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\begin{document}

\title{
{\sf Instanton Effects on the Role of the
Low-Energy Theorem for the Scalar Gluonic Correlation Function} }


\author{
D.\ Harnett, T.G.\ Steele
\thanks{email: Tom.Steele@usask.ca}
\\
{\sl Department of Physics and Engineering Physics}\\
{\sl University of Saskatchewan}\\
{\sl Saskatoon, Saskatchewan S7N 5E2, Canada.}\\[10pt]
V.\ Elias
\thanks{email: velias@julian.uwo.ca}\\
{\sl Department of Applied Mathematics}\\
{\sl University of Western Ontario}\\
{\sl London, Ontario N6A 5B7, Canada.}
}
\maketitle
\begin{abstract}
Instanton contributions to the Laplace sum-rules for correlation functions of scalar gluonic currents
are calculated. The role of the constant low-energy theorem term, whose substantial contribution is 
unique to the leading Laplace sum-rule ${\cal L}_{-1}$, is shown to be diminished by instanton contributions,   significantly increasing the resulting mass bounds for the ground state of scalar 
gluonium and improving compatibility with results from higher-weight sum-rules.  
\end{abstract}


\section{Introduction}\label{intro_sec}
In the chiral limit of $n_f$ quarks, the low-energy theorem (LET) for scalar gluonic correlation functions is 
\cite{NSVZ}
\begin{equation}
\Pi(0)=\lim_{Q\rightarrow 0} \Pi\left(Q^2\right)=\frac{8\pi}{\beta_0}\left\langle J\right\rangle\quad ,
\label{let}
\end{equation}
where
\begin{eqnarray}
& &\Pi\left(Q^2\right)=i\int d^4x\, e^{iq\cdot x}\left\langle O \vert T\left[ J(x) J(0)\right] \vert O \right\rangle
\quad ,\quad Q^2=-q^2>0
\label{corr_fn}\\
& &J(x)=-\frac{\pi^2}{\alpha\beta_0}\beta\left(\alpha \right)G^a_{\mu\nu}(x)G^a_{\mu\nu}(x)
\label{current}\\
& &\beta\left(\alpha\right) =\nu^2\frac{d}{d\nu^2}\left(\frac{\alpha(\nu)}{\pi}\right)=
-\beta_0\left(\frac{\alpha}{\pi}\right)^2-\beta_1\left(\frac{\alpha}{\pi}\right)^3+\ldots
\label{beta}\\
& &\beta_0=\frac{11}{4}-\frac{1}{6} n_f\quad ,\quad
\beta_1=\frac{51}{8}-\frac{19}{24}n_f
\quad .
\label{beta_coeffs}
\end{eqnarray}
The current $J(x)$ is renormalization group (RG) invariant for massless quarks \cite{RG}, 
and its normalization has been chosen 
so that to lowest order in $\alpha$
\begin{equation}
J(x)=\alpha G^a_{\mu\nu}(x)G^a_{\mu\nu}(x)\left[1+\frac{\beta_1}{\beta_0}\frac{\alpha}{\pi}
+{\cal O}\left(\alpha^2\right)\right]
\equiv \alpha G^2(x)\left[1+\frac{\beta_1}{\beta_0}\frac{\alpha}{\pi}+{\cal O}\left(\alpha^2\right)\right]
\quad .
\label{current_exp}
\end{equation}

Most applications of dispersion relations in sum-rules are designed to remove
dependence on low-energy subtraction constants.  However, knowledge of the LET 
for gluonic correlation functions permits the possibility of sum-rules that contain
explicit dependence on the LET subtraction constant $\Pi(0)$.  
For example, the dispersion relation appropriate to the 
asymptotic (perturbative) behaviour of the correlation function \bref{corr_fn} is
\cite{dispersion}
\begin{equation}
\Pi\left(Q^2\right)=\Pi(0)+Q^2\Pi'(0)+\frac{1}{2}Q^4\Pi''(0)
-Q^6\frac{1}{\pi}\int\limits_{t_0}^\infty
dt\,\frac{ \rho(t)}{t^3\left(t+Q^2\right)} \quad .
\label{disp_rel}
\end{equation}
where $\rho(t)$ is the hadronic spectral function with physical threshold $t_0$ 
appropriate to the quantum numbers of the
current used to construct the correlation function.



Unfortunately, direct application of the dispersion relation \bref{disp_rel} is not possible because the
theoretical (perturbative) calculation of $\Pi\left(Q^2\right)$ contains a field-theoretical divergence proportional 
to $Q^4$.  A related problem is the significant contribution of excited states and the QCD continuum 
to the integral of $\rho(t)$ in \bref{disp_rel}.  Enhancement of
the lowest-lying resonance contribution in applications to light hadronic systems  
requires greater high-energy suppression of this integral.

The established technique for dealing with these issues is the Laplace sum-rules \cite{SVZ}. A family
of  Laplace sum-rules can be obtained from the dispersion relation \bref{disp_rel} through the  
Borel transform operator $\hat B$,
\begin{equation}
\hat B\equiv 
\lim_{\stackrel{N,~Q^2\rightarrow \infty}{N/Q^2\equiv \tau}}
\frac{\left(-Q^2\right)^N}{\Gamma(N)}\left(\frac{d}{dQ^2}\right)^N
\label{borel}
\end{equation}  
which has the following useful properties in the construction of the Laplace sum-rules:
\begin{eqnarray}
& &\hat B\left[a_0+a_1Q^2+\ldots a_m Q^{2m}\right]=0\quad,\quad (m~{\rm finite})
\label{borel_poly}\\
& & \hat B \left[ \frac{Q^{2n}}{t+Q^2}\right]=\tau \left(-1\right)^nt^ne^{-t\tau}  \quad,\quad n=0,~1,~2,\ldots ~ 
(n~{\rm finite})
\label{borel_exp}
\end{eqnarray}
The theoretically-determined quantity
\begin{eqnarray}
{\cal L}_k(\tau)\equiv\frac{1}{\tau}\hat B\left[\left(-1\right)^k Q^{2k}\Pi\left(Q^2\right)\right]\quad ,  
\label{laplace}
\end{eqnarray}
leads to the following family of Laplace sum-rules, after application of  $\hat B$ to the 
dispersion relation 
\bref{disp_rel} weighted by the appropriate power of $Q^2$:
\begin{eqnarray}
{\cal L}_k(\tau)&=&\frac{1}{\tau}\hat B\left[\left(-1\right)^kQ^{2k}\Pi(0)+
\left(-1\right)^kQ^{2k+2}\Pi'(0)+\left(-1\right)^k\frac{1}{2}Q^{2k+4}\Pi''(0)\right]
\nonumber\\
& &\quad-\frac{1}{\pi}\int\limits_{t_0}^\infty dt\,\frac{1}{\tau t^3}\hat B\left[\left(-1\right)^k
\frac{Q^{2k+6}}{\left(t+Q^2\right)}\right]\rho(t)
\label{lap_sr_gen}
\end{eqnarray}
There are some important constraints on $k$ that will lead to sum-rules with predictive power.  Since 
the perturbative prediction of $\Pi\left(Q^2\right)$ contains divergent constants multiplied by $Q^4$, 
the sum-rules ${\cal L}_k(\tau)$ where this contribution is absent require $k\ge -2$.  
However, the  low-energy constants $\Pi'(0)$ and $\Pi''(0)$ are {\em not} determined by the LET [{\it i.e.} only the 
quantity $\Pi(0)$ appears in \bref{let}].  Hence the sum-rules ${\cal L}_k(\tau)$ which 
will be independent of 
$\Pi'(0)$ and $\Pi''(0)$ must satisfy $k\ge -1$, and  only the $k=-1$  sum-rule will contain dependence on the 
LET-determined quantity $\Pi(0)$: 
\begin{eqnarray}
{\cal L}_{-1}(\tau)=-\Pi(0)+\frac{1}{\pi}\int\limits_{t_0}^\infty
dt\,\frac{1}{t} e^{-t\tau}\rho(t)
\label{lap_gen_m1}\\
{\cal L}_{k}(\tau)=\frac{1}{\pi}\int\limits_{t_0}^\infty
dt\,t^k e^{-t\tau}\rho(t)\quad ,\quad k>-1
\label{lap_gen_k}
\end{eqnarray}

The ``resonance(s) plus continuum''  model is used to 
represent the hadronic physics phenomenology  contained in $\rho(t)$  in 
(\ref{lap_gen_m1}--\ref{lap_gen_k}) \cite{SVZ}.  In this model,  
 hadronic physics is (locally) dual to the theoretical QCD prediction
for energies above the continuum threshold $t=s_0$:
\begin{equation}
\rho(t)\equiv\theta\left(s_0-t\right)\rho^{had}(t)+\theta\left(t-s_0\right){\rm Im}\Pi^{QCD}(t)
\label{res_plus_cont}
\end{equation} 
The contribution of the QCD continuum to the sum-rules is denoted by
\begin{equation}
c_k\left(\tau,s_0\right)=\frac{1}{\pi}\int\limits_{s_0}^\infty
dt\,t^k e^{-t\tau}{\rm Im}\Pi^{QCD}(t)\quad .
\label{continuum}
\end{equation}
Since the continuum contribution is determined by QCD, it is usually combined with the theoretical 
quantity ${\cal L}_k\left(\tau\right)$
\begin{equation}
{\cal S}_k\left(\tau,s_0\right)\equiv {\cal L}_k\left(\tau\right)-c_k\left(\tau,s_0\right)\quad ,
\label{full_sr}
\end{equation}
resulting in the following Laplace sum-rules relating QCD to hadronic physics phenomenology:
\begin{eqnarray}
{\cal S}_{-1}\left(\tau,s_0\right)=-\Pi(0)+\frac{1}{\pi}\int\limits_{t_0}^{s_0}
dt\,\frac{1}{t} e^{-t\tau}\rho^{had}(t)
\label{lap_m1}\\
{\cal S}_{k}\left(\tau,s_0\right)=\frac{1}{\pi}\int\limits_{t_0}^{s_0}
dt\,t^k e^{-t\tau}\rho^{had}(t)\quad ,\quad k>-1
\label{lap_k}
\end{eqnarray}
The property
\begin{equation}
\lim_{s_0\to\infty}c_k\left(\tau,s_0\right)=0
\label{cont_limit}
\end{equation}
implies that the   
sum-rules \bref{full_sr} and \bref{lap_gen_k} are identical in the $s_0\rightarrow\infty$ limit.
\begin{equation}
 \lim_{s_0\to\infty}{\cal S}_{k}\left(\tau,s_0\right)=
{\cal L}_k\left(\tau\right)
\label{sr_limit}
\end{equation}


The only appearance of the $\Pi(0)$ term is in the $k=-1$ sum-rule, and as first noted in
\cite{NSVZ_glue}, this LET term comprises a significant  contribution in the $k=-1$ sum-rule.
From the significance of this scale-independent term one can ascertain the  
important qualitative role of the LET in  sum-rule phenomenology.  
To see this role, we first model the hadronic contributions 
 $\rho^{had}(t)$ using the narrow resonance approximation
\begin{equation}
\frac{1}{\pi}\rho^{had}(t)=\sum_r F_r^2m_r^2\delta\left(t-m_r^2\right)\quad ,
\label{narrow}
\end{equation}
where the  sum over $r$ represents a sum over sub-continuum resonances of mass $m_r$.  The quantity
 $F_r$ is the coupling strength of the resonance to the vacuum through the gluonic current $J(0)$, so
the sum-rule for scalar gluonic currents probes  scalar gluonium states.  
In the narrow-width approximation the Laplace sum-rules  (\ref{lap_m1}--\ref{lap_k})
become
\begin{eqnarray}
& &{\cal S}_{-1}\left(\tau,s_0\right)+\Pi(0)=
\sum_r F_r^2 e^{-m_r^2\tau}
\label{lap_m1_res}\\
& &{\cal S}_{k}\left(\tau,s_0\right)=
\sum_r F_r^2m_r^{2k+2} e^{-m_r^2\tau}\quad ,\quad k>-1\quad .
\label{lap_k_res}
\end{eqnarray}
Thus if the (constant) LET term is a significant contribution on the theoretical side of \bref{lap_m1_res}, then
the left-hand side of \bref{lap_m1_res} will exhibit reduced $\tau$ dependence relative to other theoretical 
contributions.  To reproduce this diminished $\tau$ dependence, the phenomenological 
({\it i.e.} right-hand)
side must 
contain a  light resonance with a coupling larger than or comparable to the heavier resonances.  
By contrast, the absence of the $\Pi(0)$ (constant) term in $k>-1$ sum-rules leads to stronger $\tau$ dependence
which is balanced on the phenomenological side  by suppression 
of the lightest resonances 
via the additional powers of $m_r^2$  occurring in \bref{lap_k_res}.  Thus if $\Pi(0)$ is found to dominate 
${\cal S}_{-1}\left(\tau,s_0\right)$, then one would expect qualitatively different results from analysis
of the $k=-1$ and $k>-1$ sum-rules.  

Such distinct conclusions drawn from different sum-rules can be legitimate.  
In the pseudoscalar quark sector, the lowest sum-rule is dominated by the pion, and the low  mass of the pion 
is evident from the  minimal $\tau$ dependence
in the lowest sum-rule. By contrast,  the first subsequent sum-rule has an important contribution from the 
$\Pi(1300)$ \cite{pi_1300},  as the pion contribution is suppressed by its low mass, resulting in the 
significant $\tau$ dependence of the next-to-lowest sum-rule.


In the absence of instantons \cite{instanton}, 
explicit sum-rule analyses of scalar gluonium \cite{dispersion,NSVZ_glue,glue} uphold the above 
generalization---those which include the
$k=-1$ sum-rule find a light (less than or on the order of the $\rho$ mass) gluonium state, 
and those which omit the $k=-1$  sum-rule find a state with a mass greater than $1\unt{GeV}$. 
The prediction of a light gluonium state  would have interesting 
phenomenological consequences as a state which could be identified with the $f_0(400-1200)/\sigma$ 
meson \cite{ochs}.   However, a detailed treatment of instanton contributions is essential in assessing
the viability of such a two-resonance-scale scenario (as evident in the sum-rule analysis of the $q\bar q$ 
pseudoscalar channel \cite{pi_1300}) in the scalar gluonium channel.

Seminal work by Shuryak \cite{shuryak} in the instanton liquid model \cite{ins_liquid} has indicated how
an asymptotic ($\rho_c/\sqrt{\tau}\gg 1$) expression for the instanton contribution to the $k=-1$ sum-rule
may serve to compensate for that sum-rule's LET component and bring the predicted scalar gluonium mass in
line with subsequent lattice estimates ($\sim 1.6\unt{GeV}$ \cite{lattice}).  Recent work by Forkel \cite{forkel} has addressed
in detail instanton effects on scalar gluonium mass predictions from higher-weight sum-rules and has also corroborated
lattice estimates.  However, the overall consistency of the $k=-1$ sum-rule, which is sensitive to the low-energy
theorem term, and $k\ge 0$ sum-rules, which are not, has not been addressed quantitatively.

In Section~\ref{lap_sec}, we explicitly calculate the instanton contributions to 
Laplace sum-rules of scalar gluonic currents.
We pay particular attention to the $k=-1$ sum-rule and demonstrate that
instanton contributions partially cancel against the LET constant $\Pi(0)$ and serve
to appreciably diminish its dominance of this leading order sum-rule.
The phenomenological implications of this partial cancellation
are investigated in Section~\ref{phenom_sec}, and a discussion
relating our work to other analyses of instanton effects in the scalar gluonium channel is presented
in Section~\ref{discuss_sec}.


\section{Instanton Effects in the Laplace Sum-Rules}
\label{lap_sec}  
The field-theoretical (QCD) calculation of $\Pi\left(Q^2\right)$ consists of perturbative (logarithmic) corrections 
known to three-loop order ($\overline{{\rm MS}}$ scheme) in the chiral limit of $n_f=3$ massless quarks 
\cite{glue_pt}, 
QCD vacuum effects of infinite correlation length 
parameterized by  the power-law contributions from the QCD vacuum condensates 
\cite{NSVZ_glue,condensate2}, 
\footnote{The calculation of one-loop contributions proportional to $\left\langle J\right\rangle$ 
in \protect\cite{condensate2} have been extended 
non-trivially
to $n_f=3$ from $n_f=0$, and the operator basis
has been changed from $\left\langle \alpha G^2\right\rangle$ to $\left\langle J\right\rangle$. 
}
and QCD vacuum effects of finite correlation length devolving from instantons \cite{instanton_corr}
\begin{equation}
\Pi\left(Q^2\right)=
\Pi^{pert}\left(Q^2\right)+\Pi^{cond}\left(Q^2\right)+\Pi^{inst}\left(Q^2\right)\quad ,\quad 
\label{QCD_corr_fn}
\end{equation}
with
\begin{enumerate}
\item $\ldots$ the perturbative contribution (ignoring divergent terms proportional to $Q^4$) given by
\begin{eqnarray}
& &\Pi^{pert}\left(Q^2\right)=Q^4\log\left(\frac{Q^2}{\nu^2}\right)\left[
a_0+a_1\log\left(\frac{Q^2}{\nu^2}\right)+a_2\log^2\left(\frac{Q^2}{\nu^2}\right)
\right]
\label{pi_pert}\\
& &a_0=-2\left(\frac{\alpha}{\pi}\right)^2\left[1+\frac{659}{36}\frac{\alpha}{\pi}+
247.480\left( \frac{\alpha}{\pi}\right)^2\right]
~ , ~  
a_1=2\left(\frac{\alpha}{\pi}\right)^3\left[ \frac{9}{4}+65.781\frac{\alpha}{\pi}\right]
~ , ~ 
a_2=-10.1250\left(\frac{\alpha}{\pi}\right)^4
\nonumber
\end{eqnarray}

\item $\ldots$ the condensate contributions given by 
\begin{eqnarray}
& &\Pi^{cond}\left(Q^2\right)=
\left[ b_0+b_1\log\left(\frac{Q^2}{\nu^2}\right)\right]\left\langle  J\right\rangle
+c_0\frac{1}{Q^2}\left\langle {\cal O}_6\right\rangle+d_0\frac{1}{Q^4}\left\langle {\cal O}_8\right\rangle
\label{pi_cond}\\
& &b_0=4\pi\frac{\alpha}{\pi}\left[ 1+ \frac{175}{36}\frac{\alpha}{\pi}\right]
\quad ,\quad b_1=-9\pi\left(\frac{\alpha}{\pi}\right)^2
\quad ,\quad
c_0=8\pi^2\left(\frac{\alpha}{\pi}\right)^2\quad ,\quad d_0=8\pi^2\frac{\alpha}{\pi}
\label{GG_coeffs}\\
& &\left\langle {\cal O}_6\right\rangle=
\left\langle g f_{abc}G^a_{\mu\nu}G^b_{\nu\rho}G^c_{\rho\mu}\right\rangle
\quad ,\quad
\left\langle {\cal O}_8\right\rangle=14\left\langle\left(\alpha f_{abc}G^a_{\mu\rho}G^b_{\nu\rho}\right)^2\right\rangle
-\left\langle\left(\alpha f_{abc}G^a_{\mu\nu}G^b_{\rho\lambda}\right)^2\right\rangle
\label{higher_dim_conds}
\end{eqnarray}

\item $\ldots$ and the instanton contribution given by 
\begin{equation}
\Pi^{inst}\left(Q^2\right)=
32\pi^2Q^4\int \rho^4 \left[K_2\left(\rho\sqrt{Q^2}\right)\right]^2  dn(\rho)\quad ,
\label{pi_inst}
\end{equation}
where $K_2(x)$ represents a modified Bessel function \cite{abramowitz}.
\end{enumerate}
The strong coupling constant $\alpha$ is understood to be the running coupling at the renormalization 
scale $\nu$, 
and renormalization group improvement of the Laplace sum-rules implies that $\nu^2=1/\tau$ \cite{RG_improve}. 
The instanton contributions 
represent a calculation with non-interacting instantons 
of size $\rho$, with subsequent integration over the instanton density distribution $n(\rho)$. 
\footnote{A factor of 2 to include the sum of instanton and anti-instanton contributions has  been 
included in (\protect\ref{QCD_corr_fn}).} 
The theoretical contributions to the Laplace sum-rules corresponding to \bref{QCD_corr_fn} are
\begin{equation}
{\cal L}_k(\tau)=
{\cal L}_k^{pert}(\tau)+{\cal L}_k^{cond}(\tau)+{\cal L}_k^{inst}(\tau)\quad . 
\label{laplace_ports}
\end{equation}

An alternative to the
direct calculation of the Laplace sum-rules through the definition of $\hat B$  in \bref{borel} is obtained through 
an identity relating the Borel and Laplace transform \cite{borel_ref}
\begin{eqnarray}
& &f\left(Q^2\right)=\int\limits_0^\infty d\tau F(\tau) e^{-Q^2\tau}\equiv{\cal L}\left[ F(\tau)\right]
\quad \Longrightarrow\quad \frac{1}{\tau}\hat B\left[ f\left(Q^2\right)\right]
=F(\tau)={\cal L}^{-1}
\left[ f\left(Q^2\right)\right]
\label{borel_laplace}\\
& &{\cal L}^{-1}
\left[ f\left(Q^2\right)\right]=\frac{1}{2\pi i}\int\limits_{b-i\infty}^{b+i\infty}
f\left(Q^2\right) e^{Q^2\tau}\,dQ^2
\label{inv_lap_def}
\end{eqnarray}
where the real parameter $b$ in the definition \bref{inv_lap_def} of the inverse Laplace 
transform must be chosen so that $f\left(Q^2\right)$ is analytic to the right of the 
contour of integration in the complex plane.
Using the result \bref{borel_laplace}, the Laplace sum-rules \bref{laplace} can be  
obtained from an inverse Laplace transform of 
the theoretically-determined correlation function:
\begin{equation}
{\cal L}_k(\tau)={\cal L}^{-1}\left[
\left(-1\right)^k Q^{2k}\Pi\left(Q^2\right)
\right]\quad .
\label{inv_lap}
\end{equation}


In the complex $Q^2$ plane where the inverse Laplace transform \bref{inv_lap_def} is calculated, 
the QCD expression \bref{QCD_corr_fn} for the correlation function 
$\Pi\left(Q^2\right)$ 
is analytic apart from a branch point at $Q^2=0$ with a branch cut 
extending to infinity along the negative-real-$Q^2$ axis. 
Consequently, analyticity to the right of the contour in \bref{inv_lap_def} implies that
 $b>0$.
Consider the contour
$C(R)$ in Figure \ref{cont_fig1}; $\Pi\left(Q^2\right)$ is analytic within and on $C(R)$ and so with $z=Q^2$
\begin{equation}
0=
\frac{1}{2\pi i}\oint\limits_{C(R)} \left(-z\right)^k e^{z\tau}\Pi(z) dz\quad ,
\label{cont_result}
\end{equation}
which leads to 
\begin{equation}
\frac{1}{2\pi i}\int\limits_{b-iR}^{b+iR} \left(-z\right)^k e^{z\tau}\Pi(z) dz
=-\frac{1}{2\pi i}\int\limits_{\Gamma_1+\ldots +\Gamma_4}\!\!\! \left(-z\right)^k e^{z\tau}\Pi(z) dz
-\frac{1}{2\pi i}\int\limits_{\Gamma_c+\Gamma_\epsilon}\!\! \left(-z\right)^k e^{z\tau}\Pi(z) dz
\quad .
\label{borel_inv_1}
\end{equation}  
Taking the limit as $R\to\infty$, which requires use of the asymptotic behaviour  of the modified Bessel 
function \cite{abramowitz}
\begin{equation}
K_2\left(z\right)\sim\sqrt{\frac{\pi}{2z}}e^{-z}\quad ;\quad |z|\gg 1~,~\left|{\rm arg}(z)\right|\le\frac{\pi}{2}
\end{equation}
the individual integrals over $\Gamma_{1\ldots 4}$ are found to 
vanish, resulting in the following expression for the Laplace sum-rule.
\begin{equation}
{\cal L}_{k}\left(\tau\right)=
\frac{1}{2\pi i}\int\limits_\epsilon^\infty t^ke^{-t\tau}
\left[\Pi\left(te^{-i\pi}\right)-\Pi\left(te^{i\pi}\right)\right]  dt
+\frac{1}{2\pi }\int\limits_{-\pi}^\pi \left(-1\right)^k \exp{\left(\epsilon e^{i\theta}\tau\right)}
\epsilon^{k+1}e^{i (k+1)\theta}\Pi\left(\epsilon e^{i\theta}\right)   d\theta
\label{simpl_inv_borel}
\end{equation}


Perturbative and QCD condensate contributions to the Laplace sum-rules are well known 
\cite{dispersion,NSVZ_glue,glue}, and serve as a consistency check for the conventions used to determine the 
instanton contribution
through \bref{simpl_inv_borel}.  Keeping in mind the $k\ge -1$ constraint established previously, we see 
that  the perturbative contributions to the $\theta$ integral in \bref{simpl_inv_borel}
are zero in the limit as $\epsilon\to 0$, leaving only the anticipated integral of the discontinuity 
across the branch cut [{\it i.e.} ${\rm Im}\Pi^{pert}(t)$] to determine the following perturbative contributions 
to the Laplace sum-rule.
\begin{equation}
{\cal L}^{pert}_{k}\left(\tau\right)=\int\limits_0^{\infty}
t^{k+2}e^{-t\tau} \left[-a_0-2a_1\log\left(\frac{t}{\nu^2}\right)
+a_2\left(\pi^2-3\log^2\left(\frac{t}{\nu^2}\right)\right)\right]\,dt
\label{sk_pert}
\end{equation}

The QCD condensate terms  proportional to $b_0$, $c_0$ and $d_0$ in the correlation function 
$\Pi(z)$ do not have a branch discontinuity, so their contribution to the Laplace sum-rule
arises solely from the contour $\Gamma_\epsilon$ (represented by the term in 
\bref{simpl_inv_borel} with 
the $\theta$ integral), 
and can be evaluated using the result
\begin{equation}
-\frac{1}{2\pi i}\int\limits_{\Gamma_\epsilon}\frac{e^{z\tau}}{z^n}dz=
\left\{
\begin{array}{l}
0\quad ,\quad n=0,~-1,~-2,\ldots\\
\frac{\tau^{n-1}}{(n-1)!}\quad,\quad n=1,~2,~3,\ldots 
\end{array}
\right.
\label{power_law_sk}
\end{equation}     
The QCD condensate term proportional to $b_1$ requires a more careful treatment. If $\Pi(z)$
is replaced with $\log\left(z/\nu^2\right)$ in \bref{simpl_inv_borel}  then we find
\begin{equation}
\frac{1}{2\pi i}\int\limits_{b-i\infty}^{b+i\infty}\!\!\! \left(-z\right)^k e^{z\tau}
\log\left(\frac{z}{\nu^2}\right) dz
=-\int\limits_\epsilon^{\infty} t^ke^{-t\tau}
 dt
+\frac{1}{2\pi }\int\limits_{-\pi}^\pi \!\!\left(-1\right)^k \exp{\left(\epsilon e^{i\theta}\tau\right)}
\epsilon^{k+1}e^{i (k+1)\theta}\left(\log\left(\frac{\epsilon}{\nu^2}\right) +i\theta\right)   d\theta
\label{def_cont_int2}
\end{equation}
The last term in this equation will be zero in the $\epsilon\to 0$ limit except  when $k=-1$. Similarly, 
the $t$ integral is well defined in the  $\epsilon\to 0$ limit except when $k=-1$.  With $\nu^2=1/\tau$, 
and with evaluation of the $\epsilon\to 0$ limit [which, for $k=-1$, involves cancellation between the two 
integrals in \bref{def_cont_int2}] we find
\begin{equation}
\frac{1}{2\pi i}\int\limits_{b-i\infty}^{b+i\infty} \left(-z\right)^k e^{z\tau}
\log\left(\frac{z}{\nu^2}\right) dz
=\left\{\begin{array}{l}
-\int\limits_0^{\infty} t^ke^{-t\tau}dt \quad ,\quad k>-1\\
\gamma_{_E} \quad,\quad k=-1
\end{array}
\right.
\label{b1_int}
\end{equation}
where  $\gamma_{_E}\approx 0.5772$ is Euler's constant.
It is easily verified that equations \bref{b1_int}, \bref{power_law_sk}, and \bref{sk_pert} lead to the
known results \cite{dispersion,NSVZ_glue,glue} for the non-instanton contributions to the  Laplace sum-rules for scalar gluonic
currents.

To evaluate the instanton contributions to the Laplace sum-rule, we must calculate the following integral:
\begin{eqnarray}
\frac{1}{2\pi i}\int\limits_{b-i\infty}^{b+i\infty} \left(-z\right)^k e^{z\tau}z^2
\left[K_2\left(\rho\sqrt{z}\right)\right]^2 dz
&=&\frac{1}{2\pi i}\int\limits_\epsilon^{\infty} t^{k+2}e^{-t\tau}
\left[
\left[K_2\left(\rho\sqrt{t}e^{-i\pi/2}\right)\right]^2
-
\left[K_2\left(\rho\sqrt{t}e^{i\pi/2}\right)\right]^2
\right]dt
\nonumber\\
& &
+\frac{1}{2\pi }\int\limits_{-\pi}^\pi \left(-1\right)^k \exp{\left(\epsilon e^{i\theta}\tau\right)}
\epsilon^{k+3}e^{i (k+3)\theta}\left[K_2\left(\rho\sqrt{\epsilon} e^{i\theta/2}\right)\right]^2   d\theta
\label{inst_int2}
\end{eqnarray}
Simplification of \bref{inst_int2} requires the following properties of the modified Bessel
function $K_2(z)$ \cite{abramowitz}
\begin{eqnarray}
& &K_2\left(z\right)\sim\frac{2}{z^2}\quad ,\quad z\to 0
\label{asymp_K2}\\
& &K_2\left(z\right)=\left\{
\begin{array}{l}
-i\frac{\pi}{2}H_2^{(1)}\left(ze^{i\pi/2}\right)\quad ,\quad -\pi<\arg(z)\le \frac{\pi}{2}\\[5pt]
i\frac{\pi}{2}H_2^{(2)}\left(ze^{-i\pi/2}\right)\quad ,\quad -\frac{\pi}{2}<\arg(z)\le \pi
\end{array}
\right.
\label{anal_con_K2}
\end{eqnarray}
where $H_2^{(1)}(z)=J_2(z)+iY_2(z)$ and $H_2^{(2)}(z)=J_2(z)-iY_2(z)$.
The asymptotic behaviour \bref{asymp_K2} implies that the $\theta$ integral of \bref{inst_int2}
will be zero in the $\epsilon\to 0$ limit for $k>-1$ and the identity \bref{anal_con_K2}
allows evaluation of the discontinuity in the $t$ integral of~\bref{inst_int2}, leading to the
following instanton contribution to the Laplace sum-rules: 
\begin{align}
  \mathcal{L}_{-1}^{inst}\left(\tau\right) &=
  -16\pi^3\int\nms\mathrm{d}n(\rho) \rho^4\int\limits_0^\infty tJ_2\left(\rho\sqrt{t}\right)
  Y_2\left(\rho\sqrt{t}\right)\mathrm{e}^{-t\tau}\,\mathrm{d}t -128\pi^2\int\mathrm{d}n(\rho)
  \label{sm1_inst}\\
  &= -64\pi^2 \int\nms\mathrm{d}n(\rho)\ a\mathrm{e}^{-a} \left[\;(1+a)aK_0(a)
      +(2+2a+a^2)K_1(a)\;\right]\nonumber\\
  {\cal L}_k^{inst}\left(\tau\right)    &=
  -16\pi^3\int\nms\mathrm{d}n(\rho) \rho^4\int\limits_0^\infty t^{k+2}J_2\left(\rho\sqrt{t}\right)
  Y_2\left(\rho\sqrt{t}\right)\mathrm{e}^{-t\tau} \,\mathrm{d}t \quad ,\quad k>-1
  \label{sk_inst}\\
  &= \left\{
     \begin{aligned}
       \; & 128\pi^2\int\nms\mathrm{d}n(\rho)\ \frac{a^4\mathrm{e}^{-a}}{\rho^2}
       \left[\; 2aK_0(a)+(1+2a)K_1(a)\;\right]\quad,\quad k=0  \\
       \; & 256\pi^2 \int\nms\mathrm{d}n(\rho)\ \frac{a^5\mathrm{e}^{-a}}{\rho^4}
       \left[\; (9-4a)aK_0(a) + (3+7a-4a^2)K_1(a)\;\right]\quad,\quad k=1
     \end{aligned}
     \right. \nonumber
\end{align}
where $a\equiv\rho^2/(2\tau)$ and where $K_n$ is the modified Bessel function of the
first kind of order $n$ (\textit{c.f.}~\cite{abramowitz}). 
Observe the symmetry between~(\ref{sm1_inst}) and~(\ref{sk_inst}) broken  by the
term $-128\pi^2\int\mathrm{d}n({\rho})$ appearing in~(\ref{sm1_inst})---a
term which corresponds to the second integral on the right-hand side of~(\ref{inst_int2})
and which is nonzero \emph{only} for $k=-1$. 
Conversely, we note that naively substituting $k=-1$ into~(\ref{sk_inst}) leads to
an incorrect expression for the instanton contribution to the leading order sum-rule.  
This asymmetric role  of  the instanton contributions to 
$k=-1$ and $k>-1$ sum-rules is also a property of the LET as illustrated 
in~(\ref{lap_gen_m1},\ref{lap_gen_k}). 


As discussed in Section~\ref{intro_sec}, we wish to determine whether the leading
order sum-rule $\mathcal{L}_{-1}$ might support the existence of a lowest-lying
resonance whose presence is mass-suppressed in subsequent higher order $k>-1$ sum-rules.
Such is indeed the case, for example, for the pion within sum-rules based on a
pseudoscalar $\overline q\gamma_5q$ current. Correspondingly, one might anticipate
the identification of the lowest-lying scalar gluonium state with a
$500\mbox{--}600\,{\rm MeV}$ $\sigma$ ({\it i.e.} the lower-mass range of the $f_0(400-1200)$) 
resonance~\cite{ochs}
whose contribution to higher order sum-rules is suppressed by additional
factors of $m^2_{\sigma}$ [\textit{i.e.}\ the additional factors of $m^2_r$
in~(\ref{lap_k_res})], a
scenario analogous to the $m^2_{\pi}$  suppression of pion contributions to the
pseudoscalar $\overline q\gamma_5q$ sum-rules~\cite{pi_1300}. As
already noted in Section~\ref{intro_sec}, the LET constant in the absence of instantons
supports this scenario for a sub-GeV scalar glueball.


     However, the instanton contribution to the $k = -1$ sum-rule is opposite in sign and
comparable in magnitude to the LET subtraction constant $\Pi(0)$, thereby ameliorating this term's
dominance of the lowest order sum-rule. For example, the contribution of instanton and LET
terms to the $k = -1$ sum rule in the dilute instanton liquid (DIL) model~\cite{ins_liquid},
\begin{equation}
dn(\rho)=n_c\delta\left(\rho-\rho_c\right)d\rho\quad ;\quad n_c=8\times 10^{-4}\unt{GeV^4}\quad ,\quad
\rho_c=\frac{1}{600\unt{MeV}}
\label{ins_liquid_params}
\end{equation}
 renders trivial the remaining integrations in~(\ref{sm1_inst}--\ref{sk_inst}).
If we approximate $\langle J\rangle$ by
$\left\langle \alpha G^2\right\rangle$
and employ a recently determined value of the gluon condensate \cite{GG_ref}
\begin{equation}
\left\langle \alpha G^2\right\rangle=(0.07\pm 0.01) \unt{GeV^4}
\label{GG_cond}
\end{equation}
we obtain via~(\ref{let}) and~(\ref{beta_coeffs}) an $n_f=3$ estimate of  the
LET subtraction constant:
\begin{equation}
\label{PI_approx}
  \Pi(0)\approx\frac{32\pi}{9}\left\langle \alpha G^2\right\rangle
        \approx (0.78\pm0.11)\ \unt{GeV^4}\quad.
\end{equation}
In Figure~\ref{mitigation}, we use~(\ref{sm1_inst}) and the central value of~(\ref{PI_approx}) [with $n_c$ and $\rho_c$
given in~(\ref{ins_liquid_params})] to plot
\begin{equation}
   \frac{\Pi(0) + \mathcal{L}^{inst}_{-1}(\tau)}{\Pi(0)}
\end{equation}
as a function $\tau$. We note that, as anticipated, instanton effects do indeed
significantly reduce the impact of the LET on the $k=-1$ sum-rule:
anywhere from 20--65\% for $\tau$
ranging between $0.6\ \mathrm{GeV}^{-2}$ and  $1.0\ \mathrm{GeV}^{-2}$.
Recalling that the dominance of the LET
over ${\cal S}_{-1}$ is responsible for the discrepancy in gluonium mass scales in the analysis of the
$k=-1$ and $k>-1$ sum-rules, we see that suppression of the LET by instanton effects
could reconcile this discrepancy, a possibility which is investigated further in the next section.


\section{Phenomenological Impact of Instanton Effects in the Laplace Sum-Rules}
\label{phenom_sec}
Ratios of Laplace sum-rules provide a simple technique for extracting the mass of the lightest
(narrow) resonance probed by the sum-rules. If only the lightest resonance  (of mass $m$) is included in  
\bref{lap_m1_res} and \bref{lap_k_res}, then for the first few sum-rules we see that
\begin{eqnarray}
& &\frac{{\cal S}_1\left(\tau,s_0\right)}{{\cal S}_0\left(\tau,s_0\right)}=m^2
\label{r1_r0_ratio}\\
& &\frac{{\cal S}_0\left(\tau,s_0\right)}{{\cal S}_{-1}\left(\tau,s_0\right)+\Pi(0)}=m^2\quad .
\label{r0_rm1_ratio}
\end{eqnarray}
This method of predicting the mass $m$ requires optimization of $s_0$ to minimize the  $\tau$ dependence that can occur in the 
sum-rule ratios.  However, a qualitative analysis which avoids these optimization issues
occurs in the $s_0\to\infty$ limit where bounds on the mass $m$ can also be obtained.
These bounds originate from inequalities satisfied on the hadronic physics side of the 
sum-rule because of the positivity of $\rho^{had}(t)$.  For example,
\begin{eqnarray}
& &\frac{1}{\pi}\int\limits_{t_0}^{s_0}
dt\,t e^{-t\tau}\rho^{had}(t)=\frac{1}{\pi}\int\limits_{t_0}^{s_0}
dt\,\left(t-s_0+s_0\right) e^{-t\tau}\rho^{had}(t)\le s_0
\frac{1}{\pi}\int\limits_{t_0}^{s_0}
dt\, e^{-t\tau}\rho^{had}(t)\nonumber\\
& &\Longrightarrow
{\cal S}_1\left(\tau,s_0\right)\le s_0{\cal S}_0\left(\tau,s_0\right)\quad .
\label{s1_ineq}
\end{eqnarray}
Furthermore,  positivity of ${\rm Im}\Pi^{QCD}(t)$ leads to an inequality for the
continuum.
\begin{eqnarray}
& &\frac{1}{\pi}\int\limits_{s_0}^{\infty}
dt\,t e^{-t\tau}{\rm Im}\Pi^{QCD}(t)=\frac{1}{\pi}\int\limits_{s_0}^{\infty}
dt\,\left(t-s_0+s_0\right) e^{-t\tau}{\rm Im}\Pi^{QCD}(t)\ge s_0
\frac{1}{\pi}\int\limits_{s_0}^{\infty}
dt\, e^{-t\tau}{\rm Im}\Pi^{QCD}(t)\nonumber\\
& &\Longrightarrow
c_1\left(\tau,s_0\right)\ge s_0 c_0\left(\tau,s_0\right)
\label{c1_ineq}
\end{eqnarray}  
These inequalities can be extended to include the $k=-1$ sum-rules and continuum.
\begin{eqnarray}
& &{\cal S}_0\left(\tau,s_0\right)\le s_0\left[{\cal S}_{-1}\left(\tau,s_0\right)+\Pi(0)\right]
\label{s0_ineq}\\
& &c_0\left(\tau,s_0\right)\ge s_0 c_{-1}\left(\tau,s_0\right)
\label{c0_ineq}
\end{eqnarray}
We then see that
\begin{equation}
\frac{{\cal L}_1\left(\tau\right)}{{\cal L}_0\left(\tau\right)}
=\frac{{\cal S}_1\left(\tau,s_0\right)+c_1\left(\tau,s_0\right)}{{\cal S}_0\left(\tau,s_0\right)
+c_0\left(\tau,s_0\right)}
= \frac{{\cal S}_1\left(\tau,s_0\right)}{{\cal S}_0\left(\tau,s_0\right)}
\left[
\frac{1+\frac{c_1\left(\tau,s_0\right)}{{\cal S}_1\left(\tau,s_0\right)}}{1+\frac{c_0\left(\tau,s_0\right)}{{\cal S}_0\left(\tau,s_0\right)}}
\right]
\ge \frac{{\cal S}_1\left(\tau,s_0\right)}{{\cal S}_0\left(\tau,s_0\right)} =m^2
\label{s1_bounds}
\end{equation}
where the final inequality of \bref{s1_bounds} follows from
$c_1/{\cal S}_1\ge s_0 c_0/{\cal S}_1\ge c_0/{\cal S}_0$ via
\bref{c1_ineq} and \bref{s1_ineq}.  Similarly, we find from \bref{s0_ineq} and \bref{c0_ineq} that
\begin{equation}
\frac{{\cal L}_0\left(\tau\right)}{{\cal L}_{-1}\left(\tau\right)+\Pi(0)}
=\frac{{\cal S}_0\left(\tau,s_0\right)+c_0\left(\tau,s_0\right)}{{\cal S}_{-1}\left(\tau,s_0\right)+\Pi(0)+c_{-1}\left(\tau,s_0\right)}
\ge \frac{{\cal S}_0\left(\tau,s_0\right)}{{\cal S}_{-1}\left(\tau,s_0\right)+\Pi(0)} =m^2\quad.
\label{s0_bounds}
\end{equation}
Thus the ratios of the $s_0\to\infty$ limit of the sum-rules provide bounds on the mass in this single narrow 
resonance approximation.  Extending the
analysis to many narrow resonances alters (\ref{r1_r0_ratio}--\ref{r0_rm1_ratio}) so that the sum-rule ratios are an upper 
 bound on 
the lightest resonance, upholding the bounds  (\ref{s1_bounds}--\ref{s0_bounds}) on the 
mass $m^2$ of the lightest resonance.
\begin{eqnarray}
& &\frac{{\cal L}_1\left(\tau\right)}{{\cal L}_0\left(\tau\right)}
\ge m^2
\label{L1_bounds}\\
& &\frac{{\cal L}_0\left(\tau\right)}{{\cal L}_{-1}\left(\tau\right)+\Pi(0)}
\ge m^2
\label{L0_bounds}
\end{eqnarray}

The sum-rule bounds in (\ref{L1_bounds}--\ref{L0_bounds}) can now be employed to determine the phenomenological 
impact of the instanton contributions on the sum-rule estimates of the lightest gluonium state, and to assess whether
the suppression of the LET contribution by the instanton effects is sufficient to reduce the discrepancy between 
sum-rule analyses containing or omitting the $k=-1$ sum-rule.  
Collecting results  from equations 
(\ref{pi_cond},\ref{simpl_inv_borel}--\ref{power_law_sk},\ref{sm1_inst},\ref{sk_inst}) 
the first few 
sum-rules ${\cal L}_k(\tau)$ are
\begin{align}
\mathcal{L}_{-1}(\tau) &= \frac{1}{\tau^2}\left[-a_0+a_1\left(-2+2\gamma_E\right)
  +a_2\left(\frac{\pi^2}{2}+6\gamma_E-3\gamma_E^2\right)\right]
  +\left(-b_0+b_1\gamma_E\right)\left\langle J\right\rangle
  -c_0\tau\left\langle {\cal O}_6\right\rangle
  -d_0\frac{\tau^2}{2}\left\langle{\cal O}_8\right\rangle
  \nonumber\\
  & \hspace{3em} -64\pi^2 \int\nms\mathrm{d}n(\rho)\ a\mathrm{e}^{-a} \left[\;(1+a)aK_0(a)
      +(2+2a+a^2)K_1(a)\;\right]
\label{Lm1}\\
\mathcal{L}_{0}(\tau) &= \frac{1}{\tau^3}\left[-2a_0+a_1\left(-6+4\gamma_E\right)
  +a_2\left(\pi^2-6+18\gamma_E-6\gamma_E^2\right)\right]
  -\frac{b_1}{\tau}\left\langle J\right\rangle
  +c_0\left\langle {\cal O}_6\right\rangle
  +d_0\tau\left\langle{\cal O}_8\right\rangle
  \nonumber\\
  & \hspace{3em} +128\pi^2\int\nms\mathrm{d}n(\rho)\ \frac{a^4\mathrm{e}^{-a}}{\rho^2}
       \left[\; 2aK_0(a)+(1+2a)K_1(a)\;\right]
\label{L0}\\
\mathcal{L}_{1}(\tau) &= \frac{1}{\tau^4}\left[-6a_0+a_1\left(-22+12\gamma_E\right)
  +a_2\left(3\pi^2-36+66\gamma_E-18\gamma_E^2\right)\right]
  -\frac{b_1}{\tau^2}\left\langle J\right\rangle
  -d_0\left\langle{\cal O}_8\right\rangle
\nonumber\\
  & \hspace{3em} +256\pi^2 \int\nms\mathrm{d}n(\rho)\ \frac{a^5\mathrm{e}^{-a}}{\rho^4}
       \left[\; (9-4a)aK_0(a) + (3+7a-4a^2)K_1(a)\;\right] \quad.
\label{L1}
\end{align}
Renormalization-group
improvement has been achieved by setting $\nu^2=1/\tau$ in the correlation
function and in the (three-loop, $n_f=3$, $\overline{\rm MS}$) running coupling $\alpha$:  
\begin{eqnarray}
\frac{\alpha_s(\nu)}{\pi}&=& \frac{1}{\beta_0 L}-\frac{\bar\beta_1\log L}{\left(\beta_0L\right)^2}+
\frac{1}{\left(\beta_0 L\right)^3}\left[
\bar\beta_1^2\left(\log^2 L-\log L -1\right) +\bar\beta_2\right]
\label{alpha_hl}\\
& &L=\log\left(\frac{\nu^2}{\Lambda^2}\right)\quad ,\quad \bar\beta_i=\frac{\beta_i}{\beta_0}
\quad ,
\quad
\beta_0=\frac{9}{4}\quad ,\quad \beta_1=4\quad ,\quad \beta_2=\frac{3863}{384}
\end{eqnarray}
with 
 $\Lambda_{\overline{MS}}\approx 300\,{\rm MeV}$ for three active flavours,
consistent with current estimates of $\alpha_s(M_\tau)$ \cite{PDG,pade_tau} and matching conditions through the charm threshold 
\cite{ChetyrkinKniehl}.

The nonperturbative QCD parameters are needed for further analysis of the sum-rules.
We employ the DIL model~\cite{ins_liquid} parameters summarized in~\bref{ins_liquid_params},
as well as vacuum saturation  for the dimension-8 gluon condensate~\cite{NSVZ_glue,vacuum_saturation}
\begin{equation}
\left\langle {\cal O}_8\right\rangle=14\left\langle\left(\alpha f_{abc}G^a_{\mu\rho}G^b_{\nu\rho}\right)^2\right\rangle
-\left\langle\left(\alpha f_{abc}G^a_{\mu\nu}G^b_{\rho\lambda}\right)^2\right\rangle=
\frac{9}{16}\left(\left\langle \alpha G^2\right\rangle\right)^2
\label{vac_sat}
\end{equation}
and instanton estimates of the dimension-six condensate \cite{NSVZ,SVZ} 
\begin{equation}
\left\langle {\cal O}_6\right\rangle=
\left\langle g f_{abc}G^a_{\mu\nu}G^b_{\nu\rho}G^c_{\rho\mu}\right\rangle
=\left(0.27\unt{GeV^2}\right)\left\langle \alpha G^2\right\rangle\quad .
\end{equation}
Finally,  again using the approximation $\left\langle J\right\rangle=\left\langle \alpha G^2\right\rangle$ and the
central gluon condensate value [see \bref{GG_cond}] from reference \cite{GG_ref}, we find that
the role of instanton contributions  to the sum-rules (\ref{Lm1}--\ref{L1})
is as illustrated in Figures \ref{sm1_fig}, \ref{s0_fig}, and \ref{s1_fig}.  
In particular, 
we see that the instanton contributions  
diminish ${\cal L}_{-1}(\tau)+\Pi(0)$. 
The  LET term $\Pi(0)$, which leads to the asymptotic flattening of ${\cal L}_{-1}(\tau)+\Pi(0)$
at a value substantially different from zero when instantons are absent, is clearly suppressed by 
 instanton effects in the large $\tau$ region.  As noted earlier, such flattening over the $\tau\le 1.0\,{\rm GeV^{-2}}$
region would be indicative via (\ref{lap_m1_res}) of a sub-GeV lowest-lying resonance ({\it i.e.}, $m_r^2\tau\ll 1$ 
over the physically relevant region of $\tau$)
Instanton effects no only undo this flattening, but also increase ${\cal L}_0(\tau)$ and
alter the shape of ${\cal L}_1(\tau)$. 
The corresponding effects of instantons on the sum-rule ratios (\ref{L1_bounds}--\ref{L0_bounds}) is shown in 
Figures \ref{s0sm1_fig} and \ref{s1s0_fig}.  
As expected from the instanton's impact of lowering  ${\cal L}_{-1}(\tau)$  and elevating
 ${\cal L}_{0}(\tau)$, the ratio  ${\cal L}_{0}(\tau)/\left[{\cal L}_{-1}(\tau)+\Pi(0)\right]$ is increased substantially
by inclusion of instanton effects,  increasing the corresponding upper bound on the mass of the lightest 
gluonium state.   Instanton effects also serve to  {\em lower} the ratio  
${\cal L}_{1}(\tau)/{\cal L}_{0}(\tau)$, decreasing the corresponding upper bound on the mass of the lightest 
gluonium state.  


Figures \ref{ratios_on_fig} and \ref{ratios_off_fig} summarize the ratio (mass bound) analysis in the
 presence and in the 
absence of instanton effects.  It is evident that instanton effects lead to a substantial increase in 
the mass bound on the 
lightest gluonium state, but 
other important features emerge.  For example, the instanton suppression of the LET term $\Pi(0)$ 
reduces the discrepancy between the
ratios including or omitting the $k=-1$ sum-rule.  Furthermore, 
a $\tau$-minimum stability plateau crucial for establishing a credible upper mass bound 
is seen to occur at 
reasonable energy scales ($1/\sqrt{\tau}\le 1.0\unt{GeV}$) only when instanton effects are included. 
The ratios with instanton effects included (see Figure \ref{ratios_on_fig}) are remarkably flat, suggesting
 that the mass bounds could be close to the mass prediction that would be obtained from a 
full sum-rule analysis incorporating the QCD continuum ({\it i.e.} $s_0<\infty$) in the phenomenological 
model.  


\section{Discussion}
\label{discuss_sec}
We have calculated the instanton contribution to the Laplace sum-rules of scalar gluonium
and demonstrated explicitly how, for the lowest order $k=-1$ sum-rule, this instanton
contribution cancels part of the dominant LET constant.

As noted in the Introduction, a discrepancy between the lowest lying states evident from the
lowest and from the next-to-lowest Laplace sum-rules may be indicative of two distinct states. 
Such is found to be the case, for example, in the pseudoscalar channel in which the pion
dominates the leading ($k = 0$) Laplace sum rule, but the  $\Pi(1300)$ resonance is found to dominate
the next-to-leading ($k = 1$) Laplace sum rule, because of a mass-suppression 
of the lowest-lying (pion) state
in the latter sum rule
 \cite{pi_1300}. Moreover, analyses of the scalar gluonium
channel in the absence of explicit instanton contributions seem to exhibit a similar discrepancy
between leading $k=-1$ and non-leading $k>-1$ sum-rules \cite{dispersion,NSVZ_glue,glue}.

Prior QCD sum-rule analyses of the instanton contribution to the scalar gluonium channel have
focused either on the $k = -1$ sum rule exclusively \cite{shuryak} or the LET-insensitive 
$k>-1$ sum-rules \cite{forkel}.  Although these two analyses (which are separated by almost two decades)
are both indicative of a lowest lying-resonance mass near or above $1.4\unt{GeV}$, their input content
(parameter values and levels of perturbation theory) are necessarily different, suggesting the need
for a single consistent treatment of leading $k=-1$ and non-leading $k>-1$ Laplace sum-rules in the scalar
gluonium channel that is inclusive of instanton effects. 
We have shown here that careful consideration of the contribution arising from instantons within
the $k = -1$ sum rule in this channel leads to consistency with higher sum-rules in the estimation of
lowest-lying resonance masses in the scalar gluonium channel.\footnote{Note also that we have incorporated the
significant NNLO perturbative corrections, as opposed to  LO corrections in \protect\cite{shuryak} and NLO corrections in
\protect\cite{forkel}.}   


Correspondence with the prior treatment of the $k = -1$ sum rule \cite{shuryak} can be obtained by
examining the instanton contribution \bref{sm1_inst} to this sum rule in the high-energy limit of small $\tau$. 
This contribution is obtained through evaluation of the integral in \bref{sm1_inst}
in the large-$a$ [$\tau  = \rho^2/(2a)$] limit:
\begin{eqnarray}
{\cal L}_{-1}^{inst}(\tau) &=&
-128\pi^2\int dn(\rho)
-32\pi^3\int dn(\rho) \int\limits_0^\infty x^3 J_2\left(x\right)
Y_2\left(x\right)e^{-\frac{x^2}{2a}} \, dx
\nonumber\\
& =& -64\pi^2\int dn(\rho) 
\left[
\frac{\rho^4}{4\tau^2}\left[1+\frac{\rho^2}{2\tau}\right]K_0\left(\frac{\rho^2}{2\tau}\right)+
\frac{\rho^2}{2\tau}\left[2+\frac{\rho^2}{\tau}+\frac{\rho^4}{4\tau^2}\right] K_1\left(\frac{\rho^2}{2\tau}\right)\right]
e^{-\frac{\rho^2}{2\tau}}
\label{full_inst}\\
&\rightarrow & -16\pi^{\frac{5}{2}}\int dn(\rho)  e^{-\frac{\rho^2}{\tau}}\rho^5\tau^{-\frac{5}{2}}
\left[1+{\cal O}\left(\frac{\tau}{\rho^2}\right)\right]~,~\tau\ll\rho^2~.
\label{shuryak_limit}
\end{eqnarray}
In the instanton liquid model [$\,dn(\rho ) = n_c \delta\left(\rho -  \rho_c\right)d\rho\,$],  
\bref{shuryak_limit} is consistent with eq.\ (42) of reference \cite{shuryak} for the instanton contribution 
to the $k = -1$ sum-rule, which was
utilized to anticipate the ${\cal}(1.6 \unt{GeV})$ lattice prediction of the scalar gluonium mass.  
Figure \ref{shuryak_compare_fig}
provides a comparison of this asymptotic form with the exact expression \bref{full_inst} 
for ${\cal L}_{-1}^{inst}(\tau)$              
under  the instanton-liquid assumption. Of
particular interest is the difference between the two expressions over the range $2 \lesssim  a \lesssim  3$,
corresponding to the $0.4\unt{GeV^{-2}}\lesssim\tau\lesssim 0.6\unt{GeV^{-2}}$ range in Figure \ref{ratios_on_fig} 
for which $\sqrt{{\cal L}_0/[\mathcal{L}_{-1}+\Pi(0)]}$ is flat.  Although both
the asymptotic expression of \cite{shuryak} and the exact expression provide negative
contributions which mitigate dominance of the positive LET contribution over ${\cal L}_{-1}$, the 
exact expression is substantially larger in magnitude over the region of phenomenological
interest. We speculate that such an increase relative to the analysis of \cite{shuryak} serves to 
compensate for the larger phenomenological value at present for the gluon condensate within the
low-energy theorem term \bref{PI_approx}, although it may also compensate the non-leading
perturbative contributions in \bref{pi_pert} not known at the time of \cite{shuryak}.

Of course, more sophisticated expressions for the instanton contributions have been utilized to generate
consistent scalar gluonium phenomenology both within \cite{forkel} and complementary to 
\cite{multi_instanton} a QCD sum rule framework, as noted above. The key point of the work presented
here is the reconciliation of the $k = -1$ sum rule with higher-$k$ sum-rules. We reiterate the approximate
consistency between scalar-gluonium masses obtained from the flattened regions of the 
$\sqrt{{\cal L}_0/[\mathcal{L}_{-1}+\Pi(0)]}$
and $\sqrt{{\cal L}_1/{\cal L}_0}$ curves of Figure \ref{ratios_on_fig}, 
as well as the drastic reduction of the scalar gluonium mass
evident in the curves of Figure \ref{ratios_off_fig} when instanton contributions are omitted.  
Roughly speaking, such
contributions account for half the lowest-lying scalar-gluonium mass within a sum-rule context.

\smallskip
\noindent
{\bf Acknowledgements:}  The authors  are grateful for research support
from the Natural Sciences and Engineering Research Council of Canada (NSERC).
We are also grateful to  N. Kochelev for helpful correspondence.

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\clearpage

\begin{figure}[htb]
\centering
\includegraphics[scale=0.5,angle=270]{glue_let_contour.eps}
\caption{Closed contour $C(R)$ used to 
obtain the inverse Laplace transform (\protect\ref{inv_lap}) defining the Laplace sum-rules. 
The  inner circular segment $\Gamma_\epsilon$ has a radius of $\epsilon$, and the outer circular segments 
$\Gamma_2$ and $\Gamma_3$
have a radius $R$.
The wavy line 
on the negative real axis denotes the branch cut of $\Pi(z)$, 
and the linear segments of the contour above and below the branch cut are denoted by $\Gamma_c$.
}
\label{cont_fig1}
\end{figure}
   

\clearpage

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.7]{mitigate.eps}
  \caption{\label{mitigation} 
The quantity $\left[\Pi(0)+\mathcal{L}^{inst}_{-1}(\tau)\right]/\Pi(0)$ is plotted as a function of $\tau$,
           illustrating that  instanton effects
             cancel a significant portion of the LET term in the $k=-1$ sum-rule.   }
\end{figure}


\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{sm1.eps}
\caption{
Comparison of the full field-theoretical content of ${\cal L}_{-1}(\tau)+\Pi(0)$ with  (dashed curve) and without
(solid curve) instanton effects.
}
\label{sm1_fig}
\end{figure}
   


\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{s0.eps}
\caption{
Comparison of ${\cal L}_{0}(\tau)$ with instanton effects included (dashed curve) and instanton effects excluded
(solid curve).
}
\label{s0_fig}
\end{figure}


\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{s1.eps}
\caption{
Comparison of ${\cal L}_{1}(\tau)$ with instanton effects included (dashed curve) and instanton effects excluded
(solid curve).
}
\label{s1_fig}
\end{figure}
 

\clearpage



\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{s0sm1.eps}
\caption{
Comparison of the ratio $\sqrt{{\cal L}_{0}(\tau)/\left[{\cal L}_{-1}(\tau)+\Pi(0)\right]}$ with instanton effects 
included (dashed curve) and 
instanton effects excluded (solid curve).
}
\label{s0sm1_fig}
\end{figure}



\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{s1s0.eps}
\caption{
Comparison of the ratio $\sqrt{{\cal L}_{1}(\tau)/{\cal L}_{0}(\tau)}$ with instanton effects included (dashed curve) and 
instanton effects excluded (solid curve).
}
\label{s1s0_fig}
\end{figure}
 

\clearpage

 

\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{ratios_on.eps}
\caption{
Sum-rule ratios used to obtain scalar gluonium mass bounds with inclusion of instanton effects.
The solid curve represents the ratio  $\sqrt{{\cal L}_{0}(\tau)/\left[{\cal L}_{-1}(\tau)+\Pi(0)\right]}$ 
and the dashed curve represents the ratio  $\sqrt{{\cal L}_{1}(\tau)/{\cal L}_{0}(\tau)}$. 
}
\label{ratios_on_fig}
\end{figure}


\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{ratios_off.eps}
\caption{
Sum-rule ratios used to obtain scalar gluonium mass bounds with omission of instanton effects.
The solid curve represents the ratio  $\sqrt{{\cal L}_{0}(\tau)/\left[{\cal L}_{-1}(\tau)+\Pi(0)\right]}$ 
and the dashed curve represents the ratio  $\sqrt{{\cal L}_{1}(\tau)/{\cal L}_{0}(\tau)}$. 
}
\label{ratios_off_fig}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{shur_comp_fig.eps}
\caption{
Instanton contributions to ${\cal L}_{-1}$ in the instanton liquid model as a function of $a=\rho_c^2/(2\tau)$.
The solid line represents the complete expression (\protect\ref{full_inst}) for the instanton contributions,
and the dashed line represents the lead term of the large $a$ expansion   (\protect\ref{shuryak_limit}) obtained in 
\protect\cite{shuryak}.
}
\label{shuryak_compare_fig}
\end{figure}


\end{document}






