# Magnetic States at Short Distances

Horace W. Crater1∗ 000∗Email address<EMAIL_ADDRESS>and Cheuk-Yin Wong2,3†
000†Email address<EMAIL_ADDRESS>1The University of Tennessee Space
Institute, Tullahoma, Tennessee 37388
2Department of Physics and Astronomy, University of Tennessee, Knoxville, TN
37996
3Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831  1The
University of Tennessee Space Institute, Tullahoma, Tennessee 37388
2Department of Physics and Astronomy, University of Tennessee, Knoxville, TN
37996
3Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831  The
University of Tennessee Space Institute, Tullahoma, Tennessee 37388
Department of Physics and Astronomy, University of Tennessee, Knoxville, TN
37996
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831

###### Abstract

The magnetic interactions between a fermion and an antifermion of opposite
electric or color charges in the ${}^{1}S_{0}^{-+}$ and ${}^{3}P_{0}^{++}$
states with $J=0$ are very attractive and singular near the origin and may
allow the formation of new bound and resonance states at short distances. In
the two body Dirac equations formulated in constraint dynamics, the short-
distance attraction for these states for point particles leads to a
quasipotential that behaves near the origin as $-\alpha^{2}/r^{2}$, where
$\alpha$ is the coupling constant. Representing this quasipotential at short
distances as $\lambda(\lambda+1)/r^{2}$ with
$\lambda=(-1+\sqrt{1-4\alpha^{2}})/2$, both ${}^{1}S_{0}^{-+}$ and
${}^{3}P_{0}^{++}$ states admit two types of eigenstates with drastically
different behaviors for the radial wave function $u=r\psi$. One type of
states, with $u$ growing as $r^{\lambda+1}$ at small $r$, will be called usual
states. The other type of states with $u$ growing as $r^{-\lambda}$ will be
called peculiar states. Both of the usual and peculiar eigenstates have
admissible behaviors at short distances. Remarkably, the solutions for both
sets of ${}^{1}S_{0}$ states can be written out analytically. The usual bound
${}^{1}S_{0}$ states possess attributes the same as those one usually
encounters in QED and QCD, with bound state energies explicitly agreeing with
the standard perturbative results through order $\alpha^{4}$. In contrast, the
peculiar bound ${}^{1}S_{0}$ states, yet to be observed, not only have
different behaviors at the origin, but also distinctly different bound state
properties (and scattering phase shifts). For the peculiar ${}^{1}S_{0}$
ground state of fermion-antifermion pair with fermion rest mass $m$, the root-
mean-square radius is approximately $1/m$, binding energy is approximately
$(2-\sqrt{2})m$, and rest mass approximately $\sqrt{2}m$. On the other hand,
the $(n+1)$${}^{1}S_{0}$ peculiar state with principal quantum number $(n+1)$
is nearly degenerate in energy and approximately equal in size with the
$n$${}^{1}S_{0}$ usual states. For the ${}^{3}P_{0}$ states, the usual
solutions lead to the standard bound state energies and no resonance, but
resonances have been found for the peculiar states whose energies depend on
the description of the internal structure of the charges, the mass of the
constituent, and the coupling constant. The existence of both usual and
peculiar eigenstates in the same system leads to the non-self-adjoint property
of the mass operator and two non-orthogonal complete sets. As both sets of
states are physically admissible, the mass operator can be made self-adjoint
with a single complete set of admissible states by introducing a new
peculiarity quantum number and an enlarged Hilbert space that contains both
the usual and peculiar states in different peculiarity sectors. Whether or not
these newly-uncovered quantum-mechanically acceptable peculiar ${}^{1}S_{0}$
bound states and ${}^{3}P_{0}$ resonances for point fermion-antifermion
systems correspond to physical states remains to be further investigated.

###### pacs:

25.75.-q 25.75.Dw

##  I INTRODUCTION

It is well known that for some combinations of the spin configurations and
orbital motion the magnetic interaction can be strongly attractive and
singular111 A potential is quantum mechanically singular if it is more
attractive than $-1/4r^{2}$ at the origin in the context of ($-d^{2}/dr^{2}$
$-1/4r^{2})$. See Case . at short distances Bar77 ; Bar81 ; Won86 . We can
illustrate this by a classical example as shown schematically in Fig. 1(a)
where a positive charge $q^{+}$ is making a circular orbit about a fixed
negative charge $q^{-}$ whose spin ${\ \hbox{\boldmath${s}$}}(q^{-})$ is
pointing in a direction opposite to the orbital angular momentum of $q^{+}$
Won86 . In the external field problem, (e.g., Fermi’s treatment of hyperfine
structure), the charged particle $q^{-}$ at rest with a magnetic moment
$\hbox{\boldmath${\mu}$}(q^{-})$ generates a vector potential
${\hbox{\boldmath${A}$}}={\hbox{\boldmath${\mu}$}}(q^{-})\times{\hbox{\boldmath${r}$}}/r^{3}$
which acts on the other particle, $q^{+}$. Such a “magnetic” interaction can
be very attractive when the spins and the orbital angular momentum are
oppositely aligned, as shown in the configuration of ($q^{+}q^{-}$) in Fig. 1,
where the vector potential ${A}$, arising from the $q^{-}$ magnetic dipole
moment ${\hbox{\boldmath${\mu}$}}(q^{-})$, is parallel to the $q^{+}$ orbital
momentum ${p}$. The interaction
$(-\hbox{\boldmath${p}$}\cdot\hbox{\boldmath${A}$})$ from $q^{-}$ acting on
$q^{+}$ is attractive and is proportional to
$[\hbox{\boldmath${L}$}(q^{+})\cdot\hbox{\boldmath${s}$}(q^{-})]/r^{3}$ that
is quite singular in nature. At short distances it may overwhelm the
centrifugal barrier that is proportional to $1/r^{2}$. Similarly, the
interaction from $q^{+}$ acting on $q^{-}$ will be likewise attractive and
singular if the spin of the ${\hbox{\boldmath${s}$}}(q+)$ is parallel to the
electron spin ${\ \hbox{\boldmath${s}$}}(q-)$ and pointing in the same
direction, resulting in the total spin of the $q^{+}q^{-}$ system aligning
opposite to the orbital angular momentum, as in the ${}^{3}P_{0}^{++}$ state
with $S=1,L=1$, $J=0$, $P=+1$, and $C=+1$.

The ${}^{3}P_{0}^{++}$ state is not the only state with a strong magnetic
interaction. One can envisage classically another spin configuration, the
${}^{1}S_{0}^{-+}$ state, that also has attractive and singular magnetic
interactions. As illustrated schematically in Fig. 1(b), a fermion $q^{-}$
with an electric or color charge interacts with an antifermion $q^{+}$ of
opposite electric or color charge with spins ${\
\hbox{\boldmath${s}$}}(q^{-})$ and ${\ \hbox{\boldmath${s}$}}(q^{+})$ pointing
in a opposite directions in the ${}^{1}S_{0}^{-+}$ state configuration. With
the spins opposite to each other, the magnetic moments of $q^{-}$ and $q^{+}$
are parallel to each other. The interaction between the magnetic moments is
Jac62
$H_{\mathrm{int}}=-({8\pi}/{3}){\hbox{\boldmath${\mu}$}}_{q^{-}}\cdot{\hbox{\boldmath${\mu}$}}_{q^{+}}\delta(\hbox{\boldmath${r}$}),$
which is attractive and singular at short distances. The strong and singular
magnetic interaction may overcome other repulsive interactions and may allow
the formation of bound states of the fermion and antifermion system at short
distances. For brevity of notation, the quantum numbers $P$ and $C$ in and
${}^{1}S_{0}^{-+}$ and ${}^{3}P_{0}^{++}$ will be understood.

Figure 1:  (a) The schematic picture of the ${}^{3}P_{0}$ state spin
configuration and the orbital motion of a negative charge $q^{-}$ and a
positive charge $q^{+}$ that can lead to a strong magnetic attraction at short
distances. Here, ${\hbox{\boldmath${\mu}$}}(q^{\pm})$ is the magnetic moment
of the charge $q^{\pm}$ arising from its spin
$\hbox{\boldmath${s}$}(q^{\pm})$. (b) The schematic picture of the spin
configurations of $q^{-}$ and $q^{+}$ in the ${}^{1}S_{0}$ state.

Previously, one of us (CYW), in collaboration with R. L. Becker, studied the
$(e^{+}e^{-})$ system using the Kemmer-Fermi-Yang equation Kem37  with
interactions consisting of the Coulomb interaction and the vector (magnetic)
interaction,
${\hbox{\boldmath${A}$}}_{i}={\hbox{\boldmath${\mu}$}}_{j}\times({\hbox{\boldmath${r}$}}_{i}-{\hbox{\boldmath${r}$}}_{j})/|{\hbox{\boldmath${r}$}}_{i}-{\hbox{\boldmath${r}$}}_{j}|^{3}$,
in connection with a possible scalar ${}^{3}P_{0}$ magnetic resonance Won86 .
The interest was to investigate whether there could be a resonance at the mass
of 1.579 MeV that might explain the anomalous positron peak in heavy-ion
collisions near the Coulomb barrier Sch83 . The experimental evidence for the
anomalous positron peak later turned out to be negative when greater
statistics were accumulated Ahm95 . Nevertheless, it remains of interest to
study the behavior of the two-body system at short distances and see how the
attractive magnetic interaction in the ${}^{3}P_{0}$ state may reveal itself
in some observable properties.

While the use of the Kemmer-Fermi-Yang equation with a two-body magnetic
interaction is useful to motivate an approximate description Won86 , a
consistent relativistic description of the two-body interaction at short
distances can be found in the relativistic two body Dirac equations (TBDE)
formulated in Dirac’s constraint dynamics dirac ; cnstr ; cra82 ; cww . These
relativistic two body Dirac equations give a good description to the entire
meson mass spectrum (excluding most flavor-mixed mesons) with constituent
world-scalar and vector potentials depending on just two or three invariant
functions, in previous relativistic quark-model calculations crater2 ; tmlk ;
unusual .

The application of the TBDE equations to two-body bound and resonance states
in quantum electrodynamics has intrinsic merits. In Ref. bckr , the properties
of these TBDE equations that made them work so well for the relativistic quark
model were investigated by solving them nonperturbatively (i.e. analytically
or numerically) in quantum electrodynamics (QED), where order $\alpha^{4}$
perturbative solutions are well known. The two coupled Dirac equations in the
constraint formalism depend on Lorentz-covariant potentials between the two
constituents and act on a 16-component wave function. An exact Pauli reduction
led to a second-order relativistic Schrödinger-like equation for a reduced
four-component wave function with an effective interaction containing all the
dependencies on spin, orbital angular momentum, and tensor operators. We were
able to solve the TBDE nonperturbatively (analytically or numerically) as well
as perturbatively because the spin dependent short-distance components of the
effective interaction are not singular cww ; bckr . The situation is very
different from the approximate Fermi-Breit forms, which contain singular
potentials and necessitate the introduction of arbitrary short-distance cut-
off parameters. The spin dependence of the relativistic potentials in the
exact Schrödinger-like equation arises naturally from the relativistic
reduction procedure and it incorporates detailed minimal interaction and
dynamical recoil effects, characteristic of field theory. We shall also use
the term “quasipotential” to represent this effective, non-singular
interaction.

To obtain the interaction used in the TBDE formalism, we first determined the
relativistic quasipotential to the lowest order in $\alpha$ for the Schr
ödinger-like equation in Ref. bckr  by comparing the effective interaction
with the interaction derived from the Bethe-Salpeter equation. This, in turn
led to an invariant Coulomb-like potential $A(r)=-\alpha/r$, where $\alpha$ is
the coupling constant. Insertion of this information into the minimal
interaction structures of the two body Dirac equations then completely
determined all aspects (spin-dependent as well as spin-independent) of the
interaction. (In decay  we gave a procedure to construct the full 16-component
solution to our coupled first-order Dirac equations from a solution of the
second-order equation for the reduced wave function.)

Next, we showed that both the quantum mechanical perturbative and the TBDE
non-perturbative treatments (i.e. analytic or numerical) yield the standard
spectral results for QED and related interactions through order $\alpha^{4}$ .
Such an agreement depends crucially on the inclusion of the coupling between
various components of our 16-component Dirac wave functions and on the short-
distance behavior of the relativistic quasipotential in the associated
Schrödinger-like equation. We then examined the speculations Won86  whether
the quasipotentials (including the angular momentum barrier) for some states
in the $e^{+}e^{-}$ system may become attractive enough at short distances to
yield a pure QED resonance corresponding to the anomalous positron peaks in
heavy-ion collisions Sch83 . For the ${}^{3}P_{0}$ state we found that, even
though the quasipotential becomes attractive and overwhelms the centrifugal
barrier at short distances, the spatial extension of the attraction is not
large enough to hold a resonance at the energy of 1.579 MeV bckr . This result
contradicted predictions of such states by other authors Spe91  based on
numerical solutions of three-dimensional truncations of the Bethe-Salpeter
equation, for which the entire QED bound state spectrum has been treated
successfully through order $\alpha^{4}$ only by perturbation theory.

In this paper we return to this problem of the magnetic resonance and magnetic
states, not motivated so much by new experimental data as by a discovery of an
additional peculiar solution of the TBDE overlooked in the earlier work in
Ref. bckr . Our examination of the two body Dirac equations reveals that at
short distance for both ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states, the magnetic
interactions is indeed quite strong. As a consequence, they counterbalance
other repulsive interactions to result in a quasipotential for these states
that behaves as $-\alpha^{2}/r^{2}$ at short distances.

In standard quantum mechanics for central interactions including the angular
momentum barrier $L(L+1)/r^{2}$ for states with $L\neq 0$ at short distances,
one generally retains only one of the two solutions for the radial part of the
wave function, $u=r\psi$, the one that grows with distance as ($\sim
r^{L+1})$, dropping the other solution($\sim r^{-L})$ as being too singular.
If we likewise represent the quasipotential as $\lambda(\lambda+1)/r^{2}$ with
$\lambda=(-1+\sqrt{(1-4\alpha^{2}})/2$, it leads to a short-distance solution
that behaves as $r^{\lambda+1}$, which we call the usual solution, in addition
to a solution, whose radial part grows as $r^{-\lambda}$, which we call the
peculiar solution. However, both usual and peculiar states have quantum-
mechanically acceptable behaviors at short distances, as the wave functions at
short-distances are square-integrable.

In the case of the spin singlet ${}^{1}S_{0}$ states, the eigenstates and
eigenenergies can be obtained analytically and are found to encompass both
usual and peculiar states. We find usual bound states with attributes the same
as those one usually encounters in QED and QCD, explicitly agreeing with the
standard perturbative results through order $\alpha^{4}$. In contrast, the
peculiar ${}^{1}S_{0}$ ground state of a fermion-antifermion pair with a
fermion rest mass $m$ has a root-mean-square radius approximately $1/m$, a
binding energies approximately $B_{p}$$\sim$$(2-\sqrt{2})m$, and a rest mass
approximately $\sqrt{2}m$. However, the $(n+1)$ th ${}^{1}S_{0}$ peculiar
state is nearly degenerate in energy and approximately equal in size with the
$n$th usual ${}^{1}S_{0}$.

The existence of both usual and peculiar eigenstates in the same system brings
with them conceptual and mathematical problems of the non-self-adjoint
property of the mass operator and the over-completeness of the set of
eigenstates. We resolve these problems by the introduction of a new quantum
number, the peculiarity quantum number, that makes the mass operator self-
adjoint and the combined set of usual and peculiar states a complete set in an
enlarged Hilbert space.

In the case of the ${}^{3}P_{0}$ states, both of the usual and peculiar
solutions reflect the overwhelmed centrifugal barrier and so differ
substantially from the $r^{L+1}$ and $r^{-L}$ behaviors at short distances
respectively. As a peculiar state radial wave function $u$ rises from the zero
value at the origin as $r^{-\lambda}\sim r^{\alpha^{2}}$, the strongly
attractive magnetic interaction has the tendency of bending the wave function
in such a way to allow for the possibility of a resonance. Furthermore, as the
quasipotential obtained through the relativistic reduction is sensitively
energy dependent, we can explore the behavior of the two-body system over a
larger domain of energies. We find that the usual solutions lead to no
resonant behavior, but the peculiar solution can lead to a ${}^{3}P_{0}$
resonance whose phase shift changes by $\pi$ at an appropriate energy,
depending on the description of the internal structure of the charges, the
mass of the constituent, and the coupling constant.

This paper is organized as follows. In Sec. II we give a review of the two
body Dirac equations of constraint dynamics. For those readers who are already
familiar with the constraint approach we refer them to the TBDE given in Eq.
(14) and their Schrödinger-like Pauli reduction given in Eq. (17). We
specialize to electromagnetic-like interactions only in this paper. We give in
Sec. III the single-component radial forms of Eq. (17) relevant to this paper.
In Sec. IV we examine both solutions for the ${}^{1}S_{0}$ states. In addition
to examining new bound state solutions, we show how the ${}^{1}S_{0}$ wave
functions for positive energies (and their corresponding phase shifts) can be
determined analytically in terms of Coulomb wave functions for noninteger
angular momentum. This is done for both the usual and peculiar solutions. We
explain why and how we introduce of a new quantum number, which we call the
peculiarity quantum number, to solve the problems of the non-self-adjoint
property of the mass operator and the over-completeness of the set of
eigenstates. In Sections V we examine the short distance behaviors for the
${}^{3}P_{0}$ state for the usual and peculiar solutions. In Sec. VI we
discuss the variable phase shift formalism of Calogero cal  and outline how we
use it for our phase shift analysis. Since the short distance behavior of the
${}^{3}P_{0}$ quasipotential is the same as that of ${}^{1}S_{0}$
quasipotential, we can use those same ${}^{1}S_{0}$ Coulomb wave functions as
reference wave functions in that region to compute phase shifts. There is,
however, an additional term (proportional to
$\delta({\hbox{\boldmath${r}$})}$) that does not appear in the extreme short
distance region for the ${}^{1}S_{0}$ quasipotential. Even though this term
does not contribute in the case of the phase shift for the usual solution, its
contribution to the phase shift calculations for the peculiar solution must be
considered. In Sec. VI we discuss our numerical results and in Sec. VII our
conclusions. Various technical results are presented in the appendices. In
Appendix A we give an outline of the details on the relation between the two-
body Dirac equations and their Pauli reduced Schrödinger forms. In Appendix B
we present the radial forms of those Schrödinger-like equations for a general
angular momentum coupling. In Appendix C we present details of the
${}^{1}S_{0}$ usual and peculiar bound states. In Appendix D we review the
connections between the Coulomb wave functions for noninteger angular momentum
index. Appendix E presents a review of the variable phase method of Calogero
cal  for our problem.

##  II TWO BODY DIRAC EQUATIONS

We briefly review the two body Dirac equations of constraint dynamics cra82 ;
cww ; crater2 ; tmlk ; unusual ; saz86  providing a covariant three
dimensional truncation of the Bethe Salpeter equation for the two body system.
Sazdjian saz85 ; saz92 ; saz97  has shown that the Bethe-Salpeter equation can
be algebraically transformed into two independent equations. The first yields
a covariant three dimensional eigenvalue equation which for spinless particles
takes the form

$\biggl{(}\mathcal{H}_{10}+\mathcal{H}_{20}+2\Phi\biggr{)}\Psi(x_{1},x_{2})=0,$
(1)

where $\mathcal{H}_{i0}=p_{i}^{2}+m_{i}^{2}$ . The quasipotential $\Phi$ is a
modified geometric series in the Bethe-Salpeter kernel $K$ such that in lowest
order in $K$

$\Phi=\pi i\delta(P\cdot p)K,$ (2)

where $P=p_{1}+p_{2}$ is the total momentum, $p=\mu_{2}p_{1}-\mu_{1}p_{2}$ is
the relative momentum, $w$ is the invariant total center of momentum (c.m.)
energy with $P^{2}=-w^{2}$. The $\mu_{i}$ must be chosen so that the relative
coordinate $x=x_{1}-x_{2}$ and $p$ are canonically conjugate, i.e.
$\mu_{1}+\mu_{2}=1$. The second equation, Eq. (2), overcomes the difficulty of
treating the relative time in the center of momentum system by setting an
invariant condition on the relative momentum $p$,

$(\mathcal{H}_{10}-\mathcal{H}_{20})\Psi(x_{1},x_{2})=0=2P\cdot
p\Psi(x_{1},x_{2}).$ (3)

Note that this implies
$p^{\mu}\Psi=p_{\perp}^{\mu}\Psi\equiv(\eta^{\mu\nu}+\hat{P}^{\mu}\hat{P}^{\nu})p_{\nu}\Psi$
in which $\hat{P}^{\mu}=P^{\mu}/w$ is a time like unit vector
$(\hat{P}^{2}=-1)$ in the direction of the total momentum222 We use the metric
$\eta^{11}=\eta^{22}=\eta^{33}=-\eta^{00}=1$..

One can further combine the sum and the difference of Eqs. (1) and (3) to
obtain a set of two relativistic equations one for each particle with each
equation specifying two generalized mass-shell constraints

$\mathcal{H}_{i}\Psi(x_{1},x_{2})=(p_{i}^{2}+m_{i}^{2}+\Phi)\Psi(x_{1},x_{2})=0,~{}i=1,2,$
(4)

including the interaction with the other particle. These constraint equations
are just those of Dirac’s Hamiltonian constraint dynamics for spinless
particles dirac ; cnstr ; cra84s . In order for Eq. (4) to have consistent
solutions, Dirac’s constraint dynamics stipulate that these two constraints
must be compatible among themselves,
$[\mathcal{H}_{1},\mathcal{H}_{2}]\Psi=0$, that is, they must be first class
constraints. This requires that the quasipotential $\Phi$ satisfy
$[p_{1}^{2}-p_{2}^{2},\Phi]\Psi=0$. Working out the commutator shows that for
this to be true in general, $\Phi$ must depend on the relative coordinate
$x=x_{1}-x_{2}$ only through its component, $x_{\perp},$ perpendicular to $P,$

$x_{\perp}^{\mu}=(\eta^{\mu\nu}+\hat{P}^{\mu}\hat{P}^{\nu})(x_{1}-x_{2})_{\nu}.$
(5)

The invariant $x_{\perp}^{2}\equiv r^{2}$ becomes $\mathbf{r}^{2}$ in the c.m.
frame. Since the total momentum is conserved, the single component wave
function $\Psi~{}$in coordinate space is a product of a plane wave eigenstate
of $P$ and an internal part $\psi(x_{\perp})$ cra87 333 We use the same symbol
$P$ for the eigenvalue so that the $w$ dependence of $m_{w}$ and
$\varepsilon_{w}$ in Eq. (6) is regarded as an eigenvalue dependence. The wave
function $\Psi$ can be viewed either as a relativistic 2-body wave function
(similar in interpretation to the Dirac wave function) or, if a close
connection to field theory is required, related directly to the Bethe Salpeter
wave function $\chi{\hbox{\boldmath${~{}}$}}$by saz92  $\Psi=-\pi
i\delta(P\cdot p)\mathcal{\ H}_{10}\chi=-\pi i\delta(P\cdot
p)\mathcal{H}_{20}\chi$..

We find a plausible structure for the quasipotential $\Phi$ by observing that
the one-body Klein-Gordon equation
$(p^{2}+m^{2})\psi=({\hbox{\boldmath${p}$}}^{2}-\varepsilon^{2}+m^{2})\psi=0$
takes the form
$({\hbox{\boldmath${p}$}}^{2}-\varepsilon^{2}+m^{2}+2mS+S^{2}+2\varepsilon
A-A^{2})\psi=0~{}$when one introduces a scalar interaction and time-like
vector interaction via the minimum substitutions $m\rightarrow m+S~{}$and
$\varepsilon\rightarrow\varepsilon-A$. In the two-body case, separate
classical fw  and quantum field theory saz97  arguments show that when one
includes world scalar and vector interactions then $\Phi$ depends on two
underlying invariant functions $S(r)$ and $A(r)$ ($r=\sqrt{x_{\perp}^{2}}$)
through the two body Klein-Gordon-like potential form with the same general
structure, that is

$\Phi=2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}.$ (6)

Those field theories further yield the c.m. energy dependent forms

$m_{w}=m_{1}m_{2}/w,$ (7)

and

$\varepsilon_{w}=(w^{2}-m_{1}^{2}-m_{2}^{2})/2w,$ (8)

ones that Tododov cnstr ; tod71  introduced as the relativistic reduced mass
and effective particle energy for the two-body system. Similar to what happens
in the nonrelativistic two-body problem, in the relativistic case we have the
motion of this effective particle taking place as if it were in an external
field (here generated by $S$ and $A$). The two kinematical variables (7) and
(8) are related to one another by the Einstein condition

$\varepsilon_{w}^{2}-m_{w}^{2}=b^{2}(w),$ (9)

where the invariant

$b^{2}(w)\equiv(w^{4}-2w^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2})/4w^{2},$
(10)

is the c.m. value of the square of the relative momentum expressed as a
function of $w$. One also has

$b^{2}(w)=\varepsilon_{1}^{2}-m_{1}^{2}=\varepsilon_{2}^{2}-m_{2}^{2},$ (11)

in which $\varepsilon_{1}$ and $\varepsilon_{2}$ are the invariant c.m.
energies of the individual particles satisfying

$\ \varepsilon_{1}+\varepsilon_{2}=w,\
\varepsilon_{1}-\varepsilon_{2}=(m_{1}^{2}-m_{2}^{2})/w.$ (12)

In terms of these invariants, the relative momentum appearing in Eq. (2) and
(3) is given by

$p^{\mu}=(\varepsilon_{2}p_{1}^{\mu}-\varepsilon_{1}p_{2}^{\mu})/w\mathrm{,}$
(13)

so that $\mu_{1}+\mu_{2}=(\varepsilon_{1}+\varepsilon_{2})/w=1$. In tod  the
forms for these two-body effective kinematic variables are given sound
justifications based solely on relativistic kinematics, supplementing the
dynamical arguments of fw  and saz97 .

This covariant and useful three-dimensional truncation of the Bethe-Salpeter
equation has been extended to the case of a two-fermion system where the two
constraint equations become the two body Dirac equations (TBDE) cra82 ; cra82
; cww ; crater2 ; tmlk ; unusual

$\displaystyle\mathcal{S}_{1}\psi$
$\displaystyle\equiv\gamma_{51}(\gamma_{1}\cdot(p_{1}-\tilde{A}_{1})+m_{1}+\tilde{S}_{1})\Psi=0,$
$\displaystyle\mathcal{S}_{2}\psi$
$\displaystyle\equiv\gamma_{52}(\gamma_{2}\cdot(p_{2}-\tilde{A}_{2})+m_{2}+\tilde{S}_{2})\Psi=0.$
(14)

Here $\Psi$ is a sixteen component wave function consisting of an external
plane wave part that is an eigenstate of $P$ and an internal part
$\psi=\psi(x_{\perp})$. The vector potential$\ \tilde{A}_{i}^{\mu}$ was taken
to be an electromagnetic-like four-vector potential with the time-like and
space-like portions both arising from a single invariant function $A(r)$. 444
In particular, in a perturbative context that would mean that these aspects of
$\tilde{A}_{i}^{\mu}$ were regarded as arising from a Feynman gauge vertex
coupling of a form proportional to $\gamma_{1}^{\mu}\gamma_{2\mu}A$ . The
tilde on these four-vector potentials indicate that they are not only position
dependent but also spin dependent by way of the gamma matrices. The operators
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ must commute or at the very least
$[\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0$ since they operate on the same wave
function555 The $\gamma_{5}$ matrices for each of the two particles are
designated by $\gamma_{5i}$ $i=1,2$. The reason for putting these matrices out
front of the whole expression is that including them facilitates the proof of
the compatibility condition, see cra82 ; cra87 .. This compatibility condition
gives restrictions on the spin dependencies of the vector and scalar
potentials,

$\tilde{A}_{i}^{\mu}=\tilde{A}_{i}^{\mu}(A(r),p_{\perp},\hat{P},w,\gamma_{1},\gamma_{2}),$
(15)

in addition to requiring that they depend on the invariant separation
$r\equiv\sqrt{x_{\perp}^{2}}$ through the invariant $A(r)$. The covariant
constraint (3) can also be shown to follow from Eq. (14). We give the explicit
connections between $\tilde{A}_{i}^{\mu}$ and the invariant $A(r)$ in Appendix
A. (A similar dependence occurs for $\tilde{S}_{i}$ on $S(r).$) The general
structural dependence on $A(r)$ and $S(r)$ and the spin dependence of
$\tilde{A}_{i}^{\mu},\tilde{S}_{i}$ is a consequence of the compatibility
condition $[\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0.$

The Pauli reduction of these coupled Dirac equations lead to a covariant Schr
ödinger-like equation for the relative motion with an explicit spin-dependent
potential $\Phi$,

${\bigg{(}}p_{\perp}^{2}+\Phi(S(r),A(r),p_{\perp},\hat{P},w,\sigma_{1},\sigma_{2}){\bigg{)}}\psi_{+}=b^{2}(w)\psi_{+},$
(16)

with $b^{2}(w)$ playing the role of the eigenvalue666 Due to the dependence of
$\Phi$ on $w,$ this is a nonlinear eigenvalue equation.. This eigenvalue
equation can then be solved for the four-component effective particle spinor
wave function $\psi_{+}$ related to the sixteen component spinor
$\psi(x_{\perp})$ in appendix A.

In Appendix A we outline the steps needed to obtain the explicit c.m. form of
Eq. (16). That form is liu , saz94 , crater2 ; tmlk ; unusual

$\displaystyle\\{$
$\displaystyle{\hbox{\boldmath${p}$}}^{2}+\Phi({\hbox{\boldmath${r}$},}m_{1},m_{2},w,{\hbox{\boldmath${\sigma}$}}_{1},{\hbox{\boldmath${\sigma}$}}_{2})~{}\\}\psi_{+}$
(17) $\displaystyle=$
$\displaystyle\\{{\hbox{\boldmath${p}$}}^{2}+2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}+\Phi_{D}$
$\displaystyle+{\hbox{\boldmath${L}$}\cdot(\hbox{\boldmath${\sigma}$}}_{1}{+\hbox{\boldmath${\sigma}$}}_{2}{)}\Phi_{SO}+{\hbox{\boldmath${\sigma}$}}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}L\cdot(\sigma}_{1}{\hbox{\boldmath${+}$}\sigma}_{2}{\hbox{\boldmath${)}$}}\Phi_{SOT}$
$\displaystyle+{\hbox{\boldmath${\sigma}$}}_{1}{\hbox{\boldmath${\cdot}$}\sigma}_{2}\Phi_{SS}+(3{\hbox{\boldmath${\sigma}$}}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}\sigma}_{2}{\hbox{\boldmath${\
}$}\cdot\hat{r}-\hbox{\boldmath${\sigma}$}}_{1}{\hbox{\boldmath${\cdot}$}\sigma}_{2})\Phi_{T}$
$\displaystyle+{\hbox{\boldmath${L}$}\cdot(\sigma}_{1}{\hbox{\boldmath${-}$}\sigma}_{2}{\hbox{\boldmath${)}$}\Phi}_{SOD}+i{\hbox{\boldmath${L}$}\cdot\sigma}_{1}{\hbox{\boldmath${\times}$}\sigma}_{2}\Phi_{SOX}\\}\psi_{+}$
$\displaystyle=$ $\displaystyle b^{2}\psi_{+},$

where the detailed forms of the separate quasipotentials $\Phi_{i}$ are given
in Appendix A. The subscripts of most of the quasipotentials are self
explanatory777 The subscript on quasipotential $\Phi_{D}$ refers to Darwin. It
consist of what are called Darwin terms, those that are the two-body analogue
of terms that accompany the spin-orbit term in the one-body Pauli reduction of
the ordinary one-body Dirac equation, and ones related by canonical
transformations to Darwin interactions fw ; sch73 , momentum dependent terms
arising from retardation effects. The subscripts on the other quasipotentials
refer respectively to $SO$ (spin-orbit), $SOD$ (spin-orbit difference), $SOX$
(spin-orbit cross terms), $SS$ (spin-spin), $T$ (tensor), $SOT$ (spin-orbit-
tensor).. After the eigenvalue $b^{2}$ of (17) is obtained, the invariant mass
of the composite two-body system $w$ can then be obtained by inverting Eq.
(10). It is given explicitly by

$w=\sqrt{b^{2}+m_{1}^{2}}+\sqrt{b^{2}+m_{2}^{2}}.$ (18)

For this reason we call the operator that appears to the left of Eq. (17) the
invariant mass operator. The structure of the linear and quadratic terms in
Eq. (17) as well as the Darwin and spin-orbit terms, are plausible in light of
the discussion given above Eq. (6), and in light of the static limit Dirac
structures that come about from the Pauli reduction of the Dirac equation.
Their appearance as well as that of the remaining spin structures are direct
outcomes of the Pauli reductions of the simultaneous TBDE Eq. (14). In this
paper we take the scalar interaction $S(r)=0$.

##  III TBDE SINGLE COMPONENT WAVE EQUATIONS.

The 4 component two-body wave function $\psi_{+}$ of the above Pauli-form (
17) of the TBDE can be conveniently represented by spin-singlet $S=0$ and
spin-triplet $S=1$ components with quantum numbers $\\{J,L,S\\}$ and basis
wave functions

$\langle{\hbox{\boldmath${r}$}}|wJLS\rangle\equiv\psi_{JLS}({\hbox{\boldmath${r}$}})=\frac{u_{JLS}(r)}{r}Y_{JM}(\mathbf{\hat{r})}.$
(19)

In general, the singlet and triplet states are coupled. However, we see from
Appendix B that for the case of equal masses and certain angular momentum
states, the spin singlet and spin triplet components decouple, and the TBDE
reduce to a single component equation.

Specifically, for the spin-singlet $S=0$ state with $J=L$, (the ${}^{1}J_{J}$
state), the TBDE is

$\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2\varepsilon_{w}A-A^{2}+\Phi_{D}{-}3\Phi_{SS}\\}u_{JJ0}=b^{2}u_{JJ0},$
(20)

where, using the results in Appendix B, the magnetic interaction $-3\Phi_{SS}$
is

$\displaystyle-3\Phi_{SS}$ $\displaystyle=$
$\displaystyle-3\Phi_{SS}(A,A^{\prime},{\nabla}^{2}A)=-3(\frac{1}{r^{2}}-\frac{3}{2r}\left(\frac{A^{\prime}}{w-2A}\right))((\frac{1}{\sqrt{1-2A/w}}+\sqrt{1-2A/w})-2)$
$\displaystyle-\frac{3}{2r}\left(\frac{A^{\prime}}{w-2A}\right)(\frac{1}{\sqrt{1-2A/w}}-\sqrt{1-2A/w})\mathcal{-}\frac{21}{2}\left(\frac{A^{\prime}}{w-2A}\right)^{2}-\frac{3{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}$
$\displaystyle=$ $\displaystyle-\Phi_{D}(A,A^{\prime},{\nabla}^{2}A),$

which is attractive and singular, as we discussed in the Introduction. At
large distances and for $A=-\alpha/r$ potential,
${\hbox{\boldmath${\nabla}$}}^{2}A=4\pi\alpha\delta({\hbox{\boldmath${r}$}})$
and the spin-spin interaction indeed becomes a singular interaction as
described in Jac62 . In addition to the magnetic spin-spin interaction, there
is also the repulsive Darwin quasipotential $\Phi_{D}$. In the ${}^{1}J_{J}$
state, the attractive magnetic spin-spin quasipotential in the spin-singlet
configuration exactly cancels the repulsive Darwin quasipotential,

${-}3\Phi_{SS}+\Phi_{D}=0.$ (22)

As a result of this remarkable cancellation, the eigenvalue equation for the
${}^{1}J_{J}$ state in Eq. (III) becomes simply

$\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2\varepsilon_{w}A-A^{2}\\}u_{JJ0}=b^{2}u_{JJ0}.$
(23)

Of all spin-singlet states, only in the ${}^{1}S_{0}$ states ($J=L=0)$ do the
effects of the quasipotential and the absence of a centrifugal barrier make
the combined quasipotential strongly attractive at short-distances. This, of
course, would not happen were it not for the highly attractive spin-spin
interaction discussed in the Introduction and in Eq. (III). Among the spin-
singlet states with different $J$ quantum numbers, we shall therefore focus
our attention only on the ${}^{1}S_{0}$ states.

For the spin-triplet $S=1$ states, there are two states with single component
radial equations. The first is the ${}^{3}J_{J}$ state whose radial equation
takes the form ($J\geq 1$)

$\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2\varepsilon_{w}A-A^{2}-\frac{2A^{\prime}}{r\left(w-2A\right)}+3\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}\\}u_{J1J}=b^{2}u_{JJ1}.$
(24)

The second is the ${}^{3}P_{0}$ equation which takes the form

$\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2\varepsilon_{w}A-A^{2}-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{2{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}\\}u_{011}=b^{2}u_{011}.$
(25)

Of the two spin-triplet cases, only in the ${}^{3}P_{0}$ states ($J=0,$
$L=1)~{}$ do the combined effects of the quasipotentials become so strongly
attractive at short-distances that they overwhelm the presence of the
centrifugal barrier. As discussed in the Introduction, this is due to the
highly attractive spin-orbit interaction (“magnetic” interaction) when the
total spin and the orbital angular momentum are oppositely aligned. In that
case, the competing effects of both the short-distance attraction and the
presence of the potential barrier raise the question whether the attraction is
strong enough to hold a resonance state in the continuum. Among the spin-
triplet states with different $J$ and $L$ quantum numbers, we shall therefore
focus our attention only on the ${}^{3}P_{0}$ states.

In the last term of the quasipotential in Eq. (25), the quantity $\nabla^{2}A$
is related to the particle charge density, $\rho(\hbox{\boldmath${r}$})$, seen
by each of the two particles by

$\nabla^{2}A(\hbox{\boldmath${r}$})=4\pi\alpha\rho(\hbox{\boldmath${r}$}).$
(26)

Therefore, the equation for the two-body relative wave function for the
${}^{3}P_{0}$ state becomes

$\biggl{\\{}-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2\varepsilon_{w}A-A^{2}-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{8\pi\alpha\rho(\hbox{\boldmath${r}$})}{w-2A}\biggr{\\}}u_{011}=b^{2}u_{011}.$
(27)

As we shall see in this case, the attractive magnetic interaction overwhelms
the centrifugal barrier, allowing the wave function to reach the short-
distance region where the particle charge density $\rho(r)$, if any, can be
exposed for scrutiny. This is in contrast to the situation for states in which
the centrifugal barrier dominates the short-distance region. In that case, the
centrifugal barrier will prevent the wave function from reaching the short-
distance region and the particle charge density will not make as a significant
difference in observable quantities888 Such would also be the case for
${}^{1}S_{0}$ states in which, due to the cancellation in Eq. (22), the
dependence on $\rho(r)$ is only indirect or implicit through the altered form
for $A(r)$.. We obtain the important result that the ${}^{3}P_{0}$
quasipotential depends explicitly on the particle charge density
$\rho(\hbox{\boldmath${r}$})$ at short distances. As a consequence, some
observable quantities may depend more critically on the nature of the particle
charge distribution and the forces binding the charge elements together.

For the ${}^{3}P_{0}$ state, it is convenient to separate out the centrifugal
barrier $2/r^{2}$ and the quasipotential $\Phi$ to write the above equation as

$\biggl{\\{}-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+\Phi(\hbox{\boldmath${r}$})\biggr{\\}}u_{011}=b^{2}u_{011},$
(28)

where

$\Phi(\hbox{\boldmath${r}$})=2\varepsilon_{w}A-A^{2}-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{8\pi\alpha\rho(\hbox{\boldmath${r}$})}{w-2A}.$
(29)

In our early work bckr , we limited our attention to energy regions around
1.579 MeV for the $(e^{+}e^{-})$ system and searched for ${}^{3}P_{0}$
resonances whose wave functions start from the origin in the usual way. We
found no resonance states. We return to this problem again including now an
additional (peculiar) solution of the TBDE that was overlooked in the earlier
work but has quantum-mechanically acceptable behaviors at short distances.

In Eqs. (27)), both the gauge field $A(r)$ and the gauge field source
$\rho(r)$ appear in the equation of motion for the wave function in the
${}^{3}P_{0}$ state. The appearance of the fermion charge source distribution
$\rho(r)$ brings into focus the question whether it is sufficient to describe
the magnetic interaction in the ${}^{3}P_{0}$ state completely within quantum
electrodynamics or quantum chromodynamics. Electrons in QED and quarks in QCD
are taken to be point particles with no structure. It may be necessary to go
beyond these field theories, to include additional auxiliary interactions that
hold the charge elements together, in order to properly describe the internal
structure of these particles. If these auxiliary interactions act on the
charged elements of the fermion to hold them together, they can also act on
the charged elements of the antifermion charge and will affect the
${}^{3}P_{0}$ wave function in the interior region of the charge distribution
$\rho(r)$.

The nature of these auxiliary forces holding the charged elements together is
completely unknown, although there have been many attempts to carry out such
an investigation. For example, in the Dirac’s model of an electron, a surface
tension from an unknown axillary interaction is invoked to hold the electric
charged elements of an electron together Dir48 ; Dir51 ; Dir52 ; Dir53 ; Dir62
; Dir65 . However, our knowledge on the internal structures of electrons and
quarks remain very uncertain. We shall return to examine how such a lack of
knowledge of the internal structures of these elementary quanta leads to
uncertainties in the ${}^{3}P_{0}$ magnetic resonance states in Sec. VC.

##  IV SOLUTIONS OF THE TWO BODY DIRAC EQUATIONS FOR THE ${}^{1}S_{0}$ STATE

###  IV.1  The ${}^{1}S_{0}$ quasipotential

We first consider the case of the ${}^{1}S_{0}$ state of a point fermion-
antifermion pair with electric or color charges interact through an
electromagnetic-type interaction arising from the exchange of a single photon
or gluon. The single photon annihilation diagram does not contribute because
the ${}^{1}S_{0}$ state is a charge parity even state. We thus have

$A=-\frac{\alpha}{r}.$ (30)

For brevity of notation in this subsection, we shall abbreviate the radial
wave function $u_{0JJ}$=$u_{000}$ as $u$. Equation (20) for $u$ becomes

$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}u=b^{2}u.$
(31)

with a short distance ($r<<\alpha/2\varepsilon_{w})$ behavior given by

$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{\alpha^{2}}{r^{2}}\right\\}u=0,$ (32)

with solutions

$\displaystyle u_{+}$ $\displaystyle\sim$ $\displaystyle r^{\lambda+1},$
$\displaystyle u_{-}$ $\displaystyle\sim$ $\displaystyle r^{-\lambda},$
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle(-1+\sqrt{1-4\alpha^{2}})/2,$ (33)

or

$u_{\pm}\sim r^{(1\pm\sqrt{1-4a^{2}})/2},$ (34)

both of which approach zero as $r$ approaches zero. With these behaviors, the
probability

$\psi_{\pm}^{2}d^{3}r=\frac{u_{\pm}^{2}}{r^{2}}r^{2}drd\Omega=u_{\pm}^{2}drd\Omega=r^{(1\pm\sqrt{1-4a^{2}})}drd\Omega,$
(35)

is finite for both signs. We call $u_{+}$ the usual solution, and it behaves
as $r^{\lambda+1}\sim r^{1-\alpha^{2}}$ for small $\alpha$. We call $u_{-}$
the peculiar solution, and it behaves as $r^{-\lambda}\sim r^{\alpha^{2}}$for
small $\alpha$. Both of these behaviors are physically acceptable near the
origin in the sense of (i) $u(0)$$\rightarrow 0$, and (ii) being square-
integrable in the neighborhood of $r\sim 0$. We note that if the sign in front
of $\alpha^{2}$ were positive or if we had non-zero angular momentum such that
$L(L+1)-\alpha^{2}>0$ then the second or peculiar set of solutions are not
physically admissible states.

In khel  one finds a thorough discussion on the proper boundary condition for
the radial wave function of the Schrödinger equation at the origin. They
discuss several conditions that appear in the literature: (1) Continuity of
$R=u/r$ at $r=0,$ requiring $u(0)=0$. (2) A finite differential probability in
the spherical slice $(r,r+dr$), that is $R^{2}r^{2}dr<\infty$ requiring
$u(r)\rightarrow r^{s+1},s>-1$ and again $u(0)=0$. (3) Requiring a finite
total probability inside a sphere of small radius $a$ which allows a more
singular behavior, namely $u(r)\rightarrow r^{-1/2+\varepsilon}$ where
$\varepsilon>0$ is a small positive constant, which would also include a
finite behavior of the norm. (4) Requiring time independence of the norm
leading to $u(r)\rightarrow cr^{-s+1}$, $s<1$ which again leads to $u(0)=0.$
Reference khel  furthermore shows that the radial Schrödinger equation
[$-d^{2}/dr^{2}+l(l+1)/r^{2}+2mV(r)]u(r)=2mEu(r)$ is compatible with the full
Schrödinger equation
($-\nabla^{2}+2mV(r))\frac{u(r)}{r}Y_{lm}=2mE\frac{u(r)}{r}Y_{lm}$ if and only
if the condition$~{}u(0)=0$ is satisfied. This $u(0)=0$ condition is clearly
satisfied for both solutions in Eq. (34).

In Schiff’s Quantum Mechanics schiff , a solution similar to the peculiar one
discussed here is examined for the case of the Klein-Gordon equation for the
Coulomb system. He argues that what we call the peculiar solution can be
discarded since the source of the Coulomb attraction is a finite sized nucleus
of radius $r_{0}$. In particular, he states that for $r<r_{0}$ for which the
potential is finite all the way to the origin, matching at $r_{0}$ would rule
out the peculiar solution. In our case, with point particles, the potential
does not satisfy this condition.

####  IV.1.1 ${}^{1}S_{0}$ Bound States

The solutions of the ${}^{1}S_{0}$ bound states can be obtained analytically.
In Appendix C we show how we can obtain the two sets of ${}^{1}S_{0}$ bound
state solutions that correspond to the usual and peculiar short distance
behaviors. The respective sets of eigenvalues and normalized eigenfunctions
for the state with total invariant c.m. energy (mass) $w_{\pm n}$ and the
principle quantum number $n$ is

$\displaystyle w_{\pm n}$ $\displaystyle=$ $\displaystyle
m\sqrt{2+2/\sqrt{1+{\alpha^{2}}/{(}n\pm\sqrt{1/4-\alpha^{2}}-1/2{)^{2}}}},$
$\displaystyle u_{\pm n}(r)$ $\displaystyle=$
$\displaystyle\left[\left(\frac{4\varepsilon_{w_{\pm}}\alpha
r}{n_{\pm}^{\prime}}\right)^{2}\frac{n_{r}!}{2n_{\pm}^{\prime}(n_{\pm}^{\prime}+\lambda_{\pm})!}\right]^{1/2}\exp(-\frac{\varepsilon_{w_{\pm}}\alpha
r}{n_{\pm}^{\prime}})\left(\frac{2\varepsilon_{w_{\pm}}\alpha
r}{n_{\pm}^{\prime}}\right)^{\lambda_{\pm}+1}L_{n_{r}}^{2\lambda_{\pm}+1}(\frac{2\varepsilon_{w_{\pm}}\alpha
r}{n_{\pm}^{\prime}}),$ (36)

where $n_{\pm}^{\prime}=n_{r}+\lambda_{\pm}+1=n+\lambda_{\pm}$ and

$\displaystyle\varepsilon_{w_{\pm}}=(w_{\pm}^{2}-2m^{2})/2w_{\pm}.$ (37)

For the usual states $u_{+n}$, the bound state eigenvalues $w_{+n}$ agree with
standard QED perturbative results through order $\alpha^{4}$,

$w_{+n}=2m-m{\alpha^{2}}/{4}n^{2}-m\alpha^{4}/2n^{3}(1-11/32/n)+O(\alpha^{6}),~{}n=1,2,3,...$
(38)

For the set of peculiar states $u_{-n}$, note that the peculiar ground state
$u_{-1}$ with $n=1$ has eigenenergy (mass)

$w_{-1}=m\sqrt{2+2/\sqrt{1+{\alpha^{2}}/({1/2}-\sqrt{1/4-\alpha^{2}}{)^{2}}}}\sim\sqrt{2}m\sqrt{1+\alpha},$
(39)

which represents very tight binding, with a binding energy on the order 300
keV for an $e^{+}e^{-}$ state and a root-mean-square radius on the order of a
Compton wave length instead of an angstrom. In particular we find (see
Appendix C)

$\sqrt{\langle r^{2}\rangle_{-1}}\rightarrow\frac{1}{m}.$ (40)

We note further the anti-intuitive behavior of the peculiar ground state
energy (mass), increasing with increasing coupling constant $\alpha$ instead
of decreasing. The excited states are quite near to the usual bound states. We
find the following pattern for those excited peculiar states

$w_{-n}=2m-m{\alpha^{2}}/{4(}n-1)^{2}+m\alpha^{4}/2(n-1)^{3}\left(1+11/32(n-1)\right)+O(\alpha^{6});\text{
}n=2,3,4,...$ (41)

In the nonrelativistic limit, where terms of order $\alpha^{4}$ are ignored we
find that the states are degenerate with the $n$th usual state identical to
the $(n+1)$th peculiar state. If we include the $\alpha^{4}$ corrections then
we find that

$w_{+n}-w_{-(n+1)}=-m\alpha^{4}/n^{3}.$ (42)

For all of the usual states and the remaining peculiar states we have

$\displaystyle\langle r^{2}\rangle_{+n}$ $\displaystyle=$
$\displaystyle\frac{(n+\lambda_{+})^{2}}{6\left(\varepsilon_{w_{+n}}\alpha\right)^{2}}[(n+\lambda_{+})^{2}+5\alpha^{2}+3]\text{,~{}}n=1,2,3...,$
$\displaystyle\langle r^{2}\rangle_{-n}$ $\displaystyle=$
$\displaystyle\frac{(n+\lambda_{-})^{2}}{6\left(\varepsilon_{w_{-n}}\alpha\right)^{2}}[(n+\lambda_{-})^{2}+5\alpha^{2}+3]\text{,~{}}n=2,3...,$
(43)

so that the size in the $(n+1)$th peculiar state is nearly the same as with
the $n$th usual state.

As shown in the Appendix C, the two sets of solutions, are not orthogonal with
respect to one another. For example, the two $n=1$ wave functions have the
respective forms

$\displaystyle u_{+}(r)$ $\displaystyle=$ $\displaystyle
c_{+}r^{\lambda_{+}+1}\exp(-\kappa_{+}\varepsilon_{w_{+}}\alpha r),$
$\displaystyle\kappa_{+}$ $\displaystyle=$
$\displaystyle\frac{2}{1+\sqrt{1-4\alpha^{2}}}=\frac{1}{\lambda_{+}+1},$
$\displaystyle u_{-}(r)$ $\displaystyle=$ $\displaystyle
c_{-}r^{\lambda_{-}+1}\exp(-\kappa_{-}\varepsilon_{w_{-}}\alpha r),$
$\displaystyle\kappa_{-}$ $\displaystyle=$
$\displaystyle\frac{2}{1-\sqrt{1-4\alpha^{2}}}=\frac{1}{\lambda_{-}+1},$ (44)

where for brevity of notation, we have omitted the principal quantum number
designation in $u\pm$ for the of the ground state. Clearly since they are both
zero node solutions we have

$\langle u_{-}|u_{+}\rangle=\int_{0}^{\infty}dru_{+}(r)u_{-}(r)\neq 0.$ (45)

How do we reconcile this with the expected orthogonality of the eigenfunctions
of a self-adjoint operator corresponding to different eigenvalues. In the
present context, the naive self-adjoint property requires that

$\langle
u_{+}|(-\frac{d^{2}}{dx^{2}}-\frac{2}{x}-\frac{\alpha^{2}}{x^{2}})|u_{-}\rangle=\langle
u_{-}|(-\frac{d^{2}}{dx^{2}}-\frac{2}{x}-\frac{\alpha^{2}}{x^{2}})|u_{+}\rangle.$
(46)

This boils down to

$\int_{0}^{\infty}dxu_{+}\frac{d^{2}u_{-}}{dx^{2}}=\int_{0}^{\infty}dxu_{-}\frac{d^{2}u_{+}}{dx^{2}}.$
(47)

Let us integrate by parts. Then we have

$\displaystyle\int_{0}^{\infty}dxu_{+}\frac{d^{2}u_{-}}{dx^{2}}$
$\displaystyle=$
$\displaystyle\left(u_{+}\frac{du_{-}}{dx}\right)\biggr{|}_{0}^{\infty}-\int_{0}^{\infty}dx\frac{du_{+}}{dx}\frac{du_{-}}{dx}$
$\displaystyle=$
$\displaystyle\left(u_{-}\frac{du_{+}}{dx}\right)\biggr{|}_{0}^{\infty}-\int_{0}^{\infty}dx\frac{du_{+}}{dx}\frac{du_{-}}{dx}.$

We thus have a self-adjoint operator if

$\left(u_{+}\frac{du_{-}}{dx}\right)\biggr{|}_{0}^{\infty}=\left(u_{-}\frac{du_{+}}{dx}\right)\biggr{|}_{0}^{\infty}.$
(49)

Now clearly these vanish at the upper end points. Since we have that

$\displaystyle\frac{du_{+}}{dx}$ $\displaystyle=$ $\displaystyle
u_{+}(\frac{\lambda_{+}+1}{x}-\frac{1}{\lambda_{+}+1}),$
$\displaystyle\frac{du_{-}}{dx}$ $\displaystyle=$ $\displaystyle
u_{-}(\frac{\lambda_{-}+1}{x}-\frac{1}{\lambda_{-}+1}).$ (50)

at the lower end point the LHS of Eq. (49) is

$\displaystyle\underset{x\rightarrow
0}{\lim}x^{\lambda_{+}+1}\exp(-x/(\lambda_{+}+1))x^{\lambda_{-}+1}\exp(-x/(\lambda_{-}+1))(\frac{\lambda_{-}+1}{x}-\frac{1}{\lambda_{-}+1})$
(51) $\displaystyle=$ $\displaystyle\underset{x\rightarrow
0}{\lim}x(\frac{\lambda_{-}+1}{x}-\frac{1}{\lambda_{-}+1})=\lambda_{-}+1$

whereas at the lower end point the RHS is $\lambda_{+}+1\neq\lambda_{-}+1$.
Thus, the second derivative is not self-adjoint in this context! This accounts
for the non-orthogonality of the usual and peculiar ground states in Eq. (45).

In general beginning with a set of usual and peculiar wave functions
$\\{u_{+n},u_{-n}\\}$ such that

$\displaystyle\langle u_{+n}|u_{+n^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{nn^{\prime}},$ $\displaystyle\langle
u_{-n}|u_{-n^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{nn^{\prime}},$ $\displaystyle\langle
u_{-n}|u_{+n}\rangle$ $\displaystyle\equiv$ $\displaystyle
b_{nn^{\prime}}=b_{n^{\prime}n},$ (52)

we find with

$\displaystyle H$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\left(\varepsilon_{w}\alpha\right)^{2}}\left[-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right]=\left[-\frac{d^{2}}{dx^{2}}-\frac{2}{x}-\frac{\alpha^{2}}{x^{2}}\right],$
$\displaystyle Hu_{\pm n}$ $\displaystyle=$ $\displaystyle h_{\pm n}u_{\pm
n},$ $\displaystyle h_{\pm n}$ $\displaystyle\equiv$
$\displaystyle-\kappa_{\pm n}^{2}=-1/(\lambda_{\pm}+n)^{2},~{}n=1,2,...$ (53)

where $x=\varepsilon_{w}\alpha r$, that

$\displaystyle\langle u_{+n}|H|u_{+n^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{nn^{\prime}}h_{+n}$ $\displaystyle\langle
u_{-n}|H|u_{-n^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{nn^{\prime}}h_{-n},$ $\displaystyle\langle
u_{-n}|H|u_{+n^{\prime}}\rangle$ $\displaystyle=$ $\displaystyle
b_{nn^{\prime}}h_{+n^{\prime}}$ $\displaystyle\langle
u_{+n}|H|u_{-n^{\prime}}\rangle$ $\displaystyle=$ $\displaystyle
b_{nn^{\prime}}h_{-n^{\prime}}\neq\langle u_{-n^{\prime}}|H|u_{+n}\rangle.$
(54)

In the first two terms it does not matter whether the $H$ operators operate to
the left or the right. In the last two cases we explicitly have $H$ operating
to the right. To emphasize that we write them as

$\displaystyle\langle u_{-n}|(H|u_{+n^{\prime}}\rangle)$ $\displaystyle=$
$\displaystyle b_{nn^{\prime}}h_{+n^{\prime}},$ $\displaystyle\langle
u_{+n}|(H|u_{-n^{\prime}}\rangle)$ $\displaystyle=$ $\displaystyle
b_{nn^{\prime}}h_{-n^{\prime}}.$ (55)

It is evident that with both sets of basis, $H$ is not self-adjoint since
$\langle u_{-n}|(H|u_{+n^{\prime}}\rangle)\neq(\langle
u_{-n}|H)|u_{+n}\rangle$ and $\langle
u_{+n}|(H|u_{-n^{\prime}}\rangle)\neq(\langle
u_{+n}|H)|u_{-n^{\prime}}\rangle$.

Let us see where the non-orthogonality leads us if we treat both basis on an
equal footing. In that case a general wave function for the ${}^{1}S_{0}$
system would be expanded as999 Strictly speaking we should include the
continuum states. See section below for discussion of those states. For the
purpose here the use of discrete states is sufficient.

$\Psi=\sum_{n_{+}}c_{+n}u_{+n}+\sum_{n_{-}}c_{-n}u_{-n},$ (56)

and applying the variational principle to

$\langle H\rangle=\frac{\langle\Psi|H|\Psi\rangle}{\langle\Psi|\Psi\rangle},$
(57)

and defining (we show just a finite $n\times n$ portion of the matrices)

$\displaystyle\mathbf{B}$ $\displaystyle\mathbf{=}$
$\displaystyle\begin{bmatrix}b_{11}&b_{12}&...&b_{1n}\\\
b_{21}&b_{22}&...&b_{2n}\\\ ...&...&...&...\\\
b_{n1}&b_{n2}&...&b_{nn}\end{bmatrix},$ $\displaystyle\mathbf{H}_{+}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}h_{+1}&0&...&0\\\
0&h_{+2}&...&0\\\ ...&...&...&...\\\ 0&0&...&h_{+n}\end{bmatrix},$
$\displaystyle\mathbf{H}_{-}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}h_{-1}&0&...&0\\\ 0&h_{-2}&...&0\\\
...&...&...&...\\\ 0&0&...&h_{-n}\end{bmatrix},$ (58)

then in block form we would have the eigenvalues equation

$\begin{bmatrix}\mathbf{H}_{+}&\mathbf{B\mathbf{H}}_{-}\\\
\mathbf{B\mathbf{H}_{+}}&\mathbf{H}_{-}\end{bmatrix}\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{c}_{-}\end{bmatrix}=-\kappa^{2}\begin{bmatrix}\mathbf{1}&\mathbf{B}\\\
\mathbf{B}&\mathbf{1}\end{bmatrix}\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{c}_{-}\end{bmatrix}$ (59)

(Note that the way this stands , the matrix on the left is not self-adjoint.)
Multiplying both sides on the right by

$\begin{bmatrix}\mathbf{1}&\mathbf{B}\\\
\mathbf{B}&\mathbf{1}\end{bmatrix}^{-1}=\begin{bmatrix}\mathbf{(1-B}^{2})^{-1}&-\mathbf{B(1-B}^{2})^{-1}\\\
-\mathbf{B(1-B}^{2})^{-1}&\mathbf{(1-B}^{2})^{-1}\end{bmatrix}$ (60)

we obtain

$\begin{bmatrix}\mathbf{H}_{+}&\mathbf{0}\\\
\mathbf{0}&\mathbf{H}_{-}\end{bmatrix}\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{c}_{-}\end{bmatrix}=-\kappa^{2}\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{c}_{-}\end{bmatrix}.$ (61)

It is clear that the eigenvectors corresponding to the eigenvalue sets of
$-\kappa_{+}^{2}$ and $-\kappa_{-}^{2}$ are of the form

$\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{0}\end{bmatrix},\begin{bmatrix}\mathbf{0}\\\
\mathbf{c}_{-}\end{bmatrix}.$ (62)

From Eq. (36). one recalls that the two sets of basis functions
$\\{u_{+n},u_{-n}\\}$ have distinctly different behaviors at the origin,
corresponding to the usual and peculiar solutions. In particular

$\displaystyle u_{+n}(x)$ $\displaystyle=$ $\displaystyle
c_{+n}x^{\lambda_{+}+1}\exp(-x)L_{n_{r}}^{2\lambda_{+}+1}(x),$ $\displaystyle
u_{-n}(x)$ $\displaystyle=$ $\displaystyle
c_{+n}x^{\lambda_{-}+1}\exp(-x)L_{n_{r}}^{2\lambda_{-}+1}(x),$ (63)

These generalized Laguerre polynomials are orthonormal with respect to
different weight functions $x^{\lambda_{\pm}+1}\exp(-x)$. They would each
correspond to a complete set. Together they would constitute an over-complete
set. However, that does not imply that Eq. (56) is incorrect as it allows for
the function $\Psi(x)$ to be a linear combination of functions with two
distinct behaviors at the origin. Nevertheless, the set up here is a bit
clumsy with questions of completeness and the non-self-adjoint property
remaining.

It should be realized that for the given quasipotential of the type
$-\alpha^{2}/r^{2}$ at short distances that is at hand, both the set of usual
states and the peculiar states are physically admissible states. There does
not appear to be reasons to exclude one set as being unphysical, if one is
given the attractive interaction near the origin as it is. We note however
that the peculiar states with the $r^{-\lambda}$ behavior at the origin are
excluded from existence if coefficient $\lambda(\lambda+1)$ for the $1/r^{2}$
term is greater than zero since that would lead to a $u(r)$ that is singular
at the origin. Only for interactions with sufficient attraction at the origin
(so that $-1/4\leq$ $\lambda(\lambda+1)<0)$ can these states be pulled into
existence and appear as eigenstates in the physically acceptable sheet, with
regular non-singular radial wave functions at the origin. It is desirable to
find ways to admit both types of physical states into a larger Hilbert space
to accommodate both sets of states with the mass operator to be self-adjoint
and the states to be part of a complete set. It is reasonable to assign a
quantum number which we call “peculiarity” for a states emerging into the
physical sheet in this way as physically acceptable states. The introduction
of the peculiarity quantum number enlarges the Hilbert space, allows the mass
operator to be self-adjoint, and the set of physically allowed states become a
complete set, as we shall demonstrate.

We introduce a new peculiarity observable $\hat{\zeta}$ with the quantum
number peculiarity $\zeta$ such that

$\displaystyle\hat{\zeta}\chi_{+}$ $\displaystyle=$
$\displaystyle\zeta\chi_{+}~{}~{}\mathrm{with~{}eigenvalue~{}}\zeta=+1,$
$\displaystyle\hat{\zeta}\chi_{-}$ $\displaystyle=$
$\displaystyle\zeta\chi_{-}~{}~{}\mathrm{with~{}eigenvalue~{}}\zeta=-1,$ (64)

with the corresponding spinor wave function $\chi_{\zeta}$ assigned to the
states so that a usual state is represented by the peculiarity spinor
$\chi_{+}$,

$\chi_{+}=\begin{pmatrix}1\\\ 0\end{pmatrix},$ (65)

and a peculiar state is represented by the peculiarity spinor $\chi_{-}$,

$\chi_{-}=\begin{pmatrix}0\\\ 1\end{pmatrix}.$ (66)

With this introduction, a general wave function can be expanded in terms of
the complete set of basis functions $\\{u_{+n},u_{-n}\\}$ as

$\Psi=\sum_{\zeta n}a_{\zeta n}u_{\zeta n}\chi_{\zeta},$ (67)

where $n$ represent all the spin and spatial quantum numbers of the state and
$\zeta$ the peculiarity quantum number. The variational principle applied to

$\langle H\rangle=\frac{\langle\Psi|H|\Psi\rangle}{\langle\Psi|\Psi\rangle},$
(68)

would lead to

$\displaystyle Hu_{+n}\chi_{+}$ $\displaystyle=$
$\displaystyle-\kappa_{+n}^{2}u_{+n}\chi_{+},$ $\displaystyle Hu_{-n}\chi_{-}$
$\displaystyle=$ $\displaystyle-\kappa_{-n}^{2}u_{-n}\chi_{-}.$ (69)

It is clear that in this context the usual and peculiar wave functions are
orthogonal, $H$ is self-adjoint, and the basis states are complete. That is,

$\langle i|j\rangle=\langle
u_{\zeta_{i}n_{i}}|u_{\zeta_{j}n_{j}}\rangle\equiv\int_{0}^{\infty}dru_{\zeta_{i}n_{i}}\chi_{\zeta_{i}}u_{\zeta_{j}n_{j}}\chi_{\zeta_{j}}=\delta_{\zeta_{i}\zeta_{j}}\delta_{n_{i}n_{j}}=\delta_{ij}$
(70)

and so the set of basis functions $\\{u_{+n}\zeta_{+},u_{-n}\zeta_{-}\\}$,
containing both the usual states and peculiar states in the enlarged Hilbert
space, form a complete set. We also have

$\displaystyle\langle i|H|j\rangle$ $\displaystyle=$ $\displaystyle\langle
u_{\zeta_{i}n_{i}}|H|u_{\zeta_{j}n_{j}}\rangle\equiv\int_{0}^{\infty}dru_{\zeta_{i}n_{i}}\chi_{\zeta_{i}}Hu_{\zeta_{j}n_{j}}\chi_{\zeta_{j}}$
(71) $\displaystyle=$ $\displaystyle
h_{\zeta_{i}}\delta_{\zeta_{i}\zeta_{j}}\delta_{n_{i}n_{j}}=\langle
u_{\zeta_{j}n_{j}}|H|u_{\zeta_{i}n_{i}}\rangle$ $\displaystyle=$
$\displaystyle\langle j|H|i\rangle,$

so that the mass operator $H$ in this enlarged Hilbert space is self-adjoint.
We see that the introduction of the peculiarity quantum number resolves the
problem of over-completeness property of the basis states and the non-self-
adjoint property of the mass operator.

####  IV.1.2 ${}^{1}S_{0}$ Scattering States

The ${}^{1}S_{0}$ state equation

$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w(r)}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}u=b^{2}u$
(72)

has the same form as the nonrelativistic Schrödinger equation for Coulomb
interaction

$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2m\alpha}{r}+\frac{L(L+1)}{r^{2}}\right\\}u=2mE\bar{u}=k^{2}u,$
(73)

except that the standard angular momentum term with $L(L+1)$ now take on the
value of $(-\alpha^{2})$. The two solutions the above equation are given by
the regular $F_{L}$ and irregular $G_{L}$ Coulomb wave functions,

$\displaystyle\bar{u}$ $\displaystyle=aF_{L}(\eta,kr)+cG_{L}(\eta,kr),$
$\displaystyle\eta$ $\displaystyle=-\frac{m\alpha}{k},$ (74)

with only the regular Coulomb wave function having an acceptable behavior at
the origin. The long distance behaviors of the regular and irregular solutions
are

$\displaystyle F_{L}(\eta,kr$ $\displaystyle\rightarrow$
$\displaystyle\infty)\rightarrow\mathrm{const}\times\sin(kr-\eta\log
2kr+\sigma_{L}-L\pi/2),$ $\displaystyle G_{L}(\eta,kr$
$\displaystyle\rightarrow$
$\displaystyle\infty)\rightarrow\mathrm{const}\times\cos(kr-\eta\log
2kr+\sigma_{L}-L\pi/2),$ (75)

in which $\sigma_{L}$ is the Coulomb phase shift given by

$\sigma_{L}=\arg(\Gamma(L+1+i\eta).$ (76)

Now we can solve Eq. (72) exactly for $b^{2}>0$ by analytically continuing the
above solutions to an arbitrary (non-integer) angular momentum $\lambda$ and
making a few obvious replacements by analogy,

$\displaystyle\bar{u}$
$\displaystyle=aF_{\lambda}(\eta,br)+cG_{\lambda}(\eta,br),$
$\displaystyle\lambda(\lambda+1)$ $\displaystyle=-\alpha^{2},$
$\displaystyle\eta$ $\displaystyle=-\frac{\varepsilon_{w}\alpha}{b}.$ (77)

Using the expressions for the analytically continued Coulomb wave functions to
non-integer $\lambda~{}$klein  we will presently see that we have solutions
given by the $F$ and $G$ functions in Eqs. (83) and (84) below. We emphasize
that both solutions have an acceptable behavior at the origin. Since $\lambda$
is not an integer, one can replace the irregular solution
$G_{\lambda}(\eta,br)$ by $F_{-\lambda-1}(\eta,br)$.101010 The reason that
$G_{L}$ is used in place of $F_{-L-1}$ for $L$ integer is that the latter is
not linearly independent of $F_{L}$ in that case. It is melded together with
$F_{L}$ to produce $G_{L}$ by a limited process analogous to how the Neumann
function is obtained from the Bessel functions. For $\lambda\neq$ integer,
$F_{\lambda}$ and $F_{-\lambda-1}$ are linearly independent. In particular, as
shown in Appendix D, in terms of the confluent hypergeometric function
$M(a,b;z)$

$F_{\lambda}(\rho)=C_{\lambda}(\eta)\rho^{\lambda+1}\exp(-i\rho)M(\lambda+1-i\eta,2\lambda+2;2i\rho),$
(78)

one has with

$x(\lambda,\eta)\equiv(\lambda+\frac{1}{2})\pi+\sigma_{-\lambda-1}(\eta)-\sigma_{\lambda}(\eta),$
(79)

that

$G_{\lambda}(\rho)=\frac{F_{-\lambda-1}(\rho)-\cos
x(\lambda,\eta)F_{\lambda}(\rho)}{\sin x(\lambda,\eta)},$ (80)

a linear combination of $F_{\lambda}(\rho)$ and $F_{-\lambda-1}(\rho)$. In
others words, Eq. (77) can be written as

$\bar{u}=dF_{\lambda}(\eta,br)+eF_{-\lambda-1}(\eta,br),$ (81)

where

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1+\sqrt{1-4\alpha^{2}})\equiv\lambda_{+},$ (82)
$\displaystyle-\lambda-1$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1-\sqrt{1-4\alpha^{2}})\equiv\lambda_{-},$

corresponding to the separate $\zeta=\pm 1$ sectors. As with the solutions in
Eq. (77), $F_{\lambda}(\eta,br)$ and $F_{-\lambda-1}(\eta,br)$ have acceptable
behaviors at the origin corresponding to Eq. (34). Their respective long
distance behaviors are given by

$\displaystyle F_{\lambda}(\eta,br$
$\displaystyle\rightarrow\infty)\rightarrow\mathrm{const}\times\sin(br-\eta\log
2br+\sigma_{\lambda_{+}}-\lambda_{+}\pi/2),$ $\displaystyle
F_{-\lambda-1}(\eta,br$
$\displaystyle\rightarrow\infty)\rightarrow\mathrm{const}\times\sin(br-\eta\log
2br+\sigma_{\lambda_{-}}-\lambda_{-}\pi/2).$ (83)

Alternatively we can use the related $\ G$ functions to determine the
behaviors

$\displaystyle G_{\lambda}(\eta,br$ $\displaystyle\rightarrow$
$\displaystyle\infty)\rightarrow\mathrm{const}\times\cos(br-\eta\log
2br+\sigma_{\lambda_{+}}-\lambda_{+}\pi/2),$ $\displaystyle
G_{-\lambda-1}(\eta,br$ $\displaystyle\rightarrow$
$\displaystyle\infty)\rightarrow\mathrm{const}\times\cos(br-\eta\log
2br+\sigma_{\lambda_{-}}-\lambda_{-}\pi/2).$ (84)

The respective total Coulomb phase shifts for Eq. (72) are the phase shifts
for the usual and peculiar solutions over and above those due to any angular
barrier part (absent here). They are given by

$\displaystyle\delta_{\lambda_{\pm}}$
$\displaystyle=\sigma_{\lambda_{\pm}}-\lambda_{\pm}\pi/2,$
$\displaystyle\sigma_{\lambda_{\pm}}$
$\displaystyle=\arg(\Gamma(\lambda_{\pm}+1+i\eta),$ (85)

in which

$\displaystyle\arg\Gamma(\lambda_{\pm}+1+i\eta)$
$\displaystyle=\eta\psi(\lambda_{\pm}+1)+\sum_{n=0}^{\infty}\left(\frac{\eta}{\lambda_{\pm}+1+n}-\arctan(\frac{\eta}{\lambda_{\pm}+1+n})\right),$

with the digamma function given by

$\psi(\lambda_{\pm}+1)=-\gamma+\lambda_{\pm}\zeta(2)-\lambda_{\pm}^{2}\sum_{n=1}^{\infty}\frac{1}{n^{2}(n+\lambda_{\pm})}.$
(87)

The (modified) Coulomb phase shift
$\sigma_{\lambda_{\pm}}$$-\lambda_{\pm}\pi/2$ is that for the Coulomb
$2\varepsilon_{w}A$ plus $-A^{2}$ term alone. (Again, the $\pm$ sign
corresponds to the two sectors $\zeta=\pm 1$, with usual ($+)$ and peculiar
($-)~{}$boundary conditions given in Eq. (34).) Without the $-A^{2}$ term the
phase shift would be simply $\sigma_{0}$.

##  V SOLUTIONS OF THE TWO BODY DIRAC EQUATIONS FOR THE ${}^{3}P_{0}$ STATE

###  V.1  The ${}^{3}P_{0}$ quasipotential

We now consider the case of the ${}^{3}P_{0}$ state of a fermion-antifermion
pair with electric or color charges interacting through an electromagnetic-
type interaction arising from the exchange of a single photon or gluon. As
with the ${}^{1}S_{0}$ state, the single photon annihilation diagram does not
contribute because the ${}^{3}P_{0}$ state is a charge parity even state.
Then, the two terms in Eq. (25) that precede the $\nabla^{2}A$ term precisely
cancel the barrier term $2/r^{2}$ at very short distances to give the equation
for the radial wave function

$\left\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}+\frac{8\pi\alpha
r\delta({\hbox{\boldmath${r}$}})}{wr+2\alpha}\right\\}u=b^{2}u.$ (88)

The cancellation of terms takes place in the following way. In Eq. (25), the
three terms beyond $-A^{2}$ arise from a combination of spin-orbit, spin-spin,
tensor and spin-orbit tensor interactions. From a detailed examination of Eq.
(168) in the Appendix B, we can see that the spin-orbit and tensor terms gives
rise to the first “magnetic interaction” term on the right hand side of Eq.
(168) that has a strongly attractive $-8\alpha/wr^{3}$ attractive part down to
distances on the order of $2\alpha/w$ after which this magnetic interaction
approaches $-4/r^{2}$. The dominance of the attractive magnetic interaction at
short distances that can overwhelm the centrifugal barrier is in agreement
with the simple intuitive classical picture presented in the Introduction. The
second term on the right-hand side of Eq. (168), arising from a combination of
Darwin, spin-spin and tensor terms, has a stronger repulsive
$8\alpha^{2}/w^{2}r^{4}~{}$ part down to distances on the order of $2\alpha/w$
after which it approaches $+2/r^{2}$. Together they tend to exactly cancel the
angular momentum barrier term $+2/r^{2}$ at very short distances. In addition
to the repulsive interaction containing $\delta(\hbox{\boldmath${r}$})$
arising from the assumption that the electron and positron are point
particles, the quasipotential behaves as $-\alpha^{2}/r^{2}$ at short-
distances, separated from the outside long-distance region by a barrier. The
interaction containing the delta function comes from a combination of Darwin,
spin-spin, and tensor terms. Three fourths of the repulsive term containing
$\delta(\hbox{\boldmath${r}$})$ comes from the Darwin piece while one fourth
from the combination of the spin-spin and tensor parts. For brevity of
nomenclature we shall just call it the delta function term.

One of us (HWC) examined in a previous work atk  the effects on bound state
energies due to a repulsive $\delta(\hbox{\boldmath${r}$})$ interaction by
itself, without additional radial dependence. It was found that for wave
functions $\psi$ that do not vanish at the origin and for potentials that are
less singular than $1/r^{2}$, the exact effects on the eigenvalue of including
a repulsive delta function do not agree with the results of perturbation
theory in the limit of weak coupling, when the delta function potential is
modeled as the limit of a sequence of spherically symmetric square wells. In
particular it is shown that the repulsive delta function, viewed as the limit
of square well potentials, produces no effects at all on bound state energies.
In our case here the appearance of the $\delta({\hbox{\boldmath${r}$}})$
potential differs from this reference in two aspects however. First of all the
$\delta({\hbox{\boldmath${r}$}})$ appears in conjunction with
$r/(wr+2\alpha)$, softening its repulsive effects. Secondly, the wave function
$\psi=u/r$ for the solution without the delta function term diverges at the
origin both for what we call the usual solution and what we call the peculiar
solution. If the null effects on bound state energies and phase shifts seen in
atk  should occur in our case as well, this, however, does not lead to a
problem with perturbative agreement with the spectral results.

In the case of weak potentials where the denominator $(wr+2\alpha)$ is
replaced by $wr$, we have shown previously in bckr  that the remaining terms
in Eq. (88) without the delta function term, when treated nonperturbatively,
would produce numerically the same spectral results for the ${}^{3}P_{0}$
state as the inclusion of the repulsive $\delta(\hbox{\boldmath${r}$})$
interaction treated perturbatively. The agreement of the perturbative
treatment with the delta function term for weak coupling with the
nonperturbative treatment containing no delta function term justifies the
first approximate analysis of ignoring the delta function term and treating
the remainder of the equation nonperturbatively in the following subsection.

###  V.2  Usual and Peculiar Solutions for the ${}^{3}P_{0}$ State

The wave equation (88) for the ${}^{3}P_{0}$ state without the delta function
term becomes

$\left\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}u=b^{2}u,$
(89)

with a short distance ($r<<2\alpha/w)$ form

$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{\alpha^{2}}{r^{2}}\right\\}u=0,$ (90)

the same as with the ${}^{1}S_{0}$ states. Thus, the ${}^{3}P_{0}$ states also
have the same types of solutions as the ${}^{1}S_{0}$ states, with radial wave
functions near the origin as given in Eqs. (33)-(34). Thus, there are usual
${}^{3}P_{0}$ states with peculiarity $1$, and peculiar ${}^{3}P_{0}$ states
with peculiarity $-1$.

Note that both the usual and the peculiar solutions $u_{\pm}$ $\sim
r^{(1\pm\sqrt{1-4a^{2}})/2}$ arise from the strong magnetic interaction that
significantly modifies the qualitative behavior of the interaction at short
distances, when the total spin and the orbital angular momentum are oppositely
aligned in the ${}^{3}P_{0}$ state. If the strong magnetic interaction is
absent, the $2/(r+2\alpha/w)^{2}$ term in Eq. (88) would be $2/r^{2}$, and the
wave function near the origin would be

$u_{\pm}=ar^{(1\pm\sqrt{3^{2}-4a^{2}})/2},$ (91)

with

$\psi_{\pm}^{2}d^{3}r=r^{[(1\pm\sqrt{3^{2}-4a^{2}})]}drd\Omega.$ (92)

In that case, as stated below Eq. (35), only the usual $u_{+}$ solution is
quantum-mechanically admissible, while the $u_{-}$ state becomes singular at
short distances. Such a comparison shows that the peculiar solution $u_{-}$ is
not present when there is no strongly attractive magnetic interaction at short
distances or more generally for $J\neq 0.$

###  V.3 The $\delta$ function term and the charge distribution

The discussions in the above subsection pertain to the quasipotential without
the delta function term. We now examine the full Eq. (25) for both the usual
and peculiar solutions with the $\delta({\hbox{\boldmath${r}$}})$ term
included. Consider first the perturbative treatment of taking the interaction
containing $\delta(\hbox{\boldmath${r}$})$ as a perturbation. We evaluate the
expectation value of the interaction term containing
$\delta(\hbox{\boldmath${r}$})$. Even though both usual and peculiar solutions
have a diverging $\psi_{\pm}({\hbox{\boldmath${r}$}})$ near the origin they
each are allowed as a probability amplitude since the probability
$\int_{\Delta V}\left|\psi_{\pm}({\hbox{\boldmath${r}$}})\right|^{2}d^{3}r$
for an arbitrarily small volume $\Delta V$ about the origin would be finite,
in addition to the essential boundary condition $u_{\pm}(0)=0$. With
$\left|\psi_{\pm}({\hbox{\boldmath${r}$}})\right|^{2}$ near the origin having
the behavior of $r^{(-1\pm\sqrt{1-4a^{2}})}$, the expectation value of
$\delta(\hbox{\boldmath${r}$})/(w-2A)$, after performing the angular
integration, is

$\displaystyle\int
d^{3}r\frac{r\delta({\hbox{\boldmath${r}$}})}{wr+2\alpha}\left|\psi_{\pm}({\hbox{\boldmath${r}$}})\right|^{2}$
$\displaystyle\rightarrow$ $\displaystyle\int
d^{3}rr^{\pm\sqrt{1-4a^{2}}}\frac{\delta({\hbox{\boldmath${r}$}})}{wr+2\alpha},$
(93) $\displaystyle\rightarrow$ $\displaystyle\int
drr^{\pm\sqrt{1-4a^{2}}}\frac{\delta(r)}{(wr+2\alpha)},$

which is zero for the plus sign for the usual solution but diverges for the
minus sign for the peculiar solution.

The results of Eq. (93) for the usual solution explains our previous agreement
between (i) the perturbative treatment with the delta function term for weak
coupling and (ii) the nonperturbative treatment without the delta function
term bckr . The agreement arises because in Ref. bckr  we limited our
attention only to the usual solution for which the expectation value of the
delta function term is zero.

The results of Eq. (93) for the peculiar solution indicates that the delta
function term cannot be treated as a perturbation in the present formulation,
as such a treatment will lead to a diverging energy. The delta function term
arises from the charge distribution of the interacting particles, as it is
related to the Laplacian of the gauge field, $\nabla^{2}A$, as given in Eq.
(26). A proper non-perturbative treatment of the problem of the peculiar
solution states requires the knowledge of the wave function at very short
distances. Therefore, it will require not only the knowledge of the structure
of the charge distribution but also the necessary auxiliary interactions at
even shorter distances that are needed to bind the charge elements of the
distribution together. The auxiliary interactions will affect the solutions of
the two-body wave functions at very short distances and the states of the
peculiar solution. At the present moment, we have little knowledge of the
structure of elementary charges, much less the auxiliary forces that would
bind the charge distribution together at very short distances.

The structure of the charge distribution of elementary particles at very short
distances is basically an experimental question. As the strong magnetic
interaction allows the two interacting particles to probe the short-distance
region, it is therefore useful to investigate quantities that may reveal
information on the structure of the charge distribution. While many
possibilities can be opened for examination, we shall examine the following
possibilities in the present manuscript:

(i) We shall first examine the case in which the (unknown) auxiliary
interaction that binds the charge elements of the elementary particles
together and the repulsive interaction arising from the charge density
$\rho(r)$ counteract in such a way that the total interaction at short
distances would still be dominated by the $-\alpha^{2}/r^{2}$ term. Under such
a circumstance, the effects of the auxiliary interaction would cause the delta
function term term in Eq. (88) to make no contribution at short distances.
Keeping the dominant terms, the equation of motion for the wave function
becomes Eq. (89) without the delta function term. It also must be recognized
that for the usual solution, the perturbative effect of the delta function
term (in which we ignore the effect or the potential in the denominator
$w-2A$) is accounted for by a nonperturbative (numerical) treatment of the
entire $\Phi$ without the delta function term. So, our treatment of the delta
function term in this case parallels that used in our earlier spectroscopy
calculations bckr .

(ii) We examine subsequently the case when the auxiliary interaction that
holds the charge element together leaves the gauge field $A(r)$ unchanged
while the delta function term in Eq. (88) is modified by treating the delta
function as the limit of a set of Gaussian distributions with different
widths.

(iii) We examine two additional models completely within QED (or QCD) with an
assumed basic charge distribution that generates the gauge field also in the
region interior to the charge distribution. However, the auxiliary
interactions that hold the charge together and that can interact with the
other antifermion are altogether neglected. It should be recognized that
within pure QED (or QCD), with the neglect of the auxiliary interactions that
hold the charge elements together, the charge distribution cannot be a stable
configuration.

In the next section we describe the method we use to indicate the presence or
absence of a resonance in the ${}^{3}P_{0}$ system.

##  VI PHASE SHIFT ANALYSIS

In our study of the ${}^{3}P_{0}$ state for both the usual and peculiar
solutions, we wish to find out whether or not there is an energy that will
lead to a $\pi/2$ phase shift for a given $\alpha$ and constituent mass $m$ .
Equation (28) for the ${}^{3}P_{0}$ state is a Schrödinger-like equation of
the form

$\left\\{-\frac{d^{2}}{dr^{2}}+\frac{L(L+1)}{r^{2}}+\Phi(r)\right\\}u(r)=b^{2}u(r).$
(94)

We calculate the phase shift for this problem by the variable phase method of
Calogero cal . We first describe this method generally (see Appendix E for a
more detailed review) and then later in this section describe its application
to the ${}^{3}P_{0}$ state. We take $W(r)$ to include not only the
quasipotential $\Phi(r)$ but also the angular momentum barrier.

$W(r)=\frac{L(L+1)}{r^{2}}+\Phi(r).$ (95)

Thus our equation has the form

$\left\\{-\frac{d^{2}}{dr^{2}}+W(r)\right\\}u(r)=b^{2}u(r).$ (96)

The Calogero method relies on introducing a reference potential $\bar{W}(r)$
that can be solved exactly, with two independent solutions $u_{1}$ and
$u_{2}$,

$\left\\{-\frac{d^{2}}{dr^{2}}+\bar{W}(r)\right\\}u_{i}(r)=b^{2}u_{i}(r),~{}i=1,2.$
(97)

There are many ways to choose the reference potential $\bar{W}(r)$. To display
the general idea, we consider the case in which $W(r)$ is short range. In that
case the phase shift $\delta_{L}$ is defined by

$u(r\rightarrow\infty)\rightarrow\sin(br-L\pi/2+\delta_{L}).$ (98)

The Calogero method uses two different types of $\bar{W}(r)$. In the first,
$\bar{W}(r)\equiv\bar{W}_{I}(r)$, the reference potential has the same long
and short distance behavior as $W(r)$. In the second $\bar{W}(r)\equiv
W_{II}(r)$, the reference potential does not have the same long and short
distance behavior as $W(r)$ but is especially simple.

We consider first Type I reference potential, $\bar{W}_{I}(r)=L(L+1)/r^{2}$,
the angular momentum barrier potential, for which the reference wave functions
$\bar{u}_{1}(r)$ and $\bar{u}_{2}(r)$ are the well known spherical Bessel
functions $\hat{\jmath}_{L}(br)$ and $\hat{n}_{L}(br)$ cal . The solution
$\bar{u}_{1}(r)$ is taken to be the regular solution, having the same short
distance behavior as $u(r),$ in particular, $\bar{u}_{1}(r\rightarrow 0)=0.$
The solution $\bar{u}_{2}(r)$ is taken to be the irregular solution,
$\bar{u}_{2}(r\rightarrow 0)\neq 0$. Those functions together with their long
distance behaviors are given by

$\displaystyle\bar{u}_{1}(r)$
$\displaystyle=\hat{\jmath}_{L}(br)\rightarrow\mathrm{const}\sin(br-L\pi/2),$
$\displaystyle\bar{u}_{2}(r)$
$\displaystyle=-\hat{n}_{L}(br)\rightarrow\mathrm{const}\cos(br-L\pi/2).$ (99)

We introduce the amplitude function $\alpha(r)$ and phase shift function
$\delta_{L}(r)$ to represent the wave function solutions for the $W(r)$
potential, $u(r)$ and $u^{\prime}(r)$, as

$\displaystyle u(r)$
$\displaystyle=\alpha(r)(\cos\delta_{L}(r)\bar{u}_{1}(r)+\sin\delta_{L}(r)\bar{u}_{2}(r)),$
$\displaystyle u^{\prime}(r)$
$\displaystyle=\alpha(r)(\cos\delta_{L}(r)\bar{u}_{1}^{\prime}(r)+\sin\delta_{L}(r)\bar{u}_{2}^{\prime}(r)).$
(100)

This leads to the following equation for the phase shift function (see
Appendix E)

$\tan\delta_{L}(r)=\frac{\bar{u}_{1}^{\prime}(r)u(r)-\bar{u}_{1}(r)u^{\prime}(r)}{(\bar{u}_{2}(r)u^{\prime}(r)-\bar{u}_{2}^{\prime}(r)u(r))}.$
(101)

Further manipulations lead to the differential equation for $\delta_{L}(r)$
given by

$\delta_{L}^{\prime}(r)=-\frac{[W(r)-\bar{W}(r)]}{b}\biggl{[}\hat{\jmath}_{L}(br)\cos\delta_{L}(r)-\hat{n}_{L}(br)\sin\delta_{L}(r)\biggr{]}^{2},$
(102)

To find the connection to the phase shift $\delta_{L}$ note that from Eq. (
100) and (99),

$\displaystyle u(r$
$\displaystyle\rightarrow\infty)=\mathrm{const}\left\\{\cos\delta_{L}(r\rightarrow\infty)\sin(br-L\pi/2)+\sin\delta_{L}(r\rightarrow\infty)\cos(br-L\pi/2)\right\\}$
$\displaystyle=\mathrm{const}\times\sin(br-L\pi/2+\delta_{L}(\infty)),$ (103)

and so comparison with (98) gives the solution of the phase shift $\delta_{L}$
for the $W(r)$ potential as

$\delta_{L}=\delta_{L}(\infty).$ (104)

Thus, the second order linear different equation becomes a first order non-
linear equation whose solution at $r\rightarrow\infty$ gives the phase shifts
of the scattering problem with the $W(r)$ effective potential. The boundary
condition of $\delta_{L}(0)=0$ follows from Eq. (101) when one chooses
$\bar{u}_{1}(r)$ to have the same behavior as $u(r)$ as $r\rightarrow 0.$

We consider next type II of the short-range reference potentials
$\bar{W}_{II}(r)$ which do not need to have the same long distance behavior as
$W(r)$ as long as the Schrödinger Eq. (97) containing the reference potential
$\bar{W}_{II}(r)$ has an exact solution. For example we may choose
$\bar{W}_{II}(r)=0.$ Then the two exact reference solutions of Eq. (97) are
simply

$\displaystyle\bar{u}_{1}(r)$ $\displaystyle=\sin(br),$
$\displaystyle\bar{u}_{2}(r)$ $\displaystyle=\cos(br).$ (105)

One defines a phase shift function $\gamma_{L}(r)$ as in equation (100) so
that

$\displaystyle u(r$
$\displaystyle\rightarrow\infty)=\mathrm{const}\times\\{\cos\gamma_{L}(r\rightarrow\infty)\sin(br)+\sin\gamma_{L}(r\rightarrow\infty)\cos(br)\\}$
$\displaystyle=\mathrm{const}\times\sin(br+\gamma_{L}(\infty)).$ (106)

Comparison with (98) gives

$\delta_{L}=\gamma_{L}(\infty)+\frac{L\pi}{2}.$ (107)

Since the angular momentum barrier is excluded from the equations for
$\bar{u}_{i}(r)$ one finds that the phase shift equation for integrating the
phase shift function $\gamma_{L}(r)$ includes the repulsive barrier term in
$W$ [Eq. (34)],

$\displaystyle\gamma_{L}^{\prime}(r)$
$\displaystyle=-\frac{W(r)}{b}\biggl{[}\cos\gamma_{L}(r)\sin(br)+\sin\gamma_{L}(r)\cos(br)\biggr{]}^{2}$
$\displaystyle=-\frac{W(r)}{b}\sin^{2}(br+\gamma_{L}(r)).$ (108)

Note that because of the $L(L+1)/r^{2}$ behavior of
$W(r)-\bar{W}_{II}(r)(r)=W(r)$, which dominates at large distances, one will
have to integrate quite far to obtain convergence for $\gamma_{L}(r)$.111111
Alternatively Calogero gives a formula for avoiding integrating to large
distances to build up a centrifugal phase shift. (See cal , p 92). For this
case of $\bar{W}_{II}(r)=0$, one has an equation similar to (101) with
$\delta_{L}(r)$ replaced by $\gamma_{L}(r)$. Thus even though $\bar{u}_{1}(r)$
has a different behavior than $u(r)$, we still have the boundary condition
$\gamma_{L}(0)=0.$ Eq. (107) compensates for the $-L\pi/2$ effective phase
shift due to the barrier term in $W(r)$ in Eq. ( 108). We also have the
additional boundary condition of (see Appendix E)

$\gamma_{L}^{\prime}(0)=-\frac{bL}{L+1}.$ (109)

We now turn our attention to the ${}^{3}P_{0}$ system, in particular Eq. (28)
for a general $\Phi(r)$. In this application of the Calogero method we choose
a reference potential $\bar{W}(r)\equiv\bar{W}_{III}(r)$ that in a sense is a
hybrid of the two types of reference potentials considered above. Since Eq.
(28) contains a long range Coulomb interaction $-2\varepsilon_{w}\alpha/r$ we
must include that interaction into our choice for $\bar{W}_{III}(r)$. If it
did not have the same behavior as $W(r)$ at large distances we would have to
have a way of subtracting an infinite Coulomb phase shift, $\log 2br$ . So, in
this way our application is similar to the first type $\bar{W}_{I}$ (r)above.
We also include the $-\alpha^{2}/r^{2}$ term in $\bar{W}_{III}(r)$ because as
seen in Eqs. (89), (90) and (34) the solution displays the desired short
distance peculiar as well as usual behaviors. We do not include the angular
momentum barrier term $2/r^{2}$ however, as this would prohibit a treatment of
the peculiar solution (see comments below Eq. (92)). Thus we choose

$\displaystyle\bar{W}_{III}(r)$ $\displaystyle=$
$\displaystyle-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}},$
$\displaystyle W(r)$ $\displaystyle=$ $\displaystyle\frac{2}{r^{2}}+\Phi(r),$
(110)

where $\Phi(r)$ is given in Eq. (29). In this way $\bar{W}_{III}(r)$ has some
similarities to the second type $\bar{W}_{II}(r)$ discussed above. Our choice
for $\bar{W}_{III}(r)$ permits the two exact solutions
$\bar{u}_{1}(r),\bar{u}_{2}(r)$ of Eq. (97) which becomes that of the
${}^{1}S_{0}$ state

$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w(r)}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}\bar{u}=b^{2}\bar{u}.$
(111)

Now to determine the phase shift for the actual ${}^{3}P_{0}$ state we return
to the conditions defined in Eq. (110). Then the full solution has the
asymptotic form

$u(r\rightarrow\infty)\rightarrow\mathrm{const}\times\sin(br-\eta\log
2br+\sigma_{1}-\pi/2+\delta_{1}).$ (112)

The appearance of $\sigma_{1}$ and $\delta_{1}$ includes the effects of the
angular momentum barrier term $2/r^{2}$ in the presence of the Coulomb
interaction. In Appendix E, using

$\displaystyle\bar{u}_{1}(r)$ $\displaystyle=$ $\displaystyle
F_{\lambda}(\eta,br),$ $\displaystyle\bar{u}_{2}(r)$ $\displaystyle=$
$\displaystyle G_{\lambda}(\eta,br),$ (113)

we show that the full ${}^{3}P_{0}$ phase shift $\delta$ is given by

$\delta=\delta_{1}+\sigma_{1}=\gamma_{\pm}(\infty)+\sigma_{\lambda_{\pm}}+(1-\lambda_{\pm})\pi/2,$
(114)

where (in analogy to the proof of Eq. (108) with $\bar{W}\neq 0)~{}$
$\gamma_{\pm}(r)$satisfies the nonlinear equation

$\gamma_{\pm}^{\prime}(r)=-\frac{W(r)-\bar{W}_{III}(r)}{b}\biggl{[}\cos\gamma_{\pm}(r)F_{\lambda_{\pm}}+\sin\gamma_{\pm}(r)G_{\lambda_{\pm}}\biggr{]}^{2},$
(115)

subject to the boundary condition that $\gamma_{\pm}(0)=0$ (see Appendix E).
The functions $F_{\lambda_{\pm}}$ and $G_{\lambda_{\pm}}$ are the regular and
irregular Coulomb wave functions corresponding to the negative effective
centrifugal barrier $-\alpha^{2}/r^{2}$. Again, because of the $2/r^{2}$
behavior of $W(r)-\bar{W}_{III}(r)$ which takes over at large distances, one
will have to integrate quite far to obtain convergence for $\gamma_{\pm}(r).$

We consider numerical solutions for both the usual solution with
$\lambda_{+}=(-1+\sqrt{1-4\alpha^{2}})/2$, and the peculiar solution, with
$\lambda_{-}=(-1-\sqrt{1-4\alpha^{2}})/2.~{}$In the next section we discuss
the results obtained in the numerical integration of the phase shift equation
( 115) for different behaviors at very short distances.

##  VII NUMERICAL RESULTS FOR ${}^{3}P_{0}$ RESONANCES

###  VII.1  The case without the delta function term

With the above general formalism, we can begin to examine states in the
quasipotential of Eq. (88) first without the delta function term. The
Schrödinger equation for the ${}^{3}P_{0}$ state becomes Eq. (89). In order to
gain an idea on the attractive magnetic interaction at short distances for
this ${}^{3}P_{0}$ state, we plot in Fig. 2 the corresponding quasipotential
including the angular momentum barrier,

$W(r)=\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\epsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}},$

for $w=27.85$ MeV, $\alpha=1/137$ and a constituent electron mass of 0.511
MeV. One observes that at short distances $W(r)$ becomes very attractive and
behaves as $-\alpha^{2}/r^{2}$. There is a barrier in the region between 10-2
to 10-1 GeV-1. Such a potential becomes singular at $r\rightarrow 0$ when
$\alpha$ exceeds 1/2 Case .

Figure 2: The effective potential
$W(r)=2/(r+2\alpha/w)^{2}-2\epsilon_{w}\alpha/r-\alpha^{2}/r^{2}$ for the for
the $(e^{+}e^{-})$ system in the ${}^{3}P_{0}$ state with $\alpha=1/137$ and
$w=27.85$ MeV.

We calculate the phase shift as a function of energy using the boundary
condition $\gamma_{\pm}(0)$=0, including the dependence of the potential as a
function of energy. For the usual solution ($\zeta=+1)$, our results for the
QED $e^{-}e^{+}$ system in the ${}^{3}P_{0}$ state with $\alpha=1/137$ and
$m=0.511$ MeV show no evidence whatsoever for resonances for all c.m. energies
tested (from about 1 MeV to about 100 MeV). The magnitude of the phase shifts
are of the order of $\pi/100$.

Table 1:  Variation of the resonant energy as a function of the quark mass for a fixed $\alpha_{s}=0.11$. quark | mass | $w_{R}$ |  |
---|---|---|---|---
up |  3 MeV | 27 MeV |  |
down |  5 MeV | 45 MeV |  |
strange | 135 MeV | 1220 MeV |  |
charm |  1.5 GeV | 13.6 GeV |  |
bottom |  4.5 GeV | 40.8 GeV |  |
top |  175 GeV | 1590 GeV |  |

For the peculiar solution ($\zeta=-1)~{}$with the wave functions starting with
a less positive slope, the attraction at short distances is able to bend the
wave function downward to result in a very sharp resonance at about $27.85$
MeV. In Figure 3(a) we plot the phase shift $\delta=\delta_{1}+\sigma_{1}$ as
a function of the c.m. energy $w$ and $\sin^{2}\delta$ versus $w$ in Fig. 3
(b). We start the integration at the origin and extend to about 1 angstrom. As
one observes, the phase shift undergoes a transition from near zero to $\pi$.
The resonance has a full width at half maximum of 15 KeV. We also include a
plot of the wave function in Fig. 4 from the origin up to about 1000 GeV-1.
The wave function rises as $r^{(1-\sqrt{1-4\alpha^{2}})/2}$ near the origin,
and appears nearly flat at $r\sim 10^{-3}$ GeV-1, and it slowly decreases near
the barrier. It oscillates when it emerges from the barrier at $r\sim 2\times
10^{-2}$ GeV-1.

Figure 3:  The phase shift as a function of $w$ for the $(e^{+}e^{-})$ system
with $\alpha=1/137$ and $w=27.85$ MeV. Figure 4:  The wave function $u$ of the
peculiar resonance at $w=27.85$ MeV for $\alpha=1/137$ and $m=0.511$ MeV.

Having observed a resonance for the QED interaction with the $e^{+}$ and
$e^{-}$ constituents, we turn our attention to quarks and antiquarks
interacting with a color-coulomb type interaction with an effective coupling
constant $\alpha_{s}$. We focus here only on the $\zeta=-1$ sector. In the
color-singlet $(q\bar{q})$ states of interest, the effective interaction is
then $\alpha_{\mathrm{eff}}=4\alpha_{s}/3$. To get an idea of the order of
energy for these quark-antiquark two-body resonance states, we calculate the
resonance energies for the typical case of $\alpha_{s}=0.11$ For this value,
the resonance energy varies nearly linearly with quark mass. The largest
energy resonances occur with the largest quark masses. In Table I we present
the resonance energies $w_{R}$ for the families of quarks from the up quark to
the top quark. It should be pointed out that these resonance values take into
account only the Coulomb-like portion of $A(r)=-(4/3)\alpha_{s}/r$ and ignores
any affects on the resonance values of the confining part of the potential.

To examine how the resonance energies varies with the coupling constant, we
have found that for fixed mass (e.g. 0.511 MeV) the resonance energy $w_{R}$
increases as the coupling parameter  decreases  until the coupling constant
gets to be on the order of $0.01$, when $w_{R}$ starts decreasing again.

###  VII.2  The case of representing the delta function by a Gaussian function

For the second case for the ${}^{3}P_{0}$ state given in Eqs. (88) and ( 96)
using (115), we take

$W(r)=\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}+\frac{8\pi\alpha
r\delta({\hbox{\boldmath${r}$}})}{\left(wr+2\alpha\right)},$ (116)

in which we model the three dimensional delta function by

$\delta({\hbox{\boldmath${r}$}})\rightarrow\delta_{\sigma}({\hbox{\boldmath${r}$}})=\frac{\exp(-r^{2}/2\sigma^{2})}{(2\pi)^{3/2}\sigma^{3}}.$
(117)

Table 2: Variation of resonance energy with the width of the Gaussian distribution $\sqrt{2}\sigma(\mathrm{fm})$ |  $w$(GeV)
---|---
$1000$ | $0.0279$
$100$ | $0.0278$
$10$ | $0.0279$
$1$ | $0.0398$
$0.1$ | $0.314$
$0.01$ | $3.13$
$0.001$ | $31.3$
$0.0001$ | $313$
$0.00001$ | $3130$
$0.000001$ | $31300$
$0.0000001$ | $313000$

In this treatment, we keep the point charge source term for the $A(r)$ so that
$A(r)=-\alpha/r$. What we are attempting to do is just present a mathematical
representation of the delta function that will allow a numerical solution. The
reference potential $\bar{W}_{III}(r)$ is the same as without the delta
function. With this modeling of the delta function we start off our Runge-
Kutta integration of Eq. (115) with $\gamma_{-}(0)=0$ since
$W(r)-\bar{W}_{III}(r)$ is $w^{2}2\alpha^{2}$ at the origin just as without
the delta function term. The function $\delta_{\sigma}$ does not alter the
extreme short distance behavior since it is multiplied by $r$ and vanishes at
the origin. We obtain the resonance energy results as given in Table II. It is
obvious that for small $r_{0}$ we obtain a limiting behavior of $w=3.13$ GeV-
fm/$\sqrt{2}\sigma$. There is however, a difference between what we are doing
here and what was done in atk . There the delta function was just regarded as
given, not related to other parts of the potential. Here that is not the case.
The delta function arose from the Laplacian of $A(r).$ There may therefore be
some ambiguity of, in effect, modeling $\nabla^{2}A$ in one part of the
quasipotential while leaving $A(r)$ unaffected in the other part. That leads
us then to the third case.

###  VII.3  The case of representing the charge distribution by a continuous
function

In Eq. (25), if one replaces $A(r)$ by

$A=\left(\frac{\alpha}{r}-\frac{\alpha}{r_{0}}\right)\frac{1}{1+\exp\\{(r-r_{0})/\delta
r_{0}\\}}-\frac{\alpha}{r},$ (118)

or alternatively as

$A(r)=\begin{cases}\frac{\alpha
r^{2}}{2r_{0}^{3}}-\frac{3\alpha}{2r_{0}}&\mathrm{~{}for~{}}r\leq
r_{0}\cr-\frac{\alpha}{r}&\mathrm{~{}for~{}}r\geq r_{0}.\cr\end{cases}.$ (119)

then our numerical solutions show that there is no ${}^{3}P_{0}$ resonance for
the peculiar solution for both cases, resulting in a phase shift of $\pi$ all
the way down to threshold ($w=2m$). Both of these corresponds to smeared
charge distributions from $\nabla^{2}A$ but neither have auxiliary
interactions at short distances that would bind the elements of the charge
distribution together. The reason no resonance is produced in this case is
that in the interior of the charge distribution ($r<r_{0}$), the angular
momentum barrier in Eq. (25) comes out from under the dominance of the
magnetic interaction terms as $A(r)$ tends to a finite constant. By following
steps similar to those used to determine $\gamma_{-}(0)$ in Appendix E for the
point charge one can show this results in an initial value for $\gamma_{-}(0)$
defined by

$\tan\gamma_{-}(0)=\tan x(\lambda,\eta).$ (120)

This is positive and even though small ($\sim 0.007$) is large enough to
prevent the formation of a resonance. Note that this differs from the previous
section in that here we are giving a physical connection
$\nabla^{2}A=4\pi\alpha\rho(\hbox{\boldmath${r}$})$ between the smeared delta
function and the invariant potential $A(r)$, whereas in the previous section
we simply mathematically modeled the delta function in isolation.

##  VIII DISCUSSION AND CONCLUSION

Magnetic interactions in the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states are very
attractive and singular at short distances. In the two-body Dirac equation
formulated in constraint dynamics, the magnetic interactions lead to
quasipotentials that behave as $-\alpha^{2}/r^{2}$ near the origin and admit
two different types of states. At short distances, the radial wave functions
$u(r)$ of the usual states, grow as $r^{\lambda+1}$, while the radial wave
functions of the peculiar states grow as $r^{-\lambda}$, where
$\lambda=(-1+\sqrt{1-4\alpha^{2}})/2$. They have drastically different
properties.

The existence of usual and peculiar states for the same fermion-antifermion
system poses conceptual and mathematical problems. If we keep both sets of
states in the same Hilbert space, then each set is complete by itself, but the
two sets of states are not orthogonal to each other. Our system is thus over-
complete. Furthermore, the matrix element of $H$ (the scaled invariant mass
operator for these states) between states of one type and state of the other
type are not symmetric and the $H$ operator is not self-adjoint.

Given our quasipotential of the type $-\alpha^{2}/r^{2}$ at short distances
for the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states, both the usual and peculiar
states are physically admissible. There do not appear to be compelling reasons
to exclude one of the two sets as being unphysical, if one is given the
attractive interaction $-\alpha^{2}/r^{2}$ near the origin as it is. It is
desirable to find ways to admit both types of states as physical states while
maintaining the self-adjoint property of the mass operator and the
completeness property of the set of basis states.

We are therefore motivated to introduce a quantum number $\zeta$, which we
call “peculiarity”, to specify the usual or peculiar properties of a state.
The peculiarity quantum number $\zeta$ is 1 for usual states which have
properties the same as those one usually encounters in QED and QCD. The
peculiarity quantum number is $-1$ for peculiar states which intrudes into the
physical region, when the interaction near the origin becomes very attractive,
such as the $\lambda(\lambda+1)/r^{2}$ interaction with
$-1/4\leq\lambda(\lambda+1)<0$. The introduction of the peculiarity quantum
number enlarges the Hilbert space, makes the mass operator self-adjoint, and
the enlarged physical basis states containing both usual and peculiar states
in a complete set. It is also clear from our discussions that to maintain the
self-adjoint property of the mass operator and to have a single complete set,
the presence of the peculiarity quantum number will be a general phenomenon,
when the mass operator contains very attractive interactions at short
distances such that there are more than one set of eigenstates satisfying the
boundary conditions at the origin.

It should be emphasized that the quasipotential $-\alpha^{2}/r^{2}$ has been
obtained under the assumption of a point fermion and a point antifermion for
which the gauge field potential between them is $A(r)=-\alpha/r$. The point
nature of an electron may be a good experimental concept as the lower limits
on the QED cut-off parameter $\Lambda_{\mathrm{cut}}$ with the present day
high energy accelerators exceeds the value of 250 GeV, suggesting that the
electron, muon, and tauon, behave as point particles down to 10-3 fm. The
asymptotic freedom is a good description for the interaction of quarks at
short distances. It may appear that point charge particles may be a reasonable
description. On the other hand, a finite structure of the electron or quarks
may modify significantly the short-distance attractive interactions so
substantially that the peculiar states may be pushed out of existence. The
experimental search of the peculiar states, which follows from the point
charge potential, can provide a probe of the point nature of these particles
and the interaction at short distances.

Our first focus on the attractive magnetic interaction is for the
${}^{1}S_{0}$ states, where the spins of the fermion and antifermion of
opposite electric or color charges are oppositely aligned. The usual bound
${}^{1}S_{0}$ states possess attributes the same as those one usually
encounters in QED and QCD, with bound state energies explicitly agreeing with
the standard perturbative results through order $\alpha^{4}$. In contrast, the
peculiar bound ${}^{1}S_{0}$ states, yet to be observed, not only have
different behaviors at the origin, but also distinctly different bound state
properties (and scattering phase shifts). For the peculiar ${}^{1}S_{0}$
ground state of a fermion-antifermion pair with fermion rest mass $m$, the
root-mean-square radius is approximately $1/m$, binding energies approximately
$(2-\sqrt{2})m$, and a rest mass approximately $\sqrt{2}m$. On the other hand,
the $(n+1)$${}^{1}S_{0}$ peculiar state with principal quantum number $(n+1)$
is nearly degenerate in energy and approximately equal in size with the
$n$${}^{1}S_{0}$ usual states.

Our second focus is for the ${}^{3}P_{0}$ state where the total spin and the
orbital angular momentum are oppositely aligned. The magnetic interaction
overwhelms the centrifugal repulsion at short distances and the wave function
admits a peculiar solution that grows with radial distances as $u\sim
r^{(1-\sqrt{1-4\alpha^{2}})/2}$. The particle charge density $\rho(r)$ and
auxiliary interactions that bind the charge elements together can be exposed
for scrutiny. As the structures of elementary particles are basically
experimental questions, it is useful to utilize the magnetic interaction to
probe such charge distributions at very short distances. While many
possibilities can be opened for examination, we have investigated only a few
possibilities in the present manuscript.

The ${}^{3}P_{0}$ quasipotential contains a term proportional to
$\delta(\hbox{\boldmath${r}$})$ . As the delta function term does not
contribute to the usual QED ${}^{3}P_{0}$ bound state energies, it was
plausible to ignore it as one of our explored possibilities. In that case, we
find that there is a magnetic ${}^{3}P_{0}$ resonance at 27.85 MeV for the
peculiar solution of the $(e^{+}e^{-})$ system. For various $(q\bar{q})$
systems of different flavors, we find magnetic ${}^{3}P_{0}$ resonances at
energies of the peculiar solution ranging from many tens of MeV to thousands
of GeV. It is interesting to note that these ${}^{3}P_{0}^{++}$ resonances
have the same quantum number as the vacuum.

In another one of our explored possibilities, if we mathematically model the
delta function at short distances by a sequence of Gaussians of different
widths without changing the gauge field $A(r)=-\alpha/r,$ then a completely
different behavior for the resonance energies ensues as they occur at
different energies, depending on the width of the Gaussian. In the third of
our explored possibilities, if we replace the delta function by a charge
distribution that also alters the gauge field $A(r)$, we obtain no resonance
at all.

Because of 1) the limited knowledge of the unknown auxiliary interactions and
charge distributions at very short distances, not to mention possible
alterations on the angular momentum barrier itself, and 2) the ambiguity of
treating the delta function in isolation nonperturbatively, and 3) the fact
that the delta function term does not contribute to the ${}^{3}P_{0}$ usual
bound state solution, we speculate that the first case may provided a more
reliable representation of the physics. It furthermore makes a clear
prediction of a QED resonance in a region that has not been investigated.

While we have studied the resonance ${}^{3}P_{0}$ states, future work calls
for the investigation of possible ${}^{3}P_{0}$ peculiar bound states where
the attractive interaction near the origin may allow the formation of bound
states. The presence of a delta function repulsion at the origin will also
lead to difficulties and problems similar to the ones we encounter here with
the ${}^{3}P_{0}$ peculiar resonances.

Fermion-antifermion states as we know them experimentally belong to the usual
states. Peculiar states have not been observed. Can the peculiar states be
observed? How do the usual and peculiar states interplay between them? Will
there be transitions between the usual states and peculiar states? Clearly,
the stability of peculiar states first and foremost depends on the strong
attraction near the origin, which in turn depends on the point-like nature of
the elementary particles. As we discussed earlier, substantial modification of
the attractive interaction at the origin may push the peculiar states out of
existence. Only for interactions with sufficient attraction at the origin can
the peculiar states be pulled into existence and appear as eigenstates in the
physically acceptable sheet, with non-singular radial wave functions at the
origin. This is true for both ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states. From
such a perspective, we expect that interactions at short distances have
important bearings on the existence or non-existence of the peculiar states,
and presumably also on the transition between the usual and peculiar states.
However, the interactions at short distances that may allow the peculiar
states to be stable and may effect transitions between states with different
peculiarity quantum numbers (flipping the peculiarity spinor) are not yet
known. They can only be obtained by careful experimental investigations. The
first task of such investigations should be to locate these peculiar states in
high-energy experiments where interactions at short-distance may be involved
and these strong interactions at short distances may lead us to probe short-
distance transition from the usual to the peculiar states. These new
${}^{1}S_{0}$ peculiar bound states correspond to a very tightly bound state
and a set of $(n+1$)th excited states nearly degenerate with the $n$th usual
states. It will also be of interest to search for these states as a result of
some tunneling process between the usual and peculiar states, relying on the
small probability of the usual states to explore short-distance regions where
the interaction at short distances may induce a transition from a usual state
to a peculiar state. The fact that peculiar states of $(n+1)$th${}^{1}S_{0}$
state is nearly degenerate with the usual $n$th ${}^{1}S_{0}$ state may
facilitate such a tunneling transition. Whether or not these quantum-
mechanically acceptable resonances correspond to physical states remains to be
further investigated. Future experimental as well as theoretical work on this
interesting topic will be of great interest in shedding light on the question
whether magnetic bound states and resonances play any role in the states of
fermion-antifermion systems.

Future work should include the effects of the weak interactions, in particular
the exchange of the $Z^{0}$ boson. Since the mass of the $Z^{0}$ is about 92.5
GeV the range is on the order of $10^{-2}$ GeV-1. The exchange of this
particle corresponds to not only a vector interaction but also a pseudovector
interaction. The coupling corresponding to the vector portion is wein

$\displaystyle e^{\ast}$ $\displaystyle\equiv$
$\displaystyle+\frac{g^{2}-g^{\prime 2}}{4\sqrt{g^{2}+g^{\prime
2}}}+\frac{g^{\prime}}{2},$ $\displaystyle g$ $\displaystyle=$
$\displaystyle-\frac{e}{\sin\theta},$ $\displaystyle g^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{e}{\cos\theta},$ (121)

and so

$\displaystyle e^{\ast}$ $\displaystyle=$ $\displaystyle
e[\frac{\frac{1}{\sin^{2}\theta}-\frac{1}{\cos^{2}\theta}}{4\sqrt{\frac{1}{\sin^{2}\theta}+\frac{1}{\cos^{2}\theta}}}-\frac{1}{2\cos\theta}]$
(122) $\displaystyle=$ $\displaystyle e[\frac{\cos 2\theta}{2\sin
2\theta}-\frac{1}{2\cos\theta}].$

With $\sin^{2}\theta\sim 0.23$ we find that

$e^{\ast}\sim-0.25e$ (123)

so, its coupling appears with the same sign as that of the photon. Since
$\alpha^{\ast}=e^{\ast 2}\sim 0.063$ Its effect should be small but not
negligible. There is also the question of the effects of the pseudovector
interaction, not discussed in this appendix but in jmath ,long .

Finally, there are however important mathematical and conceptual issues
associated with these two-body fermion-antifermion system at short distances
that require future careful considerations. In standard QED theory, the charge
and mass of a single charged object due to vacuum polarization and self energy
corrections need to be renormalized or regularized to render them finite for
comparison with observables. For the case with two-body magnetic bound and
resonance states, for example, how are the two-body Green’s functions
regularized, with internal lines off mass shell in a way that reflects the
Dirac constraints? How do such regularizations modify the short distance two-
body interaction? Can the regularization affects the magnetic interaction at
short distances so substantially that the peculiar states no longer survive to
intrude into the physical states? Are these peculiar states stable against
fluctuation of the vacuum in quantum field theory. These are some of the many
interesting questions associated with the two-body problem raised by the
possibility of magnetic states under consideration.

##  Appendix A Details of the equivalent Relativistic Schrödinger Equation

###  A.1
Connections between TBDE and the equivalent Relativistic Schrödinger equation
[Eq. (17)]

Here we present an outline of some details of Eq. (14) and its Pauli-
Schrödinger reduction given in full elsewhere (see cra87 ; jmath ; long ; liu
).This appendix and the one following it are specializations of Appendices A
and B given in tmlk . Each of the two Dirac equations in (14) has a form
similar to a single particle Dirac equation in an external four-vector and
scalar potential but here acting on sixteen component wave function $\Psi$
which is the product of an external part being a plane wave eigenstate of
$P~{}$multiplying the internal wave function $\psi$

$\psi=\begin{bmatrix}\psi_{1}\\\ \psi_{2}\\\ \psi_{3}\\\
\psi_{4}\end{bmatrix}.$ (124)

The four $\psi_{i}$ are each four-component spinor wave functions. To obtain
the actual general spin dependent forms of those $\tilde{A}_{i}^{\mu}$
potentials (including scalar interactions in general) which were required by
the compatibility condition $[\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0$ was a
most perplexing problem, involving the discovery of underlying supersymmetries
in the case of scalar and time-like vector interactions cra82 ,cra87 .
Extending those external potential forms to more general covariant
interactions necessitated an entirely different approach leading to what is
called the hyperbolic form of the TBDE. Their most general form for compatible
TBDE is

$\displaystyle\mathcal{S}_{1}\psi$
$\displaystyle=(\cosh(\Delta){\hbox{\boldmath${S}$}}_{1}+\sinh(\Delta){\hbox{\boldmath${S}$}}_{2})\psi=0\mathrm{,}$
$\displaystyle\mathcal{S}_{2}\psi$
$\displaystyle=(\cosh(\Delta){\hbox{\boldmath${S}$}}_{2}+\sinh(\Delta){\hbox{\boldmath${S}$}}_{1})\psi=0,$
(125)

where $\Delta$ represents any invariant interaction singly or in combination.
It has a matrix structure in addition to coordinate dependence. Depending on
that matrix structure we have either covariant vector, scalar or more general
covariant tensor interactions jmath . The operators
${\hbox{\boldmath${S}$}}_{1}$ and ${\hbox{\boldmath${S}$}}_{2}$ are auxiliary
constraints satisfying

$\displaystyle{\hbox{\boldmath${S}$}}_{1}\psi$
$\displaystyle\equiv(\mathcal{S}_{10}\cosh(\Delta)+\mathcal{S}_{20}\sinh(\Delta)~{})\psi=0,$
$\displaystyle{\hbox{\boldmath${S}$}}_{2}\psi$
$\displaystyle\equiv(\mathcal{S}_{20}\cosh(\Delta)+\mathcal{S}_{10}\sinh(\Delta)~{})\psi=0,$
(126)

in which the $\mathcal{S}_{i0}$ are the free Dirac operators

$\mathcal{S}_{i0}=\frac{i}{\sqrt{2}}\gamma_{5i}(\gamma_{i}\cdot p_{i}+m_{i}).$
(127)

This, in turn leads to the two compatibility conditions cww ; jmath ; saz86

$[\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0,$ (128)

and

$[{\hbox{\boldmath${S}$}}_{1},{\hbox{\boldmath${S}$}}_{2}]\psi=0,$ (129)

provided that $\ \Delta(x)=\Delta(x_{\perp}).$ These compatibility conditions
do not restrict the gamma matrix structure of $\Delta$. That matrix structure
is determined by the type of vertex-vertex structure we wish to incorporate in
the interaction. The three types of invariant interactions $\Delta$ that was
used in the relativistic quark model based on this approach (as most recently
discussed in unusual ,tmlk ) are

$\displaystyle\Delta_{\mathcal{L}}(x_{\perp})$
$\displaystyle=-1_{1}1_{2}\frac{\mathcal{L}(x_{\perp})}{2}\mathcal{O}_{1},\
\mathcal{O}_{1}=-\gamma_{51}\gamma_{52},~{}~{}~{}\text{ scalar}\mathrm{,}$
$\displaystyle\Delta_{\mathcal{J}}(x_{\perp})$
$\displaystyle=\beta_{1}\beta_{2}\frac{\mathcal{J}(x_{\perp})}{2}\mathcal{O}_{1},~{}~{}~{}\text{time-
like\ vector}\mathrm{,}$ $\displaystyle\Delta_{\mathcal{G}}(x_{\perp})$
$\displaystyle=\gamma_{1\perp}\cdot\gamma_{2\perp}\frac{\mathcal{G}(x_{\perp})}{2}\mathcal{O}_{1},~{}~{}\text{space-
like\ vector}{,}$ (130)

where

$\displaystyle\gamma_{5i}$
$\displaystyle=\gamma_{i}^{0}\gamma_{i}^{1}\gamma_{i}^{2}\gamma_{i}^{3},$
$\displaystyle\beta_{i}$ $\displaystyle=-\gamma_{i}\cdot\hat{P}.$ (131)

For general independent scalar, time-like vector, and space-like vector
interactions we have

$\Delta(x_{\perp})=\Delta_{\mathcal{L}}+\Delta_{\mathcal{J}}+\Delta_{\mathcal{G}}.$
(132)

The special case of an electromagnetic-like interaction (in the Feynman gauge)
applied in this paper and in bckr  corresponds to $\mathcal{J}=-\mathcal{G}$
or

$\displaystyle\Delta_{\mathcal{J}}+\Delta_{\mathcal{G}}$
$\displaystyle\equiv\Delta_{\mathcal{EM}}=(-\gamma_{1}\cdot\hat{P}\gamma_{2}\cdot\hat{P}+\gamma_{1\perp}\cdot\gamma_{2\perp})\frac{\mathcal{G}(x_{\perp})}{2}\mathcal{O}_{1}$
$\displaystyle=\gamma_{1}\cdot\gamma_{2}\frac{\mathcal{G}(x_{\perp})}{2}\mathcal{O}_{1}.$
(133)

and for scalar and electromagnetic interaction,

$\Delta(x_{\perp})=\Delta_{\mathcal{L}}+\Delta_{\mathcal{EM}}.$ (134)

This leads to121212 In short, one inserts Eq. (126) into (125) and brings the
free Dirac operator (127) to the right of the matrix hyperbolic functions.
Using commutators and $\cosh^{2}\Delta-\sinh^{2}\Delta=1$ one arrives at Eq.
(135). The structure of these equations are very much the same as that of a
Dirac equation for each of the two particles, with $M_{i}$ and $E_{i}$ playing
the roles that $m+S$ and $\varepsilon-A$ do in the single particle Dirac
equation. Over and above the usual kinetic part, the spin-dependent
modifications involving $G\mathcal{P}_{i}$ and the last set of derivative
terms are two-body recoil effects essential for the compatibility
(consistency) of the two equations jmath ; long

$\displaystyle\mathcal{S}_{1}\psi$
$\displaystyle=\big{(}-G\beta_{1}\Sigma_{1}\cdot\mathcal{P}_{2}+E_{1}\beta_{1}\gamma_{51}+M_{1}\gamma_{51}-G\frac{i}{2}\Sigma_{2}\cdot\partial(\mathcal{L}\beta_{2}\mathcal{-J}\beta_{1})\gamma_{51}\gamma_{52}\big{)}\psi=0,$
$\displaystyle\mathcal{S}_{2}\psi$
$\displaystyle=\big{(}G\beta_{2}\Sigma_{2}\cdot\mathcal{P}_{1}+E_{2}\beta_{2}\gamma_{52}+M_{2}\gamma_{52}+G\frac{i}{2}\Sigma_{1}\cdot\partial(\mathcal{L}\beta_{1}\mathcal{-J}\beta_{2})\gamma_{51}\gamma_{52}\big{)}\psi=0,$
(135)

in which $\partial_{\mu}=\partial/\partial x^{\mu}.$ With

$\displaystyle G$ $\displaystyle=\exp\mathcal{G},$
$\displaystyle\mathcal{P}_{i}$ $\displaystyle\equiv
p_{\perp}-\frac{i}{2}\Sigma_{i}\cdot\partial\mathcal{G}\Sigma_{i}.$ (136)

The connections between what we call the vertex invariants
$\mathcal{L},\mathcal{J},\mathcal{G}$ and the mass and energy potentials
$M_{i},E_{i}$ are

$\displaystyle M_{1}$ $\displaystyle=m_{1}\ \cosh\mathcal{L}\
+m_{2}\sinh\mathcal{L},$ $\displaystyle M_{2}$ $\displaystyle=m_{2}\
\cosh\mathcal{L}\ +m_{1}\ \sinh\mathcal{L},$ $\displaystyle E_{1}$
$\displaystyle=\varepsilon_{1}\ \cosh\mathcal{J}\
+\varepsilon_{2}\sinh\mathcal{J},$ $\displaystyle E_{2}$
$\displaystyle=\varepsilon_{2}\
\cosh\mathcal{J}+\varepsilon_{1}\sinh\mathcal{J}.$ (137)

Eq. (135) depends on standard Pauli-Dirac representation of gamma matrices in
block forms (see Eq. (2.28) in crater2  for their explicit forms) and
where131313 Just as $x^{\mu}$ is a four vector, so is $P^{\mu}.$ Thus, the
time-like and space-like interactions in Eq. (130) become
$\gamma_{1}^{0}\gamma_{2}^{0}$ and
${\hbox{\boldmath${\gamma}$}}_{1}\cdot{\hbox{\boldmath${\gamma}$}}_{2}$ only
in the c.m. system due to the fact that from Eq. (131),
$\beta_{i}=\gamma_{i}^{0}$ only in the c.m. frame. Likewise,
$\Sigma_{i}^{\mu}=(0,{\hbox{\boldmath${\Sigma}$})}$ only in the c.m. frame
just as is $x_{\perp}^{\mu}=(0,{\hbox{\boldmath${r}$})}$ in that frame only.

$\Sigma_{i}=\gamma_{5i}\beta_{i}\gamma_{\perp i}.$ (138)

###  A.2
Vector potentials $\tilde{A}_{i}^{\mu}$ in terms of the invariant $A(r)$

Comparing Eq. (135) with Eq. (14) we find that the spin-dependent
electromagnetic-like vector interactions of Eq. (14) are cra87 ; bckr

$\displaystyle\tilde{A}_{1}^{\mu}$
$\displaystyle=\big{(}(\varepsilon_{1}-E_{1})\big{)}\hat{P}^{\mu}+(1-G)p_{\perp}^{\mu}-\frac{i}{2}\partial
G\cdot\gamma_{2}\gamma_{2}^{\mu},$ $\displaystyle A_{2}^{\mu}$
$\displaystyle=\big{(}(\varepsilon_{2}-E_{2})\big{)}\hat{P}^{\mu}-(1-G)p_{\perp}^{\mu}+\frac{i}{2}\partial
G\cdot\gamma_{1}\gamma_{1}^{\mu},$ (139)

Note that the first portion of the vector potentials is time-like (parallel to
$\hat{P}^{\mu})$ while the next two portions are space-like (transverse to
$\hat{P}^{\mu})$. The spin-dependent scalar potentials $\tilde{S}_{i}$ are

$\displaystyle\tilde{S}_{1}$
$\displaystyle=M_{1}-m_{1}-\frac{i}{2}G\gamma_{2}\cdot\partial\mathcal{L},$
$\displaystyle\tilde{S}_{2}$
$\displaystyle=M_{2}-m_{2}+\frac{i}{2}G\gamma_{1}\cdot{\partial}\mathcal{\
L}{.}$ (140)

We have chosen a parametrization for the vertex invariants
$\mathcal{L},~{}\mathcal{J}=-\mathcal{G}$ that takes advantage of the Todorov
effective external potential forms and at the same time will display the
correct static limit form for the Pauli reduction. The logic of the choice for
these parametrizations is strengthened by the fact that for classical fw  or
quantum field theories saz97  for separate scalar and time-like vector
interactions one can show that the spin independent part of the quasipotential
$\Phi~{}$ involves the difference of squares of the invariant mass and energy
potentials

$M_{i}^{2}=m_{i}^{2}+2m_{w}S+S^{2};\
E_{i}^{2}=\varepsilon_{i}^{2}-2\varepsilon_{w}A+A^{2},$ (141)

so that

$M_{i}^{2}-E_{i}^{2}=2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}-b^{2}(w).$ (142)

Eqs. (14) and (135) involve combined scalar and electromagnetic-like vector
interactions (without the separate time-like interactions this amounts to
working in the Feynman gauge with the simplest relation between space- and
time-like parts, see Eqs. (133), (134), and cra88 ; crater2 ). In that case
the mass and energy potentials in place of Eq. (141) are respectively

$\displaystyle M_{i}^{2}$
$\displaystyle=m_{i}^{2}+\exp(2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2}),$
$\displaystyle E_{i}^{2}$
$\displaystyle=\exp(2\mathcal{G(A))(}\left(\varepsilon_{i}-A)^{2}\right),$
$\displaystyle M_{i}^{2}-E_{i}^{2}$
$\displaystyle=\exp(2\mathcal{G(A))[}2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}-b^{2}(w]$
(143)

so that from Eq. (137),

$\displaystyle\exp(\mathcal{L})$
$\displaystyle=\exp(\mathcal{L}(S,A))=\frac{\sqrt{m_{1}^{2}+\exp(2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2})}+\sqrt{m_{2}^{2}+\exp(2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2})}}{m_{1}+m_{2}},\
$ (144) $\displaystyle\exp(\mathcal{J})$ $\displaystyle=\exp(-\mathcal{G)}$

with

$\exp(2\mathcal{G(}A\mathcal{))=}\frac{1}{(1-2A/w)}\equiv G^{2},$ (145)

or

$\displaystyle-\mathcal{G}$
$\displaystyle\mathcal{=}\frac{1}{2}\log(1-2A/w)=\log\frac{E_{1}+E_{2}}{w},$
(146)

and the spin-independent minimal coupling appears like

$\Phi_{SI}=2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}.$ (147)

###  A.3 Interaction terms in the equivalent Relativistic Schrödinger Equation
[Eq. (17)]

The Klein-Gordon like potential energy terms appearing in the Pauli form ( 17)
arise from (see Eq. (143))

$M_{i}^{2}-E_{i}^{2}=\exp(2\mathcal{G)[}2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}-b^{2}(w)].$
(148)

To obtain the simple Pauli form of Eq. (16) and the subsequent detailed form
in Eq. (17) involves steps similar to those used in the Pauli reduction of the
single particle Dirac equation unusual  but with the combinations
$\phi_{\pm}=\psi_{1}\pm\psi_{4}$ and $\chi_{\pm}=\psi_{2}\pm\psi_{3}$ instead
of the upper and lower components of the single particle wave function. This
reduces the Pauli forms to 4 uncoupled 4 component relativistic Schrödinger
equations saz94 ; long ; crater2 ; liu . We work in the c.m. frame in which
$\hat{P}=(1,{\ \hbox{\boldmath${0}$})}$ and
$\hat{r}=(0,{\hbox{\boldmath${\hat{r}}$}).}$ We also define four component
wave functions $\psi_{\pm},\eta_{\pm}$ by liu

$\displaystyle\phi_{\pm}$
$\displaystyle=\exp(\mathcal{F}+\mathcal{K}\bm{\sigma}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}}\bm{\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}})\psi_{\pm}=(\exp\mathcal{F})(\cosh\mathcal{K}+\bm{\sigma}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}}\bm{\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}}\sinh\mathcal{K})\psi_{\pm},$
$\displaystyle\chi_{\pm}$
$\displaystyle=\exp(\mathcal{F}+\mathcal{K}\bm{\sigma}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}}\bm{\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}})\eta_{\pm}=(\exp\mathcal{F})(\cosh\mathcal{K}+\bm{\sigma}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}}\bm{\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}}\sinh\mathcal{K})\eta_{\pm},$
(149)

in which

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{1}{2}\log\frac{\mathcal{D}}{\varepsilon_{2}m_{1}+\varepsilon_{1}m_{2}}-\mathcal{G},$
$\displaystyle\mathcal{D}$ $\displaystyle\mathcal{=}E_{2}M_{1}+E_{1}M_{2},$
$\displaystyle\mathcal{K}$
$\displaystyle=\frac{(\mathcal{L}+\mathcal{G})}{2}.$ (150)

The substitution (149) has the convenient property that in the resultant bound
state equation, the coefficients of the first order relative momentum terms
vanish.

Using the results in liu  and unusual  we obtain for the general case of
unequal masses the relativistic Schrödinger equation ( 17) that is a detailed
c.m. form of Eq. (16). In that equations we have introduced the
abbreviations141414 Minor misprints of the equations below have appeared in
appendices in unusual  and tmlk . The ones presented here are corrected.

$\displaystyle\Phi_{D}$
$\displaystyle=-\frac{2(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}+\mathcal{F}^{\prime 2}+\mathcal{K}^{\prime
2}+\frac{2\mathcal{K}^{\prime}\sinh
2\mathcal{K}}{r}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{F}+m(r),$
$\displaystyle\Phi_{SO}$
$\displaystyle=-\frac{\mathcal{F}^{\prime}}{r}-\frac{(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}+\frac{\mathcal{K}^{\prime}\sinh 2\mathcal{\ K}}{r},$
$\displaystyle\Phi_{SOD}$ $\displaystyle=(l^{\prime}\cosh
2\mathcal{K}-q^{\prime}\sinh 2\mathcal{K}),$ $\displaystyle\Phi_{SOX}$
$\displaystyle=(q^{\prime}\cosh 2\mathcal{K}+l^{\prime}\sinh 2\mathcal{K}),$
$\displaystyle\Phi_{SS}$
$\displaystyle=\kappa(r)+\frac{2\mathcal{K}^{\prime}\sinh
2\mathcal{K}}{3r}-\frac{2(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{3r}+\frac{2\mathcal{F}^{\prime}\mathcal{K}^{\prime}}{3}-\frac{{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{K}}{3},$
$\displaystyle\Phi_{T}$
$\displaystyle=\frac{1}{3}[n(r)+\frac{(3\mathcal{F}^{\prime}-\mathcal{K}^{\prime}+3/r)\sinh
2\mathcal{K}}{r}+\frac{(\mathcal{F}^{\prime}-3\mathcal{\ \ \
K}^{\prime}+1/r)(\cosh 2\mathcal{K}-1)}{r}+2\mathcal{F}^{\prime}\mathcal{\
K}^{\prime}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{K}],$
$\displaystyle\Phi_{SOT}$ $\displaystyle=-\mathcal{K}^{\prime}\frac{\cosh
2\mathcal{K}-1}{r}-\frac{\mathcal{K}^{\prime}}{r}+\frac{(\mathcal{F}^{\prime}+1/r)\sinh
2\mathcal{K}}{r},$ (151)

where

$\displaystyle n(r)$
$\displaystyle=\nabla^{2}\mathcal{K}-\frac{1}{2}\nabla^{2}\mathcal{G}+\frac{3(\mathcal{G}-2\mathcal{K})^{\prime}}{2r}+\mathcal{F}^{\prime}\mathcal{G}^{\prime}-2\mathcal{F}^{\prime}\mathcal{K}^{\prime},$
$\displaystyle\kappa(r)$
$\displaystyle=\frac{1}{3}\nabla^{2}(\mathcal{G}+\mathcal{K})-\frac{1}{2}\mathcal{G}^{\prime
2}-\frac{2\mathcal{F}^{\prime}(\mathcal{G}+\mathcal{K})^{\prime}}{3},$
$\displaystyle m(r)$
$\displaystyle=-\frac{1}{2}\nabla^{2}\mathcal{G+}\frac{3}{4}\mathcal{G}^{\prime
2}-\mathcal{K}^{\prime 2}+\mathcal{G}^{\prime}\mathcal{F}^{\prime},$
$\displaystyle l^{\prime}$
$\displaystyle=-\frac{\mathcal{(L-G)}^{\prime}}{2r}\frac{E_{2}M_{2}-E_{1}M_{1}}{E_{2}M_{1}+E_{1}M_{2}},$
$\displaystyle q^{\prime}$
$\displaystyle=\frac{\mathcal{(L-G)}^{\prime}}{2r}\frac{E_{1}M_{2}-E_{2}M_{1}}{E_{2}M_{1}+E_{1}M_{2}}.$
(152)

(The prime symbol stands for $d/dr,$ and the explicit forms of the derivatives
are given in Eq. (153) ). For $L=J$ states, the hyperbolic terms cancel and
the spin-orbit difference terms in general produce spin mixing except for
equal masses or $J=0$. For ease of use we have listed below the explicit forms
that appear in the above $\Phi$s in Eqs. (151) -(152) in terms of the general
invariant potentials $A(r)$ and $S(r).~{}$ The radial components of Eq. (17)
are given in Appendix B.

###  A.4
Explicit expressions for terms in the relativistic Schrödinger Equation (17)
from $A(r)$ and $S(r)$

Given the functions $A(r)$ and $S(r)$ for the interaction, users of the
relativistic Schrödinger equation (17) will find it convenient to have an
explicit expression in an order that would be useful for programing the terms
in the associated equation (151). We use the definitions above given in Eqs.
(143 )-(145), and (150). In order that the terms in Eq. (151) be reduced to
expressions involving just $A(r),~{}$and $S(r)$ and their derivatives, we list
the following formulae

$\displaystyle\mathcal{F}^{\prime}$
$\displaystyle=\frac{(\mathcal{L}^{\prime}-\mathcal{G}^{\prime})(E_{2}M_{2}+E_{1}M_{1})}{2(E_{2}M_{1}+E_{1}M_{2})}-\mathcal{G}^{\prime},$
$\displaystyle\mathcal{G}^{\prime}$ $\displaystyle=\frac{A^{\prime}}{w-2A},$
$\displaystyle\mathcal{L}^{\prime}$
$\displaystyle=\frac{M_{1}^{\prime}}{M_{2}}=\frac{M_{2}^{\prime}}{M_{1}}=\frac{w}{M_{1}M_{2}}\left(\frac{S^{\prime}(m_{w}+S)}{w-2A}+\frac{(2m_{w}S+S^{2})A^{\prime}}{(w-2A)^{2}}\right),$
$\displaystyle.\mathcal{K}^{\prime}$
$\displaystyle=\frac{(\mathcal{L}^{\prime}+\mathcal{G}^{\prime})}{2}.$ (153)

Also needed are

$\displaystyle\cosh 2\mathcal{K}$
$\displaystyle=\frac{1}{2}\left(\frac{(\varepsilon_{1}+\varepsilon_{2})(M_{1}+M_{2})}{(m_{1}+m_{2})(E_{1}+E_{2})}+\frac{(m_{1}+m_{2})(E_{1}+E_{2})}{(\varepsilon_{1}+\varepsilon_{2})(M_{1}+M_{2})}\right),$
$\displaystyle\sinh 2\mathcal{K}$