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\def\vqhalf{{\textstyle{\vec{Q}\over 2}}}
\def\qhalf{{\textstyle{Q\over 2}}}
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\def\vqsix{{\textstyle{\vec{Q}\over 6}}}
\def\thalf{{\textstyle{3\over 2}}}
\def\fourth{{\textstyle{1\over 4}}}
\def\tfor{{\textstyle{3\over 4}}}
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\def\P{{\sf P}}
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\def\I{{\cal I}}
\def\Tr{{\rm Tr }}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
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\newcommand{\bra}[1]{\langle {#1} |}                        % < |
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\begin{document}

\title{Charge Form Factors of Quark-Model Pions }

\author{F. Coester\footnote{corresponding author}}
\email[]{coester@anl.gov}
\affiliation{Physics Division, Argonne National Laboratory
Argonne, IL 60439, USA}

\author{W.N. Polyzou}
\email[]{polyzou@uiowa.edu}
\affiliation{Department of Physics and Astronomy University of Iowa,
Iowa City, IA 52242 USA}



\begin{abstract}

Experimental data of the pion charge form factor are well represented
by Poincar\'e invariant constituent-quark phenomenology depending on
two parameters, a confinement scale and an effective
quark mass.  Pion states are represented by
eigenfunctions of mass and spin operators, and of the light-front
momenta.  An effective current density is generated by the dynamics
from a null-plane impulse current density. A simple shape of the wave
function depending only on the confinement scale is sufficient.
The range of quark-masses  and confinement scales consistent with both
low and high $Q^2$ data depends on the shape of the wave function.
\end {abstract}
\pacs{12.39.-x , relativistic quark model, pion form factor, null-plane 
kinematics}

\maketitle

The unitary Poincar\'e representations on the Hilbert space  of confined quark states may be
specified by mass and spin operators together with the choice of a
kinematic subgroup \cite{KP,FC}. The four-momentum operator is a function
of the mass operator $\M$ and  three kinematic quantities which depend
on the choice of a kinematic subgroup.\cite{FCR}
Single-hadron eigenstates of the mass
operator may be represented by functions, $\phi_n$, 
of constituent quark momenta, spin, flavor and
color variables. The mass operator can be defined by the spectral representation
\be
\M:= \sum_n \phi_n M_n\phi^\dagger_n\; ,
\ee
using empirical hadron masses  or by the conventional assumption that the operator 
$\M$  (or $\M^2$) is  expressed  as the sum of a kinetic term $\M_0$ (or $\M_0^2$),
and a confining potential. However,  mass operators  of confined quark need not 
involve this conventional dependence on constituent-quark masses and the formal structure
of the dynamics is simpler if it does not.\cite{FEYNMAN,CDR} The wave functions 
representing hadron states are not observable. The observable form factors are 
determined by both the wave functions and associated representations of the
Poincar\'e covariant, conserved,
current density operators $I^\mu(x)$,
\bea
U(\Lambda ,a) I^{\mu} (x) U^{\dagger} (\Lambda ,a) 
&= &I^{\nu} ( \Lambda x +a ) \Lambda_{\nu}{}^{\mu} \; ,\cr\cr
\imath [P_\mu, I^\mu(x)]&=&\partial_\mu  I^\mu(x)=0 \; .
\eea
Poincar\'e covariant conserved current operators 
may be  generated by the dynamics from any input that is covariant under the
kinematic subgroup.  
The form factors are invariant under unitary changes of the representations of 
both the state vectors and the current operators. Such transformations
 may preserve the kinematic subgroup \cite{KP,FC}.

Here we investigate the compatibility of simple shapes of the wave  function
with current operators generated by the dynamics from impulse currents.
Quark masses  of confined quark enter as scale parameters of the impulse 
currents. Their values differ with different forms of kinematics \cite{FCR}.
They are not masses of  potentially free constituent  particles.




The  null-plane 
 kinematic subgroup leaves invariant the
null-plane $n\cdot x=0$, with $n^2=0$. With a convenient
choice of the axes the components of $n$ are given by
$n=\{1,0,0,1\}$. With the null vector perpendicular to the momentum transfer, 
$n\cdot Q=0$, the invariant $Q^2=Q_\perp^2$ is kinematic.


It is an important consequence
that the form  factors do  not depend on the hadron mass.  
This feature is essential for the  quark phenomenology
of pion form factors.
 With point-form
kinematics \cite{ALLEN} form factors obtain as functions of
$\eta:=Q^2/4m_\pi^2$, which is large for moderate values of $Q^2$.
Thus, with simple wave functions, form factors are much too small for
a realistic representation \cite{BERTRAND}. 


Previously \cite{CCP} we demonstrated that the simple  quark phenomenology
sketched above   yielded charge form factors of the pion in
agreement with available data for both low $Q^2$ \cite{AMEND} and $Q^2
>1\rm{GeV}^2$. Recently new measurements \cite{VOLMER} provided more
precise data for $Q^2<2$ GeV$^2$ and new data at higher values of
$Q^2$ are expected. Assuming a simple shape of the wave function the model
depends on two scales, the confinement scale of the wave function and the quark
mass. 
The purpose of this paper is to investigate the implications of precise data
 for $Q^2>1$ GeV$^2$.
Recent  data  constrain acceptable values of the quark
mass and confinement scale to a narrow range of values, which depends on the
the shape of the wave function. 
For  $Q^2>2$ GeV$^2$ form factors consistent
with existing data decrease at different rates depending on the shape of the
 wave function.

There is no intent to
approximate features of quantum field theory in the construction of
such quark models. In particular the representations of
constituent-quark states are not meant to approximate Fock-space
amplitudes and/or satisfy features of perturbative QCD \cite{BRODSKY}.





\begin{figure}
\begin{center}
\rotatebox{270}{\resizebox{3.2in}{!}{
\includegraphics{pifig1.ps}}
}
\end{center}
\caption{Wave function dependence of pion form factors at 
low momentum transfer.
}
\end{figure}



As in ref. \cite{CCP} the models are specified by  
input currents
with the representation
\bea
&&\bra{\xi' , k_\perp', \mu',\bar \mu'} n\cdot I(0)
\ket{\xi,  k_\perp ,\mu ,\bar \mu}
= \cr\cr &&
 \delta_{\mu',\mu}  \delta_{\bar \mu',\bar \mu} \delta (\xi'-\xi)
 \delta( k_\perp'-  k_\perp- (1-\xi) Q_\perp) \; ,
\eea
and a representation of the pion state by wave functions
$\phi(\xi, k_\perp,\mu,\bar \mu)$ which is proportional to a
radial wave function $u(k^2)$ and Melosh rotation matrices,
\bea
&&\phi(\xi, k_\perp,\mu,\bar \mu):=
\sum_{\mu',\bar\mu'}
\bra{\mu} \R^\dagger(\xi,  k_\perp)\ket{\mu'}\cr\cr&\times&
\bra{\bar \mu} \R^\dagger(1-\xi, - k_\perp)\ket{\bar\mu'}
(\half,\half, \mu',\bar \mu'|0,0) u(k^2)\; .
\eea
The argument $k^2$ is related 
to the null-plane momenta by
\be
k^2+m_q^2={ k_\perp^2+m_q^2\over 4 \xi(1-\xi)}\; .
\ee
The  pion charge form factor is a functional of the radial wave function, 
$u(k^2)$,
\be
F_\pi(Q^2) = {1\over 16 \pi}\int_0^1 d\xi 
\int {d^2 \bar k_\perp\over \xi(1-\xi)} \W(\xi,\bar k_\perp)
u({k'}^2)u(k^2)
\ee 

with
\be
\bar k_\perp = k_\perp-\half (1-\xi)Q_\perp =k_\perp'+\half (1-\xi)Q_\perp
\ee
and
\be
{\W}:= \sqrt{{\xi(1-\xi)\over \sqrt{(m_q^2+k_\perp^2)(m_q^2+{k_\perp'}^2)}}}
{m_q^2+\bar k_\perp^2-\fourth(1-\xi)^2 Q^2\over \xi(1-\xi)}\; .
\ee

\begin{figure} 
\begin{center}

\rotatebox{270}{\resizebox{3.2in}{!}{
\includegraphics{pifig2.ps}}
}
\caption{Wave-function dependence of pion form factors at high
momentum transfer }
\end{center}
\end{figure}


			    \begin{figure}
\begin{center}
\rotatebox{270}{\resizebox{3in}{!}{
\includegraphics{pifig3.ps}}
}
\end{center}
\caption{Shapes of the three radial wave functions}
\end{figure}



\begin{figure}
\begin{center}
\rotatebox{270}{\resizebox{3.2in}{!}{
\includegraphics{pifig4.ps}}
}
\end{center}
\caption{Wave-function dependence of pion form factors at 
low momentum transfer.
}
\end{figure}







In \cite{CCP}  a Gaussian shape 
\be
u(k^2)= \sqrt{{4\over \sqrt{\pi}b^3}}\exp(-k^2/2b^2) 
\label{GAUSS}
\ee
was used for numerical convenience.  We expect that
a  rational  shape,  e.g. 
\be
u(k^2)= \sqrt{{32\over \pi b^3}}\left(1\over 1+k^2/b^2\right)^2
\label{FRAT}
\ee
may specify a  better model.  
For each shape of the wave function a precise fit to both
low and high $Q^2$ data tightly constrains acceptable values of the quark mass
$m_q$ and the confinement scale $b$.  A larger quark mass, requires
a different shape, illustrated by the function
\be
u(k^2)= \sqrt{{16\over \pi b^3}}\left(1\over 1+k^2/b^2\right)^\thalf\; .
\label{THALF}
\ee
The form factors  obtained with  the parameters 
listed in Table I are
shown in Figs. 1 and 2 .  

\begin{figure} 
\begin{center}

\rotatebox{270}{\resizebox{3.2in}{!}{
\includegraphics{pifig5.ps}}
}
\caption{Wave function dependence of pion form factors at high
momentum transfer }
\end{center}
\end{figure}


			    \begin{figure}
\begin{center}
\rotatebox{270}{\resizebox{3in}{!}{
\includegraphics{pifig6.ps}}
}
\end{center}
\caption{Shapes of the two radial wave functions}
\end{figure}

\begin{table}[b]
\caption{Dependence of quark masses, confinement scales  
and quark axial couplings on the shape
of the wave functions}
\begin{ruledtabular}
\begin{tabular}{lccc}
wave function&   $m_q$ (MeV)  &   $b$ (MeV) & $g_A^q $\\
\hline
eq. (\ref{GAUSS} )&  200  & 360 &  1.07\\
eq. (\ref{FRAT} ) &     230  & 430 &   .91\\
eq.  (\ref{THALF})  &   300   & 275 &  .76\\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}[b]
\caption{Dependence of quark masses, confinement scales  and quark
 axial couplings on the 
shape
of the wave functions}
\begin{ruledtabular}
\begin{tabular}{lccccc}
 $m_q$ (MeV)  &   $\alpha$ GeV$^2$ &$\beta$& $\gamma$ GeV &$\lambda 
$ GeV$^2$& $g_A^q $\\
\hline
≈ß220& .10  & .40&  .167& .299& .98\\
360 &  .03  & .155&  2.94& 1.48&    .68\\
\end{tabular}
\end{ruledtabular}
\end{table}









The  pion decay constant $f_\pi$ is related to the quark axial coupling $g_A^q$
by \cite{CALDAR,JAUS}
\bea
f_\pi&=& {\sqrt{3} g_A^q\over 8\pi^2}\int_0^1 d\xi 
\int {d^2 k_\perp\over \xi(1-\xi)}
{m_q\over \sqrt{M_0}} u(k^2)\; , \cr\cr
M_0^2&:=&{m_q^2+k_\perp^2\over \xi(1-\xi)}.
\eea
The empirical value $f_\pi=92.4$ MeV \cite{HOLSTEIN} implies
the values of $g_A^q$ listed in Table I.

The shapes of the three wave functions are compared
in Fig.~3.  All three parameterizations are in agreement with existing
data.  For larger values of $Q^2$ the form factors decrease at different rates
depending on the quark mass and the shape of the wave function.  For 
$Q^2>2$ GeV$^2$
the  form factors  obtained with eq. (10) are in agreement
with the QCD approximations of Maris and Tandy \cite{MARIS}.


Conventional QCD motivated mass
 operators \cite{CKP,ISGUR},
\be
\left( 2 \sqrt{m_q^2 + k^2} + \alpha r - {\beta \over r} +
\gamma\, \vec{s}_q \cdot \vec{s}_{\bar q}\,  e^{-\lambda^2 r^2}-m_0\right) u(r)=0
\label{MASS}
\ee
designed to fit meson spectra,  produce wave functions that require substantial
modification of the input  current \cite{CALDAR}. The application of unitary
``scattering equivalences'' \cite{EVAN}, which preserve the quark masses,
produce wave functions compatible with impulse currents.  These transformed wave
functions  may be realized adjusting the parameters in the potential.\cite {POLY, EVAN}
In Figs. 4-6  this
is illustrated for $m_q=220$ MeV \cite {ISGUR} and $m_q=360$ MeV \cite {CKP}.
The associated potential parameters  and quark axial couplings are listed 
in Table II.


 

Minimal phenomenological models of confined quark  dynamics, illustrated by 
eqs. (\ref{GAUSS}) -(\ref{THALF}), with null-plane impulse 
currents and quark masses between 200 and 300 MeV
 can easily accommodate existing experimental values  of pion form factors,  
as well  QCD based predictions for $Q^2$ in the range of several GeV$^2$ \cite{MARIS}.  Sufficiently precise data for larger values of 
$Q^2$ will  limit acceptable shapes of the wave functions and associated mass scales of 
the impulse currents.


\begin{acknowledgments}
Research supported in part by 
the U.S. Department 
of Energy, Office of Nuclear Physics  contracts  W-31-109-ENG-38 and 
DE-FG02-86ER40286 .
\end{acknowledgments}


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\end{document}










