#  Dynamics of quarks and gauge fields in the lowest-energy states in QCD and
QED

Cheuk-Yin Wong Physics Division, Oak Ridge National Laboratory111 This
manuscript has been authored in part by UT-Battelle, LLC, under contract DE-
AC05-00OR22725 with the US Department of Energy (DOE). The US government
retains and the publisher, by accepting the article for publication,
acknowledges that the US government retains a nonexclusive, paid-up,
irrevocable, worldwide license to publish or reproduce the published form of
this manuscript, or allow others to do so, for US government purposes. DOE
will provide public access to these results of federally sponsored research in
accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-
public-access-plan), Oak Ridge, Tennessee 37831, USA , Oak Ridge, TN 37831,
U.S.A. Andrew V. Koshelkin National Research Nuclear University MEPhI,
Kashirskoye shosse, 31, 115409 Moscow, Russia

###### Abstract

We examine the dynamics of quarks and gauge fields in QCD and QED interactions
in the lowest energy states with approximate cylindrical symmetry, in a flux
tube model. Using the action integral, we separate out the (3+1)D in terms of
the transverse and the longitudinal degrees of freedom and solve the resultant
equations of motion. We find that there may be localized and stable states of
QCD and QED collective excitations, showing up as particles whose masses
depend on the QCD and QED coupling constants and the flux-tube radius. Along
with known stable collective QCD excitations of the quark-QCD-QED system,
there may be stable QED collective excitations as good candidates for many
anomalous particles observed recently in the region of many tens of MeV.

###### pacs:

12.38.-t 12.38.Aw 11.10Kk

##  I Introduction

In the dynamics of quarks and antiquarks interacting with the QCD (quantum
chromodynamical) and the QED (quantum electrodynamical) interactions in
(3+1)D, it is of great interest to know what the lowest-energy states are. In
particular, we would like to know whether there may be independent collective
excitations of the quark-QCD-QED system involving the quarks interacting in
QED interaction alone, in addition to the well-known collective excitations
involving the QCD interaction. These collective excitations involving quarks
interacting in QED alone have not been unequivocally demonstrated
experimentally, although there have been intriguing pieces of evidence hinting
at their possible existence, as we shall discuss below. The purpose of the
present study is to explore whether there may be plausible theoretical basis
for the existence of these collective excitations from the viewpoints of the
dynamics of the quark-QCD-QED medium in (3+1)D.

Previous works on the simplification of the dynamics from (3+1)D to (1+1)D for
quarks interacting in gauge interactions focused on the high-energy limit with
the assumption of the longitudinal dominance, which allows the longitudinal
rescaling of the coordinates appropriate for perturbative QCD studies Ver93 ,
or the alternative description in terms of a non-perturbative longitudinal
flux tube Won10 ; Won11 ; Kos12 ; Won14 . The flux tube description can be
applied to the low-energy phenomena. We consider here the lowest energy states
in (3+1)D${}_{\\{x^{1},x^{2},x^{3},x^{0}\\}}$ with an approximate cylindrical
symmetry which allows the separation of the spatial coordinates into the
transverse $\\{x^{1},x^{2}\\}$ (or $\\{r_{\perp},\phi\\}$) and the
longitudinal $x^{3}$ coordinates. The full dynamics in
(3+1)D${}_{\\{x^{1},x^{2},x^{3},x^{0}\\}}$ can then be described as the
coupling of the transverse dynamics in the
(2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$ space-time and the longitudinal dynamics
in the (1+1)D${}_{\\{x^{3},x^{0}\\}}$ space-time, linked together by
parametric quantities that pass information from the transverse degrees of
freedom to the longitudinal degrees of freedom. We can solve the transverse
equations of motion to obtain the stationary states in the transverse degrees
of freedom. The longitudinal states and the transverse states can then be
coupled to describe a general state of the system, as in a Schmidt-type
decomposition. In particular, for the lowest-energy states of the quark-QCD-
QED system, it becomes reasonable to consider the relevant physical quantities
to possess a stationary transverse flux tube structure in the
(2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$ space-time, then the dynamics of quarks
in QCD and QED in (3+1)D can be examined in the simpler longitudinal
(1+1)D${}_{\\{x^{3},x^{0}\\}}$ space-time, as in an open string, with the
information on the transverse degrees of freedom stored as input parametric
quantities in the idealized (1+1)D${}_{\\{x^{3},x^{0}\\}}$ space-time. In
addition to a greater mathematical simplification, the separation of the 3+1
dimensional space-time into coupled (2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$
space-time and (1+1)D${}_{\\{x^{3},x^{0}\\}}$ space-time in such a manner
brings with it a greater intuitive understanding of the underlying physics. It
facilitates the investigation of the dynamics of many relevant problems. For
brevity, we shall use the label of 4D and 2D to refer to the dynamics of the
(3+1)D${}_{\\{x^{1},x^{2},x^{3},x^{0}\\}}$ space-time and the idealized open-
string (1+1)D${}_{\\{x^{3},x^{0}\\}}$ space-time respectively, if such labels
lead to no ambiguities. We shall often omit the space-time subscript labels of
$\\{x^{1},x^{2},x^{0}\\}$ and $\\{x^{3},x^{0}\\}$ when there is no ambiguity.

For the QCD interaction, the open-string idealization of the lowest energy
states of a quark-antiquark system in (1+1)D was suggested early on by the
dual-resonance model Ven68 , the Nambu and Goto meson string model Nam70 ;
Got71 , the ’tHooft’s two-dimensional meson model tH74a ; tH74b , the
classical yo-yo string model Art74 , Polyakov’s quantum bosonic string Pol81 ,
and the Lund model And83 . The open-string description was subsequently
supported theoretically by lattice gauge calculations in (3+1)D in which the
structure of a flux tube shows up explicitly Hua88 ; Bal05 ; Cos17 ; Car13 ;
Bic17 ; Bic18 . The open-string description for a quark-antiquark system in
high-energy hadron-hadron reactions and $e^{+}$-$e^{-}$ annihilations provided
the foundation for the (1+1)D inside-outside cascade model of Bjorken, Casher,
Kogut, and Susskind Bjo73 ; Cas74 , the yo-yo string model Art74 , and the
Lund model in high-energy collisions And83 . The flux tube description
receives experimental support from the limiting average transverse momentum
and the rapidity plateau of produced hadrons Cas74 ; Bjo73 ; Won91 ; Won94 ;
Won09 ; Gat92 , in high-energy $e^{+}$-$e^{-}$ annihilations Aih88 ; Hof88 ;
Pet88 ; Abe99 ; Abr99 and $pp$ collisions Yan08 .

While a confined open string in (1+1)D as the idealization of a stable quark-
antiquark system in QCD is well known Ven68 ; Nam70 ; Got71 ; tH74a ; tH74b ;
Art74 ; Pol81 ; And83 ; Hua88 ; Bal05 ; Cos17 ; Cas74 ; Bjo73 , not so well-
known is the analogous confined open string quark-antiquark system in QED in
(1+1)D with a boson mass of many tens of MeV, when we apply Schwinger’s
results for massless fermions to light quarks as massless in QED in (1+1)D
Sch62 ; Sch63 ; Won10 . Could this confined quark-antiquark open string in QED
in (1+1)D be the idealization of a stable quark-QED system in (3+1)D? We can
explore answers to such a question from two different points of view. From the
viewpoint of phenomenology, a stable quark-QED system in (3+1)D would be a
likely occurrence because quarks are confined as no fractional charged
particles have ever been observed. So, a quark interacting with a antiquark in
QED forming a confined QED bound state in (3+1)D would be an entirely
reasonable concept. However, from the purely theoretical view point,
Schwinger’s confined boson exists in the (1+1)D space-time, whereas the
physical world is the (3+1)D space-time. How can a quark-QED system be
idealized from the (3+1)D space-time to the (1+1)D space-time? The quark-QED
system must possess transverse confinement to exist as a flux tube in the
(3+1)D space-time before the flux tube can be idealized as a one-dimensional
string in the (1+1)D space-time. In this regard, we note perceptively with
great interest Polyakov’s results that electric charges interacting in compact
QED in (2+1)D are confined and that the confinement persists for all non-
vanishing coupling constants, no matter how weak Pol77 ; Pol87 . For our
problem with the separation of the (3+1)D dynamics into the
(2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$ dynamics and the associated idealization
into (1+1)D${}_{\\{x^{3},x^{0}\\}}$ dynamics, we can apply Polyakov’s results
to infer transverse confinement of quarks interacting in compact QED in the
transverse space of $\\{x^{1},x^{2}\\}$. Specifically, because a quark and an
antiquark carry opposite electric charges, and they interact in the compact
QED gauge interaction on account of the quantization of commensurate quark
charges Yan70 , it then follows from Polyakov’s results that a quark, an
antiquark, and their associated gauge fields are transversely confined in
(2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$ at the moment of the creation of a
$q\bar{q}$ eigenstate. Subsequently, the confined (2+1)D configuration in the
transverse plane can be longitudinally stretched into a flux tube in a three-
dimensional yo-yo bound-state motion in the longitudinal direction. The
stretch (2+1)D configuration will likely retain the transverse confinement,
and the dynamics of quarks in the (3+1)D space-time may occur within a
transversely confined cylindrical region as a flux tube that can be
approximated as a one-dimensional string in the idealized (1+1)D space-time.
In this idealized (1+1)D space-time, we need to examine whether quarks and
gauge fields are longitudinally confined or not. If we further consider that
the light quarks can be approximated to be massless, then according to
Schwinger’s exact solution for massless fermions in QED in
(1+1)D${}_{\\{x^{3},x^{0}\\}}$ Sch62 ; Sch63 , a light quark and its
antiparticle interacting in QED will be longitudinally confined just as well.
With the possibility of both transverse and longitudinal confinements, could
such a stable QED state be experimentally observed and be labeled as a QED
meson, in analogy to QCD mesons?

Experimentally, there are intriguing pieces of evidence hinting at the
possible existence of neutral bosons at energies of many tens of MeV that may
present themselves as good candidates for these QED mesons. For over several
decades there have been many exclusive measurements in high-energy hadron-
hadron collisions and in $e^{+}$-$e^{-}$ annhilations, in which the momenta of
the initial and final particles are properly measured so that the
bremsstrahlung contributions to the soft photons can be quantitatively
determined. It has been consistently observed that whenever hadrons are
produced, anomalous soft photons in the form of excess $e^{+}e^{-}$ pairs
about 4 to 8 times of the bremsstrahlung expectations are proportionally
produced, and when hadrons are not produced, these anomalous soft photons are
also not produced Chl84 ; Bot91 ; Ban93 ; Bel97 ; Bel02 ; DEL06 ; DEL08 ;
Per09 ; DEL10 . The transverse momenta of the excess $e^{+}$$e^{-}$ pairs lie
in the range of a few MeV/c to many tens of MeV/c, corresponding to a mass
scale from a few MeV to many tens of MeV. Owing to the simultaneous and
correlated production alongside with hadrons, the parent particle of an
anomalous soft photon is likely to contain some elements of the hadron sector,
such as a light quark and a light antiquark. The quark and the antiquark carry
color and electric charges and they interact mutually with the QCD and the QED
interactions. The parent particle of the anomalous soft photon cannot arise
from the quark and the antiquark interacting with the QCD interaction, because
such an interaction will endow the pair with a mass much greater than the mass
scale of the anomalous soft photon, in a contradictory manner. We are left
with the possibility of the quark and the antiquark interacting with the QED
interaction. Such a possibility is further reinforced by the special nature of
a confining gauge interaction, for which the lower the magnitude of the
attractive interaction coupling constant, the lower will be the mass of the
composite particle it generates, in contrast to a non-confining interaction.
According to the Schwinger’s (confined boson mass $m$)-(coupling constant $g$)
relation, $m^{2}=g^{2}/\pi$, the QED gauge interaction, upon a comparison with
the QCD gauge interaction, will bring the quantized mass $m$ of a confined
$q\bar{q}$ pair to the lower mass range of the anomalous soft photons. It was
therefore proposed in Won10 ; Won11 ; Won14 ; Won20 ; Won20a ; Won21 ; Won22
that a quark and an antiquark interacting with the QED interaction may lead to
new open string $q\bar{q}$ bound states (QED-meson states) with a mass of many
tens of MeV. These QED mesons may be produced simultaneously with the QCD
mesons in high-energy collisions Chl84 ; Bot91 ; Ban93 ; Bel97 ; Bel02 ; DEL06
; DEL08 ; Per09 ; DEL10 , and the excess $e^{+}e^{-}$ pairs may arise from the
decays of these QED mesons. The predicted masses of the isoscalar
$I(J^{\pi})=0(0^{-})$ and $I(J^{\pi})=1(0^{-})$ QED mesons are about 13-17 MeV
and 36-38 MeV respectively Won10 ; Won20 , which agree approximately with the
masses of the X17 Kra16 ; Kra21 and the E38 particles Abr12 ; Abr19 subsequent
observed. While the anomalous soft photon measurements as $e^{+}$$e^{-}$
excesses have been well established experimentally, the X17 particle and the
E38 particle needs further experimental confirmations. Ongoing experiments to
confirm the X17 and the E38 particles are continuing.

The possible existence of QED mesons raises many serious questions. Is it ever
possible for a quark and an antiquark to be produced and to interact in the
QED interaction alone, without the QCD interaction? If they can interact with
the QED interaction alone, will the QED interaction confines a quark and an
antiquark in (3+1)D? If they turn out to be confined and bound, what is the
order of magnitude of their masses? Are there pieces of experimental evidence
pointing to their possible existence in the predicted mass region?

Such questions have not been brought up until very recently for obvious
reasons. It is generally perceived that a quark and an antiquark interact with
the QCD and the QED interactions simultaneously, and the occurrence of a
stable and confined state of the quark and the antiquark interacting in the
QED interaction alone, without the QCD interaction, may appear impossible.
Furthermore, the common perception is that only the QCD interaction with its
non-Abelian properties can confine a quark and an antiquark, whereas the QED
interaction is Abelian. It has been argued that even if a quark and an
antiquark can interact in the QED Abelian interaction alone, the QED
interaction by itself cannot confine the quark and the antiquark, as in the
case of an electron and a positron, so the quark and the antiquark cannot be
confined even if they can interact with the QED interaction alone.

With regard to the first important question whether it is ever possible for a
quark and an antiquarks to be produced and to interact with the QED
interaction alone without the QCD interaction, we can consider as an example
the production of a quark and an antiquark in the reactions
$e^{+}$+$e^{-}$$\to$$\gamma^{*}$$\to$$q$+$\bar{q}$ and
$e^{+}$+$e^{-}$$\to$$\gamma^{*}\gamma^{*}$$\to$$q$+$\bar{q}$ with a center-of
mass energy $\sqrt{s}$ in the range $(m_{q}+m_{\bar{q}})<\sqrt{s}<m_{\pi}$,
where the sum of the rest masses of the quark and the antiquark is of order a
few MeV and $m_{\pi}\sim 135$ MeV PDG19 . The incident $e^{+}$+$e^{-}$ pair is
in a colorless color-singlet state, and thus the produced $q$-$\bar{q}$ pair
is also a color-singlet pair. In this range of $\sqrt{s}$, the produced $q$
and $\bar{q}$ will be in a colorless state, if they can ever be produced, and
they can only interact with the QED interaction but not with the QCD
interaction. For, if they interact with the QCD interaction, the $q$-$\bar{q}$
pair will result in a hadron state with a center-of-mass energy $\sqrt{s}$
beyond this energy range, in a contradictory manner. It is therefore possible
for a quark and an antiquark produced by $e^{+}$+$e^{-}$ annihilations to
interact with the QED interaction alone, in this range of $\sqrt{s}$ below the
pion mass.

We can proceed further to examine the cross section in such a reaction as a
function of $\sqrt{s}$. The cross section for
$e^{+}$+$e^{-}$$\to$$q$+$\bar{q}$ is the product of three factors. The first
is the kinematic factor involving the collision energy exceeding the rest mass
threshold of the quark and antiquark, and the statistical weights of spins and
angular momentum. The second is the square of the matrix element for the
transition $e^{+}$+$e^{-}$$\to$$q$+$\bar{q}$ leading to the production of the
pair, containing the final-state QED interaction $K$-factor (or Gamow factor)
between the produced quark and the antiquark Cha95 . Resonances will show up
at those energies at which the final-state QED interaction leads to bound
states. When we include all possible spin and angular momentum states in the
measurement, these two factors are non-zero. The crucial third factor is the
density of final states which depends on the energies of the quark-antiquark
bound states (such as the proposed X17 and the E38 states), and the
confinement nature of the continuum states at energies away from the bound-
state energies. If the quark and the antiquark are not confined in QED, then,
at energies away from the bound state energies, these quark-antiquark states
will appear as continuum states with a separated quark and an antiquark. The
continuum $q$-$\bar{q}$ states can then be detected as fractional charges.
Thus, the probability of producing fractional charges is non-zero in
$e^{+}$+$e^{-}$$\to$$q$+$\bar{q}$ reaction in this range of $\sqrt{s}$, if
quarks are not confined in QED interactions in (3+1)D. On the other hand, if
the quark and the antiquark are confined in QED in (3+1)D, the continuum state
of a separated quark and a separated antiquark is impossible. There will be no
continuum states at energies away from the confined bound state energies. The
density of continuum states is zero, and the probability of producing
fractional charges is zero. Therefore away from the bound state energies and
in the range of $(m_{q}+m_{\bar{q}})<\sqrt{s}<m_{\pi}$, the non-observation of
fractional charges for the $e^{+}$+$e^{-}$$\to$ $q$+$\bar{q}$ reaction will be
consistent with the confinement of quarks in QED interactions in (3+1)D. In
this case, only at the eigenenergies of the QED-confined states in this energy
range, such as those of the proposed X17 and E38 states, will show up as
resonances.

As an illustrative case for a $q$-$\bar{q}$ pair to interact in the QED
interaction alone, we have given the above reaction,
$e^{+}$+$e^{-}$$\to$$q$+$\bar{q}$, below the pion mass energy — as a good
example. There are actually other circumstances in which a $q$-$\bar{q}$ pair
can be created with a total pair energy lower than the pion mass in many other
reactions, and the produced $q$-$\bar{q}$ pair can interact in the QED
interaction alone without the QCD interaction. This may be the mechanism for
the production of the reported anomalous neutral bosons of X17, E38, and
anomalous soft photons with masses in the region of many tens of MeV, as we
described earlier Won20 . In other circumstances in the deconfinement-to-
confinement phase transition of the quark-gluon plasma, a deconfined quark and
a deconfined antiquark in close spatial proximity can coalesce to become a
$q$-$\bar{q}$ pair with a pair energy below the pion mass, and they can
interact in QED alone.

On the theoretical front, the common perception is that stable QED excitations
cannot occur without the simultaneous QCD excitations because the quark
density is a single-element quantity. So a quark and an antiquark interact
with both the QCD and QED gauge interactions simultaneously. However, because
of the three-color nature of the quarks, the quark currents $j^{\mu}$ are
3$\times$3 color matrices with 9 matrix elements. The 9 matrix elements in the
color space can be separated naturally into color-singlet and color-octet
subgroups of generators. Specifically, quarks reside in the ${3}$
representation and antiquarks reside in the $\hbox{\boldmath${3}$}^{*}$
representation, and they form a direct product of
$\hbox{\boldmath${3}$}\otimes{\hbox{\boldmath${3}$}^{*}}$=$\hbox{\boldmath${1}$}\oplus\hbox{\boldmath${8}$}$,
with a color-singlet ${1}$ subgroup and a color-octet ${8}$ subgroup. The QED
gauge fields $A_{{\rm QED}}^{\mu}$ and the QCD gauge fields $A_{{\rm
QCD}}^{\mu}$ generated by the currents are likewise 3$\times$3 color matrices.
Because the self-consistency condition for stable collective excitations, the
gauge field and the current field of each subgroup depend on each other within
the subgroup. As a consequence, the two different currents and the gauge
fields in their respective subgroups possess independent and self-consistent
stable collective QED and QCD excitations at different energies, as shown in
Won20 and described in Section VII(B-D) and VIII. It is therefore reasonable
to study the (self-consistent) collective excitations of the quark current in
the QED interaction in the color-singlet subgroup, and the QCD interactions in
the color-octet subgroup independently. The collective QCD and QED excitations
can lead to the known QCD mesons and new, unknown QED mesons. In the case when
the excitation energy of the quark-QCD-QED medium is within the range of
$(m_{q}+m_{\bar{q}})<\sqrt{s}<m_{\pi}$, then only the color-singlet excitation
of the medium is possible, and the quarks and antiquarks can interact with QED
interactions alone.

In other considerations, it has also been known for a long time since the
advent of Wilson’s lattice gauge theory that a static fermion and a static
antifermion in (3+1)D in compact QED interaction (which corresponds to a
static quark and a static antiquark with a single color and a single flavor in
(3+1)D in compact QED interaction) has a confining phase for strong coupling
and a non-confining phase for weak coupling Wil74 . The same conclusion was
reached subsequently by Kogut, Susskind, Mandelstam, Polyakov, Banks, Jaffe,
Drell, Peskin, Guth, Kondo and many others Kog75 ; Man75 ; Pol77 ; Pol87 ;
Ban77 ; Gli77 ; Pes78 ; Dre79 ; Gut80 ; Kon98 ; Mag20 . Recent numerical
lattice gauge calculations in 3+1 dimensions in compact QED interactions using
tensor networks with dynamical quarks confirm such a strong confining
confining phase and a weak coupling non-confining phase of a static fermion
and a static antifermion systems, depending on the mass of the fermion and the
strength of the coupling Mag20 . According to the present-day lattice gauge
calculations for a static fermion and a static antifermion in compact QED in
(3+1)D, the magnitude of the QED coupling constant places the compact QED
interaction as belonging to the weak coupling non-confining regime Arn03 ;
Lov21 . Consequently, the results of the present-day lattice gauge
calculations would indicate that a static quark and a static antiquark
interacting in the compact QED interaction will not be confined in (3+1)D. How
then could the non-confinement conclusion be consistent with the proposed
confinement suggestion of a $q\bar{q}$ pair confined in a QED meson in compact
QED in (3+1)D Won10 ; Won20 ? We shall discuss the answer to this puzzling
question in detail in the next section. It suffices to mention briefly here
that that we shall propose a model in the next section to resolve whether the
opposite confinement conclusions arise from the presence of the Schwinger’s
longitudinal-confinement effect Sch62 ; Sch63 in the confinement suggestion
where the quark field is treated as massless and dynamically Won10 ; Won20 ,
and the absence of the Schwinger dynamical quark effect in lattice
calculations where the quark and the antiquark are not treated as dynamical
fields but as infinitely heavy static object fixed in spatial locations Arn03
; Lov21 .

The experimental observations, the proposed interpretation of the anomalous
particles as confined $q\bar{q}$ states, and the opposite predictions on the
$q\bar{q}$ confinement in (3+1)D provide a special impetus to find out in
future lattice gauge calculations with dynamical light quarks whether quarks
with three colors and many flavors interacting in compact QED U(1) gauge
fields will be confined or not.

Clearly, whatever the theoretical predictions there may be, the confining
property of a quark interacting with an antiquark in QED in (3+1)D can only be
definitively determined experimentally. To facilitate such experimental
confrontations, it is therefore useful to study here a model of the quark-QCD-
QED system, with approximate cylindrical symmetry in a flux tube, whether
there can be confined collective QED and QCD excitations. The knowledge of the
collective QED excitations will facilitate experimental tests of the proposed
QED meson possibilities.

This paper is organized as follows. In Section II, we discuss the the lattice
gauge calculations for charges in compact QED in (3+1)D, and we note that
heretofore these calculations have treated the charge particles as static
objects fixed in space and have not treated them as dynamical fields. Because
of their absence of the important Schwinger longitudinal-confinement Sch62 ;
Sch63 , the conclusion of the present-day lattice gauge calculations regarding
the non-confinement of charges in compact QED in (3+1)D may not be as complete
and definitive as they may appear to be. We propose a model of the production
and confinement of a $q\bar{q}$ pair in compact QED interaction to highlight
the Schwinger longitudinal-confinement effect, which compliments the Polyakov
transverse confinement effect. For our investigation, we introduce our
dynamical variables and the action integral in (3+1)D in section III, . We
treat the QCD and QED gauge interactions in a single U(3) framework which
includes both the subgroups of QCD SU(3) gauge interaction and the QED U(1)
gauge interaction with different coupling constants, for our problem with
cylindrical symmetry. We apply Polyakov’s result of transverse confinement in
QED in (2+1)D Pol77 ; Pol87 and study in section IV the quark field part of
the action integral. We separate the action integral into the transverse
(2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$ Lagrangian and the longitudinal
(1+1)D${}_{\\{x^{3},x^{0}\\}}$ Lagrangian, for a system with cylindrical
symmetry as in a flux tube. There have been many previous works to separate
the longitudinal and transverse degrees of freedom of the Dirac equation. We
shall follow methods similar to those used by Wang, Pavel, Brink, Wong and
others Wan88 ; Sai90 ; Sch90 ; Pav91 ; Gat92 ; Won94a ; Won95 , with details
given in Appendix A. The relations between various quantities in 2D and 4D are
given in Appendix B. In section V, we study the gauge field part of the action
integral and separate the action integrals according to their dominance in
transverse confinement or longitudinal confinement. In section VI, we write
down the action in the idealized (1+1)D${}_{\\{x^{0},x^{3}\\}}$ space-time
which can be the starting point for the description of the longitudinal
dynamics in the flux tube environment. In section VII, we examine the proposed
model for the production of a $q\bar{q}$ pair and study the Landau level
dynamics of quarks and antiquarks in the transverse plane under the
interaction of a longitudinal magnetic field along the flux tube. We find that
the longitudinal magnetic field confines the quark and the antiquark on the
transverse plane and there is a zero mode in which the quark and the antiquark
reside in their lowest transverse energy state with the transverse mass equal
to the rest mass. With the knowledge of the quark transverse mass and the
quark transverse spatial distribution, we idealize the (3+1)D dynamics as
(1+1)D dynamics and obtain the solution to quarks and gauge fields in the
longitudinal (1+1)D dynamics. The solutions are then used to calculate the
quark currents to generate self-consistent gauge fields. We use the Maxwell
equation and obtain the masses of bound mesons as a function of the coupling
constant. In section VIII, we compare our results of many QED and QCD
excitations in quark-QCD-QED systems with low-energy experimental data. In
section IX we present our conclusions.

##  II  A Model of $q\bar{q}$ production and confinement in compact QED in
(3+1)D

It is well-known that a $q\bar{q}$ pair is confined in a QCD meson in (3+1)D,
and we do not need to belabor the question of the $q\bar{q}$ confinement in
QCD in (3+1)D again. However, the suggestion of a $q\bar{q}$ interacting in
QED in (3+1)D may be confined in a QED meson, as proposed in the present work
and in Won10 ; Won11 ; Won14 ; Won20 ; Won20a ; Won21 ; Won22 , is subject to
a serious question. For, Wilson Wil74 , Polyakov Pol77 ; Pol87 and many other
workers Dre79 ; Kog75 ; Man75 ; Pol77 ; Pol87 ; Ban77 ; Gli77 ; Pes78 ; Dre79
; Gut80 ; Kon98 ; Mag20 showed previously that a static fermion and a static
antifermion interacting in compact QED in (3+1)D are confined only for strong
coupling, and they are not confined in weak coupling. The critical coupling
constant $\beta_{c}$, below which opposite static charges are not confined in
(3+1)D, has been determined to be $\beta_{c}$=$1/e_{c}^{2}$=1.0111331(21), or
$e_{c}^{2}$=0.988989481 Arn03 ; Lov21 . As $e_{{\rm QED}}^{2}=1/137$ and
$e_{{\rm QED}}^{2}\ll e_{c}^{2}$, the QED interaction between a static quark
and a static antiquark belongs to the weak-coupling non-confining regime.
Consequently, according to present-day lattice gauge calculations, a static
fermion and a static antifermion (or a static quark and an static antiquark)
interacting in the compact QED interaction will not be confined in (3+1)D. How
could the non-confinement conclusion from present-day lattice gauge
calculations such as Arn03 ; Lov21 be consistent with the proposed confinement
suggestion of a confine quark and antiquark in compact QED in (3+1)D from
Refs. Won10 ; Won11 ; Won14 ; Won20 ; Won20a ; Won21 ; Won22 ?

It must be pointed out however that the conclusion on the non-confinement of a
static fermion ($q$) and a static antifermion ($\bar{q}$) in compact QED in
(3+1)D in present-day lattice gauge calculations comes mainly from
calculations in which the fermion ($q$) and the antifermion $(\bar{q})$ appear
only as static external probes represented by a Wilson loop, given in terms of
the product of link variables at fixed static spatial locations along time-
like world lines Arn03 ; Lov21 ; Kog75 ; Man75 ; Pol77 ; Pol87 ; Ban77 ; Gli77
; Pes78 ; Dre79 ; Gut80 ; Kon98 ; Mag20 . The fermions are in essence static
and infinitely heavy objects in fixed spatial locations. They have not been
treated as massless (or nearly massless) dynamical fermion fields. These
calculations have not included Schwinger’s dynamical-fermion longitudinal-
confinement effect that will be present when fermions, especially massless
fermions, are treated as quanta in a dynamical fermion field.

On the other hand, the confinement suggestion of Refs. Won10 ; Won11 ; Won14 ;
Won20 ; Won20a ; Won21 ; Won22 comes from Schwinger’s longitudinal confinement
effects for massless fermion and antifermions in dynamical quark and gauge
fields. Schwinger showed that a massless fermion and a massless antifermion
are confined in QED in an exactly solvable quantum field theory, and they
behave as a confined string in a boson in (1+1)D Sch62 ; Sch63 . Schwinger’s
longitudinal confinement arises from dynamical massless fermions that can form
virtual pairs in the polarization of the vacuum in quantum field theory. We
can apply Schwinger’s results directly to our quark-QED system because quarks
are fermions, light quarks are approximately massless, and a quark and an
antiquark interact in the QED. Furthermore, a quark and an antiquarks are
known phenomenologically to be confined. Therefore, direct application of
Schwinger’s QED results implies that a quark and an antiquark are confined and
they behave as a bound boson in QED in (1+1)D.

Without the important Schwinger dynamical-quark longitudinal-confinement
effect, the lattice gauge theory conclusion on the non-confinement of a static
fermion and a static antifermion (or a static quark and a static antiquark) in
compact QED in (3+1)D in present-day lattice gauge calculations may not be as
definitive as it may appear to be. Therefore, a definitive theoretical
determination on the confinement properties of quarks interacting in compact
QED in (3+1)D remains lacking and must await future lattice gauge calculations
with dynamical light quarks. In this regard, there have been many recent
advances in efficient methods of lattice gauge calculations in compact QED
with dynamical fermions (which can be taken to be quarks) in (3+1)D using the
tensor network Mag20 , dual presentation Ben20 , magnetic-field digitization
Bau21 , regulating magnetic fluctuations Kap20 , or other efficient methods.
There are also other recent advances in the studies of $q\bar{q}$ flux tubes
in lattice gauge theories in compact U(1) QED and SU(3) QCD Ama13 ; Cos17 ;
Car13 ; Bic17 ; Bic18 . They may be utilized to study the question of
confinement of dynamical quarks in compact U(1) QED in a flux tube.

Whatever the theoretical predictions on the confinement of quarks in QED in
(3+1)D may be, in the final analysis, the question whether a $q\bar{q}$ pair
is confined in QED in (3+1)D can only be settled by experiment. In the
meantime, in the presence of the two opposite theoretical conclusions with
regard to the question of quark-antiquark confinement in QED in (3+1)D, we
consider it possible that both conclusions can still be consistent with each
other, if the opposite confinement conclusions arise from the presence of the
Schwinger’s longitudinal-confinement effect Sch62 ; Sch63 in the works of
Won10 ; Won11 ; Won14 ; Won20 ; Won20a ; Won21 ; Won22 where the quark field
is treated as massless and dynamically, and the absence of the Schwinger
dynamical quark effect in lattice calculations where the quark and the
antiquark are not treated as dynamical fields but as infinitely heavy static
object fixed in spatial locations Arn03 ; Lov21 .

From the wealth of theoretical results from Schwinger, Polyakov, Drell,
Mandelstam, t’Hooft, Nambu, Jaffe, and Gribov, Sch62 ; Sch63 ; Pol77 ; Pol87 ;
Dre79 ; Kog75 ; Man76 ; tHo74 ; Nam74 ; Gli77 ; Gri82 ; Gri99 , we wish to
construct a plausible model to demonstrate the importance of Schwinger;’s
longitudinal effect in the confinement of quarks in compact QED in (3+1)D. The
proposed model can also address puzzling experimental observations, to guide
our intuition, and to suggests future tests.

In such a demonstration, we note first of all that Schwinger’s longitudinal
confinement occurs in (1+1)D space-time, whereas the physical world is in
(3+1)D. Schwinger’s longitudinal confinement is incomplete, and it needs
transverse confinement for it to be operative in the (3+1)D physical world. We
ask, what kind of configuration in the quark-QED system would it be that could
be the idealization of the one-dimensional string. We note perceptively with
keen interest that Polyakov proved earlier that in compact QED, a fermion and
an antifermion are confined in (2+1)D on the transverse plane. We can
therefore prepare a quark-QED system in (3+1)D with transverse confinement
using Polyakov’s results of transverse confinement in QED in (2+1)D as the
input, and subsequently prepare the flux tube in a ‘stretch (2+1)D’
configuration for the application of Schwinger’s longitudinal-confinement
effect for quark confinement in (3+1)D.

Accordingly, mindful of the weak coupling of the QED gauge interaction, we
combine Schwinger’s longitudinal confinement and Polyakov’s transverse
confinement to come up with a plausible model of $q\bar{q}$ production and
confinement in compact QED in (3+1)D, as described in Fig. 2 of Won22 and Fig.
1 below. We wish to show that when we prepare a produced quark and antiquark
pair system with the transverse confinement that is made possible by
Polyakov’s transverse confinement of electric charges in compact QED in
(2+1)D, and we stretch the transversely confined (2+1)D configuration
longitudinally for the longitudinal yo-yo motion as in a flux tube,
Schwinger’s longitudinal confinement in QED in (1+1)D can play important role
in the confinement of the quark and the antiquark in QED in (3+1)D. Our task
in this manuscript will be to show that the proposed model contains the
essential elements to lead to the $q\bar{q}$ confinement in compact QED in
(3+1)D and to stable QED and QCD excitations for the description of the
anomalous particles of X17, E38, and light neutral hadrons.

We would like to explain the essential concepts of the model as proposed in
Won22 . We note that there are two different types of QED U(1) gauge
interactions possessing different confinement properties Pol77 ; Pol87 ; Dre79
. There is the compact QED U(1) gauge theory in which the gauge fields
$A^{\mu}$ are angular variables with a periodic gauge field action that allows
transverse photons to self-interact among themselves. The gauge field action
in the compact QED U(1) gauge theory, in the lattice gauge units and notations
of Ref. Pol77 ; Pol87 ; Dre79 , is

$\displaystyle S=\frac{1}{2g^{2}}\sum_{x,\alpha\beta}(1-\cos
F_{x,\alpha\beta}),$ (1)

where $g$ is the coupling constant and the gauge fields $F_{x,\alpha\beta}$
are

$\displaystyle
F_{x,\alpha\beta}\\!=\\!A_{x,\alpha}\\!\\!+\\!\\!A_{x+\alpha,\beta}\\!\\!-\\!\\!A_{x+\beta,\alpha}\\!\\!-\\!\\!A_{x,\beta},{\rm
with}-\\!\\!\pi\\!\\!\leq A_{x,\alpha}\\!\\!\leq\\!\\!\pi.~{}$ (2)

There is also the non-compact QED U(1) gauge theory with the gauge field
action Pol77 ; Pol87 ; Dre79

$\displaystyle
S=\frac{1}{4g^{2}}\sum_{x,\alpha\beta}F_{x,\alpha\beta}^{2},~{}~{}~{}{\rm
with}~{}~{}~{}~{}-\infty\leq A_{x,\alpha}\leq+\infty.$ (3)

In non-compact QED gauge theories, the transverse photons do not interact with
other transverse photons. Even though the compact and the non-compact QED
gauge theories in Eqs. (1) and (3) have the same continuum limit, they have
different confinement properties. In compact QED in (2+1)D, static opposite
charges are confined for all strength of the coupling constant. In compact QED
in (3+1)D, static opposite charges are confined only for strong coupling but
un-confined for weak coupling. In non-compact QED in (3+1)D, static opposite
charges are un-confined Pol77 ; Pol87 ; Dre79 . On the other hand, massless
fermions in QED with dynamical fermions in (1+1)D are confined Sch62 ; Sch63 .

We need to ascertain the type of the QED U(1) gauge interaction between a
quark and an antiquark in a QED meson. As pointed out not the least by Yang
Yan70 , the quantization and the commensurate properties of the electric
charges of the interacting particles imply the compact property of the
underlying QED gauge theory. Because (i) quark and antiquark electric charges
are quantized and commensurate, (ii) quarks and antiquarks are confined, and
(iii) there are pieces of experimental evidence for possible occurrence of
confined $q\bar{q}$ QED meson states as we mentioned in the Introduction, it
is therefore reasonable to propose that quarks and antiquarks interact with
the compact QED U(1) interaction.

In compact QED, Polyakov Pol77 ; Pol87 showed previously that a pair of
opposite electric charges and their gauge fields in
(2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$ are confined, and that the confinement
persists for all non-vanishing coupling constants, no matter how weak. As
explained by Drell and collaborators Dre79 , such a confinement in
(2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$ arises from the angular-variable property
of $A_{\phi}$ and the periodicity of the gauge field action as indicated in
Eqs. (1) and (2). The gauge action periodicity in the neighborhood of the
produced opposite electric charges leads to self-interacting transverse gauge
photons. These transverse gauge photons interact among themselves, they do not
radiate away, and they join the two opposite electric charges and their
associated gauge fields by a confining interaction.

Figure 1: A snapshot of the configuration of the quark charge, the antiquark
charge, gauge fields $\hbox{\boldmath${A}$},\hbox{\boldmath${E}$}$, and
${B}$=$\nabla$$\times$${A}$ during the dynamical 3D yo-yo motion of a
$q\bar{q}$ pair interacting in compact QED in a QED meson, staring from a
(2+1)D transversely confined $q\bar{q}$ system. Fig. 1($a$) is the “stretch
(2+1)D” configuration. Fig. 1($b$) is the corresponding lattice transcription,
following the Hamiltonian formulation of Drell $et~{}al.$ Dre79 .

We take the above Polyakov’s result in compact QED in (2+1)D as the starting
point, join on with Schwinger’s result in QED in (1+1)D, and construct a model
of the production and confinement of a quark and an antiquark in a QED meson
in (3+1)D Won22 . We envisage the production of the nascent $q$$\bar{q}$ pair
at the origin, $(x^{1},x^{2},x^{3})$=0, at the eigenenergy
$\sqrt{s}(q\bar{q})$ of the (hypothetical) QED meson222Because quarks and
antiquarks are confined, the production of a $q\bar{q}$ pair can occur only at
CM energies, $\sqrt{s}(q\bar{q})$, of a confined and bound $q\bar{q}$ system,
such as in a QED meson or a QCD meson. At other energies away from a confined
and bound system, the produced $q$ and $\bar{q}$ would be in a continuum state
of a separated quark and antiquark of fractional charges, which however cannot
occur. Hence, a quark and an antiquark cannot be produced at energies that are
not the eigenstate energies of a bound QCD or QED meson.. and we take for
simplicity the quark charge to be positive, which can be easily generalized to
other cases of negatively-charged quark and flavor mixing. At the moment of
their creation at the QED meson energy, the quark and the antiquark must be
endowed with appropriate relative kinetic energies in all three spatial
directions. The Schwinger exact solution for a confined pair of opposite
massless charges in a spatially linear configuration Sch62 ; Sch63 suggests
that there will be a greater kinetic energy directed at one of the three
directions, which we shall choose to be the longitudinal $x^{3}$ direction.
The creation of the charge $q\bar{q}$ pair will be accompanied by the
associated creation of their gauge fields ${A}$, ${E}$, and
$\hbox{\boldmath${B}$}=\nabla\times A$, which by causality can only be in the
neighborhood of the created charges initially. We can now apply Polyakov’s
result of confinement in (2+1)D to infer that at the moment right after their
creation, the emerging quark, antiquark, and their associated QED gauge fields
are confined in the transverse $\\{x^{1},x^{2}\\}$ plane at $x^{3}$$\sim 0$ in
(2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$, with the quark and the antiquark barely
separated transversely and longitudinally, while the gauge fields are
developing and confined in the transverse plane. Identifying the quark,
antiquark, ${B}$, ${E}$, and ${A}$ configurations at the moment of creation
with Polyakov’s (2+1)D confinement configuration means that the the created
magnetic field ${B}$ lies along the longitudinal $x^{3}$-direction, and is the
magnetic field, $-F^{12}\hbox{\boldmath${e}$}^{3}$, that confines the quark,
the antiquark, and various field quantities on the transverse
$\\{x^{1},x^{2}\\}$ plane at the moment of the $q\bar{q}$ pair creation.
Subsequent to the $q\bar{q}$ creation, while the transverse confinement limit
the extent of the transverse expansion, the longitudinal kinetic energy of the
$q\bar{q}$ pair will stretch the antiquark longitudinally along the $x^{3}$
direction away from the quark at $x^{3}(q)$=0 to execute a 3D yo-yo motion
appropriate for the bound state configuration, a snapshot of which at a time
subsequent to the $q\bar{q}$ creation is shown in Fig. 1(a). The transcription
of Fig. 1(a) in terms of the lattice link and plaquette variables is shown in
Fig. 1(b), by following the Hamiltonian formulation and the notations of Drell
$et~{}al.$ Dre79 . Specifically, in the $A^{0}$=0 gauge we specify the
canonical conjugate gauge fields ${A}$ and ${E}$ at the links in Fig. 1(b),
where we display only the $A^{1},A^{2}$ and $E^{3}$ values of the conjugate
gauge fields. The ${B}$ and ${E}$ fields are aligned along the longitudinal
axis as shown in Fig. 1. They maintain the cylindrical symmetry of the stretch
(2+1)D flux tube. They will likely lead to the desirable transverse and
longitudinal confinements of the quark and the antiquark, as the magnetic
field ${B}$ sends the quark and antiquark charges into the appropriate Landau
orbitals to execute confined transverse zero-mode harmonic oscillator zero-
point motions on their respective $\\{x^{1},x^{2}\\}$ planes. At the QED meson
eigenenergy, the electric field ${E}$ and the magnetic field ${B}$ along the
longitudinal $x^{3}$ direction send the quark and the antiquark in
longitudinal 3D longitudinal yo-yo motion. The electric charge densities obey
the Gauss law associated with the divergence of the electric field ${E}$.
Consequently, the positive electric quark charge fractions (solid circles in
Fig. 1(b)) reside at the $x^{3}$=0 plaquette vertices and the negative
electric antiquark charge fractions (open circles in Fig. 1(b)) at the
antiquark plaquette vertices at their $x^{3}$ planes. In the stretch (2+1)D
configuration, the transverse gauge fields ${A}$ on the transverse links are
copies of those on the quark and the antiquark plaquettes at $x^{3}(q)$=0 and
at $x^{3}(\bar{q})$ respectively, and they are unchanged in the stretching
motion, while the longitudinal links are all
$E^{3}$=$|\hbox{\boldmath${E}$}|/4$.

Consequently, the self-interactions of the transverse gauge fields that
initially confine the charges and the gauge fields transversely may likely be
retained. The cloud of transverse gauge fields continue to interact with each
other to maintain the transverse confinement on their transverse planes. With
transverse confinement and ${E}$ and ${B}$ aligned along the $x^{3}$
direction, it remains necessary to study the longitudinal confinement,
dynamical-quark effects, and spontaneous chiral symmetry breaking. Therefore,
we study the longitudinal dynamics of quarks and gauge fields by idealizing
the flux tube in the stretch (2+1)D configuration as a longitudinal string in
(1+1)D${}_{\\{x^{3},x^{0}\\}}$ and approximate the quarks to be massless, with
the information on the transverse degrees of freedom stored as input
parametric quantities in the idealized (1+1)${}_{\\{x^{3},x^{0}\\}}$ space-
time. In the longitudinal dynamics with massless quarks in QED in
(1+1)D${}_{\\{x^{3},x^{0}\\}}$, there is a gauge-invariant relation between
the quark current $j^{\mu}$ and the gauge field $A^{\mu}$ as given by Sch62 ;
Sch63 ; Won94

$\displaystyle j^{\mu}=\frac{g_{{}_{\rm
2D}}}{{\pi}}(A^{\mu}-\partial^{\mu}\frac{1}{\partial_{\lambda}\partial^{\lambda}}\partial_{\nu}A^{\nu}),~{}~{}~{}\mu,\lambda,\nu=0,3,$
(4)

where $g_{{}_{\rm 2D}}$ is the coupling constant in (1+1)D space-time. On the
other hand, the gauge field $A^{\mu}$ depends on the quark current $j^{\nu}$
through the Maxwell equation,

$\displaystyle\partial_{\nu}(\partial^{\nu}A^{\mu}-\partial^{\mu}A^{\nu})=-g_{{}_{\rm
2D}}j^{\mu},~{}~{}~{}\mu,\nu=0,3.$ (5)

For the longitudinal motion of the quark currents $j^{\mu}$, equations (4) and
(5) lead to

$\displaystyle-\partial_{\nu}\partial^{\nu}j^{\mu}=\frac{g_{{}_{\rm
2D}}^{2}}{\pi}j^{\mu},~{}~{}~{}\mu,\nu=0,3.$ (6)

and for the longitudinal motion of the gauge field $A^{\mu}$, equations (4)
and (5) lead to

$\displaystyle-\partial_{\nu}\partial^{\nu}A^{\mu}=\frac{g_{{}_{\rm
2D}}^{2}}{\pi}A^{\mu},~{}~{}~{}\mu,\nu=0,3.$ (7)

Thus, the quark current $j^{\mu}$ and the longitudinal gauge fields $A^{\mu}$
self-interact among themselves and build a longitudinal confining interaction
between the quark and the antiquark in (1+1)D. As a consequence, in accordance
with Schwinger’s exact solution for massless fermions in QED in (1+1)D Sch62 ;
Sch63 , the light quark and the light antiquark interacting in QED will be
longitudinally confined just as well. Possessing both transverse and
longitudinal confinements as in an open-string, the quark and the antiquark
may be confined and bound in a QED meson in (3+1)D.

Although the above model has been constructed specifically with quarks
interacting in QED in our mind, Polyakov’s transverse-confinement effect and
Schwinger’s dynamical-quark longitudinal-confinement effect apply also to
quarks interacting in QCD. For this reason, we shall include both QED and QCD
in the same footing in our subsequent investigations. It can be best
implemented by separating the longitudinal and the transverse degrees of
freedom as discussed in the Introduction. It will be the task of the present
effort to investigate whether the proposed model leads to transverse
confinement and longitudinal confinement and the presence of the two types of
excitation of the quark-QED-QCD medium.

##  III  Action Integral in (3+1)D${}_{\\{x^{1},x^{2},x^{3},x^{0}\\}}$ space-
time

We would like to study the dynamics of a system of quarks and antiquarks in
QCD and QED interactions in the model of Won22 as described in the last
section. The dynamical field variables are the quark fields $\Psi_{f}^{i}$,
the QCD gauge field $A^{\mu}_{{\rm QCD}}$, and the QED gauge field
$A^{\mu}_{{\rm QED}}$. The quark fields are $\Psi_{f}^{i}$ where $i$=1, 2, 3
are the color indices and $f$ is the flavor index. The QCD gauges fields are
$A^{\mu}_{{\rm QCD}}$=$\sum_{a=1}^{8}A^{\mu}_{a}t^{a}$, where $a$=1,…,8 are
the SU(3) generator index, $\\{t^{1},t^{2},t^{3},...,t^{8}\\}$ are the
generators of the SU(3) color-octet subgroup, and $\mu=0,1,2,3$ are the
indices of the space time coordinates $x^{\mu}$ with the signature
${g}^{\mu\mu}$=(1, -1, -1, -1). The QED gauge field is $A^{\mu}_{{\rm
QED}}$=$A^{\mu}_{0}t^{0}$ where $t^{0}$ is the generator of the color-singlet
U(1) subgroup,

$\displaystyle t^{0}=\frac{1}{\sqrt{6}}\left(\begin{matrix}1&0&0\cr 0&1&0\cr
0&0&1\end{matrix}\right).$ (8)

We shall use the summation convention over repeated indices, except when the
summation symbols are needed to resolve ambiguities.

In our problem of quarks interacting with the QCD and the QED interactions, it
is useful to treat the QED and QCD gauge interactions in a single framework by
considering an enlarged U$(3)$ group, consisting of the union of the QCD SU(3)
color-octet subgroup and the QED U(1) color-singlet subgroup of generators
Won10 . This can be achieved by adding $t^{0}$ onto the eight standard
generators of the SU$(3)$ subgroup to form the nine generators
$\\{t^{0},t^{1},...t^{8}\\}$ of the U(3) group, satisfying $2\,{\rm
Tr}\\{t^{a}t^{b}\\}=\delta^{ab}$ with $a,b=0,1,..,8$ and
$A^{\mu}$=$\sum_{a=0}^{8}A^{\mu}_{a}t^{a}$. The SU(3) and the U(1) subgroups
of the U(3) group differ in their coupling constants and communicative
properties. We consider quarks with masses $m_{f}$ with the number of flavors
$N_{f}^{\lambda}$ where $f$=$u,d,s$=1,2,3 are the flavor labels, and the
superscript $\lambda$ is the interaction label with $\lambda$=0 for the QED
color-singlet U(1) subgroup, and $\lambda$=1 for QCD color-octet SU(3)
subgroup. Because of the different mass scales associated with different gauge
fields, we can choose $N_{f}^{{{\rm QED}}}$=$N_{f}^{0}$=2 and $N_{f}^{{{\rm
QCD}}}$=$N_{f}^{1}$=3. The coupling constants333We adopt here the notations
that unless stated otherwise, $g^{\lambda}$ is actually $g_{{}_{\rm
2D}}^{\lambda}$, the coupling constant in 1+1 dimensions, and $g_{{}_{\rm
4D}}^{\lambda}$ is the coupling constants in 3+1 dimensions. The superscript
$\lambda$ specifies the interaction, with $\lambda=0$ for QED and $\lambda=1$
for QCD. The strong coupling constant $\alpha_{{\rm QCD}}$=$(g^{{}_{\rm
QCD}}_{{}_{\rm 4D}})^{2}/4\pi$ and the fine structure constant
$\alpha_{c}$=$\alpha_{{\rm QED}}$=$(g_{{}_{\rm 4D}}^{{}_{\rm
QED}})^{2}/4\pi$=$(-e_{{}_{\rm 4D}}^{{}_{\rm QED}})^{2}/(\hbar c)$=1/137. As
pointed out in Won09 ; Won10 and Eq. (75), $g_{{}_{\rm 2D}}^{\lambda}$ and
$g_{{}_{\rm 4D}}^{\lambda}$ are approximately related by the flux tube radius
$R_{T}$ as $(g_{{}_{\rm 2D}}^{\lambda})^{2}=(g_{{}_{\rm
4D}}^{\lambda})^{2}/(\pi R_{T}^{2})$, when the confining flux tube is
idealized as an open string without a structure.  $g_{{{}_{\rm 4D}}f}^{a}$ are
given explicitly by

$\displaystyle g_{{{}_{\rm 4D}}u}^{0}\\!=\\!-Q_{u}^{{\rm
QED}}\\!\\!g_{{{}_{\rm 4D}}}^{{{\rm
QED}}}\\!\\!\\!\\!,\\!~{}~{}~{}~{}~{}~{}~{}g_{{{}_{\rm
4D}}d}^{0}\\!=\\!-Q_{d}^{{\rm QED}}\\!\\!g_{{{}_{\rm 4D}}}^{{{\rm
QED}}}{\rm~{}~{}~{}~{}for~{}the~{}QED~{}color}{\rm-singlet~{}subgroup},$ (9a)
$\displaystyle g_{{{}_{\rm
4D}}{\\{u,d,s\\}}}^{\\{1,..,8\\}}=Q_{\\!\\{u,d,s\\}}^{{}^{{{\rm
QCD}}}}g_{{{}_{\rm 4D}}}^{{\rm
QCD}}{\rm~{}~{}~{}~{}~{}~{}\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}for~{}the~{}QCD~{}color}{\rm-
octet~{}subgroup},$ (9b)

where we have introduced the charge numbers $Q_{u}^{{{\rm QED}}}$=2/3,
$Q_{d}^{{{\rm QED}}}$=$-$1/3, $Q_{u}^{{{\rm QCD}}}$=$~{}Q_{d}^{{{\rm
QCD}}}$=$~{}Q_{s}^{{{\rm QCD}}}$= 1. For brevity of notations, the color
and flavor indices $\\{a$,$f\\}$ and the interaction labels $\\{\lambda$, QCD,
QED} in various quantities are often implicitly understood except when they
are needed to resolve ambiguities.

We start with the 3+1 dimensional space-time $x^{\mu}$, with the dynamical
variables of the quark fields $\Psi_{f}^{i}$ and the U(3) gauge fields
$A_{\mu}^{a}$, the invariant action integral $\cal A$ Pes95 , as modified for
compact gauge interactions444There is a factor 2 on the left-hand side of the
trace relation, $2{\rm tr}\\{t^{a}t^{b}\\}=\delta^{ab}$, for the generator
$t^{a}$ and $t^{b}$. For the convenience of notation, we shall define the
operation “$Tr$ trace” over the color space as $Tr_{\rm
color}(t^{a}t^{b})$$\equiv$2tr$\\{t^{a}t^{b}\\}=\delta^{ab}$, when we
calculate the trace of the product of generators $t^{a}t^{b}$ in the color
space.  Pol77 ; Pol87 ; Dre79 , is given by

$\displaystyle{\cal A}_{{}_{\rm 4D}}=\int
d^{4}x~{}Tr\Bigg{\\{}{\bar{\Psi}}(x)\gamma^{\mu}\Pi_{\mu}\Psi(x)-{\bar{\Psi}}(x)m\Psi(x)-{\cal
L}_{A}\Bigg{\\}},$ (10)

where

$\displaystyle\gamma^{\mu}\Pi_{\mu}=iD\\!\\!\\!\\!/=\gamma^{\mu}iD_{\mu}\\!=\\!\gamma^{\mu}(i\partial_{\mu}+g_{{}_{\rm
4D}}A_{\mu})\\!=\\!\gamma^{\mu}(p_{\mu}+g_{{}_{\rm 4D}}A_{\mu}),$ (11a)
$\displaystyle{\cal L}_{A}=\frac{1}{2\pi^{2}R_{T}^{4}g_{{}_{\rm
4D}}^{2}}[1-\cos(\pi R_{T}^{2}g_{{}_{\rm 4D}}F_{\mu\nu}(x))],$ (11b)
$\displaystyle
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig_{{}_{\rm
4D}}[A_{\mu},A_{\nu}],$ (11c) $\displaystyle F_{\mu\nu}=F_{\mu\nu}^{a}t_{a}$
(11d) $\displaystyle A_{\mu}=A_{\mu}^{a}t_{a}.$ (11e)

Here the subscript label of ‘4D’ in $g_{{}_{\rm 4D}}$ and ${\cal A}_{{}_{\rm
4D}}$ is to indicate that $g_{{}_{\rm 4D}}$ is the coupling constant in 4D
space-time, ${\cal A}_{{}_{\rm 4D}}$ is the action integral over the 4D space-
time coordinates of $x^{0},x^{1},x^{2}$, and $x^{3}$, and $m$ is the quark
mass. The quantity $\sqrt{\pi}R_{T}$ in Eq. (11b) is a transverse length scale
which has been chosen to be the square root of the flux tube area.

Not all quark systems with QCD and QED interactions in (3+1)D an be simply
separated into the transverse and longitudinal degrees of freedom. The
simplification from the higher (3+1)D to a lower dimensional (1+1)D requires
the appropriate transverse confinement and the idealization of the cylindrical
flux tube as a one-dimensional string, even if it is only approximately. The
cylindrical symmetry allows the description of the (3+1)D dynamics as the
coupling of the (2+1)D${}_{\\{x^{1},x^{2},x^{0}\\}}$ transverse dynamics and
the (1+1)D${}_{\\{x^{3},x^{0}\\}}$ longitudinal dynamics linked together by
the parametric time $x^{0}$ or other input parametric quantities.

We begin with Polyakov’s result that electric charges of opposite signs are
confined in compact QED in (2+1)D Pol77 ; Pol87 and we construct the model of
Won22 as described in Fig. 1 of the last section. Our present task is to show
that quarks and antiquarks will be transversely and longitudinally confined in
such a model. Accordingly, we consider the compact gauge field interaction in
which the length scales $\sqrt{\pi}R_{T}$ in the transverse direction is small
compared to those scales along the longitudinal $x^{3}$ and temporal $x^{0}$
directions. Because of the large length scales in the longitudinal and
temporal directions, it is reasonable to expand out the cosine function in the
continuum limit for all components except the $F_{12}$ gauge field so that the
compact gauge field Lagrangian in Eq. (11b) can be approximated as

$\displaystyle{\cal L}_{A}=\frac{1}{\pi^{2}R_{T}^{4}g_{{}_{\rm
4D}}^{2}}[1-\cos(\pi R_{T}^{2}g_{{}_{\rm
4D}}F_{12})]+\sum_{\\{\mu\nu\\}\neq\\{12,21\\}}\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$
(12)

##  IV Quark part of the Action Integral

We first examine the quark part of the action integral, ${\cal A}_{Q}$, in
(10) as given by

$\displaystyle{\cal
A}_{Q}\\!=\\!Tr\\!\\!\int\\!d^{4}x\Biggl{\\{}\\!{\bar{\Psi}}(x)\gamma^{\mu}iD_{\mu}\Psi(x)\\!-\\!{\bar{\Psi}}(x)~{}m~{}\Psi(x)\\!\Biggr{\\}},$
(13)

where we choose to work with the 4D-Dirac matrices $\gamma^{\mu}$ in the Weyl
representation Ber79

$\displaystyle\gamma^{0}=\begin{pmatrix}0&I\\\ I&0\\\
\end{pmatrix},~{}~{}~{}~{}{\gamma}^{j}=\begin{pmatrix}0&-\sigma^{j}\\\
\sigma^{j}&0\\\
\end{pmatrix},~{}~{}~{}j=1,2,3,~{}~{}~{}\text{and}~{}~{}~{}~{}~{}\gamma^{5}=\begin{pmatrix}-I&0\\\
0&I\\\ \end{pmatrix}.$ (14)

In the dynamics of the quark-QCD-QED medium, the quark fields $\Psi(x)$ depend
on the gauge fields $A_{\mu}(x)$, which in turn depend on the quark fields
$\Psi(x)$, leading to a coupling problem of great complexity. Our problem is
greatly simplified because we assume cylindrical symmetry in the model of
Won22 in Fig. 1(a), with the alignment of the ${B}$ and ${E}$ fields along the
longitudinal direction. The separation of the longitudinal and transverse
degrees of freedom in the Dirac equation can be carried out by following
methods similar to those used by Wang, Pavel, Brink, Wong, and many others
Wan88 ; Sai90 ; Sch90 ; Pav91 ; Gat92 ; Won94a ; Won95 . In the Weyl
representation Ber79 , we write the quark bispinor quark field $\Psi(x)$ in
terms of transverse functions $G_{1,2}({\hbox{\boldmath${r}$}}_{\perp})$ and
longitudinal-temporal functions ${f}_{\pm}(X)$ as

$\displaystyle\\!\\!\Psi_{{}_{\rm
4D}}=\\!\\!\Psi(x)=\begin{pmatrix}G_{1}({\hbox{\boldmath${r}$}}_{\perp})f_{+}(X)\\\
G_{2}({\hbox{\boldmath${r}$}}_{\perp})f_{-}(X)\\\
G_{1}({\hbox{\boldmath${r}$}}_{\perp})f_{-}(X)\\\
-G_{2}({\hbox{\boldmath${r}$}}_{\perp})f_{+}(X)\\\ \end{pmatrix},$ (15)

where $x$=$\\{x^{1},x^{2},x^{3},x^{0}\\}$,
$\hbox{\boldmath${r}$}_{\perp}$=$\\{x^{1},x^{2}\\}$, and
$X$=$\\{x^{3},x^{0}\\}$. The transverse functions
$G_{\\{1,2\\}}({\hbox{\boldmath${r}$}}_{\perp})$ are normalized as

$\displaystyle\int\\!\\!d\hbox{\boldmath${r}$}_{\perp}\\{(G_{1}^{*}(\hbox{\boldmath${r}$}_{\bot}\\!)G_{1}(\hbox{\boldmath${r}$}_{\bot}\\!)\\!+\\!G_{2}^{*}(\hbox{\boldmath${r}$}_{\bot}\\!)G_{2}(\hbox{\boldmath${r}$}_{\bot})\\}\\!=\\!1,$
(16)

and the longitudinal functions $f_{\pm}(x^{3},x_{0})$ are normalized as

$\displaystyle\int
dx^{3}dx^{0}\\{f_{+}^{*}(x^{3},x^{0})f_{+}(x^{3},x^{0^{\prime}})+f_{-}(x^{3},x^{0^{\prime}})f_{-}^{*}(x^{3},x^{0})\\}\delta(x^{0}-x^{0^{\prime}})=1.$
(17)

The quark Lagrangian density in the Dirac representation becomes

$\displaystyle{\cal L}_{Q}$ $\displaystyle=\bar{\Psi}_{{}_{\rm
4D}}\gamma^{0}\Pi^{0}\psi_{{}_{\rm 4D}}-\bar{\Psi}_{{}_{\rm
4D}}\gamma^{3}\Pi^{3}\Psi_{{}_{\rm 4D}}-\bar{\Psi}_{{}_{\rm
4D}}\gamma^{1}\Pi^{1}\Psi_{{}_{\rm 4D}}-\bar{\Psi}_{{}_{\rm
4D}}\gamma^{2}\Pi^{2}\Psi_{{}_{\rm 4D}}-m\bar{\Psi}_{{}_{\rm 4D}}\Psi_{{}_{\rm
4D}}$ (18)
$\displaystyle=(G_{1}^{*}G_{1}+G_{2}^{*}G_{2})(f_{+}^{*}\Pi^{0}f_{+}+f_{-}^{*}\Pi^{0}f_{-})-(G_{1}^{*}G_{1}+G_{2}^{*}G_{2})(f_{+}^{*}\Pi^{3}f_{+}-f_{-}^{*}\Pi^{3}f_{-})$
$\displaystyle~{}~{}~{}-(G_{1}^{*}\Pi^{1}G_{2}+G_{2}^{*}\Pi^{1}G_{1})(f_{+}^{*}f_{-}+f_{-}^{*}f_{+})-(-G_{1}^{*}~{}i\Pi^{2}~{}G_{2}+G_{2}^{*}~{}i\Pi^{2}~{}G_{1})(f_{+}^{*}f_{-}+f_{-}^{*}f_{+})$
$\displaystyle~{}~{}~{}-m(G_{1}^{*}G_{1}-G_{2}^{*}G_{2})(f_{+}^{*}f_{-}+f_{-}^{*}f_{+})$

As shown in Appendix A, the minimization of the action integral with respect
to $G_{\\{1,2\\}}^{*}f_{\pm}^{*}$ leads to the equations of motion

$\displaystyle\delta^{2}{\cal L}/\delta f_{+}^{*}\delta G_{1}^{*}=$
$\displaystyle(\Pi^{0}-\Pi^{3})G_{1}f_{+}-mG_{1}f_{-}-(\Pi^{1}-i\Pi^{2})G_{2}f_{-}=0,$
(19a) $\displaystyle\delta^{2}{\cal L}/\delta f_{+}^{*}\delta G_{2}^{*}=$
$\displaystyle(\Pi^{0}-\Pi^{3})G_{2}f_{+}+mG_{2}f_{-}-(\Pi^{1}+i\Pi^{2})G_{1}f_{-}=0,$
(19b) $\displaystyle\delta^{2}{\cal L}/\delta f_{-}^{*}\delta G_{1}^{*}=$
$\displaystyle(\Pi^{0}+\Pi^{3})G_{1}f_{-}-mG_{1}f_{+}-(\Pi^{1}-i\Pi^{2})G_{2}f_{+}=0,$
(19c) $\displaystyle\delta^{2}{\cal L}/\delta f_{-}^{*}\delta G_{2}^{*}=$
$\displaystyle(\Pi^{0}+\Pi^{3})G_{2}f_{-}+mG_{2}f_{+}-(\Pi^{1}+i\Pi^{2})G_{1}f_{+}=0,$
(19d)

which agree with Eqs. (13a)-(13d) of Won09 using the Dirac representation. For
gauge fields $A^{1}$ and $A^{2}$ (and consequently, $\Pi^{1}$ and $\Pi^{2}$)
independent of the longitudinal coordinates as we assumed in the model of
Won22 and Fig. 1, Appendix A shows that the above set of four equations can be
separated into a set of two transverse equations for
$G_{\\{1,2\\}}(\hbox{\boldmath${r}$}_{\perp})$ and another set of two
longitudinal-temporal equations for $f_{\pm}(X)$. The set of equations for the
transverse degrees of freedom are

$\displaystyle-
m_{T}G_{1}(\hbox{\boldmath${r}$}_{\perp})+mG_{1}(\hbox{\boldmath${r}$}_{\perp})+(\Pi^{1}-i\Pi^{2})G_{2}(\hbox{\boldmath${r}$}_{\perp})=0,$
(20a) $\displaystyle-
m_{T}G_{2}(\hbox{\boldmath${r}$}_{\perp})-mG_{2}(\hbox{\boldmath${r}$}_{\perp})+(\Pi^{1}+i\Pi^{2})G_{1}(\hbox{\boldmath${r}$}_{\perp})=0.$
(20b)

where the transverse mass $m_{T}$ is the constant of the separation of
variables, which is also the eigenvalue of the set of equations. They lead to
the transverse equations,

$\displaystyle\left\\{\Pi_{T}^{2}+i[\Pi^{2},\Pi^{1}]+m^{2}-m_{T}^{2}\right\\}G_{2}(\hbox{\boldmath${r}$}_{\perp})=0$
(21a)
$\displaystyle\left\\{\Pi_{T}^{2}+i[\Pi^{1},\Pi^{2}]+m^{2}-m_{T}^{2}\right\\}G_{1}(\hbox{\boldmath${r}$}_{\perp})=0,$
(21b)

where $\Pi_{T}^{2}=(\Pi^{1})^{2}+(\Pi^{2})^{2}$. The longitudinal equations
are

$\displaystyle\int
d\hbox{\boldmath${r}$}_{\perp}(|G_{1}(\hbox{\boldmath${r}$}_{\perp})|^{2}+|G_{2}(\hbox{\boldmath${r}$}_{\perp})|^{2})(\Pi^{0}-\Pi^{3})f_{+}(X)-m_{T}f_{-}(X)=0,$
(22a) $\displaystyle\int
d\hbox{\boldmath${r}$}_{\perp}(|G_{1}(\hbox{\boldmath${r}$}_{\perp})|^{2}+|G_{2}(\hbox{\boldmath${r}$}_{\perp})|^{2})(\Pi^{0}+\Pi^{3})f_{-}(X)-m_{T}f_{+}(X)=0.$
(22b)

The operators $\Pi^{\mu}$ with $\mu=0,3$ in Eqs. (22a) and (22b) are actually
$\Pi_{{}_{\rm 4D}}^{\mu}$=$p^{\mu}+g_{{}_{\rm 4D}}A_{{}_{\rm
4D}}^{\mu}(\hbox{\boldmath${r}$}_{\perp},X)$, where we have added the label 4D
to indicate that the gauge fields $A_{{}_{\rm
4D}}^{\mu}(\hbox{\boldmath${r}$}_{\perp},X)$ are functions of the (3+1)D
coordinates $(\hbox{\boldmath${r}$}_{\perp},X)$ and the constant $g_{{}_{\rm
4D}}$ is the dimensionless coupling constant of the gauge fields in (3+1)D
space-time. In contrast, in the idealized (1+1)D${}_{\\{x^{0},x^{3}\\}}$
space-time with a structureless string, the corresponding gauge fields
$A_{{}_{\rm 2D}}^{\mu}(X)$ are functions only of the coordinates $X$, and they
satisfy the (1+1)D Maxwell equation with a coupling constant $g_{{}_{\rm
2D}}$.

We would like to cast Eqs. (22a) and (22b) into the form of a Dirac equation
for the quark fields in the idealized (1+1)D${}_{\\{x^{0},x^{3}\\}}$ space-
time by separating the transverse degrees of freedom under a coupled treatment
as discussed in the Introduction. This can be achieved by carrying out the
integrating over the transverse coordinates $\hbox{\boldmath${r}$}_{\perp}$ in
Eqs. (22a) and (22b) and identifying the term as as $A_{{}_{\rm 2D}}^{\mu}(X)$
in the (1+1)D space-time with the the coupling constant $g_{{}_{\rm 2D}}$,

$\displaystyle g_{{}_{\rm 4D}}\int
d\hbox{\boldmath${r}$}_{\perp}(|G_{1}(\hbox{\boldmath${r}$}_{\perp})|^{2}+|G_{2}(\hbox{\boldmath${r}$}_{\perp})|^{2})A_{{}_{\rm
4D}}^{\mu}(\hbox{\boldmath${r}$}_{\perp},X)\equiv g_{{}_{\rm 2D}}A_{{}_{\rm
2D}}^{\mu}(X),~{}~{}~{}\mu=0,3.$ (23)

As shown in Eq. (B10) in Appendix B, consistent gauge field solutions of the
Maxwell equations in (3+1)D and (1+1)D in the flux tube environment are
related by

$\displaystyle A_{{}_{\rm
4D}}^{\mu}(\hbox{\boldmath${r}$}_{\perp},X)=\frac{g_{{}_{\rm 4D}}}{g_{{}_{\rm
2D}}}\biggl{[}G_{1}^{*}(\hbox{\boldmath${r}$}_{\perp})G_{1}(\hbox{\boldmath${r}$}_{\perp})+G_{2}^{*}(\hbox{\boldmath${r}$}_{\perp})G_{2}(\hbox{\boldmath${r}$}_{\perp})\biggr{]}A_{{}_{\rm
2D}}^{\mu}(X),~{}~{}~{}~{}~{}\mu=0,3.$ (24)

As a consequence, the coupling constant $g_{{}_{\rm 2D}}$ and $g_{{}_{\rm
4D}}$ are related by

$\displaystyle g_{{}_{\rm 2D}}^{2}=g_{{}_{\rm 4D}}^{2}\int
d\hbox{\boldmath${r}$}_{\perp}\biggl{[}G_{1}(\hbox{\boldmath${r}$}_{\perp})^{*}G_{1}(\hbox{\boldmath${r}$}_{\perp})+G_{2}(\hbox{\boldmath${r}$}_{\perp})^{*}G_{2}(\hbox{\boldmath${r}$}_{\perp})\biggr{]}^{2},$
(25)

which shows that the coupling constant $g_{{}_{\rm 2D}}$ now acquires the
dimension of a mass. By such an introduction of $A_{{}_{\rm 2D}}^{\mu}(X)$ and
$g_{{}_{\rm 2D}}$, Eqs. (22a) - (22b) become

$\displaystyle(\Pi_{{}_{\rm 2D}}^{0}-\Pi_{{}_{\rm
2D}}^{3})f_{+}(X)-m_{T}f_{-}(X)=0,$ (26a) $\displaystyle(\Pi_{{}_{\rm
2D}}^{0}+\Pi_{{}_{\rm 2D}}^{3})f_{-}(X)-m_{T}f_{+}(X)=0,$ (26b)

where

$\displaystyle\Pi_{{}_{\rm 2D}}^{\mu}=p^{\mu}+g_{{}_{\rm 2D}}A_{{}_{\rm
2D}}^{\mu}(X),~{}~{}~{}~{}\mu=0,3.$ (27)

These equations (26a) and (26b) can be cast further into the following matrix
equation

$\displaystyle\biggl{\\{}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\Pi_{{}_{\rm
2D}}^{0}-\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}\Pi_{{}_{\rm
2D}}^{3}-m_{T}\biggr{\\}}\begin{pmatrix}f_{+}(X)\\\ f_{-}(X)\end{pmatrix}=0.$
(28)

It is then useful to introduce 2D gamma matrices $\gamma_{{}_{\rm 2D}}^{\nu}$
as in Wit84

$\displaystyle\gamma_{{}_{\rm 2D}}^{0}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},~{}~{}~{}\gamma_{{}_{\rm 2D}}^{3}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\text{ with }\gamma_{{}_{\rm 2D}}^{5}=\gamma_{{}_{\rm
2D}}^{0}\gamma_{{}_{\rm 2D}}^{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$
(29)

and the longitudinal 2D wave function $\psi_{{}_{\rm 2D}}(X)$,

$\displaystyle\psi_{{}_{\rm 2D}}(X)=\begin{pmatrix}f_{+}(X)\\\
f_{-}(X)\end{pmatrix},$ (30)

to rewrite Eq. (28) as a Dirac equation for the longitudinal (1+1)D motion,
interacting with a gauge field with a coupling constant $g_{{}_{\rm 2D}}$

$\displaystyle\\{\gamma_{{}_{\rm 2D}}^{0}\Pi_{{}_{\rm 2D}}^{0}-\gamma_{{}_{\rm
2D}}^{3}\Pi_{{}_{\rm 2D}}^{3}-m_{T}\\}\psi_{{}_{\rm
2D}}(X)=0,~{}~{}~{}\mu=0,3,$ (31)

or

$\displaystyle\\{\gamma_{{}_{\rm 2D}}^{\mu}(p_{\mu}+g_{{}_{\rm
2D}}A_{\mu}^{{}_{\rm 2D}}(X))-m_{T}\\}\psi_{{}_{\rm
2D}}(X)=0,~{}~{}~{}\mu=0,3.$ (32)

It is important to note that in idleiazing the dynamics from (3+1)D to (1+1)D
in the above manipulations, relevant information regarding the transverse
degrees of freedom is now subsumed and stored in the quark transverse mass
eigenvalue $m_{T}$ of the set of eigenvalue equations, Eq. (21), and
$g_{{}_{\rm 2D}}$ which is related to $g_{{}_{\rm 4D}}$ and the transverse
eigenfunctions $G_{\\{1,2\\}}(\hbox{\boldmath${r}$}_{\perp})$ by Eq. (25). By
such supplementary relations, the information on the transverse degrees of
freedom is packaged and passed on from the (3+1)D to the idealized (1+1)D
dynamics. In other words, upon preparing and getting the solution in the
transverse plane for a system with a cylindrical symmetry, the transverse
properties relevant for the longitudinal dynamics are contained in the
transverse mass $m_{T}$ and the gauge field coupling constant $g_{{}_{\rm
2D}}$.

The Dirac equation of motion in (1+1)D space-time corresponds to the 2D action
integral for quarks in the form

$\displaystyle\\!\\!{\cal A}_{{{}_{\rm
2D}}Q}\\!=\\!Tr\\!\\!\int\\!\\!dX\Biggl{\\{}\\!\bar{\psi}_{{{}_{\rm
2D}}}(X)\gamma_{{{}_{\rm 2D}}}^{\mu}(p_{\mu}\\!+\\!g_{{{}_{\rm
2D}}}A_{\mu}(X))\psi_{{{}_{\rm 2D}}}(X)-m_{T}{\bar{\psi}}_{{{}_{\rm
2D}}}(X)\psi_{{{}_{\rm 2D}}}(X)\Bigg{\\}}$ (33)

with the normalization condition (16) which can be re-written in terms of
$\bar{\psi}_{{}_{\rm 2D}},\psi_{{}_{\rm 2D}}$ functions as

$\displaystyle\int dx^{3}dx^{0}\bar{\psi}_{{}_{\rm
2D}}(x^{3},x^{0})\gamma^{0}_{{}_{\rm 2D}}\psi_{{}_{\rm
2D}}({x^{3}},{x^{0}}^{\prime})\delta(x^{0}-{x^{0}}^{\prime})=1.$ (34)

For brevity of notation, the descriptors 2D and 4D are often omitted unless
otherwise needed to resolve ambiguities, as the 2D and 4D nature of each
function or gamma matrix can be inferred from the arguments of the function or
from the context.

##  V Gauge field part of the action integral

The confinement properties of the quark-QCD-QED system consists of the
transverse confinement on the transverse plane and the longitudinal
confinement. The confinement pertains not only to the quarks and antiquarks,
but also to the gauge fields which link the quarks and the antiquarks. That
is, in compact QED and non-Abelian QCD, the gauge fields also self-interact
among themselves so that they do not readily radiate away. They will also
retain the confinement properties when the quark-QCD-QED system settles into
one of the meson states.

At the moment of the creation of the quark and the antiquark pair for a QCD or
a QED meson state, the gauge fields are also created. We can apply Polyakov’s
result on the confinement in (2+1)D to infer the presence of transverse
confinement of the quarks, antiquarks, and the gauge fields at the moment
immediately after the creation of the $q\bar{q}$ pair.

The occurrence of transverse confinement makes it convenient to partition the
gauge field actions into three pertinent parts,

$\displaystyle{\cal A}_{A}={\cal A}_{A}^{\rm I}+{\cal A}_{A}^{\rm II}+{\cal
A}_{A}^{\rm III}$ (35)

where ${\cal A}_{A}^{\rm I}$ and ${\cal A}_{A}^{\rm II}$ pertains to
transverse gauge fields and transverse confinement, while ${\cal A}_{A}^{\rm
III}$ is concerned mainly with longitudinal gauge field dynamics and
longitudinal confinement:

$\displaystyle{\cal A}_{A}^{\rm I}$ $\displaystyle=-\int
d^{4}x\frac{1}{\pi^{2}R_{T}^{4}g_{{}_{\rm 4D}}^{2}}~{}Tr[1-\cos(\pi
R_{T}^{2}g_{{}_{\rm 4D}}F_{12})],$ (36a) $\displaystyle{\cal A}_{A}^{\rm II}$
$\displaystyle=-\int
d^{4}xTr~{}\frac{1}{2}\left[F_{23}F^{23}+F_{13}F^{13}+F_{01}F^{01}+F_{02}F^{02}\right],$
(36b) $\displaystyle{\cal A}_{A}^{\rm III}$ $\displaystyle=-\int
d^{4}xTr~{}\frac{1}{2}F_{03}F^{03}.$ (36c)

We can alternatively express the gauge field tensors $F^{\mu\nu}$ in terms of
the gauge field vectors as
${E}$=$\sum_{i=1}^{3}E^{i}\hbox{\boldmath${e}$}^{i}$ and
${B}$=$\sum_{i=1}^{3}B^{i}\hbox{\boldmath${e}$}^{i}$. The action partition
${\cal A}_{A}^{\rm I}$ contains the magnetic field $F^{12}$=$-B^{3}$, which
lies along the longitudinal direction perpendicular to the transverse plane.
Such a magnetic field ${B}$ is essential in sending the quarks and antiquarks
into the Landau level orbitals that confine them on the transverse plane, as
we shall demonstrate in Section VIIA. The zero mode of the Landau levels
allows the quark and the antiquark to retain an effective mass $m_{T}$ that is
the same as the quark rest mass $m$. Furthermore, As discussed in detail in
Drell $et~{}al.$ Dre79 , such a magnetic field ${B}$ is also essential in
setting up the self-interaction between transverse gauge fields because the
periodicity of the action cosine function in ${\cal A}_{A}^{\rm I}$ allows the
transverse photons to interact among themselves to build up a confining
interaction between the quark, the antiquark, and the other transverse gauge
fields themselves in the transverse confinement in (2+1)D.

The second action partition ${\cal A}_{A}^{\rm II}$ pertains to
$F^{23}=-B^{1}$, $F^{31}$=$-B^{2}$, $F^{01}$=$-E^{1}$, $F^{02}$=$-E^{2}$ which
are the magnetic and electric field components lying on the transverse
$\\{x^{1},x^{2}\\}$ plane, and they participate directly in the transverse
confinement of the quarks and the transverse gauge fields on the transverse
plane.

Subsequent to the creation of the quark and the antiquark valence particles,
the quark and the antiquark will execute their longitudinal yo-yo motion. In
the proposed model in the stretch (2+1)D configuration with the gauge fields
that are copies of the gauge field configuration of the initial quark and
antiquark transverse plaquettes, the transverse confinement of the gauge field
will likely be retained.

There is the third partition ${\cal A}_{A}^{\rm III}$ pertaining to the action
integral involving the gauge fields $F^{03}$. Such a gauge field participates
in longitudinal dynamics with a linear interaction between the quark and the
antiquark in the form of a string. It is the essential ingredient in
Schwinger’s longitudinal confinement of opposite charges in QED in (1+1)D. As
Schwinger’s confinement of a pair of opposite massless charges occur
irrespective of the strength of coupling constant, we expect that it will be
the dominant driving force for longitudinal confinement. For this reason, in
our consideration of the question of longitudinal confinement, it will be
reasonable to treat ${\cal A}_{A}^{\rm I}$ as the dominant action and ${\cal
A}_{A}^{\rm I}$ and ${\cal A}_{A}^{\rm II}$ as perturbations. In a similar
manner, it is just as reasonable in the investigation of transverse
confinement to treat ${\cal A}_{A}^{\rm I}$ and ${\cal A}_{A}^{\rm II}$ as the
dominant action and ${\cal A}_{A}^{\rm III}$ as a perturbation.

###  V.1 First, the gauge field action with the ${\cal A}_{A}^{\rm I}$ and
${\cal A}_{A}^{\rm II}$ partitions

As discussed in detail in Drell $et~{}al.$ Dre79 , the periodicity of the
angular function $\cos(\pi R_{T}^{2}g_{{}_{\rm 4D}}F_{12})$ in the gauge field
action leads to the self-interaction of the transverse gauge fields. For
strong coupling such as QCD, the strong self-interaction among the transverse
gauge fields leads to a linear confining interaction joining the quark and the
antiquark by a flux tube. For weak coupling such as the compact QED
interaction, quantum fluctuations occur which randomize the coherent self-
interaction. In the case of (3+1)D, such random fluctuations disrupt the
linear confinement so greatly that for a static heavy quark without dynamical-
quark effects a static quark and a static antiquark would not be
longitudinally confined in compact QED with weak coupling Pol77 ; Pol87 ;
Dre79 . Nevertheless, for the compact QED in (2+1)D, this quantum
randomization can never be sufficient to destroy the linear confinement in
two-dimensional spatial space, no matter how weak the compact gauge
interaction may be. For our problem with the creation of a $q\bar{q}$ pair in
compact QED, the confinement of opposite charges in (2+1)D confinement can be
utilized to infer transverse confinement at the moment of the creation of the
quark and the antiquark. Thereafter, we stretch the incipient transversely
confined quark and antiquark longitudinally to execute the longitudinal yo-yo
motion appropriate for the $q\bar{q}$ at the QED meson eigenenergy, as
suggested in the model of Won22 , by copying the configurations of the
transverse links and the transverse fields longitudinally from the incipient
quark position $x^{3}(q)=0$ position to the momentary antiquark
$x^{3}(\bar{q})$ position in a coherent manner as in Fig. 1(b). Because the
transverse links are copies of those of the initial configuration, the
transverse confinement will be likely retained. The remaining problem will be
to investigate whether there will be the longitudinal confinement in compact
QED in the stretch (2+1)D configuration.

For the scratch (2+1)D single-plaquette configuration in Fig. 1(b) in the
model Won22 , the integral in Eq. (36b) will be proportional to the
longitudinal separation between the quark and the antiquark,
$|x^{3}(\bar{q})-x^{3}(q)|$. With $n=n_{1}n_{2}n_{3}$ and
$n_{3}\sqrt{\pi}R_{T}=|x^{3}(\bar{q})-x^{3}(q)|$, we have

$\displaystyle{\cal A}_{A}^{\rm I}$ $\displaystyle=-Tr\int
dt~{}\sum_{n_{1}n_{1}n_{2}}~{}\frac{1}{\pi R_{T}^{2}g_{{}_{\rm
4D}}^{2}}[1-\cos(\pi R_{T}^{2}g_{{}_{\rm
4D}}F_{12}(n({\hbox{\boldmath${r}$}}_{\perp},X)))]\bigg{|}_{X=\\{x^{0}x^{3}\\}=0},$
(37) $\displaystyle=-Tr\int
dt~{}n_{3}\sqrt{\pi}R_{T}\sum_{n_{1}n_{2}}~{}\frac{1}{\pi R_{T}^{2}g_{{}_{\rm
4D}}^{2}}[1-\cos(\pi R_{T}^{2}g_{{}_{\rm
4D}}F_{12}(n({\hbox{\boldmath${r}$}}_{\perp},X)))]\bigg{|}_{X=\\{x^{0}x^{3}\\}=0},$
$\displaystyle=-\int dt~{}~{}\kappa_{1}|x^{3}(\bar{q})-x^{3}(q)|,$

where $\kappa_{1}$ is a function of $F_{12}$

$\displaystyle\kappa_{1}=\frac{n_{3}(\sqrt{\pi}R_{T})}{\pi R_{T}^{2}g_{{}_{\rm
4D}}^{2}}\sum_{n_{1}n_{2}}Tr~{}[1-\cos(\pi R_{T}^{2}g_{{}_{\rm
4D}}F_{12}(n({\hbox{\boldmath${r}$}}_{\perp},X)))]\bigg{|}_{X=\\{x^{0}x^{3}\\}=0}.$
(38)

The above action gives a longitudinal linear interaction between the quark and
the antiquark arising from the gauge field $F^{12}$ that is favorable for the
longitudinal confinement of the quark and the antiquark.

With the stretch (2+1)D configuration in our model, the transverse gauge
fields between the quark plaquette and the antiquark plaquette are copies of
those on the quark plaquette as in Fig. 1(b), the action can be written as

$\displaystyle{\cal A}_{A}^{\rm II}$ $\displaystyle=~{}~{}-\int dtdx^{3}\int
dx^{1}dx^{2}Tr~{}\frac{1}{2}\left[F_{23}F^{23}+F_{13}F^{13}+F_{01}F^{01}+F_{02}F^{02}\right],$
(39) $\displaystyle=~{}~{}-\int dt~{}|x^{3}(\bar{q})-x^{3}(q)|\times\int
dx^{1}dx^{2}~{}Tr~{}\frac{1}{2}\left[(B^{1})^{2}+(B^{2})^{2}+(E^{1})^{2}+(E^{2})^{2}\right]\biggr{|}_{X=\\{x^{0},x^{3}\\}=0}$
$\displaystyle=~{}~{}-\int dt~{}\kappa_{2}|x^{3}(\bar{q})-x^{3}(q)|$

where $\kappa_{2}$ is

$\displaystyle\kappa_{2}=\int
dx^{1}dx^{2}~{}Tr~{}\frac{1}{2}\left[(B^{1})^{2}+(B^{2})^{2}+(E^{1})^{2}+(E^{2})^{2}\right]\biggr{|}_{X=\\{x^{0},x^{3}\\}=0}$
(40)

The action ${\cal A}_{A}^{\rm II}$ also contribute a linear interaction that
will facilitate longitudinal confinement.

###  V.2 Next, the gauge field action with the ${\cal A}_{A}^{\rm III}$
containing $F^{03}$

Having simplified the gauge field actions ${\cal A}_{A}^{\rm I}$ and ${\cal
A}_{A}^{\rm II}$, we come to examine ${\cal A}_{A}^{\rm III}$ of Eq. (36)
pertaining to $F^{03}$ of the action integral. We would like to find out what
will be the form of the gauge action ${\cal A}_{A}^{\rm III}$ when the gauge
fields $A_{\mu}(\hbox{\boldmath${r}$}_{\perp},X)$ in (3+1)D and the gauge
fields $A_{\mu}(X)$ in (1+1)D, with $\mu=0,3$, are related to each other by
Eq. (24). Equations (11c) and (24) give
$F_{03}({\hbox{\boldmath${r}$}}_{\perp},X)$ in four-dimensional space-time as

$\displaystyle F_{03}({\hbox{\boldmath${r}$}}_{\perp},X)$ $\displaystyle=$
$\displaystyle\frac{g_{4D}}{g_{2D}}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}][\partial_{0}A_{3}(X)-\partial_{3}A_{0}(X)]$
(41)
$\displaystyle-i\frac{g^{2}_{4D}}{g_{2D}}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|}][A_{0}(X),A_{3}(X)].$

On the other hand, the gauge field $F_{03}$ in (1+1)D${}_{\\{x^{3},x^{0}\\}}$
space-time is given, by definition, as

$\displaystyle F_{03}(X)$ $\displaystyle=$
$\displaystyle\partial_{0}A_{3}(X)-\partial_{3}A_{0}(X)-i{g}_{2D}[A_{0}(X),A_{3}(X)].$
(42)

As a consequence, $F_{03}(X)$ in two-dimensional space-time and
$F_{03}({\hbox{\boldmath${r}$}}_{\perp},X)$ in four-dimensional space-time are
related by

$\displaystyle F_{03}({\hbox{\boldmath${r}$}}_{\perp},X)$ $\displaystyle=$
$\displaystyle\frac{g_{4D}}{g_{2D}}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}]\\{F_{03}(X)+i{g}_{2D}[A_{0}(X),A_{3}(X)]\\}$
(43)
$\displaystyle-i\frac{g^{2}_{4D}}{g_{2D}}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}][A_{0}(X),A_{3}(X)].$

The above equation can be re-written as

$\displaystyle F_{03}({\hbox{\boldmath${r}$}}_{\perp},X)$ $\displaystyle=$
$\displaystyle\frac{g_{4D}}{g_{2D}}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}]\biggl{\\{}F_{03}(X)$
(44)
$\displaystyle+\biggl{[}i{g}_{2D}-i\frac{g^{2}_{4D}}{g_{2D}}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}]\biggr{]}[A_{0}(X),A_{3}(X)]\biggr{\\}}.$

The product $F_{03}(x)F^{03}(x)$ in Eq. (36c) becomes

$\displaystyle
F_{03}({\hbox{\boldmath${r}$}}_{\perp},X)F^{03}({\hbox{\boldmath${r}$}}_{\perp},X)=\frac{g^{2}_{4D}}{g^{2}_{2D}}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}]^{2}\biggl{\\{}F_{03}(X)F^{03}(X)$
$\displaystyle~{}~{}~{}~{}+\biggl{[}i{g}_{2D}-i\frac{g^{2}_{4D}}{g_{2D}}\\{{(|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}})\\}\biggr{]}\biggl{(}F_{03}(X)[A^{0}(X),A^{3}(X)]+[A_{0}(X),A_{3}(X)]F^{03}(X)\biggr{)}$
$\displaystyle~{}~{}~{}+\biggl{[}i{g}_{2D}-i\frac{g^{2}_{4D}}{g_{2D}}[{(|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}})]\biggr{]}^{2}\biggl{(}[A_{0}(X),A_{3}(X)][A^{0}(X),A^{3}(X)]\biggr{)}\biggr{\\}}.$
(45)

The action integral ${\cal A}_{A}^{\rm III}$ in Eq. (36c) involves the
integration of the above quantity over $x_{1}$ and $x_{2}$. Upon integration
over $x^{1}$ and $x^{2}$, the second term inside the curly bracket of the
above equation, is zero,

$\displaystyle\int
dx^{1}dx^{2}[{(|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}})]\biggl{[}i{g}_{2D}-i\frac{g^{2}_{4D}}{g_{2D}}[{(|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}})]^{1/2}\biggr{]}=0,~{}~{}~{}~{}~{}$
(46)

where we have used the relation between ${g}_{{{}_{\rm 2D}}}$ and
${g}_{{{}_{\rm 4D}}}$ as given by Eq.(25) and the normalization condition of
(16). As a consequence, the integral of $F_{03}(x)F^{03}(x)$ in Eq. (36c)
becomes

$\displaystyle\int
d^{4}xF_{03}({\hbox{\boldmath${r}$}}_{\perp},X)F^{03}(x^{0},x^{3},{\hbox{\boldmath${r}$}}_{\perp})=\int
d^{4}x[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}]\biggl{\\{}F_{03}(X)F^{03}(X)$
$\displaystyle+\biggl{[}i{g}_{2D}-i\frac{g^{2}_{4D}}{g_{2D}}[{(|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}})]\biggr{]}^{2}\biggl{(}[A_{0}(X),A_{3}(X)][A^{0}(X),A^{3}(X)]\biggr{)}\biggr{\\}}.$
(47)

For the second term in the curly bracket, the integral over $dx^{1}$ and
$dx^{2}$ is

$\displaystyle\int
dx^{1}dx^{2}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}]\biggl{[}i{g}_{2D}-i\frac{g^{2}_{4D}}{g_{2D}}[{(|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}})]^{1/2}\biggr{]}^{2},$
(48)

which can be considered as an integral over ${g}_{2D}$ in the form

$\displaystyle\frac{2}{g_{2D}}\int d(g^{2}_{2D})\int
dx^{1}dx^{2}[{|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}}][ig^{2}_{2D}-ig^{2}_{4D}[{(|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}})]]=0.$
(49)

Because of Eq. (46), the above integral gives an irrelevant constant which we
can set to zero. After these manipulations, we obtain

$\displaystyle\int
d^{4}xF_{03}({\hbox{\boldmath${r}$}}_{\perp},X)F^{03}({\hbox{\boldmath${r}$}}_{\perp},X)$
$\displaystyle=$ $\displaystyle\int
dx^{1}dx^{2}dx^{3}dx^{0}[{(|G_{1}({\hbox{\boldmath${r}$}}_{\perp})|^{2}+|G_{2}({\hbox{\boldmath${r}$}}_{\perp})|^{2}})]F_{03}(X)F^{03}(X)$
(50) $\displaystyle=$ $\displaystyle\int dx^{3}dx^{0}F_{03}(X)F^{03}(X).$

We therefore obtain the longitudinal action ${\cal A}_{A}^{\rm III}$ as

$\displaystyle{\cal A}_{A}^{\rm III}=Tr\int dX\left\\{-{1\over
2}F_{03}(X)F^{03}(X)\right\\}.$ (51)

##  VI Total action integral

Summing up all terms above in Eqs. (37), (39), and (51), we have reduced the
action integral ${\cal A}_{{}_{\rm 4D}}$ in (3+1)D, Eq. (10), to the action
integral ${\cal A}_{{}_{\rm 2D}}$ in (1+1)D given by

$\displaystyle{\cal A}_{{{}_{\rm 2D}}}$
$\displaystyle\\!=\\!Tr\\!\\!\int\\!\\!dX\Biggl{\\{}\\!\bar{\psi}(X)\gamma_{2D}^{\mu}(p_{\mu}\\!+\\!g_{2D}A_{\mu}(X))\psi(X)-m_{T}~{}{\bar{\psi}}(X)\psi(X)\Biggl{\\}}$
(52) $\displaystyle-\int
dt~{}(\kappa_{1}+\kappa_{2})|x^{3}\bar{q})-x^{3}(q)|+Tr\int dX\left\\{-{1\over
2}F_{03}(X)F^{03}(X)\right\\}.$

The subscript label of ‘2D’ is to indicate that this is the action integral
involving the integration over the 2D space-time coordinates,
$X$=($x^{3}$,$x^{0}$). This action integral can be the starting point for the
study of problems in the longitudinal (1+1)D${}_{\\{x^{3},x^{0}\\}}$ space-
time in which the properties of the gauge-field coupling constant $g_{{}_{\rm
2D}}$, and the transverse mass $m_{T}$ are input properties of the open-string
parameters fixed by the initial choice of the class of transverse states in
question, representing the ingredients that depend on the transverse degrees
of freedom. The longitudinal-temporal space-time dynamics can be treated in
the simpler (1+1)D${}_{\\{x^{3},x^{0}\\}}$ space-time, to facilitate the
examination of the complex problem of the quark-QCD-QED system.

Previously, the Schwinger model of QED in (1+1)D has been considered as an
unrealistic “toy” model incapable of quantitative connection to the realistic
fermion or quark systems in (3+1)D space-time. A major difficulty arises
because the coupling constant $g_{{}_{\rm 2D}}$ in Schwinger’s (1+1)D space-
time has the dimension of a mass. In the physical (3+1)D space-time, the
coupling constant $g_{{}_{\rm 4D}}$ is dimensionless, but the flux tube has a
structure with a radius $R_{T}$. In contrast in (1+1)D, the open-string does
not have a structure, but the coupling constant has the dimension of a mass.
It was recently realized perceptively that in reducing from the dynamics in
(3+1)D space to the dynamics in (1+1)D, the information in the structure of
the flux tube is in fact stored in the coupling constant $g_{{}_{\rm 2D}}$ and
the transverse mass $m_{T}$ in (1+1)D Won10 . The coupling constants in
different dimensional space-time systems are connected by Eq. (25) or by the
radius of the flux tube as given by Eq. (75) below Won09 ; Won10 ; Kos12 , and
the transverse mass $m_{T}$ is obtained by solving the eigenvalue equation in
the transverse plane Eqs. (20). The (1+1)D action integral in Eq. (52) shows
that the Schwinger model can be viewed as a realistic idealization of the
system in (3+1)D space-time, capable of semi-quantitative confrontation with
experiment Won20 and consistent with the string picture of hadron physics as
developed in Ven68 ; Nam70 ; Got71 ; tH74a ; tH74b ; Art74 ; Pol81 ; And83 .

In our consideration of the longitudinal confinement, we shall focus on the
longitudinal action integral ${\cal A}_{A}^{\rm III}$ and treat the transverse
action integrals ${\cal A}_{A}^{\rm I}$ and ${\cal A}_{A}^{\rm II}$ as
perturbations for many reasons. Firstly, Schwinger-type longitudinal
confinement is independent of the strength of the coupling, no matter how
weak. It is therefore a dominant force for the occurrence of longitudinal
confinement, even for weak coupling. It is therefore reasonable to treat
${\cal A}_{A}^{\rm I}$ and ${\cal A}_{A}^{\rm II}$ as perturbations in our
consideration of longitudinal dynamics. Secondly, the contributions from these
two terms, ${\cal A}_{A}^{\rm I}$ and ${\cal A}_{A}^{\rm II}$, are
proportional to the length of the flux tube and would contribute to the
confinement of the system classically. The fact that when the randomness of
the quantum field fluctuation is taking into account in lattice gauge
calculations in (3+1)D, even though they may be classically confining
longitudinally as evidenced by linear flux-tube length dependencies of ${\cal
A}_{A}^{\rm I}$ and ${\cal A}_{A}^{\rm II}$, the randomness of the quantum
fluctuation disrupts the longitudinal confinement so greatly that static
opposite charges cannot be confined in compact QED (3+1)D for weak couplings.
The non-confinement in static (3+1)D lattice gauge calculations for weak
coupling suggests that such a linear confinement effect is likely to be weak
for weak coupling in (3+1)D. Finally, the evaluation of the contributions from
the transverse action integrals ${\cal A}_{A}^{\rm I}$ and ${\cal A}_{A}^{\rm
II}$ will require the examination of the transverse lattice gauge
calculations, which will be beyond the scope of the present analysis. Future
investigation on how these terms will contribute to the longitudinal
confinement in (3+1)D will be of interest.

##  VII Dynamics of the Quark Fields and the Gauge Fields

###  VII.1  Transverse Landau levels of quarks and antiquarks in a
longitudinal magnetic field ${B}$

We shall discuss the transverse confinement of quarks in this subsection and
the longitudinal confinement of quarks and gauge fields in the next
subsection.

For quarks interacting with the non-Abelian QCD gauge field leading to bound
hadrons with energies above the pion mass, the transverse confinement arises
from the strong non-perturbative self-interaction among the gluons linking
color charges. The Feynman diagram loops involving self-interacting gluons
give rise to the dominance of planar diagrams over non-planar diagrams,
leading to the string description of the interaction between color charges
with both transverse and longitudinal confinement, as described in tHo74 . We
do not need to examine the question of the transverse confinement in QCD
again.

Focusing on the question of the transverse confinement of quarks and
antiquarks in QED, we mentioned earlier in the proposed model in Section II
that for a $q\bar{q}$ pair created at a hypothetical QED meson eigenenergy,
the produced $q\bar{q}$ pair will execute a 3D yo-yo motion directed along a
particular direction more than the other two directions, appropriate for the
hypothetical bound state configuration, and this direction can be chosen to be
the longitudinal direction. We envisage the produced $q\bar{q}$ pair of
opposite charges to emerge from the creation point and to stretch outward
transversely and longitudinally. At the moment immediately after their
creation, the $q$$\bar{q}$ charge pair will be accompanied by the simultaneous
creation of the gauge fields ${A}$, ${E}$, and
${B}$(=$\nabla$$\times$${\hbox{\boldmath${A}$}})$. We can apply Polyakov’s
result of confinement in (2+1)D to infer the transverse confinement of the
created quark, antiquark, and the gauge fields at the moment immediate after
their creation. In particular, as shown by Drell and collaborators Dre79 , the
magnetic field ${B}$=$(-F^{12}\hbox{\boldmath${e}$}^{3})$ along the
longitudinal direction will confine various gauge fields on the transverse
$\\{x^{1},x^{2}\\}$ plane at the moment of pair creation. We would like to
show explicitly here that the longitudinal ${B}$ field also will confine the
quarks and the antiquarks on the transverse planes because of Landau level
dynamics.

To study our lowest energy systems, we shall examine the simplest case with
cylindrical symmetry in the model of Won22 in which a momentary snapshot of
the configuration of the quark, the antiquark, and the gauge fields at a time
after creation is shown in Fig. 1(a). The corresponding lattice transcription
of such a “stretch (2+1)D” configuration is shown in Fig. 1(b). The transverse
gauges fields $A^{1}$ and $A^{2}$ are such that the magnetic field
${B}$=$B\hbox{\boldmath${e}$}^{3}\Theta(R_{T}-r_{\perp})$ is along the
longitudinal direction and confined within the flux tube of radius $R_{T}$,
with $A_{r_{\perp}}(r_{\perp},\phi)$=0, and $A_{\phi}(r_{\perp},\phi)$ given
by

$\displaystyle
A_{\phi}(r_{\perp},\phi)=\frac{Br_{\perp}}{2}\Theta(R_{T}-r_{\perp})+\frac{BR_{T}^{2}}{2r_{\perp}}\Theta(r_{\perp}-R_{T}),$
(53)

where $\Theta(z)$ is the step function of $z$. Represented in terms of the
components $A^{1}$ and $A^{2}$ within the flux tube of radius $R_{T}$, they
are

$\displaystyle(A^{1},A^{2})=\left(-\frac{Bx^{2}}{2},\frac{Bx^{1}}{2}\right).$
(54)

Such a gauge field are defined in (3+1)D and we shall assume that they are
approximately independent functions of the longitudinal coordinate $x^{3}$. We
have chosen such a transverse gauge field $A^{1}$ and $A^{2}$ because they
will lead to the confinement of the quarks and the antiquarks in the
transverse plane.

We can now study the transverse dynamics of quarks and antiquarks under such a
magnetic field ${B}$ treated as an external field within the flux tube. We
shall ignore the dynamics at the surface region of the flux tube where the
Gaussian quark density will be quite small. In the equations of transverse
motion for quarks, Eq. (20), the momentum operator $\Pi^{\\{1,2\\}}$ is

$\displaystyle\Pi^{\\{1,2\\}}=p^{\\{1,2\\}}-eQA^{\\{1,2\\}},$ (55)

where $e=g^{{\rm QED}}$ is the QED coupling constant and $Q$ is the quark
electric charge,

$\displaystyle Q={\rm\,sign}(Q)\,|Q|.$ (56)

The magnitude of the quark electric charge $|Q|$ and the charge sign,
${\rm\,sign}(Q)\,$, are flavor-dependent. Therefore quarks interacting in QED
interactions and in external QED fields will break isospin symmetry. In the
external magnetic field of Eq. (54), Eq. (20) becomes

$\displaystyle[(p^{1}-eQA^{1})-i(p^{2}-eQA^{2})]G_{2}(\hbox{\boldmath${r}$}_{\perp})=[m_{T}-m]G_{1}(\hbox{\boldmath${r}$}_{\perp}),$
(57a)
$\displaystyle[(p^{1}-eQA^{1})+i(p^{2}-eQA^{2})]G_{1}(\hbox{\boldmath${r}$}_{\perp})=[m_{T}+m]G_{2}(\hbox{\boldmath${r}$}_{\perp}).$
(57b)

Manipulating these two equations, we obtain

$\displaystyle[(p^{1}-eQA^{1})+i(p^{2}-eQA^{2})][(p^{1}-eQA^{1})-i(p^{2}-eQA^{2})]G_{2}(\hbox{\boldmath${r}$}_{\perp})=[m_{T}^{2}-m^{2}]G_{2}(\hbox{\boldmath${r}$}_{\perp}),$
(58a)
$\displaystyle[(p^{1}-eQA^{1})-i(p^{2}-eQA^{2})][(p^{1}-eQA^{1})+i(p^{2}-eQA^{2})]G_{1}(\hbox{\boldmath${r}$}_{\perp})=[m_{T}^{2}-m^{2}]G_{1}(\hbox{\boldmath${r}$}_{\perp}),$
(58b)

which are separate equations for $G_{2}(\hbox{\boldmath${r}$}_{\perp})$ and
$G_{1}(\hbox{\boldmath${r}$}_{\perp})$. We find that these two equations lead
to

$\displaystyle\biggl{[}p_{T}^{2}+[(\frac{eQB}{2})^{2}r_{\perp}^{2}+(-eQ)BL^{3}+eQB\biggr{]}G_{2}(\hbox{\boldmath${r}$}_{\perp})=[m_{T}^{2}-m^{2}]G_{2}(\hbox{\boldmath${r}$}_{\perp}),$
(59a)
$\displaystyle\biggl{[}p_{T}^{2}+(\frac{eQB}{2})^{2}r_{\perp}^{2}+(-eQ)BL^{3}-eQB\biggr{]}G_{1}(\hbox{\boldmath${r}$}_{\perp})=[m_{T}^{2}-m^{2}]G_{1}(\hbox{\boldmath${r}$}_{\perp}),$
(59b)

where $p_{T}^{2}$=$(p^{1})^{2}+(p^{2})^{2}$, $L^{3}$ is the angular momentum
along the longitudinal 3-axis. The eigenvalues for these two equations of
$G_{2}$ and $G_{2}$ can be obtained separately,

$\displaystyle~{}~{}\text{ for }G_{2},~{}~{}~{}$ $\displaystyle
m_{T}^{2}=m_{T2}^{2}=m^{2}+2(\hbar\omega)^{2}(2n_{2}+|\Lambda_{2}|+1-\Lambda_{2}{\rm\,sign}(Q)\,+{\rm\,sign}(Q)\,),$
(60a) $\displaystyle~{}~{}\text{ for }G_{1},~{}~{}~{}$ $\displaystyle
m_{T}^{2}=m_{T1}^{2}=m^{2}+2(\hbar\omega)^{2}(2n_{1}+|\Lambda_{1}|+1-\Lambda_{1}{\rm\,sign}(Q)\,-{\rm\,sign}(Q)\,),$
(60b)

where their eigenfunctions are

$\displaystyle
G_{1}(\hbox{\boldmath${r}$}_{\perp})=A_{1}e^{i\Lambda_{1}\phi}e^{-x^{2}/2}x^{|\Lambda_{1}|}L_{n}^{(|\Lambda_{1}|)}(x^{2}),$
(61a) $\displaystyle
G_{2}(\hbox{\boldmath${r}$}_{\perp})=A_{2}e^{i\Lambda_{2}\phi}e^{-x^{2}/2}x^{|\Lambda_{2}|}L_{n}^{(|\Lambda_{2}|)}(x^{2}),$
(61b)

$x=\omega r_{\perp}$, and $(\hbar\omega)^{2}$ =$|eQB|/2$. The Landau level
solutions are obtained when the eigenvalue $m_{T}^{2}$ is the common
$m_{T}^{2}$ solutions for both $G_{1}$ and $G_{2}$ in Eq. (60). That is, we
search for those sets of $n_{\\{1,2\\}}$, and $\Lambda_{\\{1,2\\}}$ quantum
numbers such that they yield the same $m_{T}^{2}$ values. Upon equating such
an eigenvalue condition, $m_{T1}^{2}$=$m_{T2}^{2}$ in Eq. (60), the eigenvalue
equation for the Landau level is

$\displaystyle
2n_{2}+|\Lambda_{2}|+1-\Lambda_{2}{\rm\,sign}(Q)\,+{\rm\,sign}(Q)\,=(2n_{1}+|\Lambda_{1}|+1-\Lambda_{1}{\rm\,sign}(Q)\,-{\rm\,sign}(Q)\,,$
(62)

which gives

$\displaystyle
2(n_{1}-n_{2})+|\Lambda_{1}|-|\Lambda_{2}|-(\Lambda_{1}-\Lambda_{2}){\rm\,sign}(Q)\,-2{\rm\,sign}(Q)\,=0.$
(63)

Possible solutions from the above Landau level eigenvalue equations Eqs. (63)
and 60) can be enumerated:

  1. 1.

Zero mode If $\Lambda$=0, and $n=0$, which is the solution we shall use as the
lowest-energy transverse state, then we get the solution $m_{T}^{2}=m^{2}$,

$\displaystyle~{}~{}~{}\text{ for }~{}~{}~{}~{}{\rm\,sign}(Q)\,=1,~{}~{}$
$\displaystyle m_{T}^{2}=m^{2},~{}~{}G_{1}(\hbox{\boldmath${r}$}_{\perp})\neq
0,~{}{\rm and~{}}G_{2}=0,$ (64) $\displaystyle~{}~{}~{}\text{ for
}~{}~{}~{}~{}{\rm\,sign}(Q)\,=-1,$ $\displaystyle
m_{T}^{2}=m^{2},~{}~{}G_{2}(\hbox{\boldmath${r}$}_{\perp})\neq 0,~{}{\rm
and~{}}G_{1}=0.$ (65)

We can represent the transverse wave function
$G_{\\{1,2\\}}(\hbox{\boldmath${r}$}_{\perp})$ on the $\\{x^{1},x^{2}\\}$
plane as a spinor

$\displaystyle
G(\hbox{\boldmath${r}$}_{\perp})=\begin{pmatrix}G_{2}(\hbox{\boldmath${r}$}_{\perp})\\\
G_{1}(\hbox{\boldmath${r}$}_{\perp})\\\ \end{pmatrix},$ (66)

and we can introduce the spinors with quantum numbers $\sigma^{3}=\pm 1$,
chosen to be quantized along the $x^{3}$ direction,

$\displaystyle\chi_{1/2}=\begin{pmatrix}1\\\ 0\\\
\end{pmatrix},~{}~{}~{}\chi_{-1/2}=\begin{pmatrix}0\\\ 1\\\ \end{pmatrix},$
(67)

then ,

$\displaystyle\sigma^{3}\chi_{\pm 1/2}=\pm\chi_{\pm 1/2}.$ (68)

Figure 2: The lowest-energy Landau level zero-mode solution with $m_{T}$=$m$
for the quark and the antiquark with their spins and magnetic moments aligned
with the magnetic field, with a total zero spin in a flux tube.

Thus, for a quark with a positive charge, ${\rm\,sign}(Q)\,$=1, and its spin
aligned in the longitudinal magnetic field ${B}$ direction, the quark has the
lowest energy transverse state at the zero mode, with $m_{T}=m$. The wave
function at the zero mode is the Gaussian wave function

$\displaystyle
G(\hbox{\boldmath${r}$}_{\perp})=Ae^{-x^{2}/2}\chi_{1/2}~{}~{}\text{ for
~{}~{}}{\rm\,sign}(Q)\,=1.$ (69)

For the antiquark with a negative charge, ${\rm\,sign}(Q)\,$=$-$1, and the
spin of the antiquark aligned opposite to the magnetic field ${B}$ direction,
the antiquark has the lowest energy transverse state at the zero mode, with
$m_{T}=m$. The wave function at the zero mode is the Gaussian wave function

$\displaystyle
G(\hbox{\boldmath${r}$}_{\perp})=Ae^{-x^{2}/2}\chi_{-1/2}~{}~{}\text{ for
~{}~{}}{\rm\,sign}(Q)\,=-1.$ (70)

Therefore, a quark and an antiquark can populate their transverse zero-mode
states in a longitudinal magnetic field for which their transverse mass
$m_{T}$ for longitudinal motion is the same as their rest mass $m$. The
external magnetic field ${B}$ has no effect on their effective mass for
longitudinal motion. If the mass of the quark and the antiquark is small and
can be considered to be massless, then they would remain massless. At the zero
mode, the spins of the quark and the antiquark are aligned along the external
magnetic field direction but pointing in opposite directions as shown in Fig.
2. The quark and the antiquark pair at zero mode have aligned magnetic moments
link together by the magnetic flux of the applied magnetic field as shown in
Fig. 2. The above conclusions can be generalized to quarks with a negative
charge and a different flavor, together with their antiquark counterparts.

  2. 2.

Negative $\Lambda_{1}$ and $\Lambda_{2}$ If both $\Lambda_{1}$ and
$\Lambda_{2}$ are negative, then the eigenvalue equation becomes

$\displaystyle
2(n_{2}-n_{1})+2(\Lambda_{2}-\Lambda_{1})=2{\rm\,sign}(Q)\,.~{}~{}~{}$ (71)

There can be many different solutions:

    1. (a)

For $n_{1}=n_{2}$, Landau level solutions are

$\displaystyle~{}~{}(|\Lambda_{1}|-|\Lambda_{2}|)={\rm\,sign}(Q)\,.$ (72)

    2. (b)

For $(|\Lambda_{1}|=|\Lambda_{2}|)$, Landau level solutions are

$\displaystyle n_{1}-n_{2}={\rm\,sign}(Q)\,.$ (73)

  3. 3.

Positive $\Lambda_{1}$ and $\Lambda_{2}$ If $\Lambda_{1}$ and $\Lambda_{2}$
are positive, then, Landau level solutions are

$\displaystyle~{}~{}~{}~{}~{}n_{1}-n_{2}={\rm\,sign}(Q)\,.$ (74)

In all these solutions, the harmonic oscillator wave functions of Eq. (61)
confine the quark and the antiquark on the transverse plane. The magnetic
field ${B}$ along the longitudinal direction facilitates the transverse
confinement of quarks and antiquarks on the transverse plane. The eigenvalue
equation (60) indicates that the effective mass $m_{T}$ for longitudinal
motion increases with increasing $n$ and $|\Lambda|$ quantum numbers. For non-
zero values of $n$ and $|\Lambda|$ quantum numbers, the magnetic effect
depends on the quark rest mass $m$ and decreases as the quark mass increases.
As the transverse quark current $j^{1}$ and $j^{2}$ are proportional to the
product $G_{1}^{*}G_{2}$ as well as $G_{2}^{*}G_{1}$ (see Eq. (141)), the
transverse currents $j^{1}$ and $j^{2}$ are zero for quarks and antiquarks in
the zero mode, but is non-zero for the case when the products of $G_{1}$ and
$G_{2}$ are not zero.

Our central interest is on the lowest-energy states, so we shall limit our
attention to study the longitudinal dynamics of quarks and antiquarks in the
zero mode. The transverse wave function of the quark and the antiquark in the
zero mode are Gaussian functions executing transverse harmonic oscillator
zero-point motion. Thus quarks the antiquark are transversely confined. In the
zero mode, because of the alignment of the the magnetic field ${B}$ and spins
in the longitudinal direction, the quark transverse mass $m_{T}$ is the same
as the quark rest mass $m$. Upon further approximating light quarks to be
massless, we have $m_{T}$=$m$=0. The transversely confined massless quarks and
antiquarks provide favorable ingredients for the virtual pair production and
vacuum polarization, linking the valence quark, the valence antiquark, and
gauge fields with a confining interaction. With transverse confinement well at
hand, we can idealizing the flux tube in (3+1)D as a one-dimensional string
and apply Schwinger’s model of massless fermions in QED in (1+1)D to our
problem of quarks. Consequently, we can infer that the transversely confined
massless quarks are also longitudinally confined. Possessing both transverse
and longitudinal confinement, such a quark-QED system will be confined in
(3+1)D space-time and may show up as a QED meson Won10 ; Won20 .

From Eqs. (69) and (70), the solution
$G_{\\{1,2\\}}(\hbox{\boldmath${r}$}_{\perp})$ of the transverse equations for
the zero mode is in the form of a Gaussian function which allows us to obtain
the relation between the 2D coupling constant $g_{2D}$ and the 4D coupling
mass $g_{{}_{\rm 4D}}$, according to Eq. (25),

$\displaystyle g_{2D}^{2}=g_{{}_{\rm
4D}}^{2}\frac{\omega^{2}}{2{\pi}}=\frac{g_{{}_{\rm 4D}}^{2}}{{\pi}R_{T}^{2}},$
(75)

where we have introduced

$\displaystyle R_{T}=\sqrt{2}/\omega=\sqrt{4/(e|QB|)}.$ (76)

The above relation (75) coincides with previous results relating $g_{{}_{\rm
2D}}$ and $g_{{}_{\rm 4D}}$ in Won09 ; Won10 ; Kos12 ; Won20 .

It is clear from Eq. (60) that in the zero mode, the effective quark mass
$m_{T}$ does not depend on the magnitude of the magnetic field, but the
effective flux tube radius $R_{T}$ is related to the magnitude of magnetic
field $B$ as given in Eq. (76).

It should be pointed out that the magnetic field $B$ is a dynamical variable
which is determined by the coupling constant and the electric charge
distribution in a Hamiltonian. In the lattice gauge theory of Drell $et.al.$,
the Hamiltonian in compact QED gauge fields in lattice units and notations is
given by Eq. (2.26) and (2.28) of Ref. Dre79 ,

$\displaystyle H=\frac{g_{{}_{\rm 4D}}^{2}}{2}\sum_{\rm
links}[(\nabla\phi)^{2}+(\nabla\times\hbox{\boldmath${L}$})^{2}]+\frac{1}{g_{{}_{\rm
4D}}^{2}}\sum_{\rm plaquett}(1-\cos B),$ (77)

where the electric field ${E}$ has been separated into a longitudinal
$\hbox{\boldmath${E}$}_{L}$ component and a transverse
$\hbox{\boldmath${E}$}_{T}$ component, with
$\hbox{\boldmath${E}$}_{L}=-\nabla\phi$ and
$\hbox{\boldmath${E}$}_{T}=\nabla\times\hbox{\boldmath${L}$}$. Defining
$L_{\hbox{\boldmath${p}$}}$=$L_{\hbox{\boldmath${p}$}}\hbox{\boldmath${e}$}^{3}$=$L\hbox{\boldmath${e}$}^{3}$
and
$\hbox{\boldmath${B}$}_{\hbox{\boldmath${p}$}}$=$B_{\hbox{\boldmath${p}$}}\hbox{\boldmath${e}$}^{3}$=$B\hbox{\boldmath${e}$}^{3}$
on the plaquett ${p}$, the canonical variables $E_{\hbox{\boldmath${p}$}}$ and
$L_{\hbox{\boldmath${p}$}}$ satisfy the equal-time commutation relation Kog75
; Dre79

$\displaystyle[L_{\hbox{\boldmath${p}$}},B_{\hbox{\boldmath${p}$}^{\prime}}]=i\delta({\hbox{\boldmath${p}$}},{\hbox{\boldmath${p}$}^{\prime}}).$
(78)

As a result of the above quantization, the canonical conjugate variables $B$
and $L$ has a distribution and they satisfy the uncertainty principle. The
Hamiltonian for wave function of the magnetic field $B$, as given by Eq. (3.9)
of Ref. Dre79 , is

$\displaystyle H={\cal E}_{\rm Coulomb}+2g_{{}_{\rm
4D}}^{2}(-\frac{1}{i}\frac{\partial}{\partial
B}+\epsilon)^{2}+\frac{1}{g_{{}_{\rm 4D}}^{2}}(1-\cos B),$ (79)

where ${\cal E}_{\rm Coulomb}$ is the Coulomb energy, and $\epsilon$ is the
electric charge configuration variable. The magnetic field $B$ wave function
peaks around $B_{0}(g_{{}_{\rm 4D}},\epsilon)$ that will be determined by the
coupling constant $g_{{}_{\rm 4D}}$ and the charge distribution variable
$\epsilon$, with a width that is of the order of the coupling constant
$g_{{}_{\rm 4D}}$. If we use alternatively the electric field as the dynamical
variable, then the electric field $L$ wave function peaks at $L_{0}(g_{{}_{\rm
4D}},\epsilon)$ which depends on $g_{{}_{\rm 4D}}$ and $\epsilon$ with a
distribution width in $L$ that is of order $1/g_{{}_{\rm 4D}}$ Dre79 ; Bau21 .
For compact QED with a weak coupling $g_{{}_{\rm 4D}}$, the magnetic field $B$
distribution width is small, but the electric field $L$ distribution width is
large. On the other hand, for QCD with a strong coupling, the electric field
distribution width is small but the magnetic field distribution width is
large. The solution for the a magnetic field wave equation for $\epsilon=0$ is
a Mathieu function NBS64 . In the presence of static charges, there will be
the additional $\epsilon$ term in the charge-(gauge field) coupling in Eq.
(79). The determination of the magnetic field wave function in a QED meson is
beyond the scope of the present investigation, but worthy of future
investigations. The gross features of the wave function for the magnetic field
$B$ for a weak coupling is in the form of the Mathieu function NBS64 centering
around $B\sim B_{0}$ with a narrow distribution width that increases with the
increasing strength of the coupling constant $g_{{}_{\rm 4D}}$.

The results of this section indicate that the longitudinal magnetic field
${B}$ that is needed to confine quarks and various gauge fields transversely
in Polyakov’s (2+1)D initial transverse confinement at quark-antiquark
creation continues to confine the quark and the antiquark transversely in the
stretch (2+1)D configuration of the flux tube. Furthermore, this longitudinal
magnetic field ${B}$ that is needed for Polyakov’s (2+1)D confinement also
leads to the presence of the zero mode in Landau level dynamics, for which the
quark transverse mass $m_{T}$ is the same as the quark rest mass $m$. As a
consequence, light quarks that can be considered massless remain massless in
the flux tube environment for longitudinal motion. The possibility of quarks
and antiquarks to be transversely confined in the flux tube and be massless
provide ready ingredients for the virtual pair production and vacuum
polarization, linking the valence quark, the valence antiquark, and the gauge
fields with confining interactions.

Polyakov’s confinement of charges in compact QED in (2+1)D of all coupling
strengths suggests that the transverse confinement of $q$ and $\bar{q}$ will
persist even for weak coupling. It is reasonable to idealize the traverse
configuration as a one-dimensional structureless string and study longitudinal
confinement, dynamical quark effects, and spontaneous chiral symmetry
breaking. Information on the transverse degree of freedom is now stored in the
coupling constant $g_{{}_{\rm 2D}}$ as $g_{{}_{\rm 2D}}$=$g_{{}_{\rm
4D}}/(\sqrt{\pi}R_{T})$ (Eq. (75) and in the traverse mass
$m_{T}$=$m_{T}(m,|\hbox{\boldmath${B}$}|)$ (Eqs. (60) and (63)) in such an
idealization.

###  VII.2 Quark dynamics in the longitudinal (1+1)D${}_{\\{x^{3},x^{0}\\}}$
space-time

From the last subsection, we find that the quark-QED-QCD system under
consideration possesses transverse confinement. It can be approximately
idealized as a one-dimensional string in (1+1)D. We come now to examine the
longitudinal dynamics of such an idealized system.

In the general case of quark and gauge field dynamics, the QCD gauge fields
$A^{\mu}(X)$=$\sum_{i=1}^{8}A^{\mu}_{a}(X)t^{a}$ has eight non-Abelian degrees
of freedom with $A^{\mu}_{a}(X)$, $a=1,2,...,8$ in the color SU(3) generator
space. The solution of the quark dynamics will correspondingly couple these
eight degrees of freedom in the eight $t^{a}$ directions. Such couplings in
the non-Abelian degrees of freedom will lead to color excitations, the
majority of which will not lead to stable collective excitations.

There is a known way to get stable collective excitations of the QED and QCD
gauge fields and the quark currents as carried out in Won10 ; Won20 by
introducing a unit vector $\tau^{1}$ randomly in the eight-dimensional SU(3)
generator space,

$\displaystyle\tau^{1}=\sum_{i=1}^{8}n_{a}t^{a},~{}~{}~{}\text{with}~{}~{}~{}\sqrt{n_{1}^{1}+n_{2}^{2}+...+n_{8}^{2}}=1,$
(80)

where $n_{a}$ are the components of generator vectors $\tau^{1}$ in the eight-
dimensional $t^{a}$ space, with $n_{a}={\rm tr}\\{\tau^{1}t^{a}\\}/2$ oriented
randomly. Restricting our considerations in the color-octet subspace with a
fixed orientation of $\tau^{1}$ but allowing the amplitude of the gauge fields
to vary, we can represent the dynamics of the gauge fields $A^{\mu}$ of the
quark-QCD-QED system in terms of the amplitudes $A^{\mu}_{0}$ and
$A^{\mu}_{1}$ as Won10 ; Won20 ; Won20a ; Won21

$\displaystyle A^{\mu}(X)=A^{\mu}_{0}(X)\tau^{0}+A^{\mu}_{1}(X)\tau^{1},$ (81)

where $\tau^{0}=t^{0}$ and the generators $\tau^{0}$ and $\tau^{1}$ satisfy
2Tr$(\tau^{\lambda}\tau^{\lambda^{\prime}})=\delta^{\lambda{\lambda^{\prime}}}$,
with $\lambda,{\lambda^{\prime}}=0,1$. We vary the amplitudes of the QCD gauge
fields $A_{1}^{\mu}(X)$ without varying the orientation of the QCD gauge field
in the 8-dimensional color-octet subspace. The randomness of the orientation
of such a unit vector $\tau^{1}$ in the generator space ensures that it can
cover a significant part of the available color space. Furthermore, because
$\tau^{0}$ and $\tau^{1}$ commute, the dynamics of the gauge fields in this
$\tau^{0}$ and $\tau^{1}$ generator subspace is Abelian. Fixing the
orientation of the unit vector $\tau^{1}$ is an Abelian approximation of the
non-Abelian QCD dynamics in a subspace in which the dynamics of the quark-QCD
systems can lead to stable collective excitations.

Assuming Eq.(81) everywhere below in this section, we look for the solution of
the 2D Dirac equation (30) in the form

$\displaystyle\psi(X)=\begin{pmatrix}f_{+}(X)\\\
f_{-}(X)\end{pmatrix}=\\{T_{l(M_{0};M)}\exp\\}\left\\{ig_{{{}_{\rm 2D}}}\int
dX^{\mu}A_{\mu}(X)\right\\}\exp\left(-i\omega
x^{0}+ip^{3}x^{3}\right)u;~{}~{}u=\begin{pmatrix}u_{+}\\\
u_{-}\end{pmatrix},~{}~{}~{}~{}$ (82)

where the symbol $\\{T_{l(M_{0};M)}\exp\\}$ means that the integration is to
be carried out along the line on the light-cone from the point $M_{0}$ to the
point $M$ such that the factors in the exponent expansion are chronologically
ordered from $M_{0}$ to $M$. Substituting Eq.(82) into the 2D Dirac equation
(31), we derive successively

$\displaystyle\begin{pmatrix}0&i(\partial_{0}-\partial_{3})\\\
i(\partial_{0}+\partial_{3})&0\end{pmatrix}\exp\left(-i\omega
x^{0}+ip^{3}x^{3}\right)\begin{pmatrix}u_{+}\\\
u_{-}\end{pmatrix}=m_{T}\exp\left(-i\omega
x^{0}+ip^{3}x^{3}\right)\begin{pmatrix}u_{+}\\\
u_{-}\end{pmatrix},~{}~{}~{}~{}$ (83)

where $m_{T}=m$ is the quark transverse mass given by Eq.(60) when the quark
and the antiquark are in their zero-mode, and $u$ is normalize as
${\bar{u}_{\pm p^{3}}}u_{\pm p^{3}}=\pm 2m_{T}$=$\pm m$,

$\displaystyle(\omega+p^{3})u_{-}=m_{T}u_{+},$
$\displaystyle(\omega-p^{3})u_{+}=m_{T}u_{-},$ (84) $\displaystyle
m_{T}^{2}-(\omega^{2}-(p^{3})^{2})=0;~{}~{}~{}\Rightarrow\omega=\pm\varepsilon=\pm\sqrt{m_{T}^{2}+(p^{3})^{2}},$
(85)
$\displaystyle\omega=\varepsilon;~{}~{}u_{-}=\frac{m_{T}}{\varepsilon+p^{3}}u_{+},$
$\displaystyle\omega=-\varepsilon;~{}~{}u_{-}=-\frac{m_{T}}{\varepsilon-p^{3}}u_{+}.$
(86)

Taking the normalization condition in a form

$\displaystyle Tr\int dx_{3}:(\psi^{\dagger}_{2D}\psi_{2D}):=1,$ (87)

we find the general solution of the 2D Dirac equation

$\displaystyle\psi(X)$ $\displaystyle=$
$\displaystyle\sum\limits_{p^{3}}\\{T_{l(M_{0};M)}\exp\\}\left\\{ig_{2D}\int
dX^{\mu}A_{\mu}(X)\right\\}\times$ (88) $\displaystyle\times$
$\displaystyle\frac{1}{\sqrt{2\varepsilon}}\left(~{}{\hat{q}}_{p^{3}}\begin{pmatrix}\sqrt{\varepsilon+p^{3}}\\\
\sqrt{\varepsilon-p^{3}}\end{pmatrix}\exp\left(-iPX\right)+^{\prime}~{}{\hat{\bar{q}}}^{\dagger}_{p^{3}}\begin{pmatrix}\sqrt{\varepsilon+p^{3}}\\\
-\sqrt{\varepsilon-p^{3}}\end{pmatrix}\exp\left(+iPX\right)\right),$

where $P=(\varepsilon,p^{3})$, $X=(x^{3},x^{0})$, and the hat symbols
${\hat{q}}_{p^{3}}$ and ${\hat{q}^{\dagger}}_{p^{3}}$ denote the operators of
the creation and annihilation of quarks and antiquarks of longitudinal
momentum $p^{3}$, which satisfy the standard commutative relations Pes95 .

###  VII.3 Self-consistent longitudinal quark currents and gauge fields

The quark-QCD-QED vacuum to be the lowest energy state system with quarks
filling up the (hidden) Dirac sea and interacting in QCD and QED interactions.
It is defined as the state with no (valence) quark above the Dirac sea and no
(valence) antiquark below the Dirac sea. A local disturbance with the
introduction of a perturbative gauge field
$A^{\mu}(X)=\sum_{\lambda=0}^{1}A^{\mu}_{\lambda}\tau^{\lambda}(X)$ will
generate a collective excitation of the quark-QCD-QED medium. Our task is to
find out whether there can be stable and confined collective QCD and QED
excitations showing up as stable particles that can be depicted as bound
states of (valence) quark and (valence) antiquarks.

For such a purpose, we introduce the perturbative gauge field
$A^{\mu}(X)$=$\sum_{\lambda=0}^{1}A^{\mu}_{\lambda}\tau^{\lambda}(X)$ that
acts on the quark fields. The subsequent motion of the quark fields will
generate a current which, through the Maxwell equation, will in turn generate
a gauge field. The requirement of the self-consistency of the initial and
induced gauge fields and currents leads to the stable collective excitations
of the quark-QCD-QED systems. The collective dynamics can sustain themselves
indefinitely, when the decay channels are assumed to be turned off for such an
examination. How do the quark currents and gauge fields relate to each other
when such self-consistent collective excitations are attained?

The knowledge of the solution of the longitudinal quark field $\psi(X)$
provides us the tool to obtain the quark current $j_{\lambda}^{\mu}(X)$ as a
function of the introduced gauge field $A^{\mu}(X)$. For simplicity, we shall
consider first the simple case with a single flavor $f$, whose label is
sometimes implicitly omitted for brevity of notation. The case of many flavors
and the case with flavor mixture can be easily generalized and will be
considered in section IX, as was carried out in Won10 and Won20 . The quark
current is given by

$\displaystyle j_{\lambda
f}^{\mu}(X)=Tr\left\\{{\bar{\psi}_{f}(X)}\gamma^{\mu}\tau^{\lambda}\psi_{f}(X^{\prime})\right\\},\
\ \ \ \ X^{\prime}\to X+0,~{}~{}~{}~{}~{}\lambda=0,1,$ (89)

where the trace means summing with respect to colors (but not flavors).
Substituting the $\psi$-function solution given by Eq.(88) into the above
formula, we obtain

$\displaystyle j_{\lambda f}^{\mu}(X)$
$\displaystyle\\!\\!\\!\\!=Tr\int\frac{dp}{(2\pi)}\sum_{\sigma}\Bigg{\\{}\frac{P^{\mu}\tau^{\lambda}}{\varepsilon
N_{f}}\left(n^{\lambda}_{q}(\varepsilon,f\sigma)\exp\left(-iP(X^{\prime}-X)\right)+n^{\lambda}_{\bar{q}}(\varepsilon,f\sigma)\exp\left(iP(X^{\prime}-X)\right)\right)$
(90)
$\displaystyle\times(T\exp)\left\\{i\sum_{\lambda^{\prime}}~{}g^{\lambda^{\prime}}_{{{}_{\rm
2D}}f}\int\limits_{X}^{X^{\prime}}A^{\lambda^{\prime}}_{\mu}\tau^{\lambda^{\prime}}dX^{\mu}\right\\}\Bigg{\\}},\
\ \ X^{\prime}\to X+0,~{}~{}\lambda,\lambda^{\prime}=0,1,~{}\text{no sum in
}\lambda,$

where
$n^{\lambda}_{q}(\varepsilon,f\sigma)=\langle{\hat{q}}^{\dagger}_{f\sigma,\lambda}(p,\omega){\hat{q}}_{f\sigma,\lambda}(p,\omega)\rangle$
and
$n^{\lambda}_{\bar{q}}(\varepsilon,f\sigma)=\langle{\hat{\bar{q}}}_{f\sigma,\lambda}^{\dagger}(p,\omega){\hat{\bar{q}}}_{f\sigma,\lambda}(p,\omega)\rangle$
are the quark and antiquark occupancy numbers, respectively, which are
normalized per total flavor state, and $N_{f}$ is the flavor number. The above
current (89) contains the Schwinger gauge-invariance factor

$\displaystyle(T\exp)\left\\{i~{}\sum_{\lambda^{\prime}}\int\limits_{X}^{X^{\prime}}g^{\lambda^{\prime}}_{{{}_{\rm
2D}}f}A^{\lambda^{\prime}}_{\mu}\tau^{\lambda^{\prime}}dX^{\mu}\right\\},$
(91)

where $g^{0}_{f}=g^{0}_{{{}_{\rm 2D}}\,f}=Q_{f}^{{}_{\rm QED}}g_{{{}_{\rm
2D}}f}^{{}_{\rm QED}}$ and $g^{1}_{f}$=$g^{1}_{{{}_{\rm
2D}}\,f}$=$Q_{f}^{{}_{\rm QCD}}g_{{{}_{\rm 2D}}f}^{{}_{\rm QCD}}$ are the QED
and QCD coupling constants. The quantities $Q_{f}^{{}_{\rm QED}}$ and
$Q_{f}^{{}_{\rm QCD}}$ are the electric and color charge numbers as given by
Eq. (9). We expand the operator exponent in the last equation as a series with
respect to $(X^{\prime}-X)\to+0$,

$\displaystyle(T\exp)\biggl{\\{}ig^{\lambda^{\prime}}_{{{}_{\rm
2D}}f}\tau^{\lambda^{\prime}}\int\limits_{X}^{X^{\prime}}A^{\lambda^{\prime}}_{\mu}dX^{\mu}\biggr{\\}}\\!\\!\\!$
$\displaystyle\\!\\!\\!=1+ig^{\lambda^{\prime}}_{{{}_{\rm
2D}}f}\tau^{\lambda^{\prime}}(X^{\prime}-X)^{\mu}A^{\lambda^{\prime}}_{\mu}(\xi)$
(92) $\displaystyle+\frac{i}{2}g^{\lambda^{\prime}}_{{{}_{\rm
2D}}f}\tau^{\lambda^{\prime}}(X^{\prime}-X)^{\mu}(X^{\prime}-X)^{\nu}\partial_{\nu}A^{\lambda^{\prime}}_{\mu}(\xi)$
$\displaystyle-\sum_{\lambda^{\prime\prime}}g^{\lambda^{\prime}}_{{{}_{\rm
2D}}f}g^{\lambda^{\prime\prime}}_{{{}_{\rm
2D}}f}(\tau^{\lambda^{\prime}}\tau^{\lambda^{\prime\prime}})({\tilde{X}}^{\prime}-{\tilde{X}})^{\mu}(X^{\prime}-X)^{\nu}A^{\lambda^{\prime}}_{\mu}({\tilde{\xi}})A^{\lambda^{\prime\prime}}_{\nu}(\xi)\theta({\tilde{\xi}}-\xi),~{}~{}~{}~{}~{}$

where ${\tilde{\xi}}\in[{\tilde{X}},{\tilde{X}}^{\prime}],$ and
$\xi\in[X,X^{\prime}]$ and the step function $\Theta({\tilde{\xi}}-\xi)$ comes
from the $T$-exponent. Taking the limits
$({\tilde{X}}^{\prime}-{\tilde{X}})\to+0$ and $(X^{\prime}-X)\to+0$, we have
to take into account that

$\displaystyle{({\tilde{X}}^{\prime}-{\tilde{X}})\over(X^{\prime}-X)}\to+0,$
(93)

due to the step function. As a consequence, the last term in the expansion in
Eq. (92) is equal to zero.

Upon placing the quark and the antiquark in a longitudinal magnetic field
along the transverse direction, we find earlier in the last subsection that a
quark and an antiquark in their lowest transverse energy state lie at the
zero-mode orbitals with a transverse mass $m_{T}$=$m$. We consider the
approximation to treat the quarks as massless. The idealized longitudinal
dynamics corresponds then to the dynamics of massless charges in QED as in
Schwinger QED in (1+1)D Sch62 ; Sch63 . Taking into account that
$P^{\mu}=(\varepsilon=|p|,p)$, and substituting
$(T\exp)\\{i\sum_{\lambda^{\prime}}g_{2Df}^{\lambda^{\prime}}~{}\tau^{\lambda^{\prime}}\int\limits_{X}^{X^{\prime}}A_{\lambda`f}^{\mu}dx_{\mu}\\}$
given by Eq. (92) into Eq.(89), we obtain for $(X^{\prime}-X)\to+0$

$\displaystyle j^{\mu}_{\lambda f}(X)$
$\displaystyle\\!\\!=\frac{2ig^{\lambda}_{{{}_{\rm
2D}}f}}{N_{f}}Tr\sum_{\sigma}\int\limits_{-\infty}^{+\infty}\frac{P^{\mu}dp}{(2\pi)\varepsilon}\Biggl{\\{}\left(-{\partial\exp\left(-iP(X^{\prime}-X)\right)\over\partial
P_{\nu}}\right)\left(n_{q}(\varepsilon,f\sigma)+n_{\bar{q}}(\varepsilon,f\sigma)\right)$
(94)
$\displaystyle\times\sum_{\lambda^{\prime}}\left(\tau^{\lambda}\tau^{\lambda^{\prime}}A_{\lambda^{\prime}f}^{\mu}(\xi)+{1\over
2}\tau^{\lambda}\tau^{\lambda^{\prime}}(X^{\prime}-X)_{\eta}\partial^{\mu}A_{\lambda^{\prime}f}^{\eta}(\xi)\right)\Biggr{\\}}.$

To evaluate the term
$(X^{\prime}-X)_{\eta}\partial^{\mu}_{(X)}A^{\eta}_{\lambda^{\prime}f}(\xi)$,
we note that when we apply the partial derivative
$\partial^{2}\equiv\partial_{(X)}^{2}$ onto
$(X^{\prime}-X)_{\eta}\partial^{\mu}_{{(X)}}A_{\lambda^{\prime}f}^{\eta}(\xi)$,
we get

$\displaystyle\partial_{(X)}^{2}\lim\limits_{X^{\prime}\to~{}X}\left\\{(X^{\prime}-X)_{\eta}\partial^{\mu}_{(X)}A_{\lambda^{\prime}f}^{\eta}(\xi)\right\\}=\lim\limits_{X^{\prime}\to~{}X}\partial_{(X)}^{2}\left\\{(X^{\prime}-X)_{\eta}\partial^{\mu}_{(X)}A_{\lambda^{\prime}f}^{\eta}(X)\right\\}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}=\lim\limits_{X^{\prime}\to~{}X}\left\\{-2\partial_{\eta(X)}\partial^{\mu}_{(X)}A_{\lambda^{\prime}f}^{\eta}(X)+(X^{\prime}-X)_{\eta}\partial_{\kappa(X)}~{}\partial_{(X)}^{\kappa}\\{\partial^{\mu}_{(X)}A_{\lambda^{\prime}f}^{\eta}(X)\\}\right\\}.$
(95)

Upon taking the limit $X^{\prime}\to X+0$, the second term vanishes.
Therefore, we have in the limit of $X^{\prime}\to X+0$,

$\displaystyle\lim\limits_{X^{\prime}\to~{}X}\left((X^{\prime}-X)_{\eta}\partial^{\mu}A_{\lambda^{\prime}f}^{\eta}(\xi)\right)$
$\displaystyle=$
$\displaystyle-2\partial^{\mu}\frac{1}{\partial^{2}}\partial_{\eta}A_{\lambda^{\prime}f}^{\eta}(X).$
(96)

As a consequence, after integration by parts and calculating the trace with
respect to the color indices in Eq. (95) we obtain

$\displaystyle j_{\lambda f}^{\mu}(X)=\left(A_{\lambda
f}^{\nu}(X)-\partial^{\nu}\frac{1}{\partial^{2}}\partial_{\eta}A_{\lambda
f}^{\eta}(X)\right){\cal S}^{\mu}_{\nu},$ (97) $\displaystyle{\cal
S}^{\mu}_{\nu}$ $\displaystyle=$ $\displaystyle
g^{\mu}_{\nu}~{}Q^{\lambda}_{{{}_{\rm
2D}}f}\sum_{\sigma}\int\limits_{-\infty}^{+\infty}\frac{dp}{(2\pi)\varepsilon(p)N_{f}}[n_{q}(\varepsilon,f\sigma)+n_{\bar{q}}(\varepsilon,f\sigma)],$
(98)

where $g^{\mu}_{\nu}$ is the space-time metric tensor and a bar over the
letters identifies antiparticles. To find the occupancy number we have to
specify additionally the state in which the averaging in the last formula is
carried out.

We consider first a single particle. Since it moves either along the
longitudinal axis or in the opposite direction, we have, taking into account
the normalization per total flavor states,

$\displaystyle\frac{1}{N_{f}}\sum_{\sigma}\int\limits_{-\infty}^{+\infty}\frac{dp}{\varepsilon(p)}~{}n_{q}(\varepsilon,f\sigma)=\frac{1}{N_{f}}\sum_{\sigma}\int\limits_{-\infty}^{+\infty}\frac{dp}{\varepsilon(p)}n_{\bar{q}}(\varepsilon,f\sigma)=1.$
(99)

Substituting successively Eq.(99) into Eq.(98) and in Eq.(97), we get

$\displaystyle j_{\lambda f}^{\mu}(X)=\frac{g_{{{}_{\rm
2D}}f}^{\lambda}}{\pi}~{}\left(A_{\lambda
f}^{\mu}(X)-\partial^{\mu}\frac{1}{\partial^{2}}\partial_{\nu}A_{\lambda
f}^{\nu}(X)\right),$ (100)

where $\lambda$=0 for QED, $\lambda$=1 for QCD. We note specially that in the
above Eq. (100), the induced current $j_{\lambda f}^{\mu}(x)$ of type
$\lambda$ and flavor $f$ depends only on the initial gauge field $A_{\lambda
f}^{\mu}(x)$ of same type $\lambda$ and flavor $f$, and it does not depend on
the gauge field of the other type, even though both types of gauge fields are
present in the sum over $\lambda^{\prime}$ in the Schwinger gauge-invariant
exponential factor. Such a dependence (and independence) arises because in
obtaining the above result in Eq. (100), we expand the Schwinger gauge-
invariant exponential factor in Eq. (92), and take the trace over the color
space involving the $\tau^{\lambda^{\prime}}$ generator from the Schwinger
factor, and the $\tau^{\lambda}$ generator from
$T\langle\bar{\psi}\gamma^{\mu}\tau^{\lambda}\psi\rangle$. Because of the
orthogonality of the color generators,
2tr$\\{\tau^{\lambda^{\prime}}\tau^{\lambda}\\}$=$\delta^{\lambda^{\prime}\lambda}$,
the contribution from the interaction of the other type is zero. In physical
terms, the above mathematical result means that to be observable a color-
singlet quark density oscillation must remain colorless555The physical quark
density is the time-component of the quark current, $j^{0}(x)$, and its
temporal dependence gives the quark density oscillation. In mathematical
terms, the observability of the quark density oscillation corresponds to a
non-zero value of the quark current upon taking the trace over the color
space. and cannot become colored, when the color-singlet quark density
oscillation absorbs a gauge boson. Hence, it can only absorb a QED gauge boson
$A^{\mu}_{0}$ but not a QCD gauge boson $A^{\mu}_{1}$ to make the color-
singlet quark density oscillation observable. On the other hand, a color-octet
quark density oscillation cannot become observable until it bleaches its octet
color to become colorless. Hence it can only absorb a QCD gauge boson
$A^{\mu}_{1}$ but not a QED gauge boson $A^{\mu}_{0}$ to make the color-octet
quark density oscillation observable. Therefore, in the collective dynamics of
the quark-QCD-QED medium, quark currents of the type $\lambda$ is affected
only by gauge fields of the same type $\lambda$ as shown in Eq. (100).

###  VII.4 Maxwell equation and the collective longitudinal quark and gauge
field dynamics

The equation of motion for the gauge fields $A^{\\{0,3\\}}_{\lambda f}$ can be
obtained by minimizing the action integral ${\cal A}_{{}_{\rm 2D}}^{\rm III}$
in Eq. (52) with respect to the gauge field in question. In our problem of the
longitudinal dynamics of the quark-QCD-QED medium, the dominant action
integral comes from ${\cal A}_{A}^{\rm III}$ associated with the gauge field
$F^{03}$ while the other contributions are merely perturbations and need not
be included. Such a minimization of the 2D action integral ${\cal A_{{}_{\rm
2D}}}$ in (52) gives the Maxwell equation

$\displaystyle\partial_{\nu}\partial^{\nu}A_{\lambda
f}^{\mu}(X)-\partial_{\nu}\partial^{\mu}A_{\lambda
f}^{\nu}(X)=-g^{\lambda}_{{{}_{\rm 2D}}f}j_{\lambda
f}^{\mu}(X),~{}~{}~{}\lambda=0,1.$ (101)