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%\begin{center}
\title{Asymptotics of Pion Form Factor in Relativistic Hamiltonian
Dynamics and Quark Counting Rules.}
%\vspace{5mm}
\tocauthor{A. F.~Krutov, V. E.~Troitsky}
\author{A.F.~Krutov\\
\small\em Samara State University, Samara, 443011, Russia\\
V.E.~Troitsky\\
\small\em D.V.~Skobeltsyn Institute of Nuclear Physics,
Moscow State University,\\
\small\em Moscow, 119899, Russia}
\date{}
\maketitle
%\end{center}

\begin{abstract}
The asymptotics of pion charge form factor is obtained in the
framework of relativistic Hamiltonian dynamics for
the infinite value limit of momentum transfer and zero value
limit of constituent--quark mass.
It is shown that this asymptotics is the same
as given by the quark counting rules and
is determined by relativistic kinematics only, being
independent on the constituent quarks interaction in
pion.
\end{abstract}

It is now well established that the description of the
electromagnetic structure of composite quark systems at low and
intermediate momentum transfers, i.e. in the region of so called
"soft" processes, needs nonperturbative approaches.
Moreover, in the investigation of
composite systems of light quarks (e.g., the pion) one
has to take into account relativistic effects, which give
essential contribution at low momentum transfers.

During last years the relativistic Hamiltonian dynamics
(RHD) is widely used in this nonperturbative region for
the investigation of composite quark systems (the foundation
of the approach, references to calculations and some
historical aspects can be found in the reviews
\cite{Pol89}-- \cite{Lev93}, see also
~\cite{BaK95} --~\cite{Kru97}). The RHD can be formulated in
different ways. The most widely used
forms are: the instant form dynamics, the point form dynamics
and the light--front dynamics. At present time the most popular is 
light--front
dynamics in the framework of which the many calculations of the hadronic
properties were performed. In a number of our works the instant form is used
successfully for the description of electroweak properties of the composite
systems ~\cite{BaK95} --~\cite{Kru97}. It is important to remind ourselves
that the RHD is not QCD but very successful phenomenological model.
The main feature of RHD
{\it versus} field theories is the extraction  of the
finite number of 
degrees of freedom in each
concrete case. The establishment of the relation between RHD and
fundamental principals of 
description of the quark composite systems is a
important and as yet unresolved problem.
This is especially true in regard to the principals of Quark Counting Rules
which establish the asymptotics at large transferred momenta for hadron form 
factors in fact by the model independent way 
(see ~\cite{Mat94} for the details):
\begin{equation}
F_a(Q^2)\quad\sim\quad \left(\frac{1}{Q^{2}}\right)^{(n_a-1)}\>,\quad
Q^2\to\infty 
\label{qcr}
\end{equation}
where $Q^2 = -\,t = -\,q^2$, $q$ -- transferred momentum, $n_a$ -- is a number
of the constituent quarks in basic configuration of hadron.

For the pion the asymptotics has the form:
\begin{equation}
F_\pi(Q^2)\quad\sim\quad Q^{-2}\>.
\label{as-qcd}
\end{equation}

Quark Counting Rules are based on the rather general physical assumptions:\\
-- composite quark structure of hadrons (hypothesis of quark confinement);\\
-- quasi--free behaviour of quarks bound inside hadrons (asymptotic freedom of
QCD);\\
-- revelation of point--like structure of hadrons in exclusive hard collisions
(approximate scale invariance);\\
-- decoupling of the effects of the small and large distances in hard 
   processes
(factorization hypothesis).\\
These assumptions lead to formulation of the general principle in the high
energy hadron physics so called "the principle of the automodel 
(or self--similar) behaviour".

In the present paper we show that the asymptotics
(\ref{as-qcd})
can be obtained in the frame of RHD instant form
\cite{BaK95} if $Q^2\to\infty\>, \>M\to$0 (where $M$ is the
constituent quark mass), and that the asymptotic behavior
has universal character: it does not depend on the choose
of concrete interaction model for constituent quarks and
is defined by the relativistic kinematics of two--quark
system only.

The limits defined above are connected with the following
meaning of the physical behavior at large momentum transfers. At
$Q^2\to\infty$ the quarks become free and
their mass becomes equal to the initial one (current mass). For
the pion with light quarks this means in fact that $M\to$ 0. If
a dimension parameter $b$ defines the characteristic hadron
scale (the confinement scale), then the limits have the
following meaning:  $Q^2\gg b$, $M\ll b$.

In our approach the electromagnetic pion form factor is
represented in the form of a functional, generated by the so
called free two--particle form factor
\cite{BaK95}.  This free form factor describes the
electromagnetic properties of the system of two free particles,
and in the case of pion it defines the electromagnetic current
matrix element for the system of two free quarks. If the free
quarks have the quantum numbers corresponding to pion, then the
matrix element has the form
\cite{BaK95}:
$$
<\vec P,\sqrt s,J,l,S,m_J |\,j_\mu\,| \vec
     P',\sqrt{s'} ,J',l',S',{m_J}'> =
$$
\begin{equation} =
A_\mu (s,Q^2,s')\> g_0 (s,Q^2,s')\>.
\label{param}
\end{equation}
Here $Q^2 = -(P - P')^2$,  $s = P^2$ -- system invariant mass
square,
$J = S = l =$0, $g_0 (s,Q^2,s')$ -- the free two--particle form
factor for the system of
particles without interaction. The vector $A_\mu (s,Q^2,s')$
is determined by the current transformation properties
(Lorentz--covariance and current conservation law)
\cite{BaK95}.  The explicit form of the function
$g_0(s,Q^2,s')$ in the case of point quarks was obtained in
\cite{KrT93}:
$$
g_0(s,Q^2,s')=
  \frac{(s+s'+Q^2)Q^2}{2\sqrt{(s-4M^2)
(s'-4M^2)}}\>
\frac{\theta(s,Q^2,s')}{{[\lambda(s,-Q^2,s')]}^{3/2}}\frac{1}
{\sqrt{1+Q^2/4M^2}}\times
$$
\begin{equation}
\times\left\{(s+s' +Q^2)
\cos{(\omega_1+\omega_2)}+
  \frac{1}{M}\xi(s,Q^2,s')
\>
\sin{(\omega_1+\omega_2)} \right\}\>.
\label{ff-nonint}
\end{equation}
Here $\lambda(a,b,c)=a^2+b^2+c^2-2(ab+ac+bc),
\xi=\sqrt{ss'Q^2-M^2\lambda(s,-Q^2,s')}$,\\
$\omega_1$ and $\omega_2$ - the parameters of Wigner rotation:
\begin{eqnarray}
\omega_1=\hbox{arctg}{\frac{\xi(s,Q^2,s')}{M[(\sqrt s+\sqrt
{s'})^2+Q^2]+\sqrt{ss'}(\sqrt s+\sqrt{s'})}},\nonumber\\
\omega_2=\hbox{arctg}{\frac{(2M+\sqrt s+\sqrt
{s'})\xi(s,Q^2,s')}{M(s+s'+Q^2)(2M+\sqrt s+\sqrt
{s'})+\sqrt{ss'}(4M^2+Q^2)}}\>.
\label{Wign-param}
\end{eqnarray}
 $\theta(s,Q^2,s')=
\vartheta(s'-s_1)-\vartheta(s'-s_2)$, $\vartheta$ - the step
function,
\begin{displaymath} s_{1,2}=2M^2+\frac{1}{2M^2}
(2M^2+Q^2)(s-2M^2)\mp \frac{1}{2M^2}
\sqrt{Q^2(Q^2+4M^2)s(s-4M^2).}
\end{displaymath}
In fact, the form factor $g_0(s,Q^2,s')$ in (\ref{param})
is a regular distribution (generalized function) and is a linear
continuous functional on a test functions space \cite{Vla76}.

To be used for pion form factor the functional given by
$g_0 (s,Q^2,s')$ is to be defined on the space  of functions
$\varphi(s,s') = \psi(s)\,\psi(s')$
using the following prescription \cite{BaK95}:
\begin{equation}
F_\pi (Q^2)=\int d\sqrt s\ d\sqrt{s'}\ \psi (s)\,g_0(s,Q^2,s')\,
\psi (s')
\label{ff}
\end{equation}
$\psi(s)\>,\>\psi(s')$ are functions normalized with the relativistic
density of states:
$\psi(s) = \sqrt[4]{s}\,u(k)\,k$;  $u(k)$ are the two--quark
wave functions of the relative motion $S$ -- state,
$s = 4(k^2+M^2)$.

So, to obtain the asymptotic behavior of the pion form factor
one has to evaluate the asymptotics at
$Q^2\to\infty\>,\>M\to$0
of the generalized function
$g_0(s,Q^2,s')$. This function defined on a certain class of
functions gives the pion form factor.

The constituent--quark mass $M$ and the momentum--transfer
square $Q^2$ are the parameters of generalized function
$g_0(s,Q^2,s')$. So the limits
$Q^2\to\infty$ and $M\to$ 0 are to be considered in the
generalized sense (in the sense of distributions) as the limits
of generalized functional.  The possibility of realizing the
calculation of this limit for the generalized function
$g_0(s,Q^2,s')$  can be justified physically by the fact that
the pion in the asymptotic region can be considered as a free
quark system with pion quantum numbers and the electromagnetic
properties of such system are defined just by the free
two--particle form factor $g_0(s,Q^2,s')$.

To obtain the asymptotic expansion of the generalized  function we
shall make the following assumptions.

1. Let the generalized  function be given on the
space $R^2$. During the calculation of
(\ref{ff}) we have not used such assumption, in
(\ref{ff}) the variables
$s\>,\>s'$ were in the physical region
$s\>,\>s'\ge\>4M^2$.

2.For the test function space let us take as usually the
space ${\cal S}(R^2)$ of infinitely differentiable functions,
decreasing at $|\vec s|\to \infty$ ($\vec s = (s,s')$) with all
its derivatives faster than any degree of
1/$|\vec s|$ \cite{Vla76}. This means that test
functions decrease not slowlier than
$\exp(-\,\beta\,|\vec s|^\alpha)\>,\alpha>$ 0.

3.Let us define the functional giving the regular generalized
function by
\begin{equation}
<\tilde g_0 (s,Q^2,s')\,,\varphi(s,s')> = \int\,d\mu(s,s')\,
\tilde g_0 (s,Q^2,s')\,\varphi(s,s')\>.\label{<>}
\end{equation}
Here
\begin{equation}
\tilde g_0 (s,Q^2,s') = 16\,\vartheta(s - 4M^2)\,
\vartheta(s' - 4M^2)\,g_0 (s,Q^2,s')\label{tilde g0}
\end{equation}
$$
d\mu(s,s') = \sqrt[4]{ss'}\,d\mu(s)\,d\mu(s')\>,\quad d\mu(s) = \frac{1}{4}
k\,d\sqrt{s}\>.
$$
$\vartheta(x)$ -- the standard step function. If
$\varphi(s,s') = u(k(s))\,u(k'(s'))$
then the functional (\ref{<>}) has the form
(\ref{ff}).

In fact the calculation of the
asymptotics of the generalized  function is just the calculation
of the asymptotics of the functional (\ref{<>}) when\\
$Q^2\to\infty\>,\quad M\to$0.

To estimate the asymptotics of the integral in
(\ref{<>}) let us realize the following steps:

1. Let us transform the integral in $s'$ to the integral
on the segment [0,1]. To perform this transformation we
use the fact that the free two--particle form factor
$g_0 (s,Q^2,s')$ contains the cutting functions
$$
  \vartheta(s' - s_1(s,Q^2)) - \vartheta(s' - s_2(s,Q^2))\>.
$$

2. Let us introduce two dimensionless parameters and let us
realize the limiting procedure keeping
$\eta = Q^2/b^2\to\infty$, $\xi = b/2M\to\infty$.

3. In the integral in $s$ let us introduce the new variable
$y=E/2M\>,\\s= 4M^2 + 2ME$, and let us separate explicitly
the dependence on the parameter $\eta$ introduced above.

4. Let us restrict ourselves by the main term of the
integrand in the functional (\ref{<>}) when
$\eta\to\infty$. This can be done because
the integrals do converge uniformly.

With help of the theorems from \cite{Olv90} we obtain 
the asymptotic expansion of (\ref{<>}) at $\eta\>,\>\xi\to\infty$.
Up to a preasymptotic factor this result has following form 
(see \cite{Kru97} for the details):
\begin{equation}
<\tilde g_0 (s,Q^2,s')\,,\varphi(s,s')>\quad \sim \quad
<\frac{1}{Q^2}\,g(s,s')\,,
\varphi(s,s')>\>.\label{fin fun}
\end{equation}
In the Eq.(\ref{fin fun}) the function $g(s,s')$  is given
by:
 $$
g(s,s') = M^4\,\frac{1}{\sqrt[4]{s(s - 4M^2)}}\frac{1}{(s -
2M^2)^{5/4}} \frac{\sqrt{s} + \sqrt{s - 4M^2}}{(\sqrt{s} -
\sqrt{s - 4M^2})^2}
\cdot $$ $$
\cdot\vartheta(s - 4M^2)\,\delta(\mu(s') - \mu(s))\>.\label{g(s,s')}
$$

Finally, we proved that in the sense of distributions
the following asymptotic estimate is valid:
\begin{equation}
\tilde g_0(s,Q^2,s')\>\sim\>\frac{1}{Q^2}\,g(s,s')\>,\quad
Q^2\>\to\infty\>, \quad M\>\to\>0\>.\label{g0-g} \end{equation}

Let us note that this result is also valid
in the case when the test function space
is larger than ${\cal S}(R^2)$. It is
necessary only
that the test functions ensure the uniform convergence of the
integrals entering the functional
(\ref{<>}).

Applying the result (\ref{g0-g})
to the Eq.(\ref{ff}), one can see that the asymptotics
does not depend on the choose of the functions
$u(k)$, that is on the concrete form of the quark interaction in
pion.

Thus, we obtained the pion form factor
asymptotics which
1) coincides with Quark Counting Rules asymptotics
(\ref{as-qcd}), 2) does not depend on the interaction model for
quarks in pion, 3) is defined by the free two--particle form
factor considered as generalized function. This free
two--particle form factor is obtained through relativistic
kinematics method, so the asymptotics is
due to the relativistic kinematics.

Let us compare our result with the results obtained by
other approaches. 
In the frame of light front RHD
\cite{Ter76} the asymptotic behavior of pion form
factor (\ref{as-qcd}) when
$Q^2\to\infty\>,\> M\to$0 was obtained in the case of special
power form restriction for the wave function asymptotics.
In papers \cite{ChC88}, \cite{Kei94} the asymptotics of the form
(\ref{as-qcd}) was found for the gaussian wave function.

There is a principal difference between our results and the
results of above mentioned papers. In those papers the results
are the cosequence of the structure of the argument of the wave
function for interacting quarks and of the special limitations
for wave function asymptotics. It seems difficult to interpret
this fact from the asymptotic freedom point of view which in
fact is the base of well known result (\ref{as-qcd}).
So the behavior of the two--quark system form factor
in the range $Q^2\to\infty$, where quarks are free, appears
to be a kind of "memory" about the quark interaction.

In our approach the result coinciding with (\ref{as-qcd})
follows directly from relativistic kinematics
which is valid for all energies.  Let us emphasize
that if we neglect, for example, the relativistic spin rotation
effect  (if we let the rotation parameters $\omega_{1,2}$ in
$g_0(s,Q^2,s')$ be zero) then the asymptotics of the generalized
function in (\ref{g0-g}) is immediately changed. This is in
contrast with the case of light front dynamics. It is easy to
show that the analog of the relativistic spin rotation in the
light front dynamics -- Melosh rotation \cite{ChC88} -- does
not affect the pion form factor asymptotics.

Let us give now the results for pion form factor asymptotics in
the cases of harmonic oscillator model
\cite{ChC88}:
\begin{equation}
u(k) = N_{HO}\exp(-k^2/2b^2)\>.\label{HO}
\end{equation}
and for power law wave function
\cite{Sch94b}:
\begin{equation}
u(k) =N_{PL}\,{(k^2/b^2 +
1)^{-n}}\>.\label{PL}
\end{equation}

The gaussian form of wave function makes the problem of the
calculation of asymptotics much simpler because in this case
one can use the theorems from \cite{Olv90}.

We use the model (\ref{HO}) to demonstrate the role of
relativism and, in particular, the role of the relativistic spin
rotation effect. The estimate for the pion form factor including
relativistic spin rotation effect gives
\begin{equation}
F_\pi(Q^2)\>\sim 32\sqrt{2}\frac{b^2}{Q^2}\>,\label{as s pov}
\end{equation}
while without relativistic spin rotation:
\begin{equation}
F_\pi(Q^2)\>\sim\> 4\sqrt{2}\frac{M}{Q}\>.  \label{as bez pov}
\end{equation}
The contribution of the relativistic spin rotation is
essential. If we neglect this contribution,
we do not obtain correct (i.e.coinciding with Quark Counting Rules) 
asymptotics.

To clear up the role of relativism in our calculation
let us write the nonrelativistic asymptotics obtained from
the asymptotics estimate of nonrelativistic limit of Eq.
(\ref{ff}) in the case of the model (\ref{HO}):
\begin{equation}
F_\pi(Q^2)\sim\>\exp\left(-\,\frac{Q^2}{16\,b^2}
\right)\>.
\label{as ner}
\end{equation}
The nonrelativistic asymptotics gives Gaussian decrease for the
form factor. The comparison of
(\ref{as s pov}) and (\ref{as bez pov}) with (\ref{as ner})
emphasizes the role of relativism. Let us note, that in
(\ref{as s pov})-- (\ref{as ner})
the explicit form of the normalization constant in
(\ref{HO}) is taken into account:
$N_{HO} = (4\,/\sqrt\pi\,b^3)^{1/2}$.

In the case of the model
(\ref{PL}) the form--factor asymptotics has the form:
\begin{equation}
F_\pi(Q^2) \>\sim\> 2\frac{b^2}{Q^2}\,\left(b^3\,N_{PL}^2\right)
\frac{\Gamma\left(\frac{5}{4}\right)}{n^{5/4}}\,
B\left(\frac{5}{4}\,,\,n - \frac{5}{4}\right)\>,\quad n\>>\>\frac{5}{4}
\label{n}
\end{equation}
Here $B(x\,,\,y)\>,\>\Gamma(x)$ are Euler beta-- and gamma--functions
respectively. Note, that  (\ref{n})
shows that the asymptotics (in the sense of distributions) form
(\ref{g0-g}) is really valid for the test function space larger
than ${\cal S}(R^2)$.

Let us note, that the asymptotics forms
(\ref{as s pov}) and (\ref{n}) do not contain the dimension
parameter of quark mass. This fact is connected with the
physics of asymptotically large momentum transfers when
all scales are determined by the confinement scale.
In the models (\ref{HO}) and (\ref{PL}) it is $b$ that plays
in fact the role of such parameter.

To conclude, let us formulate our main results.
1) In the frame of instant form relativistic
Hamiltonian dynamics the asymptotics  at $Q^2\to\infty$, $M\to$0
of the pion form factor coincides with that given by
Quark Counting Rules.
2) The asymptotics is determined only by the relativistic
kinematics, in particular, by the relativistic spin rotation
effect, and does not depend on the choose of quark wave
function in pion, i.e. on the model for quark interaction. \\[2mm]

The work is supported in part by the Russian Foundation for
Basic Researches (Grant No. 96 -- 02 -- 17288).

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\end{document}

