{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T01:45:26Z","timestamp":1779327926464,"version":"3.51.4"},"reference-count":63,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T00:00:00Z","timestamp":1732665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100013912","name":"Vector Foundation","doi-asserted-by":"publisher","award":["P2021-0102"],"award-info":[{"award-number":["P2021-0102"]}],"id":[{"id":"10.13039\/501100013912","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013912","name":"Vector Foundation","doi-asserted-by":"publisher","award":["SFB 1491"],"award-info":[{"award-number":["SFB 1491"]}],"id":[{"id":"10.13039\/501100013912","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004807","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["P2021-0102"],"award-info":[{"award-number":["P2021-0102"]}],"id":[{"id":"10.13039\/100004807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004807","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 1491"],"award-info":[{"award-number":["SFB 1491"]}],"id":[{"id":"10.13039\/100004807","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Based on the thermal phase structure of pure SU(2) quantum Yang\u2013Mills theory, we describe the electron at rest as an extended particle, a droplet of radius r0\u223ca0, where a0 is the Bohr radius. This droplet is of vanishing pressure and traps a monopole within its bulk at a temperature of Tc=7.95 keV. The monopole is in the Bogomolny\u2013Prasad\u2013Sommerfield (BPS) limit. It is interpreted in an electric\u2013magnetically dual way. Utilizing a spherical mirror-charge construction, we approximate the droplet\u2019s charge at a value of the electromagnetic fine-structure constant \u03b1 of \u03b1\u22121\u223c134 for soft external probes. It is shown that the droplet does not exhibit an electric dipole or quadrupole moment due to averages of its far-field electric potential over monopole positions. We also calculate the mixing angle \u03b8W\u223c30\u00b0, which belongs to deconfining phases of two SU(2) gauge theories of very distinct Yang\u2013Mills scales (\u039be=3.6 keV and \u039bCMB\u223c10\u22124 eV). Here, the condition that the droplet\u2019s bulk thermodynamics is stable determines the value of \u03b8W. The core radius of the monopole, whose inverse equals the droplet\u2019s mass in natural units, is about 1% of r0.<\/jats:p>","DOI":"10.3390\/sym16121587","type":"journal-article","created":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T10:05:56Z","timestamp":1732701956000},"page":"1587","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Electroweak Parameters from Mixed SU(2) Yang\u2013Mills Thermodynamics"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7582-3456","authenticated-orcid":false,"given":"Janning","family":"Meinert","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik, Universit\u00e4t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany"},{"name":"Institut f\u00fcr Physik, Bergische Universit\u00e4t Wuppertal, Gau\u00dfstra\u00dfe 20, D-42119 Wuppertal, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6365-0631","authenticated-orcid":false,"given":"Ralf","family":"Hofmann","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik, Universit\u00e4t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/0550-3213(71)90139-8","article-title":"Renormalizable Lagrangians for massive Yang-Mills fields","volume":"35","year":"1971","journal-title":"Nucl. Phys. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/0550-3213(72)90279-9","article-title":"Regularization and renormalization of gauge fields","volume":"44","author":"Veltman","year":"1972","journal-title":"Nucl. Phys. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/0550-3213(71)90395-6","article-title":"Renormalization of massless Yang-Mills fields","volume":"33","year":"1971","journal-title":"Nucl. Phys. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/andp.19163561702","article-title":"Zur Quantentheorie der Spektrallinien","volume":"356","author":"Sommerfeld","year":"1916","journal-title":"Annalen der Physik"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1038\/s41586-020-2964-7","article-title":"Determination of the fine-structure constant with an accuracy of 81 parts per trillion","volume":"588","author":"Morel","year":"2020","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hofmann, R. (2016). The Thermodynamics of Quantum Yang-Mills Theory: Theory and Applications, World Scientific Publishing Co Pte Ltd.. [2nd ed.].","DOI":"10.1142\/9997"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/0029-5582(59)90196-8","article-title":"The renormalizability of vector meson interactions","volume":"10","author":"Glashow","year":"1959","journal-title":"Nucl. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1007\/BF02726525","article-title":"Weak and electromagnetic interactions","volume":"11","author":"Salam","year":"1959","journal-title":"Nuovo Cim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/0031-9163(64)90711-5","article-title":"Electromagnetic and weak interactions","volume":"13","author":"Salam","year":"1964","journal-title":"Phys. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"081601","DOI":"10.1103\/PhysRevLett.95.081601","article-title":"Precision Measurement of the Weak Mixing Angle in M\u00f8ller Scattering","volume":"95","author":"Anthony","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"083C01","DOI":"10.1093\/ptep\/ptac097","article-title":"Review of Particle Physics","volume":"2022","author":"Workman","year":"2022","journal-title":"Prog. Theor. Exp. Phys."},{"key":"ref_12","unstructured":"Pich, A. (1994). The Standard Model of Electroweak Interactions. arXiv."},{"key":"ref_13","unstructured":"de Broglie, L. (1964). The Thermodynamics of the Isolated Particle, Gauthier-Villars Editor."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/BF00708650","article-title":"The reinterpretation of wave mechanics","volume":"1","year":"1970","journal-title":"Found. Phys."},{"key":"ref_15","unstructured":"Dattoli, G. (2010). The fine structure constant and numerical alchemy. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1140\/epjh\/e2014-50044-7","article-title":"Attempts at a determination of the fine-structure constant from first principles: A brief historical overview","volume":"39","author":"Jentschura","year":"2014","journal-title":"Eur. Phys. J. H"},{"key":"ref_17","unstructured":"Faessler, M.A. (2013). Weinberg Angle and Integer Electric Charges of Quarks. arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"135548","DOI":"10.1016\/j.physletb.2020.135548","article-title":"The Weinberg angle and 5D RGE effects in a SO(11) GUT theory","volume":"807","author":"Englert","year":"2020","journal-title":"Phys. Lett. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1103\/PhysRev.76.749","article-title":"The Theory of Positrons","volume":"76","author":"Feynman","year":"1949","journal-title":"Phys. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"e1601558","DOI":"10.1126\/sciadv.1601558","article-title":"Energy penetration into arrays of aligned nanowires irradiated with relativistic intensities: Scaling to terabar pressures","volume":"3","author":"Bargsten","year":"2017","journal-title":"Sci. Adv."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1088\/0004-637X\/734\/1\/5","article-title":"ARCADE 2 Measurement of the Absolute Sky Brightness at 3\u201390 GHz","volume":"734","author":"Fixsen","year":"2011","journal-title":"Astrophys. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2200517","DOI":"10.1002\/andp.202200517","article-title":"Cosmological Parameters from Planck Data in SU(2)CMB, Their Local \u039bCDM Values, and the Modified Photon Boltzmann Equation","volume":"535","author":"Hofmann","year":"2023","journal-title":"Annalen der Physik"},{"key":"ref_23","unstructured":"Meinert, J., Gavioso, R., Brida, G., Gaiser, C., Imbraguglio, D., and Hofmann, R. (2024). Search for a Rayleigh-Jeans Blackbody Anomaly at low Temperatures, in preparation."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Hofmann, R. (2017). The Isolated Electron: De Broglie\u2019s Hidden Thermodynamics, SU(2) Quantum Yang-Mills Theory, and a Strongly Perturbed BPS Monopole. Entropy, 19.","DOI":"10.3390\/e19110575"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Hofmann, R., and Grandou, T. (2022). On Emergent Particles and Stable Neutral Plasma Balls in SU(2) Yang-Mills Thermodynamics. Universe, 8.","DOI":"10.3390\/universe8020117"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1166\/qm.2012.1004","article-title":"The quantum of action and finiteness of radiative corrections: Deconfining SU(2) Yang-Mills thermodynamics","volume":"1","author":"Hofmann","year":"2012","journal-title":"Quant. Matt."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2122","DOI":"10.1103\/PhysRevD.17.2122","article-title":"Periodic Euclidean solutions and the finite-temperature Yang-Mills gas","volume":"17","author":"Harrington","year":"1978","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"036003","DOI":"10.1103\/PhysRevD.70.036003","article-title":"Quantum weights of dyons and of instantons with nontrivial holonomy","volume":"70","author":"Diakonov","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1002\/andp.201052201-210","article-title":"Spatial Wilson loop in continuum, deconfining SU(2) Yang-Mills thermodynamics","volume":"19","author":"Ludescher","year":"2010","journal-title":"Annalen der Physik"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.nuclphysb.2006.10.003","article-title":"An SU(2) KvBLL caloron gas model and confinement","volume":"760","author":"Gerhold","year":"2007","journal-title":"Nucl. Phys. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0550-3213(78)90153-0","article-title":"On the phase transition towards permanent quark confinement","volume":"138","year":"1978","journal-title":"Nucl. Phys. B"},{"key":"ref_32","first-page":"236783","article-title":"Evolving Center-Vortex Loops","volume":"2012","author":"Moosmann","year":"2012","journal-title":"ISRN Math. Phys."},{"key":"ref_33","unstructured":"Del Debbio, L., Faber, M., Greensite, J., and Olejnik, S. (1997). Center Dominance, Center Vortices, and Confinement. arXiv."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF01393973","article-title":"The shape of a figure-eight under the curve shortening flow","volume":"96","author":"Grayson","year":"1989","journal-title":"Invent. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1073\/pnas.14.7.553","article-title":"An Interpretation of Dirac\u2019s Theory of the Electron","volume":"14","author":"Breit","year":"1928","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1103\/PhysRev.28.1049","article-title":"An Undulatory Theory of the Mechanics of Atoms and Molecules","volume":"28","year":"1926","journal-title":"Phys. Rev."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1103\/PhysRev.73.416","article-title":"On Quantum-Electrodynamics and the Magnetic Moment of the Electron","volume":"73","author":"Schwinger","year":"1948","journal-title":"Phys. Rev."},{"key":"ref_38","unstructured":"Zahn, M. (2023). MIT OpenCourseWare, 2.7: The Method of Images with Point Charges and Spheres, MIT."},{"key":"ref_39","unstructured":"Ludescher, J., and Hofmann, R. (2009). CMB dipole revisited. arXiv."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"151801","DOI":"10.1103\/PhysRevLett.92.151801","article-title":"What Does a Strongly Excited \u2019t Hooft\u2013Polyakov Magnetic Monopole Do?","volume":"92","author":"Fodor","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"151802","DOI":"10.1103\/PhysRevLett.92.151802","article-title":"Resonant Excitations of the \u2019t Hooft\u2013Polyakov Monopole","volume":"92","author":"Volkov","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1126\/science.adg4084","article-title":"An improved bound on the electron\u2019s electric dipole moment","volume":"381","author":"Roussy","year":"2023","journal-title":"Science"},{"key":"ref_43","unstructured":"Hofmann, R. (2015). Electromagnetic waves and photons. arXiv."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1103\/PhysRevLett.19.1264","article-title":"A Model of Leptons","volume":"19","author":"Weinberg","year":"1967","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"170005","DOI":"10.1063\/1.5114156","article-title":"Chiral symmetry breaking out of QCD effective locality","volume":"2116","author":"Grandou","year":"2019","journal-title":"AIP Conf. Proc."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1016\/S0550-3213(98)00590-2","article-title":"Periodic instantons with non-trivial holonomy","volume":"533","author":"Kraan","year":"1998","journal-title":"Nucl. Phys. B"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"025011","DOI":"10.1103\/PhysRevD.58.025011","article-title":"SU(2) calorons and magnetic monopoles","volume":"58","author":"Lee","year":"1998","journal-title":"Phys. Rev. D"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1002\/andp.19263861802","article-title":"Quantisierung als Eigenwertproblem","volume":"386","year":"1926","journal-title":"Annalen der Physik"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1002\/andp.20095210905","article-title":"Low-frequency line temperatures of the CMB (Cosmic Microwave Background)","volume":"521","author":"Hofmann","year":"2009","journal-title":"Annalen der Physik"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1119\/1.10184","article-title":"On the statistical aspect of electron interference phenomena","volume":"44","author":"Merli","year":"1976","journal-title":"Am. J. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/j.nme.2019.02.030","article-title":"Inter-ELM pedestal localized fluctuations in tokamaks: Summary of multi-machine observations","volume":"19","author":"Laggner","year":"2019","journal-title":"Nucl. Mater. Energy"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"092508","DOI":"10.1063\/1.4820954","article-title":"Effect of ion orbit loss on the structure in the H-mode tokamak edge pedestal profiles of rotation velocity, radial electric field, density, and temperature","volume":"20","author":"Stacey","year":"2013","journal-title":"Phys. Plasmas"},{"key":"ref_53","unstructured":"Dobbs, A. (2023, December 10). Introducing MICE. Available online: https:\/\/blogs.imperial.ac.uk\/mice\/2017\/09\/01\/introducing-mice\/."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1103\/PhysRevLett.30.1343","article-title":"Ultraviolet Behavior of Non-Abelian Gauge Theories","volume":"30","author":"Gross","year":"1973","journal-title":"Phys. Rev. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1346","DOI":"10.1103\/PhysRevLett.30.1346","article-title":"Reliable Perturbative Results for Strong Interactions?","volume":"30","author":"Politzer","year":"1973","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"5632","DOI":"10.1103\/PhysRevB.23.5632","article-title":"Quantized Hall conductivity in two dimensions","volume":"23","author":"Laughlin","year":"1981","journal-title":"Phys. Rev. B"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1395","DOI":"10.1103\/PhysRevLett.50.1395","article-title":"Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations","volume":"50","author":"Laughlin","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1559","DOI":"10.1103\/PhysRevLett.48.1559","article-title":"Two-Dimensional Magnetotransport in the Extreme Quantum Limit","volume":"48","author":"Tsui","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF01090719","article-title":"Ward identities in gauge theories","volume":"10","author":"Slavnov","year":"1972","journal-title":"Theor. Math. Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1016\/0550-3213(71)90297-5","article-title":"Ward identities and charge renormalization of the Yang-Mills field","volume":"33","author":"Taylor","year":"1971","journal-title":"Nucl. Phys. B"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1086\/307221","article-title":"Measurements of \u03a9 and \u039b from 42 High-Redshift Supernovae","volume":"517","author":"Perlmutter","year":"1999","journal-title":"Astrophys. J."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1086\/300499","article-title":"Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant","volume":"116","author":"Riess","year":"1998","journal-title":"Astron. J."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1038\/s41467-020-20652-9","article-title":"Effects of plasma turbulence on the nonlinear evolution of magnetic island in tokamak","volume":"12","author":"Choi","year":"2021","journal-title":"Nat. Commun."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1587\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:40:53Z","timestamp":1760114453000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1587"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,27]]},"references-count":63,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["sym16121587"],"URL":"https:\/\/doi.org\/10.3390\/sym16121587","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,27]]}}}