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\title{SU(3) and Nonet Breaking Effects in $K_L \to \gamma \gamma$ \\
Induced by $s \to d + 2\mbox{gluon}$ due to Anomaly}
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\author{
X.-G. He$^{1}$, Chao-Shang Huang$^2$, and Xue-Qian Li$^{3,4}$}


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\address{
$^1$Department of Physics, National Taiwan University, Taipei\\
$^2$Institute of Theoretical Physics , Academia
Sinica, Beijing\\
$^3$ CCAST (World Laboratory), P.O. Box 8730, Beijing\\
$^4$ Department of Physics, Nankai University, Tianjing, 300071}
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\maketitle
\begin{abstract}
In this paper we study the effects of $s\to d +  2\mbox{gluon}$
on $K_L \to \gamma\gamma$ in the Standard Model.
We find that this interaction can induce 
new sizeable SU(3) and U(3) nonet breaking effects in 
$K_L - \eta, \eta'$ transitions and therefore in $K_L\to \gamma\gamma$ 
due to large matrix elements of  
$\langle \eta(\eta')| (\alpha_s G^a_{\mu\nu} \tilde G_a^{\mu\nu}|0 \rangle$
from QCD anomaly. 
These new effects play an important role in explaining 
the observed value. We also
study the effects of this interaction in long distance contribution to
$\Delta m_{K_L-K_S}$.
\end{abstract}
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\pacs{PACS numbers: 11.30.Hv, 13.25.Es, 14.40.AQ 
 }
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It is well known that long distance contributions 
play an important role in many low energy systems. 
Some of the notable examples are 
$K_L\rightarrow \gamma\gamma$ \cite{1,2} and 
$\Delta m_K = m_{K_L} - m_{K_S}$ \cite{3,4,5,6,7}.
For $K_L\rightarrow \gamma\gamma$, the short distance contribution alone 
accounts for only a small portion of 
the amplitude measured experimentally \cite{2,7}. 
For $\Delta m_K$, the short distance contribution
is again only a fraction of the experimental value depending on the 
value of the bag factor $B_K$\cite{3,4}.
A simple method to estimate the long distance contributions is 
the pole dominance approximation in which one assumes that 
a few low lying resonances saturate the long distance contribution. 
The commonly identified resonances in the above two cases are 
$\pi^0$, $\eta$ and $\eta'$. Combining with U(3) flavor symmetry, the
$K_L \to \gamma \gamma$ amplitude 
can be estimated\cite{2,6}.  If  U(3) nonet is a good symmetry, 
the calculations are straightforward. 
However, not only nonet but also SU(3) are known to be 
broken, there are large uncertainties in these calculations. 
In this paper we show that in the Standard Model (SM), short distance 
$s \to d + 2 \mbox{gluon}$ interaction, 
can new induce sizeable SU(3) and U(3) breaking 
effects in $K_L \to \gamma \gamma$ and $\Delta m_K$ through the 
large matrix elements $<\eta(\eta')|\alpha_s G^a_{\mu\nu}
\tilde G_a^{\mu\nu}|0>$ ($\tilde G^{\mu\nu}_a = (1/2) 
\epsilon^{\mu\nu\alpha\beta}G^a_{\alpha \beta}$) produced by 
QCD anomaly.

The short distance contribution to $K_L \to \gamma\gamma$ in the SM has been
studied before\cite{1,2,7}. 
Here we will improve the calculations by including QCD 
corrections which also serve to set up our notations.
In the SM $ s\to d \gamma \gamma$ can be generated at 
one loop level by exchanging a $W$ boson and quarks with two photons emitted 
from 
particles in the loop and particles in the external legs. 
The QCD corrected effective 
Hamiltonian for $s\to d \gamma\gamma$ is given by

\begin{eqnarray}
H_{eff}(s\to d \gamma\gamma) = M_{IR}^{\gamma\gamma} + M_R^{\gamma\gamma},
\end{eqnarray}
where $M_{IR}^{\gamma\gamma}$ is the 
irreducible contribution with the two photons
emitted from particles in the loop. $M_R^{\gamma\gamma}$ is the reducible
contribution with at least one photon emitted from 
an external $s$ or $d$ quark.

The irreducible contribution
$M_{IR}^{\gamma \gamma}$ is given by\cite{2,7,8} 

\begin{eqnarray}
M_{IR}^{\gamma\gamma}&=&-i {16\sqrt{2} \alpha_{em} G_F \over 9\pi} N a_2
\epsilon^{*\mu}(k_2) {1\over 2 k_1\cdot k_2}\nonumber\\
&\times& \sum_{i = u,c,t} 
V_{id}^*V_{is} F(x,x_i)
\bar d \gamma^\rho L R_{\mu\nu \rho} s \epsilon^{*\mu} (k_1),
\end{eqnarray}
Here $\epsilon^\mu(k)$ is the photon 
polarization vector with momentum $k$, 
$L(R) = (1\pm \gamma_5)/2$, $N = 3$ is the number of colors,
$a_2 = c_1 + c_2/N$, 
$x= 2k_1\cdot k_2/m^2_W$, $x_i = m^2_i/m_W^2$, and 
$R_{\mu\nu\rho} = k_{1\nu} \epsilon_{\mu \rho \sigma\lambda} k^\sigma_1
k^\lambda_2 - k_{2\mu} \epsilon_{\nu \rho \sigma \lambda} k_1^\sigma 
k_2^\lambda + k_1\cdot k_2 \epsilon_{\mu\nu\rho\sigma} (k_2-k_1)^\sigma$.
The function $F(x,x_i)$ is given by
\begin{eqnarray}
F(x,x_i) = {x_i\over x} \int^1_0 {\ln [1-y(1-y) x/x_i]\over y} dy.
\end{eqnarray}

The reducible contribution $M_R^{\gamma \gamma}$ is given by
\begin{eqnarray}
M_R^{\gamma\gamma}&=&{\sqrt{2} \alpha_{em}\over 6 \pi}
 \sum_{i = u,c,t} V_{id}^*V_{is} c_{12}^i 
\nonumber\\
&\times& \bar d [({1\over p_d\cdot k_1} - {1\over p_s\cdot k_2})
\sigma_{\mu\beta} \sigma_{\nu\alpha} k^\beta_1 k_2^\alpha 
+ 2i ({p_{d\mu}\over p_d\cdot k_1} - {p_{s\mu} \over p_s\cdot k_1})
\sigma_{\nu \beta} k^\beta_2\nonumber\\
& +& ( k_1 \to k_2, k_2 \to k_1; 
\mu \to \nu, \nu\to \mu) ](m_d L + m_s R) s 
\epsilon^{*\mu}(k_1) \epsilon^{*\nu}(k_2).
\end{eqnarray}

In the above $c_i$ are the Wilson coefficients
defined in the following $\Delta S=-1$ effective Hamiltonian\cite{9}

\begin{eqnarray}
&&H_{eff}(\Delta S =-1) = {4G_F\over \sqrt{2}} 
[V_{qd}^*V_{qs}(c_1 O_1 +c_2 O_2) 
- \sum_{i = u,c,t} V_{id}^* V_{is} (c_{11}^i O_{11} + c_{12}^i O_{12})],
\nonumber\\
&&O_1 = \bar q \gamma_\mu L q \bar d \gamma^\mu L s,\;\;
O_2 = \bar d \gamma_\mu L q \bar q \gamma^\mu L s,\nonumber\\
&&
O_{11} = {g_s\over 16 \pi^2} \bar d \sigma_{\mu\nu} T^a G_a^{\mu\nu}
(m_d L + m_s R)s,\nonumber\\
&&O_{12} = {e \over 16 \pi^2} \bar d \sigma_{\mu\nu} F^{\mu\nu}
(m_d L + m_s R)s,
\end{eqnarray}
where $G^{\mu\nu}_a$ and $F^{\mu\nu}$ are the gluon and photon field
strengths. Here we have also written down the operator $O_{11}$ which
is needed for the study of $s \to d gg$.


To obtain the amplitude for $K_L \to \gamma \gamma$ from the effective 
Hamiltonian $H_{eff}(s\to d \gamma\gamma)$, one needs to bound the $d$ and $s$
quarks to form a kaon. Using
$\langle 0| \bar d \gamma^\mu \gamma_5 s | \bar K^0\rangle = - i f_K P_K^\mu$,
one obtains

\begin{eqnarray}
A^{short}(\bar K^0 \to \gamma \gamma) 
&&= {2\sqrt{2} \alpha_{em} G_F\over 9 \pi}f_K [
i(N a_2 V_{ud}^*V_{us} + 3\xi c_{12}^t V_{td}^*V_{ts}) 
F_{\mu\nu} \tilde F^{\mu\nu}\nonumber\\
&&+ 3\xi c_{12}^t V_{td}^* V_{ts} F_{\mu\nu} F^{\mu\nu}],
\end{eqnarray}
where $\tilde F_{\mu\nu} = (1/2) \epsilon_{\mu\nu\alpha \beta} F^{\alpha\beta}$.
In obtaining the above result, we have used the
fact that $F(x,x_{c,t}) \approx -1/2$ (large $x_{c,t}/x$, and 
$F(x,x_u) \approx 0$ (small $x_u/x$). 
We also neglected small contributions from
$c^{u,c}_{12}$ which are proportional to $x_{u,c}$\cite{10}, but have kept 
$c_{12}^t$ which is  $-0.3$ in the SM.

The parameter $\xi$ is an average value of the quantity,
$\kappa = -(m^2_K/ 16) (1/ p_d\cdot k_1 - 1/ p_s\cdot k_2
+ 1/ p_d\cdot k_2 - 1/ p_s\cdot k_2)$. If one assumes that the
$d$ and $s$ quarks share equally the kaon momentum, then $\xi = 1$\cite{2}.
We have also estimated $\xi$ by 
calculating the quantity
$<0 | \kappa \bar d (1+\gamma_5) s |\bar K^0> using perturbative QCD
method using appropriate distribution amplitude of quarks in the 
kaon$\cite{11}. This approach also obtains a value of order one for
$\xi$. One should be aware that 
the applicability of pQCD may not be a good one here. 
We will use $\xi$ to be one in our
later discussions. We find that contribution related to $\xi$ is not 
important as long as $\xi$ is of order one. Therefore the precise number 
is not important here.
 
To estimate the irreducible contribution, one needs to know the quantity
$a_2 = c_1 + c_2/N$. This parameter behaves similarly to 
the one in hadronic $B$ and $D$
decays. It is very sensitive
to the energy scale $\mu$ becasue an accidental cancellation between $c_1$ and $c_2/N$. 
It is not well determined. In both $D$ and 
$B$ decays, the parameter $a_2$ determined from data ($|a_2|\sim 0.2 \sim 0.5$) 
is very different from factorization 
value by inserting $c_{1,2}$ at relevant scale in the expression for 
$a_2$\cite{12}. One would expect similar thing happens in kaon decays
althoug the details may be different.
To take into account uncertainties in $a_2$, we will treat it
as a free parameter allowing a range of $0.1 \sim 0.5$. One can 
obtain information about $a_2$ from 
$K_L \to \gamma \gamma$ data.

For $\xi$ of order one, and $a_2$ in the range of $0.1 \sim 0.5$, we find that
the dominant short distance $\bar K^0 \to \gamma \gamma$ 
amplitude is from the irreducible contribution. We have

\begin{eqnarray}
&&A^{short}(K_L \to \gamma \gamma)= i\tilde A^{short}  {1\over 2}
F_{\mu\nu} \tilde F^{\mu\nu},\nonumber\\
&&\tilde A^{short} = 
{8\alpha_{em} G_F \over 9 \pi} f_K N a_2 Re(V_{ud}^*V_{us}). 
\end{eqnarray}

Using $V_{ud} = 0.9735$ and $V_{us} =0.2196$ and $f_K = 1.27 f_\pi$\cite{13}, 
we obtain, 

\begin{eqnarray}
\tilde A^{short} = 2.54\times 10^{-12} a_2 \mbox{MeV}^{-1}.
\end{eqnarray} 
For $a_2 = 0.2$, it is only about 15\% of the experimental value of
$3.5\times 10^{-12}$ MeV$^{-1}$\cite{13}. Without QCD corrections $a_2 = 1/3$,
the short distance contribution is about 24\% of the total amplitude.
Long distance contributions are needed to account for the total amplitude.

Several analyses have been carried out using pole model with
$\pi^0$, $\eta$ and $\eta'$ poles to calculate the long distance contribution. 
In this model, one has\cite{6}
%
\begin{eqnarray}
\tilde A^{long} &=& \tilde A(\pi^0\rightarrow \gamma\gamma)
{\langle \pi^0|H_W| K_L\rangle \over m_K^2-m_\pi^2} \nonumber\\
& \times&\left[ 1 + {m_K^2-m_\pi^2\over m_K^2 -m_\eta^2} 
{\tilde A(\eta\rightarrow \gamma\gamma)\over \tilde 
A(\pi^0\rightarrow \gamma\gamma)} 
\left( {1+\delta \over \sqrt{3}} \cos\theta + 
{2\sqrt{2}\over \sqrt{3}} \rho \sin\theta\right)
 \right. \nonumber\\
& +&\left . {m_K^2-m_\pi^2\over m_K^2-m_{\eta'}^2} 
{\tilde A(\eta'\rightarrow \gamma\gamma)\over 
\tilde A(\pi^0\rightarrow \gamma\gamma)} 
\left({1+\delta\over \sqrt{3}} \sin\theta - {2\sqrt{2}\over \sqrt{3}} \rho \cos\theta\right)
 \right],
\end{eqnarray}
where $\theta$ is the $\eta-\eta'$ mixing angle, 
$\delta$ is the SU(3) breaking parameter \cite{6}.
The parameter $\rho$ parameterizes U(3) nonet breaking effect
and is defined as
\begin{eqnarray}
\rho = -\sqrt{{3\over 8}}{<\eta_1|H_W|K^0>\over <\pi^0|H_W|K^0>}.
\end{eqnarray}
In the nonet limit $\rho = 1$.
Chiral Lagrangian analysis gives 
$\langle \pi^0|H_W| K_L\rangle = 1.4 \times 10^{-7} m_K^2$ \cite{6}.
Using experimental values for $\pi^0, \eta,\eta' \to \gamma\gamma$, this
long distance contribution can be estimated.

We now estimate the contribution from $s\to d + gg$ to $\bar K^0 - \eta,
\eta'$ through $gg \to \eta, \eta'$ and study how the simple 
pole model calculation will be affected. 
The effective Hamiltonian $M^{gg}_{IR,R}$ for $s\to d gg$, with
color singlet $\bar d s$ bi-spinor product, 
can be obtained by some simple replacements from $M^{\gamma\gamma}_{IR,R}$. 
To obtain $M_{IR, R}^{gg}$ one first replaces
the photon polarization vectors $\epsilon^{\mu}(k_1) \epsilon^\nu(k_2)$
by the gluon polarization vector $\epsilon^\mu_a(k_1) \epsilon^\nu_a(k_2)$
with the color index $a$ summed over. Then 
one replaces $\alpha_{em}$ by
$\alpha_s (9/4)/(2N)$ and $\alpha_{em} c_{12}^i$ by $\alpha_s c^i_{11}/(2N)$ 
for $M_{IR}^{gg}$ and $M_R^{gg}$, respectively\cite{7}. The factor
$1/(2N)$ comes from picking up the color singlet part. 

Similar to the procedure in obtaining the amplitude for 
$K_L \to \gamma \gamma$,  one can obtain the amplitude for $K_L \to gg$. 
We find that with $\xi$ of order one, $a_2$ in the range of $0.1\sim 0.5$ 
and $c_{11}^t \approx -0.15$ as given in the SM, the irreducible
contribution, again, dominates the amplitude. We have

\begin{eqnarray}
A(K_L \to gg) = 
{1\over 2 N} {2\alpha_s G_F \over \pi} 
f_K N a_2 Re(V_{ud}^*V_{us}) i {1\over 2}
G^a_{\mu\nu} \tilde G_a^{\mu\nu}.
\end{eqnarray}

The above interaction can induce large $K_L - \eta, \eta'$ transitions
and therefore long distance contribution to $K_L \to \gamma\gamma$,
because QCD can induce large matrix elements for 
$\langle \eta(\eta')|\alpha_s G^a_{\mu\nu}\tilde G^{\mu\nu}_a|0\rangle$.

QCD anomaly implies that the divergence of the singlet 
current, $a^1_\mu = \bar u \gamma_\mu \gamma_5 u + \bar d \gamma_\mu \gamma_5 d
+ \bar s \gamma_\mu \gamma_5 s$, is not zero in the limit of zero
quark masses, and is given by 

\begin{eqnarray}
\langle \eta(\eta')|\partial^\mu a^1_\mu|0\rangle &=& 
\langle \eta(\eta')|2i(m_u \bar u \gamma_5 u + m_d \bar d \gamma_5 d
+m_s \bar s \gamma_5 s)|0\rangle\nonumber\\
& -& 
\langle \eta(\eta')|{3\alpha_s\over 4 \pi} G^a_{\mu\nu} \tilde G^{\mu\nu}_a|0\rangle.
\end{eqnarray}
While for the octet current, 
$a^8_\mu = \bar u \gamma_\mu \gamma_5 u + \bar d \gamma_\mu \gamma_5 d
- 2\bar s \gamma_\mu \gamma_5 s$, one obtains\cite{14}

\begin{eqnarray}
&&\langle \eta(\eta')|\partial^\mu a^8_\mu|0\rangle  = \langle \eta(\eta')|
2i(m_u \bar u \gamma_5 u + m_d \bar d \gamma_5 d
-2m_s \bar s \gamma_5 s)|0\rangle .
\end{eqnarray} 
Since $m_{u,d}$ are much smaller than $m_s$, one can neglect
terms proportional to $m_{u,d}$. One then obtains

\begin{eqnarray}
&&\langle \eta'(p)|{3\alpha_s \over 4\pi} G_{\mu\nu}^a \tilde G^{\mu\nu}_a
|0\rangle
= \sqrt{3\over 2} ( \sqrt{2} f_1 \cos\theta + f_8 \sin \theta)p^2,\nonumber\\  
&&\langle \eta(p)|{3\alpha_s \over 4\pi} G_{\mu\nu}^a \tilde G^{\mu\nu}_a
|0\rangle
= \sqrt{3\over 2} ( -\sqrt{2} f_1 \sin\theta + f_8 \cos \theta)p^2,  
\end{eqnarray}
where $f_{1,8}$ are the singlet and octet pseudo-scalar decay constants.

If there is no $\eta-\eta'$ mixing and all quark masses are equal, 
the $gg$ state being a flavor singlet can only have transition to 
$\eta_1$. However because the $\eta -\eta'$ 
mixing and the different quark masses, both
U(3) nonet and SU(3) symmetries are broken. The $K_L \to \eta, \eta'$
transitions induced by $s\to d gg$ will induce nonet and SU(3)
breaking in the total amplitude $\tilde A^{total }$. Normalizing the
signs of each contributions to theoretical calculations, we finally obtain 

\begin{eqnarray}
\tilde A^{total} &=& -\tilde A^{short} + 
\tilde A(\pi^0\rightarrow \gamma\gamma)
{\langle \pi^0|H_W| K_L\rangle \over m_K^2-m_\pi^2} \nonumber\\
& \times& \left[ 1 + {m_K^2-m_\pi^2\over m_K^2 -m_\eta^2} 
{\tilde A(\eta\rightarrow \gamma\gamma)\over \tilde A(\pi^0\rightarrow \gamma\gamma)} 
\left( {1+\delta +\delta^{gg} \over \sqrt{3}} \cos\theta + 
{2\sqrt{2}\over \sqrt{3}} (\rho+r^{gg}) \sin\theta\right)
 \right. \nonumber\\
&+ &\left . {m_K^2-m_\pi^2\over m_K^2-m_{\eta'}^2} 
{\tilde A(\eta'\rightarrow \gamma\gamma)\over \tilde 
A(\pi^0\rightarrow \gamma\gamma)} 
\left({1+\delta+\delta^{gg} \over \sqrt{3}} \sin\theta - 
{2\sqrt{2}\over \sqrt{3}} (\rho+r^{gg}) \cos\theta\right)
 \right],
\end{eqnarray}
where $\delta^{gg}$ and $r^{gg}$ are the SU(3) and nonet breaking
induced by the $s\to d gg$ interaction. They are given by

\begin{eqnarray}
&&\delta^{gg} = \sqrt{2}
f_Kf_8 m^2_K {G_F Re(V_{ud}^*V_{us}) \over \langle \pi^0| H_W| K_L\rangle 
}a_2,\nonumber\\
&&r^{gg} = - {f_1\over 2 f_8} \delta^{gg}.
\end{eqnarray} 

We find

\begin{eqnarray}
\delta^{gg} = 0.96{f_8\over f_K} a_2, 
\;\;\;\;r^{gg} = -0.48 {f_1\over f_K} a_2.
\end{eqnarray}
We see that the corrections can be sizeable and can not be neglected.

We now provide some numerical analysis. There are several parameters
involved in the long distance contributions, the mixing angle $\theta$,
the decay constants $f_{1,8}$, the SU(3) and U(3) nonet breaking parameters
$\delta$ and $\rho$, and the parameter $a_2$. Chiral perturbation
calculations and fitting data not involving $K_L \to \gamma\gamma$
have obtained $\theta \approx -20^\circ$, $\delta \approx 0.17$,
$f_8 \approx 1.28 f_\pi$ and $f_1\approx 1.10f_\pi$\cite{15}. 
We will use these values for these
parameters in the calculation of $K_L \to \gamma\gamma$. 
There is not a reliable estimate for the parameter
$\rho$. Since we are interested to see how the new $s\to d gg$ interaction
induces U(3) nonet breaking effect, we will take $\rho=1$ and attribute
nonet breaking solely to $r^{gg}$. As have been 
discussed $s\to d gg$ also induce
SU(3) breaking effect. This effect was not included in other fittings. We
therefore should include this new SU(3) breaking effect also. 

Without the $s\to d gg$ effect, we find that the amplitude $\tilde A^{total}$
is equal to $5.5\times 10^{-12}$ MeV$^{-1}$ which is considerably larger than
the experimental value $3.5\times 10^{-12}$ MeV$^{-1}$\cite{13}. 
With the new effect,
we find 

\begin{eqnarray}
\tilde A^{total} = 5.5 (1-2.56 a_2)\times 10^{-12} \mbox{MeV}^{-1}.
\end{eqnarray}
To reproduce the experimental value, $a_2$ is required to be $0.14$
which is a reasonable value to have. 

The detailed numerical results depend on
several parameters. Even with other parameters fixed, one can introduce
also a phase to $a_2$. To fit the $K_L\to \gamma \gamma$ data, 
the values for the magnitude and phase of $a_2$ can vary. We, however, would
like to emphasize that the new effect discussed can play an important role
in $K_L \to \gamma \gamma$ independent of the details. 


The new contributions for $K_L - \eta (\eta')$ transitions also change the prediction 
for long distance contribution to the $K_L$ and $K_S$ mass difference 
parameter $\, {\rm Re} (M_{12})$ in the pole dominance approximation. 
We find\cite{6}
%
\begin{eqnarray}
2m_K \, {\rm Re} (M_{12}) &=& {|\langle \pi^0|H_W|K^0\rangle|^2\over m_K^2-m_\pi^2}
\nonumber\\
&\times&
\left[ 1+ {m_K^2-m_\pi^2\over m_K^2-m_\eta^2} \left( {1+\delta+\delta^{gg} 
\over \sqrt{3}} \mbox{cos}\theta 
+{2\sqrt{2}\over \sqrt{3}}(\rho+r^{gg}) \mbox{sin}\theta\right)^2 \right .\nonumber\\
&+& \left .
{m_K^2-m_\pi^2\over m_K^2-m_{\eta'}^2} \left( {1+\delta+\delta^{gg}\over \sqrt{3}} \mbox{sin}\theta - {2\sqrt{2}
\over \sqrt{3}} (\rho+r^{gg}) \mbox{cos}\theta \right)^2 \right].
\end{eqnarray}

Without the new effects, the above would leads
to $\Delta m_K =-0.5\times 10^{-12} $ MeV 
which is a non-negligible portion of the
experimental value of $3.5\times 10^{-12}$ MeV. With the new effects and
$a_2 = 0.14$ as determined from $K_L \to \gamma\gamma$, the contribution to 
$\Delta m_K$ is $-0.8 \times 10^{-12}$ MeV, 
and again it can not be neglected. The new long distance effect
in $K_L \to \pi^0,\;\eta,\;\eta'$ transitions can have sizeable
contribution to $\Delta m_K$.

The $s \rightarrow d gg$ process can also induce $K_L$-glueball mixing, 
which would also affect $K_L\rightarrow \gamma\gamma$ 
and $\Delta m_{S-L}$, as pointed out in Ref. \cite{7} where
a light glueball mass $1.4$ GeV was used.
Recent lattice calculations indicate that the pseudo-scalar
glueball mass is about $2.3$ GeV \cite{16}. 
With such a large mass the glueball-$\eta (\eta')$ mixing contribution 
should be small and therefore the effects are 
smaller than effects discussed earlier.

In conclusion we have evaluated additional contributions to 
$K_L\rightarrow \eta (\eta')$ transitions from $s\rightarrow  d gg$ 
in the Standard Model. 
These transitions induce sizeable SU(3) and U(3) 
breaking effects and have significant
effects on long distance contributions to $K_L\rightarrow \gamma\gamma$
and $\Delta m_K$.

\noindent
{\bf\large Acknowledgments}

The work of XGH
was supported in part by 
National Science Council under grants NSC
91-2112-M-002-42,
and in part by the Ministry of
Education Academic Excellence Project 89-N-FA01-1-4-3.
The work of CSH and XQL is 
supported in part by National Natural
Science Foundation. XGH would like to thank the hospitality of
Institute for Theoretical Physics in Beijing where part of this
work was carried out. He would also like to thank Hai-Yang Cheng
for useful discussions.


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