%Paper: hep-th/9408027
%From: t32@nikhef.nl (Jan Willem v. Holten)
%Date: Thu, 4 Aug 1994 13:56:29 +0200

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\begin{document}

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\begin{flushright}
NIKHEF-H/94-22
\end{flushright}

\begin{center}

{\Large {\bf Relations between some analytic representations }} \\
\vspace{3ex}

{\Large {\bf of one-loop scalar integrals}} \\
\vspace{5ex}

{\large J.W. van Holten} \\
\vspace{3ex}

{\large NIKHEF-H, Amsterdam NL} \\
\vspace{5ex}

August 1, 1994
\vspace{7ex}



{\small {\bf Abstract}}

\end{center}

\nit
{\small
We compare several parametrized analytic expressions for an arbitrary off-shell
one-loop $n$-point function in scalar field theory in $D$-dimensional
space-time, and show their equivalence both directly and through path-integral
methods.}


\np

\pagestyle{plain}
\pagenumbering{arabic}

\section{Introduction}{\label{S1}}

Inspired by the string-based work of Bern and Kosower \ct{BK1,BK2}, Strassler
\ct{MS} has given a set of rules for computing one-loop Green functions in
field theory directly from path-integrals for relativistic point-particle
models. Similar results in this direction were already obtained earlier by
Polyakov \ct{Pol}. The procedure has been extended by various authors
\ct{McK}-\ct{CS3} so as to allow for example the computation of processes with
internal or external fermions, of higher-order terms in the
derivative-expansion of the effective potential, or of higher-loops. In many
of these applications, the emphasis is on the non-trivial structure due to
spin-polarization and non-abelian charges, and in particular on the manifest
gauge-invariance of the computational scheme, whilst the scalar loop-integrals
are merely shown to reduce to a standard form. An exception is reference
\ct{CS2}, where use of the method is made to construct the derivative expansion
of the effective action of scalar field theory.

It is the purpose of this note to show how the expressions for the scalar
loop-integrals, first obtained by the path-integral method, can also be derived
directly from the standard Feynman-diagram representation. This provides an
independent check of the correctness of the path-integral manipulations or,
if one prefers, of the equivalence between the Feynman-diagram method and
the functional-integral representation of the corresponding point-particle
theory. Indeed, a step-by-step comparison between the path-integral and Feynman
diagram calculation becomes possible. Basically it is another way of
establishing the equivalence between `first' and `second' quantization.

\section{The one-loop scalar $n$-point function}{\label{S2}}

Consider the one-loop diagrams for $g \vf^N$-theory with mass $m$ in
$D$-dimensional Minkowi\-ski space, see fig.\ 1. Since the particles in the
theory are structureless, the only effect of any vertex in the loop is a
multiplicative factor $-g /(2 \pi)^D$, an additional propagator, and the
insertion of a momentum

\be
Q^{\mu} = \sum_{r=1}^{N-2} \, k_r^{\mu},
\label{1}
\ee

\nit
into the loop; here the $k_r^{\mu}$ are the (off-shell) momenta of the $N-2$
individual external lines entering the vertex. The full one-loop contribution
to
the $n$-point function is obtained by symmetrization over all vertices, or
equivalently all external total momenta $Q_i$, $i = 1,...,n$, and taking into
account over-all momentum conservation:

\be
\sum_{i=1}^{n}\, Q_i = 0.
\label{1.1}
\ee

\thicklines

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\caption{1-loop scalar $n$-point diagram}
\end{figure}

\nit
The well-known result for the amplitude is

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{ \frac{1}{n}\, \left[ \frac{-g}{(2\pi)^D} \right]^n \,
           \del^{(D)} \lh \sum_{k=1}^{n} Q_k \rh\, \times }\\
  &  & \\
 & & \dsp{ \times \sum_{\left\{ i_k \right\}} \int_0^1 \prod_{i=1}^{n} dx_i\,
           \int d^D p \frac{\del \lh 1 - \sum_{j=1}^{n} x_j \rh }{\left[
           \sum_{l=1}^{n} x_l p_l^2\, +\, m^2\, -\, i \eps \right]^n }, }
\label{2}
\ea
\ee

\nit
where $\left\{i_k\right\}$ denotes a permutation of $(1,...,n)$, whilst

\be
p_1 = p,  \hspace{2em} \mbox{and} \hspace{2em}
p_l = p + \sum_{k=1}^{l-1} Q_{i_k}, \hspace{1em} \mbox{for} \hspace{1em}
      l = 2,...,n.
\label{3}
\ee

\nit
Clearly the last expression depends on the particular permutation of the $Q_i$.
The $x_i$ with values in the interval $[0,1]$ are the usual Feynman parameters.

The integrand in eq.(\ref{2}) can be converted to exponential form using the
Schwinger trick; introducing a factor $1/2m$ for later convenience, we write

\be
\frac{1}{\left[ \sum_{j=1}^{n} x_j p_j^2\, +\, m^2\, -\, i \eps \right]^n }\,
  =\, \frac{i^n}{(n-1)! (2m)^n}\, \int_0^{\infty} dT\, T^{n-1}
      e^{-\frac{iT}{2m}\, \sum_{j=1}^{n}\, x_j (p_j^2 + m^2 - i\eps)}.
\label{4}
\ee

\nit
The factor $T$ can be removed from the exponent by absorbing it into the
Feynman
parameters; hence we define

\be
\ag_i \equiv T x_i,
\label{5}
\ee

\nit
taking values in $[0,T]$. Thus we obtain the following expression
for the $n$-point amplitude in one loop:

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{
   \frac{1}{n!}\, \left[ \frac{-i g}{2m (2\pi)^D} \right]^n\,
   \del^{(D)} \lh \sum_{k=1}^{n} Q_k \rh \times }\\
  & & \\
  & & \dsp{ \times \sum_{\left\{ i_k \right\} }\, \int_0^{\infty} dT\,
      \int_0^T \prod_{i=1}^{n} d\ag_i \, \del \lh T - \sum_{i=1}^{n} \ag_i
      \rh\, \int d^D p e^{- \frac{i}{2m}\, \sum_{j=1}^{n}\, \ag_j (p^2_j +
       m^2)}.}
\ea
\label{6}
\ee

\nit
In the exponent we have tacitly absorbed the $i\eps$ term in the mass $m^2$,
and
we will no longer write it explicitly.

The integral over the loop-momentum $p$ has now been reduced to a Gaussian
and is therefore straightforward to perform. One finds (in Minkowski space):

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{
   \frac{i}{n!}\, \left[ \frac{-i g}{2m (2\pi)^D} \right]^n\,
   \del^D \lh \sum_{k=1}^{n} Q_k \rh \times }\\
  &  &  \\
  &  & \dsp{ \times \sum_{\left\{ i_k \right\} }\, \int_0^{\infty} dT\,
       \int_0^T \prod_{i=1}^{n} d\ag_i \, \del \lh T - \sum_{i=1}^{n} \ag_i
       \rh\, \lh \frac{-2 \pi i m }{T} \rh^{D/2}\, e^{-\, \frac{imT}{2}\, -\,
       \frac{i}{2mT}\, R_n \left[ Q_i \right] },}
\ea
\label{7}
\ee

\nit
where $R_n \left[ Q_i \right]$ is a quadratic polynomial in the external
momenta; explicitly:

\be
R_n \left[ Q_i \right]\, =\, T\, \sum_{j=2}^{n}\, \left[ \ag_j
    \lh \sum_{k=1}^{j-1}\, Q_{i_k} \rh^2 \right]\, -\,
    \sum_{j=2}^{n}\, \left[ \ag_j \sum_{k=1}^{j-1} Q_{i_k} \right]^2.
\label{8}
\ee

\nit
The sums on the right-hand side start only at $j = 2$, because $\ag_1$ is
multiplied by $\sum_{k=1}^{n} Q_{i_k}$, which vanishes acording to
eq.(\ref{1.1}) in any permutation.

We proceed to show that in the symmetric integral over the parameters $\ag_i$
the polynomial $R_n \left[ Q_i \right]$ may be replaced by

\be
R_n \left[ Q_i \right]\, =\, 2 T\, \sum_{k=2}^{n} \sum_{l=1}^{k-1}\,
    Q_{i_k} \cdot Q_{i_l}\, G\left[ \Del_{kl} \right],
\label{9}
\ee

\nit
where the function $G[\Del_{kl}]$ of the argument $\Del_{kl} =
\sum_{j=l+1}^{k}\,
\ag_j$ is defined as

\be
G[\Del_{kl}]\, =\, -\frac{1}{2}\, \left| \Del_{kl} \right|\, +\,
                   \frac{\Del_{kl}^2}{2T}.
\label{10}
\ee

\nit
This rearrangement of terms is possible, because of the momentum conservation
condition (\ref{1.1}) and the constraint

\be
\sum_{i=1}^{n}\, \ag_i = T,
\label{11}
\ee

\nit
which show the full set of variables and parameters in the polynomial $R_n
\left[ Q_i \right]$ to be redundant. Since the proof is independend of the
ordering of the external momenta, we may take any particular ordering. For
convenience of notation we take $\left\{ Q_{i_k} \right\} = \left\{ Q_{k}
\right\}$. Using the resummation formula


\be
\sum_{k=2}^{n}\, \lh \ag_k \sum_{l=1}^{k-1}\, B_l \rh\, =\,
      \sum_{l=1}^{n-1}\, \lh B_l \sum_{k=l+1}^{n}\, \ag_k \rh,
\label{12}
\ee

\nit
and splitting off the terms which contain pure squares:

\be
\sum_{k,l = 1}^{n-1}\, c_{kl}\, Q_k \cdot Q_l\, =\,
  \sum_{k=1}^{n-1}\, c_{kk}\, Q_k^2\, +\,
  2 \sum_{k=2}^{n-1} \sum_{l=1}^{k-1}\, c_{kl}\, Q_k \cdot Q_l,
\label{13}
\ee

\nit
we can rewrite the expression (\ref{8}) for $R_n \left[ Q_i \right]$ in the
form

\be
\ba{lll}
R_n \left[ Q_i \right] & = & \dsp{ \sum_{k=1}^{n-1}\, Q_k^2 \left[
    T \sum_{j=k+1}^n \ag_j\, -\, \lh \sum_{j=k+1}^n \ag_j \rh^2 \right]\, + }\\
 & & \\
 & & \dsp{  +\, 2\, \sum_{k=2}^{n-1} \sum_{l=1}^{k-1}\, Q_k \cdot Q_l \left[
    T \sum_{j=k+1}^n \ag_j\, -\, \lh \sum_{j=k+1}^n \ag_j \rh^2 \right]. }
\label{14}
\ea
\ee

\nit
Note that the mixed term has been arranged such that the sum runs only over
$k>l$,
giving rise to the factor of 2. Now we can eliminate the pure squares by using
momentum conservation:

\be
\ba{lll}
Q_1^2 & = & \dsp{ - Q_1 \cdot \sum_{l=2}^{n} Q_l , } \\
  & & \\
Q_k^2 & = & \dsp{ - Q_k \cdot \lh \sum_{l=1}^{k-1} Q_l + \sum_{l=k+1}^n Q_l
\rh,
\hspace{3em} k \geq 2,}
\label{15}
\ea
\ee

\nit
Using the constraint (\ref{11}) to expand $T$ in terms of the $\ag_i$,
eq.(\ref{14}) for $R_n \left[ Q_i \right]$ then becomes

\be
\ba{lll}
R_n \left[ Q_i \right] & = & \dsp{
  \sum_{k=2}^{n-1} \sum_{l=1}^{k-1}\, Q_k \cdot Q_l
  \left[ 2 \sum_{i=1}^{l} \ag_i\, \sum_{j=k+1}^{n} \ag_j\, -\, \sum_{i=1}^{k}
  \ag_i\, \sum_{j=k+1}^{n} \ag_j\, -\, \sum_{i=1}^{l} \ag_i\,
  \sum_{j=l+1}^{n} \ag_j \right]\, - }\\
 & & \\
 & & \dsp{ -\, Q_n \cdot \sum_{l=1}^{n-1}\, Q_l
  \left[\sum_{i=1}^l \ag_i\, \sum_{j=l+1}^n \ag_j \right].}
\ea
\label{16}
\ee

\nit
Performing the subtractions inside the double sum gives

\be
\ba{lll}
R_n \left[ Q_i \right] & = & \dsp{
    - \sum_{k=2}^{n-1} \sum_{l=1}^{k-1}\, Q_k \cdot
    Q_l \, \sum_{j=l+1}^k \ag_j \left[ \sum_{i=1}^l \ag_i +
    \sum_{i=k+1}^n \ag_i \right]\, -\, Q_n \cdot \sum_{l=1}^{n-1}\, Q_l\,
    \left[\sum_{i=1}^l \ag_i \sum_{j=l+1}^n \ag_j \right] }\\
  & & \\
  & = & \dsp{ - \sum_{k=2}^n \sum_{l=1}^{k-1}\, Q_k \cdot Q_l\,
     \sum_{j=l+1}^k \ag_j \lh T - \sum_{i=l+1}^k \ag_i \rh . }
\ea
\label{17}
\ee

\nit
To obtain the last line we have again used relation (\ref{11}). Note that the
remaining sums over $\ag_i$ are equal precisely the quantity $\Del_{kl}$
introduced before. Therefore eq.(\ref{17}) indeed reduces to eq.(\ref{9}).
Using
this relation under the integral with the constraints on the external momenta
and on the parameters $\ag_i$, we can then bring the $n$-particle amplitude
into
the form

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{
   \frac{i}{n!}\, \left[ \frac{-i g}{2m (2\pi)^D} \right]^n\,
   \del^D \lh \sum_{k=1}^{n} Q_k \rh \sum_{\left\{ i_k \right\} }\,
   \int_0^{\infty} dT\, \int_0^T \prod_{i=1}^{n} d\ag_i \, \del \lh T -
   \sum_{i=1}^{n} \ag_i \rh\, \times }\\
  &  &  \\
  &  & \dsp{  \times \lh \frac{-2 \pi i m }{T} \rh^{D/2}\,
   e^{-\frac{imT}{2}\, -\, \frac{i}{m}\, \sum_{k=2}^n \sum_{l=1}^{k-1}
   \, Q_{i_k} \cdot Q_{i_l}\, G[\Del_{kl}] }. }
\ea
\label{18}
\ee

\nit
Eqs. \ref{2}, \ref{7} and \ref{18} represent three equivalent expressions for
the one-loop scalar $n$-point function, obtained from the standard Feynman
rules
for $g \vf^N$-theory. In refs.\ct{MS}-\ct{CS3} it was shown, that similar
results are obtained from the path-integral for a relativistic point-particle
of
mass $m$ calculated in a specific way. In the following we rederive some of
these results, and show that each of our three expressions results directly
from
a single functional integral using different methods of evaluation. In the
course of these derivations we will encounter yet another analytic
parametrization of the one-loop scalar integral.


\section{The path-integral representation}{\label{S3}}

The starting point for the path-integral calculations is the expression \ct{MS}

\be
\Gam_n [Q_i]\, =\, \frac{1}{n!}\, \left[\frac{-ig}{2m(2\pi)^D} \right]^n\,
   \int_0^{\infty} \frac{dT}{T}\, \oint D\xi[\tau]\, \int_0^T \prod_{i=1}^n
   d\tau_i\, e^{\frac{im}{2}\, \int_0^T d\tau \lh \dot{\xi}_{\mu}^2 - 1 \rh\,
   +\, i \sum_{j=1}^n Q_j \cdot \xi_j },
\label{19}
\ee

\nit
which is claimed to be equal to the one-loop amplitude of eq.(\ref{2}). The
right-hand side of (\ref{19}) can be interpreted as the sum over all closed
world lines (satisfying periodic boundary conditions) of a scalar particle, of
all possible proper-time lengths $T>0$, with $n$ insertions of interactions
with
an external potential, giving the particle a kick of momentum $Q_{i_k}$ at
proper time $\tau_k$ when the particle co-ordinates are $\xi^{\mu}_k =
\xi^{\mu}(\tau_k)$. Note that in pure $g \vf^N$-theory the only point-like
interaction is the one described by one single type of vertex; in particular,
the values of $\tau_k$ for two different vertices never coincide, since this
would correspond to the introduction of a new point-like vertex. This argument
can of course be made rigorous only in a renormalizable field theory;
nevertheless in the following we take for granted that all interactions are
separated in proper time.

We can now order the interactions in proper time:

\be
0 \leq \tau_1 < \tau_2 < ... < \tau_n \leq T,
\label{20}
\ee

\nit
In principle, there are $n!$ such orderings, but the various possibilities are
all
taken into account by symmetrizing over the permutations $\left\{ i_k \right\}$
of
the external momenta $\left\{ Q_{i_k} \right\}$. Therefore eq.(\ref{19}) is
equivalent to

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{
            \frac{1}{n!}\, \left[\frac{-ig}{2m(2\pi)^D} \right]^n\, \sum_{
  \left\{ i_k \right\}}\, \int_0^{\infty} \frac{dT}{T}\, \int_0^T d\tau_n\,
  \int_0^{\tau_n} d\tau_{n-1}\, ...\, \int_0^{\tau_2} d\tau_1 \times }\\
 & & \\
 & & \dsp{ \times \oint D\xi[\tau]\, e^{\frac{im}{2}\, \int_0^T d\tau \lh
  \dot{\xi}_{\mu}^2 - 1 \rh\, +\, i \sum_{j=1}^n Q_j \cdot \xi_j }. }
\ea
\label{21}
\ee

\nit
The action integral from 0 to $T$ in the exponent can of course be decomposed
into
a sum

\be
\int_0^T d\tau = \sum_{k=1}^n\, \int_{\tau_{k-1}}^{\tau_k}\, d\tau,
\label{22}
\ee

\nit
with $\tau_0 = \tau_n$. In the following we use the standard functional
integral results for free particles in $D$-dimensional Minkowski space,
expressed by

\be
\ba{lll}
\dsp{ \int_{\xi_1}^{\xi_2} D\xi[\tau]\, e^{\frac{im}{2}\,
\int_{\tau_1}^{\tau_2}
      d\tau \lh \dot{\xi}^2_{\mu} - 1 \rh } } & = & \dsp{
      i \lh \frac{m}{2\pi i \ag} \rh^{D/2}\, e^{-\frac{im\ag}{2}\, +\,
      \frac{im}{2\ag}\, \lh \xi_2 - \xi_1 \rh^2 } }\\
 &   & \\
 & = & \dsp{ \int \frac{d^D p}{(2\pi)^D}\, e^{-\frac{i\ag}{2m}\, \lh p^2 + m^2
      \rh\, +\, i p \cdot \lh \xi_2 - \xi_1 \rh }, }
\ea
\label{23}
\ee

\nit
where $\ag = \tau_2 - \tau_1$. In eq.(\ref{21}) this result is to be used $n$
times, once for each of the functional integrals over the various sections of
the path, from $\tau_{k-1}$ to $\tau_k$. This requires the introduction of $n$
parameters

\be
\ag_1= \tau_1 - \tau_n + T, \hspace{3em} \mbox{and} \hspace{3em}
\ag_k = \tau_k - \tau_{k-1}, \hspace{1em} 2 \leq k \leq n,
\label{24}
\ee

\nit
Clearly, these parameters satisfy the constraint (\ref{11}), hence they are not
all independend; this is the result of translation invariance. The same
invariance
also makes the integral over $\tau_n$ trivial, since this amounts to an
integration over all possible choices of the origin of proper time. Thus we can
trade the multiple integral over the $\tau_k$ for a constrained integral over
the
$\ag_i$:

\be
\int_0^T d\tau_n \int_0^{\tau_{n}} d\tau_{n-1}\, ...\, \int_0^{\tau_2} d\tau_1
\, =\, T\, \int_0^T \prod_{i=1}^n d\ag_i\, \del \lh T - \sum_{j=1}^n \ag_j \rh.
\label{25}
\ee

\nit
After performing the functional integrals between each pair of successive
interaction points, and making the variable transformation (\ref{25}), one is
left only with an integral over the vertex co-ordinates $\xi^{\mu}_k$ and over
each of the momenta $p_k$ of the free particles propagating between the
vertices.
The result is

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{ \frac{1}{n!}\, \left[ \frac{-ig}{2m (2\pi)^D}
\right]^n
   \, \sum_{\left\{ i_k \right\}}\, \int_0^{\infty} dT \int_0^T \prod_{i=1}^n
   d\ag_i\, \del \lh T - \sum_{j=1}^n \ag_j \rh\, \times } \\
  &  &  \\
  &  & \dsp{ \times \int \prod_{k=1}^n \left[ \frac{d^D \xi_k d^D p_k}{
   (2\pi)^D} \right]\, e^{-\frac{i}{2m} \sum_{l=1}^n \ag_l \lh p_l^2
   + m^2 \rh\, +\, i \sum_{l=1}^n \lh Q_{i_l} + p_l - p_{l+1} \rh \cdot
   \xi_l }. }
\ea
\label{26}
\ee

\nit
As expected, the integrations over the vertex positions $\xi_k^{\mu}$ produce
$\del$-functions for momentum conservation at each vertex. The integrals over
the momenta $p_k$ eliminate all of these $\del$-functions except one: there
remains the $\del$-function expressing conservation of the external momenta.
This leaves one momentum integral, and the formula for the amplitude becomes

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{ \frac{1}{n!}\, \left[ \frac{-ig}{2m (2\pi)^D}
\right]^n
   \, \del^{(D)} \lh \sum_{k=1}^n Q_k \rh\, \sum_{\left\{ i_k \right\}}\,
   \int_0^{\infty} dT \int_0^T \prod_{i=1}^n d\ag_i\,
   \del \lh T - \sum_{j=1}^n \ag_j \rh\, \times } \\
  &  &  \\
  &  & \dsp{ \times \int d^D p\, e^{-\frac{i}{2m} \sum_{l=1}^n \ag_l
       \lh p_l^2 + m^2 \rh }, }
\ea
\label{27}
\ee

\nit
where the momenta $p_l$ are to be interpreted as in eq.(\ref{3}). Eq.(\ref{27})
is identical to eq.(\ref{6}). Hence from this result we can directly obtain
either eq.(\ref{2}) by integration over the parameter $T$, or eq.(\ref{7}) by
integrating out the loop momentum $p$. Therefore both these equations are
indeed
seen to follow in a straightforward way from the functional integral
(\ref{19}).

We proceed to show, that eq.(\ref{18}) can be derived from this same functional
integral, but using the evaluation procedure of ref.\ct{MS}, rather than
the procedure sketched above. Introducing the notation


\be
j^{\mu}(\tau)\, =\, \sum_{k=1}^n\, Q^{\mu}_{i_k}\, \del \lh \tau - \tau_k \rh,
\label{28}
\ee

\nit
eq.(\ref{21}) can be written in the form

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{
  \frac{1}{n!}\, \left[\frac{-ig}{2m(2\pi)^D} \right]^n\, \sum_{
  \left\{ i_k \right\}}\, \int_0^{\infty} \frac{dT}{T}\, \int_0^T d\tau_n\,
  \int_0^{\tau_n} d\tau_{n-1}\, ...\, \int_0^{\tau_2} d\tau_1 \times }\\
 & & \\
 & & \dsp{ \times \oint D\xi[\tau]\, e^{i \int_0^T d\tau \left[-\frac{m}{2}\,
  \xi_{\mu} \lh \frac{d^2}{d\tau^2} \rh\, \xi^{\mu}\, +j \cdot \xi -
  \frac{m}{2} \right] }. }
\ea
\label{29}
\ee

\nit
Clearly the external momenta act as external forces, and the partial
integration
in the action is allowed because of the periodic boundary conditions.
Since the action integral in the exponent is a quadratic expression in the
$\xi^{\mu}(\tau)$, we can perform the functional integration by completing
the square. This requires the introduction of the inverse of the operator
$-d^2/d\tau^2$ in the space of continuous functions on a closed line of length
$T$. As discussed in ref.\ct{MS}, such an inverse does not exist on the full
function space, because of the existence of zero modes: the constant functions.
When one restricts oneself to periodic functions $\bar{\xi}(\tau)$ which have
no
constant component:

\be
\int_0^T d\tau\, \bar{\xi}(\tau)\, =\, 0,
\label{30}
\ee

\nit
the inverse does exist, and is in fact given by $G[\tau -\tau^{\prime}]$
defined
in eq.(\ref{10}):

\be
-\frac{d^2}{d\tau^2}\, G[\tau - \tau^{\prime}]\, =\,
\del \lh \tau - \tau^{\prime} \rh\, -\, \frac{1}{T}.
\label{31}
\ee

\nit
This Green function is translation invariant, reflection symmetric and periodic
with period $T$. From eq.(\ref{31}) it follows that for functions restricted
as in (\ref{30}), one has

\be
-\frac{d^2}{d\tau^2}\, \int_0^T d\tau^{\prime}\, G[\tau - \tau^{\prime}]\,
 \bar{\xi}(\tau^{\prime})\, =\,  \bar{\xi}(\tau).
\label{32}
\ee

\nit
The procedure to be followed is therefore, as explained in \ct{CS2}, to split
off
the constant mode from $\xi^{\mu}(\tau)$:

\be
\xi^{\mu}(\tau)\, =\, \xi^{\mu}_0\, +\, \bar{\xi}^{\mu}(\tau),
\label{33}
\ee

\nit
with

\be
\xi^{\mu}_0\, =\, \frac{1}{T}\, \int_0^T d\tau\, \xi^{\mu}(\tau).
\label{34}
\ee

\nit
Then the Fourier decomposition of $\bar{\xi}^{\mu}(\tau)$ has no constant
component. Defining a new variable of integration

\be
\tilde{\xi}^{\mu}(\tau)\, =\, \bar{\xi}^{\mu}(\tau)\, +\,
  \frac{1}{m}\, \int_0^T d\tau^{\prime}\, G[\tau - \tau^{\prime}]\,
  j^{\mu}(\tau^{\prime}),
\label{35}
\ee

\nit
the action integral in the exponent in (\ref{29}) can be brought into the form

\be
\ba{lll}
\lefteqn{ \dsp{
    \int_0^T d\tau\, \left[ -\frac{m}{2}\, \xi_{\mu}\, \frac{d^2}{d\tau^2}\,
    \xi^{\mu}\, +\, j_{\mu} \xi^{\mu} \right]\, =
     \int_0^T d\tau\, \left[ -\frac{m}{2}\, \tilde{\xi}_{\mu}\,
    \frac{d^2}{d\tau^2}\, \tilde{\xi}^{\mu} \right]\, - } } & & \\
  & & \\
  & & \dsp{
     -\, \frac{1}{2m}\, \int_0^T d\tau \int_0^T d\tau^{\prime}\,
    j_{\mu}(\tau)\, G[\tau - \tau^{\prime}]\, j^{\mu}(\tau^{\prime})\, +\,
    \xi^{\mu}_0\, \int_0^T d\tau\, j_{\mu}(\tau). }
\ea
\label{36}
\ee

\nit
Note that $\xi_0^{\mu}$ has become a Lagrange  multiplier imposing the
constraint

\be
\int_0^T d\tau\, j^{\mu}(\tau)\, =\,  \sum_{k=1}^n\, Q^{\mu}_{i_k}\, =\, 0
\label{37}
\ee

\nit
In the functional integral we therefore integrate separately over $\xi_0$ and
$\tilde{\xi}$; the first integration gives the $\del$-function for momentum
conservation, the second is a pure Gaussian and can be evaluated using
eq.(\ref{23}) with $\xi_2 = \xi_1$, owing to the periodic boundary conditions.
It is straightforward to see that

\be
\int_0^T d\tau \int_0^T d\tau^{\prime}\, j_{\mu}(\tau)\,
   G[\tau - \tau^{\prime}]\, j^{\mu}(\tau^{\prime})\, =\,
   \sum_{k,l=1}^n\, Q_{i_k} \cdot Q_{i_l}\, G\left[ \tau_k - \tau_l \right].
\label{38}
\ee

\nit
Therefore we obtain

\be
\ba{lll}
\Gam_n [Q_i] & = & \dsp{
  \frac{i}{n!}\, \left[\frac{-ig}{2m(2\pi)^D} \right]^n\, \del \lh
  \sum_{k=1}^n\, Q_k \rh\, \sum_{\left\{ i_k \right\}}\, \int_0^{\infty}
  \frac{dT}{T}\, \lh - \frac{2\pi i m}{T} \rh^{D/2}\,
  \times }\\
 & & \\
 & & \dsp{ \times \int_0^T d\tau_n\, \int_0^{\tau_n} d\tau_{n-1}\, ...\,
  \int_0^{\tau_2} d\tau_1\, e^{- \frac{im}{2}T\, - \frac{i}{m}\, \sum_{k=2}^n
  \sum_{l=1}^{k-1}\, Q_{i_k} \cdot Q_{i_l}\, G \left[ \tau_k - \tau_l \right]
  }. }
\ea
\label{39}
\ee

\nit
We have used the fact that $G[0] = 0$ and the reflection symmetry of
$G[\tau_k - \tau_l]$ to resum the terms in the exponent in
such a way that the argument of $G$ is always positive: $\tau_k > \tau_l$.
This accounts for a factor of 2 difference compared to eq.(\ref{38}).

Finally, using the transformation (\ref{25}), eq.(\ref{39}) is seen to equal
the right-hand side of eq.(\ref{18}). In particular, we have

\be
\tau_k\, -\, \tau_l\, =\, \sum_{j = l+1}^{k}\, \ag_j\, =\,  \Del_{kl}.
\label{40}
\ee

\nit
Therefore the two methods of evaluating the path-integral, either directly or
using the Green function technique, give fully equivalent results.

Eq.(\ref{39}) may be used to derive yet another expression for the $n$-point
scalar amplitude. Namely, if one rescales the variables $\tau_k$ like the
$\ag_k$ in (\ref{5}):

\be
u_k\, \equiv\, \frac{\tau_k}{T},
\label{41}
\ee

\nit
taking values in the range $[0,1]$, with a simultaneous rescaling
$G[\tau_k - \tau_l] \rightarrow T \tilde{G}[u_k - u_l]$, then one obtains an
integral of $\Gam$-function type, as in the right-hand side of (\ref{4}):

\be
\ba{lll}
\lefteqn{\dsp{ \Gam_n [Q_i]\, =\, \frac{i}{n!}\, \lh -2\pi i m \rh^{D/2}\,
  \left[\frac{-ig}{2m(2\pi)^D} \right]^n\, \del \lh \sum_{k=1}^n\, Q_k \rh\,
  \sum_{\left\{ i_k \right\}}\, \int_0^{\infty} dT\, T^{n-1-(D/2)}\, \times }
  }& & \\
 & & \\
 & & \dsp{ \times  \int_0^1 du_n\,
  \int_0^{u_n} du_{n-1}\, ... \, \int_0^{u_2} du_1\,
  e^{- \frac{im}{2}T\, -\, \frac{iT}{m}\, \sum_{k=2}^n \sum_{l=1}^{k-1}\,
  Q_{i_k} \cdot Q_{i_l}\, \tilde{G}[u_k - u_l] }.}
\ea
\label{42}
\ee

\nit
Remembering the $i\eps$-prescription, the integral over $T$ can be carried out
and gives

\be
\ba{lll}
\lefteqn{\dsp{ \Gam_n [Q_i]\, =\, \frac{i\pi^{D/2}}{n!}\,
  \left[\frac{-g}{(2\pi)^D} \right]^n\,  \Gam \lh n - \frac{D}{2} \rh\,
  \del \lh \sum_{k=1}^n\, Q_k \rh\, \times }} & & \\
 & & \\
 & & \dsp{ \times \sum_{\left\{ i_k \right\}}\, \int_0^1 du_n\,
  \int_0^{u_n} du_{n-1}\, ... \, \int_0^{u_2} du_1\,
  \left[ \sum_{k,l=1}^n\, Q_{i_k} \cdot Q_{i_l}\, \tilde{G}[u_k - u_l]\,
  +\, m^2 \right]^{- n + D/2}.}
\ea
\label{43}
\ee

\section{Conclusions}{\label{S4}}

{}From the computations presented here, as well as in the earlier papers
\ct{MS}-\ct{CS3}, we can draw two main conclusions. First, different ways of
evaluating the functional integral (\ref{19}) give equivalent results. Second,
these results agree fully with a direct computation of the corresponding
Feynman
diagrams, which are derived from the appropriate relativistic quantum field
theory. In comparing the various approaches we have established four different
parametrizations of the one-loop $n$-point function\footnote{An intermediate
fifth one is the highly symmetric integral expression (\ref{26}).}
given by eqs.\ \ref{2}, \ref{7}, \ref{18} and \ref{43}.

{}From these observations, it is now possible to construct directly the
effective action for the scalar field theory in the one-loop approximation.
Let $U[\vf]$ denote a potential constructed out of the scalar fields; its
momentum-space components are defined by

\be
U[\vf(\xi)]\, =\, \int \frac{d^D Q}{(2\pi)^D}\, \tilde{U}(Q)\,
                  e^{i Q \cdot \xi}.
\label{44}
\ee

\nit
Now introduce the generating functional for the one-loop scalar $n$-point
functions:

\be
\ba{lll}
W[U] & = & \dsp{ \sum_{n=0}^{\infty}\, \int \prod_{i=1}^n d^D Q_i\:
           \Gam_n [Q_i]\, \tilde{U}(Q_n)\, ...\, \tilde{U}(Q_1) } \\
  &   & \\
  & = & \dsp{ \sum_{n=0}^{\infty}\, \frac{1}{n!}\, \left[ \frac{-ig}{2m}
        \right]^n\, \int_0^{\infty} \frac{dT}{T}\, \oint D\xi [\tau]\,
        \prod_{i=1}^n\, \lh \int_0^T  d\tau_i\, U[\vf(\xi_i)] \rh\,
        e^{\frac{im}{2}\, \int_0^T d\tau\, \lh \dot{\xi}_{\mu}^2 - 1 \rh} .}
\ea
\label{45}
\ee

\nit
In this expression the sum over $n$ can be carried out to give an exponential,
leading to the result

\be
W[U]\, =\, \int_0^{\infty} \frac{dT}{T}\, \oint D\xi [\tau]\,
           e^{i \int_0^T d\tau\, \lh \frac{m}{2}\, \dot{\xi}^2_{\mu}\, -\,
           \frac{m}{2}\, -\, \frac{g}{2m}\, U[\vf(\xi)] \rh }.
\label{46}
\ee

\nit
This is the path-integral for a relativistic scalar particle moving in an
external scalar potential $U[\vf]$. This path-integral is therefore seen to
generate by its expansion in powers of $g$ all one-loop $n$-point functions for
the scalar field theory for which $U[\vf]$ is the second derivative of the
classical potential. For example, the one-loop scalar integrals of
$g\vf^4$-theory in four dimensions are obtained by taking $U[\vf] = \vf^2/2$.
$W[U]$ thus corresponds to the one-loop approximation to the logarithm of
the generating functional for Green functions of the corresponding field theory
(modulo a constant):

\be
W[U]\, =\, - \log \det \lh - \Box + m^2 + g U[\vf] \rh .
\label{47}
\ee

\nit
In this way we reproduce, here for pure scalar field theories, the starting
point of ref.\ct{MS} and the subsequent papers cited below.
\vspace{7ex}

\nit
{\bf Acknowledgement}
\vspace{2ex}

\nit
The work described here is supported in part by the Human Capital and Mobility
Program through the network on {\em Constrained Dynamical Systems}.


\begin{thebibliography}{99}

\bit{BK1} Z.\ Bern and D.A.\ Kosower, Phys.\ Rev.\ Lett.\ 66 (1991), 1669;
          Nucl.\ Phys.\ B 362 (1991), 389;
\bit{BK2} Z.\ Bern and D.A.\ Kosower, Nucl.\ Phys.\ B379 (1992). 451
\bit{MS} M.J.\ Strassler, Nucl.\ Phys.\ B385 (1992), 145
\bit{Pol} A.M.\ Polyakov, Gauge Fields and Strings (Harwood), 1987
\bit{McK} D.G.C.\ McKeon, Ann.\ Phys.\ (NY) 224 (1993), 154
\bit{CS1} M.G.\ Schmidt and C.\ Schubert, Phys.\ Lett.\ B318 (1993), 438
\bit{CS2} D.\ Fliegner, M.G.\ Schmidt and C.\ Schubert, preprint
          HD-THEP-93-44
\bit{CS3} M.G.\ Schmidt and C.\ Schubert, Phys.\ Lett.\ B331 (1994), 69

\end{thebibliography}

\end{document}
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%
%   Draws a dashed anticlockwise arc with an arrow in the middle
%   The arc is   x_center, y_center, radius, start_angle, end_angle dsize
%
    gsw /dsize ed /arcend1 ed /arcstart1 ed /radius ed
%
    translate                               % x and y are still on stack
%
    arcend1 arcstart1 lt { /arcend1 arcend1 360 add def } if
    /arcmid1 arcend1 arcstart1 add 2 div def
%
    0 0 radius arcstart1 arcmid1 dsize width 1 dashcarc
    0 0 radius arcmid1 arcend1 dsize width 1 dashcarc
    arcmid1 rotate
    radius 0 translate
    arrowup
    grestore
} def }
%
%		#] dasharrowarc :
%		#[ arrowarcn :
%
\special{! /arrowarcn{
%
%   Draws a clockwise arc with an arrow in the middle
%   The arc is   x_center, y_center, radius, start_angle, end_angle
%
    gsw /arcend ed /arcstart ed /radius ed
%
    translate                               % x and y are still on stack
    newpath 0 0 radius arcstart arcend arcn stroke
    arcstart arcend lt {
        /arcstart arcstart 360 add def } if
    arcstart arcend add 2 div rotate        % middle of arc
    radius 0 translate                      % move to it
    arrowdown
    grestore
} def }
%
%		#] arrowarcn :
%		#[ longarrowarcn :
%
\special{! /longarrowarcn{
%
%   Draws a clockwise arc with an arrow in the end
%   The arc is   x_center, y_center, radius, start_angle, end_angle
%
    gsw /arcend ed /arcstart ed /radius ed
%
    translate                               % x and y are still on stack
    arcstart arcend lt {
        /arcstart arcstart 360 add def } if
    /arcmid 540 width 1 add mul 3.14159 div radius div def
                                            % correction for arrow
    newpath 0 0 radius arcstart arcend arcmid add arcn stroke
    arcend arcmid 2 div add rotate          % middle of arrow
    radius 0 translate                      % move to it
    arrowdown
    grestore
} def }
%
%		#] longarrowarcn :
%		#[ dasharrowarcn :
%
\special{! /dasharrowarcn{
%
%   Draws a dashed clockwise arc with an arrow in the middle
%   The arc is   x_center, y_center, radius, start_angle, end_angle
%
    gsw /dsize ed /arcend1 ed /arcstart1 ed /radius ed
%
    translate                               % x and y are still on stack
    arcstart1 arcend1 lt {
        /arcstart1 arcstart1 360 add def } if
    /arcmid1 arcstart1 arcend1 add 2 div def
    0 0 radius arcmid1 arcstart1 dsize width 1 dashcarc
    0 0 radius arcend1 arcmid1 dsize width 1 dashcarc
    arcmid1 rotate
    radius 0 translate
    arrowdown
    grestore
} def }
%
%		#] dasharrowarcn :
%		#[ arrowline :
%
\special{! /arrowline{
%
%   Draws a straight line with an arrow in the middle
%   x1,y1,x2,y2
%
    gsw p2 p1
    /dx x2 x1 sub def /dy y2 y1 sub def
    /dr dx dx mul dy dy mul add sqrt def
%
    x1 y1 translate
    newpath
        0 0 moveto
        dx dy rlineto
    stroke
    dy dx atan rotate
    dr 2.0 div 0 translate
    arrowright
    grestore
} def }
%
%		#] arrowline :
%		#[ longarrow :
%
\special{! /longarrow{
%
%   Draws a straight line with an arrow at the end
%   x1,y1,x2,y2
%
    gsw p2 p1
    /dx x2 x1 sub def /dy y2 y1 sub def
    /dr dx dx mul dy dy mul add sqrt def
%
    x1 y1 translate
    dy dx atan rotate
    newpath
        0 0 moveto
        dr width 3 mul sub 0 rlineto
    stroke
    dr width 1.5 mul sub 0 translate
    arrowright
    grestore
} def }
%
%		#] longarrow :
%		#[ dasharrowline :
%
\special{! /dasharrowline{
%
%   Draws a straight dashed line with an arrow in the middle
%   x1,y1,x2,y2
%
%   The pattern is ideally [dsize dsize] 0 setdash
%   but we want to have (2*n+1)/2 patterns, so dsize must be rounded
%   Actually we want the center to be black too so that the arrow
%   fits in nice. This means that n must be odd. So
%   r = dsize*(4*m+3)
%
    gsw /dsize ed p2 p1
    /dx x2 x1 sub def /dy y2 y1 sub def
    /dr dx dx mul dy dy mul add sqrt 2 div def
%
    x1 y1 translate
    dy dx atan rotate
%
    0 0 dr 0 dsize width 1 dashline
    dr 0 translate
    0 0 dr 0 dsize width 1 dashline
    arrowright
    grestore
} def }
%
%		#] dasharrowline :
%		#[ line :
%
\special{! /line{
%
%   Draws a straight line: x1,y1,x2,y2
%
    gsw p2 p1
%
    newpath
      x1 y1 moveto
      x2 y2 lineto
    stroke
    grestore
} def }
%
%		#] line :
%		#[ dashline :
%
\special{! /dashline{
%
%   Draws a straight dashed line: x1,y1,x2,y2,dsize
%
%   The pattern is ideally [dsize dsize] 0 setdash
%   but we want to have (2*n+1)/2 patterns, so dsize must be rounded
%
    gsw /dsize ed p2 p1
    /r y2 y1 sub dup mul x2 x1 sub dup mul add sqrt def
    /dsize r dsize 2 mul div 0.5 sub round dup 0 le { pop 0 } if 2 mul 1 add
    r exch div def
    [dsize dsize] 0 setdash
%
    newpath
      x1 y1 moveto
      x2 y2 lineto
    stroke
    grestore
} def }
%
%		#] dashline :
%		#[ carc :
%
\special{! /carc{
%
%   Draws an anti-clockwise arc segment:
%   x_center, y_center, radius, start_angle, end_angle
%
    gsw /arcend ed /arcstart ed /radius ed
%
    translate                               % x and y are still on stack
    newpath 0 0 radius arcstart arcend arc stroke
    grestore
} def }
%
%		#] carc :
%		#[ dashcarc :
%
\special{! /dashcarc{
%
%   Draws an anti-clockwise arc segment:
%   x_center, y_center, radius, start_angle, end_angle, dsize
%
    gsw /dsize ed /arcend ed /arcstart ed /radius ed
%
    translate                               % x and y are still on stack
%
%   Compute the length of the line
%
    /dr arcend arcstart sub dup 0 lt { 360 add } if
        3.14159 mul 180 div radius mul def
    /dsize dr dsize 2 mul div 0.5 sub round dup 0 le { pop 0 } if 2 mul 1 add
    dr exch div def
    [dsize dsize] 0 setdash
%
    newpath 0 0 radius arcstart arcend arc stroke
    grestore
} def }
%
%		#] dashcarc :
%		#[ vertex :
%
\special{! /vertex{
%
%   Puts a fat dot at x,y  size is the radius of the dot
%
    gs
    /dotsize ed
    translate
    newpath
        0 0 dotsize 0 360 arc
    0 setgray fill stroke
    grestore
} def }
%
%		#] vertex :
%		#[ bcirc :
%
\special{! /bcirc{
%
%   Draws an anti-clockwise blanked circle:
%   x_center, y_center, radius
%
    gsw /radius ed
%
    translate                               % x and y are still on stack
%
    1 setgray
    newpath 0 0 radius 0 360 arc fill
    0 setgray
    newpath 0 0 radius 0 360 arc stroke
    grestore
} def }
%
%		#] bcirc :
%		#[ gcirc :
%
\special{! /gcirc{
%
%   Draws an anti-clockwise blanked gray circle:
%   x_center, y_center, radius, grayscale
%
    gsw /gcolor ed /radius ed
%
    translate                               % x and y are still on stack
%
    1 setgray
    newpath 0 0 radius 0 360 arc fill
    gcolor setgray
    newpath 0 0 radius 0 360 arc fill
    0 setgray
    newpath 0 0 radius 0 360 arc stroke
    grestore
} def }
%
%		#] gcirc :
%		#[ box :
%
\special{! /box{
%
%   Draws a box x1,y1,x2,y2
%
    gsw p2 p1
    abox stroke
    grestore
} def }
%
%		#] box :
%		#[ bbox :
%
\special{! /bbox{
%
%   Draws a blanked out box x1,y1,x2,y2
%
    gsw p2 p1
    1 setgray abox fill
    0 setgray abox stroke
    grestore
} def }
%
%		#] bbox :
%		#[ gbox :
%
\special{! /gbox{
%
%   Draws a blanked out gray box x1,y1,x2,y2,color
%
    gsw /gcolor ed p2 p1
    1 setgray      abox fill
    gcolor setgray abox fill
    0 setgray      abox stroke
    grestore
} def }
%
%		#] gbox :
%		#[ btext :
%
\special{! /btext{
%
%       Makes a box that has the text centered in it
%       The center of the box is at x,y and the box is just large enough
%       for the text.
%       x,y,text,fontsize,linewidth,scale
%
    gsw /fsize ed /thetext ed translate
    /hsize thetext stringwidth pop def
    /x1 hsize fsize add 2 div neg def
    /y1 fsize 2 mul 3 div neg def
    /x2 x1 neg def /y2 y1 neg def
    1 setgray abox fill 0 setgray abox stroke
    hsize 2 div neg fsize 3 div neg moveto thetext show
    grestore
} def }
%
%		#] btext :
%		#[ b2text :
%
\special{! /b2text{
%
%       Makes a box that has two lines of text centered in it
%       The center of the box is at x,y and the box is just large enough
%       for the text.
%       x,y,text1,text2,fontsize,linewidth,scale
%
    gsw /fsize ed /text2 ed /text1 ed translate
    /hsize1 text1 stringwidth pop def
    /hsize2 text2 stringwidth pop def
    hsize1 hsize2 lt { /hsize hsize2 def } { /hsize hsize1 def } ifelse
    /x1 hsize fsize add 2 div neg def
    /y1 fsize neg 7 mul 6 div def
    /x2 x1 neg def /y2 y1 neg def
    1 setgray abox fill 0 setgray abox stroke
    hsize1 2 div neg fsize 6 div moveto text1 show
    hsize2 2 div neg fsize 5 mul 6 div neg moveto text2 show
    grestore
} def }
%
%		#] b2text :
%		#[ gtext :
%
\special{! /gtext{
%
%       Makes a gray box that has the text centered in it
%       The center of the box is at x,y and the box is just large enough
%       for the text.
%       x,y,gray,text,fontsize,linewidth,scale
%
    gsw /fsize ed /thetext ed /graysc ed translate
    /hsize thetext stringwidth pop def
    /x1 hsize fsize add 2 div neg def
    /y1 fsize 2 mul 3 div neg def
    /x2 x1 neg def /y2 y1 neg def
    graysc setgray abox fill 0 setgray abox stroke
    hsize 2 div neg fsize 3 div neg moveto thetext show
    grestore
} def }
%
%		#] gtext :
%		#[ g2text :
%
\special{! /g2text{
%
%       Makes a gray box that has two lines of text centered in it
%       The center of the box is at x,y and the box is just large enough
%       for the text.
%       x,y,gray,text1,text2,fontsize,linewidth,scale
%
    gsw /fsize ed /text2 ed /text1 ed /graysc ed translate
    /hsize1 text1 stringwidth pop def
    /hsize2 text2 stringwidth pop def
    hsize1 hsize2 lt { /hsize hsize2 def } { /hsize hsize1 def } ifelse
    /x1 hsize fsize add 2 div neg def
    /y1 fsize neg 7 mul 6 div def
    /x2 x1 neg def /y2 y1 neg def
    graysc setgray abox fill 0 setgray abox stroke
    hsize1 2 div neg fsize 6 div moveto text1 show
    hsize2 2 div neg fsize 5 mul 6 div neg moveto text2 show
    grestore
} def }
%
%		#] g2text :
%		#[ ptext :
%
\special{! /ptext{
%
%   Puts a text at x,y (focal point)
%   phi is a rotation angle
%   Mode tells how the text should be w.r.t. the focal point.
%   fsize is the fontsize
%   Then the text is given.
%   Finally the scale factor should be there
%
    gs
    /text ed
    /fsize ed
    /mode ed
    3 1 roll
    translate rotate    % We are at the focus and have the proper orientation
    mode 3 lt { 0 fsize neg translate } {
        mode 6 lt { /mode mode 3 sub def 0 fsize 2 div neg translate }
        { /mode mode 6 sub def } ifelse } ifelse
    /fsize text stringwidth pop def
    mode 1 eq { fsize 2 div neg 0 translate } {
        mode 2 eq { fsize neg 0 translate } if } ifelse
    0 0 moveto
    text show
    grestore
} def }
%
%		#] ptext :
%		#[ goval :
%
\special{! /goval{
%
%   Draws a colored oval that overwrites whatever was there.
%   x_center y_center height width rotation color linewidth scale
%
    gsw /gcolor ed /x1 ed /x2 ed /radius ed
%
    translate                               % x and y are still on stack
    x1 rotate
    x2 radius div 1.0 scale
    1 setgray      newpath 0 0 radius 0 360 arc fill
    gcolor setgray newpath 0 0 radius 0 360 arc fill
    0 setgray      newpath 0 0 radius 0 360 arc stroke
    grestore
} def }
%
%		#] goval :
%		#[ oval :
%
\special{! /oval{
%
%   Draws an oval that does not overwrite whatever was there.
%   x_center y_center height width rotation linewidth scale
%
    gsw /x1 ed /x2 ed /radius ed
%
    translate                               % x and y are still on stack
    x1 rotate
    x2 radius div 1.0 scale
%
    0 setgray   newpath 0 0 radius 0 360 arc stroke
    grestore
} def }
%
%		#] oval :
%		#[ makecurve :
%
\special{! /docurve{
    x1 2 mul x2 add 3 div
        y1 y0 sub x1 x0 sub div x2 x0 sub mul
        y2 y0 sub x2 x0 sub div x1 x0 sub mul add
        y1 add y0 2 mul add 3 div
    x1 x2 2 mul add 3 div
        y2 y3 sub x2 x3 sub div x1 x3 sub mul
        y1 y3 sub x1 x3 sub div x2 x3 sub mul add
        y2 add y3 2 mul add 3 div
    x2 y2 curveto
} def }
%
\special{! /makecurve{
%
%   Incoming stack:
%   [array of x,y pairs] width scale
%
    gsw /points ed
    /ss points length 2 idiv 2 mul def
    newpath
    ss 4 gt {
        /x1 points 0 get def
        /y1 points 1 get def
        /x2 points 2 get def
        /y2 points 3 get def
        /x3 points 4 get def
        /y3 points 5 get def
        /x0 x1 2 mul x2 sub def
        /y0 y3 y2 sub x3 x2 sub div y2 y1 sub x2 x1 sub div sub 2 mul
            x2 x1 sub dup mul x3 x1 sub div mul
            y1 2 mul add y2 sub def
        x1 y1 moveto
        docurve
        0 2 ss 8 sub { /ii ed
            /x0 points ii       get def
            /y0 points ii 1 add get def
            /x1 points ii 2 add get def
            /y1 points ii 3 add get def
            /x2 points ii 4 add get def
            /y2 points ii 5 add get def
            /x3 points ii 6 add get def
            /y3 points ii 7 add get def
            docurve
        } for
        /x0 points ss 6 sub get def
        /y0 points ss 5 sub get def
        /x1 points ss 4 sub get def
        /y1 points ss 3 sub get def
        /x2 points ss 2 sub get def
        /y2 points ss 1 sub get def
        /x3 x2 2 mul x1 sub def
        /y3 y2 y1 sub x2 x1 sub div y1 y0 sub x1 x0 sub div sub 2 mul
            x2 x1 sub dup mul x2 x0 sub div mul
            y2 2 mul add y1 sub def
        docurve
    } {
        ss 4 eq {
            points 0 get points 1 get moveto
            points 2 get points 3 get lineto
        } if
    } ifelse
    stroke
    grestore
} def }
%
%		#] makecurve :
%		#[ makedashcurve :
%
\special{! /makedashcurve{
%
%   Incoming stack:
%   [array of x,y pairs] dashsize width scale
%
    gsw /dsize ed /points ed
    /ss points length 2 idiv 2 mul def
    newpath
    ss 4 gt {
        /x1 points 0 get def
        /y1 points 1 get def
        /x2 points 2 get def
        /y2 points 3 get def
        /x3 points 4 get def
        /y3 points 5 get def
        /x0 x1 2 mul x2 sub def
        /y0 y3 y2 sub x3 x2 sub div y2 y1 sub x2 x1 sub div sub 2 mul
            x2 x1 sub dup mul x3 x1 sub div mul
            y1 2 mul add y2 sub def
        x1 y1 moveto
        docurve
        0 2 ss 8 sub { /ii ed
            /x0 points ii       get def
            /y0 points ii 1 add get def
            /x1 points ii 2 add get def
            /y1 points ii 3 add get def
            /x2 points ii 4 add get def
            /y2 points ii 5 add get def
            /x3 points ii 6 add get def
            /y3 points ii 7 add get def
            docurve
        } for
        /x0 points ss 6 sub get def
        /y0 points ss 5 sub get def
        /x1 points ss 4 sub get def
        /y1 points ss 3 sub get def
        /x2 points ss 2 sub get def
        /y2 points ss 1 sub get def
        /x3 x2 2 mul x1 sub def
        /y3 y2 y1 sub x2 x1 sub div y1 y0 sub x1 x0 sub div sub 2 mul
            x2 x1 sub dup mul x2 x0 sub div mul
            y2 2 mul add y1 sub def
        docurve
    } {
        ss 4 eq {
            points 0 get points 1 get moveto
            points 2 get points 3 get lineto
        } if
    } ifelse
    centerdash
    stroke
    grestore
} def }
%
\special{! /pathlength{
    flattenpath
    /dist 0 def
    { /yfirst ed /xfirst ed /ymoveto yfirst def /xmoveto xfirst def }
    { /ynext ed /xnext ed /dist dist ynext yfirst sub dup mul
        xnext xfirst sub dup mul add sqrt add def
        /yfirst ynext def /xfirst xnext def }
    {}
    {/ynext ymoveto def /xnext xmoveto def
        /dist ynext yfirst sub dup mul
              xnext xfirst sub dup mul add sqrt add def
        /yfirst ynext def /xfirst xnext def }
    pathforall
    dist
} def }
%
\special{! /centerdash{
    /pathlen pathlength def
    /jj pathlen dsize div 2.0 div cvi def
    /ddsize pathlen jj 2.0 mul div def
    [ddsize] ddsize 2 div setdash
} def }
%
%		#] makedashcurve :
%		#[ logaxis :
%
\special{! /logaxis{
%
%   Draws an axis from x1,y1 to x2,y2 with nl log divisions
%   size of the hashes hs, offset F
%   and width W. The stack looks like
%   x1,y1,x2,y2,nl,hs,F,W,scale
%   After the rotation the hash marks are on top if nl is positive and
%   on the bottom if nl is negative
%
    gsw /offset ed /hashsize ed /nlogs ed p2 p1
    x1 y1 translate
    /y2 y2 y1 sub def /x2 x2 x1 sub def
    y2 x2 atan rotate
    /rr x2 dup mul y2 dup mul add sqrt def
    offset 0 ne { /offset offset ln 10 ln div def } if
    /offset offset dup cvi sub def
    newpath
        0 0 moveto
        rr 0 lineto
    /lsize rr nlogs div def
    0 1 nlogs { /x2 ed
        x2 offset ge {
            /y2 x2 offset sub lsize mul def
            y2 rr le {
                y2 0 moveto
                y2 hashsize 1.2 mul lineto
            } if
        } if
    } for
    stroke
    width 0.6 mul setlinewidth
    newpath
    0 1 nlogs { /x2 ed
        2 1 9 {
            ln 10 ln div x2 add
            /xx2 ed
            xx2 offset ge {
                /y2 xx2 offset sub lsize mul def
                y2 rr le {
                    y2 0 moveto
                    y2 hashsize 0.8 mul lineto
                } if
            } if
        } for
    } for
    stroke
    grestore
} def }
%
%		#] logaxis :
%		#[ linaxis :
%
\special{! /linaxis{
%
%   x1,y1,x2,y2,num_decs,per_dec,hashsize,offset,width,scale
%
    gsw /offset ed /hashsize ed /perdec ed /numdec ed p2 p1
    x1 y1 translate
    /y2 y2 y1 sub def /x2 x2 x1 sub def
    y2 x2 atan rotate
    /rr x2 dup mul y2 dup mul add sqrt def
    newpath
        0 0 moveto
        rr 0 lineto
    /x1 rr numdec perdec mul div def
    /y1 rr numdec div def
    /offset offset x1 mul def
    0 1 numdec { y1 mul offset sub
        dup 0 ge {
            dup rr le {
                dup 0 moveto
                hashsize 1.2 mul lineto
            } if
        } if
    } for
    stroke
    width 0.6 mul setlinewidth
    newpath
    offset cvi 1 numdec perdec mul offset add {
        x1 mul offset sub
        dup 0 ge {
            dup rr le {
                dup 0 moveto
                hashsize 0.8 mul lineto
            } if
        } if
    } for
    stroke
    grestore
} def }
%
%		#] linaxis :
%
\def\axowidth{0.5 }
\def\axoscale{1.0 }
\def\axoxoff{0 }
\def\axoyoff{0 }
\def\axoxo{0 }
\def\axoyo{0 }
\def\firstcall{1}
%
\def\Gluon(#1,#2)(#3,#4)#5#6{
%
%   Draws a gluon from (x1,y1) to (x2,y2) with amplitude and number of windings
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add #5 #6 \axowidth \axoscale gluon showpage}}
}
%
\def\Photon(#1,#2)(#3,#4)#5#6{
%
%   Draws a photon from (x1,y1) to (x2,y2) with amplitude and number of
%%windings
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add #5 #6 \axowidth \axoscale photon showpage}}
}
%
\def\ZigZag(#1,#2)(#3,#4)#5#6{
%
%   Draws a zigzag from (x1,y1) to (x2,y2) with amplitude and number of zigzags
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add #5 #6 \axowidth \axoscale zigzag showpage}}
}
%
\def\PhotonArc(#1,#2)(#3,#4,#5)#6#7{
%
%   Draws a photon on an arc segment. The center of the curve is at (1,2)
%   The radius, start angle and target angle are (#3,#4,#5), 6 is the
%   amplitude of the gluon, and 7 is the number of wiggles.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 #6 #7 \axowidth \axoscale photonarc showpage}}
}
%
\def\GlueArc(#1,#2)(#3,#4,#5)#6#7{
%
%   Draws a gluon on aa arc segment. The center of the curve is at (1,2)
%   The radius, start angle and target angle are (#3,#4,#5), 6 is the
%   amplitude of the gluon, and 7 is the number of windings.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 #6 #7 \axowidth \axoscale gluearc showpage}}
}
%
\def\ArrowArc(#1,#2)(#3,#4,#5){
%
%   Draws an arc segment with an arrow in it. The center of the curve
%   is at (1,2).
%   The radius, start angle and target angle are (#3,#4,#5).
%   The arc segment runs anticlockwise
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 \axowidth \axoscale arrowarc showpage}}
}
%
\def\LongArrowArc(#1,#2)(#3,#4,#5){
%
%   Draws an arc segment with an arrow at its end. The center of the curve
%   is at (1,2).
%   The radius, start angle and target angle are (#3,#4,#5).
%   The arc segment runs anticlockwise
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 \axowidth \axoscale longarrowarc showpage}}
}
%
\def\DashArrowArc(#1,#2)(#3,#4,#5)#6{
%
%   Draws a dashed arc segment with an arrow in it. The center of the curve
%   is at (1,2).
%   The radius, start angle and target angle are (#3,#4,#5).
%   The arc segment runs anticlockwise
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 #6 \axowidth \axoscale dasharrowarc showpage}}
}
%
\def\ArrowArcn(#1,#2)(#3,#4,#5){
%
%   Draws an arc segment with an arrow in it. The center of the curve
%   is at (1,2).
%   The radius, start angle and target angle are (#3,#4,#5).
%   The arc segment runs clockwise
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 \axowidth \axoscale arrowarcn showpage}}
}
%
\def\LongArrowArcn(#1,#2)(#3,#4,#5){
%
%   Draws an arc segment with an arrow at the end. The center of the curve
%   is at (1,2).
%   The radius, start angle and target angle are (#3,#4,#5).
%   The arc segment runs clockwise
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 \axowidth \axoscale longarrowarcn showpage}}
}
%
\def\DashArrowArcn(#1,#2)(#3,#4,#5)#6{
%
%   Draws a dashed arc segment with an arrow in it. The center of the curve
%   is at (1,2).
%   The radius, start angle and target angle are (#3,#4,#5).
%   The arc segment runs clockwise
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 #6 \axowidth \axoscale dasharrowarcn showpage}}
}
%
\def\ArrowLine(#1,#2)(#3,#4){
%
%   Draws a line with an arrow in it from (x1,y1) to (x2,y2)
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add \axowidth \axoscale arrowline showpage}}
}
%
\def\LongArrow(#1,#2)(#3,#4){
%
%   Draws a line with an arrow at the end from (x1,y1) to (x2,y2)
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add \axowidth \axoscale longarrow showpage}}
}
%
\def\DashArrowLine(#1,#2)(#3,#4)#5{
%
%   Draws a dashed line with an arrow in it from (x1,y1) to (x2,y2)
%   The size of the dashes is given by the fifth argument.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add #5 \axowidth \axoscale dasharrowline showpage}}
}
%
\def\Line(#1,#2)(#3,#4){
%
%   Draws a line from (x1,y1) to (x2,y2)
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add \axowidth \axoscale line showpage}}
}
%
\def\DashLine(#1,#2)(#3,#4)#5{
%
%   Draws a line from (x1,y1) to (x2,y2) with a dash pattern of which the
%   alternating black and white pieces are approximately {5} points long
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add #5 \axowidth \axoscale dashline showpage}}
}
%
\def\CArc(#1,#2)(#3,#4,#5){
%
%   Draws an arc segment. The center of the curve
%   is at (1,2).
%   The radius, start angle and target angle are (#3,#4,#5).
%   The arc segment runs anticlockwise
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 \axowidth \axoscale carc showpage}}
}
%
\def\DashCArc(#1,#2)(#3,#4,#5)#6{
%
%   Draws a dashed arc segment. The center of the curve
%   is at (1,2).
%   The radius, start angle and target angle are (#3,#4,#5).
%   The arc segment runs anticlockwise
%   dsize (6) is the dashsize. this is rounded to make things come
%   out right.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 #5 #6 \axowidth \axoscale dashcarc showpage}}
}
%
\def\Vertex(#1,#2)#3{
%
%   Draws a fat dot at (1,2). The radius of the dot is given by 3.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoscale vertex showpage}}
}
%
\def\Text(#1,#2)[#3]#4{
%
%   Draws text at (1,2). Argument 3 is l,r or c indicating whether
%   the text is left adjusted, right adjusted or centered.
%   4 is of course the text.
%
\dimen0=\axoxoff \unitlength
\dimen1=\axoyoff \unitlength
\advance\dimen0 by #1 \unitlength
\advance\dimen1 by #2 \unitlength
\makeatletter
\@killglue\raise\dimen1\hbox to\z@{\kern\dimen0 \makebox(0,0)[#3]{#4}\hss}
\ignorespaces
\makeatother
}
%
\def\BCirc(#1,#2)#3{
%
%   Draws a circle at (1,2) and radius 3 that is blanked out.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axowidth \axoscale bcirc showpage}}
}
%
\def\GCirc(#1,#2)#3#4{
%
%   Draws a circle at (1,2) and radius 3 that is blanked out.
%   Then it fills the circle with a gray scale 4 (0 = black, 1 is white)
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 #4 \axowidth \axoscale gcirc showpage}}
}
%
\def\EBox(#1,#2)(#3,#4){
%
%   Draws a box with the left bottom at (x1,y1) and the right top
%   at (x2,y2).
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add \axowidth \axoscale box showpage}}
}
%
\def\BBox(#1,#2)(#3,#4){
%
%   Draws a box with the left bottom at (x1,y1) and the right top
%   at (x2,y2). The box is blanked out.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add \axowidth \axoscale bbox showpage}}
}
%
\def\GBox(#1,#2)(#3,#4)#5{
%
%   Draws a box with the left bottom at (x1,y1) and the right top
%   at (x2,y2). The box is blanked out and then filled with gray 5
%   (0 is black, 1 is white)
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add #5 \axowidth \axoscale gbox showpage}}
}
%
\def\Boxc(#1,#2)(#3,#4){
%
%   Draws a box with the center at (x1,y1).
%   The width and height are (3,4).
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #3 2 div sub #2 \axoyo add
 #4 2 div sub #1 \axoxo add #3 2 div add #2 \axoyo add #4 2 div add
 \axowidth \axoscale box showpage}}
}
%
\def\BBoxc(#1,#2)(#3,#4){
%
%   Draws a box with the center at (x1,y1).
%   The width and height are (3,4). The contents are blanked out
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #3 2 div sub #2 \axoyo add
 #4 2 div sub #1 \axoxo add #3 2 div add #2 \axoyo add #4 2 div add
 \axowidth \axoscale bbox showpage}}
}
%
\def\GBoxc(#1,#2)(#3,#4)#5{
%
%   Draws a box with the center at (x1,y1).
%   The width and height are (3,4). The contents are blanked out
%   Then the contents are filled with gray 5 (0 is black, 1 is white)
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #3 2 div sub #2 \axoyo add
 #4 2 div sub #1 \axoxo add #3 2 div add #2 \axoyo add #4 2 div add
 \axowidth \axoscale gbox showpage}}
}
%
\def\SetWidth#1{\def\axowidth{#1 }}
\def\SetScale#1{\def\axoscale{#1 }}
\def\SetOffset(#1,#2){\def\axoxoff{#1 } \def\axoyoff{#2 }}
\def\SetScaledOffset(#1,#2){\def\axoxo{#1 } \def\axoyo{#2 }}
%
\def\pfont{Times-Roman }
\def\fsize{10 }
%
\def\SetPFont#1#2{\def\pfont{#1 } \def\fsize{#2 }}
%
%   Now some definitions to sort out the lt etc type of options in PText
%
\makeatletter
\def\fmode{4 }
\def\@l@{l} \def\@r@{r} \def\@t@{t} \def\@b@{b}
\def\mymodetest#1{\ifx#1\end \let\next=\relax \else {
\if#1\@r@\global\def\fmodeh{-3 }\fi
\if#1\@l@\global\def\fmodeh{3 }\fi
\if#1\@b@\global\def\fmodev{-1 }\fi
\if#1\@t@\global\def\fmodev{1 }\fi
} \let\next=\mymodetest\fi \next}
\makeatother
%
\def\PText(#1,#2)(#3)[#4]#5{
%
%   Draws a postscript text in a postscript font.
%   Focal point is (1,2), rotation angle is 3, 4 is the mode (as in text)
%   and 5 is the text.
%
\def\fmodev{0 }
\def\fmodeh{0 }
\mymodetest#4\end
\put(\axoxoff,\axoyoff){\makebox(0,0)[]{\special{"/\pfont findfont \fsize
 scalefont setfont #1 \axoxo add #2 \axoyo add #3
\fmode \fmodev add \fmodeh add \fsize (#5) \axoscale ptext }}}
}
%
\def\GOval(#1,#2)(#3,#4)(#5)#6{
%
%   Draws a colored oval that overwrites whatever was there.
%   Oval(x_center,y_center)(height,width)(rotation)(color)
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3 #4 #5 #6
\axowidth \axoscale goval showpage}}
}
%
\def\Oval(#1,#2)(#3,#4)(#5){
%
%   Draws an oval that does not overwrite whatever was there.
%   Oval(x_center,y_center)(height,width)(rotation)
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3 #4 #5
\axowidth \axoscale oval showpage}}
}
%
\let\eind=]
\def\DashCurve#1#2{\put(\axoxoff,\axoyoff){\special{" [ \kromme#1] #2
\axowidth \axoscale makedashcurve showpage}}}
\def\Curve#1{\put(\axoxoff,\axoyoff){\special{" [ \kromme#1] \axowidth
\axoscale makecurve showpage}}}
\def\kromme(#1,#2)#3{#1 \axoxo add #2 \axoyo add \ifx #3\eind\else
\expandafter\kromme\fi#3}
%
\def\LogAxis(#1,#2)(#3,#4)(#5,#6,#7,#8){
%
%   Draws a line with logarithmic hash marks along it.
%   LogAxis(x1,y1)(x2,y2)(num_logs,hashsize,offset,width)
%   The line is from (x1,y1) to (x2,y2) and the marks are on the left side
%   when hashsize is positive, and right when it is negative.
%   num_logs is the number of orders of magnitude and offset is the number
%   at which one starts at (x1,y1) (like if offset=2 we start at 2)
%   When offset is 0 we start at 1. Width is the linewidth.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add #5 #6 #7 #8 \axoscale logaxis showpage}}
}
%
\def\LinAxis(#1,#2)(#3,#4)(#5,#6,#7,#8,#9){
%
%   Draws a line with linear hash marks along it.
%   LinAxis(x1,y1)(x2,y2)(num_decs,per_dec,hashsize,offset,width)
%   The line is from (x1,y1) to (x2,y2) and the marks are on the left side
%   when hashsize is positive, and right when it is negative.
%   num_decs is the number of accented marks, per_dec the number of
%   divisions between them and offset is the number
%   at which one starts at (x1,y1) (like if offset=2 we start at the second
%   small mark) Width is the linewidth.
%
\put(\axoxoff,\axoyoff){\special{"#1 \axoxo add #2 \axoyo add #3
 \axoxo add #4 \axoyo add #5 #6 #7 #8 #9 \axoscale linaxis showpage}}
}
%
\input rotate.tex
\makeatletter
%
\def\rText(#1,#2)[#3][#4]#5{
%
%   Draws rotated text at (1,2). Argument 3 is l,r or c indicating whether
%   the text is left adjusted, right adjusted or centered.
%   4 is the rotation angle and 5 is of course the text.
%
\ifnum\firstcall=1\global\def\firstcall{0}\rText(-10000,#2)[#3][]{#5}\fi
\dimen2=\axoxoff \unitlength
\dimen3=\axoyoff \unitlength
\advance\dimen2 by #1 \unitlength
\advance\dimen3 by #2 \unitlength
\@killglue\raise\dimen3\hbox to \z@{\kern\dimen2
\makebox(0,0)[#3]{
\ifx#4l{\setbox3=\hbox{#5}\rotl{3}}\else{
\ifx#4r{\setbox3=\hbox{#5}\rotr{3}}\else{
\ifx#4u{\setbox3=\hbox{#5}\rotu{3}}\else{#5}\fi}\fi}\fi}\hss}
\ignorespaces
}
\makeatother
%
\def\BText(#1,#2)#3{
%
%   Draws a box with the center at (x1,y1) and postscript text in it.
%
\put(\axoxoff,\axoyoff){\special{"/\pfont findfont \fsize
 scalefont setfont #1 \axoxo add #2 \axoyo add (#3)
 \fsize \axowidth \axoscale btext showpage}}
}
%
\def\GText(#1,#2)#3#4{
%
%   Draws a box with the center at (x1,y1) and postscript(#4) text in it.
%   The grayness of the box is given by #3
%
\put(\axoxoff,\axoyoff){\special{"/\pfont findfont \fsize
 scalefont setfont #1 \axoxo add #2 \axoyo add #3 (#4)
 \fsize \axowidth \axoscale gtext showpage}}
}
%
\def\B2Text(#1,#2)#3#4{
%
%   Draws a box with the center at (x1,y1) and two lines of postscript
%   text in it.
%
\put(\axoxoff,\axoyoff){\special{"/\pfont findfont \fsize
 scalefont setfont #1 \axoxo add #2 \axoyo add (#3) (#4)
 \fsize \axowidth \axoscale b2text showpage}}
}
%
\def\G2Text(#1,#2)#3#4#5{
%
%   Draws a box with the center at (x1,y1) and two lines of postscript
%   text (#4 and #5) in it.
%   The grayness of the box is given by #3
%
\put(\axoxoff,\axoyoff){\special{"/\pfont findfont \fsize
 scalefont setfont #1 \axoxo add #2 \axoyo add #3 (#4) (#5)
 \fsize \axowidth \axoscale g2text showpage}}
}

