\documentclass[aps]{revtex4}
%\renewcommand{\baselinestretch}{1.2}
\begin{document}


%\preprint{Preprint Universit\'{e} de Mons-Hainaut}
\draft
\title{Electromagnetic splitting for mesons and baryons \\
using dressed constituent quarks}
\author{Bernard \surname{Silvestre-Brac}}
%\thanks{}
\email[E-mail: ]{silvestre@isn.in2p3.fr}
\affiliation{Institut des Sciences Nucl\'{e}aires,
53, Av. des Martyrs, F-38026 Grenoble-Cedex, France}
\author{Fabian \surname{Brau}}
\thanks{FNRS Postdoctoral Researcher}
\email[E-mail: ]{fabian.brau@umh.ac.be}
\author{Claude \surname{Semay}}
\thanks{FNRS Research Associate}
\email[E-mail: ]{claude.semay@umh.ac.be}
\affiliation{Universit\'{e} de Mons-Hainaut, Place du Parc, 20,
B-7000 Mons, Belgium}

% shorttitle: Electromagnetic splitting for mesons and baryons
\date{\today}

\begin{abstract}
Electromagnetic splittings for mesons and baryons are calculated in a
formalism where the constituent quarks are considered as dressed
quasiparticles. The electromagnetic interaction, which contains coulomb,
contact and hyperfine terms, is folded with the quark electrical
density. Two different types of strong potentials are considered.
Numerical treatment is done very carefully and several approximations
are discussed in detail. Our model contains only one free parameter and
the agreement with experimental data is reasonable although some puzzles
still remain.
\end{abstract}

\pacs{12.39.Pn,13.40.Dk,14.20.-c,14.40.-n}

% 12.39.Pn Potential models
% 13.40.Dk Electromagnetic mass differences
% 14.20.-c Baryons (including antiparticles)
% 14.40.-n Mesons
\keywords{Potential models; Electromagnetic mass differences;
Baryons (including antiparticles); Mesons}

\maketitle

\section{Introduction}

Quantum Chromodynamics (QCD) is believed to be the good theory of
strong interaction. It has met with numerous successes in many domains.
However, in the low energy regime, it is extremely difficult to handle
because of its non perturbative character. Lattice calculations become
more and more reliable but still remain very cumbersome, time consuming
and not always transparent for the underlying physics. This explains
why, in the meson and baryon sectors, a number of alternative simpler
models were introduced. Among them, the non relativistic quark model
(NRQM) is very appealing because of its high simplicity, its ability to
treat properly the centre of mass motion, and the large number of
observables that can be described within its framework.

In NRQM formalism, the dynamical equation is the usual Schr\"{o}dinger
equation including a non relativistic kinetic energy term plus a
potential term \cite{luch91}. There exist a lot of different numerical
algorithms to solve the two-body and three-body problems with a good
accuracy (see for instance refs.~\cite{baye86,baye97,suzu98}).

Nowadays, it becomes more and more frequent to use a relativistic
expression for the kinetic energy operator. The resulting dynamical
equation is known as a spinless Salpeter equation. It has several
advantages as compared to the Schr\"{o}dinger equation
\cite{sema92,fulc94} and the corresponding numerical algorithms are now
well under control (see for instance
refs.~\cite{suzu98,fulc94,sema01,silv01}).

In NRQM (and indeed in many other QCD inspired models), the quark
degrees of freedom are no longer the bare quarks of the QCD lagrangian,
but are quasiparticles dressed by a gluon cloud and quark-antiquark
virtual pairs. They are called constituent quarks and the most visible
modification is the necessity to use in NRQM a quark mass substantially
larger than the bare mass. In principle the bare quark-quark potential
should also be modified and folded with the quark colour density to give
the final potential to be used in a Schr\"{o}dinger or in a spinless
Salpeter equation. In recent quark-quark potentials appearing on the
market, this effect is taken into account \cite{sema97,sema99,brau98,
sema01b}. The resulting spectra are in correct agreement with
experimental data. However, it seems very difficult to get, in a unified
scheme (same set of parameters), a good description of both meson and
baryon properties \cite{blas90,brau02}.

The spectra are only a part of interesting observables, and the validity
of a model should be tested on other observables, especially if they are
very sensitive to the form of the wave function. Electromagnetic
properties are best suited for such a study, because the basic QED
formalism is very well known and precise, and thus the possible
uncertainties coming from mechanisms or wave function (itself depending
on the much less known strong interactions) are more conveniently
identified.

In this paper, we focus our study on the electromagnetic splitting
between charged hadrons, both in the meson and baryon sectors. A number
of similar studies were performed in the past \cite{luch91,brau98}.
Essentially three different sources for the splitting were identified: a
small mass difference between up and down quarks, the coulomb
interaction between charged quarks, and their dipole-dipole interaction
\cite{geno98}.
All of them seem to have an important effect and the final result is a
very subtle interplay among them. This explains why a very proper and
precise treatment must be invoked, and also why this observable is very
interesting. The origin of the up and down quark mass difference is not
well understood but, presumably, this SU(2) symmetry breaking is already
effective in the genuine QCD lagrangian. Probably also the mechanisms
necessary to give the constituent mass depend in part from the
electromagnetic lagrangian, so that they contribute in addition to this
mass difference. Quarks being charged particles, the coulomb interaction
is obviously present (very often, it has been treated as a
perturbation). The dipole-dipole interaction is a consequence of
relativistic corrections to the coulomb potential.

In our paper we want to push this study further in several domains.
First we want to perform a precise and complete treatment, avoiding
perturbative expressions as much as possible. Second, we introduce the
``contact term'', which arises on equal footing as the dipole-dipole
relativistic correction, but which is neglected by most authors
\cite{luch91,geno98}. Finally, our most important improvement to our
point of
view is the use of a dressed electromagnetic interaction between quarks.
Since the constituent quarks are quasiparticles, the electromagnetic
interaction should also be modified as compared to the bare one, in a
very similar way to the quark-quark strong potential. However the
electromagnetic lagrangian is different from the QCD lagrangian and the
electromagnetic density for the quark, playing a role in the splitting,
has no reason to be identical to the colour density occurring in the
quark-quark strong potential.

In order to study the influence of the kinematics acting on the quarks,
we investigate the splitting produced with two types of wave function,
one resulting from a non relativistic hamiltonian (AL1)
\cite{silv93,Sema94} and another with a relativistic expression (called
here BSS) \cite{brau02}. Moreover, since our aim is to consider mesons
and baryons on equal footing, it is important to consider interquark
potentials that lead to a correct description of both sectors. This is
rather difficult to encounter. The first potential (AL1) relies on the
so-called funnel or Cornell potential \cite{eich75,bhad81}. It is
completely phenomenological and one can consider that the dressing of
the quarks is included and simulated in the value of the various
parameters. The second one (BSS) starts with more fundamental QCD
grounds and is based on instanton induced effects \cite{blas90,shur89};
the dressing is explicitly taken into account but there remain
nevertheless some free parameters that are adjusted on the spectra.

The paper is organized as follows. The next chapter presents in more
details the strong potentials and the way to solve the two-body and
three-body problems. The third chapter deals with the electromagnetic
interaction responsible of the splitting. The results of our
calculations are presented and compared to data in the fourth chapter.
Conclusions are drawn in the last section.

\section{Potentials and wave function}

\subsection{Strong potentials}

As stated in the introduction, we use in this paper two kinds of
interquark potentials, one that must be used in a Schr\"{o}dinger
equation (AL1) and the other in a spinless Salpeter equation (BSS). They
both depend on the relative distance $r$ between the interacting quarks
and are able to describe in a satisfactory way both the meson and the
baryon spectra. However the BSS potential is suited, for the moment, and
because of its underlying QCD basis, only for the light quark sectors
($u$, $d$, $s$ quarks or antiquarks). Their derivation and their
parameters have been reported elsewhere, and here we just want to point
out the essential features and stress their differences. In both models,
the $u$ and $d$ quarks are assumed to have the same mass. In the
following, they will be noted by the symbol $n$ (normal or non strange).

\subsubsection{AL1 Potential}

The AL1 potential \cite{Sema94}, developed for a non relativistic
kinematics, contains the minimum ingredients
necessary to get an overall reasonable description of hadronic
resonances, a central part $V_{C}$ and a hyperfine term $V_{H}$

\begin{equation}
V_{ij}(r)=-\frac{3}{16}{\bf \lambda}_{i}\cdot {\bf \lambda}_{j}
\left[V_{C}^{(ij)}(r)+V_{H}^{(ij)}(r)\right].
\label{pot}
\end{equation}

The central part is merely the Cornell potential composed of a short
range coulombic part, simulating the one-gluon exchange mechanism, and a
long range linear term,
responsible for the confinement (an additional
constant is very important to get the good absolute values)
\begin{equation}
V_{C}^{(ij)}(r)=-\frac{\kappa}{r}+ar+C.
\end{equation}
The colour dependance through the Gell-Mann matrices $\lambda$ in
relation~(\ref {pot}) comes from one gluon exchange. There is no reason,
except simplicity, that such a structure is kept for the confining and
constant parts. Nevertheless, this ansatz works well for both meson and
baryon sectors.

The hyperfine term is short range and chosen as a gaussian function
\begin{equation}
V_{H}^{(ij)}(r)=\frac{8\pi }{3m_{i}m_{j}}\kappa ^{\prime }\frac{\exp
(-r^{2}/\sigma_{ij}^{2})}{\pi ^{3/2}\,\sigma_{ij}^{3}}{\bf s}_{i} \cdot
{\bf s}_{j}.
\label{hypal1}
\end{equation}
The interesting property, as compared to Bhaduri's \cite{bhad81} or
Cornell's \cite{eich75} potential, is that the range of that force
$\sigma_{ij}$ does depend on the flavour
\begin{equation}
\sigma_{ij}=A\left( \frac{2m_{i}m_{j}}{m_{i}+m_{j}}\right)^{-B}.
\end{equation}
A kind of dressing is realized with the gaussian function, since the
theory predicts a delta contribution for the hyperfine potential.
The various parameters, including constituent masses, have been
determined on the spectra by a best fit procedure. Although very simple,
this potential does already a good job in hadronic spectroscopy.

\subsubsection{BSS Potential}

The BSS potential \cite{brau02}, developed for a semi relativistic
kinematics, contains the Cornell potential and an instanton induced
interaction. The Cornell potential has the same form as in the AL1
potential, but the constant interaction is different for meson and
baryon sectors.

The instanton induced interaction provides a suitable formalism to
reproduce well the spectrum of the pseudoscalar mesons (and to explain
the masses  of $\eta$ and $\eta'$ mesons). The interaction between one
quark and one antiquark in a meson is vanishing for $L\neq 0$ or
$S \neq 0$ states. For $L=S=0$, its form depends on the isospin of the
$q\bar q$ pair
\begin{itemize}
\item For $I=1$:
\begin{equation}
\label{ins2}
V_{I}(r)=-8\, g\, \delta(\vec{r}\,);
\end{equation}
\item For $I=1/2$:
\begin{equation}
\label{ins3}
V_{I}(r)=-8\, g'\, \delta(\vec{r}\,);
\end{equation}
\item $I=0$:
\begin{equation}
\label{ins4}
V_{I}(r)=8
\left(
\begin{array}{cc}
g & \sqrt{2}g' \\
\sqrt{2}g' & 0
\end{array}
\right)\, \delta(\vec{r}\,),
\end{equation}
in the flavour space $(1/\sqrt{2}(|u\bar{u} \rangle+|d\bar{d}
\rangle),|s\bar{s} \rangle)$.
\end{itemize}
The parameters $g$ and $g'$ are two
dimensioned coupling constants. Between two quarks in a baryon, this
interaction is written \cite{blas90,munz94}
\begin{equation}
\label{vinst}
V_{I}(r)=-4 \left(g P^{[nn]} + g' P^{[ns]} \right) P^{S=0}
\delta(\vec r\,),
\end{equation}
where $P^{S=0}$ is the projector on spin 0, and $P^{[qq']}$ is the
projector on antisymmetrical flavour state $qq'$.
We have shown in ref.~\cite{brau02} that this model is able to
describe correctly meson and baryon spectra in a consistent way
if the following interaction is added for baryons only
\begin{equation}
\label{vinstnew}
V_{I}^{\rm baryon}(r)=C_{I} \left(P^{[nn]} + P^{[ns]} \right)
P^{S=0} P^{L=0},
\end{equation}
where $C_{I}$ is a new constant.

The quark masses used in this model are the constituent masses and not
the current ones. It is then natural to suppose that a quark is not a
pure pointlike particle, but an effective degree of freedom which is
dressed by the gluon and quark-antiquark pair clouds. The form that we
retain for the colour charge density of a quark is a Gaussian function
\begin{equation}
\label{rhoi}
\rho(\vec r\,) = \frac{1}{(\Gamma\sqrt{\pi})^{3/2}} \exp(- r^2 /
\Gamma^2).
\end{equation}
It is generally assumed that the quark size $\Gamma$ depends on the
flavour. So, we consider two size parameters $\Gamma_n$ and $\Gamma_s$
for $n$ and $s$ quarks respectively.
More details are given in refs.~\cite{brau98,sema01b,brau02}

\subsection{Meson wave function}

The mesons considered in this paper are pure quark-antiquark systems, so
that the first quantized form of the wave function is
\begin{equation}
\Psi ({\bf r})=C \eta ^{I} \chi ^{S} \Phi ^{LM}({\bf r}),
\label{fom}
\end{equation}
where $C$ is the colour wave function (triplet-antitriplet coupled to
singlet), $\eta ^{I}$ the isospin wave function with total isospin $I$,
$\chi^{S}$ the spin wave function with total spin $S$, and
$\Phi ({\bf r})$ the radial wave function with a total orbital angular
momentum $L$. For all potentials considered here, $L$ and $S$ are good
quantum numbers. So the coupling to a good total angular momentum $J$ is
trivial.

There exist many numerical algorithms to compute the radial function. In
this paper, we use the method based on Lagrange mesh, which is very
simple, very precise and very fast, and for which a recent work has
shown that relativistic kinematics can be handled without any problem
\cite{sema01}. Thus the same method can be applied for both AL1 and BSS
potentials. The principle of that method is an expansion of the radial
function on basis functions
\begin{equation}
\Phi ^{LM}({\bf r})=Y_{LM}(\widehat{r})\sum_{j=1}^{N}A_{j}
\frac{f_{j}(r/h)}{\sqrt{h}r}.
\end{equation}
A scale parameter $h$ appears that allows to fasten the convergence; the
functions $f_{j}(x)$ form an orthogonal set and are related to the
Lagrange polynomials. The expansion coefficients $A_{j}$ result from an
usual diagonalization procedure. To get a very high precision, a typical
number of basis states is $N=60$. Technical details can be found in
ref.~\cite{sema01}.

\subsection{Baryon wave function}

Our method to solve the three-body problem is based on an expansion of
the space wave function in terms of harmonic oscillator functions with
different sizes. More precisely, one defines two dimensionless Jacobi
coordinates ${\bf x}$, ${\bf y}$ as
\begin{equation}
a{\bf x}={\bf r}_{2}-{\bf r}_{3}\;;\;b{\bf y}=\frac{m_{2}{\bf
r}_{2}+m_{3}{\bf r}_{3}}{m_{2}+m_{3}}-{\bf r}_{1}  \label{jac}.
\end{equation}

The baryon wave function is very similar to form~(\ref{fom}) except that
there may exist now several isospin ($\eta _{i}^{I}$) and spin
($\chi _{i}^{S}$) wave functions leading to a given set ($I,S$) (the
colour function $C$ is still unique and thus factorizes, because the
singlet can be obtained only with a quark pair in an antitriplet state)
\begin{equation}
\Psi ({\bf x},{\bf y})=C\sum_{i}A_{i}\eta _{i}^{I}\chi _{i}^{S}
\Phi_{i}^{LM}({\bf x},{\bf y})  \label{fob}.
\end{equation}
The space wave function is expressed through harmonic oscillator
functions $\phi$ by
\begin{equation}
\Phi _{i}^{LM}({\bf x},{\bf y})=\left[ \phi _{nl}({\bf x})
\phi_{\nu \lambda }({\bf y})\right] ^{LM},
\end{equation}
where $n$, $\nu$ and $l$, $\lambda$ are respectively radial and orbital
quantum numbers of harmonic oscillator functions. The number of quanta
$NQ_{i}=(2n+l+2\nu +\lambda)$ is the relevant quantity for convergence;
a very good precision is achieved if we include all basis states up to
20 quanta, $NQ_{i}\leq 20$. This may represent several hundreds or
thousands of basis states in formula~(\ref{fob}). Here again, the
expansion coefficients $A_{i}$ result from an usual diagonalization
procedure. The two scale parameters $a$ and $b$ appearing in
eq.~(\ref{jac}) are determined by minimization of the state energy and
allow to fasten the convergence. More details on the method can be found
in ref.~\cite{silv01}.

\section{Electromagnetic interaction}

\subsection{Bare potential}

The electromagnetic interaction between two pointlike particles $i$ and
$j$ of charges $Q_{i}$, $Q_{j}$ and masses $m_{i}$, $m_{j}$ is very well
known. In addition to the usual coulomb potential $U_{\rm coul}$,
relativistic corrections (at lowest order) give rise to contact,
hyperfine, tensor, symmetric and antisymmetric spin-orbit, and Darwin
terms. Tensor, spin-orbit, and Darwin terms are complicated and
presumably their effects are weak, so that people neglect them. The
hyperfine interaction $U_{\rm hyp}$ does play an important role and is
included in all serious calculations. The contact term $U_{\rm cont}$ is
usually discarded with no other justification than simplicity. In this
study we keep it, so that our total bare electromagnetic potential
writes
\begin{equation}
U_{ij}^{(b)}({\bf r})=(U_{\rm coul})_{ij}^{(b)}({\bf r}%
)+(U_{\rm cont})_{ij}^{(b)}({\bf r})+(U_{\rm hyp})_{ij}^{(b)}({\bf r})
\label{potemb},
\end{equation}
with
\begin{eqnarray}
(U_{\rm coul})_{ij}^{(b)}({\bf r})&=& \phantom{+} Q_{i}Q_{j}
\frac{\alpha }{r},
\label{coulb} \\
(U_{\rm cont})_{ij}^{(b)}({\bf r})&=&-\frac{\pi }{2}Q_{i}Q_{j}
\left(\frac{1}{%
m_{i}^{2}}+\frac{1}{m_{j}^{2}}\right) \alpha \,\delta ({\bf r}),
\label{contb} \\
(U_{\rm hyp})_{ij}^{(b)}({\bf r})&=&-\frac{8\pi Q_{i}Q_{j}}{3m_{i}m_{j}}
\alpha \,\delta ({\bf r}){\bf s}_{i} \cdot {\bf s}_{j},
\label{hypb}
\end{eqnarray}
where $\alpha $ is the fine structure constant.

\subsection{Electromagnetic quark density}

One way to introduce the very complicated mechanisms that transform the
bare quarks (pointlike) to constituent quarks is to suppose a
phenomenological density distribution for a quark. This means that a
constituent quark located at ${\bf r}$ is generated by a bare quark
located in ${\bf r}^{\prime } $ with a certain probability distribution
(or density) $\rho ({\bf r-r} ^{\prime })$. To have an appealing
physical meaning this density must be a peaked function around zero with
a certain size parameter, that depends in principle on the quark
flavour. Moreover we require the natural property that, for a vanishing
size, the constituent quark becomes pointlike and is identified to the
bare quark. Mathematically this means that the limit of the density
$\rho ({\bf u})$ for a vanishing size is the delta function
$\delta ({\bf u)}$. Another natural property is that the density is
isotropic. Lastly, the last required property is that the integral of
the density over the whole space is unity.

The most popular densities are of lorentzian, gaussian or Yukawa type.
In this study we choose a Yukawa density. There is a precise reason for
that: this density is the leading ingredient of the
meson (or baryon) charge form factor. It is an experimental fact that
the data accommodate rather nicely to a Yukawa density (giving a form
factor with an asymptotic behaviour $Q^{-2}$) \cite{brau01,silv02}.
Keeping in
mind the previous remarks, the adopted density for quark of flavour $i$
looks like
\begin{equation}
\rho _{i}({\bf u})=\frac{1}{4\pi \gamma _{i}^{2}}
\frac{e^{-u/\gamma_{i}}}{u},
\label{dens}
\end{equation}
where $\gamma _{i}$ is the electromagnetic size parameter.

The dressed potential $U$ is obtained from the bare potential $U^{(b)}$
by a double convolution over the densities of each interacting quark
\begin{equation}
U_{ij}({\bf r})=\int d{\bf u}\,d{\bf v}\,\rho _{i}({\bf u}
)\,\rho _{j}({\bf v})\,U_{ij}^{(b)}({\bf r}+{\bf v}-{\bf u}).
\label{convd}
\end{equation}
With a trivial change of variable, it is quite easy to transform this
double folding into a single one
\begin{equation}
U_{ij}({\bf r})=\int d{\bf r}^{\prime }\;U_{ij}^{(b)}({\bf r}
^{\prime })\,\rho _{ij}({\bf r}-{\bf r}^{\prime }),
\label{convs}
\end{equation}
with the definition of the new density
\begin{equation}
\rho _{ij}({\bf u})=\int d{\bf v}\,\rho _{i}({\bf v})\,\rho _{j}(%
{\bf u}-{\bf v}).
\label{densij}
\end{equation}

Applying eq.~(\ref{densij}) to the Yukawa density (\ref{dens}), one gets
\begin{equation}
\rho _{ij}({\bf u})=\frac{1}{8\pi \gamma _{i}^{3}}e^{-u/\gamma _{i}}
\quad {\rm if} \quad \gamma _{i}=\gamma _{j}
\label{dyuki}
\end{equation}
and
\begin{eqnarray}
\rho _{ij}({\bf u})&=&\frac{1}{4\pi (\gamma _{i}^{2}-\gamma _{j}^{2})}
\left( \frac{e^{-u/\gamma _{i}}}{u}-\frac{e^{-u/\gamma _{j}}}{u}\right)
\\
 &=&\frac{1}{(\gamma _{i}^{2}-\gamma _{j}^{2})}\left( \gamma _{i}^{2}
 \rho _{i}
({\bf u})-\gamma _{j}^{2}\rho _{j}({\bf u})\right) \quad {\rm if} \quad
\gamma _{i}\neq \gamma _{j}.
\label{dyukij}
\end{eqnarray}

\subsection{Dressed potential}

Starting from eq.~(\ref{potemb}) and using eqs.~(\ref{convs}) and
(\ref{dyuki}), it is easy to obtain analytically the dressed potential
\begin{equation}
U_{ij}(r)=(U_{\rm coul})_{ij}(r)+(U_{\rm cont})_{ij}(r) +
(U_{\rm hyp})_{ij}(r),
\label{potemd}
\end{equation}
in the case of two interacting constituent quarks with same size
$\gamma_{i}=\gamma _{j}=\gamma $. It writes
\begin{eqnarray}
(U_{\rm coul})_{ij}(r)&=&\phantom{+}\alpha Q_{i}Q_{j}\left[ \frac{1}{r}
\left
(1-e^{-r/\gamma}\right) -\frac{e^{-r/\gamma }}{2\gamma }\right],
\label{couldi} \\
(U_{\rm cont})_{ij}(r)&=&-\frac{\alpha Q_{i}Q_{j}}{16\gamma ^{3}}\left(
\frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}\right) \,e^{-r/\gamma},
\label{contdi} \\
(U_{\rm hyp})_{ij}(r)&=&-
\frac{\alpha Q_{i}Q_{j}}{3\gamma ^{3}m_{i}m_{j}}
\,e^{-r/\gamma }{\bf s}_{i}\cdot {\bf s}_{j}.
\label{hypdi}
\end{eqnarray}
Doing the same thing with eq.~(\ref{dyukij}), one obtains the
electromagnetic dressed potential in the case of different interacting
quarks
\begin{eqnarray}
(U_{\rm coul})_{ij}(r)&=&\phantom{+}\alpha Q_{i}Q_{j}\left( \frac{1}{r}-
\frac{\gamma _{i}^{2}}
{\gamma _{i}^{2}-\gamma _{j}^{2}}\frac{e^{-r/\gamma _{i}}}{r}+
\frac{\gamma_{j}^{2}}{\gamma _{i}^{2}-\gamma _{j}^{2}}
\frac{e^{-r/\gamma _{j}}}{r}\right),
\label{couldd} \\
(U_{\rm cont})_{ij}(r)&=&-\frac{\alpha Q_{i}Q_{j}}{8(\gamma _{i}^{2}-
\gamma_{j}^{2})}\left( \frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}\right)
\frac{e^{-r/\gamma _{i}}-e^{-r/\gamma _{j}}}{r},
\label{contdd} \\
(U_{\rm hyp})_{ij}(r)&=&-\frac{2\alpha Q_{i}Q_{j}}{3m_{i}m_{j}
(\gamma_{i}^{2}-\gamma _{j}^{2})}\frac{e^{-r/\gamma _{i}}-
e^{-r/\gamma_{j}}}{r}{\bf s}_{i}\cdot {\bf s}_{j}.
\label{hypdd}
\end{eqnarray}

It is possible to treat the electromagnetic potential $U$ as a
perturbation. In this case, we solve the dynamical equation
$[T(m_{i},m_{j})+V(m_{i},m_{j})]\left| \Psi \right\rangle =E\left| \Psi
\right\rangle$ and use the resulting wave function $\Psi$ to evaluate
$\Delta E=\left\langle \Psi \right| \Delta V+U\left| \Psi
\right\rangle$, where $\Delta V$ is the modification of the strong
potential due to the fact that the kinetic and hyperfine terms are mass
dependent. The splitting we are considering needs a slight modification
of the quark mass $m$ which becomes $\widetilde{m}$; in particular,
although the
original potentials AL1 or BSS are such that $m_{u}=m_{d}$, it is
necessary to impose the condition
$\widetilde{m}_{u}\neq \widetilde{m}_{d}$ in order to get reasonable
results. As a consequence, the kinetic energy term (including the rest
mass term) and the hyperfine term are modified as compared to the
original potentials (the rest mass term is diagonal and is treated
trivially). Consequently, we have
$\Delta V=T(\widetilde{m}_{i},\widetilde{m}_{j})+V(
\widetilde{m}_{i},\widetilde{m}_{j})-T(m_{i},m_{j})-V(m_{i},m_{j})$ and
$U=U(\widetilde{m}_{i},\widetilde{m}_{j})$.

Nevertheless, as much as possible, we will solve directly the
dynamical equation
$[T(\widetilde{m}_{i},\widetilde{m}_{j}) +
V(\widetilde{m}_{i},\widetilde{m}_{j})+U(\widetilde{m}_{i},
\widetilde{m}_{j})]\left| \Psi \right\rangle =E^{\prime }\left| \Psi
\right\rangle$.
An interesting question is whether the exact value $E^{\prime }$ is very
different from the perturbed value $ E+\Delta E$. This point is
addressed in the next section.

\section{Results}

\subsection{Determination of the parameters}

We do not want to introduce a lot of new parameters; here we restrict
the number of free parameters to the minimum unavoidable. In particular
the parameters of the strong potential are maintained without
modification. Important information is contained in the electromagnetic
size of the quarks $\gamma _{i}$. This size could depend on the flavour
and on the electrical charge of the quark. If the dependence on charge
were dominant, we could expect that sizes of quarks with the same charge
are similar. Preliminary calculations have shown that the size must be
strongly reduced for heavy quarks, showing that the dependence on mass
must be dominant. Consequently, in order to restrict again the number of
parameters, we assume that the sizes of the $u$ and $d$ quarks are the
same. Thus we impose $\gamma _{u}=\gamma _{d}=\gamma _{n}$. For AL1 and
BSS, we have $\gamma _{n}$ and $\gamma _{s}$ as free parameters. An
observable that is very sensitive to those parameters is the charge mean
square radius.

In the mesonic sector, the bare charge square radius operator for
pointlike quarks is defined by
\begin{equation}
(r^{2})^{(b)}=\sum_{i=1}^{2}e_{i}\,({\bf r}_{i}-{\bf R})^{2},
\label{r2bop}
\end{equation}
where ${\bf r}_{i}$ is the position for quark $i$ and ${\bf R}$ the
position of the centre of mass. For constituent quarks, this expression
has to be folded with quark density and should be written instead
\begin{equation}
(r^{2})=\sum_{i=1}^{2}e_{i}\int d{\bf r}^{\prime }\,({\bf r}^{\prime }-
{\bf R})^{2}\,\rho _{i}({\bf r}^{\prime }-{\bf r}_{i}).
\label{r2op}
\end{equation}
Averaging quantity (\ref{r2op}) on the meson wave function provides us
with the charge mean square radius of the meson. Performing the
calculation, one finds that this observable is a sum of a term, that can
be called the bare radius $\left\langle r^{2}\right\rangle ^{(b)}$
(which is essentially the mean value of quantity~(\ref{r2bop}) on meson
wave function), plus a term which is essentially the dipole moment of
the density. With a Yukawa density, one has explicitly
\begin{equation}
\left\langle r^{2}\right\rangle =\left\langle r^{2}\right\rangle
^{(b)}+6\sum_{i=1}^{2}e_{i}\gamma _{i}^{2}.
\label{r2}
\end{equation}
The dynamical contribution to the square radius is entirely contained in
the bare quantity, whose expression is very well known and is not
recalled here.

From relation (\ref{r2}), one sees that the pion radius depends only on
$\gamma_{n}$ and it is used to determine this parameter. The kaon
radius depends on $\gamma _{n}$ and $\gamma _{s}$; since $\gamma _{n}$
has already be determined from the pion, the kaon radius is used to
determine $\gamma _{s}$. In addition, the AL1 potential needs the
further determination of $\gamma _{c}$ and $\gamma _{b}.$ Since the
radii for $D$ and $B$ resonances are not known experimentally, we
determine the corresponding sizes by requiring a smooth behaviour versus
the mass. Fortunately the isospin splitting does not depend too much on
the precise value of the quark size. Our accepted values for the
electromagnetic sizes are summarized on table~\ref{emsize}.

Other parameters that need to be changed are the quark masses. In fact,
the only important ingredient for the splittings is the mass difference
between the down and up quarks:
$\Delta =\widetilde{m}_{d}-\widetilde{m}_{u}$. In view of this, we
choose to maintain the $s$, $c$, and $b$ quark masses at their non
perturbative value $\widetilde{m}_{i}=m_{i}$, and to keep the average
value of the isospin doublet at its non perturbative value
$(\widetilde{m}_{d}+\widetilde{m}_{u})/2=m_{n}$. The size parameters
being determined once and for all on charge radii, we have only
\emph{one free parameter} $\Delta $ at our disposal to try to reproduce
all the known electromagnetic splittings. One can imagine several
strategies to determine this parameter. We first remarked that if we fit
$\Delta $ on the mesons, the baryons were very badly reproduced, while
if we fit it on the baryons, the mesons were not spoiled dramatically.
Moreover, among the baryons, some splittings are more affected than
others by a small change of $\Delta$. Thus, we decided to fit this
parameter on one of the most sensitive and well known splitting, namely
$\Sigma^{-}-\Sigma ^{0}=$ 4.807~MeV.

One aim of our study is to see the respective influence of each
component of the electromagnetic potential (\ref{potemd}). The coulomb
part seems to us unavoidable, so we will consider in the following four
different approximations:
\begin{itemize}
\item C: Electromagnetic potential restricted to coulomb term
(\ref{couldi}) or (\ref{couldd}) alone;
\item CC: Electromagnetic potential restricted to coulomb and contact
(\ref{contdi}) or (\ref{contdd}) terms alone;
\item CH: Electromagnetic potential restricted to coulomb and hyperfine
(\ref{hypdi}) or (\ref{hypdd}) terms alone;
\item T: Total electromagnetic potential (\ref{potemd}) taken into
account.
\end{itemize}
Each approximation requires its own $\Delta $ parameter, but the
$\gamma _{i} $ parameters can be maintained to their values of table
\ref{emsize}. The corresponding results are gathered in table
\ref{delta}.

It is funny that for both potentials, increasing values of $\Delta $ are
obtained with approximations CH, C, T, CC respectively. Is it a property
independent of the strong potential (realistic enough to reproduce
baryon and meson spectra)? We have no answer.

\subsection{Experimental data}

A number of experimental data, concerning the electromagnetic
splittings, exist for both the mesonic and baryonic sectors, and for
both light quarks ($u$, $d$, and $s$) systems and for systems containing
at least a heavy quark ($c$ and $b$). The use of AL1 potential allows to
study all data, while BSS is restricted to the light quark domain. The
values for the splittings are of order of some MeV and some of them are
known with good accuracy.

Puzzling questions arise and are not solved in a satisfactory manner by
the up to date theoretical studies. Let us list some of them:
\begin{itemize}
\item $n-p=1.293$~MeV is much weaker than
$\pi ^{+}-\pi ^{0}=4.594$~MeV, despite the fact that a very naive quark
model gives $n(udd)-p(duu)=\Delta$ and
$\pi^{+}(u\bar d)-\pi ^{0}([u\bar u - d\bar d]/\sqrt{2})=0$;
\item $\pi ^{+}-\pi ^{0}>0$ while $\rho ^{+}-\rho ^{0}<0$;
\item The hierarchy $\Sigma _{c}^{+}>\Sigma _{c}^{++}>\Sigma _{c}^{0}$
is in contradiction to what one expects from naive arguments.
\end{itemize}
These remarks illustrate the fact that the mass difference between down
and up quarks cannot be the only -- or even leading -- ingredient to
explain the splitting; the internal dynamics also plays an important
role.

\subsection{Numerical precautions}

Electromagnetic splittings are of the order of some MeV or less, while
the corresponding total masses of the multiplet members are of order of
one or two GeV; this means they are small quantities resulting from
difference between large quantities. In consequence, from the physical
point of view it is important to study all the possible sources
responsible of the splitting, and from a numerical point of view one
must be very cautious and achieve a very precise calculation. An error
of $10^{-3}$ GeV on the absolute mass, which can be considered as good
for such an observable, would lead to very incorrect values concerning
the splitting. We feel that an accuracy of order of $10^{-5}$ GeV on the
absolute mass should be reached in order to insure reliable conclusions.

The Lagrange mesh method used to calculate the mesonic sector is very
fast and precise \cite{sema01}; with a number of basis states $N=60$,
the relative accuracy is of the order of $10^{-10}$, so that we are
completely
confident in our results. The only source of possible uncertainty
concerns the $\pi ^{0}$ and $\rho ^{0}$ resonances. They consist on an
equal mixing of $u\overline{u}$ and $d \overline{d}$ components. With
$\widetilde{m}_{u}\neq \widetilde{m}_{d}$, a rigorous treatment would be
very complicated for our purpose, including coupled channel
calculations. Here we consider that those resonances are
$n \overline{n}$ systems, composed of fictitious quark and antiquarks of
mass $ \widetilde{m}_{n}=(\widetilde{m}_{u}+\widetilde{m}_{d})/2 = m_n$.
Concerning the rest mass, this assumption is perfectly exact; the only
approximation concerns the mass dependent terms of the hamiltonian. The
error should be of second order in $(\Delta /m_{n})$, so that it is
quite acceptable.

The three-body problem is less favourable (whatever the numerical method
invoked!); we use a method based on expansion on harmonic oscillator
functions with different sizes \cite{silv01}. A good compromise between
speed and accuracy is to push the expansion up to $N=20$ quanta; in this
case the accuracy is of the order of $10^{-5}$ GeV and we can only
guarantee two or three significant digits in our results for the
splittings. Nevertheless, this is enough to draw reliable conclusions.

\subsection{Comparison with bare and dressed potentials}

The use of the dressed potential (\ref{potemd})-(\ref{hypdd}) instead of
the bare potential (\ref{potemb})-(\ref{hypb}) is crucial. First of all,
the bare interaction cannot be used in an exact (non perturbative)
calculation without having a collapse since the attractive delta
function is not bounded from below. Secondly, even if a perturbative
calculation, limited to the first order, produces finite results one can
ask about the relevance of these results. Nevertheless, based on
physical ground, one can argue that the delta function appearing in the
bare interaction are some artefact coming from the limited development
(lowest relativistic correction) of the electromagnetic interaction; so
a calculation at the first order of perturbation has some sense.

We have performed such calculations and found that indeed the use of the
dressed potential is necessary. For example, for the BSS potential with
the total bare electromagnetic interaction we have the following
difference of masses $\pi^+ - \pi^0=22.7$~MeV,
$\rho^+ - \rho^0=-44.2$~MeV and $K^{0*} - K^{+*}=48.7$~MeV
\cite{brau98}. In this case, the difference of
masses between the quark $d$ and $u$ is obtained by adjusting the
difference of masses of kaon multiplet and is equal to 58.3 MeV.
We see that all these results are really poor compared to the
experimental results. We note also that the motivated use of the
dressed potential improves considerably the theoretical results.

\subsection{Comparison with perturbative treatment}

Very often, the electromagnetic potential (\ref{potemd}) is treated by
perturbation. In this study we are able to make an exact calculation in
the mesonic sector, for both potentials AL1 and BSS. In the baryonic
sector, AL1 allows an exact treatment. This is not the case for BSS; the
fundamental reason for that lies in the symmetry property of the
instanton potential. It acts differently on the symmetric $(ud+du)$ and
antisymmetric $(ud-du)$ flavour states; in a framework where
$m_{u}=m_{d}$, the isospin formalism can be introduced to classify the
basis states and this property can be taken into account without
problem.
As soon as $\widetilde{m}_{u}\neq \widetilde{m}_{d}$, the up and down
quark must be considered as different and the instanton becomes very
difficult to handle; spurious states appear and the numerical treatment
does not make sense. Thus, for baryons, the three-body treatment with
BSS is done perturbatively; the wave function, with a good isospin
symmetry, is obtained from the hamiltonian including only the strong
potential and it is used to calculate the average value of the
electromagnetic hamiltonian.

To discuss the validity of the perturbative treatment, we compare the
results obtained perturbatively with those obtained with an exact
treatment. For this study we focus only on the total electromagnetic
hamiltonian (what was called T approximation previously). Let us note
that, when both perturbative and exact treatments are possible, the
parameter $\Delta$ is computed with the exact results and used, without
refitting, in the perturbative calculations.

In the mesonic sector both AL1 and BSS potentials can be used in a
perturbative and in exact treatment. The results for light mesons are
presented in table~\ref{meslgp}. The perturbative treatment appears
rather good for resonances with non zero orbital momentum or for $S=1$
states. For ground states ($L=0$, $S=0$), it is less satisfactory, the
precision being of order of 30\%. In some cases, the perturbative
treatment gives results lower than the exact treatment while in other
cases it is the opposite situation, so that no general conclusion can be
drawn. One sees also the big sensitivity to the potential, BSS and AL1
giving values that can differ a lot (for example $K^{0}-K^{+}$). In this
sector BSS appears definitively superior.
The table~\ref{mesldp} presents the results for heavy mesons obtained
with the AL1 potential. In this case the problem can be treated
perturbatively with a good accuracy.

In the first part of table~\ref{barp}, we focus on the comparison
between perturbative and exact treatment in the case of baryons for the
AL1 potential. One sees that the perturbed values can differ
dramatically from the exact values for special cases (\emph{i.e.}
$n-p$). In fact the fault must not be
attributed to the electromagnetic hamiltonian but to the strong
potential. More precisely, the hyperfine term is mass dependent
($1/(m_1 m_2)$ factor and mass dependent range of the interaction) and
it is not justified to treat by perturbation a modification of this term
due to a change of the masses. In the second part of table~\ref{barp},
the same splittings are computed with the AL1 potential in which the
strong hyperfine term is suppressed. The agreement between perturbative
and exact treatment is then very good. As the BSS potential has no mass
dependent term, we can expect that a perturbative treatment is
justified.

\subsection{Influence of various approximations}

In this part we want to study the effect of using various
approximations C, CC, CH and T of the electromagnetic hamiltonian. An
exact treatment is performed, except for the baryons with BSS potential,
as explained previously.

Let us first present the results obtained with AL1 potentials. They are
gathered on table~\ref{mesal1} for the mesons and on table~\ref{baral1}
for the baryons. Few comments are in order:
\begin{itemize}
\item  The various approximations differ significantly one from the
other. This proves again that the electromagnetic splitting is an
observable very sensitive to the physical content put in it. A more
quantitative comparison is relegated later on.
\item  Although the experimental data cannot be reproduced with a good
precision, the calculated values have the good order of magnitude and
respect more or less the hierarchy. We mean by this that the order of a
given multiplet is generally the good one and that a large (small)
theoretical splitting corresponds to a large (small) experimental one.
Let us recall that we have only one free parameter $\Delta $, which has
been fitted on the $\Sigma^{-}-\Sigma^{0}$ value.
\item  From time to time the sign is wrong (the order is the reverse),
but this effect occurs generally when compatibility is not excluded due
to error bars, or at least when the experimental uncertainty is large.
\item  None of the already mentioned puzzles (comparison of pion to
nucleon, order of the $\rho $, order of $\Sigma _{c}$) is solved in this
case.
\end{itemize}

Let us now have a look on the situation concerning the wave functions
arising from the BSS potential. The mesonic sector is presented in table
\ref{mesbs2} and the baryonic sector in table~\ref{barbs2}. The same
comments as for AL1 can be made for BSS, except that one puzzle (the
order in $\rho $ multiplet) can find a solution. Moreover the BSS
results are quite different from AL1 results, indicating that, in order
to describe the electromagnetic splitting, not only the electromagnetic
part of the interaction is important, but also the strong one via the
wave function.

In order to grasp more quantitatively the influence of each
approximation, as well as to compare the effect of the strong potential,
we calculate a chi-square value on the experimental sample (in fact a
chi-square divided by the number of data in the set). In order to avoid
a too much important weight on very precise values, a minimum
uncertainty at 0.1~MeV has been assigned arbitrarily to those values.
The results are gathered in table~\ref{chi2}. To have a more refined
analysis, we separate the sample for meson (denoted by ``M" in the
table), for baryon (``B" in the table), or the entire sample of hadrons
(``H" in the table). Moreover we also distinguish between light sector
($u$, $d$, $s$ denoted by ``l" in the table), heavy sector ($c$, $b$
denoted by ``h" in the table), or all sectors (denoted by ``a" in the
table). For instance the line ``hB" means a chi-square calculated on
heavy baryons, ``lM" calculated on light mesons, ``aH" on the entire
sample, \ldots

Interesting remarks can be emphasized:
\begin{itemize}
\item  The baryonic sector is explained in a much more satisfactory way,
whatever the chosen strong potential. This is the consequence of our
arbitrary choice to adjust the free parameter $\Delta$ on a baryon
resonance. Should we have chosen to fit $\Delta$ on a meson resonance,
the mesonic sector would have been much best reproduced, but at the
price of a dramatic spoiling of the baryonic sector, and with an overall
worse description.
\item  Concerning the BSS calculation, the exact formalism (T
approximation) is the best one for mesons, but it is the CH
approximation that is the best for baryons, and the C approximation
(presumably the crudest one) for the entire set. Anyhow, addition of the
contact term deteriorates seriously the results.
\item  Concerning the AL1 calculation, several conclusions are drawn.
For baryons, all approximations are roughly of the same quality. What is
gained on a data by one approximation versus the others is lost on
another data. For mesons, CH is always the best approximation and CC
the worse. For the totality of the sample this conclusion remains,
while the exact treatment (T) is just a bit better than the crudest one
(C). Here again the contact term has a very bad influence.
\item  AL1 and BSS calculations give results of comparable quality for
the whole set, but BSS is much better for mesons, and AL1 much better
for baryons. This may be explained by the fact that only a perturbative
treatment (and not an exact one as in AL1) can be performed with BSS in
the baryonic sector.
\end{itemize}

Just to convince you that our formalism is able to provide a good
description of meson and baryons simultaneously, we present below the
absolute masses of one member of the multiplet (the other can be
obtained using the values of the splittings given above) for both mesons
(table \ref{mesabs}) and baryons (table~\ref{barabs}). The T
approximation is used and the treatment is exact except for baryons with
the BSS potential.

The small discrepancy for light baryons in the case of AL1 potential can
be attributed to a three-body force \cite{bhad81}, which is mass
dependent. The instanton does not give rise to a three-body force for
the baryon \cite{blas90,munz94}, and the parameters have been fitted
directly to the absolute masses of mesons and baryons. The agreement
with experimental data is rather satisfactory.

\section{Conclusions}

In this paper, we calculated the electromagnetic splitting on hadronic
systems. This observable is a very sensitive test of the formalism
because it results from a very subtle and fine balance between several
physical ingredients. In order to concentrate on the physical aspects of
the problem, we were very cautious in the numerical treatment, both for
the two-body and three-body problems. Thus we are very confident with
our numerical results, and interesting conclusions can be emphasized.

As compared to previous works we considered the total electromagnetic
hamiltonian (excepted Darwin, spin-orbit, and tensor forces that we
believe
to play a very minor role). In particular we took into account the
so-called contact term that is usually neglected. Moreover, we treated
the quarks as constituent particles with an electromagnetic size that
modifies the form of the electromagnetic interaction. Lastly, we based
our calculations on wave functions resulting from two different strong
potentials: the phenomenological AL1 potential to be used with a non
relativistic kinematics energy operator and the more fundamental BSS
potential, including instanton effects, to be used with a relativistic
kinetic energy operator. The size of the quarks were determined in order
to reproduce the pion and kaon charge form factors. Only one free
parameter, the mass difference between down and up quarks, is left at
our disposal; it was fitted on the $\Sigma ^{-}-\Sigma ^{0}$ splitting.

We first showed that a perturbative treatment can be considered as good
in the mesonic sector (less good for $L=0=S$ resonances) but is not good
in the baryonic sector. The fault comes not from the electromagnetic
potential, but from a bad approximation of the strong hyperfine term.
Thus a perturbative treatment, at least for mass dependent strong
potential, must be avoided as much as possible. In this paper, most of
our calculations were performed in an exact way.

By comparison of AL1 and BSS results, we stressed that the strong
potential, via the wave function, is an important ingredient in the
description of electromagnetic splittings. In our particular case, AL1
gives a better description of baryons, and BSS a better description of
mesons.

But we proved also that the electromagnetic hamiltonian is equally
important for explaining the splittings. Adding or removing a term
(contact or hyperfine) has a non negligible influence. In particular
taking into account the contact term spoils a lot the results, and
curiously it is the approximation based on coulomb+hyperfine (the usual
ingredient for many people) which is globally the best.

In our formalism the splittings are described in a reasonable way; the
order among multiplet masses is respected and the values have the right
order of magnitude. But the agreement is far from being perfect and some
puzzles are still open questions. New improvements must be done in
future studies.

\acknowledgments

C. Semay would like to thank the FNRS for financial support (FNRS Research Associate
position).
F. Brau would like to thank the FNRS for financial support (FNRS Postdoctoral Researcher
position).
%\appendix

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\clearpage

\begin{table}[tbp] \centering
\begin{tabular}{|l|c|c|c|c|}
\hline
 & $\gamma _{n}$ & $\gamma _{s}$ & $\gamma _{c}$ & $\gamma _{b}$ \\
\hline
AL1 & $1.225$ & $0.200 $ & $0.040$ & $0.013$ \\
BSS & $1.330$ & $0.450$ & $-$ & $-$ \\
\hline
\end{tabular}
\caption{Electromagnetic sizes $\gamma_f$ of the constituents quarks (in
GeV$^{-1}$), for different flavours $f$, and for the two different
strong potentials. \label{emsize}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|l|r|r|r|r|}
\hline
    & \multicolumn{1}{|c|}{C} & \multicolumn{1}{|c|}{CC} &
    \multicolumn{1}{|c|}{CH} & \multicolumn{1}{|c|}{T} \\
\hline
AL1 & $13.2$ & $14.4$ & $12.6$ & $13.6$ \\
BSS & $7.2 $ & $9.4$ & $6.2$ & $8.4$ \\
\hline
\end{tabular}
\caption{Values of the parameter
$\Delta =\widetilde{m}_{d}-\widetilde{m}_{u}$ (in MeV), for the two
potentials, and for the four different approximations presented in the
text. \label{delta}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|l|c|r|r|r|r|}
\hline
\multicolumn{1}{|c|}{Splitting} & Exp & \multicolumn{1}{|c|}{AL1P} &
\multicolumn{1}{|c|}{AL1} & \multicolumn{1}{|c|}{BSSP} &
\multicolumn{1}{|c|}{BSS} \\
\hline
$\pi ^{+}-\pi ^{0}$ & $4.594\pm0.001$ & $1.290$ & $1.693$ &
$2.655$ & $2.974$ \\
$\rho ^{+}-\rho ^{0}$ & $-0.4\pm0.8$ & $0.559$ & $0.713$ &
$-0.033$ & $-0.008$ \\
$K^{0}-K^{+}$ & $3.995\pm 0.034$ & $8.117$ & $8.109$ &
$1.263$ & $1.165$ \\
$K^{0\ast }-K^{+\ast }$ & $6.7\pm1.2$ & $1.439$ & $1.436$ &
$4.247$ & $4.247$ \\
$K_{2}^{0}-K_{2}^{+}$ & $6.8\pm2.8$ & $-0.754$ & $-0.757$
& $3.150$ & $2.828$ \\ \hline
\end{tabular}
\caption{Electromagnetic splittings (in MeV) for light mesons obtained
with a perturbative treatment based on wave functions resulting from AL1
potential (denoted AL1P) and BSS potential (denoted BSSP) as compared to
an exact treatment (AL1 and BSS respectively). The total electromagnetic
hamiltonian is considered. For information, the experimental data,
extracted from ref.~\protect\cite{pdg}, are
given in column ``Exp". \label{meslgp}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|l|c|r|r|}
\hline
\multicolumn{1}{|c|}{Splitting} & Exp & \multicolumn{1}{|c|}{AL1P} &
\multicolumn{1}{|c|}{AL1} \\
\hline
$D^{+}-D^{0}$ & $4.79\pm 0.10$ & $2.314$ & $2.308$ \\
$D^{+\ast }-D^{0\ast }$ & $2.6\pm 1.8$ & $1.039$ & $1.037$ \\
$D_{1}^{+}-D_{1}^{0}$ & $4\pm 5$ & $-2.155$ & $-2.161$ \\
$D_{2}^{+}-D_{2}^{0}$ & $0.9\pm 3.3$ & $-2.229$ & $ -2.235$ \\
$B^{0}-B^{+}$ & $0.33\pm 0.28$ & $-1.710$ & $-1.713$ \\
\hline
\end{tabular}
\caption{Same as table~\ref{meslgp} for heavy mesons in the case of AL1
potential. \label{mesldp}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|l|c|r|r|}
\hline
\multicolumn{1}{|c|}{Splitting} & Exp & \multicolumn{1}{|c|}{AL1P} &
\multicolumn{1}{|c|}{AL1} \\
\hline
\multicolumn{4}{|c|}{AL1 with $V_H$} \\
\hline
$n-p$ & $1.293$ & $9.10$ & $1.15$ \\
$\Delta ^{0}-\Delta ^{++}$ & $2.7\pm 0.3$ & $3.58$ & $3.72$ \\
$\Delta ^{+}-\Delta ^{++}$ & $\simeq 1.9$ & $1.25$ & $1.35$ \\
\hline
\multicolumn{4}{|c|}{AL1 without $V_H$} \\
\hline
$n-p$ & $1.293$ & $3.92$ & $3.91$ \\
$\Delta ^{0}-\Delta ^{++}$ & $2.7\pm 0.3$ & $6.87$ & $6.78$ \\
$\Delta ^{+}-\Delta ^{++}$ & $\simeq 1.9$ & $2.96$ & $2.88$ \\
\hline
\end{tabular}
\caption{Same as table~\ref{meslgp} in the baryonic sector for the AL1
potential including or not its strong hyperfine term $V_H$
(\ref{hypal1}).
\label{barp}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|l|c|r|r|r|r|}
\hline
\multicolumn{1}{|c|}{Splitting} & Exp & \multicolumn{1}{|c|}{C} &
\multicolumn{1}{|c|}{CC} & \multicolumn{1}{|c|}{CH} &
\multicolumn{1}{|c|}{T} \\
\hline
$\pi ^{+}-\pi ^{0}$ & $4.594\pm 0.001$ & $1.239$ & $0.828$ & $2.111$ &
$1.693$ \\
$\rho ^{+}-\rho ^{0}$ & $-0.4\pm 0.8$ & $1.009$ & $0.857$ & $0.870$ &
$0.713$ \\
$K^{0}-K^{+}$ & $3.995\pm 0.034$ & $8.096$ & $9.290$ & $7.054$ & $8.109$
\\
$K^{0\ast }-K^{+\ast }$ & $6.7\pm 1.2$ & $1.132$ & $1.458$ & $1.139$ & $
1.436 $ \\
$K_{2}^{0}-K_{2}^{+}$ & $6.8\pm 2.8$ & $-0.794$ & $-0.793$ & $-0.763$ &
$-0.757$ \\
$D^{+}-D^{0}$ & $4.79\pm 0.10$ & $2.478$ & $1.977$ & $2.821$ & $2.308$
\\
$D^{+\ast }-D^{0\ast }$ & $2.6\pm 1.8$ & $1.513$ & $1.120$ & $1.428$ &
$1.037 $ \\
$D_{1}^{+}-D_{1}^{0}$ & $4\pm 5$ & $-2.038$ & $-2.409$ & $-1.839$ &
$-2.161$ \\
$D_{2}^{+}-D_{2}^{0}$ & $0.9\pm 3.3$ & $-2.058$ & $-2.411$ & $-1.937$ &
$-2.235$ \\
$B^{0}-B^{+}$ & $0.33\pm 0.28$ & $-1.877$ & $-1.715$ & $-1.891$ &
$-1.713$ \\
\hline
\end{tabular}
\caption{Meson electromagnetic splittings (in MeV) calculated for 4
different approximations C, CC, CH, and T, as explained in the text. The
wave functions result from the strong potential AL1 and the numerical
treatment is exact. The experimental values (Exp) come from
ref.~\cite{pdg}. \label{mesal1}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|l|c|r|r|r|r|}
\hline
\multicolumn{1}{|c|}{Splitting} & Exp & \multicolumn{1}{|c|}{C} &
\multicolumn{1}{|c|}{CC} & \multicolumn{1}{|c|}{CH} &
\multicolumn{1}{|c|}{T} \\ \hline
$n-p$ & $1.293$ & $0.89$ & $1.15$ & $0.91$ & $1.15$ \\
$\Delta ^{0}-\Delta ^{++}$ & $2.7\pm 0.3$ & $2.68$ & $3.68$ & $2.79$
& $3.72$ \\
$\Delta ^{+}-\Delta ^{++}$ & $\simeq 1.9$ & $0.57$ & $1.22$ & $0.73$
& $1.35$ \\
$\Sigma ^{-}-\Sigma ^{0}$ & $4.807\pm 0.04$ & $4.82$ & $4.82$ &
$4.81$ & $4.76$ \\
$\Sigma ^{-}-\Sigma ^{+}$ & $8.08\pm 0.08$ & $7.87$ & $8.41$ & $8.27$
& $8.55$ \\
$\Sigma ^{-\ast }-\Sigma ^{0\ast }$ & $2.0\pm 2.4$ & $3.26$ & $3.23$ &
$3.04$ & $2.94$ \\
$\Sigma ^{-\ast }-\Sigma ^{+\ast }$ & $0\pm 4$ & $1.71$ & $1.99$ &
$1.68$ & $1.96$ \\
$\Xi ^{-}-\Xi ^{0}$ & $6.48\pm 0.24$ & $7.12$ & $7.21$ & $7.38$ &
$7.38$ \\
$\Xi ^{-\ast }-\Xi ^{0\ast }$ & $3.2\pm 0.6$ & $3.01$ & $2.91$ &
$2.80$ & $2.66$ \\
$\Sigma _{c}^{++}-\Sigma _{c}^{0}$ & $0.57\pm 0.23$ & $1.00$ & $-0.02$
& $1.35$ & $0.37$ \\
$\Sigma _{c}^{+}-\Sigma _{c}^{0}$ & $1.4\pm 0.6$ & $-0.32$ & $-0.64$ &
$-0.02$ & $-0.33$ \\
$\Sigma _{c}^{++\ast }-\Sigma _{c}^{0\ast }$ & $1.9\pm 1.7$ & $1.37$ &
$0.27$ & $1.33$ & $0.19$ \\
$\Xi _{c}^{0}-\Xi _{c}^{+}$ & $5.5\pm 1.8$ & $2.81$ & $3.28$ & $3.01$
& $3.42$ \\
$\Xi _{c}^{0\prime }-\Xi _{c}^{+^{\prime }}$ & $\simeq 4.7\pm 2.1$ &
$0.20$ & $0.49$ & $-0.08$ & $0.20$ \\
$\Xi _{c}^{+\ast }-\Xi _{c}^{0\ast }$ & $\simeq 2.9\pm 2.0$ & $-0.08$ &
$-0.31$ & $-0.03$ & $-0.25$ \\
$\Xi _{c}^{0\ast \ast }-\Xi _{c}^{+\ast \ast }$ & $\simeq 4.1\pm 2.5$ &
$3.09$ & $3.42$ & $3.24$ & $3.51$ \\
\hline
\end{tabular}
\caption{Same as table~\ref{mesal1} for baryons. The theoretical
uncertainty may affect only the last digit. \label{baral1}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|l|c|r|r|r|r|}
\hline
\multicolumn{1}{|c|}{Splitting} & Exp & \multicolumn{1}{|c|}{C} &
\multicolumn{1}{|c|}{CC} & \multicolumn{1}{|c|}{CH} &
\multicolumn{1}{|c|}{T} \\
\hline
$\pi ^{+}-\pi ^{0}$ & $4.594\pm 0.001$ & $1.22$ & $-0.21$ & $4.11$ &
$2.97$ \\
$\rho ^{+}-\rho ^{0}$ & $-0.4\pm 0.8$ & $1.00$ & $0.40$ & $0.59$ &
$-0.01 $ \\
$K^{0}-K^{+}$ & $3.995\pm 0.034$ & $1.54$ & $3.68$ & $-0.97$ &
$1.17$ \\
$K^{0\ast }-K^{+\ast }$ & $6.7\pm 1.2$ & $2.88$ & $4.50$ & $2.63$ & $
4.25 $ \\
$K_{2}^{0}-K_{2}^{+}$ & $6.8\pm 2.8$ & $2.41$ & $3.49$ & $2.07$ &
$2.83$ \\
\hline
\end{tabular}
\caption{Meson electromagnetic splittings (in MeV) calculated for 4
different approximations C, CC, CH, and T, as explained in the text. The
wave functions result from the strong potential BSS and the numerical
treatment is exact. The experimental values (Exp) come from
ref.~\cite{pdg}. \label{mesbs2}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|l|c|r|r|r|r|}
\hline
\multicolumn{1}{|c|}{Splitting} & Exp & \multicolumn{1}{|c|}{C} &
\multicolumn{1}{|c|}{CC} & \multicolumn{1}{|c|}{CH} &
\multicolumn{1}{|c|}{T} \\
\hline
$n-p$ & $1.293$ & $2.78$ & $4.28$ & $2.62$ & $4.08$ \\
$\Delta ^{0}-\Delta ^{++}$ & $2.7\pm 0.3$ & $4.24$ & $8.21$ & $4.37$
& $8.34$ \\
$\Delta ^{+}-\Delta ^{++}$ & $\simeq 1.9$ & $1.18$ & $3.69$ & $1.59$
& $4.09$ \\
$\Sigma ^{-}-\Sigma ^{0}$ & $4.807\pm 0.035$ & $4.83$ & $4.81$ &
$4.81$ & $4.80$ \\
$\Sigma ^{-}-\Sigma ^{+}$ & $8.08\pm 0.08$ & $7.68$ & $8.90$ & $8.47$
& $9.69$ \\
$\Sigma ^{-\ast }-\Sigma ^{0\ast }$ & $2.0\pm 2.4$ & $4.98$ & $5.37$ &
$4.06$ & $4.43$ \\
$\Sigma ^{-\ast }-\Sigma ^{+\ast }$ & $0\pm 4$ & $3.12$ & $4.54$ &
$2.88$ & $4.33$ \\
$\Xi ^{-}-\Xi ^{0}$ & $6.48\pm 0.24$ & $4.82$ & $4.50$ & $5.87$ &
$5.55$ \\
$\Xi ^{-\ast }-\Xi ^{0\ast }$ & $3.2\pm 0.6$ & $5.09$ & $5.38$ &
$4.17$ & $4.46$ \\
\hline
\end{tabular}
\caption{Same as table~\ref{mesbs2} for baryons. The theoretical
uncertainty may affect only the last digit. Let us recall that, for
technical reasons, the theoretical values were obtained with a
perturbative treatment. \label{barbs2}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|c|r|r|r|r|r|r|r|r|}
\cline{2-9}
\multicolumn{1}{c|}{\ } & \multicolumn{4}{|c|}{AL1} &
\multicolumn{4}{|c|}{BSS} \\
\hline
$\chi^2$ & \multicolumn{1}{|c|}{C} & \multicolumn{1}{|c|}{CC} &
\multicolumn{1}{|c|}{CH} & \multicolumn{1}{|c|}{T} &
\multicolumn{1}{|c|}{C} & \multicolumn{1}{|c|}{CC} &
\multicolumn{1}{|c|}{CH} & \multicolumn{1}{|c|}{T} \\
\hline
lM & $568$ & $851$ & $317$ & $513$ & $350$ & $464$ & $500$ & $213$ \\
lB & $3.7$ & $3.9$ & $4.1$ & $5.8$ & $36$ & $155$ & $26$ & $159$ \\
lH & $205$ & $306$ & $116$ & $187$ & $148$ & $265$ & $195$ & $178$ \\
\hline
hM & $120$ & $169$ & $91$ & $134$ & $-$ & $-$ & $-$ & $-$ \\
hB & $3.6$ & $4.5$ & $4.4$ & $3.3$ & $-$ & $-$ & $-$ & $-$ \\
hH & $52$ & $73$ & $40$ & $58$ & $-$ & $-$ & $-$ & $-$ \\
\hline
aM & $344$ & $510$ & $204$ & $324$ & $-$ & $-$ & $-$ & $-$ \\ \
aB & $3.7$ & $4.2$ & $4.2$ & $4.7$ & $-$ & $-$ & $-$ & $-$ \\
aH & $135$ & $199$ & $81$ & $127$ & $-$ & $-$ & $-$ & $-$ \\
\hline
\end{tabular}
\caption{Chi-square values divided by the number of data in the sample.
The meaning of approximations C, CC, CH, and T have been explained in
the text. The meaning of the rows is related to the sub-samples taken
into account: ``l", ``h", ``a" for light, heavy and all sectors
respectively; ``M", ``B", ``H" for meson, baryon, hadron sectors
respectively. \label{chi2}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|c|r|r|r|}
\hline
System & \multicolumn{1}{c|}{Exp} & \multicolumn{1}{c|}{AL1} &
\multicolumn{1}{c|}{BSS} \\
\hline
$\pi ^{0}$ & $134.98$ & $137.3$ & $145.8$ \\
$\rho ^{0}$ & $769.3$ & $769.7$ & $756.2$ \\
$K^{+}$ & $493.68$ & $486.6$ & $491.2$ \\
$K^{+\ast }$ & $891.66$ & $902.7$ & $888.7$ \\
$K_{2}^{+}$ & $1425.6$ & $1333.4$ & $1377.5$ \\
$D^{+}$ & $1869.3$ & $1863.2$ & $-$ \\
$D^{+\ast }$ & $2010.0$ & $2016.4$ & $-$ \\
$D_{1}^{+}$ & $2422.2$ & $2419.6$ & $-$ \\
$D_{2}^{+}$ & $2459$ & $2451$ & $-$ \\
$B^{+}$ & $5279$ & $5295$ & $-$ \\ \hline
\end{tabular}
\caption{Absolute masses for some mesons. Calculations are
done with strong potentials AL1 and BSS and with the total
electromagnetic hamiltonian. The experimental values (Exp) come from
ref.~\cite{pdg}. \label{mesabs}}
\end{table}

\begin{table}[tbp] \centering
\begin{tabular}{|c|r|r|r|}
\hline
System & \multicolumn{1}{c|}{Exp} & \multicolumn{1}{c|}{AL1} &
\multicolumn{1}{c|}{BSS} \\ \hline
$p$ & $938$ & $994$ & $935$ \\
$\Delta ^{0}$ & $1234$ & $1308$ & $1260$ \\
$\Lambda $ & $1116$ & $1149$ & $1105$ \\
$\Sigma ^{-}$ & $1197$ & $1233$ & $1201$ \\
$\Sigma ^{-\ast }$ & $1387$ & $1439$ & $1395$ \\
$\Xi ^{-}$ & $1321$ & $1343$ & $1323$ \\
$\Xi ^{-\ast }$ & $1535$ & $1560$ & $1522$ \\
$\Omega $ & $1672$ & $1675$ & $1646$ \\
$\Lambda _{c}^{+}$ & $2284$ & $2290$ & $-$ \\
$\Sigma _{c}^{++}$ & $2453$ & $2466$ & $-$ \\
$\Xi _{c}^{+}$ & $2466$ & $2467$ & $-$ \\
$\Xi _{c}^{+\prime }$ & $2574$ & $2572$ & $-$ \\
$\Xi _{c}^{+\ast }$ & $2647$ & $2650$ & $-$ \\
$\Xi _{c}^{+\ast \ast }$ & $2815$ & $2788$ & $-$ \\
$\Omega _{c}^{0}$ & $2704$ & $2675$ & $-$ \\
$\Lambda _{b}^{0}$ & $5624$ & $5635$ & $-$ \\
\hline
\end{tabular}
\caption{Same as table~\ref{mesabs} for the baryons. \label{barabs}}
\end{table}

\end{document}




