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\begin{document}
\title{Relativistic Calculation of the Meson Spectrum: a Fully Covariant Treatment
Versus Standard Treatments}
\author{Horace Crater}
\affiliation{The University of Tennessee Space Institute Tullahoma, Tennessee
37388\footnote{hcrater@utsi.edu}}
\author{Peter Van Alstine}
\affiliation{12474 Sunny Glen Drive Moorpark, CA 93021}

\begin{abstract}
A large number of treatments of the meson spectrum have been tried that
consider mesons as quark - anti quark bound states. Recently, we used
relativistic quantum \textquotedblleft constraint\textquotedblright\ mechanics
to introduce a fully covariant treatment defined by two coupled Dirac
equations. For field-theoretic interactions, this procedure functions as a
\textquotedblleft quantum mechanical transform of Bethe-Salpeter
equation\textquotedblright. Here, we test its spectral fits against those
provided by an assortment of models: Wisconsin model, Iowa State model,
Brayshaw model, and the popular semi-relativistic treatment of Godfrey and
Isgur. We find that the fit provided by the two-body Dirac model for the
entire meson spectrum competes with the best fits to partial spectra provided
by the others and does so with the smallest number of interaction parameters
without additional cutoff parameters necessary to make other approaches
numerically tractable. \ We discuss the distinguishing features of our model
that may account for the relative overall success of its fits.

\end{abstract}
\eid{ }
\startpage{1}
\endpage{1}
\maketitle


\address{12474 Sunny Glen Drive, Moorpark, \\
California, 93021}

%\draft
%\today


\vspace{2cm}

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%\end{document}
\newpage

\section{ Introduction}

\setcounter{section}{1}Over 50 years after the discovery of the first meson
and over 25 years after the identification of its underlying quark degrees of
freedom, the Strong-Interaction Bound-state problem remains unsolved. Perhaps
eventually the full spectrum of mesonic and baryonic states will be calculated
directly from Quantum Chromodynamics via lattice gauge theory. This would
require use of techniques that were unknown to the founding fathers of QED.
For the present though, researchers have had to content themselves with
attempts to extend bits and pieces of traditional QED bound-state treatments
into the realm of QCD. Unfortunately, for those bound systems whose
constituent kinetic or potential energies are comparable to constituent rest
masses, nonrelativistic techniques are inadequate from the start.

In the QED bound-state problem, weakness of the coupling permitted calculation
through perturbation about the nonrelativistic quantum mechanics of the
Schrodinger equation. Using the equation adopted by Breit \cite{Breit1}%
-\cite{Breit3}(eventually justified by the Bethe-Salpeter equation
\cite{bet57}), one was faced with fact that a nonperturbative numerical
treatment of the Breit equation could not yield spectral results that agree to
an appropriate order with a perturbative treatment of the semirelativistic
form of that equation\cite{bet57}-\cite{cwyw}. \ This form of the equation
contained such terms as contact terms bred by the vector Darwin interaction
that could be treated only perturbatively, spoiling the interpretation of the
Breit equation as a bona-fide wave equation. Forays into the full relativistic
structure defined by the Bethe-Salpeter equation turned up fundamental
problems which fortunately could be sidestepped for QED due to the smallness
of $\alpha$.

In the absence of definitive guidance from QED, in recent years researchers in
QCD have felt free to jump off from any point that had proven historically
useful in QED. Some have chosen to approach the spectrum using time-honored
forms from the ``relativistic correction structure'' of atomic physics. Others
have employed truncations of field-theoretic bound-state equations in hopes
that the truncations do no violence to the dynamical structures or their
relativistic transformation properties. A third set have broken away from QED
by choosing to guess at ``relativistic wave equations'' as though such
equations have no connection to field theory.

Is there another way to attack this problem? Imagine that we could replace the
Schrodinger equation by a many-body relativistic Schrodinger equation or
improved Breit equation that could be solved numerically. One would have to
establish its validity by connecting it to quantum field theory, and its
utility by solving it for QCD. Of course such an approach would apply equally
as well to QED and so would have to recapitulate the known results of QED.
(These results might reemerge in unfamiliar forms since not originating in the
usual expansion about the nonrelativistic limit.)

Now, for the two-body bound-state problem, there is such an equation or rather
a system of two coupled Dirac equations - for an interacting pair of
relativistic spin one-half constituents. It turns out that for the two-body
case, use of Dirac's constrained Hamiltonian mechanics \cite{di64}%
-\cite{drz75} in a form appropriate for two spinning particles \cite{cra82},
\cite{saz86} (pseudo-classical mechanics using Grassmann degrees of
freedom\cite{brz},\cite{tei81})leads to a consistent relativistic quantum
description. In the two-body case, one may explicitly construct the covariant
Center of Momentum rest frame of the interacting system. In fact, the
relativistic two-body problem may be written as an effective relativistic
one-body problem \cite{tod76}, \cite{cra84},\cite{cra87}. The proper
formulation of this relativistic scheme requires the successful treatment of
the quantum ghost states (due to the presence of the \textquotedblleft
relative time\textquotedblright) that first appeared in Nakanishi's work on
the Bethe-Salpeter equation\cite{nak69}.

It might seem that although fully covariant and quantum mechanically
legitimate, such an approach would merely give a sophisticated method for
guessing relativistic wave equations for systems of bound quarks. However,
this method assumes its full power when combined with the field-theoretic
machinery of the Bethe-Salpeter equation. When used with the kernel of the
Bethe-Salpeter equation for QED, our approach combines weak-potential
agreement with QED \cite{bckr} with the nonperturbative structure of the
field-theoretic eikonal approximation\cite{tod71},\cite{saz97}. The extra
structure is automatically inherited from relativistic classical\cite{yng},
\cite{cra92} and quantum mechanics\cite{saz97}. In QED our approach amounts to
a ``quantum-mechanical transform'' \cite{saz85},\cite{saz92}of the Bethe
Salpeter equation provided by two coupled Dirac equations whose fully
covariant interactions are determined by QED in the Feynman Gauge\cite{cra88}%
,\cite{bckr}. These ``Two-Body Dirac Equations'' are legitimate quantum wave
equations that can be solved directly \cite{va86},\cite{bckr}(without
perturbation theory).

Of course there is a fly in the ointment - but one to be expected on
fundamental grounds. It turns out that the only separable interacting system
as yet explicitly constructed in a canonical relativistic mechanics is the two
body system. In practical terms, this means that we must confine the present
treatment to the meson spectrum. So far, even the relativistic treatment of
the three-body problem of QED in the constraint approach is unknown. \ No one
has been able to produce three compatible Dirac equations which include not
only mutual interactions but also necessary three body forces in closed
form\cite{ror81}.

Although still considered unusual or unfamiliar by the bulk of bound-state
researchers the structures appearing in these equations may have been
anticipated classically by J. L. Synge, the spin structures were introduced
into QED (incorrectly) by Eddington and Gaunt\cite{edd28},\cite{va97}, they
have appeared in approximate forms appropriate for weak potentials in the
works of Krapchev, Aneva, and Rizov\cite{krp79} and of Pilkuhn\cite{pilk79} .
Of greatest surprise but greatest value (to the authors), their perturbative
weak-potential versions were uncovered in QED by J. Schwinger in his virial
treatment of the positronium spectrum\cite{sch73}. The associated relativistic
mechanics transforms properly under spin-dependent generalizations of
generators found by Pryce \cite{pry48}Newton and Wigner\cite{nwt49}. The
techniques for quantization go back to those found by Dirac\cite{di64}, and
applied by Regge and Hanson to the relativistic top\cite{rge74}, by Nambu to
the string, by Galvao and Teitelboim to the single spin one-half particle
\cite{tei81}, and by Kalb and Van Alstine \cite{ka75}and by
Todorov\cite{tod76} to the pair of spinless particles. Their progenitors can
be found in the bilocal field theories of Yukawa, Markov, Feynman and
Gell-Mann as well as the myriad treatments of the relativistic oscillator
beginning with the work of Schr\"{o}dinger.

In this paper, we will compare our latest results for the meson spectrum
provided by Two-Body Dirac Equations with the corresponding results given by a
representative sample of alternative methods. The present paper is not a
detailed account of this method (already presented elsewhere-see
Refs.\cite{bckr}\cite{cra94}and references contained therein). Neither is it
an attempt to conduct an even-handed or thorough review. Rather, its purpose
is to show how such an organized scheme fares in the real world of calculation
of a relativistic spectrum and to contrast its results with those produced by
an array of approaches, each chosen on account of popularity or structural
resemblances or differences with our approach. All of the approaches that we
consider do not restrict themselves to the heavy mesons, but, as we do,
attempt fits to the light mesons as well. Where possible, we shall show how
various distinguishing features of the various approaches are responsible for
success or failure of the resulting fits to the meson spectrum. Whether our
equations ultimately prove correct or not, they have the virtue that they are
explicitly numerically solvable without additional revisions, cutoffs, etc.
unlike certain other approaches whose spectral consequences depend on ad hoc
revision necessary for numerical solution.

All of the treatments we examine attempt to describe the interactions of QCD
through the inclusion of spin-dependent interactions that in part first
appeared as small corrections in atomic physics. All include relativistic
kinematics for the constituents. One contributor to the use of such techniques
\cite{mor90} has even asserted that all of the alternative approaches that
include relativistic kinematics are actually equivalent to the nonrelativistic
quark model, so that the detailed relativistic structure of the interaction
makes no difference to the bound state spectrum. However, as we shall see in a
fully relativistic description with no extraneous parameters, the detailed
relativistic interaction structure in fact determines the success or failure
of a calculation of the full meson spectrum from a single equation.

The order of the paper is as follows:

First, in Section II, we review enough of the structures of our Two-Body Dirac
Equations and their origins in relativistic constraint dynamics to make clear
the equations that we are solving and the relativistic significances of the
potential structures appearing in them. Then, in Section III, we detail how we
incorporate the interactions of QCD into our equations by constructing the
relativistic version of the Adler-Piran static quark potential\cite{adl} that
we use when we apply our equations to meson spectroscopy. In Section IV, we
examine the numerical spectral results that are generated by this application
of the Two-Body Dirac Equations.

In Section V, we examine how the relativistic interaction structures of the
constraint approach lead even for QED to different classifications of
interaction terms than the designations used in some of the other approaches.
In Sections VI and VII, we examine two different attempts to use the Salpeter
Equation to treat the meson spectrum: the Wisconsin Model of Gara, Durand,
Durand, and Nickisch\cite{wisc} and the Iowa State Model of Spence and Vary
\cite{iowa}. Although these authors try to keep relativistic structures, they
ultimately employ weak-potential approximations and structures obtained from
perturbative QED in a non-perturbative equation (with no check to see that the
procedure even makes nonperturbative sense in QED itself). In Section VIII, we
examine the Breit Equation Model of Brayshaw \cite{bry}, which illustrates the
sort of successful fit that one can still obtain when one is allowed to
introduce interactions (into the Breit equation) whose relativistic properties
are totally obscure. In Section IX, we look at the most popular treatment -
the Semirelativistic Model of Godfrey and Isgur \cite{isgr}. This model
includes independent smearing and momentum dependent factors for each part of
the various spin-dependent interactions. Although each interaction is
introduced for apparently justifiable physical reasons, this approach breaks
up (or spoils) the full relativistic spin structure that is the two-body
counterpart of that of the one-body Dirac equation with its \textit{automatic}
relations among the various interaction terms. We examine this model to see
how well a fully covariant set of two-body Dirac equations employing only 3
potential-parameters can do versus a semirelativistic equation with
relativistic kinematics and pieces of relativistic dynamical corrections
(introduced in a patchwork manner with 10 potential parameters), when required
to fit the whole meson spectrum (including the light-quark mesons). Finally,
in Section X, we conclude the paper by reviewing some of the features of the
constraint approach that played important roles in the relative success of its
fit to the meson spectrum. We then use apparent successes of fits produced by
the nonrelativistic quark model to point out dangers inherent in judging rival
formalisms on the basis of fits to portions of the spectrum.\ At the end of
the paper, we supply sets of tables for spectral comparisons and Appendices
detailing the radial form of our Two-Body Dirac equations that we use for our
spectral calculations, a new alternative form of the equations, and the
numerical procedure that we use to construct meson wave functions. We also
include a table summarizing the important features of the various methods that
we compare in this paper.

\section{The Two-Body Dirac Equations of Constraint Dynamics}

In order to treat a single relativistic spin-one-half particle, Dirac
originally constructed a quantum wave equation from a first-order wave
operator that is the square-root of the corresponding Klein-Gordon operator
\cite{di28}. Our method extends his construction to the system of two
interacting relativistic spin-one-half particles with quantum dynamics
governed by a pair of compatible Dirac equations on a single 16-component wave
function. For an extensive review of this approach, see
Refs.\cite{cra87,bckr,cra94} and works cited therein. For the reader
unfamiliar with this approach, we present a brief review.

About 26 years ago, the relativistic constraint approach first successfully
yielded a covariant yet canonical formulation of the relativistic two-body
problem for two interacting spinless classical particles. It accomplished this
by covariantly controlling the troublesome relative time and relative energy,
thereby reducing the number of degrees of freedom of the relativistic two-body
problem to that of the corresponding nonrelativistic problem\cite{ka75}%
-\cite{drz75}. In this method, the reduction takes place through the
enforcement of a generalized mass shell constraint for each of the two
spinless particles: $p_{i}^{2}+m_{i}^{2}+\Phi_{i}\approx0$. Mathematical
consistency then requires that the two constraints be \textquotedblleft
compatible\textquotedblright\ in the sense that they be conserved by a
covariant system-Hamiltonian. Upon quantization, the quantum version of this
\textquotedblleft compatibility condition\textquotedblright\ becomes the
requirement that the quantum versions of the constraints (two separate
Klein-Gordon equations on the same wave function for spinless particles)
possess a commutator that vanishes when applied to the wave-function. In 1982,
the authors of this paper used a supersymmetric classical formulation of the
single-particle Dirac equation due to Galvao and Teitelboim to successfully
extend this construction to the \textquotedblleft
pseudo-classical\textquotedblright\ mechanics of two spin-one-half particles
\cite{tei81,cra82}. Upon quantization, this scheme produces a consistent
relativistic quantum mechanics for a pair of interacting fermions governed by
two coupled Dirac equations.

When specialized to the case of two relativistic spin-one-half particles
interacting through four-vector and scalar potentials, the two compatible
16-component Dirac equations \cite{cra87,bckr,cra94} take the form
\begin{subequations}
\begin{align}
\mathcal{S}_{1}\psi &  =\gamma_{51}(\gamma_{1}\cdot(p_{1}-A_{1})+m_{1}%
+S_{1})\psi=0\label{tbdea}\\
\mathcal{S}_{2}\psi &  =\gamma_{52}(\gamma_{2}\cdot(p_{2}-A_{2})+m_{2}%
+S_{2})\psi=0, \label{tbdeb}%
\end{align}
in terms of $\mathcal{S}_{i}$ operators that in the free-particle limit become
operator square roots of the Klein-Gordon operator.

The relativistic four-vector potentials $A_{i}^{\mu}$ and scalar potentials
$S_{i}$ are effective constituent potentials that in either limit
$m_{i}\rightarrow\infty$ go over to the ordinary external vector and scalar
potentials of the light-particle's one-body Dirac equation. Note that the
four-vector interactions enter through \textquotedblleft minimal
substitutions\textquotedblright\ inherited (along with the accompanying gauge
structure) from the corresponding classical field theory\cite{cra84,yng,cra92}%
. The covariant spin-dependent terms in the constituent vector and scalar
potentials (see Eq.(\ref{vecp} and Eq.(\ref{scalp}) below) are recoil terms
whose forms are nonperturbative consequences of the compatibility condition
\end{subequations}
\begin{equation}
\lbrack\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0. \label{cmpt}%
\end{equation}
This condition also requires that the potentials depend on the space-like
interparticle separation only through the combination
\begin{equation}
x_{\perp}^{\mu}=(\eta^{\mu\nu}+\hat{P}^{\mu}\hat{P}^{\nu})(x_{1}-x_{2})_{\nu}%
\end{equation}
with no dependence on the relative time in the c.m. frame. This separation
variable is orthogonal to the total four-momentum
\begin{equation}
P^{\mu}=p_{1}^{\mu}+p_{2}^{\mu};\ -P^{2}\equiv w^{2}.
\end{equation}
$\hat{P}$ is the time-like unit vector
\begin{equation}
\hat{P}^{\mu}\equiv P^{\mu}/w.
\end{equation}
The accompanying relative four-momentum canonically conjugate to $x_{\perp}$
is
\begin{equation}
\ p^{\mu}=(\epsilon_{2}p_{2}^{\mu}-\epsilon_{1}p_{2}^{\mu})/w;\mathrm{where}%
\ \epsilon_{1}+\epsilon_{2}=w,\ \epsilon_{1}-\epsilon_{2}=(m_{1}^{2}-m_{2}%
^{2})/w
\end{equation}
in which $w$ is the total c.m. energy. The $\epsilon_{i}$'s are the invariant
c.m. energies of each of the (interacting) particles.

The wave operators in Eqs.(\ref{tbdea},\ref{tbdeb}) operate on a single
16-component spinor which we decompose as
\begin{equation}
\psi=\left(
\begin{array}
[c]{c}%
\psi_{1}\\
\psi_{2}\\
\psi_{3}\\
\psi_{4}%
\end{array}
\right)
\end{equation}
in which the $\psi_{i}$ are four-component spinors.

Once we have ensured that the compatibility condition is satisfied,
Eqs.(\ref{tbdea},\ref{tbdeb}) provide a consistent quantum description of the
relativistic two-body system incorporating several important properties
\cite{cra87,bckr,cra94} . They are manifestly covariant. They reduce to the
ordinary one body Dirac equation in the limit in which either of the particles
becomes infinitely heavy. They can be combined to give \cite{bckr,long}
coupled second-order Schr\"{o}dinger-like equations (Pauli-forms) for the
sixteen component Dirac spinors. In the center of momentum (c.m.) system, for
the vector and scalar interactions of Eq.(\ref{vecp}) and Eq.(\ref{scalp})
below, these equations resemble ordinary Schr\"{o}dinger equations with
interactions that include central-potential, Darwin, spin-orbit, spin-spin,
and tensor terms. These customary terms are accompanied by others that provide
important additional couplings between the upper-upper ($\psi_{1}$) and
lower-lower ($\psi_{4}$) four component spinor portions of the full sixteen
component Dirac spinor. The interactions are completely local but depend
explicitly on the total energy $w$ in the c.m. frame.

The resulting coupled Schr\"{o}dinger-like equations take the general c.m.
forms
\begin{subequations}
\begin{align}
(-\mathbf{\nabla}^{2}+\Phi_{1}(\mathbf{r},\mathbf{\sigma}_{1}\mathbf{,\sigma
}_{2},w))\psi_{1}+\Phi_{2}(\mathbf{r},\mathbf{\sigma}_{1}\mathbf{,\sigma}%
_{2},w)\psi_{4}  &  =b^{2}(w)\psi_{1}\label{clpdsa}\\
(-\mathbf{\nabla}^{2}+\Phi_{3}(\mathbf{r},\mathbf{\sigma}_{1}\mathbf{,\sigma
}_{2},w))\psi_{4}+\Phi_{4}(\mathbf{r},\mathbf{\sigma}_{1}\mathbf{,\sigma}%
_{2},w)\psi_{1}  &  =b^{2}(w)\psi_{4} \label{clpdsb}%
\end{align}
These are accompanied by similar equations involving $\psi_{2}$ and $\psi
_{3.}$ However, one can use Eq.(\ref{tbdea},\ref{tbdeb}) instead to determine
$\psi_{2}$ and $\psi_{3}$ in terms of $\psi_{1}$ and $\psi_{4}$. Note that it
is not necessary to determine $\psi_{2}$ and $\psi_{3}$ to solve the coupled
eigenvalue equations Eqs.(\ref{clpdsa},\ref{clpdsb}) numerically.

In these equations, the usual invariant
\end{subequations}
\begin{equation}
b^{2}(w)\equiv(w^{4}-2w^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2}%
)^{2})/4w^{2}%
\end{equation}
plays the role of energy eigenvalue. This invariant is the c.m. value of the
square of the relative momentum expressed as a function of the invariant total
c.m. energy $w$.

Note that in the limit in which one of the particles becomes very heavy, these
Schr\"{o}dinger-like equations turn into those obtained by eliminating either
the lower or upper component of the ordinary one-body Dirac equation in terms
of the other component.

The vector potentials appearing in Eqs.(\ref{tbdea},\ref{tbdeb}) depend on
three invariant functions $G,E_{1}$ and $E_{2}$ that define the time-like
(proportional to $\hat{P}$) and space-like vector interactions
\cite{cra87,bckr} (with $\partial_{\mu}\equiv\partial/\partial x^{\mu}$)
\begin{align}
A_{1}^{\mu}  &  =\big((\epsilon_{1}-E_{1})-i{\frac{G}{2}}\gamma_{2}%
\cdot({\frac{\partial E_{1}}{E_{2}}}+\partial G)\gamma_{2}\cdot\hat
{P}\big )\hat{P}^{\mu}+(1-G)p^{\mu}-{\frac{i}{2}}\partial G\cdot\gamma
_{2}\gamma_{2}^{\mu}\nonumber\\
A_{2}^{\mu}  &  =\big((\epsilon_{2}-E_{2})+i{\frac{G}{2}}\gamma_{1}%
\cdot({\frac{\partial E_{2}}{E_{1}}}+\partial G)\gamma_{1}\cdot\hat
{P}\big )\hat{P}^{\mu}-(1-G)p^{\mu}+{\frac{i}{2}}\partial G\cdot\gamma
_{1}\gamma_{1}^{\mu}, \label{vecp}%
\end{align}
while the scalar potentials $S_{i}$ depend on $G$ and two additional invariant
functions $M_{1}$ and $M_{2}$
\begin{align}
S_{1}  &  =M_{1}-m_{1}-{\frac{i}{2}}G\gamma_{2}\cdot{\frac{\partial M_{1}%
}{M_{2}}}\nonumber\\
S_{2}  &  =M_{2}-m_{2}+{\frac{i}{2}}G\gamma_{1}\cdot{\frac{\partial M_{2}%
}{M_{1}}.} \label{scalp}%
\end{align}
Note that the terms in \ref{vecp} and \ref{scalp} which are explicitly
spin-dependent through the gamma matrices are essential in order to satisfy
the compatibility condition \ref{cmpt}. Later on, when the equation is reduced
to second-order \textquotedblright Pauli-form\textquotedblright, yet other
spin dependences eventually arise from gamma matrix terms that are independent
of the gamma matrices in the potentials. These are typical of what occurs in
the reduction of the one-body Dirac equation to the "Pauli form". In similar
fashion, the gamma matrices also give rise to spin independent terms in the
Pauli-forms. These terms emerge in a manner similar to the above two sources
of spin dependent terms in the Pauli-form of the equations.

In the case in which the space-like and time-like vectors are not independent
but combine into electromagnetic-like four-vectors, the constituent vector
interactions appear in a more compact form
\begin{align}
A_{1}^{\mu}  &  =\big(\epsilon_{1}-\frac{G(\epsilon_{1}-\epsilon_{2})}%
{2}+\frac{(\epsilon_{1}-\epsilon_{2})}{2G}\big)\hat{P}^{\mu}+(1-G)p^{\mu
}-\frac{i}{2}\partial G\cdot\gamma_{2}\gamma_{2}^{\mu}\nonumber\\
A_{2}^{\mu}  &  =\big(\epsilon_{2}-\frac{G(\epsilon_{2}-\epsilon_{1})}%
{2}+\frac{(\epsilon_{1}-\epsilon_{2})}{2G}\big)\hat{P}^{\mu}-(1-G)p^{\mu
}+\frac{i}{2}\partial G\cdot\gamma_{1}\gamma_{1}^{\mu}. \label{emvec}%
\end{align}
In that case $E_{1},E_{2}$ and $G$ are related to each other\cite{cra84,cra87}
(${\partial E_{1}/E_{2}}=-\partial G$) and for our QCD applications (as well
as for QED) are functions of only one invariant function $\mathcal{A}(r)$ in
which $r$ is the invariant
\begin{equation}
r\equiv\sqrt{x_{\perp}^{2}}.
\end{equation}
They take the forms
\begin{align}
E_{1}^{2}(\mathcal{A})  &  =G^{2}(\epsilon_{1}-\mathcal{A})^{2}),\nonumber\\
E_{2}^{2}(\mathcal{A})  &  =G^{2}(\epsilon_{2}-\mathcal{A})^{2},
\label{tvecp1}%
\end{align}
in which
\begin{equation}
G^{2}={\frac{1}{(1-2\mathcal{A}/w)}.}%
\end{equation}
In the forms of these equations used below, Todorov's collective energy
variable
\begin{equation}
\epsilon_{w}=(w^{2}-m_{1}^{2}-m_{2}^{2})/2w,
\end{equation}
will eventually appear.

In general $M_{1}$ and $M_{2}$ are related to each other\cite{cra82,cra87} and
for QCD applications are functions of two invariant functions $\mathcal{A}(r)$
and $S(r)$ appearing in the forms:
\begin{align}
M_{1}^{2}(\mathcal{A},S)  &  =m_{1}^{2}+G^{2}(2m_{w}S+S^{2})\nonumber\\
M_{2}^{2}(\mathcal{A},S)  &  =m_{2}^{2}+G^{2}(2m_{w}S+S^{2}), \label{mp}%
\end{align}
in which
\begin{equation}
m_{w}=m_{1}m_{2}/w.
\end{equation}
(In these equations, $m_{w}$ and $\epsilon_{w}$ are the relativistic reduced
mass and energy of the fictitious particle of relative motion introduced by
Todorov \cite{tod71,tod76}, which satisfy the effective one-body Einstein
condition
\begin{equation}
\epsilon_{w}^{2}-m_{w}^{2}=b^{2}(w).
\end{equation}
In the limit in which one of the particles becomes infinitely heavy, $m_{w}$
and $\epsilon_{w}$ reduce to the mass and energy of the lighter particle.) The
invariant function $S(r)$ is primarily responsible for the constituent scalar
potentials since $S_{i}=0$ if $S(r)=0$ , while $\mathcal{A}(r)$ contributes to
the $S_{i}$ (if $S(r)\not =0$) as well as to the vector potentials $A_{i}%
^{\mu}$. Originally, we derived these general forms for the scalar and vector
potentials using classical field theoretic arguments \cite{yng,cra92} (see
also \cite{tod71,cra82}). Surprisingly, the resulting forms for the mass and
energy potential functions $M_{i}$, $G$ and $E_{i}$ automatically embody
collective minimal substitution rules for the spin-independent parts of the
Schr\"{o}dinger-like forms of the equations. Classically those forms turn out
to be modifications of the Einstein condition for the free effective particle
of relative motion
\begin{equation}
p^{2}+m_{w}^{2}=\epsilon_{w}^{2}%
\end{equation}
For the vector interaction they automatically generate the replacement of
$\epsilon_{w}$ by $\epsilon_{w}-\mathcal{A}$ and for the scalar interaction
the replacement of $m_{w}$ by $m_{w}+S$. The part of Eqs.(\ref{clpdsa}%
,\ref{clpdsb}) that results from the vector and scalar interactions then takes
the form
\begin{equation}
(p^{2}+2m_{w}S+S^{2}+2\epsilon_{w}\mathcal{A}-\mathcal{A}^{2})\psi_{1,4}%
=b^{2}\psi_{1,4}.
\end{equation}
Now, we originally found these forms starting from relativistic classical
field theory. On the other hand, recently Jallouli and Sazdjian \cite{saz97}
obtained Eqs.(\ref{tvecp1}) and (\ref{mp}) in quantum field theory after
performing a necessarily laborious Eikonal summation to all orders of ladder
and cross ladder diagrams together with all constraint diagrams
(Lippmann-Schwinger like iterations of the simple Born diagram). Thus, the
structure first discovered simply in the correspondence limit has now been
verified through direct but difficult derivation from perturbative quantum
field theory.

These equations contain an important hidden hyperbolic structure (which we
could have used to introduce the interactions in the first place). To employ
it we introduce two independent invariant functions $L(x_{\perp})$ and
$\mathcal{G}(x_{\perp})$, in terms of which the invariant functions of
Eqs.(2.10,2.11) take the forms:
\begin{align}
M_{1}  &  =m_{1}\ \cosh L\ +m_{2}\sinh L\nonumber\\
M_{2}  &  =m_{2}\ \cosh L\ +m_{1}\ \sinh L \label{hyp}%
\end{align}%
\begin{align}
E_{1}  &  =\epsilon_{1}\ \cosh\mathcal{G}\ -\epsilon_{2}\sinh\mathcal{G}%
\nonumber\\
E_{2}  &  =\epsilon_{2}\ \cosh\mathcal{G}\ -\epsilon_{1}\ \sinh\mathcal{G}%
\end{align}%
\begin{equation}
G=\exp\mathcal{G}.
\end{equation}
In terms of $\mathcal{G}$ and the constituent momenta $p_{1}$ and $p_{2}$ ,
the individual four-vector potentials of Eq.(\ref{tvecp1}) take the suggestive
forms
\begin{align}
A_{1}  &  =[1-\mathrm{\cosh}(\mathcal{G})]p_{1}+\mathrm{\sinh}(\mathcal{G}%
)p_{2}-\frac{i}{2}(\partial\exp\mathcal{G}\cdot\gamma_{2})\gamma
_{2}\nonumber\\
A_{2}  &  =[1-\mathrm{\cosh}(\mathcal{G})]p_{2}+\mathrm{\sinh}(\mathcal{G}%
)p_{1}+\frac{i}{2}(\partial\exp\mathcal{G}\cdot\gamma_{1})\gamma_{1}%
\end{align}


In terms of these functions the coupled two-body Dirac equations then take the
form
\begin{subequations}
\begin{align}
\mathcal{S}_{1}\psi &  =\big(-G\beta_{1}\Sigma_{1}\cdot\mathcal{P}_{2}%
+E_{1}\beta_{1}\gamma_{51}+M_{1}\gamma_{51}-G{\frac{i}{2}}\Sigma_{2}%
\cdot\partial(\mathcal{G}\beta_{1}+L\beta_{2})\gamma_{51}\gamma_{52}%
\big)\psi=0\nonumber\\
\mathcal{S}_{2}\psi &  =\big(G\beta_{2}\Sigma_{2}\cdot\mathcal{P}_{1}%
+E_{2}\beta_{2}\gamma_{52}+M_{2}\gamma_{52}+G{\frac{i}{2}}\Sigma_{1}%
\cdot\partial(\mathcal{G}b_{2}+L\beta_{1})\gamma_{51}\gamma_{52}\big)\psi=0
\label{tbdes}%
\end{align}
in which
\end{subequations}
\begin{equation}
\mathcal{P}_{i}\equiv p-{\frac{i}{2}}\Sigma_{i}\cdot\partial\mathcal{G}%
\Sigma_{i}%
\end{equation}
depending on gamma matrices with block forms
\begin{subequations}
\[
\beta_{1}=\bigg({%
%TCIMACRO{\QATOP{1_{8}}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{1_{8}}{0}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{0}{-1_{8}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{-1_{8}}%
%EndExpansion
}\bigg),\ \ \gamma_{51}=\bigg({%
%TCIMACRO{\QATOP{0}{1_{8}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{1_{8}}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{1_{8}}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{1_{8}}{0}%
%EndExpansion
}\bigg),\ \ \beta_{1}\gamma_{51}\equiv\rho_{1}=\bigg({%
%TCIMACRO{\QATOP{0}{-1_{8}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{-1_{8}}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{1_{8}}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{1_{8}}{0}%
%EndExpansion
}\bigg)
\]%
\end{subequations}
\begin{subequations}
\[
\beta_{2}=\bigg({%
%TCIMACRO{\QATOP{\beta}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{\beta}{0}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{0}{\beta}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{\beta}%
%EndExpansion
}\bigg),\ \beta=\bigg({%
%TCIMACRO{\QATOP{1_{4}}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{1_{4}}{0}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{0}{-1_{4}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{-1_{4}}%
%EndExpansion
}\bigg)
\]%
\end{subequations}
\begin{subequations}
\[
\gamma_{52}=\bigg({%
%TCIMACRO{\QATOP{\gamma_{5}}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{\gamma_{5}}{0}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{0}{\gamma_{5}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{\gamma_{5}}%
%EndExpansion
}\bigg),\ \gamma_{5}=\bigg({%
%TCIMACRO{\QATOP{0}{1_{4}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{1_{4}}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{1_{4}}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{1_{4}}{0}%
%EndExpansion
}\bigg)
\]%
\end{subequations}
\begin{subequations}
\[
\beta_{2}\gamma_{52}\equiv\rho_{2}=\bigg({%
%TCIMACRO{\QATOP{\rho}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{\rho}{0}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{0}{\rho}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{\rho}%
%EndExpansion
}\bigg),\ \rho=\bigg({%
%TCIMACRO{\QATOP{0}{-1_{4}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{-1_{4}}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{1_{4}}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{1_{4}}{0}%
%EndExpansion
}\bigg)
\]%
\end{subequations}
\begin{subequations}
\[
\beta_{1}\gamma_{51}\gamma_{52}=\bigg({%
%TCIMACRO{\QATOP{0}{-\gamma_{5}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{-\gamma_{5}}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{\gamma_{5}}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{\gamma_{5}}{0}%
%EndExpansion
}\bigg),
\]%
\end{subequations}
\begin{subequations}
\[
\beta_{2}\gamma_{52}\gamma_{51}=\bigg({%
%TCIMACRO{\QATOP{0}{\rho}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{0}{\rho}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{\rho}{0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{\rho}{0}%
%EndExpansion
}\bigg).
\]%
\end{subequations}
\begin{equation}
\Sigma_{i}=\gamma_{5i}\beta_{i}\gamma_{\perp i}%
\end{equation}
Squaring either Dirac equation yields the unique second order
Schr\"{o}dinger-like equation \cite{bckr}
\begin{align*}
&  \big((G\Sigma_{1}\cdot\mathcal{P}_{2})^{2}-E_{1}^{2}+M_{1}^{2}+G^{2}%
{\frac{1}{4}}(\partial(\mathcal{G}+L))^{2}\\
&  +iG\Sigma_{1}\cdot\partial ln(E_{1}\beta_{1}+M_{1})(G\Sigma_{1}%
\cdot\mathcal{P}_{2})+iG\Sigma_{2}\cdot\partial ln(E_{2}\beta_{2}%
+M_{2})(G\Sigma_{2}\cdot\mathcal{P}_{1}\\
&  +[[G\Sigma_{1}\cdot\mathcal{P}_{2},G{\frac{i}{2}}\Sigma_{2}\cdot
\partial(\mathcal{G}+L\beta_{1}\beta_{2})]-{\frac{1}{2}}G^{2}\Sigma_{1}%
\cdot\partial ln(E_{1}\beta_{1}+M_{1})\partial(\mathcal{G}+L\beta_{1}\beta
_{2})\cdot\Sigma_{2}%
\end{align*}%
\begin{equation}
-{\frac{1}{2}}G^{2}\Sigma_{2}\cdot\partial ln(E_{2}\beta_{2}+M_{2}%
)\partial(\mathcal{G}+L\beta_{1}\beta_{2})\cdot\Sigma_{1}]\gamma_{51}%
\gamma_{52}\big )\psi=0. \label{sch}%
\end{equation}
(This equation is actually symmetric under interchange of particles 1 and 2.)
Performing simplifying Pauli matrix algebra and using the definition of the
mass and energy potentials, we find \cite{cra82,bckr} that the upper-upper
component of Eq.(\ref{sch}) becomes
\[
\big(p^{2}+2m_{w}S+S^{2}+2\epsilon_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\[
+{\frac{1}{4}}(\partial\mathcal{G}+\partial L)^{2}+iln^{\prime}\chi_{1}%
\chi_{2}\hat{r}\cdot p-{\frac{1}{2}}\partial^{2}\mathcal{G}+{\frac{3}{4}%
}(\mathcal{G}^{\prime})^{2}+{\frac{1}{2}}ln^{\prime}\chi_{1}\chi
_{2}\mathcal{G}^{\prime}%
\]%
\[
-{\frac{ln^{\prime}\chi_{1}}{r}}L\cdot\sigma_{1}-{\frac{ln^{\prime}\chi_{2}%
}{r}}L\cdot\sigma_{2}%
\]%
\[
+[{\frac{1}{3}}\partial^{2}\mathcal{G}-{\frac{1}{2}}(\mathcal{G}^{\prime}%
)^{2}-{\frac{1}{3}}ln^{\prime}\chi_{1}\chi_{2}\mathcal{G}^{\prime}]\sigma
_{1}\cdot\sigma_{2}%
\]%
\[
\lbrack-{\frac{1}{6}}(\mathcal{G}^{\prime\prime}-{\frac{\mathcal{G}^{\prime}%
}{r}})+{\frac{1}{6}}ln^{\prime}\chi_{1}\chi_{2}\mathcal{G}^{\prime}%
]S_{T}\big)\psi_{1}%
\]%
\[
+\big([-{\frac{1}{6}}ln^{\prime}\chi_{1}\chi_{2}(\mathcal{G}+L)^{\prime
}-{\frac{1}{2}}\mathcal{G}^{\prime}(\mathcal{G}+L)^{\prime}+{\frac{1}{6}%
}\partial^{2}(\mathcal{G}+L)]\sigma_{1}\cdot\sigma_{2}%
\]%
\begin{subequations}
\begin{equation}
\lbrack-{\frac{1}{6}}ln^{\prime}\chi_{1}\chi_{2}(\mathcal{G}+L)^{\prime
}+{\frac{1}{6}}((\mathcal{G}+L)^{\prime\prime}-{\frac{(\mathcal{G}+L)^{\prime
}}{r}})]S_{T}\big)\psi_{4}=b^{2}(w)\psi_{1} \label{cnstra}%
\end{equation}
in which $\chi_{i}\equiv(E_{i}+M_{i})/G$. This couples to the lower-lower
component of (\ref{sch}) which is
\[
\big(p^{2}+2m_{w}S+S^{2}+2\epsilon_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\[
+{\frac{1}{4}}(\partial\mathcal{G}+\partial L)^{2}+iln^{\prime}\bar{\chi}%
_{1}\bar{\chi}_{2}\hat{r}\cdot p-{\frac{1}{2}}\partial^{2}\mathcal{G}%
+{\frac{3}{4}}(\mathcal{G}^{\prime})^{2}+{\frac{1}{2}}ln^{\prime}\bar{\chi
}_{1}\bar{\chi}_{2}\mathcal{G}^{\prime}%
\]%
\[
-{\frac{ln^{\prime}\bar{\chi}_{1}}{r}}L\cdot\sigma_{1}-{\frac{ln^{\prime}%
\bar{\chi}_{2}}{r}}L\cdot\sigma_{2}%
\]%
\[
+[{\frac{1}{3}}\partial^{2}\mathcal{G}-{\frac{1}{2}}(\mathcal{G}^{\prime}%
)^{2}-{\frac{1}{3}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}\mathcal{G}^{\prime
}]\sigma_{1}\cdot\sigma_{2}%
\]%
\[
\lbrack-{\frac{1}{6}}(\mathcal{G}^{\prime\prime}-{\frac{\mathcal{G}^{\prime}%
}{r}})+{\frac{1}{6}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}\mathcal{G}%
^{\prime}]S_{T}\big)\psi_{4}%
\]%
\[
+\big([-{\frac{1}{6}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}(\mathcal{G}%
+L)^{\prime}-{\frac{1}{2}}\mathcal{G}^{\prime}(\mathcal{G}+L)^{\prime}%
+{\frac{1}{6}}\partial^{2}(\mathcal{G}+L)]\sigma_{1}\cdot\sigma_{2}%
\]%
\begin{equation}
\lbrack-{\frac{1}{6}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}(\mathcal{G}%
+L)^{\prime}+{\frac{1}{6}}((\mathcal{G}+L)^{\prime\prime}+{\frac
{(\mathcal{G}+L)^{\prime}}{r}})]S_{T}\big)\psi_{1}=b^{2}(w)\psi_{4}
\label{cnstrb}%
\end{equation}
in which $\bar{\chi}_{i}\equiv(E_{i}-M_{i})/G$. Eqs.(\ref{cnstra}) and
(\ref{cnstrb}) are two coupled eight-component Schr\"{o}dinger-like forms of
our equations that we use for our quark model bound state calculations for the
mesons in the present paper. They can be solved nonperturbatively not only for
quark model calculations but also for QED calculations since in that case
every term is quantum-mechanically well defined (less singular than
$-1/4r^{2}$). \ 

From these equations we obtain four coupled radial Schr\"{o}dinger-like
equations in the general case. But for $j=0$ or equal mass $S$ - states these
equations reduce to two coupled equations involving only upper-upper and
lower-lower components. The extra two-components for the general case arise
from orbital angular momentum mixing or spin mixing. The detailed radial forms
of these equations are given in Appendix A. For the case of QED ( $S=0$,
$\mathcal{A}=-\alpha/r),$ we have solved these coupled Schr\"{o}dinger-like
equations numerically obtaining results that are explicitly accurate through
order $\alpha^{4}$ (with errors on the order of $\alpha^{6}$)\cite{bckr}. We
have even obtained analytic solutions to the full system of coupled 16
component Dirac equations in the important case of spin-singlet positronium
\cite{va86}. For both numerical and analytic solution, the results agree with
those produced by perturbative treatment of these equations and with standard
spectral results.

Mourad and Sazdjian \cite{saz94} have recently shown how one may replace the
coupled equations (\ref{cnstra}) and (\ref{cnstrb})by single uncoupled ones
involving combinations $\psi_{1}\pm\psi_{4}$. In appendix B we present the
corresponding forms \cite{long} we have obtained from Eqs.(\ref{tbdes}). We
also present the radial forms of the corresponding equations that we have
obtained using a matrix and vector spherical harmonic decomposition of the
total wave function\cite{long}. We have repeated the earlier QED calculations
\cite{bckr} using these new forms of the equations. These new forms provide
important cross checks on our present numerical calculations of the meson spectrum.

\section{Meson Spectroscopy}

We use the constraints Eqs.(\ref{cnstra}) and (\ref{cnstrb} to construct a
relativistic naive quark model by choosing the two invariant functions
$\mathcal{G}$ and $L$ or equivalently $\mathcal{A}$ and $S$ to incorporate a
version of the static quark potential originally obtained from QCD by Adler
and Piran \cite{adl} through an intermediate effective field theory. We insert
into Eqs.(\ref{cnstra}) and (\ref{cnstrb}) invariants $\mathcal{A}$ and $S$
with forms determined so that the sum $\mathcal{A}+S$ appearing as the
potential in the nonrelativistic limit of our equations becomes the
Adler-Piran nonrelativistic $Q\bar{Q}$ potential (which depends on two
parameters $\Lambda$ and $U_{0})$ plus the Coulomb interaction between the
quark and antiquark. That is,
\end{subequations}
\begin{equation}
V_{AP}(r)+V_{coul}=\Lambda(U(\Lambda r)+U_{0})+{\frac{e_{1}e_{2}}{r}%
}=\mathcal{A}+S\ ,
\end{equation}
As determined by Adler and Piran, the short and long distance behaviors of
$U(\Lambda r)$ generate known lattice and continuum results. The Adler-Piran
potential incorporates asymptotic freedom through
\begin{equation}
\Lambda U(\Lambda r<<1)\sim1/(rln\Lambda r),
\end{equation}
and linear confinement through
\begin{equation}
\Lambda U(\Lambda r>>1)\sim\Lambda^{2}r.
\end{equation}
The long distance ( $\equiv\Lambda r>2$) behavior of the static potential
$V_{AP}(r)$ is given by
\[
\Lambda(c_{1}x+c_{2}ln(x)+c_{3}/\sqrt{x}+c_{4}/x+c_{5})
\]
in which $x=\Lambda r$ while the coefficients $c_{i}$ are given by the
Adler-Piran leading log-log model \cite{adl}. The nonrelativistic analysis
used by Adler and Piran does not determine the transformation properties of
the potential. How this potential is apportioned between vector and scalar is
therefore somewhat, although not completely, arbitrary. In earlier work
\cite{cra88} we divided the potential in the following way among three
relativistic invariants $\mathcal{V}(r),S$, and $\mathcal{A}$.(In our former
construction, the additional invariant $\mathcal{V}$ was responsible for a
possible independent time-like vector interaction.)%

\begin{align}
S  &  =\eta(\Lambda(c_{1}x+c_{2}ln(x)+c_{3}/\sqrt{x}+c_{5}+U_{0}),\nonumber\\
\mathcal{V}  &  =(1-\eta)(\Lambda(c_{1}x+c_{2}ln(x)+c_{3}/\sqrt{x}+c_{5}%
+U_{0}),\nonumber\\
\mathcal{A}  &  =V_{A}-S-\mathcal{V},
\end{align}
in which $\eta={\frac{1}{2}}$. That is, we assumed that (with the exception of
the Coulomb-like term ($c_{4}/x$)) the long distance part was equally divided
between scalar and a proposed time-like vector. In the present paper we drop
the time-like vector for reasons detailed below and assume instead that the
scalar interaction is solely responsible for the long distance confining terms
($\eta=1$). The attractive ($c_{4}=-0.58$) QCD Coulomb-like portion (not to be
confused with the electrostatic $V_{coul}$) is assigned completely to the
\textquotedblleft electromagnetic-like\textquotedblright\ part $\mathcal{A}$.
That is, the constant portion of the running coupling constant corresponding
to the exchange diagram is expected to be electromagnetic-like. Elsewhere we
have treated another model explicitly containing these features: the
Richardson potential. Its momentum space form
\begin{equation}
\tilde{V}(\vec{q})\sim1/\mathbf{q}^{2}ln(1+\mathbf{q}^{2}/\Lambda^{2})
\end{equation}
interpolates in a simple way between asymptotic freedom $\tilde{V}(\vec
{q})\sim1/\mathbf{q}^{2}ln(\mathbf{q}^{2}/\Lambda^{2})$ and linear confinement
$\tilde{V}(\mathbf{q})\sim1/\mathbf{q}^{4}$. Even though the Richardson
potential is not tied to any field theoretic base in the intermediate region
(unlike the Adler-Piran potential) and does not give as good fits to the data,
it does provide a convenient form for displaying our points about the static
quark potential. The Richardson radial form is
\begin{equation}
V(r)=8\pi\Lambda^{2}r/27-8\pi f(\Lambda r)/(27r)
\end{equation}
For $r\rightarrow0$, $f(\Lambda r)\rightarrow-1\ln(\Lambda r)$, while for
$r\rightarrow\infty$, $f(\Lambda r)\rightarrow1$. Thus, in this model, if the
confining part of the potential is a world scalar, then in the large $r$ limit
the remaining portion, regarded as an electromagnetic-like interaction
corresponding to our invariant function $\mathcal{A}(r)$, would be a Coulomb
potential with a coupling constant on the order of 1. This is in reasonable
agreement with the Adler model for which the Coulomb part of the potential is
also an attractive form of the order of $-1/r$. Support for the assumption
that the $c_{4}$ term belongs only to $\mathcal{A}$ also arises from
phenomenological considerations. We find that attempts to assign the $c_{4}$
term to the scalar potential have a drastic effect on the spin-spin and
spin-orbit splittings. In fact, using this term in $S$ through Eqs.(\ref{mp})
generates spin-spin and spin-orbit splittings that are much too small.

In our previous work, we divided the confining part equally between scalar and
time-like vector so that the spin-orbit multiplets would not be inverted. This
was done in order to obtain from our model the $a_{0}(980)$ meson which was
then considered as the prime candidate for the relativistic counterpart of the
$^{3}P_{0}$ meson. However, recent analysis indicates that that meson may be
instead a meson-meson bound state while a meson with mass of 1450 MeV may be
the correct candidate for the quark model state \cite{prtl}. Interpretation of
this other state as the $^{3}P_{0}$ meson would in fact require a partial
inversion of the spin-orbit triplet (from what one would expect based on the
positronium analog). This partial inversion is consistent with the $^{3}P_{0}$
candidate for the $u\bar{s}$ system also appearing in a position that inverts
the spin-orbit splitting. Since the sole purpose of including $\mathcal{V}$ in
our previous treatment was to prevent the inversion, we exclude it from our
present treatment. In our older treatment \cite{cra88}, we simplified the
numerics by neglecting the tensor coupling, unequal mass spin-orbit difference
couplings, and the $u-d$ quark mass differences. In the present treatment, we
treat the entire interaction present in our equations, thereby keeping each of
these effects. In our former treatment we also performed a decoupling between
the upper-upper and lower-lower components of the wave functions for
spin-triplet states which turned out to be defective but which we subsequently
corrected in our numerical test of our formalism for QED \cite{bckr}. The
corrected decoupling (appearing in both the coupled Eqs.(\ref{cnstra}) and
(\ref{cnstrb} and\ in Eq.(\ref{bliu}) in the appendix) is included in our new
meson calculations.

In the present investigation, we compute the best fit meson spectrum for the
following apportionment of the Adler-Piran potential:
\begin{subequations}
\begin{equation}
\mathcal{A}=exp(-\beta r)[V_{AP}-c_{4}/r)]+c_{4}/r+\frac{e_{1}e_{2}}%
{r},\ \ \label{apa}%
\end{equation}%
\begin{equation}
S=V_{AP}+\frac{e_{1}e_{2}}{r}-\mathcal{A}=(V_{AP}-c_{4}/r)(1-exp(-\beta r))
\label{aps}%
\end{equation}
In order to covariantly incorporate the Adler-Piran potential into our
equations, we treat the short distance portion as purely electromagnetic-like
(in the sense of the transformation properties of the potential). Through the
additional parameter $\beta$, the exponential factor gradually turns off the
electromagnetic-like contribution to the potential at long distance except for
the $1/r$ portion mentioned above, while the scalar portion gradually turns
on, becoming fully responsible for the linear confining and subdominant terms
at long distance. Altogether our potentials depend on three parameters:
$\Lambda,U_{0},$ and $\beta$.

When inserted into the constraint equations, $S$ and $\mathcal{A}$ become
relativistic invariant functions of the invariant separation $r=\sqrt
{x_{\perp}^{2}}$ . The covariant structures of the constraint formalism then
embellish the central static potential with accompanying spin-dependent and
recoil terms.

We now turn to a subject that must be dealt with whenever a relativistic wave
equation is to be solved numerically for the case of strong potentials, not
just used as a structure for low-order perturbative calculation as commonly
done. So far, we have invoked general properties of the basic Dirac equations
such as mathematical consistency along with use of the nonrelativistic limit
of our equations to identify a plausible degeneration of a relativistic
potential structure to the nonrelativistic Adler-Piran potential. But we must
now deal with effects in the two-body equation that are the counterparts of
the Klein Paradox in one-body equations. That is, as treated up to this point
in the formalism it is possible that $\mathcal{A}$ and $S$ as identified from
the nonrelativistic limit can take on all values between positive and negative
infinity. Their use as legitimate potentials in the full relativistic
equations necessarily requires modification of $E_{i},M_{i}$ and $G$ so that
these interaction functions at least remain real when one of the masses
becomes very large or when $\mathcal{A}$ becomes large and repulsive. These
modifications are not arbitrary but must maintain correct limits.

For $E_{i}=G(\epsilon_{i}-\mathcal{A})$ to be real we need only ensure that
$G$ be real which requires that $\mathcal{A}\leq w/2$. This restriction on
$\mathcal{A}$ is enough to ensure that $M_{i}=G\sqrt{m_{i}^{2}(1-2\mathcal{A}%
/w)+2m_{w}S+S^{2}}$ be real as well (so long as $S\geq0)$. (As we shall show
below in our discussion on the static limit, the case of $S<0$ does not
require any further restrictions.) In order that $\mathcal{A}$ satisfy this
inequality, we must modify it and $S$ so that $S(r)+\mathcal{A}(r)=V_{AP}%
(r)+e_{1}e_{2}/r\equiv A+\bar{S}$ with $A$ and $\bar{S}$ given by the right
hand sides of Eq.(\ref{apa}) and Eq.(\ref{aps}) respectively. Then we
reidentify $\mathcal{A}$ and $S$ such that
\end{subequations}
\begin{align}
\mathcal{A}  &  =A,\ A\leq0,\nonumber\\
\ \mathcal{A}  &  =\frac{AA_{0}}{\sqrt{A^{2}+A_{0}^{2}}},\ \ A>0\nonumber\\
S  &  =\bar{S}+A-\mathcal{A} \label{sas}%
\end{align}
where $A_{0}=w/2$. This parametrization gives $\mathcal{A}$ and $S$ that are
continuous through their first derivatives.

We next consider problems that may arise in the limit that one of the masses
becomes very large. We must modify the $M_{i}$ so that they have the correct
static limits (when say $m_{2}\rightarrow\infty$). At first sight,
Eq.(\ref{mp}) does appear to give $M_{1}\rightarrow m_{1}+S$. (Note
$G\rightarrow1$ in this heavy-mass limit). However, this limit is only true if
$m_{1}+S\geq0$. In fact, as it stands the first of the two-body Dirac
equations in (\ref{tbdea},\ref{tbdeb}) would go over to the one-body Dirac equation%

\begin{equation}
(\gamma_{1}\cdot p_{1}+|m_{1}+S|)\psi=0.
\end{equation}


The spectrum produced by this equation would probably deviate significantly
from that of the standard Dirac equation due to contributions from the region
of strong attractive scalar potential ($S<-m_{1}$). We wish to correct this
defect but in such a way that the resulting modification of $M_{i}$ not
interfere with the ability of our formalism to regenerate the correct
perturbative results in a model field theory. In \cite{bckr} we showed that
for equal masses the Todorov form of $M_{i}$ given in Eq.(\ref{mp}) for
$\mathcal{A}=0$ and $S=-\alpha/r$ yields a numerical spectrum in accord with
one derived from a perturbative treatment of the weak potential forms of our
equations. Even though the predictions of such a model field theory cannot be
checked against any experimental spectral results, the formalism should be
well defined. In contrast, it has long been known that a nonperturbative
(numerical) treatment of the Breit equation cannot yield spectral results that
agree to an appropriate order with a perturbative treatment of its own
semirelativistic form. Mathematical consistency is, or should be, a minimum
restriction on a proposed covariant bound state equation. What we neglected to
check in our constraint formalism in \cite{bckr} for scalar potentials was the
case $m_{2}>>m_{1}$ with strong coupling. We find that the parametrization
given in Eq.({\ref{mp}) fails when inserted into the Two-Body Dirac Equations.
For example, in the extreme case in which $m_{2}=10,000m_{1}$ with strong
Coulomb interaction ($\alpha=0.5$), we find that the numerical treatment of
the binding energy using the two-body constraint equations ((with Eq.
(\ref{mp})) for the case $\mathcal{A}=0$) does not yield a value for the
ground state energy level that is compatible with that predicted by the one
body Dirac equation with $M_{1}=m_{1}-\alpha/r$. To correct this problem for
the treatment of the present paper, we now take advantage of the hyperbolic
parametrization given in Eqs.(\ref{hyp}a-b). Let us assume that $L$ is a
monotonic function of $S$ for $S\geq0$ given by
\begin{equation}
\exp(L)=\frac{M_{10}+M_{20}}{m_{1}+m_{2}}\equiv\exp(L_{0}(S)),\ S\geq0,
\end{equation}
in which $M_{i0}$ are defined as the forms of $M_{i}$ given in Eqs.(\ref{mp}%
a-b). We choose this form since only in the region where $S$ is large and
negative do we expect problems. We desire a form for $M_{i}$ which goes over
to $m_{i}+S$ when $S$ becomes large and negative and one of the constituent
rest-masses is large. At the same time we require an expression that has the
correct weak potential behavior. The original forms in Eq.(\ref{mp}) for
$M_{i}$ do have the correct weak potential form. So, for weak potential, using
Eq.(\ref{hyp}a) and Eq.(\ref{mp}a) we solve for $\sinh(L)$ and obtain
\begin{equation}
\sinh(L)\approx\frac{S}{w}(1+\frac{\epsilon_{w}S}{wm_{w}})
\end{equation}
We then need a modification of this valid in regions of large negative $S$
that at $S=0$ is continuous through second derivatives with that obtained
above, yields correct strong coupling static limit spectral results, and
yields correct weak coupling equal mass results \textit{obtained
nonperturbatively (i.e. numerically)}. Although these restrictions are not
sufficient to uniquely define an extrapolated $\sinh(L)$ they are severe
enough to narrow the choices significantly. For the present treatment, we
choose
\begin{equation}
\sinh(L)=\frac{S}{w}(1+\frac{\epsilon_{w}S}{\sqrt{w^{2}+S^{2}}m_{w}})
\end{equation}
This satisfies the continuity condition and gives numerical results that
satisfy the other two restrictions. Consequently, for }$S<0$ and large $m_{2}$
we obtain a spectrum (both ground and excited states) that for this extreme
case agrees with that of the exact solution obtained from the one-body Dirac
equation
\begin{equation}
(\gamma_{1}\cdot p_{1}+m_{1}+S)\psi=0.
\end{equation}
Furthermore, \ we obtain a spectrum for equal mass and unequal mass cases that
agrees with the standard perturbative results as given in \cite{bckr}.
\ {Another choice that works just as well is
\begin{equation}
\sinh(L)=\frac{S}{w}(1+\frac{S}{\sqrt{(wm_{w}/\epsilon_{w})^{2}+S^{2}}w})
\end{equation}
while one that satisfies the continuity condition, works well numerically for
the extreme limit above, but fails the perturbative test is
\[
\sinh(L)=\frac{S}{w}(1+\frac{\epsilon_{w}S}{\sqrt{m_{w}^{2}+S^{2}}w})
\]
}

Since our applications in QCD combine both scalar and electromagnetic-like
vector interactions we must impose similar conditions for the case of
Eq.(\ref{mp}a-b) with $\mathcal{A}\neq0$. Following the same procedure
combining Eq.(\ref{hyp}a) and Eq.(\ref{mp}) for weak $S$ but arbitrary
$\mathcal{A}$ yields
\begin{equation}
\sinh(L)\approx\frac{S}{wD}(1+\frac{(\epsilon_{w}-A)S}{wDm_{w}})
\end{equation}
in which $D=1/G^{2}=1-2\mathcal{A}/w$. We then test our assumption for the
case $S=\mathcal{A}=-\alpha/r$, as before for the extreme cases of unequal
mass large coupling and equal mass small coupling. We find an extrapolation
that works reasonably well:
\begin{equation}
\sinh(L)=\frac{S}{wD}(1+\frac{(\epsilon_{w}-\mathcal{A})S}{\sqrt{w^{2}+S^{2}%
}Dm_{w}}). \label{lsa}%
\end{equation}
We note that these requirements rule out the plausible choice
\[
\sinh(L)=\frac{S}{wD}(1+\frac{S}{\sqrt{m_{w}Dw/(\epsilon_{w}-\mathcal{A}%
))^{2}+S^{2}}})
\]
whose $\mathcal{A}=0$ counterpart worked well above.

We emphasize that a crucial feature of our $\sinh(L)$ extrapolations is that
for fixed $S$, in the static limit (e.g. $m_{2}>>m_{1}$) $\sinh(L)\rightarrow
S/w$ which leads to $M_{1}\rightarrow m_{1}+S$. \ Note that as opposed to what
happens for scalar potentials, strong $\mathcal{A}$ potentials have no problem
in the static limit where the restriction $\mathcal{A}<w/2\rightarrow\infty$
on $\mathcal{A}$ is automatically satisfied.

\section{Numerical Spectral Results}

\subsection{Tabulation and Discussion of Computed Meson Spectra}

We now use our formalism as embodied in Eqs. (\ref{cnstra}), (\ref{cnstrb}),
and (\ref{apa},\ref{aps},\ref{sas},\ref{lsa}) to calculate the full meson
spectrum including the light-quark mesons. (As a check on these calculations
we have also used the more recently derived form Eq.(\ref{bliu}). Note that
the realistic Adler-Piran potential fails for light mesons when used in the
ordinary nonrelativistic Schr\"{o}dinger equation since that equation's lack
of relativistic kinematics leads to increasing meson masses as the quark
masses drop below a certain point \cite{cra81}, thereby spoiling proper
treatment of the pion, as well as other states. Here, we shall see how our
relativistic equations remedy this situation. In addition to including the
proper relativistic kinematics, our equations also contain energy dependence
in the dynamical quasipotential. Mathematically, this feature turns our
equations into wave equations that depend nonlinearly on the eigenvalue. Their
solution, which we have treated in detail elsewhere (see \cite{cra88,crcmp}),
requires an efficient iteration scheme to supplement our algorithm for the
eigenvalue $b^{2}(w)$ when our equations are written as coupled
Schr\"{o}dinger-like forms.

We display our results in Tables I at the end of the paper. In the first two
columns of each of the tables we list quantum numbers and experimental rest
mass values for 89 known mesons. We include all well known and plausible
candidates listed in the standard reference (\cite{prtl}). We omit only those
mesons with substantial flavor mixing. In the tables, the quantum numbers
listed are those of the upper-upper part of the sixteen component wave
function. To generate the fits the only significant QCD parameter that we
employ (other than the quark masses) is $\Lambda$. We merely insert the static
Adler-Piran potential into our relativistic wave equations just as we have
inserted the Coulomb potential $\mathcal{A}=-\alpha/r$ to obtain the results
of QED\cite{va86,bckr}. Note especially that we use a single $\Phi
(\mathcal{A},S)$ for all quark mass ratios - hence a single structure for all
the $\bar{Q}Q,\ \bar{Q}q,$ \textit{and} $\bar{q}q$ mesons in a single overall
fit. In the third column in Table I we present the results for the model
defined by Eqs.(\ref{apa},\ref{aps}). The entire confining part of the
potential in this model transforms as a world scalar. In our equations, this
structure leads to linear confinement at long distances and quadratic
confinement at extremely long distances (where the quadratic contribution
$S^{2}$ outweighs the linear term $2m_{w}S$). At distances at which
$\exp(-\beta r)<<1,$ the corresponding spin-orbit, Thomas, and Darwin terms
are dominated by the scalar interaction, while at short distances $\exp(-\beta
r)\sim1$ the electromagnetic-like portion of the interaction gives the
dominant contribution to the spin-orbit fine-structure. Furthermore because
the signs of each of the Thomas and Darwin terms in the Pauli-form of our
Dirac equations are opposite for the scalar and vector interactions, the
spin-orbit contributions of those parts of the interaction produce opposite
effects depending on the size of the quarkonium atom.

We obtain the meson masses given in column three as the result of a least
squares fit using the known experimental errors, an assumed calculational
error of 1.0 MeV, and an independent error conservatively taken to be 5\% of
the total width of the meson. We employ the calculational error not to
represent the uncertainty of our algorithm but instead to prevent the mesons
that are stable with respect to the strong interaction from being weighted too heavily.

The resulting best fit turns out to have quark masses $m_{b}=4.877,\ m_{c}%
=1.507,\ m_{s}=0.253,\ m_{u}=0.0547,\ m_{d}=.0580\ \mathrm{GeV}$ , along with
potential parameters $\Lambda=0.216,\Lambda U_{0}=1.865\ \mathrm{GeV}$ and
inverse distance parameter $\beta=1.936\ \mathrm{GeV}$. This value of $\beta$
implies that (in the best fit) as the quark separation increases, our
apportioned Adler-Piran potential switches from primarily vector to scalar at
about 0.1 fermi. This shift is a relativistic effect since the effective
nonrelativistic limit of the potential ($\mathcal{A}+S$) exhibits no such
shift (i.e., by construction $\beta$ drops out).

In Table I, the numbers given in parentheses to the right of the experimental
meson masses are experimental errors in $\mathrm{MeV}$. The numbers given in
parentheses to the right of the predicted meson masses are the contribution of
that meson's calculation to the total $\chi^{2}$ of 101.

The 17 mesons that contain a $b$ (or $\bar{b}$) quark contribute a total of
about 5.4 to the $\chi^{2}$, at an average of about 0.3 each. This is the
lowest contribution of those given by any family. Since the Adler-Piran
potential was originally derived for static quarks, one should not be
surprised to find that most of the best fit mesons are members of the least
relativistic of the meson families. Note, however, that five of the best fit
mesons contain highly relativistic $u$ and $s$ quarks.

The 24 mesons that contain a $c$ (or $\bar{c}$) quark contribute a total of
about 50.7 to the $\chi^{2}$ at an average of about 2.2 each. This is the
highest contribution of those given by any family. A significant part of this
contribution is due to the $\psi$ meson mass being about 32 MeV off its
experimental value. Another part of the contribution is due to fact that the
mass of the high orbital excitation of the $D^{\ast}$ tensor meson is 80 MeV
below its experimental value. In addition, the high orbital excitation of the
$D_{s}^{\ast}$ is 60 MeV low.

The 24 mesons that contain an $s$ (or $\bar{s}$) quark contribute a total of
about 46.3 to the $\chi^{2}$ at an average of about 1.3 each. This is
important because the $s$ quarks are lighter than the $c$ quarks. Part of the
reason for this effect is that our $\chi^{2}$ fitting procedure accounts for
the fact that our meson model ignores the level shifts due to the instability
of many of the mesons that contain an $s$ quark through the introduction of a
theoretical error on the order of 5\% of the width of the unstable mesons.

The 36 mesons that contain a $u$ (or $\bar{u}$) quark contribute a total of
about 54.6 to the $\chi^{2}$ at an average of about 1.5 each while the 16
mesons on our list that contain a $d$ (or $\bar{d}$) quark contribute a total
of about 18.6 to the $\chi^{2}$ at an average of about 1.2 each.

The worst fits produced by our model are those to the $\psi$ and the $D^{*}$
and $D_{s}^{*}$ high orbital excitations. Although two of these mesons contain
the light $u$ and $d$ quarks, in our fit the more relativistic bound states
are not in general fit less well. In fact, the $\pi,K,D$ and $\rho$ mesons are
fit better than these two excited $D^{*}$ and $D_{s}^{*}$ mesons.

We see that over all, the two-body Dirac equations together with the
relativistic version of the Adler-Piran potential account very well for the
meson spectrum over the entire range of relativistic motions, using just the
two parametric functions $\mathcal{A}$ and $S$.

We now examine another important feature of our method: the goodness with
which our equations account for spin-dependent effects (both fine- and
hyperfine- splittings). Table I shows that the best fit versus experimental,
ground state singlet-triplet splittings for the $b\bar{u},\ b\bar{s}%
,\ c\bar{c},\ c\bar{u},\ c\bar{d},\ c\bar{s},\ s\bar{u},\ s\bar{d},\ u\bar{d}$
systems are 48 vs 46, 59 vs 47, 151 vs 117, 134 vs 142, 132 vs 142, 147 vs
144, 418 vs 398, 418 vs 398, and finally 648 vs 627 MeV. We obtain a uniformly
good fit for all hyperfine ground state splittings except for the $\eta
_{c}-\psi$ system. The problem with the fit for that system of mesons occurs
because the $D^{\ast}\ ^{3}P_{2}$ and $D_{s}^{\ast}\ ^{3}P_{2}$ states are
significantly low while the $\psi$ is significantly high. Furthermore, the
singlet and triplet $P$ states are uniformly low. An attempt to lower the $c$
quark mass by correcting the $\psi$ mass while raising the charmonium and
$D^{\ast},D_{s}^{\ast}\ P$ state masses would require raising the $c$ quark
mass. Reducing one discrepancy would worsen the other. Below, we will uncover
what we believe is the primary cause for this discrepancy as we examine other
aspects of the spectrum.

For the spin-orbit splittings we obtain values for the $R$ ratios $(^{3}%
P_{2}-^{3}P_{1})/(^{3}P_{1}-^{3}P_{0}))$ of 0.71,0.67,0.42,-0.19,-0.58,-3.35
for the two $b\bar{b}$ triplets, and the $c\bar{c},s\bar{s},u\bar{s},u\bar{d}$
spin triplets compared to the experimental ratios of
0.66,0.61,0.48,0.09,-0.97,-0.4. This fit ranges from very good in the case of
the light $\Upsilon$ multiplet to miserably bad for the two lightest
multiplets. From the experimental point of view some of the problem may be
blamed on the uncertain status of the $^{3}P_{0}$ light quark meson bound
states and the spin-mixing in the case of the $K^{\ast}$ multiplet. From the
theoretical point of view, the lack of any mechanism in our model to account
for the effects of decay rates on level shifts undoubtedly has an effect.
Another likely cause is that as we proceed from the heavy mesons to the light
ones, the radial size of the meson grows so that the long distance
interactions, in which the scalar interactions become dominant, play a more
important role. The Thomas precession terms due to scalar interactions are
opposite in sign to and tend to dominate those due to vector interactions.
This results in partial to full multiplet inversions as we proceed from the
$s\bar{s}$ to the $u\bar{d}$ mesons. This inversion mechanism may also be
responsible for the problems of the two orbitally excited $D^{\ast}$ and
$D_{s}^{\ast}$ mesons described above. It may be responsible as well for the
problem of the singlet and triplet $P$ states since the scalar interaction
tends to offset the dominant shorter range vector interaction, at least slightly.

We also examine the effect of the hyperfine structure of our equations on the
splitting between the $^{1}P_{1}$ and weighted sum $[5(^{3}P_{2})+3(^{3}%
P_{1})+1(^{3}P_{0})]/9$ of bound states. We obtain pairs of values equal to
3.520,3.520;1.408,1.432; 1.392,1416 for the $c\bar{c},u\bar{s},u\bar{d}$
families versus the experimental pairs of 3.526,3.525;1.402,1.375;1.231,1.303.
The agreement of the theoretical and experimental mass differences is
excellent for the $\psi$ system, slightly too large and of the wrong sign for
the $K$ system and too small and of the wrong sign for the $u\bar{d}$ system.
Part of the cause of this pattern is that pure scalar confinement worsens the
fit for the light mesons because of its tendency to reverse the spin-orbit
splitting, thereby shifting the center of gravity. The agreement, however, for
the light systems is nevertheless considerably better than that in the case of
the fine structure splitting $R$ ratios. Another part of the discrepancy may
be due to the uncertain status of the light $^{3}P_{0}$ meson as well as the
spin-mixing in the case of the $K^{\ast}$ multiplet. Note that in the case of
unequal mass $P$ states, our calculations of the two values incorporate the
effects of the $\vec{L}\cdot(\vec{s}_{1}-\vec{s}_{2})$ spin-mixing effects.
(The use of nonrelativistic notation is only for convenience.)

These differences between heavy and light meson systems also occur in the
mixing due to the tensor term between radial $S$ and orbital $D$ excitations
of the spin-triplet ground states. This mixing occurs most notably in the
$c\bar{c},u\bar{s}$ and $u\bar{d}$ systems. The three pairs of values that we
obtain are 3.808,3.688;1.985,1.800;1.986,1.775 respectively versus the data
3.770,3.686;1.714,1.412;1.700;1.450. Our results are quite reasonable for the
charmonium system but underestimate considerably the splitting for the light
quark systems. As happened for the significant disagreement in the case of the
fine structure, our results here worsen significantly for the light meson
systems. The spectroscopy of the lighter mesons is undoubtedly more complex
due to their extreme instability (not accounted for in our approach). Note,
however, that for the spin-spin hyperfine splittings of the ground states the
more relativistic (lighter quark) systems yield masses that agree at least as
well with the experimental data as do the heavier systems. This same mixed
behavior shows up again for the radial excitations.

The incremental $\chi^{2}$ contributions for the six $^{3}S_{1}$ states of the
$\Upsilon$ system is just 1.8. It is 12.9 over three states for the triplet
charmonium system (primarily due to the $\psi$ deviation), 3.0 for the two
$\phi$ states, 1.6 for the three $^{1}S_{0}$ states of the $K$ system (note,
however that these fits include expected errors due to the lack of level shift
mechanisms and are thus reduced), 7.3 for the two $^{3}S_{1}$ states of for
the $K^{\ast}$ system, 2.2 for the three triplet $u\bar{d}$ states and 8.2 for
the three singlet $u\bar{d}$ states . The $\chi^{2}$ contribution at first
increases, then decreases with the lighter systems. Overall, the masses are
much too large for the radially excited light quark mesons. These
discrepancies may be due both to neglect of decay-induced level shifts and to
the increased confining force for large $r$ from linear to quadratic (there is
no term to compensate for the quadratic $S^{2}$ term).

The isospin splitting that we obtain for the spin singlet $B$ meson system is
1 MeV. Our calculation includes the contribution from the $u-d$ mass
difference of 3.3 MeV as well as that due to different charge states. The
effect of the latter tends to offset the effects of the former since the $b$
and $\bar{u}$ have the same sign of the charge while the $b$ and $\bar{d}$
have the opposite while $m_{d}>m_{u}$. In the experimental data this offset is
complete (0). In the case of the $D^{+}-D^{0}$ splitting our mass difference
of 7 MeV represents the combined effects of the $u-d$ mass difference and the
slightly increased electromagnetic binding present in the case of the $D^{0}$
and the slightly decreased binding in the case of the $D^{+}$. The
experimental mass difference is just 4 MeV. These effects work in the same way
for the spin-triplet splitting resulting in the theoretical value 5 MeV
compared with the experimental value 3 MeV. For the $^{3}P_{2}$ isodoublet we
obtain 4 MeV versus about 0 for the experimental values. Our isospin
splittings are enhanced because of the large $u-d$ quark mass difference that
gives the best overall fit. For the $K-K^{\ast}$ family the experimental value
for the isospin splitting is $4\ $MeV for the singlet and triplet ground
states. This splitting actually grows for the orbital excitation ($K_{2}%
^{\ast}$) to 7 MeV. The probable reason for this increase is that at the
larger distances, the weak influence of the Coulomb differences becomes small
while only the actual $u-d$ mass difference influences the result (although it
does seem rather large). It is difficult to understand why our results stay
virtually zero for all three isodoublets. Note that as with the $B$ doublets,
the theoretical contributions of the combined effects of the $u-d$ mass
differences and the electrostatic effects tend to cancel. However, the
experimental masses do not show this expected cancellation.

Overall, comparison with the experimental data shows that the primary strength
of our approach is that it provides very good estimates for the ground states
for all families of mesons and for the radial excitation and fine structure
splittings for the heavier mesons. On the other hand, it overestimates the
radial and orbital excitations for the light mesons. Its worst results are
those for the fine structure splittings for the light mesons. Both weaknesses
are probably due to long distance scalar potential effects. Below, we shall
discuss other aspects of our fit to the spectrum when we compare its results
to those of other approaches to the relativistic two-body bound state problem.

\subsection{Explicit Numerical Construction of Meson Wave Functions\bigskip}

There are 89 mesons in our fit to the meson spectrum. \ An important advantage
of the constraint formalism is that its local wave equation provides us with a
direct way to picture the wave functions. As examples, we present the wave
functions that result from our fits to the three mesons: the $\pi$ , for which
we present the radial part of $\psi_{1}+\psi_{4}$ that solves Eq.(\ref{bliu}%
)$;$%

%TCIMACRO{\FRAME{ftbpFU}{4.0465in}{2.5054in}{0pt}{\Qcb{The $\pi$ wave function
%plotted against $x=\log(r/r_{0})$ }}{\Qlb{pi}}{plttthdr3.eps}%
%{\special{ language "Scientific Word";  type "GRAPHIC";
%maintain-aspect-ratio TRUE;  display "USEDEF";  valid_file "F";
%width 4.0465in;  height 2.5054in;  depth 0pt;  original-width 3.9972in;
%original-height 2.4639in;  cropleft "0";  croptop "1";  cropright "1";
%cropbottom "0";  filename 'plttthdr3.eps';file-properties "XNPEU";}} }%
%BeginExpansion
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.5054in,
width=4.0465in
]%
{plttthdr3.eps}%
\caption{The $\pi$ wave function plotted against $x=\log(r/r_{0})$ }%
\label{pi}%
\end{center}
\end{figure}
%EndExpansion
the $\rho$ for which we present the radial part of the wave functions
$\psi_{1}+\psi_{4}$ for both $S$ and $D$ states that solve Eq.(\ref{bliu} ;
%TCIMACRO{\FRAME{ftbpFU}{4.0465in}{2.5054in}{0pt}{\Qcb{The $\rho(S)$ wave
%functions plotted against $x=\log(r/r_{0})$ }}{\Qlb{rhos}}{plttthdr1.eps}%
%{\special{ language "Scientific Word";  type "GRAPHIC";
%maintain-aspect-ratio TRUE;  display "USEDEF";  valid_file "F";
%width 4.0465in;  height 2.5054in;  depth 0pt;  original-width 3.9972in;
%original-height 2.4639in;  cropleft "0";  croptop "1";  cropright "1";
%cropbottom "0";  filename 'plttthdr1.eps';file-properties "XNPEU";}} }%
%BeginExpansion
\begin{figure}
[ptbptb]
\begin{center}
\includegraphics[
height=2.5054in,
width=4.0465in
]%
{plttthdr1.eps}%
\caption{The $\rho(S)$ wave functions plotted against $x=\log(r/r_{0})$ }%
\label{rhos}%
\end{center}
\end{figure}
%EndExpansion
%TCIMACRO{\FRAME{ftbpFU}{4.0465in}{2.5054in}{0pt}{\Qcb{The $\rho(D)$ wave
%function plotted against $x=\log(r/r_{0})$ }}{\Qlb{rhod}}{plttthdr2.eps}%
%{\special{ language "Scientific Word";  type "GRAPHIC";
%maintain-aspect-ratio TRUE;  display "USEDEF";  valid_file "F";
%width 4.0465in;  height 2.5054in;  depth 0pt;  original-width 3.9972in;
%original-height 2.4639in;  cropleft "0";  croptop "1";  cropright "1";
%cropbottom "0";  filename 'plttthdr2.eps';file-properties "XNPEU";}} }%
%BeginExpansion
\begin{figure}
[ptbptbptb]
\begin{center}
\includegraphics[
height=2.5054in,
width=4.0465in
]%
{plttthdr2.eps}%
\caption{The $\rho(D)$ wave function plotted against $x=\log(r/r_{0})$ }%
\label{rhod}%
\end{center}
\end{figure}
%EndExpansion
and the $\psi/J$ for which we present the radial part of the wave functions
$\psi_{1}+\psi_{4}$ for both $S$ and $D$ states that solve Eq.(\ref{bliu}).%
%TCIMACRO{\FRAME{ftbpFU}{4.0465in}{2.5054in}{0pt}{\Qcb{The $\psi(S)$ wave
%function plotted against $x=\log(r/r_{0})$ }}{\Qlb{psis}}{plttthdr4.eps}%
%{\special{ language "Scientific Word";  type "GRAPHIC";
%maintain-aspect-ratio TRUE;  display "USEDEF";  valid_file "F";
%width 4.0465in;  height 2.5054in;  depth 0pt;  original-width 3.9972in;
%original-height 2.4639in;  cropleft "0";  croptop "1";  cropright "1";
%cropbottom "0";  filename 'plttthdr4.eps';file-properties "XNPEU";}} }%
%BeginExpansion
\begin{figure}
[ptbptbptbptb]
\begin{center}
\includegraphics[
height=2.5054in,
width=4.0465in
]%
{plttthdr4.eps}%
\caption{The $\psi(S)$ wave function plotted against $x=\log(r/r_{0})$ }%
\label{psis}%
\end{center}
\end{figure}
%EndExpansion
%TCIMACRO{\FRAME{ftbpFU}{4.0465in}{2.5054in}{0pt}{\Qcb{The $\psi(D)$ wave
%function plotted against $x=\log(r/r_{0})$ }}{\Qlb{psid}}{plttthdr5.eps}%
%{\special{ language "Scientific Word";  type "GRAPHIC";
%maintain-aspect-ratio TRUE;  display "USEDEF";  valid_file "F";
%width 4.0465in;  height 2.5054in;  depth 0pt;  original-width 3.9972in;
%original-height 2.4639in;  cropleft "0";  croptop "1";  cropright "1";
%cropbottom "0";  filename 'plttthdr5.eps';file-properties "XNPEU";}} }%
%BeginExpansion
\begin{figure}
[ptbptbptbptbptb]
\begin{center}
\includegraphics[
height=2.5054in,
width=4.0465in
]%
{plttthdr5.eps}%
\caption{The $\psi(D)$ wave function plotted against $x=\log(r/r_{0})$ }%
\label{psid}%
\end{center}
\end{figure}
%EndExpansion
In each plot the scale $r_{0}$ is proportional to the Compton wave length
corresponding to the reduced mass of the two quark system.

Using a scheme outlined in Appendix C, we obtain an analytic approximation to
the meson wave functions in terms of harmonic oscillator wave functions. \ The
two primary parameters we use for each meson are the scale factor $a$ and the
leading power (short distance behavior) exponent $k$. \ In addition we take as
parameters the coefficients of the associated Laguerre polynomials. We write
the radial wave function for each meson in the form
\begin{equation}
u(r)\doteq\sum_{n=0}^{N}c_{n}v_{n}(r)
\end{equation}
where
\begin{equation}
v_{n}(r)\ =\sqrt{\frac{2(n!)}{(n+k-1/2)!}}\exp(-y^{2}/2)y^{k}L_{n}%
^{k-1/2}(y^{2})
\end{equation}
in which $y=r/a=\alpha e^{x}$ and (with $z=y^{2}$)%

\begin{equation}
L_{n}^{k-1/2}(z)=\frac{e^{z}z^{-k+1/2}}{n!}\frac{d^{n}}{dz^{n}}(e^{-z}%
z^{k+n-1/2}).
\end{equation}
We then vary the two parameters $a$ and $k$ to obtain the best fit$.$ The
coefficients are fixed by
\begin{equation}
c_{n}=\int_{0}^{+\infty}v_{n}(r)u(r)dr
\end{equation}
For meson radial wave functions with more than one component \ (like the
$\psi/J)$ we fit each component separately. \ In the table below we give a
typical list for parameters $a,k,c_{n}$ for the $\pi$, $\rho$, and $\psi/J $. \ \ %

\begin{equation}%
\begin{tabular}
[c]{llll}
& $\pi$ & $\rho$ & $\psi/J$\\
$k$ & 2.3073423E-001 \  & 9.8578925E-001 \  & 9.2724750E-001 \ \\
$\alpha^{2}$ & 1.2210572E--004 \ \ \ \  & 2.0470760E-001 \  & 5.8594729E--002
\ \ \\
$c_{0}$ & -9.7061324E-001 \  & 5.6829022E-001 \  & 8.6340073E-001 \ \\
$c_{1}$ & 1.9718831E-001 \  & -5.5426675E-001 \  & -3.7785095E-001 \ \\
$c_{2}$ & -1.1892580E-001 \  & 4.5564711E-001 \  & 2.7011136E-001 \ \\
$c_{3}$ & 3.9323176E--002 \ \  & -2.9596894E-001 \  & -1.4488800E-001 \ \\
$c_{4}$ & -4.7493498E--002 \ \  & 2.1194455E-001 \  & 1.0562063E-001 \ \\
$c_{5}$ & 1.5951941E--002 \ \  & -1.2990096E-001 \  & -5.8554918E--002 \ \ \\
$c_{6}$ & -2.2163829E--002 \ \  & 8.8770721E--002 \ \  & 4.4652224E--002
\ \ \\
$c_{7}$ & 9.3538756E--003 \ \ \  & -5.3653737E--002 \ \  & -2.4410075E--002
\ \ \\
$c_{8}$ & -1.1299709E--002 \ \  & 3.5773104E--002 \ \  & 1.9878119E--002
\ \ \\
$c_{9}$ & 5.7479922E--003 \ \ \  & -2.1618467E--002 \ \  & -1.0316735E--002
\ \ \\
$c_{10}$ & -6.2419493E--003 \ \ \  & 1.4216745E--002 \ \  & 9.2491337E--003
\ \ \ \\
$c_{11}$ & 3.4486176E--003 \ \ \  & -8.5738108E--003 \ \ \  & -4.3412998E--003
\ \ \ \\
$c_{12}$ & -3.6367315E--003 \ \ \  & 5.6769781E--003 \ \ \  & 4.4967508E--003
\ \ \ \\
$c_{13}$ & 2.0430725E--003 \ \ \  & -3.3134948E--003 \ \ \  & -1.7708593E--003
\ \ \ \\
$c_{14}$ & -2.1601903E--003 \ \ \  & 2.3390141E--003 \ \ \  & 2.2926598E--003
\ \ \ \\
$c_{15}$ & 1.2286995E--003 \ \ \  & -1.1943145E--003 \ \ \  & -6.6351556E--004
\ \ \ \ \\
$c_{16}$ & -1.2691935E--003 \ \ \  & 1.0380630E--003 \ \ \  & 1.2316978E--003
\ \ \ \\
$c_{17}$ & 7.7202984E--004 \ \ \ \  & -3.4274069E--004 \ \ \ \  &
-1.9315837E--004 \ \ \ \ \\
$c_{18}$ & -7.1625503E--004 \ \ \ \  & 5.2085665E--004 \ \ \ \  &
6.9778820E--004 \ \ \ \ \\
$c_{19}$ & 5.1870027E--004 \ \ \ \  & -5.0260349E-006 \  & -3.6467729E-007
\ \\
$c_{20}$ & -3.7115575E--004 \ \ \ \  &  & \\
$c_{21}$ & 3.7723292E--004 \ \ \ \  &  & \\
$c_{22}$ & -1.5671750E--004 \ \ \ \  &  &
\end{tabular}
\end{equation}
We note several features. \ First, the fit to the $\pi$ wave function appears
to converge significantly more slowly than those for the $\rho$ and $\psi/J$.
\ (We do not present plots comparing the numerical wave function with the
harmonic oscillator wave function fits since there are no visible
differences). Also note \ that the $\pi$'s short distance behavior is
distinctly different from those of the other two. All three wave functions
possess polynomial coefficients that exhibit an oscillatory behavior.

\subsection{Numerical Evidence for Goldstone Boson Behavior}

In our equations, the pion is a Goldstone boson in the sense that its mass
tends toward zero numerically in the limit in which the quark mass numerically
goes toward zero. This may be seen in the accompanying plot (units are in $%
%TCIMACRO{\unit{MeV}}%
%BeginExpansion
\operatorname{MeV}%
%EndExpansion
)$. Note that the $\rho$ meson mass approaches a finite value in the chiral
limit. This non-Goldstone behavior also holds for the excited pion states.
\ None of the alternative approaches discussed in the following sections have
displayed this property. Another distinction we point out is that our $u$ and
$d$ quark masses (on the order of 55-60 $%
%TCIMACRO{\unit{MeV}}%
%BeginExpansion
\operatorname{MeV}%
%EndExpansion
$) are significantly smaller than the constituent quark masses appearing in
most all other models (on the order of 300 $%
%TCIMACRO{\unit{MeV}}%
%BeginExpansion
\operatorname{MeV}%
%EndExpansion
$) - closer to the small current quark masses of a few $%
%TCIMACRO{\unit{MeV}}%
%BeginExpansion
\operatorname{MeV}%
%EndExpansion
$. \ Note, however, that the shape of our pion curve is not what one would
expect \ from the Goldberger-Trieman relation%
\begin{equation}
m_{q}=m_{\pi}^{2}F_{\pi}.
\end{equation}
%

%TCIMACRO{\FRAME{ftbpFU}{6.6314in}{5.1249in}{0pt}{\Qcb{$\pi$ and $\rho$ masses
%versus quark mass in MeV}}{}{f50.ps}{\special{ language "Scientific Word";
%type "GRAPHIC";  maintain-aspect-ratio TRUE;  display "USEDEF";
%valid_file "F";  width 6.6314in;  height 5.1249in;  depth 0pt;
%original-width 11.0056in;  original-height 8.4968in;  cropleft "0";
%croptop "1";  cropright "1";  cropbottom "0";
%filename 'f50.ps';file-properties "XNPEU";}} }%
%BeginExpansion
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=5.1249in,
width=6.6314in
]%
{f50.ps}%
\caption{$\pi$ and $\rho$ masses versus quark mass in MeV}%
\end{center}
\end{figure}
%EndExpansion


Thus this aspect our model requires further investigation. \ 

\section{\textbf{Comparison of Structures of Two-Body Dirac Equations with
Those of Alternative Approaches}}

So far, we have obtained spectral results given by our equations when solved
in their own most convenient form. In Sections (VI-IX) we shall compare our
results with recent universal fits to the meson spectrum produced by a number
of other authors. These approaches employ equations whose structures appear
radically different from ours. However, as we have shown elsewhere
\cite{va86}, because our approach starts from a pair of coupled but compatible
Dirac equations, these equations can be rearranged in a multitude of forms all
possessing the same solutions. Among the rearrangements are those with
structures close to those of the authors whose spectral fits we shall shortly
examine. In order to see how structural differences in each case may lead to
differences in the resulting numerical spectra, we shall begin by considering
relevant rearrangements of the two-body Dirac equations.

The first two alternative approaches which we shall discuss use truncated
versions of the Bethe-Salpeter equation while the third uses a modified form
of the Breit equation. In order to relate the detailed predictions of our
approach to these alternatives, we need to relate our minimal substitution
method for the introduction of interactions to the introduction of interaction
through the use of kernals that dominates the older approaches. The
field-theoretic kernal employs a direct product of gamma matrices times some
function of the relative momentum or coordinate. What is the analog of the
kernal in our approach? In earlier work we found that we could obtain our
\textquotedblleft external potential\textquotedblright\ or \textquotedblleft
minimal interaction\textquotedblright\ form of our two-body Dirac equations
from yet another form displaying a remarkable hyperbolic structure. We were
able to recast our compatible Dirac equations (\ref{tbdea},\ref{tbdeb}) as
\begin{align}
\mathcal{S}_{1}\psi &  =(\cosh(\Delta)\mathbf{S}_{1}+\sinh(\Delta
)\mathbf{S}_{2})\psi=0,\label{cnhyp}\\
\mathcal{S}_{2}\psi &  =(\cosh(\Delta)\mathbf{S}_{2}+\sinh(\Delta
)\mathbf{S}_{1})\psi=0,\nonumber
\end{align}
in which \cite{jmath}
\begin{align}
\mathbf{S}_{1}\psi &  \equiv(\mathcal{S}_{10}\cosh(\Delta)+\mathcal{S}%
_{20}\sinh(\Delta))\psi=0,\nonumber\\
\mathbf{S}_{2}\psi &  \equiv(\mathcal{S}_{20}\cosh(\Delta)+\mathcal{S}%
_{10}\sinh(\Delta))\psi=0, \label{cnmyp}%
\end{align}
with
\begin{align}
\mathcal{S}_{10}\psi &  =\big(-\beta_{1}\Sigma_{1}\cdot p+\epsilon_{1}%
\beta_{1}\gamma_{51}+m_{1}\gamma_{51}\big)\psi\nonumber\\
\mathcal{S}_{20}\psi &  =\big(\beta_{2}\Sigma_{2}\cdot p+\epsilon_{2}\beta
_{2}\gamma_{52}+m_{2}\gamma_{52}\big)\psi\label{s0}%
\end{align}
and
\begin{equation}
\Delta={\frac{1}{2}}[\gamma_{51}\gamma_{52}](L(x_{\perp})+\gamma_{1}%
\cdot\gamma_{2}\mathcal{G}(x_{\perp})). \label{del}%
\end{equation}


We then recover the explicit \textquotedblleft external
potential\textquotedblright\ forms of our equations, (\ref{tbdea},\ref{tbdeb})
from (\ref{cnhyp},\ref{cnmyp}) by moving the free Dirac operators to the right
to operate on the wave function. This rearrangement produces the derivative
recoil terms apparent in Eqs.(\ref{tbdea},\ref{tbdeb}a)). $\Delta$ may take
any one of (or combination of) eight invariant forms. In terms of
\begin{equation}
\mathcal{O}_{1}=-\gamma_{51}\gamma_{52},
\end{equation}
these become $\Delta(x_{\perp})=-L(x_{\perp})\mathcal{O}_{1}/2,\gamma_{1}%
\cdot\hat{P}\gamma_{2}\cdot\hat{P}J(x_{\perp})\mathcal{O}_{1}/2,\gamma
_{1\perp}\cdot\gamma_{2\perp}\mathcal{G}(x_{\perp})\mathcal{O}_{1}/2$ or
$\alpha_{1}\cdot\alpha_{2}\mathcal{F}(x_{\perp})\mathcal{O}_{1}/2$ for scalar,
time-like vector, space-like vector, or tensor (polar) interactions
respectively. Note that in our $\Delta(x_{\perp})$ in Eq.(\ref{del}) above,
$\mathcal{G}(x_{\perp})$ enters multiplied by the electromagnetic-like
combination $\gamma_{1}\cdot\gamma_{2}=-\gamma_{1}\cdot\hat{P}\gamma_{2}%
\cdot\hat{P}+\gamma_{1\perp}\cdot\gamma_{2\perp}$ of time and space-like
parts. The axial counterparts to the constraints with polar interactions are
given by (note the minus sign compared with the plus sign in Eqs.(\ref{cnhyp}%
a-b)) \cite{jmath}
\begin{align}
\mathcal{S}_{1}\psi &  =(\cosh(\Delta)\mathbf{S}_{1}-\sinh(\Delta
)\mathbf{S}_{2})\psi=0\\
\mathcal{S}_{2}\psi &  =(\cosh(\Delta)\mathbf{S}_{2}-\sinh(\Delta
)\mathbf{S}_{1})\psi=0,\nonumber
\end{align}
in which $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$ are still given by
(\ref{cnmyp}) with axial counterparts to the above $\Delta$'s given by
$C(x_{\perp})/2,\gamma_{51}\gamma_{1}\cdot\hat{P}\gamma_{52}\gamma_{2}%
\cdot\hat{P}H(x_{\perp})\mathcal{O}_{1}/2$,$\gamma_{51}\gamma_{1\perp}%
\cdot\gamma_{52}\gamma_{2\perp}I(x_{\perp})\mathcal{O}_{1}/2$ and $\sigma
_{1}\cdot\sigma_{2}Y(x_{\perp})\mathcal{O}_{1}/2$ respectively. The advantage
of the hyperbolic form is that with its aid we may first choose among the 8
interaction types in an unambiguous way to introduce interaction (without
struggling to restore compatibility) and then, for computational convenience,
transform the Dirac equations to \textquotedblleft external
potential\textquotedblright\ form. In the weak-potential limit of our
equations, the coefficients of $\gamma_{51}\gamma_{52}$ in the expansion of
our $\Delta$ interaction matrix in Eq.(\ref{del}) directly correspond to the
interaction kernals of the Bethe-Salpeter equation. Note however, that because
of the hyperbolic structure, what we call a \textquotedblright vector
interaction\textquotedblright\ actually corresponds to a particular
combination of vector, pseudoscalar, and pseudovector interactions in the
older approaches (see Eq.(\ref{eff}) below).

This difference in classification of interactions becomes apparent when we put
our equations into a Breit-like form. Consider the linear combination
\begin{equation}
\beta_{1}\gamma_{51}\mathbf{S}_{1}+\beta_{2}\gamma_{52}\mathbf{S}_{2}
\label{add}%
\end{equation}
For later convenience, form the interaction matrix:
\begin{equation}
\mathcal{D}(x_{\perp})={\frac{1}{2}}\beta_{1}\gamma_{51}\beta_{2}\gamma
_{52}\Delta(x_{\perp}).
\end{equation}
After simplification, the linear combination Eq.(\ref{add}) of our two
hyperbolic equations becomes
\begin{equation}
w\Psi=[H_{10}+H_{20}+V(x_{\perp},\alpha_{1},\alpha_{2},\beta_{1},\beta
_{2},\gamma_{51},\gamma_{52})]\Psi
\end{equation}
in which
\begin{equation}
\Psi=exp(-\mathcal{D})\psi
\end{equation}
and
\begin{equation}
H_{10}=\alpha_{1}\cdot p_{\perp}+\beta_{1}m_{1},\ H_{20}=-\alpha_{2}\cdot
p_{\perp}+\beta_{2}m_{2}.
\end{equation}
For the electromagnetic vector kernal $\Delta(x_{\perp})={\frac{1}{2}}%
[\gamma_{51}\gamma_{52}]\gamma_{1}\cdot\gamma_{2}\mathcal{G}(x_{\perp}),$
$\mathcal{D}$ then becomes
\begin{equation}
\mathcal{D}={\frac{1}{2}}\mathcal{G}(x_{\perp})(\alpha_{1}\cdot\alpha_{2}-1),
\end{equation}
so that the relativistic Breit-like equation takes the c.m. form
\begin{equation}
w\Psi=[\mathbf{\alpha}_{1}\cdot\mathbf{p}-\mathbf{\alpha}_{2}\cdot
\mathbf{p}+\beta_{1}m_{1}+\beta_{2}m_{2}+w(1-exp[\mathcal{G}(x_{\perp
})(\mathbf{\alpha}_{1}\cdot\mathbf{\alpha}_{2}-1)])]\Psi
\end{equation}


In lowest order this equation takes on the familiar form for four-vector
interactions (seemingly missing the traditional Darwin interaction piece

\noindent$\sim\mathbf{\hat{r}}\cdot\mathbf{\alpha}_{1}\mathbf{\hat{r}}%
\cdot\mathbf{\alpha}_{2})$.%

\begin{equation}
w\Psi=(\mathbf{\alpha}_{1}\cdot\mathbf{p}-\mathbf{\alpha}_{2}\cdot
\mathbf{p}+\beta_{1}m_{1}+\beta_{2}m_{2}-w(\mathcal{G}(x_{\perp}%
)(\mathbf{\alpha}_{1}\cdot\mathbf{\alpha}_{2}-1)))\Psi.
\end{equation}


However, as we first showed in \cite{cra94}, expanding this simple structure
to higher order in fact generates the correct Darwin dynamics. As a
consequence, our unapproximated equation yields analytic and numerical
agreement with the field theoretic spectrum through order $\alpha^{4}$.
Explicitly, our full interaction is
\begin{align}
exp[\mathcal{G}(x_{\perp})(\mathbf{\alpha}_{1}\cdot\mathbf{\alpha}_{2}-1)]  &
={\frac{e^{-\mathcal{G}}}{4}}[3\cosh(\mathcal{G})+\cosh(3\mathcal{G}%
)+\gamma_{51}\gamma_{52}(3\sinh(\mathcal{G})-\sinh(3\mathcal{G}))\nonumber\\
&  +\mathbf{\alpha}_{1}\cdot\mathbf{\alpha}_{2}(\sinh(3\mathcal{G}%
)+\sinh(\mathcal{G}))+\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}(\cosh
(\mathcal{G})-\sinh(3\mathcal{G}))] \label{eff}%
\end{align}
so that our Breit-like potential contains a combination of \textquotedblright
vector\textquotedblright\ and \textquotedblright
pseudovector\textquotedblright\ interactions originating from the four-vector
potentials of the original constraint equations in \textquotedblright
external-potential\textquotedblright\ form. \cite{qed}

In this section we have seen how the two-body Dirac equations with
field-theoretic interaction structure automatically retain the correct Darwin
structure of QED. Such a demonstration should be carried out for each
alternative treatment (if possible) in order to check that truncations and
numerical procedures have not destroyed its own version of the field-theoretic
Darwin structure for its treatment of the vector interaction of QED.

\section{ The Wisconsin Model of Gara, Durand, Durand, and Nickisch}

\subsection{Definition of The Model and Comparison of Structure with Two-Body
Dirac Approach}

The authors of reference \cite{wisc} base their analysis of quark-antiquark
bound states on the reduced Salpeter equation containing a mixture of scalar
and vector interactions between quarks of the same or different flavors. When
rewritten in a notation that aids comparison to our approach, their bound
state equation takes the c.m. form
\begin{equation}
\lbrack w-\omega_{1}-\omega_{2}]\Phi(\mathbf{p})=\Lambda^{+}(\mathbf{p}%
)\gamma^{0}\int{\frac{d^{3}p^{\prime}}{(2\pi)^{3}}}[\mathcal{A}(\mathbf{p}%
-\mathbf{p}^{\prime})\gamma_{\mu}\Phi(\mathbf{p}^{\prime})\gamma^{\mu
}+S(\mathbf{p}-\mathbf{p}^{\prime})\Phi(\mathbf{p}^{\prime})]\gamma^{0}%
\Lambda^{-}(-\mathbf{p}) \label{wisc}%
\end{equation}
in which $\mathcal{A}$ and $S$ are functions that parametrize the
electromagnetic-like and scalar interactions, $\Lambda^{\pm}$ are projection
operators, $w$ is the c.m. energy, $\omega_{i}=(\mathbf{p}^{2}+m_{i}%
^{2})^{1/2}$, while $\Phi$ is a four by four matrix wave function represented
in block matrix form as
\begin{equation}
\Phi=\bigg[{%
%TCIMACRO{\QATOP{\phi^{+-}}{\phi^{--}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{\phi^{+-}}{\phi^{--}}%
%EndExpansion
}{%
%TCIMACRO{\QATOP{\phi^{++}}{\phi^{-+}}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{\phi^{++}}{\phi^{-+}}%
%EndExpansion
}\bigg]
\end{equation}
They obtain this equation from the full Bethe-Salpeter equation by making an
assumption equivalent to using a position-space description in which they
calculate the interaction potential with the equal time constraint, neglecting
retardation. (These are the usual ad-hoc assumptions that in our approach are
automatic consequences (in covariant form) of our two simultaneous, compatible
Dirac equations.) These restrictions turn Eq.(\ref{wisc}) into the standard
Salpeter equation. In addition the Wisconsin group employs what we call the
\textquotedblleft weak potential assumption\textquotedblright: $(w+\omega
_{1}+\omega_{2})>>V$. This assumption turns Eq.(\cite{wisc}) into the reduced
Salpeter equation which, because of the properties of the projection operator,
allows the Wisconsin group to perform a Gordon reduction of its equation to
obtain a reduced final equation in terms of $\phi^{++}$ alone. In our approach
we make no such \textquotedblleft weak potential assumption\textquotedblright%
\ and therefore must deal directly with the fact that our Dirac equations
themselves relate components of the sixteen component wave function to each
another. Unlike what happens in the reduced Salpeter equation, in our method
this coupling leads to potential dependent denominators, a strong potential
structure that we found crucial in demonstrating that our formalism yields
legitimate relativistic two-body equations. Just as we do, however, the
Wisconsin group works in coordinate space where the dynamical potentials are
local and easy to handle. However, in their method upon Fourier transformation
the kinetic factors $\omega_{i}$ then become nonlocal operators. In contrast,
the entire dynamical structure of our two-body Dirac equations, is local as
long as the potentials are local.

The Wisconsin group uses local static potentials that play the role of our
Adler-Piran potential:
\begin{align}
\mathcal{A}(r)  &  =-{\frac{4}{3}}\frac{\alpha_{s}(r)}{r}e^{-\mu^{\prime}%
r}+\delta(-\frac{\beta}{r}+\Lambda r)(1-e^{-\mu r})\nonumber\\
S(r)  &  =(1-\delta)(-\frac{\beta}{r}+Br)(1-e^{-\mu r})+(C+C_{1}r+C_{2}%
r^{2})(1-e^{-\mu r})e^{-\mu r} \label{wsint}%
\end{align}


Note that Gara et al introduce a confining electromagnetic-like vector
potential proportional to a parameter $\delta$. This differs from our approach
in which the (dominant) linear portion of the confinement potential has no
electromagnetic part. Like Adler's potential, theirs has a long range $1/r$
part (the so-called Luscher term). Its short range part is
electromagnetic-like just as is ours, and like Adler's is obtained from a
renormalization group equation.

They base their analysis on a nonperturbative, numerical solution of the
reduced Salpeter equation Eq.(\ref{wisc}) with interaction Eq.(\ref{wsint}).

\subsection{Comparison of Wisconsin Fit with that of Two-Body Dirac Equations}

Initially, we present the Wisconsin results for $\delta=0$ in Table II. This
table includes many of the $b\bar{b},c\bar{c}$, and $s\bar{s}$ states. We also
give in this table the fit we would have obtained with our equations if we had
limited our best fit just to these 18 mesons. In Table III we include the
Wisconsin variable-$\delta$ (vector-confinement) best fit results, and the
best fit results our method gives when restricted to these 25 mesons alone.
For uniformity of presentation we give all of the Wisconsin results in terms
of absolute masses (rather than the mass differences and averages these
authors presented for the spin-orbit triplets). Although Gara et al. did not
perform the same $\chi^{2}$ fit that we do, we present (in parentheses) the
incremental $\chi^{2}$ contribution for each meson so that we can easily
compare the results of the two methods. We also compare their $R$ values and
$^{3}P\ avg.$ to ours directly in the discussion below.

Consider first the results given in Table II. Overall, the results from our
fits to the entire spectrum are closer in 13 of the 18 mesons to the
experimental results than theirs. Their $R$ values of 0.73,0.64,0.60 for the
two $\Upsilon$ and the single $\chi$ spin-orbit multiplets are closer to the
experimental values of 0.66,0.61,0.48 only for the second multiplets than the
R values given by the fit of our model: 0.67,0.66,0.40. Their $^{3}P $ average
$[5(^{3}P_{2})+3(^{3}P_{1})+1(^{3}P_{0})]/9$ for these three multiplets of
9.895, 10.273, 3.512 are set against our three values of 9.901, 10.264, 3.515
and the experimental numbers of 9.900, 10.260, 3.525. Their hyperfine
splitting for the two charmonium multiplets are 194 MeV and 39 MeV compared
with our values of 150 MeV and 79 MeV. Since the experimental splittings are
117 and 92 MeV here the constraint results are more accurate. The experimental
radial excitation energies of 563, 332, for the two $\Upsilon$ excitations and
614, and 589 MeV for the singlet and triplet charmonium excitations are
accounted for significantly better by our values of 569, 336, 636, and 564 MeV
than by the Wisconsin results of 593,364, 645, and 491 MeV.

We now consider Table III, corresponding to their variable combination of
scalar and vector confinement. Our results are closer to the experimental
results for 16 of the 25 mesons. In detail, their $R$ values of 0.83,0.78, and
0.60 are less accurate than two of our values of 0.64,0.68, 0.35 respectively.
Their $^{3}P$ averages $[5(^{3}P_{2})+3(^{3}P_{1})+1(^{3}P_{0})]/9$ of 9.902
,10.262, 3.513 represent an improvement relative to ours (9.901, 10.264,
3.513) compared to the results of Table II. Their hyperfine splittings for the
two charmonium multiplets of 200 and 47 MeV are significantly worse than our
fits of 150 and 79 MeV. Their hyperfine splittings for the mesons with one $d$
or $s$ quark are 27, 51, and 127 MeV. Our fits of 128,138, 420 MeV
respectively are much closer to the experimental results of 141, 141, 398 MeV.

The radial excitation energies for the two lowest $\Upsilon$ excitations and
the singlet and triplet charmonium excitations are again accounted for
significantly better by three of four of our values of 569,335,636,568 MeV for
the results in the last column than the Wisconsin results of 602,331,654,491
MeV. In summary, the major strength of our approach is reflected in its better
fits to the hyperfine splittings and radial excitations. The Wisconsin group's
results for the fine structure splitting are overall about the same as ours.
Moreover, even a casual glance at the results shows one glaring discrepancy
that results from their approach - their hyperfine splittings for the light
quark mesons. The cause of this is probably the fact that their reduced
Salpeter approach does not include coupling of the upper-upper piece to the
other 12 components of the 16 component wave function. In fact, the lighter
the meson, the worse is their result. In our QED numerical investigations we
found that couplings to the lower-lower components of the wave function were
essential in order to obtain agreement with the standard perturbative spectral
results of QED. We conjecture that the same strong-potential effects that led
to our successful results in QED are responsible for the goodness of our
hyperfine splitting, particularly for the mesons containing the light quarks.

Gara et al. point out that in their approach the straight line Regge
trajectories ($j$ versus $w^{2}$) for the light quark systems are much too
steep, with slopes greater than twice the observed slopes for pure scalar
confinement. The best fit experimental slope and intercept values for the
$\rho,a_{2},\rho_{3}$ trajectory are (0.88,0.48). The slope and intercept
values that we obtain for our model in Table I are (0.87,0.47), in excellent
agreement with the best experimental fit. For the $\phi_{1},f_{2},\phi_{3}$
trajectory the experimental values are (0.83,0.11) while our model of Table I
produces the set of values (0.85,0.095). The intercepts are not as accurate as
those for the $\rho$ trajectory although our results actually produce a
tighter fit to a straight line trajectory than do the experimental results.
Finally we come to the $\pi,b_{1},\pi_{2}$ trajectory. We obtain the values
(0.57,-0.04). Compared to the experimental values of (0.72,-0.04) our slopes
are about 25-30\% small, although our fit to the straight line is just as
tight. The probable reason for the relative advantage of our results over
those of the Wisconsin group is that our bound state equations include a
strong potential structure, and are not limited by the weak potential
approximation built into the reduced Salpeter equation.

\section{ The Iowa State Model of Spence and Vary}

\subsection{Definition of The Model and Comparison of Structure with Two-Body
Dirac Approach}

In contrast to the Wisconsin group, Spence and Vary \cite{iowa} use the
Salpeter equation in its unreduced form, including the crucial coupling
between positive and negative frequency states, particularly for the light
mesons. The Iowa State group considers several different types of confinement
models obtaining the best results when using a scalar linear confinement plus
massless vector boson exchange-potential including the projected Breit
interaction. Spence and Vary work in momentum space, using the kernel
\begin{equation}
\frac{-4\pi a\gamma_{0}\gamma_{\mu}\times\gamma_{0}\gamma_{\mu}}%
{-(q-q^{\prime})^{2}}+4\pi b{%
%TCIMACRO{\QATOP{lim}{\mu\rightarrow0}}%
%BeginExpansion
\genfrac{}{}{0pt}{}{lim}{\mu\rightarrow0}%
%EndExpansion
}\big[{\frac{\partial}{\partial\mu}}\big ]^{2}\frac{\gamma_{0}\times\gamma
_{0}}{-(q-q^{\prime})^{2}+\mu^{2}}%
\end{equation}
They emphasize that in addition to the instantaneous approximation they must
include the retardative Breit term. They accomplish this by expanding the
quantity $1/[-(q-q^{\prime})^{2}]$ in powers of $(q_{0}-q_{0}^{\prime})^{2}$
where $q=(q_{1}-q_{2})/2,\ q^{\prime}=(q_{1}^{\prime}-q_{2}^{\prime})/2$, with
$q_{i}=(q_{0i},\mathbf{q}_{i})=((m_{i}^{2}+\mathbf{q}_{i}^{2})^{1/2}%
,\mathbf{q}_{i})$. Thus they obtain
\begin{equation}
\frac{1}{-(q-q^{\prime})^{2}}\approx{\frac{1}{(\mathbf{q}-\mathbf{q}^{\prime
})^{2}}}+{\frac{(q_{0}-q_{0}^{\prime})^{2}}{(\mathbf{q}-\mathbf{q}^{\prime
})^{4}}}%
\end{equation}
However, they perform a weak potential approximation for the quarks on this
expanded term by using the on-shell free particle Dirac equation for each
quark to obtain the Breit term
\begin{equation}
V_{B}={\frac{-4\pi a\mathbf{\alpha}_{1}\cdot(\mathbf{q}-\mathbf{q}^{\prime
})\times\mathbf{\alpha}_{2}\cdot(\mathbf{q}-\mathbf{q}^{\prime})}%
{(\mathbf{q}-\mathbf{q}^{\prime})^{4}}}%
\end{equation}
In order to treat the mixing of positive and negative frequencies correctly,
they then insert projection operators. They do not include the corresponding
Breit term for the scalar interaction since their numerical tests show it to
be very small.

\subsection{Comparison of Fit with that of Constraint Approach}

In Table IV, we give their results for a set of mesons together with our
results for the same set of mesons. In the fourth column of this table we
present the results we would obtain if we limited our fit just to the 42
mesons Spence and Vary used listed in this table. Note that unlike our fitting
procedure theirs does not use any of the light mesons to determine the
parameters in their interaction.

Of the 42 mesons in their table, our fits are closer to data in 32. Only a
small part of this disparity is due to the differences in fitting procedure.
Probably a larger part is due to their relatively crude approximation to the
short distance part of the QCD potential. We proceed now with a detailed
comparison. Their $R$ values for the two upsilon and one charmonium multiplet
are 3.44,-3.88,1.13 demonstrating the weakest portion of their dynamics. Their
value for the mass of the (experimental) 10.268 meson - 10.165 may be a
misprint (although the error they list uses this value). If it is a misprint
and should read 10.265 then their $R$ value for that multiplet is 2.38.
Something clearly still would be amiss in their approach, however. Since the
upsilon system is nearly nonrelativistic, we cannot blame their choice of
relativistic wave equation. More likely, the source is either the lack of
asymptotic freedom behavior in their short distance potential, or perhaps
their treatment of the Breit term (without the self-regulating short distance
behavior of potential energy terms in the denominator possessed by the Dirac
equation, such terms should only be treated perturbatively). If there is
indeed no misprint, then the switching of the sign indicates that their scalar
potential contribution is too large (its spin orbit contribution would be of
the opposite sign from the vector contribution and of longer range, hence
affecting more the radially excited multiplet than the lower one.) Our R
values of 0.68,0.72,0.44 are considerably closer to the experimental ratios of
0.66,0.61,0.48. For the two light quark multiplets ($s\bar{s},u\bar{d}$) they
obtain 0.34 and -0.45. The negative sign indicates that the scalar interaction
is becoming dominant for the $^{3}P_{2}$ state. Our multiplets are also
partially inverted for the $\phi$ and totally inverted in the $\rho$ system.
We obtain the values -0.14,-2.48 (the sign and large value indicate near
inversion). The experimental values of 0.09,-1.00 (if not due to a
misidentification of the $^{3}P_{0}$) show that while the $s\bar{s}$ multiplet
is nearly inverted at the top, the $u\bar{d}$ multiplet is drastically
inverted. Both approaches founder in these treacherous waters. Comparing their
$^{3}P$ averages $[5(^{3}P_{2})+3(^{3}P_{1})+1(^{3}P_{0})]/9$ of
9.859,3.497,1.433,1.015 GeV for the lowest lying spin-orbit multiplets listed
in the table with our values of 9.902,3.516,1.470,1.386 and the experimental
results of 9.900,3.527,1.503,1.303 GeV we see that ours are closer in each
case to the experimental results. We see also that for charmonium, our average
is nearly equal to our $^{1}P_{1}$ level while the Iowa State results are 75
MeV higher than their $^{1}P_{1}$ level. For the $u\bar{d}$ system, our
average is 25 MeV higher than our $^{1}P_{1}$ level while theirs is 122 MeV
above their calculated $^{1}P_{1}$ level. Their values of the hyperfine
splittings are 117, 60, 333,379,403,581 MeV for the two charmonium multiplets,
and the $D-D^{\ast},D_{s}-D_{s}^{\ast},K-K^{\ast},\pi-\rho$ pairs. Comparison
with the experimental splittings of 117,92,142,144,398,627 MeV shows a mixed
pattern of excellent to good results for the heaviest and lightest to very
poor results for the intermediate more hydrogen-like mesons. Our results are
151,78,133,146,408,644 MeV. Our ground state charmonium result is not nearly
as good as theirs, since we fit the entire spectrum while they fit the $c$
mass to the $psi$ particle. However, being 34 MeV high leaves much to be
desired, with problems arising in part from the scalar interaction at longer
distances. The difference between our light meson results and theirs is also
probably due to the fitting procedure. However, our results for the
intermediate hydrogen-like mesons are decidedly superior to the Iowa state
results, less than 0.5\% off the experimental data. The origin of the relative
unevenness of fits in their approach is unclear. (We have commented earlier on
the origin of our $\psi$ fit deviation.)

Finally, we compare the radial excitations. The four upsilon states in the
experimental column of the table occur at intervals of 563,332,225 MeV while
the three charmonium triplet states and the two charmonium singlet states
occur at intervals of 589,354,614 MeV. The corresponding Spence and Vary
intervals are 614,287,315,599,394,657 MeV while our intervals are
568,335,254,555,390,628 MeV. With the exception of the first radial triplet
charmonium excitation our model fits the data more closely than theirs. In
summary, our wave equation when used with the Adler-Piran potential gives
significantly better fits to the data across all aspects of the spectrum. We
suspect that the short-comings of the Iowa State approach originate in the
overly simple choice of vector potential with no asymptotic freedom behavior
and in the weak potential treatment of their Breit term. A more serious
criticism is that even though their choice of wave equation is a standard one
for perturbative use, no one has tested its nonperturbative reliability. In
particular, it is unknown if this approach will reproduce the correct QED fine
and hyperfine structure for that dynamics when treated numerically. Without
this check, one cannot be certain that the wave equation does not distort the
dynamical input.

\section{The Breit Equation Model of Brayshaw}

\subsection{Definition of The Model and Comparison of Structure with Two-Body
Dirac Approach}

Brayshaw \cite{bry}treats quarkonium with the aid of the Breit equation and an
interaction Hamiltonian with five distinct parts, four of which are
independent. As usually done for the Breit equation the times associated with
each particle are identified or related in some favored frame (normally c.m.)
selected so that the relative time does not enter the potential. In that frame
Brayshaw uses the equation
\begin{equation}
H\Psi=(H_{0}+H_{C}+H_{B}+H_{S}+H_{I}+H_{L})\Psi=w\Psi\label{bry}%
\end{equation}
in which $H_{0}$ is the free Breit Hamiltonian
\begin{equation}
H_{0}=\mathbf{\alpha}_{1}\cdot\mathbf{p}-\mathbf{\alpha}_{2}\cdot
\mathbf{p}+\beta_{1}m_{1}+\beta_{2}m_{2}%
\end{equation}
while $H_{C}$ and $H_{B}$ are a Coulomb and an associated Breit interaction
\begin{align}
H_{C}  &  ={\frac{c_{1}}{r}}\nonumber\\
H_{B}  &  =-{\frac{c_{1}(\mathbf{\alpha}_{1}\cdot\mathbf{\alpha}%
_{2}+\mathbf{\alpha}_{1}\cdot\mathbf{\hat{r}\alpha}_{2}\cdot\mathbf{\hat{r}}%
)}{2r}}%
\end{align}
As indicated in our discussion about the Salpeter equation in Section(VI),
this part of the interaction comes from the vector portion of the kernel. The
author acknowledges the difficulties associated with the Breit interaction,
pointing out that the radial equation has a singularity at a radial separation
of $r_{0}=-c_{1}/w>0$. He bypasses Breit's proposal that this interaction be
used only in first order perturbation theory by using only positive energy
spinors in his variational procedures. We point out that this was not
necessary in our approach since the hyperbolic structure of our eight basic
interactions avoids problems inherent in Breit's formulation \cite{cwyw}. In
particular, it avoids appearance of midpoint singularities. Another criticism
of his approach shared with those of Vary and Spence and the Wisconsin group
is that having avoided the pitfalls of the Breit equation, he uses his
replacement without testing whether or not his formalism would yield the
standard QED results numerically if he limited his interaction to the usual
Coulomb interaction. Such a test, as we have said before, would (if
successful) help eliminate the possibility of the wave equation introducing
spurious physics.

In Eq.(\ref{bry}), $H_{L}$ is a long range confining portion which
incorporates the requirement that the wave function vanish identically for
radial separations $r>a$ with a boundary condition at $r=a$. Brayshaw argues
for this term over and above a linear confinement piece on the grounds that at
some separation $r_{p}$ corresponding to a threshold energy $E_{p}$ ,
production of $q\bar{q}$ pairs should become energetically favorable. His
radial parameter $a$ plays the role of $r_{p}$ in specifying the range at
which such effects (among others) dominate confinement. He expects that $a$ is
on the order of $\langle r\rangle$ for the light quark mesons while wave
functions for the heavy quark mesons would have fallen to zero for $r<<a$.
When introducing the explicit form of his linear confinement potential, the
author finds that it cannot simply be added as a Lorentz scalar to the
Hamiltonian since such a term produces far too large a mass shift for the
light quark systems. Instead he chooses
\begin{equation}
H_{I}=c_{2}(\beta_{1}+\beta_{2})r.
\end{equation}
which he shows contributes very weakly for the light quark systems, while
contributing significantly for the heavy quark systems with an intermediate
contribution for the hydrogen-like intermediate mass mesons. Unfortunately,
however, note the important fact that the Lorentz transformation character of
his interaction is ambiguous, being neither scalar ($\sim\beta_{1}\beta_{2}$)
or (time-like) vector ($\sim1_{1}1_{2}$).

Finally Brayshaw introduces a special short range attractive piece solely in
order to obtain a good fit to the pion and kaon. Instead of a spin-dependent
contact term used in a number of semirelativistic approaches
[\cite{licht,rob,isgr}] he uses
\begin{equation}
H_{S}=H_{B}(1_{1}1_{2}+\beta_{1}\beta_{2})\frac{c_{4}r\theta(b-r)}%
{2(m_{1}+m_{2}+c_{4})}%
\end{equation}
This term resembles a cross term between a linear confinement piece and the
Breit term that might emerge from some sort of iteration. The short range
character of this part-scalar, part-vector interaction is specified through
taking $b<<a$. In contrast, our approach possess a short range spin-spin
interaction that is quantum mechanically well defined and which arises
straightforwardly from the Schr\"{o}dinger reduction of our Dirac equations.
We do not need to add it in by hand.

\subsection{Comparison of Fit with that of Constraint Approach}

In spite of its ad hoc nature, we have included the procedure of Brayshaw
among our comparisons because it turns out that his resultant fit for the 56
mesons (that overlap with our fit) is, in contrast to the two previous
examples, just slightly worse than our fit. In Table V we include in the
fourth column the fit we would obtain with our model if we included only the
56 mesons that our fit has in common with Brayshaw's. On a meson by meson
basis we compare by using incremental $\chi^{2}$ values.

Of the 56 mesons in the table, our fits are closer to data in only 26,
although overall our fit is better. However, this overall difference may not
be as significant as in the previous examples because here we did not use
identical fitting procedures for both models. Brayshaw's $R$ values for the
two upsilon, the one charmonium, the $K^{\ast}$, $\phi$ and $\rho-\pi$ triplet
$P$ multiplets are 0.47,0.34,0.32,0.55,0.25,0.19 and are distinctly different
from our values of 0.66,0.69,0.39,-0.71,-0.25,-5.67 and the experimental
numbers of 0.66,0.61,0.48,0.09,-0.97,-0.4. Although the constraint/Adler-Piran
combination is distinctly better than the Breit/Brayshaw approach for the
heavier mesons, both give poor results for the lighter mesons. None of his
light multiplets are inverted, whereas although ours are inverted they are not
inverted in the same way as the experimental numbers. Again, our inversions
are due to the scalar potential. His approach includes (see $H_{S}$) a partial
Hamiltonian that governs intermediate range behavior, in which time-like and
scalar interactions contribute equally. This may be responsible for his lack
of the partial inversion shown by the data.

Comparing his $^{3}P$ averages $[5(^{3}P_{2})+3(^{3}P_{1})+1(^{3}P_{0})]/9$ to
the $^{1}P_{1}$ mesons for the charmonium, $K^{\ast}$, and $\rho-\pi$ systems
we find the following three pairs of numbers:
3.517,3.498;1.335,1.355;1.251,1.202. Comparison to our numbers of
3.519,3.520;1.435,1.421;1.434,1.411 and the experimental numbers of
3.526,3.525;1.402,1.375;1.231,1.303 shows that our approach gives better
agreement for the heavier mesons, his somewhat better for the lighter while
both do about the same for the $K^{\ast}$.

His values of the hyperfine splittings are 118,100,143,158,410,636 MeV for the
two charmonium multiplets, and the $D-D^{\ast},D_{s}-D_{s}^{\ast},K-K^{\ast
},\pi-\rho$ pairs. Comparing with the experimental splittings of
117,92,142,144,398,627 MeV shows a clear pattern of excellent to good results
for the heaviest, lightest, and the intermediate more hydrogen-like mesons.
Our results are 151,79,133,145,416,647 MeV. Our ground state charmonium result
is not nearly as good as Brayshaw's but for the others we have about the same
quality of fit. It may be that his choice of $H_{S}$ rectifies the problem our
treatment encounters. But, the disadvantage of this is that his $R$ values for
the heavy mesons are worse. This effect appears to be similar to the trouble
we encountered, mentioned in our discussion of Table I .

For the radial excitations, the four upsilon states in the data portion of the
table occur at intervals of 563,332,225 MeV while the three charmonium triplet
states and the two charmonium singlet states occur at intervals of 589,354,614
MeV. The pion excitation is 1160 MeV. The corresponding Brayshaw intervals are
555,335,320,551,566,569,888 MeV while our intervals are
572,337,257,564,395,636,1403 MeV. With the exception of the second radial
triplet upsilonium and charmonium excitation intervals, the fits of both
models are of about the same quality. Note that excited pion predictions
bracket the experimental results. This appears to be a common thread in the
radial and orbital excitations of the light quark mesons, with his results on
average closer to the experimental values. Our results are, on average, better
for the heavier mesons.

However, his apparently good fit emerges from a structurally flawed
theoretical approach. His potentials are chosen in a patchwork manner using
the 5 parameters $a,c_{1},c_{2},c_{3},c_{4}$(he sets $b=a/10$). In terms of
Lorentz transformation properties he uses four invariant functions (scalar,
time-like, electromagnetic like and mixed ($H_{S},H_{B}$ and $H_{I}$)). The
Adler-Piran potential that we use has only two invariant functions
corresponding to scalar and electromagnetic like interactions. The constraint
approach is not a patchwork; indeed, its wave equation itself (once
$\mathcal{A}$ and $S$ are chosen) fixes the spin, orbital and radial aspects
of its potential and its spectra. Again as in the case of Spence and Vary, a
serious criticism of this approach is that Brayshaw has not tested the
nonperturbative reliability of his equation.

The most important warning provided by Brayshaw's approach is that an ad hoc
structure with ambiguous Lorentz properties can do so well at fitting the spectrum.

\section{The Semirelativistic Model of Godfrey and Isgur}

\subsection{Definition of The Model and Comparison of Structure with Two-Body
Dirac Approach}

We begin with a general discussion of Semirelativistic Quark Models (with and
without Relativistic Kinematics). We term a \textquotedblleft semirelativistic
quark model\textquotedblright\ one that uses a two-body wave equation that has
one of the following three c.m. forms.
\begin{align}
(\mathbf{p}^{2}+\Phi(\mathbf{r},\mathbf{s}_{1},\mathbf{s}_{2}))\psi &
=(w-m_{1}-m_{2})\psi\nonumber\\
(\sqrt{\mathbf{p}^{2}+m_{1}^{2}}+\sqrt{\mathbf{p}^{2}+m_{2}}+\Phi
(\mathbf{r},\mathbf{s}_{1},\mathbf{s}_{2}))\psi &  =w\psi\nonumber\\
(\mathbf{p}^{2}+\Phi(\mathbf{r,s}_{1},\mathbf{s}_{2}))\psi &  =b^{2}(w)\psi
\end{align}
In each of these equations $\mathbf{p}^{2}$ is the square of the c.m. relative
momentum while $\Phi(\mathbf{r},\mathbf{s}_{1},\mathbf{s}_{2})$ is an
effective potential which includes central, spin-orbit, spin-spin, tensor and
possibly Darwin terms. In each, the wave function has four components with no
coupling to lower-lower components. The most important difference between the
first form and the others is that the latter two have exact relativistic
kinematics. The former is almost always called a nonrelativistic quark model
although strictly speaking almost all spin dependences (at least those that
arise from vector and scalar interactions) vanish in the nonrelativistic
limit. These equations differ from the Two-Body Dirac equations and the Breit
and instantaneous Bethe-Salpeter approaches primarily in that their
spin-dependences are put in by hand, abstracted from the Fermi-Breit
reductions of the Breit and instantaneous Bethe-Salpeter approaches. For
Coulomb-like potentials originating in the Coulomb Gauge, these terms contain
singular potentials. Consequently they must either be treated purely
perturbatively (thus ruling out application to the light quark mesons) or
through the introduction of smoothing parameters that may or may not be
features of the actual potential. The two-body Dirac equations of constraint
dynamics, like their one-body cousin, have a natural smoothing mechanism -
potential dependent denominators in the spin-dependent and Darwin terms of the
resultant Schrodinger-like form - that eliminates the necessity for ad hoc
introduction of such terms. The Breit equation may also possess a natural
smoothing mechanism, but a nonperturbative treatment of it leads to erroneous
results in QED \cite{kro81}. The instantaneous Salpeter equation may have a
natural smoothing mechanism, but has not been tested for QED even though the
equation is over 50 years old. Authors who have attempted to use these types
of semi-relativistic equations to treat the entire meson spectrum include
Lichtenberg \cite{licht}(the third type), Stanley and Robson \cite{rob} and
Godfrey and Isgur \cite{isgr} (the second type), and Morpurgo, Ono, and
Sch\"{o}berl \cite{mor90}(the first type) . Each of these authors ignore the
spin-independent part of the Fermi-Breit interaction. This neglect is not
justifiable since this part of the interaction will have an effect on $S$
states that is significantly different from its effect on non $S$-states,
being normally short ranged compared with the rest of the central force part
of the problem. In this paper, we select one of these models for our final
comparison, the model of Godfrey and Isgur, since this model is most often
cited in recent experimental works and theoretical papers on rival approaches.

As we have said, Godfrey and Isgur assume a semi-relativistic wave equation of
the second type possessing exact relativistic kinematics but through the
inconvenient sum-of-square-roots form. They then determine the form of
interaction in the following way. They assume that the confining piece of the
interaction is a world scalar (according to the discussion in their appendix).
However, according to their discussion of the nonrelativistic limit in their
introduction they appear to treat the confining piece as a time-like four
vector in that their spin-orbit piece includes a Thomas term and has no color
magnetic portion. Note that in our Dirac treatment, if we were to treat the
confining piece as pure time-like four-vector, the accompanying $-\mathcal{V}%
^{2}$ term in the Schr\"{o}dinger-like form of our equation would lead to
deconfinement. On the other hand, in the Dirac equation, for a scalar
confining piece the $+S^{2}$ term in the Schr\"{o}dinger-like form of the
equation reinforces confinement. We have shown elsewhere \cite{yng} that these
quadratic local terms ($S^{2}$ and $\mathcal{V}^{2}$) lead through a canonical
transformation \cite{cra84d} to the Darwin interactions for scalar and
time-like vector interactions, respectively. In the discussion in their
appendix, Godfrey and Isgur explain how they modify the Coulomb potential with
the aid of a smoothing function. At the same time they appear to ignore the
Darwin term (e.g. the spin independent contact term) in the on-shell reduction
of the $q\bar{q}$ scattering amplitude. Although they modify the short range
part of their interaction with the aid of a smearing function, this
modification does not compensate for the ignored Darwin term. Since the
authors have ignored this part of the interaction, their inclusion of linear
confinement does not carry with it any $S^{2}$ or $-\mathcal{V}^{2}$ addition.
This patchwork way of including the physics blurs the relativistic
significance of their quark model. The attempt to parametrize this part of the
potential separately from the rest of the Fermi-Breit interaction is no more
justifiable than parametrizing the spin-orbit term separately from the
spin-spin and tensor terms (which they have not done). In our two-body Dirac
equations the Darwin portion is tied directly to the Lorentz form of the
interaction, including the spin-dependent portions of the interaction.

In addition to bypassing the problems of singular spin-dependent terms by
assuming a smoothing parameter Godfrey and Isgur include nonlocal
(momentum-dependent) potentials by replacing the mass dependent $m_{i}^{-1}$
in the Fermi-Breit term by $(\mathbf{p}^{2}+m_{i}^{2})^{-1/2}$. They claim
that this is necessary because the Fermi-Breit reduction (or the on-shell
$q\bar{q}$ scattering amplitude in c.m.) does not adequately express the full
momentum dependence (or nonlocal nature) of the potential. While this might be
true, we have found that such nonlocal behavior is not necessary to obtain
excellent results either in lowest order QED or in the quark model.

Like the Adler-Piran potential that we use in our approach, their potential
includes a running coupling constant. In fact, by convolving a parametric
Gaussian fit to the running coupling constant with the ${\frac{1}%
{\mathbf{q}^{2}}}$ , they obtain their desired smoothing of the Coulomb
potential, thus killing two birds with one stone. In addition, they are able
to treat the zero isospin mesons like the $\eta$ and $\eta^{\prime}$ by
including a phenomenological annihilation term. We leave out this term in our
results of Table I-V and in our comparison with the results of Godfrey and
Isgur in Table VI. Lichtenberg \cite{licht} has compared an earlier version of
our quark model for the meson spectrum with that of Godfrey and Isgur. The
potential we used in that earlier version was the one-parameter Richardson
potential, with the confinement piece chosen to be one-half time-like vector
and one-half scalar. As Lichtenberg pointed out, Godfrey and Isgur obtained
significantly better agreement with the data than we did. He states that this
is because they use significantly more parameters than we do including four in
the potential and six to describe relativistic effects, ten altogether,
compared to our one. However, in fairness to Godfrey and Isgur, we do not
believe that as a general rule the number of parameters that appear in the
potential is, in itself, of as much significance as how these parameters are
distributed. For example, in our present and previous models there are two
invariant functions, $\mathcal{A}$ and $S$ related to the single
nonrelativistic (Adler-Piran) $V_{AP}$ that itself depends on two parameters.
These functions are independent. Specifying their form fixes both
spin-independent and spin-dependent parts of the quasipotential $\Phi_{w}$. We
might say that our formalism has 5 quark mass parameters and two parametric
functions. Increasing the number of parameters that $\mathcal{A},S$ depends on
may or may not increase the goodness of the fit. According to our way of
counting, Godfrey and Isgur have independent parametric functions for the two
spin-orbit parts of the potential, the spin-spin contact part, the tensor
part, the scalar potential, and the spin-independent part of the vector
potential, altogether 6 parametric functions. From our way of counting the
number of parameters the number of parametric functions would not increase no
matter how many parameters are included in fixing the functional form of each
of these six functions. Likewise, in our case, no matter how many parameters
we use in fixing $\mathcal{A},S$ there are only two independent parametric
functions. Our approach is distinct from that of Godfrey and Isgur in that we
do not alter the functional form at the level of the spin-dependence but
rather at the level of the kernels.

Finally, before we compare our present work with that of Godfrey and Isgur, we
note that our present model differs from our earlier one used by Lichtenberg
in his comparison of the two approaches. Our present treatment differs in its
replacement of the Richardson potential by the Adler-Piran potential. The
intermediate range form of the A-P potential is closely tied to an effective
field theory related to QCD and is superior to Richardson's ansatz.
Furthermore, in calculations based on our earlier treatment we ignored the
tensor coupling and unequal mass spin-orbit difference couplings which we
explicitly include in calculations based on the present treatment. We have
also corrected a defect in the decoupling we used between the upper-upper and
lower-lower components of the wave functions for spin-triplet states in our
older treatment.

\subsection{Comparison of Fit with that of Constraint Approach}

We now compare the fit given by our model to that provided by the model of
Godfrey and Isgur.

In Table VI we display in the fourth column the fit we would obtain with our
model if we included only the 77 mesons that our fit has in common with that
of Godfrey and Isgur. We then compare the fits by examining the incremental
$\chi^{2}$ values for each meson.

For the 77 mesons in their table, our fits are closer to data in only 32;
overall their fit is better. Generally speaking our results are better on the
newer mesons and their fit is better on the older mesons. A detailed
comparison reveals the following. Their $R$ values for the two upsilon, the
one charmonium, the $K^{\ast}$, $\phi$ and $\rho-\pi$ triplet $P$ multiplets
are 0.29,0.50,0.57,0.36,0.42,0.47 and are distinctly different from our values
of 0.68,0.76,0.41,-0.66,-0.21,-4.00 and the experimental numbers of
0.66,0.61,0.48,0.09,-0.97,-0.4. As was true for the Brayshaw analysis, the
constraint/Adler-Piran combination gives a distinctly better fit than the
Isgur-Wise approach for the heavier mesons, while both give poor results for
the lighter mesons. As was the case for Brayshaw's spectrum, none of their
light multiplets are inverted, whereas although ours are inverted they are not
inverted in the same way as the experimental numbers are. Again, our
inversions are due to the action of the scalar potential. Godfrey and Isgur
include a time-like contribution in the spin-orbit part of their Hamiltonian.
This may be responsible for their lack of the partial inversion that appears
in the data.

Computing their $^{3}P$ averages $[5(^{3}P_{2})+3(^{3}P_{1})+1(^{3}P_{0})]/9$
along with the $^{1}P_{1}$ mesons for the charmonium, $K^{\ast}$ and $\rho
-\pi$ system we find the following three pairs of numbers:
3.524,3.520;1.392,1.340;1.262,1.220. Comparison with our numbers of
3.519,3.520;1.424,1.411;1.419,1.397 and the experimental numbers of
3.526,3.525;1.402,1.375;1.231,1.303 shows the constraint approach giving
slightly better numbers for the heavier mesons and the $K^{\ast}$ while the
Godfrey-Isgur results are somewhat better for the lighter mesons. Their
$^{3}D$ average $[7(^{3}D_{2})+5(^{3}D_{1})+3(^{3}D_{1})]/15$ and their
$^{1}D_{2}$ meson for the $K^{\ast}$ are 1.795,1.780 MeV while our results and
the experimental results are 1.873,1.879 and 1.774,1.773 MeV respectively. Our
results are relatively closer to one another while theirs are closer to the
data in an absolute sense. This is indicative of the general trend of our
orbitally excited light mesons being somewhat high. We suspect that this is
due to the $S^{2}$ behavior becoming dominant at longer distance, changing the
behavior of the confining potential in the effective Schrodinger-like equation
from linear to quadratic.

Their values of the hyperfine splittings are
130,60,160,150,430,130,620,150,120 MeV for the two charmonium multiplets, and
the $D-D^{\ast},D_{s}-D_{s}^{\ast},$ two $K-K^{\ast},$ and three $\pi-\rho$
pairs. Comparing with the experimental splittings of
117,92,142,144,398,-48,627,165,354 MeV and our results of
150,78,133,145,403,208,645,239,166 MeV demonstrates that while our results are
closer than theirs for most of the newer mesons and the $K-K^{\ast}$, their
results are more in line for most of the older mesons. Again this shows a
pattern of our method overestimating the radially excited states of the light mesons.

Let us see if this trend of overestimation continues for the radial
excitations of fixed quantum numbers. The six upsilon states in the data
portion of the table occur at intervals of 563,332,225,285,154 MeV while the
three charmonium triplet states and the two charmonium singlet states occur at
intervals of 589,354,614 MeV while the three singlet $K$ and the two triplet
$K^{\ast}$ states occur at intervals of 977,370 and 520 MeV. Finally the three
pion and three rho excitations occur at 1160,495 and 698,654 MeV. The
corresponding Isgur-Wise intervals are
540,350,280,250,220,580,420,650,980,570,680,1150,580,680,550 MeV compared to
570,336,256,213,186,561,393,633,1099,495,894,1383,634,986,561 MeV. Again we
encounter a pattern of our results being more accurate overall for the newer
mesons while theirs are more accurate for the older ones (with our results too
large for all of the older ones).

Primarily what we learn from this comparison is that not only does the scalar
interaction lead to partial triplet inversions for the lighter mesons but also
yields radial and orbital excitations that are too high for a related reason:
the presence of the $S^{2}$ term in the effective potential.

On the other hand, as Godrey and Isgur themselves point out, their treatment
of the relativistic effects is schematic, with no wave equation involved,
allowing an uncontrolled approach in which there are no tightly fixed
connections among the various spin-dependent and spin independent parts of the
effective potential $\Phi$

\section{Conclusion and Warnings About the Dangers of ''Relativistic'' and
''Nonrelativistic'' Spectral Fits}

In this paper, we have investigated how well the relativistic constraint
approach performs in comparison with selected alternatives when used to
produce a single fit of experimental results over the whole meson spectrum.
Our approach is distinguished from others by its foundation - a set of
coupled, compatible, fully covariant wave equations whose nonperturbative
numerical solution yields the mass spectrum along with wave functions for the
$q\bar{q}$ meson bound states. Its virtue - generation of fully covariant spin
structures - also serves to restrict and relate plausible interaction terms
just as the ordinary single-particle Dirac equation determines relations among
Pauli spin dependences and fixes the proper strength of the Thomas precession
term in electrodynamics. The dynamical structures of the constraint approach
were originally discovered in classical relativistic mechanics but have since
been verified for electrodynamics through diagrammatic summation in quantum
field theory in the field-theoretic eikonal approximation \cite{saz97}.

To use such relativistic equations to treat the phenomenological chromodynamic
$q\bar{q}$ bound-state, one must construct a relativistic interaction that
possesses the limiting behaviors of QCD. In our approach we have done this by
using the nonrelativistic static Adler-Piran potential to construct a
plausible relativistic interaction that regenerates the AP potential as its
nonrelativistic limit. In our equations, this process generates a host of
accompanying interaction terms. When describing these interactions, one must
guard against a semantic difficulty in the verbal classification of the
various parts of the interaction as \textquotedblleft scalar\textquotedblright%
, \textquotedblleft vector\textquotedblright, \textquotedblleft
pseudovector\textquotedblright\ etc. The various formalisms classify these in
different ways but in our equations, the meaning of these terms can be readily
determined through examining their roles in the defining equation
Eq.(\ref{tbdes},\ref{cnhyp},\ref{cnmyp},\ref{del}). Once these terms have been
introduced, the constraint formalism automatically produces a system of
important accompanying terms like quadratic terms that dominate at long
distance (reinforcing or undermining confinement) or spin dependences that
accompany chosen interactions producing level splits that agree or disagree
with the experimental results in various parts of the spectrum.

After identification of the relativistic transformation properties of
interaction terms the constraint method leaves almost no leeway for fiddling
with (unnecessary) cutoffs, etc. Some years ago, when applied to the
$e^{-}e^{+}$ system, its structure proved restrictive enough to rule out
within it the presence of postulated anomalous resonances\cite{bckr,spence}.
In recent work on the relation of our equations to the Breit and earlier
Eddington-Gaunt equations for electromagnetic bound-states, the method has
explicitly demonstrated the importance of keeping spin couplings among pieces
of the full 16-component wave-functions whose counterparts are often truncated
or discarded in alternative treatments \cite{cwyw,va97}.

The fits that we have examined as alternatives fall into different classes:
motivated relativistic fits ( constraint vs truncations of standard
field-theoretic), ad-hoc relativistic fits, and cautious semirelativistic fits.

Among the relativistic ones, there is a danger exemplified by the Brayshaw
model which achieves relative success despite the dubious relativistic nature
of its interaction. As always, what makes fits hard to judge at this stage is
the ease with which one can achieve apparent success over limited regions of
the spectrum using highly-parametrized interactions.

Finally, some authors have even produced unabashedly nonrelativistic fits.
They claim to obtain good fits to the meson spectrum through the use of
variants of the nonrelativistic quark model (NRQM). \ \cite{mor90},
\cite{martin}. \ These authors even claim success at fitting the light quark
mesons for which the assumptions $T<<mc^{2}$,
%TCIMACRO{\TEXTsymbol{\vert}}%
%BeginExpansion
$\vert$%
%EndExpansion
$V|<<mc^{2}$ of the nonrelativistic Schr\"{o}dinger equation are patently
false. What can account for the apparent success of the NRQM? \ 

Morpurgo states \cite{mor90} that the various potential models, including the
nonrelativistic quark model, are merely different parametrizations of an
underlying exact QCD Lagrangian description. That is, all use essentially the
same spin and flavor structures. For example, for the mesons one can derive a
\textquotedblleft parametrized mass\textquotedblright\ with general form (for
the present discussion restricted to $\pi,K,\rho,K\ast$)%

\begin{equation}
``\mathrm{parametrized\ mass}^{\prime\prime}=A+B(P_{1}^{s}+P_{2}^{s}%
)+C\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}+D(P_{1}^{s}+P_{2}^{s})\vec{\sigma
}_{1}\cdot\vec{\sigma}_{2} \label{mor}%
\end{equation}
in which $P_{i}^{s}$ is the projector onto the strange quark sector. These
authors say that this structure although typical of an NRQM description,
follows from QCD itself. They state that the form Eq.(\ref{mor}) is common to
all of the relativistic or semirelativistic quark models. They assert that any
one of them can be successful but not superior to any other, if it merely
reproduces the spin flavor structure of the general parametrization. Thus,
from their point of view selection of the \textquotedblleft
best\textquotedblright\ model is entirely a matter of taste and simplicity. We
disagree with this assessment for the following reasons. First, the kinetic
and potential parameters have significances beyond simply producing a fit for
the two-body bound-state sector in isolation. When the spin-flavor structure
in (\ref{mor}) appears in the constraint approach, its accompanying
constituent quark masses turn out to be closer to the current-quark masses
than those produced by most other approaches while the constraint method
requires only two, or at most three parametric functions to be used beyond the
parameters of the constituent quark masses. The constraint scheme successfully
uses one set of these parametric functions for the entire spectrum of meson
states including the radial as well as orbital excitations. But most
importantly, within the bound-state spectrum itself, in our relativistic
approach even though superficially sharing the basic spin-flavor structure
(\ref{mor}), all potentials do not fare equally well. The essential point is
that even in the simplest form of our equations, the parametrization is
different from that given in the Morpurgo form in that its parameters A,B,C,D,
are themselves dependent on the energy operator on the left hand side. When
that happens, some relativistic potentials do better than others. In
particular, of those we investigated, the potential that works the best (the
Adler-Piran potential) is one possessing many of the features important in
lattice QCD calculations (e.g. linear and subdominant confining pieces). The
combination of the constraint approach with the Adler-Piran potential embodies
more of the important physical effects contained in QCD-related effective or
numerical field theories.

Can one understand the apparent successes of the NRQM fits by starting from
the relativistic treatments? Some authors \cite{martin88} and \cite{jaczko}
have used bounds on the kinetic square-root operator $\sqrt{\mathbf{p}%
^{2}+m^{2}}$ to attempt to understand the apparent success of the
nonrelativistic potential models for relativistic quark-antiquark states.
Instead, we will give an explanation that starts directly from the
relativistic constraint approach.

Some years ago, Caswell and Lepage \cite{cas} rewrote a relativistic
constraint equation in an effective nonrelativistic Schrodinger-like form.
Here, we do the opposite and recast the NRQM Schrodinger equation in a form
resembling the constraint equation. As we have seen our two-body Dirac
equations lead to an effective Schr\"{o}dinger-like equation of the form
\begin{equation}
\lbrack p^{2}+\Phi_{w}(x_{\perp},\sigma_{1},\sigma_{2})]\psi=b^{2}(w)\psi
\end{equation}
In the c.m. system this becomes
\begin{equation}
\lbrack\mathbf{p}^{2}+\Phi_{w}(\mathbf{r},\mathbf{\sigma}_{1},\mathbf{\sigma
}_{2})]\psi=b^{2}(w)\psi
\end{equation}
Even though the stationary state nonrelativistic Schr\"{o}dinger equation
\begin{equation}
\lbrack\frac{\mathbf{p}^{2}}{2\mu}+V(\mathbf{r},\mathbf{\sigma}_{1}%
,\mathbf{\sigma}_{2})]\psi=E_{B}\psi\label{nre}%
\end{equation}
has a similar form, the corresponding structures in each have entirely
different physical significances. For example, in Eq.(\ref{nre}), the vectors
$\mathbf{p}$ and $\mathbf{r}$ are nonrelativistic quantities in contrast with
their counterparts in the constraint approach that appear in the relativistic
equation in the c.m.. system. One can easily manipulate the nonrelativistic
Schr\"{o}dinger equation into a form similar in appearance to the constraint
Schr\"{o}dinger form by multiplying both sides of the equation by $2\mu$ and
adding $b^{2}(w)-2\mu E_{B}$ to both sides. The result is
\begin{equation}
\lbrack\mathbf{p}^{2}+\Phi_{w}(\mathbf{r},\mathbf{\sigma}_{1},\mathbf{\sigma
}_{2})]\psi=b^{2}(w)\psi
\end{equation}
in which
\begin{equation}
\Phi_{w}(\mathbf{r},\mathbf{\sigma}_{1},\mathbf{\sigma}_{2})=2\mu
V(\mathbf{r},\mathbf{\sigma}_{1},\mathbf{\sigma}_{2})+b^{2}(w)-2\mu E_{B}%
\end{equation}
In numerical calculations the $\mathbf{p}$ operator and $\mathbf{r}$ variable
are treated the in the same manner in calculations based on both the
relativistic constraint equation and the nonrelativistic equation. \ But as we
have seen, they have different physical significances in each equation. \ When
used to fit parts of the meson spectrum, the apparent success of the NRQM from
this point of view is then due to its incorporation of variables numerically
indistinguishable from their covariant versions together with a potential that
fortuitously coincides (for a limited range of states) with a covariant one
modified by an energy dependent constant term that varies from state to state.

\appendix\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}

\section{ \ Radial Wave Equations Corresponding to Eqs.(\ref{cnstra}) and
(\ref{cnstrb})}

We obtain the radial forms of the coupled constraint equations (\ref{cnstra})
and (\ref{cnstrb}) that we use for our numerical solution for the general
fermion-antifermion system by forming standard matrix elements of
spin-dependent operators (see Appendix C of Ref.(\cite{cbwv})). We start from
the general wave function of the form
\begin{equation}
\psi_{ijm}=\sum_{l,s}c_{ils}R_{ilsj}\mathcal{Y}_{lsjm};\ i=1,2,3,4
\end{equation}
in which $R_{ilsj}={\frac{u_{ilsj}}{r}}$ is the associated radial wave
function and $\mathcal{Y}_{lsjm}$ is the total angular momentum eigenfunction.
The resultant Schr\"{o}dinger-like equation \ref{cnstra} for the singlet
states ($j=l,s=0$) $u_{1j0j}$ which couples this upper-upper component to
$u_{1j1j}$ and $u_{4j0j}$ is then given by
\[
\{-\frac{d^{2}}{dr^{2}}+\frac{j(j+1)}{r^{2}}+2m_{w}S+S^{2}+2\epsilon
_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\[
+ln^{\prime}\chi_{1}\chi_{2}\frac{d}{dr}-{\frac{3}{2}}\partial^{2}%
\mathcal{G}+{\frac{9}{4}}(\mathcal{G}^{\prime})^{2}+{\frac{3}{2}}ln^{\prime
}\chi_{1}\chi_{2}\mathcal{G}^{\prime}+{\frac{1}{4}}(\mathcal{G}^{\prime
}+L^{\prime})^{2}-\frac{ln^{\prime}\chi_{1}\chi_{2}}{r}\}u_{1j0j}%
\]%
\[
-\frac{ln^{\prime}(\chi_{1}/\chi_{2})}{r}\sqrt{(j(j+1)}u_{1j1j}%
\]%
\begin{equation}
+\{[{\frac{1}{2}}ln^{\prime}\chi_{1}\chi_{2}(\mathcal{G}+L)^{\prime}+{\frac
{3}{2}}\mathcal{G}^{\prime}(\mathcal{G}+L)^{\prime}-{\frac{1}{2}}\partial
^{2}(\mathcal{G}+L)]\}u_{4j0j}=b^{2}(w)u_{1j0j}.
\end{equation}
The corresponding equation \ref{cnstrb} for the lower-lower component
$u_{4j0j}$ which couples it to $u_{4j1j}$ and $u_{1j0j}$ then takes the form
\[
\{-\frac{d^{2}}{dr^{2}}+\frac{j(j+1)}{r^{2}}+2m_{w}S+S^{2}+2\epsilon
_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\begin{subequations}
\[
+ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}\frac{d}{dr}-{\frac{3}{2}}\partial
^{2}\mathcal{G}+{\frac{9}{4}}(\mathcal{G}^{\prime})^{2}+{\frac{3}{2}%
}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}\mathcal{G}^{\prime}+{\frac{1}{4}%
}(\mathcal{G}^{\prime}+L^{\prime})^{2}-\frac{ln^{\prime}\bar{\chi}_{1}%
\bar{\chi}_{2}}{r}\}u_{4j0j}%
\]%
\end{subequations}
\begin{subequations}
\[
-\frac{ln^{\prime}(\bar{\chi}_{1}/\bar{\chi}_{2})}{r}\sqrt{(j(j+1)}u_{4j1j}%
\]%
\end{subequations}
\begin{equation}
+\{[{\frac{1}{2}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}(\mathcal{G}%
+L)^{\prime}+{\frac{3}{2}}\mathcal{G}^{\prime}(\mathcal{G}+L)^{\prime}%
-{\frac{1}{2}}\partial^{2}(\mathcal{G}+L)]\}u_{1j0j}=b^{2}(w)u_{4j0j}.
\end{equation}
For $S$ states or equal mass systems, these equations lose their couplings to
$u_{1j1j}$ and $u_{4j1j}$ since $\sqrt{j(j+1)}ln(\chi_{1}/\chi_{2}%
)=\sqrt{j(j+1)}ln(\bar{\chi}_{1}/\bar{\chi}_{2})=0$. However, for the general
unequal mass case, these equations are coupled to those for the $j=l,s=1$
components $u_{1j1j}$ and $u_{4j1j}$. For those triplet states the remaining
coupled Schr\"{o}dinger-like equations in the full set of four are
\begin{subequations}
\[
\{-\frac{d^{2}}{dr^{2}}+\frac{j(j+1)}{r^{2}}+2m_{w}S+S^{2}+2\epsilon
_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\end{subequations}
\begin{subequations}
\[
+ln^{\prime}\chi_{1}\chi_{2}\frac{d}{dr}-{\frac{1}{2}}\partial^{2}%
\mathcal{G}+{\frac{1}{4}}(\mathcal{G}^{\prime})^{2}+{\frac{1}{2}}ln^{\prime
}\chi_{1}\chi_{2}\mathcal{G}^{\prime})+{\frac{1}{4}}(\mathcal{G}^{\prime
}+L^{\prime})^{2}-{\frac{ln^{\prime}\chi_{1}\chi_{2}}{r}}%
\]%
\end{subequations}
\begin{subequations}
\[
+{\frac{ln^{\prime}\chi_{1}\chi_{2}+\mathcal{G}^{\prime}}{r}}\}u_{1j1j}%
-\frac{ln^{\prime}(\chi_{1}/\chi_{2})}{r}\sqrt{(j(j+1)}u_{1j0j}%
\]%
\end{subequations}
\begin{equation}
+\{[-{\frac{1}{2}}ln^{\prime}\chi_{1}\chi_{2}(\mathcal{G}+L)^{\prime}%
-{\frac{1}{2}}\mathcal{G}^{\prime}(\mathcal{G}+L)^{\prime}+{\frac{1}{2}%
}\partial^{2}(\mathcal{G}+L)]-{\frac{(\mathcal{G}+L)^{\prime}}{r}}%
\}u_{4j1j}=b^{2}(w)u_{1j1j}%
\end{equation}
and
\begin{subequations}
\[
\{-{\frac{d^{2}}{dr^{2}}}+{\frac{j(j+1)}{r^{2}}}+2m_{w}S+S^{2}+2\epsilon
_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\end{subequations}
\begin{subequations}
\[
+ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}{\frac{d}{dr}}-{\frac{1}{2}}%
\partial^{2}\mathcal{G}+{\frac{1}{4}}(\mathcal{G}^{\prime})^{2}+{\frac{1}{2}%
}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}\mathcal{G}^{\prime})+{\frac{1}{4}%
}(\mathcal{G}^{\prime}+L^{\prime})^{2}-{\frac{ln^{\prime}\bar{\chi}_{1}%
\bar{\chi}_{2}}{r}}%
\]%
\end{subequations}
\begin{subequations}
\[
+{\frac{ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}+\mathcal{G}^{\prime}}{r}%
}\}u_{4j1j}-{\frac{ln^{\prime}(\bar{\chi}_{1}/\bar{\chi}_{2})}{r}}%
\sqrt{(j(j+1)}u_{4j0j}%
\]%
\end{subequations}
\begin{equation}
+\{[-{\frac{1}{2}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}(\mathcal{G}%
+L)^{\prime}-{\frac{1}{2}}\mathcal{G}^{\prime}(\mathcal{G}+L)^{\prime}%
+{\frac{1}{2}}\partial^{2}(\mathcal{G}+L)]-{\frac{(\mathcal{G}+L)^{\prime}}%
{r}}\}u_{1j1j}=b^{2}(w)u_{4j1j}.
\end{equation}
Next we write out the four coupled equations for the two triplet states
$j=l\pm1$. Eq.(\ref{cnstra}) for the triplet states ($s=1,l=j-1$) $u_{1j-11j}%
$, which couples this upper-upper component wave function to $u_{1j+11j}%
,u_{4j+11j},$ and $u_{4j-11j}$, becomes
\[
\{-{\frac{d^{2}}{dr^{2}}}+{\frac{j(j-1)}{r^{2}}}+2m_{w}S+S^{2}+2\epsilon
_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\[
+ln^{\prime}\chi_{1}\chi_{2}{\frac{d}{dr}}-{\frac{1}{2(2j+1)}}\partial
^{2}\mathcal{G}+{\frac{1}{4}}(\mathcal{G}^{\prime})^{2}+{\frac{1}{2(2j+1)}%
}ln^{\prime}\chi_{1}\chi_{2}\mathcal{G}^{\prime}+{\frac{1}{4}}(\mathcal{G}%
^{\prime}+L^{\prime})^{2}-{\frac{ln^{\prime}\chi_{1}\chi_{2}}{r}}%
\]%
\[
-(j-1){\frac{ln^{\prime}\chi_{1}\chi_{2}+\mathcal{G}^{\prime}/(2j+1)}{r}%
}\}u_{1j-11j}%
\]%
\[
+{\frac{\sqrt{(j(j+1)}}{2j+1}}[-(\mathcal{G}^{\prime\prime}-{\frac
{\mathcal{G}^{\prime}}{r}})+ln^{\prime}\chi_{1}\chi_{2}\mathcal{G}^{\prime
}]u_{1j+11j}%
\]%
\[
+\{-({\frac{1}{(2j+1)}}ln^{\prime}\chi_{1}\chi_{2}+\mathcal{G}^{\prime}%
){\frac{(\mathcal{G+}L)^{\prime}}{2}}+{\frac{1}{2(2j+1)}}\partial
^{2}(\mathcal{G}+L)+{\frac{(j-1)}{(2j+1)}}{\frac{(\mathcal{G}+L)^{\prime}}{r}%
}\}u_{4j-11j}%
\]%
\begin{equation}
+{\frac{\sqrt{(j(j+1)}}{2j+1}}[-ln^{\prime}\chi_{1}\chi_{2}(\mathcal{G}%
+L)^{\prime}+((\mathcal{G}+L)^{\prime\prime}-{\frac{(\mathcal{G}+L)^{\prime}%
}{r}})]\}u_{4j+11j}=b^{2}(w)u_{1j-11j}%
\end{equation}
The corresponding equation \ref{cnstrb} for the lower-lower component becomes
\begin{subequations}
\[
\{-{\frac{d^{2}}{dr^{2}}}+{\frac{j(j-1)}{r^{2}}}+2m_{w}S+S^{2}+2\epsilon
_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\end{subequations}
\begin{subequations}
\[
+ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}{\frac{d}{dr}}-{\frac{1}{2(2j+1)}%
}\partial^{2}\mathcal{G}+{\frac{1}{4}}(\mathcal{G}^{\prime})^{2}+{\frac
{1}{2(2j+1)}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}\mathcal{G}^{\prime
}+{\frac{1}{4}}(\mathcal{G}^{\prime}\mathcal{+}L^{\prime})^{2}-{\frac
{ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}}{r}}%
\]%
\end{subequations}
\begin{subequations}
\[
-(j-1){\frac{ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}+\mathcal{G}^{\prime
}/(2j+1)}{r}}\}u_{4j-11j}%
\]%
\end{subequations}
\begin{subequations}
\[
+{\frac{\sqrt{(j(j+1)}}{2j+1}}[-(\mathcal{G}^{\prime\prime}-{\frac
{\mathcal{G}^{\prime}}{r}})+ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}%
\mathcal{G}^{\prime}]u_{4j+11j}%
\]%
\end{subequations}
\begin{subequations}
\[
+\{-({\frac{1}{(2j+1)}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}+\mathcal{G}%
^{\prime}){\frac{(\mathcal{G}+L)^{\prime}}{2}}+{\frac{1}{2(2j+1)}}\partial
^{2}(\mathcal{G}+L)+{\frac{(j-1)}{(2j+1)}}{\frac{(\mathcal{G}+L)^{\prime}}{r}%
}\}u_{1j-11j}%
\]%
\end{subequations}
\begin{equation}
+{\frac{\sqrt{(j(j+1)}}{2j+1}}[-ln^{\prime}\bar{\chi}_{1}\bar{\chi}%
_{2}(\mathcal{G}+L)^{\prime}+((\mathcal{G}+L)^{\prime\prime}-{\frac
{(\mathcal{G}+L)^{\prime}}{r}})]\}u_{1j+11j}=b^{2}(w)u_{4j-11j}.
\end{equation}
These two equations are coupled to the corresponding two equations for the
triplet $s=1,l=j+1$ states which take the forms
\begin{subequations}
\[
\{-{\frac{d^{2}}{dr^{2}}}+{\frac{(j+1)(j+2)}{r^{2}}}+2m_{w}S+S^{2}%
+2\epsilon_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\end{subequations}
\begin{subequations}
\[
+ln^{\prime}\chi_{1}\chi_{2}{\frac{d}{dr}}+{\frac{1}{2(2j+1)}}\partial
^{2}\mathcal{G}+{\frac{1}{4}}(\mathcal{G}^{\prime})^{2}-{\frac{1}{2(2j+1)}%
}ln^{\prime}\chi_{1}\chi_{2}\mathcal{G}^{\prime}+{\frac{1}{4}}(\mathcal{G}%
^{\prime}+L^{\prime})^{2}-{\frac{ln^{\prime}\chi_{1}\chi_{2}}{r}}%
\]%
\end{subequations}
\begin{subequations}
\[
+(j+2){\frac{ln^{\prime}\chi_{1}\chi_{2}-\mathcal{G}^{\prime}/(2j+1)}{r}%
}\}u_{1j+11j}%
\]%
\end{subequations}
\begin{subequations}
\[
+{\frac{\sqrt{(j(j+1)}}{2j+1}}[-(\mathcal{G}^{\prime\prime}-{\frac
{\mathcal{G}^{\prime}}{r}})+ln^{\prime}\chi_{1}\chi_{2}\mathcal{G}^{\prime
}]u_{1j-11j}%
\]%
\end{subequations}
\begin{subequations}
\[
+\{-(-{\frac{1}{(2j+1)}}ln^{\prime}\chi_{1}\chi_{2}+\mathcal{G}^{\prime
}){\frac{(\mathcal{G}+L)^{\prime}}{2}}-{\frac{1}{2(2j+1)}}\partial
^{2}(J+L)+{\frac{(j+2)}{(2j+1)}}{\frac{(\mathcal{G}+L)^{\prime}}{r}%
}\}u_{4j+11j}%
\]%
\end{subequations}
\begin{equation}
+{\frac{\sqrt{(j(j+1)}}{2j+1}}[-ln^{\prime}\chi_{1}\chi_{2}(\mathcal{G+}%
L)^{\prime}+((\mathcal{G}+L)^{\prime\prime}-{\frac{(\mathcal{G}+L)^{\prime}%
}{r}})]\}u_{4j-11j}=b^{2}(w)u_{1j+11j}%
\end{equation}
and
\begin{subequations}
\[
\{-{\frac{d^{2}}{dr^{2}}}+{\frac{(j+1)(j+2)}{r^{2}}}+2m_{w}S+S^{2}%
+2\epsilon_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\end{subequations}
\begin{subequations}
\[
+ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}{\frac{d}{dr}}+{\frac{1}{2(2j+1)}%
}\partial^{2}\mathcal{G}+{\frac{1}{4}}(\mathcal{G}^{\prime})^{2}-{\frac
{1}{2(2j+1)}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}\mathcal{G}^{\prime
}+{\frac{1}{4}}(\mathcal{G}^{\prime}+L^{\prime})^{2}-{\frac{ln^{\prime}%
\bar{\chi}_{1}\bar{\chi}_{2}}{r}}%
\]%
\end{subequations}
\begin{subequations}
\[
+(j+2){\frac{ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}-\mathcal{G}^{\prime
}/(2j+1)}{r}}\}u_{4j+11j}%
\]%
\end{subequations}
\begin{subequations}
\[
+{\frac{\sqrt{(j(j+1)}}{2j+1}}[-(\mathcal{G}^{\prime\prime}-{\frac
{\mathcal{G}^{\prime}}{r}})+ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}%
\mathcal{G}^{\prime}]u_{4j-11j}%
\]%
\end{subequations}
\begin{subequations}
\[
+\{-(-{\frac{1}{(2j+1)}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}+\mathcal{G}%
^{\prime}){\frac{(\mathcal{G}+L)^{\prime}}{2}}-{\frac{1}{2(2j+1)}}\partial
^{2}(\mathcal{G}+L)+{\frac{(j+2)}{(2j+1)}}{\frac{(\mathcal{G}+L)^{\prime}}{r}%
}\}u_{1j+11j}%
\]%
\end{subequations}
\begin{equation}
+{\frac{\sqrt{(j(j+1)}}{2j+1}}[-ln^{\prime}\bar{\chi}_{1}\bar{\chi}%
_{2}(\mathcal{G}+L)^{\prime}+((\mathcal{G}+L)^{\prime\prime}-{\frac
{(\mathcal{G}+L)^{\prime}}{r}})]\}u_{1j-11j}=b^{2}(w)u_{4j+11j}.
\end{equation}
For the $^{3}P_{0}$ states there are only two coupled equations:
\[
\{-{\frac{d^{2}}{dr^{2}}}+{\frac{2}{r^{2}}}+2m_{w}S+S^{2}+2\epsilon
_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\[
+ln^{\prime}\chi_{1}\chi_{2}{\frac{d}{dr}}+{\frac{1}{2}}\partial
^{2}\mathcal{G}+{\frac{1}{4}}(\mathcal{G}^{\prime})^{2}-{\frac{1}{2}%
}ln^{\prime}\chi_{1}\chi_{2}\mathcal{G}^{\prime}+{\frac{1}{4}}(\mathcal{G}%
^{\prime}+L^{\prime})^{2}-{\frac{ln^{\prime}\chi_{1}\chi_{2}}{r}}%
\]%
\[
+2{\frac{ln^{\prime}\chi_{1}\chi_{2}-\mathcal{G}^{\prime}}{r}}\}u_{1101}%
\]%
\begin{subequations}
\[
-\{-{\frac{1}{2}}ln^{\prime}\chi_{1}\chi_{2}(\mathcal{G}+L)^{\prime}+{\frac
{1}{2}}\mathcal{G}^{\prime}(\mathcal{G}+L)^{\prime}+{\frac{1}{2}}\partial
^{2}(\mathcal{G}+L)-2{\frac{(\mathcal{G}+L)^{\prime}}{r}}\}u_{4101}%
\]%
\end{subequations}
\begin{equation}
=b^{2}(w)u_{1101}%
\end{equation}
and
\begin{subequations}
\[
\{-{\frac{d^{2}}{dr^{2}}}+{\frac{2}{r^{2}}}+2m_{w}S+S^{2}+2\epsilon
_{w}\mathcal{A}-\mathcal{A}^{2}%
\]%
\end{subequations}
\begin{subequations}
\[
+ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}{\frac{d}{dr}}+{\frac{1}{2}}%
\partial^{2}\mathcal{G}+{\frac{1}{4}}(\mathcal{G}^{\prime})^{2}-{\frac{1}{2}%
}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}\mathcal{G}^{\prime}+{\frac{1}{4}%
}(\mathcal{G}^{\prime}+L^{\prime})^{2}-{\frac{ln^{\prime}\bar{\chi}_{1}%
\bar{\chi}_{2}}{r}}%
\]%
\end{subequations}
\begin{subequations}
\[
+2{\frac{ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}-\mathcal{G}^{\prime}}{r}%
}\}u_{4101}%
\]%
\end{subequations}
\begin{subequations}
\[
-\{-{\frac{1}{2}}ln^{\prime}\bar{\chi}_{1}\bar{\chi}_{2}(\mathcal{G}%
+L)^{\prime}+{\frac{1}{2}}\mathcal{G}^{\prime}(\mathcal{G}+L)^{\prime}%
+{\frac{1}{2}}\partial^{2}(\mathcal{G}+L)-2{\frac{(\mathcal{G}+L)^{\prime}}%
{r}}\}u_{1101}%
\]%
\end{subequations}
\begin{equation}
=b^{2}(w)u_{4101}.
\end{equation}


Note that in each of the equations above, the quasipotential couples the
upper-upper component to the lower-lower component.

\section{\ Simpler Pauli-form of the Two-Body Dirac Equations for $\psi
_{1}+\psi_{4}$ and their Radial Forms}

Reference \cite{long} sets out Two-Body Dirac Equations containing general
covariant interactions along with their accompanying Schr\"{o}dinger-like
forms. The general interactions consist of the eight Lorentz invariant forms
corresponding to scalar, time and space-like vector studied here along with
five others: pseudoscalar, time- and space-like pseudovector, axial and polar
tensor. When Eq.(\ref{tbdes}) is written in terms of the four four-component
spinors $\psi_{1...4}$ it decomposes into eight coupled equations. In
\cite{long} Long and Crater showed how these may be rearranged in Pauli-form
or Schrodinger-like equations in terms of the combination $\phi_{+}=\psi
_{1}+\psi_{4}$ in the process providing a simpler coupling scheme than that
contained in Eq.(\ref{sch}). Eq.(4.24) of that reference yields the following
equation (simplified here for electromagnetic-like interactions ($\partial
J\equiv{\frac{\partial E_{1}}{E_{2}}}=-\partial G$) and scalar interactions
alone):
\[
\lbrack E_{1}D_{1}^{-+}{\frac{1}{E_{1}M_{2}+E_{2}M_{1}}}(M_{2}D_{1}^{++}%
-M_{1}D_{2}^{++})
\]%
\[
+M_{1}D_{1}^{--}{\frac{1}{E_{1}M_{2}+E_{2}M_{1}}}(E_{2}D_{1}^{++}+E_{1}%
D_{2}^{++})]\phi_{+}%
\]%
\begin{equation}
=(E_{1}^{2}-M_{1}^{2})\phi_{+}%
\end{equation}
in which the kinetic-recoil terms appear through the combinations:
\[
D_{1}^{++}=\exp\mathcal{G}\Bigl[\sigma_{1}\cdot{p}+{\frac{i}{2}}{\sigma
_{2}\cdot\partial}\bigl[L+\mathcal{G}(1-{\sigma_{1}\cdot\sigma_{2}%
})\bigr]\Bigr]
\]%
\begin{subequations}
\[
D_{2}^{++}=\exp\mathcal{G}\Bigl[\sigma_{2}\cdot{p}+{\frac{i}{2}}{\sigma
_{1}\cdot\partial}\bigl[L+\mathcal{G}(1-{\sigma_{1}\cdot\sigma_{2}%
})\bigr]\Bigr]
\]%
\end{subequations}
\[
D_{1}^{-+}=\exp\mathcal{G}\Bigl[\sigma_{1}\cdot{p}+{\frac{i}{2}}{\sigma
_{2}\cdot\partial}\bigl[-L+\mathcal{G}(1-{\sigma_{1}\cdot\sigma_{2}%
})\bigr]\Bigr]
\]%
\begin{equation}
D_{1}^{--}=\exp\mathcal{G}\Bigl[\sigma_{1}\cdot{p}+{\frac{i}{2}}{\sigma
_{2}\cdot\partial}\bigl[L-\mathcal{G}(1+{\sigma_{1}\cdot\sigma_{2}%
})\bigr]\Bigr].
\end{equation}
Manipulations using both sets of Pauli-matrices then lead to the c.m. form
\[
\lbrack\mathbf{p}^{2}+2m_{w}S+S^{2}+2\epsilon_{2}\mathcal{A}-\mathcal{A}^{2}%
\]%
\[
-i(2\mathcal{G}^{\prime}-\frac{E_{2}M_{2}+E_{1}M_{1}}{E_{2}M_{1}+E_{2}M_{1}%
}(L-\mathcal{G})^{\prime}\hat{r}\cdot\mathbf{p}-{\frac{1}{2}}\nabla
^{2}\mathcal{G}-{\frac{1}{4}}{(\mathcal{G})^{\prime}}^{2}-(\mathcal{G}%
^{\prime}+L^{\prime})^{2}+\frac{E_{2}M_{2}+E_{1}M_{1}}{E_{2}M_{1}+E_{2}M_{1}%
}{\frac{1}{2}}\mathcal{G}^{\prime}(L-\mathcal{G}^{\prime})
\]%
\[
+{\frac{\vec{L}\cdot(\vec{\sigma}_{1}+\vec{\sigma}_{2})}{r}}(\mathcal{G}%
^{\prime}-{\frac{1}{2}}\frac{E_{2}M_{2}+E_{1}M_{1}}{E_{2}M_{1}+E_{2}M_{1}%
}(L-\mathcal{G})^{\prime}-{\frac{\vec{L}\cdot(\vec{\sigma}_{1}-\vec{\sigma
}_{2})}{2r}}\frac{E_{2}M_{2}-E_{1}M_{1}}{E_{2}M_{1}+E_{2}M_{1}}(L-\mathcal{G}%
)^{\prime}%
\]%
\[
\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}({\frac{1}{2}}\nabla^{2}\mathcal{G}%
+{\frac{1}{2r}}L^{\prime}+{\frac{1}{2}}(\mathcal{G}^{\prime})^{2}-{\frac{1}%
{2}}\mathcal{G}^{\prime}(L-\mathcal{G})^{\prime}\frac{E_{2}M_{2}+E_{1}M_{1}%
}{E_{2}M_{1}+E_{2}M_{1}})
\]%
\[
\vec{\sigma}_{1}\cdot\hat{r}\vec{\sigma}_{2}\cdot\hat{r}({\frac{1}{2}}%
\nabla^{2}L-{\frac{3}{2r}}L^{\prime}+\mathcal{G}^{\prime}L^{\prime}-{\frac
{1}{2}}L^{\prime}(L-\mathcal{G})^{\prime}\frac{E_{2}M_{2}+E_{1}M_{1}}%
{E_{2}M_{1}+E_{2}M_{1}})
\]%
\begin{equation}
+{\frac{i}{2}}(L+\mathcal{G})^{\prime}(\vec{\sigma}_{1}\cdot\hat{r}\vec
{\sigma}_{2}\cdot\mathbf{p}+\vec{\sigma}_{2}\cdot\hat{r}\vec{\sigma}_{1}%
\cdot\mathbf{p})+{\frac{i}{2}}(L-\mathcal{G}){\frac{E_{1}M_{2}-E_{2}M_{1}%
}{E_{2}M_{1}+E_{2}M_{1}}}{\frac{\vec{L}\cdot(\vec{\sigma}_{1}\times\vec
{\sigma}_{2})}{r}}]\phi_{+} \label{bliu}%
\end{equation}%
\[
=b^{2}(w)\phi_{+}%
\]
In terms of $\mathcal{D}=E_{1}M_{2}+E_{2}M_{1}$ the corresponding radial forms
then become%

\[
s=0, \ \ j=l
\]
%

\[
\big\{-{\frac{d^{2}}{dr^{2}}}+{\frac{j(j+1)}{r^{2}}}+2m_{w}S+S^{2}%
+2\epsilon_{2}\mathcal{A}-\mathcal{A}^{2}%
\]%
\[
-(2\mathcal{G}-\log(\mathcal{D})+\mathcal{G}+L)^{\prime}({\frac{d}{dr}}%
-{\frac{1}{r}})
\]%
\[
-{\frac{1}{2}}\nabla^{2}(L+4\mathcal{G})-{\frac{1}{4}}(-L-2\mathcal{G}%
+2\log(\mathcal{D}))^{\prime}(-L-4\mathcal{G})^{\prime}\big\}u_{j0j}%
\]%
\begin{equation}
+\mathrm{\exp}(-\mathcal{G}-L){\frac{w(m_{1}-m_{2})}{\mathcal{D}}%
}(-\mathcal{G}+L)^{\prime}{\frac{\sqrt{j(j+1)}}{r}}u_{j1j}=b^{2}(w)u_{j0j},
\end{equation}
coupled to
\begin{subequations}
\[
s=1,\ \ j=l
\]%
\end{subequations}
\begin{subequations}
\[
\big\{-{\frac{d^{2}}{dr^{2}}}+{\frac{j(j+1)}{r^{2}}}+2m_{w}S+S^{2}%
+2\epsilon_{2}\mathcal{A}-\mathcal{A}^{2}%
\]%
\end{subequations}
\begin{subequations}
\[
-(\mathcal{G}-L-\log(\mathcal{D}))^{\prime}{\frac{d}{dr}}+{\frac{-L^{\prime}%
}{r}}%
\]%
\end{subequations}
\begin{subequations}
\[
+{\frac{1}{2}}\nabla^{2}L+{\frac{1}{4}}(2\log(\mathcal{D})+(-L+2\mathcal{G}%
))^{\prime}L^{\prime}\big\}u_{j1j}%
\]%
\end{subequations}
\begin{equation}
+\mathrm{\exp}(-\mathcal{G}-J){\frac{(\epsilon_{1}-\epsilon_{2})(m_{1}+m_{2}%
)}{\mathcal{D}}}(-\mathcal{G}+L)^{\prime}{\frac{\sqrt{j(j+1)}}{r}}%
u_{j0j}=b^{2}(w)u_{j1j},
\end{equation}
and $s=1,j=l+1$
\[
\bigl\{(-{\frac{d^{2}}{dr^{2}}}+{\frac{j(j-1)}{r^{2}}})+2m_{w}S+S^{2}%
+2\epsilon_{2}\mathcal{A}-\mathcal{A}^{2}%
\]%
\[
+[\log(\mathcal{D})-2\mathcal{G}+{\frac{1}{2j+1}}(\mathcal{G}+L)]^{\prime
}{\frac{d}{dr}}%
\]%
\[
\lbrack-j\log(\mathcal{D})+{\frac{1}{2j+1}}\big((4j^{2}+j+1)\mathcal{G}%
-\mathcal{G}-L\big
)]^{\prime}{\frac{1}{r}}%
\]%
\[
+{\frac{1}{4}}(-{(\mathcal{G}+L)^{\prime}}^{2})+{\frac{1}{2j+1}}%
\big(({\frac{1}{2}}\nabla^{2}L+\mathcal{G}^{\prime}({\frac{2j-3}{4}%
}\mathcal{G}+\mathcal{G}+L)^{\prime}-{\frac{1}{2}}\log^{\prime}(\mathcal{D}%
)L^{\prime}\big)\bigl\}u_{j-11j}%
\]%
\[
+{\frac{\sqrt{j(j+1)}}{2j+1}}\bigl\{2[\mathcal{G}+L]^{\prime}{\frac{d}{dr}%
}+[(-\mathcal{G}-L)(1-2j)+3\mathcal{G}]^{\prime}{\frac{1}{r}}%
\]%
\begin{equation}
+\nabla^{2}(L)-L^{\prime}(\log(\mathcal{D})-2\mathcal{G})^{\prime
}\bigl\}u_{j+11j}=b^{2}(w)u_{j-11j},
\end{equation}
coupled to $s=1,j=l-1$
\begin{subequations}
\[
\bigl\{(-{\frac{d^{2}}{dr^{2}}}+{\frac{(j+1)(j+2)}{r^{2}}})+2m_{w}%
S+S^{2}+2\epsilon_{2}\mathcal{A}-\mathcal{A}^{2}%
\]%
\end{subequations}
\begin{subequations}
\[
+[\log(\mathcal{D})-2\mathcal{G}-{\frac{1}{2j+1}}(\mathcal{G}+L)]^{\prime
}{\frac{d}{dr}}%
\]%
\end{subequations}
\begin{subequations}
\[
\lbrack(j+1)\log(\mathcal{D})-{\frac{1}{2j+1}}\big((4j^{2}+7j+4)\mathcal{G}%
-\mathcal{G}-L\big )]^{\prime}{\frac{1}{r}}%
\]%
\end{subequations}
\begin{subequations}
\[
+{\frac{1}{4}}(-{(\mathcal{G}+L)^{\prime}}^{2})-{\frac{1}{2j+1}}%
\big(({\frac{1}{2}}\nabla^{2}L+\mathcal{G}^{\prime}({\frac{2j+5}{4}%
}\mathcal{G}-\mathcal{G}-L-C)^{\prime}+{\frac{1}{2}}\log^{\prime}%
(\mathcal{D})L^{\prime}\big)\bigl\}u_{j+11j}%
\]%
\end{subequations}
\begin{subequations}
\[
+{\frac{\sqrt{j(j+1)}}{2j+1}}\bigl\{2[\mathcal{G}+L]^{\prime}{\frac{d}{dr}%
}+[(-\mathcal{G}-L)(2j+3)+3\mathcal{G}]^{\prime}{\frac{1}{r}}%
\]%
\end{subequations}
\begin{equation}
+2\nabla^{2}L+L^{\prime}(\mathrm{\log}(\mathcal{D})-2\mathcal{G})^{\prime
}\bigl\}u_{j-11j}=b^{2}(w)u_{j+11j}.
\end{equation}


\section{ Numerical Construction of Meson Wave Functions}

We obtain from our computer program a numerical wave function $\bar{u}(x)$
normalized so that%

\begin{equation}
\int_{-\infty}^{+\infty}\bar{u}(x)^{2}dx=1.
\end{equation}
The radial variable is related to $x$ by $r=r_{0}e^{x}$ and the radial wave
function $u(r)=\bar{u}(x)e^{-x/2}$. \ Hence
\begin{equation}
\int_{0}^{+\infty}u(r)^{2}dr=r_{0}\int_{-\infty}^{+\infty}\bar{u}(x)^{2}dx.
\end{equation}
Now let $v_{n}(r)$ be some radial basis functions that are orthonormalized so
that
\begin{equation}
\int_{0}^{+\infty}v_{n}(r)v_{n^{\prime}}(r)dr=\delta_{nn^{\prime}}.
\end{equation}
Thus
\begin{equation}
u(r)=\sum_{n=0}^{\infty}u_{n}v_{n}(r)
\end{equation}
where
\begin{equation}
u_{n}=\int_{0}^{+\infty}v_{n}(r)u(r)dr=r_{0}\int_{-\infty}^{+\infty}\bar
{v}_{n}(x)\bar{u}(x)dx.
\end{equation}
Note that $\bar{v}_{n}(x)=v_{n}(r)e^{x/2}$ so that we can compute the $u_{n}$
in a straightforward way. \ Thus we have as an approximation
\begin{align}
u(r)  &  \doteq r_{0}\sum_{n=0}^{N}v_{n}(r)\int_{-\infty}^{+\infty}\bar{v}%
_{n}(x)\bar{u}(x)dx\nonumber\\
&  =\sum_{n=0}^{N}c_{n}v_{n}(r)\equiv w_{N}(r).
\end{align}


Now we use a least squares fit to determine the $c_{n}$ .\ In the limit of
large $N$ we have $c_{n}\rightarrow u_{n}$ since we minimize the quantity
\begin{equation}
\chi^{2}\equiv\int_{-\infty}^{+\infty}|\bar{u}(x)-\bar{w}_{N}(x)|^{2}dx
\end{equation}
For the $v_{n}(r)$\ we use harmonic oscillator functions defined by%

\begin{equation}
\psi_{n}^{k}(y)=c(n,k)e^{-y^{2}/2}y^{k}L_{n}^{k-1/2}(y^{2})
\end{equation}
in which $c(n,k)=\sqrt{\frac{2(n!)}{(n+k-1/2)!}}$ is a normalization constant
and (with $z=y^{2}$)%

\begin{equation}
L_{n}^{k-1/2}(z)=\frac{e^{z}z^{-k+1/2}}{n!}\frac{d^{n}}{dz^{n}}(e^{-z}%
z^{k+n-1/2}).
\end{equation}
So for example%

\begin{align}
L_{0}^{k-1/2}(z)  &  =1\nonumber\\
L_{1}^{k-1/2}(z)  &  =k+1/2-z\nonumber\\
L_{2}^{k-1/2}(z)  &  =\frac{1}{2}[(5/2+k-z)L_{1}^{k-1/2}(z)-(1/2+k)L_{0}%
^{k-1/2}(z)\nonumber\\
&  =[(k+3/2)(k+1/2)-2(k+3/2)z+z^{2}]/2\nonumber\\
&  ...\nonumber\\
L_{n+1}^{k-1/2}(z)  &  =\frac{1}{n+1}[(2n+1/2+k-z)L_{n}^{k-1/2}%
(z)-(n+k-1/2)L_{n-1}^{k-1/2}(z)]
\end{align}
Thus letting $\ y=r/a=\alpha e^{x}$ we have%

\begin{align}
\bar{v}_{0}(x)  &  =c(0,k)\alpha^{k}\exp(x(2k+1)/2)\exp(-\alpha^{2}%
e^{2x}/2)\nonumber\\
\bar{v}_{1}(x)  &  =\sqrt{\frac{1}{k+1/2}}\bar{v}_{0}(x)(k+1/2-\alpha
^{2}e^{2x})\nonumber\\
\bar{v}_{2}(x)  &  =\sqrt{\frac{2!}{(k+1/2)(k+3/2)}}\bar{v}_{0}%
(x)[(k+3/2)(k+1/2)-2(k+3/2)\alpha^{2}e^{2x}+\alpha^{4}e^{4x}]/2.\nonumber\\
&  ...\nonumber\\
\bar{v}_{n}(x)  &  =\sqrt{\frac{n!}{(k+1/2)..(k+n-1/2)}}\bar{v}_{0}%
(x)\sum_{m=0}^{n}(-)^{m}\frac{(n+k-1/2)!}{(n-m)!(k-1/2+m)!m!}(\alpha
e^{x})^{2m}%
\end{align}


\bigskip

\section{\bigskip\ Table Comparing Features of Approaches Treated in this
Paper}%

\[%
\begin{tabular}
[c]{llllll}
& HC-PVA & Durand et al & Spence , Vary & Brayshaw & Godfrey,Isgur\\
Wave Eqn & Two-Body Dirac & Reduced SE & Truncated BSE & Breit & None\\
Covariance & Explicit & Implicit & Implicit & Implicit & Implicit\\
Nonperturb Tests & Str. ptnl -QED & Wk ptnl. & Str. ptnl. & Str. ptnl. & Str.
ptnl.\\
\# of \ Parametric fns & 2 & 2 & 2 & 3 & 6\\
$\chi^{2}$ & 101 & 1332 vs 116 & 2625 vs 69 & 204 vs 111 & 85 vs 105\\
Locality & Local & Non-local & Non-local & Local & Non-local
\end{tabular}
\ \ \ \ \
\]


\begin{thebibliography}{99}                                                                                               %


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\bibitem {crcmp}H. Crater, J. Comp. Phys. \textbf{115,} 470 (1994)

\bibitem {wisc}A. Gara, B. Durand, and L. Durand, Phys. Rev D1 \textbf{40},843
(1989), \textbf{42}, 1651 (1990).

\bibitem {iowa}J.R. Spence and J.P. Vary, Phys. Rev C, \textbf{47}, 1282 (1993).

\bibitem {bry}D.D. Brayshaw, Phys. Rev. D \textbf{36}, 1465 (1987)

\bibitem {licht}D.B. Lichtenberg et al, Z. Phys. \textbf{C19,} 19 (1983)

\bibitem {rob}D.P. Stanley and D. Robson, D\textbf{21 }31\textbf{ }(1980)

\bibitem {isgr}S. Godfrey and N. Isgur, Phys. Rev. \textbf{D1} \textbf{32},
189 (1985)

\bibitem {di28}P. A. M. Dirac, Proc. R. Soc. London \textbf{A 117}, 610 (1928).

\bibitem {long}P. Long and H. W. Crater, J. Math. Phys. \textbf{39}, 124 (1998)

\bibitem {cnl}For positronium our results agree with the standard results of
in E.E. Salpeter, Phys. Rev. \textbf{87} 328,(1952). M. A. Stroscio, Phys.
Rep. C\textbf{22}, 215(1975) without the annihilation diagram. This reference
also gives the spectrum for the general hydrogenic (unequal mass) case but in
the $j-j$ coupling scheme, rather than the $LS$ coupling scheme present in
\cite{bckr} here. The results for the $LS$ scheme is given in \cite{krp79} but
with a spectrum that does not have the correct $j=l$ spin mixing
fine-structure even though the weak potential bound state equations given
there are correct. The corrected spectrum appears in \cite{bckr} and in J.
Connell, Phys. Rev. \textbf{D43}, 1393, (1991).

\bibitem {saz94}J. Mourad and H. Sazdjian, Journal of Physics G, \textbf{21},
267 (1995).

\bibitem {prtl}Review of Particle Physics, D. E. Groom et al., The European
Physical Journal \textbf{C15} (2000) 1

\bibitem {cbwv}H. W. Crater, R. L. Becker, C. Y. Wong, and P. Van Alstine,
Report No. ORNL/TM-12122, 1992.

\bibitem {misce1}These theta matrices have algebraic properties that permit
more efficient calculation of the commutation relations appropriate to two
spinning bodies permitting simplification of otherwise complicated
consequences of compatibility ($[\mathcal{S}_{1},\mathcal{S}_{2}]_{-}\psi=0$)

\bibitem {jmath}H. W. Crater, and P. Van Alstine, J. Math. Phys.\textbf{31},
1998 (1990).

\bibitem {qed}All this extra structure makes possible straight-forward
nonperturbative solution of the constraint equations and may account for the
differences between the resulting (correct field-theoretic) spectrum and the
incorrect spectrum produced by nonperturbative solution of the usual Breit
equation for QED \cite{va97}.

\bibitem {spence}J.R. Spence and J.P. Vary, Phys. Lett \textbf{B254}, 1 ,(1991)

\bibitem {martin}A. Martin, Phys. Lett. \textbf{B100}, 511 (1981)

\bibitem {martin88}Phys. Lett. \textbf{B214}, 561 (1988)

\bibitem {jaczko}G. Jazcko and L. Durand, Phys. Rev. D 58 1998, 114017-1,114017-9

\bibitem {cas}W.E. Caswell and G.P. Lepage, Phys. Rev. \textbf{A18, }863 (1977)
\end{thebibliography}

\newpage

\textbf{TABLE I - MESON MASSES FROM COVARIANT CONSTRAINT DYNAMICS}
\halign{#\hfil&\qquad\hfil#&\qquad\hfil#\cr
NAME & EXP. & THEORY\cr\cr
$\Upsilon : b \overline b \ 1^3S_1$ & 9.460( 0.2)& 9.453( 0.6)\cr
$\Upsilon : b \overline b \ 1^3P_0$ & 9.860( 1.3)& 9.842( 1.4)\cr
$\Upsilon : b \overline b \ 1^3P_1$ & 9.892( 0.7)& 9.889( 0.1)\cr
$\Upsilon : b \overline b \ 1^3P_2$ & 9.913( 0.6)& 9.921( 0.5)\cr
$\Upsilon : b \overline b \ 2^3S_1$ & 10.023( 0.3)& 10.022( 0.0)\cr
$\Upsilon : b \overline b \ 2^3P_0$ & 10.232( 0.6)& 10.227( 0.2)\cr
$\Upsilon : b \overline b \ 2^3P_1$ & 10.255( 0.5)& 10.257( 0.0)\cr
$\Upsilon : b \overline b \ 2^3P_2$ & 10.269( 0.4)& 10.277( 0.8)\cr
$\Upsilon : b \overline b \ 3^3S_1$ & 10.355( 0.5)& 10.359( 0.1)\cr
$\Upsilon : b \overline b \ 4^3S_1$ & 10.580( 3.5)& 10.614( 0.9)\cr
$\Upsilon : b \overline b \ 5^3S_1$ & 10.865( 8.0)& 10.826( 0.2)\cr
$\Upsilon : b \overline b \ 6^3S_1$ & 11.019( 8.0)& 11.013( 0.0)\cr
$B: b \overline u \ 1^1S_0$ & 5.279( 1.8)& 5.273( 0.1)\cr
$B: b \overline d \ 1^1S_0$ & 5.279( 1.8)& 5.274( 0.1)\cr
$B^*: b \overline u \ 1^3S_1$ & 5.325( 1.8)& 5.321( 0.1)\cr
$B_s: b \overline s \ 1^1S_0$ & 5.369( 2.0)& 5.368( 0.0)\cr
$B_s: b \overline s \ 1^3S_1$ & 5.416( 3.3)& 5.427( 0.1)\cr
$\eta_c : c \overline c \ 1^1S_0$ & 2.980( 2.1)& 2.978( 0.0)\cr
$\psi: c \overline c \ 1^3S_1$ & 3.097( 0.0)& 3.129( 12.6)\cr
$\chi_0: c \overline c \ 1^1P_1$ & 3.526( 0.2)& 3.520( 0.4)\cr
$\chi_0: c \overline c \ 1^3P_0$ & 3.415( 1.0)& 3.407( 0.4)\cr
$\chi_1: c \overline c \ 1^3P_1$ & 3.510( 0.1)& 3.507( 0.2)\cr
$\chi_2: c \overline c \ 1^3P_2$ & 3.556( 0.1)& 3.549( 0.6)\cr
$\eta_c : c \overline c \ 2^1S_0$ & 3.594( 5.0)& 3.610( 0.1)\cr
$\psi: c \overline c \ 2^3S_1$ & 3.686( 0.1)& 3.688( 0.1)\cr
$\psi: c \overline c \ 1^3D_1$ & 3.770( 2.5)& 3.808( 2.0)\cr
$\psi: c \overline c \ 3^3S_1$ & 4.040( 10.0)& 4.081( 0.2)\cr
$\psi: c \overline c \ 2^3D_1$ & 4.159( 20.0)& 4.157( 0.0)\cr
$\psi: c \overline c \ 3^3D_1$ & 4.415( 6.0)& 4.454( 0.4)\cr
$D: c \overline u \ 1^1S_0$ & 1.865( 0.5)& 1.866( 0.0)\cr
$D: c \overline d \ 1^1S_0$ & 1.869( 0.5)& 1.873( 0.1)\cr
$D^*: c \overline u \ 1^3S_1$ & 2.007( 0.5)& 2.000( 0.4)\cr
$D^*: c \overline d \ 1^3S_1$ & 2.010( 0.5)& 2.005( 0.3)\cr
$D^*: c \overline u \ 1^3P_1$ & 2.422( 1.8)& 2.407( 0.6)\cr
$D^*: c \overline d \ 1^3P_1$ & 2.428( 1.8)& 2.411( 0.5)\cr
$D^*: c \overline u \ 1^3P_2$ & 2.459( 2.0)& 2.382( 11.3)\cr
$D^*: c \overline d \ 1^3P_2$ & 2.459( 4.0)& 2.386( 3.5)\cr
$D_s: c \overline s \ 1^1S_0$ & 1.968( 0.6)& 1.976( 0.5)\cr
$D_s^*: c \overline s \ 1^3S_1$ & 2.112( 0.7)& 2.123( 0.9)\cr
$D_s^*: c \overline s \ 1^3P_1$ & 2.535( 0.3)& 2.511( 6.2)\cr
$D_s^*: c \overline s \ 1^3P_2$ & 2.574( 1.7)& 2.514( 9.6)\cr
$K: s \overline u \ 1^1S_0$ & 0.494( 0.0)& 0.492( 0.0)\cr
$K: s \overline d \ 1^1S_0$ & 0.498( 0.0)& 0.492( 0.4)\cr
$K^*: s \overline u \ 1^3S_1$ & 0.892( 0.2)& 0.910( 0.6)\cr
$K^*: s \overline d \ 1^3S_1$ & 0.896( 0.3)& 0.910( 0.3)\cr
$K_1: s \overline u \ 1^1P_1$ & 1.273( 7.0)& 1.408( 3.2)\cr
$K_0^*: s \overline u \ 1^3P_0$ & 1.429( 4.0)& 1.314( 0.7)\cr
$K_1: s \overline u \ 1^3P_1$ & 1.402( 7.0)& 1.506( 1.0)\cr
$K_2^*: s \overline u \ 1^3P_2$ & 1.425( 1.3)& 1.394( 0.5)\cr
$K_2^*: s \overline d \ 1^3P_2$ & 1.432( 1.3)& 1.394( 0.6)\cr
$K^*: s \overline u \ 2^1S_0$ & 1.460( 30.0)& 1.591( 0.2)\cr
$K^*: s \overline u \ 2^3S_1$ & 1.412( 12.0)& 1.800( 6.7)\cr
$K_2: s \overline u \ 1^1D_2$ & 1.773( 8.0)& 1.877( 0.8)\cr
$K^*: s \overline u \ 1^3D_1$ & 1.714( 20.0)& 1.985( 1.4)\cr
$K_2: s \overline u \ 1^3D_2$ & 1.816( 10.0)& 1.945( 1.3)\cr
$K_3: s \overline u \ 1^3D_3$ & 1.770( 10.0)& 1.768( 0.0)\cr
$K^*: s \overline u \ 3^1S_0$ & 1.830( 30.0)& 2.183( 1.4)\cr
$K_2^*: s \overline u \ 2^3P_2$ & 1.975( 22.0)& 2.098( 0.2)\cr
$K_4^*: s \overline u \ 1^3F_4$ & 2.045( 9.0)& 2.078( 0.1)\cr
$K_2: s \overline u \ 2^3D_2$ & 2.247( 17.0)& 2.373( 0.5)\cr
$K_5^*: s \overline u \ 1^3G_5$ & 2.382( 33.0)& 2.344( 0.0)\cr
$K_3^*: s \overline u \ 2^3F_3$ & 2.324( 24.0)& 2.636( 1.9)\cr
$K_4^*: s \overline u \ 2^3F_4$ & 2.490( 20.0)& 2.757( 1.6)\cr
$\phi: s \overline s \ 1^3S_1$ & 1.019( 0.0)& 1.033( 2.2)\cr
$f_0: s \overline s \ 1^3P_0$ & 1.370( 40.0)& 1.319( 0.0)\cr
$f_1: s \overline s \ 1^3P_1$ & 1.512( 4.0)& 1.533( 0.3)\cr
$f_2: s \overline s \ 1^3P_2$ & 1.525( 5.0)& 1.493( 0.3)\cr
$\phi: s \overline s \ 2^3S_1$ & 1.680( 20.0)& 1.850( 0.8)\cr
$\phi: s \overline s \ 1^3D_3$ & 1.854( 7.0)& 1.848( 0.0)\cr
$f_2: s \overline s \ 2^3P_2$ & 2.011( 69.0)& 2.160( 0.1)\cr
$f_2: s \overline s \ 3^3P_2$ & 2.297( 28.0)& 2.629( 1.6)\cr
$\pi: u\overline d \ 1^1S_0$ & 0.140( 0.0)& 0.144( 0.2)\cr
$\rho: u\overline d \ 1^3S_1$ & 0.767( 1.2)& 0.792( 0.1)\cr
$b_1: u\overline d \ 1^1P_1$ & 1.231( 10.0)& 1.392( 2.1)\cr
$a_0: u\overline d \ 1^3P_0$ & 1.450( 40.0)& 1.491( 0.0)\cr
$a_1: u\overline d \ 1^3P_1$ & 1.230( 40.0)& 1.568( 0.7)\cr
$a_2: u\overline d \ 1^3P_2$ & 1.318( 0.7)& 1.310( 0.0)\cr
$\pi: u\overline d \ 2^1S_0$ & 1.300( 100.0)& 1.536( 0.1)\cr
$\rho: u\overline d \ 2^3S_1$ & 1.465( 25.0)& 1.775( 1.4)\cr
$\pi_2: u\overline d \ 1^1D_2$ & 1.670( 20.0)& 1.870( 0.9)\cr
$\rho: u\overline d \ 1^3D_1$ & 1.700( 20.0)& 1.986( 1.9)\cr
$\rho_3: u\overline d \ 1^3D_3$ & 1.691( 5.0)& 1.710( 0.0)\cr
$\pi: u\overline d \ 3^1S_0$ & 1.795( 10.0)& 2.166( 7.9)\cr
$\rho: u\overline d \ 3^3S_1$ & 2.149( 17.0)& 2.333( 0.7)\cr
$\rho_4: u\overline d \ 1^3F_4$ & 2.037( 26.0)& 2.033( 0.0)\cr
$\pi_2: u\overline d \ 2^1D_2$ & 2.090( 29.0)& 2.367( 0.5)\cr
$\rho_3: u\overline d \ 2^3D_3$ & 2.250( 45.0)& 2.305( 0.0)\cr
$\rho_5: u\overline d \ 1^3G_5$ & 2.330( 35.0)& 2.307( 0.0)\cr
$\rho_6: u\overline d \ 1^3H_6$ & 2.450( 130.0)& 2.547( 0.0)\cr
$\chi^2$ & 0.0& 101.0\cr}

\newpage\textbf{TABLE II - COMPARISON OF MESON MASSES FROM }

\textbf{WISCONSIN MODEL I and COVARIANT CONSTRAINT DYNAMICS }

\halign{#\hfil&\qquad\hfil#&\qquad\hfil#&\qquad\hfil#\cr
NAME & EXP. & WISC1 & CTBD \cr\cr
$\Upsilon : b \overline b \ 1^3S_1$ & 9.460( 0.2)& 9.434( 60.2)& 9.453( 4.4)\cr
$\Upsilon : b \overline b \ 1^3P_0$ & 9.860( 1.3)& 9.852( 1.9)& 9.843( 9.7)\cr
$\Upsilon : b \overline b \ 1^3P_1$ & 9.892( 0.7)& 9.885( 2.7)& 9.889( 0.4)\cr
$\Upsilon : b \overline b \ 1^3P_2$ & 9.913( 0.6)& 9.909( 1.0)& 9.920( 3.3)\cr
$\Upsilon : b \overline b \ 2^3S_1$ & 10.023( 0.3)& 10.027( 1.3)& 10.023(
0.0)\cr
$\Upsilon : b \overline b \ 2^3P_0$ & 10.232( 1.1)& 10.246( 13.1)& 10.228(
1.1)\cr
$\Upsilon : b \overline b \ 2^3P_1$ & 10.255( 0.6)& 10.268( 12.3)& 10.257(
0.3)\cr
$\Upsilon : b \overline b \ 2^3P_2$ & 10.269( 0.6)& 10.282( 13.2)& 10.276(
3.8)\cr
$\Upsilon : b \overline b \ 3^3S_1$ & 10.355( 0.5)& 10.381( 49.2)& 10.359(
1.2)\cr
$\eta_c : c \overline c \ 1^1S_0$ & 2.979( 1.9)& 2.981( 0.1)& 2.973( 0.7)\cr
$\psi: c \overline c \ 1^3S_1$ & 3.097( 0.1)& 3.175( 553.4)& 3.123( 62.8)\cr
$\chi_0: c \overline c \ 1^3P_0$ & 3.415( 1.0)& 3.404( 5.3)& 3.405( 4.5)\cr
$\chi_1: c \overline c \ 1^3P_1$ & 3.510( 0.1)& 3.492( 29.6)& 3.504( 4.0)\cr
$\chi_2: c \overline c \ 1^3P_2$ & 3.556( 0.1)& 3.545( 10.5)& 3.544( 13.2)\cr
$\eta_c : c \overline c \ 2^1S_0$ & 3.594( 5.0)& 3.626( 3.6)& 3.609( 0.8)\cr
$\psi: c \overline c \ 2^3S_1$ & 3.686( 0.1)& 3.665( 39.8)& 3.687( 0.1)\cr
$\phi: s \overline s \ 1^3S_1$ & 1.019( 0.0)& 1.098( 535.2)& 1.020( 0.1)\cr
$\phi: s \overline s \ 2^3S_1$ & 1.680( 50.0)& 1.616( 0.2)& 1.361( 5.5)\cr
$\chi^2$ & 0.0&1332.6& 115.7\cr} \newpage

\textbf{TABLE III COMPARISON OF MESON MASSES FROM }

\textbf{WISCONSIN MODEL II and COVARIANT CONSTRAINT DYNAMICS }

\halign{#\hfil&\qquad\hfil#&\qquad\hfil#&\qquad\hfil#\cr
NAME & EXP. & WISC2 & CTBD\cr\cr
$\Upsilon : b \overline b \ 1^3S_1$ & 9.460( 0.2)& 9.426( 62.6)& 9.454( 2.0)\cr
$\Upsilon : b \overline b \ 1^3P_0$ & 9.860( 1.3)& 9.862( 0.1)& 9.845( 4.5)\cr
$\Upsilon : b \overline b \ 1^3P_1$ & 9.892( 0.7)& 9.892( 0.0)& 9.890( 0.1)\cr
$\Upsilon : b \overline b \ 1^3P_2$ & 9.913( 0.6)& 9.917( 0.7)& 9.919( 1.6)\cr
$\Upsilon : b \overline b \ 2^3S_1$ & 10.023( 0.3)& 10.028( 1.3)& 10.024(
0.1)\cr
$\Upsilon : b \overline b \ 2^3P_0$ & 10.232( 1.1)& 10.238( 1.5)& 10.229(
0.4)\cr
$\Upsilon : b \overline b \ 2^3P_1$ & 10.255( 0.6)& 10.256( 0.0)& 10.257(
0.2)\cr
$\Upsilon : b \overline b \ 2^3P_2$ & 10.268( 0.6)& 10.270( 0.2)& 10.276(
3.1)\cr
$\Upsilon : b \overline b \ 3^3S_1$ & 10.355( 0.5)& 10.359( 0.7)& 10.359(
0.7)\cr
$B: b \overline d \ 1^1S_0$ & 5.279( 2.1)& 5.381( 137.2)& 5.274( 0.3)\cr
$\eta_c : c \overline c \ 1^1S_0$ & 2.979( 1.9)& 2.967( 1.4)& 2.975( 0.1)\cr
$\psi: c \overline c \ 1^3S_1$ & 3.097( 0.1)& 3.167( 272.4)& 3.120( 28.8)\cr
$\chi_0: c \overline c \ 1^3P_0$ & 3.415( 1.0)& 3.402( 5.1)& 3.412( 0.2)\cr
$\chi_1: c \overline c \ 1^3P_1$ & 3.510( 0.1)& 3.493( 17.5)& 3.505( 1.8)\cr
$\chi_2: c \overline c \ 1^3P_2$ & 3.556( 0.1)& 3.548( 4.0)& 3.538( 18.1)\cr
$\eta_c : c \overline c \ 2^1S_0$ & 3.594( 5.0)& 3.621( 1.5)& 3.611( 0.6)\cr
$\psi: c \overline c \ 2^3S_1$ & 3.686( 0.1)& 3.668( 17.9)& 3.688( 0.3)\cr
$D: c \overline d \ 1^1S_0$ & 1.869( 0.5)& 1.983( 574.6)& 1.875( 1.5)\cr
$D^*: c \overline d \ 1^3S_1$ & 2.010( 0.6)& 2.010( 0.0)& 2.003( 1.9)\cr
$D_s: c \overline s \ 1^1S_0$ & 1.969( 0.7)& 2.097( 671.1)& 1.968( 0.1)\cr
$D_s^*: c \overline s \ 1^3S_1$ & 2.110( 2.0)& 2.148( 52.7)& 2.106( 0.6)\cr
$K: s \overline d \ 1^1S_0$ & 0.498( 0.0)& 0.743(3340.4)& 0.498( 0.0)\cr
$K^*: s \overline d \ 1^3S_1$ & 0.896( 0.3)& 0.870( 5.1)& 0.918( 3.5)\cr
$\phi: s \overline s \ 1^3S_1$ & 1.019( 0.0)& 1.019( 0.0)& 1.020( 0.0)\cr
$\phi: s \overline s \ 2^3S_1$ & 1.680( 50.0)& 1.510( 0.9)& 1.424( 2.1)\cr
$\chi^2$ & 0.0&5168.9& 72.8\cr} \newpage\textbf{TABLE IV -COMPARISON OF MESON
MASSES FROM}

\textbf{SPENCE-VARY MODEL and COVARIANT CONSTRAINT DYNAMICS}
\halign{#\hfil&\qquad\hfil#&\qquad\hfil#&\qquad\hfil#\cr
NAME & EXP. &SPENCE \& VARY & CTBD\cr\cr
$\Upsilon : b \overline b \ 1^3S_1$ & 9.460( 0.2)& 9.463( 0.2)& 9.455( 0.7)\cr
$\Upsilon : b \overline b \ 1^3P_0$ & 9.860( 1.3)& 9.809( 27.4)& 9.843( 3.0)\cr
$\Upsilon : b \overline b \ 1^3P_1$ & 9.892( 0.7)& 9.827( 80.7)& 9.890( 0.1)\cr
$\Upsilon : b \overline b \ 1^3P_2$ & 9.913( 0.6)& 9.889( 12.3)& 9.922( 1.5)\cr
$\Upsilon : b \overline b \ 2^3S_1$ & 10.023( 0.3)& 10.077( 76.0)& 10.023(
0.0)\cr
$\Upsilon : b \overline b \ 2^3P_0$ & 10.232( 0.6)& 10.211( 9.3)& 10.227(
0.5)\cr
$\Upsilon : b \overline b \ 2^3P_1$ & 10.255( 0.5)& 10.227( 17.9)& 10.256(
0.0)\cr
$\Upsilon : b \overline b \ 2^3P_2$ & 10.268( 0.4)& 10.165( 261.3)& 10.277(
2.0)\cr
$\Upsilon : b \overline b \ 3^3S_1$ & 10.355( 0.5)& 10.364( 1.9)& 10.358(
0.2)\cr
$\Upsilon : b \overline b \ 4^3S_1$ & 10.580( 3.5)& 10.679( 19.1)& 10.612(
2.0)\cr
$B: b \overline u \ 1^1S_0$ & 5.279( 1.8)& 5.003( 512.2)& 5.273( 0.2)\cr
$\eta_c : c \overline c \ 1^1S_0$ & 2.980( 2.1)& 2.979( 0.0)& 2.977( 0.0)\cr
$\psi: c \overline c \ 1^3S_1$ & 3.097( 0.1)& 3.097( 0.0)& 3.128( 27.6)\cr
$\chi_0: c \overline c \ 1^1P_1$ & 3.526( 0.2)& 3.422( 291.4)& 3.517( 2.4)\cr
$\chi_0: c \overline c \ 1^3P_0$ & 3.415( 1.0)& 3.437( 6.8)& 3.403( 2.2)\cr
$\chi_1: c \overline c \ 1^3P_1$ & 3.510( 0.1)& 3.477( 31.6)& 3.503( 1.5)\cr
$\chi_2: c \overline c \ 1^3P_2$ & 3.556( 0.1)& 3.522( 32.5)& 3.547( 2.6)\cr
$\eta_c : c \overline c \ 2^1S_0$ & 3.594( 5.0)& 3.636( 1.9)& 3.605( 0.1)\cr
$\psi: c \overline c \ 2^3S_1$ & 3.686( 0.1)& 3.696( 2.8)& 3.683( 0.2)\cr
$\psi: c \overline c \ 1^3D_1$ & 3.770( 2.5)& 3.735( 4.0)& 3.802( 3.4)\cr
$\psi: c \overline c \ 3^3S_1$ & 4.040( 10.0)& 4.090( 0.7)& 4.073( 0.3)\cr
$\psi: c \overline c \ 2^3D_1$ & 4.159( 20.0)& 4.119( 0.1)& 4.150( 0.0)\cr
$\psi: c \overline c \ 3^3D_1$ & 4.415( 6.0)& 4.404( 0.1)& 4.445( 0.6)\cr
$D: c \overline u \ 1^1S_0$ & 1.865( 0.5)& 1.814( 58.3)& 1.864( 0.0)\cr
$D^*: c \overline u \ 1^3S_1$ & 2.007( 5.0)& 2.147( 443.4)& 1.997( 2.5)\cr
$D_s: c \overline s \ 1^1S_0$ & 1.968( 0.6)& 1.902( 90.9)& 1.974( 0.8)\cr
$D_s^*: c \overline s \ 1^3S_1$ & 2.110( 0.7)& 2.281( 555.3)& 2.120( 2.0)\cr
$K: s \overline u \ 1^1S_0$ & 0.494( 0.0)& 0.496( 0.2)& 0.491( 0.3)\cr
$K^*: s \overline u \ 1^3S_1$ & 0.892( 0.2)& 0.899( 0.2)& 0.899( 0.2)\cr
$\phi: s \overline s \ 1^3S_1$ & 1.019( 0.0)& 1.022( 0.2)& 1.026( 1.2)\cr
$f_0: s \overline s \ 1^3P_0$ & 1.370( 40.0)& 1.260( 0.2)& 1.293( 0.1)\cr
$f_1: s \overline s \ 1^3P_1$ & 1.512( 4.0)& 1.340( 42.3)& 1.512( 0.0)\cr
$f_2: s \overline s \ 1^3P_2$ & 1.525( 5.0)& 1.523( 0.0)& 1.480( 1.4)\cr
$\phi: s \overline s \ 2^3S_1$ & 1.680( 20.0)& 1.746( 0.3)& 1.829( 1.4)\cr
$\pi: u\overline d \ 1^1S_0$ & 0.140( 0.0)& 0.144( 0.6)& 0.143( 0.3)\cr
$\rho: u\overline d \ 1^3S_1$ & 0.767( 1.2)& 0.715( 1.3)& 0.777( 0.0)\cr
$b_1: u\overline d \ 1^1P_1$ & 1.231( 10.0)& 0.893( 21.5)& 1.361( 3.2)\cr
$a_0: u\overline d \ 1^3P_0$ & 1.450( 40.0)& 1.085( 2.1)& 1.435( 0.0)\cr
$a_1: u\overline d \ 1^3P_1$ & 1.230( 40.0)& 0.997( 0.8)& 1.534( 1.3)\cr
$a_2: u\overline d \ 1^3P_2$ & 1.318( 7.0)& 1.191( 15.8)& 1.288( 0.9)\cr
$\pi: u\overline d \ 2^1S_0$ & 1.300( 100.0)& 1.298( 0.0)& 1.518( 0.1)\cr
$\pi_2: u\overline d \ 1^1D_2$ & 1.670( 20.0)& 1.527( 1.0)& 1.844( 1.5)\cr
$\chi^2$ & 0.0&2624.7& 68.6\cr}

\newpage\textbf{TABLE V - COMPARISON OF MESON MASSES FROM }

\textbf{BRAYSHAW MODEL and COVARIANT CONSTRAINT DYNAMICS}
\halign{#\hfil&\qquad\hfil#&\qquad\hfil#&\qquad\hfil#\cr
NAME & EXP. & BRAYSHAW & CTBD\cr\cr
$\Upsilon : b \overline b \ 1^3S_1$ & 9.460( 0.2)& 9.452( 1.3)& 9.451( 1.7)\cr
$\Upsilon : b \overline b \ 1^3P_0$ & 9.860( 1.3)& 9.866( 0.3)& 9.842( 2.5)\cr
$\Upsilon : b \overline b \ 1^3P_1$ & 9.892( 0.7)& 9.910( 4.5)& 9.889( 0.1)\cr
$\Upsilon : b \overline b \ 1^3P_2$ & 9.913( 0.6)& 9.926( 2.5)& 9.920( 0.7)\cr
$\Upsilon : b \overline b \ 2^3S_1$ & 10.023( 0.3)& 10.007( 4.8)& 10.023(
0.0)\cr
$\Upsilon : b \overline b \ 2^3P_0$ & 10.232( 0.6)& 10.214( 4.9)& 10.229(
0.1)\cr
$\Upsilon : b \overline b \ 2^3P_1$ & 10.255( 0.5)& 10.252( 0.1)& 10.258(
0.1)\cr
$\Upsilon : b \overline b \ 2^3P_2$ & 10.268( 0.4)& 10.265( 0.2)& 10.278(
1.8)\cr
$\Upsilon : b \overline b \ 3^3S_1$ & 10.355( 0.5)& 10.342( 2.8)& 10.360(
0.4)\cr
$\Upsilon : b \overline b \ 4^3S_1$ & 10.580( 3.5)& 10.662( 9.4)& 10.617(
1.9)\cr
$B: b \overline u \ 1^1S_0$ & 5.279( 1.8)& 5.332( 13.7)& 5.270( 0.3)\cr
$B^*: b \overline u \ 1^3S_1$ & 5.325( 1.8)& 5.377( 13.2)& 5.317( 0.3)\cr
$\eta_c : c \overline c \ 1^1S_0$ & 2.980( 2.1)& 3.011( 3.5)& 2.976( 0.0)\cr
$\psi: c \overline c \ 1^3S_1$ & 3.097( 0.1)& 3.129( 21.0)& 3.127( 17.8)\cr
$\chi_0: c \overline c \ 1^1P_1$ & 3.524( 0.2)& 3.498( 13.0)& 3.520( 0.3)\cr
$\chi_0: c \overline c \ 1^3P_0$ & 3.415( 1.0)& 3.410( 0.3)& 3.409( 0.4)\cr
$\chi_1: c \overline c \ 1^3P_1$ & 3.510( 0.1)& 3.514( 0.2)& 3.508( 0.2)\cr
$\chi_2: c \overline c \ 1^3P_2$ & 3.556( 0.1)& 3.540( 5.2)& 3.547( 1.5)\cr
$\eta_c : c \overline c \ 2^1S_0$ & 3.594( 5.0)& 3.580( 0.2)& 3.612( 0.3)\cr
$\psi: c \overline c \ 2^3S_1$ & 3.686( 0.1)& 3.680( 0.7)& 3.691( 0.4)\cr
$\psi: c \overline c \ 1^3D_1$ & 3.770( 2.5)& 3.773( 0.0)& 3.811( 4.0)\cr
$\psi: c \overline c \ 3^3S_1$ & 4.040( 10.0)& 4.246( 8.0)& 4.086( 0.4)\cr
$\psi: c \overline c \ 2^3D_1$ & 4.159( 20.0)& 4.288( 0.8)& 4.163( 0.0)\cr
$D: c \overline u \ 1^1S_0$ & 1.865( 0.5)& 1.903( 24.2)& 1.864( 0.0)\cr
$D^*: c \overline u \ 1^3S_1$ & 2.007( 1.4)& 2.046( 24.5)& 1.997( 1.7)\cr
$D^*: c \overline u \ 1^3P_1$ & 2.422( 1.8)& 2.428( 0.1)& 2.413( 0.3)\cr
$D^*: c \overline u \ 1^3P_2$ & 2.459( 2.0)& 2.458( 0.0)& 2.383( 18.8)\cr
$D_s: c \overline s \ 1^1S_0$ & 1.969( 0.6)& 1.976( 0.8)& 1.974( 0.4)\cr
$D_s^*: c \overline s \ 1^3S_1$ & 2.112( 2.0)& 2.134( 6.6)& 2.119( 0.7)\cr
$D_s^*: c \overline s \ 1^3P_1$ & 2.535( 0.3)& 2.515( 7.2)& 2.515( 7.0)\cr
$D_s^*: c \overline s \ 1^3P_2$ & 2.574( 1.7)& 2.546( 3.6)& 2.513( 17.0)\cr
$K: s \overline u \ 1^1S_0$ & 0.494( 0.0)& 0.495( 0.0)& 0.492( 0.1)\cr
$K^*: s \overline u \ 1^3S_1$ & 0.892( 0.2)& 0.905( 0.5)& 0.908( 0.7)\cr
$K_1: s \overline u \ 1^1P_1$ & 1.273( 7.0)& 1.355( 1.1)& 1.421( 3.6)\cr
$K_0^*: s \overline u \ 1^3P_0$ & 1.430( 4.0)& 1.086( 10.8)& 1.349( 0.6)\cr
$K_1: s \overline u \ 1^3P_1$ & 1.402( 7.0)& 1.294( 3.4)& 1.524( 4.3)\cr
$K_2^*: s \overline u \ 1^3P_2$ & 1.425( 1.3)& 1.409( 0.2)& 1.399( 0.5)\cr
$K^*: s \overline u \ 1^3D_1$ & 1.714( 20.0)& 1.690( 0.0)& 2.004( 2.6)\cr
$K_2: s \overline u \ 1^3D_2$ & 1.816( 10.0)& 1.764( 0.4)& 1.892( 0.8)\cr
$K_3: s \overline u \ 1^3D_3$ & 1.770( 10.0)& 1.770( 0.0)& 1.780( 0.0)\cr
$\phi: s \overline s \ 1^3S_1$ & 1.019( 0.0)& 1.022( 0.1)& 1.030( 2.1)\cr
$f_0: s \overline s \ 1^3P_0$ & 1.370( 40.0)& 1.185( 0.4)& 1.345( 0.0)\cr
$f_1: s \overline s \ 1^3P_1$ & 1.512( 4.0)& 1.446( 4.5)& 1.546( 1.2)\cr
$f_2: s \overline s \ 1^3P_2$ & 1.525( 5.0)& 1.511( 0.1)& 1.496( 0.4)\cr
$\phi: s \overline s \ 2^3S_1$ & 1.680( 20.0)& 1.778( 0.4)& 1.860( 1.4)\cr
$\phi: s \overline s \ 1^3D_3$ & 1.854( 7.0)& 1.922( 1.4)& 1.856( 0.0)\cr
$\pi: u\overline d \ 1^1S_0$ & 0.140( 0.0)& 0.140( 0.0)& 0.143( 0.2)\cr
$\rho: u\overline d \ 1^3S_1$ & 0.767( 1.2)& 0.776( 0.0)& 0.790( 0.2)\cr
$b_1: u\overline d \ 1^1P_1$ & 1.231( 10.0)& 1.202( 0.1)& 1.411( 4.4)\cr
$a_0: u\overline d \ 1^3P_0$ & 1.450( 40.0)& 0.990( 2.4)& 1.542( 0.1)\cr
$a_1: u\overline d \ 1^3P_1$ & 1.230( 40.0)& 1.253( 0.0)& 1.590( 1.3)\cr
$a_2: u\overline d \ 1^3P_2$ & 1.318( 7.0)& 1.302( 0.2)& 1.318( 0.0)\cr
$\pi: u\overline d \ 2^1S_0$ & 1.300( 100.0)& 1.028( 0.1)& 1.543( 0.1)\cr
$\pi_2: u\overline d \ 1^1D_2$ & 1.670( 20.0)& 1.593( 0.2)& 1.883( 1.6)\cr
$\rho: u\overline d \ 1^3D_1$ & 1.700( 20.0)& 1.741( 0.1)& 1.998( 3.4)\cr
$\rho_3: u\overline d \ 1^3D_3$ & 1.691( 5.0)& 1.680( 0.0)& 1.722( 0.2)\cr
$\chi^2$ & 0.0& 204.2& 111.0\cr} \newpage\textbf{TABLE VI - COMPARISON OF
MESON MASSES FROM }

\textbf{ISGUR-WISE MODEL and COVARIANT CONSTRAINT DYNAMICS}
\halign{#\hfil&\qquad\hfil#&\qquad\hfil#&\qquad\hfil#\cr
NAME & EXP. & ISGUR\&WISE& CTBD\cr\cr
$\Upsilon : b \overline b \ 1^3S_1$ & 9.460( 0.2)& 9.460( 0.0)& 9.453( 0.8)\cr
$\Upsilon : b \overline b \ 1^3P_0$ & 9.860( 1.3)& 9.850( 0.5)& 9.842( 1.6)\cr
$\Upsilon : b \overline b \ 1^3P_1$ & 9.892( 0.7)& 9.880( 1.4)& 9.889( 0.1)\cr
$\Upsilon : b \overline b \ 1^3P_2$ & 9.913( 0.6)& 9.900( 1.8)& 9.921( 0.6)\cr
$\Upsilon : b \overline b \ 2^3S_1$ & 10.023( 0.3)& 10.000( 6.9)& 10.023(
0.0)\cr
$\Upsilon : b \overline b \ 2^3P_0$ & 10.232( 0.6)& 10.230( 0.0)& 10.228(
0.2)\cr
$\Upsilon : b \overline b \ 2^3P_1$ & 10.255( 0.5)& 10.250( 0.3)& 10.257(
0.0)\cr
$\Upsilon : b \overline b \ 2^3P_2$ & 10.269( 0.4)& 10.260( 1.0)& 10.277(
0.8)\cr
$\Upsilon : b \overline b \ 3^3S_1$ & 10.355( 0.5)& 10.350( 0.3)& 10.359(
0.2)\cr
$\Upsilon : b \overline b \ 4^3S_1$ & 10.580( 3.5)& 10.630( 2.4)& 10.615(
1.2)\cr
$\Upsilon : b \overline b \ 5^3S_1$ & 10.865( 8.0)& 10.880( 0.0)& 10.828(
0.2)\cr
$\Upsilon : b \overline b \ 6^3S_1$ & 11.019( 8.0)& 11.100( 1.2)& 11.014(
0.0)\cr
$B: b \overline u \ 1^1S_0$ & 5.279( 1.8)& 5.310( 3.3)& 5.272( 0.2)\cr
$B^*: b \overline u \ 1^3S_1$ & 5.325( 1.8)& 5.370( 6.9)& 5.319( 0.1)\cr
$B_s: b \overline s \ 1^1S_0$ & 5.369( 2.0)& 5.390( 1.2)& 5.368( 0.0)\cr
$B_s: b \overline s \ 1^3S_1$ & 5.416( 3.3)& 5.450( 1.4)& 5.426( 0.1)\cr
$\eta_c : c \overline c \ 1^1S_0$ & 2.980( 2.1)& 2.970( 0.2)& 2.978( 0.0)\cr
$\psi: c \overline c \ 1^3S_1$ & 3.097( 0.0)& 3.100( 0.1)& 3.128( 14.1)\cr
$\chi_0: c \overline c \ 1^1P_1$ & 3.526( 0.2)& 3.520( 0.5)& 3.520( 0.5)\cr
$\chi_0: c \overline c \ 1^3P_0$ & 3.415( 1.0)& 3.440( 4.4)& 3.408( 0.4)\cr
$\chi_1: c \overline c \ 1^3P_1$ & 3.510( 0.1)& 3.510( 0.0)& 3.507( 0.2)\cr
$\chi_2: c \overline c \ 1^3P_2$ & 3.556( 0.1)& 3.550( 0.5)& 3.548( 0.9)\cr
$\eta_c : c \overline c \ 2^1S_0$ & 3.594( 5.0)& 3.620( 0.4)& 3.611( 0.2)\cr
$\psi: c \overline c \ 2^3S_1$ & 3.686( 0.1)& 3.680( 0.5)& 3.689( 0.1)\cr
$\psi: c \overline c \ 1^3D_1$ & 3.770( 2.5)& 3.820( 4.2)& 3.809( 2.5)\cr
$\psi: c \overline c \ 3^3S_1$ & 4.040( 10.0)& 4.100( 0.5)& 4.082( 0.2)\cr
$\psi: c \overline c \ 2^3D_1$ & 4.159( 20.0)& 4.190( 0.0)& 4.159( 0.0)\cr
$\psi: c \overline c \ 3^3D_1$ & 4.415(6.0)& 4.450( 0.4)& 4.456(0.6)\cr
$D: c \overline u \ 1^1S_0$ & 1.865( 0.5)& 1.880( 2.7)& 1.865( 0.0)\cr
$D^*: c \overline u \ 1^3S_1$ & 2.007( 0.5)& 2.040( 12.6)& 1.998( 0.8)\cr
$D^*: c \overline u \ 1^3P_1$ & 2.422( 1.8)& 2.440( 0.9)& 2.408( 0.6)\cr
$D^*: c \overline u \ 1^3P_2$ & 2.459( 2.0)& 2.500( 3.8)& 2.381( 13.6)\cr
$D_s: c \overline s \ 1^1S_0$ & 1.968( 0.6)& 1.980( 1.4)& 1.976( 0.6)\cr
$D_s^*: c \overline s \ 1^3S_1$ & 2.112( 0.7)& 2.130( 3.0)& 2.121( 0.8)\cr
$D_s^*: c \overline s \ 1^3P_1$ & 2.535( 0.3)& 2.530( 0.4)& 2.512( 6.7)\cr
$D_s^*: c \overline s \ 1^3P_2$ & 2.574( 1.7)& 2.590( 0.9)& 2.513(11.6)\cr
$K: s \overline u \ 1^1S_0$ & 0.494( 0.0)& 0.470( 8.0)& 0.494( 0.0)\cr
$K^*: s \overline u \ 1^3S_1$ & 0.892( 0.2)& 0.900( 0.1)& 0.907( 0.5)\cr
$K_1: s \overline u \ 1^1P_1$ & 1.273( 7.0)& 1.340( 0.5)& 1.411( 2.2)\cr
$K_0^*: s \overline u \ 1^3P_0$ & 1.429( 4.0)& 1.240( 2.3)& 1.323( 0.7)\cr
$K_1: s \overline u \ 1^3P_1$ & 1.402( 7.0)& 1.380( 0.1)& 1.509( 2.3)\cr
$K_2^*: s \overline u \ 1^3P_2$ & 1.425( 1.3)& 1.430( 0.0)& 1.393( 0.5)\cr
$K^*: s \overline u \ 2^1S_0$ & 1.460( 30.0)& 1.450( 0.0)& 1.593( 0.2)\cr
$K^*: s \overline u \ 2^3S_1$ & 1.412( 12.0)& 1.580( 1.5)& 1.801( 7.9)\cr
$K_2: s \overline u \ 1^1D_2$ & 1.773( 8.0)& 1.780( 0.0)& 1.879( 1.1)\cr
$K^*: s \overline u \ 1^3D_1$ & 1.714( 20.0)& 1.780( 0.1)& 1.988( 1.6)\cr
$K_2: s \overline u \ 1^3D_2$ & 1.816( 10.0)& 1.810( 0.0)& 1.947( 1.5)\cr
$K_3: s \overline u \ 1^3D_3$ & 1.770( 10.0)& 1.790( 0.0)& 1.770( 0.0)\cr
$K^*: s \overline u \ 3^1S_0$ & 1.830( 30.0)& 2.020( 0.5)& 2.188( 1.7)\cr
$K_2^*: s \overline u \ 2^3P_2$ & 1.975( 22.0)& 1.940( 0.0)& 2.098( 0.3)\cr
$K_4^*: s \overline u \ 1^3F_4$ & 2.045( 9.0)& 2.110( 0.3)& 2.080( 0.1)\cr
$K_2: s \overline u \ 2^3D_2$ & 2.247( 17.0)& 2.260( 0.0)& 2.377( 0.7)\cr
$K_5^*: s \overline u \ 1^3G_5$ & 2.382( 33.0)& 2.390( 0.0)& 2.350( 0.0)\cr
$\phi: s \overline s \ 1^3S_1$ & 1.019( 0.0)& 1.020( 0.0)& 1.031( 1.9)\cr
$f_0: s \overline s \ 1^3P_0$ & 1.370( 40.0)& 1.360( 0.0)& 1.329( 0.0)\cr
$f_1: s \overline s \ 1^3P_1$ & 1.512( 4.0)& 1.480( 0.7)& 1.536( 0.4)\cr
$f_2: s \overline s \ 1^3P_2$ & 1.525( 5.0)& 1.530( 0.0)& 1.493( 0.4)\cr
$\phi: s \overline s \ 2^3S_1$ & 1.680( 20.0)& 1.690( 0.0)& 1.852( 0.9)\cr
$\phi: s \overline s \ 1^3D_3$ & 1.854( 7.0)& 1.900( 0.4)& 1.849( 0.0)\cr
$f_2: s \overline s \ 2^3P_2$ & 2.011( 69.0)& 2.040( 0.0)& 2.162( 0.1)\cr
$\pi: u\overline d \ 1^1S_0$ & 0.140( 0.0)& 0.150( 1.6)& 0.143( 0.1)\cr
$\rho: u\overline d \ 1^3S_1$ & 0.767( 1.2)& 0.770( 0.0)& 0.788( 0.1)\cr
$b_1: u\overline d \ 1^1P_1$ & 1.231( 10.0)& 1.220( 0.0)& 1.397( 2.6)\cr
$a_0: u\overline d \ 1^3P_0$ & 1.450( 40.0)& 1.090( 1.0)& 1.507( 0.0)\cr
$a_1: u\overline d \ 1^3P_1$ & 1.230( 40.0)& 1.240( 0.0)& 1.573( 0.8)\cr
$a_2: u\overline d \ 1^3P_2$ & 1.318( 0.7)& 1.310( 0.0)& 1.309( 0.0)\cr
$\pi: u\overline d \ 2^1S_0$ & 1.300( 100.0)& 1.300( 0.0)& 1.535( 0.1)\cr
$\rho: u\overline d \ 2^3S_1$ & 1.465( 25.0)& 1.450( 0.0)& 1.774( 1.6)\cr
$\pi_2: u\overline d \ 1^1D_2$ & 1.670( 20.0)& 1.680( 0.0)& 1.871( 1.0)\cr
$\rho: u\overline d \ 1^3D_1$ & 1.700( 20.0)& 1.660( 0.0)& 1.986( 2.2)\cr
$\rho_3: u\overline d \ 1^3D_3$ & 1.691( 5.0)& 1.680( 0.0)& 1.711( 0.1)\cr
$\pi: u\overline d \ 3^1S_0$ & 1.795( 10.0)& 1.880( 0.5)& 2.169( 9.4)\cr
$\rho: u\overline d \ 3^3S_1$ & 2.149( 17.0)& 2.000( 0.5)& 2.335( 0.8)\cr
$\rho_4: u\overline d \ 1^3F_4$ & 2.037( 26.0)& 2.010( 0.0)& 2.036( 0.0)\cr
$\pi_2: u\overline d \ 2^1D_2$ & 2.090( 29.0)& 2.130( 0.0)& 2.372( 0.6)\cr
$\rho_3: u\overline d \ 2^3D_3$ & 2.250( 45.0)& 2.130( 0.1)& 2.307( 0.0)\cr
$\rho_5: u\overline d \ 1^3G_5$ & 2.330( 35.0)& 2.340( 0.0)& 2.311( 0.0)\cr
$\chi^2$ & 0.0& 84.5& 104.7\cr}

\vfill\eject


\end{document}
