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\begin{document}
\begin{titlepage}
\hspace*{8cm} {UGVA-DPNC 1997/7-171 July 1997}
\newline
\hspace*{10.5cm} 
%{\it Corrected version 16/11/98}
\begin{center}
\vspace*{2cm}
{\large \bf
Convergence and Gauge Dependence Properties of the
Resummed One-loop Quark-Quark Scattering
Amplitude in Perturbative QCD }
\vspace*{1.5cm}
\end{center}
\begin{center}
{\bf J.H.Field }
\end{center}
\begin{center}
{ 
D\'{e}partement de Physique Nucl\'{e}aire et Corpusculaire
 Universit\'{e} de Gen\`{e}ve . 24, quai Ernest-Ansermet
 CH-1211 Gen\`{e}ve 4.
}
\end{center}
\vspace*{2cm}
\begin{abstract}
The one-loop QCD effective charge $\alpha_s^{eff}$ for quark-quark scattering
is derived by diagrammatic resummation of the one-loop amplitude using an 
arbitary covariant gauge. Except for the particular choice of gauge parameter
$\xi = -3$, $\alpha_s^{eff}$ is found to {\it increase} with increasing scale
as $\ln Q$ or $\ln^2 Q$. For $\xi = -3$, $\alpha_s^{eff}$  decreases with 
increasing $Q$ and satisfies a renormalisation group equation. Experimental
measurements of $\alpha_s$ indicate, however, that in this case, $\alpha_s^{eff}$
is convergent only for scales $\le$ 300 GeV.                      
\end{abstract}
\vspace*{1cm}
PACS 12.38-t, 12.38.Bx, 12.38.Cy
\newline
{\it Keywords ;} Quantum Chromodynamics
Renormalisation Group Invariance,
Asymptotic Freedom.
\end{titlepage}
 
 
\SECTION{\bf{Introduction}}
 Quark-quark scattering in next-to-leading order QCD has been calculated
 by several different groups~[1-4]. 
 Coqueraux and De Rafael~\cite{x1}, calculated the one-loop corrections to the
 invariant amplitude in the Feynman gauge using an on-shell renormalisation
 scheme~\cite{x5}. In Refs.[2-4] complete expressions for the squared invariant
 amplitude were given in the dimensional regularisation scheme~\cite{x6}.
 The calculations presented in the present paper generalise those of Ref.[1]
 in two ways:  
 \begin{itemize}
 \item[(i)] An arbitary covariant gauge is considered.
 \item[(ii)] The one-loop Ultra-Violet (UV) divergent loop and 
  vertex diagrams are resummed to all orders in $\alpha_s$.
  \end{itemize}
  (ii) yields a scale-dependent `effective charge' as a factor in the
  invariant amplitude. The resummation is done, not by solving a 
  renormalisation group equation, but by an exact sum of the relevant 
  diagrams to give the QCD analogue of the Dyson sum of QED.  
  \par The results are very surprising. The effective charge is 
  gauge dependent at $O(\alpha_s^3)$ and beyond, and except for one
  specific choice of gauge, does not display `asymptotic freedom' but
  instead {\it increases} as $\ln Q$ or $\ln^2 Q$ at large scales Q. 
  Only for the same special choice of gauge, where contributions from
  vertex diagrams vanish, does the effective charge satisfy a 
  renormalisation group equation of the type that is valid in 
  QED~[7-11]. For this special choice of gauge (called `loop gauge')
  the effective charge decreases with increasing scale, but only to 
  a fixed limit $Q_L$, determined by the convergence radius of the 
  geometric sum of gluon and fermion loops in the dressed gluon 
  propagator. The measured value of $\alpha_s$
  suggests that $Q_L \simeq$ 300 GeV, already of phenomenological 
  importance at the Fermilab $\rm{p} \overline{\rm{p}}$ collider. The similar 
  restrictions imposed by convergence of the Dyson sum in QED for an
  arbitary subtraction scale have been considered by the author in 
   another paper~\cite{x12}.
  \par The plan of the paper is as follows. In the next Section the Feynman
  gauge calculation of the quark-quark scattering amplitude~\cite{x1} is 
  generalised to an arbitary covariant gauge, using results of one-loop
  calculations reported in Ref.[13]. In Section 3 the one-loop corrections
   are diagrammatically resummed to yield the QCD analogue of the Dyson sum
   of QED. In Section 4 the self-similarity and renormalisation group
   properties of the effective charge derived in Section 3 are discussed.
   In the final Section the classical proofs of the asymptotic
   freedom property of QCD in the literature are critically examined 
   in the light of the results obtained in the previous Sections. Also briefly
   discussed are : (i) `renormalons' (ii) the generalisation to higher
   loop order 
   vacuum polarisation and vertex corrections and (iii)
    pinch technique and related calculations of proper self energy and 
   vertex functions, both in QCD and in the Standard Electroweak Model.
    
\SECTION{\bf{The quark-quark scattering amplitude to $O(\alpha_s^2)$
in an arbitary covariant gauge}}
 The process considered is the scattering of two equal mass quarks through
 an angle of 90$^{\circ}$ in their CM system. The lowest order diagram is shown
 in Fig.1a. The four-vectors of the incoming (outgoing) quarks are $p_1,p_2$
 ($p_3,p_4$). In this configuration the exchanged gluon has a virtuality
 $t = -s/2$ where:
 \begin{eqnarray}
 t & \equiv & (p_1-p_3)^2 = u \equiv (p_1-p_4)^2  \\
 s & \equiv & (p_1+p_2)^2 = (p_3+p_4)^2
 \end{eqnarray}
 Denoting the invariant amplitude corresponding to the diagram in Fig.1a by 
 ${\cal M}^{(0)}$ (quark spin and colour indices are suppressed), then the
 $O(\alpha_s^2)$ amplitude may be written as:
 \begin{equation}
  {\cal M}^{(1)} =  {\cal M}^{(0)} \sum_{i} {\cal A}_i
 \end{equation}
 where the one-loop corrections ${\cal A}_i$ are given by the diagrams shown
 in Figs.1b-i. Three topologically distinct types of diagrams occur: 
 \begin{itemize}
 \item vertex corrections as in Figs.1b,c and the two similar diagrams
 given by the exchange $13 \leftrightarrow 24$;
 \item loop corrections (Figs.1d-g);
 \item box diagrams (Figs.1h,i).
 \end{itemize} 
 The corresponding corrections ${\cal A}_V$, ${\cal A}_L$ and ${\cal A}_B$
 have been calculated in leading logarithmic approximation and in Feynman
 gauge in Ref.[1] :
 \begin{eqnarray}
 {\cal A}_V & = & \frac{\alpha_s^0}{\pi}\left[ -3 \ln (\frac{Q}{m_g})
 + 3 \ln^2(\frac{Q}{m})-6 \ln(\frac{Q}{m})\ln (\frac{Q}{m_g})\right]  \\
 {\cal A}_L & = & \frac{\alpha_s^0}{\pi}\left[-\frac{5}{2}
 \ln (\frac{Q}{m_g}) + \frac{n_f}{3} \ln(\frac{Q}{\overline{m}})\right]  \\
 {\cal A}_B & = & \frac{\alpha_s^0}{\pi}\ln 2 \left[\ln(\frac{Q^2}
 {\sqrt{2} m^2}) \right]
 \end{eqnarray}
 Here $Q = \sqrt{-t} = \sqrt{s/2}$ is the physical scale, $m$ is the mass of
 the scattered quarks, $m_g$ is a regulator gluon mass and $\overline{m}$
 is the average mass: 
 \begin{equation}
 \overline{m}= (m_1 m_2 ... m_{n_f})^{\frac{1}{n_f}}
 \end{equation}
 of the $n_f$ quark flavours contributing to the vacuum polarisation loops
 in Fig.1d. $\alpha_s^0$ is the square of the
 renormalised on-shell strong coupling
 constant. The singly logarithmic terms in ${\cal A}_V$ and ${\cal A}_L$
 may be combined to yield a term proportional to the first coefficient in
 the perturbation series in $\alpha_s$ of the beta function of QCD. Denoting
 these terms by ${\cal V}^0$ and ${\cal L}^0$, respectively, and generalising to an arbitary
 covariant gauge specified by the parameter $\xi$, in which the gluon 
 propagator is written as :
 \begin{equation}
 P^{\mu \nu}(q^2) = - \frac{i}{q^2}\left[ g^{\mu \nu}-(1-\xi)
 \frac{q^{\mu}q^{\nu}}{q^2} \right]
 \end{equation}
 then~\cite{x13}:
 \begin{eqnarray} 
 {\cal V}^0(\xi) & = &  -\frac{\alpha_s^0}{\pi} \frac{3}{4} (3 +\xi)
  \ln (\frac{Q}{m_g})  \\
{\cal L}^0(\xi) & = &  \frac{\alpha_s^0}{\pi}\left[-\frac{3}{4}
(\frac{13}{3}-\xi) \ln (\frac{Q}{m_g}) + \frac{n_f}{3} 
\ln(\frac{Q}{\overline{m}})\right]
\end{eqnarray}
Note that the box diagrams in Figs. 1h,i and the abelian (QED-like) vertex
digrams of Fig. 1b, do not contribute, in an arbitary covariant gauge, to the
 UV divergent (before renormalisation) singly logarithmic terms.
See reference [14] for a complete diagrammatic discussion of gauge cancellations
in the one-loop corrected quark-quark scattering amplitude. 
Adding Eqns.(2.9) and (2.10):
\begin{equation}
{\cal V}^0+{\cal L}^0 = -\frac{\alpha_s^0}{\pi}\left[\frac{11}{2}
 \ln (\frac{Q}{m_g}) - \frac{n_f}{3} 
\ln(\frac{Q}{\overline{m}})\right]
\end{equation}
so the dependence of ${\cal M}^{(1)}$ on the gauge parameter $\xi$ 
cancels in the sum of vertex and loop contributions.
Eqn.(2.11) may be written as:
\begin{equation}
{\cal V}^0+{\cal L}^0 = -\frac{\alpha_s^0}{\pi}(\frac{11}{2}
  - \frac{n_f}{3}) \ln(\frac{Q}{\mu_0}) =
-\frac{\alpha_s^0}{\pi} \beta_0 \ln(\frac{Q}{\mu_0})
\end{equation} 
where $\beta_0$ is the first coefficient of the QCD $\beta$ function 
 [ $a_s \equiv \alpha_s/\pi$] :
\begin{equation} 
\mu \frac{\partial a_s}{\partial \mu} = \beta(a_s) = -\beta_0 a_s^2 + ...
\end{equation}
Comparing Eqns.(2.11) and (2.12) gives :
\begin{equation}
\mu_0 = \exp \left(\frac{33\ln m_g - 2 n_f \ln \overline{m}}
{33-2 n_f} \right)
\end{equation}
It will be shown below that the Renormalisation Group Equation (RGE)
(2.13) is satisfied by the resummed one-loop effective charge for the
quark-quark scattering amplitude only for the particular choice of
gauge parameter $\xi = -3$ where the vertex contribution ${\cal V}^0$
vanishes.
\par In the subsequent discussion, the same renormalisation subtraction
 point $t = -\mu^2$ is chosen for all the UV divergent diagrams
 in Fig.1. In this case Eqns.(2.9),(2.10) are replaced by:
 \begin{eqnarray} 
 {\cal V}^{\mu}(\xi) & = &  -\frac{\alpha_s^{\mu}}{\pi} \frac{3}{4} (3 +\xi)
  \ln (\frac{Q}{\mu})  \\
{\cal L}^{\mu}(\xi) & = &  \frac{\alpha_s^{\mu}}{\pi}\left[-\frac{3}{4}
(\frac{13}{3}-\xi) + \frac{n_f}{3}\right]\ln(\frac{Q}{\mu})
\end{eqnarray}  
 where $\alpha_s^{\mu}$ is the QCD coupling constant at scale $-\mu^2$.
 To $O(\alpha_s^2)$, $\alpha_s^{Q}$ may be identified with the solution,
 $\alpha_s^{RGE}(Q)$, 
 of the RGE (2.13):
 \begin{eqnarray} 
 {\cal M}^{LO} & = & {\cal M}^{(0)} + {\cal M}^{(1)} = {\cal M}^{(0)}
 (1+{\cal V}^{\mu}+ {\cal L}^{\mu}) \nonumber \\
 & = & \frac{{\cal M}^{(0)}}{\alpha_s^{\mu}}\left[\alpha_s^{\mu}
 [1- \frac{\alpha_s^{\mu}}{\pi}\beta_0 \ln(\frac{Q}{\mu})]\right]
 \nonumber \\
 & = & \frac{{\cal M}^{(0)} \alpha_s^{RGE}(Q) }{\alpha_s^{\mu}}+
 O\left( (\alpha_s^{\mu})^3\right)
\end{eqnarray} 
where
\begin{equation}
 \alpha_s^{RGE}(Q) = \alpha_s^Q = \frac{ \alpha_s^{\mu}}
{1+ \frac{\alpha_s^{\mu}}{\pi}\beta_0 \ln(\frac{Q}{\mu})}
\end{equation}
${\cal M}^{LO}$ denotes the amplitude including Leading Order
vertex and vacuum polarisation corrections. In the following section the
amplitude ${\cal M}^{(\infty)}$, in which these corrections are summed to
all orders in $\alpha_s$, is derived.
%

\SECTION{\bf{The resummed quark-quark scattering amplitude}}
The topographical structures \footnote{The use of the word `topographical structure' to indicate
    a particular disposition of vertex and (possibly resummed) self-energy
    insertions in a diagram is deliberate. Diagrams with internal lines in
    the vertex or self-energy insertions have a different topology but the same
    topography as the one-loop diagrams shown in Fig.2} of the diagrams that modify the gluon propagator
in the quark-quark scattering amplitude at $O(\alpha_s^2)$, $O(\alpha_s^3)$
and $O(\alpha_s^4)$ are shown in Figs.2a,b,c respectively. $V$ and $L$ denote
vertex and loop (vacuum polarisation) contributions:
\begin{eqnarray}
 V & = & V_1 \\
 L & = & \sum_{i=1}^{n_f} F_i + G_1 +G_2 +G_3 
\end{eqnarray}
$V_1$ corresponds to the diagram in Fig.1c ;  $\sum_{i=1}^{n_f} F_i$
to Fig.1d and $G_1$, $G_2$, $G_3$ to Figs.1e,f,g.
  In the case that only one vertex insertion occurs there is a factor 2 for
the two ends of the gluon propagator. Since the propagator has only two ends
the vertex corrections are never higher than quadratic in the perturbation
series for the amplitude. Although the topographical structure of diagrams
containing vertex corrections is different at $O(\alpha_s^3)$ than at
$O(\alpha_s^2)$ it remains the same at all higher orders. The all--orders
resummed amplitude is:
\begin{eqnarray}
{\cal M}^{(\infty)} & = & {\cal M}^{(0)}\left[ 1 + 2V + L \right. \nonumber \\
                  &   &~~~~~+V^2 +2VL + L^2  \nonumber \\ 
                  &   &~~~~~\left.+V^2L +2VL^2 + L^3 + ...~~ \right]  \nonumber \\ 
                  & = & {\cal M}^{(0)}\left[ 1 + L + L^2 + ... +V^2 \right.
                        (1 + L + L^2 + ... \nonumber \\  
                  &   &~~~~~\left.+2V(1 + L + L^2 + ...~~ \right]  \nonumber \\ 
                  & = & \frac{{\cal M}^{(0)}(1+V)^2}{1-L} 
\end{eqnarray}  
In an arbitary covariant gauge with momentum subtraction at scale $\mu$,
and in leading logarithmic approximation, 
$V = {\cal V}^{\mu}(\xi)/2$, $L = {\cal L}^{\mu}(\xi)$ so that                   
\begin{equation}
{\cal M}^{(\infty)} = \frac{{\cal M}^{(0)}  \alpha_s^{eff}(Q)}{\alpha_s^{\mu}}
= \frac{{\cal M}^{(0)}(1+\frac{1}{2}{\cal V}^{\mu}(\xi))^2}{1-{\cal L}^{\mu}(\xi)}
\end{equation}
leading to the following exact expression for the resummed one-loop 
effective charge:
\begin{equation}
\alpha_s^{eff}(Q) = \alpha_s^{\mu}\frac{\left[1-\frac{3 \alpha_s^{\mu}}{8 \pi}
(3+\xi) \ln(\frac{Q}{\mu})\right]^2}
{1+\frac{\alpha_s^{\mu}}{4 \pi}\left[13-3\xi-\frac{4 n_f}{3}\right]
\ln(\frac{Q}{\mu})}
\end{equation}
The conventional one-loop QCD running coupling constant Eqn.(2.18)
 is recovered only for the special choice of gauge parameter $\xi = \xi_L=-3$
 (`loop gauge'). Only in this case does $\alpha_s^{eff}(Q)$ decrease monotonically
  with increasing $Q$. For any other choice of gauge $\alpha_s^{eff}(Q)$ does not
  show `asymptotic freedom' as $Q \rightarrow \infty$ but instead {\it increases}
  as $\ln Q$ when $\xi \ne \xi_V$ and as $\ln^2 Q$ when $\xi = \xi_V$. The
   choice $\xi = \xi_V$ (`vertex gauge') where
   \[ \xi_V = (39-4 n_f)/9  \]
   corresponds to a vanishing coefficient of $\ln(Q/\mu)$ in the denominator
 of Eqn.(3.5). As disussed in more detail below, only in loop gauge is the
 equation for the effective charge `self similar' like the effective charge in
 QED or the solution (2.18) of the RGE Eqn.(2.13). Even in loop gauge the
  effective charge of Eqn.(3.5) does not decrease without limit as 
$Q \rightarrow \infty$. The maximum possible scale $Q_L$ (Landau scale) is
determined by the convergence properties of the geometric sum that yields the
denominator of Eqn.(3.5). The geometric series is convergent  
provided that $|L|<1$~\cite{x15}. This implies that:  
\begin{equation}
\left|\frac{\alpha_s^{\mu}}{4 \pi}[13-3 \xi- \frac{4 n_f}{3}]
\ln(\frac{Q}{\mu})\right| < 1
\end{equation}
The corresponding Landau scale is then:
\begin{equation}
Q_L = \mu \exp \left[{\frac{4 \pi}{\alpha_s^{\mu}}
[13-3 \xi- \frac{4 n_f}{3}]}\right]
\end{equation} 
and
\begin{equation}
\alpha_s^{eff}(Q_L) = \frac{\alpha_s^{\mu}}{2}
\left[1-\frac{3 \alpha_s^{\mu}}{8 \pi}
(3+\xi) \ln(\frac{Q_L}{\mu})\right]^2
\end{equation}
For $\xi = -3$:
\begin{equation}
\alpha_s^{eff}(Q_L) = \frac{\alpha_s^{\mu}}{2}
\end{equation}
so the convergence condition (3.6) implies that in this case $\alpha_s^{eff}$
cannot evolve down by more that a factor of $\frac{1}{2}$ of its initial value
before Eqn.(3.5) diverges
--- there is no `asymptotic freedom'.
\par Numerical values of $Q_L$ for $\mu = 5$ GeV,  $\alpha_s^{\mu} = 0.2$
 and $n_f=5$ (corresponding, approximately, to the experimental
 value of $\alpha_s^{\mu}$)  
 and four different choices of gauge parameter are presented in Table 1.
 For loop gauge ($\xi = -3$) $Q_L$ is only 300 GeV, with phenomenological 
 consequences perhaps already at the Fermilab $ \rm{p} \overline{\rm{p}}$ collider, 
 but certainly at the future LHC pp collider. It may be noted that, in all
 cases except vertex gauge, $Q_L$ lies well below the Grand Unification (GUT)
 scale of $\simeq 10^{15}$ GeV. Since for any gauge choice except $\xi = -3$,
 $\alpha_s^{eff}$ diverges as $\ln Q$ or $\ln^2 Q$ at large $Q$, there can be
 no `unification'~\cite{x16,x17} of the strong and electromagnetic interactions
 at large scales $Q$, for any choice of gauge parameter, at least if the running
 strong coupling constant is identified with an effective charge such as that
 in Eqn.(3.5). In fact all studies, to date, of Grand Unification have implicitly
 used loop gauge where the maximum scale $Q_L$ is only $\simeq$ 300 GeV.
 Details of the divergent pathologies of geometrical series when $|R| \ge 1$,
 where $R$ is the common ratio of the series,
 may be found in Refs.[12,15]. In vertex gauge, since all vacuum polarisation
 contributions vanish, there is no convergence limitation on $Q$ in Eqn.(3.5). 
 For all gauge choices except $\xi = -3$ and with $Q > \mu$ the qualitative
 features of the evolution of $\alpha_s^{eff}$ with $Q$ are similar. At first
 $\alpha_s^{eff}$ decreases until the scale $Q_0$ given by:
 \begin{equation}
Q_0 = \mu \exp \left[{\frac{8 \pi}{3 \alpha_s^{\mu}}
(3+\xi)}\right]
\end{equation} 
 At $Q_0$, $\alpha_s^{eff}$ vanishes and has vanishing first derivative. For 
 $Q > Q_0$, $\alpha_s^{eff}$ then increases with asymptotic behaviour 
 $ \propto \ln Q$ ( or $\ln^2 Q$ in vertex gauge). Numerical values of $Q_0$ for the
  same parameter choices as in Table 1 are presented in Table 2. The evolution
  of $\alpha_s^{eff}$ with $Q$ for different gauge choices is shown, for
   the interval 5-300 GeV in Fig 3. and, with an expanded logarithmic scale in
 the interval 10-10$^4$ TeV, in Fig 4. The vanishing of $\alpha_s^{eff}$ 
 occurs at respective scales of 18.1, 176 and 5800 TeV for vertex, Feynman 
 and Landau gauges. In Figs 3 and 4, $\mu = 5$ GeV, $\alpha_s^{\mu} = 0.2$ and
 $n_f = 5$ are assumed.
%  
\SECTION{\bf{Self-Similarity Properties of $\alpha_s^{eff}$ and the
Renormalisation Group}}
Introducing the abbreviated notation:
\[ v(\xi) \equiv \frac{3}{8} (3 + \xi)~~~,~~ l(\xi) = \frac{1}{4}\left[
13-3\xi-\frac{4 n_f}{3}\right] \] 
\[ \alpha_s^{eff}(Q)/\pi \equiv a_Q~~~,~~ \alpha_s^{\mu}/\pi \equiv a_{\mu}~~~,
~~\lambda \equiv \ln (Q/\mu) \]
Eqn.(3.5) may be written as:
\begin{equation}
a_Q = \frac{a_{\mu}[1-a_{\mu}v(\xi)\lambda]^2}{1+a_{\mu}l(\xi)\lambda}
\end{equation}
With $\xi = -3$, $v(-3) = 0$, Eqn.(4.1) becomes 
\begin{equation}
a_Q = \frac{a_{\mu}}{1+a_{\mu}l(-3)\lambda}
\end{equation}  
Eqn.(4.2) is the solution of a one-loop RGE similar to that for the QED 
effective charge~\cite{x10,x11}:
\begin{equation}
\frac{Q}{a_Q}\frac{\partial a_Q}{\partial Q} = -l(-3) a_Q = -\beta_0 a_Q
\end{equation}
where
\begin{equation}
\beta_0 = l(-3) = \frac{11}{2}-\frac{n_f}{3}
\end{equation}
 For a gauge choice such that $v \ne 0$ the partial differential equation
  satisfied by $a_Q$ is:
\begin{equation}
\frac{Q}{a_Q}\frac{\partial a_Q}{\partial Q} = -a_Q \left[\frac{l(\xi)}
{(1-a_{\mu}v(\xi)\lambda)^2}+
2\frac{v(\xi)(1+a_{\mu}l(\xi)\lambda)}{(1-a_{\mu}v(\xi)\lambda)^3}\right]
\end{equation}
so that, in this case, the effective charge $a_Q$ does not satisfy the RGE 
 (2.13).
Expanding in powers of $a_{\mu}$ on the right hand side of Eqn(4.5) gives:
\begin{eqnarray}
\frac{Q}{a_Q}\frac{\partial a_Q}{\partial Q}& = &-a_Q \left[l(\xi)+2 v(\xi)
+ O(a_{\mu})\right] \\
& = &-a_Q \left[\beta_0
+ O(a_{\mu})\right] 
\end{eqnarray}
Thus, neglecting terms of $O(a_Q a_{\mu}) \simeq O(a_{\mu}^2)$ so that only
 the first term in the QCD perturbation series is retained , $a_Q \rightarrow
 a_Q^{(1)}$ and it can be seen that $a_Q^{(1)}$, for an arbitary gauge choice,
 satisfies the same partial differential equation as the one-loop 
 all orders resummed $a_Q$ in loop gauge. The relation of this result to 
 previous derivations of the QCD running coupling constant, where it has 
 generally been conjectured that a gauge invariant result is obtained to all
 orders in perturbation theory, is discussed in the following Section.
 \par The equation (4.2) is self-similar in the sense that , for any values of
 $\mu$ and $Q$ the equation defined by the exchange $\mu \leftrightarrow Q$
 is identical to the original equation. A consequence of this symmetry 
 property is that $a_Q$ in Eqn(4.2) is independent of $\mu$ (with the 
 important caveat that, since the denominator is the sum to infinity of a
 geometric series, $\mu$ must be such that $|a_{\mu}l(-3)\lambda|<1$),
 and that in the equation with $\mu \leftrightarrow Q$, $a_{\mu}$ is 
 independent of Q. This is the mathematical basis of the Renormalisation
  Group~\cite{x7,x8,x9}.
 Such a universal self-similarity property is not, however, shared by Eqn.(4.1)
 when $v(\xi) \ne 0$. 
 \par Eqn.(4.1) is self-similar under the exchange $\mu \leftrightarrow Q$
 provided that the equation: 
\begin{equation}
a_{\mu} = \frac{a_{Q}[1+a_{Q}v(\xi)\lambda]^2}{1-a_{Q}l(\xi)\lambda}
\end{equation}
and Eqn.(4.1) are both valid.
Simultaneous solution of Eqns(4.1),(4.8) in the case that $v(\xi) \ne 0$ leads
to a quadratic equation for $a_{\mu}$ with the solution:
\begin{equation}
a_{\mu} = \frac {1}{v(\xi)\lambda}+
\frac{a_Q}{2}\pm\frac{1}{2}\sqrt{(\frac{2}{v(\xi)\lambda}+a_Q)
(\frac{2}{v(\xi)\lambda}-3a_Q)}
\end{equation}
Real solutions of Eqn.(4.9) exist provided that either
\begin{equation}
\frac{2}{3v(\xi)\lambda}> a_Q > -\frac{2}{v(\xi)\lambda}
\end{equation}
(both factors under the square root positive) or
\begin{equation}
\frac{3v(\xi)\lambda}{2}> \frac{1}{a_Q} > -\frac{v(\xi)\lambda}{2}
\end{equation} 
(both factors under the square root negative) 
\par The solution for $a_{\mu}$, Eqn.(4.9), is independent of $l(\xi)$. Thus for fixed
values of $a_Q$, $Q$, $\mu$, $v(\xi) \ne 0$, satisfying the conditions (4.10) or
(4.11) there are, in general, two values of $a_{\mu}$ such that Eqn.(4.1) is
self-similar. This value of $a_{\mu}$ has however no relation to the effective
charge at the scale $\mu$ give by Eqn.(4.1) when $Q=\mu$. Choosing $Q = 90$ GeV
and Landau gauge then (see Fig.3) $a_Q = 0.095/\pi = 0.0302$. The choice 
$\mu = 5$ GeV gives $2/(v(0)\lambda) = 1.933$. The $a_Q$ terms under the square root
 of Eqn.(4.9) may then be neglected, leading to the solutions:
 \[ a^+_{\mu} \simeq \frac{2}{v(0)\lambda}+\frac{a_Q}{2} = 1.9648 \]
 \[ a^-_{\mu} \simeq \frac{a_Q}{2} = 0.0152 \]
  to be compared with the physical value:
  \[ a_{\mu} = \alpha^{eff}(5 GeV)/\pi =0.2/\pi = 0.064 \]
 \par Unlike for the special case $v = 0$, the first derivative of $a_Q$ in 
 general depends on the scale $\mu$. For $Q=\mu$ the derivative 
 is a negative constant 
 fixed by the first coefficient in the perturbation series for the beta function
 as in Eqn.(4.3). For any other choice of $\mu$ when $v \ne 0$ the derivative
 varies with $Q$ and $\mu$ according to Eqn.(4.5). Thus, if the effective charge is 
 parametrised in terms of $a_{\mu}$, $a_{\mu'}$ at the reference scale $Q_R$:
\begin{equation}
a_{Q_R} = a_{\mu}\frac{[1-a_{\mu}v(\xi)\lambda_R]}{[1+a_{\mu}l(\xi)\lambda_R]}
= a_{\mu'}\frac{[1-a_{\mu'}v(\xi)\lambda_R']}{[1+a_{\mu'}l(\xi)\lambda_R']}
\end{equation}
\[ \lambda_R \equiv \ln(Q_R/\mu)~~~,\lambda_R' \equiv \ln(Q_R/\mu') \]
then for $Q \ne Q_R$ and $v(\xi) \ne 0$ the effective charge $a_Q$
 predicted by the formula containing $\mu$ (the second member of Eqn.(4.12))
 will be different to that predicted by that containing $\mu'$ (the third member
 of Eqn.(4.12)). 
 Renormalisation group invariance with respect
  to the choice of the scale $\mu$ is
 therefore not respected unless $v(\xi)=0$, i.e. $\xi =-3$.
    
\SECTION{\bf{Discussion}}
 In the original derivations of the `asymptotic freedom' property of QCD
 `\cite{x18,x19} no calculations were performed beyond the lowest non-trivial
 order, $O(\alpha_s)$, and no actual amplitudes for physical processes were 
 considered. It was conjectured (without any check by direct diagrammatic
 calculation) that the QCD running coupling constant (RCC) could, in general
 (for any choice of gauge) be identified with the solution of the differential
 equation (2.13). The one-loop  QCD beta function was calculated by considering
 the Callan-Symanzik~\cite{x10,x11} equation for an irreducible n-point function
 (typically the gluon-quark-quark vertex or the triple gluon vertex). Calculation
 of the anomalous dimensions of the quark and gluon fields then yields the
 (gauge invariant) expression for the first beta function coefficient $\beta_0$
 in Eqn.(2.12) above. An analogous result is obtained above by considering
 the unresummed  one-loop correction to the physical quark-quark scattering
 amplitude. The true high order behaviour, however, corresponds to the sum
 of {\it all} the possible  amplitudes for the process of interest.
 For this the actual topographical structure of the diagrams contributing
 to the amplitude must be properly taken into account. 
  Renormalisation scale invariance of the RCC, and the asymptotic freedom
   property, for an arbitary choice of covariant gauge,
   are not confirmed by the 
 diagrammatic calculation of higher order corrections in the case of the
  quark-quark scattering amplitude considered in this paper. 
  \par Gauge dependence of the RCC in QCD has been considered previously in the literature,
  but usually as an effect only at the two-loop level and at higher orders. It was pointed
  out that, in the case when the bare parameters of the theory are held fixed, the 
  gauge parameter becomes scale dependent, and for certain momentum subtraction
  renormalisation schemes, the second coefficient of the beta function is both
  scheme and gauge dependent~\cite{x20}. For an arbitary covariant gauge specified
  by the fixed parameter $\xi$, as in the one-loop discussion above, the second
  beta function coefficient is, however, both gauge and renormalisation scheme
  invariant. Assuming that the RCC, mass and gauge parameter each satisfy
  renormalisation group equations similar to (2.13), and solving the coupled
  system of differential equations, solutions were found for the RCC that
  strongly depended on the initial conditions imposed on the running gauge
  parameter~\cite{x20}. These solutions exhibit either asymptotic freedom-like behaviour
  or increase with increasing scale until an ultraviolet fixed point is reached
  ~\cite{x21,x22}. As commonly done in the literature, the RCC was treated as an
  independent mathematical object, without reference to any actual physical process,
  and the renormalisation group equations were assumed to hold without specific
  diagrammatic justification.
  It is shown above that, if the RCC is identified with the effective charge
  of the quark-quark scattering amplitude, the renormalisation group equation
  holds only for the specific gauge choice $\xi = -3$. The gauge parameter can 
  then neither vary nor satisfy a RGE.
 \par When quark mass effects are taken into account, the one loop beta function 
  coefficient is also gauge dependent, and has a value which depends on the
  particular n-point Green function considered for its derivation. The mass
  dependent corrections to the triple gluon vertex~\cite{x23} and the gluon
  ghost ghost vertex~\cite{x24} are different. A detailed discussion may
  be found in Reference ~\cite{x25}. A corollary is that the RCC in physical amplitudes
  is both gauge and process dependent at physical scales where quark mass effects
  (other than those contained in the asymptotically dominant logarithmic
  terms ) are important. 
  \par The effective charge (3.7) has been calculated here for the simple case
  of quark-quark scattering with a unique physical scale $Q =\sqrt{-t}$.
   In this case the direct physical interpretation as the strength
    of the interaction between two
  currents varying as a function of their separation ($\simeq 1/Q$) is 
  particularly transparent.
   However, since every dressed propagator has just two ends, similar 
   expressions for the RCC (in general a function of some running
   loop 4-momentum $k$) are expected in all physical amplitudes
   containing virtual gluon lines.
   Two examples are shown in Fig.5. Fig.5a shows the three topographically
   distinct classes of diagrams that contribute to the anomalous magnetic
   moment of a heavy quark at $O(\alpha_s^3)$. In Fig.5b the same classes
   of diagrams are shown for the four-loop photon proper self energy function
   due to radiatively corrected quark vacuum polarisation loops. As for the
   quark-quark scattering case the same topographical structures (giving at most
   a quadratic dependence on the vertex corrections ) is found at all higher orders
   in the `dressed gluon propagator'. The diagrams of Fig.5b are related via the
   optical theorem and analytical continuation to the process: 
   \[ e^+e^- \rightarrow \gamma^* \rightarrow q \overline{q}+X \]
   where $X$ denotes $g$, $gg$ or $q' \overline{q}'$.
   \par There has been considerable recent interest in the structure in high orders
   of perturbation theory of diagrams containing chains of vacuum polarisation loops
   in internal gluon propagators (for example the generalisation to higher orders
   of the $O(\alpha_s^3)$  diagrams with two vacuum polarisation loops shown in 
   Fig.5). The anstatz used for these so-called `renormalon chains'
   ~\cite{x26} is to replace $n_f/3$ in a calculation considering only $n_f$ different
   flavours of fermion vacuum polarisation loops
    with $n_f/3-11/2= -\beta_0$. This is
   clearly a good approximation in the limit $n_f \rightarrow \infty$. As inspection
   of Fig.5 shows, however, this will result, in any gauge in which the vertex correction
   is non-vanishing, in a miscounting of the contribution 
   of the latter, which are included at order
   $n$ in terms of the form $\beta_0^n$, but actually should never appear at higher order than
   quadratic in the perturbation series. With however the gauge choice 
   $\xi = -3$ (loop gauge) all vertex corrections vanish and the renormalon
   chains are correctly given by the above replacement. This gauge choice is,
   in any case, the one universally (although tacitly) made in all phenomenological
   applications of the QCD running coupling constant, where renormalisation
   group invariance is assumed.
   Thus the `Naive Non-Abelianization' ansatz~\cite{x27} or `Large $\beta_0$ Limit'
   that is typically assumed in phenomenological studies of renormalon effects is 
   a correct one only in loop gauge.   
  The remark that the choice of gauge $\xi = -3$
   implies the vanishing of all vertex corrections, has previously been made in the 
   context of two-loop corrections to heavy quark production in 
   $e^+e^-$ annihilation~\cite{x28}.
   \par The classification of diagrams as in Figs.2 and 5, according to the 
   categories (in an obvious notation) $L^n,~VL^n,~V^2L^n$, will remain when
   L, V are calculated with an arbitary number of internal lines. The global
   structure of Eqn(3.5) will then remain the same for calculations including an
   arbitary number of loops, though additional non-leading terms in $\ln Q$ will
   result from integration over internal loops in the basic one-loop 
   vacuum polarisation and vertex diagrams shown in Fig.1
   \par As in Ref.(1), only the leading logarithmic terms in the one-loop
   correction (and hence in the resummed effective charge) have been taken into
   account in the above discussion. Constant terms in $V$ and $L$ 
   have been neglected. For a general renormalisation scheme however, constant
   gauge and renormalisation scheme dependent terms also occur, so that
   Eqn.(4.1) is replaced by the expression:
   \begin{equation}
   a_Q = a_{\mu}\frac{\{1-a_{\mu}[v(\xi)\lambda+c_v(\xi)]\}^2}
   {1+a_{\mu}[l(\xi)\lambda+c_l(\xi)]}
   \end{equation}
   In the $\overline{MS}$ scheme~\cite{x14}:
   \[ c_l(\xi) =
    \frac{10 n_f}{9}-\frac{97}{12}-\frac{3}{8}\xi -\frac{3}{16} \xi^2 \]
    By a suitable scale choice $\mu = \mu'$ and with $\xi = -3$ Eqn.(5.1) may
    be written as:
   \begin{equation}
   a_Q = \frac{a_{\mu'}}
   {1+a_{\mu'}[l(-3)\ln\frac{Q}{\mu'}+c_l(-3)]}
   \end{equation}
   So in this (non asymptotic) case even in loop gauge the effective charge
   does not correspond exactly to  the solution (2.18) of the one-loop 
   RGE (2.13). Numerically $l(-3) = 3.833$, $c_l(-3) = -3.090$ for 
   $n_f = 5$. Thus, in the $\overline{MS}$ scheme in loop gauge, the
   resummed one-loop
   invariant charge is only asymptotically a renormalisation group invariant
   when constant terms in the one loop correction are retained.
   \par Following the observation that the UV divergent part of the vertex 
   correction in Fig. 1c may be associated with a related diagram in which
   the virtual quark propagator is shrunk to a point (or `pinched') it was 
   suggested~\cite{x29,x30} to redefine a gluon proper self energy function
   by adding to the contributions of Figs. 1d-g, that of the pinched
   vertex diagram. At one loop order the resulting gluon proper self energy
   function is then gauge invariant. It was then (incorrectly) stated that
   a gauge invariant resummed gluon propagator may be trivially derived
   from the one loop result (for example, Eqn.(2.19) of Ref.[30]).
   In fact it is easy to show that if the one loop corrected quark-quark 
   scattering amplitude is gauge invariant (the correct initial assumption
   of the pinch technique calculations of Refs.[30,31]) then resummed 
   amplitudes at all higher orders must be gauge dependent.
   Introducing the gauge invariant one-loop quantity:
   \begin{equation}
   B \equiv L + 2V
   \end{equation}
   The resummed amplitude at O($\alpha_s^{n+2}$) may then be written for 
   $n \ge 1$ (see Fig. 2 and Eqn.(3.3)) as:
  \begin{equation}
  {\cal M}^{(n+2)} = {\cal M}^{(0)}(L+V)^2L^{n-1} 
   \end{equation}     
   Expressing ${\cal M}^{(n+2)}$ in terms of the gauge invariant quantity $B$
   and the gauge dependent quantity $L(\xi)$ gives:
\begin{equation}
  {\cal M}^{(n+2)} = {\cal M}^{(0)}\frac{1}{4}(B+L(\xi))^2L(\xi)^{n-1} 
   \end{equation}
  which is manifestly gauge dependent.
  The Dyson sum in Eqn.(2.19) of Ref.[30] correctly describes the all orders
  resummed amplitude, not for an arbitary gauge parameter $\xi$, but only for
  the special choice $\xi =-3$ when $V(\xi) = 0$, $B=L(-3)$ and
\begin{equation}
  {\cal M}^{(n+2)} = {\cal M}^{(0)}B^{n+1}~~~~~(\xi = -3) 
   \end{equation}
   Clearly, the above argument for manifest gauge dependence, shown to be valid
   at the resummed one-loop level must also hold at arbitary loop order
   if the vertex and self-energy insertions satisfy a generalised Ward identity
   giving, at each order of perturbation theory, a condition such as (5.3).
   It has been shown~\cite{x32}, by the application of background
   field techniques, that Ward identities relating vertex and self-energy 
   contributions may indeed be derived that are valid to all orders in perturbation 
   theory. The gauge independence of the Ward identity at each fixed order 
   then necessarily implies gauge dependence when the corresponding vertex
   and self-energy diagrams are resummed.   
   \par By consideration of a sub-set of n-loop diagrams for the
   off-shell gluon-gluon scattering amplitude in a non-covariant gauge, it has been
   claimed~\cite{x14} to demonstrate that the RCC of QCD is both gauge invariant
   and process independent, and that it may be identified with the solution (2.18) of the
   RGE (2.13). The n-loop diagrams considered are those that may be constructed as
   a formal `product' of n+1 tree level four point functions. The diagrams contain
   both resummed one loop gluon vacuum polarisation and vertex diagrams and a sub
   set of irreducible n-loop diagrams. This set of diagrams cannot, as claimed, 
  be identified with the one loop RCC in Eqn.(2.18), which results
   solely from the resummation of one loop (one particle irreducible) diagrams.
   At any order in the perturbation series these resummed one loop diagrams
   contribute the leading powers of both $\ln Q$ and $n_f$. They give in fact
   the `renormalon' contribution~\cite{x26} (see above) that determines the
   high order behaviour of the perturbation series. The irreducible n-loop
   ($n>1$) diagrams of the sub-set considered in Ref~\cite{x14} will contribute
   constant terms or non-leading powers of $\ln Q$, and therefore cannot be identified
   with terms in the diagrammatic expansion of the RCC in (2.18).
   Similarly, it has been conjectured (without explicit calculation) in
    Ref.~\cite{x31} that the `missing' vertex contributions needed to make, say
    ${\cal M}^{(2)}$ in Eqn.(5.5) above gauge invariant may be derived from `pinch
    parts' of two-loop irreducible diagrams. This is not possible since the 
    required `missing' contributions contain the factor $(\alpha_s \ln Q)^2$, (see 
    Eqn.(2.15), whereas, as is well known, in both QED~\cite{x33} and QCD
    ~\cite{x34} irreducible two-loop vacuum polarisation and vertex diagrams have, at
    most, next-to-leading logarithmic behaviour $\approx \alpha_s^2 \ln Q$. 
% insertion 7/7/97
    This is easily demonstrated by considering the two-loop solution 
    of the renormalisation group equation for the effective charge.
    In QED, or in QCD in the $\xi = -3$ gauge, Eqn.(4.2) generalises to:
    \begin{equation}
a_Q = \frac{a_{\mu}}{1+a_{\mu}\beta_0\lambda+\frac{\beta_1 a_{\mu}}{\beta_0}
     \ln(1+a_{\mu}\beta_0\lambda)}
    \end{equation} 
    where $\beta_1$ is the second $\beta$-function coefficient. Expanding the 
    right side of Eqn.(5.7) up to $O(a_{\mu}^3)$ yields:
 \begin{equation}
a_Q = a_{\mu} \left[ 1- a_{\mu}\beta_0\lambda+a_{\mu}^2\beta_0^2\lambda^2
- a_{\mu}^2\beta_1\lambda-a_{\mu}^3\beta_0^3\lambda^3 
+\frac{3}{2}a_{\mu}^3\beta_0^3\beta_1\lambda^2+ O(a_{\mu}^4)\right]
    \end{equation}
  It can be seen that $\beta_1$, given by two particule irreducible
  vacuum polarisation or vertex diagrams, occurs only in sub-leading
  logarithmic terms of the form $\beta_1 a_{\mu}^n \lambda^{n-1}$.
  No possible re-arrangement of these terms can compensate the manifest
  gauge dependence of the leading-logarithmic terms of the form
  $(\beta_0 a_{\mu} \lambda)^n$.                  
   \par The property exhibited above, for QCD, of gauge dependence of amplitudes
   on resumming one-loop corrections that, at lowest order, are gauge invariant, 
   is expected to be a general property of non-abelian gauge theories.
   In such theories the gauge boson propagator, in an arbitary covariant gauge
   is written as~\cite{x35}:
    \begin{equation}
 P^{\mu \nu}(q^2) = - \frac{i}{q^2-M^2}\left[ g^{\mu \nu}-(1-\xi)
 \frac{q^{\mu}q^{\nu}}{q^2-\xi M^2} \right]
 \end{equation}
 where M is the renormalised gauge boson mass.
    The topographical structure
     of diagrams contributing to, say, neutrino-neutrino
    scattering via Z exchange is the same as that for quark-quark scattering
    shown in Fig.1. The one-loop vertex correction containing the non-abelian
    $ZW^+W^-$ coupling is gauge dependent~\cite{x36}.
     Since the gauge dependence cancels at lowest order (without resummation)
    then, just as for QCD, it cannot cancel at any higher order in the resummed
    one loop amplitude.
    Indeed a similar conclusion as to the necessity of the $\xi = -3$ gauge
    in order to obtain an effective charge that satisfies a RGE, reached in this
    paper for QCD, has previously been obtained, for the case of the Weinberg-
    Salam model, by Baulieu and Coqueraux~\cite{x37}. These authors pointed out
    that with the special gauge choice (in the notation of the present paper)
    $\xi = -3$, the renormalisation constant of the $Z-\gamma$ mixing term
    vanishes, so that effective charges satisfying separate (decoupled) RGE's
    may be associated the one-loop resummed photon and Z-boson propagators.
    The case of W exchange was not considered, but (as may be seen by
    inspection of the relevant formulae given, in an arbitary covariant gauge,
    in Ref.~\cite{x38}) the choice $\xi = -3$ results in the vanishing of the
    renormalisation constants associated with the one-loop vertex corrections
    to both the $Z$ and $W$ exchange fermion-fermion scattering amplitudes.
    As for the QCD case considered in the present paper, it is then expected
    that, only for this special choice of gauge, an effective charge satisfying
    a RGE may be associated with the resummed $W$ propagator. 
     \par The pinch technique, and 
    similar methods to formally shift gauge dependent pieces between diagrams,
    have also been applied to 
    electroweak amplitudes~[32,36,38-41]. Although gauge invariant
    boson proper self energy functions may be defined at one-loop level,
    any resummed higher order amplitude is demonstrably gauge dependent,
    by the same argument as that given above for QCD. So, although for a 
    particular choice of gauge parameter (such that the sum of all vertex
    corrections vanish) resummed W and Z running propagators may be
    defined that satisfy a RGE, they cannot, contrary to the claim
    of Refs.[32,36,38-41], be so defined in a gauge invariant manner.          
    The diagrammatic inconsistency of these procedures 
     is made manifest by the inclusion in the modified vector
     boson self energy function of box diagram contributions. If the effective
     charge is expanded as a perturbation series, the box diagram contributions
     at each order will give terms of a geometric series. There is no way that 
     such a series can be meaningfully interpreted in terms of
     a sum of such diagrams. In fact, box 
     diagrams can be systematically resummed~[42-44], but the 
     result found is typically the exponential of a double logarithm
     of the relevant physical scale, not a geometric series. For the 
     case of the fermion-fermion scattering amplitude the contribution
     of box diagrams is expected to be important only in the $|t| \rightarrow
     0$ limit and to vanish~\cite{x38} in the $|t| \rightarrow \infty$ limit.
     \par A discussion of the gauge dependence, beyond one-loop order, of the 
     resonant $Z$ boson amplitude, may be found in Ref.[45]. 
    \par The limited convergence domain imposed by requiring finiteness of the
    geometric series, which occurs in all theories in which the RCC decreases
    with increasing scales, can be avoided by choosing a very high renormalisation 
    scale\footnote{I am indebted to W.Beenakker for this remark}.
    This is equivalent, for such theories, to the choice, in QED, of on-shell
    renormalisation yielding a RCC that is convergent for all scales below
    the Landau scale~\cite{x12}. Although such a choice guarantees convergence
    for all physical scales below the the chosen renormalisation point, it appears
    artificial from a physical viewpoint. If a
    strong interaction process at, say, the scale of the
    mass of the charm quark is to be described using a renormalisation point at the
    GUT scale $Q_{GUT}$, the formula for the RCC at scale $m_c$ will depend upon
    the masses of all strongly interacting elementary particles
    below $Q_{GUT}$ and  above $m_c$. There will be a phenomenon of `inverse decoupling'
    whereby the lower the scale the more high mass particles must be taken into
    account. Feynman amplitudes using such a renormalisation scale would loose
    their corresponence (valid in the on-shell scheme) with space-time processes.
    With on-shell renormalisation the decoupling of heavy particles at low
    scales is intuitively understood in terms of the Heisenberg Uncertainty
    principle. It seems natural that the physics of the strong interaction
    at the scale $m_c$ should be independent of the value of $m_t$ when
    $m_t \gg m_c$. This is no longer the case if the RCC is renormalised at scales
    $ \gg m_t$.
    \par An enormously successful phenomenology of the strong interaction
    has been developed over the past three decades based on 
    perturbative QCD ~\cite{x46}.
    However, it might be hoped that the physical predictions of a candidate 
    gauge {\it theory} would be gauge invariant at all orders in
    perturbation theory, as is the case in QED. Explicit calculation for
    the current
    non-abelian gauge theories (both QCD and electro-weak theory) shows,
    however, that this is not the case. The point with error bars at
     $Q = 90$GeV on the loop gauge curve in Fig.3 shows the error on 
     $\alpha_s$ ($\pm0.005$) measured in hadronic Z decays at LEP~\cite{x47}.
     The input value $\alpha_s(5$GeV$) = 0.2$ has been chosen to be consistent
     with deep inelastic scattering measurements~\cite{x48}. If the asymptotic running coupling
     constant with 
     time-like argument measured at LEP at the scale $Q=M_Z$ has
     a similar value to  $\alpha_s^{eff}(Q)$ for space-like
     argument considered in this paper, it is clear from Fig.3 that the {\it measured}
     value of the gauge parameter must be close to -3 and that the Feynman, Landau and
     vertex gauges are {\it experimentally} excluded.
     \par Many aspects of QCD, such as: the existence of the colour quantum
     number, of spin one gluons with self-coupling, the predicted values of
     colour factors, and a coupling constant that decreases with increasing
     scales in the experimentally accessible region, are confirmed, beyond doubt,
     experimentally~\cite{x46}. Other conjectured properties of the theory
     such as `asymptotic freedom' in the $Q \rightarrow \infty$ limit and
     general gauge invariance of physical amplitudes
      are not substantiated by a close examination
     of the predictions of the theory itself, without reference to experiment.
     It should also be stressed that the manifest gauge dependence of the 
     quark-quark scattering amplitude found, by direct calculation, in this
     paper, is in contradiction with formal proofs~\cite{x49,x50} of the gauge
     invariance of S-matrix elements in non-abelian gauge theories.
     It seems however, that what is actually proved in these papers is
     the gauge invariance, at all orders in perturbation theory, of 
     generalised Ward identities. The consequences of {\it resumming} diagrams
     of fixed loop order, which as shown above, necessarily generates
     gauge dependence, were not considered. The breakdown of the convergence of the RCC, for the
     special gauge choice $\xi = -3$ (which, as previously pointed out for the
     case of the Wienberg-Salam model, is the only one which permits 
     effective charges satisfiying RGE's to be defined~\cite{x37}), has obvious and 
     serious negative consequences for the physical relevance
     of the discussions of Grand
     Unified Theories, based on RCC evolution, in the literature 
     ~\cite{x16,x17}. The usual assumption made in such studies
     is that RCC evolution according to RGE's, is valid up to the Planck
     Scale.

     \section{Acknowledgements}
     I wish to thank especially W.Beenakker for his careful reading of the manuscript
     and for constructively critical comments. Discussions with M.Consoli are also 
     gratefully acknowledged. I thank M.Veltman for pointing out to me the work 
     contained in Ref.~\cite{x50}.     
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\end{thebibliography}
%\pagebreak
%{\bf \Large Figure Captions}
%\newline
%{\bf Fig.1~~~} Diagrams contributing to ${\cal M}^{(LO)}$. Solid lines denote quarks,
% wavy lines gluons and the thin line in g) ghosts.
%\newline
%{\bf Fig.2~~~} The topographical structure of diagrams contributing to the resummed
% quark-quark 
%scattering amplitude: a) O($\alpha_s^2$), b) O($\alpha_s^3$), c) O($\alpha_s^3$).
%In diagrams containing only one vertex insertion the contribution given by the
%exchange $13 \leftrightarrow 24$ (see Fig.1) is understood to be included.
%\newline
%{\bf Fig.3~~~}The variation of the Effective Charge (3.5) with the scale $Q$
%for different choices of the gauge parameter $\xi$. $5 GeV < Q < 300 GeV$
% ($\alpha_s^{eff}(5GeV) = 0.2$, $n_f = 5$).
%\newline
%{\bf Fig.4~~~}The variation of the Effective Charge (3.5) with the scale $Q$
%for different choices of the gauge parameter $\xi$. $10 TeV < Q < 10000 TeV$
%($\alpha_s^{eff}(5GeV) = 0.2$, $n_f = 5$).
%\newline
%{\bf Fig.5~~~} The topographical structure of diagrams contributing to: a) the
%  O($\alpha_s^3$)
%contribution to the anomalous magnetic moment of a quark, b) the four--loop
%photon proper self energy function. 
 \pagebreak
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
Gauge & Feynman ($\xi=1$) & 
 Landau ($\xi=0$)  & Loop ($\xi=-3$)  & Vertex ($\xi=19/9$)  \\ \hline
$Q_L$(GeV) & 7.68$\times10^8$ & 1.0$\times10^5$  & 301 & $\infty$  \\
\hline
\end{tabular}
\caption[]{ Values of the Landau scale $Q_L$ (convergence limit) of the QCD effective charge
for $\mu = 5$ GeV, $\alpha_s^{\mu}= 0.2$, $n_f = 5$ }      
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
Gauge & Feynman ($\xi=1$) & Landau ($\xi=0$)  & Loop ($\xi=-3$)  & Vertex ($\xi=19/9$)  \\ \hline
$Q_0$(TeV) & 176 & 5800  & Undefined &  18.1  \\
\hline
\end{tabular}
\caption[]{ Values of the $Q_0$ (the scale at which $\alpha_s^{eff}(Q)$ vanishes ). Parameters as in
Table~1. }      
\end{center}
\end{table}
\newpage
\vspace*{4cm}
\begin{figure}[htbp]
\begin{center}
%\hspace*{-0.5cm}\mbox{
%\epsfysize10.0cm\epsffile{test_new.eps}}
\mbox{\epsfig{file=qcdf1.eps,height=12cm}}
%\mbox{\epsfig{file=test_new.eps,height=6cm}}
\caption{ Diagrams contributing to ${\cal M}^{(LO)}$. Solid lines denote quarks,
 wavy lines gluons and the closed loop in g) ghosts.}
\label{fig-fig1}
\end{center}
 \end{figure}  
\begin{figure}[htbp]
\begin{center}\hspace*{-0.5cm}\mbox{
\epsfysize10.0cm\epsffile{qcdf2.eps}}
\caption{The topographical structure of diagrams contributing to the resummed
 quark-quark 
scattering amplitude: a) O($\alpha_s^2$), b) O($\alpha_s^3$), c) O($\alpha_s^3$).
In diagrams containing only one vertex insertion the contribution given by the
exchange $13 \leftrightarrow 24$ (see Fig.1) is understood to be included.}
\label{fig-fig2}
\end{center}
 \end{figure}  
\begin{figure}[htbp]
\begin{center}\hspace*{-0.5cm}\mbox{
\epsfysize10.0cm\epsffile{qcdf3.eps}}
\caption{The variation of the Effective Charge (3.5) with the scale $Q$
for different choices of the gauge parameter $\xi$. $5~GeV < Q < 300~GeV$
 ($\alpha_s^{eff}(5GeV) = 0.2$, $n_f = 5$). The error bars ($\pm$ 0.005) on the 
 point at $Q = 90 ~GeV$ on the loop gauge curve are typical of those on an
 $\alpha_s$ measurement using hadronic Z decays~\cite{x47}.}
\label{fig-fig3}
\end{center}
 \end{figure}  
\begin{figure}[htbp]
\begin{center}\hspace*{-0.5cm}\mbox{
\epsfysize10.0cm\epsffile{qcdf4.eps}}
\caption{The variation of the Effective Charge (3.5) with the scale $Q$
for different choices of the gauge parameter $\xi$. $10~TeV < Q < 10000~TeV$
($\alpha_s^{eff}(5GeV) = 0.2$, $n_f = 5$).}
\label{fig-fig4}
\end{center}
 \end{figure} 
\begin{figure}[htbp]
\begin{center}\hspace*{-0.5cm}\mbox{
\epsfysize10.0cm\epsffile{qcdf5.eps}}
\caption{ The topographical structure of diagrams contributing to: a) the
  O($\alpha_s^3$)
contribution to the anomalous magnetic moment of a quark, b) the four--loop
photon proper self energy function.}
\label{fig-fig5}
\end{center}
 \end{figure} 
  
\end{document}
 
 
 
 
 
 
\bye
 
 
 


