On the Stability of the Open-String QED Neutron and Dark Matter

manuscript No. (will be inserted by the editor)

On the stability of the open-string QED neutron and dark matter

Cheuk-Yin Wongb 1Physics Division, Oak Ridge National Laboratorya, Oak Ridge, Tennessee 37831, USA

Abstract We study the stability of a hypothetical QED deconfinement-to-confinement transition of the quark neutron, which consists of a color-singlet system of two gluon plasma in high-energy heavy-ion collisions. Be- d quarks and a u quark interacting with the quantum cause of the long lifetime of the QED dark neutron, self- electrodynamical (QED) interactions. As a quark can- gravitating assemblies of QED dark neutrons or dark not be isolated, the intrinsic motion of the three quarks antineutrons of various sizes may be good candidates in the lowest-energy state of the QED neutron may lie for a part of the primordial dark matter produced dur- predominantly in 1+1 dimensions, as in a d-u-d open ing the deconfinement-to-confinement phase transition string. In such an open string, the attractive d-u and of the quark gluon plasma in the evolution of the early u-d QED interactions may overcome the weaker repul- Universe. sive d-d QED interaction to bind the three quarks to- Keywords Anomalous soft photons · Schwinger gether. We examine the stability of the QED neutron QED2 · Open string · Dark matter in a phenomenological three-body problem in 1+1 dimensions with an effective interaction between electric charges extracted from Schwinger’s exact QED solu- 1 Introduction tion in 1+1 dimensions. The phenomenological model in a variational calculation yields a stable QED neutron Recent experimental observations of the anomalous soft energy minimum at a mass of 44.5 MeV. The analo- photons [1,2,3,4,5,6,7,8,9], the X17 particle at about gous QED proton with two u quarks and a d quark 17 MeV [10,11,12], and the E38 particle at about 38 has been found to be too repulsive to be stable and MeV [13,14,15], have generated a great deal of inter- does not have a bound or continuum state, onto which ests [16]-[29],[30,31,32,33,34,35,36,37,38,39,40,41,42, the QED neutron can decay via the weak interaction. 43]. With a mass in the region of many tens of MeV, Consequently, the lowest-energy QED neutron is stable the produced neutral anomalous particles appear to lie against the weak decay, has a long lifetime, and is in outside the domain of the Standard Model. Many spec- fact a QED dark neutron. Such a QED dark neutron ulations have been proposed for these objects, includ- and its excited states may be produced following the ing the cold quark-gluon plasma, QED mesons, the fifth arXiv:2010.13948v3 [hep-ph] 9 May 2021 force of Nature, the extension of the Standard Model, aThis manuscript has been authored in part by UT-Battelle, QCD axion, dark matter and many others. LLC, under contract DE-AC05-00OR22725 with the US De- Among the suggested descriptions, we wish to focus partment of Energy (DOE). The US government retains our attention on the quantized QED mesons descrip- and the publisher, by accepting the article for publication, tion of [25,26,27,28], which links the anomalous parti- acknowledges that the US government retains a nonexclu- sive, paid-up, irrevocable, worldwide license to publish or cles together in a coherent framework. We note that the + − reproduce the published form of this manuscript, or al- anomalous soft photons are produced as excess e e low others to do so, for US government purposes. DOE pairs when hadrons are produced, and are absent when will provide public access to these results of federally spon- hadrons are not produced [1,2,3,4,5,6,7,8,9]. A par- sored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan), ent particle of the anomalous soft photons is likely to Oak Ridge, Tennessee 37831, USA contain some elements of the hadron sector, such as a be-mail: [email protected] light quark-antiquark pair. A quark and an antiquark 2 interact mutually with the quantum chomodynamical only to the cases when the total available energy of (QCD) and quantum electrodynamical (QED) interac- the interacting quarks (in their center-of-mass system) tions. The parent particle of the anomalous soft photons exceeds the pion mass threshold (of about 134 MeV). cannot arise from the quark-antiquark pair interacting Light quarks have masses of a few MeV, and a light with the QCD interaction, because such an interaction quark and an antiquark pair can be readily produced will endow the pair with a mass much greater than the with an available energy much below the pion mass mass scale of the anomalous soft photons. We are left threshold. Theoretically, there is no known physical law with the possibility of the quark and the antiquark in- that forbids the light quarks to interact with QED interacting with the QED interaction. The QED inter- teractions alone. According to Gell-Mann’s Totalitarian action of the quark-antiquark pair can be described in Principle, what is not forbidden is allowed [59]. Experi- 1+1 dimensional space-time because quarks cannot be mentally, whether quarks can interact with QED inter- isolated, as in an open string. The possibility of the actions alone can only be answered by testing its theo- composite light quark pair interacting in QED inter- retical consequences with experiments, and such a test actions is further reinforced by the special nature of a in connection with the aforementioned anomalous par- confining gauge interaction in 1+1 dimensions as was ticles demonstrates that under appropriate conditions first shown by Schwinger [73,74], for which the greater when the available energy for a light quark-antiquark the strength of the attractive confining interaction, the pair is much below the pion mass threshold, quarks can greater will be the mass of the composite particle it interact with QED interactions alone. generates. Relative to the QCD interaction, the QED interaction will bring the quantized mass of a qq¯ pair to the lower mass range of the anomalous soft photons. It was therefore proposed in [25,26,27] that a quark and an antiquark in a qq¯ system interacting with the QED interaction may lead to new open string bound + _ states (QED-meson states) with a mass of many tens of MeV. These QED mesons may be produced simultaneously with the QCD mesons in the string fragmentation process in high-energy collisions [1,2,3,4,5,6,7, 8,9], and the excess e+e− pairs may arise from the de- (a) Electric field lines of force of two equal cays of these QED mesons. By using the method of and isolatable charges of opposite signs bosonization [44,45,46,47,48,49,50,51,52,53,54,55,56, 57,58], the mass of the I(J π)=0(0−) isoscalar QED meson was predicted to be 17.9±1.8 MeV and the mass of q q π − the isovector (I(J )=1(0 ),I3=0) QED meson to be 36.4±3.8 MeV [28]. These state masses match those of (b) Electric field lines of force of a qq¯ QED meson the X17 particle, the E38 particle, and the possible parent particles of the anomalous soft photons, indicating that these anomalous particles are consistent with their d u d description as confined composite qq¯ systems interacting in QED interactions. Q = -1/3 2/3 -1/3 The above experimental observations and theoreti- (c) Electric field lines of force of a d-u-d QED neutron cal interpretation of the anomalous particles raise many important questions concerning quarks in QCD and Fig. 1 The electric field lines of force. (a) For two isolatable charges of opposite signs such as an electron and a positron. QED interactions. Can quarks interact in QED inter- (b) For a light quark q and a light antiquark q¯ of the same actions alone? If so, how do a quark and an antiquark flavor interacting in QED interactions in a color-singlet qq¯ interact in QED interactions? What are the observable QED meson. (c) For a color-singlet QED neutron in the consequences that can be tested by experiments? d-u-d configuration. In answering these questions, it is often argued that We can understand how this occurs by finding out quarks experience QCD and QED interactions simulta- how the quarks interact in QED interactions and how neously and thus it may appear at first sight that quarks the mass scales of qq¯ composite particles depend on the cannot interact in QED interactions alone, without the gauge interaction coupling constant. Quarks carry elec- QCD interactions. However, it should be realized that tric charges and cannot be isolated. The non-isolation the simultaneous QCD and QED interactions pertain property requires the quarks to interact in QED inter- 3 actions in ways that are differently from the ways in In the above discussions on QED interactions in which isolatable electric charges such as electrons and 3+1 dimensions, the proposal in Figure 1(a) and Fig- positrons interact. For, if the QED interaction between ure 1(b) suggests interaction laws for quarks different a quark and its antiquark were the same as the QED from the interaction laws for electrons in QED. The use interaction between two equal and isolatable charges of different electrodynamical interaction laws for elec- of opposite signs, the electric field lines of force would tric charges of a quark and an antiquark versus elec- extend to very large transverse distances as shown in tric charges of an electron and positron may appear Fig. 1(a) [62]. The quark and antiquark pair would extraordinary at first sight, but it is in fact related to form a positronium-like bound state that would be eas- the question on the confinement or non-confinement ily dissociated, resulting in isolated charged particles properties of different fermions when they interacting with a fractional charge. No such isolated quarks have in QED gauge interactions. In the lattice implemen- ever been observed. Thus, the non-isolation property of tation of fermions in QED U(1) gauge theory in 3+1 quarks requires the quarks to interact in QED interac- dimensions, there can be the compact version of QED tions in ways different from isolatable electric charges gauge interaction in which the vector potentials can be such as electrons and positrons. angular variables with the topology of a circle. There can also be the noncompact version in which the vector potentials can assume values from −∞ to +∞. As The non-isolation of a light quark and a light an- emphasized not the least by Yang [75] and described in tiquark means that quarks interact in QED in such a detail by Polyakov [76], the compactness of the QED way that its electric field lines of force do not spread U(1) gauge group has a direct bearing on charge quan- out as in Fig. 1(a) and must be arranged so as to sat- tization and the confinement of the fermion systems. isfy the condition of non-isolation as in Fig. 1(b). The If the QED U(1) gauge interaction is to be embedded non-isolation property of a light quark will be consistent into a large group with the hope of a grand unification, with Schwinger’s idealized solution in 1+1 dimensions, the version realized in Nature will likely be compact if the electric field lines of forces emanating from a light because of charge quantization. quark with a positive electric charge in 3+1 dimensions are bundled together in the form of a flux tube to be Using the compact version of the U(1) lattice gauge connected to the electric field lines of force entering into theory in 3+1 dimensions, Wilson [77] found that the another light quark with the opposite sign of charge Wilson loop expectation value for a large loop with as in Fig. 1(b). Such a behavior of bundling and con- strong coupling behaves in accordance with the area necting of the electric field lines of force is anticipated law, which corresponds to the confinement of the fermions. from Schwinger’s exact confining solution for massless Wilson therefore conjectured that fermion-antifermion fermions with electrical charges of opposite signs in the systems interacting in the QED U(1) gauge interac- idealized 1+1 dimensional QED gauge field where the tion in 3+1 dimensions have a phase transition with electric field line of force is directed along the string. a confined phase for strong coupling and a nonconfin- Schwinger’s massless fermions in 1+1 dimensional QED ing phase for weak coupling. t’Hooft [78] and Polyakov gauge field fit the interacting massless quark system [79,80] studied the same problem and noted the oc- well because a confined pair of interacting quark and currence of non-trivial topological configurations of the antiquark can be described naturally as the two ends of vector A field in a circle as magnetic monopoles that a one-dimensional open string [63,64,65,66,67,68,47, serve to confine a fermion and an antifermion as in a 48,49,69,70,71]. The bundling and connecting of the string. By studying the density and the condensate of electric field lines of forces of opposite-sign charges as the monopoles as a function of the coupling in 3+1 di- in a flux tube will lead to a consistent description of mensions, Polyakov concluded that fermions under the the non-isolation of interacting light quarks, the con- compact QED U(1) gauge interaction in 3+1 dimen- finement of interacting light quarks, the properties of sions have a confining phase for strong coupling and a the electric field lines of force, the binding between light non-confining phase for weak coupling. The same con- quark and antiquark of opposite-sign charges, the con- clusion of a confining phase for strong coupling and non- sistency with the Schwinger exact solution in 1+1 di- confining for weak coupling for fermions interacting in mensions, and the consistency with the experimental compact QED U(1) gauge interactions in 3+1 dimen- observations of the aforementioned anomalous parti- sions was reached by many workers including Kogut, cles. From the above perspectives, Schwinger’s 1+1 di- Susskind, Mandelstam, Banks, Jaffe, Peskin, Guth, Kondo mensional string is just an idealization of the bundling and many others [81,82,83,84,85,86,87,88,89], and by of the connecting electric field lines in a physical 3+1 recent lattice gauge calculations using tensor networks dimensional flux tube as depicted in Fig. 1(b). with dynamical quarks [89]. 4

The above QED U(1) lattice gauge results in 3+1 pothesis that under appropriate conditions, light quarks dimensions indicate the presence of the confining and can interact in QED interactions alone, when they are the nonconfining phases in fermion systems in 3+1 di- produced with an invariant mass in the region of many mensions. It is therefore reasonable to consider the pos- tens of MeV. sibility that under the QED interactions in 3+1 dimensions, quarks reside in the confining phase if they interact in QED interactions alone while electrons reside in the nonconfining phase. Hence, there can be different QED interaction laws for quarks and for electrons as depicted in Figure 1(a) and Figure 1(b) respectively. Such a description will be consistent with experimental observations of the confinement of quarks. Even though the electromagnetic coupling constant is the same for the electric charges of quarks and electrons, how the quark properties (of masses, quark colors, quark frac- There are interesting consequences when light quarks tional electric charges, unknown quark intrinsic electro- can interact with QED interactions alone. Light quarks magnetic structure, and quark gauge group topological carry electric charges and the QED interaction between properties) may lead to the bifurcation to the confined elementary charges of opposite signs is attractive. The phase for quarks and the non-confined phase for elec- attractive QED attraction between light quarks with trons in 3+1 dimensional QED gauge theories will be a opposite-sign charges leads to composite light quark qq¯ subject worthy of future investigations. Clearly, what- QED mesons, as was discussed previously [25,26,27,28]. ever the theoretical predictions there may be, the con- In a similar way, in a color-singlet QED neutron of d-u- fining property of quarks in QED interactions alone in d with three different colors, the attractive QED forces 3+1 dimensions must be tested by experiments. The may lead to a stable composite system. present investigation on the stability and the masses of composite quark systems under QED interactions alone serves to facilitate the examination of such an important test. The light quark fields themselves are approximately massless, and massless quarks do not setup any mass scale. The mass scales of composite particles of light quarks are set up by the coupling constants of the exchange gauge interactions, as one can infer from the work of Schwinger for confined massless fermions in √ 1+1 dimensions for which m=e/ π, where m is the Our understanding of the structure of the electric composite boson mass and e is the gauge interaction field lines of force between a light quark and an anti- coupling constant [73,74]. If the attractive gauge inter- quark allows us to study similar electric field lines of actions arise from the non-perturbative interaction of force in the QED neutron in 3+1 dimensions in Fig. the light quarks with gluons, then the mass scale will 1(c). The non-isolation property of the d quarks finds be of the order of hundreds of MeV [25,26,27,28]. On its consistent expression if the electric field lines of force the other hand, if the attractive gauge interactions arise entering into the two d quarks are bundled and con- from the non-perturbative interaction between the light nected to those emanating from the u quark in the mid- quarks and the QED gauge fields, then the mass scale dle. With a charge of the opposite sign and a magnitude as inferred from the QED coupling constant will be of twice as large as each the two d quarks, the electric field the order of tens of MeV [25,26,27,28]. Consequently, in lines of forces emanating from the u quark can join on circumstances in which the available energy of the com- and accommodate the electric field lines of forces en- posite interacting light quark system is of order tens of tering into both d quarks on the left and on the right MeV, light quarks can interact with the QED interac- in Fig. 1(c). The binding of the three quarks in the tions alone. neutron can arise from such joining of the electric field Based on the above experimental observations of the lines. Hence, the intrinsic motion of the three quark anomalous particles and our theoretical understanding system in the lowest-energy state of the QED neutron of the dependence of the mass scales on the gauge in- lie predominantly in 1+1 dimensions, as in an idealized teractions, it is reasonable to presume as a working hy- d-u-d open string. 5

finement is limited only to constituents in the presence (a) QED neutron of the QCD gauge fields, or it suffices to involve only the down quark up quark down quark light quark fields and the QED gauge fields by themselves without the QCD interactions. Clearly, this is a fundamental question that can be answered only by ex- Q= (electric ) e -1/3 charge 2/3 -1/3 periments. The observation of the anomalous soft photons, the X17, and the E38 particles indicate that the (b) QED proton quark confinement is an intrinsic property of quarks. The investigations presented here and elsewhere [25,26, up quark down quark up quark 27,28] will allow us to carry out further experimental tests on such a basic property.

electric 2/3 1/3 There is also the additional interest in the QED neu- Q= (charge ) e - 2/3 tron, along with the QED mesons and their correspond- Fig. 2 The schematic picture of a color-singlet three-quark ing antiparticle counterparts, as candidate particles for system in 1+1 dimensions in (a) a QED neutron, and (b) a the primordial dark matter because they are massive, QED proton. The lengths of the arrows represent schemati- they may be produced during the deconfinement-to- cally the forces acting on each of the quarks arising from the confinement stage of the quark gluon plasma phase tran- QED interactions. The vector sums of the forces acting at each quark reveal the attractive forces binding the QED d-u- sition, and the quark-gluon plasma phase may occur in d neutron and the repulsive forces disrupting the QED u-d-u the early history of the Universe. proton. To study the stability of the QED neutron and proton with three light quarks, we need to develop the tools for the relativistic three-body problem by calibrating We can now study the u-d-u open-string configura- the effective QED interaction with Schwinger’s exact tion of the QED neutron in 1+1 dimensions in Fig. 2(a). QED solution for massless fermions in 1+1 dimensions, We display the different QED forces acting on the three which we proceed to carry out in the next few sections. quarks, with the magnitude proportional to |QiQj| for the force between quark i and quark j, an attractive 2 Schwinger’s boson and massless fermions in force for negative Q Q , and a repulsive force for posi- i j 1+1 dimensional QED tive QiQj. The attractive d-u and u-d QED interactions between the two d quarks and the center u quark in Fig. Our goal is to study the stability of color-singlet states 2(a) may overcome the weaker repulsive d-d QED in- involving three light quarks in QED interactions. To teraction between the two d quarks, to bind the three pave the way for such an investigation, we would like quarks in the QED neutron. However, the analogous to sharpen our theoretical tools by examining the analo- configuration of the QED proton in Fig. 2(b) with two gous two-body problem of a massless fermion-antifermion u quarks and one d quark in the u-d-u configuration pair in QED, for which the exact solution from Schwinger may likely be unstable because of the stronger repul- is already known [73,74]. sion between the two u quarks in comparison with the Schwinger showed previously that in 1+1 dimen- weaker attractive interaction between the d and the u sions, massless fermions and antifermions interacting quarks. We would like to show quantitatively in section with the QED gauge interaction with a coupling con- 6 that this may indeed be the case. The conclusions we stant e give rise to a bound boson with a mass m, given will reach concerning the stability of the QED neutron by [73,74] and proton will also be applicable to the stability of e QED antinucleons. m = √ , (1) π It is worth pointing out that theoretical and experimental investigations on the QED neutron are inter- where the QED coupling constant e has the dimension 1 esting on many accounts. First is the possibility of its of a mass in 1+1 dimensions . In terms of the descrip- being a new exotic member in the family of particles of tion of e as a unit of charge and Q as the charge number, the Standard Model. Its properties probably set it apart 1 We adopt here the notations that e is actually e2D , the QED from other particles because it is a special combination coupling constant in 1+1 dimensions, and e4D is the QED of the light quark fields and the QED gauge fields that coupling constants in 3+1 dimensions, with the fine structure constant defined by α=α =e2 /4π = 1/137. As pointed out has not yet been known up to now. It also calls for a bet- 4D 4D in [94,25,95], e2D and e4D are related by the flux tube radius ter understanding on the role of the interplay between R as e2 = e2 /(πR2 ), when the confining flux tube is T 2D 4D T different fields in the confinement process, whether con- approximated as an open string without a structure. 6 the fermion can be described as possessing a charge e field Aµ leads to the gauge field satisfying the Klein- √ and a charge number Q = 1 and the antifermion a Gordon equation with a mass m = e/ π, charge (−e) and a charge number Q = (−1) in the e2 Schwinger model. − Aµ − Aµ = 0. (6) π A derivation of Schwinger’s exact solution of (1) can be found in [74] and explained in details in Chapter Therefore, the non-perturbative coupling between a gauge field and a fermion field in 1+1 dimensions results in a 6 of [90]. Recent generalizations and extensions of the √ Schwinger model can be found in [91,92,93]. It is il- boson field with a quanta of mass m=e/ π as given by luminating to review its salient points here to see in (1). what way we may treat Schwinger’s boson as a rela- It is clear from the above review that the exact so- tivistic two-body problem. Schwinger’s exact solution lution of the boson state does not lend itself readily can be obtained self-consistently as a many-body field to a simple quantum mechanical two-body problem of theory problem involving the response of all fermions in valence fermion (quark) and valence antifermion (anti- the presence of a perturbing gauge field Aµ. We start quark) involving a simple fundamental two-body inter- by considering a vacuum state in which all the nega- action, because it involves concepts and operations of tive energy states of the massless fermions in the Dirac gauge invariance, gauge field self-consistency, the can- sea are occupied. A disturbance in the fermion density cellation of the fermion current singularities, and mass- and/or fermion current jµ will generate a perturbing less fermions that are beyond the simple two-body in- gauge field Aµ. The presence of the perturbing Aµ interactions in conventional two-body problems. In spite duces a change of the gauge phases of fermion field op- of this being the case, it is desirable to construct a phe- erator ψ through the Dirac equation nomenological two-body model for the valence fermion and antifermion with an effective interaction Φ that µ γ (i∂µ − eAµ)ψ(x) = 0, (2) can be calibrated to contain the basic properties of the theory and to yield Schwinger’s exact result. Ex- µ where γ are the gamma matrices in 1+1 dimensions. amples of such an approach can be found in the suc- The change of the gauge phases of fermion field op- cesses of relativistic and non-relativistic hadron spec- erator ψ in turn lead to a change of the fermion cur- troscopy where the non-perturbative QCD solution in- rent jµ. By imposing the Schwinger modification fac- R x µ volving the lattice gauge theory is approximated by ie 0 Aµ(ξ)dξ tor, e x , to ensure the gauge invariance of the a two-body theory with phenomenological effective in- fermion Green’s function, the induced fermion current teractions (see for example, [97,98,99,100,101,102,103, µ j is given implicitly as a function of the perturbing 104,105,106,107,108,109,110]). Being a phenomenolog- µ gauge field A by ical two-body theory, such a theory and its generalizations will need to be worked out with considerable µ −e ie R x A (ξ)dξµ j (x) = lim + lim tr e x0 µ 0 0 theoretical support and persistent confrontation with 2 x0=x0 x0=x0 x1=x10− x1=x10+ experiments so that it can be refined and readjusted, ×hT (ψ(x0)γ0γµψ(x)i , (3) should new experimental data and new theoretical predictions become available. The present investigation on where T is the time-order operator. Upon evaluating the stability of the QED neutron represents an explo- the above limits from the left and from the right at ration along such lines. the space-time point x = (x0, x1), the fermion current An additional advantage of a successful phenomeno- singularities from the left and from the right cancel and logical two-body problem treatment rests on it ability the induced gauge-invariant fermion current jµ is found to simplify the calculations, to retain the essential fea- to be related explicitly to the perturbing gauge field Aµ tures, to provide an intuitive understanding, and to help by solve problems that may not be solvable in a full treatment of the field theory, paving the way for our analysis 2 µ e µ µ 1 ν on the stability of the three-quark system in section 6. j = − A − ∂ λ ∂νA . (4) π ∂ ∂λ µ The resultant fermion current j in turn leads to a new 3 Relativistic two-body problem gauge field A˜µ, through the Maxwell equation,

µν µ ν ν µ µ Relativistic many-body treatment of bound states have ∂ν F = ∂ν (∂ A˜ − ∂ A˜ ) = −j . (5) been carried out in QCD and QED with a high degree of From (4) and (5), the self-consistency of the resultant successes [102,103,104,105,106,107,108,109,110,111,112, gauge ﬁeld A˜µ matching the initial perturbing gauge 113,114]. In the formulation of Dirac, Todorov, Crater, 7

Van Alstine, and Sazdjian, and many others, the basic The above equation cannot be satisfied if Φ12(x12) 6= ingredients consist of treating particles and antiparti- Φ21(x21). The simplest way to satisfy the above equa- cles as independent positive-energy entities with effec- tion is to take tive interactions between them. Each particle obeys a Φ (x ) = Φ (x ) = Φ(x ), (12) mass-shell constraint on: (i) the momentum, (ii) the 12 12 21 21 12 particle mass, and (iii) the effective interactions from which is the relativistic analogue of Newton’s third law. the other particles. The effective interactions can be The compatibility condition (11) then requires the effec- obtained by matching with the perturbative or non- tive interaction Φ(x12) to depend only on the coordinate perturbative counterparts of the field theory or by phe- x12⊥ transverse to the total momentum P =p1 + p2, nomenological considerations. In accordance with Dirac’s Φ(x12) = Φ(x12⊥), (13) constraint dynamics [111], the mass-shell constraints must however be compatible with each other, result- (x − x ) · P where x = (x − x ) − 1 2 P. (14) ing in additional functional requirements or additional 12⊥ 1 2 P 2 terms in the equivalent Schrödinger-type equations whose We shall work in the CM system where the total mo- eigenvalues lead to the eigenstates and the masses of the mentum P is (P 0,P 1)=(M, 0), M is the invariant mass composite particle in question. of the composite system, and the relative coordinate For simplicity, we neglect particle spins whose ef- xij⊥=(x1 −x2) involves only spatial coordinates x1 and fects are expected to be small and consider a two-body x2. The particle momentum pi can be separated out effective interaction Φij(xij) arising from the particle j into a component i parallel to P and a component qi at xj acting on the particle i at xi, depending on the transverse to P as relative coordinate x = x − x . The relativistic two- ij i j P body wave equations for the wave function Ψ for QED pi = (i, qi) = i √ + qi, i = 1, 2, (15) 2 interactions in 1+1 dimensions consist of two mass-shell P pi · P constraints on each of the interacting particles [102,103, where i = √ and q1 + q2 = 0. (16) 104,105,106,107,108,109,110,111,112,113,114], P 2 In terms of i, the invariant mass of the composite sys- 2 2 H1|Ψi = p1 − m1 − Φ12(x12) |Ψi = 0, (7a) tem M is given by 0 M = P = 1 + 2. (17) 2 2 H2|Ψi = p − m − Φ21(x21) |Ψi = 0. (7b) 2 2 The two-body wave equations (7a) and (7b) in the CM system becomes We would like to calibrate the effective interaction Φij by comparing the solution of the above two-body prob- 2 2 2 1|Ψi = q1 + m1 + Φ(x12⊥) |Ψi, (18a) lem with Schwinger’s exact QED solution in 1+1 di- 2|Ψi = q2 + m2 + Φ(x ) |Ψi. (18b) mensions. We construct the total Hamiltonian H from 2 2 2 12⊥ these constraints by Because q2 = (−q1), the second equation of the above is simply 2 X 2 2 2 2 H = Hi. (8) (2 − 1)|Ψi = (m1 − m2)|Ψi. (19) i=1 It is only necessary to solve for the eigenstate of the In order that each of these constraints be conserved in first Schrödinger-type equation (18a) to obtain 1, and time we must have the quantity 2 can be obtained as an algebraic equation from (19). The knowledge of 1 and 2 (taken to dHi [Hi, H]|ψi = i |Ψi = 0. (9) be positive) then gives the invariant mass M of the in- dτ teracting two-body system. As a consequence, the above equation leads to the com- The non-perturbative two-body problem of Eqs. (18) patibility condition between the two constraints [102, can be represented diagrammatically as in Fig. 3. 103,104,105,106,107,108,109,110,111,112,113,114], 4 Schwinger’s QED boson as a relativistic [Hi, Hj]|Ψi = 0. (10) two-body problem in 1+1 dimensions

Since the mass commutes with the operators, this im- The brief summary presented in section 2 makes it plain plies that the Schwinger boson that is conﬁned and bound 2 2 in 1+1 dimensions is in fact a non-linear self-consistent [p1Φ21(x21)] − [p2Φ12(x12)] |Ψi = 0. (11) solution of a many-body system of great complexity. It 8

q¯ 2 relativistic two-body wave equation. We have however used the particle energy i in lieu of the rest mass m , to make it applicable also to the mass- Φ X i less limit. The reduced mass factor depends on the ¡ eigenvalues which should be self-consistently de- i q termined by the wave equations (18a) and (18b). 1 Fig. 3 Diagrammatic representation of the non-peturbative two-body equation (18) for the valence quark q1 and the va- 6 lence antiquark q¯2 interacting with a QED effective interac- ε κ tion Φ leading to the composite qq¯ bound state X. 5 √ Φ(y)/ 4  ε /√κ is desirable to construct a phenomenological two-body 3.26 3 2.87 problem for QED in 1+1 dimensions involving a va- 2.42 2 lence fermion and a valence antifermion with an effec- 1.89 tive phenomenological interaction Φ(x12⊥) that can be 1 1.01  calibrated to contain the basic properties of the the- y=√κ(x -x ) 0 1 2 ory and to yield Schwinger’s exact QED result in 1+1 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 dimensions. -1 Accordingly, we consider the two-body wave equa- Fig. 4 The solid√ curve gives the phenomenological effective tions (18a) and (18b) (or (19)) in the CM system for potential Φ(y)/ κ for a fermion and an antifermion interacting in the QED two-body problem in 1+1 dimensions. The two charge particles with charge numbers Q1 and Q2, energies of the lowest five eigenstates are indicated by hori- interacting with a phenomenological effective QED in- zontal lines with their corresponding wave functions Ψ(y) as teraction Φ(x12⊥) curves on these lines. The parameter√ κ in the phenomenological effective potential Φ(y)/ κ is obtained by matching the 212 solution for the lowest-energy state of the two-body problem Φ(x12⊥) = (−Q1Q2)κ|x1 − x2|. (20) 1 + 2 with Schwinger’s exact solution.

The effective QED interaction Φ(x12⊥) in 1+1 dimensions has been chosen such that: Our task is to obtain the eigenstates for the wave 1. In the Coulomb gauge, the interaction energy be- equation (18a) with the effective potential of (20). tween charges Q1e and Q2e in 1+1 dimensional QED Schwinger’s case of massless fermion and antifermion 2 is (−Q1Q2e /2)|x1 − x2| [45], which is indeed con- corresponds to Q1=1, Q2 = −1, m1 = m2 = 0, 1 = fining for the attractive interaction between unlike 2 = , and M = 2. Using the dimensionless variable charges. It is reasonable to use such a spatially lin- √ y = κ(x1 − x2), (21) ear interaction in the phenomenological two-body problem. The value of the coefficient parameter κ the effective interaction Φ(y)/κ is then of such a linear potential in (20) will be affected Φ(y) = √ |y|, (22) by the use of the phenomenological reduced mass κ κ factor, Schwinger’s self-consistency condition on the gauge field, and the gauge invariance constraint. It and the wave equation (18a) becomes is therefore appropriate to extract κ phenomenolog- ∂2 2 − + √ |y| − Ψ(y) = 0. (23) ical from Schwinger’s exact solution. ∂y2 κ κ 2. The quantity κ is proportional to the square of the coupling constant e2. The exact value of κ will be We show in Fig. 4 the dimensionless effective potential, √ √ √ chosen to give Schwinger’s solution of m = e/ π Φ(y)/ κ, expressed in units of κ. It is a linearly- rising function of the dimensionless spatial separation for a massless fermion-antifermion pair interacting √ in the QED interaction. |y| between the charges, expressed units of of 1/ κ. 3. The effective interaction contains the charge factor The solution of the wave equation with a linearly rising interaction is the Airy function. The wave equation (23) (−Q1Q2), which leads to an attractive interaction if becomes the Airy equation Q1Q2 < 0, and a repulsive interaction if Q1Q2 > 0, as in standard QED interaction in quantum electro- ∂2 − |z| − (√ )4/3 Ψ(z) = 0, dynamics. ∂z2 κ 4. The reduced mass factor 2(12)/(1 + 2) has been where z = (√ )1/3|y|. (24) chosen to give the proper reduced mass in the non- κ 9

The solution satisfying the boundary condition at large physical coupling constants e4D in 3+1 dimensions by |y| is the flux tube radius RT , [25,28,94,95,96] 1 4α √ 4/3 2 2 4D Ψ(z) = Ai(|z| − ( ) ). (25) (e2D ) = 2 (e4D ) = 2 , (29) κ πRT RT with α = 1/137 and R = 0.4 fm [28], which yields The eigenstate is obtained by matching the wave func- √ 4D √T tion and its derivative at z=0. There are two types of κ = 23.8 MeV, and ~/ κ = 8.3 fm. eigenstates with even and odd parities: As a final check of a faithful representation of the phenomenological two-body model for the Schwinger 0 0 4/3 even parity: Ψ (z)|z=0 = Ai (−(√ ) ) = 0, exact field theory in 1+1 dimensions, we note that they κ share the distinct property that the mass of the sys- odd parity: Ψ(z)| = Ai(−(√ )4/3) = 0. (26) z=0 κ tem increases with the increase in the magnitude of the coupling constant, in contrast to a non-confining We label the locations where the wave function or its interaction such as the positronium where the mass of 0 derivative are zero as (−as) with Ai(−as) = 0 or Ai (−as)=0.the composite system decreases with the increase in the Then the eigenvalues of the wave equations are given magnitude of the coupling constant. by √ (√ )4/3 = a , or = a3/4 κ. (27) Table 1 Solution of the two-body problem with the effec- κ s s √ tive interaction Φ(y)/ κ for the lowest states. Here n is√ the number of nodes of the two-body wave function, and M/ κ Table I gives the values of a , energy , mass M, √ s is the dimensionless measure of the composite particle mass. p 2 p 2 hy i, and hq1i/ κ of the lowest ten states. Fig. 4 displays the energies as the horizontal lines, with √ √ √ n Parity a / κ M/ κ phy2i phq2i/ κ their corresponding wave functions Ψ(y) exhibiting dif- s 1 ferent number of nodes. In the two-body problem with 0 even 1.02 1.01 2.02 0.862 0.60 1 odd 2.34 1.89 3.98 1.38 1.09 the phenomenological two-body interaction, the mass of 2 even 3.25 2.42 4.84 1.78 1.41 the lowest state for the phenomenological potential is √ 3 odd 4.09 2.87 5.74 2.10 1.66 M = 2.02 κ. On the other hand, the Schwinger’s exact 4 even 4.82 3.25 6.50 2.38 1.89 √ solution (1) in field theory gives M = e/ π. Therefore, 5 odd 5.52 3.60 7.20 2.63 2.08 6 even 6.16 3.92 7.84 2.86 2.27 by matching the mass of the lowest eigenstate from the 7 odd 6.78 4.20 8.40 3.07 2.43 phenomenological two-body theory with the mass from 8 even 7.37 4.47 8.94 3.27 2.60 Schwinger’s field theory, we find κ to be 9 odd 7.94 4.73 9.46 3.46 2.73 e2 e2 κ = ∼ , (28) 4.08π 4π We note from the above solutions of the two-body where for simplicity, we shall approximate the denomi- problem that a fermion-antifermion composite system nator 4.08π to be 4π. We have thus obtained the phe- with the phenomenological QED interaction possesses nomenological interaction Φ in Eq. (20) between two excited states with a higher number of nodes n, in ad- electric charges interacting in QED in 1+1 dimensions. dition to the lowest state with n = 0. They represent Such a knowledge of the effective QED interaction be- higher vibrational excitations of the fermion-antifermion tween two electric charges will enable us to study the system as an open string. As a rough guide for future states of three quarks interacting in 1+1 dimensional searches of QED meson states in (qq¯) composite sys- QED. tems, we can treat the observed X17 state at E = 16.70

We need the value of the coupling constant e = e2D MeV [10] to be the lowest band heads of the (I = 0, n = which can be obtained from e4D . In the physical world of 0) state, and the E38 state at E = 37.38 MeV [14] to 3+1 dimensions, the one-dimensional open string with- be the lowest band heads of the (I = 1,I3 = 0, n = 0) out a structure is in fact an idealization of a flux tube state respectively. We can then use the QED meson so- with a transverse radius RT , as depicted schematically lutions as displayed in Fig. 4 and Table I to build on in Figs. 1(b) and 1(c). The masses calculated in 1+1 these two band heads semi-empirically an approximate dimensions can represent physical masses, when the theoretical energy level diagram for QED meson states structure of the flux tube is properly taken into ac- with higher numbers of nodes n, as shown in Fig. 5. count. Upon considering the structure of the flux tube On the right panel of Fig. 5, we also display the decay in the physical 3+1 dimensions, we find that the cou- energy E(I = 0, n) − E(I = 0, n − 1) in the transition pling constant e2D in 1+1 dimensions is related to the from the (I = 0, n) state to the (I = 0, n−1) state. This 10

? − q¯ γ q¯ γ e ? 80 I=0, n=9 80 2 2 q¯ γ e− 2 2 2 2 I=0, n=8 + I=1, n=1 e 70 70 I=0, n=7 X q X q X q e I=0, n=6 ¡ ¡e− ¡ 60 I=0, n=5 60 q γ q γ? 1 1 1 1 + q γ? I=0, n=4 e 1 1 e+ 50 50 I=0, n=3 (a) (b) (c) 40 I=0, n=2 40 I=1, n=0 Fig. 6 Fig. 6(a) depicts the diagram for the decay of the

Energy (MeV) I=0, n=1 QED meson X into two real photons (γ1γ2), Fig. 6(b) the 30 30 ∗ ∗ decay of the QED meson X into two virtual photons (γ1 γ2 ), and Fig. 6(c) the decay of the QED meson X into a dilepton 20 (I,n+1) → (I,n) 20 + − I=0, n=0 (e e ) pair. (0,1) → (0,0)

10 (0,2) → (0,1) 10 (0,3) → (0,2) (0,4)... → (0,3) 0 0 Semi-empirical Transition energy QED meson E(I=0,n+1) - E(I=0,n) the quarks and additional interactions will add further energy level E(I,n) to the complexity of the spectrum and the number of Fig. 5 The left panel gives the semi-empirical QED meson the composite particles X. In the 1+1 dimensional de- energy level E(I, n) where I is the isopin quantum number scription, the composite particle X cannot decay, as the and n is the number of nodes, as obtained from Fig. 4 and Table I by assuming the observed X17 state at 16.70 MeV quarks execute a yo-yo motion along the string in an [10], as the band heads of the (I = 1, n = 0) and and the E38 idealization of the flux tube, and the photon is repre- state at E = 37.38 MeV [14] as the band head of the (I = sented by an effective interaction binding the quarks. 1,I3 = 0, n = 0) states respectively. The right panel gives the In the physical 3+1 dimensions where the structure of transition energy E(I=0, n + 1) − E(I=0, n) as dashed lines. the flux tube is taken into account and the photon decay channel opens up, the quark and the antiquark at decay energy corresponds to the diphoton energy if the different transverse coordinates in the tube traveling (I = 0, n) state decays into the (I, n − 1) state with the from opposing longitudinal directions can make a sharp emission of two real or virtual photons as indicated in change of their trajectories turning to the transverse di- the diagram Fig. 7(b) and 7(c) below. Future experi- rection to annihilate, leading to the emission of photons mental searches for these two-body excited states and as depicted in Fig. 6(a), 6(b), and 6(c). The number their diphoton decays in QED mesons will be of great of emitted photons depends on the spin and parity of interest. the decaying system [115,116]. We have illustrated the Before we apply the effective interaction to the three case of two photon decay in Fig. 6(a), 6(b), and 6(c). quark problem, we wish to examine whether a varia- The emitted photon can be two real photons (γ1γ2) in ∗ ∗ tional calculation using the effective interaction (20) 6(a), two virtual photons (γ1 γ2 ), in 6(b), or a dilepton will also lead to the same bound state mass for the (e+e−) pair in 6(c). lowest-energy state. We find indeed that the variational The above diagrams in Fig. 5 pertains to QED me- calculation can give the correct lowest-energy bound son states which can decay directly into elementary par- state mass, as shown in Appendix A. This justifies the ticles. There can be excited QED meson states X∗ with use of the variational calculations in the three-body nodal number n which can de-excite by decaying to problem to obtain the lowest-energy state of a QED lower QED meson state X0 with a smaller nodal num- neutron in section 6. ber n0 < n, with the emission of elementary γ, γγ, and γ∗γ∗ as shown in Fig. 7. 5 The decay of composite qq¯ QED mesons

q q γ q γ∗ e− We would like to review and extend here our knowledge γ e+ on the decay of the QED meson [25,28] to facilitate γ γ∗ e− ∗ X∗ ∗ X X + their experimental detection. The composite system X X0 e 0 X X0 formed by a valence quark q1 and a valence antiquark q¯ ¯ ¡q ¡q¯ q¯2 interacting with the eﬀective QED interaction Φ can ¡ (a) (b) (c) be described by the diagram in Fig. 3. As shown in the Fig. 7 Diagrams for the de-excitation of the excited QED last section, there are many eigenstate solutions of the meson X∗ to the QED meson X0 with particle emissions: Fig. relativistic two-body equations for the composite sys- 7(a) X∗ → X0 + γ , Fig. 7(b) X∗ → X0 + γ + γ , and Fig. tem X. The additional intrinsic degrees of freedom of 7(c) X∗ → X0 + γ∗ + γ∗ . 11

~  For the decay from a bound state X, the decay am- ψ(q/2√κ) plitude needs to be folded in with the bound state mo- 1 n=0 (a) 1 2 mentum wave function. For X → k1 + k2 where the 3 4 ∗ ∗ + − 0.5 ﬁnal states (k1, k2) are (γ1, γ2), (γ1 , γ2 ), or (e , e ) as in diagrams 6(b), 6(c), and 6(d), the decay amplitudes 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 are [105,106]  q/2√κ Z -0.5 3 M(X→k1 + k2)= d q1Ψ˜(q1)Γ (q1 +q ¯2 →k1 +k2), (30)

~  2 where Ψ˜(q1) is the bound state momentum wave func- |ψ(q/2√κ)| tion of a constituent, q1 = qpq pq¯ /2, and Γ (q1 +q ¯2 → 1 2 (b) n=0 k1 + k2) is the Feynman amplitude for the diagram of 0.6 q1 +q ¯2 → k1 + k2, (see e.g. Eqs. (3.6) of [105] or Eq. 1 0.4 2 (2.2) of [106] for the case of two-photon decay). 3 The two-body spatial wave function Ψ(y) for the 4 0.2 QED meson X in Eqs. (23) has been expressed in terms of the dimensionless relative spatial coordinate y = √ 0 κ(x − x ). The corresponding wave function in the -4 -3 -2 -1 0 1 2 3 4 q1 q¯2 √ q  momentum space, Ψ˜(q/2 κ) is the Fourier transform /2√κ of Ψ(y) of Eqs. (22) and (25), Fig. 8 Fig. 11(a) gives√ the two-body wave function in momentum space Ψ˜(q/2 κ) of the composite q-q¯ system as a Z √ √ √ 1 function of the relative momentum q = q1 − q2 = 2q1 in units ˜ ˜ −i(q1/ κ)y √ √ Ψ(q/2 κ)=Ψ(q1/ κ)= √ e Ψ(y)dy, (31) ˜ 2 2π of 2 κ, and Fig. 11(b) gives |Ψ(q/2 κ)| . where q = q −q = 2q , and the constituent momentum 1 2 1 √ q1 is expressed in units of κ. The momentum distri- the two virtual photons, bution of the composite q q¯ system X with a mass M 1 2 2 2 P ∗ ∗ = (pγ∗ + pγ∗ ) , is γ1 γ1 1 2 2 2 dN(P, q) √ Qγ∗γ∗ = −(pγ∗ − pγ∗ ) , (33) = δ(P 2 − M 2)|Ψ˜(q/2 κ)|2. (32) 1 1 1 2 dP dq q 2 q 2 and P ≡ Pγ∗γ∗ = Pγ∗γ∗ , Q ≡ Qγ∗γ∗ = −qγ∗γ∗ . We show in Figs. 11 (a) the momentum wave func- 1 1 1 2 1 1 1 2 √ tions Ψ˜(q/2 κ), and in Fig. 11(b) the corresponding Experimental measurement of the virtual diphoton pair √ momentum probability function |Ψ˜(q/2 κ)|2, for the distribution ∗ lowest five states of the composite system. We can un- dN(Pγ∗γ∗ , q ∗ ∗ ) 1 2 γ1 γ2 dN(P,Q) derstand the contents of Eq. (32) intuitively as follows: = (34) dPγ∗γ∗ dQγ∗γ∗ dP dQ the invariant square of the momentum sum, P 2, probes 1 2 1 2 the invariant mass M 2 of the composite QED meson, will provides useful information on the composite qq¯ whereas the invariant square of the momentum differ- particles of massless light quarks from virtual diphoton ence, q2, measures the inverse spatial size of the decay- decay measurements. Specifically, one makes a scatter ing QED meson as /phq2i, and the number of nodes plot of diphoton events on the two dimensional plane √ ~ of |Ψ˜(q/2 κ)|2 in Eq. (32) and Fig. 11(b) reveals the of P and Q. In actual experiments with the presence of cuts and windows in various kinematical regions, one internal structure of the composite q1q¯2 system. Previously, the lowest-energy (I = 0) QED meson may resort to the use of the event mixing method to has been considered to be the candidate for the X17 normalize the distribution as particle and the (I = 1,I3 = 0) QED meson for the E38 h dN(P,Q) i dN(P,Q) dP dQ particles [28]. By analogy with the corresponding QCD = correlated . (35) h dN(P,Q) i mesons, they are expected to decay into two photons dP dQ norm dP dQ either as real photons as in Fig. 6(a) or as two virtual mixed events photons as in Fig. 6(b). The above normalized distribution dN(P,Q)/dP dQ|norm The decay via two virtual photons as in diagram is expected to cluster sharply around the invariant mass Fig. 6(b) may provide an interesting probe to yield ad- M = (P + Q)/2 of the composite particles and spread ditional information on the decaying QED meson par- out in width in the direction of Q. One can alternative ent particle. One can construct the invariant momenta display the distribution in terms of the rotated coordi- square of the sum and differences of the 4-momenta of nates (P + Q)/2 and a small stripe of P − Q > 0. 12

6 Stability of the QED neutron with three reasonable to neglect such velocities so that we can ap- 0 quarks interacting in QED interactions proximate the xij⊥ that is transverse to momentum Pij to be the relative coordinate xij⊥ that is transverse to Returning to the three-body problem and the QED neu- the total center-of-mass momentum P instead. In such 0 tron, we construct the composite system of three quarks an approximation, the relative coordinate xij⊥ in the 0 by selecting quarks of three different colors to form a effective interaction Φij(xij⊥) becomes just the relative color-singlet state. We consider the d, u, and d quarks coordinate xij⊥ of i and j in the three-quark center- to be placed on the x-axis with coordinate labels x1 of-mass system. Corrections for such an approximation and x3 for the two d quarks and x2 for the u quark. By will be of order Vij, which can be taken as velocity- allowing all xi coordinates to assume both positive and dependent corrections in future refinements. negative values, while fixing the center of mass position In the wave equations (39a)-(39c), the phenomeno- (see Eq. (45) below), we allow all possible arrangement logical effective QED interaction extracted from of the ordering of the positions of the three quarks in the Schwinger’s exact QED solution is variations. We wish to find out quantitatively whether 2 e2 the attractive QED interactions between the d quarks Φ (x )= i j (−Q Q )κ|x − x | with κ= .(40) ij ij⊥ + i j i j 4π and the u quark can overcome the repulsive and weaker i j QED interaction between the two d quarks so as to sta- Equations (39) and (40) constitute the system of bilize the QED neutron, as discussed schematically in relativistic three-body equation for three quarks inter- Fig. 2(a). action in the effective QED interactions, where in our We work in the three-quark CM system in which first survey, we have neglect of the spin degrees of free- 0 1 P = p1 + p1 + p3 = (P ,P ) = (M, 0), and the relative dom. We would like to investigate whether there is a coordinate xij⊥=(xi − xj) involves only spatial coordi- lowest-energy equilibrium state of the QED neutron by nates xi and xj. The three particle momenta in the CM using a variational wave function. It is convenient to system are choose a Gaussian variational wave function of the spatial dimensionless spatial variables y1, y2, y3 with stan- pi = (i, qi), i = 1, 2, 3, (36) dard deviations σ1, σ2, and σ3 as variational parame- pi · P ters, where i = √ , (37) 2 P 2 2 2 y1 y2 y3 and we consider the particles to be of positive energy Ψ(y1, y2, y3) = N exp − 2 − 2 − 2 , (41) 4σ1 4σ2 4σ3 only, with i > 0. The rest mass M of the composite √ particle is where yi = κxi. The charge numbers of the quarks are Q1 = Q3 = −1/3, and Q2 = 2/3. The expectation 0 M = P = 1 + 2 + 3. (38) values of (39a)-(39c) using the variational wave function We generalize the two-body equations of (7a) and (7b) Ψ are 2 2 2 to the three-body problem by imposing three mass-shell 1 1 y1 m1 hΨ| |Ψi=hΨ| 2 − 4 + constraints relating the momenta, the masses, and their κ 2σ1 4σi κ interactions in the form 212 2 213 −1 + ( )|y1 − y2|+ ( )|y1 − y3| |Ψi, (42a) 2 2 1 + 2 9 1 + 3 9 H1|Ψi = p1 − m1 − [Φ12(x12) + Φ13(x13)]|Ψi = 0, (39a) 2 2 2 2 2 2 1 y2 m2 H2|Ψi = p2 − m2 − [Φ21(x21) + Φ23(x23)]|Ψi = 0,(39b) hΨ| |Ψi=hΨ| − + κ 2σ2 4σ4 κ H |Ψi = p2 − m2 − [Φ (x ) + Φ (x )]|Ψi = 0.(39c) 2 2 3 3 3 31 31 32 32 221 2 223 2 The compatibility conditions on the mass-shell constraints + ( )|y2 − y1|+ ( )|y2 − y3| |Ψi, (42b) 2 + 1 9 2 + 3 9 lead to the requirement that Φ (x ) = Φ (x ) and ij ij ji ij 2 1 y2 m2 the variable x in the effective interaction Φ (x ) be hΨ| 3 |Ψi=hΨ| − 3 + 3 ij ij ij κ 2σ2 4σ4 κ x = x0 which is transverse to the total momentum 3 3 ij ij⊥ of the combined momentum P =p + p . This x0 co- 231 −1 232 2 ij i j ij⊥ + ( )|y3 − y1|+ ( )|y3 − y2| |Ψi. (42c) ordinate should be the relative spatial coordinate in 3 + 1 9 3 + 2 9 the frame in which the center-of-mass of the system of Because of the symmetry of the two d quarks we can constituents i and j is at rest. For the three-body prob- assume for the lowest-energy state lem, the center-of-mass motion of any two constituents 1 1 σ1 = σ3, (43) i and j has a velocity Vij=(pi + pj )/(i + j) and may not be at rest. Even though Vij may not be zero, it so that the variational parameters consist only of σ1 will be constrained and limited in a bound state. It is and σ2. We look for the state with the lowest composite 13 √ mass M in the variations of σ and σ , imum, M = 44.5 MeV, at σ = 2.40 / κ=19.9 fm and 1 2 √ 1 ~ σ = 1.05 / κ=8.71 fm as shown in Fig. 9. δ2M(σ , σ ) 2 ~ 1 2 = 0. (44) δσ1δσ2 The motion of the three quarks should maintain a fixed center of mass for the composite system. It is necessary 7 Properties of the lowest-energy QED neutron for the coordinates of the three quarks to satisfy the center-of-mass condition on the spatial coordinates, Table 2 lists the physical properties of the QED neutron 3 at its energy minimum. The two d quark energies 1 and X iyi = 0. (45) 3 are smaller than the u quark energy 2 because the i=1 two effective interactions in Eq. (42) for each of the d The variational wave function Ψ is normalized accord- quarks have opposite signs while those for the u quark ing to have the same sign. The root-men-squared separation between the two d quarks is 28.2 fm and thus the QED Z 2 neutron spans a length of order many tens of fermi. The dy1dy2dy3|Ψ(y1, y2, y3)| δ(1y1 + 2y2 + 3y3)=1. (46) wave function of the d quarks have a larger value of Because of the CM condition, there are actually only the standard deviation σ1 as compared to the standard two independent spatial variables which can be chosen deviation σ2 of the u quark. In the classical description of an open string [67,68], the d quarks shuttle about to be y1 and y3. However, we need to treat all three spatial variables as independent in the beginning, and the u quark in a yo-yo motion back and forth from the impose the CM constraint (45) only when we evaluate left to the right of the u quark and back. When a d quark comes to one side of the u quark, the other d the expectation values in (42) to calculate i and M at the end. quark goes to the other side to balance the center-of- mass motion. The u quark itself also makes excursions about the geometrical center, as indicated by a smaller 46.2 M(σ ,σ ) of the QED neutron 46 1 2 45.8 value of the standard deviation σ . 45.6 2 45.4 M(MeV) 45.2 45 44.8 44.6 44.4 Table 2 Properties of the lowest-energy QED neutron M(MeV) 2 46 2.2 45.5 Quantity Value 2.4 1.3 45 1/2 1.2 σ1/κ 2.6 1.1 M (mass of the QED neutron) 44.5 MeV 2.8 1 44.5 0.9 1/2 3 0.8 σ2/κ p 2 p 2 h1i= h3i (the d quark) 11.3 MeV p 2 h2i (the u quark) 21.8 MeV p 2 h(x1 −x2) i(between d quark and u quark) 20.4 fm Fig. 9 The mass M of the QED neutron as a function of the p 2 √ h(x3 − x1) i (between two d quarks) 28.2 fm variation parameters σ , σ in units of / κ=8.29 fm. The 1 2 ~ σ1 (of wave function for the d quarks) 19.9 fm QED neutron has an energy minimum at M = 44.5 MeV at √ √ σ2 (of wave function for the u quark) 8.71 fm σ1/ κ = 2.40 and σ2/ κ = 1.09.

The force vectors in Fig. 2(a) give a qualitative de- In the evaluation of the QED neutron mass M, the scription of the various forces leading to the binding of unknown quantities i are needed to defined the effec- the three quarks in a QED neutron. The variational cal- tive interactions. They can be obtained self-consistently culations demonstrate the stability of the QED neutron and iteratively with initial guesses. Knowing the effec- in a quantitative analysis. It is illuminating to see how tive interactions and the given variational parameters the effective interactions between the three quarks can σ1 and σ2, we evaluate the expectation values on the bind them together into a QED neutron from a more right hand sides of (42a)-(42c) numerically. The calcu- quantitative viewpoint as an illustration. For such a lated values of i on the left hand sides of (42a)-(42c) purpose, we add the wave equations in (39) and we get can form the basis of the next iteration until conver- the total mass-shell condition gence is achieved. In the numerical calculations, we have 3 3 used quark masses m = 2.16 MeV and m = 4.67 MeV X 2 2 X 2 u d (i − mi ) − qi (47) [117]. i=1 i=1 By such variational calculations, we find that the −2[Φ12(x12) + Φ13(x13) + Φ23(x23)] Ψ(x1, x2, x3)=0. mass M as a function of σ1 and σ2 has an energy min- 14

This is just a three body system with a total effective At the minimum energy point, the values of and √ √ 1 2 interaction are 1 = 0.4767 κ, 2 = 0.9163 κ, and so 2/1 = 1.922 and the above dependencies can be evaluated. We show Φtot(x1, x2, x3)=2[Φ12(x12)+Φ13(x13)+Φ23(x23)]. (48) the effective interactions Φtot and Φij between differ- We can acquire a better understanding how the three ent quarks, as a function of y1 for the QED neutron at y = 0, y = −y in Fig. 10. The attractive u-d in- 2 3 √ 1 √ teractions 2Φ / κ and 2Φ / κ are shown as the Effective interactions as a function of y 12 1 23 1 √ 1 dashed curve. The repulsive d-d interaction 2Φ / κ at y2=0 and y3=-y1 13 1   2Φ √ κε Φ √ κε is shown as the dashed-dot curve in Fig. 10. The total d u d 12 / 1 , 2 23 / 1 5  2Φ √ κε effective interaction Φtot is displayed as the solid curve 13 / 1  y y Φ /√ κε which is a confining interaction that binds the three 1 y tot 1 2 3 4 quarks together. Hence there is a stable QED neutron QED neutron arising from the balances of the mutual electrostatic 3 forces between the quarks. κε 1 √ / κε 1 With the above confining potential in Fig. 10 as 2 Φ tot /√ Φ 23 an illustrative example, we can understand the energy , 2 κε 1 minimum of the QED neutron in two intuitive ways. 1 /√ Φ 12 In the description of the classical string [67,68], the 2 y1 two d quarks execute yo-yo motion shuttling about the 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 u quark back and forth after reaching the longitudinal turning points in the confining potential of Fig. 10. -1 2Φ  In the quantum mechanical description, the attractive 13 /√ κ ε net confining QED interaction as shown in Fig. 10 be- -2 1 tween the quarks is counterbalanced by the quantum stress pressure [118] that arises from the derivatives of Fig. 10 The effective interaction 2Φ12, √2Φ23, 2Φ31, and Φtot = 2Φ12 + 2Φ23 + 2Φ31 in units of κ1 for a QED the single-particle wave function, reaching the lowest- neutron, where Φij is between quarks at yi and yj and Φtot energy equilibrium between the attractive QED inter- is the total effective interaction, for a selected sample d-u- action and the quantum stress pressure. d arrangement of the three quarks shown at the upper left corner. The potentials are obtained as a function of y1, at y2 = 0, y3 = −y1, where y1, y2 and y3 are the positions of the d, u, and d quarks respectively, 8 The stability of the QED proton and the QED neutron weak decay quarks can bind together in the QED neutron when We would like to study next whether QED color-singlet we examine various components of the effective inter- proton with two u quarks and a d quark can be stable. actions between different pairs of quarks as a function For such a calculation, we carry out the variational cal- of a set of representative spatial coordinates. We can culations as in the above QED neutron case, with the u choose a sample set of representative coordinates such and d quarks in the QED neutron replaced by d and u that the first d quark coordinate is y1, the u quark co- quarks respectively. That is, we consider the u, d, and u ordinate is at y2=0, and the second d quark coordinate quarks to be placed on the x-axis with coordinate labels is at y3=−y1 because of the CM constraint. For such a x1 and x3 for the two u quarks and x2 for the d quark. sample set of representative coordinates, we can study By allowing all xi coordinates to assume both positive the behavior of various effective interactions which can and negative values, while fixing the center of mass po- be expressed as functions of a single variable y . The √ 1 sition (Eq. (45)), we allow all possible arrangement of various effective interactions 2Φij/ κ1 between quark the ordering of the positions of the three quarks in the i at yi and quark j at yj and the total Φtot as a function variations, including both the linear u-d-u configura- of y1 are tion as in Fig. 2(b) and the u-u-d. In this case of QED proton, we have Q = 2/3, Q = −1/3, and Q = 2/3. 2Φ12(y1, y2) 2Φ23(x23) 42/1 2 1 2 3 √ = √ = |y1|, κ κ 1+ / 9 Our variations over a very large range of σ1 and σ2 1 y2=0,y3=−y1 1 y2=0,y3=−y1 2 1 values fail to find an energy minimum. Extending the 2Φ13(y1, y3) 83/1 −1 −4 √ = ( )|y1| = ( )|y1|, range of σ will only drive the total energy of the sys- 1 κ 1 + 3/1 9 9 y2=0,y3=−y1 tem lower with the u quarks farther and farther apart Φtot(y1, y2, y3) 82/1 2 −4 without the energy turning to a minimum. The condi- √ = ( ) + ( ) |y1|. κ 1 + / 9 9 tion of (44) cannot be satisfied for this case. We can 1 y2=0,y3=−y1 2 1 15

Effective interactions as a function of y1 decay onto, the density of ﬁnal state for the weak decay at y2= 0, y3= -y1 of a QED neutron onto a QED proton is zero, conse-   2Φ √ κε Φ √ κε quently the rate of the QED weak decay into a QED u d u 12 / 1 , 2 23 / 1 5  2Φ √ κε 13 / 1 proton is zero. The QED neutron can only decay by  y y y Φ /√ κε a baryon-number non-conserving transition which pre- 1 2 3 4 tot 1 sumably has a very long life time. Therefore, the lowest energy QED neutron is a stable particle with a very 3 QED proton ε κ 1 long lifetime. /√ 2 Φ 23 ε , 2 κ 1 /√ 9 Other favorable conﬁgurations of quarks 1 Φ 12 2 interacting in QED y1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Although the QED proton with two up quarks and

2 Φ a down quark cannot have a stable bound or contin- -1 Φ tot /√  13 κε / uum state, other neutral QED quark and antiquark sys- √  1 κ ε tems and their corresponding charge-conjugate coun- -2 1 terparts may have stable configurations arising from quarks and antiquarks interacting in QED interactions Fig. 11 The effective interaction 2Φ12, √2Φ23, 2Φ31, and Φtot = 2Φ12 + 2Φ23 + 2Φ31 in units of κ1 for a QED alone. Whether heavy quarks can also interact in QED proton, where Φij is between quarks at yi and yj and Φtot interactions alone has not been excluded, and it is of is the total effective interaction, for a selected sample u-d- interest to examine such a possibility theoretically so u arrangement of the three quarks shown at the left upper corner. The potentials are obtained as a function of y1, at as to facilitate future experimental assessments on the y2 = 0, y3 = −y1, where y1, y2 and y3 are the positions probability of its occurrence. By including heavy quarks of the u, d, and u quarks respectively, The total effective also among the quarks interacting in QED alone in Fig. interaction Φtot shown as the solid curve is a linear repulsive 12, there can be a large number of favorable configura- interaction, indicating that the QED proton is not stable. tions of neutral open-string systems in which the forces acting on the quarks and antiquarks are subject to a linear QED force that is attractive between two charges understand the failure by looking at the total potential of opposite signs and repulsive for two charges of the 2Φ as a sample arrangement shown in Fig. 11. The tot same sign. effective potentials at y = 0, y = −y for the sample 2 √ 3 1 √ case with σ1 = σ3 = 2.09 κ and σ2 = 2.80 κ, which give 2/1 = 3.71 are shown in Fig. 11. The magni- u,c,t u,c,t d,s,b d,s,b tude of the sum of the attractive effective interactions (a) 2Φ12=2Φ23 between the d quark and the two u quarks is 2/3 -2/3 -1/3 -1/3 smaller than the magnitude of the repulsive interaction 2Φ13 between the two u quarks. The total effective interaction Φtot is repulsive; it decreases as |y1| increases. d,s,b u,c,t d,s,b Hence, a QED proton does not possess a stable bound (b) state. The QED proton also does not possess a contin- -1/3 2/3 -1/3 uum state with isolated quarks because the isolation of quarks as color-triplet quarks is forbidden. Therefore, the QED proton does not exist either as a stable bound (c) d,s,b u,c,t u,c,t u,c,t d,s,b state nor a continuum state with isolated quarks. There is no QED proton state. -1/3 2/3 -2/3 2/3 -1/3 The absence of a QED proton state has an impor- Fig. 12 For quarks interacting in QED forces alone, some tant consequence on the weak decay of the QED neu- favorable neutral open-string configurations for the lowest en- tron. The weak decay of the QED neutron occurs when ergy states involving (a) a quark and an antiquark, (b) three a d quark in the three-quark system decays into an u quarks, and (c) three quarks plus a quark-antiquark pair. quark. Such a QED neutron weak decay would result Their electric field lines of force and the quark electric charge numbers are indicated. These open-string configuration may in a possible QED proton final state, if a QED proton be stabilized by the linear QED forces which are attractive be- state could exist. Because there is no final bound or tween charges of opposite signs and repulsive between charges continuum QED proton state for the QED neutron to of the same sign. 16

It is of interest to compare and contrast the favor- three quark combination, and a color and anticolor for able lowest energy configurations shown in Fig. 12 for the additional quark-antiquark pair in Fig. 12. (c). It the QED interactions and the QCD interactions. The will be of interest to study the stability of these neu- favorable QED configurations in Fig. 12 are charge- tral color-singlet quark systems both theoretically and neutral and are dominantly linear as in an open string, experimentally. arising from the attractive linear forces between electric charges of same signs and repulsive forces between electric charges of opposite signs. In contrast, the QCD 10 The detection of a QED neutron color forces between a quark and an antiquark as well as that between two quarks in a color singlet state are at- A QED neutron is a composite object containing two tractive in nature. Thus, the composite system in QCD d quarks and one u quark interacting in QED inter- can be electrically-charged. The intrinsic shape of a sys- actions. The relativistic three-body equations of (39a), tem of a pair of quark and antiquark for both the QCD (39b), (39c) with a confining potential such as shown and QED is linear, as in the open string, as discussed in in Fig. 10 are expected to have many eigenstates and [28]. However, the QED three-quark composite is pre- eigenvalues. The inclusion of spin-spin, spin-orbit, and dominantly linear, while the QCD three-quark compos- other interactions between the quarks in a future fully ite is not. The QED quark composite particles have a three-dimensional calculation will add a greater degree linear length dimension, /(mass), of order ten fermi, ~ of complexity to the spectrum of the QED neutron. which is much greater than the linear length dimen- As the detection of a QED neutron will depend on sion of QCD quark composite particles, of order 0.5- its decay products, we would like to examine how a 1 fermi [119]. If we place QCD and QED composite QED neutron decays from an excited eigenstate n∗ . particles in the language of the energy surfaces of the QED We note first of all what it will not do. It will not disso- system as a function of the system dimensions, simi- ciate itself into isolated quark constituents because of lar to the perspective advanced in [120,121], we can the non-isolation nature of the quarks. It will not decay view the energy surface of the composite quark sys- by weak interactions into a positively charged entity be- tem as possessing both the QCD energy minima as cause there is no bound or continuum final QED proton well as the lower-energy QED energy minima located state. at much larger composite particle dimensions. As the energy minima arise from different types of interactions belonging to different symmetry groups, the QCD and n0 (final) γ n0 (final) QED meson QED QED QED minima are separated by barriers which prevent z }| { z }| { z }| { the transition from one type of minima to the other. In Fig. 12, we list different choices of flavors for each of the quarks. The quark and the antiquark in Fig. 12(a) ¡ ¡ have electric charges of opposite signs, and the attrac- V 1 V tive QED forces between them can stabilize the system. 2 V There can additional flavor linear combinations if one 1 further assumes flavor SU(2) and SU(3) symmetries as | {z } | {z } discussed in [28]. ∗ n∗ (initial) n (initial) QED Fig. 12(b) shows three quarks, with two quarks of QED electric charges of (−1/3) and a quark of charge (2/3). (a)(b) They are the analogue of the QED neutron and are Fig. 13 The decay modes of a QED neutron from an excited ∗ 0 likely to be stable. Fig. 12(c) shows a linear chain of five eigenstate nQED(initial) to the final eigenstate nQED(final): quarks. An example of such a configuration is d-u-u¯-u- (a) by the emission of a photon, and (b) by the emission of d which can be built even longer as d-u-(u-u¯)n-d, with a QED meson which subsequently decays into real or virtual photons, or a dilepton pair as described in Fig. 6(a), 6(b), n = 0, 1, 2... It have electric charges with alternating and 6(c). signs such that the QED forces between them may be attractive and balanced to stabilize the system. In the configurations in Fig. 12, one can construct In 1+1 dimensions, an excited state of QED neu- color-singlet states by choosing color-anticolor combi- tron cannot decay as the photon is represented in effect nations for the quark-antiquark combinations in Fig. by an effective potential Φ as discussed in sections 3 12(a), three different colors for the three-quark combi- and 4, and the quarks do not radiate photons. In the nation in Fig. 12(b), and three different colors for the physical 3+1 dimensions, the transverse structure of 17 the flux tube must be taken into account and the pho- provide information on the QED neutron structure. We ton emission channel opens up. A quark can make a may rely on the presence of these emitted photons and sharp change of its trajectory turning to the transverse QED mesons to reconstruct the complete spectrum of direction with the emission of a photon. By such an the QED neutron. The de-excitation may also go through emission process at the vertex V1 as depicted in dia- many steps with sequential emissions of QED mesons grams in Fig. 13(a), a quark in a QED neutron at an and/or photons. Accordingly, we can look for unknown ∗ initial excited eigenstate nQED (initial) can de-excite to photons and QED mesons that accompany the produc- 0 reach the final eigenstate nQED (final). The multipolar- tion of other photons and QED mesons. ity of the photon transition will depend on the spins and the parities of the initial and final states in ques- We note with keen interest that as QED neutrons tion. Alternatively, a valence quark in an initial excited can be produced only by the coalescence of deconfined ∗ quarks in high-energy heavy-ion collisions during the eigenstate nQED (initial) can de-excite by the emission phase transition of the quark gluon plasma, the pro- a photon at the vertex V1 in Fig. 13(b) leading to the duction of the QED neutron, if it can be so identified, production of a quark-antiquark pair at the vertex V2. The produced quark can join up with the remaining can be used as a signature for the quark-gluon plasma ∗ production. It is necessary to identify the QED neu- two quarks of the initial QED neutron nQED (initial) to 0 tron if it is produced. A QED neutron can be identified become the final QED neutron eigenstate nQED (final), while the produced antiquark can combine with the va- by its QED meson emission spectral lines which ex- lence quark to form a QED meson. In such a decay with hibit its own characteristic structure. With the mass of the emission of a QED meson as depicted in Fig. 13(b), the lowest energy QED neutron state predicted to be 44.5 MeV, and if a harmonic oscillator spectrum for the the flavor of the produced pair at V2 must agree with the flavor of the valence quark emitting the virtual pho- QED neutron can be an order-of magnitude guide, we expect the masses of the emitted QED mesons in the ton at V1 so that the flavor and the charge of the final QED neutron remains unchanged. decay of an excited QED neutron state to be of order 50 MeV. We therefore estimate the production of dipho- It is easy to envisage that successive emissions of ton resonances with an invariant mass of order 50 MeV photons and QED mesons will allow an excited QED to accompany the production of the QED neutron. A neutron n∗ to de-excite, and eventually to reach the QED way to distinguish those QED mesons as arising from lowest energy QED neutron state. Through out the de- the QED neutron decay or from a qq¯ pair production excitation process, the three quarks constituents remain is to study the differences of the QED meson emission bound to each other as an entity, in the conservation spectral lines as a function of the probability for the of the QED neutron number, while the emitted QED occurrence of deconfinement, which is correlated with mesons will decay into real photons, virtual photons in the mass, energy, and centrality of the high-energy col- the form of dileptons, or a dilepton pair as described by lision process. For those collisions with a low probabil- diagrams 6(a), 6(b), and 6(c) in section 5. At the end of ity leading to deconfinement, the detected QED mesons the de-excitation, only the ground state QED neutron emission spectral lines will not contain those lines emit- remains and it does not radiate because it is the lowest ted from the excited QED neutrons. On the other hand, energy QED neutron state. Because the lowest-energy for collisions with a high likelihood of attaining decon- QED neutron ground state does not decay or radiate, finement, there will be excited QED neutrons with the it is therefore a QED dark neutron. accompaniment of the emission of QED mesons with With the exception of the QED dark neutron which these spectral lines from their de-excitation. By judi- has no decay products, the detection of a QED neutrons cial correlations with the probability of such collisions, can be carried out by searching for their decay prod- those QED meson spectral lines associated with the de- ucts of photons and QED mesons arising from the de- excitation of the excited QED neutrons may be sepa- excitation of the excited QED neutron states with the rated. The on-set of these extra QED meson lines aris- emitted QED mesons detected as diphoton resonances. ing from the decay of excited QED neutrons may be We envisage that by the coalescence of the quarks of dif- a good signature of the on-set of deconfinement and a ferent colors, QED neutrons at the lowest-energy state signature of the quark-gluon plasma formation. as well as the excited states may be produced during the deconfinement-to-confinement phase transition of Another method to detect of the QED neutron, in- the quark gluon plasma. The de-excitation of the ex- cluding the QED dark neutron, may be carried out by cited QED neutron states will yield photons and QED elastic and inelastic collisions with electrons or nuclei in mesons of various energies exhibiting the spectrum of various materials, such as semiconductors, scintillators, the QED neutron system. The de-excitation energies and superconductors [122]. 18

11 Summary, Conclusions and discussions composite particle mass is proportional to the gauge field coupling constant. Such a proportionality places The observations of the anomalous soft photons, the the mass of the composite particles containing mass- X17 particle, the E38 particle, and the associated the- less quarks interacting in QED alone in the region of oretical interpretations in terms of QED mesons bring many tens of MeV [25,26,27,28], in the gap between with them three outstanding questions: (I) Can quarks the quark masses and the pion mass. Thus, in physical interact with the QED interactions alone, without the circumstances when the available energy of the quark presence of the QCD interactions? (II) If so, how do the system in its CM system is much lower than the pion quarks interact in QED interactions? (III) If quarks can mass threshold in the region of many tens of MeV, light interact with the QED interactions alone, what will be quarks interacting in QED interactions alone are per- the consequences and implications? mitted without the presence of the QCD interactions. Heretofore, our usual experience of quarks interact- If light quarks can interact with QED interactions ing with gauge fields have been confined mainly to the alone, a new frontier will be opened up for exploration region where the energy of the interacting quark system because quarks carry electric charges, electric charges of (in its CM system) is above the pion mass threshold. opposite signs attract, and attractive interactions result In this energy region, quarks interact with both QCD in stable and confined composite quark states. There and QED gauge interactions simultaneously, and they will be many physical arrangements of light quarks and are invariably accompanied by gluon exchange interac- antiquarks in which the attractive QED forces allow the tions. It is impossible for quarks to interact only with composite system to form bound and confined states. the QED interactions alone, when we restrict ourselves The many quantum numbers that characterize the quarks to this energy region. will also add complexities to the spectrum of these com- The situation may be different if we venture into posite particles. For example, for light quarks with two the lower energy region. We note that quarks carry flavors and S = 0 moving in phase and out of phases electric charges and can interact in QED interactions. with each other, we show earlier that there can be There is no physical law that forbids quarks to inter- the (I = 0,I3 = 0) state at 17.9±1.8 MeV and the act in QED interactions alone. If we focus our atten- (I = 1,I3 = 0) state at 36.4±3.8 MeV. QED mesons tion on light quarks, we can find out the mass scales formed by a light quark and a light antiquark of the of composite particles containing light quarks interact- same flavor are also possible. By the use of the semi- ing non-perturbatively in different gauge interactions. empirical formula for the QED meson state energy de- The light quark masses are of order a few MeV, and veloped in [28], one locates theoretically the S = 0 ¯ a light quark-antiquark pair can be readily produced single-flavor excitation (I = 0,I3 = 0) dd QED meson at energies much below the pion mass threshold. The state at 21.2±2.1 MeV and the uu¯ QED meson state at non-isolation property of quarks requires the produced 34.7±3.5 MeV. quark and antiquark to interact in QED interactions in The possible occurrence of the QED mesons can ways that are differently from the ways in which isolat- be tested by searching for the decay products of two able electric charges interact. To be non-isolatable, the real photons, two virtual photons in the form of dou- electric field lines of force of the produced quark and an- ble dilepton pairs, or a single dilepton pair. The X17 tiquark do not spread out and must be bundled together particle observed in the decays of the 4He∗ and 8Be∗ in the form of a flux tube for which the Schwinger model [10,11] with an e+e− invariant mass of 17 MeV and of massless fermions in QED in 1+1 dimensions can be the state at 19±1 MeV in emulsion studies [12] match an idealized approximation. A produced light quark and the predicted mass of the isoscalar 0(0−) QED meson a light antiquark can be approximately treated as mass- [28]. The E38 MeV particle, observed in high-energy pC, less. Schwinger’s 1+1 dimensions fits the interacting dC, dCu collisions at Dubna [13,14] with a γγ invariant light quark-antiquark pair well because both Schwinger mass of about 38 MeV, matches the predicted mass of massless fermions and the interacting massless quarks the isovector QED meson [28]. These are encouraging are confined, and a confined massless quark and anti- experimental observations. The QED (uu¯) and QED quark pair can be described naturally as the two ends (dd¯) states have yet to be located and identified exper- of a one-dimensional open string [47,48,49,63,64,65,66, imentally. Further experimental measurements in the 67,68,69,70,71]. low invariant mass region will be of great interest. Massless quarks do not set up a mass scale. The The QED mesons are not the only color-singlet com- mass scales of the composite particles containing light posite states arising from quarks interacting in QED in- quarks in a gauge interaction can be estimated from teractions alone. The QED neutron with two d quarks the Schwinger model [73,74], which reveals that the and one u quark with three different colors can form 19 a color-singlet composite system. The QED neutron dark neutron. Such QED dark neutrons and its excited can be stable because the attractive QED interactions states may occur at the deconfinement-to-confinement between two d quarks and the u quark overcome the phase transition of the quark-gluon plasma and may be weaker repulsion between the two d quarks. With a a signature of the deconfinement-to-confinement transi- phenomenological three-body model in 1+1 dimensions tion of the quark gluon plasma in high-energy heavy-ion with an effective interaction between electric charges collisions. Self-gravitating assemblies of QED dark neu- extracted from Schwinger’s exact QED solution, we find trons may be stable astrophysical objects. Because of quantitatively in a variational calculation that there is its long lifetime, self-gravitating QED dark neutron as- a QED neutron energy minimum at a mass of 44.5 MeV. semblies (and similarly QED dark antineutron assem- The analogous QED proton with two u quarks and a d blies) of various sizes may be good candidates for a quark has been found to be too repulsive to be stable part of the primordial dark matter produced during and does not have a bound or continuum state. the deconfinement-to-confinement phase transition of Because of its composite nature, a QED neutron the quark gluon plasma in the evolution of the early can exist in different excited states. The excited states Universe. are expected to decay by emitting photons and/or QED In another matter, LIGO recently observed the merger mesons to make transitions to lower QED neutron states. of two neutron stars through the detection of their grav- One of the two d quarks may decay into an u quark by itational waves in 2017 [123]. The merging of the two way of the weak interaction. However, because the QED neutron stars will likely lead to the production of a proton does not possess a stable bound state nor a con- quark matter with deconfined quarks [124,125,126]. The tinuum state of isolated quarks, the rate of the QED authors of [124,125] proposed a new signature for a neutron weak decay into a QED proton is zero. first-order hadron-quark phase transition in merging Among all QED neutron states, the ground QED neutron stars, which may provide the opportunity to neutron state located at 44.5 MeV distinguishes itself study the properties of the post-merger quark matter. from higher excited QED neutron states as a stable As the quark matter cools and undergoes the deconfinement- particle without decay products. It can only decay by to-confinement phase transition during the merging pro- a baryon-number non-conserving transition, which pre- cess, the coalescence of deconfined quarks to become sumably has a very long lifetime. As a consequence, the confined quarks will produce excited QED neutrons and lowest state QED neutron is a dark neutron. The QED the QED dark neutrons in the post-merger environ- antineutron ground states is likewise a dark antineu- ment. So, QED dark neutrons may be copiously pro- tron. The only mode of destruction for a QED dark duced in the post-merger environment of merging neu- neutron and a QED dark antineutron is their mutual tron stars. annihilation, with the production of photons and QED mesons. As it is suggested here that the confinement to de- On account of their being predicted to be stable confinement phase transition at the early history of the particles with a very long lifetime without decay prod- Universe in the quark-gluon plasma phase may generate ucts, the QED dark neutrons and QED dark antineu- the QED dark neutron assemblies as seeds for the pri- trons may be good candidate particles for a part of the mordial dark matter, it will be of great interest to study dark matter. We envisage that in the early evolution of whether QED dark neutrons and/or its excited states the Universe after the big bang, the Universe will go may be produced in high-energy heavy-ion collisions through the quark-gluon plasma phase with deconfined where quark gluon plasma may be produced. The detec- quarks and gluons. As the primordial matter expands tion of the QED dark neutrons may be made by search- and cools down the quark-gluon plasma undergoes a ing for the photons and/or QED mesons during the phase transition from the deconfined phase to the con- de-excitation from its excited states. The de-excitation fined phase, deconfined quarks of three different colors of the excited QED neutron states will yield photons may coalesce to form color-singlet states. While many and QED mesons with its own characteristic QED me- of the color-singlet systems of three quarks have suffi- son emission spectrum. The capability of precise dilep- cient energy to form hadrons, there may be some proton measurements with large magnets in high-energy duced color-singlet three-quark systems in which their heavy-ion collisions may make it possible to study the total energy is below the QCD neutron energy of about spectrum of the produced QED mesons through their 1 GeV. These three-quark systems may form QED neu- decays into two virtual photons, as discussed in section tron states that are bound by QED interactions. The 5. We may rely on the presence of these emitted photons de-excitation of the excited QED neutron state will find or produced QED mesons to reconstruct the spectrum its way down to the lowest energy state of the QED of the QED neutron. 20

For simplicity in the present first survey of the QED We introduce a Gaussian variational wave function with neutron, we have neglected the spin degree of freedom. the variational parameter σ, While the spin will not likely affect the the stability, 1 y2 y2 the quark confinement, and the gross structure of the Ψ(y)=(√ )1/2 exp{− }=N exp{− }. (A.3) 2πσ 4σ2 4σ2 QED neutron, it will play an significant role in the fine structure and the spectrum of the QED neutron. We obtain The spin degree of freedom, along with the orbital an- 1 2σE hH0i(σ) = + √ . (A.4) gular momentum, the collective rotation, and the col- 4σ2 2π lective vibration should be taken into account in fu- From the requirement of δhH0i(σ)/δσ = 0, we get ture studies. Theoretical investigations on the internal √ !1/3 structure and the energy spectrum of the QED neu- 2π σ = , (A.5) tron will be valuable to assist the detection of the pro- 4E duced QED neutrons. How the QED mesons and QED neutron interact with themselves and with hadrons will at which open up another avenue to explore the interplay be- 3E hH0i = √ . (A.6) tween species from the same branch and from different 2π branches of the quark family of the Standard model. From Eq. (A.2), the value of E2 at the equilibrium value Furthermore, the possibility of the many-body inter- of σ becomes action between QED dark neutrons forming a bound 2 3σE multi-QED-neutron system and the interaction between E = hH0i = √ . (A.7) 2π the QED neutron matter and other standard QCD matter will add other dimensions to the complexity of mat- Eliminating σ from Eqs. (A.5) and (A.7) we get ter associated with the QED dark neutron. It will be of 3 3/4 great interest to extend the frontier of QED neutrons E = √ = 1.034 (A.8) ( 2π)2/341/3 into new regions both theoretically and experimentally. √ !1/3 2π σ = = 0.876. (A.9) 4E 12 Acknowledgments √ The variational calculation gives E = / κ ∼ 1 as The author would like to thank Profs. H. Georgi, Y. given in (A.8), and thus Jack Ng, Lai-Him Chan, A. Koshelkin, Gang Wang, √ e = κ, and M = 2 = √ , (A.10) H. Sazdjian, G. Wilk, Pisin Chen, Larry Zamick, Jia- π Chao Wang, Scott Willenbrock, and J. Stone for help- ful communications and discussions. The research was which agrees with the lowest eigenenergy obtained by supported in part by the Division of Nuclear Physics, solving the wave equation directly, indicating the valid- U.S. Department of Energy under Contract DE-AC05- ity of the variational calculation for the lowest-energy 00OR22725. two-body state.

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