\section{INTRODUCTION}

Some time ago Sharpe and Bernard and Goltermann\cite{Sharpe}
showed that the quark mass dependence of certain
physical quantities near the chiral limit should serve as a
particularly incisive test of the difference between quenched
and full QCD. In particular, the behavior of the pion
mass as a function of the quark mass in quenched QCD is expected
to be non-analytic at $m_q=0$, with an anomalous exponent $\delta$,
\begin{equation}
\label{eq:chlog}
m_{\pi}^2 \propto (m_q)^{\frac{1}{1+\delta}}
\end{equation}
The value of the chiral log exponent in the pion mass should
be related to the coefficient of the $\eta'$ hairpin insertion by
\begin{equation}
\label{eq:delta}
\delta = \frac{m_0^2}{48\pi^2f_{\pi}^2}
\end{equation}
Calculations we have reported previously\cite{Lat96} showed no evidence
for chiral logs in the pion mass at $\beta=5.7$ over the range of hopping
parameters used in that calculation, from .1610 to .1680
(pion masses of .650 to .254 in lattice units). The value
of the exponent $\delta = .015(47)$ was obtained, i.e.
consistent with zero. The results reported here are
completely consistent with the previous negative results. We
find that the value of $\delta$ obtained in this region
from the hairpin coefficient varies from $\delta<.01$ at
$\kappa=.161$ to $\delta = .045(5)$ at $\kappa=.168$.
Even when extrapolated to the chiral limit, the exponent
is only $\delta=.053(7)$.

In this present calculation,
we have used the pole-shifting ansatz of the modified
quenched approximation (MQA) \cite{MQA} to resolve the
exceptional configuration problem. This allows us to
greatly improve our statistics at light quark masses
and to extend the calculations much closer to critical
hopping parameter.
The pion mass at these lighter quark mass values is much more
sensitive to the chiral logs and our errors are small
enough to see a significant effect even though $\delta\approx .05$.
At the lightest three quark masses (.1683, .1685, and .1687)
we find clear evidence for a deviation from linearity
at just the level and functional form
expected from the hairpin calculation. The agreement between the hairpin
result and the pion mass behavior provides significant numerical
support for the theoretical picture of chiral
logarithms and their connection to $\eta'$ loops in the quenched
approximation.

Both the overall smallness of the chiral log parameter
and the strong quark mass dependence are somewhat unexpected,
in light of the fact that (a) the physical $\eta'$ mass gives
$\delta\approx.17$, and (b) in the chiral limit,
the hairpin mass insertion is
related by the Witten-Veneziano formula to the topological
susceptibility of pure glue (without quarks) and should
therefore be rather insensitive to the quark mass (at
least for quark masses much smaller than the QCD scale).
In an effort to understand the origin of this suppression
and quark mass dependence, we carried out another calculation of
the hairpin diagram using a clover improved quark action.
(The clover coeffient used here is $C_{sw}=1.57$.) The two
main effects of clover improvement are (1) to increase the
hairpin coefficient $m_0^2$ by more than a factor of 2,
and (2) to decrease the slope of $m_0$ as a function of
quark mass by about a factor of 2 (where quark mass for
the clover case is rescaled so that equal quark mass
corresponds to equal pion mass). This strongly suggests
that both the suppression and the strong quark mass dependence
of $m_0$ are lattice artifacts.
Thus we would expect the chiral log parameter $\delta$
to be quite sensitive
to lattice spacing and to increase substantially at higher
$\beta$ values. Recent results on the pion mass from
the $CP-PACS$ collaboration\cite{CP}
indicating that $\delta\approx .1$ at $\beta=6.47$ are quite consistent
with this conclusion.\cite{future}

\begin{table}
\caption{$\eta'$ mass $m_0$, predicted chiral log parameter $\delta$,
and pion mass.}
\label{tab:deltalimits}
\begin{tabular}{|c|c|c|c|}
\hline
$\kappa$& $m_0$ & $\delta$ & $m_{\pi}$ \\
\hline
$.1687$ & $        $ & $       $ & $.165(5)$ \\
$.1685$ & $.383(22)$ & $.050(6)$ & $.195(4)$ \\
$.1683$ & $        $ & $       $ & $.221(3)$ \\
$.1680$ & $.371(20)$ & $.045(5)$ & $.254(2)$\\
$.1675$ & $.346(16)$ & $.038(3)$ & $.298(2)$\\
$.1667$ & $.316(14)$ & $.030(3)$ & $.356(2)$\\
$.1650$ & $.268(13)$ & $.019(2)$ & $.458(1)$\\
$.1630$ & $.221(13)$ & $.011(1)$ & $.558(1)$\\
\hline
\end{tabular}
\end{table}

In Table I the results for $m_0$ are presented. Also listed
are the resulting values of the chiral log exponent $\delta$,
using Eq. (\ref{eq:delta}), with $f_{\pi}$ calculated on the same
ensemble. In the last column, we list the
values of the pion mass obtained from the valence pion propagator.

As a function of $\kappa^{-1}$, the values of $m_{\pi}^2$ obtained
from Table I are quite linear for the heavier masses, but show
a significant deviation from linearity for $\kappa > .1680$.
This deviation has both the form and the magnitude predicted
by the chiral log formula (\ref{eq:chlog}) combined with the
hairpin values for $\delta$. A direct fit of $m_{\pi}^2$
to the formula (\ref{eq:chlog}), including values from $\kappa
=.165$ to $\kappa=.1687$ gives $\delta=.058(39)$, i.e.
significantly different from zero and consistent with the
hairpin results. However, this direct fitting procedure with constant
$\delta$ has
several drawbacks. The rather large error reflects the observed fact
that the value of $\delta$ changes significantly over the fit
range. Also, the value of $\kappa_c$ must be taken as one of
the fitting parameters, and there is a strong correlation
between $\kappa_c$ and $\delta$.

A more accurate way to observe the presence and magnitude of the
chiral logs is to consider the ratio of differences:
\begin{equation}
\label{eq:ratio}
\frac{\Delta\left(m_{\pi}^{2(1+\delta)}\right)}{\Delta\left(
\kappa^{-1}\right)}
\end{equation}
where the differences are computed between adjacent values of
$\kappa$. With the correct choice of $\delta$, this ratio
should be constant. Note that calculating this ratio does not require a
determination of $\kappa_c$. Another advantage is that the
values of $m_{\pi}$ at different $\kappa$'s are positively
correlated, so that errors on the difference are typically
30-50\% smaller than on the (mass)$^2$ itself. The errors
are computed by a single elimination jackknife. To see the
chiral log effect, we first plot in Fig.1 the ratio
(\ref{eq:ratio}) for $\delta = 0$. The horizontal dashed line
represents the result of the best linear fit to $m_{\pi}^2$
for the $\kappa$ range $.163$ to $.168$. The three points
at the lightest quark mass, $<25$ MeV, are obtained from the
three differences computed between the four kappa values
.1680, .1683, .1685, and .1687. These points show a
significant chiral log effect which is consistent with
the hairpin expectations. To see this we plot in Fig. 2
the ratio (\ref{eq:ratio}) with values of $\delta$
taken from interpolating the hairpin results in Table I. With this
choice of $\delta$, the ratio's are consistent with
a constant. To see how sensitive this plot is to the
chosen values of $\delta$, we plot in Fig. 3 the ratios
(\ref{eq:ratio}) with a mass independent value of
$\delta=.1$. The behavior of the ratio is now completely
inconsistent with a constant, illustrating that a
value as large as $\delta=.1$ is ruled out by the data.

\begin{figure}
\vspace*{3.6cm}
\special{psfile=chlog_delta=0.ps
           angle=270 hscale=30 vscale=30 hoffset=-25 voffset=135}
\vspace{0.5cm}
\caption[]{The ratio $\Delta\left(m_{\pi}^2\right)/\Delta\left(
\kappa^{-1}\right)$, i.e. Eq. \ref{eq:ratio} with $\delta=0$.}
\end{figure}

\begin{figure}
\vspace*{3.6cm}
\special{psfile=chlog_delta=hairpin.ps
           angle=270 hscale=30 vscale=30 hoffset=-25 voffset=135}
\vspace{0.5cm}
\caption[]{The ratio $\Delta\left(m_{\pi}^{2(1+\delta)}\right)/\Delta\left(
\kappa^{-1}\right)$ with $\delta$ from Table I.}
\end{figure}

\begin{figure}
\vspace*{4.0cm}
\special{psfile=chlog_delta=.1.ps
           angle=270 hscale=30 vscale=30 hoffset=-25 voffset=135}
\vspace{0.5cm}
\caption[]{The ratio $\Delta\left(m_{\pi}^{2(1+\delta)}\right)/\Delta\left(
\kappa^{-1}\right)$ with $\delta=0.1$.}
\end{figure}

Finally, we present in Fig. 4 the values of $m_0$ (in
lattice units) obtained from the clover improved action,
compared with those obtained from the unimproved Wilson
action.
As discussed above, the clover improved value is
substantially larger in magnitude and the quark mass
dependence is significantly reduced.\cite{future}

\begin{figure}
\vspace*{4.0cm}
\special{psfile=m0_a_and_bc.ps
           angle=270 hscale=30 vscale=30 hoffset=-25 voffset=135}
\vspace{0.5cm}
\caption[]{Comparison of $m_0$ for Wilson action (lower line) and
clover action with $C_{sw}=1.57$ (middle line). The upper line is the result of a linear
extrapolation to zero slope.}
\label{fig:hp_raw}
\end{figure}


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\hyphenation{author another created financial paper re-commend-ed}

\title{The running QCD coupling in the pre-asymptotic region}

\author{G. Burgio\address{Dipartimento di Fisica, Universit\`a di Parma
 and INFN, Gruppo Collegato di Parma, Parma, Italy},
        F. Di Renzo$^{\rm b}$, C. Parrinello\address{Dept. of
Mathematical Sciences, University of Liverpool,
Liverpool L69 3BX, U.K. \\
 (UKQCD Collaboration)}, C. Pittori\address{Institut de Physique
Nucl\'eaire Th\'eorique, Universit\'e de Li\`ege au Sart
Tilmann, B-4000 Li\`ege, Belgique}}

\begin{document}

\begin{abstract}
We study deviations from the perturbative asymptotic behaviour
in the running QCD coupling by analysing non-perturbative measurements
of $\alpha_s(p)$ at low momenta ($p \approx 2 \ {\rm GeV}$) as
obtained from the lattice three-gluon vertex.
Our exploratory study provides some evidence for power corrections to
the perturbative running proportional to $1/p^2$.
\end{abstract}

\maketitle

\section{INTRODUCTION}
\label{sec:intro}

The standard procedure to parametrise non-perturbative QCD effects
in terms of power corrections to perturbative results
is based on the Operator Product Expansion (OPE). In this framework,
the powers involved in the expansion are uniquely
fixed by the symmetries and the dimension of the relevant operator
product.
The above picture has recently been challenged
\cite{Akhoury,etc,Ceccobeppe}, when it was pointed out that
 power corrections which are not {\it a priori} expected from OPE
 may in fact appear in physical observables.
 Such terms may arise from (UV-subleading) power corrections to
$\alpha_s(p)$, corresponding to non-analytical contributions to the
$\beta$-function.
Clearly, the existence of OPE-independent
power corrections, if demonstrated,
 would have a major impact on our
understanding of non-perturbative QCD effects and would affect
QCD predictions for several processes.
For example, $\frac{\Lambda^2}{p^2}$ contributions
may be relevant for the analysis of
$\tau$ decays
\cite{alt,Akhoury}.

It would be highly desirable to
develop a theoretical framework where the occurrence of these effects
is demonstrated and estimates are obtained from first principles QCD
calculations. The results in \cite{Lepage,Ceccobeppe} can be considered
as a first step in this direction: some evidence for
an unexpected $\frac{\Lambda^2}{Q^2}$ contribution to the gluon
condensate was found by means of lattice calculations.

The aim of the present work is to test a method to detect
the presence of power corrections in the running QCD coupling.
Non-perturbative lattice estimates of the
coupling at low momenta are compared with perturbative formulae. The
final goal is to investigate the conjecture that OPE-independent power
corrections to physical observables are linked to power terms in the
running coupling.
Although at this stage our work is exploratory in nature and further
simulations will be required to obtain a conclusive answer,
our analysis provides some preliminary evidence for power corrections
to $\alpha_s(p)$ for a particular definition of the coupling.

The paper is organised as follows: in Section \protect\ref{sec:relev}
we briefly review some theoretical arguments in support of power
corrections to $\alpha_s (p)$, illustrating the special role that may be
played by $\frac{\Lambda^2}{p^2}$ terms.
In Section \protect\ref{sec:lattice}
we analyse the lattice data and present some preliminary
evidence for power corrections.
Finally, in Section \protect\ref{sec:conc} we draw our conclusions.

\section{WHY POWER CORRECTIONS?}\protect\label{sec:relev}

Power corrections to $\alpha_s (p)$ can be shown to
arise naturally in many physical schemes \cite{pino,maclep}.
Such corrections cannot be excluded {\em a priori} in any
renormalisation scheme.
Clearly, the non-perturbative nature of such effects makes it very hard
to assess their dependence on the renormalisation scheme.
 A term of order ${\Lambda^2}/{p^2}$ is a strong candidate for a
power correction to $\alpha_s(p)$. To see why, consider
 the interaction of two heavy quarks in the static limit
and in the one-gluon-exchange approximation
(for a more detailed discussion see \cite{zakEQ}).
The static potential $V(r) $ can be written as
\begin{equation}
\protect\label{HQP}
V(r) \, \propto \, \alpha_s \ \int d^3p \,
\frac{\exp^{i \vec{p} \cdot \vec{r}}}{|\vec{p}|^2}.
\end{equation}
If one inserts in the above formula a running coupling of the form
$\alpha_s(p^2) \approx {\Lambda^2}/{p^2}$, this results in a linearly
confining potential.
Similarly, consider the ``force" definition of the running
coupling:
\begin{equation}
\alpha_{q\bar{q}}(Q) = \frac{3}{4} r^2 \frac{dV}{dr} \;\;\;\;\;
(Q = \frac{1}{r}),
\end{equation}
where again $V(r)$ represents the static interquark potential.
A linear confinement term in $V(r)$ generates a
$1/Q^2$ contribution to the coupling, whose order of magnitude is given
by the string tension.
This can be interpreted as a clue for the existence of a
$\frac{\Lambda^2}{p^2}$ contribution, providing an estimate
for its expected order of magnitude, at least in one (physically
sound) scheme.
Finally, power corrections to $\alpha_s(p)$ also
emerge if one assumes that the
singularities appearing in the perturbative formulae for the running
coupling are ``removed" by non-perturbative effects \cite{RedBog}.

\section{ANALYSIS AND RESULTS}
\protect\label{sec:lattice}

We shall compare non-perturbative lattice data for $\alpha_s(p)$ with
simple models where a power correction term is added to the perturbative
formula at a given order. The first problem is the possible
interplay between power corrections and our ignorance about higher
orders of perturbation theory. In particular, for the scheme that we
will consider, the three-loop coefficient of the $\beta$-function is not
known. Knowledge of such a coefficient would allow a more reliable
comparison of our estimates for the $\Lambda$ parameter in our scheme
with lattice determinations  of $\Lambda$ in a different scheme,
for which the three-loop result is available \cite{Lusch}.
In fact, although matching the $\Lambda$ parameter between different
schemes only requires a one-loop computation, the reliability of such a
comparison rests on the assumption that the  value of $\Lambda$ in each
scheme is fairly stable with respect to the inclusion of higher order
terms in the definition of $\Lambda$.
In practice, when working at two- or three-loop order, the value of
$\Lambda$ is still quite sensitive to the order of the calculation.
 Even within such limitations, we will argue that it is
possible to estimate the impact of three-loop effects and that
a description with power corrections seems relevant even at that order.

\subsection{Choice of the coupling}

We need to measure
$\alpha_s (p)$ at low momenta (where power-like terms may be sizeable) and
in a relatively wide momentum range.
For this purpose, the best method is one where $\alpha_s (p)$ can be
measured for several momentum values from a single Monte Carlo data set.
One suitable method is the determination of
the coupling from the renormalised lattice three-gluon vertex
function \protect\cite{io,cpcp}.
By varying the renormalisation scale $p$, one can determine $\alpha_s(p)$
for different momenta from a single simulation. Obviously the
renormalisation scale must be chosen in a range such that finite volume
effects and discretisation  errors are both under control.
The numerical results for $\alpha_s(p)$ used in this work were
obtained by applying such a method on a sample of 150 Monte Carlo
configurations on a $16^4$ lattice at $\beta=6.0$. The calculation was
performed in the Landau gauge.
For full details of the method we refer the reader to Ref.
\protect\cite{cpcp}, where such results were first presented.
In order to detect violations of rotational invariance, different
combinations of lattice vectors have sometimes been used
for a fixed value of $p^2$, which accounts for the graphical
``splitting" of some data points.

\subsection{Two-loop analysis}

At the two-loop level, we consider the following formula:

\begin{eqnarray}
\protect\label{2lp}
\alpha_s(p) \, = \, \frac{1}{b_0 \, \log(p^2/\Lambda_{2l}^2)} \, - \,
\frac{b_1}{b_0}
\frac{\log(\log(p^2/\Lambda_{2l}^2))}{(b_0 \,
\log(p^2/\Lambda_{2l}^2))^2}
\, \nonumber \\ + \, c_{2l} \, \frac{\Lambda_{2l}^2}{p^2}
\qquad \qquad \qquad \qquad \qquad \qquad \end{eqnarray}

By fitting the data to (\protect\ref{2lp}) we obtain two
estimates for ($\Lambda_{2l}$,$c_{2l}$), namely
($0.84(1)$,$0.31(3)$) and ($0.73(1)$,$0.99(7)$), with comparable
values for $\chi^2_{dof} \leq 1.8$.
The momentum range for the fit corresponds to  $p \sim 1.8 - 3$ {\rm GeV}.
We take the first set of values as our best estimate of the parameters
as the corresponding value of $\Lambda_{2l}$ is close to
what is obtained from a ``pure" two-loop fit, i.e. $\Lambda_{2l}$ is
stable with respect to the introduction of power
corrections. This choice will be supported also by
independent considerations at the three-loop level.
In summary, a two-loop description with power corrections based on
(\protect\ref{2lp}) fits well the data in a consistent momentum range.
Our best fit is shown in Figure 1.

\begin{figure}[htb]\protect\label{fig:2loopot}
\vspace{9pt}
\ewxy{2loopot.ps}{70mm}
\caption{The best fit to (\protect\ref{2lp}).}
\end{figure}

\subsection{Three-loop analysis}

A major obstacle for a three-loop analysis is our ignorance of
the first non-universal  coefficient $b_2$ of the perturbative
$\beta$-function.
In order to gain insight, we perform a two-parameter fit to
the ``pure" three-loop formula, taking $\Lambda_{3l}$ and the unknown
coefficient $b_2$ as the fitting parameters. We call
$b_2^{eff}$ the fit estimate for $b_2$.
We obtain $\Lambda_{3l} =
0.72(1)$, $b_2^{eff} = 1.3(1)$, with $\chi^2_{dof} \approx 1.8$
(see dashed curve in Fig. 2).
The momentum range where we obtain the best description of the
data is $p \sim 2 - 3$ {\rm GeV}. Our result for
$\Lambda_{3l}$ provides (via perturbative
matching) an estimate for $\Lambda_{\overline{MS}}$ which is in very good
agreement with the estimate in \cite{Lusch}, which was obtained from
the computation of the $\Lambda$ parameter in a completely different
scheme. Although our estimate depends on the extra parameter
$b_2^{eff}$, the agreement between the two results is remarkable.

So far, the success of the ``pure" three-loop fit suggests that the
power term in the two-loop formula merely provides an effective
description of three-loop effects. However, it turns out that there
is room for a power correction even at the three-loop level. To see
this, we consider a three-loop formula with a power correction:

\begin{eqnarray}
\protect\label{3lp}
\alpha_s(p) =  \frac{1}{b_0 \, L}  \, - \,
\frac{b_1}{b_0}
\frac{\log(L)}{(b_0 \, L)^2} \qquad \qquad \qquad \nonumber \\
+ \, \frac{1}{(b_0 \, L)^3} \,
\left( \frac{b_2^{eff}}{b_0} + \frac{b_1^2}{b_0^2}
(\log^2(L) - \log(L) + 1 ) \right) \nonumber \\
+ \, c_{3l} \, \frac{\Lambda_{3l}^2}{p^2}, \qquad \qquad
\qquad \qquad \qquad \qquad
\end{eqnarray}

\begin{figure}[htb]\protect\label{fig:3loopot}
\vspace{9pt}
\ewxy{3l.ps}{70mm}
\caption{Fits to (\protect\ref{3lp})
(solid line) versus a pure three-loop fit (dashed line).}
\end{figure}
where $L = \log(p^2/\Lambda_{3l}^2)$ and
 $b_2^{eff}$ is again to be determined from a fit.
Fitting the data to (\protect\ref{3lp}), we obtain
$\Lambda_{3l} = 0.72(1)$, $b_2^{eff} = 1.0(1)$
and $c_3 = 0.41(2)$, with $\chi^2_{dof} \approx 1.8$, in a momentum
range $1.8 \, {\rm GeV} < p < 3 \, {\rm GeV}$ (see Fig. 2).
We note that the value for $\Lambda_{3l}$
is fully consistent with the previous determination from the ``pure"
three-loop description. The value for $b_2^{eff}$ is also reasonably
stable with respect to the previous determination.
By comparing results from fits to (\protect\ref{2lp}) and
(\protect\ref{3lp}), it emerges that
\begin{equation}
c_2 \Lambda_{2l}^2 = 0.22(2) \, {\rm GeV}^2 \sim c_3 \Lambda_{3l}^2 = 0.21(2)
 \, {\rm GeV}^2.
\protect\label{eq:tuttotiene}
\end{equation}
In other words, the power terms providing the best fit to
(\protect\ref{2lp}) and (\protect\ref{3lp}) are numerically the same,
so that there seems to be no interplay between
the indetermination connected to the perturbative terms and the
power correction term, within the precision of our data.
We take this fact as an indication that a description in terms of power
corrections is still relevant at the three-loop level.
Notice that the numerical value of the power correction
is comparable to the standard estimate for the string tension.

One could object that at the two-loop level we had  chosen between
two sets of values for ($\Lambda_{2l}$,$c_{2l}$), and that our choice
is crucial for the validity of (\protect\ref{eq:tuttotiene}).
 An {\it a posteriori} justification for our choice is obtained
from the following test: we plot a few values for $\alpha_s(p)$ as
generated by the ``pure'' three-loop formula
for $\Lambda_{3l}= 0.72$ and $b_2 = 1.0$. Then, by fitting such
points to the ``pure'' two-loop formula, one gets $\Lambda_{2l}
\approx 0.84$, i.e. the value for which (\protect\ref{eq:tuttotiene})
holds. Again, the above test seems to confirm that perturbative and
non-perturbative contributions do not mix in our formulae
when upgrading from a two-loop to a three-loop description, thus
suggesting that a genuine $\frac{\Lambda^2}{p^2}$
correction is present in  the data.

\section{CONCLUSIONS}
\protect\label{sec:conc}

We have discussed an exploratory investigation of power corrections
in the running QCD coupling $\alpha_s(p)$ by comparing non-perturbative
lattice results with theoretical models. Some evidence was
found for $1/p^2$ corrections, whose size would be consistent
with what is suggested by simple arguments from the static potential.

Our results need to be further tested by the analysis of a larger data
set and by a study of the dependence of the fit coefficient on the
ultraviolet and infrared lattice cutoff.
A very delicate issue is the assessment of the scheme dependendence of
our results. In particular, we note that the definition of the coupling
that we adopted is {\it a priori} gauge-dependent. This point will
be the focus of our future work.

\section{ACKNOWLEDGEMENTS}
We thank B. Alles, H. Panagopoulos and D. G. Richards for allowing us
to use data files containing the results of Ref. \protect\cite{cpcp}.
C. Parrinello acknowledges the support of PPARC
through an Advanced Fellowship.
C. Pittori thanks J. Cugnon and the ``Institut de
Physique de l'Universit\'e de Li\`ege au Sart Tilman"
and acknowledges the partial support of IISN.
We thank C. Michael for stimulating discussions.