transverse momentum distributions of the anomalous soft photons which provide
valuable information on the masses of the parent bosons of the anomalous soft
photons. They are exclusive measurements in which the momenta of all
participating charged particles are measured. The knowledge of the momenta of
all initial and final charged particles allows an accurate determination of
the QED bremsstrahlung $dN/dp_{T}$ distributions shown as triangular points in
Fig. 8. As shown in Fig. 8, the observed yields of soft photons exceed the
bremsstrahlung contributions substantially in both $pp$ collisions in 8(a) and
$e^{+}e^{-}$ annihilations in 8(b). The solid circular points in Fig. 8 show
the experimental $dN/dp_{T}$ data of soft photons after subtracting the
experimental background. Fig. 8(a) gives the WA102 data for $pp$ collisions at
$p_{\rm lab}$=450 GeV/c from Belogianni $et~{}al.$ Bel02 , and Fig. 8(b) is
the DELPHI results for $e^{+}e^{-}$ annihilation at the $Z_{0}$ mass of 91.18
GeV DEL06 .

Figure 8:  (a) Anomalous soft photon $dN/dp_{T}$ data from $pp$ collisions at
$p_{\rm lab}$=450 GeV/c obtained by Belogianni $et~{}al.$ Bel02 . (b)
Anomalous soft photon $dN/dp_{T}$ data from the DELPHI Collaboration for
$e^{+}e^{-}$ annihilation at the $Z_{0}$ mass of 91.18 GeV DEL06 . The solid
circular points represent the experimental data after subtracting the
experimental background, and triangle points represent the deduced
bremsstrahlung contributions. The component yields from different masses of
the thermal model are shown as separate curves. The total theoretical yields
from the different masses and the QED bremsstrahlung contribution are shown as
the solid curves.

We would like to inquire how the production of the QED mesons may be
consistent with the transverse momentum distribution of the anomalous soft
photons. For such an investigation, we rely on the thermal model which
describes well the transverse momentum distributions in the production of
hadrons of different masses in high-energy $pp$ collisions Hag65 ; Abe07 ;
Abe09 ; Ada11 . We assume that the thermal model can be extended from the
production of QCD mesons to the production of QED mesons whose decay products
are assumed to appear as anomalous soft photons. In such a thermal model, the
transverse momentum distribution of the produced QED mesons is related to the
produced QED meson mass $m$ by Hag65

$\displaystyle\frac{dN}{p_{T}dp_{T}}=Ae^{-\sqrt{m^{2}+p_{T}^{2}}/T}.$ (70)

The contribution to the total $dN/dp_{T}$ from each boson of mass $m$ is
proportional to $p_{{}_{T}}e^{-\sqrt{m^{2}+p_{T}^{2}}/T}$ which is zero at
$p_{T}$=0 and has a peak at the location $p_{T}$ given by

$\displaystyle p_{T}^{2}=\frac{1}{2}[T^{2}+\sqrt{T^{4}+4T^{2}m^{2}}].$ (71)

If $m=0$, then $dN/dp_{T}$ peaks at $p_{T}=T$. If $m$ is much greater than
$T$, then $dN/dp_{T}$ peaks at $p_{T}\sim$ $\sqrt{mT}$. Hence, for each
contributing boson mass component, the thermal model gives a distribution that
starts at zero at $p_{T}$=0 and reaches a peak of $dN/dp_{T}$ and decreases
from the peak. The total $dN/dp_{T}$ is a sum of contributions from different
QED mesons and boson components,

$\displaystyle\frac{dN}{p_{T}dp_{T}}\\!=\\!\sum_{i}A_{i}e^{-{\sqrt{m_{i}^{2}+p_{T}^{2}}/T_{i}}}.$
(72)

There can be as many contributing bosons as the number of underlying peaks in
the $dN/dp_{T}$ spectrum. While different decompositions of the spectrum into
different masses (and peaks) are possible, the structure of the $dN/dp_{T}$
data appears to require many components in Fig. 8(a) and only a single
component in Fig. 8(b). In the thermal model analysis of the $pp$ data in Fig.
8(a), we note that there appears to be a boson component of real photons with
$m_{1}$=0. Because of the $m_{i}T_{i}$ ambiguity777In fitting the thermal
model, the $m_{i}T_{i}$ ambiguity gives different values of $m_{i}$ for
different values of $T_{i}$ without changing significantly the overall quality
of the fitting. associated with the product of the meson mass $m_{i}$ and the
temperature $T_{i}$, we are content with only a consistency analysis. We
assume that QED isoscalar and isovector mesons with masses as given by Table 1
are produced in the collision, and their subsequent decay into $e^{+}e^{-}$
pairs give rise to the excess $e^{+}e^{-}$ pairs observed as anomalous soft
photons. Allowing other parameters to vary, the thermal model fit in Fig. 8(a)
is obtained with parameters $A_{1}$=$3.85\times 10^{4}$$/(2{\rm MeV}/c$),
$T_{1}$=2.20 MeV, $A_{2}$=6.65$\times 10^{4}$$/(2({\rm MeV}/c$), $T_{2}$=5.62
MeV, $A_{3}$=0.266$\times 10^{4}$$/(2({\rm MeV}/c$), and $T_{3}$=11.8 MeV,
where the different components also shown as separate curves. Adding the
contributions from the three components onto the bremsstrahlung contributions
yields the total $dN/dp_{T}$ shown as the solid curve. The comparison in Fig.
8(a) indicates that the $pp$ data are consistent with a photon component and a
boson component with a mass around 17 MeV. The magnitude of the $m_{3}$=36.4
MeV component is of the same order as the bremsstrahlung contribution or the
noise level, and is rather uncertain in Fig. 8($a$). In Fig. 8($b$), the
addition of the single component with $A_{1}$=2.7$\times$10-3$/(16{\rm
MeV}$/c), $m_{1}$=36.4 MeV, and $T_{1}$=28.3 MeV onto the bremsstrahlung
contributions gives a consistent description of the soft photon data in
$e^{+}e^{-}$ annihilations shown as the solid curve.

The component with $m_{1}$=0 in Fig. 8(a) may be associated with the decay of
the QED mesons into two photons. If so, it will be of interest to measure the
$\gamma\gamma$ invariant mass to look for diphoton resonances, as carried out
in Abr12 ; Abr19 ; Ber11 ; Ber14 ; Sch11 ; Sch12 . The $m_{2}$=17.9 MeV
components in Fig. 8(a) and the $m_{3}$=36.4 MeV component in Fig. 8(b) may be
associated with the predicted isoscalar and isovector QED mesons of Table 1.
If so, a measurement on the invariant masses of the $m_{2}$ and $m_{3}$
components will be of great interest to confirm the existence of these QED
mesons. The recent reports of the observation of a hypothetical E38 boson at
38 MeV and the structures in the $\gamma\gamma$ invariant masses at 10-15 MeV
and 38 MeV Abr12 ; Abr19 ; Ber11 ; Ber14 ; Sch11 ; Sch12 provide encouraging
impetus for further studies.

In a recent study, D’yachencho and collaborators re-analyzed the transverse
momentum distributions of anomalous soft photons of the WA102 Collaboration
Bel02 and the momentum distribution of the reaction of $p+C\to 2\gamma+X$
Abr12 and reached similar conclusions on possible production of neutral X17
and E38 bosons in these reactions Dya21 ; Dya21a .

###  V.9 Observation of the anomalous X17 particle in
3H($p$,$e^{+}e^{-}$)4Heg.s. and 7Li($p$,$e^{+}e^{-}$)8Beg.s.

The anomalous soft photons discussed in the last subsection are produced in
highly-relativistic reactions in which the produced particles travel with
velocities close to the speed of light with a transverse momentum $p_{T}$ of
the order of many tens of MeV/c. Hence the parent particles of the anomalous
soft photon that decay into electron-positron pairs would have a rest mass of
order $m\sim p_{T}/$(speed of light), or many tens of MeV/c2. They indicate
possible existence of neutral particles in the mass regions of many tens of
MeV. The extraction the anomalous particle masses from the $p_{T}$ spectrum in
the thermal model analysis has some degrees of uncertainty. The particle
masses can be better ascertained by direct measurements. Independent of the
anomalous soft photons, two other methods have been employed in the search of
light neutral bosons. One uses the decay of the neutral boson into an
$e^{+}e^{-}$ pair deB96 ; deB97 ; deB01 ; Vit08 ; deB11 ; Kra16 ; Kra19 ;
Kra21 ; Kra21a ; Sas22 ; Kra22 , and the other uses the decay into two real
photons Abr09 ; Abr12 ; Abr16 ; Abr19 .

In search of possible candidate particles for the axion Don78 , de Boer and
collaborators initiated a program to study the $e^{+}e^{-}$ spectrum in low-
energy proton fusion of light nuclei deB96 ; deB97 ; deB01 ; Vit08 . The E1
$e^{+}e^{-}$ decay of the 17.2 MeV state in 12C, and the M1 $e^{+}e^{-}$ decay
of the 17.6 MeV state in 8Be to their respective ground states were examined
to look for short-lived neutral bosons with masses between 5 and 15 MeV/c2.
Whereas for the E1 decay at large correlation angles exhibits no deviation
from internal pair conversion, the M1 angular correlation in the decay of the
excited 1+ state of 8Be at 18.15 MeV surprisingly deviates from internal pair
conversion at the 4.5$\sigma$ level deB96 ; deB97 ; deB01 . In collaboration
with ATOMKI, the group reported a neutral particle at 12 $\pm$ 2.5 MeV in the
decay of the excited 1+ isovector 17.64 state of 8Be Vit08 .

Krasznarhokay and collaborators at ATOMKI has been continuing the search for a
neutral boson with an improved $e^{+}e^{-}$ spectrometer. In a subsequent
experiment in the reaction of $p$+7Li$\to e^{+}$+$e^{-}$+8Be${}_{\rm
ground\,state}$ at proton energies about 1 MeV, the 18.15 MeV
$J^{\pi}I$=$1^{+}0$ excited 8Be∗ state was observed to decay to the 8Be ground
state by the emission of a hypothetical neutral “X17” boson with a mass of
16.70$\pm$0.35(stat)$\pm$0.5(syst) MeV Kra16 . The ATOMKI observation of such
an X17 boson has generated a great deal of interest. Even though a neutral
isoscalar QED-confined $q\bar{q}$ state was predicted earlier to have a mass
of 12.8 MeV in Table I of Won10 888 The prediction of the mass of 12.8 MeV for
the isoscalar QED meson in Table I of Won10 was calculated in the massless
quark limit. When the correction of the non-zero quark mass has been taken
into account, the predicted mass of the isoscalar QED meson increases to 17.9
MeV Won20 , the observed neutral boson with a mass of 17 MeV led to many
speculations inside and outside the known families of particles of the
Standard Model Won10 ; Won20 ; Zha17 ; Fen16 ; Fro17 ; Bat15 ; Ros17 ; Bor19 ;
Bor19a ; Cha22 ; Ell16 ; Alv18 ; Mun18 ; Ban18 ; Pad18 ; Kub22 ; Viv22 ;
Viv22a ; Bar22 .

Supporting experimental evidence for the X17 particle in the decay of the
excited 18.05 MeV $J^{\pi}I$=$0^{-}0$ state of 4He Kra19 ; Kra21 and the decay
of the excited 17.23 MeV $J^{\pi}I$=$1^{-}1$ state of 12C Kra22 have been
reported by the ATOMKI Collaboration. Earlier observations of similar
$e^{+}$$e^{-}$ pairs with invariant masses between 3 to 20 MeV in the
collision of nuclei with emulsion detectors have been reported El88 ; El96 ;
Jai07 ; deB11 . There are furthermore possible $\gamma\gamma$ invariant mass
structures at an energy around 10 to 15 MeV and 38 MeV in $pp$, and $\pi^{-}p$
reactions in COMPASS experiments Ber11 ; Ber14 ; Bev12 ; Sch11 ; Sch12 ; Ber12
; Bev20 . Different theoretical interpretations, astrophysical implications,
and experimental searches have been presented, including the fifth force of
Nature, the extension of the Standard Model, the QCD axion, dark matter, the
QED mesons, 12-quark-state and many others Zha17 ; Fen16 ; Fro17 ; Bat15 ;
Ros17 ; Ell16 ; Alv18 ; Mun18 ; Ban18 ; Pad18 ; Won10 ; Won20 ; Kub22 ; Viv22
; Viv22a ; Bar22 . As reviewed in the Proceedings to the X17 Workshop X1722 ,
the confirmation of the X17 particle is actively pursued by many laboratories,
including ATOMKI x17Kra , Dubna x17Abr , STAR x17STAR , MEGII x17MEG , TU
Prague x17Prague , NTOF x17NTOF , NA64 x17NA64 , INFN-Rome x17INFNRome , NA48
x17NA48 , Mu3e x17Mu3e , MAGIX/DarkMESA x17MAGIX , JLAB PAC50 x17JLAB ;
x17JLAB1 , PADME x17PADME , DarkLight ARIEL ; Tre22 , LUXE Hua22 , and
Montreal Tandem Mon22 .

In the ATOMKI experiments, proton beams at a laboratory energy of 0.5 to 2.5
MeV were used to fuse with 3H, 7Li, and 11B target nuclei to form excited
states of 4He, 8Be, and 12C, respectively Kra16 ; Kra19 ; Kra21 ; Kra21a ;
Kra22 ; Sas22 . The target nuclei have been so chosen that the ground states
of the product nuclei by proton fusion are closed-shell nuclei. Specifically,
the 4He ground state in the reaction of 3H$(p,e^{+}e^{-})^{4}$Heg.s. is a
spherical closed-shell nucleus. The 8Be ground state in the reaction of
7Li$(p,e^{+}e^{-})^{8}$Beg.s. is an prolate close-shell nucleus with a
longitudinal to transverse radius ratio of about 2:1. The 12C ground state in
the reaction of 11B$(p,e^{+}e^{-})^{12}$Cg.s. is an oblate closed-shell
nucleus with a longitudinal to transverse radius ratio of about 1:2 Won70 .
There is consequently a large single-particle energy gap between particle
states above the closed shell and the hole states at the top of the fermi
surface below the closed shell. The captured proton in the product nucleus in
the proton fusion reaction populate proton single-particle states above the
large closed-shell energy gap, with the proton hole single-particle states of
the triton core 3H much below at the top of the Fermi surface. The transition
of the proton from the proton particle states above the closed-shell energy
gap to the proton hole state at the top of the fermi surface to reach the
closed-shell 4He ground state in the reaction will release the large closed-
shell-gap energy that is of order about 17-20 MeV, as shown in Fig. 9($a$) and
9($b$) . Such a large closed-shell-energy gap may be sufficient to produce a
neutral boson, if there would exist such a stable neutral boson particle with
the proper energy, quantum numbers, and other conditions appropriate for its
production.

Figure 9:  (a) The energy levels and widths of 4He states showing the
transition in the decay of the 21.01 MeV $J^{\pi}I$=$0^{-}0$ 4He excited state
to the 4He ground state with the emission of the X17 particle at $\sim$17 MeV,
as interpreted in Kra19 . (b) The energy levels and widths of 8Be states
showing the transition in the decay of the excited 18.15 MeV
$J^{\pi}I$=$1^{+}0$ state to the 8Be ground state with the emission of the X17
particle, as interpreted in Kra16 .

The picture is simplest for the $p$+3H$\to^{4}$He∗$\to$
$(e^{+}$+$e^{-})$+4Heg.s. reaction for which the ground state of the product
4He nucleus is spherical. The captured proton in the lowest few excited 4He∗
state must occupy a $p$-shell proton single-particle state that is
significantly stretched outside the triton 3H core with a proton and two
neutrons in the lower occupied $s$-states. In the collision of the incident
proton on the target 3H nucleus, there is the fusion barrier which comprises
of the Coulomb and centrifugal barrier for a single-particle with an angular
momentum $l$,

$\displaystyle E_{l}=\frac{Z_{p}Z_{T}\,e^{2}}{R}+\frac{l(l+1)\hbar^{2}}{2\mu
R^{2}},$ (73)

where $R=r_{0}(A_{p}^{1/3}+A_{T}^{1/3})$,
$\mu=[(A_{p}A_{T})/(A_{p}+A_{T})]m_{\rm nucleon}$, $(Z_{p},A_{p})$, and
$(Z_{T},A_{T})$ are the projectile and target charge and atomic numbers,
respectively. For $r_{0}$=1.3 fm, the $p$+3H$\to^{4}$He∗ fusion barrier for
$l=1$ is about 6 MeV which places the low energy beam of around 1 MeV to lie
below the fusion barrier for the $l=1$ partial waves.

Figure 10:  (a) ATOMKI experimental $e^{+}$ and $e^{-}$ angular correlations
at different proton energies in the reaction of 4He$(p,e^{+}e^{-})$4Heg.s.
Kra19 ; Kra21 . The solid curves are contributions from known sources of the
$e^{+}e^{-}$ cross sections from the decay of the 20.21 MeV $0^{+}0$ 4He∗
excited state to $0^{+}0$(ground state) and the E1 decay from non-resonant
capture into an excited 4He continuum state, displaying an excess of
$e^{+}e^{-}$ production at large correlation angles Kra19 ; Kra21 . ($b$) The
data of an excess $e^{+}e^{-}$ production at large correlation angles can be
described by the de-excitation of the 21.02 MeV $0^{+}0$ 4He∗ state to the
ground state with the emission of a hypothetical X17 particle with a mass of
16.94 MeV, as described by the solid histograms Kra19 ; Kra21 . (c) Similar
$e^{+}e^{-}$ excess occurs in the reaction of 7Li$(p,e^{+}e^{-})$8Beg.s.
Kra21a . The data of excess $e^{+}e^{-}$ production at large correlation
angles can be described as arising from the decay of the excited 18.15 MeV
$J^{\pi}I$=$1^{+}0$ state of 8Be to the ground state with the emission of the
X17 particle at $\sim$17 MeV (solid histogram) Kra21a .

The capture of a proton of $E_{\rm lab}$ will lead to an excited 4He∗ nucleus
at an excitation of $E_{x}$=$(A_{T}/(A_{T}+1))E_{\rm lab}+Q$ where the $Q$
value for the p+3H$\to^{4}$He∗ reaction is 19.815 MeV. The ATOMKI experimental
proton energies of $E_{\rm lab}$=$\\{$510, 610, and 900$\\}$ keV in Kra21 in
Fig. 10(a), correspond to the 4He∗ excitation energies of
$E_{x}$=$\\{20.20,20.27,20.49\\}$ MeV. We can compare these excitation
energies $E_{x}$ with the energy levels of 4He∗ states in Fig. 9($a$) Til92 ,
where the lowest states are likely pocket resonances in the potential pocket
Lee20 , as evidenced by their narrow proton widths: The $J^{\pi}I=0^{+}0$
state at 20.21 MeV has a proton width of 0.64 MeV while the $J^{\pi}I=0^{-}0$
state at 21.02 MeV has a proton width of 0.50 MeV.

Depending on the collision energy, there will be probabilities for capturing
into the pocket resonances inside the potential pocket with various proton
capture widths. These resonances will subsequently decay to the ground state
by internal conversion with the emission of an $e^{+}e^{-}$ pair. For example,
capturing to the $0^{+}0$ resonance state at 20.21 MeV will decay to the
ground state by internal conversion with the emission of an $e^{+}e^{-}$ pair,
as indicated in Fig. 9($a$). There is also a finite probability for the direct
non-resonant $l$=1 capture into a continuum state which tunnels through the
Coulomb and centrifugal barrier to the interior region and subsequently decays
into the ground state, with the emission an $e^{+}e^{-}$ pair by the E1
electromagnetic radiation. The range of the collision energies in the ATOMKI
experiment lead to the population of both the $J^{\pi}I=0^{+}0$ state at 20.21
MeV and the $J^{\pi}I=0^{-}0$ state at 21.02 MeV. The de-excitation of the
$J^{\pi}I=0^{+}0$ state at 20.21 MeV can be accounted for by the internal pair
conversion from $0^{+}0\to 0^{+}0$ final state. The solid curves in Fig.
10($a$) represent the sum of (i) the $0^{+}0\to 0^{+}0$ internal conversion
contribution and (ii) the E1 non-resonant capture contribution to $e^{+}e^{-}$
Kra19 . As one observes in Fig. 10($a$), there appears an excess of the
$e^{+}e^{-}$ counts at large correlation angles. The excess $e^{+}e^{-}$ yield
can be described as arising from the decay of the $J^{\pi}I$=$0^{-}0$ excited
state of 4He∗ at 21.02 MeV to the 4He ground state with the emission of a
neutral “X17” boson with a mass of 16.94$\pm$0.12$\pm$ MeV as indicated in
Fig. 10($b$) Kra21 . In such an interpretation, the reaction for the
production of the X17 state is

${}^{4}{\rm He}^{*}{\rm(21.02~{}MeV,}J^{\pi}I=0^{-}0)\to~{}{\rm
X17}~{}+~{}^{4}{\rm He}({\rm ground~{}state},J^{\pi}I=0^{+}0),$ (74)

which implies that the quantum numbers of the emitted X17 particle is
$J^{\pi}I=0^{-}0$.

In another experiment in the reaction of 7Li+$p\to^{8}$Be∗$\to$
$(e^{+}$+$e^{-})$\+ 8Beg.s. at ATOMKI Kra16 ; Kra19 at a proton beam energy of
a few MeV, the 18.15 MeV $J^{\pi}I$=$1^{+}0$ state of 8Be∗ was populated
together with its neighboring states. From the nuclear reaction viewpoints,
the captured proton populates single-particle states in the deformed Nilsson
single-particle $p$-shell states. The fusion barrier for $l=1$ partial waves
lies at about 4.5 MeV if the target nucleus were spherical. Even though the
fusion barrier will be modified because of the target deformation Won73 , the
fusion barrier for $l=1$ would likely place the low energy beam of around 1 to
2 MeV to be below the fusion barrier for $l=1$ partial waves, as evidenced by
their narrow resonance widths. The capture of a proton of $E_{\rm lab}$ will
lead to an excited 8Be∗ nucleus at an excitation energy of
$E_{x}=(A_{T}/(A_{T}+1))E_{\rm lab}+Q$ where the $Q$ value for s the
p+7Li$\to^{8}$Be∗ reaction is 17.255 MeV. The proton fusion experiments at
ATOMKI with energies $E_{\rm lab}$=$\\{$0.800,1.04,1.10,1.20$\\}$ MeV in
$p$+7Li$\to$8Be∗ in Kra16 correspond to the production of an excited 8Be∗
nucleus at $E_{x}=$$\\{$17.955,18.165,18.2175,18.305$\\}$ MeV, respectively.

We can compare these excitation energies $E_{x}$ with the energy levels of
8Be∗ states in Fig. 9($b$) Til04 . Of particular interest are the 17.640 MeV
$J^{\pi}I=$$1^{+}1$ isovector state with a total width of 0.011 MeV decaying
by $\gamma$ and $\alpha$ emissions, and the 18.150 MeV $J^{\pi}I=$$1^{+}0$
isoscalar state with a total width of 0.138 MeV decaying by $\gamma$ and
$\alpha$ emissions. Depending on the collision energy, there will be
probabilities for capturing into the pocket resonance states inside the
potential pocket with various proton capture widths. These resonances will
subsequently decay into the ground state by internal conversion with the
emission of an $e^{+}e^{-}$ pair. There will also be a finite probability for
the non-resonant $l$=1 capture into a continuum state which tunnels through
the Coulomb and centrifugal barrier and decays to the ground state with an
$e^{+}e^{-}$ pair by an E1 electromagnetic radiation. The $J^{\pi}I=1^{+}0$
state at 18.15 MeV can make a transition to the $0^{+}0$ final ground state by
the M1 $e^{+}e^{-}$ internal pair conversion. The dotted curve in Fig. 10(c)
represents the sum of (i) the M1 $e^{+}e^{-}$ internal conversion contribution
and (ii) the E1 non-resonant capture contribution to $e^{+}e^{-}$ Kra21a . As
one observes in Fig. 10($c$), there appears an excess of the $e^{+}e^{-}$
counts at large correlation angles around $\theta\sim 140^{o}$. The excess
$e^{+}e^{-}$ yield can be described by the solid curve as arising from the
decay of $J^{\pi}I$=$1^{+}0$ excited state of 8Be∗ at 18.15 MeV to the 8Be
ground state with the emission of a neutral “X17” boson with a mass of
17.11$\pm$0.12 MeV Kra21a .

The approximate equality of the masses of the hypothetical neutral boson in
4He∗ and 8Be∗ decays suggests that they are likely to be the same particle,
the X17 particle with $J^{\pi}I=0^{-}0$, emitted in the decay of the excited
$0^{-}0$ state at 21.02 MeV of 4He∗, in the $l=1$ partial wave from the
excited $1^{+}0$ state at 18.05 MeV of 8Be∗. From Wheeler’s molecular
viewpoints of the nuclei structure of light nuclei Whe37 , the ground state of
8Be is likely to be in the form of two $\alpha$ clusters because of the strong
binding of the alpha particle. So, in the excited 18.05 MeV $J^{\pi}I=$1+0
excited 8Be∗ system, the emission would likely come from one of the two
$\alpha$ clusters of 8Be∗ and the emitting cluster has a non-zero angular
momentum relative to the 8Be∗ nucleus center of mass. From this viewpoint, it
is reasonable to assign the quantum numbers of the X17 to be $J^{\pi}I=0^{-}0$
from the 8Be∗ decay, the same as those from the 4He∗ decay.

Recently, a question was raised concerning the background subtraction of the
$e^{+}e^{-}$ signals from the M1 and E1 contributions in the analysis of the
anomalous X17 particle in the 7Li+$p$$\to$Be∗ decay Kra16 . Hayes $et~{}al.$
Hay22 re-examined the angular correlations in the $e^{+}e^{-}$ decay of those
excited states in 8Be in the ATOMKI experiment. In the range of ATOMKI proton
energies, the ratio of the E1 and M1 contributions to the $e^{+}e^{-}$
production was found to be a sensitive function of energy. They questioned the
earlier assumption in Kra16 of an energy-independent admixture of the E1 and
M1 contributions for the $e^{+}e^{-}$ production. They found that the
existence of a ‘bump’ in the measured angular distribution depended strongly
on the assumed M1/E1 ratio and the measured large-angle contributions to the
$e^{+}e^{-}$ angular distribution to be lower than expectation. They cast
doubts on the evidence for the existence of the anomaly in the present
analysis.

However, there was another recent measurement of the angular correlation of
$e^{+}e^{-}$ pairs produced in the 7Li($p,\gamma$)8Beg.s reaction at the sharp
$E_{p}$= 441 keV resonance and 650 keV, 800 keV and 1100 keV proton beam
energies Sas22 . The spectra measured at the resonance could be understood
through the M1 internal pair creation process, but in the case of the off-
resonance regions (direct proton capture) significant anomalies were observed
in the $e^{+}e^{-}$ angular correlations supporting the X17 hypothetical
particle creation and decay. Clearly, many more measurements in search of the
X17 particles and many theoretical analyses need to be carried out to clarify
the situation.

###  V.10 Observation of the anomalous X17 particle in
11B(p,$e^{+}e^{-}$)12Cg.s.

Subsequent to the observation of the X17 particle, it was suggested that the
electromagnetic E1 decay of the excited $J^{\pi}I$=$1^{-}1$ 12C∗ state at
17.23 MeV to the ground state should be studied in order to determine whether
the X17 particle has a vector or axial-vector characteristics Fen20 . ATOMKI
measurements with proton energies between 1.5 to 2.5 MeV showed that the X17
particle was generated predominantly by the E1 radiation. It was concluded in
Kra22 that the association of the observation of the X17 particle with the
decay of the 17.23 MeV $J^{\pi}I$=$1^{-}1$ 12C∗ state supports the vector
character of the X17 particle, as suggested by Feng and collaborators Kra22 .
In such an interpretation, the reaction for the production of the X17 state is

${}^{12}{\rm C}^{*}~{}{\rm(17.23~{}MeV,}J^{\pi}I=1^{-}1)\to~{}{\rm
X17}~{}+~{}^{12}{\rm C}({\rm ground~{}state},J^{\pi}I=0^{+}0),$ (75)

which implies that the quantum numbers of the emitted X17 particle is
$J^{\pi}I=1^{-}1$.

In the fusion of $p$ with 11B target nucleus, the captured proton populates a
Nilsson oblate orbital originating from the spherical $p$-shell states. The
Coulomb and centrifugal barrier for a spherical $l=1$ partial waves would lie
at about 4.3 MeV. Even though the fusion barrier will be modified because of
the target deformation, the collision energy of $E_{\rm lab}$ from 1.5 to 2.5
MeV are much below the fusion barrier. So most of the the lowest 12C∗ excited
states populated by the proton fusion reaction are likely pocket resonances in
the potential pocket with small proton decay widths. The capture of a proton
of $E_{\rm lab}$ will lead to an excited 12C∗ nucleus at an excitation of
$E_{x}=(A_{T}/(A_{T}+1))E_{\rm lab}+Q$ where the $Q$ value for the
$p$+11B$\to^{12}$C∗ reaction is 15.957 MeV Kel17 . The proton-11B fusion
experiments at ATOMKI with energies of $E_{\rm lab}$=$\\{$1.5,1.70,1.88, 2.10,
2.5$\\}$ MeV in Kra22 correspond to a the production of an excited 12C at
energies of $E_{x}$=$\\{17.33,17.52,17.68,17.88,18.25\\}$ MeV, respectively.
They can be compared with the energy levels and their widths of 12C∗ states
listed in Table IV Kel17 .

Table 4:  Listed are the 12C energy levels accessible to low-energy proton fusion with 11B and their widths, from Kel17 . The $Q$ value for the fusion of $p$+11B$\to$12C∗ is 15.975 MeV. $E_{x}$ in 12C |  $J^{\pi}I$ | $\Gamma$ | Decay
---|---|---|---
(MeV) |  | (MeV) |
16.1060$\pm$0.0008 |  $2^{+}$ 1 | 0.0053 | $\gamma,p,\alpha$
16.62$\pm$0.050 |  $2^{-}$ 1 | 0.28 | $\gamma,p,\alpha$
17.23 |  $1^{-}$ 1 | 1.150 | $\gamma,p,\alpha$
17.76$\pm$0.020 |  $0^{+}$ 1 | 0.096 | $p,\alpha$
18.16$\pm$0.070 |  $(1^{+}$ 0) | 0.24 | $\gamma,p$
18.35$\pm$0.050 |  $3^{-}$ 1 | 0.22 | $\gamma,p,\alpha$
18.35$\pm$0.050 |  $2^{-}$ 0+1 | 0.35 | $n,p,\alpha$
18.60$\pm$0.100 | $(3^{-})$ | 0.30 |
18.71 | $(I=1)$ | 0.10 |
18.80$\pm$0.040 | $2^{+}$ | 0.10 | $\gamma,n,p$
19.20$\pm$0.600 | $(2^{-},1)$ | 0.49 | $\gamma,p,\alpha$
19.40 $\pm$0.025 |  $(1^{-}$ 1) | 0.10 | $\gamma,p,\alpha$
19.555$\pm$0.025 |  $(1^{-}$ 1) | 0.10 | $\gamma,p,\alpha$
19.69 | $1^{+}$ | 0.23 | $n,p$
20.00$\pm$0.100 | $2^{+}$ | 0.375 | $\gamma,n,p$

One observes that there are five 12C∗ excited states which will be populated
within the experimental range of proton energies of the ATOMKI experiments.
Three of these states, the 17.23 MeV $1^{-}1$ state, the 18.15 MeV $1^{+}0$
state, and the 18.35 MeV $3^{-}1$ state emits gamma radiation which will
contribute to the $e^{+}e^{-}$ internal conversion.

The interpretation of the X17 as possessing the $J^{\pi}I=1^{-}1$ isovector
characteristics as presented in Kra22 is subject to serious questions. First
of all, such an interpretation of the X17 as an isovector $J^{\pi}I=1^{-}1$
particle differs from the the $J^{\pi}I=0^{-}0$ isoscalar interpretation of
the X17 particle as suggested earlier in 4He∗ and 8Be∗ decays Kra16 ; Kra19 .
The observed boson or bosons in the two measurements have about the same mass,
and are likely the same particle. If there is only a single neutral boson with
a mass of about 17 MeV, then one the two assignments of the $J^{\pi}I$ quantum
numbers of the neutral X17 particle may be incorrect. Secondly, the large
number of states in the neighborhood of the 17.23 MeV $1^{+}$1 12C state
(Table 4) that are populated within the experimental range of proton energies
of the ATOMKI experiments make it clear that it may be necessary to take into
account the possibility that the X17 may arise from other states. Of
particular interest is the 18.16 MeV $1^{+}0$ state with a width of 0.24 MeV,
which decays by $\gamma$ and $p$ emissions. This 18.16 MeV $1^{+}0$ 12C state
may be the analogue of the 18.15 MeV $1^{+}0$ 8Be state because they have the
same quantum numbers, excitation energies, decay widths, decay channels, and
they may represent a proton populating a $p$ orbital away from a triton core
in one of the $\alpha$ particles in the picture of Wheeler’s alpha-particle
model. The X17 particle observed in the 11Li$(p,e^{+}e^{-})^{12}$Cg.s.
reaction in Kra22 may arise also from the decay of the 18.15 MeV 1${}^{+}0$
state in 11Li$(p,e^{+}e^{-})^{12}$Cg.s. collisions. If the transition from the
18.15 MeV $J^{\pi}I=1^{+}0$ state to the ground state contributes dominantly
to the X17 production instead of the transition from the 17.23 MeV
$J^{\pi}I=1^{-}1$ state as suggested in Kra22 , the X17 data of 4He∗ decay,
the 8Be∗ decay and the 12C∗ decay will be consistent with each other, whereas
the $J^{\pi}I=1^{-}1$ interpretation of 12C contradicts the the
$J^{\pi}I=0^{-}0$ interpretation of 4He and 8Be.

In order to discriminate the two possible assignments of $J^{\pi}I=1^{-}1$ or
$0^{-}0$, we can suggest the search for two-photon decay of the X17 particle.
Such a decay is possible if the X17 is a $J^{\pi}I=0^{-}0$ particle. However,
because of the Landau-Yang theorem Lan48 ; Yan50 , the decay of a $1^{-}$
state into two photons is impossible. If X17 is found to decay into two
photons, then the X17 can only be a $0^{-}0$ particle and not a $1^{-}1$
particle.

Another way to study the 12C∗ system and the X17 particle is to excite a 12C
target ground state by direct photon excitation to various states, such as the
$1^{-}1$ at 17.23 MeV by E1 excitation and the $1^{+}0$ at 18.15 MeV by M1
excitation. After the nucleus excited to these specific states, their decay by
the emission of an X17 particle or a internal conversion photon in the form of
$e^{+}e^{-}$ can be examined.

There is one more way to study the X17 particle and its quantum number
assignments, if the X17 indeed has the $J^{\pi}I=1^{-}1$ property as proposed
by Fen20 ; Kra22 . Because the strength of the giant dipole state of 12C
resides at 25 MeV and not at 17.23 MeV and the 17.23 represents only a very
small fraction of the giant dipole strength Ber75 ; He16 , the X17 particles
would be copious produced at $E_{x}$=25 MeV, and and such a copious X17
production would allow the study of the X17 properties, if the X17 indeed has
the $J^{\pi}I=1^{-}1$ property.

###  V.11 The observation of the E38 Particle

Abraamyan and collaborators at Dubna have been using the two-photon decay of a
neutral boson to study the resonance structure, the dynamics, and the
interaction of the lightest hadrons near their thresholds. Their experiments
have been carried out by using the proton, deuteron, and other light-ion beams
of the Nucleotron at JINR, Dubna on fixed internal fixed targets of C, Cu, and
other nuclei, with the production of hadrons and photons. Their PHONTON2
photon detector consists of two arms placed at 26o and 28o degrees from the
beam direction, with each arm equipped with 32 lead-glass photon detectors as
shown in Fig. 11($a$). The photon detectors measure the energies and the
emission angles of the photons. From the energies and the opening angle
$\theta_{12}$ of two photons $\gamma_{1}$ and $\gamma_{2}$ in coincidence, the
invariant mass of the pair, $M_{\gamma_{1}\gamma_{2}}=E_{\gamma 1}E_{\gamma
2}(1-\cos\theta_{12})$, can be evaluated and the invariant mass distribution
determined. Neutral bosons with large invariant masses can be searched by
selecting photon pairs with large opening angles from different arms. On the
other hand, by selecting photon pairs from the same arm with small opening
angles, it is possible to study neutral bosons with small invariant masses
such as those below the pion mass gap $m_{\pi}$, if they ever could be stable.
Thus, the PHOTON2 detector can be used to probe the possible existence of
neutral bosons over a large dynamical range. Previously, along with the
detection of the $\pi^{0}$ and $\eta$ mesons, the Dubna Collaboration
uncovered a new resonance structure at $M_{\gamma\gamma}$=360 MeV (the R360
resonance) Abr09 . The resonance was subsequently confirmed by repeated
measurements Abr16 with higher statistics. We shall explore a possible model
for R360 in Section 7F in terms of a molecular state of two pions and two E38
QED mesons.

Figure 11:  ($a$) Dubna PHOTON2 detector arrangement to study the two-photon
decay of reaction products in the collision of proton and light ions on fixed
internal targets. ($b$) The invariant mass distribution of photon pairs from
the same event are shown as the solid curves with the invariant mass
distribution of photon pairs from the mixed events shown as the shaded
regions. The signal of the invariant mass of the correlated pair after
subtracting the mixed-event gubernatorial background shows a resonance
structure at 38 MeV.

In 2011, Van Beveren and Rupp studied $e^{+}e^{-}$$\to\pi^{+}\pi^{-}$ Aul05 ,
$p\bar{p}$$\to J/\psi\pi^{+}\pi^{-}$ Aal09 , and
$e^{+}e^{-}$$\to\Upsilon\pi^{+}\pi^{-}$ cross sections and observed that the
deviation of the cross section from a global fit oscillates with a period of
73 to 79 MeV Bev11 . An analysis of the invariant mass distribution of
$M_{\Upsilon(1S)}$ as determined from
$e^{+}e^{-}\to\Upsilon(2^{3}S_{1})\to\pi^{+}\pi^{-}\to\pi^{+}\pi^{-}\mu^{+}\mu^{-}$
and the invariant mass distribution of $M_{\Upsilon(1S)}$ as determined from
$\Upsilon(2^{3}S_{1})\to\mu^{+}\mu^{-}$ Gui09 suggested a shift of the
excitation function by 38 MeV. Van Beveren and Rupp therefore speculated on
the possible existence of a light boson with a mass of 38 MeV Bev11 . Their
examination of the two photon spectrum of the COMPASS Collaboration data Sch11
also suggested a possible resonance at 38 MeV. There ensued a debate on the
background subtraction of the COMPASS data and the reliability of the
identification of the E38 resonance using the raw COMPASS data Ber11 ; Ber14 ;
Bev12 ; Sch11 ; Sch12 ; Ber12 ; Bev20 .

Stimulated by the possibility of a light neutral boson with a mass of about 38
MeV, the Dubna Collaboration undertook a search for the E38 particle using the
two-photon decay. The search was carried out in the $d$(2.0 GeV/n)+C, $d$(3.0
GeV/n)+Cu and $p$(4.6 GeV)+C reactions with internal targets at the JINR
Nuclotron. It was reported in Version 1 of Ref. Abr12 that the invariant
masses of the two-photon distribution in these reactions exhibit a resonance
structure at around 38 MeV. Unaware of the earlier theoretical prediction of
an isoscalar $J^{\pi}I=0^{-}0$ QED-confined $q\bar{q}$ state at 38.4 MeV in
Table I of Won10 that could be a good candidate for such a particle, Version 2
of Ref. Abr12 stated, however, “due to non-ordinariness of the obtained
results (standing out of The Standard Model) and at the request of co-authors,
the first version of the article is withdrawn for further verification and
more detailed description of the experiment and data analysis.” Subsequent
repeated checking of the results of their analysis over a period of many years
confirmed their findings, and the updated results were then reported in Abr19
.

In the latest Dubna experiment, the two-photon invariant masses spectra were
studied in nuclear reactions of $d$C, $d$Cu, and $p$C at 2.75, 3.83, and 5.5
GeV/c per nucleon, respectively. In different momentum windows, photon pairs
in the same event from the PHOTON2 detector are selected and their invariant
masses calculated to construct the invariant mass distribution shown as the
solid curves in Fig. 11($b$). The corresponding combinatorial backgrounds are
obtained in the event-mixing method by calculating the invariant masses of two
photons from two different events, shown as the shade regions in Fig. 11($b$).
The difference between the correlated invariant mass distribution and the
mixed-event invariant mass distribution gives the signal invariant mass
distribution, as shown in the lower panels of Fig. 11($b$) for different phase
space windows. The resonance structure in Fig. 11(b) indicates a neutral
boson, the E38 particle, at about 38 MeV Abr19 . Signals from other phase
space windows show a similar resonance structure.

The observation of E38 at Dubna completes an important piece of the anomalous
particle puzzle as the isoscalar X17 and the isovector E38 come in a pair, and
they are orthogonal linear combinations of the $|u\bar{u}\rangle$ and
$|d\bar{d}\rangle$ components. The agreement of their masses with those
predicted by the phenomenological open-string model of QED-confined $q\bar{q}$
model of Section V lends support to the description that a quark and an
antiquark can be confined and bound as stable QED-confined mesons interacting
in the Abelian U(1) QED interaction. This is a rather unusual and unfamiliar
concept. The confirmation of these anomalous particle observations will be
therefore of great interest by detecting the X17, the E38, and $\pi^{0}$ in
the same experimental set up.

###  V.12 Anomalous Particle Production mechanisms

It is necessary to understand the mechanisms how the anomalous particles may
be be produced in order to fully support the interpretation on their
production as QED mesons. For the low-energy production of the X17 particle in
the decays of the excited states of 4He∗ and 8Be∗ at ATOMKI Kra16 ; Kra21 , we
envisage the scenario as described in Section 2 that the excited states of
8Be(1${}^{+}\,$18.15 MeV) and ${}^{4}{\rm He}(0^{-}\,$20.02 MeV) are formed by
pulling a proton out of one of the alpha-particles of the
$(\alpha)^{n}$-nucleus core and by placing the proton on an orbital that is
considerably outside the corresponding tritium core as shown in Fig. 3($a$).
The stretched string-like interaction between the proton and the tritium core
polarizes the vacuum so much that the proton may emit a virtual gluon which
fuses with the virtual gluon from the 3H core to lead to the production of a
color-singlet $q\bar{q}$ pair by the reaction $g+g\to q+\bar{q}$ as shown in
Fig. 3($a$). At the appropriate $\sqrt{s}(q\bar{q})$ eigenenergy, the QED
interaction between the $q$ and the $\bar{q}$ may result in the formation of
the QED-confined $q$$\bar{q}$ bound state X17 Won10 ; Won20 , which
subsequently decays into $e^{+}$-$e^{-}$.

For the production of E38 in high-energy nucleus-nucleus collisions at Dubna
Collaboration Abr12 ; Abr19 , and the anomalous soft photons in the high-
energy hadron-hadron collisions at CERN Chl84 ; Bot91 ; Ban93 ; Bel97 ;
Bel02pi ; Bel02 ; Per09 , we envisage the production of many $q\bar{q}$ pairs
as described in Fig. 3($b$) of Section I2. In such high-energy processes, many
$q\bar{q}$ pairs are produced. Even though most of the produced $q\bar{q}$
will materialize as QCD mesons, some of the $q\bar{q}$ pairs may materialize
as QED mesons, as the E38 particle that decays into diphotons at Dubna Abr19
and as the X17 particle and the E38 particle that decay into the anomalous
soft photons in the WA102 experiment Bel02 .

In the other process for the production of the anomalous soft photons in high-
energy $e^{+}e^{-}$ annihilation at $Z^{0}$ mass at DELPHI, we envisage the
production $q\bar{q}$ pairs as described in Fig. 1($c$) of Section 2 in which
the $e^{+}e^{-}$ annihilation process leads to the production of a large
number of $q\bar{q}$ pairs. While most produced $q$$\bar{q}$ pairs will lead
to hadron production, there may however be $q$$\bar{q}$ pairs with
$(m_{q}+m_{\bar{q}})$$<$$\sqrt{s}$$(q\bar{q})$$<$$m_{\pi}$ for which the QED
interaction between the quark and the antiquark may lead to the production of
the X17 and E38 particles at the appropriate energies and the anomalous
particles subsequently decay into $e^{+}e^{-}$ pairs.

##  VI Questions on quark confinement in compact QED in (3+1)D from lattice
gauge calculations

###  VI.1 Lattice gauge calculations prediction of deconfined static quark and
antiquark in compact QED in (3+1)D

We mentioned in Section 4 that in lattice gauge theory a static fermion and a
static antifermion in (3+1)D in compact QED interaction has a strong-coupling
confined phase and a weak-coupling deconfined phase, as shown by Wilson,
Kogut, Susskind, Mandelstam, Polyakov, Banks, Jaffe, Drell, Peskin, Guth,
Kondo and many others Wil74 ; Kog75 ; Man75 ; Pol77 ; Pol87 ; Ban77 ; Gli77 ;
Pes78 ; Dre79 ; Gut80 ; Kon98 ; Mag20 . The transition from the confined phase
to the deconfined phases occurs at the coupling constant $\alpha_{\rm
crit}=g_{\rm crit}^{2}/4\pi$=0.988989481 Arn03 ; Lov21 . The magnitude of the
QED coupling constant as given by the fine-structure constant
$\alpha_{c}$=1/137 is less than $\alpha_{\rm crit}$. That is, the QED
interaction between a quark and an antiquark belongs to the weak-coupling
deconfined regime. Therefore, a static quark and a static antiquark are
deconfined in lattice gauge calculations in compact QED in (3+1)D.

The deconfined static quark and static antiquark in the lattice gauge results
in (3+1)D poses a serious question. There are experimental circumstances in
which a quark and an antiquark can be produced and they can interact in QED
alone, without the QCD interaction, as we discussed in Section 2. For example,
we can study 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 range $(m_{q}+m_{\bar{q}})<\sqrt{s}(q\bar{q})<m_{\pi}$,
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 $q$ and $\bar{q}$
pair and the quanta mediating their interactions must also combine together
into a color-singlet final state. In this energy range below the collective
QCD excitation mass gap of $m_{\pi}$, there is insufficient energy to excite a
collective QCD excitation to produce a QCD meson. A $q$ and $\bar{q}$ can be
produced and interact in the QED interaction alone. In the color-singlet
$(q\bar{q})^{1}$ configuration, the produced $q$ and $\bar{q}$ can interact
with the colorless Abelian U(1) QED interaction to form a color-singlet
$[(q\bar{q})^{1}\gamma^{1}]^{1}$ final state, if there is a QED meson
eigenstate at this eigenenergy. At energies other than the QED meson
eigenenergies in this energy range below $m_{\pi}$, the $e^{+}$ \+ $e^{-}$
collision will probe the dynamics of a quark and antiquark interacting in QED
alone, without the QCD interaction. The absence of fractional charges in
collisions in this energy range in $e^{+}$ \+ $e^{-}$ collisions indicates the
absence of the continuum states of an isolated quark and an antiquark in the
interaction of a quark and an antiquark in the QED interaction.

On the other hand, the solution of deconfined static quark and static
antiquark in the lattice gauge calculations in QED in (3+1)D predicts that the
$q$ and $\bar{q}$ produced in $e^{+}$ \+ $e^{-}$ collisions in the range of
energy below the QCD collective excitation mass gap $m_{\pi}$ will not be
confined and would appear as fraction charges, when the quark interact with
the antiquark in QED alone in (3+1)D. However, no such fractional charges have
ever been observed. Furthermore, the phenomenological open-string QCD and QED
meson model with the hypothesis of a confined $q\bar{q}$ pair in QED in (3+1)D
leads to QED meson and QCD meson spectra in agreement with experimental data,
as discussed in Section 5. There is a confined regime in (1+1)D QED for
dynamical massless quarks in the Schwinger confinement mechanismSch62 ; Sch63
, and there is also a confined regime for quarks in compact QED in (2+1)D
Pol77 ; Pol87 in Polyakov’s transverse confinement, for all gauge coupling
interaction strengths. They indicate that the present-day lattice gauge
calculations for compact QED in (3+1)D may not be complete and definitive,
because the important Schwinger dynamical quark effects associated with light
quarks has not been included. Future lattice gauge calculations with dynamical
quarks in compact QED interactions in (3+1)D will be of great interest in
clarifying the question of quark confinement in QED.

###  VI.2 Compact and non-compact U(1) QED gauge interactions in lattice gaige
calcualtions

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 opposing theoretical conclusions on quark
confinement in QED in (3+1)D and the agreement of the experimental spectrum of
the anomalous particles with the QED meson predictions, it is possible that
the two conclusions can still be consistent with each other, if the
confinement conclusions arise from the inclusion of the Schwinger confinement
mechanism Sch62 ; Sch63 in the works of Won10 ; Won11 ; Won14 ; Won20 ; Won22
; Won22a ; Won22b ; Won22c , while and the deconfinement conclusion arises
from the absence of the Schwinger confinement mechanism in lattice
calculations Arn03 ; Lov21 . Therefore, it is worth constructing a plausible
“stretch (2+1)D” flux tube model to demonstrate the importance of the
Schwinger confinement mechanism in compact QED in (3+1)D.

In such a demonstration, we note first of all that the Schwinger confinement
mechanism occurs in (1+1)D space-time, whereas the physical world is in
(3+1)D. For the Schwinger confinement mechanism to be operative, the quark-QED
system must possess transverse confinement 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 from Polyakov’s previous results
that electric charges of opposite signs 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 . We can combine Schwinger’s
longitudinal confinement in (1+1)D QED with Polyakov’s transverse confinement
in (2+1)D compact QED to study QED confinement in a flux tube environment in
(3+1)D.

Before we come to the “stretch (2+1)D” flux tube model of $q\bar{q}$
production in compact QED in (3+1)D, it is necessary to clarify the concept of
compactness in the QED interaction.

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 which 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

$\displaystyle S=\frac{1}{2g^{2}}\sum_{x,\alpha\beta}(1-\cos
F_{x,\alpha\beta}),$ (76)

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

$\displaystyle
F_{x,\alpha\beta}\\!=\\!A_{x,\alpha}\\!\\!+\\!\\!A_{x+\alpha,\beta}\\!\\!-\\!\\!A_{x+\beta,\alpha}\\!\\!-\\!\\!A_{x,\beta},{\rm
with}-\\!\\!\pi\\!\\!\leq A_{x,\alpha}\\!\\!\leq\\!\\!\pi.~{}$ (77)

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

$\displaystyle
S=\frac{1}{4g^{2}}\sum_{x,\alpha\beta}F_{x,\alpha\beta}^{2},~{}~{}~{}{\rm
with}~{}~{}~{}~{}-\infty\leq A_{x,\alpha}\leq+\infty.$ (78)

In non-compact QED gauge theories, the photons do not interact with other
photons and the fermions in non-compact QED gauge theories are always de-
confined. In compact QED in (2+1)D, the photons interact among themselves
because the gauge fields are angular variables. Static charges of opposite
signs are confined for all strength of the coupling constant in compact QED in
(2+1)D. In compact QED in (3+1)D, however, static opposite charges are
confined only for strong coupling but de-confined for weak coupling Pol77 ;
Pol87 ; Dre79 . Even though the compact and the non-compact QED gauge theories
in Eqs. (76) and (78) have the same continuum limit, they have different
confinement properties. On the other hand, in the Schwinger confinement
mechanism in QED in (1+1)D, massless fermions are confined for all coupling
strengths in the continuum limit, which does not distinguish between compact
and non-compact QED 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 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 in 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
Eq. (76). 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.

###  VI.3 The stretch (2+1)D model

To construct a “stretch (2+1)D” flux tube model Won22b ; Won22c for the
production of a quark and an antiquark in a QED meson in (3+1)D, we take the
Polyakov’s transverse confinement configuration in compact QED in (2+1)D as
input. We envisage the production of the nascent $q$$\bar{q}$ pair at the
origin $(x^{1},x^{2},x^{3})$=0 in the center-of-mass system, at the
eigenenergy $\sqrt{s}(q\bar{q})$ of a QED meson999 As discussed in the
Introduction, states of a quark and an antiquark do not exist except in a
confined $q\bar{q}$ eigenstate, so we can study the confinement dynamics of a
quark and an antiquark only when the quark and the antiquark under
consideration here are constituents of a confined system such as a QED meson.
Such a logical circularity arises because of the peculiar property of quark
confinement. . 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 birth and born to be in a QED meson state, the quark and the
antiquark must possess an equal and opposite momenta in the center-of-mass
system8 as shown in Fig. 12(a). The quark momentum defines the longitudinal
direction from which we can define the transverse coordinates $(x^{1},x^{2})$
and the transverse planes at $x^{3}$=(constant). The quark and the antiquark
produced at birth are located in the vicinity of the origin, and they are
separated by an infinitesimal longitudinal separation $\Delta x^{3}$ and an
infinitesimal transverse separation $\Delta{\hbox{\boldmath${r}$}}_{\perp}$ as
in Fig. 12($a$). Polyakov’s transverse confinement of the quark and the
antiquark in compact QED in (2+1)D is realized on the transverse
$(x^{1},x^{2})$-plane at $x^{3}\sim 0$, at the birth of the $q$-$\bar{q}$
pair. 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 with the created ${E}$ and ${B}$
fields lying along the longitudinal $x^{3}$ direction.

Figure 12:  Fig. 12(a) depicts the initial production of the $q\bar{q}$ pair
of a QED meson, with an infinitesimal longitudinal separation $\Delta x^{3}$.
Fig. 12($b$) gives 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 longitudinal yo-yo motion
of the $q\bar{q}$ pair interacting in compact QED in a QED meson, starting
from a (2+1)D transversely confined $q\bar{q}$ system. Fig. 12($b$) is the
“stretch (2+1)D” configuration. Fig. 12($c$) is the corresponding lattice
transcription, following the Hamiltonian formulation of Drell $et~{}al.$ Dre79
.

Subsequent to their births, the quark and the antiquark will execute
stretching and contracting “yo-yo” motion along the longitudinal $x^{3}$
direction, appropriate for the QED meson bound state in question. As the quark
and antiquark stretch outward in the longitudinal $x^{3}$ directions, we can
construct a longitudinal tube structure of gauge fields in the stretch (2+1)D
configuration by duplicating longitudinally the transversely-confined gauge
fields that exist on the transverse $(x^{1},x^{2})$-plane at $x^{3}\sim 0$
initially at their birth, for the longitudinal region between the stretching
quark and antiquark. A snap shot of the stretch (2+1)D flux tube
configurations at an early moment in the longitudinal stretching motion is
shown in Fig. 12($b$). The transcription of Fig. 12($b$) in terms of the
lattice link and plaquette variables is shown in Fig. 12($c$), 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. 12($c$), 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 $x^{3}$ axis as shown in Fig.
12($b$). These gauge field configurations are the ones that lead to Polyakov’s
transverse confinement of the quarks. They maintain the cylindrical symmetry
of the stretch (2+1)D flux tube. They will lead to the desirable full
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 ${E}$ along the longitudinal $x^{3}$ direction sends the
quark and the antiquark in longitudinal 3D longitudinal stretching and
contracting “yo-yo” motion. The electric charge densities obey the Gauss law
associated with the divergence of the electric field ${E}$. The positive
electric quark charge fractions (solid circles in Fig. 12($c$)) reside at the
-$x^{3}$/2 plaquette vertices and the negative electric antiquark charge
fractions (open circles in Fig. 1($c$)) at the antiquark plaquette vertices at
their $x^{3}$/2 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 $x^{3}(q)$=$-x^{3}/2$ and at
$x^{3}(\bar{q})=x^{3}/2$ respectively, and they are unchanged in the
stretching motion, while the longitudinal links are all
$E^{3}$=$|\hbox{\boldmath${E}$}|/4$.

By the duplication of the initial transversely-confined gauge fields along the
longitudinal direction, we obtain a longitudinal tube with a cylindrical
symmetry in (3+1)D. The property of the transverse confinement at one
longitudinal coordinate at birth is thereby extended to the whole tube, and
the longitudinal tube will likely be a transversely-confining flux tube.

We show in the stretch (2+1)D model Won22c that the longitudinal ${B}$ field
that is present initially to confine quarks and antiquarks on the transverse
plane at their birth continues to confine quarks and antiquarks transversely,
because of the Landau level dynamics. The cloud of transverse gauge fields
continue to interact with each other to maintain the transverse confinement on
their transverse planes. As a consequence, quarks, antiquarks, and gauge
fields will be transversely confined in the stretch (2+1)D flux tube.

With the attainment of transverse confinement and ${E}$ and ${B}$ aligned
along the longitudinal direction in the flux tube, it remains necessary to
examine the question of longitudinal confinement. Therefore, we idealize 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

$\displaystyle j^{\mu}=\frac{g_{{}_{\rm
2D}}}{{\pi}}(A^{\mu}-\partial^{\mu}\frac{1}{\partial_{\lambda}\partial^{\lambda}}\partial_{\nu}A^{\nu}),$
(79)

where $g_{{}_{\rm 2D}}$ is the coupling constant in (1+1)D space-time which is
related to the coupling $g_{{}_{\rm 4D}}$ in (3+1)D bt Eq. (11). On the other
hand, the gauge field $A^{\mu}$ depends on the quark current $j^{\nu}$ through
the Maxwell equation,

$\displaystyle\partial_{\nu}(\partial^{\nu}A^{\mu}-\partial^{\mu}A^{\nu})=-g_{{}_{\rm
2D}}j^{\mu}.$ (80)

For the longitudinal motion, Eqs. (79) and (80) lead to the Klein-Gordon
equations in $j^{\mu}$ and $A^{\mu}$ for a boson with a mass $m=g_{{}_{\rm
2D}}/\sqrt{\pi}$ ,

$\displaystyle-\partial_{\nu}\partial^{\nu}j^{\mu}=\frac{g_{{}_{\rm
2D}}^{2}}{\pi}j^{\mu},~{}~{}~{}~{}\text{and}~{}~{}~{}~{}-\partial_{\nu}\partial^{\nu}A^{\mu}=\frac{g_{{}_{\rm
2D}}^{2}}{\pi}A^{\mu}.$ (81)

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 the Schwinger confinement mechanism 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 will be confined and bound in a QED meson in (3+1)D.

##  VII Implications of quark confinement in the QED interaction

As the theoretical expositions and the accompanying experimental evidence
presented in the preceding sections strongly suggest possible quark
confinement in the QED interaction, it is worth examining the implications of
such a new concept in order to explore further new physics on the frontier.
Such an exploration will necessarily be speculative in nature. However, some
educated expectations may stimulate deeper explorations into the unknown
frontier, and some other expectations have measurable experimentally
consequences to merits further considerations.

###  VII.1 Confinement may be an intrinsic property of quarks

The experimental observations, if definitively confirmed under further
scrutiny, will indicate that the attribute of quark confinement is not be the
sole property of the QCD interaction alone and that quarks are also confined
in the QED interaction. Consequently, an interesting possibility is that the
confinement attribute may be an intrinsic property of quarks. This possibility
is reinforced by the observational absence of free quarks, which indicates
further that in the interaction of a quark and an antiquark in their many
mutual interactions, there may be a quark confinement principle which holds
that in the dynamics of quarks in different interactions, each interaction
always leads to the confinement of quarks. That is, a quark and its antiquark
may be confined and bound as a neutral boson in their many mutual
interactions. Specifically, a quark and its antiquark certainly interact
mutually in the weak-interaction and also in the gravitational interaction.
There is no physical law that forbids a quark and an antiquark to exchange a
$Z^{0}$ boson or a graviton to interact in the weak or the gravitational
forces alone. What is not forbidden is allowed, in accordance with Gell-Mann’s
Totalitarian Principle Gel56 . Consequently, a quark and its antiquark may be
confined and bound as a neutral boson in the weak interaction and the
gravitational interaction with the exchange of a $Z^{0}$ boson or a graviton.

It is of interest to estimate the mass of such a confined neutral boson with a
weak strength of the coupling constant. The quark and it antiquark reside
predominantly in (1+1)D. If the work of Coleman Col76 on confined fermions in
QED in (1+1)D can be an approximate analogous guide, such a confined boson
falls into the 2D-weak-coupling regime. In the 2D-weak-coupling regime the
mass of the composite boson would be approximately their rest masses, with
additional contributions from the weak coupling interaction that can be
calculated in the mass perturbation theory Col76 . For an interaction with a
weak strength of coupling constant, we may expect the mass of the confined and
bound boson to be close to the rest mass of the quark and the antiquark. Such
a particle however would decay by quark-antiquark annihilations into two real
photons, two virtual photons, or an $e^{+}e^{-}$ pair. The search for such a
confined $q\bar{q}$ pair may be signaled by a boson with a mass close to the
sum of the rest mass of the quark and its antiquark in the region of 4-10 MeV.
Experiments in high-energy hadron-hadron, nucleus-nucleus, and $e^{+}e^{-}$
collisions in search of anomalous particles decaying into two real or virtual
photons or $e^{+}e^{-}$ in the region of a 4-10 MeV will shed light on
possible $q\bar{q}$ interactions with a weak strength of coupling constants.
Rigorous theoretical work on the question of quark confinement in the weak and
gravitational interactions and the possibility of these weak-interaction-
confined and gravitation-confined composite $q\bar{q}$ particles as dark
matter material would be of great interest.

The possibility of quarks confined in the QED interaction also implies that
the QED interaction between a quark and an antiquark differs from the QED
interaction between an electron and a positron. It will be of great interest
to find out all the differences there can be and why are they different. For
example, the QED interaction between an electron and a positron may belong to
the non-compact QED theory while the QED interaction between a quark and an
antiquark may belong to the compact QED theory. The possibilities of the
compact and non-compact QED bring with them the question whether the QED
interaction is unique or endowed with a multitude of experimentally testable
possibilities with different topological properties. A related question is
whether the QED interaction between quarks in a nucleon may also contain the
linear QED confinement component that depends on the magnitudes and signs of
the electric charges in addition to the standard Coulomb component as in Eq.
(84) below.

###  VII.2 New family of QED-confined particles and dark matter

The success of the open-string description of the QCD and QED mesons leads to
the search for other neutral quark systems stabilized by the QED interaction
between the constituents in the color-singlet subgroup, with the color-octet
QCD gauge interaction as a spectator field. Of particular interest is the QED
neutron with the $d$, $u$, and $d$ quarks Won22 ; Won22a . They form a color
product group of ${3}$ $\otimes$ ${3}$ $\otimes$ ${3}$ =
${\hbox{\boldmath${1}$}}\oplus{\hbox{\boldmath${8}$}}\oplus{\hbox{\boldmath${8}$}}\oplus{\hbox{\boldmath${10}$}}$,
which contains a color singlet subgroup ${1}$ where the color-singlet currents
and the color-singlet QED gauge fields reside. In the color-singlet
$d$-$u$-$d$ system with three different colors, the attractive QED interaction
between the $u$ quark and the two $d$ quarks overwhelms the repulsion between
the two $d$ quarks to stabilize the QED neutron at an estimated mass of 44.5
MeV Won22 . The analogous QED proton has been found theoretically to be
unstable because of the stronger repulsion between the two $u$ quarks, and it
does not provide a bound state nor a continuum state for the QED neutron to
decay onto by way of the weak interaction. Hence the QED neutron may be stable
against the weak interaction. It may have a very long lifetime and may be a
good candidate for the dark matter. Because QED mesons and QED neutrons may
arise from the coalescence of deconfined quarks during the deconfinement-to-
confinement phrase transition in different environments such as in high-energy
heavy-ion collisions, neutron-star mergers Bau19 ; Wei20 , and neutron star
cores Ann20 , the search of the QED bound states in various environments will
be of great interest.

In a related matter, an assembly of the QED-confined $q\bar{q}$ mesons can
also be a good candidate for a part of the dark matter Won20 . Depending on
its mass, the assembly can be an $e^{+}e^{-}$ emitter, a $\gamma$ emitter, or
the dark matter with no particle emission if the mass exceeds their respective
threshold values Won20 . The dark matter models of the QED meson and the QED
neutron utilize the material and the elements of the Standard Model, but with
a new confining combination. Theoretical and experimental studies will be
needed to examine the condense state of an assembly of QED mesons or QED
neutrons.

In addition to new particles, the fact that the QED mesons are complex objects
brings with them additional degrees of freedom to lead to many excited QED
mesons states. For example, there can be vibrational and rotational states
formed by these QED mesons. We can get some idea on the vibrational states
from the spectrum of a stretched string as shown in Fig. 7 of Won22 . The
possibility of adding quarks with different flavors, angular momentum, and
spin quantum numbers will add other dimensions to the number of species of the
QED-confined $q\bar{q}$ composite particles.

###  VII.3 Beyond the confining interaction of a quark and an antiquark in
(3+1)D

A quark and an antiquark reside predominately in (1+1)D, in which the
interaction between a quark and an antiquark is the linear confining
interaction for both QED and quasi-Abelian QCD, as discussed in Section 3 and
4. In the physical (3+1)D space-time, such a linear interaction is only the
dominant part of the full interaction between the quark and the antiquark.
There will be additional residual interactions arising from the presence of
the transverse degrees of freedom. There are also contributions from the spin-
spin, spin-orbit, tensor, and other higher-order terms of the Breit
interaction Bar92 ; Won00 ; Won01 ; Cra09 .

For a confining string with a string tension $\sigma$, Lüscher Lus81 ; Lus80
considered the fluctuations in the transverse direction of the flux tube as a
massless bosonic field theory in two dimensions with a classical action, for
which the action can be integrated out to lead to a potential between a static
quark at ${\bf r}_{1}$ and an antiquark at $\bf r_{2}$ in the large string
length limit as

$\displaystyle V({\bf r}_{1}{\bf r}_{2})=\sigma|{\bf r}_{1}-{\bf
r}_{2}|+c-\frac{\alpha}{|{\bf r}_{1}-{\bf r}_{2}|}+O\left(\frac{1}{{|\bf
r}_{1}-{\bf r}_{2}|^{2}}\right),$ (82)

where $\alpha$ depends on the coupling constant, and $c$ is a constant. These
are therefore long range residual interactions in both the confined QCD and
QED mesons. They represent corrections that arise from expanding the potential
between a quark and an antiquark in powers of their separation $|{\bf
r}_{1}-{\bf r}_{2}|$. A powerful tool to study the non-perturbative behavior
of the interquark potential is the “Effective String Theory” in which the
confining tube contains the quark and the antiquark at the two ends Nam70 ;
Nam74 ; Got71 ; Lus81 ; Lus80 ; Pol91 ; Hel14 ; Aha13 ; Bon21 . The Nambu-Goto
action can be integrated exactly in all geometries that are relevant for
lattice gauge theories: the rectangle (Wilson loop) in Bil13 , the cylinder
(Polyakov loop correlators) in Lus04 ; Lus05 and the torus (dual interfaces)
in Bil06 .

For quarks with color charge numbers $Q_{1}^{{\rm QCD}}$ and $Q_{2}^{{\rm
QCD}}$ interacting in the QCD interaction, we can match the above equation
(82) with the Cornell potential Eic75 and the phenomenological quark-antiquark
potentials in Bar92 ; Won00 ; Won01 ; Won20 and Cra09 . Upon neglecting the
spin-spin, spin-orbit, other higher order terms, and an unimportant potential
constant, we have the linear-plus-color-Coulomb interaction of QCD

$\displaystyle V^{{\rm QCD}}({\bf r}_{1}{\bf r}_{2})=Q_{1}^{{\rm
QCD}}Q_{2}^{{\rm QCD}}\left[-\sigma^{{\rm QCD}}|{\bf r}_{1}-{\bf
r}_{2}|+\frac{\alpha_{s}}{|{\bf r}_{1}-{\bf r}_{2}|}\right].$ (83)

The quark and the antiquark also interact in the QED interaction. We can
generalize the above to include both QCD ($\lambda=1$) and QED ($\lambda=0$)
interactions to give

$\displaystyle V({\bf r}_{1}{\bf r}_{2})$
$\displaystyle=\sum_{\lambda=0}^{1}Q_{1}^{\lambda}Q_{2}^{\lambda}\left[-\sigma^{\lambda}|{\bf
r}_{1}-{\bf r}_{2}|+\frac{\alpha_{\lambda}}{|{\bf r}_{1}-{\bf
r}_{2}|}\right]\tau^{\lambda},~{}~{}~{}\lambda=\begin{cases}0&\text{QED}\cr
1&\text{QCD}\cr\end{cases},$ (84)

where $\tau^{0}$=$t^{0}$ is the generator of the U(1) gauge subgroup as
defined in Eqs. (14), and $\tau^{1}$ is a fixed generator of the SU(3)
subgroup oriented randomly in the eight-dimensional color-octet generator
space in the quasi-Abelian approximation of the non-Abelian QCD, as defined in
(23) and discussed in Section 3\. The generators $\tau^{0}$ and $\tau^{1}$
satisfy $2{\rm
tr}(\tau^{\lambda}\tau^{\lambda^{\prime}})=\delta^{\lambda\lambda^{\prime}}$.

The above equation is for a single flavor. In the case with many flavors and
flavor mixing, their effects can be taken into account by replacing
$Q_{i}^{\lambda}$ by the effective charge $\tilde{Q}_{i}^{\lambda}$, where
$\tilde{Q}_{i}^{\lambda}$ is defined by Eq. (47) in subsections 4C and 4G. It
can be further generalized to the case when the quark constituent and the
antiquark constituent possess different flavors. For a composite
$q_{1}\bar{q}_{2}$ particle with many flavors and flavor mixing, the above
interaction between the quark $q_{q}$ and the antiquark $\bar{q}_{2}$ becomes

$\displaystyle V({\bf r}_{1}{\bf r}_{2})$
$\displaystyle=\sum_{\lambda=0}^{1}\tilde{Q}_{q_{1}}^{\lambda}\tilde{Q}_{\bar{q}_{2}}^{\lambda}\left[-\sigma^{\lambda}|{\bf
r}_{1}-{\bf r}_{2}|+\frac{\alpha_{\lambda}}{|{\bf r}_{1}-{\bf
r}_{2}|}\right]\tau^{\lambda}.$ (85)

When there is no flavor mixing, as in the case of the charm and the beauty
quarks, the effective charge are just those of the standard quark model, with
$Q_{\\{u,d,c,s,t,b\\}}^{{\rm QCD}}$=1 and $Q_{\\{u,c,t\\}}^{{\rm QED}}$=2/3,
$Q_{\\{d,s,b\\}}^{{\rm QED}}$=$-1/3$, and
$Q_{\bar{q}}^{\lambda}$=$-Q_{q}^{\lambda}$.

In addition to the above linear-plus-Coulomb-type interaction, one can include
spin-spin, spin-orbit, tenor, and the full Breit interaction to study the
spectroscopy of hadrons, as carried out for a $q\bar{q}$ system for example in
Bar92 ; Won00 ; Won01 ; Cra09 . Such an interaction will result in an open-
string-type solution of the bound meson states.

It is of interest to examine the meson-meson polarization in a many-meson
system, assuming that the bound state solutions for the quark and antiquark
system within each meson have already been obtained. The task is to study the
color-Coulomb and electric-Coulomb interaction between constituents of
different mesons to see how such perturbative interactions may affect the
dynamics of the many-meson system101010 The linear interaction between
constituents of different mesons are screened and can be included by assuming
a screening length as is done in Won04 . We shall not included them in the
present survey. .

When we work with pions as a possible molecular component, we should keep in
mind however that $\tilde{Q}_{q}^{{\rm QCD}}(\pi)$=$\tilde{Q}_{\bar{q}}^{{\rm
QCD}}(\pi)$=0. That is, the effective color charge of a quark or an antiquark
in a pion is zero, as discussed in subsection 4G and in Table 2. This arises
because of the cancellation of color charges for two-flavor isovector QCD
$q\bar{q}$ composite particles. With the absence of the effective color
charges, there is no reactionary response to the confining interaction in
$\pi^{0}$. The confinement of the constituents of $\pi^{0}$ comes only from
the quark condensate and quark masses $m_{q}$ and $m_{\bar{q}}$ (see Section
4D). There is consequently no color-polarization of the pion under the color-
Coulomb interaction with neighboring color mesons. So, in our subsequent
discussions on the color-polarization of QCD mesons, such a zero effective
quark and antiquark color charges of a pion need to be kept in our mind. On
the other hand, for the discussions in electric charges and electric-
polarization in the QED interaction between mesons, pions contain effective
electric charges, and can still be polarized by QED forces between mesons. So,
pions can be included when we examine QCD-QED molecular states arising from
the electric-Coulomb polarization between mesons.

###  VII.4  Dipole-dipole interaction between neutral mesons

In a single neutral meson (color-neutral with $\lambda=1$ or electric-neutral
with $\lambda=0$, as the case may be), the confining interaction between the
quark and its antiquark causes the constituents to execute a yo-yo motion,
whose classical trajectories for massless quarks are schematically depicted in
Fig. 4. There will be no net static dipole moment for a composite neutral
$q\bar{q}$ system.

We consider a system with $N$ neutral mesons and study the long-range color-
Coulomb or electric-Coulomb interaction between the constituents from
different mesons as perturbations, starting with the $N$=2 system of two
neutral mesons. Owing to the two composite systems carrying color and electric
charges, the action of the long-range color-Coulomb and electric-Coulomb
interactions will polarize the mesons Won04 . As a consequence, they acquire
dynamical dipole moments, leading to a dipole-dipole interaction between the
composite systems as discussed in detail by Peskin and Bhanot Pes79 ; Bha79 .
Under appropriate conditions, the polarization can lead to meson molecular
states as discussed previously in Won04 . Among many other descriptions, meson
molecular states have been studied experimentally and theoretically since the
earlier works of Cho03 ; Won04 ; Tor04 ; Clo04 ; Pak04 ; Swa04 ; Vol04 , and
in references cited in the recent works of Guo18 ; Yan20 ; Don21 ; Luo22 .

A proper way to examine the meson polarization and meson molecular states in
the $N$-meson system is to investigate each of the neighboring 2-meson pair as
a four-body problem, with the interaction of the type in Eq. (85), and reduce
the problem to a two-body bound state problem as discussed in Won04 . It will
also be necessary to take into account the antisymmetry of the quarks if they
are identical. The problem can then be generalized from the 2-meson system to
the case of the $N$-meson system.

Following Wheeler’s First Moral Principle111111 Wheeler’s First Moral
Principle states “‘Never make a calculation until you know the answer’. Make
an estimate before every calculation, try a simple physical argument before
every derivation. Guess and answer to every puzzle. A right guess reinforces
the instinct. A wrong answer brings the refreshment of surprises.” Tay92 .
Tay92 , it is worth studying a simple massive dipole-dipole model before we
carry out extensive calculations. We can gain new insight as to interesting
geometrical configurations of the $N$-meson phase space where interesting
physics may lie, to merit further considerations.

Figure 13: The quark and antiquark configurations in the interaction between
meson $A$ and $B$ with quark charges $\pm q_{A}$ and $\pm q_{B}$,
respectively. The two mesons are separated by the radius vector
$\hbox{\boldmath${r}$}_{AB}$ from meson $A$ to meson $B$. As a result of the
long-range Coulomb interaction, mesons $A$ and $B$ acquire dynamical dipole
moments $\hbox{\boldmath${d}$}_{A}$ and $\hbox{\boldmath${d}$}_{B}$,
respectively. In Figs. (13$a$) and 13($b$), $\theta_{A}$ is the angle between
$\hbox{\boldmath${d}$}_{A}$ and $\hbox{\boldmath${r}$}_{AB}$, $\theta_{B}$ is
the angle between $\hbox{\boldmath${d}$}_{B}$ and
$\hbox{\boldmath${r}$}_{AB}$, and $\phi_{AB}$ is the azimuthal angle between
$\hbox{\boldmath${d}$}_{A}$ and $\hbox{\boldmath${d}$}_{B}$ when we place
$\hbox{\boldmath${d}$}_{A}$ and $\hbox{\boldmath${d}$}_{B}$ in the polar
coordinates with $\hbox{\boldmath${r}$}_{AB}$ as the polar axis. Figs. 13($c$)
and 13($d$) are the linear and orthogonal equilibrium configurations with
respect to angular variations at a fixed $\hbox{\boldmath${r}$}_{AB}$.

We consider in Fig. 13 a meson $A$ with charges $\pm|q_{A}|$, and meson $B$
with charges $\pm|q_{B}|$, which can be color or electric, static or
dynamical, pure or mixed flavor, as the case may be. For each of the mesons
$A$ and $B$, we presume that their linear-plus Coulomb interactions of (85)
lead to their individual known bound states. In the neighborhood of each
other, they are subject to the polarization interaction from the long-range
interaction of the constituents of other mesons as perturbations. As a
consequence, the mesons acquires dynamical dipoles
$\hbox{\boldmath${d}$}_{A}$=$|q_{A}|\hbox{\boldmath${a}$}_{A}$ and
$\hbox{\boldmath${d}$}_{B}$=$|q_{B}|\hbox{\boldmath${a}$}_{B}$, respectively,
where the dynamical dipole $\hbox{\boldmath${d}$}_{i}$ is directed from
$-|q_{i}|$ to $+|q_{i}|$. The potential generated by the dynamical dipole
$\hbox{\boldmath${d}$}_{A}$ at a radius vector $\hbox{\boldmath${r}$}_{AB}$
from meson $A$ to meson $B$ is given by

$\displaystyle V(\hbox{\boldmath${r}$}_{AB})$
$\displaystyle=-\alpha\hbox{\boldmath${d}$}_{A}\cdot\nabla(\frac{1}{r_{AB}}),$
(86)

where $\alpha$=$(g^{\lambda})^{2}/4\pi$. The interaction energy between
dipoles $\hbox{\boldmath${d}$}_{A}$ and $\hbox{\boldmath${d}$}_{B}$ is

$\displaystyle W$
$\displaystyle=(-|q_{B}|)V(\hbox{\boldmath${r}$}_{AB}-\frac{\hbox{\boldmath${a}$}_{B}}{2})+|q_{B}|V(\hbox{\boldmath${r}$}_{AB}+\frac{\hbox{\boldmath${a}$}_{B}}{2})$
(87)
$\displaystyle=\alpha\left[\frac{\hbox{\boldmath${d}$}_{A}\cdot\hbox{\boldmath${d}$}_{B}}{r_{AB}^{3}}-3\frac{(\hbox{\boldmath${d}$}_{A}\cdot\hbox{\boldmath${r}$}_{AB})(\hbox{\boldmath${d}$}_{B}\cdot\hbox{\boldmath${r}$}_{AB})}{r_{AB}^{5}}\right].$

In terms of the angle $\theta_{A}$ between $\hbox{\boldmath${d}$}_{A}$ and
$\hbox{\boldmath${r}$}_{AB}$, the angle $\theta_{B}$ between
$\hbox{\boldmath${d}$}_{A}$ and $\hbox{\boldmath${r}$}_{AB}$, and the
azimuthal angle $\phi_{AB}$ between $\hbox{\boldmath${d}$}_{A}$ and
$\hbox{\boldmath${d}$}_{B}$, when we place $\hbox{\boldmath${d}$}_{A}$ and
$\hbox{\boldmath${d}$}_{B}$ in the polar coordinate system with
$\hbox{\boldmath${r}$}_{AB}$ as the polar axis, (Fig. 13($a$) and ($b$)), the
interaction energy $W$ between $A$ and $B$ is

$\displaystyle W(r_{AB},\theta_{A},\theta_{B},\phi_{AB})$
$\displaystyle=\alpha\frac{d_{A}d_{B}}{(r_{AB})^{3}}\bigl{(}\sin\theta_{A}\sin\theta_{B}\cos\phi_{AB}-2\cos\theta_{A}\cos\theta_{B}\bigr{)}.$
(88)

For a fixed $\hbox{\boldmath${r}$}_{AB}$, we can get the equilibrium
configurations with respect to angular variations by taking the first and
second derivatives with respect to $\theta_{A}$, $\theta_{B}$ and $\phi_{AB}$.
Assuming that at the angular equilibrium, the dynamical dipole moments
$\hbox{\boldmath${d}$}_{A}$ and $\hbox{\boldmath${d}$}_{B}$ are weak functions
of the angles, we find that for a fixed $\hbox{\boldmath${r}$}_{AB}$, there
are two equilibrium configurations with respect to the angular variations:

  1. 1.

The linear equilibrium configuration in which the dynamical dipoles are
linearly aligned along the radius vector $\hbox{\boldmath${r}$}_{AB}$ between
$A$ and $B$ as in a linear array:

$\displaystyle\theta_{A}=0,~{}~{}\theta_{B}=0,~{}~{}\phi_{AB}={\rm
ignorable}.$ (89)

The dipole configurations shown in Fig. 13($a$), with a small $\theta_{A}$ and
$\theta_{B}$, approach the linear equilibrium configuration shown in Fig.
13($c$).

  2. 2.

The orthogonal equilibrium configuration in which the dynamical dipoles are
orthogonal to the radius vector $\hbox{\boldmath${r}$}_{AB}$ between mesons
$A$ and $B$, with opposing dynamical dipoles:

$\displaystyle\theta_{A}=\frac{\pi}{2},~{}~{}\theta_{B}=\frac{\pi}{2},~{}~{}\phi_{AB}=\pi.$
(90)

The dipole configurations shown in Fig. 13($b$), with opposing dipoles
orthogonal to the radius vector $\hbox{\boldmath${r}$}_{AB}$, and $\theta_{A}$
and $\theta_{B}$ close to right angles, approach the orthogonal equilibrium
configuration shown in Fig. 13($d$).

These equilibrium configurations for a fixed $\hbox{\boldmath${r}$}_{AB}$ can
also be figured out by elementary physics arguments.

To obtain the molecular state of mesons $A$ and $B$ for a given linear or an
orthogonal configuration, it is necessary to solve the bound state problem
involving $A$ and $B$ as a two-body problem in the relative coordinate
$r_{AB}$, with the reduced mass of the two mesons in the two body potential
$W(r_{AB})$ Won04 . The dipole-dipole interaction $W(r_{AB})$ at the
equilibrium configurations are attractive at large separations $r_{AB}$, at
which the dipole-dipole interactions are good approximations. However, at
small $r_{AB}$, it is necessary to forgo the dipole approximation and study
the Coulomb-plus-linear interaction as a four-body problem, as is done in
Won04 , where the two-body potential for mesons $A$ and $B$ is found to be
negative and contain a potential pocket. In the case of mesons containing
constituents of identical particles, it is necessary to take into account the
antisymmetry of these constituents. The effects of antisymmetry is small when
the mesons are far apart and thus such effects are not important for large
$r_{AB}$ when the dipole approximation can be approximately valid. At small
$r_{AB}$, the antisymmetry gives rise to an effective repulsion between the
mesons, and consequently a 6-12 type potential of $(a/r^{12}-b/r^{6})$ with
$a,b>0$ containing a potential pocket. The potential obtained from these
considerations will also be attractive at large distance and a potential
pocket at short distances that will have bound meson molecular states in the
limit of large meson mass.

Many QCD meson molecular states have been studied in Cho03 ; Won04 ; Tor04 ;
Clo04 ; Pak04 ; Swa04 ; Vol04 and in references cited in Guo18 ; Yan20 ; Don21
; Luo22 . With QED obeying similar dynamics as the QCD mesons, there should be
similar linear and orthogonal molecular states for a system of two or many QED
mesons.

From the Bohr-Summerfeld quantization rule, the interaction potential such as
the attractive $W(r_{AB})$ with a potential pocket would lead to a bound state
if the action integral along the radial $r_{AB}$ degree of freedom is equal to
$2n\pi$. The condition for molecular bound states depends crucially on the
reduced mass of the system, which depends in turn on the rest masses of the
quarks. Bound molecular states will begin to appear, when the reduced mass
exceeds a threshold. Quantitative calculations will need to be carried out to
find out where the quark mass thresholds lie for the existence of the
molecular QCD meson and QED meson states.

Given the large range of the quark masses of different flavors from a few MeV
to hundreds of GeV, it is useful to consider a massive dipole-dipole model in
the large mass limit such that the masses of interacting dipoles are assumed
to exceed the threshold for bound molecular states, for systems with an
attractive interaction potential $W(r_{AB})$ as a function of the dipole-
dipole separation $r_{AB}$, with the mesons oriented at angles of angular
equilibrium (such as the linear or the orthogonal equilibrium of Figs. 13($c$)
and ($d$)). Our interest is to have an idea on the geometric configurations of
the meson dipoles in the large mass limit so as to point to possible
configurations that merit further analysis. Such a massive dipole-dipole model
is not the ultimate definitive determination of stable configurations. On the
other hand, the large range of quark masses ensure however that it is not just
of academic interest. The model serves as a guide to interesting meson
geometrical configurations that may be possible when the quark masses are
large enough to exceed the mass threshold for molecular formation, and
therefore worthy of further considerations with more rigorous analysis.

###  VII.5  Interesting molecular configurations in the massive dipole-dipole
model

We can consider $N$ neutral mesons in the massive dipole-dipole model to
explore different molecular configurations of interest. The mesons will
interact with each other to lead to dynamical dipole moments, resulting in
many different equilibrium configurations. The interactions are operative in
the $\tau^{0}$ and $\tau^{1}$ color space, as the color charge densities
generates color potentials while the electric charge densities generates
electric potentials. The dipole-dipole interaction energy $W$ of $N$ mesons is
given by

$\displaystyle W(\\{{\bf
r}_{i,j}^{\lambda}\\},\\{\theta_{i}^{\lambda}\\},\\{\phi_{i,j}^{\lambda}\\})=\frac{1}{2}\sum_{\lambda=0}^{1}\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\left[\frac{\alpha_{\lambda}d_{i}^{\lambda}d_{j}^{\lambda}}{r_{ij}^{3}}(\sin\theta_{i}^{\lambda}\sin\theta_{j}^{\lambda}\cos\phi_{i,j}^{\lambda}-2\cos\theta_{i}^{\lambda}\theta_{j}^{\lambda})\right]\tau^{\lambda}.$
(91)

If we consider the approximation of only pairwise dipole-dipole interaction
between neighboring dipoles, then the potential energy $W$ of $N$ mesons is
given by

$\displaystyle W(\\{{\bf
r}_{i,i+1}^{\lambda}\\},\\{\theta_{i}^{\lambda}\\},\\{\phi_{i,i+1}^{\lambda}\\})=\sum_{\lambda=0}^{1}\sum_{i=1}^{N}\left[\frac{\alpha_{\lambda}d_{i}^{\lambda}d_{i+1}^{\lambda}}{r_{i,i+1}^{3}}(\sin\theta_{i}^{\lambda}\sin\theta_{i+1}^{\lambda}\cos\phi_{i,i+1}^{\lambda}-2\cos\theta_{i}^{\lambda}\theta_{i+1}^{\lambda})\right]\tau^{\lambda},$
(92)

where in the case of a closed figure such as a polygon, the index $N+1$
reverts to be 1.

With the linear and the orthogonal equilibrium configurations, there can be
many different configurations for a system of many neutral mesons (color-
neutral or electric neutral, as the case may be). Some examples of the
possible configurations of molecular states are depicted in Fig. 14.

Figure 14:  Interesting configurations of QCD and QED neutral mesons in the
massive dipole-dipole model that may merit further considerations. In these
configurations, a meson is depicted with its positive charge constituent in a
solid eclipse, and its negative charge constituent in an open eclipse.
Examples shown here include: a linear chain in Fig. 14($a$), an orthogonal
chain as in Fig. 14($b$), the extension both longitudinally and transversely
in Fig. 14($c$), a linear chain turning in a right angle $\theta_{1}=\pi/2$ in
Fig. 14($d$), a triangle in Fig. 14($e$), and a square in Fig. 14($f$).

One can build the linear chain as in Fig. 14($a$), the orthogonal chain as in
Fig. 14($b$), the extension both longitudinally and transversely, as in Fig.
14($c$), or turning in a right angle $\theta_{1}=\pi/2$ at a corner, as in
Fig. 14($d$). By extending such chains and building up an orthogonal
configuration into a third direction, it is possible to construct molecular
states in three dimensions as in organic chemistry.

In addition to these linear and orthogonal configurations in two dimensions
and perhaps also in three dimensions, there may be molecular polygon meson
systems. There may be regular polygons if all the meson dipoles are the same,
with $d_{i}$ = $d$. There may also be irregular if some of the meson dipoles
are different. We can consider the regular polygon as an simple illustration.
Such regular polygon configurations arise in the massive dipole-dipole model
because the interaction energy $W^{(N)}(r)$ as a function of the separation
$r$ between the nearest neighboring dipoles are always negative and attractive
for $N\geq 2$. For example for $N=3$, the molecular state in the form of a
regular triangle as shown in Fig. 14($e$) has an interaction energy

$\displaystyle W^{(3)}(r)=-\frac{3}{4}\frac{d^{2}}{r^{3}}.$ (93)

For $N=4$, the molecular state in the form of a square as shown in Fig.
14($f$) has the interaction energy

$\displaystyle W^{(4)}(r)=(-9-\frac{1}{\sqrt{2}})\frac{d^{2}}{r^{3}}.$ (94)

These interaction energies are always negative and attractive, and in the
massive dipole-dipole model, a negative interaction energy with a potential
pocket will lead to a bound state in the large mass limit. For the case of a
polygon with $N$ dipoles, the interaction energy from the contributions from
the nearest neighboring dipoles is

$\displaystyle W^{(N)}({\rm
all~{}nearest~{}neighboring~{}pairs})=\frac{N(N-1)}{2}\frac{d^{2}}{r^{3}}[1-5\cos^{2}(\frac{\pi}{N})],$
(95)

which is also always negative, and the contribution from next-to-next
neighbors diminishes in strength. In the massive dipole-dipole model, such a
negative interaction energy will lead to a stable bound molecular state of a
polygon with $N$ mesons in the massive dipole-dipole model.

It is interesting to take note the recent results on the stable
$D^{*}D^{*}D^{*}$ triangular molecular state from a quantitative analysis in
Ref. Luo22 . The $D^{*}D^{*}D^{*}$ triangular state for $N=3$, together with
the molecular state X(3872) for $N=2$ , may indicate that the molecular state
mass threshold in the large mass limit of the dipole-dipole model may have
been reached already with the charm quark mass. If this is the case, we should
expect the massive dipole-dipole model to be approximately valid for other
$N\geq$ 3 cases for open charm mesons. Open charm mesons in the form of
square, pentagon, hexagon, and higher polygons may be possible. Future
experimental and theoretical research on these exotic open charm meson
polygons will be of interest.

###  VII.6 Molecular states with both QCD and QED mesons

The QCD and QED configurations in the massive dipole-dipole model considered
in the last subsection deal with color-neutral QCD mesons or electric-neutral
QED mesons by themselves in their respective $\tau^{1}$ and $\tau^{0}$ color
subspace. In addition to these purely QCD and purely QED mesons, there may be
molecular states in the $\tau^{0}$ sector comprising of both QCD and QED
mesons interacting the QED interaction in the massive dipole-dipole model.

In such a mixture of QCD and QED mesons, we can include both electric-neutral
and charged QCD mesons. A charged QCD meson arises from a quark and an
antiquark possessing electric charges of the same sign. For such a charged QCD
meson, we assume that the forces leading to quark confinement in the QCD meson
is dominated by the strong QCD interaction. Thus, even though there occurs a
repulsive linear electric interaction between electric constituent charges of
the same sign, such a QED electric-linear repulsion will be overwhelmed by the
attractive QCD confining color-linear interaction between the quark and the
antiquark and the confining force from the quark condensate. The QCD
interaction is expected to lead to the confined QCD meson state. As a
consequence, a charged QCD meson $A$ with constituents $q_{A}$ and
$\bar{q}_{A}$ possesses a charge electric monopole with a net electric charge
$Q_{A}$

$\displaystyle Q_{A}=\frac{Q_{q_{{}_{A}}}+Q_{\bar{q}{{}_{{}_{A}}}}}{2}.$ (96)

The charged meson also possesses electric dipole charges
$\pm|Q_{q_{{}_{A}}}+Q_{{\bar{q}}_{A}}|/2$ and a static electric dipole
$\hbox{\boldmath${d}$}_{A}({\rm static})$,

$\displaystyle\hbox{\boldmath${d}$}_{A}({\rm
static})=\frac{|Q_{q_{{}_{A}}}+Q_{{\bar{q}}_{A}}|}{2}{\hbox{\boldmath${a}$}_{A}},$
(97)

where $\hbox{\boldmath${a}$}_{A}$ is the open-string length vector of meson
$A$ from the lesser-charged quark to the greater-charged quark of $A$. In the
presence of neighboring mesons with electric dipole moments, the charged meson
will also be polarized to acquire an additional dynamical dipole moment. While
the quantitative magnitude of the total dipole moment needs to be worked out
in detail, it suffice to consider here that the QCD meson possesses a total
electric dipole vector sum $\hbox{\boldmath${d}$}_{A}$ in its interaction with
the neighboring QED meson. The effect of the electric monopole leads to an
additional electric monopole-dipole interaction energy with a neighboring
dipole $\hbox{\boldmath${d}$}_{B}$, and the total electric interaction energy
$W$ is given by

$\displaystyle
W=\alpha_{c}\left[\frac{Q_{A}\hbox{\boldmath${d}$}_{B}\cdot\hbox{\boldmath${r}$}_{AB}}{r_{AB}^{3}}+\frac{\hbox{\boldmath${d}$}_{A}\cdot\hbox{\boldmath${d}$}_{B}}{r_{AB}^{3}}-3\frac{(\hbox{\boldmath${d}$}_{A}\cdot\hbox{\boldmath${r}$}_{AB})(\hbox{\boldmath${d}$}_{B}\cdot\hbox{\boldmath${r}$}_{AB})}{r_{AB}^{5}}\right].$
(98)

With an additional monopole-dipole interaction, the orientation angles at the
linear equilibrium configuration are unchanged. That is, they remain to be
$\theta_{A}$=$\theta_{B}$=0, with ignorable $\phi_{AB}$. So, the linear
molecular state configurations for mixed QCD and QED mesons are unchanged as
in the case of neutral QCD and QED mesons. However, the orientation angles at
the orthogonal equilibrium configuration are modified by the presence of the
monopole-dipole interaction. For this reason, we shall not consider the
orthogonal configuration in the present survey.

An interesting (QED meson)-(QCD quarkonium)-(QED meson) combination as
depicted in Fig. 15($a$). In such a configuration, the positive charge of the
QCD quarkonium will attract a QED meson dipole pointing away from the
quarkonium while the negative charge of the quarkonium will attract a QED
dipole pointing into the heavy quarkonium. Such a chain of QED-QCD-QED mesons
is a molecular state in the massive dipole-dipole model at large mass limit.
Whether such a structure with the X17 or E38 as the QED meson have reduced
masses exceeded the mass thresholds for the occurrence of molecular states
will need to be worked out quantitatively in detail.

Figure 15:  Examples of interesting linear molecular configurations in the
massive dipole-dipole model of QCD-QED mesons that may merit further
considerations. In these configurations, a a solid eclipse stands a positive
charge and an open eclipse for a negative charge. Examples shown here include
a linear chain for (QED meson)-(QCD quarkonium)-(QED meson) in Fig. 15($a$),
the (QCD meson)-(QED meson)-(QED meson)-(QCD meson) in Fig. 15($b$), and the
(QED meson)-(QCD meson)-(QCD meson)-(QED meson) in Fig. 15($c$) and 15($d$).
Pions as examples of the QCD mesons are indicated schematically.

Other interesting combinations involve charged or neutral QCD and QED mesons
as depicted in Fig. 15 ($b$)-($d$). In the meson chain in $(b$)-($d$), the QCD
meson can be either neutral or charged with electric charge of the appropriate
sign. These configurations are the equilibrium configurations in the massive
dipole-dipole model.

Even though quarks in pions does not carry color charges because of the
cancellation of their color charges in the isovector state, they carry
electric charges. As a consequence, pion can be electrically polarized by
other mesons and they in turn can polarize other mesons. The possible
equilibrium configurations in Fig. 15($b$)-($d$) in the massive dipole-dipole
model provides impetus to explore whether the $\pi$-(QED meson)-(QED
meson)-$\pi$ or the (QED meson)-$\pi$-$\pi$-(QED meson) molecules may be
related to the perplexing ABC anomaly Aba60 ; Boo61 ; Ban73 ; Adl11 ; Bas17 ;
Kom18 and the R360 anomaly Abr09 ; Abr19 . If the molecular states can resolve
these anomalies, they will provide additional support for the description of
the anomalous X17 and E38 particles as QED mesons.

In exploring the molecular description of the ABC and R360 resonances, we need
to estimate the masses in these linear QCD-QED meson molecules. In a molecular
state of the type in Figs. 15, the molecular binding energy of the mesons
would be of order of a few MeV (see for example the molecular state in Ref.
Won04 ). So, the mass of the molecular states consisting of

$\displaystyle(\text{mass of molecular state})=\sum_{i}m_{i}-(\text{molecular
binding energy})$ (99)

where $m_{i}$ are the mass of the meson $i$ constituents of the molecular
state. Within an error of the order of a few MeV, the mass of a molecular
state is approximately

$\displaystyle(\text{mass of a molecular state})\approx\sum_{i}m_{i}.$ (100)

We can use the above approximate molecular-state relation to estimate the
masses of the molecular states in Figs. 15($b$)-($d$). If the QED mesons in
these figures are the X17 particles each with a mass of 17 MeV, then the
masses in these linear QCD-QED meson molecules will be

$\displaystyle m_{\pi^{+}}+m({\rm X17)}+m({\rm X17})+m_{\pi^{-}}$
$\displaystyle=314{\rm~{}MeV},$ $\displaystyle{\rm
and}~{}~{}~{}m_{\pi^{0}}+m({\rm X17)}+m({\rm X17})+m_{\pi^{0}}$
$\displaystyle=304{\rm~{}MeV},$ (101)

which fall within the energy of the ABC resonance at $M_{\rm ABC}$=310 MeV
with a width of $\Gamma_{\rm ABC}$=10 MeV Aba60 . Other measurements gave the
masses as $M_{\rm ABC}$=316 MeV$\pm$10 with a width $\Gamma_{\rm
ABC}$=55$\pm$10 MeV in $d+p\to^{3}$He+ X collisions with a deuteron beam at
2.83 MeV at a scattering angle of 0.3o of the fused 3He Ban73 . Another
measurement gives $M_{\rm ABC}$= 298$\pm 5$ MeV to 319$\pm 8$ MeV with a width
from $\Gamma_{\rm ABC}$=38$\pm 5$ MeV to 44+$\pm 9$ MeV Kom18 . There is thus
an approximate relation

$\displaystyle m(X17)\approx\frac{1}{2}(m_{\rm ABC}-2m_{\pi}).$ (102)

If the QED mesons in Fig. 15($b$) are E38 particles each with a mass of 38
MeV, then the masses in these linear QCD-QED meson molecules will be

$\displaystyle m_{\pi^{+}}+m({\rm E38)}+m({\rm E38})+m_{\pi^{-}}$
$\displaystyle=$ $\displaystyle 354{\rm~{}MeV},$
$\displaystyle\text{and}~{}~{}~{}m_{\pi^{0}}+m({\rm E38)}+m({\rm
E38})+m_{\pi^{0}}$ $\displaystyle=$ $\displaystyle 344{\rm~{}MeV},$ (103)

which fall within the energy of the R360 resonance of $M_{\rm R360}$=360$\pm
7\pm 9$ MeV with a width of $\Gamma_{\rm R360}$=$63.7\pm 17.8$ MeV observed by
Dubna Abr09 . The SACLAY group also observed similarly an anomalous resonance
at $M_{X}$=365$\pm 23$ MeV with a width of $\Gamma_{\rm X}$=$51\pm 10$ MeV in
the reaction of $d+p\to^{3}$He+ X for a deuteron beam at 3.82 MeV at a
scattering angle of 0.3o Ban73 . There is thus an approximate relation

$\displaystyle m(E38)\approx\frac{1}{2}(m_{\rm R360}-2m_{\pi}).$ (104)

Equations (102) and (104) relating the ABC and R360 resonances with the X17
and E38 resonances suggests that the proposed description of QED-confined
mesons have the prospect of linking the five anomalies of (i) the anomalous
soft photons, (ii) the X17 particle, (iii) the E38 particle, (iv) the ABC
resonance, and (v) the R360 resonance in as single consistent QED meson
framework.

As the masses of the configurations in Fig. 16($c$) and ($d$) falling within
the measured ABC and R360 resonances, it is thus worth exploring whether the
molecular states may provide the appropriate descriptions for these two
anomalous resonances. The ABC resonance is produced in nuclear collisions with
the following characteristics occurring near the threshold of two pion
production Aba60 ; Boo61 ; Ban73 ; Adl11 ; Bas17 ; Kom18 :

  1. 1.

the ABC resonance occurs at energies near but slightly beyond the threshold
for two pion production (of 270-280 MeV) at $M_{\rm ABC}\sim$ 310 MeV with a
relatively narrow width of order 50-60 MeV

  2. 2.

most likely observed in the forward and backward directions

  3. 3.

isoscalar nature of the $\pi\pi$(X) pair

  4. 4.

the occurrence of the ABC resonance is accompanied by the fusion of the
colliding nuclei into a fused nuclear system Aba60 ; Boo61 ; Ban73 ; Adl11 ;
Bas17 ; Kom18 as in $d$+$p$$\to$3He$+$X0, $d+p\to^{3}$H+X+ Aba60 ; Ban73 , and
$d+d\to^{4}$He+$X^{0}$ Adl11

  5. 5.

no occurrence of the ABC resonance when there is no fusion of the colliding
nuclei, as in $pn\to pp\pi^{0}\pi^{-}$ and $pn\to pn\pi^{0}\pi^{0}$ even
though the $d^{*}(2380)$ resonance appears in these reactions. Thus, the
occurrence of the ABC resonance may be independent of the $d^{*}({2380})$
resonance Bas17 .

Figure 16:  The possible mechanism of the production of the molecule state
$\pi$-(QED meson)-(QED meson)-$\pi$ in the collision of $n$ and $p$ near the
two-pion threshold by the fusion of two virtual gluons, with the subsequent
string fragmentation of the $q_{1}$-$\bar{q}_{2}$ open string into the four
mesons, $X_{1},\pi_{1},\pi_{2},X_{2}$, near the $\pi$$\pi$ threshold.

We can attempt to interpret the molecular state description of the ABC and the
R360 resonances by describing the possible production mechanism in the
following way. The basic process may be the collision of a projectile nucleon
$p$ and a target nucleon $n$ as depicted in Fig. 16, entering as a sub-process
in the more complicated nuclear collisions. In such a $p+n$ collision, we may
envisage the emission of a gluon from $n$ at the vertex $V$ and and a gluon
from $p$ at the vertex $V^{\prime}$ as in Fig. 16. The fusion of the two
gluons lead to a $q_{1}$ and $\bar{q}_{2}$ string which pull apart from each
other. The fragmentation of the receding $q_{1}\bar{q}_{2}$ string in the
inside-outside cascade picture of Bjorken, Casher, Kogut, and Susskind Bjo73 ;
Cas74 produces the pions and other particles, $X_{1},\pi_{1},\pi_{2},X_{2}$,
as depicted in Fig. 16. Because the produced particles arise by pulling the
receding $q_{1}\bar{q}_{2}$ open string apart from the colliding nucleons, the
fragmentation process will be facilitated in the forward and backward
direction of the colliding baryon system. Hence, the productions of the ABC
and R360 resonances may be favored in the forward and backward directions.

The isospin properties and the parity of the X17 as an isoscalar particle and
E38 particles as an isovector particle require that they must be produced in
pairs in order to have the quantum number of the vacuum. The produced
$\pi^{0}\pi^{0}$ or $\pi^{+}\pi^{-}$ pair can also form a coupled isoscalar,
even-parity state in the QCD sector. The pair of X17 or E38 QED mesons can
combine with two pions to form an isoscalar, even-parity object, depicted
schematically in Fig. 16. The narrowness of the widths of the ABC and R360
resonances indicates that they may likely be molecular resonances with a weak
binding and long-range interactions, with the mesons far separated. They may
not likely be resonances from the exchange of the strong-interaction scalar
QCD meson for which the width would be much greater.

The requirement that the ABC resonance must necessarily accompany the fusion
reaction may perhaps be understood in terms of the color flow of the
projectile and target nucleons in Fig. 16. The incident $p$ and $n$ are
initially in a colorless color-single state. After $p$ and $n$ each emits a
virtual gluon at $V$ and $V^{\prime}$, the scattered nucleons $p^{\prime}$ and
$n^{\prime}$ will be colored objects. With the produced $q\bar{q}$ open string
materializing as the ABC resonance of the colorless molecular state of
$\pi$-X17-X17-$\pi$, the process would not be observable if the scattered
$p^{\prime}$ and $n^{\prime}$ remain as colored objects and cannot combine
with other colored objects to form a colorless state. The scattered colored
$p^{\prime}$ and $n^{\prime}$ can bleach their colors by exchanging a virtual
gluon and fusing together into a colorless bound state of the fused nuclei.
Hence, the fusion reaction always accompany the production of the molecular
state of the ABC resonance. Such a fusion of the baryons and the fusion of the
gluons with the subsequent production of the ABC resonance can take place in
conjunction because they occur in different parts of the three-dimensional
space.

It is interesting to note the similarity of the proposed mechanism for the
production of the ABC resonance in Fig. 16 and the proposed mechanism for the
production of the X17 particle in $p+^{3}$H${}_{\rm ground~{}state}$ in Fig.
3($a$), because the process of color bleaching leading to a fused nucleus are
the same, as discussed in Section 2.

If the molecular states of $\pi$-X17-X17-$\pi$ and $\pi$-E38-E38-$\pi$ are
proper descriptions of such resonances, we would expect that X17 particles may
appear as decay products of the ABC resonance, and E38 particles may appear as
decay products of the R360 resonance. A search for these particles
accompanying the ABC resonance and the R360 resonance in these reactions in
the forward and backward directions will be of interest. In this regard, it is
interesting to note the possible mutual evidential support that reactions
involved in the production of R360 are also the reactions where the E38
particle has been observed Abr09 ; Abr19 . Further research on these
interesting suggestions with regard to the ABC and the R360 resonances will be
of great interest.

###  VII.7  Molecular states with baryons

The consideration of molecular mesons can be generalized to include the
discussions on baryons when we approximate a baryon as a confined state of a
quark and a diquark, with the diquark playing the role of an antiquark. In
such a picture, a charged baryon possesses an electric monopole in addition to
an electric dipole moment. The color and electric charges in the presence of
neighboring mesons or baryons will be polarized and they will acquire
additional dynamical dipole moments. Likewise, we need to consider both the
dynamical color and electric polarization in the molecular states involving
baryons. Because a baryon that carry an electric charge repel another baryon
with electric charges of the same sign, molecular states with baryons would be
favorable with a charged baryon with a neutral baryon or an antibaryon with a
charge of opposite signs. In this regard, molecular states consisting of a
baryon and an antibaryon opens a new degree of freedom to the construction of
hadron molecular states as discussed in Don21 .

An example of particular interest is the $d^{*}(2380)$ resonance Adl14 ;
Adl14a ; Wor14 ; Wor16 that occur often with the ABC effect. Although a
6-quark description is possible Bas13 and chiral quark model Gol89 ; Pin08 is
used to describe the baryon-baryon resonance, it can also be depicted
equivalently as a molecular state of
$\Delta^{+}[IJ=(1/2)(3/2)^{+}]$-$\Delta^{0}[IJ=(1/2)(3/2)^{0}]$ state with a
$D_{03}$($I=0$,$J^{\pi}$=$3^{+}$) dipole-dipole molecule. The molecular state
binds its two $\Delta$ resonances by a binding energy of $E_{\rm
binding}$=2$\times 1232$\- 2380=82 MeV. Because of the molecular binding, the
two $\Delta$ constituents lie below their rest mass energies, and as a result,
the decay life time is lengthened with a narrower width of about 70 MeV, much
smaller than the expected width of twice the $\Delta$ decay width of 110 MeV.
They are formed in the collision of a proton and a neutron so that they can
occur as a stretched configuration between the $\Delta^{+}$ and $\Delta^{0}$
predominantly in the forward and backward directions. It will be interesting
to examine the molecular properties of the $d^{*}(2380)$, in addition to the
other descriptions.

##  VIII Conclusion and discussions

As advised by the late John Archibald Wheeler on many occasions, “In the
exploration on the new frontiers of physics, we make progress by walking on
two legs, with one leg on firm grounds, and the other in a venturing spirit”.
On the question of quark confinement in QED, on firm grounds are (i) the non-
isolation of quarks, (ii) a quark and its antiquark interaction in the QED
interaction, and (iii) a quark and its antiquark are confined in the QED
interaction in (1+1)D (Schwinger). In venturing spirit is the question whether
they might be confined in the physical world of (3+1)D, when they interact in
the QED interaction alone.

A prerequisite for asking such a question is to inquire whether there are
experimental circumstances in which a $q\bar{q}$ pair may be produced and may
interact in QED interaction alone. We find that in hadron-hadron, $AA$,
$e^{+}e^{-}$, and $e^{-}A$ collisions, there can be situations in which a
quark and an antiquark may be produced with a center-of-mass energy below the
pion mass gap $m_{\pi}$ for collective QCD excitation, and the quark and its
antiquark can interact non-perturbatively in the QED interaction alone.

It is well known that in the Schwinger confinement mechanism, massless
fermions interacting in the Abelian QED gauge interactions in (1+1)D are
confined for all strengths of the gauge interaction, as in an open string,
leading to a confined and bound neutral boson with a mass proportional to the
magnitude of the coupling constant.

Light quarks have masses of only a few MeV and they can be approximated as
massless. A quark and an antiquark cannot be isolated, so they reside
predominantly in (1+1)D. They can be produced and interact in the QED
interaction alone. The conditions under which the Schwinger confinement
mechanism can be applicable are met when a light quark and a light antiquark
are produced and interact with the QED interaction alone. Consequently, we can
apply the Schwinger mechanism to quarks to conclude that a light quark and its
antiquark are confined in the QED interaction in (1+1)D. The requirement of
the massless condition for quarks is actually not as restrictive as it may
appear to be because Coleman, Jackiw, and Susskind showed that the Schwinger
confinement persist even when quarks are massive in (1+1)D. Therefore, a quark
and its antiquark are confined in the QED interaction for all flavors and
gauge interaction strength in (1+1)D.

On questions of quark confinement and $q\bar{q}$ QCD bound states, the non-
Abelian QCD interaction can also be approximated as a quasi-Abelian
interaction. As a consequence, the Schwinger confinement mechanism can be
applied to quarks interacting in both the QED interaction and the QCD
interaction, leading to confined $q\bar{q}$ pairs in QED and QCD open string
states in (1+1)D, with the composite boson masses depending on the magnitudes
of the QCD and QED coupling constants. Such a viewpoint is consistent with the
QCD string description of hadrons in the NambuNam70 and GotoNam70 string
model, the string fragmentation models of particle production of Bjorken,
Casher, Kogut, and Susskind Cas74 ; Art74 ; And83 , the Abelian projection
model tHo80 , and the Abelian dominance model Sei07 ; Suz08 .

In a phenomenological analysis, we inquire whether an open string in the
phenomenological open string model of QCD and QED $q\bar{q}$ systems in (1+1)D
can be the idealization of a flux tube in (3+1)D and can show up as a bound
and confined boson. In such a phenomenological open string model, we need an
important relationship to ensure that the boson mass calculated in the lower
(1+1)D can properly represent the mass of a physical boson in (3+1)D. The open
string in (1+1)D can describe a physical meson in (3+1)D if the structure of
the flux tube is properly taken into account. This can be achieved by relating
the coupling constant in (1+1)D with the coupling constant in (3+1)D and the
flux tube radius $R_{T}$ Won09 ; Won10 ; Kos12 ; Won22a . Using such a
relationship, we find that that $\pi^{0},\eta$, and $\eta^{\prime}$ can be
adequately described as open string $q\bar{q}$ QCD mesons. By extrapolating
into the $q\bar{q}$ QED sector in which a quark and an antiquark interact with
the QED interaction, we find an open string isoscalar $I(J^{\pi})$=$0(0^{-})$
QED meson state at 17.9$\pm$1.5 MeV and an isovector
$(I(J^{\pi})$=$1(0^{-}),I_{3}$=0) QED meson state at 36.4$\pm$3.8 MeV.

On the experimental front, it has been observed that anomalous soft photons
with a transverse momentum of many tens of MeV/c are proportionally produced
when hadrons are produced and are not produced when hadrons are not produced,
indicating that the production of hadrons are always accompanied by the
production of neutral particles with masses in the region of many tens of
MeV/c2. In search of axion with a mass of many tens of MeV, anomaly pointing
to the production of an X17 particle with a mass of about 17 MeV has been
observed in the decay of He4, 8Be, and 12C excited states. There have been
also observation of the E38 particle at Dubna with a mass of 38 MeV. The
predicted masses of the isoscalar and isovector QED mesons in the open-string
model of the QCD and QED mesons are close to the masses of the reported X17
and E38 particles observed recently, making them good candidates for these
particles. Experimental confirmation of the reported X17 and the E38 particles
in the same experimental setup will shed light on the question of quark
confinement for quarks interacting in the Abelian U(1) QED interaction.

On the theoretical front, there is a need for theoretical clarification on the
question of confinement with regard to lattice gauge calculations. The lattice
gauge calculations indicate that a static quark and a static antiquark
interacting in the compact QED interaction will not be confined in (3+1)D.
However, the deconfined solution for static quark and static antiquark in
compact QED in (3+1)D in lattice gauge calculations contradicts the
experimental absence of fractional charges. This indicates that the present
lattice gauge calculations for compact QED in (3+1)D may not be complete and
definitive, because the important Schwinger dynamical quark effects associated
with light quarks has not been included.

We have constructed a “stretch (2+1)D” flux tube model to investigate the
importance of the Schwinger mechanism on quark confinement in QED in (3+1)D
Won22 by utilizing the Polyakov’s transverse confinement in conjunction with
Schwinger’s longitudinal confinement, We find that the stretch (2+1)D flux
tube model leads to quark confinement in compact QED in (3+1)D Won22a . Such a
quark confinement result of the stretch (2+1)D flux tube model is consistent
with the experimental absence of fractional charges. Furthermore, it gives
predictions on the masses of neutral QED mesons and QED mesons in agreement
with the experimental QCD and QED meson spectra. It is therefore worthy of
further considerations. It is important to find out whether future lattice
gauge calculations with dynamical light quarks, in a configuration such as the
stretch (2+1)D flux tube configuration, will lead to confined quarks in
compact QED in (3+1)D.

The success of the open-string description of the QCD and QED mesons leads to
the search for other neutral quark systems stabilized by the QED interaction
between the constituents in the color-singlet subgroup, with the color-octet
QCD gauge interaction as a spectator field. Of particular interest is the QED
neutron with the $d$, $u$, and $d$ quarks Won22c . They form a color product
group of ${3}$ $\otimes$ ${3}$ $\otimes$ ${3}$ =
${\hbox{\boldmath${1}$}}\oplus{\hbox{\boldmath${8}$}}\oplus{\hbox{\boldmath${8}$}}\oplus{\hbox{\boldmath${10}$}}$,
which contains a color singlet subgroup ${1}$ where the color-singlet current
and the color-singlet QED gauge field reside. In the color-singlet $d$-$u$-$d$
system with three different colors, the attractive QED interaction between the
$u$ quark and the two $d$ quarks overwhelms the repulsion between the two $d$
quarks to stabilize the QED neutron at an estimated mass of 44.5 MeV. The
analogous QED proton has been found theoretically to be unstable, and it does
not provide a bound state nor a continuum state for the QED neutron to decay
onto by way of the weak interaction. Hence the QED neutron may be stable
against the weak interaction. It may have a very long lifetime and may be a
good candidate for the dark matter. Because QED mesons and QED neutrons may
arise from the coalescence of deconfined quarks during the deconfinement-to-
confinement phrase transition in different environments such as in high-energy
heavy-ion collisions, neutron-star mergers Bau19 ; Wei20 , and neutron star
cores Ann20 , the search of the QED bound states in various environments will
be of great interest.

It is necessary to address the question why the QCD mesons and the QCD neutron
have not been observed to decay to the lower energy QED counterparts, even
though the QCD hadrons lie higher in energy than those of the QED mesons and
the QED neutron. The QCD meson and neutron and their analogous QED
counterparts lie at different local energy minima in the configuration space..
The QCD mesons have a flux tube radius of about 0.4 fm and a longitudinal
length $L_{{\rm QCD}}$ of about a fermi, as one can infer from the lattice
gauge calculations and phenomenological considerations. For the QED meson, a
flux tube radius about the same as that of the QCD meson appears to be a
reasonable concept, because such a flux tube radius gives the QED meson masses
in agreement with the observed X17 and E38 masses. The longitudinal lengths
$L_{{\rm QED}}$ of the QED mesons are an order of magnitude greater than those
of the QCD mesons Won22c . The QCD meson and their analogous QED meson lie at
different local energy minima in the longitudinal length space $L$. In between
the two energy minima, there is a barrier between the QCD meson and the
analogous QED counterpart. To make a transition from a QCD meson energy
minimum at $L_{\rm QCD}$ so as to come to the region of the analogous QED
meson in the local energy minimum at $L_{\rm QED}$, the QCD meson needs to
tunnel under a barrier from $L_{{\rm QCD}}$ to $L_{{\rm QED}}$ with a barrier
height of order $(L_{{\rm QCD}}-L_{{\rm QED}})\times 1$ GeV/fm, where 1 GeV/fm
is the magnitude of the QCD string tension coefficient. This gives a barrier
height of order 10 GeV. Such a high barrier makes the transition probability
prohibitively small. Similarly, a QCD neutron has a spatial length scale of
order 1 fm, whereas the QED neutron has a spatial length scale of order 20-30
fm Won22c . They lie in different local energy minima in the length scale
space $L$. The high barrier between the QCD neutron energy minimum and the QED
neutron energy minimum makes the transition probability prohibitively small.

Experimentally, in the environment in which a pair of color-singlet quark and
an antiquark are produced, such as by the fusion of two virtual photons or two
virtual gluons at the eigenenergy of a QED meson (e.g. 17 or 38 MeV, with the
proper quantum numbers), as in Figs. 1, 2 and 3, the color-singlet $q$ and
$\bar{q}$ pair created at a local point will expand to the full longitudinal
extension to become a QED meson and will carry out a longitudinal yo-yo motion
appropriate for the QED meson eigenstate, as described in Fig. 12 and in Ref.
Won22c . Similarly, in the surface region of a quark-gluon plasma during the
confinement-to-deconfinement phase transition in high-energy heavy-ion
collisions, deconfined $u$ and $d$ quarks with low energies may search for
partners to form a low-energy $d$-$u$-$d$ bound state. A color-singlet
combination of low-energy $u$ and $d$ quarks may form a $d$-$u$-$d$ QED
neutron eigenstate by their mutual QED interactions, if these quarks posses
the proper eigenenergy. A search for the QED neutron in high-energy heavy-ion
collisions may be of great interest.

###### Acknowledgements.

The author is indebted to Prof. V. F. Perepelitsa whose talk at the
International Symposium on Multiparticle Dynamics, in 2009 introduced the
author to the subject of anomalous soft photons which raised author’s interest
on the question of quark confinement in the QED interaction. The author would
like to thank Profs. Y. Jack Ng, A. Koshelkin, H. Sazdjian, Soren Sorensen, D.
Blaschke, Kh. U. Abraamyan, Gang Wang, Xi-Guang Cao, G. Wilk, Y. Sharon, L.
Zamick, and I-Yang Lee for helpful communications. The research was supported
in part by the Division of Nuclear Physics, U.S. Department of Energy under
Contract DE-AC05-00OR22725 with UT-Battelle, LLC.

## References

  * (1) J. Schwinger, Gauge invariance and mass II, Phys. Rev. 128, 2425 (1962).
  * (2) J. Schwinger, Gauge theory of vector particles, in Theoretical Physics, Trieste Lectures, 1962 (IAEA, Vienna, 1963), p. 89.
  * (3) S. Coleman, R. Jackiw, and L. Susskind, Charge shielding and quark confinement in the massive Schwinger model, Ann. Phys. 93, 267 (1975).
  * (4) S. Coleman, More about the massive Schwinger model, Ann. Phys. 101, 239 (1976).
  * (5) A. M. Polyakov, Quark confinement and topology of gauge theories Nucl. Phys. B120, 429 (1977).
  * (6) A. M. Polyakov, Gauge Fields and Strings, Hardwood Academic Publishers, Switzerland, 1987.
  * (7) K. G. Wilson, Confinement of quarks, Phys. Rev. D19, 2445 (1974).
  * (8) J. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev D11, 395 (1975).
  * (9) S. Mandelstam, Vortices and quark confinement in non-Abelian gauge theories, Phys. Lett. 53B, 476 (1975).
  * (10) T. Banks, B. Myerson, and J. Kogut, Phase transitions in Abelian lattice gauge theories, Nucl. Phys. B129, 493 (1977).
  * (11) J. Glimm and A. Jaffe, Instantons in a U(1) lattice gauge theory: A Coulomb dipole gas, Comm. Math. Phys. 56, 195 (1977).
  * (12) M. E. Peskin, Mandelstam-’t Hooft duality in Abelian lattice models, Ann. Phys. (N. Y. ) 113, 122 (1978).
  * (13) A. Guth, Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory, Phys. Rev. D 21, 2291 (1980).
  * (14) K-I Kondo, Existence of confinement phase in quantum electrodynamics, Phys. Rev. D58, 085013 (1998).
  * (15) G. Magnifico, T. Felser, P. Silvi, S. Montangero, Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks, Nature Com. 12, 3600 (2021), [arXiv:2011.10658].
  * (16) S. D. Drell, H. R. Quinn, B. Svetitsky, and M. Weinstein, Quantum electrodynamics on a lattice: A Hamiltonian variational approach to the physics of the weak-coupling region, Phys. Rev. D19, 619 (1979).
  * (17) G. Arnold, B. Bunk, T. Lippert and K. Schilling, Compact QED under scrutiny: it’s first order, Nucl. Phys. Proc. Suppl. 119 (2003) 864 [arXiv:hep-lat/0210010]
  * (18) L. C. Loveridge , O. Oliveira , and P. J. Silva, Lattice pure gauge compact QED in the Landau gauge: photon propagator, phase structure, Dirac string, Phys. Rev. D 104, 114511 (2021), and references cited therein.
  * (19) J. Schwinger, On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664 (1951).
  * (20) C. Y. Wong, Introduction to High-Energy Heavy-Ion Collisions, World Scientific Publisher, 1994.
  * (21) H. Georgi, The Schwinger point, JHEP11 (2019) 057, [arxiv:1905.09632].
  * (22) H. Georgi and B. Noether, Non-perturbative Effects and Unparticle Physics in Generalized Schwinger Models, [arxiv:1908.03279v3].
  * (23) H. Georgi and B. Warner, Generalizations of the Sommerfield and Schwinger models, JHEP01, 047 (2020), [arXiv:1907.12705v2].
  * (24) H. Georgi, Automatic Fine-Tuning in the Two-Flavor Schwinger Model, Phys. Rev. Lett. 125, 181601 (2020), [arXiv:2007.15965].
  * (25) H. Georgi, Mass perturbation theory in the 2-flavor Schwinger Model with opposite masses, JHEP 10 (2022) 119, [arXiv:2206.14691].
  * (26) R. Dempsey, I. R. Klebanov, S. S. Pufu, B. Zan, Discrete Chiral Symmetry and Mass Shift in Lattice Hamiltonian Approach to Schwinger Model, [arXiv:2206.05308].
  * (27) C. Y. Wong, Anomalous soft photons in hadron production, Phys. Rev. C81, 064903 (2010), [arXiv:1001.1691].
  * (28) C. Y. Wong, Anomalous soft photons associated with hadron production in string fragmentation, Talk presented at the IX International Conference on Quark Confinement and Hadron Spectrum, Madrid, Spain, Aug 30-Sept 3, 2010, AIP Conf.Proc.1343:447 (2011), [arXiv:1011.6265].
  * (29) C. Y. Wong, An overview of the anomalous soft photons in hadron production, Talk presented at International Conference on the Structure and the Interactions of the Photon, 20-24 May 2013, Paris, France, PoS Photon 2013, 002 (2014), [arXiv:1404.0040].
  * (30) C. Y. Wong, Open string QED meson description of the X17 particle and dark matter, JHEP 08 (2020) 165, [arxiv:2001.04864].
  * (31) C. Y. Wong, On the stability of the open-string QED neutron and dark matter, EPJA 58, 100 (2022), [arxiv:2010.13948].
  * (32) C. Y. Wong, QED mesons, the QED neutron, and the dark matter, in Proceedings of the 19th International Conference on Strangeness in Quark Matter, EPJ Web of Conferences 259, 13016 (2022), [arXiv:2108.00959].
  * (33) C. Y. Wong, QED meson description of the X17 and other anomalous particles, in Proceedings of the Workshop of “Shedding Light on X17”’ September 6-8, 2021, Centro Ricerche Enrico Fermi, Rome, Italy, [arXiv:2201.09764].
  * (34) C. Y. Wong and A. Koshelkin, Dynamics of quarks and gauge fields in the lowest-energy states in QCD and QED, arXiv:2111.14933 (2022).
  * (35) A. Koshelkin and C. Y. Wong , Dynamics of quarks and gauge fields in the lowest-energy states in QCD and QED, in Proceedings of the 41st International Conference in High Energy Physics, 6-13 July, 2022 Bologna, Italy, PoS 414, 302 (2022), [arXiv:2212.11749].
  * (36) P.V. Chliapnikov $et~{}al.$, Observation of direct soft photon production in $\pi^{-}p$ interactions at 280 GeV/c, Phys. Lett. B 141, 276 (1984).
  * (37) F. Botterweck $et~{}al.$ (EHS-NA22 Collaboration), Direct soft photon production in $K^{+}p$ and $\pi^{+}p$ interactions at 250 GeV/c, Z. Phys. C 51, 541 (1991).
  * (38) S. Banerjee $et~{}al.$ (SOPHIE/WA83 Collaboration), Observation of direct soft photon production in $\pi^{-}p$ interactions at 280 GeV/c, Phys. Lett. B305, 182 (1993).