On the question of quark confinement in the Abelian <i>U</i>(1) QED gauge interaction

doi:10.1007/s11467-023-1288-0

Front. Phys.

2023, Vol. 18

Issue (6) : 64401 https://doi.org/10.1007/s11467-023-1288-0

VIEW & PERSPECTIVE

On the question of quark confinement in the Abelian U(1) QED gauge interaction

Cheuk-Yin Wong(

)

Physics Division, Oak Ridge National Laboratory*, Oak Ridge, TN 37831, USA

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Abstract

If we approximate light quarks as massless and apply the Schwinger confinement mechanism to light quarks, we will reach the conclusion that a light quark $q$ and its antiquark $q ¯$ will be confined as a $q q ¯$ boson in the Abelian U(1) QED gauge interaction in (1+1)D, as in an open string. From the work of Coleman, Jackiw, and Susskind, we can infer further that the Schwinger confinement mechanism persists even for massive quarks in (1+1)D. Could such a QED-confined $q q ¯$ one-dimensional open string in (1+1)D be the idealization of a flux tube in the physical world in (3+1)D, similar to the case of QCD-confined $q q ¯$ open string? If so, the QED-confined $q q ¯$ bosons may show up as neutral QED mesons in the mass region of many tens of MeV [Phys. Rev. C 81, 064903 (2010) & J. High Energy Phys. 2020(8), 165 (2020)]. Is it ever possible that a quark and an antiquark be produced and interact in QED alone to form a confined QED meson? Is there any experimental evidence for the existence of a QED meson (or QED mesons)? The observations of the anomalous soft photons, the X17 particle, and the E38 particle suggest that they may bear the experimental evidence for the existence of such QED mesons. Further confirmation and investigations on the X17 and E38 particles will shed definitive light on the question of quark confinement in QED in (3+1)D. Implications of quark confinement in the QED interaction are discussed.

Keywords quark confinement QCD interaction QED interaction Schwinger model open string model of mesons QCD molecular states

Issue Date: 05 June 2023

Cite this article:

Cheuk-Yin Wong. On the question of quark confinement in the Abelian U(1) QED gauge interaction[J]. Front. Phys. , 2023, 18(6): 64401.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1288-0
https://academic.hep.com.cn/fop/EN/Y2023/V18/I6/64401

Fig.1 (a) The production of a

q q ¯

pair by the

e +

e −

collision with a single intermediary virtual photon at low energies, (b) the production of a

q q ¯

pair by the

e +

e −

collision with two intermediary virtual photons at low energies, and (c) the production of many

q q ¯

pairs by the

e +

e −

collision at high energies.

Fig.2 In high-energy electron scattering off an atomic nuclei with charge

Z

, the bremsstrahlung-like reaction may lead to the emission of a photon, with the creation of a

q q ¯

pair [66, 67]. A QED meson may be produced if the

s

of the emitted

q q ¯

pair coincides with the eigenenergies of the QED meson,

X

, by a single photon emission as in (a), or by the fusion of two photons, as in (b).

Fig.3 (a)

q q ¯ (X)

production by the fusion of two virtual gluons in the de-excitation of a highly-excited

4 H e ∗ (A B C D) → 4 H e

(ground state) (

A ′ B ′ C D

) +

q

q ¯

(X) with the fusion of two virtual gluons between

B

and

A

. (b)

q q ¯ (X)

production in hadron

q q ¯

-hadron or a nucleus−nucleus collision by

A

B

→

A ′

B ′

(q q ¯) n →

A ′

B ′

∑ i h i + q q ¯ (X)

		$J π I$	Experimental mass (MeV)	Semi-empirical mass formula (MeV)	Meson mass in massless quark limit (MeV)

QCDmeson	$π 0$	$0 − 1$	134.9768 $±$ 0.0005	134.9^‡	0
	$η$	$0 − 0$	547.862 $±$ 0.017	498.4 $±$ 39.8	329.7 $±$ 57.5
	$η ′$	$0 − 0$	957.78 $±$ 0.06	948.2 $±$ 99.6	723.4 $±$ 126.3
QEDmeson	isoscalar	$0 − 0$		17.9 $±$ 1.5	11.2 $±$ 1.3
	isovector	$0 − 1$		36.4 $±$ 3.8	33.6 $±$ 3.8
	X17	(1⁺)	16.70 $±$ 0.35 $±$ 0.5^†
PossibleQEDmesoncandidates	X17	( $0 −$ )	16.84 $±$ 0.16 $±$ 0.20^#
	X17	( $1 −$ )	16.86 $±$ 0.17 $±$ 0.20^□
	E38		37.38 $±$ 0.71 $⊕$
	E38		40.89 $±$ $0.91 ⊖$
	E38		39.71 $±$ $0.71 ⊗$

Tab.1 Comparison of experimental and theoretical masses of neutral,

I 3

= 0, and

S

= 0 QCD and QED mesons obtained with the semi-empirical mass formula (58) for QCD mesons and (64) for QED mesons, with

α s

= 0.68

±

0.08,

R T

= 0.40

±

0.04 fm, and

α Q E D

= 1/137.

	Particle $i$	$J π I$	$Q ~ q Q C D (i)$	$Q ~ q Q C D (i)$ $θ p = − 24.5 o$	$Q ~ q Q E D (i)$	$Q ~ q Q E D (i)$ $θ p = − 24.5 o$

QCDmeson	$π 0$	$0 − 1$	0	0	$1 / 2$	0.7071
	$η$	$0 − 0$	$− 3 sin ? θ p$	0.7182	$3 cos ? θ p / 6$	1.1144
	$η ′$	$0 − 0$	$3 cos ? θ p$	1.576	$sin ? θ p / 6$	−0.1693
QEDmeson	isoscalar (X17)	$0 − 0$			$1 / (32)$
QEDmeson	isovector (E38)	$0 − 1$			$1 / 2$

Tab.2 The effective color charge

Q ~ q Q C D (i)

and the effective electric charge

Q ~ q Q E D (i)

of the quark constituent in the QCD or QED meson

i

, in the presence of flavor mixing. The mixing angle

θ p = − 24.5 o

is from Ref. [74].

Fig.4 (a) Classical space-time trajectories of massless quarks in the yo−yo motion of a meson with two flavors in the isospin singlet state with

I = 0, I 3 = 0

. The motion of the

u

and

d

quarks are in phase, and the quark charges effectively add together. (b) Classical space-time trajectories of massless quarks in yo−yo motion of a meson with two flavors in the isospin triplet state with

I = 1, I 3 = 0

. The motion of the

u

and

d

quarks are out of phase, and the quark charges effectively subtract from each other.

Fig.5 (a) A QED meson

X

can decay into two real photons

X → γ 1 + γ 2

, (b) it can decay into two virtual photons each of which subsequently decays into an

(e + e −)

pair,

X → γ 1 ∗ + γ 2 ∗ → (e + e −) + (e + e −)

, and (c) it can decay into a single

(e + e −)

pair,

X → γ 1 ∗ + γ 2 ∗ → e + e −

Fig.6 Feynman diagrams for

p 1

p 2

→

p 3

p 4

k

Experiment	Collision energy	Photon $k T$	Photon/Brem ratio

$K + p$ , CERN, WA27, BEBC (1984)	70 GeV/c	$k T <$ 60 MeV/c	4.0 $±$ 0.8
$K + p$ , CERN, NA22, EHS (1993)	250 GeV/c	$k T <$ 40 MeV/c	6.4 $±$ 1.6
$π + p$ , CERN, NA22, EHS (1997)	250 GeV/c	$k T <$ 40 MeV/c	6.9 $±$ 1.3
$π − p$ , CERN, WA83, OMEGA (1997)	280 GeV/c	$k T <$ 10 MeV/c	7.9 $±$ 1.4
$π + p$ , CERN, WA91, OMEGA (2002)	280 GeV/c	$k T <$ 20 MeV/c	5.3 $±$ 0.9
$p p$ , CERN, WA102, OMEGA (2002)	450 GeV/c	$k T <$ 20 MeV/c	4.1 $±$ 0.8
$e +$ $e −$ $→$ hadrons, CERN, DELPHI with hadron production (2010)	~91 GeV (CM)	$k T <$ 60 MeV/c	4.0
$e +$ $e −$ $→$ $μ +$ $μ −$ , CERN, DELPHI with no hadron production (2008)	~91 GeV (CM)	$k T <$ 60 MeV/c	1.0

Tab.3 The ratio of the soft photon yield associated with hadron production to the bremsstrahlung yield in high-energy hadron−hadron collisions and

e +

−

e −

annihilations, compiled by Perepelitsa [42].

Fig.7 Number of soft photons

N γ

in units of

(10 − 3 γ

/jet), shown as solid circular points, as a function of (a) neutral hadron multiplicity

N n e u

, (b) charged hadron multiplicities,

N c h a r g e

, and (c) the total hadron multiplicity,

N t o t a l

, from the DELPHI Collaboration [45]. The number of soft photons from the bremsstrahlung process is shown as triangular data points.

Fig.8 (a) Anomalous soft photon

d N / d p T

data from

p p

collisions at

p l a b

= 450 GeV/c obtained by Belogianni

e t a l .

[41]. (b) Anomalous soft photon

d N / d p T

data from the DELPHI Collaboration for

e + e −

annihilation at the

Z 0

mass of 91.18 GeV [43]. 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.

Fig.9 (a) The energy levels and widths of ⁴He states showing the transition in the decay of the 21.01 MeV

J π I

0 − 0

⁴He excited state to the ⁴He ground state with the emission of the X17 particle at ~17 MeV, as interpreted in [47]. (b) The energy levels and widths of ⁸Be states showing the transition in the decay of the excited 18.15 MeV

J π I

1 + 0

state to the ⁸Be ground state with the emission of the X17 particle, as interpreted in Ref. [46].

Fig.10 (a) ATOMKI experimental

e +

and

e −

angular correlations at different proton energies in the reaction of ⁴He

(p, e + e −)

⁴He_g.s. [47, 48]. 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 [47, 48]. (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 [47, 48]. (c) Similar

e + e −

excess occurs in the reaction of ⁷Li

(p, e + e −)

⁸Be_g.s. [49]. 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 π I

1 + 0

state of ⁸Be to the ground state with the emission of the X17 particle at ~17 MeV (solid histogram) [49].

Fig.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 is 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.

Fig.12 (a) depicts the initial production of the

q q ¯

pair of a QED meson, with an infinitesimal longitudinal separation

Δ x 3

. (b) gives a snapshot of the configuration of the quark charge, the antiquark charge, gauge fields

A, E

, and

B

∇

A

during the dynamical longitudinal yo−yo motion of the

q q ¯

pair interacting in compact QED in a QED meson, starting from a (2+1)D transversely confined

q q ¯

system. (b) is the “stretch (2+1)D” configuration. (c) is the corresponding lattice transcription, following the Hamiltonian formulation of Drell

e t a l .

[16].

Fig.13 The quark and antiquark configurations in the interaction between meson

A

and

B

with quark charges

± q A

and

± q B

, respectively. The two mesons are separated by the radius vector

r A B

from meson

A

to meson

B

. As a result of the long-range Coulomb interaction, mesons

A

and

B

acquire dynamical dipole moments

d A

and

d B

, respectively. In (a) and (b),

θ A

is the angle between

d A

and

r A B

θ B

is the angle between

d B

and

r A B

, and

ϕ A B

is the azimuthal angle between

d A

and

d B

when we place

d A

and

d B

in the polar coordinates with

r A B

as the polar axis. (c) and (d) are the linear and orthogonal equilibrium configurations with respect to angular variations at a fixed

r A B

Fig.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 (a), an orthogonal chain as in (b), the extension both longitudinally and transversely in (c), a linear chain turning in a right angle

θ 1 = π / 2

in (d), a triangle in (e), and a square in (f).

Fig.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 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 (a), the (QCD meson)−(QED meson)−(QED meson)−(QCD meson) in (b), and the (QED meson)−(QCD meson)−(QCD meson)−(QED meson) in (c) and (d). Pions as examples of the QCD mesons are indicated schematically.

Fig.16 The possible mechanism of the production of the molecule state

π

−(QED meson)−(QED meson)−

π

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

−

q ¯ 2

open string into the four mesons,

X 1, π 1, π 2, X 2

, near the

π

π

threshold.

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