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 di- mensions 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 in- teracting 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 simulta- neously with the QCD mesons in the string fragmen- tation 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 me- son 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 par- ent particles of the anomalous soft photons, indicating that these anomalous particles are consistent with their d u d description as confined composite qq¯ systems interact- ing 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.