QED2 boson description of the X17 particle and dark matter

The oscillations of the color charges and electric charges of the quark and antiquarks of the underlying vacuum in a flux tube generate QCD2 bosons (mesons) and QED2 bosons (Phys.Rev.C81,064903(2010)). The predicted mass of the isoscalar QED2 boson in the flux tube environment is close to the X17 mass, leading to the suggestion that the X17 particle may be the isoscalar $I(J^\pi)$=0(0$^-$) QED2 boson arising from the oscillation of the electric charges of the quarks and antiquarks of the vacuum in the flux tube that mediates the meson-exchange interaction between the nucleon and the nuclear core in the excited 0(0$^-$) state of $^4$He. The isoscalar 0(0$^-$) QED2 boson can decay into an electron-positron pair or two photons in free space but the decays will be inhibited if the gravitational binding energy of the QED2 boson bound in a QED2 boson assembly exceeds its rest mass. Consequently, a self-gravitating isoscalar QED2 boson assembly whose mass $M$ and radius $R$ satisfy $(M/M_\odot)/(R/R_\odot) \gtrsim 4.71 \times 10^5$ will not produce electron-positron pairs or photons and may be a good candidate for a primordial dark matter.


I. INTRODUCTION
Recent observation of a light, neutral boson with a mass of about 17 MeV in the electromagnetic M1 decay of the excited I(J π )=0(1 + ) state of 8 Be has generated a great deal of interest [1]. Supporting evidence for the existence of the hypothetical X17 particle has been reported recently in the decay of the excited 0(0 − ) state of 4 He [2]. Earlier investigations on possible observations of similar e + -e − pairs with masses between 3 to 20 MeV by energetic nuclear projectile collisions in emulsion have also been reported [3][4][5]. Different theoretical interpretations, astrophysical implications, and further experimental searches concerning such a particle (or particles) have been presented [6][7][8][9][10][11][12][13][14][15]. However, a definitive description of the X17 particle has not yet emerged.
We note that both the excited 8 Be and 4 He nuclei of interest have the common property of a nucleon occupying a highly excited state, of about 20 MeV, outside a nuclear core. The interaction between the single nucleon and the core can be described by the exchange of a virtual meson that can be represented by the presence of a flux tube. There is thus a flux tube environment between the single-particle and the nuclear core in the X17 production process. We would like to ask whether the presence of the flux tube at the excitation energy of about 20 MeV may provide the environment to induce the occasional production of the X17 particle at the appropriate energy.
The association of a flux tube with the production of hadrons and soft photons occurs in a different context in an earlier physical phenomenon. Previously, anomalous soft photons, in excess of what are expected from electromagnetic bremsstrahlung, have been observed in conjunction with the production of hadrons, mostly mesons, in high-energy K + p [16,17], π + p [17], π − p [18,19], pp collisions [20], and e + -e − annihilations * wongc@ornl.gov [21][22][23][24]. The anomalous soft photons are proportionally produced whenever hadrons are produced [23,24], and they are absent in high-energy e + + e − → µ + + µ − bremsstrahlung when hadrons are not produced [22]. The transverse momenta of these anomalous soft photons lie in the low range of a few MeV/c to many tens of MeV/c. As quarks and antiquarks carry both color charges and electric charges, a "QED2 boson" model [25][26][27] was proposed to describe the simultaneous production of mesons and anomalous soft photons as arising from the oscillations of both color charges and electric charges of the quarks and antiquarks of the underlying vacuum in a flux tube environment. Other different interpretations of the anomalous soft photons have also been presented [28][29][30][31][32][33][34][35][36][37]. We note with special interest that the predicted mass of the isoscalar QED2 boson from electric charge oscillations in Table I of Ref. [25] is close to the mass of the X17 particle. It is natural to ask whether the X17 particle may indeed be the isoscalar "QED2 boson" arising from the soliton-like oscillation of the electric charges of quarks and antiquarks of the vacuum in the flux tube that is present in the excited 8 Be and 4 He nuclei under consideration.
To carry out such an exploration, we review in Section II the quantum field theory of stable QCD and QED bosons and their simultaneous production in a flux tube environment. We start with the previous result connecting the coupling constants in two-dimensional space-time and those in the physical four-dimensional space-time. Such a connection allows us to write down the mass formula for QCD2 and QED2 bosons. In previous work in Ref. [25], the mass formula was applied to mesons with large transverse masses in Z 0 decay. In the present study, we are interested in low-energy nuclear spectroscopy and meson rest masses for which we re-calibrate the massivequark correction term and make a small adjustment on the phenomenological parameters. In Section III, we specialize to spin S=0 bosons and obtain a corrected mass formula for QCD2 and QED2 bosons that matches well with experimental meson masses. The mass formula yields an isoscalar QED2 boson whose mass and quantum numbers agree with those of the X17 particle in the decay of the 4 He 0 − excited state observed in [2]. In Section IV, we study spin S=1 mesons in QCD2, to gain information on the multiplicity and quantum numbers of spin S=1 X17-like particles to be expected. In Section V, reexamine the experimental transverse momentum distribution of anomalous soft photons using a thermal model to see if the possible presence of the 17 MeV component in the transverse momentum distribution may be a robust result. In Section VI, we examine isoscalar QED2 bosons as possible constituents of dark matter and study the relationship between the mass and the radius of the QED2 boson assembly when the decay of a QED2 boson will be inhibited. In Section VI, we present our conclusions and discussions.

II. QCD AND QED BOSONS IN A FLUX TUBE ENVIRONMENT
From the work of Schwinger [38,39], it is well known that in quantum electrodynamics of 1 space and 1 time dimensions (QED2), as occurs in a one-dimensional string, the electric charge oscillation of massless fermion fields generates a non-interacting, bound electrodynamical boson, the QED2 boson. The oscillation of the QED2 gauge field induces an oscillation of the fermion field, which in turn generates an oscillation of the gauge field. The non-linear, non-pertubative, and self-consistent dependency of the fermion field current and the gauge field lead to a stable boson field satisfying a Klein-Gordon equation with a a mass m, given in terms of the QED2 coupling constant by (1) (For a pedagogical derivation of such a remarkable result, see for example, Chapter 6 of [40].) Schwinger's QED2 pertains to a system of electric charges. The application of Schwinger's concepts to the color charges of quarks and antiquarks was pioneered by Bjorken [41], Casher, Kogut, Suskind, Neuberger, Nussinov [42][43][44], Coleman [45,46], Halpern [47], Witten [48] and many others [25-27, 37, 40, 49-58], who suggested that the quantum field theory of stable boson particles and their particle production in the physical four-dimensional space time (QCD4) can be considered to be analogous to those in two-dimensional space-time in a string. QCD bosons (mesons) that are stable within the QCD2 theory are solitons and antisolitons [45][46][47] and they will be produced along a QCD string, when a fermion and antifermion (quark and an antiquark) at the two ends of the string pull apart at high energies [42][43][44][45][46]. The rapidity distribution of these produced mesons exhibits the property of boost invariance in the limit of infinite energies [41][42][43][44]. For a finite energy system, the boost-invariant solution turns naturally into a rapidity plateau, whose width increases with energy as ln( √ s) [40,50].
The q-q string is an idealization of a flux tube with a transverse profile, which reveals itself as the transverse momentum distribution of the produced particles [51]. Experimentally, the presence of a flux tube is evidenced by the limiting average transverse momentum and a rapidity plateau [40-42, 50, 51, 57] as in high-energy e + -e − annihilations [59][60][61][62][63] and pp collisions [64]. The classical model of hadrons as yo-yo strings and the fragmentation of high-energy strings have been successfully applied to the phenomenology of the hadronization process in highenergy hadron-hadron collisions and e + -e − annihilations [40,65].
Quarks and antiquarks carry both color charges and electric charges. In the problem of the simultaneous production of hadrons and anomalous photons, both QCD and QED bosons are produced. It is necessary to treat QED and QCD in a single framework. The simultaneous production of QCD and QED bosons can be studied by considering an enlarged U(3) QCD and QED group that is the union of the color SU(3) subgroup of QCD and the electromagnetic U(1) subgroup of QED [25]. This can be achieved by introducing the generator t 0 for the U(1) subgroup, which adds on to the eight standard generators of the SU(3) subgroup, {t 1 , ...t 8 }, to form the nine generators of the U(3) group. They satisfy tr{t α t β } = δ αβ /2 for α, β = 0, 1, .., 8. The two subgroups differ in their coupling constants and communicative properties. Limiting our consideration to quarks with two light flavors, we examine the QCD4×QED4 system in fourdimensional space-time x µ , with µ=0,1,2,3. The dynamical variables are the quark fields, ψ i f , and the U(3) gauge fields, A ν =A α ν t α , where i is the color index with i=1,2,3, f is the flavor index with f =u, d, and α is the U(3) generator index with α=0,1,..,8. The coupling constants g α f depend on α and f and are given explicitly by with Q u =2/3, Q d =−1/3, α s =(g QCD4 ) 2 /4π, and α=(g QED4 ) 2 /4π=1/137. It was shown previously that under plausible assumptions of longitudinal dominance and transverse confinement, QCD4×QED4 in 3 space and 1 time coordinate in a flux tube can be approximated by QCD2×QED2 in 1 space and 1 time coordinates with a quark transverse mass m T arising from the transverse confinement of the quarks [25,26,57,58]. The coupling constants of the 2D dimensional space-time in QCD2 and QED2 are found to be related to those in 4D dimensional space-time by [25,57] where R T is the radius of the flux tube. We can check the qualitative consistency of such a relation from dimensional analysis. The coupling constant in two dimensional space-time, g QED2 and g QCD2 , have the dimension of a mass, as Eq.
(1) indicates. The strong interaction coupling constant α s and the fine-structure constant α in four dimensional space-time are dimensionless. The only physical attribute that can is called upon to turn a string in 2 dimensional space-time into a flux tube in four dimensional space-time is the radius of the flux tube R T . Hence Eq. (4) involving R T (in standard =c=1 units) is qualitatively consistent from the dimensional points of view. As a result of the compactification, our task is then to search for stable boson solutions in QCD2×QED2 with the above effective coupling constants g 2 2D and a transverse mass m T .
Following the work on massive Schwinger QED2 and QCD2 model as carried out by Coleman, Casher, Kugut, Susskind, Witten and many others using the method of bosonization [42][43][44][45][46][47][48][49][50]55], we show in Ref. [25] how we can bosonize the fermion fields in of the U(3) group to obtain stable QED2 and QCD2 bosons for the case with two flavors. The up and down quark fields combine to form the isoscalar I=0 boson fields and the isovector I=1 boson fields. We limit our attention to boson fields with the color-neutral and charge-neutral I 3 =0 component. The square of the mass m I of these I=0 or 1 bosons is found to be [25] (m a (I) where the superscript a is the QED or the QCD label, γ = 0.5772 is the Euler constant, and µ a is a mass scale, which arises from the bosonization of the scalar densitȳ ψψ. The scalar densityψψ diverges in perturbation theory and has to be renormalized such that ψ ψ =0 in a free theory. It will need to be re-normal-ordered again in an interacting theory [46]. The scalar density and the corresponding mass scale µ a therefore depends on the interaction. The above result of Eq. (5) represents a QCD2×QED2 generalization of previous results [40,[45][46][47][48][49][50][51][53][54][55][56], where QED2 and QCD2 have been examined separately.

III. PHENOMENOLOGICAL SEMI-EMPIRICAL MASS FORMULA FOR SPIN S=0 BOSONS
The application of the mass formula (5) requires the knowledge of transverse mass m T and the mass scale µ which depend on the physical system under consideration. They are therefore solution-dependent and interaction-dependent. While rigorous investigations based on first principles of QCD4×QCD4 can continue to proceed, the approximate validity of the mass formula Eq. (5) can be the basis for a phenomenological semi-empirical description of the lowest QCD2 and QED2 bosons in a flux tube environment. Because the masses of the current quarks are small in comparison with the QCD hadron masses, it is reasonable to hold the view that the massless quark limit as given by the first term of the mass formula (5) is the basic non-perturbative result. The second term of massive quark corrections can be considered phenomenologically based on our knowledge of the physics of the problem under consideration.
The quark and antiquark fermion fields possess the spin degree of freedom for which they can combine to form the spin S=0 and S=1 boson fields. We treat the spin degree of freedom as a perturbation and consider first the spin S=0 bosons in this section because they have the lowest masses.
Phenomenologically, the lowest-lying spin S=0 neutral QCD bosons are the π 0 particle, with a mass m π =134.98 MeV and I(J π ) = 1(0 − ), and the η 0 particle, with a mass m η =547.3 MeV and I(J π ) = 0(0 − ). In previous work in Ref. [25], the mass formula was applied to mesons with large transverse masses in Z 0 decay. In the present study, we are interested in low-energy nuclear spectroscopy and meson rest masses for which we need to re-calibrate the massive-quark correction term and make a small adjustment on the phenomenological parameters. We observe that the first term of the mass formula Eq. (5) gives a massless pion in the massless-quark limit, in agreement with the concept of the pion being a Goldstone boson in the standard QCD theory and the pion acquires a mass due to non-zero quark mass and transverse confinement. It is therefore reasonable to calibrate the second massivequark correction term in Eq. (5) by m 2 π . The magnitude of this correction term should depend on the square of the two-dimensional coupling constant, as in the first term in Eq. (5). We are therefore led to the following corrected semi-empirical mass formula for the lowest spin S=0 bosons, By setting α s =0.50 and R T =0.42 fm, the above mass formula describes well the lowest I=0 π 0 and η 0 mesons and the X17 particle masses, as indicated in Table I. The value of α s is close to the value of 0.58 used in the study of meson spectroscopy and reactions [66,68], and the flux tube radius R T is close to the value of 0.35 fm inferred previously in [25] from the transverse momentum of produced hadrons in the Z 0 hadronic decay. The I(J π ) quantum numbers of the QCD2 bosons as mesons are known. We can therefore infer the quantum numbers of the QED2 bosons by analogy. In Column 4 of Table I, we list the quantum numbers I(J π ) of the QED2 bosons taken to be the same I(J π ) as the corresponding QCD2 particles of the same I and S. Table I shows that the S=0 isoscalar electric charge oscillation of the quarks and antiquarks from the vacuum in a flux tube yields an isoscalar I(J π )=0(0 − ) QED2 boson with a mass and quantum numbers that agree with those of the X17 particle observed in the decay of 4 He nucleus from the excited 0(0 − ) state to the ground 0(0 + ) state. It may therefore be a good candidate for the X17 particle [2]. An S=0 QED2 isovector boson with a mass of 34.68 MeV in a flux tube is also predicted and is yet to be definitively observed, although the high momentum tail of the transverse momentum spectrum of the anomalous soft photons provides a weak hint of its possible existence [20,25].  It is interesting to note that for the S=0 bosons, the mass ordering of the isovector and isoscalar bosons in QED2 is the reverse of that in QCD2. That is, in a flux tube, the QED2 isovector I(J π )=1(0 − ) boson lies higher in mass than the QED2 isoscalar I(J π )=0(0 − ) boson, whereas the QCD2 isovector I(J π )=1(0 − ) boson (the π meson) lies lower in mass than the QCD2 isoscalar I(J π )=0(0 − ) boson (the η meson). This arises because in QED there is a difference in the magnitudes and the signs of the electric charges of the up and down quark, whereas in QCD there is no difference in the magnitudes and the signs of the color charges of the up and down quark.

IV. SPIN S=1 QCD2 AND QED2 BOSONS
The oscillations of the quark and antiquark currents lead not only to spin S=0 bosons but also to spin S=1 bosons. Our understanding of the S=1 QCD2 bosons in a flux tube will provide useful information on the multiplicity and the I(J π ) quantum numbers of analogous QED2 bosons to be expected in the flux tube environment. We treat the splitting due to the spin degree of freedom perturbatively as arising from additional residual interactions, similar to earlier treatments in meson spectroscopy and meson reactions [66][67][68]. We can generalize the mass formula in Eq. (6) to include the additional residual spin-dependent interactions, and write the mass formula as m a (I, S)= m a (I, S=0) + v a S (s 1 · s 2 + 1/4) (7) + v a ST (s 1 · s 2 + 1/4) τ 1 · τ 2 , where m a (I, S=0) for spin S=0 bosons are given by Eq. (6), s 1 , s 1 , τ 1 ,τ 2 are the spin and isospin operators of the quark and antiquark fields, respectively. The interaction strengths v a S and v a ST are phenomenological parameters that can be fixed when the masses of the S=1 particles are known.  In QCD2, the S=1 boson with I(J π )= 1(1 − ) is the ρ meson, and the S=1 boson with I(J π )=0(1 − ) is the ω meson. From their experimental masses, the QCD residual interaction parameters can be determined to be v QCD S = 399.04 MeV, v QCD ST =341.00 MeV, yielding a good agreement between the semi-empirical mass formula and the experimental masses, as shown in Table II. The good description of the lightest QCD bosons by the semiempirical mass formula suggests that in addition to the S=0 QED2 bosons discussed in the last section, S=1 QED2 bosons with quantum numbers I(J π )= 1(1 − ) and 0(1 − ), analogous to the ρ and ω mesons in QCD, are also expected. The interaction parameters v QED S,ST are not known.

V. TRANSVERSE MOMENTUM DISTRIBUTION OF ANOMALOUS SOFT PHOTONS
The isoscalar QED2 boson was first identified theoretically in quantum field theory in Ref. [25]. It was proposed as one of the components of the anomalous soft photons produced in the flux tube environment in high energy collisions. However, for its unequivocal identification, it is necessary to observe the 17 MeV QED2 boson from the invariant mass spectrum of anomalous photons. The agreement of theoretical mass and quantum numbers of the isoscalar QED2 boson with those of the experimental X17 particle provides special impetus for such an invariant mass measurement in the future. Additional encouraging support comes from the transverse momentum distribution of anomalous soft photons of [20] where a mass 17 MeV component appears to be present in the earlier analysis in [25]. As the decomposition of the transverse momentum distribution is not a unique procedure, we would like to re-examine other ways to decompose the p T distribution and see if the possible presence of the 17 MeV component in the transverse momentum distribution may indeed be a robust result.
In the present analysis, we rely on the empirical systematics of particle production in high-energy pp collisions where a thermal model appears to provide a good description of the transverse momentum distribution of QCD2 bosons (mesons) [69]. We assume that the thermal model also provide a good description for the production of QED2 bosons which materialize as anomalous soft photons. In the thermal model, the transverse momentum distribution depends on the produced boson mass as [69] where E= m 2 + p 2 T , m is the boson mass, and e q can be a Boltzmann or a Tsallis distribution. We choose the simpler Boltzmann distribution, e −E/T , instead of the Tsallis distribution that contains an additional unknown parameter q. With the Boltzmann distribution, the peak of the transverse momentum distribution dN/dp T that is proportional to p T e −E/T occurs at If m is much greater than T , then dN/dp T peaks at p T ∼ √ mT . If m = 0, then dN/dp T peaks at p T = T . In the thermal model for anomalous soft photons, there will be a peak in dN γ /dp T for each thermal contribution from each boson mass. The total dN γ /dp T should be the sum of many peaks as there can be in the number of contributing boson mass components.
In Fig. 1, we show the dN γ /dp T data of anomalous soft photons in pp collisions at 450 GeV/c from Belgianni et al. [20]. It is difficult to fit the distribution with a single thermal component. There appears to be a sharp peak at p T ∼2-4 MeV, indicating that it is necessary to include a real photon component with m γ 3 =0. It has a temperature of approximately 2 MeV, the p T location of the peak of dN γ /dp T . The origin of the real photon component with m γ 3 =0 is not known, but it shows up prominently in the transverse momentum distribution. component is hard to detect as it is nearly at the noise level. The solid curve in Fig. 1 is obtained by adding the contributions from three components, +A 3 e −p T /T3 , with m γ 1 =17 MeV, m γ 2 =35 MeV, T 1 =6 MeV, m γ 3 =0, and T 3 =2.3 MeV, A 1 =4.4 × 10 4 , A 3 =3.4 × 10 4 . In the same plot, the contributions from the three components are also separately displayed. The decomposition is similar to the earlier results in [25] and lends support to the possible presence of the m γ 1 ∼17 MeV component in the anomalous soft photons in pp collisions observed in [20]. A direct measurement on the invariant masses of the anomalous soft photons to confirm the 17 MeV boson mass component will be of great interest.

VI. BEHAVIOR OF A MASSIVE QED2 BOSON ASSEMBLY
The QED2 isoscalar I(J π )= 0(0 − ) boson is the lowestlying QED2 bosons. It mass of 17 MeV exceeds the mass of an electron-positron pair. In free space, it can decay into an electron-positron pair, or it can annihilate into two photons. There are situations when such decays are forbidden, when an aggregate of the QED2 bosons form a massive assembly (as in a dark matter substructure or subhalo) whose gravitational binding energy for an isoscalar QED2 boson exceeds its rest mass. Such assemblies of QED2 bosons present themselves as good candidates for the primordial cold dark matter. We would like to make an estimate on the mass and radius of such assemblies where they may be found.
We consider an assembly of A QED2 bosons of mass M A ≡M and we place a test QED2 boson of mass m X bound at the surface of the assembly at radius R, the mass M A+1 of the combined system is where G is the gravitational constant. The Q values for the QED2 boson at the surface of the (A+1) assembly to decay into an electron-positron pair or 2γ are Thus, the QED2 boson m X will not decay into an electron-positron pair when the mass and radius of the assembly satisfy and the QED2 boson m X will not decay into two photons when M and R satisfies As shown by Schwinger [39], the QED2 bosons are noninteracting in the massless-quark limit in a string in one space and one time dimensions [39]. Coleman examined how such non-interacting property may be modified in a massive-quark theory in two-dimensional space-time [45]. He found that soliton-like bound states and trapping of the charges persist, as in the massless theory. In the physical 4-dimensional free space-time, the QED2 bosons may scatter or interact with the external fields, leading to the production of lower-mass particles. For example, the interaction of the QED2 bosons with the Coulomb field in the detector leads to the production of electron-positron pairs for the detection of the anomalous soft photons in hadron production. However, in a gravitating assembly of QED2 bosons, the difference in the gravitational energy of the initial QED2 bosons and the gravitational energy of the final lower-mass particle states will affect the properties of the QED2 assembly. Referring to the mass and the radius of the sun, it is convenient to define a dimensionless boundary value B 0 given by A QED2 boson assembly will behave differently depending on its M/R values as follows: 1. The QED2 boson assembly will emit electronpositron pairs and gamma rays if 2. The QED2 boson assembly will emit only gamma rays but no e + e − pairs, if 3. The QED2 boson assembly will not emit e + e − pairs nor gammas rays , if which is essentially the condition for a QED2 boson black hole.
The above boundaries characterize the properties of a QED2 boson assembly where dark matter may be found. An assembly of QED2 bosons satisfying Eq. (13) (which is the same as or (17)) can be a good candidate for a primordial cold blackhole dark matter, as it is non-baryonic, created at the hadronization of the quark-gluon plasma phase, and not from a stellar collapse.

VII. CONCLUSIONS AND DISCUSSIONS
Quarks and antiquarks of different flavors carry both color charges and electric charges. Their oscillations in the environment of a flux tube generate QCD2 and QED2 boson quanta of various masses and quantum numbers. These quanta can be considered to be solitons in the flux tube environment [45][46][47]. The QCD2 bosons correspond to π, η, ρ and ω mesons whose masses obey a simple mass formula. The generalization of the semi-empirical mass formula from the color charges to electric charges yields an isoscalar QED2 boson with a mass of 17 MeV, as it was approximately predicted in Ref. [25]. The agreement of the mass and the quantum numbers of the isoscalar QED2 boson with those of the X17 particle makes it a good candidate for the X17 particle recently observed in decay of the excited 0(0 − ) state of 4 He to the ground 0(0 + ) state of 4 He. The isoscalar QED2 boson can be produced from the decay of the 0(0 − ) state of 4 He because of the presence of the flux tube of the virtual meson mediating the interaction between the nucleon at the appropriate energy and quantum numbers outside the core of the 4 He nucleus.
Whether the isoscalar QED2 boson is related to the 1 + X17 particle observed in the decay of 8 Be remains to be investigated. If the 1 + excited state of 8 Be emits a particle in the l = 1 state, then the emitted particle can be a 0 − particle, the same as the X17 particle observed in the 4 He decay. It will be necessary to check experimentally how the 8 Be nucleus in the excited 1 + state decays.
The success of the QCD2 description of the light meson masses suggests that there may be additional spin S=1 analogous QED2 bosons. In the spectroscopy of the X17 particles, the possibility of these additional particles should be kept in mind.
The proposed interpretation of the X17 particle receives support from the occurrence of the anomalous soft photons consistently produced in conjunction with mesons in high-energy hadron-nucleon and e + -e − collisions [16][17][18][19][20][21][22][23][24]. Their simultaneous production may indicate theoretically their common origin as QCD2 and QED2 oscillation of the quarks and antiquarks in the underlying vacuum [25]. The study of the X17 particle and the anomalous soft photons may therefore be intimately linked together.
We would like to address the relevance of the QED2 bosons with regard to the production and the stability of dark matter. We can envisage that in the early evolution of the universe after the big bang, the universe will go through the stage of quark-gluon plasma production with deconfined quarks and gluons. As the primordial matter cools down the quark-gluon plasma undergoes a phase transition from the deconfined phase to the confined phase, and hadronization occurs. Hadrons are produced from the quark-gluon plasma by way of flux tube production. As quarks and antiquarks carry both color charges and electric charges, the hadron production in a flux tube as QCD2 bosons will necessarily be accompanied by QED2 boson production, just as the production of hadrons is accompanied by the production of anomalous soft photons in high-energy hadron-nucleon collisions and e + -e − annihilations observed experimentally [16][17][18][19][20][21][22][23][24].
These produced QED2 bosons are presumably tightly bound and non-interacting in the Schwinger's picture of two-dimensional space-time. As solitons, they likely maintain their strong binding in the presence of scattering. In the physical four dimensional space-time, they can however decay or scatter inelastically into lower mass stable particles in free space. On the other hand, if they find themselves in spatial locations where their gravitational binding energies exceeds their rest masses, then their decay or scattering into two photons or an electronpositron pair will be inhibited. There can be QED2 boson assemblies produced at this stage where the gravitational binding energy of a QED2 boson exceeds its rest mass. For such a QED2 boson assembly, the QED2 bosons will be stable against particle decay, annihilation, and inelas-tic collisions. They may form the primordial cold dark matter that may be the source of gravitational attraction for other objects.
In addition to nuclear experiments to confirm the results of the 8 Be and 4 He experiments, anomalous soft photon production in pp collisions and e + e − annihilations may be an additional tool to examine these QED2 bosons. [16][17][18][19][20][21][22][23][24]. Thus, they can act as QED2 boson factories where the spectroscopy of soft QED2 bosons may be examined. It is important to measure the invariant masses of the anomalous soft photons to see what kind of mass spectrum there can be, as they are predicted to be QED2 bosons, the same as the X17 particles.
The recent inclusive experiment of the NA64 Collaboration [14] of finding no "dark" soft photon excesses needs to reconcile with the earlier finding of anomalous soft photon excesses in high energy hadron-p, pp, and e + -e − collisions [16][17][18][19][20][21][22][23][24]. In the detection of soft photons in the DELPHI Collaboration [24], the photon detection is carried out by studying the electron and positron tracks in a TPC, while in the NA64 experiment, the produced soft photon needs to penetrate a calorimeter and is detected downstream in a separate calorimeter. It is not known whether the difference in the detection setups and techniques may account for the presence or absence of excess soft photons in the two measurements.
Future work also call for experimental and theoretical studies of the properties of the observed X17 particles and their reactions. Much theoretical work will need to be done to study the decays and reactions of QED2 particles in a strong gravitational field to shed more lights on the fate of the QED2 boson assembly in the possible primordial dark matter environment.
As it is suggested here that the hadronization of the early history of the universe in the quark-gluon plasma phase generates simultaneously the QED2 boson assemblies as seeds for dark matter, it will be of great interest to study whether indeed QED2 bosons as anomalous soft photons are produced in high-energy heavy-ion experiments where quark gluon plasma may be produced.

ACKNOWLEDGMENTS
The author wishes to thank Prof. Y. Jack Ng for helpful discussions. The research was supported in part by the Division of Nuclear Physics, U.S. Department of Energy under Contract DE-AC05-00OR22725.