{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:19:38Z","timestamp":1760239178797,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T00:00:00Z","timestamp":1602547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The equation of state of SU(3) Yang\u2013Mills theory can be modelled by an effective Z3\u2212symmetric potential depending on the temperature and on a complex scalar field \u03d5. Allowing \u03d5 to be dynamical opens the way to the study of spatially localized classical configurations of the scalar field. We first show that spherically symmetric static Q-balls exist in the range (1\u22121.21)\u00d7Tc, Tc being the deconfinement temperature. Then we argue that Q-holes solutions, if any, are unphysical within our framework. Finally, we couple our matter Lagrangian to Einstein gravity and show that spherically symmetric static boson stars exist in the same range of temperature. The Q-ball and boson-star solutions we find can be interpreted as \u201cbubbles\u201d of deconfined gluonic matter; their mean radius is always smaller than 10 fm.<\/jats:p>","DOI":"10.3390\/sym12101668","type":"journal-article","created":{"date-parts":[[2020,10,17]],"date-time":"2020-10-17T07:23:22Z","timestamp":1602919402000},"page":"1668","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Gravitating Bubbles of Gluon Plasma above Deconfinement Temperature"],"prefix":"10.3390","volume":"12","author":[{"given":"Yves","family":"Brihaye","sequence":"first","affiliation":[{"name":"Service de Physique de l\u2019Univers, Champs et Gravitation, UMONS Research Institute for Complex Systems, Universit\u00e9 de Mons, Place du Parc 20, 7000 Mons, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7801-036X","authenticated-orcid":false,"given":"Fabien","family":"Buisseret","sequence":"additional","affiliation":[{"name":"Service de Physique Nucl\u00e9aire et Subnucl\u00e9aire, UMONS Research Institute for Complex Systems, Universit\u00e9 de Mons, Place du Parc 20, 7000 Mons, Belgium"},{"name":"CeREF, Chauss\u00e9e de Binche 159, 7000 Mons, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1016\/0370-2693(78)90737-2","article-title":"Thermal Properties of Gauge Fields and Quark Liberation","volume":"72","author":"Polyakov","year":"1978","journal-title":"Phys. Lett. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2610","DOI":"10.1103\/PhysRevD.20.2610","article-title":"Lattice Models of Quark Confinement at High Temperature","volume":"20","author":"Susskind","year":"1979","journal-title":"Phys. Rev. D"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.nuclphysa.2005.02.130","article-title":"Quark gluon plasma and color glass condensate at RHIC? The Perspective from the BRAHMS experiment","volume":"757","author":"Arsene","year":"2005","journal-title":"Nucl. Phys. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1038\/s41567-018-0360-0","article-title":"Creation of quark\u2013gluon plasma droplets with three distinct geometries","volume":"15","author":"Aidala","year":"2019","journal-title":"Nat. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1103\/PhysRevD.26.963","article-title":"First Order Phase Transition in the SU(3) Gauge Theory at Finite Temperature","volume":"26","author":"Yaffe","year":"1982","journal-title":"Phys. Rev. D"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/0550-3213(82)90172-9","article-title":"Critical Behavior at Finite Temperature Confinement Transitions","volume":"210","author":"Svetitsky","year":"1982","journal-title":"Nucl. Phys. B"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2667","DOI":"10.1103\/PhysRevD.25.2667","article-title":"The Wilson Line in Finite Temperature Gauge Theories","volume":"25","author":"Weiss","year":"1982","journal-title":"Phys. Rev. D"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/aphy.1994.1073","article-title":"Are Z(N) bubbles really there?","volume":"234","author":"Smilga","year":"1994","journal-title":"Ann. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/S0370-2693(99)01242-3","article-title":"Spatial\u2019t Hooft loop, hot QCD and Z(N) domain walls","volume":"469","author":"Kovner","year":"1999","journal-title":"Phys. Lett. B"},{"key":"ref_10","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","author":"Hooft","year":"1978","journal-title":"Nucl. Phys. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/0550-3213(96)00170-8","article-title":"Thermodynamics of SU(3) lattice gauge theory","volume":"469","author":"Boyd","year":"1996","journal-title":"Nucl. Phys. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"125006","DOI":"10.1103\/PhysRevD.72.125006","article-title":"Higher representations: Confinement and large N","volume":"72","author":"Sannino","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"014020","DOI":"10.1103\/PhysRevD.87.014020","article-title":"Q-ball formation at the deconfinement temperature in large-Nc QCD","volume":"87","author":"Brihaye","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"074020","DOI":"10.1103\/PhysRevD.82.074020","article-title":"Simulation of Z(3) walls and string production via bubble nucleation in a quark-hadron transition","volume":"82","author":"Gupta","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"015202","DOI":"10.1103\/PhysRevC.93.015202","article-title":"Nontopological Soliton in the Polyakov Quark Meson Model","volume":"93","author":"Jin","year":"2016","journal-title":"Phys. Rev. C"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"182302","DOI":"10.1103\/PhysRevLett.87.182302","article-title":"Explosive decomposition in ultrarelativistic heavy ion collision","volume":"87","author":"Scavenius","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.physletb.2005.03.079","article-title":"Can dissipation prevent explosive decomposition in high-energy heavy ion collisions?","volume":"614","author":"Fraga","year":"2005","journal-title":"Phys. Lett. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"125016","DOI":"10.1103\/PhysRevD.86.125016","article-title":"Effects of Quarks on the Formation and Evolution of Z(3) Walls and Strings in Relativistic Heavy-Ion Collisions","volume":"86","author":"Gupta","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"044901","DOI":"10.1103\/PhysRevC.88.044901","article-title":"Domain growth and fluctuations during quenched transition to quark-gluon plasma in relativistic heavy-ion collisions","volume":"88","author":"Mohapatra","year":"2013","journal-title":"Phys. Rev. C"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/JHEP12(2016)032","article-title":"Q-holes","volume":"12","author":"Nugaev","year":"2016","journal-title":"J. High Energy Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1143\/PTP.128.925","article-title":"The separation of the chiral and deconfinement phase transitions in the curved space-time","volume":"128","author":"Sasagawa","year":"2012","journal-title":"Prog. Theor. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"041501","DOI":"10.1103\/PhysRevD.88.041501","article-title":"Deconfinement transition and Black Holes","volume":"88","author":"Flachi","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1103\/PhysRevLett.61.141","article-title":"Particle-Like Solutions of the Einstein Yang-Mills Equations","volume":"61","author":"Bartnik","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6","DOI":"10.12942\/lrr-2012-6","article-title":"Dynamical Boson Stars","volume":"15","author":"Liebling","year":"2012","journal-title":"Living Rev. Relativ."},{"key":"ref_25","first-page":"054","article-title":"Boson Stars and QCD Boson Stars","volume":"LC2019","author":"Kulshreshtha","year":"2020","journal-title":"PoS"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1140\/epjc\/s10052-006-0065-x","article-title":"Thermodynamics of the PNJL model","volume":"49","author":"Ratti","year":"2007","journal-title":"Eur. Phys. J. C"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"085003","DOI":"10.1103\/PhysRevD.66.085003","article-title":"Spinning Q balls","volume":"66","author":"Volkov","year":"2002","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"016009","DOI":"10.1103\/PhysRevD.85.016009","article-title":"A large-Nc PNJL model with explicit ZNc symmetry","volume":"85","author":"Buisseret","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/S0370-2693(01)00286-6","article-title":"Event-by-event fluctuations from decay of a Polyakov loop condensate","volume":"504","author":"Dumitru","year":"2001","journal-title":"Phys. Lett. B"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Sarkar, S., Satz, H., and Sinha, B. (2010). The Physics of the Quark-Gluon Plasma, Springer. Lecture Notes in Physics.","DOI":"10.1007\/978-3-642-02286-9"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"114510","DOI":"10.1103\/PhysRevD.71.114510","article-title":"Static quark anti-quark interactions in zero and finite temperature QCD. I. Heavy quark free energies, running coupling and quarkonium binding","volume":"71","author":"Kaczmarek","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"014019","DOI":"10.1103\/PhysRevD.73.014019","article-title":"Phases of QCD: Lattice thermodynamics and a field theoretical model","volume":"73","author":"Ratti","year":"2006","journal-title":"Phys. Rev. D"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1090\/S0025-5718-1979-0521281-7","article-title":"A Collocation Solver for Mixed Order Systems of Boundary Value Problems","volume":"33","author":"Ascher","year":"1979","journal-title":"Math. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"064002","DOI":"10.1103\/PhysRevD.72.064002","article-title":"Rotating boson stars and Q-balls","volume":"72","author":"Kleihaus","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"084048","DOI":"10.1103\/PhysRevD.89.084048","article-title":"Charged Q-balls and boson stars and dynamics of charged test particles","volume":"89","author":"Brihaye","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.physletb.2004.04.027","article-title":"Chiral effective model with the Polyakov loop","volume":"591","author":"Fukushima","year":"2004","journal-title":"Phys. Lett. B"},{"key":"ref_37","unstructured":"Biswal, M., Digal, S., and Saumia, P. (2019). Z3 meta-stable states in PNJL model. arXiv."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1134\/S1063776120020077","article-title":"Review of Nontopological Solitons in Theories with U(1)-Symmetry","volume":"130","author":"Nugaev","year":"2020","journal-title":"J. Exp. Theor. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1668\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:20:39Z","timestamp":1760178039000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1668"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,13]]},"references-count":38,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["sym12101668"],"URL":"https:\/\/doi.org\/10.3390\/sym12101668","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,10,13]]}}}