q-condensate %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
 \vspace{-0.060\textheight}
  \begin{center}\mbox{\psboxto(8cm;0cm){fig4q.ps}}\end{center}
   \vspace{-0.070\textheight}
    \caption{Diagram with insertion of four-quark condensate.}
     \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The same reasoning is valid almost for all OPE diagrams, those
with a gluon condensate, with a vector quark condensate, and with
a quark-gluon-quark condensate. Only the diagram with four-quark
scalar condensates is different (see Fig.~2): in that case there
are 2 $\gamma$-matrices on one line between two external vertices
(1 from the fermion propagator and 1 from the quark-gluon vertex)
because the scalar condensate cancels one $\gamma$-matrix. Thus,
we realize that the OPE contribution involves two parts, one being
anti-self-dual ($\ASD$) and the other one self-dual ($\SD$)
\ba
 \Pi^{\mu\nu;\alpha\beta}_{\ope}(q)
 &=& \ASD^{\mu\nu;\alpha\beta}(q)
  + \SD^{\mu\nu;\alpha\beta}(q) \ ,\label{eq-19} \\
 (-\hat D \ASD)^{\mu\nu;\alpha\beta}(q)
 &=& \ASD^{\mu\nu;\alpha\beta}(q)
\ \equiv \ \Pi^{\asd}(q^2)\left( P_1^{\mu\nu;\alpha\beta}
 - P_2^{\mu\nu;\alpha\beta} \right) \ ,\label{eq-ASD}\\
 (\hat D \SD)^{\mu\nu;\alpha\beta}(q)
 &=& \SD^{\mu\nu;\alpha\beta}(q)
\ \equiv \ \Pi^{\sd}(q^2)\left( P_1^{\mu\nu;\alpha\beta}
 + P_2^{\mu\nu;\alpha\beta} \right) \ .\label{eq-SD}
\ea
The appearance of the $\SD$-diagrams breaks the anti-duality
of the two correlators $\Pi$ and $\left(\hat D \Pi \right)$.


  We can rewrite formulae~(\ref{eq-19})--(\ref{eq-SD}) to obtain
the following representation for the OPE-induced part of
correlator:
\be
 \Pi^{\mu\nu;\alpha\beta}_{\ope}(q)
  = P_1^{\mu\nu;\alpha\beta}
    \left[\Pi^{\sd}(q^2)+\Pi^{\asd}(q^2)
    \right]
 + P_2^{\mu\nu;\alpha\beta}
    \left[\Pi^{\sd}(q^2)-\Pi^{\asd}(q^2)
    \right]\ .
 \label{eq-OPE-D}
\ee
As a simple consequence of this representation and
Eq.(\ref{project}) we have a useful relation
\be
 \Pi^{\mu\nu;\mu\nu}_{\ope}(q)
  = 6\ \Pi^{\sd}(q^2)\ .
\label{eq-mnmn}
\ee

Using relations (\ref{eq-ASD})--(\ref{eq-SD}), one can easily
calculate the OPE coefficients for different diagrams.
For example, let us consider the $\va{\bar qGq}$-condensate
and its contribution to the coefficient $b_\pm$.
Indeed, we know that this contribution is of the $\ASD$-type,
that is
$$ \Pi_{\qq{G}}^{\mu\nu;\alpha\beta}
 = c(q^2)\left( P_1^{\mu\nu;\alpha\beta}
              - P_2^{\mu\nu;\alpha\beta}\right)\ . $$
Therefore, for a light-like vector $z$, one has $$\Pi_{\qq{G}}
 = \Pi_{\qq{G}}^{\mu\nu;\alpha\beta} g_{\mu\alpha} z_\nu z_\beta
 = c(q^2)\frac{2 (q \cdot z)^2}{q^2}\ . $$
This quantity reduces to the linear combination of $\va{\bar qGq}$
condensate contributions to the correlator for vector currents
(see~\cite{bib-MR:92,bib-BM:98}). In this way, we get the formula
$$\Pi_{\qq{G}} = \frac{-320(q\cdot z)^2}{81q^6}\pi\as\va{\bar
qq}^2$$ from which we then obtain the fraction $80/81$ appearing
in Eqs.~(\ref{eq-bq-})--(\ref{eq-bq+}).

If $\SD_{\mu\nu;\alpha\beta}(q) = 0$, then we would have the same
SRs both for $\rho$- and $b_1$-mesons. We process this
hypothetical SR within the standard approach without
$\alpha_s$-corrections in the perturbative contribution and obtain
the following values for the low-energy parameters of a
hypothetical $\rho b_1$-meson in Anti-Dual Nature:
\be
 m_{\rho b_1} = (0.865 \pm 0.030)~\Gev\ ; \quad
 f_{\rho b_1} = (0.162\pm 0.005)~\Gev\ ; \quad
 s_{\rho b_1} = 1.58~\gev{2}\ . \label{eq-20}
\ee
We see that the mass and the decay constant of the
$\rho$-meson are not so much affected by this (anti-)duality
breakdown (10\% for the mass). The case of the $b_1$-meson is
quite opposite. Here the mass falls down to 45\% (the decay
constant, to 16\%, see~(\ref{eq-fbas_loc})). This seems to be
quite natural. In the case of the $\rho$-meson, the deformation of
the SR is large (the quark condensate contribution is enhanced by
a factor of $3.25$), but its functional dependence on the Borel
parameter $M^2$ is almost the same. This is not the case for the
$b_1$-meson. The deformation of the SR due to the opposite sign of
the quark condensate contribution is essential and this results in
such a large effect for the mass of the $b_1$-meson.


\section*{QCD vacuum tensor susceptibility}

 The QCD vacuum tensor susceptibility $\chi$  has been introduced
in~\cite{bib-HJ:95,bib-HJ:96} in order to analyze, in the QCD SR
approach, the nucleon tensor charges $g_T^u$ and $g_T^d$. It is
defined through the correlator~(\ref{eq-corr_mnab})
\be
 \chi = \frac{\Pi_\chi(0)}{6\qq{}}\ , \quad
  \Pi\chi(q^2)\equiv \Pi^{\mu\nu;\mu\nu}(q^2)\ .
 \label{eq-21}
\ee
He and Ji~\cite{bib-HJ:95} obtained for $\Pi_\chi(0)$ the value
\be
 \frac1{12}\Pi_\chi(0) \approx 0.002~\gev{2}\ .
 \label{eq-22}
\ee

  In order to obtain a reliable estimate in our approach
we substitute the decomposition (\ref{eq-4}) in (\ref{eq-21}),
use the relation (\ref{eq-mnmn}) and arrive at the expression 
\ba
 \Pi_\chi(q^2)
  = 3 \left(\Pi_+(q^2) +  \Pi_-(q^2)\right)
  = 6 \ \Pi^{\sd}(q^2)\ .
  \label{eq-31}
\ea 
This relation clearly demonstrates that $\Pi_{\chi}$ 
is formed by the \SD part of OPE, \ie, 
by the {\bf four-quark condensate} contribution. 
Using the dispersion relations~(\ref{eq-Disp})
\be
 \frac1{12}\Pi_{\chi}(0)
  = \frac{1}{4\pi}\int_0^\infty
     \frac{\rho_{+}^\phen(s)+\rho_{-}^\phen(s)}{s} d s\ ,
 \label{eq-PiDS}
\ee
and the phenomenological models for spectral densities 
$\rho_\pm(s)$ in~(\ref{eq-ro_phen}),
the value of $\Pi_\chi(0)$ can be
expressed in terms of mesonic static characteristics
(the analogous formula has been published in~\cite{bib-Bochum:98}),
\be \label{eq-Pi0_our}
 \frac1{12}\Pi{_\chi}(0) =
  \frac{(f^T_{b_1})^2 - (f^T_\rho)^2 - (f^T_{\rho'})^2}2
+ \frac{s_{\rho,\rho'}-s_{b_1}}{16\pi^2}
   = \left\{
     \begin{array}{ll}
        - 0.0055 \pm 0.0008~\gev{2}&[ \mbox{\small NLC} ]\\
        - 0.0053 \pm 0.0021~\gev{2}&[ \mbox{\small Standard} ]
     \end{array}
     \right.\ ,
\ee
presented in (\ref{eq-fras})--(\ref{eq-fbas}) for NLC SR and
(\ref{eq-fras_loc})--(\ref{eq-fbas_loc}) for the standard SR
respectively\footnote{%
Depicted errors are obtained by a special invented $\chi^2$-criterium
and take into account only the SR stabiliy.}.
Note here, both the resuts are very close one to
another due to strong cancellations in the difference (\ref{eq-Pi0_our}). 
So, just this combination accumulates the
effect of the four-quark part of the whole condensate
contribution. If we return to the example of an anti-dual Nature 
(see the end of the previous section,~(\ref{eq-20})) where this
contribution is absent, we obtain the exact cancellation
in~(\ref{eq-Pi0_our}), \ie $\Pi{_\chi}(0) = 0$.

B\&O in~\cite{bib-BO:97} have used the specific representation
that leads to the decomposition
\be
 \Pi_\chi(q^2) = 12\ \Pi_1(q^2) + 6 q^2\Pi_2(q^2)\ ,
 \label{eq-24}
\ee
where $\Pi_1(q^2) = \frac12 \Pi_+(q^2)$
and $q^2\Pi_2(q^2) = \frac12 \left(\Pi_-(q^2) - \Pi_+(q^2)\right)$.
Erroneously suggesting that
$\Ds \lim_{q^2 \to 0}\left[q^2\Pi_2(q^2)\right]=0$
and using the trick suggested in~\cite{bib-BK:93}
based on the dispersion relation\footnote{%
Here $\rho^\phen(s)$ and $\rho^\pert(s)$ are 
the corresponding spectral densities; 
the difference of these functions validates
the usage of unsubtracted dispersion relation}
\be
 \Pi^{\nonp}_{\chi}(0) \equiv \frac1{12}\Pi_{\chi}(0)
  = \frac{1}{\pi}\int_0^\infty
     \frac{\rho^\phen(s)-\rho^\pert(s)}{s} d s\ ,
 \label{eq-PiDS}
\ee
they concluded that
\be
 \Pi_\chi(0)^{\nonp} = \Pi_1^{\nonp}(0)
  = (f^T_{b_1})^2 - \frac{s_{b_1}}{8\pi^2}
    \approx - 0.008~\gev{2} \ .
 \label{eq-34}
\ee
But we see from our analysis that the value of 
$\left(\Pi_-(q^2) - \Pi_+(q^2)\right)$ is identically equal to $0$
only in an absolutely self-dual world, which is definitely not
realized in QCD
\be \label{eq-Pi0}
\frac{\Pi_-(0) - \Pi_+(0)}{2}
  = \frac{s_{\rho,\rho'}+s_{b_1}}{8\pi^2}
  - (f^T_{\rho})^2 - (f^T_{\rho'})^2 - (f^T_{b_1})^2
   = - 0.0060 \pm 0.0017~\gev{2}\  [\mbox{\small NLC}]\ .
 \label{eq-35}
\ee
This value is comparable with the value of the B\&O
estimate~(\ref{eq-34}) for $\Pi_1^{\nonp}(0)$ and should be
definitely taken into account. Comparing two estimates, our
(\ref{eq-Pi0_our}) and B\&O (\ref{eq-34}), one sees not so large
deviation from one another. It should not be taken by surprise
because radiative corrections significantly reduce the B\&O value
to $\Pi_1^{\nonp}(0)\approx - 0.003~\gev{2}$. For this reason the
actual magnitude of our total correction to this estimate is of an
order of 100\%. 
When our paper was finished we find the paper~\cite{bib-Bochum:98}
which contains an estimate of the correlator, 
$\frac1{12}\Pi_{\chi}(0)=-(0.0083-0.0104)~\gev{2}$,
using the constituent quark model. 
The authors of the paper have determined also a rather wide window
for VTS by analog of Eq.(\ref{eq-Pi0_our})
using QCD SR results from different literature sources:
$\frac1{12}\Pi_{\chi}(0)=-(0.0042-0.0104)~\gev{2}$.
As we pointed out in the Introduction, 
since these meson constants appear in VTS in a form of a difference, 
one has to define them more precisely and 
in the framework of a unified approach. 
So the large width of this window is not surprise for us.

Finally, let us briefly discuss the effect of our estimate of VTS
on the nucleon tensor charges. Here we follow to pioneering paper
by He and Ji~\cite{bib-HJ:96} where these charges were roughly
estimated using two types of SRs. Our result~(\ref{eq-Pi0_our})
increases the lower (decreases the upper) boundary for the $g_T^u$
($g_T^d$) charge approximately by a factor of $1.4$: \ba
 g_T^u &=& 1.47 \pm 0.76 \ ; \label{eq-gTu}\\
 g_T^d &=& 0.025 \pm 0.008 \ . \label{eq-gTd}
\ea
(The results of He and Ji
 $g_T^u = 1.33 \pm 0.53$ and $g_T^d = 0.04 \pm 0.02$
 have been obtained for too low, in our opinion, value
 of $\Lambda_\qcd = 100 \Mev $.
 We, instead, use the value of $\Lambda_\qcd = 250 \Mev $.)
\vspace{5mm}

\noindent\textbf{\large Acknowledgments} \vspace*{3mm}

This work was supported in part by the COSY Forschungsprojeht
J\"ulich/Bochum. We are grateful to O.~V.~Teryaev, discussions
with whom inspired this work, to R.~Ruskov and N.~G.~Stefanis for
fruitful discussions. One of us (A.B.) is indebted to Prof.
K.~Goeke and N.~G.~Stefanis for warm hospitality at Bochum
University.

\begin{appendix}
\vspace*{12mm}
\appendix
\hspace*{2mm}{\Large \bf Appendix}
\vspace{-3mm}
\section{Expressions for nonlocal contributions to SR}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\label{subs-A.1}
 \setcounter{equation}{0}
The form of contributions of NLCs to OPE~(\ref{eq-OPE}) depends on
a model of NLC. At the same time the final results of SR
processing demonstrate stability to the variations of the NLC
model provided the scale of the average vacuum quark virtuality
$\lambda_q^2$ is fixed. Here we use the model (delta-ansatz)
suggested in~\cite{bib-MR:92} and used extensively
in~\cite{bib-BM:98}; the model leads to Gaussian decay for scalar
quark condencate \footnote{Here $E(0,z)=P\exp(i \int_0^z dt_{\mu}
A^a_{\mu}(t)\tau_a)$ is the Schwinger phase factor required for
gauge invariance.}, $\Ds \va{ \bar{q}(0)E(0,z) q(z)} \sim  \va{
\bar{q} q} \exp\left( -|z^2|\lambda_q^2 /8 \right) $ (see details
in~\cite{bib-MR:92}), dominated in $b_\pm$ via $b_4$. Here the
factorization hypotheses is applied for the four-quark condensate
reducing its contribution to a pair of scalar condensates. In the
NLC approach this may leads to an overestimate of the four-quark
condensate contribution due to an evident neglecting of a
correlation between these scalar condensates, see Fig.2.

In this model we obtain the ``coefficients" for OPE in the SR
(\ref{eq-SRrho})--(\ref{eq-SRb1})
\be
 \frac{b_\mp\left(M^2\right)}{\mp 2}
  = b_2\left(M^2\right)
  + b_3\left(M^2\right)
 \pm b_4\left(M^2\right) \ ,
\ee
where $b_2$ corresponds to the vector ($\va{\bar q \gamma_m
q}$), $b_3$ to the quark-gluon-quark ($\va{\bar q G_{\mu\nu} q}$)
and $b_4$ to the four-quark ($\va{\bar q q \bar q q}$) vacuum
condensate contributions (here $\Delta \equiv
\lambda_q^2/(2M^2)$), \ba
 b_2\left(M^2\right) &=& - 16\ ; \label{eq:b_2} \\
 b_3\left(M^2\right) &=&
  \frac{4\left(60 - 273\Delta +359\Delta^2 -134\Delta^3
         \right)}
       {3\left(1-\Delta
         \right)^3}\ ; \label{eq:b_3} \\
 b_4\left(M^2\right) &=&
  24\left(\Delta-7\right)
     \frac{\log\left(1-\Delta
               \right)}{\Delta}
 + 4\frac{25\Delta^2-21\Delta-6}
         {\left(1-\Delta
          \right)^2}\ . \label{eq:b_4}\
\ea The gluonic contribution $a_\pm$ coincides in this model with
the standard expression~(\ref{eq-aGG}). For quark and gluon
condensates we use the standard estimates (for ``renorm-invariant"
quantities in~(\ref{eq-vc_AB}) we do not refer to any
normalization point)
\ba
 \va{\frac{\as}{\pi} G^2}  = 1.2 \cdot 10^{-2}~\gev{4}\ ,~~
 \va{\sqrt{\as}\bar{q}q}^2 = 1.83\cdot 10^{-4}~\gev{6}\ ,
 \label{eq-vc_AB} \\
 \lambda_q^2\left(\mu^2\approx 1~\gev{2}\right)
  \equiv \frac{\va{\bar{q}\nabla^2 q}}{\va{\bar{q}q}}
       = \frac{\va{\bar{q}\left(ig\sigma_{\mu\nu}G^{\mu\nu}\right)q}}
              {2\va{\bar{q}q}}
   = 0.4 \pm 0.1~\gev{2}\ .
 \label{eq-lq_AB}
\ea

\section{Input parameters in B\&B and B\&O papers}
\label{subs-A.2}
 \setcounter{equation}{0}
Both groups of the authors of~\cite{bib-BB:96} and
\cite{bib-BO:97} used different definitions of initial parameters
for processing the SRs. Namely, B\&O used the following set of
values (without any indication on the scale at which
renormalization non-invariant quantities are determined):
\be
 \as \approx 0.1 \pi = 0.314\ , \quad
   4\pi^2 \va{\bar qq} = -0.55~\gev{3} \ , \quad
   \va{\sqrt{\as} \bar qq}^2 \approx 0.61 \cdot 10^{-4}~\gev{6}\ ,
 \label{eq-vc_BO}
\ee
whereas B\&B\footnote{%
Let us remind that the standard value is $\va{\sqrt{\as} \bar q
q}^2 \approx 1.83\cdot 10^{-4}~\gev{6}$~\cite{bib-SVZ:79}.} (on
the scale $\mu^2 \approx 1~\gev{2}$)
\be \as = 0.56 , \quad
   \va{\bar qq} = (- 0.250)^3~\gev{3}\ , \quad
   \va{\sqrt{\as} \bar qq}^2 \approx 1.37 \cdot 10^{-4}~\gev{6}\ .
 \label{eq-vc_BB}
\ee
This resulted in approximately the same overall contributions
of the quark condensate in both papers,
see the end of Sect.~2.

\end{appendix}

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\bibitem{bib-BM:98}
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\bibitem{Book98}
Particle Data Group Booklet, July 1998.                       %4

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\bibitem{piv91} V.~M.~Belyaev, and B.~L.~Ioffe,                 %10
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\end{thebibliography}

\end{document}

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  \message{PSBOX(\temp) loading}%
\else
    \ifdim\temp cm>\psboxversion cm
      \message{PSBOX(\temp) loading}%
    \else
      \message{PSBOX(\psboxversion) is already loaded: I won't load
        PSBOX(\temp)!}%
      \let\temp=\psboxversion
      \let\tempp=\endinput
    \fi
\fi
\tempp
\let\psboxversion=\temp
\catcode`\@=11
% Every macro likes a little privacy...
%
%Trying to tame the variety of \special commands for Postscript: the
%  universal internal command \PSspeci@l##1##2 takes ##1 to be the
%  filename and ##2 to be the integer scale factor*1000 (as for usual
%   TeX \scale commands)
%
\def\psfortextures{%     For TeXtures on the Macintosh
%-----------------
\def\PSspeci@l##1##2{%
\special{illustration ##1\space scaled ##2}%
}}%
\def\psfordvitops{%      For the DVItoPS converter on IBM mainframes
%----------------
\def\PSspeci@l##1##2{%
\special{dvitops: import ##1\space \the\drawingwd \the\drawinght}%
}}%
\def\psfordvips{%      For DVIPS converter on VAX, UNIX and PC's
%--------------
\def\PSspeci@l##1##2{%
%    \special{/@scaleunit 1000 def}% never read dox without trying!
\d@my=0.1bp \d@mx=\drawingwd \divide\d@mx by\d@my% BUG! for large \drawingwd
\special{PSfile=##1\space llx=\psllx\space lly=\pslly\space%
urx=\psurx\space ury=\psury\space rwi=\number\d@mx
}}}%
\def\psforoztex{%        For the OzTeX shareware on the Macintosh
%--------------
\def\PSspeci@l##1##2{%
\special{##1 \space
      ##2 1000 div dup scale
      \number-\psllx\space \number-\pslly\space translate
}}}%
\def\psfordvitps{%       From the UNIX TeXPS package, vers.>3.12
%---------------
% Convert a dimension into the number \psn@sp (in scaled points)
\def\psdimt@n@sp##1{\d@mx=##1\relax\edef\psn@sp{\number\d@mx}}
\def\PSspeci@l##1##2{%
% psfig.psr contains the def of "startTexFig": if you can locate it
% and include the correct pathname, it should work
\special{dvitps: Include0 "psfig.psr"}% contains def of "startTexFig"
\psdimt@n@sp{\drawingwd}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\drawinght}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psllx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\pslly bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psurx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psury bp}
\special{dvitps: Literal "\psn@sp\space startTexFig\space"}
\special{dvitps: Include1 "##1"}
\special{dvitps: Literal "endTexFig\space"}
}}%
\def\psfordvialw{%   Try for dvialw, a UNIX public domain
%---------------
\def\PSspeci@l##1##2{
\special{language "PostScript",
position = "bottom left",
literal "  \psllx\space \pslly\space translate
  ##2 1000 div dup scale
  -\psllx\space -\pslly\space translate",
include "##1"}
}}%
\def\psforptips{%   For MS-DOS; LUOMA@brandeis.bitnet
%---------------
\def\PSspeci@l##1##2{{
\d@mx=\psurx bp
\advance \d@mx by -\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\incm{\d@mx}
\let\tmpx\dimincm
\d@my=\psury bp
\advance \d@my by -\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\incm{\d@my}
\let\tmpy\dimincm
\d@mx=-\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\d@my=-\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\at(\d@mx;\d@my){\special{ps:##1 x=\tmpx, y=\tmpy}}
}}}%
\def\psonlyboxes{%     Draft-like behaviour if none of the others works
%---------------
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1})}}\hss}}}}
}}%
\def\psloc@lerr#1{%
\let\savedPSspeci@l=\PSspeci@l%
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1}) #1}}\hss}}}}
\let\PSspeci@l=\savedPSspeci@l% restore normal output for other figs!
}}%
%\def\psfor...  add your own!
%
% Some common defs
%
\newread\pst@mpin
\newdimen\drawinght\newdimen\drawingwd
\newdimen\psxoffset\newdimen\psyoffset
\newbox\drawingBox
\newcount\xscale \newcount\yscale \newdimen\pscm\pscm=1cm
\newdimen\d@mx \newdimen\d@my
\newdimen\pswdincr \newdimen\pshtincr
\let\ps@nnotation=\relax
{\catcode`\|=0 |catcode`|\=12 |catcode`|%=12 |catcode`~=12
|catcode`#=12 |catcode`*=14
|xdef|backslashother{\}*
|xdef|percentother{%}*
|xdef|tildeother{~}*
|xdef|sharpother{#}*
}%
% useful to display special chars in \tt; fails for \,#,%
\def\R@moveMeaningHeader#1:->{}%
\def\uncatcode#1{%
\edef#1{\expandafter\R@moveMeaningHeader\meaning#1}}%
%
\def\execute#1{#1}% NOT stupid: cs in #1 are then identified BEFORE execution
\def\psm@keother#1{\catcode`#112\relax}% borrowed from latex
\def\executeinspecs#1{%
\execute{\begingroup\let\do\psm@keother\dospecials\catcode`\^^M=9#1\endgroup}}%
\def\@mpty{}%
% \if\matchin#1#2<=> \iftrue if #1 contains #2, <=>\iffalse otherwise:
% \if\matchexpin: idem, but #1 & #2 are first fully expanded (no \if
% inside!)
% \tmpa & \tmpb contain what's before and after the occurence of #2
\def\matchexpin#1#2{
  \fi%
%\message{(#1>#2)}
  \edef\tmpb{{#2}}%
  \expandafter\makem@tchtmp\tmpb%
  \edef\tmpa{#1}\edef\tmpb{#2}%
  \expandafter\expandafter\expandafter\m@tchtmp\expandafter\tmpa\tmpb\endm@tch%
  \if\match%
}%
\def\matchin#1#2{%
  \fi%
  \makem@tchtmp{#2}%
  \m@tchtmp#1#2\endm@tch%
  \if\match%
}%
\def\makem@tchtmp#1{\def\m@tchtmp##1#1##2\endm@tch{%
  \def\tmpa{##1}\def\tmpb{##2}\let\m@tchtmp=\relax%
  \ifx\tmpb\@mpty\def\match{YN}%
  \else\def\match{YY}\fi%
}}%
% converts any dimen in cm, with 1E-4 cm precision
\def\incm#1{{\psxoffset=1cm\d@my=#1
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\number\d@mx.}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \edef\dimincm{\dimincm\number\d@mx}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\dimincm\number\d@mx}
}}%
%
%  \ReadPSize{PSfilename} reads the dimensions of a PostScript drawing
%      and stores it in \drawinght(wd)
\newif\ifNotB@undingBox
\newhelp\PShelp{Proceed: you'll have a 5cm square blank box instead of
your graphics (Jean Orloff).}%
\def\s@tsize#1 #2 #3 #4\@ndsize{
  \def\psllx{#1}\def\pslly{#2}%
  \def\psurx{#3}\def\psury{#4}%  needed by a crazyness of dvips!
  \ifx\psurx\@mpty\NotB@undingBoxtrue% this is not a valid one!
  \else
    \drawinght=#4bp\advance\drawinght by-#2bp
    \drawingwd=#3bp\advance\drawingwd by-#1bp
%  !Units related by crazy factors as bp/pt=72.27/72 should be BANNED!
  \fi
  }%
\def\sc@nBBline#1:#2\@ndBBline{\edef\p@rameter{#1}\edef\v@lue{#2}}%
\def\g@bblefirstblank#1#2:{\ifx#1 \else#1\fi#2}%
{\catcode`\%=12
\xdef\B@undingBox{%%BoundingBox}}%
%% is not a true comment in PostScript, even if % is!
\def\ReadPSize#1{
 \readfilename#1\relax
 \let\PSfilename=\lastreadfilename
 \openin\pst@mpin=#1\relax
 \ifeof\pst@mpin \errhelp=\PShelp
   \errmessage{I haven't found your postscript file (\PSfilename)}%
   \psloc@lerr{was not found}%
   \s@tsize 0 0 142 142\@ndsize
   \closein\pst@mpin
 \else
% each entry in \GlobalInputList should be unique
   \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
   \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
     \immediate\write\psbj@inaux{\lastreadfilename,}%
   \fi%
   \loop
     \executeinspecs{\catcode`\ =10\global\read\pst@mpin to\n@xtline}%
     \ifeof\pst@mpin
       \errhelp=\PShelp
       \errmessage{(\PSfilename) is not an Encapsulated PostScript File:
           I could not find any \B@undingBox: line.}%
       \edef\v@lue{0 0 142 142:}%
       \psloc@lerr{is not an EPSFile}%
       \NotB@undingBoxfalse
     \else
       \expandafter\sc@nBBline\n@xtline:\@ndBBline
       \ifx\p@rameter\B@undingBox\NotB@undingBoxfalse
         \edef\t@mp{%
           \expandafter\g@bblefirstblank\v@lue\space\space\space}%
         \expandafter\s@tsize\t@mp\@ndsize
       \else\NotB@undingBoxtrue
       \fi
     \fi
   \ifNotB@undingBox\repeat
   \closein\pst@mpin
 \fi
\message{#1}%
}%
%
% \psboxto(xdim;ydim){psfilename}: you specify the dimensions and
%    TeX uniformly scales to fit the largest one. If xdim=0pt, the
%    scale is fully determined by ydim and vice versa.
%    Notice: psboxes are a real vboxes; couldn't take hbox otherwise all
%    indentation and all cr's would be interpreted as spaces (hugh!).
%
\def\psboxto(#1;#2)#3{\vbox{%
   \ReadPSize{#3}%
   \advance\pswdincr by \drawingwd
   \advance\pshtincr by \drawinght
   \divide\pswdincr by 1000
   \divide\pshtincr by 1000
   \d@mx=#1
   \ifdim\d@mx=0pt\xscale=1000
         \else \xscale=\d@mx \divide \xscale by \pswdincr\fi
   \d@my=#2
   \ifdim\d@my=0pt\yscale=1000
         \else \yscale=\d@my \divide \yscale by \pshtincr\fi
   \ifnum\yscale=1000
         \else\ifnum\xscale=1000\xscale=\yscale
                    \else\ifnum\yscale<\xscale\xscale=\yscale\fi
              \fi
   \fi
   \divide\drawingwd by1000 \multiply\drawingwd by\xscale
   \divide\drawinght by1000 \multiply\drawinght by\xscale
   \divide\psxoffset by1000 \multiply\psxoffset by\xscale
   \divide\psyoffset by1000 \multiply\psyoffset by\xscale
   \global\divide\pscm by 1000
   \global\multiply\pscm by\xscale
   \multiply\pswdincr by\xscale \multiply\pshtincr by\xscale
   \ifdim\d@mx=0pt\d@mx=\pswdincr\fi
   \ifdim\d@my=0pt\d@my=\pshtincr\fi
   \message{scaled \the\xscale}%
 \hbox to\d@mx{\hss\vbox to\d@my{\vss
   \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
      \kern-\psyoffset
      \PSspeci@l{\PSfilename}{\the\xscale}%
      \vss}\hss\ps@nnotation}%
   \global\wd\drawingBox=\the\pswdincr
   \global\ht\drawingBox=\the\pshtincr
   \global\drawingwd=\pswdincr
   \global\drawinght=\pshtincr
   \baselineskip=0pt
   \copy\drawingBox
 \vss}\hss}%
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm %should not be necessary
}}%
%
% \psboxscaled{scalefactor*1000}{PSfilename} allows to bypass the
%   rounding errors of TeX integer divisions for situations where the
%   TeX box should fit the original BoundingBox with a precision
%   better
%   than 1/1000.
%
\def\psboxscaled#1#2{\vbox{%
  \ReadPSize{#2}%
  \xscale=#1
  \message{scaled \the\xscale}%
  \divide\pswdincr by 1000 \multiply\pswdincr by \xscale
  \divide\pshtincr by 1000 \multiply\pshtincr by \xscale
  \divide\psxoffset by1000 \multiply\psxoffset by\xscale
  \divide\psyoffset by1000 \multiply\psyoffset by\xscale
  \divide\drawingwd by1000 \multiply\drawingwd by\xscale
  \divide\drawinght by1000 \multiply\drawinght by\xscale
  \global\divide\pscm by 1000
  \global\multiply\pscm by\xscale
  \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
     \kern-\psyoffset
     \PSspeci@l{\PSfilename}{\the\xscale}%
     \vss}\hss\ps@nnotation}%
  \advance\pswdincr by \drawingwd
  \advance\pshtincr by \drawinght
  \global\wd\drawingBox=\the\pswdincr
  \global\ht\drawingBox=\the\pshtincr
  \global\drawingwd=\pswdincr
  \global\drawinght=\pshtincr
  \baselineskip=0pt
  \copy\drawingBox
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm
}}%
%
%  \psbox{PSfilename} makes a TeX box having the minimal size to
%      enclose the picture
\def\psbox#1{\psboxscaled{1000}{#1}}%
%------------------------------------------------------
%  \joinfiles file1, file2, ...n \into joinedfilename .
%     makes one file out of many
%  \splitfile joinedfilename
%     the opposite
\newif\ifn@teof\n@teoftrue
\newif\ifc@ntrolline
\newif\ifmatch
\newread\j@insplitin
\newwrite\j@insplitout
\newwrite\psbj@inaux
\immediate\openout\psbj@inaux=psbjoin.aux
\immediate\write\psbj@inaux{\string\joinfiles}%
\immediate\write\psbj@inaux{\jobname,}%
%
% INPUT REDEFINITION
%
% works if #1 is a single character
\def\toother#1{\ifcat\relax#1\else\expandafter%
  \toother@ux\meaning#1\endtoother@ux\fi}%
\def\toother@ux#1 #2#3\endtoother@ux{\def\tmp{#3}%
  \ifx\tmp\@mpty\def\tmp{#2}\let\next=\relax%
  \else\def\next{\toother@ux#2#3\endtoother@ux}\fi%
\next}%
%
% \readfilename defs:
%
\let\readfilenamehook=\relax
\def\re@d{\expandafter\re@daux}% spares typing 10 \expandafter's...
\def\re@daux{\futurelet\nextchar\stopre@dtest}%
\def\re@dnext{\xdef\lastreadfilename{\lastreadfilename\nextchar}%
  \afterassignment\re@d\let\nextchar}%
\def\stopre@d{\egroup\readfilenamehook}%
\def\stopre@dtest{%
  \ifcat\nextchar\relax\let\nextread\stopre@d
  \else
    \ifcat\nextchar\space\def\nextread{%
      \afterassignment\stopre@d\chardef\nextchar=`}%
    \else\let\nextread=\re@dnext
      \toother\nextchar
      \edef\nextchar{\tmp}%
    \fi
  \fi\nextread}%
\def\readfilename{\bgroup%
  \let\\=\backslashother \let\%=\percentother \let\~=\tildeother
  \let\#=\sharpother \xdef\lastreadfilename{}%
  \re@d}%
%
% redefines \input using \readfilename
%
\xdef\GlobalInputList{\jobname}%
\def\psnewinput{%
  \def\readfilenamehook{% each entry in \GlobalInputList should be unique
    \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
    \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
      \immediate\write\psbj@inaux{\lastreadfilename,}%
    \fi%
    \ps@ldinput\lastreadfilename\relax%
    \let\readfilenamehook=\relax%
  }\readfilename%
}%
\expandafter\ifx\csname @@input\endcsname\relax    % then Plain
  \immediate\let\ps@ldinput=\input\def\input{\psnewinput}%
\else
  \immediate\let\ps@ldinput=\@@input
  \def\@@input{\psnewinput}%
\fi%
%
\def\nowarnopenout{%
 \def\warnopenout##1##2{%
   \readfilename##2\relax
   \message{\lastreadfilename}%
   \immediate\openout##1=\lastreadfilename\relax}}%
\def\warnopenout#1#2{%
 \readfilename#2\relax
 \def\t@mp{TrashMe,psbjoin.aux,psbjoint.tex,}\uncatcode\t@mp
 \if\matchexpin{\t@mp}{\lastreadfilename,}%
 \else
   \immediate\openin\pst@mpin=\lastreadfilename\relax
   \ifeof\pst@mpin
     \else
     \errhelp{If the content of this file is so precious to you, abort (ie
press x or e) and rename it before retrying.}%
     \errmessage{I'm just about to replace your file named \lastreadfilename}%
   \fi
   \immediate\closein\pst@mpin
 \fi
 \message{\lastreadfilename}%
 \immediate\openout#1=\lastreadfilename\relax}%
% % will have an unusual catcode below; use * instead
%\vbox
{\catcode`\%=12\catcode`\*=14
\gdef\splitfile#1{*
 \readfilename#1\relax
 \immediate\openin\j@insplitin=\lastreadfilename\relax
 \ifeof\j@insplitin
   \message{! I couldn't find and split \lastreadfilename!}*
 \else
   \immediate\openout\j@insplitout=TrashMe
   \message{< Splitting \lastreadfilename\space into}*
   \loop
     \ifeof\j@insplitin
       \immediate\closein\j@insplitin\n@teoffalse
     \else
       \n@teoftrue
       \executeinspecs{\global\read\j@insplitin to\spl@tinline\expandafter
         \ch@ckbeginnewfile\spl@tinline%Beginning-Of-File-Named:%\endcheck}*
       \ifc@ntrolline
       \else
         \toks0=\expandafter{\spl@tinline}*
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \fi
   \ifn@teof\repeat
   \immediate\closeout\j@insplitout
 \fi\message{>}*
}*
\gdef\ch@ckbeginnewfile#1%Beginning-Of-File-Named:#2%#3\endcheck{*
 \def\t@mp{#1}*
 \ifx\@mpty\t@mp
   \def\t@mp{#3}*
   \ifx\@mpty\t@mp
     \global\c@ntrollinefalse
   \else
     \immediate\closeout\j@insplitout
     \warnopenout\j@insplitout{#2}*
     \global\c@ntrollinetrue
   \fi
 \else
   \global\c@ntrollinefalse
 \fi}*
\gdef\joinfiles#1\into#2{*
 \message{< Joining following files into}*
 \warnopenout\j@insplitout{#2}*
 \message{:}*
 {*
 \edef\w@##1{\immediate\write\j@insplitout{##1}}*
\w@{% This collection of files was produced with CERN psbox package}*
\w@{% To decompose and tex it:}*
\w@{%-save this with a filename CONTAINING ONLY LETTERS and a .TEX}*
\w@{% extension (say, JOINTFIL.TEX), in some uncrowded directory;}*
\w@{%-make sure you can \string\input\space psbox.tex (version>=1.3);}*
\w@{%  (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get}*
\w@{%  and tex the file psboxall.tex; more info in psbREAD.ME)}*
\w@{%-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by}*
\w@{%  the first file in the joining (after splitting JOINTFIL.TEX into}*
\w@{%  it's constituents, TeX will try to process it as it stands).}*
\w@{\string\input\space psbox.tex}*
\w@{\string\splitfile{\string\jobname}}*
\w@{\string\let\string\autojoin=\string\relax}*
}*
 \expandafter\tre@tfilelist#1, \endtre@t
 \immediate\closeout\j@insplitout
 \message{>}*
}*
\gdef\tre@tfilelist#1, #2\endtre@t{*
 \readfilename#1\relax
 \ifx\@mpty\lastreadfilename
 \else
   \immediate\openin\j@insplitin=\lastreadfilename\relax
   \ifeof\j@insplitin
     \errmessage{I couldn't find file \lastreadfilename}*
   \else
     \message{\lastreadfilename}*
     \immediate\write\j@insplitout{%Beginning-Of-File-Named:\lastreadfilename}*
     \executeinspecs{\global\read\j@insplitin to\oldj@ininline}*
     \loop
       \ifeof\j@insplitin\immediate\closein\j@insplitin\n@teoffalse
       \else\n@teoftrue
         \executeinspecs{\global\read\j@insplitin to\j@ininline}*
         \toks0=\expandafter{\oldj@ininline}*
         \let\oldj@ininline=\j@ininline
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \ifn@teof
     \repeat
   \immediate\closein\j@insplitin
   \fi
   \tre@tfilelist#2, \endtre@t
 \fi}*
}%
% To be put at the end of a file, for making a tar-like file containing
%   everything it used.
\def\autojoin{%
 \immediate\write\psbj@inaux{\string\into{psbjoint.tex}}%
 \immediate\closeout\psbj@inaux
 \expandafter\joinfiles\GlobalInputList\into{psbjoint.tex}%
}%
%----------------------------------------------------------------
%  Annotations & Captions etc...
%
%
% \centinsert{anybox} is just a centered \midinsert, but is included as
%    people barely use the original inserts from TeX.
%
\def\centinsert#1{\midinsert\line{\hss#1\hss}\endinsert}%
\def\psannotate#1#2{\vbox{%
  \def\ps@nnotation{#2\global\let\ps@nnotation=\relax}#1}}%
\def\pscaption#1#2{\vbox{%
   \setbox\drawingBox=#1
   \copy\drawingBox
   \vskip\baselineskip
   \vbox{\hsize=\wd\drawingBox\setbox0=\hbox{#2}%
     \ifdim\wd0>\hsize
       \noindent\unhbox0\tolerance=5000
    \else\centerline{\box0}%
    \fi
}}}%
% for compatibility with older versions, but \psfig is a bad name!
%\def\psfig#1#2#3{\pscaption{\psannotate{#1}{#2}}{#3}}
%\def\psfigurebox#1#2#3{\pscaption{\psannotate{\psbox{#1}}{#2}}{#3}}
%
% \at(#1;#2)#3 puts #3 at #1-higher and #2-right of the current
%    position without moving it (to be used in annotations).
\def\at(#1;#2)#3{\setbox0=\hbox{#3}\ht0=0pt\dp0=0pt
  \rlap{\kern#1\vbox to0pt{\kern-#2\box0\vss}}}%
%
% \gridfill(ht;wd) makes a 1cm*1cm grid of ht by wd whose lower-left
%   corner is the current point
\newdimen\gridht \newdimen\gridwd
\def\gridfill(#1;#2){%
  \setbox0=\hbox to 1\pscm
  {\vrule height1\pscm width.4pt\leaders\hrule\hfill}%
  \gridht=#1
  \divide\gridht by \ht0
  \multiply\gridht by \ht0
  \gridwd=#2
  \divide\gridwd by \wd0
  \multiply\gridwd by \wd0
  \advance \gridwd by \wd0
  \vbox to \gridht{\leaders\hbox to\gridwd{\leaders\box0\hfill}\vfill}}%
%
% Useful to measure where to put annotations
\def\fillinggrid{\at(0cm;0cm){\vbox{%
  \gridfill(\drawinght;\drawingwd)}}}%
%
% \textleftof\anybox: Sample text\endtext
%   inserts "Sample text" on the left of \anybox ie \vbox, \psbox.
%   \textrightof is the symmetric (not documented, too uggly)
% Welcome any suggestion about clean wraparound macros from
%   TeXhackers reading this
%
\def\textleftof#1:{%
  \setbox1=#1
  \setbox0=\vbox\bgroup
    \advance\hsize by -\wd1 \advance\hsize by -2em}%
\def\textrightof#1:{%
  \setbox0=#1
  \setbox1=\vbox\bgroup
    \advance\hsize by -\wd0 \advance\hsize by -2em}%
\def\endtext{%
  \egroup
  \hbox to \hsize{\valign{\vfil##\vfil\cr%
\box0\cr%
\noalign{\hss}\box1\cr}}}%
%
% \frameit{\thick}{\skip}{\anybox}
%    draws with thickness \thick a box around \anybox, leaving \skip of
%    blank around it. eg \frameit{0.5pt}{1pt}{\hbox{hello}}
% \boxit{\anybox} is a shortcut.
\def\frameit#1#2#3{\hbox{\vrule width#1\vbox{%
  \hrule height#1\vskip#2\hbox{\hskip#2\vbox{#3}\hskip#2}%
        \vskip#2\hrule height#1}\vrule width#1}}%
\def\boxit#1{\frameit{0.4pt}{0pt}{#1}}%
%
%
\catcode`\@=12 % cs containing @ are unreachable
%
% CUSTOMIZE YOUR DEFAULT DRIVER:
%    Uncomment the line corresponding to your TeX system:
%\psfortextures%     For TeXtures on the Macintosh
%\psforoztex   %     For OzTeX shareware on the Macintosh
%\psfordvitops %     For the DVItoPS converter for TeX on IBM mainframes
\psfordvips   %     For DVIPS converter on VAX and UNIX
%\psfordvitps  %     For dvitps from TeXPS package under UNIX
%\psfordvialw  %     For dvialw, UNIX public domain
%\psonlyboxes  %     Blank Boxes (when all else fails).

