\documentstyle[12pt]{article}
\newcommand{\reali}{\hbox{\rm I\hskip-2pt\bf R}}
\newcommand{\interi}{\hbox{\rm{Z\hskip-8.2pt Z}}}
\newcommand{\fo}{F\,(\omega, t)}
\newcommand{\fmo}{F\,(- \omega, t)}
\newcommand{\fos}{F^{*}\,(\omega, t)}
\newcommand{\fs}{F\,(s, t)}
\newcommand{\fz}{F\,(z, t)}
\newcommand{\fzi}{F\,(z_i, t)}
\newcommand{\fzz}{F\,(z_0, t)}
\newcommand{\fzs}{F^{*}\,(z, t)}
\newcommand{\fmz}{F\,(-z, t)}
\newcommand{\kf}{$ln \,F\,(s,t)$ }
\newcommand{\vo}{\varphi \, (\omega, t)}
\newcommand{\vop}{\varphi \, (\omega', t)}
\newcommand{\vz}{\varphi \, (0, t)}
\newcommand{\vzt}{\varphi \, (z, t)}
\newcommand{\vbz}{\varphi \, (\bar z, t)}
\newcommand{\vzi}{\varphi \, (z_i, t)}
\newcommand{\vzp}{\varphi \, (z', t)}
\newcommand{\vzs}{\varphi ^{*} \, (z, t)}
\newcommand{\om}{\omega}
\newcommand{\omt}{\tilde \omega}
\newcommand{\os}{\omega^{*}}
\newcommand{\omp}{\omega^{'}}
\newcommand{\zs}{z^{*}}
\newcommand{\zp}{z^{'}}
\newcommand{\zh}{\hat z}
\newcommand{\de}{\delta}
\newcommand{\eps}{\varepsilon}
\newcommand{\vf}{\varphi}

\newcommand{\intdp}{{\int_{0}^{2 \pi}}}
\newcommand{\intinf}{{\int_{-\infty}^{\infty}}}
\newcommand{\intoz}{{\int_{\omega_0}^{\infty}}}
\newcommand{\intzz}{{\int_{z_0}^{\infty}}}
\newcommand{\inttz}{{\int_{\tilde z}^{\infty}}}
\newcommand{\suminf}{{\sum_{n= -\infty}^{\infty}}}
\newcommand{\sumN}{{\sum_{n= -N}^{N}}}
\newcommand{\sumn}{{\sum_1^{n}}}
\newcommand{\sumni}{{\sum_{n+ 1}^{\infty}}}
\newcommand{\sumin}{{\sum_{1}^{\infty}}}
\newcommand{\sumzi}{{\sum_{n = 0}^{\infty}}}

\newcommand \norm {{1 \over 2\pi}}
\newcommand \norh {{1 \over \pi}}
\newcommand \norme {{\eps \over 2\pi}}
\newcommand \normi {{i \over 2\pi}}
\newcommand \normim {{-i \over 2\pi}}

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\ifx\toctitle\undefined\def\toctitle#1{\relax}\fi
\begin{document}
%\begin{center}
\title{Absolute upper bounds on $\pi^0\pi^0$ elastic scattering
  amplitude at arbitrary energies}
\tocauthor{M. N. Mnatsakanova and Yu. S. Vernov}
\author{M.N.Mnatsakanova\\
\small\it Institute for Nuclear Physics of Moscow State University,
Russia\\
\small and \\
Yu.S.Vernov \\
\small\it Institute for Nuclear Research, Russian Academy of Sciences,
Russia}

\date{}
\maketitle

%\end{center}
\begin{abstract}
Absolute upper bounds on the modulus of $\pi^0\pi^0$ elastic scattering
amplitude are obtained for arbitrary physical energies. These bounds are
the consequence of analyticity, unitarity and crossing symmetry only.
\end{abstract}

One of the remarkable achievements of axiomatic quantum field theory
is the derivation of absolute (numerical) bounds on
$\pi ^0\pi^0$ elastic scattering amplitude. These bounds
are rigorous consequences of general principles of quantum field
theory only. Precisely they are consequences of analyticity in $s$
and $t$ planes, crossing symmetry and unitarity.
The first bound of this
kind has been derived by Martin [1].  The improved bounds were
obtained in paper of \L ukash\'uk and Martin [2] and then in papers
[3-8]. Above mentioned bounds are bounds in nonphysical region,
where $\mid s\mid <4m_\pi ^2, \mid t\mid <4m_\pi ^2$ and $\mid u\mid <4m_\pi ^2$ or bounds on
some integrals over low energies intervals. As it has been shown by
A.\,Martin [1], such bounds, in fact, imply the restrictions on the
physical amplitudes, precisely, on the integrals of the scattering
amplitude over all physical region.

The distinguished role of $\pi^0\pi^0$ scattering follows from the
full symmetry of its elastic scattering amplitude $F \,(s,t,u)$ in $s,t$
and $u$-variables. According to the quark structure of hadrons
$\pi^0\pi^0$ total cross section $\sigma _{tot}^{\pi^0\pi^0}$ has
the same order as $\sigma _{tot}^{\pi N}$ and $\sigma _{tot}^{N N}$.

In our paper [9] absolute upper bounds on $\pi^0\pi^0$ total cross
section at arbitrary physical energies have been derived. The present
report is the continuation of this paper.
Here we show that absolute bounds  on $\mid  \fs
\mid $ in the region:  $\mid s\mid  < 4\, m_\pi ^2, \; 0 < t < 4\, m_\pi
^2$ lead to the absolute bounds on  $\mid  \fs \mid $ at physical $s$ and
the same $t$.

Precisely we obtain
bounds on $\mid  F (\bar s, t) \mid $, where  $\bar s $ is some point in the
interval $(s, \nu \, s), \; \nu > 1$ and in other respects is arbitrary.
The dependence of the obtained bounds on $\nu$ is rather weak, so we have
reasonable bound even in case $\nu - 1 \ll 1$  (almost local bound).

Bounds on $\mid \fs \mid $ at $s >  4\, m_\pi ^2$ and $t > 0$ implies absolute
bounds on  $\mid  \fs \mid $ at physical $s$ and  $t$. At $t = 0$ this bound has
the same form as  Froissart-Martin bound  [2, 10], but this
"absolute Froissart-Martin bound" has not unknown constant and so is valid
at finite energies.


To derive upper bound on $|\fs|$ at arbitrary physical $s$ and positive
$t$ inside Martin ellipse we use DR for the $ln F \,(s,t)$ (LDR). It is
convenient to change $s$ on $z\equiv \left (s- 2m^2 + {t/ 2}\right
)^2$. Function $\fz $ has only one right cut $(z_0, \infty), \;
z_0 = \left(2m^2 + {t/ 2}\right )^2 $ and DR has the simplest form:

$$
F \,(z, t)=F \,(0,t) + \frac{z}{\pi}\int\limits_{z_0}^\infty  \frac
{A \,(z',t)d\,z'}{z'(z' - z)}                                \eqno (1),
$$
where $A \,(z,t)\equiv Im F \,(z,t)$.

It is well known that according to the result of Jin and Martin [11]
DR require only two subtractions if $t > 0$ and belongs to Martin ellipse.
As $A \,(z,t) > 0$ at $0 \leq t < 4m^2$, then $\fz$ has at least one zero for
the same reason as at $t = 0$ (see e.g. [12]). As
$$ F \,(\zs,t) = \fzs  , $$
then this zero must be in a real point.

If $\fz$ has zero at point $\zh$:
$$
\fz = (z - \zh) \,\vzt                      \eqno (2),
$$
then LDR have to be written for $\vzt$:
$$
ln \left | \frac {\vzt}{\vf (z_i,t)} \right | = \frac {z - z_i}{\pi} P
\intzz \frac{\de \,(z',t) d\,z'}{(z' - z)\,(z' - z_i)}
\eqno (3),
$$
where $\de\,(\zp,t)$ is the phase of function $\vzp, \; z_i$ belongs to the
interval $(0, z_0)$. Let's point out that $ln \,\vf\,(\zs,t) = (ln \,
\vzt)^*$.  In order to show this it is sufficient to show that $\de\,(\zp
- i \eps, t) = - \de \,(\zp + i \eps, t)$. First, in accordance with
positiveness of $A \,(\zp, t) \quad \de \,(\zp , t)$ satisfies the following
double inequality:  

$$ 
0 \leq \de \,(\zp, t) \leq \pi                      \eqno (4) 
$$ 
From DR for $\vzt$ we have:  

$$ 
\vz = \frac 1 \pi \intzz \frac {Im \vzp d\,\zp} \zp >0 
$$ 
So $\vf(z_0, t) >0,\; \de \,(z_0, t) = 0$
and  as $ \vf(\zs, t) = (\vzt)^*$, the necessary result is evident.

If $\fz$ has no zeroes, then eq.(3) is valid for $ln \fz$ as well. Only if
$F \,(z_0, t) <0, \; \de \,(\zp, t)$ must be substituted by $\tilde \de
\,(\zp, t) \equiv \de \,(\zp, t) - \pi. $ Evidently

$$ -\pi\leq \tilde \de \,(\zp , t) \leq 0           \eqno (4')
$$
To obtain the desirable bound
in the same way both in case of existence and absence of zeroes of $\fz$,
let's consider function

$$
\psi_{\beta} (z, t) = \mid {z -z_0}\mid ^\beta \vzt;  \qquad \beta \in
\reali  \qquad z > 0      \eqno (5)
$$

It can be checked  that
$$
ln \frac {z - z_0} {z_0 - z_i} = (z_i - z) P \intzz \frac {d\,\zp}{(\zp -
z_i)(\zp - z)}   \qquad z_i \in (0, z_0)                  \eqno (6)
$$
Thus

$$
ln \left | \frac {\psi_{\beta} (z, t)}  {\psi_{\beta} (z_i, t)}   \right |
= (z - z_i) P \intzz \frac {d\,\zp \de_\beta (\zp)}{(\zp - z_i)(\zp - z)}
\eqno (7)
$$
where

$$ \de_ \beta (z) = \de \,(z) - \beta \pi $$ So $$ -\beta\pi
\leq  \de_ \beta (z) \leq \pi (1 - \beta)        \eqno (8)
$$
Below for
simplicity we consider the case $z \gg z_0$. As $t\leq 4m_\pi^2$, $z_0
\leq 16m_\pi^4$, then ${z/ z_0} >100$, if $\sqrt s >1 Gev$. So, actually
all energies higher then 1Gev satisfy our condition.

If $z\gg z_0$, then $\psi_1 (z, t) \cong \fz  $ (we can suppose that
$0 < \zh < 4 m_\pi^2$ as in this case obtained bound are found
to be absolute).

If $\fz$ has no zeroes and $\fzz > 0$, then LDR for $\fz$ coincide with
LDR for $\vzt $.

Now let's choose some interval $(z, \mu z), \: z>z_0, \: \mu >1$.

To use efficiently eq.(7) let's integrate both parts of this equation with
some positive normed weight function $\rho \,(z): \; \int _z^{\mu z} \rho
(\zp)d\,\zp = 1$. Using the low of the mean in the left part of the
equality and changing the order of integration in the right part of it, we
obtain:

$$
ln\, \left | \frac {\psi_\beta (\bar z,t)}{\psi_\beta  (z_i,t)} \right | =
\norh \intzz \frac {\de _\beta (z')}{z' - z_i} P \int_z^{\mu z} \frac
{(z'' - z_i)\,\rho \,(z'')\,d\,z''}{z' - z''}                     \eqno (9)
$$
Upper bound for ${\psi_\beta (\bar z,t)}/ {\psi_\beta  (z_i,t)}$
follows from condition (8) and depends on the choice of function $\rho \,
(z)$. We consider sufficiently wide class of functions  $\rho \, (z)$,
namely $\rho_\alpha \, (z) = C_\alpha \, z^{\alpha - 1},\; C_\alpha  =
{\alpha / {z^\alpha \, (\mu^\alpha  - 1)}}, {}\;{} \alpha \in \reali$.

We obtain explicit result at arbitrary $\alpha$. It gives the possibility
for any given $\mu$ to choose $\alpha$ in such a  way, that resulting
bound would be optimal.

Using condition $z \gg z_0$ we can rewrite eq.(9):

$$
ln\, \left | \frac {\psi_\beta \,(\bar z,t)}{\psi_\beta  \,(z_i,t)} \right
| = \frac {\alpha}{\pi (\mu^\alpha - 1)} \int_{u_0}^\infty \frac {\de
\,(u) d\,u}{u - u_i} \Phi_\alpha\,(u)                         \eqno (10)
$$

$$
\Phi_\alpha\,(u) =  \int_1^{\mu} \frac {y^\alpha d\,y}{y - u}
\qquad u_0 = {{z_0}\over z}; \qquad u_i = {{z_i}\over z}.
$$
It can be easily shown
that for any $\alpha$ there exists such $\tilde u _\alpha$,
that
$$
\Phi_\alpha\,(\tilde u _\alpha) =  0,
$$
$$
\Phi_\alpha\,(u) < 0,\qquad u < \tilde u _\alpha;
$$
$$
\Phi_\alpha\,(u) > 0  \qquad u > \tilde u _\alpha
$$
It is evident that $1 < \tilde u _\alpha < \mu$. So in
accordance with condition (8):

$$
ln\, \left | \frac {\psi_\beta \,(\bar z,t)}{\psi_\beta \,
(z_i,t)} \right | \leq (1 - \beta) \,I_\alpha;   \qquad  \beta \leq 0
\eqno (11)
$$

$$ ln\, \left | \frac {\psi_\beta \,(\bar z,t)}{\psi_\beta \,
(z_i,t)} \right | \leq \beta \, \tilde I_\alpha;   \qquad  \beta \geq 1
\eqno (12)
$$

$$
ln\, \left | \frac {\psi_\beta \,(\bar z,t)}{\psi_\beta \,
(z_i,t)} \right | \leq \beta \tilde I_\alpha + (1 - \beta)\,I_\alpha ;
\qquad  0 \leq \beta \leq 1
\eqno (13),
$$
where

$$
I_\alpha  = \frac {\alpha}{\mu^\alpha - 1}
\int_{\tilde u}^\infty \frac {d\,u}{u} \Phi_\alpha \,(u)     \eqno (14)
$$

$$
\tilde I_\alpha  = \frac {\alpha}{\mu^\alpha - 1}
\int_{u_0}^{\tilde u} \frac {d\,u}{u - u_i} \Phi_\alpha \,(u)     \eqno
(15) $$

$I_\alpha$ and $ \tilde I_\alpha $ can be easily obtained if $\alpha  \neq
0$.

Actually, changing the order of integrations we can show that
$$
I_\alpha = ln\, \tilde u_\alpha  -\frac \alpha {\mu^\alpha  - 1}
\int_1^\mu y^{\alpha - 1}ln\, |\tilde u_\alpha  -y| d\,y
\eqno (16)
$$
The last integral  can be calculated by using integration by parts
and taking into account that $\Phi_\alpha \,(\tilde u _\alpha) =  0$.

Final result is:
$$
I_\alpha = ln\, \frac {\tilde u_\alpha }{\mu - \tilde u_\alpha } + \frac 1
{\mu^\alpha - 1} ln\, \frac {\tilde u_\alpha - 1}{\mu - \tilde u_\alpha }
\eqno (17)
$$

By the same way it is easy to show that
$$
\tilde I_\alpha  = I_\alpha + ln\, \frac {z}{z_0 - z_i } +
\frac {\mu ^\alpha \,ln\, \mu}{\mu^\alpha - 1}  - \frac {1}{\alpha}
\eqno (18)
$$
$I_0$ can be estimated directly or as a $lim_{\alpha \to 0}I_\alpha $.

We make use of a second possibility as in this way we obtain also
approximate expression for $I_\alpha $ in the case $\mu^\alpha - 1 \ll 1$.

If $y^\alpha \cong 1 + \alpha \; ln\, y, \; 1 \leq y \leq \mu$, then

$$
{1\over \alpha} \, ln\, \frac {\tilde u_\alpha - 1}{\mu - \tilde u_\alpha}
\cong P \int_1^\mu \frac{ln\, y \,d\,y} {\tilde u_\alpha  - y}
\eqno (19)
$$
Calculating the last integral we obtain:

$$
\begin{array}{rcl} \displaystyle

{ln\, \mu} \;{ } \lim_{\alpha \to 0} \;{ } \frac 1 \alpha \;{ } ln\, \frac
{\tilde u_\alpha - 1}{\mu - \tilde u_\alpha} & \displaystyle = &
{\pi^2\over 3} - {1\over 2} \;(ln\,^2\mu + ln\,^2\tilde
u_\alpha ) + ln\, (\mu - \tilde u_\alpha ) - \\ & & \displaystyle -
Li_2\left({1\over \tilde u_\alpha }\right ) - Li_2\left({\frac {\tilde
u_\alpha }{\mu}}\right ),
\end{array}
\eqno (20)
$$
where
$$ Li_2 (x) \equiv \sumin   \frac {1}{n^2} \,x\,^n; \qquad 0 < x < 1
$$

In accordance with eqs. (15) and (18):
$$
I_0 = \frac {1}{ln\, \mu} \left [{\pi^2\over 3} - {1\over 2}
\; ln\,^2\frac {\mu} {\tilde u_0} -  Li_2\left({1\over \tilde u_0}\right )
- Li_2\left({\frac {\tilde u_0 }{\mu}}\right )\right ]  \eqno (21)
$$
It is easy to see that
$$
\tilde u_0 = \frac {1 +\mu} 2                           \eqno (22)
$$
Eq.(21) (with the evident substitution  $\tilde u_0  \to  \tilde u_\alpha
$ ) is also an approximate expression for $I_\alpha $ in case $\mu^\alpha
- 1 \ll 1.$

In accordance with eq.(16)
$$
\tilde I_0 = I_0 +  ln\,\frac {\sqrt \mu \,z} {z_0 -z_i}     \eqno (23)
$$
and

$$
\tilde I_\alpha = I_\alpha +  ln\,\frac {\sqrt \mu \,z} {z_0 -z_i} + \frac
{\alpha\, ln\,^2 \mu} {12} \qquad   \mbox{if} \qquad
\alpha\, ln\, \mu \ll 1                                   \eqno (24)
$$

Let's describe now how to calculate $ \tilde u_\alpha $.
$\tilde u_\alpha $ is determined by the condition
$\Phi_\alpha \,(\tilde u_\alpha) = $ (see eq. (10)). This condition can be
written as equality:

$$
\int_1^{\tilde u_\alpha } \frac {y^\alpha \,d\,y}{\tilde u_\alpha  - y} =
\int_{\tilde u_\alpha } ^{\mu} \frac {y^\alpha \,d\,y}{y - \tilde u_\alpha
} \eqno (25)
$$

Calculating integrals in both sides of eq. (21) we come to the equation
for $\tilde u_\alpha $:

$$
\begin{array}{l} \displaystyle
\tilde u_\alpha \:\sumzi \frac {(2\alpha + 1)} {(n - \alpha)(n + \alpha +
1)} = \\
\qquad {} \displaystyle =  \mu\,^\alpha \sumzi \frac 1 {(n - \alpha)}
\left (\frac {\tilde u_\alpha } {\mu} \right )^n - \sumzi \frac 1
{(\tilde u_\alpha) ^{n + 1} (n + \alpha + 1)}; \\
\qquad {} \displaystyle\alpha \not\in
\interi _\pm, \qquad \alpha \ne 0
\end{array}
\eqno (26)
$$
If $\alpha \in
\interi _{\stackrel {+} - },$ then it is more simply to make computation
directly.

After elementary calculation we obtain the equations:

$$
ln\, \frac {\tilde u_n }{\mu - \tilde u_n } = \sumn \frac 1 k \left
(\frac \mu {\tilde u_n } \right )^k + \sumni \frac 1 {k \, (\tilde u_n) ^k}
\eqno (27)
$$

$$
ln\, \frac 1 {\tilde u_{-n} - 1 } = \sumn \frac 1 k
\left (\tilde u_{-n } \right )^k + \sumni \frac 1 k \left (\frac{ \tilde
u_{ -n}} {\mu} \right )^k                            \eqno (28)
$$

If $\mu \gg 1$, then infinite sums in (26) and (27) are unessential. For
$\alpha = 0$ see eq.(20). It is easy to show that $\tilde u_{- \frac 1 2} =
\sqrt \mu, \; \tilde u_{- 1} = 2\mu / (\mu + 1) $.

As we derived the explicit formulae for $ I_\alpha $ and $ \tilde I_\alpha
$ and equation for $ \tilde u_\alpha$, we can obtain by numerical
calculations the optimal value of $\alpha \equiv \alpha _{opt}$ for any
given $\mu$.

In case $\alpha = -1$ we have very simple expression for $ \tilde I_\alpha
$:

$$
\tilde I_{-1}  = ln\, \frac {z}{z_0 - z_i } +
\ln \,\frac {2\, e\, \mu}{\mu - 1}
\eqno (18')
$$



Now let's proceed to numerical bounds.
We need to use some bounds at nonphysical region. To this end we use the
bounds  on $F \,(s, t, u )$ obtained by Lopez and Mennessier [7]:

$$
\begin{array}{ccc} \mid F \,(1,3,0)\mid &<& 3.3\\
\mid F \,(2,3,0)\mid &<& 14.5
\end{array}
\eqno (29)
$$
We put $m_\pi^2 = 1$, so $z_0 = \frac {49} 4$.
In our variables $(z, t)$ these results mean:
$$
\begin{array}{ccc}
\mid F \,({1 / 4}, 3)  \mid & < & 3.3 \\
\mid F \,({9 / 4}, 3)  \mid & < & 14.5
\end{array}
\eqno (29')
$$
So if $F (z, 3)$ has no zero, then

$$
\mid F (\bar z, 3)\mid < 0.275 \, z \, C\,(\alpha, \mu)      \eqno (30),
$$
where

$$
C\,(\alpha, \mu) = exp \,\left [ I_\alpha +  \frac {\mu ^\alpha \,ln\,
\mu}{\mu^\alpha - 1}  - \frac {1}{\alpha}  \right ]
$$
If $F \,(z, 3)$ has a zero, then it is easy to see that the weakest, e.g.
absolute, bound corresponds the point of zero, which satisfies the
following condition: $3.3 \,{(\zh - 1/4)}^{-1} = 14.5\,{(9/4 -
\zh)}^{-1}$.  So absolute bound for $\mid F \,(\bar z, 3)\mid $ is:

$$
\mid F \,(\bar z, 3)\mid  < 8.9 \, z \, C\,(\alpha, \mu)        \eqno (31)
$$
For $\alpha = -1 \; C\,(\alpha, \mu)  =  \frac {2\, e\, \mu}{\mu - 1}$
(see eq.(18$'$)).

The absolute bound (31) implies the analogous bound on $A \,(\bar z, 3)$.

Let's point out that obtained bound for $\mid F \,(\bar z, t)\mid $
depends very weakly on the behavior of the scattering amplitude at $\zp >
\mu z$.  In fact it can be improved only by the factor $ exp\,
(-I_\alpha).$

We can obtain for $A \,(\bar z, 3)$ bound stronger then for $\mid F
\, (\bar z, t)\mid $. To this end we consider the behavior of $\de \,(\zp,
t)$ which leads to $max \,A\,(\zh, t)$. In fact the behaviour of $\de
\,(\zp, t)$ corresponding $max \, |F \,(\bar z, t)|$ implies $A\,(\bar z, 
t) = 0$.  Evidently we have to change $\de \,(\zp, t)$  only in the 
interval $(z, \mu z)$. The necessary behaviour of $\de \,(\zp, t)$ can be 
easily obtained by standard variational methods.


Let's point out that upper bound on $A \,(\bar z, t)$ can be derived
directly from DR [9].

In conclusion let's see what absolute bound on $A \,(\bar z, 0)$ follows
from the analogous bound on $A \,(\bar z, t)$ (for details see [9]).
It is well known that at given $A \,(\bar z, t) \; max \,A \, (\bar z, 0) $
is realized if partial amplitudes $a_l\,(z)= 1$ till some $l = L$ and then
$a_l\,(z)= 0$.  Thus the sum over $a_l\,(z)$ at $t > 0$ and $t = 0$ can be
easily expressed in terms of $L$ and $L$ is determined by our absolute
bound.

The final result is:
$$
\sigma_{tot}^{max} \leq \frac {2\pi}{t} \, ln^2\, (B s)   \eqno (32),
$$
where $B$ is a known constant.

By the same way, using the standard methods, we can obtain the absolute
bounds on $\mid \fs\mid , t \leq 0$ and other quantities, for example  on
$a_l\,(s) $ at $l > L$.



\begin{thebibliography}{99}
\bibitem{1} A.\,Martin, High-energy physics and elementary particles
(IAEA, Vienna, 1965), p. 165.
\bibitem{2} L.\L ukash\'uk and  A.\,Martin,  Nuovo Cim.  {\bf  52A}
(1967) 122.
\bibitem{3} B.\,Bonnier and R.\,Vinh-Mau,  Phys. Rev.  {\bf 165}
(1968) 1923.
\bibitem{4} J.B.\,Healy, Phys. Rev.  {\bf D8}
(1973) 1904.
\bibitem{5} G.\,Auberson, L.\,Epele, G.\,Mahoux and F.R.A.\,Sima\~o,
Nucl.  Phys.  {\bf B73} (1974) 314;  Nucl. Phys.  {\bf B94} (1975)
311.
\bibitem{6} C.\,Lopez,  Nucl. Phys.  {\bf B88} (1975) 358;
Nuovo Cim. Lett.    {\bf 13} (1975) 69;  {\bf B96}
(1975) 515; C.\,Lopez and G.\,Mennessier,  Nucl. Phys.  {\bf B96}
(1975) 515.
\bibitem{7} C.\,Lopez and G.\,Mennessier,  Nucl. Phys.
{\bf B118} (1977).
\bibitem{8}
A.D.\,Gangal and S.M.\,Roy, Nuovo Cim.  {\bf  76A}
(1983) 52.
\bibitem{9}
M.N.\,Mnatsakanova and Yu.S.\,Vernov, "Absolute upper on $\pi^0\pi^0$
total cross sections at arbitrary energies " in: {\it Proc. of the XIth
Workshop on High Energy Physics and Quantum Field Theory},
Sankt-Petersburg, Russia, 1996, pp. 385-390.
\bibitem{10}
M.\,Froissart, Phys. Rev. {\bf 123} (1961) 1053;\\
A.\,Martin, Phys. Rev. {\bf 129} (1963) 1432.
\bibitem{11}
Y.S.\,Jin and A.\,Martin, Phys. Rev.  {\bf B135} (1964)
1375.
\bibitem{12}
Yu.S.\,Vernov,
Particles and  Nuclei {\bf 6} (1975) 601 (in Russian).

\end{thebibliography}
\end{document}

