Electromagnetic Form Factors and the Hypercentral CQM
M. De Sanctis
New results about the electromagnetic form factors of the nucleon are obtained with a semirelativistic version of the hypercentral constituent quark model (hCQM) and a relativistic current. The complex structure of the constituent quarks is taken into account implicitly by means of phenomenological constituent quark form factors. We obtain a detailed reproduction of the experimental data up to $5~GeV^2$, moreover our findings about constituent quark root mean square radii are of the same order than the recent ones obtained analyzing the proton structure functions.
form_factors.tex
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\begin{document}

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\def\epsls{\varepsilon'\!\cdot\!l'}

\title{Electromagnetic Form Factors and the Hypercentral CQM}
\author{M. De Sanctis}  
\affiliation{INFN,Sezione di Roma1, Piazzale Aldo Moro, Roma (Italy) } 
\author{M.M. Giannini, E. Santopinto and A.Vassallo}
\affiliation{Universit\'a di Genova e INFN, Sezione di Genova, 
via Dodecaneso 33, 16142 Genova (Italy)}
%
\date{\today}
%
\begin{abstract}
New results about the electromagnetic form factors of the 
nucleon are obtained with a semirelativistic version of the hypercentral 
constituent quark model (hCQM) and a relativistic current. The complex 
structure of the constituent quarks is taken into account implicitly by means 
of phenomenological constituent quark form factors. We obtain 
a detailed reproduction of the experimental data up to $5~GeV^2$, moreover our 
findings about constituent quark root mean square radii are of the same order 
than the recent ones obtained analyzing the proton structure functions.
\end{abstract}
\pacs{13.40 Gp Electromagnetic form factors}
\maketitle
 
The new data on the ratio of the electric and magnetic form factors of the
proton \cite{Jones:1999,Gayou:2001,Gayou:2002}, showing an unexpected decrease 
with $Q^2$, have triggered again the interest in the description
of the internal nucleon structure in terms of various effective 
models: bag models, chiral soliton models, constituent
quark models, etc.$~$. 

Already in 1973 Iachello {\em et al.} \cite{Iachello:1972nu} were able to 
obtain a good reproduction of all the existing nucleon form factor data using 
a Vector Meson Dominance 
(VMD) model introducing an
intrinsic form factor to describe the internal structure of the nucleon.
If one plots the ratio $G_E/G_M$, the results of the original fit decrease 
with $Q^2$ and cross zero at about $8~GeV^2$.  
In 1996 Holzwarth \cite{Holzwarth} has shown
that the simple Skyrme soliton model, with vector meson corrections and with
the
nucleon initial and final states  boosted to the
Breit Frame, leads to $G^p_E$ that decreases with $Q^2$ and  
crosses zero at $10~GeV^2$. 
In this case the crossing is due to a zero in the Skyrme model form factor as 
explained also in Ref.~\cite{Holzwarth02}.
In 1995 using a constituent quark model Cardarelli {\em et al.}
\cite{Cardarelli:1995dc} have
calculated 
the e.m. form factors of the nucleon in a
light front approach fitting the SLAC data ~\cite{Hohler,Walker} by means
of form
factors for the constituent quarks.
Frank {\em et al.}
\cite{Frank:1995pv} in 1996 have
constructed a relativistic light cone constituent quark model and 
calculated the electric and magnetic form factors of the proton. 
If one plots their calculations as a ratio of
the electric and magnetic form factors one  can see a strong decrease with
$Q^2$ due to the presence of a zero in the 
electric form factor at $Q^2~=~6~GeV^2$ \cite{Miller:2002}. 
In 1999~ \cite{ts99} with a simple non relativistic quark model, the hCQM~
\cite{pl,sig},
boosting the
initial and final state to the Breit Frame and considering relativistic
corrections to the non relativistic current \cite{mds} we have shown
explicitly that
the
decrease is a relativistic effect \cite{ts99,rap} and it disappears without
these
corrections
\cite{ts99,rap,fra99}. This calculation makes use of the nucleon form
factors previously determined in \cite{mds}. Using a chiral CQM and a point
form dynamics the Pavia-Graz
group \cite{Wagenbrunn:2000es,Boffi:2001zb} has shown a good
reproduction of the form
factors and a decrease of 
the ratio up to $4~GeV^2$. A similar reproduction \cite{Merten:2002nz}
 is obtained also within 
   a Bethe Salpeter approach to a constituent quark model with an 
instanton based interaction \cite{Loring:2001kx}. Using the MIT Bag model 
a sharp decrease with $Q^2$ of the ratio is expected and a change of sign at 
$Q^2~=1.5~GeV^2$. The inclusion of the pion cloud not only improves the
static properties of the model and restores the chiral symmetry, but also
the behavior of the ratio $G^p_E/G^p_M$
\cite{Lu:1997sd,Lu:1999np,Lu:yc}. Interesting results are also obtained from 
LQCD calculations extrapolated to the chiral limit \cite{Thomas4}. Finally we 
can say that the VMD fit by Iachello \cite{Iachello2004,Bijker:2004yu} the 
extended VMD model by Lomon \cite{Lomon:2002jx}, the soliton
model calculation by Holzwarth \cite{Holzwarth,Holzwarth02}, 
the calculation by Miller \cite{Miller:2002}
 and the relativistic two quark spectator model calculation by Ma \emph{et
al.}\cite{Ma} describe the new Jlab data quite well. Moreover the electric 
form factor of the proton is well described 
also by an explicit quark-diquark model \cite{elena}. For reviews on the 
subject the readers are referred to ~\cite{Gao:2003ag,dejager}.

In the following we present new results
obtained with the hCQM, using a semirelativistic Hamiltonian and a 
relativistic quark current. Preliminary 
results have been presented at various Conferences \cite{trieste,tasco,baryon}.

First we briefly review the hCQM \cite{pl,sig}. 
The experimental $4$ and $3$ star non strange resonances can be arranged
in  $SU_{sf}(6)$ multiplets. This means that the quark dynamics
has a dominant $SU_{sf}(6)-$ invariant part, which accounts for the average
multiplet energies. In the hCQM it is assumed to be
\cite{pl}
\begin{equation}\label{eq:pot}
V(x)= -\frac{\tau}{x}~+~\alpha x,
\end{equation}
where $x$ is the hyperradius 
\begin{equation}
x=\sqrt{\mbox{\boldmath $\rho$}^2+{\mbox{\boldmath $\lambda$}^2}}~~,
\end{equation}
with {\boldmath $\rho$} and {\boldmath $\lambda$} being the Jacobi coordinates
describing the internal quark motion. \\
\noindent Interactions of the type linear plus Coulomb-like have been used 
since long time for the meson sector, e.g. the Cornell potential. Moreover this
form has been supported by recent Lattice QCD calculations \cite{bali}.\\
\noindent In the case of
baryons a so called hypercentral approximation has been
introduced \cite{has,rich}; this approximation amounts to average 
any two-body 
potential for the three quark system over the hyperangle 
$t=arctg(\frac{{\rho}}{{\lambda}})$ and the angles $\Omega_{\rho}$ and 
$\Omega_{\lambda}$, and it works quite well especially for the lower part of the 
spectrum \cite{hca}.  In this respect, the hypercentral potential of 
Eq.(\ref{eq:pot}) can also be considered as the hypercentral approximation of 
the two-body linear plus Coulomb-like potential.
The splittings within the multiplets are produced by a perturbative term
breaking the $SU_{sf}(6)$ symmetry, 
which, as a first approximation, can be assumed to be the
standard hyperfine interaction $H_{hyp}$ \cite{is}. We consider the 
non strange baryons as bound states of three constituent quarks; 
in the baryon rest frame, the three quark Hamiltonian can be written as:
\begin{equation}\label{eq:ham}
H_{nr} = 3m+\frac{\mbox{\boldmath $p$}_\rho^2+
\mbox{\boldmath $p$}_\lambda^2}{2m}-\frac{\tau}{x}~
+~\alpha x+H_{hyp},
\end{equation}
where $m$ is the quark mass (taken equal to $1/3$ of the nucleon mass) and 
{\boldmath $p$}$_\rho$, {\boldmath $p$}$_\lambda$ are the conjugate momenta of 
the Jacobi coordinates. 
The strength of the hyperfine interaction is determined in order to
reproduce the $\Delta-N$ mass difference, the remaining two free
parameters are fitted to the spectrum 
leading to the values $\alpha= 1.61~fm^{-2}$ and $~\tau=4.59~$.


Keeping these parameters fixed, this non relativistic constituent quark model 
has been used to calculate 
various physical quantities of interest: the photocouplings 
\cite{aie}, the
electromagnetic transition amplitudes \cite{aie2} and, introducing 
relativistic 
corrections to the one-body non relativistic current, also the elastic nucleon
form factors \cite{mds} and the ratio between the electric and magnetic form
factors of the proton \cite{rap}. We have shown that kinematical relativistic 
corrections (like boosts and a relativistic one body current with an expansion 
in the quark momenta up to the first order, keeping the exact dependence on 
the  momentum transfer $Q^2$)  are very important for the elastic form factors 
\cite{rap,mds}
but give only minor corrections in the transition ones \cite{mds2}.


As a further step towards a full relativistic description, we have
 developed a semirelativistic version of the hypercentral constituent quark 
model. The resulting wave functions, boosted to the Breit frame, have been 
used to  calculate the matrix elements of the relativistic one body current 
without any expansion in the quark momenta.

The potential of Eq. (\ref{eq:pot}) has been fitted to all the 3 and 4 stars 
resonances using the correct relativistic kinetic energy \cite{traini}
\begin{equation}\label{eq:hrel}
H = \sum_{i=1}^3 \sqrt{p_{i}^2+m^2}-\frac{\tau}{x}~
+~\alpha x+H_{hyp}.
\end{equation}
The eigenequation of the Hamiltonian (\ref{eq:hrel}) is solved in the nucleon 
rest frame, the resulting spectrum is not much different from the non 
relativistic
one and the parameters of the potential are only slightly modified, confirming 
that kinematic relativistic corrections are not very important for the 
spectrum.   
A complete description of a
variational solution of this equation, using a hyperspherical formalism will
be published elsewhere \cite{traini}.
A first semirelativistic version of our hCQM, but without SU(6) breaking and 
so without the possibility of reproducing the spectrum, 
was first introduced by Traini and Faccioli 
\cite{Faccioli:1998aq} and the wave functions obtained in this way 
have given realistic results for structure functions 
\cite{Faccioli:1998aq,Cano:1999kx,Biasioli:1998cv,Traini:1997jz} and 
also for the generalized parton distributions \cite{Boffi:2002yy,Boffi:2003yj,
Pasquini:2004gc}.

The baryon current is chosen to be the sum of the three elementary quark 
currents and it corresponds to the impulse approximation \cite{mds,rap}:
\begin{equation}
j_{\mu}^{(q)}=\bar{u}(\mbox{\boldmath $p$}')\gamma_\mu u(\mbox{\boldmath $p$})
\end{equation}
and the nucleon state in the rest frame is assumed to be \cite{mds}: 
\begin{equation}
  \Psi_{\mbox{\boldmath ${\scriptstyle P}$}=0}(\mbox{\boldmath $p$}_{\rho},
 \mbox{\boldmath $p$}_{\lambda})=u(\mbox{\boldmath $p$}_1)
    u(\mbox{\boldmath $p$}_2)u(\mbox{\boldmath $p$}_3) 
    \varphi(\mbox{\boldmath $p$}_\rho,\mbox{\boldmath $p$}_\lambda)
    \label{eq:relwf}
\end{equation}
where $\varphi(\mbox{\boldmath $p$}_\rho,\mbox{\boldmath $p$}_\lambda)$ 
with $\mbox{\boldmath $p$}_\rho=(\mbox{\boldmath $p$}_1-
\mbox{\boldmath $p$}_2)/\sqrt{2}$ and $\mbox{\boldmath $p$}_\lambda=
(\mbox{\boldmath $p$}_1+
\mbox{\boldmath $p$}_2-2\mbox{\boldmath $p$}_3)/\sqrt{6}$, 
is the
eigenfunction of the Hamiltonian of equation (\ref{eq:hrel}); 
$u(\mbox{\boldmath $p$}_i)$ is the Dirac spinor of the i-th quark and 
$  \mbox{\boldmath $p$}_i $ is momentum of the i-th quark in the nucleon rest 
frame.
The boosts to the Breit frame have been applied to the initial and final wave 
functions.
With respect to our previous papers \cite{mds,rap}, both the boosts on 
the spatial variables and on the Dirac indices are performed without any
approximation. Moreover another important improvement is given by the use of 
semirelativistic wave functions. 

The resulting theoretical form factors of the nucleon, calculated without 
free parameters, and without any expansion in the quark momenta, 
can be seen in FIG. \ref{fig:ff_nudo_mds} and \ref{fig:rap_nudo_mds}.  

\begin{figure}[!ht]
  \centering
  \subfigure{\includegraphics[width=8.cm]{gep_nudo_mds.eps}} 
  \subfigure{\includegraphics[width=8.cm]{gmp_nudo_mds.eps}}\\
  \subfigure{\includegraphics[width=8.cm]{gen_nudo_mds.eps}} 
  \subfigure{\includegraphics[width=8.cm]{gmn_nudo_mds.eps}}
  \caption{(Color online) Elastic form factors of the nucleon. The solid line 
    corresponds to the semirelativistic hCQM calculation. The experimental 
    data for proton are taken from the reanalysis made by Brash {\em et al.}
    \cite{Brash:2004} of the data from \cite{Andivahis:1994,Bartel:1973,
      Janssens:1966,Litt:1970,Sill:1993}; 
    for the neutron electric form factor, the experimental data 
    are taken from \cite{Becker:1999,Eden:1994,Golak:2001,Herberg:1999,
      Meyerhoff:1994,Ostrick:1999,Passchier:1999,Plachkov:1990,Rohe:1999,
      Zhu:2001}, and for the neutron magnetic form 
    factor, the experimental data are taken from 
    \cite{Anklin:1998,Golak:2001,Kubon:2002,Lung:1993,Markowitz:1993,Rock:1982,
      Xu:2000}.}
  \label{fig:ff_nudo_mds}
\end{figure}
\begin{figure}[!ht]
  \begin{center}
    \includegraphics[width=8cm]{rap_nudo_mds.eps}
    \caption{(Color online) The measured ratio $\mu_pG_E^p/G_M^p$ compared 
      with the semirelativistic hCQM calculation (solid line). 
      The experimental data are taken from 
      \cite{Jones:1999,Gayou:2001,Gayou:2002}.}
    \label{fig:rap_nudo_mds}
  \end{center}
\end{figure}

The results reported in Figure 1 show a quite good reproduction of the data 
even if some problems are still present especially at low $Q^2$. Nonetheless 
there is a great improvement in comparison with the non relativistic 
calculations of Refs \cite{mds,rap}. 

 
Since we consider constituent quarks, which can be in principle considered 
composite objects \cite{petronzio}, we parametrize phenomenologically their 
structure by means of constituent quark form factors, as already done by other 
authors \cite{Cardarelli:1995dc}.

If the constituent quarks have electromagnetic form factors, the quark 
current can be written as:
\begin{equation}
  j_\mu^{(q)}=\bar{u}(\mbox{\boldmath $p$}')\left [ 
    (F_1^q(Q^2)+\kappa_q F_2^q(Q^2))\gamma_\mu - \frac{1}{2m} \kappa_q 
    (p+p')_\mu F_2^q(Q^2) \right ] u(\mbox{\boldmath $p$})
\end{equation}
where $F_1^q$ and $F_2^q$ are the Dirac and Pauli form factors and $\kappa_q$ 
is the anomalous magnetic moment of the $q$ $(=u,d)$ quark. 
The constituent quark form 
factors are chosen as a linear combination of monopole and dipole $Q^2$ 
behavior.
Fitting the free parameters to the reproduction of $G_M^p$, $G_M^n$, $G_E^n$ 
and $\mu_pG_E^p/G_M^p$ we obtain the curves shown in FIG. \ref{fig:ff_comp} and 
\ref{fig:rap_comp}

\begin{figure}[!ht]
  \centering
  \subfigure{\includegraphics[width=8cm]{gepgdip_comp.eps}} 
  \subfigure{\includegraphics[width=8cm]{gmpgdip_comp.eps}}\\
  \subfigure{\includegraphics[width=8cm]{gen_comp.eps}} 
  \subfigure{\includegraphics[width=8cm]{gmngdip_comp.eps}}
  \caption{(Color online) Elastic form factors of the nucleon. The solid line 
    corresponds to the semirelativistic hCQM calculation with constituent 
    quark form factors. The experimental 
    data for proton are taken from the reanalysis made by Brash {\em et al.}
    \cite{Brash:2004} of the data from \cite{Andivahis:1994,Bartel:1973,
      Janssens:1966,Litt:1970,Sill:1993}; 
    for the neutron electric form factor, the experimental data 
    are taken from \cite{Becker:1999,Eden:1994,Golak:2001,Herberg:1999,
      Meyerhoff:1994,Ostrick:1999,Passchier:1999,Plachkov:1990,Rohe:1999,
      Zhu:2001}, and for the neutron magnetic form 
    factor, the experimental data are taken from 
    \cite{Anklin:1998,Golak:2001,Kubon:2002,Lung:1993,Markowitz:1993,Rock:1982,
      Xu:2000}.}
  \label{fig:ff_comp}
\end{figure}
\begin{figure}[!ht]
  \begin{center}
    \includegraphics[width=8cm]{rap_comp.eps}
    \caption{(Color online) The measured ratio $\mu_pG_E^p/G_M^p$ compared 
      with the semirelativistic hCQM calculation (solid line). 
      The experimental data are taken from  \cite{Jones:1999,Gayou:2001,
        Gayou:2002}.}
    \label{fig:rap_comp}
  \end{center}
\end{figure}

The resulting values of the constituent quarks mean square radii are:
\begin{equation}
  <r^2>_u=0.10~\mbox{fm}^2 \qquad \qquad <r^2>_d=-0.09~\mbox{fm}^2
\end{equation}
in good agreement with the findings of Petronzio {\em et al.} \cite{petronzio},
who have been able to identify complex objects inside the proton.
As it can be seen in FIG \ref{fig:ff_comp} and  \ref{fig:rap_comp}
the experimental data are very well reproduced. The goodness of the 
reproduction is emphasized by plotting the form factors divided by the dipole 
form.

With respect to the non relativistic case, the semirelativistic wave 
functions have more high momentum components.
This fact, together with the application of exact boosts to the Breit Frame, 
leads to an improvement in the reproduction of the existing data on the 
electromagnetic form factors.  
However a good description of the data is obtained only if phenomenological 
constituent quark form factors are introduced in the electromagnetic current.  
In this way we have a  very nice agreement with the available experimental data
up to $5~ GeV^2$ . 
Moreover the dimensions of these composite constituent quarks are found in 
agreement with the recent analysis of the nucleon structure functions 
performed by Petronzio {\em et al}.

Finally, it results that for a good reproduction of the elastic form 
factor data both the relativistic effects and the composite nature of the 
constituent quarks have to be taken into account. 
We observe that such constituent quark form factors actually parametrize not 
only  their structure but also all the relativistic effects which have not yet 
been included in our calculations.


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%35
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%36
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%42
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%43
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%44 
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%46
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%%%%%%%%%% EXP DATA %%%%%%%%%%%%%

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%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{thebibliography} 

\end{document}






















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%35
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%44 
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%49
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%50
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%52
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%%%%%%%%%% EXP DATA %%%%%%%%%%%%%

%53
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%75
\bibitem{petronzio}
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%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{thebibliography}